Mean-field theory of the phase diagram of ultrasoft, oppositely charged polyions in solution

Abstract

We investigate the phase separation of the "ultrasoft restricted primitive model" (URPM), a coarse-grained representation of oppositely charged, interpenetrating polyelectrolytes, within a mean-field description based on the "chemical picture." The latter distinguishes between free ions and dimers of oppositely charged ions (Bjerrum pairs) which are in chemical equilibrium governed by a law of mass action. Interactions between ions, and between ions and dimers are treated within linearized Poisson-Boltzmann theory, at four levels of approximation corresponding to increasingly refined descriptions of the interactions. The URPM is found to phase separate into a dilute phase of dimers, and a concentrated phase of mostly free (unpaired) ions below a critical temperature T(c). The phase diagram differs, however, considerably from the predictions of recent simulations; T(c) is about three times higher, and the critical density is much lower than the corresponding simulation data [D. Coslovich, J. P. Hansen, and G. Kahl, Soft Matter 7, 1690 (2011)]. Possible reasons for this unexpected failure of mean-field theory are discussed. The Kirkwood line, separating the regimes of monotonically decaying and damped oscillatory decay of the charge-charge correlation function at large distances is determined within the random phase approximation.

Gas-liquid critical point of the ultrasoft restricted primitive model from analytic theory

“…In general, the simulation results show a strong dependence of the coexistence envelope (its location and shape) on the system size compared to the case of the RPM. The gas-liquid phase coexistence in the URPM has been also predicted by the mean-field (MF) like theories [7,8], although with the critical point being considerably distant from the simulations. As expected, the MF theories predict a classical critical behavior near the critical point.Motivated by the above mentioned simulation studies, we address the issue of the gas-liquid criticality in the URPM using the theory that exploits the method of collective variables (CVs) [9,10].…”

“…In order to make some contact with the results obtained for the RPM, hereafter we use the same reduced units for the density as in [6], i.e., ρ * = ρσ 3 where σ is the diameter of the polyion. Our choice of the reduced temperature, T * = k B T /u 0 with u 0 being the maximum strength of the attractive interaction, coincides with that of [1,2,6,7]. In particular, one gets u 0 = 2Q 2 /( πσ) for the URPM and u 0 = Q 2 /σ in the case of the RPM where σ is the diameter of the polyion in the former case and the diameter of the hard sphere/ion in the latter case.…”

“…Expanding the entropic part of Hamiltonian (4) in powers of ρ N and ρ Q (more exactly, in powers of deviations of ρ N and ρ Q from their MF values), we arrive at the expression similar to that obtained in [11] (see equation (7) in [11]). The main difference is that in the case of the URPM, the contributions from the hard sphere reference system to series expansion coefficients a (i n ) n reduce to the ideal gas terms.…”

“…As expected, our estimates of the critical parameters of the URPM coincide with the results obtained in [8] in the RPA. It should be noted that the RPA, like other mean-field theories [7], predicts a far too high critical temperature and a far too low critical density compared with the available simulation data [1,2,6]. Some possible reasons for such a situation have been discussed in [7].…”

Salt-induced effective interactions and phase separation of an ultrasoft model of polyelectrolytes

“…Under such conditions it is not unreasonable to adopt a highly coarse-grained description of polyelectrolytes modelled as penetrable spherical objects characterized by a continuous gaussian charge distribution with a width of the order of the radius of gyration R g of the coil. Such a representation was recently introduced to investigate solutions of oppositely charged polyelectrolytes, within the so-called "ultrasoft primitive model" (UPM) [4][5][6][7]. In the present paper we adapt the model to solutions of equally charged polyelectrolytes in the presence of added salt.…”