CRITICALPHENOMENA IO.T THERMODYNAMICS IN THE NEIGHBORHOOD OF THE CRITICAL POINT The entire structure of thermodynamics,as describedin the preceding ters, appearedat mid-century to be logically complete,but the re founderedon one ostensiblyminor detail.That "detail" had to with the propertiesof systemsin the neighborhoodof the critical point. ical thermodynamicscorrectly predicted that various "generalized :ptibilities" (heat capacities,compressibilities, magnetic susceptibili- etc.) should divergeat the critical point, and the gerreralstruciure of ssicalthermodynamics strongly suggested the analytic lbrm (or "shape") those divergences.The generalizedsusceptibilities do diverge,but the rlytic form of the divergencesis not as expected. In addition the yergencesexhibit regularities indicative of an underlying integrative inciple inexplicableby classicalthermodynamics. Observationsof the enormousfluctuations at critical points date back 1869,when T. Andrewsrreported the "critical opalescenceo'of fluids. scattering of light by the huge density fluctuations renders water ilky" and opaqueat or very near the critical temperatureand pressure .29 K, 22.09 MPa). Warming or cooling the water a fraction of a in restoresit to its normaltransparent state. Similarly, the magnetic susceptibility divergesfor a magnetic system r its critical transition, and again the fluctuations in the magnetic t are divergent. A variety of other types of systemsexhibit critical or "second-order" msitions; severalare listed in Table 10.1 along with the corresponding parameter" (the thermodynamicquantity that exhibits divergent ions,analogous to the magneticmoment). rT. Andrews, Phil. Trans. Royal Soc.f59, 575 (1869). 255 256 critical Phenomena TABLE 10.1 Exarnplesof Critical Points and Their Order Parameters* Crilical Point Order Parameter Example Tr, (K) Liquid-gas Molar volume HrO 647.05 Ferromagnetic Magneticmoment Fe 1044.0 Antiferromagnetic Sublatticemagnetic FeF, 78.26 moment )\-line in aHe aHe quantummechanical aHe 1.8-2.1 amplitude Superconductivity Electron pair amplitude Pb 7.L9 Binary fluid mixture Fractional segregation ccl4-c7Fr4 301.78 of components Binary alloy Fraction of one atomic Cu-Zn 739 specieson one sublattice Ferroelectric Electric dipole moment Triglycinesulfate 322.5 'Adapted from Shang-Keng lvla, Modern Theory of Critical Phenomeza (Addison-Wesley Advanced Book Program, CA,1976. Used by permission) In order to fix thesepreliminary ideasin a specificway we focus on the gas-liquid transition in a fluid. Consider first a point P, Z on the coexistencecurve; two local minima of the underlying Gibbs potential then compete,as in Fig. 10.1(page 205). If the point of interestwere to move off the coexistencecurve in either direction then one or the other of the two minima would becomethe lower. The two physical states,corre- sponding to the two minima, have very different valuesof molar volume, ? -----> FIGURE IO.T Competition of two minima of the underlying Gibbs potential near the coexistence curve. Thermodynamicsin the Neighborhood of the Critical Point 257 {.t [:/ T-> FIGURE IO.2 Coalescenceof minima of the underlying Gibbs potential as the critical point is approached. molar entropy, and so forth. Thesetwo statescorrespond, of course,to the two phasesthat competein the first-orderphase transition. Supposethe point P,T on the coexistencecurve to be chosencloser to the critidal point. As the point approachesthe critical T and P the two minimaof the underlyingGibbs potential coalesce (Fig. 10.2). For all points beyondthe critical point (on the extendedor extrapolated coexistencecurve) the minimum is single and normal (Fig. 10.3).As the critical point is reached(moving inward toward the physical coexistence curve) the single minimum developsa flat bottom, which in turn develops e "bump" dividing the broadenedminimum into two separateminima. The singleminimum "bifurcates" at the critical point. The flattening of the minimum of the Gibbs potential in the region of the critical stateimplies the absenceof a "restoring force" for fluctuations eway from the critical state (at least to leading order)-hence the diver- gent fluctuations. This classicalconception of the developmentof phase transitions was formulated by Lev Landau,z and extended and generalizedby Laszlo Tisza,' to form the standardclassical theory of critical phenomena.The cssential idea of that theory is to expand the appropriate underlying thermodynamic potential (conventionallyreferred to as the "free energy functional") in a power seriesin Z - 7", the deviation of the temperature from its value 7"(P) on the coexistencecurye. The qualitative features described here then determine the relative signs of the first several 2cf . L. D. Landau and E. M. Lifshitz, StatisticalPhysics, MIT Press,Cambridge, Massachuserts end London, 1966. 3cf . L. Tisza, GeneralizedThermodynamics, MIT Press, Cambridge, Massachusettsand London, 1966 (see particularly papers 3 and 4). 258 Critical Phenomena li v' \? 4\ "l\J \' l \v/ \\ i9 Gl \^/ \L / \ v/ T ----+ FIGURE 10 3 The classicalpicrure of the developmentof a first-orderphase transition. The dotted curve is the qxtrapolated(non-physical) coexistence curve. coefficients,and these terms in turn permit calculation of the analytic behavior of the susceptibilitiesas T" approachesthe critical temperature 7,,. A completely analogoustreatment of a simple mechanicalanalogue model is given in the Example at the end of this section,and an explicit thermodynamic calculation will be carried out in Section 10.4. Ar this point it is sufficient to recognize that the Landau theory is simple, straightforward, and deeply rooted in the postulates of macroscopic thermodynamics;it is basedonly on thosepostulates plus the reasonable assumptionof analyticity of the free energyfunctional. However,a direct comparison of the theoreticalpredictions with experimentalobservations was long bedeviledby the extremedifficulty of accuratelymeasuring and controlling temperature in systemsthat are incipiently unstable, with gigantic fluctuations. ln 1944Lars Onsageraproduced the flrst rigorousstatistical mechanical solution for a nontrivial model (the "two-dimensional lsing model',), and it exhibited a type of divergencevery different from that expected.The scientific community was at first loath to accept this disquieting fact, particularly as the model was two-dimensional(rather than three-dimen- sional), and furthermore as it was a highly idealized construct bearing little resemblanceto real physical systems.ln 1945 E. A. Guggenheims aL. Onsager, Phys. Reu.65,177 (1944). sE. A Guggenheim, "I. Chem Phys.l3,253 (1945) Thermodynamics in the Neighborhood of the Critical Point 259 observedthat the shapeof the coexistencecurve of fluid systemsalso cast doubt on classicalpredictions, but it was not until the early 1960sthat precisemeasurements6 forced confrontation of the failure of the classical Landau theory and initiated the painful reconstructiod that occupiedthe decadesof the 1960sand the 1970s. Deeply probing insightsinto the nature of critical fluctuationswere developedby a number of theoreticians,including Leo Kadanoff, Michael Fischer,G. S. Rushbrooke,C. Domb, B. Widom, and many others.8'eThe construction of a powerful analytical theory ("renormalization theory") wasaccomplished by KennethWilson, a high-energytheorist interested in statistical mechanicsas a simpler analogue to similar difficulties that plaguedquantum field theory The sourceof the failure of classicalLandau theorycan be understood relativelyeasily, although it dependsupon statisticalmechanical concepts yet to be developedin this text. Neverthelesswe shall be able in Section 10.5 to anticipatethose results sufficiently to describethe origin of the difficulty in pictorial terms.The correction of the theory by renormaliza- tion theory unfortunatelylies beyond the scopeof this book, and we shall simply describethe generalthermodynamic consequences of the Wilson theory. But first we must developa frameworkfor the descriptionof the analytic form of divergen\ quantities,and we must review both the classicalexpectations and t\e (very different)experimental observations. To all of this the following mechanicalanalogue is a simpleand explicit introduction. Example The mechanicalanalogue of Section9.1 providesinstructive insights into the flatteningof theminimum of thethermodynamic potential at thecritical point as that minimum bifurcatesinto two competingminima below {.. We again considera length of pipe bent into a semicircle,closed at both ends,standing verticallyon a tablein the shapeof an invertedU, containingan internalpiston. On either side of the piston there is 1 mole of a monatomicideal gas.The metal balls that were insertedin Section9.1 in order to break the symmetry(and thereby to produce a first-order rather than a second-ordertransition) are not present. If d is the angle of the piston with respectto the vertical, ^ft is the radius of curvature of the pipe section, and Mg is the weight of the piston (we neglect gravitational effectson the gas itself), then the potential energyof the piston is 'cl. P Heller and G B. Benedek, Phys. Reu. Let.8,428 (1962) 1cf . H. E. Stanley, Introduction to Phase Transitions und Critical Phenomena, Oxford Univ. Press, New York and Oxford, 1971 *./. H. E. Stanley, Ibid. eP. Pfeuty and G. Toulouse, Introduction to the Renormalization Group and Critical Phenomenu, John Wiley and Sons, NY 1977. 260 Critical Phenomena (MgR)cosl, and the Helmholtz potential is F = IJ _ ZS: (USR)cos0+