Journal of the Virtual Explorer

A summary of the published values for solid-solid-solid dihedral angles of geological relevance is presented in Table 4, with an indication of whether the results are the medians of populations of apparent angles measured on 2-D sections or whether they pertain to true 3-D values. Angle populations in single-phase materials have a median value of 120˚ (the only possible result when all three angles at each triple junction are included), with a range dependent on the extent of anisotropy of grain boundary energy (e.g. Kruhl, 2001; Leibl et al., 2007). For two-phase junctions, angle distributions tend to have a standard deviation in the region of 10-15˚.

The sensitivity of equilibrium angles to pressure and temperature has been little studied. The population of true equilibrium dihedral angles in quartz aggregates (the quartz-quartz-quartz angle) is not sensitive to temperature (Kruhl, 2001), whereas the median of equilibrium two-phase (apparent) dihedral angles in sulphides (e.g. at sphalerite-galena-galena junctions) decreases significantly with increasing temperature, with potential use as a geothermometer (Stanton and Gorman, 1968; Lusk et al. 2002).

A summary of the published values for melt-solid-solid dihedral angles of relevance to melt-bearing rocks is given in Table 5. Most of the published measurements have been made on 2-D sections; the reported values for these studies are the medians of the measured population. Only two studies to date have reported measurements of true 3-D angles (Cmíral et al. 1998, Holness, 2006). Typical frequency distributions of the true 3-D angles are shown in Figure 3.

Figure 3. Texturally equilibrated melt-solid-solid dihedral angles

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Frequency distributions of true dihedral angles measured for texturally equilibrated natural examples of rapidly quenched melt-bearing systems. From Holness (2006).

The variation of the liquid-solid-solid equilibrium dihedral angle with pressure, temperature and fluid composition is controlled by the layer of adsorbed species on the interfaces (Holness, 1993; Brenan and Rose, 2002; Takei and Shimizu, 2003). In general, the higher the extent of surface activity the more sensitive is the equilibrium angle to changes in pressure and temperature (Holness, 1993; Holness and Graham, 1995).

If the liquid has a similar composition and structure to the solid then the energy of the liquid-solid interface, and thus the dihedral angle, is low. The smallest angles are generally found in single component systems such as ice-water (Walford and Nye, 1991; although see Mader (1992)) or systems with very high solubility of the solid phase (e.g. sucrose-H₂O, Pharr and Ashby, 1983; NaCl-H₂O ice, Spetzler and Anderson, 1968; olivine-H₂O at high pressure, Yoshino et al, 2007). In simple binaries, the dihedral angle is a function of temperature (and thus liquid composition), with the lowest angles when the liquid composition is closest to that of the solid phase (Takei, 2000; Ikeda et al., 2002; Takei and Shimizu, 2003).

Conversely, the highest angles occur where the solid and liquid have the most disparate compositions and structures. Examples of the latter include argon-calcite, argon-quartz, CO₂-quartz (Holness 1993), metal-olivine (see list of references in Table 5). Fe-S liquids in silicate matrices also have generally high dihedral angles (e.g. Ballhaus and Ellis, 1996; Minarik et al., 1996; Gaetani and Grove, 1999; Terasaki et al., 2007; again see Table 5 for a more comprehensive list) although these are sensitive to changes in oxygen fugacity and pressure (Gaetani and Grove, 1999; Rose and Brenan, 2001; Takafuji et al., 2004; Terasaki et al. 2008).