1. Introduction In these lectures some of the general features of the phenomena of phase transitions in matter will be examined. Wewill first reviewsome of the experimental phenomena. We then turn to a discussion of simple thermodynamic and so-called mean-field theoretical approaches to the problem of phase transitions in general and critical phenomena in particular,showing what theyget right and what theyget wrong. Finally,wewill examine modern aspects of the problem, the scaling hypothesis and introduce the ideas behind a renormalization group calculation. FIG. 1. Phase Diagram of Water.1 FIG. 2. Liquid-Vapor P-V phase diagram isotherms near the critical point.2 Consider the twowell known phase diagrams shown in Figs. 1 and 2. Along anyofthe coexistence lines, thermodynamics requires that the chemical potentials in the coexisting phases be equal, and this in turn givesthe well known Clapeyron equation: ∆ dP = H ∆ ,(1.1) dT coexistence T V where ∆H and ∆V are molar enthalpyand volume changes, respectively,and T is the tempera- ture. Manyofthe qualitative features of a phase diagram can be understood simply by using the Clapeyron equation, and knowing the relative magnitudes and signs of the enthalpyand volume changes. Nonetheless, there are points on the phase diagram where the Clapeyron equation can- not be applied naively,namely at the critical point where ∆V vanishes. The existence of critical points was controversial when it was first considered in the 19th century because it means that you can continuously transform a material from one phase (e.g., a liquid) into another (e.g., a gas). Wenow hav e manyexperimental examples of systems that 1G. W.Castellan, Physical Chemistry,3rd ed.,(Benjamin Pub.Co., 1983), p. 266. 2R.J. Silbeyand R.A. Alberty, Physical Chemistry,3rd ed.,(John Wiley&Sons, Inc. 2001) p. 16. Winter Term 2009 -2- Chemistry 593 have critical points in their phase diagrams; some of these are shown in Table 1. In each case, the nature of the transition is clearly quite different (from the point of viewofthe qualitative symmetries of the phases involved). TABLE 1. Examples of critical points and their order parameters3 Critical Order Example T (o K) Point Parameter c Liquid-gas Density H2O 647.05 Ferromagnetic Magnetization Fe 1044.0 Anti-ferromagnetic Sub-lattice FeF2 78.26 magnetization Super-fluid 4 He-amplitude 4 He 1.8-2.1 Super-Electron pair Pb 7.19 conductivity amplitude Binary fluid Concentration CCl4-C7 F14 301.78 mixture of one fluid Binary alloyDensity of one Cu − Zn 739 kind on a sub-lattice Ferroelectric Polarization Triglycine 322.5 sulfate The cases in Table I are examples of so-called 2nd order phase transitions, according to the naming scheme introduced by P.Eherenfest. More generally,annth order phase transition is one where, in addition to the free energies, (n − 1) derivativesofthe free energies are continuous at the transition. Since the first derivativesofthe free energy give entropyand volume, all of the freezing and sublimation, and most of the liquid-vapor line would be classified as first-order tran- sition lines; only at the critical point does it become second order.Also note that not all phase transitions can be second order; in some cases, symmetry demands that the transition be first order. At a second order phase transition, we continuously go from one phase to another.What differentiates being in a liquid or gas phase? Clearly,both have the same symmetries, so what quantitative measurement would tell us which phase we are in? We will call this quantity (or quantities) an order parameter,and adopt the convention that it is zero in the one phase region of the phase diagram. In some cases there is a symmetry difference between the phases and this makes the identification of the order parameters simpler,inothers, there is no obvious unique choice, although we will showlater that for manyquestions, the choice doesn’tmatter. 3S.K. Ma, Modern Theory of Critical Phenomena,(W.A. Benjamin, Inc., 1976), p. 6. Winter Term 2009 -3- Chemistry 593 Forexample, at the liquid-gas critical point the density (molar volume) difference between the liquid and vapor phases vanishes, cf. Fig. 2, and the density difference between the two phases is often used as the order parameter.Second order transitions are also observed in ferro- magnetic or ferroelectric materials, where the magnetization (degree or spin alignment) or polar- ization (degree of dipole moment alignment) continuously vanishes as the critical point is approached, and we will use these, respectively,asthe order parameters. Other examples are giveninTable 1. At a second order critical point, manyquantities vanish (e.g., the order parameter) while − −1 ∂ ∂ others can diverge(e.g., the isothermal compressibility, V ( V/ P)T,N cf. Fig. 2). In order to quantify this behavior,weintroduce the idea of a critical exponent.For example, consider a fer- romagnetic system. As we just mentioned, the magnetization vanishes at the critical point (here, this means at the critical temperature and in the absence of anyexternally applied magnetic field, H), thus near the critical point we might expect that the magnetization, m might vanish like β m∝|Tc − T| ,when H = 0, (1.2) or at the critical temperature, in the presence of a magnetic field, m∝H 1/δ .(1.3) The exponents β and δ are examples of critical exponents and are sometimes referred to as the order parameter and equation of state exponents, respectively; we expect both of these to be posi- tive.Other thermodynamic quantities have their own exponents; for example, the constant mag- netic field heat capacity (or CP in the liquid-gas system) can be written as −α CH ∝|Tc − T| ,(1.4) while the magnetic susceptibility, χ ,(analogous to the compressibility) becomes −γ χ ∝|Tc − T| .(1.5) Non-thermodynamic quantities can also exhibit critical behavior similar to Eqs. (1.2)−(1.5). Perhaps the most important of these is the scattering intensity measured in light or neutron scattering experiments. As you learned in statistical mechanics (or will see again later in this course), the elastic scattering intensity at scattering wav e-vector q is proportional to the static structure factor NS(q) ≡<|N(q)|2 >, (1.6) where N(q)isthe spatial Fourier transform of the density (or magnetization density), and < ...> denotes an average in the grand canonical ensemble. In general, the susceptibility or compress- ibility and the q → 0limit of the structure factor4 are proportional, and thus, we expect the scat- tered intensity to divergewith exponent γ as the critical point, cf. Eq. (1.5). This is indeed observed in the phenomena called critical opalescence. At T = Tc and non-zero wav e-vectors, we write, 1 S(q)∝ .(1.7) q2−η 4See, e.g., http://ronispc.chem.mcgill.ca/ronis/chem593/structure_factor.1.html. Winter Term 2009 -4- Chemistry 593 Finally,weintroduce one last exponent, one that characterizes the range of molecular cor- relations in our systems. The correlation-length is called ξ ,and we expect that −ν ξ ∝|Tc − T| ,(1.8) where we shall see later that the correlation-length exponent, ν >0. Some experimental values for these exponents for ferromagnets are giveninTable 2. The primes on the exponents denote measurements approaching the critical point from the two-phase region (in principle, different values could be observed). What is interesting, is that eventhough the materials are comprised of different atoms, have different symmetries and transition tempera- tures, the same critical exponents are observed, to within the experimental uncertainty. TABLE 2. Exponents at ferromagnetic critical points5 Material Symmetry T(o K) α , α ′ βγ, γ ′ δη Fe Isotropic 1044.0 α = α ′=0. 120 0.34 1.333 0.07 ±0.01 ±0.02 ±0.015 ±0.07 Ni Isotropic 631.58 α = α ′=0. 10 0.33 1.32 4.2 ±0.03 ±0.03 ±0.02 ±0.1 EuO Isotropic 69.33 α = α ′=0. 09 ±0.01 γ = YFeO3 Uniaxial 643 0.354 1. 33 ±0.005 ±0.04 γ ′=0. 7 ±0.1 Gd Anisotropic 292.5 γ = 1. 33 4.0 ±0.1 Of course, this behavior might not be unexpected. After all, these are all ferromagnetic transitions; a phase transition where "all" that happens is that the spins align. What is more inter- esting are the examples shown in Table 3. Clearly,the phase transitions are very different physi- cally; nonetheless, universal values for the critical exponents seem to emerge. 5S.K. Ma, op. cit.,p.12. Winter Term 2009 -5- Chemistry 593 TABLE 3. Exponents for various critical points6 Critical o Material Symmetry Tc( K) α , α ′ β γ , γ ′ δ η Points ⋅ α ≤ Antiferro- CoCl2 6H2O Uniaxial 2.29 0. 11 0.23 magnetic α ′≤0. 19 ± 0. 02 α = α ′= FeF2 Uniaxial 78.26 0. 112 ± 0. 044 α = α ′=− γ = RbMnF3 Isotropic 83.05 0. 139 0.316 1. 397 0.067 ± 0. 007 ± 0. 008 ± 0. 034 ± 0. 01 = α ∼ γ = γ ′= Liquid-gas CO2 n 1304.16 1/8 0.3447 1. 20 4.2 ± 0. 0007 ± 0. 02 Xe 289.74 α = α ′=0. 08 0.344 γ = γ ′=1. 203 4.4 ± 0. 02 ± 0. 003 ± 0. 002 ±0. 4 3 He 3.3105 α ≤ 0. 3 0.361 γ = γ ′=1. 15 α ′≤0. 2 ±0. 001 ±0. 03 4 He 5.1885 α = 0. 127 0.3554 γ = γ ′=1. 17 α ′=0. 159 ±0. 0028 ±0. 0005 Super-fluid 4 He 1.8-2.1 0. 04 ≤ α = α ′ <0 − = γ = ∼ Binary CCl4 C7 F14 n 1301.78 0.335 1. 2 4 Mixture ±0. 02 Binary Co − Zn n = 1739 0.305 γ = 1.