Universal $L^{-3}$ finite-size effects in the viscoelasticity of amorphous systems

We present a theory of viscoelasticity of amorphous media, which takes into account the effects of confinement along one of three spatial dimensions. The framework is based on the nonaffine extension of lattice dynamics to amorphous systems, or nonaffine response theory. The size effects due to the confinement are taken into account via the nonaffine part of the shear storage modulus $G^{\prime }$. The nonaffine contribution is written as a sum over modes in $k$-space. With a rigorous argument based on the analysis of the $k$-space integral over modes, it is shown that the confinement size $L$ in one spatial dimension, e.g., the $z$ axis, leads to a infrared cutoff for the modes contributing to the nonaffine (softening) correction to the modulus that scales as $L^{-3}$. Corrections for finite sample size $D$ in the two perpendicular dimensions scale as $\sim {(L/D)}^4$, and are negligible for $L\ll D$. For liquids it is predicted that $G^{\prime }\sim L^{-3}$ is in agreement with a previous more approximate analysis, whereas for amorphous materials $G^{\prime }\sim G_{\text{bulk}}^{\prime }+\beta L^{-3}$. For the case of liquids, four different experimental systems are shown to be very well described by the $L^{-3}$ law. The theory can also explain previous simulation data of confined jammed granular packings.

Figure 1

(a) Schematic section in real space of the confined cylindrical sample. (b) Geometry of the different regions over which the $k$-space integral (3) can be taken; see full explanation in text. This is not to scale; in fact $k_D\gg 2\pi /L$. Both parts of this diagram have full rotational symmetry about the $z$ axis.

Figure 2

3D rendering of the geometry of integration in $k$-space for the confined system of Fig. 1.

Figure 3

The same construction as Fig. 1, but allowing for the finite diameter $D$ of the cylinder. Here $D/L=4$. Compared to Fig. 1, panel (b) has been enlarged for clarity.

Figure 4

Experimental data of low-frequency shear modulus $G^{\prime }$ versus confinement length $L$ for different systems: (a) short-chain (nonentangled) polybutylacrylate [31]; (b) an ionic liquid [32]; (c) short-chain (nonentangled) polystyrene melts [33]; (d) nanoconfined water [34]. Circles represent experimental data while the solid line is the law $G^{\prime }\sim L^{-3}$, with a prefactor determined by fitting to the data.