Microscopic Phase-Field Modeling

Abstract

Phase-field methods offer a versatile computational framework for simulating large-scale morphological evolution. However, the applicability and predictability of phase-field models are inherently limited by their ad hoc nature, and there is currently no version of this approach that enables truly first-principles predictive modeling of large-scale non-equilibrium processes. Here, we present a bottom-up framework that provides a route to the construction of mesoscopic phase-field models entirely based on atomistic information. Leveraging molecular coarse-graining, we describe the formulation of an order parameter-based free energy functional appropriate for a phase-field description via the enhanced sampling of rare events. We demonstrate our approach on ice nucleation dynamics, achieving a spatiotemporal scale-up of nearly $10^8$ times compared to the microscopic model. Our framework offers a unique approach for incorporating atomistic details into mesoscopic models and systematically bridges the gap between microscopic particle-based simulations and field-theoretic models.

Figures (5)

• Figure 1: Schematic illustration of the hierarchical coarse-graining (CG) strategy for constructing a mesoscopic representation of the water-ice interface directly from atomistic simulations. To derive microscopic phase-field dynamics from fully atomistic data, the approach involves two key steps: (1) molecular coarse-graining and (2) CG-level sampling.
• Figure 2: Systematic construction of a molecular OP for ice growth at the ice-water interface ($31.50\,\mathrm{\AA}\,\times\,29.84\,\mathrm{\AA}\times\,68.58\,\mathrm{\AA}$). (a) The local bond OP $\tilde{q}_6$ can distinguish ice (red) from water (blue), demonstrating that $\phi$ can be systematically derived from $\tilde{q}_6$. (b) Time evolution of $\langle \tilde{q}_6\rangle$ during FG and CG simulations. The CG simulation (green) enables enhanced sampling of ice-like configurations, increasing $\tilde{q}_6$ from around 0.4 (FG, blue) to 0.8. After crystallization, an additional CG run (red) was utilized to estimate the absolute free energy density of the solid phase. (c) $\tilde{q}_6$ distribution in the initial configuration ($\tau_\mathrm{CG}=0$) and the final configuration ($\tau_\mathrm{CG}=2\times10^6$) from the molecular CG simulation.
• Figure 3: Microscopic bulk free energy landscape underlying ice growth at the interface, expressed in terms of the OP $\phi$. The CG PMF (solid red line) is obtained from histogram analysis using MBAR and represents a relative free energy, shifted such that the solid-phase value is zero. The dashed line shows a fit to the CG PMF using the relative free energy $(1-p(\phi))(f_l-f_s) + q(\phi)f_w$, from which the interpolating polynomials $p(\phi)$ and $q(\phi)$ are determined, cf. Fig. \ref{['fig:fig4']}.
• Figure 4: Microscopically derived interpolating polynomials (solid lines): (a) $q(\phi)$ for the double-well free energy and (b) $p(\phi)$ for the chemical free energy, which exhibit slight deviations from the conventionally used phenomenological forms (dots). These microscopic polynomials were obtained by fitting the bulk free energy $F_I^\mathrm{bulk}$ to the CG PMF derived from a histogram analysis of the CG sampling (Fig. \ref{['fig:fig3']}).
• Figure 5: Microscopically-derived phase-field model for ice growth. (a) Scaling up the molecular CG interface to the phase-field level. (b) Equilibrium profile of phase-field $\phi$ from the center of interface. (c) Time evolution of $\phi$ driven by microscopically-derived free energy functional using Model A dynamics [Eq. (2)], where $\tau_{PF}$ is the phase-field timestep ($\phi=1$: ice, $0$: water).