Contact Value Theorem Research Articles

Explicit solvent effects on macromolecular interactions from a solvent-augmented contact value theorem.

The Derjaguin-Landau-Verywey-Overbeek (DLVO) theory has been a remarkably accurate framework for the characterization of macromolecular stability in water solvent. In view of its solvent-implicit nature neglecting the electrostatics of water molecules with non-negligible charge structure and concentration, the precision of the DLVO formalism is somewhat puzzling. In order to shed light on this issue, we derive from our earlier explicit solvent formalism [S. Buyukdagli et al., Phys. Rev. E 87, 063201 (2013)1539-375510.1103/PhysRevE.87.063201] a solvent-augmented contact value theorem and assess the contribution of solvent molecules to the interaction of charged membranes. We find that in the case of hydrophobic membranes with fixed charges embedded in the membrane surface, the nearly exact cancellation of various explicit solvent effects of substantially large magnitude but opposite sign keeps the intermembrane pressure significantly close to the double-layer force of the DLVO theory. Then, in the case of hydrophilic surface charge groups within the aqueous region, due to the spatial separation of the membrane substrate from the location of the fixed charges where the nonlocal dielectric response of the structured solvent is sharply localized, the interfacial field energy and the contact charge densities remain unaffected by the explicit solvent. As a result, the hydration of the lipid head groups suppresses the signature of the solvent molecules from the membrane interaction force.

Electric double layers with modulated surface charge density: exact 2D results

Electric double layers (EDL) with counterions only, say electrons with the elementary charge −e, in thermal equilibrium at the inverse temperature β are considered. In particular, we study the effect of the surface charge modulation on the particle number density profile and the pressure. The mobile particles are constrained to the surface of a 2D cylinder and immersed in vacuum (no dielectric image charges). An EDL corresponds to the end-circle of the cylinder which carries a (fixed) position-dependent line charge density. The geometries of one single EDL and two EDLs at distance d are considered; the particle density profile is studied for both geometries, the effective interaction of two EDLs is given by the particle pressure on either of the line walls. For any coupling constant Γ ≡ βe 2 = 2 × integer, there exists a mapping of the 2D one-component Coulomb system onto the 1D interacting anticommuting-field theory defined on a chain of sites. Using specific transformation symmetries of anticommuting variables, the contact value theorem is generalized to the EDL with modulated line charge density. For the free-fermion coupling Γ = 2 it is shown that, under certain conditions, the matrix of interaction strengths between anticommuting variables diagonalizes itself which permits one to obtain exact formulas for the particle density profile as well as the pressure. The obtained results confirm the previous indications of weak-coupling and Monte Carlo observations that the surface charge inhomogeneity implies an enhancement of the counterion density at the contact with the charged line and a diminution of the pressure between two parallel lines in comparison with the uniformly charged ones (with the same mean charge densities).

Osmotic pressure of permeable ionic microgels: Poisson-Boltzmann theory and exact statistical mechanical relations in the cell model.

Ionic microgels are soft colloidal particles, composed of crosslinked polymer networks, which ionize and swell when dispersed in a good solvent. Swelling of these permeable, compressible particles involves a balance of electrostatic, elastic, and mixing contributions to the single-particle osmotic pressure. The electrostatic contribution depends on the distributions of mobile counterions and coions and of fixed charge on the polymers. Within the cell model, we employ two complementary methods to derive the electrostatic osmotic pressure of ionic microgels. In Poisson-Boltzmann (PB) theory, we minimize a free energy functional with respect to the electrostatic potential to obtain the bulk pressure. From the pressure tensor, we extract the electrostatic and gel contributions to the total pressure. In a statistical mechanical approach, we vary the free energy with respect to microgel size to obtain exact relations for the microgel electrostatic osmotic pressure. We present results for planar, cylindrical, and spherical geometries. For models of membranes and microgels with fixed charge uniformly distributed over their surface or volume, we derive analogs of the contact value theorem for charged colloids. We validate these relations by solving the PB equation and computing ion densities and osmotic pressures. When implemented within PB theory, the two methods yield identical electrostatic osmotic pressures for surface-charged microgels. For volume-charged microgels, the exact electrostatic osmotic pressure equals the average of the corresponding PB profile over the gel volume. We demonstrate that swelling of ionic microgels depends on the variation of the electrostatic pressure inside the particle and discuss implications for interpreting experiments.

A correlation-hole approach to the electric double layer with counter-ions only

ABSTRACTWe study a classical system of identically charged counter-ions near a planar wall carrying a uniform surface charge density. The equilibrium statistical mechanics of the system depends on a single dimensionless coupling parameter. A new self-consistent theory of the correlation-hole type is proposed which leads to a modified Poisson–Boltzmann integral equation for the density profile, convenient for analytical progress and straightforward to solve numerically. The exact density profiles are recovered in the limits of weak and strong couplings. In contrast to previous theoretical attempts of the test-charge family, the density profiles fulfil the contact-value theorem at all values of the coupling constant and exhibit the mean-field decay at asymptotically large distances from the wall, as expected. We furthermore show that the density corrections at large couplings exhibit the proper dependence on coupling parameter and distance to the charged wall. The numerical results for intermediate values of the coupling provide accurate density profiles which are in good agreement with those obtained by Monte Carlo simulations. The crossover to mean-field behaviour at large distance is studied in detail.

Asymmetric Coulomb fluids at randomly charged dielectric interfaces: anti-fragility, overcharging and charge inversion.

We study the distribution of multivalent counterions next to a dielectric slab, bearing a quenched, random distribution of charges on one of its solution interfaces, with a given mean and variance, both in the absence and in the presence of a bathing monovalent salt solution. We use the previously derived approach based on the dressed multivalent-ion theory that combines aspects of the strong and weak coupling of multivalent and monovalent ions in a single framework. The presence of quenched charge disorder on the charged surface of the dielectric slab is shown to substantially increase the density of multivalent counterions in its vicinity. In the counterion-only model (with no monovalent salt ions), the surface disorder generates an additional logarithmic attraction potential and thus an algebraically singular counterion density profile at the surface. This behavior persists also in the presence of a monovalent salt bath and results in significant violation of the contact-value theorem, reflecting the anti-fragility effects of the disorder that drive the system towards a more "ordered" state. In the presence of an interfacial dielectric discontinuity, depleting the counterion layer at the surface, the charge disorder still generates a much enhanced counterion density further away from the surface. Likewise, the charge inversion and/or overcharging of the surface occur more strongly and at smaller bulk concentrations of multivalent counterions when the surface carries quenched charge disorder. Overall, the presence of quenched surface charge disorder leads to sizable effects in the distribution of multivalent counterions in a wide range of realistic parameters and typically within a distance of a few nanometers from the charged surface.

On the contact values of the density profiles in an electric double layer using density functional theory

A recently proposed local second contact value theorem [Henderson D., Boda D., J. Electroanal. Chem., 2005, 582, 16] for the charge profile of an electric double layer is used in conjunction with the existing Monte Carlo data from the literature to assess the contact behavior of the electrode-ion distributions predicted by the density functional theory. The results for the contact values of the co- and counterion distributions and their product are obtained for the symmetric valency, restricted primitive model planar double layer for a range of electrolyte concentrations and temperatures. Overall, the theoretical results satisfy the second contact value theorem reasonably well, the agreement with the simulations being semi-quantitative or better. The product of the co- and counterion contact values as a function of the electrode surface charge density is qualitative with the simulations with increasing deviations at higher concentrations.

Equation of state of charged colloidal suspensions and its dependence on the thermodynamic route

The thermodynamic properties of highly charged colloidal suspensions in contact with a salt reservoir are investigated in the framework of the renormalized Jellium model (RJM). It is found that the equation of state is very sensitive to the particular thermodynamic route used to obtain it. Specifically, the osmotic pressure calculated within the RJM using the contact value theorem can be very different from the pressure calculated using the Kirkwood-Buff fluctuation relations. On the other hand, Monte Carlo simulations show that both the effective pair potentials and the correlation functions are accurately predicted by the RJM. It is suggested that the lack of self-consistency in the thermodynamics of the RJM is a result of neglected electrostatic correlations between the counterions and coions.

Effects of counterion size on the attraction between similarly charged surfaces

Interaction between similarly charged surfaces can be attractive at high electrostatic coupling constants Ξ = l(B)Z(2)/μ(GC), where l(B) is the Bjerrum length, μ(GC) the Gouy-Chapman length, and Z the valency of counterions. While this effect has been studied previously in detail, as a function of surface charge density and valency of the pointlike counterions, much less is known about the effect of counterion size. We apply the Wang-Landau sampling Monte Carlo (MC) simulation method to compute the free energy F as a function of the scaled distance between the plates D̃=D/μ(GC) for a range of Ξ and scaled counterion radii R̃=R/μ(GC). We find that for large Ξ and small ion radius, there is a global equilibrium distance D̃=D̃(eq)=2(1+R̃), correctly giving the expected value at the point counterion limit. With increasing R̃ the global minimum in F(D̃) changes to a metastable state and finally this minimum vanishes when R̃ reaches a critical value, which depends on Ξ. We present a state diagram indicating approximate boundaries between these three regimes. The Wang-Landau MC method, as it is applied here, offers a possibility to study a wide spectrum of extended problems, which cannot be treated by the use of contact value theorem.

A MODIFIED POISSON-BOLTZMANN STUDY OF THE SINGLET ION DISTRIBUTION AT CONTACT WITH THE ELECTRODE FOR A PLANAR ELECTRIC DOUBLE LAYER.

The properties of the singlet ion distributions at and around contact in a restricted primitive model double layer are characterized in the modified Poisson-Boltzmann theory. Comparisons are made with the corresponding exact Monte Carlo simulation data, the results from the Gouy-Chapman-Stern theory coupled to an exclusion volume term, and the mean spherical approximation. Particular emphasis is given to the behaviour of the theoretical predictions in relation to the contact value theorem involving the charge profile. The simultaneous behaviour of the coion and counterion contact values is also examined. The performance of the modified Poisson-Boltzmann theory in regard to the contact value theorems is very reasonable with the contact characteristics showing semi-quantitative or better agreement overall with the simulation results. The exclusion-volume-treated Gouy-Chapman-Stern theory reveals a fortuitous cancellation of errors, while the mean spherical approximation is poor.

Structure of penetrable sphere fluids and mixtures near a slit hard wall: A modified bridge density functional approximation

The modified density functional theory, which is based both on the bridge density functional and the contact value theorem, has been proposed for the structural properties of penetrable sphere fluids and their mixtures near a slit hard wall. The Verlet-modified bridge function proposed by Choudhury and Ghosh [J. Chem. Phys. 119, 4827 (2003)] for one-component system has been extended for fluid mixtures. The radial distribution functions obtained from the Verlet-modified bridge function are in excellent agreement with computer simulations over a wide range of density and temperature and better than those obtained from the standard integral theories including the Percus-Yevick and hypernetted-chain closures. The calculated particle density distributions confined in a slit pore are also found to be reasonably good compared to the simulation data. Even for high density systems the accuracy of the hypernetted-chain and the mean-field approximation functionals increase with increasing temperature. However, the agreement between theory and simulation slightly deteriorates in the systems of low temperature.

Evidence from Monte Carlo simulations for a second contact value theorem for a double layer formed by 2:1/1:2 salts at low electrode charges

A generalization of a recently advanced second contact theorem [Henderson and Boda, J. Electroanal. Chem. 582, 16 (2005)], a local contact condition for the charge profile of a symmetric valency restricted primitive model planar double layer at low electrode charge, is proposed to cater to asymmetric 2:1/1:2 electrolytes. Monte Carlo simulations performed for such asymmetric electrolytes at a set of physical states with different concentrations and temperatures present convincing evidence that the generalization of the Henderson–Boda expression is valid at low electrode charge. Theoretical support is provided by the corresponding modified Poisson–Boltzmann results, which are seen to follow the simulation data closely.

Some simulation and modified Poisson–Boltzmann theory results for the contact values of an electrolyte near a charged electrode

Abstract Since the development of a contact value theorem by Henderson et al. it has been known that the prediction of the venerable Gouy–Chapman (GC) theory for the contact value of the density profile of an electrolyte near an electrode is incorrect. The more recent semi-empirical work of Fawcett and Henderson and the simulations of Boda and Henderson have shown that the contact value of the charge profile at low electrode charge is also given incorrectly by the GC theory, at least for a binary symmetric restricted primitive model electrolyte. In the present study we examine the contact values of the profiles of the double layer formed by the electrolyte species in a binary symmetric restricted primitive model electrolyte as a function of the electrode charge by means of simulation. It is difficult to extract detailed information from simulation values of the density and charge profiles at large electrode charge because the contact values of these profiles are dominated by the large quadratic term in the electrode charge. However, we find from our simulations for a fairly wide range of concentrations and electrode charges, that the product of the counterion and coion profiles is a sensitive probe with which to study the double layer because it is a direct test of the basis of the GC approximation that the counterion and coion profiles are exponentials of ± zeψ ( x )/ kT and so have a product that is identically one. We find that this product is not one or even a constant as the electrode charge is varied and, thus, the GC contact values do not agree even qualitatively with the simulation results. In contrast, the modified Poisson–Boltzmann theory is quite successful in accounting for the behavior of this product.

Structure of a freely jointed tangent hard-sphere chain in hard slit pores

A density functional theory (DFT), which is based both on the density functional expansion of the first-order direct correlational functional with respect to the bulk molecular density and on the contact-value theorem at the hard wall, has been developed for studying the structure of a freely jointed tangent hard-sphere chain in hard slit pores. Two different second-order direct correlation functions (DCFs) based on the Polymer Reference Interaction Site model (PRISM) coupled with the Percus–Yevick closure relation and the Wertheim's first-order perturbation theory modified by Yu and Wu (modified TPT1) were used as inputs. The calculated results show that (i) the PRISM theory is in excellent agreement with the computer simulations even for high densities, and that (ii) the calculated local density distribution strongly depends on the second-order DCFs as well as the DFT version employed.

Field theoretic derivation of the contact value theorem in planar geometries and its modification by the Casimir effect.

The contact value theorem for Coulomb gases in planar or filmlike geometries is derived using a Hamiltonian field theoretic representation of the system. The case where the film is enclosed by a material of different dielectric constant to that of the film is shown to contain an additional Casimir-like term which is generated by fluctuations of the electric potential about its mean-field value.

The capacitance of the solvent primitive model double layer at low effective temperatures

The effect of the solvent, modeled as a hard sphere fluid, on the behavior of the double layer at low temperatures is studied by means of Monte Carlo simulations. We find that the density profiles are strongly affected by the addition of solvent molecules. The profiles pass from near monotonic functions that exhibit partial drying to oscillatory functions that exhibit partial wetting as the bulk density of the solvent is increased in the system. However, the capacitance, as in the case of the primitive model, increases at low effective temperatures, while after a maximum it decreases at higher effective temperatures as the temperature increases. We find that the adsorption of the ions decreases and changes sign from positive to negative with the decreasing temperature at low electrode charges. Some of these features can be explained by means of a contact value theorem.

Highly charged colloidal suspensions in planar slits

A simple density functional approximation, which is based both on the density functional Taylor series expansion of the one-particle direct correlation function (DCF) and on the exact contact value theorem for a structureless hard wall with infinity wall distance, has been developed to study the structural properties of a highly charged colloidal suspension and compared with the computer simulation. Two kinds of two-particle DCFs, which are the rescaled mean spherical approximation (RMSA) and the Rogers-Young (RY) closure relation, have been used to calculate the local concentration profiles of a highly charged colloidal suspension confined in charged silt walls as input. The calculated results show that for a structureless hard wall the present approximation is better than the hypernetted chain (HNC) functional approximation and describes well the structural properties of a confined colloidal suspension compared with the computer simulation. However, the agreement between the computer simulation and the theory for the density profiles deteriorates with increasing the wall-particle interaction.

Density functional approximations for confined classical fluids

A density functional approximation, which is based on both the density functional Taylor series expansion of the one-particle direct correlation function and the exact contact value theorem for a hard wall, has been proposed to study the structural properties of confined classical fluids. The approximation has been applied to calculate the density profiles of sticky hard-sphere fluids confined in structureless hard walls. The calculated density profiles have shown that the present approximation compares very well with the results from the computer simulation. Furthermore, a density functional perturbative approximation, which is based on both the weighted-density approximation for the repulsive part of potential and the present approximation for the attractive part of potential, has been developed to predict the density profiles of model fluids with the attractive part of potential and has been applied to calculate the density profiles of hard-sphere Yukawa fluids near a planar slit. The calculated results also show that the proposed perturbative approximation is a significant improvement upon those of the modified version of the Lovett-Mou-Buff-Wertheim, and compares very well with the computer simulation.

Hard-sphere Yukawa fluid near a planar slit

A density functional perturbative approximation, which is based on both the weighted-density approximation (WDA) of Tarazona and the density functional approximation of Rickayzen et al., has been employed to predict the density profiles of the hard-sphere attractive Yukawa fluid near a planar slit. The calculated density profiles show that the density functional perturbative approximation is a significant improvement upon those of the modified version of the Lovett–Mou–Buff–Wertheim (LMBW-1), which uses the exact contact value theorem, and compares very well with the computer simulation even for the low temperatures in the vicinity of the liquid–vapor fluid bulk transition.

Grand canonical Monte Carlo and modified singlet integral equations for the density profile of a Yukawa fluid near a planar wall

Results for the density profile for Yukawa molecules near a hard wall and an exponential attractive wall are presented for Grand Canonical Monte Carlo (GCMC) simulations, for the singlet hypernetted chain (HNC) integral equation and for a modified version of the Lovett–Mou–Buff–Wertheim (LMBW-1) which uses the exact contact value theorem. The results of the standard singlet HNC are quite poor. If the LMBW equation is modified (but still using the bulk direct correlation function) the results at high temperature become reasonable. However, the results at low temperatures, close to the bulk coexistence curve, are only a partial improvement. The contact value of the density profile is now quite good but the profile has the incorrect oscillatory behavior predicted by the unmodified approximate equation. The GCMC simulation results show a clear effect of drying at low temperatures.

Mesoscopic description of an electrochemical interface

A theory describing mesoscopic equilibrium and near-equilibrium behavior of polar solutions near electrochemical interfaces is presented. It is based on an expression for the stress tensor found within the framework of a non-local Landau order parameter approach which takes into account the possibility of continuous surface phase transitions in the double layer. Some applications of the theory are discussed, including the generalization of the contact value theorem used in the theory of colloids and thin films and the interpretation of the singular behavior of the surface viscosity and capacitance.