Faraday Symp. Chem. SOC.,1983 18 173-188 Studies of the Concentration Dependence of the Conformational Dynamics of Solutions containing Linear Star or Comb Homopolymers BY C. J. T. MARTEL M. G. DIBBS R. L. SAMMLER, T. P. LODGE,? $ T. M. STOKICH C. J. CARRIERE AND J. L. SCHRAG* Department of Chemistry and Rheology Research Center University of Wisconsin Madison Wisconsin 53706 U.S.A. Received 26th August 1983 The concentration dependence of the conformational dynamics of polymer solutions as revealed by measurements of oscillatory flow birefringence (0.f.b.) has been obtained for narrow-distribution linear comb and regular-star molecules for concentrations in the range c[q] 5 11. The data obtained show that the relaxation-time spectrum is affected markedly by concentration.The longest relaxation time is affected most; for the linear comb or 3-armed star polystyrenes studied to date z exhibits an exponential dependence on concentration for c[q] < 3. The Muthukumar theory correctly predicts the dependence of z on concentration for solutions containing linear molecules for which c[q] < 5. There is some indication that z1as a function of concentration exhibits two different concentration regimes one above and one below the onset of significant entanglement effects. The shortest relaxation times are almost unaffected by concentration. The measured frequency dependences of 0.f.b. properties for finite concentrations have also been compared with the predictions of the Muthukumar and Freed and the Muthukumar theories for the concentration dependence of the relaxation times for the bead-spring model.Extensive studies of the oscillatory flow birefringence (0.f. b.)l-I0 and linear visco- elastic (v.e.)11-22 properties of polymer solutions have provided considerable insight into the dynamics of conformational change in both the dilute and the semi-dilute regimes. Both experiments frequently have utilized high viscosity solvents (such as the Aroclors) and time-temperature superposition to extend the experimentally accessible effective frequency range to as many as seven decades thus providing information about polymer dynamics ranging from the slowest overall shape-change modes down to quite local rearrangements for which motions of a small number of monomer units form the basic motional repeat unit.Both techniques are sufficiently sensitive and precise to permit extrapolation of low-concentration data to obtain infinite-dilution properties;8-10v11 14-199 21 however the 0.f.b. experiment is generally sufficiently more sensitive and precise that most of the dynamics information discussed here has been obtained from birefringence measurements only. Both experiments provide the same dynamics information except in the high-frequency regime. Fig. 1 illustrates the excellent agreement normally observed at low and intermediate frequencies ; the solution contained 0.0304 g cmP3of 390 000molecular-weight linear atactic polystyrene in Aroclor 1248. Here the frequency dependence (w is the angular frequency) of phasing of the birefringence with respect to the sinusoidally time-varying shear rate t Present address Department of Chemistry University of Minnesota Minneapolis Minnesota 55455 U.S.A.$ Present address Dow Chemical Company Midland Michigan 48640 U.S.A. 173 100 90 80 70 60 50 X 40 30 20 10 0 0 1 2 3 4 5 6 log (w aT) Fig. 1. Comparison of the frequency dependence of -(0 -0,J (birefringence data solid line) and the equivalent angle x and lq* -qLIR from viscoelastic measurements for 0.0304 g cm-3 solution of 390000 molecular-weight PS in Aroclor 1248 reduced to 25.0OOC. (Os0 is the low-frequency limiting value of 0,.) 0,T = 24.97 “C aT = 1.00; 0, T = 10.00 “C aT = 14.61; a,T = 0.02 “C aT = 233.51.~100= 3.44 poise. is represented by the solid line; the individual data points have been replaced by the smooth curve for clarity. The viscoelastic data are shown as individual data points; the quantities plotted are the magnitude of the complex viscosity coefficient q*(y~*= q’ -ir”) minus the high-frequency frequency-independent value of v’ usually denoted as &,and the angle x defined by x = tan-l [q”/(q’-&)]. Time-temperature superposition has been employed to superpose data obtained at three different temperatures; the subscript R on the quantity Iq* -q’,lR indicates reduction by multiplication by the factor T,p,/Tpa where the T are absolute temperature the p are the solution densities the subscript 0 refers to the reference temperature 25.00 OC and aT is the shift factor.22 (It is necessary to use q’ rather than the bulk solvent viscosity vssuggested by most theoretical treatments since the polymer-chain dynamics contribution to the v.e.properties is being sought; the use of r’ apparently removes additional contributions from various sources including a substantial solvent modi- fication in the immediate neighbourhood of the chain due to polymer-solvent interactions.23) There have been several studies of the initial concentration dependence of 0.f.b. and v.e. properties of linear monodisperse polymers.9* 22 Except for very low molecular weights bead-spring-model prediction^^^-^^ (isolated-molecule theories) have been shown to be in very good agreement with infinite-dilution properties in both good and 8 solvents at low and intermediate frequencies when exact eigenvalues are employed (8 solvent agreement is poorer).For dilute solutions the relaxation times of the bead-spring model are expected to exhibit some concentration dependence while the corresponding relaxation strengths are usually assumed to be independent of c.J. T. MARTEL et al. concentration.22 Experimental work has shown that the intermolecular interactions occurring in dilute solutions affect relaxation times significantly and the longest relaxation time more than the others.g- l1 Recently explicit equations for the initial concentration dependence of the relaxation times of the bead-spring model have been obtained by Muthukumar and Freed2i and by Muthukumar.28 This paper reports observed concentration dependences for extensive 0.f.b.measurements for solutions of polystyrene (PS) and poly(a-methylstyrene) (PMS) for concentrations in the range c[q]5 11 and compares the observed dependences with theoretical predictions. All comparisons assume that relaxation strengths are essentially independent of concen-tration for the solutions studied. EXPERIMENTAL MATERIALS The studies reported here have been carried out with narrow-distribution (Mw/Mn< l.lO) atactic polymer samples linear polystyrenes 6a (M,= 860000) 3b (Mw= 390000) 4b (Ew 111000) 60817 (E,= 53700) and 8b (M,= 10000) (manufacturer's data; Pressure = Chemical Co.); a linear poly(a-methylstyrene) PMS no. 5 (Mw=400000) generously provided and characterized by L.J. Fetters Exxon Corp.; regular (equal-arm molecular weight) star polystyrenes E-3 (three arms total M = 192600) F-3 (3 arms total a,= 224300) 5-12 (12 arms total M = 805000) and 10-12 (12 arms total Mw= 2090000) also provided and characterized by Dr Fetters ;and a regular 25-branch (equal-length branches) comb polystyrene C632 (total M =913000; Mwof backbone = 275000; M of branch = 25700) generously provided and ~haracterized~l by J. E. L. Roovers of the National Research Council of Canada. All solutions were prepared in the chlorinated biphenyl solvent Aroclor 1248 lot KM 502 (Monsanto Chemical Co.). This solvent was selected for its large dependence of viscosity on temperature22 and the close match of its index of refraction with that of PS and PMS to eliminate form birefringence effect^.^ i* All solutions were prepared by weight; concentrations were converted to g cmP3 assuming additivity of volumes (assumed densities are 1.445 1.060 and 1.080 at 25 "C for Aroclor 1248 PS and PMS respectively).In general the initial (highest concentration) solution was prepared by direct addition of polymer to solvent. Solvation was assisted by moderate heating (<60 "C) and occasional gentle stirring; total solvation time ranged from 6 to 9 weeks depending on the sample molecular weight. The initial poly(a- methylstyrene) solution was prepared by first dissolving the polymer in analytical-reagent-grade benzene which was subsequently stripped out in uacuo after the addition of Aroclor 1248. All other solutions were prepared by direct dilution and were subjected to low heating (< 40 "C) and occasional gentle stirring for at least 2 weeks prior to use.METHOD The second-generation thin-fluid-layer 0.f.b. apparatus and the measurement technique have been described el~ewhere.~. The transducers have been interfaced to second- and third- 32 generation computerized data-acquisition and processing systems which increase the effective frequency range of the sets of apparatus and improve their sensitivity; these systems are also described el~ewhere.~~-~~ All data reported here were obtained at an optical wavelength (in air) of 5770 A. Solution temperatures were controlled to within kO.01 "C and were determined by means of thermistors calibrated against an N.B.S.calibrated standard platinum resistance thermometer. The Aroclors are themselves weakly birefringent when subjected to a shearing deformation ;thus the total birefringence observed has contributions from both the polymer and the solvent. The polymer contribution is obtained by correcting the measured values for the solvent contribution based on the assumption of simple additivity of the polarizability tensors for the various constituents (procedure of Sadron).i,8 Thus the tensor sum of the polarizability contributions of the volume fractions of pure solvent and polymer is assumed to correspond to the solution properties. For the low-shear-rate conditions employed in the thin-fluid-layer 0.f.b. instrument the principal polarizability directions remain at &45 "with CONFORMATIONAL DYNAMICS OF POLYMER SOLUTIONS respect to the streamline direction throughout the sinusoidally time-varying cycle of deformation which leads to a particularly simple vector-subtraction correction procedure which is discussed el~ewhere.~.As mentioned previously in connection with the viscoelastic properties apparently there is a substantial solvent modification in the immediate neighbourhood of a PS or PMS chain which also causes the above birefringence correction procedure to be detectably in error for molecular weights below ca. lo5. RESULTS AND DISCUSSION The 0.f.b. data are reported in terms of the frequency dependence of the magnitude SMand the phase angle 8 of the complex mechano-optic coefficient S* defined as (phasor notation) -An* S* = S exp (i8,) = S’+ is” = -(1) 3* where An the real part of An* is the difference between indices of refraction n and n corresponding to the principal polarizability directions and j the real part of ?* is the sinusoidally time-varying shear rate as defined The quantities SM and 8 are utilized rather than S’ and S” since 0 is by far the most sensitive function of the four and the 0.f.b.experiment is sufficiently precise to be able to utilize this sensitivity. Measurements obtained at various temperatures for each solution were reduced to the 25.00 OC reference temperature resulting in reduced-variable plots of bg(&/aT) and 8 against logfa, where f is the frequency in Hz and a is the time-temperature superposition factor., Scatter in measured SMand 0 before superposition are estimated to be 0.3% and f0.3’ respectively throughout the working frequency range except for the lowest molecular weight the lowest concentrations and the highest frequencies where the solvent contribution becomes large.Infinite-dilution properties are reported as [SM] = limSM/c and [O,]= limo,. c-0 c+o The general character of the concentration dependence of the 0.f.b. properties is illustrated in fig. 2. The finite-concentration results for five solutions of PS 3b are displayed; the individual data points have been replaced by smooth curves for clarity and these have been overlayed directly. The location on the reduced-frequency .axis of the initial departure of 8 from the -180’ low-frequency limit is governed by the longest relaxation time z,.Thus for PS 3b z increases by a factor of ca. 40 as the Concentration is increased by a factor of ca. 10. At high reduced frequencies (log fa 25) however 8 depends only weakly on concentration. In terms of the concentration dependence of the bead-spring-model relaxation times {zp)these results suggest that the effect of concentration is strongly mode-dependent ;with increasing mode number p (pincreasing corresponds to zp decreasing) the effect of concentration is decreased. Thus z has a strong concentration dependence whereas zN (the shortest relaxation time for the model) is almost independent of concentration. For frequencies above log fa x 5,8 exceeds the -270’ limit given by simple chain-dynamics theories for all PS and PMS samples studied to date.This anomalous behaviour is the principal subject of another article.3s Comparisons of the frequency dependence of infinite-dilution v.e. and 0.f. b. properties for linear molecules with the theoretical (isolated-molecule) bead-spring- model predictions indicate that in this concentration regime the observed properties tend to correspond more closely to the theoretical non-free-draining (dominant hydrodynamic interaction) predictions although the behaviour is generally inter- c. J. T. MARTEL et al. I'l'l'l"'!'~'! l 1 -6 -300 -i -7 -280 3 \ --8 -260 h -b -? 5 -9 -240 M 0 -2 ---220 -10 --200 -11 --180 111111111,!,111 -1 0 1 2 3 4 5 6 log UQ,) Fig.2. Plots of log(S,/a,) and 8 against logfu at various concentrations for 390000 molecular weight PS in Aroclor 1248 reduced to 25.00 OC. A = 5770 A. Values of c/g as follows (1) 0.1115 (2) 0.0713 (3) 0.0356 (4) 0.0215 (5) 0.0109. mediate between the free-draining and non-free-draining limits ;the appropriate value of the theoretical hydrodynamic interaction parameter h* required to fit experimental results is a function of the goodness of the solvent.8~9~22 In general excellent quantitative agreement has been found between experiment and bead-spring-model predictions8* 22 in the low- and intermediate-frequency regimes. The values of the two g9 theoretical parameters N and h* (N is the number of modes or gaussian subchains representing the polymer molecule and is proportional to the molecular weight) used to fit the 0.f.b.infinite-dilution properties of linear PS for all molecular weights studied correspond to a subchain molecular weight of ca. 5200 and h* ranging from 0.125 at high molecular weight to 0.175 at low molecular weight while for the linear-PMS data the subchain molecular weight is ca. 8000 and h* is 0.15. Analysis of the results for the PS comb and stars with small numbers of arms produces Nand h* values that agree with the linear-PS values. Fig. 3 illustrates the excellent agreement usually obtained; for the 390000 molecular-weight PS solution N = 75 and h* = 0.15. However for stars with large numbers of arms the agreement between model predictions and infinite-dilution properties is substantially poorer in the frequency regime where branching effects are most visible; fig.4 illustrates the trends generally observed. In general the 8 data show a 'crossover' behaviour. At low frequency ($a < 2.1) the observed behaviour is like that of linear polymers while for higher frequencies (fa > 3) the 8 data agree with the predicted values; the frequency regime 2.1 <faT < 3 shows a depression of the predicted 8 branching peak. The same trends are seen in v.e. data. Model calculations too extensive to be described here suggest that the crossover behaviour results from a suppression of the contribution from the centre of such stars; the central regions of any molecular geometry contribute by far 178 CONFORMATIONAL DYNAMICS OF POLYMER SOLUTIONS -6 -7 I-8 O.2 - u) L M -0 -9 -10 -11 0 1 2 3 4 5 6 log Cfar1 Fig. 3. Plots of log ([&&]/a,)and [e,] against logfa for 390000 molecular-weight Ps in Aroclor 1248 reduced to 25.00 OC. Curves Zimm theory for N = 75 h* = 0.15. Lo = 5770 A. Values of T/OC as follows @ 25.00; 0 15.88; 0, 2.81; 0 -1.42. -5 - l ' l ' l l ' T l I 1 d-300 i --280 -6- - I -260 2 - -7 - -240 0, -.-c.-P m -2 &a-00 -220 ---9 ---200 --10 ---180 -1l.III1I1~I1~'' 2 3 4 5 6 C. J. T. MARTEL et al. the major part of low-frequency v.e. or 0.f.b. properties. The reasons for such a suppression are not clear yet but excluded-volume effects caused by the high segment density in the centre of such stars would produce changes in the right direction.No quantitative evaluations of the effect on v.e. and 0.f.b. properties are available to date but Miyake and Freed have recently evaluated the effect of excluded volume on intersegment distances for linear and star molecules;37 their predictions appear to be consistent with this picture. Because of this discrepancy between predictions and infinite-dilution properties for stars no comparisons between finite-concentration predictions and measured frequency-dependent 0.f. b. properties will be presented here. The theoretical bead-spring-model predictions employed here (linear star or comb geometries) are calculated from1? 35 N (3) [Sol P-1 and (4) where 6kTllp NkT C T;/T~ P-1 is the infinite-dilution value of the relaxation time for the pth mode N is the number of modes (or gaussian subchains) for the model q' is an optical factor N is Avogadro's number k is Boltzmann's constant M is the polymer molecular weight o is the radian driving frequency [q]is the steady-flow intrinsic viscosity and A are the exact eigenvalues of the Zimm H-A or Lodge-Wu B matrix.38 For regular (equal-molecular-weight arms) stars eqn (3)-(5) may be written in a particularly useful form owing to the degeneracies introduced in the H*A or B matrices by geometrical symmetry.P-1 P-2 odd modes even modes Odd even t wherefis the number of arms and Nb is the number of gaussian subchains in one arm of the star.Thus for regular stars with several arms the contributions of the odd-numbered modes are enhanced markedly and steady-flow properties such as [So] or [q] will be dominated almost totally by the slowest mode. Also from the degenerate H-A matrix it is clear that T? will be controlled by 2Nb or twice the arm molecular weight hereafter designated as the 'span moleciilar weight ' rather than total molecular weight. (For a linear molecule the two are equal.) CONFORMATIONAL DYNAMICS OF POLYMER SOLUTIONS Fig. 5 shows the general character of the concentration dependence (at 25.00 "C) of z for the 400000 molecular-weight PMS and the 390000 molecular-weight PS for which the most extensive concentration range has been studied.The z1 values are obtained by superposing theoretical curves and experimental data. Assuming that all mode strengths are essentially concentration independent there is a unique relaxation- time distribution that will produce a given curve shape. The error bars shown for the higher concentrations are not a result of experimental uncertainty but are a reflection 0 0.05 0.10 c/g crn+ Fig. 5. Plot oft against c for 390000 molecular-weight PS (0) and 400000 molecular-weight PMS (0)in Aroclor 1248 reduced to 25.00 "C.(--) z = zy[1 +cA -~'2(cA)~+2(cA)~], M; (-) z1= zyexp(cA); (--) z = zy(l+cA) M.F.zY = 2.02ms A = 41. of inadequate theoretical fits. The highest concentration corresponds to c[q]x 11 for which zl/zy x 36. For c[q]c 3 the dependence of z on c is nearly exponential; how- ever for higher c an interesting change is observed that is readily seen in a semi- logarithmic plot for the same 390000 molecular-weight PS sample in fig.6 (which also shows z1values for four other molecular weights of linear PS). For this sample the low-c region (c[q]< 3) clearly shows an exponential concentration dependence while at intermediate concentrations (4 c c[q]c 9) there is a sigmoidal curvature apparently followed by a second regime showing a nearly exponential concentration dependence (c[q]from 9 to 11). This change in character in the 4 < c[q]< 9 region suggests the existence of two different concentration regimes centred about c[q]= 6 or 7. Interestingly from v.e. data one would expect entanglement effects to become dominant for this molecular weight at a concentration of 0.085 g ~m-~, suggesting that this transition may be caused by entanglement^.^^ None of the curves for the other molecular weights shown in fig.5 extends to sufficiently high concentrations to show c. J. T. MARTEL et al. this transition clearly although the data for the 860000 molecular-weight PS may show its onset; for this sample v.e. data suggest that entanglement effects should become dominant at a concentration of 0.048 g ~m-~. The z1for all other molecular weights (1 11 000 53 700 and 10000) show an exponential concentration dependence for the entire range of concentrations studied to date. The initial slope of these curves is a strong function of molecular weight particularly at higher molecular weights as I I1 10-61 I I1 I1 1 I I I I I I 1 1 1 1 11 11 I 0 0.0 5 0.10 0.1 5 0.20 clg cm-' Fig.6. Concentration dependence of z for several molecular weights of linear polystyrene in Aroclor 1248 reduced to 25.00 "C. (a) M = 860000 initial slope = 32.5 A = 75; (b) M = 390000 initial slope = 17.8 A = 41 ;(c) M = 11 1000 initial slope = 9.7 A = 22; (d) M = 53700 initial slope = 7.4 A = 17; (e)M = 10000 initial slope = 5.6 A = 13. noted in the fig. 6. Fig. 7 and 8 present values of log(z,/zy) plotted against c for linear and star molecules; each figure shows results for molecules having nearly the same span molecular weight. Fig. 7 illustrates that linear and 3-armed stars show the same exponential character with the slope of the curves increasing with span molecular weight as expected.For PS 5-12 a 12-armed star the dependence is not strictly exponential and the initial slope is larger than expected. Fig. 8 presents the results for molecules with a larger span molecular weight; here the span molecular weight CONFORMATIONAL DYNAMICS OF POLYMER SOLUTIONS 10 0-.-c- 1 0 0.01 0.02 0.02 0.03 0.05 0.06 0.07 c/gcme3 Fig. 7. Concentration dependence of (T,/z:) for several polystyrene stars and a linear polystyrene all of similar span molecular weights in Aroclor 1248 reduced to 25.00 OC.(-) Linear span M = 111000 T = 0.27 ms; (8,PS 5-12 (12-arm star) span M = 140000 T = 0.35 ms; 0 PS F-3 (3-arm star) span M = 135200 T = 0.31 ms; 0,PS E-3 (3-arm star) span M = 116400 z = 0.29 ms.clg ~rn-~ Fig. 8. Concentration dependence of (T/z:) for linear comb and 12-armed star polystyrenes of similar span molecular weights in Aroclor 1248 reduced to 25.00 OC. 0,Linear span M = 390000 T; = 2.0 ms; 0,PS 10-12 (12-arm star) M = 336000 T = 0.83 Ins; 0,PS C632 (comb) span M = 326400 T = 3.1 ms. estimate for PS C632 the 25-armed comb is probably too high since it is assumed to be the sum of the backbone molecular weight and twice the arm molecular weight. The results for the linear and comb samples seem reasonable but the 12-armed star shows a marked deviation. If the deviation is due to normal interchain interactions this may suggest that theoretical evaluations of the concentration dependence of relaxation times for star molecules will be more difficult than for linear polymers; the c.J.T. MARTEL et al. observed dependences are clearly different. However the infinite-dilution 0.f. b. properties for the 12-armed stars also show anomalous behaviour as noted earlier and whatever is responsible for this may also cause the different concentration behaviour shown in fig. 7 and 8. Fig. 9 presents semilogarithmic plots of measured values of the low-frequency (‘steady-flow’) limit of S, denoted as SMo divided by c plotted against c for the linear and star samples with span molecular weights of 1 1 1 000-140000. Again as for the z plots an exponential concentration dependence is observed for (S,,/c) for the linear and 3-armed stars.The initial slope of the curves is consistent with the span 10-4 I M m 5 v1 . n 0 -. 0 m 2 10-7 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 cJg~rn-~ Fig. 9. Concentration dependence of (S,,/c) for a linear polystyrene and several polystyrene stars all of similar span molecular weights in Aroclor 1248 reduced to 25.00 OC. (-@-) linear span M = I1 1000; 0,PS 5-12 (12-arm star) span M = 140000; 0,PS F-3 (3-arm star) span M = 135200; 0, PS E-3 (3-arm star) span M = 116400. molecular weights. The 12-armed star results show curvature as did the z values plotted for this molecule in fig. 7. Note that [SMO] is nearly the same for all samples although the total molecular weights differ by up to a factor of 8; the span molecular weight is the major controlling factor for [SMO].Fig. 10 is a semilogarithmic plot of values of z and K(S,,/c) as a function of c where K is an arbitrary constant introduced so that the shape and slope of the concentration dependence of these quantities can be compared for the same linear and star molecules (span molecular weights of 111000-140000). Clearly the same dependence is observed for both z and S, for this concentration regime for these molecules even for the lower-molecular- weight 12-armed star. This is also observed for all other molecules studied which have lower span molecular weights as well as for the comb sample but not for the 390000 and 860000 molecular-weight linear PS or the PS 10-12 star except at the highest concentrations for the 390000 molecular-weight linear PS.If one assumes that the concentration dependence of relaxation times is mode-dependent as the observed 0.f.b. frequency dependence indicates then from eqn (3) and (4) [and (6) and (7)] one would predict a difference in concentration dependence for z and S,,/c of the type exhibited by the high-span-molecular-weight samples except at the highest concentrations. The lower-molecular-weight (S,,/C) results may be misleading owing CONFORMATIONAL DYNAMICS OF POLYMER SOLUTIONS o-2 c L ---,OR - - - - - 1 ti4 I l l 1 l l 1 I I 1 0 0.05 0.10 c/g ~rn-~ Fig. 10. Comparisonofthe concentration dependences of z (0) polystyrene and several polystyrene stars all of similar span molecular weight in Aroclor 1248 and K(S,,/c) (:) for a linear reduced to 25.00 "C.(a)linear M = 111 000; (b)PS E-3 (3-arm star) span M = 116400; (c) PS F-3 (3-arm star) span M = 135200; (d)PS 5-12 (12-arm star) span M = 140000. to concentration-dependent contributions caused by modifications of solvent in the neighbourhood of the polymer chains as was noted previously. Recently Muthukumar and Freed2' and Muthukumar28 have obtained explicit expressions for the initial concentration dependence of the bead-spring-model relaxation times by considering the effect of intermolecular hydrodynamic interaction (quasi-static limit). The Muthukumar-Freed (M.F.) result for the pth relaxation time is zp = zO,(l +Acp-"+ ...) (8) and the Muthukumar (M) expression is zP = T;[ 1 +AcP-"-~'2(Ac/l-~)$+~(AcP-~)~-...] (9) where A is a positive constant obtained from the initial slope of the lnz against c curve and K is a positive exponent with a value of 0.5 in 8 solvents and 0.65-0.80 in good solvents.The M.F. expression was expected to be restricted to concentrations for which c[q] 5 1 while the M result would be expected to apply for a larger range of concentration. In order to compare predictions based on eqn (8) and (9) with experimental results values of N,h*,A and K are required. Nand h* are determined by fits to infinite-dilution properties as noted previously. IC has been arbitrarily selected to be 0.65 for 185 c. J. T.MARTEL et al. PS/Aroclor solutions (moderately good solvent conditions) ;the resulting predictions are insensitive to small changes in IC so its value is relatively unimportant.Fig. 5 includes curves corresponding to the predicted 2 concentration dependences from the M.F. and M theories and also shows an empirical exponential dependence (solid line) zp = zpexp (AcpK). (10) The M.F. theory clearly does not describe the observed dependence for the linear 390000 molecular-weight PS and the 400000 molecular-weight PMS for c[q] L 1 while the M result works fairly well for c[q] up to nearly 5. However when one looks at the shapes of predicted and experimental 0.f.b. frequency dependences for these molecules it is clear that although the M.F. result predicts quite accurately the observed relaxation-time spacings for concentrations such that c[q] s 2,9 the M expression and the arbitrary exponential form work equally well in this regime.However as one goes to higher concentrations the predicted 0.f.b. frequency dependences no longer match the measured properties. For c[q] > 3 for the 390000 molecular weight PS the M.F. result slightly underestimates the 8 peak and the breadth of the relaxation-time spectrum while the M equation leads to an overesti- mation of the 8 peak and predicts a correct relaxation-time spectrum breadth. The empirical exponential form also overestimates both the 8 peak and the breadth of the relaxation time spectrum at high concentrations. For c[q] = 1.9 for the 53700 molecular weight PS the M.F. result gives a good match in 8 peak shape but an incorrect spectral breadth.The M result and the exponential form overestimate the Bs peak but predict the correct spectral breadth. Fig. 11 and 12 illustrate the character of the disagreement between the M.F. and M predictions and the observed properties at even higher concentrations that are well -5 m-300 -6 ' 1-280 -7 -2 60 -<b g -8 -240 M --9 -220 -10 -200 -1 1 -1 80 -1 0 1 2 3 4 5 6 lo€! cf4,) Fig. 11. Plots of log(S,/a,) and 8 against logfa for 0.1000 g cmP3 solution of 53700 molecular weight PS in Aroclor 1248 reduced to 25.00 OC. & = 5770 A. T/OC as follows 8 25.00; 0, 15.88; (D 2.81 0,-1.42. Theoretical curves (N =9 h* = 0.175 A = 17 K = 0.65) 1 solid line M.F.; 2 dashed line M exponential. CONFORMATIONAL DYNAMICS OF POLYMER SOLUTIONS -4I j -320 t -5 -300 -6 -280 3 2 -0" -7 -260 -8 -240 -10 -200 -11 -180 ~111l11l1,1,1l1l ~ -2 -1 0 1 2 3 4 5 6 log Va,) Fig.12. Plots of log(S,/a,) and Os against logfa for 0.1115 gcmP3 solution of 390000 molecular-weight PS in Aroclor 1248 reduced to 25.00 OC Lo = 5770 A. T/OC as follows 0 45.00; 0,25.00; a,15.88; @ 2.81; 0 -1.42. Theoretical curves (N= 75 h* = 0.15 A = 41 K = 0.65) (--) rpaexp (AcP-~),(1,-) M.F. (2,--) M. A Fig. 13. Molecular-weight dependence of A for linear polystyrene in Aroclor 1245 reduced to 25.00 OC. beyond the expected range of applicability. Interestingly the M.F. form appears to produce a better relaxation-time spacing for lower-molecular-weight polymers in this regime (c[q]z3) and the M result works surprisingly well for the 390000 molecular-weight PS (c[q]= 11) while the empirical exponential form does not.It is clear that the experimental establishment of ranges of applicability for such theories must await completion of extensive molecular-weight-dependence and concentration-dependence studies. However fig. 13 presents a log-log plot of A against molecular weight for the linear polystyrenes studied to date. c. J. T. MARTEL et al. 187 CONCLUSIONS 0.f.b. and v.e. data clearly demonstrate that the relaxation times for slow modes have a strong concentration dependence while the fast motions are nearly independent of concentration. For the linear comb and 3-armed regular-star molecules studied to date z exhibits an exponential dependence on concentration for c[q]c 3.For linear molecules the Muthukumar theory correctly predicts the dependence of z on concentration for c[q]< 5. For the 390000 molecular-weight linear PS solutions for which the most extensive studies are available z as a function of concentration appears to exhibit two regimes the second appearing at concentrations just above that at which entanglement effects are predicted to become dominant from v.e. studies. 12-armed star PS does not exhibit the same exponential concentration dependence. Contrary to expectations the low-frequency frequency-independent values of SM denoted as SM0,show the same concentration dependence as z for linear and 3-armed stars for which the span molecular weight is below 150000.However the higher- molecular-weight linear PS samples show different concentration dependences for z (stronger dependence) and S, in the low-concentration regime and the same dependence at higher concentrations; this is predicted by any theory in which concentration effects are mode dependent. Regular stars with several arms would be expected to show more nearly the same concentration dependence for both z1and S, due to enhancement of the contributions from odd-numbered relative to even-numbered modes; this appears to be the case for the 12-armed stars studied to date. G. B. Thurston and J. L. Schrag J. Chem. Phys. 1966 45 3373. G. B. Thurston J. Chem. Phys. 1967 47 3582. G. B. Thurston and J. L. Schrag J. Polym.Sci. Part A-2 1968 6 1331. G. B. Thurston and J. L. Schrag Trans. SOC. Rheol. 1962,6 325. J. W. Miller and J. L. Schrag Macromolecules 1975 8 361. A. L. Soli and J. L. Schrag Macromolecules 1979 12 1159. M. G. Minnick and J. L. Schrag Macromolecules 1980 13 1690. T. P. Lodge J. W. Miller and J. L. Schrag J. Polym. Sci. Polym. Phys. Ed. 1982 20 1409. T. P. Lodge and J. L. Schrag Macromolecules 1982 15 1376. lU T. P. Lodge and J. L. Schrag Macromolecules in press. l1 R. M. Johnson J. L. Schrag and J. D. Ferry Polym. J. 1970 1 742. l2 D. J. Massa J. L. Schrag and J. D. Ferry Macromolecules 1971 4 210. l3 K. Osaki and J. L. Schrag Polym. J. 1971 2 541. l4 Y. Mitsuda K. Osaki J. L. Schrag and J. D. Ferry Polym. J. 1973 4 354. l5 K. Osaki Y. Mitsuda R.M. Johnson J. L. Schrag and J. D. Ferry Macromolecules 1972 5 17. l6 K. Osaki J. L. Schrag and J. D. Ferry Macromolecules 1972 5 144. T. C. Warren J. L. Schrag and J. D. Ferry Macromolecules 1973 6 467. N. Nemoto Y. Mitsuda J. L. Schrag and J. D. Ferry Macromolecules 1974 7 253. l9 R. W. Rosser N. Nemoto J. L. Schrag and J. D. Ferry J. Polym. Sci. Polym. Phys. Ed. 1978 16 1031. 2o B. G. Brueggeman M. G. Minnick and J. L. Schrag Macromolecules 1978 11 119. 21 K. Osaki Adu. Poly. Sci. 1973 12 1. 22 J. D. Ferry Viscoelastic Properties of Polymers (Wiley-Interscience New York 1980). 23 T. M. Stokich and J. L. Schrag to be published. 24 P. E. Rouse Jr. J. Chem. Phys. 1953 21 1272. 25 B. H. Zimm J. Chem. Phys. 1956 24 269. z6 G. B. Thurston and A.Peterlin J. Chem. Phys. 1967 46,4881. 27 M. Muthukumar and K. F. Freed Macromolecules 1978. 11. 843. 28 M. Muthukumar personal communication ; Macromolecules submitted for publication. N. Hadjichristidis A. Guyot and L. J. Fetters Macromolecules 1978 11 668. 3o N. Hadjichristidis and L. J. Fetters Macromolecules 1980 13 191. 31 J. E. L. Roovers Polymer 1979 20 843. 3z J. W. Miller Ph.D. Thesis (University of Wisconsin 1979). 188 CONFORMATIONAL DYNAMICS OF POLYMER SOLUTIONS 33 A. L. Soli and J. L. Schrag Macromolecules 1979 12 1159. 34 T. M. Stockich M. G. Dibbs T. P. Lodge and J. L. Schrag to be published. 35 M. G. Dibbs Ph.D. Thesis (University of Wisconsin 1983). 36 T. P. Lodge and J. L. Schrag Macromolecules in press. 37 A. Miyake and K.F. Freed Macromolecules submitted for publication; personal communication. 38 R. L. Sammler J. L. Schrag and A. S. Lodge Rheology Research Center Report no. 82 (University of Wisconsin Madison Wi 1982). 39 K. Osaki K. Nishizawa and M. Kurata Macromolecules 1982 15 1068.