A continuum framework for phase field with bulk-surface dynamics

This continuum mechanical theory aims at detailing the underlying rational mechanics of dynamic boundary conditions proposed by Fischer et al. (Phys Rev Lett 79:893, 1997), Goldstein et al. (Phys D Nonlinear Phenom 240:754–766, 2011), and Knopf et al. (ESAIM Math Model Numer Anal 55:229–282, 2021). As a byproduct, we generalize these theories. These types of dynamic boundary conditions are described by the coupling between the bulk and surface partial differential equations for phase fields. Our point of departure within this continuum framework is the principle of virtual powers postulated on an arbitrary part $$\mathcal {P}$$ P where the boundary $$\partial \mathcal {P}$$ ∂ P may lose smoothness. That is, the normal field may be discontinuous along an edge $$\partial ^2\mathcal {P}$$ ∂ 2 P . However, the edges characterizing the discontinuity of the normal field are considered smooth. Our results may be summarized as follows. We provide a generalized version of the principle of virtual powers for the bulk-surface coupling along with a generalized version of the partwise free-energy imbalance. Next, we derive the explicit form of the surface and edge microtractions along with the field equations for the bulk and surface phase fields. The final set of field equations somewhat resembles the Cahn–Hilliard equation for both the bulk and surface. Moreover, we provide a suitable set of constitutive relations and thermodynamically consistent boundary conditions. In Knopf et al. (2021), a mixed (Robin) type of boundary condition for the chemical potentials is proposed for the model in Fischer et al. (1997), Goldstein et al. (2011). In addition to this boundary condition, we also include this type of mixed boundary condition for the microstructure, that is the phase fields. Lastly, we derive the Lyapunov-decay relations for these mixed type of boundary conditions for both the microstructure and chemical potential.