Diffusion in Solids and Phase Transitions

Motivation

Many technologically important materials — hydrogen-storage alloys, lithium-ion electrode materials, gels, shape-memory alloys — undergo coupled deformation and species diffusion. The diffusion drives compositional changes that modify the elastic modulus and natural shape of the solid; conversely, stress alters the chemical potential and thus the kinetics of diffusion. At low temperatures or high concentrations, these systems also exhibit phase separation, described by Cahn–Hilliard-type free energies with double-well structure. Understanding the interplay between mechanics, diffusion, and phase transitions at the PDE level is both mathematically rich and practically important.

This research thread started from thermomechanical models for shape-memory alloys and expanded to cover hydrogen storage, doubly nonlinear Cahn–Hilliard systems, and large-strain poromechanics, through a long-running collaboration with T. Roubíček (Prague) and E. Bonetti, P. Colli, and L. Scarpa.

Details

Shape-memory alloys under electromagnetic loading (2009–2013)

Shape-memory alloys (SMAs) are intermetallics that undergo a reversible martensitic phase transformation. When the alloy is also ferromagnetic (as in Ni–Mn–Ga), the transformation can be triggered or guided by a magnetic field. The thermomechanical analysis of such systems requires coupling elastoplasticity, magnetism, heat conduction, and electric conduction.

Roubíček & Tomassetti (2010) — Thermodynamics of shape-memory alloys under electric current. The first paper in the collaboration with Roubíček, formulating the full coupled system and establishing a global existence result via a Galerkin approximation. The electric current provides both Joule heating and a body force through the Lorentz effect.
Roubíček & Tomassetti (2013) — Phase transformations in electrically conductive ferromagnetic shape-memory alloys, their thermodynamics and analysis. The most technically demanding paper of this phase, incorporating a full micromagnetics description of the ferromagnetic order together with the SMA transformation. The analysis established existence of weak solutions in the large-deformation regime using compensated compactness and monotone operator theory.

Hydrogen storage: stress effects and hysteresis (2014–2016)

Metal hydrides are attractive materials for solid-state hydrogen storage because of their high volumetric capacity. However, their practical use is hampered by two effects: (i) the large volumetric expansion during hydrogen absorption induces stresses that alter the absorption thermodynamics; (ii) the absorption/desorption cycle exhibits hysteresis, with different equilibrium pressures on the way in and out. Modeling both effects requires coupling diffusion to mechanics and phase transitions.

Roubíček & Tomassetti (2014) — Thermomechanics of hydrogen storage in metallic hydrides: modeling and analysis. A large-deformation model for hydrogen diffusion in an elastic solid undergoing a $\alpha$–$\beta$ phase transition was formulated. The model couples a Cahn–Hilliard equation (for the hydrogen concentration) to a momentum balance (for the stress). Global existence was established by combining an energy argument with a compensated-compactness estimate.
Duda & Tomassetti (2015) — Stress effects on the kinetics of hydrogen adsorption in a spherical particle: an analytical model. While the 2014 paper addressed the full PDE system, this paper derived closed-form analytical solutions for a spherical hydride particle under the assumption of linear elasticity and a sharp interface. The result is an explicit formula relating the absorption plateau pressure to the particle radius, the elastic modulus, and the transformation strain — directly testable against experiments.
Duda & Tomassetti (2016) — On the effect of elastic distortions on the kinetics of diffusion-induced phase transformations. Extending the sharp-interface model to a more general class of configurations, this paper showed that elastic distortions systematically shift the transformation kinetics in a manner that can produce or suppress hysteresis, providing a mechanical explanation for the observed pressure hysteresis in metal hydrides.
Roubíček & Tomassetti (2015) — Thermomechanics of damageable materials under diffusion: modelling and analysis. A parallel development considering materials that can also damage under large stresses, combining the diffusion/phase-transition framework with a damage mechanics model. The analysis required new estimates to control the interplay between damage softening and species diffusion.

Cahn–Hilliard systems with nonlinear viscosity (2017–2020)

The Cahn–Hilliard equation with a nonlinear (logarithmic or degenerate) mobility describes phase separation in fluids and solids. A particularly challenging variant — the doubly nonlinear Cahn–Hilliard system — arises when both the chemical potential and the time derivative of the concentration are related to the concentration by nonlinear operators. This is the setting for the following series of papers with Bonetti, Colli, and Scarpa:

Bonetti, Colli & Tomassetti (2017) — A non-smooth regularization of a forward-backward parabolic equation. The doubly nonlinear Cahn–Hilliard system degenerates in certain parameter regimes into a forward-backward parabolic equation, which is ill-posed. This paper introduced a regularization based on a non-smooth (maximal monotone) perturbation and proved convergence of solutions as the regularization parameter vanishes.
Tomassetti (2017) — Smooth and non-smooth regularizations of the nonlinear diffusion equation. A companion paper analyzing the two regularization strategies (smooth Yosida approximation vs. non-smooth maximal monotone inclusion) and comparing their convergence rates.
Bonetti, Colli, Scarpa & Tomassetti (2018) — A doubly nonlinear Cahn–Hilliard system with nonlinear viscosity. The full doubly nonlinear system was analyzed by variational techniques, establishing existence of weak solutions with uniform bounds via a Galerkin scheme with a priori estimates derived from an entropy-like functional.
Bonetti, Colli, Scarpa & Tomassetti (2020) — Bounded solutions and their asymptotics for a doubly nonlinear Cahn–Hilliard system. This paper proved that solutions remain $L^\infty$-bounded for all time, and characterized their long-time behavior: as $t \to \infty$, solutions converge to stationary states that satisfy a nonlinear elliptic problem. The key tool is an Alikakos-type iteration adapted to the doubly nonlinear structure.

Large-strain poromechanics and continuum thermodynamics (2018–2022)

Alongside the Cahn–Hilliard program, a parallel effort extended the diffusion–mechanics coupling to fully nonlinear (large-strain) poroelastic bodies:

Roubíček & Tomassetti (2018) — A thermodynamically consistent model of magneto-elastic materials under diffusion at large strains and its analysis. This paper unified the magnetoelastic and diffusion threads: a magneto-elastic body is allowed to absorb or desorb a solvent species, which modifies both its magnetic response and its elastic stiffness. Existence was proved at finite strain.
Roubíček & Tomassetti (2020) — Dynamics of charged elastic bodies under diffusion at large strains. The previous analysis was extended to charged species (ions), adding electrostatic effects. The charged-diffusion model is relevant to lithium-ion batteries and biological membranes.
Tomassetti (2022) — Modeling the diffusion of a fluid in a strained solid: a comparison between different formats. A pedagogical paper that systematically compared three formulations of coupled fluid–solid diffusion (referential, spatial, and convective), clarifying the conditions under which they are equivalent and identifying the sources of confusion in the literature.
Roubíček & Tomassetti (2021) — A convective model for poro-elastodynamics with damage and fluid flow towards Earth lithosphere modelling. An application-motivated paper extending the poroelastic framework to geomechanics: the solid skeleton can damage, the fluid is compressible, and inertia is retained. Well-posedness is established for the resulting hyperbolic–parabolic system.
Roubíček & Tomassetti (2024) — Inhomogeneous thermoplasticity. The most recent paper in this thread studies thermoelastic–thermoplastic materials with inhomogeneous properties, establishing existence and uniqueness for the coupled heat/deformation problem and exploring the role of inhomogeneity in shear localization.

Swelling, dehydration, and shape morphing in soft materials (2022–2026)

A parallel research direction addresses the coupled chemo-mechanical and thermal response of swollen soft materials — hydrogels and edible composites — using Flory–Rehner/Flory–Huggins swelling theory. Unlike the Cahn–Hilliard thread above, the emphasis here is on moisture-driven deformation, differential shrinkage across material domains, and buckling instabilities arising from inhomogeneous dehydration.

Curatolo, Tomassetti, van der Sman & Teresi (2026) — Thermal dehydration of swollen heterogeneous soft materials. Soft Matter, in press. A multiphysics model for bi-domain soft materials (e.g., edible bilayers) subject to intensive heating. The model couples heat transfer, moisture diffusion, and finite-strain mechanics through a temperature-dependent Flory–Huggins free energy. Simulations reveal critical temperature thresholds, wet-bulb equilibrium stages, and the role of the domain interface in controlling differential shrinkage and buckling. The work extends previous results on isothermal dehydration of single-domain bodies to the fully coupled, heterogeneous, thermal setting.

Summary and open questions

The thread traces a progression from SMA thermodynamics (2009–2013) → hydrogen storage with stress effects (2014–2016) → abstract Cahn–Hilliard analysis (2017–2020) → large-strain poromechanics and charged diffusion (2018–2022) → geological and thermoplastic applications (2021–2024) → thermal dehydration and shape morphing in soft materials (2026). The unifying theme is the rigorous analysis of coupled nonlinear PDE systems arising at the mechanics–chemistry interface. Open questions include the derivation of effective diffusion coefficients in heterogeneous elastic media and the analysis of phase-field fracture coupled to species diffusion.