Not All Things Freeze

Posted by jns on July 20, 2005

Some time ago I started reading¹ Robert Wolke’s What Einstein Told His Cook 2. It is a collection of very short pieces about food and cooking from a chemist’s point of view, assembled from his Washington Post columns.

Rather early on though, he made a small error of fact. I point this out not to chastise the author, but as an excuse to talk about helium, one of my favorite science topics.²

He wrote:

[In answer to a reader's question about why frozen cola exhibits separated ice crystals:]

All liquids turn into solids — that is, they freeze — when they get cold enough.

[Robert L. Wolke, What Einstein Told His Cook 2: The Sequel -- Further Adventures in Kitchen Science (W. W. Norton & Company, New York, 2005) p. 5.]

Before I make my point, there are a couple of preliminaries to discuss about freezing.

“Freeze” itself is easy enough: it just means turning from a liquid or a gas — better to say “from a fluid”, because “fluid” encompasses both³ — into a solid. “Solid” generally implies some sort of crystaline structure, but we can be generous and include amorphous solids like glasses.⁴ In other words, we can take “freeze”, as a synonym for “solidify”, to mean much the same as it means in casual English.

“When they get cold enough” is also without serious hidden landmines, although usually a physicist will want to know under what conditions “they get cold enough”. One common condition is that “they get cold” just sitting around in the air, under normal atmospheric pressure. This is frequent and familiar to us: we are very familiar with water freezing under normal atmospheric conditions.

However, thermodynamically speaking, freezing under normal atmospheric conditions is not terribly interesting scientifically, even if it can be well defined. A particular condition of thermodynamic interest is when a substance freezes “under its own vapor pressure”.

Imagine a closed vessel — make it clear glass so we can see what’s going on inside — filled only with pure substance of interest and then sealed off. We put enough stuff into the vessel and lowered the temperature enough, that there is liquid stuff and gaseous stuff in the vessel, both visible at the same time. On the earth, all the liquid will be at the bottom of the vessel, and all the gas will be at the top, with the two phases separated by a “menicus”, or interface.⁵ Let’s also say, to avoid misunderstanding, that the meniscus is exactly in the middle (by volume) of the vessel. That way, as we vary the temperature there will always be gas and liquid in the vessel, and the meniscus will always be right in the middle. If we keep the temperature of the vessel constant for awhile, so that all the stuff is at the same temperature, then the two phases are said to be “coexisting in thermodynamic equilibrium”.

Now, lower the temperature of the stuff in the vessel and, sooner or later, solid stuff will appear and the material is said to have frozen “under its own vapor pressure”. In fact, there will be a unique temperature, called the “triple-point temperature”, at which gas, liquid, and solid phases can all coexist in thermodynamic equilibrium.⁶

To be precise then, our author probably meant by his statement that all liquids freeze under their own vapor pressure if they get cold enough.

However, this isn’t strictly true. The element helium has many interesting and surprising properties. Among them, helium is the only element that will not freeze under its own vapor pressure. It will indeed freeze, but it must be under at least 25 atmospheres of pressure to do so. This also implies that helium has no triple point, unlike any other elemental substance you can think of.

For many years around the turn of the 19th to 20th century, there was thought to be a class of substances called “permanent gasses”. These were gases that could not be caused to condense into droplets of liquid, no matter how much pressure was applied to them. Then it was discovered that they would condense, provided they were cooled to low enough temperatures first. The temperature below which each must be cooled before condensation is even possible is its “critical temperature”.⁷

The critical temperature is usually not the same as the “boiling point” temperature. “Boiling” usually implies that the substance is at atmospheric pressue. For example, nitrogen boils at 77 K⁸ but its critical temperature is about 126 K; oxygen has a critical temperature 155 K, but boils at 90 K.

The critical temperature of helium is 4.2 K, which is really, really, really cold. Until helium is cooled at least to that temperature, condensation is impossible and liquid cannot be produced under any pressure. As it turns out, the critical pressure is low, so helium will liquefy rather easily at atmospheric pressure, if one can get it cold enough.

Helium, the last of the “permanent gases”, was finally liquefied by the Dutch physicist Heike Kamerlingh Onnes in 1908 at his laboratory in Leiden.⁹ Some of us, who have been low-temperature physicists in previous lives, think of Kamerlingh Onnes as a sort of scientific grandfather, since we date the beginning of low-temperature physics to his liquefaction of helium. We also esteem the memory of Sir James Dewar(1842–1923), inventor of the Dewar flask (commercially known as a Thermos bottle), without a couple of which my own thesis experiment would not have been possible. Like atomic physics, low-temperature physics is a distinctly 20th-century discipline.

Helium does not freeze under its own vapor pressure, but it does do very odd things when it is cooled further below its critical temperature. At 2.17 K (at vapor pressure, i.e., at liquid-vapor coexistance), pure helium-4 (by far the most abundant isotope of helium¹⁰) undergoes what is know as a “superfluid transition”, also called the “lambda line” (the reason why to be explained in another essay sometime). The superfluid phase of helium exhibits many wondrous properties, like the lack of viscosity — the ability to flow through microscopic channels unimpeded¹¹ — and various other unusual behaviors.

However, we must save those topics for another time.¹²

———-
¹I’ve long since finished, too, and moved on to the prior volume in the series. I find them a little on the light-weight side, but considering the audience and the venue, that’s not altogether surprising. Regardless, they have been fun and informative reading.

²I have no doubt that this is because in my formative years, i.e., when I was a graduate student, I did low-temperature work (cryogenics) at liquid-helium temperatures (about 2–5 K, or 2–5 degrees above “absolute zero”) measuring properties of helium itself.

³Operationally, “fluid” is anything that “flows”, i.e., any substance which is subject to the equations of motion from fluid dynamics.

⁴There is controversy over whether “glassy solids” are a different form of matter from gas, liquid, and solid, but for the present purposes it’s not necessary to choose sides.

⁵Observing the meniscus, for example its curvature, can tell us many interesting things about the properties of the substance.

⁶Practically speaking, triple-points are very useful since they occur at a unique, well-defined and reproducible temperature. If one can contrive, say, to have a glass vessel filled with pure water in equilibrium with all three phases present, then one knows exactly what the temperature is of the entire system. This procedure is actually used in the definition of the “International Practical Scale of Temperatures”, a set of standard procedures for establishing nearly thermodynamic absolute temperature calibrations in the laboratory. Temperature, though, is a whole other story.

⁷Since “critical phenomena”, the study of elemental properties very near the critical point (i.e., near the critical temperature and critical pressure) was my area of research for some 15 years or more, there’s much more I could say about it, but this isn’t the place.

⁸“K” is the abbreviation for Kelvins, the units of the thermodynamic temperature scale. A Kelvin (NB, not “degrees Kelvin”!) is the same size as a Celsius degree; “0″ on the Kelvin scale is “absolute zero”, which is about -273 centigrade degrees, or -459 Farenheit degrees.

⁹Read a fascinating essay about this called “Heike Kamerlingh Onnes and the Liquefaction of Helium“, written by Jedtsada Laucharoen, a student at the Horace Mann School, The Bronx. His essay was the 1st place prize winner in the physics category of the “Laureates of Tomorrow Nobel Essay Contest”.

¹⁰The only other naturally occuring isotope is helium-3; I forget offhand what the relative abundances are. But, this does give me an excuse to quote (from memory, so my precision may only be close) my favorite first line from a book, J.D. Wilke’s Properties of Liquid and Solid Helium (my bible in graduate school): “Helium exists in three naturally occuring isotopes: helium-4, helium-3, and helium-6; as the latter has a half-life of only 0.67 seconds, it need concern us no further.” And, indeed, in the ensuing 700+ pages, helium-6 is never mentioned again.

¹¹Every low-temperature physicist’s nightmare is to develop a microscopic “superleak” in his apparatus. Below the superfluid transistion helium will pour out of the apparatus, but it is exceedingly difficult to do leak detection at such low temperatures, so the usual response is to throw the thing out and start over. Fortunately, I never faced encountered that problem.

¹²As a closing treat, I will thrill you with the title of my Ph.D. dissertation: Shear Viscosity and Thermal Conductivity in Liquid Helium-4 and Dilute Mixtures of Helium-3 in Helium-4 near the Lambda Transition

Statistical Fluctuations

Posted by jns on July 2, 2005

Abraham Pais, a physicist who wrote what is generally regarded as the definitive scientific biography of Einstein, said of his subject that there are two things at which he was “better than anyone before or after him; he knew how to invent invariance principles and how to make use of statistical fluctuations.” Invariance principles play a central role in the theory of relativity. Indeed, Einstein had wanted to call relativity the “theory of invariants”.
["Miraculous Visions: 100 Years of Einstein", The Economist, 29 December 2004.]

By way of explanation for the quotation: I came across it a few months ago and wanted to make note of it 1) because it’s quite true, and gives a remarkable insight into Einstein’s mode of thinking; and 2) because fluctuations loom large in my own way of looking at the physical world — because of my working experience in science — and because invariance principles are an interesting and important concept in physics. I’d like to discuss both of them sometime, but it will require far more presence of mind, and time, than I have to give it right now. So, I’ll preserve the quotation here and maybe get to it later.