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It's my understanding that the colder liquids get (or anything else for that matter) the slower the constituting particles move. That being the case, why is H$_2$O either 'water' or 'ice'? Given that temperature is continuous, why isn't there a continual physical change depending on temperature?

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    $\begingroup$ Why should there be a "continuous change"? Different phases are mostly defined by such discontinuous changes - look up what a phase transition is. $\endgroup$ – ACuriousMind Aug 8 '15 at 12:52
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    $\begingroup$ @ACuriousMind That's a textbook example of begging the question. $\endgroup$ – DanielSank Aug 10 '15 at 3:00
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Water, like most liquids does indeed get more viscous as its temperature approaches freezing point. See the graph below, which I took from the "Engineering Toolbox"

However, what's interesting about this curve is that it does not diverge as $T\to 0^\circ{\rm C}$. The reason is that a phase change really is for all effective purposes a discontinuous phenomenon: at $0+\epsilon^\circ{\rm C}$ the water molecules have enough kinetic energy to avoid binding to one another in a solid lattice: at $0-\epsilon^\circ{\rm C}$ they do not and this argument holds for any $\epsilon>0$. Otherwise put: the freezing point is an energy threshold in the same way that the behavior of a body in the neighborhood of the Earth shows behaviors that are similarly discontinuous functions of the body's total energy: if the body's speed energy exceeds the escape velocity at a point by an arbitrarily small amount, the body will follow a hyperbolic path and leave the neighborhood forever: lower the speed to an arbitrarily small amount below escape velocity and the path will be elliptic and the motion periodic: the path topologies change discontinuously from noncompact to compact.

enter image description here

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    $\begingroup$ About 325 BC Pytheas of Massalia sailed into the North Atlantic, a region "where land properly speaking no longer exists, nor sea nor air, but a mixture of these things, like a [jellyfish], in which it is said that earth and water and all things are in suspension as if this something was a link between all these elements, on which one can neither walk nor sail. (en.wikipedia.org/wiki/Pytheas). Apparently he sailed into temperatures approaching 0 degC. $\endgroup$ – Ernie Aug 8 '15 at 17:28
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    $\begingroup$ @Ernie Very interesting, although the account in Wikipedia seems to suggest instead slush: a mixture of liquid water and ice particles and vapor (in the air above) at its triple point. $\endgroup$ – WetSavannaAnimal Aug 9 '15 at 5:29

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