Polymer Journal, Vol. 17, No. 1, pp 225-238 (1985)
Light Scattering Studies of Polymer Solutions and Melts*
Benjamin CHU
Departments of Chemistry and of Materials Science and Engineering, State University of New York, Stony Brook, New York 11794-3400, U.S.A.
(Received July 16, 1984)
ABSTRACT: Static and dynamic properties of polymer solutions and melts can be investigated by means of modern scattering techniques. While small angle X-ray scattering (SAXS) and small• angle neutron scattering (SANS) have made advances particularly in studies related to the static structure factor S(K), laser light scattering including the use of Fabry-Perot interferometry and photon correlation spectroscopy has become a standard tool in studying polymer molecular motions. In polymer solutions, the main technique is to use measurements of angular distribution of integrated scattered intensity by means of visible light, SAXS or SANS for S(K) and measurements of angular distribution of the spectrum of scattered light by means of photon correlation spectroscopy for the dynamic structure factor, S(K, w). Recent advances have been made in the method of data analysis related to the ill-posed Laplace inversion problem. The new approaches include the singular value decomposition technique and methods of regularization with different criteria for the smoothing operator. By combining static and dynamic light scattering measure• ments with appropriate algorithms for the Laplace inversion of the time correlation function, a new analytical technique has been developed for polymer molecular weight characterizations. The non• intrusive method has been applied to determine the molecular weight distributions of linear and branched polyethylene in 1,2,4-trichlorobenzene at 135°C and of poly(l,4-phenylenetere• phthalamide) in concentrated sulfuric acid. In addition, a new prism light scattering cell is being developed to integrate the above capabilities with chromatographic and other separation techniques. Aside from translational motions of the center of mass of polymers in dilute solution, photon correlation spectroscopy also permits us to investigate rotational, flexual and internal segmental motions. Polymer molecules entangle in semidilute solution. Light-scattering spec• troscopy measures a cooperative diffusion coefficient and a slow mode which has been shown to be different in magnitude from the self-diffusion coefficient. The entanglement behavior varies from coils to rod-like polymers. Static and dynamic properties of polymer solutions in semidilute and semiconcentrated regimes can be related to those of bulk polymer melts where measurements of polarized Rayleigh-Brillouin spectra and depolarized Rayleigh spectra yield information on localized structural relaxation and collective segmentallmolecular orientational motions. Relaxation times covering a very broad frequency range will be discussed. KEY WORDS Light Scattering I Polymer Dynamics I Photon Correlation I Interferometry I Melts I Molecular-Weight Distribution I
In the low K regime, the long probing wavelength combining angular distribution of light scattering of visible light, such as .!c0 = 632.8 nm from a He-Ne intensity measurements with small angle X-ray scat• laser, permits us to reach macroscopic spatial length tering (SAXS) and small angle neutron scattering 4 scales with K[ = (4nnj},0 ) sin (0/2)] of the order of 10 (SANS), the static structure factor S(K) can be em - 1 . If we change the probing radiation to X-rays measured over a large range of spatial scales from or neutrons, the much shorter X-ray or neutron atomic spacings to micron sizes. wavelengths, of the order of angstroms, cannot The dynamic structure factor can also be mea• reach a comparably low K value by decreasing the sured using visible light, X-rays or neutrons. In scattering angle with 10-3 radians. Thus, by SANS, only the spin echo technique1 offers suf-
225 B.CHU
ficient high resolution and covers a time scale concentrated sulfuric acid. Brief outlines of the comparable to polymer dynamics. Time-resolution characterization procedures are presented here in experiments using the X-ray synchrotron radiation2 order to show that laser light scattering is a com• have also been performed on a wide variety of dy• plementary analytical technique to many of the namic phenomena including those related to poly• more standard methods, such as osmometry, ultra• mer systems, such as the kinetics of ligand binding centrifugation and size exclusion chromatography. in proteins. However, the temporal and spectral In semidilute solutions, two characteristic decay properties of X-ray synchrotron radiation have times have been observed: a fast cooperative dif• barely been utilized in SAXS for studies of polymer fusion coefficient De and a slow-mode diffusion dynamics. In this article, we are concerned mainly coefficient Dslow' which has been found to mimic
with laser light scattering3.4 including the use of properties of the self-diffusion coefficient D 5 • A brief Fabry-Perot interferometry and photon correlation review of its present status and the interconnection spectroscopy for investigating polymer molecular of diffusive motions in semi dilute solutions, gels and motions in solutions and melts. polymer melts will be presented. Many aspects of laser light scattering have been Finally, for bulk polymer melts, measurements of discussed in a series of NATO ASI's on photon polarized Rayleigh-Brillouin spectra and depolar• correlation and scattering techniques. 5 -s Parallel to ized Rayleigh spectra by means of Fabry-Perot the NATO ASI's, there have also been conferences interferometry and polarized Rayleigh spectra by on photon correlation techniques in fluid mech• means of photon correlation spectroscopy will yield anics.9 Light-scattering spectroscopy has been information on localized structural relaxation, col• made into a powerful routine method for study• lective segmental/molecular orientational motions ing the dynamics of polymer solutions10•11 and and dynamics of concentration fluctuations. meltsY·13 Instrumentation in photon correlation Implications of an extremely broad distribution of spectroscopy has been well developed. 14 The only relaxation times will be discussed. In my presen• recent modification being the availability of dual tation, I emphasize only recent developments, delay time increments for a single-clipped cor• mostly performed by my research group, partly relator15 and of commercial digital correlators with because reviews of earlier results exist1 _, 3 and logarithmically spaced or variable delay time incre• partly because I am more familiar with our own ments so as to increase the bandwidth of the mea• work. sured time correlation function without increasing the available number of correlator channels. With I. EXPERIMENTAL TECHNIQUES applications of the singular value decomposition technique and the method of regularization to the Aside from bandwidth improvements in digital ill-posed Laplace' inversion problem, 16 we are able correlator designs/5 recent advances include ac• to retrieve characteristic delay time distributions of cessibility of light scattering measurements to small interest from the measured band-width limited time scattering angles18•19 and improvements of com• correlation function which contains noise. Under plementary techniques in fluorescence photobleach• favorable conditions, we are able to estimate the ing recovery (FPR)29 and forced Rayleigh scattering characteristic delay time distribution without an a (FRS).22 - 24 In particular, a small angle light scat• priori assumption on its form. Thus, information tering prism cell and the use of a moving Ronchi other than the average characteristic delay time grating in FPR will be discussed. becomes accessible. In dilute solutions, measurements of angular I. Small Angle Light Scattering Prism Cell distribution of absolute scattered intensity and of In dilute polymer solutions, by combining time correlation functions permit us to characterize measurements of angular distribution of absolute a variety of polymers which are often difficult to scattered intensity with Rayleigh linewidths we are investigate because of experimental obstacles. For able to measure the weight-average molecular example, polyethylene is soluble in trichlorobenzene weight Mw, the z-average radius of gyration only at high temperatures ( 130°C) and poly(l,4- the second virial coefficients A 2 and kd, and the "z• phenylene terephthalamide)/PPTA is soluble in average" hydrodynamic radius, (Ri: 1 ); 1 using the
226 Polymer J., Vol. 17, No. I, 1985 Light Scattering of Polymer Solutions and Melts
expressions based on S(K) Rvv(K) and S(K, w ): P7 P6 HC Rvv (K,C)
1 2 +2A2 C )[1 +(1 +2A2 MwC)- (Rg )zK2/3]
(1)
(Rf: 1 )z- 1 = kBTj(6ni](D0 )z) (2) where
Figure 1. Schematic diagram of a prism light scatter• and ing spectrometer. Laser, Spectra Physics model 124 HeNe laser, A0 =632.8nm operating at IOmW; M-1, 2, (T)[ =(D 0 )Jl +kctC)K2] = JTG(r)dr 3, dielectric mirrors; P-1, · · · 7, pinholes; L-1, 2, lenses. Total angular divergence be= 0.2° at small scattenng with G(r) being the normalized linewidth distri• angles, and <50= 0.3° at large scattering angles. butions; r( = r c-l ), the characteristic linewidth or the reciprocal of the characteristic decay time rc; 1], permit us to estimate the characteristic linewidth the solvent viscosity; kB, the Boltzmann constant; distribution G(r) by means of a Laplace inversion and N. the number of molecules of species i. In eq 2: have assumed that of the z• lgO>(r)l= G(r)e-r'dT (4) average translational diffusion coefficient (D)z f" satisfies the condition KR < 1 with R being a char• where, in the self-beating mode, G< 2>(K, r) = acteristic length. For random coils, with KRg < 1 1 A(l b 1 g< >(K, r) 1 2) with A and b being the re• the particle scattering factor [or the static struc• + spective background and spatial coherence factor. ture factor S(K)] 1. At higher scattering angles, Access of dynamic linewidth measurements at small as K is increased, internal molecular motions begin and large scattering angles also provides us with an to contribute to (T) with a leading term propor• experimental approach to separate the characteris• tional to K4 : tic decay time due to translation from those due to (T) = (D)zK2(l + f(R;)zK2 + · · ·) (3) internal motions. (2) The temperature-controlled light-scattering where f is a dimensionless number depending upon prism cell has an effective small cell volume which is chain structure, polydispersity and solvent power.17 a contiguous part of a capillary flow system. Thus, if we were trying to avoid the complications Therefore, such a cell can be adapted as a detector introduced by eq 3, we need to make linewidth for size exclusion chromatography. The essential measurements at small scattering angles whenever features of the prism light scattering cell are its two the polymer size becomes large. Figure 1 shows a accessible surfaces of the prism as exits for the schematic diagram of such a prism light-scattering scattered light and the use of the prism as a cell18 which can measure the angular distribution of reference in differential refractometry whereby the excess absolute scattered intensity, RvvCK), down to refractive index of the solution is measured by a scattering angle(} of The main advantages of relative changes in the direction between the in• the prism cell are as follows. cident and the transmitted laser beam. (I) It permits simultaneous measurements of Other developments for light-scattering spec• light scattering intensity and time correlation func• trometers using optical fibers 19 and a time interval tion. Therefore, we have access to determinations digitizer20 •21. have been reported. It should be em• of A , Mw, (R;)z, (Rf: 1 )z- 1 and kct in dilute so• 2 phasized that techniques related to photon cor• lutions of very high molecular weight polymers with relation spectroscopy, either in optical arrange• molecular weights ;:S 108 . Precise measurements of ments or in hardware correlator designs, have ma• the intensity time correlation function, G<2>(K, r), tured. Routine commercial instruments exist.
Polymer J., Vol. 17, No. I, 1985 227 B.CHU
Nevertheless, we should also realize that there have FRS to studies of polymer solutions in semidilute been recent developments in related experimental and concentrated regimes as well as of gels will techniques which can be used to study polymer undoubtedly play an important role because of dynamics. specificity and dynamic range made available by such techniques even though we must be careful to 2. Advances in Fluorescence Photobleaching pay particular attention to the chemistry of photo• Recovery ( FPR) and Forced Rayleigh Scattering chromic32 and fluorescent probes. (FRS) Complementary methods such as forced Rayleigh 3. Laplace Inversion scattering22 - 24 (or holographic relaxation spectros• Equation 4 is a Fredholm integral equation of the copy (HRS)) and fluorescence photobleaching re• first kind. The problem associated with recovering coveryl5·26 are capable of measuring translational the normalized characteristic linewidth distribution diffusion coefficients many orders of magnitude function G(T) is due to the ill-conditioned nature of smaller ( -10- 12 cm2 s-1) than photon correlation eq 4 since the experimental data, 1g 11 >(r) I, is band• spectroscopy. Although FPR and FRS (or HRS) width limited and contains noise. There have been require the use of respective fluorescent and photo• chapters of two recent books dealing specifically chromic probe molecules, the probes have the ad• with the linewidth polydispersity analysis.33 ·34 By vantage of selectivity, i.e., observing only systems of means of eq 4 the amount of information about interest by design. There are experimental com• G(T) which can be recovered from bandwith limited plications associated with both techniques. and noisy 1 g11 >(r) 1 is limited. Essential mathematics However, in FPR, the introduction of periodic allowing implementation of the singular value de• pattern photobleaching26 •27 and of a scanning de• composition technique and a linearized method of tection system28 has simplified both theory and regularization have been described. 16 practice as well as improved the signal processing In the singular value decomposition technique, technique. Lanni and Ware29 combined the sim• we first introduce a reasonable and mathematically plicity of the periodic pattern photobleaching with tractable initial approximation for G(T), such as the scanning detection and were able to virtually N eliminate sensitivity due to long-term motion of the G(T)= I PJ!(r-n (5) specimen and data distortion due to continued i=1 photobleaching by the monitoring beam. where the Dirac delta functions form an N sum of Furthermore, the decay at the fundamental fre• fixed discrete characteristic linewidths, r;'s with quency is observed in real time and the measure• amplitudes, P;'s, which are determined by a linear ment is insensitive to de drift components. In FPR, least squares fitting procedure. In matrix form, a small fraction of the probe molecules is decom• (6) posed while in HRS, which was pioneered by Pohl et al. 30 and Eichler et al., 31 the probe molecules are where Cis the (m x n) curvature matrix with cii= not destroyed. In their initial scheme, Pohl et al. e -ri'', Pis the parameter vector oflength n, and b is used two intersecting coherent pulsed laser beams to the data vector of length m with b; = g11 >( r;). The induce a transient thermal grating in a slightly symbol solution in a least-squares sense. absorbing medium and measured thermal diffu• Prior to entering the singular value decomposition sivity by monitoring the decay of the reading beam, routine,35 the columns of Care scaled to unit norm as diffracted by the thermal grating which was in order to improve the numerical stability of the gradually smeared out by thermal diffusion. By inversion. We have using photochromic dye molecules, a periodic con• (7) centration grating of photoexcited dye molecules is created.22 - 24 If the dye molecules are attached to where A=CH, x=H- 1 P, and macromolecules, the self-diffusion behavior of la• belled chains is observed provided that there is no concentration dependence of the labelled dye on the observed relaxation time. Applications of FPR and Equation 7 is decomposed by orthogonal transfor-
228 Polymer J., Vol. 17, No. I, 1985 Light Scattering of Polymer Solutions and Melts mations to the equivalent problem position technique which has been coupled with the use of eq 5 can retrieve the optimum information on usv- 1 x""b (8) G(T) without additional constraints such as posi• with U and v- 1 being orthogonal matrices and S tivity, while we have taken an identity operator for J being a diagonal matrix whose non-zero elements in order to retain linear techniques for the method are monotonically decreasing. By defining x = Vy of regularization. There are other approaches to the and UTb = g, we have method of regularization, such as Fisher's test used by Provencher37 with origin from the Tikhonov Sy"='g (9) regularization. 38 which yields The Laplace inversion problem can be refined further. In order to optimize the amount of infor• (10) mation which can be retrieved, it is important to with s; being the ith singular value, S;;· The elements understand the physical basis and limitations re• of S contain the essential elements of information of lated to the constraints which can be incorporated eq 6. A rank reduction step is usually employed to into the regularization parameters, i.e., the more we limit the amount of information which can be know the system, the more information we can extracted above the noise level of the measured retrieve from a given set of bandwidth limited data. We have successfully selected k such that measured data which contain noise. Thus, the llr G = [K* K + c:.:J* J]- 1 K*b (15) where the asterisk denotes the adjoint operator. The N'h iteration estimate of G has the form: (17) arNJ=K*b+[(l-cx)I-K*K]G Polymer J., Vol. 17, No. l, 1985 229 B. CHu scattering angle. At finite concentration and scatter• Q 0. 10 ing angle, and > o.o s.?(msio. 15. Q) a ar:dfld"r:6Pl" 2 Ti:::o:DJ(i +kdC)K [i + +2A2 MwC)- 1 K 2 ] O 0.00 ___. __ OJ (19) > where we can accept the approximations that A +' 2 0 0.05 and kd are relatively independent of the molecular OJ 0:: 0.0 6. 0 12.0 18.0 weight over limited ranges of Fw(M) and f; in t (ms) dilute solutions. With D 0 =k0 M-"0 , we can compute the molecular weight distribution from Figure 2(a). Plot of relative deviation in Ig 111(r) I for laser light scattering measurements. From eq 18 the two characteristic linewidth distributions: G(r) with and 19, we have two extrapolations based on and without the I% shaded section. The shaded contri• eq 5. On the intensity (P /) axis, we have bution of G(r) corresponds to a base-line adjustment of A plot of I g11 >(r) I versus r is inserted to i/Pt :::o: l +(f + l/3)RUC)K2 +0(K4 ) (20) demonstrate the precision required in the time cor• relation function measurements, i.e., the fittings can where we have set with always be achieved such that the measured and the 1 G + (D j)[ = P n and D J being the unnormalized computed 1 gl >( r) 1 are indistinguishable within the ex• scattered intensity of the J'h representative fraction perimental error limits. Measured and computed base• line must agree within 0.1%- and T)K2 , respectively. On the characteristic line• width (r) axis, we have DJ=T)K2 = l+fRUC)K2 (21) 9. 0 where +2A2 MwC). By com• bining eq 20 and 21, we can estimate f and R i as r. 6.0 well as P t values. L As an illustration, a determination of molecular lJ weight distribution (MWD) of PPTA or Kevlar is N 3.0 shown in Figure 2. The details have been described 0 elsewhere. 39 - 41 It should be noted that PPTA is soluble mainly in sulfonic acids which are highly 0.0 corrosive and that the polyelectrolytes may exhibit 0 125 250 375 500 long-range electrostatic interactions. With experim• I (s-1) ental values of Rg, f, A , kd, and k , which remain 2 0 Figure 2(b). Plot of G(r) versus r. The shaded area essentially constant for a given polymer-solvent represents I% of the area of JG(r)dT. system, we can determine the MWD of PPT A in 96% H2 S04 plus 0.05 M K2 S04 using one time correlation function measured at one fixed scatter• shown in Figure 2(c). In the molecular weight ing angle (e.g., 8=45°) and concentration (e.g., c= transform, we have used D 0 =2.07x w-s M- 0 -75 4 1 2.8 x l0- g ml- ). Figure 2(a) shows a typical plot with M expressed in g mol- 1, A 2 = 3.23 x lO- 3 cm3 of I g11 >(t) I versus delay time t for the PPTA solution. mol g- 2 , kd=60 cm3 g- 1 , 6=0.2 and The corresponding G(r) as shown in Figure 2(b) 35 nm where 6 is the molecular anisotropy. For can be determined using methods described in PPTA solutions, eq 18 and 19 need to be modified Section 1.3. In our case, G(r) was obtained by the because of molecular anisotropy: singular value decomposition technique [eq 5-11]. +462 /5)/(1-46/5+462/7)] Using the procedures outlined in Section II. LA, we could obtain a MWD for PPTA with Mw=4.26 x X [J +2A2MwC/(i (22) l04 g mol- 1 and Mz : Mw: Mn=6.2: 1.8 : l as 230 PolymerJ., Vol. 17, No. I, 1985 Light Scattering of Polymer Solutions and Melts shaded area of G(r) in Figure 2(b ). The correspond• 0.8 ing uncertainty in MWD is illustrated by the shaded area in Figure 2(c). Thus, if there is a high molecular weight tail with weight fractions corresponding to 'J ;: -0.01 and M>5 x 105g mol-1, the light scattering lL 0. 4 N method cannot be used to confirm either its pres• 0,_., ence or its absence, unless we are able to improve our experimental precision another order of magni• 0. 0 tude, which is unlikely. Nevertheless, determi• 10' 10 4 105 10 6 107 nations of MWD by laser light scattering have tv1Cdolton) unique advantages, i.e., the technique is non• invasive, the measurements can be performed in a Figure 2(c). Molecular weight distribution of PPTA closed system, and the Laplace inversion, although based on G(r) from Figure 2(b). The shaded area delicate, has a solid mathematical foundation. Since represents the baseline uncertainty in the time cor• it can readily be used to determine MWD's of relation function measurement which can be translated polymers such as polyacrylamide,42 polyphos• to uncertainties in the high molecular weight tail. Fw(M) phazine,43 polyethylene44.45 and PPTA,39 - 41 its is the weight distribution. eventual utilization as an analytical tool, especially for speciality polymers, is expected. Furthermore, its utility as a detector for chromatography cannot 0.8 be overemphasized. B. Coil-Globule Transition Another area of interest is the coil-globule tran• E 3 0. 4 sition.U·45 The temperature and molecular weight ;: dependence of the hydrodynamic radius Rh of lL polystyrene with Mw varying from 3.8 x 106 to 20.6 x 106 dissolved in two different solvents, cy• 10' 10' clopentane and cyclohexane, have been studied by analytical ultracentrifugation.46 All data can be tv1 (do 1 ton) represented on a single master curve in a plot of the Figure 2(d). Cumulative molecular weight distribution expansion factor ah[=Rh(T)/Rh(O)] versus the re• of PPTA from Figure 2(c). Molecular weight fractions duced variable N/ N, = M wl T 12/(naMa) where IT I= vary from I 04 to - 5 x 105 . The high molecular weight 11-0/TI and naMa is an adjustable parameter tail corresponding to polymers with M > 5 x 105 is seen ""' 150 ± 30 g mol-1. In the 0-domain (N/ N, < 3), as within the experimental error limits. ah- 1. In the collapsed domain (N/ N, > 20), ah varies as M,";; 116 lrl- 113 and The intrinsic viscosity [17] (- R3 I M) is related to the particle volume. Therefore, viscometry should At M> 5 x 105 g mol- 1, the amount of polymer in Fw.oum has become negligibly small. be a more sensitive technique than quasielastic light scattering (where D- Rh- 1) in evaluating the globule-coil transition behavior. However, the spe• cific viscosity defined as IJsp = (IJ!IJa)- I, requires a (23) differential or ratio measurement of viscosity of the where R;. app denotes the effective radius of gyration solution 17 and that of the solvent IJa· In the col• at finite concentrations without the (j correction. lapsed domain, highly dilute solutions must be In the data analysis, it is particularly important to used11 so that the difference between 11 and IJa are realize the need for precise baseline measurements. necessarily quite small. Nevertheless, viscosity A 0.1% baseline fluctuation can be translated to measurements with sensitivity approaching ±0.01 <:; 1% of fG(r)dr in the low frequency limit of the sec or bry/ry-10-4 are feasible,47 even though at characteristic linewidth distribution as shown by the high molecular weights (M <:;107) and extremely low Polymer J., Vol. 17, No. I, 1985 231 B.CHU concentrations ( C;:;; 10-7g em- 3), further improve• correctly as the self-diffusion coefficient of a single ments in precision in the absence of shear rate polymer chain reptating through the entangled dependence become desirable. polymer coils. However, its molecular weight and Light scattering can measure both Rg and Rh. concentration dependence, nevertheless, mimics With photon correlation and photon counting it those of the self-diffusion coefficient. Consequently, appears to be a good technique to study the coil its behavior could probably be related to reptation globule transition. Indeed, many studies have been of a clustering of polymer chains through the reported.48 - 53 A comparison of the literature data46 entangled coils. The cluster concept came about shows two notable exceptions on the hydrodynamic because Dslow <'iDs. radius in attempts to reach the collapsed regime by Polymer self-diffusion in entangled systems has using light scattering.51 - 53 In particular, the ah been reviewed by Tirrell58 who has listed 277 ref• variation in the large Nj N, region is much too rapid erences. The extensive review is instructive and for the Prague group. 53 Although they tried to take contains a lucid account of the present status. advantage of the high viscosity of dioctyl phthalate Dynamics of entangled systems has been reviewed as their solvent, the same advantage may become a by Graessley, 59 and of the reptation model60 by de difficulty in reaching the required thermodynamic Gennes61 and others.62 ·63 The polymer chain mov• equilibrium. The discrepancy of the sedimentation ing along the contour of the tube64 made up of data with the MIT group is only qualitative. neighboring polymer chains has a "tube diffusion Nevertheless, light scattering experiments studying coefficient," Dtube: the coil globule transition using a range of narrow (24) molecular weight distribution polymers should be repeated especially in approaching and reaching the where M 0 and (0 are the molecular weight of a collapsed domain. Extension of such investigations monomer unit and the monomer friction coefficient, using star polymers would also be very interesting. respectively. The polymer chain of length L(=bMjM0 ) will escape from its tube over a timer C. Internal Motions such that Dtube = L 2/2r or Analysis of internal motions in dilute polymer (25) solutions has remained to be a difficult problem" because the dynamic structure factor is model de• and pendent. A more generalized approach based on the (26) first cumulant of g<' 1(K, r) by Akcasu et al. 54•55 and Han and Akcasu56 suggests that eq 3 is an extremely The concentration dependence of Ds can be for• useful expression to experimentally account for mulated from D,(C, M)=D,(O, M)(C/C*)' where internal motions of polymers in solution. More C*( -M1 - 3v with Rg-Mv) is the overlap con• detailed analysis on translational diffusion and centration. As Ds(C,M)-M-2 , D,(O,M)-R;'• internal motion of high molecular weight polysty• M-v, we have renes in benzene at infinite dilution has been re• D,(C,M)-M-2c 232 Polymer J., Vol. 17, No.1, 1985 Light Scattering of Polymer Solutions and Melts tron microscopy, 71 and Rutherford back scatter• ing72 have been summarized in Table I of reference 58 by M. Tirrell. In quasielastic light scattering (QELS), the slow mode diffusion coefficient Dsiow mimics, but is smaller in magnitude than, the self• diffusion coefficient. 11 •73 Ternary isorefractive QELS uses an isorefractive polymer-solvent pair to ::;: 10 measure the self-diffusion coefficient of a trace amount of a miscible, "visible" second polymer.74•75 It does not require the introduction of probe mol• 0. 1 10 ecules as in FPR and FRS, but has limited applica• * bility because few miscible polymer pairs with one C/C of which being isorefractive to the solvent exist. Figure 3. Log-log plot of M(onjoC)/RT versus CjC* Further studies on the slow mode are needed before with C* = Mj(NAR;) for PMMA in MMA. we can have a clearer concept on its origin as to why 102 C* Temp it mimics, but is slower than, the self-diffusive Symbol motions of a single polymer coil. One of the practical experimental difficulties in light-scattering studies of polymer solutions in the Thermal polymerized PMMA semidilute regime is the preparation of homo• Plus signs 2.00 52.2 2.33 90 geneous dust-free solutions which become highly Hollow squares 3. 70 74.5 1.49 50 Inverted hollow 4.54 84.5 1.24 50 viscous at high concentrations. We have succeeded triangles in preparing such solutions by taking advantage of thermal polymerization processes. 73 During the Prepolymerized PMMA -by interpolation- thermal polymerization of methyl methacrylate Large hollow 2.36 circles MMA, we have been able to investigate both the Large solid 0.08 static and dynamic properties of polymethyl circles methacrylate PMMA dissolved in MMA. As MMA can be clarified readily, we can prepare a dustfree, homogeneous polymer solution at concentrations weights in toluene and methyl ethyl ketone. 78 In beyond the usual concentration ranges made avail• addition to M(8nj8C)/RT, the reduced length able by dissolving a polymer in a solvent. From is also a universal function of CjC*, inde• eq I, we have pendent of molecular weight or solvent quality where is the static correlation length obeying HC 1 (an) (28) Rvv = RT ac r.P where (on/8Ch,P is the osmotic compressibility. (29) With the polymer concentration detected by Raman However, the reduced dynamic length79 is a spectroscopy, Figure 3 shows a log-log plot of universal function of kdC, instead of A 2 C which is M(8nj8C)/RT versus C/C* for PMMA in MMA. related to the reduced concentration CjC*. is The universal curve starts at values slightly higher the dynamic correlation length obeying eq 2 with (D0 )z being replaced by (D)z =lim (r)/K 2 for the than I in dilute solutions, shows a broad cross-over region and eventually approaches theta behavior high-frequency component of the time correlation with a slope of 2. The results are in reasonale function (expressed in terms of DJ in the absence agreement when we repeat our studies using narrow of the slow mode (Dsiow). Theories of cross-over molecular weight distribution PMMA dissolved in behavior of the osmotic compressibility and re• MMA,76 as shown by hollow circles in Figure 3. duced static length Rg in semi dilute polymer There is a small polydispersity effect. 77 The uni• solutions in a good solvent have been develop• versal curve holds for narrow molecular weight ed. 77 ·80 - 82 For dynamic properties less is known distribution polystyrene of different molecular but further theoretical developments are forth- Polymer 1., Vol. 17, No. I, 1985 233 B. CHU coming. It should be noted that in the evaluation analyzed by Fabry-Perot interferometry. At much of (8n/8Ch,p, and in semidilute solutions, the longer characteristic decay times, i.e., for presence of two modes, due to cooperative (DJ l/F:<:;I0- 6 s, the dynamics of slowly relaxing fluc• and "self-diffusive" (Dsiowl motions, could influ• tuations in density and optical anisotropy of bulk ence the interpretation of the proposed universal polymers near the glass transition can be analyzed curves. At this time, there is good agreement on by photon correlation spectroscopy. For polarized the cooperative diffusion coefficient. The slow light scattering, the scattered intensity includes both mode requires further studies. There have been isotropic and anisotropic components, e.g., (at 8= reports suggesting (I) the absence of a slow 90o) fvv=liso+(4/3)/vH· If fvH-4,_/iso• So mode, 79 i.e., quasielastic light scattering sees only the polarized spectrum of scattered light in a simple the fast mode, (2) a slow mode which is orders viscous fluid, due to longitudinal density fluc• of magnitude lower than the fast mode,83 and tuations, has a central Rayleigh peak and two (3) a slow mode which is comparable in magni• shifted Brillouin peaks where the total scattered 85 tude to the fast mode.84• The difference in magni• intensity is proportional to pk8 TjK0 with p and Ko tude between De and D, 1ow could be related to being the mass density and the static modulus of solvent quality and there are many experiments compression, respectively. K0 = 1/ f3T with {JT being showing the presence of two modes for polymer the isothermal compressibility. The ratio of the solutions in the semidilute regime even though the central peak intensity to the total Brillouin intensity origin of D,1ow from quasielastic light scattering is Ic/218 in the purely viscous limit is Ic/218 = y -1 still uncertain. If both modes were present and had where y( = Cr/Cv) is the ratio of the specific heats at comparable amplitude, the composite nature of constant pressure and volume. The linewidth of (8n/8Ch.r, and could be the source of devi• the central Rayleigh peak, rR =KK2jpCp with K ation from the present experimental confirmation being the thermal conductivity. The Brillouin on the universal curves, e.g., could depend shift is determined by the adiabatic longitu• on polymer coil size in addition to the entan• dinal sound velocity v1(K): = Kv1(K) where glement correlation length because of the presence v1(K=0)=(yKofp)112 and the Brillouin linewidth, of two intensity sources for the excess scattered r B = [K2 j(2p )](IJv + 41Jsf3) with IJv and IJs being the light. volume and shear viscosities. In simple viscous Light scattering measurements of (8n/8Ch,r, Rg, liquid, relaxing times that contribute toward visco• Rh, A 2 , and kd and intrinsic viscosity [IJ] measure• sities are fast compared to ments from collapsed to theta and good solvent For viscoelastic fluids with finite relaxation times regimes in dilute solution and of (8n/8Ch,r, which contribute to both elastic and viscous re• for each individual mode as well as IJ from dilute sponses, the frequency dependent adiabatic longitu• to semidilute solutions will undoubtedly elucidate dinal modulus M( = Ko + (4/3)G with G being the our understanding of static and dynamic properties shear modulus) has the form 12•86 of polymer solutions including entangled systems w p(r)iwr such as gels and polymer melts. Dynamic measure• M*(w)=yKo+MR I -dr (30) o 1 +lWT ments in semi dilute solutions at KR > 1 have been ignored so far for lack of comprehensive theoretical where MR is the total relaxation strength for all models even though we have, in fact, touched upon, processes which are coupled to the longitudinal but ignored its complexity in evaluating stress and p(r) is the distribution of relaxation times 87 (r). With the decrease in / 8 , a Mountain peak 3. POLYMER MELTS appears to account for the constant total intensity and Ic/218 . The shape of the Mountain peak A. Theoretical Background SJ._w)=i"" p(r)r dr (31) Rayleigh-Brillouin and depolarized Rayleigh 0 1 +(wr)2 spectra reveal effects of segmental and orientational does not depend on K. motions and cover a broad range of characteristic More generally, the time correlation of the iso• decay times (10- 8-10- 12 s) which can be frequency tropic component of the scattered light has the 234 Polymer J., Vol. 17, No. I, 1985 Light Scattering of Polymer Solutions and Melts form: 13 are often compared with theoretical spectra which have been convoluted to include the effects of the ciso(K, t)=rxz/ I N±exp{iK·[r Polymer J., Vol. 17, No. I, 1985 235 B. 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