Langmuir 1994,10, 1627-1629
Dip i n G”(w) of Polymer Melts and Semidilute Solutions Rony Granekt Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot 76100, Israel, and Department of Chemistry and Biochemistry, University of California, 405 Hilgard Avenue, Los Angeles, California 90024 Received January 10,1994
Introduction The long time relaxation of stress and diffusion in polymer melts and semidilute solutions are well described by de Gennes’s’ and Doi-Edwards’ reptation model. However, at times shorter than the chain’s Rouse time T R , the model predicts an intermediate “breathing”regime,ZJ where a chain segment (n) follows anomalous Rouse diffusion along the tube, ( (A&J2) = t’l2 (with 1, denoting the segment position along the tube). Since the chain is Gaussianly distributed in space (( r2) = l ) , the real-space displacement of a chain segment, in particular that of the chain ends, follow^^^^ ( (ArJ2)
31
( (A1n)2)’/2b
(LeL)’/2b(t/?R)1’4 (1)
where Le is the entanglement length and b is the segment (or persistence) length. A similar effect has been suggested for the stress relaxation f ~ n c t i o n . ~According ,~ to the reptation idea, stress is relaxed when the chain ends pass (for the first time) through an entanglement point. It follows that the stress relaxation function Gbrth(t)(with the plateau modulus normalized to unity) obeys4p5
where a and d are numerical constants resulting from the first-passage-time nature of this anomalous diffusion problem9 a N 0.508. At further shorter times, times shorter than the entanglement time Te, stress relaxes by free Rouse motion. In “living”polymer systems one has to consider another time scale, the lifetime of an average chain Tb.5,7 This is the average time for a chain to keep its integrity before breaking into two chains or combining with another chain. These large dynamic fluctuations in chain length were incorporated analytically using a Poisson renewal mode1.8~9 Numerical evaluation of the complex modulus G*(w) based on this combined Poisson renewal/Doi-Edwards model was carried out.8 The results showed a dip in G”(w),whose position on a log-log plot occurs roughly in the middle of the breathing regime. The value of G” at the dip, G”min, was found sensitive to the ratio LdL. For “dead”polymers, t Present and permanent address: Weizmann Institute of Science. (1) de Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornel1 University: Ithaca, NY, 1979. (2) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon: Oxford, 1986. (3) de Gennes, P . 4 . J . Chem. Phys. 1971,55,572. (4) Doi,M. J. J . Polym. Sci.,Polym.Lett. Ed. 1981,19,265; J.Polym. Sci., Polym. Phys. Ed. 1981, 21, 667. (5) Cates, M. E. Macromolecules 1987,20, 2289. (6) Granek, R. Unpublished work. (7) For a review, see: Cates, M. E.; Candau, S.J. J.Phys. C Condens.
Matter 1990,2, 6869. (8)Granek, R.; Cates, M. E. J. Chem. Phys. 1992, 96,4758. (9) Lequeux, F.; Courty, F. Phys. Lett. A 1991, 158, 197.
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and for living polymers whose lifetime (usually referred to as the “breaking time”) is sufficiently longer than T,, a preliminary rough estimate, based on a limited set of numerical data, was obtained8 as G”min/G‘m = LdL. Remarkably, this ratio has been found very useful for systems of wormlike micelles in semidilute solutions for obtaining chain length values and scaling properties from experimental Cole-Cole plots.8JOJl In this short paper I reexamine this estimate for the dip, since a small correction to this estimate may lead to very different conclusions. I consider both dead and living polymers, but for simplicity I shall limit the discussion to monodisperse systems (although living systems are widely polydisperse); the effect of polydispersity is primarily in determining the numerical constants for the results (such as the dip value), which is not my main purpose here. Dead Polymers Analytical Approach. My approach is based on the self-consistent assumption that the dip in G”(w) appears well inside the breathing regime of frequencies, T R - ~ 1, we get from eq 13 for the kinetically controlled breathing relaxation G''b,.th,i
G*(w) have been evaluated according to eq 3 for several values of the ratio L,/L. A typical result (Le/L = 0.01) is shown in Figure 1. (Frequencies are measured in units of (~27,&1.) The dip in G" has been extracted and is plotted in Figure 2 against L,/L; it shows a clear power-law behavior. The exponent obtained from the fit is 0.78; the small discrepancy from the expected value, 0.8, is attributed to numerical errors. (The prefactor C in eq 8 is obtained as C N 0.62, and the reason for the discrepancy from the value C N 0.81 obtained above is also suspected to be numerical.) Living Polymers Living (or equilibrium, self-assembled) polymers are characterized by a certain lifetime, 7b, after which they loose their For example, a living polymer can
N
1/*Tv
The Rouse relaxation contribution remains the same as in the dead polymer case, eq 10. Following the same procedure as in the previous section, I arrive at the results
and
G'"i,, thus increases from its dead polymer value as 7 b decreases beyond am,,-'and reaches the value (Le/L)lI2for 7 b = 7,. For experimental purposes it might be easier to (12)Turner, M. S.;Cates, M. E.Langmuir 1991, 7, 1590. (13)Kern, F.; Zana, R.; Candau, S. J. Langmuir 1991, 7, 1344.
Langmuir, Vol. 10, No. 5, 1994 1629
Notes 10
1
0.1
G’ 0.01
0.001
0.001 1 10.’~ 1 o6
1 o8
10’0
10’2
10j4
0
Figure 3. Imaginary (G”) and real (G’) parts of the complex modulus of living polymers for L./L = lo-‘and for { = Tb/(1r2Td) =3X The frequency is measured in units of (T2Td)-’.
. . . . . . .*’
. . . . . . . .‘
10’2
10”
. .
....Id 10-’0
6 Figure 4. Dip value G’”h against the ratio f in the very fast reaction regime { < 10-lO. The line is a power-law fit to the data obeying G”,,,h = 1.74 X lCb7P388.
Discussion In this paper I have refined the estimate for the dip value of G” previously suggested in ref 8, and showed that it is proportional to (LJL)O.*. I have also investigated more quantitatively the fast kinetic regime where the dip value is determined also by Tb. Since I have normalized G ( t )to unity in the t 0 limit of the breathing regime (c.f. eq lo), these results for G”min have to be divided by the extrapolated plateau modulus G’, when applied to experimental results. It is interesting to reconsider experimental reports for wormlike micellar systems where the less accurate estimate
(G“,iJG’cc = Le/L) was previously used to analyze the results. For semidilute solutions of cetylpyridinium chloride (CPyC1) in brine studied recently by Berret et a1.,’0 I obtain using eq 9 (and the relation1 Le = 15/9 different dependences of the averaged chain length on increasing surfactant concentration 4, according to the method used to find the correlation length E (in parentheses): L = 4°.72(light scattering, t = and L = 4°.60 (rheology,t = The latter result-L = q5°.60-is very close to the one predicted by Cates for self-assembly polymers with excluded volume interaction in the semidilute regimea7 For aqueous solutions of hexadecyltrimethylammonium bromide (CTAB) in the presence of KBr studied by Khatory et al.,” the length obtained using this procedure shows a very weak dependence on surfactant concentration, with the exponent varying from 0.07 (rheology, 5 = 4a.62)to -0.15 (theoretical, 5 = qk3l4). Indeed, this latter system was described as anomalous, and it was even suggested that it corresponds to a branched gel-like network.” It is worthwhile mentioning here another method to obtain the chain length from rheo1ogydatas2 This involves the plateau regime of G’(w), which begins at w 1 = Td-l on the low-frequency side and terminates at w2 = T , - ~ on the (L/LJ3 for dead high-frequency side. Thus, w2/w1 polymers, but this is no longer true for living polymers whose breaking time (lifetime) is shorter than the reptation time. The advantage of using the value of G”min rather than this plateau regime method is 2-fold. First, it is more practical, as rheometers are often unable to reach the highl . living polymers another frequency regime w = ~ ~ - For apparent advantage is the possibility to obtain the chain length when the breaking time is much shorter than the reptation timegJ0Jlor even the Rouse time (but not shorter than wm0-l), since the result in eq 8 does not depend on any time scale. Finally, I would like to note that the Doi-Edwards model used here for the short time, prereptation, relaxation builds on a Rouse-type motion. This should be well applicable for melts in which the screening length for the hydrodynamic interaction is believed to be of the order of the persistence length. However, for semidilute solutions effective medium theory2J5 shows that sufficiently high relaxation modes are Zimm-like, which suggests that the short time behavior should be controlled by Zimm motion. These issues will be addressed in future work.
(14)The lifetime is in fact set to 1.11Tb so that it coincides with the effective lifetime of an average chain used in ref 8.
(15)Perico, A.;Freed, K. F. J. Chem. Phys. 1984,81,1466,1475.
use instead of eq 15 the equivalent expression
For example, in systems of wormlike micelles of ionic surfactants, an increase of salt concentration leads to an increase in chain length and thus changes TV,but Te remains (almost) unaltered, so one can directly check the scaling G’lmin ~v-1/2, To verify eq 15, I have used the Poisson renewal model outlined in refs 8 and 9, assuming that the stress relaxation function of a dead system ( T b -) is given by eq 10. This model uses a Poisson distribution for the lifetime of a chain (Le.,the renewal time) and is therefore more realistic than the deterministic renewal model described above.14 For simplicity however I have assumed that the polymers are monodisperse. This is quite inadequate an assumption for living polymers; however, the effect of the commonly used Poisson distribution for chain length in the results in eqs 14 and 15 is expected to have influence only on numerical prefactors. This is because the averaging (assuming also T b = L-l as for the breaking-recombination reactions) over these results is essentially independent of the minimum chain length cutoff. The numerical results were obtained for the ratio LJL = lo4 and for different values of T b given in terms of the ratio { = Tb/(?r2Td) (used throughout refs 5,8, and 12). The range of {which corresponds to the regime under discussion (Te < Tb < Uno-’) is < { < 10-lo. A typical result = 3 x 10-12) of G* is plotted in Figure 3. In Figure 4 we plot the numerical values of G” at the dip against 5; fitted by a power law. The value obtained for the exponent is 0.386, indeed very close to the expected value 318 = 0.375.
-
(r
-
