Nonlinear Thermodynamic Relaxation in Living Polymer Systems

We present the complete (nonlinear) solution for the response of a system of l1vlng polymers to an arbitrary thermodynamic perturbation of its equilibrium polymer size distribution. Our results are relevant for the interpretation of T-jump experiments on wormhke mlcelles m the concentrated regime, where very large perturbations to the equilibrium Size distribution can be easily obtained. In a recent work lbmer and Cates [I] analyzed the linear relaxation spectrum of the polymer lengtlt distribution of a system of "living" polymers [2]. These polymers can break and recombine reversibly; well studied examples arise in viscoelastic surfactants phases. According to simple tlteory, the polymers or micelles have an equilibrium eTponential length distribution (in suitable unfits) Co(L) = (exP I-L/Lol (1) characterized by an average length Lo which depend on rite thermodynamic variables of the system such the temperature, rite pressure and the volume fraction of monomers, # In a simple model [2] Lo ~ 4"~exp (-E/(2kT)) where E is the energy to create two end-caps. Under a sudden modification of one of the thermodynamic variables (most commonly, of the temperature) the distribution co(L) relaxes to a new equilibrium exponential distribution c(L), characterized by a new average length L The characteristic ttrne and the functional form of the relaxation of the size distribution provides information about the microscopic factors which control the kinetics of the system. The simplest description of living polymer kinetics assumes that a chain can only change mass either by breaking in two new shorter chains or by recombining with anotlter to form a new larger chain. This scission-recombination scheme provides a integro-dit§erential equation for the time 493 JOURNAL DE PHYSIQUE II N°5 evolution of c(t, L) d(t, L) = kLc(t, L) + 2k /~ c(t, L')dL'+ + k~ /2 /~ c(t, ')~t, L L')dL~ k'c(t, L) /~ c(t, L')dL' ~~~