Kinetic Theory of Soft Matter. The Penetrable-Square-Well Model

Abstract

The penetrable-square-well (PSW) pair interaction potential is defined as $ϕ(r)=ε_r$ if the two interacting particles are overlapped ($r<σ$), $ϕ(r)=-ε_a$ inside a corona ($σ λ$). Thus, the potential reduces to the conventional square-well (SW) one in the limit $ε_r\to\infty$ and to the penetrable-sphere (PS) potential if $ε_a\to0$ or $λ\toσ$. This paper aims at studying the temperature dependence of the Navier--Stokes transport coefficients of a dilute gas of particles interacting via the PSW model. By exploiting the fact that the PSW scattering process is analogous to that of a light ray passing through two concentric spherical media with different refractive indices, the scattering angle is analytically derived as a function of the impact parameter and the relative velocity of the colliding particles; depending on the values of those two quantities, collisions can be soft, hard, or grazing. Next, by standard application of known general results from the Chapman--Enskog method, the Navier--Stokes transport coefficients in the first-order approximation are numerically evaluated. It is found that the PSW coefficients are practically indistinguishable from the SW ones for temperatures low enough ($k_BT\lesssim 0.2 ε_r$), there exists a transition regime ($0.2 ε_r\lesssim k_BT\lesssim 10ε_r$) where the transport coefficients interpolate between the SW and the PS ones, and finally the PSW coefficients are comparable to the PS ones for high enough temperatures ($k_BT\gtrsim 10ε_r$). The results are applied to the temperature profiles of the planar Fourier flow.