Chemistry - Why is ice cubical in structure at very low temperature?
Solution 1:
First, a little clarification. The kind of ice you see at ambient pressure, ice I, exists as stacks of bilayers of water molecules. If they are stacked in an ABAB sequence, they form hexagonal ice, and cubic ice in an ABCABC sequence. The difference between the two allotropes is just the stacking of the molecules, so in fact they are structurally quite similar. The structure of cubic ice is analogous to the diamond structure of carbon, with hexagonal ice being analogous to lonsdalite. This picture is complicated by the fact that ice is generally proton-disordered, i.e. the molecules can have different orientations that break the crystal symmetry, subject to certain rules (the "ice rules").
There is evidence that ice crystallises as cubic ice in the upper atmosphere, namely the angle subtended by solar haloes and the angles formed in nascent snowflakes, for example. However, more recent evidence suggests that cubic ice doesn't not exist as a pure phase, but only in stacking-disordered hexagonal ice (http://pubs.rsc.org/en/content/articlelanding/2015/cp/c4cp02893g#!divAbstract). Curiously, there is also evidence that the lonsdalite form of carbon doesn't exist in reality, and is just stacking-faulted diamond (http://www.nature.com/articles/ncomms6447).
This is all very well, but does not tell us why hexagonal ice forms in preference under certain conditions, or cubic ice (?) under others. If you were to perform density functional theory calculation (essentially approximately solve the Schrodinger equation for a many atom system), you would find that hexagonal and cubic ices have almost identical energies, and are essentially degenerate at zero temperature, which makes it even more confusing why hexagonal ice is almost exclusively seen on Earth. I don't think there is a convincing answer to your question, since the best calculations do not show why the hexagonal structure of ice is lower in energy than cubic, but it could be due to a number of reasons: orientational disordering, which complicates modelling ices, zero point energy or free energy considerations, or a breakdown in the approximations used to perform such calculations. In any case, it looks like cubic ice doesn't exist under terrestrial conditions.
Solution 2:
This is a difficult question. Apparently, researchers don't know yet (or I didn't manage to find an article which completely solves the problem).
The preference for cubic structure at low temperatures was long thought to be caused by a lower liquid-ice surface tension in cubic ice nucleus (see G. P. Johari, J. Chem. Phys., 2005, 122, 194504). However, some authors point out that this assertion is not supported by strong evidence, and note that thermodynamically speaking, there is no reason why cubic should be preferred over hexagonal (E. B. Moore and V. Molinero, Phys. Chem. Chem. Phys., 2011, 13, 20008), that is to say that cubic ice is not more stable. These authors performed molecular dynamics simulations, and concluded that a the cubic ordering is kinetically controlled, though they left the elucidation of the kinetic factors an open question.
Of course, this dates back to 2011, any update would be welcome (it is not exactly my field of expertise...).
Solution 3:
As explained in the other answers, water ice may not actually become cubic at low temperature. In fact, if formed at higher temperature as the usual hexagonal phase, water ice will remain so when cooled. There is no thermodynamic driving force to rearrange the hexagonal structure to a cubic one.
We can expand on the earlier answers in two, make that three, ways:
1) Ice at very low temperatures can undergo a proton ordering process, forming a lower symmetry phase.
2) Cubic ice is doubtful as Ice I, but is known in a higher pressure phase, Ice VII.
3) Update: Cubic Ice I, as $\ce{D2O}$, has been reported from careful warming of a metastable low-density phase, Ice XVII.
Proton ordering: Ice XI
The ice structure presented in this answer is somewhat of an idealization. The hydrogen atoms there are perfectly ordered. However, in real Ice I (and other molecular phases, which share this characteristic), the hydrogen atoms can migrate around the hydrogen-bonded rings so that the molecules are still $\ce{H2O}$ but the protons are randomized. Only the oxygen atoms are really fixed in Ice I, and the familiar hexagonal symmetry comes from the arrangement of oxygen atoms in this randomized environment.
However, the proton-ordered form is slightly lower in energy and, at low temperature, this overcomes the favorable entropy of the random arrangement and the idealized structure now does apply. With this change comes a lowered symmetry of the oxygen atoms so that now the ice is orthorhombic instead of hexagonal (or cubic). This is Ice XI. Ice XI is believed to be stable up to 72 Kelvins, so it may exist on colder bodies in our Solar System and in interstellar space.
When the Pressure's On: Ice VII
While the cubic form of Ice I appears to be of doubtful existence, it does have the property that if we expand it slightly another identical cubic structure can be intertwined with it. This is what happens in Ice VII and its low-temperature, proton-ordered form Ice VIII (again proton ordering lowers the symmetry, so Ice VIII is tetragonal rather than cubic). Both are formed under several GPa pressure. Ice VII has been identified in diamonds, indicating that Earth's water is not just on the surface; it penetrates deep into the mantle and thus broadly affects the structure and physics of Earth's rock. The finding of Ice VII on Earth is discussed in more detail in this answer from Space Exploration Stack Exchange.
This Just In: Cubic $\ce{D2O}$ Ice I
Leo del Rossi at al. report in Nature Materials that purely cubic Ice I may be obtained from careful warming of $\ce{D2O}$ Ice XVII between $110$ and $180$ Kelvins. Ice XVII itself is metastable and was made in this case as a clathrate with hydrogen (or deuterium). Carefully removing the hydrogen at low temperature resulted in purified Ice XVII, which when warmed transformed to the cubic Ice I phase before going on to hexagonal Ice I. Multiple analytical techniques corroborate the existence of the cubic ice.