Amorphous solids: their structure, lattice dynamics and elasticity

Abstract Amorphous solids fall into any reasonable definition of a solid. This means that it should be possible to describe amorphous solids in terms of an expansion in deviations from a well defined rigid microscopic reference frame — as in the Cauchy-Born lattice-dynamics of periodic crystals. The continuum limit of this microscopic expansion is a field theory — continuum elasticity. It follows that different types of amorphous solids can differ from each other and from crystalline solids only because of differences in the nature of their microscopic reference states. A theory of amorphous solids which implements this general point of view must satisfy two very general restrictions: (1) translation-rotation invariance of the expansion and (2) the reference state must be a stable equilibrium state. We construct such a general theory and also describe some results of its application to specific types of amorphous solids. The monograph consists of three parts. Part I. The Cauchy-Born Theory of Solids , describes a general and explicitly translation-rotation invariant formulation of the Cauchy-Born expansion. This is done by using the fact that the translation-rotation invariant energy of a many particle system can be regarded as a function of the interparticle distances. The resulting formalism leads to local expressions for the continuum limit which can be used to derive microscopic expressions for the local elastic constants and for the stresses from the coefficients in the microscopic expansion around a specific reference state. One finds that the initial stresses in the microscopic reference state lead to special stress induced terms in the harmonic expansion whose continuum limit is the second order strain. Their effect on the bulk stability of the reference states of tenuous solids is closely analogous to the role of stresses in continuum stability theory. Part II. The Rigidity of Floppy Bonded Networks , studies the stability of tenuous reference states. The theory developed in this section extends the standard considerations of the effect of stresses on the elastic stability of thin rods and shells to the complex internal structures which describe the bulk of solids. There is a geometric aspect. The interaction scheme of a physical model, the bond structure, defines a graph — the bonded network. When this network can be deformed continuously in d -dimensions even when all its bonded distances are fixed is geometrically floppy. A model which is described by such a network has a manifold of free degrees of freedom which have no rigidity for unstressed reference states. Mostly the free modes describe collective deformations of the reference state. We show that their number can be very large and that they often constitute a significant fraction of all eigenmodes. Like the bending of a thin rod these free modes are sensitive to stresses in the reference state around which one is expanding. When there are stresses they can become unstable, leading to structural buckling instabilities, or stable — depending on the sign of the relevant stresses. The theory of floppy networks thus allows us to study structural buckling, the bulk analog of the Euler buckling of rods and shells, and the stabilization of bulk shears by stretching which is the origin of the shear rigidity of most soft solids. Part III. The Role of Stresses in Amorphous Solids , applies these results. The emphasis is on the stability of the reference state. We require stability against structural buckling and on the role of stresses in stabilizing the soft modes of tenuous bonding structures. The shear rigidity of rubbers and wet gel-likes is that of floppy bonded networks which are stretched. Like the shear rigidity of stretched membranes it is that of a network of stretched springs. We show that this is possible only because such solids are not rigid down to the atomic level. We discuss the stability of stressed granular packings emphasizing the fact that the packing has to be stable against structural buckling. This gives considerable insight into the internal mechanics of granular packings because it not only considers the equilibrium conditions on the individual grains but also the collective linear stability of a stressed packing. We then describe a new model of quenched glasses which emphasizes the role of the internal stresses in the glass. The essence of this model is the distinction between the molecular configuration in the quenched liquid, the “snapshot state” and the stable equilibrium reference state of the glass which emerges from it even for the most rapid ideal quench. We argue that structural buckling dominates the restructuring in the ideal quench and show that this predicts strong correlations between the local structure of the reference state and the internal stresses. Some of the most striking properties of glasses appear naturally in this microscopic model.