Ginzburg-Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations

Boltzmann–Gibbs Random Fields with Mesh-free Precision Operators Based on Smoothed Particle Hydrodynamics

Boltzmann–Gibbs random fields are defined in terms of the exponential expression exp ⁡ ( − H ) \exp \left (-\mathcal {H}\right ) , where H \mathcal {H} is a suitably defined energy functional of the field states x ( s ) x(\mathbf {s}) . This paper presents a new Boltzmann–Gibbs model which features local interactions in the energy functional. The interactions are embodied in a spatial coupling function which uses smoothed kernel-function approximations of spatial derivatives inspired from the theory of smoothed particle hydrodynamics. A specific model for the interactions based on a second-degree polynomial of the Laplace operator is studied. An explicit, mesh-free expression of the spatial coupling function (precision function) is derived for the case of the squared exponential (Gaussian) smoothing kernel. This coupling function allows the model to seamlessly extend from discrete data vectors to continuum fields. Connections with Gaussian Markov random fields and the Matérn field with ν = 1 \nu =1 are established.

A Note on the Solutions for a Higher-Order Convective Cahn–Hilliard-Type Equation

The higher-order convective Cahn-Hilliard equation describes the evolution of crystal surfaces faceting through surface electromigration, the growing surface faceting, and the evolution of dynamics of phase transitions in ternary oil-water-surfactant systems. In this paper, we study the H3 solutions of the Cauchy problem and prove, under different assumptions on the constants appearing in the equation and on the mean of the initial datum, that they are well-posed.

On the Well-Posedness of A High Order Convective Cahn-Hilliard Type Equations

High order convective Cahn-Hilliard type equations describe the faceting of a growing surface, or the dynamics of phase transitions in ternary oil-water-surfactant systems. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation.

Scattering intensity of bicontinuous microemulsions and sponge phases

Monte Carlo simulations of dynamically triangulated surfaces of variable topology are used to investigate the scattering intensities of bicontinuous microemulsions. The bulk scattering intensity is shown to follow the Teubner-Strey expression. The domain size and the correlation length are extracted from the scattering peaks as a function of the bending rigidity, saddle-splay modulus, and surfactant density. The results are compared to earlier theories based on Ginzburg-Landau and Gaussian random field models. The ratio of the two length scales is shown to be well described by a linear combination of logarithmically renormalized bending rigidity and saddle-splay modulus with universal prefactors. This is in contrast to earlier theoretical predictions in which the scattering intensity is independent of the saddle-splay modulus. The equation of state, and the asymptotics of the bulk and film scattering intensities for high and low wave vectors are determined from simulations and compared with theoretical results.

Lamellar phases confined in quasicylindrical pores: lattice model results

A two-dimensional (2D) vector lattice model of microemulsions is applied to study the structure of lamellar phases confined in long rectangular pores. One-point distribution functions are calculated within mean field approximation. The effects of pore geometry and surface fields are considered. A 2D analog of an onion phase is favored by a pore with strongly hydrophilic walls. For neutral walls, far from the phase boundaries, the lamellar phase is stable inside the pore. By contrast, close to the lamellar-tubular phase boundary a pore with neutral walls favors a 2D tubular phase. This is the analog of capillary condensation. In all cases the excess pressure is calculated as a function of the pore geometry.

Topological lifshitz line, off-specular scattering, and mesoporous materials

Ordered phases formed by surfactants in water solutions, and used in technological processes as templates for the synthesis of mesoporous materials, exhibit topological fluctuations. From the results of the Monte Carlo simulations of the lamellar phase we have established a relation between topological fluctuations and the behavior of the off-specular scattering intensity. We have defined the topological Lifshitz line. At this line the peak position in the off-specular scattering intensity moves from the zero (lamellar phase with fixed topology) to the nonzero value of the scattering wave vector (lamellar phase with fluctuating topology).

Systematic approach to bicontinuous cubic phases in ternary amphiphilic systems

The Fourier approach and theories of space groups and color symmetries are used to systematically generate and compare bicontinuous cubic structures in the framework of a Ginzburg-Landau model for ternary amphiphilic systems. Both single and double structures are investigated; they correspond to systems with one or two monolayers in a unit cell, respectively. We show how and why single structures can be made to approach triply periodic minimal surfaces very closely, and give improved nodal approximations for G, D, I-WP, and P surfaces. We demonstrate that the relative stability of the single structures can be calculated from the geometrical properties of their interfaces only. The single gyroid G turns out to be the most stable bicontinuous cubic phase since it has the smallest porosity. The representations are used to calculate distributions of the Gaussian curvature and 2H-nuclear-magnetic-resonance band shapes for C(P), C(D), S, C(Y), and F-RD surfaces.