Viscous constitutive relations of solid‐liquid composites in terms of grain boundary contiguity: 2. Compositional model for small melt fractions

<jats:p>We develop a compositional model for the grain boundary (GB) diffusion creep of melt‐bearing polycrystalline aggregates. This model is an extension of the GB diffusion control model by taking into account the effects of finite liquid diffusivity and finite reaction rates, which are important at small melt fractions (ϕ). For shear viscosity <jats:italic>η</jats:italic>, there exists a critical melt fraction ϕ<jats:sub><jats:italic>c</jats:italic></jats:sub><jats:sup><jats:italic>η</jats:italic></jats:sup> below which the rate‐limiting process changes from diffusion through GB to diffusion through the liquid, and for bulk viscosity <jats:italic>ξ</jats:italic>, there exists a critical melt fraction ϕ<jats:sub><jats:italic>c</jats:italic></jats:sub><jats:sup><jats:italic>ξ</jats:italic></jats:sup> below which the rate‐limiting process changes from diffusion through GB to reaction at the pore surface. The implication of the model results is that different processes may limit the kinetics of bulk and shear viscosities at small ϕ. The model also predicts that stress can cause compositional and hence volumetric asymmetry in pores. As melt fraction increases from zero to the critical melt fractions, rapid decreases occur in <jats:italic>η</jats:italic>, from <jats:italic>η</jats:italic><jats:sub><jats:italic>cc</jats:italic></jats:sub> to 0.2<jats:italic>η</jats:italic><jats:sub><jats:italic>cc</jats:italic></jats:sub>, and in <jats:italic>ξ</jats:italic>, from ∞ to 0.4<jats:italic>η</jats:italic><jats:sub><jats:italic>cc</jats:italic></jats:sub>, where <jats:italic>η</jats:italic><jats:sub><jats:italic>cc</jats:italic></jats:sub> represents <jats:italic>η</jats:italic> for Coble creep. In the GB diffusion control model, these decreases occur discontinuously at zero melt fraction, but this singularity is removed in the compositional model. For a typical grain size in the mantle (3 mm), the critical melt fractions are estimated semiquantitatively as ϕ<jats:sub><jats:italic>c</jats:italic></jats:sub><jats:sup><jats:italic>η</jats:italic></jats:sup> = 10<jats:sup>−4</jats:sup> and ϕ<jats:sub><jats:italic>c</jats:italic></jats:sub><jats:sup><jats:italic>ξ</jats:italic></jats:sup> < 6 × 10<jats:sup>−8</jats:sup>, demonstrating the significant effect of very small amount of melt on the viscosities in the mantle.</jats:p>