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We investigate the thermal equation of state, bulk modulus, thermal expansioncoefficient, and heat capacity of MH-III (CH$_4$ filled-ice Ih), needed for thestudy of CH$_4$ transport and outgassing for the case of Titan andsuper-Titans. We employ density functional theory and ab initio moleculardynamics simulations in the generalized-gradient approximation with a van derWaals functional. We examine the finite temperature range of $300$K-$500$K andpressures between $2$GPa-$7$GPa. We find that in this P-T range MH-III is lessdense than liquid water.  There is uncertainty in the normalized moment of inertia (MOI) of Titan; itis estimated to be in the range of $0.33-0.34$. If Titan's MOI is $0.34$,MH-III is not stable at present in Titan's interior, yielding an easier pathfor the outgassing of CH$_4$. However, for an MOI of $0.33$, MH-III isthermodynamically stable at the bottom of a ice-rock internal layer capable ofstoring CH$_4$. For rock mass fractions $\lessapprox 0.2$ upwelling melt islikely hot enough to dissociate MH-III along its path. For super-Titansconsidering a mixture of MH-III and ice VII, melt is always positively buoyantif the H$_2$O:CH$_4$ mole fraction is $>5.5$. Our thermal evolution model showsthat MH-III may be present today in Titan's core, confined to a thin ($\approx10$km) outer shell. We find that the heat capacity of MH-III is higher thanmeasured values for pure water-ice, larger than heat capacity often adopted forice-rock mixtures with implications for internal heating.

The Equation of State of MH-III: a possible deep CH$_4$ reservoir in Titan, Super-Titan exoplanets and moons