A Preliminary Examination of End Effects in Polyelectrolyte Theory: The Potential of a Line Segment of Charge

The potential, ICT, of a line segment of charge is calculated in the Debye-Huckel approximation. We determine ICT as a function of segment length and the position of the test charge. If the test charge is near the line segment and the length of the line segment of charge is large relative to the screening length, qT is well approximated by the electrostatic potential of an infinite line of charge. When the test charge is far from the line segment, qT reduces to the point charge limit. Recently, considerable attention has been devoted to elucidating the properties of the line of charge model of polyelectrolyte The infinitely long line of charge model has been employed to calculate the colligative properties of dilute polyelectrolyte the diffusion constant of a mobile ion in the presence of a polyel e c t r ~ l y t e , ~ , ~ and the expansion parameter, 2, of the excluded volume theory.6 We note, however, that real polyions are of finite size; as such, it would be quite useful to determine the magnitude of end effects on the electrostatic potential arising from a finite line of charge. Hence, a partial motivation of the present work is to determine when the replacement of the potential of the line segment of charge by that of an infinite line is justified and when it is not. We also note that the line segment of charge may perhaps be a plausible model for low molecular weight DNA and helical poly(g1utamic acid) a t low to moderate ionic strengths. Thus, this paper is a preliminary step toward understanding the dependence of polyelectrolyte behavior on chain length. If the length of the line segment of charge is large relative to the screening length and if the test charge is near the source, it seems intuitively reasonable that the electrostatic potential should be well approximated by the infinite line of charge result. However, when the test charge is far from the line segment of charge, the line segment should appear as a point charge. In the context of the Debye-Huckel approximation, verification of the above conjectures will be presented in what follows. Consider a uniformly charged line segment of length L immersed in bulk solvent. It is assumed that each infinitesmal piece interacts with the test charge via a screened Coulomb potential. For discussions concerning the applicability of an effective charge density and counterion condensation, we refer to the l i t e r a t ~ r e . ’ ~ , ~ ~ Let the line segment of charge lie on the z axis, in the cylindrical coordinate system (r,O,z), and let one end of the line segment be located at z = 0. Whereupon, the potential, #T, felt at a point r = (r,B,z) by a test charge is given by @ L exp(-K[r2 + ( z z ’ ) ~ ] ~ / ~ ) (1) [ r2 + ( z ~ ’ ) ~ ] 1 ’ 2 1c~(r) = K J dz‘ *Address correspondence to this author at the Department of Chemistry, Louisiana State University, Baton Rouge, LA 70803. with @ 5 charge per unit length of the line segment and D2 --= bulk dielectric constant. K-’ is the Debye screening length and is defined by Here, e is the protonic charge. The summation extends over all the ionic species ‘‘2’ in solution; Ci is the concentration of species “i’’ in ions per cm3; p i is the valence of the ith species. kg is Boltzmann’s constant, and T is the absolute temperature. It is convenient to define the dimensionless interaction energy !bd; $‘d is related to $T through the relation J/d = The test charge can either be positioned between the two ends of the line segment, i.e., 0 5 z I L, or below or above an end of the segment, i.e., z < 0 or z > L. By symmetry, it is obvious that the two latter situations are physically equivalent. t-le!bT/kB? 5 = JePl/kBTD2. Let z” = z z’, it then follows from eq 1 that Similarly when z < 0 1cdb = &”’“’ dz!/ expj-K[r2 + z”2]1/21 [r2 + 2”2]1 /2 expj-K[r2 + z ’ ’ ~ I (4) [r2 + ~ ” 2 ] 1 / 2 i‘z’ dz While it is apparent from eq 3 and 4 that the potential $d must vary continuously from the 0 I z 5 L case to that when z < 0, we find it conceptually useful to examine the two situations separately. As the 0 I z I L configuration is most closely related to the infinite line result, it is treated first. At this juncture, a few qualitative observations on certain limiting forms of +d are necessary. First of all, it must 0024-9297/80/2213-0335$01.00/0