Monte Carlo studies of polyelectrolyte solutions. Effect of polyelectrolyte charge density

The influence of the polyion charge density on the osmotic behavior of linear polyelectrolytes is studied by the Monte Carlo method and by the modified Poisson-Boltzmann equation. The polyion is assumed to be either a uniformly or discretely charged cylinder placed along the axis of the cylindrical cell containing only counterions. As a result, osmotic saturation is observed at high polyion charge density. In simulations of solutions with divalent counterions the maximum in the osmotic pressure versus degree of ionization is observed in analogy with recent studies of spherical and planar systems. The correlation between the osmotic coefficient and the acceptance rate observed in simulations is reported.     

Electro-osmotic equilibria for a semipermeable shell filled with a solution of polyions

The authors study theoretically the electrostatic equilibria for a shell filled with a suspension of polyions (e.g., colloids, polyelectrolytes, etc.) and immersed in an infinite salt-free reservoir. The shell is treated as impermeable for polyions, but allowing free diffusion of counterions. From the solution of the linearized Poisson-Boltzmann equation we obtain the distribution of the potential and concentration profiles for polyions. The authors then derive explicit formulas for the excess electro-osmotic pressure of a polyion solution exerted by the shell. This is shown to be due to a concentration of polyions at the inner shell boundary and can be very different from the pressure of a corresponding bulk polyion solution.     

Counterion-counterion correlation in the double layer around cylindrical polyions: counterion size and valency effects

Monte Carlo simulation and Poisson-Boltzmann results on some aspects of structure and thermodynamics of aqueous polyelectrolyte solutions are presented. The polyelectrolyte solution is described by an infinitely long cylindrical polyion surrounded by counterions modeled as rigid ions moving in a continuum dielectric. Ion-ion correlations in the form of volume average of the counterion-counterion distribution function in the double layer surrounding the polyion are reported for mono- and divalent counterions and for a range of polyion concentrations and charge density parameters in each case. These results confirm again strong influence of the charge density parameter of polyions on properties of polyelectrolyte solutions. The structural information is supplemented by the calculated thermodynamic properties such as osmotic coefficients and heats of dilutions; the latter quantity has not been examined yet in detail by computer simulations. The results are discussed in view of the existing experimental data from the literature for these properties.     

Finite volume solution of the modified Poisson-Boltzmann equation for two colloidal particles

The double layer forces between spherical colloidal particles, according to the Poisson-Boltzmann (PB) equation, have been accurately calculated in the literature. The classical PB equation takes into account only the electrostatic interactions, which play a significant role in colloid science. However, there are at, and above, biological salt concentrations other non-electrostatic ion specific forces acting that are ignored in such modelling. In this paper, the electrostatic potential profile and the concentration profile of co-ions and counterions near charged surfaces are calculated. These results are obtained by solving the classical PB equation and a modified PB equation in bispherical coordinates, taking into account the van der Waals dispersion interactions between the ions and both surfaces. Once the electrostatic potential is known we calculate the double layer force between two charged spheres. This is the first paper that solves the modified PB equation in bispherical coordinates. It is also the first time that the finite volume method is used to solve the PB equation in bispherical coordinates. This method divides the calculation domain into a certain number of sub-domains, where the physical law of conservation is valid, and can be readily implemented. The finite volume method is implemented for several geometries and when it is applied to solve PB equations presents low computational cost. The proposed method was validated by comparing the numerical results for the classical PB calculations with previous results reported in the literature. New numerical results using the modified PB equation successfully predicted the ion specificity commonly observed experimentally.     

Ionic correlations in the inhomogeneous atmosphere surrounding cylindrical polyions: catalytic effects of polyions

The structural properties of linear polyelectrolyte solutions in the presence of a salt as evidenced through ionic correlations in the inhomogeneous atmosphere around a polyion and their consequence such as the catalytic potential are studied by using Monte Carlo simulation techniques. The simulations are performed on the cylindrical cell model where a uniformly charged hard cylinder mimics the linear polyion, which is caged in its own cylindrical cell containing counterions and salt. The cell (volume) average of the interionic correlations is presented as a function of the polyion and salt concentrations and ion radius. These results are utilized to study the catalytic effects of polyions as manifested through the changes in the collision frequency between ions in the double layer surrounding the polyion relative to that in the pure electrolyte solution. The reported results suggest a strong influence of the added salt/polyelectrolyte concentration ratio on the structural properties of the solution and hence on ion-ion collision frequency. The machine simulations are supplemented by nonlinear Poisson-Boltzmann results. Fair agreement between two different theoretical methods of calculating the collision frequency is obtained.     

Density functional theory for planar electric double layers: closing the gap between simple and polyelectrolytes

We report a nonlocal density functional theory (NLDFT) for polyelectrolyte solutions within the primitive model; i.e., the solvent is represented by a continuous dielectric medium, and the small ions and polyions by single and tangentially connected charged hard spheres, respectively. The excess Helmholtz energy functional is derived from a modified fundamental measure theory for hard-sphere repulsion, an extended first-order thermodynamic perturbation theory for chain connectivity, and a quadratic functional Taylor expansion for electrostatic correlations. With the direct and cavity correlation functions of the corresponding monomeric systems as inputs, the NLDFT predicts the segment-level microscopic structures and adsorption isotherms of polyelectrolytes at oppositely charged surfaces in good agreement with molecular simulations. In particular, it faithfully reproduces the layering structures of polyions, charge inversion, and overcharging that cannot be captured by alternative methods including the polyelectrolyte Poisson-Boltzmann equation and an earlier version of DFT. The NLDFT has also been used to investigate the influences of the small ion valence, polyion chain length, and size disparity between polyion segments and counterions on the microscopic structure, mean electrostatic potential, and overcharging in planar electric double layers containing polyelectrolytes.     

The density distributions of the counterions and the coions confined in two similarly charged plates

By using the field-theoretic method, we established a unified systematic formulation of a model of counterions and coions confined in two similarly charged plates, and calculated the density distributions of counterions and coions with various coupling parameters by the two methods: Poisson-Boltzmann (PB) approach and the strong coupling (SC) theory, respectively. We also performed Monte Carlo simulations, and obtained the density distributions of counterions and coions with several different coupling parameters. Comparing our theoretical results with those from Monte Carlo simulation, we find that the PB approach is valid when the coupling parameter Xi is smaller than 1, but, as Xi > or = 1, the results by the PB approach deviate from the corresponding Monte Carlo simulation data, and the deviation gets larger with the coupling parameter increasing. This shows that the PB approach is completely invalid when the coupling parameter is equal to 1 or larger than 1. For the latter case, the development trend of the distribution curve calculated by SC theory agrees with that from Monte Carlo simulation as the coupling parameter increases. This demonstrates that the SC theory can give a qualitative available explanation on the density distribution of the counterions in the system in which the coupling parameters are strictly confined.     

聚电解质溶液热力学性质的研究

本文采用柱形胞腔模型以及Vink的近似方法, 求解聚电解质溶液的Poisson-Boltzmann方程, 得到不同条件下聚离子周围静电势的分布。进而得到了不同条件下聚离子、反离子和同电荷离子的活度系数及溶剂的渗透系数。所得结果与实验值能较好地吻合。     

Preferential interaction between DNA and small ions in mixed-size counterion systems: Monte Carlo simulation and density functional study

Competitive binding between counterions around DNA molecule is characterized using the preferential interaction coefficient of individual ion in single and mixed electrolyte solutions. The canonical Monte Carlo (MC) simulation, nonlinear Poisson-Boltzmann (PB) equation, and density functional theory (DFT) proposed in our previous work [Wang, Yu, Gao, and Luo, J. Chem. Phys. 123, 234904 (2005)] are utilized to calculate the preferential interaction coefficients. The MC simulations and theoretical results show that for single electrolyte around DNA, the preferential interaction coefficient of electrolyte decreases as the cation size is increased, indicating that the larger cation has less accumulation ability in the vicinity of DNA. For the mixed electrolyte solution, it is found that cation diameter has a significant effect on the competitive ability while anion diameter has a negligible effect. It proves that the preferential interaction coefficients of all ions decrease as the total ionic concentration is increased. The DFT generally has better performance than the PB equation does when compared to the MC simulation data. The DFT behaves quite well for the real ionic solutions such as the KCl-NaCl-H2O, NaCl-CaCl2-H2O, and CaCl2-MgCl2-H2O systems.     

Towards an exact theory of polyelectrolytes

Within the framework of the gas approximation an exact method is developed for calculating the electric potential distribution around a polyion. The polyion is modelled as an infinitely long impermeable cylinder and the mobile ions are assumed to be impermeable spheres. Equations are derived for two potentials ψ_min and ψ_max so that the genuine potential ψ lies in between. The equations are numerically solved for different parameter values. The results are compared with data obtained on the basis of the standard Poisson-Boltzmann equation.     

Highly accurate biomolecular electrostatics in continuum dielectric environments

Implicit solvent models based on the Poisson-Boltzmann (PB) equation are frequently used to describe the interactions of a biomolecule with its dielectric continuum environment. A novel, highly accurate Poisson-Boltzmann solver is developed based on the matched interface and boundary (MIB) method, which rigorously enforces the continuity conditions of both the electrostatic potential and its flux at the molecular surface. The MIB based PB solver attains much better convergence rates as a function of mesh size compared to conventional finite difference and finite element based PB solvers. Consequently, highly accurate electrostatic potentials and solvation energies are obtained at coarse mesh sizes. In the context of biomolecular electrostatic calculations it is demonstrated that the MIB method generates substantially more accurate solutions of the PB equation than other established methods, thus providing a new level of reference values for such models. Initial results also indicate that the MIB method can significantly improve the quality of electrostatic surface potentials of biomolecules that are frequently used in the study of biomolecular interactions based on experimental structures.     

Efficient solution technique for solving the Poisson-Boltzmann equation

The Poisson-Boltzmann (PB) equation has been extensively used to analyze the energetics and structure of proteins and other significant biomolecules immersed in electrolyte media. A new highly efficient approach for solving PB-type equations that allows for the modeling of many-atoms structures such as encountered in cell biology, virology, and nanotechnology is presented. We accomplish these efficiencies by reformulating the elliptic PB equation as the long-time solution of an advection-diffusion equation. An efficient modified, memory optimized, alternating direction implicit scheme is used to integrate the reformulated PB equation. Our approach is demonstrated on protein composites (a polio virus capsid protomer and a pentamer). The approach has great potential for the analysis of supramillion atoms immersed in a host electrolyte.     

Aqueous phase behavior of polyelectrolytes with amphiphilic counterions modulated by cyclodextrin: the role of polyion flexibility

Polyelectrolytes with amphiphilic counterions, PEACs, are water insoluble because the amphiphiles self-assemble into highly charged micelles that strongly associate with the equally highly charged polyions. However, in the presence of water soluble cyclodextrins (CDs) that form inclusion complexes with the amphiphiles and prevent micellization, PEACs become soluble as the dispersed amphiphiles behave essentially as simple monovalent counterions. In this paper, we illustrate, by example, how strongly the ternary phase behavior of PEAC:CD:water depends on the polyion flexibility; for a highly flexible polyion (polyacrylate) the amphiphilic aggregates dictate the phase behavior, whereas a much stiffer polyion (DNA) itself dictates liquid crystalline ordering.     

Osmotic coefficient of a polyelectrolyte solution with a mixture of monovalent counterions

An analytical solution of the Poisson-Boltzmann equation is presented for the cell model of a polyelectrolyte solution with two species of monovalent counterions of different size. On this basis the osmotic coefficient is calculated as a function of the mole fraction of counterions and their radii. It is shown that the degree of binding of the smaller ion species increases as its mole fraction decreases.     

Electric double layer for spherical particles in salt-free concentrated suspensions including ion size effects

The equilibrium electric double layer (EDL) that surrounds colloidal particles is essential for the response of a suspension under a variety of static or alternating external fields. An ideal salt-free suspension is composed of charged colloidal particles and ionic countercharges released by the charging mechanism. Existing macroscopic theoretical models can be improved by incorporating different ionic effects usually neglected in previous mean-field approaches, which are based on the Poisson-Boltzmann equation (PB). The influence of the finite size of the ions seems to be quite promising because it has been shown to predict phenomena like charge reversal, which has been out of the scope of classical PB approximations. In this work we numerically obtain the surface electric potential and the counterion concentration profiles around a charged particle in a concentrated salt-free suspension corrected by the finite size of the counterions. The results show the high importance of such corrections for moderate to high particle charges at every particle volume fraction, especially when a region of closest approach of the counterions to the particle surface is considered. We conclude that finite ion size considerations are obeyed for the development of new theoretical models to study non-equilibrium properties in concentrated colloidal suspensions, particularly salt-free ones with small and highly charged particles.     

An application of the modified Poisson- Boltzmann equation in studies of osmotic properties of micellar solutions

The spherical cell model of colloidal solutions is applied in calculations of the osmotic pressure of micellar systems. The predictions of the nonlinear Poisson-Boltzmann equation (MPB) and of the Modified Poisson-Boltzmann equation (MPB) containing the leading terms of the fluctuation potential and the exclusion volume corrections to the mean potential acting on simple ions are compared with the results of recent computer simulations. Both PB and MPB seem satisfactory for solutions with monovalent counterions while the MPB is preferable for studies of the solutions containing divalent countenons.On leave from the University of Ljubljana, Ljubljana, Yugoslavia     

Theoretical study of catalytic effects in micellar solutions

The catalytic effect of charged micelles as manifested through the increased collision frequency between the counterions of an electrolyte in the presence of such micelles is explored by the Monte Carlo simulation technique and various theoretical approaches. The micelles and ions are pictured as charged hard spheres embedded in a dielectric continuum with the properties of water at 298 K with the charge on micelles varying from zero to z(m) = 50 negative elementary charges. Analytical theories such as (i) the symmetric Poisson-Boltzmann theory, (ii) the modified Poisson-Boltzmann theory, and (iii) the hypernetted-chain integral equation are applied and tested against the Monte Carlo data for micellar ions (m) with up to 50 negative charges in aqueous solution with monovalent counterions (c; z(c) = +1) and co-ions (co; z(co) = -1). The results for the counterion-counterion pair correlation function at contact, g(cc)(sigma(cc)), are calculated in a micellar concentration range from c(m) = 5 x 10(-)(6) to 0.1 mol/dm(3) with an added +1:-1 electrolyte concentration of 0.005 mol/dm(3) (for most cases), and for various model parameters. Our computations indicate that even a small concentration of a highly charged polyelectrolyte added to a +1:-1 electrolyte solution strongly increases the probability of finding two counterions in contact. This result is in agreement with experimental data. For low charge on the micelles (z(m) below -8), all the theories are in qualitative agreement with the new computer simulations. For highly charged micelles, the theories either fail to converge (the hypernetted-chain theory) or, alternatively, yield poor agreement with computer data (the symmetric Poisson-Boltzmann and modified Poisson-Boltzmann theories). The nonlinear Poisson-Boltzmann cell model results yield reasonably good agreement with computer simulations for this system.     

Counterion release from adsorbed highly charged polyelectrolyte: an electrooptical study

The adsorption of sodium poly(4-styrene sulfonate) on oppositely charged beta-FeOOH particles is studied by electrooptics. The focus of this paper is on the release of condensed counterions from adsorbed polyelectrolyte upon surface charge overcompensation. The fraction of condensed Na+ counterions on the adsorbed polyion surface is estimated according to the theory of Sens and Joanny and it is compared with the fraction of condensed counterions on nonadsorbed polyelectrolyte. The relaxation frequency of the electrooptical effect from the polymer-coated particle is found to depend on the polyelectrolyte molecular weight. This is attributed to polarization of the layer from condensed counterions on the polyion surface, being responsible for creation of the effect from particles covered with highly charged polyelectrolyte. The number of the adsorbed chains is calculated also assuming counterion condensation on the adsorbed polyelectrolyte and semiquantative agreement is found with the result obtained from the condensed counterion polarizability of the polymer-coated particle. Our findings are in line with theoretical predictions that the fraction of condensed counterions remains unchanged due to the adsorption of highly charged polyelectrolyte onto weakly charged substrate.     

Critical coagulation concentration of a salt-free colloidal dispersion

Both exact and approximate analytical solutions of the Poisson-Boltzmann equation for two planar, parallel surfaces are derived for the case when a dispersion medium contains counterions only, and the results obtained are used to evaluate the critical coagulation concentration of a spherical dispersion. A correction factor, which is a function of the valence of counterions, the surface potential of a particle, and the potential on the midplane between two particles at the onset of coagulation, is derived to modify the classic Schulze-Hardy rule for the dependence of the critical coagulation concentration on the valence of counterions. The correction factor is found to increase with the increase in the valence of counterions and/or with the increase in the surface potential. However, it approaches a constant value of 0.8390 if the surface potential is sufficiently high.