


\section{Introduction}

Dynamic boundary conditions for phase segregation are ubiquitous in mathematical biology, geology and industrial processes. Conversely, spontaneous phase segregation of binary mixtures have been modeled by the Cahn--Hilliard equation \cite{Cah61}. As for the underlying mechanics, Fried \& Gurtin and Gurtin \cite{Fri93,Gur96} proposed the original continuum framework for the study of these types of equations, namely the Allen--Cahn/Ginzburg--Landau and Cahn--Hilliard equations. Additionally, Espath, Calo \& Fried \cite{Esp20} and Espath \& Calo \cite{Esp21a} generalized these ideas to encompass second gradient theories, namely the Swift--Hohenberg/Brazovski\v{\i} and phase-field crystal equations.

As for the dynamic boundary conditions, a continuum mechanical theory has not yet been proposed to the best of our knowledge. Nonetheless, to account for the types of interactions in the presence of solid walls in confined systems while focusing on the early stage of the demixing kinetics, Fischer, Maass, \& Dieterich \cite{Fis97} proposed a set of dynamic boundary conditions for flat walls. These dynamic boundary conditions are characterized by an evolution equation on the boundary coupled with the bulk evolution equation. Many phenomena may fit into this scenario, including polymer mixtures, metallic alloys, and metamorphic rock formation, among other physical and industrial processes.

In this work, we aim at exploring the underlying mechanical principles of the bulk-surface connection for phase-field theories. Our continuum framework is constructed based on the work by Fried \& Gurtin \cite{Fri93,Gur96}, Espath \& Calo \cite{Esp21a}, and Espath \cite{Esp21b,Esp21c} to generalize the models proposed by Fischer, Maass, \& Dieterich \cite{Fis97}, Goldstein, Miranville \& Schimperna \cite{Gol11}, and Knopf, Lam, Liu \& Metzger \cite{Kno21a}. To this end, we provide a generalized version of the principle of virtual powers 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. Finally, through this version of the partwise free-energy imbalance, we propose the constitutive relations and a set of thermodynamically consistent boundary conditions. Lastly, this continuum framework has two different bulk-surface types of couplings, one through the principle of virtual powers and another through the species balance (and consequently through the free-energy imbalance).

The remainder of this work is organized as follows. In section \S\ref{sc:principle.virtual.powers}, we provide a generalized version of virtual powers for bulk-surface dynamics, derive the surface and edge microtractions along with the field equations. In section \S\ref{sc:conserved.species}, we postulate the species balance of the bulk-surface system. In \S\ref{sc:free.energy.imbalance}, we postulate a generalized version of the partwise free-energy imbalance. In \S\ref{sc:constitutive.relations}, we provide suitable constitutive relations along with a set of thermodynamically consistent boundary conditions.

\subsection{Synopsis of purely variational models}

The model proposed in \cite{Fis97,Gol11,Kno21a} on a body $\cl{P}$ with boundary $\partial\cl{P}$ for the underlying free-energy functional
\begin{equation}\label{eq:free.energy.functional}
\Psi = \int\limits_{\prt} \Big( \fr{1}{\epsilon} f(\varphi_{\scriptscriptstyle\cl{P}}) + \fr{\epsilon}{2} |\mathrm{grad}\mskip2mu\varphi_{\scriptscriptstyle\cl{P}}|^2 \Big) \,\mathrm{d}v + \int\limits_{\dprt} \Big( \fr{1}{\delta} f(\varphi_{\scriptscriptstyle\partial\cl{P}}) + \fr{\iota \delta}{2} |\Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\varphi_{\scriptscriptstyle\partial\cl{P}}|^2 \Big) \,\mathrm{d}a,
\end{equation}
reads
\begin{equation}\label{eq:original}
\left\{
\begin{aligned}
\dot{\varphi}_{\scriptscriptstyle\cl{P}} &= m_{\mskip-2mu\scriptscriptstyle\cl{P}}\triangle\mu_{\scriptscriptstyle\cl{P}}, && \text{in } \cl{P}, \\[4pt]
\mu_{\scriptscriptstyle\cl{P}} &= -\epsilon \triangle\varphi_{\scriptscriptstyle\cl{P}} + \fr{1}{\epsilon} f^\prime (\varphi_{\scriptscriptstyle\cl{P}}), && \text{in } \cl{P}, \\[4pt]
\dot{\varphi}_{\scriptscriptstyle\partial\cl{P}} &= m_{\mskip-2mu\scriptscriptstyle\partial\cl{P}}\triangle_{\mskip-2mu\scriptscriptstyle\cl{S}}\mu_{\scriptscriptstyle\partial\cl{P}} - \beta m_{\mskip-2mu\scriptscriptstyle\cl{P}} \partial_n\mu_{\scriptscriptstyle\cl{P}}, && \text{on } \partial\cl{P}, \\[4pt]
\mu_{\scriptscriptstyle\partial\cl{P}} &= - \iota \delta \triangle_{\mskip-2mu\scriptscriptstyle\cl{S}}\varphi_{\scriptscriptstyle\partial\cl{P}} + \fr{1}{\delta} g^\prime (\varphi_{\scriptscriptstyle\partial\cl{P}}) + \epsilon \partial_n \varphi_{\scriptscriptstyle\cl{P}}, && \text{on } \partial\cl{P}, \\[4pt]
\varphi_{\scriptscriptstyle\cl{P}} &= \varphi_{\scriptscriptstyle\partial\cl{P}}, && \text{on } \partial\cl{P}, \\[4pt]
\partial_n \mu_{\scriptscriptstyle\cl{P}} &= \fr{1}{L} (\beta \mu_{\scriptscriptstyle\partial\cl{P}} - \mu_{\scriptscriptstyle\cl{P}}), && \text{on } \partial\cl{P}.
\end{aligned}
\right.
\end{equation}
Here, the superposed dot represents the time derivative, and $\triangle$ and $\triangle_{\mskip-2mu\scriptscriptstyle\cl{S}}$ are the Laplace and Laplace--Beltrami operators, respectively. $\varphi_{\scriptscriptstyle\cl{P}}$ and $\varphi_{\scriptscriptstyle\partial\cl{P}}$ are the bulk and surface conserved phase fields, $\mu_{\scriptscriptstyle\cl{P}}$ and $\mu_{\scriptscriptstyle\partial\cl{P}}$ are the bulk and surface chemical potentials, $f$ and $g$ are the bulk and surface potentials, and $m_{\mskip-2mu\scriptscriptstyle\cl{P}}$ and $m_{\mskip-2mu\scriptscriptstyle\partial\cl{P}}$ are the bulk and surface mobility coefficients. Lastly, $\epsilon$, $\delta$, $\iota$, $\beta$ and $L$ are real positive parameters. It is important to what follows to note that \eqref{eq:original} has the Cahn--Hilliard type of structure for both the bulk and surface.

\subsection{Synopsis of this work}

In Figure \ref{fg:geometry}, $\cl{B}$ denotes a region of a three-dimensional point space $\cl{E}$ where $\cl{P}\subseteq\cl{B}$ is an arbitrary subregion of $\cl{B}$ with a closed surface boundary $\partial\cl{P}$ oriented by an outward unit normal $\bs{n}$ at $\bs{x}\in\partial\cl{P}$. The surface $\partial\cl{P}$ may lose smoothness along a curve, namely an edge $\partial^2\cl{P}$. In a neighborhood of an edge $\partial^2\cl{P}$, two smooth surfaces $\partial\cl{P}^{\pm}$ are defined. The limiting unit normals of $\partial\cl{P}^{\pm}$ at $\partial^2\cl{P}$ are denoted by the pair $\{\bs{n}^+,\bs{n}^-\}$. The pair of unit normals characterizes the edge $\partial^2\cl{P}$. Similarly, the limiting outward unit tangent-normal of $\partial\cl{P}^{\pm}$ at $\partial^2\cl{P}$ are $\{\bs{\nu}^+,\bs{\nu}^-\}$. Additionally, $\partial^2\cl{P}$ is oriented by the unit tangent $\bs{\sigma}\coloneqq\bs{\sigma}^+$ such that $\bs{\sigma}^+\coloneqq\bs{n}^+\times\bs{\nu}^+$. Furthermore, the body $\cl{B}$ and all its parts are open sets in $\cl{E}$.
\begin{figure}[h]
  \centering
  \includegraphics[width=0.65\textwidth]{geometry.pdf}
  \caption{Part $\cl{P}$ with nonsmooth boundary surface $\partial\cl{P}^\pm$ oriented by the unit normal $\bs{n}$ with the outward unit tangent-normal $\bs{\nu}^\pm$ at the smooth boundary-edge $\partial^2\cl{P}$ oriented by the unit tangent $\bs{\sigma}\coloneqq\bs{n}\times\bs{\nu}$. The surface $\partial\cl{P}$ lacks smoothness at an edge $\partial^2\cl{P}$.}
  \label{fg:geometry}
\end{figure}

In this work, we propose a continuum theory with two kinematical processes, a bulk $\varphi_{\scriptscriptstyle\cl{P}}$ and a surface $\varphi_{\scriptscriptstyle\partial\cl{P}}$ fields on $\cl{P}$ and $\partial\cl{P}$, respectively. Within this framework, each of
\begin{equation}
\begin{gathered}
\int\limits_{\prt} \gamma \dot{\varphi}_{\scriptscriptstyle\cl{P}} \,\mathrm{d}v, \qquad \int\limits_{\dprt} \xi_{\scriptscriptstyle\cl{S}} \dot{\varphi}_{\scriptscriptstyle\cl{P}} \,\mathrm{d}a, \\[4pt]
\int\limits_{\dprt} \zeta \dot{\varphi}_{\scriptscriptstyle\partial\cl{P}} \,\mathrm{d}a, \qquad \int\limits_{\dprt} \xi_{\scriptscriptstyle\cl{S}} \dot{\varphi}_{\scriptscriptstyle\partial\cl{P}} \,\mathrm{d}a, \qquad \text{and} \qquad \int\limits_{\ddprt} \tau_{\scriptscriptstyle\partial\cl{S}} \dot{\varphi}_{\scriptscriptstyle\partial\cl{P}} \,\mathrm{d}\sigma,
\end{gathered}
\end{equation}
represents an external form of power expenditure, where $\gamma$ is the external bulk microforce, $\xi_{\scriptscriptstyle\cl{S}}$ is the surface microtraction, $\zeta$ is the external surface microforce, and $\tau_{\scriptscriptstyle\partial\cl{S}}$ is the edge microtraction. These power expenditures may be described as follows.
\begin{itemize}
  \item $\gamma \dot{\varphi}_{\scriptscriptstyle\cl{P}}$ represents the power expended on the atoms of $\cl{P}$ by sources external to the body $\cl{P}$; \\[4pt]
  \item $\xi_{\scriptscriptstyle\cl{S}} \dot{\varphi}_{\scriptscriptstyle\cl{P}}$ represents the power expended across $\partial\cl{P}$ by configurations neighboring the boundary of the body $\partial\cl{P}$ and exterior to $\cl{P}$; \\[4pt]
  \item $\zeta \dot{\varphi}_{\scriptscriptstyle\partial\cl{P}}$ represents the power expended on the atoms of $\partial\cl{P}$ by sources external to the boundary of the body $\partial\cl{P}$ and not originated from $\cl{P}$; \\[4pt]
  \item $\xi_{\scriptscriptstyle\cl{S}} \dot{\varphi}_{\scriptscriptstyle\partial\cl{P}}$ represents the power expended on the atoms of $\partial\cl{P}$ by sources external to the boundary of the body and originated from $\cl{P}$; \\[4pt]
  \item $\tau_{\scriptscriptstyle\partial\cl{S}} \dot{\varphi}_{\scriptscriptstyle\partial\cl{P}}$ represents the power expended across $\partial^2\cl{P}$ by configurations neighboring the boundary of the boundary of the body $\partial^2\cl{P}$ and exterior to $\partial\cl{P}$.
\end{itemize}
Conversely, the internal power expenditure is given by the contribution of the following terms
\begin{equation}
\begin{gathered}
\int\limits_{\prt} \bs{\xi} \cdot \mathrm{grad}\mskip2mu \dot{\varphi}_{\scriptscriptstyle\cl{P}} \,\mathrm{d}v, \qquad - \int\limits_{\prt} \pi \dot{\varphi}_{\scriptscriptstyle\cl{P}} \,\mathrm{d}v, \\[4pt]
\int\limits_{\dprt} \bs{\tau} \cdot \Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\dot{\varphi}_{\scriptscriptstyle\partial\cl{P}} \,\mathrm{d}a, \qquad \text{and} \qquad - \int\limits_{\dprt} \varpi \dot{\varphi}_{\scriptscriptstyle\partial\cl{P}} \,\mathrm{d}a,
\end{gathered}
\end{equation}
where $\bs{\xi}$ is the bulk microstress, $\pi$ is the internal bulk microforce, $\bs{\tau}$ is the surface microstress, and $\varpi$ is the internal surface microforce. Through a suitable principle of virtual powers, we arrive at the microtractions presented in the external power, $\xi_{\scriptscriptstyle\cl{S}}$ and $\tau_{\scriptscriptstyle\partial\cl{S}}$, and the field equations.

Next, given the bulk and surface species fluxes, $\bs{\jmath}_{\scriptscriptstyle\cl{P}}$ and $\bs{\jmath}_{\scriptscriptstyle\partial\cl{P}}$, and the bulk and surface external rates of species production, $s_{\scriptscriptstyle\cl{P}}$ and $s_{\scriptscriptstyle\partial\cl{P}}$, we postulate the partwise species balances for $\cl{P}$ and $\partial\cl{P}$, where the balance on $\partial\cl{P}$ is supplemented by a contribution originated from $\cl{P}$ and given by
\begin{equation}
- \int\limits_{\dprt} \beta \mskip3mu \bs{\jmath}_{\scriptscriptstyle\cl{P}} \cdot \bs{n} \,\mathrm{d}a.
\end{equation}
Then, with a suitable free-energy imbalance, we account for the rate at which energy is transferred to $\cl{P}$ and $\partial\cl{P}$ due to species transport to determine the constitutive relations and arrive at the following set of equations
\begin{equation}\label{eq:intro.governing}
\left\{
\begin{aligned}
\dot{\varphi}_{\scriptscriptstyle\cl{P}} &= s_{\scriptscriptstyle\cl{P}} - \mathrm{div}\mskip2mu \bs{\jmath}_{\scriptscriptstyle\cl{P}}, && \text{in } \cl{P}, \\[4pt]
\mu_{\scriptscriptstyle\cl{P}} &= - \mathrm{div}\mskip2mu\bs{\xi} - \gamma + \partial_{\vphiP} \psi_{\scriptscriptstyle\cl{P}}, && \text{in } \cl{P}, \\[4pt]
\dot{\varphi}_{\scriptscriptstyle\partial\cl{P}} &= \beta \mskip3mu \bs{\jmath}_{\scriptscriptstyle\cl{P}} \cdot \bs{n} + s_{\scriptscriptstyle\partial\cl{P}} - \Div_{\mskip-6mu\scriptscriptstyle\cl{S}} \bs{\jmath}_{\scriptscriptstyle\partial\cl{P}} + 2K \bs{\jmath}_{\scriptscriptstyle\partial\cl{P}} \cdot \bs{n}, && \text{on } \partial\cl{P}, \\[4pt]
\mu_{\scriptscriptstyle\partial\cl{P}} &= - \Div_{\mskip-6mu\scriptscriptstyle\cl{S}}\bs{\tau} + 2K \bs{\tau} \cdot \bs{n} - \zeta - \bs{\xi} \cdot \bs{n} + \partial_{\vphidP} \psi_{\scriptscriptstyle\partial\cl{P}}, && \text{on } \partial\cl{P},
\end{aligned}
\right.
\end{equation}
and
\begin{equation}\label{eq:intro.constitutive}
\left\{
\begin{aligned}
\bs{\xi} &= \partial_{\Grad\vphiP} \psi_{\scriptscriptstyle\cl{P}}, && \text{in } \cl{P}, \\[4pt]
\bs{\jmath}_{\scriptscriptstyle\cl{P}} &= - \bs{M}_{\mskip-4mu\scriptscriptstyle\cl{P}} \mskip3mu \mathrm{grad}\mskip2mu\mu_{\scriptscriptstyle\cl{P}}, && \text{in } \cl{P}, \\[4pt]
\bs{\tau} &= \partial_{\Grads\vphidP} \psi_{\scriptscriptstyle\partial\cl{P}}, && \text{on } \partial\cl{P}, \\[4pt]
\bs{\jmath}_{\scriptscriptstyle\partial\cl{P}} &= -\bs{M}_{\mskip-4mu\scriptscriptstyle\partial\cl{P}} \mskip3mu \Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\mu_{\scriptscriptstyle\partial\cl{P}}, && \text{on } \partial\cl{P},
\end{aligned}
\right.
\end{equation}
where $K \coloneqq - \fr{1}{2} \Div_{\mskip-6mu\scriptscriptstyle\cl{S}} \bs{n}$ is the mean curvature, $\partial_{\vphiP} \coloneqq \partial/\partial\varphi_{\scriptscriptstyle\cl{P}}$, $\partial_{\vphidP} \coloneqq \partial/\partial\varphi_{\scriptscriptstyle\partial\cl{P}}$, $\partial_{\Grad\vphiP} \coloneqq \partial/\partial(\mathrm{grad}\mskip2mu\varphi_{\scriptscriptstyle\cl{P}})$, and $\partial_{\Grads\vphidP} \coloneqq \partial/\partial(\Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\varphi_{\scriptscriptstyle\partial\cl{P}})$.

Also, note that the appearance of bulk microstress $\bs{\xi}$ in the surface chemical potential $\mu_{\scriptscriptstyle\partial\cl{P}}$, in Equation \eqref{eq:intro.governing}$_4$, results from the coupling at the principle of virtual powers' level whereas the presence of the bulk species flux $\bs{\jmath}_{\scriptscriptstyle\cl{P}}$ in the surface species time derivative $\dot{\varphi}_{\scriptscriptstyle\partial\cl{P}}$, in Equation \eqref{eq:intro.governing}$_3$, results from the coupling at the surface species balance's level.

Aside from the fact that we present a new version of the principle of virtual powers and free-energy imbalance, there are three key differences between our continuum framework and previous works on dynamic boundary conditions. First, our theory is based on underlying mechanical principles. Second, our theory generalizes the resulting equations in \cite{Gol11,Kno21a}. Third, we consider that the boundary $\partial\cl{P}$ may be endowed with a discontinuous normal field, allowing the assignment of edge microtractions.

\subsection{Differential tools}

In this subsection, we present helpful mathematical tools, from \cite{Esp21a,Esp21b}, to be used in the remainder of this study. We derive the relevant differential relations on a body $\cl{B}$ and a surface $\cl{S}$.

Let $\cl{P}$ be an arbitrary part embedded in a region $\cl{B}$ of a three-dimensional point space $\cl{E}$. With the coordinates $\tau^i$ $(i=1,2,3)$, the $i$th contravariant basis $\bs{g}^i$, and the conventional partial derivative $\partial_i\coloneqq\partial/\partial\tau^i$, let $\kappa$ and $\bs{\kappa}$ be, respectively, a smooth and a vector fields on $\cl{B}$. Then, within this setting, the gradient differential operator is defined as
\begin{equation}\label{eq:gradient.scalar}
\mathrm{grad}\mskip2mu\kappa \coloneqq \partial_i\kappa\mskip3mu\bs{g}^i.
\end{equation}

Next, consider a smooth surface $\cl{S}\subset\cl{P}$ oriented by the unit normal $\bs{n}$ at $\bs{x}\in\cl{S}$. Let $\cl{S}$ be parameterized by coordinates $\tau^p$ with $p=1,2$ and $\bs{z}$ be a smooth extension of $\cl{S}$ along its normal $\bs{n}$ at $\bs{x}$ such that
\begin{equation}\label{eq:parameterization.surface.extension}
\bs{z}(\bs{x},\tau)\coloneqq\bs{x}+\tau\bs{n}(\bs{x}),\qquad\forall\,\bs{x}\in\cl{S},
\end{equation}
with $\tau$ representing the normal coordinate $n$ and taking values in an open interval of zero so that there exists a one-to-one mapping $\bs{z}\leftrightarrow(\bs{x},\tau)$. Such parameterization induces the local covariant basis
\begin{equation}\label{eq:covariant.basis}
\bs{g}_p\coloneqq\partial_p\bs{z}=\partial_p\bs{x}+\tau\partial_p\bs{n},\qquad\text{and}\qquad\bs{g}_n\coloneqq\partial_3\bs{z}=\bs{n}.
\end{equation}
With expression \eqref{eq:covariant.basis} at $\bs{x}=\bs{z}(\bs{x},0)$, we define
\begin{equation}
\bs{e}^p\coloneqq\bs{g}^p|_{\tau=0}.
\end{equation}
Thus,
\begin{equation}
\bs{e}_p=\partial_p\bs{x}.
\end{equation}
Furthermore, the contravariant $\bs{g}^p$ and covariant $\bs{g}_q$ bases satisfy
\begin{equation}\label{eq:covariant.contravariant.basis}
\bs{g}^p\cdot\bs{g}_q=\delta^{p}_{\cdot q}.
\end{equation}

Bearing in mind the parameterization \eqref{eq:parameterization.surface.extension}, consider the differential operators as follows. The gradient definition given in expression \eqref{eq:gradient.scalar} takes the form
\begin{equation}\label{eq:gradient.scalar.surface.extension}
\mathrm{grad}\mskip2mu\kappa=\partial_n\kappa\mskip3mu\bs{n}+\partial_p\kappa\mskip3mu\bs{e}^p.
\end{equation}
Next, let $\Prj{\bs{n}}\coloneqq\Prj{\bs{n}}(\bs{n})$ be the projector onto the plane defined by $\bs{n}$ at $\bs{x}\in\cl{S}$ such that
\begin{equation}\label{eq:tan.projector}
\Prj{\bs{n}}\coloneqq\bs{1}-\bs{n}\otimes\bs{n}=\Prj{\bs{n}}^{\scriptscriptstyle\mskip-1mu\top\mskip-2mu}.
\end{equation}
In view of the expression \eqref{eq:gradient.scalar.surface.extension} along with \eqref{eq:tan.projector}, the surface gradient is given by
\begin{equation}\label{eq:surface.gradient}
\Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\kappa \coloneqq \partial_p\kappa\mskip3mu\bs{e}^p=\Prj{\bs{n}} \mathrm{grad}\mskip2mu\kappa,
\end{equation}
and the surface divergence by
\begin{equation}\label{eq:surface.divergence}
\Div_{\mskip-6mu\scriptscriptstyle\cl{S}}\bs{\kappa}=\partial_p\bs{\kappa}\cdot\bs{e}^p=\mathrm{grad}\mskip2mu\bs{\kappa}\mskip+2mu\colon\mskip-2mu\Prj{\bs{n}}.
\end{equation}
Then, the Laplace--Beltrami operator may be written as
\begin{equation}
\triangle_{\mskip-2mu\scriptscriptstyle\cl{S}}\kappa \coloneqq \Div_{\mskip-6mu\scriptscriptstyle\cl{S}}\Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\kappa = \mathrm{grad}\mskip2mu(\Prj{\bs{n}}\mathrm{grad}\mskip2mu\kappa)\mskip+2mu\colon\mskip-2mu\Prj{\bs{n}}.
\end{equation}

Lastly, for any smooth vector field $\bs{\kappa}$ on a smooth closed oriented surface $\cl{S}$, the surface divergence theorem states that
\begin{equation}\label{eq:smooth.divs.theo.closed.S}
\int\limits_{\cl{S}}\Div_{\mskip-6mu\scriptscriptstyle\cl{S}}(\Prj{\bs{n}} \bs{\kappa})\,\mathrm{d}a = 0,
\end{equation}
whereas, owing to the lack of smoothness at an edge $\cl{C}$, on a nonsmooth closed oriented surface $\cl{S}$ with limiting outward unit tangent-normals $\bs{\nu}^+$ and $\bs{\nu}^-$ at $\cl{C}$, the surface divergence theorem exhibits a \emph{surplus}, that is,
\begin{equation}\label{eq:nonsmooth.divs.theo.closed.S}
\int\limits_{\cl{S}}\Div_{\mskip-6mu\scriptscriptstyle\cl{S}}(\Prj{\bs{n}} \bs{\kappa})\,\mathrm{d}a=\int\limits_{\cl{C}}\surp{\bs{\kappa}\cdot\bs{\nu}}\,\mathrm{d}\sigma,
\end{equation}
where $\surp{\bs{\kappa}\cdot\bs{\nu}}\coloneqq\bs{\kappa}\cdot\bs{\nu}^++\bs{\kappa}\cdot\bs{\nu}^-$. Conversely, for open nonsmooth surfaces, we have to consider an extension of the surface divergence theorem \eqref{eq:nonsmooth.divs.theo.closed.S}, that is,
\begin{equation}\label{eq:nonsmooth.divs.theo.open.S}
\int\limits_{\cl{S}}\Div_{\mskip-6mu\scriptscriptstyle\cl{S}}(\Prj{\bs{n}} \bs{\kappa})\,\mathrm{d}a=\int\limits_{\partial\cl{S}}\bs{\kappa}\cdot\bs{\nu}\,\mathrm{d}\sigma+\int\limits_{\cl{C}}\surp{\bs{\kappa}\cdot\bs{\nu}}\,\mathrm{d}\sigma.
\end{equation}

\section{Virtual power principle}
\label{sc:principle.virtual.powers}

We are now in a position to postulate the principle of virtual powers. Considering the power expenditures discussed in the previous section, the principle reads
\begin{equation}\label{eq:virtual.power.principle}
\cl{V}_{\mathrm{ext}}(\cl{P},\partial\cl{P};\chi_{\scriptscriptstyle\cl{P}},\chi_{\scriptscriptstyle\partial\cl{P}}) = \cl{V}_{\mathrm{int}}(\cl{P},\partial\cl{P};\chi_{\scriptscriptstyle\cl{P}},\chi_{\scriptscriptstyle\partial\cl{P}}),
\end{equation}
where $\chi_{\scriptscriptstyle\cl{P}}$ and $\chi_{\scriptscriptstyle\partial\cl{P}}$ are two sufficiently smooth virtual fields defined, respectively, on $\cl{P}$ and $\partial\cl{P}$. The external and internal virtual power are, respectively, given by
\begin{equation}\label{eq:external.virtual.power}
\cl{V}_{\mathrm{ext}}(\cl{P},\partial\cl{P};\chi_{\scriptscriptstyle\cl{P}},\chi_{\scriptscriptstyle\partial\cl{P}}) = \int\limits_{\prt} \gamma \chi_{\scriptscriptstyle\cl{P}} \,\mathrm{d}v + \int\limits_{\dprt} (\zeta + \xi_{\scriptscriptstyle\cl{S}}) \chi_{\scriptscriptstyle\partial\cl{P}} \,\mathrm{d}a + \int\limits_{\dprt} \xi_{\scriptscriptstyle\cl{S}} \chi_{\scriptscriptstyle\cl{P}} \,\mathrm{d}a + \int\limits_{\ddprt} \tau_{\scriptscriptstyle\partial\cl{S}} \chi_{\scriptscriptstyle\partial\cl{P}} \,\mathrm{d}\sigma,
\end{equation}
and
\begin{equation}\label{eq:internal.virtual.power}
\cl{V}_{\mathrm{int}}(\cl{P},\partial\cl{P};\chi_{\scriptscriptstyle\cl{P}},\chi_{\scriptscriptstyle\partial\cl{P}}) = \int\limits_{\prt} \bs{\xi} \cdot \mathrm{grad}\mskip2mu \chi_{\scriptscriptstyle\cl{P}} \,\mathrm{d}v - \int\limits_{\prt} \pi \chi_{\scriptscriptstyle\cl{P}} \,\mathrm{d}v + \int\limits_{\dprt} \bs{\tau} \cdot \Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\chi_{\scriptscriptstyle\partial\cl{P}} \,\mathrm{d}a - \int\limits_{\dprt} \varpi \chi_{\scriptscriptstyle\partial\cl{P}} \,\mathrm{d}a.
\end{equation}
Next, we aim at deriving the explicit forms of the surface microtraction $\xi_{\scriptscriptstyle\cl{S}}$ and the edge microtraction $\tau_{\scriptscriptstyle\partial\cl{S}}$. Noting that $\bs{\tau} \cdot \Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\chi_{\scriptscriptstyle\partial\cl{P}} = \Prj{\bs{n}} \bs{\tau} \cdot \Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\chi_{\scriptscriptstyle\partial\cl{P}}$ while combining \eqref{eq:external.virtual.power} and \eqref{eq:internal.virtual.power} through \eqref{eq:virtual.power.principle} along with the divergence theorem and the surface divergence theorem for nonsmooth closed surfaces \eqref{eq:nonsmooth.divs.theo.closed.S}, we are led to
\begin{multline}
\int\limits_{\prt} \chi_{\scriptscriptstyle\cl{P}} (\mathrm{div}\mskip2mu\bs{\xi} + \pi + \gamma) \,\mathrm{d}v + \int\limits_{\dprt} \chi_{\scriptscriptstyle\cl{P}} (\xi_{\scriptscriptstyle\cl{S}} - \bs{\xi} \cdot \bs{n}) \,\mathrm{d}a \\[4pt]
+ \int\limits_{\dprt} \chi_{\scriptscriptstyle\partial\cl{P}} (\Div_{\mskip-6mu\scriptscriptstyle\cl{S}}(\Prj{\bs{n}}\bs{\tau}) + \varpi + \zeta + \xi_{\scriptscriptstyle\cl{S}}) \,\mathrm{d}a + \int\limits_{\ddprt} \chi_{\scriptscriptstyle\partial\cl{P}} (\tau_{\scriptscriptstyle\partial\cl{S}} - \surp{\bs{\tau} \cdot \bs{\nu}}) \,\mathrm{d}\sigma = 0.
\end{multline}
Then, by variational arguments, the microtractions read
\begin{equation}\label{eq:microtractions}
\xi_{\scriptscriptstyle\cl{S}} = \bs{\xi} \cdot \bs{n}, \qquad \text{and} \qquad \tau_{\scriptscriptstyle\partial\cl{S}} = \surp{\bs{\tau} \cdot \bs{\nu}},
\end{equation}
while the bulk and surface field equations are given by
\begin{equation}\label{eq:field.equations}
\mathrm{div}\mskip2mu\bs{\xi} + \pi + \gamma = 0, \qquad \text{and} \qquad \Div_{\mskip-6mu\scriptscriptstyle\cl{S}}(\Prj{\bs{n}}\bs{\tau}) + \varpi + \zeta + \xi_{\scriptscriptstyle\cl{S}} = 0.
\end{equation}
Note that the bulk microforce balance \eqref{eq:field.equations}$_1$ has the standard form proposed by Fried \& Gurtin \cite{Fri93}. However, the surface microforce balance \eqref{eq:field.equations}$_2$ has a contribution from the bulk, namely $\xi_{\scriptscriptstyle\cl{S}}$. Additionally, the term $\Div_{\mskip-6mu\scriptscriptstyle\cl{S}} (\Prj{\bs{n}}\bs{\tau})$ may be split as $\Div_{\mskip-6mu\scriptscriptstyle\cl{S}} (\Prj{\bs{n}}\bs{\tau}) = \Div_{\mskip-6mu\scriptscriptstyle\cl{S}} \bs{\tau} - 2K \bs{\tau} \cdot \bs{n}$. Then, the surface microforce balance \eqref{eq:field.equations}$_2$ may be written as
\begin{equation}\label{eq:surface.field.equation}
\Div_{\mskip-6mu\scriptscriptstyle\cl{S}}\bs{\tau} - 2K \bs{\tau} \cdot \bs{n} + \varpi + \zeta + \xi_{\scriptscriptstyle\cl{S}} = 0,
\end{equation}
for each smooth part of $\partial\cl{P}$.

\section{Conserved species}
\label{sc:conserved.species}

We now account for the case where the bulk and surface phase fields, $\varphi_{\scriptscriptstyle\cl{P}}$ and $\varphi_{\scriptscriptstyle\partial\cl{P}}$, represent the concentration of a conserved species. We therefore supplement the field equations \eqref{eq:field.equations} by two partwise species balances, that is, the bulk species balance
\begin{equation}
\dot{\overline{\int\limits_{\prt}\varphi_{\scriptscriptstyle\cl{P}}\,\mathrm{d}v}} = \int\limits_{\prt} s_{\scriptscriptstyle\cl{P}} \,\mathrm{d}v - \int\limits_{\dprt} \bs{\jmath}_{\scriptscriptstyle\cl{P}} \cdot \bs{n} \,\mathrm{d}a,
\end{equation}
and the surface species balance
\begin{equation}
\dot{\overline{\int\limits_{\dprt}\varphi_{\scriptscriptstyle\partial\cl{P}}\,\mathrm{d}a}} = - \int\limits_{\dprt} \beta \mskip3mu \bs{\jmath}_{\scriptscriptstyle\cl{P}} \cdot \bs{n} \,\mathrm{d}a + \int\limits_{\dprt} s_{\scriptscriptstyle\partial\cl{P}} \,\mathrm{d}a - \int\limits_{\ddprt} \surp{\bs{\jmath}_{\scriptscriptstyle\partial\cl{P}} \cdot \bs{\nu}} \,\mathrm{d}\sigma.
\end{equation}

Using the divergence theorem and the surface divergence theorem for nonsmooth closed surfaces \eqref{eq:nonsmooth.divs.theo.closed.S} followed by localization, these partwise species balances lead us to
\begin{equation}\label{eq:pointwise.bulk.species.balance}
\dot{\varphi}_{\scriptscriptstyle\cl{P}} = s_{\scriptscriptstyle\cl{P}} - \mathrm{div}\mskip2mu \bs{\jmath}_{\scriptscriptstyle\cl{P}},
\end{equation}
and
\begin{equation}\label{eq:pointwise.surface.species.balance}
\dot{\varphi}_{\scriptscriptstyle\partial\cl{P}} = - \beta \mskip3mu \bs{\jmath}_{\scriptscriptstyle\cl{P}} \cdot \bs{n} + s_{\scriptscriptstyle\partial\cl{P}} - \Div_{\mskip-6mu\scriptscriptstyle\cl{S}}(\Prj{\bs{n}}\bs{\jmath}_{\scriptscriptstyle\partial\cl{P}}).
\end{equation}
Note that the bulk species balance has a standard form. However, the surface species balance has a contribution from the bulk, namely $- \beta \mskip3mu \bs{\jmath}_{\scriptscriptstyle\cl{P}} \cdot \bs{n}$. Additionally, the term $\Div_{\mskip-6mu\scriptscriptstyle\cl{S}} (\Prj{\bs{n}}\bs{\jmath}_{\scriptscriptstyle\partial\cl{P}})$ may be split as $\Div_{\mskip-6mu\scriptscriptstyle\cl{S}} (\Prj{\bs{n}}\bs{\jmath}_{\scriptscriptstyle\partial\cl{P}}) = \Div_{\mskip-6mu\scriptscriptstyle\cl{S}} \bs{\jmath}_{\scriptscriptstyle\partial\cl{P}} - 2K \bs{\jmath}_{\scriptscriptstyle\partial\cl{P}} \cdot \bs{n}$. Then, the surface species balance \eqref{eq:pointwise.surface.species.balance} may be written as
\begin{equation}\label{eq:pointwise.surface.species.balance.alt}
\dot{\varphi}_{\scriptscriptstyle\partial\cl{P}} = - \beta \mskip3mu \bs{\jmath}_{\scriptscriptstyle\cl{P}} \cdot \bs{n} + s_{\scriptscriptstyle\partial\cl{P}} - \Div_{\mskip-6mu\scriptscriptstyle\cl{S}}\bs{\jmath}_{\scriptscriptstyle\partial\cl{P}} + 2K \bs{\jmath}_{\scriptscriptstyle\partial\cl{P}} \cdot \bs{n},
\end{equation}
for each smooth part of $\partial\cl{P}$.

\section{Free-energy imbalance}
\label{sc:free.energy.imbalance}

First, note that the actual power is given by
\begin{equation}
\cl{W}_{\mathrm{ext}}(\cl{P},\partial\cl{P}) \coloneqq \cl{V}_{\mathrm{ext}}(\cl{P},\partial\cl{P};\dot{\varphi}_{\scriptscriptstyle\cl{P}},\dot{\varphi}_{\scriptscriptstyle\partial\cl{P}}).
\end{equation}
In the free-energy imbalance, together with the external power expenditure, we account for the rate at which energy is transferred to $\cl{P}$ and $\partial\cl{P}$ due to species transport. Thus, the free-energy imbalance reads
\begin{align}\label{eq:partwise.free.energy.imbalance}
\dot{\overline{\int\limits_{\prt}\psi_{\scriptscriptstyle\cl{P}}\,\mathrm{d}v}} + \dot{\overline{\int\limits_{\dprt}\psi_{\scriptscriptstyle\partial\cl{P}}\,\mathrm{d}a}} \le {}& \cl{W}_{\mathrm{ext}}(\cl{P},\partial\cl{P}) \nonumber\\[4pt]
&+ \int\limits_{\prt} \mu_{\scriptscriptstyle\cl{P}} s_{\scriptscriptstyle\cl{P}} \,\mathrm{d}v - \int\limits_{\dprt} \mu_{\scriptscriptstyle\cl{P}} \bs{\jmath}_{\scriptscriptstyle\cl{P}} \cdot \bs{n} \,\mathrm{d}a \nonumber\\[4pt]
&- \int\limits_{\dprt} \beta \mu_{\scriptscriptstyle\partial\cl{P}} \bs{\jmath}_{\scriptscriptstyle\cl{P}} \cdot \bs{n} \,\mathrm{d}a + \int\limits_{\dprt} \mu_{\scriptscriptstyle\partial\cl{P}} s_{\scriptscriptstyle\partial\cl{P}} \,\mathrm{d}a - \int\limits_{\ddprt} \surp{\mu_{\scriptscriptstyle\partial\cl{P}} \bs{\jmath}_{\scriptscriptstyle\partial\cl{P}} \cdot \bs{\nu}} \,\mathrm{d}\sigma.
\end{align}
Noting that $\Prj{\bs{n}}\bs{\jmath}_{\scriptscriptstyle\partial\cl{P}} \cdot \Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\mu_{\scriptscriptstyle\partial\cl{P}} = \bs{\jmath}_{\scriptscriptstyle\partial\cl{P}} \cdot \Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\mu_{\scriptscriptstyle\partial\cl{P}}$ and uncoupling $\partial\cl{P}$ from $\cl{P}$, for the sake of simplicity, we have that
\begin{equation}\label{eq:bulk.free.energy.imbalance.plain}
\dot{\psi}_{\scriptscriptstyle\cl{P}} + (\pi - \mu_{\scriptscriptstyle\cl{P}}) \dot{\varphi}_{\scriptscriptstyle\cl{P}} - \bs{\xi} \cdot \mathrm{grad}\mskip2mu\dot{\varphi}_{\scriptscriptstyle\cl{P}} + \bs{\jmath}_{\scriptscriptstyle\cl{P}} \cdot \mathrm{grad}\mskip2mu\mu_{\scriptscriptstyle\cl{P}} \le 0,
\end{equation}
and
\begin{equation}\label{eq:surface.free.energy.imbalance.plain}
\dot{\psi}_{\scriptscriptstyle\partial\cl{P}} + (\varpi - \mu_{\scriptscriptstyle\partial\cl{P}}) \dot{\varphi}_{\scriptscriptstyle\partial\cl{P}} - \bs{\tau} \cdot \Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\dot{\varphi}_{\scriptscriptstyle\partial\cl{P}} + \bs{\jmath}_{\scriptscriptstyle\partial\cl{P}} \cdot \Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\mu_{\scriptscriptstyle\partial\cl{P}} \le 0.
\end{equation}
Additionally, assuming that the bulk and surface free-energy densities $\psi_{\scriptscriptstyle\cl{P}}$ and $\psi_{\scriptscriptstyle\partial\cl{P}}$ are, respectively, given by constitutive response functions that are independent of $\mu_{\scriptscriptstyle\cl{P}}$, $\mu_{\scriptscriptstyle\partial\cl{P}}$, $\mathrm{grad}\mskip2mu\mu_{\scriptscriptstyle\cl{P}}$, and $\Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\mu_{\scriptscriptstyle\partial\cl{P}}$
\begin{equation}
\psi_{\scriptscriptstyle\cl{P}} \coloneqq \psi_{\scriptscriptstyle\cl{P}}(\varphi_{\scriptscriptstyle\cl{P}},\mathrm{grad}\mskip2mu\varphi_{\scriptscriptstyle\cl{P}}), \qquad \text{and} \qquad \psi_{\scriptscriptstyle\partial\cl{P}} \coloneqq \psi_{\scriptscriptstyle\partial\cl{P}}(\varphi_{\scriptscriptstyle\partial\cl{P}},\Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\varphi_{\scriptscriptstyle\partial\cl{P}}),
\end{equation}
we have that
\begin{equation}\label{eq:bulk.free.energy.total.derivative}
\dot{\psi}_{\scriptscriptstyle\cl{P}} = \partial_{\vphiP} \psi_{\scriptscriptstyle\cl{P}} \dot{\varphi}_{\scriptscriptstyle\cl{P}} + \partial_{\Grad\vphiP} \psi_{\scriptscriptstyle\cl{P}} (\mathrm{grad}\mskip2mu\varphi_{\scriptscriptstyle\cl{P}})^{\bs{\dot{}}},
\end{equation}
and
\begin{equation}\label{eq:surface.free.energy.total.derivative}
\dot{\psi}_{\scriptscriptstyle\partial\cl{P}} = \partial_{\vphidP} \psi_{\scriptscriptstyle\partial\cl{P}} \dot{\varphi}_{\scriptscriptstyle\partial\cl{P}} + \partial_{\Grads\vphidP} \psi_{\scriptscriptstyle\partial\cl{P}} (\Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\varphi_{\scriptscriptstyle\partial\cl{P}})^{\bs{\dot{}}}.
\end{equation}
Then, combining \eqref{eq:bulk.free.energy.imbalance.plain}, \eqref{eq:surface.free.energy.imbalance.plain}, \eqref{eq:bulk.free.energy.total.derivative}, and \eqref{eq:surface.free.energy.total.derivative}, we are led to two pointwise free-energy imbalances
\begin{equation}\label{eq:bulk.free.energy.imbalance}
(\mu_{\scriptscriptstyle\cl{P}} - \pi - \partial_{\vphiP} \psi_{\scriptscriptstyle\cl{P}}) \dot{\varphi}_{\scriptscriptstyle\cl{P}} + (\bs{\xi} - \partial_{\Grad\vphiP} \psi_{\scriptscriptstyle\cl{P}}) \cdot \mathrm{grad}\mskip2mu\dot{\varphi}_{\scriptscriptstyle\cl{P}} - \bs{\jmath}_{\scriptscriptstyle\cl{P}} \cdot \mathrm{grad}\mskip2mu\mu_{\scriptscriptstyle\cl{P}} \ge 0,
\end{equation}
and
\begin{equation}\label{eq:surface.free.energy.imbalance}
(\mu_{\scriptscriptstyle\partial\cl{P}} - \varpi - \partial_{\vphidP} \psi_{\scriptscriptstyle\partial\cl{P}}) \dot{\varphi}_{\scriptscriptstyle\partial\cl{P}} + (\bs{\tau} - \partial_{\Grads\vphidP} \psi_{\scriptscriptstyle\partial\cl{P}}) \cdot \Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\dot{\varphi}_{\scriptscriptstyle\partial\cl{P}} - \bs{\jmath}_{\scriptscriptstyle\partial\cl{P}} \cdot \Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\mu_{\scriptscriptstyle\partial\cl{P}} \ge 0.
\end{equation}
The bulk and surface free-energy imbalance equations, \eqref{eq:bulk.free.energy.imbalance} and \eqref{eq:surface.free.energy.imbalance}, serve to devise additional constitutive response functions in what follows.

\section{Additional constitutive response functions}
\label{sc:constitutive.relations}

We now assume that the set of independent variables is given by $\{\varphi_{\scriptscriptstyle\cl{P}},\varphi_{\scriptscriptstyle\partial\cl{P}},\mathrm{grad}\mskip2mu\varphi_{\scriptscriptstyle\cl{P}},\Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\varphi_{\scriptscriptstyle\partial\cl{P}},\mu_{\scriptscriptstyle\cl{P}},\mu_{\scriptscriptstyle\partial\cl{P}}\}$ while the set of dependent variables is $\{\pi,\varpi,\bs{\xi},\bs{\tau},\bs{\jmath}_{\scriptscriptstyle\cl{P}},\bs{\jmath}_{\scriptscriptstyle\partial\cl{P}}\}$. Thus, we find that the local inequality \eqref{eq:bulk.free.energy.imbalance} and \eqref{eq:surface.free.energy.imbalance} are satisfied in all processes if and only if:
\begin{itemize}
\item The bulk and surface microstress $\bs{\xi}$ and $\bs{\tau}$ are, respectively, given by
\begin{equation}\label{eq:microstress}
\bs{\xi} \coloneqq \partial_{\Grad\vphiP} \psi_{\scriptscriptstyle\cl{P}}, \qquad \text{and} \qquad \bs{\tau} \coloneqq \partial_{\Grads\vphidP} \psi_{\scriptscriptstyle\partial\cl{P}}.
\end{equation}
\item The internal bulk and surface microforces $\pi$ and $\varpi$ are, respectively, given by constitutive response functions that differs from the bulk and surface chemical potential by a contribution derived from the response functions $\psi_{\scriptscriptstyle\cl{P}}$ and $\psi_{\scriptscriptstyle\partial\cl{P}}$
\begin{equation}\label{eq:internal.microforces}
\pi \coloneqq \mu_{\scriptscriptstyle\cl{P}} - \partial_{\vphiP} \psi_{\scriptscriptstyle\cl{P}}, \qquad \text{and} \qquad \varpi \coloneqq \mu_{\scriptscriptstyle\partial\cl{P}} - \partial_{\vphidP} \psi_{\scriptscriptstyle\partial\cl{P}}.
\end{equation}
\item Granted that the bulk and surface species fluxes $\bs{\jmath}_{\scriptscriptstyle\cl{P}}$ and $\bs{\jmath}_{\scriptscriptstyle\partial\cl{P}}$ depend smoothly on the gradient of the bulk chemical potential, $\mathrm{grad}\mskip2mu\mu_{\scriptscriptstyle\cl{P}}$, and the surface gradient of the surface chemical potentials, $\Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\mu_{\scriptscriptstyle\partial\cl{P}}$, these fluxes are, respectively, given by a constitutive response function of the form
\begin{equation}
\bs{\jmath}_{\scriptscriptstyle\cl{P}} \coloneqq - \bs{M}_{\mskip-4mu\scriptscriptstyle\cl{P}} \mathrm{grad}\mskip2mu\mu_{\scriptscriptstyle\cl{P}}, \qquad \text{and} \qquad \bs{\jmath}_{\scriptscriptstyle\partial\cl{P}} \coloneqq -\bs{M}_{\mskip-4mu\scriptscriptstyle\partial\cl{P}} \Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\mu_{\scriptscriptstyle\partial\cl{P}},
\end{equation}
where the mobility tensors $\bs{M}_{\mskip-4mu\scriptscriptstyle\cl{P}}$ and $\bs{M}_{\mskip-4mu\scriptscriptstyle\partial\cl{P}}$ must obey the residual dissipation inequalities
\begin{equation}
\mathrm{grad}\mskip2mu\mu_{\scriptscriptstyle\cl{P}} \cdot \bs{M}_{\mskip-4mu\scriptscriptstyle\cl{P}} \mathrm{grad}\mskip2mu\mu_{\scriptscriptstyle\cl{P}} \ge 0, \qquad \text{and} \qquad \Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\mu_{\scriptscriptstyle\partial\cl{P}} \cdot \bs{M}_{\mskip-4mu\scriptscriptstyle\partial\cl{P}} \Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\mu_{\scriptscriptstyle\partial\cl{P}} \ge 0.
\end{equation}
\end{itemize}
For the sake of simplicity, we let $\bs{M}_{\mskip-4mu\scriptscriptstyle\cl{P}} \coloneqq m_{\mskip-2mu\scriptscriptstyle\cl{P}} \bs{1}$ and $\bs{M}_{\mskip-4mu\scriptscriptstyle\partial\cl{P}} \coloneqq m_{\mskip-2mu\scriptscriptstyle\partial\cl{P}} \bs{1}$. With this choice for $\bs{M}_{\mskip-4mu\scriptscriptstyle\partial\cl{P}}$, the surface flux $\bs{\jmath}_{\scriptscriptstyle\partial\cl{P}}$ remains proportional to $\Grad_{\mskip-2mu\scriptscriptstyle\cl{S}}\mu_{\scriptscriptstyle\partial\cl{P}}$ and therefore tangential to $\partial\cl{P}$. Thus, the normal component of $\bs{\jmath}_{\scriptscriptstyle\partial\cl{P}}$ appearing in expression \eqref{eq:pointwise.surface.species.balance.alt} vanishes.

Important to what follows is the explicit form of the bulk and surface chemical potentials when using \eqref{eq:field.equations}$_1$ and \eqref{eq:surface.field.equation} in \eqref{eq:internal.microforces}. That is,
\begin{equation}
\mu_{\scriptscriptstyle\cl{P}} = - \mathrm{div}\mskip2mu\bs{\xi} - \gamma + \partial_{\vphiP} \psi_{\scriptscriptstyle\cl{P}},
\end{equation}
and
\begin{equation}
\mu_{\scriptscriptstyle\partial\cl{P}} = - \Div_{\mskip-6mu\scriptscriptstyle\cl{S}}\bs{\tau} + 2K \bs{\tau} \cdot \bs{n} - \zeta - \bs{\xi} \cdot \bs{n} + \partial_{\vphidP} \psi_{\scriptscriptstyle\partial\cl{P}},
\end{equation}
which take the following form when considering \eqref{eq:microstress},
\begin{equation}\label{eq:bulk.chemical.potential}
\mu_{\scriptscriptstyle\cl{P}} = \partial_{\vphiP} \psi_{\scriptscriptstyle\cl{P}} - \mathrm{div}\mskip2mu\Big(\partial_{\Grad\vphiP} \psi_{\scriptscriptstyle\cl{P}}\Big) - \gamma,
\end{equation}
and
\begin{equation}\label{eq:surface.chemical.potential}
\mu_{\scriptscriptstyle\partial\cl{P}} = \partial_{\vphidP} \psi_{\scriptscriptstyle\partial\cl{P}}- \Div_{\mskip-6mu\scriptscriptstyle\cl{S}}\Big(\partial_{\Grads\vphidP} \psi_{\scriptscriptstyle\partial\cl{P}}\Big) - \zeta - \partial_{\Grad\vphiP} \psi_{\scriptscriptstyle\cl{P}} \cdot \bs{n},
\end{equation}
for each smooth part of $\partial\cl{P}$. Note that the normal component of $\bs{\tau}$ vanishes for the free-energy function \eqref{eq:free.energy.functional} with \eqref{eq:microstress}$_2$.

In what follows, consider that $\cl{P} \coloneqq \cl{B}$.

\subsection{Further connections: boundary conditions}

To be slightly more general, let us define different parts of the boundary $\partial\cl{P}$. Let $\partial\cl{P}^{\mathrm{dyn}}$ be the boundary with the dynamic bulk-surface interplay and $\partial\cl{P}^{\mathrm{sta}}$ the static boundary such that $\partial\cl{P} \coloneqq \partial\cl{P}^{\mathrm{dyn}}\cup\partial\cl{P}^{\mathrm{sta}}$ and $\partial\cl{P}^{\mathrm{dyn}}\cap\partial\cl{P}^{\mathrm{sta}}=\varnothing$. Also, let ${}_\circ\partial^2\cl{P}$ denote the boundary of the dynamic boundary $\partial\cl{P}^{\mathrm{dyn}}$ while $\partial^2\cl{P}$ still denotes the edge along which the normal field is discontinuous.

Next, given the boundary conditions derived based upon thermodynamical principles in \cite{Esp20,Esp21a,Dud21}, we stipulate that the boundary conditions for the phase fields may be prescribed as follows. The essential boundary conditions, that is, the assignment of microstructure, read
\begin{equation}\label{eq:dirichlet.phi.dP}
\varphi_{\scriptscriptstyle\cl{P}} (\bs{x},t) = \varphi_{\scriptscriptstyle\partial\cl{P}} (\bs{x},t), \qquad \forall \bs{x} \in \partial\cl{P}^{\mathrm{dyn}}_{\mathrm{ess}},
\end{equation}
and
\begin{equation}\label{eq:dirichlet.phi.ddP}
\varphi_{\scriptscriptstyle\partial\cl{P}} (\bs{x},t) = \varphi_{\scriptscriptstyle\partial^2\cl{P}}^{\mathrm{env}} (\bs{x}), \qquad \forall \bs{x} \in \partial^2\cl{P}^{\mathrm{sta}}_{\mathrm{ess}} \qquad \text{and} \qquad {}_\circ\partial^2\cl{P}^{\mathrm{sta}}_{\mathrm{ess}},
\end{equation}
where the surface phase-field $\varphi_{\scriptscriptstyle\partial\cl{P}}$ is the action of the dynamic environment on $\partial\cl{P}^{\mathrm{dyn}}_{\mathrm{ess}}$ and $\varphi_{\scriptscriptstyle\partial^2\cl{P}}^{\mathrm{env}}$ is the action of the static environment on $\partial^2\cl{P}^{\mathrm{sta}}_{\mathrm{ess}}$. On a static environment, expression \eqref{eq:dirichlet.phi.dP} takes the form
\begin{equation}\label{eq:dirichlet.phi.dP.sta}
\varphi_{\scriptscriptstyle\cl{P}} (\bs{x},t) = \varphi_{\scriptscriptstyle\partial\cl{P}}^{\mathrm{env}} (\bs{x}), \qquad \forall \bs{x} \in \partial\cl{P}^{\mathrm{sta}}_{\mathrm{ess}},
\end{equation}
where $\varphi_{\scriptscriptstyle\partial\cl{P}}^{\mathrm{env}}$ is the action of the static environment on $\partial\cl{P}^{\mathrm{sta}}_{\mathrm{ess}}$.

Instead, we may opt for the natural boundary conditions, that is, the assignment of microtractions. Then, we have
\begin{equation}
\xi_{\scriptscriptstyle\cl{S}} (\bs{x},t) = \xi_{\scriptscriptstyle\cl{S}}^{\mathrm{env}} (\bs{x}), \qquad \forall \bs{x} \in \partial\cl{P}^{\mathrm{sta}}_{\mathrm{nat}},
\end{equation}
and
\begin{equation}
\tau_{\scriptscriptstyle\partial\cl{S}} (\bs{x},t) = \tau_{\scriptscriptstyle\partial\cl{S}}^{\mathrm{env}}(\bs{x}), \qquad \forall \bs{x} \in \partial^2\cl{P}^{\mathrm{sta}}_{\mathrm{nat}} \qquad \text{and} \qquad {}_\circ\partial^2\cl{P}^{\mathrm{sta}}_{\mathrm{ess}},
\end{equation}
where $\xi_{\scriptscriptstyle\cl{S}} = \bs{\xi} \cdot \bs{n}$, $\tau_{\scriptscriptstyle\partial\cl{S}} = \surp{\bs{\tau} \cdot \bs{\nu}}$ on $\partial^2\cl{P}^{\mathrm{sta}}_{\mathrm{nat}} $, $\tau_{\scriptscriptstyle\partial\cl{S}} = \bs{\tau} \cdot \bs{\nu}$ on ${}_\circ\partial^2\cl{P}^{\mathrm{sta}}_{\mathrm{nat}}$ due to \eqref{eq:nonsmooth.divs.theo.open.S}, and $\xi_{\scriptscriptstyle\cl{S}}^{\mathrm{env}}$ and $\tau_{\scriptscriptstyle\partial\cl{S}}^{\mathrm{env}}$ are the actions of the static environment, respectively, on $\partial\cl{P}^{\mathrm{sta}}_{\mathrm{nat}}$ and $\partial^2\cl{P}^{\mathrm{sta}}_{\mathrm{nat}} \cup {}_\circ\partial^2\cl{P}^{\mathrm{sta}}_{\mathrm{nat}}$.

As for the bulk species balance, as an essential boundary condition, we may prescribe
\begin{equation}\label{eq:dirichlet.mu.dP}
\mu_{\scriptscriptstyle\cl{P}} (\bs{x},t) = \beta \mu_{\scriptscriptstyle\partial\cl{P}}(\bs{x},t), \qquad \forall \bs{x} \in \partial\cl{P}^{\mathrm{dyn}}_{\mathrm{ess}},
\end{equation}
where $\beta \mu_{\scriptscriptstyle\partial\cl{P}}$ is the action of the dynamic environment on $\partial\cl{P}^{\mathrm{sta}}_{\mathrm{nat}}$. Expression \eqref{eq:dirichlet.mu.dP}, on a static environment, takes the form
\begin{equation}\label{eq:dirichlet.mu.dP.sta}
\mu_{\scriptscriptstyle\cl{P}} (\bs{x},t) = \mu_{\scriptscriptstyle\partial\cl{P}}^{\mathrm{env}}(\bs{x}), \qquad \forall \bs{x} \in \partial\cl{P}^{\mathrm{sta}}_{\mathrm{ess}},
\end{equation}
where $\mu_{\scriptscriptstyle\partial\cl{P}}^{\mathrm{env}}$ is the action of the static environment on $\partial\cl{P}^{\mathrm{sta}}_{\mathrm{ess}}$.

Instead, as a natural boundary condition, we may opt for
\begin{equation}
\bs{\jmath}_{\scriptscriptstyle\cl{P}} (\bs{x},t) \cdot \bs{n} = \jmath_{\scriptscriptstyle\partial\cl{P}}^{\mathrm{env}} (\bs{x}), \qquad \forall \bs{x} \in \partial\cl{P}^{\mathrm{sta}}_{\mathrm{nat}},
\end{equation}
where $\jmath_{\scriptscriptstyle\partial\cl{P}}^{\mathrm{env}}$ is the action of the static environment on $\partial\cl{P}^{\mathrm{sta}}_{\mathrm{nat}}$ through the normal component of the flux $\bs{\jmath}_{\scriptscriptstyle\cl{P}}$. Conversely, for the surface species balance, as an essential boundary condition, we may prescribe
\begin{equation}
\mu_{\scriptscriptstyle\partial\cl{P}} (\bs{x},t) = \mu_{\scriptscriptstyle\partial^2\cl{P}}^{\mathrm{env}}(\bs{x}), \qquad \forall \bs{x} \in \partial^2\cl{P}^{\mathrm{sta}}_{\mathrm{ess}} \qquad \text{and} \qquad {}_\circ\partial^2\cl{P}^{\mathrm{sta}}_{\mathrm{ess}},
\end{equation}
where $\mu_{\scriptscriptstyle\partial^2\cl{P}}^{\mathrm{env}}$ is the action of the static environment on $\partial^2\cl{P}^{\mathrm{sta}}_{\mathrm{ess}} \cup {}_\circ\partial^2\cl{P}^{\mathrm{sta}}_{\mathrm{ess}}$, whereas, as a natural boundary condition, we may opt for
\begin{equation}
\surp{\bs{\jmath}_{\scriptscriptstyle\partial\cl{P}} (\bs{x},t) \cdot \bs{\nu}} = \jmath_{\scriptscriptstyle\partial^2\cl{P}}^{\mathrm{env}} (\bs{x}), \qquad \forall \bs{x} \in \partial^2\cl{P}^{\mathrm{sta}}_{\mathrm{nat}},
\end{equation}
and, due to \eqref{eq:nonsmooth.divs.theo.open.S},
\begin{equation}
\bs{\jmath}_{\scriptscriptstyle\partial\cl{P}} (\bs{x},t) \cdot \bs{\nu} = \jmath_{\scriptscriptstyle\partial^2\cl{P}}^{\mathrm{env}} (\bs{x}), \qquad \forall \bs{x} \in {}_\circ\partial^2\cl{P}^{\mathrm{sta}}_{\mathrm{nat}},
\end{equation}
where $\jmath_{\scriptscriptstyle\partial^2\cl{P}}^{\mathrm{env}}$ is the action of the static environment on $\partial^2\cl{P}^{\mathrm{sta}}_{\mathrm{nat}}$ through the tangent-normal component of the flux $\bs{\jmath}_{\scriptscriptstyle\partial\cl{P}}$.

Additionally, in view of the partwise free-energy imbalance \eqref{eq:partwise.free.energy.imbalance}, using a Colleman--Noll type of argument, we have that
\begin{equation}\label{eq:mix.boundary}
\int\limits_{\dprt} (\dot{\varphi}_{\scriptscriptstyle\cl{P}} + \dot{\varphi}_{\scriptscriptstyle\partial\cl{P}}) \mskip3mu \xi_{\scriptscriptstyle\cl{S}} \,\mathrm{d}a \ge 0, \qquad \text{and} \qquad \int\limits_{\dprt} (\mu_{\scriptscriptstyle\cl{P}} - \beta \mu_{\scriptscriptstyle\partial\cl{P}}) \mskip3mu \bs{\jmath}_{\scriptscriptstyle\cl{P}} \cdot \bs{n} \ge 0\footnotemark.
\end{equation}
\footnotetext{This type of condition was derived by Espath \& Calo \cite[Equation (157)]{Esp21a} based upon mechanical and thermodynamical arguments.}Moreover, the terms in \eqref{eq:mix.boundary} may be prescribed with a dissipative nature. That is, as a mixed boundary condition, expressions \eqref{eq:mix.boundary} read
\begin{equation}
\bs{\xi} (\bs{x},t) \cdot \bs{n} = \fr{1}{K} (\dot{\varphi}_{\scriptscriptstyle\cl{P}} + \dot{\varphi}_{\scriptscriptstyle\partial\cl{P}}), \qquad \forall \bs{x} \in \partial\cl{P}^{\mathrm{dyn}}_{\mathrm{mix}},
\end{equation}
and
\begin{equation}
\bs{\jmath}_{\scriptscriptstyle\cl{P}} (\bs{x},t) \cdot \bs{n} = \fr{1}{L} (\mu_{\scriptscriptstyle\cl{P}} - \beta \mu_{\scriptscriptstyle\partial\cl{P}}), \qquad \forall \bs{x} \in \partial\cl{P}^{\mathrm{dyn}}_{\mathrm{mix}},
\end{equation}
where $K, L > 0$.

\subsection{Specialized equations}

In view of the free-energy functional \eqref{eq:free.energy.functional}, and expressions \eqref{eq:pointwise.bulk.species.balance}, \eqref{eq:pointwise.surface.species.balance.alt}, \eqref{eq:bulk.chemical.potential}, and \eqref{eq:surface.chemical.potential}, our theory renders the following set of equations
\begin{equation}\label{eq:recovering.original}
\left\{
\begin{aligned}
\dot{\varphi}_{\scriptscriptstyle\cl{P}} &= s_{\scriptscriptstyle\cl{P}} + m_{\mskip-2mu\scriptscriptstyle\cl{P}}\triangle\mu_{\scriptscriptstyle\cl{P}}, && \text{in } \cl{P} \\[4pt]
\mu_{\scriptscriptstyle\cl{P}} &= -\epsilon \triangle\varphi_{\scriptscriptstyle\cl{P}} + \fr{1}{\epsilon} f^\prime (\varphi_{\scriptscriptstyle\cl{P}}) - \gamma, && \text{in } \cl{P}, \\[4pt]
\dot{\varphi}_{\scriptscriptstyle\partial\cl{P}} &= s_{\scriptscriptstyle\partial\cl{P}} + m_{\mskip-2mu\scriptscriptstyle\partial\cl{P}}\triangle_{\mskip-2mu\scriptscriptstyle\cl{S}}\mu_{\scriptscriptstyle\partial\cl{P}} - \beta m_{\mskip-2mu\scriptscriptstyle\cl{P}} \mskip3mu \partial_n\mu_{\scriptscriptstyle\cl{P}}, && \text{on } \partial\cl{P}, \\[4pt]
\mu_{\scriptscriptstyle\partial\cl{P}} &= - \iota \delta \triangle_{\mskip-2mu\scriptscriptstyle\cl{S}}\mu_{\scriptscriptstyle\partial\cl{P}} + \fr{1}{\delta} g^\prime (\varphi_{\scriptscriptstyle\partial\cl{P}}) + \epsilon \mskip3mu \partial_n \varphi_{\scriptscriptstyle\cl{P}} - \zeta && \text{on } \partial\cl{P}.
\end{aligned}
\right.
\end{equation}

The boundary conditions, may be summarized as follows:
\begin{equation}
\forall \bs{x} \in \partial\cl{P}^{\mathrm{dyn}}
\left\{
\begin{aligned}
\varphi_{\scriptscriptstyle\cl{P}} &= \varphi_{\scriptscriptstyle\partial\cl{P}}, \qquad \text{or} \qquad \epsilon \mskip3mu \partial_n \varphi_{\scriptscriptstyle\cl{P}} = \fr{1}{K} (\dot{\varphi}_{\scriptscriptstyle\cl{P}} + \dot{\varphi}_{\scriptscriptstyle\partial\cl{P}}), \\[4pt]
\mu_{\scriptscriptstyle\cl{P}} &= \beta \mu_{\scriptscriptstyle\partial\cl{P}}, \qquad \text{or} \qquad - m_{\mskip-2mu\scriptscriptstyle\cl{P}} \mskip3mu \partial_n \mu_{\scriptscriptstyle\cl{P}} = \fr{1}{L} (\mu_{\scriptscriptstyle\cl{P}} - \beta \mu_{\scriptscriptstyle\partial\cl{P}}),
\end{aligned}
\right.
\end{equation}
and
\begin{equation}
\forall \bs{x} \in \partial\cl{P}^{\mathrm{sta}}
\left\{
\begin{aligned}
\varphi_{\scriptscriptstyle\cl{P}} &= \varphi_{\scriptscriptstyle\partial\cl{P}}^{\mathrm{env}}, \qquad \text{or} \qquad \epsilon \mskip3mu \partial_n \varphi_{\scriptscriptstyle\cl{P}} = \xi_{\scriptscriptstyle\cl{S}}^{\mathrm{env}}, \\[4pt]
\mu_{\scriptscriptstyle\cl{P}} &= \mu_{\scriptscriptstyle\partial\cl{P}}^{\mathrm{env}}, \qquad \text{or} \qquad - m_{\mskip-2mu\scriptscriptstyle\cl{P}} \mskip3mu \partial_n \mu_{\scriptscriptstyle\cl{P}} = \jmath_{\scriptscriptstyle\partial\cl{P}}^{\mathrm{env}},
\end{aligned}
\right.
\end{equation}
and
\begin{equation}
\forall \bs{x} \in \partial^2\cl{P}^{\mathrm{sta}}
\left\{
\begin{aligned}
\varphi_{\scriptscriptstyle\partial\cl{P}} &= \varphi_{\scriptscriptstyle\partial^2\cl{P}}^{\mathrm{env}}, \qquad \text{or} \qquad \iota \delta \mskip3mu \surp{\partial_{\nu} \varphi_{\scriptscriptstyle\partial\cl{P}}} = \tau_{\scriptscriptstyle\partial\cl{S}}^{\mathrm{env}}, \\[4pt]
\mu_{\scriptscriptstyle\partial\cl{P}} &= \mu_{\scriptscriptstyle\partial^2\cl{P}}^{\mathrm{env}}, \qquad \text{or} \qquad - m_{\mskip-2mu\scriptscriptstyle\partial\cl{P}} \surp{\partial_\nu \mu_{\scriptscriptstyle\partial\cl{P}}} = \jmath_{\scriptscriptstyle\partial^2\cl{P}}^{\mathrm{env}},
\end{aligned}
\right.
\end{equation}
and
\begin{equation}
\forall \bs{x} \in {}_\circ\partial^2\cl{P}^{\mathrm{sta}}
\left\{
\begin{aligned}
\varphi_{\scriptscriptstyle\partial\cl{P}} &= \varphi_{\scriptscriptstyle\partial^2\cl{P}}^{\mathrm{env}}, \qquad \text{or} \qquad \iota \delta \mskip3mu \partial_{\nu} \varphi_{\scriptscriptstyle\partial\cl{P}} = \tau_{\scriptscriptstyle\partial\cl{S}}^{\mathrm{env}}, \\[4pt]
\mu_{\scriptscriptstyle\partial\cl{P}} &= \mu_{\scriptscriptstyle\partial^2\cl{P}}^{\mathrm{env}}, \qquad \text{or} \qquad - m_{\mskip-2mu\scriptscriptstyle\partial\cl{P}} \mskip3mu \partial_\nu \mu_{\scriptscriptstyle\partial\cl{P}} = \jmath_{\scriptscriptstyle\partial^2\cl{P}}^{\mathrm{env}}.
\end{aligned}
\right.
\end{equation}

\section{Acknowledgements}

The authors would like to thank Gabriel Nogueira de Castro for rendering the image in Figure \ref{fg:geometry}. This work was partially supported by the Flexible Interdisciplinary Research Collaboration Fund at the University of Nottingham Project ID 7466664.

\section{Declarations: Funding and/or Conflicts of interests/Competing interests \& Data availability}

The authors have no conflicts to disclose.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.
