Coarsening dynamics of ternary amphiphilic fluids and the self-assembly of the gyroid and sponge mesophases: Lattice-Boltzmann simulations

Diffusion-Driven Dissolution or Growth of a Liquid Drop Embedded in a Continuous Phase of Another Liquid via Phase-Field Ternary Mixture Model

We simulate the diffusion-driven dissolution or growth of a single-component (resp. two-component) drop embedded in a continuous phase of a binary (resp. single-component) liquid. Our theoretical approach follows a standard diffuse-interface model of partially miscible ternary liquid mixtures, which is based on a regular solution model assumption together with a Flory-Huggins and Cahn-Hilliard representation of the excess and nonlocal components of the Gibbs free energy of mixing. Based on 2D simulation results, we show that for a single-component drop embedded in a continuous phase of a binary liquid (which is highly miscible with either one component of the continuous phase but essentially immiscible with the other) the size of the drop can either shrink to zero or reach a stationary value, depending on whether the global composition of the mixture is within the one-phase region or the unstable range of the phase diagram. On the other hand, for an isolated two-component drop embedded in a continuous phase of a single-component liquid (which is essentially immiscible with either one component of the drop but miscible with the other) the size of the drop can either grow or shrink and, in particular, it will eventually go to zero if the global composition of the mixture is within the one-phase region; otherwise, for system locations in the unstable range the size of the drop tends to a constant value as the composition within the drop reaches its final equilibrium value.

Amplitude-modulated atomic force microscopy reveals the near surface nanostructure of surfactant sponge (L(3)) and lamellar (L(α)) phases

Amplitude-modulated atomic force microscopy (AM-AFM) has been used to study the nanostructure of cetylpyridinium chloride (CPCl)-hexanol-0.2 M NaCl sponge (L3) and lamellar (Lα) phases near a mica surface. For both phases, membrane volume fractions of 22, 27, and 32 vol % were investigated, with the L3 or Lα phase selected by adjusting the co-surfactant/surfactant ratio (hexanol/CPCl). For the L3 phase, the presence of the surface flattens the three-dimensional bulk structure. AM-AFM clearly resolves the membrane and solvent passages in the near surface layer. Increasing the membrane volume fraction decreases the size of the image features because of the lower solvent content. Within error, the average passage sizes in the near surface layer are the same as those in the bulk at the same concentration. Images of the Lα phase reveal undulating near surface sheets. At the highest membrane concentration, the image is very smooth, because the lamellar sheet is confined between the surface and the next near surface layer, which is in close proximity as a result of the low solvent content. As the membrane concentration is reduced, the space between layers is increased and undulations appear in the near surface lamellar structure. Undulations are more pronounced at the lowest membrane volume fraction.

Mesoscale models of dispersions stabilized by surfactants and colloids

In this paper we discuss and give an outlook on numerical models describing dispersions, stabilized by surfactants and colloidal particles. Examples of these dispersions are foams and emulsions. In particular, we focus on the potential of the diffuse interface models based on a free energy approach, which describe dispersions with the surface-active agent soluble in one of the bulk phases. The free energy approach renders thermodynamic consistent models with realistic sorption isotherms and adsorption kinetics. The free energy approach is attractive because of its ability to describe highly complex dispersions, such as emulsions stabilized by ionic surfactants, or surfactant mixtures and dispersions with surfactant micelles. We have classified existing numerical methods into classes, using either a Eulerian or a Lagrangian representation for fluid and for the surfactant/colloid. A Eulerian representation gives a more coarse-grained, mean field description of the surface-active agent, while a Lagrangian representation can deal with steric effects and larger complexity concerning geometry and (amphiphilic) wetting properties of colloids and surfactants. However, the similarity between the description of wetting properties of both Eulerian and Lagrangian models allows for the development of hybrid Eulerian/Lagrangian models having advantages of both representations.

Towards a mesoscopic model of water-like fluids with hydrodynamic interactions

We present a mesoscopic lattice model for non-ideal fluid flows with directional interactions, mimicking the effects of hydrogen bonds in water. The model supports a rich and complex structural dynamics of the orientational order parameter, and exhibits the formation of disordered domains whose size and shape depend on the relative strength of directional order and thermal diffusivity. By letting the directional forces carry an inverse density dependence, the model is able to display a correlation between ordered domains and low density regions, reflecting the idea of water as a denser liquid in the disordered state than in the ordered one.

Derivation and thermodynamics of a lattice Boltzmann model with soluble amphiphilic surfactant

We derive a lattice Boltzmann model of a ternary fluid mixture, one component of which consists of amphiphilic molecules with a rotational degree of freedom. The model was first introduced elsewhere where it was constructed on the basis of the earlier Shan-Chen type lattice Boltzmann model for binary fluids. We provide a rigorous derivation of the model from an underlying continuum kinetic theory. In particular, we show how the model can be interpreted as a discretization of a Vlasov-Boltzmann type kinetic theory for a fluid composed of two species of oppositely charged monopoles and a composite, dipolar molecule. We also derive a free-energy functional for the model, including the contribution from the amphiphiles, and confirm that the free energy is lowered during phase separation and during the formation of a stable emulsion.

Long-living intermediates during a lamellar to a diamond-cubic lipid phase transition: a small-angle X-ray scattering investigation

To generate nanostructured vehicles with tunable internal organization, the structural phase behavior of a self-assembled amphiphilic mixture involving poly(ethylene glycol) monooleate (MO-PEG) and glycerol monooleate (MO) is studied in excess aqueous medium by time-resolved small-angle X-ray scattering (SAXS) in the temperature range from 1 to 68 degrees C. The SAXS data indicate miscibility of the two components in lamellar and nonlamellar soft-matter nanostructures. The functionalization of the MO assemblies by a MO-PEG amphiphile, which has a flexible large hydrophilic moiety, appears to hinder the epitaxial growth of a double diamond (D) cubic lattice from the lamellar (L) bilayer structure during the thermal phase transition. The incorporated MO-PEG additive is found to facilitate the formation of structural intermediates. They exhibit greater characteristic spacings and large diffusive scattering in broad temperature and time intervals. Their features are compared with those of swollen long-living intermediates in MO/octylglucoside assemblies. A conclusion can be drawn that long-living intermediate states can be equilibrium stabilized in two- or multicomponent amphiphilic systems. Their role as cubic phase precursors is to smooth the structural distortions arising from curvature mismatch between flat and curved regions. The considered MO-PEG functionalized assemblies may be useful for preparation of sterically stabilized liquid-crystalline nanovehicles for confinement of therapeutic biomolecules.

Fluid-bicontinuous gels stabilized by interfacial colloids: low and high molecular weight fluids

Carefully tuned composite materials can have properties wholly unlike those of their separate constituents. We review the development of one example: colloid-stabilized emulsions with bicontinuous liquid domains. These non-equilibrium structures resemble the sponge mesophase of surfactants; however, in the colloid-stabilized case the interface separating the liquid domains is itself semi-solid. The arrangement of domains is created by arresting liquid-liquid phase separation via spinodal decomposition. Dispersed colloids exhibiting partial wettability become trapped on the newly created interface and jam together as the domains coarsen. Similar structures have been created in polymer blends stabilized using either interfacial nanoparticles or clay platelets. Here it has been possible to create the domain arrangement either by phase separation or by direct mixing of the melt. The low molecular weight liquid and polymer based structures have been developed independently and much can be learnt by comparing the two.

Emulsification of partially miscible liquids using colloidal particles: nonspherical and extended domain structures

We present microscopy studies of particle-stabilized emulsions with unconventional morphologies. The emulsions comprise pairs of partially miscible fluids and are stabilized by colloids. Alcohol-oil mixtures are employed; silica colloids are chemically modified so that they have partial wettability. We create our morphologies by two distinct routes: starting with a conventional colloid-stabilized emulsion or starting in the single-fluid phase with the colloids dispersed. In the first case temperature cycling leads to the creation of extended fluid domains built around some of the initial fluid droplets. In the second case quenching into the demixed region leads to the formation of domains which reflect the demixing kinetics. The structures are stable due to a jammed, semisolid, multilayer of colloids on the liquid-liquid interface. The differing morphologies reflect the roles in formation of the arrested state of heterogeneous and homogeneous nucleation and spinodal decomposition. The latter results in metastable, bicontinuous emulsions with frozen interfaces, at least for the thin-slab samples, investigated here.

Structural transitions and arrest of domain growth in sheared binary immiscible fluids and microemulsions

We investigate spinodal decomposition and structuring effects in binary immiscible and ternary amphiphilic fluid mixtures under shear by means of three-dimensional lattice Boltzmann simulations. We show that the growth of individual fluid domains can be arrested by adding surfactant to the system, thus forming a bicontinuous microemulsion. We demonstrate that the maximum domain size and the time of arrest depend linearly on the concentration of amphiphile molecules. In addition, we find that for a well-defined threshold value of amphiphile concentration, the maximum domain size and time of complete arrest do not change. For systems under constant and oscillatory shear we analyze domain growth rates in directions parallel and perpendicular to the applied shear. We find a structural transition from a sponge to a lamellar phase by applying a constant shear and the occurrence of tubular structures under oscillatory shear. The size of the resulting lamellae and tubes depends strongly on the amphiphile concentration, shear rate, and shear frequency.

Chirality and domain growth in the gyroid mesophase

We describe the first dynamical simulations of domain growth during the self-assembly of the gyroid mesophase from a ternary amphiphilic mixture, using the lattice Boltzmann method. The gyroid is a chiral structure; we demonstrate that, for a symmetric amphiphile with no innate preference for left- or right-handed morphologies, the self-assembly process may give rise to a racemic mixture of domains. We use measurements of the averaged mean curvature to analyse the behaviour of domain walls, and suggest that diffusive domain growth may be present in this system.

Solving the Fokker-Planck kinetic equation on a lattice

We propose a discrete lattice version of the Fokker-Planck kinetic equation in close analogy with the lattice-Boltzmann scheme. Our work extends an earlier one-dimensional formulation to arbitrary spatial dimension D. A generalized Hermite-Gauss procedure is used to construct a discretized kinetic equation and a Chapman-Enskog expansion is applied to adapt the scheme so as to correctly reproduce the macroscopic continuum equations. The linear stability of the algorithm with respect to the finite time step Delta t is characterized by the eigenvalues of the collision matrix. A heuristic second-order algorithm in Delta t is applied to investigate the time evolution of the distribution function of simple model systems, and compared to known analytical solutions. Preliminary investigations of sedimenting Brownian particles subjected to an orthogonal centrifugal force illustrate the numerical efficiency of the Lattice-Fokker-Planck algorithm to simulate nontrivial situations. Interactions between Brownian particles may be accounted for by adding a standard Bhatnagar-Gross-Krook collision operator to the discretized Fokker-Planck kernel.

Stress response and structural transitions in sheared gyroidal and lamellar amphiphilic mesophases: Lattice-Boltzmann simulations

We report on the stress response of gyroidal and lamellar amphiphilic mesophases to steady shear simulated using a bottom-up lattice-Boltzmann model for amphiphilic fluids and sliding periodic (Lees-Edwards) boundary conditions. We study the gyroid per se (above the sponge-gyroid transition, of high crystallinity) and the molten gyroid (within such a transition, of shorter-range order). We find that both mesophases exhibit shear thinning, more pronounced and at lower strain rates for the molten gyroid. At late times after the onset of shear, the skeleton of the crystalline gyroid becomes a structure of interconnected irregular tubes and toroidal rings, mostly oriented along the velocity ramp imposed by the shear, in contradistinction with free-energy Langevin-diffusion studies which yield a much simpler structure of disentangled tubes. We also compare the shear stress and deformation of lamellar mesophases with and without amphiphile when subjected to the same shear flow applied normal to the lamellae. We find that the presence of amphiphile allows (a) the shear stress at late times to be higher than in the case without amphiphile, and (b) the formation of rich patterns on the sheared interface, characterized by alternating regions of high and low curvature.

Simulated annealing study of gyroid formation in diblock copolymer solutions

Conditions for the formation of gyroid structures in diblock copolymer solutions are examined using a simulated annealing technique. The simulations were performed on diblock copolymer systems of A(NA)-b-B(NB) (with NA<NB) in solvents that are selective to the A blocks. It is shown that gyroid structures form in a narrow range of block copolymer concentrations between the hexagonally packed cylindrical and the lamellar phases and at an almost constant B-monomer concentration. It is also shown that the gyroid structure is especially sensitive to the B-solvent interaction (epsilonBS) and the length of the B block (NB). Phase diagrams for the diblock copolymer solutions are constructed. These predicted results are consistent with previous experimental observations. The three-dimensional isosurface contour plots of the simulated gyroid structure shows two interpenetrating strut networks. The projection along the [111] direction of the simulated gyroid structure and the spherically averaged structure factor are in good agreement with previous experimental results.

Modelling biological complexity: a physical scientist's perspective

We discuss the modern approaches of complexity and self-organization to understanding dynamical systems and how these concepts can inform current interest in systems biology. From the perspective of a physical scientist, it is especially interesting to examine how the differing weights given to philosophies of science in the physical and biological sciences impact the application of the study of complexity. We briefly describe how the dynamics of the heart and circadian rhythms, canonical examples of systems biology, are modelled by sets of nonlinear coupled differential equations, which have to be solved numerically. A major difficulty with this approach is that all the parameters within these equations are not usually known. Coupled models that include biomolecular detail could help solve this problem. Coupling models across large ranges of length- and time-scales is central to describing complex systems and therefore to biology. Such coupling may be performed in at least two different ways, which we refer to as hierarchical and hybrid multiscale modelling. While limited progress has been made in the former case, the latter is only beginning to be addressed systematically. These modelling methods are expected to bring numerous benefits to biology, for example, the properties of a system could be studied over a wider range of length- and time-scales, a key aim of systems biology. Multiscale models couple behaviour at the molecular biological level to that at the cellular level, thereby providing a route for calculating many unknown parameters as well as investigating the effects at, for example, the cellular level, of small changes at the biomolecular level, such as a genetic mutation or the presence of a drug. The modelling and simulation of biomolecular systems is itself very computationally intensive; we describe a recently developed hybrid continuum-molecular model, HybridMD, and its associated molecular insertion algorithm, which point the way towards the integration of molecular and more coarse-grained representations of matter. The scope of such integrative approaches to complex systems research is circumscribed by the computational resources available. Computational grids should provide a step jump in the scale of these resources; we describe the tools that RealityGrid, a major UK e-Science project, has developed together with our experience of deploying complex models on nascent grids. We also discuss the prospects for mathematical approaches to reducing the dimensionality of complex networks in the search for universal systems-level properties, illustrating our approach with a description of the origin of life according to the RNA world view.

Large-scale lattice Boltzmann simulations of complex fluids: advances through the advent of computational Grids

During the last 2.5 years, the RealityGrid project has allowed us to be one of the few scientific groups involved in the development of computational Grids. Since smoothly working production Grids are not yet available, we have been able to substantially influence the direction of software and Grid deployment within the project. In this paper, we review our results from large-scale three-dimensional lattice Boltzmann simulations performed over the last 2.5 years. We describe how the proactive use of computational steering, and advanced job migration and visualization techniques enabled us to do our scientific work more efficiently. The projects reported on in this paper are studies of complex fluid flows under shear or in porous media, as well as large-scale parameter searches, and studies of the self-organization of liquid cubic mesophases.

Towards performance control on the Grid

Advances in computational Grid technologies are enabling the development of simulations of complex biological and physical systems. Such simulations can be assembled from separate components--separately deployable computation units of well-defined functionality. Such an assemblage can represent an application composed of interacting simulations or might comprise multiple instances of a simulation executing together, each running with different simulation parameters. However, such assemblages need the ability to cope with heterogeneous and dynamically changing execution environments, particularly where such changes can affect performance. This paper describes the design and implementation of a prototype performance control system (PerCo), which is capable of monitoring the progress of simulations and redeploying them so as to optimize performance. The ability to control performance by redeployment is demonstrated using an assemblage of lattice Boltzmann simulations running with and without control policies. The cost of using PerCo is evaluated and it is shown that PerCo is able to reduce overall execution time.

Grid-based steered thermodynamic integration accelerates the calculation of binding free energies

The calculation of binding free energies is important in many condensed matter problems. Although formally exact computational methods have the potential to complement, add to, and even compete with experimental approaches, they are difficult to use and extremely time consuming. We describe a Grid-based approach for the calculation of relative binding free energies, which we call Steered Thermodynamic Integration calculations using Molecular Dynamics (STIMD), and its application to Src homology 2 (SH2) protein cell signalling domains. We show that the time taken to compute free energy differences using thermodynamic integration can be significantly reduced: potentially from weeks or months to days of wall-clock time. To be able to perform such accelerated calculations requires the ability to both run concurrently and control in realtime several parallel simulations on a computational Grid. We describe how the RealityGrid computational steering system, in conjunction with a scalable classical MD code, can be used to dramatically reduce the time to achieve a result. This is necessary to improve the adoption of this technique and further allows more detailed investigations into the accuracy and precision of thermodynamic integration. Initial results for the Src SH2 system are presented and compared to a reported experimental value. Finally, we discuss the significance of our approach.

Finite-difference lattice-Boltzmann methods for binary fluids

We investigate two-fluid Bhatnagar-Gross-Krook (BGK) kinetic methods for binary fluids. The developed theory works for asymmetric as well as symmetric systems. For symmetric systems it recovers Sirovich's theory and is summarized in models A and B. For asymmetric systems it contributes models C, D, and E which are especially useful when the total masses and/or local temperatures of the two components are greatly different. The kinetic models are discretized based on an octagonal discrete velocity model. The discrete-velocity kinetic models and the continuous ones are required to describe the same hydrodynamic equations. The combination of a discrete-velocity kinetic model and an appropriate finite-difference scheme composes a finite-difference lattice Boltzmann method. The validity of the formulated methods is verified by investigating (i) uniform relaxation processes, (ii) isothermal Couette flow, and (iii) diffusion behavior.