Two-dimensional computer studies of crystal stability and fluid viscosity

Abstract

The mathematical analogy between the elastic stress due to particle displacements in Hooke's law solids and the viscous stress due to velocity gradients in incompressible fluids correlates two interesting phenomena. In a two-dimensional crystal the elastic restoring force opposing particle displacements approaches zero with increasing crystal size, leading to a logarithmically diverging rms displacement in the large-system limit. The vanishing of the solid-phase force is mathematically analogous to the lack of viscous damping for a particle moving slowly through a two-dimensional incompressible fluid. These two continuum results are compared with discrete-particle computer simulations of two-dimensional solids and fluids. The divergence predicted by macroscopic elasticity theory agrees quantitatively with computer results for two-dimensional harmonic crystals. These same results can also be correlated with White's experimental study of the viscous resistance to a cylinder (a falling wire) moving slowly through a viscous fluid. The agreement is good. Finally, we carried out a computer study of a two-dimensional fluid confined between two moving walls (plane Couette flow). Despite theoretical predictions that transport coefficients in two-dimensional systems diverge, no viscosity anomalies were observed under the conditions of the computer simulation.