Electrostatic Persistence Length of a Wormlike Polyelectrolyte

The worm model of Kratky and Porod has been extended to “locally stiff’ polyelectrolytes. Neglecting excluded volume effects, the electrostatic persistence length, Pel, has been obtained for a continuous, uniform charge distribution in which both charge rearrangements due to bending and fluctuations due to thermal motion are not allowed. Comparisons of experimentally determined dimensions of carboxymethylcellulose in aqueous NaCl with theoretical results reveals good agreement between theory and experiment. In the Appendix, the relationship between the discrete and continuous model is examined. Furthermore we treat the continuous charge distribution with rearrangements with and without fluctuations. If the polyelectrolyte is assumed to be locally stiff, the results of the latter two cases reduce identically to the continuous, uniform case in which the charges are frozen in place. (I) Introduction Polyelectrolyte excluded volume theories assume that the unperturbed mean-square end-to-end distance (ho2) is independent of the supporting electrolyte concentration, CS.lp2 The basis of this assumption comes either from the use of Stockmayer-Fixman, S-F, plots, which give a slight ionic strength dependence for the unperturbed dimensions,3 even though S-F plots assume ( ho2) is independent of solvent, or from direct measurements in relatively high salt concentration theta solvent^. ^^^ However, for sufficiently low C,, one would intuitively expect that local electrostatic forces exert a significant influence on (ho2). The model of Rice and Harris6 takes account of local electrostatic interactions by considering an equivalent Kuhn chain with charges concentrated a t the midpoints of the statistical elements; if nearest neighbor segment interactions are assumed, the polymer behaves as a random chain. Thus, in the absence of long-range interactions, the somewhat artificial Rice-Harris model gives unperturbed chain dimensions that depend in a complicated fashion on CS. In the low salt limit, the Debye screening length, K ~ , is much larger than the distance between charges on the polyelectrolyte chain, so that the replacement of a discrete charge distribution by a continuous one should be a good approxim a t i ~ n . ~ The polyelectrolyte can therefore be viewed as a structureless, charged space curve, i.e., a wormlike polymer with a continuous charge distribution. In this paper, we shall calculate the electrostatic persistence length of a charged wormlike polymer which is sufficiently stiff that no excluded volume effects are present. The electrostatic persistence length, Pel, is approximately related to ( ho2) by? where L = contour length of the chain; Po persistence length in the absence of electrostatic forces (Le., C, + a); and PT = total persistence length; Pel is obtained for (i) a continuous, uniform charge distribution without charge rearrangements due to bending and without fluctuations due to thermal motion. Our results are then compared with experimental data on carboxymethylcellulose dimensions and reasonably good agreement is demonstrated. Furthermore, in the Appendix we consider three additional calculations relating to Pel: (ii) the discrete model with no charge rearrangements or fluctuations; (iii) the continuous charge distribution with charge rearrangements, but no fluctuations; and (iv) the continuous charge distribution with charge rearrangements and fluctuations. The results of case (ii) reduce to the continuous charge distribution result (i) if KU 0. (Here a is length of a monomer unit.) Finally, cases (iii) and (iv) reduce to case (i) if the polymer is assumed to be locally stiff; the exact definition of local stiffness will be presented in the body of the paper. (11) T h e Charged Wormlike Polymer (A) General Formalism. Consider a charged space curve whose infinitesimal elements interact via a screened Coulomb potential. We wish to calculate the electrostatic persistence length, Pel. V , the increase in potential energy per unit length due to electrostatic repulsions relative to the reference configuration of a straight rod, is given by3 V = ‘12 t Rc-2 (11.1) t = bending constant of the rod and R , is the radius of curvature of the element of space curve at which V is evaluated. I t then follows immediately from the worm model that (11.2) L kBT k g is Boltzmann’s constant. Thus, we direct our attention to determining the explicit form oft = t ( ~ ) in eq 11.1. Let us choose the origin at an arbitrary point somewhere in the middle of the space curve, and let us parameterize the space curve by s, the contour length relative to the origin. If F(s) is the location of a point on the space curve relative to the origin, then F(s) = f(s)l’+ g ( s ) j + h(s)/l (11.3) where i, j , h are unit vectors in the x , y , z directions, respectively. Define Fo(s) to be the location of the point in the straight rod reference configuration. We shall choose the reference configuration to lie along i so that we can write FO(S) = S; = fo(s)l’ (11.4) Now, the length of the space curve must remain invariant, i.e., 2t 2P0 = 2P,1= (ho2) -s ( b ) = J b [ ( f ’ ( ~ ) ) ~ + (g’(s))2 + ( h ’ ( ~ ) ) ~ ] ~ / ~ ds = J b [ ( f ~ ’ ( s ) ) ~ ] ~ / ~ ds (11.5) for any arbitrary b. The prime denotes differentiation with respect to s. Hence, ( f ’ ( s ) I2 + (g’(s))2 + ( h ’ ( ~ ) ) ~ = (f0’(s)l2 = 1 (11.6) Settingf’(s) = 1 6(s), where 6(s) > 0, we find on direct substitution into eq 11.6 and on solving the quadratic that results Vol. 10, No. 5, September-October 1977 Electrostatic Persistence Length of a Wormlike Polyelectrolyte 945 6(s) = 1 [l {(g’(s))2 + (h’(s))””2 (11.7) We now introduce the concept of local stiffness; Le. , g;(s)2 + h’(s)2 << 1 (we see later this is equivalent to neglecting t e rns of order R,-4) f ’ ( s ) = 1 6 ( ~ ) = 1 ‘12 ((g’(S))’ + (h‘(~))‘) (11.8) Furthermore, the unit tangent vector u(s) is giver? by u(s) = ( f ’ ( s ) , g’b), h’b ) ) (11.9) A general property of unit tangent vectors and their derivatives follows from u(s)a(s) = l. d u b 1 u(s) -= f ’ ( s ) f ” ( s ) + g’(s)g”(s) + h’(s)h”(s) = 0 dS (II,10) From eq 11.8, it follows that f”’(s) = -. {g’”)g’’! s) + h’(s)h“(s)) and eq 11.10 becomes {g’(s)g”(s) + h’(s)h”(s) l ( (g’(s))2 + (h’(s))2) = 0 (11.11)