Two-phase fluid deformable surfaces with constant enclosed volume and phase-dependent bending and Gaussian rigidity

In this work I derive analytic expressions for the curvature-dependent fluid-substrate surface tension of a hard-sphere fluid on a hard curved wall. In the first step, the curvature thermodynamic properties are found as truncated power series in the activity in terms of the exactly known second- and third-order cluster integrals of the hard-sphere fluid near spherical and cylindrical walls. These results are then expressed as packing fraction power series and transformed to different reference regions, which is equivalent to considering different positions of the dividing surface. Based on the truncated series it is shown that the bending rigidity of the system is non-null and that higher-order terms in the curvature also exist. In the second step, approximate analytic expressions for the surface tension, the Tolman length, the bending rigidity, and the Gaussian rigidity as functions of the packing fraction are found by considering the known terms of the series expansion complemented with a simple fitting approach. It is found that the obtained formulas accurately describe the curvature thermodynamic properties of the system; further, they are more accurate than any previously published expressions.

We consider the renormalization of the bending and Gaussian rigidity of model membranes induced by long-range interactions between the components making up the membrane. In particular we analyze the effect of a finite membrane thickness on the renormalization of the bending and Gaussian rigidity by long-range interactions. Particular attention is paid to the case where the interactions are of a van der Waals type.

We use virial series to study the equilibrium properties of confined soft-spheres fluids interacting through the inverse-power potentials. The confinement is induced by hard walls with planar, spherical, and cylindrical shapes. We evaluate analytically the coefficients of order two in density of the wall-fluid surface tension γ and analyze the curvature contributions to the free energy. Emphasis is in bending and Gaussian rigidities, which are found analytically at order two in density. Their contribution to γ(R) and the accuracy of different truncation procedures to the low curvature expansion are discussed. Finally, several universal relations that apply to low-density fluids are analyzed.

Following the route of the stress tensor we study the free energy of a fluid liquid-vapor interface in the van der Waals approximation for planar, cylindrical and spherical interfaces. By performing a systematic expansion in powers of the inverse of the curvature radii, and appropriately defining the Gibbs dividing surface, we find unambiguous expressions for the surface tension, the spontaneous curvature, the bending rigidity and the Gaussian rigidity.

We study the RG flow of two dimensional (fluid) membranes embedded in Euclidean D-dimensional space using functional RG methods based on the effective average action. By considering a truncation ansatz for the effective average action with both extrinsic and intrinsic curvature terms we derive a system of beta functions for the running surface tension, bending rigidity and Gaussian rigidity. We look for non-trivial fixed points but we find no evidence for a crumpling transition at $T\neq0$. Finally, we propose to identify the $D\rightarrow 0$ limit of the theory with two dimensional quantum gravity. In this limit we derive new beta functions for both cosmological and Newton's constants.

Through the continuity of the DREIDING force field, we propose, for the first time, the finite-deformation plate theory for the single-layer hexagonal boron nitride (h-BN) to clarify the atomic source of the structure against deformations. Divergent from the classical Föppl-von Karman plate theory, our new theory shows that h-BN’s two in-plane mechanical parameters are independent of two out-of-plane mechanical parameters. The new theory reveals the relationships between the h-BN’s elastic rigidities and the atomic force field: (1) two in-plane elastic rigidities come from the bond stretching and the bond angle bending; (2) the bending rigidity comes from the inversion angle and the dihedral angle torsion; (3) the Gaussian rigidity only comes from the dihedral angle torsion. Mechanical parameters obtained by our theory align with atomic calculations. The new theory proves that two four-body terms in the DREIDING force field are necessary to model the h-BN’s mechanical properties. Overall, our theory establishes a foundation to apply the classical plate theory on the h-BN, and the approach in this paper is heuristic in modelling the mechanical properties of the other two-dimensional nanostructures.

We give a rigorous proof of the approximability of the so-called Helfrich's functional via diffuse interfaces under a constraint on the ratio between the bending rigidity and the Gaussian rigidity.

Two-Phase Fluid Deformable Surfaces with Constant Enclosed Volume and Phase-Dependent Bending and Gaussian Rigidity

Bending rigidities of tensionless balanced liquid-liquid interfaces as occurring in microemulsions are predicted using self-consistent field theory for molecularly inhomogeneous systems. Considering geometries with scale invariant curvature energies gives unambiguous bending rigidities for systems with fixed chemical potentials: the minimal surface Im3m cubic phase is used to find the Gaussian bending rigidity κ[over ¯], and a torus with Willmore energy W=2π^{2} allows for direct evaluation of the mean bending modulus κ. Consistent with this, the spherical droplet gives access to 2κ+κ[over ¯]. We observe that κ[over ¯] tends to be negative for strong segregation and positive for weak segregation, a finding which is instrumental for understanding phase transitions from a lamellar to a spongelike microemulsion. Invariably, κ remains positive and increases with increasing strength of segregation.

We investigate the formation of two-phase lipidic tubes of membrane in the framework of the Canham-Helfrich model. The two phases have different elastic moduli (bending and Gaussian rigidity), different tensions and a line tension prevents the mixing. For a set of parameters close to experimental values, periodic patterns with arbitrary wavelength can be found numerically. A wavelength selection is detected via the existence of an energy minimum. When the chemical composition induces an important enough size disequilibrium between both phases, a segregation into two half infinite tubes is preferred to a periodic structure.

We consider how membrane fluctuations can modify the miscibility of lipid mixtures, that is to say how the phase diagram of a boundary-constrained membrane is modified when the membrane is allowed to fluctuate freely in the case of zero surface tension. In order for fluctuations to have an effect, the different lipid types must have differing Gaussian rigidities. We show, somewhat paradoxically, that fluctuation-induced interactions can be treated approximately in a mean-field type theory. Our calculations predict that, depending on the difference in bending and Gaussian rigidity of the lipids, membrane fluctuations can either favor or disfavor mixing.

We investigate hexatic membranes embedded in Euclidean D-dimensional space using a reparametrization invariant formulation combined with exact renormalization group equations. An XY model coupled to a fluid membrane, when integrated out, induces long-range interactions between curvatures described by a Polyakov term in the effective action. We evaluate the contributions of this term to the running surface tension, bending, and Gaussian rigidities in the approximation of vanishing disinclination (vortex) fugacity. We find a non-Gaussian fixed point where the membrane is crinkled and has a nontrivial fractal dimension.

Prefusion and postfusion states of the biological fusion process between lipid bilayer vesicle membranes are studied in this paper. Based on the Helfrich-type continuum theory, a diffuse interface model is developed which describes the phase changes on the deformable vesicles via a scalar phase field function, and incorporates the adhesion effect between the different phases of the vesicles through a nonlocal interaction potential. Various equilibrium configurations in the prefusion and postfusion states are examined. The effects of spontaneous curvatures, bending, and Gaussian rigidities on the fusion process are discussed. Instead of considering only the regions in close contact as in many previous studies, the present approach allows us to include the energetic contributions from all parts of the vesicles. By carrying out simulations based on the gradient flow of the associated energy functional, we are also able to elucidate the dynamic transitions between the prefusion and postfusion states.

We study the shapes of pored membranes within the framework of the Helfrich theory under the constraints of fixed area and pore size. We show that the mean curvature term leads to a budding- like structure, while the Gaussian curvature term tends to flatten the membrane near the pore; this is corroborated by simulation. We propose a scheme to deduce the ratio of the Gaussian rigidity to the bending rigidity simply by observing the shape of the pored membrane. This ratio is usually difficult to measure experimentally. In addition, we briefly discuss the stability of a pore by relaxing the constraint of a fixed pore size and adding the line tension. Finally, the flattening effect due to the Gaussian curvature as found in studying pored membranes is extended to two-component membranes. We find that sufficiently high contrast between the components' Gaussian rigidities leads to budding which is distinct from that due to the line tension.

It is argued that to arrive at a quantitative description of the surface tension of a liquid drop as a function of its inverse radius, it is necessary to include the bending rigidity k and Gaussian rigidity in its description. New formulae for k and in the context of density functional theory with a non-local, integral expression for the interaction between molecules are presented. These expressions are used to investigate the influence of the choice of Gibbs dividing surface, and it is shown that for a one-component system, the equimolar surface has a special status in the sense that both k and are then the least sensitive to a change in the location of the dividing surface. Furthermore, the equimolar value for k corresponds to its maximum value and the equimolar value for corresponds to its minimum value. An explicit evaluation using a short-ranged interaction potential between molecules shows that k is negative with a value around minus 0.5–1.0 kBT and that is positive with a value that is a bit more than half the magnitude of k. Finally, for dispersion forces between molecules, we show that a term proportional to log(R)/R2 replaces the rigidity constants and we determine the (universal) proportionality constants.