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James Kovalcin has addressed this question here: http://en.allexperts.com/q/Physics-1358/2009/1/freeze-point-water-under.htm

I have examined phase diagrams for water, but I am interested in the melting point for ice under pressures of 1 to 100 Atmospheres. And the phase diagrams aren't very quantitative.

James Kovalcin does include:

The formula for predicting the freezing point is given by

P=-395.2*((T/273.16)^9-1)

where T is absolute temperature in Kelvin and P is pressure in MPa.

This is valid for pressures up to 209.9 MPa. At higher pressures water freezes to ice-three, ice-five, ice-six or ice-seven at increasing temperatures.

This is most helpful. But I still wonder the orgins of the formula. Can you help me on this?

Further obviously the answer to the equation for T = 273.16 is 0. Clearly this is wrong, unless this is the difference from one atmosphere. In the reference James Kovalcin includes: http://www.lsbu.ac.uk/water/phase.html there is an equation for the melting curve for hexagonal ice here: http://www1.lsbu.ac.uk/water/hexagonal_ice.html that includes the following equation:

Pressure=6.11657x10^-4 -414.5x((Temperature/273.15)^8.38 -1).

Thanks for any answers, suggestions or references you can give me

So if you go to the website he mentioned: http://www1.lsbu.ac.uk/water/water_phase_diagram.html and check out the phase diagram of water, when you mouse-over the various lines you get pop-up equations that define the lines. These equations are polynomial fits to lab data. That one is mostly nearly vertical, so it makes sense that it be defined as pressure in terms of temperature and not the other way around. There are several more, but that one definitely applies over your requested pressure range and beyond. It starts at the triple point, which means that the equation definitely is correct if you look at the data. The people who put that information together have obviously done WAY more ice physics than I have. Perhaps you could contact the site author directly about it, Martin Chaplin: martin.chaplin@btinternet.com is from the site.

The origins of the formula itself will be a detailed, microscopic calculation of free energy based on molecule configuration. That's pretty much the origin of all thermal physics. With water, the situation is quite complex, as the molecules are polarized, have symmetric and asymmetric aspects, and there's hydrogen bonding involved. That's graduate-school level thermodynamics for further details, way beyond the scope of this forum.

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