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\title{Supplementary Note for ``A structural model for the \emph{in vivo} human cornea including collagen-swelling interaction" }

\author{Xi Cheng \and Steven J. Petsche \and Peter M. Pinsky }
\maketitle

\section*{Triphasic equations}

Modeling of polyelectrolyte gels requires a suitable description of
interactions between the fluid, solid and ionic phases. Mow \emph{et al}.
\cite{Mow1980} and Lai \emph{et al}. \cite{Lai1991} proposed a triphasic
framework based on a continuum representation of interpenetrating phases and
their interactions. The solid, fluid and ions are treated as distinct phases
with individual velocity fields. This framework extends Biot's poroelastic
theory \cite{Biot1941} to model the transport of ionic species and their
interactions with fixed charges, and is able to predict the swelling of
tissues. The theory has been generalized to finite deformations and multiple
electrolyte systems \cite{Huyghe1997,Gu1998} and applied to corneal swelling
under conditions of thermodynamic equilibrium by Bryant and McDonnell \cite%
{Bryant1998}. Wilson et al. \cite{Wilson2005} simplified the triphasic
theory to a biphasic swelling model based on Lanir's hypothesis \cite%
{Lanir1987} in which the transport of ions is postulated to be much faster
than that of water. The Donnan osmotic pressure is employed to avoid solving
the fields of ionic concentrations, although the fluid pressure and
displacement have to be solved in a coupled system.

Consider a binary electrolyte gel, consisting of a mixture of solid, fluid,
and ionic phases, immersed in a ionic bath at constant electrostatic
potential $\varphi _{0}$, at hydrostatic pressure $p_{0},$ and with ionic
concentrations $C_{0+}=C_{0-}=C_{0}$. We start by recalling the familiar
triphasic equations for the polyelectrolyte gel written for conditions of
thermodynamic equilibrium \cite{Bryant1998,Huyghe1997,Lai1991}. \ Momentum
balance, conservation of mass, and Poisson's equation, require%
\begin{eqnarray}
\mathbf{\nabla }\cdot \left( \mathbf{\boldsymbol{\sigma }-}p\mathbf{1}%
\right) &=&\mathbf{0}  \label{g1} \\
\mathbf{\nabla }\cdot \mathbf{J}_{w} &=&0  \label{g2} \\
\mathbf{\nabla }\cdot \mathbf{J}_{i} &=&0,\quad i=+,-  \label{g3} \\
\mathbf{\nabla }\cdot \epsilon _{w}\mathbf{E} &=& F\left( z_{f}C_{f}+%
\sum_{i=+,-}z_{i}C_{i} \right) \label{g4}
\end{eqnarray}%
where $\boldsymbol{\sigma }$ is the effective stress generated by the solid
phase and $p$ is the fluid pressure, $\mathbf{J}_{w}$ and $\mathbf{J}%
_{i}(i=+,-)$ are the fluid flow and ionic fluxes, respectively, $\mathbf{E}$
is the electrical field, $\epsilon _{w}$ is the dielectric permittivity of
the electrolyte solvent, $C_{i}(i=+,-)$ and $C_{f}$ are the concentration of
the mobile ions and fixed charges, respectively, $z_{i}(i=+,-)$ and $z_{f}$
denote the valence value associated with each ionic and charge species. At
equilibrium, the fluid flow and ionic fluxes vanish (i.e $\mathbf{J}_{w}=%
\mathbf{J}_{i}=\mathbf{0}$) in the gel, which implies the gradients of the
electrochemical potentials vanish \cite{Bryant1998,Lai1991} and allowing (%
\ref{g2}) and (\ref{g3}) to be replaced with
%$C_i (i = +, -)$ denote the concentration of the mobile ions and $C_f$ is the
%fixed charge concentration. $z_i (i = +, -)$ and $z_f$ are the valence value
%associated with the mobile ions and fixed charges, respectively.
%For most biological tissues, no appreciable fluid
%flow or ionic fluxes exist in steady state. Therefore it may be further assumed that the water flow
%and ionic fluxes vanish which implies the gradients of the electrochemical
%potentials vanish \cite{Bryant1998} and allowing (\ref{g2}) and (\ref{g3}) to be
%replaced with%
\begin{eqnarray}
\mathbf{\nabla }\mu _{w} &=&\mathbf{0}  \label{g5} \\
\mathbf{\nabla }\mu _{i} &=&\mathbf{0},\quad i=+,-  \label{g6}
\end{eqnarray}%
where the water and ionic chemical potentials are given by,%
\begin{align}
\mu _{w}& =\mu _{w}^{\text{ref}}+\left[ p-RT\left( C_{+}+C_{-}\right) \right]
/\rho _{w}  \label{mu_w} \\
\mu _{i}& =\mu _{i}^{\text{ref}}+\frac{RT}{M_{i}}\ln C_{i}+z_{i}\frac{%
F\varphi }{M_{i}},\quad i=+,-  \label{mu_i}
\end{align}%
where $\rho _{w}$ is the water density, $M_{i}(i=+,-)$ are the atomic weight
for the anion and cation, respectively, $R,T$ and $F$ are the gas constant,
temperature and Faraday constant, respectively, $\mu _{w}^{\text{ref}}$ and $%
\mu _{i}^{\text{ref}}$ are the water potential and ionic chemical potentials
at the reference state, $\varphi $ is the electrostatic potential.
Integrating (\ref{g5}) and (\ref{g6}) between any point in the bath and
electrolyte gives the electrolyte fluid pressure and Boltzmann distribution,%
\begin{eqnarray}
p &=&p_{0}+\Pi  \label{g7} \\
C_{i} &=&C_{0}\exp \left( -\frac{z_{i}F}{RT}(\varphi -\varphi _{0})\right)
\label{g8}
\end{eqnarray}%
where
\begin{equation}
\Pi =RT\left( C_{+}+C_{-}-2C_{0}\right)  \label{g9}
\end{equation}%
is the osmotic pressure difference between the electrolyte and bath. \ Note
that (\ref{g8}) implies%
\begin{equation}
C_{+}C_{-}=C_{0}^{2}  \label{g20}
\end{equation}%
Employing (\ref{g7}) in (\ref{g1}) and (\ref{g8}) in (\ref{g4}) allows the
governing coupled equations to be expressed as,%
\begin{align}
& \boldsymbol{\nabla }\cdot \left( \mathbf{\boldsymbol{\sigma }-}\left( \Pi
\left( \varphi \right) +p_{0}\right) \mathbf{1}\right) =\mathbf{0}
\label{g10} \\
-& \nabla ^{2}\varphi =\frac{F}{\epsilon _{w}}\left( z_{f}C_{f}-2C_{0}\sinh
\left( \frac{F\varphi }{RT}\right) \right)  \label{g11}
\end{align}%
equation (\ref{g10}) gives the momentum balance condition for the
solid-fluid mixture, and (\ref{g11}) is the Poisson-Bolztmann equation for
the electrostatic potential $\varphi .$ \ These equations require the
specification of $p_{0}$ and $C_{f}$ (typically dependent on the volume
dilation $J$) as well as suitable boundary conditions. \ The coupled
solution provides the pointwise osmotic pressure $\Pi $ which may vary
rapidly at the nanoscale.

\section*{Tangent operator}

The finite element tangent operator requires evaluation of two material
tensors given by%
\begin{eqnarray*}
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
_{\text{elastic}} &=&4\frac{\partial ^{2}\mathcal{W}_{\text{elastic}}\left(
\mathbf{C}^{\prime }\right) }{\partial \mathbf{C}^{\prime }\partial \mathbf{C%
}^{\prime }} \\
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
_{\text{electrolyte}} &=&4\frac{\partial ^{2}\mathcal{W}_{\text{electrolyte}%
}\left( J\right) }{\partial \mathbf{C}\partial \mathbf{C}}+4\frac{\partial
^{2}\mathcal{W}_{\text{bath}}\left( J\right) }{\partial \mathbf{C}\partial
\mathbf{C}}
\end{eqnarray*}%
The tensor $%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
_{\text{elastic}}$ may be derived based on the above given definition of $%
\mathcal{W}_{\text{elastic}};$ a detailed description can be found in \cite%
{Petsche2014}. \ \ The tensor $%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
_{\text{electrolyte}}$, is evaluated as follows,
\begin{align}
\mathbb{C}_{\text{electrolyte}}& =J\left( J\frac{\partial ^{2}\mathcal{W}_{%
\text{electrolyte}}}{\partial J^{2}}+\frac{\partial \left( \mathcal{W}_{%
\text{electrolyte}}+\mathcal{W}_{\text{bath}}\right) }{\partial J}\right)
\mathbf{C}^{-1}\otimes \mathbf{C}^{-1}  \notag \\
& -2J\frac{\partial \left( \mathcal{W}_{\text{electrolyte}}+\mathcal{W}_{%
\text{bath}}\right) }{\partial J}\mathbf{C}^{-1}\odot \mathbf{C}^{-1}  \notag
\\
& =J\left( JK_{os}-P_{os}-p_{0}\right) \mathbf{C}^{-1}\otimes \mathbf{C}%
^{-1}+2J\left( P_{os}+p_{0}\right) \mathbf{C}^{-1}\odot \mathbf{C}^{-1}
\label{C_electro}
\end{align}

\bigskip

\noindent where $\partial \mathbf{C}^{-1}/\partial \mathbf{C=-C}^{-1}\odot
\mathbf{C}^{-1}$ and $\left[ \mathbf{C}^{-1}\odot \mathbf{C}^{-1}\right]
_{ABCD}=\frac{1}{2}\left[ C_{AC}^{-1}C_{BD}^{-1}+C_{AD}^{-1}C_{BC}^{-1}%
\right] $, and $P_{os} = -\partial\mathcal{W}_{\text{electrolyte}}/\partial J$
and $K_{os} = -\partial P_{os}/\partial J$ are unit cell osmotic pressure
and osmotic compressibility, respectively.

The spatial form of $%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
_{\text{electrolyte}}$ is given by $c_{\text{electrolyte}}=\phi _{\ast
}\left(
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
_{\text{electrolyte}}\right) $, where $\phi _{\ast }$ is the fourth-order
push-forward operator based on $\mathbf{F.}$ \ It is found that,%
\begin{equation}
c_{\text{electrolyte}}\left( J\right) =\left( JK_{os}-P_{os}-p_{0}\right)
\mathbf{1}\otimes \mathbf{1+}2\left( P_{os}+p_{0}\right) \mathbf{I}
\label{c_electro}
\end{equation}
It is clear from equations (\ref{C_electro}) and (\ref{c_electro}) that the current 
electrolyte swelling model may be easily implemented in a finite element code since the 
electrolyte (volumetric stiffness) tensors can be directly computed from the volume dilation $J$ 
in a straightforward manner.

\bigskip 

\begin{thebibliography}{99}

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compression: theory and experiments. \textit{J. Biomech. Eng.} \textbf{102},
73-84. %doi:10.1115/1.3138202

\bibitem[Lai et al.(1991)]{Lai1991} Lai WM, Hou JS, Mow VC. 1991 A triphasic
theory for swelling and deformation behaviors of articular cartilage.
\textit{J. Biomech. Eng.} \textbf{113}, 245-258. %DOI: 10.1115/1.2894880

\bibitem[Biot(1941)]{Biot1941} Biot MA. 1941 General theory of
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155-164.

\bibitem[Huyghe et al.(1997)]{Huyghe1997} Huyghe JM, Janssen JD. 1997
Quadriphasic theory of swelling incompressible porous media. \textit{Int. J.
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\bibitem[Gu et al.(1998)]{Gu1998} Gu WY, Lai WM and Mow VC. 1998 A mixture
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120}, 169-180. %DOI: 10.1115/1.2798299

\bibitem[Wilson et al.(2005)]{Wilson2005} Wilson W, Donkelaar CCV, Huyghe
JM. 2005 A comparison between mechano-electrochemical and biphasic swelling
theories for soft hydrated tissues. \textit{Trans. ASME} \textbf{127},
158-165. %doi:10.1115/1.1835361

\bibitem[Lanir(1987)]{Lanir1987} Lanir Y. 1987 Biorheology and fluid flux in swelling tissues.
I. Bicomponent theory for small deformations, including concentration effects.
\textit{Biorheology} \textbf{24}, 173-187.

\bibitem[Bryant and McDonnell(1998)]{Bryant1998} Bryant MR, McDonnell PJ.
1998 A triphasic analysis of corneal swelling and hydration control. \textit{%
Trans. ASME} \textbf{120}, 370-381. %DOI: 10.1115/1.2798004

\bibitem[Petsche et al.(2014)]{Petsche2014} Petsche SJ.
2014 Multiscale Mechanics and Experimental Investigation of the
Human Cornea. Thesis (Ph.D.)--Stanford University.

\end{thebibliography}

\end{document}
