= Solution
Let the electrolyte <permittivity> be $\varepsilon$ and let $\kappa_D=1/\ell_D$ be the inverse <Debye–Hückel screening length>. We reserve $k$ for the imposed lateral <wavevector>. In the <Debye–Hückel approximation>, linearization about an electrically neutral bulk electrolyte gives
$$
(\nabla^2-\kappa_D^2)\phi=0
$$
away from fixed charges. For ionic species of valence $z_i$ and bulk number density $n_i$, $\kappa_D^2=\sum_i n_i z_i^2e^2/(\varepsilon k_BT)$; the approximation requires $|z_ie\phi|\ll k_BT$. Substitution of a lateral cosine into this screened <Poisson equation> leaves a vertical decay constant
$$
\boxed{p=\sqrt{k^2+\kappa_D^2}.}
$$
The <screened sinusoidal surface-charge mode> thus decays faster than a laterally uniform charge mode. A nonzero $k$ retains a finite decay length even in the zero-salt limit.
The exterior media and their boundary conditions are not specified in the paper. Specifying the two <surface charge densities> alone does not fix the normal derivative on the inside unless the outside response is also fixed. We first take the confined-gap idealization, with the prescribed charge supplying all the displacement flux into the electrolyte between the sheets. The <electrostatic interface boundary conditions> are then
$$
\varepsilon\phi_z(x,d/2)=\sigma_+(x),\qquad
-\varepsilon\phi_z(x,-d/2)=\sigma_-(x).
$$
This is, for example, the limit of negligible exterior displacement admittance. The <electrostatic potential> solving these conditions and the <Debye–Hückel approximation> equation is
$$
\boxed{\phi(x,z)=\frac{\alpha}{\varepsilon p\sinh(pd)}
\left[\cos(kx)\cosh\bigl(p(z+d/2)\bigr)
+\cos(kx+\theta)\cosh\bigl(p(z-d/2)\bigr)\right],\quad |z|<d/2.}
$$
Differentiation checks the two surface signs directly. This solution allows a sine as well as a cosine lateral component when the phase is nonzero.
In linear screened <electrostatics>, the quadratic charging free energy is $\tfrac12\int\sigma\phi\,dA$. It is equivalently $\tfrac\varepsilon2\int[(\nabla\phi)^2+\kappa_D^2\phi^2]\,dV$ with the stated boundary conditions. The second term represents the linearized ionic response; integrating only the bare <electric field> energy would omit it. If $\mathcal A$ denotes membrane area, the wavelength-averaged energy density is
$$
\frac E{\mathcal A}=\frac12\left\langle\sigma_+\phi(x,d/2)+\sigma_-\phi(x,-d/2)\right\rangle_x.
$$
Using $\langle\cos^2(kx)\rangle_x=1/2$ and $\langle\cos(kx)\cos(kx+\theta)\rangle_x=\cos\theta/2$ gives the <phase registration of screened charged sheets> energy
$$
\boxed{\frac E{\mathcal A}=\frac{\alpha^2}{2\varepsilon p}
\left[\coth(pd)+\frac{\cos\theta}{\sinh(pd)}\right].}
$$
The positive cosine coefficient shows that, for nonzero charge amplitude and finite separation,
$$
\boxed{\theta_{\min}=\pi\pmod{2\pi},\qquad
\frac{E_{\min}}{\mathcal A}=\frac{\alpha^2}{2\varepsilon p}\tanh(pd/2).}
$$
At the minimum the <electrostatic potential> simplifies to $\phi=\alpha\cos(kx)\sinh(pz)/[\varepsilon p\cosh(pd/2)]$. Positive and negative charge patches face opposite signs across the gap. The two charge patterns therefore favor a half-wavelength lateral offset rather than like-charge alignment.
For comparison, if identical electrolyte fills all of space on both sides of infinitesimally thin charge sheets, the <electrostatic interface boundary conditions> are continuity of $\phi$, decay at infinity, and the charge-induced derivative jumps
$$
\phi_z(z_s^+)-\phi_z(z_s^-)=-\sigma_s(x)/\varepsilon.
$$
Superposing the two <screened sinusoidal surface-charge modes> gives, in particular between the sheets,
$$
\boxed{\phi(x,z)=\frac{\alpha}{2\varepsilon p}
\left[\cos(kx)e^{-p(d/2-z)}+\cos(kx+\theta)e^{-p(d/2+z)}\right].}
$$
Its full-space continuation replaces each vertical distance by the corresponding absolute distance. Evaluating the <electrostatic potential> on both sheets now yields
$$
\boxed{\frac E{\mathcal A}=\frac{\alpha^2}{4\varepsilon p}
\left[1+e^{-pd}\cos\theta\right],\qquad \theta_{\min}=\pi\pmod{2\pi}.}
$$
The constant term is the two isolated-sheet self energies, and the cosine term is their screened interaction. The energy depends on the exterior convention, but both stated physical idealizations give the same <phase registration of screened charged sheets>: unlike charge patches oppose each other. In the full-space convention the optimum interaction energy is $-\alpha^2e^{-pd}/(4\varepsilon p)$, giving an attractive normal <force> density $-\alpha^2e^{-pd}/(4\varepsilon)$. Phase sensitivity becomes exponentially weak for $pd\gg1$; if $\alpha=0$ or the separation tends to infinity there is no selected phase.
One can also display how an exterior dielectric response interpolates between these cases. Let $Y_o\ge0$ be its normal displacement admittance for this lateral <Fourier mode>: $Y_o=0$ for the confined-gap idealization, $Y_o=\varepsilon p$ for identical exterior electrolyte, and $Y_o=\varepsilon_o|k|$ for an ion-free exterior dielectric. Define
$$
D_s=\varepsilon p\tanh(pd/2)+Y_o,\qquad
D_a=\varepsilon p\coth(pd/2)+Y_o.
$$
The symmetric and antisymmetric surface-charge combinations give
$$
\frac E{\mathcal A}=\frac{\alpha^2}{4}
\left[\frac{1+\cos\theta}{D_s}+\frac{1-\cos\theta}{D_a}\right].
$$
Since $D_s<D_a$, its cosine coefficient is positive. This makes the phase-minimizing conclusion robust while exposing the boundary information needed for an absolute <electrostatic energy>.
Back to article page
