Non-equilibrium Interfacial Transfer

• Non-equilibrium interfacial transfer is the dynamic, time-dependent exchange of mass, energy, or momentum across phase boundaries when systems are out of equilibrium.
• Advanced modeling approaches, including phase-field models and hydrodynamic DFT, provide quantitative insights into interface kinetics and scaling behaviors.
• Applications span rapid solidification, colloidal phase separation, and thermal management in nanostructures, where non-equilibrium effects crucially drive material performance.

Non-equilibrium interfacial transfer refers to the time-dependent exchange of mass, energy, or momentum across the interface between two phases under conditions where the adjacent bulk phases (and often the interface itself) are out of local or global thermodynamic equilibrium. Such non-equilibrium states arise naturally during rapid phase transformations, nucleation, solidification, interfacial chemical reactions, transient heat or mass fluxes, or whenever external driving forces (e.g., electric fields, imposed currents, laser pulses) disrupt equilibrium at or near an interface. The phenomenon spans a wide range of materials classes and physical regimes, including molecular fluids, colloidal suspensions, crystalline solids, porous media, and nanostructured device interfaces. Central to the discipline is the rigorous description—via continuum, statistical, or atomistic models—of (i) how constitutive interfacial properties (e.g., tension, free energy, transfer resistances) evolve dynamically under non-equilibrium, (ii) how fluxes and relaxation rates are set by the underlying microscopic or mesoscopic mechanisms, and (iii) how observable signatures (e.g., temperature jumps, mass fluxes, entropy production) differ qualitatively and quantitatively from equilibrium behavior.

1. Theoretical Foundations and Governing Frameworks

Non-equilibrium interfacial transfer is formulated in the language of non-equilibrium thermodynamics, combining local conservation laws (mass, energy), dynamical equations of state, and boundary/interface conditions encoding Onsager reciprocity, statistical kinetics, or chemical-mechanical drive.

A prototypical example is the early-stage relaxation of a non-equilibrium interface in a phase-separating colloidal fluid (Bier et al., 2013). Here, a dynamic density functional theory (DDFT) for the conserved colloidal density $\varrho(\mathbf r, t)$ is utilized: $$\frac{\partial \varrho}{\partial t} = -\nabla \cdot \mathbf{j}(\mathbf r,t)$$ with a flux given by

$$\mathbf{j}(\mathbf r, t) = -D\,\varrho(\mathbf r, t) \nabla \left[ \frac{\delta F}{\delta \varrho(\mathbf r)} \right]$$

where $F$ is a non-local free-energy functional encoding hard-sphere and square-well attractions. The local chemical potential acts as the driving force for interfacial flux, even when the bulk phases remain out of two-phase coexistence.

In metallic alloys and solidification, generalized phase-field models incorporate non-equilibrium interface conditions by extending the classical Allen–Cahn and Cahn–Hilliard formalism. Here, the interface is rendered as a diffuse region with continuous fields for phase and composition, and both short-range (atomic-transfer) and long-range (diffusive) mechanisms are treated within an Onsager-matrix formalism (Li et al., 2023).

In solid-state systems, interfacial transfer can often be described by kinetic rate theories based on transition-state statistical mechanics, as recently formulated for interstitial transfer across crystalline-crystalline interfaces (Weissmüller, 29 Jan 2026): $$J = \nu_0 \rho_T \theta^A \theta^B e^{-\epsilon^T/k_BT} \left[ e^{\mu_S^A/k_B T} - e^{\mu_S^B/k_B T} \right]$$ where the flux depends explicitly on the absolute chemical potentials and vacancy fractions in each phase, rather than just the chemical potential difference.

2. Mechanisms and Dynamical Regimes of Interfacial Relaxation

Non-equilibrium interfacial transfer underpinning phenomena (evaporation, condensation, nucleation, phase growth, heat transfer, charge/displacement waves) can be grouped according to dominant mechanisms and relevant timescales:

• Diffusion-Controlled Relaxation: The interface rapidly adopts a local equilibrium profile even when bulk phases are out of equilibrium. For example, in oversaturated colloidal fluids, the chemical potential at the interface flattens to its coexistence value over a region whose width grows as $R(t) \sim t^{1/2}$, and the interfacial tension $\gamma(t)$ converges to its equilibrium value with power-law decay ($t^{-1/2}$ or $t^{-3/2}$) depending on the proximity to phase coexistence (Bier et al., 2013).
• Kinetic (Reaction- or Barrier-Limited) Exchange: Where activated processes set the rate of transfer, such as interstitial atomic hopping or charge transfer across interfaces, the rate equations can differ fundamentally from common Butler–Volmer forms—exhibiting explicit dependence on absolute chemical potentials and reactive-site occupancies, and yielding dramatic slow-downs near phase boundaries or critical points (Weissmüller, 29 Jan 2026).
• Phonon or Electron Non-Equilibrium: At solid-solid or metal-insulator interfaces under transient heating or current pulses, bottlenecks arise when non-equilibrium phonon or electronic distributions must relax by energy exchange processes with strong spectral mismatch between constituent phases. The resulting non-equilibrium resistances often outweigh equilibrium interface scattering contributions (Li et al., 2022, Han et al., 2023).
• Mode-Selective Interfacial Coupling: Ultrafast energy transfer across a metal–insulator interface can proceed via direct coupling between hot electrons and selected high-frequency interface vibrational modes (electronic Kapitza process), followed later by “ordinary” thermalization via acoustic-phonon channels (Rothenbach et al., 2019).

3. Quantitative Descriptions: Power Laws, Scaling, and Kinetic Rate Laws

Many non-equilibrium interface phenomena display universal scaling regimes before global equilibrium is reached:

• In colloidal and molecular fluid interfaces, the deviations of local chemical potential and flux from equilibrium evolve as

$$\Delta \widetilde{\mu}(\tilde z, t) \sim M_1(\tilde z t^{-1/2}) + t^{-1} M_2(\tilde z t^{-1/2}),\qquad \tilde j(\tilde z, t) \sim t^{-1/2} J_1(\tilde z t^{-1/2}) + t^{-3/2} J_2(\tilde z t^{-1/2})$$

with leading order determined by the far-from-equilibrium state of the bulk (Bier et al., 2013).

• For phase transformations, the “solute-trapping” coefficient $K(V)$ (partition ratio across a moving interface) interpolates between equilibrium and complete trapping,

$K(V) = \frac{K_e + V/V_D}{1+V/V_D}$

with drag and interface friction scaling with interface velocity, as derived in phase-field and Cattaneo-type relaxational models (Li et al., 2023).

• In heat transfer at solid–solid interfaces, the total interfacial resistance $R_{\text{int}}$ includes both the Landauer/DMM equilibrium prediction and significant (often dominant) non-equilibrium corrections $R_\text{neq}$ due to the relaxation of out-of-equilibrium phonon populations. These extra resistances typically scale with Debye temperature mismatch and acoustic-branch velocity differences, with relaxation lengths often exceeding hundreds of nanometers (Li et al., 2022, Han et al., 2023).
• Kinetic models for interstitial transfer predict equilibration rates that diverge near bulk spinodal (critical) points, which rationalizes the dramatic experimental slow-downs in metal hydride (e.g. Pd–H) charging near miscibility gaps (Weissmüller, 29 Jan 2026).

4. Applications and Experimental Manifestations

Non-equilibrium interfacial transfer is critically relevant in:

• Colloidal and molecular phase separation: Interface relaxes toward local coexistence much before bulk equilibration, providing a theoretical foundation for early-stage measurements of interfacial tension and evaporation rates (Bier et al., 2013).
• Rapid solidification and solid-state transformations: Phase-field models that incorporate explicit non-equilibrium interface kinetics capture solute trapping, dendritic growth, and interface drag in alloys, supporting quantitative comparisons of experiment and simulation in regimes inaccessible to sharp-interface Stefan-type models (Li et al., 2023).
• Thermal management in semiconductors and nanostructures: The interfacial thermal resistance in, e.g., Si–Ge or III–V/III–V interfaces, is controlled by non-equilibrium phonon relaxation (not merely interface scattering), necessitating continuum kinetic models (PBE) for device-scale predictions. Engineered phonon-matching or interface grading can minimize the non-equilibrium resistance (Li et al., 2022, Han et al., 2023).
• Electrochemical interfaces: Stability and patterning of electrode–electrolyte interfaces, critical to battery and catalysis technologies, are governed by non-equilibrium free energy and kinetic rate laws that can suppress or promote interfacial instabilities (island formation, phase separation) depending on current, voltage, and reaction mechanism (Fraggedakis et al., 2020, Cheng et al., 2015).
• Polymer/water and porous media systems: Atomistically-resolved MD studies reveal that interfacial thermal conductance in soft matter depends chiefly on temperature-driven dynamical effects rather than vibrational density-of-states overlap, while porous media models clarify when explicit interfacial resistance must be included to capture local thermal non-equilibrium (Babaei et al., 2 Dec 2025, Kostelecky et al., 8 Apr 2025).

5. Advanced Modeling Approaches and Thermodynamic Consistency

Recent advances emphasize the importance of consistent thermodynamic and kinetic coupling between bulk and interfacial regions, including:

• Generalized Onsager Matrices: For multi-component and multi-mechanism systems, such as rapid solidification or surface reactions, full Onsager–Casimir reciprocal relations yield coupled boundary conditions for mass, heat, momentum, and surface fluxes, capturing both direct and reciprocal effects (e.g., diffusiophoresis and interfacial slip) (Gaspard et al., 2018, Li et al., 2023).
• Hydrodynamic Density Functional Theory (h-DFT): Combines classical DFT-derived free energy functionals with generalized Maxwell–Stefan diffusion and Navier–Stokes momentum conservation, providing a seamless framework for predicting non-equilibrium mass transfer at vapor–liquid and liquid–liquid interfaces, and accurately reproducing NEMD simulations for both ideal and strongly non-ideal mixtures (Bursik et al., 2 Jul 2025).
• Curvature-Corrected Non-Equilibrium Transfer: In nucleation and small-scale morphology evolution, curvature introduces corrections to both the local and excess (Gibbs) resistances to heat and mass transfer, leading to significant quantitative differences from planar-interface models in sub-100 nm droplets/bubbles (Glavatskiy et al., 2013).
• Non-Equilibrium Free-Energy Surfaces: Simulation protocols based on collective variables linked to non-equilibrium extensions of interfacial free energy allow the direct calculation of $\gamma(T, \Delta\mu)$ and its impact on nucleation/growth kinetics, resolving ambiguities in classical models (Cheng et al., 2015).

6. Key Quantitative Relations and Regimes

Interfacial Transfer Mechanism	Governing Law(s) / Key Results	Reference
Colloid/molecular interface	DDFT, $\partial_t\phi = -\partial_z j$; $\gamma(t)\to \gamma_\text{eq}$	(Bier et al., 2013)
Solidification, alloys	Onsager-matrix phase-field, $K(V)$, drag force, dissipation rates	(Li et al., 2023)
Interstitial transfer (lattice)	$$J = \nu_0 \rho_T \theta^A \theta^B e^{-\epsilon^T/k_BT}[e^{\mu^A/k_BT}-e^{\mu^B/k_BT}]$$	(Weissmüller, 29 Jan 2026)
Heat transfer in nanostructures	Peierls–Boltzmann, $R_\text{int} = R_\text{eq} + R_\text{neq}$	(Li et al., 2022, Han et al., 2023)
Porous media (heat)	Two-temperature model, $q_\text{int} = hA_{fs}(T_s-T_f)$	(Kostelecky et al., 8 Apr 2025)
Hydrodynamic DFT (mass)	$$\partial_t\rho_i + \nabla\cdot(\rho_i \mathbf v + \mathbf j_i^{\mathrm{diff}})=0$$	(Bursik et al., 2 Jul 2025)
Interfacial curvature correction	$R_{ab}(1/R) = R_{ab,\infty}[1 + c_1/R + c_2/R^2]$	(Glavatskiy et al., 2013)
Non-equilibrium $\gamma(T,\Delta\mu)$	Simulations with collective variables, interface free energy profile	(Cheng et al., 2015)

The broad implication is that failure to account for non-equilibrium effects—whether in the interface kinetics, bulk relaxation, resistance partitioning, or thermodynamic driving forces—can result in qualitative failure of modeling predictions across scales and systems. Only by explicitly treating the time- and mechanism-dependent transport and energetic characteristics of interfaces under non-equilibrium does theory align quantitatively with experiment and simulation.