We consider the possibility of smectic and columnar phases in fluids and colloidal suspensions of aligned rods and disks interacting through excluded-volume forces. After briefly reviewing previous work, and applying known cell model techniques to compare the phase behaviors of disks and rods, we present and discuss the results from new Monte Carlo simulations of perfectly oriented rods (spherocylinders) and disks (torocylinders). We conclude tentatively that columnar phases are stable only in the case of disks.
Pore formation in unilamellar lipid vesicles is believed to occur when the concentration of membrane-bound drug molecules exceeds a certain value. We treat this phenomenon in analogy with that of the micellization of surfactant in bulk aqueous solutions, thereby relating the threshold concentration of drug molecules to the free energy associated with transferring a molecule to a pore from its uniformly-dispersed state in the membrane. Incorporating the effect of lateral tension induced by osmotic pressure, we calculate the lowering of the pore-formation threshold with increasing tension. These predictions are tested by direct measurements on liposomal dispersions involving the antifungal drug amphotericin B.
The angular intensity distribution of He beams scattered from compact clusters and from diffusion limited aggregates, epitaxially grown on metal surfaces, is investigated theoretically. The purpose is two-fold: to distinguish compact cluster structures from diffusion limited aggregates, and to find observable signatures that can characterize the compact clusters at the atomic level of detail. To simplify the collision dynamics, the study is carried out in the framework of the sudden approximation, which assumes that momentum changes perpendicular to the surface are targe compared with momentum transfer due to surface corrugation. The diffusion limited aggregates on which the scattering calculations were done, were generated by kinetic Monte Carlo simulations, It is demonstrated, by focusing on the example of compact Pt heptamers, that signatures of structure of compact clusters may indeed be extracted from the scattering distribution. These signatures enable both an experimental distinction between diffusion limited aggregates and compact clusters, and a determination of the cluster structure, The characteristics comprising the signatures are, to varying degrees, the rainbow, Fraunhofer, specular and constructive interference peaks, all seen in the intensity distribution, It is also shown, how the distribution of adsorbate heights above the metal surface can be obtained by an analysis of the specuIar peak attenuation. The results contribute to establishing He scattering as a powerful tool in the investigation of surface disorder and epitaxial growth on surfaces, alongside with STM.
This article describes briefly several applications of a molecular theory of lipid organization in membranes to systems of biophysical interest. After introducing the basic concepts of this mean field theory we outline three of its recent applications. i) Calculations of lipid chain conformational statistics in membrane bilayers, and comparison of the results (e.g. bond orientational order parameters) to experiment and molecular dynamics simulations. Good agreement is found. ii) A molecular model for lipid-protein interactions, which explicitly considers the effects of a rigid hydrophobic protein on the elastic (conformational) properties of the lipid bilayer. We also analyze the role of the `hydrophobic mismatch' between the protein and lipid bilayer thickness. iii) A molecular level calculation of the vesicle to micelle transition, attendant upon the addition of ('curvature loving') surfactant to a lipid bilayer vesicle. Future applications, e.g. to the calculation of the free energy barriers involved in membrane fusion are briefly mentioned.
A two-dimensional lattice model, originally introduced by Granek et al. [J. Chem. Phys. 101, 4331 (l994)], is used to demonstrate the intricate coupling between the intramicellar interactions that determine the optimal aggregation geometry of surfactant molecules in dilute solution, and the intermicellar interactions that govern the phase behavior at higher concentrations. Three very different scenarios of self-assembly and phase evolution are analyzed in detail, based on Monte Carlo studies and theoretical interpretations involving mean-field, Landau-Ginzburg, Bethe-Peierls, and virial expansion schemes. The basic particles in the model are `'unit micelles'' which, due to spontaneous self-assembly or because of excluded area interactions, can fuse td form larger aggregates; These aggregates are envisaged as hat micelles composed of a bilayerlike body surrounded: by a curved semitoroidal rim. The system's Hamiltonian involves one- through four-body potentials between the unit micelles, which account for their tendency to form aggregates of different shapes, e.g., elongated vs disklike micelles. Equivalently the configurational energy of the system is a sum of micellar self-energies involving the packing free energies of the constituent molecules in the bilayer body and in rim segments of different local curvature. The rim energy is a sum of a line tension term and a 1D curvature energy which depends on the rim spontaneous curvature and bending rigidity. Different combinations of these molecular parameters imply different optimal packing geometries and hence different self-assembly and phase behaviors. The emphasis in this paper Is on systems of `'curvature loving'' amphiphiles which, in our model, are characterized by negative line tension. The three systems studied are: (i) A dilute solution of stable disklike micelles which, upon increasing the concentration, undergoes a first-order phase transition to a continuous bilayer with isolated hole defects. An intermediate modulated `'checkerboard'' phase appears under certain conditions at low temperatures. (ii) A system of unit micelles which in dilute solution tend to associate into Linear micelles. These micelles are rodlike gt low temperatures, becoming increasingly more flexible as the temperature increases.-Upon increasing the concentration the micelles grow and undergo (in 2D) a continuous transition into nematic and `'stripe'' phases of long rods. At still higher concentrations the micellar stripes fuse into continuous sheets with line defects. (iii) A system in which, already in dilute solution, the micelles favor the formation of branched aggregates, analogous to the branched cylindrical micelles recently observed in certain surfactant solutions, As the concentration increases the micelles associate into networks (''gels'') composed of a mesh of linear micelles linked by `'T-like'' intermicellar junctions. The network may span the entire system or phase separate and coexist with a dilute micellar phase, depending on the details of the molecular packing parameters. (C) 1995 American Institute of Physics.
The elastic behavior of mixed bilayers composed of two amphiphilic components with different chain length (and identical head groups) is studied using two molecular level models. In both, the bilayer free energy is expressed as a sum of chain, head group and interfacial contributions as well as a mixing entropy term. The head group and interfacial terms are modeled using simple phenomenological but general expressions. The models differ in their treatment of the chain conformational free energy. In one it is calculated using a detailed mean-field molecular theory. The other is based on a simple `'compression'' model. Both models lead to similar conclusions. Expressing the bilayer free energy as a sum of its two monolayer contributions, a thermodynamic stability analysis is performed to examine the possibility of spontaneous vesicle formation. To this end, we expand the bilayer free energy as a power series (up to second order) in terms of the monolayer curvatures, their amphiphilic compositions and the average cross sectional areas per molecule; all variables are coupled, with the molecular composition and areas treated as degrees of freedom which are allowed to relax during bending. Using reasonable molecular interaction parameters we find that a second order transition from a planar to a curved (vesicle) geometry in a randomly mixed bilayer is unlikely. Most of our analysis is devoted to calculating the spontaneous curvature and the bending rigidity of the bilayer as a function of its amphiphile chain composition. We find that adding short chain amphiphiles to a layer of long chain molecules reduces considerably its bending rigidity, as already known from calculations involving only the chain contributions. However, we find that inclusion of head group and interfacial interactions moderates the effect of the added short chains. We also find that the bending rigidity Of pure monolayers is approximately linear in chain length, as compared to the nearly cubic dependence implied by the chain free energy alone (at constant head group area). Our main result involves the calculation of the spontaneous curvature as a function of composition. We find, for different chain mixtures, that upon adding short chains to long chain monolayers, the spontaneous curvature first increases nearly Linearly with composition and then (beyond mole fraction of about 0.5) begins to saturate towards the spontaneous curvature of a pure short chain layer. Qualitative arguments are provided to explain this behavior. (C) 1995 American Institute of physics.
A molecular model is used to calculate the free energy of mixed vesicles and cylindrical micelles, composed of lipid molecules and short chain surfactants. The free energy of both aggregates (modeled as an infinite planar bilayer and an infinite cylindrical aggregate) is represented as a sum of internal free energy and mixing entropy contributions. The internal free energy is treated as a sum of chain (conformational), head group, and surface tension terms. Calculating the free energy of each aggregation geometry as a function of lipid/surfactant composition and using common tangent construction we obtain the compositions of the bilayer and the micelle at the phase transition. By varying certain molecular parameters (such as the `'hard core'' area of the surfactant head group or the length of the surfactant tail) we study the role of molecular packing characteristics in determining the compositions at phase coexistence. We find, as expected, that upon increasing the preference of the surfactant for the micellar geometry (larger spontaneous curvature) the bilayer is solubilized at lower surfactant/lipid concentration ratios. For some typical values of the parameters used, reasonable agreement with experimental results for mixtures of egg phosphatidylcholine and octylglucoside is obtained.
A molecular, mean-field theory of chain packing statistics in aggregates of amphiphilic molecules is applied to calculate the conformational properties of the lipid chains comprising the hydrophobic cores of dipalmitoyl-phosphatidylcholine (DPPC), dioleoyl-phosphatidylcholine (DOPC), and palmitoyl-oleoyl-phosphatidylcholine (POPC) bilayers in their fluid state. The central quantity in this theory, the probability distribution of chain conformations, is evaluated by minimizing the free energy of the bilayer assuming only that the segment density within the hydrophobic region is uniform (liquidlike). Using this distribution we calculate chain conformational properties such as bond orientational order parameters and spatial distributions of the various chain segments. The lipid chains, both the saturated palmitoyl (-(CH2)(1)4-CH3) and the unsaturated oleoyl (-(CH2)(7)-CH=CH-(CH2)(7)-CH3) chains are modeled using rotational isomeric state schemes. All possible chain conformations are enumerated and their statistical weights are determined by the self-consistency equations expressing the condition of uniform density. The hydrophobic core of the DPPC bilayer is treated as composed of single (palmitoyl) chain amphiphiles, i.e., the interactions between chains originating from the same lipid headgroup are assumed to be the same as those between chains belonging to different molecules. Similarly, the DOPC system is treated as a bilayer of oleoyl chains. The POPC bilayer is modeled as an equimolar mixture of palmitoyl and oleoyl chains. Bond orientational order parameter profiles, and segment spatial distributions are calculated for the three systems above, for several values of the bilayer thickness (or, equivalently, average area/headgroup) chosen, where possible, so as to allow for comparisons with available experimental data and/or molecular dynamics simulations. In most cases the agreement between the mean-field calculations, which are relatively easy to perform, and the experimental and simulation data is very good, supporting their use as an efficient tool for analyzing a variety of systems subject to varying conditions (e.g., bilayers of different compositions or thicknesses at different temperatures).
We study the two-dimensional (2-D) structural and thermodynamic changes in smectic-A/lamellar phases of self-assembling surfactant systems, in which the rim associated with a bilayer edge has a preferred curvature. This property was not considered in previous studies of 2-D aggregation, where an infinite bilayer emerges already at very low concentrations. A lattice Hamiltonian is used to describe the bending energy of the rim: An occupied lattice site corresponds to a minimum, disklike, micelle, and a bending energy penalty is associated with corners and straight edges depending on the value of the spontaneous curvature. When the spontaneous radius of curvature of the rim is small and the bending modulus is large, we find that the `'condensation'' transition-i.e., the `'collapse'' of the finite aggregates into a continuous bilayer-is postponed to high concentrations. At low concentrations the bending energy leads to an effective repulsive interaction between the aggregates, which in turn can result in ordered (modulated) structures for not too large ratios of thermal energy to bending energy (which is the expected situation in most systems of interest). Our model should be applicable to the systems of decylammonium chloride and cesium perflourooctanoate studied by Boden and co-workers (NMR and conductivity measurements) and Zasadzinski and co-workers (freeze fracture), where monodisperse micellar disks are observed to layer in stacked planes. In the latter system a 2-D order of disk-shaped aggregates appears within the smectic-A layers, which is also consistent with our theory. Experimental studies of the structural evolution under further condensation of the system are not yet available.
The interaction free energy between a hydrophobic, transmembrane, protein and the surrounding lipid environment is calculated based on a microscopic model for lipid organization. The protein is treated as a rigid hydrophobic solute of thickness d(P), embedded in a lipid bilayer of unperturbed thickness d(L)o. The lipid chains in the immediate vicinity of the protein are assumed to adjust their length to that of the protein (e.g., they are stretched when d(P) > d(L)o) in order to bridge over the lipid-protein hydrophobic mismatch (d(P) - d(L)o). The bilayer's hydrophobic thickness is assumed to decay exponentially to its asymptotic, unperturbed, value. The lipid deformation free energy is represented,as a sum of chain (hydrophobic core) and interfacial (head-group region) contributions. The chain contribution is calculated using a detailed molecular theory of chain packing statistics, which allows the calculation of conformational properties and thermodynamic functions (in a mean-field approximation) of the lipid tails. The tails are treated as single chain amphiphiles, modeled using the rotational isomeric state scheme. The interfacial free energy is represented by a phenomenological expression, accounting for the opposing effects of head-group repulsions and hydrocarbon-water surface tension. The lipid deformation free energy DELTAF is calculated as a function of d(P) - d(L)o. Most calculations are for C-14 amphiphiles which, in the absence of a protein, pack at an average area per head-group a0 congruent-to 32 angstrom2 (d(L)o congruent-to 24.5 angstrom), corresponding to the fluid state of the membrane. When d(P) = d(L)o, DELTAF > 0 and is due entirely to the loss of conformational entropy experienced by the chains around the protein. When d(P) > d(L)o, the interaction free energy is further increased due to the enhanced stretching of the tails. When d(P) < d(L)o, chain flexibility (entropy) increases, but this contribution to DELTAF is overcounted by the increase in the interfacial free energy. Thus, DELTAF obtains a minimum at d(P) - d(L)o congruent-to 0. These qualitative interpretations are supported by detailed numerical calculations of the various contributions to the interaction free energy, and of chain conformational properties. The range of the perturbation of lipid order extends typically over few molecular diameters. A rather detailed comparison of our approach to other models is provided in the Discussion.
The two successive fluid-fluid phase transitions in surfactant Langmuir monolayers are described using a highly simplified molecular model: a `reactive' mixture of inter-converting squares of two different sizes. The model is solved by a mean-field lattice approach and by Monte Carlo simulations. The mean-field scheme involves a re-division of the original lattice into `cells' which can contain either one large square representing the (projection on the lattice of) an amphiphilic molecule in a conformationally disordered ('expanded') state, or clusters consisting of 1-4 small squares, each representing an ordered ('stretched') molecule. This procedure circumvents some of the difficulties associated with the size disparity of the adsorbed particles. In spite of its simplicity, the model can explain some major, as well as some subtle, characteristics of experimental monolayer phase diagrams. These include the conditions under which the monolayer exhibits one phase transition, two or none; the decrease of the triple point temperature with increasing chain length, and the gradual decrease with temperature of the liquid-condensed phase density.
We consider the effect of shear velocity gradients on the size (L) of rodlike micelles in dilute and semidilute solution. A kinetic equation is introduced for the time-dependent concentration of aggregates of length L, consisting of ``bimolecular'' combination processes L + L' –> (L + L') and ``unimolecular'' fragmentations L –> L' + (L - L'). The former are described by a generalization (from spheres to rods) of the Smoluchowski mechanism for shear-induced coalescence of emulsions, and the latter by incorporating the tension-deformation effects due to flow. Steady-state solutions to the kinetic equation are obtained, with the corresponding mean micellar size (LBAR) evaluated as a function of the Peclet number P, i.e., the dimensionless ratio of flow rate-gamma and rotational diffusion coefficient D(r). For sufficiently dilute solutions, we find only a weak dependence of LBAR on P. In the semidilute regime, however, an apparent divergence in LBAR at P congruent-to 1 suggests a flow-induced first-order gelation phenomenon.
In this paper we present a rigid-rod model (involving a restricted set of orientations) which is solved first with mean-field theory and then by Monte Carlo simulation. It is shown that both interparticle attractions and anisotropic adsorption energies are necessary in order for two successive fluid-fluid transitions to occur. The first is basically a gas-liquid condensation of ``lying down'' rods in the plane of the surface, and the second involves a ``standing up'' of the particles. A close qualitative correspondence is established between the results obtained in the mean-field and Monte Carlo descriptions. The role of biaxial states, i.e., in-plane orientational ordering, is also discussed in both contexts. To this end, we develop an analogy between our one-component rod monolayer and a bidisperse system of interconverting isotropic particles.
Within the framework of two complementary models, we show that the densities and patterns of defects in amphiphile-water systems with lamellar organization are coupled to the strength of the bilayer-bilayer interactions and hence to the overall surfactant concentration. We consider defects which introduce curvature (i.e., larger head-group area per molecule) while preserving the integrity of stacked bilayers at surfactant volume fractions of several tenths. These features are favored if the molecules comprising the lamellae are preferentially packed with a nonplanar aggregate-water interface: curvature defects lower the local free energy in systems constrained by aggregate-aggregate interactions to lamellar geometry. As the amphiphile volume fraction is increased-and the bilayer-bilayer spacing thereby decreased-we predict phase transitions between lamellar phases of different defect patterns on the bilayer surface, with concurrent decrease in the defect area fraction per bilayer. Specifically, there is a progression from a stripe-like pattern of parallel channels to a random network of line defects to a pore phase, with the latter appearing at the highest amphiphile concentrations but characterized by the lowest density of defects. Connection is made with experimental work which has recently suggested various departures from classical lamellar structure.
The steady-state bimolecular annihilation reaction A + B –> 0 on two-dimensional surfaces is studied via computer simulations. In the simulations A and B are randomly adsorbed on vacant sites, and reaction takes place whenever A and B reach nearest-neighbor sites, either directly following adsorption or through diffusion. It is found that both with and without diffusion the reactants segregate into separate islands of A's and B's. The islands vary in size and exhibit highly ramified shapes. Moreover, the islands are self-similar with a fractal dimension D = 1.89 (similar to percolation, but also other clusters). D is found to be independent of the diffusion rate K. Other fractal dimensions, e.g., of the ``hull'' differ from those of percolating clusters. The steady-state coverage theta* = theta*A + theta*B decreases with K, as expected (theta*A = theta*B, corresponding to equal fluxes of A and B is the only physical solution). For systems with immobile particles (K = 0) we find theta* congruent-to 0.59 and theta* congruent-to 0.49 for the square and the triangular lattices, respectively, similar to the percolation thresholds on these lattices. The long-time characteristics of the system (D, theta*, etc.) are independent of the initial conditions of the simulation, indicating that the system reaches a stable steady state. Furthermore, for the large systems simulated (typically 500 x 500 lattice sites) it was found that the long-time behavior is independent of the input mode. Namely, the same results are obtained for conserved (i.e., exactly balanced) and nonconserved (statistically balanced) A,B input mechanisms, indicating that on the time scale of the simulations (approximately 10(4) Monte Carlo steps) the apparent steady state (for nonconserved input) is essentially identical with the true steady state (for the conserved input).
A mean field theory of chain packing in amphiphilic aggregates is used to calculate conformational and thermodynamic properties of the inverse hexagonal phase. These properties are compared with those for planar bilayers and curved monolayers. Calculated bond order parameters reveal that chains packed in the hexagonal arrangement have more conformational freedom than chains packed in a bilayer. The calculated order parameters are in good agreement with recent experimental results. Free energy calculations are also presented. It is found that for small areas per head group the packing free energy of amphiphiles in a bilayer is considerably higher than in the hexagonal phase.