B. H. Bradshaw-Hajek

Dynamic Dielectric Response of Concentrated Colloidal Dispersions: Comparison between Theory and Experiment†

The cell-model electrokinetic theory of Ahualli et al. Langmuir 2006, 22, 7041; Ahualli et al. J. Colloid Interface Sci. 2007, 309, 342; and Bradshaw-Hajek et al. Langmuir 2008, 24, 4512 is applied to a dense suspension of charged spherical particles, to exhibit the systems dielectric response to an applied electric field as a function of solids volume fraction. The models predictions of effective permittivity and complex conductivity are favorably compared with published theoretical calculations and experimental measurements on dense colloidal systems. Physical factors governing the volume fraction dependence of the dielectric response are discussed.

A robust cubic reaction-diffusion system for gene propagation

Continuum modelling of gene frequencies during spatial dispersion of a population arrives at a reaction-diffusion equation with cubic source term, rather than the quadratic equation that Fisher proposed in 1937. For the case of three possible alleles at one diploid locus, with general degrees of fitness for the six genotypes, we derive a new system of coupled cubic reaction-diffusion equations for two independent gene frequencies. When any number of preexisting alleles compete for a single locus, in the important case of partial dominance and shared disadvantage of preexisting alleles, the new mutant allele is described by a single equation if the total population is known. In the case of Mendelian inheritance considered by Fisher, this equation is the Huxley equation, a reaction-diffusion equation whose source term is degenerate cubic with two real roots. Some practical analytic solutions of the genetic dispersion equation are constructed by the method of nonclassical symmetry reduction. The obtained solutions satisfy specific boundary conditions and they are different from previously derived travelling wave solutions.

Frequency-dependent electrical conductivity of concentrated dispersions of spherical colloidal particles.

This paper outlines the application of a self-consistent cell-model theory of electrokinetics to the problem of determining the electrical conductivity of a dense suspension of spherical colloidal particles. Numerical solutions of the standard electrokinetic equations, subject to self-consistent boundary conditions, are implemented in formulas for the electrical conductivity appropriate to the particle-averaged cell model of the suspension. Results of calculations as a function of frequency, zeta potential, volume fraction, and electrolyte composition, are presented and discussed.

Exact non-classical symmetry solutions of Arrhenius reaction-diffusion

Exact solutions for nonlinear Arrhenius reaction–diffusion are constructed in n-dimensions. A single relationship between nonlinear diffusivity and the nonlinear reaction term leads to a non-classical Lie symmetry whose invariant solutions have a heat flux that is exponential in time (either growth or decay), and satisfying a linear Helmholtz equation in space. This construction also extends to heterogeneous diffusion wherein the nonlinear diffusivity factorizes to the product of a function of temperature and a function of position. Example solutions are given with applications to heat conduction in conjunction with either exothermic or endothermic reactions, and to soil–water flow in conjunction with water extraction by a web of plant roots.

The extended-domain?eigenfunction method for solving elliptic boundary value problems with annular domains

Standard analytical solutions to elliptic boundary value problems on asymmetric domains are rarely, if ever, obtainable. In this paper, we propose a solution technique wherein we embed the original ...

Exact solutions for logistic reaction–diffusion equations in biology

Reaction–diffusion equations with a nonlinear source have been widely used to model various systems, with particular application to biology. Here, we provide a solution technique for these types of equations in N-dimensions. The nonclassical symmetry method leads to a single relationship between the nonlinear diffusion coefficient and the nonlinear reaction term; the subsequent solutions for the Kirchhoff variable are exponential in time (either growth or decay) and satisfy the linear Helmholtz equation in space. Example solutions are given in two dimensions for particular parameter sets for both quadratic and cubic reaction terms.

The Actual Dielectric Response Function for a Colloidal Suspension of Spherical Particles

In this paper, we present a theoretical analysis of the dielectric response of a dense suspension of spherical colloidal particles based on a self-consistent cell model. Particular attention is paid to (a) the relationship between the dielectric response and the conductivity response and (b) the connection between the real and imaginary parts of these responses based on the Kramers-Kronig relations. We have thus clarified the analysis of Carrique et al. (Carrique, F.; Criado, C.; Delgado, A. V. J. Colloid Interface Sci. 1993, 156, 117). We have shown that both the conduction and displacement current components are complex quantities with both real and imaginary parts being frequency dependent. The dielectric response exhibits characteristics of two relaxation phenomena: the Maxwell-Wagner and the alpha-relaxations, with the imaginary part being the more sensitive instrument. The inverse Fourier transform of the simulated dielectric response is compared with a phenomenological, two-exponential response function with good agreement obtained. The two fitted decay times also compare well with times extracted from the explicit simulations.

Computation of Extensional Fall of Slender Viscous Drops by a One-Dimensional Eulerian Method

We develop a one-dimensional Eulerian model suitable for analyzing the behavior of viscous fluid drops falling from rest from an upper boundary. The method allows examination of development and behavior from early time, when a drop and filament begin to form, out to large times when the bulk of the fluid forms a drop at the bottom of a long thin filament which connects it with the upper boundary. This model overcomes problems seen in Lagrangian models, caused by excessive stretching of grid elements, and enables a better examination of the thin fluid filament.

Numerical implementation of the EDEM for modified Helmholtz BVPs on annular domains

In a recent paper by the current authors a new methodology called the Extended-Domain-Eigenfunction-Method (EDEM) was proposed for solving elliptic boundary value problems on annular-like domains. In this paper we present and investigate one possible numerical algorithm to implement the EDEM. This algorithm is used to solve modified Helmholtz BVPs on annular-like domains. Two examples of annular-like domains are studied. The results and performance are compared with those of the well-known boundary element method (BEM). The high accuracy of the EDEM solutions and the superior efficiency of the EDEM over the BEM, make EDEM an excellent alternate candidate to use in the animation industry, where speed is a predominant requirement, and by the scientific community where accuracy is the paramount objective.

High-Frequency Behavior of the Dynamic Mobility and Dielectric Response of Concentrated Colloidal Dispersions

A matched asymptotic analysis of the system of equations governing the electrokinetic cell model of ref 4 (Ahualli, S.; Delgado, A.; Miklavcic, S.; White, L. R. Langmuir 2006, 22, 7041) is performed. Asymptotic expressions are obtained for the dynamic mobility and complex conductivity response of a dense suspension of charged spherical particles to an applied electric field. The asymptotic expressions are compared with full numerical calculations of the linear response functions as a function of surface (zeta) potential, electrolyte strength, and particle density. We find that the numerical procedure used is robust and highly accurate at a very high frequency under a wide range of double-layer conditions. The asymptotic form for the dielectric response of the system is accurate to megahertz frequencies. The asymptotic formulas for the other response functions have limited viability as predictive tools within the current range of experimentally accessible frequencies but are useful as checks on numerical calculations.