Bragg-Williams Theory for Particles with a Size-Modulating Internal Degree of Freedom.

The field of soft matter teems with molecules and aggregates of molecules that have internal size-modulating degrees of freedom. Proteins, peptides, microgels, polymers, micelles, and even some colloids can exist in multiple-often just two dominating-states with different effective sizes, where size can refer to the volume or to the cross-sectional area for particles residing on surfaces. The size-dependence of their accessible states renders the behavior of these particles pressure-sensitive. The Bragg-Williams model is among the most simple mean-field methods to translate the presence of inter-particle interactions into an approximate phase diagram. Here, we extend the Bragg-Williams model to account for the presence of particles that are immersed in a solvent and exist in two distinct states, one occupying a smaller and the other one a larger size. The basis of the extension is a lattice-sublattice approximation that we use to host the two size-differing states. Our model includes particle-solvent interactions that act as an effective surface tension between particles and solvent and are ignorant of the state in which the particles reside. We analyze how the energetic preference of the particles for one or the other state affects the phase diagrams. The possibility of a single phase-two phases-single phase sequence of phase transitions as a function of increasing temperature is demonstrated.

Incorporation of Molecular Reorientation into Modeling Surface Pressure-Area Isotherms of Langmuir Monolayers.

Langmuir monolayers can be assembled from molecules that change from a low-energy orientation occupying a large cross-sectional area to a high-energy orientation of small cross-sectional area as the lateral pressure grows. Examples include cyclosporin A, amphotericin B, nystatin, certain alpha-helical peptides, cholesterol oxydation products, dumbbell-shaped amphiphiles, organic-inorganic nanoparticles and hybrid molecular films. The transition between the two orientations leads to a shoulder in the surface pressure-area isotherm. We propose a theoretical model that describes the shoulder and can be used to extract the energy cost per molecule for the reorientation. Our two-state model is based on a lattice-sublattice approximation that hosts the two orientations and a corresponding free energy expression which we minimize with respect to the orientational distribution. Inter-molecular interactions other than steric repulsion are ignored. We provide an analysis of the model, including an analytic solution for one specific lateral pressure near a point of inflection in the surface pressure-area isotherm, and an approximate solution for the entire range of the lateral pressures. We also use our model to estimate energy costs associated with orientational transitions from previously reported experimental surface pressure-area isotherms.

Saddle-curvature instability of lipid bilayer induced by amphipathic peptides: a molecular model.

Amphipathic peptides that partition into lipid bilayers affect the curvature elastic properties of their host. Some of these peptides are able to shift the Gaussian modulus to positive values, thus triggering an instability with respect to the formation of saddle curvatures. To characterize the generic aspects of the underlying mechanism, we employ a molecular lipid model that accounts for the interfacial tension between the polar and apolar regions of the membrane, for interactions between the lipid headgroups, and for the energy to stretch or compress the hydrocarbon chains. Peptides are modeled as cylinders that partition into the host membrane in a parallel orientation where they diminish the space available to the lipid headgroups and chains. The penetration depth into the membrane is determined by the angular size of the peptide's hydrophilic region. We demonstrate that only peptides with a small angular size of their hydrophilic region have an intrinsic tendency to render the Gaussian modulus more positive, and we identify conditions at which the Gaussian modulus adopts a positive sign upon increasing the peptide concentration. Our model allows us to also incorporate electrostatic interactions between cationic peptides and anionic lipids on the level of the linear Debye-Hückel model. We show that electrostatic interactions tend to shift the Gaussian modulus toward more positive values. Steric and electrostatic lipid-peptide interactions jointly decrease the effective interaction strength in the headgroup region of the host membrane thus suggesting a generic mechanisms of how certain amphipathic peptides are able to induce the formation of saddle curvatures.

Influence of spontaneous curvature on the line tension of phase-coexisting domains in a lipid monolayer: A Landau-Ginzburg model.

The line tension between two coexisting phases of a binary lipid monolayer in its fluid state has contributions not only from the chemical mismatch energy between the two different lipid types but also from the elastic deformation of the lipid tails. We investigate to what extent differences in the spontaneous curvature of the two lipids affect the line tension. To this end, we supplement the standard Landau-Ginzburg model for the line tension between coexisting phases by an elastic energy that accounts for lipid splay and tilt. The spontaneous curvature of the two lipids enters into our model through the splay deformation energy. We calculate the structure of the interfacial region and the line tension between the coexisting domains numerically and analytically, the former based on the full non-linear model and the latter upon employing an approximation in the free energy that linearizes the resulting Euler-Lagrange equations. We demonstrate that our analytical approximation is in excellent agreement with the full non-linear model and use it to identify relevant length scales and two physical regimes of the interfacial profile, double-exponential decay, and damped oscillations. The dependence of the line tension on the spontaneous curvatures of the individual lipids is crucially dependent on how the bulk phases are affected. In the special case that the bulk phases remain inert, the line tension decreases when the difference between the spontaneous curvatures of the two lipid types grows.

Integral Representation of Electrostatic Interactions inside a Lipid Membrane.

Interactions between charges and dipoles inside a lipid membrane are partially screened. The screening arises both from the polarization of water and from the structure of the electric double layer formed by the salt ions outside the membrane. Assuming that the membrane can be represented as a dielectric slab of low dielectric constant sandwiched by an aqueous solution containing mobile ions, a theoretical model is developed to quantify the strength of electrostatic interactions inside a lipid membrane that is valid in the linear limit of Poisson-Boltzmann theory. We determine the electrostatic potential produced by a single point charge that resides inside the slab and from that calculate charge-charge and dipole-dipole interactions as a function of separation. Our approach yields integral representations for these interactions that can easily be evaluated numerically for any choice of parameters and be further simplified in limiting cases.

Curvature Elasticity of the Electric Double Layer.

Mean-field electrostatics is used to calculate the bending moduli of an electric double layer for fixed surface charge density of a macroion in a symmetric 1∶1 electrolyte. The resulting expressions for bending stiffness, Gaussian modulus, and spontaneous curvature refer to a general underlying equation of state of the electrolyte, subject to a local density approximation and the absence of dipole and higher-order fields. We present results for selected applications: the lattice-gas Poisson-Fermi model with and without asymmetric ion sizes, and the Poisson-Carnahan-Starling model.

Macroscopic Freestanding Nanosheets with Exceptionally High Modulus.

Macroscopic single-wall carbon nanotube (SWCNT) films of nanoscale thickness have significant potential for an array of applications that demand thin, transparent, conductive coatings. Using macroscopic micrometer thick polystyrene sheets as a reference, we characterize the elastic response of freestanding multifunctional SWCNT nanosheets possessing both exceptionally high Young's modulus and good durability. Thin SWCNT films (20-200 nm thick) asymmetrically "doped" with dilute concentrations of superparamagnetic colloids were suspended in ethanol as freestanding nanosheets. Through repeated and controlled deformation in an external magnetic field, we measure the temporal relaxation of nanosheet curvature back to equilibrium. From the relaxation time and its dependence on nanosheet thickness and length, we extract the SWCNT nanosheet modulus through a simple viscoelastic model. Our results are consistent with nearly ideal SWCNT rigidity percolation with moduli approaching 200 GPa and limited plasticity for sufficiently thick sheets, which we attribute to the screening of van der Waals interactions by the surrounding solvent and the macroscopic nature of the deformation.

Microscopic details of a fluid/thin film triple line.

In recent years, there has been a considerable interest in the mechanics of soft objects meeting fluid interfaces (elasto-capillary interactions). In this work we experimentally examine the case of a fluid resting on a thin film of rigid material which, in turn, is resting on a fluid substrate. To simplify complexity, we adapt the experiment to a one-dimensional contact geometry and examine the behaviour of polystyrene and polycarbonate films directly with confocal microscopy. We find that the fluid meets the film in a manner consistent with the Young-Dupré equation when the film is thick, but transitions to what appears similar to a Neumann-like balance when the thickness is decreased. However, on closer investigation we find that the true contact angle is always given by the Young construction. The apparent paradox is a result of macroscopically measured angles not being directly related to true microscopic contact angles when curvature is present. We model the effect with an Euler-Bernoulli beam on a Winkler foundation as well as with an equivalent energy-based capillary model. Notably, the models highlight several important lengthscales and the complex interplay of tension, gravity, and bending in the problem.

Adhesion of like-charged lipid vesicles induced by rod-like counterions.

Adhesion of electrically charged lipid vesicles and subsequent formation of multi-vesicle aggregates can be induced by multivalent rod-like counterions. Motivated by recent experimental observations we calculate the equilibrium conformation of two identical vesicles that adhere onto each other. The degree of adhesion reflects the competition between predominantly electrostatic attraction and vesicle bending. Our model assumes the enclosed vesicle volume is allowed to freely adjust and the area of the vesicle membrane is fixed and remains constant. We describe the electrostatic attraction, which arises from the bridging of the rod-like counterions between the two like-charged vesicles, using a recently developed mean-field theory. Bending fluctuation-induced entropic repulsion, depletion forces between the apposed vesicle membranes induced by the rod-like counterions, and van der Waals attraction between the vesicles are estimated to induce only minor shifts in the equilibrium vesicle conformation. Our model predicts the dependence of vesicle adhesion (including its onset) exclusively from material or molecular parameters such as vesicle size and charge, bending stiffness of the membrane, effective length and net charge of the added rod-like counterions, as well as concentrations of rod-like counterions and additional salt content. We demonstrate that the demixing of charged lipids between the adhesion region and the uncomplexed parts of the vesicles has only a minor influence on the degree of adhesion. Our predictions are in qualitative agreement with recent experimental findings.

Modeling the camel-to-bell shape transition of the differential capacitance using mean-field theory and Monte Carlo simulations.

Mean-field electrostatics is used to calculate the differential capacitance of an electric double layer formed at a planar electrode in a symmetric 1:1 electrolyte. Assuming the electrolyte is also ion-size symmetric, we derive analytic expressions for the differential capacitance valid up to fourth order in the surface charge density or surface potential. Our mean-field model accounts exclusively for electrostatic interactions but includes an arbitrary non-ideality in the mixing entropy of the mobile ions. The ensuing criterion for the camel-to-bell shape transition of the differential capacitance is analyzed using commonly used mixing models (one based on a lattice gas and the other based on the Carnahan-Starling equation of state) and compared with Monte Carlo simulations. We observe a reasonable agreement between all our mean-field models and the simulation data for the camel-to-bell shape transition. The absolute value of the differential capacitance for an uncharged (or weakly charged) electrode is, however, not reproduced by our mean-field approaches, not even upon introducing a Stern layer with a thickness equal of the ion radius. We show that, if a Stern layer is introduced, its thickness dependence on the ion size is non-monotonic or, depending on the salt concentration, even inversely proportional.

Differential capacitance of ionic liquids according to lattice-gas mean-field model with nearest-neighbor interactions.

The Bragg-Williams free energy is used to incorporate nearest-neighbor interactions into the lattice gas model of a solvent-free ionic liquid near a planar electrode. We calculate the differential capacitance from solutions of the mean-field consistency relation, arriving at an explicit expression in the limit of a weakly charged electrode. The two additional material parameters that appear in the theory-the degree of nonideality and the resistance to concentration changes of each ion type-give rise to different regimes that we identify and discuss. As the nonideality parameter, which becomes more positive for stronger nearest-neighbor attraction between like-charged ions, increases and the electrode is weakly charged, the differential capacitance is predicted to transition through a divergence and subsequently adopt negative values just before the ionic liquid becomes structurally unstable. This is associated with the spontaneous charging of an electrode at vanishing potential. The physical origin of the divergence and the negative sign of the differential capacitance is a nonmonotonic relationship between the surface potential and surface charge density, which reflects the formation of layered domains alternatingly enriched in counterions and coions near the electrode. The decay length of this layered domain pattern, which can be many times larger than the ion size, is reminiscent of the recently introduced concept of "underscreening."

Investigations of the influence of liposome composition on vesicle stability and drug transfer in human plasma: a transfer study.

Liposomal delivery constitutes a promising approach for i.v. administration of temoporfin (mTHPC) because lipid membranes can host these drug molecules. This study investigates the transfer and release of mTHPC to plasma proteins and stability of various liposomal formulations. To this end, we employed traces of radioactive markers and studied the effects of fatty acid chain length and the degree of saturation in the lipophilic tail, addition of cholesterol and PEGylation of the membrane surface and different drug-to-lipid ratios (DLRs). Liposomes were incubated in human plasma for various incubation times. Drawn samples were separated by asymmetrical flow field-flow fractionation (AF4). Drug was recovered in four fractions identified as albumin, high-density lipoprotein (HDL), low-density lipoprotein (LDL) and liposomes. Our results suggest that mTHPC fits best into fluid, unmodified bilayers when the drug-to-lipid ratio is low. Membrane rigidification as well as the presence of cholesterol and PEGyated lipids reduced the ability of the membrane to accommodate the drug but simultaneously improved the vesicle stability in plasma. Both mechanisms jointly affect the total degree of mTHPC release. We analyzed our data using a kinetic model that suggests the drug to be associated with the host membrane in two distinct states of which only one interacts directly with the plasma compartment.

Vesicle aggregates as a model for primitive cellular assemblies.

Primitive cell models help to understand the role that compartmentalization plays in origin of life scenarios. Here we present a combined experimental and modeling approach towards the construction of simple model systems for primitive cellular assemblies. Charged lipid vesicles aggregate in the presence of oppositely charged biopolymers, such as nucleic acids or polypeptides. Based on zeta potential measurements, dynamic light scattering and cryo-transmission electron-microscopy, we have characterized the behavior of empty and ferritin-filled large unilamellar POPC vesicles, doped with different amounts of cationic (DDAB, CTAB) and anionic (sodium oleate) surfactants, and their aggregation upon the addition of anionic (tRNA, poly-l-glutamic acid) and cationic (poly-l-arginine) biopolymers, respectively. The experimental results are rationalized by a phenomenological modeling approach that predicts the average size of the vesicle aggregates as function of the amount of added biopolymers. In addition, we discuss the mechanism of vesicle aggregation induced by oppositely charged biopolymers. Our study complements previous reports about the formation of giant vesicle clusters and thus provides a general vista on primitive cell systems, based on the association of vesicles into compartmentalized aggregates.

Differential capacitance of an electric double layer with asymmetric solvent-mediated interactions: mean-field theory and Monte Carlo simulations.

The differential capacitance of an electrical double layer is directly affected by properties of the electrolyte solution such as temperature, salt concentration, ionic size, and solvent structure. In the present work, we employ a mean-field approach and Monte Carlo simulations to investigate how the inclusion of asymmetric solvent-mediated ion-ion and ion-surface interactions affects the differential capacitance. We focus on a charged flat electrode immersed in an electrolyte solution of monovalent ions at physiological concentration in a uniform dielectric background. Solvent-mediated anion-anion, anion-cation and cation-cation interactions are modeled on the basis of Yukawa potentials with three independent strengths that add to Coulomb and excluded volume pair-potentials, the latter accounted for through a lattice gas approach. We use the three interaction strengths to produce and analyze asymmetric profiles of the differential capacitance as function of the electrode's surface charge density. While solvent-mediated anion-anion and cation-cation interactions mainly affect the behavior at medium charge densities of the electrode, anion-cation repulsion increases the differential capacitance of a weakly charged electrode. We present a simple phenomenological model to rationalize this finding. Most importantly, because the added solvent-mediated interaction potential is comparatively soft, our mean-field model is able to qualitatively - and in some cases quantitatively - reproduce all Monte Carlo simulation results, even at high surface charge densities of the electrode.

Surface tension of a Yukawa fluid according to mean-field theory.

Yukawa fluids consist of particles that interact through a repulsive or attractive Yukawa potential. A surface tension arises at the walls of the container that encloses the fluid or at the interface between two coexisting phases. We calculate that surface tension on the level of mean-field theory, thereby either ignoring the particle size (ideal Yukawa fluid) or accounting for a non-vanishing particle size through a nonideal contribution to the free energy, exemplified either on the level of a lattice gas (lattice Yukawa fluid) or based on the Carnahan-Starling equation of state (Carnahan-Starling Yukawa fluid). Our mean-field results, which do not rely on assuming small gradients of the particle concentrations, become exact in the limit of large temperature and large screening length. They are calculated numerically in the general case and analytically in the two limits of small particle concentration and close to the critical point for a phase-separating system. For a sufficiently small particle concentration, our predicted surface tension is accurate whereas for a phase boundary, we expect good agreement with exact calculations in the limit of a large screening length and if the mean-field model employs the Carnahan-Starling equation of state.

The apparent charge of nanoparticles trapped at a water interface.

Charged spherical nanoparticles trapped at the interface between water and air or water and oil exhibit repulsive electrostatic forces that contain a long-ranged dipolar and a short-ranged exponentially decaying component. The former are induced by the unscreened electrostatic field through the non-polar low-permittivity medium, and the latter result from the overlap of the diffuse ion clouds that form in the aqueous phase close to the nanoparticles. The magnitude of the long-ranged dipolar interaction is largely determined by the residual charges that remain attached to the air- (or oil-) exposed region of the nanoparticle. In the present work we address the question to what extent the charges on the water-immersed part of the nanoparticle provide an additional contribution to the dipolar interaction. To this end, we model the electrostatic properties of a spherical particle - a nanoparticle or a colloid - that partitions equatorially to the air-water interface, thereby employing nonlinear Poisson-Boltzmann theory in the aqueous solution and accounting for the propagation of the electric field through the interior of the particle. We demonstrate that the apparent charge density on the air-exposed region of the particle, which determines the dipole potential, is influenced by the electrostatic properties in the aqueous solution. We also show that this electrostatic coupling through the particle can be reproduced qualitatively by a simple analytic planar capacitor model. Our results help to rationalize the experimentally observed weak but non-vanishing salt dependence of the forces that stabilize ordered two-dimensional arrays of interface-trapped nanoparticles or colloids.

Role of ion hydration for the differential capacitance of an electric double layer.

The influence of soft, hydration-mediated ion-ion and ion-surface interactions on the differential capacitance of an electric double layer is investigated using Monte Carlo simulations and compared to various mean-field models. We focus on a planar electrode surface at physiological concentration of monovalent ions in a uniform dielectric background. Hydration-mediated interactions are modeled on the basis of Yukawa potentials that add to the Coulomb and excluded volume interactions between ions. We present a mean-field model that includes hydration-mediated anion-anion, anion-cation, and cation-cation interactions of arbitrary strengths. In addition, finite ion sizes are accounted for through excluded volume interactions, described either on the basis of the Carnahan-Starling equation of state or using a lattice gas model. Both our Monte Carlo simulations and mean-field approaches predict a characteristic double-peak (the so-called camel shape) of the differential capacitance; its decrease reflects the packing of the counterions near the electrode surface. The presence of hydration-mediated ion-surface repulsion causes a thin charge-depleted region close to the surface, which is reminiscent of a Stern layer. We analyze the interplay between excluded volume and hydration-mediated interactions on the differential capacitance and demonstrate that for small surface charge density our mean-field model based on the Carnahan-Starling equation is able to capture the Monte Carlo simulation results. In contrast, for large surface charge density the mean-field approach based on the lattice gas model is preferable.

Poisson-Boltzmann model of electrolytes containing uniformly charged spherical nanoparticles.

Like-charged macromolecules typically repel each other in aqueous solutions that contain small mobile ions. The interaction tends to turn attractive if mobile ions with spatially extended charge distributions are added. Such systems can be modeled within the mean-field Poisson-Boltzmann formalism by explicitly accounting for charge-charge correlations within the spatially extended ions. We consider an aqueous solution that contains a mixture of spherical nanoparticles with uniform surface charge density and small mobile salt ions, sandwiched between two like-charged planar surfaces. We perform the minimization of an appropriate free energy functional, which leads to a non-linear integral-differential equation for the electrostatic potential that we solve numerically and compare with predictions from Monte Carlo simulations. Nanoparticles with uniform surface charge density are contrasted with nanoparticles that have all their charges relocated at the center. Our mean-field model predicts that only the former (especially when large and highly charged particles) but not the latter are able to mediate attractive interactions between like-charged planar surfaces. We also demonstrate that at high salt concentration attractive interactions between like-charged planar surfaces turn into repulsion.

Emergence of a Stern Layer from the Incorporation of Hydration Interactions into the Gouy-Chapman Model of the Electrical Double Layer.

In one of the most commonly used phenomenological descriptions of the electrical double layer, a charged solid surface and a diffuse region of mobile ions are separated from each other by a thin charge-depleted Stern layer. The Stern layer acts as a capacitor that improves the classical Gouy-Chapman model by increasing the magnitude of the surface potential and limiting the maximal counterion concentration. We show that very similar Stern-like properties of the diffuse double layer emerge naturally from adding a nonelectrostatic hydration repulsion to the electrostatic Coulomb potential. The interplay of electrostatic attraction and hydration repulsion of the counterions and the surface leads to the formation of a diffuse counterion layer that remains well separated from the surface. In addition, hydration repulsions between the ions limit and control the maximal ion concentration and widen the width of the diffuse double layer. Our mean-field model, which we express in terms of electrostatic and hydration potentials, is physically consistent and conceptually similar to the classical Gouy-Chapman model. It allows the incorporation of ion specificity, accounts for hydration properties of charged surfaces, and predicts Stern layer properties, which we analyze in terms of the effective size of the hydrated counterions.

Modeling Lipid-Lipid Correlations across a Bilayer Membrane Using the Quasi-chemical Approximation.

Mixed fluid-like lipid membranes exhibit interactions not only among the lipids within a given leaflet but also across the bilayer. The ensuing collective interleaflet coupling of entire membrane domains has been modeled previously using various mean-field approaches. Yet, also on the level of individual lipids have correlations across the bilayer been observed experimentally for binary mixtures of charged/uncharged lipids with mismatching combinations of short and long acyl chain lengths. The present study proposes a lattice gas model to quantify these correlations. To this end, we represent a macroscopically homogeneous lipid bilayer by two coupled two-dimensional lattice gases that we study using the quasi-chemical approximation. We demonstrate that the rationalization of previous experimental results is only possible if besides two-body lipid-lipid interactions within and across the bilayer our model also accounts for an additional multibody interaction mechanism, namely the local hydrophobic height mismatch created by pairing short and long chain lipids together. The robustness of the quasi-chemical approximation is verified by comparison with Monte Carlo simulations.