A model for the electric field-driven deformation of a drop or vesicle in strong electrolyte solutions

In the boundary conditions adopted for a drop, the interface has no electrical capacitance and no monopole surface charge. It is also impervious or impenetrable to ions, and is therefore referred to as ideally polarizable. This gives the interfacial boundary conditions

respectively. Here $S$ denotes the sharp interface between the two phases, with outward unit normal $\bm{n}$, and $[\cdot]\equiv[\cdot]^{\partial\Omega_{2}}_{\partial\Omega_{1}}$ denotes the jump across $S$, with the convention that it is the limit as $S$ is approached from the exterior domain $\Omega_{2}$ minus the limit as $S$ is approached from the interior domain $\Omega_{1}$.

Continuity of velocity of the solvent and the kinematic boundary condition imply that

where $\bm{x}_{s}$ is any point on $S$. The stress-balance boundary condition is that

Here $\bm{T}_{H}$ is the stress tensor for a Newtonian fluid with viscosity $\mu$ and $(\bm{e})_{ij}=(\partial_{x_{j}}u_{i}+\partial_{x_{i}}u_{j})/2$ is the symmetric part of the velocity gradient. $\bm{T}_{M}$ is the part of the Maxwell stress tensor due to the electric field, which has been written in terms of the potential $\phi$, $\sigma$ is a constant surface tension, and $\kappa_{1}$ and $\kappa_{2}$ are the principal curvatures of $S$. With $\bm{n}$ pointing outward on $S$ the principal curvatures are taken to be positive when the curve of intersection of S by a plane containing $\bm{n}$ and a principal direction is convex on the side to which $\bm{n}$ points, and is negative otherwise.

Under equilibrium conditions there is typically a difference in the electric potential between distinct phases that occurs at and near the interface between them. It is caused by a difference in the affinity for charge carriers of the phases and is referred to as an inner or Galvani potential [5]. The boundary conditions

where $l^{(+)}$ and $l^{(-)}$ are constant partition coefficients, imply the presence of a non-zero Galvani potential when $l^{(+)}\neq l^{(-)}$. The notation $c^{(\pm)}|_{\partial\Omega_{i}}$ denotes evaluation of the ion concentration $c^{(\pm)}$ in the limit as $S$ is approached from the domain $\Omega_{i}$ for $i=1,2$. Boundary conditions of this type have been considered by, for example, Zholkovskij et al. [17] and by Mori & Young [15].

Mori et al. [18] introduce a set of phenomenological boundary conditions for the behavior of a cell or vesicle, where the interior and exterior phases are separated by a membrane, and these are summarized here. The membrane has an initial equilibrium reference configuration $S_{ref}$ with a local coordinate system $\bm{\theta}$ that serves as a Lagrangian or material coordinate. At subsequent times the membrane, which is treated as a sharp interface $S$ between the phases $\Omega_{i}$ that nonetheless has electromechanical structure, is described by $\bm{x}=\bm{X}(\bm{\theta},t)$.

The same notation for interfacial quantities is used for both a drop and a vesicle. Here we have: $[\cdot]$ for the jump in a quantity across the membrane, $\cdot|_{\partial\Omega_{i}}$ for evaluation as the side $\partial\Omega_{i}$ of the membrane is approached, or simply on $S$ for a quantity that is continuous.

If the membrane is semi-permeable or porous to the aqueous solvent it allows a flow of solvent in the normal direction that occurs by osmosis and is assumed to be proportional to the jump in the solvent’s partial pressure across the membrane. Hence

where $\pi_{m}\geq 0$ is an effective membrane porosity and the partial pressure of the ions is given by the ideal gas law.

The trans-membrane flux for each ion species is taken to be proportional to the jump in its electrochemical potential across the membrane surface. Mori et al. [18] give the boundary condition

where $g^{(\pm)}\geq 0$ is a gating constant for each ion species, and the velocity on the left hand side, $\bm{v}^{(\pm)}-\partial_{t}\bm{X}|_{\bm{\theta}}$, is the velocity of ions relative to the membrane material.

In the analysis that follows, a boundary condition is needed for the normal component of the molecular ion flux $\bm{j}^{(\pm)}\cdot\bm{n}$ in the electrolyte solution immediately adjacent to the membrane, where the flux $\bm{j}^{(\pm)}$ is given by equation (2.3), i.e., $\bm{j}^{(\pm)}=c^{(\pm)}(\bm{v}^{(\pm)}-\bm{u})$. If the normal ion flux $\bm{j}^{(\pm)}\cdot\bm{n}$ in the electrolyte and the normal ion flux across the membrane given by (2.14) are not equal then charge is not conserved at the membrane boundaries $\partial\Omega_{i}$ but accumulates there. We note that this occurs if both (2.13) and (2.14) are applied to a membrane that is semi-permeable to solvent, i.e., when $\pi_{m}>0$. This is verified by subtracting equation (2.13) multiplied by the ion concentration $c^{(\pm)}$ from (2.14) to form the electrolyte ion flux $\bm{j}^{(\pm)}\cdot\bm{n}$, and noting that the concentration of each ion species is not continuous but has a jump across the membrane.

This difficulty is easily resolved by modifying the trans-membrane flux boundary condition (2.14) to read

which conserves charge across the membrane and its boundaries and holds in general, that is for all $\pi_{m}\geq 0$. When the membrane is impermeable to solvent (i.e., $\pi_{m}=0$) the boundary condition (2.13) reduces to no-slip between the electrolyte and membrane surface, and (2.14) and (2.15) are identical, but when the membrane is semi-permeable to solvent (i.e., $\pi_{m}>0$) the solvent and ion species cross the membrane at different rates via distinct pores or channels, and (2.15) must be applied instead.

The membrane has electrical capacitance because it can maintain a jump in the potential across its outer faces, between which the potential varies linearly with normal distance. The jump in the potential is referred to as the trans-membrane potential and is denoted by $[\phi]$. The membrane has no net monopole charge, in the sense that at distinct points along the normal the surface charge densities on opposite faces have equal magnitude and opposite polarity, cancelling each other. The charge on each face is related to the trans-membrane potential by the membrane capacitance per unit area $C_{m}$. Analogous to the first two boundary conditions of (2.8) for a drop, but modified to express continuity of electric displacement and the capacitance of the membrane, we have the vesicle boundary conditions

Here $\varepsilon_{m}$ is the membrane permittivity, and $d$ is its thickness.

Two specific membrane capacitance models of Mori et al. [18] are

These hold if, for example, the bulk material of the membrane is incompressible, and then (2.17a) applies when the membrane surface is locally inextensible and (2.17b) applies when it is locally extensible. The capacitance per unit area in the initial undeformed or reference state is $C_{m}^{0}$, which is constant for a homogeneous membrane. The ratio of the local surface area in the deformed state to its value in the initial reference state following a material point $\bm{x}=\bm{X}(\bm{\theta},t)$ is denoted by $\delta A(\bm{X})$. Bulk incompressibility implies that the volume of a membrane material element $d(\bm{X})\delta A(\bm{X})$ is conserved during deformation, so that for the locally inextensible surface of (2.17a) $\delta A(\bm{X})=1$ with $d(\bm{X})$ and $C_{m}^{0}$ conserved, while for (2.17b) $d(\bm{X})^{-1}\propto\delta A(\bm{X})$.

The stress-balance boundary condition is

on $S$. The vesicle membrane has a bending stiffness that is expressed in terms of the bending modulus $\kappa_{b}$, the mean curvature $H=(\kappa_{1}+\kappa_{2})/2$, the Gaussian curvature $K=\kappa_{1}\kappa_{2}$, and a spontaneous curvature $C_{0}$. The spontaneous curvature $C_{0}$ is twice the mean curvature of the membrane material in a traction-free equilibrium state, see for example Vlahovska et al. [19] and Seifert [20], and it is usually set to zero. Here, $\nabla_{s}$ denotes the surface gradient operator, $\nabla_{s}^{2}$ is the Laplace-Beltrami or surface Laplacian operator, and the convention for the sign of the principal curvatures $\kappa_{1}$ and $\kappa_{2}$ is the same as that in (2.10).

The tension in the membrane surface is denoted by $\sigma$ in (2.18) but, in contrast to its appearance in (2.10) for a drop, the value of $\sigma$ is not a given constant. For a locally inextensible membrane $\sigma$ is determined by imposing inextensibility as a constraint, namely

or equivalently $\delta A(\bm{X})=1$, for all $t>0$. Here $\partial_{t}\bm{X}_{s}$ denotes the tangential projection of the velocity $\partial_{t}\bm{X}$ of a material particle onto $S$. For a locally extensible membrane a stress-strain equation of state must be introduced, see for example [21]. The membrane tension is initially constant, but during deformation it can vary from point to point on $S$, leading to the term $\nabla_{s}\sigma$ in (2.18).

We note that Mori et al. [18] consider a membrane bending stress given by the variational derivative of a general energy functional. The specific choice of the functional due to Helfrich [22] has been adopted in many studies of cell and vesicle deformation, and leads to the bending stress term on the right hand side of (2.18). Barthès-Biesel [21] gives a concise description of the different mechanical properties of vesicle, cell, and capsule membranes.

The electric field that induces flow and deformation is applied at time $t=0$. It can be spatially uniform and constant, or more general, so that

where $\phi_{\infty}$ is a solution of Laplace’s equation. Also

A drop is assumed to have relaxed to a spherical shape for $t<0$, with no applied field and equilibrium initial conditions given by

However, when there is a nonzero Galvani potential difference between the two phases the $t<0$ equilibrium relations for the potential and ion concentrations are revised to accommodate an electrical double layer - this will be considered later, in §5.3.

The quantities used to put the equations in nondimensional form are listed in Table 1. The length scale $a$ is the initial drop radius, or a similar length scale for a vesicle. The velocity scale $U_{c}=D_{*}/\lambda_{*}$ is the ion-diffusive scale, where $D_{*}$ is a representative ion diffusivity and $\lambda_{*}$ is the Debye layer thickness. The time scale $a/U_{c}$ is the scale on which the Debye layers acquire charge after $t=0$, i.e., the charge-up time scale. The scale for the pressure $p$ is the scale of the Stokes flow regime, and the scale for the potential is the thermal voltage $k_{B}T/e$, where $k_{B}$ is the Boltzmann constant and $T$ is the temperature.

All material and rate parameters in the two phases are made nondimensional by a single representative value. For example, the ion concentrations are made nondimensional by a representative ambient ion concentration $c_{*}$, and the salt concentration is made nondimensional by $s_{*}$. Similarly, $D^{(\pm)},k_{d},k_{a},\varepsilon$, and $\mu$ are made nondimensional by $D_{*},k_{d*},k_{a*},\varepsilon_{*}$, and $\mu_{*}$ respectively. To observe equilibrium between the ion and salt concentrations, the relation

holds. The Debye layer thickness $\lambda_{*}$ is given by

Dimensionless groups that appear are

The role of $\tau$ and $\alpha$ is considered below, in §3. Except for this, the analysis is carried out in the limit of small Debye layer thickness, $\epsilon\rightarrow 0$, with $\Delta,\nu$, and $\Psi$ all of order $O(1)$. The quantity $\Delta$ is the ratio of the capillary velocity, $\sigma/\mu_{*}$ to the ion-diffusive velocity $U_{c}$. The group $\nu\epsilon$ is the ratio of electrostatic stress to viscous stress, and is referred to either as the inverse of a Mason number or as an electrical Hartmann number. If the Helmholtz-Smoluchowski slip velocity $U_{HS}$ is based on the thermal voltage, then $U_{HS}=\varepsilon_{*}(k_{B}T/e)^{2}/\mu_{*}a$, and the group $\nu\epsilon=U_{HS}/U_{c}$ is also the ratio of the two velocity scales. The parameters $\epsilon,\Delta$, and $\nu$ are all fixed for a given choice of electrolytes and initial drop size. In contrast, the parameter $\Psi$ is the ratio of the applied potential difference $\Phi_{\infty}$ to the thermal voltage, and is a control parameter.

In the scaling regime of this study, the Peclet number $Pe=U_{c}a/D_{*}$ does not appear independently, since the velocity scale for $U_{c}$ implies that $Pe=\epsilon^{-1}$.

The same variable and parameter names are now used to express the governing equations and boundary conditions in nondimensional form. The molecular ion flux $\bm{j}^{(\pm)}=c^{(\pm)}(\bm{v}^{(\pm)}-\bm{u})$ of (2.3), recast via (LABEL:NPdimb) and the relation $D^{(\pm)}=b^{(\pm)}k_{B}T$, is

in nondimensional form. Conservation of ions (2.1) and salt (2.2), together with the expression (2.4) for the ion kinetic rate term $\cal{R}$ and the incompressibility condition of (2.7) give

The Poisson equation (2.6) is

so that $q$ is a normalized charge density. The continuity and Stokes equation are

where the body force is written in terms of the potential via (2.5).

Drop Boundary Conditions. In nondimensional form, from (2.5) and (2.26), the interfacial boundary conditions (2.8) on $S$ become

Continuity of velocity and the kinematic condition are formally unchanged, namely

The stress-balance boundary condition becomes

The boundary conditions (2.12) for a Galvani potential (when $l^{(+)}\neq l^{(-)}$) are formally unchanged, namely

Vesicle Boundary Conditions. In nondimensional form the trans-membrane ion flux $\bm{j}^{(\pm)}\cdot\bm{n}$ is given by

where the gating constant $g^{(\pm)}$ of (2.15) has been made nondimensional and scaled by $g_{*}=\frac{c_{*}D_{*}}{k_{B}Ta}$. The flow of solvent across the membrane is given by

where the effective porosity $\pi_{m}$ of (2.13) has been made nondimensional and scaled by $\pi_{*}=\frac{D_{*}}{k_{B}Tc_{*}a}$. These relations contrast with the boundary conditions (2.32) and (2.33) for an ideally polarizable drop with a sharp impervious interface; the cell has a small $O(\epsilon)$ flux of ions and solvent across the membrane when it is semi-permeable to these species.

The relations (2.16) that express continuity of electric displacement across the membrane become

Since a lipid bilayer or similar biomembrane has a thickness of the order of $5$nm to $8$nm, which is the same order of magnitude as the thickness of the neighboring Debye layers, the membrane thickness has been made nondimensional by $\lambda_{*}$ and the membrane capacitance per unit area, $C_{m}$ or $C_{m}^{0}$ of (2.17), has been made nondimensional by $\varepsilon_{*}/\lambda_{*}$. This accounts for the $\epsilon$-scaling in (2.39).

The stress-balance boundary condition is now

where the tension in the membrane and the bending modulus have been made nondimensional by $\mu_{*}U_{c}$ and $\mu_{*}U_{c}a^{2}$ respectively, and the spontaneous curvature $C_{0}$ has been set to zero. The membrane inextensibility constraint is formally unchanged, namely

Initial and Far-Field Conditions. The initial conditions are

except that the first and third relations are revised for a drop when it has a nonzero Galvani potential, as considered in §5.3, and we have assumed a uniform applied field. Here $c_{i}$ is the initial dimensionless ion concentration on $\Omega_{i}$ for $i=1,2$. A drop is initially spherical with radius 1, whereas a vesicle has a known initial equilibrium configuration $S_{ref}$. Far from a drop or vesicle,

The applied potential and constant electric field are related by $\Phi_{\infty}=E_{\infty}a$. For a more general applied field $\phi$ approaches a specified solution $\phi_{\infty}$ of Laplace’s equation that has magnitude $\Psi$.

To resolve the dynamics of the Debye layers we introduce a local surface-fitted or intrinsic orthogonal curvilinear coordinate system. This has tangential coordinates $\xi_{1}$ and $\xi_{2}$ that are aligned with the principal directions on $S$ and normal coordinate $n$, with $n=0$ on $S$ and $n>0$ in $\Omega_{2}$. The origins of the Eulerian and intrinsic coordinate systems are $O$ and $O^{\prime}$, respectively, and the position vector $\bm{x}$ of a point $P$ in space relative to $O$ is written in the two coordinate systems as

Here $\bm{x}=\bm{X}(\xi_{1},\xi_{2},t)$ is the parametric equation of $S$ and $\bm{n}$ is its outward unit normal. Since $\xi_{1}$ and $\xi_{2}$ are principal directions on $S$ they define an orthogonal coordinate system on it with associated unit tangent vectors $\bm{e}_{i}=\frac{1}{a_{i}}\frac{\partial\bm{X}}{\partial\xi_{i}}$, where $a_{i}=|\frac{\partial\bm{X}}{\partial\xi_{i}}|$ for $i=1,2$. With the convention that $\bm{n}=\bm{e}_{1}\times\bm{e}_{2}$, Rodrigues’ formula implies that the principal curvatures $\kappa_{i}$ satisfy $\frac{\partial\bm{n}}{\partial\xi_{i}}=\kappa_{i}\frac{\partial\bm{X}}{\partial\xi_{i}}$, and the change in $\bm{x}$ corresponding to increments in the intrinsic coordinates with time fixed is

Expressions for vector differential operators and the rate of strain tensor in a general orthogonal curvilinear coordinate system can be found in, for example, [23], Appendix 2.

Dependent variables within the Debye layers are denoted by upper case letters, with terms in an expansion in integer powers of $\epsilon$ denoted by subscripts, so that for the ion concentrations $c^{(\pm)}=C^{(\pm)}=C_{0}^{(\pm)}+\epsilon C_{1}^{(\pm)}+\ldots$ and for the potential $\phi=\Phi=\Phi_{0}+\epsilon\Phi_{1}+\ldots$, for example. The analysis of the Debye layers is based on a local normal coordinate $N$ defined by

where $N=O(1)$ as $\epsilon\rightarrow 0$.

The leading order ion concentrations and potential are given by the Gouy-Chapman solution. Reviews of this are given by, for example, [4] and [5], and it is summarized here because it is the foundation for the analysis that follows. In the present context, all dependent variables depend parametrically on the tangential coordinates $\xi_{1}$ and $\xi_{2}$, and on time $t$.

In terms of local variables, at leading order, the ion conservation equations (3.2) and the interface condition (2.32) for a drop, or (2.37) with (2.26) for a vesicle, imply that

An integration immediately gives

This has a simple interpretation in terms of the molecular ion flux (2.26), which has local expansion $\bm{j}^{(\pm)}=\bm{J}_{0}^{(\pm)}+\epsilon\bm{J}_{1}^{(\pm)}+O(\epsilon^{2})$ in the Debye layers, and where the leading order flux $\bm{J}_{0}^{(\pm)}$ is in the normal direction. Equation (5.5) states that $\bm{J}_{0}^{(\pm)}$ is zero throughout the layers, with diffusion and electromigration of ions in equilibrium at this level of approximation.

The requirement of matching with the outer regions is that

where an $S_{+}$ or $S_{-}$ subscript denotes, respectively, the limit of a quantity in the outer regions as $S$ is approached from the exterior of the drop (as $n\rightarrow 0^{+}$) or from the interior (as $n\rightarrow 0^{-}$). We have just shown at equation (4.5) that the ion concentration $c_{0}|_{S_{\pm}}=c_{i}$ ($i=1,2$) is constant over regions where the flow enters the Debye layers, at least at early times, but it is to be determined and satisfies (4.4) elsewhere.

Integration of (5.5) with the matching conditions (5.6) gives the Boltzmann distribution between the ion concentrations and the potential, namely

Here $\eta_{0}$ is the first term in the expansion of $\eta=\Phi-\phi|_{S_{\pm}}$, which is the excess potential in the Debye layers.

From equation (2.29), the charge density $q$ in the layers is therefore given at leading order by $Q_{0}=-c_{0}|_{S_{\pm}}\sinh\eta_{0}$, and the potential $\Phi_{0}$ satisfies the Poisson-Boltzmann equation

This can be integrated once on multiplying by $\partial_{N}\Phi_{0}$ and using the matching conditions (5.6) to find

For a drop, the first of the interfacial boundary conditions (2.31), that $[\phi]=0$, states that the potential is continuous across $S$ with a surface value denoted by $\lim_{N\rightarrow 0^{\pm}}\Phi_{0}=\Phi_{0}|_{S}$. A second integration then gives the Gouy-Chapman solution in the form

where the upper choice of signs hold for $N>0$ and the lower choice for $N<0$. For a vesicle, its membrane capacitance implies that there can be a non-zero jump in the potential across the vesicle membrane, with $\lim_{N\rightarrow 0^{-}}\Phi_{0}\equiv\Phi_{0}|_{\partial\Omega_{1}}\neq\lim_{N\rightarrow 0^{+}}\Phi_{0}\equiv\Phi_{0}|_{\partial\Omega_{2}}$. This gives the Gouy-Chapman solution