DIFFUSION COEFFICENTS 0F POLYSTYRENE AND ACRYLONITRILE COPOLYMERS IN DILUTE SOLUTIONS BY LIGHTBEATING SPECTROSCOPY Disseflmea for m Degree of P11. 8. Micmm STATE WEVERSITY STEVEN WtLLlAM GYESZLY 1 9 7 4 5." 5..) 3"] 3’3 LIBRARY ' i l‘,r'.v‘}?z‘_,§ai: Stw L. Umv ersity . u ‘ ,mmLi .-‘.‘: 'QM‘ "’.I. " . ." . ‘ 31...? This is to. certify thag‘w ’ .. A. ' ’1',- _,._., . . sums? DIFFUSI,L;.V £5: .:.v_ 0- <>v fl , (sf '5‘? ,1iN..-‘-tl . ‘ r‘ Baeilne SPECTROSCOPY presented by STEVEN WILLIAM GYESZLY has been accepted towards fulfillment of the requirements for Ph. D. degreein Materials Science .‘._. "v".: [M W Major professor Dgi'eFM 29; 177% 0-7639 ABSTRACT DIFFUSION COEFFICIENTS OF POLYSTYRENE AND ACRYLONITRILE COPOLYMERS IN DILUTE SOLUTIONS BY LIGHTBEATING SPECTROSCOPY BY Steven William Gyeszly Self-diffusion coefficients (D) were measured of polystyrene (PS) and polystyrene-acrylonitrile copolymer (SAN) in a range of molecular weights and in several solvents in order to demonstrate the effectiveness and reliability of the light beating spectroscopic technique as well as to extend the diffusion coefficient data for these systems. Agreement between D values determined by lightbeating spectroscopy and calculated from Stokes-Einstein equation for PS spheres in water showed useability and reliability of the used instrumentation. The determined values of D for P8 in methylethylketone (MEK) and benzene were in agree- ment with literature values obtained by other methods proving validity of the new technique and calculation originated by this work. The diffusion of SAN in different solvents was deter- mined for the first time in this work. Steven William Gyeszly It was found that the diffusion coefficient for PS and SAN is the highest in MEK compared with benzene and dimethylformamide (DMF). The value of D for PS is lowest in decalin compared to MEK, DMF and benzene. Concentration dependence of diffusion of PS and SAN are varied in different solvents. Lower values of D were obtained for higher molecular weights. All of determined values of D generally agree with the theoretical expectation except for PS-decalin system. The very low diffusion for PS in decalin can be explained by a relatively simple hypothesis dealing with agglomer- ation of PS molecules. DIFFUSION COEFFICIENTS OF POLYSTYRENE AND ACRYLONITRILE COPOLYMERS IN DILUTE SOLUTIONS BY LIGHTBEATING SPECTROSCOPY BY Steven William Gyeszly A DISSERTATION Submitted to .Michigan.State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Metallurgy, Mechanics, and Materials Science 1974 To my parents and to my wife. ii ACKNOWLEDGMENTS The author wishes to express his sincere gratitude to his major thesis advisor Dr. Robert F. Blanks for his generous assistance and guidance which ultimately led to the completion of the work reported in this dissertation. The author is also grateful to the chairman of his doctoral committee Dr. Robert Summitt and members thereof Dr. James W. Goff and Dr. Gary L. Cloud, who with their helpful and understanding attitude greatly encouraged him in the course of his research. Grateful acknowledgement is also extended to the Chemistry Department, and especially for the cooperation of Dr. Jack B. Kinsinger who provided needed laboratory and equipment without which this thesis could not have been completed. Thanks are also due Dr. Jerry A. Cowen for the use of his instrument, and to Edward F. Grabowski and Edwin L. Doak for their assistance, and deep appreciation to Dr. Wayne H. Clifford for his valuable advice. Finally, and above all, I wish to express my deep felt thanks and gratitude to my wife, Zsuzsanna, for her understanding, help, and encouragement, throughout the course of this study. Without her help this work would not have been started nor finished. iii TABLE OF CONTENTS I I INTRODUCTION 0 O O O O O O O O O O O O O 0 0 II 0 THEORY O O O O O O ' O O O O O O O O I O O O O A. Relationship Between the Spectral Dis- B. tribution of the Intensity of Scattered Light from a Dilute Polymer Solution and the Diffusion Coefficient . . . . Lightbeating Spectroscopy . . . . . . III. EXPERIMENTAL. . . . . . . . . . . . . . . . A. B. C. D. E. Instrumentation . .-. . . 1. Light Source. . . . . Incident Optics . . . Collection Optics . . Photomultiplier . . . Spectrum Analyzer . UTDUJN 00.00. .0000. o o o calibration O O O O O O O O O O O O 0 Sample Preparation--Dust Particles in the Diluted Polymer Solution. . . . . Calculation . . . . . . . . . . . . . Diffusion Constants of the Polymer Molecules in Dilute Solution. . . . . Iv. RESULTS 0 O O O O O O O O O O O O O O O O O Polystyrene-Methylethylketone. . . Polystyrene-Benzene. . . . . . . . Polystyrene-Dimethylformamide. . . Polystyrene-Decalin. .r. . . . Polystyrene-Acrylonitrile in Methylethyl- ketone O O O O O O O O O O O O O Polystyrene-Acrylonitrile Copolymer in Benzene O O O O ‘ O O O O O O O O O O 0 iv 11 16 16 18 19 21 21 22 28 39 53 57 57 64 70 72 72 TABLE OF CONTENTS-~Continued Page Polystyrene-Acrylonitrile Copolymer in Dimethylformamide. . . . . . . . . . 72 Comparison of the Diffusion Coefficients for a Constant Concentration of a Given Polymer (molecular weight is constant) in Different Solvents. . . . 79 Comparison of Diffusion Coefficient of Polystyrene and Polystyrene-acryloni— trile (the same concentration and molecular weight) in Different Solvents . . . . . . . . . . . . . . . 79 V. DISCUSSION . . . . . . . . . . . . . . . . . 84 APPENDIX. . . . . . . . . . . . . . . . . . . . . 98 LIST OF REFERENCES. . . . . . . . . . . . . . . . 102 TABLE LIST OF TABLES Range of halfwidths of lightbeating spectra of light scattered from polystyrene spheres in water. O O O O O O O O O O O O O O O O O O Ratio of halfwidth of lightbeating spectra calculated from single and three Lorentzians. A typical computer printout used for calcula- tionO O O O O O O O O O ‘O O O O O O O O O O O Molecular weights of polystyrene and poly- styrene-acrylonitrile copolymer used in this work. O O O O O O O O O O O O O O O O O O O O Density and refractive index of the solvent used in this work . . . . . . . . . . . . . . Diffusion coefficient of polystyrene in dif- ferent solVents . . . . . . . . . . . . . . . Diffusion coefficient of polystyrene-acryloni- trile in different solvents . . . . . . . . . Diffusion coefficient of polystyrene in decalin O O O O O O O O O O HO O O O O O O O O vi Page 26 46 48 58 59 60 61 63 FIGURE 1. 2. 3. 10. 11. 12. LIST OF FIGURES Lightbeating spectrometer. . . Optical detection train. .~. . Angular dependence of halfwidth of the lightbeating spectrum of polystyrene spheres (d = 1090A). o e o o o o o o o Angular dependence of halfwidth of the.light- beating spectrum of polystyrene spheres. (d = 1090A). . . . . . . . . . Photocurrent spectrum of light scattered ' benzene (scattering angle 25°) Photocurrent spectrum of light scattered.’ methylethylketone (scattering angle 25°) Photocurrent spectrum of light scattered dimethylformamide (scattering angle 25°) Photocurrent spectrum of light scattered decalin (scattering angle 25°) Photocurrent spectrum of light scattered water (scattering angle 25°) . Sum of the square error of calculated half- = 250,000 widths. Monsanto SAN copolymer in methylethylketone (C = scattering angle is 25°. . . . 0.3 g/ 009) Angular dependence of calculated halfwidth. Monsanto copolymer (MW = 250, 000) in methyl- ethylketone. . . . . . . . . Diffusion coefficient of polystyrene ( 80, 000) in different solvents. vii MW Page 17 20 24 27 33 34 35 36 37 50 51 65 LIST OF FIGURES--Continued FIGURE 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. Diffusion coefficient 185,000) in different Diffusion coefficient 338,000) in different Diffusion coefficient 130,000) in different Diffusion coefficient 271,000) in different Diffusion coefficient decalin . . . . . . . Diffusion coefficient Page of polystyrene (MW. = solvents. . . 66 of polystyrene (MW. = solvents. . . . . 67 of polystyrene (MW= solvents. . . . . . 68 of polystyrene (MW= solvents. . 69 of polystyrene in O O O O O O O O O O O 71 of polystyrene- acrylonitrile copolymer ( = 203,000, 23% acrylonitrile content) in ifferent sol- vents . . . . . . . . Diffusion coefficient O O O O O O O O O O O 73 of.polystyrene- acrylonitrile copolymer ( =.247,000, 25% acrylonitrile content) in ifferent solvents. . . . . . . Diffusion coefficient O O O O O O O O O O O 74 of polystyrene- acrylonitrile copolymer ( = 275,000, 14% acrylonitrile content) in ifferent solvents. . . . . . . Diffusion coefficient O O O O O O O O O O O 75 of*polystyrene- acrylonitrile copolymer ( = 325,000, 25% acrylonitrile content) in ifferent solvents. . . . . . . Diffusion coefficient O O O O O O O O O O O 76 of polystyrene- acrylonitrile copolymer ( = 634,000, 22% acrylonitrile content) in ifferent solvents. . . . . . . Diffusion coefficient O O O O O O O O O O O 77 of polystyrene— acrylonitrile copolymer ( . = 332,000, 38% acrylonitrile content) in ifferent solvents. . . . . . . O O O O O O O O O O O 78 viii ILIST OF FIGURES--Continued FWIGURE 24. 25. 26. 27. 28. 29. 30. Comparison of diffusion coefficient of poly- styrene and polystyrene-acrylonitrile copolymer as function of molecular weight.in methylethylketone at 0.1 g/lOOg concentra- tion. O O O O O O O O O O O ' O O O O O O O O O Comparison of diffusion coefficients of poly- styrene and polystyrene-acrylonitrile copolymer as function of molecular weight in benzene at 0.1 g/lOOg concentration . . . . . Comparison of diffusion coefficients of poly- styrene and polystyrene-acrylonitri1e copolymer as function of molecular weight in dimethylformamide at 0.1 g/lOOg concentrar tionO O O O O O O O O O O O O O O O O O O O O Relationship between the diffusion.coeffi- cient and the intrinsic viscosity of differ— ent polystyrene solutions (MW = 680,000). . . Photocurrent spectrum of light scattered from polystyrene latex at 35° scattering angle . . Photocurrent spectrum of light scattered from polystyrene-acrylonitrile‘copolymer(MW = 332,000, acrylonitrile content 38%) in dimethylformamide (c = 1.0 g/lOOg).. Scatter- ing angle 25°. Halfwidth 255 Hz. (D = 1.44 x 10-7 cmz/SeC) O O O O O O O O O O O O O O O O Photocurrent spectrum of light scattered from polystyrene-acrylonitrile copolymer ( = 332,000, acrylonitrile content 38%) in di- . methylformamide (c = 1.0 g/lOOg). Scattering angle 35°. Halfwidth 496 Hz. (D = 1.46 x 10-7 cmz/seCo) e o e e o o o o e o o o o e o 0 ix Page 80 82 83 87 95 96 97 I . INTRODUCTION Lightscattering'first discussed theoretically by Ilayleigh (1871).has become an important tool in polymer ascience in the last few decades. From measurements of ZLightscattering on dilute polymer solutions, the molecular Vveights of the polymers can be calculated and important :information about the size and shape of polymer molecules can be obtained . The use of the laser as a light source made it pos- ssible to determine the motion of polymer molecules in asolution by lightbeating spectroscopy. This measurement Inethod is relatively new. Pecora (1964) gave the theoreti- <2a1 relationship between the spectral distribution of Slight scattered from a polymer solution and the diffusion (:oefficient of the polymer molecules in the same solution. In the last seven years a number of experimental astudies have been published which verify this theory. The simplest way to do it is by measuring the lightbeating Spectrum of light scattered from polystyrene spheres in ‘Nater, and calculating the diffusion coefficient from this Ineasurement based on the theory. The calculated diffusion <:oefficient from the lightbeating spectroscopy measurement (:an be compared with the calculated diffusion coefficient fiErom the Stokes-Einstein's diffusion equation, if the (Siameter of the polystyrene spheres is known. This Was (done by several researchers (Cummin, 1964; Arecchi, 1967; IDubin, 1967; Dunning, 1970; Reed, 1970; Kramer, 1971; (Dhbayoshi, 1972; Lee, 1972; and others). Good.agreement was .found, hence, this method .may' be used as a calibration Igrocedure for lightbeating spectroscopy measurements. Most diffusion measurements by lightbeating spectros- <:opy have involved biological macromolecules (Dubin, 1967; (Summins, 1969; Carew, 1969; Dubin, 1969; Nada, 1969; and (others). Only a few experiments have been done in study- :ing the diffusion of synthetic polymers in solution. Lightbeating spectrum measurements of light scattered ;from polystyrene molecules in dilute solutions of cyclo— hexane were made by White (1966) and Chu' (1969) . Determin— iation of the diffusion coefficient of polystyrene"in“cyclo- hexane by lightbeating was reported by French‘.('1969) , Reed (1970), and for the polystyrene—methylethylketone system by Ford (1970) and by Kramer (1971). Stutesman and Blanks (1973) have reported the diffu— sion constant of polyacrylamide in aqueous solution Ineasured by this method. All of the diffusion coefficients reported in these. ‘Norks were in good agreement with diffusion'coefficients Jneasured by other methods. Therefore it is assumed in ‘this work that the theoretical relationship between the .lightbeating spectrum of the light scattered from polymer Imolecules in dilute solution and their diffusion in the .same solution is valid. Hence this work uses this rela- 'tionship to obtain information about the diffusion of «different synthetic macromolecules in various solvents. Because lightbeating spectroscopy is a relatively new Ineasurement.method for the determination of the diffusion . (Angular brackets denote a time average.) Using the autocorrelation funCtion of the optical field with the following assumptions--the scatterers are statistically independent and identical, position and- orientation are statistically independent and identical, position and orientation are statistically independent therefore factor amplitudes and phases also are inde- pendent--the optical spectrum of the scattered light is given by +oo I(w) = (N/Zfl) f [exp{i(w-wo)T}][CA(r)][C¢(T)JdT, 3. —00 where the amplitude correlation function for spherical scatterers is ICA(T)] = A? and the phase autocorrelation function [C¢(T)] is given as <[exp{-16}f]t)}][exp{i§;§yt+t)}]>; In The phase autocorrelation function was analyzed by Pecora (1964) for lightscattering of dilute solutions of macro- molecules. As a result of thermal motion, the local density of a polymer solution is not equal to its average macroscopic density at a given point in time and at a given position in the fluid. This density fluctuation results in a spectral distribution of the intensity of the light scat- tered from a polymer solution. The density fluctuation is related to the change of the dielectric constant of the medium. The study of change of the dielectric constant of the medium leads to results analogous to Van Hove's (1954) space time correlation function. This space time corre- lation function can be divided into two parts: a self and a distinct correlation. The self-correlation function for the center of mass of a polymer molecule can be calculated from Langevin's equation of Brownian motion, d1? where E'is velocity of the center of the polymer molecule, 8 is the friction coefficient per unit mass and Af(t) is the fluctuating acceleration of molecule. The relationship between the diffusion coefficient of a molecule and its friction coefficient is given by Einstein's diffusion equation, 5. where D is the diffusion coefficient of the molecule, k is the Boltzman constant, T is the absolute temperature, In is the mass of molecule and B is the friction coefficient per unit mass. The basic assumptions of light scattering from dilute polymer solution are the following: Scattering from a single molecule does not depend on the presence of other molecules; the internal rearrangements of polymer molecules do not influence the movement of the mass; and no inter- action exists among the polymer molecules in the solution. Considering the above assumptions Pecora (1964) gives the following relationship between the spectral distribution of intensity of scattered light and the translational diffusion Coefficient of the polymer molecules in their solution: I(w)¢20°D/(m2+Q“DZ), 6. were I is the intensity of the scattered light, Q is the s<-"a-‘|:t.ering vector; on is the light frequency, and D is the translational diffusion coefficient. Analysis of Equation 6 shows that a plot of intensity aga-inst frequency is Lorentzian and that its half-width at half of maximum intensity Am;5 is equal to QZD. A similar restllt is given by Cummins e_t 31. (1964)- The half-width at half of maximum intensity is given in units of radian/sec. Changing this to Hertz and using Equation 1 and the relationship between Awg and D we can obtain Av = l61r(n/).o)2D sin2(-(22-) 7. Where AV is the full width at half maximum height of the Lorentzian curve in Hertz. From Equation 7 it can be seen that plotting AV against sin2(%) results in a straight line which goes through the origin. For a given light source and solution, (Ii/10)2 is constant, therefore the direction tangent, 16n(n/Ao)2D is directly proportional to the trans- lational diffusion coefficient of polymer molecules in their solution. This equation has been verified adequately by the experiments of a number of investigators as mentioned earlier. Pecora (1964 and 1968a) gave the spectral distribution of light scattered from a dilute solution of monodisperse optically isotropic rigid rods. It was found that rota- tional diffusion becomes an important part along with the translational diffusion for scattering from long rods at large scattering angles. Pecora (1969) has also investi- gated theoretically the spectral distribution of light scattered frOm once broken rods. His results are not dis- cussed here, since the polymer molecules being dealt with 10 in this work do not exist in solution as either rigid or once broken rods. Pecora (1965b and 1968b) has developed a theoretical expression to predict the Rayleigh spectrum of light scattered from flexible macromolecules in dilute solutions considering intramolecular effects, which are related to ~the size of the macromolecules. Pecora and Tagami (1969a and 1969b) have discussed the effects of polydispersity of polymers on the Rayleigh spectrum. It was shown that if x = K2 is small, particularly smaller than 0.5, the Rayleigh spectrum con- sists of a single Lorentzian only, which is related to trans- lational diffusion. There K is the scattering vector and is the radius of gyration. In dilute solution it is assumed that the polymer molecules are acting independently. Each molecule can be considered to be a string of beads with a tendency to coil itself to form a spherical cloud of chain segments having radial symmetry (Rodriguez, 1970). The diameter of this spherical cloud is not known, but the root mean square distance of elements of the chain from its center of gravity can be calculated Naz. <52) e 6 , 8. where <82> is the root mean.square distance of elements of chain from its center of gravity, which is called "radius of gyration”, N is the number of links in the chain which 11 can be calculated from the molecular weight of each seg- ment, and a is the length of links, which for molecules with carbon backbone is equal to the C-C single bond dis- tance. Considering the bond angle and the rotation of the segments of polymer molecules, the right side of Equation 8 should be multiplied by corresponding values, approximately two to four, to obtain a more realistic result. If X is larger than 1.0, then the Rayleigh spectrum consists of terms other than the translational one, which are related to intramolecular effects or poly- dispersity. Values of X were calculated for all polymer-solvent and copolymer—solvent systems which have been used in this work. It was found that X is always smaller than 0.1 if the scattering angle is equal to or less than 45°. Therefore the Rayleigh spectrum which is obtained should consist of only one Lorentzian, so intramolecular effects and polydispersity of macromolecules are not discussed here. B. LIGHTBEATING SPECTROSCOPY An excellent review of light beating spectroscopy was published by Cummins and Swinney (1970), so this subject will be discussed only briefly here. The first demonstra- tion of lightbeating was done by Forrester‘§£;§1. (1955). '1) ‘LJ _‘). 12 Photoelectric mixing, called lightbeating, is similar to mixing of alternating current electrical signals in non- linear circuit elements. In the above mentioned experi- ment, a Mercury vapor lamp was used as the light source. A major improvement in the light source came with the introduction of the laser, which made the technique a relatively common laboratory procedure. Two approaches were applied to develop standard laboratory techniques; namely, heterodyne and homodyne detection. Cummins‘gt;§l. (1964) used heterodyne detection for studying diffusion of polystyrene latex spheres in a dilute solution by Rayleigh line broadening.' With the laser a light source the unscattered light was mixed with the scattered light on the detector. The unscattered light Was used as a local oscillator. Ford and Benedek (1965) utilized homodyne detection for the examination of light scattering in sulfur hexa— . fluoride, SF ‘Homodyne detection is different than hetero— 6. dyne detection in that no local oscillator is used, which means the scattered light is detectedMalone on the.photo— detector. Considering the view of Forrester (1961), Cummins and Swinney (1970) give an'analysis of homodyne and heterodyne detection. Because this work is based on homodyne detection only their final results for homodyne detection will be 13 given below. For an optical field which is a narrow band Gaussian random process, the following equations are valid. The optical spectrum of'a field is given by - .. ‘ Y/TT . .. 1(0)) -' YZ+ (w-wo‘)‘ I 9. a Lorentzian function, where Y is Awk (optical) half width. at half maximumheight; mo is the center, is the total intensity, I.is the intensity, and w is the frequency. The photocurrent spectrum for thesame field is 61. Pi+ (w)q:¢ = it) + 2 0(w) + 2 2.§§¥ZIZE;—7 , 10. e where is the shot noise term from instrumentation, ?o(w) is the d.c. component, 2 fi¥£:.(2f)z is the beating part of Lorentizian spectrum, 2 is the total power centered at w = 0, and 27 is the Aw}: (photocurrent) halfwidth. It can be seen that the halfwidth of a Lorentzian spectrum is equal to twice the halfwidth of the original optical spectrum, i.e., it is equal to the full width of the optical spectrum. .Therefore, if one measures the half— width of the Lorentzian spectrum, one may obtain diffusion coefficients from Equation 7. It is important to note that these equations are valid only for fields with Gaussian statistics, otherwise in the-photocurrent spectrum, Equa- tion 10, no lightbeating term.exists. 14 The development of the theories of light scattering 'and lightbeating is based on several assumptions: (1) the scattered radiation field on the surface of a photo- detector is spatially coherent; (2) a pure monochromatic field is given by the light source; (3) the time period of the photocurrent spectrum measurement is long enough so that the average value of the total effect of the photo- current spectrum can be measured at any frequency. Cummins and swinney (1970) made a theoretical investi- gation of how the experimental results will be affected when the above assumptions are not quite true during the actual experiment. Their conclusions briefly are the following: 1. The ratio of the photocurrent signal to the square of some constant total photocurrent is inversely propor- tional to the area of the detector when the area is much larger than lz/Q, where A is the wavelength of the light and Q is the solid angle which the source subtends at the detector. If the area of detector is much smaller than 12/0, then this ratio approaches a constant value, which is independent of the area of the detector. Since 0 is pro— portional to the scattering volume, it is important to minimize the scattering volume. In practice this can be done if the light beam is focused into the scattering volume. 2. Phase effect on the 3. There if we compare source of the 15 fluctuations of the exciting source have no result. is no difference in the light beating spectrum a multimode laser source to a monochromatic same intensity. III. EXPERIMENTAL A. INSTRUMENTATION A schematic illustration of the instrumentation which was used is shown in Figure l. The spectrometer was de- signed and constructed for Brillouin spectrum analysis in the Chemistry Department at Michigan State University by Gaumer (1972), and used by Toth (1973), Nordhaus (1973), and Kumar (1973). Only one.analysis of Rayleigh spectra was made previously with this spectrometer by Stutesman (1973), who analyzed the Rayleigh spectrum with a General Radio l900-A swept-frequency.spectrum analyzer. “This analyzer measures only one bandwidth at a time and does not analyze all of the signal simultaneously. 'This method is very time-consuming and the reading of the data pointS'from the spectra is very difficult and has a high probability of error. Another problem with this approach is that the use of high laser power for obtaining a high signal to noise ratio is not possible because the sample may be heated by the laser beam during the lengthy exposure, about 45 minutes, which is required to obtain a spectra with the General Radio 1900—A analyzer. This heating effect results in an unwanted diffusion process due to thermal gradients 16 17 .Hmumeouuoomm mcfiumoounmfiq .H whomflm uncumeaa_umc xmaxeoux xum>admm=m swig; sum¢d ¢m~_d_nuzu=auxm 18 inside the solution, so low laser power must be used, which gives a low signal to noise ratio. To avoid any heat affect, a different kind of spectrum analyzer should be used, which measures the full spectrum during a relatively short time. At the present time there are two possibili- ties, use of a corrlation function computer or a "real time" analyzer. Both instruments analyze all of the signal at one time. The choice for this work was a "real time" analyzer, which was borrowed from the Physics Department of Michigan State University. The spectrometer with a "real time" analyzer for analysis of the Rayleigh spectrum was used for the first time at Michigan State University; The description of the spectrometer is given in detail in Gaumers dissertation (1973) so it is not repeated here. 1. Light Source The light source was a SpectrafiPhysics Argon ion laser model 165-03. Some basic data about this Argon ion laser are; the beam diameter is 15mm at 5145A wavelength, the beam divergence is 0.5 milliradians, the bore material is BeO, the output power stability with power stabilizer onris‘i0.5% over 10 hours, the input requirements are 190 to 225V, three phases, 35 amps per line, the water flow required is 2.2 gallons per minute minimum at 25 psi, and the water temperature is max. 35°C. l9 2. Incident Optics The incident optical system includes a tiltable mirror and a 100mm focal length lens. The mirror can be posi- tioned in order to send the incident light beam into the sample from different angles. As mentioned earlier for the photocurrent, the signal to shot noise ratio of the photocurrent is inversely pro- portional to the solid angle which the source subtends at the detector. This angle is directly proportional to the scattering volume, which means a minimum scattering volume results in a maximum signal to shot noise ratio. The scattering volume can be minimized if the laser beam is focused into the sample by a short focal length lens. 3. Collection Optics Part of the light scattered by the sample has been ool— lected and focused onto the surface of photomultiplier by the collection optics, which are illustrated in Figure 2. and A were about 1mm which 1 2 determined the cone angle in which the scattered light was The diameters of aperture A collected. The lens L focused the scattered light into 1 the A3 pinhole, which diameter was approximately 0.01mm. The scattered light which passed through the pinhole was focused by lens L onto the surface of the photomultiplier 2 tube. This system of collection optics combined with the incident optics gives a very high signal to shot noise ratio. 20 .sflmuu moon Howamwuasaouosm msoaaon mnmfiso mmxon EswswEdad cofluomuoc Honeymo .m mHsmHm momsoq I; oaocsfim ~95 mousuummm magmas“; «nun—n Hamo mamfimm «dud 3 £6 a I; 21 4. Photomultiplier An EMI 9558B tube was employed as a photomultiplier. Excitation of the photomultiplier was done by a power supply. To reduce the thermionic emissions which result in a high dark current, the photomultiplier tube was placed inside a Products for Research, Inc. Model TE—104 refrigerated chamber. 5. Spectrum AnalyZer The Signal Analysis Industries Corp. Model SAI—SlA real spectrum analyzer/digital integrator was used for spectrum analysis. This analyzer is able to measure ten frequency scales from 0-20Hz to 0-lMHz. Flat weighting is 200 lines. With no signal applied the noise level is 60dB below the full scale. The fast display made to an oscil- loscope is 40 msec and the slow one to a recorder is 8 sec. The output of the spectrum analyzer was a logarithmic current power which was connected to a Hewlett-Packard recorder. The spectrum was recorded on 2 cycle semilog graph paper. The output of the analyzer was connected also to an oscilloscope which showed the full spectrum immediately. 22 B. CALIBRATION The calibration of any system can be done either by calibration of each element of the system or calibration of the total system at once. If possible the calibration should be done in both ways. In this work the alignment procedure of the spectrometer followed the procedure given by Gaumer (1972) and Nordhaus (1973). The spectrum analyzer was cali- brated following the procedure given by its instruction manual. The calibration of the total system was done by use of polystyrene latex as a scattering sample. Dunning (1970), Reed (1970), Bloomfield (1972), Chu .et a1. (1972), and Ohbayashi (1972), have measured the diffu- sion constants of different polystyrene spheres in water by lightbeating spectroscopy. They have found very good general agreement between the calculated diffusion-coefficients from the Stokes-Einstein equation and the experimentally determined one from lightbeating spectroscopy. The Stokes- Einstein equation gives a relationship between the diffusion coefficient of spheres in a liquid and their sizes: _ kT D - W- 11. Where IC is the Boltzman constant, T is the absolute 23 temperature, ns is the viscosity of the system, and r is the radius of the spheres. If each element of Equation 11 is known, the diffusion coefficient can be calculated. The calculated diffusion coefficient from the Stokes- Einstein equation for polystyrene latex (diameter of 1090A) is equal to 4.1 x 10"8 cm2/sec. at 21°C temperature. The angular dependence of the halfwidth of the light— beating spectrum of scattered light from polystyrene latex is shown in Figure 3. It can be seen that the halfwidths (”best fit") are on a straight line which goes through the origin as the theory predicts. But the direction tangent of this line is about 20% larger than the predicted one from the combination of Equations 7 and 10. This difference can not have arisen from the dust effect (to be discussed in the next section), because the dust effect results in a decrease in the diffusion coefficient instead of an increase. The diffusion coefficient is inversely proportional to the radius of the spheres, a smaller radius gives a higher dif— fusion coefficient, which means a higher halfwidth in the lightbeating spectrum. This does not seem to be the prob— lem however, because Reed (1970) and others used the same latex and they claimed good agreement between the Calculated and measured diffusion coefficient. There is apossibility that the polystyrene latex sample was too old, so the sus— pension agent did not work satisfactorily which possibly 24 fiiigb shié?) auuéh Figure 3. Angular dependence of halfwidth of the light— beating spectrum of polystyrene spheres (d = 1090A). 25 caused sedimentation of the polystyrene spheres, which means the concentration of spheres in upper regions in the sample decreased and, considering the polydispersity of the latex, smaller spheres stayed in upper region of the sample. Both effects would increase the diffusion coefficient. The concentrated polystyrene latex was purchased from Dow Chemi- cal Company about ten months before its usage. It was kept in a refrigerator. The final solution was made four weeks in advance and it was kept at a constant temperature of 21°C. Estimating an average of 3% reading error for each reading point (this is reasonable because of sharp changes in the curve), a range of halfwidth of lightbeating spectrum can be given (Table l and Figure 4). The calculation method is explained in its own section. Each Lorentzian curve with halfwidth in the ranges given in Table 1 fits the light? beating spectrum curve, so that the average error for each reading point is less than 3%. It can be seen in Table 1 that the range of the total.noise is very narrow, for each scattering angle, so the halfwidth ranges with almost con— stant error do not come from the difference of the "total noises". (The photocurrent of a.lightbeating spectrum to the "total noise" ratio is about 2-35 depending on the fre— quency.) Considering the above facts, especially that using the halfwidth method range brings together the values of the 26 Table 1. Range of halfwidths of lightbeating spectra of light scattered from polystyrene spheres in water. Scattering Range of Range of "Best fit" angle (°). _halfwidth (Hz) ."total.noisef .halfwidth (Hz) 35 140-190 0.65-0.69 165 45 230-310 0.95-1.14 270 60 390-530 0.63-1.11 470 27 In" width 0 theoretical halfwidth from diffusion an _ coefficient calculated from Stokes- Einstein's equation 4m» JL :fifegl imigb 'mié!) Figure 4. Angular dependence of halfwidth of the light- beating Spectrum of polystyrene spheres (d = 1090A). 28 calculated and the experimental diffusion coefficients, it was assumed that the lightbeating spectrometer described earlier with the spectrum analyzer can be employed for diffusion measurements of polymers in dilute solution. » C. SAMPLE PREPARATION--DUST PARTICLES IN THE DILUTED POLYMER SOLUTION It is believed that fornaccurateflightbeating:measure— wants, a "dust free" solution is required. Dust may be introduced into solution in three ways: from the solvent; from the surface of the polymer; and during the solution preparation. The solvent can be distilled under special conditions, it can be microfiltered or it can be centri- fuged to make it presumably dust free. The procedure of preparing the solution may be done in principle in a "dust free" environment.' These methods are not'practicalyfithey are very time-consuming and the results are‘doubtful.‘ To remove the dust from the surface of a polymer (meaning com— mercial polymers) is possible only by making a.dilute solu— tion and either centrifuging or microfiltering it.- This means a cleaning procedure for the solvent alone is not necessary, and the "dust free" environment is not important, because all of the dust particles can be removed from the final solution. 29 . Reedr(l970) used 0.45u pore size Millipore filters for filteration of polystyrene/cyclohexane solutions. Kramer '(1971) reported that he removed dust particles by centrifu- gation. The author of this work recognizes the problem of the presence of dust particles in the solution during a light— beating experiment but he argue3“that:the filtration or centrifuging of the solution is not a practical answer to this problem. The argument is the following: The pore size of the microfilter.must be large enough to pass the largest of the polymer molecules,*otherwise‘somefipolymer molecules would be filtered out.‘ The exact size of polymer molecules in solu— tion can only be estimated because accurate methodse-either experimental or calculation--for their determination do not exist.. Therefore, it is necessary to use some kind of safety factor to determine the minimum size of the filter pore. Particles with a much larger size than polymer molecules remain in the solution after filtration. It means that the solution is not "dust free".‘ Only the average size of the dust particles is reduced by the filtration. In this way, to assume the solution to be "dust free"-and give no consid- eration to the particles in the solution leads to error. This work followsranother,.neWVapproach'tO'overcome the problem of the presence of dust particles in the solution. It was mentioned earlier that the dust can be introduced 30 . into the solution by the solvent, polymer or the environ- ment. In the environment two factors, air and glassware, should be considered. An air filtered room minimizes the dust in the air. Good cleaning procedures combined with rinses by solvents should provide "dust free" glassware, or more exactly, the dust concentration should be related to the dust concentration in the solvent. The other possibil- ity of introducing dust into.the solution is from the polymer. Dust particles cover the surface of the polymer grains. (Commercial polymers, used in this'work, were in granular form.)W Therefore the surface to volume ratio is small, which means the volume of dust particles, which is directly proportional to the surface of polymer, is also small compared With the volume of polymer. The copolymers were made at the.Chemical Engineering Department of Michigan State University, they‘were kept away from dust during the preparation process, hence only a very low'amount*of'dust was present. The process of solution of polymers and dilu— tion of the solutions reduced the concentration of“dust originating from the polymer. The weighed polymers were put into about 80 cc of solvent and were shaken well in a volumetric flask.‘ The polymer solution was then left to stand overnight and then it was diluted to 100 cc and shaken well. After shaking, 31 the solution was not moved for a minimum of 24 hours to permit the sedimentation of any larger dust particles which might be present. The required solution volumes were carefully withdrawn using a pipette so that the solu- tion on the bottom (about 25% of total volume) was not disturbed or moved into the other flasks where the final diluted solutions were made. The original solutions were further diluted by solvents 2 to 25 times depending on the required concentration. The final solutions in the sample cells were left undisturbed for a minimum of 24 hours prior to the light scattering measurements so that additional particle sedimentation might take place. The height of the scattered volume from the bottom of a sample cell was the same for each measurement--about 10% below top level of the solution. According to Stoke's sedimentation law for non-uniform size particles, the smaller ones would be found in the top layers of the solution. Because of the two effects--sedimentation and dilu- tion--it may be assumed that the volume of the particles of dust in the solution which came from the polymers was negligible compared with the volume of dust particles coming from the solvents, and that the dust particles in the scat— tering volume came from the solvents. Therefore, it‘would seem that solvent filtration would reduce the problem of dust particles in the scattering volume. But since absolute 32 filtration does.nctvexist,fifiltration“of the solvent is not the best answer to the dust problem. The author's feeling is that the following situation during centrifuging of the polymer solution will result in changes in the polymer-solvent system: During centrifuging sedimentation of both dust particles and polymer molecules can occur as a result of dragging. The probability that this will happen depends on the relative sizes of the dust particles and polymer molecules. For a given size of dust particle the probability would be higher for larger polymer molecules. Therefore for solutions of polydisperse polymer, molecules with lower molecular weights might be expected in the higher region of the solution after centrifuging, hence the measured diffusion coefficient will be higher than that which corresponds to the average molecular weight. In the following, the method is shown which was used in this work to compensate for the presence of dust particles in the scattering volume. Using unfiltered pure solvents without any polymer as a scattering sample, the total photocurrent spectra have been obtained, and they are shown in Figures 5, 6, 7, and 8 for different solvents. It can be seen that for benzene and decalin, except below 100 Hz, the photo- current spectrum is approximately a horizontal line. In the same way for methylethylketone and dimethylformamide the 33 PIIOTIICIIIIEIY POWER RELATIVE l I l I .1 l. _J l I zoo 400 so: :00 1000 1200 1400 I: Figure 5. Photocurrent spectrum of light scattered in benzene (scattering angle 25°). 34 I ruorocuutut POWER I RELATIVE l I I I I I I I zoo too :00 so. 1000 1200 1400 1000 N1 Figure 6. Photocurrent spectrum of light scattered in methylethylketone (scattering angle 25°). Figure 7. PROTOCURREIT POWER RElRTIVE 35 I I 1 1L 1 I I I no ooo coo o o mu :2» mo moo Hz Photocurrent spectrum of light scattered in dimethylformamide (scattering angle 25°). 36 h; I z I 2 I E’l .. I §_ r . I, 1 I I I I I I I I I 400 800 1200 1000 Figure 8. Photocurrent spectrum of light scattered in decalin (scattering angle 25°). 37 PIOTOCURRERT nttorlv: I I I .11 I I I I zoo ooo ooo :00 1000 1200 nooo loco Figure 9. Photocurrent spectrum of light scattered in water (scattering angle 25°). 38 photocurrent spectrum is approximately a horizontal line at frequencies above 200 Hz. This means the total photo- current spectrum is constant over 200 Hz for all the solvents of interest here, even with the assumed presence of dust particles. Earlier, it was explained that the dust particles in solution come mostly from the solvent. Hence, those horizontal line spectra of the solvents can be considered as "total shot noise" which arises from the . instrumentation and the solvent. .Therefore, to subtract this "total shot.noisef'from the photocurrent.spectrum of scattered light of the polymer solution, the lightbeating term of the spectrum can.be obtained. Kramer (1971) has investigated the presence of dust particles in methylethylketone.. He measured the total photocurrent spectrum of scattered light from unfiltered, filtered, and centrifuged methylethylketone, and found that the microfiltered (45002 pore size filter) methylethylketone has a spectral component below 200 Hz, which cannot be recorded for centrifuged (2 hours at 100 x G) samples; His data shows agreement to data.of this work. (For unfiltered, uncentrifugated methylethylketone no spectral component over 300 Hz was found in the total.photocurrent. The subtraction of."total noise" from the total.photo— current spectrum is not simple for.the following“reasons. 39 The absolute value of the photocurrent spectrum is not known, it depends on the intensity of the laser beam, the power current of the photomultiplier and the attenuation of the input to the spectrum-analyzer for a constant scatter- ing angle for a given sample. Input and output limitation of spectrum-analyzer require changes in photomultiplier current and the input attenuator of the spectrum-analyzer for each measurement. It is not necessary to know the absolute value of the photocurrent spectrum, because the information desired the shape, and thus the halfwidth of the Spectrum, does not depend on the absolute value of the spectrum. Therefore, the calculation of the halfwidth and the estimation of "total noise" can be done by computer which is discussed later. D. CALCULATION As mentioned earlier the photocurrent spectrum for homodyne detection of a Lorentzian optical spectrum con« sists of three terms: shot noise, the d.c. component and the lightbeating spectrum. If the resulting current in the load resistor is fed through a capacitor, then the d.c. component is blocked out, and the photocurrent spectrum is the sum of the lightbeating spectrum and shot noise. To obtain the lightbeating spectrum, shot noise must be sub- tracted from the total photocurrent spectrum. The shot 40 noise level, sometimes called white shot noise level, can be determined in two ways: (1) By measuring the spectrum at high frequencies, i.e., frequencies beyond which the signal level is significant. The shot noise level is inde— pendent of frequency, therefore, practically, measurement of the photocurrent spectrum at high frequencies where the photocurrent is constant gives the level of shot noise. (2) The sample is replaced byAa light bulb, which is used as a light source. The light bulb serves as a white noise generator because the signal spectrum for the light bulb as a light source is negligible compared to the shot noise. In the following the method used to calculate the diffu- sion coefficient in this work is discussed. It can be;seen in Figures 5, 6, 7, and 8-that the spectral component below 300 Hz is a smooth curve. Assuming uniform dust particles size (which is a reasonable assump— tion for a small scattering volume considering the process of sedimentation), this spectral component could arise from light scattered from dust particles. These dust particles are present in the polymer solution also. Therefore, the spectrum of light scattered from a polymer solution consists of the sum of two single Lorentzians; one corresponds to 41 the polymer, the other to the dust particles. If the spec- trum of the scattered light consists of the sum of two single Lorentzians for homodyne detection the lightbeating part of the photocurrent spectrum consists of the sum of three Lorentzians. Cummins gt_al. (1969) present the equation = 2 2P :fi (F‘+F‘)ffl‘ PLB 2(1) (Bo +B2)z ('fi m+4Fo + 2B°Bzw +(F0+F2) + B2 2. 2.1-. . :TI'. > 12 . w +4112 where PLB is the photocurrent of the lightbeating part of the spectrum To and P2 are the spectral halfwidths of the two Lorentzians in the scattered light spectrum and Bo and B2 are the corresponding intensities. It can be seen that there are four variables B0, B2, F0, and T2 in Equation 12. Considering the total photo- current spectrum which is the sum of the lightbeating spectrum and the "total noise" there is a fifth variable, the "total shot noise". Hence to calculate the halfwidth of the scattered light from the polymer molecules the curve— fitting computer program must include five unknown para— meters. Using only 15-20 data points, the uncertainty of these parameters would be very high and a large error in 42 the analysis of polymer diffusion would result. Before this difficult problem can be discussed, a simple case should be investigated; yig.,a single Lorentzian in the spectrum of the scattered light. In this case, the total photocurrent spectrum of scattered light is given by Equation 10, which may be written P (w) = 4757”“ 13. 1+? where A is a constant, w is the frequency, B is the half- width of the lightbeating spectrum, and D is the "total shot noise". The photocurrent is known for each frequency from the experimental spectrum, so there are only three unknown parameters: A, B, and D. The least squares estimate of halfwidth of the light- beating spectrum of scattered light can be calculated by the computer program given in the Appendix, provided the scattered light spectrum consists only of a single Lorentzian. Assuming that a uniform size of dust particle is present in the scattering volume, the spectrum of light scattered from dust particles will consist of only a single Lorentzian, provided the sample consist of solvent and dust only. It means the halfwidth of the lightbeating spectrum of scattered light from the dust particles can be calcu— lated, so the number of the unknown variables in Equation 12 is reduced from five to four. But four variables still 43 are too many to calculate an accurate halfwidth of the lightbeating spectrum of light scattered from polymer mole- cules. (Kramer [1971] was unable to obtain reasonable halfwidths by fitting the lightbeating spectrum of filtered [dust remained] dilute methylethylketone solution of poly- styrene.) Therefore, the number of variables should be reduced from four to three. The simplest way to do this is to calculate the halfwidth of the lightbeating spectrum of light scattered from polymer molecules without considering the presence of dust particles. The question is--how much error is introduced by this simplification? In other words, what is the difference between the halfwidth calculated using Equation 12 and that which is calculated by Equation 10, considering the same shot noise? It can be calculated by setting the two equations equal to each other. It means calculating the halfwidths from each equation at a point of the spectrum, which is obtained by actual measurement. Dividing both sides by 2 2/n 1 2 2r n+1"2 yer) manna.)2 (3° w2+4r02 + ZB°Bsz+(ro+r2)‘ + 32 w2+4'P_L7-2 _ 2.x ' 7.171%! 14- Where 2Fo and 270 are the halfwidths of the lightbeating spectra of the light scattered from polymer molecules calcu- lated from the different equations. 44 Equation 14 may be simplified by use of the following assumptions: 1. The intensity of-the light scattered from polymers, B0, is much higher than the intensity of the light scattered from dust particles, B2. This is true because the intensity is a function of the concen- tration and is independent of the halfwidth of the spectrum. It was mentioned earlier, that the dust concentration is much lower than the concentration of polymers.- This conclusion was supported by observation of the relative intensity and analyzer attenuator settings while recording the spectra of solvent and polymer solution. 2. The frequencies under consideration are much higher than the halfwidth of the lightbeating spectrum of the light scattered from dust particles. For fre- quencies over 300 Hz, the light beating spectrum of light scattered from dust particles is essentially zero, as shown earlier. Using these assumptions Bzzaf%%%:2 becomes negligibly small compared with other terms in Equation 14. If 82 = BB0, T2 = GPO and w = GPO then Equation 14 is reduced to . 1. .1. + J(l+a) o): .......... 15 (1+8; (3 +4SF0 (5 jig(1'4"!)er 3 1o +4Yo . Next it is necessary to determine the maximum values for a B and 6. 45 Using a relatively very low diffusion constant, D = 1.0 x 10-7 cmz/sec in methylethylketone at 25° scat- tering angle, the theoretical halfwidth of the lightbeat— ing spectrum of light scattered from polymers is about 200 Hz. The estimated halfwidth of the lightbeating spectrum of light scattered from dust particles in methyln ethylketone is about 20 Hz at 25° scattering angle, so the value of a is not larger than 0.1. To obtain the same photocurrent a minimum of 6 dB attenuation had to be used, when the sample was a dilute polymer solution, compared with dust so the input was reduced at least by a factor of four. Therefore, the intensity of light scattered from a polymer solution is at least 16 times larger than the intensity of light scattered from dust particles only hence so 8 is less than 0.1. 6 can be varied between 0.5 and 8, depending on the value of the halfwidth of the lightbeating spectrum of the light scattered from the polymer molecules. The yo/Fo ratio was calculated using the above assump- tion, and typical values are shown in Table 2. It was found that the halfwidth calculated from Equation 10 is smaller by less than 5% than the halfwidth calculated from Equation 12. It was concluded therefore that the experi- mental light spectrum of light scattered from dilute polymer solution can be fitted by a single Lorentzian, 46 Table 2. Ratio of halfwidth of lightbeating spectra calcu- lated from single and three Lorentzians. Yo a B 6 T? 0.05 0.02 0.5 0.972 1.0 0.984 3.0 0.968 5.0 0.977 7.0 0.979 0.05 0.05 0.5 0.939 1.0 0.966 3.0 0.925 5.0 0.946 7.0 0.950 0.10 0.02 0.5 0.974 1.0 0.985 3.0 0.970 5.0 0.978 7.0 0.980 0.10 0.05 0.5 0.943 1.0 0.967 3.0 0.930 5.0 0.949 7.0 0.953 0.50 0.10 0.5 0.959 1.0 0.971 3.0 0.928 5.0 0.944 7.0 0.947 .............. 47 without a significant loss in precision. In this way all of the spectra were fitted by Equation 10, using a FORTRAN program for the CDC 6500 Computer. This conclu- sion is further justified by the reasonably good answers obtained for the polystyrene latex calibration runs using only a single Lorentzian calculation. The computer printout, Table 3, gives the different halfwidths, with the "total shot noise", which results in a minimum for the function i .PTi,r.PEi Z (———-) i=1 PEi where PEi is the experimental photocurrent at the ith frequency and PTi is the value calculated from Equation 13. Most authors average the diffusion coefficients obtained from measurements at different scattering angles. Each dif- fusion coefficient for a given scattering angle is calcu- lated from Equation 7, using the halfwidth of the best fitting curve calculated by computer. This work follows another approach to calculate the diffusion coefficient from the halfwidth of the lightbeating spectrum of light scattered from polymer molecules. The reason for using the alternative approach will become apparent below. Using 2 cycle semilog graph paper to record the total photocurrent spectra, an average of 12% reading error is 48 Table 3. A typical computer printout used for calcula- tion. Columns from left to right: 1. halfwidth, current power at zero frequency, 3. 4. sum of the least square errors. 2. photo- "total noise", OMEGA = frequency l-EXP = experimental photocurrent power l-CALC = calculated photocurrent power E(l) = the least square error 20.0000 3300.990] 1.297510 .010137 50.0000 543.1131 1.209901 .014905 00.0000 216.0151 1.276117 .012815 110.0000 117.2141 1.256311 .010190 140.0000 74.736? 1.230973 .007453 170.0000 52.6567 1.20060? .005017 200.0000 39.7103 1.165724 .003200 230.0000 31.4811 1.126860 .002590 260.0000 25.9074 1.004502 .003172 290.0000 21.9555 1.039000 .005170 320.0000 19.0484 .991002 .008665 350.0000 16.8453 .940563 .013621 350.0000 15.1347 .088037 .019972 410.0000 13.7794 .533634 .027510 440.0000 12.0075 .777517 .030399 470.0000 11.7953 .719813 .046191 500.0000 11.0570 .660613 .050837 N0. OF 0010 GEARCHES AT FACH 0F... 230.000 232.0650 31.0277 1.124051 .002587 ourna 1-Exo I-CALC F111 (600.00 9.10 80°“ -0017819 500.00 6.50 0.02 .018971 600.00 5.10 5.16 .012007 700.00 4.10 4.20 .023541 000.00 3.00 3.53 -.018805 1300.00 2.10 7.08 -.0os448 1400.00 1.95 1.95 .001944 lqOOOOO ‘opq Ions -.000393 1000.00 1.73 1.76 .019204 49 reasonable. Therefore for 13 data points the total error 13 PT. - PE. 1. J. 2 ( )2 i=1 PE: can be as much as 0.0052. This means that if an experi- mental spectrum is a perfect match to the theoretical spectrum, reading the points and calculating the halfwidth can cause an error as great as 0.0052. Therefore, each and any halfwidth with error equal to or less than 0.0052 for 13 data points may be the reguired halfwidth; more exactly, instead of one value of halfwidth a range of values of halfwidth should be used. Of course, when the total error is larger than 0.0052 for all of the calculated half- width the halfwidth with minimum error should be used. As was explained in a previous chapter, plotting the half- widths of lightbeating spectra of light scattered from polymer molecules against the sine squared of half of their corresponding scattering angle should give a straight line which goes through the origin. Figure 10 shows the first step of a typical calcula- tion. The halfwidth with minimum total error is 392 Hz. But between 340 and 440 Hz the total error is less than 0.0052; therefore, this is the range of the halfwidth. Figure 11 shows the next step in the calculation. The ranges at 25° and 35° were plotted and a range was found 50 ll! F’ L— 0.005 ..----: """"" """" _ 5 0.002 - 5 _ 5‘340-«0 3: l in l 31 I J 300 400 500 Ill Figure 10. Sum of the square error of calculated halfwidths. Monsanto SAN copolymer MH 250,000 in methylethylketone ( 0.3 g/lOOg) scattering angle is 25°C. 51 In! ... error less than 2% for each data point "r“ at one angle. 1 -—-error less than 2% for each data point at two angles. 8" O halfwidth calculated from "best fitting". 60' g i : 3: I”. 5 Ifli siI1(-z—) sin ( 2) Figure 11. Angular dependence of calculated halfwidth. Monsanto copolymer (MW = 250,000) in methyl- ethylketone. 52 for which the average error for each point at both angles was no greater than 2%. The range was 340-410 Hz for 25° scattering angle and 680-790 for 35°. The calculated range for diffusion coefficient was 1.68-2.05 x 10-7 cm2/sec. The diffusion coefficients calculated from the "best fitting" halfwidth minimum error) were 1.28 x 10.7 cm2/sec for 35° and 1.92 x 10"7 cmz/sec for 25° scattering angle, and the average is 1.60 x 10-7 cmZ/sec. The original value differ from the average by i 20%. The author feels that using this range method where it is possible, gives more realistic results. Therefore, a range is shown for most of the diffusion coefficients. Where this is not possible, because of reasons mentioned earlier, only the average diffusion coefficients are given. In these cases the difference between the average diffu- sion coefficient and the original one is less than :8%. To eliminate the possibility of error arising from the optical anisotropy of the sample cell (dirt or fault) each measurement was done twice, with the cell turned between measurements. (This changed the horizontal posi— tion of the scattering volume also a little bit, so different polymer molecules were in the scattering volume during the second measurement.) Both spectra were recorded and the two curves were matched. Only those spectra have been used in this work which matched each other perfectly. 53 E. DIFFUSION CONSTANTS OF THE POLYMER MOLECULES IN DILUTE SOLUTION Investigating the concentration dependence of the diffusion coefficient of polystyrene (MW = 3.5 x 106 and 5 x I06) in tetrachloromethane and methylethylketone Tsvetkov and Klenin (1958) obtained an S shaped curve plotting the diffusion constant against concentration. They found for very low concentration (lower than 0.07 g/lOOg) there is no change in the diffusion coefficient. In the region of concentrations from 0.012 to 0.6 g/100g, the diffusion coefficient shows a very sharp increase. For higher concentrations the rate of growth decreases and the curve approaches a constant value. The concentra— tion dependence of the diffusion coefficient is much less for poor solvents such as methylethylketone than for good solvents such as tetrachloromethane. Ford gt_gl. (1970) found that the diffusion coeffi— ,cient of polystyrene in methylethylketone above a concen- tration of about 0.35 g/lOOg is approximately constant. Below this the concentration dependence of the diffusion coefficient is described by the equation, D(c) = Do (1 + KDc) 16. Where Do is the value of the diffusion coefficient at the limit of zero polymer concentration and KD is a molecular 54 weight dependent constant. The KD was found to be posi- tive for molecular weights above 100,000 and negative for lower molecular weights. Schick and Singer (1950) published negative values of KD for MW = 9.5 x 104 and MW = 2.4 x 105, and positive values for MW equal or higher than 5.6 x 105 for polystyrene- methylethylketone systems. Using the same sample of poly- styrene in tetrachloromethane, KD was found to be negative for Mw = 9.5 x 104, zero for Mw = 2.4 x 105 and MW = 5.6 x 105, and positive for Mw = 6.8 x 105 and MW = 9.1 x 105. Schick and Singer (1950) observed negative KD values for polystyrene of Mw = 6.8 x 105 in decalin. Meyerhoff (1960) reported positive KD for polystyrene of MW.= 5.28 x 105 in methylethylketone. Both Singer and Tsvetkov show the theoretical deriva- tion of Equation 16. Ford 22 31. (1970) derived a theo— retical relationship between KD and the molecular weight for polystyrene-methylethylketone systems, but the agreement between the theoretically calculated‘KD and that calculated from actual measurement was very poor. The diffusion coefficient at the limit of zero polymer concentration, 00’ for polymers with different molecular weights was investigated by FordngLEl. (1970). They showed a relationship between DO and-the molecular weight in the form of 55 D = K M. 17. Where K0 is a constant and M is the molecular weight. For polystyrene-methylethylketone systems K0 was found to be 4 cmz/sec. and b was found to be equal to 3.1 i 0.2 x 10' equal to 0.53 i 0.02 at 298°K (average molecular weights were used). Equation 17 is similar in form to Staudinger intrine sic viscosity equation Where In] is the intrinsic viscosity, and K and a are constants. But because the coefficient of b is —1, Do is inversely proportion to In]. For polystyreneemethylethyl— ketone systems a is in the range of 0.58 to 0.635. Schick and Singer (1950) found b equal to 0.53 for a methyl— ethylketone system which is in exact agreement with the b value found by Ford‘gg‘gl. (1970), and for polystyrene— tetrachloromethane b was reported to be 0.59 by Schick and Singer (1950). Reed (1970) gave a value of b = 0.51 and Ko = 1.20 for a polystyrene—cyclohexane system. Ford 25‘31. (1970) showed a method to calculate the unperturbed dimensions of polymer molecules in dilute solu- tions. Two equations are given (the difference is whether the expansion factor molecular weight dependence is calcu- lated by the Flory (1953) or the Kurata and Stockmayer 56 (1960) equation). Plotting (Dom-Il against DO3M2 or .r.2 g(af)MDo the solvent independent ratio (—§f0% can be calculated,(g(af) is a function of the expansion factor), (E62) is called the unperturbed dimensions of the polymer molecules. The (;%)8 ratio was calculated for a poly- styrene-methylethylketone solution and was found to be (800 $.40) x 10.11 cm at 298°K. This value is higher than, but of the same order of magnitude as, the result calculated from viscosity measurements which was 670 x 10-ll cm. IV. RESULTS The principal goal of this study was to investigate the diffusion coefficient for different polymer solvent and copolymer-solvent systems but without a detailed study of any of them, and to develop the measurement procedure and calculation method for lightbeating spectroscopy, which could be a base for further investigation. Therefore, the diffusion coefficients of five concentra- tions for each of five specific monopolymer and six co- polymer samples each in three or four solvents were determined. Although complete calculations, such as a determination of the diffusion coefficient at the limit of zero concentration could not be done, estimated values for certain systems are given. All of the measurements were done at 21°C temperature. The necessary basic in- formation about the polymers and solvents are given in Tables 4 and 5. All of the measured diffusion coeffi- cients are given in Tables 6, 7, and 8. Polystyrene-Methylethylketone Below a concentration of 0.2 g/lOOg the diffusion coefficient in methylethylketone definitely increases with increasing concentrations of polystyrene in the solution 57 58 Table 4. Molecular weights of polystyrene and polystyrene- acrylonitrile copolymer used in this work. ............. M x 10'3 MW x 10'3* Polystyrenes UC 000 57.7 130.0 UC 010 37.9 80.0 UC 030 102.7 271.0 UC 040 117.8 338.0 MS-l90 103.0 185.0 Polystyrene-acrylonitrile copolymers. MS-14 141.0 275.0 MS-23 120.0 203.0 MS-37 205.0 332.0 MS-22 339.0 634.0 MoI-25 80.0 247.0 MoII-25 107.0 325.0 .......... UC Union Carbide Michigan State University Mo .Monsanto E from Gel Permeation Chromatography Number average molecular weight Weight average molecular weight 59 Table 5. Density and refractive index of the solvents used in this work. Solvent Density g/cm3 Refractive 20°C Index ... Benzenel 0.879 1.501 Methylethylketonez 0.805 1.380 Dimethylformamide3 0.944 1.427 Decalin (cis/trans)4 0.896 1.475 1 Baker (Reagent) 2Fisher (Certified) 3Mallinckrodt (Reagent) 4 Mateson-Coleman_(Practical) Table 6. 60 Diffusion coefficient of polystyrene in different solvents. Concentration Difquion coeffi¢ient3c1057‘cm?[sec. (9/1009) ._fBenzene. MEK ..,DMF 0.02 3.60 0.05 4.35 UC 010 0.10 3.89 2.47 0.20 2.65 4.90 2.30 0.50 3.84 4.65 2.58 0.12 2.80 3.02 1.38-1.89 0.30 2.12 2.44 1.88 UC 000 0.60 2.22 2.84 2.00 1.20 2.25 2.85 2.08 3.00 3.10 2.46 2.25 0.02 1.01-1.18 MS-190 0.10 2.18 1.74-3.80 0.20 1.40-3.10 1.94-3.42 0.50 1.60 1.89-3.05 0.12 1.23-2.12 1.68-2.05 1.15 0.30 1.28-1.87 1.31-1.89 1.31 UC 030 0.60 2.68 1.47-2.22 1.20 1.92 1.31-1.95 1.34 3.00 1.42-2.46 1.31-1.93 1.38-1.78 0.02 . 1.46 0.55-0.97 0.05 1.18 1.58-1.80 1.15-1.18 UC 040 0.10 1.01-1.48 1.43-1.55 1.07-1.36 0.20 1.14-1.42 1.88-2.13 0.50 1.14-1.34 l.88-2.19 1.01-l.16 61 Table 7. Diffusion coefficient of polystyrene-aorylonitrile in different solvents. ...... Concentration Diffusion coefficient x107 cmzjs'e‘c; (g/100g). .Benzene . MEK... ....... DMF 0.02 1.60-1.90 1.34 0.05 2.38-3.05 1.02-1.38 MS-14 0.10 0.90-1.89 2.62-3.20 1.24-2.00 0.20 1.10-2.10 2.74 0.95-1.66 0.50 1.40-1.80 2.79-3.37 1.51 0.02 2.24 0.05 2.22 1.67 MS-23 0.10 1.86 2.84-3.66 1.34—1.57 0.20 1.71 4.20 1.12—1.33 0.50 1.84 3.61 1.93 0.04 2.30-2.95 1.25-1.67 0.10 2.08 1.48 MS-37 0.20 2.24-3.06 0.84-1.70 0.40 1.98 1.40 1.00 2.93 1.45 0.10 1.42 MS-22 0.50 1.57-1.73 0.04 1.15-1.52 0.64—1.15 0.10 1.37-1.98 0.43-1.15 MS-22 0.20 1.67-2.56 0.78-1.21 0.40 1.64-2.56 0.61-1.98 1.00 2.40 1.30-1.84 0.06 1.59-2.08 0.15 1.38-2.90 MoI-ZS 0.30 1.68-2.09 0.60 1.72 1.50 1.78 0.12 3.25 l.01—1.56 0.30 3.70 1.64 MoI-25 0.60 3.66 2.03 1.20 3.41 2.28 3.00 4.00 2.80 continued 62 Table 7--continued ..................................... Concentration Diffusion coefficients x1‘0‘7' cm'zj'se‘c. (g/lOOg) Benzene .MEK .. . ... ..DMFT 0.02 1.55-1.75 0.05 1.44-1.50 0.20 1.75-2.30 0.12 2.21 1.13-2.00 0.30 2.70-3.12 2.18 1.20 2.58 1.95 3.00 3.02-3.18 63 Table 8. Diffusion coefficient of_polystyrene in deCalin. ............................................. Concentration Diffusionrgoefficient x 107‘cmzjsec. (g/lOOg) UC 000 UC 010 UC 030 . .UC.040. 0.05 0.8-l.37 0.91 0.10 1.02 1.23 0.96 0.50 0.20 0.56 0.56 0.76 0.37-0.73 0.50 0.63 0.75 0.66 0.32-0.68 64 (Figures 12, 13, and 14). Over 0.2 g/lOOg concentration the diffusion coefficient of polystyrene in methylethyl- ketone seems to be approximately constant (Figures 15 and 16). The exact determination of the diffusion coefficient at the limit of zero polystyrene concentration, Do is very difficult because only five concentrations were used, and the extrapolation to zero concentration involves a large uncertainty. An estimated range of Do can be given: Polystyrene MW Do x 107 cmZ/Sec UC 010 80,000 3.60-4.00 UC 000 130,000 2.70-3.00 UC 030 271,000 1.65-1.85 Plotting log Do against log MW, the directional tangent range is 0.63-0.66 with a negative sign, which is equiva- lent to -b in Equation 17. This value is higher than the value of b reported in the literature (0.53), but con— sidering the small amount of data and the use of Do ranges instead of fixed values, the agreement probably is satisfactory. Polystyrene-Benzene It may be concluded that the diffusion constant is not a function of the concentration for the polystyrene—benzene system in the range 0.12 to 3.0 g/100 9 concentration (Figures l3, 14, 15, and 16). Do may be estimated only for 65 5.0 " 0m1 0 «fine 0 4-0 - O O 3.0 - )0 A A 2.0 - 1-0 '- x benzene A dimethylformamide O methylethylketone l I l L J l I 0.02 0 l 0.2 0.3 0.4 0.5 one. gnoog Figure 12. Diffusion coefficient of polystyrene (MW = 80,000) in different solvents. 66 fim- 0'10 on 2I000 9 33 b 15 2.0 )- 6 10 X ...1 X benzene A dimethylformamide O methylethylketone L, I I I I I 002 04 04 03 04 ms c0nc.g1100¢ Figure 13. Diffusion coefficient of polystyrene (Mw = 185,000) in different solvents. 67 LS Oil. 0.1! 10 9 ° 6 4 1.0 L L A X benzene A dimethylformamide O methylethylketone .1 l I .1 I J 0.05 0.1 0.2 0.4 0.5 cone. gllOOg Figure 14. Diffusion coefficient of polystyrene (MW = 338,000) in different solvents. 68 10"9’ 01:101 0 Inc K o 0 0 X x 40 I 2 o - b A» 1.0 ‘- X benzene A dimethylformamide O methylethylketone | | | | l I I . 0-3 -6 I. 2.0 3.0 cone. glloog Figure 15. Diffusion coefficient of polystyrene (MW = 130,000) in different solvents. 69 30" 01107 cullsOfi K 201 I? X benzene A dimethylformamide 0 methylethylketone I I I I 0 0 0 L0 20 10 0000.311003 Figure 16. Diffusion coefficient of polystyrene (Mw = 271,000) in different solvents. 70 UC 040 which is about 1.15-1.35 x 10-7 cmZ/sec. and for UC 030 which is about 1.60-1.90 x 10..7 cmz/sec. In this case the exact molecular weight dependence of Do cannot be obtained but it does decrease with increasing molecular weight as expected. Polystyrene-Dimethylformamide The data show that the diffusion coefficient for poly? styrene—dimethylformamide tends to increase with increasing concentration below 3.0 g/lOOg (Figures 15 and 16), but for Mw = 3.38 x 105 it is nearly constant (Figure 14). Ranges of D0 which can be estimated 7 2 Polystyrene MW ‘ f .. Do x110 ‘cm /sec UC 000 130,000 1.65-1.85 UC 030 271,000 1.00-1.25 Polystyrene-Decalin The available data indicate the diffusion coefficient for the polystyrene-decalin system is unchanged or decreases with increasing concentration, and definitely does not increase in the range of concentration from 0.05 g/lOOg to 0.5 g/100g (Figure 17). A precise determination of Dd_is not possible, but the data show positively that-Do is de- creasing with an increase of molecular weight. The range of Do is about 0.5-1.3 x 10-7 cmz/sec. for the molecular weight range of 80,000 to 338,000. 71 1.4 ' ‘ X MW = 130,000 '3x107 0 MW = 80,000 0 law 1.2 _ 0 A MW - 271,000 El MW = 338,000 LOP- X II 0.8- i ‘ o 015- ‘3 0 II 04- 02 F' I I II I I 0.1 0.2 0.3 0,4 o_5 conc. glloog Figure 17. Diffusion coefficient of polystyrene in decalin. 72 Polystyrene-Acrylonitrile in Methylethylketone As in the polystyrene-methylethylketone system the diffusion coefficient for the polystyrene-acrylonitrile in methylethylketone below of 0.2 g/lOOg concentration increases with increasing concentration of copolymer and over 0.2 g/lOOg concentration approaches some constant value as can be seen in Figures 18, 19, 20, 21, and 22. The molecular weight dependence of Do cannot be given because Do cannot be estimated with the required accuracy. Polystyrene—Acrylonitrile CopOlymer in Benzene It seems that below 1.5 g/lOOg concentration the diffusion coefficient for polystyrene-acrylonitri1e co- polymer in benzene does not change with increasing concen- tration (Figures 18, 19, 20, and 21). Although a precise Do determination is not possible because of the small number of data points, it can be seen that Do is decreasing with increases of molecular weight. But because of the relatively small range of molecular weights (2.03 x 105 — 6.34 x 105) this decrease seems small. Polystyrene-AcrylonitrilefiCopolymer'in Dimethylformamide The diffusion coefficient slightly increases with in— creasing polystyrene-acrylonitrile copolymer concentration (Figures 18, 19, 20, 21, 22, and 23). The molecular weight dependence of Do can be estimated using the ranges of Do' 73 “13' 0.12!ch 40" 30- “ I 1.0 - X benzene A dimethylformamide O methylethylketone I I I I iL I 0.02 0.1 0.2 0.3 M 0.5 cone. glloog Figure 18. Diffusion coefficient of polystyrene— acrylonitrile copolymer ( = 203,000, 23% acrylonitrile content) in different solvents. 74 L0 ' 0 0x101 cmzlsea ° 0 o O 30 F L 1 a A» 2.0 i ‘ E L x 0. L0 X benzene A dimethylformamide O methylethylketone I I I I I I -l 03 06 L0 10 $0 conc.glloog Figure 19. Diffusion coefficient of polystyrene— acrylonitrile copolymer = 247,000, 25% acrylonitrile content) in different solvents. 75 35" 0'10 cuzluc a, IA A» L0 - X benzene A dimethylformamide O methylethylketone $07 I I I I I I 0.1 0.2 0.3 0.4 0.5 00010. (I100; Figure 20. Diffusion coefficient of polystyrene- acrylonitrile copolymer (MW = 275,000, 14% acrylonitrile content) in different solvents. 3.0 mm7 cmzlsan 10 L0 76 Figure 0 O o A I (A I: X benzene A dimethylformamide O methylethylketone I I I I I I L, I I I I L-I 0.1 0.3 0.5 0.7 0.9 1.1 To conc.1llooi 21. Diffusion coefficient of polystyrene- acrylonitrile copolymer = 325,000, 25% acrylonitrile content) in different solvents. 77 25 '— 0x107 0.1/sec to h- )l 1" + ”I . 11 I I I I 002 02 04 05 x benzene A dimethylformamide 0 methylethylketone L0 cone. glloog Figure 22. Diffusion coefficient of polystyrene— acrylonitrile copolymer ( = 634,000, 22% acrylonitrile content) in different solvents. 78 U 30 0:101 CIIZISQQ I) O 2.0 - O ‘P A. A 10 h A X benzene A dimethylformamide O methylethylketone I I I I I 004 0.2 0.4 0.5 10 cone. gllooi Figure 23. Diffusion coefficient of polystyrene- acrylonitrile copolymer ( = 332,000, 38% acrylonitrile content) in different solvents. 79 M. 0. -MS-14 275,000 1.10-1.30 MS-23 203,000 1.40-1.60 MS-22 634,000 0.70-1.00 MOI-25 247,000 1.20-1.50 Comparison of the Diffusion Coefficients for a Constant ConCentration'Of a Given Pol er (molecular weight'is constant) '1n Different Solvents Considering the four solvents benzene, decalin, di- methylformamide, and methylethylketone, it can be seen in Figures 12, 13, 14, 15, and 17 that the diffusion coeffi- cient for a given molecular weight and concentration of polystyrene is always the highest in methylethylketone and definitely the smallest in decalin. The second highest diffusion coefficient was measured in benzene and a slightly smaller diffusion constant was found in dimethylformamide. For acrylonitrile-polystyrene copolymer the diffusion co? efficient is also highest in methylethylketone. The diffu- sion coefficients measured in benzene and dimethylformamide are approximately equal. Comparison of Diffusion Coefficient of Polystyrene and‘Polystyreneeacrylonitrile‘(the‘same concentra— tion and molecular weight)'in'Different'so1vents Figure 24 shows the diffusion coefficient of polystyrene and polystyrene-acrylonitri1e as a function of molecular 80 .sowumnusmosoo mooa\m H.o um wsoumxawnno lawnuofi cw usmflwB “masooHOE mo sowuocsm mm HmEMHomoo mafiuuflcoamuom Imcwuhummaom paw msoumumxaom mo usoflowwmooo c0flmDMMflo mo somwummfioo .vm muomflm ...: 1.: vac mun gum flaw on“ an . _ u _ _ — a. an I cm .X .I O l_9m x Hoshaomoo x mswnhumaaom o * A; :u‘N.o 0 menu.— .1 Q6 81 weight for one concentration (0.1 g/lOOg) in methylethyl- ketone. It can be seen that the diffusion coefficient of the copolymer is higher by about 40-60% than the diffusion coefficient of the polystyrene. In Figure 25 it can be seen that the diffusion coefficient of the c0polymer and polystyrene are about the same in benzene. While the copolymer exhibits only a slight decrease in its diffusion coefficient with increasing molecular weight, the change is much larger for polystyrene. In dimethylformamide (Figure 26), the diffusion coefficient seems to be a little bit higher for the co- polymer than for polystyrene. 82 .coflumuusmocoo mooa\m H.o um momusmn cw uanoB HmHoooHofi mo soauossm mm Hmfihaomoo maflnuwsoaauom Iocmumumhaom use msmumumaaom mo musoflowmmmoo GOHmDMMHU mo somflHmmEoo .mm mhsmflm mural 3.... mun ans 3— a2 _ 4 1 n 41 . . IF: 2 t .. .I.Y~ 1.. O. mafiaaomoo 0HHHDHGOHNHUMImcmHMUmMHom x wamnmpmmHom o k 0 cans 9 n9: 83 .cowumuusmocoo mooa\m H.o um mowamahom Iamnpmsflo ca unmwos HmHsomHOS mo GOwuossm mm HmESHomoo wafluuwslouom Imsmuwummaom can msmnmummaom Mo musowowmmooo sowdeMflv mo GOmHHmmEoo ...: .. ene mun can can and .4 q . . _ _ _ Hoahaomoo mHHHUHCOHwnomumsmumummHom x wcoumummaom 0 .mm musmam Q— efi suxuln ”Guufl ea V. DISCUSSION The presence of dust particles in dilute polymer solutions and their effects on lightbeating spectroscopy measurements have been discussed in detail in a previous section. The calibration procedure was also covered earlier. This discussion focuses on the results of these measurements and their comparison with published data. As mentioned earlier, it was not the purpose of this study to obtain a precise determination of the diffusion coefficient at the limit of zero polymer or copolymer con- centration, but rather to develop a standard measurement procedure and calculation technique for the diffusion coefficient in order to give a base for further investi- gation. To the author's best knowledge no measurement of the diffusion coefficient of polystyrene-acrylonitrile copolymer by lightbeating spectrosc0py has been made prior to this work. As was mentioned in the previous section, the diffu- sion coefficient of polystyrene decreases or remains con- stant with increasing concentration in decalin (Figure 17). This result shows similarity to the data published by Schick and Singer (1950), who found the diffusion coefficient 84 85 concentration dependence of polystyrene at Mw = 6.8 x 105 can be described by the equation D(c) = 0.5 (l - 0.15c) where D(c) is the diffusion coefficient at a given concen- tration and c is the concentration of polystyrene in decalin in units of g/100 cm3. It can be seen because the coefficient of concentration (~0.15) is negative and small, the diffusion coefficient decreases very slowly with increasing concentration. The results of this work also show that KD in equation 16 is negative for a range of small molecular weights (80,000 to 332,000). This is similar' to the concentration dependence of the diffusion coefficient of polystyrene in methylethylketone found by Schick and Singer (1950) and Ford (1970). However, the sign of KD ‘ turns out to be positive at molecular weight larger than 5 x 105 in methylethylketone according to Schick and Singer (1950), Meyerhoff (1960), and Ford (1970) but remains nega- tive in decalin according to Schick and Singer (1950) and this work. When a polymer molecule is dissolved in a solvent the molecule will be extended depending on the degree of associ- ation between the solvent and the polymer. Dissolving the polymer in a "good" solvent causes the segments of the polymer to associate better with the solvent molecules than 86 with each other, so the total volume occupied by a single polymer cloud will be extended according to Rodriguez (1970). The intrinsic viscosity is directly proportional to the volume of the polymer molecule in solution according to Flory (1953), and Do is inversely proportional to the volume of the polymer in solution according to Tsetkov and Klenin (1958). These facts lead to the commonly known conclusion that the intrinsic viscosity is inversely pro- portional to Do' Schick and Singer (1950) give the in- trinsic viscosity and D0 of polystyrene.(Mw = 680,000) in different solvents. It can be seen that if values of In] are plotted against Do' the points are in a line for four solvents, but the point for decalin is considerably off the line (Figure 27). An examination of the diffusion coefficients of poly— styrenes with.different molecular weights, which is part of this work, reveals that higher values are found in methyl- ethylketone (relatively poor solvent) than in benzene (good solvent for polystyrene). A determination of the rank of the solvent (good or bad) can.be based on a comparison of the solubility parameters of the polymers and solvents. A smaller absolute value of the difference between the solubility parameter of polymer and solvent means a better solvent. Hansen (1967) reported solubility parameter values for different solvents.. A comparison of the 87 “Nil 0 toluene cull/$00 \ X ethylbenzene A carbon tetrachloride :Imethylethylketone O decalin 20 —' L0 - Figure 27. Relationship between the diffusion coefficient and the intrinsic viscosity of different polystyrene solutions (MW = 680,000). 88 solubility parameters of polystyrene and methylethylketone and.benzene shows that benzene is a very "good" solvent. and methylethylketone is a relatively "poor" solvent. The dissolved polymer molecules in "good" solvents are expanded more than in "poor" solvents, because the 'segments of polymers associate better with the solvent molecules than with each other. This expansion can be . defined as: ' I-2 8 a=<:———2 ’ 19. r o where (5'2)Li is the actual root-mean-square end-to-end distance, (35:)8 is the root-mean-square end-to-end distance for the unperturbed or unswollen dimension, and a is the expansion factor (Rodriguez, 1970). A relationship between the intrinsic viscosity and the expansion factor is given .. 2 3 .r 7 [n] = <-§—) M15 83 . 20. where [n] is the intrinsic viscosity, 0 is a universal con- by Flory (1953): stant and M is the molecular weight. It can be seen for a polymer with a given molecular weight, Equation 20 can be written as In] = C 03 , 21. where C is constant and independent of the solvent. 89 Therefore, the ratio of two intrinsic viscosities measured in two different solvents at the same tempera- ture is equal to the cube of the ratio of the correspond— ing expansion factors. Using the intrinsic viscosity data for polystyrene published by Kurata (1963), it was found that a in benzene is about 1.2 times larger than in methyl- ethylketone, which means the volume of a polymer molecule in benzene is about 1.7 times larger than in methylethyl- ketone. The diffusion coefficient is inversely proportional to the volume of the polymer molecules (similar to Stokes- Einstein equation shown earlier), so a higher diffusion coefficient is expected for polystyrene in methylethylketone than in benzene. LThe data of this work agree with this expectation. Generally speaking, the diffusion coefficient in a "poor" solvent for a given polymer is larger than in a "good" solvent. Decalin is a poor solvent for polystyrene. The theta temperature of a mixture of trans and gig decalin, 19.3°C, is given by Okada 22'31. (1963). (The theta temperature, sometimes called the Flory temperature, is approximately the temperature at which a polymer of infinite molecular weight would precipitate from the solvent.) Close to the 0 temperature the volume of the polymer molecules in solun tion is the smallest (unperturbed or unswollen'dimensions).' The experimental temperature of this work was 21°C, which 90 is very close to the 0 temperature, so it can be said that the volumes of the polystyrene molecules in decalin had to be a minimum. Therefore, the diffusion coefficient of polystyrene measured in decalin at that temperature should have been the highest among the diffusion coefficients obtained from measurements in different solvents. Looking at Tables 6 and 8, it can be seen that this is not true, the diffusion coefficient of polystyrene measured in decalin is extremely low, the lowest one among all of the diffusion coefficients measured in different solvents, which seems to be contradictory to the theoretical interpretation of the diffusion of polymers in dilute solution. Considering the fact that two diffusion measurements of polystyrene in decalin, based on different test methods (diffusion cell, Schick, 1950; lightbeating spectroscopy, this work) gave approximately the same results, the prob- ability of error in the measurements is reduced. Therefore, a simple hypothetical explanation of this problem will be given here without detailed proof, future study may improve our understanding of this phenomenon. The extremely low diffusion coefficient of polystyrene in decalin may have arisen from the fact that the volume of polystyrene molecules is large in decalin. But this is the opposite of what the theory says: the volume of a polymer molecule is the smallest in a poor solvent or very 91 close to the 9 temperature. However, this is true for one polymer molecule only. It is proposed that agglomeration occurs in polystyreneedecalin system, so the total volume of an average agglomerate is greater than the volume of one molecule in the best solvent. This results in a very low diffusion coefficient for polystyrene in decalin. Many polystyrene molecules stick together (the solvent molecules are not able to separate them), so they move much slower, and their slow movement gives a very low diffusion coeffi- cient. Increasing the concentration of p01ystyrene results in an increase in the number of the molecules per agglom-... erate, which leads to a lower diffusion constant and results in a negative KD. The low viscosity also can be explained. Any viscosity measurements unlike the diffusion measurement, are related to the solution properties instead of those of one polymer molecule itself. Comparing two solutions, polystyrene-benzene and polystyrene-decalin (close to 0 temperature) for the same concentration the number of polymer molecules is the same in both solutions. Regardless of whether the polymer molecules may agglomerate, their total volume in decalin is equal to the number of molecules times their minimum volume. In benzene the poly- styrene molecules are extended, so their total volume is ' equal to number of molecules times the extended volume of each molecule and this total volume is much greater than 92 the total volume in decalin. Since the viscosity is related to the solution, this higher total volume of polymer molecules in benzene results in a higher vis- cosity. This means that the simple model given here, which says the polystyrene molecules agglomerate in decalin, explains the extraordinary behavior of dilute decalin solutions of polystyrene close to the 0 temperature. Furthermore, the external shearing imposed upon the system, when measuring viscosity, would tend to break up the polymer agglomerates. General comments about the relationship between sol- vents and diffusion coefficients of polymers were given earlier, so the results of this work will now be compared with literature data. The diffusion coefficient of poly— styrene in decalin was discussed earlier. In the previous section avrangeofDo of 1.15 to 1.35 7 cmz/sec is given for polystyrene MW = 3.38 x 105 in x 10— benzene. Elias (l961)reported Do==1J50 x 10'.7 cmz/sec for Mw = 6.06 x105 for the same system. A comparison of the two values shows the DC from the literature considering the molecular weight difference is a little bit higher than the value calculated in this work. Do data for polystyrene in methylethylketone at 20°C given by Gralen and Lagermalin (1952), Mayerhoff (1960), and Schick and Singer (1950) com- pared with the Dd range of this work are shown below: 93 7 2 -5 Do x 10 cm /sec. MW x 10 Source 3.25 5.28 Meyerhoff 6.40 1.8 Gralen 5.14 2.4 Schick 2.70-3.00 1.3 This work It can be seen that the Do given by this work is lower than the literature data. This difference may have arisen from the fact that lower concentrations were used in this work than in others, which means to determine Do’ points closer to zero concentration were used. In other studies in the future the diffusion coefficient below 0.01 g/lOOg concentration should be measured for a more accurate determination of Do' Duffusion coefficient data for polystyrene-dimethyl- formamide were not found in the literature. According to the author's best knowledge no diffusion coefficient measure- ments have been made for polystyrene-acrylonitrile copolymer in dilute solution. Therefore the data of this work cannot be compared with others. The exact relationship between the diffusion coeffi- cient of polystyrene-acrylonitrile copolymer in dilute solution and the acrylonitrile content of the copolymer cannot be established based on this study, because of the difference in molecular weights of the samples. But comparing 94 diffusion data for polystyrene and copolymer, as was men- tioned in the previous chapter, it can be concluded that the arcylonitrile content of the copolymer increases the diffusion in dilute solutions of methylethylketone. In benzene the acrylonitrile content of the copolymer reduces the molecular weight dependence of the diffusion coeffi- cient of polystyrene. It is very interesting to compare the value of DO for polystyrene, polystyrene-acrylonitrile copolymer and poly- acrylonitrile in dimethylformamide. 7 2 Molecular Do x 10 cm /sec. Weight Source Polystyrene 1.00-1.25 271,000 This work Polyacrylo- 1.25 270,000 Bisschops nitrile (1955) Polystyrene/ 1.10-1.30 275,000 This work acrylonitrile copolymer It can be seen that D0 is about the same for poly- styrene and polyacrylonitrile in dimethylformamide and Do remains about the same for their c0polymer. Of course there is a possibility of a coincidence, and this problem can be the subject of future work. PIOTOCUIIEIT POWER IEllTIVE 95 Figure 28. 000 000 1200 1000 2000 ll Photocurrent spectrum of light scattered from polystyrene latex at 35° scattering angle. MI PIOIOCURIEIT POWER IElIIIVE 96 I I I I I I I I 'l I 400 000 1200 1000 2000 Hz Figure 29. Photocurrent spectrum of light scattered from polystyrene-acrylonitri1e copolymer (MW = 332,000, acrylonitrile content 38%) in dimethylformamide (c = 1.0 g/ 100g). Scattering angle 25°. Halfwidth 255 Hz. (D = 1.44 x 10' cmz/sec) PIOTOCUIIEIT POWER RELATIVE 97 Figure 30. J I 7% 0:0 I 12 :0 I 10]” I 200% Photocurrent spectrum of light scattered from polystyrene-acrylonitrile copolymer (MW = 332,000, acrylonitrile content 38%) in dimethylformamide (c = 1.0 g/lOOg). Scattering angle 35°. Halfwidth 496 Hz. (0 = 1.46 x 10'7 cmz/sec.) APPENDIX APPENDIX Program IG4 is used to determine the best values of A, B, and D in Equation 13 from a set of photocurrent power versus frequency data. Briefly, a simple algebraic tech- nique was devised to find the least squares solution for A and D given a specific value of B, the halfwidth. This is done in subroutine FUNC. The program steps through the whole range of possible B values to bracket the optimum, and then goes into a golden search, using subroutine GOLD, to find the best value of B to any desired precision. 98 80 90 100 105 110 115 120 125 130 140 150 99 PROGRAM 164IINPUTnOUTPUToTAPE2=INpUT0TADE3=OUTPUTI DIMENSION BMINIIO) COMMON AI(50)0 WISOIo N CALL NORLANK READ(29300) NSETS HRITE(30310) NSETS DO 190 ISET=10NSETS READI293201 BMIN REA01293001 N9WZR0 DELW "9115(39330) BMIN WRITEI39340) N9WZQO DFLW N2=IN+961/49 IFINZ - 2) 80990980 CALL EXIT READI29350) (A1III0I319N) DO 100 1:19N WIII=WZP 9 DELW*II-l) CONTINUE READI202101 B708M.OB.PRINT BMX=BM WPITEI392401 WRIIEI39220) B79BM9DB9PRINT WPITEI39240) 8:82-08 RM=8H + 0.5908 FLAG=1.0 E=1.0£6 I=l R38 0 DB EP=E IFIB-RMI 11591609160 CALL FUNCIBvoDoEI IFIPRINT) 12591250120 WRITEI39230) B9A9D9E 1F(FLAG*IEP-EI) 13091109110 FLAG3-FLAG IFIFLAGI 14091109110 BMINII)=B - DB 131*1 IFII'II) 11091509150 WRITEI39370) BMIN I31 100 GO TO 110 160 K=I'I WRITEI39240) M=200*ALOG(200000*OR1 * 200 HRITEI39390) M9 (RMINIJIOJ=19K) DO 185 J=I9K BM=BMINIJI BL=BM - DB BH=8M + 08 CALL GOLDIRL9BH9BM9E9M) URITEI39240) OEL=IRH-BL)*005 BM=BL 0 DEL CALL FUNCIBM9A9O9E) WRITEI30230) BM9A909E9DEL HRITEI39360) 62:000 00 170 I=19N X3WIII/RM STAR=A/(100 * X*X) 0 D EX=ISTAR - AIIIII/AIII) 52:52 + Exoex WRITEI39370) ”(1)9 AIIII9 STAR9 EX 170 CONTINUE IFIABSIE2-EI - IoOE-SI 13991809180 180 URITEI39380) E29 E 135 CONTINUE 190 CONTINUE CALL EXIT 210 FORMATI8F1003) 220 FORMATI10X04Floo3) 230 FORMATI10X02F15.403F15.6) 240 FORMATIII 300 FORMATII397X92F10.3) 310 FORNATI/II/9I10913H SETS OF DATA) 320 FORMATIIOAB) 330 FORMATIIH19l/910X010A8) 340 FORMATII/919921H POINTS FROM OMEGA = 9F6ol0 4 14H IN STEPS OF9F8.1) 350 FORMATIIOF6.ZI 360 FORMAT(///915X95HOMEGA95X05HI-EXP04X9 6 6HI-CALC95X94HEIIII 370 FORMAT(10X03F10.20F12.6) 380 FORMATI/l910X915HERROR IN E-CALC92F15.10) 390 FORMATII910X9IS924H = NO. 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