Derivation of expressions for the spontaneous curvature, mean and Gaussian bending constants of thermodynamically open surfactant monolayers and bilayers

Line tension of multicomponent bilayer membranes

The line tension or edge energy of bilayer membranes self-assembled from binary amphiphilic molecules is studied using self-consistent-field theory (SCFT). Specifically, solutions of the SCFT equations corresponding to an infinite membrane with a circular pore, or an open membrane, are obtained for a coarse-grained model in which the amphiphilic species and hydrophilic solvents are represented by ABandED diblock copolymers and C homopolymers, respectively. The edge energy of the membrane is extracted from the free energy of the open membranes. Results for membranes composed of mixtures of symmetric and cone- or inverse cone-shaped amphiphilic molecules with neutral and/or repulsive interactions are obtained and analyzed. It is observed that an increase in the concentration of the cone-shaped species leads to a decrease of the line tension. In contrast, adding inverse cone-shaped copolymers results in an increase of the line tension. Furthermore, the density profile of the copolymers reveals that the line tension is regulated by the distribution of the amphiphiles at the bilayer edge.

Bending elasticity of nonionic surfactant layers

A novel approach to evaluate the bending elasticity of monolayers formed by nonionic surfactants with a rigid head group is introduced by means of considering head group repulsion as derived from the free energy of mixing rigid hydrophilic head groups with surrounding solvent molecules as well as contributions related to the hydrophobic tails. Explicit expressions for the spontaneous curvature (H0), bending rigidity (kc) and saddle-splay constant (kc) have been derived for the constraint of constant chemical potential of free surfactant (thermodynamically open layers) as well as the constraint of constant aggregation number (thermodynamically closed layers). Most interestingly, it is demonstrated that kc for thermodynamically open layers formed by a nonionic surfactant with rigid tail and head group always must be zero. However, kc for surfactants with a flexible tail as a function of the head group-to-tail volume ratio is found to go through a maximum at some large, positive value of kc and H0 approximately 0. Eventually, kc falls below zero as the head group volume increases above a certain value. Hence, we may conclude that nonionic surfactants with a rigid head group may form thermodynamically stable fluid layers or aggregates only insofar the hydrophobic part is flexible with respect to chain conformational degrees of freedom and the head group is not too voluminous. It is found that the head group repulsion contribution to kcH0 is always positive whereas the corresponding contribution to kc may be positive or negative depending on whether the hydrophobic layer of the film is thicker or thinner than the hydrophilic layer.

Vesicle self-reproduction: the involvement of membrane hydraulic and solute permeabilities

Conditions for self-reproduction are sought for a growing vesicle with its growth defined by an exponential increase of vesicle membrane area and by adequate flow of the solution across the membrane. In the first step of the presumed vesicle self-reproduction process, the initially spherical vesicle must double its volume in the doubling time of the membrane area and, through the appropriate shape transformations, attain the shape of two equal spheres connected by an infinitesimally thin neck. The second step involves separation of the two spheres and relies on conditions that cause the neck to be broken. In this paper we consider the first step of this self-reproduction process for a vesicle suspended in a solution whose solute can permeate the vesicle membrane. It is shown that vesicle self-reproduction occurs only for certain combinations of the values of membrane hydraulic and solute permeabilities and the external solute concentration, these quantities being related to the mechanical properties of the membrane and the membrane area doubling time. The analysis includes also the relaxation of a perturbed system towards stationary self-reproduction behavior and the case where the final shape consists of two connected spheres of different radii.

Bending Energetics of Tablet-Shaped Micelles: A Novel Approach to Rationalize Micellar Systems

A novel approach to rationalize micellar systems is expounded in which the structural behavior of tablet-shaped micelles is theoretically investigated as a function of the three bending elasticity constants: spontaneous curvature (H0), bending rigidity (k(c)), and saddle-splay constant (k(c)). As a result, experimentally accessible micellar properties, such as aggregation number, length-to-width ratio, and polydispersity, may be related to the different bending elasticity constants. It is demonstrated that discrete micelles or connected cylinders form when H0 > 1/4xi, where xi is the thickness of a surfactant monolayer, whereas various bilayer structures are expected to predominate when H0 < 1/4xi. Our theory predicts, in agreement with experiments, a transition from discrete globular (tablet-shaped) micelles to a phase of ordered, or disordered, connected cylinders above a critical surfactant concentration. Moreover, a novel explanation for the mechanism of growth, from small globular to long rodlike or wormlike micelles, follows as a consequence from the theory. In accordance, polydisperse elongated micelles (large length-to-width ratio) form as the bending rigidity is lowered, approaching the critical point at k(c) = 0, whereas monodisperse globular micelles (small length-to-width ratio) are expected to be present at large k(c) values. The spontaneous curvature mainly determines the width of tablet-shaped or ribbonlike micelles, or the radius of disklike micelles, whereas the saddle-splay constant primarily influences the size but not the shape of the micelles.

Bending elasticity of charged surfactant layers: the effect of mixing

Expressions have been derived from which the spontaneous curvature (H(0)), bending rigidity (k(c)), and saddle-splay constant (k(c)) of mixed monolayers and bilayers may be calculated from molecular and solution properties as well as experimentally available quantities such as the macroscopic hydrophobic-hydrophilic interfacial tension. Three different cases of binary surfactant mixtures have been treated in detail: (i) mixtures of an ionic and a nonionic surfactant, (ii) mixtures of two oppositely charged surfactants, and (iii) mixtures of two ionic surfactants with identical headgroups but different tail volumes. It is demonstrated that k(c)H(0), k(c), and k(c) for mixtures of surfactants with flexible tails may be subdivided into one contribution that is due to bending properties of an infinitely thin surface as calculated from the Poisson-Boltzmann mean field theory and one contribution appearing as a result of the surfactant film having a finite thickness with the surface of charge located somewhat outside the hydrophobic-hydrophilic interface. As a matter of fact, the picture becomes completely different as finite layer thickness effects are taken into account, and as a result, the spontaneous curvature is extensively lowered whereas the bending rigidity is raised. Furthermore, an additional contribution to k(c) is present for surfactant mixtures but is absent for k(c)H(0) and k(c). This contribution appears as a consequence of the minimization of the free energy with respect to the composition of a surfactant layer that is open in the thermodynamic sense and must always be negative (i.e., k(c) is generally found to be brought down by the process of mixing two or more surfactants). The magnitude of the reduction of k(c) increases with increasing asymmetry between two surfactants with respect to headgroup charge number and tail volume. As a consequence, the bending rigidity assumes the lowest values for layers formed in mixtures of two oppositely charged surfactants, and k(c) is further reduced in anionic/cationic surfactant mixtures where the surfactant in excess has the smaller tail volume. Likewise, the reduction of k(c) is enhanced in mixtures of an ionic and a nonionic surfactant where the ionic surfactant has the smaller tail. The effective bilayer bending constant (k(bi)) is also found to be reduced by mixing, and as a result, k(bi) is seen to go through a minimum at some intermediate composition. The reduction of k(bi) is expected to be most pronounced in mixtures of two oppositely charged surfactants where the surfactant in excess has the smaller tail in agreement with experimental observations.

Bending elasticity of charged surfactant layers: the effect of layer thickness

The bending properties of charged one-component surfactant films of finite thickness have been theoretically investigated. It is demonstrated that finite thickness effects are of crucial importance for layers formed by an ionic surfactant with a flexible hydrophobic tail, whereas the influence on layers formed by a surfactant with a rigid tail is less pronounced. As a matter of fact, in the former case, the spontaneous curvature and mean and Gaussian bending constants all become significantly modified as compared to an infinitely thin surface and assume identical values as if the surfactant layer were bent at constant layer thickness. As a result, the spontaneous curvature is found to decrease, whereas the magnitudes of the mean and Gaussian bending constants both increase with increasing layer thickness as well as with increasing hydrophobic-hydrophilic interfacial tension. All of these trends are consistent with experimental observations. In addition, it is demonstrated that separating the hydrophilic-hydrophobic interface and the surface of charge a certain distance from each other tends to increase the spontaneous curvature and the mean bending constant, whereas the Gaussian bending constant becomes increasingly negative. It is also found that the work of bending a bilayer into a geometrically closed vesicle is substantially raised to large positive values for surfactants with flexible aliphatic chains, whereas the corresponding quantity is negative for surfactants with rigid tails, indicating that stable bilayer structures may only be formed by the former surfactant. Furthermore, each of the bending elasticity constants for monolayers formed by a double-chain ionic surfactant are found to assume lower values as compared with layers formed by the corresponding single-chain surfactant.

Model calculations of the spontaneous curvature, mean and Gaussian bending constants for a thermodynamically open surfactant film

The spontaneous curvature (H(0)), mean and Gaussian bending constants (k(c) and k (c)), as defined in the well-known Helfrich expression, have been calculated from a detailed model for a thermodynamically open surfactant layer. The effect of head group cross-section area, surfactant tail length and electrolyte concentration for monovalent ionic surfactants have been investigated. Geometrical packing constraints subjected to the aggregated hydrocarbon tails and electrostatics are found to be the dominant contributions to H(0), k(c) and k (c). In addition, the transition from spherocylindrical micelles to vesicles were investigated in terms of the three parameters and the following simple expressions were derived as criteria for coexistence between micelles and vesicles H(0)=1/4 xi and N(ves)/N(mic)=exp[4 pi(k(c)+k (c))/kT], where xi is the thickness of the hydrocarbon part of the film and N(mic) and N(ves) the average aggregation numbers of micelles and vesicles, respectively. However, it is found that the ratio N(ves)/N(mic) is order of magnitudes too large for vesicles to form at all in charged single-surfactant systems where the surfactant head is of moderate size.

Statistical mechanics of bilayer membrane with a fixed projected area

The equilibrium and fluctuation methods for determining the surface tension, sigma, and bending modulus, kappa, of a bilayer membrane with a fixed projected area are discussed. In the fluctuation method the elastic coefficients sigma and kappa are measured from the amplitude of thermal fluctuations of the planar membrane, while in the equilibrium method the free energy required to deform the membrane is considered. The latter approach is used to derive new expressions for sigma and kappa (as well as for the saddle-splay modulus), which relate them to the pair-interactions between the amphiphiles forming the membrane. We use linear response theory to argue that the two routes lead to similar values for sigma and kappa. This argument is confirmed by Monte Carlo simulations of a model membrane whose elastic coefficients are calculated using both methods.