Dynamics of amorphous solids and viscous liquids

This thesis consists of fifteen publications (P1-P15) published between 1987 and 1996 and a summary. In this abstract an overview of the main results is given by following the summary's three Chapters. The first Chapter with the title "AC Conduction in Disordered Solids" reviews and comments P1-P7. In P1 from 1988 a phenomenological model for AC conduction in disordered solids is proposed. It is shown that the model explains the characteristic AC properties of disordered solids, summarized in 8 points. P3-P7 is an attempt to explain the success of the model of P1, which is based on a number of ad hoc hypotheses. In this line of reasoning, P2 from 1991 is a digression; in P2 - mainly for pedagogical reasons and with few original contributions - a number of common misunderstandings and misconceptions are cleared up. P3 and P4 from 1993 discuss a acroscopic model for AC conduction; P4 details the brief paper P3, that is included in the thesis mainly because its figures are more pedagogical than those of P4. By means of analytical approximations it is shown that at sufficiently low temperatures - corresponding to extreme disorder - all disordered solids with thermally activated conduction exhibit the same AC conductivity independent of the energy barrier probability distribution; the AC conductivity is "universal". This result is confirmed by computer simulations in two and three dimensions. The universal AC conductivity is very close to that of the phenomenological model studied in P1. P5 from 1994 derive universality for the microscopic so-called hopping model. The universality prediction is confirmed by computer simulations in two dimensions, but the quantitative agreement is not quite as good as for the macroscopic model. In P6 from 1995, the mean-square displacement as function of time for a charge carrier "hopping" in an extremely disordered solid is calculated from the expression for the universal AC conductivity derived in P5. P7 published in 1996 proposes a new expression for the universal AC conductivity in hopping models, derived by assuming that electrical conduction in extremely disordered solids is essentially a one-dimensional process. It is shown from computer simulations in two and three dimensions that the new expression is more realistic than that of P5. Still, the new expression is relatively close to that of the phenomenological model proposed in P1. The second Chapter with the title "Viscous Liquids and the Glass Transition" reviews and comments P8-P10. In P8 from 1987 a simple model for the glass transition is proposed in which there is only one relevant degree of freedom, the potential energy of a region in the liquid. The model was originally constructed to explain the non-Arrhenius temperature-dependence of the average relaxation time in viscous liquids, an approach that is also followed in Chapter 2. However, in P8 itself the focus was on the prediction that there are two different types of glass transitions. In P9 from 1995, B„ssler's random walk model for viscous liquids is studied. It is argued that at low temperatures this model is well described by the simple model of P8. Thus, a clear physical picture of the low-temperature behavior of B„ssler's random walk model is established. In P10 published in 1996, an alternative model for explaining the non-Arrhenius temperature-dependence of the average relaxation time in viscous liquids is proposed. In the new model, the short-time (or high-frequency) elastic properties of the liquid determine the activation energy for the average relaxation time. It is shown that the new model agrees well with experiment on a number of organic molecular liquids. The third Chapter with the title ``Extensions of Linear Response Theory'' reviews and comments P11-P15. P11 from 1988 deals with electrical 1/f noise. This, in a sense, is a linear phenomenon, but as shown in P11 the magnitude of the noise is determined by {\it fourth order} cumulant averages of the equilibrium current fluctuations (while the ordinary linear response - the AC conductivity - is determined by the time autocorrelation function, a {\it second order} cumulant average). P12 from 1989 discusses a maximum entropy ``ansatz'' for nonlinear response theory. This ``ansatz'' makes it possible to predict the nonlinear response in a static external field from a complete knowledge of the equilibrium fluctuations of the quantity of interest. P13 and P14 both deal with nonlinear viscoelasticity. P13 from 1990 suggests a simple formula, a so-called ``constitutive relation'', for calculating the stress for an arbitrary shear flow. In P14 from 1993 it is shown that, if the equilibrium fluctuations of the stress are described by a Langevin equation, there is only one way to extend this equation to deal with stress fluctuations in a nonlinear flow. The extension is shown to be consistent with linear response theory, ensuring consistency. P15 from 1994 concerns the calculation of time autocorrelation functions (that via the fluctuation-dissipation theorem determine the linear response). An approximation is suggested, in which the calculation is reduced to calculating a ``double'' canonical average as well as the mean-square displacement as function of time. The 15 publications are related to each other in the following way. P1-P7 is a continuously progressing attempt to explain the AC properties of extremely disordered solids (with P2 as a digression). P8 discusses a simple model for viscous liquids and the glass transition. In P9 it is shown that this model at low temperatures results from B„ssler's random walk model, a model that is very similar to the hopping model studied in P1 and P5-P7. P10 discusses an alternative phenomenological model for viscous liquids. In many respects, the P10 model is complementary to the P8-P9 models. Thus, one may conjecture that these two models more or less span the ``universe'' of possible phenomenological models. P11 concerns a frequency-dependent electrical property different from the conductivity, namely the electrical excess noise. P12 is related to P11, because the main result of P11 is a special case of the general formalism developed in P12. P13 and P14 both concern the same subject, nonlinear viscoelasticity. This subject is relevant for any viscous liquid close to the glass transition, where one expects the liquid to become nonlinear at shear rates larger than the inverse Maxwell relaxation time. In the linear limit, the constitutive relation suggested in P13 predicts a frequency-dependent viscosity that varies as one over the universal AC conductivity studied in Chapter 1. Thus, it is proposed that the atoms in a viscous liquid have the same mean-square displacement as function of time as that of charge carriers in an extremely disordered solid. P14, though only dealing with nonlinear viscoelasticity, is closely related to P12, because P14 gives an alternative recipe for estimating the nonlinear response from knowledge of the equilibrium fluctuations. In P15 the mean-square displacement as function of time is assumed known; this quantity is calculated in P6, if the liquid on the relevant time-scale may be regarded as a disordered solid, as suggested in P13. P15 is motivated by P8-P10: In these papers the focus is on the temperature-dependence of the mean relaxation time in a viscous liquid, and P15 starts from asking why quite different physical quantities in viscous liquids have roughly the same mean relaxation time.