= Fixed-charge response to added divalent chloride
{title2=$\delta\phi=(k_BT/e)\delta c(e^{-2v}+2e^v-3)/(c_{1+}+2c_{2+}-c_{1-}-2c_{2-})$}
For an ideal planar <electric double layer> at fixed <surface charge density>, let $v=e\phi_0/(k_BT)$. Adding bulk divalent chloride concentration $\delta c$ adds two monovalent coions per counterion. Differentiating the squared <Grahame equation> gives the displayed expression, with all denominator concentrations evaluated at the surface. The numerator concentrations are bulk increments. Dropping depleted coions at strongly negative potential gives the leading counterion approximation. A large finite increment instead requires solving the fixed-charge equation at the new composition; exponentiating a first-order voltage estimate does not make that estimate exact.
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