Partially Wetted Crystalline Interfaces

• Partially wetted crystalline interfaces are configurations where incomplete wetting yields finite contact angles due to a balance of anisotropic surface energies and substrate adhesion.
• Methodologies span from atomistic-to-continuum models and Winterbottom constructions to molecular simulations that quantify interfacial tensions and validate anisotropic Young’s law.
• Findings inform applications in nanofabrication and crystal growth by linking geometric confinements, surfactant effects, and dynamic pinning to real-world material behaviors.

Partially wetted crystalline interfaces are interfacial configurations where a crystalline phase in contact with another phase or a substrate achieves incomplete wetting, resulting in a finite macroscopic contact angle determined by energetic and geometric constraints. These phenomena appear across hard and soft condensed matter, in both equilibrium and dynamic contexts, and manifest robustly in both isotropic and strongly anisotropic (i.e., crystalline) interfacial energy landscapes. Partially wetted crystalline interfaces are central to the morphology of supported nanocrystals, the kinetics of nucleation and growth, and interfacial phase transitions in soft matter and lattice-gas models. Their properties are governed by a balance between bulk thermodynamics, anisotropic surface and interface energies, and possible additional degrees of freedom such as surfactants or geometrical confinement.

1. Microscopic to Continuum Paradigms: Winterbottom Construction

A rigorous passage from atomistic models to continuum interfacial mechanics for crystalline drops on substrates is achieved by $\Gamma$-convergence of discrete interaction energies. For particles arranged on a triangular lattice with Heitmann–Radin sticky-disc potentials on a rigid substrate, the system energy scales to a continuum functional of the form

$$\mathcal{E}(D) = \int_{\partial^* D \setminus \partial S} \gamma(\nu) \,d\mathcal{H}^1 - \sigma\,\mathcal{H}^1(\partial^* D \cap \partial S)$$

where $\gamma(\nu)$ encodes surface anisotropy, and $\sigma$ is a substrate adhesion parameter microscopically determined by potential strengths. The minimization of this functional yields truncated Wulff shapes (Winterbottom shapes), where the drop exhibits a finite contact angle at the solid–substrate interface when substrate adhesion falls below a critical threshold. The transition from complete to partial wetting is governed by the adhesion parameter: for $\sigma \leq \sigma_c = -\min_\nu \gamma(\nu)$, the drop fully wets the substrate (zero contact angle); for $\sigma > \sigma_c$, partial wetting prevails and a finite contact angle is set by the anisotropic Young law $\nabla\gamma(\nu_D)\cdot\nu_S = -\sigma$ (Piovano et al., 2020).

2. Molecular Simulations and Thermodynamic Integration Approaches

Numerical studies of wall-attached crystalline nuclei and crystal–substrate wetting often use models such as hard spheres (Asakura–Oosawa), Lennard–Jones (LJ), or colloid–polymer mixtures. In such systems, the wall–liquid ($\gamma_{wl}$), wall–crystal ($\gamma_{wc}$), and crystal–liquid ($\gamma_{cl}$) interfacial tensions are quantified either by local pressure tensor calculations or thermodynamic integration. Young’s equation,

$$\gamma_{wl} - \gamma_{wc} = \gamma_{cl} \cos\theta,$$

relates these quantities to the equilibrium (macroscopic) contact angle $\theta$. Simulations confirm that, by tuning wall potential strength, surface structure, and interaction range, a regime is routinely accessed where $\theta$ is finite, i.e., the crystal only partially wets the wall. Molecular-dynamics studies show, for instance, that in Lennard–Jones systems, the (111) crystal face can consistently achieve $90^\circ<\theta<120^\circ$ across a wide range of wall–particle interaction strengths and commensurabilities (Benjamin et al., 2012), and MC simulations of confined AO mixtures directly measure a robust $\theta\approx 70^\circ$, closely matching Young’s prediction (Deb et al., 2012).

System	Measured Contact Angle ($\theta$)	Wetting Regime
AO colloid–polymer	$70^\circ\pm2^\circ$	Partial wetting
Lennard–Jones (111)	$97^\circ$–$114^\circ$	Partial wetting

The partial wetting regime is physically realized when $\gamma_{wc}>\gamma_{wl}$ and both are less than $\gamma_{cl}$, balancing the cost of creating a crystal–substrate interface against that of wall–liquid and liquid–crystal interfaces (Benjamin et al., 2012).

3. Anisotropy, Thermodynamic Transitions, and Pinning

Crystalline interfaces exhibit strong anisotropy in interfacial energy, leading to faceting and orientation-dependent contact angles. Anisotropic Young’s law generalizes the classic relation by incorporating the gradient of the interfacial energy density, and in the case of $\pi/3$ periodic surface tension (e.g., triangular lattice), the critical adhesion determines a rigorous wetting–dewetting boundary. Under partial wetting, the equilibrium shape is a truncated Wulff crystal whose facets meet the substrate at a finite angle (Piovano et al., 2020).

In lattice-based models with surfactants or other interface-active agents, mass conservation or limited surfactant availability triggers pinning phenomena and coexistence of moving and immobilized facets. The Blume–Emery–Griffiths (BEG) model, discretized on $\varepsilon\mathbb{Z}^2$, exhibits three-flow regimes controlled by surfactant mass conservation: (1) pinned facets (partial wetting), (2) facet-dependent motion with nonlocal surfactant redistribution, and (3) transition to complete wetting as surfactant mass becomes sufficient (Cicalese et al., 28 Dec 2025). Metastable, faceted, quasi-octagonal shapes and partial pinning emerge from constrained minimization, connecting discrete lattice kinetics to continuum evolution by crystalline mean-curvature flow with additional conservation or redistribution laws (Cicalese et al., 21 Oct 2025, Cicalese et al., 28 Dec 2025).

4. Surface Freezing, Premelting, and Soft Matter Interfaces

In soft-matter systems modeled by mean-field density functional theory (DFT), partially wetted crystalline interfaces can occur in the form of surface freezing (prefreezing) and surface melting (premelting) transitions. For weak or short-range wall–fluid repulsion, a wall can promote crystallization (prefreezing): as the chemical potential approaches the solid–liquid coexistence value from below, a finite crystalline layer nucleates at the wall in a first-order (symmetry-breaking) fashion, with a free-energy barrier and logarithmic divergence of the crystalline film thickness. Conversely, strong or long-range repulsion can induce surface premelting, where a liquid film intrudes at the wall–crystal interface as coexistence is approached from above, exhibiting either continuous or first-order behavior with the possibility of a surface critical point (Archer et al., 2016). These interfacial phenomena constitute concrete realizations of partially wetted crystalline configurations in soft-matter.

5. Structured and Nanotextured Crystalline Interfaces

Structured interfaces, such as arrays of surface-attached nanoparticles or spheres, exhibit stable partially wetted (Cassie–Baxter) states under externally applied fields or pressure. The geometry (sphere radius $R$, spacing $L$), local contact angle $\theta$, and imposed pressure $\Delta P$ collaboratively determine the threshold for impalement, where the meniscus transitions from a suspended to a fully wetted (Wenzel) configuration. Closed-form expressions for the critical impalement pressure

$$\Delta P_c = \frac{2\sqrt{2}\gamma_{LV}}{L}\sin\left[\theta + \arcsin\left(\frac{L}{\sqrt{2}R}\right)\right]$$

are consistent with molecular-dynamics simulations, which confirm the continuum-predicted stabilization of the partially wetted state up to $\Delta P_c$ (Bhattarai et al., 2017). This corroborates the crucial role of geometric re-entrance and local curvature in pinning the three-phase line, delaying complete wetting.

6. Dynamic Evolution and Metastability

Nontrivial dynamic behavior is observed at partially wetted crystalline interfaces, especially in regimes with incomplete surfactant coverage or conserved phase fields. Minimizing-movements schemes in discrete lattice models reveal a sequence of regimes: initial pinning (no movement of diagonal facets), subsequent surfactant-mediated mobility (with reduced local surface tension), and eventual transition to uniform motion as interfacial coverage increases. Such dynamics are characterized by nonlocal redistribution, velocity quantization, and persistent metastable octagonal geometries with both moving and stationary facets (Cicalese et al., 28 Dec 2025). These findings are the first rigorous connection between discrete surfactant lattice models and experimentally observed pinning and wetting transitions.

7. Methodological Connections and Broader Physical Implications

The study of partially wetted crystalline interfaces synthesizes discrete-to-continuum variational principles, molecular dynamics and Monte Carlo simulation, density functional theory, and geometric evolution approaches. Equilibrium shapes and contact line behavior arise as minimizers of anisotropic energies rooted in atomistic or mesoscopic models, while dynamic evolution reflects dissipative mechanisms subject to mass conservation or surfactant redistribution. These phenomena underpin crystal growth, nanofabrication, heterogeneous nucleation, interfacial pinning, and the design of superhydrophobic and wetting-resistant materials.

Partially wetted crystalline interfaces, in all these contexts, exemplify the interplay of geometry, anisotropy, competing interactions, and nonlocal constraints in the formation, stability, and evolution of interfacial structures in condensed matter and soft materials (Piovano et al., 2020, Deb et al., 2012, Bhattarai et al., 2017, Cicalese et al., 28 Dec 2025, Archer et al., 2016, Benjamin et al., 2012, Cicalese et al., 21 Oct 2025).