Critical phenomena: 150 years since Cagniard de la Tour

Critical phenomena were discovered by Cagniard de la Tour in 1822, who died 150 years ago. In order to mark this anniversary, the context and the early history of his discovery is reviewed. We then follow with a brief sketch of the history of critical phenomena, indicating the main lines of development until the present date. Os fen\'omenos cr\'{\i}ticos foram descobertos pelo Cagniard de la Tour em Paris em 1822. Para comemorar os 150 anos da sua morte, o contexto e a hist\'oria initial da sua descoberta \'e contada. Conseguimos com uma descri\c{c}\~ao breve da hist\'oria dos fen\'emenos cr\'{\i}ticos, indicando as linhas principais do desenvolvimento at\'e o presente.

Born in Paris in 1777, Charles Cagniard was educated at l' École Polytechnique, and went on to become a prolific scientist and inventor.Besides his discovery of critical phenomena, Cagniard de la Tour investigated the nature of yeast and its role in the fermentation of alcohol and was interested the physics of the human voice as well as bird flight.His interest in acoustics led to the invention of the siren (see figure 2), which he named after sea creatures from Greek mythology who lured sailors to their doom.Experiments on steam engines in the late 17th and early 18th centuries motivated interest in the behaviour of fluids at high temperatures and pressures.Denis Papin (1647 -1712) who invented the "steam digester" -a forerunner of the steam engine -noticed that when heated under pressure, water remains in its liquid phase at temperatures far greater than the usual boiling point of 100 • C: the temperature of the boiling point increases with increasing pressure.
The term "latent heat", for the energy required to complete a solid-liquid or liquid-vapour phase transition, was introduced around 1750 by Joseph Black (1728 -1799).In 1783 James Watt (1736 -1819) analysed its dependency on pressure, and found that the latent heat of vaporisation decreases as the temperature is increased.At this time, gases were considered to be distinct from vapours (produced by evaporating liquids)."Elastic fluids" which were not reducible to liquid form were termed gases.It was in the second half of the 18th century that Antoine-Laurent de Lavoisier (1743 -1794) showed gases and vapours to be one and the same, and a third state of matter beside solids and liquids.He also suggested that gases could be liquefied at sufficiently low temperature and high pressure [1].
The first successful experiments on liquefaction of gases took place in 1784, when Jean-François Clouet (1751 -1801) and Gaspard Monge (1746 -1818) achieved the liquefaction of gaseous sulphur dioxide by cooling and compression.There followed a sequence of successful experiments, including by chemist and physicist Michael Faraday (1791 -1867), in which gases were liquefied, thus removing the distinction between vapour and gas [2,3].Hydrogen, oxygen, nitrogen, and carbon monoxide, which were previously thought to be incondensably gaseous and were called "permanent gases" were eventually liquefied 1877.
The discovery of what we now call the critical point came about with Cagniard de la Tour's experiments with Papin's digester.In 1822, in the context of his interests in acoustics, he placed a flint ball in a digester partially filled with liquid.Upon rolling the device, a splashing sound was generated as the solid ball penetrated the liquid-vapour interface.Cagniard de la Tour noticed that upon heating the system far beyond the boiling point of the liquid, the splashing sound ceased above a certain temperature.This marks the discovery of the supercritical fluid phase.In this phase there is no surface tension as there is no liquid-gas phase boundary.The supercritical fluid can dissolve matter like a liquid and can diffuse through solids like a gas.
In two articles in the Annales de Chimie et de Physique [4], Cagniard de la Tour described how he heated a sealed glass tube of alcohol under pressure, see figure 3.He observed that the liquid expanded to approximately twice its original volume, and then vanished, having been converted to a vapour so transparent that the tube appeared completely empty.On re-cooling the system a thick cloud appeared.We now recognise this as an observation of critical opacity and the discovery of the critical point.He also observed that beyond a certain temperature, increasing the pressure did not prevent the evaporation of the liquid.
In a following paper, Cagniard de la Tour reported upon a series of related experiments with a variety of substances [5].Desiring to demonstrate that the existence of a limiting temperature above which a liquid vapourises irrespective of pressure is a general phenomenon, he experimented on water, alcohol, ether and carbon bisulphide.He measured the critical temperature at which the interface tension vanished, as determined by the disappearance of the meniscus, and discovered that for each substance, there is a certain temperature beyond which total vaporisation of the liquid occurs and where no increase in pressure will liquefy the gas.In the case of water, this critical temperature was estimated to be 362 • C, a remarkably accurate result (modern measurements give 374 • C).His experiments demonstrated that this "état particulier" requires high temperatures, almost independent of the volume of the tube: "... cet état particulier exige toujours une température très-élevée, presque indépendante de la capacité du tube" [5].We now know that the état particulier marks the critical end-point of a line of first-order phase transitions, where the transition becomes continuous.
While many of Cagniard de la Tour's contemporaries regarded his results as being particular to the substances involved rather than a general phenomenon [6], Faraday recognised the significance of his work [3].In a letter to William Whewell in 1844, Faraday wrote Tour's state" and "the Cagniard de la Tour point" [8].In 1861, Dmitri Mendeleev (1834Mendeleev ( -1907)), referred to it as the "absolute Siedetemperatur", or absolute boiling point [9].
In 1869, the term we now use -the critical point -was eventually coined by Thomas Andrews (1813 -1885), who further elucidated the meaning of Cagniard de la Tour's état particulier [10].
Andrews studied the pressure-volume curve of the liquid-vapour coexistence line of carbonic acid and clarified that a gas may only condense to a liquid, or a liquid evaporate to a gas, below certain values of the temperature and pressure -the état particulier .Beyond this point lies the supercritical phase, where the distinction between liquid and vapour disappears.
In what followed, the early experiments of Cagniard de la Tour blossomed into a large-scale intellectual adventure.In 1873, van der Waals (1837 -1923) showed in his doctoral thesis [11] that Andrews' experimentally based equation of state may be explained qualitatively using an extension of the ideal gas law which modelled molecular attraction and hard-core repulsion in a simple manner.This in turn suggested to Heike Kamerlingh Onnes (1853 -1926) how to estimate the critical points for 'permanent gases', which gave the conceptual bases for the eventual liquefaction of helium, followed soon after by the discovery of superconductivity.On the other hand, the simple mean-field-like values of the "critical exponents" obtained from his equations are not adequate for a quantitative description of real systems, as realised experimentally in 1896 by Jules-Émile Verschaffelt (1870 -1955).Mean-field-like treatments were systematised in the phenomenological theory of Lev Davidovich Landau (1980Landau ( -1968)), where phase transitions in all spatial dimensions were predicted [12].
On the other hand, the important concept of "universality" of critical phenomena was introduced by Pierre Curie (1859 -1906), who discovered that ferromagnetic materials become demagnetised above a critical temperature [13] which is often referred to as a "Curie point".Formal analogies between a priori unrelated physical systems have been of great usefulness in trying to understand critical phenomena and were also one of the motivations when Wilhelm Lenz (1888Lenz ( -1957) ) introduced the simple many-body system now usually called "Ising model" [14].Ernst Ising (1900Ising ( -1998) solved the one-dimensional case in his doctoral thesis (1924) and the absence of a phase transition there clearly showed that a conceptual explanation of the critical point beyond the level of mean-field theories had to be sought.This conclusion was further strengthened by the achievements of Lars Onsager (1903 -1976), who in 1944 calculated exactly the specific heat of the two-dimensional Ising model in the absence of an external magnetic field and in 1949 announced the correct formula for the spontaneous magnetisation, proven by C.N. Yang (1922 -) in 1952.
By a tour de force and combining techniques of conformal field-theory with integrable systems, Alexander Zamolodchikov (1952 -) showed in 1989 that the two-dimensional Ising model in an external magnetic field, but with the temperature fixed to the critical temperature, is integrable [15].
In view of the absence of an exact solution for the three-dimensional Ising model, numerical techniques came to the fore.These are based either on systematic expansions around the known extreme cases of very high or very low temperatures as suggested by Cyril Domb (1920 -) in his doctoral thesis in 1949 [16], or else are based on large-scale simulations which go under the name of "Monte Carlo method" and suggested in 1949 by Nicholas Metropolis (1915Metropolis ( -1999) ) and Stanislaw Ulam (1909 -1986) [17].In the 1960s it was realised by Leo Kadanoff (1937 -) and Michael Fisher (1931 -) that a general theoretical framework for phase transitions would have to be formulated in terms of a "scaling theory" which in particular led to "scaling relations" between the critical exponents which describe the behaviour of the various measurable quantities close to a critical point.This opened the way to a full theoretical description of critical phenomena through the "renormalisation group" by Kenneth Wilson (1936 -) in 1971.This has been the basis for very precise predictions of the values of the critical exponents in two and three dimensions.
On the other hand, since the days of Cagniard de la Tour experimental techniques have been continuously refined.Very precise estimates for the values of the critical exponents can nowadays be obtained.For a long time, however, while experimentalists were busy measuring the critical behaviour of three-dimensional bulk systems, theorists could only calculate exactly the critical behaviour of two-dimensional systems, which can only be realised at the surface of some substrate.It took surprisingly long until phase transitions for systems confined to a surface were experimentally observed.The first confirmed example seems to have been found in Nancy by André Thomy in his thèse 3ème cycle (1959) for krypton adsorbed on graphite [19].Fittingly, this discovery arose exactly a century after the death of Cagniard de la Tour.
Nowadays, the most precise experiments are carried out on board the space shuttle, the space station MIR and the International Space Station.As an example, we quote the result for the specific-heat exponent α = 0.11 ± 0.03 obtained for the critical point in the simple fluid SF 6 during the Spacelab D2 mission (1999), in good agreement with the current theoretical estimate α = 0.109 ± 0.002 [18].
(Mean-field theory would have predicted α = 0.) In the 150 years since its inception, the field of critical phenomena has blossomed and now forms a cornerstone of modern physics, both experimental and theoretical and this development nicely illustrates how a topic of purely fundamental research, given enough time, can diversify into initially unforeseeable directions.Its founder, Charles Cagniard de la Tour died in Paris on the 5 th of July 1859.

Figure 1 :
Figure 1: Portrait of Cagniard de la Tour, courtesy of the Universidade do Minho, Portugal.

Figure 2 :
Figure 2: The improved siren, invented and named by Charles Cagniard de la Tour.Photograph courtesy of the École Polytechnique Paris, France.

Figure 3 :
Figure 3: The first page of Cagniard de la Tour's article, in which the discovery of critical phenomena is reported.

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"Cagniard de la Tour made an experiment some years ago which gave me occasion to want a new word".Referring to what we now call the critical point, he continued, "how am I to name this point at which the fluid & its vapour become one according to a law of continuity.Cagniard de la Tour has not named it; what shall I call it?"Whewell suggested to call it the point of vaporiscience or the point at which fluid is disliquified or the Tourian state, and in a later publication Faraday refers to "Cagniard de la