## Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

### 16. Some Special Topics: Phase Transitions

### 16.3. Continuous Transitions and the Critical Point

For continuous phase transitions, considerable effort has been paid to adequately describe and understand the process, and this has led to an image that captures the essentials well. We recall that thermodynamic equilibrium is determined by minimization of the appropriate potential with respect to the internal variable or variables. Because *η* = Δ*ρ* = *ρ* − *ρ*_{cri} is single-valued above *T*_{cri} and has two values below *T*_{cri}, this difference is conventionally used as the internal variable (see __Chapter 2__) characterizing the liquid–gas transition. In connection with transitions, the internal variable used is normally called the *order parameter*. In the sequel of this section we first discuss some experimental facts, and thereafter mean field theory.

**16.3.1 Limiting Behavior**

Both, single compounds and mixtures exhibit continuous transitions. In __Figure 16.6__ the relative density (*ρ*_{L} − *ρ*_{V})/2*ρ*_{cri} of CO_{2} versus the relative temperature *t* ≡ (*T* − *T*_{cri})/*T*_{cri} is given, while __Figure 16.7__ shows the difference in volume fraction (*ϕ*^{(1)} − *ϕ*^{(2)}) for CCl_{4} in C_{7}F_{14} versus *t*. In both cases, the behavior is described by a power law [3].

** Figure 16.6** The relative density (

*ρ*

_{L}−

*ρ*

_{G})/2

*ρ*

_{cri}of CO

_{2}versus the relative temperature −

*t*= (

*T*

_{cri}−

*T*)/

*T*

_{cri}with

*T*

_{cri}= 304.18 K, as described by (

*ρ*

_{L}−

*ρ*

_{G})/2

*ρ*

_{cri}=

*B*(−

*t*)

*with*

^{β}*B*= 1.85 and

*β*= 0.340 ± 0.015.

** Figure 16.7** The difference in volume fraction

*ϕ*

^{(1)}in the upper phase and

*ϕ*

^{(2)}in the lower phase for CCl

_{4}in C

_{7}F

_{14}versus the relative temperature −

*t*= (

*T*

_{cri}−

*T*)/

*T*

_{cri}with

*T*

_{cri}= 301.786 K, as described by

*ϕ*

^{(1)}−

*ϕ*

^{(2)}=

*B*(−

*t*)

*with*

^{β}*B*= 1.81 and

*β*= 0.335 ± 0.020.

This type of experiment has led to the definition of various critical exponents. A critical exponent of a function, say *a* for a function *f*(*x*) of *x*, is defined by

(16.25)

and we write *f*(*x*) ∼ *x ^{a}*. This does not imply that

*f*(

*x*) =

*Ax*, but rather that

^{a}*f*(

*x*) =

*Ax*(1 +

^{a}*Bx*+ …) with

^{b}*b*> 0, and where

*Ax*is called the

^{a}*singular part*of

*f*(

*x*) and (1 +

*Bx*+ …) the

^{b}*regular part*or

*background function*. The exponent

^{7)}*a*defines the rate of approach of

*f*(

*x*) to zero for

*a*> 0 or to infinity for

*a*< 0. If

*a*= 0, the result is ambiguous. In this case

*f*(

*x*) has either a discontinuity or else a logarithmic singularity, that is,

*f*(

*x*) behaves like ln(

*T*−

*T*

_{cri}).

The order parameter, that is, Δ*ρ* along the saturation curve for *T* < *T*_{cri}, is described by

(16.26)

where *B* and *β* are parameters and (1 + …) is the background function to which the critical exponent *β* is insensitive. Experimentally, the range for *β* is 0.3 < *β* < 0.4.

The response function – that is, the compressibility *κ _{T}* – behaves similarly but becomes infinite at

*T*=

*T*

_{cri}. For

*T*<

*T*

_{cri}(for

*ρ*=

*ρ*

_{liq}or

*ρ*=

*ρ*

_{gas}on the saturation curve) and

*T*>

*T*

_{cri}(for

*ρ*=

*ρ*

_{cri}),

*κ*is described by, respectively,

_{T}(16.27)

where *C*, *C*′, *γ*, and *γ*′, are parameters and is the compressibility of an ideal gas with molecules of mass *m* at of density *ρ*_{cri} at *T*_{cri}. Experimentally, the range for *γ* and *γ*′ is 1.2 < *γ* < 1.4.

Such a power-law relation also holds for the heat capacity *C _{V}*. This is given in a two-phase system for

*T*<

*T*

_{cri}(for

*ρ*

_{liq}+

*ρ*

_{gas}= 2

*ρ*

_{cri}) and

*T*>

*T*

_{cri}(for

*ρ*=

*ρ*

_{cri}) by, respectively,

(16.28)

where *A*, *A*′, *α* and *α*′ are parameters. Experimentally, the range for *α* and *α*′ is −0.2 < *α* < 0.2.

For the pressure *P* along the critical isotherm *T* = *T*_{cri}, data are described by

(16.29)

where sgn(*I*) = 1 if *I* > 0 and sgn(*I*) = −1 if *I* < 0. Experimentally, we have 4 < *δ* < 5.

Finally, we examine the (total) pair correlation function *h*(*r*) = *g*^{(2)}(*r*) − 1 (see Section 7.2). For large *r*, *h *→ 0 and we expect, and also from Ornstein–Zernike theory it appears, that *h*(*r*) for *T *> *T*_{cri} behaves like

__(16.30)__

where *f*(*r*) is a weakly varying function__ ^{8)}__ of

*r*,

*D*is the dimension of space, and where the

*coherence*(or

*correlation*)

*length*

*ξ*measures the range of order in the liquid. In essence,

*ξ*is a measure of the greatest distance from a given particle at which this particle can still cause nonrandom effects on the particle density. Under normal conditions the order of magnitude for

*ξ*is a few molecular diameters, but for

*T*→

*T*

_{cri}and

*ρ*=

*ρ*

_{cri}, the value of

*ξ*diverges to macroscopic size. In this case, high- and low-density regions appear on an increasingly larger scale until macroscopic regions of different density in the fluid lead to the two-phase region for

*T*≤

*T*

_{cri}. However, the small(er) scale fluctuations do not disappear and within high-density regions there will be smaller regions of lower density, and vice versa. As soon as

*ξ*approaches the wavelength of visible light, say 0.5–0.8 μm, these fluctuations lead to the phenomenon of

*critical opalescence*. Using experimental data it appears that

*ξ*also scales with

*T*and for

*T*<

*T*

_{cri}(for

*ρ*=

*ρ*

_{liq}or

*ρ*=

*ρ*

_{gas}on the saturation curve) and

*T*>

*T*

_{cri}(for

*ρ*=

*ρ*

_{cri}) is given by, respectively,

(16.31)

Using the result from __Eq. (16.30)__, Fisher [4] showed that for *D* = 2, *h*(*r*) ∼ ln*r*, that is, increases with increasing *r*, which is physically impossible. He therefore introduced the additional exponent *η* (obeying 0 < *η* < 2) for *r *→ ∞ at (*ρ* = *ρ*_{cri}) and (*ρ* = *ρ*_{cri} and *T* = *T*_{cri}), respectively, by

(16.32)

It appears that in 3D the value of *η* is small, typically about 0.03. Moreover, *η* is related to *ν* and *γ* (__Example 16.1__).

Example 16.1: The Fisher relation

In __Chapter 6__ we derived for the compressibility

Using *h*(*r*) → *H _{D}*exp(−

*r*/ξ)/

*r*

^{D}^{−2+η}we obtain

with *x* = *r*/*ξ*. If *η* < 1, the integral converges and it has value 1 for *η* = 0. Because κ* _{T}* ∼

*t*

^{−γ}, and

*ξ ∼ t*

^{−ν}, we have

There appear to be more relations between the critical exponents. They were originally derived as (thermodynamic) inequalities between the various critical exponents, and usually named after their discoverer. We have for example,

__(16.33)__

__(16.34)__

__(16.35)__

Later, we will see that they actually are equalities and, moreover, that *α*′ = *α* and *γ*′ = *γ*. The derivation of some of them is easy, but for others it is quite involved. As an example we derive the Rushbrooke inequality. To do so we need *Stanley's lemma* that states that for *f*(*x*) ∼ *x ^{a}* and

*g*(

*x*) ∼

*x*with

^{b}*f*(

*x*) ≤

*g*(

*x*) and for 0 <

*x*< 1, so that ln

*x*< 0, we have

*a*≥

*b*. From thermodynamics, we know that

*C*−

_{P}*C*=

_{V}*TVα*

^{2}/

*κ*(all quantities positive, see Section 2.1), so that

_{T}*C*≥

_{P}*TVα*

^{2}/

*κ*. Because we have for the heat capacity

_{T}*C*∼ (−

_{P}*t*)

^{−α′}, for the compressibility

*κ*∼ (−

_{T}*t*)

^{−γ′}, and for the thermal expansion coefficient

*α*∼ (−

*t*)

^{β}^{−1}, we obtain at once −

*α*′ ≤

*γ*′ + 2(

*β*− 1) or

*α*′ + 2

*β*+

*γ*′ ≥ 2 (for 0 <

*α*′ < 1, while for

*α*′ ≤ 0, the most we can say is that 2

*β*+

*γ*′ ≥ 2).

Furthermore, it appears that the critical exponents for a number of phenomena are essentially the same (__Table 16.2__). This happens to be the case if, for two phenomena, both the spatial dimensionality, for example, 2D or 3D, and the dimensionality of the order parameter, for example, a scalar or vector, are the same. These phenomena are said to belong to the same *universality class*. For the phenomena listed in __Table 16.2__ the lattice gas model (see __Appendix C__), with a scalar order parameter, forms the prototype. The liquid–gas transition also belongs to this class (__Problem 16.9__). We limit ourselves entirely to this class, and some experimental results are given in __Table 16.3__.

** Table 16.2** Critical points and their order parameters.

^{a)}

** Table 16.3** Experimental values of critical exponents for various fluids.

**16.3.2 Mean Field Theory: Continuous Transitions**

Remember that in thermodynamics the equilibrium situation is conveniently obtained by minimizing the proper thermodynamic potential, say the “generalized” Helmholtz energy *F*(*η*;*T*,*V*), dependent on temperature *T*, volume *V*with respect to the order parameter (internal variable) *η*. We used the designation “generalized” because the “generalized” *F* only becomes the thermodynamic *F* after minimization with respect to *η*. In general, *η* varies with position, that is, is a field quantity, and we have to use the energy __density__. For a uniform distribution of *η* we can use the energy itself, but we will still use the energy density, here denoted by or by just or . The order parameter *η* can be a local average of an underlying microscopic parameter describing the physics of the system. For example, for a ferromagnetic material the order parameter is the magnetization *m* which in the lattice model (see __Appendix C__) results as the average of the magnetic spins *σ* with value ±1. Above *T*_{cri}, *m* = 0, while below *T*_{cri} the magnetization is ±*m*. The order parameter *η* can also be purely macroscopic. For example, for a fluid the order parameter is conveniently taken as the density difference Δ*ρ* between liquid and gas. In __Figure 16.8__ the qualitative shape of the *F*(*η*;*T*,*V*) curves as a function of temperature for various conditions of *V* and *T* for a fluid are given. The simplest approach to describe these curves is to use the so-called mean field approach, originally due to Landau.

** Figure 16.8** The shape of the Helmholtz energy curve

*F*. Upper row: Continuous transition along the coexistence curve where (from left to right) the images represent the shape changing to

*T*

_{cri}(mid-right) and above

*T*

_{cri}(outer-right); Lower row: Discontinuous transition across the coexistence curve where images (from left to right) represent the shape change from liquid via coexistence line (mid-right) to gas (outer-right).

In this approach, typically represents the grand potential density (see __Chapter 2__), and we assume that the Helmholtz part *F* (*η*) of near *T*_{cri} can be expanded in a power series in *η*, reading

(16.36)

The driving force *h* is the conjugate variable to the order parameter *η*. For a fluid the driving force is the chemical potential *μ* (with order parameter Δ*ρ*), while for a ferromagnetic material mentioned above it is the external magnetic field *H* (with order parameter *m*). In principle, all coefficients *a _{n}* depend on temperature

*T*. The parameter

*a*

_{0}is a reference energy and can be omitted. Equilibrium requires that , and in the absence of an external driving force, this requires that

*a*

_{1}is absent. Further note that

*a*

_{4}should be > 0 if the system is to be stable. In cases the order parameter

*η*can have only the two values ±

*η*, as for the magnetization case, the third-order terms must also be absent because should equal . For the fluid case this is obviously not the case if we use

*η*= Δ

*ρ*= (

*ρ*−

*ρ*

_{cri}). Nevertheless, we assume this to be true for the moment, that is, we essentially discuss the magnetic problem, and come back later to the case

*a*

_{3}≠ 0. For the moment we also consider the case when

*h*= 0. So, we are left with . As usual, equilibrium is reached when and the solutions are simply found to be

(16.37)

The coefficient *a*_{2} can be expanded with respect to temperature reading , where (generally ). Since for *T* < *T*_{cri}, *η* should be non-zero for *η* arbitrarily close to *T*_{cri}, we must have *a*_{2,0} = 0 and hence *a*_{2} = *a*_{2,1}Δ*T*. A similar expansion could be used for *a*_{4}, but as this will lead to second-order temperature effects, we take *a*_{4} as a constant. This implies that for *T* < *T*_{cri}, the solution is *η** = ±(−*a*_{2,1}Δ*T*/2*a*_{4})^{1/2}. To ease the notation slightly, we write

(16.38)

with *a* and *b* as the parameters, *t* as before, and omitting the asterisk from now on.

We now calculate the critical exponents. For *T* < *T*_{cri} the solution is *η* = (−*at*/2*b*)^{1/2}, so comparing with *η *∼ (*T*_{cri} − *T*)* ^{β}* results in

*β*= ½.

For *T *> *T*_{cri}, *η* = 0 and thus . For *T* < *T*_{cri}, *η* = (−*at*/2*b*)^{1/2} and thus . Because , we easily obtain

(16.39)

The heat capacity thus changes discontinuously at *T*_{cri}, which can be represented by taking the exponent *α* = 0.

To compute the other exponents we need to let *h* ≠ 0. Equilibrium is obtained from and, using , reads *h* = 2*atη *+ 4*bη*^{3}. On the critical isotherm *t* = 0 and therefore *h *∼ *η*^{3} so that *δ* = 3.

The response function, the isothermal susceptibility *X _{T}* in this case, is given by

(16.40)

where *η*(*h*) is a solution of *h* = 2*atη *+ 4*bη*^{3}. Since for *T *> *T*_{cri} we have *η* = 0, *X _{T}* = (2

*at*)

^{−1}. Similarly, for

*T*<

*T*

_{cri}at

*h*= 0, we have

*η*

^{2}= −

*at/b*,

*X*= (−4

_{T}*at*)

^{−1}, and thus

*γ*=

*γ*′ = 1. Note that

*X*(

_{T}*T*<

*T*

_{cri}) = −½

*X*(

_{T}*T*>

*T*

_{cri}) always in this model.

**16.3.3 Mean Field Theory: Discontinuous Transitions**

Let us consider briefly whether the Landau approach can be applied to discontinuous transitions. So far, we have used . We omitted a linear term because we required that *η* = 0 for *T *> *T*_{cri}, and a cubic term on the basis of symmetry. We now allow the cubic term so that we have

(16.41)

Considering only the case that *h* = 0, we obtain equilibrium from and the solutions are found to be

(16.42)

with *c* ≡ 3*c*′/8*b*. The solution for *T* < *T*_{cri} is acceptable (i.e., real) if *c*^{2} − 2*at*/*b* > 0, that is, for a temperature *t* < *t** ≡ *bc*^{2}/2*a*. Because *t** > 0, the temperature *T** > *T*_{cri}, the latter being the temperature where the coefficient of *η*^{2}vanishes. Recall that for the continuous transition an acceptable solution becomes available only for *t* < 0 or *T* > *T*_{cri}. __Figure 16.8__ (lower row) shows a schematic of the behavior for the discontinuous transition. For *t* > *t**, shows only one minimum. For *t* < *t** a secondary minimum appears which at *t* = *t*_{1} is equal to the one at *η* = 0. For *t* < *t*_{1} this minimum becomes the global minimum and the value of *η* jumps discontinuously to a nonzero value, representing the discontinuous transition. Note that *η*(*t*_{1}) is not arbitrarily small, as required by Landau theory. Generally – but not always – the presence of a cubic term in *η* will give rise to a discontinuous transition.

**16.3.4 Mean Field Theory: Fluid Transitions**

Let us now return to our case of the liquid–gas transition for which *a*_{1}, *a*_{3} ≠ 0. In this case, reads (using the chemical potential *μ* as the proper driving force)

__(16.43)__

For the general case a solution of __Eq. (16.43)__ is rather difficult to obtain. For the fluid case, however, we apply a trick by writing *η* = *η*_{0} + Δ*η*. If we do so we obtain

(16.44)

(16.45)

(16.46)

(16.47)

(16.48)

Each of the functions *B*(*η*_{0}), … , *E*(*η*_{0}) can be considered as a function of temperature, which we expand to first order in *T*. The function *A*(*η*_{0},*μ*) depends on *η*_{0} and *μ*, so that a first-order expansion in *T* and *μ* with respect to *T*_{cri}and *μ*_{cri} reads with Δ*T* = *T *− *T*_{cri} and Δ*μ* = *μ* − *μ*_{cri} using the notation . We can eliminate the linear and cubic terms in Δ*η* by setting them equal to zero. First, solving the cubic term *D*(*η*_{0}) for *η*_{0} leads to *η*_{0} = −*a*_{3}/4*a*_{4} = −*a*_{3,1}Δ*T*/4*a*_{4}. Substitution in the linear term *B*(*η*_{0}) results, to first order in Δ*T*, in

(16.49)

where the last step can be made because Δ*Tη*_{0} is of second order in Δ*T*. Since *B*(*η*_{0}) = 0 requires Δ*μ* = *a*_{1,1}Δ*T*, we see that this condition is already satisfied. Therefore, along the line determined by *η*_{0} = −*a*_{3,1}Δ*T*/4*a*_{4}, the Landau function will read

(16.50)

which has the same form as for the magnetic problem. The solution accordingly is also the same and reads for *T* < *T*_{cri}

(16.51)

All critical exponents are thus also the same as before. The complete solution is obtained by superposing the Δ*η* solution on the *η*_{0} behavior. To interpret *η*_{0} we recall that *η* = *η*_{0} + Δ*η* = *ρ* − *ρ*_{cri}. Using *η*_{liq} = *η*_{0} + Δ*η* and *η*_{gas} = *η*_{0} − Δ*η*, we see that *ρ*_{liq} + *ρ*_{gas} = 2*η*_{0}, so that *η*_{0} equals the average density and represents the *law of rectilinear diameters* (see __Chapter 4__). Because *η*_{0} = −*a*_{2,1}Δ*T*/4*a*_{4}, the average density is a linear function of temperature, as approximately confirmed by experimental evidence. However, expanding all functions to second order leads to nonlinear behavior [6].

In conclusion to this part, we note that the order of magnitude of the critical exponents as obtained from the mean field approach is correct, but that quantitative agreement is missing. The Landau expression is approximate and we neglected fluctuations, and further steps are therefore required. This includes scaling, using the idea of generalized homogeneous functions applied to thermodynamic potentials, from which it becomes clear that the thermodynamic inequalities become equalities, and renormalization, a theory that states that phenomena behave the same at different length scales near the critical point and provides accurate numerical values for the critical exponents.

Problem 16.6

Show that *f*(*x*) ∼ *x ^{a}* ≤

*g*(

*x*) ∼

*x*for 0 <

^{b}*x*< 1 and

*a*≥

*b*.

Problem 16.7

Show, by evaluating d(*F*/*V*) and using the Euler expression *F* = *G *− *PV*, that the differential of the Helmholtz density reads *μ*d*ρ* − *s*d*T*, with *s* = *S*/*V*.

Problem 16.8

Verify that the inequalities for the critical exponents as given by __Eqs (16.33)–(16.35)__ are satisfied as equalities for the mean field critical exponent values.

Problem 16.9: The van der Waals critical exponents

The vdW expression, [*P *+ *a/V*^{2}](*V *− *b*) = *NkT*, rewritten as polynomial in *V* reads *V*^{3} − [*b *+ (*NkT*/*P*)]*V*^{2} + (*a*/*P*)*V* − (*ab*/*P*) = 0.

**a)** Comparing with the expression , show that *T*_{cri} = 8*a*/27*Nkb*, *P*_{cri} = *a*/27*b*^{2}, and *V*_{cri} = 3*b*.

**b)** Show that *P*_{cri}*V*_{cri}/*RT*_{cri} = 3/8.

**c)** Defining *T*_{red} ≡ *T*/*T*_{cri}, *P*_{red} ≡ *P*/*P*_{cri} and *V*_{red} ≡ *V*/*V*_{cri}, show that vdW expression reduces to .

**d)** Further defining *ω* ≡ (*ρ* − *ρ*_{cri})/*ρ*_{cri}, *p* ≡ (*P *− *P*_{cri})/*P*_{cri} and *t* ≡ (*T *− *T*_{cri})/*T*_{cri}, show that the vdW expression reads *p*(2 − *ω*) = 8*t*(1 + *ω*) + 3*ω*^{3}.

**e)** To calculate the exponent *β*, consider the behavior near the critical point. Show that the vdW expression reduces to *ω*(3*ω*^{2} + 12*t*) = 0, using that for *ω* = 0, *p *= 4*t*.

**f)** Show that this solution leads to *ω* = ±2(−*t*)^{1/2} = ±(1 − *T*_{red})^{1/2} or *β* = ½.

**g)** To calculate the exponent *δ*, consider the behavior at *T*_{cri}, that is, *t* = 0. Show that the vdW expression becomes *p*(2 − *ω*) = 3*ω*^{3}.

**h)** Show that this solution leads to *p* ≅ (3*ω*^{3}/2)[1 + (*ω*/2) + …]) or *δ* = 3.

**i)** To calculate the critical exponent *γ*, show that the compressibility *κ _{T}* ∼ (∂

*p*/∂

*ω*)

*is given by*

_{T}*κ*∼ (24

_{T}*t*+ 18

*ω*

^{2}− 6

*ω*

^{3})/(2 −

*ω*)

^{2}. Show that this solution, for

*t*> 0, leads to

*κ*∼ 6

_{T}*t*or

*γ*= 1.