Shear-transformation-zone theory of linear glassy dynamics

We present a linearized shear-transformation-zone (STZ) theory of glassy dynamics in which the internal STZ transition rates are characterized by a broad distribution of activation barriers. For slowly aging or fully aged systems, the main features of the barrier-height distribution are determined by the effective temperature and other near-equilibrium properties of the configurational degrees of freedom. Our theory accounts for the wide range of relaxation rates observed in both metallic glasses and soft glassy materials such as colloidal suspensions. We find that the frequency-dependent loss modulus is not just a superposition of Maxwell modes. Rather, it exhibits an $\alpha$ peak that rises near the viscous relaxation rate and, for nearly jammed, glassy systems, extends to much higher frequencies in accord with experimental observations. We also use this theory to compute strain recovery following a period of large, persistent deformation and then abrupt unloading. We find that strain recovery is determined in part by the initial barrier-height distribution, but that true structural aging also occurs during this process and determines the system’s response to subsequent perturbations. In particular, we find by comparison with experimental data that the initial deformation produces a highly disordered state with a large population of low activation barriers, and that this state relaxes quickly toward one in which the distribution is dominated by the high barriers predicted by the near-equilibrium analysis. The nonequilibrium dynamics of the barrier-height distribution is the most important of the issues raised and left unresolved in this paper.

Figure 1

Experimental data for Vitreloy 4 and theoretical comparisons for the storage modulus $G^{\prime }(\omega )$ (red squares) and the loss modulus $G^{″}(\omega )$ (blue circles). The data points are taken directly from Fig. 2 of Gauthier et al. [2].

Figure 2

Experimental data and theoretical comparisons for the storage modulus $G^{\prime }(\omega )$ (red squares) and the loss modulus $G^{″}(\omega )$ (blue circles), for three different suspensions of thermosensitive particles, as reported in [9]. The values of the parameters are listed in the text.

Figure 3

Strain recovery as a function of time. The data points (blue circles) are taken directly from Fig. 4b of [8]. The strain is measured from its value after unloading and after an initial, almost instantaneous, elastic relaxation. The red theoretical curve has been computed using the parameter values discussed in the text.

Figure 4

Strain increments induced by small stress steps at waiting times of $t_w=30,600,3000\hspace{0.5em}\mathrm{and}\hspace{0.5em}10\hspace{0.5em}000$ sec, indicated respectively, from top to bottom on the left-hand side, by blue down triangles, green up triangles, red squares, and black circles. The data points are taken directly from Fig. 5(a) of [8]. The parameters used in computing the theoretical curves are discussed in the text.