Nanoparticles: Optical and Ultrasound Characterization 9783110267341, 9783110265910

Citation preview

Anatol M. Brodsky Nanoparticles

Anatol M. Brodsky

Nanoparticles Optical and Ultrasound Characterization

DE GRUYTER

Anatol M. Brodsky, Professor Emeritus University of Washington Department of Chemistry 36 Bagley Hall Seattle, WA 98195 USA

ISBN 978-3-11-026591-0 e-ISBN 978-3-11-026734-1 Library of Congress Cataloging-in-Publication Data Brodsky, Anatol M.   Nanoparticles : optical and ultrasound characterization / by Anatol M. Brodsky.    p. cm.   Includes bibliographical references and index.   ISBN 978-3-11-026591-0 (alk. paper)   1.  Nanoparticles.  I. Title.   TA418.9.N35B76 2011620.5   620’.5—dc23 2011032052 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de. © 2012 Walter de Gruyter GmbH & Co. KG, Berlin/Boston Typesetting: Apex CoVantage, LLC. Printing: Hubert & Co. GmbH & Co. KG, Göttingen U Printed on acid-free paper Printed in Germany www.degruyter.com

Preface

Recent advances in the experimental and theoretical description of light and ultrasonic waves scattering in media with hard and soft nano nonuniformities (objects with nanometer dimensions) are considered here. Described results are important for the perfection of the fast in-line control of the industrial processes, medical diagnostics, biological studies, construction of quantum information devices and environmental monitoring. The development of the theory described in this book would be impossible without experimental results of graduate students B. Anderson, S. Smith, G. Mitchell, M. Hamad, and S. Ziegler under the supervision of Professor L. Burgess at the University of Washington Chemistry Department. The central results in ultrasonic diffraction spectroscopy of media with nanoparticles were received and described in articles by Professor M. Stautberg Greenwood at Pacific Northwest National Laboratory and her coauthors, including the author of this book, who proposed the ­described theory of the corresponding effects. For help in the difficult work of the preparation of the manuscript the author is obliged to J. Forster, S. Brodsky, and M. Oakley. Seattle, August 2011

Anatol M. Brodsky

About the Author

Professor emeritus Dr. Anatol M. Brodsky received his BS in physical chemistry at the Moscow State University (USSR) in 1948; his PhD in physical chemistry at the Institute of Petrochemistry of the USSR Academy of Science, Moscow, in 1953; and his DSc in chemical physics at the Institute of Chemical Physics of the USSR Academy of Science, Moscow, in 1960. Professor Brodsky was the leading scientist at the Institute of Electrochemistry of the USSR Academy of Sciences and professor at the Moscow State University until he emigrated to the United States in 1989. From 1991 to 2007 he taught at the University of Washington in Seattle. He is currently professor emeritus.

Contents

Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v About the Author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1 1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References and notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

 oherence loss in light backscattering by media with nanoscale C nonuniformities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Comparison of the theory with the results of experiments with the Michelson interferometer . . . . . . . . . . . . . . . . . . . . . . 2.5 The influence of external static electric fields E0 on the light scattering by nonspherical nanopaticles. . . . . . . . . . . 2.6 The dynamics of phase transition in media with randomly distributed nanoparticles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 The spectroscopy of carbon nanotube networks . . . . . . . . . . . 2.8 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 References and notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 4

2

 ptical diagnostics based on coherent light transport effects in O media with mesoscopic nonuniformities. . . . . . . . . . . . . . . . . . . 3.1 Outline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Introduction and background. . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Grating wave reflection spectroscopy . . . . . . . . . . . . . . . . . . . . 3.4 Low coherence interferometry of nonuniform media. . . . . . . . 3.5 Theory of coherent transport effects in GLRS and LCI . . . . . 3.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 References and notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 7 14 19 23 24 30 31 33 49

3

53 53 54 56 61 63 65 67

x     Contents

4

 ltrasonic grating diffraction spectroscopy and reflection U techniques for characterizing slurry properties. . . . . . . . . . . . . . 4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Description of UDGS experiments. . . . . . . . . . . . . . . . . . . . . . 4.2.1 Scan-over frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Scan-over angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Measurements with the blank. . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Grating equation and critical frequency calculation. . . . . . . . . 4.4 Experimental measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Novel method for measuring the velocity of sound. . . . . . . . . 4.4.2 Possible effect of viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Shear wave reflection techniques and the measurement of viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Data analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Slope and reflection coefficient . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Data obtained during a 3-hour interval . . . . . . . . . . . . . . . . . . 4.7 Sensor calibration and reliability. . . . . . . . . . . . . . . . . . . . . . . . 4.8 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 References and notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



71 71 73 76 76 76 76 80 87 88



89 93 94 95 95 97 97

Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

1  Introduction

Coherence effects in the propagation of optical and other classical and quantum waves in nonuniform on molecular and mesascopic scales media have attracted a great deal of attention during the past 20 years. Because both constructive and destructive interference of multiple scattered waves can occur in such media, a variety of peculiar effects, including fluctuational waveguiding and wave localization, can take place. The consequences of such effects in random systems and especially in media with randomly distributed nanononuniformities (nonuniformities with nanometer dimensions) are qualitatively different from those occurring in wave scattering in uniform media. The propagation of electromagnetic waves in random structures had traditionally been described using a photon diffusion model. In typical diffusion models it is assumed that the phase information is partly or completely lost after a finite number of scattering events, as described by the transport mean free path, lmp, which is the distance electromagnetic waves travel in the medium before their phase characteristics are randomized and photon diffusion approximation can be applied. However, phase effects can in fact survive in random media after multiple light-scattering events at relatively long distances and can lead to nontrivial phenomena not predicted in the framework of diffusion theory. A possibility of manifestation of such phase-dependent interference effects in multiscattering media was suggested in pioneering works by Watson [1] and remains a subject of great interest [2–17]. The realization of such effects in optics was first experimentally confirmed in 1984 by Kuga and Ishimaru [4] and in 1985 in the works by Van Albada and Langedijk and Wolf and Maret [5,6]. In these experiments the characteristics of the back-scattered light were measured using incoherent intensity detection methods. Our study of the coherence loss in backscattered light in nanononuniformation media described in Chapters 2 and 3 can be considered as a further advancement of the previously mentioned works using coherent signal detection. Coherence effects are especially important in the study of media with both randomly and regularly distributed nanoparticles.

2   1  Introduction

In our works described in Chapter 2, the detailed coherence structure of light backscattered from a bulk system with nanoscale nonuniformities was investigated. Aqueous suspensions of noninteracting mesoscopic spherical particles were used as a simple experimental model. The measurements, collected with the help of the specifically modified Michelson interferometer, allow us to compare two parts of a split incident wave packet: a reference wave packet and a signal wave packet that has been reflected and partially destroyed after traveling in a bulk sample as a result of scattering by large-scale and short-scale nonuniformities. By changing the optical length of the travel distance for a reference wave packet before mixing both packets, it is possible to examine the characteristics of light coherently reflected from the nonuniform sample bulk after different dwell distances inside the sample. Such examinations open new possibilities for a detailed in situ optical evaluation of statistical and dynamic characteristics of multiscattering systems of nanoparticles and other mesoscopic irregular structures with dimensions from a few to hundreds of nanometers. The first theoretical problem considered in Chapter  2 is to relate light propagation characteristics with individual scatterer properties in a medium containing randomly distributed scatterers with nanometer dimensions. This difficult problem, known as the radiative transfer problem, lacks an exact analytical solution in the case of electromagnetic waves in threedimensional multiscattering media [5]. A solution for the problem of light scattering by the individual spherical hard particle in the form of infinite series over angular momentums was found more than 100 years ago by Mie [13,14]. There are three limits after approximate summations of previously mentioned series. One of the particles with the mean dimension R smaller then the wavelength is shape independent and depends in addition to R only on dielectric characteristics of the considered media. The other limit, in cases where the particle dimension is much larger then wavelength, has also a relatively simple form with only one shape-dependent numerical parameter in addition to R. We call the intermediate interval between small and large R mesoscopic, or resonance, interval because the scattering cross-sections have nonmonotonic resonant behavior with maxima and minima in this interval. The light scattering in the resonance interval could in principal provide important information about the structure and dynamic of the studied systems with nanoparticles.

1  Introduction   3

Our approach described in detail in Chapter  2 is analogous in some aspects to the introduction of Fermi’s pseudopotential in neutron scattering. It is based on an approximate expression for the effective dielectric function of the system, which is valid in the case of large numbers of randomly distributed particles. An important characteristic of this approximation is that it expresses dielectric functions through the sum of individual scattering amplitudes while taking into account interference effects. In order to compare theoretical calculations with experimental results it is necessary to have expressions for scattering amplitudes of individual particles. Especially informative is the resonance frequency interval where the wavelength of light is comparable to the particle perimeter. In Chapter 2, approximate expressions based on the Mie solution for the amplitudes of scattering by individual uniform spherical particles are used. One of the important properties of such amplitudes is an appearance of resonances calculated for spherical particles [13,14]. Such resonances, which can be considered as the specific realization of so-called Redge poles when analyzed in general wave scattering theory, also have to be present, albeit with specific structures, during the scattering by nanoparticles of different shapes and structures. They have been observed in the cross section of light scattered by individual particles. The drastic fluctuations of electromagnetic field, near both nonmetallic and metallic nanoparticles in the frequency intervals corresponding to Mie resonances where particles behave as individual resonant antennas, opens a possibility for observing and exploiting different surface-enhanced nonelastic optical transitions. Such observation has not yet fully exploited all possibilities for the detection of shape and anisotropy of nanoparticles. Chapter  3 is devoted to applications of light scattering by a special device with periodic structures (grating light reflection spectroscopy  – GLRS) for the measurement of nanononuniformity characteristics. In Chapter 4 the ultrasound scattering method for the characterization of particles with nanoscale and larger dimensions of the order of ultrasound wave lengths is described. In this chapter we take into account the experimental advantage of ultrasonic wave scattering. The relative slowness of acoustic waves (commonly between 102 and 104 m/s, which is much slower than the 3 x 108 m/s for light) and their relatively low frequency (ranging from about 1 kHz for audible sounds to 10 MHz for ultrasound, compared to 1014 Hz in the case of visible

4   1  Introduction

light) have deep consequences for the perception of these phenomena. Unlike light waves, acoustic phenomena are slow enough for the detectors to record their real-time variations, whereas optical detectors are only intensity-sensitive. Reciprocally, for transmitters, controllable wide-band ultrasonic sources (piezocomposite arrays) exist that can create any waveform, whereas optical wave packets always depend on how light emission is performed by individual nanoparticles. In a way, acoustics have the same capabilities that optics would offer if a 1015 Hz oscilloscope and a perfectly controllable light source existed. Possible applications for techniques described in this book include the analysis and control of particle suspensions, colloidal dispersions, and polymer solutions. Corresponding industries amount to tens of billions of dollars per year [5,7–11]. This analysis is also important in developing of a number of new nanotechnologies, biological applications, and medical diagnostics and therapies. To realize this potential it is necessary to solve the nontrivial, theoretical inverse problem; that is, to find theoretical expressions for the relationship between observed signals and studied system properties. We consider solutions of corresponding theoretical models in Chapters 2 through 4.

1.1  References and notes 1. K. Watson (1969). Multiple scattering of electromagnetic waves in an under dense plasma. J. Math. Phys. 10, 688. 2. V. Shalaev (1996). Electromagnetic properties of small-particle composites. Physics Reports 272, 61. 3. POAN Research Group, ed. (1998). New Aspects of Electromagnetic and Acoustic Wave Diffusion, Springer Tracts in Modern Physics 144 (Berlin: Springer Verlag). 4. Y. Kuga and A. Ishimaru (1984). Retroreflectance from dense distribution of spherical particles. J. Opt. Soc. Am. 48, 831. 5. M. P. Van Albada and A. Langedijk (1985). Observation of weak localization of light in a random medium. Phys. Rev. Lett. 55, 2692. 6. P. E. Wolf and G. Maret (1985). Weak localization and coherent backscattering of photos in disordered media. Phys. Rev. Lett. 55, 2696–2699. G. Maret and P. E. Wolf (1987). Multiple light scattering from disordered media: The effect of Brownian motion of scatterers. Z. Phys. B Condens. Matter 65, 409 – 413. 7. L. Boguslavsky (2010). The methods of nanoparticles synthesis and their size sensitive parameters. Vestnik, MITCT (in Russian) 5, N5, 3. 8. S. Eliezer, N. Elias, E. Grossman, D. Fisher, I. Gauzman, Z. Henis, et al. (2004). Synthesis of nanoparticles with fermo second laser pulses. Phys. Rev. B 69, 44 – 46.

1.1  References and notes   5

  9. L. Madler, H. Kammler, R. Muller, and S. Pratsinis (2002). Control synthesis of nanostructured particles by flame spray pyrolisis. J. Aerosol. Sci. 33, 369–389. 10. M. J. Cocero, A. Martin, F. Mattea, and S. Varona (2009). Encapsulation and co precipitation processes with supercritical fluids: Fundamentals and applications. J. Supercrit Fluids 47, 546–555. 11. Z. Y. Ma, D. Dosev, M. Nichkova, S. J. Gee, B. D. Hammock, and I. M. Kennedy (2009). Synthesis and bio-functionalization of multifunctional magnetic Fe2. J. Materials Chemistry 19, 4695. 12. Z. B. Wang, B. S. Luk’yanchuk, M. Hong, Y. Lin, and C. Chong (2004). Energy flow around a small particle investigated by classical Mie theory. Phys. Rev. B70, 035418. 13. G. Meeten (1997). Refraction by spherical particles in the intermediate scattering region. Optic Comm. 134, 233. 14. H. C. van de Hulst (1981). Light Scattering by Small Particles (New York: Dover). 15. P. Snabre and A. Arhabiass (1998). Anisotropic scattering of light in random media: Incoherent backscattering spotlight. Appl. Optics 37, 4017. 16. M. Born and I. Wolf (1999). Principles of Optics, 7th ed. (Cambridge: Cambridge University Press). 17. M. Greenwood, A. Brodsky, L. Burgess, and L. Bond (2006). Investigating Ultrasonic Diffraction Grating Spectroscopy and Reflection Techniques for Characterizing Slurry Properties. In: Nuclear Waste Management: Accomplishment of the Environmental Management Science Program. ACS Symposium Series 943, 100 –132.

2  Coherence loss in light backscattering   by media with nanoscale   nonuniformities 2.1  Introduction This chapter is organized as follows: the central experimental techniques based on application of a modified Michelson interferometer are described in Section 2.2, where the example of a characterization of model nanosystems is also considered [1]. The theory developed in the framework of the nonperturbative method is provided in Section 2.3 [2–5], with calculation details described in the Appendix. The comparison between the experimental results and theory is considered in Section 2.4. The influence of the static external electric field on light scattering by nanoparticles is considered in Section 2.5. Section 2.6 is devoted to the important problem of phase transition dynamics in systems with nanoparticles. The general description of existing optical studies of concentrated carbon nanoparticle solutions is discussed in Section 2.7. In the concluding Section 2.8 the general results are summarized and possible future applications and generalizations of the described experimental and theoretical techniques are discussed.

2.2  Experiment The Optiphase, Inc., optical coherence domain interferometer [1,2] is the main element of the experimental device shown schematically in Figure 2.1. The instrument light source is a superluminescent diode (SLD) with a narrow wave-packet bandwidth of approximately 40 nm centered around 1310 nm. A corning SMF28 single-mode optical fiber with a 9 µm core is used throughout the instrument. A circulator routes wave packets from the source to the probe and heterogeneous sample, and then directs the scattered wave packets collected by the probe to an interferometer. A triangular potential is applied to piezoelectric fiber stretchers, stretching 40 m of optical fiber wrapped around each stretcher to achieve varying

8   2  Coherence loss in light backscattering  

Fig. 2.1  Block diagram of the Optiphase, Inc., interferometer with arrows demonstrating the direction of light through single-mode optical fibers. The Piezoelectric Transducers (PZTs) and Faraday Reflector Mirrors (FRMs) effectively form two arms of a interferometer by varying the optical path distance the wave packets travel, which allows us to observe various wave-packet dwell distances, ℓ , in the suspension. E0 is the imposed constant electric field [3].

optical path differences between the interferometer arms on the order of a few millimeters. When the optical path difference between the arms is within the coherence length λc of the source (approximately 28 µm) and the wave packets from each arm are in phase, constructive interference occurs. Signal processing of the interferogram results in discrete peaks, followed by a decay pattern, as shown in Figure 2.2. The main variable in the measurements defining the signal value is the wave-packet dwell distance, ℓ , which is indicative of the optical path that ­backscattered wave packets travel in the sample matrix. The two chosen model systems, described in this section, were composed of suspensions of polymer particles at 0.01 and 0.10 volume fractions purchased from Duke Scientific Corporation [6]. The standards at

2.2  Experiment   9  

Fig. 2.2  Signal profile obtained from a 0.01 volume fraction aqueous suspension of 298 nm diameter polystyrene particles. The initial peak (A) corresponds to Fresnel reflection from the inside of the probe window, while peak (B) is the reflection from the outside of the probe window. Only photons that are coherently backscattered from the polystyrene particles and return to the interferometer are detected. In the case of highly scattering systems, this results in a tailing signal decay immediately following the reflection off the outside of the probe window [2].

0.01 volume fraction were monodispese polystyrene spheres in an aqueous solution with diameters ranging from 21 nm to 1.745 µm. The mean diameters of the nanospheres were measured using transmission electron microscopy (TEM) and are provided in Table 2.1. The deviation of sphericity for each standard is less than 1%. The standards at 0.10 volume fractions were narrowly dispersed latex spheres with particle diameters ranging from 30 nm to 2.0  µm, as shown in Table 2.2. Each suspension contained less than 0.1% of a Duke proprietary preservation and less than 0.2% of a Duke proprietary surfactant to prevent coalescence.

10   2  Coherence loss in light backscattering  

Tab.  2.1  Size characteristics of the Duke Scientific 3000-series aqueous polystyrene particle suspensions used in this work. Sample concentration was 0.01 solids by volume.

Particle diameter (nm)

Relative standard deviation of particle diameter (%)

  21   41   60   81   102   199   269   350   404   453   491   519   596   701   799   895 1101 1361 1588 1745

7.1 4.4 4.2 3.3 2.9 3.0 2.6 2.0 1.0 0.9 0.8 1.0 1.0 0.9 1.1 0.9 2.1 1.8 1.6 1.4

2.2  Experiment   11  

Tab. 2.2  Size characteristics of the Duke Scientific 5000-series aqueous latex particle suspensions used in this work. Sample concentration was 0.10 solids by volume. The particle size distribution is much broader for the 0.01 volume fraction polystyrene samples of comparable particle dimensions. Particle diameter (nm)

Relative standard deviation of particle diameter (%)

   30    60    80    90   100   160   300   340   360   430   600   670   740   870 1000 1300 2000

≤18 ≤18 ≤18 ≤15 ≤15 ≤6 ≤3 ≤3 ≤3 ≤3 ≤3 ≤3 ≤3 ≤3 ≤3 ≤5 ≤4

12   2  Coherence loss in light backscattering  

Prior to measurement, each standard suspension was inverted three times, sonicated for 30 s, and inverted three additional times. After discarding several drops of waste, approximately 0.5 mL of each sample was placed in a stirred vial for triplicate measurements following a random block experimental design. The probe was inserted directly into the suspensions to ensure measurement of the bulk of the media. The probe is composed of an optical fiber sheathed in a 0.25  in. diameter stainlesssteel tube slightly offset from a thin glass window set at an 8° angle to minimize the Fresnel reflection from the probe-medium interface. Nevertheless, sharp peaks from both the inside and the outside of the probe window appear before the tailing decay.

Fig. 2.3  Detail from decay profiles of a series of solvents. The two sharp peaks result from specular reflection off the inside and outside of the probe window, respectively. The intensity of these peaks depends on the change in effective refractive index, which occurs at each window boundary. The inset shows a calibration curve constructed from the ratio of outer to inner window peak intensities, which allows the determination of the effective refractive index of samples placed in contact with the probe [2].

2.2  Experiment   13  

Fig. 2.4  Decay profiles of a series of 0.01 volume fraction suspensions of polystyrene particles in water. Particle radii range from 10.5 to 872.5 nm. The sharp peak at the beginning of each profile is caused by the reflection off the outside of the probe window [2].

Fig. 2.5  Decay profiles of a series of 0.10 volume fraction suspensions of latex particles in water. Particle radii range from 15 to 59 to 1,000 nm [2].

14   2  Coherence loss in light backscattering  

The peaks from the inside and the outside of the probe window are labeled, respectively, as peaks A and B in Figure 2.2. Due to the probe design utilized in this work, a ratio between the previously mentioned probe window peaks can be used to determine the refractive index difference between the inner window-air and outer window-sample interfaces. Analysis of a series of solvents with refractive indices ranging from 1.3270 to 1.4502 was performed to establish a calibration curve of the relationship between the peak ratios resulting from the probe design and the effective mean refractive index of the sample, as shown in Figure 2.3. This calibration was then applied to the suspension systems to determine the effective mean refractive index of the particle suspensions, which was used to find ℓ, the relative photon dwell distance in the sample matrix. The decay portions of the signal profiles obtained from the 1% and 10% series of suspensions are shown in Figures 2.4 and 2.5, respectively. These decay profiles are the basis for our comparison of theory and experiment and will be more fully discussed in Section 2.4.

2.3  Theory In the theoretical description of experiments with the application of the Michelson interferometer, we consider here the relatively simple case of nonmagnetic systems with the scalar and pseudoscalar (imaginary) parts of refractive index n(ω,x), which can be presented in the following form: ,

(1)

where ω is the light frequency, x is the coordinate vector with the sample boundary at x1 = 0, and θ (x1) is the step function. For the sake of simplicity in presentation, the dielectric fluctuations outside the sample (at x1 < 0) and the differences between the properties of sample bulk and its surface layer are neglected. The sample is assumed to be statistically uniform and the local refractive index n2 (ω;x) is taken in the following form:

,

2.3  Theory   15  

, ,

(2)

where nm (ω) is the refractive index of media surrounding the particle and δ N(x) describes the particle fluctuations in the sample. The angular brackets denote averaging over the studied sample volume, and the sum over β is taken over all types of particles. The main theoretical problem to solve is to relate light propagation characteristics with individual scatterer properties in a medium containing randomly distributed scatterers. This difficult problem, known as the radiative transfer problem, lacks an exact analytical solution in the case of electromagnetic waves in threedimensional multiscattering media [7–15]. Our approach, analogous in some aspects to the introduction of the Fermi pseudopotential in neutron scattering [14,16–20], is based on an approximate expression for the effective dielectric function of the system, which is valid in the case of large numbers of randomly distributed particles. An important characteristic of this approximation is that it expresses dielectric functions through the sum of individual scattering amplitude with the help of the so-called coherent phase approximation, according to which: ,

,

(3)

− where Nβ is the mean concentration of particles of the type β, Aβ (0) is the complex amplitude of forward light scattering by such individual particles, and c is the velocity of light. The imaginary component of Aβ (0) describes the effects of coherence loss in individual scattering events, which lead to the exclusion of scattered wave components from the measured intensity of the main course of traveling coherent wave packets. According to the general optical theorem [21], Im Aβ (0) is proportional to σt the sum of all cross sections of light scattering with frequency ω by the particle β [22]. We will suppose that σt is averaged over all directions. This

16   2  Coherence loss in light backscattering  

corresponds to the realistic systems with random nanoparticle distributions when the external static field E0 is equal to zero. The possible influence of nanoparticle orientations when E0 ≠ 0 will be discussed in Section 2.5. We will also suppose that considered light frequencies are less than the frequency of ionization threshold and are not equal to or near the frequencies of optical resonance transitions. According to Berestetsky, Lifshitz, and Pitaevsky [23] in the considered case,

.

(4)

In our works, where experiments were performed with monodisperse spherical particles [10], we use the special approximation for Mie’s solution for light scattering by uniform dielectric spheres. For the sake of simplicity, we disregard in Equation 3 the effects of direct nonelastic processes and correspondingly set Im n2(ω) equal to Im κ (ω). The generalization that takes into account the possibility of bulk and surface inelastic processes is straightforward. The Expression 3 is valid when .

(5)

This validity condition is less stringent in the following calculations when only contributions of the imaginary part of κ (ω) are important. The expressions for Aβ (0) are relatively simple and general in the following two limits. 1. In the high wavelength (small particle) limit when it is possible to use the Raleigh-Gans approximation [24,25]:



,

(6)

2.3  Theory     17

where Rβ is the spherically averaged particle dimension, F is a dimensionless form factor of the order of unity, and np and nm are, respectively, the refractive indices of the particle and the surrounding media. 2. In the short wavelength (Fraunhofer diffraction limit) when

,

(7)

where Sβ is the so-called radar cross section of the particle β, which is two times larger than the geometrical cross section [24,25]. In the case of a spherical particle, Sβ does not depend on the direction of light propagation and is equal to 2πR2β . For the theoretical description, the most nontrivial is the intermediate wave interval between Limits 6 and 7, which we will call the resonance scattering interval. In this interval the scattering amplitudes are not monotonic and are critically dependent on shape and other specific particle characteristics with a number of maxima. For this reason, experiments in the resonance scattering interval can provide the most detailed information about the characteristics of studied systems. The theoretical and experimental study of this interval is very active and an unfinished part in the light scattering theory [13,14]. In our works [10] we have considered and compared with experiments the approximate expression for κ (ω) for spherical particles valid in resonance scattering interval if the refractive indices of media and particles, nm and np, do not substantially differ from each other and the following inequality holds: .

(8)

In such a case it is possible to describe light scattering by uniform spheres with the van de Hulst approximation to the Mie solution, ­according to which [15,16,24]

(9)

18   2  Coherence loss in light backscattering  

The Expression 9 provides, with the accuracy defined by Condition 8, the correct limits both for small and large radius limits for Mie theory, as well as the description of resonance interval, if to substitute in Expression 9 .

(10)

In this case we get the exact expression in the small radius limit and at the same time retain correct expression for Aβ (0) for larger Rβ up to terms of the higher relative order of γ. In Mie calculations, the parameter γ defines the boundary condition at a particle-medium interface with a jump across the boundary of the normal component of the vector potential. In real experiments, particles can be surrounded by a narrow flexible dipole boundary layer and an electrochemical double layer (both atomic dimensions are much less than the wavelength of incident light) responsible for the repulsion between particles. Both of these layers can contribute to the previously mentioned jump. At the particle-polar solvent interface the potential drop at atomic distances can reach 0.1–2.0 V. Therefore, dielectric properties and the boundary condition for the electric field at the particle interface can be substantially altered, and γ has to be considered as an adjustable parameter. It follows from Equations 3 and 9 that for random monodisperse samples with spherical nanoparticles we have the following expressions for |κ (ω)|2 and Im κ (ω):

(11)

2.4  Comparison of the theory with the results of experiments     19

In Equation 11 ρβ is a volume fraction of particles with the dimension Rβ . The rather lengthy more general calculations of the expression for the signal in the considered experiments with Michelson interferometer are presented in the Appendix.

2.4  Comparison of the theory with the results   of experiments with the Michelson   interferometer We will compare theory with experimental measurements with the Michelson interferometer in the interval of optical length ℓ˜ : ,

(12)

where σ is wave-packet width and ℓ˜m with spherical polisterene and latex nanoparticles in water [2] corresponds to the value of ℓ˜, after which the ­backscattering signal becomes undistinguishable from the background. According to our calculations, described in the Appendix, the averaged experimental signals S (ℓ˜  ) presented in Figures 2.4 and 2.5 must have for small ℓ˜ and R the following form: 

(13)

where S0 is the background signal and

,





(14)

The constant C in Equation 13 characterizes the experimental device response and is independent of the individual sample properties.

20   2  Coherence loss in light backscattering  

Expression 13 was compared into the experimental data in Brodsky, Mitchell, et al. [2], using a nonlinear least-squares algorithm from the curve-fitting toolbox of MATLAB (The MathWorks, Inc., Natick, MA). The values for the |κ (ω0)|2 and Im κ (ω0) were extracted from the fit of the experimental data, which we will refer to as α and β, respectively. The parameters A and B, which are related to overall backscatter intensity and the rate at which backscatter intensity decreases as a function of ℓ , can also be derived from theoretical expressions set forth in the Appendix. Figure 2.6 contains a comparison of theoretical and experimental values for A and B as a function of particle radius. The experiments performed elucidate the signal dependence on the particle radii R = Rβ for samples with constant particle volume fraction ρ = ρβ = const. As seen in Figures 2.6 and 2.7, the dependence of the

Fig. 2.6  Plot of fitting parameters obtained by fitting experimental data (dashed lines) with theoretical expressions for the same parameters from Mie theory (solid lines). (A) The α and β terms for the 0.01 volume fraction sample and (B) the α and β terms for the 0.10 volume fraction sample. As explained in Chapter 4, because we have neglected terms with g (ω0,ℓ ) from Equation A.38, the expression for β used to calculate the presented theoretical values is only valid for relatively small values of R [2].

2.4  Comparison of the theory with the results of experiments   21  

experimental signal on ℓ˜ and the values of α and β on R reasonably fit the theoretical expressions. The dependence on R shown in Figure 2.7 reproduced the general shape of Mie resonances. The differences between detailed experimental and theoretical curves can be attributed to differences between real particle structures and their uniformity supposed in Mie theory, as well as to our application of the approximate version of Mie theory. In Figure 2.7 we see a comparison of the α terms for the 0.01 and 0.10 volume fraction sample series. The position of the Mie resonances do not shift with the changing particle concentration and depend mostly on the light scattering characteristics of individual particles, while the intensity of the resonances varies with sample concentration. This allows us to separate information about the particle density and individual particle characteristics, including absorption, in both monodisperse and polydisperse particle suspensions.

Fig. 2.7  Comparison of the α fitting parameter for the 0.01 and 0.10 volume fraction sample sets. The Mie resonance maxima occur at the same particle radius regardless of sample concentration, allowing the deconvolution of particle density and particle scattering characteristics. Highlighted in the inset are two reproducible anomalies in signal intensity observed near the first Mie resonance maximum in both sample sets. We attribute this increase in intensity to surface-enhanced inelastic scattering events [2].

22   2  Coherence loss in light backscattering  

It is also worth noting that the factor ∆2, which enters in Expression 14 for − A (ℓ˜  ), decreases with increasing particle concentration N and coherence length λc , which is accompanied with decreasing fluctuation effects. An analysis of intensity fluctuation (Figure 2.8) shows that they are much larger than the optical polarization fluctuation in the uniform polar media.

Fig. 2.8  Analysis of signal fluctuations. (A) and (B) are, respectively, the relative standard deviation (RSD) of the coherent backscatter signal near the beginning of the decay profiles of the 0.01 and 0.10 volume fraction samples. The fluctuations are a convolution of instrumental noise and signal changes caused by Brownian motion of the particles in the matrix. A general trend of increasing fluctuation intensity with increasing particle radius is observed [2].

2.5  The influence of external static electric fields E0   23  

2.5  The influence of external static electric   fields E0 on the light scattering by nonspherical nanopaticles In the previous sections of this chapter (Section 2.4) that considered light scatting effects, it was supposed that studied nanoparticles (nanononuniformities) are spherical or almost spherical. The effects of nonspherity can be observed by imposing the Michelson interferometer or other lightscattering measurement devices on the additional static electric field E0 shown in Figure 2.1. The most interest in the literature gather the statics and dynamics in soft systems: the complex material that can self-assemble into ordered structures with typical lengths on mesoscopic scales as well as effects of electric field on the deformation of the shape of suspended liquid drops and drops on solid surfaces. These phenomena are fundamentally interesting and have numerous practical applications. The corresponding active field of research, which is only at its infancy, is described in the very informative review article by Tsori [26]. It is established in the considered results that the initially spherical soft droplet, after being put in an external electric field, elongates in the direction of the field. The degree of elongation is calculated as a result of competition between electrostatic energy, preferring in the external field a long-direction drop, and surface tension, preferring a spherical drop. The solution of the full electrodynamical problem for pure dielectric drops in external constant electric field led to a ­discriminating function Φ obeying the equation .

(15)

Here D, R, and M are the ratios of dielectric constants, resistivity, and viscosity of the outer liquid to those of the nonconducting drops, such drops are prolate when Φ > 0, oblate when Φ < 0, and spherical if Φ = 0. Qualitatively, the result of the solution of the electrostatic problem is that it is possible to interpret that there is an electrostatic free-energy penalty for dielectric interfaces perpendicular to the uniform electrical field. In the preceding discussion we have assumed that we considered pure dielectric materials. However, even the small conductivity due to a small salt content

24   2  Coherence loss in light backscattering  

can change the interfacial interactions between nanoparticles and the electric fields. In the previously considered expression for the description of the light scattering of solid nanoparticles it was supposed the performing of averaging on particles orientations. Correspondingly there has been no observable effects of external field change, because in such cases the averaged amplitude A(0) does not depend on the particle’s orientation. However, when the particles are noticeably nonspherical, their orientation along the external static electric field E0 will change the value of the forward scattering amplitude A(0). In the simplest case of the constant in a plane condenser we have the typical change

electric field

of the free energy ∆F of individual particles equal approximately to ,

(16)

where ε ′′ is the optical contrast between the particle material and that of the environment media, ν0 ~ 10-28m3 is the order of magnitude of nanoparticle volume oriented in the direction of field E0. The change of free energy in Equation 16 can be comparable with κT in real cases [27]. In the considered problem of hard nanoparticle experiments, the signal change with changing direction of E0 can establish at least qualitatively the ­deviation of the typical particle’s shape from a spherical one.

2.6  The dynamics of phase transition in media   with randomly distributed nanoparticles In this section we follow the dynamical effect description in Brodsky [28]. For the description of dynamical processes, including the phase transition at the initial moment of the nanoparticals formation with the help of coherent phase approximation 3, we have to find expressions for dynamic − changes of concentrations Nβ. In solving this problem we follow to the approach proposed by Lifshitz and coauthors and described in [29]. The only difference from Lifshitz et al. [29] is that we consider the phase transition dynamics from liquid or gas to a solid state but not, as in Lifshitz et al. [29], from one solid state to another.

2.6  The dynamics of phase transition in media   25  

In the considered systems the change from a metastable to a stable phase occurs as the result of fluctuation in a homogeneous medium, which forms small quantities of a new phase, or embrions. The energetically unfavorable process of creation of an interface, however, has the result that when the nucleus is less than a certain size it is unstable and disappears again. Only embrions whose size R is greater than a definite value Rcr (for a given state of the metastable phase) are stable; this value is called the critical size, and embrions of this size will be called critical embrions. The central element of the calculations is the expression for critical radius Rcr , which according to Lifshitz et al. [29] is equal to ,

(17)

where ν′ and ρ are correspondingly the molecular volume and concentration of the embrion substance in the considered system. ρ0∞ is the concentration in saturated solution near the plane surface of solute, α is the surface tension (or, more generally, surface energy density), and κ is the Boltzman constant. We will consider separately and compare with experiments two opposite limits. First we will address the case when the mean particle (nonuniformities) radius R is less than Rcr and then the case when the mean radius R is of the order of or larger than Rcr. In the final part of this section, the results are summarized and possible applications of the described theoretical and experimental techniques are considered. In the small particle radius (long wavelength limit) we will take the ­expression Aβ  (0) in the Rayleigh-Gans approximation (see Equation 6):



(18)

26   2  Coherence loss in light backscattering  

where Rβ is the averaged particle dimension with Rβ < Rcr . In this case, embrions become larger with increasing time t and

. For the ex-

pression Nβ ≡ N(R) we can use a purely thermodynamic approach. In such an approach one can consider only the problem of calculating the probability of occurrence in a medium of fluctuational nuclei of various sizes, the medium being regarded as in equilibrium. Since the state of the metastable phase does not actually correspond to complete statistical equilibrium, this treatment applies only to times that are much less than the critical nucleus formation time (reciprocal probability per unit time), after which the change to the new phase occurs in practice, and the metastable state ceases to exist. For the same reason, the thermodynamic calculation of the formation probability is feasible only for embrions with size Rβ < Rcr ; larger nuclei develop into the new phase. That is, such large fluctuations are not among the group of microscopic states that correspond to the ­(metastable) macroscopic state under consideration. Instead of the thermodynamic probability of nucleation, we shall refer here to a quantity proportional to this, the “equilibrium” distribution faction for nuclei of various sizes existing in the medium, denoted by f0 (R); f0, is the number of nuclei per unit volume of the medium with sizes in the range dR. According to the thermodynamic theory of fluctuations, ,

(19)

where Fmin (R) is the minimum work needed to form an embrion with mean radius R. This work is made up of volume and surface parts and for an embrion with the averaged radius R equal [28] to ,

(20)

where Rcr is given by Expression 17. The maximum value of Fmin (R) at R = Rcr corresponds to a sharp minimum in f0 (R). Disregarding the contribution of particles with radii larger than Rcr , we found from Equations 3 and 18–20 that in the considered case we can use the following ­approximations for κ (ω):

2.6  The dynamics of phase transition in media   27  



(21)

The described treatment of the particle growth kinetics was related only to the initial stage of the transition: the total volume of all nuclei of the new phase has to be so small that their formation and growth have no appreciable effect on the “degree of metastability” of the main phase, and the critical size of the nuclei determined by the degree of metastability may be regarded as a constant. In this stage there is a fluctuational formation of nuclei of the new phase, and the growth of each embrions is independent of the behavior of the others. In the later stage, when the supersaturation of the solution becomes very slight and mean radius of the embrions are relatively large, the growth process is different. The fluctuational formation of new nuclei has now practically ceased, as the critical size is great. The increase in the critical size accompanying the steady decrease in the degree of supersaturation has the result that the smaller among the grains of the new phase already formed fall below the critical range and redissolve. Thus, a decisive role at this stage is played by the “swallowing up” of small embrions by large ones, which grow as the result of the dissolution of the small ones (coalescence). The equilibrium concentration ρ0R of the solution at the surface of an embryo with radius R is given by the thermodynamic formula ,

(22)

where ρ0∞ is the concentration of a saturated solution above a plane surface of the solute, α is the surface tension coefficient at the phase interface,

28   2  Coherence loss in light backscattering  

and ν ′ is the molecular volume of the solute. The concentration ρ0R is defined in terms of the volume of the substance dissolved in unit volume of the solution. With this definition, the diffusive flux

at a grain

surface is equal to the rate of change as the embrion radius R: .

(23)

In the cases of liquid or gas solutions, the diffusion coefficient D in Equation 23 can in many cases be put equal to Brownian diffusion coefficient D = DBr , which is much larger than molecular diffusion coefficient DM . Because the concentration ρ is assumed to be so small that the concentration distribution around the grain can be regarded as equal, at each instant, to the steady distribution ρ (r) corresponding to the relevant value of R: , where ρ is the mean concentration in the solution and the diffusive flux i(R) is equal to ,

(24)

where the parameter

. The important quantity δ is equal to

.

(25)

It is the time-dependent supersaturation of the solution. The quantity 

(26)

2.6  The dynamics of phase transition in media   29  

is the critical radius. When R > Rcr , the grain becomes larger and when R < Rcr it dissolves

,

. In the case of the light wave-

length smaller than the mean particle radius, we can use the expressions for Aβ (0) in the ­Fraunhofer diffraction limit [7], when

.

(27)

In (27) Sβ is the so-called radar cross section of the particle β, which is two times larger than the geometrical cross section. In the case of randomly oriented particles, with the shape not drastically different from spherical ones, we have the following relation: . The description of the considered coagulation process is to some extent simplified, because at large R, according to Brodsky [28], after a long enough time the particle radii distribution does not depend on time and is described by the universal function P(u) equal to:

,

, 

(28)

30   2  Coherence loss in light backscattering  

where . In this case the overall particle density N(t) decreases with increasing time by the law ,

(29)

where Q is the total initial concentration  ,

(30)

(31)

and the mean particle radius R increased with time t according to the following law .

(32)

2.7  The spectroscopy of carbon nanotube networks The previous analysis in this chapter has been devoted mostly to the consideration of materials with relatively low concentrations of randomly distributed nanoparticles. In the past decades there have been substantial efforts in preparing methods and applications of samples with relatively high concentrations of nanoparticles. Special attention was devoted to the preparation and applications of single-wall carbon nanotube networks of nanoparticles built from carbon atoms [30–33]. The structure of the ­elements of such carbon nanotubes is shown in Figure 2.9.

2.8  Conclusions   31  

Fig. 2.9  Single-walled carbon nanotube. Five unit cells are shown [31]. Reprinted figure with permission from E. Richter and K. Subbaswamy. Phys. Rev. Lett. 79, 2738 (1997). Copyright 1997 by the American Physical Society. http://prl.aps.org/abstract/ PRL/v79/i14/p2738_1

Optical experiments, including light absorption, Raman spectroscopy, and photoluminescence, have been used for single-wall carbon nanotube study. Results of these experiments have been interpreted as evidence of the formation of one-dimensional nanotubes with different diameters in concentrated solution of carbon nanoparticles. The detailed analysis of a corresponding very active field is beyond the scope of this book.

2.8  Conclusions The quantitative analysis of coherence loss effects in light scattering by the media with nanoscale nonuniformities can be based on the experimentally measured values of dielectric characteristics κ and ∆. In such analysis the most direct approach is to compare the decrease with the increasing distance of the amplitudes of the narrow coherent wave packets traveled on the distance ℓ in nanononuniform media with the decrease of analogous wave packets in the uniform media. The corresponding ­expressions have the following structure

 (33)

32   2  Coherence loss in light backscattering  

In Equation 33 ω0 is the center frequency and sigma is the width of the wave packet with the initial strength E0, n1 is the real part of the refractive is the volume of the detector where a

index of the media, and

comparison of R wave-packet transformation is measured. The expression for E(ω0,ℓn) can be constructed with the help of Equations 23 and 26. The example of such construction is provided in Brodsky, Mitchell, et al. [2], where the corresponding experiments were done with the help of a modified Michelson interferometer. The expression for E(ω0;ℓn), according to previous analysis, has the following general structure: ,

(34)

where the parameter B = Imκ0 characterizes the coherence loss in the individual scattering and A(ℓn1) is proportional to ∆2|κ0|, which takes into account the coherence loss due to the interference effects in multiple scattering. The quantitative analysis of coherence loss effects can be based on dielectric characteristics of the nanononuniform media developed in this chapter. In such analysis the most direct approach is to follow the decrease of a coherent wave packet with the increase of its traveling length ℓ in the nonuniform media. Substantial problems remain in the consideration of scattering by randomly oriented and distributed nonspherical and polarizable scatterers such as polycrystals and liquid crystals. They do not require a qualitatively different treatment. As shown by Placzek, the cross sections of incoherent light scattering by the freely oriented systems can contain three independent parts: scalar, asymmetric, and symmetric [23,24,29]. However, the cross sections of coherent scattering contain only a scalar part and thus the effective dielectric function introduced for the description of the considered coherence effect remains a scalar with nonscalar effects contributing to an effective light absorption. The effects of nonscalar scattering components can, in principle, be separated and measured by the modification of the considered techniques by imposing external electric fields E0 on studied samples containing randomly distributed orientable or polarizable particles and measuring the

2.9  Appendix   33  

signal dependence on the angle between the light polarization and the external field. The understanding of corresponding effects is important for the progress in the rapidly growing field of optofluidics [34]. Of special interest for electrochemistry is an observation of the influence of the electric field E0 on the phase transitions from liquid to ­crystalline in the double layers near electrode surfaces [35]. The majority of this chapter was devoted to the case of materials with relatively low concentrations and randomly distributed nanoparticles. In the past two decades there was considerable progress in developing methods of synthesizing samples containing high concentrations of nanoparticles, particularly, synthesizing carbon atoms forming one-dimensional nanotube crystals (Figure 2.9) [30–33]. The results of light scattering from such nanotubes were interpreted as evidence of the formation of one-dimensional carbon nanotubes with different diameters.

2.9  Appendix To obtain an expression for E(ω;x) convenient for experimental data averaging we introduce Debye’s potentials F(1,2) [15]:

,

(A.1)

where E(ω,x) and H(ω,x) are, respectively, the electric and magnetic fields. The Maxwell equations for F(1,2) have the form of the two coupled linear equations:

,

(A.2)

where the square brackets designate the vector product. In order to perform the calculations in a compact form, we introduce the algebra of operators in a six-dimensional space, which is the direct product (i × α) of the three-dimensional space i = 1, 2, 3 and the two-dimensional

34   2  Coherence loss in light backscattering  

photon spin space α = 1, 2 with the following matrix elements of relevant operators I, Ikℓ, Sk, and σ1,3: , , , ,

(A.3)

where eijk is the completely asymmetric tensor in three-dimensional space with e123 = 1 and δij is Kroneker’s symbol. Equation A.2 can now be ­rewritten in a matrix form as follows: , (A.4)

where F ≡ F (ω;x) is a column:

.

(A.5)

The vector F (ω;x) obeys the following relation: .

(A.6)

2.9  Appendix   35  

We introduce the retarded matrix Green function G (ω,x,x′), which is the solution of the equation:



.

(A.7)

It follows from Equation A.7 and the definition of the retarded Green ­functions that

.

(A.8)

This result corresponds to the continuity of the tangential components of E and H across the sample boundary. With the help of G (ω,x,x′), we can now write expression 33 for T(ℓ ) as follows:

 (A.9)

36   2  Coherence loss in light backscattering  

where I0 is the electromagnetic energy flow entering the studied sample bulk, F0 describes the polarization structure of the incident light, and σ is considered wave-packet width. In the following, we suppose that ℓ is and, correspondingly, it is possible to separate the scatterlarger than ing effects at the sample surface interval with the width smaller than We consider the case of linear polarization when 



.

(A.10)

Now we introduce, following Fock’s idea [36], the fifth, timelike ­coordinate ν and present the Green function G in the following form: ,

(A.11)

where U (ν ) is the matrix function of ν, ω, x, x′ equal to ,

(A.12)

with the matrix operator K defined in Equation A.7. In Equation A.11 we have taken into account that . The matrix U (ν ) has to satisfy the linear differential equation ,

(A.13)

2.9  Appendix   37  

with the boundary condition . We seek the expression for U(ν ;x,x′) in the following form:

 (A.14)

.



According to Equations A.12 and A.14 expΛ(x) has to satisfy the­ following equation:

,

(A.15)

where the square brackets denote the commutator: . The Relation A.15 can be reduced, taking into account Equation A.4 to the linear equation for Λ:



(A.16)

38   2  Coherence loss in light backscattering  

It follows from Equation A.16 that

 where

(A.17)





(A.18)



The exponent follows:

in Equation A.12 can be rewritten as



(A.19)

The matrix I˜(q) in Equations A.17 and A.18 is defined by

. The expression A.12 for Λ satisfies the following relations:

(A.20)

2.9  Appendix   39  

,

.

(A.21)

It follows from Equations A.8–A.21 that

(A.22)



To simplify the calculations we will take into account that in cases considered in our works the following quantity (ζ ) is small: .

(A.23)

Neglected in the following calculation are terms of the order of (ζ ) < 1, which correspond to far tails of the fluctuation distribution of optical

40   2  Coherence loss in light backscattering  

properties where random deviations from the applied Gaussian distribution law and irregular speckle effects become significant. The inequality A.23 allows us to not impose the condition of nonnegativity of N(x) and make the following substitution: .

(A.24)

After such substitution we find from Equation A.22 the following ­expression for T(ℓ ):



, , 

.

(A.25)

In transforming Equations A.22–A.23 we have made the substitution

and taken into account Equation A.19 and the identity

2.9  Appendix   41  

. This allows us also to present the Λ (ω;ν ;x) in the following form: ,

(A.26)

 (ω ;ν ;q) equal to with the matrix Λ



(A.27)



In Equations A.23 and A.25 there enters the following Fourier ­transform δη (ω,q):





, (A.28)

where δ N(q) is the Fourier transform of

. The averaged-over fluc-

tuations of δ N(q, ω) in the following calculations enter only under integrals over dω and dq(1)...dq(n) with the integrands invariant under the inversion

42   2  Coherence loss in light backscattering  

dq(n)→ – dq(n), ω → – ω combined with a complex conjugation. This allows us to make the following ­substitution for the factor under the integrals in Equation A.28:

and to put δη (q,ω) equal to



(A.29)

When averaging-over fluctuations δ N(x) in a sample bulk we assume that the fluctuations are the stationary isotropic Gaussian random process defined by the basic moments:



(A.30)

where ∆2 characterizes the intensity of fluctuations in the volume with dimensions defined by the coherence length λc , with q being the correla  tion length. In the case of the noninteracting spherical particles with the radii R we have .

Appendix   43  

It follows from Equation A.30 that



(A.31)

The factor δ −ω′ ω in previous expression follows from the time-­dependence of the correlator 〈δη (t′;q) δη (t;q)〉 only on the difference t′–t.

.

(A.32)



Note that both quantities A.31 and A.32 do not depend on x. It follows from Equation A.17 that



,



(A.33)

44   2  Coherence loss in light backscattering  

The matrix structure of Equation A.33 can be resolved by taking into account the identities

with the following result:



(A.34)

After integration in Equation A.34 over dz and integration by parts over dν we find that



Appendix   45  

where



The integration over dp in the expression for T(ℓ ) can be made ­analytically with the following result:

(A.35) 

Definitions of the functions g(ω,ν) and f± (ω,ν) are given in Equations A.36 and A.37. In the derivation of Equation A.35 we have taken into account the invariance of the integrand under complex conjugation with simultaneous substitution ω → – ω and for simplicity we do not take into account explicitly the effects of frequency dispersion in g(ω,ν) and f±(ω,ν). These effects lead to the substitution in Equation A.35 of the group velocities for the phase one:

46   2  Coherence loss in light backscattering  

.

In the considered experiments, where n(ω) varies relatively slowly in the frequency interval ω0 ± σ, and the particle concentration is relatively low, the disregarded dispersion effects are unimportant. In Equation A.35 we have the following expression for g (ω,ν)

Appendix   47  



,

.

(A.36)

∼ In Equation A.36 we have used the definition in A.28 of Λ and Equation A.26. The quantities f+,− (ω0,ν), which enter in the expression for T(ℓ ) in the most important case of small ν, are equal to . The function ­approximately to νm , where

(A.37)

has the maximum at ν equal

. The remaining integral over dν in Equation A.31 can be taken in the asymptotic (stationary phase) approximation because we have supposed that the considered values of ℓ satisfy the inequality . This inequality allows us to separate and exclude from consideration surface effects. We find after asymptotic integration over dν that

48   2  Coherence loss in light backscattering  



.

(A.38)



We have supposed that . The structure of the exponential factor in Equation A.38 deserves ­special attention. It is equal to , where g(ω0 ,ℓ˜ ) is proportional to . For the larger

.

Such quadratic dependence on ℓ˜  for the wave-packet decomposition and the light intensity distribution homogenization is dominating at large ℓ˜. It is characteristic for light diffusion processes with effective light diffusion coefficient D proportional to .

(A.39)

In the estimation A.39 we have used Equation 3 and performed a Fourier transform over ω to find the signal time dependence. Note the dependence of the experimentally measured quantities on the wave-packet width σ and Lc.

2.10  References and Notes   49  

2.10  References and Notes   1. K. Watson (1969). Multiple scattering of electromagnetic waves in an underdence plasma. J. Math. Phys. 10, 688.   2. A. Brodsky, G. T. Mitchell, S. L. Zeigler and L. Burgess (2007). Coherence loss in light backscattering by random media with nanoscale nonuniformities. Phys. Rev. E 75, 046605.   3. S. Randall, A. Brodsky, and L. Burgess (2005). Manifestation of Mie resonances in coherent light backscattering from random media. Mod. Phys. Lett B 19, 181 (World Scientific Publishing).   4. POAN Research Group, ed. (1998). New Aspects of Electromagnetic and Acoustic Wave Diffusion, Springer Tracts in Modern Physics 144 (Berlin: Springer Verlag).   5. P. Sheng, ed. (1990). Scattering and Localization of Classical Waves in Random Media (London: World Scientific). The latest works on localization of electromagnetic waves in random media have been summarized in S. Balog, P. Zakharov, F. Scheffold, and S. E. Skipetrov (2006). Photocount statistic in mesoscopic optics. Phys. Rev. Lett. 97, 103901.   6. 3000 Series Nanosphere Size Standards, 5000 Series Latex Microsphere Suspensions, Duke Scientific Corporation, Palo Alto, CA.   7. M. van Rossum and T. Nieuwenhyzien (1999). Multiple scattering of classical waves: Microscopy and diffusion. Rev. Mod. Phys. 71, 313.   8. P. de Vries, D. van Coevorden, and A. Langendijk (1998). Point scatteres for classical waves. Rev. Mod. Phys. 70, 447.   9. V. Apalkov, M. Raikh, and B. Shapiro (2004). Almost localized photon modes in continuous and discrete models of disordered media. J. Opt. Soc. Am. B 21, 132. 10. I. Dodd and I. McCarthy (1964). Scattering of energy-time wave packets from many-body system. Phys. Rev. 134, A1136. 11. A. Kalugin, A. Bronshtein, and R. Mazar (2004). Propagation of the twofrequency coherence function in an inhomogeneneous background random medium. Waves in Random Media 14, 389, and references therein. 12. D. J. van Manen, J. O. A. Robertsson, and A. Curtis (2005). Modeling of wave propagation in homogeneous media. Phys. Rev. Lett. 94, 164301. 13. Y. Kuga and A. Ishimaru (1984). Retroreflectance from a dense distribution of spherical particles. J. Opt. Soc. Am. A 8, 831; M. P. Van Albada and A. Lagendijk (1985). Observation of weak localization of light in a random media. Phys. Rev. Lett. 55, 2692; P. E. Wolf and G. Maret (1985). Multiple light scattering from disordered media: The effect of Brownian motion of scatterers. Phys. Rev. Lett. 55, 2696. 14. See the description of Fermi pseudopotentials in M. Goldberger and K. Watson (1964). Collision Theory (New York: Wiley). 15. For the seminal work in applications of light scattering for random media diagnostics see F. V. Ignatovich and L. Novotny (2006). Real-time and backgroundfree detection of nanoscale particles. Phys. Rev. Lett. 96, 013901; S. Randall, A. Brodsky, L. Burgess, and R. Green (2003). Optical Low-Coherence Reflectometry for Nondestructive Process Measurements. In: Review of Progress in Quantitative Nondestructive Evaluation 22 (New York: Plenum).

50   2  Coherence loss in light backscattering  

16. For the same work see M. Hamad, A. Brodsky, and L. Burgess (2000). Appl. Spectrosc. 54, 1506; M. Hamad, S. Kailasam, A. Brodsky, R. Han, J. Higgins, D. Thomas, et al. (2005). Monitoring of a pharmaceutical nanomilling process using grating light reflection spectroscopy. Appl. Spectrosc. 59, 16, and references therein. 17. Control of nanoparticle characteristics is important for a number of new optoelectronic technologies, in particular for quantum information devices where such particles are used as single photon emitters. See Z. Yuan, B. Kardynal, R. Stevenson, A. Shields, C. Lobo, K. Cooper, et al. (2002). Electrically driven single-photon source. Science 295, 102. 18. In the past few years, imaging techniques based on coherence loss in backscattering measurements have demonstrated considerable potential for qualitative in vitro study of biological tissues and blood. See V. Tuckin, X. Xu, and R. Wang (2002). Dynamic optical coherence tomography in studies of optical clearing, sedimentation, and aggregation of immersed blood. Appl. Opt. 41, 258. 19. A. Brodsky and L. Burgess (2003). Theoretical study of the coherent light backscattering by random media. Int. J. Mod. Phys. B 12, 337. 20. A. Kim and A. Ishimaru (1999). A Chebyshev Spectral Method for radiative transfer equations applied to electromagnetic wave propagation and scattering in a discrete random mwdia. J. Comput. Phys. 152, 264, and references therein. The effect of Mie resonances in multiple scattering was recently considered by N. Ghosh, A. Pradhan, P. K. Gupta, S. Gupta, V. Jaiswal, and R. P. Singh (2004). Depolarization of light in multiply scattering: medium: effect of the refractive index of a scatterer. Phys. Rev. E 70, 066607; see also G. Schweiger and M. Horn (2006). Effect of changes in size and index of refraction on the resonance wavelength of microspheres. J. Opt. Soc. Am. B 23, 212. Among the different attempts to solve the radiative transfer problem for multiple scattering of classical waves, we mention the diagramic approach with partial diagram summation, the random matrix method, and the steady-state Langevin approach in G. Spivak and A. Zyuzin (1998). Fluctuations of coherent light transmission trough disordered media. Solid State Commun. 65, 311. The fact that most of the approximations used for solutions to the considered problems have unknown validity limits makes their practical application problematic. The most advanced is the theory of wave propagation in the case of one-dimensional randomness, since the corresponding Green functions have a relatively simple exponential structure and it is possible to rely directly on the Fokker-Planck equation. 21. I. Lifshitz, S. Gredescul, and L. Pasteur (1998). Introduction to the Theory of Disordered Systems (New York: Wiley). 22. For a discussion of light propagation in optically active random media see M. Xu and R. R. Alfano (2005). Random walk localization of light. Phys. Rev. Lett. 95, 213901; M. Matsuo, S. Miyoshi, M. Azuma, Y. Nakano, and Y. Bin (2005). Polarized small-angle light scattering from gels estimated in terms of a statistical approach. Phys. Rev. E 72, 041403. 23. B. Berestetsky, E. Lifshitz, and L. Pitaevsky (1982). Quantum Electrodynamics, 2nd ed. (Oxford: Pergamon Press). 24. R. Newton (1982). Scattering Theory of Waves and Particles (New York: Springer Verlag).

2.10 References and Notes   51  

25. H. C. van de Hulst (1957). Light Scattering by Small Particles (New York: Wiley); G. H. Meeten (1997). Refraction by spherical particles in the intermediate scattering region. Opt. Commun. 134, 233. 26. Y. Tsori (2009). Colloquium: Phase transitions in polymers and liquids in electric fields. Rev. Mod. Phys. 81, 1971. 27. P. C. Das and A. Puri (2002). Energy flow and fluorescence near a small metal particale. Phys. Rev. B 65, 155416; A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, et al. (2005). Time-resolved reflectance at null source-detector separation: Improving contrast and resolution in diffuse optical imaging. Phys. Rev. Lett. 95, 078101. 28. A. Brodsky (2011). J. of Optics; in press. 29. E. M. Lifshitz and L. P. Pitaevskii (1989). Physical Kinetics (Oxford: Pergamon Press). 30. A. Thess, R. Lee, P. Nikolaev, H. Dai, P. Petit, J. Robert, et al. (1996). Crystalline ropes of metallic carbon nanotubes. Science 273, 483. 31. E. Richter and K. Subbaswamy (1997). Theory of size-dependent resonance raman scattering from carbon nanotubes. Phys. Rev. Lett. 79, 2738. 32. M. Dresselhans, G. Dresselhans, and P. Eklung (1996). Sciens of Fullerenes and Carbon Nanotubes (New York: Academic Press). 33. D. Hecht, L. Hu, and G. Gruner (2006). Conductivity scaling with bundle length and diameter in single walled carbonnanotube networks. Appl. Phy. Lett. 89, 133112. 34. Y. Lamhot, A. Barak, C. Rotschild, M. Segev, M. Safar, and E. Lifshitz (2009). Optical control of thermocapillary effects in complex nanofluids. Phys. Rev. Lett. 103, 264503, and literature therein. 35. M. Watanabe, A. Brodsky, and W. Reinhardt (1991). Dielectric properties and phase transitions of water between conducting plates. J. Phys. Chem. 95, 4593. 36. The Fock’s method is described in the book by W. Bogolubov and D. Shirkov (1997). Introduction to the Theory of Quantized Fields (New York: Wiley). It is applied for the solution of the Block-Nordsiek model in quantum field theory.

3  Optical diagnostics based on coherent light transport effects in media with mesoscopic nonuniformities 3.1  Outline As mentioned in the Introduction, the coherence effects in a propagation of optical waves in highly scattering nonuniform media with characteristic dimensions of nonuniformities of the order of or less than wavelength has attracted a great deal of attention by both experimentalists and theoreticians in recent years. The interest in this problem was stimulated by the fact that in such media there can occur in some areas both the constructive and destructive interference of multiple scattered optical waves. That gives rise to a number of nontrivial wave coherence effects. The observation of these effects opens new possibilities for the solution of the important problem of in situ optical diagnostics of highly scattering systems of nanoparticles and other mesoscopic structures with characteristic dimensions from 1 to 100 nm. Such diagnostics include the evaluation of the particle shapes and the characteristics of their distribution in space, including clustering and void formation, as well as particle movement effects, including Brownian motion and flow-induced stresses in nanosuspences. The diagnostics have important perspectives not only in scientific studies but also in practical applications, including the control of important industrial processes, environment monitoring, and medical diagnostics. In particular, the coherent, nonionizing optical scattering may serve as an effective substitute for X-ray-based techniques in medical imaging. Optical coherent scattering effects can be exploited for the detection and characterization of aerosols present in the atmosphere, including the detection of hazardous bioaerosols. The measurement of particle size and particle distribution by optical coherent scattering can serve for the rapid and accurate characterization of mesoscopic colloidal suspensions, which is important, in particular, in the industrial production of polymer latexes. This chapter describes the experimental and theoretical study of coherent transport effects (CTE) in the visible light scattering in systems with volume or surface mesoscopic disorder. The progress in the study and applications of CTE critically depends on the advances in the theory of

54   3  Optical diagnostics based on coherent light transport effects

the light propagation in highly nonuniform media. In the development of the theory we have used our previous results and take into account the recent results of classical wave transport theory in nonuniform media, developed mainly for the geophysical application and the interpretation of radio-wave scattering in the atmosphere. In Section 3.2 we briefly outline the basic concepts of coherent wave scattering in nonuniform media and formulate the goals of the proposed work. In Sections 3.2–3.5 we describe our works in the field. In our concluding remarks (Section 3.6) we discuss the wide practical perspectives of optical CTE measurements.

3.2  Introduction and background The propagation of classical electromagnetic, acoustic, and elastic waves in nonuniform media has been the subject of experimental and theoretical investigations for several decades in various areas of physics and physical chemistry (see [1–11] and the literature therein). Despite intensive efforts the subject still presents a challenge and the experiment continues to generate new and often unexpected results. Interest in this problem is stimulated by the fact that, in nonuniform media, a constructive interference of multiple scattered fields can occur in some spatial regions. This interference gives rise to such intriguing effects, manifested in multiple light scattering in random systems, as strong (quasi) localization, fluctuational waveguiding, weak localization (enhanced back scattering), and local enhancement of nonlinear processes such as second harmonic generation. Constructive and destructive interference also lead to the appearance of specific anomalies in optical scattering on regular structures, such as regular gratings [12]. Of special interest is the specific behavior of light in random gain media (such as pumped laser materials) where the laser type optical emission enhancements can take place even in mirrorless devices [13]. Among the listed coherence effects in random nonuniform media, weak localization (enhanced backscattering) has been verified experimentally beyond any doubt for light scattering. In the published experimental works the weak localization effect manifests itself in large amplitude fluctuations in the scattering intensity as a function of the backscattering angle, with a sharp peak in the narrow angle around the 180° backscattering direction.

3.2  Introduction and background   55  

However, despite a number of experimental efforts reported in the literature, there is yet no generally accepted experimental confirmation of the theoretical predictions that strong classical wave localization or quasilocalization is possible in three-dimensional random media. Thus, the verification of strong localization is still an unanswered question in the modern optics of highly scattering media. The experimental indications strongly imply that the realization of optical quasi-localization effect can be found in our work [11]. We will address both experimentally and theoretically the problem of the behavior of the part of classical optical waves (wave packets), which is remembered in the process of multiple scattering the initial phase characteristics and polarization after distances comparable with or longer than the free path. The coherent transport effects are directly related to the equilibrium and dynamical characteristics of the studied media and can be used for the diagnostics of the static and dynamic properties of mesostructures. To develop CTE methodology, it is important to develop sensitive measurement methods since the predominant part of the wave intensity distributions at distances longer than the mean free path is governed by the photon diffusion-type equations, and forgets the phase and polarization properties of the original wave packets. In the discussed work we will further develop and apply to different systems with nanoscale dimensions, two new sensitive methods for following CTE, which were developed recently in our works grating wave reflection spectroscopy (GWRS) and low coherence interferometry (LCI) of nonuniform media. Both methods are described in detail in Sections 3.3 and 3.4. For the full evaluation of the information contained in CTE measurements it is necessary to rely on the theory of wave scattering in inhomogeneous media. In deterministic cases with simple geometries of nonuniformities distribution, a number of approximate analytical and numerical techniques have been developed for solving the linear macroscopic equations describing classical electromagnetic wave propagation. Even in these cases the realistic imaging procedures have generally required that individual scattering objects be only weakly scattering (e.g. when Born approximation is valid). In the presence of randomly distributed homogeneities, direct and inverse scattering problems become especially complicated since the wave equations acquire a complex stochastic character, and direct analytical and numerical computations become impossible

56   3  Optical diagnostics based on coherent light transport effects

for particularly important situations when the characteristic distances between nonuniformities become comparable with a wavelength. Even in the most sophisticated of the existing theoretical approaches to the problem of multiple wave scattering in nonuniform media, based on the introduction of functional integrals with the application of Anderson’s replica method or the supersymmetry approach (in analogy of techniques exploited in analysis of electronic properties of disordered metals), it is necessary to use one or the other versions of perturbative techniques with partial summation of perturbative terms [9,14]. Such approximations reduce the problem to phenomenological equations of the wave energy diffusion in which important coherence effect, including constructive interference, are at least partly neglected. The specific problem that cannot be addressed at all in the framework of such diffusion-type approximations is the description of the small part of light intensity that remembers the initial phase and polarization characteristics on distances much longer than free-path ones. The theory of one-dimensional wave propagation in random media is an exception because the corresponding Green functions have a relatively simple structure and one might use directly the Fokker-Planck equation. In Section 3.5 we will describe our approach proposed for the theoretical description of CTE. This approach is based on an analogy with the exact solution of the Block-Nordsiek model in quantum field theory with the help of, described in Chapter 2, Fock’s techniques of an introduction of the fifth, time-like coordinate [15]. This technique has some common features with the “imbedding technique” used previously by Rytov, Kravstov, and Tatarskii [4], and was recently exploited for the analysis of the scalar Helmholtz equation in random media by Samelson and Mazar [16].

3.3  Grating wave reflection spectroscopy We have developed a new optical diagnostic method, grating light reflection spectroscopy (GLRS), based on measurements of intensity and phase in light reflected from a transmission diffraction grating in contact with a diagnostic sample near specific critical points. The theory and experimental studies of the method are described in the articles [12,17–19], and are the subject of two U.S. patents [20].

3.3  Grating wave reflection spectroscopy   57  

A general schematic diagram of GLRS operation is shown in Figure 3.1. The technique exploits anomalies in the reflection of waves from a diffraction grating at special threshold values of parameters (singular points) where one of the beams with diffraction order mcr in the analyte medium transforms itself from a traveling wave to an evanescent wave. We call this beam a critical one. At these thresholds, the energy of the transforming beam is redistributed among all other reflected and transmitted beams. Such redistribution has a singular character, and it can be theoretically described in a relatively simple manner in analogy to the method previously used in the general theory of multichannel wave ­scattering [17]. The redistribution occurs at a specific wavelength for a given grating period incident angle and sample refractive index. It generates singular features in the reflected spectrum, and changes in the sample bulk

Fig. 3.1  Illustration of GLRS sample  /grating substrate  interface. (A) The transmitted diffraction order m = 1, containing a wavelength λ that lies along the grating surface, fulfilling the singularity condition; (B) the resulting exponentially  at the singularity decaying evanescent wave with its tangential wave vector   [12]. condition 

58   3  Optical diagnostics based on coherent light transport effects

refractive index will be manifested as changes in the incident angle and wavelength at which the corresponding transformation occurs. The features contained in the reflection spectrum near the thresholds allow for separation of surface and bulk effects in a sample and for simultaneous determination of the real and imaginary (absorptive) parts of the dielectric function of the sample. For the nanostructure diagnostics it is especially important that we have proven that GLRS can provide analytically relevant information regarding mesoscopic suspensions and colloids in liquid and gas samples. For example, using existing GLRS devices, it is possible to measure particle sizes continuously from less than 10 nm up to 10 mm in diameter. Such measurements are based on the effect of the coherence loss during wave scattering by particles and other nonuniformaties in a sample matrix,

Fig.  3.2  Derivative plots for a series of polystyrene microspheres in water, each at a 2.5% solids concentration. The water solvent or reference peak lies at the uppermost of the series of peaks on the left [12].

3.3  Grating wave reflection spectroscopy   59  

where scattering events that change the properties of evanescent waves near the threshold lead to changes in the threshold characteristics. These changes manifest themselves by the appearance of a specific imaginary component in the effective dielectric constant. This result is due to the coherence distortion of transmitted waves, which takes place even if the particles do not absorb waves in the exploited frequency interval. The measurement of dielectric properties and mesoscopic particle characteristics by GLRS techniques has some similarity to attenuated total reflection (ATR) and “frustrated total internal reflection” (FTR) methods [21]. FTR has been used previously for the measurement of static and dynamic properties of colloid particles in electrolyte solutions [22]. However, ATR and FTR provide less information than GLRS and are less convenient because they require the application and maintenance of a high-refractive-index substrate. Also the sensitivity of GLRS is

Fig.  3.3  Close-up of derivative plots for the small scattering sampling series. (A) 42 nm radius particles; (B) 48 nm radius particles; (C) 63 nm radius particles; (D) 102 nm radius particles. The insert covers the full wavelength range of the samples, showing variance outside the singularity region [12].

60   3  Optical diagnostics based on coherent light transport effects

substantially better than that of ATR and FTR, which is equal, in existing devices, according to Brodsky et al. [23], not to more than one part in 105 of dielectric function. Our published results indicate that GLRS allows us to measure one part in 106 even with our current suboptimal apparatus. Furthermore, this precision represents only a current experimental limitation and by no means indicates a fundamental barrier for the GLRS method. To date, our research has focused on expanding the applicability of GLRS as much as its sensitivity. With better detector resolution, source stability, temperature control, and so on, we predict that measurements of one part in 107 can be easily achieved. GLRS also has advantages in comparison with different methods based on the formation of surface plasmons, which have a sensitivity approximately the same as ATR. In addition, the surface plasmon technique is significantly limited by the requirement of the application of highly conductive

Fig.  3.4  Typical LCI signals of highly scattering systems (b). The averaged reflectometer scans for different concentrations (by weight) and diameters of polystyrene microspheres in deionized water [12].

3.4  Low coherence interferometry of nonuniform media   61  

metal films, for example, silver or gold. The sensitivity of the corresponding measurements depends heavily on the details of surface structure, including adsorption. Therefore, it is very difficult to construct robust industrial sensors to measure bulk sample properties based on this technique. In conclusion, we provide Figures 3.2 and 3.3, which show examples of dependencies of GLRS signals on particle characteristics, and Figure 3.4, which shows a comparison of the observed signals with the theoretical calculations. These figures are taken from [18,19]. In Brodsky et al. [12] one can also find experimental results demonstrating the high sensitivity phase changes of scattered light near critical points to the characteristics of studies media.

3.4  Low coherence interferometry of nonuniform media In our work [11], substantial coherence effects were observed in low coherence (white light) interferometry of nonuniform dielectric media with characteristic nonuniformity dimensions less than, or of the order of, the wavelength. LCI, which was originally developed to resolve sharp discontinuities along guided wave structures [24], is based on a classical optical measurement technique first proposed by Sir Isaac Newton. A block diagram of the LCI device, which allows direct measurements of coherence effects in backscattering, is shown in Figure 3.5. In this device, light from the source is split evenly via an optical fiber splitter. One of the fibers directs light to the test sample and the other to a moving reference mirror. The wave packets reflected from the sample and the wave packets reflected from the reference mirror are recombined at the coupler. Interference between two light signals is observed when their optical path lengths match to within the corresponding coherence length of these packets. Unlike in cases of sharp discontinuities along the waveguide, where a processed signal (interferogram) looks like a single peak, in cases of highly scattering systems, the processed signal (and the underlying interferogram) takes a much more elaborate shape. Typical LCI signals of highly scattering systems are shown in Figure 3.6. Note that these figures clearly show that the intensity of the initial peak (reflection) is proportional to the particle size and concentration in the medium. The observed positive deviation from Beer’s Law at longer photon dwell times (see Figure 3.6) indicates the realization of some mechanism of the effective coherent

62   3  Optical diagnostics based on coherent light transport effects

Fig.  3.5  Block diagram of LCI device used in [11(2)].

light storage. Our work [11] showed that LCI measurements provided information not only about the concentration of nonuniformities (particles) in a medium, but also about the degree of uniformity of their space distribution. The intensity of the initial peak (reflection) in the LCI signal is proportional to the particle size and concentration in the medium. The intensity of the high frequency fluctuations in an averaged signal is related to the Brownian motion of the particles and of nonuniformities, whereas the positive deviation from Beer’s Law of the LCI signal at longer photon dwell times indicates the realization of some mechanisms of the effective coherent light storage (quasi­localization) specific for random particle distributions. The main conclusion of the detailed analysis of our experimental data on LCI [11,24] is that this method allows us to get detailed information about the structural characteristics of nonuniformities in nanostrucrures with the length of these nonuniformities substantially less than the wavelength. This result is important for both scientific and practical applications. In particular, it follows from our results that coherent nonionizing optical scattering may serve as an efficient substitute for x-ray-based techniques in medical imaging. The problem of finding such substitute has been actively discussed in literature (see reviews [25]). The preliminary experiments also show that LCI can be used for the control of polymerization and many other industrial processes where it is necessary to have the information about the properties of nanostructures.

3.5  Theory of coherent transport effects in GLRS and LCI   63  

Fig.  3.6  Typical LCI signal of highly scattering systems (a). A single reflectormeter scan (the highly fluctuating line) and an average of 50 individual scans (bold line) for a 2.5% (by weight) solution of 308 nm in diameter polystyrene microspheres in deionized water. It is the distance difference between the mirror position and the probe/ sample interface, with respect to the instrument coupler. The dotted line represents a Beer’s Law type of decay curve [11(2)].

3.5  Theory of coherent transport effects in GLRS and LCI To extract maximum information from GLRS and LCI experiments it is necessary to use and advance further the theory of light propagation in nonuniform nanostructures. The general theory of wave propagation in nonuniform media is different for the following three regimes: 1. The short wave limit, when the distances between nonuniformities are much bigger than the wavelength and it is possible to use for the inter-

64   3  Optical diagnostics based on coherent light transport effects

pretation of the coherent potential approximation or even geometric optics. 2. The opposite limit, where the wavelength is much longer than both the interparticle distances and the distances between their density fluctuations and, correspondingly, the wave cannot resolve the detailed structure of inhomogeneities and the effective medium concept cannot be effective. 3. The intermediate regime, defined as the one in which the wavelength is comparable to or not very much less than the distances between the inhomogeneities. The intermediate regime is the most difficult one to treat theoretically, since both perturbation and homogenization approximations are, in that case, inapplicable. However, this regime is rich in physical phenomena, such as resonant scattering, coherent and diffusive transport localization, and so on. It is impossible to extract the most theoretically and practically valuable information about nanostructures from CTA measurements without its theoretical understanding. We have proposed [23] the general method of solutions of the problem of electromagnetic wave propagation in random media, valid for wide parameter intervals including the intermediate regime, which is based on analogy with the exact solution of the Bloch-Nordsiek model in quantum field theory with the help of Fock’s techniques of an introduction of the fifth time-like coordinate. The substantial element of the developed theory is the exact reduction of the Maxwell wave equations to the system of the first order differential equations. The theory was compared in Yodh and Chance [25] with LCI results for the mean scattering amplitude (coherence function of the first order) for the case of Gaussian distribution of non-small particles in studied dielectric media. There is a possibility of straightforward generalization of the theory to the case of the coherence functions of the second and fourth order that are necessary, in particular, for analysis of information contained in the fluctuations observed in LCI. Generalization of the theory to the case of the presence of surface nonuniformities, including the surface roughness, will require solving only the surmountable technical problems. Much more difficult is the problem of the description of the non-­Gaussian, multiscale nonuniformity distributions. At the same time, the analysis of such systems is especially important for many practical applications.

3.6  Concluding remarks   65  

Substantial unresolved problems are connected with the analysis of nonstationary particle distributions. It was shown in Brodsky et al. [11] that, for example, the Brownian motion changes the LCI signal fluctuations. In the nonstationary case, the complications are connected with the specific character of granular matter movements, characterized by nonlinear inelasticity effects [26–28], cluster formation, and collapse. The analysis of CTE measurements has a substantial potential to provide data necessary for the understanding of the previously mentioned processes. The obtained experimental data confirms this conclusion.

3.6  Concluding remarks The possible future experimental and theoretical analysis of GLRS from matrices with random and regular nanoparticle distributions confined by the walls described is stated in [18,19]. The sensitivity can be increased by at least three orders of magnitude, in particular, by measuring the phase and polarization changes in scattered light. Direct phase changes in the light waves, backscattered from the systems of nanoscatterers, can be followed by low coherence interferometry. The method can be applied to the study of the equilibrium and dynamics of dense systems of the scatterers. Special attention has to be paid to the quasi-localization and memory effects noticed in our previous experiments and especially to the evaluation of information provided by fluctuation patterns in the reflected light beams. We will combine data received by LCI and GLRS in the theoretical analysis of experiments. It is important to measure effective birefringence, which can arise even in optically nonactive inhomogeneous media in experiments with polarized light waves (both in GLRS and LCI). The theoretical analysis of corresponding experimental results allows the extraction of the maximal amount of information about inhomogeneities. Special scattering experiments with polarized light will be made with optically active media and distributed in inactive medium optically active particles. The information on the media optical activity and the optically active particle characteristics is important in particular for pharmaceutical industry and medical diagnostics, as well as in environmental monitoring and detection of hazardous biosuspenses in the atmosphere. The measurement of nonlinear effects, such as second harmonic generation and wave mixing in the

66   3  Optical diagnostics based on coherent light transport effects

case of light scattering from random systems, is of special interest, since in these systems there are possibilities of drastic local wave intensity increase due to stochastic resonances and local caustic formation [29,30]. The measurements of such nonlinear effects can provide important information about the uniformity of studied systems including void and cluster formations. With the help of GLRS it is possible to measure mesoscopic fluctuations of nanoparticle concentrations near electrochemical interfaces. Analysis of the fluctuation and the decrease of backscattering enhancement will provide additional information about particle concentration and movements, especially in optically active media [26]. It is possible to measure the influence of nanoparticle dynamics, including the influences of Brownian motion and rheology effects, in both GLRS and LCI. In this chapter, the described works have shown that GLRS is a simple, rugged, versatile technique for monitoring the properties of heterogeneous matrices with nanoscale nonuniformities. GLRS has been used as a tool for monitoring particle size changes in drug particles at high solids concentrations induced by attrition milling processes. These results indicate that GLRS has performed successfully under conditions that inhibit the use of most other particle sizing methodologies, that is, at high solids concentration, with a large range of particle sizes, and with on-line/ in-line and real-time measurement capabilities. Although this study was performed off-line, the samples were analyzed at full concentration and since GLRS is a real-time characterization tool, it is expected that similar results would be obtained if the study were to be performed on-line. In fact, a fiber-optically coupled GLRS immersion probe suitable for on-line/ in-line monitoring and control in industrial processes has already been fabricated. Grating light reflection spectroscopy is a direct method that can be applied quite easily for the measurement of particle size in concentrated slurries (e.g. see Table 3.1). It is anticipated that GLRS will find many uses in the future for real-time monitoring of the optical properties of concentrated heterogeneous samples undergoing large changes in particle size. Finally, it is necessary to mention that the potential exists to use GLRS to monitor particle size distributions, which may be estimated by observing changes in the Mie resonances.

3.7  References and notes   67  

Tab.  3.1  Comparison of observed and predicted values for particle size distribution with Δ. Mean Diameter R0 (nm)

Particle size distribution width, Δ (nm): Manufacturer specification

Particle size distribution width Δ (nm): Best-fit theory prediction (5% CI, n = 49)

15.0 30.0 40.0 45.0 50.0

  6.3 12.7 16.9 15.9 17.6

  6.0 ± 0.5 12.5 ± 0.9 15.5 ± 1.2 18.3 ± 1.6 21.6 ± 1.3

Both GLRS and LCI should be considered as parts of a new general and flexible coherent wave “technology” that can be effectively applied to a variety of diagnostic problems of nonuniform media with nanometer characteristic dimensions including surfaces and photonic crystal characterizations [31].

3.7  References and notes 1. A. Ishimaru (1978). Wave Propagation and Scattering in Random Media (New York: Academic Press). The possibility of the manifestation of interference effects in multiple scattering has been suggested by K. Watson (1969). Multiple scattering of electromagnetic waves in an underdense plasma. J. Math. Phys. 10, 688. The realization of such effects in the electromagnetic wave backscattering was first determined in 1984 in experiments by Y. Kuga and A. Ishimaru (1984). Retroreflectance from a dense distribution of spherical particles. J. of Opt. Sco. Am. A8, 831. 2. V. Klyatskin (1980). Stochaotic Equations and Waves in Random Inhomogeneous Media (Moscow: Nauka). 3. I. Lifschitz, S. Gredescul, and L. Pastur (1988). Introduction to the Theory of Disordered Systems (New York: Wiley). 4. S. Rytov, V. Kravstov, and V. Tatarskii (1988). Principles of Statistical Radiophysics (Berlin: Springer Verlag). 5. M. Nieto-Vesperinas and J. Dainty, eds. (1990). Scattering in Volumes and Surfaces (Amsterdam: North-Holland). 6. V. Tatarsky, A. Ishimaru, and V. Zavorotny, eds. (1993). Wave Propagation in Random Media (Bellingham, WA: SPIE Press); P. Sheng (1995). Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (New York: Academic Press). 7. Yu. N. Barabanenkov, Yu. A. Kravtsov, V. D. Ozrin, and A. I. Saichev (1991). Enhanced Back-Scattering in Optics. Progress in Optics, 29, 65–197 (Amsterdam:

68   3  Optical diagnostics based on coherent light transport effects

Elsevier), 66; V. Shalaev (1996). Electromagnetic properties of small-particle composites. Physics Report 272, 61.   8. F. Scheffold and G. Maret (1998). Universal conductance fluctuations of light. Phys. Rev. Lett. 81, 5800.   9. R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk (1997). Observations of anomalous transport of strongly multiple scattered light in disordered thin slabs. Phys. Rev. Lett. 79, 4369. P. de Vries, D. van Coevorden, and A. Lagendijk (1998). Point scatteres for classical waves. Rev. Mod. Phys. 70, 1447. These works show the breakdown of radiative transport theory in the description of light propagation at short distances in strongly scattering samples. 10. P. Brouwer (1998). Transmission trough a many-channel random wavegide with adsorption Phys. Rev. B57, 10526. 11. A. Brodsky, P. Shelley, S. Thurber, and L. Burgess (1997). Low coherence interferometry of particles distributed in a dielectric medium. J. Opt. Soc. Am. A14, 2263. S. R. Thurber, A. M. Brodsky, and L. W. Burgess (2000). Characterization of random media by low-coherence interferometry. App. Spect. 54, 1506–1514. 12. A. Brodsky, L. Burgess, and S. Smith (1998). Grating light reflection spectroscopy. App. Spect. 52, 332A. 13. A. Burkov and A. Zyuzin (1996). Correlations in transmission of light through a disordered amplifying medium. Phys. Rev. B55, 5736. N. Lawandy, R. Balachandran, A. Gomes, and E. Sauvain (1994). Laser action in strongly scattering media. Nature 368, 436. P. de-Oliviera, A. Perkins, and N. Lawandy (1996). Coherent backscattering from high-gain scattering media. Optics Let. 12, 1685. 14. K. Efetov (1997). Supersymmetry in Disorder and Chaos (New York: Cambridge University Press). 15. The solution of Bloch–Nordsiek model with the help of Fock’s techniques is described in the Introduction to W. Bogolubov and D. Shirkov, eds. (1997). Quantum Field Theory (New York: Wiley). 16. G. Samelson and R. Mazar (1996). Path-integral analysis of scalar wave propagation in multiple-scattering random media. Phys. Rev. E54, 5497. 17. B. Anderson, L. Burgess, and A. Brodsky (1996). Grating light reflection spectroscopy for determination of bulk refractive index and absorbance. Anal. Chem. 68, 1081. 18. B. Anderson, L. Burgess, and A. Brodsky (1996). Threshold effects in light scattering from gratings. Phys. Rev. E54, 912. 19. B. Anderson, L. Burgess, and A. Brodsky (1997). Grating light reflection spectroscopy of colloids and suspensions. Langmuir 15, 4273. 20. B. Anderson, A. Brodsky, and L. Burgess. Analytical sensor using grating light reflection spectroscopy: Patent Number 5,502,560 (March 1996); Patent Number 5,610,708 (March 1997). 21. N. Harrick (1979). Internal Reflection Spectroscopy (Ossinging, NY: Harrick Scientific Corp.); G. J. Sprokel and J. D. Swalen (1991). The Attenuated Total Reflection Method: In Handbook of Optical Constants of Solids, edited by E. Palik (New York: Academic Press), Chap. 4.

3.7  References and notes   69  

22. D. Prieve, S. Bilke, and N. Frej (1990). Brownian motion of a single microscopic sphere in a colloidal force field. Faraday Disc. Chem. Soc. 90, 209, and the literature therein. 23. A. Brodsky, S. Thurber, P. Shelley, and L. Burgess (1997). The theory was described at the PIER Symposium; see the Proceedings, Cambridge, MA, July 7–11. 24. D. Danielson and C. Whittenberg (1987). Guided-wave reflectometry with micrometer resolution. App. Optics 26, 2836. 25. A. Yodh and B. Chance (1995). Spectroscopy and imaging with diffusing light. Phys. Today 48, W3, 34. 26. S. Torquato (1994). Unified methodology to quantify the morphology and properties of inhomogeneous media. Physica A207, 79. 27. N. Wagner, R. Kraus, A. Rennie, B. D’Auguanno, and J. Goodwin (1991). The microstructure of polydisperse, charged colloidal suspensions by light and neutron scattering J. Chem. Phys. 95, 494. 28. H. Jaeger, S. Nagel, and R. Behringer (1996). Granular solids, liquids and gases. Rev. Mod. Phys. 68, 1259. T. van Noije and M. Ernst (1997). Mesoscopic theory of granular fluids. Phys. Rev. Lett. 79, 411. 29. V. Markel, V. Shalaev, E. Stechel, W. Kim, and L. Armstrong (1996). Smallparticles composites: I. Linear optical particles. Phys. Rev. B53, 2425–2436. 30. V. Shalaev, E. Poliakov, and V. Markel (1996). Small-particles composites: II. Nonlinear optical properties. Phys. Rev. B53, 2437–2449, and the literature therein. 31. A. A. Maradudin, ed. (2011). Structured Surface as Optical Metamaterials (Cambridge: Cambridge University Press).

4  Ultrasonic grating diffraction spectroscopy and reflection techniques for characterizing slurry properties 4.1  Introduction The objective of this chapter is to describe the possibility of the new ultrasonic method – ultrasonic diffraction grating spectroscopy (UDGS) – to measure the speed of sound in liquids and slurries and to measure the particle size of a slurry. The first set of experimental results for liquids described in this chapter has been reported in [1–3]. UDGS is analogous to the method in optics that uses the grating light reflection spectroscopy (GLRS) described in Chapter 3. The optical method has been successful in determining the particle size of slurry in the range from about 2 to 200 nm and also in measuring the index of refraction. Ultrasonic allows us, in principle, to measure larger but still mesoscopic particle sizes. For the review of ultrasonic measurements techniques for liquids see [3,4]. The main element of the ultrasonic diffraction grating device described in [1,2] was formed by machining parallel triangular-shaped grooves, spaced 300 microns apart with a 120° angle, on the flat surface of a half cylinder (diameter = 5.08 cm, height = 3.8 cm) (Figure 4.1). As shown in Figure 4.2, the grating is placed in the immersion chamber and the send and receive transducer are mounted in brackets at an angle of 30° with respect to the normal direction. A toneburst signal of desired frequency is sent to the send transducer, producing a longitudinal wave that strikes the back of the grating at an incident angle of 30°. The receive transducer measures the reflected diffracted m = 0 longitudinal wave (see Figure 4.3). As the frequency decreases, the m = 1 transmitted longitudinal wave in the liquid moves to larger angle. For water, the critical angle becomes 90° at a frequency of 5.67 MHz. At this frequency, called the critical frequency, the wave becomes evanescent. At a slightly smaller frequency, the evanescent wave disappears. In order to conserve energy, the energy is distributed to all other waves, and an increase in amplitude of the reflected signal is expected. Such increase in amplitude was observed in the GLRS optics experiments and the first objective of the experiments in [2] was to observe a similar increase in amplitude for the UDGS experiments.

72   4  Ultrasonic grating diffraction spectroscopy and reflection techniques

Fig.  4.1  Stainless-steel ultrasonic diffraction grating [1]. Reprinted with permission from: M. S. Greenwood, A. M. Brodsky, L. Burgess, and L. J. Bond. ACS Symposium Series 943 (2006). Copyright 2006 American Chemical Society.

Fig.  4.2  Immersion chamber showing grating and transducer configuration [1]. Reprinted with permission from: M. S. Greenwood, A. M. Brodsky, L. Burgess, and L. J. Bond. ACS Symposium Series 943 (2006). Copyright 2006 American Chemical Society.

4.2  Description of UDGS experiments     73

Fig.  4.3  Schematic diagram of computer-controlled data acquisition system [1]. Reprinted with permission from M. S. Greenwood, A. M. Brodsky, L. Burgess, and L. J. Bond. ACS Symposium Series 943 (2006). Copyright 2006 American Chemical Society.

As the evanescent wave interacts with the particles, the attenuation of the signal occurs. As a result, the signal of the reflected m = 0 wave, which is measured by the receive transducer, decreases in amplitude. Since the attenuation is dependent on particle size, a corresponding algorithm can be developed to determine these sizes. Such an algorithm for the analysis of the data from the GLRS experiments has been developed in [1,2] and is described in this chapter. The set of measurements, obtained by observing the m = 0 reflected longitudinal wave, has been reported [1–3]. Data were obtained using stainless-steel gratings with a grating spacing of 200 microns and 300 microns for a variety of incident angles. A peak in the amplitude was observed at the expected critical frequency in all cases, but it was not large. One of the first objectives of the described research was to modify the experimental apparatus so that larger peaks are produced in the data at the critical frequency. A second objective was to investigate how the energy of the m = 1 transmitted wave is redistributed at frequencies below the critical frequency.

4.2  Description of UDGS experiments Two sets of experiment data for liquids using UDGS [1,2] and measurements of viscosity using reflection techniques [3] have been reported. The research discussed in this chapter has been centered on experiments

74   4  Ultrasonic grating diffraction spectroscopy and reflection techniques

with slurries of polystyrene spheres to determine the effect of varying the particle size. The first set of measurements was carried out with a stainless-steel grating at a critical frequency of 7.0 MHz. Notable effects were observed and have been reported in [2]. In order to increase the ultrasound transmitted into the slurry, experiments have been carried out with an aluminum grating at a critical frequency of 3.5 MHz for slurries of polystyrene spheres having diameters of 215 microns, 275 microns, 363 microns, and 463 microns, as a function of the weight percentage of the slurry. At 11% the signal for these four slurries has widely separated amplitudes. Additional data for the measurement of viscosity have been obtained using the 70° wedge. The measurement of viscosity has been carried out using a fused silica wedge with 45° angles and shows that the self-calibrating method gives good agreement to measurements obtained using a laboratory viscometer [5]. The experimental data reported in [1,2] were obtained using diffraction grating described in the introduction, formed by machining parallel triangular-shaped grooves on the flat surface of a stainless-steel halfcylinder, as shown in Figure 4.1. The half-cylinders have a diameter of 5.08 cm, with a groove spacing of 200 microns on one and 300 microns on the other. The included angle of the triangular groove is 120°. The grating is placed in the immersion chamber (27.9 cm diameter) in a recess in the mounting plate to fix its position and angle relative to the send transducer, as shown in Figure 4.2, and the chamber is filled with the desired liquid. The mounting plate can rotate in order to change the incident angle θ, and is then fastened at the desired (fixed) angle to the base of the chamber with set screws. The send and receive transducers are mounted in brackets that are placed in the aluminum housings. The send transducer housing is fastened to the base of the chamber. The chamber itself is mounted on a turntable, which is motorized and computer-controlled to an angular placement with an accuracy of 0.1°. The receive transducer is mounted to the (nonrotating) base of the turntable and, thus, the receive transducer does not move. The desired angle between the send and receive transducers is obtained by rotating the turntable with the immersion chamber (and send transducer) attached. In Figure 4.2 the send and receive transducers make equal angles with respect to the normal grating surface; the receive transducer is pointed at the back surface of the diffraction grating in this photograph. The system also permits the receive transducer to be pointed to the front surface of the diffraction grating.

4.2  Description of UDGS experiments     75

The data acquisition system consists in [1–3] of a commercially available pulsar-receiver card and a digitizer card mounted in a personal computer. The pulsar sends out a sinusoidal toneburst signal having an amplitude of about 330 volts peak-to-peak. The receiver has a maximum gain of 70 dB. The digitizer has 8 bits of resolution with a sampling rate of 100 MHz. A schematic diagram is shown in Figure 4.3. Custom software was written to control the two cards and consists of five modules: instrument setup, data acquisition and display, parameter measurement, data storage, and control of a motorized turntable to change angles between send and receive transducers. The software was written in “C” and operates on a DOS-based platform on a PC. Instrument setup consists of setting parameters for the pulser (frequency and number of cycles), receiver (gain), and digitizer (sample rate). Figure 4.4 shows the receive signal and cursors displayed on the computer monitor. The operator sets the software cursors: a time-of-flight gate labeled TOF, cursor G1, and cursor G2, and the threshold (TH) shown slightly above the time axis. The peak amplitude is found by examining the peak-to-peak amplitude between cursors G1 and G2 and selecting the largest value. The time of flight can also be obtained and is found by first locating the time point after the TOF gate where the waveform first exceeds the set threshold. Next, the software finds the preceding zero-voltage crossover point, which

Fig.  4.4  Toneburst signal obtained by receive transducer [1]. Reprinted with permission from M. S. Greenwood, A. M. Brodsky, L. Burgess, and L. J. Bond. ACS Symposium Series 943 (2006). Copyright 2006 American Chemical Society.

76   4  Ultrasonic grating diffraction spectroscopy and reflection techniques

becomes the resulting time-of-flight. Averaging is also used to factor out random noise. Two types of measurements are possible with this data acquisition system: scan-over frequency and scan-over angle.

4.2.1  Scan-over frequency In the setup of the data acquisition system, the receive transducer is positioned at a desired fixed angle, and the initial frequency, step in frequency Δf, and the final frequency are chosen. The system operates by sending a sinusoidal toneburst signal with the initial frequency. The maximum amplitude between the cursors G1 and G2 is determined and stored in a file. Then the frequency is increased by Δf and the maximum amplitude is determined again. This process is repeated until the final frequency is reached. The data file contains the amplitude for each frequency. It also contains the amplitude corrected for receiver gain, which is defined as the amplitude that would have occurred if there were no gain.

4.2.2  Scan-over angle In this case, a constant toneburst frequency is chosen. The initial angle of the turntable, the change in angle Δϕ, and the number of steps are chosen by the operator. The system operates by positioning the turntable at the initial angle and sending a toneburst signal of the constant frequency. The maximum amplitude between cursors G1 and G2 is determined and recorded to a file. Then the angle is increased by Δϕ, and the process is repeated for the desired number of steps. The data file contains the amplitude for each angle. The amplitude corrected for receiver gain is also provided.

4.2.3  Measurements with the blank To account for the transducer response as a function of frequency, measurements were made with the blank, which has the same dimensions as the grating except that the flat surface of the half-cylinder is smooth.

4.3  Grating equation and critical frequency calculation Figure  4.5 shows an ultrasonic beam of frequency f traveling through a solid and striking the grating-liquid interface at an incident angle θ.

4.3  Grating equation and critical frequency calculation     77

Fig.  4.5  Definition of angles and path lengths [1]. Reprinted with permission from M. S. Greenwood, A. M. Brodsky, L. Burgess, and L. J. Bond. ACS Symposium Series 943 (2006). Copyright 2006 American Chemical Society.

As a result of constructive interference, a refracted beam in the liquid is produced at angle ϕm. The distance between adjacent grooves in the ultrasonic diffraction grating is d. (For simplicity, the grooves are shown schematically here as “slits.”) In the solid the speed of sound is CL and the wavelength is λ1. In the liquid the speed of sound is c and the wavelength is λ. Constructive interference occurs when:

78   4  Ultrasonic grating diffraction spectroscopy and reflection techniques







(1)

where m is zero, or a positive or negative integer. When m = 0, Snell’s Law is obtained: .

(2)



Using the results of Equation 2, Equation 1 becomes .



(3)

Equation 3 is the so-called grating equation and determines the angle ϕm. Note that as the frequency decreases, the angle ϕ1 is 90° and Equation 3 becomes .

(4)



To understand how UDGS can be used to measure the speed of sound, consider a longitudinal wave striking the back of a 300-micron stainlesssteel grating at an incident angle of 30°, as shown in Figure  4.6A. The specularly reflected m = 0 longitudinal wave is measured by the receive transducer. In Figure 4.6B the incident and reflected waves are shown, as well as the reflected m = 0 and m = −1 shear waves, and the transmitted m = 0 and m = 1 longitudinal waves. The positions of the m = 0 waves are, of course, unchanged when the frequency changes. However, the diffracted m = 1 longitudinal wave in water does change with frequency. Figure 4.6B shows the position of the m = 1 transmitted longitudinal wave at 7 MHz, 6 MHz, and 5.65 MHz. As the frequency decreases, the angle increases, and at a frequency of 5.65 MHz, the angle becomes 90°. At this frequency, called the critical frequency, the wave transforms from a traveling wave and becomes evanescent. This means that it is an exponentially decaying wave in the liquid or slurry. At a slightly smaller frequency, the evanescent

4.3  Grating equation and critical frequency calculation     79

Fig.  4.6A  Grating and transducer configuration for obtaining scan-over-frequency data [1]. Reprinted with permission from M. S. Greenwood, A. M. Brodsky, L. Burgess, and L. J. Bond. ACS Symposium Series 943 (2006). Copyright 2006 American Chemical Society.

Fig.  4.6B  Diagram showing reflected and transmitted diffracted waves [1]. Reprinted with permission from M. S. Greenwood, A. M. Brodsky, L. Burgess, and L. J. Bond. ACS Symposium Series 943 (2006). Copyright 2006 American Chemical Society.

80   4  Ultrasonic grating diffraction spectroscopy and reflection techniques

wave disappears. To conserve energy, the energy is redistributed to all other waves, and an increase in amplitude of the specularly reflected signal is expected. This increase in amplitude has been observed [1,2], and one objective in the current research is to obtain the critical frequencies with a smaller experimental error and also for two different liquids. Figure 4.6B also shows the m = −1 reflected shear wave in the stainless steel. As the frequency decreases, this wave approaches −90° and also becomes evanescent at a frequency of 8.6 MHz. The energy redistribution at this frequency should also be observed in the receive transducer.

4.4  Experimental measurements Figure 4.7 shows a photograph of the stainless-steel grating in which the transducers are exposed directly to the stainless steel. The triangular grooves are spaced 243 microns apart, with a 1200 included angle. The longitudinal

Fig.  4.7  Photograph of a 243-micron stainless-steel grating [1]. Reprinted with permission from M. S. Greenwood, A. M. Brodsky, L. Burgess, and L. J. Bond. ACS Symposium Series 943 (2006). Copyright 2006 American Chemical Society.

4.4  Experimental measurements     81

send and receive transducers (0.95 cm diameter) make an angle of 33° with respect to a normal to the grating surface. A shear wave receive transducer is also positioned at 17°, but data are not presented here. To account and correct for the transducer response, the stainless steel part was fabricated with a smooth front face, and it is called a “blank.” The transducers are exposed in place, and data for a scan-over frequency are obtained for the blank. Then the grating is machined on the front face. At each frequency in the scan-over frequency, the data for the grating are corrected by the following: Vgrating corrected for transducer response = Vgrating/Vblank.

(5)

In the described work, data were obtained for: water; sugar water (SW) solutions having weight percentages of 5, 10, 15, 17.5, 20, 22.5, 30, 35, 40, 43, 46, 50, 52, 54, and 56; and NaCl solutions having weight percentages of 1.07, 2.75, 4.44, 6.31, 10.2, 12,3, 16.0, 19.0, and 22.2. Two or more trials were taken for each solution. The density of each one was measured using a pychnometer, and the velocity of sound was measured by using time-of-flight method. The weight percentages of the NaCl solutions were chosen so that the velocity was very close in value to a given SW solution. For example, 15% SW has a velocity of sound of 1538 m/s, and 4.44% NaCl, 1535 m/s. Fifty percent SW has a velocity of 1720 m/s, and 22.2% NaCl, 1744 m/s. The viscosity of SW solutions increases with its concentration. For example, 15% SW has a viscosity of 1.6 cP, while 50% SW has a viscosity of 15.4 cP at 20 °C. The sodium chloride solutions have a nearly constant viscosity over the range of interest with only a viscosity of 2.0  cP at a concentration of 26 wt %. Data were obtained for a scan-over frequency for all of the solutions. The objective is to compare the experimental value of the critical frequency with the theoretical value obtained from Equation 4, using the measured velocity of sound. Figures 4.8 to 4.11 show the peak that occurs at the critical frequency. The frequency listed in each figure is the maximum value for the trial and is, of course, close in value to the critical frequency. The Savitsky-Golay smoothing and differentiation procedure was used to analyze the data to obtain the critical frequency. This procedure smoothes the data over a desired number of points (filter) and then fits a polynomial of desired order through these points. The value of the

82   4  Ultrasonic grating diffraction spectroscopy and reflection techniques

Fig.  4.8  Scan-over frequency data for 10% sugar water and 2.75% NaCl. The velocity of sound in 10% sugar water is 1,515 m/s and in 2.75% NaCl is 1,513 m/s [1]. Reprinted with permission from M. S. Greenwood, A. M. Brodsky, L. Burgess, and L. J. Bond. ACS Symposium Series 943 (2006). Copyright 2006 American Chemical Society.

4.4  Experimental measurements     83

Fig.  4.9  Scan-over frequency data for 20% sugar water and 6.32% NaCl. The velocity of sound in 20% sugar water is 1,554 m/s and in 6.31% NaCl is 1,554 m/s [1]. Reprinted with permission from M. S. Greenwood, A. M. Brodsky, L. Burgess, and L. J. Bond. ACS Symposium Series 943 (2006). Copyright 2006 American Chemical Society.

84   4  Ultrasonic grating diffraction spectroscopy and reflection techniques

Fig.  4.10  Scan-over frequency data for 30% sugar water and 10.2% NaCl. The velocity of sound in 30% sugar water is 1,603 m/s and in 10.2% NaCl is 1,604 m/s [1]. Reprinted with permission from M. S. Greenwood, A. M. Brodsky, L. Burgess, and L. J. Bond. ACS Symposium Series 943 (2006). Copyright 2006 American Chemical Society.

4.4  Experimental measurements     85

Fig.  4.11  Scan-over frequency data for 40% sugar water and 16% NaCl. The velocity of sound in 40% sugar water is 1,663 m/s and in 16.0% NaCl is 1,666m/s [1]. Reprinted with permission from M. S. Greenwood, A. M. Brodsky, L. Burgess, and L. J. Bond. ACS Symposium Series 943 (2006). Copyright 2006 American Chemical Society.

86   4  Ultrasonic grating diffraction spectroscopy and reflection techniques

derivative is obtained by differentiating the polynomial. Studies [1,2,5] have shown that the critical frequency is located at the peak value of the grating data, corrected for transducer response in Equation 5. Therefore, determining the frequency at which the first derivative is zero is the method used to determine the critical frequency. The first-derivative method is difficult when the peak of interest is close to the m = −1 shear wave peak in the data. It is more precise to use the grating data that are not corrected for transducer response. In these data one can see the amplitude decreasing as the frequency increases. Near the critical frequency, a small peak appears in the data, which is identified most easily by using the second derivative to note a change in curvature. Two sets of data were obtained for SW. In each set, two trials were obtained for each solution, although the SW concentrations were not the same in both data sets. The critical frequency for each data set is the average value for the two trials. Figure 4.12 shows the experimental critical frequencies for the SW solutions. The asterisks show data set 1 analyzed using

Fig.  4.12  Comparison of experimental critical frequencies for sugar water solutions with theoretical values [1]. Reprinted with permission from M. S. Greenwood, A. M. Brodsky, L. Burgess, and L. J. Bond. ACS Symposium Series 943 (2006). Copyright 2006 American Chemical Society.

4.4  Experimental measurements     87

Fig.  4.13  Comparison of experimental frequencies for NaCl solutions with theoretical values [1]. Reprinted with permission from M. S. Greenwood, A. M. Brodsky, L. Burgess, and L. J. Bond. ACS Symposium Series 943 (2006). Copyright 2006 American Chemical Society.

the first derivative and the diamonds using the second derivative. Data set 2 was analyzed using only the second derivative and is shown by the square symbol. Figure  4.12 shows that the first- and second-derivative analysis methods are in good agreement. The theoretical values of the critical frequency are calculated using Equation 4 and are shown by the solid line. Figure  4.13 shows the analysis for the NaCl solutions, using the first and second derivative, and comparison with the theory using Equation 4.

4.4.1  Novel method for measuring the velocity of sound Equation 4 can be inverted to obtain the velocity of sound from a measurement of the critical frequency. However, the measurement of peak height above background, as defined in Figure  4.9, provides another method of determining the velocity of sound (or, with slightly larger error, the acoustic impedance). Figure 4.14 shows the peak height above background plotted versus the velocity of sound. For a given velocity of sound, both the SW and NaCl solutions yield values of the peak height

88   4  Ultrasonic grating diffraction spectroscopy and reflection techniques

Fig.  4.14  Peak height above background versus velocity of sound for sugar and NaCl solutions [1]. Reprinted with permission from M. S. Greenwood, A. M. Brodsky, L. Burgess, and L. J. Bond. ACS Symposium Series 943 (2006). Copyright 2006 American Chemical Society.

above background that are very close in value. The dependence on density alone is ruled out because of plot of peak height above background versus density and shows poor agreement.

4.4.2  Possible effect of viscosity Comparison of the graphs of the critical frequency versus the velocity of sound for SW solutions and SW solutions and NaCl solutions shows that the data and theory seem to fit somewhat better for the NaCl solutions than for the SW solutions. The data for the SW solutions show that the experimental critical frequency is smaller at larger concentrations than the theory. However, 50% SW has a viscosity of 15  cP at 20 °C, while 56% has 32 cP, but there is not a corresponding deviation shown between the measured and theoretical value for these two points. It is possible that the experimental error is the cause of the deviation. However, the first and second derivative analyses are in good agreement. At this point, the SW

4.5  Shear wave reflection techniques and the measurement of viscosity      89

data offer a suggestion of the role of the viscosity. Extending Equation 4 to include the effects of viscosity is being considered by collaborators at the University of Washington, and that work is in progress. Therefore, UDGS offers the possibility of measuring the velocity of sound, and the deviation between the experimental and theoretical values of the critical frequency can be used to see the effect of the viscosity.

4.5  Shear wave reflection techniques and the measurement of viscosity Figure 4.15 shows a fused quartz wedge, with the base immersed in the liquid, for the measurement of viscosity. The horizontal shear wave transducer has a frequency of 7.5 MHz. In 1949 Mason et al. [6] showed that when ultrasound strikes the base at an angleθ with respect to a normal to the base, the effective acoustic impedance is given by

.

(6)

Fig.  4.15  Diagram of an apparatus for observing multiple reflections of a horizontal shear wave [1]. Reprinted with permission from M. S. Greenwood, A. M. Brodsky, L. Burgess, and L. J. Bond. ACS Symposium Series 943 (2006). Copyright 2006 American Chemical Society.

90   4  Ultrasonic grating diffraction spectroscopy and reflection techniques

Thus, using a large angle, such as 70°, makes the measurement of viscosity more efficient by causing a greater change in the reflection coefficient for various liquids. The plan for the research was to use the selfcalibrating method developed for the density measurements (Equation 6) for the measurement of viscosity. A shear wave transducer produces ultrasound in which the vibrations are perpendicular to the direction of motion, in contrast to the longitudinal transducer in which the vibrations are along the direction of motion. If one imagines an arrow pointing in the direction of the vibrations, then, in Figure 4.15, the arrow would be perpendicular to the plane of a paper, producing so-called horizontal shear wave. The electronics for this measurement are quite different from those used for UDGS. In this case, a square wave pulse is sent to the transducer with a voltage of 140  V and a width of 67 nanoseconds. The output signal from the receiver was input to a 12-bit digitizer board in the computer. The data acquisition was set up to analyze the signals from the digitizer (12-bit, 100 Megasamples/sec) automatically using Matlab software and digitizer software. Figure 4.15 shows that the ultrasound strikes the quartz-liquid interface, where some of it is reflected to the opposite side; 100% is reflected by the quartz-air interface. It then strikes the quartz-liquid interface again, and is reflected to the transducer, where a pulse is recorded. Because shear waves do not travel easily in a liquid, a very small amount of ultrasound travels into the liquid. The reflection coefficient at the quartz-liquid interface is defined as the ratio of the amount of ultrasound reflected to the amount of incident ultrasound. As shown in this section, the reflection coefficient is dependent on the viscosity. The reflection coefficient decreases as the viscosity increases because more energy is transmitted into the liquid. That is, a “stiffer” liquid can more easily support a shear wave. The first set of data, after the electronics and computer code were all deemed to be working correctly, was obtained for water, 10% SW, 30% SW, and 50% SW. These data, shown in Figures 4.16–4.18, were obtained in a 3-hour interval and are used to illustrate the type of data acquired. Figure 4.16 shows the signal obtained by the transducer, and Figure 4.17 shows the expansion of one of the echoes and the fast Fourier transform (FFT) of that signal. The analysis is carried out by obtaining the peak value of the voltage signal and the maximum value of the FFT amplitude.

4.5  Shear wave reflection techniques and the measurement of viscosity      91

Fig.  4.16  Signal obtained by receive transducer showing echoes produced by multiple reflections [1]. Reprinted with permission from M. S. Greenwood, A. M. Brodsky, L. Burgess, and L. J. Bond. ACS Symposium Series 943 (2006). Copyright 2006 American Chemical Society.

Fig.  4.17  Voltage signal for echo 4 (top) and its fast Fourier transform [1]. Reprinted with permission from M. S. Greenwood, A. M. Brodsky, L. Burgess, and L. J. Bond. ACS Symposium Series 943 (2006). Copyright 2006 American Chemical Society.

92   4  Ultrasonic grating diffraction spectroscopy and reflection techniques

Fig.  4.18  Data obtained for water and thee sugar water solutions during a 3-hour time interval [1]. Reprinted with permission from M. S. Greenwood, A. M. Brodsky, L. Burgess, and L. J. Bond. ACS Symposium Series 943 (2006). Copyright 2006 Amer­ican Chemical Society.

Fig.  4.19  Data plotted on a logarithmic scale versus echo number [1]. Reprinted with permission from M. S. Greenwood, A. M. Brodsky, L. Burgess, and L. J. Bond. ACS Symposium Series 943 (2006). Copyright 2006 American Chemical Society.

4.6  Data analysis     93

The data for four liquids are shown in Figure 4.18, where the FFT amplitude is plotted versus the echo number. These data were obtained during a 3-hour time interval at room temperature (22 °C). The important point is that the data for the four samples are separated, although the separation for water and 10% SW is not obvious on this scale. The analysis of these data is shown in Figure 4.19, using water as reference. The point here is that the water data and the data for the SW solutions were obtained in a relatively short period of time –3 hours. The experimental data in Figure 4.19 show a very good fit with a straight line.

4.6  Data analysis The data are analyzed by dividing the amplitude for the liquid by the amplitude for water for each echo. The “amplitude” can refer to the FFT amplitude or the peak voltage amplitude. Thus, water is being used as the calibration liquid. The only difference between the liquid and water data is the result of the reflection at the interface. In the division, all other factors cancel out. The natural logarithm of the ratio is plotted versus the echo number, as shown in Figure  4.20, and the slope of the line is obtained for each liquid.

Fig.  4.20  Percent error in an average of five sets of data [1]. Reprinted with permission from M. S. Greenwood, A. M. Brodsky, L. Burgess, and L. J. Bond. ACS Symposium Series 943 (2006). Copyright 2006 American Chemical Society.

94   4  Ultrasonic grating diffraction spectroscopy and reflection techniques

4.6.1  Slope and reflection coefficient To obtain the relationship between the slope and the reflection coefficient, we use the fact that the voltage of the Nth echo is given by: Voltage = α (reflection coefficient for liquid)N.

(7)

Water is used for calibration of the slopes. Using Equation 7 for water and another liquid, we obtain the following for the echo N1: .

(8)

A similar equation can be written for echo N2. Taking the logarithm of both equations and subtracting yields the desired relationship between slope and reflection coefficient: 



(9)

The term slope refers to the slope on the logarithmic plot shown in Fig­ure 4.19. To determine the reflection coefficient for the liquid using the slope, the reflection coefficient for water must be determined. The relationship between viscosity and the reflection coefficient is given in Greenwood and Bamberger [3] as: ,





(10)

where ρ and η are the density and viscosity of the liquid, ρs is the density of the solid, cTS is the shear wave velocity in the solid, θ is the angle of incidence with respect to the normal, R is the reflection coefficient, and ω is the angular frequency expressed in radians per second. Solving Equation 10 for the reflection coefficient R:



(11)

where K is given by

.

(12)

4.7  Sensor calibration and reliability     95

Tab. 4.1  Results for Data Obtained during a 3-hour Interval Liquid

Slope

liquid Sensor Product Density by Viscosity of Density at 22 °C Weight and Viscosity (g/cm3) (centipoises) (cP-g/cm3)

Rc

10% SW -1.28E-03 0.99594 1.401 30% SW -6.15E-03 0.99352 3.587 54% SW -2.83E-02 0.98257 26.313

1.038 1.121 1.253

1.35 ± 0.04 3.20 ± 0.04 21.0 ± 0.2

Handbook Viscosity at 20 °C (centipoises) 1.33 3.18 24.6

Since the viscosity of water is 1 cP, Equations 11 and 12 can be used to determine the reflection coefficient for water. Then, using the slope on logarithmic plot, Equation 9 can be used to determine the reflection coefficient for the liquid. The next step is to determine the product of the density and viscosity using Equation 10. For fused quartz, the shear velocity is 3,760 m/s, and the density is 2,200 kg/m3.

4.6.2  Data obtained during a 3-hour interval Table  4.1 shows the results of the data illustrated in Figure  4.19. The agreement between the data and theory is excellent, and the viscosity values are also in very good agreement with handbook values.

4.7  Sensor calibration and reliability The data presented previously were obtained at a room temperature of about 22 °C. The sensor would be calibrated by obtaining data for water, similar to that shown in Figure 4.18, for several temperature values over the desired temperature range. The data for each temperature would be stored in a file. For the reliable operation of the sensor, the water reference file must not change appreciably at a given temperature day after day. To test the reliability over time, five sets of data were obtained for water during a 5-day period, usually one each day. For each echo, average values of the peak voltage and FFT amplitude were obtained, as well as the standard deviation of the five values. The standard deviation was converted to a percent error, which is plotted versus the echo number in Figures 4.19 and 4.20 for the peak voltage values. The five

96   4  Ultrasonic grating diffraction spectroscopy and reflection techniques

Tab. 4.2  Viscosity Measurements Obtained for Sugar Water Solutions Using Average Water Reference File Liquid

Number of Trials

Sensor Viscosity (cP)

Standard Deviation

Handbook Viscosity at 20 °C

  5% SW 10% SW 15% SW 20% SW 25% SW 30% SW 35% SW 40% SW 45% SW 50% SW 55% SW 60% SW

3 2 3 3 2 1 2 2 2 2 2 2

  1.35   1.64   1.72   2.11   2.49   2.78   3.32   6.53   7.41 12.50 24.00 45.10

0.19 0.80 0.33 0.72 0.46 ? 0.03 0.92 0.31 0.26 3.70 0.93

  1.14   1.33   1.59   1.94   2.45   3.18   4.33   6.15   9.43 15.40 28.40 42.70

Fig.  4.21  Comparison of viscosity obtained by sensor with handbook values [1]. Reprinted with permission from M. S. Greenwood, A. M. Brodsky, L. Burgess, and L. J. Bond. ACS Symposium Series 943 (2006). Copyright 2006 American Chemical Society.

4.8  Conclusion     97

sets of data for water were used to obtain an average water reference file for the FFT amplitude and the peak voltage values. The data for 11 SW solutions were analyzed using both the FFT amplitude water reference file and the peak voltage water reference file. The results showed that the most accurate values of the viscosity were obtained using the peak voltage water reference file. Table 4.2 shows the results using the peak voltage water reference file and using echoes 2 through 20 to obtain the slope on the logarithmic plot. Figure 4.21 compares the experimental measurements with the handbook values with quite good agreement.

4.8  Conclusion The theory and experiments described in this chapter confirm that ultrasonic diffraction grating spectroscopy (UDGS) can be used as the effective method of measurement of particle sizes in liquid and gas with dimensions larger than those of nanoparticles, which is based on the GLRS analysis described in Chapter 3. The combination of GLRS and UDGS measurements opens the possibility of the effective application of coherence lost in light and acoustic spectroscopy for the characterization of statistics and dynamics of a number of practically important systems. The considered theoretical approach can be used in the description of a number of nontrivial effects in liquids, in particular in the description of sonoluminescence [4], and can be applied to the description of dynamical processes in electrochemistry and microbiology (see the latest works on the subject by A. Kornyshev et al. [5]). For the general discussions of the perspectives of acoustic studies see Brodsky [4] and Panetta et al. [6].

4.9  References and notes 1. M. S. Greenwood, A. M. Brodsky, L. Burgess, and L. J. Bond (2006). Investigating Ultrasonic Diffraction Grating Spectroscopy and Reflection Techniques for Characterizing Slurry Properties. In: Nuclear Waste Management. Accomplishments of the Environmental Management Science Program. ACS Symposium Series 943. 2. M. S. Greenwood, A. M. Brodsky, L. Burgess, L. J. Bond, and M. Hamad (2004). Ultrasonic diffraction grating spectroscopy and characterization of fluids and slurres. Ultrasonics 42, 531.

98   4  Ultrasonic grating diffraction spectroscopy and reflection techniques

3. M. S. Greenwood and J. Bamberger (2002). Ultrasonic sensor to measure the density of liquid or slurry during pipeline transport. Ultrasonics 40, 413. 4. A. M. Brodsky (1999). What is the source of light in sonoluminescence? Mod. Phys. Letts. B 13, 1019–1025. 5. M. E. Flatté, A. A. Kornyshev, and M. Urbakh (2009). Nanoparticles at electrified liquid-liquid interfaces: New options for electro-optics. FARADAY Discussions 143, 109. 6. P. D. Panetta, B. J. Tucker, R. A. Pappas, and S. Ahmed (2003). Characterization of solid liquid suspensions utilizing ultrasonic measurements. AIP Conf. Proc. 657, 1644–1650. See also the review article by POAN Research Group (1998). New Aspects of Electromagnetic and Acoustic Wave Diffusion: In Springer Tracts in Modern Physics (Berlin: Springer Verlag), 144.

Index

acoustic waves 3–4 Anderson’s replica method 56 anisotropy of nanoparticles 3 attenuated total reflection (ATR) 59–60 Beer’s law 61, 62 Bloch-Nordsiek model 56, 64 Boltzman constant 25 Brownian motion 22, 53, 62, 65, 66 carbon nanotube networks, spectroscopy of 30–31 coagulation process 29–30 coalescence 9, 27 coherence effects 1, 32, 53, 54, 61 coherence loss, effects of 15–16, 31, 32 coherence loss in light backscattering 7–48 – Boltzman constant 25 – carbon nanotube networks, spectroscopy of 30–31 – coagulation process 29–30 – coalescence 27 – coherent backscatter signal, relative standard deviation (RSD) of 22 – compact form calculations 33 – concluding remarks 31–3 – critical embrions 25 – critical radius, central nature of 25 – Debye’s potentials 33 – droplet elongation, calculation of 23–4 – electric field, influence on phase transition from liquid to crystalline form 33 – electromagnetic energy flow 35 – equilibrium solution concentration 27–8 – experimental results and theory, comparison of 19–22 – experimental technique 7–14

– Fock’s idea 35 – Fraunhofer diffraction limit 17, 29 – Gaussian distributions 40, 64 – Green functions 35, 36, 56 – inversion of suspensions 12 – Kroneker’s symbol 33 – light propagation characteristics, relation with individual scatterer properties 15–19 – liquid or gas solutions, cases of 28–9 – MATLAB (MathWorks, Inc.) software 20 – Maxwell equations 33 – metastable, change to stable phase 25 – Michelson interferometer 19–22, 23 – non-Gaussian multiscale nonuniformity distributions 64–5 – nonscalar scattering components, effects of 32–3 – particle growth kinetics 26–7 – particle-medium interface, boundary condition at 18 – particle suspensions, decay profiles of 13 – phase transmission dynamics 24–30 – polarization structure 35 – polymer particles, suspensions of 8–9 – probe design 14 – quantitative analysis of coherence loss effects, dielectric characteristics of nanononuniform media and 32 – Rayleigh-Gans approximation 16–17, 25–6 – resonance scattering interval 17 – signal density, reproducible anomalies in 21 – signal dependence on particle radii 20–21 – signal fluctuations, analysis of 22 – single-walled carbon nanotube 31 – small particle radius (long wavelength limit) 25–6

100   Index

– solvents, decay profiles of 12 – static external electric field, influence of 23–4 – theory development 14–19 – thermodynamic theory of fluctuations 26–7 – van de Hulst approximation 17 coherent transport effects (CTE), optical diagnostics based on 53–67 – Anderson’s replica method 56 – attenuated total reflection (ATR) 59–60 – background information 54–6 – Beer’s law 61, 62 – Bloch-Nordsiek model 56, 64 – concluding remarks 65–7 – dielectric properties, measurement of 59–60 – effective birefringence, measurement of 65 – experimental and theoretical analysis of GLRS, future for 65 – Fock’s idea 64 – Fokker-Planck equation 56 – frustrated total internal reflection (FTR) 59–60 – GLRS technique, versatility of 66 – grating wave reflection spectroscopy (GWRS) 56–61 – interferograms 61–2 – low coherence interferometry of nonuniform media 61–3 – Maxwell wave equations 64 – measurement methods 55 – mesoscopic fluctuations, measurement of 66 – mesoscopic particle characteristics, measurement of 59–60 – nanostructure diagnostics 58–9 – nonlinear effects, measurement of 65–6 – one-dimensional wave propagation in random media, theory of 56 – particle size distributions, use of GLRS for monitoring of 66–7 – quazilocalization 62, 65 – surface plasmon technique 60–61 – theory of CTE in GLRS and LCI 63–5

– wave propagation in nonuniform media, differences from general theory 63–4 – wave scattering in inhomogeneous media, theory of 55–6 computer-controlled acquisition system in UGDS 73 concluding remarks – coherence loss in light backscattering 31–3 – coherent transport effects (CTE), optical diagnostics based on 65–7 – ultrasonic grating diffraction spectroscopy (UGDS) and reflection techniques 97 constructive interference of multiple scattered fields 54 critical embrions 25 critical frequency in UGDS 71, 73, 74, 81, 86, 87, 88, 89 – calculation of 76, 77, 78, 80 critical radius, central nature of 25 data acquisition system for UGDS 75 data analysis in UGDS 93–5 – data analysis algorithm 73 Debye’s potentials 33 destructive interference of multiple scattered fields 54 dielectric properties, measurement of 59–60 diffraction grating, construction of 71, 74 droplet elongation, calculation of 23–4 Duke Scientific Corporation 9–11 effective birefringence, measurement of 65 electric field, influence on phase transition from liquid to crystalline form 33 electromagnetic energy flow 35, 36 electromagnetic field fluctuations 3 electromagnetic waves 1, 2, 15, 54, 55, 64 embrions see critical embrions equilibrium solution concentration 27–8

Index     101

experimental and theoretical analysis of GLRS, future for 65 experimental critical frequencies for sugar water solutions, comparison with theoretical values 86 experimental data (UGDS) 74 experimental frequencies for NaCI solutions, comparison with theoretical values 87 experimental measurements (UGDS) 80–89 experimental results and theory of coherence loss in light backscattering, comparison of 19–22 experimental techniques – coherence loss in light backscattering 7–14 – ultrasonic grating diffraction spectroscopy (UGDS) and reflection techniques 71–3, 73–6 external field change, observable effects on 24 Faraday reflector mirrors (FRMs) 8 Fermi, Enrico 3, 15 first derivative method, difficulty with 86 fluctuations 3, 25, 26, 41–2 – amplitude 54 – dielectric 14 – fluctuation effects 22 – fluctuational waveguiding 1, 54 – high frequency fluctuations 62 – LCI signal 64–5 – mesoscopic 66 – particle fluctuations 15 – signal fluctuations 22 – thermodynamic theory of fluctuations 26–7 – wavelength density 64 Fock’s idea – coherence loss in light backscattering 35 – optical diagnostics based on CTE 64 Fokker-Planck equation 56 Fraunhofer diffraction limit 17, 29 frequency dispersion effects 45–6

Fresnel reflection 9, 12 frustrated total internal reflection (FTR) 59–60 Gaussian distributions 40, 64 grating equation and critical frequency calculation 76–80 grating light reflection spectroscopy (GLRS) – experimental and theoretical analysis of GLRS, future for 65 – particle size distributions, use in monitoring of 66–7 – technique of, versatility of 66 – theory of CTE in GLRS and LCI 63–5 grating wave reflection spectroscopy (GWRS) 55, 56–61 Green functions 35, 36, 56 immersion chamber 71, 72 instrument setup (UGDS) 75–6 interference effects 1, 3, 32 interferograms 61–2; see also low coherence interferometry (LCI); Michelson interferometer interferometer (Optiphase, Inc.) 8 invertion of suspensions 12 Kroneker’s symbol 33 light propagation characteristics, relation with individual scatterer properties 15–19 light scattering effects 23 liquid or gas solutions, cases of coherence loss for 28–9 low coherence interferometry (LCI) 55, 60, 64, 65, 66, 67 – of nonuniform media 61–3 MATLAB (MathWorks, Inc.) software – coherence loss in light backscattering 20 – reflection techniques and UGDS 90 Maxwell wave equations – coherence loss in light backscattering 33

102   Index

– optical diagnostics based on CTE 64 memory effects 65 mesoscopic fluctuations, measurement of 66 mesoscopic particle characteristics, measurement of 59–60 metastable, change to stable phase 25 Michelson interferometer 19–22, 23 Mie resonances 3, 21, 66 Mie theory 2, 3, 17, 18, 20, 21 nanononuniformities 1, 23 nanoparticles – distributed 1 – electric fields and, interfacial interactions between 24 – high concentrations of, spectroscopy of carbon nanotube networks with 30–31 – light emission by individual particles 4 – metallic and nonmetallic 3 – multiscattering systems of 2, 3, 53 – nonsphericity of, effects of 23 – phase transition dynamics in systems with 7, 24–30 – shape and anisotropy of 3 – spherical, random monodispersal with 18–19 – static external electric field, influence on light scattering by 7, 23–4 nanostructure diagnostics 58–9 non-Gaussian multiscale nonuniformity distributions 64–5 nonlinear inelasticity effects, measurement of 65–6 nonscalar scattering components, effects of 32–3 nonsphericity effects 23 one-dimensional wave propagation in random media, theory of 56 optical coherent scattering effects 53 optical wave packets 3–4, 53, 55 particles 19, 24 – aqueous latex particle suspensions, size characteristics of 11

– aqueous polystyrene particle suspensions, size characteristics of 13 – Brownian motion of 22, 62 – colloid 59 – concentration of 62 – evanescent wave interaction with 73 – Gaussian distribution of non-small particles 64 – growth kinetics of 27 – individual particles, free energy of 24 – individual particles, scattering of 3, 15–16, 21, 65 – latex particles in water 13 – nonspherical 24 – orientation along external static electric field 24 – particle growth kinetics 26–7 – particle-medium interface, boundary condition at 18 – particle movement effects 53 – particle size distributions, use of GLRS for monitoring of 66–7 – particle suspensions, decay profiles of 13 – polarizable 32–3 – polystyrene 13 – radar cross-section of 17, 29 – randomly distributed 3, 15, 29 – refractive indices of media and 17 – signal dependence on particle radii 20–21 – size changes 66 – spherical 2, 3, 16, 17, 42 – surrounding of, in experimental circumstances 18 – wave scattering by 58–9 phase effects 1 phase transmission dynamics 24–30 photoluminescence 31 photons 9, 14 – dwell times and distances 14, 61, 62 – photon diffusion 1, 55 – photonic crystal 67 – two-dimensional spin space 33–4 piezocomposite arrays 4 piezoelectric transducers (PZTs) 8 polarization structure 35 polymer particles, suspensions of 8–9

Index     103

polystyrene particles 9 polystyrene spheres, slurries of 74 probe design, coherence loss in light backscattering 14 pseudopotentials (Fermi) 3, 15

static external electric field, influence of 23–4 surface plasmon technique 60–61 suspended liquid drops, effects of electric field on deformation of 23

quantitative analysis of coherence loss effects, dielectric characteristics of nanononuniform media and 32 quazilocalization effects 62, 65

theory development – coherence loss in light backscattering 14–19 – coherent transport effects (CTE) in GLRS and CTI 63–5 thermodynamic theory of fluctuations 26–7 transducer response – correction for 81 – measurement with blank to account for 76 transmission electron microscopy (TEM) 9

Rayleigh-Gans approximation 16–17, 25–6 Redge poles 3 resonance frequency interval 3 resonance scattering interval 17 rheology effects 66 Savitsky-Golay smoothing 81 scan-over angle (UGDS) 76 scan-over frequency (UGDS) 76, 81, 82, 83, 84, 85 scattering effects, separation of 36 sensor calibration, reliability and 95–7 signal attenuation 73 signal density, reproducible anomalies in 21 signal dependence on particle radii 20–21 signal fluctuations, analysis of 22 single-walled carbon nanotube 31 small particle radius (long wavelength limit) 25–6 solid surfaces, effects of electric field on deformation of drops on 23 solvents, decay profiles of 12 sonoluminescence 97 speckle effects 40 spectroscopy – of carbon nanotube networks 30–31 – grating light reflection spectroscopy (GLRS) 3, 56–7, 58, 59, 60, 61, 63, 65, 66, 67, 71, 73, 97 – grating wave reflection spectroscopy (GWRS) 55, 56–61 – light and acoustic spectroscopy 97 – Raman spectroscopy 31 – ultrasonic grating diffraction spectroscopy (UDGS) 71–97

ultrasonic grating diffraction spectroscopy (UGDS) and reflection techniques 71–97 – computer-controlled acquisition system 73 – concluding remarks 97 – critical frequency 71, 73, 74, 81, 86, 87, 88, 89 – critical frequency, calculation of 76, 77, 78, 80 – data acquisition system 75 – data analysis 93–5 – data analysis algorithm 73 – diffraction grating, construction of 71, 74 – experimental critical frequencies for sugar water solutions, comparison with theoretical values 86 – experimental data 74 – experimental frequencies for NaCI solutions, comparison with theoretical values 87 – experimental measurements 80–89 – experimental techniques 71–3, 73–6 – first derivative method, difficulty with 86 – grating equation and critical frequency calculation 76–80 – immersion chamber 71, 72

104   Index

– instrument setup 75–6 – interval of three hours, data obtained during 95 – main element, construction of 71, 74 – MATLAB (MathWorks, Inc.) software 90 – polystyrene spheres, slurries of 74 – Savitsky-Golay smoothing 81 – scan-over angle 76 – scan-over frequency 76, 81, 82, 83, 84, 85 – sensor calibration, reliability and 95–7 – signal attenuation 73 – transducer response, correction for 81 – transducer response, measurement with blank to account for 76 – velocity of sound, measurement of 87–8 – viscosity 73, 74, 81, 90, 95 – possible effect of 88–9 – reflection coefficient and, relationship between 94 – shear wave reflection techniques and measurement of 91–3

– sugar water solutions, measurements for 96–7 ultrasonic sources, wide-band 4 ultrasound scattering 3–4 van de Hulst approximation 17 velocity of sound, measurement of 87–8 viscosity 73, 74, 81, 90, 95 – possible effect of 88–9 – reflection coefficient and, relationship between 94 – shear wave reflection techniques and measurement of 91–3 – sugar water solutions, measurements for 96–7 wave localization 1, 55 wave propagation in nonuniform media, differences from general theory 63–4 wave scattering in inhomogeneous media, theory of 55–6 weak localization (enhanced backscattering), experimental verification of 54–5