Textural equilibrium melt geometries

In textural equilibrium, partially molten materials minimise the surface energy bound up in grain boundaries and grain-melt interfaces. I have calculated textural equilibrium melt geometries for an idealised problem which considers an infinite tesellation of tetrakaidecahedral grains. The geometries are described in detail in the following manuscript:

Rudge J.F. Textural equilibrium melt geometries around tetrakaidecahedral grains (2017) in review [preprint] [arXiv]

The calculations were performed using Ken Braake’s Surface Evolver software. Links to the Surface Evolver scripts for calculating the geometries are provided below. Summary data is also provided giving the energy; pressure; areas of interfaces and contacts; and permeabilities for a range of porosities and dihedral angles. Units are such that the each edge in the unit cell has length 2.

Tetrakaidecahedral grains

Two key parameters control the nature of the melt topology: the porosity and the dihedral angle. The images below show the melt quadruple junction (yellow, left), the grain (pink, middle), and a combination of grain and melt (right).


Connected topology, dihedral angle 30°, porosity 3%. [Surface Evolver script] [Summary data]


Isolated topology, dihedral angle 120°, porosity 5%. [Surface Evolver script] [Summary data]


Square-wetted topology, dihedral angle 70°, porosity 8%. [Surface Evolver script] [Summary data]


Hex-wetted topology, dihedral angle 130°, porosity 2%. [Surface Evolver script] [Summary data]

Quadruple junction with tetrahedral symmetry

I also provide a Surface Evolver script which reproduces a simpler textural equilibrium problem: that of a quadruple junction with tetrahedral symmetry. This problem was studied by von Bargen and Waff (1986); Cheadle (1989); and Nye (1989). However, there is no space-filling tesselation of grains consistent with such a quadruple junction.


Tetrahedral junction, dihedral angle 30°, porosity 3%. [Surface Evolver script]

3D Printed Models

3D models of a selection of the melt geometries are available from shapeways.

If you have access to a 3D printer, you can use the orignal stl files.


Cheadle M.J. Properties of texturally equilibrated two-phase aggregates (1989) Ph. D. thesis, University of Cambridge

von Bargen N. and Waff H.S. Permeabilities, interfacial-areas and curvatures of partially molten systems - results of numerical computation of equilibrium microstructures (1986) J. Geophys. Res. Solid Earth 114(6) 1-19 [AGU]

Nye J.F. The geometry of water veins and nodes in polycrystalline ice (1989) J. Glaciol. 35(119) 17-22 [CUP]