Chapter 4: Critical phenomena in fluidsvan der Waals, Maxwell, and the milky glow of a critical fluid

4.1Thermodynamics and phase diagrams

Thermodynamic potentials

Consider a system \(S\) exchanging energy \(\Delta E\) and particles \(\Delta N\) with a reservoir \(R\) at temperature \(T\) and chemical potential \(\mu\): the grand canonical ensemble. The thermodynamic potentials are the energy \(E\), the Helmholtz free energy \(F = E-TS\), the Gibbs free energy \(G = E-TS+pV\), and the enthalpy \(H = E+pV\). They are functions of different combinations of the entropy \(S\), volume \(V\), particle number \(N\), chemical potential \(\mu\), pressure \(p\), and temperature \(T\):

From \(dS = \frac1T(dE + p\,dV - \mu\,dN)\), which defines \(T\), \(p\), and \(\mu\),

Phase diagrams and the Clausius–Clapeyron relation

The Gibbs free energy is minimized in equilibrium with a reservoir that imposes \(T\) and \(p\). The coexistence line between two phases I and II is the locus where \(G_{\rm I} = G_{\rm II}\), i.e. \(\mu_{\rm I} = \mu_{\rm II}\), since \(\mu = (\partial G/\partial N)_{T,p}\) and so \(G = N\mu\): the chemical potential is the Gibbs free energy per particle.

Take the liquid–solid coexistence line in the \(p\)–\(T\) plane, \(G_l(T,p,N) = G_s(T,p,N)\), and move along it by \(T\to T+\delta T\), \(p\to p+\delta p\). Expanding to first order, with \(\partial G/\partial T = -S\) and \(\partial G/\partial p = V\),

Typically \(S_l>S_s\) and \(V_l>V_s\) at fixed particle number, so the coexistence line has positive slope. Exceptions: water, where \(V_l\lt V_s\) (ice floats) and the melting line has negative slope, and \(^3\)He, where \(S_l\lt S_s\) at low temperature. Since \(S_l\ne S_s\), latent heat is released at this first-order (discontinuous) transition. At a critical point, \(S_l\to S_g\) and \(V_g\to V_l\), and the latent heat becomes vanishingly small.

Landau's symmetry principle

Two phases with different symmetry must be separated by a line of transitions, because a symmetry cannot change continuously: it is either present or absent. A liquid and a solid are therefore believed always to be separated by a line of transitions, while the liquid–gas line can end at a critical point because there is no symmetry difference between liquid and gas. The principle does not say how the transition occurs: the liquid–solid line could be first or second order, or both, depending on pressure, and in rubber the transition to the solid state at fixed temperature, as cross-links increase, is thought to be continuous. Although a symmetry is either present or absent, it can emerge either continuously or discontinuously as the coupling constants change.

Two-phase coexistence

Heat a liquid at constant pressure. It expands, then begins to boil: the temperature stays constant (latent heat), and so does the pressure, while the volume increases as the gas fraction grows from 0 to 1. Once all the liquid is gas, further heating expands the gas. Now follow an isotherm instead. Mechanical stability requires \(\partial p/\partial V\lt 0\), i.e. a positive compressibility \(\kappa_T\equiv -\frac1V\big(\frac{\partial V}{\partial p}\big)_T > 0\), so that the system pushes back when compressed. Compressing a gas below \(T_c\), droplets of liquid form at constant pressure until the system is all liquid, after which the pressure rises again. The behavior is non-analytic at the boundaries of the two-phase region.

Model equations of state such as van der Waals's are analytic all the way through the two-phase region: the approximation behind them does not ensure that the equilibrium state globally minimizes the Gibbs free energy. The fix uses coexistence, \(\mu_l = \mu_g\), at constant \(T\) and \(p\). From \(G = F + pV = \mu N\), \(dG = \mu\,dN + N\,d\mu\), and comparing with \(dG = -S\,dT + V\,dp + \mu\,dN\) gives \(d\mu = -\frac SN\,dT + \frac VN\,dp\). Along an isotherm (\(dT=0\)),

the two areas enclosed between the horizontal coexistence isotherm and the analytic continuation of \(p(V)\) must be equal. This is Maxwell's equal-area construction.

4.2The van der Waals equation

With \(v = V/N\) the volume per particle, the ideal gas has \(pv = k_BT\). Van der Waals took into account the hard core of the atoms, which reduces the available volume by an amount \(b\) (the excluded volume), and their attraction, which lowers the energy per particle by an amount proportional to the density and hence, via \(p = -\partial F/\partial V\), lowers the pressure by an amount \(\propto V^{-2}\):

where \(a>0\) measures the attraction and \(b\) the hard-core volume; both can be fitted to experiment. As \(T\to T_c\), \(p(v)\) develops an inflection point with zero slope and zero curvature, \(\partial p/\partial v = \partial^2p/\partial v^2 = 0\). Rearranging,

a cubic with one real root for \(T>T_c\) and three for \(T\lt T_c\). At \(T_c\) the three merge, so the cubic must equal \((v-v_c)^3 = v^3 - 3v_cv^2 + 3v_c^2v - v_c^3\). Matching coefficients,

The last ratio is a universal number, independent of \(a\) and \(b\) and thus the same for all fluids. Experimentally it is 0.292 for argon, 0.23 for water, and 0.31 for \(^4\)He: reasonable, in that the number does not vary by orders of magnitude or even by factors of 2.

The law of corresponding states

Rescale by the critical values, \(\pi\equiv p/p_c\), \(\nu\equiv v/v_c\), \(\tau\equiv T/T_c\). Substituting \(p_c = a/27b^2\), \(v_c = 3b\), \(k_BT_c = 8a/27b\),

When scaled by \(p_c\), \(v_c\), \(T_c\), all fluids are predicted to have the same equation of state, with no other parameters: the law of corresponding states. All thermodynamic properties that follow from the equation of state should then be universal too. This is a form of universality, but quite different from universality at the critical point: it applies everywhere in the phase diagram. It is in fact a simple consequence of dimensional analysis and more general than van der Waals suggests. If the pair potential has the form \(U(r) = \epsilon\,u(r/\sigma)\), as for the Lennard-Jones potential of the inert gases, then all such fluids share one equation of state \(p^* = \Pi(v^*,T^*)\) in units set by \(\epsilon\) and \(\sigma\), and \(p_cv_c/k_BT_c\) is a constant (Exercise 4-1). Experimentally the law holds well, even for fluids that do not obey the van der Waals equation: \(pv/k_BT\) against reduced pressure at fixed reduced temperature falls on one curve for a wide variety of fluids.

Critical behavior

Since \(p = -(\partial F/\partial V)_T\), integrating the equation of state gives

where at fixed \(N\) the function \(f(T)\) absorbs every term independent of \(V\), including \(-Nk_BT\log N\) and the thermal-wavelength factors of the ideal gas. With \(a=b=0\), \(-F_{\rm ideal} = Nk_BT\log V + f(T)\), so

The entropy difference is independent of \(T\), so \(C_V = T\,\partial S/\partial T\) is that of the ideal gas, \(C_V^{\rm vdW} = \tfrac32Nk_B\) (in three dimensions, from \(U^{\rm ideal} = \tfrac32Nk_BT\)). \(C_V\) does not diverge at the critical point: \(\alpha = 0\). On the other hand, the standard identity

gives \(C_p - C_V = Nk_B\big/\big[1 - 2a(v-b)^2/(v^3k_BT)\big]\). At \(v = v_c = 3b\) and \(k_BT_c = 8a/27b\), the bracket is \(1 - T_c/T\):

So \(C_p\ge C_V\) in equilibrium, because \(\kappa_T\ge0\), and \(C_p\) diverges in the same way as \(\kappa_T\): \(\gamma = 1\). The divergence of \(\kappa_T\) has a direct physical meaning. It is a response function, describing how the volume changes under applied pressure, and its divergence means the system becomes extraordinarily sensitive to pressure near the critical point. This is reasonable: for \(T\) slightly below \(T_c\) the system is unstable toward phase separation, so the critical point is a state of marginal stability.

Near the critical point, write \(t\equiv\tau-1 = (T-T_c)/T_c\) and \(\phi\equiv\nu-1 = (V-V_c)/V_c\). Expanding the reduced equation of state,

(The expansion assumes \(\phi\sim t^{1/2}\), justified after the fact.) Apply Maxwell's construction at fixed \(t\lt 0\): with \(d\pi = (-6t - \tfrac92\phi^2)\,d\phi\) and \(v = v_c(1+\phi)\),

using \(\int_l^g dp = p_g - p_l = 0\). For this to hold for all \(t\), \(\int\phi(-6t-\tfrac92\phi^2)\,d\phi\) must vanish, so \(\phi_l^2 = \phi_g^2\) and \(\phi_l = -\phi_g\). Equal pressures, \(\pi_l = \pi_g\), then give \(12t\phi_g + 3\phi_g^3 = 0\):

The exponents \(\alpha\), \(\beta\), \(\gamma\), and \(\delta\) are the same as those of the mean-field theory of a ferromagnet (Table 3.1). The van der Waals equation is a mean-field theory.

4.3Vicinity of the critical point

The width in \(V\) of the two-phase region, for fixed particle number and temperature, is the difference between the equilibrium volumes of liquid and gas, and it vanishes as \(T\to T_c^-\). Well above \(T_c\) the isotherms are close to the ideal gas law; as \(T\to T_c^+\) they flatten, with a horizontal tangent at \((p_c, V_c, T_c)\), so the compressibility diverges. We will see that it is the analog of the magnetic susceptibility. The critical exponents of a fluid are defined by

with \(v_{l(g)} = V_{l(g)}/N\), and primes denoting \(T\lt T_c\). The width of the coexistence curve plays the role of the spontaneous magnetization.

4.4Spatial correlations

In Chapter 3 we related the two-point correlation function to the isothermal susceptibility through the static susceptibility sum rule. For a fluid, the two-point function describes statistical fluctuations in density, so we start by relating fluctuations in the number of particles in a fixed volume to the compressibility.

Embed a volume \(V\) in a much larger system \(\Omega\) and use the grand canonical ensemble, \(\Xi = \operatorname{Tr}\,e^{-\beta(\mathcal H-\mu N)}\). Then

To express this in measurable quantities, use Jacobians, \(\partial(u,v)/\partial(x,y)\equiv\det\begin{pmatrix}\partial u/\partial x & \partial u/\partial y\\ \partial v/\partial x & \partial v/\partial y\end{pmatrix}\), all at fixed \(T\):

Since \(G(T,p,N) = \mu(T,p)N\), \(\partial\mu/\partial N|_p = 0\) and the first term vanishes. The Maxwell relation from \(dG\), \(\partial\mu/\partial p|_N = \partial V/\partial N|_p = 1/\rho\), involves the particle density \(\rho(T,p) = \partial N/\partial V|_p\). With \(\kappa_T = -V^{-1}\partial V/\partial p\),

Now define the dimensionless two-point function of the density,

which measures fluctuations about the mean density \(\langle\rho(\vec r)\rangle = \rho\) and vanishes as \(|\vec r-\vec r'|\to\infty\). With \(N = \int\rho(\vec r)\,d^dr\) in the volume \(V\) and translational invariance,

the compressibility sum rule, the fluid version of the susceptibility sum rule. Well away from the critical regime \(G(\vec r)\sim e^{-|\vec r|/\xi}\big/|\vec r|^{(d-1)/2}\xi^{(d-3)/2}\), so as in Chapter 3, a diverging compressibility implies a diverging correlation length, with \(\nu = \tfrac12\) in mean-field theory.

Critical opalescence

Density fluctuations scatter light. For light of wavelength \(\lambda\) scattered through an angle \(\theta\), the intensity is proportional to the structure factor

Fourier transforming the form of \(G\) above gives the Ornstein–Zernike result

Measuring \(I(k)\) is an accurate way to determine \(\xi(T)\), and at the critical point it gives \(\eta\). The divergence of \(\kappa_T\) appears as the divergence of \(\int d^dr\,G(\vec r) \propto S(0)\), so as \(T\to T_c\) scattering grows dramatically at small angles. When correlated density fluctuations reach \(\xi\sim\lambda_{\rm light}\), the light is strongly scattered, multiple scattering becomes important, and the fluid turns milky and opaque: critical opalescence.

4.5Measuring critical exponents

Critical exponents are only defined as limiting power laws as \(t\to0\): \(f(t)\sim t^\lambda\) means

An exponent quoted as zero may mean a discontinuity at \(T_c\) or a logarithmic divergence, since \(\log t = \lim_{\alpha\to0}\big[(1-t^{-\alpha})/\alpha\big]\) up to sign. In general an exponent describes only the leading behavior. There are usually subdominant corrections to scaling; for the heat capacity,

The correction vanishes as \(|t|\to0\) but can matter a great deal at small nonzero \(|t|\). When \(\theta\lt 1\) the correction is itself singular, a confluent singularity; such terms are important near the superfluid transition of \(^4\)He, and their origin is explained by the RG in Chapter 9. There is usually also a smoothly varying background that is not singular at \(T_c\). In the van der Waals gas, for example, the free energy has an ideal-gas part and an interaction part; the former is always smooth, and the latter gives the critical behavior, as in \(C_p = C_V + Nk_BT/(T-T_c)\). Besides exponents there are amplitudes: \(C_V = A t^{-\alpha}\) for \(t>0\) and \(A'(-t)^{-\alpha'}\) for \(t\lt 0\). The exponents satisfy \(\alpha = \alpha'\) and are universal; the critical amplitudes \(A\), \(A'\) are not, but the amplitude ratio \(A/A'\) is.

It is notoriously difficult to measure critical exponents. In a theorist's dream, one would measure \(C_V(T)\), plot \(\log C_V\) against \(\log t\), and read \(\alpha\) off the slope. In practice:

• \(C_V\) is measured as \(\Delta E/\Delta T\), which approximates the heat capacity only for \(\Delta T\ll|T-T_c|\), while the thermometer's resolution \(\delta T\) requires \(\Delta T>\delta T\) and hence \(|t|\gg\delta T/T_c\). Since we want the limit \(t\to0\), very high resolution thermometry is needed.
• The analytic background must be subtracted, which requires modeling it. Eliminating it fully means making \(t\) small enough that the divergence dominates.
• Close to \(T_c\), instrumental resolution, impurities, and the finite size of the system round off the divergence. Away from \(T_c\), corrections to scaling and the background matter. One is lucky to get two decades of convincing power-law behavior.
• \(T_c\) is not known a priori; it is an adjustable parameter, along with the background and correction terms.
• As \(T\to T_c\) the system takes longer and longer to equilibrate: critical slowing down. Larger and larger fluctuating regions relax by slower and slower mechanisms (often diffusion). The relaxation time of the zero-wavenumber mode diverges as \(\tau_0\sim\xi(T)^z\), defining the dynamic critical exponent \(z\), between about 2 and 4 or 5 depending on the dynamic universality class (Chapter 8).

Further reading

• N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (Addison-Wesley, 1992), Chapter 4.
• E. A. Guggenheim, "The principle of corresponding states," J. Chem. Phys. 13, 253 (1945).
• D. Beysens, in Phase Transitions, Proceedings of the Cargèse Summer School 1980 (Plenum, 1982), p. 25.
• G. Ahlers, in Phase Transitions, Proceedings of the Cargèse Summer School 1980 (Plenum, 1982), p. 1.
• H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, 1971).
• B. Smit and D. Frenkel, "Vapor–liquid equilibria of the two-dimensional Lennard-Jones fluid(s)," J. Chem. Phys. 94, 5663 (1991).