Temperature anisotropy at equilibrium reveals nonlocal entropic contributions to interfacial properties

Density gradient theory for fluids has played a key role in the study of interfacial phenomena for a century. In this work, we revisit its fundamentals by examining the vapor-liquid interface of argon, represented by the cut and shifted Lennard-Jones fluid. The starting point has traditionally been a Helmholtz energy functional using mass densities as arguments. By using rather the internal energy as starting point and including the entropy density as an additional argument, following thereby the phenomenological approach from classical thermodynamics, the extended theory suggests that the configurational part of the temperature has different contributions from the parallel and perpendicular directions at the interface, even at equilibrium. We find a similar anisotropy by examining the configurational temperature in molecular dynamics simulations and obtain a qualitative agreement between theory and simulations. The extended theory shows that the temperature anisotropy originates in nonlocal entropic contributions, which are currently missing from the classical theory. The nonlocal entropic contributions discussed in this work are likely to play a role in the description of both equilibrium and nonequilibrium properties of interfaces. At equilibrium, they influence the temperature- and curvature-dependence of the surface tension. Across the vapor-liquid interface of the Lennard Jones fluid, we find that the maximum in the temperature anisotropy coincides precisely with the maximum in the thermal resistivity relative to the equimolar surface, where the integral of the thermal resistivity gives the Kapitza resistance. This links the temperature anisotropy at equilibrium to the Kapitza resistance of the vapor-liquid interface at nonequilibrium.

Figure 1

(Top) A zoom-in at the interfacial region, showing the spatial contributions to the configurational temperatures perpendicular (${\xi }_{\perp }$) and parallel (${\xi }_{\parallel }$) to the vapor-liquid interface of argon/LJ-fluid at equilibrium ($T=102$ K, $T^{*}=0.85$). The left figure plots Eqs. (19) and (20) using results from extended SGT and the right figure plots the terms in Eq. (23) using results from MD with ${\xi }_{\parallel }$=(${\xi }_x+{\xi }_y$)/2. The vertical dash-dot line shows the position of the equimolar surface and the horizontal solid line, the system temperature. The vertical solid line shows the position of maximum temperature anisotropy, $z_T$. Vapor is located at $z<-2$ and liquid at $z>2$. (Bottom) Simulation snapshot illustrating the different regions (vapor, interface, and liquid bulk) and directions in space.

Figure 2

The spatial contributions to the configurational temperatures perpendicular, ${\xi }_{\perp }$ (blue solid lines), and parallel, ${\xi }_{\parallel }$ (red dashed lines), to the vapor-liquid interface of argon at equilibrium at different temperatures. The figures to the left (a, c, e) show results from extended SGT and the figures to the right (b, d, f) show results from MD at $T$ = 84 K, $T^{*}$ = 0.7 (a, b), $T$ = 108 K, $T^{*}$ = 0.9 (c, d) and $T$ = 120 K, $T^{*}$ = 1 (e, f). The vertical dash-dot lines show the position of the equimolar surface.

Figure 3

The spatial contributions to the kinetic temperature defined in Eq. (22) from the direction perpendicular, ${\xi }_{\text{md(k)},\perp }\left(z\right)$ (blue solid line), and parallel, ${\xi }_{\text{md(k)},\parallel }\left(z\right)$ (red dashed and black dash-dot lines), to the vapor-liquid interface of argon at equilibrium at $T$ = 102 K, $T^{*}=0.85$.

Figure 4

The magnitude of the spatial contributions (a, b) and the configurational temperature (c) in equilibrium MD simulations with $T=102$ K, $T^{*}=0.85$ as a function of the number of particles in the bin at $z_T$ [see Fig. 1 from MD simulations (red crosses)] and as predicted by Eqs. (25, 26, 27) with parameters fitted to minimize the least square distance between the MD-results and the equations. The bin-width and $L_z$ were kept unchanged, and $N_{\text{bin}}$ was changed by modifying $N, L_x$, and $L_y$.

Figure 5

The difference between the spatial contributions to the temperature (blue dash dot line) and the parallel and perpendicular components of the pressure tensor (solid line). The surface tension equals the integral given by the shaded area. The results come from extended SGT-theory or MD simulations at equilibrium at $T=108$ K, $T^{*}=0.9$ as described in Sec. 3. Bulk vapor is at negative values of $z$ and bulk liquid is at positive values of $z$.

Figure 6

The thermal resistivity across the interface obtained from NEMD simulations using the methodology described in Sec. 3, where the temperature of the cold thermostat was $T_c=108\mathrm{K}, T^{*}=0.9$. The shaded area gives the Kapitza resistance.

Figure 7

Position of the maximum of the temperature anisotropy, $z_T$, and the maximum of the thermal resistivity, $z_K$, plotted as functions of the surface temperature for argon/the LJ fluid. The shaded area is the interfacial region, where the solid lines and the surface temperature have been calculated using the approach of Xu et al. in Ref. [57].