Osmotic Coefficient Calculations for Dilute Solutions of Short Stiff-Chain Polyelectrolytes Dmytro Antypov ² and Christian Holm* ,²,‡ Max-Planck-Institut fu ¨r Polymerforschung, Ackermannweg 10, 55128, Mainz, Germany, and Frankfurt Institute for AdVanced Studies, Max-Von-Laue-Strasse 1, 60438 Frankfurt, Germany ReceiVed September 19, 2006; ReVised Manuscript ReceiVed NoVember 23, 2006 ABSTRACT: Osmotic properties of dilute salt-free solutions of stiff-chain polyelectrolytes are studied using computer simulations. A number of factors affecting the charge distribution around the macroions are analyzed quantitatively for parameters of an aqueous solution of a typical synthetic strongly charged rodlike polyelectrolyte. Departing from a mean-field treatment of an infinitely long linear charged rod, we significantly refine the model description. We found that incorporation of effects such as electrostatic correlations between counterions, chain flexibility, and addition of salt decreases the osmotic coefficient slightly. On the other hand, a model of a finite macroion with a noncentral charge distribution predicted significantly higher pressure. We quantify all these contributions separately and show what deviation one should expect from the prediction of the Poisson-Boltzmann solution of the infinite rod cell model. We further comment on the discrepancy between our simulation results and experimentally measured values. I. Introduction Strongly charged rodlike polyelectrolytes are an ideal model system to study the phenomenon of counterion condensation in dilute solutions. Unlike flexible-chain polyelectrolytes, they do not change their conformation upon changing the polymer concentration or adding salt. In a polar solvent, such macro- molecules dissociate into highly charged rodlike macroions and oppositely charged counterions. If no salt is added, bulk properties of such solution directly depend on the distribution of the counterions which is defined by a delicate balance between electrostatic interactions and the translational entropy. While all counterions equally explore space, they can be hypothetically divided into two groups. Those counterions close to the macroion and localized by strong Coulombic forces are often considered as condensed, 1,2 whereas the others are free and osmotically active. This qualitatively explains why the experimentally measured osmotic pressure Π of such a system is far lower than the one predicted by assuming that all counterions are free. The reduced thermodynamic activity of the counterions can be expressed in terms of the osmotic coefficient where Π id ) c c k B T is the ideal gas pressure at given counterion concentration c c and temperature T, and k B is the Boltzmann constant. Experimental studies of strongly charged polyelec- trolytes show that φ is a weak function of polymer concentration, and typically measured values are of the order of 0.2-0.3 for univalent counterions in the dilute regime; i.e., see references in ref 3. Similar values of the osmotic coefficient are also obtained within a Katchalsky cell model 4 by solving the Poisson-Boltzmann (PB) equation in cylindrical geometry. 5 Within this approach the counterion and polymer degrees of freedom in the solution are decoupled by considering only one fixed and completely rigid macroion confined together with its counterions in an electrically neutral cell. A further assumption, namely that the macroion is infinitely long, allows for an analytical solution of this system at zero salt. 6,7 Compared to a bulk polyelectrolyte solution, a mean-field (mf) assumption is employed twice in the standard PB approach: once, by decomposing the solution into a product of noninteracting cells, and posing only the problem of one macroion confined to a surrounding cell, and second, by employing the PB functional to find the counterion distribution within this cell. The first mf approximation can only be tested for finite-length charged rods, since in a solution all polyelectrolytes are of finite length. To this end we introduced recently an optimal cell model 8 showing that in this context the mf approximation works excellently from dilute up to semidilute conditions but that the differences induced by the finiteness can really become large for short rods. Recent theoretical and simulation studies of dilute polyelectro- lytes also confirmed the validity of the cell model approximation provided an appropriate cell geometry is used. 9,10 The second mf assumption of applying the PB equation is known to be a good approximation only for a moderately charged rod in the presence of monovalent counterions (see references cited in refs 11 and 12). If discrepancies are observed with either experiments or simulations, it is far from obvious to which degree they are due to the first and/or second mean-field approximation. Under experimental conditions there are a number of other factors that may affect the osmotic properties in various ways. These include electrostatic, excluded volume, and macroion-macroion cor- relations, residual salt, chain flexibility, backbone charge distribution, and nonelectrostatic interactions between the coun- terions and the backbone. It is one of the goals of this article to quantify the contribution of the above effects separately. In a previous study 5 we compared the influence of electrostatic correlations within the infinite-rod cell model and found them to only weakly influence the osmotic coefficientsthey yielded a lower value on the order of a few percent. Experimentally measured values of the osmotic coefficient were still about 10% lower. This motivated us to investigate which variations of the * Corresponding author. E-mail: c.holm@fias.uni-frankfurt.de. ² Max-Planck-Institut fu ¨r Polymerforschung. ‡ Frankfurt Institute for Advanced Studies. φ ) Π Π id (1) 731 Macromolecules 2007, 40, 731-738 10.1021/ma062179p CCC: $37.00 © 2007 American Chemical Society Published on Web 01/10/2007