Von Neumann's Law Research Articles

An accurate von Neumann's law for three-dimensional foams.

The diffusive coarsening of 2D soap froths is governed by von Neumann's law. A statistical version of this law for dry 3D foams has long been conjectured. A new derivation, based on a theorem by Minkowski, yields an explicit analytical von Neumann's law in 3D which is in very good agreement with detailed simulations and experiments. The average growth rate of a bubble with F faces is shown to be proportional to F1/2 for large F, in contrast to the conjectured linear dependence. Accounting for foam disorder in the model further improves the agreement with data.

Shell Analysis and Effective Disorder in a 2D Froth

The static and evolutionary properties of two-dimensional cellular structures, or froths, are discussed in the light of recent work on structuring of the froth into concentric shells. Of interest is the dual role of a topological dislocation (“defect”) in an otherwise uniform froth, considered both as a source of disorder and also as a source generating a shell-structured froth. We present simulations on an initially uniform hexagonal froth. A defect is introduced by forcing either a T1 or T2 process in the stable structure, after which the froth is allowed to evolve according to von Neumann's law. In the first case, topological inclusions are found in the first few layers early in the evolution. In the second case, no inclusions appear over the entire evolutionary period. The growing disorder (as measured by the second moment of the side distribution, μ2) is isotropic. For the special case of a T2-formed froth in a uniform network, the SSI structure is retained with μ2≠0 only for the zeroth, first, and second layers. The ratio between topological perimeter and radius of the shells is close to 6, the value for a hexagonal froth.

A multiphase-field model: sharp interface asymptotics and numerical simulations of moving phase boundaries and multijunctions

In this paper we bring together, compare and extend two recent developments in a special formulation of an anisotropic multiphase-field model; namely the results of sharp interface asymptotic analysis by Nestler and Wheeler [B. Nestler, A.A. Wheeler, Phys. Rev. E 57 (3) (1998) 2602.] and numerical simulations of moving phase boundaries and multijunctions by Garcke et al. [H. Garcke, B. Nestler, B. Stoth, Physica D 115 (1998) 87; SIAM J. Appl. Math., in press] First, we present the formulation of the multiphase-field model, which includes surface energy anisotropy. Then we state, how the leading order conditions at both interfaces and junctions can succinctly be derived in the sharp interface limit by introducing a generalized Cahn–Hoffman ξ -vector formalism and by using the motion of a stress tensor. These analytical results contain that the force balance at multijunctions is recovered, which comprises Young's law and, in the anisotropic case, additional shear forces. Next, we present numerical simulations of evolving phase boundaries and junctions, concentrating on the case of isotropic phases. We find that our numerical solutions of the multiphase-field model compare favorably with the exact solutions of the sharp interface analytical results. We observe that the classical angle conditions at trijunctions are obtained numerically. Finally, we perform simulations of grain growth evolution and numerically verify the validity of the qualitative features of the von Neumann law.

Scaling state in froths deduced from a corrected von Neumann's law

A corrected and generalized von Neumann's law is proposed, so that its statistical consequences agree with all experiments reported in the literature. It is shown that a froth's scaling state, concerning the surface area distribution, may be deduced from the stochastical independence between the number of sides distribution and the bubbles' surface area, and on the simple change of von Neumann's law from dA/ dt = k( n − 6) to dA/ dt = kA( n − 6).

Geometry and foams: 2D dynamics and 3D statics.

We discuss the implications of the classical Gauss-Bonnet formula to foam in two and three dimensions. For a two-dimensional foam it gives a generalization of the von Neumann law for the coarsening of foams to curved surfaces. As a consequence of this we find that the stability properties of stationary bubbles of such a froth depend on the Gaussian curvature of the surface. For three-dimensional foam we find a relation between the average Gaussian curvature of a soap film and the average number of vertices for each face.

Two-dimensional bubble rafts

Abstract We show that magnetic bubbles and lipid monolayer bubbles belong to the same family of cellular pattern systems, whose best studied member is the two-dimensional soap froth. We analyse the family resemblances by showing that these systems share the phase space structure of the ideal planar froth. The dynamics of these systems is studied with the goal of seeing how their topological structure might arise and unravel. Monolayer bubbles evolve in time, while magnetic bubbles do not; we show that this is due to a symmetry breaking in the interactions that form the domains. We propose the Gibbs-Thompson effect as responsible for this breaking in monolayers, and show that it induces a dynamics which is, to lowest order, von Neumann's law.

Coarsening in Two-Dimensional Soap Froths and the Large-Q Potts Model

ABSTRACTA detailed comparison between the experimental evolution of a two-dimensional soap froth and the large Q state Potts model is presented. The pattern evolution starting from identical initial conditions will be compared as well as a variety of distribution functions and correlations of the two systems. Simulations on different lattices show that the discrete lattice of the Potts model causes deviations from universal domain growth by weakening the vertex angle boundary conditions that form the basis of von Neumann's law. We show that the anisotropy inherent in a discrete lattice simulation, which masks the underlying ‘universal’ grain growth, can be overcome by increasing the range of the interaction between spins or increasing the temperature. Excellent overall agreement between the kinetics, topological distributions and domain size distributions between the low lattice anisotropy Potts-model simulations and the soap froth suggests that the Potts model is useful for studying domain growth in a wide variety of physical systems.

Structure of random cellular networks and their evolution

The methods of statistical mechanics are applied to the structure of random, space-filling, cellular structures (foams, metallurgical grain aggregates, biological tissues). Microreversibility of the structural properties under elementary transformations is demonstrated. Maximum entropy inference under a few constraints yields structural equations of state, relating the size of cells to their topological shape. These equations of state classify the structures, and are criteria for their randomness. Quasistatic evolution is obtained if the structure remains in statistical equilibrium at all times. In 2D, it is governed by von Neumann's law.