Criticality of colloids with distinct interaction patches: The limits of linear chains, hyperbranched polymers, and dimers

We use a simple model of associating fluids which consists of spherical particles having a hard-core repulsion, complemented by three short-ranged attractive sites on the surface (sticky spots). Two of the spots are of type A and one is of type B; the bonding interactions between each pair of spots have strengths epsilon(AA), epsilon(BB), and epsilon(AB). The theory is applied over the whole range of bonding strengths and the results are interpreted in terms of the equilibrium cluster structures of the phases. In addition to our numerical results, we derive asymptotic expansions for the free energy in the limits for which there is no liquid-vapor critical point: linear chains (epsilon(AA)not equal 0, epsilon(AB)=epsilon(BB)=0), hyperbranched polymers (epsilon(AB)not equal 0, epsilon(AA)=epsilon(BB)=0), and dimers (epsilon(BB)not equal 0, epsilon(AA)=epsilon(AB)=0). These expansions also allow us to calculate the structure of the critical fluid by perturbing around the above limits, yielding three different types of condensation: of linear chains (AA clusters connected by a few AB or BB bonds); of hyperbranched polymers (AB clusters connected by AA bonds); or of dimers (BB clusters connected by AA bonds). Interestingly, there is no critical point when epsilon(AA) vanishes despite the fact that AA bonds alone cannot drive condensation.

Equilibrium self-assembly of colloids with distinct interaction sites: Thermodynamics, percolation, and cluster distribution functions

We calculate the equilibrium thermodynamic properties, percolation threshold, and cluster distribution functions for a model of associating colloids, which consists of hard spherical particles having on their surfaces three short-ranged attractive sites (sticky spots) of two different types, A and B. The thermodynamic properties are calculated using Wertheim's perturbation theory of associating fluids. This also allows us to find the onset of self-assembly, which can be quantified by the maxima of the specific heat at constant volume. The percolation threshold is derived, under the no-loop assumption, for the correlated bond model: In all cases it is two percolated phases that become identical at a critical point, when one exists. Finally, the cluster size distributions are calculated by mapping the model onto an effective model, characterized by a-state-dependent-functionality (f) over bar and unique bonding probability (p) over bar. The mapping is based on the asymptotic limit of the cluster distributions functions of the generic model and the effective parameters are defined through the requirement that the equilibrium cluster distributions of the true and effective models have the same number-averaged and weight-averaged sizes at all densities and temperatures. We also study the model numerically in the case where BB interactions are missing. In this limit, AB bonds either provide branching between A-chains (Y-junctions) if epsilon(AB)/epsilon(AA) is small, or drive the formation of a hyperbranched polymer if epsilon(AB)/epsilon(AA) is large. We find that the theoretical predictions describe quite accurately the numerical data, especially in the region where Y-junctions are present. There is fairly good agreement between theoretical and numerical results both for the thermodynamic (number of bonds and phase coexistence) and the connectivity properties of the model (cluster size distributions and percolation locus).

The shape of soap films and Plateau borders

We have calculated the shapes of flat liquid films, and of the transition region to the associated Plateau borders (PBs), by integrating the Laplace equation with a position-dependent surface tension γ(x), where 2x is the local film thickness. We discuss films in either zero or non-zero gravity, using standard γ(x) potentials for the interaction between the two bounding surfaces. We have investigated the effects of the film flatness, liquid underpressure, and gravity on the shape of films and their PBs. Films may exhibit 'humps' and/or 'dips' associated with inflection points and minima of the film thickness. Finally, we propose an asymptotic analytical solution for the film width profile.

Invariant integrals applied to nematic liquid crystals with small Ericksen number and topological defects

Invariant integrals are derived for nematic liquid crystals and applied to materials with small Ericksen number and topological defects. The nematic material is confined between two infinite plates located at y = -h and y = h (h is an element of R+) with a semi-infinite plate at y = 0 and x < 0. Planar and homeotropic strong anchoring boundary conditions to the director field are assumed at these two infinite and semi-infinite plates, respectively. Thus, a line disclination appears in the system which coincides with the z-axis. Analytical solutions to the director field in the neighbourhood of the singularity are obtained. However, these solutions depend on an arbitrary parameter. The nematic elastic force is thus evaluated from an invariant integral of the energy-momentum tensor around a closed surface which does not contain the singularity. This allows one to determine this parameter which is a function of the nematic cell thickness and the strength of the disclination. Analytical solutions are also deduced for the director field in the whole region using the conformal mapping method. (C) 2013 Elsevier Ltd. All rights reserved.

A numerical analysis of generalized Boussinesq-type equation using continuos/discontinuous FEM

An improved class of Boussinesq systems of an arbitrary order using a wave surface elevation and velocity potential formulation is derived. Dissipative effects and wave generation due to a time-dependent varying seabed are included. Thus, high-order source functions are considered. For the reduction of the system order and maintenance of some dispersive characteristics of the higher-order models, an extra O(mu 2n+2) term (n ??? N) is included in the velocity potential expansion. We introduce a nonlocal continuous/discontinuous Galerkin FEM with inner penalty terms to calculate the numerical solutions of the improved fourth-order models. The discretization of the spatial variables is made using continuous P2 Lagrange elements. A predictor-corrector scheme with an initialization given by an explicit RungeKutta method is also used for the time-variable integration. Moreover, a CFL-type condition is deduced for the linear problem with a constant bathymetry. To demonstrate the applicability of the model, we considered several test cases. Improved stability is achieved.

Synchronization in Von Bertalanffy’s models

Many data have been useful to describe the growth of marine mammals, invertebrates and reptiles, seabirds, sea turtles and fishes, using the logistic, the Gom-pertz and von Bertalanffy's growth models. A generalized family of von Bertalanffy's maps, which is proportional to the right hand side of von Bertalanffy's growth equation, is studied and its dynamical approach is proposed. The system complexity is measured using Lyapunov exponents, which depend on two biological parameters: von Bertalanffy's growth rate constant and the asymptotic weight. Applications of synchronization in real world is of current interest. The behavior of birds ocks, schools of fish and other animals is an important phenomenon characterized by synchronized motion of individuals. In this work, we consider networks having in each node a von Bertalanffy's model and we study the synchronization interval of these networks, as a function of those two biological parameters. Numerical simulation are also presented to support our approaches.

Soliton-type and other travelling wave solutions for an improved class of nonlinear sixth-order Boussinesq equations

An improved class of nonlinear bidirectional Boussinesq equations of sixth order using a wave surface elevation formulation is derived. Exact travelling wave solutions for the proposed class of nonlinear evolution equations are deduced. A new exact travelling wave solution is found which is the uniform limit of a geometric series. The ratio of this series is proportional to a classical soliton-type solution of the form of the square of a hyperbolic secant function. This happens for some values of the wave propagation velocity. However, there are other values of this velocity which display this new type of soliton, but the classical soliton structure vanishes in some regions of the domain. Exact solutions of the form of the square of the classical soliton are also deduced. In some cases, we find that the ratio between the amplitude of this wave and the amplitude of the classical soliton is equal to 35/36. It is shown that different families of travelling wave solutions are associated with different values of the parameters introduced in the improved equations.