Diffusion and Electrophoretic NMR 9783110551525, 9783110551532, 9783110551655

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Peter Stilbs Diffusion and Electrophoretic NMR

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Peter Stilbs

Diffusion and Electrophoretic NMR

Author Prof. em. Peter Stilbs Royal Institute of Technology (KTH) Applied Physical Chemistry Teknikringen 36 SE-100 44 Stockholm Sweden [email protected]

ISBN 978-3-11-055152-5 e-ISBN (PDF) 978-3-11-055153-2 e-ISBN (EPUB) 978-3-11-055165-5 Library of Congress Control Number: 2019933230 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2019 Walter de Gruyter GmbH, Berlin/Boston Typesetting: Integra Software Services Pvt. Ltd. Printing and binding: CPI books GmbH, Leck Cover image: Peter Stilbs/John Wiley and Sons. Magnetic field gradient and shim coils of an iron-magnet geometry JEOL FX-100 NMR probe. www.degruyter.com

Preface With great initial hesitation, I considered a surprising invitation in early 2017 by De Gruyter Publishers to write a book on NMR diffusometry in their series of graduateoriented textbooks. A related one by Bernhard Blümich and coworkers on Compact NMR had already set a high standard indeed [1]. Three advanced monographs of excellent quality on translational motion and NMR also had appeared less than a decade ago: by William S. Price in 2009 [2], by Paul T. Callaghan in 2011 [3] and by Geir Humborstad Sørland in 2014 [4]. Being professor emeritus since 2012, I felt that there was not much more I could add to all this and to my own previous reviews on the subject and some families of applications. The first one (in Progress in NMR Spectroscopy) formally dates back to 1987 [5], and was widely influential in the past (it was actually authored in late 1985 and appeared in print in late 1986). At the time of writing this book, it had about 1,600 citations. However, in later years, it seems that much of the actual information in it is not noticed or actually read; especially numerous application studies listed and outlined there seem widely ignored or overlooked. My most recent NMR diffusometry review dates back to 2011 [6]. It is similar in terms of outline to that in this book, although more compact. However, this text appears unnoticed by many potential NMR users too, most likely because it constitutes a chapter (among many other NMR-related ones with similar fate) in Encyclopedia of Analytical Chemistry. This would not be the first choice for looking for introductory information on NMR techniques and similar topics. Finally, an essaylike one (written in 2015, but finally published in paper form in 2017) describes personal historical recollections on the early evolution of multicomponent diffusometry techniques in high-resolution NMR [7]. Oleg Lebedev, series editor at De Gruyter, gradually convinced me that a book of primarily educational format and focus should indeed serve a purpose, especially for persons who want to get started with hands-on NMR diffusometry. That the intended book need and should not be “complete” with regard to choice of references and topics, and there was virtually no limit to the number of color illustrations that contributed a lot to my final positive decision. I just hope that colleagues around the globe understand the scope and format, and do not feel offended that links to their important contributions to the field might have somewhat arbitrarily been left out. I want to clearly stress that this book is not intended to compete in any way with the detailed and quite complete texts mentioned earlier. It is simply meant to be an introductory and rather nonmathematical textbook for beginners, with some basic NMR experience. A main objective for a book of this kind is to introduce the subject to those who want to proceed quickly to a “hands-on” situation. It is easy to get misled by older literature and descriptions of numerous ingenious experimental procedures and subsequently get lost into unnecessarily complicated or awkward https://doi.org/10.1515/9783110551532-201

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measurement procedures. Modern equipment is also intrinsically much more artifact-free than it was some decades ago, and there is less or no need for numerous “corrective” measurement procedures that were common in the past; some of which could even be quite contra productive today. Nowadays, necessary equipment for NMR diffusometry is readily available in many labs around the globe, also because “gradient-enhanced” procedures became popular in high-resolution NMR. This has, in turn, promoted manufacture and sales of NMR probes with integral magnetic field gradient coils as a standard option. Such probes may also have multinuclear capability, through interchangeable inserts and generally have quite sufficient performance for many diffusion-related applications. Users who need higher specifications regarding gradient strengths and similar will find dedicated probes and gradient drivers available commercially. Some of that equipment may have also been designed for NMR microscopy (i.e., magnetic resonance imaging (MRI) on small objects). Here, the probe would have three gradient coils for the x-, y- and z-spatial dimensions and three independent gradient drivers. Coil design and gradient driver efforts in MRI have undoubtedly also promoted vastly superior instrumental performance in modern-day NMR diffusometry. A companion method, electrophoretic NMR (eNMR), has been around for quite some time [8, 9]. It later found its way into high-resolution NMR, primarily through efforts by Charles S. Johnson, Jr. and coworkers around 1990 [10]. I personally spent some months in his lab in 1992 to learn some tricks. Later I realized that practical and reliable routine implementation of eNMR remained deceptive as a concept, at least in our hands – and over many years. In particular, we found that the competing and mostly irreproducible phenomenon of electro-osmosis was difficult to suppress or correct for. Various glass coatings (meant to suppress electro-osmosis) could work for a single sample, but were not stable enough to survive washing and sample changing during longer actual measurement series. In this context, I feel a need to apologize at this stage to my own past Ph.D. students who often had to suffer in despair with irreproducible experiments, especially Kim Paulsen, Erik Thyboll-Pettersson and Fredrik Hallberg. Primarily through later-year innovative ideas by my local colleague István Furó and our common former Ph.D. student Pavel Yushmanov (nowadays electronics and fine mechanical wizard at the department), it appears that the eNMR technique has finally reached a reasonable state of maturity and stability. I have, therefore, included a final section on eNMR in this book. Despite a much more limited system applicability than NMR diffusometry, high-resolution eNMR is a technique with great and still largely unexploited potential in chemistry, even after three decades of existence. Before finally accepting the offer to compose and write this book on my own, I contacted a few distinguished colleagues regarding possible coauthorship. Lack of time within the foreseeable future was the common reason to decline. Also, I

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remember very well a previous joint book project efforts by three colleagues – after more than a year of ambitious authoring, differing opinions on the content, level and style made it impossible to finalize the work. Well, there is also a saying “too many cooks spoil the soup.” I do not really think that could have been the case with the coauthors I myself had in mind, but some practical, stylistic and logistic aspects of writing likely got simpler. However, I hope that a related saying, “cook a soup on a nail,” will not characterize the final result (a stone soup version of this saying also circulates). Before finally deciding to undertake this task, I also consulted Bernhard Blümich, who already contributed to this book series with the volume Compact NMR some years earlier. Bernhard advised me that authoring such a book requires a lot of devotion and daily discipline – of course, he was right. However, on many occasions during this intermittent two-year authoring, I rather got personal associations to the frantically writing Dr Mabuse in the 1933 movie classic The Testament of Dr. Mabuse, directed by Fritz Lang. Anyway, I now want to thank all former collaborators in the topic areas of this book. In recent decades, contact and discussions with persons on the other side of the globe have been particularly valuable – special thanks go to William S. Price (University of Western Sydney), who has been a great support and a good friend for decades. He promptly assisted with various matters on numerous occasions while authoring this book. I have also had great “across the Globe” contact with Philip W. Kuchel and Paul T. Callaghan. Paul tragically passed away in 2012, and is sadly missed. Help and advice from local collaborators István Furó, Pavel Yushmanov and Sergey Dvinskikh have also been of immense importance for me during later years. István and Pavel also kindly provided a number of illustrations and related material for the eNMR sections of this text. In the past, I also enjoyed very fruitful and stimulating collaboration with Michael E. Moseley, Wyn Brown, Olle Söderman, Björn Lindman, Terence Cosgrove, Peter C. Griffiths, Lars Nordenskiöld, Aatto Laaksonen, Magnus Nydén and Harald Walderhaug, in particular. I also thank all former Ph.D. students, postdocs and visiting scientists over the years. Thanks also go to Klaus Zick of Bruker Biospin and Gareth Morris and Mathias Nilsson for kindly providing information, documentation and figure material related to Bruker and Varian/Agilent PGSE instrumentation and the GNAT data processing package. Andrew Coy and Bertram Manz made available information, text and figure material related to PGSE operation on recent-year “bench-top” spectrometers manufactured by Magritek. Daniel Topgaard and Daniel Gallichan generously provided graphic tools and computer code for simulating echo formation, probably saving me weeks of time and effort. Special thanks go to Oleg Lebedev who originally persuaded me to write this book, and to Lena Stoll of De Gruyter for innumerable consultations regarding technical typesetting and graphical matters. They all patiently and promptly dealt with. During the final months of writing and actual preproduction, their respective roles

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at De Gruyter were taken over by Mareen Pagel and Ria Fritz, with a continued pleasant, nonformal, constructive and very helpful atmosphere. I am also very grateful that they gladly accepted a considerably longer manuscript text than initially agreed on. Finally, I want to thank my former Ph.D. supervisor Sture Forsén for directing me into the field of NMR diffusometry more than 40 years ago, when I moved from Lund to a different and indeed highly molecular transport-oriented scientific environment in Uppsala [7]. His encouragement and moral support for authoring this book is greatly appreciated as well. Åkersberga, December 31, 2018

Peter Stilbs

References [1] [2]

Blümich, B., Haber-Pohlmeier, S., & Zia, W. Compact NMR. Berlin/Boston: De Gruyter, 2014. Price, WS. NMR Studies of Translational Motion – Principles and Applications. Cambridge: Cambridge University Press, 2009. [3] Callaghan, PT. Translational Dynamics & Magnetic Resonance, Principles of Pulsed Gradient Spin Echo NMR. Oxford: Oxford University Press, 2011. [4] Sørland, GH. Dynamic Pulsed-Field-Gradient NMR. Berlin, Heidelberg: Springer, 2014. [5] Stilbs, P. Fourier transform pulsed-gradient spin-echo studies of molecular diffusion. Prog Nucl Magn Reson Spectrosc. 1987; 19: 1–45. [6] Stilbs, P. Diffusion and Electrophoretic Studies Using Nuclear Magnetic Resonance. In: RA Meyers, ed. (Nuclear Magnetic Resonance and Electron Spin Resonance Spectroscopy). Chichester, UK: John Wiley & Sons Ltd, 2011:1439–1461. [7] Stilbs, P. Historical: Early multi-component FT-PGSE NMR self-diffusion measurements-some personal reflections. Magn Reson Chem. 2017; 55: 386–394. [8] Holz, M., Lucas, O., & Müller, C. NMR in the presence of an electric current. Simultaneous measurements of ionic mobilities, transference numbers, and self-diffusion coefficients using an NMR pulsed gradient experiment. J Magn Reson. 1984; 58: 294–305. [9] Holz, M. Electrophoretic NMR. Chem Soc Rev. 1994; 23: 165–174. [10] Johnson, C., Jr, & He, Q. Electrophoretic Nuclear Magnetic Resonance. Adv Magn Reson. 1989; 13: 131–159.

Contents Preface

V

About the Author 1 1.1

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Historical perspectives and basic pulsed NMR concepts 1 Early NMR work was field sweep based – Hahn conceived and developed pulsed NMR excitation and detection 1 1.2 Hahn discovered spin echo formation in pulsed NMR 2 1.3 The introduction of FT-NMR led to a transition from sweep to predominantly pulsed NMR detection 3 1.4 The somewhat dormant technique of magnetic field gradient spin echo NMR diffusometry gradually achieves greater popularity in the 1970s 5 1.5 Pulsed field gradient spin echo NMR opens doors for diffusometry and makes magnetic field gradient-based imaging possible 8 1.6 Frequency-resolved multicomponent NMR diffusometry emerged in the 1970s 9 1.7 Computer and NMR spectrometer developments make FT-PGSE easier and a more powerful technique 9 1.8 NMR imaging (MRI and NMR microscopy) techniques emerged in the early 1970s 10 1.8.1 Further reading 13 1.9 Electrophoretic NMR emerges as a componentresolved technique 13 1.10 Outline, scope and required background for the present book 13 1.10.1 Further reading 14 1.11 Basic pulsed NMR concepts – a pictorial vector model crash course 15 1.11.1 Energy levels and frequencies in NMR spectroscopy 15 1.11.2 The comparably very low frequency of NMR spectroscopy has several consequences 15 1.11.3 The “rotating frame” description of excitation and emission in NMR 17 1.11.4 Spin magnetization dephases with time in the x′–y′-plane and constitutes the time-domain NMR signal – the “free induction decay” 18 1.11.5 Restoration of thermal equilibrium and loss of phase coherence in the x′–y′ plane are two types of spin relaxation processes 20

X

1.11.6 1.11.7 1.11.8 1.11.8.1 1.11.9 1.12 2 2.1 2.2 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 2.3.6 2.4 2.4.1 2.5 2.5.1 2.5.2 2.5.3 2.6 2.6.1 2.6.2 2.6.3 2.6.3.1 2.6.3.2 2.6.3.3 2.6.3.4 2.6.4

Contents

Detection of transverse magnetization is made in a phasesensitive manner, followed by discrete digitization 20 Rotating frame magnetization is monitored through quadrature detection and complex Fourier transformation 21 Actual pulsed NMR hardware and detection procedures 25 Further reading 26 Correcting basic non-idealities of pulse generation and data acquisition 27 References 28 Basic concepts 30 Molecular transport processes and the NMR time window 30 Rotational diffusion 32 Self-diffusion and other modes of translational motion 36 Estimating self-diffusion coefficients from a macroscopic perspective 38 Estimating self-diffusion coefficients from a molecular perspective 40 General and experimental methodological historical considerations 41 Flow and related modes of non-Brownian motion 42 Influence of barriers to molecular transport 42 Further reading 44 Mutual diffusion 44 Further reading 46 Why study self-diffusion and other transport processes by NMR? 47 Main Objectives 47 What can favorably be studied and by which approach? 47 The most basic experimental strategy based on self-diffusion data 48 Spin-echo formation under the influence of a steady magnetic field gradient: the key to NMR diffusometry 49 Labeling nuclei with positional information 50 The intrinsic “time-reversal” effect of spin-echo formation 50 Illustrating spin-echo formation 52 52 The basic 90o–90o Hahn echo 54 A 90o–180o pulse pair 55 A 90o–90o–90o sequence – the stimulated echo and others Spin phase graphing, magnetization helices and diffraction analogies 59 Effects of J-modulation on spin-echo formation 61

Contents

2.6.5 2.6.6 2.6.7 2.6.8 2.6.9 2.7 2.7.1 2.7.2 2.7.3 2.7.4 2.7.5 2.7.6 2.8 2.8.1 2.8.2 2.8.3 2.8.4 2.8.5 2.8.5.1 2.8.5.2 2.8.5.3 2.8.5.4 2.8.5.5 2.9 2.9.1 2.9.2 2.9.3 2.9.4 2.9.5 2.10 3 3.1 3.2

Quantifying self-diffusion using spin echoes – a simplified model 63 Influence of flow or on spin-echo formation 65 Self-diffusion effects alongside other NMR parameters in spin-echo experiments 66 Spin echoes and steady-gradient diffusional effects in multipulse NMR experiments 67 Quantifying self-diffusion using spin echoes – deriving echo attenuation expressions more rigorously 69 Spin-echo formation under the influence of pulsed magnetic field gradients 70 The basic PGSE experiment 71 The stimulated echo (PGSTE) variant of the PGSE experiment 74 Basic advantages in using PGSTE rather than PGSE 75 Multipulse CPMG-based PGSE 76 Background magnetic field gradient influence on PGSE experiments 77 Further reading 78 Optimization and evaluation of basic PGSE and PGSTE experiments 79 Dealing with intrinsic nuclear spin factors 79 Dealing with quadrupolar nuclei 80 PGSE, as applied to carbon-13 nuclei 81 Dealing with instrumental factors and other nonidealities 82 Brief summary of common complications in PGSE studies 83 Convection 83 Eddy currents and sample vibration 84 Radiation damping 85 Chemical exchange and cross-relaxation 85 General advice on sample size and length in PGSE studies 86 Phase cycling 86 Software design of selective phase cycles 90 Separating out a three-pulse generated stimulated echo 91 Using “crusher” magnetic field gradient pulses to destroy transverse magnetization within a pulse sequence 93 “Multiplexed” NMR experiments 94 Further reading 94 References 94 PGSE NMR diffusometry example applications Further reading 100 Strategies and constraints 100

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3.3 3.4 3.4.1 3.4.1.1 3.4.1.2 3.4.1.3 3.5 3.5.1 3.5.2 3.5.2.1 3.5.2.2 3.5.3 3.5.3.1 3.5.3.2 3.5.4 3.5.4.1 3.5.4.2 3.5.4.3 3.5.5 3.5.5.1 3.5.5.2 3.5.6 3.5.6.1 3.5.6.2 3.5.7 3.5.7.1 3.5.7.2 3.5.7.3 3.6 3.6.1 3.6.1.1 3.6.2 3.6.2.1 3.6.2.2 3.6.2.3 3.6.3 3.6.3.1 3.6.3.2 3.6.4 3.6.4.1

Contents

Systems suitable for diffusion-related investigation by NMR 101 Diffusion in single or binary liquids 103 Monitoring self-diffusion of gases dissolved in water 104 Objectives 104 Experimental aspects 104 Further reading 105 Diffusion-based NMR signal detection and monitoring of miscellaneous aggregation processes in aqueous solution 105 General objectives 107 Rhodamine 6G self-aggregation in aqueous solution 107 Objectives 107 Experimental aspects 107 Dissolved organic matter in natural waters 108 Objectives 108 Experimental aspects 109 Substrate binding to supramolecular structures in aqueous solution 110 Objectives 110 Experimental aspects 111 Further reading 111 Nucleotide aggregation in aqueous solution 112 Objective 112 Experimental aspects 113 Protein aggregation in aqueous solution 114 Objectives 114 Experimental aspects 115 PGSE in organometallic chemistry 116 Objectives 116 Experimental aspects 117 Further reading 117 Surfactant systems 117 Physicochemical characteristics 117 Further reading 122 Surfactant aggregation 123 Objectives 123 Experimental aspects 123 Further reading 125 Micellar solubilization 125 Objectives 126 Experimental and evaluational aspects 126 Mixed micellization 129 Objectives 129

Contents

3.6.4.2 3.6.4.3 3.6.5 3.6.5.1 3.6.5.2 3.6.5.3 3.6.6 3.6.6.1 3.6.6.2 3.6.7 3.6.7.1 3.6.7.2 3.6.8 3.7 3.7.1 3.7.2 3.7.2.1 3.7.2.2 3.7.3 3.7.3.1 3.7.3.2 3.7.3.3 3.7.4 3.7.4.1 3.7.4.2 3.7.4.3 3.8 4 4.1 4.1.1 4.2 4.2.1 4.3 4.4 4.4.1

Experimental aspects 129 Further reading 131 “Microemulsions” and microemulsion structure 132 Objectives 132 Experimental aspects 132 Further reading 134 Surfactant adsorption from aqueous solution onto surfaces and large particles 134 Objectives 135 Experimental aspects 135 Surfactant and polymer binding to suspended carbon nanotubes in aqueous solution 138 Objectives 139 Experimental aspects 139 Further reading 140 Polymer systems in solution; general 140 Further reading 144 Diffusion in mixed polymer systems in solution 144 Objective 144 Experimental aspects 145 Counterion diffusion in polyelectrolyte systems in solution 145 Objectives 145 Experimental aspects 146 Further reading 147 Block copolymer and polymer–surfactant interaction in aqueous solution – associative thickeners 147 Objectives 148 Experimental aspects 148 Further reading 150 References 150 Getting started 155 The experimental environment to aim for 155 Further reading 159 PGSE on a conventional NMR supercon spectrometer with a commercial field gradient probe and gradient current driver 159 Further reading 161 PGSE on a newer generation permanent magnet bench-top spectrometers 162 PGSE on a single-sided or otherwise compact “NMR-mouse” type of instrumentation 165 Further reading 166

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4.5

Contents

Basic preoperation checklist on “noncompact” spectrometer types 166 References 173

5

Nonobvious pitfalls and other potentially confusing elements in PGSE studies 175 5.1 Nonlinearity between relative echo amplitudes and component concentrations 175 5.2 Eddy current and sample vibration disturbances 176 5.3 Dealing with concentrated samples in proton PGSE experiments 179 5.3.1 Radiation damping effects 179 5.4 Convective transport in the sample 179 5.4.1 Special ways of suppressing macroscopic convection physically 182 5.4.2 Suppressing convection effects in PGSE by using more complex pulse sequences 182 5.5 Effects of chemical exchange during the PGSE sequence 184 5.6 Influence from restricted diffusion and inherent approximations in the Stejskal–Tanner relation 185 5.6.1 Restricted diffusion in general 185 5.6.2 The GPD approximation 187 5.6.3 The SGP approximation 187 5.6.4 Further reading 188 5.7 Less obvious or widely misconceived concepts in the context of NMR diffusometry 188 5.7.1 “Affinity NMR” 188 5.7.1.1 Further reading 189 5.7.2 “DOSY” and “DOSY Maps” 189 5.7.3 “NMR chromatography” and “matrix-assisted NMR” 190 5.7.4 Further reading 191 References 191 6 6.1 6.2 6.3 6.3.1 6.4 6.5

Data preparation, evaluation and presentation 195 Further reading 199 The actual fitting problem in the context of PGSE data sets 199 Some existing data processing strategies in similar context 200 Further reading 205 Prior knowledge in PGSE experimental analysis 206 Exponential-type fitting of PGSE bandshape data – some basic examples 207

Contents

6.6 6.6.1 6.7 6.8 6.8.1 6.8.2 6.8.3 6.8.4 6.9 6.9.1 6.9.1.1 6.9.1.2 6.9.1.3 6.9.2 6.10 6.10.1 6.10.2 6.11 6.11.1 6.12 6.12.1 6.12.1.1 6.12.1.2 6.12.2 6.12.3

7 7.1 7.2 7.3 7.3.1

Step-wise unraveling of more complex spectral overlap in PGSE data 211 Achieving PGSE signal separation by increasing experimental dimensionality 215 Automated overview scanning PGSE data – “HR-DOSY,” “RECORD” processing and similar 215 Analyzing PGSE data sets by applying distributions of self-diffusion coefficients 219 Further reading 223 Stretched exponentials of Kohlrausch–Williams–Watts type 223 Gamma function modeling of polydispersity in PGSE data 224 Inverse Laplace transform modeling 226 Which data analysis procedure to choose for analyzing PGSE data sets? 227 Existing software for off-line processing and deeper analysis of PGSE data 228 CORE, described technically 229 The SCORE modification of CORE 230 Comparing the CORE and SCORE variants 230 Summarizing and presenting results – DOSY displays and related 231 Error estimation in PGSE studies 232 Monte Carlo estimation of parameter confidence limits 234 Error estimation, as graphically or numerically implied in HR-DOSY or RECORD-type data analysis outcomes 235 PGSE and NMR data processing in retrospect 235 Further reading 239 NMR data processing software sources 239 Dedicated “PGSE software” 240 The DOSY toolbox and GNAT 240 CORE 240 Other software of general interest in the context of PGSE studies 243 Supporting mathematic and numeric software 244 References 247 Specialized measurements and systems 250 Pulsed radiofrequency (B1) rather than B0 field gradient diffusometry 250 Static superconducting magnet stray field diffusometry Modulated magnetic field gradient methods 252 Further reading 253

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7.4 7.5 7.5.1 7.5.2 7.5.3 7.5.4 7.6 7.6.1

Contents

HR-MAS PGSE diffusometry studies on semisolids 253 PGSE NMR diffusometry in heterogeneous or microheterogeneous systems 255 Biological systems 258 Probing periodic structures through “diffusive diffraction” effects in PGSE 258 Suppressing “natural” background magnetic field gradients in heterogeneous systems 260 Further reading 261 Spatially localized NMR diffusometry and NMR Microscopy 261 Further reading 263 References 263

8 8.1 8.2 8.3 8.4 8.5

Electrophoretic NMR (eNMR) 266 Further reading 269 Evaluating echo signal phase data in eNMR 269 eNMR reviews 270 Basic experimental implementation of eNMR 271 Beginner’s lessons learned by using the originally common U-tube eNMR sample geometry 272 8.6 General considerations in experimental eNMR 274 8.6.1 Safety 274 8.6.2 Basic physics 274 8.6.3 Sample heating in eNMR 275 8.6.4 Magnetic field gradients and motion induced by currents in the sample 275 8.6.5 Further reading 276 8.7 Why study electrophoretic mobility rather than self-diffusion? 276 8.7.1 Transference numbers and electrophoretic mobility 277 8.8 Electro-osmosis and electrophoresis in a historical perspective 278 8.8.1 Electro-osmosis and the “double layer” concept 278 8.8.2 Electrophoresis 279 8.8.3 Further reading 280 8.9 More recent methodological developments in eNMR 281 8.9.1 Pulse sequences and sample cells 282 8.9.2 Hardware evolution 283 8.9.3 Partial performance specifications for the eNMR-1000 unit 8.10 eNMR equipment used by other researchers 286 8.11 General considerations on suitability of eNMR applications 8.11.1 Further reading 289

286 288

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8.12 8.12.1 8.12.1.1 8.12.1.2 8.12.1.3 8.12.2 8.12.2.1 8.12.2.2 8.12.2.3 8.12.3 8.12.3.1 8.12.3.2 8.12.3.3 8.13 8.14

9 9.1 9.2 9.3 9.3.1 9.4 9.5 9.6 9.7 9.8 9.9

10

Some eNMR example studies 289 Ion binding to poly(ethylene oxide) polymer chains in aqueous solution 290 Objective 291 Experimental aspects 292 Further reading 294 Surfactant/nonionic polymer interaction 294 Objectives 294 Experimental aspects 294 Further reading 296 Investigating “ionic liquids” by eNMR 296 Objectives 297 Experimental aspects 297 Further reading 298 Spatially resolved NMR-based monitoring of electrically induced transport 299 Further reading 299 References 300

Building home-brew PGSE and eNMR instrumentation 303 Further reading 303 PGSE on older-generation resistive electromagnet-type spectrometers 303 Home-brew gradient coils for superconducting magnet geometries 307 Further reading 308 Gradient current generation and safety considerations 308 Home-brew gradient control 309 Electrophoretic NMR (eNMR) 310 Considering complete home-brew NMR systems 311 Further reading 312 Internet resources 312 References 314 Other more recent texts and reviews on NMR diffusometry References 316

315

Appendix: Quantifying self-diffusion-related spin echo magnetic field gradient-induced spin echo attenuation and deriving the basic pulsed-gradient Stejskal–Tanner relation 318 Index

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About the Author Peter Stilbs is Professor Emeritus of Physical Chemistry at the Royal Institute of Technology (KTH), Stockholm, Sweden. He received his doctorate degree in Physical Chemistry (TeknD) in 1974 at the Lund Institute of Technology under the guidance of secondgeneration nuclear magnetic resonance (NMR) pioneer Sture Forsén. After a postdoc year at Brock University (St. Catharines, Canada), he has held positions at Uppsala University (Sweden) and University of Bergen (Norway), before being appointed to the KTH chair in 1986.

https://doi.org/10.1515/9783110551532-202

1 Historical perspectives and basic pulsed NMR concepts The interest in studying transport processes such as self-diffusion, flow and electrophoresis via nuclear magnetic resonance (NMR) methods has grown rapidly in the past decades. There are several reasons involved, including recent availability of commercial high-performance instrumentation. The use of similar equipment and methodology for NMR microscopy (magnetic resonance imaging [MRI] on small objects) has also been utilized in these processes. Both techniques can also be combined for creating spatially resolved images that quantify transport processes in heterogeneous or confined systems. Data available on lateral transport processes such as self-diffusion do not normally require further interpretation, unlike many other techniques that require elaborate theoretical frameworks and simplifying models for converting a measured quantity into something physically conceivable. Nowadays, the precision and accuracy of NMR-based diffusometry is good (1% is routinely achievable). The measuring range spans over five decades; however, it is often limited by other factors, such as rapid spin relaxation and convective overturning in the sample. NMR techniques for self-diffusion studies originated in the year 1949 by the pioneering work of Erwin Hahn (1921–2016). At the time he was an unsupervised postdoc, who also invented, analyzed and experimentally implemented the concept of pulsed NMR spectroscopy on which the modern NMR rests upon [1, 2].

1.1 Early NMR work was field sweep based – Hahn conceived and developed pulsed NMR excitation and detection Pulsed NMR is based on the concept of “free nuclear precession” of nuclear spins, originally put forward by Felix Bloch in a companion paper to one of the two independent first detections of NMR [3]. The original experimental study was done through a magnetic field sweep approach. His group included William Hansen and Martin Packard [4]. Edward Purcell made another pioneering study together with coworkers Henry Torrey and Robert Pound [5]. They independently detected NMR via a slightly different but still sweep-based set up. Purcell shared the 1952 Nobel Prize with Bloch, and their original announcement papers were published in the same issue of Physical Review in 1946 (see http://mriquestions.com/who-discovered-nmr.html for more historic information). Several reviews in the earlier days of NMR detection have shown that this field was highly influenced by the recent personal experiences with radio and radar equipment in the military services during World War II, and by general electronics instrumental development at the time. In addition, there was abundance of surplus military electronics. My own speculation is that Erwin Hahn (Figure 1.1) was also https://doi.org/10.1515/9783110551532-001

2

1 Historical perspectives and basic pulsed NMR concepts

Figure 1.1: The author standing proudly next to Erwin Hahn at a conference lunch break in Chamonix, France in 2005 (private photo).

militarily influenced in his pulsed NMR approach, and its subsequent evolution for generation and detection of spin echoes. He had previous service in the US Navy, teaching pulsed techniques such as sonar and radar for two years (see https://www. aip.org/history-programs/niels-bohr-library/oral-histories/4652). Hahn’s 2006 Russell Varian Prize lecture also has historical information from that time [6]. In 1990 he also wrote a highly informative but often overlooked article on the development of NMR and magnetic resonance-based imaging (MRI and NMR microscopy) [7].

1.2 Hahn discovered spin echo formation in pulsed NMR In a brilliant follow-up study to his less-cited single-pulse paper introducing the pulsed NMR concept experimentally, Hahn correctly described and interpreted spin echo formation effects he had noted when using two or more radiofrequency (rf) pulses [2]. He also showed that they provide a pathway to information for self-diffusion in a sample. The key concept is that echo amplitudes are affected quantitatively through molecular motion in a spatially varying magnetic field. Supported by another NMR pioneer, Charles P. Slichter, Hahn also derived a proper quantifying equation that linked echo attenuation to self-diffusion for a constant magnetic field gradient (i.e., a linearly changing magnetic field along a chosen direction).

1.3 The introduction of FT-NMR largely made sweep NMR methods obsolete

3

1.3 The introduction of FT-NMR led to a transition from sweep to predominantly pulsed NMR detection The basic signal response in pulsed NMR (the so-called free induction decay [FID]) is a composite signal received from induced transverse sample magnetization, which decays exponentially with time. The FID is thus an unresolved time-domain signal, originating from all pulse-excited nuclei in the sample. Hence, it was not originally very useful outside physics-related studies of single components. However, Fourier transformation of the FID results in a normal frequency-domain spectrum. Such procedures became realistic only through the rapid progress in the field of digital electronics, that is, through the introduction of laboratory computers from around 1970 and onwards, and a faster Fourier transform algorithm (FFT) developed by Cooley and Tukey in 1965 [8] (Figure 1.2). RF pulse Field/frequency lock coils

Magnet region Some milliseconds or seconds

Shim coils Gradient coils

“FID” Radiofrequency transmitter Frequency synthesizer Pulse programmer

Computer and human interface

Time-domain NMR spectrum

Preamplifier Phase-sensitive detector Audio frequency low-pass filters

Receiver electronics

Radiofrequency pulse generation

Various electronic control circuits

Probe

Some microseconds

Fourier transformation

Frequency-domain NMR spectrum

Analog-digital converter Some Hz or kHz

Figure 1.2: A simplified block diagram of a pulsed Fourier transform NMR spectrometer. The basic detection mode is a time-domain FID signal response from the sample, that follow a sequence of one or more relatively strong (e.g., 100 W) rf pulses. The FID is further digitized and Fourier transformed into a frequency domain spectrum. Normally one uses two detection channels rather than one, as further described in Section 1.11. Currently in use are permanent as well as superconducting magnet systems (as illustrated in Figure 4.1). Resistive electromagnets similar to those implied in Figure 1.3 and illustrated in Figure 6.33 were the standard electromagnets used in the 1950s to the late 1970s, but are quite rare nowadays. In a superconducting magnet geometry, the diffusion-detecting magnetic field gradients used in NMR diffusometry are normally applied along the sample axis and in the main magnetic field direction. In a permanent or resistive magnet system, this direction is normally transverse to the tubular-shaped sample axis.

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1 Historical perspectives and basic pulsed NMR concepts

The key step toward pulsed Fourier Transform NMR (FT-NMR) was taken in the late 1960s. Richard R. Ernst and Weston A. Anderson then demonstrated the feasibility of Fourier transformation of digitized time-domain FID’s into frequency-domain NMR spectra [9]. They were both working for the NMR manufacturer Varian Associates at the time. Fourier transformation computations of punched paper-tape digitized data were done off-line on a mainframe computer. Ernst was later awarded the 1991 Nobel Prize for FT-NMR, other methodological development and subsequent creation of many new NMR techniques within his research group. It should be noted that pulsed NMR detection and spin echo formation effects like those discovered by Hahn four decades earlier are a central component of most modern NMR techniques as well. These do include multidimensional NMR (the main tool used in the protein structure-related shared Nobel Prize of 2002), and also MRI, a field in which the Nobel Prize was awarded for physiology or medicine in 2003. Indeed, many feel that it is strange that Hahn did not even share any of these prizes. Apart from the already mentioned spin echo-related basic methodology Hahn and Maxwell also discovered and studied even smaller proton homonuclear and heteronuclear spin–spin couplings through pulsed spin-echo NMR [10, 11]. Similar studies through sweep NMR at the same time also observed this phenomenon [12, 13], although for quite large heteronuclear spin–spin couplings involving 31P and 19F. Along with Hartmann, Hahn also invented and experimentally tested the concept of the so-called Hartmann–Hahn condition of polarization transfer, which makes use of various types of solid-state NMR possible [14]. Many of Hahn’s contributions are described in a multiauthor monograph titled “Pulsed Magnetic Resonance: NMR, ESR and Optics. A Recognition of E.L. Hahn” [15]. It appeared a year after the main modern NMR Nobel Prize in chemistry was awarded to Richard R. Ernst in 1991, and a decade before the shared MRI Nobel Prize in medicine or physiology was awarded to Paul Lauterbur and Peter Mansfield. Sweep NMR remained the standard NMR approach for three early decades, and “high-resolution” 1H NMR methodologies of various types grew hugely in popularity among chemists (Figure 1.3). Their applications were mostly routine analysis and determination of molecular structure in the context of synthetic work in organic chemistry. Other nuclei than protons were not normally accessible for typical commercial instruments. Pulsed NMR became a comparably peripheral technique for decades, which was mostly utilized in the realm of physics and nearby fields. Freeman and Morris described early development of commercially available NMR in a recent paper [16]. As a sideline, it is somewhat amusing to recall that both 1952 Nobel laureates are said to firmly have claimed that NMR was a technique for physicists, and was of no use whatsoever in chemistry. In the early 1950s, the Swedish biological/medical NMR pioneer Erik Odeblad approached Felix Bloch with a

1.4 Spin-echo NMR diffusometry steeply gains larger popularity in the 1970ies

5

Rf transmitter

Rf receiver and amplifier Sweep and shim coils

Magnet pole

Sweep and shim coils

Magnet pole

Control console and recorder

Sweep generator Figure 1.3: Schematic geometry and layout of the magnet/probe environment of a resistive iron magnet NMR system. Rf coils for excitation and detection could be applied transverse to the magnetic field in the hybrid variant drawn here, or both along the sample tube axis (the normal situation). Magnetic field gradients in such geometries were normally applied along the main magnetic field, but transverse to the sample axis.

request to also study biological materials by proton NMR on his equipment, but it was rejected. Later, Odeblad successfully started such studies on his own, building an existing equipment designed and built by Gunnar Lindström (who some years earlier was very close to make the first discovery of chemical shifts in NMR) [17].

1.4 The somewhat dormant technique of magnetic field gradient spin echo NMR diffusometry gradually achieves greater popularity in the 1970s The number of pulsed NMR spectrometers gradually increased over the years, but they were relatively rare in comparison with the number of continuous wave (CW), magnetic field swept ones. One particular and unique application of pulsed NMR – self-diffusion studies, as pioneered by Hahn – stood out over the years to come and seemed to fascinate a small but diverse range of researchers in physics and chemistry. Such measurements were originally made in static (constant) magnetic field gradients similar to those present in Hahn’s “non-perfect” original instrument set up. At the time, magnetic field gradients could be achieved either by using a “poor” magnet or by intentionally making the magnetic poles of a typical resistive iron magnet slightly nonparallel, using a thin metal plate shim spacer for

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1 Historical perspectives and basic pulsed NMR concepts

alignment. It’s true that the concept of “shimming” in NMR indeed originates from early procedures for geometrical alignment of electromagnetic pole shoes that also used a huge wrench or Insex/Allen tool for tightening the supporting bolts and nuts. A later development was the introduction of extra current-driven “(Golay) shim coils,” which was meant to be designed to minimize magnetic field gradients across the sample – or to create a correcting field gradient in a chosen direction [19]. Ideally, individual gradients generated in this approach are describable by one of a set of spherical harmonics, and the orthogonal properties of these render possible substantially independent adjustments (Figures 1.4 and 1.5). By proper design, real-world shim coils can be manufactured that also well-approximate such gradient sets over relatively large sample volumes, rather than an ideal point target. An innovative variant was suggested in 2010, in the form of many (originally 24) small, localized coils around the sample area [18]. Of course, modern numeric computational tools help immensely in such context, including design of gradient coils intended for selfdiffusion measurement generation. Turner [20] and Hidalgo-Tobon [21] have described basic gradient coil design principles in considerable detail.

30 20

Z (mm)

10

Hz 8,000

A 1.5

6,000

1

4,000 2,000

0

0

–10

–2,000

–20

–6,000

0.5 0 –0.5

–4,000 –8,000 –30 –20

0 Y (mm)

20 20

10

–1 –1.5

–10 –20 0 X (mm)

Figure 1.4: (left) Golay invented the concept of ideally orthogonal and independent shim coils in NMR in 1958 based on adding together magnetic field gradients from suitably shaped and oriented current-carrying coils. They should ideally be of the general shape of (real) spherical harmonics, illustrated in the picture. Blue regions are positive and yellow negative. The top one would correspond to the main magnetic field (B0) and those in the second row to first-order x, y and z gradients, respectively. Those in the bottom rows illustrate the shape of second- and thirdorder gradients. (From Wikipedia. Figure courtesy of Inigo Quilez, Wikipedia Creative Commons Attribution Share-Alike 3.0 Unported license). (right) Juchem and coworkers [18] have presented an alternative gradient shaping concept that is based on many small, circular and individually controllable gradient coils around the sample. Reprinted with permission from Juchem et al., J Magn Reson. 2010; 204: 281–28; Copyright (2010) Elsevier.

To current supply

gy

Figure 1.5: (left) Golay-type “shim coils” (one on each side) on the sidewalls of a late 1970s generation multinuclear iron magnet NMR probe (for the JEOL FX100). Here, the sample axis is formally thought to be in the vertical magnet “y-direction” and the main magnetic field is in the z-direction. As seen, it was like a dozen individual coils on the probe sidewalls, of varying direction and “order” (i.e., first-, second- and third-order [cf. Figure 1.4]). The probe picture was previously published in Historical: early multi-component FT-PGSE NMR self-diffusion measurements-some personal reflections. Magn Reson Chem, 2017; 55: 386–394, and is reproduced here by permission; Copyright (2017) Wiley. (right) Schematic design of z and y Golay-type “gradient coils” for superconducting magnet geometry. The current leads are omitted for the gy drawing for better clarity (see also Figures 4.2, 4.3 and 9.2–9.5 for actual coil designs). The sample axis is oriented along the main magnetic field direction (z), which on a normal NMR spectrometer type is vertically oriented. In MRI on larger objects, it is horizontal for practical reasons. y- and x-direction gradient arrangements are identical, although oriented at 90° differently along the cylinder axis. Note that in the context of NMR diffusometry there are normal “shim coil sets” (coarse superconducting ones [adjusted upon magnet installation] as well as finer adjustment user-accessible resistive ones for optimization of magnetic field homogeneity at the sample location) and also specialized ones (z-only, or x,y,ztype) with high current capability for actual generation of magnetic field gradients used in NMR diffusometry or imaging context.

Front

Back

Back

Front

z-direction 1.4 Spin-echo NMR diffusometry steeply gains larger popularity in the 1970ies

7

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1 Historical perspectives and basic pulsed NMR concepts

1.5 Pulsed field gradient spin echo NMR opens doors for diffusometry and makes magnetic field gradient-based imaging possible The next methodological step for NMR-based self-diffusion measurements was the idea of generating pulsed magnetic field gradients in the sample through dedicated gradient coils that were directed along the main magnetic field direction. The key concept seems to originate from a somewhat vague suggestion about using “alternating magnetic field gradients” by McCall, Douglass and Anderson in a remarkable 1963 contribution [22]. Here the authors also outline a large family of future application directions for NMR self-diffusion measurements. Stejskal (1932–2011) and his two years older PhD student Tanner achieved an instrumental, theoretical and methodological implementation of the so conceived pulsed magnetic field gradient spin echo (PGSE) NMR technique, as described in a seminal and widely cited 1965 paper [23]. It is part of the content in Tanner’s 1966 thesis. Many pioneering application topics were covered here as well; among them were restricted diffusion using PGSE methodology [24–26] (Figure 1.6) and practical utilization of the three-pulse stimulated echo [27].

y-displacement

Free diffusion

Time

Obstructed diffusion Time

Restricted diffusion Time

Figure 1.6: Illustrating translational diffusion in two dimensions, under three different conditions. Quantification of these through NMR diffusometry will be further discussed in later chapters.

Stejskal and Tanner did not use the acronym PGSE. Together with my former PhD student Michael E. Moseley, I introduced it in the late 1970s. It has subsequently been accepted and is used in main recent monographs on NMR diffusometry. Previously “PFG NMR” and similar were the most common abbreviations, and are still seen. Many feel

1.7 Computer and NMR spectrometer developments

9

that this and similar acronyms do not properly reflect that spin echo formation is at least as important a component of the experiment as the magnetic field gradient one. Of course, main MRI or NMR microscopy techniques also would not be possible without the use of pulsed magnetic field gradients. Originally, computer control of these was not available as a methodological component in either PGSE diffusometry or NMR-based imaging. Today, it is the natural way – combined with computer-controlled digital data acquisition and data processing.

1.6 Frequency-resolved multicomponent NMR diffusometry emerged in the 1970s A few years after the introduction of pulsed FT-NMR, Vold and coworkers in 1969 pointed out that multicomponent spin relaxation can be studied using Fourier transformation of signal responses in multi-pulse NMR [28]. They also mentioned that frequency-resolved FT-PGSE should be a feasible tool for quantifying self-diffusion of individual components of a mixture or solution. The first FT-PGSE demonstration experiment appeared almost five years later [29], and a number of chemically significant ones began to emerge in the late 1970s. A mid-1980s review of FT-PGSE [30], already lists almost 50 applications of NMR-based multicomponent self-diffusion studies in diverse areas of chemistry. With the introduction of specialized commercial NMR instrumentation for this purpose around 1990, the popularity of FT-PGSE has further accelerated and established a valuable, multifaceted and widely available tool. Instrumental FT-PGSE performance is nowadays impressive indeed, and post-processing of data is also much facilitated through the availability of personal computer systems with easy data transfer and data interchange options. Such matters are discussed in greater detail in Chapter 6.

1.7 Computer and NMR spectrometer developments make FT-PGSE easier and a more powerful technique Off-line data processing of FT-NMR data was initially very awkward and difficult, since spectrometer computers and spectrometer vendor software had no standard options for data transfer for off-line processing. One usually had to be content with basic data output like “peak height” or “peak integral” results for the experiment in question, until like the mid-1980s. Today, such operations are truly trivial, which has paved the way for more advanced data processing options, further described in Chapter 6.

10

1 Historical perspectives and basic pulsed NMR concepts

Che m

ical

shi ft

nt fficie

e on co iffusi d f l Se

Figure 1.7: A 3D-like DOSY-mode FT-PGSE results display, summarizing y-axis self-diffusion and x-axis NMR frequency information through contour interpolation. A 2D-like vertical projection contour variant is normally preferred, since peaks then do not visually obscure each other.

In later decades, also a FT-PGSE 2D-type results display mode, generically named “DOSY” (Diffusion Ordered SpectroscopY), have significantly promoted and popularized the use of FT-PGSE for the purpose of quantifying molecular self-diffusion. Charles S. Johnson, Jr. and coworkers introduced it in the early 1990s [31, 32]. In their simplest form, DOSY displays are 2D frequency/self-diffusion contour interpolations over raw one-dimensional PGSE results for successive NMR frequency increments (Figure 1.7). The full spectral frequency dimension resolution (e.g., like 8k or 16k) is normally kept in such data displays, while that in the diffusion dimension is much more limited and is often presented on a logarithmic axis. More complex data processing procedures are discussed or listed in Chapter 6. It should be noted that only some more complex “DOSY variants” would rank as multidimensional NMR in the normal sense (i.e., COSY, NOESY). The introduction of a generic DOSY acronym has unfortunately caused considerable nomenclature and conceptual confusion in the context of NMR diffusometry. “DOSY” (instead of FTPGSE, or PGSE, and similar) is nowadays absurdly common as a method-implying notation in the scientific literature, even in situations where no 2D- or 3D-type results display was ever generated.

1.8 NMR imaging (MRI and NMR microscopy) techniques emerged in the early 1970s Followed by the pioneering work of Lauterbur [33], using a sweep spectrometer and magnetic field gradients in selected directions to coarsely produce a 2D-image slice of water-containing capillaries inside a normal NMR tube, magnetic-resonance– based imaging procedures later developed enormously in complexity and scope. By far the largest application volume is in the medical field and can sample whole

1.8 NMR imaging (MRI and NMR microscopy) techniques emerged in the early 1970s

11

body volumes. In such context it is named “MRI” – magnetic resonance imaging – to avoid the word “nuclear,” which was thought to be scary. Detection in NMR imaging is almost exclusively based on proton NMR, and the signal-generating tissue components are primarily water and fat. Image contrast appears naturally through concentration effects, and the presence of less mobile (and more rapidly spin relaxing) bone structures. Further contrast is achievable by making excitation and detection spatially sensitive to nuclear spin relaxation effects. Those may be “naturally” occurring, or be generated through added “contrast agents,” which are normally based on complexes of paramagnetic ions such as Gd3+ that affect spin relaxation of water and fat in tissue. A special type of brain MRI is called functional MRI (fMRI), where water spin relaxation effects naturally occur as a result of localized metabolic processes in brain tissue that propagate via alteration of levels of oxygen-binding hemoglobin during “mind work” at a particular brain location. When dealing with chemical or physical applications on small (centimeter or millimeter-like) objects, one normally uses the notation “NMR microscopy,” rather than “MRI.” A spatial resolution like 10 µm is achievable in NMR microscopy, while routine whole-body MRI spatial image resolution is limited to 1 mm or so. NMRbased imaging techniques are today based on computer-controlled pulsed magnetic field gradient generation (commonly in three dimensions) as well as pulsed rf excitation and detection similar to that of the NMR-based self-diffusion measurements described in this book. Much of the instrumentation components and detection procedures can thus be used interchangeably between PGSE and MRI/NMR microscopy. Further mutual benefit is achieved through shared use and development of measurement procedures and hardware components, such as gradient coil designs and controlling electronics for gradient generation. Many basic NMR imaging operations and methodological building blocks were originally developed within basic methodological research on small-volume samples. Later it was found as a niche within larger-volume medical MRI. Examples include combining chemical shifts of normal spectroscopic NMR with spatial detection of NMR microscopy, thereby producing chemically separated images of multicomponent materials or tissues. Particularly useful concepts are to spatially detect diffusion and flow, and even to separate them detection-wise, also in heterogeneous materials. “Diffusion MRI” is nowadays a hugely important and advanced technique in medical radiology [34]. Its original medical application relates to (self-diffusion related) contrast in brain tissue after a stroke at the spatial location in question. Notably, corresponding contrast based on spin relaxation is rather poor or nonexistent. The composite experimental procedure is in essence to combine a PGSE sequence with the imaging capabilities of MRI. Flow patterns in blood vessels can similarly be spatially detected and quantified even in three dimensions. Such methodology originates from a very seminal paper by Callaghan, Eccles and Xia [35], where flow and self-diffusion of a laminarly flowing liquid in a thin

12

1 Historical perspectives and basic pulsed NMR concepts

tube were separated experimentally and quantified spatially as well. At the time I personally found the implications of the image in Figure 1.8 quite stunning. As evident in many sections of this book, Paul Callaghan (Figure 1.9) has been a giant in its field of context.

Figure 1.8: Simultaneous NMR microscopy imaging of self-diffusion (left) and laminar flow (right) of CuSO4-doped water in a 0.7 mm inner diameter Teflon tube. For instrumental and data processing reasons at the time, the circular tube appears elliptical and there is a higher relative noise level in the diffusion profile. The flow profile fits almost perfectly to a Poiseuille distribution in space (From Callaghan et al, J Phys E: Sci Instrum. 1988;21:820–822 [35], reproduced with permission, Copyright (1988) IOP Publishing).

Figure 1.9: Sir Paul T. Callaghan (1947–2012), repeating Sir Isaac Newton’s echo experiment that was aimed at quantifying the speed of sound. The study was made at the very same location, in Cambridge, September 2011 (private photo). Callaghan was a giant in both NMR microscopy and PGSE methodology, as clearly manifested in some 300 scientific publications and 2 monographs of huge significance [36, 37]. He also acted as an advisor to the New Zealand government in various matters, and was a great public, as well as scientific communicator. Callaghan founded the company Magritek, which presently makes bench-top, portable and educational-type NMR spectrometer systems, as mentioned elsewhere in this book. Apart from his NMR work, the titles of two of his other books illustrate to some extent his wide areas of influence; “Wool to Weta: Transforming New Zealand’s Culture and Economy” and the educationally oriented “Are Angels OK?: The Parallel Universes of New Zealand Writers and Scientists.” His educational lectures that relate to NMR diffusometry are highly recommended: http://www.magritek.com/support/videos/.

1.10 Outline, scope and required background for the present book

13

1.8.1 Further reading Topgaard D. Multidimensional diffusion MRI. J Magn Reson. 2017; 275: 98–113.

1.9 Electrophoretic NMR emerges as a component-resolved technique In contrast to the simplicity of basic PGSE studies, the closely related electrophoretic NMR (eNMR) technique still remains quite challenging and problematic one in comparison, even several decades after its conceptual introduction [38]. Saarinen and Johnson made the first high-resolution eNMR study in 1988 [39]. Suitable commercial eNMR equipment in the form of a third-party spectrometer add-ons are available and remedies for minimizing the influence of disturbing side effects such as electro-osmosis have also been developed. So, the eNMR feasibility situation is also now much brighter than it was a decade ago. The topic is described in greater detail in Chapter 8.

1.10 Outline, scope and required background for the present book In the following sections, these families of methodologies will be described in more detail, together with sample applications from the literature and preferred experimental and methodological strategies for achieving optimal results. With some sidelines, the text coverage will focus on liquid samples. It is primarily written for newcomers who themselves consider using, or have already tried NMR diffusometry techniques. Some basic knowledge about NMR theoretical and practical experience with experimental NMR in solution will be assumed. Mathematical formalism and duplicate derivations of already existing key relations will be kept to a minimum. Familiarity with the simple vector NMR model suffices for comprehension of the text in this book. Readers who feel unfamiliar with any NMR concepts such as 90o pulse, rotating frame, vector model, dephasing, transverse and longitudinal spin relaxation, phase correction, phase cycling and so on should first consult some updating textbook or web resource on basic NMR techniques in solution (see the list “Further reading”). A “pictorial vector model NMR crash course” is provided as well in Section 1.11 of this chapter. There is also a huge and wide volume of literature in the general area of magnetic field gradient methodology in NMR in the context of MRI and microscopy. Unfortunately, medically MRI-oriented treatises may be misleading and unfair regarding certain aspects. After all, the main methodological MRI pioneers were indeed physicists and chemists, and most key concepts existed well before the

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1 Historical perspectives and basic pulsed NMR concepts

widespread use of MRI. A quote I once saw in a popularized article in the newspaper Newsweek around 1990 is strikingly diagnostic regarding oddly differing perspectives here. It read something like “Nuclear Magnetic Resonance used to be an obscure technique, utilized by chemists and physicists. But now the medical community has developed the field into . . ..”

1.10.1 Further reading The main recent NMR diffusometry texts were already listed in the Preface of this book. For general updates on the NMR background, freely available lecture notes by Keeler at www.keeler.ch.cam.ac.uk/index.html look almost perfectly matched with the present chapter. The latest printed book version is more detailed: Understanding NMR Spectroscopy. Nuclear Magnetic Resonance: Concepts and Methods by Canet is also highly recommended. A well-received and relatively new nonmathematical textbook is by Jacobsen: NMR Spectroscopy Explained: Simplified Theory, Applications and Examples for Organic Chemistry and Structural Biology. Considerably more advanced NMR textbooks are Spin Dynamics: Basics of Nuclear Magnetic Resonance by Levitt and Principles of Nuclear Magnetic Resonance in One and Two Dimensions by Ernst, Bodenhausen and Wokaun. A rather unique pictorial textbook by Blümich; Essential NMR; is highly recommended as a companion book to the present text. Other suitable textbooks, web resources and NMR-related software are listed as text or download links at www.ebyte.it/library/StansNmrLinks.html; “Stan’s NMR, MRI, NQR and ESR links.” An excellent source of links for various NMR resources is www.spincore.com/nmrinfo/ and somewhat similar The Open NMR Project – NMR Wiki can be found at nmrwiki.org. An excellent web-based and partly interactive introduction to basic NMR by Joseph P. Hornak has been available via this link: https://www.cis.rit.edu/htbooks/nmr/ for quite some time. Paul T. Callaghan recorded a remarkable series of demonstration lectures on basic NMR and central concepts in NMR diffusometry available at http://www.magritek.com/support/ videos/ In addition to these, there are various chapters in the multi-volume set Encyclopedia of Nuclear Magnetic Resonance (Wiley, New York, Grant, D.M. and Harris, R.K., original Eds.). It first appeared in 1996 and was later been supplemented with some additional contributions in 2002. If you are fortunate, this hardcover resource could be found at your local library. The original issues appear to be mostly out of print, and are hugely expensive nowadays. This Encyclopedia series morphed in 2007 into an electronic resource, eMagRes, which is still very active. It lists more than 700 articles at the time of writing, written by experts and pioneers in the respective areas (https://onlinelibrary.wiley.com/doi/book/10.1002/ 9780470034590). The journal Concepts in Magnetic Resonance, Series A and B continuously publishes reviews on numerous specialized topics in NMR, and conceptually bridges a gap between an educational and a research level reading.

1.11 Basic pulsed NMR concepts – a pictorial vector model crash course

15

1.11 Basic pulsed NMR concepts – a pictorial vector model crash course An attempt to provide a short summary of text is included here in the form of mainly visual illustrations and brief explaining text. It is meant to serve for basic understanding of the most central elements of NMR spectroscopy, as described within the framework of the semiclassical vector model. The content should be sufficient as background for those with at least basic knowledge of spectroscopy concepts in general and who have had at least some previous experience with NMR. The particular subject of spin relaxation is covered in some additional detail in Chapter 2. For more complete and deeper understanding of various NMR concepts, readers should consult the literature and electronic resources listed in Section 1.10.

1.11.1 Energy levels and frequencies in NMR spectroscopy NMR spectroscopy is based on quantization of nuclear spin states for nuclei with spin quantum number (I) greater than 0, and corresponding energy differences of these in an applied magnetic field (B0 ). The most favorable NMR nucleus is the proton, which has high natural abundance and occurrence and I = 1=2 as well as a high magnetogyric ratio (γ). Spin-1/2 nuclei have two spin states, denoted with red and blue spinning symbols in Figure 1.10. The resonance frequency (ν0 ) that matches the energy gap between these (ΔE) is thus linearly dependent on the applied magnetic field. In the vector model, the “spins” are thought of as tiny magnets themselves, which precess along the applied magnetic field axis (z) at a rate equal to the NMR frequency, but at a random phase relation in the x–y plane. So, at equilibrium and for too many spins (there would be typically more than 1018 in a sample) there will just be a net nuclear magnetization component along the applied field direction, as illustrated in red in the “arrow bundle” part in the upper right section. Note that the vector model view has elements of quantization (energy levels), as well as from classical mechanics (rotation, precession).

1.11.2 The comparably very low frequency of NMR spectroscopy has several consequences The basic NMR signal is quite attenuated in some respects relative to those of other spectroscopy forms. At thermal equilibrium, relative energy-level populations are given by the Boltzmann distribution. This difference is quite small in NMR – only some parts per million. Spectroscopic NMR frequencies are very low as well, and

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1 Historical perspectives and basic pulsed NMR concepts

Net magnetization z

𝜈0 = 𝛾B0/2π

B0

y

Zero field

ΔE in B0

n n

e–ΔE/kBT

Relative energy

x

Figure 1.10: Illustrating quantized nuclear magnetization energy levels in an applied magnetic field (see text).

Zero field

Relative energy

similar to other electromagnetic transitions such as Einstein coefficients for “stimulated absorption and emission,” respectively describe the transition probabilities in general and are denoted as B12 and B21 . As a result of their equality, only the small excess fraction of spins indicated contributes to the actual net energy absorption process (Figure 1.11).

ΔE in B0

“2” B12

A21

n n

e–ΔE/kBT≈ 1.0 ‒

1.6 × 10–5

Thermal equilibrium

General relations; Einstein coefficients

B21

Protons ‒ 100 MHz

ΔE

Fraction actually contributing to absorption signal

“1” Figure 1.11: Illustrating basic energy-related factors that influence NMR spectroscopy very differently than higher-frequency spectroscopy forms (optical or other).

1.11 Basic pulsed NMR concepts – a pictorial vector model crash course

17

“Spontaneous emission,” represented by A21 arrow in the figure is the normal mechanism for maintaining thermal equilibrium of energy level populations. This typically occurs at sub-nanosecond time scales for energy levels relevant to much higher frequency optical spectroscopy. However, the spontaneous emission probability and kinetic rate depend on the frequency or energy gap cubed, and spontaneous emission is totally negligible at NMR frequencies. Instead, various types of non-radiative spin relaxation mechanisms coupled to molecular motion operate, albeit at low rates. Typically re-equilibration after perturbation of the thermal Boltzmann equilibrium typically occurs at millisecond to second time spans. As discussed in the following section, such “spin relaxation” studies are an excellent pathway for measuring molecular reorientational rates rather than translational diffusion, which is the focus of this textbook.

1.11.3 The “rotating frame” description of excitation and emission in NMR “Excitation” in the vector model NMR description is achieved through a rf electromagnetic signal perpendicular to the main magnetic field. It is considered to be taking place through the magnetic component of the electromagnetic field turning the magnetization along that perpendicular axis, i.e., along some axis in the x–y plane in the coordinate system depicted in Figure 1.10. Visualization of its effect on the spin system here becomes very much simplified by introducing a rotating coordinate system – “the rotating frame.” The z-axis is unaffected, but the x–y plane is now thought to rotate at the nominal NMR frequency (in the MHz range), which is set within some kHz of the signal location of interest. In such context one relabels the original “laboratory” axes (x and y) to x′ and y′, respectively (Figure 1.12). Now the MHz oscillating/rotating rf field instead becomes stationary or lowfrequency (typically Hz to kHz) and is denoted as B1 . It can be thought of as being applied in any direction in the x′–y′-plane by changing its phase. Note that one uses coherent excitation in NMR unlike many other types of spectroscopies (exceptions are many laser-based variants). Signal detection is made in a coherent manner as well, and corresponds to measuring spin magnetization in some selectable direction in the x′–y′-plane – also by selecting a “receiver phase” setting. Tipping the magnetization from the equilibrium of z-direction can be done by applying a short rf pulse of chosen phase perpendicular to the z-direction. In the rotating frame this corresponds to applying a static B1 magnetic field in the same direction rather than an MHz rf pulse. The net magnetization will be rotated around the same axis, and a receiver coil can pick it up. In the figure and according to convention, the B1 field is thought to be oriented along the positive x′-axis and the receiver along the negative y′-axis. The angle rotated depends

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1 Historical perspectives and basic pulsed NMR concepts

𝜈1 = 𝛾B1 / 2𝜋 z

z

B1

xʹ

yʹ

xʹ

yʹ

Figure 1.12: NMR excitation and detection related to an oscillating electromagnetic rf field is simplified through introduction of a rotating coordinate system along its direction. A “90° pulse” is illustrated. It corresponds to rotation by a quarter of a turn around the B1 axis at the corresponding frequency ν1, as indicated in the figure. Sign conventions and rotational directions in NMR are such that for an rf pulse along the positive x′-axis and a positive magnetogyric ratio γ (like for protons) the initial rotation direction is actually toward the negative y-axis [40], according to a “right-hand rule” or convention. The net magnetization in the x′–y′-plane is thought to correspond to the “NMR signal” in general and the primary “magnetization detector” is thought to be in the y′-direction. In reality such a 90° signal, for example, from protons would thus be negative in reality. Its phase and sign are further manipulated and transformed electronically and digitally to achieve a “normal, positive” signal.

on the duration and strength of the B1 rf pulse, but as seen a maximum signal (maximum y′-magnetization) occurs for a “90° pulse.” In practice, this setting is calibrated by an experiment. For a normal set up and NMR sample probe, appropriate conditions typically become like 10 µs at common rf power settings of some 100 W. The rf power is not normally changed during experiments, only the rf pulse length. Through frequency analysis (by Fourier transformation) one can deduce that an rf pulse of p seconds excites an NMR frequency region of 1/p Hz. A 10 µs pulse length would thus affect a 100 kHz wide frequency range (although beyond some 10 kHz from the “center frequency” the effect gets attenuated and even reversed in sign). When needed, so-called shaped rf pulse envelopes correspond to more even or tailored frequency coverage and are used in specialized applications.

1.11.4 Spin magnetization dephases with time in the x′–y′-plane and constitutes the time-domain NMR signal – the “free induction decay” One should clearly note that what matters for the magnetization rotation direction is the phase of the exciting rf signal pulse, a concept that can be misunderstood. Its phase is not related to something like a sinusoidal or cosinusoidal first half period,

1.11 Basic pulsed NMR concepts – a pictorial vector model crash course

19

but rather to a long-term signal coherence to a reference frequency, as illustrated in Figures 1.13 and 1.14.

𝜈1 = 𝛾B1/2𝜋 z

z

z B1

yʹ

xʹ

yʹ

xʹ

yʹ

xʹ

At 󰜈0 NMR “0”

Negative Exactly at reference frequency

20 10 0 20

Frequency

Close to reference frequency

20 0

0

2

4

6

8

10

Away from reference frequency

0 –20

Positive

0

2

4

6

8

10

–20 60 40 20 0 –20

0

2

4

6

8

10

(Sum, can be analyzed via FT)

0

2

4

6

8

10

Time Figure 1.13: Illustrating magnetization changes in the rotating frame shortly after the initial 90° pulse for spins with NMR frequencies at a nominal rf frequency and slightly “off resonance.” Magnetization vectors of different frequencies will fan out in the x′–y′ plane and mutually cancel through destructive interference, as well as through transverse (T2 ) spin relaxation processes. The initial “free induction” signal thus decays with time. As illustrated, implying a single “NMR detector” that nominally monitors y′ magnetization, one cannot distinguish between signals with positive and negative NMR frequency deviations from the center frequency.

Figure 1.14: These four “rf-pulses” have the same phase, which they share with the continuous (reference) sine wave as shown.

20

1 Historical perspectives and basic pulsed NMR concepts

1.11.5 Restoration of thermal equilibrium and loss of phase coherence in the x′–y′ plane are two types of spin relaxation processes Transverse spin relaxation also manifests itself in the width of the NMR bandshape of a given group of otherwise identical nuclei; fast relaxation corresponds to broad signals and slow relaxation to narrow ones. Apart from “real” changes in spin states, transverse relaxation of spins is additionally affected by the inhomogeneity in the applied magnetic field, artificially making signals from identical nuclei appear at different NMR frequencies. This basically results in a broadening of the “true” NMR bandshape. Under such conditions, the “apparent” bandwidth can be formally translated into an “effective transverse spin relaxation rate,” and quantified in terms of a corresponding transverse spin relaxation time that is denoted T2* (T2* ≤ T2 ≤ T1 ). The “time constant” characterizing the decay of the FID is T2* , and in most NMR applications one strives to remove the artificial field-gradient–related contributions to it. The common procedure is named “shimming”, that is, careful adjustments of the magnetic field homogeneity at the sample location through currents in “shim coils” of various geometries. Magnetic field homogeneity of 1 part in 100 million or better is routinely achievable on commercial high-resolution spectrometers. A magnitude of further degree of homogenization can be effectively achieved by the averaging effect of spinning the sample along its axis. Note that an inhomogeneously broadened signal only approximately becomes Lorentzian shaped (similar to as discussed in Section 1.11.7). Controllable signal “broadening” of signal bandshapes through an inhomogeneously applied magnetic field is actually the very basis for NMR diffusometry, as discussed in the following sections. Other transverse spin relaxation factors are merely limiting and disturbing factors in the same context. Spin relaxation (Figure 1.15) is discussed in further detail in Chapter 2.

1.11.6 Detection of transverse magnetization is made in a phase-sensitive manner, followed by discrete digitization Magnetization detection on early pulsed FT-NMR spectrometers was made in a way corresponding to only one “detector,” similar to the the y′-direction. As illustrated in Figure 1.13, one cannot distinguish between “positive” and “negative” frequencies in the rotating frame, and had to place the reference frequency on one side of signal-containing region of the sample, making all signals appear as being of the same “sign”. The “two detector” concept discussed below is a more recent invention that solves this problem (Figure 1.16). The misrepresentation artifacts of insufficiently well-sampled periodical signals are often described as “aliasing” or “foldback” in the NMR literature. Such “ghost peaks” or other manifestations would occur if the “sweep width” (as it is called

1.11 Basic pulsed NMR concepts – a pictorial vector model crash course

Non-Boltzmann distribution after rf absorption

z

Thermal equilibrium

yʹ

xʹ

21

z

yʹ

xʹ

z

Restoration of Boltzmann energy distribution – longitudinal, or T1 spin relaxation xʹ

yʹ

Loss of phase coherence in the x ʹ‒y ʹ plane ‒ transverse, or T2 spin relaxation

z

z

yʹ xʹ

xʹ

yʹ

Figure 1.15: The upper part illustrates application of a 180° pulse, rather than a 90° pulse. Ideally, no magnetization would remain in the x′–y′ plane, and thus there would be no time-domain (FID) NMR signal. “Longitudinal” magnetization would recover back to equilibrium, characterized by the longitudinal spin relaxation time, T1 . Loss of phase coherence and net magnetization in the x′–y′ plane is called transverse spin relaxation, and is similarly characterized by a transverse spin relaxation time, T2 . A maximum transverse magnetization can be achieved with a “90° pulse,” as illustrated. Ideally, both recovery curves are exponential and both relaxation processes occur in parallel. Relaxation rates differ for different nuclei in the sample, chemical or physical. Transverse spin relaxation has more contributions than longitudinal, and is thus characterized by faster equilibration, that is, in general T2 ≤ T1 for a given group of nuclei.

even on non-sweep, pulsed FT spectrometers) is made too small, or if the reference frequency is not centered in the region of interest.

1.11.7 Rotating frame magnetization is monitored through quadrature detection and complex Fourier transformation The time evolution of magnetization corresponding to a single signal (“1”) after an x′-directed 90° pulse is visualized as a low-frequency rotating vector in the x′-y′-plane with an angular difference frequency ν1 from the reference frequency of the rotating frame. Note clearly that this quantity or effect should not be mistaken for that previously illustrated in Figures 1.12 and 1.13 above, although both use the same traditional notation. In more mathematical terms its components can be summarized mathematically as follows:

22

1 Historical perspectives and basic pulsed NMR concepts

Detecting time evolution of magnetization vectors after a 90° pulse z

z

z

B1

xʹ

yʹ

yʹ

xʹ

yʹ

xʹ

y ʹ-signal

x ʹ-signal

1 0

0

–1

–1

0

2

4 6 y ʹ-signal

1

8

10

–1

Discrete digital detection – the Nyquist frequency: A signal must be sampled at more than twice its own frequency

0

2

4 6 x ʹ-signal

8

10

0

2

4

8

10

1

0

0 0

2

4

Signal value

Using two “detectors” (in the xʹ and the yʹ directions) one can distinguish between positive and negative frequencies

1

6

8

f=1,845.3 Hz 1

10

–1

6

ɸ=0° fs = 2,000 Hz fs/2=1,000 Hz

0 –1 0

5

10 Time (ms)

15

20

Figure 1.16: Thinking in terms of two orthogonal detectors (like x′ and y′ oriented), one can logically separate positive and negative frequencies in the rotating frame (although the actual detection mode differs somewhat, as described in the following figure). Placing the nominal NMR frequency in the middle of the spectrum is then no longer a problem. On a one-detector system, signals would have incorrectly been “reflected” from one side and appeared at an artificial frequency with the wrong sign. In addition, there is a more general and fundamental issue regarding sampling of periodic signals; the Shannon theorem and the Nyquist sampling rate, as illustrated and described in the lower right hand part of the Figure. The example 1845.3 Hz “high frequency” sine wave (red) is not sampled well enough here (only at 1,000 Hz) and instead appears as representing a lower frequency (blue), being 2000 − 1845.3 = 154.7 Hz (illustration prepared through Matlab code by Costas Vlachos; “Interactive demo of Nyquist’s sampling theorem”, available through Mathworks File Exchange).

My′ = M0 expð − t=T2* Þ · cosð2πν1 · tÞ

(1:1)

Mx′ = M0 expð − t=T2* Þ · sinð2πν1 · tÞ

(1:2)

Mz = M0 expð − t=T1 Þ

(1:3)

For many signal contributions of various frequencies one cannot directly measure or analyze the individual contributions. For further analysis one often introduces the concept of complex magnetization describing magnetization evolution in the rotating frame through Euler’s formula (with x expressed in radians); eix = cosðxÞ + i · sinðxÞ. This view of a rotating vector in a mixed real/imaginary coordinate system may initially look unfamiliar to those non-mathematically minded, but it is the common

1.11 Basic pulsed NMR concepts – a pictorial vector model crash course

23

mathematical tool to further describe and analyze such matters. The conceptual components are graphically visualized in Figure 1.17. 4 3 2 1 0 –1 –2 –3 –4

y = exp(ix)

0

1

2

Phase sin(x) angle

3

4

cos(x)

5 x-value

6

7

8

9

10

Figure 1.17: Rotation, as described in terms of a complex exponential is equivalent to that of a sum of complex trigonometric cosine and sine functions.

Equations (1.1) and (1.2) can then be combined to describe the overall magnetization time evolution more compactly: Mt = M0 expð − t=T2* Þ · expð2i · πν1 · tÞ

(1:4)

For retrieving the actual NMR spectrum in the frequency domain one applies a complex Fourier transform, the result of which (FTðMt Þ) becomes FTðMt Þ =

M0 T2* 1 + 4π2 T2*2 ðν − ν1 Þ2

−i

M0 T2* 2πðν − ν1 Þ 1 + 4π2 T2*2 ðν − ν1 Þ2

(1:5)

at a given frequency location (ν) in the so-created actual NMR frequency-domain spectrum. The first term is named the real part of the frequency domain spectrum, the absorption part or spectrum (A) or the in-phase magnetization. For a single absorption line it corresponds to a Lorentzian shaped function with a maximum at ν = ν1 and has a width at half height of 1=πT2* . The second term is called the imaginary part, the dispersion part (D) or the out-of-phase magnetization. Ideal bandshapes like those illustrated in Figure 1.18 are not normally the direct result of Fourier transformation of the detected complex spin magnetization. Rather, the outcome looks like that illustrated in Figure 1.19. One normally also finds phase deviations in a multiline spectrum that change linearly in magnitude along the frequency axis. They can be similarly adjusted mathematically by introducing linearly frequency-dependent signal phase adjustments – again by corresponding admixture of original absorption and dispersionmode intensity contributions. This is named “first-order phase correction” and is done in practice in an alternating manner, and in parallel with the zero-order phase correction (Figure 1.20).

24

1 Historical perspectives and basic pulsed NMR concepts

1 0.5 0 –0.5

0

20

40

60

80

100

Figure 1.18: The basic NMR bandshape components, as quantified in eq. (1.5). The absorption bandshape is blue and the dispersion bandshape is red. Note that the latter has considerable intensity far from the center. This particular pair corresponds to a signal that is thought to be centered at 50 Hz on the frequency x-axis and characterized by a T2* of 1 s.

0.8 0.6 0.4 0.2 0 –0.2 –0.4

0

20

40

60

80

100

Figure 1.19: The following bandshapes result from “borrowing” 30% dispersion mode signal to the nominal absorption mode signal and equally “borrowing” 30% absorption mode signal into the dispersion one. This can be though in terms of having a wrong reference phase. The result becomes “incorrectly phased signals,” which one adjusts mathematically (“phase corrects”) through interactive or partially automatized computer operation before further use. A basic (zeroorder) phase correction affects signals of all frequencies in the spectrum equally.

1

1

0.5

0.5

0

0

–0.5

–0.5

–1

0

0.1

0.2

0.3

–1

0

2

4

6

8

10

12

14

16

Figure 1.20: One reason for a linearly changing signal phase along the frequency axis can be the introduction of incorrect phase information, simply through deviations from the true zero reference time during data acquisition. Data acquisition is never started immediately after a strong rf pulse, as it ideally is thought to be. The figure illustrates five FID signal components with varying frequencies. The left part is an enlargement of the right part, and shows the varying phase deviation that occurs with a time origin that is not ideally zero, but rather 0.1 on this arbitrary time scale (dashed vertical line).

1.11 Basic pulsed NMR concepts – a pictorial vector model crash course

25

Folded signals (as described in Section 1.11.6) may appear to have a signal phase that differs from those that are actually present, and cannot both be “corrected” with the same “phasing settings.” Their origin is that spectrometer detection systems additionally have filters that steeply dampen (and phase-twist) frequencies that originate from outside the nominal “sweep width” or frequency cutoff. The main function of such filtering procedures is to prevent noise from outside the region of interest to fold back and become added to the actually existing spectral noise in the frequency range studied. With digital rather than analog filters of this same kind, such phase anomalies are less pronounced. “Magnitude spectra” should finally be mentioned in this context. They are generated mathematically by simply adding squared contributions from the real (Re) and imaginary (Im) parts, that is, T2* M = ðRe2 + Im2 Þ1=2 = ðA2 + D2 Þ1=2 = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 + 4π2 T2* ðν1 − νÞ2

(1:6)

All phase information is thus discarded in this procedure. The basic bandshape becomes somewhat odd with rather wide wings, as a result of mixing the wide dispersion mode signal into the more narrow normal absorption one. Magnitude representation is useful under certain circumstances, although one should note that such individual bandshapes are not additive in overlapping regions of multisignal spectra (Figure 1.21).

1

0.5

0

0

20

40

60

80

100

Figure 1.21: Lorentzian absorption (blue) and Lorentzian magnitude spectra (magenta), as compared to the same frequency and intensity axes. The underlying data are the same as in previous figures.

1.11.8 Actual pulsed NMR hardware and detection procedures An NMR spectrometer is a fairly complex device of high sensitivity and stability. Modern high-field, high-resolution ones are designed to have a resolution of less than a tenth of an Hz, for example, at 500 MHz absolute frequency, which

26

1 Historical perspectives and basic pulsed NMR concepts

corresponds to two parts in 1010. The stability and the magnetic field homogeneity of the system must also match such standards. Deuterium-based field/frequency lock on a typically solvent signal in the sample is one key step. Careful magnet shimming and sample spinning must be applied as well to achieve performance at this level. In NMR diffusometry procedures described in later chapters, resolution demands are lower. A spectral resolution of a few Hz on a non-spinning sample should still be considered as being normal measuring conditions, unless the sample basically has broad NMR signals to begin with. As discussed earlier, the relatively weak NMR signal response from the probe, following something like a powerful 90° rf pulse is called a FID. It is basically a damped cosinusoidal signal, which contains information on its NMR frequency and bandwidth. In the Subsection 1.11.7 we described the procedure of complex Fourier transformation to extract these data from the raw signal. The two signal components required here are generated as illustrated in Figure 1.22. It is left as an exercise to the reader to confirm the partial results shown when multiplying cosð2πν0 tÞ with cosð2πν1 tÞ or sinð2πν1 tÞ, as done electronically in the mixer blocks. A table of trigonometric identities will be helpful. cos(2𝜋𝑣0t) Preamplifier Signal

cos(2𝜋𝑣1t)

cos(2𝜋(𝑣1–𝑣0)t)

Mixer

Re

cos(2𝜋𝑣1t) cos(2𝜋𝑣0t)

Im 90° Phase shifter

sin(2𝜋𝑣1t)

Mixer

exp(2i𝜋(𝑣1–𝑣0)t)

sin(2𝜋(𝑣1–𝑣0)t)

Figure 1.22: Basic principle and building blocks for extracting signals for the quadrature detection mode. The mixer blocks additionally create equally large (ν1 + ν0) high frequency components, which are electronically or digitally removed by low-pass filtering before digitization of the Re and Im signals. The mixer blocks can also be conceptually seen in terms of the more general concept “phase sensitive detector” (see e.g., Wikipedia entries on “Phase detector” and “Lock-in amplifier”). Referring again to potential notation confusion in this context, a traditional figure of this kind (and in Figure 1.23) refers to actual NMR frequencies, not difference ones, as in Section 1.11.7.

1.11.8.1 Further reading Traficante DD. Phase-sensitive detection Part I: Phase, gates, phase-sensitive detectors, mixers, and the rotating frame. Concepts Magn Reson. 1990; 2: 151–167. Traficante DD. Phase-sensitive detection Part II: Quadrature phase detection. Concepts Magn Reson. 1990; 2: 181–195.

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27

1.11.9 Correcting basic non-idealities of pulse generation and data acquisition Considering significant demands on electronic circuits and other matters in the detection scheme in the previous section 1.11.8, it seems likely that its straightforward application may lead to artifacts. Two sources of non-idealities would be phase relations that are not being spaced exactly 90° and pick-up and amplification channel unbalance for x′ and y′ signals. By suitable phase cycling, the effects and artifacts arising from such origin can largely be cancelled out. The original idea seems traceable in the PhD thesis of D.I. Hoult (Oxford, 1973), and later appeared also in a journal paper [41]. The basic phase cycling scheme was named CYCLOPS, and has been followed by numerous others, that were designed for various purposes (see Section 2.9) (Figure 1.23).

Scan # Pulse phase 1

+xʹ

Initial magnetization direction

Mxʹ

Myʹ

–yʹ

sin(2𝜋(𝜈1–𝜈0)t)

–1.1 cos(2𝜋(𝜈1–𝜈0)t)

cos(2𝜋(𝜈1–𝜈0)t)

1.1 sin(2𝜋(𝜈1–𝜈0)t)

2

+yʹ

+xʹ

3

–xʹ

+yʹ

–sin(2𝜋(𝜈1–𝜈0)t)

1.1 cos(2𝜋(𝜈1–𝜈0)t)

–xʹ

–cos(2𝜋(𝜈1–𝜈0)t)

–1.1 sin(2𝜋(𝜈1–𝜈0)t)

4

–yʹ

Cycled and added Re FID = –Myʹ(1) + Mxʹ(2) + Myʹ(3) – Mxʹ(4) = 4.2 cos(2𝜋(𝜈1–𝜈0)t) Cycled and added Im FID = +Mxʹ(1) + Myʹ(2) – Mxʹ(3) – Myʹ(4) = 4.2 sin(2𝜋(𝜈1–𝜈0)t) Figure 1.23: Illustrating phase and amplifier unbalance cancellation through the CYCLOPS scheme. It is a four-step cycle where pulse phase and receiver phase both follow the sequence +x, +y, −x, −y, in which their relative phase difference remains constant. Each step in the cycle therefore results in the same bandshapes, which are incrementally added together, as required. Individual outcomes are summarized in the table graph, also together with a demonstration of amplifier unbalance cancellation, where it is assumed that the y′-channel has a 10% higher signal gain than the x′-channel. After four scans, imbalances are cancelled out. (Largely after an example in Jacobsen, “NMR Spectroscopy Explained” (Wiley, 2007), page 211).

Especially in the context of PGSE diffusometry discussed later, 180° “inversion pulses” are important building blocks in pulse sequences. Artifacts and nonideal behavior is expected if they do not fully work as intended. A suitable procedure was suggested quite some time ago in the context of early 2D NMR methodological development and was named EXORCYCLE [42]. Similarly to the CYCLOPS outcome, 180y and 180-y pulses should lead to the same result, as would 180x and 180-x pulses, except for an opposite sign.

28

1 Historical perspectives and basic pulsed NMR concepts

In any normal application a 180° inversion would be preceded (or followed) by something like a 90° pulse to create an actual NMR signal. A “90–180-(spin-echo) experiment” based on these ideas could thus be composed by co-adding and subtracting four subsequent scans where τ represents some chosen time delay. 90x′ − τ − 180y′ − τ − ðAcqÞ + 90x′ − τ − 180 − y′ − τ − ðAcqÞ + 90x′ − τ − 180x′ − τ − ðAcqÞ − 90x′ − τ − 180 − x′ − τ − ðAcqÞ −

1.12 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

Hahn, EL. Nuclear induction due to free Larmor precession. Phys Rev. 1950; 77: 297–298. Hahn, EL. Spin echoes. Phys Rev. 1950; 80: 580–594. Bloch, F. Nuclear induction. Phys Rev. 1946; 70: 460–474. Bloch, F., Hansen, WW., & Packard, ME. Nuclear induction. Phys Rev. 1946; 69: 127. Purcell, EM., Torrey, HC., & Pound, RV. Resonance absorption by nuclear magnetic moments in a solid. Phys Rev. 1946; 69: 37. Hahn, EL. Lille conference talk. J Magn Reson. 2006; 179: 9–10. Hahn, EL. NMR and MRI in retrospect. Philos Trans R Soc London. 1990; A 333: 403–411. Cooley, JW., & Tukey, JW. An algorithm for the machine calculation of complex Fourier series. Math Comput. 1965; 19: 297–301. Ernst, RR., & Anderson, WA. Application of Fourier transform spectroscopy to magnetic resonance. Rev Sci Instrum. 1966; 37: 93–102. Hahn, EL., & Maxwell, DE. Chemical shift and field independent frequency modulation of the spin echo envelope. Phys Rev. 1951; 84: 1246–1247. Hahn, EL., & Maxwell, DE. Spin echo measurements of nuclear spin coupling in molecules. Phys Rev. 1952; 88: 1070–1084. Gutowsky, HS., McCall, DW., & Slichter, CP. Coupling among nuclear magnetic dipoles in molecules. Phys Rev. 1951; 84: 589–590. Gutowsky, HS., McCall, DW., & Slichter, CP. Nuclear magnetic resonance multiplets in liquids. J Chem Phys. 1953; 21: 279–292. Hartmann, SR., & Hahn, EL. Nuclear double resonance in the rotating frame. Phys Rev. 1962; 128: 2042–2053. Pulsed Magnetic Resonance: NMR, ESR and Optics. A recognition of E.L. Hahn. Oxford: Oxford University Press, 1992. Freeman, R., & Morris, GA. The Varian story. J Magn Reson. 2015; 250: 80–84. Xia, Y., & Stilbs, P. The first study of cartilage by magnetic resonance: A historical account. Cartilage. 2016; 7: 293–297. Juchem, C., Nixon, TW., McIntyre, S., Rothman, DL., & de Graaf, RA. Magnetic field modeling with a set of individual localized coils. J Magn Reson. 2010; 204: 281–289. Golay, MJE. Field homogenizing coils for nuclear spin resonance instrumentation. Rev Sci Instrum. 1958; 29: 313–315.

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[20] Turner, R. Gradient Coil Systems. In: Encyclopedia of NMR Spectroscopy. DM Grant & RK Harris, eds. New York: Wiley, 1996:2223–2233. [21] Hidalgo-Tobon, SS. Theory of gradient coil design methods for magnetic resonance imaging. Concepts Magn Reson A. 2010; 36A: 223–242. [22] McCall, DW., Douglass, DC., & Anderson, EW. Self-diffusion studies by means of nuclear magnetic resonance spin-echo techniques. Ber Bunsenges Phys Chem. 1963; 67: 340–366. [23] Stejskal, EO., & Tanner, JE. Spin diffusion measurements. spin echoes in the presence of a time-dependent field gradient. J Chem Phys. 1965; 42: 288–292. [24] Tanner, JE., & Stejskal, EO. Restricted self-diffusion of protons in colloidal systems by the pulsed-gradient, spin-echo method. J Chem Phys. 1968; 49: 1768–1777. [25] Tanner, JE. Transient diffusion in a system partitioned by permeable barriers. application to NMR measurements with a pulsed field gradient. J Chem Phys. 1978; 69: 1748–1754. [26] Tanner, JE. Transient diffusion in a system partitioned by permeable barriers. Application to NMR measurements with pulsed field gradient. J Phys Chem. 1978; 69: 1748–1754. [27] Tanner, JE. Use of the stimulated echo in NMR diffusion studies. J Chem Phys.1970; 52: 2523–2526. [28] Vold, RL., Waugh, JS., Klein, MP., & Phelps, DE. Measurement of spin relaxation in complex systems. J Chem Phys. 1968; 48: 3831. [29] James, TL., & McDonald, GG. Measurement of the self-diffusion coefficient of each component in a complex system using pulsed – gradient Fourier transform NMR. J Magn Reson. 1973; 11: 58–61. [30] Stilbs, P. Fourier transform pulsed-gradient spin-echo studies of molecular diffusion. Prog Nucl Magn Reson Spectrosc. 1987; 19: 1–45. [31] Johnson, CS. Diffusion ordered nuclear magnetic resonance spectroscopy: Principles and applications. Prog Nucl Magn Reson Spectrosc. 1999; 34: 203–256. [32] Morris, KF., & Johnson, CS,Jr. Diffusion-ordered two-dimensional nuclear magnetic resonance spectroscopy. J Am Chem Soc. 1992; 114: 3139–3141. [33] Lauterbur, PC. Imaging formation by induced local interactions: Examples employing nuclear magnetic resonance. Nature. 1973; 242: 190–191. [34] Jones, DK. Diffusion MRI – Theory, Methods and Applications. Oxford: Oxford University Press, 2011. [35] Callaghan, PT., Eccles, CD., & Xia, Y. NMR microscopy of dynamic displacements: k-space and q-space imaging. J Phys E: Sci Instrum. 1988; 21: 820–822. [36] Callaghan, PT. Principles of Nuclear Magnetic Resonance Microscopy. Oxford: Oxford University Press, 1991. [37] Callaghan, PT. Translational Dynamics & Magnetic Resonance, Principles of Pulsed Gradient Spin Echo NMR. Oxford: Oxford University Press, 2011. [38] Holz, M., Lucas, O., & Müller, C. NMR in the presence of an electric current. Simultaneous measurements of ionic mobilities, transference numbers, and self-diffusion coefficients using an NMR pulsed gradient experiment. J Magn Reson. 1984; 58: 294–305. [39] Saarinen, T., & Johnson, C,Jr. High-resolution electrophoretic NMR. J Am Chem Soc. 1988; 110: 3332–3333. [40] Levitt, MH. The signs of frequencies and phases in NMR. J Magn Reson. 1997; 126: 164–182. [41] Hoult, DI., & Richards, RE. Critical factors in the design of sensitive high resolution nuclear magnetic resonance spectrometers. Proc R Soc Lond A. 1975; 344: 311–340. [42] Bodenhausen, G., Freeman, R., & Turner, D. Suppression of artifacts in two-dimensional J spectroscopy. J Magn Reson. 1977; 27: 511–514.

2 Basic concepts 2.1 Molecular transport processes and the NMR time window When heating a system, the energy added increases its internal kinetic energy, leading to overall increased rates of molecular and particle motion. Apart from macroscopic convection and convection-like phenomena, caused by thermal or density gradients in the system, motional partitioning into internal molecular motions (rotation about bonds and vibration), overall reorientational diffusion and translational (lateral) diffusion of molecules and aggregates will also result. The history of translational diffusion is particularly fascinating, and the concept has been fundamental for the general development of natural sciences, starting more than 100 years ago [1, 2]. Temperature itself equates to the average kinetic energy in a system, and the transfer of energy through collisions between molecules also determines the rate of chemical reactions. For these to occur, molecules must get into close proximity on the molecular level as well – and transport processes of the above kinds again enter in [3, 4]. Many types of motions are indeed of fundamental importance for chemical processes in Nature – from biological ones to industrial and geological ones. Time scales may span from picoseconds to billions of years. Radiation-induced rather than thermally related collisional molecular transformations and reactions are exceptions, of course. Basically, intermolecular encounters lead to molecular transport characterized by “random walks” in space, originally seen and described by the botanist Robert Brown in 1827, albeit on macroscopic objects – water-suspended particles in pollen grains. The term “Brownian motion” was later coined, and nowadays generically refers to reorientational as well as translational motion, resulting from molecular and particle collisions (Figure 2.1). Importantly, molecular transport rates like those characterizing self-diffusion also contain information on molecular size, molecular aggregation and molecular interactions of various kinds. As such, they are very valuable sources of physicochemical information. The time scale of diffusometry measurements depends on the method chosen. For the normal pulsed magnetic field gradient NMR techniques and variants focused on in this book, it is to some extent selectable in a millisecond to second range, depending on the instrumental performance and system spin relaxation characteristics. Modulated gradient methods described in some detail in Section 7.3 push it considerably further toward short observation time windows. One should note at this point that even large macromolecules in solution normally diffuse several magnitudes further than their own radii during the corresponding time window. For normal liquid systems, the observation time does not really matter over many magnitudes of time. The situation may get very different for complex liquids, porous media, https://doi.org/10.1515/9783110551532-002

2.1 Molecular transport processes and the NMR time window

31

40 30 20 10 0 –10 –20 –30 –40 40 30 40

20 10

20

0 –10

0 –20

–20

–30 –40 –40

Figure 2.1: The outcome of 50 random walks of 10,000 steps each in three-dimensional space. The x-, y- and z-axes have equal and arbitrary dimensions. Mark Hoyle’s code for “Random Walks in Matlab” (Mathworks File Exchange web site) was used for this illustration and that in Figure 1.6. Note that the cluster center statistically remains at the origin. In the molecular world and in an NMR sample, there would be about 1020 trajectories of a trillion steps each during a second. Statistical averaging is then close to exact, and the actual distribution will be smooth, as the one in Figure 2.6.

plastic crystals and concentrated polymer solutions and melts. By varying the “diffusion observation time” in such a context, as combined with suitable often-complex theoretical interpretation, transport conditions, mechanisms and confinement dimensions can be quantified and mapped in considerable detail (Figure 2.2). One also considers three basic types of diffusion processes – self-diffusion (also named tracer diffusion in older literature), mutual diffusion (also named interdiffusion) and rotational diffusion. Self-diffusion is the actual topic of this book, and it can be studied by the magnetic field gradient NMR methods described in the following sections. Mutual diffusion refers to relaxation of concentration gradients in nonequilibrium systems, and basically results from selfdiffusion processes as well. It has traditionally been studied by optical methods, spatially monitoring overall concentration in various ways (through color or refractive index changes). All types of diffusional processes result from molecular collisions, which lead to random exchange of positions in space as well as molecular reorientation.

32

2 Basic concepts

Quasielastic neutron scattering

10–12

10–9

Field-cycling NMR relaxometry

Field gradient spin-echo NMR variants

NMR microimaging 10–3

10–6

Isotopic tracer diffusion

100

103

106

Diffusional time scale/s Figure 2.2: Schematic illustration of the time scales of alternative techniques for studies of translational molecular diffusion. The full ranges indicated are not necessarily coverable in practice for a particular method, system and instrumentation setup (largely after Kimmich and Fatkullin [5] and information modified).

Collision frequencies and intensities are directly linked to the system temperature, and so do average translational displacements and molecular reorientation rates.

2.2 Rotational diffusion Molecular and particle collisions also result in random stepwise reorientation, apart from translational motion in space. In the gas phase, reorientational rates are quantized and well described only from inertial considerations. The gasphase experimental tool is primarily microwave spectroscopy (also specifically called rotational spectroscopy), which operates in frequency ranges that match the quantized transitions between various rotational states (i.e., GHz and THz bands). Depending on the frequency range chosen, internal motions such as vibration modes and rotation around single or partially single bonds also affect the spectra. Furthermore, rotational and vibrational state lifetimes are pressure dependent, reflecting that probabilities for collisions to occur vary with concentration. Intermolecular forces and other interactions are normally of no concern in the gas phase. In the liquid state, a macroscopic classical view can be applied. Collisions still lead to reorientation, but are more frequent than in the gas phase. They also occur in much smaller steps and typically at picosecond intervals. In analogy with the Stokes–Einstein–Sutherland equation discussed later (2.5), rotational diffusion of a sphere of radius r in a medium with viscosity η, the pertinent relation is described by Drot = kB T=8πηr3 . It further characterizes the average angle squared hθ2 i rotated in a given time t through an Einstein-like (2.4) rotational analogue: that is, hθ2 i = 2Drot t. Here kB represents Boltzmann’s constant and T the absolute temperature. For example, for a water molecule, broadly an average reorientation of one radian occurs at closely 1 ps at the room temperature.

2.2 Rotational diffusion

33

Rotation rates around various axes will differ for various reasons. For a real molecule rather than a sphere, intermolecular interactions also operate and will affect rotation rates along given molecular axes. Generally, such a motion is not inertia related, as in the gas phase, but instead is governed by energetic barriers toward geometric displacement of other molecules during rotation (Figure 2.3). A disk-like molecule will thus reorient more rapidly in the plane of the disk, rather than transverse to it. The so-called Perrin frictional factors exist for rotation and quantitatively characterize geometry-related macroscopic reorientational retardation factors to good approximation.

Figure 2.3: Molecular reorientation also originates from molecular collisions and can be seen as a diffusional process, separate from molecular translation. In the gas phase (and as studied through rotational spectroscopy in the microwave region) it can be described in the form of rotational motion around the center of gravity, and along its mass/position-related principal axes of inertia. In solution, rotational motion is not inertial (see text) and intermolecular interactions such as hydrogen bonding also alter the situation and modify the position of the rotational center as well as the directions of the principal axes. This complicates interpretation of spin relaxation-based NMR data and those of other techniques. The figure was generated with the Avogadro software and schematically illustrates rotational motion of phenol around arbitrarily chosen axis directions, originating from the center of gravity.

Reorientational molecular motion cannot be studied by the magnetic field gradient NMR methods, which are the actual topic of this book. Instead, such processes lend themselves well to NMR quantification through measurements of nuclear spin relaxation, provided the dominant spin relaxation mechanism actually is purely intramolecular, and hence directly linked to reorientation (Figure 2.4). This is nicely the case for carbon-13 with directly bound protons (dipolar mechanism) and for deuterium spin relaxation (quadrupolar mechanism). The other main experimental technique for studying rotational diffusion in solution is dielectric relaxation. This technique is macroscopic and not component resolved, like NMR spin relaxation.

34

2 Basic concepts

10 1

Small molecules

Higher field

Lower field

T1 T2

T1

10–1 T1, T2 10–2

Large molecules polymers, proteins

10–3 10–4

T2 2πν0𝜏c = ω0𝜏c ≈ 1

Solids

10–5 10–12 10–11 10–10 10–9 10–8 10–7 10–6 𝜏c (correlation time)

10–5

Increasing viscosity or molecular size Figure 2.4: Schematic overview of the relation between (dipolar) nuclear spin relaxation times T1 and T2 and characteristic “correlation times” τ c for molecular reorientation (broadly, the average time it takes for a bond vector in a molecule to reorient one radian; both are expressed in seconds). The basic “BPP” theory for spin relaxation stems back to work by Bloembergen, Purcell and Pound in 1948 [6], who considered spin relaxation induced by motionally fluctuating magnetic dipolar interactions between protons. The numeric y-axis figures and other details of the diagram here are only indicative. Actual spin relaxation times depend on the nucleus studied, as well as the actual relaxation mechanism. Those indicated in the diagram should be relatively near the magnitude to be expected for dipolar proton–proton relaxation (observing protons) or proton–carbon-13 relaxation (observing carbon-13) at normal NMR frequencies. The corresponding relaxation rates for, for example, quadrupolar deuterium would be like a magnitude more rapid. This figure is a slightly redrawn version of a NMR web resource created by Hans J. Reich, at the University of Wisconsin (https://www. chem.wisc.edu/areas/reich/nmr/), who kindly allowed to use in this book. The diagram is basically similar to that in the original BPP paper. The leftmost overlapping T1 and T2 curves have been separated here for graphic clarity.

In almost all contexts of the main topics of this book (i.e., NMR diffusometry), spin relaxation is merely a destructively limiting complication from a measurement point of view. But translational diffusion indeed contributes significantly to intermolecular dipolar spin relaxation in, for example, proton-rich samples such as pure water. Here, modulation of dipolar interaction is induced through varying proximity and orientation between protons in separate molecules. However, inter- and intramolecular contributions to proton–proton dipolar spin relaxation are entangled and interdependent. Torrey wrote the pioneering paper in this field in 1953 [7] (Figure 2.5). Separation of these two contributions to the observed spin relaxation rate (i.e., 1=T1 ðobsÞ = 1=T1 ðinterÞ + 1=T1 ðintraÞ) may still be possible, albeit with some difficulty. Tools include isotopic dilution with deuterium, or the addition of paramagnetic

2.2 Rotational diffusion

m

35

(a) Intramolecular and intermolecular N

B 3cos2θ –1 r3

S

r

1H

r 12C

B0

r

1H 1H

r

r

1H

r

1H

r r

12C

1H

12C

θ r

(b) 13C

1H

(c)

2

r

Intramolecular

12C

H Electric field gradient along bond

Intramolecular

Figure 2.5: Illustrating dipolar interactions between two nearby spins and quadrupolar interaction with an electric field gradient. Brownian motion, induced by collisions, causes such interaction to vary randomly. The general dipole intensity factor is proportional to gyromagnetic ratio squared of the nucleus in question, and its geometrical dependence is given by ð3cos2 ðθÞ − 1Þ=r 3 . Intramolecular reorientation (rotational diffusion) has no distance influence in a case like (b), and the interpretation of spin relaxation data in terms of molecular reorientation becomes easy. Relative intermolecular molecular motion (like in (a)) causes both the angular and the distance term vary. Frequency components of varying dipolar fields that are multiples or fractions of the actual NMR frequency are those that cause nuclear spin relaxation. (c) Quadrupolar spin relaxation is exclusively intramolecular, and is related to electric field gradients along internal chemical bonds. Therefore, it only relates to rotational diffusion. Many nonspin-1/2 nuclei have quadrupole moments that are too large for wide NMR applicability in this context. Notable clear-cut candidates for rotational diffusion studies are carbon-13 bound to protons, under wide band spin decoupling conditions and covalently bound deuterium. Combining carbon-13 and deuterium spin relaxation data make the approach even more significant for studying rotational diffusion of organic molecules, especially when supplemented by data from varying NMR frequencies. Dipolar field drawing, courtesy of Wikipedia (by Maschen – Own work, Public Domain, https://commons. wikimedia.org/w/index.php?curid=16179829).

substances. Both primarily affect the intermolecular contribution to nuclear spin relaxation. Such diffusional studies on multicomponent systems through nuclear spin relaxation cannot be made component separated, in the normal sense of NMR field gradient methods or other spin relaxation studies. Reasonably detailed information on intermolecular motion can still be extracted from spin relaxation, especially when varying the NMR frequency. Normal NMR has relatively limited frequency range in this respect; most nonspecialist facilities today would not cope with more than a factor of 2 or so. Much wider and different ranges can be investigated through the relatively exotic tool of field cycling NMR relaxometry; see, for example, Kimmmich et al. [5, 8, 9]. The lateral transport time scale of field cycling relaxometry is correspondingly wide and differs from that of magnetic field gradient techniques discussed in this book. The time window also

36

2 Basic concepts

bridges the gap to quasielastic neutron scattering based data (see Figure 2.2) and samples motion at time scales that are very significant and decisive regarding certain types of molecular motion. Korb recently summarized the general topic of magnetic relaxation dispersion and its applications in an extensive review [10]. Deeper understanding of spin relaxation can be found in several texts. Kowalewski and Mäler provided a general and modern treatise on nuclear spin relaxation in liquids, which covers many links to chemical applications. A second and updated edition of their book appeared in late 2017 [11]. A similar book by Kruk also appeared in 2018 [12]. Many NMR textbooks do have chapters that treat the subject in some detail and there are also several relevant sections in Encyclopedia of Magnetic Resonance. Canet and Mutzenhardt wrote a compact and particularly recommendable introduction to the subject of nuclear spin relaxation some time ago [13].

2.3 Self-diffusion and other modes of translational motion The concept of “self-diffusion” refers to an average random transport in space of molecules, ions, aggregates and particles through thermal Brownian motion. For unconfined systems the resulting displacement probability distribution is Gaussian, isotropic and widens with time. The “self-diffusion coefficient” D (normally in units of m2s−1) quantifies the distribution, and is of the order of 10−9 m2 s−1 for small molecules in normal solution at room temperature. This corresponds to an average threedimensional displacement of about 0.1 mm in 1 s. In older literature, self-diffusion coefficients are often expressed and tabulated in cm2 s−1 units. Some biophysical texts also may use Fick units (1 Fick = 10−7 cm2 s−1), to make the typical data range less numerically awkward. A self-diffusion coefficient of 10−9 m2 s−1 thus corresponds to 100 Fick units or 10−5 cm2 s−1. For one dimension, the diffusional displacement probability distribution from a zero origin reads:  2 −z 1 (2:1) pðz, tÞ = pffiffiffiffiffiffiffiffiffiffiffi e 4Dt 4πDt Average diffusional translation can thus be calculated directly from this Gaussian space–time distribution. It is functionally identical to the so-called normal distribution in statistics, where μ represents the mean value, σ the standard deviation and σ2 the variance. So, in diffusional context, 2Dt functionally corresponds to σ2 . ðz − μÞ 1 − pðz, μ, σ2 Þ = pffiffiffiffiffiffiffiffiffiffi e 2σ2 2πσ2

2

(2:2)

Semi-quantitative considerations regarding diffusion are facilitated through the timeaverage for the squared displacement hΔz2 i. When applied to eq. (2.1), it leads to

37

2.3 Self-diffusion and other modes of translational motion



 Δz2 =

∞ ð

z2 pðz, tÞ dt

(2:3)

0

which simplifies to the convenient, so-called Einstein relation: hΔz2 i = 2Dt

(2:4)

For a three-dimensional case, the factor 2 is simply replaced by 6, and analogous hΔx2 i and hΔy2 i terms are added within the brackets to quantify the overall displacement in three-dimensional space hΔr2 i, which becomes 6Dt (Figure 2.6).

350 300 250 200 150 100 50 3 2

1

0

–1

–2

–3

–3

–2

–1

0

1

2

3

Figure 2.6: The spatial probability distribution for two-dimensional self-diffusion, as a function of 2Dt in each dimension. The vertical amplitude is arbitrary. Suray Shankar’s code for “Diffusion in 1D and 2D” (Mathworks File Exchange web site) was used for generating this graph.

The single parameter D, the self-diffusion coefficient characterizes the spatial transport distribution fully in isotropic solution. Adolf Eugen Fick introduced the concept as such in 1855, in his studies of flow of a solute along its concentration gradient. He also stated his first and second laws of (mutual) diffusion. The singleparameter averaged description contained in a numerical diffusion coefficient is no longer strictly valid under certain circumstances, like for concentrated solutions or melts of high polymers. Statistical considerations must also be taken into account if the system has very few particles or if the observation time is very short. As further discussed in the following sections, different manifestations of restricted diffusion (wall effects and effects of diffusion through semipermeable barriers) are often significant in micro- and macroheterogeneous systems. Anisotropic diffusion rates are a characteristic feature of many liquid crystalline systems, as well as for local orientation of their domains. For general reference on diffusional transport in normal solutions, one can note that a “typical protein” like Lysozyme has a self-diffusion coefficient of 10−10 m2 s−1 in aqueous solution at ambient temperature (in the monomeric form, see Section 3.5.6).

38

2 Basic concepts

It would thus diffuse 7 μm in three-dimensional space in 100 ms. Its geometrical dimensions are textbook listed as ellipsoidal with a 26 × 45 Å shape. 10 Å units correspond to 1 nm, which is thus 7,000 times less than the average Lysozyme selfdiffusion distance in 100 ms.

2.3.1 Estimating self-diffusion coefficients from a macroscopic perspective To a reasonable approximation, the size-, temperature- and solution viscosity dependence of self-diffusion for a spherical object in isotropic solution is predictable through the Stokes–Einstein–Sutherland relation (2.5), where self-diffusion rates (D) result from a balance between thermal forces (kT) and frictional ones (here given by f = 6πηr). In such a context, f is referred to as the friction coefficient [14–16]. Sutherland’s name is often left out when referring to this relation, but it was he who first combined Einstein’s molecularly related relation with Stokes’ earlier macroscopic relation for the frictional coefficient in the context of transport of objects in solution through gravitational influence. D=

kB T kB T = f 6πηr

(2:5)

kB represents the Boltzmann constant (J K−1), η the solvent viscosity (P, Poise), r the molecular or particle hydrodynamic radius and T the absolute temperature. Friction coefficients for other idealized geometries such as prolate or oblate ellipsoids or rod-like objects also have been derived (for an overview, see Price [17] pp. 9–14 or Cantor and Schimmel [18]). A review by Macchioni and coworkers specifically covers applications such as size determination based on eq. (2.5) in the context of NMR diffusometry [19]. First-order hydrodynamic theories also consider concepts like “sticking” or “slipping” boundary conditions between the moving object and the imaginary continuum solution implied. In such a context, one may see a factor 4 (“slip”) instead of 6 (“stick”) in (2.5). This means that under “stick” conditions, the solvent–solute interaction is strong enough for the “surface layer” of the solvent to move with the much larger solute one. Stokes (1819–1903) originally based his contribution to (2.5) on observations of falling or moving macroscopic objects in liquids of varying viscosity. Of course, in a molecular world perspective, things are slightly different. A real-life solvent is not continuous or infinitely small on the molecular scale, and moving molecules in solution may have functional groups (e.g., –OH or –NH2) that interact with the solvent in specific ways. A recurring point of focus in this context is the existence of and influence from “hydration layers” or the presence of “immobile water” around solute molecules, ions and biological macromolecules. Conclusions of this kind should be considered judged with great caution. Some

2.3 Self-diffusion and other modes of translational motion

39

statements and findings in the literature may be justified and relevant, others are not. Also note that the concept of viscosity in (2.5) is a macroscopic one. It does not describe transport reality for small molecules in a solution of high polymers or similar. A typically very high macroscopic viscosity (as for a gel-like system) may be irrelevant at the molecular level. Alternative concepts such as “microviscosity” or “obstruction effects” (for diffusional transport) have instead been introduced, together with various suggestions for “corrections” of experimental self-diffusion coefficients. Equation (2.5) is often a fairly good approximation to reality, and implies a fairly weak third root inverse dependence of D on molecular mass for near-spherical molecules. For example, a medium-sized protein in solution should have a D-value only 10–20 times lower than that of a small molecule in the solution. Evans and coworkers recently compiled and tested with remarkable improvement (Figure 2.7) an extensive set of small-molecule self-diffusion data against a modified Stokes–Einstein equation that accounts for additional and relatively simple hydrodynamic factors introduced by Gierer and Wirtz back in 1953 [20]. After some numerical transformations regarding the functional form of the friction factor expression, they arrived at the following expression: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D = kB Tð3α=2 + 1=ð1 + αÞÞ=6πη 3 3MW=4πρeff NA

(2:6)

where α = rS =r =

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 MWS =MW

(2:7)

The “effective density” (ρeff ) in eq. (2.6) is a single adjustable parameter, common to the whole data set. MW and MWS denote the molar masses of the solute and solvent, respectively, forming a basis for approximate estimation of their respective Estimation of diffusion coefficients SEGWE

Measured

>600 measurements >20 solvents Variable temperature Predicted

Measured

Stokes–Einstein

Predicted

Figure 2.7: The graphical abstract from Evans et al. [21], illustrating a much improved predictive ability for small molecules in the solution using the SEGWE approach, compared to that obtained using the traditional Stokes–Einstein–Sutherland model (eq. (2.5)). Reproduced with permission (published under a ACS Creative Commons Attribution (CC-BY) license).

40

2 Basic concepts

radii, r and rS . They named the model SEGWE [21], and also provided a “self-diffusion” predictor based on eq. (2.6). Intermolecular interactions or molecular self-aggregation can affect molecular self-diffusion to a considerable extent, which is the key to make diffusion studies potentially useful for elucidating a variety of physicochemical or analytically relevant questions. A general strategy in science is to strive for comparison between theory and actual observations, which should ultimately lead to better understanding of Nature. For NMR diffusometry, the situation is quite uniquely favorable. The parameter measured has a simple and direct interpretation, and does not normally require additional model considerations regarding details of molecular motion. Molecular interactions that affect self-diffusion are normally easy and uncomplicated to quantify and interpret physically.

2.3.2 Estimating self-diffusion coefficients from a molecular perspective The link from the microscopic to the macroscopic world is made within the framework of statistical mechanics. The many-body problem regarding Brownian motion in a macroscopic sample with >1020 molecules or particles (i) is reduced to a single number, the self-diffusion coefficient (D), through the relation D = lim

t!∞

E 1D ½ri ðtÞ − ri ð0Þ2 6t

(2:8)

where ri ðtÞ represents the location of particle i at time t and the hi represent the ensemble average. In this context, ½ri ðtÞ − ri ð0Þ2 is known as the mean square displacement. Today, self-diffusion rates can be modeled and computed via the so-called molecular dynamics simulations. The basic underlying ideas date back more than half a century, but their applications have hugely benefited from recent-decade computing advances. Molecular motion is here modeled for an ensemble of molecules (typically a “box” of some 1,000) within a framework of molecular interactions of various kinds, and is governed by temperature regarding intensity of motion and Newton’s laws. Software for such studies is widely available nowadays, together with extensive computer resources. Simple simulations on few molecules can be made on desktop computers – upscaling to larger systems will steeply create a need for high-performance supercomputer facilities. In molecular dynamics simulations, the so-calculated self-diffusion coefficient ðDυ Þ of a constituent is evaluated from its averaged velocity (υ) autocorrelation results in a simulation [22] (eq. (2.9)). For example, see Laaksonen and Stilbs [23] and Moktan et al. [24], who describe such procedures in great detail. Deviations between observed behavior and simulation results constitute the basis for better

2.3 Self-diffusion and other modes of translational motion

41

understanding and parameterization of assumed molecular interaction parameters in future studies: ^ υ ðtÞ = < υðtÞ · υð0Þ > C < υð0Þ · υð0Þ >

(2:9)

This quantity that relates molecular or particle velocity at a given time (0) and a time t later (or earlier) is thought to decay approximately exponentially for an average over many molecules. In this normalized form, it would go from 1 to zero with time. The summarizing self-diffusion coefficient becomes kB T Dυ = m

∞ ð

Cυ ðtÞdt =

kB T · τυ m

(2:10)

0

Here, τυ denotes the velocity autocorrelation time. For an exponential relation, it would correspond to a point in time where the relation in (2.9) has decayed to 1/e of its value at t = 0. An brief discussion on related matters and their links to diffusional aspects is found in the PGSE monograph by Price [17] (Section 1.3.2).

2.3.3 General and experimental methodological historical considerations Historically, the first available method for quantifying self-diffusion was by spatially monitoring the spread of trace quantities of system components of isotopically labeled molecules, ions or particles. This approach could be based on trace quantities of molecules labeled with isotopes, detecting diffusional transport through suitable position-sensitive detector systems. Such studies are the origin of the older notation “tracer diffusion” for self-diffusion, where normally radioactive isotopes were used. Another approach was based on diffusional mixing of isotopically labeled molecules, as analyzed using mass spectrometry. Both approaches were cumbersome and time-consuming indeed and are not much in use today. The NMR approach is much simpler, and since NMR is molecularly resolved it can probe diffusion issues even for complex mixtures in solution, without any need for isotopic labeling. Examples of experimentally determined self-diffusion coefficients are listed in Table 2.1. There are as many self-diffusion coefficients in a system as there are components. In a solution of, for example, NaCl in water, there are in principle five coefficients, one each for Na+, Cl– and three for H2O and its autoionization components H+ and OH−. Note that the sodium and chloride ion self-diffusion coefficients differ, but for reasons of electroneutrality, no charge separation occurs macroscopically over time. Halle and Karlström extensively discussed proton migration mechanisms in water in their 1983 study [25], a paper that has later been followed up by others (see e.g., Chen and coworkers [26]).

42

2 Basic concepts

Table 2.1: Approximate self-diffusion coefficients at 25 °C (in 10−9 m2 s−1; modified, with permission from Stilbs P, Progr Nucl Magn Reson Spectrosc, 19, pp. 1–47, 1987; Copyright (1987) Elsevier). Gases

Oxygen in air



Solids

Helium in Pyrex

 .–

Aqueous solution

Oxygen Ethanol Benzoic acid Urea Sucrose Hemoglobin Surfactant micelle Li+ Na+ Cl− Acetate ion

Organic solution

Benzene in ethanol Cyclohexane in benzene

Polymer systems

Polystyrene (M=) in CCl n-octane in cis-polyisoprene

. . . . . . .–. . . . . . . . .

2.3.4 Flow and related modes of non-Brownian motion This section primarily refers to measurements of translational self-diffusion in isotropic solution at thermal equilibrium. Competing macroscopic flow effects like thermally induced convection may at times represent significant and sometimes troublesome complications, which are not easily controllable. Intentional or forced molecular transport in the form of electrophoresis or electro-osmosis can be induced through, for example, application of an electric field across the sample. Electrophoretic transport (further covered in Chapter 8) is a molecularly specific phenomenon, while electro-osmosis or other types of flow affect all components in a sample. Experimentally separating such effects from molecular self-diffusion will be discussed in more detail.

2.3.5 Influence of barriers to molecular transport Anisotropic diffusion rates characterize solids and many liquid crystalline systems and are a result of structural barriers to molecular transport. Such phenomena have successfully been studied by normal or specialized diffusion NMR techniques for decades, and been summarized in reviews [27, 28].

2.3 Self-diffusion and other modes of translational motion

43

In suspensions or in the presence of macromolecules or supramolecular aggregates in solution, micro- or nanoheterogenities may also occur even in macroscopically isotropic solution systems, posing impenetrable barriers (obstructions) or restrictions like semipermeability with regard to molecular diffusion trajectories (such as for cell membranes). Handling such effects at the experimental design as well as at the evaluational stages is crucial for proper NMR-based diffusion studies on systems of this type. Around 10 vaguely similar “obstruction correction relations” have been suggested since the original one by Wang. Most are listed and discussed in Price [17], Section 1.8.6. In general, the suggested corrections do not differ much and are usually hard to confirm, since the basic problem is quite diffuse. Furthermore, it is in a borderline between a structureless macroscopic and a molecular world with localized interactions of various kinds. Wang’s original relation reads Dobs = Df ð1 − 1.5ϕÞ, where Df corresponds to the diffusion coefficient without obstructing entities, and ϕ is their volume fraction to be corrected for [29] (Figure 2.8).

Liquid

Liquid

Inner volume Liquid

Obstructed diffusional path

Impenetrable obstacles

Reduced diffusional rate through membrane

Membrane

Figure 2.8: Obstructive influence on self-diffusion rates relates to a longer effective transport distance past impenetrable objects. Restricted diffusion rates may occur as a result of a semipermeable region or “membrane” that separates two liquid-like volumes, and may involve “carrier” molecules of other kinds. This is the case for, for example, living cells or vesicles in aqueous suspension. Self-diffusion rates may differ significantly between inner and outer regions of such structures. One should note that the obstructing objects in the left figure part are meant to be three-dimensional and the zig-zag patterns in the drawing are highly schematic and simplified. In reality, the segments should be many magnitudes more tightly spaced than in this illustration. One should also be aware of substantial influence on NMR-based self-diffusion studies from differing magnetic susceptibility between regions in such heterogeneous systems. Obstruction, as well as susceptibility-related effects, is further discussed in some detail in Chapter 7.

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2.3.6 Further reading The above introductory outline text in Sections 2.1–2.3.5 primarily refers to simple liquids. Conditions differ in what is called as “complex fluids”. Self-diffusion in such systems need not be Gaussian, as probed at the timescale of NMR methods. To some extent, this more advanced subject will be discussed in Chapters 3 and 5. Recommended discussions of in this context are as follows: A compact introduction to the general subject of diffusion: Jost W. Fundamental aspects of diffusion processes. Angew Chem Int Ed Engl. 1964; 3: 713–722. Chapter 1 in Callaghan’s monograph; Callaghan PT. Translational Dynamics & Magnetic Resonance, Principles of Pulsed Gradient Spin Echo NMR. Oxford: Oxford University Press, 2011. Grebenkov DS. Use, misuse, and abuse of apparent diffusion coefficients. Concepts Magn Reson. 2010; 36A: 24–35. Koay CG, Özarslan E. Conceptual Foundations of Diffusion in Magnetic Resonance. Concepts Magn Reson. 2013; 42A: 116–129. Phillies GDJ. Diffusion on a molecular scale as observed using PGSE NMR. Concepts Magn Reson. 2015; 44A: 1–15. Phillies GDJ. Diffusion in Crowded Solutions. Adv Chem Phys. 2016; 161: 277–358. Burada PS, Hanggi P, Marchesoni F, Schmid G, Talkner P. Diffusion in Confined Geometries. ChemPhysChem. 2009; 10: 45–54.

2.4 Mutual diffusion It is important also to note that “self-diffusion” relates to the average random motion of molecules or particles – normally one refers to systems in equilibrium. Brownian-type motion is merely a reflection of thermal energy exchange in the system and, of course, also operates in nonequilibrium situations – such as for macroscopic relaxation of a nonequilibrium concentration gradient. The relevant quantifying quantity here is a conceptually different one – “mutual diffusion.” (Figure 2.9).

t = initial

t = intermediate

t = long

Figure 2.9: Illustrating mutual diffusion of a layer of a colored compound in a colorless solvent in a cuvette at different times. The initial concentration gradient will relax with time toward an equal distribution, as described by Fick’s first law (eq. (2.11)). An actual measuring cell for optical detection of mutual diffusion is also shown. It probably dates back to the 1960s or earlier, and was built in the local mechanical workshop at the Section of Chemistry KTH, Stockholm. The observation was made through a narrow vertical window in the middle region of both short sides. Not much of such local mechanical-optical skills typically remain in such environments today.

2.4 Mutual diffusion

45

The mutual diffusion coefficient can be visualized to characterize the relaxation of concentration gradients according to Fick’s first law. For two components (solute and solvent) in one dimension, it reads J = − D′ðdc=dxÞ

(2:11)

Here dc=dx represents the solute concentration gradient, D′ the mutual diffusion coefficient and J the flow of solute molecules per unit cross-sectional area. The underlying net molecular motion here originates from the same thermal motion that causes self-diffusion. Eventually, the once sharp boundary becomes blurred and the concentration difference disappears and a single homogeneous solution of lower concentration remains. Concentration levels in such a context are normally monitored optically. For colored solutes, this is straightforward through normal absorption spectroscopy-like procedures. Concentration changes of others can be quantified through their influence on the refractive index of the solution. Interference-type optical setups of, for example, Schlieren type are often used in such a context. The monographs listed in Section 2.4.1 are highly recommended readings that summarize the heydays of pioneering work aimed at understanding molecular transport and solution structure. In a two-component system, there is only one mutual diffusion coefficient. Depending on the composition, it may approach either of the self-diffusion coefficients of the components. In an infinitely dilute “two-component” solution, the (single) mutual diffusion coefficient approaches the self-diffusion coefficient of the solute and is not directly related to that of the solvent (cf. Figure 2.10). At intermediate concentration ranges, the distinction between mutual and self-diffusion coefficients is important. The general theoretical framework in this context is found in texts on “irreversible thermodynamics.” Basically, one describes solute mutual diffusion (Ds ) in a two-component system as being determined by   kB T ∂ ln γ ð1 − ϕÞ 1+ Ds = f ∂ ln c

(2:12)

where γ denotes the solute activity, c its concentration and ϕ its volume fraction. The situation rapidly gets considerably more complicated and entangled with the growing number of components. Among other aspects, the number of mutual diffusion coefficients grows to (n−1)2 in an n-component system. Self-diffusion remains the same simple statistical/geometrical and individually component-related quantity, regardless of the system’s complexity.

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Diffusion coefficient/m2 s–1

–11 2.5 × 10

2 1.5 1 0.5 0

0

2

4

6 8 10 Concentration / kg m–3

12

14

16

Figure 2.10: Diffusion coefficients from dynamic light scattering (blue crosses, mutual diffusion) and early-generation FT-PGSE (red diamonds, self-diffusion) for relatively well-fractionated and monodisperse poly(ethylene oxide) of nominal molecular weight 280.000 in 50:50 H2O/D2O at 25 oC. Redrawn from data in Brown et al. [30]. Trends are consistent with an extrapolated common mutual and self-diffusion coefficient close to 10−11 m2 s−1 at infinite dilution.

2.4.1 Further reading This 1,120 page book describes in great detail numerous diffusion-related concepts and alternative techniques for studies of molecular transport: Serdyuk IN, Zaccai NR, Zaccai J. Methods in Molecular Biophysics – Structure, Dynamics, Function. Cambridge: Cambridge University Press, 2007. An older classic of related conceptual content is: Cantor CR, Schimmel PR. Biophysical Chemistry, Part II: Techniques for the Study of Biological Structure and Function. New York: W.H. Freeman, 1980. (Part I of Cantor and Schimmel covers The Conformation of Biological Macromolecules and Part III The Behavior of Biological Macromolecules) Classic texts on diffusion: Bird RB, Stewart WE, Lightfoot EN. Transport phenomena. New York: Wiley, 1960. Crank J. The Mathematics of Diffusion. Oxford: Oxford University Press, 1975. Crank J, McFarlane NR, Newby JC, Paterson GD, Pedley JB. Diffusion Processes in Environmental Systems. London: Macmillan, 1981. Tyrell HJ, Harris KR. Diffusion in liquids. London: Butterworths, 1984. Cussler EL. Diffusion: Mass Transfer in Fluid Systems. Cambridge: Cambridge University Press, 1984. Hansen JP, McDonald IR. Theory of Simple Liquids. London: Academic press, 1976. de Groot SR, Mazur P. Non-Equilibrium Thermodynamics. New York: Dover, 1984. Truskey GA, Yuan F, Katz DF. Transport Phenomena in Biological Systems. New York: Prentice Hall, 2003. Dill KA, Bromberg S. Molecular Driving Forces. New York: Garland Science, 2003.

2.5 Why study self-diffusion and other transport processes by NMR?

47

2.5 Why study self-diffusion and other transport processes by NMR? 2.5.1 Main Objectives The simple answer is that quantitative information regarding transport-related quantities is often easily and directly linked to other physically relevant parameters such as molecular interactions – without the use of intermediate model considerations needed for many alternative techniques. With the wide availability of NMR instrumentation in academic and industrial laboratories and commercially available add-ons (e.g., NMR probes and magnetic field gradient drivers), the step-in barrier to this field is also rather low. Especially for collecting self-diffusion information, there is no realistic and generally applicable alternative to pulsed magnetic field gradient spin-echo NMR methods. Using classic radioactive or nonradioactive isotopic tracer techniques is difficult, time-consuming and often “impossible.” Magnetic field cycling spin-lattice relaxometry was mentioned earlier, and has some unique but otherwise limited applicability. The same also holds good for the neutron spin-echo-based detection techniques, which can only be applied in national or international research facilities. A handful of rather exotic fluorescence and light-scattering techniques also exist, such as fluorescence photobleaching recovery, confocal fluorescent microscopy, number fluctuation in dynamic light scattering (DLS), fluorescence correlation spectroscopy and forced Rayleigh scattering. They are well described in the book Methods in Molecular Biophysics, listed in Section 2.4.1. Some are quite powerful in certain limited ranges of applications, but may require quite complex and expensive instrumentation and assistance of skilled expert experimentalists. DLS techniques are relatively common, and may appear to be a pathway to diffusion information in, for example, colloidal systems. In reality, the quantity measured is not self-diffusion of the colloidal particle or macromolecule. It is actually mutual diffusion, and relates to microscopic local concentration fluctuations through Brownian motion. These occur even in a sample in macroscopic equilibrium, and can be translated into a diffusion coefficient via numerical evaluation of the time autocorrelation of the scattered light intensity. However, in dilute solution of a macromolecule, the experimental (mutual) diffusion coefficient from DLS should approach to the macromolecular self-diffusion one (c.f. Figure 2.10).

2.5.2 What can favorably be studied and by which approach? NMR diffusometry methods are easiest to apply to solutions, but may with some difficulty and through more complex procedures be applied to gases, some solids, liquid crystals, soft matter and heterogeneous systems of various kinds. In addition to

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diffusion, the same basic concepts are adaptable for studying transport through flow, for example, electrophoresis. The latter companion technique, electrophoretic NMR (eNMR), and some of its applications will be described in the concluding sections of this book. Not all nuclei are suitable for NMR detection in the context of self-diffusion measurements. Protons are the “best,” owing to their relatively high magnetogyric ratio (i.e., high NMR frequency) and relatively low spin relaxation rates. About 10 others are suitable, but to lesser extent. McGregor has provided a good compact overview on “NMR properties of other spin-½ nuclei” [31]. Only a few quadrupolar nuclei can realistically be studied, owing to rapid quadrupolar spin relaxation, mostly combined with a low NMR frequency. Deuterium is the most commonly studied one, and such studies are quite easy on enriched samples. A relatively low deuteron quadrupole moment and an NMR frequency above average are detection-favorable characteristics.

2.5.3 The most basic experimental strategy based on self-diffusion data The following generic application example may serve to illustrate a typical application for NMR-based self-diffusion measurements: Assume you want to check or quantify whether molecule type “A” binds to molecule type “B” in solution, and perhaps also with what stoichiometry such binding occurs? This would normally (but not always) constitute an obvious family of applications that should be eminently suitable for diffusion-based investigation in certain circumstances. Referring to Figure 2.11, one should note that, for example, association of two similar-sized molecules should not lead to a particularly significant increase in hydrodynamic radius over those of the constituents. For an aggregate that remains spherical and only grows in size, the change in self-diffusion should only be of the pffiffiffi order of 3 2 ≈ 1.25. Therefore, quantification of self-diffusion coefficients will require high measurement precision and normally transport-related corrections of various

(a)

(b)

Figure 2.11: In case (a) of aggregating or binding entities of similar size, both self-diffusion coefficients decrease, but to a marginal extent. In case (b) the self-diffusion coefficient of the larger entity is almost unaffected, while that of the smaller one may change by more than a magnitude and become equal to that of the larger entity. In the case of partial and time-averaged binding, an intermediate site-weighted value approximately given by eq. (2.13) applies.

2.6 Spin-echo formation under the influence of a steady magnetic field gradient

49

kinds would also be justified. The situation is much more favorable if one of the constituents is a macromolecule or similar, and the other is a “small” one. Binding influence on self-diffusion behavior on the “small” one can then be substantial, and becomes robustly quantifiable into equilibrium constants and similar. At this point one should clearly note that NMR-based self-diffusion coefficients are almost always time-averaged quantities – that is, over the “free” and the “bound” states, and as based on a single NMR signal. This is because of the fairly long spectroscopic NMR time scale in component frequency-separation context, and the similarly long one for PGSE diffusion experiment as such. Both are normally of the typical crude order of 100 ms, depending on instrumental settings. The lifetime of a dimer or for a bound state is normally much shorter than that, except in relatively rare circumstances. In a fast exchange two-site situation, self-diffusion will quantitatively be given as follows: Dobs = pDbound + ð1 − pÞDfree

(2:13)

which can be rewritten in the form p=

Dfree − Dobs Dfree − Dbound

(2:14)

Here one assumes that the “macromolecular” self-diffusion is unchanged upon aggregation. The degree of binding ð0 < p < 1Þ will then be readily amenable for evaluation through the implied simple comparison of the experimentally accessible selfdiffusion coefficients in eq. (2.14). By also varying the concentrations, we can obtain information about the mode of aggregation and its stoichiometry, in terms of parameters such as equilibrium or association constants. This proven two-site bound-free approach has been the starting point for numerous PGSE investigations ever since its implied generic use, building on studies of metal ion hydration, way back in the 1960s [32]. With modern instrumentation, signal collection and data processing need not take more than 10 min per self-diffusion coefficient in routine applications.

2.6 Spin-echo formation under the influence of a steady magnetic field gradient: the key to NMR diffusometry Before proceeding, one should clearly note at this point that magnetic field gradient NMR-based techniques do not actually detect and quantify self-diffusion coefficients and related quantities directly. What is actually observed is rather an average movement of individual molecules in solution along a chosen direction in the sample (normally along the long axis of a common NMR tube, also denoted as the z-axis in a superconducting magnet geometry). The concept of

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self-diffusion is considered through an assumed Gaussian displacement probability distribution of spin-bearing molecules. In the absence of disturbing phenomena such as convection and system-related presence of diffusional barriers (and a few simplifying assumptions), such a Gaussian transport model is closely precise, due to statistical averaging over the large number of molecules or particles involved. Other transport situations can be dealt with as well. In principle, one just has to specify and evaluate the appropriate spin phase distribution in the x′–y′ plane in the rotating frame, in analogy with the procedures discussed in the next sections. A general echo amplitude attenuation evaluation procedure for arbitrary modes of motion was originally described by Carr and Purcell [33], who extended the original analysis by Hahn [34].

2.6.1 Labeling nuclei with positional information The normal NMR approach for studying diffusion and transport by NMR is founded on the same basic concepts as MRI (NMR imaging and NMR microscopy). The initial methodological step is to note that the angular NMR frequency γ (rad s−1) and the normal frequency ν (Hz) are proportional to the applied magnetic field B0 (T) and the magnetogyric ratio γ (rad T−1 s−1) of the nuclei observed: ω = 2πν = γB0

(2:15)

Hence, in a magnetic field that varies over the sample, there will be a linear relation between location in space and the NMR frequency. Normally a constant gradient (i.e., a linearly varying one along the sample axis) is applied to achieve the second step. ωz = γB0 + γgz z

(2:16)

This suffices for basic one-dimensional imaging of an object along the z-axis, by a standard pulsed NMR experiment (Figure 2.12). – It should be noted at this point that magnetic field gradients should be expressed in units of T m-1, although the older CGS unit of Gauss cm−1 is still widely used in the context of PGSE. Note that 1 T unit amounts to 10,000 G units, so 1 T m−1 corresponds to 100 G cm−1, and 1 G cm−1 to 10 mT m−1.

2.6.2 The intrinsic “time-reversal” effect of spin-echo formation In essence, an NMR spin-echo sequence based on two or more radiofrequency pulses brings back and refocuses the individual phase components of the magnetization originally contained in the FID to some extent, depending on the sample and

2.6 Spin-echo formation under the influence of a steady magnetic field gradient

51

Bz Magnetic field gradient z x or y Objects

z Intensity Frequency spectrum Hz Figure 2.12: An illustration of one-dimensional NMR field gradient-based imaging of objects in the z-direction. By combining such “spectra” from various gradient directions in the x–y plane as well, a projection-reconstruction-based two-dimensional image of the object can be generated mathematically. This was the basic procedure used in early NMR imaging studies.

the measurement conditions. Dispersion in frequency space by chemical or magnetic field gradient generated NMR frequency shifts can in principle be fully refocused into an echo. Transverse spin relaxation is irreversible, on the other hand, and cannot be refocused at all. As such, it constitutes a competing and limiting factor for NMR diffusometry, based on magnetic field gradients and spin-echo formation (Figure 2.13). Generally, refocusing will occur with any pulse angles in a two-pulse combination, unless their spacing in time is such that irreversible transverse spin relaxation dominates completely. Ideally, the echo amplitude (relative to that corresponding to the initial nuclear magnetization) in any two-pulse sequence of this kind should be given by sinðϕ1 Þð1 − cosðϕ2 ÞÞ=2 (or some trigonometric identity such as sinðϕ1 Þsin2 ðϕ2 =2Þ), where ϕ1 represents the first tip angle and ϕ2 the second [36]. Hahn’s original experiments were for instrumental and practical reasons made with two or more 90o pulses. The relation predicts that the echo signal “loss” thus should amount to 50% of the signal contained in the FID, whose amplitude corresponds to the initial spin magnetization. In the following section, we shall see why. Kingsley provided an excellent discussion and well-drawn illustrations of related matters in part II of a series of papers in Concepts in Magnetic Resonance [36]. Part I of a formally NMR imaging-related discussion by Hennig in the same journal is very highly recommended reading as well [37]. The same goes for two other papers [38, 39], which also consider echo formation from an NMR imaging perspective. Of course, a late educational paper by Hahn himself is highly relevant and historically essential in this context [40]. While a correct mathematical and

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A

B

C

D

E

F

Glitches from rf pulses

Figure 2.13: Hahn made a “time-reversal” analogy, picturing a situation of runners of differing ability on a racetrack. It appears on the cover of the November issue of Physics Today of 1953, featuring a later summary article on the concept and discovery of spin echoes [35]. At a particular point in time after the beginning of the race, participants are instructed to stop and return to the starting point. Assuming that their return speed remains the same as that during the first leg of the race, each one of them should reach the original starting line at the same point in time, in analogy with the concept of spin-echo formation. An actual original proton spin-echo experiment oscilloscope trace for glycerol from the original article is illustrated here as well. The FID and the “Hahn echo” signals are seen, together with barely discernible and much narrower rf pulse responses. Phase-sensitive detection in NMR was not yet used, so the FID and the echoes appear positive. Reproduced from [35], both with the permission of the American Institute of Physics.

graphical analysis of the 90o–90o spin-echo experiment was made by Hahn in his original paper, its visual and conceptual visualization of the refocusing stage remains problematic and has been potentially counterintuitive for many. Also, some existing literature graphics are difficult to understand, are cluttered or have some errors or contradictions. Note that in the following illustrations and paragraphs in this section, diffusional influence on spin-echo formation is not yet introduced. Also, magnetization recovery through longitudinal spin relaxation in the z-direction and irreversible transverse dephasing in the x′–y′ plane are also left out for clarity.

2.6.3 Illustrating spin-echo formation 2.6.3.1 The basic 90o–90o Hahn echo The basic procedure is to initially consider and map dispersion in the x′–y′ plane by labeling individual spin vectors as “fast” and “slow” relative to the rotating frame, and note that they normally stay “fast” and “slow” throughout the whole pulse sequence. Initially, just spin vector dispersion in the x′–y′ plane results after the first 90o rf pulse, as a consequence of differing precession frequencies (“fast” and “slow”). This corresponds to the normal NMR FID, and its decay is a result of spin phase coherence loss. The second 90o rf pulse tilts the remaining x′–y′ magnetization “slice” into the zdirection with a second dimension that can be either x′ or y′, depending on the relative 90o–90o pulse pair phase relationship and the definition of the phase of the original

2.6 Spin-echo formation under the influence of a steady magnetic field gradient

53

90o rf pulse. Basically, one uses the same phase for both, and considers them to be “x′-pulses”. Refocusing will then occur in the negative “y′-direction,” ultimately resulting in a “negative” spin echo, as compared to the signal corresponding to the FID. The following two graphs (Figures 2.14 and 2.15) illustrate computer simulation of 90o–90o spin-echo formation, based on the basic “fast” and “slow” labeling of individual spin vectors. For clarity, a suitable number of spin vectors, precession frequencies and timing parameter have been chosen, so as to just fill the x′–y′-plane after maximally 180o dispersion for the “fastest” and “slowest” spin vectors. At that point in time, the second rf pulse is applied to initiate the “reverse” process of spinecho formation. For larger (or smaller) degrees of dispersion, visualizations of this kind can otherwise become very cluttered and confusing to the reader.

Figure 2.14: Hahn 90o–90o spin-echo formation using perspective-accounting graphic visualization that relatively enlarges components closer to the observer. Individual frames are selected for illustration and are not equally spaced in time. The initial rf pulse initiates the transition between frames 1 and 2. Spins then fan out in the x′–y′-plane, “red” corresponding to “fast” and “white/ silver” to “slow.” The second rf pulse is applied exactly at the point in time where the “fastest” spin vectors have evolved 180°, causing the flip of magnetization illustrated in frames 5–7. Spin vectors then partially refold into an eight-shaped echo pattern illustrated in the last frame (#15). Its vertical projection is a circle, resulting in a negative magnetization in the x′–y′-plane, which has half the magnitude of the initial FID. These frames from a computer visualization sequence are reproduced here, courtesy of their creator Daniel Topgaard, Lund (Sweden).

Figures 2.15 (a) and (b) further illustrate the echo formation processes. In the calculations behind the data illustrated in b), step 1 calculates the individual spin vector precession “rate” (in arbitrary units), that are for simplicity is set to vary linearly

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SE

90°

120

90°

1

150

SE 180

90 1 0.8 0.6 0.4 0.2 0

60 30

0

90° SE

210

330 240

(a)

(b)

270

300

Figure 2.15: (a) A composite image is seen of frames 7 and 15 in the series in Figure 2.14, together with arrows indicating the spin dispersion that results in the x′–y′-plane for three selected “fast” spin vectors in the upper quadrant. Again, remember that the individual spin vectors keep their “speed” and “direction” in the x′–y′-plane throughout the whole sequence, up to and beyond the point of echo formation. What happens is perhaps easiest to see, considering the behavior of three specific spin vectors, just after the second rf pulse (frame 7 in the previous figure). The very “fastest” (deep red) and “slowest” (white/silver) point straight upward into the z-direction, and having no component in the x′–y′-plane they will not disperse further in it at all. The same goes for the spin vectors that initially were exactly on-resonance. They now point in the negative z-direction, and similarly will remain stationary up until and beyond the echo formation point in time (frame 15 in the previous figure). (b) Final outcome of a simple Matlab echo formation simulation, summarizing the deep red to gray “fast” spin vector behavior in the two right-hand quadrants in Figure 2.14 (see text). Note that the spin vector ensemble forms a circle at the maximum echo point in time, reflecting what is incomplete echo formation. The circle center of gravity is at a relative coordinate 0.5 instead of the full 1.0 magnetization, corresponding to a 50% reduction in signal amplitude.

from 0 at the bottom to π at the top along the right-hand semicircle. The important insight is again to remember that individual vector (slow/fast) precession “rates” are constant throughout the experiment. Step 2 calculates the individual vector projections (proj) onto the vertical axis. They simply relate to their individual directions and “rate” as sin(rate). In step 3 one recalls that spin vector precession angles are constant and known already, and that precession “angles” are proportional to their “rate.” When the “fastest” spin vectors in the top region precess, for example, half a turn, the intermediately rapid ones at the “equator” will only have transversed a quarter one, and the bottom ones have hardly moved at all. Spin vectors in the two “slow” quadrants (gray/white/silver) to the left behave analogously, but become mirror images to the “red/gray set.” Step 4 is to make a polar plot of these data with arguments (angle, proj) (or (rate, proj)), resulting in the right-hand Matlab figure (b). It corresponds to a vertical projection of data in Figure 2.14. 2.6.3.2 A 90o–180o pulse pair A later 90o–180o pulse variant is optimal regarding the ease of visualization as well as actual measurement efficiency. Carr and Purcell [33] introduced it a few years

2.6 Spin-echo formation under the influence of a steady magnetic field gradient

55

after Hahn’s original study, together with the conceptual illustration that has generically been redrawn in Figure 2.16 and in numerous earlier publications on NMR diffusometry. It should also be noted that echo formation could ideally become complete, instead of intrinsically attenuated by 50%, as in the original Hahn echo sequence. Note that the latter variant is still referred to as a “Hahn echo” in current literature, and that the less efficient 90o–90o variant is rarely used, except as a building block for more complex pulse sequences.

Excitation z (a)

B1

Dephasing z (b)

90°

y′

x′

Inversion z (c) 180°

y′ x′

z

(f)

(e)

y′

y′ x′

Refocusing

z

z

(d)

x′

y′ x′

y′ x′

Echo peak

Dephasing

Figure 2.16: Spin-echo generation through the use of a 90o–180o -pulse sequence (or a π=2 − π one, if angles are expressed in radians), neglecting transverse spin relaxation (T2 ) effects. (With present-day sign conventions, both pulses would be seen as “negative” x-ones [41], as implied here through the added spiral symbol at the x′-axis). Note that when using the same radiofrequency (rf) phase (here along the x′ direction in the rotating frame) the detected echo-related magnetization along the y′-axis becomes negative. Using instead a y′-phase for the second 180-pulse results in a positive echo instead along the same axis. Here, the spin vector groups marked in “blue” and “red” represent spins that have a slower and faster precession frequency than the reference one for the rotating frame. In the context of spin-echo-based self-diffusion studies and as described in the following sections, such frequency variation is intentionally achieved by applying a magnetic field gradient across the sample. In other words, it represents different locations in the sample. In the absence of spin relaxation effects and with perfectly homogeneous rf fields for excitation and refocusing, echo refocusing at the indicated point in time will ideally be complete, unless the spins/molecules move (and thus change NMR frequency) during the experiment.

2.6.3.3 A 90o–90o–90o sequence – the stimulated echo and others Back in 1949 Hahn also tested the use of three pulses (Figure 2.17) and discovered that no less than five echoes can occur, one of which was named the “stimulated echo.” The three “other echoes” have found very little use in NMR. However, papers

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by Kimmich et al. [42–44] touch on such issues. Note that some of the echoes and the FID’s in Figure 2.17 would not occur in a case without any partly restored or remaining z-magnetization to “re-excite.” Longitudinal spin relaxation during the pulse sequence restores magnetization. If desired, one could intentionally leave magnetization in the z-direction by using pulses that deviate from exact 90° ones. Similar effects will also occur to some extent if the “90-degree” pulses are nonideal. Actual relative echo intensities among these five may thus vary with pulse spacing, pulse angles and relaxation rates for the system. According to text at mriquestions. com/stimulated-echoes.html (no reference was listed), the number of echoes after the last pulse in a four-pulse 90° sequence would be 13, and for six it would be 121. Generally, for an n-pulse sequence, the number given is [3(n−1)−1]/2 (Figure 2.17).

90x FID1

90x

90x FID3

FID2 “Hahn” echo Echo_12

2󰜏 T+󰜏

󰜏 0

T

2T 2T–󰜏 2T–2󰜏

0

𝜏1

Stimulated RE echo Echo_23 Echo_13 𝜏2

2𝜏2–2𝜏1 2𝜏2–2𝜏1

Time 2𝜏2

2𝜏1

𝜏2+𝜏1

Figure 2.17: Three-pulse rf sequence fivefold echo formation, as observed by Hahn [34] (left, reprinted with permission, Copyright (1950) American Physical Society). All echoes were at the time observed as positive as well as relatively sharp and separated, because of the slightly inhomogeneous background magnetic field and phase-insensitive detection scheme in Hahn’s original spectrometer setup. Note that for ideal 90° pulses, complete dephasing and in the absence of longitudinal spin relaxation, there will be no net magnetization left in the z-direction. The last two echoes would then never form. Also note that actual echoes and FID’s will extensively overlap and interfere in a high-resolution NMR situation where signals decay more slowly in the time domain. Selective FID and echo cancelling through phase cycling procedures is then used for separating desired signal components from the nonwanted ones (cf. Section 2.9). The right-hand of the figure shows actual echo phases, assuming use of +x-pulses throughout, and a –x receiver phase setting. Echo_12 is the normal two-pulse Hahn echo; the stimulated echo arises from magnetization that the second rf pulse stored in the z-direction; the RE (refocusing echo) forms through refocusing of the Hahn echo by the third rf pulse. Finally, echoes Echo_23 and Echo_13 result from rf pulses 2 and 3 and 1 and 3, respectively. Procedures for separating the stimulated echo from the others are further discussed in Section 2.9.

As further discussed in the folllowing sections, the stimulated echo is the preferred basic observable in the context of NMR diffusometry. It is therefore of some interest to try to visualize its “shape.” Spatial formation of the 90o–90o Hahn echo in the rotating frame was previously discussed in Section 2.6.3.1. Here, a rather clear and well-defined eight-shape was seen to form, with a net magnetization that maximally becomes 50% of that of the original FID. Already Hahn graphically visualized stimulated echo formation [34]. His drawings still leave many confused, like those

57

2.6 Spin-echo formation under the influence of a steady magnetic field gradient

simpler eight-shape ones for the original Hahn echo formation. For example, similar ones also appear in Figure 4.22 on page 155 of the monograph by Callaghan [45] and in part II of the article series by Kingsley [36]. There are few explicit illustration attempts for the “the stimulated echo shape” in current literature. One can initially deduce stimulated echo variant formation that vaguely resembles that for the two-pulse 90o–90o echo sequence (Figure 2.14) and that it is visually and spatially much more complex than for the 90o–180o two-pulse spinecho, although the general principles remain the same. Through tools of computer z

1

z x

y

y

z y

z

2

x

z x

y

11 z

90°x

1 2

y

4 6 7 5

y

z

4

z

y

9

y

13 x

90°x

3

x

8

12 z x

z

z

7

6

y

y

3

z

14

y

x

y

x

y

5

10

z x

y

y

z x

15

z

x

90°x

8

910

11

12

13

14

15

Figure 2.18: Visualizing stimulated echo formation for an ensemble of spins, not including longitudinal or real transverse spin relaxation. Numbers indicate selected magnetization frames and their location in time. The 90o–90o–90o sequence used the same rf phase (x) for all three pulses. Note the initial formation of a “Hahn echo” at frame 8, with subsequent dephasing, before the second rf pulse at frames 9 and 10. The stimulated echo occurs very closely after frame 12, and magnetization then progresses to create the echo that forms via the first and last rf pulse very closely near frame 14. Note that the Hahn echo and the last echo have the same basic eight-figure appearance. For visual clarity, a minimal initial dephasing (just below half a turn for the fastest/slowest spins) was chosen here, so dephasing is not “complete,” and phase memory and interference somewhat affects the simulation. Also operating is interference in the echo formation processes, which depend on timing and dephasing rates, additionally causing slight time displacements of the echo maxima from their nominal positions. As seen in the lower summarizing magnetization graph, all three echoes become negative, and the stimulated and last echoes have lower amplitudes than the initial Hahn echo. They have magnetization components also in the positive y′-direction, reducing overall signal strength. The simulation and visualization code (in Matlab) was kindly provided by Daniel Gallichan, Cardiff. I thank Daniel for several rounds of generous advice and almost instant feedback, even including software modifications.

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2 Basic concepts

z

x

y

90°x 90°x

z

y

90°x 90°x

90°x

90°x

z

z

y

90°x 90°x

x

y

90°x

90°x

90°x

90°x

Figure 2.19: Magnetization in a stimulated echo experiment (lower part), also specifically highlighting stimulated echo shape changes (upper part) at four incrementally varying simulation-timing settings. A five times higher degree of dephasing compared to the previous figure was chosen here, so echoes then become correspondingly sharper and their mutual interference is much reduced. Again, these simulations do not account for any longitudinal or irreversible transverse spin relaxation, which would occur in a real-life situation. The stimulated echo location in time is indicated with a red arrow. The initial Hahn echo is marked with a green one, the refocusing one of the Hahn echo through the third rf pulse is blue and the echo that refocuses the initial FID through the third rf pulse is black.

simulation and computer graphics, things get simpler. The following figures illustrate individual frames of computer animations of echo formation. In summary, one must conclude that the general “shape” of a stimulated echo is not as clearly defined as the always-recurring Hahn echo eight-shape, and that it will furthermore vary with

2.6 Spin-echo formation under the influence of a steady magnetic field gradient

59

experimental settings. So, a similarly specific “stimulated echo shape” does not seem to exist. One should note that almost all historical and recent simulations/visualizations of this kind in the literature are made with a rectangular distribution of “slow– fast” spin vectors, rather than with that corresponding to a real NMR bandshape (a Lorentzian-like frequency distribution). Actual echo shapes would slightly differ for this reason (Figures 2.18 and 2.19). 2.6.3.4 Spin phase graphing, magnetization helices and diffraction analogies Singer presented a simple educational graphical approach back in 1978, which he named “spin phase graphing” [46], visually illustrating self-diffusion and flow NMR concepts (Figure 2.20). Dephasing

90x

Rephasing

g

x’

Phase angle

x’ +

– FID

a f

a

y’

b

b c

g

e

y’

d

c

Echo

x’

f

e

d

‘d’

y’

x’

a

g

b c

f e

“e”

d e f g

d c b a

“d”

y’

‘e’

a b c d e f g

Echo Position and frequency exchange

90x

180y

Figure 2.20: Spin-echo formation and diffusional attenuation, as demonstrated in a spin phase graph (after the concept introduced by Singer[46]).

Currently most popular way of additionally illustrating the actual effects of a magnetic field gradient and the refocusing process is in terms of a “magnetization helix” being created in the sample under the influence of a magnetic field gradient. The pitch of the helix will depend on its strength and time evolution. The present variant is traceable to a paper by Saarinen and Johnson [47], which also describes a method for measuring the strength of magnetic field gradients in NMR diffusometry. The helix-grating view is common to other concepts used in various types of laser-based light-scattering techniques, a field Johnson worked in before returning to NMR in the late 1980s. He later described in retrospect the steps behind the introduction of such optical analogy concepts in NMR [48].The basic first step is to visually connect a

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2 Basic concepts

magnetic field gradient on an NMR sample to its influence on the appearance of its NMR signal. For a constant gradient in the z-direction in a superconducting magnet geometry, a situation as in Figure 2.21 will result. Combining that with a Hahn spinecho sequence, one arrives at a situation as depicted in Figure 2.22. Magnetic field gradient

Active sample volume

0

Figure 2.21: Schematic NMR spectrum from a single line type of molecule (like water) without and under the influence of a constant magnetic field gradient in the z-direction. By “active” here it means the volume that is within the radiofrequency pickup region of the coil. NMR sensitivity would gradually fall toward its edges. The rf field constancy volume is normally close to what is designwise also the region of magnetic field gradient constancy. Both factors contribute in reality to make wings of the spectrum look less ideal than depicted in the figure.

Immediately after initial 90° rf pulse

At the echo point in time

180° rf pulse Dephasing

Rephasing

Figure 2.22: Spin-echo formation and winding/unwinding of a magnetization helix during a 90ox–180ox spin-echo sequence.

2.6 Spin-echo formation under the influence of a steady magnetic field gradient

61

The pitch of the helix (commonly denoted Λ in the literature), in a steady-gradient situation, evolves with time (t) up to the unwinding, and ultimately the refocusing rf pulse as Λ=

2π γgzt

(2:17)

In NMR imaging and some PGSE literature, this pulse sequence component is referred to as the encoding step. The denominator in eq. (2.17) is seen to correspond to the wave number (the number of waves per unit length) of the helical pattern (here λz ) and actually constitutes what is commonly referred to as “k-space” for mapping locations in space in NMR imaging or microscopy. Mansfield and Grannell introduced the optical and X-ray analogous concept in NMR magnetic field gradient context [49]. Referring to the example magnetization helix illustrated here, the z-component kz = 2π=λz and the position at the helix origin ðz0 Þ actually constitute a Fourier pair (i.e., are interrelated through a Fourier transform). Nowadays NMR-based images are indeed generated through gradient mapping into phase-related data sets, followed by multidimensional Fourier transformation. There is also a related concept of “q-space” that was introduced by Callaghan et al. [50] for spatially mapping displacements, like diffusion and flow, which can be combined with k-space NMR imaging. Both concepts are lucidly described in Chapter 5 of Callaghan’s monograph [45]. The q-space is also the focus of an educational article by Koay and Özarslan [51]. One should be aware that the pitch of a helix of this kind (and thus the number of turns wound before refocusing starts) is quite large, and very much larger than in conceptually drawn figures such as Figure 2.22. For protons, 10 mm upward from the sample center in Figure 2.21 and under the influence of an 1 T m−1 gradient in that direction, the helix pitch (Λ) actually becomes 2.4⋅10−5 m, meaning that 10 mm above the center, spin vectors would have rotated over 40,000 turns during 100 ms. It might appear quite strange that refocusing could work under such conditions, but evidently it does. Obvious sources of serious problems in this context would primarily originate from magnetic field gradient instability or mismatch of any kind. The basic “phase graph” concept by Singer [46] (Figure 2.20) has later been considerably extended, as described in a more advanced review paper by Weigel [52]. Although its focus is MRI applications, the tools described are highly relevant also for NMR diffusometry. 2.6.4 Effects of J-modulation on spin-echo formation When applying a basic Hahn echo sequence in proton NMR, one will often note oddlooking absorption bandshapes after Fourier transforming the second half of the spin echo, compared to those for normal single-pulse detection. For example, in single-

62

2 Basic concepts

pulse NMR, “first-order” multiplets, like 1:3:3:1 and 1:2:1 binomial patterns for –CH2– and –CH3 signals, would be expected for protons in an ethyl group. The relative multiplet total integrals/intensities should be 2:3. In spin-echo NMR, one instead observes cosinusoidal intensity variation of such signal components with the echo-forming time parameter (τ). It is called J-modulation and relates to fractions of periods of 1/J, where J is the homonuclear spin coupling in Hertz. For alkyl protons, J is of the order 7 Hz for three-bond couplings. As a consequence, for τ-values near multiples of 1/7 s (approximately 140 ms) the multiplet appearance may look relatively “normal,” since spin vectors have precessed full or half turns before the detection starting point in time (cf. Figure 2.23), although components may alternate between being positive or negative.

(a)

Excitation z

End of initial dephasing z (b)

90°

B1

yʹ

yʹ

xʹ

xʹ z

Start of reversed dephasing

yʹ z

(e)

yʹ

180°

xʹ z

(d)

xʹ

(c)

Inversion z

(f)

yʹ

yʹ xʹ Normal echo point in time

xʹ Continued dephasing

Figure 2.23: In spin-echo experiments signal frequency dispersion due to differences in chemical shifts ideally get fully refocused in the same direction in the rotating frame. In case of frequency differences due to homonuclear spin coupling, spin vectors at the nominal time of the echo will point in other directions as those corresponding to single-line ones. This will manifest itself in signal phases that vary with the rf pulse interval. The underlying reason is that the 180° pulse will also invert the spin state of the coupling partner so that the population that was “slow” before the 180° refocusing pulse becomes “fast” afterward, and vice versa. In this variant of the classic Carr–Purcell illustration (cf. Figure 2.16), it is thought that the reference frequency is placed in the midpoint of a signal doublet with “red” and “blue” components, and an –x pulse is applied to it, which tips all magnetization into the x′–y′-plane (a). Then it is thought to evolve for a time that dephases both components 45° degrees in the respective directions (b), before applying a –x 180° pulse (c), which instead makes them point 135° in the same respective directions. At that point in time, they also reverse spin states and precession directions, and then precess for 45° more in this reversed direction (d). The situation in (e) would have been the normal echo point in time, but now the respective spin vectors are 90° out of phase, as compared to signals that were not homonuclearly coupled. Normal spin-echo data acquisition starts at point (e) in time, and continues during period (f). In this schematic example, the multiplet components will appear to be 90° or −90° out of phase, as compared to other signals from the sample that refocus “normally” (as in Figure 2.16).

2.6 Spin-echo formation under the influence of a steady magnetic field gradient

63

For intermediate or longer τ values, multiplet components become distinctly phase shifted from other echo signals in the spectrum. For higher-order spin systems (i.e., for multiline spectra that result when the spin coupling size and the frequency difference between participating nuclei are similar in magnitude), complex interference effects occur. In effect, they can be seen as a very efficient transverse signal relaxation pathway [53, 54]. From a practical PGSE viewpoint, the overall result on spin-echo amplitudes is a considerable reduction in spin-echo intensity of such spectral bands. For “stimulated echo” signals discussed later, effects of J-modulation are typically much smaller than for basic spin-echo signals, since the time spin vectors spend in the x′–y′-plane is typically much shorter – only about 10 ms or so. Under these conditions, spin dispersion effects become only marginally evolved. Consequently, spectral intensities do not attenuate much, nor do they differ much in shape from those for single-pulse absorption-mode NMR. In fact, the effect of spin–spin couplings on echo formation in NMR was discovered in Hahn’s early experiments. He found a periodical amplitude modulation of proton-based echoes for ethyl alcohol with varying rf pulse spacing and similar effects for 19F spin echoes for some fluorine-containing organic molecules [34]. The phenomenon of spin–spin couplings and their influence on echo refocusing was further clarified in a later study with Maxwell [55]. One should note that the typical frequency spacing for J-couplings (e.g., 7 Hz for three-bond proton ones, as in an ethyl group) was very much smaller than the actual frequency resolution of Hahn’s spectrometer. Even the much larger effects of chemical shifts in proton NMR remained unknown at the time. Recall that chemical shift frequency dispersion does not manifest itself in spin-echo experiments. Unlike spin–spin coupling influence, it is ideally fully refocused in spin-echo context. Proctor and Yu first discovered chemical shifts in NMR [56], but for nuclei with larger chemical shift effects than for protons. Anecdotally, NMR community physicists were initially very annoyed by any such “disturbing chemical effects,” making more accurate determinations of magnetogyric ratios and similar quantities rather pointless. Several studies by others followed, and the “first” chemical shift-resolved NMR spectrum of an organic molecule (again, ethanol(!)) is ranked to be that in a 1951 paper by Arnold et al. [57]. The frequency resolution of their instrument was still insufficient to resolve the one-magnitude smaller spin–spin couplings already discovered by Hahn. Actual spin–spin couplings in ethanol were resolved later in the same year by Gutowsky, McCall and Slichter [58, 59], which opened up for NMR to become a hugely important tool in chemistry.

2.6.5 Quantifying self-diffusion using spin echoes – a simplified model To achieve detection motion along the direction of the magnetic field gradient (diffusion, flow, etc.), one must combine the magnetic field mapping described earlier

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with the concept of spin-echo formation in the rotating frame or reference. Singer presented a nice semi-intuitive derivation of the effect of random motion on I = 1=2 spins in a constant magnetic field gradient back in 1978 [46]. Proceeding from eq. (2.16) and the geometry described in the precious section, we can consider angular phase changes linked to spatial motion of spin-bearing molecules. Each molecule “sees” a slightly different field value due to a field gz gradient across the sample. The “excess” phase angle of a molecule at an initial position z1 due to the gradient becomes ϕ1 = γgz z1 t

(2:18)

Here z1 represents the position of the molecule in the gradient along the z- direction and t the time from the start of the experiment to the time of observation of the phase angle. We now suppose that a diffusion “event” occurs to that molecule, and that such an event is the sudden jump of the molecule at some arbitrary time t1 to another position z2 . The phase angle after a time t would then become ϕ2 = γgz1 ðt − t1 Þ + γgz2 t1

(2:19)

The difference in phase angles for a stationary and so-displaced molecule becomes ϕ2 − ϕ1 = Δϕ = γgt1 ðz2 − z1 Þ

(2:20)

Δϕ = γgðΔzÞt1

(2:21)

or

The time t1 at which a diffusion jump occurs varies between 0 and t. Proceeding to evaluate the average of the square of the phase shift, we obtain D E D E ðτ D E ðΔzÞ2 ðΔzÞ2 τ3 (2:22) ðΔϕÞ2 = γ2 g2 t12 dt1 = γ2 g2 3 t t 0

hðΔzÞ2 i

Here t is the diffusion rate, that is, the average of the squared distance jumped per time unit. Replacing this quantity with the self-diffusion coefficient D by using the Einstein relation for one-dimensional diffusion (eq. (2.4)) leads to D E ðΔzÞ2 = 2D (2:23) t and the mean-squared phase deviation will be D E ðΔϕÞ2 = 2γ2 g2 Dτ3 =3

(2:24)

65

2.6 Spin-echo formation under the influence of a steady magnetic field gradient

The pertinent attenuation factor for the overall magnetization vector (M) thus becomes M = M0 expð − 2γ2 g2 Dτ3 =3Þ

(2:25)

Using more rigorous arguments (like in similar context here in Section 2.6.9 and later), Carr and Purcell [33] and Torrey [60] originally arrived at the same relation.

2.6.6 Influence of flow or on spin-echo formation Similarly, for, for example, plug flow in the direction of the magnetic field gradient, one can derive an equation that quantifies the relation between flow rate and spinecho formation. It is easy to see that refocusing will here not occur along the x′- or y′-axes, but rather somewhere else in the x′–y′ plane. The echo will thus be phase shifted, rather than appearing purely positive or negative (Figure 2.24).

Gradient or time

Figure 2.24: Schematic Fourier-transformed echo effects resulting from a spin-echo pulse sequence under the influence of a magnetic field gradient and transport conditions of random diffusion or plug flow, respectively.

The relationship between plug flow velocity (υ), flow time (t), gradient strength (g) and magnetization phase shift (ϕ) turns out to be ϕðt, g, υ, γÞ = γgυt2

(2:26)

with angular units depending on those chosen for the right-hand side parameters. Steady-gradient studies are today rather uncommon, except in the form of the socalled fringe field experiments, which use the huge and very stable magnetic field gradient that surrounds a normal superconducting magnet coil (see Section 7.2

66

2 Basic concepts

below). Procedures of this kind have found use for investigations of, for example, polymer melts and similar. The normal procedure is to pull the probe some distance out from its normal location, and perform the steady-gradient experiment at a lower than nominal NMR frequency [61]. Variants that use two supercon magnets sandwiched upon each other have also been described. Fringe-field procedures lead to a slice-selected detection area, since the rf pulses will excite only part of the sample – the rest of it will be out of resonance. Also, frequency resolution is not possible under the influence of the strong magnetic field gradient. However, Wu and Johnson have described a pneumatic shuttle system for bringing the sample back and forth between a high-gradient position and the normal one to bypass this limitation [62].

2.6.7 Self-diffusion effects alongside other NMR parameters in spin-echo experiments In the absence of spin relaxation effects, homonuclear spin–spin couplings, chemical exchange and so on, refocusing is ideally complete as long as the nuclei remain stationary in the sample during the sequence. Transport along the gradient direction by, for example, flow, convection or self-diffusion will also modify the situation. Hahn correctly noted this in his original paper [34] and also derived their basic quantitative relation to self-diffusion in the case of a steady background magnetic field gradient. He found that random self-diffusion leads to an exponentialtype attenuation of the echo amplitude relative to the magnetic field gradient strength. It also exponentially depends on the third power of rf pulse interval (τ), the magnetic field gradient and the magnetogyric ratio of the nuclei squared and the transverse spin relaxation rate, T2 : Að2τÞ=Að0Þ = expð − 2τ=T2 Þ · expð − 2γ2 Dg2 τ3 =3Þ

(2:27)

From a basic NMR point of view, the third power dependence of echo attenuation relative to the diffusional contribution is obviously troublesome, when using the two-pulse echo sequence for T2 -determination purposes, especially for a poorly homogeneous magnet system and rapidly diffusing species. Here, more complex multipulse PGSE methods described elsewhere (see Chapter 7) are often much preferred or even necessary in heterogeneous systems. By manipulating the pulse interval and the gradient strength (if possible), one can, to some extent, optimize the diffusional attenuation of the echo relative to the destructively irreversible transverse relaxation one. Signal attenuation through transverse spin relaxation cannot be brought back in the echo form, which is a generally limiting problem for NMR diffusometry methods. To make things worse, rapid transverse spin relaxation is highly correlated with slow molecular diffusion. For most “typical” systems in solution, spin relaxation

2.6 Spin-echo formation under the influence of a steady magnetic field gradient

67

attenuation need not be any huge obstacle in practice on modern high-performance instrumentation when using appropriate pulse sequences and reasonably optimized timing parameters.

2.6.8 Spin echoes and steady-gradient diffusional effects in multipulse NMR experiments In early investigations on echo formation using many rf pulses, one noted additional types of diffusional influence. In 1954, Carr and Purcell suggested and tested a method to measure the “true” transverse spin relaxation time (T2 ), uninfluenced by diffusion and magnetic field inhomogeneity effects that seriously may influence T2 -results, alternatively based on a sweep-recorded NMR signal width [33]. The original pulse method for T2 -determination, in the form of the “Hahn echo” amplitude at varying echo times, may partly fail, because of diffusional influence on the experiment for “long” T2 -values. Carr and Purcell aimed to bypass this effect by applying a train of more or less closely spaced 180o pulses, following an initial 90° one (Figure 2.25).

90x

180y 󰜏

180y 2󰜏

exp(–time/T2*)

180y 2󰜏

180y 2󰜏

󰜏

etc

exp(–time/T2) Time

Figure 2.25: Initial part of a Carr–Purcell–Meiboom–Gill (CPMG) pulse train [33, 63]. Normally one would use tens or more of refocusing 180° pulses. The intermediate echo maximum envelope with time between the 180° pulses is affected by T2 , self-diffusion and homonuclear spin–spin couplings, with proportions that can be tuned by changing the pulse spacing.

As expected, exponentially decaying echo maxima formed between subsequent echoes at the midpoints between the 180o pulses in the train. This decay rate more closely corresponds to the true T2 -value than the apparent one from an inhomogeneitybroadened signal line width, but is still to some extent influenced by self-diffusion of the molecules studied, as well as through molecular transport due to convective overturning in the sample. Depending on the “background” magnetic field inhomogeneity, the echoes become more or less sharp as well. Carr and Purcell observed that even echoes were much less influenced by convection than odd-numbered ones. Importantly too, they noted that the diffusional influence could be

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diminished and ideally brought to zero by spacing the refocusing 180o pulses tightly together in time (t), and derived the following quantitative echo attenuation envelope relation for a situation of n consecutive echoes:

(2:28) A = A0 exp ð − t=T2 Þ + ð − γ2 g2 Dt3 =12n2 Þ At the time, spin-echo experiments had been carried out with arbitrarily or identically phased rf pulses. In 1958 Meiboom and Gill suggested the use of coherent and controlled excitation in an improvement of the Carr–Purcell experiment [63]. Here the phase of the initial 90° pulse is set to differ by 90° to those of the 180o train. In the original Carr–Purcell form, cumulative effects otherwise occur that seriously deteriorate the quality of the measurements. The Meiboom–Gill modification creates self-compensation of deviations from the true 180o level. Nonidealities of this kind will also inevitably result from even small rf inhomogeneity characteristics in the probe system, especially affecting noncentral sample areas. This modified pulse sequence and detection mode later came to be named a Carr–Purcell–Meiboom–Gill (CPMG) train. It has found considerable use in basic NMR as well as in MRI and NMR microscopy, for quantifying T2 and for primarily contrast purposes, respectively (Figure 2.26).

z

(a)

yʹ

z

(b)

yʹ xʹ

yʹ

yʹ xʹ

z

(d)

z

(c)

xʹ z

(e)

yʹ xʹ

xʹ

Figure 2.26: Self-compensation of minor nonidealities in the Carr–Purcell 180° rf pulse train, through the Meiboom–Gill modification regarding rf signal phase relations in it. Redrawn with permission from Meiboom and Gill, Rev Sci Instrum. 1958;29:688–691 [63], Copyright (1958), American Institute of Physics. The odd handedness of the rotating frame in the original publication has been kept.

It is important to note that what was discovered in this context was general building blocks for pulse sequence self-compensation in some sense for hardware nonidealities and to allow separation of simultaneously mixed forms of molecular transport (diffusion, convection, flow, etc.) in PGSE-type applications. Some later implementations of such building blocks will be discussed in more detail later and

2.6 Spin-echo formation under the influence of a steady magnetic field gradient

69

some (CYCLOPS and EXORCYCLE) are discussed already in Section 1.11. Today, computerized numeric optimization methods are generally recommended way to design very complex pulse trains of various kinds, compensating for instrumental nonidealities and other problems [64, 65]. J-modulation effects on echo formation in CPMG trains can actually be eliminated, simply by adding extra 90° pulses at the midpoint between the 180° ones. The simplifying consequences in this context were unnoticed for two decades, until their rediscovery by Aguilar et al. [66]. Torres and coworkers also have designed pulse schemes that eliminate J-modulation influence [67] (Figure 2.27).

90x

󰜏

180y

󰜏

90y

󰜏

180y

󰜏 Time

rf n

Figure 2.27: Eliminating J-modulation in a CPMG train, by adding a 90° pulse at the formation point in time for the intermediate echoes (after Aguilar et al. [66]).

2.6.9 Quantifying self-diffusion using spin echoes – deriving echo attenuation expressions more rigorously The classical analysis is founded on an early Bloch–Torrey analysis [60], including spin relaxation terms and accounting for Fick’s second law of diffusion. Modernday computational schemes originate from a paper by Karlicek and Lowe [68]. A key simplifying concept here was to replace 180° rf pulse influence by a reversal of the magnetization dispersion by the gradient pulse. After several analysis steps, they arrived at a master expression for the integrated y′-magnetization (i.e., the relative echo amplitude) ψðtÞ of this form: 2 2 32 3 ðt ðt′ 6 7 (2:29) ψðtÞ = ψð0Þexp4 − Dγ2 4 gðt′′Þdt′′5 dt′5 0

0

where g(t″) represents the total effective magnetic field gradient, external and internal. eq. (2.29) has become the standard master equation used for evaluating even quite complex gradient and rf pulse schemes. Returning to the previous semi-intuitive derivation of echo attenuation for a Hahn spin echo in a constant one-dimensional magnetic field gradient (eqs. (2.18) – (2.25)), we can use eq. and the 180° rf pulse magnetic field gradient sign inversion reversal trick to provide a more rigorous derivation of the same end result. In this view, one also introduces the concept of the “effective gradient,” which accounts

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for the sign reversal. Instead of the traditional rotating frame view of the combined effects of rf pulses and gradient dispersion of spin vectors, one uses a “rotating flipflop” frame of reference. For t < τ, the integral becomes ðt

gðt′Þdt′ = − gt

(2:30)

gðt′Þdt′ = gðt − 2τÞ

(2:31)

0

and for t > τ ðt 0

Finalizing the integration, summing these together and taking the logarithms of the resulting echo amplitudes, we arrive at the expression "ðτ # 2ðτ  Að2τÞ − 2γ2 Dg2 τ3 2 2 2 2 2 = − γ D g t dt + g ðt − 2τÞ dt = (2:32) ln 3 A0 0

τ

The same analytical scheme is directly also applicable for the more complex pulsed magnetic field gradient sequences described in the next section and elsewhere. Integration, summation and simplification steps involved are rather straightforward, but quickly become quite tedious and error-prone when carried out manually. Numerous explicit integration steps for various example pulse sequences are listed in detail and tabulated in Sørland’s book [69]. Price and Sørland also provide Maple or Mathematica code for this general purpose in their respective monographs [17, 69]. See also Chapter 5.5 in Callaghan’s book [45] and the educational paper by Kuchel et al. [70] that also provides related Mathematica code and describes derivational steps in detail.

2.7 Spin-echo formation under the influence of pulsed magnetic field gradients As mentioned in the introduction, the incorporation of pulsed rather than steady gradients was a key methodological improvement in the context of NMR diffusometry. The most important practical one was probably that the spin echo could be detected in a magnetic field that was homogeneous in the normal sense. This paved the way for future frequency-resolved PGSE methodology, via Fourier transformation of the echo signal (i.e., FT-PGSE). In addition, it should be noted that the spin echo gets narrower in the time domain with stronger gradients. For this reason, its amplitude could become difficult

2.7 Spin-echo formation under the influence of pulsed magnetic field gradients

71

to detect and quantify and demand high-performance electronics and digitizing systems. Under the influence of a steady gradient, the spin-echo method also becomes contradictory. If the NMR spectrum of the spins in question is spread out over several tens of kilohertz or more, it is no longer possible to excite every region of the sample area evenly with a single rf pulse. As also evident from eq. (2.27), transverse relaxation affects both the transverse relaxation term and the diffusional one, and they are not readily separable by varying the τ-value. Mechanically varying the gradient would be a way, but not really a feasible one in practice. Current-controlled generation of magnetic field gradients through extra coils around the sample is easy in principle, but under constant-gradient conditions heating effects of various kinds will quickly become a problem with increasing gradient strength.

2.7.1 The basic PGSE experiment

90x

rf

g(t)

180y

𝜏

𝜏 Time

𝛿

𝛿 Δ

Signal

Figure 2.28: The 90°–180° “Hahn echo”-based PGSE experiment variant. The initial FID signal after the first rf pulse is not sampled here. Actual data acquisition starts at the echo peak and includes the half-echo that follows.

The methodological introduction of pulsed magnetic field gradients in 1965 constitutes an important step indeed. In its basic application, field gradients are applied in pairs within the Hahn echo sequence, preceding and following the magnetization-inverting second rf pulse. For ideally rectangular gradient pulses, the so-called Stejskal–Tanner relation (eq. (2.33)) will describe the echo attenuation. Using the computational scheme outlined in Section 2.6.9, it is straightforward to derive this relation (see the Appendix in the final section of this book).

(2:33) A = A0 · expð − 2τ=T2 Þ · exp − ðDγ2 g2 δ2 ðΔ − δ=3ÞÞ

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2 Basic concepts

It should be noted that the δ=3 − term here is basically a “correction” for the finite duration of the gradient pulses (some implications and consequences are discussed elsewhere in this book) (Figure 2.29).

90x rf

𝛿

g(t)

g

𝛿

g

Time

∫g(t')dt' ∫∫g(t")dt"dt'

–g𝛿Δ

Figure 2.29: Effective gradient sequence for the 90°–180° Stejskal–Tanner experiment (the 180° phase inversion has been substituted with a sign reversal of the second gradient pulse), illustrating the quantities in eq.(2.29). Redrawn and slightly modified, with permission, from Figure 5.18 in Callaghan [45], page 210; Copyright (2011) Oxford University Press.

1

0.8

0.8

0.6 0.4

0.6 0.4 0.2

0.2 0

log(Amplitude)

1

Amplitude

Amplitude

Provided the rf pulse interval (τ) is kept constant, echo attenuation will only depend on diffusion-related parameters and can be separated from transverse spin relaxation. However, when transverse spin relaxation rates become similar or shorter than the duration of the pulse sequence, the signal amplitude attenuation can become significant or overwhelming. Remedies include increasing the gradient strength, while shortening the overall duration of the pulse sequence, or instead using the stimulated echo variant (PGSTE) of the basic “Hahn echo” PGSE sequence.

0

2

4

6 –1

Gradient (Tm )

8

0

0

20

40

60 2

–2

Gradient squared (T m )

0 –0.5 –1 –1.5 –2 –2.5 –3 –3.5 –4 –4.5

0

20

40

60

Gradient squared (T2m–2)

Figure 2.30: Basic evaluation of a model proton-based PGSE experiment at fixed rf pulse interval settings (τ), using nonlinear and linearized forms of eq. (2.33). The gradient strength was thought to be varied in 16 equidistant steps, from 0 to 7.5 T m−1, at fixed gradient timing settings of δ = 1 ms and Δ = 100 ms, at a D-value of 10−11 m2 s−1.

The standard procedure is to increment the gradient strength ðgÞ, while keeping the gradient timing parameters ðΔ, δÞ constant. The echo amplitude will then ideally de

pend on expð − D γ2 g2 δ2 ðΔ − δ=3Þ Þ. Traditionally one would proceed by analyzing the data via semilogarithmic plots of echo amplitude ðAÞ against the quantity

2.7 Spin-echo formation under the influence of pulsed magnetic field gradients

73

within the bracket. It is often denoted “b” or “k” in this context; “b” is especially common in the medically oriented MRI literature, and “k” was so in physicochemical work some decades back. The slope of such ideally linear plot thus corresponds to the experimental D-value (Figure 2.30). However, nonlinear fitting procedures of data of the above type are trivial with today’s computing power and are much preferred over such linearized evaluation. Rationales include more correct statistical data point weighting of the experimental raw data, especially at high echo attenuation levels. It has also been suggested in this context that one should space individual gradient settings individually to add more statistical weight to curve areas where they change more steeply. When sampling diffusion information for many components with widely different self-diffusion coefficients in the same experiment, this could be a recommendable procedure, and may already be available as a standard option in standard spectrometer software. One should be aware that overdoing such biased global data point mapping may instead add unjustified statistical weight to lower gradient settings. Related ideas were recently also discussed by Masuda and coworkers [71], but from the general viewpoint that less attenuated peaks in a data set like a spin relaxation measurement or a PGSE study are already well sampled and characterized by good signal/noise, compared to more attenuated ones. It was demonstrated that a quite significant experimental time saving could be achieved with only minor loss in measurement quality through arrayed experiments where the number of scans (NS) are varied throughout the data set with appropriate normalization of equivalent spectral amplitudes. The matching settings for the standard case of exponentially decaying peaks would thus be an exponentially rising NS number in such an approach. Recent educational papers by Kuchel et al. [70] and Sinnnaeve [72] describe very detailed and alternative reanalyses of the derivational steps behind the analogously derivable standard PGSE attenuation relation (eq. (2.33)), also for other gradient shapes than just the rectangular ones. In some context the “Stejskal–Tanner relation” (2.33) may be a approximation. It assumes using ideally rectangular gradient pulses normally with a short duration ðδÞ compared to their separation ðΔÞ. The term δ=3 accounts for diffusion during the gradient pulses. The limiting case (where gradient pulses are treated as infinitely short, while the gradient strength and gradient pulse length product is finite) is normally referred to the SGP (short gradient pulse) limit in the literature. The concepts are normally of no concern in normal solution PGSE studies, but may become quite significant when evaluating data from specialized PGSE-based studies at experimental settings required for heterogeneous systems. The subject is further discussed in Section 5.6.1 and later, and in the Appendix. The use of digitally controlled, nonrectangular gradient pulses is otherwise quite common nowadays. Sinusoidal or ramped rectangular shapes are the most common, and primarily thought to lessen influence of steep gradient changes that may cause eddy current disturbances. Such gradient pulse shape options have already been around as standard settings in spectrometer software for many years already, ever

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since Gross and Kosfeld introduced sinusoidal ones in PGSE in 1969 [73]. Several papers have also appeared over the years that derive “Stejskal–Tanner” equation variants for numerous alternative gradient shapes than truly rectangular ones [72, 74–76]. One should note that the standard procedure of experimental calibration by basing gradient influence on PGSE diffusional echo attenuation on a “calibration substance” with known self-diffusion coefficient (at closely the same experimental settings) compensates for a variety of systematic errors in PGSE. An highly recommended list of “calibration substances” is that compiled by Holz and Weingärtner [77].

2.7.2 The stimulated echo (PGSTE) variant of the PGSE experiment 90x

90x 󰜏1

90x 󰜏2

󰜏1 Time

rf

g(t)

𝛿

𝛿

Δ

Signal

Figure 2.31: The stimulated echo variant of the basic PGSE experiment. Relative rf pulse phase and receiver phase cycling is required to cancel out other echoes (there are four others). For example, the first rf transmitter pulse phase and also simultaneously the receiver phase should at least be cycled four times, in turn along the x-, −x-, y- and –y-directions, while keeping the second and third pulse phases constant. Proper phase cycling schemes for this and other considerably more complicated pulse sequences is normally provided in the vendor’s spectrometer pulse libraries already. One particularly recommended and simplifying variant uses a comparably weaker “crusher gradient pulse” after the second rf pulse, intended to destroy any transverse magnetization at this point (schematically drawn in orange color in the figure).

The attenuation relation for the stimulated echo sequence illustrated in Figure 2.31 reads:

1 A = A0 · expð − ð2τ1 =T2 + τ2 =T1 ÞÞ · exp − ðDγ2 g2 δ2 ðΔ − δ=3ÞÞ 2

(2:34)

Here, and in what is nowadays referred to as a PGSTE experiment, “additional echoes” are removed through phase cycling and influence of somewhat weaker interdispersed “crusher gradient pulses” within the pulse sequence. The intention is to

2.7 Spin-echo formation under the influence of pulsed magnetic field gradients

75

destroy unwanted magnetization components while being in the transverse magnetization plane. Only half of the initial magnetization gets refocused, and therefore a factor ½ commonly appears in eq. (2.34). Accounting for competing spin relaxation effects, the signal attenuation disadvantage is actually often much less. In practice, the overall effect normally becomes relative enhancement rather than attenuation. Stimulated echo, rather than “Hahn echo” detection in NMR diffusometry, is for this and other reasons the normally preferred and recommendable way to proceed in basic NMR diffusometry. In the magnetization helix visualization variant of Saarinen and Johnson [47], the situation at the point in time where the stimulated echo forms appears as in Figure 2.32.

mm from z-center

5

0

–5 1 0.5 0.5

0

y'-Magnetization 0

–0.5

x'-Magnetization

Figure 2.32: Schematic magnetization helix along the z-sample axis in a superconducting magnet z-gradient geometry, at the point in time where the stimulated echo forms. As for a 90°–90° Hahn echo, the helix is displaced 0.5 units in the y-direction, and the net x′-component is zero. The echo amplitude thus amounts to 0.5 of that in the original FID signal. Before and after the stimulated echo center point in time, the helix along the sample axis oscillates in the x′–y′-plane, like a “twister tornado,” and fans out toward a net zero magnetization situation in the x′–y′-plane.

2.7.3 Basic advantages in using PGSTE rather than PGSE As discussed earlier in Section 2.6.4, for homonuclearly coupled spin systems Hahn echo signals appear “phase distorted” for first-order multiplets in proton NMR. They can also be practically nullified after about τ = 100 ms for strongly coupled spin

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systems (as for protons in long alkyl chain). Echoes in a PGSTE experiment of the same total duration are only moderately affected. The reason is relatively easy to see. As mentioned already, J-modulation only occurs while magnetization is transverse rather than longitudinal, so “storing magnetization” in the z-direction for most of the duration of the pulse sequence (as for PGSTE) suppresses destructive J-modulation effects to a large extent. More complex pulse sequences developed in later decades can actually remove spin couplings (and thus also J-modulation) altogether in PGSE context, at the expense of longer experimental duration and complexity [67, 78]. Especially for macromolecular or supramolecular systems in solution, also differing spin relaxation effects enter in. As illustrated in Figure 2.4, transverse spin relaxation tends to be much more rapid in slowly tumbling systems than longitudinal, and irreversible signal loss during the echo sequence thus becomes less pronounced, as a direct effect of longitudinal “storage” during the sequence. The concept of temporarily “hiding” magnetization in the longitudinal (z)-direction during the pulse sequence is utilized in several more complex variants of the basic PGSE experiments. Its main purpose is to avoid or minimize the effects of several destructive influences on echo formation, which operate on transverse magnetization during the pulse sequence. One should finally note that refocusing could be brought about only through gradients, without a reversing 180° radiofrequency pulse. This concept is named a “gradient echo,” and in its basic form uses a bipolar pair of gradient pulses. While conceptually important as a building block in more complex pulse sequences and widely used in NMR imaging, the basic effect illustrated in Figure 2.33 is only of marginal interest in the context of NMR diffusometry.

90x rf g(t)

Time

Figure 2.33: Spin-echo creation through the use of a bipolar gradient pair. Refocusing is analogous to that illustrated in Figure 2.22, except for the mirroring effect of the 180° rf pulse.

2.7.4 Multipulse CPMG-based PGSE Applying the CPMG sequence in PGSE diffusometry context, gradient pulses are inserted between the 180° pulses at times mτ and ðm + 2Þτ, where m is an odd integer,

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77

and a second gradient pulse between the 180° pulses at times ðm + 4n + 2Þτ and ðm + 4n + 4Þτ, where n is an integer. By setting ΔCPMG = ð4n + 2Þτ one thus insures that the second gradient pulse occurs after an odd number of 180° pulses [79, 80]. Following Price (p. 90) [17] and referring to Figure 2.34, the spin-echo attenuation will then be given by Aðtecho = 2NτÞ = A0 expð − techo =T2 Þ · expð − γ2 g2 Dδ2 ðΔCPMG − δ=3ÞÞ

180y

90x 180y 2

180y

180y

2

(2:35)

180y 2

g

g

Time

Δ = (4n+2) Echo at 2N Figure 2.34: CPMG-based PGSE timing sequence, where only the last echo is collected. It is relatively seldom used as compared to normal PGSE or PGSTE.

2.7.5 Background magnetic field gradient influence on PGSE experiments In the presence of significant gradients in the static field (symbolized g0 ), the echo attenuation relations become more complicated, even in a strictly linear and orthogonal two-dimensional case. For a Hahn spin-echo PGSE experiment under such conditions, its influence would generally be described by [81] " Að2τÞ = Að0Þ exp

− 2τ=T2 − ðγgδÞ2 DðΔ − δ=3Þ − 2γg02 Dτ3 =3 + γ2 g · g0 Dδðt12 + t22 + δðt1 + t2 Þ + 2δ2 =3 − 2τ2 Þ

# (2:36)

where t1 represents the time between the first rf pulse and the leading edge of the first gradient pulse and t2 the time span from the trailing edge of the second gradient pulse to the echo at 2τ. In practice, such external gradients would hardly be constant, and would be difficult to correct for in practice. Increasing the pulsed gradient influence by suitable experimental parameter changes can make the second and third factors negligible, compared to the controllable first one. On normal high-resolution spectrometer setups with very low basic magnet field inhomogeneity, external gradients would normally not be a source for complications. As discussed in Section 7.6 of the textbook by Price [17] and in an original paper [82], the cross term in eq. (2.36) would result in the echo attenuation

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depending not only on δ and Δ but also on other timing parameters. Semilogarithmic plots of Að2τÞ versus γ2 g2 δ2 ðΔ − δ=3Þ then become nonlinear, with significant contributions from this cross term. Deviation patterns from linearity would depend on the polarity of the pulsed gradients relative to the background gradient, especially at low-amplitude-pulsed gradient settings. This can be used for diagnosing this particular experimental problem. A first-order compensation remedy would be to collect data for two experiments (i.e., with positive as well as negative pulsed gradient settings) and then use a suitable averaging procedure to cancel out the relatively symmetrical deviations. Like all sound data fitting, this should be done on nonlinearized data. Linearized data plots are primarily only useful for graphical illustration of various concepts and effects. An equation similar to (2.36) describes the situation for a PGSTE sequence. Considering typical rf and gradient timing settings for the two alternative experiments, a general conclusion can be drawn that the cross-term effect for a PGSTE sequence will generally be less important than for an equivalent PGSE one [17]. As already stated, situations where there are significant background gradients in PGSE studies today would be rare, and would primarily occur when using very low-pulsed gradient settings and/or unusual and inappropriate sample shapes and sizes. Potentially more significant echo attenuation effects, induced by internal gradients in samples of heterogeneous systems of various kinds, often occur for several reasons. However, some families of more complex pulse sequences, originally devised by Karlicek and Lowe [68], can compensate for such extraneous echo attenuation. Sørland’s textbook, in particular, contains extensive discussions of experimental studies, design and use of such pulse sequences in this context [69]. Generally, there are several other complications in heterogeneous systems, some of which are further outlined in Section 7.5.

2.7.6 Further reading Hennig J. Echoes-How to Generate, Recognize, Use or Avoid Them in MR-Imaging Sequences Part I: Fundamental and Not So Fundamental Properties of Spin Echoes. Concepts Magn Reson. 1991; 3: 125–143. Hennig J. Echoes-How to Generate, Recognize, Use or Avoid Them in MR-Imaging Sequences Part II: Echoes in Imaging Sequences. Concepts Magn Reson. 1991; 3: 179–192. Burstein D. Stimulated Echoes: Description, Applications, Practical Hints. Concepts Magn Reson. 1996; 8: 269–278.

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2.8 Optimization and evaluation of basic PGSE and PGSTE experiments 2.8.1 Dealing with intrinsic nuclear spin factors First, one should consider the implications of the information in Table 2.2 on diffusional echo attenuation and competing transverse spin relaxation, as computed from the Stejskal–Tanner eq. (2.33). Evidently echo attenuation significantly depends on the magnetogyric ratio of the nuclei under investigation via a expð − γ2 Þ factor. This magnetogyric echo attenuation effect related to self-diffusion is illustrated in Figure 2.35 and is quite substantial (being an exponent of something squared).

Table 2.2: NMR parameters of relevance for some nuclei that can possibly be studied by PGSE diffusometry in solution (adapted by permission, from Stilbs P, Progr Nucl Magn Reson Spectrosc, 19, pp. 1–47, 1987, and corrected, Copyright (1987) Elsevier). Nucleus 

H H  Li  Be  C  F  Na  Mg  P  Cl  Ag  Cd  Cs 

Spin quantum number (I)

Magnetogyric ratio (γ, 10–7 rad T−1 s−1)

(γ=γ H )2

/  / / / / / / / / / / /

. . . . . . . . . . −. . .

. . . . . . . . . . . . .

Typical T2 range in solution (s−1) .– .– .–* .–* .–† .– Below .* Below .* .– Below .*,‡ Very long* .–* .–*

* As aqueous ions in solution. † Instrumentally determined and dependent on proton decoupling conditions. ‡ Perchlorate ions in solution; the figure for chloride ions is below 25 ms.

Echo intensities also exponentially depend on nuclear spin relaxation rates, but unless spin relaxation so fast that this effect dominates or even “kills” the echo, this need not be a problem in practice. If experimental signal/noise is too low one could simply go for a longer time accumulation, but definitely it will be advisable to reconsider modify one’s experimental PGSE or PGSTE settings before doing so. A massive signal gain may result even when correcting a single nonoptimal parameter or pulse program setting. If transverse spin relaxation times are of the same

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Relative amplitude

1 0.8 0.6

Protons Deuterium Carbon-13

0.4 0.2 0

0

0.5 1 Magnetic field gradient/Tm– 1

1.5

Figure 2.35: Expected diffusion-related echo attenuation effects during PGSE experiments on a molecule type with self-diffusion coefficient 10−9 m2 s−1, as based on proton, carbon-13 and deuteron detection, respectively. The gradient strength is thought to be varied between 0 and 1.5 T m−1, while keeping Δ and δ constant at 100 and 1 ms, respectively. Of course, absolute signal amplitudes for the three nuclei would be quite different, because of their differing intrinsic NMR sensitivities and natural abundance. The actual “NMR” receptivities of carbon-13 and deuterium like 4 and 6 magnitudes are lower than that for protons, respectively. Studies are still quite realistic and easy on isotopically enriched sample components. In actual experiments, one would obviously benefit strongly with regard to measurement quality by lowering the gradient setting range for protons, and to increase them for deuterium- and carbon-13-based measurements, as compared to the “settings” illustrated in the figure.

magnitude as the duration of the pulse sequence, shortening it and correspondingly increasing the magnetic field gradient strength will likely do wonders. Prolonged pffiffiffi accumulation only helps with a factor n, where n represents the number of echoes acquired. The absolute majority of NMR diffusion measurements to date have been made on protons – the most abundant “NMR nucleus” and also the most facile nucleus for the purpose, owing to its high NMR frequency and relatively low spin relaxation rates. Selecting from otherwise feasible NMR nuclei, actually just a handful, is for spin relaxation reasons amenable for PGSE studies. This holds even in favorable situations on concentrated systems. PGSE detection on 1H, 2H (enriched), 7Li, 13C, 19F and 31P should generally be quite realistic on any reasonable system in solution. Nuclei with lower magnetogyric ratios are much more difficult, and studies are often only possible in favorable circumstances.

2.8.2 Dealing with quadrupolar nuclei In addition, those with I >1/2 have a quadrupole moment, which mostly leads to spin relaxation rates that are way too fast to be compatible with requirements of PGSE-type experiments. However, if the quadrupole moment is small and

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particularly if the nucleus also chemically resides in a electric charge symmetric environment, studies on such nuclei may still be feasible. Deuterium is a particularly favorable exception among quadrupolar nuclei. It has a low quadrupole moment and a moderately low magnetogyric ratio. Here, even with covalently bound deuterium (i.e., despite an asymmetric electrostatic environment) spin relaxation remains moderately rapid. Studies turn out to be relatively easy on isotopically enriched sample components of low molecular weight. A very high selectivity is achievable as well, facilitating targeted studies on single components in very complex systems indeed, or when otherwise facing high dynamic range problems in proton-based PGSE. Multicomponent self-diffusion investigations via deuterium monitoring get more problematic since the chemical shift dispersion in deuterium frequency-domain spectra is small. It is typically only 1 % of that of 13C on the same compound, when accounting for differences in spin relaxation rates. By using spectral deconvolution methods described in Chapter 6, signal overlap need not be a problem anymore, especially on high-field spectrometer systems. Most other quadrupolar nuclei are more or less inaccessible for FT-PGSE due to rapid spin relaxation. However, exceptions are seen when the quadrupole moment is small and there is a symmetric electronic environment around the nucleus, as for hydrated or otherwise symmetrical ions in aqueous solution. Examples include 35 ClO4, 6Li+, 7Li+, 133Cs+ and 9Be2+. Spin-lattice relaxation rates for the latter ions in D2O solution are several seconds, and they are seldom below 1 s, even in aqueous solutions containing macromolecules. One should be aware that 133Cs ions in aqueous solution have a quite temperature-dependent chemical shift. This may cause problems of various kinds in PGSE studies of cesium ions in aqueous solution. Also note that the original analyses of the PGSE experiment were made for spin-1/2 nuclei. It is not self-evident that this should also apply for quadrupolar nuclei. Callaghan et al. demonstrated that it does for isotropic systems, but not for anisotropic ones like liquid crystals [83]. Furó and Dvinskikh have later summarized deuterium-based self-diffusion studies in anisotropic systems in a review paper [84].

2.8.3 PGSE, as applied to carbon-13 nuclei 13

C has a natural abundance as 1% and a magnetogyric ratio at 25% of that of protons. A large number of isotopically enriched compounds are commercially available, but often at a relatively high cost. On the other hand, natural-abundance 13C FT-PGSE measurements on concentrated systems are actually quite easy. Very complex systems may then become accessible for study as a benefit of the large l3C chemical shift dispersion and the relatively low spin relaxation rates.

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In the context of experimental design, one should remember that 13C T2-relaxation under proton broadband decoupling is artificially dependent on the efficiency of the proton decoupling. The residual line broadening corresponds to instrumentally related apparent 13C T2-values that are less than those for the corresponding T1-values, even for compounds of low-molecular weight (Figure 2.36). For this reason, three-pulse stimulated-echo PGSTE appears preferable in 13Cbased self-diffusion studies. Relatively few such studies have appeared in the literature, although the influence of the approach was demonstrated (using Hahn echoes rather than stimulated ones) on low-performance instrumentation already in the early 1980s [85–87].

Original 10,000

Arbitrary intensity

8,000 6,000 4,000 2,000 0 5 10 Trace # 15

3

2.5

2

1.5

1

0.5 × 104

Frequency channel Figure 2.36: A carbon-13-based PGSTE data set on a 20% mixture of ethyl and dodecyl benzene in CDCl3. A standard 5 mm NMR tube was used, for gradient settings 3–55 G cm−1 in 16 equidistant steps, each time-averaging 16 half-echoes. Δ and δ were kept constant at 200 and 6 ms, respectively.

2.8.4 Dealing with instrumental factors and other nonidealities Such matters will be further covered in later chapters (see especially Chapter 5), but it is appropriate to emphasize a few key points also at this introductory stage. Connecting to the discussion in the previous section, it is particularly transverse magnetization that is vulnerable, while longitudinal one is unaffected. This is a generally valid argument to routinely avoid the basic PGSE sequence in favor of a PGSTE-like one, where magnetization is dominantly longitudinal throughout.

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There are several potentially disturbing external factors to consider, and it is advisable to eliminate or minimize them, rather than to try to compensate through more or less complicated pulse sequence modifications. Such procedures may actually degrade measurement quality, unless they are properly tuned or implemented. Even then, they may degrade detection sensitivity and be otherwise time consuming, with no benefit. Some pulse sequences in the literature also look ingenious indeed, but in reality have been used in a single study by the authors themselves. – “Keep it simple,” in that sense is perhaps the most important advice I can give to the intended audience of this book. It is so easy to be misled by reading some inspiring and pioneering paper that might suggest otherwise. Another everyday advice that conveys much the same message is “If it ain’t broke, don’t fix it.” Modern commercial PGSE NMR instrumentation has far better performance than that was two decades ago and many original PGSE artifact types are practically eliminated. Trying to duplicate previously published experimental procedures that were described decades ago is not a good idea. Instead, one should do some quick test experiments to check the performance of one’s own setup and characteristics of the system under study. The default procedure should be to use a simple PGSTE sequence, unless there are well-based indications or suspicions of experimental complications.

2.8.5 Brief summary of common complications in PGSE studies This section will briefly summarize some sources of problems, which continue to plague the PGSE type of techniques. They will be described in some more detail in Chapter 5, together with counteracting procedures to lessen their degrading influence. The recommended primary strategy is to apply suitable experimental hardware or sample modifications. Partial compensation through modified pulse sequences or postprocessing of data should be considered as secondary options. Chapter 7 in Price’s book [17] constitutes particularly recommended reading in this context, where several types of potential measurement problems in NMR diffusometry are analyzed in detail. 2.8.5.1 Convection Sample temperature control has always been a problem in NMR, since is normally achieved with a flow of air or nitrogen along the lower part of the sample tube. To some extent, this will create a thermal gradient in the sample volume and convection effects thus become inevitable. This is true even for the ambient temperature range. Experimental influence in PGSE-type measurements may furthermore not appear in an obvious way – it may just result in higher apparent self-diffusion coefficients than that in reality, especially at elevated temperatures. If apparent D-values change downward

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when testing the same experiment with longer Δ-values, convection influence is likely significant. Partial remedies include the use more narrow NMR tubes (like 3 mm i.d. heavywall 5 mm ones instead of standard thin-wall 5 or 10 mm ones) or concentric tube arrangements and/or short or otherwise geometrically confined samples. Fairly efficient bipolar PGSTE-type pulse sequences have also been developed that considerably lessen convection effects on echo formation at the cost of increased experimental complexity and time required for equivalent signal/noise in the experiment. 2.8.5.2 Eddy currents and sample vibration Applying pulsed field gradients in or around an NMR probe may create counteracting electromagnetic and mechanical responses in nearby metallic objects, which may propagate to the NMR sample region. The historically most troublesome ones originate from the so-called eddies of electric current in nearby metal materials, causing secondary and transient magnetic field gradients. They used to be a quite significant source of disturbance in PGSE-type experiments in superconducting magnet systems. Especially aluminum probe housing components or inner magnet bore walls were prone to such counter response. In contrast, resistive iron magnets of older generation NMR were rather immune in such respect. Of course, eddy disturbances get enhanced with stronger and more steeply applied and decayed pulsed magnetic field gradients. So, first-order indications of eddy current influence in PGSE would thus be expected to result in downward-bent semilogarithmic Stejskal–Tanner plots (or systematically negative deviations in nonlinear fitting of PGSE data) when increasing the strength of the magnetic field gradient. It is actually quite easy to diagnose if one’s system is significantly affected by eddy current disturbances by simply recording a normal FID delayed at selected time increments such as 0–10 ms after a gradient pulse (Figure 2.37). τ

90 Time

rf

g(t)

𝛿

Signal

Figure 2.37: A simple and straightforward procedure for testing potential eddy current influence in one’s setup. Eddy current influence manifests itself in signal attenuation, phase distortion and slight frequency shift.

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2.8.5.3 Radiation damping This is the third basic source of experimental problems. It may occur as a result of electromagnetic coupling between the nuclei studied and the exciting rf field. Conceptually, it has been described as being analogous to microphone–amplifier–loudspeaker feedback in an audio system. Chaotic NMR influence has been observed to persist even after 10 s [88]. Normally radiation damping is only troublesome in proton-based PGSE on highly tuned high-resolution probes, especially when using large-volume samples with normal water or some other proton-rich solvent. Detuning the probe from optimum settings (and readjusting the nominal length of the rf pulses) may be an easy brute-force cure in PGSE studies. For “dedicated,” less highly tuned and dedicated diffusion probes, radiation-damping problems might not be obvious, but can still seriously affect measurements [82]. Radiation damping influence in PGSE studies can manifest themselves in many different ways, including even initially rising echo amplitudes with increased gradient strength. Avoiding large-volume NMR tubes in addition to routinely using deuterated solvents easily eliminates unnecessary radiation-damping-related disturbances. As mentioned, large volume NMR tube samples also have considerably elevated sensitivity for convection disturbances in PGSE NMR, and should be avoided for that reason as well. 2.8.5.4 Chemical exchange and cross-relaxation There are several types of chemical exchange and at very varying rate ranges. Effects related to signal broadening from intramolecular chemical exchange, such as between molecular conformations or for labile protons on ionizable functional groups, may affect PGSE studies. Johnson and coworkers analyzed their influence on PGSE experiments [89, 90]. It results in peaks or bandshapes that differ in apparent self-diffusion behavior from others for the same molecule. Depending on the actual PGSE-type pulse sequence used, odd effects like oscillating signal amplitudes for chemically exchanging groups of nuclei may also occur [90]. One should also be aware that cross-relaxation between spin groups may produce similar effects in NMR diffusometry as chemical exchange [91, 92]. Particularly important families of exchange processes are ligand exchange or substrate binding to larger macromolecules. Here echo amplitudes for the ligand or substrate deviate from the normal “Stejskal–Tanner” behavior in PGSE experiments and can become bimodal in a semilogarithmic results visualization, depending on the time scales. A very valuable further quantitative analysis in terms of kinetics and binding can be achieved by analyzing echo attenuation behavior through the so-called Kärger equations [93, 94] (see also the discussion in Section 3.5 and the application examples in Sections 3.6.6 and 3.6.7).

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2.8.5.5 General advice on sample size and length in PGSE studies It is advisable to keep the sample volume constant and optimally minimal during, for example, measurement series on several consecutive samples. For several reasons, including those just mentioned, large volume NMR tubes (i.e., > 5 mm outer diameter ones) should be avoided. Exceptions are when the solution is quite viscous, or if the basic sensitivity is for some reason low, like when studying nuclei other than protons or 19F. Some “high-performance NMR diffusion probes” may have quite limited spatial field gradient constancy, since they have been optimized for maximum magnetic field gradient generation at a given current flow. The maximum recommended sample length could be about 10 mm, and the rf homogeneity may also be significantly degraded outside about the same length. More standard magnetic field gradient-equipped probes have about the same sample volume recommendations as normal NMR probes in general – like filling it to something about 30 mm height, centering it well at the rf and gradient coil midpoints and keeping conditions constant in measurement series on several samples. To extend the effective sample length beyond the recommended limit could result in several types of experimental artifacts in PGSE studies. Making the sample considerably smaller may instead cause other problems. For example, shimming the magnetic field for good resolution might become difficult too for short samples. Spherical or cylindrical bulb inserts for normal NMR tubes have existed for decades and may help in this context. They are also less prone to convection-related problems, but one should be aware of the risk of creating local magnetic susceptibility-related magnetic field gradients around small samples and at the meniscus [82] (cf. Figure 2.38).

2.9 Phase cycling This concept means adding together FID’s or spin echoes that are acquired with systematically incremented spin-exciting rf phase and receiver phase setting. Usually one uses 90° steps, combined with the already-mentioned CYCLOPS procedure (see Section 1.11), which is meant to average out and compensate for hardware nonidealities in one’s spectrometer setup. Recalling the outcomes of original pulsed NMR experiments such as the threepulse-stimulated echo (as detected by Hahn), we can also note that the FID’s as well as the echoes were quite well separated in time. This is because they were created in a rather inhomogeneous magnetic field and samples had relatively rapid transverse spin relaxation. Hahn’s work was done with signal detection that was not phase sensitive, nor was rf pulse excitation made with pulses of any explicit phase, like “x” or “y” or similar. Conditions at the time would seem to not have mattered much if detection electronics (then commonly of “diode type”) were anyway only sensitive to signal amplitude and not to its phase.

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0.999997

0.999996

0.999995

0.999994

0.999993

0.999992 Figure 2.38: FEMLAB (today’s Comsol Multiphysics) simulation of background gradients for an undersized sample of water in a standard NMR tube. Since the shape of the meniscus varies according to surface tension and the tube radius, it was arbitrarily modeled as a half sphere. The simulations were performed using a sample radius of 4.5 mm and a sample length of 6.2 mm discretized into 10,000 mesh points and setting the respective magnetic susceptibilities to χ air = 1 and χ water = 0.999991. Each contour line reflects a 0.5 ppm magnetic field increment. Reproduced, with permission, from Price et al. J Magn Reson. 2001; 150: 49–56 [82] (Copyright (2001) Elsevier).

Meiboom and Gill introduced phase-sensitive or synchronous excitation and detection in pulsed NMR [63] when describing and testing a modified Carr–Purcell echo train sequence [33]. In their variant (later named a CPMG train), the initial rf excitation pulse is set to differ by 90° in phase from subsequent and equally phased 180° echo refocusing phases in the sequence. Today’s instrumentation electronics is vastly different from the vacuum tubebased circuits used by those early pioneers. Among other improved characteristics, and if desired, nowadays, rf and detection phase settings can be controlled with subdegree accuracy through digital electronic circuitry. NMR magnets also have more homogeneous field profiles and greater stability than in the early days. In the past 50 years, concepts related to multidimensional NMR have also become quite dominant. “Signal phase” is almost always a central building and shaping block in the multitude of pulse sequences that have been devised lately, especially in the field of multidimensional NMR (e.g., COSY, NOESY, etc.).

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Keeler educationally discusses such matters in considerable detail in his textbook [95] and in lecture handouts available from his website at http://www-keeler. ch.cam.ac.uk. One should particularly note that with two or more pulses, an NMR pulse sequence could have several simultaneous outcomes. Through phase cycling, individual ones can be selected or suppressed in the final acquired and arithmetically summed and averaged data set. Figure 2.39 illustrates optional information content in the last FID of a three-pulse sequence. Here it constitutes one of the time domains in multidimensional NMR spectroscopy. The stimulated echo signal that is relevant to PGSTE-type echo-based diffusometry is selected in a similar manner through alternatively designated phase cycling.

Why we need coherence selection (a)

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Figure 2.39: Illustrating three possible two-dimensional NMR outcomes of a pulse sequence that consists of three 90° pulses (figures reproduced with kind permission by of James Keeler – they are from his Euromar lecture in 2011). Note that for a three-pulse sequence of this kind, there are additionally two more FID’s (after rf pulses 1 and 2) as well as the five echo signals discussed before (cf. Figure 2.17). These initial two FID’s and later echoes are not considered or used in the 2D NMR graphical context of the figure, only the final FID and the other time domain. “Experiment” selection through suppression and cancellation of FID’s and echoes is done through phase cyclingbased selection procedures briefly outlined in the following section. The central concept is called coherence pathway selection [96]. The reader is referred to references listed in further reading ( Section 2.9.5) for details on these procedures. Note that the further normal “stimulated echo experiment” (that is particularly relevant to NMR diffusometry experiments) is just one of many additional selection alternatives within a quite large number of seemingly unrelated experimental outcomes of a three-pulse sequence.

Phase cycling-based selection procedures are also necessary for unwanted-echo suppression in the context of PGSE-based NMR diffusometry and related context. On modern high-resolution NMR spectrometers and on normal solution-phase samples, FID’s and echoes would normally overlap in the time domain, which can become troublesome already for 3 rf pulses. Individual echoes and FID’s may intrinsically differ in phase by 180° or can individually be forced to do so through suitable cyclic rf phase alterations.

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The good news is that normal NMR users do not have to care about the details of such procedures, since they are normally preprogrammed by the spectrometer vendor for a large variety of standard pulse sequences, including those for NMR diffusometry. The only thing to be kept in mind is to apply the recommended number of experimental repeats and data acquisitions (typically multiples of 4, 8 or 16). When observing data acquisition progress visually on the spectrometer monitor, one initially notes that unwanted echo suppression and echo baseline effects are most pronounced during the first four data acquisitions, where CYCLOPS-type corrections have also acted [97]. Thereafter, composite FID+echo shape changes become virtually unobservable, except for a continuously time-averaged increase in the signal/noise ratio. Porting some more exotic and specialized literature procedure into one’s system should not normally pose any insurmountable problems either. Careful “proofreading” would be recommended before actually activating and using such a self-authored or manually modified procedure. The preprogrammed phase cycling sequence for Bruker system stimulated echo PGSE sequence is illustrated as an example in Figure 2.40. The bad news is that commonly used theoretical tools for designing phase cycles [96, 100] [101, 102] introduce new concepts and nomenclature (such as coherence order, pathways and selection) and rest on mathematical formalism and quantum physics concepts that may be unfamiliar to many. The coherence pathway selection procedures have an initially steep learning curve, also for seasoned older-generation NMR spectroscopists. Personally, I must admit that I have never had a need to design a phase-cycling sequence myself, during 50 years of almost exclusively NMR-based research. Most readers of this introductory book will face a similar situation in practice. In case of actual demand or through general interest, one can consult appropriate texts and resources among references in this section, and in the Section 2.9.5. Attempting to describe the subject here in sufficient detail does not feel justified, considering the intended “non-mathematical” type of text and the rather extensive space required (several tens of pages). It would have been just a repetition of already existing introductions of high quality and excellent educational quality standards. The chapter in Keeler’s textbook, as supplemented by downloadable material from his website, seems to be the best recommendable starting point. An Internet search for “coherence transfer pathways” also leads to several useful texts and presentations. A more recent methodological development is a family of phase cycle selection procedures called cogwheel cycling, which may reduce the required number of cycling steps substantially [103]. The educational review on this new concept by Zuckerstätter and Müller is highly recommended [104].

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ph11 = 0 ph12 = 1 #ifndef LED ph1 = 0 0 0 ph2 = 1 3 0 ph3 = 1 3 0 ph31 = 1 0 2 #endif #ifdef LED ph1 = 0 0 0 ph2 = 0 0 0 ph3 = 0 0 0 ph4 = 0 0 0 ph5 = 0 1 2 ph31 = 0 1 2 #endif

0 2 2 2 2 1 1 1 1 3 3 3 3 2 2 2 2 2 0 0 3 3 1 1 1 1 3 3

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Figure 2.40: The phase cycling section of a Bruker PGSTE three-pulse or PGSTE-LED [98] five-pulse program sequence code, including CYCLOPS-type averaging cycles. The notations 0, 1, 2, 3 stand for pulse angles (in multiples of π=2) and correspond to x, y, −x and –y rf excitation pulses and receiver phases, respectively. The upper rows refer to the respective pulse numbers and the bottom row to the receiver phase. For a normal PGSTE, the recommended minimum cycle is 16 stimulated echo data acquisitions. For the five-rf pulse PGSTE-LED variant, the number here grows to 32. A seven-rf pulse bipolar gradient pulse pair variant (PGSTE-BPP-LED) [99] has largely replaced the original PGSTE-LED one, although it requires twice as many phase cycles.

2.9.1 Software design of selective phase cycles Phase selection procedures in this context are conceptually reminiscent of the socalled Sudoku puzzles and should lend themselves to be reformulated into computer algorithm code. In 1990, Wu and Shen [105] presented and coded in BASIC a simple and direct phase-cycling selection procedure, which they named “PHASE” (cf. Figure 2.41). Automated procedures also minimize risks for accidental mistakes or for typos to propagate. The best starting point to search for related computer code would be the extensive and active website compilation by Stan Sykora, at www.ebyte.it/library/StansNmrLinks.html. For key literature references and specific software, see Jerschow and Müller [106]. They initially described Matlab

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code for the purpose and named it CCCP, which was later translated into C++ and made available from Jerschow’s webpage (or via Sykora’s links). Their original article also discusses actual PGSE-type self-diffusion applications in the context of CCCP-based phase cycles.

Figure 2.41: Partial dialogue and output from the PHASE program [105], when calculating the proper phase cycle for the five-rf pulse LED experiment (cf. Figure 2.40). The resulting output had 64 phase cycles, but only the 16 first are listed in the figure.

One frequently finds that many older Internet literature search hits for NMR software lead to nowadays-stale resources, with software that is no longer under active development or being accessible at all. The extremely powerful and multifaceted SpinDynamica package ( www.spindynamica.soton.ac.uk) is a notable exception. It has capabilities way beyond coherence pathway selection, but requires a Mathematica installation and some reasonable prior experience regarding somewhat complex and unforgiving syntax of Mathematica.

2.9.2 Separating out a three-pulse generated stimulated echo As discussed earlier (cf. Figure 2.17), three 90° pulses may lead to five echo signals, as well as an initial FID. Slightly weaker FID’s will in reality also occur after the second and third pulses, since some z-magnetization will recover within the duration of the pulse sequence. There are brilliant experimental illustrations of selection procedures in Chapter 3 of Sørland’s book [69]. The selection scheme was originally devised and illustrated by Fauth et al. [107] in the context of electron paramagnetic (spin) resonance (EPR and ESR), rather than nuclear magnetic resonance. The principles remain the same.

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In a high-resolution situation, echoes and FID’s would extensively overlap and interfere. For visual clarity, again a condition is depicted in Figure 2.42 (as in Figure 2.17), where the echoes and FID’s are well separated in time, like they would in a somewhat inhomogeneous magnetic field. Externally generated

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Figure 2.42: A stimulated echo separation scheme, based on the strategy originally devised by Fauth et al. [107] and modified variants of graphical illustrations of the same in Chapter 3 of Sørland’s book [69]. The lower part is basically identical to Figure 2.17 and in the upper traces bars symbolically illustrate the signs of the FIDs and echoes that arise for four selected combinations of transmitter and receiver phases. The remaining signal after digitally adding these four outcomes together as indicated is ideally only the stimulated echo. Note that this simplified illustrative scheme does not involve any +y or –y pulses or receiver settings, as needed for CYCLOPS-type corrections (cf. Section 1.11 and Figure 2.40).

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magnetic field gradient pulses are left out as well. One normally refers to the stimulated echo three-pulse sequence as being composed of three “90x-pulses.” Note that in reality, the transmitter phase (here +x or –x) as well as the receiver phase (also +x or –x) is systematically covaried in the selection scheme illustrated. Again, as pointed out in Section 1.11 and later, there is some confusion in the literature regarding sign conventions of rf pulses and receiver phases. At times, they do not really matter, but can still be confusing or ambiguous in graphical illustrations. The effect of a 90+x pulse on nuclei with positive magnetogyric ratio actually is to create a negative magnetization in the y′-direction in the rotating frame, as in Figure 2.42. The article “The signs of frequencies and phases in NMR” by Levitt [41] is highly recommended reading in this context.

2.9.3 Using “crusher” magnetic field gradient pulses to destroy transverse magnetization within a pulse sequence This is a standard procedure in this context, which supplements rf transmitter/ receiver phase cycling procedures. Typically, it reduces the number of phase cycling steps needed by half and thus speeds up experiments. Crusher pulses also may reduce the influence of other measurement nonidealities. They are highly efficient and recommendable building blocks in PGSTE and more complex NMR diffusometry experiments, even at high gradient power settings. In the case of a stimulated echo selection phase selection procedure as in Section 2.9.2, the proper location of a crusher pulse is between rf pulses 2 and 3 (cf. Figure 2.31). Provisions for such magnetic field gradient pulse generation existed in JEOL iron magnet-based spectrometers in the mid-1970s. At that time, they were named “homospoil” (magnet field homogeneity) rather than “crusher” pulses. Bruker and Varian deemed at the time that rf phase cycling procedures were sufficient in the context of the emerging field of multidimensional spectroscopy. Only decades later, and now on supercon magnet systems, did they introduce magnetic field gradient coils and modestly powerful gradient drivers as a standard options. Initially, such “crusher” pulse facilities were still intended for signal selection procedures and not for PGSE diffusometry. However, these spectrometer components were eminently suitable for many types of NMR diffusometry studies, as later discovered by many. Crusher pulses should be strong enough to fully eliminate transverse magnetization, but preferably not stronger. A typical setting would be one Gauss/cm (10 mT m−1) for one millisecond. Often, crusher gradient shapes are made half-sinusoidal rather than rectangular to lessen disturbances of eddy current type. With modern digitally controlled gradient amplifiers, shape alterations are routinely easy and crusher pulses

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are normally applied through the same field gradient coils as the diffusion-charting gradient pulses.

2.9.4 “Multiplexed” NMR experiments Several suggestions toward NMR strategies where numerous experiments could be done simultaneously and later selected computationally have been presented over the years; for an overview, see, for example, Schlagnitweit et al. [108]. Early on, ideas were based on noise rather than coherent rf excitation, followed by Hadamard rather than Fourier transformation [109, 110]. Postexperimental phase selection procedures of “raw experimental data” have also been performed [111]. The combined concepts still appear a bit far-fetched in the context of future PGSE-type methodology.

2.9.5 Further reading Zur Y. Analysis of the multi-echo spin-echo pulse sequence. Concepts Magn Reson Part A. 2017; e21402. Price WS. Gradient NMR. In: Annual Reports on NMR Spectroscopy, New York: Academic Press, 1996:51–142. Kingsley PB. Combining CYCLOPS with other phase cycles. J Magn Reson 1994; 110: 102–105. Kingsley P. Product Operators, Coherence Pathways, and Phase Cycling. Part I: Product Operators, Spin-spin Coupling, and Coherence Pathways. Concepts Magn Reson. 1995; 7: 29–47. Kingsley P. Product Operators, Coherence Pathways, and Phase Cycling. Part II: Coherence Pathways in Multipulse Sequences: Spin Echoes, Stimulated Echoes, and Multiple-Quantum Coherences. Concepts Magn Reson. 1995; 7: 115–136. Kingsley P. Product Operators, Coherence Pathways, and Phase Cycling. Part III: Phase Cycling. Concepts Magn Reson. 1995; 7: 167–192. Frahm J, Hänicke W. Phase Cycling. In: Encyclopedia of Nuclear Magnetic Resonance, Grant DM, Harris RK, eds. New York: Wiley, 1996:4407–4423.

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[29] Wang, J. Theory of the Self-Diffusion of Water in Protein Solutions. A New Method for Studying the Hydration and Shape of Protein Molecules. J Am Chem Soc. 1954; 76: 4755–4763. [30] Brown, W., Stilbs, P., & Johnsen, R. Friction Coefficients in Self-Diffusion, Velocity Sedimentation, and Mutual Diffusion for Poly(ethylene Oxide) in Aqueous Solution. J Polym Sci, Polym Phys. 1983; 21: 1029–1039. [31] Meyers, RA. Solution Nuclear Magnetic Resonance: Spin-1/2 Nuclei Other than Carbon and Proton. John Wiley & Sons Ltd, 2009. [32] McCall, DW., Douglass, DC., & Anderson, EW. Self-diffusion studies by means of nuclear magnetic resonance spin-echo techniques. Ber Bunsenges Phys Chem. 1963; 67: 340–366. [33] Carr, HY., & Purcell, EM. Effects of Diffusion on Free Precession in Nuclear Magnetic Resonance Experiments. Phys Rev. 1954; 94: 630–638. [34] Hahn, EL. Spin Echoes. Phys Rev. 1950; 80: 580–594. [35] Hahn, EL. Free nuclear induction. Physics Today. 1953; 6: 4–9. [36] Kingsley, P. Product Operators, Coherence Pathways, and Phase Cycling. Part II: Coherence Pathways in Multipulse Sequences: Spin Echoes, Stimulated Echoes, and Multiple-Quantum Coherences. Concepts Magn Reson. 1995; 7: 115–136. [37] Hennig, J. Echoes-How to Generate, Recognize, Use or Avoid Them in MR-Imaging Sequences Part I: Fundamental and Not So Fundamental Properties of Spin Echoes. Concepts Magn Reson. 1991; 3: 125–143. [38] Bowtell, R., Bowley, RM., & Glover, P. Multiple Spin Echoes in Liquids in a High Magnetic Field. J Magn Reson. 1990; 88: 643–651. [39] Scheffler, K. A pictorial description of steady-states in rapid magnetic resonance imaging. Concepts Magn Reson. 1999; 11: 291–304. [40] Hahn, E. Echo Phenomena and Non-Linearity. Concepts Magn Reson. 1994; 6: 193–199. [41] Levitt, MH. The signs of frequencies and phases in NMR. J Magn Reson. 1997; 126: 164–182. [42] Ardelean, I., Kimmich, R., Stapf, S., & Demco, DE. Multiple Nonlinear Stimulated Echoes. J Magn Reson. 1997; 127: 217–224. [43] Ardelean, I., & Kimmich, R. Diffusion measurements using the nonlinear stimulated echo. J Magn Reson. 2000; 143: 101–105. [44] Fischer, E., & Kimmich, R. Constant time steady gradient NMR diffusometry using the secondary stimulated echo. J Magn Reson. 2004; 166: 273–279. [45] Callaghan, PT. Translational Dynamics & Magnetic Resonance, Principles of Pulsed Gradient Spin Echo NMR. Oxford: Oxford University Press, 2011. [46] Singer, JR. NMR diffusion and flow measurements and an introduction to spin phase graphing. J Phys E: Sci Instrum. 1978; 11: 281–291. [47] Saarinen, T., & Johnson, C, Jr. Imaging of Transient Magnetization Gratings in NMR. Analogies with Laser-Induced Gratings and Applications to Diffusion and Flow. J Magn Reson. 1988; 78: 257–270. [48] Johnson, CS. The Evolution of Ideas about Optical Analogies to PFGNMR and the Visualization of PFGNMR Experiments. Encyclopedia of Magnetic Resonance. 2007; [49] Mansfield, P., & Grannell, PK. NMR ‘diffraction’ in solids. J Phys C: Solid State Phys. 1973; 6: L422. [50] Callaghan, PT., Eccles, CD., & Xia, Y. NMR microscopy of dynamic displacements: k-space and q-space imaging. J Phys E:Sci Instrum. 1988; 21: 820–822. [51] Koay, CG., & Özarslan, E. Conceptual Foundations of Diffusion in Magnetic Resonance. Concepts Magn Reson. 2013; 42A: 116–129.

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[52] Weigel, M. Extended phase graphs: dephasing, RF pulses, and echoes – pure and simple. J Magn Reson Imaging. 2015; 41: 266–295. [53] Vold, RL., & Vold, RR. Transverse relaxation in homonuclear coupled spin systems. J Am Chem Soc. 1974; 96: 4043–4045. [54] Vold, RL., & Vold, RR. Nuclear magnetic relaxation in coupled spin systems. Progr Nucl Magn Reson Spectrosc. 1978; 12: 79–133. [55] Hahn, EL., & Maxwell, DE. Spin echo measurements of nuclear spin coupling in molecules. Phys Rev. 1952; 88: 1070–1084. [56] Proctor, WG., & Yu, FC. On the Nuclear Magnetic Moments of several Stable Isotopes. Phys Rev. 1951; 81: 20–30. [57] Arnold, JT., Dharmatti, SS., & Packard, ME. Chemical Effects on Nuclear Induction Signals from Organic Compounds. J Chem Phys. 1951; 19: 507. [58] Gutowsky, HS., McCall, DW., & Slichter, CP. Coupling among Nuclear Magnetic Dipoles in Molecules. Phys Rev. 1951; 84: 589–590. [59] Gutowsky, HS., McCall, DW., & Slichter, CP. Nuclear Magnetic Resonance Multiplets in Liquids. J Chem Phys. 1953; 21: 279–292. [60] Torrey, HC. Bloch equations with diffusion terms. Phys Rev. 1956; 104: 563. [61] Kimmich, R., & Fischer, E. One and two-dimensional pulse sequences for diffusion experiments in the fringe field of superconducting magnets. J Magn Reson. 1994; 107: 229–235. [62] Wu, D., & Johnson, CS, Jr. Diffusion-Ordered 2D NMR in the Fringe Field of a Superconducting Magnet. J Magn Reson. 1995; A 116: 270–272. [63] Meiboom, S., & Gill, D. Modified Spin‐Echo Method for Measuring Nuclear Relaxation Times. Rev Sci Instrum. 1958; 29: 688–691. [64] Borneman, TW., Hürlimann, MD., & Cory, DG. Application of optimal control to CPMG refocusing pulse design. J Magn Reson. 2010; 207: 220–233. [65] Anand, CK., Bain, AD., Curtis, AT., & Nie, Z. Designing optimal universal pulses using secondorder, large-scale, non-linear optimization. J Magn Reson. 2012; 219: 61–74. [66] Aguilar, JA., Nilsson, M., Bodenhausen, G., & Morris, GA. Spin echo NMR spectra without J modulation. Chem Commun (Camb). 2012; 48: 811–813. [67] Torres, AM., Zheng, G., & Price, WS. J-compensated PGSE: an improved NMR diffusion experiment with fewer phase distortions. Magn Reson Chem. 2010; 48: 129–133. [68] Karlicek, RF, Jr.., & Lowe, IJ. A modified pulsed gradient technique for measuring diffusion in the presence of large background gradients. J Magn Reson. 1980; 37: 75–91. [69] Sørland, GH. Dynamic Pulsed-Field-Gradient NMR. Berlin, Heidelberg: Springer, 2014. [70] Kuchel, PW., Pagès, G., Nagashima, K. et al. Stejskal-Tanner equation derived in full. Concepts Magn Reson Part A. 2012; 40A: 205–214. [71] Masuda, R., Gupta, A., Stait-Gardner, T., Zheng, G., Torres, A., & Price, WS. Shortening NMR experimental times. Magn Reson Chem. 2018; 56: 847–851. [72] Sinnaeve, D. The Stejskal-Tanner equation generalized for any gradient shape-an overview of most pulse sequences measuring free diffusion. Concepts Magn Reson. 2012; 40A: 39–65. [73] Gross, B., & Kosfeld, R. Anwendung der Spin-Echo-Methode bei der Messung der Selbstdiffusion. Messtechnik. 1969; 7/8: 171–177. [74] Price, WS., & Kuchel, PW. Effect of Nonrectangular Field Gradient Pulses in the Stejskal and Tanner (Diffusion) Pulse Sequence. J Magn Resonance. 1991; 94: 133–139. [75] Merrill, MR. NMR Diffusion measurements Using a Composite Gradient PGSE Sequence. J Magn Reson. 1993; 103: 223–225. [76] Röding, M., & Nydén, M. Stejskal-Tanner Equation for Three Asymmetrical Gradient Pulse Shapes Used in Diffusion NMR. Concepts Magn Reson Part A. 2015; 44: 133–137.

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[77] Holz, M., & Weingärtner, H. Calibration in accurate spin-echo self-diffusion measurements using proton and less-common nuclei. J Magn Reson. 1991; 92: 115–125. [78] Foroozandeh, M., Castañar, L., Martins, LG. et al. Ultrahigh-Resolution Diffusion-Ordered Spectroscopy. Angew Chem Int Ed Engl. 2016; 55: 15579–15582. [79] Packer, KJ., Rees, C., & Tomlinson, DJ. Modification of the pulsed magnetic field-gradient spin echo method of studying diffusion. Mol Phys. 1970; 18: 421–423. [80] Packer, KJ., Rees, C., & Tomlinson, DJ. Studies of Diffusion and Flow by Pulsed NMR Techniques. Adv Mol Relax Proc. 1972; 3: 119–131. [81] Stejskal, EO., & Tanner, JE. Spin Diffusion measurements. Spin Echoes in the presence of a Time-Dependent Field Gradient. J Chem Phys. 1965; 42: 288–292. [82] Price, WS., Stilbs, P., Jönsson, B., & Söderman, O. Macroscopic background gradient and radiation damping effects on high-field PGSE NMR diffusion measurements. J Magn Reson. 2001; 150: 49–56. [83] Callaghan, PT., Le Gros, MA., & Pinder, DN. The measurement of diffusion using deuterium pulsed field gradient nuclear magnetic resonance. J Chem Phys. 1983; 79: 6372–6381. [84] Furó, I., & Dvinskikh, SV. Field gradient NMR of liquid crystals. Modern Magnetic Resonance. 2006; 1: 113–118. [85] Stilbs, P., & Moseley, ME. Carbon-13 pulsed-gradient spin-echo studies, a method for the elimination of J-modulation and proton exchange effects in self- diffusion measurements. Chem Scr. 1980; 15: 215–216. [86] Moseley, ME., & Stilbs, P. Fourier transform carbon-13 pulsed-gradient spin-echo studies. The self-diffusion of trans-decalin in the polystyrene/trans- decalin system. Chem Scr. 1980; 16: 114–116. [87] Lindman, B., Ahlnäs, T., Söderman, O., Walderhaug, H., Rapacki, K., & Stilbs, P. Fourier transform carbon-13 relaxation and self-diffusion studies of microemulsions. Faraday Discuss Chem Soc. 1983; 76: 317–329. [88] Abergel, D., & Louis-Joseph, A. Generating spin turbulence through nonlinear excitation in liquid-state NMR. J Magn Reson. 2009; 196: 115–118. [89] Johnson, CS, Jr. Effects of chemical exchange in diffusion-ordered 2D NMR spectra. J Magn Reson. 1993; 102: 214–218. [90] Chen, A., Johnson, CS, Jr.., Lin, M., & Shapiro, MJ. Chemical Exchange in Diffusion NMR Experiments. J Am Chem Soc. 1998; 120: 9094–9095. [91] Dvinskikh, SV., & Furó, I. Cross-Relaxation Effects in Stimulated-Echo-Type Pgse NMR Experiments by Bipolar and Monopolar Gradient Pulses. J Magn Reson. 2000; 146: 283–289. [92] Pagès, G., Dvinskikh, SV., & Furó, I. Suppressing magnetization exchange effects in stimulated-echo diffusion experiments. J Magn Reson. 2013; 234: 35–43. [93] Kärger, J. NMR Self-Diffusion Studies in Heterogeneous Systems. Adv Colloid Interface Sci. 1985; 23: 129–148. [94] Wijesekera, D., Stait-Gardner, T., Gupta, A. et al. A Complete Derivation of the Kärger Equations for Analysing NMR Diffusion Measurements of Exchanging Systems. Concepts Magn Reson. 2019; 47A, e21468. https://doi.org/10.1002/cmr.a.21468 [95] Keeler, J. Understanding NMR Spectroscopy, 2nd Edition. Hoboken, NJ: Wiley, 2010. [96] Bodenhausen, G., Kogler, H., & Ernst, RR. Selection of coherence-transfer pathways in NMR pulse experiments. J Magn Reson (1969). 1984; 58: 370–388. [97] Kingsley, PB. Combining CYCLOPS with other phase cycles. J Magn Reson. 1994; 110: 102–105. [98] Gibbs, S., & Johnson, C,Jr. A PFG NMR Experiment for Accurate Diffusion and Flow Studies in the Presence of Eddy Currents. J Magn Reson. 1991; 93: 395–402. [99] Wu, D., Chen, A., & Johnson, CS. An Improved Diffusion-Ordered Spectroscopy Experiment Incorporating Bipolar-Gradient Pulses. J Magn Reson, Ser A. 1995; 115: 260–264.

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[100] Mitschang, L., Pontsingl, H., Grindrod, D., & Oschkinat, H. Geometrical Representation of Coherence Transfer Selection By Pulsed Field Gradients in High-Resolution Nuclear Magnetic Resonance. J Chem Phys. 1995; 102: 3089–3098. [101] Xiang, BS. Coherence-transfer table: A convenient method for deriving coherence-transfer pathways of an NMR pulse sequence. I. Coherence-transfer table in I-x, I-y, and I-z system. Concepts Magn Reson. 1999; 11: 393–408. [102] Xiang, BS. Coherence-transfer table: A convenient method for deriving coherence-transfer pathways of an NMR pulse sequence. II. Coherence-transfer table in I+, I-, and I-z system. Concepts Magn Reson. 2000; 12: 43–53. [103] Levitt, MH., Maduh, PK., & Hughes, CE. Cogwheel phase cycling. J Magn Reson. 2002; 155: 300–306. [104] Zuckerstätter, G., & Müller, N. Coherence pathway selection by cogwheel phase cycling in liquid-state NMR. Concepts Magn Reson A. 2007; 30A: 81–99. [105] Wu, D., & Shen, L. Computer-Aided Design of Phase Cycling in PFT-NMR Experiments. Chinese J Magn Reson. 1990; 7: 445–457. [106] Jerschow, A., & Müller, N. Efficient Simulation of Coherence Transfer Pathway Selection by Phase Cycling and Pulsed Field Gradients in NMR. J Magn Reson. 1998; 134: 17–29. [107] Fauth, J-M., Schweiger, A., Braunschweiler, L., Forrer, J., & Ernst, RR. Elimination of Unwanted Echoes and Reduction of Dead Time in Three-Pulse Electron Spin-Echo Spectroscopy. J Magn Reson. 1986; 66: 74–85. [108] Schlagnitweit, J., Zuckerstätter, G., & Müller, N. Toward multipurpose NMR experiments. Magn Reson Chem. 2010; 48: 1–8. [109] Blümich, B., & Ziessow, D. Nonlinear noise analysis in nuclear magnetic resonance spectroscopy. 1D, 2D, and 3D spectra. J Chem Phys. 1983; 78: 1059–1076. [110] Kupce, E., Nishida, T., & Freeman, R. Hadamard NMR spectroscopy. Prog Nucl Magn Reson Spectrosc. 2003; 42: 95–122. [111] Ivchenko, N., Hughes, CE., & Levitt, MH. Multiplex phase cycling. J Magn Reson. 2003; 160: 52–58.

3 PGSE NMR diffusometry example applications As mentioned in the Introduction, this chapter focuses on diffusion-related NMR measurement procedures for primary “chemical” problems in isotropic solution, and most of the examples stem from this area. Some others will be touched on in the context of more complex procedures described in Chapter 7. With some modification and tuning of experimental parameter settings, PGSEtype studies are applicable in wide fields, including gases, solids, semisolids, liquid crystals, emulsions and a variety of heterogeneous systems. Variants thereof can also monitor and quantify flow effects and other types of forced transport, like in electrophoretic NMR (eNMR). This subject is covered later in this book, together with other families of application examples. Combined with NMR microscopy or MRI techniques, one can actually achieve spatial resolution and quantification of NMR-based diffusometry, as well as electrically or flow-driven transport processes. Apart from eNMR, the three recent textbooks by Price, Callaghan and Sørland that were mentioned in the Preface cover such topics with great in depth and detail. They are somewhat differently organized and also vary in their focus. In short, Price’s book is the most chemistry-related one, Callaghan’s book connects to NMR microscopy and deeper physics and mathematics of PGSE techniques and Sørland’s book leans toward technical and industrial applications, heterogeneous systems, as well as home brew design of specialized equipment and procedures in such context. Of course, there are also many other older general monographs or book chapters on the subject, some of which were listed in the Preface. One should note that such sources might be technically misleading, due to instrumental and other progress since the original publication date. The somewhat sketchy solutions of PGSE example applications in this chapter have been selected for illustration, and are by no means intended to be complete, nor wholly representative. The intention has primarily been to provide guidance and possible inspiration for future studies of related nature. A number of reviews that contain extensive listings of PGSE applications were already listed in the Preface.

3.1 Further reading Zubkov M, Dennis GR, Stait-Gardner T et al. Physical characterization using diffusion NMR spectroscopy. Magn Reson Chem. 2017; 55: 414–424.

3.2 Strategies and constraints Diffusion NMR is nowadays a proven tool for quantitative investigation of a wide variety of chemical systems or physicochemical phenomena. Strategies would normally https://doi.org/10.1515/9783110551532-003

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rely on changes in component self-diffusion behavior that can be translated into parameters related to intermolecular interaction or similar. Further quantification and characterization would be based on a comparison of component self-diffusion behavior, as a function of parameters like relative concentrations or system temperature. A separate category of methodologically oriented applications has also emerged in later years, without any apparent goals besides analytically separating signal contributions in multicomponent systems. Mostly, little or no concern about actual measurement precision or accuracy is seen in such studies, and rarely are any actual physicochemical questions addressed either. In general, a measurement precision and accuracy of 1% or better is normally achievable in the majority of PGSE studies made on modern instrumentation. The measuring range may cover five decades in D. Additionally, the time window of the experiment can be selected to be milliseconds to seconds, all depending on the sample and the performance of the instrumentation. The great advantage of PGSE NMR diffusometry techniques is that each system component can normally be investigated separately. This is nowadays true even in quite complex mixtures and over quite a great span of relative concentrations and signal intensities. The experimental parameter – the self-diffusion coefficient – also does not require any further physical interpretation or model relation in isotropic solution. It simply represents geometrical translation probability along the applied magnetic field gradient, according to the Einstein relation (hΔz2 i = 2Dt) or through the translational distribution function itself (c.f. eqs. (2.4 and 2.1)). One should also keep in mind that the situation will differ in anisotropic and heterogeneous systems, or when the experimental diffusional NMR timescale does not significantly exceed that of local dynamics or chemical exchange processes in the system. For normal solution studies, these factors are rarely a complication.

3.3 Systems suitable for diffusion-related investigation by NMR The most common situation would be that an investigator actually has a physicochemical question, like interaction between two molecular kinds or general transport conditions in general in the sample. Would NMR diffusometry be a suitable technique for the objective in question? The simple answer could be that it is eminently suitable or unsuitable, depending on the range of conditions. The main selective criterion would be that a clear effect on component’s self-diffusion should be anticipated when varying some suitable system parameters. Often that is the case if macromolecules, polymers or supramolecular aggregates are components of solution systems. Individual diffusion rate may even change a magnitude or more, upon varying experimental conditions, providing clear-cut information through a structurally and dynamically related parameter – the self-diffusion coefficient. Self-aggregating surfactant and polymer-containing families of systems in solution stood out as potentially promising application systems already in the early

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days of Fourier transform (FT)-PGSE diffusometry. Many pending questions and controversies were also swiftly answered or settled as well. System and physicochemical characteristics that were previously subject to speculation or inaccessible by other techniques and tools became routinely quantifiable. Indeed, FT-PGSE opened new doors, at minor effort levels indeed. As mentioned in Section 2.5.3, association between two similarly sized molecules in solution into some 1:1 complex will not lead to dramatic changes in time-averaged self-diffusion coefficients of either one – likely, the decrease will only be of the order of 20–40% or so, and the aggregation process may require high-precision measurements for robust quantification. The situation will get quite a lot more favorable from a measurement point of view if one of the components is a considerably more slowly diffusing one, like a macromolecule or colloidal entity such as a surfactant micelle. The two-site, time-averaged freebound model underlying eq. (2.13), that is, Dobs = pDbound + ð1 − pÞDfree describes the situation with great success and has been used in numerous studies in the literature. Target systems and objectives that are predictably more or less problematic for investigation by FT-PGSE include: – Binding, dimerization and association of small molecules. Still, much useful information can be extracted from NMR diffusometry in certain situations. – Binding phenomena for biological macromolecules, like proteins. Complications include low experimental signal/noise due to relatively rapid spin relaxation and generally broad and spectrally dispersed signal bandshapes, in the presence of a dominant water peak band shape. Macromolecule–macromolecule binding studies are therefore normally unfavorable. In contrast, diffusional changes of small molecules or counterions upon binding to macromolecules occasionally are very easy to measure. Their proper interpretation may be more difficult to achieve, however. – Samples with many components. Signals may overlap or interfere, making data evaluation more difficult. – Poorly characterized, heterogeneous or very polydisperse ones. Technical formulations are often of this kind, and PGSE measurements may become difficult to interpret. – Low-concentration solutions in proton-rich solvents, like water or organic liquids. Solvent peaks may interfere, and disturbing radiation-damping effects may be substantial. “Less feasible” or “unfavorable” may at times translate into “more difficult” rather than “hopeless” or “futile.” With good experimental design, appropriate countermeasures, proper data evaluation and good tuning of experimental parameters, the prospects for meaningful studies on systems of high chemical or other significance may still be good.

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General aspects, not to be forgotten: – Even for soundly determined self-diffusion data, many types of studies may turn out to become rather pointless or misleading in some sense, unless the investigators do have proper physicochemical or biological understanding of their systems and theoretical background for their interpretation. Seeking supporting expertise through collaboration is always recommendable. In retrospect, it seems that the initially slow and geographically isolated exploitation of PGSE-based NMR diffusometry during the years 1975–1985 was to a large extent traceable to two complementary factors. One was unfamiliarity with the existence of this new tool and/or lack of proper NMR instrumentation and related skills – for many researchers with actual physicochemical system expertise and interest. Others, who had great NMR expertise and good NMR facilities instead lacked curiosity and interest in “chemistry-related” aspects, and did not see the chemical potential in PGSE-type strategies. Only through mutual collaboration could one pass this type of “transition state.” Today, instrumentation suitable for PGSE studies by relatively inexperienced NMR users is widely available – but wider collaboration is still always recommended when in doubt. In the following sections, a number of examples of “families” of physicochemical FT-PGSE applications will be listed and discussed from feasibility and other viewpoints. Many of them stem from our own studies, which began in the late 1970s. One of the leading incentives at the time was indeed to chart the feasibility of the new tool of FT-PGSE-based multicomponent self-diffusion measurements on a wide range of systems and physicochemical issues in solution. Most of the studies were made using comparably low-performance instrumentation, but still produced conclusions and results of significance and good precision.

3.4 Diffusion in single or binary liquids With NMR diffusometry, new doors had been opened to areas that were virtually uncharted in 1950. Additionally, there was a great interest in concepts like the structure and dynamics of liquids at the time, and one was aware that self-diffusion data would provide a quite direct tool for elucidation of such matters. An illustrative time document is a gas diffusion paper by Winn [1], which covers several 100 degrees in temperature. One might wonder, “How on Earth did he do that”? Tracer techniques were the only available tools; here the experimental design was mixing isotopically different (but not radioactive) liquids and gases, while monitoring the process with a mass spectrometric setup. The Discussion part touches on concepts like intermolecular interaction potentials and radial distribution functions for molecular arrangement and orientation. Before the emerging development of multicomponent NMR diffusometry in the 1970s, NMR-based diffusion studies on even the majority of binary or more complex

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systems had to be made on samples where one or more components were “NMRinvisible.” This could be achieved through isotopic labeling with deuterium or “naturally,” like in the case of solvents like CCl4 or per fluorinated liquids. A pioneering time document of this kind is a study by McCall and Douglass [2], where data are also referenced to theories of simple solutions and binary mutual diffusion information within the realm of irreversible thermodynamics formalism.

3.4.1 Monitoring self-diffusion of gases dissolved in water A few studies of this type (on methane, neon and xenon) appeared some decades ago, with rather common exploratory NMR-methodological and physicochemical aims, similar to those just outlined. 3.4.1.1 Objectives Central topics in all three studies were comparison between simulations, theory of solutions and experimental data for translational diffusion (and also rotational diffusion, for methane). Also discussed were hydration shells (more tightly bound water) around methane and the noble gases. It was also concluded that relative Xe and Ne self-diffusion coefficients match the ratio of the inverse square root of their masses, rather than the ratio of van der Waals radii (Figure 3.1).

M = 16

Methane D = 2·10–9 m2 s–1

21

Neon D = 4·2·10–9 m2 s–1

129

Xenon D = 1·9·10–9 m2 s–1

Figure 3.1: Methane, Neon and Xenon, approximately drawn to (van der Waals) scale, together with experimental aqueous solution self-diffusion coefficients at 298 K.

3.4.1.2 Experimental aspects For methane in water [3], the obvious initial obstacles were a low concentration of solute, against a huge water background signal in proton-based PGSE. They could be overcome, however, by using 10 mm heavy-wall NMR tubes with an integral Teflon valve, and switching to deuterated methane instead of normal. Deuterium is a quadrupolar nucleus, and resides in an asymmetric environment in methane, but overly rapid spin relaxation was not an actual problem in this study. Actually, deuteromethane spin relaxation was measured and considered here as well, when interpreting

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the overall experimental data through molecular dynamic simulations. The samples were pressurized to a few bars and equilibrated. The actual study was done on relatively low-performance PGSE instrumentation (at 2.3 T and 15 MHz on an iron magnet system, with somewhat insufficient gradient strength and sensitivity). Two Xenon-129 [4] and Neon-21 [5] based studies by other groups followed, as based on two rather exotic NMR nuclei. Of course, some kind of sealed NMR tube arrangement had to be used here as well. Xenon-129 has I = 1/2 and a relatively high NMR frequency, but in aqueous solution its spin relaxation times are very long – over 100 s, so time averaging to enhance experimental signal/ noise becomes very time consuming. Neon-21 has I = 3/2, but in aqueous environment, its transverse spin relaxation time in the study was of the order of 1 s, which is in a quite convenient range. The study was made using (expensive) 21Ne-enriched neon gas, and heavy instead of normal water. Likely, spin relaxation rates also get longer in heavy water, despite an isotope-related increase in viscosity. A side intention here was to calibrate the common magnetic field gradient strength of the 21Ne PGSE setup based on previously known D2O reference self-diffusion data, using the same probe and gradient components. 3.4.1.3 Further reading Moktan and coworkers [6] have later discussed some of these data, and others in terms of parameters related to solution structure. That paper is quite detailed and pedagogical, regarding the experimental-computational feedback one can achieve through combined use of computer simulations and actual experimental self-diffusion data. Van der Waals radii of the noble gases have more recently been reconsidered and compiled by Vogt and Alvarez [7].

3.5 Diffusion-based NMR signal detection and monitoring of miscellaneous aggregation processes in aqueous solution This would likely be the most obvious type of PGSE applications, and numerous applications of this kind have indeed been made over the years. A large number of them were already summarized in a 1987 review [8], and many more in Price’s 2009 book [9]. A 2017 review by Pagès et al. [10] additionally lists a large number of more recent ones. In most situations, a PGSE-type self-diffusion approach to binding phenomena would be routinely successful, unless there are complications like huge water solvent peaks in the spectrum, obstructing NMR signals of actual interest and causing destructive radiation damping-related influence on signal acquisition. Switching to deuterated solvents and using high-performance instrumentation and using recently developed solvent-suppression techniques in PGSE can do wonders in such situations, as illustrated in Examples 3.5.2 and 3.5.3. Depending on molecular types

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and system composition, the basic situation would range from simple dimerization to much more complex equilibrium ones. It also matters if chemical exchange is “fast” or not on the NMR and PGSE timescales. Consider a simple two-site system, where a ligand or “drug” (L) exchanges between being free in solution and n equivalent sites on a macromolecular or supramolecular entity, or alternatively a protein in solution (P). The coupled differential equations describing the echo signal intensities (S) at the free and bound sites are dSf Sf Sb = − γ2 g2 Df δ2 Sf − + dt τf τb

(3:1)

dSb Sb Sf + = − γ2 g2 Db δ2 Sb − dt τb τf

(3:2)

where τf and τb represent the lifetimes of the free and bound states, respectively. Initially, the relative Sf and Sb are given by their respective populations, pf and pb = ð1 − pf Þ. At a time t = Δ and under conditions of fast exchange, this reduces to the simple exponential form A = Sb + Sf = expð − γ2 g2 Dobs δ2 ΔÞ

(3:3)

where the time-averaged Dobs is described through the by now familiar relation Dobs = pb Db + ð1 − pb ÞDf

(3:4)

The combined information is further transformable into summarizing equilibrium constant-type information and similar quantities through relations like pffiffiffiffiffiffiffiffiffiffiffiffi (3:5) pb = α − α2 − β where α=

ðCL + nCP + Kd Þ 2CL

β=

nCP CL

(3:6)

Here CP and CL represent the total concentration of supramolecular/protein-type object and ligand/drug, respectively. Their respective diffusion coefficients in eq. can be measured in separate experiments, or could be evaluated via further curve fitting of titration-like experimental outcomes. The most common situation would be that Db is monitorable throughout such a titration experiment and would remain essentially constant anyway, since the hydrodynamic radius of the combined aggregate would not change much. Equations (3.1)–(3.6) also summarize limiting forms of the so-called Kärger equations [11, 12]. They apply also in more complex context than simple aggregation

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processes in solution and at conditions where chemical exchange occurs at a similar or longer timescale than the PGSE experiment. The equations and their implications are nicely summarized and discussed in Price’s book (pp. 150–162). In simple solution applications, chemical exchange is almost always rapid, and when aiming for quantification of binding phenomena it normally suffices to just consider eq. (3.4), together with an appropriate binding model and binding constant expression for the investigated system in question (analogous to (3.5)–(3.6)).

3.5.1 General objectives Aggregation and binding are, of course, fundamental issues in almost all branches of chemistry, biology and medicine. The diffusion-based path to such information is to translate binding-related influence on experimentally determined selfdiffusion coefficients into global characterization in terms of binding models and their equilibrium constants. As side information, one can also extract information, specifically related to molecular diffusion in the “free” and “bound” states in question. In a further step, one can attempt to deduce aggregate size from self-diffusion data, assumed molecular shape and macroscopic parameters like solution viscosity. Such issues were reviewed in considerable detail by Macchioni and coworkers [13].

3.5.2 Rhodamine 6G self-aggregation in aqueous solution 3.5.2.1 Objectives Rhodamine 6G (Rh6G) is a dye compound, which finds its use together with others in many biophysical and biotechnological applications, such as fluorescence microscopy. Fluorescence correlation spectroscopy is a tool in that family, which can provide somewhat indirect, but specific self-diffusion information in complex biological systems. The diffusion coefficient Rh6G is commonly used as a reference point in such context, and for that reason it was of great interest to make a more absolute determination of it using PGSE, since such NMR techniques do not rest on indirect calibrations or model assumptions in this context [14]. 3.5.2.2 Experimental aspects The task was not entirely trivial, since the target concentration range is very low indeed (1–400 µM). Even with highly enriched D2O as solvent, one had to use solvent suppression techniques like PGSE-WATERGATE [15, 16]. For the lowest concentrations, up to 10,000 cycles of experimental time averaging were needed as well, before an acceptable level of signal/noise could be reached, even though on a highfield NMR instrument with a particularly sensitive cryocooled probe system was

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used. A major contributing reason is that the NMR signal from Rh6G does not have any really “strong” and sharp proton NMR peaks, so signal intensity becomes low for this reason as well. Figure 3.2 illustrates the actual experimental data, which were subject to further processing in terms of assumed monomer–dimer aggregation (i.e., a dimerization equilibrium constant and estimates of monomer and dimer self-diffusion coefficients). As expectable, the time-averaged Rh6G self-diffusion coefficient does not vary more than 20% between a monomeric situation at the lowest concentrations and a 70% dimeric one at 400 μM Rh6G. Testing aggregation models beyond dimerization (see Section 3.5.5) would not be feasible here, considering the already marginal information content of the data.

D/10–10 m2 s–1

3.8 3.6 3.4 3.2 3.0 2.8

0

0.1

0.2

0.3 0.4 [Rh6G]/mM

Figure 3.2: Graphical summary of raw experimental data from Majer and Zick [14] for rhodamine 6G self-diffusion in D2O. At the very lowest concentration, at 1 µM concentration of solute, error limits are of the order of 10%. The molecular structure of rhodamine 6G (normally supplied in hydrochloride form) is also shown. This dye has four rings, and would be expected self-aggregate in aqueous solution already at low concentrations, due to hydrophobic forces as well as intermolecular stackingtype ones between the relatively flat ring systems.

3.5.3 Dissolved organic matter in natural waters 3.5.3.1 Objectives Detecting and identifying trace quantities of a multitude of poorly characterizable and likely polydisperse compounds in the environment through NMR diffusometry would not normally be a primary option. However, the study of Zheng and Price [17] on “natural pond water” (Figure 3.3) using NMR diffusometry methods is of considerable PGSE methodological interest. It pushes the limits of solute detection and demonstration of water suppression in such context considerably. Environmental monitoring is also a currently widely prioritized area of science, and results and conclusions may be of great interest for many. Actually, several NMR-based related investigations of related systems have been made previously, and some tens of others are listed among the references of this chapter.

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Figure 3.3: A nowadays almost stagnant channel and pond system, some 150 km north of Stockholm, Sweden. It was once a key component of a small-scale iron manufacturing facility at Strömsberg (active 1643–1920, private photo).

3.5.3.2 Experimental aspects Personally, I remember attempting PGSE studies on “humic acids” (quite highly colored, polydisperse high-molecular-weight degradation products from wood, etc.) in the early 1980s, and attempting to do primitive “solvent suppression” of water by simply moving the gradient setting window high enough for the otherwise very dominant water signal to decay to almost zero intensity. The intention was to leave only PGSE signals from the more slowly diffusing humic acids. The attempts were unsuccessful on that generation of spectrometers; the signal/noise was simply too low. The project was initiated by the request of people that did dynamic light scattering on related systems. Their studies instead failed because of the high laser light absorbance of the humic acids. As I vaguely recall, the humic acid content got essentially “charred” even in aqueous solution. Zhang and Price could not detect any trace of dissolved organic matter (DOM) signals in their normal NMR spectra. This is not unexpected, since the overall DOM concentration range was typically at a level of only 1 mg L−1 in water, and the corresponding NMR signals are spread over a wide frequency range. However, using the stimulated-echo PGSTE-WATERGATE approach they had previously developed [16], an almost totally water-suppressed data set with adequate signal/noise (like 20) resulted after 16k time-averaged echo acquisitions. Classes of compounds (carbohydrates, carboxyl-rich alicyclic molecules and aliphatic ones) could be identified, together with diffusion-based estimates of their average hydrodynamic radii (using the Stokes–Einstein–Sutherland eq. (2.4)). Some components experimentally had closely similar selfdiffusion constants, possibly indicating mutual aggregation and binding. The

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DOM content in these ponds appears entirely “natural” and only marginally contributed by human influence.

3.5.4 Substrate binding to supramolecular structures in aqueous solution Cyclodextrins are a family of compounds made up of amylose-type sugar molecules, bound together in a ring. They find its use in food, pharmaceutical, drug delivery and chemical industry, as well as in agriculture and environmental engineering. Cyclodextrins can be topologically represented as toroids, where the center and edge expose hydroxyl groups toward aqueous solution. The interior of the toroids is intermediately hydrophilic, but able to host hydrophobic molecules. The exterior hydroxyls of cyclodextrins make them water soluble. Cyclodextrins are able to form host–guest complexes (inclusion compounds) with hydrophobic molecules given the unique nature imparted by their structure. As a result, such molecular structures have found a number of applications in a wide range of fields. This ability of forming complexes with hydrophobic molecules has led to their usage in supramolecular chemistry and as a supramolecular carrier molecule; see Chen and Jiang [18] (Figure 3.4).

Figure 3.4: Alpha-cyclodextrin (6 units) in stick and Van der Waals filled graphic form, and beta-cyclodextrin (7 units) in stick form. It is thought that for many cyclodextrins in host–guest complex situations in aqueous solution, the host cavity in the center becomes somewhat conically shaped.

3.5.4.1 Objectives The first PGSE NMR feasibility study of this kind was made on cyclodextrins in the early 1980s [19] and has been followed by a multitude of later ones. A 2005 review by Cohen et al. [20] is particularly cited in this context. Indeed, there is a widespread interest in a multifaceted “general chemistry” area of this kind. Therefore, this type of application for PGSE diffusometry was likely an eye-opener for many. There is still largely unexploited potential for PGSE diffusometry in specialized areas of organic and inorganic chemistry and biochemistry.

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3.5.4.2 Experimental aspects Proton spectral characteristics for this class of molecules are relatively favorable in PGSE context, regarding sharpness of peaks and spin relaxation rates. Determinations of self-diffusion coefficients of cyclodextrins and guest molecules in D2O solution (where hydroxyls are predominantly deuterated) are routinely easy. Data analysis in terms of aggregate stoichiometry, binding constants and “titration curves” when gradually adding “guest” molecules to cyclodextrins in solution is relatively trouble-free as well. The standard two-site relation Dobs = pDbound + ð1 − pÞDfree would hold, with a monitorable value for Dbound throughout a “titration,” and an expected “break” in the “guest” diffusion curve at a concentration corresponding to the formation of a 1:1 host–guest complex. These data were subsequently converted into equilibrium constants (Kc ) for the inclusion process (cf. Figures 3.5 and 3.6).

20

–ΔG°/kJ mol–1

10

0

nc 2

4

6

8

Figure 3.5: The change in standard free energy − ΔG0 ð = RT ln Kc Þ versus the number of carbon atoms (nC ) for the inclusion of a homologous series of n-alcohols into α-cyclodextrin (blue line and circles) and β-cyclodextrin (red line and circles). Based on data from Rymdén et al., J Inclusion Phenom. 1983;1:159–167 [19], and redrawn with permission, Copyright (1983) Springer.

Figure 3.6: AMP and caffeine molecular models and schematic visualization of stacking-type association in aqueous solution.

3.5.4.3 Further reading Price WS. Recent Advances in NMR Diffusion Techniques for Studying Drug Binding. Aust J Chem. 2003; 56: 855–860.

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Cohen Y, Avram L, Evan-Salem T, Frish L. Diffusion NMR in supramolecular chemistry. Anal Meth Supramol Chem. 2007; 163–219. Thordarson P. Determining association constants from titration experiments in supramolecular chemistry. Chem Soc Rev. 2011; 40: 1305–1323. Gossert AD, Jahnke W. NMR in drug discovery: A practical guide to identification and validation of ligands interacting with biological macromolecules. Prog Nucl Magn Reson Spectrosc. 2016; 97: 82–125. Barhoum S, Palit S, Yethiraj A. Diffusion NMR studies of macromolecular complex formation, crowding and confinement in soft materials. Prog Nucl Magn Reson Spectrosc. 2016; 94–95: 1–10.

3.5.5 Nucleotide aggregation in aqueous solution Mononucleotides like adenosine monophosphate and cytosine, guanine and uridine analogs have partly hydrophilic and partly hydrophobic structural elements, in addition to having flat ring components and a phosphate group. They are essential components in biochemical context and known to self-aggregate in aqueous solution – but a central question is how. A commonly suggested aggregation mode in such context is “stacking” the relatively flat ring-like structures into somewhat flexible rod-like aggregates that grow with concentration. A model compound here is caffeine, which has a similar, but simpler structure. The stacking geometry constraints would likely include relatively lateral and rotational geometrical displacements, that is, leading to a more “spiral-like” aggregate structure than the schematically linear one depicted in Figure 3.6. 3.5.5.1 Objective The purpose of the study was to chart the feasibility of diffusion-based NMR investigations of systems of this kind, and to test variants of the “stacking” mode for molecular aggregation. The following and some others were considered: 1. Direct n-merization from monomers according to an equilibrium is the type as in eq. (3.7), optimizing values for n through computer fitting procedures. This aggregation mode is formally the same as that used for modeling micellization, discussed later in Section 3.6.1: nMÐMn 2.

(3:7)

Indefinite aggregation, all steps equal (this is often referred to as an isodesmic aggregation model), or variants thereof:

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M + MÐM2 M2 + MÐM3 M3 + MÐM4

(3:8)

M4 + MÐM5 ... 3.

Indefinite aggregation, where the first dimerization step has a different equilibrium constant than the remaining ones (but otherwise the same formal equilibria steps as in eq. (3.8)).

Additional model studies were made on caffeine binding to mononucleotide systems in aqueous solution. 3.5.5.2 Experimental aspects The original study of this kind [21] was made on equipment with low performance (sensitivity and gradient strength), compared to what is available today. To avoid a huge water solvent signal and to avoid disturbing influence of intermediate chemical exchange on signals from ionizable groups, heavy water was used as solvent. Since it would be expected that the different aggregation models would create only marginal relative differences at the fitting stage, it felt necessary to collect a large number of self-diffusion data at varying concentrations (typically 0–0.5 M). Probably, only marginally better basic experimental data quality, extending to lower concentrations could be collected on a modern PGSE-equipped spectrometer system. Since 0.5 M is a relatively high concentration, it also felt justified and necessary to correct measured self-diffusion data for obstruction effects (see Figure 2.8) from the aggregates themselves. The simplest relation in this context was once suggested by Wang [22], and reads Dobs = Df ð1 − 1.5ϕÞ

(3:9)

where Df would correspond to the diffusion coefficient without obstructing entities. At a volume fraction of ϕ of these in solution, Wang’s model predicts the expected apparent level of decreased self-diffusion. Refined variants, which also include the shape of obstructing elements, have later been derived and discussed. Some of them are listed and discussed in Price [9], Section 1.8.6; see also Jönsson et al. [23]. In general, they do not differ much in their predictions, as the problem is pretty vague anyway. Further complicating matters, the typical decrease in self-diffusion upon aggregation is visually similar and not numerically very different from

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suggested “obstruction corrections.” A further recurring aspect in similar context is whether to additionally account for increased solution viscosity. This would likely correspond to accounting twice for basically the same factor. The relative influence on aggregate self-diffusion from geometrical growth upon n-merization was modeled to first order for each aggregation step, going from the monomer value from monomer (Dð1Þ) as follows: DðnÞ = Dð1Þ=½ðn0.3333 Þðpolynomð′′n′′Þ

(3:10)

where the n0.3333 factor is supposed to correct for the general increase in hydrodynamic radius, and the polynom(“n”) one was based on a least-squares fitting of the so-called Perrin F-factors (ratios between frictional factors for spheres and ellipsoids of the same volume, see Cantor and Schimmel, p 580 [24]), assuming that a sphericalprolate crossover point within a vertical stacking model is reached at the dimer stage. In models 2 and 3, self-diffusion influences of about 25% will result on these grounds. As already anticipated, there were no great differences between various association models. The only clear indication throughout was that model 2 (indefinite aggregation, all steps equal) did not describe reality well at all, and that therefore there are clear indications of significant cooperative factors in the association process. Overall, PGSE measurements of this kind are easy, albeit somewhat tedious. The data evaluation stage was more problematic, including various corrective factors and quite extensive computer programming and simulation efforts for overall modeling and fitting of the data. Essentially the same type of study was repeated later, but with divalent metal ions added. These influence the aggregation processes significantly [25] (Figure 3.7).

D/10–9 m2 s–1

0.4

0.3

0.2 0.0

0.1

0.2 0.3 [AMP]/M

Figure 3.7: Concentration dependence of AMP self-diffusion in D2O, and fitted curves pertaining to model 1 (solid black line), an extension of model 1 where n-mers associate with m-mers (dashed magenta line) and model 2 (blue, dashed line). Redrawn by permission, based on data and analyses in Rymdén and Stilbs Biophys Chem; 1985;21: 145–156 [21], Copyright (1985) Elsevier.

3.5.6 Protein aggregation in aqueous solution 3.5.6.1 Objectives The feasibility of such PGSE applications has been systematically studied by Price, as applied to the common protein lysozyme (see Price’s monograph, Section 2.5, or

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the original paper [26]). The general issues as well as the data analysis strategy are the same as just discussed in Section 3.5.5.2 for mononucleotide aggregation – finding the proper aggregation mode through appropriately corrected PGSE self-diffusion studies, and the equilibrium constant(s) that characterize it. 3.5.6.2 Experimental aspects Some additional problems quickly become apparent. The protein has few reasonably sharp or intense peaks in basic proton NMR, since there are no repeating units. Since lysozyme is also quite large (molecular weight 14.3 kDa), it tumbles significantly slower than small- to medium-sized molecules in solution, affecting PGSE studies negatively through shortened transverse spin relaxation times. Further aggregation from the monomeric state would shorten transverse relaxation even further. Various studies were made on solutions containing 0–10 mM lysozyme, in 90:10 H2O/D2O solution, with added NaCl. About 10 mM may not sound much. However, when translating this number to concentration by weight, it corresponds to 143 g L−1. Concepts like obstruction effects and solution viscosity evidently become important to consider, as in Section 3.5.5. No spectra are shown in the paper, but it seems the presence of a “water signal” from 90% H2O solvent was not too disturbing (Figure 3.8).

D/10–10 m2 s–1

1.2 1.1 1.0 0.9 0.8

0

1

2

3

4 5 6 [Lysozyme]/mM

Figure 3.8: Hen egg-white lysozyme in three different graphical representations; stick, space-filling and ribbon, respectively (underlying data and graphical display options courtesy of the Protein Data Bank website https://www.rcsb.org/pdb/home/home.do) and Lysozyme self-diffusion as a function of total concentration (based on data from Price et al. [26]).

The authors applied a very ambitious obstruction-corrected isodesmic model (eq. (3.8)) and tested several variants through computer simulation. Experimental data on self-diffusion of 0–6 mM lysozyme in 0.5 M NaCl fitted well,

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using Dmonomer = 1.19 ± 0.01⋅10−10 m2 s−1 and a common equilibrium constant for indefinite aggregation steps of 118 ± 12 M−1. Monomer–dimer equilibria were tested too, but the isodesmic model fitted the data significantly better. It is not at all unreasonable that this particular protein aggregation proceeds in this indefinite manner, although it is known that other protein systems do aggregate via discrete dimerization (like bovine serum albumin) and tetramerization steps (like sorbitol dehydrogenase). The authors noted issues like nonideal behavior that was sensitive to both the salt concentration and pH. Lysozyme has an isoelectric point of pH 11, and at normal pH it has a net positive charge. Being a charged species, changes in the diffusion coefficients must be interpreted within a framework that includes electrostatic repulsion and aggregation. A later high-concentration study on lysozyme aggregation by Barhoum and Yethiraj may be of interest in this context [27].

3.5.7 PGSE in organometallic chemistry There has been relatively little use of NMR diffusometry methods in fields like inorganic and organic chemistry. Since many “inorganic” problems intrinsically concern atoms and ions with experimentally inaccessible or “difficult” NMR properties, this is mostly understandable. Potential applications in “organic” chemistry appear relatively common and easy to do experimentally, but still PGSE-type techniques are rarely seen here. A few are listed among other references in the relatively recent review by Pagès [10], and also in the previously mentioned reviews on PGSE in supramolecular chemistry (see Section 3.5.4). The situation would likely change, if some more “eye-opening” NMR diffusometry studies were to appear in areas of high general interest. One such area that caught on early was ion-pairing in the field of organometallic chemistry, that is, in the borderline between inorganic and organic chemistry. It seems traceable to work by Pregosin, in particular, who has written several reviews on the subject, as well as a textbook on NMR in organometallic chemistry. 3.5.7.1 Objectives The objective was to investigate aggregation processes of organometallic compounds and ligands, including solvation shells. Many constituents are ionic, so electrostatic considerations and the solvent type (primarily through its dielectric constant) become important for complex formation. Most solvents in this context are “inert” and “nonpolar” and have much lower dielectric constants than water. A common tool in such context is quantum chemical computation of molecular properties, and numerous software packages exist (Figure 3.9).

3.6 Surfactant systems

Contact ion pair

Solvent-separated ion pair

Solvent-shared ion pair

117

Penetrated ion pair

Figure 3.9: Schematic organizational modes for actual ions and “solvent shells” in the vicinity of ion pairs (after Pregosin in references in Section 3.5.7.3).

3.5.7.2 Experimental aspects Normally, there should be no great experimental problems to measure self-diffusion in systems encountered in this context. Low solvent viscosity (like for dichloromethane) may be a potential problem through increased risk of convection-related artifacts during measurements. Organometallic compounds also often contain 19F and 31 P; two quite favorable types of spin-1/2 nuclei, which can provide complementary information to that derived from protons and 13C NMR. Some classes of organometallic compounds or complexes may have constituents that are paramagnetic. That will normally make PGSE self-diffusion studies impossible, because of echoattenuation competing efficient electron-nuclear spin relaxation. 3.5.7.3 Further reading Pregosin PS, Rueegger H. Nuclear magnetic resonance spectroscopy. Compr Coord Chem II. 2004; 2: 1–35. Pregosin PS. Ion pairing using PGSE diffusion methods. Prog Nucl Magn Reson Spectrosc. 2006; 49: 261–288. Bellachioma G, Ciancaleoni G, Zuccaccia C, Zuccaccia D, Macchioni A. NMR investigation of noncovalent aggregation of coordination compounds ranging from dimers and ion pairs up to nano-aggregates. Coord Chem Rev. 2008; 252: 2224–2238. Pregosin PS. NMR in Organometallic Chemistry. Weinheim: Wiley-VCH, 2012.

3.6 Surfactant systems 3.6.1 Physicochemical characteristics The so-called surfactants are typically molecules with a nonpolar alkyl chain and a polar terminal end group, which can be anionic, cationic, zwitterionic or nonionic. In aqueous solution such molecules (monomers) typically aggregate into spherical micellar aggregates above a certain concentration, called the critical micelle concentration (cmc). It is typically in the 1–50 mM range, but can get much lower for nonionic surfactants. Packing constraints broadly determine the actual number of surfactant

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monomers in a spherical micelle and many other types or related structures [28]. Micellar geometry would typically be spherical, corresponding to an aggregation number of 50, molecular weight of 10 kDa and a radius of 2 nm or 20 Å (Figures 3.10 and 3.11).

(a)

(b)

(c)

(d)

(e)

Figure 3.10: Some of the most common surfactant molecules used in basic surfactant system research. A common structural feature is a hydrophobic tail of hydrocarbon type, and a polar head group, that can be ionic, zwitterionic or nonionic. Ionic surfactants of course include counterions (typically sodium, potassium, chloride or bromide ions; not shown in the figure). (a) Stearate (anionic, commonly used with sodium or potassium counterions); (b) “C12E8” [a C12 alkane chain with a head group made up by eight ethylene oxide fragments, and terminated by an OH group (nonionic)]; (c) dodecyl sulfate (anionic, almost always used with sodium counterions); (d) cetyl trimethyl ammonium (cationic, almost always with bromide counterions); and (e) “Aerosol OT” (dioctyl sulfosuccinate, anionic, but with two alkyl chains, and almost always with sodium counterions). A particular characteristic of C12E8 is a very low cmc, compared to others. Aerosol OT does not form normal micelles in aqueous solution, but readily forms inverted ones (“water droplets”) in hydrocarbon-rich solution. Graphic representations were made using the computer freeware “Avogadro.”

There is also another central critical quantity, the so-called Krafft point. This is the lowest temperature at which aqueous monomer solubility is high enough to reach the cmc concentration where micelles can actually form (see Figure 3.12). For many practical uses of surfactants, this should be well below room temperature. At elevated surfactant concentrations rod-like micellar structures and various liquid

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Figure 3.11: A schematic representation of spherical and short rod-like surfactant micelles in water. Counterions are not included in the drawing. These slightly redrawn figures originate from graphic illustrations in a thesis by Jan Ulmius (Lund Institute of Technology, 1978; reproduced and graphically modified, courtesy of the Author). Unlike most other schematic illustrations of surfactant micelles, these drawings convey a proper view of a dynamic structure and chemical exchange between micellar and aqueous environments. In an NMR context one should note that chemical exchange between micellized and monomeric surfactants in the aqueous phase is rapid on the NMR timescale, and that quantities like chemical shifts, spin relaxation rates and self-diffusion coefficients will be observed as time-averaged ones.

Concentration

Hydrated crystals

Total solubility

Micelles

CMC curve

Krafft temperature 10

Unimers

20 30 Temperature/°C

40

Figure 3.12: Most surfactants in practical use have a Krafft point below the room temperature (see the text). It increases with alkyl chain length and also depends on the surfactant head-group and the counterions. A well-known everyday manifestation of Krafft temperature behavior is the poor function of sodium- or potassium-based soaps in “hard” water, where “calcium soaps” may precipitate. Technical and cosmetic formulations often use mixtures of nonionic and anionic surfactants to optimize general performance, and to avoid formation of unwanted phase structures.

crystalline phases form. Adding salts (like NaCl) to the system initially tends to promote aggregate growth, and also alters the phase behavior. At very low system water content, typically reversed micellar structures form (see Figure 3.15 below). NMR spin relaxation studies are proven to be very useful for investigating the micellar interior. In short, a typical surfactant micellar radius is close to that of the surfactant monomer, and the aggregation number closely matches with “packing constraints”, to form such a spherical entity. As for many other types of surfactant

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Slow (tumbling) correlation time

θ H

Order parameter, S

0.3

Rapid local reorientational H motion correlation time and order

0.2

0.1

0.0

“Fast” correlation time/ps

aggregates, approximate dimensions and geometries are predictable on such geometric grounds, although deeper understanding must include factors like hydrophobic and electrostatic interaction. Still, such a simple geometrical packing view holds well for a number of more complex surfactant-containing structures and phases. The polar groups form a more rigid “palisade layer” toward the aqueous environment. The interior hydrocarbon-like one is much less rigid, and macroscopically appears close to being made up by a “droplet” of hydrocarbons of the same length, regarding “microviscosity” and similar concepts [29]. Internal reorientational dynamics and order with reference to the more rigid micellar surface area has also been mapped, and indeed shows gradually higher motional freedom inward (see Figure 3.13).

20 10 0

Micellar surface

Figure 3.13: Actual motional and structural characterization of a micellized surfactant entity (dodecyl trimethyl ammonium chloride, DOTAC). It was based on combined carbon-13 and deuterium spin relaxation studies at several magnetic fields [30]. The “order parameter” is here defined as the time   average 21 3cos2 θ − 1 fast , with reference to the leftmost part in the figure. The common “slow reorientation” correlation time for overall micellar tumbling was globally evaluated to be 4 ns. Partly based on original figures and redrawn by permission from Söderman et al, J Phys Chem. 1985; 89: 3693–3701 Copyright (1985) American Chemical Society.

Micelle formation from monomers in aqueous solution is a cooperative, rather than a gradual stepwise (isodesmic) phenomenon. This also leads to a “critical micelle concentration,” rather than to a continuous range of intermediate aggregates of varying sizes. To a good approximation, micellization often macroscopically described as a “pseudophase separation,” using concepts like “the micellar phase” and “the aqueous phase.” This is a limiting case for high n-values of a straightforward equilibrium model like nðmonomersÞÐðmicellesÞ and thus Kn = ½micelles=½monomersn . The cmc becomes = ðnKn Þ − 1=n and the concentration of micellized monomers grows almost linearly with total surfactant concentration above the cmc, while the monomer concentration remains almost constant, and closely equal to the cmc (Figure 3.14). The same basic view is also valid when describing the partial incorporation of a third, partly water-soluble compound (the solubilizate) into this pseudo-two-phase framework (referred to as “solubilization”). Relative compartmentation fractions of the

Monomer and micellized concentration/M

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121

0.018 0.016 0.014 0.012 0.01 Monomer

0.008

Micellized

0.006 0.004 0.002 0

0

0.005

0.01 0.015 Total surfactant concentration/M

0.02

0.025

Figure 3.14: Micellization, modeled as a cooperative process, for a case of n = 55 and cmc = 15 mM (like for dodecyl ammonium chloride in water). The figure was created using a short Matlab script written by Bengt Jönsson, Lund University. It is based on a computational scheme originally described by Persson et al. [31].

solubilizate are treated in the same way as partition equilibria between macroscopic phases (e.g., commonly used octanol–water partition coefficients for characterizing the hydrophobicity of a particular molecular type). See Figure 3.18 below and the example application in Section 3.6.3 for a more detailed and quantitative discussion of micellar solubilization. Outside a “micellar phase boundary” (often referred to as the L1 phase in this context), the situation becomes more complex and various liquid crystalline phases form at elevated surfactant and solubilizate concentrations. Even for just three components, intermediate phase behavior may get quite complicated and is typically illustrated in triangular (ternary) phase diagrams (Figure 3.15). Prior mapping of phase behavior is very important in the context of technical or cosmetic uses of surfactant systems. In this context, it should be stressed that “rigid” phase behavior observed in model systems is commonly undesired in actual technical formulations (like paints, creams, shampoos, etc.), since it may likely reduce factors like their useful temperature range or other characteristics. Some liquid crystalline phases may be mechanically quite rigid as well. Once formed, they can cause clogging of pipes or pumps in the manufacturing process or similar problems. For these and other reasons, technical formulations are often intentionally composed of mixtures of constituents, having smoother or rather undefined phase boundaries. Already upon initial surfactant aggregation to micelles in aqueous solution, huge changes in self-diffusion may occur, which provide detailed and quantifiable links to system characterization of various binding phenomena, including those of counterion binding. Interactions in more complex systems containing polymers

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Decanol

20 80

Reversed micellar

40 60

Reversed hexagonal

60 40

Lamellar 80

20

Water

20

40

Micellar

60

80

Sodium octanoate

Hexagonal

Figure 3.15: Schematic ternary phase diagram for a typical water, surfactant and medium- to long-chain alcohol system near room temperature. Outside the pure phase regions indicated in color various mixed phases or crystalline surfactants coexist.

or solubilized molecules are well suited for detailed quantification through selfdiffusion measurements as well. As already mentioned, so-called lyotropic liquid crystal phases are often formed at elevated surfactant concentrations in water, with or without further system components. Such systems were investigated early on in NMR diffusometry history, through unresolved PGSE techniques, involving monitoring of water diffusion only. That liquid crystal phase structure reflects itself in anisotropic self-diffusion rates seemed to fascinate physicists as well as chemists. Extending the concepts to use lyotropic liquid crystals as model systems for biological membrane function and structure is a highly relevant and obvious approach as well, which has attracted a lot of experimental effort. In retrospect, early NMR selfdiffusion studies on liquid crystals really helped to pave the way for later more widespread use in many other areas of chemistry, physics and biomedicine. 3.6.1.1 Further reading Söderman O, Stilbs P. NMR studies of complex surfactant systems. Progr Nucl Magn Reson Spectrosc. 1994; 26: 445–482. Griffiths PC, Cheung AYF, Davies JA, Paul A, Tipples CN, Winnington AL. Probing interactions within complex colloidal systems using PGSE-NMR. Magn Reson Chem. 2002; 40: S40–S50. Furó I. NMR spectroscopy of micelles and related systems. J Mol Liquids. 2005; 117: 117–137. Holmberg K, Lindman B, Jönsson B, Kronberg B. Surfactants and Polymers in Aqueous Solution, 2nd Edition. Chichester, UK: John Wiley & Sons, 2002. Söderman O, Stilbs P, Price WS. NMR studies of surfactants. Concepts Magn Reson. 2004; 23A: 121–135.

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Evans DF. Wennerström H. The Colloidal Domain: Where Physics, Chemistry, Biology, and Technology Meet, 2nd Edition. New York: Wiley-VCH, 1999. Israelachvili J. Intermolecular and Surface Forces, Third Edition. Academic Press, Inc.: San Diego, 2011. Christian SD, Scamehorn JF, Eds. Solubilization in Surfactant Aggregates. Surfactant Science Series. 1995; 55: 1–547 (Marcel Dekker, New York).

3.6.2 Surfactant aggregation As outlined in Section 3.6.1, surfactants are known to initially self-aggregate into micellar structures in aqueous solution, as driven by a variety of forces. Hydrophobic interaction to minimize hydrocarbon–water contact and general geometrical packing constraints are the most obvious ones. For surfactants with ionic head groups also electrostatic interactions become relevant; there are repulsive ones between head groups and attractive ones toward counterions in solution. Just adding simple salts (like NaCl) to anionic or cationic surfactant solutions may alter aggregation characteristics heavily. 3.6.2.1 Objectives To chart the feasibility of PGSE-based monitoring of surfactant aggregation, including counterion binding, a previously unknown (and probably otherwise useless) model surfactant was synthesized by mixing equal amounts of dichloroacetic acid and n-decylamine in aqueous solution. This will thus transform into a cationic decylammonium one, with dichloroacetate as counterion. After purification through recrystallization it turned out to have a cmc in the 20 mM concentration range, and a Krafft point somewhat above room temperature; all as anticipated. Dichloroacetate and decylammonium proton PGSE signals in D2O solutions of 16 different total surfactant concentrations were monitored [32]. Tetramethyl silane (TMS) was added in small amounts to each solution, to provide a signal from an entity that was hopefully totally solubilized (see Section 3.6.3) and would directly reflect “the micellar self-diffusion” throughout the PGSE experimental series (apart from its normal use as a chemical shift standard). 3.6.2.2 Experimental aspects The PGSE Hahn-echo-based experiments were performed at 40 oC on an old-generation iron magnet system at 100 MHz NMR frequency, using internal deuterium lock on the D2O solvent peak. Apart from the Krafft-point-related need to depart from ambient temperature to a higher one, there were no experimental or evaluation problems of any kind. Obstruction correction attempts were not deemed necessary, since the total concentration of surfactant was below 0.25 M (about 70 g L−1) and this was an exploratory study. Also, the water self-diffusion coefficient did not decrease significantly at elevated total surfactant concentrations.

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Data evaluated using the simple two-site relation (Dobs = pDbound + ð1 − pÞDfree ) were already discussed, which resulted in an aggregation number of about 42. TMS diffusion data represented the “micellar self-diffusion coefficient,” although curve fitting like that used in the later paper on mononucleotide aggregation (Section 3.5.5) could alternatively have been used. Such procedures were not yet developed locally at the time, however. The evaluation results were above expectation, since they confirm a textbook-like picture of micellization (Figure 3.16). The cmc was evaluated to be at 26 mM total concentration. From here on the fraction of micellized decylammonium as well as micellarly bound dichloroacetate counterions grow linearly. The respective fractions in the monomeric state thus undergo linear changes as well. An observed slight decrease in monomeric decylammonium concentration above the cmc was found to agree with electrostatically extended theories of micellization of ionic surfactants at the time. C10H21ND3+

HDO

2·10–9

200

5·10–10

CHCl2COO

2·10–10

C10H21ND3+

0.8

150

–

c/mM

D/m2 s–1

0.9

Degree of counterion binding

10–9

0.7 CHCl2COO–

Micellar 100

CHCl2COO–

10–10

50

Free

TMS

5·10–11 50

100

150

200 ctot/mM

50

100

150

C10H21ND3+

200 ctot/mM

Figure 3.16: Experimental component self-diffusion coefficients in the decylammonium dichloroacetate system as a function of total concentration in heavy water solution at 40 oC, together with an evaluation in terms of a simple and time-averaged two-site micellar/free binding situation. Redrawn from data in Stilbs and Lindman J Phys Chem. 1981; 85: 2587–2589 [32], by permission, Copyright (1981) American Chemical Society.

The most satisfactory finding was the constancy of the “degree of counterion binding,” that is, the ratio of micellarly bound dichloroacetate to micellized decylammonium. Such behavior is predicted from electrostatic considerations for counterion binding in polyelectrolyte systems in general. Here, it was also at an unusually high level (80%) compared to counterion binding conditions for inorganic ions (typically like 30%). The high value is attributable to binding force contributions from additional hydrophobic interactions, as confirmed in more extensive studies that included dichloroacetate counterion analogues [33].

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Finally, it should be noted that as a general consequence of cooperative binding equilibria like for micellization (nMÐMn , at high n) quantities like the cmc can be evaluated from a graph of a pertinent quantity (like a self-diffusion coefficient or chemical shift) against 1/C, where C represents the total concentration of the aggregating compound [34]. Two linear relations should result, intersecting at the cmc, as illustrated in Figure 3.17. For high aggregation n-values, such a simplified procedure works almost as well as a full nonlinear fitting one to the actual equilibrium expressions. For lower n, the “knee” at the intersection becomes more rounded. Several studies have appeared in the literature where one instead plots the logarithm of the self-diffusion coefficient (or chemical shift) against the total surfactant concentration. Relatively linear graphs normally result here as well, also with a knee near the actual cmc value. This approach is misguided, since it has no underlying physicochemical basis or relevance through an aggregation model.

0.8

D/10–9 m2 s–1

0.7

Dmonomer

0.6 0.5 0.4 0.3 0.2 0.1 0

Dmicelle 0

20

1/cmc 40

60

80

1/C/M–1 Figure 3.17: Illustrating the proper linearized way of analyzing micellization “pseudophase transition” equilibria, via the decylammonium self-diffusion coefficients underlying Figure 3.16. This general linear approach was originally developed by Persson and coworkers [34]. The original application concerned experimental carbon-13 NMR chemical shift changes upon micellization.

3.6.2.3 Further reading Rusanov AI, Shchekin AK, Volkov NA. Diffusion in micellar systems: theory and molecular modelling. Russ Chem Rev. 2017; 86: 567–588.

3.6.3 Micellar solubilization Original techniques for studying micellar solubilization, that is, incorporation of more or less water-soluble small- to medium-sized molecules (the “solubilizate”)

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into micellar entities include vapor pressure studies by gas–liquid chromatography on the air–vapor mixture above the micellar solution. Quite substantially enhanced incorporation into a mainly aqueous macroscopic environment of otherwise only sparingly water-soluble compounds can be achieved through solubilization. At elevated solubilizate concentration, micellar distortions of various kinds would occur, ultimately leading to phase transitions of various kinds, as was illustrated for n-decanol in sodium octanoate micelles in Figure 3.15. In the early 1980s, FT-PGSE NMR was found to be an excellent way to quantify micellar solubilization, and possibly provide a path to more detailed molecular understanding of it. It must be admitted that the solubilization application of NMR diffusometry was discovered accidentally. While attempting to map sodium dodecyl surfactant (SDS) aggregation using FT-PGSE NMR diffusometry, an “extra” peak with an odd apparent D-value in between those of water and SDS appeared in the spectra. It was quickly diagnosed to originate from acetone that was used for cleaning the NMR tubes, and evidently had not dried out properly. But the methodological implications became immediately obvious [35]. Based on the hydrophobicity, solubilizates would reside in the hydrocarbonlike micellar interior or partly in the surface “palisade layer.” For example, solubilized cyclohexane would strongly prefer the micellar interior environment, while n-pentanol likely would locate and orient in a way where its –OH groups would reside in the palisade layer, posing motional constraints also on the interconnected alkyl chain. Spin relaxation-based techniques like those behind the illustrations in Figure 3.13 can provide direct and quantitative answers to such issues; PGSE self-diffusion based ones can provide only indirect information. 3.6.3.1 Objectives To systematically and quantitatively study micellar solubilization for classes of surfactants and solubilizate types within a pseudophase aqueous/micelle partition equilibrium model framework [35–37]. 3.6.3.2 Experimental and evaluational aspects Incorporation of small- to medium-sized solubilizate molecules into micelles in aqueous solution will affect solubilizate self-diffusion quite strongly, since the micellar entities are like 10 times larger in diameter and thus diffuse like 10 times slower. Again, the same and recurring two-site equation (Dobs = pDbound + ð1 − pÞDfree ) will describe the situation well, especially at low relative solubilizate concentrations. Technically, experimental data of this kind are normally trivially easy to collect in the form of proton PGSE-based self-diffusion coefficients, even on low-performance instrumentation. However, one should use heavy water as solvent and strive to hold the relative solubilizate concentration below 10% of that surfactant, to avoid distorting the micellar structure by saturating it with solubilizate. Note, however, that on

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the molecular level, the lowest possible actual “reporter” concentration would be discretely one solubilizate molecule per micelle, so some micellar perturbation cannot be avoided. It would correspond to something like a 1% concentration of micellarly bound solubilizate. For evaluation, one requires numeric values for free diffusion of solubilizate (without surfactant present), micellar diffusion and time-averaged ones for solubilizate, as partitioned between the micellar and the aqueous pseudophase. Obstruction corrections would self-cancel, since a relative comparison of self-diffusion coefficients made here. The above relation can be transformed into a more direct form in this context: p=

Dfree − Dobs Dfree − Dmicellar

(3:11)

providing a numeric value for the relative fraction of solubilizate in micellarly solubilized form. It should be evident that experimental data would be most accurate for p-values around 0.5 and that very low and very high p-values are hard to quantify. Since we are formally dealing with partition equilibria between micellar and aqueous pseudophases (cf. Figure 3.18) one must then proceed to an estimation of a formal solubilizate concentration-based partition coefficient (Kc ) for such an equilibrium situation Kc =

Cmicellar p Vaqueous = Caqueous 1 − p Vmicellar

(3:12)

Micellar headgroup region (“palisade layer”) Aqueous phase “Octanol” Core region

Water Micellar pseudophase

KSolubilization =

[ SMicellar ] [ SAqueous ]

Kc =

[ SOctanol ] [ SWater ]

Figure 3.18: Incorporation of a third, partly water and oil-soluble compound into a micellar solution is treated like a “classical” macroscopic partition equilibrium (like partitioning between octanol and water phases), regarding the micelles and the aqueous environments as pseudophases.

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where V’s correspond to the estimated macroscopic aqueous and micellar volumes in a pseudophase view or the system. The aqueous one is easy to estimate to a good precision; the micellar one is somewhat more problematic [36]. One particular issue is whether to include an estimate of the palisade layer volume or not, regarding what is the actual solubilization volume. Fortunately, for practical and most other considerations, a relative comparison of partition coefficients is what really matters (Figure 3.19). 8

1 In(partition cofficient)

p. degree of binding

7 0.8

0.6

0.4

0.2

2

4

6

8

Number of carbons

10

6 5 4 3 2 1

2

4

6

8

10

Number of carbons

Figure 3.19: Illustrating p-value and partition coefficient dependence on alkyl chain length for solubilization of n-alcohols in SDS micelles (loosely based on data in Stilbs P, J Colloid Interface Sci. 1982; 87:385–394). Redrawn by permission; Copyright (1982) Elsevier.

Relative Kc -values with a homologous series of compounds (like alcohols from npropanol to n-octanol) can be further compared, by applying standard thermodynamic relations like ΔGo = − RT ln Kc and evaluating quantities like the free energy increment for transfer of –CH2– entities from the aqueous to the micellar and hydrocarbon-like environment. The numerical values for both n-alcohols and methyl n-alkyl ketones were found to be close to –2.6 kJ mol−1 [36]. That molecular self-diffusion in aqueous solution is strongly affected by the presence of surfactants, has recently been rediscovered and become classified as a form of “NMR chromatography” (see Section 5.7.3). The first such 2D-like “DOSY map” based on alcohol solubilization in SDS micellar solution appeared already in 1994 [38], well before the introduction of the term “NMR chromatography,” however. The purpose at the time was merely to illustrate DOSY data presentation. Numerous publications of similar type have appeared recently, but the underlying concepts and diffusion-retarding mechanism of “solubilization” are sadly not even mentioned or seemingly understood.

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3.6.4 Mixed micellization Since FT-PGSE provides molecularly resolved information, issues for more complex surfactant systems are amenable for study through component-resolved selfdiffusion information. Long-standing discussions regarding molecular organization in mixed surfactant systems have benefited from such data, since macroscopic and thermodynamic information from many alternative techniques (like calorimetry, conductivity or surface tension) can only provide molecularly unresolved clues in the form of “nonideal behavior” or similar. One such topic is general miscibility of fluorocarbons and hydrocarbons, all the way down to the molecular level. It is known, for example, that a mixture of n-hexane and perfluoro-n-hexane separates into two phases below a critical temperature of 23 oC. For longer alkyl chain lengths, the corresponding critical temperature is higher [39]. Would mixed micelles form from hydrocarbon and fluorocarbon surfactants in aqueous solution? Or will some segregation occur, and what form would it take? Such questions were posed already in 1974 [40], and possibly earlier. An FT-PGSE study from mixed hydrocarbon–fluorocarbon surfactant micellization was done some decade later and led to the conclusion that mixed micelles do form, and no clear indications of segregation were found [41]. Also, micellar solubilization of hydrocarbon alcohols in micelles formed by a perfluorinated surfactant was made, indicating similar solubilization partition coefficients and binding trends between the aqueous and micellar fluorocarbon pseudophases as for normal hydrocarbon surfactants [42]. A notable anomaly in this respect was observed for benzene. 3.6.4.1 Objectives The objective was to reinvestigate the still controversial issue by FT-PGSE-based self-diffusion, with better equipment than was available in 1984. The study was complemented with chemical shift information, since it has previously been established that 19F chemical shifts in NMR for alkyl chains are quite sensitive to their chemical environment. Again, sodium perfluoro octanoate (SPFO) and sodium decyl sulfate (SDeS) were chosen for study [43]. The basic literature cmc values of the two surfactants at 25 oC were closely similar; 33 mM for SPFO and 31 mM for SDeS, and their degrees of counterion binding too (0.55 and 0.63, respectively). 3.6.4.2 Experimental aspects It would appear that this could be a relatively simple task – measure the self-diffusion coefficients of both surfactants (one by proton PGSE and the other by 19F PGSE) and analyze the data so as to find the respective “micellar self-diffusion coefficients.” If they are equal, both are in a common aggregate – or? In reality, things are considerably more complicated. Figure 3.20 iillustrates basic self-diffusion data for the single surfactant components. Mixed system behavior

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3 PGSE NMR diffusometry example applications

Self-diffusion coefficient/m2 s–1

5

×10–10

4 3 2 1

0

20

40

60 1/C (M–1)

80

100

Self-diffusion coefficient (m2 s–1)

Figure 3.20: Micellization of SPFO (orange) and SDeS (green) was studied separately, and analyzed via the linear graphic approach described previously. For a better determination of the cmc, one should have chosen more data points between 20 and 100 mM surfactant concentration, but that parameter is of minor actual concern here (cf. Figure 3.21). Based on data and graphics in ref. [43] and redrawn by permission from Nordstierna et al. J Am Chem Soc 2006;128: 6704–6712 Copyright (2006) American Chemical Society.

2

× 10–10 xH = 0.75 xH = 0.25 xH = 0.5 xH = 0.5 xH = 0 xH = 1

1.5

1

0.5

0

0

1

2

3 4 1/C (M–1)

5

6

7

Figure 3.21: Selected data from a larger set of mixed micelle PGSE self-diffusion studies, based on proton (green lines) or 19F (orange line) NMR. Based on data and graphics in ref. [43] and redrawn by permission from Nordstierna et al. J Am Chem Soc 2006;128: 6704–6712 Copyright (2006) American Chemical Society.

in the 150–250 mM total concentration range is shown in Figure 3.21, where deviations from regular and ideal behavior are apparent. Figure 3.22 summarizes quantitatively the asymmetry regarding surfactant monomer amount that is left out from micellization.

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131

cmonomer (mM)

50 40

Total

30 20 SPFO

SDeS

10 0 0.0

0.2

0.4

0.6

0.8

1.0

xH Figure 3.22: Data fitted amount of surfactant monomer in nonmicellized form in the total concentration range 150–250 mM. Based on data and graphics in ref. [43] and redrawn by permission from Nordstierna et al. J Am Chem Soc 2006;128: 6704–6712 Copyright (2006) American Chemical Society.

After applying theoretical frameworks like regular solution theory and incorporating supplementary information from 19F chemical shift, data interpretation was still difficult. Without high-quality self-diffusion and chemical shift data, this would not have been possible at all, since various aggregation models only manifest themselves in subtle ways when comparing experimental data with outcomes of theoretical predictions. It was concluded that the most likely aggregation model was one of mixed micellization, but with internally demixed micelles – that is, the schematically illustrated alternative III in Figure 3.23.

(I)

(II)

(III)

(IV)

Figure 3.23: Mixed micellization alternatives (adapted and redrawn from Nordstierna et al. [43]). (I) Complete demixing, (II) complete mixing, (III) internal demixing in a single type of micelles and (IV) partially demixed micelles. Based on data and graphics in ref. [43] and redrawn by permission from Nordstierna et al. J Am Chem Soc 2006;128: 6704–6712 Copyright (2006) American Chemical Society.

The study was later followed up with additional NMR and small-angle neutron scattering experiments on additional type of surfactant pairs. Here, also information on the micellar shapes was added to the array of experimental data [44]. The general conclusions on surfactant demixing remained similar. 3.6.4.3 Further reading The Wikipedia article on Regular solution Kissa E. Fluorinated surfactants and repellents. New York: Marcel Dekker, 2001.

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3.6.5 “Microemulsions” and microemulsion structure Microemulsions are a class of clear, isotropic and thermodynamically stable liquids of relatively low viscosity, containing significant and similar amounts of water, hydrocarbon, surfactant and normally also a “co-surfactant.” The latter is commonly a medium-chain alcohol, like n-butanol. In the mid-late 1970s there was a lot of agitated controversy regarding the microscopic structure of microemulsions, and several types of structural suggestions had been presented. Some of these were clearly at odds with other physicochemical parameters, like observed macroscopic rigidity and viscosity. Normal NMR spectra of microemulsions also have sharp and single peaks for all constituents, suggesting a considerable motional freedom and rapid chemical exchange between local environments on the NMR timescale. The emerging use of multicomponent FT-PGSE diffusometry at about the same point in time could not have been timelier. It proved to conveniently provide detailed and organizationally related information on all constituents of such systems [45–47]. 3.6.5.1 Objectives As described in previous sections, organization of molecules of surfactant type into larger aggregates like micelles in aqueous solution is coupled to a considerably lower constituent diffusional rate – typically by a magnitude. Any microemulsion component that gets incorporated into aggregates of significant size will exhibit a markedly lower self-diffusion rate. The limiting cases would appear as depicted in Figure 3.24. Intermediate behavior is conceptually more important. Consequently, systematic studies of self-diffusion over wide ranges of relative constituent concentrations were undertaken. Of great help here was previous charting by others of phase boundaries in such complex systems. This basically requires four dimensions and has to be graphically presented in a framework of tetrahedral or component-ratio constrained variants of the phase diagram example illustrated in Figure 3.15. 3.6.5.2 Experimental aspects Such studies are basically very easy. Concentrations are high, and normally there would be at least one isolated proton NMR signal for each compound. For field shimming convenience, dynamic range compression and field/frequency lock purposes, and use of heavy water rather than normal is recommended. In some of the studies that will be described later, monitoring of surfactant diffusion became problematic or impossible because of relatively short transverse spin relaxation in Hahn-echo-type measurements with relatively low magnetic field gradient capability. Possibly, the more rapid spin relaxation is an indication for the formation of larger aggregates than simple

“Water”

“Oil”

“Oil”

“Water”

D/10–10 m1 s–1

3.6 Surfactant systems

8 7 6 5 4 3 2 1

133

Toluene Water

Alcohol SDS 1 2 3 4 5 6 7 8 Number of alcohol carbons

Figure 3.24: Limiting compartmentation and resulting motional constraints for water/oil (W/O) and oil/water (O/W) organized systems in solution. Components making up or are mainly situated inside the “droplets” would diffuse at a low rate; those in the “continuous environment” would diffuse relatively freely. Those that reside in both environments to some significant extent will acquire self-diffusion coefficients that are time averaged with population factors just like micellar solubilization, as discussed in Section 3.5.3. Components located primarily at a distinct interface would be expected to diffuse slowly. The right-hand part shows proton PGSE-based self-diffusion data at 25 °C for SDS/water/toluene/n-alcohols of various lengths, at relative weight fractions 17.5:35:12.5:35; see the original paper for actual data and more details [49]. It is redrawn by permission from the original publication; Stilbs, P. et al. J Colloid Interface Sci. 1983; 95:583–585 Copyright (1983) Elsevier.

spherical micelles. On a modern PGSE setup, more detailed measurements could likely be made at more compressed timing parameter settings, using a stimulated-echo sequence and higher gradient settings. It was initially found that adding short- to medium-chain alcohols to micellar ones, containing trace amounts of TMS as a marker and “hydrocarbon diffusion reference” caused considerable structural changes from distinct micelles [48]. TMS diffusion in SDS micellar solution increased tenfold after gradually adding an equal weight of n-butanol to the SDS already present in aqueous solution, while butanol diffusion rates remained essentially constant. SDS diffusion increased two to five times. Water diffusion decreased to some 30%, which here will represent a time-averaged value of water and butanol diffusion. This initial study strongly indicated that solution structure had been altered from distinct “oil-in-water” micellar droplets to something best described as “disrupted” micelles with less rigidity toward structural deformation, resulting in lower barriers toward translational motion of components. Continuing from this micellar system to a microemulsion situation with simultaneously high concentrations of hydrocarbon, water, surfactant and alcohol cosurfactant confirmed this general view. Of interest was also to chart the influence of alcohol chain length. Intuitively, being more “water-like,” methanol and ethanol should be less effective “structure-breakers” than butanol. On the other hand, longer alcohols like n-heptanol and n-octanol would likely assist in stabilizing micelles with regard to structural rigidity. In this relative concentration range, the system would then rather turn into inverted micellar form, that is, become “water-in-oil”-

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like in a structural perspective. This was all nicely confirmed by experiments [49] (see Figure 3.25). Note that intermediate structural fragments may remain large and be even larger than micelles. This is despite an apparent increase in self-diffusion rate also for the primary structure-forming component: the surfactant. In a highly dynamic situation, with low structural barriers between regions, self-diffusion might possibly be less restricted than for the pure O/W or W/O situations. Complementary studies on component molecular reorientation (rather than overall translation) in similar systems were later made through multifield NMR spin relaxation measurements. It was concluded that the proposed “microemulsion” model was indeed valid, and that such microemulsions within their isotropic phase boundary limits can span a continuous range of structures from normal micellarlike to inverted micellar structures. The balance of forces may still be quite delicate, and a total structural inversion from O/W to W/O type can be brought about also by relatively minor changes in salinity (e.g., adding a few percentage of NaCl to the system). Other types of microemulsions than the oil/water/surfactant/alcohol ones described earlier also exist, and show similarly complex structural behavior. 3.6.5.3 Further reading Kahlweit M, Strey R, Haase D. et al. How to study microemulsions. J Colloid Interface Sci. 1987; 118: 436–453. Chevalier Y, Zemb T. The Structure of Micelles and Microemulsions. Rep Progr Phys. 1990; 53: 279–371. Söderman O, Stilbs P. NMR studies of complex surfactant systems. Progr Nucl Magn Reson Spectrosc. 1994; 26: 445–482.

3.6.6 Surfactant adsorption from aqueous solution onto surfaces and large particles Many NMR parameters like chemical shifts and spin relaxation rates undergo rather large changes upon binding phenomena of this kind [50, 51]. Adsorption of surfactants or polymers also may change the physical properties of surfaces in many ways, and has wide technical relevance. Of course, also self-diffusion characteristics of a molecule would change dramatically between solution and surface environments. PGSE should therefore be a valuable tool for quantification of such processes. Unlike some other “classical” and macroscopic techniques, NMR diffusometry would be applicable to systems in “equilibrium.” In reality, several complicating factors affect such studies. With proper understanding of such processes and their influence on NMR parameters, a wider and more detailed picture of adsorption emerges, on the other hand.

3.6 Surfactant systems

Micelle Surfactant monomer

135

Aqueous solution

Bilayer Hemimicelle

Adsorbed surfactant Solid surface Figure 3.25: Schematic illustration of various basic surfactant adsorption modes from solution. There is more or less motional freedom of alkyl chains in such a situation (investigable through nuclear spin relaxation, particularly via deuterium NMR on enriched compounds). In case of ionic surfactants, there would also be correspondingly organized counterions.

3.6.6.1 Objectives The objective was to test the feasibility of PGSE studies for quantification of adsorption phenomena from aqueous solution. The design, outcome and general discussion in the model study by Schönhoff and Söderman [52] on nonionic surfactant adsorption on relatively monodisperse polystyrene latex particles will be used as an illustration. 3.6.6.2 Experimental aspects From an NMR perspective, low signal/noise compared to other forms of spectroscopy is a recurring problem. In the context of adsorption studies, the area per weight unit may appear very large, and amount to like 100 m2 g–1 of adsorbent. In a molecular perspective, the actual adsorption area is actually quite small and could potentially become blocked with “impurities.” Anyway, one must thus deal with solutions that are less concentrated than one might wish for. The influence of a huge water peak from the sample is a further annoyance. Still other complications result through increased spin relaxation rates upon adsorption (Figure 3.25). Selecting a molecule to be adsorbed that has a “good” proton NMR spectrum generally helps a lot in NMR-based investigations, especially so in the context of PGSE self-diffusion studies. In this study, the nonionic surfactant “C12E5” (dodecyl penta(ethylene oxide)) was chosen. It is available commercially at excellent purity levels from Nikko, Japan. The ethylene oxide part has a nice and narrow proton signal, with fairly long transverse spin relaxation rates (about 460 ms). From a surfactant function point of view, the cmc in water is quite low (2.6 mg mL−1). Spherical micelles form above the cmc, but on increased concentration there is a shape shift to rod-like structures.

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Similarly, going for a well-defined adsorbent as well would be important from a data interpretation perspective. Samples studied were aqueous suspensions of (previously) surfactant-free polystyrene latex spheres (r = 101 nm), at a concentration like 1.8 vol%. Full surface coverage was judged to occur at around 1 mg mL−1 surfactant concentrations. Surfactants in the system under study occur in three different states: in an adsorption layer around the latex particles, in micellar and in monomeric form – the exchange between monomeric and micellar surfactant is fast on the NMR and PGSE timescales. Therefore, one can consider two surfactant sites A and B: Site A is constituted by monomeric and micellar surfactant, characterized by an average diffusion coefficient, DA , and a mean residence time of a surfactant in this site, τA . In site B, the adsorbed surfactant is described by a mean residence time τB in the adsorption layer and a diffusion coefficient DB , given by the diffusion of the latex particle, which in turn is predictable from the Stokes–Einstein–Sutherland eq. (2.5). The contribution from lateral diffusion of surfactant along the latex surface would be negligible, since the maximum displacement of molecules along the surface is small compared to the mean displacement of the whole particle. Kärger has described a mathematical model describing the effect of two-site exchange on the echo decay in PGSE diffusion experiments [11, 12]. The simplest limiting case of the equations was recently discussed earlier in this chapter. For a fuller treatment, see for example, sections in Price [9] (Chapter 4) and Callaghan [53] (pp. 283–292). In the short gradient pulse limit (δ  Δ), and taking into account that relaxation times might differ in both sites, the intensity of a Hahn echo is given by the still somewhat complex relations [52]: IðkÞ = pA expð − aA Þ + pB expð − aB Þ

(3:13)

where 9

 > kðDA + DB Þ + Δ T1 + T1 + τ1 + τ1 ∓ > > A B A B = 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h  aA, B =

i2 2 > 2> > > kðDA − DB Þ + Δ T1 − T1 + τ1 − τ1 + τ4Δτ > > : B A B A A B ; 8 > > >
1Þ all particles feel the effects of restriction. Here the displacement of the particle (and the echo attenuation in a PGSE experiment) becomes independent of Δ and depends only on R. The graphs at the lower section schematically depict the corresponding echo amplitude outcomes in PGSE experiments under these three conditions, where some degree of “free diffusion” would occur at lowΔ values also under conditions as in (b) and (c).

The situation rapidly becomes quite complicated indeed, when considering other geometries and effects of semipermeable barriers to diffusion, as discussed in the same publication [45] and elsewhere (see Further reading, Section 5.6.4). The SGP and GPD approximations discussed in the following sections also enter in, as well as spin relaxation influence from wall contact within the confining geometry. It is beyond the scope of this introductory book to discuss this

5.6 Influence from restricted diffusion and inherent approximations

187

complex family of matters in appropriate detail. Specialized aspects and systems are covered in Chapter 7. Selected Further reading matter is summarized in Section 5.6.4.

5.6.2 The GPD approximation The Gaussian phase approximation concept originates from the method of phase accumulation when calculating echo attenuation effects, as originally suggested by Douglass and McCall [46]. It is of no concern for studies on freely diffusing normal liquid systems, but may be a significant factor for confined liquids at diffusion restricting conditions as shown in Figure 5.6.

5.6.3 The SGP approximation The SGP limit approximation means that experimental gradient pulse lengths (δ) are so short that diffusion during those intervals can be neglected in the analysis. It is normally of no concern in PGSE studies on liquid systems. For normal experimental rf pulse timing settings (typically below 10 ms gradient durations at a 100 ms separation) and for normal solution systems, the SGP approximation is already excellent (see e.g., Jerschow and Müller [47] as was confirmed by Röding and Nydén in a recent educational paper [48]). SGP approximation violation often becomes a significant factor for a liquid phase confined in porous systems, as pointed out by Blees [49]. If the length scale of pores corresponds to the molecular mean square displacement during the gradient pulse, the measured echo intensities become a function of the gradient pulse length, and provide a tool for characterizing such structures in some detail (see e.g., Malmborg et al. [50] or the textbook by Sørland [38] for some actual experimental studies). Violation of the SGP approximation will occur if the following relation is not met [51]. R broadly represents the “pore radius” and D0 the bulk self-diffusion coefficient of the pore-filling liquid under study. δ

R2 D0

(5:5)

Price et al. conclude that in the application of NMR diffusometry to ‘‘real’’ systems, achieving finite gradient pulses are probably not the major difficulty. Rather, effects because of inhomogeneities in pore size and interference from background gradients are more likely to be a more serious problem, since these tend to wash out the details in the echo decays from which structural information can be derived [52]. Price and Söderman further reviewed the SGP and GPD subjects in this context

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5 Nonobvious pitfalls and other potentially confusing elements in PGSE studies

and considered their complications through model simulations [53]. The concepts are fundamentally important for PGSE studies on porous and heterogeneous systems as further discussed in Chapter 7.

5.6.4 Further reading Wang LZ, Caprihan A, Fukushima E. The narrow-pulse criterion for pulsed-gradient spin-echo diffusion measurements. J Magn Reson Ser A. 1995; 117: 209–219. Stepišnik J. Violation of the gradient approximation in NMR self-diffusion measurements. Z Phys Chem. 1995; 190: 51–62. Appropriate Sections and Chapters of the monographs by Price [37], Callaghan [21] and Sørland [38]. Diffusion NMR of Confined Systems: Fluid Transport in Porous Solids and Heterogeneous Materials., Ed: Valiullin R. London: The Royal Society of Chemistry, 2017. Diffusion in Condensed Matter. Methods, Materials, Models., Eds: Heitjans P and Kärger J, Springer (electronic edition) 2012; 1–965 Kärger J, Ruthven DM, Theodorou DN. Diffusion in Nanoporous Materials. Wiley, 2012, pp.1–872. Vogel M. NMR studies on simple liquids in confinement. Eur Phys J-Spec Top. 2010; 189: 47–64. Valiullin R, Kärger J, Glaser R. Correlating phase behaviour and diffusion in mesopores: Perspectives revealed by pulsed field gradient NMR. Phys Chem Chem Phys. 2009; 11: 2833–2853. Waldeck AR, Kuchel P, Lennon A, Chapman BE. NMR Diffusion Measurements to Characterise Membrane Transport and Solute Binding. Progr Nucl Magn Reson Spectrosc. 1997; 30: 39–68.

5.7 Less obvious or widely misconceived concepts in the context of NMR diffusometry Over the years, various strange concepts and “re-inventions of the wheel” have been introduced in NMR and also MRI. The realm of PGSE-based self-diffusion studies is no exception, and some examples have already been discussed previously. At times, they seem unintentional and a consequence of not being familiar with older literature. Others can be suspected to originate from attempts to patent some procedure, making it appear to be “new.”

5.7.1 “Affinity NMR” In the late 1990s several papers using this notation appeared [54–59]. Basically, the idea relates to simultaneous screening a mixture of “drug binding candidates” for

5.7 Less obvious or widely misconceived concepts in the context of NMR diffusometry

189

strong binding affinity for a particular substrate. Basically, this refers to a variant of PGSE binding studies, intended for qualitative ‘‘drug discovery.’’ Here one often studies multicomponent mixtures of small molecules (‘‘substrates’’), binding to macromolecules or some semisolid support of chromatography-like structure, containing the binding environment of interest on its surface. One then proceeds to choose experimental PGSE parameters that are carefully adjusted, so that in the absence of the target-binding environment, all of the substrate molecule PGSE intensities are suppressed to the baseline noise level (but preferably no further). Then, for the same PGSE settings, the experiment is repeated on a sample with the target-binding environment present. In the case of actual binding of any of the substrates to the macromolecule, their effective (time-averaged) self-diffusion rate of that substrate type will thus become lower, and the diffusion-related attenuation will be smaller. The corresponding signal(s) will consequently rise from the PGSE baseline noise for this reason, allowing identification. An obviously possible complication to this approach is a parallel decrease in T2 of the substrate molecule NMR signals upon binding, causing spin-relaxation–related PGSE signal attenuation, rather than enhancement through reduced self-diffusion rate. With properly tuned PGSE timing and gradient settings, such competing and oppositely acting echo attenuation effects can be suitably eliminated or minimized. A more recent paper of this kind uses suitably covered nanoparticles as binding substrate [60]. One can anticipate that related and unpublished experimental designs are in use within the general field in pharmaceutical industry, as a tool under under the generic concept “combinatorial chemistry,”which nowadays is extremely important for the pharmaceutical industry. However, from a basic PGSE NMR perspective, there is nothing really new. 5.7.1.1 Further reading Garrido L, Beckmann N, Eds. New Applications of NMR in Drug Discovery and Development. The Royal Society of Chemistry, 2013.

5.7.2 “DOSY” and “DOSY Maps” The concept of “DOSY” in spectroscopic NMR diffusometry has already been mentioned, and will be further examined in Chapter 6. Especially newcomers in later decades have became confused to believe that DOSY is “the” method for measurement of molecular self-diffusion – and they want “to do DOSY” on some system. DOSY is a display mode for PGSE results, rather than a method – and is basically founded on statistical constructs. Increasing numbers of DOSY studies have also emerged, where the authors seem totally unaware of the fact that the closely the same or similar systems were

190

5 Nonobvious pitfalls and other potentially confusing elements in PGSE studies

studied by PGSE NMR self-diffusion techniques decades ago. Probably contributing to such unfortunate mistakes would be too limited literature search scans, for example, only combining DOSY and “the system in question,” excluding “selfdiffusion,” “PGSE,” “PFG–NMR” or similar as search words. In the same context, as a newcomer one can lead to conceive that NMR-based self-diffusion studies do require more complex measurement procedures than in reality. As discussed in previous sections and chapters, basic NMR diffusometry on solution samples can be trivially simple, once the equipment has been set up – provided certain experimental pitfalls are avoided. In my own opinion, spectrometer manufacturers are largely to blame, probably using “DOSY capability” and catchy 2D-like illustrations as sales arguments. In addition, actual spectrometer manuals for pulsed magnetic field gradient studies may be titled “DOSY procedures”, or something similar. Vendor-supplied preprogrammed measurement procedures and pulse sequences also include complex DOSY variants of various kinds, mixing FTPGSE hybrids with conventional actual real multidimensional (e.g., COSY) signalseparating building blocks. The notation “DOSY map” has more lately appeared, referring to DOSY-type results display with some kind of self-diffusion related data on the y-axis and NMR chemical shift values on the x-axis. Without further analysis of such data, one would imply that there is some kind of “fingerprint” information in such a “map” for a particular system under study. With altered experimental settings, it may look quite different and published ones are frequently so tiny that little detail is seen anyway. They would thus seem useless for any “analytical identification” purposes or similar. The general “DOSY” concept family will likely continue to be misleading to many readers for many years to come.

5.7.3 “NMR chromatography” and “matrix-assisted NMR” NMR diffusometry studies on semisolid chromatographic material will be further discussed in discussed in Chapter 7. Indeed, there are very direct links between classic analytical or preparative fractionating chromatographic separation and adsorbent self-diffusion. Relations between separation and diffusion concepts have been known for a century or so for chromatographic media such as solids, micelles and the gas phase, and also manifest themselves in broadening of chromatographic elution peaks. In such context it was noted that adsorbent self-diffusion rates differ, primarily depending on binding affinity to the chromatographic material. Armstrong and Henry first introduced the use of micelles in high performance liquid chromatography (HPLC) in 1980 [61]. The technique is used mainly to enhance retention and selectivity of various solutes that would otherwise be inseparable or poorly resolved. Separation mechanisms were discussed in greater detail in a later paper [62].

References

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Terms such as “NMR chromatography” [63–66] or “matrix-assisted NMR” [67–70] have recently emerged, implying new concepts in PGSE NMR. The actual basic diffusion-retarding one here is (small-molecule) solute solubilization into much larger and slowly diffusing surfactant micelles, as previously described in the applications chapter 3 in this book, in Section 3.6.3. Beginning with the work by McBain a century ago, the concept of solubilization is well known and established. PGSE NMR-based studies on solubilization were made already in 1980. Some decade later they were supplemented by a merely illustrative graphical DOSY display of multicomponent NMR diffusometry data [71]. The systems chosen at the time were medium-chain alcohols in DTAB or SDS micellar solution (c.f. Fig. 3.10 and text in that context). As already known and demonstrated [72, 73], further selfdiffusion retardation as well as the enhanced “DOSY-separation” result from higher degrees of micellar incorporation for longer alcohol hydrocarbon moieties. The concept of “solubilization” is still not even mentioned in any of the “NMR chromatography” or “matrix-assisted” labeled references just cited.

5.7.4 Further reading Christian SD, Scamehorn JF, Eds. Solubilization in Surfactant Aggregates. Surfactant Science Series. 1995; 55: 1–547 (Marcel Dekker, New York).

References [1]

Antalek, B. Accounting for spin relaxation in quantitative pulse gradient spin echo NMR mixture analysis. J Am Chem Soc. 2006; 128: 8402–8403. [2] Price, W., & Kuchel, P. Effect of nonrectangular field gradient pulses in the stejskal and tanner (Diffusion) pulse sequence. J Magn Reson. 1991; 94: 133–139. [3] Price, WS. Pulsed‐field gradient nuclear magnetic resonance as a tool for studying translational diffusion: Part 1. Basic theory. Concepts Magn Reson. 1997; 9A: 299–336. [4] Merrill, MR. NMR diffusion measurements using a composite gradient PGSE sequence. J Magn Reson. 1993; 103: 223–225. [5] Sinnaeve, D. The Stejskal-Tanner equation generalized for any gradient shape-an overview of most pulse sequences measuring free diffusion. Concepts Magn Reson. 2012; 40A: 39–65. [6] Nixon, TW., McIntyre, S., Rothman, DL., & De Graaf, RA. Compensation of gradient-induced magnetic field perturbations. J Magn Reson. 2008; 192: 209–217. [7] Mansfield, P., & Chapman, B. Active magnetic screening of coils for static and timedependent magnetic field generation. J Phys. E 1986; 19: 540–545. [8] Mansfield, P., & Chapman, B. Active magnetic screening of gradient coils in NMR imaging. J Magn Reson. 1986; 66: 573–576. [9] Poole, M., & Bowtell, R. Novel gradient coils designed using a boundary element method. Concepts Magn Reson. 2007; 31B: 162–175. [10] Poole, M., Lopez, HS., & Crozier, S. Adaptively regularized gradient coils for reduced local heating. Concepts Magn Reson. 2008; 33B: 220–227.

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[11]

Goora, FG., Colpitts, BG., & Balcom, B. Arbitrary magnetic field gradient waveform correction using an impulse response based pre-equalization technique. J Magn Reson. 2014; 238: 70–76. Wu, D., Chen, A., & Johnson, CS. An improved diffusion-ordered spectroscopy experiment incorporating bipolar-gradient pulses. J Magn Reson, Ser A. 1995; 115: 260–264. Gibbs, SJ., & Johnson, CS,Jr. A PFG NMR experiment for accurate diffusion and flow studies in the presence of eddy currents. J Magn Reson. 1991; 93: 391–402. Price, WS., Elwinger, F., Vigouroux, C., & Stilbs, P. PGSE-WATERGATE, a new tool for NMR diffusion-based studies of ligand-macromolecule binding. Magn Reson Chem. 2002; 40: 391–395. Zheng, G., Stait-Gardner, T., Kumar, PGA., Torres, AM., & Price, WS. PGSTE-WATERGATE: An STE-based PGSE NMR sequence with excellent solvent suppression. J Magn Reson. 2008; 191: 159–163. Zheng, G., Torres, AM., & Price, WS. Solvent suppression using phase-modulated binomial-like sequences and applications to diffusion measurements. J Magn Reson. 2008; 194: 108–114. Hahn, EL. Radiation damping of an inhomogeneously broadened spin ensemble. Concepts Magn Reson. 1997; 9: 65–67. Mao, XA., & Ye, CH. Understanding radiation damping in a simple way. Concepts Magn Reson. 1997; 9: 173–187. Price, WS., Stilbs, P., Jönsson, B., & Söderman, O. Macroscopic background gradient and radiation damping effects on high-field PGSE NMR diffusion measurements. J Magn Reson. 2001; 150: 49–56. Price, WS., & Walchli, M. NMR diffusion measurements of strong signals: the PGSE-Q-switch experiment. Magn Reson Chem. 2002; 40: S128–S132. Callaghan, PT. Translational Dynamics & Magnetic Resonance, Principles of Pulsed Gradient Spin Echo NMR. Oxford: Oxford University Press, 2011. Swan, I., Reid, M., Howe, PW. et al. Sample convection in liquid-state NMR: why it is always with us, and what we can do about it. J Magn Reson. 2015; 252: 120–129. Carr, HY., & Purcell, EM. Effects of diffusion on free precession in nuclear magnetic resonance experiments. Phys Rev. 1954; 94: 630–638. Kiraly, P., Swan, I., Nilsson, M., & Morris, GA. Improving accuracy in DOSY and diffusion measurements using triaxial field gradients. J Magn Reson. 2016; 270: 24–30. Hedin, N., & Furó, I. Temperature imaging by H-1 nmr and suppression of convection in nmr probes. J Magn Reson. 1998; 131: 126–130. Hayamizu, K., & Price, WS. A new type of sample tube for reducing convection effects in PGSE-NMR measurements of self-diffusion coefficients of liquid samples. J Magn Reson. 2004; 167: 328–333. Kato, H., Saito, T., Nabeshima, M., Shimada, K., & Kinugasa, S. Assessment of diffusion coefficients of general solvents by PFG-NMR: Investigation of the sources error. J Magn Reson. 2006; 180: 266–273. Lounila, J., Oikarinen, K., Ingman, P., & Jokisaari, J. Effects of thermal convection on nmr and their elimination by sample rotation. J Magn Reson. 1996; 118: 50–54. Esturau, N., Sanchez-Ferrando, F., Gavin, JA., Roumestand, C., Delsuc, MA., & Parella, T. The use of sample rotation for minimizing convection effects in self-diffusion NMR measurements. J Magn Reson. 2001; 153: 48–55. Jerschow, A., & Müller, N. Convection compensation in gradient enhanced nuclear magnetic resonance spectroscopy. J Magn Reson. 1998; 132: 13–18. Sørland, GH., Seland, JG., Krane, J., & Anthonsen, HW. Improved convection compensating pulsed field gradient spin-echo and stimulated-echo methods. J Magn Reson. 2000; 142: 323–325.

[12] [13] [14]

[15]

[16] [17] [18] [19]

[20] [21] [22] [23] [24] [25] [26]

[27]

[28] [29]

[30] [31]

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[32] Momot, KI., & Kuchel, PW. Convection-compensating PGSE experiment incorporating excitation-sculpting water suppression (CONVEX). J Magn Reson. 2004; 169: 92–101. [33] Momot, KI., & Kuchel, PW. Convection-compensating diffusion experiments with phasesensitive double-quantum filtering. J Magn Reson. 2005; 174: 229–236. [34] Momot, KI., & Kuchel, PW. PFG NMR diffusion experiments for complex systems. Concepts Magn Reson. 2006; 28A: 249–269. [35] Nilsson, M., & Morris, GA. Improving pulse sequences for 3D DOSY: Convection compensation. J Magn Reson. 2005; 177: 203–211. [36] Zheng, G., & Price, WS. Simultaneous convection compensation and solvent suppression in biomolecular NMR diffusion experiments. J Biomol NMR. 2009; 45: 295–299. [37] Price, WS. NMR Studies of Translational Motion – Principles and Applications. Cambridge: Cambridge University Press, 2009. [38] Sørland, GH. Dynamic Pulsed-Field-Gradient NMR. Berlin, Heidelberg: Springer, 2014. [39] Johnson, CS,Jr. Effects of chemical exchange in diffusion-ordered 2D NMR spectra. J Magn Reson. 1993; 102: 214–218. [40] Aguilar, JA., Adams, RW., Nilsson, M., & Morris, GA. Suppressing exchange effects in diffusion-ordered NMR spectroscopy. J Magn Reson. 2014; 238: 16–19. [41] Pagès, G., Dvinskikh, SV., & Furó, I. Suppressing magnetization exchange effects in stimulated-echo diffusion experiments. J Magn Reson. 2013; 234: 35–43. [42] Woessner, DE. N.M.R. Spin-Echo Self-Diffusion Measurements on fluids undergoing restricted diffusion. J Phys Chem. 1963; 67: 1365–1367. [43] Stejskal, EO. Use of spin echoes in a pulsed magnetic-field gradient to study anisotropic restricted diffusion and flow. J Chem Phys. 1965; 43: 3597–3603. [44] Tanner, JE., & Stejskal, EO. Restricted self-diffusion of protons in colloidal systems by the pulsed-gradient, spin-echo method. J Chem Phys. 1968; 49: 1768–1777. [45] Price, WS. Gradient NMR. Ann Rep NMR Spectrosc. 1996; 32: 51–142. [46] Douglass, D., & McCall, D. Diffusion in paraffin hydrocarbons. J Chem Phys. 1958; 62: 1102–1107. [47] Jerschow, A., & Müller, N. Efficient simulation of coherence transfer pathway selection by phase cycling and pulsed field gradients in NMR. J Magn Reson. 1998; 134: 17–29. [48] Röding, M., & Nydén, M. Stejskal-Tanner equation for three asymmetrical gradient pulse shapes used in diffusion NMR. Concepts Magn Reson. 2015; 44A: 133–137. [49] Blees, MH. The effect of finite duration of gradient pulses on the pulsed- field-gradient NMR method for studying restricted diffusion. J Magn Reson. 1994; 109: 203–209. [50] Malmborg, C., Topgaard, D., & Söderman, O. NMR diffusometry and the short gradient pulse limit approximation. J Magn Reson. 2004; 169: 85–91. [51] Callaghan, PT. A simple matrix formalism for spin echo analysis of restricted diffusion under generalized gradient waveforms. J Magn Reson. 1996; 129: 74–84. [52] Price, WS., Stilbs, P., & Söderman, O. Determination of pore space shape and size in porous systems using NMR diffusometry. Beyond the short gradient pulse approximation. J Magn Reson. 2003; 160: 139–143. [53] Price, WS., & Söderman, O. Some “reflections” on the effects of finite gradient pulse lengths in PGSE NMR experiments in restricted systems. Israel J Chem. 2003; 43: 25–32. [54] Lin, M., Shapiro, MJ., & Warening, JR. Diffusion-edited NMR-affinity NMR for direct observation of molecular interactions. J Am Chem Soc. 1997; 119: 5249–5250. [55] Lin, MF., Shapiro, MJ., & Wareing, JR. Screening mixtures by affinity NMR. J Org Chem. 1997; 62: 8930–8931. [56] Shapiro, MJ. Affinity NMR. Am Lab. 1998; 30: 20–20. [57] Gonnella, N., Lin, MF., Shapiro, MJ., Wareing, JR., & Zhang, XL. Isotope-filtered affinity NMR. J Magn Reson. 1998; 131: 336–338.

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[58] Chen, A., & Shapiro, MJ. Affinity NMR. Anal Chem. 1999; 71: 669A–675A. [59] Anderson, RC., Lin, MF., & Shapiro, MJ. Affinity NMR – decoding DNA-binding. J Comb Chem. 1999; 1: 69–72. [60] Diez-Castellnou, M., Salvia, M-V., Springhetti, S., Rastrelli, F., & Mancin, F. NanoparticleAssisted affinity NMR spectroscopy: high sensitivity detection and identification of organic molecules. Chem Eur J. 2016; 22: 16957–16963. [61] Armstrong, DW., & Henry, SJ. Use of an aqueous micellar mobile phase for separation of phenols and polynuclear aromatic hydrocarbons via HPLC. J Liq Chromatogr. 1980; 3: 657–662. [62] Armstrong, DW., Ward, TJ., & Berthod, A. Micellar effects on molecular diffusion: theoretical and chromatographic considerations. Anal Chem. 1986; 58: 579–582. [63] Kavakka, JS., Parviainen, V., Wahala, K., Kilpelainen, I., & Heikkinen, S. Enhanced chromatographic NMR with polyethyleneglycol. A novel resolving agent for diffusion ordered spectroscopy. Magn Reson Chem. 2010; 48: 777–781. [64] Pemberton, C., Hoffman, R., Aserin, A., & Garti, N. New insights into silica-based NMR “chromatography”. J Magn Reson. 2011; 208: 262–269. [65] Pemberton, C., Hoffman, RE., Aserin, A., & Garti, N. NMR chromatography using microemulsion systems. Langmuir. 2011; 27: 4497–4504. [66] Heisel, KA., Goto, JJ., & Krishnan, VV. NMR chromatography: molecular diffusion in the presence of pulsed field gradients in analytical chemistry applications. AJAC. 2012; 03: 401–409. [67] Tormena, CF., Evans, R., Haiber, S., Nilsson, M., & Morris, GA. Matrix-assisted diffusionordered spectroscopy: mixture resolution by NMR using SDS micelles. Magn Reson Chem. 2010; 48: 550–553. [68] Adams, RW., Aguilar, JA., Cassani, J., Morris, GA., & Nilsson, M. Resolving natural product epimer spectra by matrix-assisted DOSY. Org Biomol Chem. 2011; 9: 7062–7064. [69] Cassani, J., Nilsson, M., & Morris, GA. Flavonoid mixture analysis by matrix-assisted diffusion-ordered spectroscopy. J Nat Prod. 2012; 75: 131–134. [70] Evans, R., Hernandez-Cid, A., Dal Poggetto, G. et al. Matrix-assisted diffusion-ordered NMR spectroscopy with an invisible matrix: a vanishing surfactant. RSC Adv. 2017; 7: 449–452. [71] Morris, KF., Stilbs, P., & Johnson, CS. Analysis of mixtures based on molecular size and hydrophobicity by means of diffusion-ordered 2D NMR. Anal Chem. 1994; 66: 211–215. [72] Stilbs, P. Fourier transform NMR pulsed-gradient spin-echo (FT-PGSE) self- diffusion measurements of solubilization equilibriums in SDS solutions. J Colloid Interface Sci. 1982; 87: 385–394. [73] Söderman, O., & Stilbs, P. NMR studies of complex surfactant systems. Prog Nucl Magn Reson Spectrosc. 1994; 26: 445–482.

6 Data preparation, evaluation and presentation

[A] = 12.4 mM in D2O [B] = 50.3 mM in D2O DA = 1.95 ± 0.05·10–11 m2 s –1 DB = 4.95 ± 0.07·10–11 m2 s –1 @ T = 298.5 ± 0.5 K

(a)

Self-diffusion coeff icient / m2 s–1

Based on the actual application, PGSE strategies may differ somewhat in appearance or aim. Two main families of PGSE-type applications have emerged (Figure 6.1): (a) To determine one or more sample self-diffusion coefficients to good precision, in well-defined systems – for further physicochemical interpretation in terms of theoretical considerations or quantification through association constants and related quantities. (b) To achieve spectral signal separation in PGSE data sets, based on or aided with differing component self-diffusion. Samples here may have partly unknown composition, actual experimental precision is of minor concern and there are rarely implied physicochemical questions. Often the data analysis is here presented in 2D graphical form: variants of a so-called DOSY (Diffusion Ordered (NMR) SpectroscopY) display.

(b)

Chemical shift

Figure 6.1: PGSE studies of type “(a)” and “(b)”, respectively (see text), are illustrated. For case “(a)”, the information could constitute a basis for something like a determination of an association constant between A and B, or some other physicochemical quantity. For “(b)”, the main or only goal may have been to achieve an analytically related component signal separation, based on or aided by PGSE NMR.

Of course, “(a)” and “(b)” would share some elements of proper experimental design and data processing fundamentals. Key problems and limitations for both may also become apparent (or overlooked) already for two sample components. Unless there is significantly bad signal/noise in the experimental data set, routine PGSE data evaluation in a case of type “(a)” should often be a simple matter. The basic procedures were already illustrated in Figure 2.30. An analysis using standard spectrometer-operating software is normally based on peak heights or integrals for a single spectral peak, under conditions of negligible or no overlap with other spectral components. Already vendor-supplied software should properly cope up with https://doi.org/10.1515/9783110551532-006

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6 Data preparation, evaluation and presentation

nonlinear data evaluation in such simple situations, using nonlinear fitting procedures. Linearized semilogarithmic “Stejskal–Tanner plots” are obsolete alternatives within today’s computerized spectrometer environment, but remain useful for illustrating PGSE concepts of various kinds. In a multicomponent situation, things may get more or less complicated. By tuning instrumental and pulse program settings, some problematic factors may get virtually eliminated. For a solvent diffusion rate a magnitude or so greater than that that of solutes’ “peak elimination” is often simply achievable by extending the timing and gradient settings of the basic PGSE or PGSTE sequences somewhat. When such an approach is out of range, or if there is too much signal loss from competing spin relaxation echo damping, PGSE-type WATERGATE variants [1] may still do the job very efficiently. Such an approach works even with relatively similar component–solvent diffusion rates. A third and always recommended variant is to simply exclude irrelevant spectral areas (like the solvent region) in the analysis. With increasing component complexity and spectral overlap in a mixed system under investigation, the situation rapidly gets worse with regard to reliable determination of individual self-diffusion coefficients. Going for a nonconstrained, full data analysis of a composite peak may be problematic already in a situation with only two overlapping components of similar size. It has been known for ages (but still seems to be rediscovered periodically by individuals working in various disciplines) that such data fitting will only work if the two “time constants” differ by a magnitude or so, and that data sampling and signal/noise are good as well. It also helps if the component contributions are of similar amplitude. The situation regarding data analysis in PGSE context is brighter than for “single-channel” information in the form of time-varying peak height data only, influenced by random additional noise. One key factor is that signals are more or less frequency resolved and can be analyzed separately. Also, individual component signal information is not contained in single-frequency data (like peak height), but occurs across its entire bandshape (Figure 6.2). Figure 6.3 illustrates the basic difficulties. An extensive review on the general subject appeared some decades ago [2], as applied to purely exponential function fitting of single-channel data (like peak heights in spectroscopy). “PGSE exponentials” discussed here are analytically different (typically exp(–t2) instead of exp(–t)), but behave analogously in a data-fitting context, producing the same type of artifacts and having similar noise sensitivity. A typical PGSE data set may typically comprise some 100 thousands of data points, in a readily available digitized form. The spectral amplitude of a peak at its central frequency could basically represent its “intensity.” Using only the “peak heights” would be a waste of information, since individual PGSE peak data would normally include like 10–100 or so “intensities” along its entire bandshape. Peak amplitude finding spectrometer software may also introduce systematic errors, by basing peak data on the largest amplitude in a small frequency range near the

6 Data preparation, evaluation and presentation

197

12,000 10,000

Arbitrary intensity

8,000 6,000 4,000 2,000 0

0

20

40

60 80 Arbitrary frequency

100

120

Figure 6.2: Typical sampling of an isolated NMR peak (synthetic).

Arbitrary intensity

Difference

100 50 0 –50

0

500

1,000

1,500

2,000 Data point #

2,500

3,000

3,500

4,000

500

1,000

1,500

2,000 Data point #

2,500

3,000

3,500

4,000

10,000 5,000 0 0

Figure 6.3: PGSE overlap in an unresolved synthetic case of three equally large signal contributions (with imaginary “diffusion coefficients” 1.5, 2.0 and 2.5⋅10−11 m2 s−1), a common bandwidth, some slight white noise and reasonably correct sampling. The data consist of 128-point bandshape attenuations at 32 gradient settings (blue) and are displayed in 4,096-point sequential form. The situation corresponds to that of the frequency-resolved PGSE cases discussed later, except that only peak heights at the peak maximum (red) are used in the data fitting. Component separation of such unresolved data is not possible, already at such minute contributions from signal noise ( Χi

Figure 7.6: Magnetic field lines around suspended spherical cell-like structures in a continuous medium in a strong external magnetic field. This Gauss’s law for magnetism Mathematica-based simulation figure is a kind contribution to this book by P.W. Kuchel, Sydney. Note that there are significant magnetic field gradients near the outer surface and ideally none inside. χ denotes the respective magnetic susceptibilities. Also, signals from inner and outer regions of a given solution constituent will differ with regard to chemical shifts.

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7 Specialized measurements and systems

7.5.1 Biological systems Although there may be interconnecting channels, most “walls” in the context of “heterogeneous systems” are normally impenetrable on the PGSE NMR time scale. For biological systems this need not be the case, and inflow and outflow kinetics can be highly desirable system parameters to investigate from a chemical point of view. An early study by Brindle and coworkers on kinetics for transmembrane transport was based on the “natural magnetic field gradient” effects [37] just mentioned (cf. Figure 7.6). Here model compounds like alanine were added to a solution of a suspension of erythrocytes in water, which gradually made their way through the cell membranes into the cell interior. The growth of those signals was followed through proton spin-echo NMR at 2-min intervals for approximately half an hour, and thus quantifying cell influx rates [37, 38]. These particular applications look easy in principle, but are plagued by problems such as the need for oxygenation of cell suspensions to keep them alive, and related measurement problems from stirring-induced flow in the sample. Kuchel was one of the coauthors of the Brindle paper, who later characterized many aspects of structure, shape, binding and inflow/outflow kinetics of cell-like systems through a number of diffusion-related NMR techniques. This area has been summarized in an overview in the year 1997 [39] and in a recent review paper focused on the “split peak effect” of solutes in biological cell-like systems [40] and kinetic considerations.

7.5.2 Probing periodic structures through “diffusive diffraction” effects in PGSE As discussed earlier and illustrated in Figure 5.6, geometric confinement to a space that is smaller than the average diffusional displacement for unrestricted diffusion during the same time corresponds to non-Gaussian diffusion and requires modified data evaluation in PGSE experiments. Such data provide a direct relationship to some geometric characterization of the system in question, but the physics and mathematics is not altogether straightforward. Usually, one considers experimental conditions where the short gradient pulse (SGP) approximation is reasonably valid. Here the gradient pulse lengths (δ) are thought to be much shorter than their separation (Δ). The (normalized) echo attenuation (EΔ ðqÞ) is given by ð (7:3) EΔ ðqÞ = PðZ, ΔÞ expði2πqZÞdZ where PðZ, ΔÞ represents the “average propagator” [41], that is, the probability that a molecule has moved a displacement Z during a time Δ, irrespective of its starting position. In analogy with light and neutron scattering [42] one here introduces a wave vector q(m−1), given as follows [4, 43]:

7.5 PGSE NMR diffusometry in heterogeneous or microheterogeneous systems

259

Figure 7.7: Two-dimensional representations of periodic structures of two schematic kinds, reflecting planes and dispersed spheres (yellow, liquid phase; orange, impenetrable solid phase). Common variants in nature are pores with interconnecting channels or partly penetrable walls.

q=

γgδ 2π

(7:4)

Even under the SGP approximation, the evaluation of eq. (7.3) becomes quite complicated or analytically intractable, also for quite simple geometries. A case studied already by Tanner and Stejskal in 1968 was diffusion between planes, separated by a distance a with the gradient direction perpendicular to the planes [44]: EΔ ðqÞ =

2½1 − cosð2πqaÞ ð2πqaÞ

2

+ 4ð2πqaÞ2 ×

∞ X n=1

expð −

n2 π2 DΔ 1 − ð − 1Þn cosð2πqaÞ Þ h i2 (7:5) a2 ð2πqaÞ2 − ðnπÞ2

At long Δ, the second term in becomes negligible compared to the first one, and periodic (“diffractive”) minima corresponding to the first term are seen to appear at q = n=aðn = 1, 2, 3, ...Þ (Figure 7.8). The corresponding expression including effects of wall relaxation can be found elsewhere [36]. Price and coworkers also considered PGSE interpretation of various similar systems, beyond the SGP approximation [45]. It should also be noted that the observed “diffraction patterns” could be numerically Fourier transformed to yield the actual PðZ, ΔÞprobability distribution, as illustrated in Figure 7.9. Highly valuable general discussions on the Fourier transform relationship between PGSE echo attenuation curves and molecular displacement probabilities was given by Callaghan et al. and Cory and Garroway in seminal papers [46, 47]. One should also note that the experimental outcome of (Gaussian) free diffusion in PGSE follows the form expected, that is, a Gaussian relation like E = expð − γ2 g2 δ2 D · ðΔ − δ=3ÞÞ, since the Fourier transform of Gaussian-related data is also generally a Gaussian-shaped curve.

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7 Specialized measurements and systems

E(q, Δ)

1 0.5 0 0

0.5

1

1.5

2

2.5

3

3.5

4

0

0.5

1

1.5

2 q

2.5

3

3.5

4

E(q, Δ)

100 10–5 10–10

100

1.0 P(Z,t) (normalized)

EΔ(q)

Figure 7.8: “Diffraction patterns” simulated based on q-variation for the first term in eq. (7.5), when using a nominal model value of a = 1, and displayed on linear and logarithmic y-axis scales. In a real-world situation, factors like surface relaxation and size polydispersity also affect these patterns and generally tend to average out distinct details in them.

10–1 10–2 10–3 0.0 –4

10

0.0 0.5

1.0

1.5 2.0

q(105 × m–1)

2.5

–200

–100

0

+100

+200

Z (μm)

Figure 7.9: Diffusive diffraction patterns observed for a circular water-filled slit in a Shigemi NMR tube (c.f. Figure 5.4). It was externally measured to have 128 μm thickness, well corresponding to the pattern observed and fitted under the SGP approximation (solid curve). Δ was kept constant at 2 s, while applying various gradient strength settings 318 K (left, blue circles). The average propagator PðZ, ΔÞobtained by numerical Fourier inversion of data is closely similar to those in the left Figure section is illustrated to the right. Here the separation was 112 μm and δ was varied as well. The curves correspond to δ= 2 (blue), 100 (magenta) and 200 ms (green). Redrawn, with permission from Price et al. J Magn Reson. 2003; 160:139–143 [45]; Copyright (2003) Elsevier.

7.5.3 Suppressing “natural” background magnetic field gradients in heterogeneous systems Internal magnetic field gradients in samples of heterogeneous systems of various types often occur for several reasons, including diamagnetic susceptibility

7.6 Spatially localized NMR diffusometry and NMR Microscopy

261

variations and contributions from paramagnetic impurities. However, some families of more complex pulse sequences, originally devised by Karlicek and Lowe [48] can compensate for such extraneous echo attenuation. Common variants are suggested by Cotts and coworkers [49] and Latour and coworkers [50]. In general, the key internal gradient-reducing building blocks are bipolar gradient pulse pairs that are sandwiched around 180° rf pulses. Sørland’s textbook, in particular, covers extensive discussions of experimental studies, design and use of such pulse sequences in the context of studies of technical–industrial systems [51].

7.5.4 Further reading Callaghan P, Stepišnik J. Generalized analysis of motion using magnetic field gradients. Adv Magn Opt Reson. 1996; 19: 325–388. Porous media-related and associated methodological Chapters in the NMR diffusometry monographs by Price, Callaghan and Sørland, mentioned in the Preface. The Wikipedia entry on Nuclear Magnetic Resonance in porous media. Diffusion NMR of Confined Systems: Fluid Transport in Porous Solids and Heterogeneous Materials. Ed: Valiullin R., London: The Royal Society of Chemistry, 2017. Sen PN. Time-dependent diffusion coefficient as a probe of geometry. Concepts Magn Reson. 2004; 23A: 1–21. Burada PS, Hanggi P, Marchesoni F, Schmid G, Talkner P. Diffusion in confined geometries. ChemPhysChem. 2009; 10: 45–54. Koay CG, Özarslan E. Conceptual foundations of diffusion in magnetic resonance. Concepts Magn Reson. 2013; 42A: 116–129. Jeener J. Macroscopic molecular diffusion in liquid NMR, revisited. Concepts Magn Reson. 2002; 14: 79–88. Chapters on porous systems in Blümich B, Haber-Pohlmeier S, Zia W. Compact NMR. Berlin/Boston: De Gruyter, 2014.

7.6 Spatially localized NMR diffusometry and NMR Microscopy Stray field diffusometry, briefly described in Section 7.2 is inherently localized, since only a sample slice at a time contributes to an NMR signal. Similarly, localized excitation can be achieved by using pulsed gradients, perhaps in combination with frequency-selective rf pulses (usually sinc-shaped). Such procedures can be combined with diffusion-sensitive or other elements, as discussed in Section 7.2. They may be needed in specialized applications, although such a case is relatively rare and inherently leads to substantial signal loss compared to normal full sample PGSE detection. For single-component systems at high concentrations (like polymer

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melts or single liquids confined to a porous matrix), from a sensitivity point of view, this need not be any significant problem. A recent and conceptually odd application of slice-selected and diffusion-related NMR is to monitor diffusion in solution through spatial quantification of solute concentrations in a vertical tube while relaxing concentration gradients in it – just like in “classical” optical diffusion measurements based on light absorption or refractive index changes [52]. A four-component resolution was achieved in this study and the results were summarized in DOSY-type form. The concepts could be useful in specialized applications (liquid phase separation was an example study in the paper) (Figure 7.10).

(a)

τ

1H

G1

2τ

π G2

τ

π G1

G3

G2

G3

Gz B0 Slice thickness Δ z = Δω 𝛾G2

15 10

Ω1 Ω2

z-position Ωn–1 Ωn Ω z= 𝛾G2

5 0 –5 –10

Z-position (mm)

(b)

–15 0.00.2 0.4 0.6 0.8 1.0 Normalized Intensity

Figure 7.10: A figure from Pantoja et al. [52], illustrating the slice-selective pulse sequence used, and the conceptual measurement idea. Slice selection is achieved in the pulse sequence via the shaped π-pulses during the gradient influence denoted G2 in the figure, but self-diffusion sensitive elements are not used. Reproduced with permission from Pantoja et al. Magn Reson Chem. 2017; 55: 519–524 Copyright (2017) Wiley.

Having access to triple-gradient (x, y, z) facilities, self-diffusion (as well as flow) can also be imaged and quantified spatially [46, 53]. Later, such measurements have in become hugely important in medical MRI, for 3D-like monitoring blood flow in human brains and other organs. This topic was reviewed in a multiauthor book that extensively covers historical and methodological developments in NMR magnetic field gradient techniques as well [54].

References

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7.6.1 Further reading Callaghan PT. Principles of Nuclear Magnetic Resonance Microscopy. Oxford: Oxford University Press, 1991. Callaghan PT. Translational Dynamics & Magnetic Resonance, Principles of Pulsed Gradient Spin Echo NMR. Oxford: Oxford University Press, 2011. Topgaard D. Multidimensional diffusion MRI. J Magn Reson. 2017; 275: 98–113.

References [1] [2]

[3]

[4] [5] [6] [7] [8] [9] [10] [11] [12]

[13] [14] [15] [16]

Canet, D. Radiofrequency field gradient experiments. Prog Nucl Magn Reson Spectrosc. 1997; 30: 101–135. Guendouz, L., Leclerc, S., Retournard, A., Hedjiedj, A., & Canet, D. Single-sided radiofrequency field gradient with two unsymmetrical loops: Applications to nuclear magnetic resonance. Rev Sci Instrum. 2008; 79: 23704. Kimmich, R., Unrath, W., Schnur, G., & Rommel, E. NMR Measurement of Small SelfDiffusion Coefficients in the Fringe-Field of Superconducting Magnets. J Magn Reson. 1991; 91: 136–140. Callaghan, PT. Translational Dynamics & Magnetic Resonance, Principles of Pulsed Gradient Spin Echo NMR. Oxford: Oxford University Press, 2011. Hurlimann, MD. Diffusion and relaxation effects in general stray field NMR experiments. J Magn Reson. 2001; 148: 367–378. Geil, B. Measurement of translational molecular diffusion using ultrahigh magnetic field gradient NMR. Concepts Magn Reson. 1998; 10: 299–321. Mcdonald, PJ. Stray field magnetic resonance imaging. Prog Nucl Magn Reson Spectrosc. 1997; 30: 69–99. Smith, K., Burbidge, A., Apperley, D. et al. Stray-field NMR diffusion q-space diffraction imaging of monodisperse coarsening foams. J Colloid Interface Sci. 2016; 476: 20–28. Wu, D., & CS, Johnson., Jr. Diffusion-ordered 2D NMR in the fringe field of a superconducting magnet. J Magn Reson Ser A. 1995; 116: 270–272. Wu, DH., & CS, Johnson., Jr. Fresh spins for nmr signal enhancement through programmed sample translation cycles. J Magn Reson. 1997; 127: 225–228. Packer, KJ., Rees, C., & Tomlinson, DJ. Modification of the pulsed magnetic field-gradient spin echo method of studying diffusion. Mol Phys. 1970; 18: 421–423. Stepišnik, J., Mohorič, A., Mattea, C., Stapf, S., & Serša, I. Velocity autocorrelation spectra in molten polymers measured by NMR modulated gradient spin-echo. EPL (Europhysics Letters). 2014; 106: 27007. Stepišnik, J., Mattea, C., Stapf, S., & Mohorič, A. Molecular velocity auto-correlation of simple liquids observed by NMR MGSE method. Eur. Phys. J. B (2018); 91: 293. Callaghan, PT., & Stepišnik, J. Frequency-domain analysis of spin motion using modulatedgradient NMR. J Magn Reson. 1995; 117: 118–122. Callaghan, P., & Stepišnik, J. Generalized Analysis of Motion Using Magnetic Field Gradients. Adv Magn Opt Res. 1996; 19: 325–388. Stepišnik, J., & Callaghan, PT. Low-frequency velocity correlation spectrum of fluid in a porous media by modulated gradient spin echo. Magn Reson Imaging. 2001; 19: 469–472.

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[17] Stepišnik, J., Lasic, S., Mohorič, A., Serša, I., & Sepe, A. Spectral characterization of diffusion in porous media by the modulated gradient spin echo with CPMG sequence. J Magn Reson. 2006; 182: 195–199. [18] Stepišnik, J., Mohorič, A., Serša, I., & Lahajnar, G. Analysis of Polymer Dynamics by NMR Modulated Gradient Spin Echo. Macromol Symp. 2011; 305: 55–62. [19] Caprihan, A., Wang, LZ., & Fukushima, E. A multiple-narrow-pulse approximation for restricted diffusion in a time-varying field gradient. J Magn Reson. 1996; 118: 94–102. [20] Andrew, ER., & Szczesniak, E. A historical account of NMR in the solid state. Prog Nucl Magn Reson Spectrosc. 1995; 28: 11–36. [21] Andrew, ER. Magic Angle Spinning. In: eMagRes http://dx.doi.org/10.1002/9780470034590. emrstm0283, John Wiley & Sons, Ltd, 2007. [22] Maas, WE., Laukien, FH., & Cory, DG. Gradient, high resolution, magic angle sample spinning NMR. J Am Chem Soc. 1996; 118: 13085–13086. [23] Viel, S., Ziarelli, F., & Caldarelli, S. Enhanced diffusion-edited NMR spectroscopy of mixtures using chromatographic stationary phases. Proc Natl Acad Sci USA. 2003; 100: 9696–9698. [24] Pages, G., Delaurent, C., & Caldarelli, S. Investigation of the chromatographic process via pulsed-gradient spin-echo nuclear magnetic resonance. Role of the solvent composition in partitioning chromatography. Anal Chem. 2006; 78: 561–566. [25] Viel, S., Ziarelli, F., Pages, G., Carrara, C., & Caldarelli, S. Pulsed field gradient magic angle spinning NMR self- diffusion measurements in liquids. J Magn Reson. 2008; 190: 113–123. [26] Caldarelli, S. Chromatographic NMR: a tool for the analysis of mixtures of small molecules. Magn Reson Chem. 2007; 45: S48–S55. [27] Viel, S., Ziarelli, F., Pages, G., Carrara, C., & Caldarelli, S. Pulsed field gradient magic angle spinning NMR self-diffusion measurements in liquids. J Magn Reson. 2008; 190: 113–123. [28] Zimmerman, JR., & Brittin, WE. Nuclear magnetic resonance studies in multiple phase systems: Lifetime of a water molecule in an adsorbing phase on silica gel. J Phys Chem. 1957; 61: 1328–1333. [29] Resing, HA. Nuclear magnetic resonance relaxation of molecules absorbed on surfaces. Adv Mol Relax Proc. 1968; 1: 109–154. [30] Brown, RJS., Chandler, R., Jackson, JA. et al. History of NMR well logging. Concepts Magn Reson. 2001; 13: 335–413. [31] Mitchell, J., Webber, JB., & Strange, JH. Nuclear magnetic resonance cryoporometry. Phys Rep. 2008; 461: 1–36. [32] Sagidullin, AI., & Furó, I. Pore size distribution measurements in small samples and with nanoliter volume resolution by NMR cryoporometry. Langmuir. 2008; 24: 4470–4472. [33] Petrov, OV., & Furó, I. NMR cryoporometry: Principles, applications and potential. Prog Nucl Magn Reson Spectrosc. 2009; 54: 97–122. [34] Brownstein, KR., & Tarr, CE. Spin-Lattice Relaxation in a System Governed by Diffusion. J Magn Reson. 1977; 26: 17–24. [35] Brownstein, KR., & Tarr, CE. Importance of Classical Diffusion in NMR Studies of Water in Biological Cells. Phys Rev. 1979; A 19: 2446–2453. [36] Callaghan, PT. Pulsed-Gradient Spin-Echo NMR for Planar, Cylindrical, and Spherical Pores Under Conditions of Wall Relaxation. J Magn Reson Ser A. 1995; 113: 53–59. [37] Brindle, KM., Brown, FF., Campbell, ID., Grathwohl, C., & Kuchel, PW. Application of spinecho nuclear magnetic resonance to whole-cell systems. Biochem J. 1979; 180: 37–44. [38] Brindle, KM., & Campbell, ID. NMR studies of kinetics in cells and tissues. Quart Rev Biophys. 1987; 19: 159–182.

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[39] Waldeck, AR., Kuchel, PW., Lennon, AJ., & Chapman, BE. NMR diffusion measurements to characterize membrane transport and solute binding. Prog Nucl Magn Reson Spectrosc. 1997; 30: 39–68. [40] Kuchel, PW., Kirk, K., & Shishmarev, D. The NMR ‘split peak effect’ in cell suspensions: Historical perspective, explanation and applications. Progr Nucl Magn Reson Spectrosc. 2018; 104: 1–11. [41] Kärger, J., & Heink, W. The Propagator Representation of Molecular Transport in Microporous Crystallites. J Magn Reson. 1983; 51: 1–7. [42] Johnson, CS. The Evolution of Ideas about Optical Analogies to PFGNMR and the Visualization of PFGNMR Experiments. eMagRes https://doi.org/10.1002/9780470034590.emrhp0088 2007. [43] Callaghan, PT. Principles of Nuclear Magnetic Resonance Microscopy. Oxford: Oxford University Press, 1991. [44] Tanner, JE., & Stejskal, EO. Restricted self-diffusion of protons in colloidal systems by the pulsed-gradient, spin-echo method. J Chem Phys. 1968; 49: 1768–1777. [45] Price, WS., Stilbs, P., & Söderman, O. Determination of pore space shape and size in porous systems using NMR diffusometry. Beyond the short gradient pulse approximation. J Magn Reson. 2003; 160: 139–143. [46] Callaghan, PT., Eccles, CD., & Xia, Y. NMR microscopy of dynamic displacements: k-space and q-space imaging. J Phys E: Sci Instrum. 1988; 21: 820–822. [47] Cory, DG., & Garroway, AN. Measurement of Translational Displacement Probabilities by NMR - An Indicator of Compartmentation. Magn Reson Med. 1990; 14: 435–444. [48] RF, Karlicek.,Jr, & Lowe, IJ. A modified pulsed gradient technique for measuring diffusion in the presence of large background gradients. J Magn Reson. 1980; 37: 75–91. [49] Cotts, RM., Hoch, MJR., Sun, T., & Markert, JT. Pulsed field gradient stimulated echo methods for improved NMR diffusion measurements in heterogeneous systems. J Magn Reson. 1989; 83: 252–266. [50] Latour, L., Li, L., & Sotak, C. Improved PFG Stimulated-Echo Method for the Measurement of Diffusion in Inhomogeneous Fields. J Magn Reson Ser B. 1993; 101: 72–77. [51] Sørland, GH. Dynamic Pulsed-Field-Gradient NMR. Berlin, Heidelberg: Springer, 2014. [52] Pantoja, CF., Bolaños, JA., Bernal, A., & Wist, J. Mutual Diffusion Driven NMR: a new approach for the analysis of mixtures by spatially resolved NMR spectroscopy. Magn Reson Chem. 2017; 55: 519–524. [53] Callaghan, PT., & Xia, Y. Velocity and Diffusion Imaging in Dynamic NMR Microscopy. J Magn Reson. 1991; 91: 326–352. [54] Jones, DK. Diffusion MRI - Theory, Methods and Applications. Oxford: Oxford University Press, 2011.

8 Electrophoretic NMR (eNMR) Electrophoresis means a plug flow-like displacement of ions or charged entities in a solution under the influence of an externally applied electric field. A steady-state velocity, balancing the electrostatic driving force and frictional ones in a solution, is typically reached within nanoseconds. In this context, it is appropriate to recall a seminal multiscience article by one of the original NMR pioneers, Edward Purcell. It is titled “Life at low Reynolds number” [1]. With hand-drawn and sketchy figures and entertaining text, he illustrates the world that molecules and also very small organisms have to cope with. Originally, the presentation was in the form of a symposium talk, in honor of another great physicist, Victor F. Weisskopf. The take-home message for eNMR and factors therein is that inertial effects for motion are totally irrelevant in such a molecular world – apart from Brownian motion, objects stop once an external force is removed. Of course, concepts like solution conductivity are also closely related to electrophoretic mobility, although this is an unresolved quantity, which involves motion of all charged species in solution. There are a number of additional related concepts and effects, mostly found under the generic heading “Electrokinetic phenomena” (see, e.g., the corresponding Wikipedia article, or virtually any textbook on physical chemistry) (Figure 8.1). In spin-echo NMR experiments, such coherent transport processes lead to a phase shift of the signal (c.f. Figure 2.24), instead of the echo attenuation that results from random self-diffusion. A common eNMR pulse sequence used today is illustrated in Figure 8.3. Although there had been early experimental considerations regarding eNMR-like studies, as well as some preliminary experimental measurements [3–5], Holz is generally credited for the first basic study of NMR-detected ion transport under the influence of an applied electric field by pulsed magnetic field gradient spin-echo methods [6]. He studied inorganic ions from a classic electrochemical perspective, using relatively high electric currents and ion concentrations. Saarinen and Johnson modified and extended the concept into a high-resolution proton version some years later [7]. The basic approach is very similar to normal FT-PGSE, albeit with a superimposed electric field. It is primarily applicable to aqueous solutions containing ions, or molecules with ionizable functional groups, but many other types of systems can be studied as well. One should also note that it is perfectly possible to do combined PGSE and eNMR measurements in succession, without removing the sample or changing the probe setup. Overall molecular motion in eNMR will depend on flow-like transport, depending on molecular charge and applied electric field, as well as parallel random https://doi.org/10.1515/9783110551532-008

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Hydrodynamic friction “Upstream” friction Surrounding field asymmetry

Electrostatic force

Figure 8.1: Electrophoretic motion of an ion in aqueous solution under the influence of an electric field, retarding frictional forces and shielding via counterions. A steady-state velocity is reached on a nanosecond timescale. The hydrodynamic frictional force is conceptually the same as in self-diffusion and could be described by Stokes’ law, although one would have to account for partial common motion of counterions and hydration “shells.” The “upstream” one reflects that counterions move in the other direction and exert an always-counteracting hydrodynamic force. The “distortion” one is attributed to the dipole moment that is established with the redistribution of the surrounding counterions by the external electric field.

motion through self-diffusion. Neglecting transverse spin relaxation and other disturbing factors like convection (that could become severe, due to Joule heating of the sample) and electro-osmosis (further discussed later), the overall expression describing a Hahn echo-based eNMR experiment (including the phase changes) will ideally read as follows: Signal ∝ exp½ − DðγGδÞ2 ðΔ − δ=3Þ · exp½iϕ

(8:1)

Here the phase shift angle of a given signal is given by ϕ = γδgΔE Eμ

(8:2)

The electrophoretic mobility (μ) in question is proportional to the molecular charge, and of course also depends on size- and viscosity-related factors. The latter also reflect on the overall solution conductivity (which is normally denoted κ). Such experiments are done at constant gradient-related parameters, nominally keeping the diffusion-related attenuation constant and leaving only the phase shift factor. In reality, the overall signal amplitude does not remain constant under such conditions, as implied in eq. (8.1). It will be irreversibly dampened over time through increasing electric field and current influence on other factors than only electrophoresis. Overall, such effects affect the signal in a slightly irregular exponential decay manner. Some of the great challenges in eNMR are to minimize such effects to achieve the ideal experimental behavior implied in eq. (8.1). One should clearly note that the basic underlying experimental approach in eNMR is the same as in the diffusion-sensitive PGSE experiment. Consequently, self-diffusion will affect the echo signal as well – but still in the form of a general

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8 Electrophoretic NMR (eNMR)

Uncharged reference TMA+ phase by 1H eNMR z(TMA+) = +1.03 PF6– phase by 19F eNMR

180 PF6– anion

120

cm –1 ) Phase shift (degree)

TMA+ cation

0

108 HDO reference

Elect ric fie

65

ld (V

22

μ(TMA+) = 3.45×10–8 m2 V–1 s–1

60 0 –60

–120 –180 0

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20

40 60 80 100 Electric field (V cm–1)

120

Figure 8.2: Illustrating the echo signal phase modulation effect in eNMR, which provides the quantitative link to electrophoretic mobility of charged species in solution. Reprinted and adapted with permission from Bielejewski et al. J Magn Reson. 2014; 243:17–24 [2] (Copyright (2014) Elsevier).

90 90

󰜏1

90

󰜏2

90 2󰜏1

90

󰜏2

󰜏1

rf

󰛿

󰛿

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g(t) Δ/2

Δ/2 ΔE/2

E ΔE/2 Figure 8.3: A double stimulated echo pulse sequence is often used in eNMR. The main purpose of the double variant is to achieve basic suppression of bulk convection and electro-osmosis flow effects on echo formation, as discussed in Chapter 5. The electrophoretic voltage is typically stepped from 0 to some 100–1,000 V, at fixed magnetic field gradient and rf timing durations (δ, Δ, τ). Its length can extend over the whole pulse sequence (symmetrically, from the first to the last rf pulse), and one would also use crusher gradient pulses, as in Figure 2.31. rf phase cycling is also required, as well as sign alternation of the electric field, primarily to suppress polarization phenomena at the electrodes and to reverse the direction of electro-osmotic flow. This is best done within a pulse sequence (like here), but for polarization reasons also should at least be done between successive scans (together with appropriate data acquisition sign alternations).

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and constant signal attenuation, rather than through flow-related eNMR phase shift effects. A closer examination of the relevant equations indicates that one should evidently strive for maximally long magnetic field gradient pulse intervals ðΔÞ to maximize the electrophoretic flow effect over the signal-damping parallel random diffusion effect on the spin echo. A compromise regarding timing settings must always be done, so as to really achieve measurable electrophoretic flow while still not attenuating the studied eNMR signal too much through diffusional influence in the experiment. It would be recommendable to make some short script or program to coarsely model the various influences on the spin-echo signal before proceeding to make actual eNMR experiments

8.1 Further reading The Wikipedia or physical chemistry textbook entries on electrophoresis and electrokinetic phenomena.

8.2 Evaluating echo signal phase data in eNMR Constrained mixing of the real (absorption) and imaginary (dispersion) parts of the spectrum (cf. Figure 1.18) is the actual mathematical procedure behind “phasing” of an NMR spectrum and for measuring phase angles in eNMR data sets (Figure 8.4). Compared to recording spectral amplitudes (like in normal PGSE), measuring phase shifts in raw eNMR data could be relatively straightforward – provided the NMR signal to be studied is well separated from others. It can be manually done “on line” via the standard spectrometer software spectral phasing procedures (see Section 1.11). The first step is to adjust the common overall and the linearly frequency-dependent phasing parameters (often denoted p0 and p1) to optimal values for signals that do not exhibit phase shifts from electrophoretic motion. Then, one sets “p0” phase to nominally read zero, and repeats the phasing section, but now only varying “p0,” to make the electrophoretically studied signal phase appear purely “absorption mode,” and note down the “p0” increment for further evaluation via eq. (8.2). Such a tedious procedure can likely be automatized to some extent by using existing software features like macro command scripts (Figure 8.5). If there is some signal overlap with other sample component spectra, things may get more difficult, since one has to deal with mixed absorption and dispersion mode NMR signals. Even if the signal of interest appears to be properly numerically phased to pure absorption mode, it will still have contributions from dispersion mode signal intensities from nearby signals. The situation will get worse (a) in the presence of high intensity signals compared to the one(s) of actual interest, like in

8 Electrophoretic NMR (eNMR)

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0.5 Dispersion intensity

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–0.5 0

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Figure 8.4: NMR bandshapes are basically Lorentzian, and a normal NMR spectrum corresponds to the (real, blue) absorption spectrum. The (imaginary, red) dispersion counterpart is rarely used in normal NMR, except as spectral component in “phasing.” Note that component dispersive bandshapes are not additive in the normal sense. For various instrumental and detection reasons, the originally Fouriertransformed FID or half-echo signal is not properly phased, and is adjusted manually so as to provide a “clean” absorption mode NMR spectrum. After such “corrections,” the dispersion mode NMR signal counterpart is normally discarded. Note that for a purely Lorentzian bandshape with exact phasing, a graph of dispersion mode intensity against absorption mode one at the same frequency is an exact circle. Such a graph (yellow) is denoted a DISPA plot, and has been used in various NMR contexts, primarily to diagnose slight deviations from exactly Lorentzian bandshape form, and also for “automatic spectral phasing.” Note that this DISPA plot is not closed near the 0,0 coordinates, because there is still spectral intensity at the edges of the absorption and dispersion bandshapes. The actual schematic bandshape functions calculated for the (arbitrary) frequency (f) interval 0–100. were (real, blue) 1./((f-50.)2 +1.) and (imaginary, red) – (f−50.)2/((f−50.)2 +1.), corresponding to a Lorentzian centered at frequency 50., with nominal intensity = 1 and bandwidth-determining “spin relaxation rate” of 1.0.

the case of a water solvent signal (Figure 8.6) and (b) in case of overlap with signals belonging to a component with opposite charge, that is, having an opposite signal phase change with increasing electrophoretic current.

8.3 eNMR reviews The general field of electrophoretic NMR and its applications have been a subject of several reviews in the past [9–16]. Indeed, there are also very significant eNMR-related considerations in the early and wider flow topic review by Packer et al. [4]. Packer historically re-reviewed this area in an encyclopedia chapter in 1996 [5]. In general, older texts are recommended regarding the physical basis behind eNMR, and newer ones regarding actual application examples and later steps toward methodological robustness.

271

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Absorption and dispersion intensities

8.4 Basic experimental implementation of eNMR

3 2.5 2 1.5 1 0.5 0 –0.5

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Figure 8.5: Illustrating potential influences on measurements of spectral phase angles for a synthetic absorption mode adjusted eNMR-observed signal (basically blue) with a phase that differs from that of a nearby and larger signal (absorption mode violet, dispersion mode orange-brown). It has arbitrarily been set to differ maximally by 90 degrees, like for a pure dispersion mode signal. Unlike the corresponding absorption mode intensity of the larger signal (violet), the dispersion mode one (red, or orange-brown) extends over a wide frequency range, and could affect manual eNMR phase angle measurements quite significantly, as emphasized in the right-hand detail, where the smaller “eNMR-observed” signal is added onto the presently fully dispersive bandshape of the bigger signal. The situation is potentially worsened through varying or opposite sign spectral phase increments throughout the eNMR data set, when incrementing the eNMR current setting. The bandshape expressions here are basically the same as in Figure 8.4, except that the larger signal at frequency 75 was set to be 50 times larger than the eNMR-observed one at frequency 25.

8.4 Basic experimental implementation of eNMR In a typical application to solution systems, the electric field is applied for like 100 ms duration (corresponding to ΔE in (8.2)) via high-voltage leads (should be designed to be safe at a 2,000 V DC voltage) that enter the sample. The lead sections in contact are platinum or palladium wire, bent to suitable shapes. The most straightforward electrical sample compartment access is from above, through an isolated cable down the superconducting magnet bore. Connecting through connectors in the probe’s lower section requires high-voltage leads through the probe body, all the way past the rf and gradient coils. Sample change then requires probe removal and repositioning, which is a quite difficult procedure to deal with during any type of measurement series. Various sample geometries have been suggested. The originally most common was a U-tube like one, placed inside a normal 10 mm thin-wall NMR tube. As an advantage, such a setup has an escape path for bubbles that may form during the experiment. High-voltage sources and DC amplifiers are available commercially and are adequate in many respects, but one still faces problems to properly control and time the

Intensity

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×108

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Data point # Figure 8.6: Even signals well outside the nominal “sweep range” or spectral window may significantly contribute signal amplitudes within it, although it may appear that electronic and digital signal filtering should massively dampen them. This is particularly true for the dispersion mode contribution, whose wings decay with frequency in a 1/ω-type manner. The illustration example in the figures here is a proton PGSE, rather than eNMR data set (containing a 1 mM solution of an analog “AHEI” of the water-soluble aromatic molecule type “luminol” in a mixed H2O/D2O solution [8]), but the issue is the same. The upper part shows the whole data passed and baseline-corrected data set of sixteen 16k spectra (262,144 data points in total); the lower part shows the first 16k spectrum in that set. Absorption mode signals are represented in blue and dispersion ones in red. Additionally, there is a relatively strong water signal to the right of the spectral window. Its absorption might appear absent, but the dispersion bandshape counterpart is still there and still appears in the form of a general and very significant amplitude contribution within the active spectral window that leans toward the water signal. In PGSE, this does not matter, since the absorption mode signals that are actually used are well phased and baseline corrected (except for some oddities at the edges). They do not change shape within the PGSE data set. Reading signal phase, through nominal spectrometer software phase corrections in an eNMR variant, as illustrated in Figure 8.5 will use varying contributions of the evidently “distorted” dispersion bandshape, which may cause problems.

electric field pulses digitally. This could be done with some simple logic circuit or computer interface, triggered by some spectrometer external timing pulse output. It is important to understand that “control” here should relate to actual sample current, rather than the much simpler task of digitally setting and timing a nominal applied voltage for the current source. Conductors, connectors and electrodes added for eNMR purposes in the sample area also become rf interference sources or “antennas” and magnetic field homogeneity disturbances that must be properly dealt with.

8.5 Beginner’s lessons learned by using the originally common U-tube eNMR sample geometry The U-tube geometry offers easy electrode access, but has one intrinsic disadvantage in that the overall phase information of the oppositely evolving magnetization vectors

8.5 Beginner’s lessons learned since the U-tube era

273

of the two tube parts gets added in their composite NMR signal. The phase evolutions instead translate into a common cosine-like amplitude modulation of the eNMR signal, and the actual sign of the electrophoretic mobility is lost. For one or possibly two charged components, its sign could be self-evident or logically deducible, but this would not always be so. Additional problems are a poor sample-filling factor (which inevitably leads to a low sensitivity) and a shape that is awkward adjust the background magnetic field homogeneity for. The relatively narrow capillaries of U-tube shape are difficult to clean, and prior surface treatment against electro-osmosis (see below) tends to be washed away in such a process, leading to frustratingly irreproducible

–

Electro-osmosis

+

– +

+

–

– +

+

– +

–

+

–

+

–

+

–

Electro-osmosis

+

–

+

Electro-osmosis

+

–

–

Figure 8.7: Some schematic eNMR tube variants; front and top cross sections views are illustrated: the U-tube configuration (left), the cylindrical configuration (middle) and the concentric configuration (right). The electrodes are indicated with green for positive and red for negative. The NMR-sensitive region is inside the marked dotted square. Ions moving in the field are shown in red (anions) and green (cations). Electro-osmosis may also cause migration in the system as indicated; the direction depends on the wall charge (dark grey). Note that two different types of solutions may be used in a concentric geometry. This is indicated by the additional dark blue inner solution with orange (anionic) and yellow (cationic) molecules. A conducting polymer plug (violet) separates the two different solutions. Either of these should be composed so as to not contribute to the eNMR signal. The picture originates from a local year 2006 version of an eNMR procedure manual at our department. The figure is included here, courtesy of the manual authors Erik Pettersson Thyboll and Christoph Weise.

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experiments. We later abandoned U-tube capillaries for designs similar to the cylindrical middle one as shown in Figure 8.7, as discussed in the following section.

8.6 General considerations in experimental eNMR 8.6.1 Safety Of course, careless handling of components carrying high voltages of several 100 V (and perhaps over 1 kV) from power sources with considerable current capability (several tens of milliamperes) is dangerous for sure, and may even be lethal. All necessary precautions must be taken to avoid risks to personnel and instrumentation. Suitable cabling and connectors have to be used too, and they must have high mechanical integrity and proper grounding. No reconnections whatsoever should be attempted or considered at live voltage status conditions.

8.6.2 Basic physics Actual feasibility of actual eNMR studies depends on many fundamental factors. Some appear relatively obvious when examining and considering implications of relations like these, in relation to eqs (8.1) and (8.2): Ie κσ

(8:3)

V =E·l

(8:4)

σ l

(8:5)

υ = μE

(8:6)

E=

ρ=R

Where E, applied electric field [V m−1]; Ie , electrophoretic current [A]; κ, sample conductivity [S m−1 (= Ω − 1 m−1)]; R, sample resistance [Ω]; ρ, sample resistivity [Ωm]; σ, cross sectional area of the electrophoretic cell [m2]; V, electrophoretic voltage [V]; l, distance between the electrodes in the electrophoretic cell [m]; μ, electrophoretic mobility of the ion [m2 V−1s−1]; υ, velocity of electrophoretic transport in the electric field [m s−1]. Due to polarization phenomena at the electrodes and other nonequilibrium conditions that develop when applying an electric field across an electrolyte solution, parameters like sample conductivity are measured with laboratory desktop electronic devices (conductometers) that operate with ac voltages and very low measurement currents, typically in the kHz range.

8.6 General considerations in experimental eNMR

275

8.6.3 Sample heating in eNMR Joule heating of the sample is an important limiting factor in eNMR. It can be shown to develop at a rate of Ie2 =σ 2 κ

(8:7)

per unit volume [9]. Consequences include temperature shifts that would be physicochemically unacceptable and also induction of convective flow. The severity of such effects would depend on sample geometry, its surroundings, efficiency of thermostating airflow, the actual measurement time window and other factors. In a specific investigation of actual sample heating in capillary U-tube setups, temperature effects of several degrees could be noted, even at intermittent eNMR-timing matched electrophoretic currents of a few milliamperes [12]. Note that for a given acceptable heating rate, the only way to increase the velocity-generating electric field (E) is to reduce the sample conductivity. This may be inherently impossible, and is the prime factor that excludes many systems from eNMR applicability.

8.6.4 Magnetic field gradients and motion induced by currents in the sample Moving charges in a magnetic field influence the field pattern and are also influenced by it. Potential complications relevant for eNMR were already discussed by Packer et al. [4] and elaborated on in the later review by Johnson and He [9]. Basically, there are two effects to consider – the first one is the possibility of creating a significant magnetic field gradient near the sample through the current that flows though it. However, such an induced magnetic field gradient should be linear, minor and correctable. The other is diversion of the current flow by the main magnetic field in NMR. Its implications were also discussed by Packer, Holz and Johnson and He, and were considered to be of marginal practical importance for eNMR in a superconducting magnetic geometry. The situation will vary, depending on the relative geometric arrangement of the main magnetic field and the current direction. It was deemed to be potentially troublesome in iron magnet geometries, where the main field is oriented horizontally across the sample and the eNMR-driving electric field is along the vertical axis of the sample cell. Cineastes may recall 1990 movie The Hunt for Red October (www.imdb.com/ title/tt0099810/), where a new nuclear submarine type is on its maiden tour. It was equipped with an imagined totally silent propelling device where (conductive) seawater was forced to flow under the influence of such electromagnetic effects.

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8.6.5 Further reading The Wikipedia article on Lorentz force.

8.7 Why study electrophoretic mobility rather than self-diffusion? Full interpretation of eNMR data requires self-diffusion information as well. However, complementing that with electrophoretically related data may open new doors. As an example of a possibly artifact-prone self-diffusion transport-related application, one can consider partial binding (to a degree p) of a charged molecule ðAÞ to an uncharged one ðBÞ. A two-site binding analysis regarding electrophoretic mobility results in the following: p=1−

μA, obs − μB, obs μA, free

(8:8)

An important point to note is that in comparison with the analogous diffusion approach implied in an equation of the form Dobs = pDbound + ð1 − pÞDfree , the electrophoretic variant is a null experiment and becomes sensitive even to small degrees of binding (cf. Figure 8.22). Recall that PGSE diffusion-based studies are only weakly sensitive to binding effects, unless they involve “small-molecule” binding to macromolecules or formation of large aggregates. The other important point to note is that if one combines electrophoretic mobility with PGSE-based self-diffusion coefficient data on the same system, one can also arrive at quantitative information on the stoichiometry of the complex, using the Nernst–Einstein relation: μ=

zeD kB T

(8:9)

Here z denotes the (nominal) charge number of the complex and e the elementary charge. Recall that both electrophoretic mobility and self-diffusion can be measured in consecutive eNMR and PGSE experiments, without removing the sample from the NMR probe. Of course, electrophoretic mobility and binding information collected experimentally (c.f. Figure 8.8) may be of great interest for comparison with predictions, and should be complemented by corresponding self-diffusion data. Modeling comparisons can be made based on various theories and relations under general headings such as molecular electrostatics, polyelectrolyte theory, electrokinetic phenomena and battery science. As such, experimental data could be of great significance for experts in these areas, who are perhaps even unaware of the existence of eNMR as a tool. Interdisciplinary collaboration could consequently be very rewarding.

8.7 Why study electrophoretic mobility rather than self-diffusion?

277

Figure 8.8: A proton eNMR experiment on a 2 mM solution of tetramethylammonium chloride (TMAC) in perdeuterated ethanol. The proton–deuteron exchange rate is slow on the NMR time scale, so there is a separate peak from HDO at 4.8 ppm. The TMA+ signal is the only one that is affected by the applied electrostatic field across the sample, as manifested in a linear phase modulation relative to the others with increasing voltage. Modified, from Fredrik Hallberg’s Thesis [17], courtesy of the author.

8.7.1 Transference numbers and electrophoretic mobility Globally important parameters in this context are also the ion transport or transference numbers of an electrolyte system, that is, how large a fraction of the current is actually carried by the positive or negative ions of an electrolyte, respectively. Their ratio can be determined by macroscopic electrochemical techniques, originally developed in the nineteenth century. Basically, individual transference numbers (t ± ) are closely related to the respective individual ionic self-diffusion coefficients. It is easy to see that this must be so, since t± =

μ± μ+ + μ−

(8:10)

and individual electrophoretic mobilities (μ ± ) given by eq. (8.9) have all parameters except the self-diffusion coefficients in common. This would basically hold for simple ions with defined charges. The condition gets more complex for macroions where screening effects like those further described in Section 8.8.1 become more important. As an illustrative example, one can consider a dilute aqueous sodium chloride solution. The sodium ion transference number is here about 0.4 (and thus it is 0.6 for the chloride ion; the ratio is therefore 0.66). This is in good agreement with their

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experimental self-diffusion coefficients in similar conditions (e.g., listed in Table 2.1), that have a ratio of 0.65. Of course, transference numbers may slightly vary with temperature and electrolyte concentration, but less so than for self-diffusion coefficients.

8.8 Electro-osmosis and electrophoresis in a historical perspective 8.8.1 Electro-osmosis and the “double layer” concept F. F. Reuss first reported electro-osmotic flow in 1809. He showed that water could be made to flow through a plug of clay by applying an electric voltage. Clay is composed of closely packed particles of silica and other minerals, and water is forced through the narrow spaces between these particles just as it would through a narrow glass tube. Consider a liquid electrolyte, consisting of positive and negative particles (“ions”) in solution, such as NaCl salt dissolved in water. In the interface between a solid surface and such an electrolyte solution, a net fixed electrical charge, a layer of mobile ions, known as an electrical double layer or Debye layer forms, leading to a screening of charges. The double layer is effectively a capacitor skin at the interface, which has a small voltage across it, the so-called zeta-potential (typically 0.01–0.1 V). It is defined as the potential of the surface minus the potential just outside the double layer. Its extension is characterized by the concept of Debye length, which is a measure of a charge carrier’s net electrostatic effect in solution and how far its electrostatic effect persists. With each Debye length, charges are increasingly electrically screened and the electric potential will decrease in magnitude by 1/e, where e represents the electronic charge. In an electrolyte or a colloidal suspension, the Debye length (normally denoted κ − 1 , and not to be mixed-up with a similar use of symbol for electric conductivity) for a monovalent electrolyte is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εr ε0 kB T κ−1 = (8:11) 2NA e2 I Here I represents the ionic strength of the electrolyte (should be in moles m−3 and similarly the notation should not be mixed-up with the same common symbol for electrical current strength), ε0 the permittivity of free space, εr the dielectric constant of the medium, kB the Boltzmann constant, T the absolute temperature, NA the Avogadro number and e the elementary charge. When an electric field is applied to the fluid (usually via electrodes placed at inlets and outlets), the net charge in the electrical double layer is induced to move by the resulting Coulomb force, dragging the bulk electrolytic fluid along. The

8.8 Electro-osmosis and electrophoresis in a historical perspective

279

resulting transport effect is termed electro-osmotic flow. To some extent, coating the surface in contact with the electrolyte with a molecular layer of polymer “brushes” or gel-like structures can dampen it. Recipes for such procedures have been described by, for example, Hjertén [18] and Cifuentes et al. [19].

(a)

(b)

(c)

Figure 8.9: Illustrating the “double layer” concept at a solid interface in contact with water or an electrolyte, causing a gradually changing electrostatic potential outward toward the solution phase. To first approximation, the charge profile is exponential and is characterized at different points by concepts like the Stern layer and the zeta-potential (a). For amorphous systems like glass, it forms as a consequence of its ionic composition and somewhat mobile sodium counterions at the interface. In fact, it is very difficult to find materials without such surface “defects” (b). Electro-osmotic flow is even present in nominally electro-neutral Teflon tubes or quartz capillaries. To some extent, it can be suppressed by chemically or physically coating the surface with polymer “brushes,” to lessen surface transport of the counterions through a “gel-like” situation (c). Nasty chemicals, such as acrylamide, initiated with ammonium persulfate, and assisted by a glass pretreatment with even nastier and covalently binding methacryloxypropyltrimethoxysilane surface-coat were typically used. Similarly designed methylcellulose coatings were tried too. In our experience, the procedures may at times work, although reproducible coatings and reasonably stable layers were found to be frustratingly hard to achieve in practice.

8.8.2 Electrophoresis Traditionally, electrophoresis always had its main application field in bioanalysis and biochemical separations. Its present main methodology emerged from pioneering work on protein separation and analysis in the 1930s by the 1948 year Nobel laureate Arne Tiselius. M. Smoluchowski, E. Hückel and D.C. Henry had earlier laid the basic foundations. Tiselius was a student of The (Theodor) Svedberg – the 1926 Nobel Prize laureate in chemistry. His actual prize topic was colloid chemistry-related studies on metal sols. However, in the light of later findings and in the perspective of time, the award appears somewhat dubious.

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But already in initial use in 1925 was Svedberg’s hugely significant invention of the ultracentrifuge. It was the actual topic in Svedberg’s Nobel Prize acceptance lecture as well. By monitoring macromolecular sedimentation rates during ultracentrifugation, it could be demonstrated that biological macromolecules have distinct molecular weights, within experimental error. Randomizing diffusion is an opposing force to ordered transport like sedimentation under gravity influence, and mutual diffusion coefficients and sedimentation rates both appear in equations describing sedimentation. In fact, a variant called equilibrium ultracentrifugation exists. Here an exponential concentration profile develops with time, manifesting a Boltzmann-like energy distribution (just like the well-known barometric pressure formula for the atmosphere). Sedimentation rate data (with buoyancy information in the form of solution density and molecular partial molecular volume) are thus combined with measurements of mutual diffusion coefficients in such studies. Initial investigations were made with colored protein molecules like myoglobin and hemoglobin, whose sedimentation rates were monitored optically and were found to prove that these proteins were distinct entities. Many consider that work to be the starting point of what was later named “molecular biology.” In 1926, J.B. Sumner made a similarly pivotal discovery in this area, through his finding that the crystallizable enzyme urease was a protein. Notably, Svedberg’s discovery was based on the general concept of molecular transport properties, just like Tiselius’ later work on electrophoresis. Today, ultracentrifugation is much less used than electrophoresis, but still remains a very powerful and developing technique (see Chapter D4 in Methods in Molecular Biophysics, listed in Section 8.8.3). It is also much more routine than it was originally, through self-balancing rotor designs, using a single, somewhat bendable rotor axis. In contrast, various electrophoresis-based procedures have instead grown hugely in practical importance and have become virtually indispensable everyday tools in modern biochemistry and for analytical aspects of medicine. Through later collaborators and students, the work by Svedberg and Tiselius also paved the way for creation and expansion of several biomedical/biophysical companies (like Pharmacia) in the Uppsala-Stockholm area. They have later merged in several steps with today’s globally dominant pharmaceutical giants and GE Healthcare.

8.8.3 Further reading This 1,120 page book has a wide coverage of historical and modern methods of all kinds and constitutes highly recommended general reading in this context: Serdyuk IN, Zaccai NR, Zaccai J. Methods in Molecular Biophysics – Structure, Dynamics, Function. Cambridge: Cambridge University Press, 2007. Physical Chemistry textbook or Wikipedia entries on double layer, Stern layer, electrostatic screening, Debye length and zeta-potential.

8.9 More recent methodological developments in eNMR

281

8.9 More recent methodological developments in eNMR Some initial beginner’s lessons learned by using the originally common U-tube sample holders were mentioned in Section 8.4. They were followed by others, and have led to continuous modifications of hardware and methodology over the years. Various PGSE-related experimental complications have been discussed in earlier chapters, and some of them might be also of importance in eNMR. The three main added problems are now Joule heating of the sample (leading to temperature drift and convective overturning), bubble formation at the electrodes and electroosmotic flow. The last one is by far the most difficult factor. As already mentioned, ion movement at the double layer of solid–electrolyte interface causes electro-osmosis and drags along the bulk solution. It is easy to realize that the build-up time-scale must be quite long in a molecular perspective. In an NMR-like tube, it is actually of the order of tens to hundreds of milliseconds [20, 21]– to be compared with a nanosecond-like steady-state evolution period for electrophoretic movement (Figure 8.10).

50 ms

100 ms

500 ms

1500 ms

3000 ms

1,500 ms Figure 8.10: Previously unpublished material from the electro-osmotic time-evolution imaging work of Manz et al. [20]. The sample tube (an open normal 5 mm o.d. tube with 3.4 mm i.d., containing a 1 mM KCl aqueous electrolyte solution) had upper and lower Teflon-plug embedded electrodes. Under the influence of an applied electric field, a Poiseuille-type parabolic flow profile gradually develops over a period of a second or so. Initially, the center region has a more flat velocity profile. At longer evolution times, the flow becomes chaotic, likely because of Joule heating induced convection. Note that since the volume is closed, there will be no net flow. The maximum positive and negative flow velocities here were of the order of 0.1 mm s−1.

The great time separation of electro-osmotic and electrophoretic build-up times forms a basis for separation and possible elimination of either of them by spin manipulation in suitable Carr–Purcell-type modified eNMR-experimental form. At the time or the work by Manz et al. [20], charting electro-osmotic flow build-up was one of the main incentives behind this investigative study. The ideas could not be brought to finalization, since the eNMR high-voltage unit could not be controllably operated at what was deemed to be a high enough electric field reversal rate. More successful later steps in the same direction are discussed in the following sections.

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8.9.1 Pulse sequences and sample cells Many initial eNMR obstacles have gradually been overcome or lessened in severity through later year methodological modifications. Key steps include a. Steps toward convection and electro-osmotic flow compensating PGSTE-based eNMR pulse sequence variants [12, 22, 23]. A basic one is illustrated in Figure 8.3. b. The use of standard 5 mm NMR tubes (instead of U-tube capillaries) with insulated palladium electrodes. For various reasons, Pd electrodes have lower electrolytic hydrogen/oxygen bubble formation tendencies than Pt, or are surface catalytically more efficient then “Pt-black” ones [23]. c. Development of more convenient sample holders and connectors, which include electronic rf noise suppressing filter circuitry [23] (Figure 8.11). d. Mathematical and numeric reference compensation for electro-osmotic and conductive flow effects through back-calculation, based on parallel eNMR behavior of water in the same sample [23] (Figure 8.12). e. Basic eNMR pulse sequence progress regarding separation of electrophoretic transport from electro-osmotic and convection-related flow, as based on their vastly different build-up time scales [2, 23]. CPMG-like eNMR pulse sequences can achieve a basic separation on these grounds. f. Extending eNMR applicability to samples of higher conductivity [2] by modified pulse sequences (Figure 8.13).

Figure 8.11: A partly disassembled eNMR sample holder (top). The electrodes to the left are intended to dip into 5 mm tube, as in middle variant in Figure 8.7. The metal part has the outer shape of a normal NMR spinner sample holder for a Bruker spectrometer. Its interior contains part of the rf filter components needed for proper operation. The sample is lowered into place, using the highvoltage cable (not shown) attached to the other part of the high-voltage connector seen to the right. A later variant, using concentric electrodes, is also shown in assembled form (bottom). The pictures are provided here courtesy of Pavel Yushmanov, P & L Scientific (www.plscientific.se), who also designed and manufactured the units.

283

8.9 More recent methodological developments in eNMR

Phase (degrees)

HDO

TMA+

5.5

5.0

4.5

4.0

3.5

3.0

350 300 250 200 150 100 50 0 ‒50 ‒100

TMA+, Corrected

TMA+ HDO 0

Chemical shift (ppm)

20

40

60

80 100 120

Electric field (V cm‒1)

Figure 8.12: Illustrating numeric compensation for electro-osmotic and conductive flow in eNMR, as based on echo phase reference behavior of the residual water signal in the sample (10 mM TMABr in D2O, with some residual water proton signal). Reprinted and adapted from Hallberg et al. [23], where the actual procedures are described in more detail (with permission from J Magn Reson. 2008; 192: 69–77 Copyright (2008), Elsevier).

Times 2n 90° 𝜙1

rf

180° 𝜙2,6 󰜏

󰜏

180° 𝜙3,7 󰜏

󰜏

180° 𝜙4,8

180° 𝜙5,9

󰜏

󰜏

󰜏

󰜏

󰛿 G Δ/8n

E Δ/8n

No such pulse in the last loop

Figure 8.13: A (truncated) CMPG-type eNMR pulse sequence, intended to achieve a first-order separation of electrophoretic motion from much more slowly developing electro-osmotic flow. Extensive rf phase cycling is required. Reprinted with permission from Bielejewski et al. J Magn Reson. 2014; 243: 17–24 [2] (Copyright (2014) Elsevier).

8.9.2 Hardware evolution Locally, we have gradually changed our hardware and experimental control procedures. When starting out in 1993, we used the same U-tube sample approach, probe type and gradient/electrophoretic voltage unit as in the original studies by the Johnson group [24]. He very kindly made the latest variant available to us, through

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his local electronics engineer at the time (Steve Woodward). The unit was controlled by a single trigger pulse from the spectrometer, and had its own timing circuitry, controlled by a QuickBasic software interface on a Windows PC. The unit did have proper current control, through agile adjustment of the applied eNMR electric field strength. We also bought eNMR and PGSE probes that were very similar to those used by the Johnson group. Craig Bradley of Cryomagnet Systems (Indianapolis, USA; today’s Anasazi Instruments) designed and manufactured them to our specifications. They were z-gradient only, but had first-order self-shielded gradient coils and one was multinuclear as well. As designed for the original U-tube geometry, such probes therefore had to be removed and repositioned with each sample change. Apart from the Bradley probes themselves, we soon abandoned this setup in favor of a home-brew PC Windows-based National Instruments (NI) LabView software and hardware control unit. It amplified voltage information from an NI digitalto-analog converter card in the PC into a Trek high-voltage amplifier that in turn connected to the actual sample electrodes. Parallel gradient control was exclusively handled by normal Bruker hardware and software, which merely provided a trigger pulse for the LabView control circuitry. We also abandoned the U-tube-type sample capillaries and developed designs that were much more convenient to use and had higher NMR sensitivity. They also had their voltage leads entering through the top bore of the superconducting magnet and thus there was no need to remove and reposition the probe when changing samples. In addition, these later setups had current-controlled electrophoretic voltage pulses. Most of the procedures and the LabView GUI were tested out and implemented by graduate student Erik Thyboll Pettersson and postdoc Christoph Weise, with technical assistance by Pavel Yushmanov and under general supervision by István Furó. The LabView/Trek amplifier setup had a fine and elaborate GUI – but also had a number of drawbacks, including software and hardware incompatibility issues for later LabView, NI cards, PC interfacing and Windows versions. With external USB-based NI units, rather than the originally available specific controller cards for PCI PC motherboard slots, things would have been simpler. Later modification of locally written LabView software modules was also deemed to be quite problematic, without support of the original programmer(s) (Figure 8.14). We finally decided to abandon this NI LabView/Trek voltage amplifier-based setup as well and to go for a self-contained external unit. It was to be controlled by Bruker software only. Pavel Yushmanov designed and built a prototype and now offers production quality units commercially through the company P & L Scientific (www.plscientific.se). The package includes hardware, cabling, software, sample holder and an eNMR measuring cell (Figure 8.11). The main unit housing is illustrated in Figure 8.15, and it is basically connected to the spectrometer as in Figure 8.16.

Figure 8.14: A screenshot of one of three control and monitoring modes of our LabView eNMR setup during operation.

8.9 More recent methodological developments in eNMR

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8 Electrophoretic NMR (eNMR)

8.9.3 Partial performance specifications for the eNMR-1000 unit

Figure 8.15: The P & L eNMR-1000 controller and electrophoretic current source. At the time of writing, it is to my knowledge the only commercially available one.

Output voltages: Digital-to-analogue converter size: Output current at ± , V (HV mode): at ±  V (HC mode): Output power: Peak power Mean power Minimum/maximum pulse length: Duty cycle: Output pulse shapes: Slew rate (amplifier): Settling time:

 to ±, V  ×  bit  to ± mA  to ±  mA  W  W  µs to  s % (absolute sum of positive and negative voltages) Rectangular of arbitrary polarity Greater then  V µs− Less than  µs, to within less than % of set voltage value

At the time of writing, a handful of eNMR-1000 units have been sold globally, and there were no commercial competitors.

8.10 eNMR equipment used by other researchers The eNMR technique remains a quite exotic one. The original pioneers (the Holz and the Johnson groups) left the area for other topics already in the 1990s. Others have locally successfully or more or less unsuccessfully designed setups of varying performance. eNMR-methodological efforts and hardware options used by others are described in various detail by Scheler and coworkers [16, 25–31] and by He (former Ph.

Tr2

Ch‒

Ch+

Programming error protection

Encoder interface

Encoder interface

Power supply

eNMR setup

∑

High V. Amp

High-voltage power supply

Filter box

Figure 8.16: Basic connection of the unit in Figure 8.15 to external logic outputs of a Bruker-type spectrometer.

Programming TTL pulse sequences from the conventional trigger outputs

Tr2

NMR specrometer

eNMR cell

Filter assembly

8.10 eNMR equipment used by other researchers

287

High-voltage cable

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8 Electrophoretic NMR (eNMR)

D. student with Johnson) and coworkers [22, 32–34]. The thesis work by Zhang [35] and one of the publications based on it [36] are quite detailed regarding local eNMR hardware details and solutions. Some of these were later used in a study by Gouverneur and coworkers [37]. The reviews already listed in Section 8.3 include additional hardware and methodological information, together with a more complete list of earlier eNMR applications. Apart from “traditional” eNMR applications, a few battery-related ones have also appeared [36, 38–41]. Direct NMR-based imaging of some types of ionic transport in battery-like context is also possible [42]. Rather different measurement cells than that in basic solution eNMR are generally required, as further outlined in Section 8.13.

8.11 General considerations on suitability of eNMR applications Much of the early work in eNMR was methodologically oriented, and of little or no actual physicochemical relevance. Still, it provided inspiration and encouragement for using this new tool in a number of fields. Electrophoretic motion is basically a more complex phenomenon than molecular self-diffusion, and interdisciplinary collaboration with theoretical chemists or other science branches is generally even more highly recommended for eNMR than for PGSE. Almost all routine laboratory studies using electrophoresis today are biologicalmedical in nature, such as separating proteins or DNA fragments. For basic NMR detection reasons, most biological macromolecules are poor candidates for ultimately successful eNMR-based investigations. The basic limiting factors are sensitivity related, as a consequence of their relatively broad and structureless proton NMR spectra and relatively rapid spin relaxation. On the other hand, and as demonstrated in the earlier PGSE chapters, the mere presence of macromolecules in solution may very strongly affect transport properties for smaller molecules or ions that bind to, or generally associate with them in more or less distinct ways. However, PGSE self-diffusion and eNMR-related transport effects might be quite different. Self-diffusion rates in a series of increasingly large macromolecules would continuously decrease with molecular weight. On the other hand, electrophoretic mobility of an entity would depend on its total charge and could even be positive or negative in sign. Even small charges could be detectable by eNMR, being a “null” detection technique (Figure 8.17). Size-related factors in eNMR may operate in ways that may not be immediately intuitive. The classic example here is separation of proteins via electrophoresis. This is normally preceded by denaturation into more open and linear structural forms by standard denaturation techniques, including disulfide bridge breaking with mercaptoethanol. Then a binding surfactant like sodium dodecyl sulfate (SDS) is added, followed by buffering the pH to some suitable

8.12 Some eNMR example studies

H2O

Acetonitrile

Electrophoretic voltage

PEO

289

Chemical shift Figure 8.17: Detecting electrophoretic motion of an otherwise uncharged polymer entity (here poly (ethylene oxide), PEO) via proton-based eNMR indicates that partial ion (anion) binding to it occurs in the presence of a dissolved metal salt (here Li+) in solution. Uncharged entities such as trace amounts of HDO and acetonitrile are unaffected. This presentation drawing was kindly provided by István Furó (10 mM LiClO4 and 10 mM EO units in Mw = 22,000 low polydispersity PEO dissolved in d3-acetonitrile).

value (to achieve a stable protein overall charge). In normal solution, one finds that electrophoretic mobilities then become largely independent of molecular weight. This is because the charge per amino acid unit is about the same (on average) and size-related and charge-related factors balance out. Only when repeating the experiment in a gel matrix (typically a poly(acrylamide) one) does one achieve a molecular weight-related separation of the desired kind. Here forced electrophoretic protein motion becomes hindered, to an extent that is closely linear with molecular weight. This is a routine procedure for analytical purposes, named SDS-PAGES electrophoretic separation. Here one should note that eNMR studies on gel-like or viscous systems would normally be free from disturbing effects from electro-osmosis or heat-induced convective flow, at least at moderate electrophoretic current levels.

8.11.1 Further reading The Wikipedia entries on electrophoresis, electro-osmosis, polyacrylamide gel electrophoresis and isoelectric point.

8.12 Some eNMR example studies The examples chosen below mostly stem from our own laboratory, and like for the PGSE examples in Chapter 3 are not intended to be anywhere near a complete or

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fully representative compilation. Rather, they are meant to exemplify what can be done, and illustrate how eNMR-based studies can complement and extend information from PGSE-based self-diffusion studies. – It was already mentioned earlier that it is perfectly possible to do PGSE self-diffusion measurements and eNMR-based electrophoretic mobility determinations in direct succession, without removing the sample, or changing the probe setup. Diffusion data are also needed in the context of data evaluation (cf. μ = zeD=kB T (eq. (8.9))). This is a key to successful eNMR studies of many kinds. In some circumstances, it could additionally be useful and justified to summarize combined results in a multidimensional plot, like in “DOSY displays.” Morris and Johnson already introduced the more basic concept of “MOSY” (mobility-ordered spectroscopy) back in 1992 [43, 44], at about the same time as the introduction of various “DOSY” variants. MOSY displays are analogous to DOSY ones, but instead constitute statistical constructs for eNMR-based electrophoretic mobilities rather than self-diffusion data. As for basic DOSY displays, MOSY visualizations do not normally add any actual information to already tabulated results, but may similarly appear visually attractive (Figure 8.18). One should again note that reading actual signal phases in more complex and overlapping eNMR data sets may be difficult and artifact-prone, especially for bandshapes belonging to components with oppositely directed electrophoretic mobilities (i.e., having opposite charges). These signals thus also have opposing NMR signal phases, which may be difficult to read or estimate accurately because of the long-range NMR frequency influence of the dispersive components of the bandshapes.

8.12.1 Ion binding to poly(ethylene oxide) polymer chains in aqueous solution It has long been known that small poly(ethylene oxide)-based ring structures (crown ethers) bind metal ions quite selectively in aqueous solution. For example, the 6-unit 18-Crown-6 member compound has a high affinity for K+ ions and a lower one for Na+. Such systems have been used as model ion carriers across membrane-like structures, since they are soluble in water as well as some organic liquids (Figure 8.19). Open-chain and partly or fully methyl terminated variants of the crown ethers find use as solvents in, for example, organic synthesis and technical formulations, under the common name “glymes” or glycol ethers. Longer variants are classified as poly(ethylene oxide) polymers (PEO) and are commercially available in various molecular weight range fractions. Such compounds attract wide current interest and are exploited as solid polymer electrolytes.

Electrophoretic mobility (10–8 m2 V–1 s–1)

8.12 Some eNMR example studies

assl 6

l

aa

assl

ll l 6

(a)

4

4

2

2

0

0

–2

–2

–4

–4

–6

4

3 2 1 Chemical shift (ppm)

0

–6

a/l

291

lll

(b)

4

3 2 1 Chemical shift (ppm)

0

Figure 8.18: A proton eNMR-based “MOSY” display of a model system of three amino acids in heavy water at pD=6 and pD=3, respectively. The purpose of the study was to investigate the feasibility of the approach for drug discovery and metabolomics type applications. The “bandshape” in the electrophoretic mobility dimension of a “MOSY-map” is a somewhat arbitrary one. It should basically relate to an error estimation or experimental uncertainty, rather than to an actual distribution of electrophoretic mobility (there should be no such thing here). Like for most DOSY and MOSY applications, it is not clear how these bandshapes were actually made up, and they appear too wide, considering likely levels of experimental uncertainty. A corresponding “DOSY-map” based on self-diffusion data was also shown, but was much less component resolved, because of closely similar self-diffusion coefficients of the constituents. Redrawn with permission from Fang et al. Magn Reson Chem. 2017; 55: 584–588 [45]; Copyright (2017) Wiley.

Figure 8.19: Crown ethers 21-Crown-7, 18-Crown-6 and 15-Crown-5: 18-Crown-6 has a high affinity for aqueous potassium ions over sodium ions; the situation is reversed for 15-Crown-5. And 21-Crown-7 has great affinity for cesium and rubidium ions. Evidently, the size of the center cavity has pivotal significance in this context to match the size of the ion. The drawing was created with the Avogadro software.

8.12.1.1 Objective After earlier studies on cesium and sodium ion binding to 15-Crown-5 and 18Crown-6 structures by PGSE and eNMR [46], we were interested in a long-standing

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question whether similar binding could occur for alkali metal and other ions also to open-chain poly(ethylene oxide) systems in solution [47]. In such a case, one could speculate that transient ring-like PEO loops could form and create crown ether-like cavities, possibly stabilized under the influence of metal ion binding. However, it would be hard to prove such a mechanism. 8.12.1.2 Experimental aspects The PEO backbone has excellent proton NMR properties in the context of PGSE or eNMR investigations, with a relatively narrow single bandshape region and unusually long spin relaxation times for a polymer structure in solution. One would not expect that proton PGSE-based self-diffusion studies would lead to anything significant, since any ion-binding effect on PEO self-diffusion would be small. PGSEbased studies on nuclei of ions with suitable NMR properties (e.g., 7Li and 133Cs) could be more rewarding, since slightly larger binding effects on their self-diffusion would be expected. eNMR studies appear much more feasible, since even minor charge accumulation at the PEO chain would make it move electrophoretically in solution. Interaction between PEO and cations has previously been studied by conductivity methods. Those results were model dependent and not fully systematic. Ion-specific effects have been noted for a long time in various systems, and are commonly discussed in relation to an empirical Hofmeister (or lyotropic) classification and ordering that has been around for well over 100 years. He originally studied the solubility of proteins in various electrolyte solutions. One often observes great binding and other differences between monovalent and di- and trivalent metal anions in similar context. Of course, both the cation and the anion type are of interest in such a study, aimed at a reasonably wide coverage of system variation. After some consideration, we chose iodide, acetate and perchlorate salts of the anions as shown in Figure 8.20. Deciding factors were commercial availability and actual solubility in methanol. Some hesitation regarding perchlorate salts in contact with combustible material such as methanol arose, but we chose to go ahead, considering the relatively low perchlorate concentrations aimed for. We would recommend others to consider and judge the risks for themselves, and not regard our study as proof of safe procedures and handling of such suspect substance combinations. Indeed the studies show quite significant variations in ion-binding characteristics for various salts, but it was also clear that the anion did not matter much. Equation (8.9) (μ = zeD=kB T) was used for evaluating the eNMR-based electrophoretic mobilities, making use of previously measured self-diffusion information in Figure 8.21. When varying the cation type, eNMR-detected changes look quite dramatic, as seen in Figure 8.21. However, one should bear in mind that the strongest binding

8.12 Some eNMR example studies

Li+

Na+

K+

293

Rb+ Cs+ Mg2+ Zn2+ Cd2+ Ca2+ Ba2+ Al3+ Sc3+

DPEO (10–11 m2 s–1)

10 9 8 7 6 5 4 3 2 1 0 Li Na K Rb Cs Mg Zn Cd Ca Ba Al Sc Cation Figure 8.20: The cations chosen for study, at a 2 mM concentration in methanol-d4. The PEO was a low polydispersity fraction of molecular weight 22,000, at a nominal concentration that also corresponds to a concentration of 2 mM, formally referring to ethylene oxide (EO) units (not to actual PEO molarity). As anticipated, PGSE-based data confirmed that self-diffusion of PEO is indeed insignificantly affected by any ion-binding effects. Drawings are from the doctorate thesis of M. Giesecke (KTH, 2014), courtesy of the author.

8 7

Perchlorate salts Iodide salts Acetate salts

ZPEO

6 5 4 3 2 1 0 Li Na K Rb Cs Mg Zn Cd Ca Ba Al Sc Cation Figure 8.21: eNMR-based estimations of the total PEO polymer charge in various 2 mM electrolyte solutions in methanol. Not all combinations were studied. Reprinted and adapted from Giesecke et al. J Phys Chem B. 2016; 120: 10358–10366 [47], with permission, Copyright (2016) American Chemical Society.

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(corresponding to a total polymer charge of ca 5 e) still only corresponds to an average one of only 0.01 e per monomer unit. 8.12.1.3 Further reading Zhang Y, Cremer PS. Chemistry of Hofmeister anions and osmolytes. Annu Rev Phys Chem. 2010; 61: 63–83.

8.12.2 Surfactant/nonionic polymer interaction The interaction (or lack of interaction) between water-soluble nonionic polymers and surfactants has been the subject of large interest over the years. The topic is of considerable technological interest as well. In many formulations containing surfactants, polymers are added to enhance the performance, such as in adjusting rheological properties. Polymer–surfactant interaction is also interesting from a basic scientific point of view. What is the nature of the interaction (?) – and what are the properties of any surfactant–polymer complexes (?) – are typical questions posed. 8.12.2.1 Objectives Surfactant aggregation into micellar form in aqueous solution had previously been subject to several PGSE-based studies (see the example studies in Section 3.6). The effects on surfactant self-diffusion when reaching the critical micelle concentration (cmc) are relatively large, and allow clear and quantitative conclusions on the aggregation details. Basic polymer diffusion is relatively slow to begin with and has no significant concentration dependence. The situation differs for certain functional polymer types that can self-aggregate into three-dimensional networks in aqueous solution, as exemplified in Section 3.7. Self-diffusion rates then become significantly concentration-dependent. In the additional presence of surfactants, such aggregation can be enhanced or lessened with concentration, depending on system type and concentration ranges. Ionic polymers and ionic surfactants in some combinations are also anticipated to show strong indications of binding and to even form rather waterinsoluble structures. 8.12.2.2 Experimental aspects For polymers that are not of such self-associating type, basic polymer diffusion and surfactant micellar diffusion are relatively similar in range. Binding phenomena would not be expected to lead to any significant and easily quantifiable changes either. An obvious candidate polymer for PGSE- or eNMR-based investigation is PEO, because of excellent proton NMR properties and commercial availability in relatively

8.12 Some eNMR example studies

295

monodisperse form. Such characteristics make PGSE and eNMR measurements and data interpretation considerably easier than otherwise. A combined PGSE and eNMR study was thus made with PEO and four types of surfactants (SDS, dodecyl trimethylammonium bromide (DoTAB), potassium dodecanoate (KC12) and a nonionic “green” surfactant made up by an octanoate chain, coupled to a glucose unit (C8G1)). The main focus was to chart the feasibility of eNMR-detected binding of surfactants to a nonionic polymer [48], in the light of existing theories for polymer–surfactant interaction . A full study was only made for SDS/PEO. The other (reference) surfactant/PEO pairs were studied only through PGSE-based self-diffusion data. As expected, additionally indicated surfactant binding to PEO was only seen for the KC12/PEO pair. DoTAB and C8G1 surfactant self-diffusion data indicated no signs of further binding to PEO in aqueous (D2O) solution. In this context, there is a parallel parameter to the cmc, named the critical association concentration (cac). Mutual interaction between surfactant and polymer leads to significant surfactant self-assembly already below the normal cmc. The structure of the surfactant–polymer complexes had been extensively discussed in the past. Cabane and Duplessix established, on the basis of small-angle neutron scattering data, a model for the complex in the PEO/SDS case that had not since been challenged [49, 50]. Here the complex is described as micelles (one or several, depending on the concentrations of surfactant and polymer) “decorated” by the polymer, which is adsorbed to the surface of the micelle (or, the reverse). Later studies indicated that initial micelles formed in the polymer solution are smaller than those formed in the absence of polymer, although the aggregation number increases upon further addition of surfactant. After various considerations on the raw data in Figure 8.22, one arrives at the summarizing graphs illustrated in Figure 8.23. Obviously, by combining self-

4

D/m2s–1

Electrophoretic mobility

10–9

10–10

3 2 1 0

10–11 0.1

SDS [wt%]

1.0

0.1

SDS [wt%]

1.0

Figure 8.22: Experimental self-diffusion and electrophoretic mobility data (in units of 10−8 m2 V−1s−1 (negative)). Circles and squares represent SDS data, in the absence and presence of 1% PEO, respectively. Diamond symbols refer to PEO data. Adapted and redrawn, with permission from Pettersson et al. Langmuir. 2004; 20: 1138–1143 [48], Copyright (2004), American Chemical Society.

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1

pb

NAgg

250 200

0.8

150

0.6

100

0.4

50

0.2

0

0 0.1

SDS [wt%]

0.1

SDS [wt%]

Figure 8.23: Illustrating the evaluated number of SDS molecules (Nagg) per PEO molecule versus SDS concentration (left) and the fraction of SDS-decorated PEO molecules versus the SDS concentration in the concentration ranges specified in the text and indicated in Figure 8.23. Adapted with permission from Pettersson et al. Langmuir. 2004; 20: 1138–1143 [48], Copyright (2004), American Chemical Society.

diffusion data and electrophoretic mobilities, one can arrive at a quite detailed quantification of polymer–surfactant interaction for PEO/SDS in aqueous solution. 8.12.2.3 Further reading Holmberg K, Lindman B, Jönsson B, Kronberg B. Surfactants and Polymers in Aqueous Solution, 2nd Edition. Chichester, UK: John Wiley & Sons, 2002. Walderhaug H, Söderman O, Topgaard D. Self-diffusion in polymer systems studied by magnetic field-gradient spin-echo NMR methods. Prog Nucl Magn Reson Spectrosc. 2010; 56: 406–425.

8.12.3 Investigating “ionic liquids” by eNMR This is a class of compounds that have attracted high and general interest in recent years, although their history spans more than a century. Two “typical pairs” of constituents are illustrated in Figure 8.24 below, but it should be noted that there might be a thousand variants at the time of writing. In general, they are electrically conducting fluids (i.e., electrolytes), having a surprisingly wide fluid state temperature range. Functional use in battery applications seems to be an obvious candidate field. Physicochemical factors like those discussed in Section 8.8 for aqueous solutions of simple electrolytes (such as “double layer”) have little or no relevance in the context of ionic liquids, such as for concepts of “ion pairing.” However, mutual steric charge shielding of between charge-carrying molecules appears to be an important factor. Ionic liquids are often powerful solvents that have found use in organic synthesis. At times, some bizarrely brand them as “green solvents” in such

8.12 Some eNMR example studies

297

context. Applications also may make use of their basically very low vapor pressure at ambient temperatures. 8.12.3.1 Objectives Charge transport in ionic liquids is a phenomenon of utmost interest for electrochemical (e.g., battery) applications, but also of high complexity, involving transport of ion pairs, charged clusters and single ions. Understanding of conductivity details at a molecular level is limited due to unknown individual conductivity contributions of cations, anions and clusters. A 1H- and 19F-based electrophoretic NMR investigation was undertaken, meant to determine electrophoretic mobilities of cations and anions in seven different ionic liquids [37]. Some related eNMR studies on ionic liquids had previously been made by Zhang et al. [35, 36]. 8.12.3.2 Experimental aspects The authors made this study with locally designed sample holders and current driver arrangements. Some information regarding adaptations from Hallberg et al. [23] and Zhang et al. [35, 36] is given. Ionic liquid systems have considerably higher conductivity than that in previously described studies on dilute aqueous electrolyte systems, so measuring procedures were correspondingly adjusted, since sample heating and convective flow would be likely problems. One countermeasure was to use capillary bundles in a concentric sample tube arrangement similar to the middle one in Figure 8.7. Further problems were moisture and air sensitivity, requiring sample seal arrangements. 1 H- as well as 19F-based measurements were made. NMR sensitivity would not be a problem, since the systems are 100% concentrated, but measurements cannot be made under field/frequency internal lock NMR conditions. The local equipment additionally did not seem to allow measurement conditions where only the electrophoretic current was varied. This potentially makes the situation more uncertain and artifact-prone. The eNMR study was actually made at a constant (pulsed) electrophoretic voltage setting, while varying the magnetic field gradient strength. In reality, it becomes a mixed self-diffusion and eNMR one (cf. eq. (8.1)), where the self-diffusion-related signal attenuation is discarded, and the electrophoretic mobility is extracted from the signal phase change. Since there is no reference signal from a noncharged entity, one must simply assume that signal phase changes upon varying the magnetic field gradient current are strictly eNMRrelated. Using an arbitrary reference value (i.e., phase reading at a zero magnetic field gradient setting) will not matter here (Figure 8.24). Relatively large phase change error limits ( ± 3°) are quoted. They seem even larger and somewhat arbitrary in the figures of the publication. Such uncertainties would appear reasonable, considering the rather complex and mutually interfering out-of-phase proton NMR spectra that will occur. No actual eNMR data sets were

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8 Electrophoretic NMR (eNMR)

+

N

O

CF3

+

O

N

CF3

N

N F F

CH3

F P F

CH3 F F

50

Phase shift / Degree symbol

Phase shift / Degree symbol

15 55

BMATFSI (a) cation U = 50 V; ΔDrift = 400 ms

45 40 35 30 25 –3

1 2 –2 –1 0 Gradient strength (T m−1 )

3

4

BMATFSI (b) anion U = 50 V; ΔDrift = 400 ms

10 5 0 –5 –10 –15 –20 –4

–3

–2 –1 0 1 2 Gradient strength (T m−1)

3

Figure 8.24: Example types of ionic liquids, together with results from two measurement series from the study by Gouverneur et al. Phys Chem Chem Phys. 2015; 17: 30680–30686 [37]. Ionic mobilities were measured by varying the magnetic field gradient current (see the text), at relatively low electrophoretic voltages. Reproduced with permission of The Royal Society of Chemistry, Copyright (2015).

understandably shown in the publication or its supplementary material. The basic linear eNMR-related linear signal phase influence of the applied electrophoretic voltage (or rather, the assumed electrophoretic current) still seems experimentally robust enough for the purpose – charting ionic transference numbers for a variety of ionic liquids. The conclusions drawn were that electrophoretic mobilities as well as transference numbers vary considerably between systems. 8.12.3.3 Further reading The Wikipedia entry on ionic liquid. Earle MJ, Seddon KR. Ionic liquids. Green solvents for the future. Pure & Appl Chem. 2000; 72: 1391–1398. Magnetic Resonance in Chemistry; Special issue on NMR of Ionic Liquids, Vol 56(2) 2018 – which includes a mini-review by R. Nanda and K. Damodaran on “NMR methods used in the study of the structure and dynamics of ionic liquids”.

8.14 Further reading

299

8.13 Spatially resolved NMR-based monitoring of electrically induced transport Like for PGSE-based self-diffusion, eNMR and related flow effects like electro-osmosis can be spatially detected when combined with the normal procedures of NMR imaging and microscopy. A remarkable study was made as early as in 2002 by Weber and Kimmich [51], where channels in a synthetic porous or partly hollow object that was manufactured from layers of thin sheets. It had holes at random locations that were drilled under computer control. After assembly of a pack of such sheets, the holes connected as channels of random shape transverse to the individual layers. But the exact locations and shapes were known from design and formed a basis for parallel simulations of electrically induced flow and NMRbased experimental imaging of it. Battery-related research is currently of high interest for various reasons, including also electrical field-induced transport of lithium, phosphorous and fluorine constituents, apart from proton-bearing ones. These nuclei do have reasonably “good” NMR properties that permit eNMR and related studies. Experimentally, one would first have to build specialized sample compartments, which likely may require some skills and tricks – especially since battery liquids or gels may be quite moisture-sensitive. For an example study of this kind, see Klett et al. [42], as summarized in Figure 8.25. Insitu current

e–

NMR tube

LiPF6 battery electrolyte

9.5 mm

Li electrode

Time Li electrode

Electrochemical mass transport parameters

0 1 2 NMR images of 7Li concentration profiles

Figure 8.25: The graphical abstract picture from Klett et al. [42], summarizing the basic concepts and issues of this of battery-related transport study. Reprinted with permission from Klett et al. J Am Chem Soc. 2012; 134: 14654–14657, Copyright (2012) American Chemical Society.

8.14 Further reading The eNMR-related KTH doctorate theses of Erik Pettersson, Fredrik Hallberg, Marianne Giesecke and Yang Fang can be freely downloaded in PDF format by searching the link www.avhandlin gar.se for the respective names.

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References [1] [2] [3] [4] [5] [6]

[7] [8]

[9] [10] [11] [12] [13] [14] [15]

[16] [17] [18] [19]

[20] [21] [22] [23]

Purcell, EM. Life at low Reynolds number. Am J Phys. 1977; 45: 3–11. Bielejewski, M., Giesecke, M., & Furó, I. On electrophoretic NMR. Exploring high conductivity samples. J Magn Reson. 2014; 243: 17–24. Packer, KJ. The study of slow coherent molecular motion by pulsed nuclear magnetic resonance. Mol Phys. 1969; 17: 355–368. Packer, KJ., Rees, C., & Tomlinson, DJ. Studies of diffusion and flow by pulsed NMR techniques. Adv Mol Relax Proc. 1972; 3: 119–131. Packer, K. Diffusion & Flow in Liquids. In: Encyclopedia of NMR Spectroscopy, DM Grant & RK Harris, eds. New York: Wiley, 1996: 1615–1626. Holz, M., Lucas, O., & Müller, C. NMR in the presence of an electric current. Simultaneous measurements of ionic mobilities, transference numbers, and self-diffusion coefficients using an NMR pulsed gradient experiment. J Magn Reson. 1984; 58: 294–305. Saarinen, TR., & Johnson, CS., Jr. High-resolution electrophoretic NMR. J Am Chem Soc. 1988; 110: 3332–3333. Maeztu, R., Gonzalez-Gaitano, G., Tardajos, G., & Stilbs, P. Chemiluminescence of phthalhydrazide derivatives in organized media: Interactions with surfactants and cyclodextrins. J Lumin. 2011; 131: 662–668. Johnson, C, Jr., & He, Q. Electrophoretic nuclear magnetic resonance. Adv Magn Reson. 1989; 13: 131–159. Holz, M. Electrophoretic NMR. Chem Soc Rev. 1994; 23: 165–174. Johnson, C, .Jr. Electrophoretic NMR. In: Encyclopedia of NMR Spectroscopy, DM Grant & RK Harris, eds. New York: Wiley, 1996:1886–1895. Pettersson, E., Furó, I., & Stilbs, P. On experimental aspects of electrophoretic NMR. Concepts Magn Reson. 2004; 22A: 61–68. Griffiths, PC., Paul, A., & Hirst, N. Electrophoretic NMR studies of polymer and surfactant systems. Chem Soc Rev. 2006; 35: 134–145. Stilbs, P., & Furó, I. Electrophoretic NMR. Curr Opin Colloid Interface Sci. 2006; 11: 3–6. Stilbs, P. Diffusion and electrophoretic studies using nuclear magnetic resonance. Encyclopedia of Analytical Chemistry, Wiley On-line Library, https://doi.org/10.1002/ 9780470027318.a9052 Wiley, 2009. Scheler, U. Electrophoretic NMR (update of C.S. Johnson’s 1996 Encycl Magn Res article). eMagRes. 2012; Hallberg, F. Molecular interactions studied by electrophoretic and diffusion NMR. Physical chemistry KTH. Stockholm 2010; PhD Thesis: 1–67. Hjertén, S. High-performance electrophoresis: Elimination of electroendosmosis and solute adsorption. J Chromatogr A. 1985; 347: 191–198. Cifuentes, A., Canalejas, P., Ortega, A., & Dı́ez-Masa, JC. Treatments of fused-silica capillaries and their influence on the electrophoretic characteristics of these columns before and after coating. J Chromatogr A. 1998; 823: 561–571. Manz, B., Stilbs, P., Jönsson, B., Söderman, O., & Callaghan, PT. NMR Imaging of the time evolution of electroosmotic flow in a capillary. J Phys Chem. 1995; 99: 11297–11301. Wu, D., & Johnson, C, .Jr. Flow imaging by means of 1D pulsed-field-gradient NMR with application to electroosmotic flow. J Magn Reson. 1995; A 115: 123–126. He, QH., & Wei, ZH. Convection compensated electrophoretic NMR. J Magn Reson. 2001; 150: 126–131. Hallberg, F., Furó, I., Yushmanov, PV., & Stilbs, P. Sensitive and robust electrophoretic NMR: instrumentation and experiments. J Magn Reson. 2008; 192: 69–77.

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[24] Saarinen, TR., & Woodward, WS. Computer-controlled pulsed magnetic field gradient NMR system for electrophoretic mobility measurements. Rev Sci Instrum. 1988; 59: 761–763. [25] Wong, S., & Scheler, U. Electrophoresis of macromolecules in solution detected by electrophoresis-NMR. Colloids Surf A. 2001; 195: 253–257. [26] Scheler, U. Determination of effective size and charge of polyelectrolytes by diffusion and electrophoresis Nuclear Magnetic Resonance (NMR). Handbook of Polyelectrolytes and Their Applications. 2002; 2: 173–188. [27] Bohme, U., & Scheler, U. Effective charge of poly(styrenesulfonate) and ionic strength-an electrophoresis NMR investigation. Colloids Surf. 2003; 222: 35–40. [28] Böhme, U., & Scheler, U. Counterion mobility and effective charge of polyelectrolytes in solution. Macromol Symp. 2004; 211: 87–92. [29] Bohme, U., & Scheler, U. Effective charge of bovine serum albumin determined by electrophoresis NMR. Chem Phys Lett. 2007; 435: 342–345. [30] Bohme, U., & Scheler, U. Effective charge of polyelectrolytes as a function of the dielectric constant of a solution. J Colloid Interface Sci. 2007; 309: 231–235. [31] Scheler, U. NMR on polyelectrolytes. Curr Opin Colloid & Interface Sci. 2009; 14: 212–215. [32] He, QH., Liu, YM., & Nixon, T. High-field electrophoretic nmr of protein mixtures in solution. J Am Chem Soc. 1998; 120: 1341–1342. [33] He, QH., Liu, YM., Sun, HH., & Li, EC. Capillary array electrophoretic NMR of proteins in biological buffer solutions. J Magn Reson. 1999; 141: 355–359. [34] He, QH., Lin, W., Liu, YM., & Li, EC. Three-dimensional electrophoretic NMR correlation spectroscopy. J Magn Reson. 2000; 147: 361–365. [35] Zhang, Z. Probing transport of Ion dense electrolytes using electrophoretic NMR. Virginia Polytechnic Institute. 2013; PhD Thesis: 1–134. [36] Zhang, Z., & Madsen, LA. Observation of separate cation and anion electrophoretic mobilities in pure ionic liquids. J Chem Phys. 2014; 140: 084204. DOI doi.org/10.1063/1.4865834 [37] Gouverneur, M., Kopp, J., van Wüllen, L., & Schönhoff, M. Direct determination of ionic transference numbers in ionic liquids by electrophoretic NMR. Phys Chem Chem Phys. 2015; 17: 30680–30686. [38] Dai, H., Sanderson, S., Davey, J., Uribe, F., & Zawodzinski, TA, .Jr. Electrophoretic NMR measurements of lithium transference numbers in polymer gel electrolytes. Proc Electrochem Soc. 1997; 96–17: 111–120. [39] Walls, HJ., & Zawodzinski, TA. Anion and cation transference numbers determined by electrophoretic NMR of polymer electrolytes sum to unity. Electrochem Solid-State Lett. 2000; 3: 321–324. [40] Kataoka, H., Saito, Y., Miyazaki, Y., & Deki, S. Ionic mobilities of PVDF-based polymer gel electrolytes as studied by direct current NMR. Solid State Ionics. 2002; 152: 175–179. [41] Zhao, JS., Wang, L., He, XM., Wan, CR., & Jiang, CY. Determination of lithium-ion transference numbers in LiPF6-PC solutions based on electrochemical polarization and NMR measurements. J Electrochem Soc. 2008; 155: A292–A296. [42] Klett, M., Giesecke, M., Nyman, A. et al. Quantifying mass transport during polarization in a Li Ion battery electrolyte by in situ 7Li NMR imaging. J Am Chem Soc. 2012; 134: 14654–14657. [43] Morris, KF., & Johnson, CS, .Jr. Mobility-ordered 2-dimensional nuclear magnetic resonance spectroscopy. J Am Chem Soc. 1992; 114: 776–777. [44] Morris, KF., & Johnson, CS. Mobility-ordered 2D NMR spectroscopy for the analysis of ionic mixtures. J Magn Reson. 1993; 100: 67–73. [45] Fang, Y., Yushmanov, PV., & Furó, I. Assessing 2D electrophoretic mobility spectroscopy (2D MOSY) for analytical applications. Magn Reson Chem. 2017; 55: 584–588.

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[46] Price, WS., Hallberg, F., & Stilbs, P. A PGSE diffusion and electrophoretic NMR study of Cs+ and Na+ dynamics in aqueous crown ether systems. Magn Reson Chem. 2007; 45: 152–156. [47] Giesecke, M., Hallberg, F., Fang, Y., Stilbs, P., & Furó, I. Binding of monovalent and multivalent metal cations to polyethylene oxide in methanol probed by electrophoretic and diffusion NMR. J Phys Chem B. 2016; 120: 10358–10366. [48] Pettersson, E., Topgaard, D., Stilbs, P., & Söderman, O. Surfactant/Nonionic polymer interaction. A NMR diffusometry and NMR electrophoretic investigation. Langmuir. 2004; 20: 1138–1143. [49] Cabane, B., & Duplessix, R. Organization of surfactant micelles adsorbed on a polymer molecule in water: a neutron scattering study. J Physique. 1982; 43: 1529–1542. [50] Cabane, B., & Duplessix, R. Decoration of semidilute polymer solutions with surfactant micelles. J Phys(Paris). 1987; 48: 651–662. [51] Weber, M., & Kimmich, R. Maps of electric current density and hydrodynamic flow in porous media: NMR experiments and numerical simulations. Phys Rev A. 2002; 6602: 6306.

9 Building home-brew PGSE and eNMR instrumentation Building “home-made” instrumentation is hardly a recommended procedure today, but may be the only option in the case of economic constraints and related circumstances. If so, one may still require access to a good mechanical workshop and a good electronics engineer or technician with rf electronics experience. Today, few scientists have those skills, equipment and time to spare themselves. Gradient coils and gradient drivers would normally be the most essential components needed. One should be aware that adding gradient coils and extra conductors to an existing spectrometer system will most likely degrade its performance for “normal measurements” in a number of ways. A positive factor today is that there is a second-hand market for NMR probes. Prices may be quite reasonable indeed and may amount less than a few percent of the original price tag. Used probes constitute a natural starting point for initial design attempts aimed at creating local home-brew PGSE capability. Similarly, you can even get probes for free from some facility, when they upgrade their NMR equipment. If you are really lucky, the probe may even have integral magnetic field gradient coils to begin with. Some gradient coil design texts are listed in Section 4.1.1. Home-brew retuning of probes, to match a different NMR frequency, is realistic, provided one has some basic knowledge of rf electronics. Various small-scale commercial companies around the world do make such modifications and can also assist in designing and adding gradient coils to probe assemblies.

9.1 Further reading Fukushima and Roeder wrote the classic text on NMR tinkering some decades ago, but it may be hard to find at a reasonable price nowadays. Even the electronic Kindle edition was quite expensive at the time of writing: Fukushima E, Roeder S. Experimental Pulse NMR – A Nuts and Bolts Approach. London: Addison-Wesley, 1981. Sørland describes several homemade PGSE setups and components in his relatively recent book: Sørland GH. Dynamic Pulsed-Field-Gradient NMR. Berlin, Heidelberg: Springer, 2014. Much general advice and background material useful for mechanical, electronic and optical tinkering is found in this rather unique book: Moore JH, Davis CC, Coplan MA, Greer SC. Building Scientific Apparatus. Cambridge, U.K.: Cambridge University press, 2009.

9.2 PGSE on older-generation resistive electromagnet-type spectrometers Today, this topic would mostly be of historical interest, since such instruments are rarely in operation anymore. Still, it may be educational to consider and learn from https://doi.org/10.1515/9783110551532-009

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+

–

Fig. 583 (h = 45). Figure 9.1: Magnetic field gradient coil designs of various kinds were conceived already in the nineteenth century. Par A. Ganot reproduced this one in the 12th edition of his textbook Traité de Physique (published in 1866). Physics and universal scientific giants James Clerk Maxwell (1831–1879) and Hermann Helmholtz (1821–1894) formulated the mathematics and physics of optimal coil design at the time., See the respective entries for Maxwell coil and Helmholz coil in Wikipedia, for example.

the condition some decades ago, when virtually no commercial instrumentation for PGSE (or eNMR) was available. Some design hints of this period in time appear conceptually useful even today, and will be briefly described in the following sections. To some extent, particulars of the iron magnet generation of PGSE equipment is also shared with the emerging low-field permanent magnet NMR discussed in Section 4.4. Older-generation computer and pulse programming facilities were understandably very limited, and have little actual relevance today. Our own JEOL FX-100-type systems were taken out of service in the year 2000, after almost 25 years of operation. The spectrometer was still fully functional, but general and PGSE NMR performance was clearly beginning to get substandard. Other main problems faced were high electricity consumption, about 10 kW. Cooling water needs were correspondingly high too (like tens of liters per minute), although we were not explicitly charged for this. When using normal tap water, there were corrosion problems of various kinds and were further amplified by the high flow rates. Repairs were costly and could be complicated indeed. Fortunately, by using distilled water in a closed-circuit heat exchanger and circulation pump configuration, one could lessen corrosion to a good extent. The system was designed to deal with power dissipation of about 10 kW, and local office and computer system cooling water was circulated in the external branch of the heat exchanger. Probably the use of distilled water in the cooling circuits prolonged the lifetime of the spectrometer system by almost 10 years. The cost of the pump and the heat exchanger were considered as an excellent investment.

9.2 PGSE on older-generation resistive electromagnet-type spectrometers

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Apart from the JEOL FX series, no commercial gradient coil systems were offered for high-resolution NMR spectrometer variants in the early 1980s. This was several years after the widespread introduction of pulsed FT-NMR a decade earlier. JEOL had fortunately kept a combined probe coil design for field sweep purposes on the earlier sweep NMR “PS series.” On the pulsed FT-NMR “FX-series,” the coil pair was instead connected antiphase to as an alternative create “homospoil” (magnetic homogeneity spoil) pulses via a tiny gradient driver under pulse sequence command. The original purpose of these gradient coils was not PGSE-type applications, but rather to nullify residual transverse magnetization during multipulse NMR experiments. Today, one would call them “crusher” rather than “homospoil” pulses (Figure 9.2).

Figure 9.2: Left: The tiny original JEOL FX series “homospoil driver” (basically a small gated transistor (black), in series with a current-limiting resistor (the silver-colored potentiometer), all operating at 5 V). Gradient current capability was of the order of 1 A. Right: A multinuclear JEOL FX-100 series probe, as illustrated in Figure 1.5. As for the FX-60 series, its sidewalls had like 10 gradient (shim) coils wound in various patterns, as well as the central heavier-gauge homospoil coil. They were integral parts of each probe, and (of course) there were identical, but mirrored, ones on each probe side. Together with the gradient driver in (a) one could typically achieve gradient strengths like 10 mT m−1 (1 G cm−1), which actually suffices for many low-molecular system studies. Probably, much higher current levels could have been used, but we never tried. The probes were meant for normal analytical-type NMR as well, and we could not risk damaging them. The probe photo was previously published in Stilbs, P. Historical: early multi-component FT-PGSE NMR self-diffusion measurements-some personal reflections. Magn Reson Chem, 2017; 55: 386–394 and is reproduced here with permission; Copyright (2017) Wiley.

Bruker policy at the time was explicitly to regard magnetic field gradient coils in the probe or shim coil design as unnecessary in a high-resolution NMR context. The only meaningful purpose seen in this context was to “destroy” transverse magnetization in multipulse NMR experiments, a branch of pulsed NMR that “exploded” with the introduction and development of multidimensional NMR in the mid-1970s [1]. Magnetization cancelling effects needed could generally be achieved through phase cycling procedures [2]. Notably, Bruker did indeed sell (non-FT) resistive magnet-pulsed NMR spectrometers before 1970. Those did have PGSE capability, albeit at a fairly poor level of spectral

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resolution. In retrospect, it seems that there must have been some kind of “cultural” difference and perhaps other barriers between the high-resolution, chemistry-oriented and the high-power solid-state physics NMR branches of this company. The very first FT-PGSE study was actually not done on a resistive electromagnet system. James and McDonald used the normal shim coil system on a Varian HR-220 superconducting spectrometer, applying fairly low extra gradient current pulses through the z-gradient shim coils [3]. The same approach could, in principle, have been applied on any resistive or superconducting magnet system at the time, with relatively minor spectrometer modifications. It would still be limited to quite low gradient currents, because of the relatively thin coil wire diameters used. In fact, Bangerter also described electronically how to create pulsed gradients through commercial spectrometer shim coils about at the same time as James and McDonald. He did not seem to have any PGSE interest in mind, just the “homospoil/crusher pulse” application [4]. Although such weak “shim-generated” gradient pulses would suffice for studies on low-molecular weight solution system, further progress in this area was based on the much higher current capacity and excellent gradient constancy of the aforementioned JEOL “homospoil coils” (by us [5, 6] and independently by Callaghan et al.[7, 8]). The Callaghan group went on to heavily modify JEOL probes to achieve higher gradients and make possible multigradient NMR microscopy experiments [9] (Figure 9.3).

Figure 9.3: Left: The quadrupole-type x- and y-gradient coil designed by Moseley and Loewenstein [10], for a Bruker 90 MHz iron magnet probe. It is based on a design by Zupancic and Pirs [11], and allows gradient generation in two orthogonal directions transverse to the sample axis. Only one turn of one of the coils is shown in the middle drawing. Capillary tube guides for the coil bundles (ca. 10 turns each of lacquered 0.6 mm diameter copper wire (AWG 22 type)) were attached onto the normal probe insert using fast-curing cyanoacrylate glue. Right: An actual prototype gradient coil of this kind. Here the coil assembly is held in place using normal document tape, and placed for display on a Teflon bobbin, rather than being wrapped around the target probe insert. Note the quite thin copper wires and capillary tube guides.

In the early 1980s, Moseley and Loewenstein found an elegant and easy way to add gradient coils to Bruker-type iron magnet probes [10]. It was based on a previous design suggestion by Zupancic and Pirs [11].

9.3 Home-brew gradient coils for superconducting magnet geometries

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Eddy current disturbances for resistive iron magnet PGSE setups are only occasionally mentioned or considered in older PGSE literature [12, 13]. Colleagues, with whom I have discussed PGSE with in past years, shared my view that eddy current disturbances in resistive magnet probe systems are indeed much lower than that for supercon ones. Iron magnet probe systems are also generally less well tuned than the modern supercon ones, so radiation-damping disturbances are correspondingly reduced too. Since gradients are normally applied in a transverse manner across the sample, convective overturning tendencies are lessened as well, when compared to a vertically oriented, long cylindrical sample in a superconducting magnet PGSE setup.

9.3 Home-brew gradient coils for superconducting magnet geometries Locally building something like the laminated “printed circuit”-type coils illustrated in Figures 4.2 and 4.3 do not seem realistic, especially when considering also the need for water-cooling channels in the assembly. Instead, one would have to copy and apply design procedures of the past, and possibly include basic active magnetic field gradient shielding in the design. Figures 9.4 and 9.5 illustrate what one could hope for, with the help of local mechanical and electronic workshop assistance. Unless there is lots of space in the probe housing, fitting anything beyond a simple Maxwell-pair z-gradient only coil would be difficult. Even then, space may not always permit geometrically optimal design for it.

Figure 9.4: (Left) Unshielded x-, y-, z-gradient coils wound with copper wire and fastened through embedding with epoxy resin onto a circular Perspex/Lucite/Plexiglass tube, meant to fit outside the inner rf assembly of the probe illustrated in Figure 9.5. The outer aluminum probe cover is also shown. (Right) Inner rf part and Perspex tube guide of the same probe assembly as in Figure 9.4. The actual probe was a 63 MHz 23Na one, intended for medical “mouse-size” MRI research. As seen, there was adequate space here to house all three (x, y, z) gradient coil pairs needed for imaging purposes. Pictures reproduced here, courtesy of Michael E. Moseley presently at the Stanford University Radiology branch.

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Figure 9.5: (Left) Some of the (unshielded, z-gradient only) gradient coil variants designed and tested by William S. Price in his 1990 thesis work. (Right) The most optimal coil design found at the time. The coil windings here are further away from the aluminum probe housing than that in other variants, and eddy current influence in PGSE experiments was far less than for some of the others seen in. Its gradient capability was postcalibrated to be like 12 G cm−1 A−1. Eddy current influence in PGSE was found to be negligible after 14 ms, when using 50 G cm−1 gradient pulses. (Reproduced here courtesy of William S. Price; the probe was a 400 MHz narrow-bore Varian one).

9.3.1 Further reading Turner R. Gradient Coil Systems. In: Encyclopedia of NMR Spectroscopy, Grant DM, Harris RK, eds. New York: Wiley, 1996:2223–2233. Hidalgo-Tobon SS. Theory of gradient coil design methods for magnetic resonance imaging. Concepts in Magnetic Resonance Part A. 2010; 36A: 223–242. Gibbs SJ, Morris KF, Johnson CS, Jr. Design and Implementation of a Shielded Gradient Coil for PFG NMR Diffusion and Flow Studies. J Magn Reson. 1991; 94: 165–169. Carlson JW. Compact transverse magnetic gradient coils and dimensioning method therefor. United States Patent #4,755,755, July 5, 1988. Schenck JF, Hussain MA, Edelstein WA. Transverse gradient magnetic field coils for nuclear magnetic resonance imaging. United States Patent #4,646,024, Feb. 24, 1987. Golay MJE. Field homogenizing coils for nuclear spin resonance instrumentation Rev Sci Instrum. 1958; 29:313–316. While PT, Forbes LK, Crozier S. Designing gradient coils with reduced hot spot temperatures. J. Magn. Reson 2010; 203: 91–99. Hegde S, Maneesha K, Supreetha S, Mahipal R. Simulation, design and implementation of magnetic field gradient coils for earth’s field MRI. IRJET. 2018; 5: 1370–1373.

9.4 Gradient current generation and safety considerations Throughout the years, gradient currents have been generated in many ways. Early on, one used a lead-acid car battery as a monopolar current source, due to their quick, stable and massive current supply capability (a few 100 A “instantaneous” current is typically listed). A MOS FET transistor arrangement under digital control in some timer or computer form would typically be used for gating the battery current.

9.5 Home-brew gradient control

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Normally there is absolutely no need for such high gradient currents; a few amperes of current is often more than adequate for the purpose. Stability and reproducibility are much more important factors. In our early work around 1980, we used less than 1 A gradient currents, corresponding to about 1 G cm−1 gradient strength. The original gradient driver is illustrated in Figure 9.2. Today, one should strive for at least a basic 5–10 A current capability or similar. Top of the line units may provide 60 A at the time of writing. An important issue is the actual gradient coil vulnerability to overcurrent damage. Consider, for example, Price’s coil illustrated in Figure 9.5. It had a resistance (R) of 0.2 Ω, and it seems it was typically used with a current (I) of 5 A. The power dissipation then becomes R × I2 = 0.2 × 52 = 5 W. Similarly, AWG 22 copper wire, used in the gradient coils in Figure 9.3, is listed to have a resistance of 50 mΩ per meter. Putting 5 A through 1 m of it would thus dissipate 1.25 W. Altogether, this does not sound much, but for a tightly wound many-turn coil in a confined space like a probe, heat build-up could quickly become an issue. Also, the coil resistance would change with temperature, meaning that if the current source is voltage controlled, the current (and thus the PGSE gradient) will change with time as a consequence of coil heating. As a rule of thumb, many consider a continuous gradient coil current level of 1 A to be “safe” from a burnout or damage point of view. Note, however, that the relation R × I2 implies a squared current dependence, and that for a 10 A current level, one can only allow a 1% gradient duty cycle, relying on that same criterion. Having a protective series fuse in the coil circuit should be considered a mandatory precaution. Apart from gated gradient current sources, there are also “gradient amplifiers,” meaning that an initial signal shape of some form gets amplified into voltage over or current through gradient coils, all dependent on design. The current-controlled variant would obviously be the best, but historically voltage-driven ones have been much more common. Modified (DC-coupled or other) powerful audio amplifiers, originally designed for rock concerts and similar have found use here. Such a design was described by Galvosas et al.[14], which was capable of producing bipolar gradient currents in excess of 100 A at 300 V across the gradient coils. Problems introduced with such an approach include balancing and matching positive and negative gradient pulses and to blank the output current to exactly zero between actual gradient pulses. This may not be so easy in practice and requires additional circuitry.

9.5 Home-brew gradient control Commercial spectrometers generally should have some digital BNC-type outputs that are addressable through the normal pulse program code, providing 5 V trigger or gating signals to external gradient units. This would provide timing information for the gradient pulses. Changing the gradient strength is more

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complicated and so is shaping pulses to the desired form – steep rectangular ones are generally not recommendable. Also, actual gradient current flowing through the coils rather than applied voltage across them really matters. Provisions for bipolar gradient capability while preserving gradient pulse matching and truly blanking gradient output between gradient pulses in such a set up are even more challenging. Personally, I originally used spectrometer timing-gated low-gradient monopolar equipment back in the 1970–1980s. Pulse imperfections or eddy current disturbances were not noticeable in practice. Gradient control was limited to duration changes at a fixed gradient strength. Later, we gradually merged to more powerful gradient driver unit, which was triggered by spectrometer software logic pulses to an external connector, further driving PC software, which in turn controlled the duration and strength of actual gradient pulses. The unit was still limited to basically rectangular pulse output, but had bipolar capability. The current stage is standard spectrometer software digital control of optionally bipolar gradient pulse shape, eddy current corrections, gradient blanking, strength and duration – including hybrid spectrometer control of an optional nonvendor supplied electrophoretic current generator. Realistically available home-brew alternatives today would be to (a) go for the relatively simple original option of timing control of gated gradient pulses from an external unit. With current capability of a few amperes, it just requires a few transistors, resistors, connectors, power supply and a simple integrated circuit. Alternative (b) would be to use spectrometer triggering of a software-controlled external gradient unit. A suitable external front-end environment is National Instruments LabView. We previously used it for electrophoretic current control, as illustrated in Figure 8.14. Academic environments may have site licenses for this very powerful, but otherwise slightly expensive software. A basic software license is otherwise bundled with controller and data acquisition card or unit purchases. A lower-cost alternative is to use microcontrollers of various kinds. Presently, the Arduino family is highly popular for various hobbyist and commercial applications. It interfaces to, for example, Matlab for higher-level control, and is also supported by very active user forums on the Internet.

9.6 Electrophoretic NMR (eNMR) Except for specialized measurements that can be done at voltages below 100 V, any home-brew design and assembly attempts must strongly be discouraged. It is simply very dangerous and potentially lethal to deal with voltages that can reach over 1,000 V, at current capabilities of several tens of milliamperes. Professionally designed individual building blocks like voltage amplifiers and computer-controllable high-voltage sources are available commercially, but still one has to consider safety in all steps of interconnection and actual use.

9.7 Considering complete home-brew NMR systems

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9.7 Considering complete home-brew NMR systems As touched upon in several sections earlier, building or assembling complete NMR systems is much more realistic than it used to be, except for high-resolution, highmagnetic field ones. Modules such as mixers, amplifiers, duplexers and so on or larger building blocks like Halbach-type magnets are nowadays readily available from many sources, and at quite a reasonable price tag. There is also a considerable second-hand market on the Internet for units such as otherwise very expensive lock-in amplifiers and similar. As discovered by many, new doors were also opened through software-defined radio (SDR) units of various kinds. Figure 9.6 illustrates a possible software/hardware realization of such a unit.

Linux

Fast IO bus

Ethernet

FPGA Generate digital rf pulse, phase amplitude recycle

Decimation filter Quadrature detection

SDR Generate analogue rf

Digital time Domain signal

N Power amp

S Pre amp

Figure 9.6: A possible logistic interconnection of modules for a basic SDR-type digitally controlled pulsed NMR spectrometer unit.

The advance in digital data processing through devices based on readily available field programmable gated array hardware (FPGA) combined with suitable analog hardware has made it possible to design sophisticated rf devices. An FPGA can be incorporated with a single-board computer and programmed to produce digital outputs

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and receive digital inputs with nanosecond resolution. It typically supports on-chip memory, external memory interfaces and a rich set of peripheral connectivity interfaces. The fast input and output can be programmed to generate digital rfs with accurate phase, pulse length and amplitudes in a range useful for NMR and also to analyze incoming digital information in real time. Combined with suitable digital-to-analog and analog-to-digital converters, a complete RF system can be built. In combination, these devices are known as SDR. An example where these devices can be used to great effect is in low-field NMR, originally in the frequency range 1–50 MHz [15, 16]. Nowadays, analog data from the NMR spectrometer can be digitized directly at the Larmor frequency by using a 12 bit 250 MHz ADC (analog-to-digital converter). It can then be filtered and demodulated digitally with code embedded in the FPGA and further be linked directly to a Linux host by a fast internal bus. Preferably, the NMR machine should be run seamlessly with code in a higher level-programming environment such as Matlab or Visual Studio.

9.8 Further reading Michal CA. A low-cost multi-channel software-defined radio-based NMR spectrometer and ultra-affordable digital pulse programmer. Concepts Magn Reson. Part B. 2018; e21401. https://doi.org/10.1002/cmr.b.21401 Webber JBW, Demin P. Credit-card sized field and benchtop NMR relaxometers using field programmable gate arrays. Magn Reson Imaging. 2018; in press: https://doi.org/10.1016/j.mri.2018. 09.018 Ariando D, Chen C, Greer M, Mandal S. An autonomous, highly portable NMR spectrometer based on a low-cost System-on-Chip (SoC). J Magn Reson. 2019; 299: 74–92.

9.9 Internet resources Numerous sources of information regarding scientific instrumentation of various kinds exist on Internet in the form of discussion groups and user forums. Participants are often very helpful in answering questions and may have deep knowledge of relevant areas. Links to very valuable information indeed can also be found via a simple Google search. Often there are complete construction details regarding advanced hobbyist or truly scientific equipment, including NMR or MRI (Figure 9.7). There are also digests and organized information in perhaps unexpected locations like Pinterest, provided one has registered a user profile of relevant areas. Two hardware-related forums and general information sources that are quite active at the time of writing are: http://www.sciencemadness.org and http://www.appro pedia.org/Category:Open_source_scientific_hardware

OpenLabTools - University of Tekla Lab - berkley’s Cambridge initiative in open- initiative for a library of opensource hardware for science source DIY quality scienctific lab equipment Red Pitaya - Opensource measurement and control tool

Raspberry Pi - Creditcard sized computer running Linux

Arduino - A class of open source microcontrollers useful for automating equipment

Figure 9.7: Partial screenshot of a graphical menu page at http://www.appropedia.org/Category:Open_source_scientific_hardware.

ECHO Open – Echo stethescope

Plant tissue culture systems from U. of Guelph

UPB smartphone spectrometer

Openly Published Environmental Sensing Lab at OSU -rain catchment, wind vane, soil moisture, etc.

Libre Space Foundation has both DIY satellite ground station and OS satellite UPSat

EEZ OS bench power supply

Open Source Imaging NMR, MRI, EMF

Marchetto Lab - opensource sensors and instrumentation

free loader mechanical testing machine

Cambridge JIC 3D printable programmable digital microscope

9.9 Internet resources

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One should also be aware of software-oriented Internet resources such as SourceForge and Github, found at https://sourceforge.net and https://github.com. Also in the NMR field, potentially very useful material can be found here at no cost.

References [1] [2] [3]

[4] [5]

[6]

[7] [8] [9] [10]

[11] [12] [13] [14]

[15] [16]

Aue, WP., Bartholdi, E., & Ernst, RR. Two-dimensional spectroscopy. Application to nuclear magnetic resonance. J Chem Phys. 1976; 64: 2229–2246. Price, WS. Gradient NMR. Ann Rep NMR Spectrosc. 1996; 32: 51–142. James, TL., & McDonald, GG. Measurement of the self-diffusion coefficient of each component in a complex system using pulsed-gradient Fourier transform NMR. J Magn Reson. 1973; 11: 58–61. Bangerter, BW. A simple pulsed field gradient circuit for high resolution NMR spectrometers. J Magn Reson. 1974; 13: 87–93. Stilbs, P., & Moseley, ME. Nuclear spin-echo experiments on standard Fourier transform NMR spectrometers. Application to multicomponent self-diffusion studies. Chem Scr. 1979; 13: 26–28. Stilbs, P., & Moseley, ME. Multicomponent self-diffusion measurement by the pulsedgradient spin-echo method on standard Fourier transform NMR spectrometers. Chem Scr. 1980; 15: 176–179. Callaghan, PT., Trotter, CM., & Jolley, KW. A pulsed field gradient system for a Fourier transform spectrometer. J Magn Reson. 1980; 37: 247–259. Callaghan, PT., Jolley, KW., & Trotter, CM. A pulsed field gradient system for use with a JEOL FX60. JEOL News. 1980; 16A: 48–50. Callaghan, PT. Pulsed field gradient nuclear magnetic resonance as a probe of liquid state molecular organization. Austr J Phys. 1984; 37: 359–387. Moseley, ME., & Loewenstein, A. Anisotropic translational diffusion of methane and chloroform in thermotropic nematic and smectic liquid crystals. Mol Cryst Liq Cryst. 1982; 90: 117–144. Zupancic, I, & Pirs, J. Coils producing a magnetic field gradient for diffusion measurements with NMR. J Phys E, Sci Instrum. 1976; 9: 79–80. Hrovat, MI., & Wade, CG. NMR pulsed gradient diffusion measurements. II. residual gradients and lineshape distorsions. J Magn Reson. 1981; 45: 67–80. Von Meerwall, ED., & Kamat, M. Effect of residual field gradients on pulsed-gradient NMR diffusion measurements. J Magn Reson. 1989; 83: 309–323. Galvosas, P., Stallmach, F., Seiffert, G., Kärger, J., Kaess, U., & Majer, G. Generation and application of ultra-high-intensity magnetic field gradient pulses for NMR spectroscopy. J Magn Reson. 2001; 151: 260–268. Tang, W., & Wang, W. A single-board NMR spectrometer based on a software defined radio architecture. Meas Sci Technol. 2011; 22: 015902. Chen, HY., Kim, Y., Nath, P., & Hilty, C. An ultra-low cost NMR device with arbitrary pulse programming. J Magn Reson. 2015; 255: 100–105.

10 Other more recent texts and reviews on NMR diffusometry Apart from those already mentioned in the Preface and in various chapters, there are additional valuable books and specialized or general reviews in this field. More recent ones are listed below; reference links to many older ones can be found in my 1987 review [1], and among references listed later. Apart from his year 2009 book already mentioned in the Preface, Price and coworkers have provided extensive general reviews on the subject a decade earlier [2–4], as well as some more brief ones [5, 6]. Excellent chapters related to and covering NMR diffusion techniques and concepts are also found in Kimmich’s unique monograph from 1997 [7] and in a multiauthor chemical engineering-oriented book, edited by Stapf and Han in 2006 [8]. Claridge wrote in 2009 a good and well-balanced introduction to NMR diffusometry, which is particularly suitable for those working in organic chemistry [9]. The 2002 review by Weingärtner and Holz leans more toward physicochemical issues, and appears perfectly balanced and informative regarding both NMR and their application focus [10]. Many who seek information and guidance related to NMR diffusometry methodology would likely overlook these three sources. A previous general review by Charles S. Johnson in Encyclopedia of Magnetic Resonance was updated in 2011,

Figure 10.1: The author (left), in company with prominent NMR diffusionists and book authors (left to right) Rainer Kimmich, William S. Price and Jörg Kärger, awaiting refreshments after a session at the Leipzig 15th European Experimental NMR Conference (EENC) in the year 2000 (Private photo).

https://doi.org/10.1515/9783110551532-010

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through coauthorship with Donghui Wu [11]. It is published in the electronic continuation of that encyclopedia set, which is named eMagRes and constitutes brief but still quite complete coverage of the field. A decade earlier, Johnson also wrote a review that specifically discusses the DOSY concept [12]. Kärger and coauthors have covered general diffusion-related concepts and NMR diffusometry on porous and heterogeneous materials in great detail [13, 14]. A 2005-year review by Cohen et al. is focused on supramolecular chemistry and lists a few 100 references to application papers and basic PGSE background material [15]. A biomedically oriented review by Nicolay and coworkers appeared in 2001 [16]. Among recent PGSE-related reviews, a chapter in a review on food structures by Gross et al. [17] and another analytically focused one by Pagès et al. [18] are particularly relevant and updated. The latter one has an extensive list of later-year applications of NMR-based diffusion measurements. Several background articles for PGSE and NMR in general can also be found in the educationally oriented journal Concepts in Magnetic Resonance. Diffusion is covered in a wide range of specialized open-access proceedings articles for the semiannual conference series Diffusion Fundamentals at the web site http://diffusion.uni-leipzig.de, in a much wider sense than NMR-based alone. Several Chapters of ref [13] (Diffusion in Condensed Matter, 3rd Ed) also describe the concept of diffusion in a broader sense, as well as alternative methods for study than NMR. There is also a nice review here on electrophoretic NMR (Ch 17, pp 717–744), by its pioneer Manfred Holz.

References [1] [2] [3] [4]

[5] [6]

[7] [8] [9]

Stilbs, P. Fourier transform pulsed-gradient spin-echo studies of molecular diffusion. Prog Nucl Magn Reson Spectrosc. 1987; 19: 1–45. Price, WS. Gradient NMR. Ann Rep NMR Spectrosc. 1996; 32: 51–142. Price, WS. Pulsed‐field gradient nuclear magnetic resonance as a tool for studying translational diffusion: Part 1. Basic theory. Concepts Magn Reson Part A. 1997; 9: 299–336 Price, WS. Pulsed‐field gradient nuclear magnetic resonance as a tool for studying translational diffusion: Part II. Experimental aspects. Concepts Magn Reson Part A. 1998; 10: 197–237 Price, WS. NMR Diffusometry. In: GA Webb, ed. Modern Magnetic Resonance. Springer International Publishing, 2017:1–17. Stait-Gardner, T., Torres, A., Zubkov, M., Willis, SA., Zheng, G., & Price, WS. Diffusion NMR: A tool to investigate the dynamics of organic systems. Current Organic Chemistry. 2018; 22: 758–768. Kimmich, R. NMR Tomography Diffusometry Relaxometry. Berlin, Heidelberg, New York: Springer, 1997. Stapf, S., & Han, S-I. NMR Imaging in Chemical Engineering. Weinheim: Wiley-VCH, 2006. Claridge, TDW. Tetrahedron Organic Chemistry Series- High-Resolution NMR Techniques in Organic Chemistry, 3rd Ed – Chapter 10 – Diffusion NMR spectroscopy (pp 303–344). Elsevier Science, 2016.

References

317

[10] Weingärtner, H., & Holz, M. NMR studies of self-diffusion in liquids. Ann Rep Progr Chem, Sect C. 2002; 98: 121–155. [11] Johnson, CS., & Wu, D. Diffusion measurements by magnetic field gradient methods. eMagRes. https://doi.org/10.1002/9780470034590.emrstm0118.pub2 2011; 1–24. [12] Johnson, CS. Diffusion ordered nuclear magnetic resonance spectroscopy: principles and applications. Progr Nucl Magn Reson Spectrosc. 1999; 34: 203–256. [13] Heitjans, P., & Kärger, J. Diffusion in Condensed Matter. Methods, Materials, Models, 3rd Ed. Springer, Berlin heidelberg, 2005. [14] Kärger, J., Ruthven, DM., & Theodorou, DN. Diffusion in Nanoporous Materials. Wiley-VCH Verlag, 2012. [15] Cohen, Y., Avram, L., & Frish, L. Diffusion NMR spectroscopy in supramolecular and combinatorial chemistry: An old parameter – New insights. Angew Chem-Int Ed. 2005; 44: 520–554. [16] Nicolay, K., Braun, KP., Graaf, RA., Dijkhuizen, RM., & Kruiskamp, MJ. Diffusion NMR spectroscopy. NMR Biomed. 2001; 14: 94–111. [17] Groß, D., Zick, K., & Guthausen, G. Recent MRI and diffusion studies of food structures. Ann Rep NMR Spectrosc., 2017; 90: 145–197. [18] Pagès, G., Gilard, V., Martino, R., & Malet-Martino, M. Pulsed-field gradient nuclear magnetic resonance measurements (PFG NMR) for diffusion ordered spectroscopy (DOSY) mapping. Analyst. 2017; 142: 3771–3796.

Appendix: Quantifying self-diffusion-related spin echo magnetic field gradient-induced spin echo attenuation and deriving the basic pulsed-gradient Stejskal–Tanner relation In this context one often uses various approximations, related to self-diffusion during “short” magnetic field gradient pulses. Either one mathematically neglects pulse duration (like δ=3) in comparison with pulse separation ðΔÞ, when deriving expressions that quantify their influence on echo attenuation, or physically and mathematically one “neglects” self-diffusion during the “comparably short” gradient pulses in comparison with their separation, possibly by including some firstorder correction to the Δ-related attenuation effect. Some nomenclature confusion is seen in this area. As discussed in Chapter 2, the time probability distribution for Gaussian diffusion in one spatial dimension ðxÞ is given by 1 2 pðx, tÞ = pffiffiffiffiffiffiffiffiffiffiffi e − x =4Dt 4πDt

(A:1)

For a situation like in the upper part of Figure A.1 with (infinitely) short magnetic field gradient pulses (often named the SGP (short gradient pulse) approximation), a spin-echo signal attenuates in a relative sense with x-gradient ðgÞ induced phase dispersion ðϕÞ during a selected time period ðΔÞ as ð ð 1 2 (A:2) A = pðx, tÞϕðxÞdx = pffiffiffiffiffiffiffiffiffiffiffiffi e − x =4DΔ eiγgδx dx 4πDΔ x

x

For zero gradient strength, the second factor in the integral becomes 1, and so does the overall expression. For a finite value of the gradient strength, phase-dispersionrelated signal attenuation will be described by ð 1 2 A = pffiffiffiffiffiffiffiffiffiffiffiffi e − x =4DΔ eiγgδx dx = 4πDΔ x "ð # (A:3) ð 1 − x2 =4DΔ − x2 =4DΔ pffiffiffiffiffiffiffiffiffiffiffiffi e · cosðγgδxÞdx − i e · sinðγgδxÞ 4πDΔ x

x

The imaginary term becomes zero, since it contains an asymmetric function that is integrated between − ∞ and + ∞. Evaluating the remaining real term leads to the result https://doi.org/10.1515/9783110551532-011

Appendix

319

Ðt Figure A.1: The time course of the integral gðt′Þdt′ which generally describes the area of applied 0

gradient and describes the amount of signal phase gradation. The actual pulse sequence considered is a basic two-pulse PGSE one, using notation as in the text below. The example echo attenuation derivations in this appendix largely follow the outline by Mori and Tournier [2], pp. 20–21. Also note a supplementary visualization in Figure 2.29 of Chapter 2.

A = e−γ

2 g2 δ2 DΔ

(A:4)

In the derivation above one has really assumed that self-diffusion monitoring starts at time t1 and finishes at time t3, referring to Figure A.1 (since the gradient pulses were assumed infinitely narrow). The SGP approximation step was thus sneaked in in eq. (A.2) by replacing a variable time (t) with a constant ðΔÞ. Gradient gradation effects on spin dispersion were in reality contained in the product of g and δ, which is regarded as a “gradient area.” This is consequently considered being a finite quantity, although the gradient shape is regarded as infinitely narrow. PGSE experiments on heterogeneous and ordered systems are normally interpreted within such a simplified framework, using more or less complex equations that are still based on the same type of approximation. Also, spin gradation from other than rectangular magnetic field gradient pulse shapes (ramped, sinusoidal, etc.) may often be quantified in terms of their g(t) “gradient areas,” regardless of their actual shape. See also Section 5.6.3 in general, and Section 7.5.2 for a specific example of a heterogeneous system, which was structurally characterized within the PGSE SGP approximation. PGSE experiments on isotropic solutions, as well as on heterogeneous systems, are typically run under conditions of gradient pulse lengths ðδÞ like 2–10 ms, at time separations ðΔÞ of 20–100 ms. Nowadays, gradient shapes other than the originally truly rectangular (e.g., ramped or sinusoidal) are often used. Often one here applies another type of “short gradient pulse” approximation; again equating the gradation effect of the pulses to be equivalent to that of their g(t) “gradient areas.” Effectively one then inserts a corresponding fictive rectangular gradient pulse into

320

Appendix

the normal Stejskal–Tanner equation (2.33) and proceeds from there during the evaluation of an experiment or a simulation. For large pulse separations, this turns out to be an excellent approximation. If desired, one can instead derive and use the corresponding exact analytical expressions through the general computational scheme described below, as has already been done in several publications (see the References below). Phase gradation effects of the gradient pulses actually are a function of both location ðxÞ and time ðtÞ Ð (A:5) ϕðx, tÞ = e − iγ gðtÞtx where gðtÞ represents the time-domain shape of the gradient pulse. The proper Gaussian self-diffusion term remains the same as above, but the time variable t is no longer substituted by the constant Δ. The basic nonapproximated variant of eq. (A.2) then is ð ð 1 2 (A:6) A = pðx, tÞ · ϕðx, tÞdx = pffiffiffiffiffiffiffiffiffiffiffi e − x =4Dt eiγgðtÞtx dx 4πDt x

x

To calculate the spin-echo signal attenuation, we now need to integrate not only over location ðxÞ but also over time ðtÞ. In a very simplifying computational scheme suggested by Karlicek and Lowe [1], inverting radiofrequency pulse effects on spin precession do not directly occur. They are instead included in a concept of an “effective magnetic field gradient” where pulse effects on dephasing and rephasing by, for example, a 180° rf pulse is equated to an inversion of the sign of the actual gradient. After a somewhat lengthy derivation, Karlicek and Lowe [1] arrived at the master equation (in their own notation) 2 2 32 3 ðt ðt′ 6 7 (A:7) AðtÞ = Að0Þexp4 − Dγ2 4 gðt′′Þdt′′5 dt′5 0

0

where g(t″) represents this “effective magnetic field gradient,” external and internal. Reformulated, reindexed and specifically referring to a basic PGSE experiment with two rectangular gradient pulses and the timing notations in Figure A.1, eq. (A.7) would read t4 ð

lnðA=Að0ÞÞ = − Dγ

ðt

2 t1

0

!2 gðt′Þdt′

dt

(A:8)

References

321

Note that we are now dealing with a double integral – the inner one concerns the actual gradient pulse duration and strength shape and the outer one concerns the total gradient time influence. For illustration, we will carry through the integration steps for the original case of truly rectangular pulses. Corresponding expressions for other pulse shapes can be derived analogously (see e.g. [3] [2, 4–6]). To achieve the implied integration, one simply divides the time period t1 − t4 in Figure A.1 into three separate time segments, where inner integrals can simply be Ðt added together. For the time period t1:t2 the inner integral gðt′Þdt′ = gt, for t2:t3 it 0

becomes a constant gδ and for t3:t4 it is gδ − gðt − t3Þ. We then arrive at the final integration/summation steps through ! t4 ðt3 ð ðt2 t1

ðgδ − gðt − t3ÞÞ2 dt

g2 δ2 +

g2 t2 dt +

lnðA=Að0ÞÞ = − Dγ2

t2

(A:9)

t3

We can simplify the calculations by defining t1 as the origin of the time axis; that is, t1 = 0. Then let the time interval t1:t3 equate to Δ, so that t4 becomes Δ + δ. These three integral terms then result ðδ

1 g2 t2 dt = g2 δ3 3

(A:10)

g2 δ2 dt = g2 δ2 ðΔ − δÞ = g2 δ2 Δ − g2 δ3

(A:11)

0

ðΔ δ Δð+ δ

Δ

1 ðgδ − gðt − ΔÞÞ2 dt = g2 δ3 3

(A:12)

Adding them together directly leads to the now familiar Stejskal–Tanner relation (cf. eq. (2.33)) 1 lnðA=Að0ÞÞ = − γ2 g2 δ2 ðΔ − δÞD 3

(A:13)

Extension to more complex pulse shapes and combinations of pulses is relatively straightforward; see, for example, the references below for further details.

References [1] [2]

Karlicek, R.F. Jr., Lowe, I.J. A modified pulsed gradient technique for measuring diffusion in the presence of large background gradients. J Magn Reson.1980; 37: 75–91. Mori, S., Tournier, J.-D. Introduction to diffusion tensor imaging, 2nd Ed. Oxford: Elsevier and Academic press, 2014.

322

[3] [4] [5] [6]

Appendix

Price, W., Kuchel, P. Effect of nonrectangular field gradient pulses in the Stejskal and Tanner (diffusion) pulse sequence. J Magn Reson. 1991; 94: 133–139. Merrill, M.R. NMR Diffusion measurements using a composite gradient PGSE Sequence. J Magn Reson. 1993; 103: 223–225. Kuchel, P.W., Pagès, G., Nagashima, K. et al. Stejskal-Tanner equation derived in full. Concepts Magn Reson. 2012; 40A: 205–214. Röding, M., Nydén, M. Stejskal-Tanner equation for three asymmetrical gradient pulse shapes used in diffusion NMR. Concepts Magn Reson. 2015; 44A: 133–137.

Index 111

Cd2+ 147 113 Cd2+ 147 133 Cs 292 133 Cs+, 147 13 C FT-PGSE 81 15-Crown-5 291 18-Crown-6 290 19 F 117, 129 – eNMR 297 1D imaging 50 2D imaging 51 2D-maps of diffusion vs spin relaxation 227 2rr file 161 31 P 117 3-dimensional networks 147, 149, 294 7 Li 292 7 Li+ 147 9 Be2+ 147 absorption 23 absorption and dispersion mode 269 – mixed 269 absorption mode 269 adsorption competition 140 adsorption phenomena 135 adsorption studies 135 affinity NMR 189 aggregation number 124 Agilent 155 aliasing 20 ambient temperature 180 AMP (adenosine monophosphate) 112 amplifier 26 – lock-in 26 ANSYS 247 Arduino 310 – microcontroller 310 Arduino microcontroller 165 association 102 associative thickeners 147 attack and decay form 157 average propagator 258 background gradients 172, 175 bandshape constancy 200 bandshape distortions 205 https://doi.org/10.1515/9783110551532-012

barriers to diffusion 186 – semi-permeable 186 baseline inconsistencies 198 baseline-flattening algorithms 171 baseline-flattening procedures 199 battery science 276 Battery-related research 299 Bench-top spectrometers 162 biased data fitting 73 bilinear 203 binary systems 103 Binding 102 binding affinity 190 binding constants 111 biological systems 258 biomedical applications 316 bipolar gradient capability 310 bipolar gradient currents 309 bipolar gradient pulse pairs 261 Bipolar (gradient) Pulse Pair (BPP) 178 blanking gradient output 310 Bloch-Torrey analysis 69 block copolymer micelles 150 blood flow 262 Boltzmann distribution 15 Boltzmann-like energy distribution 280 BPP-LED pulse sequence 178 Brownian motion 30, 36 Bruker 155 Bruker Minispec 165 Bruker-type iron magnet probes 306 – adding gradient coils to 306 C12E5 (dodecyl penta(ethylene oxide)) 135 C12E6 surfactant 167 calibration substances 74 Carr-Purcell echo train 87 cell membranes 258 cell suspensions 258 cell-like structure 257 chaotic instabilities 180 characterization 101 – diffusometry strategies 101 chemical engineering 315 chemical exchange 85, 137, 184 – effect of 184 – effect on PGSE experiments 85

324

Index

– time scale 137 chemical shift dispersion 81 chromatographic media 190 chromatographic separation 190 circularly dependent results 200 cmc 117 coherence pathway selection 89 coherent excitation 17 coherent transport processes 266 collaboration 276 – interdisciplinary 276 colloidal suspension 278 combinatorial chemistry 189 combining eNMR and PGSE diffusometry 276 commercial equipment 155 Compact NMR 165 compartmentation 120, 133 complex fluids 44 complex magnetization 22 complex pulse shapes 321 Component separation 206 Comsol Multiphysics 246 concentric double tube arrangements 181 conductivity 266 – of solutions 266 confidence intervals 235 CONTIN 226, 243 contrast 11 convection 66 convection compensation 182 – double stimulated echo sequence 182 – spinning the sample tube 182 convection-compensation 181 convective flow 275 convective overturning 170, 180 Cooley-Tukey FFT algorithm 238 cooperative binding equilibria 125 Copolymers 140 – block 140 – random 140 CORE 200 – download and manual 240 CORE algorithm 229 correlation time 33 cosine-like amplitude modulation 273 Coulomb force 278 counterion binding 121, 141, 145 counterion distribution 145

counterion-polyion association 145 CPMG 87 CPMG train 68 critical association concentration (cac) 295 critical micelle concentration 117 critical phenomenon 181 cross-relaxation 180 crown ethers 290 crusher gradient pulses 74 crusher pulses 93, 305 cryoporometry 256 Cyclodextrins 110 CYCLOPS 27 Debye length, 278 DECRA 228 DECRA code 244 decylammonium 123 degree of counterion binding 124 deuterium NMR 141 deuterium spin relaxation 149 deuteromethane 104 dextran 144 diamagnetic susceptibility variations 261 dichloroacetate 123 dielectric constant 278 dielectric relaxation 33 diffusion 32, 45, 144, 185 – in gels 144 – mutual – self-diffusion, distinction 45 – mutual 45 – restricted 185 – rotational, sphere 32 diffusion between planes 259 diffusion coefficient distribution 220 diffusion ordered spectroscopy 206 diffusive diffraction 259 diffusometry 30 – time scale 30 dimerization 102 dimerization step 113 DISCRETE 243 DISPA plot 270 dispersion 23 dispersion mode 269 displacement probability 50 disrupted micelles 133 distributions 143 – molecular weight 143

Index

dodecyl trimethylammonium bromide (DoTAB), 295 DOSY 10, 189 DOSY data presentation 128 DOSY display 195 DOSY map 190 DOSY toolbox 240 DOSY variants 190 double layer 278 – electrical 278 double PGSTE sequence 181 double-exponential fitting 217 drug discovery 189 dummy gradient pulses 171 dummy scans 171 dynamic-range problems 179 Earth’s magnetic field 165 echo 8, 55, 56, 65, 68 – 8-shape 56 – flow influence 65 – multi-pulse 68 – stimulated 8, 55 echo amplitude 51 – two-pulse sequence 51 echo proportionality with relative component concentrations 176 echoes 55 – other 55 eddy current disturbances 175 – check for 177 – diagnosing 84 – resistive iron magnet PGSE 307 eddy current influence 205 eddy current suppression 156 eddy currents 176 – influence influence of 84 effective magnetic field gradient 320 EHEC, Ethyl (hydroxyethyl) cellulose) 147 Einstein coefficients 16 Einstein relation 37, 101 electric field 266 – applied 266 electrochemical (e.g. battery) applications 297 electrochemical perspective 266 – classical 266 electrochemical techniques 277 – macroscopic, classical 277 electrokinetic phenomena 266, 276

electrolyte concentration 278 Electron Paramagnetic Resonance (EPR) 155 electro-osmosis 13, 267, 281 – build-up time 281 – spatially detected 299 electro-osmotic and conductive flow reference-compensation for 282 electro-osmotic flow 278 electrophoresis 266, 279 electrophoretic flow 269 electrophoretic mobilities 277 electrophoretic mobility 266, 267 – from signal phase change 297 electrophoretic mobility and binding information 276 electrophoretic movement 281 electrophoretic NMR 270 electrostatic considerations 116 electrostatic driving force 266 electrostatic interaction 120 electrostatic interactions 123 electrostatic repulsion 116 electrostatics 276 – molecular 276 elementary charge 278 ellipsoids 38 emulsion 175 emulsion droplets 186 endianess 237 eNMR 13 – CPMG variant of 281 – electrophoretic NMR 266 eNMR measuring cell 284 eNMR-applicability 275 entanglement 142 Environmental monitoring 108 error estimation 234 – Monte Carlo type 234 error estimation in PGSE studies 232 erythrocytes 258 Euler’s formula 22 exchange lifetimes 137 EXORCYCLE 27 exponential decay functions 200 exponential fitting 207 exponential function fitting 196 external gradients 77 extrapolation procedures in PGSE 220

325

326

Index

Fast Fourier Transformation (FFT) algorithms 237 FFT 3 Fick 37 – first and second law 37 Fick unit 36 FID 3, 26, 52 field gradient 2, 5, 8, 70, 77, 94, 176 – background 77 – crusher pulses 94 – magnetic 2, 5 – pulsed magnetic 8 – pulsed 70 – ramped or sinusoidal pulses 176 field gradients 257, 260 – internal 257, 260 field programmable gated array (FPGA) 311 field/frequency 168 – drift and oscillations 168 field/frequency lock 26, 168 first-order flow patterns 181 flow 11 flow reaction monitoring 162 flow-like transport 266 fMRI 11 foldback 20 Folded signals 25 food structures 316 Fourier transform 3 free diffusion in PGSE 259 free induction decay 26 frequency analysis 18 frequency cutoff 25 frequency-selective rf pulses 261 friction coefficient 38 friction factor 39 frictional force 266 fringe field experiments 65 fringe field NMR diffusometry 251 FT-NMR 9 FT-PGSE 9, 306 – Historical 306 gamma distribution 225 gamma distribution based exponentials 240 gamma distributions of self-diffusion coefficients 224 gases in water 104 gated gradient pulses 310

Gauss unit 50 Gaussian diffusion 185, 318 Gaussian diffusion coefficient distribution 220 Gaussian phase approximation 187 Gaussian transport model 50 geometric characterization 258 geometric confinement 258 geometrical information through PGSE 185 ghost peaks 20 GNAT (General NMR Analysis Toolbox) 240 Golay 6 – coils 6 gradient amplifiers 309 gradient areas 319 gradient blanking 157 gradient calibration 169 – reference compound 169 gradient coil 156, 169, 309 – design 156 – heating 309 – water cooling 169 gradient coil calibration 169 gradient coil overcurrent 168 – protective circuitry 168 gradient coils 155, 303 – adding 303 – Maxwell-pair z-gradient 307 – self-shielded 155 gradient current 169 – duty cycle limits 169 gradient driver 309 – design considerations 309 gradient drivers 303 gradient echo 76 gradient enhanced techniques 155 gradient pulse matching 310 gradient pulse shapes 73 – ramped 73 – rectangular 73 – sinusoidal 73 gradient pulses 309 – bipolar, matching 309 – blanking 309 gradient settings 172 – logarithmically incremented 172 graphical user interface (GUI) 159 Hadamard transformation 94 Hadley convection 181

Index

Hahn echo 52, 55, 71 Halbach magnet assemblies 162, 164 Halbach magnets 311 hardware considerations 159 heterogeneous systems 78, 257, 260, 319 HEUR type (Hydrophobically modified urethane −ethoxylate) 147, 319 high performance liquid chromatography (HPLC) 190 High Resolution (HR) MAS NMR 255 higher dimensional PGSE data sets 215 highly tuned probe-sample combinations 179 – problems 179 High-Resolution DOSY 217 high-resolution probes 156 – gradient coil equipped 156 High-voltage sources 271 Hofmeister (or lyotropic) classification 292 home-brew PGSE 303 homospoil pulses 305 host-guest complex 111 HR-DOSY 217 HR-MAS PGSE 255 hydration layers 38 hydrocarbon-fluorocarbon 129 hydrodynamic radius 38 hydrodynamic theories 38 hydrogen-bonding conditions 257 hydrogen/oxygen bubble formation 282 hydrophilic/hydrophobic 147 – alternating blocks 147 hydrophobic interaction 120, 123, 124 hydrophobically modified cellulose 147 ILT approach 226 imaginary part 23 immobile water 38 Indefinite aggregation 112, 113, 114 inertial effects for motion 266 inflow/outflow kinetics 258 inhomogeneity 20 Internet discussion groups and user forums 312 inverse Laplace transform (ILT) 226 inverted micellar structures 134 ion condensation 145 ion pairs 297 ion transport 266 Ionic liquid systems 297

Ionic liquids 296 ionic strength 278 iron magnet generation PGSE 304 irreversible echo attenuation 175 Irreversible Thermodynamics 45 isodesmic aggregation model 112 isoelectric point 116, 289 JCAMP-DX data format 237 JEOL 155 JEOL FX series 305 J-modulation 62, 175 – elimination 69 Joule heating 267, 275 kinetic data 199 kinetics 200 – analogy to PGSE experiments 200 Kohlrausch–Williams–Watts (KWW) distribution 223 Krafft point 118, 123 Kärger equations 85, 106, 136, 185 k-space 61 LabView 310 – gradient pulse control 310 LabView control circuitry 284 LabView software and hardware control unit 284 large-diameter sample tubes 180 – problems 180 lead-acid car battery 308 – current source 308 Lenz’s law 177 ligand exchange 85 light and neutron scattering 258 light scattering 47 – dynamic 47 liquid crystalline phases 119, 121 liquid crystalline systems 42 living cell suspension 175 localized excitation 261 LOCODOSY 218 log-normal distributions 224 Longitudinal Eddy current Delay (LED) 178 long-range influence of the dispersive components of the bandshapes 290 long-range nonspecific interactions 145 Lorentz force 276

327

328

Index

Lorentzian 23 low-viscosity solutions 180 – problems 180 lyotropic liquid crystal 122 Lysozyme 114 – self-diffusion 37 macro-ions 277 Macromolecule-macromolecule binding 102 magic angle 253 magnetic field gradient (MFG) NMR diffusometry 252 magnetic field gradient shielding 307 magnetization helix 59, 75 magnetogyric ratio 15 Magnitude spectra 25 Magritek 164 MAP thickeners 150 Maple 246 Mathcad 246 Mathematica 246 Matlab 244 matrix division 230 measurement artifacts 172, 179 mechanical vibrations 176 Metabolab 239 metal ion binding 292 mica stacks 186 micellar aggregates 117 micellar droplets 133 micellar incorporation 191 micellar interior 126 micellar phase 120 micellization 112, 124 Microemulsions 132 – structure 132 micro-imaging setup 159 microviscosity 39, 120 miniaturized NMR spectroscopic equipment 256 mixed micelles 129 mixed micellization 131 mixture analysis 200 Mnova NMR 239 model associative polymer (MAP) 149 modulated gradient approach 253 molecular dynamics simulations 40 molecular transport 46 – alternative techniques 46

moments 183 – zeroth and first order 183 monodisperse polymers 143 Mononucleotides 112 MOSY (mobility ordered spectroscopy) 290 MOSY-visualizations 290 MRI 9, 11 – diffusion 11 MRI or NMR Microscopy 177 multi-component FT-PGSE diffusometry 132 multi-component NMR diffusometry 191 multi-component PGSE data sets 207 multidimensional NMR 305 multilinear 203 multivariate analysis 203 mutarotation of glucose 202 mutual diffusion 31, 44, 262 – studied by PGSE 262 mutual diffusion coefficients 280 Mw/Mn-based polydispersity characterization 224 myoglobin and haemoglobin 280 nanoparticles 189 nanotubes 138 – carbon 138 narrow NMR tubes 84 Navier-Stokes equations 180 negative curvature of s semilogarithmic graph 172 Neon-21 105 Nernst-Einstein relation 276 neutron scattering 143 n-merization 112 NMR 4, 5, 8, 9, 13, 30 – continuous wave 5 – electrophoretic 13 – Fourier Transform 4 – imaging 9 – microscopy 9 – PFG 8 – pulsed 4 – sweep 4 – time scale 30 – vector model 13 NMR chromatography 128, 190 NMR diffusometry 101, 102, 190 – suitability 101 – technical formulations 102

Index

– time window 101 NMR imaging and microscopy 299 NMR mouse 165, 253 NMR probes 303 – retuning 303 – second-hand 303 NMR spectrometer on a chip 165 NMR-invisible 104 NMRlab 239 Nobel Prize 279 noble gases 105 noise map 199 non-Gaussian diffusion 258 non-linear data evaluation 196 non-linear fitting 73, 125 non-linear fitting procedures 196 non-linear least squares methods 207 non-linear set of gradient settings 171 non-perfect peak phasing 198 non-radiative 17 N-way toolbox 244 Nyquist sampling rate 22 obstruction 185 obstruction correction 43 obstruction corrections 114 obstruction effects 39, 113 Octave, FreeMat and Scilab 245 – Matlab alternatives 245 oil well logging 256 oil-in-water 133 organic chemistry 315 organometallic chemistry 116 Origin 246 O/W or W/O 134 Oxford MRX Diffusion analyzer 165 packing constraints 119, 123 palisade layer 120, 126 palladium electrodes 282 PARAFAC – parallel factor analysis 204 PARAFAC download 244 paramagnetic impurities 261 paramagnetic ions as dopants 167 partial binding 185, 256 partition equilibria 121, 127 partition equilibrium 126 PDI - polydispersity index 223 PDI (polydispersity) index 143

peak elimination 196 peak integrals vs peak amplitudes 198 PEO 145 PEO backbone 292 – proton NMR properties 292 perchlorate salts 292 – caution 292 perfluorinated surfactant 129 Permanent magnet 162 – systems 162 permanent magnet type NMR 304 permittivity of free space 278 Perrin frictional factors 33 PGSE 8, 72, 76 – CPMG variant 76 PGSE-type WATERGATE 196 PGSE-WATERGATE 107 PGSTE 72 PGSTE-LED 90 PGSTE-WATERGATE 109 phase anomalies 25 phase correction 13 – first order 23 – zero order 23 phase cycling 13, 27, 74, 86 – cogwheel 89 – selection procedures 88 phase cycling requirements 184 Phase detector 26 phase sensitive detection 87 phasing parameters 269 – p0 and p1 269 physico-chemical issues 315 platinum or palladium wire 271 plug flow 266 Pluronic® F-127 139 point by point bandshape 229 polarization phenomena 274 polydisperse polymers 176 polydispersity 219 – influence on PGSE experiments 219 polyelectrolyte systems 124 polyelectrolyte theory 276 polyelectrolytes 141, 145 polymer 140 – decorated by surfactants 295 – dynamics 141 polymer brushes or gel-like structures 279 – dampening electro-osmosis 279

329

330

Index

polymer diffusion 294 polymer melts 262 polymer polydispersity 221 polymers 135, 141 – adsorption of 135 – extended chain 141 – random coil 141 polymer-surfactant interaction 294 – theories for 295 poly(acrylic acid) (PAA) 146 poly(ethylene oxide), 141 Poly(ethylene oxide) 145 poly(methacrylic acid) (PMA) 146 poly(methyl metacrylate) 141 polystyrene latex particles 135 populations 15 pore radius 187 pore-filling liquid 187 porous and heterogeneous materials 316 porous matrix 262 positive amplitude bias 198 positive curvature of a semilogarithmic graph 172 potassium dodecanoate (KC12) 295 precision and accuracy 101 pre-emphasis 157, 177 principal component analysis (PCA). 203 prior knowledge 200, 206 probability distribution 36 projection-reconstruction 51 prolate 38 proton migration 41 pseudo phase separation 120 pseudophase view 128 Pt-black 282 q-space 61 quadrupolar nuclei 81 quadrupole moment 80 – effect of 80 Quantification 101 – diffusometry strategies 101 quantification through association constants and related quantities 195 radiation damping 85, 172, 179 radiofrequency gradient diffusometry 250 ramped 319 random walks 30

Rayleigh-Bénard number 180 real part 23 rectangular pulses 321 RECORD processing 218 regular solution theory 131 relaxation 20 – transverse 20 relaxivity 256 – of surfaces 256 relaxometry 35 – field cycling 35 reorientational dynamics 120 – chain 141 reorientational rates 17 restricted diffusion 186 reversed micellar structures 119 reviews 315 – specialized or general 315 Reynolds number 266 rf interference sources 272 Rhodamine 6G 108 rod-like micellar structures 118 rod-like objects 38 rotating flip-flop frame 70 rotating frame 13, 17 rotational ambiguity 200, 203 Rouse type motion 142 sample conductivity 275 sample filling factor 273 sample temperature 170 Sample vibration 176 sample viscosity 181 sample volume 86 – advice 86 sapphire NMR tubes 181 scaling and polydispersity 143 scaling parameters 221 scaling relations 142, 221 SCORE (Speedy COmponent REsolution) 230 screening effects 277 SDR design 163 – NMR spectrometer 163 SDS 149, 295 SDS micellar solution 133 SDS-PAGES electrophoretic separation 289 sedimentation rate 280 self-aggregation 40 self-diffusion 2, 31, 47, 105, 122, 143

Index

– anisotropic 122 – binding, association 48 – computer simulations 105 – fast exchange, binding 49 – polydispersity influence 143 – restricted 8 self-diffusion approach to binding phenomena 105 self-shielding 156 semilogarithmic plots 72, 78 semi-logarithmic Stejskal-Tanner plots 196 sinusoidal 319 separating proteins or DNA fragments 288 separation of proteins via electrophoresis 288 SGP approximation 187, 319 SGP (short gradient pulse) limit 73 Shannon theorem 22 Shielded gradient coil 177 Shigemi NMR tubes 182 shimming 6 SigmaPlot 246 sign conventions 93 – frequencies and phases 93 signal loss 261 SILT-DOSY (Simultaneous Inversion of Laplace Transform-DOSY) 227 single board NMR spectrometer 165 Size-resolved NMR 206 slipping 38 – boundary condition 38 slow exchange 257 slow exchange situation 137 sodium decyl sulphate (SDeS) 129 sodium dodecyl surfactant (SDS) 150 sodium perfluoro octanoate (SPFO) 129 Software Defined Radio (SDR) 162 software-defined radio (SDR) 311 solid phase 256 solubilizate 125, 126 solubilization 125, 128, 139, 191 – alcohols 128 – micellar 125 solution conductivity 267 solution theory 252 solvent peak problems 179 solvent suppression remedies 179 sonication 139 spatial quantification 262 spatially localized excitation 261

331

spectral noise 25 spectral off-line processing 236 spectral phasing procedures 269 spectral signal separation in PGSE data sets 195 spectrometer data formats 239 spin echo 2 – Hahn 2 spin phase graphing 59 spin relaxation 13, 33, 34, 36, 56, 72, 176 – dipolar 33 – intermolecular 34 – intramolecular 33 – longitudinal 56 – quadrupolar 33 – relation to molecular size and tumbling rate 176 – theory 36 – transverse, longitudinal 13 – transverse 72 spin-spin couplings 4, 63 split peak effect 258 SPLMOD 243 Spontaneous emission 17 squared current dependence 309 stacking 112 static-gradient NMR diffusometry 250 statistical averaging 50 statistical weight 73 Steady-gradient studies 65 steady-state velocity 266 Stejskal-Tanner relation 71, 168 – derivation 73 step-wise (isodesmic) phenomenon 120 sticking 38 – boundary condition 38 stimulated echo 56, 93 – phase cycling 93 – shape 56 Stokes-Einstein-Sutherland equation 136, 137 Stokes-Einstein-Sutherland relation 38 STRAFI 250 – stray field diffusometry 250 Stray field diffusometry 261 stretched KWW exponentials 240 stroke 11 structure-breakers 133 substrate binding 85 supramolecular chemistry 316

332

Index

surface relaxation 256 surface to volume (S/V) ratio 256 surface treatment against electro-osmosis 273 surfactant aggregation 294 surfactant-polymer complexes 294, 295 surfactants 117 – adsorption of 135 susceptibility differences 257 susceptibility matched glass 182 susceptibility-related artifacts 167 suspension 257 sweep width 25

tracer diffusion 41 transference numbers 277 transferring spectrometer data 237 transition probabilities 16 translational diffusion 17 trans-membrane transport 258 triple gradient amplifier/coil 158 triple-gradient (x,y,z) facilities 262 tube/reptation model 142 tune and match 167 two-site relation 124 two-site system 106

technical formulations 121, 139 technical-industrial systems 261 temperature control 83 temperature control in NMR 180 temperature difference across the sample 180 – problems 180 ternary phase diagrams 121 Tesla unit 50 test sample 167 – choice 167 tetramethyl ammonium ion (TMA+) 146 Tetramethyl silane 123 thermal equilibrium 15 thermal gradients 181 thermostating airflow 275 theta-solvent 141 three-way decomposition 204 time-averaged free-bound model 102 time-gradient shape 73 time-reversal 52 titration curves 111 TMS 123 TMS as a marker 133

ultracentrifugation 280 Ultrahigh-Resolution DOSY 217 universal NMR experiments 239 unrestricted diffusion 258 U-tube geometry 271, 272 Varian 4, 155 vector model 13, 17 velocity 40 – autocorrelation 40 velocity autocorrelation 252 viscosity 38 voltage-controlled 309 water-in-oil 133 Xenon-129 105 yeast cells 186 zeta potential 278 z-only diffusion probes 158