Linearized Poisson-Boltzmann Theory Research Articles

The crystallization enthalpy and entropy of protein solutions: microcalorimetry, van't Hoff determination and linearized Poisson-Boltzmann model of tetragonal lysozyme crystals.

During a first-order phase transition, a thermodynamic system releases or absorbs latent heat. Despite their fundamental importance, the heat or enthalpy change occurring during protein crystallization has been directly measured only in a few cases, and the associated entropy change can only be determined indirectly. This work provides an experimental determination and theoretical analysis of the dependence of the molar crystallization enthalpy of lysozyme solutions, ΔHxtal, on the physicochemical solution parameters. Its value is determined directly by isothermal microcalorimetry and indirectly by a van't Hoff analysis of solubility data, which quantitatively agree. This suggests a two-state crystallization process, in which oligomeric intermediates play a minor role. ΔHxtal is found to be negative on the order of few tens of the thermal energy per molecule. It is independent of protein concentration and stirring speed, but weakly depends on salt (NaCl) concentration and solution pH. Assuming that crystals are electrostatically neutral, these trends are explained by a linearized Poisson-Boltzmann theory. In addition, the molar crystallization entropy, ΔSxtal, is analyzed. The dependence of the van't Hoff entropy on salt concentration and pH is captured by the model, complementing the analysis of crystallization thermodynamics.

Poisson-Boltzmann study of the effective electrostatic interaction between colloids at an electrolyte interface

The effective electrostatic interaction between a pair of colloids, both of them located close to each other at an electrolyte interface, is studied by employing the full, nonlinear Poisson-Boltzmann (PB) theory within classical density functional theory. Using a simplified yet appropriate model, all contributions to the effective interaction are obtained exactly, albeit numerically. The comparison between our results and those obtained within linearized PB theory reveals that the latter overestimates these contributions significantly at short inter-particle separations. Whereas the surface contributions to the linear and the nonlinear PB results differ only quantitatively, the line contributions show qualitative differences at short separations. Moreover, a dependence of the line contribution on the solvation properties of the two adjacent fluids is found, which is absent within the linear theory. Our results are expected to enrich the understanding of effective interfacial interactions between colloids.

Analytic Model for the Dipole Potential of a Lipid Layer.

The larger permeability of anions than cations through a lipid bilayer can be rationalized by the positive sign of the bilayer's dipole potential. That is, upon crossing the lipid headgroups toward the hydrocarbon chain region, the electrostatic potential increases by several hundred millivolts. We derive an analytic expression for the dipole potential of a single lipid layer using an electrostatic model that is based on an extended version of the linearized Poisson-Boltzmann theory. The model highlights the ability of the lipid headgroups to render the dipole potential positive by inducing an orientational ordering of the solvent molecules. The positive contribution of the solvent overcompensates the negative dipole potential due to the bare lipids. Our theoretical prediction compares accurately with measurements of the dipole potential that we have conducted for mixed anionic-zwitterionic lipid monolayers at the air-water interface.

Effective electrostatic interactions in mixtures of charged colloids

We present a theory of effective electrostatic interactions in polydisperse suspensions of charged macroions, generalizing to mixtures a theory previously developed for monodisperse suspensions. Combining linear response theory with a random phase approximation for microion correlations, we coarse grain the microion degrees of freedom to derive general expressions for effective macroion-macroion pair potentials and a one-body volume energy. For model mixtures of charged hard-sphere colloids, we give explicit analytical expressions. The resulting effective pair potentials have the same general form as predicted by linearized Poisson-Boltzmann theory, but consistently incorporate dependence on macroion density and excluded volume via the Debye screening constant. The volume energy, which depends on the average macroion density, contributes to the free energy and so can influence thermodynamic properties of deionized suspensions. To validate the theory, we compute radial distribution functions of binary mixtures of oppositely charged colloidal macroions from molecular dynamics simulations of the coarse-grained model (with implicit microions), taking effective pair potentials as input. Our results agree closely with corresponding results from more computationally intensive Monte Carlo simulations of the primitive model (with explicit microions). Simulations of a mixture with large size and charge asymmetries indicate that charged nanoparticles can enhance electrostatic screening of charged colloids. The theory presented here lays a foundation for future large-scale modeling of complex mixtures of charged colloids, nanoparticles, and polyelectrolytes.

Interaction between heterogeneously charged surfaces: Surface patches and charge modulation

When solid surfaces are immersed in aqueous solutions, some of their charges can dissociate and leave behind charged patches on the surface. Although the charges are distributed heterogeneously on the surface, most of the theoretical models treat them as homogeneous. For overall non-neutral surfaces, the assumption of surface charge homogeneity is rather reasonable since the leading terms of two such interacting surfaces depend on the nonzero average charge. However, for overall neutral surfaces the nature of the surface charge distribution is crucial in determining the intersurface interaction. In the present work we study the interaction between two charged surfaces across an aqueous solution for several charge distributions. The analysis is preformed within the framework of the linearized Poisson-Boltzmann theory. For periodic charge distributions the interaction is found to be repulsive at small separations, unless the two surface distributions are completely out-of-phase with respect to each other. For quenched random charge distributions we find that due to the presence of the ionic solution in between the surfaces, the intersurface repulsion dominates over the attraction in the linear regime of the Poisson-Boltzmann theory. The effect of quenched charge heterogeneity is found to be particularly substantial in the case of large charged domains.

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

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.

Solvated calcium ions in charged silica nanopores

Hydroxyl surface density in porous silica drops down to nearly zero when the pH of the confined aqueous solution is greater than 10.5. To study such extreme conditions, we developed a model of slit silica nanopores where all the hydrogen atoms of the hydroxylated surface are removed and the negative charge of the resulting oxygen dangling bonds is compensated by Ca(2+) counterions. We employed grand canonical Monte Carlo and molecular dynamics simulations to address how the Ca(2+) counterions affect the thermodynamics, structure, and dynamics of confined water. While most of the Ca(2+) counterions arrange themselves according to the so-called "Stern layer," no diffuse layer is observed. The presence of Ca(2+) counterions affects the pore filling for strong confinement where the surface effects are large. At full loading, no significant changes are observed in the layering of the first two adsorbed water layers compared to nanopores with fully hydroxylated surfaces. However, the water structure and water orientational ordering with respect to the surface is much more disturbed. Due to the super hydrophilicity of the Ca(2+)-silica nanopores, water dynamics is slowed down and vicinal water molecules stick to the pore surface over longer times than in the case of hydroxylated silica surfaces. These findings, which suggest the breakdown of the linear Poisson-Boltzmann theory, provide important information about the properties of nanoconfined electrolytes upon extreme conditions where the surface charge and ion concentration are large.

Interaction of a Spherical Colloid and a Porous Membrane in a Bulk Electrolyte

A systemic computation of an electrostatic interaction between a charged spherical colloid and a charged porous membrane with a fixed potential is made under the linear Poisson-Boltzmann theory. The colloid moves along the symmetry axis of the membrane and they are both immersed in a bulk electrolyte. In the calculation, a significant attraction between the colloid and the membrane is found. The orifices on or around the centre of the membrane play a major role in the attraction. The effect of the reduced orifice sizes of the membrane on the interaction is taken into account. Furthermore, the electrostatic interaction energies are significantly changed by the variation of ionic strengths (concentration and valence relating the Dybe length).

Crucial roles of charged saccharide moieties in survival of gram negative bacteria against protamine revealed by combination of grazing incidence x-ray structural characterizations and Monte Carlo simulations

Grazing incidence x-ray scattering techniques and Monte Carlo (MC) simulations are combined to reveal the influence of molecular structure (genetic mutation) and divalent cations on the survival of gram negative bacteria against cationic peptides such as protamine. The former yields detailed structures of bacterial lipopolysaccharide (LPS) membranes with minimized radiation damages, while the minimal computer model based on the linearized Poisson-Boltzmann theory allows for the simulation of conformational changes of macromolecules (LPSs and peptides) that occur in the time scale of ms. The complementary combination of the structural characterizations and MC simulation demonstrates that the condensations of divalent ions (Ca2+ or Mg2+) in the negatively charged core saccharides are crucial for bacterial survival.

Ions and Inhibitors in the Binding Site of HIV Protease: Comparison of Monte Carlo Simulations and the Linearized Poisson-Boltzmann Theory

Proteins can be influenced strongly by the electrolyte in which they are dissolved, and we wish to model, understand, and ultimately control such ionic effects. Relatively detailed Monte Carlo (MC) ion simulations are needed to capture biologically important properties of ion channels, but a simpler treatment of ions, the linearized Poisson-Boltzmann (LPB) theory, is often used to model processes such as binding and folding, even in settings where the LPB theory is expected to be inaccurate. This study uses MC simulations to assess the reliability of the LPB theory for such a system, the constrained, anionic active site of HIV protease. We study the distributions of ions in and around the active site, as well as the energetics of displacing ions when a protease inhibitor is inserted into the active site. The LPB theory substantially underestimates the density of counterions in the active site when divalent cations are present. It also underestimates the energy cost of displacing these counterions, but the error is not consequential because the energy cost is less than k B T, according to the MC calculations. Thus, the LPB approach will often be suitable for studying energetics, but the more detailed MC approach is critical when ionic distributions and fluxes are at issue.

Curvature dependence of the electrolytic liquid-liquid interfacial tension

The interfacial tension of a liquid droplet surrounded by another liquid in the presence of microscopic ions is studied as a function of the droplet radius. An analytical expression for the interfacial tension is obtained within a linear Poisson-Boltzmann theory and compared with numerical results from nonlinear Poisson-Boltzmann theory. The excess liquid-liquid interfacial tension with respect to the pure salt-free liquid-liquid interfacial tension is found to decompose into a curvature-independent part due to short-ranged interfacial effects and a curvature-dependent electrostatic contribution. Several curvature-dependent regimes of different scalings of the electrostatic excess interfacial tension are identified. Symmetry relations of the interfacial tension upon swapping droplet and bulk liquid are found to hold in the low-curvature limit, which, e.g., lead to a sign change of the excess Tolman length. For some systems a low-curvature expansion up to the second order turns out to be applicable if and only if the droplet size exceeds the Debye screening length in the droplet, independent of the Debye length in the bulk.

Finite-Thickness-Enhanced Attractions for Oppositely Charged Membranes and Colloidal Platelets

Within linearized Poisson-Boltzmann theory, we study the disjoining pressure of two oppositely charged parallel objects (membranes and colloidal platelets) in a 1:1 electrolyte, with a focus on the effects of their finite thickness. This extension of the standard Gouy-Chapman model from an interacting pair of double layers to a quartet (one on each side of the two interacting objects) is shown to enhance the regime of attractive interactions significantly, in particular, when the separation and the thickness are on the order of the Debye length of the solvent, provided the dielectric mismatch between objects and solvent is not too extreme. The enhancement of attractions occurs for objects with fixed charge as well as for those that exhibit charge regulation but not for those with a constant surface potential. The underlying mechanism for this enhancement for thin objects is the transfer of net ionic charge from the electrolyte in between to the other sides. For biological membranes in water, this effect is small; however, it is due to strong image charge effects.

Volume terms for charged colloids: A grand-canonical treatment

We present a study of thermodynamic properties of suspensions of charged colloids on the basis of linear Poisson-Boltzmann theory. We calculate the effective Hamiltonian of the colloids by integrating out the ionic degrees of freedom grand canonically. This procedure not only yields the well-known pairwise screened-Coulomb interaction between the colloids, but also additional volume terms that affect the phase behavior and the thermodynamic properties, such as the osmotic pressure. These calculations are greatly facilitated by the grand-canonical character of our treatment of the ions and allow for relatively fast computations compared to earlier studies in the canonical ensemble. Moreover, the present derivation of the volume terms are relatively simple, make a direct connection with Donnan equilibrium, yield an explicit expression for the effective screening constant, and allow for extensions to include, for instance, nonlinear effects.

The interplay between screening properties and colloid anisotropy: Towards a reliable pair potential for disc-like charged particles

The electrostatic potential of a highly charged disc (clay platelet) in an electrolyte is investigated in detail. The corresponding non-linear Poisson-Boltzmann (PB) equation is solved numerically, and we show that the far-field behaviour (relevant for colloidal interactions in dilute suspensions) is exactly that obtained within linearized PB theory, with the surface boundary condition of a uniform potential. The latter linear problem is solved by a new semi-analytical procedure and both the potential amplitude (quantified by an effective charge) and potential anisotropy coincide closely within PB and linearized PB, provided the disc bare charge is high enough. This anisotropy remains at all scales; it is encoded in a function that may vary over several orders of magnitude depending on the azimuthal angle under which the disc is seen. The results allow to construct a pair potential for discs interaction, that is strongly orientation dependent.

Optimal linearized Poisson-Boltzmann theory applied to the simulation of flexible polyelectrolytes in solution.

Optimal linearized Poisson-Boltzmann (OLPB) theory is applied to the simulation of flexible polyelectrolytes in solution. As previously demonstrated in the contexts of the cell model [H. H. von Grunberg, R. van Roij, and G. Klein, Europhys. Lett. 55, 580 (2001)] and a particle-based model [B. Beresfordsmith, D. Y. C. Chan, and D. J. Mitchell, J. Colloid Interface Sci. 105, 216 (1985)] of charged colloids, OLPB theory is applicable to thermodynamic states at which conventional, Debye-Huckel (DH) linearization of the Poisson-Boltzmann equation is rendered invalid by violation of the condition that the electrostatic coupling energy of a mobile ion be much smaller than its thermal energy throughout space, |nu(alpha)e psi(r)|<<k(B)T. As a demonstration of its applicability to flexible polyelectrolytes, OLPB theory is applied to a concentrated solution of freely jointed chains. The osmotic pressure is computed at various reservoir ionic strengths and compared with results from the conventional DH model for polyelectrolytes. Through comparison with the cylindrical cell model for polyelectrolytes, it is demonstrated that the OLPB model yields the correct osmotic pressure behavior with respect to nonlinear theory where conventional DH theory fails, namely at large ratios of mean counterion density to reservoir salt density, when the Donnan potential is large.

Effective charge saturation in colloidal suspensions

Because micro-ions accumulate around highly charged colloidal particles in electrolyte solutions, the relevant parameter to compute their interactions is not the bare charge, but an effective (or renormalized) quantity, whose value is sensitive to the geometry of the colloid, the temperature or the presence of added-salt. This nonlinear screening effect is a central feature in the field of colloidal suspensions or polyelectrolyte solutions. We propose a simple method to predict effective charges of highly charged macro-ions, that is reliable for monovalent electrolytes (and counterions) in the colloidal limit (large size compared to both screening length and Bjerrum length). Taking reference to the non linear Poisson–Boltzmann theory, the method is successfully tested against the geometry of the macro-ions, the possible confinement in a Wigner–Seitz cell, and the presence of added salt. Moreover, our results are corroborated by various experimental measures reported in the literature. This approach provides a useful route to incorporate the nonlinear effects of charge renormalization within a linear theory for systems where electrostatic interactions play an important role.

Molecular dynamics simulation of interfacial electrolyte behaviors in nanoscale cylindrical pores

Molecular simulations have been carried out on aqueous electrolytes in cylindrical pores a few nanometers in diameter, with uncharged wall or with dispersed, discrete charges modeling silica. The results show a classical Stern layer of adsorbed counterions near the wall, and then a diffuse layer with depleted ion concentration in the interfacial region. The depletion region coincides roughly with the more ordered water structure in the interface. Examination of the interaction energy shows that hydration energy of the ions disfavors the interior, suggesting the importance of the solvent structure near the wall. Comparison with the linearized Poisson–Boltzmann theory indicates qualitative differences in the predicted interfacial behavior.

Dynamic linear Poisson–Boltzmann theory

We construct an extended-Lagrangian Brownian dynamics algorithm for charged polymers in simple electrolyte solutions which explicitly couples fluctuations in polyion conformation to charge density fluctuations in the ion atmosphere without explicit simulation of solvent and counterions. The algorithm is applicable in the Debye–Hückel limit of low solvent ionic strength. The theoretical description of the system has the appearance of conventional linearized Poisson–Boltzmann theory, except that the screening lengths become space- and time-dependent, hence we consider the algorithm a “dynamic” extension of linearized Poisson–Boltzmann theory. Use of the algorithm is illustrated in a simple model system, and some polyion properties are compared to exact results obtained by conventional Brownian dynamics simulation.

Screened electrostatic interactions between clay platelets

An effective pair potential for systems of uniformly charged lamellar colloids in the presence of an electrolytic solution of microscopic co- and counterions is derived. The charge distribution on the discs is expressed as a collection of multipole moments, and the tensors which determine the interactions between these multipoles are derived from a screened Coulomb potential. Unlike previous theoretical studies of such systems, the interaction energy may now be expressed for discs at arbitrary mutual orientation. The potential is shown to be exactly equivalent to the use of linearized Poisson–Boltzmann theory.

Thermodynamics of association to a molecule immobilized in an electric double layer

A thermodynamic theory of association to a molecule immobilized near a surface has been developed. Exact equations for the binding enthalpy, entropy and equilibrium reaction constant for an immobilized complex are derived. Using linear Poisson–Boltzmann theory of the electric double-layer interaction between an ion-penetrable sphere and a hard plate allows a closed form evaluation. We briefly discuss application of the theory to a DNA chip at high (1 M NaCl) and low (0.01 M NaCl) ionic strength for dielectric and metallic substrates. Predicted strong electrostatic effects suggest the feasibility of electronic control of DNA hybridization and design of chips avoiding the DNA folding problem.