An approximation scheme for model disordered solids is proposed that leads to the fully analytical evaluation of the elastic constants under explicit account of the inhomogeneity (nonaffinity) of the atomic displacements. The theory is in quantitative agreement with simulations for central-force systems and predicts the vanishing of the shear modulus at the isostatic point with the linear law $\mu ~(z-2d)$, where $z$ is the coordination number. The vanishing of rigidity at the isostatic point is shown to be a consequence of the canceling out of positive affine and negative nonaffine terms.

The vibrational spectra of solids, both ordered and amorphous, in the low-energy regime, control the thermal and transport properties of materials, from heat capacity to heat conduction, electron-phonon couplings, conventional superconductivity, etc. The old Debye model of vibrational spectra at low energy gives the vibrational density of states (VDOS) as proportional to the frequency squared, but in many materials the spectrum departs from this law which results in a peak upon normalizing the VDOS by frequency squared, which is known as the “boson peak.” A description of the VDOS of solids (both crystals and glasses) is presented starting from first principles. Without using any assumptions whatsoever of disorder in the material, it is shown that the boson peak in the VDOS of both ordered crystals and glasses arises naturally from the competition between elastic mode propagation and diffusive damping. The theory explains the recent experimental observations of boson peak in perfectly ordered crystals, which cannot be explained based on previous theoretical frameworks. The theory also explains, for the first time, how the vibrational spectrum changes with the atomic density of the solid, and explains recent experimental observations of this effect.

Melting is well understood in terms of the Lindemann criterion, which essentially states that crystalline materials melt when the thermal vibrations of their atoms become so vigorous that they shake themselves free of the binding forces. This picture does not necessarily have to hold for glasses, where the nature of the solid–liquid cross-over is highly debated. The Lindemann criterion implies that the thermal expansion coefficients of crystals are inversely proportional to their melting temperatures. Here we find that, in contrast, the thermal expansion coefficient of glasses decreases more strongly with increasing glass temperature, which marks the liquid–solid cross-over in this material class. However, this proportionality returns when the thermal expansion coefficient is scaled by the fragility, a measure of particle cooperativity. Therefore, for a glass to become liquid, it is not sufficient to simply overcome the interparticle binding energies. Instead, more energy must be invested to break up the typical cooperative particle network that is common to glassy materials. The thermal expansion coefficient of the liquid phase reveals similar anomalous behaviour and is universally enhanced by a constant factor of approximately 3. These universalities allow the estimation of glass temperatures from thermal expansion and vice versa.

Using an approximation scheme within the convective diffusion (two-body Smoluchowski) equation framework, we unveil the shear-driven aggregation mechanism at the origin of structure formation in sheared colloidal systems. The theory, verified against numerics and experiments, explains the induction time followed by explosive (irreversible) rise of viscosity observed in charge-stabilized colloidal and protein systems under steady shear. The Arrhenius-type equation with shear derived here, extending Kramers’ theory in the presence of shear, clearly demonstrates the important role of shear drive in activated-rate processes as they are encountered in soft condensed matter.

The mechanical response of solids depends on temperature, because the way atoms and molecules respond collectively to deformation is affected at various levels by thermal motion. This is a fundamental problem of solid state science and plays a crucial role in materials science. In glasses, the vanishing of shear rigidity upon increasing temperature is the reverse process of the glass transition. It remains poorly understood due to the disorder leading to nontrivial (nonaffine) components in the atomic displacements. Our theory explains the basic mechanism of the melting transition of amorphous (disordered) solids in terms of the lattice energy lost to this nonaffine motion, compared to which thermal vibrations turn out to play only a negligible role. The theory is in good agreement with classic data on melting of amorphous polymers (for which no alternative theory can be found in the literature) and offers new opportunities in materials science.