Thermodynamics of Phase Transitions

Thermodynamics of Phase Transitions

Phase transitions describe the transformation of matter between different states (solid, liquid, gas) or between different structural phases (e.g., polymorphs, magnetic phases). The thermodynamics governing these transitions is rooted in the minimization of Gibbs free energy ($G=H-TS$).

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1. Fundamental Thermodynamic Framework

Gibbs Free Energy & Phase Stability

• At constant temperature and pressure, the equilibrium phase is the one with the lowest Gibbs free energy.
• A phase transition occurs when the Gibbs free energies of two phases become equal: $G_1=G_2$.
• The chemical potential ${\mu }_i=(\mathrm{\partial }G\mathrm{/}\mathrm{\partial }{n}_i{)}_{T,P}$ drives mass transfer between phases.

First-Order vs. Second-Order Transitions (Ehrenfest Classification)

Feature	First-Order	Second-Order (Continuous)
Derivative of $G$	Discontinuity in first derivatives ($S$, $V$)	Discontinuity in second derivatives ($C_P$, $\alpha$, ${\kappa }_T$)
Latent heat	Yes ($\mathrm{\Delta }H\mathrm{\ne }0$)	No ($\mathrm{\Delta }H=0$)
Volume change	Yes ($\mathrm{\Delta }V\mathrm{\ne }0$)	No ($\mathrm{\Delta }V=0$)
Examples	Melting, boiling, sublimation	Ferromagnetic → paramagnetic, superconducting transition
Phase coexistence	Two phases coexist at transition T	Single phase with gradually changing order parameter

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2. Key Thermodynamic Equations

Clausius–Clapeyron Equation

Describes the pressure-temperature relationship along a first-order phase boundary:

$$\frac{dP}{dT}=\frac{\mathrm{\Delta }S}{\mathrm{\Delta }V}=\frac{\mathrm{\Delta }H}{T\mathrm{\Delta }V}$$

• Application: Predicting melting/boiling point changes with pressure.
• Example: Ice melting — $\mathrm{\Delta }V<0$ (ice contracts upon melting), so $dP\mathrm{/}dT<0$, explaining why ice melts under pressure.

Gibbs–Thomson Effect (at Nanoscale)

For nanoparticles, the melting temperature is depressed due to curvature:

$$\mathrm{\Delta }{T}_m=T_m^{\mathrm{\infty }}-T_m(r)=\frac{2{\gamma }_{sl}{V}_m{T}_m^{\mathrm{\infty }}}{r\mathrm{\Delta }{H}_m}$$

Where:

• $r$ = particle radius
• ${\gamma }_{sl}$ = solid-liquid interfacial energy
• $V_m$ = molar volume
• $\mathrm{\Delta }{H}_m$ = bulk enthalpy of fusion

Nanoscale implication: This is critical for nanoscale growth — smaller particles melt at significantly lower temperatures.

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3. Phase Transition Thermodynamics at the Nanoscale

Nanoscale systems exhibit deviations from bulk thermodynamic behavior due to:

a) Surface Energy Dominance

The total Gibbs free energy includes a surface term:

$$G_{\text{total}}=G_{\text{bulk}}+\gamma A$$

Where $\gamma$ is surface tension and $A$ is surface area. At small $r$, the surface contribution becomes comparable to the bulk term.

b) Size-Dependent Phase Diagrams

• Melting point depression: $T_m(r)=T_m^{\mathrm{\infty }}(1-\frac{2{\gamma }_{sl}}{r{\rho }_s\mathrm{\Delta }{H}_m})$
• Polymorph stability reversal: A metastable bulk phase can become thermodynamically stable at the nanoscale (e.g., anatase TiO₂ is stable below ~14 nm, while rutile is stable in bulk).

c) Coexistence & Hysteresis

• First-order transitions in finite systems exhibit suppressed discontinuous jumps and smoother transitions due to surface premelting or prewetting.
• Hysteresis arises from kinetic barriers (nucleation energy).

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4. Nucleation Theory — The Gateway to Transitions

Phase transitions begin with nucleation, described by Classical Nucleation Theory (CNT):

Homogeneous Nucleation

The Gibbs free energy change for forming a spherical nucleus of radius $r$:

$$\mathrm{\Delta }G(r)=\frac{4}{3}\pi r^3\mathrm{\Delta }{G}_v+4\pi r^2\gamma$$

• First term: Volume free energy (driving force, negative below $T_c$)
• Second term: Surface energy barrier (positive)
• Critical radius: $r^{\ast }=-{\displaystyle \frac{2\gamma }{\mathrm{\Delta }{G}_v}}$
• Activation barrier: $\mathrm{\Delta }{G}^{\ast }={\displaystyle \frac{16\pi {\gamma }^3}{3(\mathrm{\Delta }{G}_v{)}^2}}$

Heterogeneous Nucleation

• Occurs at surfaces, defects, or impurities — lowers $\mathrm{\Delta }{G}^{\ast }$ significantly.
• Wetting angle $\theta$ reduces the barrier: $\mathrm{\Delta }{G}_{\text{het}}^{\ast }=\mathrm{\Delta }{G}_{\text{hom}}^{\ast }\cdot f(\theta )$

At the Nanoscale

• Nucleation barriers are smaller due to higher supersaturation.
• Two-step nucleation is often observed (e.g., dense liquid-like precursor → crystal), deviating from CNT.

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5. Landau Theory of Phase Transitions

A phenomenological approach using an order parameter $\eta$ (e.g., magnetization, density difference):

$$G(T,P,\eta )=G_0+a(T-T_c){\eta }^2+b{\eta }^4+c{\eta }^6+\dots$$

• Second-order transition: $b>0$, minimum shifts continuously from $\eta =0$ to $\eta \mathrm{\ne }0$ below $T_c$.
• First-order transition: $b<0$, the free energy develops a local minimum at $\eta \mathrm{\ne }0$ before crossing below the $\eta =0$ minimum.

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6. Classification by Phase Transition Type

Type	Thermodynamic Driver	Examples
Melting / Freezing	$\mathrm{\Delta }G=0$ at $T_m$; entropy-driven ($\mathrm{\Delta }{S}_m>0$)	Ice ↔ Water
Vaporization / Condensation	Large $\mathrm{\Delta }V$; Clausius–Clapeyron	Water ↔ Steam
Sublimation / Deposition	Direct solid ↔ gas	Dry ice, frost formation
Allotropic / Polymorphic	Lattice stability change; small $\mathrm{\Delta }H$	Graphite ↔ Diamond, FCC ↔ BCC
Magnetic	Order-disorder of spins (Landau)	Ferro ↔ Para (Curie point)
Superconducting	Condensation of Cooper pairs (2nd order)	Normal ↔ Superconducting
Martensitic	Diffusionless shear; athermal	Steel austenite ↔ martensite
Order-Disorder	Configurational entropy	Alloy ordering (Cu₃Au)

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7. Key Thermodynamic Quantities at Transition

• Latent heat: $\mathrm{\Delta }{H}_{tr}=T_{tr}\mathrm{\Delta }{S}_{tr}$
• Entropy change: $\mathrm{\Delta }{S}_{tr}=\mathrm{\Delta }{H}_{tr}\mathrm{/}{T}_{tr}$ (for first-order)
• Heat capacity discontinuity: $\mathrm{\Delta }{C}_P=T{\left(\frac{\mathrm{\partial }S}{\mathrm{\partial }T}\right)}_P$ — diverges or jumps at $T_c$
• Compressibility & thermal expansion: Divergent near critical points (second-order)

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8. Critical Phenomena & Scaling

Near a continuous (second-order) phase transition, thermodynamic quantities follow power laws:

• $C_P\sim \mathrm{\mid }T-T_c{\mathrm{\mid }}^{-\alpha }$
• $\chi \sim \mathrm{\mid }T-T_c{\mathrm{\mid }}^{-\gamma }$ (susceptibility)
• $\xi \sim \mathrm{\mid }T-T_c{\mathrm{\mid }}^{-\nu }$ (correlation length)
• $M\sim (T_c-T{)}^{\beta }$ (order parameter)

Universality: Exponents depend only on symmetry, dimensionality, and range of interactions — not microscopic details.