Abstract
In textural equilibrium, partially molten materials minimize the total surface energy bound up in grain boundaries and grain–melt interfaces. Here, numerical calculations of such textural equilibrium geometries are presented for a space-filling tessellation of grains with a tetrakaidecahedral (truncated octahedral) unit cell. Two parameters determine the nature of the geometries: the porosity and the dihedral angle. A variety of distinct melt topologies occur for different combinations of these two parameters, and the boundaries between different topologies have been determined. For small dihedral angles, wetting of grain boundaries occurs once the porosity has exceeded 11%. An exhaustive account is given of the main properties of the geometries: their energy, pressure, mean curvature, contiguity and areas on cross sections and faces. Their effective permeabilities have been calculated, and demonstrate a transition between a quadratic variation with porosity at low porosities to a cubic variation at high porosities.
Highlights
The physical properties of partially molten materials depend crucially on the geometry of melt at the scale of individual grains
One of the key features of textural equilibrium is that different kinds of melt topology are possible for different values of the dihedral angle and at different porosities [2,3]
The equation above explains the limiting behaviour seen in figure 19 for low porosities and low dihedral angles, which tend to finite values of k/(φ2d2) as φ → 0, and whose geometries can be well described by tubes along the grain edges with approximately uniform cross section
Summary
The physical properties of partially molten materials depend crucially on the geometry of melt at the scale of individual grains. In the late 1980s was the simplest three-dimensional problem—four grains meeting at a junction with tetrahedral symmetry—solved fully numerically by von Bargen & Waff [5], Cheadle [6] and Nye [7] This simple model with tetrahedral symmetry provides important insights into when a melt network is expected to be connected, and provides constraints on the expected permeability [5,6,8] and electrical conductivity [6,9] of such networks. This article provides an exhaustive account of textural equilibrium melt geometries around a particular choice of solid grains which do fill space. Appendices provide more detail on the numerical methods, and provide some analytical solutions for special cases
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