Learning the bulk and interfacial physics of liquid-liquid phase separation with neural density functionals

TL;DR

Simulation-based supervised machine learning and classical density functional theory are used to investigate bulk and interfacial phenomena associated with phase coexistence in binary mixtures and determine the contact angles at fluid-fluid interfaces along the line of triple-phase coexistence.

Abstract

We use simulation-based supervised machine learning and classical density functional theory to investigate bulk and interfacial phenomena associated with phase coexistence in binary mixtures. For a prototypical symmetrical Lennard-Jones mixture our trained neural density functional yields accurate liquid-liquid and liquid-vapour binodals together with predictions for the variation of the associated interfacial tensions across the entire fluid phase diagram. From the latter we determine the contact angles at fluid-fluid interfaces along the line of triple-phase coexistence and confirm there can be no wetting transition in this symmetrical mixture.

Figures (4)

• Figure 1: To exemplify the generation of the training dataset, a specific realization of a randomized inhomogeneous environment is shown. This is characterized by the species-resolved external potentials $V_{{\rm ext}}^{(1)}(z)/\epsilon$ and $V_{{\rm ext}}^{(2)}(z)/\epsilon$ (top panel), chemical potentials $\mu_1/\epsilon =\mu_2/\epsilon=0.237599$ and temperature $k_BT/\epsilon=1.118673$. GCMC simulation results for the partial density profiles $\rho_1(z)$ and $\rho_2(z)$ are used in the local training of the species-resolved one-body direct correlation functional $c_1^{(i)}(z;[\rho_1,\rho_2],T)$, $i = 1, 2$; see text for details of the neural network. Whilst the training data consists of qualitatively similar profiles, the realization shown was not included in the training set thereby enabling a direct test of neural network predictions. The self-consistent solution of the Euler-Lagrange equations \ref{['EQselfConsistent']} using the trained neural density functional yields results for the partial density profiles (labelled "DFT") that are identical on the scale of the plot to data generated by direct GCMC simulations ("sim") for the specific external potentials; see bottom panel.
• Figure 2: Density profiles at the different types of fluid-fluid interfaces predicted by the neural DFT: $\beta\gamma$ liquid-liquid (top panel) and $\alpha\beta$ gas-demixed liquid (second panel) at the scaled temperature $k_BT/\epsilon=0.93$. Shown are the scaled partial density profiles $\rho_1(z)a^3$ and $\rho_2(z)a^3$, as well as the scaled total density profile ${\cal N}(z)a^3=[\rho_1(z) + \rho_2(z)]a^3$. The $\alpha\gamma$ interface (not shown) is identical to the $\alpha\beta$ interface upon exchanging species 1 and 2. At the increased temperature $k_BT/\epsilon=1.0$ the system displays $\alpha$--$\beta\gamma$ coexistence (bottom panel) between $\alpha$ gas and mixed $\beta\gamma$ liquid with identical partial density profiles, $\rho_1(z) = \rho_2(z)$.
• Figure 3: Top panel: neural density functional results for the bulk $\beta\gamma$ liquid-liquid binodal at fixed total bulk density $(\rho_1+\rho_2)a^3=0.663$ shown as a function of bulk composition $X_1=\rho_1/(\rho_1+\rho_2)$ and scaled temperature $k_BT/\epsilon$. The results (circles) are obtained from the plateau values of equilibrium interfacial density profiles (see the top panel in Fig. \ref{['FIGinterfaceStructure']}). The fit according to Eq. \ref{['EQfitFunctionShortPaper']} is obtained using only data below and including cutoff temperature $k_BT/\epsilon=1.06$ (brown circles), with the resulting numerical values for $k_BT_\lambda/\epsilon$ and exponent $\beta$ given in the legend. For comparison, the black cross denotes the $\lambda$ point obtained from simulation by wilding1997. Middle panel: variation of the scaled chemical potential $\mu / \varepsilon = \mu_1 / \varepsilon = \mu_2 / \varepsilon$ with respect to temperature along the specified path of constant total density. Bottom panel: neural density functional results for the scaled tension $\sigma_{\beta\gamma} a^2 / \epsilon$ of the $\beta\gamma$ interface obtained from functional line integration \ref{['EQfunctionalIntegral']}.
• Figure 4: Top panel: bulk fluid phase diagram of the symmetrical binary Lennard-Jones system as a function of scaled total density $(\rho_1+\rho_2)a^3$ and scaled temperature $k_BT/\epsilon$ together with simulation data of wilding1997. Binodals and the $\lambda$ line are plotted. The vertical dotted red line indicates the path of constant total density for which the liquid-liquid binodal is shown in Fig. \ref{['FIGliqliqBinodal']}. Middle panel: scaled interfacial tensions between gas and demixed liquid, $\sigma_{\alpha\beta}a^2/\epsilon = \sigma_{\alpha\gamma}a^2/\epsilon$ and between liquid and liquid, $\sigma_{\beta\gamma}a^2/\epsilon$, for $T< T_{\rm CEP}$ and between the gas and the mixed liquid, $\sigma_{\alpha-\beta\gamma}a^2/\epsilon$ (green dots), for $T>T_{\rm CEP}$. The dashed vertical line indicates $T_{\rm CEP}$. Bottom panel: contact angles $\theta_\alpha$ and $\theta_\beta=\theta_\gamma$ on the triple phase line, obtained from Eq. \ref{['EQcosThetaBeta']}. The sketch shows the lens of (demixed) liquid $\beta$ together with the contact angles and tensions.