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University 3 \ll lllllllll\\\\\\\\\l\\l\\\\ll EV llllllllll Hill 3 1293 10421 9310 This is to certify that the thesis entitled STUDY OF DIFFUSION 0F POLYDISPERSE POLYSTYRENE AND STYRENE—ACRYLONITRILE COPOLYMERS IN SOLUTION BY LIGHT BEATING SPECTROSCOPY AND INTERFEROMETRY presented by P. V. S. R. Krishnam Raju has been accepted towards fulfillment of the requirements for Ph. D. degree in Chem1ca1 Engmeermg {M rflwg Major professor Date May 10, 1978 0-7639 STUDY OF DIFFUSION OF POLYDISPERSE POLYSTYRENE AND STYRENE-ACRYLONITRILE COPOLYMERS IN SOLUTION BY LIGHT BEATING SPECTROSCOPY AND INTERFEROMETRY By P. V. S. R. Krishnam Raju A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemical Engineering 1978 I I I <- _. I X 0’ ('. ’t I ABSTRACT STUDY OF DIFFUSION OF POLYDISPERSE POLYSTYRENE AND STYRENE-ACRYLONITRILE COPOLYMERS IN SOLUTION BY LIGHT BEATING SPECTROSCOPY AND INTERFEROMETRY By P. V. S. R. Krishnam Raju A systematic study was made of diffusion in dilute and moderately concentrated solutions of polystyrene in benzene and decalin, and styrene-acrylonitrile copolymer in dimethyl formamide, methyl ethyl ketone and benzene. Diffusion data were obtained at ambient temperature in the concentration range of 0.01 to 10% by weight of polymer. Studies were made with polydisperse polystyrene samples having weight average molecular weights in the range of 38,000 to 350,000 and for polydisperse styrene-acrylonitrile copolymer samples having weight average molecular weights in the range of 200,000 to 800,000. The styrene-acrylonitrile copolymers of azeotropic composition, 24% by weight of acrylonitrile, were synthesized by free radical polymerization. Translational diffusion coefficients were obtained using a laser homodyne spectrometer in dilute polymer solutions and using an interferometric method in moderately concentrated polymer solutions. An expression was derived for the relationship between the experimental average diffusion coefficient obtained from the interferometer and the __._-.~._.__VV_— . ._ P. V. S. R. Krishnam Raju distribution of diffusion coefficients for the individual species of a polydisperse polymer. The concentration dependence of the diffusion coefficient is linear over the entire concentration range investigated by both these experimental methods. The data are fit with an equation of the form 0 = 00(l + kdc). The value of kd is always positive in good solvents. In other solvents the value of kd is negative at low molecular weights and positive at high molecular weights. Observed 7 cmZ/sec. values of D0 are l to 7xl0' Data obtained in this work are compared with available theoretical treatments for diffusion in monodisperse homopolymer solutions. A semi-empirical relation, based upon a modification of the Kirkwood-Riseman approach, is proposed for describing diffusion in infinitely dilute polymer solutions, and is tested against data from this work and from the literature. The observed diffusional behavior can be explained for all the polymer-solvent pairs inves- tigated in this work using the modified Kirkwood-Riseman expression for DO and a method of Duda and Vrentas for evaluating kd. The difference in the numerical values of the diffusion coefficients obtained for a polydisperse polymer-solvent pair from interferometry and from light beating spectroscopy, in the concentration range where the two methods overlap, can be explained by the influence of polydispersity on the results of each method. Dedicated to my parents. ii ACKNOWLEDGMENTS The author would like to express his deep appreciation to his major professor, Dr. Robert F. Blanks, for his constant guidance and assistance throughout the course of this work, and also for his painstaking review of the manuscript. Gratitude is expressed to Dr. Jack B. Kinsinger and Dr. Donald K. Anderson for their generous guidance and active help in performing the experiments. Thanks are also due to Dr. Martin C. Hawley for his participation in this work. Special thanks are due to Dr. Thomas V. Atkinson for help with the computer interfacing. The author is greatly indebted to the Department of Chemical Engineering and the Division of Engineering Research for providing financial support during his graduate work. A Sincere gratitude is due to my parents for their moral sup- port and encouragement throughout my graduate studies. Special thanks are expressed to my wife, Udaya, for providing encouragement and for her patience while I spent countless hours on this work. TABLE OF CONTENTS LIST OF TABLES . LIST OF FIGURES Chapter I. INTRODUCTION . II. SCOPE Objectives of the Research Polymers and Solvents Used Experimental Methods . Terminology in Polymer Solutions Principles of Diffusion III. THEORY OF DIFFUSION IN POLYMER SOLUTIONS Diffusion in Infinitely Dilute Polymer Solutions Kirkwood-Riseman Theory . . . . . . Flory's Theory . Johnston' 5 Theory . . . Semi- Empirical Relation . . Diffusion in Dilute and Moderately Concentrated Polymer Solutions . Two Parameter Theory . . Modified Pyun and Fixman Theory . . . Comparison of Predicted Values of kd with experimental data IV. POLYMERIZATION AND FRACTIONATION . Copolymerization Theory Rate of Copolymerization . . Chemically- Controlled Termination Diffusion- Controlled Termination . Synthesis of Copolymers Initiator Monomers Polymerizations . Molecular Weights and Molecular Weight. Distributions . iv Page vii ix Chapter Fractionation of Copolymers Small-Scale Fractionations Large-Scale Fractionations V. EXPERIMENTAL METHODS FOR MEASURING DIFFUSION COEFFICIENTS . Light Beating Spectroscopy . Background . . Theory. Beat Frequency . Theory of Brownian Motion . Application to Polymer Solutions Experimental Apparatus . . Computer Interfacing of the Averager . Procedure for Experimental Run . Data Analysis Calibration . Sample Preparation . Error Analysis Interferometry. Background Experimental Apparatus . . Procedure for Experimental Run . Theory and Calculations . Extension to Polymer Solutions Calibration . . . . Error Analysis VI. PRESENTATION OF EXPERIMENTAL DATA Light Beating Spectroscopy . Interferometry . . . . VII. DISCUSSION OF THE RESULTS AND COMPARISON WITH THEORY . Comparison of Light Beating Data for Polystyrene in Methyl Ethyl Ketone . Comparison of Light Beating Data for. Polystyrene in Tetrahydrofuran . . Comparison of Interferometry Data for Polystrene in Toluene . . Comparison of Data Obtained from Interferometry with the Data Obtained from Light Beating for Polystyrene in Toluene . . . Discussion of the Results Obtained from Light Beating Spectroscopy . . . . Page 59 64 154 I61 T64 T66 T67 Chapter Polystyrene in Various Solvents Styrene- -Acrylonitrile Copolymer in Various Solvents . Discussion of the Results Obtained from Interferometry . . Monodisperse Polystyrene in Benzene . Polydisperse Polystyrene in Benzene . . Styrene- Acrylonitrile Copolymer in Various Solvents Comparison of Light Beating and Interferometry . Data . . Comparison of Thermodynamic Parameters VIII. CONCLUSIONS AND RECOMMENDATIONS NOMENCLATURE Appendix A. SAMPLE CALCULATION . B. MARK-HOUNINK CONSTANTS D. LIGHT BEATING DATA . E. DATA FROM INTERFEROMETRY . G. CARDS USED IN COMPUTER INTERFACING H. PROGRAM FOR DATA TRANSFER J. DERIVATION FOR FUNCTION I (m,n) BIBLIOGRAPHY vi Page T69 T75 l79 l79 182 l86 192 194 196 201 208 215 217 220 223 231 234 238 Table 2-1. 4-2. 4-3. 5-1. 5-2. 5-4. 6-1. 6-2. 7-2. 7-3. LIST OF TABLES Molecular weights and molecular weight distributions for polystyrene Details of bulk polymerization at 60°C using AIBN Molecular weight and molecular weight distribution of copolymers by GPC . . . . . Results of small-scale fractionations Results of large-scale fractionations Comparison of half widths obtained from the recorder and computer . Ratios of Dav/D(Mw) Results of calibration runs for polystyrene spheres in water . . . . . . . . . . Results of the calibration runs on the interferometer Values of D and k calculated from light beating data 9 . . . . . . Values of Do and kd obtained from interferometric data for monodisperse polystyrenes and their mixtures . . . . . Comparison of King, et al. and Ford, et al. data for polystyrene in methyl ethyl ketone at 25°C, obtained by light beating spectroscopy Comparison of Mendema, et al. and Jamieson, et al. data for polystyrene in tetrahydrofuran . Comparison of experimental and theoretical values of D 0 vii Page 60 61 63 65 100 106 108 135 143 151 156 163 170 Table 7-9. 7-10. 7-12. 7-13. 7-16. Comparisons for the values of B for polystyrene . Comparisons of experimental and theoretical values of 00 for the copolymers . . Comparisons of experimental and theoretical values of kd for the copolymer Concentration dependence of D, for monodisperse polystyrenes and their mixtures obtained from interferometry Comparison of experimental and theoretical values of 00 for the monodisperse polystyrenes and their mixtures obtained by interferometry Comparison of experimental and theoretical values of Do for polydisperse polystyrene (PS-2) obtained from interferometry . Comparison of experimental and theoretical values of kd for polydisperse polystyrene (PS-2) obtained from interferometry Comparison of experimental and theoretical values of kd for polystyrene (PS-2) in benzene using higher values of B . Comparison of experimental and theoretical values of D for the copolymers obtained from inter- ferometry . . . . Comparisons of experimental and theoretical values of kd for copolymers obtained from interferometry . Comparison of experimental and theoretical values of kd for copolymers obtained from interferometry with higher values of B . Comparison of experimental and theoretical values of the thermodynamic parameters . . . viii Page 174 177 178 181 183 184 184 186 188 189 191 195 Figure 3-1A. 3-18. 3-10. LIST OF FIGURES A free-draining molecule during translation through solvent Translation of a chain molecule with perturbation of solvent flow relative to the molecule Molecular weight dependence in cyclohexane at 35°C Molecular weight dependence in cyclohexane at 35°C Molecular weight dependence in methyl ethyl ketone at Molecular weight dependence in toluene at 20°C Molecular weight dependence in benzene at 25°C Molecular weight dependence in methyl ethyl ketone at Molecular weight dependence in toluene at 20°C Molecular weight dependence in benzene at 25°C Comparisons of theoretical predictions for kd (cm 3/g) with experimental and Ovm msmz .mcmcxamxpoa cow mcompznwgpmwc agave: empaumpos use magmwoz empaumpoz--.Pum m4m

Y. This quantity is also called the unperturbed dimension of the polymer, because the statistical analysis is based on the assumption that the polymer chain configuration is completely free of outside influences. The configuration of the polymer molecule will also depend on its environment, which is often a solvent. In a good solvent, where the energy of interaction between a polymer molecular segment and a solvent molecule adjacent to its exceeds the mean of the energies of interaction between the polymer-polymer and solvent-solvent pairs, the molecule will tend to expand further compared with its unper- turbed dimension so as to reduce the frequency of contacts between pairs of polymer segments. This expansion is characterized by a parameter, a, which is called the linear expansion factor, and is defined as 3 = 7 a (2-1) where 7 is the mean square end-to-end distance for the polymer in any particular environment. In a poor solvent, on the other hand, where the energy of interaction between polymer segment and solvent is more repulsive, smaller configurations in which polymer- polymer contacts occur more frequently will be favored. In the limit where the solvent is so poor that the polymer assumes its 10 minimum or unperturbed dimensions, the polymer chain is described as being in its "theta" state as will be discussed below. It may be better understood to rank the thermodynamic quality of a solvent for a polymer (good solvent or bad solvent), based upon thermodynamic arguments using the Flory-Huggins equation (2A-l). This equation gives the free energy of mixing of polymer (2) with solvent (1) as TTF'g n1 1n 4] + n2 1n ¢2 + X ¢1 ¢2 (n1 + m n2) (2-2) where n, is the moles of component i, ¢i is the volume fraction of component i, m is the ratio of molar volumes of polymer to solvent, and x is the Flory-Huggins thermodynamic interaction parameter. In order for a given solvent to dissolve polymer the free energy of mixing should be negative. Since the first two terms in the above equation are always negative, this means the smaller the value of x, the better the thermodynamic quality of the solvent for the polymer. The thermodynamic quality of a solvent for a polymer can be investi- gated experimentally. The osmotic pressure of a dilute polymer solution (for which the partial molar volume of the solvent is indistinguishable from its molar volume) can be expressed as (2A-2) n = (R T / M2) c2 where n is the osmotic pressure, R is the gas constant, T is the absolute temperature, M2 and c2 are the molecular weight and 11 concentration of polymer. At higher concentrations, where binary and higher order interactions of polymer segments are present n = R T [ (c / M ) + A c2 + A c3 + ..... ] 2 2 2 2 3 2 ~ where A2, A3 etc. are the second, third, and higher osmotic virial coefficients. According to Flory's theory the second virial coefficient A2, can be defined in terms of x and a as AZ=(Vp2/VS)(8-X)F(X) <2-3) where 2 3 F(x)=l- x.+ x .-——,§" +--- 2'21 3'31 414 x = 2 (a2 - 1) Here vp is the specific volume of polymer and vS is the molar volume of solvent. The second virial coefficient for polymer solutions can also be obtained from light scattering measurements. The linear expansion factor, a, may be measured by intrinsic viscosity both at theta and non-theta conditions. Alternatively the unperturbed dimension and the value of ;5 may be measured by light scattering 2>k 0 many systems and are available in the literature (3A-9). measurements. The values of 7 have been measured for The osmotic swelling of the polymer by the polymer-solvent interactions in good solvents is often referred to as the ”excluded volume effect.“ Two or more polymer segments remote from one another 12 along the chain cannot occupy the same volume element at the same time, because of their finite volumes. In other words repulsive forces will act between these segments when they are close to one another. In addition, this repulsive force will, to some extent, be altered by the presence of solvent molecules. Intermolecular interactions of this sort are associated with the "excluded volume effect." The excluded volume effect vanishes under a special condition of temperature or solvent, which is known as the Flory "theta" temperature or theta solvent, and the condition is called the theta condition. The theta condition arises because of the apparent can- cellation at this condition, of the effect of volume exclusion of segments which tend to enlarge the molecule, and the effect of Vander walls attraction between segments which contracts the mole- cule. At the theta condition, a must equal unity irrespective of the molecular weight of the polymer. When x = 1/2, Flory defined this as the theta condition in terms of the interaction parameter x. From equation 2-3, we can conclude that at the theta condition, A2, the second virial coefficient is zero. This background material is provided as an aid in under- standing the results and conclusions to be described in this work. In a thermodynamically good solvent, or at any condition other than the theta condition for a polymer solvent pair, excluded volume effects give rise to the linear expansion of the polymer molecule. It is only at the theta condition, when excluded volume effects are 13 absent, that the linear expansion factor is unity, and the second virial coefficient is numerically equal to zero. Principles of Diffusion Diffusion is the movement of an individual component through a mixture. Although the most common driving force for diffusion is a concentration gradient of the diffusing component, it can also be caused either by a pressure gradient or by a temperature gradient. In this section diffusion caused only by concentration gradients will be discussed. Diffusion may result from molecular motion only or by a combination of molecular and turbulent motion. In the absence of turbulence the rate of diffusion of component A is given by Fick's Law dx ——‘1 (2-4) JA""° DAB dz Where JA is the molar flux for component A, c is the molar density of solution, DAB is the diffusion coefficient of A in solution in B, xA is the mole fraction of component A, and z is the direction of diffusion. The negative sign emphasizes that diffusion occurs in the direction of a drop in concentration. The flux JA was defined with respect to molar average velocity. In engineering process calculations it is usually desirable to refer to a coordinate system fixed in the equipment. Therefore, Fick's first law in terms of NA, the molar flux relative to stationary coordinates, becomes: 14 d XA (2-5) N + N AB TE? A = xA ( A B) ' C D N This equation shows that the flux defined in terms of NA is the result of two quantities: xA (NA + NB) wglch is the molar flux of a resulting bulk motion of fluid and - c DAB 7Eé° which is the molar flux of A resulting from the diffusion superimposed on the bulk flow. For binary systems DAB = DBA' All these relationships are based on the assumption that DAB is not dependent on concentration. This may not be true for con- centrated solutions. Dependence of DAB on concentration is the result of change of mobility of the solute with concentration and deviations of the mixture from ideal behavior. For non ideal mix- alna alnxA activity coefficient of species A (2A - 2a). Therefore to evaluate tures DAB can be corrected by a factor of where aA is the the molar flux NA’ it is essential to know the concentration dependence of DAB' In this work the concentration dependence of the diffusion coefficient for polymer molecules in solution is examined. The equation of continuity for polymer molecules in solution is obtained by making a mass balance over an arbitrary differential fluid element. The diffusion equation (2-6) is obtained by the insertion of the expression for molar flux into equation of contin- uity with the assumptions of constant molar density, constant dif- fusivity and zero mass average velocity. dCA 2 7H? ‘ DAB V cA (2‘5) 15 The diffusion equation is called Fick's second law of diffusion. In multicomponent mixtures the diffusion contribution to the mass flux is seen to depend in a complicated way on the concentration gradients of the substances present. For multicomponent ideal-gas mixture a relation is known (2E-l) between Dij (the diffusivity of pair i-j in the multicomponent mixture) and19ij (the diffusivity of pair i-j in the binary mixture). For ideal-gas multicomponent mixture the flux equations are known as Stefan-Maxwell equations (2E-l). CHAPTER III THEORY OF DIFFUSION IN POLYMER SOLUTIONS Diffusion in binary systems of large polymer molecules and small solvent molecules exhibits markedly different behavior as the relative proportions of the two species are varied over the entire concentration range. Most of the research in this area leads to the conclusion that for diffusion in dilute polymer solutions in good solvents, the value of the diffusion coefficient, 0, generally increases with polymer concentration in the region of low polymer concentration. On the other hand, it has been shown that for dif- fusion in polymer films or solids, in the region near undiluted polymer the value of 0 increases quite sharply with increasing diluent concentration. These facts lead to the idea that the 0 versus concentration curve for a polymer solvent system should exhibit a maximum at an intermediate concentration in the range from pure solvent to pure polymer. Although existing data for 0 covering a wide range of polymer concentrations are still quite few, this prediction is widely confirmed (3C-l, 3C-2) in thermodynamically good solvents. For the purposes of this work, the total concentration range is considered in five sub-regions. They are: (l) infinitely dilute region, (2) dilute region (up to 1% by weight of polymer), 16 17 (3) intermediate concentration region (up to 10% by weight of polymer), (4) concentrated region (up to about 90% by weight of polymer) and (5) bulk polymer region. Only the first three ranges are studied in this work. Diffusion in the infinitely dilute range has been investigated most thoroughly and is best understood. This is discussed in the section on "Diffusion in Infinitely Dilute Polymer Solutions" in Chapter III. As the polymer concentration increases slightly from the limit of infinite dilution, the dif- fusion coefficient may be expected to vary as D = 00 [l + kd c] D0 is the diffusion coefficient at the limit of zero polymer con- centratibn, and c is the polymer mass concentration. The parameter kd is a function of both thermodynamic and hydrodynamic factors. The section on "Modified Pyun and Fixman Theory" in Chapter III discusses the relations for obtaining kd from the combination of two parameter theory, and a modified Pyun and Fixman (3B-3) theory. In the last section the value of kd predicted by the above theories is compared with the experimentally available values in the literature for polymer-solvent systems. Diffusion in Infinitely Dilute Polymer Solutions The most important quantity obtained from a diffusion study in infinitely dilute polymer solutions is Do’ the value of D at the limit of zero polymer concentration. For this quantity the well known Einstein formula (3A-l) is 18 D0 = k T / f0 (3-1) In this equation k is the Boltzman constant, T is the absolute temperature of the solution, and f0 is the value of f at the limit of zero polymer concentration. Here f stands for the frictional coefficient of the polymer molecule, which is defined as the force experienced by the polymer molecule when it moves with a velocity of one centimeter per second relative to the solvent. The value of f0 is influenced both by the size and shape of the polymer molecule as well as by the viscosity no, of the solvent. For a rigid spheri- cal molecule of radius Ra’ the Stokes formula is fo = 6 n no Ra Most linear polymer molecules assume a randomly coiled form in solu- tion. The derivation of an expression for f0 for such molecules was first made by Kirkwood and Riseman (3A-2). They did not take into account the excluded volume effects between polymer segments. These effects were considered later by Flory (3A-3) and Johnston (3A-4), in two different approaches toward deriving relations for f0. The rest of this section contains a brief description of the theories of Kirkwood-Riseman, Flory, and Johnston. These theories are com- pared with the existing experimental data in the literature. This section concludes with the derivation of a semi-empirical model for predicting 00’ based on Kirkwood-Riseman theory and proposed by the author. 19 Kirkwood-Riseman Theory The Kirkwood-Riseman theory of transport processes in polymer solutions provides a convenient method for predicting the transla- tional diffusion coefficient at infinite dilution. The theory is applicable under theta conditions only because excluded volume effects were not considered in the derivation. The frictional coefficient at infinite dilution of the polymer is developed on the basis of a random coil model with hindered internal rotation. The theory is based on the notion that the peripheral elements of the polymer chain perturb the flow in the neighborhood of the interior elements in such a manner that they are partially shielded from hydrodynamic interactions with the exterior fluid. At high molecular weights, the hydrodynamical shielding of the interior elements may become so effective that their contribution to the resistance offered by the molecule to the external fluid is negligibly small. Using this approach Kirkwood-Riseman derived the following equation for the translational diffusion coefficient of a chain-like molecule at infinite dilution (3A-5d, 3A-6) -51 a D 7n; (1+3X) (3-2) Nhere =.LZ_Q___£_ nOL 712115 20 L is the effective bond length, n is the number of effective bonds or segments in a chain, C is the translational friction coefficient of a segment, and n0 is solvent viscosity. The parameter X is a measure of the hydrodynamic interactions between segments. The parameter c is not directly observable or predictable by a simple method for polymer solutions. In the limiting case of X >> 1, the parameter c drops out. The two limiting cases X = O and X >> 1, for the above equation, have special significance for polymer solutions. In the case X = 0, there is no hydrodynamic interaction between segments, and the velocity of the medium everywhere is approximately the same as though the polymer molecule were not present. The solvent streams through the molecule almost (but not entirely) unperturbed by it, hence this is called the free draining case. Figure 3-1A is illustrative of this case. The case X >> 1, is illustrated in Figure 3-18. In this case the velocity of the solvent relative to the molecule increases from zero at the center to a value approaching its external value at some distance from the center. For this case the intrinsic viscosity is equivalent to that for a rigid sphere molecule, therefore, flexible polymer chains in this limit behave hydrodynamically as rigid sphere molecules. This limit corresponds to very large hydrodynamic interactions between segments, and the polymer molecule is treated as an hydrodynamically equivalent sphere. Thus the variable X represents the degree of drainage of the solvent through the polymer molecule domain, and is called the draining parameter. Yamakawa (BA-5d) compared the experimental intrinsic viscosity data for polyisobutylene in benzene 21 Figure 3-lA.--A free-draining molecule during translation through solvent.* *Arrows indicate flow vectors of the solvent relative to the polymer chain. 22 Figure 3-lB.--Translation of a chain molecule with perturbation of solvent flow relative to the molecule.* *Arrows indicate flow vectors of the solvent relative to the polymer chain. 23 with that of the predictions from the theory with X = O and X >> 1. From the comparisons Yamakawa concluded that the case with X >> 1 describes the behavior of the experimental data well. There is evidence that the case with X>o»1 describes the polymer solution behavior at infinite dilution (BA-6). Consequently, for X >>1 equation 3-2 reduces to the following form Do = 0.196 kkT (3_3) 8 no A M where is the mean square end-to-end distance of the unperturbed chain, M is the polymer molecular weight, and [00)6 is the value of 00 under theta conditions. . In order to test the validity of equation 3-3, we have chosen data from the literature obtained on the polystyrene- cyclohexane system at the theta temperature (235°C) and compared them with the predictions from equation 3-3. Figure 3-1 is a plot of [Do]e versus Mw’ the weight average molecular weight of the polymer, on a log-log paper. Since the molecular weight distribu- tion of the polystyrene used in the various studies was not reported in the literature (except for the data of King, et al.) we have assumed that the characteristic molecular weight of the polymer is the weight average molecular weight. Values of the parameter A for 11 -11 this system vary from 645 x 10' to 775 x 10 cm, in the Apr/:9 .pc ct .627: Cr. 5:50 L Cubism 24 .Uomm um mcmxwzopozo c_ mcmcxuw>_on Loo on we mocmucmomu agave: Lm_:umpoz--.F-m meam_d no_ mo_ mo_ ¢o_ qm “cmwmz 2 III] I 1 I I. I, ’I 1 II/O/ U I ’l” .1 pm / III”/ II OI I’mI.// a 14r.,/;, 11“]? . I, I IIQd/ . II I .,I .I 4 .< «.3858 .i- I II . we once; ms“ sum: mnm :owpmacm -1- / [IMP/.a . II I . :22 .2 2 52226: co 88 a 110/ L A~_-m acmemz no 11. . u“ [I .11. a m. m. U L 3 0 l a . w. 1 top w. a .u. 14 D» .4 L m. I: u” .1: w. 11.1: . m. .1. . m. .J .L a m. Aoum :o_pm:cmv zcomgu m.:oumccow a m. “mum comumacmv xuomcu m.zcoFu 1|..11...1. .m nu Ampum usmwmz / I 0 iAl/j 1_A o ‘Uollnllp alIUlJUl 19 1UBIDIJJ903 “OISOJJLO Amum cowumacmv xcomcu m.:oum:;ow Amum cowpwacmv xcomgu m.>copm .11..|1..11 Ampu_oa com on we mucmwcmamu ucmwmz Lm_:um_ozn-.<-m mesmwd op mo— mop op c .4111: 1 1 1d .4 ‘4 4 q ‘ 1 .4“4 S 3: .unuwmz empaom_os mmmcm>m ucmwmz Amnm covpmscwv meowgu m.:oumc;ow Amum cowpmacmv xeomgu m.>co_a 11. 374%; .3 pm .tocgmzmz mo 330 L ILAI L. ml 0.! o—xm op o ‘Uollnllp allUlJUl 12 1U9l3lJJ303 UOISPJJIO opxm 3O .UOmN um mcmNcwo :_ mcmczpm>_oq co» co co mocmucmamu p;o_m3 empzompoz--.m-m mesmwd op mop c o— Amlm comumzcmv xgomcu m.:oum:;ow Am1m cowumzcmv xeomgu m_xco~d :22 .2 pm .22.“. .6 88 o i1i 1 “14" d d 4i J ‘ z .uzmmmz Lm_:om_os mmmem>m ucmwmz -_l__A L L 0L2“ . P ”1 ml OFxN op o ‘u011nlip 311014“! 19 1U313111903 "OLSOJJLO CFO opxm 31 (M = Mw)' Data were needed for [n], intrinsic viscosity, for all the systems. These were obtained from the Polymer Handbook (3A-9). Even though there were many relations for predicting the variation of [n] with molecular weight in the handbook, the relations for the molecular weight range of interest that were recommended by the editors of the book were used. These relations are shown in Appendix 8. From the Figures 3-2 through 3-5, it can be concluded that Flory's theory as given by equation 3-5 does not agree with experimental data for diffusion in infinitely dilute polymer solu- tions. Flory's theory predicts that o should be a universal constant independent of the nature of the polymer and independent of the solvent medium. There exists still a controversy on the validity of o being an universal constant (3A-18, 3A-19, 3A-20). The deviation of the experimental data from Flory's theory may be attributed to his assumption that all linear dimensions of a flexible coil change by the same factor when it is transferred from one solvent to another. Even though Flory's approach is theoretically sound, the value to be used for the universal constant 4 was not exactly known. Johnston's Theory Recently Johnston (3A-4) combined some known expressions to offer a rather simple view of diffusion in infinitely dilute polymer solutions. Based on the concept of an equivalent hydrodynamic sphere, impenetrable to solvent, he obtained the expression for fo as 32 f0 = 6 n no Ra (3-6) where Ra is the radius of the hydrodynamically equivalent sphere. He used an intrinsic viscosity expression for dilute polymer solu- tions based on Einstein's viscosity relation (3A-21) 2.5 N V [n] = -—"—]fL—Ji (3-7) u where N0 is Avogadro's number, Ve is the volume of the equivalent hydrodynamic sphere and Mu is the viscosity average molecular weight. By eliminating the radius of the equivalent hydrodynamic sphere, Ra’ between equations 3-6 and 3-7, and with the use of the Mark-Houwink expression, [n] = KV Ma (3-8) Johnston derived an expression for 00 as = k T a+l 1/3 00 6 n no [(10 n No)/3 Kv Mu ] (3-9) where Kv and a are called the Mark-Houwink constants and are con- stants for a particular polymer-solvent pair. Once again the validity of equation 3-9 is put to test by comparing the calculated values for 00 from equation 3-9, with the same experimental values that were used for comparing Flory's theory, both for theta and non-theta conditions. The comparisons are shown in Figures 3-2 through 3-5. To obtain Do values from 33 Johnston's theory we assumed that Mu is equal to Mw’ This assumption is valid if the polymer molecular weight distributions are not broad. The values of Kv and a used in equation 3-9 are tabulated in Appendix 8. From the Figures 3-2 through 3-5, one can easily conclude that Johnston's theory agrees well with the available experimental data to within ten percent for all the cases where the Mark-Houwink parameters are well established. If we compare Flory's theory with that of Johnston's, it can be seen that both have the same molecular weight dependence. The difference is in the so-called universal constants of P'1 ¢]/3. According to Johnston's theory P'] ¢1/3 is [10 n NO/3]]/3 equivalent to 6 N This does not imply that Johnston's theory is better than Flory's theory. For the few systems compared here Johnston's theory seems to be predicting diffusional behavior adequately. In this study, Johnston's theory was used for predicting 00 for all the polymer-solvent pairs when the viscometric parameters were well established. Semi-Empirical Relation Johnston's and also Flory's theories require the avail- ability of accurate Mark-Houwink constants, Kv and a,'h1equation 3—8. Even though these constants are available for the most common polymer-solvent pairs, there are many polymer-solvent pairs where these constants are not available. Van Krevelen (3A-22) derived empirical relationships for predicting Kv and a. Using his method 7* r‘.’ 34 the mean difference between experimental and calculated values for Kv and a was about 30%, for more than three hundred polymer-solvent pairs. In this work a semi-empirical relation for 00 was obtained using the predicted value of a. The value of a was chosen rather than both a and Kv’ because a is easier to predict and its range for all the commercially available polymers is small, 0.5 to 0.75, compared to the range of Kv‘ The semi-empirical relation, is a modified form of the Kirkwood-Riseman equation, obtained by multiplying equation 3-3 on the right hand side with 2(l-a), which gives = 0.195 k T 2(1_a) D ---wr- 0 n0 A M /2 (3-10) This relation should predict 00 both in theta and non-theta conditions, because of the incorporation of the excluded volume effects through the parameter a. The factor 2 O-alwas designed such that at theta condition (a = 0.5) it is equal to one. Thus at the theta condition the value of Do is still predicted by the Kirkwood- Riseman equation, however under non-theta conditions the value of Do is corrected by a factor less than one (since a lies between 0.5 and 0.75 for most of the commercial polymers). The correction for Do in non-theta conditions is in the appropriate direction since it is known that the polymer molecule expands under non-theta conditions compared to its size at the theta condition, and the diffusion coefficient should therefore decrease. 35 The validity of equation 3-10 is put to test by comparing the available data for polystyrene in different solvents. The experimental data used for comparison in the earlier theories are also used here. The theoretical predictions fOr the polystyrene- cyclohexane system by equation 3-10, under theta conditions would be exactly the same as those shown in Figure 3-1, because a = 0.5. Under non-theta conditions, for all the other solvents, the compari- sons are shown in Figures 3-6 through 3-8. The two dotted lines in these figures correspond to the range of the parameter A available in the literature, as mentioned in the section on "Kirkwood- Riseman Theory" earlier in Chapter III. The values of the parameter a used are shown in Appendix 8. From the comparisons in Figures 3-1 and 3-6 through 3-8 it may be concluded that the relation for Do,given by equation 3-10, predicts the diffusion coefficient at infinite dilution under both theta and non-theta conditions surprisingly well. In this work whenever the viscometric parameters were not well established for the polymer-solvent pair, equation 3-10 was used to predict Do' Diffusion in Dilute and Moderately Concentrated Polymer Solutions The concentration dependence of the translational diffusion coefficient in dilute polymer solutions is given by D = 00 [l + k c + ....J (3-10A) d The concentration dependence of D has been the subject of a large 36 .UOmN pm mcoumx chum ngums cw mcmgzum>Foa com on we mucmucmamu “comm: empaom_oz-1.oum «Lamp; o_ m mop mop «op j~aqau « «i ~4.av.fi a . . ~q|qd1q1 W wlO—XM 3 o 2 “some; Lm—zompoe mmmcm>m pgmwmz . G , m 11 II. w. 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W 87m 5.5253 < US was? 0 161 III 00 53 5.528 EoZEEqumm ”Hun Ill 1 nv 1; $22 .2 2 .28 .2 33 4 2:5 .2 2 .95. 2 38 o 2 2% ml 37 .UOON um mem:_ou cw mcmcxumeoq com on mo mucmucmamc pgmwmz Lm—aum_osu-.uum mczmwd nop mop ‘41.!‘1- ‘1 ‘ m op _ .. - G ‘ 3: .psmwmz cmfizow_oe momem>m acmwmz Aopum cowumscmv < co ..... mace; sow: cowumFmL .mowcvaEmqumm ..... $35 .2 2 59.2.8: 2 380 i1 v A LLLLI o— w op op iopxm uo—xm o ‘u011nlip allULJUl 19 1ua19111903 UOISOJJIO O 38 .UOmN um mcmNcmn cw mcmczum>_oa Lee co co mocmocmamu unm_wz LmF=um_oz--.m-m mcamwm op -‘ ‘ q ‘ m Ao_-m comumacmv < we once; mcu :pwz covumpmc pmuwcrasmuwsmm :35 .2 2 .25 .6 38 o op .1 1 d I 4 J G 1 m ‘ I1 I ‘ 3 z .pzmwmz cmpzumpoe momcm>m pcmwmz cop 1-. 1 'ALL‘I wuo—xm o_ o ‘u011nlip a11U14U1 19 1ua11111903 UOISOJJIO 39 number of experimental and theoretical investigations. However, there still exist striking differences between the various results with regard to the concentration region within which equation 3-10A can be approximated by the first two terms (BB-l, 3A-15, 3A-7). Recently Duda, et al. (3B-2) showed that the coefficient kd can be determined from the following equation k = 2 A M - kS - b - 2 V (3-11) d 2 1 20 Here A2 is the thermodynamic second virial coefficient, M is the molecular weight of the polymer and V20 is the partial specific volume of the polymer in the limit of zero polymer concentration. The quantity, b], is defined by the series expansion v1 = v10 [1 + 61 c + ---] where V] is the partial specific volume of the solvent, and VlO is its value at infinite dilution. Similarly the quantity kS is defined by the series expansion f12 = (f12)o [1 + k5 c + ---] (3-11A) where f12 is the friction coefficient defined by the following equa- tion: Force on a polymer molecule = f12 (u2 - u1) Here, u1 and u2 are the velocity of solvent and polymer respectively, with respect to a convenient reference frame. The parameter (f12)0 40 is the value of the friction coefficient, f12’ in the limit of zero polymer concentration, which in theory is the same as fo defined in equation 3-1. Prediction of diffusion coefficients for polymer-solvent systems in dilute solutions of polymer depend on both 00 and kd. In the previous section, theories for prediction of 00 were given; the objective of this section is to devise a method for the prediction of kd. The quantities b] and V20 in equation 3-11 can be determined experimentally. Predictions of A2 are obtained utilizing the two parameter theory of dilute polymer solution thermodynamics. Duda, et al. (BB-3) modified the results of Pyun and Fixman (3B-4) to yield a method for predicting ks. The rest of this section consists of a description of two parameter theory and a description of modifications to the Pyun and Fixman equations for predicting ks' Finally the experimental values of kd available in literature are compared with theoretical predictions. Two Parameter Theory A commonly accepted thermodynamic theory of polymer solutions actually consists of the results of a group of early theoretical papers which are now collectively referred to as the two parameter theory. Within the framework of the two parameter theory, the properties of dilute polymer solutions such as average molecular dimensions, second virial coefficients, etc., may be expressed in terms of two basic parameters. One is the mean square end-to-end distance of a chain in the theta state, and the other is the 41 excluded volume parameter, which is usually designated by z. The excluded volume, and hence the parameter 2, vanish at the theta condition. This is, indeed, the definition of the theta state. Therefore the heart of the two parameter theory are the inter- relations between dilute solution properties and the two parameters and z. A fundamental difficulty that arises in the two para- meter theory is that z is not directly observable by experimental techniques. It is therefore impossible to make an explicit compari- son of theory with experiment. This difficulty is circumvented by the technique discussed later in this section. An approximate relationship for A2 is described by Yamakawa (3B-5b), and can be expressed by the following equations: N B h (2) A2 = ° 2 ° (3-12) 2=§ (an) 1/2 2 = (3/2n)3/2 L343. (3-14) A The parameter B is a measure of the effective volume excluded to one segment by the presence of another and is related to the binary cluster integral 8, for a pair of segments. _ 2 B - B/MS MS is the molecular weight of a segment. 8 is defined by the 42 equation s=([1-mm1m where g(R) is the pair correlation function between segments with R the separation at infinite dilution. The parameter 8 represents the molecular interactions between segments, and can be obtained once the intermolecular potential is known. The function g(R) is a compli- cated function of R and is discussed in detail by Yamakawa (BB-5b). The two parameter theory requires that a and ho be functions of 2 only. The function hO arises from intermolecular interactions. Duda, et a1. (BB-3) suggest the use of Yamakawa-Tanaka (BB-6) expression for a and the Kurata-Yamakawa (SB-7, 3B-8) expression for ho: 0.46 0.541 + 0.459 (1 + 6.04 2) (3-16) Q 11 0.547 [1 - (l + 3.903 21'9'46831 (3-17) me1= Z The above theoretical expressions are obtained by series expansion, and neglect higher order terms. The theory does predict that A2 = O and a = l at the theta condition, where B = O. The second virial coefficient, A2, increases with B, the greater the solvent power, the larger the second virial coefficient. The coefficient A2 decreases with increasing molecular 43 weight. By knowing B, and M, the second virial coefficient A2 can be predicted using equations 3-12 to 3-17. Modified Pyun and Fixman Theory Pyun and Fixman (38-4) have calculated kS in the expression for the frictional coefficient f = f0 (1 + kS c + ----) where f0 is the frictional coefficient at infinite dilution, and c is the concentration of polymer in the solution. For the sake of clarity the same notation as used by Pyun and Fixman is also used here. They chose the following procedure for calculating the frictional coefficient: (1) They assumed in their model that any solvent inside the spherical polymer domain is trapped there and will be considered part of the sphere for the purpose of calculating the mean velocities, (2) They chose a particular reference point in the solution, (3) They computed the velocity of the sphere and the solvent at that point for a given configuration, (4) These quanti- ties are then averaged over all possible configurations. Pyun and Fixman define the friction coefficient E as Force on the polymer molecule = 5 (vS - vf) vS is the average velocity of the spherical polymer cloud including the trapped solvent, and vf is the average velocity of the untrapped solvent. The analysis of Pyun and Fixman yields the expressions 44 f = a (1 + 9v) (3-19) -§1 = 1 + [7.16 - k(A)] 1v + - - (3-20) 0 1 2 1/2 k(A) = 24 2 ln[l + x + (22x];2x ) jg] x2 exp (2 x + x ) 0 (3-21) [-A](l - x2) (2 + x)] dx A1 = 3 n2 XS/8 n a: (3—22) where ¢v is the volume fraction of spheres, n is the number of seg- ments per molecule, XS is the second virial coefficient for segment- segment interactions, and a5 is the radius of a sphere composed of solute and trapped solvent. ‘ Duda, et al. (3B-3) modified the results of Pyun and Fixman based on the assumption that the radius aS in equation 3-22 and hence the quantity A], depend on polymer concentration. They first obtained a relation between E of Pyun and Fixman and f12 of equation 3-llA to facilitate the utilization of the Pyun and Fixman theory. Secondly they wrote series expansions for as, A], and k(A) in terms of increasing powers of c, the polymer mass concentration, and sub- stituted these expressions into 3-20, to arrive at ks = [7.16 - k(A*)] 4nag NO/3M - v -b (3-23) 20 1 45 21n[l + x +(2x + x2)1/2;l k(A*) = 24 (2x + ;2)1/2 X exp J (3-24) 0 2 [-A(l - x ) (2 + x)]dx _ 4096 2 A* " 72 TI a (3'25) , V6FM' A a where A, z, and a are defined in equations 3-3, 3-14 and 3-16. Therefore kS can be predicted using equations 3-23 to 3-26. It is clear that if A, B, V20 and b1 are known or can be estimated for a particular polymer-solvent pair, then equations 3-12 to 3-17 and 3-23 to 3-26 can be used to predict kd as a function of molecular weight of the polymer for that particular polymer-solvent pair. Comparison of Predicted Values of kd with Experimental Data A FORTRAN program was written using equations 3-12 to 3-17 and 3-23 to 3-26. The inputs to the program were polymer molecule weight, M, the parameter A, and the excluded volume parameter B. Since 8 is not known precisely for any polymer solvent pair, a range of values of B were used. Using the program the values of kd were generated as a function of molecular weight for polystyrene. The results of the computer program are displayed in Figures 3-9 through 46 3-11. All these figures contain theoretical predictions of kd for polystyrene as function of polymer molecular weight and the excluded volume parameter, 8. Figure 3-9 illustrates molecular weight dependence of kd for the values of B‘in the range of 0 to 1 x 10'27cm3, and polymer molecular weight in the range of 2 x 104 to 106. Figure 3-10 is similar to Figure 3-9, except the molecular weight range is from 106 to 7 x 106. Figure 3-11 is similar to Figure 3-9, with the values of B in the range of l x 10'27 to -27cm3. 10 x 10 The value of A used was 700 x 10'11cm. (BA-9), b = 0, and V20 = 0.9cm3/gram. The value of A in the literature '1 to 755 x 10'H varies from 645 x 10' cm, so the mid value in the range was chosen. The above value of V20 is for pure polystyrene. Since D1 is not known for the polymer-solvent pairs used in this work, it was assumed to be zero. The choice of V20 and b1 are not critical until the value of kd is less than 10 cm3/gram. Even at this low value of kd, V20 and b1 contribute only about 10% of the total value of kd. From Figures 3-9 to 3-11, it can be concluded that the value for kd under theta conditions is always negative (the curve cor- responding to B = O), and decreases significantly as molecular weight is increased. As the polymer-solvent systems move away from the theta condition (increasing values of B), kd has negative values for low molecular weights, but becomes positive as the molecular weight is increased. For diffusion in good solvents (large values of B), kd is always positive for all molecular weights in the range of 104 to 107. Therefore, for diffusion in dilute polymer solutions, 0 47 ._V c_xmvo new eo_vxzteo. B so» 32. 328.233 5.: 53%:qu .3» 233—3.... 333.89: no 23235316-.” v.53... 25 00—. mo— 1 1 d n u a d V cm.- .idl 11 q 1 q q q d d o u S x m 3: .2393 2223.8. unto: 2.3m: R . .6 4 4 mm- Ill-[i o C H”!!! . 36 11. o ex mo.o Av . mm 4 mud . S o; .m ea .82ch we as. ca 23368... :ufiwcwwuw a .2 2 . o a 9.3m. 25¢ 358. E ocucxumxpoa .8» 375; ”—o 2. .36... a. .w MW we 2.3m“ ”Paw :22... 5 2833—3 so» 37% pa 2. .95. mo 38“ o .Uomn an mcoxuco—oao :. ocmgxumzpoa Lou 5~1m u:a.m3 o.. u opxm KN . . oo_- o.~ o 8— gm can one . com .m be =o*uu==u we vx no co*uu_umga .mu_uwgomgh .UOmN um macaw“ _>;um .agums c. ucmgxgma_cq ‘0‘ Aep-3 1 2 2 ‘ (2 ktZZ/kZZ) l/2 _ 2 1/2 A1 ‘ (2 ktll/kll) ¢k = ktlZ/2(ktll kt22 )1/2 [I] represents initiator concentration, moles/liter; k tii and k.. are 11 termination and propagation reaction rate constants for monomer i; 56 fkde is the effective initiator decomposition rate constant, and ktl2 contained in ok, is a cross termination rate constant. Values of ¢kl favors cross-termination. Diffusion-controlled termination. A kinetic expression for the rate of diffusion-controlled copolymerization was obtained by North and Atherton (4A-2) by considering the termination reaction as * + * l ”1 M1 . . "t(12) M1 + M2 l ----d> dead polymer (4-ll) + * M2 M2 J where the termination rate constant kt(12) is a function of copolymer composition. Then the rate of copolymerization was found to be R _ (r1 [M]]2 + 2 [M1] [M2] + r2 [M2]2) R1/2 (4 ‘2) klfiz) (r1 [”11/k11 + r2 [”21/k22) where [M1] and [M2] are the concentrations of the two monomers. Synthesis of Copolymers Styrene-acrylonitrile copolymers used in the diffusion mea- surements in this work were synthesized by free radical polymeriza- tion in bulk. To do the synthesis it was necessary to analyze which of the two kinetic mechanisms, equations 4-l0 or 4-12, is useful for 57 predicting the rates of styrene—acrylonitrile copolymerizations. Blanks and Shah (4A-3) showed that neither the kinetic ¢k factor alone, nor the diffusion parameter kt(l2) alone, satisfactorily describe the data for copolymerization of styrene and acrylonitrile. Since the theoretical rate expressions could not be relied upon to determine the time of reaction for required conversion, it was decided to use the kinetic data of Shah (4A-4), which he obtained from small scale experiments. The three copolymers that were synthesized were all of azeotropic composition. This was done in order to ensure copolymers of uniform chemical composition, so that chemical hetrogenity corrections may be neglected in the diffusion measurements. Initiator The initiator used in this work for the synthesis of styrene- acrylonitrile copolymers is o-a'-Azo-Bis-Isobutyronitrile (AIBN). The reasons for using AIBN are: (l) The rate of initiation is independent of monomer composition, because AIBN releases primary radicals that combine efficiently with both monomers; (2) the spontaneous decomposition rate of AIBN is substantially independent of the reaction medium; and (3) unlike benzoyl peroxide, AIBN is not susceptible to induced decomposition. The AIBN,obtained from East- man Kodak Company, was purified by recrystallization from acetone. A large quantity was dissolved in acetone at room temperature till saturation. The solution was filtered, and cooled in an ice water bath until a crop of crystals were obtained. The procedure was 58 repeated twice and the crystals were dried under vacuum at room temperature. The purified AIBN crystals were stored in a refrigera- tor. Monomers Both the monomers used in this work, styrene (ST) and acry- lonitrile (ACN) were of high purity when they were obtained from the manufacturers. Styrene was obtained from Dow Chemical Company and acrylonitrile from Eastman Kodak Company. The containers were stored in a refrigerator and only the approximate amounts needed for each run were withdrawn at one time. The required monomers for an experiment were withdrawn and passed through columns of activated alumina to remove the dissolved inhibitor. The inhibitor-free monomers were used in the polymerization reactions. Polymerizations Each polymerization reaction was carried out in a two-liter, round-bottomed flask at 60°C under nitrogen atmosphere. Cold monomer mixture was heated up to 60°C in the reactor as quickly as possible, and then the initiator AIBN was added. After completion of the reaction, the contents of the flask were poured into chilled methanol in a waring blender to precipitate the polymer. The volume of methanol used for each precipitation was four times the volume of the reaction mixture. The polymers were then redissolved in methyl ethyl ketone, filtered, and reprecipitated in methanol. The polymers were dried to constant weight in a vacuum oven at 30°C, for 59 approximately ten hours. Table 4-l gives the details of bulk polymerization at 60°C using AIBN. Molecular Weights and Molecular Weight Distribution Samples of all the polymers that were synthesized were sent to the analytical laboratories of Dow Chemical Company for determina- tion of molecular weight and molecular weight distribution by Gel Permeation Chromatography (GPC). Table 4-2 contains the GPC results. The GPC results were cross checked against results from viscometry. Fractionation of Copolymers Polymer fractionation experiments were performed for pre- paring copolymers of narrow molecular weight distribution. The method used was fractional precipitation. Fractional precipitation offers the best opportunity for a close approach to equilibrium and, thereby, the greatest efficiency in each step. One of the practical difficulties with the method is the long time required for the settling of the precipitate with the result that about one day is required for the separation of each fraction. The large volumes of solution that must be handled in this method also pose a problem. For efficient fractionation, precipitation must be carried out at low concentration, about one percent for low molecular weight polymer and one tenth of a percent for polymer of one million molecular weight. This means that in fractionation of a 25 gram 60 oe Fmp Nmep.o mooo.o coop m.on o.~m muz_omm.u=: mg» no: emumxm mg» . co.em o—pm.— pocaguoz wcwapo» maze—u no: ma: :o.u compo ac: -apom we» can uo_uuwm mmwcvu:o_u 3.. 332:8: 2: .3 85:82 8.3 5.8 $8; .2555: 2.352 xenopu ——_um we: go.» gnwpu ace lapom use use oopuuom mace—vao—u was wuuu.a.uoea ugh ea mucmgooqg< oo.mm e~.~m soco.— .ccasum: ucw~=om saw—u xcw> gm:.aucou mzu be use om.uwga Eouuoa use a» umpuumm mm: mmmcpuac—u 3.. 33.383 e: t 85:82 2.3 8.8 $8.. .2355: 589.35 mesa; cw so» w—uuom ouaumqpuwga cu vmzop_u mm: u, oumwwnvumua «we.» so coco n=:V.vom= gas» on quvu muou.a.umea use oucawoomuauw u -Lumaan mg» p..u aco>—Om c we”; aeo>pom coz uea>—0m EB .8333 we 3. co 3:226“ .53; £823 do £582 5 a -meu use co mucmesou :c: be acaosc 4.-" 0'... il- D.“ a I" ‘II. II: 'III I- .II'JIAI. 1'10; lIl.|l.1.Ihl|".nlJPlul .mcc_u~=o.uomsw upmum1—pm5m we mu—amom--.m-e uam<~ 64 needed compared to other systems. The amount of solvent used was calculated from the following equation (4C-4) Volume fraction of copolymer in solution = l/XJI/2 (4-l3) Large-Scale Fractionations The aim of the fractionation was to obtain copolymer of low polydispersity. It was decided to obtain only the low molecular weight fraction of SAN-l, the middle fraction of SAN-2, and the high molecular weight fraction of SAN-3. All the large-scale fractionations were performed at 25°C in a 4.5 liter flask. The volume fraction of polymer dissolved in chloroform was calculated using equation 4-l3. This clear solution was filtered and methanol added until a cloudiness appeared. The solution was then warmed to 35°C and allowed to cool to 25°C. The precipiate was .removed and redissolved in chloroform and reprecipitated in chilled methanol in a waring blender. The fraction obtained was dried to constant weight in a vacuum oven at 30°C. Table 4—4 contains the results of the final three fractions for the three copolymers that were fractionated. Small amounts of these fractionations were sent to the analytical laboratory of Dow Chemical Company for determina- tion of molecular weights and molecular weight distributions by Gel Permiation Chromatography. Table 4-4 also contains the Gel Permiation Chromatography results. Comparison of the polydispersity ratios (ratio of Mw/Mn) of the copolymers in Table 4—2, with those of the fractions in 65 mnz .mcowuawom cmexwoa cow acmwp umgmumem wo Escuomam--.F-m mesmwu mxmmn cwzowpwcm E + o» o» S .. o» . . . xmmn cuwmpzmm Illllnlmv 70 the Brillouin peaks can be obtained with a pressure scanning Fabry- Perot interferometer, which is used to study viscoelastic relaxation process in polymers. The center peak,called the Rayleigh peak (refer to Figure 5-l), is often of a Lorentzian functional form with half width proportional to the diffusion coefficient (SA-6). For solutions of polymers the Rayleigh peak contains information about the rates of motions as well as the types of motions of the polymer molecules. Pecora has derived theoretical equations which relate the shape of the spectra to translational diffusion of rods, spheres and gaussian coils; rotational diffusion of rods; and intermolecular motions of flexible coil polymers (SA-7 to 5A-12), to molecular parameters. Experimental techniques have been developed apace with the theoretical work of Pecora. As a result, during the past decade, light beating spectroscopy has developed into a major new method for analyzing optical fields with an effective resolution orders of magnitude greater than was available with traditional spectroscopic techniques. Forrester, et al. (SA-l3) proposed that two beams of light with slightly differing frequencies could be mixed (heterodyned) resulting in a beat note which could be detected in a nonlinear detector. This concept was accomplished experimentally with the aid of lasers. Since lasers have an extremely narrow line width of a few Hz or less, it is possible to detect frequency shift as small as l0 Hz. This high level of sensitivity makes possible the study of thermodynamic properties and transport coefficients that constantly fluctuate about mean values. 71 The first experimental application of the principle of light beating spectroscopy to study polymer solutions was made by Cummins, et al. (SA-l4). They developed an optical heterodyne technique shown in Figure 5-2. The scattered beam from the solution and the reflected beam from the laser follow parallel paths to the surface of a photo- multiplier tube. The photomultiplier tube observes the beating of the scattered light with the reflected laser light. Later the optical self beat method,shown in Figure 5-2, was developed (SA-l5). The scattered light at the photodetector has a frequency distribution. The components of this spectrum beat with each other causing fluctuations in the output current of the photo- multiplier tube which are analyzed by a spectrum analyzer. The optical self beat spectrometer is superior to the heterodyne system, in that it is much simpler from the experimental point of view, the half width is twice as large as that of the optical heterodyne method resulting in improved accuracy, and it does not detect any uniform motion of the solution (i.e. convection does not affect the measurement). The optical self beat method was used in this work. A survey of the literature shows that the technique of light beating spectroscopy has been used to obtain the spectrum of scattered light from many types of polymer solutions and solutions containing biologically interesting molecules, with components whose molecular weight ranges from l04 to 108. The theory and experimental aspects of light beating spectroscopy are thoroughly discussed in the literature; for example, reference may be made to recent review articles (SA-l6, 5A-l7, 5A-l8, 5A-l9, SA-ZD), several of which 72 A. Optical heterodyne method: \ffi B. Optical homodyne method: Figure 5-2.--Differences in heterodyne and homodyne methods. L - laser light source HM - half silvered mirror M - full reflection mirror C - sample cell PM - photomultiplier tube SA - spectrum analyzer e - scattering angle 73 contain extensive bibliographies. Other recent work of interest in polymer solutions are studies of polystyrene in the following solvents: cyclohexane (3A-7, 5A-Zl, 5A-22), methyl ethyl ketone (3A—l4, 3A-l5, 5A-25), tetrahydrofuran (33-10, 33-11) and benzene (SA-27). Theory Since the technique of light beating spectroscopy has been thoroughly discussed in the literature, only the relevant portion of the theory will be discussed here. In order to measure the spectrum of scattered light centered at the incident laser frequency l4 of about 5.83 x 10 Hz (5l45 °A), and to obtain a measurement of the half width of the spectrum, in the order of 50 to l0,000 Hz, a 10 to 101] resolution of about 10 Hz is required. This very high resolution is achieved by optical beating leading to the name "Light Beating Spectroscopy." Beat frequency.-—For the sake of clear understanding of what a beat frequency is, let us consider a simple example that illus- trates the self beating technique. Suppose that the light incident on the photocathode surface of a photomultiplier tube contains only two component waves with only two discrete frequencies, w1 and "2' The electric field of this light spectrum is represented by E(t) = A Cos wlt + B Cos wzt (5-l) where A and B are constants and t is time. The light beating 74 technique employs a unique property of the photomultiplier tube, namely that the current output of the tube is proportional to the square of the incident electric field (or power) of the light striking the photocathode. Therefore i(t) = c E2(t) (5-2) where C is a constant. Substitution of equation 5-l into equation 5-2 gives i(t) = c [A2(l + Cos 2w1t)/2 + 32(1 + Cos 2w2t)/ (5-3) 2 + AB Cos(w1 + w2)t + AB Cos(w1 - w9)t] ‘— The photomultiplier tube does not have an unlimited frequency response and the highest frequency that it can follow is limited to approximately one kilomegahertz. Therefore the first three terms in equation 5-3 result in a D.C. electrical component. The fourth term, however, is a low frequency component and the frequency difference is referred to as the "beat frequency." It is this component which is measured and resolved by light beating spectroscopy. Polymer solutions contain many molecules in the scattering volume, and the spectrum of the scattered light is a continuous spectrum, E(w), and more complicated than the discrete two component example discussed above. If the scattered field incident on the photomultiplier tube is not a discrete frequency but a spectrum, then the beat signal from the tube will not be a single discrete 75 frequency, but it too will exhibit a spectrum. The relation between the beat signal spectrum in the photomultiplier tube current, i(w), and the incident field E(w) is given by the convolution integral cc i(w) = C E(A) E(A - w) d) '(I where C is a constant. It was shown (SA-28) that if the power spectrum of the light scattered from a source is Lorentzian, centered at w = wo, and with a half width F, the self-beat power spectrum of the photocurrent from a photomultiplier tube detector is also Lorentzian, but with its center frequency at w = O, and with a half width of 2?. Theory of Brownian motion.--It has been well established that the spectral distribution of scattered light yields information about the Brownian motion of the molecules in a solution responsible for scattering (SA-6). The motion of a Brownian particle in solu- tion will appear to be irregular and random. The force exerted on such a Brownian particle consists of two parts. The first is the frictional force due to the drag exerted on the particle by the fluid. In this case, if u is the velocity of the particle, then this force is given by yu, where v is the friction constant. The second part of the force is the fluctuating force, A'(t), represent- ing the constant molecular bombardment exerted on a particle by the surrounding fluid. It is assumed that A'(t) varies extremely 76 rapidly compared to the variations in u. The equation of motion for such a Brownian particle is __ = — Cu-I- A(t) , (5'30) where c = v/m and A(t) = A'(t)/m. The differential equation 5-30 is called a stochastic differential equation because A(t) is a randomly varying function. The solution to the above differential equation can be obtained by finding the probability that the particle has velocity u at time t, given that u = uoat t = o. This is given by the probability density function N(u,‘t;uo). The probability density function w (r, t; r0, ”0) written in terms of the displacement of the particle,r3 instead of the velocity u has some important properties. For such a Brownian particle it has been shown (SA-35) that the mean-square displacement of the particle, for large times, 15 6 k T m C < lr- r0|2> = t = 6Dt (5-31) where-%% = D, the translational diffusion coefficient. Using the above expression for the mean square displacement of the particle, it was shown, (SA-35) for large times, that lr-rlz O . z I w (Y's t9 Y'Os U0) m7 EXP { - T} (5'32) 77 This is the well known solution to the diffusion equation = szc (5-33) (DO) (*0 which becomes 6(r - r0) as t + o. The solution'UJequation 5-33 with the initial condition that C(r, o) = C0 6(r) is ( ) C° { r2 } ( ) C r, t = -—-—————— exp - -——- 5-34 8(nDt)3/2 4Dt Therefore, the probability density function W(r, t; r , uo), which 0 is a solution to the equation of motion for a Brownian particle, is also the solution to the diffusion equation. In a dilute polymer solution, the macromolecule is constantly bombarded by the solvent molecules, which leads to the translation of the macromolecule. The probability P(r, t) of finding a molecule at position r at time t, if it is at the origin at time zero is given by the diffusion equation m : Dv2P(r’ t) at where D is the translational diffusion coefficient of the macro- molecule. The light wave monitors the translation of the molecule in the solution through the molecular polarizability which trans- lates with the molecule. Application to polymer solutions.--Consider a polymer solu- tion composed of identical, isotropic, polymer segments in a solvent. 78 The density of the segments in a particular volume under observation fluctuates with time and hence scatters light. Fluctuations in the density of the solvent itself will be ignored. Let c be the excess dielectric constant of polymer solution, the dielectric constant of the solution minus that of the pure solvent. The excess dielectric constant fluctuations Be, in turn, correspond to a high degree of approximation to local segment density fluctuation, or to concentra- tion fluctuations of polymer segments, Go. Local concentration is a function of time t, and of position within the scattering medium, r. Therefore we may write 68 5—6 OCU‘, t) (5-4) 68(r, t) = Based on the theory of Brownian motion, we assume that the micro- scopic concentration fluctuations obey, on the average, the macro- scopic translational diffusion equation, Fick's Law (SA-36, 5A-37, 5A-38) m = ovzadr. t) (5-51 dt where D is the translational diffusion coefficient. Since the concentration fluctuations cannot be observed with the naked eye, the fluctuations are observed by observing the scattered light. The scattered light from the polymer solutions contains two major components. One component arises from the inci- dent light on the polymer solution, represented by [exp(-iwot)], where w0 is the frequency of incident light. The second component 79 arises from the concentration fluctuations represented as 5c(r,t). Therefore the scattered light field E(t) can be represented as E(t) is proportional to 6c(r,t).[exp(-iwot)] (5-6) The detailed expression for E(t) may be found in other works (SA-18, 5A-19). The scattered electric field is analyzed using a photo- multiplier tube. The spectral composition of the photoelectric current from the photomultiplier is obtained after substituting the value of E(t) from equation 5-6 into equation 5-2 and then taking the Fourier transform of the equation 5-2. Since E(t) is a function of both r and t, the Fourier transformation is done in two steps. The first step would be transforming the r dependence to the Fourier spatial form, and the second step would be transforming the time domain into the frequency domain. In E(t) only 6c(r,t) is dependent on r. Let us transform this into the Fourier space domain. Since in Fourier space domain, all the positions are treated as vectors, let us define a scattering vector K as where k0 is the wave vector for the incident light and kS represents the vector for the scattered light. The next step is to relate K to the scattering angle, 9, and the wavelength of incident light A0. The wavelength of the scattered field is related to the wavelength 80 of the incident field by the Bragg Law (SA-l9) A 7%- = 2 x sin(B/2) (5-7) f where n is the refractive index of the scattering medium. The relation between the scattering vector K and the wavelength of the scattered light Af, is given by from equation (5-7) 4n sin(e/2) K = -(———-)-)\O/n (5-7A) Now equation 5-5 in Fourier space domain is written as “SC K t = ov2 6c(K,t) (5-8) dt From equation 5-7 we know that, by fixing the scattering angle a, and the wavelength of the incident light A0, we fix the spatial Fourier component from which scattering is being observed. By solving equation 5-8, at t=0, it can be shown that (SA-28) c(K,t) = c(K,O) exp (-KZDt) (5-9) combining equations 5-6 and 5-9 we find that E(t) is proportional to exp(-K20t) exp(-iwot) (5-lO) 8) Now, to transform equation S-lD into the frequency domain, principles of autocorrelation functions are utilized. The spectrum of the scattered light may be related to the autocorrelation function, C(T), of the electric field of light. The autocorrelation function is the time average of the product of the signal, at any time t, with the signal at any time t + T. C(T) = (S-ll) The power spectrum of the scattered light can be obtained from the autocorrelation function of the scattered light by using the Wiener- Khintchine theorem (tA-lB) o: P(w) = g; C(T) eth d1 (5-12) -0: P(w) is the power spectrum of the scattered light. The autocor- relation function of the scattered field is generally expressed in terms of the correlation function as Nfl= = total intensity of the scattered light, and g(])(t) is the correlation function of the scattered field. The correlation function of the scattered field is simply an expression that characterizes the optical field incident upon the photomultiplier tube surface. For dilute polymer solutions it has 82 been shown that 9(1) (T) is of the form (SA-l8) 9(1) (T) = exp (-iwot) exp (-DK2T) (5-l4) <1)( T) is obtained from the proportionality shown (2’ (T). This form for g in equation 5-lO. The photocurrent correlation function, 9 corresponds to the photocurrent power spectrum which results from the response of the photomultiplier tube to the incident scattered light field. The correlation function of the photocurrent 9(2) (T), is related to the correlation function of the scattered field by (SA-l8) 9(2) (T) = l + 9(1) (T) l 2 . (5-15) Using equations S-ll to 5-l5, and performing the integration, the photocurrent power spectrum associated with the scattered field is 1 given as (SA-18) 2 ZDKZ/n w2 + (201(2)2 e 2 2n (5-l6) + P(w) = 5(w) + The photocurrent consists of three components. The first term in the above equation §7§%3-is the shot noise term. Shot noise is the outcome of the random time behavior of the anode pulses as a result of incident radiation on the photomultiplier tube. The shot noise level can be determined by examining the spectrum at high frequencies, beyond the range in which the beat signal is significant. The second term in equation 5-l6 is the D.C. component. The third term is a 83 Lorentzian of half width Awl/Z’ and centered at w = 0. Thus, measurement of the photocurrent spectrum from w = O to l0 x Aw”2 permits accurate determination of the half width of the optical spectrum. If half width is measured in Hertz (cycles per second) 2 Aw = 2 K D (5-17) l/2 Using 5-l7 and 5-7A one obtains 2 Aw1/2 (AC/n) 2 (5-l8) l6 sin (6/2) The spectral half width is proportional to the square of sin (9/2). The analysis developed till now in this section holds only for noninteracting systems of monodisperse macromolecules, which are small compared to AD, the incident light wavelength. For poly- disperse polymers g(]) (T) of equation 5-14 consists of a sum or distribution of single exponentials o: ) 9(1) (I) l = [ G (T) e'FT dr (5-19) 0 where T = 2 D K2. The distribution function of decay rates, G (F) may be a broad continuous distribution. G (P) dP is the fraction of the total intensity scattered, on the average by molecules for which F = 0K2, within dP. In studying polydisperse systems, one must 84 adopt a procedure of data analysis that recognizes this aspect. The procedure used for data analysis in this work is covered in detail in the data analysis section. Experimental Apparatus In this work diffusion coefficients in the dilute solution range were obtained using a laser homodyne spectrometer. A diagram of this spectrometer is shown in Figure 5-3. It consists of a laser light source, the scattering cell, light collecting optics, photomultiplier tube, spectrum analyzer and averager, an X - Y recorder, oscilloscope, and a computer along with its peripherals. The laser was a Spectra Physics model 165 argon ion, operating on a single mode at 5l45°A. It had also a polarizer which permitted only plane polarized light to pass through. The light beam from the laser was reflected from its path by a mirror and directed through the center of a cylindrical sample cell. The light beam was focused into the cell by using an appropriate lens. The sample cell was situated on a rotating table which was used to select the desired scattering angle. The incident laser light beam could be redirected through the center of the sample cell at any scattering angle by rotation and translation of the reflecting mirror on its moveable mount. Scattering angles from O to 180° were possible with this arrangement. A Spectra Physics model l32 He-Ne laser was used for align- ing the optics and the light collecting system. The light collection optics are shown in Figure 5-4. The light scattered from the sample .cmpmsocuumam mcwummaugmw411.m-m mczmwm eaape_ep ceuzaeoe mewwnwuwmep 85 388: .88— . _ .8388. “MWMHMDM “mummy.“ cmwww BET, .5 _,m0\ - meww1mwxm weowuqo mpqsmm muse cmw— wuwas — . mowuao - 111 mwnwu opoca :owumuoc\ 111 . . . ‘ »_aa=m . Luzon cmmmp . mec moweao ucmuwocw 111 . 111 . .IIIIIIIuIIIIIIINTI .l mem_. Loccws - 86 .muwwao cowpom__oo acmw411.¢1m mesmwm wees we was» cew_awp_320eo;a 2a e_o;ewa we mzoppwn mcmEmu Nu .wu mmczucmam mpnewcm> N< .P< maxee Ezewe=P< mm .Nm .Fm F_ee weasem um . _m - , mm , , mm F< N< we _ _ _ _ws mm l.l10| Ilil. .IIAT.I.II.II.II.I.II ...l...ll.|l...ll.|l.ll.l 2m um _ no _ _mu_ A: 87 cell at the desired angle was collected by a series of apertures, lenses and pinholes and focused upon the surface of the photo- multiplier tube. For more details about the collection system and alignment refer to the work of Gyeszly (SA-4) and Stutesman (SA-29). The photomultiplier tube was an EMI model 9558 B. It was placed in a refrigeration chamber to reduce the level of dark cur- rent. The output of the photomultiplier tube was connected to the spectrum analyzer-averager system. The spectrum analyzer was a Federal Scientific model UA - 14A, and the averager was also Federal Scientific Model lOl4. This combination provided "real time" analysis of the scattered light spectrum. The spectrum analyzer is capable of measuring spectra on l2 frequency ranges from 0 - l0 Hz to 0 - 50,000 Hz. It also pro- vides 400 line resolution and a variety of output options. The averager decreases the random noise in the signal by averaging the instantaneous spectra as many times as desired. In this work all spectra were averaged l024 times. The output of the averager was a voltage versus frequency spectrum, which was connected to the oscilloscope for instantaneous display of the full spectrum at all times. The spectrum could also be plotted on a Varian Associates F - 80 X - Y recorder. To make data handling easier, quicker and more accurate, the spectrum averager was interfaced to a PDP B/E mini computer. The details of the interfacing will be described in the next section. 88 Computer Interfacing of the Averager When a device or instrument is electrically connected to a computer so that it provides data to the computer or receives data from it, it is said to be interfaced to the computer. The device or instrument thus interfaced becomes a computer peripheral. Inter- facing an instrument to a computer is accomplished by connection of the data source to the computer input bus. Frequently the form and level of data must be adjusted to suit the output and input require- ments of the computer and peripheral, also data transfer timing information must be provided. These functions are performed by the interfacing circuit. As was mentioned earlier, the spectrum averager was interfaced to a PDP B/E mini computer. The main pur- poses of interfacing the averager are to obtain accurate data and to make data handling and analysis fast and simple. The spectrum averager is used in conjunction with the spectrum analyzer and receives three timing signals from the anal- yzer in addition to the output spectrum. These timing signals are: the averager sweep gate, the circulation pulses, and the averager start trigger. The function of these signals are important for a clear understanding of the operation of the interfacing circuit. The averager sweep gate is high during every spectrum read- out from the analyzer. Each of the 400 frequency elements is timed to the circulation pulses. Within the high interval of the sweep gate, 400 circulation pulses are emitted. The first corresponds to frequency element l, the last corresponds to frequency element 400 89 as shown in Figure 5-5. The pulses which occur during the interval that the sweep gate is low correspond to no frequency elements stored in the averager; they are ignored. The circulation pulses generated during every sweep gate are used in the averager to read out the contents of each memory cell location and to write this information back into memory. The data written back may be modified, such as during an averaging cycle, when new data are added to the contents of the memory, or an erase cycle, when the data re-entered is forced to zero. However, the operation of reading every cell location and writing data back into the same location is uncondi- tional. The averager start trigger is used to permit loading of spectrum data into the memory. It is used only during an averaging cycle. Thereafter, it has no further function. After the averaging has been completed all the averager is doing is reading out the con- tents of each memory cell location and writing them back when ever the sweep gate is high. The two timing signals, the averager sweep gate and the circulation pulses, are used for generating the data transfer timing information between the computer and the averager. The form and level of data was the same both in the computer and the averager (both of them had TTL logic). During the read out cycle the amplitude corresponding to each of the 400 frequency locations was available at the averager outputs in digital form (10 binary bits), when the circulation pulse went high. Since 400 circulation pulses occur in 100 msec., the time between two pulses is around 250 usec. The circulation pulses and one bit of the digital data were observed on a dual beam dual 90 _ n awco. . mFQAU. . mwoxu mcoummcuummm. . umop co mcoumwcuummm. _ . . _ _ - . — . . _ _ . n . a . . . _ . . .Hcmmmwcu pcmum cwumcm><. . . n . . . . _ . . u . u . . n u " mmmpz cow ” _ . . . . _ . 11111.1f111 11 .11 .I_.1 ..11.J ....I_ 1111 111.11 1.1 . . . . . w "mmmwaq :owpmwzucwu Tl. U Q) m E O O P ”mama ammZm Lemmcm>< HI .Emcmeu mewswu uwmmm11.m1m «gamma 0; 91 trace oscilloscope, to find out the exact time of the availability of digital data at the output of the averager. The results of this observation are shown in Figure 5-6. The circulation pulse was high for l2 psec. and the digital data was available around 24 psec. before the circulation pulse went high and around 64 pSEC. after the circulation pulse went low. Thus, we had around 60 usec. to transfer the data once the circulation pulse was high. The connections to the computer which are used for programmed data transfer are shown in Figure 5-7. All data are transferred into or out of the accumulator as lZ-bit words during an input/output (here after written as I/O) instruction. The bit assignments of an I/O instruction word are shown in Figure 5-8. Bits 0-2 must be octal 6, the operation code for an I/O transfer. The operation decoder, upon detecting a 6 enables the IOP generator, which gener- ates pulses, to be used to synchronize the input or output data with the computer cycle. Three IOP pulses can be generated, designated IOP l, IOP 2, and IOP 4 in that sequence. Bits 9-ll of the instruction word control which of the IOP pulses will be gen- erated. The middle six bits of the instruction word are used to identify the external device which is to provide or accept the data. During an IOP cycle, the accumulator input connections AC O-ll are active so that the data connected to them at that time will appear in the accumulator. A few connections to the operation controller of the computer are also available and are very useful. These are: the skip line (SKP) which is active during an IOP and which can be used to cause the computer to skip the next instruction in the 92 .mumu wmuwmwc wo awn xcmaa wco can mmpza cowumwaucwo cow Emcmmwu mcweww11.c1m mcamwm meme Peewmwe we pwn acmcwn mco » , J .2 _. (ILI - empaa cowumpsucwo oom— uwm : ow. : om com 1 NF _ Alflni. -_. u, . _ E _ . . 1111081031. 93 IACI I I Accuvuunon _ | I I o 11 — BAC 0-11 I MEMORY x BUFFER (MB) 0.11 _ OMB 0111 REGISTER I T 09111111011 I o 3 ) oscoosa I ‘ no I MEMORY I l —: IOP l IOP 9 II GEN I ; IOP 2 1 I ; 10p 4 $111? 1111: I , SW1 CLEAR 11c # ETA 1111111110191 4 fi OPERATION INITIALIZE I CONTROLLER A "m nun — A nun 11111115 51511111 A BIS 1 111111115 51511111 A 81$ 3 COMPUTER <— —> wonw Figure 5-7.--Computer connections for programmed data transfer. 0 11 1 I 1] o11101110111011/011/011/0 1/o]1/o 1/0 I OPERATION DEVICE IOP CDDE=6 ADDRESS GENERATOR CONTROL Figure 5-8.-—I/0 instruction word. 94 program; the clear accumulator line (CLA) which is convenient for clearing the accumulator before new information is read into it. In order to facilitate the connections of devices with TTL logic level inputs and outputs to the computer I/O lines, an interface buffer box is used. This unit protects the computer I/0 lines from damage from erroneous connections, and provides I/0 line drivers and buffers so that ordinary TTL circuits can be used for data inputs and outputs. An I/O patch card is used to bring the I/O connections from the buffer box into the Analog Digital Designer (A00) for convenience in building the interface circuit (see Figure 5-9). The interface circuit was built using the following cards: I/0 patch card, gated driver card, dual flag card, octal decoder card and a NAND gate card. All these cards consist of 32 pins, and they sit in the sockets provided in the ADD.‘ For details of these cards refer to Appendix G. The arrows pointing towards the pins on the cards indicate the signal is an input to the card. The arrows pointing out of the pin indicate that the signals may be obtained out of the card. All of the input and output connections of the cards are brought to the top of the card for patch wiring. All the connections between the cards in the interfacing circuit are shown in Figure 5-lD. For the sake of clarity the patch wiring connections are not shown completely, but only the connec- tions on each card are shown. The 10 bit digital data from the averager is connected to data inputs (pins 3-l2) on the gated driver card. The pins 1 and 2 are grounded, because they are not being 95 cmmmcm>m Echuwam V .mesasou ecu op cmmmcm>m as» Eocw mean we zopu11.m1m mesmww Aooqv pwzocwo momwcmpcw xon cewwzn muwwcmucw gene: cmuzasoo m\w mom 96 co>wcu uwuom 33.83 W Eocw mama w vcaocm 1'1 :2. N ma—w mxm mu 0: .210: 110.: oz sauna o\_ .uwaucwu acwuewcmu:_11.op1m mcamww nu~n 39 O _ m 3.0 Con cmuouoc .ouuo 7. all. cowuapaugwo om_=a «wow .A. 325 . comocm>n .80? .3 1w... RAOQ _ion :2 On w C~ .uvnu_ nu s r11111111 nu . , acme uguu mmpw post muem tea: 97 used. The data outputs of the gated driver card (pins l3-24) are connected to accumulator inputs (pins l3-24) on the I/O patch card. Device addresses of 44 and 45 are used in this work. The octal decoder card was wired to give device addresses 44 and 45. These addresses along with the IOP pulses were used as the timing informa- tion for the computer. In order for the computer to know that the averager is ready to transmit data, a dual flag card is used. Flag 1 is used for averager sweep gate and flag 2 for circulation pulses. Since the flag is set only on the falling edge of the signal, the averager sweep gate was inverted using a NAND gate, and this signal was connected to pin 2 on the dual flag card. Device address 44 and IOP l were used to check when the sweep gate went HI. To clear flag l device address 45 and IOP 2 were used. Since the digital data from the averager was available till 60 u sec after the circulation pulse went L0, the circulation pulse signal was directly connected to pin l6 on the dual flag card. Device address of 44 and IOP 2 and IOP 4 were used to clear the flag 2 after it was set. The operation of the interface circuit is easily understood by following the computer program used for data transfer which is attached in Appendix H. The program is a combination of FORTRAN and SABER languages. The letter S in column one indicates it is a SABER statement. The software of the PDP 8/E is set up such that if a variable is defined in the common statement, the storage locations for that variable are in ascending order in field l beginning at location 200. The first 7 steps in the program are 98 directed towards achieving this object of writing this data at field 1 and location 200, so that it can be retrieved very easily later. After the averaging function was completed in an experi- mental run, the program was executed. The program first clears the sweep gate flag (flag 1) by using the instruction 6452. After clearing flag I it enters a DO loop where it waits until the sweep gate goes HI. This is done in order that the sweep gate is at the beginning of its cycle. 50 when the sweep gate goes HI, the SKP line from flag 1 that is connected to the SKP on I/O patch card goes L0 or is grounded momentarily. This L0 on the SKP on I/O patch card produces a jump to the next instruction in the program. The next instruction 6452 clears the flag 1. In any of the above instructions there is no data transfer because the signals that control data transfer to the computer are device address 44 and IOP 2. This combination has not been used till now, in either clearing or checking flag l. Once the sweep gate goes HI, the program next clears the circulation pulse flag (flag 2) with the instruction 6444. Then it checks if the circulation pulse 1 is HI, by instruction 6451 in a DO loop. Once the circulation pulse l goes HI, it produces a momentary ground on SKP line connected to the I/O card at pin 27. This produces a jump to the next instruction in the program which is 6442; this produces HI logic levels at pins 27 and 28 on the gated driver card, which controls data transfer. The instruction 6442 is used both to transfer data into the computer and also to clear the flag of the circulation pulses. The program next transfers data from the accumulator to the prescribed storage 99 location. Then it enters the DO loop to check for the second cir- culation pulse. This process is repeated until all 400 circulation pulses are found. After reading all 400 locations the program asks for information on shot noise, the scattering angle, and the experi- mental number. Then it writes all this information on to the Floppy disk, under file name read in, which is used later for data analysis. After successful interfacing, several experiments were run whereby half widths obtained from the recorder graphs and the com- puter were compared. The results are shown in Table 5-l. From the table it can be seen that the half widths are very close to each other. After the interfacing the time required for running the experiment and to obtain the half width was reduced from around two hours to around two minutes. Procedure for Experimental Run l. A careful alignment of the optics in the homodyne spectrometer is necessary in order to achieve accurate results. The beam from the sighting laser is used to align pin holes and lenses so that they define a straight optical path. 2. The refrigeration chamber for the photomultiplier tube should be turned on at least twelve hours before the start of the experimental run. This is done to insure that a stable temperature of -lO°C is attained in the chamber. 3. The laser power supply is turned on, and the laser is allowed to lase at the lowest output power, for at least 30 minutes. lOO _.m~ em.mo oooo__ om _.wm wo.om ooom ~m.mmw w.¢¢~ oooop me ¢¢.wmw m.mmp ooom nw.p—N me.¢- coco. om «c.0om mo.mom ooom m~.mmm Pw.omm oooow om wo.m~m um.¢~m ooom N: N: N: sauce memo _muwmwo mmcmm mwmc< soc; Eocm mwmxwmc< Esceeeem we mguewz w_e: .cmuagsou vcm cwucoumc esp Eocw uwzwmuno mgpuwz wpm; wo :omwcmasoU11.w1m m4mee ; e: 3: geez_ee .Azzve\>ee we meweem--.w-m mem< eep mew mew me e“. mew ee_ ee ee_ mew mm” me me_ _ep ee, em Ae~\ew\ev Aee\e~\mv Ae~-m .ee eeemev e_ee< ._eexe eeeez w_e= ._eexm eeeez w_ez seeeew eeeez wee: .cmpez cw mmemgam mcmczumawoa cow was; :owuegnwpeu wo mupzmmm11.m1m u4m0 1 ‘3‘ \8 \ \: m - x=0 \\ 0 .mb ————— b\- \ 1 dx €9“ I - 8 ,_\x<0 \ 'E\ \ u\l. \ Figure 5-16.--Diffusion cell coordinates. Case I (x 2 O) (i) x + a t 3 O c = c1 (11) t = 0 c = c1 a > x > 0 (iii) x = 0 c = (c14-c2)/2 t 2 0 Case II (x 5 0) (i) x + - a t 2 O c = c2 (ii)t=0 C=C2 >x>..oc (iii) x = O c = (c14-c2)/2 t 2 0 where x is defined in Figure 5-16. In order to solve equation 5-27 with the above boundry condi- tions it is necessary to assume that (l) the concentration dependence of the diffusion coefficient 0 is neglibible over the small concen- tration range involved, and (2) the diffusion gradient has the properties of normal distribution curves. These assumptions are valid if c1 and c2 are nearly equal. The solution of equation 5-27 may be obtained with Laplace transforms to give the following identical solution for both Case I and Case II. erf.[ x ] (5-28) where co is the concentration at the zero position in the cell and as a result of assumption (2) above is equal to (c1 + c2)/2, and erf. (z) is the error function of variable 2. The refractive index, n, in the small concentration range used may be assumed to be pro- portional to the concentration, so that -—————2-=1% erf. [ x ] (5-29) The fringe pattern obtained from the interferometer is equivalent to a plot of the refractive index versus distance in the diffusion cell. The distances on the photographic plate are not equal to the distances in the cell because the camera magnifies the image by the factor M, which is the magnification factor of the camera. A repre- sentation of the fringe pattern is shown in Figure 5-17. Because the fringe pattern is a representation of refractive index versus distance in the cell the refractive index difference may be repre- sented by the number of fringes displaced. In traversing from point A to point B on Figure 5-17, the total number of fringes crossed will be the number of fringes displaced because of the dif- ference in refractice index between solutions at point A and B. Each fringe will correspond to a change in refractive index by an amount An. Let J be the total number of fringes from top to bottom; k is the local fringe number in the top half of the cell and j is the local fringe number in the bottom half of the cell. Let xj be the measured distance corresponding to fringes j and k respectively. Therefore, where x > 0 ‘II . k e .J1IV L , , 1 _ fl 127 and equation 5-29 becomes x L = erf" I 2" J" J I (5-30) V4Dt . Similarly, where x < O n - 110 = J _ 2. n - n 2d 2 l and x. . —-11— = erf’] [———-J-J 'J 2 I (5-31) /4Dt It is very difficult to determine the exact midpoint of the dif- fusion zone; however, the distance, xk + xj, is very easily deter— mined by difference measurements. Therefore x. x . 1* + k = erf'1 [ g—ET—J-J + erf"l [ Zk‘; J ] (5-32) 1741515 1/4Dt The measurements taken from the photographic plate are different from the cell distances because of the magnification by the camera. The image is magnified by a factor M. Therefore X- +X X.+X .11_L=_11__.'£ (5-33) /4Dt M/4Dt where xj and xk are distances on the photographic plate. Hence, X. + x k Dt ='_l§ 1 J 2°J 1 2k 0 (5'34) 4M - - g - - erf [ J ]+ EY‘f [TI The value 1 1 l x. + Xk erf-1 [ J 3 21 ] + erf-1 { 2k‘; J ] 1 is obtained for several j's and k's for each exposure and averaged. The averages for several exposures are plotted versus exposure time. The slope of the resulting line is determined and thus, 0 = E1999 (5-35) 4M This diffusion coefficient is assumed equal to the mutual dif- fusivity at the average concentration co. As mentioned previously, M = 1.929, so that the factor 4112 equals 14.884. The distances on the photographic plate were measured with an optical comparator made from Gaertner microscope fitted with a traveling eyepiece. The traveling eyepiece could scan a total distance of five centimeters by turning a crank and the distance traveled was indicated on a vernier scale accurate to 0.0001 centimeters. See Appendix A for details of a sample experimental run, and data analysis. 129 Extension to polymer solutions.--If there is more than one solute, as in polydisperse polymer solutions, the theory developed in the last section is still valid, however the measured diffusion coefficient is an average value representing the average 0 for all the individual species in the solution. In this section a relation between the average diffusion coefficient measured Davg’ and the diffusion coefficient of the individual species, Di’ is determined. Before deriving relations for a multicomponent system, let us take a closer look at the two-component system. If the refractive index, n, of a two-component system is a linear function of the solute concentration over the concentration range encountered in the cell, we may write n = n(co) + R(c-co) (5-36) where n(co) is the refractive index at concentration-co and the constant R, which is the differential refractive index increment, is the change in refractive index corresponding to unit change of solute concentration. Substitution of equation 5-36 into 5-28 yields the refractive index distribution for free diffusion for the case of 0 independent of c, -1 ] e d1 (5_37) 130 Here A n denotes RAc, the total difference in refractive index across the initial boundry. Equation 5-37 is another representation of equation 5-29. An expression for the refractive index gradients is obtained by differentiating equation 5-37 %%-= —é—£—-exp(-x2/4Dt) (5-38) ZJnDt One of the methods of obtaining diffusion coefficients from refrac- tive index gradients is called the Reduced Height-Area ratio, where _ (A 1112 (5-39) avg 4 t (an/3x)2 For the case of a two-component system in which 0 is independent of c and n is linear in c, it may be shown by substitution of equation 5-38 into equation 5-39, that D = D (5-40) When two or more solutes are present in the cell, the situa- tion is sufficiently complicated so that we consider here only the case of a linear dependence of n on the concentration of the N solutes. It has been shown in the literature that for solutions of polymer of sufficiently high molecular weight, the refractive index of the solution is independent of the molecular weight of polymer (SB-ll). This makes the analysis less complicated, therefore we can write 131 N n = "(Clo’ c20, ---, cNo) + X RiICi - cio) (5-41) i=1 where n(c]o, C20, ----, CNo) denotes the refractive index of a solution in which the concentration of the N solutes are Clo ---— CNo’ We also assume that there is no interaction of solute flows and that each Ci is sufficiently small that the concentration dependence of the several diffusion coefficients may be neglected. It has been shown (SB-12) that equation 5-28 can be used for multicomponent systems by replacing c by Ci and D by Di’ where c1 and Di correspond to c and D of the l'Ul species. Substitution of equation 5-41 into equation 5-28, gives x/J4D.t - 9.2 n - ”(C1 '.--9 C ) + 2 -x o. no w. (2M?) e d). (5-42) IIMZ a i where A n is the total difference of refractive index across the starting boundry and w. = -i——i- (5-43) are solute fractions on the basis of refractive index. Since the value of R1 is the same for every solute, wi is the weight fraction of solute i. The refractive index gradient is obtained by differentiat- ing equation 5-42. 132 exp(-x2/4Dit) (5-44) 090) X: II II M2 .1 .1. fl 1 Therefore for systems containing N solutes without interacting flows, and with each Di independent of every concentration and n a linear function of concentration, substitution of equation 5-44 into 5—39 leads to the relation (5-45) l m —l < 4.0 .1. ll M2 ._I fill. Equation 5-45 gives a relation between individual diffusion coef- ficients of the N solutes and the average diffusion coefficient measured by the interferometer. It is frequently convenient to introduce distributions for wi in equation 5-45 and change the sum into an integral. Most of the commerical polymers are generally represented by a Schulz (SB-l3) distribution. The Schulz distribu- tion for weight fraction of polymer of length y is given by - h h Wy - ;;TTF:T7' (DY/X") EXPI-DY/Xn) (5-45) where iflzighfl Mn xn h 133 To solve equation 5-45 with the distribution of 5-46, we need a relation between Dy and My, where My is the molecular weight of polymer chain of length y. The only simple relation available is _ -a Dy - K (My) (5-47) where K and a are constants. This relation is strictly valid for D of polymers only at infinite dilution. However, due to the lack of any relations, it was decided to use 5-47. If equation 5-47 and 5-46 are substituted into equation 5-45, and the integration per- formed we obtain - 1L, - Davg - Mn (5 48) where A [P(h + l)/F(h + o/z + 1)]2 K h A is a constant for a particular distribution. The result of equation 5-48 is very important because it leads to the conclusion that the average diffusion coefficient measured using interferometry depends on Mn’ the number average molecular weight. Since Mn is biased towards the small chains in the polydisperse polymer, the 0 measured by interferometry would also be biased towards the avg small chains. Calibration In this study, the accuracy of the interferometric technique was established by measurement of the concentration-dependent diffusion coefficient for the binary system sucrose-water at 25°C. 134 This particular system was chosen because accurate, widely accepted diffusion data over a concentration range are available for compari- son. The accuracy of the method used in this work was tested by comparing diffusion coefficients at 25°C for five aqueous solutions of sucrose with those reported by Gosting and Morris (58-14). The data of Gosting and Morris have been carefully checked by several investigators (SB-15 to 58-17) and are thought to be accuarate to i 0.2%. Gosting and Morris fit their data to the following empirical relationship using the method of least squares. 6 DS = 5.226(1 — 0.0148 c0) x 10' i 0.002 (5-49) where DS is the binary diffusion coefficient for sucrose-water system 3 of solu- at 25°C and co is the concentration of sucrose in 100 cm tion. The results of comparison are summarized in Table 5-4. The results deviated by only 1% or less from equation 5-49. From Table 5-4 it can be concluded that the precision of the inter- ferometer is no worse than about i 1%. Error Analysis The source of errors in interferometry can be attributed to the following factors: 1. The accuracy in determining the fraction of a fringe when measuring the total fringes on a particular exposure. 2. The assumption that the concentration dependence of D is negligible over the small concentration difference involved in the experimental run. 135 oew.o 1 www.m www.m v.o m omo.o 1 mww.m www.m ¢.F e om~.o + mmp.m wep.m o.~ m omm.o 1 mm—.m mwp.m v.0 N mmw.o + www.m vow.m w.o P cowpew>mo agapeconeg mwgw Av—1mmv wucmcmwmm Eu oop\mmoco=m wo mew .oz cam mmeucmucma umm\mso mop x o .ucmwuwwwwoo cowmzwwwo m mmocuam wo .ucou Peucmewcwqu .LmHmEOLmMLwHCw ms“ :0 mp5.» 20.5.92:me $5. $0 mu. Pamwm11.v1m m1_m<._. 136 3. The magnification factor of the camera. 4. Errors in measurement of distances on the photographic plate. The average diffusion coefficient is calculated by using equation 5-34, and requires that the average of the right-hand side of the equation 5-34 for several exposures when plotted versus exposure time must be linear. From the slope of this line the average dif- fusion coefficient is obtained. The estimated standard error in determining the slope is a quantity that contains some of the above mentioned errors, although perhaps not all of them. Given the unavailability of any other direct method for determining the error, the estimated standard error in the slope is used. From this, the diffusion coefficients obtained here by the interferometric technique are accurate to within 2.5%. CHAPTER VI PRESENTATION OF EXPERIMENTAL DATA Light Beating Spectroscopy The value of the diffusion coefficients measured for all the polymer-solvent systems studied in this work are presented in Appendix D. The concentration dependence of the diffusion coef- ficients for all the polymers in various solvents is illustrated in Figures 6-1 through 6-5. As previously described the diffusion coefficients are accurate to within 3.5%. For each of the polymer- solvent systems, linear least squares extrapolations were performed to obtain the value of the diffusion coefficient at infinite dilution, 00’ and also the slope of the diffusion coefficient- concentration line at low concentration. In performing the linear extrapolations each experimental diffusion coefficient is attributed equal weight. From the slope of the calculated line the value of kd as represented by equation 3-10 is obtained. The resulting values of kd and D0 are displayed in Table 6-1. It is evident from the experimental results in Figures 6-1 through 6-5, that the concentration dependence of 0 may be approximated by a linear relationship over a relatively large concentration range, at least up to about 20 grams per liter, for all the molecular weights used in this work. These results confirm the recent findings on polymer systems, such as polystyrene in butanone (3A-l4) and poly- styrene in tetrahydrofuran (3B-10, 38-11) where the linear dependency 137 138 D x 10 cmz/sec. Concentration of polymer, (g/lOO cm3) 1.0 A _L 4 L L O 0.5 1.0 1.5 2.0 2.5 Figure 6-l.-—Concentration dependence of diffusion coefficients obtained from light beating for PS-l in benzene (O) and decalin (CD at 25°C. 139 7 0x10 cm2/sec. .8 , .6 . .4 r .2 L Concentration of polymer (g/lOO cm3) .0 A 1 AL 1 1 __L O 0.4 0.8 1.2 1.6 2.0 2.4 Figure 6-2.--Concentration dependence of diffusion coefficients obtained from light beating for PS-2 in benzene (o) and decalin (:1) at 25°C. 140 0x107 2 8 _ cmZ/sec. Concentration of polymer (g/100 cm3) 1.0 . - - L 4 0 0.5 1.0 1.5 2.0 2.5 Figure 6-3.--Concentration dependence of diffusion coefficients obtained from light beating for SAN-l in dimethyl . formamide (a), methyl ethyl ketone (9) and benzene (O) at 25°C. 141 1‘2 Concentration of polymer (g/lOO cm3) Ho 0.5 1.0 1.5 2.0 Figure 6.4.--Concentration dependence of diffusion coefficients obtained from light beating for SAN-2 in dimethyl formamide (a), methyl ethyl ketone (D), and benzene (9) at 25°C. 142 D x 107 IcmZ/sec. 2.4 ' l. 1. l. 0.8 . 3 Concentration of polymer (g/lOO cm ) 0./* ‘ ‘ f 4 ‘ 0 0.5 1.0 1.5 2.0 2.5 Figure 6-5.--Concentration dependence of diffusion coefficients obtained from light beating for SAN-3 in benzene (0), methyl ethyl ketone (O), and dimethyl formamide (O) at 25°C. 143 TABLE 6.l.--Values of Do and kd calculated from light beating data. Polymer Solvent 00 x 107 cmZ/sec kd’ cm3/gram PS-l Decalin 1.688 t 0.042 -ll.845 i 2.17 PS-2 Decalin 1.159 1 0.004 8.450 t 0.28 PS-l Benzene 3.505 i 0.011 - 5.960 t 0.29 PS-2 Benzene 1.911 t 0.013 20.840 : 0.61 SAN-l Benzene 2.338 t 0.032 -13.350 i 1.33 SAN-l Dimethyl Formamide 1.342 i 0.023 23.050 1 1.72 SAN-1 Methyl Ethyl Ketone 2.393 t 0.005 4.320 t 0.22 SAN-2 Benzene 2.073 t 0.021 - 1.540 t 1.00 SAN-2 Dimethyl Formamide 1.205 t 0.036 31.400 i 3.63 SAN-2 Methyl Ethyl Ketone 2.312 t 0.049 14.480 i 2.22 SAN-3 Benzene 1.302 t 0.003 9.410 t 1.82 SAN-3 Dimethyl Formamide 0.789 1 0.028 43.41 t 3.49 SAN-3 Methyl Ethyl Ketone 1.759 t 0.053 14.56 i 5.08 144 was also observed. Our data do not yield any evidence suggesting a more complicated behavior with a concentration independent region at very low concentration (3B-l). The relative change of the diffusion coefficient with concentration is represented by the single para- meter kd. The diffusion coefficients of polystyrene polymers in decalin were consistently lower than those recorded in benzene. For the low molecular weight polystyrene the diffusion coefficients decreased with increase in concentration, for high molecular weights the diffusion coefficient increased with increase in concentration. This was observed in both solvents used for polystyrene. For copolymers, the diffusion coefficients in dimethyl formamide always increased with increase in concentration for all the molecular weights. The same phenomena as in dimethyl formamide is also observed in methyl ethyl ketone. The results.in benzene are a classical example of concentration dependence. At low molecular weights the diffusion coefficient decreased with increase in con- centration, at medium molecular weights, the diffusion coefficients are approximately concentration independent, and at high molecular weights diffusion coefficients increased with increasing concentra- tion. Interferometry The average value of the diffusion coefficients measured by interferometry for the polymer-solvent systems studied in this work are presented in Appendix E. The concentration dependence of the 145 diffusion coefficients is illustrated in Figures 6-6 through 6-10. It was mentioned earlier that the diffusion coefficients measured by this method are accurate to within 2.5%. The main purpose of obtaining data by interferometry was to measure diffusion coef- ficients in the intermediate concentration range. We were not very successful in obtaining data at higher concentrations because of flow problems encountered in the diffusion cell at high concentra- tions. For the low molecular weights data were obtained up to a concentration of ten grams of polymer in one hundred cm3 of solu- tion. For the intermediate molecular weights the experiments could be performed only up to a concentration of five grams of polymer in one hundred cm3 of solution, and for high molecular weights only up to two grams of polymer in one hundred cm3 of solution. Diffusion coefficients of monodisperse polystyrenes and mix- tures of these monodisperse samples were obtained over a concentra- tion range to check the validity of equation 5-45. The results of these experimental runs are shown in Figure 6-6. For these polymers diffusion coefficients were obtained at low concentrations in order that linear extrapolations can be performed to zero polymer con- centrations. The data show significant scatter, which might be due to small concentration gradients producing relatively few fringes. It is not advisable to measure diffusion coefficients at such low concentrations with interferometry. Linear extrapolations were performed for all the polymers in Figure 6-6, and the values of kd and 00 obtained from a least-squares fit are displayed in Table 6—2. 146 0x10 cmzlsec. 10.0 F Polymer concentration (g/lDO cm3) 1.0 1 I I L l L O 0.2 0.4 0.6 0.8 1.0 1.2 Figure 6.6.--Concentration dependence of diffusion coefficients obtained from interferometry for PS-3 (a), PS-4 uh), PS-S (o) and PS-6 (a) in benzene at 25°C. 147 3.0 Polymer concentration (g/lOO cm3) A A 1 l 0 2 4 6 8 10 12 Figure 6-7.--Concentration dependence of diffusion coefficients obtained from interferometry for PS-Z in benzene (0) at 25°C. 2.5 148 D x 10 12 r cm2/sec. Polymer concentration (g/100 cm3) 3 I l J 1 l 1 O l 2 3 4 5 6 Figure 6-8.--Concentration dependence of diffusion coefficients obtained from interferometry for SAN-l in dimethyl formamide (a), and methyl ethyl ketone (O) at 25°C. 149 D x 107 11.0 p cm2/sec. /’ /’ /' o ./ 10.0 ’ z’ 1’ 1’ / 9.0 r 1’ 1’ /' / 8.0 b // o / /' / 7.0 r /. 6.0 5.0 Polymer concentration (g/100 cm3) 4.0 _L A 1 l A n 0 1 2 3 4 5 6 Figure 6-9.--Concentration dependence of diffusion coefficients obtained from interferometry for SAN-2 in dimethyl formamide (D), and methyl ethyl ketone (O) at 25°C. 150 D x 107 10.0 —' cm2/sec. 8.0 7.0 I' / 6.0 1- / 5.0 4.0 Polymer concentration (g/100 cm3) 3.0 ‘ 4 4~ t‘ i 0 1.0 2.0 3.0 4.0 5.0 Figure 6-10.-—Concentration dependence of diffusion coefficients obtained from interferometry for SAN-3 in dimethyl formamide (D), and methyl ethyl ketone (O) at 25°C. 151 Nw.m h mmo.om wop.o w wwm.m oco~com m1me m~.mw e om~.mm mmm.o h emm.o mooNcom m1me Pm.mm h oem.mm mmm.o H mm_.m o=o~cmm ¢1mo m_.P— H mmw.wm mmm.o H wmm.m ocoNcom m1mo .Eocm\mao .ox .omm\mso .wow x on pco>—om cmsxwoo .mocoust ewes» oco mmcocxpmapoo omcmomwoocos cow memo owcuoEocowcoucw Eocw omcwopno ox one on wo moopo>11.m1m m4m oo wo mnomwnenEou11.Pom cmexpoa Fmowgwnsmuwsmm agowzu m.:oumc;oo .oo to mmapm> pmuPumgomcu ucm Faucmswgmaxm to comwgmasoouu.m-m m4mFom cmszpom umm: m mo mzpw> ax to m=Fm> ax mo mapm> .vx mo mwapm> Pmuwpmeomgu use Poucmewgmaxm mo comwcmaaoo--.¢-n m4m

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Eu m|$m1Hlm~1 x m mgamoaxm e mgzmoqu V p mgzmoaxm mmegm>< o-~.o o~o~.o m¢m~.o NmmN.o pnum.o mmegm>< momm.o mpvm.o Pmmm.o emvw.o upm~.o mmegm>< mom~.o NmPN.o mopm.o mmpw.o nmmm.o Eu .wa + xxv 213 Exposure \lC35(.J"|--§(Jt)i—'I (x: + XL12 (A + B)2 Slope of the plot of versus time, t is 1.2745 x 10'5 (See Figure A-l for plot.) 0 = Slope = 1.2745 x 10'5 avg 4M2 14.884 = 8.56 x 10'7 cmz/sec l I 2 (xk + xj) cmz/sec. (A + B)2 , cm 0.051826 0.059161 0.066567 0.074726 0.081575 0.086568 2 214 .wpm cog —eo:mewgooxo go» :owuopoopeu11.F1< mgzmmm ooom ooem comp comp com o q q a a 1 -m 32883 as: \\\ \\\ \ \o. \\ LO \\ \\\ \\. \\\ IN \.\\ ‘\\ \ O\\ ._ m \\ \ x m mac x o_ oom o . n \Ne m-og x meew _ oee_m m + < op x 1111143. m + < N xx + .X P: :. u momgo> .11111d1 go pope N . . v_x+.x N . . APPENDIX B MARK—HOUWINK CONSTANTS 215 216 The Mark-Houwink relation is [r1]=KMa Where [n] is the intrinsic viscosity, M is the molecular weight of the polymer and K and a are Mark-Houwink constants. Polymer Solvent K, cm3/gram a Reference PS Benzene 11.3 x 10"3 0.73 AB-l PS MEK 39.0 x 10'3 0.58 AB-2 PS Toluene 10.5 x 10'3 0.73 AB-3 PS Cyclohexane 84.6 x 10"3 0.5 AB-4 SAN DMF 16.2 x 10'3 0.73 7E-1 SAN MEK - 0.68 75-1 SAN Benzene - 0.52 3A-22 PS Decalin - 0.50 3A-22 MEK - for methyl ethyl ketone Here PS - stands for polystyrene; SAN — for azeotropic styrene-acrylonitrile copolymer; DMF - for dimethyl formamide; and APPENDIX D LIGHT BEATING DATA 217 218 Concentration D x 107 Polymer Solvent of Polymer 2 gms/lOO cm3 cm /sec PS-l Decalin 2.003 1.312 1.001 1.454 0.5001 1.551 0.1012 1.716 PS-2 Decalin 1.9987 1.357 0.9994 1.251 0.4997 1.209 0.1050 1.171 PS-2 Benzene 1.9959 2.714 0.9979 2.288 0.4989 2.117 0.0983 1.955 PS-l Benzene 1.9995 3.087 0.9975 3.297 0.0994 3.467 0.0497 3.51 SAN-1 Dimethyl formamide 1.7044 1.857 0.8522 1.622 0.4261 1.495 0.1031 1.348 SAN-1 Methyl ethyl ketone 1.7771 2.575 0.8886 2.485 0.4443 2.447 0.1016 2.394 0.0508 2.401 SAN-1 Benzene 1.9996 1.748 0.9998 1.975 0.4999 2.138 0.1003 2.336 0.0502 2.354 SAN-2 Dimethyl formamide 1.6031 1.774 0.8016 1.589 0.4008 1.344 0.1063 1.252 0.0106 1.173 219 Concentration 7 Polymer Solvent of Polymer D g 10 gms/lOO cm3 cm /sec SAN-2 Methyl ethyl ketone ‘ 1.6664 2.843 0.8332 2.632 0.4166 2.494 0.1050 2.292 SAN-2 Benzene 2.0106 2.016 1.005 2.037 0.0503 2.028 0.1072 2.115 0.0536 2.051 SAN-3 Dimethyl formamide 1.999 1.476 0.9995 1.107 0.4998 1.007 0.4998 1.007 0.1074 0.776 0.0537 0.832 SAN-3 Methyl ethyl ketone 1.0015 1.994 0.5008 1.921 0.1067 1.845 0.0107 1.693 SAN-3 Benzene 2.0028- 1.545 1.0014 1.429 0.1005 1.314 0.0101 1.301 APPENDIX E DATA FROM INTERFEROMETRY 220 221 Concentration D x 10 Polymer Solvent of Polymer in 2 gms/lOO cm3 cm /sec PS-3 Benzene 0.063 5.877 0.1265 6.528 0.406 7.88 0.701 7.56 1.054 9.441 PS-4 Benzene 0.0595 1.854 0.1192 2.546 0.411 3.152 0.694 4.036 1.0517 3.818 PS-S Benzene 0.065 3.869 0.1318 4.489 0.405 5.691 1.0663 5.558 PS-6 Benzene 0.0891 2.972 0.1782 3.362 0.6414 4.443 PS-2 Benzene 0.2113 3.127 0.528 - 3.531 0.702 4.126 1.001 4.441 2.007 5.375 5.00 9.503 9.93 12.886 SAN-1 Dimethyl formamide 0.6458 4.075 1.09 4.524 1.96 5.149 5.01 7.489 SAN-1 Methyl ethyl ketone 1.067 8.467 1.98 8.561 4.86 11.981 SAN-2 Dimethyl formamide 1.111 4.704 1.98 5.083 4.85 6.353 222 Concentration D x 107 Polymer Solvent of Polymer in 2 gms/lOO cm3 cm /sec SAN-2 Methyl ethyl ketone 1.04 6.087 1.964 7.704 4.987 10.340 SAN-3 Dimethyl formamide 1.063 3.619 1.989 4.151 SAN-3 Methyl ethyl ketone 1.085 4.831 1.98 6.945 APPENDIX G CARDS USED IN COMPUTER INTERFACING 223 224 I/O PATCH CARD: This card serves as an input output line from the computer interface buffer box to the com- puter interface ADD. This is a 32 pin patch card. The card t0p diagram is shown in Figure AI-l. Pin numbers 1 through 6 are used for device address. In an I/O instruc- tion, the middle six bits of the instruction word are used to identify the device which is to provide or accept data. The pins 1 through 6 give out signals during an I/O instruction corresponding to the middle six bits of the instruction word. These signals are decided by the octal decoder card. Pins 7 to 12 are the IOP pulses. They are the pulses that are available from the computer during an I/O instruction. Pins 13 to 24 are the lines that are connected to the accumulator whenever the skip (SKP) pin (pins 26 or 27) goes low or is grounded momentarily. The data that are available to the pins 13 to 24 will be handed to the computer when- 0 10.3410 20 .4121: 30pm: 40:63.41" 50.1121? 6O ”11!) I I ’0 s: 1 1 160001 170 004: 180005} 190°“I 200001? 210000 220 on: 230010 240°'_‘_ 310911111 320’“; Q Figure AI-l ever the SKP line is grounded momentarily. The other pins are not used in this work. 225 OCTAL DECODER CARD: The card provides a system for decoding a six bit binary coded input into a two digit octal output from 00 to 77 in octal number base.- The outputs of the two independent binary to octal converter circuits can be combined with the two NOR gates included to provide a decoded signal for any given two digit octal number. The card top diagram is shown in Figure AI-2. Pins 1 through 6 are for device address. These six pins receive signals from the pins 1 through 6 on the I/O patch card. During an I/O instruction the computer sends signals to these six pins in binary form cor- responding to the middle six bits of the instruction word. Pins 8 through 14 are one octal decoder designated 81 and pins 15 to 22 are the other decoder designated 80, for the two octal digits in the device address num- ber. Pins 24 to 28 are the two NOR gates. 0 10‘0“"? 20.04 2" 3000510? 40356.17: 50.07:: 60.00.11 70-? 80' 90' 10C). “0' 12C)- 130' 14051,]; 150" c MOD" o 110.? i 18CD>T 190'7“o ZOCDv? 210'? 220.7 230an 24C) 25C) 260 270 28C) 29C) 3000* 3100+» 32C) ,__@ u 10.0-I -j ul N' —| 0 C l I l 0 .- Figure AI-2 As an example say the middle six bits of the instruction word are 45 octal. So when the computer encounters the I/O instruction, it sends binary signals to the I/O patch card pins 1 through 6. If these are connected to pins 1 through 6 of the octal decoder card, then the signals are in the octal decoder. If pins 4 x 81 and 226 0 5 x 8 are connected to pins 25 and 26, then pin 24 would go HI when the I/O instruction is executed, the rest of the time it is L0. 227 GATED DRIVER CARD: It is necessary that each device output signal be connected to the accumulator 0 - 11 lines only when the data are to be transferred from that device. For this purpose, external data source registers are connected to the accum- ulator 0 - 11 lines through gates which have outputs that are active only at the appropriate times. The appropriate time is when the device is addressed and an IOP pulse is generated by the execution of an I/O instruction. The card top diagram of the gated driver card is shown in Figure AI-3. Pins 1 - 12 are data inputs and pins 13 - 24 are data outputs from the gated driver. Pins 27 and 28 are called strobe (STR) and select (SEL) signals. These signals control data transfer between the inputs and outputs of the gated driver card. When both the STR and SEL signals are HI, the signal levels at the inputs determine the C) 10.00 20m 3001‘? 40103 ‘12 50004 0 60.0501 £8156:2 90 008‘ 10()«u 11C3«w 12C)m 130915 0 14095? lacofi‘ 16c)»0' 17095 18095 190061 20C)°51 7.10953; 2208!?9 23(325 24C)°W ZSCD'FTT 260-1717 270 '5" 280 csu 290 5M)°* 310“" 330 .‘DC‘ u-e — we a..- ,____9 Figure AI-3 signals at the output. When either SEL or STR are low, the data inputs are disconnected from the data outputs. 228 DUAL FLAG CARD: ON The synchronization of the appearance of the 1C3‘n7 2C)‘11- data transfer instruction in the program ZE;::‘ . '3 operation with the external devices readiness zggsif' 1Com for that transfer is accomplished with "skips" 3c) 903‘" and "flags." The flag is a circuit that pro- 10C)¢v- 11C).fl‘: instruction that allows a branch in the pro- ::E§;;Lf , , 200.”: gram depend1ng upon the state of read1ness of 21c).",. 22() the external device. The card top diagram of 230 3'“ 240010 the dual flag is shown in Figure AI-4. The 25033: 26C) : dual flag card consists of two flags. Pins 33%; i' - - 29C) 1 14 are for flag 1 and 15 28 are for 300 .4“ flag 2. Each flag circuit is usually a flip— 3:8 ’“ flop which is set by the falling edge of the , (3‘ ready signal. Each flag also has clear pins, whereby the flag can be cleared. The pins Figure AI-4 for clearing flag 1 are 10 to 14 and for flag 2 are 24 to 28. As an example suppose we want the computer to know when a particular device 45 is ready to transmit data. We take the signal that indicates the readiness of the device 45 into pin 2. A 6451 instruction will test the state of the flag. A device select of 45 is connected to pin 8 or 9, and the IOP l is connected to pin 7. 229 When the computer encounters the instruction 6451, it gives out an IOP 1 pulse and a device select pulse for 45 through an octal decoder card. At this time if the readiness of the device signal is also HI, then the SKP line (pin 6) will provide a momentary LO signal (ground). If this SKP line is connected to the SKP on the I/O patch card, then the next instruction in the program is skipped. By using a flag card, we can design our software such that the com— puter loops around checking to see if the device is ready to trans- mit data; when it finds it is ready it goes out of the loop and reads the data. Once the flag is set it must be cleared. Flag 1 is cleared by HI signals at pins 10 and 11 or 12, and flag 2 by HI signals at 24 and 25 or 26. When the computer is first turned on, it is desirable to have all the device flags cleared. An initialize pulse is generated (called INIT) when the computer is turned on or when start is pushed. The INIT signal from the I/O patch card is usually wired to pins 13 or 14 for flag 1 and to pins 27 or 28 on flag 2 for clearing the flags. 230 NAND GATE CARD: A card top diagram of the NAND gate card is shown in Figure AI-S. One of the NAND gate integrated circuit is a quadruple two inpUt gate, while the other is a dual four input NAND gate. Each of the six gates can be used independently to perform NAND gate functions. The NAND gate is a negative or inverted output AND gate. As such, the NAND gate output is LO only when all inputs are HI. An unused input of a NAND gate acts like a HI input and has no effect on the gate function. If only one input is used the NAND gate output is always the opposite in logic level of the input. Thus a single input NAND gate is a logic level inverter. Figure AI-5 APPENDIX H PROGRAM FOR DATA TRANSFER 231 mmmmmmm mmmm mmmm CDFI, COUNT, POINT, OPDEF COMMON DIMENSION CLA CLL 6214 TAD DCA 6452 6441 0MP 6452 CLA TAD DCA 6444 6451 JMP 6442 6211 DCAI ISZ ISZ JMP 0 JMP NOP 232 DCAI 3400 /Defining mnemonic code for DCAI IDATA IDATA(400) 6201 DCFI (200) POINT POINT POINT COUNT A -620 /C1ear the accumulator and link /Read data field /Two's complement add 6201 (OCTAL) with contents of accumulator /Deposit the contents of accumulator at DCFI and clear the accumulator /Clear the sweepgate flag /Is the sweepgate HI /No, check it again /Clear the sweepgate flag /Clear accumulator /Two's complement add literal 200 with contents of accumulator /Deposit the contents of accumulator at point and clear accumulator /C1ear the circulation pulse flag /Is the circulation pulse HI /No, check it again /Read data and clear circulation pulse flag /Change to data field 1 /Deposit the contents of accumulator at point and clear accumulator /Increment point /Increment count and skip next instruction if count is zero /Go to A /Go to original data field /Go to 0 /Count initially is 620 (OCTAL) or 400 digital /Point initially is zero /No operation 233 READ (1, 5) COMNT, SNOIC, THETA FORMAT (8H COMNT = ,A6/8HSNOIC = ,F10.8/ 8H THETA = ,F4.1) READ (1, 6) FILE FORMAT (7H FILE = ,A6) CALL OOPEN ('FLP2', FILE) WRITE (4, 7) COMNT, SNOIC, THETA FORMAT (5X, A6, 5X, F10.8, FX, F4.1) WRITE (4, 8) (IDATA(I), I = 1,400) FORMAT (1415) CALL OCLOSE END APPENDIX J DERIVATION FOR FUNCTION I (m,n) 234 235 Benbast and Bloomfield showed (SA-33) that the I function was uvm m e-(u + v) I( (m, n) = dudv (J-l) (u + v)n related to Beta function and are defined in terms of the Gamma func- tion as J? P(m + 1) P(2 m + n + 2) (J_2) + 1 P(m + 3/2) I(m,n) = 2m where P(x) is a Gamma function of x. They performed numerical calculations from equation 0-1 for various values of m and n and tabulated the results. Their results do not correspond to the values obtained from J-2 for the same values of m and n. Therefore the relation between J-l and J-2 was examined here. Let u = r cos2 0, v = r sin2 0 dudv = 2 r sin ¢ cos ¢ dr do n/2 (r c0520)m (r sinzg)m -r ‘.I(m,n)= 2 e r sin 0 cos 0 drdo rn 236 n/2 “ '.I(m,n)= 2 [ (c052¢)m (sin2¢)m sin cos 4 d { O O r2m—n+1 e-rdr (J-3) The second integral in J-3 is a Gamma function of (2m - n + l), and the first integral can be related to the Beta function as follows. The Beta function is defined as (AJ-l) 1 B(m, n) = I tm-1 (l-t)n'1 dt 0 Let sin2 0, 1-t=cos2 0, dt = 2 sin 0 cos 0 do t = n/2 B(m, n) = 2 I (sin2 ¢)m-1 (cosz ¢)n-1 sin ¢ cos ¢ d 0 O n/2 .'.B(m+l,m+l) = 2 I (sin2 ¢)m (cos2 ¢)m sin ¢ cos ¢ d ¢ (J-4) 0 Comparing J-3 and J-4 we obtain I(m, n) = B(m+1, m+1) P(2m-n+2) (J—S) The relation between Beta and Gamma functions is (AJ-l) 237 B(m+1, am) e P(wgmigeu Using the above relation and J-5 it is found F(m+1) T(m+1) F(2m-n+2) P(2m+2) I(m, n) = This relation gives identical values corresponding to m and n in the tables of Benbast and Bloomfield. This relation for I(m, n) is therefore the correct one and was used in this work. BIBLIOGRAPHY 238 1A-1 2A-1 2A-2 2E-1 3A-1 3A-2 3A-3 3A-4 3A-5 3A-6 3A-7 3A-8 3A-9 Secor, Robert M. A. I. Ch. E. Journal 11 (1965):452. Flory, P. J. Principles of Polymer Chemistry. New York: Cornell University Press, 1953, ch. XIII. Morawetz, H. Macromolecules in Solution. New York: John Wiley 8 Sons, 1966, (a) p. 276. Bird, R. 8.; Stewart, W. E.; and Lightfoot, E. L. Transport Phenomenon. New York: John Wiley 8 Sons, 1960. Einstein, A. Ann Physik 17 (1905):549. Kirkwood, J. G., and Riseman, J. J. Chem. Phys. 16 (1948):565. Mandelkern, L., and Flory, Paul J. J. Chem. Phys. 20 (1952): 212. Johnston, H. K., and Rudin, A. Polymer Letters 9 (l971):55. Yamakawa, H. Modern Theory of Polymer Solutions. New York: Harper & Row, 1971, (a) ch. 3; (b)*ch. 4; (c) ch. 5; (d) ch. 6; (e) ch. 7. Vrentas, J. S., and Duda, J. L. J. App. Poly: Sci. 20 (1976): 1125. King, T. A.; Knox, A.; Lee, W. 1.; and Adam, J. D. G. Polymer 14 (1973):151. Cowie, J. M. G., and Cussler, E. L., J. Chem. Phys. 46 (1967): 4886. Brandrup, J., and Immergut, E. H” eds. Polymer Handbook, 2nd ed. New York: John Wiley & Sons, 1975. 3A-10 Jacob, M.; Varoqui, R.; Kenine, S.; and Danue, M. J. Chem. 3A-11 3A-12 Phys. 59 (1962):865. Matsuda, H.; Aonuma, H.; and Kuroiwa, S. J. Appl Polymer Sci. 14 (1970):335. Rept. Prog. in Polymer Sci. (Japan) 11 (1968):25. 3A-13 Meyerhoff, G., Makromol Chem. 37 (1960):97. 239 3A-14 3A-15 3A-16 3A-17 240 King, T. A.; Knox, A.; and McAdam, J. D. G. Polymer 14 (1973):293. Ford, N. C.; Karasaz, F. E.; Owen, J. E. M. Disc Faraday Soc. 49 (1970):228. Nachtiga11,and Meyerhoff, G. Phys. Chem. 30 (1961):35. Elias, H. G. Macromol Chem. 50 (1961):l. 3A-18 Auer, R. L., and Gardner, C. S. J. Chem. Phys. 23 (1955):1546. 3A-19 3A-20 3A-21 3A-22 38-1 3B-2 3B-3 3B-4 38-5 3B-6 3B-7 3B-8 38-9 Ptitsyn, O. 8., and Eizner, Yu. E. Soviet Phys. Tech. Phys. 4 (1960):1020. Zimm, B. H. J. Chem. Phys. 24 (1956):269. Einstein, A. Theory of Brownian Motion. New York: Dover, 1956. VanKrevelen, D. W. Properties of Polymers, Correlations with Chemical Structure. New York: Elsevier Publishing Co., 1972. Tsvetkov, V. N., and Klenin, S. I. J. Polymer Sci. 30 (1958): 187. Duda, J. L., and Vrentas, J. S. J. App. Polymer Sci. 20 (1976): 2569. J. Polymer Sci.; Polymer Phys. ed. 14 (l976):101. Pyun, C. W., and Fixman, M. J. Chem. Phys. 41 (1964):937. Refer to 3A-5. Yamakawa, H., and Tanaka, G. J. Chem. Phys. 47 (l967):3991. Kurata, M.; Fukatsu, M.; Sotobayashi, H.; and Yamakawa, H. J. Chem. Phys. 41 (1964):139. Yamakawa, H. J. Chem. Phys. 48 (1968):2103. Paul, D. R.; Mavichak, V.; and Kemp, D. R. J. App. Polymer Sci. 15 (1971):1553. 38-10 McDonnell, M. E., and Jamieson, A. M. J. Macromol Sci. Phys. 3B-11 B13 (1977):67. Mandema, W., and Zeldenrust, H. Polymer 8 (1977):835. 3B-12 3C-1 3C-2 4A-1 4A-2 4A-3 4A-4 4C-1 4C-2 4C-3 4C-4 5A-3 5A-4 5A-5 5A-6 5A-7 5A-8 5A-9 5A-10 5A-11 5A-12 5A-13 241 Berry, G. C., and Casassa, E. F. Macromol Rev. 4 (1970):l. Hayes, M. J., and Park, G. S. Trans Faradpy Soc. 52 (1956): 949. Chalykh, A. Ye., and Vasenin, R. M. 8 (1966):2107. ‘ Polymer Science U.S.S.R. 0dian, George. Principles of Polymerization. New York: McGraw-Hill, 1970, ch. 6, (a) pp.20L21. Atherton, J. N., and North, A. M. Trans Farad Soc. 58 (1962): 2049. Blanks, Robert F., and Shah, B. N. Chem. ed. 14 (1976):2589. J. Polymer Sci., Polymer Shah, B. N. Ph.D. thesis, Michigan State University, 1975. Shimura, Y.; Mita, 1.; and Kambe, H. J. Polymer Sci. B,2 (1964):403. Mino, G. J. Polymer Sci. 22 (1956):369. Ljerka, Lovric. J. Polymer Sci. A2,7 (1969):1357. Private communication from Dr. J. B. Kinsinger, Chemistry Dept., Michigan State University, East Lansing. Vrancken, E. Ph. D. thesis, University of Akron, 1975. Gyesezly, S. W. Ph.D. thesis, Michigan State University, 1974. Nature 126 (1930):201, 400, 603. Chem. Phys. 43 (1965):1562. Gross, E. Pecora, R. J. J. Chem. Phys. 48 (1968):4126 J. Chem. Phys. 49 (1968):1032. J. Chem. Phys. 49 (1968):1036. Pecora, R. and Tagami, Y. J. Chem. Phys. 51 (1969):3293. J. Chem. Phys. 51 (1969):3298. Pecora, R. Macromolecules 2 (1969):31. Forrester, A. T.; Parkins, W. E.; and Gerjuoy, E. Phys. Rev. 72 (1947):728. 5A-14 5A-15 5A-16 5A-17 5A-18 5A-19 5A-20 5A-21 5A-22 5A-25 5A-27 5A-28 5A-29 5A-30 5A-31 5A-32 5A-33 5A- 34 5A-35 5A-36 242 Cummins, H. Z.; Knable, N.; and Yeh, Y. Phys. Rev. Lett. 12 (1964):150. Ford, N. C., and Benedek, G. 8. Phys. Rev. Lett. 15 (1965): 649. Sheperd, I. W. Rep. Prog. Phys. 38 (1975):565. Ford, N. C.; Gabler, R.; Karasz, F. E. Adv. in Chem. Ser 125 (1974):25. Cummins, H. G., and Swinney, H. L. Progress in Optics 8, ed. Wolf. Amsterdam: North Holland Publishing Co., 1970. Chu, 8. Laser Light Scatteripg. New York: Academic Press, 1974. Jamieson, A. M., and Maret, A. R. Chem. Soc. Rev. 2 (1973): 325. Frederick, J. E., and Reed, T. F. Macromolecules 4 (197l):72. Brown, J. C., and Pusey, P. N. J. Chem. Phys. 62 (1975):1136. Frederick, J. E.; Reed, T. F.; and Kramer, 0. Macromolecules 4 (1971):242. Adam, M.; and Delsanti, M. J. de Physiqpe 37 (l976):1045. Fujime, Satoru. J. Phy. Soc. Japan 29 (1970):416. Stutesman, W. D. M.S. thesis, Michigan State University, 1972. Thompson, 0. S. Rev. Sci. Instr. 41 (1970):1228. J. Chem. Phys. 54 (1971):1411. J. Phys. Chem. 75 (197l):789. Benbast, J.A., and Bloomfield, V. A. Phys. ed. 10 (l972):2475. J. Polymer Sci.; Polymer Nelson, T. J. M.S. thesis, Michigan State University, 1974. McQuarrie, D. A. Statistical Mechanics. New York: Harper 8 Row, 1976, ch. 20. Gulari, E.; Brown, R. J.; and Pinas, C. J. A. I. Ch. E. Journal 19 (1973):1196. 243 5A-37 Mountain, R. 0. Rev. of Modern Physics 38 (1966):205. 5A-38 . J. Chem. Phys. 50 (1969):1103. SB-l Weissberger and Rossiter, eds. Physical Methods of Chemistry, Part IV. New York: Wiley Interscience, 1972, ch. 4. 58-2 Paul, D. R.; Mavichak, V.; and Kemp, D. R. J. App. Polymer Sci. 15 (1971):1553. 58-3 Duda, J. L.; Sigelko, W. L.; and Vrentas, J. S. J. Phys. Chem. 73 (1969):141. 58-4 Pau1, D. R. I. and E. C. Fundm. 6 (1967):217. 58-5 Secor, R. M. A. I. Ch. E. Journal 11 (1965):452. 58-6 Rehage, G., and Ernst, 0. Disc Faraday Soc. 49 (l970):208. 58-7 Zehnder, L. Z. Instrumentenk 11 (1891):275. 58-8 Caldwell, C. S.; Hall, J. R.; and Babb, A. L. Rev. Sci. Instr. 28 (l957):816. 58-9 Bidlack, D. L. Ph.D. thesis, Michigan State University, 1964. 58-11 Shimura, Y. J. Polymer Sci., Part A2 4 (1966):423. 58-12 Sundelof, L. O. Arkiv for Kemi. 25 (1965):l. 58-13 Schulz, G. V. Z. Physik Chem. 843 (1939):25. 58-14 Gosting, L. J., and Morris, M. S. J. Am. Chem. Soc. 71 (1949):1998. 58-15 Akeley, D. F., and Gosting, J. L. J. Am. Chem. Soc. 75 (1953): 5685. 58-16 Creeth, J. M. J. Am. Chem. Soc. 77 (1955):6428. 58-17 Cussler, E. L., and Dunlop, P. J. J. Phys. Chem. 70 (1966): 1880. 7B-l Coleman, M. F., and Fuller, R. E. J. Macromol Sci., Phy. ed. 14 (1976):101. 7C-3 Chitrangad. Ph. D. thesis, University of Rochester, 1973. 70—4 Pusey, P. N.; Vaughan, M. J.; and Williams, G. J. J. Chem. Soc. London, Faraday trans. 70 (1974):l696. 70-6 7E-1 7F-1 AB-1 AB-2 AB-3 AB-4 AJ-1 244 Berry, G. C. J. Chem. Phys. 44 (1966):4540. Rami Reddy, G., and Kalpagam, V. J. Polymer Sci., Polymer Phys. ed. 14 (l976):759. Noda, I.; Kitano, T.; and Nagasawa, M. J. Polymer Scipy Polymer Phys. ed. 15 (1977):1129. Bawn, C. E. H.; Freeman, C.; and Kamaliddin, A. Trans. Faradgy Soc. 46 (1950):1107. Outer, P.; Carr, C. I.; and Zimm, B. H. J. Chem. Phys. 18 (1950):830. Breitenbach, J. W.; Gabler, H.; and Olaj, 0. F. Macromol Chem. 81 (1964):32. Inagaki, H.; Suzuk, H.; Fuiji, M.; and Matsuo, T., J. Phys. Chem. 70 (1966):17l8. Spiegel, Murray R. Mathematical Handbook. New York: McGraw- Hill, 1968. "1111111'111'1111141111113