Phase Transitions

Phenomenon

Phase Transitions and Critical Phenomena — Equilibrium, Interfaces, and Fluctuations

Scope: rigorous first-principles derivation of phase transitions, coexistence, and critical behavior. Integrates classical thermodynamics, statistical mechanics, and continuum descriptions of interfacial phenomena.

1. Classification of Phase Transitions

1.1 Ehrenfest Classification

Phase transitions are characterized by discontinuities in derivatives of the Gibbs free energy:

OrderDiscontinuityExample
First-orderFirst derivative ((\partial G/\partial T = -S), (\partial G/\partial P = V))Melting, boiling, sublimation
Second-orderSecond derivative (heat capacity, compressibility)Superconductivity, ferromagnetic Curie point

First-order transitions: latent heat present, coexistence of phases.

Second-order transitions: continuous first derivatives, but discontinuous or divergent second derivatives.

2. Fundamental Relation for Coexistence: Gibbs Criteria

For two phases α\alpha and β\beta of a single substance at equilibrium: T(α)=T(β),P(α)=P(β),μ(α)=μ(β).T^{(\alpha)} = T^{(\beta)}, \quad P^{(\alpha)} = P^{(\beta)}, \quad \mu^{(\alpha)} = \mu^{(\beta)}.

Differentiating the equality of chemical potentials: dμ=−s dT+v dP⇒(vβ−vα)dP=(sβ−sα)dT.d\mu = -s\,dT + v\,dP \Rightarrow (v_\beta - v_\alpha) dP = (s_\beta - s_\alpha) dT.

This gives the Clapeyron equation: dPdT=ΔsΔv=hfgT vfg.\boxed{\frac{dP}{dT} = \frac{\Delta s}{\Delta v} = \frac{h_{fg}}{T\,v_{fg}}.}

For vaporization where vg≫vlv_g \gg v_l and vapor behaves ideally: dln⁡PsatdT=hfgRvT2.\frac{d\ln P_{sat}}{dT} = \frac{h_{fg}}{R_v T^2}. This is the Clausius–Clapeyron equation.

3. Gibbs Free Energy Surfaces and Phase Coexistence

For a pure substance, G(T,P)G(T,P) is continuous and convex. At a fixed T:

  • Stable phase minimizes G.
  • At coexistence: Gl=GvG_l = G_v.
  • The slope of G–P curve: (partialG/partialP)T=V.(\\partial G/\\partial P)_T = V.

Coexistence line corresponds to locus of equal Gibbs free energies.

4. Phase Rule and Coexistence Regions

Gibbs Phase Rule: F=C−P+2.F = C - P + 2. For single component (C = 1):

  • Two-phase region: F=1F = 1 (one degree of freedom ⇒ line in T–P plane).
  • Three-phase (triple point): F=0F = 0 ⇒ fixed T and P.

5. Critical Point and Equations of State

The critical point is defined where liquid and vapor become indistinguishable: \left( rac{\partial P}{\partial v}\right)_T = 0, \quad \left( rac{\partial^2 P}{\partial v^2}\right)_T = 0.

For van der Waals equation: (P+av2)(v−b)=RT.\left(P + \frac{a}{v^2}\right)(v - b) = RT. Solving the above conditions gives: vc=3b,Pc=a27b2,Tc=8a27Rb.v_c = 3b, \quad P_c = \frac{a}{27b^2}, \quad T_c = \frac{8a}{27Rb}.

Reduced form: Pr=8Tr3vr−1−3vr2.P_r = \frac{8T_r}{3v_r - 1} - \frac{3}{v_r^2}. This predicts the existence of a critical isotherm (inflection point) and smooth transition.

6. Order Parameter and Landau Theory

Near the critical point, distinguishable phases differ by an order parameter (e.g., density difference rhol−rhov\\rho_l - \\rho_v).

6.1 Landau Expansion

Gibbs potential expanded in powers of order parameter m: G=G0+a(T−Tc)m2+bm4+…G = G_0 + a(T - T_c)m^2 + b m^4 + \dots

Minimizing GG: ∂G∂m=2a(T−Tc)m+4bm3=0.\frac{\partial G}{\partial m} = 2a(T - T_c)m + 4bm^3 = 0. Solutions:

  • m=0m = 0 (T>TcT > T_c, single phase)
  • m2=−a2b(T−Tc)m^2 = -\frac{a}{2b}(T - T_c) for T<TcT < T_c (two coexisting phases)

Thus, m∼(Tc−T)1/2m \sim (T_c - T)^{1/2}, yielding critical exponent beta=1/2\\beta = 1/2 (mean-field result).

7. Critical Exponents and Scaling Laws

Empirical power laws near TcT_c:

QuantityBehaviorExponent
Order parameter (density difference)$m \simT - T_c
Isothermal compressibility$κ_T \simT - T_c
Heat capacity$C_V \simT - T_c
Surface tension$σ \simT - T_c
Correlation length$ξ \simT - T_c

Scaling relations connect them (Rushbrooke, Widom, etc.): α+2beta+γ=2,γ=β(delta−1).\alpha + 2\\beta + \gamma = 2, \quad \gamma = \beta(\\delta - 1).

These exponents are universal—they depend only on dimensionality and symmetry, not microscopic details.

8. Statistical-Mechanical View of Phase Transitions

The canonical partition function: Z=∑ie−Ei/(kBT).Z = \sum_i e^{-E_i/(k_B T)}.

For large N, phase transitions correspond to nonanalyticities in G=−kBTln⁡ZG = -k_B T \ln Z.

8.1 Mean-Field Approximation

Interactions replaced by average field; yields critical behavior similar to van der Waals and Landau theories.

8.2 Fluctuations and Correlation Functions

Two-point correlation function: g(r)=⟨δrho(0)δrho(r)⟩.g(r) = \langle \delta\\rho(0) \delta\\rho(r) \rangle.

Correlation length xi\\xi diverges near critical point: xi∝∣T−Tc∣−nu\\xi \propto |T - T_c|^{-\\nu}. Divergence implies long-range fluctuations and critical opalescence.

9. Real-Fluid Behavior and Experimental Correlations

9.1 Reduced Properties and Law of Corresponding States

Fluids with similar reduced variables Pr,Tr,vrP_r, T_r, v_r follow approximately universal relations: Z=f(Tr,Pr).Z = f(T_r, P_r).

9.2 Benedict–Webb–Rubin and Extended EoS

Empirical refinements of van der Waals capture real critical behavior: P=RTρ+(B0RT−A0−C0/T)ρ2+…P = RTρ + (B_0RT - A_0 - C_0/T)ρ^2 + \dots

9.3 Experimental Observations

Near TcT_c, sharp anomalies occur in density, compressibility, and light scattering. Opalescence arises from large density fluctuations.

10. Interfacial Thermodynamics

10.1 Surface Tension

Define excess Helmholtz free energy per unit area: σ=(frac∂A∂As)T,V,n.\sigma = \left(\\frac{∂A}{∂A_s}\right)_{T,V,n}. Mechanical definition from normal and tangential stress components: σ=∫−inftyinfty(Pn−Pt) dz.\sigma = \int_{-\\infty}^{\\infty} (P_n - P_t)\,dz.

10.2 Gibbs Adsorption Equation

For interface with species i: dσ=−∑iΓidμi,dσ = -\sum_i \Gamma_i dμ_i, where Gammai\\Gamma_i is surface excess concentration. Adsorption modifies surface tension.

11. Nucleation and Phase Transformation Kinetics

11.1 Homogeneous Nucleation

Formation of new phase nucleus of radius rr within metastable phase: ΔG(r)=4πr2σ+43πr3Δgv.\Delta G(r) = 4πr^2σ + \frac{4}{3}πr^3Δg_v. Critical radius where dΔG/dr=0dΔG/dr = 0: r∗=−2σΔgv.r^* = -\frac{2σ}{Δg_v}.

Critical nucleation barrier: ΔG∗=16πσ33(Δgv)2.ΔG^* = \frac{16πσ^3}{3(Δg_v)^2}.

Nucleation rate: J=J0exp⁡(−ΔG∗/kBT).J = J_0 \exp(-ΔG^*/k_B T).

11.2 Heterogeneous Nucleation

Barrier reduced by geometric factor depending on contact angle theta\\theta: f(θ)=(2+cos⁡θ)(1−cos⁡θ)24.f(θ) = \frac{(2 + \cos θ)(1 - \cos θ)^2}{4}. ΔGhet∗=f(θ)ΔGhom∗.ΔG^*_{het} = f(θ) ΔG^*_{hom}.

12. Spinodal Decomposition

Within miscibility gap, second derivative of Gibbs energy negative: \left( rac{\partial^2 G}{\partial x^2}\right)_{T,P} < 0. Small fluctuations grow spontaneously without barrier. Described by Cahn–Hilliard equation: ∂x∂t=∇⋅(M∇μ),μ=δG/δx.\frac{\partial x}{\partial t} = ∇·(M ∇μ), \quad μ = δG/δx. This governs phase separation dynamics and coarsening kinetics.

13. Summary Equations

ConceptKey Relation
Clapeyron equationdP/dT=Δs/Δv=hfg/(Tvfg)dP/dT = Δs/Δv = h_{fg}/(Tv_{fg})
Clausius–Clapeyron (ideal)dln⁡Psat/dT=hfg/(RvT2)d\ln P_{sat}/dT = h_{fg}/(R_v T^2)
Critical conditions(partialP/partialv)T=0,(partial2P/partialv2)T=0(\\partial P/\\partial v)_T = 0, (\\partial^2P/\\partial v^2)_T = 0
Landau expansionG=G0+a(T−Tc)m2+bm4G = G_0 + a(T - T_c)m^2 + b m^4
Critical exponentsbeta,gamma,alpha,nu\\beta, \\gamma, \\alpha, \\nu etc., universal scaling laws
Nucleation barrierΔG∗=16πσ3/[3(Δgv)2]ΔG^* = 16πσ^3/[3(Δg_v)^2]
Spinodal condition(partial2G/partialx2)T,P=0(\\partial^2G/\\partial x^2)_{T,P} = 0
Surface tension (Gibbs)dσ=−∑Γidμidσ = -∑Γ_i dμ_i
  • pure-substances.md — equations of state and phase diagrams.
  • mixtures-phases.md — phase equilibria and Gibbs–Duhem consistency.
  • 10_NonEquilibrium_Thermodynamics.md — Cahn–Hilliard and transport coupling for phase-change kinetics.