Thermo-Coalescence: Thermal, Capillary, Phase Dynamics

• The thermo-coalescence model is a theoretical and computational framework that integrates capillarity, thermodynamics, kinetics, and phase change to predict drop, grain, or cluster merging.
• It employs coupled equations for capillary flow, vapor diffusion, and heat transport to capture temperature-driven morphological transitions like freezing-pinning and asymmetric sintering.
• The model provides quantitative criteria via dimensionless groups and entropy production metrics for controlled coalescence in systems ranging from nanoparticle synthesis to astrophysical cold gas clouds.

The thermo-coalescence model encompasses a diverse class of theoretical and computational frameworks unifying capillarity, thermodynamics, kinetics, and phase change to describe the coalescence of drops, grains, clusters, and multiphase bodies under temperature gradients or during phase transitions. Across physical systems—from surface freezing of sessile drops to non-isothermal grain sintering, atomic nanoclusters, and catastrophic coalescence in close-packed emulsions—these models capture the interplay of thermal-driven morphological change, evaporation/condensation, latent-heat release, capillary flow, and thermodynamic non-equilibrium, providing quantitative criteria for predicting coalescence kinetics, arrest, and morphological outcomes.

1. Fundamental Equations and Mechanistic Principles

Thermo-coalescence models are constructed by coupling capillary flow, vapor diffusion, heat transport, and phase evolution under rigorous mass, momentum, and energy conservation. For two volatile sessile drops on a cold substrate, the core equations in lubrication limit ($\varepsilon=H/L\ll1$) are:

• Evolution of the liquid–gas interface $h(x,t)$ and the ice–liquid front $s(x,t)$:

$$\partial_t h + \partial_x \{ (h^3/3\mu)\,\partial_x [\gamma\partial_{xx} h] \} = -J_v/\rho_l$$

or, non-dimensionally,

$$\partial_t h + \partial_x (h^3\,\partial_x \kappa) = -J(x,t), \quad \kappa = -\partial_{xx}h$$

• Vapor diffusion governs the evaporative flux:

$$J(x,t)= -D_m\, \partial_z c_v|_{h}, \quad c_v|_{z\to\infty}=RH\cdot c_{ve}, \quad c_v|_{z=h}=c_{ve}(T_l)$$

• Heat-transfer with Stefan condition for the freezing front:

$$Ste~[\Lambda_s\,\partial_z T_s - \partial_z T_l]_{z=s} = \partial_t s, \quad Ste=\rho_s L_f H/(k_l\Delta T)$$

Evaporation leads to an asymmetric flux $J_v(x)$ due to vapor shielding, generating a curvature difference and thus a capillary-driven migration of the drops. This attractive capillary flux ($q\sim h^3 \partial_x \kappa$) competes against the upward freezing front $s(x,t)$—the phase-line for coalescence is set where migration and freezing time-scales are commensurate ($h(x,t)$0) (Kavuri et al., 2024).

Thermo-coalescence in grain sintering invokes phase-field models with temperature-dependent free-energy functionals and kinetic relations for conserved (density) and non-conserved (orientation) order parameters, coupled to dynamic heat transfer. The free energy includes contributions from thermal, configurational, interface-gradient, and entropy terms, with kinetic mobilities that may be strongly Arrhenius in temperature (Yang et al., 2018).

Coalescence of metallic nanoparticles—e.g., Au—features a melting point $h(x,t)$1 depressed by the Gibbs–Thomson effect, but the coalescence temperature $h(x,t)$2 is distinctly lower, set by the onset of surface premelting (Lindemann criterion) and exacerbated by surface-energy reduction and local temperature spikes at the neck ($h(x,t)$3) (Kamachali, 2019).

2. Key Dimensionless Groups and Phase Diagrams

Thermo-coalescence behavior is governed by dimensionless numbers quantifying viscous, thermal, kinetic, and capillarity processes:

• Capillary number: $h(x,t)$4
• Stefan number: $h(x,t)$5
• Péclet number (vapor): $h(x,t)$6
• Knudsen number (evaporation): $h(x,t)$7
• Scaled latent heat: $h(x,t)$8
• Relative humidity: $h(x,t)$9

Phase diagrams delineate the regions of merging and non-merging as functions of initial gap $s(x,t)$0, $s(x,t)$1, and $s(x,t)$2. For example, coalescence is achieved for $s(x,t)$3 and $s(x,t)$4 at $s(x,t)$5K, but these boundaries shrink sharply with increasing $s(x,t)$6 or $s(x,t)$7. The regime structure quantifies the competition between capillary pull (fast at low RH) and freezing-pinning (accelerated at high RH or $s(x,t)$8), leading to distinct operational windows for controlled coalescence (Kavuri et al., 2024).

3. Thermodynamic Non-Equilibrium, Entropy Production, and Morphological Kinetics

Thermodynamic non-equilibrium (TNE) is central to modern thermo-coalescence models. The D2V33 discrete Boltzmann framework tracks central kinetic moments:

• Non-organized momentum flux (NOMF):

$s(x,t)$9

• Non-organized energy flux (NOEF):

$$\partial_t h + \partial_x \{ (h^3/3\mu)\,\partial_x [\gamma\partial_{xx} h] \} = -J_v/\rho_l$$0

TNE metrics, including $$\partial_t h + \partial_x \{ (h^3/3\mu)\,\partial_x [\gamma\partial_{xx} h] \} = -J_v/\rho_l$$1 (global TNE strength), are tightly correlated with bridge growth, interface relaxation, and entropy production. In non-isothermal coalescence, latent-heat release at the neck increases pressure gradient resistance, delaying bridge formation ($$\partial_t h + \partial_x \{ (h^3/3\mu)\,\partial_x [\gamma\partial_{xx} h] \} = -J_v/\rho_l$$2), suppresses TNE peak intensity, and produces oscillatory entropy-production dynamics with extended duration and greater integrated dissipation, despite a lower peak rate (Sun et al., 24 Feb 2025, Sun et al., 2023).

A concise encapsulation is that $$\partial_t h + \partial_x \{ (h^3/3\mu)\,\partial_x [\gamma\partial_{xx} h] \} = -J_v/\rho_l$$3 and $$\partial_t h + \partial_x \{ (h^3/3\mu)\,\partial_x [\gamma\partial_{xx} h] \} = -J_v/\rho_l$$4 reach maxima when the interface decreases fastest, while valleys mark maximal droplet elongation. Surface-tension gradients drive coalescence, while NOMF acts as resistance, with these forces peaking sequentially and governing the multi-stage relaxation process.

4. Temperature-Dependent Kinetics and Arrested/Morphological Outcomes

Thermal gradients and transitions enable or inhibit coalescence by tuning driving and resisting forces at interfaces. Notable outcomes include:

• Freezing-pinning: For sessile evaporating drops, rapid freezing arrests capillary migration, pinning the contact line before gap closure and preventing coalescence at large $$\partial_t h + \partial_x \{ (h^3/3\mu)\,\partial_x [\gamma\partial_{xx} h] \} = -J_v/\rho_l$$5 or high $$\partial_t h + \partial_x \{ (h^3/3\mu)\,\partial_x [\gamma\partial_{xx} h] \} = -J_v/\rho_l$$6. Conversely, strong evaporation or optimal $$\partial_t h + \partial_x \{ (h^3/3\mu)\,\partial_x [\gamma\partial_{xx} h] \} = -J_v/\rho_l$$7 accelerates merging (Kavuri et al., 2024).
• Non-isothermal sintering: In grain coalescence, a lateral temperature gradient causes asymmetric neck growth, tilting mass flux from cold to hot regions and resulting in curved grain boundaries and asymmetric evolution. For non-identical grains, morphologies may undergo three-stage relaxation: neck-dominated growth, intermediate stabilization, and late-stage boundary migration (Yang et al., 2018).
• Nanocluster coalescence and bifurcation: The transition between solid-state sintering and liquid-like coalescence in Au nanoparticles is governed by a sharp bifurcation at $$\partial_t h + \partial_x \{ (h^3/3\mu)\,\partial_x [\gamma\partial_{xx} h] \} = -J_v/\rho_l$$8, with stochastic selection of outcome; above $$\partial_t h + \partial_x \{ (h^3/3\mu)\,\partial_x [\gamma\partial_{xx} h] \} = -J_v/\rho_l$$9, fluidic coalescence occurs, below $$\partial_t h + \partial_x (h^3\,\partial_x \kappa) = -J(x,t), \quad \kappa = -\partial_{xx}h$$0 locked bicrystal dumbbells form (Puytov et al., 2021).
• Arrested coalescence in phase-field models: Temperature-dependent kinetic functions employing higher-derivative geometric invariants enable programmable arrest or merging behavior for solid–solid boundaries, sensitive to local temperature and curvature (Khodadad et al., 18 Feb 2026).

5. Applications: Multiphase Matter, Nucleation, and Astrophysical Contexts

Thermo-coalescence models have been deployed across a wide range of systems:

• Alloy and ceramic sintering, including spark plasma and laser-assisted techniques, through quantitative phase-field models (Yang et al., 2018).
• Nanoparticle synthesis, growth, and fragmentation—connecting excess heat (enthalpic and entropic) in cluster coalescence to melting, morphological transitions, and latent-heat release, with direct implications for synthesis/fragmentation balances in reactive atmospheres (Yang et al., 2019).
• Catastrophic destabilization in crystalline emulsions: Synchronized, system-wide coalescence of hexagonally packed droplets is triggered at a critical temperature, with a phase-line determined by interfacial tension, network elasticity, and crystal content, culminating in percolation-driven catastrophic coalescence (Abedi et al., 2020).
• Astrophysical cold gas clouds: Coagulation of cold droplets under radiative cooling is governed by a cooling-driven inverse-square force law $$\partial_t h + \partial_x (h^3\,\partial_x \kappa) = -J(x,t), \quad \kappa = -\partial_{xx}h$$1, directly analogous to Newtonian gravity but with surface area as the monopole. This mechanism controls cold mass growth and fragmentation-coagulation thresholds in the multiphase interstellar medium (Gronke et al., 2022).

6. Universal Model Features and Open Directions

Essential features of thermo-coalescence frameworks include:

• Joint, dynamically evolving couplings: Capillarity, phase change, vapor diffusion, latent heat, and TNE all directly enter via free-energy gradients, kinetic laws, or closure relations.
• Control by dimensionless groups: Thermo-coalescence phase behavior is organized in closed multidimensional spaces (gap, humidity/chemical potential, temperature, characteristic timescales), which enables predictive, experiment-driven phase mapping.
• Role of non-equilibrium entropy production: Quantitative tracking of irreversible dissipation via entropy production provides both a physical and diagnostic criterion for stage transitions and relative thermal efficiency.
• Temperature-tunable morphological regimes: By programmatically varying $$\partial_t h + \partial_x (h^3\,\partial_x \kappa) = -J(x,t), \quad \kappa = -\partial_{xx}h$$2, the onset, progression, or arrest of coalescence can be kinetically controlled, with clear, model-based criteria for transition zones and kinetic bottlenecks.
• Relevance for dynamic materials design, multiphase process engineering, and the interpretation of coalescence in non-equilibrium or spatially inhomogeneous thermodynamic environments.

Thermo-coalescence models thus constitute a unifying and extensible theoretical formalism for understanding, predicting, and controlling the merging of bodies under the action of coupled capillary, thermal, and kinetic driving forces, with broad applicability from micro/nanoscale devices to astrophysical structure formation (Kavuri et al., 2024, Yang et al., 2018, Kamachali, 2019, Puytov et al., 2021, Khodadad et al., 18 Feb 2026, Yang et al., 2019, Sun et al., 24 Feb 2025, Gronke et al., 2022).