J . Phys. Chem. 1990, 94, 371-375
371
Nonlinear Viscoelasticity of Wormlike Micelles (and Other Reversibly Breakable Polymers) M. E. Cates Cavendish Laboratory, Madingley Road, Cambridge CB3 OHE, U.K. (Received: February 16, 1989)
A model, previously introduced to describe the linear dynamics of entangled, reversibly breakable polymers, is extended to describe nonlinear effects. This extension of the model requires that (1) the nonlinear effects of a flow enter solely through its influence on polymer diffusion (not through any direct effect on the reaction rates for chain scission and recombination processes) and (2) the stress is dominated by polymer entropy contributions. With these assumptions, the viscoelastic functions are predicted to closely resemble those of the Doi-Edwards/Marrucci theory for monodisperse unbreakable polymers. The status of each assumption is discussed, with reference to recent experiments on viscoelastic solutions of polymer-like surfactant micelles.
1. Introduction The experimental study of reversibly breakable polymers, known as “equilibrium polymers” or “living polymers”, has been an established field for many years. Examples include molten sulfur,14 p ~ l y ( a - m e t h y l s t y r e n e ) , ~poly(tetrahydrofuran),’ ,~ and liquid selenium.8 These systems contain subunits that polymerize reversibly into extended linear chains. Recently, there has been intense renewed theoreticaleI3 and interest in equilibrium polymers in the context of viscoelastic surfactant solutions. These solutions exhibit rubbery flow behavior at only a few percent surfactant; many of them are stable at room temperature and are therefore of considerable interest in various rheological application^.^^ There is good evidence, from both light and freeze-fracture2* experiments, that the rubbery flow behavior is directly related to the presence in equilibrium
of very long, flexible, entangled wormlike micelles. The dynamic equilibrium of scission and recombination of the polymeric species in these systems has two interesting effects. Firstly, the molecular weight distribution (MWD) of the polymers is in thermal equilibrium. (This contrasts with ordinary polymer solutions, in which the MWD is quenched at the time of synthesis, and thermal equilibrium only applies to the remaining configurational degrees of freedom.) It is readily shown that under most conditions of interest the equilibrium MWD is extremely broad; indeed, in the semidilute or entangled regime it is e~ponential.~**’~ A second effect of the dynamical equilibrium is that breakage and recombination reactions, if sufficiently fast, can influence the relaxation of chain entanglements and thus completely alter chain diffusion properties and viscoelasticity. At the level of linear response theory, these phenomena have been discussed in the context of a simple model,I2 itself a generalization of the reptation mode1,2s-28which successfully describes the dynamics of unbreakable polymers. These developments are reviewed in sections 2 and 3. The extension of the model to describe nonlinear phenomena follows in sections 4 and 5. For simplicity of presentation, we consider only molten systems (in which the volume fraction of polymeric material is unity). The extension of the linearized model to systems with a solvent present was given in ref 13; the nonlinear theory for melts to be presented in sections 4 and 5 may be generalized to solutions in a parallel fashion.
Bacon, R.; Fanelli, R. J . A m . Chem. Soc. 1943.65, 639. Eisenberg, A. Macromolecules 1969, 2, 44. Petschek, R.; Pfeuty, P.; Wheeler, J. C. Pfiys. Rev. E 1986, 34, 2391. Cates, M. E. Europhys. L e f f .1987, 4, 497. Worsfold, D. J.; Bywater, S. J . Polym. Sci. 1957, 26, 229. (6) Kennedy, S. J.; Wheeler, J. C. J . Cfiem. Pfiys. 1983, 78, 953; 1983, 78. 1523. (7) Dreyfuss, M. P.; Dreyfuss, P. Forfschr. Hochpolym.-Forsch. 1967,4, 528. ( 8 ) Faivre, G.; Gardissat, J.-L. Macromolecules 1986, 19, 1988. (9) Mukerjee, P. J . Phys. Chem. 1972, 76, 565. Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J . Chem. SOC.,Faraday Trans. I 1976, 72, 1525. (IO) Porte, G.J . Pfiys. Cfiem. 1983, 87, 3541. ( I I ) Safran, S. A.; Turkevich, L. A.; Pincus, P. J . Pfiys. Lett. 1984, 45, L69. (12) Cates, M. E. Macromolecules 1987, 20, 2289. (13) Cates, M. E. J . Phys. (Les Ulis, Fr.) 1988, 49, 1593. (14) Ikeda, S.; Ozeki, S.; Tsunoda, M. A. J . Colloid Inferface Sci. 1980, 73, 27. Imae, T.; Kamiya, R.; Ikeda, S. J . Colloid Interface Sci. 1985, 108, 215. (15) Porte, G.; Appell, J.; Poggi, Y. J . Phys. Chem. 1980,84,3105. Porte, G.;Appell, J. J . Phys. Chem. 1981, 85, 251 I . Appell, J.; Porte, G.; Poggi, Y.J . Colloid Interface Sci. 1982, 87, 492. (16) Candau, S. J.; Hirsch, E.; Zana, R. J. Phys. (Les Ulis, Fr.) 1984, 45, 1263; J . Colloid Interface Sci. 1985, 105, 521. (17) Candau, S. J.; Hirsch, E.;Zana, R.; Adam, M. J . Colloid Interface Sci. 1988, 122, 430. ( I 8) Messager, R.; Ott, A.; Chatenay, D.; Urbach, W.; Langevin, D. Phys. Rev. L e f f .1988, 60, 1410. ( I 9) Hoffmann, H.; Loebl, H.; Rehage, H.; Wunderlich, 1. Tenride Deferg. 1985, 22, 290 and references cited therein. (20) Wunderlich, 1.; Hoffmann, H.; Rehage, H. Rheol. Acfa 1987,26,532. Rehage, H.; Hoffmann, H.; Wunderlich, I. Ber. Bunsen-Ges. Phys. Chem. 1986, 90, 1071. (21) Rehage, H.; Hoffmann, H. J . f f i y s . Cfiem. 1988, 92,4712. (22) Sakaiguchi, Y.; Shikata, T.; Urakami, H.; Tamura, A,; Hirata, H. Colloid Polym. Sci. 1987, 265, 750. (23) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1987, 3, 1081; Langmuir 1988, 4, 354. (24) Shikata, T.; Hirata, H.; Takatori, E.;Osaki, K. J . Non-Newtonian Fluid Mech. 1988, 28, 17 I .
(I) (2) (3) (4) (5)
0022-3654/90/2094-037 1$02.50/0 , , I
2. Reptation Model The dynamics of entangled, unbreakable polymers is quite well understood in terms of the reptation model.25-28 In this model, relaxation of chain conformations occur by the gradual disengagement of any given chain, by curvilinear diffusion along its own contour, from a tubelike environment. The tube consists of neighboring chains; these present a set of obstacles to diffusion normal to the chain contour. The curvilinear diffusion constant of the chain in its tube is denoted D J L ) , which under melt conditions varies as Le/L, where L is the length of the chain and Le is the entanglement length. For sections of chain of length L T~,,,, for this initial break to occur; this takes a time T E l / ( k l ) = CT T ~ Lwhich ~ , is the longest relaxation time as calculated in the (?breakTrep)i/2.These arguments assume that the dynamics of a Rouse model. chain end during its lifetime is dominated by the pure reptation process, Le., curvilinear diffusion of the chain as a whole. This 3. Role of Reversible Scission Reactions which we assume from now on, is valid so long as Tbreak >> T,,,, We next consider how these results are modified in the presence returning to the case of ‘Tbre& IT,,,,~ in section 6 . of a dynamic equilibrium involving scission and recombination The single-exponential stress decay can be explained by noting of the chains. The first effect is to enforce equilibrium of the that the relaxation mechanism, described above, is “democratic” MWD as described in section I . Under general c o n d i t i o n ~ , ~ J ~ J ~among tube segments. Before a given tube segment relaxes (in we find for the number density c(L) of chains of L monomers time E T ) , the chain occupying it typically undergoes many scission and recombination reactions, so that there is no memory of either c(L) = const L-2 exp(-L/L) (3.1) the initial length of the chain or the position on the chain initially corresponding to that tube segment. Thus, all tube segments relax L = exp(E/2kBn (3.2) at the same rate; there is no dispersion of relaxation times. The where E is the scission energy of the chain (Le., the creation energy single-exponential decay may therefore be viewed as a kind of for a pair of chain ends). If the reaction kinetics are exceedingly motional narrowing: there is a strong separation of time scales slow, so that negligible breaking occurs on the time scale of between the fast process (breaking/recombination), which dedisengagement of a typical chain (of length L), then the only effect termines the “averaging time” ?break for variables such as the chain on linear viscoelasticity is through the molecular weight distrilength, and the much slower process of stress relaxation. bution. At a first approximation, we can construct the relaxation The above results were found for a unimolecular scission model function p ( t ) as the weighted average of (2.1) over the chain length with uniform breaking probability. An alternative reaction involves distribution c(L) (3.1). At the level of steepest descents, one finds “end interchange” whereby a free end attacks an interior bond of another chain. This bond breaks to form two new ends, one p(t) = exp[-const ( t / ~ ~ ~ , ) l / ~ ] (3.3) of which combines immediately with the attacking end, the other where here and below, T , , ~denotes T,(L), obeying (2.2). This remaining free. If Tbreak is again defined as the lifetime of a chain represents an extremely nonexponential decay of stress. This before such a reaction occurs somewhere along its length, then prediction contrasts with experiments on several of the entangled the previous analysis remains applicable with only trivial modiwormlike surfactant systems mentioned earlier, in which an exfications. However, a third possible reaction involves bimolecular tremely pure exponential stress-decay curve is seen.”-24 This exchange of two interior bonds via a four-armed intermediate. demonstrates that the kinetics whereby micelles break and reform In contrast to either of the previous reactions, this mechanism can (or otherwise exchange material) is fast enough to directly affect lead to the relaxation of stress without the involvement of a chain the stress relaxation process. end. In cases when such a reaction predominates, the treatment To study this new regime, it was found helpful in ref 12 to based on an underlying tube is perhaps inapplicable and an apintroduce a specific model for the scission and recombination proach reminiscent of the classical “transient network” theory of reactions (although the qualitative conclusions turn out to be more polymer viscoelasticity might be more promising. A model of this general). In this model we assume the following: kind has been developed in ref 23. What is unclear at present ( I ) Scission of a chain is a unimolecular process, which occurs is whether or not such a model can really explain the observed with equal probability per unit time per unit length on all chains. single-exponentialrelaxation. Certainly, one can choose to assume The rate of this reaction is a constant k for each chemical bond; a single lifetime for the transient c r o ~ s l i n k s but , ~ ~ in principle, we introduce the parameter different cross-links ought to have different local environments and hence (in principle at least) a spread in intrinsic relaxation (3.4) rates. which is the lifetime of a chain of the mean length ( L ) before 4. Nonlinear Flows breaking into two pieces. The basic ideas of the model have been laid out in the previous (2) Recombination occurs as a bimolecular process, with a rate constant that is indepsndent of the molecular weights of the two section in the context of a linear theory. The regime of interest is when Tp,,,k T , , ~ ) , the stress relaxation function p(t) indeed obeys (3.3) as expected. Houever, for smaller Tbreak, p ( t ) becomes more like a k,T. Since at these deformations there are already strong non-
The Journal of Physical Chemistry, Vol. 94, No. 1. 1990 373
Nonlinear Viscoelasticity of Wormlike Micelles linear effects on the motion of unbreakable polymers,28we expect that for breakable chains strongly nonlinear effects will arise long before there is any direct influence of flow on reaction rates. With this assumption, progress can be made by slightly adapting the nonlinear theories of Doi and Edwards26s28and Marrucci2’ as formulated for unbreakable polymers. In the unbreakable case, the main influence of strong flows is as follows. Suppose we consider a tube that is initially a random walk. In a finite deformation E is applied, the average length of the tube increases. (The effect is second order in deformation and thus unimportant in the linear regime.) This is followed by a rapid retraction of the chain down its tube, which occurs in a time T ,
