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Physical Optics of Dynamic Phenomena and Processes in Macromolecular Systems
Physical Optics of Dynamic Phenomena and Processes in Macromolecular Systems Proceedings 27th Microsymposium on Macromolecules Prague, Czechoslovakia, July 16-19,1984 Editor Blahoslav Sedlàcek
W G DE
Walter de Gruyter • Berlin • New York 1985
Editor Blahoslav SedldCek, PhD.,DSc. Institute of Macromolecular Chemistry Czechoslovak Academy of Sciences Heyrovsky sq. 2 CS-162 06 Prague 616 Czechoslovakia
Library of Congress Cataloging in Publication Data Prague IMPAC Microsymposium on Macromolecules (27th : 1984) Physical optics of dynamic phenomena and processes in macromolecular systems. „Under the sponsorship of the International Union of Pure and Applied Chemistry (IUPAC)"~Pref. Includes bibliographies and index. 1. Macromolecules-Analysis-Congresses. 2. Spectrum analysis-Congresses. I. Sedl&cek, B. (Blahoslav) II. International Union of Pure and Applied Chemistry. III. Title. QD380.P73 1984 547.7'046 84-28793
CIP-Kurztitelaufnahme der Deutschen
Bibliothek
Physical optics of dynamic phenomena and processes in macromolecular systems : proceedings / 27th Microsymposium on Macromolecules, Prague, Czechoslovakia, July 16-19,1984. Ed. Blahoslav Sedlàcek. Berlin ; New York : de Gruyter, 1985. ISBN 3-11-010234-X (Berlin) ISBN 0-89925-011-4 (New York) NE: Sedlàcek, Blahoslav [Hrsg.]; Microsymposium on Macromolecules
311010234 X Walter de Gruyter • Berlin • New York 0-89925-011-4 Walter de Gruyter, Inc., New York Copyright © 1985 by Walter de Gruyter& Co., Berlin 30. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form - by photoprint, microfilm or any other means nor transmitted nor translated into a machine language without written permission from the publisher. Printing: Gerike GmbH, Berlin. Binding: Dieter Mikolai, Berlin. - Printed in Germany.
PREFACE This volume includes the majority of the short special lectures and about h a l f of the poster contributions presented at the 27th IUPAC Microsymposium "Physical Optics of Dynamic Phenomena and Processes in Macromolecul a r Systems", held in Prague (Czechoslovakia) from July 16 to 19, 1984. The Prague Meetings on Macromolecules (including Microsymposia, D i s c u s s i o n Conferences and Summer Schools) have been organized, almost from t h e i r very beginning in 1967, under the sponsorship of the International of Pure and Applied Chemistry (IUPAC). Due to t h e i r s p e c i f i c ,
Union
usually
monothematic character, these meetings would seem to be in contrast to the regular IUPAC Symposia and Congresses, which have a broad, polythemat i c program: in fact, however, the micro-Symposia supplement the macroSymposia by bringing together s p e c i a l i s t s in the respective f i e l d ( s )
for
an e f f e c t i v e exchange of ideas and experience. Thus, the volume also o f fers the reader a good opportunity to inform himself about recent progress in the respective
disciplines.
This book deals with a problem area which could also be c a l l e d "Dynamics of Polymers and Biopolymers as Studied by Physico-Optical
and Related
Methods", acting in s o l u t i o n s , c o l l o i d d i s p e r s i o n s , gels and s o l i d s . The subject (macromolecules) and a tool for t h e i r study (method) are 'percolated'
to an extent which makes the papers rather complex to read,
but ' h o l o g r a p h i c a l l y '
i l l u s t r a t i v e nonetheless; at the same time, however,
t h i s makes any systematic c l a s s i f i c a t i o n of papers into sections
diffi-
c u l t or a r b i t r a r y . Therefore, only methods (and theories) not d i r e c t l y connected with the application to a given polymer are grouped together in a special
section.
The contents are divided into f i v e s e c t i o n s : Section A deals with the (hydro)dynamic behaviour of (preferably model) polymers in d i l u t e and semidilute s o l u t i o n s , followed by l i g h t s c a t t e r i n g and other physicooptical methods. In Section B s i m i l a r problems connected with concentrated systems (such as dense s o l u t i o n s and g e l s ) are discussed; a l s o , some new
VI
or l e s s well-known methods for their research are introduced. Sections C (Intramolecular Phenomena and Processes) and D (Intermolecular Phenomena and
Processes) s t r e s s processes
which occur in macromolecular systems
either i n d i v i d u a l l y or, in most cases, simultaneously. For several
papers,
no s t r i c t borderline can be drawn between the individual s e c t i o n s : the i n c l u s i o n in one or another section i s made, for the r e a d e r ' s convenience, with respect to the main emphasis and s i m i l a r i t y to subsequent papers. Section E deals with theoretical and/or experimental methods outlined or used for a general or, sometimes, special purpose. Occasionally, here and in other sections of t h i s volume, a theory, technique or application may depart somewhat from the scope of the book, but i t has been retained as i t might be of i n t e r e s t for comparison, application or as additional
in-
formation. Due to the very limited time a v a i l a b l e , pressing demands were sometimes made upon the more than 140 authors, whose e f f i c i e n t cooperation in preparing nearly seventy camera-ready papers i s highly appreciated. This form of p u b l i s h i n g has the indisputable advantage of an early publication of papers, which i s of great value, e s p e c i a l l y in the case of conference proceedings. With regard to the e d i t i n g , compromises are in some cases unavoidable in view of the 'least-change p r i n c i p l e ' barrier':
and the 'language
I hope that the papers w i l l be found at l e a s t understandable,
i f not well formulated. My thanks are a l s o due to de Gruyter Publishers and t h e i r leading coworkers for a very e f f i c i e n t and helpful approach to a l l problems which arose during our cooperation. I hope that these f r u i t f u l contacts in the Prague Meetings on Macromolecules w i l l also continue in the future. Blahoslav Sedlácek Chairman and Editor
VII
CONTENTS
Preface
A.
Dilute and Semidilute Solutions
Hydrodynamics of Polymer Chain in Dilute Solution
1
C.J.C. Edwards and R.F.T. Stepto QELS from Very Long and Semiflexible Filaments in
17
Solution S. Fujima and T. Maeda Theoretical and Experimental Studies of Anisotropy in
33
Translational Diffusion with Special Reference to Tobacco Mosaic Virus T. Maeda, S. Fujime, K. Kubota, H. Urabe and Y. Tominaga Molecular Weight and Second Virial Coefficient
37
Influences on the Concentration Dependence of Diffusion Coefficient of Polystyrene Molecules in Dilute Solution K. Witkowski, L. Wolinski, and B. Kolodziej The Concentration Dependence of Dimensions of Flexible Polymer Molecules in Solution J.S. King, W. Boyer, G.D. Wignall, and R. Ullman
43
Vili Diffusion and Relaxation in Polymer-Solvent Systems
49
by Photon Correlation Spectroscopy C. Cohen and D.H. Hwang Quasi-Elastic Light Scattering in Semi-Dilute
59
Solutions at the ©-Temperature M. Adam and M. Delsanti Light Scattering Observations of the Plateau Modulus in Semi-Dilute Theta Solutions
67
A.-M. Hecht, H. Bohidar, and E. Geissler Dynamic Light Scattering on Semidilute Solutions of
71
Polystyrene as a Function of Solvent Quality W. Brown Dynamic Properties of Entangled Polymer Solutions by
83
Forced Rayleigh Light Scattering M.F. Millet, H. Hervet, and L. Léger Dynamics of Polyelectrolytes in Aqueous Solutions
87
Probed by Electro-Optical Relaxation S. Wijmenga, F. van der Touw, and M. Mandel Dynamic Light Scattering from a Rod-Like Polymer in
107
Semidilute Solution K.H. Langley and P.S. Russo Laser Light Scattering of Dilute and Semi-Dilute Hydrosoluble Biopolymer and Synthetic Polyelectrolyte Solutions G. Muller
117
IX
B.
Concentrated Systems: Gels and Solids
Statistical Theory of Streaming Birefringence in
131
Temporary Polymer Networks E. Kröner and R. Takserman-Krozer Quasi-Elastic
Light Scattering of an Irreversible
145
Sol-Gel Transition S.J. Candau, M. Ankrim, J.P. Munch, P. Rempp, G. Hild, and R. Okasha Microphase Separation in Poly(acrylamide-
157
bisacrylamide) Copolymerized Gels E. Geissler and A.-M. Hecht Rayleigh and Brillouin Scattering from Linear
165
Polymer During Gelation T. Igarashi and S. Kondo Quasi-Elastic Light Scattering Studies of the
177
Diffusion of Compact Macromolecules through Gels D. B. Sellen Slow Diffusion of Labeled Macromolecules Studied
191
by a Holographic Grating Technique M. Antonietti, J. Coutandin, D. Ehlich, and H. Sillescu Mechanical Stability of Gels Undergoing Large Swelling Y. Hirokawa, J. Kucera, S.-T. Sun, and T. Tanaka
197
X Photon Correlation Spectra of Amorphous Low
205
Molecular Weight Polyethyl Methacrylate above the Glass Transition Temperature G. Fytas Rayleigh-Brillouin Scattering of Amorphous Polymers
217
C.H. Wang and B. Stuhn Brillouin Scattering Study of Segmental Motions of
229
Polyalkyl Methacrylates (PMMA, PEMA and PnBMA) at Low Temperature E. Kato and Y. Saji Time Resolved Small-Angle X-Ray Scattering Studies
2 33
on Kinetics and Molecular Dynamics of Order-Disorder Transition of Block Polymers T. Hashimoto Thermostimulated Luminescence in the Relaxation
245
Regions of Polymers J. Pospisil, A. Havranek and I.A. Tale
C.
Intramolecular Phenomena and Process
Dynamic Phenomena, Interactions, Conformational
251
Changes and Chemical Processes in Macromolecular Systems as Studied by Polarized Luminescence E. Anufrieva and M. Krakovyak Intramolecular Exciraer Formation and Polymer Dynamics: The Coil-Globule Transition C. Cuniberti and A. Perico
267
XI
Excimer Formation in Polystyrene Solutions:
271
Effect of the Coil-Globule Transition P. Stepänek, C. Konäk, and B. Sedläcek The Influence of Chromophoric Groups Rotation on
275
the Decay of Fluorescence of Polystyrene in Solution K. Sienicki and C. Bojarski Small-Angle Scattering of Polyelectrolyte Solutions
279
R. Koyama Light-Induced Conformational Changes of Macromolecules
287
in Solution as Detected by Flash Photolysis in Conjunction with Light-Scattering Measurements M. Irie and W. Schnabel Photon Correlation Spectroscopy of Polymethacrylic Acid
301
P. Brak and A. Persoons Photon Correlation Spectroscopy of Macromolecular
305
Solutions under High Pressure B. Nyström and J. Roots Intramolecular Motility in Pig Immunoglobulin G
317
Studied by Neutron Spin Echo Technique S. Borbely, Y.M. Ostanevich, L. Cser, B. Farago, F. Mezei, and F. Franek The Conformation of a DNA-Protein Complex
321
M.A. Scheerhagen, H. van Amerongen, M.E. Kuil, R. van Grondelle, and J. Blok Electro-Optical Kerr Effect Studies of Hemoglobin Z. Biaszczak, M. Witrh, and B. Norden
325
XII
Molecular Mobility in Liquid Polymers and
329
Polymerization Process Investigated by Acoustical Methods P. Hauptmann
D.
Intermolecular Phenomena and Processes
Theory of Light Scattering from Systems of
335
Interacting Spherical Macromolecules in Solution R. Klein, G. Nägele and W. Hess Cooperative Growth of Molecular Aggregation in
349
Semiconcentrated PBLG Solutions Investigated by the Dynamic Electro-Optical Method H. Watanabe, T. Nakano, and Y. Fukuda Light Scattering Anomaly Observed in Dilute
353
Solutions of Poly(vinyl alcohol) Aged at High Concentration L. Mrkvickovä, C. Konäk, and B. Sedläcek Light Scattering from Carrageenan Solutions
359
C. De Jonghe, H. Reynaers, K. Bloys van Treslong, and F. Varkevisser Studies of Protein-Nucleic Acid Interactions by
363
Photon Correlation Spectroscopy. II. tRNA-BSA Interactions at Intermediate and High Ionic Strength A. Patkowski and B. Chu Solvation of Protein Molecules Studied by Combined QELS and SAXS Investigations K. Gast, D. Zirwer, P. Plietz, J.J. Müller, G. Damaschun, and H. Welfle
367
XIII Light Scattering Studies of Bovine Eye Lens
371
a -Crystallin at Higher Concentrations: Short and Lt
Long-Range Order C. Andries, M. Van Laethem, and J. Clauwert Magneto-Optics of Colloids
377
R.V. Mehta Characterization of Inverse Polyacrylamide Latices
397
by Quasi-Elastic Light Scattering F. Candau, C. Holtzcherer, and S. Candau Interaction between Micelles of Poly(styrene-b-
401
hydrogenated butadiene-b-styrene) in 1,4-Dioxane/n-Heptane Mixtures C. Konak, P. Stepanek, and Z. Tuzar Properties of Block Copolymer Micelles Near the
405
C.M.C. and C.M.T. Z. Tuzar, P. Stepanek, and C. Konak Lateral Diffusion of Micelles Measured by
409
Fluorescence Recovery after Photobleaching W. Van de Sande and A. Persoons
E.
Theoretical and Experimental Methods
Molecular Dynamics Simulation of the n-Alkanes
413
Rotator Phase M.A. Mazo, E.F. Oleinik, N.K. Balabaev, L.V. Lunevskaya, and A.G. Grivtsov The Kerr Effect Relaxation in High Electric Fields H. Watanabe and A. Morita
427
XIV
Dielectric Relaxation of a Flexible Molecule
443
A. Morita and H. Watanabe Photon Statistics in Light Scattering
447
J. Perina Attenuation of a Coherent Field in a Dense
457
Dispersion of Particles C. Konäk, P. Stepanek, J. Krepelka, and J. Perina Determination of Polydispersity Index by
461
Quasi-Elastic Light Scattering P. Stepanek, Z. Tuzar, and C. Konäk Application of Picosecond Laser Pulses in Light
465
Scattering Studies B. Van Wonterghem and A. Persoons Application of Experimental Techniques for
469
Obtaining Photon Correlation Functions at High Pressure J. Roots and B. Nyström Bimodal Analysis of QELS Data
473
R. Johnsen Application of Speckle Techniques in
477
Macromolecular Physics J. Holoubek, J. Mikes, and B. Sedläcek Light Scattering Speckle Photography: Determination of Slow Correlation Times J. Holoubek
481
XV
Study of Polymer Solution/Solid Interfaces by
485
Evanescent Wave Spectroscopy H. Hervet, D. Ausserr^, and F. Rondelez Singular Value Decomposition in the Analysis of
493
Spectra W. Curtis Johnson, Jr. Flow Birefringence of Macromolecular Solutions in
507
Various Elongational Flows R. Cressely, R. Hockquart, J.-P. Decruppe, and T. Wydro Use of Light Scattering Methods for the Study of
511
Dynamic Processes of Macromolecules Coagulation in Hydrocarbon Medium G.F. Bolshakov Association of Alcohols Investigated by Rayleigh
517
Light Scattering B.M. Fechner Refractometric Study of Pressure Effect on
521
Preferential Sorption of Mixed Liquids on Gels under Dynamic Conditions T. Macko, D. Berek, and M. Chalinyova Compatibilization through Hydrogen Bonding of Pairs of Copolymers
525
0. Aouadj, A. Lassoued, and S. Djadoun
Contributors
529
Index
541
HYDRODYNAMICS OF POLYMER CHAINS IN DILUTE SOLUTION
Christopher J C Edwards and Robert F T Stepto Department of Polymer Science and Technology, U n i v e r s i t y of Manchester I n s t i t u t e of Science and Technology, Manchester M60 1QD, UK
Introduction The t r a n s l a t i o n a l
d i f f u s i o n c o e f f i c i e n t , D, of a molecule i s related to
i t s e f f e c t i v e hydrodynamic radius, r D , by the Stokes-Einstein D = kT/f = kT/6irnorD
U)
where f i s the molecular f r i c t i o n c o e f f i c i e n t and n 0 viscosity.
equation:
i s
the
solvent
The r e l a t i o n between D and the d e t a i l e d molecular
structure
of a polymer chain i s complicated, mainly because of hydrodynamic interactions
between the centres of f r i c t i o n
the equation due to Kirkwood
„
=
kT
+
kT_
represents the best
in the chain.
(1,2)
[ £ ]
(2)
analytical s o l u t i o n to the problem.
i s the segmental
Currently,
Here £ - 6-nTiQ-3
f r i c t i o n c o e f f i c i e n t with a the radius of a segment and
x the number of segments in the chain [R" 1 ] = i K r ^ 7 > where i s the mean r e c i p r o c a l
separation of segments i and j and
the angled brackets denote a c o n f i g u r a t i o n a l
average.
Eq(2) can be
w r i t t e n in a form which i s independent of solvent v i s c o s i t y
(3), namely:
Physical Optics of Dynamic Phenomena and Processes in Macromolecular Systems © 1985 Walter de Gruyter & Co., Berlin • New York - Printed in Germany
The f i r s t term on the righthand side (rhs) of Eq(3) i s the s o - c a l l e d f r e e - d r a i n i n g term associated with the flow of solvent through the molecular domain and t h i s term becomes i n s i g n i f i c a n t as x-»-».
The
second term on the rhs of Eq(3) takes account of hydrodynamic i n t e r a c t i o n s between segments and relates to the effective impermeable d i f f u s i o n radius of the chain. The derivation of Eq(2) makes several important assumptions which have been discussed recently in d e t a i l ( 4 ) .
For example, the chain i s assumed
to d i f f u s e as a r i g i d body in a solvent continuum.
In a d d i t i o n , hydro-
dynamic interactions between segments are assumed to be weak and the i n t e r a c t i o n s are assumed not to perturb the equilibrium configurations of the chain. In the present paper, previous interpretations in terms of Eq(2) of r^ for short polymethylene (PM) (3,5), polyoxyethylene (POE) (5,6) and l i n e a r and c y c l i c poly(dimethylsiloxane) together and compared.
(PDMS) (7,8) chains are brought
For a given chain structure c i s found to vary
with solvent, temperature and chain length, and possible reasons for these v a r i a t i o n s are discussed.
In a d d i t i o n , for systems with strong
s p e c i f i c interactions Eq(2) i s shown not to apply with free-draining being of n e g l i g i b l e consequence.
F i n a l l y , the predicted errors (4) in
Eq(2) compared with the f u l l Oseen evaluation of D for high molar-mass PM chains are discussed.
A key point in t h i s comparison and in the
c o r r e l a t i o n s of r D for short chains i s the use of r e a l i s t i c chain models and Monte Carlo calculations
(4,9) to evaluate
3 Interpretation
of
experimental
data
M u l t i p l i c a t i o n of each term in Eq(3) by x^ y i e l d s :
i i i ir - = r + r^ D F E
V
(4) '
where rp, the e f f e c t i v e f r e e - d r a i n i n g r a d i u s , i s equal to x.a and rp, the e f f e c t i v e impermeable r a d i u s , i s equal to
For unperturbed
c h a i n s , x^/r D and x f y r ^ tend to the same constant value and x*/ rp tends to zero as x+». the s u m - i n - p a r a l l e l
Thus the e f f e c t i v e d i f f u s i o n r a d i u s , rp, i s
of the f r e e - d r a i n i n g r a d i u s , rp, and the e f f e c t i v e
impermeable radius rp.
Eq(4) has been used p r e v i o u s l y
(3,10,11) to
i l l u s t r a t e g r a p h i c a l l y the r e l a t i v e magnitudes of the c o n t r i b u t i o n s of rp and rp to rg.
Values of r F = X 2 / [ R _ 1 ] have been evaluated
f o r PM, POE and c y c l i c and l i n e a r PDMS using a Monte Carlo technique which employs Metropolis sampling (12) and r o t a t i o n a l - i s o m e r i c - s t a t e models of the polymer chains given by Flory £ t a i 1.
Short polymethylene chai ns.
(13).
In Figure 1 x /r^, x 2 /rp and
x=/rp are plotted versus x for short PM chains ( 3 ) .
Also shown in
Figure 1 i s the Gaussian curve, calculated by assuming that rp=3ir* /8. a
The f r e e - d r a i n i n g curve xfyrp has been calculated for
= 0.08 nm which i s a reasonable estimate of the s i z e of a - C H 2 -
segment.
(4,9)
4
Figure 1 Variation of reciprocal diffusion £adii of polymethylene at 298K with chain length (x). [R ] and 2 calculated using Hill-Stepto model (5) . a = 0.08 nm. Curves: 1. Gaussian approximation to StokesEinstein radius, 2. Stokes-Einstein radius, 3. impermeable radius and 4. free-draining radius.
Inspection of Figure 1 shows that the effects of free-draining are still significant for x = 50.
The Gaussian approximation to x ^ / r E is poor
at low x but tends with increasing chain length to the same value as the Monte Carlo curve rather faster than the effects of free-draining disappear.
Figure 2 shows the same plots with experimental
short PM chains in benzene and quinoline also included (10).
data for It is
5 apparent from Figure 2 that a constant value of the segment radius, a, will not explain the experimental
data and that, furthermore, the value
of a depends on solvent.
Figure 2 Reciprocal diffusion radii versus chain length (x) for (•) polymethylene/ quinoline/298K and (0) polymethylene/benzene/298K. Curves: 1. Freedraining radius, x*/rp with rp = x.a and a = 0.08 nm, 2. impermeable radius, x | / r E witlj r E = x ^ / [ R _ 1 ] , 3. Kirkwood-Riseman diffusion radius, x V r D = x 2 / r p + x 2 / r p , 4. and 5. calculated diffusion radius with a =.b and (4) < c b > = 0.216 and (5) < c b > = 0.121.
Previously, this increase in the apparent segment radius with increasing x has been rationalised by allowing a to vary in proportion to the effective bond length of the chain b=( generally tend
to decrease as the molar volume of the solvent and/or the solvent viscosity
increase. Table 1
Values of Ck required to give agreement between calculated and experimental S t o k e s - E i n s t e i n r a d i i for polymethylene chains in various solvents
x
b/nm
benzene 25°C c
5 6 1 8 9 10 12 14 16 18 19 20 24 28
0.228 0.247 0.263 0.277 0.289 0.300 0.318 0.331 0.343 0.352 0.356 0.359 0.371 0.380
11 cP, , V cm mol"
b
carbon tetrachloride 25°C c
b
0.267 U723U 0.221 0.218 0.217 0.215 0.204
0.287 W7ZN 0.225 0.204
0.203 0.203 0.220 0.226
0.196 0.213
0.216 0.608 88.9
0.193 0.182
tetralin 22.2°C c
b
0,.157 0..139 0,.127 -
0,.127 -
0,.105 0,.112 0..108
0.185 0.178
0..134
0.201 0.938 96.5
0..122 2.,1 Ì6
quinoline 25°C C
b
0..148 0..121 0..115 0..116 0..116 0..118 0..120 0..125 0.,136
decalin 22.2°C c
b
0,.134 0,.118 0,.104 0,.098 0,.099 0,.092 0,.092 0,.091 0,.096
0,.109 0. 121 3.,44 118
0..100 2..64 159
7 The values of in benzene and quinoline have been used to recalculate x V r p , with rp = x.b, and the curves obtained in this way are those shown dashed in Figure 2.
The agreement between the
calculated curves and the experimental data is now good except at the shortest chain lengths, where end-effects become important.
The general
decrease in with increasing molar volume and/or viscosity of solvent may be thought of as resulting from the assumption of a solvent continuum.
< % > is a measure of the effective friction experienced by
a "hydrodynamic segment" and this decreases as slippage due to voids in the solvent structure (or molar volume) increases and as the perturbation of solvent flow for a given shear gradient decreases (i.e. viscosity increases).
The constancy of , or the concomitant increase in a
with chain length is not so easily rationalised.
It may result from the
assumption of rigid-body hydrodynamics in the derivation of the Kirkwood equation^).
Given that segmental velocities depend less on chain
length than the centre-of-mass velocity, the difference between these velocities will increase with chain length giving an effective increase in
2.
The proportionality of e to b remains unexplained.
Polyoxyethylene chains.
Figure 3 shows plots according to Eq(4) for
POE in water and quinoline with x*/rp calculated for a = 0.1 nm (10). In this case the experimental data for POE in quinoline agree well with the calculated curve, x 2 /Tq, suggesting that a is independent of x. However, the reason for this behaviour, which contrasts with that found previously for PM, becomes apparent when data for methoxy terminated POE chains are considered (points shown as "+" in Figure 3).
Similar
behaviour is now observed to that found previously for PM and it is clear that quinoline molecules hydrogen-bond to the terminal
-OH groups
of the POE chain and thereby increase the effective chain length.
It is
probable that a doubly end-blocked POE chain would show an even larger increase in x^/Rq as x decreases similarly to that found previously for PM.
nm'
15
10
00
20
to
60
t
80
100
Figure 3 Reciprocal diffusion radii versus chain length (x) for (•) POE/quinoline/ 298K and (•) P0E/water/298K. The points marked (+) refer to POE terminated with one -OH and one -QMe group in quinoline at 298K. Curves: 1. Free-draining radius x^/rp, a = 0.1 nm, 2. Impermeable radius x V f p , 3. Kirkwood-Riseman diffusion radius x i / r D . The data for POE in water (5,6) l i e well below the calculated curve for x*/r D at a l l values of x.
In t h i s case water molecules are
hydrogen-bonded to the skeletal oxygen atoms as well as the terminal -OH groups, thereby increasing the e f f e c t i v e chain cross-section.
Thus
POE chains in quinoline and water provide examples of systems which show non-Kirkwood behaviour due to strong s p e c i f i c interactions between polymer and solvent. 3.
Linear and c y c l i c poly(dimethylsiloxane)
(1-PDMS and r-PDMS).
In
Figure 4 an analysis is presented for 1-PDMS in toluene and bromocyclohexane (10).
In t h i s case the data can only be reproduced by consider-
ing the f r i c t i o n centres to be fSi ( C H 3 ) u n i t s
with a = 0.25 nm.
nnf s
i
__ _ — — —
f
_
—
!
~
/
/
A
2
\
°0
10
20
30
to
50
Figure 4 Reciprocal diffusion radii versus chain length (x) for (0) 1-PDMS/toluene/ 298K and (0) l-PDMS/bromocyclohexane/301K. Curves: 1. Free-draining radius, x'/rp, a = 0.25 nm, 2. Impermeable diffusion radius x^/r^, 3. Ki rkwood-Ri seman diffusion radius x^/rn, 4. and 5. calculated with a = b and (4) < c b > = 0.509 and (5) = 0.405.
Again at low x the effective segment size varies with chain length, although this effect is less marked for 1-PDMS than for PM, possibly due to the larger segment radius of 1-PDMS.
The values of required to
reproduce the experimental data are 0.509 in toluene and 0.405 in broniocyclohexane.
Curves recalculated using those values of
are
shown dashed in Figure 4 and as for PM the agreement with the experimental data is good. shown in Figure 5.
An equivalent analysis (11) for data on r-PDMS is Since 1-PDMS and r-PDMS differ only in topology it
would be expected that the same value of a would reproduce the data for both species.
However, Figure 5 shows that this is not the case and
that for r-PDMS a = 0.4 nm as compared with the value of 0.25 nm found for 1-PDMS.
The difference between the values of «i
0.4 nm and
ca 0.25 nm for the segment radius of r-PDMS compared with 1-PDMS is consistent with less free-draining flow in the cyclic molecules as would be expected from their higher segment densities.
10
Figure 5 Reciprocal d i f f u s i o n radii v e r s u s chain length (x) for cyclic PDMS in (•) t o l u e n e at 298K and (•] b r o m o c y c l o h e x a n e at 301K. Curves: 1. and 2. free draining radius x 2 / r p w i t h rp = x.a and a = 0.25 nm and a = 0.4 nm r e s p e c t i v e l y , 3. i m p e r m e a b l e d i f f u s i o n radius x * / r E , 4. and 5. K i r k w o o d - R i s e m a n d i f f u s i o n radii w i t h (4) the sum of c u r v e s (1) and (3) ( a = 0 . 2 5 nm) and (5) t h e sum of curves (2) and (3) (a=0.4 nm). (x) c o m p l e t e e n u m e r a t i o n ( + ) Monte Carlo c a l c u l a t i o n s .
Again a g r e e m e n t w i t h the experimental
d a t a is improved
(11) if the
segment radius is a l l o w e d to vary in p r o p o r t i o n to an e f f e c t i v e length. br
bond
For c y c l i c species t h e e f f e c t i v e bond length m a y be redefined as
= (6/x)i b e c a u s e the m e a n - s q u a r e end separation
is u n d e f i n e d for such molecul es. to d e f i n e values of
Values of bp have been used in Eq(5)
> for the data in t o l u e n e and
The improvement in a g r e e m e n t w i t h the experimental
bromocyclohexane.
data at short
chain
lengths using a c o n s t a n t v a l u e of < c b > is not so m a r k e d as for 1 - P D M S , but t h e d i f f e r e n c e significant.
in v a l u e s of < c b > b e t w e e n the two solvents is still
11 4.
Characteristic values of segment radii at Infinite chain length.
use of Eq(5) to define the segmental
The
friction coefficient means that at
infinite chain length the effective segment radius tends to a constant value as the effective bond length b reaches its limiting value for the unperturbed chain, viz
= .b
(6)
with independent of chain length, and evaluated as described from data on short chains.
The parameter a ^ is useful for summarising
differences in segmental
friction for different polymer/solvent
systems
at given temperatures.
Table 2 gives the resulting values of a n for
PM, 1-PDMS and r-PDMS.
Dependences of a ^ on solvent are clearly seen
and relate to solvent viscosity and molar volume as previously discussed. The values of
a^ increase in the order PM < 1-PDMS < r-PDMS showing that
free-draining decreases in the same order. Table 2 Values of and a for 'Kirkwood-Riseman 1 systems, from experimental dafa on finite chains.
PPM/CC1 4 /298K PM/benzene/298K PM/tetralin/295K PM/quinoline/298K PM/decalin/298K o 1-PDMS/toiuene/298K 2
» 0 r-PDMS/to!uene/298K 1-PDMS/BrC,H../301K o 11 r-PDMS/BrCgH .J/301K
/So
evaluated
b
b /nm
a oo /nm
0.225 0.213 0.122 0.121 0.100
0.435
0.098 0.093 0.053 0.052 0.044
0.509
0.566
0.288
0.559
II
0.316
1.31
0.402
0.527
0.405
0.566
0.229
0.449
II
0.254
0.923
0.402
0.371
12 Oseen analysis and rat
infinite chain length
Notwithstanding the r i g i d body approximation, the Kirkwood equation known to be approximate.
For t h i s reason Edwards, Kaye and Stepto
is (4)
have recently examined for PM chains the magnitudes of the e r r o r s introduced by the assumptions of pre-averaging and weak hydrodynamic interactions.
Using a development of the equations for the
frictional
f o r c e s on a polymer chain which i s related to that o r i g i n a l l y used by Kirkwood and Riseman ( 1 ) , i t has been shown that the Kirkwood equation r e l i e s only on the assumption of weak hydrodynamic i n t e r a c t i o n s and not on the use of a pre-averaged Oseen t e n s o r .
In a d d i t i o n , ensembles of
r i g i d - b o d y c o n f i g u r a t i o n s of r o t a t i o n a l - i s o m e r i c - s t a t e model PM chains were generated using the Monte Carlo procedure described
previously.
The d i f f u s i o n tensor equations were then solved numerically for two d i f ferent c a s e s .
F i r s t l y , using the pre-averaged form of the Oseen t e n s o r :
when i i s the bond l e n g t h , r^j i s the magnitude of the distance between f r i c t i o n centres i and j and
denotes the unit m a t r i x .
the approximation of weak hydrodynamic i n t e r a c t i o n s the i n t e r a c t i o n s are pre-averaged.
In t h i s case
i s not p r e s e n t , but
Secondly, the f u l l form of the Oseen
tensor
8-irrio Ar i j has been used, where Q i j Q i j denotes the d i r e c t product of the unit vector connecting segments i and j .
This approach g i v e s the exact
Oseen s o l u t i o n for the case of a r i g i d body moving with constant and random o r i e n t a t i o n
velocity
( i . e . Brownian motion i s assumed to be
overwhelming) in a solvent continuum.
For ease of comparison of
results
from the two cases with those from the Kirkwood equation, Eq(2) can be expressed a s :
13
i
=
f
where [R
I + x
(9)
x2
] represents the sum of reciprocal
normalised to unit bond length and
K =c/6im 0 Ji
separations of segments characterises
strength of the hydrodynamic interactions between segments. previous observation that s is related to b implies that to a limiting value of x-m».
the The
< increases
The lower limit of < = 0 represents
complete free-draining. Figure 6 shows a plot for PM in benzene at 298 (K) of the percentage reduction in c/f relative to the Kirkwood _1 result, for the pre-averaged Oseen solution, as a function of x 2 .
50
J /
4
*-.x= 0.566 '*»
A • \ , • x=0.S23 2.0
x
X x
\
0.1,36
\
•
*=0.31.9
\
V
*
\ \ \ \
V '
V \
\
*=0.17i.
s * ^ V V • * CM *~r — A
o.e
Figure 6 Percentage difference between values o£ c/f for PM chains from «Oseen> and K-R treatments as a function of x - 2 . (+) percentage differences at percentage constant K , (•) percent differences derived from experimental diffusion coefficients of PM in benzene at 298K.
14 Several
values of
to the values of Eq(9).
are employed and the points marked (0) correspond
K
derived by interpreting the experimental
K
data using
It can be seen that the limiting reduction in c/f at high
molar-mass lies in the range 2-3%.
Corresponding plots
(4) are shown
in Figure 7 for the results from the full Oseen treatment with those from the Kirkwood equation. c/f as x+co is of the order of 8%.
compared
Here the limiting reduction in
The overestimate of 8% in t/f from
Eq(9) compared with the full Oseen treatment is characteristic of the PM chain in benzene at 298K.
Since
(or 5) depends on chain
K
structure, solvent and temperature, the overestimate in ?/f from Eq(9) will also depend on these variables. Figure 7 (and 6) that actual
In addition, it can be seen from
polymer/solvent systems must be used in
order to determine the values of
K
to be considered.
at present of determining < independently.
There is no way
The overestimate in 5/f
for a given value of < is the same as the overestimate in the parameter
(3,10). 10.0
50
10
5
(X *=0.566
c
o
T/T
1 , the diffusion coefficient D
v
= T
- 1 2 /q
follows the same law as
that measured by gradient concentration (10) : D c ( c m 2 / s ) • = (1.25 + 0.10)X10" 6 C; it is proportional to the concentration. At T/T„ < 1, the diffusion coefficient D = T g R The ratio
-1
/q n
2
is larger than D . " c
of the two diffusion coefficients follows the relation : D /D = 1 + 5.22X10" 2 S c
C-'-
04+0
-2
63 Discussion
The observations can be explained as follows. The diffusion coefficient D^ is proportional to the osmotic bulk modulus
K
=
C
3TT/3C
and inversely pro-
portional to the friction coefficient f per unit volume of the solvent through the polymer : D
c
= —. This relation can be directly demonstrated by f
comparison with intensity light scattering and sedimentation measurements from which it was found that (11-12) : K (dyn/cm 2 )
= (2.93 + 0.14)X10 7 C 3
and s
- 1.15X10-' 4
•96+0.04
The sedimentation coefficients, s, is related to the friction coefficient f per unit volume by s = (1-vp) C/f
(pv = 0.703 for polystyrene in cyclo-
hexane at 35°C). Following scaling laws (1-3)the reduced diffusion coefficient D /D , where c o D q is the diffusion coefficient of an isolated chain, is a function only of C/C . In Fig.4 we can see that the scaling law is fully respected.
Figure 4 : Triangles are values obtained from ref.(13) : AM =1.3Xl0 5 ,VM =2X106. For the meaning of the other symbols see Fig.3, D /D values have been cal= 1.30X10-4 / K (14)^ ° culated using : D (cm 2 /s)
64 The diffusion coefficient D^ is the diffusion coefficient of the polymer pseudo-gel (7) :D^ = (K+ M)/f ,tohereM is the longitudinal elastic modulus of the pseudo-gel. Following this hypothesis D /D - 1 must be proportional £ c to M/K. From the present measurements and the measurements done inref/l'), we deduce that M (dyn/cm 2 )
= 1.53x10® c'" 9 ^. The elastic longitudinal mo-
dulus of the pseudo-gel is proportional to the volume number density of 2 entanglements (> C ). This point is in agreement with the theoretical predictions (1-3) and shear elastic modulus behaviour deduced from viscoelastic measurements (5) . The observation by quasi-elastic light scattering, in a semi-dilute 9 polymer solution, of either a liquid or pseudo-gel behaviour, depending on the experimental time scale (2,3), is possible because : M and K are of the same order of magnitude. The time t , which is independent of the momentum transfer , is a structural relaxation time and is eaual to the stress viscoelastic relaxation time T . T. as well K .L as T_ is related to the disentanglement of one chain. Figure 5 shows that the reduced relat ie . R . * xation timers does not obey a scaling law in C/C as is the case in a good solvent (4) .
Figure 5 : T R /T j as a function of C/C* where T j is the characteristic time of the first mode of a single chain in the dilute regime (C
(1984).
(1976).
1 5 9 5
(1980).
11. S t e p a n e k , P . , P e r s y n s k i , R . , D e l s a n t i , M . , A d a m , M . , M a c r o m o l e c u l e s , to a p p e a r in (1984). 12. Vidakovic, P. , A l l a i n , C. , R o n d e l e z , F. , J. P h y s i q u e 42^, L 3 2 3 13. M u n c h , J . P . , H i l d , G . , C a n d a u , S. , M a c r o m o l e c u l e s j_6, 71 14. S c h m i d t , M. , B u r c h a r d , W. , M a c r o m o l e c u l e s , J4-, 210
(1981).
(1983).
(1981).
LIGHT SCATTERING OBSERVATIONS OF THE PTATEAU MODULUS IN SEMI-DILUTE THETA SOLUTIONS § Anne-Marie Hecht, Himadri Bohidar
and Erik Geissler
Laboratoire de Spectrométrie Physique*, Université de Grenoble I, B.P. 68, 38402 St Martin d'Hères Cedex, France. Abstract Dynamic light scattering measurements are described for semi-dilute solut-
7
ions of polystyrene (M w = 2.6X10
Daltons) in cyclohexane at the theta temp-
erature. Observations as a function of scattering angle allow the effects of the shear modulus to be distinguished from those of the osmotic modulus. Introduction Theta solutions of polymers of sufficiently high molecular weight are characterized by two distinct cross-over concentrations that separate the dilute from the semi-dilute régime (Figure 1): the usual overlap concentr-0.5 ation (proportional to M
, where M is the degree of polymerization),
and a critical entanglement concentration c e (proportional to M - 1 ) .
Figure 1 Schematic diagram of the behaviour of the shear (G) and osmotic (K ) moduli as a function of concentration in theta solutions of high O S molecular weight polymers. § Present address: Department of Physics, Indian Institute of Technology, New Delhi 110016, India. * CNRS associate
laboratory
Physical Optics of Dynamic Phenomena and Processes in Macromolecular Systems © 1985 Walter de Gruyter & Co., Berlin • New York - Printed in Germany
68 Brochard (1) has noted that in theta solutions the osmotic modulus is = K° c 3 , (1) OS OS is a constant. The polymer-solvent friction coefficient f varies K
where K
OS
(2, The shear modulus, G, however, is determined by the probability that two polymer chains encounter one another, and hence G
=
G c2. o
(3)
Brochard's suggestion is that in semi-dilute theta solutions (c* < G »
K
os
c 1 (Fig.1) except for a divergence at the highest concentrations vAiich is more pronounced the greater the value of M. The data in Fig.4 correspond to:
where the Devalues are measured values (D) corrected for solvent backflcw according to: D = Dc(1-0) and 0 is the volume fraction of polystyrene. This exponent is substantially lower than the value y = 0.75 predicted by de Gennes1. Furthermore, there is no tendency for y to increase with M. This inability of the model to describe the observed variation of D^ with C should serve as a reminder of the extent to which entanglements influence the concentration dependence and which is an aspect ignored in the simple model. "Marginal" solvent (ethyl acetate) Results for PS 950 in ethyl acetate are shewn in Fig.5. The main feature is the presence of two distinct modes of relaxation which are separable using bimodal analysis. Thus, at all but the lowest concentrations, a single exponent plus baseline gave a very poor fit (at the highest concentration it was appropriate to add a baseline term to the bimodal expression for a marginally improved fit). The slew relaxation has a slightly negative dependence on C whereas thefesthas a positive dependence given by:
This exponent is much lewer than predicted for cooperative motions of the pseudogel, and there is evidence that this may be the case in poorer solvents. However, it is not suggested that a complete separation of the two nodes has been achieved (or can be using the present irethod) and the points at low concentration in particular scatter considerably making the slope uncertain.
76
The relative intensity amplitude of the fast mode is strongly concentration dependent (Fig.5 a), the correlation function being close to single exponential at the lowest concentration and rapidly beccming a birrodal distribution as C increases. Marginal and theta solvents reveal a complexity which is not found in good solvents and bimcdal autocorrelation 12 13 functions in ethyl acetate have been noted previously ' _ Amplitude
la) 2 U q10/m-2
_ ir/qW
C % I w/w)
10
12
Fig. 5. PS 950/ethyl acetate (25°C): (a) Relative intensity airplitude of the fast mode.(b) Fast (I) and slew (0) modes derived using bimodal analysis. Cumulant data (A) (=weighted average of fast and slow modes) and D^p (•) data from classical gradient diffusion. Fig.6. PS 950/ethyl acetate (C = 3.29% w/w): (a) The relative intensity 2
anplitude of the fast node depicted in Fig. 5. (b) (q -dependence of) fast and slow components in Fig. 5. We observe that the slow relaxation decreases strongly in intensity amplitude with increasing concentration, a finding which contrasts with the exponential increase in the amplitude of a slew relaxation described by 12 Mathiez et al . This suggests, instead, the presence of local concentration inhomogeneities in their solutions particularly since their slow mode amplitude was found to decrease on standing. The amplitude of the
77
fast node should eventually decrease at even higher concentrations as constraints increase due to the greater entanglement density as shown by 10
Hwang and Cohen
. A slow node as found for good solvents should then
became evident. The angular dependence of the relaxation frequency is linearly dependent 2 on q (Fig.6) suggesting that both modes are diffusional processes. The intensity amplitude of the fast mode increases strongly with increasing angle (Fig.6a). It is presumed here that the slow relaxation describes the centre of mass translational motions while the fast corresponds to the cooperative motions of the developing pseudogel. At this intermediate molecular weight (M = 950,000) both processes are features of the autocorrelation function. This situation persists over a wide range in concentration, well in excess of C* where it is usually assumed that the transient gel motions are the only ones of importance. Valussof D c u m (= ) have an intermediate concentration dependence, reflecting the relative weighting of the two nodes. 5 Data for the low molecular weight (M = 10 ) sample are typified by single exponential (Sf.
If.
• th » «!>
• llh F i g u r e 4. A n g u l a r E l e c t r i c f i e l d of The appearance o f sity fluctuations
d i s t r i b u t i o n of I r / 2 I n i n the h o r i z o n t a l plane the i n c i d e n t l i g h t is in the v e r t i c a l d i r e c t i o n . d i r e c t i v i t y i s a s s o c i a t e d w i t h f o r m a t i o n o f denw i t h the s i z e of the wavelength of the l i g h t . 2
of intermolecular potential ( i . e . energetic e l a s t i c i t y ) .
On t h e o t h e r hand,
hydrodynamics t e l l s us t h a t damping c o e f f i c i e n t o f sound i s g i v e n by r = -J—J-i-j-n + c)|k|2
(8)
where ti i s t h e shear v i s c o s i t y and ? i s t h e volume v i s c o s i t y . The v a l u e o f aX s (=r/Vg)=0.4 observed f o r phonon ( F i g u r e 2) g i v e s t h e v a l u e o f 4 / 3 - r i + ? = 0 . 6 p o i s e a c c o r d i n g t o e q u a t i o n ( 8 ) , i f t h e v a l u e o f X i s r e g a r d e d as 3000A, 5 - 1 i . e . k = 2 x l 0 cm . T h i s v a l u e o f 0 . 6 p o i s e cannot be t h e v a l u e deduced f r o m m o t i o n o f l o n g c h a i n m o l e c u l e s i n r u b b e r y s t a t e . Phonon damping a l s o may be l e d from a n h a r m o n i c i t y o f i n t e r m o l e c u l a r
potential.
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173 ally perfect dielectric crystals under the condition w s t « 1 , 2 aA
=
2ttC y Trio V * S 3pvs2
(10)
where C y is the isochoric specific heat, T is the absolute temperature and p is the density. In gas-kinetic theory one finds the following
expression
for the thermal conductivity,!«, of solids: k = 4-CvVs2X
(11)
f
otXs, therefore, becomes
5lOO-
20
40
115
channel number
9-8 .
40
20 (a)
115
channel number
or
Ih ' OT5*
Figure 2. Correlation functions obtained from light scattered from a 7.5% polyacrylamide gel (bis/acrylamide ratio 1%) using a multibit correlator whilst scanning angles of scatter at 16s per degree. Incident light 50mw, 633nm. The broken lines are the computer baselines. a) scan from 85 to 95 . sample time 3ys. b) scan from 80 to 115 s i x times with gel in s l i g h t l y different positions (see text), sample time 0.1s, prescaling factor 128, light attenuation 170.
where de/dt is the scanning rate and 9 S p ^ ( 9 ) i s the f i r s t order normalized spatial correlation function. at N (
)
2
There is an apparent short term baseline
1 + ß{ 1 - R ) n whilst the zero time value i s N ( s
S
)2(l+e)
s sp —> 2 /Ii P Thus T and both the and the computer baseline i s Ns(. ) • 9 ( ) spati a l l y integrated total count rate and the count rate associated with the broadened component may be found from one experiment. In equation 6, r i s replaced by R. Figure 2a shows a correlation function obtained with a Polyacrylamide gel by scanning from 85° to 95°. From the zero time value relative to the computer baseline ß=0.43. I t has been found convenient with gels to work with this relatively low coherence factor in order to avoid
182 critical alignment of the system.
Distortion of the correlation function
due to systematic variation in
and g ^ ( - r ) over the angular range
is negligible, and the data is effectively valid for e=90 . to investigate g
^(e)
It is possible
by using much longer sample times, and the result
of such an experiment is shown in Figure 2b where the equivalent values of e are indicated.
The problem here is to obtain a sufficiently large sample
of the fluctuations in for the correlation function to be properly defined, without introducing systematic variations in . The correlao o tion was therefore taken over six scans between 80 and 115 , and between each scan the cylindrical sample was twisted about its axis slightly so as to change the speckle pattern.
On the assumption of Gaussian statistics
however, this correlation function contains no information about the gel. It is entirely a function of the arrangement and dimensions of the stops in the scattered light receiver system. In all work with gels the mounting of the apparatus is important.
If the
gel is set in oscillation due to external vibrations, the otherwise stationary component of the random speckle changes, and oscillations, usually at the frequency of oscillation of the gel, appear in the correlation function.
This phenomenon has been used to investigate the mechanical
properties of gels (3,4), but in the type of work being discussed here it is important to avoid these oscillations.
This may be achieved most
effectively by mounting the apparatus on a heavy base which is then placed on an air bed.
Polysaccharide Gels Agarose and calcium alginate gels have been investigated (2,5,6).
The
Rayleigh ratio of light scattered from both types of gel is some two orders of magnitude greater than that from a "clear" gel such as polyacrylamide. Nevertheless provided that the dimensions of the specimen are reduced sufficiently to avoid secondary scatter, the Debye theory is obeyed, i.e. 4 2 2 if X Rg is plotted against Sin ( e / 2 ) / A for different wavelengths, using incoherent light, a single plot is obtained, indicating that the scatter does not arise from macroscopic fluctuations in gel density.
183 10D D0
•
»
0-5-
\ nm
2-5
Figure 3. Diffusion coefficient, D, relative to that in dilute solution, D , of dextran fractions within agarose gels of various concentration, C, as a function of dextran hydrodynamic diameter, d. The plots at the various concentrations have been superimposed by displacing horizontally (C =0.7%). 0 0.3% • 0.7% v 1% A 2% X 4% Degrees of spectral broadening have been found to be very small,one percent or less, falling as low as 0.01% for a 4% agarose gel.
Correlation funct-
ions are non-exponential and relaxation times are of the order of tens of milliseconds and do not vary as [sin^(9/2)j observed.
\
much less variation being
It is not appropriate therefore to describe g ^ ( i )
scattered from these gels in terms of diffusion coefficients.
for the light However when
dextran fractions of suitable concentration are allowed to diffuse within these gels, separate relaxation times which vary as [sin^(e/2j]
^ can be
easily identified and measured, and the diffusion coefficients of the dextrans within the gels determined.
Because of the high level of unbroadened
scatter from these gels it is safe to assume that the optical beating is entirely heterodyne, so that it is not necessary in this case to know 6 in order to determine diffusion coefficients. order to determine partition coefficients.
However e must be known in
The zero time value of that part
of the correlation function associated with the diffusing marker macromolecule (after subtracting the baseline and contributions from the gel) is
184 2gN .
The partition coefficient is found by comparing the counts
per sample time associated with the diffusing macromolecule > with the count rate observed when the cell contains a solution of the diffusing macromolecule only.
A correction must be made however for attenuation of
the incident and scattered light due to the turbidity of the gel.
Figure 3 shows diffusion coefficients of dextran within agarose gel, D, relative to corresponding diffusion coefficients in water, D q , as a function of hydrodynamic diameter, D, and agarose concentration, from D q using the Stokes-Einstein equation.
d is calculated
D/D Q has been plotted against
log d for each agarose concentration and the plots displaced horizontally relative to the data at 0.7% concentration, (C Q ), to yield a single "master" plot.
The fact that this can be done shows that:
D/D q = f[g(C)d]
(8)
g(C) is a function which can be found from the displacements and f is the function generating the master curve.
A double logarithmic plot of the
displacement against gel concentration shows that: 9(C) = a C b with b = 0.5. 1 /2
(9) Since by dimensional analysis D/D
must be a function of
p-| ' d where p-j is the length of gel fibre per unit volume, and C=p-]m where m is the mean mass per unit length of fibre, b=0.5 is just the value expected if m is independent of concentration.
A plot of very similar shape was
obtained with calcium alginate gel except that values of d for given values of D/D Q were seven times smaller, indicating that the mass per unit length of gel fibre is some fifty times smaller.
The only available theory
for the function f is due to Ogston et al (7) who give the relation: D/D q = E x p ^ 1 / 2 P l 1 / 2 d / 2 ]
(10)
If plotted as a function of log d, an exponential
is similar in shape to
the curve shown in Figure 3, but the transition from the freely diffusing to the completely immobilized state takes place much more slowly.
The val-
idity of equation 10, and possible modifications to it, have already been discussed extensively(2).
It is thought that it is valid for conditions
near complete immobilization, and on this basis molecular weights per unit length of gel fibre of 25,000 and 590g.mol"^nm~^ have been calculated for agarose and calcium alginate gels respectively.
The latter figure is in
excellent agreement with the model proposed for calcium alginate gel by
185
o •
»
ö
4H
cm 10^ total intensity
\
o\ total intensity
1Ö4-*—g—*
r
broadened component
ï
3cr
90° (a)
T
T
130*
angle of scatter
-o-
J
1öl
broadened component
90°
30* (b)
130*
angle of scatter
Figure 4. Diffusion c o e f f i c i e n t s and i n t e n s i t i e s of l i g h t scattered from a 7.5% Polyacrylamide gel (bis/acrylamide ratio=l%). X indicates the gel alone, 0 indicates the gel surrounded by, a) a 4% solution of bovine plasma albumin, b) a 4% solution of a dextran f r a c t i o n of weight average molecular weight 70,000 g.mol~l. a) and b) represent d i f f e r e n t gel samples.
Morris et a l ( 8 ) . The f i g u r e f o r agarose suggests that there are of the order of f i f t y agarose chains associated together in each gel f i b r e .
Partition
c o e f f i c i e n t s determined f o r agarose are consistent with the d i f f u s i o n coeffi c i e n t s on the basis of the theory of Ogston et al for gel concentrations above 1%. At lower concentrations smaller p a r t i t i o n c o e f f i c i e n t s are found, possibly due to dextrans becoming temporarily trapped by mobile agarose chains (2).
Polyacrylamide Gels Polyacrylamide gels were made up as cylinders of 5mm in diameter using the
186 standard procedure ( 9 ) , except that 0.01M sodium azide was added during the preparation of the g e l s , which were subsequently stored in 0.01M sodium azide s o l u t i o n .
The bis/acrylamide r a t i o was 1% and gel concentrations ran-
ged from 4% to 13%. Light s c a t t e r i n g measurements were made with each sample held with i t s a x i s v e r t i c a l in the centre of a 20mm diameter c y l i n d r i c a l l i g h t s c a t t e r i n g c e l l , in which i t was surrounded by 0.01M azide s o l u t i o n , or either 4% bovine plasma albumin or 4% dextran (M w =70,000) s o l u t i o n with 0.01M azide added. Even without d i f f u s i n g macromolecules present degrees of spectral broadening up to 30% were observed and,in general, d i f f u s i o n c o e f f i c i e n t s were obtained with equation 6 by means of an appropriately weighted l e a s t squares f i t going up to the t h i r d term of the polynomial.
Usually
the scanning method (from - 5 ° to +5° of the desired angle) described above was used, although preliminary experiments were carried out in the normal manner in order to s e l e c t an appropriate sample time.
The s p a t i a l l y
integ-
rated total count rate and the count rate associated with the broadened component were found with equation 7, and these were converted to Rayleigh r a t i o s by c a l i b r a t i n g the l a s e r l i g h t s c a t t e r i n g apparatus a g a i n s t a conventional Aminco l i g h t s c a t t e r i n g apparatus which in turn had been c a l i b r a ted with Ludox.
The usual s c a t t e r i n g volume correction was applied. Small
corrections for systematic v a r i a t i o n s i n the various parameters with angle of scatter within the scanning range were made for angles below 50°.
A
Malvern Instruments 64 channel m u l t i b i t c o r r e l a t o r was used for a l l measurements with polyacrylamide g e l s . Figure 4 shows data obtained at various angles of scatter f o r a 7.5% gel and Figure 5 shows data for various gel concentrations.
Diffusion coeffic-
ients were converted to 20°C assuming a v a r i a t i o n proportional to the r a t i o of absolute temperature to the v i s c o s i t y of water. Considering f i r s t p o l y acrylamide gels in the absence of d i f f u s i n g macromolecules; calculated d i f f u s i o n c o e f f i c i e n t s were found to be independent of angle of scatter,and the deviation of the c o r r e l a t i o n function from a s i n g l e exponential was not u s u a l l y more than that expected from the fact that the optical beating i s neither completely heterodyne nor completely homodyne. The maximum value of found was 1.02.
Whereas the total Rayleigh r a t i o increased
markedly at 1 ow angles, the Rayleigh r a t i o of the s p e c t r a l l y broadened component was found to be independent of angle of s c a t t e r . Measurements of
187
Figure 5. a) Diffusion coefficients and, b) intensities of l i g h t scattered from polyacryl amide gels of various concentration. X gel alone, 0 gel surrounded by 4% bovine plasma albumin solution. In b) the upper plots are the total Rayleigh ratios at 90 angle of scatter, whilst the lower plots are the Rayleigh ratios of the spectrally broadened component which were found to be independent of angle of scatter.
the total Rayleigh ratio were also made with incoherent l i g h t in an Aminco 4 apparatus using the wavelengths 546nm and 436nm. Plots of X R, against 2
sin ( e / 2 ) A
2
yielded a single plot for the two wavelengths showing that the
stationary component in the fluctuations in gel density does not in the main arise from macroscopic causes and that there are genuine static fluctuations in polyacrylamide density at a molecular level. A double logarithmic plot of diffusion coefficient against gel concentration yielded a straight line of slope 0.8. However the concentrations indicated in Figure 5 have not been corrected for swelling and as this i s l i k e l y to increase with decreasing concentration i t is l i k e l y that 0.8 is too high. Similarly the slope of the double logarithmic plot of the Rayleigh ratio of the spectrally broadened component against concentration 0.55, i s also probably too high.
188 Laser l i g h t s c a t t e r i n g from polyacrylamide gels has already been extensively investigated by G e i s s l e r and Hecht (10,11) and the r e s u l t s discussed in terms of current t h e o r i e s .
The r e s u l t s for the gels in the absence of
d i f f u s i n g macromolecules w i l l not therefore be discussed further here. Bovine plasma albumin and dextran f r a c t i o n of molecular weight 70,000 have -7 2 - 1 -7 2 -1 d i f f u s i o n c o e f f i c i e n t s in water of 5.8x10 tively
cm s
and 3.4x10
cm s
respec-
(no buffers or s a l t s other than sodium azide were added in these
experiments). Thus they have respectively higher and lower d i f f u s i o n c o e f f i c i e n t s than that of a 7.5% polyacrylamide gel ( 4 . 4 x l 0 - 7 c m 2 s _ 1 ) and 4% s o l u t i o n s have Rayleigh r a t i o s r e s p e c t i v e l y f i v e times and twice that of the s p e c t r a l l y broadened component.
S u r p r i s i n g l y therefore, e s s e n t i a l l y
the same d i f f u s i o n c o e f f i c i e n t as that of the gel was measured after these macromolecules had been allowed to d i f f u s e in for twenty four hours, and »
there was no evidence of more than one relaxation time in the c o r r e l a t i o n functions.
There was a small increase in the magnitude of the s p e c t r a l l y
broadened component with bovine plasma albumin, but none with the dextran fraction.
The most marked effect in the case of a 7.5% gel was a f a l 1 of
some 30% in the total scattered i n t e n s i t y .
The f a l l in Rayleigh r a t i o i s
of the same order as the Rayleigh r a t i o of the s o l u t i o n s surrounding the g e l , and i t was checked by making measurements with incoherent l i g h t at two wavelengths as described above. The effect of a 4% s o l u t i o n of bovine plasma albumin on gels of various concentration i s shown in Figure 5.
The measured d i f f u s i o n c o e f f i c i e n t
f a l l s with increasing gel concentration until i t equals the d i f f u s i o n c o f f i c i e n t of the g e l , a f t e r which i t r i s e s as the d i f f u s i o n c o e f f i c i e n t of the gel. Even when the measured d i f f u s i o n c o e f f i c i e n t i s markedly d i f f erent from that of the gel at the lower gel concentrations, there was no evidence of more than one relaxation time i n the c o r r e l a t i o n f u n c t i o n s . The proportionate increase in the s p e c t r a l l y broadened component of the scattered l i g h t increases with decreasing gel concentration, w h i l s t the f a l l in the total scattered i n t e n s i t y seems to be greatest at intermediate concentrations.
At gel concentrations where no change i n d i f f u s i o n c o e f f -
i c i e n t was measured, evidence that the protein had a c t u a l l y permeated the gel i s provided by the fact that there was a marked effect on the total
189
scattered intensity.
This was however confirmed by dyeing the protein
within the gel with aniline blue.
Discussion and Conclusion I t i s obvious that polysaccharide gels and polyacrylamide gels are quite different as regards both the dynamics of their structure, and the mechanisms by which compact macromolecules diffuse within them.
In the case of
polysaccharide gels the picture we have i s one of an almost stationary fibrous structure, in which compact macromolecules can diffuse subject to being impeded by the geometry of the structure, but with well defined d i f f usion coefficients which can be clearly identified and measured by quasie l a s t i c light scattering.
In the case of polyacrylamide gels we have a
highly mobile structure, which i t s e l f has a diffusion coefficient corresponding to the diffusion of fluctuations in the density of the gelling material.
I t appears that whereas diffusing compact macromolecules are impe-
ded by the gel structure their diffusion coefficient does not fall below that of the gel i t s e l f and that at higher gel concentrations the diffusing macromolecules are carried along by the diffusion fluctuations in the density of the gelling material, presumably following the migration of the less dense regions, provided of course that they are small enough to enter the gel in the f i r s t instance.
Thus the diffusion processes are not indep-
endent and i t is not possible to assign a well defined diffusion coefficient to the diffusing macromolecules. Thus partition coefficients cannot be determined by quasi-elastic light scattering. The fact that the diffusing macromolecules actually give r i s e to a marked fall in the overall scattered intensity, but not the broadened component, indicates that they show a preference for the regions of the gel which are permanently less dense, so rendering the system more homogeneous.
Further studies of diffusion within
polyacrylamide gels are in progress.
Acknowledgements The author wishes to thank Mr S. Hunter of the Astbury Department of
190 Biophysics at Leeds, who prepared the polyacryalmide gels and c a r r i e d out the protein dyeing experiments.
The photon c o r r e l a t i o n equipment was
obtained with a grant from the Science and Engineering Research Council.
References 1.
Tanaka, T . , Hocker, 0 . , Benedeck, G.B.: J.Chem.Phys.
59, 5151-5159
(1973), and many subsequent p u b l i c a t i o n s by these and other authors. 2.
Key, P.Y., S e l l e n , D.B.: J.Polym.Sci.Polym.Phys.Ed. 20, 659-679 (1982).
3.
Brenner, S . L . , Gelman, R.A., Nossal, R.: Macromolecules 11 , 202-212 (1978).
4.
Gelman, R.A., Nossal, R.: Macromolecules
311-316 (1979).
5.
Mackie, W., S e l l e n , D.B., S u t c l i f f e , J . : P o l y m e r ] ^ , 9-16 (1978).
6.
S e l l e n , D.B.; Polymer Jj), 1110 (1978).
7.
Ogston, A.G., Preston, B.N., Wells, J . D . ; Proc.R.Soc. London A333, 297-316 (1973).
8.
M o r r i s , E.R., Rees, D.A., Thom, D., Boyd, J . : Carbohydrate Research 66, 145-154 (1978).
9.
Shapiro, A . L . , Vinuela, E . , Maizel, J . V . : Biochem.Biophys.Res.Comm. 28, 815-820 (1967).
10. Hecht, A.M., G e i s s l e r , E.: Journal de Physique 39, 631-638 (1978). 11. G e i s s l e r , E . , Hecht, A.M.: Macromolecules J 4 , 185-188 (1981).
SLOW DIFFUSION OF LABELED MACROMOLECULES STUDIED BY A HOLOGRAPHIC GRATING TECHNIQUE
Markus Antonietti, Jochen Coutandin, Dietmar Ehlich, Hans S i l l e s c u I n s t i t u t für Physikalische Chemie der Universität Mainz Jakob-Welder-Weg 15, D-6500 Mainz
Introduction Experimental methods for investigating slow translational diffusion are subject to limits given by the Einstein relation D = ^ r ( t ) 2 ^
r
/6 t where
( t ) 2 ^ is the mean square average displacement of the diffusant in the
time interval t. Most techniques imply an interface between two parts of the sample one containing the diffusant at the beginning of the experiment. Examples are the radioactive tracer (1), the IR-scanning (2), and the Rutherford backscattering (3) techniques. Although i t has been possible to prepare thin polymer films of thickness down to a few um the d i f f i c u l t i e s of obtaining a well defined interface are considerable and have so far limited most applications to diffusion coefficients D ^ 1 0 ^
cm^s
The
NMR pulsed gradient techniques (4) are applicable to homogeneous samples, however, are limited by the requirement that the diffusion time t must be shorter than the longest spin relaxation time which again yields a limit of D 2>10" 1 0 cm^s~*. This limitation can be overcome by the holographic grating technique in which, by interference of two coherent beams, a grating is induced in a sample where the diffusant molecules are labeled with a photosensitive dye. The diffusive decay of this grating i s monitored by measuring the intensity of forced Rayleigh scattering (FRS) from the sample. The f i r s t application of this technique to polymer diffusion was an investigation of labeled polystyrene (PS) in solution (5). We have used a similar experimental setup for measuring self diffusion coefficients in molten PS (6, 7, 8). In order to extend the range of accessible D values we are using photolabels having a
lifetime of some days with respect to
thermal decay. Furthermore, we have increased the angular range of the two
Physical Optics of Dynamic Phenomena and Processes in Macromolecular Systems © 1985 Walter de Gruyter & Co., Berlin • New York - Printed in Germany
192
intersecting beams to 9 ~ 90° thus reducing the grid spacing d = X/ 2 sin(9/2)
to ~ 300 nm for X = 458 nm. Since the FRS intensity is 2
2
roughly (cf.ref.7) proportional to exp(- 8ir Dt/d ) i t decays to 1% of i t s -14 2 -1 i n i t i a l value in t < 1 h for D £ 10
cm s
. Thus, diffusion of macro-
molecules in polymer melts can easily be investigated within acceptable diffusion times. The present contribution provides experimental data for testing de Gennes1 reptation model (10) of polymer chain diffusion.
Experimental The fluorescein label used in our previous applications (6-8) gives r i s e to secondary reactions that may influence the FRS decay curves (7,11). Furthermore, i t exists as a dimer in PS, and affects the diffusion of labeled PS in the region of
low
molecular weights (7,8). Thus, we have
prepared a stilbene label by the following procedure (12,13). p-Tolunitrile is treated with a 3:1 mixture (by volume) of H 2 S0 4 (98%) and HN03 (65%) at T < 30°C. Condensation of the resultant o - n i t r o - p - t o l u n i t r i l e with an equimolar quantity of p-dimethylamino-benzaldehyde occurs within 2 h at 120-130°C in a mixture containing ~ 5% piperidine as catalyst. One obtains a black s o l i d , 2-nitro-4-cyano-4'-dimethylamino-stilbene, which is recrystallized from acetic acid (m.p. 178°C). The n i t r i l e is hydrolysed by refluxing over night in a water-ethanol-K0H mixture. The Cs salt of the product is reacted with the chloromethyl end group of PS as described in ref. (7). The main label reaction
193
PS—0— c
occurs in two steps where the first photochemical step leads to a colourless intermediate which reacts thermally to the blue final product. The thermal step is very rapid above ~ 150°C, and very slow below«50°C. In between, it occurs on the same time scale as the diffusion process which can, however, easily be separated since the FRS decay by diffusion depends upon the grid spacing d. Crosslinked PS was obtained by a Friedel-Crafts reaction with p-dichloroxylylene in C ? H.C1 solution (14):
The soft gel obtained from a 20% solution was passed through a fine sieve, treated with THF/HC1, precipitated from CH 3 0H, and dried at 175°C in order to obtain particles of crosslinked PS having a size of
~ 100 um (14).
194 These particles were mechanically mixed with 2-5% of a fine powder of labeled linear PS, pressed to a 0.3 mm thick pellet, and annealed in vacuo for 4 days at 180°C. During this annealing period the labeled chains diffuse into the particles and form a solution in the crosslinked PS matrix. For details see ref.(14). The diffusion coefficients could be determined as described for linear PS in ref.(7).
Results and Discussion The molecular weight dependence of polymer chain diffusion is usually discussed in terms of a power law D = D o (T)M" a
(1)
where a = 2 is predicted by the reptation model. In order to test this prediction, we have measured D values of 9 samples of linear PS labeled with fluorescein having a number average degree of polymerization P R between 177 and 1 550 (18 000 6
£ 161 000; M w /M n 5 1.02). Except for the
highest P n values of 1 100 and 1 550, we have dissolved the labeled PS in a PS matrix having M
= 110 000. At temperatures between 135 and 208°C, -14 -10 2 -1 the D values ranging between 2 x 10 and 3 x 10 cm s could be fitted by the WLF equation 0 = 0^ exp [ - CjCg In 10/(c2 + T - T )]
(2)
where T^ = 104°C is the glass transition temperature of the matrix and D^, c^, c2 are fit parameters that depend upon M n - The values c^ and c 2 (8) are approximately constant for P n £
400 and similar to those found for the
zero shear gradient viscosities. At higher P n > our c 2 values increase in a manner that cannot be understood in terms of the usual free volume assumptions. In a plot of log D versus log P n we find straight lines for Pn
Z 300. A least square fit of the power law, Eq.(l), yields a slope a
that varies from 2.0 at 208°C to values above 2.4 at temperatures below 167°C. We have checked whether this behavior originates from a label influence by doing some experiments using the stilbene label described above. For short chains, P n = 177, we find D values increased by 20-50% in comparison with fluorescein labeled PS. However, for P n > 300 the differences become smaller, and the label influence upon D can safely be neglected. At
195
present, we have no explanation for the unexpected temperature dependence of a. The fact that a approaches the reptation value of 2 at the highest temperature confirms the reptation model for diffusion of s u f f i c i e n t l y flexible chains in an entangled polymer melt. Polymer chain diffusion can be forced to curvilinear motion along the chain contour by chemical crosslinking of the surrounding matrix. In particular, the diameter of the "tube" defined in the reptation model (10) can be varied by changing the average number N c of monomer units between crosslinks. I f one assumes about 200 monomer units between entanglements in molten linear polymers (10,15) a choice of N c < 200 provides a test of whether the reptative motion depends c r i t i c a l l y upon a tube diameter defined by the root mean square distance between entanglements. Since entanglements are largely eliminated in dilute solutions one can prepare networks without physical entanglements by performing the crosslinking reaction in dilute solution as described above. In our experiments (14) we have varied N c between 16 and 400, and we have measured D values of PS chains having chain lengths P n between 177 and 572 at temperatures of 177, 185 and 194°C. The D values were found reduced with respect to diffusion in uncrosslinked PS by factors less than 4. The reduction is smaller in the microgel systems prepared from dilute solution. Apparently, the fixation of physical entanglements by crosslinking in concentrated entangled polymer solutions leads to networks where reptative motion is more d i f f i c u l t . The measured D values in a net-1 8 work with N c = 50 were proportional to P~ "
at a l l 3 temperatures. The
main result of our study is the remarkably small decrease of D in crosslinked as compared with linear molten PS which provides strong support of the reptation model.
References 1.
Bueche, F.: J. Chem. Phys. 48, 1410 (1968)and references cited therein.
2.
Klein, J . : Nature 271, 143 (1978).
3.
Kramer, E.J., Green, P., Palmstrom, Ch.J.: Polymer 25, 473 (1984).
4.
von Meerwall, E.D.: Adv. Polym. S c i . 54, 1 (1983).
5.
Hervet, H., Leger, L., Rondelez, F.: Phys. Rev. Lett. 42, 1681 (1979).
196 6.
Coutandin, J . , S i l l e s c u , H., Voelkel, R.: Makromol. Chem. Rapid Commun. 3, 649 (1982).
7.
A n t o n i e t t i , M., Coutandin, J . , S i l l e s c u , H.: Macromolecules 17, 798 (1984).
8.
A n t o n i e t t i , M., Coutandin, J . , S i l l e s c u , H.: Makromol. Chem. Rapid Commun., in press.
9.
E h l i c h , D.: Diplomarbeit, Mainz 1984.
10.
de Gennes, P.G.: " S c a l i n g Concepts in Polymer P h y s i c s " , Cornell Univ e r s i t y P r e s s , Ithaca 1979.
11.
Coutandin, J . : PhD-thesis, Mainz 1984.
12.
P f e i f f e r , P.; Ber. 48, 1777, 1808 (1915).
13.
S p l i t t e r , J . S . , C a l v i n , M.: J. org. Chem. 20, 1086 (1955).
14.
A n t o n i e t t i , M., S i l l e s c u , H.: submitted f o r p u b l i c a t i o n . rd Ferry, J . D . : V i s c o e l a s t i c Properties of Polymers, 3 Ed., Wiley, New York 1981.
15.
MECHANICAL INSTABILITY OF GELS UNDERGOING LARGE SWELLING Yoshitsugu Hirokawa*, John Kucera, Shao-Tang Sun§ and Toyoichi Tanaka Massachusetts
Institute of Technology, Cambridge, MA
02139,
USA *Permanent Address: Research and Development Center, Nippon Zeon Company, Kawasaki, Japan ^Hercules Company, Wilmington, Delaware, USA
Abstract When
an
ionic
polymer
gel
undergoes
extensive
swelling,
a regular buckling pattern temporarily appears on its surface due to a mechanical instability. scale
of
the pattern
comparable
to
disappears
leaving the gel exactly
shape.
the
The characteristic
increases with
overall
size
of
length
time until it becomes the
gel
and
then
it
similar to its original
In this paper we demonstrate a method
for creating
the buckling pattern in equilibrium and study its dependence on gel size and osmotic pressure.
I.
Introduction
The kinetics of swelling and shrinking gels is an important scientific and technological problem which is not yet fully understood. gel
It was
expansion
can
demonstrated be
described
process of a polymer network diffusion the
of
previous,
a
polymer but
(1).
network
incorrect,
that, as
a
for
small
swelling,
collective
diffusion
The concept of collective is
idea
totally that
different
the
gel
from
swelling
and shrinking is caused by the diffusion of solvent molecules. The new point of view on the kinetics of gel swelling was supported by the quantitative agreement between the collective
Physical Optics of Dynamic Phenomena and Processes in Macromolecular Systems © 1985 Walter de Gruyter & Co., Berlin • New York - Printed in Germany
198 diffusion laser
coefficient
light
scattering
experiments ments
(1,2).
two
a
polymer
network
spectroscopy
determined
and macroscopic
2
3(±0.5)xl0" cm /s
orders
of
7
and
magnitude
experi-
2
3.2xl0" cm /s
smaller
than
by-
swelling
T h e v a l u e s o b t a i n e d by i n d e p e n d e n t 7
were
were
of
which
that
of
water
c o m p a r e d to the o r i g i n a l v o l u m e the
linear
molecules. For
large
kinetic
swelling
theory
no
behavior
emerges
surface.
First
gels,
this
thin
slabs
We
describe
seen
applies. form
as
now
to
how
of
Interesting,
mechanical
a transitory
has
affixed
here
and osmotic Gel
the
buckling
in
II.
longer
in
a
base
the
phenomenon
been
produced
by
(crosslink 240
pi
and
4 0mg
0.133
molecule)
of
initiate
7.5cm
on
film.
thickness
grams and
were
of
(main
ammonium
the
0-0.64
grams
dissolved
a
to
hour
in
in
glass 2mm
persulfate
gelation. a
glass
in
of 100
sodium
bulbs
4cm
were
glass
and
was
allowed
made
plates
in
pouring
with
set
for
Slabs pre-gel
spacers
and
added
to
immediately
Spherical
diameter.
by
water.
(accelerator) were
then
to
atmosphere.
acrylate
ml of p u r e
(initiator)
solution
mold
nitrogen
thick
square
The
constitu-
N,N'-methylenebisacrylamide
N,N,N',N'-tetramethylethylenediamine of
transferred
to
treated
copolymerization
5.00 g r a m s of a c r y l a m i d e
molecule)
(ionizable
0.2
spherical
Preparation
molecule),
cast
the
equilibrium
depends
S a m p l e s of g e l s w e r e p r e p a r e d in free r a d i c a l
one
on
pressure.
at room temperature. ent
in
in
chemically
instability
non-linear
buckling
at
least
gels
were
gel
from
solution
of
onto
covering
with
®
commercially
available
onto
acrylic
square
treated backings.
film The
(GelBond film
PAG)
glued
chemically
bonds
to t h e g e l d u r i n g g e l a t i o n , p r e v e n t i n g t h a t side from when
immersed
immersed reach
in
in
pure
equilibrium.
solvent. water
and
After kept
for
curing, 24
the
hours
to
swelling
gels 3
were
days
to
199 III. K i n e t i c Figure in
Experiments
1 s h o w s the t i m e e v o l u t i o n of a s p h e r i c a l gel
pure
water.
glossy.
Initially
Immediately
regular
pattern
similar
of
pattern
increased
the
day
a
single
the
original
engrossed Growth
pattern radius
the
the
gel
until
diameter
that and
of
gel. gel
became
The
gel
swells
pressure
of
Na
Immediately osmotic it
a
a
regular
layer
of
while
pattern
decreasing
the
P a t t e r n F o u n d in
Slab
gels
of
into
This with
to
an
shows
acrylic some
of Shape
of
is
its
growth
the
osmotic
remain
causes
layer.
fixed limited
osmotically
square
until
As
buckling
scale
The
the
from
solvent.
length
as
acrylate.
released
layers
gradient
thicknesses
f o r m e d w i t h one as
gels
similarity
but
root
finally
of the
disappears.
after
same
sodium
acrylate
face m e c h a n i c a l l y
previously
to e q u i l i b r i u m in p u r e w a t e r . remains.
times the
Equilibrium
square these
sphere.
the
shear a
thickens
expansion
various
concentration were
to
pattern
sodium
gel b e c o m e s h o m o g e n e o u s a g a i n a n d the b u c k l i n g IV.
size
the
of
the
deeper
expanded
diffusively
one
glossy four
30%
surface
out
stress.
the
by
its
"diffuses"
shear
in
size
disappeared.
introduced
expands,
thickness
released time
it
a
extent because
immersion,
and
large
forms
the
ions
after
diffuses,
by
+
stress
creating which
to s u c h a large
The after
about
to
and fine,
until
Eventually, again
a
glass.
comparable
20%
o c c u r r i n g a f t e r the b u c k l i n g h a d
smooth
appeared
deeper
became
about
was
there frosted
forming
the
with
surface
became
element of
gel
immersion
to
entire
continued
original
the
after
immersed
they
described. were
anchored Figure
allowed
to
W i t h one side f i x e d the
is n o t m a i n t a i n e d
between
2
swell pattern initial
200
Figure 1: Swelling of a spherical gel. Swelling time 0 min, (b) 0 min, (c) 3 min, (d) 11 min, (e) 30 min, 90 min, (g) 26 hours, (h) 3 days.
(a) (f)
201
d
e
icm
j.cm
Figure 2: Swelling of slab gels with different thicknesses and same concentration of sodium acrylate (0.15g/100ml water). Thickness of gels: (a) 2.00mm, (b) 1.65mm, (c) 1.33mm, (d) 1.01mm, (e) 0.83mm, (f) 0.66mm, (g) 0.50mm, (h) 0.32mm. Note scale difference for (e), (f), (g) and (h).
202
1.0
d0 (mm) Figure 3: Relation between initial thickness of gel.
and final states. size
with
characteristic
wavelength
and
One can clearly see the increase in pattern
increasing
thickness.
The
characteristic
length
scale ( A) , obtained by taking the square root of the average area per cell, is plotted in Figure 3 as a function of the original thickness (dQ).
The length scale is linearly related
to original thickness in this instance with slope 1.95 (±0.04) and intercept 0.19 (±0.03).
203
a >
h
Figure 4: Swelling of slab gels with same thickness (2.00mm) and different concentrations of sodium acrylate. Concentrations (g/lOOml water): (a) 0.12, (b) 0.10, (c) 0.09, (d) 0.08, (e) 0.07, (f) 0.06, (g) 0.05, (h) 0.04.
204
Figure
4 shows
sodium
acrylate.
The
sodium
acrylate
(high
buckling
and
effect
large
concentrations less
the
gels
having
swelling
Those
varying
osmotic
(low osmotic
swelling.
of
the concentration of
high
concentrations
pressure)
whereas
the
pressure)
intermediate
show gels
swelling
higher
and
osmotic
the
pressure.
values of osmotic between
depth
of
the
There
of
regular
having
low
show no buckling and
show
the
of buckling but with an irregular pattern. of
a
stages
Both the amount
buckling
seems
early
to
increases
be
two
with
critical
stress one of which defines the boundary
linear and non-linear behavior, and the other which
determines
if
the
pattern
is
regular.
Further
study
is
necessary to understand this phenomenon.
References 1.
Tanaka, T, (1979) .
Fillmore,
D.J.:
J.
2.
Tanaka, T., Hocker, L. , Benedek, G.B.: J. Chem. 59, 5151 (1973)
Chem.
Phys.
7_0,
1214 Phys.
PHOTON CORRELATION SPECTRA OF AMORPHOUS LOW MOLECULAR WEIGHT POLYETHYL METHACRYLATE ABOVE THE GLASS TRANSITION TEMPERATURE
George F y t a s * M a x - P l a n c k - I n s t i t u t f ü r Polymerforschung Mainz c/o I n s t i t u t f ü r P h y s i k a l i s c h e Chemie der U n i v e r s i t ä t Mainz Postfach 3980, D-6500 Mainz, FRG
Abstract
The photon c o r r e l a t i o n f u n c t i o n s o f the p o l a r i z e d component o f the s c a t poly(ethyl methacrylate) (PEMA) of 4 4 weight average molecular weight M 1 . 6 x 10 and 7 . 3 x 10 were measured t e r e d l i g h t from two f r a c t i o n s o f
A
i n the temperature range from 70 t o 107 C and a t pressure from 1 to 1000 b a r . An a n a l y s i s o f the data using o n l y one r e l a x a t i o n a-process y i e l d s an unexpected t e m p e r a t u r e , pressure and Mw dependent d i s t r i b u t i o n o f l a x a t i o n t i m e s . I n c o n t r a s t to t h i s , a double r e l a x a t i o n process
re-
repre-
s e n t a t i o n o f the experimental time c o r r e l a t i o n f u n c t i o n s leads to
relax-
a t i o n parameters which have p h y s i c a l l y meaningful values f o r both PEMA f r a c t i o n s . The c h a r a c t e r i s t i c time o f the two processes approach each o t h e r a t high t e m p e r a t u r e s , low pressures and hence the s e p a r a t i o n
is
l a r g e r f o r the higher Mw PEMA f r a c t i o n . The present r e s u l t s are i n agreement w i t h the f i n d i n g s o f d i e l e c t r i c and mechanical experimental
•Permanent address: Department o f Chemistry, U n i v e r s i t y o f C r e t e , I r a k i i o n , Greece
Physical Optics of Dynamic Phenomena and Processes in Macromolecular Systems © 1985 Walter de Gruyter & Co., Berlin • New York - Printed in Germany
methods.
206 Introduction
During the last few years, photon correlation spectroscopy (PCS) has been extensively applied to study bulk polymer dynamics above the glass transition temperature T g (1). Several general features emerge from these studies, which indicate universality of the dynamics of the relaxation process probed by PCS. 1. A main glass-rubber relaxation, also known as primary or a-mode from dielectric relaxation measurements (which has a tremendous effect on the mechanical
properties), dominates the time corre-
lation function of the polarized scattered Intensity arising from slow density fluctuations. 2. The experimental correlation functions have a highly non-exponential shape and can be,represented well by the empirical W i l H a m s - W a t t s function (2) ( e x p ( - t / T Q ) ^ ) with distribution parameter B
0.4 for all studied polymers except poly(alkyl methacrylates). 3. The T0
.
mean relaxation time t(= - y r(B
1
) with r being the gamma function) con-
forms to the VFT or the equivalent WLF temperature equation which describe well mechanical shift factors (3). While the shape of the time correlation functions is insensitive to pressure (B is constant), the relaxation time t exhibits a strong pressure dependence and the activation volume corresponds to the volume of few monomer units (4). For poly (propylene glycol), t
f and B are found to be independent of molecular weight and of the probing wavelength (5). These findings suggest that localized segmental motions of the polymer chain grossly affect the density fluctuations in undiluted polymers near T . For poly(alkyl methacrylates), however, where two relaxation processes have been observed in dielectric and mechanical measurements (3,6) the light scattering results do not conform to the previous scheme (8-10). The relatively low value of B (^ 0.2), which is found to be temperature and pressure dependent distorts significantly the mean time T . One could fit, of course, a very broad distribution with two less distributed functions; however, the solution should be unique and physically meaningful
results
should be obtained. Since we do not yet have a theoretical model
describ-
ing the measured correlation spectra of bulk polymers near T , a significant contribution to this problem can be achieved by selecting suitable experimental conditions i.e. temperature, pressure and molecular weight,
207 and by comparison with other methods. In poly(butyl methacrylate)
(PBMA)
(10) we have shown that the most appropriate analysis of the data near T g and at ambient pressure involves two distinct relaxation
processes.
In this paper we attempt to explore the applicability of PCS on broad distributions in amorphous polymers and report a photon correlation study of the polarized scattered light from two fractions of poly(ethyl (PEMA) of weight average molecular weight (M w ) 1 . 6 x 1 0
4
methacrylate)
(T g =47°C) and
7 . 3 x 1 0 4 (Tg=65°C) in the temperature range from 70°C to 107°C and at a pressure from 1 to 1000 bar. It is known, mainly from dielectric measurements ( 6 ) , that the separation on the time scale of the two processes varies rapidly with temperature and pressure. Since similar response is also expected by changing T , the present pressure,temperature and molecular weight dependent study is further justified.
Theoretical
Background
The quantity measured in PCS is the auto correlation function o f the scattered electric field g(q,t)=/ where q 1s the
scattering wave vector and the angular brackets denote an ensemble average. The function g(q,t) depends on the polarization of the scattered light relative to the incident beam. In the VV-geometry, the scattered Includes both isotropic
Intensity
and anisotropic Iy^ contributions (V and H
denote perpendicular and parallel polarization to the scattering phase respectively). For weakly anisotropic scatterers, which 1s the case for poly(alkyl methacrylates), the VV intensity is almost equal to the Isotropic component. This arises m a i n l y from density fluctuations in undiluted polymers and has the correlation
function:
g(q.t) = < M q , t ) S p * ( q , o ) > / < | 6 p ( q , o ) | 2 >
(1)
where 6p(q,t) is the q th mode o f the density fluctuations. Equation (1) can also be written in the form (5): fl(q.t) = J m , n i,j where
(2)
is the position of the center of mass of scattering unit j of
208 polymer n at time t. Only the segment center of mass motion affects the isotropic scattering component in eq.(2). However, due to the intra- and interchain Interactions in bulk polymers, there is no rigorous calculation of the general expression entering in eq.(2). The normalized first order correlation function g(t) is related to the directly measured single clipped intensity correlation function G ^ t ) in an homodyne experiment through the equation: G k (t) = A|l+b|g(t)| 2 |
(3)
where A is the baseline and b is an unknown parameter used in the fitting procedure which depends on the relaxation strength i.e. the fraction of the - fi
scattered intensity associated with the slow (t>10" s) density fluctuations.
Experimental Photon correlation functions at different temperatures (70-107°C) and pressures (1-1000 bar) were obtained at a scattering angle of 6=90° with the apparatus described elsewhere (4). The light source was an argon ion laser operating at A=514.5 nm and power up to 300 mW. The incident beam was polarized perpendicular to the scattering plane and since there was very little depolarized intensity, no polarizer was used in the scattered light. The single-clipped photocount auto-correlation function G ^ t ) was measured by means of a 96-channel Malvern correlator. The method of matching the sections with different delay time AT to construct the composite correlation function over 6 decades in time was described in ref.(5). The two samples used in this study (kindly provided by Dr.W.Wunderlich, Rohm & Haas GmbH Darmstadt, W.Germany) were poly(ethyl methacrylate) (PEMA) 4 4 having M w =1.6x10
and 7.3x10
respectively. The ratio
M w / M n for both sam-
ples amounts to 2.36. The glass transition temperature determined from DTA data amount to 48
and 65°C for the low and high molecular weight fraction
respectively. We have prepared clear, optically homogeneous, dust free and strain free samples using the procedure described elsewhere (10). A strong confirmation of the suitability of the bulk PEMA samples for photon correlation measurements was provided by the low value of the Landau-Plazcek intensity ratio
(11).
209 Data analysis
The highly
nonexponential
time correlation functions of PEMA can be re-
presented by the Williams-Watts function (2): g(t) = exp ( - t / T 0 ) 6
(4)
however, the distribution parameter 3 is found to decrease with decreasing temperature (T) at constant pressure (P) and with increasing P
at
constant
T. For given P,T,B is smaller for the higher molecular weight PEMA fraction and varies systematically in the range 0.17-0.4. For the other bulk polymers investigated so far, 3 is found to be
0.4 insensitive to T and P
provided that a single relaxation mechanism dominates g(t). The present results mirror those obtained for poly(methyl methacrylate) (PMMA) (9) and indicate the presence of at least two relaxation processes, as is the case of PBMA (10). Moreover, we know from dielectric and mechanical studies of poly(alkyl methacrylate) that the separation on the time scale of the two processes is larger 1n PEMA than in PBMA at given P,T. For these reasons we performed a double-relaxation process fit of the equation: B B 1/9 1 ? ( ( G k ( t ) - A ) / A ) 1 / Z = ^ e x p i - t / T , ) 1 + b 2 exp(-t/T 2 ) L
(5)
to the experimental time correlation functions as described elsewhere (10). The numerical subscripts are used to denote the relaxation parameters of the fast (1) and slow (2) component respectively. A further support for the present type of data analysis is provided by the internal
consistency
of the obtained relaxation parameters for the two PEMA fractions.
Results and Discussion
1. Molecular weight dependence First
we consider the time correlation functions of the two PEMA frac-
tions as a function of temperature at atmospheric pressure. Flg.1
shows
two normalized correlation functions g(t) at 85.5°C for PEMA.It Is apparent, that g(t) of the higher M -fraction (hereafter referred as II) w
210
lines represent f i t s of eq.(4) and eq.(5) to the experimental functions for the low and high molecular weight fractions respectively. The experimental points (340 for the low and 430 for the high molecular weight) are within the thickness of the lines. i s shifted to long delay times and shows a broader shape as compared to PEMA-I (low Mw f r a c t i o n ) . Whilst the former effect should be expected because of the higher T g value of PEMA-II, the change of the relaxation time d i s t r i b u t i o n can be ascribed to the double relaxation d i s t r i b u t i o n feature of g ( t ) . For polystyrene (12), where one relaxation process dominates the slow density fluctuations, the shape of g ( t ) i s indeed insensitive to the molecular weight and the correlation functions are therefore superposable. The mean r e l a x a t i o n times T^ and
obtained from the f i t of eq.(5) to the
experimental correlation functions at atmospheric pressure, are plotted vs 1/T in F i g . ( 2 ) . Up to about 80°C for sample I and 100°C for sample I I the deviation plot of the f i t to eq.(4) shows a systematic error and the data have consequently been represented by eq.(5). In t h i s temperature
211
range, the d i s t r i b u t i o n parameters 3j and ¿ 2
are
equal to 0.36+0.2 and
0.34+0.2 for the fast and slow process respectively and the ratio b^/bg amounts to 0.43+0.04. At higher temperatures the relaxation time t^ i s close to t 2 ( F i g . ( 2 ) ) , so that the s i n g l e Williams-Watts equation (4) represents the experimental functions yielding the slow relaxation time x 2 (10).
The slow time x 2 for the two PEMA fractions exhibit a large temperature dependence typical for a - d i e l e c t r i c times and mechanical s h i f t factors of amorphous polymers near and above T . Hence in these systems t 2 conforms to the WLF equation (3): t2 g
C,(T-TJ 2
g
where C< and C 0 are the WLF constants and t_ i s the mean relaxation time 1 2 g at T . For PEMA the superposition of dynamic shear compliance data y i e l d s 0 ^ 1 8 and C 2 =65K for T g =335K (3). Using the measured values for T g (321 and 338K for PEMA I and I I respectively) and molecular weight Independent C 2 (= 65K) (12,13) we obtain C,, =21 .4+0.4 from the f i t of eq.(6) to the times t 2 of Fig.2. This finding suggests that the difference between t 2 for the two PEMA fractions i s mainly due to the difference in the fractional
free
volume. Concerning the temperature dependence of t 2 , there i s a f a i r agreement between the results derived from l i g h t scattering and mechanical relaxation data. As for the absolute value of t 2 > one has to compare the photon correlation functions g(t) with compressional compliance measurements (14). A new pertinent aspect of the relaxation times in Fig.2 i s the invarlance, within experimental uncertainty, of the fast relaxation time t ^ , on the molecular weight and hence on T . We feel therefore j u s t i f i e d 1n f i t t i n g an Arrhenius temperature dependence to the times t^. The obtained activation energy amounts to 22+5kcal/mol, which l i e s in the reported range of the activation energy for the B-d1electric process. On the grounds of t h i s agreement we ascribe the fast l i g h t scattering mode to the 3-relaxation. For poly(alkyl methacrylates),the B - d i e l e c t r i c mode i s assigned to the hindered rotation of the -COOR side group around the bond l i n k i n g 1t to the main chain (6,15). The slow and fast relaxation processes in Fig.2 seem to merge at high temperatures where the shape of the experimentalcorrela-
212
110
90 i
i
PEMA OM^^-IO4 AM^I^-IO*
1 x/s
T/°C
/
/
10-
10"-
y
70 i
/
/
/ 1
I T
•
27
1000. ~T~ K
2.9
F i q . 2 : Temperature dependence o f the mean r e l a x a t i o n time o f the slow process (o,A) and the f a s t process ( s o l i d symbols) i n PEMA. t i o n f u n c t i o n become narrower.
Finally,
the d i f f e r e n c e between B f o r the
two PEMA functions and the strong temperature dependence o f 6 as well strong evidence o f the a d d i t i o n a l
is
f a s t mode.
2. Pressure dependence
While the h y d r o s t a t i c pressure has a pronounced e f f e c t on the mean ar e l a x a t l o n time f o r poly(ethyl acrylate) (PEA) ( 4 ) , PS (1 6), poly(methyl acrylate)
(PMA) (17) and poly(phenylmethylsiloxane)
(18) the shape o f
the time c o r r e l a t i o n functions i s e s s e n t i a l l y i n s e n s i t i v e to, pressure (B i s c o n s t a n t ) .
For PEMA however, the shape o f g ( t )
i s changed with
pressure i n a s i m i l a r manner as with decreasing temperature. illustrates
Fig. 3
t h i s s i t u a t i o n f o r PEMA-II at 100°C, 1 bar and 800 bar,
213
Fig.3: Normalized composite correlation functions for PEMA at 373K at two different pressures. The thick lines represent fits of eq.(5) to the experimental functions. The 340 and 430 experimental points at 1 and 800 bar respectively are within the thickness of the lines. which is a further support for the double relaxation distribution 1n the experimental correlation functions for PEMA. We have consequently fitted eq.(5) to the experimental data and the obtained values of times t^ and t 2 at 373K are plotted vs pressure in Fig.4. If we consider the relaxation process as an activated process, the slope of the lines in Fig.4 is proportional to the activation volume AV (»2.303 RT(81ogx/3P)j), which represents the difference between the molar volume of the transition state and the molar volume of the initial state. M & for the fast relaxation process is much smaller then A V 2
We see that AV^
for the slow process in both PEMA fractions. The relaxation time x 2 exhibits a strong pressure dependence, which resembles strongly the results obtained for the glass-rubber (a) relaxation In bulk polymers above T . a 3 " b S a t 373K amounts to 204 and 234 cm /mo1 for PEMA-I and II respectively. This small difference is due to the decrease of the cooperativity 1n the
214
F i g . 4 : Pressure dependence of the mean r e l a x a t i o n time of the slow process (o,A) and the f a s t process ( s o l i d symbols) in PEMA. high molecular weight f r a c t i o n g r o s s l y r e p r e s e n t e d by T / ( C , + T - T J ( 4 ) . The 9 u value of AV2 corresponds to the molar volume of aifewimonomer r e p e a t u n i t s ( t h e monomer volume i s 103 cm^/mol a t 20°C). Thus, the slow (a)mode can be understood on a molecular b a s i s as a process involving i s o m e r i z a t i o n of the main chain r e s t r i c t e d to only a small number of monomer r e p e a t u n i t s ( 5 , 1 9 ) . A l t e r n a t i v e l y , the weak p r e s s u r e dependence of the f a s t r e l a x a t i o n time
in Fig.4 i s in accord with the physical p i c t u r e of the hindered
s i d e group r o t a t i o n . F i n a l l y , the a n a l y s i s o u t l i n e d above has y i e l d e d unique values f o r the r e l a x a t i o n parameters of the two processes and hence led to an adequate d e s c r i p t i o n of the slow d e n s i t y f l u c t u a t i o n s in amorphous PEMA above T .
215
Acknowlegement This work i s based on the experimental data obtained in the Department of Chemistry at the University of Bielefeld, West Germany. The financial support from the Max-Planck-Gesellschaft I s gratefully acknowledged. Thanks are due to Dr.I.G.Voigt-Martin for reading the manuscript and to Mrs. I . S c h i l l e r for typing i t .
References 1. Patterson, G.D.: Adv.Polym.Sei.: 48, 125 (1983). 2. Williams, G., Watts, D.C.: Trans.Faraday Soc. 66, 80 (1970). 3. Ferry, J.D.: Viscoelastic Properties of Polymers, John Wiley & Sons, New York 1980. 4. Fytas, G., Patkowski, A., Meier, G., Dorfmüller, Th.: Macromolecules 870 (1982). 5. Wang, C.H., Fytas, G., L i l g e , D., Dorfmüller, Th.: Macromolecules 1_4, 1363 (1981 ). 6. Mc Crum, N.G., Read, B.E., Williams, G.: Anelastic Properties and Dielectric Effects in Polymeric S o l i d s , John Wiley & Sons, New York 1967. 8. Patterson, G.D., Stevens, J.R..Lindsey, C.P.: J.Macromol.Sei., Phys. B18, 641 (1980). 9. Patterson, G.D., Carrol, P.J., Stevens, J.R.: J.Polym.Sci., Polym.Phys. Ed. 21_, 613 (1983). 10. Meier,G., Fytas,G., Dorfmüller, Th.: Macromolecules 1_7, 957 (1984). 11. Fytas, G., L i , B.Y., Wang, C.H.: to be published. 12. Mittag, U.
Diplomarbeit, Universität Bielefeld (1 983).
13. Plazek, D.J., 0'Rourke, V.M.: J.Polym.Sci. A-2, 9, 209 (1971 ). 14. Fytas, G., Wang, C.H., Fischer, E.W., Meier, G., to be published 15. Heijboer, J.: Intern.J.Polymeric Mat. 6, 11 (1 977). 16. Patterson, G.D., Caroll, P.J., Stevens, J.R.: J.Polym.Sci., Polym.Phys. Ed. 21_, 605 (1983). 17. Fytas, G., Patkowski, A., Meier, G., Dorfmüller, Th.: J.Chem.Phys. 80, 2214 (1984). 18. Fytas, G., Dorfmüller, Th., Chu, B.: J.Polym.Sci., Polym.Phys.Ed. 22, 000 (1984). 19. Hall, C.K., Helfand, E.: J.Chem.Phys. 77, 3275 (1982).
RAYLEIGH-BRILLOUIN SCATTERING OF AMORPHOUS POLYMERS
C.H.Wang Department of Chemistry, University of Utah, Salt Lake City, Utah 84112 / USA
B. Stuhn
*
Institut für Physikalische Chemie der Universität Mainz, 6500 Mainz, FRG
* present address: Chemische Werke Hüls, Marl, FRG
Abstract The technique of determining the longitudinal stress modulus over a wide dynamic range using Rayleigh-Brillouin
scatter-
ing has been studied. A correct relation between the RayleighBrillouin spectrum owing to density fluctuations and the mechanical relaxation spectrum of linear viscoelastic
polymer
is provided. The method of computing the longitudinal modulus and the compliance spectra from the time correlation of density fluctuations of PPG, as determined by the photon correlation spectroscopic technique, is illustrated.
Physical Optics of Dynamic Phenomena and Processes in Macromolecular Systems © 1985 Walter de Gruyter & Co., Berlin • New York - Printed in Germany
218 Introduction Rayleigh-Brillouin
s c a t t e r i n g in bulk p o l y m e r s
o c c u r s as a
r e s u l t of f l u c t u a t i o n s of local d e n s i t y a n d o r i e n t a t i o n c h a i n s e g m e n t s . The d y n a m i c s o f the long w a v e l e n g t h a t i o n s in the f r e q u e n c y range of 0.1 - 1 0 ^
Hz can be
i n v e s t i g a t e d by the d y n a m i c l i g h t s c a t t e r i n g t e c h n i q u e Fabry-Perot interferometry and optical mixing At the 90° s c a t t e r i n g a n g l e ,
of
fluctuusing
spectroscopy.
the s c a t t e r e d l i g h t o b s e r v e d
in
the VV s c a t t e r i n g c o n f i g u r a t i o n is a s u p e r p o s i t i o n of the density fluctuation and orientational
fluctuation
w h e r e a s the s c a t t e r e d l i g h t o b s e r v e d in the VH c o n f i g u r a t i o n s is due
to o r i e n t a t i o n a l
by m e a s u r i n g b o t h the VV a n d VH s p e c t r a ,
components,
scattering
fluctuations.
Thus,
it is p o s s i b l e
o b t a i n the c o m p o n e n t o w i n g to d e n s i t y f l u c t u a t i o n s . the R a y l e i g h - B r i l l o u i n s p e c t r u m o w i n g to d e n s i t y
to
To study
fluctuations
is e q u i v a l e n t to s t u d y i n g the F o u r i e r t r a n s f o r m of
C(q,t)
g i v e n by C ( q j t) = < 6 p ( q , t ) 6 p * ( q ) >
where
6p(q,t)
(1)
is the s p a t i a l F o u r i e r t r a n s f o r m of the
f l u c t u a t i o n of the n u m b e r d e n s i t y f r o m the e q u i l i b r i u m w i t h i n s c a t t e r i n g v o l u m e V. T h e q u a n t i t y q is the
value
scattering
v e c t o r a n d its a m p l i t u d e is r e l a t e d to the s c a t t e r i n g a n g l e 9 by B r a g g r e l a t i o n : q = 4im s i n ( 6 / 2 ) / A , w h e r e
A
is
the
w a v e l e n g t h of the i n c i d e n t l i g h t in v a c u u m , a n d n is the r e f r a c t i v e index of the s c a t t e r i n g m e d i u m . In the
amorphous
p o l y m e r s y s t e m the l i g h t s c a t t e r i n g s p e c t r u m d e p e n d s only
on
the a m p l i t u d e of the s c a t t e r i n g v e c t o r a n d not on its d i r e c t i o n . In the l i g h t s c a t t e r i n g e x p e r i m e n t u s i n g 5 -1 l i g h t , q is of the o r d e r o f 10
cm
visible
; thus, in an
a m o r p h o u s m e d i u m the f l u c t u a t i o n w a v e l e n g t h of
density
f l u c t u a t i o n s g i v e n by A„ = 2ir/q is long c o m p a r e d w i t h
the
219
intermolecular distance. This permits using the method of continuum mechanics to calculate C(q,t), the time correlation 1-4 function of density fluctuactions.In our previous work we were mainly interested in the interpretation of the Rayleigh-Brillouin spectrum in terms of the longitudinal modulus. In this paper we consider the reverse problem and we hope to extract information about the longitudinal modulus spectrum in the 0.1 to 10^ Hz frequency range from the time correlation function of density fluctuations, a quantity which can be measured by the photon correlation
technique.
Although our theoretical result is valid above and below the glass transition temperature, we argue that above T O the longitudinal modulus as determined by the photon correlation technique mainly reflects the bulk modulus, due to the fact that above T g the shear modulus is smaller than the bulk modulus by one to two orders of magnitude. Since the bulk modulus is difficult to measure by the usual mechanical technique, light scattering provides an effective method for determining the bulk modulus. In this paper we shall first provide the relationship between the imaginary part of the longitudinal modulus M"
and the Rayleigh-Brillouin
spectrum
using the method of continuum mechanics. Computation of M" from the dynamic light scattering spectrum is illustrated by the photon correlation function data of poly(propylene c glycol) above T g published previously . Longitudinal Modulus and the Dynamic Light Scattering Spectrum The most general constitutive equation describing the stressstrain relation of an amorphous linear viscoelastic system is given w by 6 (2) t
a.,(t)=-P • Pjlr)
1
i ' O
'l
4.
\
ft. t't, junction point
segmental distribution
)
Figure 1. Molecular mechanism of order-disorder t r a n s i t i o n . and the spatial d i s t r i b u t i o n of the chemical junction points h(r) becomes broader with time as shown in Figure 1(b). by a random stochastic process.
This process may be described
As the reptation takes place the segments
loose t h e i r memories with respect to t h e i r position and orientation, resulting in mixing of the unlike segments, increased i n t e r f a c i a l
thick-
ness and eventually in a e s s e n t i a l l y uniform segmental density d i s t r i b u tion c h a r a c t e r i s t i c for the disordered mixture as shown in Figure 1(c). We further simplify our treatment by assuming that the temperature T i s raised s u f f i c i e n t l y high above T^,, so that the thermodynamic interaction between A and B polymers does not s i g n i f i c a n t l y affect the d i f f u s i o n of polymers. Under these conditions, the change of the segmental density p r o f i l e with time P K ( r , t ) should be simply given by Fickian (9) 3p K (r, t)/3t
=
D c V2 p K ( r , t ) ,
K = A or B
(2)
where D c i s the s e l f - d i f f u s i v i t y f o r the block polymer as a whole which
236 should be a function of the self-diffusivity of the constituent polymers Dc
In case when T i s not sufficiently high above T , eq. 2 should be
modified to take into account an effect of f i n i t e value of x(T).
I f the
i n i t i a l segment density profile is given by a periodic function with a repeat distance 21, having domain size 2a and 2£-2a for the A and B domains, respectively, and the domain boundary thickness as characterized by the characteristic
inter facial
thickness
(1, 10) Ttn aQ as shown in
Figure 1(c) (lamellar mi crodomain), i . e . , P A (x,t=0)
= p A (x,t=0)/p Ao
•f+ £ *
00 2 2 a 2 y ( m tt o \ 1 L " exp( 72~ — J I T m=l v
s i.„ n
( mira \ „ „ „ ( mirx \ b H c o s J
(3)
where p Ao i s the number density of the A segment in the homopolymer A, then the reduced segment density profile p A ( x , t ) at time t i s given by (from eqs. 2 and 3 (9)) 9 00 a . 2 v 1 7 ^ m=l
~ i pA{x>t]
"FT
sin
(—A—J
/ J J„„„ / m it (" — 2
exP
cos
o
+ 2D t c 2
—i—J
(4)
= pA(x,t=0,ao=0)*h(x,t)
(5)
where h(x, t ) represents the spatial distribution of the chemical junctions at time t and is a solution of eq.2 with the i n i t i a l condition of h(x, t = 0)
= (2mJ 0 2 )" 1 / 2 exp(-x2/2ao2)
(6)
for the spatial distribution of junctions (Gaussian approximation).
h(x,
t ) is given by (9) h(x,t)
= [2w(a Q 2 + a 1 2 )]" 1 / 2 exp[-x 2 /2(a o 2 + a / ) ]
(7)
where a
2
1
= 2D t c
Similar equation for p B (x, t ) is obtained by replacing x by x-2a.
(8) Now
knowing the segmental density profile P K (X, t ) , one can calculate the
237 scattering contrast for X-ray and therefore the change of the scattering intensity with time I(q, t), I(q, t)
=
I(q, t = 0) exp[-2R(q)t]
R(q)
=
q2Dc
q
(4TT/A)
(9)
where
=
(10) sin
9
(11)
where X and 26 are the wavelength of X-ray and the scattering angle, respectively.
Based on eqs. 9 to 11, one can study kinetics of the order-
disorder transition by the time-resolved SAXS technique.
Equilibrium Aspects of Order-Disorder Transitions
Effects of the order-disorder transition on the SAXS profiles were investigated by Roe et al (5) and Hashimoto et al- (6).
Strong scattering maxima
in the ordered state arises from a long-range spatial order of the microdomains and the spacing calculated from the maximum generally decreases with increasing T and decreasing p, polymer volume fraction in neutrally good solvents, due to decreasing segregation power (6), D
~
(p/T)1/3
(in ordered state)
(12)
A weak scattering maximum is observed even for block polymers in bulk and concentrated solution in disordered state (5, 6) and is interpreted as the maximum arising from correlation
hole effect
as predicted by deiGennes (11)
and Leibler (3) based on the random phase approximation (RPA) and recently by Benoit and Benmouna (12).
The scattering formula presented by Leibler (3) for the bulk disordered block polymers may be rewritten as follows (6) I_1(q) —
{F(x)/N - 2A} - 2B/T
if one can write x as A + (B/T).
(13)
We applied eq. 13 to the block polymer
solutions with neutral solvents by replacing x for bulk by x (eV)
0 .9 0.7
< s> (Hz)
io21 io20
dilat. + DTA + +
T x(l are excluded vofume exponents, is the solvent viscosity,w =l/n and, (x) is the Riemann zeta function. We have shown(6) eq.2 to hold also in case of randomly distributed chromophores of long lifetime at low w by studying dilute solutions of pyrenyl-labeled polyvinyl acetate with w=0.01. At fixed molecular weight,
269 the solvent dependence of K was given at T^ © by k — T/1^0^ (3) To the purpose of studying the behavior below 6, two features of the pyrenyl excimers method are important :(a)the spectroscopic characteristics of the pyrenyl moieties affording the use of very dilute polymer solutions,(b)the inverse proportionality of Jc to the volume swelling factor providing a sensitive probe of any departure from the coil behavior. Results and discussion The results here presented refer to two samples of the same pyrenyl-labeled polyvinyl acetate (PVA-PBA) used in the investigation of the dynamics above 0 (MWl=l.08x10 ,MW2=6.2xl05). The relevant properties of PVA-PBA solutions in the three solvents employed are listed in Table 1( T is the light scattering phase separation point of lxlO -4 g/dl lolutions,TE is the point that separates diffusion from equilibrium controlled regime in the F £ /F m VS. T diagram, T m a x IS the maximum temperature value at which the k were calculated in each solvent).Mixtures of MOH-EOH and MOH-POH at 20°C were used as well.The 6-temperature of solvents and solvents: mixtures were obtained by the turbidimetric method(7),the viscosities measured by capillary viscometry and fluorescence spectra obtained with a Perkin-Elmer MPF44 spectrofluorimeter. The temperature and solvent dependence of the proportionality constant in eq.3 result negligible in respect of the friction and swelling effects.A reduced average relaxation time t X> re( j = =(K r e d ) - 1 =2.46(F e /F m ) C^/T)10~12seconds,is obtained taking at reference T//r\o for MOH at 6°C(0). Figure 1 shows t , relative to its value at 9 plotted against the reduced temperature. Two regions with very distindt behaviors are evident. Table 1.Characterization of PVA-PBA solutions in single solvents Solvent Methanol(MOH) Ethanol (EOH) Propanol(POH)
e CC) 6 57 81
a
T p C C) T E 1.08xÌ0b 6.2xl05 ("C) T-j where x . is the time of energy localization on the phenyl group. We have shown recently C5) that in case of PS, x• J is of order of picoseconds. Therefore rotation of the phenyl groups at room temperature has a secondary meaning in the description of PS monomer fluorescence decay and the factor deciding about the character of decay is the E,1 process leading to the formation of excimers. Figure 1b shows the time dependence of InCi-gCt)) computed from 8q. (5) at a fixed energy transport rate and concen_2 tration of F3S (q = 1.72x10 ) . As one can see, in this case too, the rotation of chromophoric groups only negligibly modifies the excimer fluorescence decay. The especially characteristic feature (.Pig. 1b) is the shift of
278
maximum of ijj(t) to shorter times with increasing rotation rate. Similar changes of ijjrt) were shown recently by Predrickson and Prank (3) on the basis of a model utilising the t-matrix approximation in solving the problem of energy transport in a macromolecule. Prom the fitting of Eq. (.4) to the fluorescence decay of poly (1-vinylnaphthalene) and poly (2-vinylnaphthalene) we have determined values of PES which are equal to 6.26x10"^ and 8.02x10 ^ , respectively.
This work was carried out under Research Project MR.1-5.
References 1. Papers presented at this Conference. 2. Anufrieva, E.V., Gotlib, Y.Y.: Adv. Polym. Sci. 40 , 1 11981). — 3. Predrickson, G.H., Pranck, C.W.: Macromolecules 16 , 572577 (1983). — 4. Itagaki, H., Horie,K., Mita, I.: Macromolecules 16 , 1395-1397 (1983). — 5. Sienicki, K., Bojarski, C. in preparation. 6. Soutar, I., Phillips, D., Roberts, A.J., Rumbles, G.: J. Polym. Sci., Polym. 3d. 20 , 1759-1770 (1982) . 7. Sienicki, K., Bojarski, 0.: Polym. Photochem., 4 , 000 (1984) . ~ 8. Friedrich, C., Laupretre, P., Noel, C., Monnerie, L.: Macromolecules , 1119-1125 (.1981) . 9. Gelles, R., Pranck, C.W. : Macromolecules , 741-748
SMALL-ANGLE SCATTERING OF POLYELECTROLYTE SOLUTIONS
Ryuzo Koyama College of General Education, Kyoto University Kyoto 606, Japan
Introduction The small-angle neutron and X-ray scattering curves of polyelectrolyte solutions have various characteristic forms according to the physical conditions(1 — 5). tions without added salts
In the solu-
the scattering intensity I(q) has —2 —1 —1
a broad peak in the region 10" ~ 10
A
, where q is the
absolute value of the scattering vector q q = (WA)sin(6/2) (A is the
wavelength
(1)
and 6 is the scattering angle).
But when some salts are added to the solution, I(q) increases at small q, and becomes monotonically decreasing with q at sufficiently high concentration of the salts.
These scatt-
ering curves have been calculated in a previous paper(6), assuming a strong electrostatic repulsion between rodlike segments in a random coil molecule.
The following calcula-
tion derives these by a more realistic chain polymer model(7).
Calculation of Scattering Intensity In this calculation we neglect the internal structure of the monomer unit, and regard it as a point scatterer of the radiation.
The scattering intensity per unit volume of the
polymer solution can be written as the Fourier transform of the radial distribution function of the monomers g(R) ,
Physical Optics of Dynamic Phenomena and Processes in Macromolecular Systems © 1985 Walter de Gruyter & Co., Berlin • New York - Printed in Germany
280 r
I(q) = C-j. n [ 1 + n
(g(R) - l ) e x p ( - i q . R ) d R ]
(2)
where C^ is a constant and n is the average number of monomers in unit volume.
Since
ng(R) gives the probability of find-
ing a monomer at the distance R from another monomer at the origin, this is the sum of the probability of finding a monomer of the same polymer with that at origin
ng^(R)
and that
of finding a monomer of different polymer from that at origin n g 2 (R) , i.e. g(R) =
gl(R)
+ g 2 (R)
(3)
When using the normalized intramolecular distribution function f..(R) of the i-th and j-th monomers in the polymer, ng-^(R) can be written as N N £ E f..(R), (i + J) (1) J i j Therefore the Fourier transform of g-^(R) can be given by the ng (R) = N
molecular scattering factor of the polymer P(q) g]_(q) = n _ 1 ( N P(q) - 1 ) _
P
N
(5)
N
P(q) = N
E E fM(q) J i j where f..(q) is the Fourier transform of f..(R). ij ij 5 do not contain the special terms with i = j f 1 ± ( R ) = 6(R), but eq 6 includes them.
f
ii(q) =
1
(6) Eqs 4 and
(i = 1J 2..,N)
(7)
Here 6(R) is the 6 function.
Next, considering the strong electrostatic repulsion between the polyelectrolyte chains, we assume that the probability of finding two monomers from two different polymer molecules within a distance R q is negligible, but beyond this distance this probability is almost constant.
Then if a
monomer of a polymer A is somewhere in a spherical region of radius R , any monomer from polymers other than A is scarcely
281 f o u n d at the c e n t e r of this ical region.
Therefore,
spherafter
i n t r o d u c i n g the o r i g i n of a c o o r dinate s y s t e m at the i - t h m o n o m e r of p o l y m e r A, one can w r i t e x i m a t e l y the p r o b a b i l i t y
appro-
of not
f i n d i n g the m o n o m e r of the
other
p o l y m e r s at R as (cf. P i g . l )
Fig. 1
C where i
o
(8)
f. .(R + S) d S ij " ~ (S(S) = G o exp(-S 2 /R o 2 ) , Using eqs 6 — 1 3 »
(12)
Gq = 1
(13)
one can calculate the Fourier transform of
- p(R) = g 2 (R) - 1 as - p(q) = n _ 1 N h (q) P(q) h
=
TT3/2C
G
n l? R
o o
(14)
2
(15)
*-o o
0(q) = exp(-X 2 /4),
X = Roq
(16)
By eqs 3 and 11, I(q) of eq 2 can be given by 1 + n(g^(q) - p(q)). Therefore using eqs 5 and 14, we obtain the equation for I(q) I(q) = C I n N P(q) [ 1 - h (q) ]
(17)
For the particular case q = 0 we have P(0) = (J>(0) = 1 (from eqs 6, 7 and 16), and eq 17 becomes 1(0) = C I n N [ 1 - h ]
(18)
This 1(0) should not be negative, hence h < 1
(19)
For various polymer models, P(q) is generally a decreasing function of q, but l-(q) is a monotonically increasing function converging to unity at large q.
Therefore for
very small h ( > 1)
w h e r e L can be g i v e n by (22)
L = N ec Kql/Cjn N
I(q)/CT n N H=.8B
Fig. 2
284
When this P(q) is introduced into eq 17, we obtain an approximate equation valid for large L q I (q) = tt C I n HQ R q [ 1 - h exp(-X 2 /4) ] /X
(23)
The maximum point X m (h) as a function of h of this I(q) can be determined numerically X (h) = R Mq m o m
/0,,n (24)
where R q is given by eq 15 R
o
= ir~3/1,v(h/C G d n)1/2 o 0K0
(25)
Therefore from eqs 2 3 ~ 2 5 , the maximum point q m and the maximum intensity I(q ) can be written as •"m q
m
=
7T3 / \„C „ G_ g, ) 1 / 2 n 1 / 2 h " 1 / 2 X o o o
(h)
I(q m )/no« R q [ 1 - h e x p ( - X m ( h ) 2 / ^ ) ] /X m (h)
(26] (2?)
Particularly for h = l, the numerical calculation gives X
(1)
=
2.2118..
(28)
Since C , G and 1 are constants, in this case we have from o' o o ' eqs 25 ~ 27, the qualitative results n"1/2
Rq
q
m
~n
(29)
1 / 2
(30)
I(q m )/n - n " 1 / 2
(31)
In ordinary polymers with single carbon bonds in the main chain
the monomer length is Qq - 2.5 A.
and eq 28
Using this value
in eq 26, and also assuming C Q = G
= 1 , Q M for h = l
becomes q m = 8.36 n 1 / 2 X" 1
(32)
These results are compared with the experimental results of small-angle neutron scattering of deuterated sodium poly(styrenesulfonate) solutions without added salts by Nierlich and others(l).
In Figures 4 and 5 the straight lines show the
285 I(qm)/n
(A_1)
q
.1
, 01
-I L. 10
10 Fig. 4
10
k
n
5
3
(T )
10,-5
10 - 3
10 n
Fig-.5
c a l c u l a t e d v a l u e s by eqs 32 a n d 31 r e s p e c t i v e l y , a n d the dots show the e x p e r i m e n t a l
small
values.
Results F i g u r e 4 shows that eqs 30 a n d 32 a g r e e experimental results.
w e l l w i t h the
The f o r m e r q u a l i t a t i v e
result
has
b e e n o b t a i n e d by H a y t e r and o t h e r s ( 9 ) , b u t eq 32 g i v e s a n a b solute
value.
In p o l y e l e c t r o l y t e
solutions without added salts,
the
e l e c t r o s t a t i c r e p u l s i o n b e t w e e n p o l y m e r chains is s t r o n g , Rq
can be a s s u m e d v e r y large c o m p a r e d w i t h
polymer solutions. assume h s 1
Therefore
irrespectively
o b t a i n e d in the c a l c u l a t i o n .
nonelectrolyte
f r o m eqs 15 a n d 19 we
reases R
can
of n, a n d I(q) has a m a x i m u m as However
it is
considered
that the a d d e d salts ions s c r e e n the e l e c t r o s t a t i c between polyelectrolyte
and
forces
c h a i n s , a n d this e f f e c t p r o b a b l y
So in the s o l u t i o n s w i t h a d d e d salts we
dec-
can
o a s s u m e a s m a l l e r v a l u e of h by eq 15. that as h d e c r e a s e s I(q) i n c r e a s e s at m u m d i s a p p e a r s at h = 0 . 8 5 . I(q) is m o n o t o n i c a l l y
Figure
2 a n d 3 show
s m a l l aq, a n d the
maxi-
A n d for h s m a l l e r t h a n t h i s ,
d e c r e a s i n g w i t h q.
Therefore
c a l c u l a t i o n c a n e x p l a i n the e x p e r i m e n t a l r e s u l t
this
(1~4).
286
The assumption h ~ 1 for very strong interaction means that I(q) has a very small value at q ~ 0
(eq 18).
Thermo-
dynamically 1(0) is related to the osmotic compressibility of the solution K = n-"*" ( 9n/3ir 1(0) oc k B T n 2 k
(33)
where IT is the osmotic pressure, T is the absolute temperature and k_ is the Boltmann constant. Consequently this assumpB tion agrees with the known fact that k of polyelectrolyte solutions without added salt
is very small compared with that
for nonelectrolyte polymer solutions(10).
References 1.
Nierlich, M.,Williams, C.E., Boue, P., Cotton, J.P., Daoud , M., Parnoux, B., Jannink, G., Picot, C., Moan, M., Wolf, C., Rinaudo, M., de Gennes, P.G.rJ. Phys.(Paris) 40, 701704 (1979).
2.
Plestil, J., Mikes, J., Dusek, L.: Acta Polym. 30, 29-32 (1979)•
3.
Ise, N., Okubo, T., Yamamoto, K., Kawai, H., Hashimoto, T. , Fujimura, M., Hiragi, Y.,: J. Am. Chem. Soc. 102, 79017906 ( 1 9 8 0 ) . Patokowsky, A., Gulari, E., Chu, B.: J. Chem. Phys. 73, 4178-4184 (1980).
4. 5. 6.
Benmouna, M., Weill, G., Benoit, H., Akcasu, A.Z.: J. Phys. (Paris) 43, 1679-1685 (1982). Koyama, R.,: Physica B 120, 418-421 (1983).
7.
Koyama, R.,: Macromolecules (to be published 1984).
8.
Koyama, R.,: J. Phys. Soc. Jpn. 34, 1029-1038 (1973), 36, 1409-1417 (1974).
9.
Hayter, J., Jannink, G., Brochard-Wyart, F., de Gennes, P. G.: J. Phys.(Paris), Lett. 41, L-451-454 (1980).
10. Lifson, S., Katchalsky, A.: J. Polym. Sei. 13, 43-55 (195 4).
LIGHT-INDUCED CONFORMATIONAL CHANGES OF MACROMOLECULES IN SOLUTION AS DETECTED BY FLASH PHOTOLYSIS IN CONJUNCTION WITH LIGHT SCATTERING MEASUREMENTS
Masahiro Irie Institute of Scientific and Industrial Research, Osaka University, Osaka, Japan Wolfram Schnabel Hahn-Meitner-Institut für Kernforschung Berlin, Bereich Strahlenchemie, D-1000 Berlin 39, Federal Republic of Germany
Introduction Recently, flash photolysis in conjunction with the light scattering detection method has been applied for investigations concerning the dynamics of macromolecules in solution. The dynamics of disentanglement diffusion were studied by measuring the rate of the change of the light scattering intensity (LSI) after very fast main-chain scission. In this case the diminution of the average molecular weight gave rise to a decrease of the LSI after irradiation of the polymer solution with a 20 ns flash [1] . Flash photolysis in conjunction with the LS-detection method is also applicable to measure the rate of conformational changes of macromolecules at constant chain length. As can be seen from the Debye equation (1) [2] —
l s 2 > s i n 2 ( V 2 ) + 2A 2
= j- + 3A
0
M
(1)
w
the light scattering intensity R^ (Rayleigh ratio) is correlated to the weight average molecular weight M w , the mean square radius of gyration and the second virial coeffi-
Physical Optics of Dynamic Phenomena and Processes in Macromolecular Systems © 1985 Walter de Gruyter & Co., Berlin • New York - Printed in Germany
288 cient A 2 - Here, K = U n ^ n ^ / N ^ )
(dn/dc) 2 , c is the polymer
concentration, n^ is the refractive index of the solvent, dn/dc is the specific refractive index increment, Ag is the wavelength of the incident light and N a is Avogadro's number. 2 Expansion of the polymer chain leads to an increase in and A, which causes a decrease in the LSI, whereas upon 2 shrinkage of the chain, leading to a decrease in and A the LSI increases. Advantage was taken of this behavior by measuring the rate of the photo-induced conformational change of a polyamide having backbone azobenzene groups [3]: COOH NH
~ O ^ "
N=
N
~ 0 ~
N H
_
c o
C
(I)
°""
COOHJn
Upon irradiation with UV light (350 nm < A
mnm'
J J trans, extended stable
The reverse cis
J
J
U k
R
,
(2)
cts. compact Instable
trans reaction can be achieved either by
heating or by irradiation with light of A
> 470 nm. It causes
unfolding of the compact chain as indicated by a decrease of the light scattering intensity due to an increase of the radius of gyration. The rate of unfolding of polyamide I (M = 3 7x10 ) was measured after pretreatment by UV irradiation for trans ->• cis isomerization. In these experiments, dilute polymer solutions
(0.3 g/1) in various solvents were irradiated
with 15 ns flashes of 530 nm light at 22°C. The halflife of the decrease of the LSI in N,N-dimethylformamide (DMF) was -4 5.7x10 s. Additional data are given in Table 1.
289
Table 1
Conformational Relaxation of Polyamide Chains Subsequent to cis -»• trans isomerization of Azobenzene Groups in the Backbone
Solvent N,N-dimethylacetamide
5.8x10
N,N-dimethylformamide (DMF)
5.7x10
4:1 b) DMF-ethanol
7.2x10
3:.2b) DMF-ethanol
11.0x10
4:1
b)
DMF-water
8.8x10
-4 -4 -4 -4 -4
Halflives of the decrease of the light scattering intensity at 514 nm and 22°C. Volume ratio. From the results obtained with mixtures of DMF and ethanol it was concluded that upon worsening
the solvent quality the
rate of unfolding was retarded. In the following a new experiment concerning light-induced coil contraction of macromolecules containing pendant azobenzene groups will be presented. Moreover, an experiment concerning coil expansion of a polyelectrolyte due to a light-induced pH jump will be described. Results 1) Coil contraction of polystyrene having pendant azobenzene groups.
Coil contraction due to trans -»- cis isomerization was
studied recently [4] with copolymers containing pendant azogroups: These copolymers,consisting of styrene and 4-(methacryloylamino) azobenzene
(see Table 2 ) ,
290
were found to precipitate in dilute cyclohexane solution at temperatures above the critical miscibility temperature upon irradiation by UV light [5] . Because the optical absorption spectra of the two isomers differ appreciably (^ m a x of the Table 2.
Sample
Characterization of copolymers of styrene and 4-(methylacryloylamino)azobenzene Content of Azobenzene Groups (mol %)
m Nw
a
>
PS-A-2. 2
2.2
3.1x104
PS-A-4. 3
4.3
2.7x104
PS-A-5. 6 PS-A-6. 5
5.6 6.5
2.0x104 1.8x104
Estimated from GPC using a calibration curve for polystyrene. trans form: 353 nm. \ of the cis form: 440 nm) the trans ' max cis isomerization induced by a 15 ns flash of 347 nm light could readily be detected by absorption measurements. It occurred during the flash, i.e. with a rate constant greater 8
— 1
than 10 s
. From the very fast trans •*• cis isomerization of
pendant azo groups it was inferred that neighbouring phenyl groups did not interact strongly with trans azobenzene groups. Light scattering measurements revealed the following: With all copolymers except PS-A-2.2, an increase of the LSI after the flash with halflives of several hundred us was observed. A typical oscillogram demonstrating the LSI change is shown in Fig. 1, trace (a). It can be seen that the LSI decreased initially. This decrease is due to the concurrent increase of the optical absorption at 514 nm (see oscillogram (b) ) and of the decrease of dn/dc as a consequence of trans •*• cis isomerization. The refractive index increment was measured with copolymer PS-A-2.2 in cyclohexane solution at 28°C and 546 nm: dn/dc = 0.125 ml/g before and dn/dc = 0.100 ml/g after trans •*• cis isomerization. Due to the low content of azo-
291
a
L.S.
2mV 200jus U0'-168mV b
Fig.1. Oscillograms illustrating the increase in the light scattering intensity (a) and in the optical absorption (b) both at 514 nm after irradiation of copolymer PS-A-4.3 in cyclohexane solution (0.13 g/1) with a 15 ns flash of 347 nm light at 25°C.
0. A.
H
H
200>JS
J.
5mV T
Uo*230mV benzene groups copolymer PS-A-2.2 did not precipitate after irradiation. The relative slow increase in the LSI is considered to reflect the conformational change involving a decrease in both the radius of gyration of the coils and the second virial coefficient [6,7]. The halflives of the LSI increase for the various copolymers are compiled in Table 3. The fact that no increase in the LSI after the laser flash was observed with copolymer PS-A-2.2 is consistent with a result obtained by stationary irradiations: precipitation did not occur when the content of azobenzene units was less than 3 mol %[5]. Table 3. Sample PS-A-2.2 PS-A-4.3 PS-A-5.6 PS-A-6.5
Conformational Relaxation of Polystyrene with Pendant Azobenzene Groups -1/2
a>
(s)
—
-4 2.1x10 4.0x10 -4 8.5x10 -4
Halflives of the increase in the light scattering intensity at 514 nm and 2 5°C.
292 It is interesting to note that the halflives of coil contraction of polystyrene with pendant azobenzene groups are of the same order of magnitude as the halflives of coil expansion measured for coil expansion of polyamides having azobenzene groups in the main chains. However, it is to early to arrive at general conclusions at this stage of the studies. Quantitative parameter studies are planned in order to measure the rate of conformational changes as a function of chain length, content of azobenzene groups etc. Preliminary experiments have shown that the halflife coil contraction decreases with increasing temperature. Experiments carried out with PS-A-4.3 at three different polymer concentrations (0.11, 0.076 and 0.055 g/1) yielded the same halflife of LSI change. This result is clear evidence that the process observed in the range of several hundred us is
an
intramolecular reaction. The fact that the
isomerization of only a few azobenzene groups per chain induces significant conformational changes is attributed to a severe perturbation of the balance of polymer-solvent and polymer-polymer interaction by trans
cis isomerization of
pendant azobenzene groups. In a former paper [5], it was pointed out that the dipole moment increases from 0.5 to 3.1 D upon trans
cis isomerization of the azo groups. It
is feasible, therefore, that cis-azobenzene groups are more prone to interact with styrene base units than trans-azobenzene groups because of dipole-induced interactions. The phenomena observed with the copolymers dissolved in cyclohexane can be, then, interpreted as follows: trans-azobenzene groups interact more favorably with solvent molecules than with styrene base units, whereas the reverse situation is true for cis-azobenzene groups which interact less strongly with solvent molecules than with styrene base units. The enhanced capability of cis-azobenzene groups of interacting with other segments of the chain is considered to give rise to a shrinkage of the polymer chain in the initial stage
293
after the trans •*• cis isomerization, i.e. at a time when this interaction is purely intramolecular (in very dilute solution). When different macromolecules collide at a later stage, intermolecular interaction will also become operative. As a consequence, the polymer chains will aggregate and precipitation will be observed. These processes are schematically illustrated in Fig. 2. This model is substantiated by the fact that the increase in the LSI after the flash was not inhibited by low molecular weight azobenzene compounds that were added to the polymer solution. The idea that interaction between cis-azo groups causes coil contraction can be discarded, therefore. Intramolecular Interaction
Fig.2.
Schematic illustration of chain contraction and precipitation of polystyrene with pendant azobenzene groups.
Experiments with solutions of copolymer PS-A-4.3 in different solvents yielded results that are in accordance with this model. With decahydronaphthalene solutions similar effects were observed as with cyclohexane solutions. With benzene and methylene chloride solutions, on the other hand, no slow change in the LSI (in the y,s range) was observed, although the trans -+• cis isomerization proceeded to the same extent and at the same rate as in cyclohexane solution and caused a very rapid decrease in the LSI during the flash. This result correlates very well with the findings that photo-stimulated precipitation occurred in cyclohexane and decahydronaphthalene but not in benzene and methylene chloride solutions [53.
294
Actually, these results were expected because benzene and methylene chloride are envisaged to interact with polar cis azobenzene groups. Therefore, this kind of interaction will predominate over the interaction of styrene base units with cis-azobenzene groups in this case. On the other hand, decahydronaphthalene cannot be polarized and will not interfere with the interaction between styrene base units and cis-azo groups. 2) Coil expansion of poly(methacrylic acid) induced by rapid neutralization^]. The method employed to measure the rate of coil expansion of polyelectrolytes is based on a rapid change in the degree of ionization induced by a pH-jump which is achieved by flash photolysis of triphenylmethane leuco hydroxide derivates
1. Upon irradiation of 1 with a 20 ns laser
flash of ultraviolet light, hydroxide ions can be produced "instantaneously", i.e., within the duration of the flash according to reaction (3)
^•""O-J^O-"'
V
} ( d ,
d j
. )
(5)
Since g. .(r) = 0 for r < d. . , eq. (4) together with (5) determines c - . for IJ 'J 'J r < d.. and, more important, g. . and h. . for r > d. .. From the solution of the system (4) and (5) one gets the partial structure factors by S i j ( k ) = 61d where h ^ k )
+
(n^)*
i s the Fourier transform of
hij(k) .(r).
(6)
338 The S.jj(k) determine the i n t e n s i t y I ( k ) of scattered l i g h t , x-rays or neutrons through
I(k) -
m 1/ I (N.N ) 7 2 f (k) f . ( k ) S. (k) 1 J 1 J 1J i,j=l
(7)
where N^ i s the total number of p a r t i c l e s of species i , and f.(k) = f.(0) b ^ k )
(8)
denotes the s c a t t e r i n g amplitude of a p a r t i c l e of species i . The form factor b.j(k) for a homogeneous spherical p a r t i c l e i s 3j (kd./2) b
where j
i(k)=
(9)
kdi/2
i s the f i r s t spherical Bessel
function.
The procedure t o ^ o l v e the equations (4) and (5) in MSA follows e a r l i e r work by Blum [7], Blum and Hoye [8] and Hiroike
[9] and w i l l be published
elsewhere together with a systematic representation of the r e s u l t s . Here, some of them w i l l be presented and discussed.
(a) Influence of the f i n i t e s i z e of the small ions S.jj(k) and 9-jj( r )
are
calculated from a three-component PM. The f i r s t spe-
cies are the macroions of valency z^, number concentration sphere diameter d . The counterions
and hard-
and the ions of an additional
1-1
e l e c t r o l y t e form the other two components. We assume for s i m p l i c i t y that a l l small ions have the same diameter d„sm N(which i s varied in the c a l c u l a t i o n ) and that they carry one elementary charge ( \ z 2 \ = |z | =1). F i r s t we consider the case without s a l t . Fiq. 1 shows the g . . ( r ) for a s i tuation which might be typical f o r a m i c e l l a r system. The c o r r e l a t i o n s between the macroions are given by g
and the
counterion
distribution
around a central macroion i s described by j 3g12 . Due to the a t t r a c t i v e teraction the counterions
in-
accumulate in the extreme neighborhood of the
central macroion, the maximum of g 1 2 being at r/dj = 0.55 as expected i f
339
Fig. 1. Macroion-macroion ( g u ) > macroion-counterion (g x ) and counterioncounterion ( g 2 2 ) radial distribution functions of a two-component PM with number concentration n, = 4.6xl0 1 8 cnf 3 , valency z i = - 20 and hard-sphere diameter dj = 50 A of the macroions. The counterions diameter d is set equal to 5 A. sm ^ d s m = 5 % . A shallow minimum of g J 2 appears at the position of a " s h e l l " of nearest neighbor macroions, which is followed by a smaller maximum a r i sing from the correlations between the macroions. In Fig. 2 the structure factor S n ( k ) and the radial distribution function g n ( i " ) of the macroions are shown for point-like d $m = 5 % . As expected, S
and g
n
counterions and for
exhibit somewhat stronger structure
for dsm = 5 since f i n i t e size small ions are not as effective in screening the interaction among macroions as point-like ions. This excluded volume effect is less pronounced at smaller number densities n
of the macro-
ions. The influence of the f i n i t e size of the small ions is of particular importance i f one adds s a l t . At the rather large total volume concentrations of the systems in Figs. 1 and 2 the hard-core part of the interaction more and more determines S,, and 3g,, . Fig. 3 shows the effect of added s a l t re3 11 il lative to the s a l t - f r e e case. The structure in S ^ t k ) is being reduced and the increase at k=0 reflects the increased osmotic compressibility with
340
Fig. 2. Macroion-macroion structure factor S ^ k ) and radial distribution function 9 u ( r ) of a two component PM. The macroion species parameters are the same as in Fiq. 1. 1( ) d = 5 X; 1( — ) d = 0 A. 3 ' sm ' sm increasina salt content.
(b) Binary mixtures of macroions So far the scattered radiation was assumed to arise from a single species of macroions (denoted by the index 1). It is an advantage of the multicomponent primitive model to generalize easily to the case of several species of macroions scattering radiation with different scattering powers f
.
The measured structure factor is of the form of eq. (7) and can be defined as S M (k) = ^ L y Nf 2 a B
(N N J l 2 ;
f f Q S Q (k) a 6 '
(10) '
v
where LI N —r a f = -
I L
a
fa 2
(11)
Na
M The normalization in (10) is chosen such that S (k) = 1 for noninteracting particles. For the case of pure scattering polydispersity one assumes that all macroions behave identically with regard to their structural properties but that there are two types of particles differing in their scattering power (fx \ f 2 ). Then g a o = g for all a and 3 and (10) reduces to
341
Fig. 3. Structure factor S n ( k ) of the macroions for a three-component PM, which models a system with added s a l t . n : = 5.0xl0 1 8 cm~ 3 , z 1 = - 2 0 , dj = 50 A and d s m = 5 A . n, i s the cation respectively anion number concentration of the added 1-1 electrolyte. ( ) n 3 /nj = 0, that i s without added s a l t ; ( — ) n 3 /n t = 50; ( - • - ) n 3 /nj = 100.
S M (k) = y S i d ( k ) + (1-y) = S D ( k )
(12)
y = f2 / ¥
(13)
where
S^k)
and
f = (J N f )/(£ N ) a a
in (12) denotes the structure factor of an effective one-component
system. Whereas the simple result (12) is exact for pure scattering polydispersity i t has been proposed to use i t also as an approximation for size polydispersity [10] . The usefulness of this decoupling approximation for charged systems i s now being tested. In the following we r e s t r i c t ourselves to a binary mixture and assume both species to have equal refractive indices. As long as to the wavelength of l i g h t we have f„ = const x d„
d. i s small compared with the same constant
342 f o r both species (a = 1,2). Defining the r a t i o of diameters X = d 1 /d 2 with d
d 2 and x. = N / ( N x + N 2 ) , the quantity y ( x 2 ) i n eq. (13) has i t s mini-
mum at the concentration x* = X 3 / ( l + X 3 ) , where i t has the value y ( x 2 * ) s y * = 4X 3 /(1+X 3 ) . The r a t i o of incoherent to coherent s t a t i c s c a t t e r i n g i s given from eq. (12) as I i n c / I c o h = R/sid(k)
(14)
with R = ( l - y ) / y . Therefore, R has i t s maximum at the concentration x * 1 - x * , so that the r a t i o ^ n c / l c 0 h
1S
=
l a r g e s t at x * .
We have used X = 0.8, so that x * = 0.339, y * = 0.896 and R = 0.116. The expression (10) i s then calculated for the following three-component systems: n
= 3xl018cm"3, d
= 40 ft, d 2 = 50 8, z
= z 2 = - 15 f o r the compositions
n /n 2 = 1; 0.3 and 0.05. In addition there are corresponding numbers of 0 univalent counterions with d = 4 A. These r e s u l t s are compared with sm n id S ( k ) , eq. (12), where S (k) i s the structure factor S (k) of a two-component system, which c o n s i s t s of macroions of number concentration ( n 1 + n 2 ) , valency z = - 15 and a volume f r a c t i o n equal to the volume f r a c t i o n of the 3
3
two species of macroions in the real system, that i s n = ^ ( n ^ +n 2 d 2 )/6. The diameter d of the macroions in the e f f e c t i v e system i s then given by _ 3
(n +n 2 )d
3
=
(n d
3
+n 2 d 2 ). The second component of the e f f e c t i v e system con-
s i s t s of n e u t r a l i z i n g 3
counterions
with d„
= 4 %.
sm
I t i s found that S D ( k ) i s a very good approximation to S M ( k ) near the main maximum of the structure f a c t o r . But near k=0 large discrepancies are found in p a r t i c u l a r at higher r e l a t i v e concentrations n 2 /nj> S^(0) overestimates the true structure factor S ^ ( 0 ) . F i g . 4 shows the r e s u l t s for n
2
=
V
decoupling approximation at small wavevectors
has already been noticed by V r i j [ l l ] and by Pusey, Fijnaut and V r i j
[12]
within the Percus-Yevick approximation for hard-core systems. We have i n cluded the corresponding hard-sphere r e s u l t s in F i g . 4, since in t h i s case the MSA treatment i s identical to the Percus-Yevick approximation. I t i s seen that the s l i g h t difference between S D ( k ) and S M ( k ) for the hard-core system near k m a x i s reduced by charging up the p a r t i c l e s . For the i n v e s t i g a t i o n of charge p o l y d i s p e r s i t y e f f e c t s we have taken
343
F i g . 4. Structure factors S (k) and S (k) of a binary mixture of macroions of composition n . / n ^ l , together with the corresponding functions of the equivalent uncharged system. The parameters are n,= 3 x l 0 1 8 cm" 3 , d.= 40 A, d = 50 A, z =z 2 = - 15 and d = 4 A. ( — ) S M ( k ) ; (•••) S " ( k ) of the equivalent uncharged system; ( — ) S u ( k ) ; ( — ) S&(k) of the equivalent uncharged system.
n
i p2 , z
= - 15, a f i x e d diameter r a t i o A = 0.8 and have varied the
charge r a t i o from z 2 /z
= 0.66 to 1.33 and have compared the r e s u l t s f o r
S M ( k ) and S D ( k ) for e f f e c t i v e systems with z = (z +z )/2. I t i s found that S^(k) for k values near k m a x i s s t i l l
an acceptable approximation to S M ( k )
within the chosen parameter i n t e r v a l s . With regard to the long-wavelength M region the gross discrepancy remains; in addition S (0) changes from 0.075 to 0.030 in varying z 2 /z
within the above i n t e r v a l , whereas S ^ ( 0 ) = 0 . 1 1
for a l l charges z .
I I I . Results for Dynamical
Properties
Photon c o r r e l a t i o n experiments on polystyrene spheres [13,14,15] detect f l u c t u a t i o n s o f the system on the scale which i s of the same order as the average distance between macroions. Therefore, the s c a t t e r i n g experiments probe the system on the length scale of i t s short-range structure and de-
344 tect the temporal changes of this structure. For this reason it cannot be expected that macroscopic hydrodynamic equations of motion are sufficient for a theoretical description of the dynamical behavior of macromolecular suspensions as seen in light and neutron scattering. The basic transport equation, appropriate for N interacting Brownian particles, is the Fokker-Planck equation for the distribution function f(F;t) i f ^ , . . .
,...
of the momenta^- and coordinates r^ >
i=l,...,N of the macroions ^
f(r;t) = Q f(r;t)
(15)
where the Fokker-Planck operator is given by
5
"
£i -3F7 + li
+
?
'(kgT 4
+
^
£i )
'
(16)
Here, F. is the force acting on particle i by all other macromolecules and Ç q is the one-particle friction coefficient of the macroions. Hydrodynamic interactions can be neglected for the highly charged and rather dilute systems of polystyrene spheres. This is not justified for more concentrated and less charged systems. Various collective and one-particle properties have been calculated on the basis of eq. (15) by Hess and Klein [16]. Here, we will only mention some of the results, which are of particular importance for dynamic light scattering. The dynamic structure factor S(k,t)orits Laplace transform S(k,z) can be calculated with the help of the projection operator formalism of Mori and Zwanzig [17] . In the experimentally accessible time regime of the correlator one obtains S(k,z) =
^
c T (k)k m z + - - — C(k.z)
(17)
Here, S(k) = S(k,t=0) is the static structure factor, Cy (k) = kgT/m S(k) and ç(k,z) is a generalized dynamic friction function, which generalizes the friction coefficient ç
of a single macroion to a frequency and wave-
345 vector dependent function ç ( k , z ) . This quantity i s found to c o n s i s t of two p a r t s , namely the one-particle f r i c t i o n due to the presence of the solvent and a dynamical part which a r i s e s from the direct i n t e r a c t i o n s between the moving macroions: (18) Here, c i s the concentration of macroions and n ( k , z ) i s the Laplace t r a n s form of the longitudinal dynamic v i s c o s i t y of the i n t e r a c t i n g macroions. One of the most i n t e r e s t i n g r e s u l t s of the above mentioned l i g h t s c a t t e r i n g experiments on nearly monodisperse samples i s the fact that outside the extreme forward d i r e c t i o n the dynamic structure factor S ( k , t ) cannot be r e presented by a s i n g l e exponential function of time. From our general res u l t (17) i t i s evident that t h i s behavior i s due to the z-dependence of C(k,z) ; the deviations of S ( k , t ) from a s i n g l e exponential therefore a r i se from the v i s c o e l a s t i c properties of the suspension. This has been checked by c a l c u l a t i n g the dynamic longitudinal
viscosity
function n ( k , t ) . The Mori-Zwanzig formalism provides an expression for n ( k , t ) in the form of a time-dependent c o r r e l a t i o n function. This function has been evaluated [16] by using the mode-coupling approximation [17]. The result is 2 n(k,t) =
( 2tt )
1 /d 3 k' T [ S ( k ' )] S ( i k+k 1 , t ) S ( i k - k ' , t )
(19)
Here, * y [ S ( k ) ] i s a known function of the s t a t i c structure factor and S ( k , t ) i s the dynamic structure f a c t o r , whose Laplace transform i s expressed in terms of ri(k,z) in eq. (17) and eq. (18). Therefore, eqs. (17) to (19) form a closed set of equations which was solved numerically, where one takes S ( k ) either from a s t a t i c l i g h t s c a t t e r i n g experiment or from a theory of the type described in the f i r s t part of t h i s paper. For a comparison with experiment we have proceeded in the following way: From the experimentally determined S ( k , t ) one can define a mean-relaxation time through
346 T ( k ) =
S?W/S(k,t)dt 1 ' o
(20)
2
If y (k) = D Q k /S(k) denotes the first cumulant of S(k,t), where D q is the diffusion coefficient of a non interacting particle, then A(k) = (y (k) - - r ' ^ k ^ / y ^ k ) is an appropriate measure of the deviation of S(k,t) from a single exponential in time. On the other hand, from eqs. (17), (18) and (20)
t(k) can also be expressed as T(k) = [co + | n(k) k 2 ]/(c T 2 (k) k 2 )
where CO
n(k)
=
/ n(k,t) dt o
is the (k dependent) longitudinal viscosity. Using the numerical solution of the self-consistent set of eqs. (17) and (19), one obtains a theoretical result for A(k). The agreement with the light scattering data of Grliner and Lehmann [15] is within 20 % , see ref. [16] . It should be noted that this agreement is obtained without the fitting of an adjustable parameter. From this discussion it becomes clear why the measured correlation functions in QELS are non-exponential at finite k. The concentration fluctuations, which give rise to light scattering, are coupled to stress fluctuations of the highly correlated system of macroions. The stress fluctuations reflect the non-trivial viscoelastic behavior of the system, as described by n(k,t). If S(k,t) were a single exponential, x(k) would be equal to the inverse of the first cumulant and A(k) would vanish. The fact that the suspension exhibits viscoelasticity, leads to elastic behavior, if the system is probed on a sufficiently short time scale. The corresponding "high frequency elastic constants" have been calculated [18]. It is found that the longitudinal constant varies as the square of the concentration c, whereas the shear elastic constants increases with c / 3 . Finally it is pointed out that various other physical properties characterizing the dynamical behavior of suspensions of charged spherical macromolecules have been calculated on the same basis (16,18,19,20).
347 IV. Conclusion The aim of t h i s paper has been to demonstrate that structural and dynamical properties of suspensions of charged spherical macromolecules can be d e scribed and understood on a microscopic b a s i s . I t has been shown that in f a i r l y concentrated suspensions l i k e micellar systems the role of the small ions should be taken into account for the c a l c u l a t i o n of S ( k ) . Furthermore, various p o l y d i s p e r s i t y e f f e c t s have been investigated for s t a t i c propert i e s of charged hard spheres. With regard to the dynamical structure factor i t was found that even a monodisperse system of charged hard spheres can in general not be described by a s i n g l e exponential function, since at f i n i t e s c a t t e r i n g vectors the v i s c o e l a s t i c s t r e s s f l u c t u a t i o n s give r i s e to a more complicated behavior.
References 1.
Pusey, P.N., Tough, R.J.A., i n : Dynamic Light Scattering and V e l o c i metry: Applications of Photon Correlation Spectroscopy, R. Pecora, ed., Plenum, New York 1984.
2.
Hayter, J . B . , Penfold, J . , J. Chem. Soc. Faraday Trans. 1, 77,1851 (1981).
3.
Kalus, J . , Hoffmann, H., R e i z l e i n , K., U l b r i c h t , W., I b e l , K., Ber. Bunsenges. Phys. Chem. 86, 37 (1982).
4.
Cebula, D.J., Goodwin, J.W., J e f f r e y , G.C., O t t e w i l l , R.H., Parentich, A., Richardson, R.A., Faraday Disc. No. 76, 37 (1983).
5.
Hayter, J . B . , Penfold, J . , Mol. Phys. 42, 109 (1981).
6.
Hansen, J . P . , Hayter, J . B . , Mol. Phys. 46, 651 (1982).
7.
Blum, L . , Mol. Phys. 30, 1529 (1975).
8.
Blum, L . , Hoye, J . S . , J. Phys. Chem. 81, 1311 (1977).
9.
H i r o i k e , K., Mol. Phys. 33, 1195 (1977).
10. See the contribution by Pusey, P.N., in Faraday D i s . No. 76, 93 (1983). 11. V r i j , A . , J. Chem. Phys. 69, 1742 (1978). 12. Pusey, P.N., F i j n a u t , H.M., V r i j , A., J. Chem. Phys. 77, 4270 (1982). 13. Brown, J . C . , Pusey, P.N., Goodwin, J.W., O t t e w i l l , R.H., J. Phys. A 8, 664 (1975). 14. Dahlberg, P . S . , Boe, A . , Strand, K.A., Sikkeland, T., J. Chem. Phys. 69, 5473 (1978).
348 15. Grüner, F., Lehmann, W., J. Phys. A 12, L 303 (1979); 15, 2847 (1982). 16. Hess, W., Klein, R., Adv. Phys. 32, 173 (1983). 17. See, for instance, Boon, J.P., Yip, S., Molecular Hydrodynamics, McGraw Hill, New York, 1980. 18. Klein, R., Hess. W., Faraday Disc. No. 76, 137 (1983). 19. Hess, W., Klein, R., J. Phys. A 15, L 669 (1982). 20. Klein, R., Hess, W., in: Ionic Liquids, Molten Salts and Polyelectrolytes, K.H. Bennemann, F. Brouers, D. Quitmann, eds., Lecture Notes in Physics, vol. 172, Springer-Verlag, Berlin, 1982.
COOPERATIVE GROWTH OF MOLECULAR CONCENTRATED PBLG SOLUTIONS ELECTRO-OPTICAL
Hiroshi
University
of T o k y o ,
molecular
solutions methods.
was
used
solution
investigated
and 2 8 . 0 ) x l 0 0 0 0
College
SEMIDYNAMIC
weights
(sample
I,
and Y o u j i r o
of A r t s
of P B L G
in
dynamic
Sciences,
liquid
crystalline
electro-optical are
II,
IV) and
and
applied pulsed range
are
and b e n z e n e (very poor).
rectangular
sine w a v e s available
with
in
pulses
with
amplitude
this
(2.1,
10.3,
the
Electric
amplitude
4 KV(p-p).
experiment
was
from
semi-concentrated
are m - c r e s o 1 ( g o o d ) , 1 , 2 - d i c h l o r o e thane (fairly
dioxane(poor),
JAPAN
transition
of P B L G u s e d III,
FUKUDA
Tokyo,
of
to a l y o t r o p i c
by
and
Meguroku,
the m e c h a n i s m
aggregation
Molecular
NAKANO,
Komaba,
to e l u c i d a t e
semi—concentrated phase,
Tatsuo
of C h e m i s t r y ,
In o r d e r
IN
BY THE
METHOD
WATANABE,
Department
AGGREGATION
INVESTIGATED
solvents good),
fields
up
The
18. G,
to 2 0 K V
and
frequency
from 2 0 0 H z
to
20
KHz. For quency
the D C
AC
fields,
oscillates frequency tude
of
fields
with of
field.
the o s c i l l a t i n g dipole
component
of
to n e g a t i v e of
electric
a frequency
the A C
permanent
value
the
(rectangular
the
increasing
signals
signal
increase
s ignals i n c r e a s e s on
of
the
sign
changes and
increasing
fre-
of
from the
value
and
angular ampli-
as e x p e c t e d
the
the«/,
low
the at , the
decreases
However,
the
is p o s i t i v e
of 2 ui w h e r e ui is On
orientation.
further
and
birefringence
the b i r e f r i n g e n c e on
pulse)
in
the
the
static
positive absolute oiUJ.
Physical Optics of Dynamic Phenomena and Processes in Macromolecular Systems © 1985 Walter de Gruyter & Co., Berlin • New York - Printed in Germany
A
350 typical
example
Fig.1,
in w h i c h
amplitude
of
the A a r e
the
appears
the
tive
of
depends
weight
the
reported
posi-
in
semi-
et
al(l)
which
relaxation
after
the
plate, and
to c h e c k
signals.
its
the
of
observed
in h i g h
region
is
shown
in F i g . 2 a .
signal
is
positive
AC p u l s e
but
negative
and
state. tion
Just of
for
goes
first
An EB s i g n a l
decomposition
into
after
to
a
to
steady
annihilathe
and
signal
then
and
its
and
decay
(b)
positive
components.
the
frequency
the
2 0 KHz AC p u l s e ,
negative
The
down s l o w l y
reaches after
in
the
frequency
just
a high
t h e AC p u l s e ,
down s h o o t s Fig.2.
of
of
axis
birefringence
signal
application
removal
analyzer
sign
weak
enabled
in-between
the
A typical
DC ( A )
optical
was p l a c e d
the p o l a r i z e r order
dependencies
by u s i n g
field
direction,
Angular of
f 1Hz)
1
1.5
frequency AC C O )
components
and
of
EB.
been
high
birefringent
Fig.1.
solutions
already
by M o r i
we a p p l i e d
log
birefrin-
PBLG has
3.5
Such
from the
the n e g a t i v e
in m - c r e s o l
and
solvents.
of
concentrated
free
critical
which
signals to
values.
some
the m o l e c u l a r
aninversion
the and
birefringence
concentration
gence
in
o are
static
beyond
the n a t u r e
shown
oscillation
The n e g a t i v e
on
is
its and
us
fields.
In
to o b s e r v e
the p u l s e d parallel
to
this
work
the
field
fields.
A
the
field
351 gradually presents
decays. how
The
the
complicated
c a n be
interpreted
signal
and a s l o w
The the al
of
birefringence w h e r e So i s
Fig.3a.
phase
is
and £
positive
the
The
decay
annihilation
of
the
values
o f u> a r e the
These
at
AC
is
formed orient
between
t=0,are
is
It
is
in
Fig.2a
clear
the
the
and
with
negative
a
the
the
•
ones
of
in
sudden
oscillating
that
8/So
abnormal
AC f i e l d s by
in
remov-
signal.
for
In
several
relaxation
is
to.
of
that
a sort
semi-concentrated the
the
retardation
compared
of
of
birefringence
suggest
under
after
normal
obtained
a peak
the v a l u e
strongly
aggregates
in
positive
birefringence
relaxation
positive
compared.
fast
negative
DC and 200 c / s
at
of
Fig.2b
signal
a
normalized
field
field
larger
results
aggregation the
for
of
relaxation
free
the
for
relaxations
faster
as
the
slow
field
value
the
of
difference
birefringence
Fig.3b
the
The
slow.
birefringence
a very
expressed
the
very
signal.
feature is
AC f i e l d .
is
a superposition
negative
frequency
the
light
as
remarkable
high
relaxation
high
of
large
solutions
frequency
AC
and
field.
I i
V\ ^
. 5
.5
\
0
••.'••"••«.^•pfiW.
3 0
50
100
0
5
10
Tims)
Fig.3.
Normalized
solution. denoted others is
on e a c h are
part
Inset of
curves
of
Frequencies
curve.
positive.
different.
initial
decay
c=70g/dm3.
EB i s
Note in
negative
is
EB by
the
fast
in
PBLG(IV)/EDC
AC f i e l d
for
time
a close a
EB
the
negative
that
Ca)
the of
20 KHz
scale
in
observation sampling.
and DC and
the
(a)
and
of
the
are
(b)
352 The o r i e n t a t i o n the
direction
field,
as
mechanism
of
the
suggested
perpendicular measurement dichroism
is
the
of
the m i d d l e
chain benzyl
group,
is
absolute
value
however,
indicating
side
chains
negative
of
the
at
is
electric
dichroism
t h e movement free with
of
the low
the
is
very
respect
of
side The
small, groups
to
the
sign
frequency
AC f i e l d s . benzyl
and
The
band of
t h e DC and
frequency
that
is,
absorption
at
to
type
c o n f i r m e d by
That
high
relatively
is
of
moment
perpendicular
PBLG h e l i x
257nm,
positive
dipole
birefringence.
dichroism.
at but
is
the n e g a t i v e
electric
AC f i e l d s
induced
orientation by
orientation
of
of
on
the b a c k
the bone
helix. The n e g a t i v e obeys
the K e r r ' s
birefringence law
in
the
in h i g h
range
of
concentration
dependence
of
birefringence
for
IV/m—cresol
estimate
the
threshold
polymer-solvent negative
sample
concentration
system
is
birefringence
The
results
the K e r r
about
suggest
solutions,
consequently
relevant
from the K e r r the
induced
dicular
to
the
region
and
the
moment w h i c h
field
of
this
From the
we
the
negative
could
specific
beyond growth
of
that
the
direction
for
different
of
the
from
aggregates
and p o s s i b l e
orients will
be
the
lyotropic
is
decay
of
semi-concentrated
a new m e c h a n i s m The v o l u m e
constant
dipole
of
solutions
14g/dm3
f o r m a t i o n which
explanations.
constant
a cooperative
in p r e - t r a n s i t i o n phase
observation.
region
appears.
aggregates crystalline
frequency
liquid the estimated
origins
the h e l i x discussed
PBLG
elsewhere.
Re f e f e n c e s
1.
Mori,
Y. , O o k u b o ,
Sci.,
Polym.
Phys.
N. , H a y a k a w a , Ed.,
20,
2111
R. , Wada, (1982)
of
perpen-
Y. : J .
Polym.
LIGHT SCATTERING ANOMALY OBSERVED IN DILUTE SOLUTIONS OF POLY (VINYL ALCOHOL) AGED AT HIGH CONCENTRATION
Libuse Mrkvickova, Cestmlr Konak, Blahoslav
Sedlacek
Institute of Macromolecular Chemistry, Czechoslovak Academy of Sciences, 162 06 Prague 6, Czechoslovakia
Introduction When studying the ageing of poly(vinyl alcohol) by integral light scattering
(ILS) we observed an anomaly consisting in a
strong decrease in ILS intensity at lower angles. This phenomenon was first reported by Doty and Steiner
(1). It was re-
cognized to be characteristic of the strict "liquid-like type" structure
(2). Under such conditions, the effect is caused by
ordering of particles
(leading to a considerably
large excluded
volume) and not by the aggregation itself - this would
indicate
an opposite effect. Therefore, we decided to try to elucidate this problem by using also other LS methods.
Experimental Samples. After dissolution alcohol), PVA
(5 hours at 80°C) of poly(vinyl
(BHD Chemical LTD., England; M ^ = 2 2 300), sol-
utions of concentrations c = 1 . 6 ,
6.4, 10, 16% w/v were left to
age at 25°C. Before measurements the samples were diluted considerably
(< 1%w/v). Aqueous 0.1 N K C 1 was used as solvent.
Methods. ILS measurements were performed with a Fica 40 000 apparatus A =546.1
(vertically polarized primary beam of wavelength nm, angle interval 30-150°). For quasielastic
scattering
light
(QELS) measurements a homodyne spectrometer with
96 channel correlator
(He-Ne laser, A = 632.8 nm, interval
Physical Optics of Dynamic Phenomena and Processes in Macromolecular Systems © 1985 Walter de Gruyter & Co., Berlin • New York - Printed in Germany
18-
354
105°) was used. Turbidity was measured with a Perkin-ElmerHitachi UV-VIS spectrometer 340 (A = 435. 8, 546. 1 , 684,3 nm) . For the characterization of supermolecular particles (SMP), both integral (ITR) and differential (DTR) turbidity ratio methods were applied (3). The solution were purified with a Beckman L8-55 ultracentrifuge (rotor SW.1, 15 000 r.p.m., 1 hour).
Results and Discussion The minimum on the reciprocal particle scattering function (9) was observed by ILS if the solution was aged at a higher -2 -3 concentration than the critical one, c = 3.3x10 gem , i.e.,
P
when the space was filled with molecularly dispersed PVA (4). The magnitude and position of this extreme are practically stable in time and depend on the given concentration after dilution (
dn
l
(3)
380
The intensity and degree of polarization of the scattered light are given by I = Trc^
(4)
P = 1 - 4k S |/Tr]
(21)
and (C
sca>S = ^
[e
2S2
+ L
1(X
+ Y)]
(22)
2. Rayleigh-Gans Particles When the colloidal particles satisfy the conditions for R-G scattering (54) all the above parameters can be modified by incorporating a suitable form factor R(0,(|>) . This factor will be different for 'T' and 'L' configurations. Thus ( ^ „ ' J r = R(6,$) T ' L (^) T ' L o U'R_G o U,dipole
(23)
Similarly one can define other parameters. It is straightforward to show that the relation (14) will be satisfied in this case while the equation (19) will not be obeyed (46).
383
3. Small Conducting Spheroids When particles are metalic and are much smaller then the wavelength of light the scattering may not be pure electric dipole scattering. In the limit when m = 00, the scattering consists of electric and magnetic dipole radiation (54). In this case it has been shown that (20) i = ei (C
ext>i "
(C
ext»r
L = l " o =
or in the concentration c: Dc /D o =1 + k_< DtY> = 1 + k' D c, ' where Do is the diffusion coefficient at infinite dilution, k D and k^ are interaction parameters; k D = k]i)/v, v being the partial specific volume of the particles. The simplest model system for theoretical calculation of k D is a dispersion of hard spheres. Two types of interactions are considered in the calculation of k D : static ones, proportional to the second virial coefficient, A2, and hydrodynamic ones, which are responsible for the scatter of calculated k D values (cited, e.g., in (1)) ranging from -6 to +3. Most of the model systems for experimental testing (lattices, colloid dispersions of metals, etc.) are unsuitable due to electrostatic interactions. For silica particles with hydrocarbon chains on their surfaces (so far the best approximation of hard spheres) in cyclohexane, k Q = 1.3 + 0.2 (2) . In this study, block copolymer micelles, for which geometrical and hydrodynamical dimensions coincide (3), were used as an experimental model for hard spheres. A2 and k Q were measured by integral and quasielastic light scattering and compared with theoretical values.
Physical Optics of Dynamic Phenomena and Processes in Macromolecular Systems © 1985 Walter de Gruyter & Co., Berlin • New York - Printed in Germany
402
Experimental Micellar system: Spherical micelles with aliphatic cores and polystyrene shells of three-block copolymer Kraton G-1650 (Shell product) in mixtures 1 ,4-dioxane/0-30 vol.-% n-heptane. Methods: Quasielastic light scattering (QELS), Ar laser A = = 514.5 nm, 0= 90', 96 channel digital correlator. D c values were evaluated from autocorrelation functions (in all cases by the perfect single-exponential fit, indicating that micelles were practically monodisperse). Hydrodynamic radius, R H , was calculated from the Stokes formula R H = kT/67rriDc, where n is the solvent viscosity. Integral light scattering (ILS): Sofica instrument, X = 546 nm, 6 = 30°- 150'. M from K'c/R o = 1/M w + 2A_c, where K 2 R is the Rayleigh ratio at 0->O.
1
and A„ were evaluated w 2 is the optical constant and r
Results QELS and ILS data on block copolymer micelles (Figs. 1 a and 1b) provide values of basic parameters like relative micellar mass, MW . hydrodynamic radius, R„, and interaction parameters, AZ0 N and k
(Table 1).
Table 1: Light scattering data for Kraton micelles in 1,4-dioxane/n-heptane mixtures at 26°C. For the definitions of measured and calculated quantities see text. n-Heptane
Calculated values
Measured values Mw x10~6
A 2 x10 6 3
vo1.— %
RH
-2
nm
2. 1
19.5 22.3
cm g mol
0
4.80
10
4.44
4.2
20
4. 20
7.5
30
3.77
12.7
K ' D
AHSX1Q6
2 cm3 g _J mol
KD
3.07
1 .09
6.63
5.66
1
7.54
7.19
1 .01 1.15
4.02
23 . 2 24.9 1 1 .95
10.97
.05
403
c.W 2 [g cm 3 ]
Fig. 1. Concentration dependences of D C
(1a) and KC/R q
(1b)
for Kraton micelles in 1,4-dioxane/n-heptane solvent mixtures at 26"C. Curves are labelled by vol.-% of n-heptane.
404
1,4-Dioxane is a better solvent for polystyrene, while n-heptane is a better solvent for hydrogenated polybutadiene. Although solvent mixtures used (i.e., up to 30 vol.-% n-heptane) are good solvents for polystyrene and poor ones for the aliphatic blocks, an increase of the n-heptane content improves the thermodynamic quality of a solvent mixture towards micellar aliphatic cores and deteriorates that for the polystyrene shell. As a consequence, M^ decreases with n-heptane content. The increase of R^ means that the effect of swelling of micellar aliphatic cores prevails over that of deswelling of polystyrene shells. A2 values for the hard sphere model system is given (4) by Aip = 4N A V/M^ , where Nft is the Avogadro number o and V = 4ttR^/3 is the volume of a spherical particle. Since the HS
values of A2 from ILS (Fig.1a) and A2
obtained using experi-
mental R H values are similar, the assumption that the block copolymer micelles under study behave like hard spheres seems to be correct. The calculated values of k_ L)(= k'N,, D A V/MW ) in Table 1 (1.01-1.15) are smaller than that for modified silica particles (2). No effect of solvent composition exceeding the experimental error (i.e., solvent quality) has been observed.
References 1.
Stepinek, P., Konak, C.: Adv. Colloid Interface Sci. to be published.
2.
Kops-Werkhoven, M.M., Fijnaut, H.M.: J. Chem. Phys. 74, 1618 (1981).
3.
Tuzar, Z., Plestil, J., Konak, C., Hlavatd, D., Sikora, A.: Makromol. Chem. 184, 2111 (1983).
4.
Yamakawa, H.: Modern Theory of Polymer Solutions, Harper and Row, New York 1980.
PROPERTIES OF BLOCK COPOLYMER MICELLES NEAR THE C.M.C. and C.M.T.
Zdenek Tuzar, Petr Stepinek, Cestmir Konak Institute of Macromolecular Chemistry, Czechoslovak Academy of Sciences, 162 06 Prague 6, Czechoslovakia
Introduction Micellization of block copolymers in selective solvents (i.e. good solvents for one block, which are poor solvents for the other block) obeys the model of closed association (1). This model is characterized by an equilibrium between unimer (molecularly dissolved copolymer) and spherical micelles (having a core formed of the insoluble blocks and a shell containing the soluble blocks), with a sharp mass and size distribution. Closed association model also assumes the existence of a critical micelle concentration (c.m.c.), below which only unimer and above which also micelles can be detected by a given method. In analogy to c.m.c., a critical micelle temperature (c.m.t.) can be defined, above which, at a given copolymer concentration, no micelles would be present. The objective of this study has been to detect and describe block copolymer micelles near c.m.c. and c.m.t. by light scattering methods.
Experimental Copolymer: Fraction of a three block copolymer poly(styreneb-hydrogenated butadiene-b-styrene) (Kraton G-1650, Shell 4 product), free of homopolystyrene (M = 7x10 , 29 wt.-% w styrene). In selective solvents used, 1,4-dioxane and 1,4dioxane/30 vol.-% n-heptane mixture, micelles are formed with aliphatic core and polystyrene shell (2).
Physical Optics of Dynamic Phenomena and Processes in Macromolecular Systems © 1985 Walter de Gruyter & Co., Berlin • New York - Printed in Germany
406 Methods. Quasielastic light scattering
(QELS): homodyne spec-
trometer, 96 channel digital correlator, Ar laser (X = 514 nm), 0 = 90*. Collective diffusion coefficient, D c , was evaluated from the autocorrelation function G(t) = A e - ^ + B, where 2 r = 2 D C K r K being the scattering vector, t time delay and A,B constants. - Integral light scattering
(ILS): Sofica
goniometer, X = 546 nm, © = 30°-150°. Processed data provide J.-L. -I ,, (av) ,, (m) (m) ,„ (u) (u) , . the average molar mass M = M w + M w , w being w w ' the mass fraction, (m) and (u) pertain to micelles and unimer, respectively. Results Micelles near c.m.c. C.m.c. for any block copolymer micelles can be expected much smaller than for, e.g., soap micelles; this has not been determined by light scattering yet. Assuming the closed association model, the ILS data should follow the idealized plot on Fig.1
(1). 1
Fig.1. Schematic plot 1 / M ( a v ) = f(c) for w closed association.
,(ov) M'
cm c Experimental data (Fig.2) show a different behavior of micelles at low c. Both QELS and ILS failed to record c.m.c. The perfect single exponential fit in QELS measurements indicates practically monodisperse micelles even at the lowest measured and 1/M^ a V ' with c w — —3 dilution, a decrease of both values below c = 1x10 g cm is concentration. Instead of an increase of D
407
Q..10P (cm2s
\o
^ctflgcm3)30 Fig. 2. Concentration dependences of D c and 1/M^ aV ' for micellar system Kraton/1,4-dioxane at 26°C.
i 7
(m)
i
i
i
(u-m)
!
(u)
D{u)
-
0
dt'-l
3
c
1 25
i 35
• •
-
|
:
cmt i -—45
i 55
T (*C)
65
Fig. 3. Temperature dependence of D for Kraton solution c - 3 - 3 (c = 5x10 g cm ) in 1,4-dioxane/30 vol.-% n-heptane. For explanation of symbols see text.
408 observed, implicating an increase of micellar size and mass. This finding resembles a similar effect described in the literature (cf. 3). There is no conclusive explanation of this effect. Micelles near c.m.t. Three temperature regions can be seen in Fig.3. D c values in the two outer regions corresponding to unimer and micelles (•) were evaluated from QELS data by a perfect single exponential fit. In the central region, contributions of unimer and micelles to the scattered light intensity are comparable. D ^ 3 ^ values (o) are based on the forced single exponential fit of poor quality. D ^ a p p ' values, evaluated by the four-parameter two-exponential fit (•) using extrapolated values (*) for the unimer contribution, must not be identified with DUpp)
£
;
D 0n) ^ Un f or tunately, above 46 °C in the vicinity of
the c.m.t. micelles contribute to the total intensity of scattered light so little that
cannot be evaluated. The
complex problem of the evaluation of D ( u ) and D ( m ) in the central region is the object of a further study.
References 1. Elias, H.-G., in: "Light Scattering from Polymer Solutions", Huglin, M.B., ed., Academic Press, London 1972 2. Tuzar, Z., Plestil, J., Konäk, C., Hlavatd, D., Sikora, A.: Makromol.Chem. 184, 2111 (1983) 3. Lally, T.P., Price, C.: Polymer 15, 325 (1974) 4. Berne, B.J., Pecora, R.: Dynamic Light Scattering, J.Wiley and Sons, New York 1976, p.41
LATERAL DIFFUSION OF MICELLES MEASURED BY FLUORESCENCE RECOVERY AFTER PHOTOBLEACHING Preliminary results on micelles labeled with a solubilized fluorescent dye
Werner Van De Sande (°), André Persoons Laboratory for Chemical and Biological Dynamics University of Leuven, Celestijnenlaan 200 D, B-3030
Leuven
Introduction Fluorescence Recovery After Photobleaching (FRAP) is a well established technique to measure the lateral diffusion of fluorescent molecules and macromolecules labeled with a fluorophore (1,2,3). Our preliminary results show that the FRAP technique is also applicable to micellar solutions. The labeling of the micelles is achieved by solubilizing a fluorescent dye.
Materials and Methods We used the cationic surfactant N,N-dimethyldodecylammonium chloride (DDAC). The preparation and purification is discribed in (4). The fluorescent dye was an N (10)-alkylated acridine orange derivative ( 3,6-bis(dimethylamino)-10dodecyl acridinium bromide, Fig. 1 ). The preparation is discribed in (5). A new and simple purification method yielded a purity of 99% (to be compared to 93% obtained earlier (6)). The dye is very slightly soluble in water. Solutions were prepared by mixing aqueous solutions of the three components (surfactant, NaCl and dye).
Physical Optics of Dynamic Phenomena and Processes in Macromolecular Systems © 1985 Walter de Gruyter & Co., Berlin • New York - Printed in Germany
410
Fig.l.
The fluorescent dye 3,6-bis(dimethylamino)-10-dodecyl acridinium bromide
[ xxxx CHj) N
I R
( \ m = v = 4 92 nm ).
Br N(CH1))
R = (CH 2 ) 11 CH 3
The main features of the FRAP apparatus were as follows. An argon-ion laser was operated at 488 nm. The laser intensity of the bleaching pulse was about 200 mW. To monitor the fluorescence recovery this intensity was optically attenuated by 4 a factor 10 . Typical bleaching times were 5 ms The radius of the uniform circular beam, focussed through a microscope objective, was 6 ym.
Theory and Data Analysis The experimental curve F(t) was normalized in the form of the fractional fluorescence recovery curve f(t): =
F (t) - F(0) F - F (0)
U
'
The fractional recovery curve was fitted to the exact closed formula (7): f(t) = exp(-x) x { I Q (x) + I 1 (x) } IQ and
(2)
are modified Bessel functions, x = 2x D /t,
t d = u^/4D, oj is the radius of the circular beam, D is the diffusion coefficient. The value of the intensity F(0), immediately after bleaching, was obtained by fitting the initial part of F(t) to a poly-
nomial of second degree. The initial points after the bleaching pulse
were omitted.
The value of the intensity F(