Sharp Droplet-Air Interface Model

• Sharp droplet–air interface models are mathematical representations treating the liquid–air boundary as an abrupt discontinuity with distinct physical properties.
• They integrate the augmented Young–Laplace equation and disjoining pressure terms to capture contact-line dynamics, energy barriers, and wetting transitions in multiphase flows.
• Advanced numerical techniques like Galerkin finite elements, ghost-fluid methods, and level-set tracking are employed for efficient simulation of complex wetting phenomena.

A sharp droplet–air interface model refers to the mathematical and computational representation of multiphase flows where the liquid–air interface is considered to be a discontinuity, with distinct physical properties and sharp changes in fields such as density, pressure, and stress. Such models enforce capillary and wetting physics at the interface, accurately capture contact line dynamics and phase transitions, and deliver high-fidelity predictions for dynamics, equilibrium shapes, and surface energy barriers in applications ranging from wetting on patterned substrates to aerosol evaporation and droplet deformation under external forces.

1. Fundamental Equations and Physical Principles

The core formulation is the augmented Young–Laplace equation. In its dimensional form,

$\gamma_{LV} C(s) + p_{LS}(\delta(s)) = \Delta p$

where $\gamma_{LV}$ is the liquid–vapor interfacial tension, $C(s)$ is local mean curvature, $\Delta p$ is the pressure jump across the interface, and $p_{LS}(\delta)$ denotes the disjoining pressure (liquid–solid interaction) as a function of local distance $\delta$ from the substrate (Chamakos et al., 2013). The disjoining pressure incorporates microscale attraction and repulsion, commonly parameterized via a generalized Lennard–Jones form: $$\Pi_{LS}(\delta) = w_{LS}\left[\left(\frac{\sigma}{\delta+\varepsilon}\right)^{C_1} - \left(\frac{\sigma}{\delta+\varepsilon}\right)^{C_2}\right]$$ with $w_{LS}$ controlling hydrophilicity, and $\sigma, \varepsilon, C_1, C_2$ tuning interaction range and regularization.

The interface curvature is given by

$C(s) = \frac{d\phi}{ds} + \frac{\sin \phi}{r(s)}$

and the profile is constrained by the arc-length relation

$$\left(\frac{dr}{ds}\right)^2 + \left(r(s)\frac{d\phi}{ds}\right)^2 = 1$$

and global volume conservation

$\int_{0}^{S_{max}} r^2(s)\frac{d\phi}{ds}ds = T$

where $S_{max}$ is the unknown total arc-length and $T$ is the prescribed volume (or area in 2D).

Energy barriers between coexisting stable wetting states are computed as

$$\Delta F_{S\to S'} = \gamma_{LV} \left[ (A_{LV}^U - A_{LV}^S) - \cos\theta_Y (A_{LS}^U - A_{LS}^S) \right]$$

and inform transition kinetics and hysteresis across Cassie–Baxter/Wenzel regimes.

In dynamical settings, the full incompressible or compressible hydrodynamics (Navier–Stokes, Euler) is solved in both phases, with interface jump conditions enforcing kinematic continuity and Young–Laplace stress balance (Fechter et al., 2015, Demont et al., 2023). In evaporation, sharp models treat mass and energy transfer via discontinuous interfacial source terms (Salimnezhad et al., 2024).

2. Treatment of Three-Phase Contact Lines and Substrate Interactions

A prominent advantage of sharp interface models is the implicit and unified treatment of contact-line physics. The disjoining pressure term in the augmented Young–Laplace equation enforces the equilibrium (Young) contact angle and captures dynamic wetting behavior without imposing explicit boundary conditions at the three-phase contact line (Chamakos et al., 2013, Chamakos et al., 2016). For patterned or rough substrates, the local distance field $\delta(x,y)$ is computed via a one-time Eikonal equation, tracing complex topographies and enabling multiple (unknown) contact lines and air pockets beneath the drop.

The resulting approach bypasses the need to pre-specify the number or location of contact lines, automatically handling transitions, film bridges, and lateral surface detachment accommodated by the geometry (Chamakos et al., 2013). The friction forces and stick–slip dynamics at the contact line arise from the interplay between the rapidly varying disjoining pressure and viscous stresses; the model predicts macroscopic resistance without ad-hoc slip lengths or angle laws (Chamakos et al., 2016).

3. Numerical Discretization and Solution Methods

Sharp interface implementation leverages parameterizations where the arc-length $s$ is the sole variable, with unknowns $r(s), \phi(s), K, S_{max}$ forming a one-dimensional boundary value problem (Chamakos et al., 2013). Discretization is performed using Galerkin finite elements, and nonlinear systems are solved by Newton–Raphson iteration. Stability and solution branches (saddle, coexisting states) are traced via pseudo-arc-length continuation, with eigenvalue analysis of the Jacobian $\partial R/\partial u$ differentiating stable and unstable equilibria.

For hydrodynamic problems, ghost-fluid methods are utilized. These involve exact multi-phase Riemann solvers for local jump conditions and level-set methods for tracking the moving interface (Fechter et al., 2015). The interface itself is maintained as a discontinuity, with explicit reconstruction of velocity, pressure, and thermodynamic quantities in each phase.

Advanced methods couple Lattice Boltzmann (LB) and Immersed Boundary (IB) techniques to simulate droplets of arbitrary surface stiffness, proceeding via mesh-interpolated velocity and force spreading between Lagrangian interface nodes and Eulerian fluid grids. Membrane elasticity is parameterized by isotropic prestress, area-dilation, and shear moduli, enabling traversal of droplet-to-elastic capsule limits (Pelusi et al., 2023).

In studies of sedimentation and evaporation in aerosols, sharp interface population-balance equations govern the time evolution of droplet-size distributions and viral load, with characteristic line analysis yielding exact solutions under physical assumptions (sphericity, diffusion-limited evaporation, negligible interactions) (Zendehroud et al., 9 Jan 2026).

4. Wetting Transitions, Energy Barriers, and Surface Free Energy Calculations

Sharp interface approaches accelerate the computation of equilibrium and transition energy barriers central to wetting hysteresis and reversibility (Chamakos et al., 2013). The total free energy is reduced to

$F = \gamma_{LV}(A_{LV} - \cos\theta_Y A_{LS})$

given the Young relation among interfacial tensions, and barrier heights $\Delta F$ follow from saddle-point configurations in the profile space. Pseudo-arc-length continuation maps out the entire solution branch, capturing energy maxima (unstable intermediates) and minima (stable wetting states).

This framework thus quantifies the response of droplets to external disturbances, thermal fluctuations, and patterned substrate transitions, providing predictive capacity for technological applications such as dewetting control and design of superhydrophobic surfaces.

5. Handling of Compressibility, Phase Change, and Mass/Energy Transfer

For compressible multiphase scenarios (e.g., liquid–vapor systems with phase transition), sharp interface methods treat the interface as a shock-like discontinuity. The coupling states for numerical solution in the bulk are determined by local multi-phase Riemann problems across the interface, enforcing mass, momentum, and energy jump conditions, including surface tension and latent heat (Fechter et al., 2015). The local normal speed of the interface is computed from the Riemann solution itself, and interface tracking is executed via level-set evolution.

Evaporation and sedimentation of droplet ensembles are represented by sharp interface population-balance PDEs, with parameter sensitivities to ambient RH, diffusion constants, and fall height directly informing viral load predictions in airborne transmission models (Zendehroud et al., 9 Jan 2026). Limitations (neglect of solute effects, vapor–field overlap, perfect molecular accommodation) are noted, and diffuse-interface extensions handle finite-thickness and solute-driven transitions.

In phase-field and diffuse-interface models, the sharp-interface limit is approached by tuning mobility scaling and interface thickness. Asymptotic analysis yields convergence to the classical jump conditions for stress, velocity, and mass transport; optimal mobility scalings (e.g., $M \propto \varepsilon^{1.7}$) are found for quantitative agreement with reference sharp interface solutions (Demont et al., 2023, Chen et al., 17 Jan 2025).

6. Model Limitations, Extensions, and Comparative Performance

Sharp interface models, by construction, treat the liquid–air interface as infinitely thin, with abrupt changes in physical fields. Consequently, they neglect sub-micron interface phenomena and molecular details that can be important in cases involving surfactants, soluble species, or high curvature. The traditional approach assumes noninteracting droplets, fixed shape (spherical), and relies on steady-state vapor diffusion; extensions to account for interface slip, solute crystallization, and Marangoni flows use hybrid or phase-field representations (Falavarjani et al., 2022, Chen et al., 17 Jan 2025, Salimnezhad et al., 2024).

Benchmarks demonstrate that, for physically realistic domains and parameter ranges, sharp interface models capture equilibrium shapes, wetting transitions, dynamic contact angle evolution, and inertia–viscous regimes in excellent agreement with experimental data and full three-dimensional simulations (Chamakos et al., 2016, Pelusi et al., 2023). In advection-dominated regimes, schemes such as MULES employ algebraic flux limiting to preserve interface sharpness, with error metrics (IAE, MCE) quantifying trade-offs in mass conservation and interface resolution (Sharifi, 2024).

7. Computational Advantages and Application Domains

Sharp droplet–air interface models offer significant computational advantages for simulation of droplet wetting on complex, patterned, or rough surfaces. The avoidance of explicit boundary conditions at unknown, multiply-connected contact lines simplifies mesh generation, solver implementation, and stability analysis. This enables rapid tracing of both stable and unstable equilibrium states, mapping of free energy landscapes, and efficient prediction of wetting transition barriers in surfaces of arbitrary geometric complexity (Chamakos et al., 2013).

Applications span microfluidics, spray combustion, airborne viral transmission modeling, liquid metal droplet wetting, and soft matter materials where control of the droplet–air interface dictates performance. The flexibility of sharp interface algorithms, and their well-established mathematical foundation, position them as a robust methodology for high-fidelity study and engineering of multiphase systems with rich interface physics.