Slowing down of relaxation in a complex system by constraint dynamics

Abstract

In this paper we view a relaxing complex system such as entangled polymer melt to consist of three parts: (1) an individual primary species PS of interest; (2) a heat bath (HB) whose interaction with the PS provides the primary mechanism of relaxation; (3) other relaxing species whose interactions with the PS, the PS-C coupling, are for us the principal characteristic of complexity. The PS-C coupling is represented by time dependent constraints whose effect begins only after the primary relaxation process due to the PS-HB interaction is already underway. The overall process is described both physically and theoretically. The latter is described classically by means of time dependent Dirac constraint theory applied to a Liouville operator formalism. The physical and theoretical discussion leads to a time dependent relaxation rate W(t). The specific form of W(t) is adduced based on the requirement of time-temperature equivalence or thermorheological simplicity. The result is a time independent relaxation rate W0 for times short compared to the onset of the effect of the time dependent constraints at tc=ω−1c, and a time dependent rate W0(ωct)−n for times long compared to tc. The case W0tc<1 is of most interest because the relaxation process then reveals the effect of complexity empirically. The empirically observed result is then just the Kohlrausch form. Furthermore, a second relation between τ0≡W−10 and the effective relaxation time in the Kohlrausch form follows immediately. It is also noted that the present framework can be applied more generally in relaxation phenomena if thermorheological simplicity is viewed as a special case of correlation or constraint scaling in which dW/W=−ndt/t for t>tc.