Langmuir 1991, 7, 159G1594
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Linear Viscoelasticity of Living Polymers: A Quantitative Probe of Chemical Relaxation Times M.S.Turner* and M.E.Cates Cavendish Laboratory, Madingley Road, Cambridge CB3 OHE,U.K. Received November 9, 1990. In Final Form: February 26, 1991
Stress relaxation in wormlike micelles and other "living polymers" ie governed by an interplay between simple reptation and the ability of the micelles to break and recombine. A simple model of the relaxation is available for which the linear viscoelastic functions can be written as averages over a one-dimensional stochastic process. Detailed results from a numerical simulationof this procesa are presented for varioue values of { - / T ~ ~ , where 7bob.ll is the time taken for a micelle of the mean length to break and T , . ~ is ita (hypothetical) reptation time. First the stress relaxation modulus G ( t ) is calculated and then thls is transformed to give the frequency-dependentmodulus G*(w). The terminal stress relaxation time for the system T is also found. When { is small, the stress relaxation is exponential. However, for {of order unity, there are clear departures from exponential relaxation which are most clearly seen in a Cole-Cole representation. Compariso_nof our calculated plots with those measured experimentally allow a direct . enables the chemical relaxation time TM (which is hard estimate of the parameter { = T ~ / T This to measure directly in concentrated systems) to be estimated purely from linear viscoelastic data. In one systemwhere an independentmeasurementof rbrak is available,we find agreementto within the experimental error. 1. Introduction
The dynamics of entangled reversibly breakable polymers, which can break and recombine end to end, has received recent and experimentalbl6 attention, especially in the context of viscoelastic phases of surfactant solutions. Aqueous systems such as CTAB/ KBr and CTAC/NaSal, under appropriate conditions,are known2 to assemble reversibly into flexible wormlike micelles. These are, in favorablecases,extremelylong (many thousand angstroms) and flexible and undergo scission and recombination reactions on a relatively rapid time scale T b d , which we define as the mean waiting time for a micelle of the average length to break into two pieces. Some recent experiments" concern the frequency-dependent linear shear modulus G*(w) for these systems. The results can be presented in the form of a Cole-Cole plot, in which the imaginary part of the complex modulus (the loss modulus G") is plotted against the real part (the storage modulus G'). The theoretical model of ref 1,described in more detail below, predicts a single exponential stress relaxation and (1) Cata, M.E. Macromolecules 1987,20,2289. Cates, M. E. J. Phys. (Park) 1988,49,1693. Cates, M. E. J. PhyS. Chem., in prese. ( 2 ) For a rmnt review of theory and experiment see Cates, M. E., Candau, S.J. J. Phys.: Condem. Matter 1990,2,6869, (3) Turner, M. S.; C a b , M. E. J. Phys., (Paris) 1990,61, 307. (4) Lmg, J.; h a , R. Chemical Relaxat~onMethode. In Surfactant solutio^; Zana, R., Ed.;Marcel Dekker: New York, 1987; p 405 and reference therein. (6) Ken, F.;Collin, D.; Candau, S.J. To be submittedfor publication. (6) SKkata, T.; Hirata, H.; Kotaka,T. Longmuir 1987,3,1081. (7) Shikata, T.; Hirata, H.; Kotaka,T. Longmuir 1988,4,354. (8) Rehage, H.; Hoffmann, H. J. Phys. Chem. 1988,92,4712. (9) Hoffmann, H.: Loebl, M.; Rehaae, - H.;Wunderlich. I. Tenside Deterg. 1986,22,290. (10) Hoffmaun, H. Rehage, H.; Wunderlich, I. Rheol. Acta 1987,26, cnn 004.
(11) Porte,G.;Appell, J. J. Phye. Chem. 1981,85,2511. (12) Appell, J.; Porte,G.; Poggi,Y. J. Colloid Interface Sci. 1981,87, 492. (13) Porta, 0.; Marignan, J.; Baasereau, P.; May, R. J. Phys. (Park) 1988,49, 611. (14) Candau,S. J.; Hirsch, E.; Zana, R. J. PhYS. (Paris)1984,45,1263. (15) Candau, S. J.; Hirsch, E.; Zana, R. J . Colloid Interface Sci. 1985, 105,521. (16) Candau, S. J.; Merikhi, F.; Wateon, G.;Lemarechal, P.J . Phys. (Paris) 1990,51,977.
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hence a semicircular C o l d o l e plot, in the regime when scission reactions are rapid on the time scale of reptation17 of an average chain. This prediction is well confirmed in experiments on several types of wormlike micellar system." In the present work, we give a detailed numerical analysis of the model in the crossover regime where significantdepartures from the semicirclearise.'&" Below we present a quantitative method for estimating the chemical relaxation time T b d in any given experimental system, purely by analyzing its linear viscoelastic response. This is of great interest for two separate reasons. Firstly, given a reaction scheme for breakdown and fusion of aggregates! 7~ can in principle be measured indeThus where pendently by temperature jump method~.2*~J~ viscoelastic and T-jump data are available for the same experimental system, a direct quantitative test of the model of ref 1becomes possible for the first time. This is done in section 6 below, with encouraging results. Secondly,the T-jump signal itself tends to decrease rapidly withconcentration(sincethe static propertiesof the system become less and less sensitive to the chain length distribution in the semidilute regime).2 In highly entangled materials, therefore, we cannot get an independent test of the model of ref 1;but if that model is presumed correct, our work allowsquantitative determination of the breaking time in a regime where this was not previously possible. 2. Model
In the entangled regime, each polymer is constrained by a *tube" (consistingof the entanglementaof the polymer with its neighbors).17J8 We consider first the imposition of a small step strain on the system, which is taken to be at equilibrium initially. The effect of this small strain is to distort each tube, which in turn means that the polymer is constrained to have a nonequilibriumconformation. The stress associated with this is relaxed by the curvilinear diffusion (reptation) of the chain out of its initial tube and into a new tube, which is at equilibrium. In the limit of (17) Doi, M.; Edwards, 5. F. The Theory of Polymer Dynamics; Clarendon Preee: Oxford, 1986. (18) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Preee: Ithaca, NY, 1979.
0 1991 American Chemical Society
Linear Viscoelasticity of Living Polymers
Langmuir, Vol. 7, No. 8, 1991 1591
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unbreakable chains ( T b h m) this process is well ~nderstood.~~JS When the time scale for breaking ceases to be very much greater than the typical reptation time ( T ~ ~the ) , dynamics of stress relaxation are complicated by the processes of chain scission and recombination, Upon making the following assumptions, it is possible to proceed with a quantitative analysis of the coupled processes of reptation and reversible scission, as shown in ref 1. Assumption 1. We suppose that when a chain breaks, the two daughter chains become uncorrelated and evolve seperately. This is equivalent to imposing the mean field condition that each chain end, when it recombines with another end at a later time, does so with a randomly chosen chain end and not its previous partner. The conditions under which this assumption holds are discussed in ref 1; it should be valid for flexible chains at moderately high concentration. Assumption 2. We suppose that the onlymodeof stress relaxation is reptation out of the stressed tube (albeit by a chain whose length is allowed to make discrete jumps due to the processes of scission and recombination) and not the disappearance of the tube itself due to the transient nature of the constraints. Thus the fraction of stress p ( t ) remaining after a time t is the fraction of original (stressed) tube through which no chain end has passed up to that time. (Clearlyp(0) = 1and p ( t ) 0 as t -1. The tube will in fact be "evolving" as the constraints making up the tube reptate away; this should not be significant qualitatively but could be a source of error at the quantitative level, especially in systems that are close to the overlap threshold rather than strongly entangled. Assumption 3. As a model of the chemical relaxation processes we take the f~llowing:l*~ (a) Chain scission is a random process occurring at any point on a chain with probability k per unit length per unit time. (b) End to end chain recombination occurs with equal probability between any two chains: thus we write the probability of a given chain fusing with a second chain of length L (per unit time) as k' $(L)were $(L)is the concentration of chains with length L. In steady state the chain length distribution is exponential with mean L (k'/2k)ll2, in accordance with the predictions of equilibrium statistical mechanicsS2 The linear response of the chain length distribution in a system governed by these kinetics can (in conjunction with assumption 1)be calculated analytic all^.^ There is a well-defined characteristic time for scission and recombination reactions which we write as 7 b 4 = l/kL, with L the mean chain length. In particular, it was shown3that the res onse to a small perturbation in the average chain length is an exponential decay with relaxation time -/ 2. This perturbation is the one expected in temperature jump experiments; hence these can provide direct access to 7- for systems undergoing random scission and endto-end fusion. (The situation is more complex when "endinterchange" reactions are present;3 we do not consider this case in the present work.) In linear viscoelasticity one measures properties such as G*(o),which are related through integral transforms to the stress relaxation function G ( t ) . We can define G ( t ) in terms of p ( t ) as
of a "tube" in which the chain resides. This tube limits the lateral movement of the chain but allows curvilinear diffusion in much the same way as a snake moves through tall grass. The stress relaxation mechanism model of ref 1, which combines reptation and chain scission/recombination, can be cast as a one-dimensional stochastic process in the following way. The curvilinear diffusion constant of a chain in its tube D,(L) vaires as L-l. Since stress is associated with the deformationof tube segments, it is easier to imagine the chain stationary and the tube diffusing relative to it. In this case, a given tube segment relaxes when it reaches the end of the chain. The motion is that of a hypothetical "particle" (representing a tube segment) of diffusivity D,(L) on a line of length L (representing the chain) with absorbing boundary conditions at the chain ends. However, the chain breaks and recombines a t random with others on the time scale + b d . These processes are included by allowing the absorbing ends of the line segment to make random jumps with appropriate transition probabilities. By averaging over all starting positions of the particle on the line (equivalent to averaging over all tube segments), and all initial chain lengths, one obtains the survivalprobability of the particle to time t , which is the stress relaxation function p ( t ) .
G ( t ) e Go A t )
a reasonable level of efficiency, the time discretization step lengths were optimized separately for each value of S: It should be noted that 7reqisnot amenable to separate experimental measurement in living polymer systems. Therefore a knowledge of how the shape of ~ * ( odepends ) on { cannot in itaelf provide quantitative predictions of the chemical relaxation time T b d . For experimental comparison it is appropriate instead to introduce a new
- -
-
e
(1)
Where Go is called the plateau modulus (the stress imposed per unit strain at t = 0). The motion of a chain in a concentrated environment of other chains (i.e. when the chains are well overlapped) is well understood in terms of the reptation model." This model describes the spatial constraints imposed by the neighboring chains in terms
3. Numerical Study A suitable numerical algorithm for the calculation of p ( t ) was presented in ref 1. However, in that paper it was not implemented with the precision required to obtain quantitative accuracy at relatively short times, t I w, which is the crucial regime if the departures from monoexponential relaxation are to be studied, nor was the transform into the frequency domain attempted. In the present work we have used a similar algorithm to compute p ( t ) for various values of { defined as f = 7 b r d / T r r p . The program starts with a particle on a line whose left and right segments are chosen independently from an exponential distribution of unit mean. (Thus we choose units so L = 1;note that the average length of the initial chain is 2, which is, correctly, the weight-average.) The program then evolves the particle by a random walk with appropriate diffusivity along the line, coupled with the random end-hop processes corresponding to chain scission and recombination. The frequency of these processes is governed by the particular value of f. The time taken for the particles to relax (reach an absorbing end), rj, is recorded and a continuous approximation to the function N ( t ) is constructed as the following sum over N independent trials
With e(t C
ti)
defined as e(t
ift 1. The subtle departures from exponential behavior in the crossoverregion are most easily seen in the Cole-Cole representation. We have presented a convenient method for parameterizing Cole-Cole plots that are close to semicircular and shown how this allows the key parameter for chemical relaxation processes, TW, to be estimated quantitativelyfrom the linear viscoelastic spectra without recourse to other information. Our
Turner and Cates
analysisis only appropriate when reptation is the dominant mechanism of chain motion on the time scale of n&. If this is not the case (with instead breathing or Rouse modes playing a role), the departures from single exponential take a different form that cannot be calculated at all simply.' However, the high frequency asymptote can be predicted in each case, and is different from that found above for reptation, in which the scaled Cole-Cole plot (for r*) should approach the point (1,O) along a l i e of asymptotic slope -1. In systems where this asymptote is observed, the model we have used to extract an estimate of the breaking time is probably appropriate.
Acknowledgment. We wish to thank F. Kern, D. Collin, and S. J. Candau for permission to use their experimental results shown in Figure 3. We are grateful to Professor S. J. Candau for suggesting the usefulness of the Cole-Cole representation and for numerous helpful comments. M.S.T. also thanks C. Marques and D. Shim for their advice and acknowledges the support of SERC and the Food Research Institute (Norwich) in the form of a CASE studentship. This work was funded in part by under EC Grant SC10288-C.
