Kinematics of motion
We propose a continuum framework to capture the evolution of a multicomponent elastic solid undergoing spinodal decomposition under multiple reversible chemical reactions. In our framework, the deformations induced across the solid boundaries and compositional changes drive the stress generation process. Henceforth, we refer to this mechanism as stress-assisted volume changes. Following the notation proposed by Fried and Gurtin [9], we treat the solid as a continuum body that occupies an open subset B of the Euclidean space \(\mathcal {E}\). A time-dependent deformation field \(\varvec{\chi }: \ \) B\( \ \textsf {x} \ ]0,\text {T} [ \rightarrow \mathcal {B}_{t} \subset \mathcal {E}\) describes the motion from a configuration B onto another configuration \(\mathcal {B}_{t}\). We refer to \(\mathbf{B }\) as the reference configuration and to \(\mathbf{X }\) as the particles in \(\mathbf{B }\). The reference configuration \(\mathbf{B }\) represents an undeformed state of the solid. The deformation field characterizes the kinematics of motion in the body, and after deformation, it assigns to each material particle \(\mathbf{X }\) at a given \(t \in \text {T}\) a spatial particle \(\mathbf{x }\) in the current configuration \(\mathcal {B}_{t}\). Then, we express the deformation field as
$$\begin{aligned} \mathbf{x }{\mathop {=}\limits ^{{\mathrm{def}}}}\varvec{\chi }(\mathbf{X }, t) = \varvec{\chi }_{t} (\mathbf{X }), \end{aligned}$$
(1)
and abusing notation
$$\begin{aligned} \mathcal {B}_{t} = \varvec{\chi }_{t} (\mathbf{B }). \end{aligned}$$
(2)
The deformation field is invertible; namely, there exists an inverse deformation field \(\varvec{\chi }^{-1}: \mathcal {B}_{t} \ \textsf {x} \ T \rightarrow \ \) B\( \ \subset \mathcal {E}\) such that
$$\begin{aligned} \mathbf{x }= \varvec{\chi }_{t} (\varvec{\chi }^{-1} (\mathbf{x }, t)), \end{aligned}$$
(3)
which renders
$$\begin{aligned} \mathbf{X }{\mathop {=}\limits ^{{\mathrm{def}}}}\varvec{\chi }^{-1} (\mathbf{x }, t). \end{aligned}$$
(4)
Measure of strain
In deforming bodies undergoing mass transport and chemical reactions, the particles move relative to each other as a result of external forces and compositional changes. A description of this movement measures the relative displacement of the particles. We use a Lagrangian description of the displacement field \(\mathbf{u }\) which defines the kinematics of the motion, that is,
$$\begin{aligned} \mathbf{u } = \mathbf{x }(\mathbf{X }, t)- \mathbf{X }, \end{aligned}$$
(5)
and the deformation gradient
$$\begin{aligned} \mathbf{F } = \nabla \varvec{\chi }_{t} = \nabla \mathbf{u } + \mathbf{I }, \end{aligned}$$
(6)
where I denotes the second-order identity tensor. To ensure an admissible deformation, that is, a continuum body cannot penetrate itself, the Jacobian of the deformation gradient must fulfill the following constraint
$$\begin{aligned} J {\mathop {=}\limits ^{{\mathrm{def}}}}\text {det} \ \mathbf{F } > 0. \end{aligned}$$
(7)
The velocity of a material particle \(\mathbf{X }\) as a function of the motion is
$$\begin{aligned} \varvec{V}{\mathop {=}\limits ^{{\mathrm{def}}}}\frac{\partial \varvec{\chi }(\mathbf{X }, t)}{\partial t}, \end{aligned}$$
(8)
and its counterpart in the current configuration is
$$\begin{aligned} \varvec{v}{\mathop {=}\limits ^{{\mathrm{def}}}}\frac{\partial \varvec{\chi }(\mathbf{X }, t)}{\partial t} \Big \vert _{\mathbf{X }=\varvec{\chi }^{-1} (\mathbf{x }, t)}. \end{aligned}$$
(9)
Thus, the spatial velocity \(\varvec{v}\) describes a material particle located at \(\mathbf{x }= \varvec{\chi }_{t} (\mathbf{X })\) at time t.
Given the definition of the deformation gradient and the spatial velocity, the right Cauchy-Green stress, is given by
$$\begin{aligned} \mathbf{C } = \mathbf{F }^{\top } \mathbf{F }, \end{aligned}$$
(10)
We apply the change of variables theorem to relate the reference and current configurations an infinitesimal area and volume elements, that is,
$$\begin{aligned} \, \text {d}a= & {} J \mathbf{F }^{-\top } \, \text {d}a_{\text {R}}, \end{aligned}$$
(11)
$$\begin{aligned} \, \text {d}v= & {} J \, \text {d}v_{\text {R}}. \end{aligned}$$
(12)
Fundamental balances
We derive a set of balance equations in the form of partial differential equations that define how the mass, linear and angular momenta, internal energy, and entropy vary in time as the solid-species system endures mechanical and chemical processes. As suggested in [9,10,11,12], three primary fields govern the coupled chemo-mechanical responses of the solid: the deformation \(\varvec{\chi }(\mathbf{X }, t)\), the species concentration \(\varphi ^{\alpha }_{\text {R}}(\mathbf{X }, t)\) per unit of reference volume, and the chemical potential \(\mu ^{\alpha }_{\text {R}}(\mathbf{X }, t)\) per unit of reference volume where \(\alpha \) denotes the \(\alpha \)-th species that composes the solid.
Let \(\mathbf{P } \subset \mathbf{B }\) be an arbitrary control volume in conjunction with its boundary \(\text {S} = \partial \mathbf{P }\); analogously, consider \(\mathcal {P}_{t}\) as a bounded control volume of \(\mathcal {B}_{t}\) such that \(\mathcal {P}_{t} = \varvec{\chi }(\mathbf{P })\) with boundary \(\mathcal {S}= \partial \mathcal {P}_{t}\). According to Cauchy’s theorem, the traction \(\mathbf{t }\) on a surface \(\, \text {d}a\subset \mathcal {S}\) and whose normal \(\varvec{n}\) points outwards is \(\mathbf{t } = \mathbf{T } (\mathbf{x }, t) \varvec{n}\), this traction characterizes the force exerted by the rest of the body \(\mathcal {B}_{t} \setminus \mathcal {P}_{t}\) on \(\mathcal {P}_{t}\) through \( \, \text {d}a\subset \mathcal {S}\) [10, 11], where t depends linearly pointwise on the normal \(\varvec{n}\) through Cauchy’s stress tensor T [16]. Applying (11) to the identity \(\mathbf{t }_{\text {R}} \, \text {d}a_{\text {R}}= \mathbf{t } \, \text {d}a\), we find the force acting on the surface element \(\, \text {d}a\) as a function of the surface element \(\, \text {d}a_{\text {R}}\) [9, 10]. This identity leads to the nominal stress tensor \(\mathbf{T }_{\text {R}}\), that is, the first Piola-Kirchhoff,
$$\begin{aligned} \mathbf{T }_{\text {R}}\mathbf{N } \, \text {d}a_{\text {R}}= \mathbf{T } \varvec{n}\, \text {d}a\ \ \ \ \text {with} \ \ \ \ \mathbf{T }_{\text {R}} = J \mathbf{T } \mathbf{F }^{-\top }. \end{aligned}$$
(13)
As mentioned above, the chemo-mechanical interactions take place through an elastically deforming solid composed by a network and constituent species. Consequently, we formulate balances of mass conservation for both the solid and the constituent species. Thus, we define \(\varphi ^{\alpha }_{\text {R}}\) as the local concentration of the \(\alpha \)-th species per unit of undeformed configuration together with a spatial species outflux \(\varvec{\jmath }^{\alpha }\). In agreement with the balance of mass conservation, the rate of mass change of the \(\alpha \)-th species in the control volume \(\mathbf{P }\) has to be equal to the contribution from the mass supply, typically caused by chemical reactions between the species, and the net mass flux through the boundary \(\mathcal {S}\), that is,
$$\begin{aligned} \dot{\overline{\int _{\mathbf{P }} \varphi ^{\alpha }_{\text {R}}\, \text {d}v_{\text {R}}}} = \int _{\mathbf{P }} s^{\alpha } \ \, \text {d}v_{\text {R}}- \int _{\mathcal {S}} \varvec{\jmath }^{\alpha } \cdot \varvec{n}\, \text {d}a, \end{aligned}$$
(14)
where \(s^{\alpha } \) is the mass supply expressed in the reference configuration. The mass supply is composed of two terms, an external contribution due to external agents and internal contributions caused by chemical reactions. Thereby,
$$\begin{aligned} s^\alpha = s^\alpha _{\text {int}} + s^\alpha _{\text {ext}}. \end{aligned}$$
(15)
Using the divergence theorem, we transform the surface integral of the species flux into a volume integral of the divergence of the species flux as follows
$$\begin{aligned} \dot{\overline{\int _{\mathbf{P }} \varphi ^{\alpha }_{\text {R}}\, \text {d}v_{\text {R}}}} = \int _{\mathbf{P }} s^{\alpha } \ \, \text {d}v_{\text {R}}- \int _{\mathcal {P}_{t}} {\mathrm{div}}\varvec{\jmath }^{\alpha } \, \text {d}v. \end{aligned}$$
(16)
The Lagrangian description of (16) is
$$\begin{aligned} \dot{\overline{\int _{\mathbf{P }} \varphi ^{\alpha }_{\text {R}}\, \text {d}v_{\text {R}}}} = \int _{\mathbf{P }} s^{\alpha } \ \, \text {d}v_{\text {R}}- \int _{\mathbf{P }} {\mathrm{Div}}\varvec{\jmath }^{\alpha }_{\text {R}}\, \text {d}v_{\text {R}}, \end{aligned}$$
(17)
where we use the Piola transform. Thus, the material species flux is then \(\varvec{\jmath }^{\alpha }_{\text {R}}= \mathbf{F }^{-1} (J \varvec{\jmath }^{\alpha })\). Finally, the localized version of (17) is
$$\begin{aligned} \dot{\varphi ^{\alpha }_{\text {R}}} = s^{\alpha } - {\mathrm{Div}}\varvec{\jmath }^{\alpha }_{\text {R}}. \end{aligned}$$
(18)
The concentration of each species is linearly dependent on the other, via the following constraint,
$$\begin{aligned} \sum _{\alpha =1} ^{n} \varphi ^{\alpha }_{\text {R}}= 1, \end{aligned}$$
(19)
which renders
$$\begin{aligned} \sum _{\alpha =1} ^{n} \dot{\varphi ^{\alpha }_{\text {R}}} = 0 \ \ \text {and} \ \ \sum _{\alpha =1} ^{n} \nabla \varphi ^{\alpha }_{\text {R}}= 0, \end{aligned}$$
(20)
where n stands for the total number of species. The mass constraint that (19) expresses must hold when the solid is solely composed of the diffusing species. Herein, we restrict our attention to the case where mass transport by vacancies is not feasible.
Henceforth, a superimposed dot \((\ \dot{} \ )\) stands for the material time derivative, for instance, \(\dot{\varphi ^{\alpha }_{\text {R}}}\) is the material time derivative of the concentration species. Given the conservation of the solid mass, we define \(\rho \) and \(\rho _{0}\) as the solid density in the current and reference configuration, respectively. Then, the balance of solid mass reads
$$\begin{aligned} \int _{\mathcal {P}_{t}} \rho \, \text {d}v = \int _{\mathbf{P }} \rho _{0} \, \text {d}v_{\text {R}}, \end{aligned}$$
(21)
In (21), we convert the volume integral in the current configuration into its counterpart in the reference configuration by employing the relation (12). Finally, we use the localization theorem that leads to the local conservation of solid mass
$$\begin{aligned} \rho _{0}= J\rho . \end{aligned}$$
(22)
Neglecting all inertial effects to focus on quasi-static processes, i.e., we assume the spatial velocity \(\varvec{v}\) is nearly constant through the time, the balance of conservation of linear momentum reads
$$\begin{aligned} \int _{\mathcal {S}} \varvec{t}\, \text {d}a+ \int _{\mathbf{P }} \mathbf{b } \, \text {d}v_{\text {R}} = \mathbf {0}. \end{aligned}$$
(23)
The balance of linear momentum relates forces to changes in the motion of the body. Such balance involves the traction \(\mathbf{t }\) acting on a surface element \(\, \text {d}a\) as well as a body force \(\mathbf{b }\). Conventionally, the body force \(\mathbf{b }\) accounts for forces resulting from gravitational effects. Through the divergence theorem, we express the surface integral in (23) as a volume integral
$$\begin{aligned} \int _{\mathcal {P}_{t}} {\mathrm{div}}\mathbf{T } \, \text {d}v+ \int _{\mathbf{P }} \mathbf{b } \, \text {d}v_{\text {R}}= \mathbf {0}, \end{aligned}$$
(24)
and after some straightforward manipulations in (24), the localized Lagrangian form of the balance of linear momentum is
$$\begin{aligned} {\mathrm{Div}}\mathbf{T }_{\text {R}}+ \mathbf{b } = \mathbf {0}. \end{aligned}$$
(25)
The balance of conservation of angular momentum is
$$\begin{aligned} \int _{\mathcal {P}_{t}} \mathbf{x }\times \varvec{t}\, \text {d}v+ \int _{\mathbf{P }} \mathbf{x }\times \mathbf{b } \, \text {d}v_{\text {R}}= \mathbf{0 }. \end{aligned}$$
(26)
After using the definition of the balance of linear momentum, the divergence theorem, and the localization theorem, this implies that \(\mathbf{T }^{\top } = \mathbf{T }\) . The previous relation implies the symmetry of the Cauchy’s tensor [17, 18]. Finally, the localized Lagrangian form of the balance of angular momenta is
$$\begin{aligned} \text {skw} \mathbf{T }_{\text {R}}\mathbf{F }^{\top } = \mathbf{0 }. \end{aligned}$$
(27)
Following the line of thought introduced by Gurtin and Fried [19,20,21,22], we separate balances of conservation laws from constitutive equations. As a consequence, we include a balance of microforces, that is
$$\begin{aligned} \int _{\mathbf{P }} (\pi ^\alpha + \gamma ^\alpha ) \, \text {d}v_{\text {R}}= - \int _{\mathcal {S}} \varvec{\xi }^{\alpha } \cdot \varvec{n}\, \text {d}a, \end{aligned}$$
(28)
where the vector \(\varvec{\xi }^{\alpha }\) and the scalar \(\pi ^\alpha \) (\(\gamma ^\alpha \)) correspond to the \(\alpha \)-th microstress and \(\alpha \)-th the internal (external) microforce, respectively. In general, the microstresses and microforces are quantities associated with the microscopic configuration of atoms. We express the microforce balances in a Lagrangian form
$$\begin{aligned} \int _{\mathbf{P }} (\pi ^\alpha + \gamma ^\alpha ) \, \text {d}v_{\text {R}}= - \int _{\mathbf{P }} {\mathrm{Div}}\varvec{\xi }^{\alpha }_{\text {R}} \, \text {d}v_{\text {R}}, \end{aligned}$$
(29)
and after applying the locatization theorem, the microforce balances read
$$\begin{aligned} \pi ^\alpha + \gamma ^\alpha = - {\mathrm{Div}}\varvec{\xi }^{\alpha }_{\text {R}}, \end{aligned}$$
(30)
where \(\varvec{\xi }^{\alpha }_{\text {R}} = \mathbf{F }^{-1} (J \varvec{\xi }^{\alpha })\).
Laws of thermodynamics and free-energy inequality
To describe the thermodynamics of this system, we introduce a power expenditure \(\mathcal {W}_{\text {ext}} = \mathcal {W}_{\text {ext}}(\mathbf{P }) + \mathcal {W}_{\text {ext}}(\mathcal {P})\) externally to \(\mathbf{P }\) and \(\mathcal {P}\) done by the external microforce and force on \(\mathbf{P }\), and the microtraction and traction on \(\mathcal {S}\)
$$\begin{aligned} \mathcal {W}_{\text {ext}}(\mathbf{P })&=\sum _{\alpha =1}^{n}\left\{ \int \limits _{\mathbf{P }}\gamma ^\alpha \dot{\varphi }^\alpha _{\text {R}}\, \text {d}v_{\text {R}}\right\} + \int \limits _{\mathbf{P }}\mathbf{b } \cdot \varvec{v}\, \text {d}v_{\text {R}}, \end{aligned}$$
(31a)
$$\begin{aligned} \mathcal {W}_{\text {ext}}(\mathcal {P})&=\sum _{\alpha =1}^{n}\left\{ \int \limits _{\mathcal {S}}\xi _{\scriptscriptstyle \mathcal {S}}^\alpha \dot{\varphi }^\alpha _{\text {R}}\, \text {d}a\right\} + \int \limits _{\mathcal {S}} \varvec{t}\cdot \varvec{v}\, \text {d}a. \end{aligned}$$
(31b)
By neglecting all inertial effects and body forces, we use the first law of thermodynamics to characterize the balance between the rate of internal energy \(\dot{\varepsilon }\) and the expenditure rate of the chemo-mechanical power, caused by external forces, species transport, and chemical reactions. The first law is then,
$$\begin{aligned} \dot{\overline{\int _{\mathbf{P }} \varepsilon \, \text {d}v_{\text {R}}}} = \mathcal {W}_{\text {ext}} -\int _{\mathcal {S}} \varvec{q}\cdot \varvec{n}\ \, \text {d}a+ \int _{\mathbf{P }} r \, \text {d}v_{\text {R}}- \sum _{\alpha =1}^{n} \left\{ \int _{\mathcal {S}} \mu ^{\alpha }_{\text {R}}\varvec{\jmath }^{\alpha } \cdot \varvec{n}\, \text {d}a- \int _{\mathbf{P }} \mu ^{\alpha }_{\text {R}}s^{\alpha }_{\text {ext}} \, \text {d}v_{\text {R}}\right\} . \end{aligned}$$
(32)
There is no contribution of \(s^{\alpha }_{\text {int}}\) to the energy balance (32). The entropy imbalance, in the form of the Clausius–Duhem inequality, states that the rate of growth of the entropy \(\eta \) is at least as large as the entropy flux \(\varvec{q}/ \vartheta \) plus the contribution from the entropy supply \(q/\vartheta \), that is,
$$\begin{aligned} \dot{ \overline{\int _{\mathbf{P }} \eta \, \text {d}v_{\text {R}}}} \ge - \int _{\mathcal {S}} \frac{\varvec{q}\cdot \varvec{n}}{\vartheta } \, \text {d}a+ \int _{\mathbf{P }} \frac{r}{\vartheta } \, \text {d}v_{\text {R}}, \end{aligned}$$
(33)
where \(\varvec{q}\), r, and \(\vartheta \) stand for the spatial heat flux, heat supply, and temperature, respectively. The localized Lagrangian version of (32) and (33) read
$$\begin{aligned} \dot{\varepsilon } = \mathcal {W}_{\text {ext}} - {\mathrm{Div}}\varvec{q}_{\text {R}}+ r - \sum _{\alpha =1}^{n} \left\{ {\mathrm{Div}}\mu ^{\alpha }_{\text {R}}\varvec{\jmath }^{\alpha }_{\text {R}}- \mu ^{\alpha }_{\text {R}}s^{\alpha }_{\text {ext}}\right\} , \end{aligned}$$
(34)
and
$$\begin{aligned} \dot{\eta } \ge - {\mathrm{Div}}\vartheta ^{-1} \varvec{q}_{\text {R}}+ \vartheta ^{-1} r, \end{aligned}$$
(35)
where \(\varvec{q}_{\text {R}}= \mathbf{F }^{-1} (J \varvec{q})\) is the material heat flux. Moreover, \(\mathcal {W}_{\text {ext}}\) is
$$\begin{aligned} \mathcal {W}_{\text {ext}} =\sum _{\alpha =1}^{n} \left\{ ( {\mathrm{Div}}\varvec{\xi }^{\alpha }_{\text {R}} + \gamma ^{\alpha } ) \dot{\varphi }^\alpha _{\text {R}}+ \varvec{\xi }^{\alpha }_{\text {R}} \cdot \nabla \dot{\varphi }^\alpha _{\text {R}}\right\} + ({\mathrm{Div}}\mathbf{T }_{\text {R}}+ \mathbf{b })\cdot \varvec{v}+ \mathbf{T }_{\text {R}}:\dot{\mathbf{F }} . \end{aligned}$$
(36)
Rewriting (34) and (35), and multiplying (35) by \(\vartheta \), we obtain
$$\begin{aligned} \dot{\varepsilon } = \mathcal {W}_{\text {ext}} - {\mathrm{Div}}\varvec{q}_{\text {R}}+ r - \sum _{\alpha =1}^{n} \left\{ \nabla \mu ^{\alpha }_{\text {R}}\cdot \varvec{\jmath }^{\alpha }_{\text {R}}+ \mu ^{\alpha }_{\text {R}}{\mathrm{Div}}\varvec{\jmath }^{\alpha }_{\text {R}}- \mu ^{\alpha }_{\text {R}}s^{\alpha }_{\text {ext}} \right\} , \end{aligned}$$
(37)
and
$$\begin{aligned} \vartheta \dot{\eta } \ge \vartheta ^{-1} \nabla \vartheta \cdot \varvec{q}_{\text {R}}- {\mathrm{Div}}\varvec{q}_{\text {R}}+ r. \end{aligned}$$
(38)
We obtain Helmholtz free energy from applying the Legendre transform to the internal energy while replacing the entropy of the system by the temperature as an independent variable., i.e., \(\dot{\psi } = \dot{\varepsilon } - \dot{\vartheta } \eta - \vartheta \dot{\eta }\). Consequently, we obtain
$$\begin{aligned} \dot{\psi } \le \mathcal {W}_{\text {ext}} - \sum _{\alpha =1}^{n} \left\{ \nabla \mu ^{\alpha }_{\text {R}}\cdot \varvec{\jmath }^{\alpha }_{\text {R}}+ \mu ^{\alpha }_{\text {R}}{\mathrm{Div}}\varvec{\jmath }^{\alpha }_{\text {R}}- \mu ^{\alpha }_{\text {R}}s^{\alpha }_{\text {ext}}\right\} - \vartheta ^{-1} \nabla \vartheta \cdot \varvec{q}_{\text {R}}- \dot{\vartheta }\eta , \end{aligned}$$
(39)
Introducing the balances of both mass conservation and microforces into (39), the free-energy inequality under isothermal conditions is
$$\begin{aligned} \dot{\psi } \le \mathbf{T }_{\text {R}}:\dot{\mathbf{F }} + \sum _{\alpha =1}^{n} \left\{ (\mu ^{\alpha }_{\text {R}}- \pi ^{\alpha }) \dot{\varphi ^{\alpha }_{\text {R}}} + \varvec{\xi }^{\alpha }_{\text {R}} \cdot \nabla \dot{\varphi ^{\alpha }_{\text {R}}} - \varvec{\jmath }^{\alpha }_{\text {R}}\cdot \nabla \mu ^{\alpha }_{\text {R}}- \mu ^{\alpha }_{\text {R}}s^{\alpha }_{\text {int}}\right\} . \end{aligned}$$
(40)
The principle of material frame indifference
Throughout the derivation of the constitutive behavior of the multicomponent solid, we use the Larché–Cahn derivative for both scalar and gradient fields as expressed by [23], together with the mass constraint given by (19). We assume the following constitutive dependence of the free energy \(\psi \)
$$\begin{aligned} \psi = \hat{\psi }(\varvec{\varphi }_{\text {R}}, \nabla \varvec{\varphi }_{\text {R}}, \mathbf{F }) = \hat{\psi }^{ch}(\varvec{\varphi }_{\text {R}}, \nabla \varvec{\varphi }_{\text {R}}) + \hat{\psi }^{el}(\mathbf{{F} }^{e}(\mathbf{F },\varvec{\varphi }_{\text {R}})). \end{aligned}$$
(41)
The objectivity principle requires the constitutive relation (41) to be invariant under a superposed rigid body motion or equivalently, independent of the observer. We can relate two different displacement fields \(\chi \) and \(\chi ^{*}\) as follows
$$\begin{aligned} \chi ^{*} (\mathbf{X }, t) = \mathbf{Q }(t)\chi (\mathbf{X },t) + \mathbf{c }(t), \end{aligned}$$
(42)
where \(\mathbf{Q }(t)\) represents a rotation tensor and \(\mathbf{c }(t)\) the relative translations. Therefore, the transformation of the potential (41) following (42) implies
$$\begin{aligned} \psi = \hat{\psi }(\varvec{\varphi }_{\text {R}}, \nabla \varvec{\varphi }_{\text {R}}, \mathbf{F }) = \overline{\psi }(\varvec{\varphi }_{\text {R}}, \nabla \varvec{\varphi }_{\text {R}}, \mathbf{C }), \end{aligned}$$
(43)
which ensures consistency with the dissipation inequality (40) and the principle of frame-indifference.
Constitutive equations
By using the Coleman-Noll procedure [24], we find a set of constitutive equations as a pair for each kinematic process. We then rewrite (40) following 41 as
$$\begin{aligned} \Bigg (\mathbf{T }_{\text {R}}- \frac{\partial \hat{\psi }}{\partial \mathbf{F }}\Bigg ) :\dot{\mathbf{F }} + \sum _{\alpha =1}^{n} \Bigg (\mu ^{\alpha }_{\text {R}}- \pi ^{\alpha } - \frac{\partial \hat{\psi }}{{\partial \varphi ^{\alpha }_{\text {R}}}} \Bigg ) \dot{\varphi ^{\alpha }_{\text {R}}}+ \sum _{\alpha =1}^{n} \Bigg (\varvec{\xi }^{\alpha }_{\text {R}} - \frac{\partial \hat{\psi }}{\partial {\nabla \varphi ^{\alpha }_{\text {R}}}} \Bigg ) \cdot \nabla \dot{\varphi ^{\alpha }_{\text {R}}} - \sum _{\alpha =1}^{n} \left\{ \varvec{\jmath }^{\alpha }_{\text {R}}\cdot \nabla \mu ^{\alpha }_{\text {R}}+ \mu ^{\alpha }_{\text {R}}s^{\alpha }_{\text {int}}\right\} \ge 0 . \end{aligned}$$
(44)
We seek to enforce (44) for arbitrary values of arbitrary values for \(\dot{\mathbf{F }}\), \(\dot{\varphi }^{\alpha }_{\text {R}}\), \(\nabla \dot{\varphi }^{\alpha }_{\text {R}}\), and \(\nabla \mu ^{\alpha }_{\text {R}}\) at any instant and position.
Following the notation and the definition for the Larché–Cahn derivatives for both scalar and gradient fields as proposed by [23], the relative chemical potential \(\mu ^{\alpha }_{\text {R}\sigma }\) results from the Larché–Cahn derivative as a consequence of incorporating the mass constraint given by (19). According to Larché–Cahn [15], the relative chemical potential expresses the chemical potential of \(\alpha \)-th species measured relative to the chemical potential of \(\sigma \)-th species. This definition entails that, for saturated systems, the mass constraint given by (19) must always hold. Analogously, the relative microforce \(\varvec{\xi }^{\alpha }_{\text {R}\sigma }\) emerges from the constraint imposed in the concentration gradients by (20). As a consequence, we rewrite (44) in the Larché–Cahn sense the following terms: \(\pi ^{\alpha } := \pi ^{\alpha }_{\sigma }\), \(\mu ^{\alpha } := \mu ^{\alpha }_{\text {R} \sigma }\) and \(\varvec{\xi }^{\alpha }_{\text {R}} := \varvec{\xi }^{\alpha }_{\text {R}\sigma }\) as well as the material mass fluxes \(\varvec{\jmath }^{\alpha }_{\text {R}}:= \varvec{\jmath }^{\alpha }_{ \text {R} \sigma }\) as all these quantities are expressed relative to the \(\sigma \)-th reference species. Thus, the free-energy inequality is
$$\begin{aligned} \Bigg (\mathbf{T }_{\text {R}}- \frac{\partial \hat{\psi }}{\partial \mathbf{F }}\Bigg ) :\dot{\mathbf{F }} + \sum _{\alpha =1}^{n} \Bigg (\mu ^{\alpha }_{\text {R} \sigma } - \pi ^{\alpha }_{\sigma } - \frac{\partial ^{(\sigma )} \hat{\psi }}{{\partial \varphi ^{\alpha }_{\text {R}}}} \Bigg ) \dot{\varphi ^{\alpha }_{\text {R}}} + \sum _{\alpha =1}^{n} \Bigg (\varvec{\xi }^{\alpha }_{\text {R}\sigma } - \frac{\partial ^{(\sigma )} \hat{\psi }}{\partial {\nabla \varphi ^{\alpha }_{\text {R}}}} \Bigg ) \cdot \nabla \dot{\varphi ^{\alpha }_{\text {R}}} - \sum _{\alpha =1}^{n} \left\{ \varvec{\jmath }^{\alpha }_{\text {R} \sigma } \cdot \nabla \mu ^{\alpha }_{_{\text {R}}\sigma } +\mu ^{\alpha }_{_{\text {R}}\sigma } s^{\alpha }_{\text {int}}\right\} \ge 0 . \end{aligned}$$
(45)
The latter implies that the following relations must hold to keep consistency with the dissipation imbalance
$$\begin{aligned} \mathbf{T }_{\text {R}}&= \frac{\partial \hat{\psi }}{\partial \mathbf{F }}, \end{aligned}$$
(46a)
$$\begin{aligned} \pi ^{\alpha }_{\sigma }&= \mu ^{\alpha }_{\text {R} \sigma } - \frac{\partial ^{(\sigma )} \hat{\psi }}{{\partial \varphi ^{\alpha }_{\text {R}}}}, \end{aligned}$$
(46b)
$$\begin{aligned} \varvec{\xi }^{\alpha }_{\text {R}\sigma }&= \frac{\partial ^{(\sigma )} \hat{\psi }}{\partial {\nabla \varphi ^{\alpha }_{\text {R}}}}. \end{aligned}$$
(46c)
We use a logarithmic multi-well potential together with a multi-gradient-type potential for the chemical energy, that is,
$$\begin{aligned} \hat{\psi }^{ch}(\varvec{\varphi }_{\text {R}}, \nabla \varvec{\varphi }_{\text {R}}) = N_v k_B \vartheta \left( \sum _{\alpha =1} ^{n} \varphi ^{\alpha }_{\text {R}}\ln \varphi ^{\alpha }_{\text {R}}\right) + N_v \sum _{\alpha =1}^{n} \sum _{\beta =1} ^{n} \Omega ^{\alpha \beta } \varphi ^{\alpha }_{\text {R}}\varphi ^{\beta }_{\text {R}}+ \dfrac{1}{2} \sum _{\alpha =1} ^{n} \sum _{\beta =1} ^{n} \Gamma ^{\alpha \beta } \, \nabla \varphi ^{\alpha }_{\text {R}}\cdot \nabla \varphi ^{\beta }_{\text {R}}. \end{aligned}$$
(47)
This expression corresponds to the extension of the Cahn–Hilliard equation towards multicomponent systems [25, 26]. The Ginzburg–Landau free energy governs the dynamics of the phase separation process undergoing spinodal decomposition. In (47), \(N_{v}\) is the number of molecules per unit volume, \(k_B\) is the Boltzmann constant, and \(\Omega ^{\alpha \beta }\) represents the interaction energy between the mass fraction of the \(\alpha \)-th and \(\beta \)-th species. The interaction between the species \(\alpha \) over \(\beta \) is reciprocal, thus \(\Omega ^{\alpha \beta }\) is symmetric. The interaction energy is positive and is related to the critical temperature for each pair of species, \(\vartheta _{c}^{\alpha \beta }\), (between the \(\alpha \)-th and \(\beta \)-th species). Following standard convention, we adopt that \(\Omega ^{\alpha \beta } = 0\) when \(\alpha = \beta \) and \(\Omega = 2k_{B}\vartheta _{c}^{\alpha \beta }\) when \(\alpha \ne \beta \) [25,26,27]. Furthermore, \(\Gamma ^{\alpha \beta } = \sigma ^{\alpha \beta }\ell ^{\alpha \beta }\) [force] (no summation implied by the repeated indexes) represents the magnitude of the interfacial energy between the \(\alpha \)-th and \(\beta \)-th species. The parameters \(\sigma ^{\alpha \beta }\) and \( l^{\alpha \beta }\) are the interfacial tension [force/length] and the interfacial thicknessFootnote 2 for each pair of species (between the \(\alpha \)-th and \(\beta \)-th species) [length], respectively. In [25], the authors define the force \(\Gamma ^{\alpha \beta }\) as \(N_{v}\Omega ^{\alpha \beta } (\ell ^{\alpha \beta })^{2}\).
Following [11], we assume that the elastic solid behaves as a compressible neo-Hookean material whose elastic energy is given by
$$\begin{aligned} \hat{\psi } ^{el} (\mathbf{F }^{e}) = \dfrac{G}{2} \left[ \mathbf{F }^{e} :\mathbf{F }^{e} - 3 \right] + \dfrac{G}{\beta } \left[ (\text {det} \mathbf{F }^{e})^{-\beta } - 1 \right] , \end{aligned}$$
(48)
where G and \(\beta \) are material parameters that relate to the shear modulus and the weak compressibility of the material. \(\beta \) is a function of the Poisson ratio \(\nu \) such that \(\beta = 2 \nu / 1- 2\nu \). In line with treatments of thermoelasticity, we assume a multiplicative decomposition of the deformation gradient [11], that is,
$$\begin{aligned} \mathbf{F }^{e}&= \mathbf{F }^{\varphi } \mathbf{F }, \end{aligned}$$
(49a)
$$\begin{aligned} \mathbf{F }^{\varphi }&= \Bigg (1 + \sum _{\alpha =1} ^{n} \omega ^{\alpha }(\varphi ^{\alpha }_{\text {R}}- \varphi ^{\alpha }_{\text {{R}} 0} ) \Bigg )^{-\frac{1}{3}} \mathbf{I } , \end{aligned}$$
(49b)
$$\begin{aligned} \mathbf{F }^{\varphi }&= \text {J}^{-\frac{1}{3}}_\varphi \mathbf{I }. \end{aligned}$$
(49c)
This expression suggests that when the local species concentrations change relative to their initial distribution, the solid must undergo elastic deformation. The swelling material parameter \(\omega ^{\alpha }\) is associated with the molar volume of the solute, the volume occupied by a mol of each species scaled by the maximum concentration [10, 11]. More recent works suggest that \(\omega ^{\alpha }\) can be modelled as a dilation tensor or as a function of the overall reaction rate [28, 29].
The evolution of the conserved field \(\varphi ^{\alpha }_{\text {R}}\) obeys a non-Fickian diffusion driven by the chemical potential differences between the species. We combine (46b) and (46c) using the balance of microforces (30) and the constitutive relation for the free energy (41) to express the relative chemical potential of the \(\alpha \)-th species as
$$\begin{aligned} \mu ^{\alpha }_{\text{ R } \sigma } = \frac{\partial ^{(\sigma )} \hat{\psi }}{\partial \varphi ^{\alpha }_{\text{ R }}} - {\mathrm {Div}}\frac{\partial ^{(\sigma )} \hat{\psi }}{\partial \nabla \varphi ^{\alpha }_{\text{ R }}}-(\gamma ^{\alpha }+\gamma ^{\sigma }), \end{aligned}$$
(50)
and therefore,
$$\begin{aligned} \mu ^{\alpha }_{\text{ R } \sigma }&= N_v k_{B} \vartheta \left( \ln \dfrac{\varphi ^\alpha _{\text{ R }}}{\varphi ^\sigma _{\text{ R }}} \right) + 2 N_v \sum _{\beta =1}^{n} (\Omega ^{\alpha \beta } - \Omega ^{\sigma \beta }) \varphi ^\beta _{\text{ R }}\nonumber \\ {}&\quad - \sum _{\beta =1} ^N (\Gamma ^{\alpha \beta } - \Gamma ^{\sigma \beta }) \, {\mathrm {Div}}\nabla \varphi ^\beta _{\text{ R }}- \frac{1}{3} \omega ^{\alpha }_{\sigma } \text{ J}_{\varphi }^{-1} \text{ tr }[\mathbf {T }_{\text{ R }}\mathbf {F }^{\top }]-(\gamma ^{\alpha }+\gamma ^{\sigma }), \end{aligned}$$
(51)
where
$$\begin{aligned} \omega ^{\alpha }_{\sigma } = \omega ^{\alpha } - \omega ^{\sigma }. \end{aligned}$$
(52)
The constitutive relation for the first Piola-Kirchhoff stress tensor is
$$\begin{aligned} \mathbf{T }_{\text {R}}= G \text {J}_{\varphi }^{-1/3}[\mathbf{F }^{e} - (\text {det} \mathbf{F }^{e})^{-\beta } \mathbf{F }^{e-\top }]. \end{aligned}$$
(53)
We also consider the off-diagonal terms in the Onsager reciprocal relations and thus, we describe the species fluxes as
$$\begin{aligned} \varvec{\jmath }^{\alpha }_{\text {R} \sigma } {\mathop {=}\limits ^{{\mathrm{def}}}}- \sum _{\beta =1} ^{n} \varvec{M}^{\alpha \beta } \, J \mathbf{C }^{-1} \nabla \mu ^{\beta }_{\text {R} \sigma }, \end{aligned}$$
(54)
where \(\varvec{M}^{\alpha \beta }\) are the Onsager mobility coefficients. We use the standard assumption that the mobility coefficients depend on the phase composition. In particular, we express this dependency in terms of the concentration of each species. We use the definition \(\varvec{M}^{\alpha \beta } = M^{\alpha \beta } \varphi ^{\alpha }_{\text {R}}(\delta ^{\alpha \beta } - \varphi ^{\beta }_{\text {R}}) \mathbf{I }\) (no summation implied by the repeated indexes) where \(\delta ^{\alpha \beta }\) and \(M^{\alpha \beta }\) are the Kronecker delta of dimension n and the mobility coefficients [26], respectively.