Neumann's Law Research Articles

Growth Laws and Self-Similar Growth Regimes of Coarsening Two-Dimensional Foams: Transition from Dry to Wet Limits

We study the topology and geometry of two-dimensional coarsening foam with an arbitrary liquid fraction. To interpolate between the dry limit described by von Neumann's law and the wet limit described by Marqusee's equation, the relevant bubble characteristics are the Plateau border radius and a new variable: the effective number of sides. We propose an equation for the individual bubble growth rate as the weighted sum of the growth through bubble-bubble interfaces and through bubble-Plateau border interfaces. The resulting prediction is successfully tested, without an adjustable parameter, using extensive bidimensional Potts model simulations. The simulations also show that a self-similar growth regime is observed at any liquid fraction, and they also determine how the average size growth exponent, side number distribution, and relative size distribution interpolate between the extreme limits. Applications include concentrated emulsions, grains in polycrystals, and other domains with coarsening that is driven by curvature.

Deformation of Platonic foam cells: Effect on growth rate

The diffusive growth rate of a polyhedral cell in dry three-dimensional foams depends on details of shape beyond cell topology, in contrast to the situation in two dimensions, where, by von Neumann's law, the growth rate depends only on the number of cell edges. We analyze the dependence of the instantaneous growth rate on the shape of single foam cells surrounded by uniform pressure; this is accomplished by supporting the cell with films connected to a wire frame and inducing cell distortions by deforming the wire frame. We consider three foam cells with a very simple topology; these are the Platonic foam cells, which satisfy Plateau's laws and are based on the trivalent Platonic solids (tetrahedron, cube, and dodecahedron). The Surface Evolver is used to model cell deformations induced through extension, compression, shear, and torsion of the wire frames. The growth rate depends on the deformation mode and frame size and can increase or decrease with increasing cell distortion. The cells have negative growth rates, in general, but dodecahedral cells subjected to torsion in small wire frames can have positive growth rates. The deformation of cubic cells is demonstrated experimentally.

The Voronoi Implicit Interface Method for computing multiphase physics.

We introduce a numerical framework, the Voronoi Implicit Interface Method for tracking multiple interacting and evolving regions (phases) whose motion is determined by complex physics (fluids, mechanics, elasticity, etc.), intricate jump conditions, internal constraints, and boundary conditions. The method works in two and three dimensions, handles tens of thousands of interfaces and separate phases, and easily and automatically handles multiple junctions, triple points, and quadruple points in two dimensions, as well as triple lines, etc., in higher dimensions. Topological changes occur naturally, with no surgery required. The method is first-order accurate at junction points/lines, and of arbitrarily high-order accuracy away from such degeneracies. The method uses a single function to describe all phases simultaneously, represented on a fixed Eulerian mesh. We test the method's accuracy through convergence tests, and demonstrate its applications to geometric flows, accurate prediction of von Neumann's law for multiphase curvature flow, and robustness under complex fluid flow with surface tension and large shearing forces.

On two-dimensional foam ageing

The present study aims at documenting, making use of an original set-up allowing to acquire well-converged data, the coarsening of foams in two dimensions. Experiments show that a foam behaves quite differently depending on the way it has been prepared. We distinguish between an initially quasi-monodisperse foam and a polydisperse foam. The coarsening laws are initially different, although both foams reach a common, time-dependent asymptotic regime.The ageing process relies on exchanges between adjacent foam cells (von Neumann's law), and on topological rearrangement (1 and 2 processes) whose rates are measured in all regimes. We attempt to make their contribution to the evolution of the area S and facet number n distribution of probability P(S, n, t) quantitative. The corresponding mean field theory predictions represent well the phenomenon qualitatively, and are sometimes in quantitative agreement with the measurements. A simplified version of this theory, taking the form of a Langevin model, explains in a straightforward manner the different scaling laws in the different regimes, for the different foams.

Unified paradigm for interface dynamics

In this paper we develop a common theoretical framework for the dynamics of thin featureless interfaces. We explicitly demonstrate that the same phase field and velocity dependent one-scale models characterizing the dynamics of relativistic domain walls, in a cosmological context, can also successfully describe, in a friction dominated regime, the dynamics of nonrelativistic interfaces in a wide variety of material systems. We further show that a statistical version of the von Neumann's law applies in the case of scaling relativistic interface networks, implying that, although relativistic and nonrelativistic interfaces have very different dynamics, a single simulation snapshot is not able to clearly distinguish the two regimes. We highlight that crucial information is contained in the probability distribution function for the number of edges of domains bounded by the interface network and explain why laboratory tests with nonrelativistic interfaces can be used to rule out cosmological domain walls as a significant dark energy source.

Self-similar grain size distribution in three dimensions: A stochastic treatment

Abstract In this paper, a stochastic formulation of three-dimensional grain growth is presented. This formulation employs the recent extension of the von Neumann law to three dimensions, and leads to a Fokker–Planck equation for the size distribution. The self-similar solutions of the Fokker–Planck equation presented here are based on the assumption of quasi-stationary distributions reached in the long time limit. The resulting grain size distributions, obtained both numerically and analytically, are shown to be in good agreement with each other and also with those obtained from computer simulations, indicating the validity of the stochastic approach.

Bone Marrow Adipocytes Prevent Hematopoietic Expansion in Homeostasis and in Bone Marrow Transplantation

In mammalian bone marrow (BM), osteoblasts and endothelium constitute functional niches that support hematopoietic stem cells (HSC). Adult BM also contains numerous adipocytes, whose numbers correlate inversely with the hematopoietic activity of the marrow. As described by Neumann's law in 1882, distal skeletal regions are adipocytic and thus non-hematopoietic in the adult. Fatty infiltration of the hematopoietic red marrow also occurs following irradiation or chemotherapy and is a diagnostic feature in biopsies from patients with marrow aplasia. However, whether adipocytes participate in hematopoietic regulation or simply expand to fill marrow space is unclear. We have found that murine hematopoiesis is reduced in adipocyte-rich marrow during homeostasis, and that adipocytes antagonize marrow recovery post-irradiation. By flow cytometry, colony forming assay, and competitive repopulation, we found a reduced frequency of HSCs and short-term hematopoietic progenitors in the adipocyte-rich vertebrae of the tail compared to the adipocyte-free vertebrae of the thorax. In A-ZIP/F1 “fatless” mice, which are genetically incapable of forming adipocytes, post-irradiation marrow engraftment is accelerated relative to wild type mice. Likewise, pharmacologic inhibition of adipocyte formation with the PPARg inhibitor Bisphenol-A-DiGlycidyl-Ether (BADGE) enhances hematopoietic recovery after BM transplant. Moreover, we have found that mice deficient in the adipocyte-specific protein adiponectin, which has been described to inhibit hematopoietic progenitor expansion in vitro, have increased progenitors in fatty marrow both during homeostasis and after BM transplant. Our data implicate adipocytes as negative regulators of the bone marrow microenvironment, and demonstrate that antagonizing adipogenesis is advantageous for enhancing hematopoietic recovery in the setting of bone marrow transplantation.

How foam-like is the shear-induced lamellar phase of an ionic liquid crystal?

In a recent article [M.H. Godinho et al., Liq. Cryst. 35, 103 (2008)] we reported that sheared films of two n-alkylimidazolium salts exhibit liquid crystalline behaviour below their bulk equilibrium freezing temperature. The resulting morphologies are strongly reminiscent of two-dimensional liquid foams: the materials partition into dark domains (cells) separated by brighter (birefringent) walls, which are approximately arcs of circle and meet at vertices (“Plateau borders”) with three or more sides. Here we investigate whether they satisfy known quantitative results for foams [see, e.g. D. Weaire and S. Hutzler, The Physics of Foams (Oxford University Press, Oxford, 1999)]. We find that: (i) where three walls meet, they do so at approximately 120° angles, for all times studied; (ii) Lewis's law of linear relation between cell area and number of sides is approximately satisfied at late times; (iii) the morphological patterns coarsen in time, both T1 and T2 processes are observed and, at late times, evolution is consistent with von Neumann's law; and (iv) relatively large numbers of 5-sided cells survive up to fairly late times. Results (i) and (iii) suggest that surface tension may play a key role in determining the physics of this system, as it does in low-viscosity liquid foams.

Shear‐induced lamellar phase of an ionic liquid crystal at room temperature

The phase behaviour of a number of N‐alkylimidazolium salts was studied using polarizing optical microscopy, differential scanning calorimetry and X‐ray diffraction. Two of these compounds exhibit lamellar mesophases at temperatures above 50°C. In these systems, the liquid crystalline behaviour may be induced at room temperature by shear. Sheared films of these materials, observed between crossed polarisers, have a morphology that is typical of (wet) liquid foams: they partition into dark domains separated by brighter (birefringent) walls, which are approximately arcs of circle and meet at “Plateau borders” with three or more sides. Where walls meet three at a time, they do so at approximately 120° angles. These patterns coarsen with time and both T1 and T2 processes have been observed, as in foams. The time evolution of domains is also consistent with von Neumann's law. We conjecture that the bright walls are regions of high concentration of defects produced by shear, and that the system is dominated by the interfacial tension between these walls and the uniform domains. The control of self‐organised monodomains, as observed in these systems, is expected to play an important role in potential applications.

Simulation of Ideal Grain Growth Using the Multi-Phase-Field Model

The kinetics and topology of ideal grain growth were simulated using the phase-field model. Large scale phase-field simulations were carried out where ten thousands grains evolved into a few hundreds without allowing coalescence of grains. The implementation was first validated in two-dimensions by checking the conformance with square-root evolution of the average grain size and the von Neumann-Mullins law. Afterwards three-dimensional simulations were performed which also showed fair agreement with the law describing the evolution of the mean grain size against time and with the results of S. Hilgenfeld et al. in 'An Accurate von Neumann's Law for Three-Dimensional Foams', Phys. Rev. Letters, 86(12)/2685, March 2001. Finally the steady state grain size distribution was investigated and compared to the Hillert theory.

Prediction and Relation of Thermal Conductivity with Average Relative Atomic Mass and Density for Semiconducting Compound Functional Crystal Materials

Thermal conductivity plays an important role in dominating a functional material's behavior in heat conduction processes and its applicability. Thus it is important to develop a prediction method for thermal conductivity to improve the thermal design and manufacturing processes of functional materials. According to the microscopy theories of heat conduction and the Koop–Neumann Law, a new method of predicting the thermal conductivity in close relationship to the average relative atomic mass and density of semiconducting compound functional crystals materials is developed in this paper. A new formula for calculating thermal conductivity that results in a simple but accurate approach is proposed. The thermal conductivities calculated using the formula are compared with the values obtained by experimental measurement and found to be in agreement for many semiconducting compound functional crystals and ceramic materials. It is discovered in further analysis that the larger the average relative atomic mass and density are, the more accurately the thermal conductivity is predicted. Such a prediction method for thermal conductivity is likely to be more useful in the computerized analogy practice of heat transfer, where the objective is to provide reference data of thermal conductivity to improve the fabrication and thermal design of semiconducting compound functional crystals and ceramic materials.

Static wetting on flexible substrates: a finite element formulation

AbstractIn static wetting on an elastic substrate, force exerted by the liquid–vapour surface tension on a solid surface deforms the substrate, producing a capillary ridge along the contact line. This paper presents a finite element formulation for predicting elastic deformation, close to the static wetting line (with angle of contact=90o and σSV=σSL).The substrate deformation is modelled with the Mooney–Rivlin constitutive law for incompressible rubber‐like solids. At the contact line, a stress singularity is known to arise, due to the surface tension acting on a line of infinitesimal thickness. To relive the stress singularity, either (i) the surface tension is applied over a finite contact region (of macroscopic thickness), or (ii) the solid crease angle is fixed. These two options suggest that normal component of Neumann's triangle law of forces, for the three surface tensions, is not applicable for elastic substrates (as for rigid ones). The vertical displacement of the contact line is a strong function of liquid/vapour surface tension and shear modulus of the solid. Copyright 2004 John Wiley & Sons, Ltd.

Modeling of grain growth in two dimensions

A detailed consideration of the time-dependent bivariate distribution P R ,n ;t in both the grain size R, and the number of sides n of the grain in two dimensions is given. Using this formalism and the von Neumann law, an explicit form of the Fokker–Planck equation for the marginal distribution F R ,t is provided. Based on available experimental evidence an explicit form of bivariate distribution is also constructed and its consequences are examined. Bivariate analysis shows the crucial importance of the correlation between the grain size and its number of sides and provides a more fundamental underpinning to modeling of Grain Growth.

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.

Three-Phase Effective Contact Angle in a Model Pore

We consider a model pore (2D) in which a sharp interface between two fluids contact a third fluid which wets the solid boundary. If the configuration is capillary dominated, the geometry can be determined analytically in terms of the effective contact angle. This angle depends not only on the interfacial tensions, but also on the capillary pressures. However, if the height of the cusp formed by the wetting fluid is much smaller than the pore width, the effective contact angle is a simple function of the interfacial tensions. It turns out to be the same function as in the case of an undeformed wetting layer of molecular thickness. The analytical expression for the effective contact angle has been confirmed by a numerical technique, known as the lattice-Boltzmann method. This method, in turn, has been validated with Neumann's law for the three-phase contact angles.

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.

Capillarity-driven interface dynamics: Application to grain growth phenomenon

Abstract The dynamics of interfaces separating two regions with small and large grain sizes are studied by both computer simulations and a theoretical approach in the framework of a zero-temperature Potts model. Flat and circular interfaces are considered. It is shown that the interface is advancing in the direction of smaller grains with dynamics predicted by a theory based on Neumann's law for the circular interface and energy considerations for the flat interface. In both geometries, the simulation results are correctly described by these analytical models.

Computer simulation of topological evolution in 2-D grain growth using a continuum diffuse-interface field model

The local kinetics and topological phenomena during normal grain growth were studied in two dimensions by computer simulations employing a continuum diffuse-interface field model. The relationships between topological class and individual grain growth kinetics were examined, and compared with results obtained previously from analytical theories, experimental results and Monte Carlo simulations. It was shown that both the grain-size and grain-shape (side) distributions are time-invariant and the linear relationship between the mean radii of individual grains and topological class n was reproduced. The moments of the shape distribution were determined, and the differences among the data from soap froth, Potts model and the present simulation were discussed. In the limit when the grain size goes to zero, the average number of grain edges per grain is shown to be between 4 and 5, implying the direct vanishing of 4- and 5-sided grains, which seems to be consistent with recent experimental observations on thin films. Based on the simulation results, the conditions for the applicability of the familiar Mullins-Von Neumann law and the Hiller's equation were discussed.

Topological correlations in Bénard-Marangoni convective structures.

B\'enard-Marangoni convection displays a two-dimensional (2D) hexagonal pattern. A topological analysis of these structures is presented. We describe the elementary topological transformations involved in such patterns (neighbor switching process, cell disappearance or creation, and cellular division) and the typical defects [pentagon-heptagon pair, ``flower'' (a cluster of more than three polygons incident on the same vertex), etc.]. Usual topological laws (Lewis's, Aboav-Weaire's, Peshkin's, and Lema\^{\i}tre's laws) are satisfied. For Von Neumann's law, a modification which takes into account a physical effect specific to the dynamics of B\'enard-Marangoni structures (selection of average cell size in steady regime) has been introduced. We also compare topological correlations in B\'enard-Marangoni structures with those derived from biological tissues or simulated structures (2D hard disk tessellations, Ising mosaics, distributions of maximum entropy formalism, etc.). The agreement is fair for pentagons, hexagons, and heptagons. \textcopyright{} 1996 The American Physical Society.