Counterion Release from Macroion Assemblies of Planar Geometry.

Macroions such as nanoparticles, polyelectrolytes, ionic gels, and amphiphiles can form condensed, often self-assembled, phases that are embedded in a solvent region. The condensed phase contains not only the partially or fully immobile charges of their macroions but also corresponding counterions that are mobile and thus free to migrate out of their confinement into the solvent region where they benefit from high translational entropy. Based on the nonlinear Poisson-Boltzmann model for monovalent ions, we quantify the corresponding fraction of released counterions for a planar slab geometry of the macroion phase. Slab thickness, extension of the solvent phase, fixed background charge density provided by the macroions, and dielectric constants inside slab and solvent combine into three dimensionless parameters that the fraction of released counterions depends on. We calculate that fraction and analyze the limits of a thin macroion phase, a large solvent phase, and linearized theory, where simple analytic results become available. Of particular interest is the presence of a single-planar interface that separates a bulk macroion phase from an extended solvent region. We calculate the apparent surface charge density that emerges due to the released counterions. Our model yields a comprehensive description of counterion partitioning between a planar macroion phase and a solvent region on the level of mean-field electrostatics in the absence of added salt ions.

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.

Monte Carlo simulations and mean-field modeling of electric double layers at weakly and moderately charged spherical macroions.

Monte Carlo simulations are employed to determine the differential capacitance of an electric double layer formed by small size-symmetric anions and cations in the vicinity of weakly to moderately charged macroions. The influence of interfacial curvature is deduced by investigating spherical macroions, ranging from flat to moderately curved. We also calculate the differential capacitance using a previously developed mean-field model where, in addition to electrostatic interactions, the excluded volumes of the ions are taken into account using either the lattice-gas or the Carnahan-Starling equation of state. For both equations of state, we compare the mean-field model for arbitrary curvature with a recently developed second-order curvature expansion. Our Monte Carlo simulations predict an increase in the differential capacitance with growing macroion curvature if the surface charge density is small, whereas for moderately charged macroions the differential capacitance passes through a local minimum. Both mean-field models tend to somewhat overestimate the differential capacitance as compared with Monte Carlo simulations. At the same time, they do reproduce the curvature dependence of the differential capacitance, especially for small surface charge density. Our study suggests that the quality of mean-field modeling does not worsen when weakly or moderately charged macroions exhibit spherical curvature.

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.

Debye-Hückel Free Energy of an Electric Double Layer with Discrete Charges Located at a Dielectric Interface.

Poisson-Boltzmann theory provides an established framework to calculate properties and free energies of an electric double layer, especially for simple geometries and interfaces that carry continuous charge densities. At sufficiently small length scales, however, the discreteness of the surface charges cannot be neglected. We consider a planar dielectric interface that separates a salt-containing aqueous phase from a medium of low dielectric constant and carries discrete surface charges of fixed density. Within the linear Debye-Hückel limit of Poisson-Boltzmann theory, we calculate the surface potential inside a Wigner-Seitz cell that is produced by all surface charges outside the cell using a Fourier-Bessel series and a Hankel transformation. From the surface potential, we obtain the Debye-Hückel free energy of the electric double layer, which we compare with the corresponding expression in the continuum limit. Differences arise for sufficiently small charge densities, where we show that the dominating interaction is dipolar, arising from the dipoles formed by the surface charges and associated counterions. This interaction propagates through the medium of a low dielectric constant and alters the continuum power of two dependence of the free energy on the surface charge density to a power of 2.5 law.

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.

Modeling hydration-mediated ion-ion interactions in electrolytes through oscillating Yukawa potentials.

Classical Poisson-Boltzmann theory represents a mean-field description of the electric double layer in the presence of only Coulomb interactions. However, aqueous solvents hydrate ions, which gives rise to additional hydration-mediated ion-ion interactions. Experimental and computational studies suggest damped oscillations to be a characteristic feature of these hydration-mediated interactions. We have therefore incorporated oscillating Yukawa potentials into the mean-field description of the electric double layer. This is accomplished by allowing the decay length of the Yukawa potential to be complex valued. Ion specificity emerges from assigning individual strengths and phases to the Yukawa potential for anion-anion, anion-cation, and cation-cation pairs as well as for anions and cations interacting with an electrode or macroion. Excluded volume interactions between ions are approximated by replacing the ideal gas entropy by that of a lattice gas. We derive mean-field equations for the Coulomb and Yukawa potentials and use their solutions to compute the differential capacitance for an isolated planar electrode and the pressure that acts between two planar, like-charged macroion surfaces. Attractive interactions appear if the surface charge density of the macroions is sufficiently small.

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.

Role of Transmembrane Proteins for Phase Separation and Domain Registration in Asymmetric Lipid Bilayers.

It is well known that the formation and spatial correlation of lipid domains in the two apposed leaflets of a bilayer are influenced by weak lipid-lipid interactions across the bilayer's midplane. Transmembrane proteins span through both leaflets and thus offer an alternative domain coupling mechanism. Using a mean-field approximation of a simple bilayer-type lattice model, with two two-dimensional lattices stacked one on top of the other, we explore the role of this "structural" inter-leaflet coupling for the ability of a lipid membrane to phase separate and form spatially correlated domains. We present calculated phase diagrams for various effective lipid-lipid and lipid-protein interaction strengths in membranes that contain a binary lipid mixture in each leaflet plus a small amount of added transmembrane proteins. The influence of the transmembrane nature of the proteins is assessed by a comparison with "peripheral" proteins, which result from the separation of one single integral protein into two independent units that are no longer structurally connected across the bilayer. We demonstrate that the ability of membrane-spanning proteins to facilitate domain formation requires sufficiently strong lipid-protein interactions. Weak lipid-protein interactions generally tend to inhibit phase separation in a similar manner for transmembrane as for peripheral proteins.

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."

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.

Debye-Hückel theory of weakly curved macroions: Implementing ion specificity through a composite Coulomb-Yukawa interaction potential.

The free energy of a weakly curved, isolated macroion embedded in a symmetric 1:1 electrolyte solution is calculated on the basis of linear Debye-Hückel theory, thereby accounting for nonelectrostatic Yukawa pair interactions between the mobile ions and of the mobile ions with the macroion surface, present in addition to the electrostatic Coulomb potential. The Yukawa interactions between anion-anion, cation-cation, and anion-cation pairs are independent from each other and serve as a model for solvent-mediated ion-specific effects. We derive expressions for the free energy of a planar surface, the spontaneous curvature, the bending stiffness, and the Gaussian modulus. It is shown that a perturbation expansion, valid if the Yukawa interactions make a small contribution to the overall free energy, yields simple analytic results that exhibit good agreement with the general free energy over the range of experimentally relevant interaction parameters.

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.

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.

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.

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.

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.

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.