Abstract
At high temperature and pressure, solid diffusion and chemical reactions between rock minerals lead to phase transformations. Chemical transport during uphill diffusion causes phase separation, that is, spinodal decomposition. Thus, to describe the coarsening kinetics of the exsolution microstructure, we derive a thermodynamically consistent continuum theory for the multicomponent Cahn–Hilliard equations while accounting for multiple chemical reactions and neglecting deformations. Our approach considers multiple balances of microforces augmented by multiple component content balance equations within an extended Larché–Cahn framework. As for the Larché–Cahn framework, we incorporate into the theory the Larché–Cahn derivatives with respect to the phase fields and their gradients. We also explain the implications of the resulting constrained gradients of the phase fields in the form of the gradient energy coefficients. Moreover, we derive a configurational balance that includes all the associated configurational fields in agreement with the Larché–Cahn framework. We study phase separation in a three-component system whose microstructural evolution depends upon the reaction–diffusion interactions and to analyze the underlying configurational fields. This simulation portrays the interleaving between the reaction and diffusion processes and how the configurational tractions drive the motion of interfaces.
Highlights
We assume that a mass density α, a diffusive flux j α, and a reactive mass supply rate sα characterize the instantaneous state of each component α = 1, . . . , n
We require that α, j α, and sα evolve subject to a pointwise component content balance in the formα = −div j α + sα, ∀ 1 ≤ α ≤ n, (1)
We introduce constitutive relations for the diffusive flux j α, the reactive mass supply rate sα, the internal microforce density π α, and the microstress ξ α for each component α = 1, . . . , n, which allow us to close the system of evolution equations for the phase fields φα, α = 1, . . . , n
Summary
We require that α, j α, and sα evolve subject to a pointwise component content balance in the formα = −div j α + sα, ∀ 1 ≤ α ≤ n,. Stipulating that the mass supply rates and the diffusive fluxes satisfy constraints of the form n n sα = 0 and j α = 0,. Α=1 α=1 we sum the component content balance (1) over α from 1 to n to find that the total mass density n. N, from expressions (3) and (4) together with the requirement that the total mass density is a fixed constant, we have that the following constraint n φα = 1. We express the microforce balances of Fried and Gurtin [8, §IV] in its pointwise form as div ξ α + π α + γ α = 0.
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