Polymer Journal, Vol. 20, No. 10, pp 875-881 (1988)

Self Diffusion of Polymers in the Concentrated Regime I. Temperature Dependence of the Self Diffusion Coefficient and the Steady Viscosity of Polystyrene in Dibutyl Phthalate

Norio NEMOTO, Takaharu KOJIMA, Tadashi INOUE, and Michio KURATA

Institute for Chemical Research, Kyoto University, Uji, Kyoto 611, Japan (Received May 11, 1988)

ABSTRACT: The role of the segment friction coefficient in slow chain dynamics in con­ centrated polymer solutions was studied from temperature dependence of the self diffusion coefficient D, and the steady viscosity I'/- Concentrated solutions of two polystyrene samples (code F4; Mw=43900 and F40; Mw=355000) in dibutyl phthalate with concentration of 40 and 50wt% were used as test samples. Solutions of their blend containing I wt% of labeled F40 were also studied at total concentration of 40wt%. For unentangled PS (F4) solutions, temperature dependence of D, and I'/ at each concentration was found to be in good agreement with each other and the functional form was well described in terms of the free volume theory. The function with the same values of the free volume parameters at each concentration is also applicable for description of T dependence of D, of entangled PS (F40) solutions and also of D, of F40 in the blend solution. The same segment friction coefficient was concluded to be operative in mass and momentum transports in concentrated solutions irrespective of the entanglement coupling. KEY WORDS Self Diffusion Coefficient / Steady Viscosity / Concentrated Solution/ Polystyrene/ Temperature Dependence/ Free Volume Theory/ Forced Rayleigh Scattering /

In recent publications,1 •2 we postulated that tion with surrounding polymer chains and the self diffusion coefficient Ds of flexible depends on M and c in a very complicated polymers in the semidilute regime as well as manner. 3 - 14 Results on temperature depen­ in melts can be given by the product of two dence of Ds of a polyisoprene melt support contributions as eq 1, eq 1. 1 In the semidilute regime, data analysis on Ds of polystyrene in tetrahydrofuran2 and (1) also recent reports on the translational fric­ Here .fi(O represents contribution from micro tion coefficient of poly(methyl methacrylate) Brownian motion of a segment and is a simple in polystyrene solutions15 have shown that function of the segment friction coefficient C their behavior can be successfully interpreted

i.e., / 2 oc Tl(. The ( is supposed, from its def­ by the same idea on which eq 1 is based. Thus inition, to be independent of molecular weight a similar study on the concentrated regime of the polymer and to be a function of con­ seems to be all that remains for establishing centration c and temperature T. gi(c, M) re­ the validity of eq 1 over the whole concen­ presents the effect of the topological interac- tration range from the semidilute regime to

875 N. NEMOTO et al. pure melt. By fully specifying dependences of compared with one other and are analyzed on gz(c, M) on the c and M, we may get deeper the basis of the free volume concept. insight on the effects of the topological in­ teraction on slow chain dynamics in condensed EXPERIMENTAL systems, which is still under extensive study. In this paper, we report the results of forced Materials Rayleigh scattering measurements on concen­ Two narrow distribution polystyrene sam­ trated solutions of two polystyrene samples ples (code F4; Mw=43900 and F40; Mw= and their binary blends in dibutyl phthalate at 355000) were labeled with a photobleachable c=40 and 50wt% over a wide range of tem­ dye of 2-nitro-4-carboxy-4' -dimethylamino­ perature. We also report results of steady stilbene by following the labelling procedure viscosity measurements on solutions of the low described in an earlier report. 16 Concentrated molecular weight PS. Solutions of the higher solutions of the labeled homopolymers and molecular weight PS form entanglement net­ their binary blend in dibutyl phthalate (DBP) works at these concentrations, but those of the were prepared for two different concentra­ lower molecular weight PS are not entangled. tions of 40 and 50 wt% by exactly the same Their temperature dependences are critically procedure as described earlier. 17 The solu-

0 500 1000 (a) tis

C 0 ~of*ie.:~~~~~?!-W~~~7;?~~~~ > GI "C

0 500 1000 (b) tis

Figure 1. (a) Decay of diffracted light intensity of /d due to self diffusion of labeled PS with Mw =43900 in DBP at T = 60°C. Polymer concentration is 40 wt% and the grating spacing is 8 µm. The solid curve is a least-square fit with eq 2. (b) Normal residuals obtained from the fit in the upper figure.

876 Polymer J., Vol. 20, No. 10, 1988 T Dependence of D, and 11 of PS Solutions tion of the blend contained I wt% of the la­ d as q=2n/d. beled higher molecular weight PS(F40) and (3) 39% of the unlabeled lower molecular weight PS(F4). Good linearity, as is shown in Figure 2, assures that D, was determined to an accuracy of 10%. Methods The steady viscosity 11(fl of solutions of the Forced Rayleigh scattering measurements lower molecular weight PS(F4) has been mea­ were performed with an instrument recently sured as a function of the shear rate }; with a constructed at our laboratory. 16 Diffracted cone and plate type of viscometer. Details of light intensity /d at the Bragg angle from the the viscometer and experimental procedure are solutions was measured as a function of time described elsewhere. 18 The zero shear rate over a temperature range from 25 to 120°C value 1J was determined to an accuracy of I 0%. under temperature control of ± 0.1 °C. A typi­ cal example of raw data is shown in Figure I. RESULTS AND DISCUSSION The time profile of /d could be fitted by eq 2 with a single decay rate rd to all data. Temperature Dependence of D, and 1J of Unentangled PS-DBP Solutions Id= {A exp( -I'dt)+ B} 2 + C 2 (2) In Figure 3, D, and 1J of the lower molecular Here A is amplitude, B and C are contri­ weight PS(F4)-DBP solutions at c=40 and butions from coherent and incoherent back­ 50 wt% are semilogarithmically plotted against ground optical field, respectively. The rd the reciprocal of temperature r- 1• The so­ was estimated from a least-square fit to an lutions are not entangled at these concen­ accuracy of 10% at all temperatures. The self trations, if the extent of entanglement coupling diffusion coefficient D, of the polymers in is estimated by the relation, Mee= 1.8 x 104 solutions was calculated as the slope in the (gcm- 3 ),19 where Me is the molecular weight linear plot of rd against square of the wave between entanglements. It is evident from the vector q which is related to the grating spacing figure that T dependence of either Ds or '1 can not be represented with the Arrehnius type of equation. The behavior, which is convex to the 02 abscissa for D, and concave for 17, may be interpreted in terms of the free volume theory as will be shown later. It is well known that dependences of 17 on c, M, and T can be discussed with eq 4 in analogy I with eq I for D,. 20 Ill 0.1 C' (4)

Here f., (O is a function of c and T, and is

supposed to be proportional to (, i.e., f 1 (0 oc 3 T/fi(O. 91(c, M) is a function which expresses the effect of the topological 1J. 5 10 interaction on q2/ µm-2 If 91 and 92 are independent of T to PS solu­ tions as is postulated here, then T/Ds should Figure 2. Plot of the decay rate rd against square of have the same temperature dependence as 17 the wave vector q obtained for the 40 wt% solution of F4 in DBP at three temperatures of I, 70°; 2, 40°; and 3, does at fixed c and M. In Figure 4, reduced 25°C. quantities D,,,T/D,T, and 17/17, are compared.

Polymer J., Vol. 20, No. 10, 1988 877 N. NEMOTO et al.

r,•c r,•c -8 120 90 60 30 120 90 60 30

s- • • -.. 0- C' • 2 0- • -C' E1- 0- -OI • 0 0- 1:1- -'II) • • E -10 0- 0 u • + • 0- 0- Vl • -0 • 1:1- OI 0- • B- • - vi -1 0 _g C s- + 0- -OI • 0 0- • -1 2 .__....__.____._ _.__....J.....__,__.1..._....L.__J - - 2 .__....__.____.•_ _.__....J.....__,__.1..._....L.__J -1 2.5 2.8 3.1 3.4 2.5 • 2.8 3.1 3.4 (a) r-1110-3K-1 r- 1110-3K-1 Figure 4. Comparison of temperature dependence be­ r,•c tween D, ,T/D,T, and IJ/IJ, calculated from the data in 120 90 60 30 Figure 3. ·The reference temperature T, is 60°C. Symbols are(.) and<•) for D, and 1J at c=40wt% and (0-) 4 0 and (D) for D, and IJ at c=50wt%, respectively.

0 the reduced data at each concentration. 0 3 Thus, we can conclude that separation of local and global motions by f and g as is -QI 0 • II) • given by eq 1 and 4 is valid for D. and I'/, and 0 0 a. that the same segment friction coefficient ( 2 0 • may be used to mass and momentum trans­ C' - 0 • ports at least in unentangled solutions. -OI 0 0 0 • - • Effect of Entanglement Coupling on Tem­ • • perature Dependence of D • • The higher molecular weight PS sample (F40; Mw=355000) forms an entanglement 0 '--~---'--~-~~~-~~~ ~.5 2.8 3.1 3.4 network at c=40 and 50wt%. Thus, com­ ( b) r- 1110-3K-1 parison of D. data of these solutions with those Figure 3. Temperature dependence of the self diffusion of the unentangled PS(F4) solutions in the coefficient D, and the steady viscosity 1J of F4 solutions. previous section is quite suitable for exam­ (a) Plot of D, vs. r- 1. Concentration is (.), 40 and ination of the effect of entanglement coupling (0-), 50wt%. (b) Plot of 1J vs. r- 1. Concentration is (.), 40 and (D), 50wt%. on T dependence of D •. In addition, we studied T dependence of the tracer diffusion coefficient of F40 (c= I wt%) in the F4 solution (c= Here 60°C is chosen as a reference tempera­ 39wt%). Values of D. obtained are tabulated ture T,, and D.,, and I'/, are corresponding with D. and I'/ values of the F4 solutions in values at T,. Agreement looks excellent for Table I. Figure 5 gives a semilogarithmic plot

878 Polymer J., Vol. 20, No. 10, 1988 T Dependence of D, and I'/ of PS Solutions

Table I. Self diffusion coefficient D, and I'/ TJ•c of PS in DBP 90 60 30

Sample D, X 1011 / D,,,T • Tj°C cm2 s - 1 1J /poise I'/ XI], • code D,T, • • • 0- 532 21.6 • F440% 25 1.87 19.6 - • 0- CII 315 12.8 • 30 3.30 11.3 1:-10' • 0- 40 8.39 4.59 123 5.00 u • • 50 20.0 1.99 48.0 1.95 • 1/l • 60 41.0 24.6 -C -0 76.4 0.553 12.8 0.520 -0 70 -Cl ' 80 131 0.332 7.75 0.315 0 -0 ' 90 212 0.221 5.10 0.207 - -12 -0 ' 100 347 0.132 3.60 0.146 -0 ' F40 in F4 25 0.292 19.3 40% 30 0.513 11.2 2.5 2.8 3.1 3.4 40 1.27 4.66 r- 1110-3K-1 50 2.92 2.09 60 6.30 1.00 Figure 5. Semilogarithmic plot of D, vs. r- 1• Symbols 70 11.4 0.569 are:(.), F4, 40%; (,), F40, 40%; (e), F40 in F4, 40%; 80 19.4 0.344 (0-), F4, 50%; and (-0), F40, 50wt%. 90 32.4 0.212 F40 40% 50 0.102 2.04 60 0.214 1.00 I I I I 70 0.419 0.526 80 0.755 0.300 90 1.34 0.174 1.2 ------100 2.17 0.110 ,11 l"'I • 110 3.07 0.0802 1.0 --~---! - -Qi-~---- 120 3.90 0.0648 F4 50% 25 0.152 52.7 9140 44.4 0. 8 __ a. ______--- 30 0.327 24.9 4780 23.2 40 1.22 6.90 1530 7.43 50 3.31 2.62 527 2.56 I I I I 60 8.95 I 206 20 50 80 110 70 19.7 0.468 94.0 0.456 r,•c 80 38.2 0.248 48.7 0.236 Figure 6. Temperature dependence of D, and I'/ was 90 68.9 0.142 26.2 0.127 critically examined by plotting a ratio J x, defined by eq 100 114 0.0879 16.0 0.0777 5, against T. Symbols are the same as in Figures 3 and 5. F40 50% 60 0.0382 1.00 70 0.0837 0.470 80 0.164 0.247 and _2.6 at c=40 and 50wt%, respectively. 90 0.386 0.108 Such a strong M dependence in the transition 100 0.536 0.0798 110 0.981 0.0448 region from the unentangled to entangled state 120 1.51 0.0299 is consistent with the self diffusion behavior of the same PS molecule m melts around M/ Me~ 1 where the value /3 has been found to of D. vs. r- 1 . The most pronounced feature in be close to 2.5. 21 •22 On the other hand, de­ the figure is that D. decreased drastically with crease in D. with increasing c may be mainly the increase in both c and M at constant T. If related to increase m local friction. More the dependence of D. on Mat constant c and T detailed studies on dependences of D. on c and were boldly expressed by the power law of M will be reported in forthcoming papers. D.ocM-P, the exponent /3 would become 2.5 In spite of the strong dependence of D. on c

Polymer J., Vol. 20, No. 10, 1988 879 N. NEMOTO et al. and M, relative changes in Ds values with be expressed by the product of the local fric­ varying temperature look almost the same for tion factor f and the global structure factor g two homopolymer solutions and one blend as given by eq 1 and 4, reduced quantities such solution with the same total concentration. as D,,,T/D,T, and r,/11, are equal to a ratio of For a quantitative comparison, a reduced the segment friction coefficient ( at T and T,. quantity ax=[D,,,T/DJ,lx was first calculated Their temperature dependences may be ex­ from D, data in Figure 5 by taking the ref­ pressed in terms of the WLF type of equation, erence temperature T, as 60°C. Here the suffix log(D,,,T/DJ,) = log(r,/r,,) x denotes the F40 or F4 solutions, or the blend solution. Then a ratio Llx, defined by eq 5, has = log(((T)!((T,)) been calculated and is plotted against T in C/(T-T,) Figure 6. (6) Cz'+(T-T,) (5) Here Ci' and Cz' are constants and related to In the figure, we also show Llx of r, of the F4 the free volume parameters, the fractional free solutions estimated by the same procedure. volume, f,, and the thermal expansion coef­ The Ll x = 1 in the ordinate means that the ficient of the free volume, ci.r, by eq 7. temperature dependence of D, or r, of the C1' = B/2.303!,, (7) solution denoted by the suffix x is identical with that of Ds of the F4 solution at the where B is a constant of an order of unity. corresponding concentration. If there is a 10% Equation 6 implies that a plot of (T-T,)/ difference between ax(T) and aF4 (T), it results log(((T)/((T,)) against T-T, should give a in Llx= 1.1 or 0.9. All Llx values are located in the range from 0.8 to 1.2 except only one Llx value for Ds of the 50% F40 solution at 90°C.

In consideration of an experimental accuracy 50 of 10% for both quantities of D, and r,, agreement seems satisfactory. Thus the tem­ ...... --... perature dependence of D, is unaffected by the -t- C=40%: presence of entanglement coupling. The above UI - 40 results may be regarded as strong support of • • --t- •• 0- the applicability of eq I to D, over a wide - UI • if- concentration range from the semidilute re­ '----'- gion to pure polymer melts. 'Ol --0 30 Analysis of the Data in Terms of the Free C=50% -t-.. Volume Theory I It was pointed out in a previous section that t- T dependence of D, and r, (Figures 3 and 5) did - 20 not obey the simple Arrehnius type of equa­ tion. We here apply the free volume theory for -40 20 0 20 40 60 interpretation of the data, because the theory (T-Tr)/K has been demonstrated to be quite successful Figure 7. Applicability of WLF equation to tempera­ for explaining the T dependence of viscoelastic ture dependence of D, and "I in Figures 3 and 5 was properties of polymeric materials in concen­ tested. Symbols are the same as in Figures 3 and 5. Solid trated solutions and in bulk. 23 If D, and r, can lines are the best fit from eq 6.

880 Polymer J., Vol. 20, No. 10, 1988 T Dependence of D, and 11 of PS Solutions

Table II. Free volume parameters 2. N. Nemoto, M. R. Landry, I. Noh, T. Kitano, J. A. Wesson, and H. Yu, Macromolecules, 18, 308 (1985). C Ci' 1Xr 3. L. Leger, H. Hervet, and F. Rondelez, Macro­ C1' 1; molecules, 14, 1732 (1981). wt% K 10-4K-1 4. P. T. Callaghan and D. N. Pinder, Macromolecules, 17, 431 (1984). 40 4.06 143. 0.107 7.49 5. J. A. Wesson, I. Noh, T. Kitano, and H. Yu, Macromolecules, 17, 782 (1984). 50 4.84 134. 0.0898 6.71 6. H. Kim, T. Chang, J. H. Yohanan, L. Wang, and H. Yu, Macromolecules, 19, 2737 (1986). 7. H. Deschamps and L. Leger, Macromolecules, 19, straight line. When the Ds and 17 data are 2760 (1986). plotted in such a manner, we obtain a straight 8. C. R. Bartles, B. Christ, and W. W. Graessley, Macromolecules, 17, 2702 (1984). line at each c, as is shown in Figure 7, from 9. (a) M. Antonietti, J. Coutandin, R. Grutter, and H. which C1' and C2 ' are easily evaluated. The Clr Sillescu, Macromolecules, 17, 798 (1984). and fr were calculated from eq 7 by putting (b) M. Antonietti, J. Coutandin, and H. Sillescu, B= 1, and are listed with values of C/ and Macromolecules, 19, 793 (1986). 10. (a) B. A. Smith, Macromolecules, 15, 469 (1982). C/ in Table II. The thermal expansion cof­ (b) B. A. Smith, E.T. Samulski, L.-P. Yu, and M.A. ficient of DBP and PS above Ts is 8.6 x 10-4 Winnik, Macromolecules, 18, 1901 (1985). and 6.5x 10-4 K- 1 , respectively. 24 The Clr (c) B. A. Smith, S. J. Mumby, E.T. Samulski, and L.­ values of the two PS~DBP solutions in Ta­ P. Yu, Macromolecules, 19, 470 (I 986). 11. (a) P. F. Green, P. J. Mills, C. J. Palmstron, J. W. ble II are intermediate between them. The Mayer, and E. J. Kramer, Phys. Rev. Lett., 58, 2145 glass transition temperature Ts of the so­ (I 984). lutions was roughly estimated from DSC (b) P. F. Green and E. J. Kramer, Macromolecules, measurements as - 55° for c = 40 wt% and 19, 1108 (1986). 12. G. Fleisher, Polym. Bull., 11, 75 (1984). - 38°C for 50 wt%, respectively. Assuming 13. D.S. Pearson, G. VerStrate, E. von Meerwall, and F. linear dependence of f on T, the fractional G. Schilling, Macromolecules, 20, 1133 (1987). free volume at Ts was calculated as 0.021 14. Q. Trang-Cong, T. Chang, C. C. Han, and Y. Nishijima, Polymer, 27, 1705 (1986). and 0.024 for the solutions with c = 40 and 15. N. Nemoto, S. Okada, T. Inoue, and M. Kurata, 50 wt%, respectively. The values are in rea­ Macromolecules, 21, 1502 (I 988). sonable agreement with the accepted value 16. T. Inoue, N. Nemoto, T. Kojima, Y. Tsunashima, of /g, = 0.025 ± 0.003. 23 and M. Kurata, Nihon Reorogi Gakkaishi, 16, 72 (1988) (in Japanese). 17. T. Inoue, N. Nemoto, T. Kojima, and M. Kurata, Acknowledgements. We are very grateful the preceeding paper. to Mr. T. Yamagishi of this institute for DSC 18. N. Nemoto, K. Okawa, and H. Odani, Bull. Inst. measurements. We also thank to Toyo Soda Chem. Res., Kyoto Univ., 51, 118 (1973). Co., Ltd. for supplies of PS samples. This 19. K. Osaki, K. Nishizawa, and M. Kurata, Macro­ molecules, 15, I 068 (I 982). work was partly supported by a Grant-in-Aid 20. G. C. Berry and T. G. Fox, Adv. Polym. Sci., 5, 261 for Scientific Research (No. 62550653) of (1968). Ministry-of Culture, Science and Education of 21. H. Watanabe and T. Kotaka, Macromolecules, 20, Japan. 530 (I 987). 22. N. Nemoto, S. Okada, T. Inoue, and M. Kurata, Macromolecules, 21, 1509 (1988). REFERENCES 23. J. D. Ferry, "Viscoelastic Properties of Polymers," 3rd ed, John Wiley, New York, N. Y., 1980. l. N. Nemoto, M. R. Landry, I. Noh, and H. Yu, 24. J. A. Riddick and W. B. Bunger, "Organic Solvents," Polym. Commun., 25, 141 (1984). 3rd ed, John Wiley, New York, N. Y., 1980.

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