Shear modulus of simulated glass-forming model systems: effects of boundary condition, temperature, and sampling time

The shear modulus G of two glass-forming colloidal model systems in d = 3 and d = 2 dimensions is investigated by means of, respectively, molecular dynamics and Monte Carlo simulations. Comparing ensembles where either the shear strain γ or the conjugated (mean) shear stress τ are imposed, we compute G from the respective stress and strain fluctuations as a function of temperature T while keeping a constant normal pressure P. The choice of the ensemble is seen to be highly relevant for the shear stress fluctuations μ(F)(T) which at constant τ decay monotonously with T following the affine shear elasticity μ(A)(T), i.e., a simple two-point correlation function. At variance, non-monotonous behavior with a maximum at the glass transition temperature T(g) is demonstrated for μF(T) at constant γ. The increase of G below T(g) is reasonably fitted for both models by a continuous cusp singularity, G(T) ∝ (1 - T∕T(g))(1∕2), in qualitative agreement with recent theoretical predictions. It is argued, however, that longer sampling times may lead to a sharper transition.