Nonequilibrium thermodynamics of an interface

Interfacial thermodynamics has deep ramifications in understanding the boundary conditions of transport theories. We present a formulation of local equilibrium for interfaces that extends the thermodynamics of the “dividing surface,” as introduced by Gibbs, to nonequilibrium settings such as evaporation or condensation. By identifying the precise position of the dividing surface in the interfacial region with a gauge degree of freedom, we exploit gauge-invariance requirements to consistently define the intensive variables for the interface. The model is verified under stringent conditions by employing high-precision nonequilibrium molecular-dynamics simulations of a coexisting vapor-liquid Lennard-Jones fluid. We conclude that the interfacial temperature is determined using the surface tension as a “thermometer,” and it can be significantly different from the temperatures of the adjacent phases. Our findings lay foundations for nonequilibrium interfacial thermodynamics.

Figure 1

(a) The simulations' unit cell is $213\times 21\times 21$ (in Lennard-Jones unit) and is shown here truncated in the vapor region. It periodically repeats in the $y,z$ directions, and it contains 16 192 particles interacting through a spline-interpolated Lennard-Jones potential [39]. The reflecting boundaries in the $x$ direction (red in the vapor; blue in the liquid) and the blue volume are thermostatted via velocity rescaling (see the Appendix). (b) Equilibrium surface tension as a function of temperature; the data points follow the curve $\gamma \left(T\right)={\gamma }_0{(1-T/T_{\text{c}})}^{\nu }$, where $T_{\text{c}}$ is the critical temperature. (c) Equilibrium chemical potential as a function of temperature, fitted with the formula $\mu \left(T\right)={\mu }_0+aTlogT+bT$. Uncertainties in (b) and (c) are below 2% of the range of reported values.

Figure 2

Profiles of mass (a) and internal energy (b) densities in the vicinity of the interface subjected to a heat flux; uncertainties are below 0.5% of the observed ranges. (c) The corresponding excess densities ${\rho }^{\text{s}}$ and $u^{\text{s}}$, calculated from (a) and (b), respectively, as the shaded area between the profiles (colored curves) and their sharpened extrapolations (black lines) for various positions $x^{\text{s}}$ of the dividing surface [see Eq. (1)]; uncertainties are below 1% of the range of values in this panel.

Figure 3

Kinetic temperature profiles in the vicinity of the interface subjected to a heat flux (a) or a mass flux (b). The red and blue lines are linear extrapolation of the bulk profiles and used to define the liquid and vapor interfacial temperatures, as shown by the dashes and compared with $T^{\text{s}}$ (green dash). The shaded area indicates the interfacial region, with the equimolar and equienergetic dividing surfaces shown by the green lines.

Figure 4

(a) Relative deviations of the vapor, liquid, and interface temperatures from the values $T_u^{\text{coex}}$ for which Eq. (3a) is verified. Parts (b) and (c) are, respectively, the relative deviations of $T^{\text{s}}$ from the temperature $T_s^{\text{coex}}$ that verifies Eq. (3b), and from the temperature $T_u^{\text{stru}}$ that verifies Eq. (4a). On the right side of each panel, the gray chart shows the corresponding box-and-whisker diagram in each thermodynamics situation investigated here.