Statistical mechanics of electrolytes and polyelectrolytes. III. The cylindrical Poisson–Boltzmann equation

Abstract

The Poisson-Boltzmann equation for the potential due to an infinitely long, cylindrical polyion in a dilute ionic solution is studied and certain properties of its solution are proved. It is shown that when the charge density on the polyion exceeds a critical value, the surrounding ionic solution separates into two essentially independent phases. The inner phase which is close to the polyion consists predominantly of counterions. An explicit solution to the distribution function in this phase is obtained to the leading order and its validity is proved. This function depends only on the polyion charge density, and not on the concentration of the ionic solution. The distribution function in the outer phase depends upon the ionic strength but, to the leading order, not on the polyion charge density as long as it exceeds the critical value. It is shown that as the charge density on the polyion is increased, a proportionate number of counterions appears in the inner phase so as to neutralize the effect of this increase on the outer phase. It is proved that the potential, to its leading order, in the outer phase for any polyion charge density higher than the critical value, is the same as that due a polyion carrying a charge density equal to the critical value. The nonlinear dependence of the surface value of the potential on the polyion charge density is proved.

Growth of linear charged micelles

“…Actually, the micelle is not a line but a thin cylinder,of radius a; in the limit UK -=c 1, arguments advanced for lines [12] should remain asymptotically valid for cylinders [14]. The dimensionless potential can be expressed by…”

“…(12); the third inequality allows us to neglect the ideal mixing contribution so that $(s) is virtually independent of s and equal to about -2 log KU as for infinite lines [12] or thin infinite cylinders [14]. The leading term becomes…”

“…Note that in eqs. (13)(14)(15) the cut-off a has to be introduced in order to conform to eq. (12) which converges for nonzero a. Equations (l), (7) and (15) lead to an anomalous growth law [13] (n + 1) log L = constant + ~/KQ (16) The effect of counterions condensing nonuniformly is remarkable: it changes the sign of eq.…”

Effective charge saturation in colloidal suspensions

“…Importantly, κa → 0 is the asymptotic regime where the celebrated Manning limiting law [11,32] happens to be exact, and the condensation criterion holds. In this limit, above the condensation threshold, the electrostatic potential solution of the full non-linear PB equation is indistinguishable from that of a cylinder carrying a line charge density λ equiv = 1/ℓ B [11].…”

“…Here, the object is an infinitely long and thin rod bearing λ charges per unit length. At infinite dilution and in the absence of salt, it can be shown at the PB level that the polyelectrolyte is electrostatically equivalent to a rod carrying λ equiv charge per unit length, where the equivalent charge density saturates to a critical value λ equiv = 1/ℓ B when λ increases [9][10][11]. In general however, PB theory can be solved analytically in very few geometries only and the difficulty remains to predict Z eff for a given colloidal system [4,5,7,12,13].…”