Phase Transitions

A homogeneous equilibrium state of matter is the most natural one, given the fact that the interparticle interactions are translationally invariant. Nevertheless there is no contradiction with the fundamentals of the Statistical Mechanics if under some conditions we have—at a fixed volume— different phases of one and the same substance: liquid and vapor, solid and liquid, liquid and vapor, etc. What are the conditions for such a situation to occur, and why do different phases exist in principle? First, let us take the existence of different phases for granted, and consider the conditions for their coexistence. Each phase is assumed to be homogeneous and characterized by its equation of state in the form µ = µ(P,T ) , (1) in which all the three variables are intensive. Suppose we have two coexisting phases, a and b. Then, each of them, with respect to the other one, simultaneously plays the following three roles: (i) a piston, (ii) a heat bath, and (iii) a particle reservoir. This means that (i) the pressures, (ii) the temperatures, and (iii) the chemical potentials of the two phases coincide, leading to the following condition of the phase coexistence in terms of the equations of state:

µa(P,T ) = µb(P,T ) . (2)

Eq. (2) implies that for a given temperature, T , there can exist only one special critical pressure, Pc(T ), at which the coexistence of the two phases is possible. Equivalently, for a given pressure, P , there can exist only one critical temperature, Tc(P ). The functions Pc(T ) and Tc(P ) can be found from (2) in an implicit form: µa(Pc(T ),T ) ≡ µb(Pc(T ),T ) , (3)

µa(P,Tc(P )) ≡ µb(P,Tc(P )) . (4) Hence, the relation (2) deﬁnes a critical line in the (P,T )-plane on which—and only on which—the coexistence of two phases is possible. Clearly, coexistence of three diﬀerent phases is possible only at an isolated point—so-called triple point—on the (P,T )-plane, as in this case we have one more independent constraint, say,

µb(P,T ) = µc(P,T ) . (5)

Eqs. (2) and (5) can be simultaneously satisfied only at an isolated point: two independent equations for two unknown variables. In a general case, the coexistence of more than three different phases is impossible because we will get three or more independent equations for just two variables. We see that at a fixed pressure there is only one special temperature, Tc(P ), at which two (or three) phases can coexist. Hence, if we pass this special temperature the system will change its phase in a jump-like way. This phenomenon is known as the first-order phase transition. Now we are in a position to understand the reason for a phase transition to occur, and, in particular, the reason for existing different phases, since there is no phase transitions without phases. Fixed pressure and varying temperature are very convenient variables, because the state of the system remains homogeneous up to a single special point Tc. The reason for different phases to exist is thermodynamic favorability. In the space of states of a macroscopic system there can be two or more different regions corresponding to qualitatively different properties of the system (liquid, vapor, solid, ferromagnetic/paramagnetic, etc.). In accordance with Gibbs distribution, the system can be found in each of these regions with the probability proportional

1 to the contribution to the total partition function, associated with the corresponding region. Given two phases, a and b, we thus write the corresponding probabilities as

Za Zb Pa = , Pb = . (6) Za + Zb Za + Zb Here we face a technical problem: We know how to calculate the partition function at a fixed volume, as the volume is a natural parameter fixing the spectrum of the energy eigenvalues. But now we want to fix pressure, not volume. A trick is to treat the pressure as resulting from a constant force (equal to pressure times piston area) applied to our system. We thus get a situation of an external potential equal to −force × coordinate = P × surface area × coordinate = PV. (7) That is for a system of the volume V we get the extra contribution of PV to the energy (and Helmholtz free energy), or, equivalently, the factor of exp(P V/T ) to the partition function. Recalling that

F + PV = G, (8) where G is the Gibbs free energy, we conclude that at a ﬁxed pressure the eﬀective partition function for the phase a reads −Ga/T Za = e , (9) and the same for the phase b. Hence,

Pa Za = = e(Gb−Ga)/T . (10) Pb Zb The probabilities become equal at the point where

Ga(P,T ) = Gb(P,T ) , (11) and are radically diﬀerent away from this point, since the diﬀerence Gb − Ga is an extensive quantity, proportional to the total number of particles.—If it is non-zero, then it is macroscopically large. Hence, at the critical point the systems “jumps” from one phase into the other one. In accordance with Eq. (10), the most favorable phase is the phase with the lowest Gibbs free energy. Is there any connection between the condition (11) and the condition (2) of phase coexistence? Yes, and quite a trivial one, because G ≡ µN . (12)

Problem 47. Establish the identity (12).

Rewriting (10) as Pa Za = = eN(µb−µa)/T , (13) Pb Zb we see that at the ﬁxed pressure the most favorable phase is the phase with a smaller chemical potential. The Gibbs free energies of the two phases are equal at the phase transition point.—What about the rest of quantities? Taking into account that G = E − TS + PV, (14) and that, by deﬁnition, the pressure and temperature remain the same at the phase transition point, from (11) we get Ea − TSa + PVa = Eb − TSb + PVb . (15)

2 Note that neither energy, nor entropy, nor volume is supposed to be the same in the two phases. Eq. (16) can be rewritten as Eb − Ea = T (Sb − Sa) + P (Va − Vb) . (16) In this form it describes the energy balance. The left-hand side is the change of energy. The second term in the right-hand side is the work performed over the system. Correspondingly the term

Lab = T (Sb − Sa) (17) is the heat transferred to the system from the heat bath during the transition. It is called the latent heat. During the I-order phase transition the heat transfer does not result in any temperature change. This means that the heat capacity behaves like a δ-function of (T − Tc). Problem 48. Derive the Clapeyron equation: dP L c = ab . (18) dT T (Vb − Va) Hint. Diﬀerentiate the identity (3), remembering (12).

Consider some point on the phase transition line T = Tc(P ). At the given temperature and pressure the system can have two different volumes, Va, and Vb. What happens if starting from, say, volume Va < Vb, we will be keeping temperature constant while increasing volume, V ? The system cannot be totally in the phase b, until V reaches the value Vb. It cannot be totally in the phase a either, since V > Va. The only possibility is the phase separation, when a part of the system volume corresponds to the phase b, while the rest of the volume is in the phase a. And this is precisely the situation of the phase coexistence, described by the relation (2). From (2) it follows that if T is fixed, then the pressures of the two coexisting phases correspond to the critical pressure Pc(T ) and thus are independent of volume. The same is true for the densities, since the temperature and pressure unambiguously define the density of each of the two phases. Hence, the only quantities that are sensitive to V in the region V ∈ [Va,Vb] are the volumes V1 and V2 of the components a and b, respectively. These volumes are easily found from the relation V V 1 + 2 = 1 , (19) Va Vb and an obvious equation V1 + V2 = V. (20)

Problem 49. Show that Eq. (19) follows from the requirement that the total number of particles be ﬁxed.

The Critical Point. Second-Order Phase Transitions

Suppose we take a fixed volume with coexisting liquid and vapor, and start to increase the temperature. At high enough T , there should be no phase separation: in the limit of T → ∞ the system behaves like an ideal gas, because the kinetic energy of particles is much larger than the potential energy of their interaction. Hence, with increasing temperature we always cross a phase transition line. There are two generic scenarios of what will be happening: (i) the liquid component expands until the vapor component vanishes; (ii) the liquid component evaporates and vanishes. By continuity argument, we conclude that for the given number of particles, N, there should exist such a critical volume, V∗(N)—it is a bit more physical to talk of a critical density, n∗, because V∗ ≡ N/n∗, that the volumes of both coexisting phases remain finite up to the very phase transition point. The phase transition point at n = n∗ turns out to be different from the generic (first-order) liquid-vapor phase transition point. At this point there is no latent heat. All the thermodynamic potentials remain continuous. When approaching this point, the densities of liquid and vapor approach each other, so that beyond this point it becomes fundamentally impossible to tell one from another.

3 The critical point in the liquid-vapor system corresponds to what is known as the second-order phase transition. A generic feature of the second-order phase transitions, as opposed to the first-order transitions, is the absence of jumps in thermodynamic potentials and densities. Instead of jumps there take place anomalous fluctuations with fractal structure. In terms of liquid and vapor, the fractal structure of fluctuations looks as follows. In the liquid domain of the phase separated system there occur and disappear large regions of vapor. In these regions of vapor there are somewhat smaller regions of liquid, and so on, down to the microscopic scale. Staring from the interatomic distances, the fractal structure of fluctuations ranges to some of distance called correlation radius, rc. At distances smaller than rc it is simply impossible to tell liquid from vapor, and, correspondingly, whether the temperature is below or above the critical point! Correlation radius is a function of the vicinity to the critical point, which blows up at the critical point. For example, with temperature approaching T∗ at fixed n = n∗, the correlation radius behaves like 1 rc ∝ ν , (21) |T − T∗| where the critical exponent ν is one and the same for both T > T∗ and T < T∗ regions, while the proportionality coefficients are different. Another generic feature of the second-order phase transitions is a non-analytic behavior (often—divergence) of the heat capacity: 1 C ∝ α . (22) |T − T∗| The difference between the density of vapor (liquid) and the critical density behaves like

β nliq − n∗ = n∗ − nvap ∝ (T∗ − T ) , (23)

For the liquid–vapor critical point: ν ≈ 2/3, α ≈ 0.1, β ≈ 0.3. The relation (23) completely describes the behavior of the system in the phase separation region in the vicinity of the critical point. Indeed, suppose the volume of the system is a bit diﬀerent from the V∗, which is equivalent to saying that the total number of particles is a bit diﬀerent from its critical value N∗. We are in a phase separation region, so we have two phases with the volumes V1 and V2 for the liquid and vapor, respectively. We have a straightforward relation of the particle balance:

nliqV1 + nvapV2 = N. (24)

An important fact about this relation—see previous discussion—is that nliq and nvap depend only on temperature, and not on V1 and V2. This means that we can use (23) for nliq and nvap even if we are not exactly at the critical volume [but still rather close to it, since this relation works only when T∗ − T T∗]. So, identically rewriting (24) as (nliq − n∗)V1 + (nvap − n∗)V2 = N − N∗ , (25) using (23), and eliminating V2 by V1 + V2 = V , we ﬁnd the relation

V1 1 n − n∗ − ∝ β (26) V 2 (T∗ − T ) which yields the fraction of the volume occupied by the liquid at given (T, n, V ).

Problem 50. Show that Eq. (23) implies that the boundary of the phase coexistence region in the (V,T ) plane obeys—in the vicinity of the point (V∗,T∗)—the equation:

1/β Tc − T∗ ∝ −|Vc − V∗| . (27)

Here Tc = Tc(Vc) is the equation for the boundary [which can be equivalently read as Vc = Vc(Tc)]. Hint 1. Note that boundary actually consists of two separate curves meeting each at the critical point. One of the curves corresponds to V1 = V (and thus V2 = 0) and another one is the curve where V2 = V (and thus V1 = 0).

Hint 2. Do not forget to take into account that all our relations are valid only when |T −T∗| T∗, |V −V∗| V∗,

4 so that everywhere except for the diﬀerences |T − T∗| and |V − V∗| one can replace T with T∗ and V with V∗.

Scale Invariance. Hyperscaling Relation

Consider a sequence of linear system sizes, {Ln},(n = 0, 1, 2,...), that differ by some given factor, say, n 2 for definiteness. That is Ln = 2 l0 (with l0 being some large as compared to interatomic distances, but still microscopic scale). Let Zn be the partition function for the system size Ln. For a “normal” macroscopic system we would have 2d Zn+1 = (Zn) (normal) , (28) where d is the spatial dimension. [Note that in d dimensions the system of the linear size 2L consists of 2d subsystems of the size L.] Eq. (28) is basically the definition of a macroscopic system as an ensemble of independent subsystems. At the critical point of a second-order phase transition we can formally write

2d Zn+1 = (Zn) ξ (critical) , (29) where the dimensionless quantity ξ is a correction to Eq. (28) coming from the fact that there are macroscopically large ﬂuctuations. A crucial assumption about ξ, that holds true for all “standard” second-order phase transitions is that ξ is one and the same for all n’s. This assumption follows from a bit more general assumption of scale invariance of the ﬂuctuations at the second-order critical point. Scale invariance here means independence of the length scale. For the free energies Fn = −T ln Zn we then have

Fn+1 = aFn + f , (30) with a = 2d , f = −T ln ξ . (31)

The recursive relation (30) allows one to explicitly express Fn in terms of F0 = −T ln Z0 and f: an − 1 F = anF + f . (32) n 0 a − 1 Now if we are in a close vicinity of the critical point, but not exactly at the critical point, then expressions (29)-(32) work only up to n = n∗ corresponding to the correlations radius:

n∗ = log2(rc/l0) . (33)

For L > rc normal scaling takes place, with F directly proportional to the system volume. Hence, the free energy F corresponding to a system size L rc can be written as

d F (L) = (L/rc) Fn∗ . (34) In accordance with (32)-(33) this means

f L d f L d F = F0 + − . (35) a − 1 l0 a − 1 rc

The ﬁrst term in the r.h.s. of Eq. (35) does not contain rc and thus is not sensitive to approaching the criticality. The second term, which we denote as Fsing, contains rc and thus demonstrates a certain singularity at the critical point. It is convenient to characterize the vicinity to the critical point Tc by the dimensionless parameter |T − T | τ = c . (36) Tc ν Then, in accordance with (21) we have rc ∝ 1/τ and thus

dν Fsing ∝ τ . (37)

5 Now we readily ﬁnd the singular parts of entropy and heat capacity ∂F S ∝ sing ∝ τ dν−1 , (38) sing ∂τ ∂S C ∝ sing ∝ τ dν−2 . (39) sing ∂τ Comparing Eq. (39) to the deﬁnition (22) of the critical exponent α, we arrive at the celebrated hyperscaling relation α = 2 − dν (40) which establishes a dependence between the critical exponents ν and α.