Michael Murat

Discrete network models for the low-field Hall effect near a percolation threshold: Theory and simulations

The critical behavior of the weak-field Hall effect near a percolation threshold is studied with the help of two discrete random network models. Many finite realizations of such networks at the percolation threshold are produced and solved to yield the potentials at all sites. A new algorithm for doing that was developed that is based on the transfer matrix method. The site potentials are used to calculate the bulk effective Hall conductivity and Hall coefficient, as well as some other properties, such as the Ohmic conductivity, the size of the backbone, and the number of binodes. Scaling behavior for these quantities as power laws of the network size is determined and values of the critical exponents are found.

Current distributions in a two-dimensional random-resistor network

AbstractThe current and logarithm-of-the-current distributionsn(∣i∣) andn(∣ln ∣i∣∣) on bond diluted two-dimensional random-resistor networks at the percolation threshold are studied by a modified transfer matrix method. Thek th moment (−9⩽k⩽8) of n(∣ln ∣i∣∣) i.e., 〈∣ln ∣i&∣∣k〉, is found to scale with the linear sizeL as (InL)β(k). The exponents β(k) are not inconsistent with the recent theoretical prediction β(k)=k, with deviations which may be attributed to severe finitesize effects. For small currents, ln n(y)≈−γγ, yielding information on the threshold below which the multifractality of

Molecular-Dynamics Simulations of Polymer Surfaces and Interfaces

\hat n

Diffusion-limited aggregation near the percolation threshold

(∣i∣) breaks down. Our numerical results for the moments of the currents are consistent with other available results.

Polymer Surfaces and Interfaces: A Continuum Simulation Approach

The dielectric breakdown model is applied to the simulation of the breakdown of a 2-D dielectric material placed between two lines held at a high potential difference. The resulting discharge patterns closely resemble diffusion limited depositions on surfaces. The resemblance is further confirmed by comparing the geometrical properties of the two structures.

Growth and Viscous Fingers on Percolating Porous Media

From a single chain in a dilute solution to an entangled polymer melt, from bulk systems to more complex interfacial problems, computer simulations have played a critical role not only in testing the basic assumptions of various theoretical models but also in interpreting experimental results. Early computer simulations of polymers were mostly carried out on a lattice using Monte Carlo methods. This approach has led to significant progress in recent years and will continue to do so in many areas. In some cases however, for example in the study of shear, lattice models have serious limitations. For this reason and also due to the availability of more powerful computers, continuum, off-lattice polymer models have recently become popular. In this article, we review some of the recent progress in studying polymers at surfaces and interfaces using continuum models.

Molecular Dynamics Study of Dendrimer Molecules in Solvents of Varying Quality

We derive expressions for determination of the stress and the elastic constants in systems composed of particles interacting via noncentral two-body potentials as thermal averages of products of first and second partial derivatives of the interparticle potentials and components of the interparticle separation vectors. These results are adapted to hard potentials, where the stress and the elastic constants are expressed as thermal averages of the components of normals to contact surfaces between the particles and components of vectors separating their centers. The averages require knowledge of the simultaneous contact probabilities of two pairs of particles. We apply the expressions to particles for which a contact function can be defined, and demonstrate the feasibility of the method by computing the stress and the elastic constants of a two-dimensional system of hard ellipses using Monte Carlo simulations.

Interaction between grafted polymeric brushes: A molecular-dynamics study.

Diffusion-limited aggregation (DLA) and dielectric breakdown (DB) models have been used to simulate growth controlled by a Laplacian field on a square lattice network. A fraction ƒ (near the percolation threshold ƒc) of the bonds had a high conductivity (equal to 1), while the others had a low conductivity, equal to R. We used 10-5 < R < 1 for DLA and R = 10-8 for DB. We find crossover from growth on an incipient percolation cluster, with fractal dimensionality D ≅ 1.3, for small length scales, to that on a uniform substrate (D ≅ 1.7), for a large length scales. The crossover length behaves as LR ≈-a, with the crossover exponent a ≅ 0.25. The results were using a scaling theory.