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3.4.6 Electric Potential and Partition Coefficient Derivatives
When the mixture is charged it is necessary to solve for the electric potential using the electroneutrality condition in (2.7.1-4). This equation may be rewritted as a polynomial in , where and Here, and the polynomial degress is where . Since more than one solute may carry the same charge , the coefficients should be evaluated from the summation of over all such solutes. Only real positive roots are valid, since according to (3.4.6-2). Using Descartes' rule of signs, an inspection of the coefficients shows tht there is only one sign change in the polynomial, regardless of the sign of , implying that there will always be only one positive root , which must thus be real. Therefore, there cannot be any ambiguity in the calculation of , irrespective of the polynomial degree. Newton's method is used to solve for the positive real root when .
Using the above relations, it follows that . An examination of the equations resulting from the linearization of the internal virtual work shows that it is necessary to evaluate derivatives of with respect to and , which are given by In these expressions, the derivatives of are obtained from the user-defined constitutive relations for the solubility. Derivatives of may be evaluated by differentiating the electroneutrality condition to produce The derivative may be evaluated from where is the referential solid volume fraction (volume of solid in current configuration per volume of the mixture in the reference configuration) and is the referential fixed charge density (equivalent charge in current configuration per volume of the mixture in the reference configuration).