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Starting from the triple point, is the melting line between solid-phase and liquid-phase infinite? If not, why does it end? Because pressures are so high that classical inter-molecular interactions break down? Is this related in any way to the concept of degenerate matter? Can such pressure (at which the line ends) be easily computed?

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No, the boundary doesn't suddenly "end" or "fade away", as the liquid-gas boundary fades away near the critical point.

Instead, the sudden end indicates that many other things may happen in the region of these extremely high pressures and the diagram doesn't want to discuss those because they're outside the limits of interest of the author of the diagram. By other things, I mean primarily new triple junctions that separate new phases of "ice", like those on this more detailed phase diagram of water:

enter image description here

Fifteen phases of ice are known today and the other triple junctions are counted as "triple points", too.

The required pressures to see these new phase transitions are extreme – thousands of atmospheres or much more – but the new phases are still much less extreme than the matter of white dwarfs or neutron stars. It's still some kind of ice that just wants to arrange a bit differently than normal ice.

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That's interesting although raises quite a few new questions. Your phase diagram shows that it becomes very difficult to have liquid phase above 10^9 Pa. Is that behavior (vertical solid-liquid line becoming nearly horizontal) specific to water? Can this 10^9 limit be estimated a priori from atomic quantities? – FrenchKheldar Dec 10 '12 at 8:24
Hi, whether you will have a liquid at these high pressures also depends on the temperature. You may have liquids - well, supercritical fluids - at pretty much arbitrarily high pressure if you also increase the temperature. The temperature of the transition increases with the pressure because pressure is a form of potential energy, so for qualitative things to change, a huge pressure (huge potential energy) must also be accompanied by a comparable kinetic energy etc. – Luboš Motl Dec 10 '12 at 10:36
It's pretty hard to calculate the shapes of those things from the first principles (of atomic physics) but even if one does so, $10^{9}$ is only relevant at "near-room temperatures". At much higher temperatures, the corresponding pressure is much higher, too. Note that the temperature scale above is linear but the pressure scale is logarithmic. ... The basic qualitative behavior at very high pressures is rather universal for various compounds. – Luboš Motl Dec 10 '12 at 10:37
While the log-linear scale does exaggerate the horizontalness of the liquid-ice VII transition, that line is markedly different from the liquid-ice I line, which appears almost exactly vertical. Is there some specific physical meaning to this pressure? Since ice I stops being stable, I would imagine it has to do with such loose packing being unable to cope with such pressures no matter the temperature. – Emilio Pisanty Dec 10 '12 at 14:39

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