Chapter 3. Phase transitions. 3.1 Introduction. 3.2 Thermodynamics of phase transitions

Transcription

1 Chater 3 Phase transitions 3.1 Introduction Phase transitions are ubiquitous in Nature. We are all familiar with the different hases of water (vaour, liquid and ice), and with the change from one to the other. Changes of hase are called hase transitions. hese henomena are very imortant not only in natural rocesses, but also in industry. o name just a few everyday examles: the evaoration of liquids, formation of ice or liquid water sheets on surfaces, construction of different materials in metallurgy... Here we give an elemantary introduction to the sub ject of hase transitions. his is a vast subject, and a lot of research effort is still being dedicated to hase transitions, both from exerimental and theoretical oints of view. he first microscoically-based understanding of hase transitions is due to van der Waals, who in 1873 resented a rimitive theory of the gas-liquid transition. In his doctoral thesis he resented the van der Waals equation of state, and linked its arameters to a molecular model. Even though it was a first attemt, it laid down the basic ideas on which the modern theoretical understanding is based. Later, Landau (1937) roosed a henomenological (not fully microscoic) aroach that was crucial to understand second-order hase transitions and later develoments. Another aramount ste forward was given by Wilson (1971) and others, who develoed a owerful and unifying set of concets (basically the idea of scaling close to critical oints and the renormalisation grou) for second-order hase transitions. First-order transitions are less well understood, and current work relies almost exclusively on mean-field theory, which we shall review and aly in various contexts. 3.2 hermodynamics of hase transitions Let us start with the so-called hase diagram of a simle substance. A hase diagram is a reresentation, tyically in a lane, of the regions where some substance is stable in a given hase. he axes reresent external control variables (intensive arameters), such as ressure, temerature, chemical otential or an external field, or sometimes one extensive variable (volume, magnetisation, etc.) Density is sometimes used in the case of fluids. he different hases are searated by lines, indicating hase transitions, or regions where the system is unstable. 49

2 Let us review some of the most imortant features of the hase diagram of a simle substance. In Fig. 3.1(a) the hase diagram is reresented in the ressure-temerature (-) lane. he three ossible hases (solid, liquid and gas) are searated by first-order hase transition lines (continuous lines in the grah) where two hases coexist at the same time (hence the name coexistence lines ). Phases are indicated by their names, and two secial oints are called (trile oint) and C (critical oint). Figure 3.1: (Colour on the Web). Schematic hase diagram in the (a) ressuretemerature, and (b) temerature-density lane. he three coexistence lines are: sublimation line: here the solid coexists with the gas. his line exists from zero temerature u to the trile-oint temerature, 3. On lowering the temerature at constant ressure starting from the gas side, the gas would reach the sublimation line, at which crystallites would begin to form until the whole system becomes a crystal. Conversely, when crossing the line from the crystal side, gas would begin to sublimate from the crystal until the system has assed to gas entirely. Any other trajectory on the diagram would lead to a similar behaviour. melting line, also called fusion line: here the solid coexists with the liquid. On crossing this line from the liquid side the system would begin to crystallise; conversely, the crystal would melt in the other direction. his line exists from 3 u to infinite temerature (there is no uer limit in the melting line). his means that one cannot ass from liquid to crystal (or the other way round) without crossing the transition line, a result coming from the essentially different symmetries of these hases (full translational and rotational invariance in the liquid turn into discrete symmetries in the case of the crystal hase). condensation line, also called vaour-ressure line: here the gas coexists with the liquid. On crossing the line from the gas side the system begins to form drolets of 50

3 liquid on the condensation line, which grow and coalesce until all the material has assed to the liquid hase. Conversely, from the liquid side, gas bubbles form at the line, which grow and coalesce until all the material has transformed into gas. his is also called vaour-ressure line because it gives the maximum ressure the system can stand as a gas for a given temerature. he line begins at the trile oint and ends at the critical oint, C, with c the critical temerature and c the critical ressure. Above this oint C there is no distinction between gas and liquid, and one may continually ass from gas to liquid, without ever crossing the transition line, by going into the suercritical region. his indicates that there is basically one fluid hase, which can be understood from the oint of view of symmetry since both hases, liquid and gas, have the same symmetries (rotational and translational invariance). Figure 3.2: Original grah by Guggenheim (1945) where the law of corresonding states was first shown with exerimental data on a reduced temerature vs. reduced density hase diagram for a number of substances. All these transitions are first-order hase transitions, which are accomanied by a latent heat (the system absorbs or gives energy to the heat bath) and a change in density (or volume). A different henomenon is the critical oint, where no such discontinuities occur. A critical oint is an examle of a continuous hase transition. In Fig. 3.1(b) the hase diagram is reresented in the temeraturedensity lane. Shaded regions corresond to coexistence regions where the system resents two hases at the same time. In dashed line different isobars [i.e. the curves = (ρ, ) for fixed ressure] at different temeratures are reresented. Below the critical ressure c the isobars have two horizontal sectors corresonding to the gas-liquid and liquid-solid hase transitions. Above c only one lateau (that corresonding to the liquid-solid hase transition at c ) can occur. Exactly at the critical ressure the isobar develos an instability where the inverse comressibility becomes zero, a behaviour that is reroduced by the simle 51

4 statistical-mechanical theories develoed so far (extended second-virial coefficient and erturbation theory). An intriguing feature of the liquid-gas coexistence curve is that it aroximately follows the law of corresonding states: When the coexistence line ρ is lotted in scaled units, using c and the critical density ρ c as units of temerature and density, the exerimental data aroximately lie on a single universal line. his is followed by a large number of substances, excet a few ones (tyically very light elements). In Figs. 3.2 and 3.3 the hase diagrams of some noble gases (Ar, Ne, Kr and Xe) are dislayed in reduced units, i.e. units scaled by the values of the critical arameters: ρ = ρ ρ c, = c, = c. (3.1) he different data collase into a single line when lotted is reduced units; therefore the liquid-gas coexistence line becomes universal. Some other molecular substances also follow this law. he law is rather remarkable considering the disarate values of critical arameters of the substances; some data are given below: Gibbs hase rule c (K) c (atm) ρ c (g cm 3 ) H 2 O CO O Xe Ar Let us now discuss the coexistence conditions, i.e. the conditions that two hases have to satisfy in order to coexist on a firstorder transition line and therefore with the same thermodynamic conditions. wo coexisting hases have the same thermodynamic Gibbs free energy er article 1 µ = G/N at the coexistence line; outside of this line one or the other is the most stable hase. hermodynamics says that in order for two systems in contact to be in equilibrium (in our case two hases of the same substance) there must be mechanical, thermal and chemical equilibrium. Let us denote by a and b the two hases; then a = b, a = b, µ a = µ b. (3.2) But the chemical otential is a function of (, ) (remember that according to the Gibbs- Duhem equation, µ, and are not indeendent). herefore, at two-hase coexistence (i.e. on a coexistence line, for examle, the condensation line), µ a (, ) = µ b (, ). (3.3) his gives a relation between and, e.g. = () which is a line on the - hase diagram. Is it ossible to have three (a, b and c), instead of two, hases coexisting at the same time? We should have µ a (, ) = µ b (, ) = µ c (, ), (3.4) 1 Here we should remember that Euler s theorem alied to the Gibbs free energy gives G = Nµ, see Eqn. (1.27). 52

5 Figure 3.3: (Colour on the Web). (a) Reduced ressure = / c and (b) equilibrium coexistence densities ρ = ρ/ρ c versus reduced temerature = / c along the hase coexistence curves for Ar, Ne, Kr and Xe. Figure taken from Vorobev, Chem. Phys. Lett. 383, 359 (2004). which leads to a single oint, the trile oint, on the hase diagram. We can see that it is not ossible to have more than three hases coexisting at the same time for a simle one-comonent substance, excet in very secial circumstances. Let us now consider a more general system, namely a mixture of l comonents, characterised by the arameters (N,,, x (i) ), where x (i) = N (i) /N [with N (i) the number of articles of secies i and N the total number of articles] is the molar fraction or comosition of the i-th secies. Note that i x (i) = 1 so that only l 1 mole fractions are indeendent. Let us denote by µ (i), i = 1, 2,, l the chemical otential of the i-th secies. In order to have coexistence between r different hases a 1, a 2, a r, we must have µ (1) a 1 (,, {x (i) }) = µ (1) a 2 (,, {x (i) }) =... = µ ((1) a r (,, {x (i) }), µ (2) a 1 (,, {x (i) }) = µ (2) a 2 (,, {x (i) }) =... = µ ((2) a r (,, {x (i) }), µ (l) a 1 (,, {x (i) }) = µ (l) a 2 (,, {x (i) }) =... = µ ((l) a r (,, {x (i) }). (3.5) 53

6 his is a set of (r 1) l equations. he number of unknowns is 2 + r (l 1), corresonding to, and the l 1 mole fractions in each of the r hases. hen we must necessarily have: (r 1)l 2 + r(l 1) r l + 2. (3.6) his is the famous Gibbs hase rule. For l = 1 (one-comonent system) r 3, so that we have at most three hases coexisting at the same time (trile oint). For l = 2 (binary mixture) we have r 4 so that we may have u to a quadrule oint where four hases coexist simultaneously, and so on. In general, for l comonents, the hase diagram has to be lotted in the l + 1-dimensional sace (,, x (1), x (2),, x (l 1) ) [note that N is not relevant since G, which determines all roerties of the system, is simly roortional to N, i.e. G(N,, ) = Nµ(, )] and in this diagram we may have subsaces of dimensionality d = l +1, l, l 1,..., 2, 1 and 0, with r = l +2 d hases coexisting in each subsace, corresonding resectively to volumes (one stable hase, r = 1, d = l + 1), areas (two coexisting hases, r = 2, d = l), lines (three coexisting hases, r = 3, d = l 1),..., and finally oints (r = l + 2 coexisting hases in subsaces of dimension d = 0). he Gibbs hase rule relates hase coexistence to the toology of the hase diagram. Clausius-Claeyron equation his is again a result for first-order hase transitions. Suose two hases of a onecomonent system coexist on a coexistence line (e.g. liquid and gas on the condensation line). We have µ a (, ) = µ b (, ). If we move along the coexistence line by differential amounts d and d, we will have µ a ( + d, + d) = µ b ( + d, + d), since we are still at coexistence. hen: µ a (, ) + ( µa ) d + ( ) ( ) µa d + d coex ( ) µa d = µ b (, ) + ( ) µa = ( ) µb d + ( ) ( ) µb d + d coex But using the Gibbs-Duhem equation, Ndµ V d + Sd = 0, we have ( ) µ = 1 ( ) µ ρ, = S N ( ) µb d, ( ) µb. (3.7) s (3.8) (where s is the entroy er article), so that we arrive at the Clausius-Claeyron equation: ( ) d = s a s b = s a s b = S d v coex a v b V a V b V, (3.9) where v = 1/ρ = V/N is the secific volume (volume er article), ρ a and ρ b the densities of the coexisting hases, and S and V the jums of the entroy and volume at the transition line at (, ). Incidentally, in view of the finite value of the derivative d/d at coexistence, the Clausius-Claeyron equation exlains why a first-order hase transition is characterised by discontinuous changes in entroy and volume (or density). S gives 54

7 the heat L that is transferred from the system to the environment as L δq = S, called the latent heat. We can aly this equation to the liquid-gas coexistence line. he derivative of the coexistence curve at any oint is given by (3.9). Since this derivative is ositive, S and V must have the same sign. he sign deends on the identification of a and b. Let us associate a with the gas and b with the liquid. Since V a > V b (because the gas has a larger volume) we must have S a > S b : the gas has more entroy, as it should be if we are to identify entroy with disorder. When going from a low-temerature hase to a hightemerature hase entroy always increases, S > 0, because C = ( S/ ) > 0. But the sign of V is more uncertain. Generally it is also ositive, but in some cases it is negative; a aradigmatic examle is that of freezing in water: when going from ice to liquid water S > 0 but V < 0 (since ice is less dense than liquid water at coexistence) which imlies that the sloe of the freezing line is negative. We see that the Clausius-Claeyron equation allows us to identify some basic roerties of firstorder hase transitions on the basis of the sloe of the coexistence line. Figure 3.4: (Colour on the Web). Schematic ressure-secific volume hase diagram for different temeratures above, equal to and below the critical temerature c. Coexistence conditions are reresented by the horizontal lines. he red line is the enveloe of coexistence secific volumes. Dashed lines reresent metastable states (see later in the chater). Lever rule Let us reconsider the liquid-gas hase transition which will allow us to introduce a few new concets. Fig. 3.4 illustrates the hase diagram in the V lane. Again we have drawn isotherms for < c, = c and > c, but let us focus on the first. he coexistence line is reresented by a heavy line. he dotted sectors of the gas and liquid branches re- 55

8 resent the metastable states, i.e. mechanically stable (with ositive comressibility) but thermodynamically unstable states (since it is more favourable energetically to roceed with the hase transition); in fact it is ossible to reare metastable states in very ure substances by secial rocedures. We will comment on these states later. What haens as we comress the gas at fixed temerature below the critical temerature? he ressure follows the isotherm until it reaches oint A; here liquid drolets begin to form in the system. As the system is further comressed, it moves along the horizontal line AB (at constant ressure) until the whole amount of gas has condensed into liquid at B. From there on the system follows the same isotherm with increasing values of the ressure. At all oints between A and B the system is a mixture of gas and liquid. ake oint D, with global density ρ D (the density of the liquid regions will be ρ A and that of the gas regions will be ρ B ). ake v D = 1/ρ D to be the total secific volume, and v A = 1/ρ A and v B = 1/ρ B the secific volumes of gas and liquid. hen v D = N A N v A + N B N v B = x A v A + x B v B, (3.10) where N A, N B are the number of molecules in the gas and liquid regions, resectively, x A, x B the mole fractions, and N = N A + N B the total number of molecules. Since x A + x B = 1, multilying: (x A + x B )v D = x A v A + x B v B x A x B = v B v D v D v A. (3.11) his is the so-called lever rule: the ratio of mole fractions is equal to the inverse ratio of the distance between the secific volumes of the hases to the global secific volumes. Behaviour of thermodynamic otentials: tyes of hase transitions In discussing hase transitions it is convenient to use the Gibbs free energy, G = G(N,, ) since it deends on two intensive variables, (, ), which are the same for the two hases that coexist at a first-order hase transition line. G satises the stability criterion d 2 G > 0 which, as exlained in Chater 1, leads to a number of stability conditions on the resonse functions. For examle: κ = 1 V ( ) V = 1 V ( ) G 2 hese conditions, together with the relations V = > 0, C = ( ) ( ) S G = > 0. (3.12) 2 ( ) ( ) G G, S =, (3.13) are sufficient to obtain some conclusion as to the form of the function G/N = µ = µ(, ) as a function of and. In Fig. 3.5 the volume vs. ressure is lotted in a region where there is no hase transition. In art (a) the volume V (which of course is ositive) is lotted, as a monotonically decreasing function of ressure, as it should be in view of the stability condition κ > 0; 56

9 Figure 3.5: (Colour on the Web). Behaviour of Gibbs free energy G(N,, ) and its first derivatives V and S for fixed N with resect to [at fixed ], (a) and (c), and with resect to [at fixed ], (b) and (d), when there is no hase transition. these two facts allow one to lot the function µ(, ) for fixed in art (b) as a monotonically increasing function of, but with negative second derivative (that is, as a concave function). In anels (c) and (d) the deendence of µ(, ) with is discussed; note that S > 0, that C > 0 suggests that S increases monotonically with and finally that µ(, ) is also a concave function with resect to at fixed. Note that µ (and therefore G) is always continuous. When a first-order transition is resent, the situation changes. We have seen that there is an associated latent heat at the transition, so that the entroy must be discontinuous. Also, the volume is discontinuous. his is reflected in Fig he effect of all this on µ is the existence of a kink at the transition, both with resect to and. Again G (but not its first derivatives, reflecting jums in entroy and volume) is a continuous function of its variables. A similar analysis can be made based on the Helmholtz free energy F = F(N, V, ). But here is relaced by V, which is discontinuous at the transition. Also, one exects > 0 so that F/ V < 0, and κ = 1 V ( ) V = 1 V ( ) V = 1 V ( 2 ) F V 2 > 0, (3.14) 57

10 Figure 3.6: (Colour on the Web). Same as Fig. 3.5 but when there is a first-order hase transition. he discontinuity of G at the transition is indicated by lotting in dashed lines the tangents on both sides of the transition. so that F is an overall convex function of V. With this roviso the deendence of F on V at constant temerature when a hase transition is resent can be lotted as in Fig Note the linear sector joining the two coexisting states, along which the hase transition takes lace (i.e. the two hases exist at the same time in varying roortions). But there are other tyes of hase transitions. Older classification schemes (e.g. that due to Ehrenfest) have been revised and nowadays all transitions not being of first order are called second-order or (referably) continuous hase transitions. Here not only G but also its first derivatives are continuous functions (therefore there is no latent heat or volume discontinuities); it is the second derivatives of G (i.e. the resonse functions) that may exhibit anomalies. he anomalies may be of two tyes: either discontinuities, or divergencies (logarithmic, ower-law or otherwise). For examle, Fig. 3.8 shows the divergence of the constant-ressure heat caacity of a samle of YBa 2 Cu 3 O 7 x, which is a high- c suerconductor, at the suerconductor hase transition. 3.3 Mean-field theory: van der Waals theory From now on we consider the roblem of hase transitions from the oint of view of statistical mechanics, seeking an exlanation in terms of microscoic roerties. We have already obtained van der Waals equation essentially by two methods: calculation of virial 58

11 Figure 3.7: (Colour on the Web). Behaviour of F when there is a first-order hase transition. he horizontal sector in the = (V ) curve is transformed into a linear sector in F = F(V ). In the case when there is no hase transition this linear sector is absent. equation of state u to second order and resummation of the hardshere contribution to the ressure (Clausius equation) and direct use of erturbation theory. here is yet a third way, which is more general in the sense that it contains the idea of a mean field which ervades the field of statistical mechanics. Also, we will study the instability redicted by van der Waals theory in more detail, which will hel us get acquainted with many of the basic features of hase transitions. Consider the configurational artition function of a classical simle fluid: Z N = dr 1 dr N e βu(r 1,,r N ). (3.15) Suose that the effect of articles on a given one can be reresented by means of an effective one-article otential φ acting on each article searately but being otherwise identical for all the articles: N U(r 1,,r N ) = φ(r i ). (3.16) i=1 he configurational artition function can then be factorised: Z N = dr 1 dr N e [ β i φ(ri) = V dre βφ(r) ] N. (3.17) his is the general exression of the mean-field aroximation for a classical fluid. Different choices for φ(r) give different imlementations of the theory. In essence, the mean-field aroach is an ideal aroximation in the sense that it aarently decoules the degrees of freedom. But there is more than meets the eye: the mean field is meant to contain not only the contribution of any external field but also, indirectly and in an average way, the effect on a given article of the remaining ones. 59

12 Figure 3.8: Constant-ressure heat caacity divided by temerature, c /, of a samle of YBa 2 Cu 3 O 7 x at the suerconductor hase transition. In this case the divergence is logarithmic. More details to be found in Regan et al., J. Phys.: Condens. Matter 3, 9245 (1991). he temerature axis is t ( c )/ c. So all we are left to do is choosing an adequate effective otential φ(r). A simle choice can be made if we bear in mind that the real air otential has reulsive and attractive contributions: in the integral over r (the ositions of a given article, all the others being equivalent) only the regions where there is no article overla contribute. So we subtract the volume excluded to a given article by the remaining N 1 articles, V exc, coming from the reulsive (exclusion) interactions, and leave a constant (attractive) contribution u 0 < 0 (constant in the sense that it does not deend on the ositions of the other articles but in an average way). hen and Q = Z N = [ (V V exc ) e βu 0] N, (3.18) Z N N!Λ 3N = he Helmholtz free energy is then: [ (V Vexc)e βu 0] N N!Λ 3N. (3.19) F = k log ( N!Λ 3N) Nk [log (V V exc ) βu 0 ]. (3.20) he ressure is = ( ) F V = Nk V V exc N ( ) u0. (3.21) V 60

13 Since it is reasonable to assume that u 0 be roortional to the density (i.e. to the number of articles), u 0 = aρ, and that there will be a contribution to V exc from each of the N 1 articles, V exc = bn (we take N 1 N), we obtain = ρk 1 bρ aρ2, (3.22) which is van der Waals equation of state. Connection of the energy constant a and the volume constant b with molecular arameters of a articular model, the squarewell fluid, is given in section 2.3.2, where we found b = 2πσ 3 /3 and a = 2πλσ 3 /3, with σ the hardshere diameter, and ǫ the well deth; λ is related to the breadth of the well δ by Eqn. (2.32). his identification is not crucial at the moment and we will kee on using a and b as the system constants. We already made the observation that the second term in the righthand side of (3.22) can make the comressibility to be negative. o see this more exlicitely, let us calculate the derivative ( / ρ) N, : ( ) (k) 1 ρ N, = ρ { } ρ 1 bρ aρ2 = k 1 (1 bρ) 2 2aρ k. (3.23) From here the inverse comressibility is ( ) 1 κ 1 = (1 bρ) 2aρ 2 k. (3.24) κ 0 here will be an instability (i.e. κ 1 In Fig. 3.9 the function = 0) when 1 (1 bρ) 2aρ 2 k = 0. (3.25) g(ρ) ( ) 1 κ = κ 0 1 (1 bρ) 2aρ 2 k (3.26) is reresented for different values of (for simlicity we have set some articular values for the constants a, b and k in the figure since the actual values of ρ and g(ρ) are not imortant for the argument). Note that the excess of g(ρ) over unity reflects the contribution of interactions to the inverse comressibility. he function g(ρ) has a root for the first time at some temerature c, called critical temerature, and it is a double root (a minimum). he values of c and the critical density ρ c follow by calculating the location of the minimum, searching for the condition that the minimum is zero, i.e. ( ) ( 2 ) = 0, = 0. (3.27) ρ ρ c, c ρ 2 ρ c, c Differentiating g(ρ) [which is equivalent to calculating the second derivative ( 2 / ρ 2 )] and equating to zero: [ g b (ρ) = 2 (1 bρ) a ] = 0, (3.28) 3 k 61

14 Figure 3.9: he function g(ρ) (inverse comressibility normalised by same function for ideal-gas, defined by Eqn. (3.26) for three different values of. from which the minimum is at ρ = 1 ( ) 1/3 kb 1. (3.29) b a Substituting, the value of g(ρ) at the minimum is which is zero at a temerature c such that ( ) a 2/3 ( ) a g(ρ) = 3 2, (3.30) kb kb ( ) a 2/3 ( ) a g(ρ) = 3 2 = 0 k c = 8a kb kb 27b, (3.31) From here the critical density is ρ c = 1 ( ) 1/3 kc b 1 = 1 b a 3b, (3.32) and the critical ressure c = ρ ck c aρ 2 c 1 bρ = a c 27b2. (3.33) he law of corresonding states can be exlained by van der Waals equation. If we scale ρ, and with the corresonding critical values, it is easy to arrive at = 8ρ k 3 ρ 3ρ 2. (3.34) 62

15 van der Waals theory then rovides a universal equation of state and in fact it gives lausibility to the law, but the universal equation rovided by van der Waals theory is not the correct one. For examle, the socalled comressibility factor, defined by Z = v/k (a non-dimensional number), adots a more or less universal value at the critical oint, a value which exerimentally is observed to be Z c = c v c /k c 0.29 for a wide range of substances, whereas van der Waals theory redicts Z c = cv c k c = c = 3 = (3.35) ρ c k c 8 Figure 3.10: (Colour on the Web). (a) Pressure-volume curves for different temeratures. he one for < c shows a loo with extrema at a and b. he true thermodynamic hase transition is indicated by the red horizontal line at ressure coex. 1 and 2 are the coexisting liquid and vaour hases, resectively. Shaded regions A and B have the same area according to Maxwell s construction. In (b), a linear term in the volume has been added to better visualise the loo (this does not affect the calculation of the coexistence arameters, as can be easily checked). In Fig. 3.10(a) three isotherms of the scaled van der Waals equation are reresented for temeratures > c, = c and < c (scales in axes are not given as they are irrelevant for the resent discussion). he first two isotherms are as exected. But that for < c (thick black line) looks eculiar: it does not redict a horizontal sector signalling coexistence between the gas and the liquid, but a loo with a maximum and a minimum, denoted in the figure by a and b. his is strange because at these extrema κ 1 = 0, so 63

16 they are a kind of critical oints: they define the sinodal line, more visible in a V hase diagram, Fig. 3.11(a) [in this case the extensive variable has been chosen as density ρ]. Inside the sinodal line the fluid is unstable, since κ 1 < 0, which imlies / V > 0, which is not ossible for a stable material. If we disregard these unstable states (between a and b ), redicted by the theory, we are left with the gas and liquid branches, from small volumes u to a, and from large volumes u to b (i.e. the regions where κ > 0). But still we have to exlain the hase transition. he correct interretation is that: the hase transition occurs at some ressure coex, i.e. at some horizontal line connecting the two outer branches at oints 1 and 2, as exerimentally observed [see for examle Fig. 3.5(a)] the remaining arts of the two branches, between 1 and a and between 2 and b, reresent metastable (i.e. mechanically stable but termodynamically unstable) states. he existence of metastable states is a genuine (but in fact not 100 % correct) rediction of van der Waals theory (actually of any mean-field theory) In order for this interretation to be oerational, we must somehow be able to calculate coex. he way the transition ressure is calculated is based on the Maxwell construction. We roceed as follows: Using the relation F(N, V, ) = V (N, V, )dv (3.36) (we are of course assuming that N is fixed and constant) we can obtain the free energy as a function of V for fixed and, Fig. 3.10(b). he analytical exression is: F(N, V, ) = Nk [ log ( ρλ 3) 1 ] Nk [ log (1 bρ) + aρ ]. (3.37) k Figure 3.11: Schematic hase diagrams V (a) and (b) of the van der Waals model. he loo visible in F = F(V ) is traced back to the loo in the equation of state = (V ). 64

17 Note that there is an instability region where F is concave (between a and b ); this corresonds to the unstable sector in (V ) [ositive derivative (V )]. Now, searching for the horizontal ( = coex ) sector associated with the hase transition in (V ) is equivalent to searching for the straight line in F(V ), joining oints 1 and 2 (i.e. the two coexisting hases), that corresonds to the true, thermodynamically ermissible trajectory followed by the fluid, reresented in Fig. 3.10(b) in red. his sector is linear in the grah F = F(V ) since, in this region, F/ V = coex. In order to identify the oints 1 and 2 (i.e. the volumes of the coexisting gas and liquid, V 1 and V 2 ), we first note that they must have the same ressure [i.e. same sloe in the F(V ) grah], so that ( ) ( ) F F = V V 1 2 Also, the straight line must have a common tangent at 1 and 2, which imlies In fact this condition can be written (3.38) ( ) ( ) F F = = F 1 F 2, (3.39) V V V V 2 1 (V 2 V 1 ) = F 1 F 2 = V2 V 1 dv. (3.40) he integral is the area under the curve = (V ) between 1 and 2. his equation imlies that the areas of A and B in the grah of (V ) must be equal. his grahical criterion fixes the ressure 1 = 2 = coex at the transition, and we have Maxwells construction. Note that the condition (3.39) reflects in fact the equality of chemical otentials, since G = µn = F + V = F and from (3.39) we have, rearranging terms: F 1 ( ) F1 V 1 = F 2 V 1 herefore, an equivalent statement of the Maxwell construction is ( ) F V, (3.41) V ( ) F2 V 2. (3.42) V 2 (ρ 1, ) = (ρ 2, ), µ(ρ 1, ) = µ(ρ 2, ), (3.43) as we should have exected since this is a hase transition roblem where two hases, at 1 and 2, coexist in equilibrium at the same temerature. Finally, in Fig. 3.11(b), a schematic - hase diagram for the van der Waals model is deicted. It only contains a liquid-vaour coexisting line, since the transitions involving the solid cannot be described by the theory. Critical exonents One of the most interesting and imortant toics in the field of hase transitions is critical 65

18 oints. A critical oint exhibits very eculiar roerties; for instance, we already know that comressibility becomes infinite, which means that the material does yield when an arbitrarily small ressure is exerted on it. his and in fact all roerties of critical oints are due to the so-called scale invariance of critical hases. According to this hyothesis, thermodynamic roerties of the system exhibit divergencies or become zero according to very simle mathematical functions, usually ower laws with well-defined exonents, called critical exonents. he remarkable thing is that the values of critical exonents are the same for a large class of very different exerimental systems, deending only on very general roerties, such as dimensionality, and not on secific details of the interactions. his is because in the vicinity of a critical oint there are fluctuations of all length scales. Systems sharing the same values of critical exonents are said to belong to the same universality class. In this section we will simly calculate the critical exonents of the van der Waals model for the liquid-vaour hase transition. he subject will be covered more deely in a later chater. We begin with the isothermal comressibility obtained from the scaled van der Waals equation (we omit the suerscrits * for the sake of brevity in notation): = 8ρ ( ) 3 ρ 3ρ2 κ 1 = ρ = 24ρ ρ (3 ρ) 2 6ρ2. (3.44) At the critical density ρ c = 1 (but for general ) and since the critical temerature is c = 1: κ 1 = 6( 1) κ ( c ) γ, (3.45) where γ is the critical exonent associated with the comressibility. he comressibility diverges as a ower law, as indeed observed exerimentally in the liquid-vaour transition, and van der Waals theory redicts the exonent to be γ = 1. Let us now examine the heat caacity at constant volume. Exerimentally C V c α. o see the value of α redicted by van der Waals theory we first calculate the energy. Using (3.37): E = 2 ( F/ ) V = 3 Mk Naρ. (3.46) 2 herefore ( ) E C V = = 3 2 Nk c 0, (3.47) V so that van der Waals theory redicts α = 0. Another critical exonent is δ, defined as const. + const.(ρ ρ c ) δ, k c = c, (3.48) i.e. it says how the ressure tends to its critical value at the critical temerature. Setting = 1 and exanding about ρ = ρ c = 1: = 8ρ 3 ρ 3ρ2 = (ρ 1)3 + (3.49) 66

19 exonent value vdw (mean field) exerimental value α β γ δ so that δ = 3. Finally, aonther very imortant exonent is the β exonent, associated with the variation of the density difference between the coexisting liquid and vaour = ρ 1 ρ 2 below c : It can be shown that van der Waals redicts β = 1/2. ρ ( c ) β, < c. (3.50) able 3.3 summarises the values of the exonents redicted by van der Waals theory. In fact, all mean-field theories give the same values for the exonents, and these values are called classical exonents. he differences with resect to the real (exerimental) values come from the neglect of fluctuations inherent to mean-field theories. Inclusion of fluctuations is crucial to imrove the theory of critical henomena beyond mean field. 3.4 Lattice models he statistical mechanics of simle liquids and gases of interacting articles is extremely difficult. Sometimes severe aroximations are necessary. It is therefore useful to consider crude models that are amenable to analytical treatment, and for which sometimes even exact solutions can be obtained. By considering these models much can be learned on the effect of interactions and on the nature of hase transitions. hese models are tyically lattice models, where microscoic variables are discretised and translational coordinates are frozen, normally on a crystalline lattice. Analysis of lattice models has led in many cases to a dee understanding of quite general concets in the theory of hase transitions. An imortant model is the so-called lattice gas: a system of N atoms restricted to lie on the sites of a D-dimensional lattice, with the condition that there is at most one atom on each site. yically only nearest-neighbour interactions are considered. he model is aroriate for understanding real roblems, such as the liquid-vaour transition, adsortion henomena, etc. We will have more to say about this model later. In fact, the lattice gas can be shown to be isomorhic to another discrete model, the Ising model, which was roosed to understand the henomenon of ferromagnetism. We turn to discuss this model in some detail in the next few sections. First, let us define the Hamiltonian. he model is defined on a one-, two-, three- (or more) dimensional lattice, on each node of which there lies a sin with discrete orientations. For examle, for a sin-1/2 Ising model, the i-th sin could oint u or down, and this is accounted for by a sin variable 67