The spatiotemporal correlations of the local stress tensor in supercooled liquids are studied both theoretically and by molecular dynamics simulations of a two-dimensional (2D) polydisperse Lennard-Jones system. Asymptotically exact theoretical equations defining the dynamical structure factor and all components of the stress correlation tensor for low wave-vector q are presented in terms of the generalized (q-dependent) shear and longitudinal relaxation moduli, G(q, t) and K(q, t). We developed a rigorous approach (valid for low q) to calculate K(q, t) in terms of certain bulk correlation functions (for q = 0), the static structure factor S(q), and thermal conductivity κ. The proposed approach takes into account both the thermostatting effect and the effect of polydispersity. The theoretical results for the (q, t)-dependent stress correlation functions are compared with our simulation data, and an excellent agreement is found for (with being the mean particle diameter) both above and below the glass transition without any fitting parameters. Our data are consistent with recently predicted (both theoretically and by simulations) long-range correlations of the shear stress quenched in heterogeneous glassy structures.
KT(t), KA(t) are universal functions by their definition, implying a perfect temperature adjustment to keep constant either T or the entropy S. Therefore, KT(t) or KA(t) cannot be directly obtained by, say, simulations with imperfect thermostats (allowing for some undefined T-variations). Still, they can be calculated even in this case based on the non-universal correlation functions, but it requires application of certain transformations to the correlation data, as described in Ref. 10. By contrast, this problem is absent for G(q, t) since shear stress fluctuations are insensitive to small T-variations.
An alternative assumption is that the periodic boundary conditions (PBCs) are applied.
Note that condition (15) does not imply any jump of at t = 0+ if the center-of-mass velocity of the system is kept equal to 0.
We do not use subscript “L” in K(q, t) since a q-dependent compression can be only longitudinal; it can be isotropic only for q = 0.
Here and below, the brackets 〈⋯〉 mean ensemble average supplemented with the averaging over t′, and for any X = T(t), p(t), etc.
Simultaneous correlations corresponding to the time-shift t = 0 are not altered by the thermostat due to its canonical nature.
Note that here we deal with nondimensional specific heat per particle, cv, which is equal to the total heat capacity of the system divided by NkB.
The glass-transition temperature Tg ≈ 0.26 was determined in prior Monte Carlo (MC) and MD studies using a continuous cooling protocol with a (standard choice for the) finite cooling rate.10 Therefore, Tg corresponds to the temperature where the α-relaxation time becomes comparable to the cooling time (in the MC simulations with local moves or in MD simulations), and the system falls out of equilibrium. However, thanks to the particle-swap MC technique,58 we succeed in equilibrating our 2D pLJ system down to T = 0.16.10 Starting from these equilibrated configurations, we performed MD simulations to explore the equilibrium dynamics of the supercooled liquid, also below Tg ≈ 0.26 (cf. Ref. 10 for a detailed discussion). Here, we continue to report Tg as a relevant temperature for our model because for T ≲ Tg, dynamic correlation functions show pronounced plateaux for times beyond the initial short-time relaxation. Upon cooling, these plateaux increase in length and extend over the entire sampling time of the MD simulation at low T.
The importance of interdiffusion processes for glass-forming systems with size polydispersity was discussed in Ref. 61 and for binary mixtures in the framework of mode-coupling theory in Ref. 62. However, the general form of the magnitude of the interdiffusion term [cf. prefactor in Eq. (52)] was not recognized there.
Note that the agreement at long times slightly deteriorates at low temperatures, T ≲ Tg, which is a natural effect of the poorer statistics in the regime of long structural relaxation time τα.
Note that the region t ≲ 2τL provides a negligible contribution to the integral in Eq. (97) since Δtmax ≫ τL.
Note that while Nq = 2 for all q’s we consider (with nq2 = 1, 8, 32, 64), it typically gets higher for larger q.