Phase Transformations

A phase transformation (or phase transition) is a discontinuous change in the properties of a substance as its environment is changed only infinitesimally: mathematically, a derivative that diverges or fails to exist. The different forms, such as ice, water, and steam, are its phases, and the variables that select among them, such as temperature and pressure, are control parameters. Steam condenses either by lowering the temperature or by raising the pressure. A phase diagram shows the most stable phase in each region (metastable phases can still exist for a while); its lines are where two phases coexist in equilibrium, and at the triple point all three do. The liquid–gas line is the vapor pressure, ending at the critical point, beyond which liquid and gas are no longer distinct. Pick a temperature and pressure:

Water’s melting line slopes backward: raising the pressure lowers the melting point, because ice is less dense than water. Most substances are like carbon dioxide instead, where pressure raises the melting point; at atmospheric pressure CO₂ has no liquid at all, and solid “dry ice” sublimates directly. Helium is stranger still: ⁴He stays liquid down to absolute zero unless compressed to about 25 bar, and below 2.2 K it becomes a superfluid.

Elemental carbon has two familiar solid phases. At ordinary pressure graphite is more stable, so diamonds slowly (very slowly) turn into graphite. Since \((\partial G/\partial P)_{T,N} = V\), the denser diamond’s \(G\) rises more slowly with pressure, and above about 15 kbar diamond wins; since \((\partial G/\partial T)_{P,N} = -S\), graphite’s larger entropy pushes that crossover higher as the temperature rises.

Transitions were long classified by their abruptness. Melting and boiling are “first-order,” since the first derivatives of \(G\), namely \(S\) and \(V\), jump at the boundary. Gentler transitions, such as at a critical point or helium’s superfluid transition, were called “second-order” and so on, by how many derivatives it takes to reach a discontinuity. Today they are simply called continuous. Magnetic systems show both: in a type-I superconductor, a field above \(B_c\) destroys superconductivity (for lead, \(T_c = 7.2\) K and \(B_c = 0.08\) T); in a ferromagnet, flipping the field across zero flips the magnetization, a first-order jump, until the Curie temperature, a critical point above which it becomes continuous.

On a phase boundary, the liquid and gas are in diffusive equilibrium, so their Gibbs free energies (per molecule, their chemical potentials) are equal, and must stay equal as you move along it: \(dG_l = dG_g\), or \(-S_l\,dT + V_l\,dP = -S_g\,dT + V_g\,dP\). With \(S_g - S_l = L/T\), the latent heat,

A large entropy change means a small temperature change shifts the equilibrium a lot: a steep curve. A large volume change makes it shallow. For water at 100 °C, \(dP/dT \approx 3.6\) kPa/K; for melting ice, \(\Delta V < 0\) and \(dP/dT \approx -135\) bar/K, the backward-sloping line above. Neglecting the condensed phase’s volume, treating the vapor as an ideal gas, and taking \(L\) constant gives the vapor pressure equation (Problem 5.35), for one mole:

Test it on water’s measured vapor pressures: on a plot of \(\ln P\) against \(1/T\) it is a straight line of slope \(-L/R\).

For a magnetic system at constant \(T\) and external field \(\mathcal{H}\), the quantity minimized is \(G_m\) with \(dG_m = -S\,dT - \mu_0M\,d\mathcal{H}\), giving the analogue (Problem 5.47)

For the ferromagnet, the boundary is flat because up and down have equal entropy by symmetry. For the superconductor, which expels the field (the Meissner effect, \(M_{\text{II}} \propto -\mathcal{H} < 0\)), the negative slope means the superconducting phase always has less entropy than the normal phase, except at \(T = 0\), where both vanish by the third law and the boundary is flat.

An ideal gas never changes phase. The van der Waals equation adds molecular interactions approximately: molecules repel at contact, so each takes up volume \(b\); and they attract when close but not touching, giving a potential energy \(-aN^2/V\) and an extra (negative) pressure \(-aN^2/V^2\):

Below a certain temperature its isotherms develop a wiggle where compressing the fluid would lower its pressure, which is unstable. The true equilibrium is decided by the Gibbs free energy: from \((\partial G/\partial V)_{T} = V(\partial P/\partial V)_T\),

For pressures where the isotherm gives three volumes, \(G(P)\) has three branches forming a triangular loop, and the system takes the lowest: gas at low pressure, liquid at high, jumping between them where the branches cross. Going around the loop, \(\oint dG = \oint V\,dP = 0\), so the transition pressure is where the two shaded areas are equal: the Maxwell construction. In reduced variables \(t = T/T_c\), \(p = P/P_c\), \(v = V/V_c\), the constants disappear (Problem 5.51):

where the critical point, at which liquid and gas become identical, is where \(\partial P/\partial V = \partial^2P/\partial V^2 = 0\) (Problem 5.48).

Near the critical point, expanding the reduced equation about \(v = 1\) gives (Problem 5.55)

From it, the boundary’s slope at the critical point is \(dP/dT = 4P_c/T_c\), and the liquid–gas volume difference shrinks as \(V_g - V_l \propto (T_c - T)^{\beta}\) with \(\beta = \tfrac12\) (experiments find a universal value of about \(\tfrac13\)), so the latent heat vanishes as \(L \approx 6Nk_{\mathrm{B}}T_c\sqrt{1 - T/T_c}\). The critical isotherm is \(P - P_c \propto (V - V_c)^{\delta}\) with \(\delta = 3\) (experiments: about 4 or 5), and the compressibility diverges as \((T - T_c)^{-\gamma}\) from above and \((T_c - T)^{-\gamma'}\) from below with \(\gamma = \gamma' = 1\). These critical exponents express a general pattern near continuous transitions:

with \(x\) a control parameter and \(x_c\) its value at the critical point.

where the order parameter may be an energy, volume, magnetization, susceptibility, or heat capacity, and the control parameter a temperature, pressure, or field. A power law \(y \propto x^\alpha\) is scale-invariant: rescaling \(x \to ax\) only rescales \(y\). On a log–log plot it is a straight line, so you can measure the exponent from the van der Waals model directly:

Why real exponents are universal, the same for fluids and magnets with very different microscopic physics, was explained by the renormalization group, building on Leo Kadanoff’s block spins; Kenneth G. Wilson received the 1982 Nobel Prize in Physics “for his theory for critical phenomena in connection with phase transitions.” Phase transitions have become a model for complex systems in general: global behavior that is similar despite very different local interactions. But a word of caution: a straight line on a log–log plot is not proof of a power law, and many reported ones lack statistical support (M. P. H. Stumpf and M. A. Porter, “Critical Truths About Power Laws,” Science 335, 665 (2012)).

This concludes Statistical Physics 1. Starting from temperature and energy, the course counted microstates, defined entropy as the logarithm of their number, and found that temperature, pressure, and chemical potential are all derivatives of it. Engines met their Carnot limit, the free energies took over when systems touch their surroundings, and here the same free energies decided between ice, water, and steam. Along the way, three model systems, the two-state paramagnet, the Einstein solid, and the ideal gas, showed how the laws of microscopic physics turn into measurable thermodynamics, and the van der Waals fluid showed how even a crude model of interactions can produce a phase transformation and a critical point.