A Kinetic Phase-Field Model of Diffusion Bonding: A Nonlocal Approach to Interface Coalescence

TL;DR

The paper tackles the challenge that conventional phase-field models inherently drive coalescence of nearby solid–solid interfaces, which is undesirable for diffusion bonding. It introduces a kinetic phase-field framework grounded in a geometric conservation law, where a nonlocal coalescence function $g$ modulates interface mobility based on curvature-based invariants, enabling arrest or persistence of interlayers. The model is formulated in 1D, 2D, and 3D with a mixed finite element implementation, and its thermodynamic consistency is ensured via a nonincrease of the free energy $\mathcal{P}$ under $0\le g\le1$. Numerical results demonstrate controlled coalescence, including 1D suppression, 2D preservation of parallel and diagonal interfaces, and 3D-like behavior with curvature-based arrest, while coupling to elasticity is explored for Ti–ZrC diffusion bonding with an interlayer Ti. The work shows how geometry-driven kinetics can predict and tune interlayer thickness and morphology under thermomechanical conditions, offering a scalable alternative to concentration-driven phase-field approaches for diffusion bonding and related interfacial phenomena.

Abstract

Conventional phase-field models often drive solid-solid interfaces to coalesce when in close proximity. This feature limits their use for processes like diffusion bonding, where the interfaces might need to remain distinct under certain thermodynamic conditions. We develop a kinetic phase-field model to address this problem, using an evolution equation based on a geometric conservation law for interfaces, rather than the gradient descent evolution that is typical in phase-field modeling. This formulation enables us to specify complex kinetic laws, and we use this to incorporate a physically-motivated geometric criterion to control interface merging. This criterion, based on nonlocal higher-derivative curvature invariants of the phase field, can be temperature-dependent, allows for a range of behaviors from complete coalescence to the preservation of distinct boundaries. Simulations show controlled bonding kinetics, demonstrating capabilities that are not available with existing methods for modeling interfaces that must remain distinct under given thermodynamic conditions.

Figures (18)

• Figure 1: Comparison of a scalar field $\phi(x_1,x_2)$ (top row) and the corresponding activation of the function $g(\phi)$ (bottom row), before (left) and during (right) a potential coalescence event. The function $g$ localizes to regions satisfying the geometric criteria of a persistent minimum.
• Figure 2: Visualization of a 3D phase field with interface detection based on Sylvester's criterion. The function $g_{3D}$ accurately identifies the interior region where $\phi$ approaches a local minimum.
• Figure 3: Space-time contour plots of $\phi(x,t)$ evolution in 1D. (a) Conventional model where interfaces merge, and the domain becomes $\phi \approx 1$. (b) With the coalescence functional active, interfaces approach but a stable $\phi \approx 0$ region is preserved.
• Figure 4: Initial condition for the parallel vertical interfaces case.
• Figure 5: Equilibrium state for the parallel vertical interfaces, showing a persistent gap. (a) Phase-field $\phi$. (b) Interface contour.