The emergence of the solid state in glass-forming materials
upon
cooling is accompanied by changes in both thermodynamic and viscoelastic
properties and by a precipitous drop in fluidity. Here, we investigate
changes in basic elastic properties upon cooling in a family of simulated
polymer fluids, as characterized by a number of stiffness measures,
such as the “glassy plateau shear modulus” G
p, the “non-ergodicity parameter” f
s,q*, the bulk modulus B, the Poisson ratio ν, and the “Debye–Waller
parameter” ⟨u
2⟩,
where G
p, f
s,q*, and ⟨u
2⟩
correspond to the shear stress relaxation function G(t), the self-intermediate scattering function F
s(q*, t),
and the mean square displacement on a ps timescale, respectively.
The time dependence of G(t) at elevated
temperatures (T) resembles the power-law decay predicted
by the Rouse model, but stress relaxation transitions to a stretched
exponential form in the low-T liquid regime dominated
by glassy segmental dynamics. In this “glassy dynamics”
regime, the relaxation times from G(t) and F
s(q*, t) closely track each other for all polymer models investigated,
thereby justifying the identification of the α-relaxation time
τα from F
s(q*, t) with the structural relaxation time
τ
G
from G(t). We show that τα can be expressed
quantitatively both in terms of measures of the material “stiffness”, G
p, and ⟨u
2⟩, and the extent L of cooperative particle
exchange motion in the form of strings, establishing a direct relation
between the growth of emergent elasticity and collective motion. Moreover,
the macroscopic stiffness parameters, G
p, B, and f
s,q*, can all be expressed quantitatively in terms of the molecular
scale stiffness parameter, k
B
T/⟨u
2⟩, with k
B being Boltzmann’s constant, and we discuss the
thermodynamic scaling of these properties. We also find that G
p is related to the cohesive energy density
ΠCED, pointing to the critical importance of attractive
interactions in the elasticity and dynamics of glass-forming liquids.
Finally, we discuss fluctuations in the local stiffness parameter
as a quantitative measure of elastic heterogeneity and their significance
for understanding both the linear and nonlinear elastic properties
of glassy materials.