Electrostatic Persistence Length of a Polyelectrolyte Chain

We calculate the electrostatic persistence length I, both for flexible and for stiff polyelec- trolytes using a self-consistent variational theory. For the case of intrinsically rigid polyelectrolytes we recover the classical results due to Odijk, Skolnick, and Fixman (OSF), namely, 1, = K-~ where K-~ is the Debye screening length. In contrast, 1, for intrinsically flexible polyelectrolytes is found to be proportional to K-~ in the limit of large screening. This is in accord with simulations, experiments, and numerical estimates. We also provide a criterion for the crossover from the OSF result to the behavior observed for intrinsically flexible polyelectrolytes. Despite numerous theoretical studies of charged polymers,l-1° a theory for the description of their conformations, at the level comparable to that for neutral polymer solutions,ll does not exist. The pres- ence of multiple length scales (six) makes a scaling description of polyelectrolytes difficult to construct. The behavior of a dilute polyelectrolyte solution has been largely analyzed in terms of the electrostatic persistence length advanced by Odijk3 and independently by Skolnick and Fi~man. ~ In the Odijk-Skonick-Fixman (hereaf- ter called OSF) theories, the polyion is described by a semiflexible chain near the rod limit having N charged monomers separated by a distance A along its contour. The electrostatic interaction between the charged seg- ments is assumed to be given the Debye-Huckel potential, u(r) = ZBe-Vr. The strength of the interaction is measured in terms of the Bjerrum length Ig = e2/4mkT where e is the charge per segment, and E is the dielectric constant of the solvent. The Debye screening length, r~ = K-~, is roughly the distance beyond which the electrostatic interaction is screened. It depends on the total concentration, n, of the count- erions (assumed to be monovalent) and any added electrolytes through the relation K~ = 4nZgn. OSF calculated the increase in free energy due to electrostatic interaction and elastic bending energy for a slightly bent configuration with reference to a rodlike configuration. This was used to analyze the effect of the charges on the chain stiffness, leading naturally to the crucial concept of the electrostatic persistence length, I,. Following Odijk, the configuration of the polyion is described by the unit tangent vector u(s) = &/as, 0 I s 5 L, where rb) is the radius vector of a segment of the chain at the curvilinear position s, and L is the contour length of the chain. If the chain is near the rod limit, only those paths which minimize the elastic free energy, Le. , paths for which the angle 8 defined by cos(B(s)) = u(s). u(O) varies linearly with s, contribute significantly to the partition function.12 The electrostatic contribution to the chain stiffness due to the departure from the rod limit allows for the calcula- tion of I, which in the limit of KL >> 1 is given by3~4 Abstract published in Advance ACS Abstracts, December 15, 1994. 0024-9297/95/2228-0577$09.0Qf0 The configurational properties of the chain are deter- mined by the total persistance length which is the sum of ZOSF and the bare persistence length IO. Odijk's result is based on the assumption that 8(s) is small, which is appropriate for stiff chains. However, this approximation is expected to break down for intrinsically flexible chains. In this case fluctuations in chain configurations could make a substantial con- tribution to the free energy, which in turn may result in the alteration of the electrostatic persistence length. Recently, Barrat and Joanny13 argued that these fluc- tuations contribute to (8(s)2) where (. 0) indicates the average over all paths weighted by the appropriate Boltzmann factor. The criterion for the validity of Odijk's theory can be established by examining the behavior of (8(s)2) as a function of s. Barrat and Joanny (BJ) noted that Odijk's estimate for los~ is valid at length scales larger than a certain crossover length sc given by13