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I'm aware of the critical constants for gas-liquid transition but is there such constants for liquid-solid transition? Like the maximum temperature above which a liquid cannot be solidified.

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See Elenius and Dzugutov "Evidence for a liquid-solid critical point in a simple monatomic system" J. Chem. Phys. 131, 104502 (2009). http://arxiv-web3.library.cornell.edu/abs/0906.4947

"It is commonly believed that the transition line separating a liquid and a solid cannot be interrupted by a critical point. This opinion is based on the traditional symmetry argument that an isotropic liquid cannot be continuously transformed into a crystal with a discrete rotational and translational symmetry. We present here a molecular-dynamics simulation of a simple monatomic system suggesting the existence of a liquid-solid spinodal terminating at a critical point. We show that, in the critical region, the isotropic liquid continuously transforms into a phase with a mesoscopic order similar to that of the smectic liquid crystals. We argue that the existence of both the spinodal and the critical point can be explained by the close structural proximity between the mesophase and the crystal. This indicates a possibility of finding a similar thermodynamic behavior in gelating colloids,liquid crystals, and polymers."

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In non-technical terms, No. This is because the existence of critical constants in case of liquid-gas transition is because they essentially become a single phase beyond the critical point. There, liquid and gas is indistinguishable thermodynamically as can be visualized from the phase diagrams (region beyond the critical point). But for liquid-solid transition, this equivalency of phase does not hold true and hence it does not have an indistinguishable phase like the gas-liquid transition.

But solid-liquid transition can have eutectic points and complicated phase diagrams in case of a solution. Moreover, the mixture of two liquids also has a critical solution point beyond which the miscibility of the liquids break down.

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