Mechanical and thermodynamic routes to the liquid-liquid interfacial tension and mixing free energy by molecular dynamics

TL;DR

This work extends Bakker's equation to liquid–liquid interfaces by computing interfacial tension from stress anisotropy in a quasi-1D MD setup and compares it with thermodynamic TI routes using extended-DS and phantom-wall schemes. For immiscible LJ liquids, mechanical interfacial tension $\gamma_{\alpha\beta}$ agrees with the TI isolation work, while for miscible cases a substantial extra cost from mixing energy appears, revealed by the PW path where the osmotic pressure and separation energetics are disentangled. The analysis introduces detachment and separation components, showing that in the ideal mixture limit the separation work corresponds to a purely entropic mixing contribution, and in non-ideal mixtures the PW method captures the full free energy of mixing (enthalpic plus entropic). Overall, the results clarify how interfacial and mixing free energies partition in LL systems and demonstrate the utility of phantom walls to resolve mixing contributions from interfacial energetics.

Abstract

In this study, we carried out equilibrium molecular dynamics (EMD) simulations of the liquid-liquid interface between two different Lennard-Jones components with varying miscibility, where we examined the relation between the interfacial tension and isolation free energy using both a mechanical and thermodynamic approach. Using the mechanical approach, we obtained a stress distribution around a quasi-one-dimensional (1D) EMD systems with a flat LL interface. From the stress distribution, we calculated the liquid-liquid interfacial tension based on Bakker's equation, which uses the stress anisotropy around the interface, and measures how it varies with miscibility. The second approach uses thermodynamic integration by enforcing quasi-static isolation of the two liquids to calculate the free energy. This uses the same EMD systems as the mechanical approach, with both extended dry-surface and phantom-wall (PW) schemes applied. When the two components were immiscible, the interfacial tension and isolation free energy were in good agreement, provided all kinetic and interaction contributions were included in the stress. When the components were miscible, the values were significantly different. From the result of PW for the case of completely mixed liquids, the difference was attributed to the additional free energy required to separate the binary mixture into single components against the osmotic pressure prior to the complete detachment of the two components, i.e., the free energy of mixing.

Figures (5)

• Figure 1: (a) Setup of the equilibrium simulation system with a liquid-liquid interface controlled by a miscibility parameter $\eta$. A pressure control piston and a semipermeable phantom wall (PW) were located on each end of the system in the $x$-direction (left and right). (b) Side snapshots of the equilibrium systems for the free-energy calculation by the thermodynamic integration using the (i) extended dry surface (DS), and (ii) extended PW methods. For the extended DS method (i), the miscibility parameter $\eta$ was varied while keeping the PWs at $x^\mathrm{pw}_{1}$ and $-x^\mathrm{pw}_{1}$ far from the liquid; thus, the PWs as well as the PCs are shown only on the top panel of (i). For the PW method (ii), the PW positions $x^\mathrm{pw}$ and $-x^\mathrm{pw}$ were varied while keeping $\eta$ unchanged.
• Figure 2: (a) Distributions of the diagonal stress components $\tau_{yy}$ (black) and $\tau_{xx}$ (green) calculated by the VA, and the densities $\rho_{\alpha}$ (blue) and $\rho_{\beta}$ (red) of $\alpha$ and $\beta$ components for the systems at $\eta =0.01$, 0.4, 0.8 and 1. Enlarged snapshots of the systems around the interface are also shown. (b) Interfacial tension $\gamma_{\alpha \beta}$ calculated from the stress distribution by Eq. (\ref{["eq:ex_Bakker's eq"]}) as the mechanical route (MR). The value of $2\gamma_\mathrm{LV}$ obtained in an independent system is also displayed.
• Figure 3: Comparison between the relative interfacial tension $-(\gamma_{\alpha \beta} - 2\gamma_\mathrm{LV})$ obtained with the mechanical route and work for isolation $W_\mathrm{iso}$ for various $\eta$. Error bars for $W_\mathrm{iso}^\mathrm{DS}$ are not shown for better visualization here (see Fig. \ref{['fig:result_DS']}).
• Figure 4: (a) Force on the right phantom wall (PW) per unit area upon the calculation of the work for isolation $W_\mathrm{iso}^\mathrm{PW}(\eta)$ by the extended-PW method in the completely miscible case ($\eta=1$). The inset corresponds to Fig. \ref{['fig:system']} (a-ii). (b) Schematic of the force balance among the force on the left and right PWs, the pressure of the two single component liquids, and that of the mixed liquid in the center between the two PWs.
• Figure 5: (a) Work for isolation $W_\mathrm{iso}^\mathrm{DS}$ obtained by the DS method in Eq. (\ref{['app_eq:w_iso_DS']}), and (b) average potential energy between different components $\alpha$ and $\beta$ for various miscibility parameter $\eta$. The inset corresponds to Fig. \ref{['fig:system']} (a-i). The error bars for (b) were smaller than the size of the symbol.