Polarization of counterions in polyelectrolytes.

A theory of the polarization of counterions bound to a polyion, such as a DNA, in low and high electric field strengths is developed using statistical mechanics of inhomogeneous systems. For low fields, one finds that the polarizability p is (Zq)2rho0betaL3/(12[l + Lrho0sigma(L, b, xi, Z, I, rho0)]), where sigma = integral1 0(lambda' - lambda0) ¿dc(lambda - lambda')/dlambda¿lambda = lambda0 dlambda']), Z and L are the valence and the length of the polyion, respectively, q is the proton charge, beta = 1/kBT, T is the temperature, kB is the Boltzmann constant, I is the ionic strength, lambda = x/L and lambda0 = x0/L are scaled distances, x0 is a reference point such that the inhomogeneous counterion density at x0 is equal to rho0--the uniform density in the absence of an electric field E--and c(x) is the direct correlation function of the homogeneous counterion-polyion phase, which includes attractive and repulsive interactions. If Lsigma(L, ...) is much less than one, then the polarizability is proportional to L3. If the term Lsigma(L, ...) is much larger than one, the polarizability scales as L2. The induced dipole moment saturates and its value is the same as that of Mandel-Manning theories. The onset of the saturation, however, depends critically on the direct correlation function and hence polyelectrolyte effects. In the formalism, the polarization of the counterions is the equilibrium response to an electric field provided E is less than Esaturated. A dynamical scheme that incorporates the fact that in high fields the bound counterions conduct is discussed.