6.1 First-order phase transitions

Definition of phase transitions

Phase transitions occur when a system shifts from one thermodynamic phase to another, producing changes in its macroscopic behavior. In statistical mechanics, studying these transitions reveals how collective particle interactions give rise to dramatically different states of matter.

First-order vs second-order transitions

The key distinction comes down to how thermodynamic quantities behave at the transition point.

• First-order transitions involve discontinuous jumps in thermodynamic variables like volume and entropy. The system absorbs or releases latent heat as it crosses the transition. Think of ice melting: the temperature holds steady at 0°C while energy goes into breaking the crystal structure.
• Second-order (continuous) transitions have no discontinuity in volume or entropy themselves, but their derivatives (specific heat, compressibility) show discontinuities or divergences. There's no latent heat. The system instead displays critical phenomena near the transition point, such as diverging correlation lengths.

Gibbs free energy discontinuity

At a first-order transition, two distinct phases coexist, each with its own Gibbs free energy. The system always occupies whichever phase minimizes $G$, so at the transition temperature the two free energy curves cross. On either side of that crossing, one phase is thermodynamically favored over the other.

The first derivative of $G$ with respect to temperature or pressure jumps discontinuously at the transition:

$\Delta G = G_2 - G_1 \neq 0$

This discontinuity in the first derivative is precisely what defines a transition as "first-order" in the Ehrenfest scheme.

Thermodynamic properties

Several measurable quantities characterize first-order transitions and distinguish them from continuous ones.

Latent heat

Latent heat is the energy absorbed or released during the transition while the temperature stays constant. It quantifies how much energy is needed to convert one phase entirely into the other.

$L = T(S_2 - S_1)$

where $S_1$ and $S_2$ are the entropies of the two coexisting phases. For water, the latent heat of fusion is about 334 J/g, while the latent heat of vaporization is roughly 2260 J/g, reflecting the much larger entropy change involved in breaking free of the liquid phase entirely.

Volume discontinuity

First-order transitions produce an abrupt change in the system's specific volume:

$\Delta V = V_2 - V_1 \neq 0$

Water expanding by about 9% when it freezes is a familiar example. In liquid-gas transitions, the volume change is far more dramatic: steam at 100°C and 1 atm occupies roughly 1600 times the volume of the same mass of liquid water.

Entropy change

Entropy jumps discontinuously at the transition point, directly related to the latent heat by:

$\Delta S = \frac{L}{T}$

Entropy increases during melting and vaporization (the system becomes more disordered) and decreases during freezing and condensation. The magnitude of $\Delta S$ reflects how structurally different the two phases are.

Ehrenfest classification

The Ehrenfest classification categorizes phase transitions by identifying which derivative of the Gibbs free energy first shows a discontinuity.

Discontinuities in derivatives

• First-order transitions: discontinuities appear in the first derivatives of $G$, namely entropy ($S = -\partial G/\partial T$) and volume ($V = \partial G/\partial P$).
• Second-order transitions: first derivatives are continuous, but second derivatives like specific heat ($C_p$), isothermal compressibility ($\kappa_T$), and thermal expansion coefficient ($\alpha$) show discontinuities or divergences.
• The "order" of the transition equals the lowest-order derivative of $G$ that is discontinuous.

Note that the Ehrenfest scheme works cleanly for first-order transitions but becomes less useful for continuous transitions, where derivatives often diverge rather than simply jump. Modern treatments typically distinguish only between "first-order" and "continuous" transitions.

Order parameter behavior

An order parameter is a quantity that is zero in one phase and nonzero in the other, capturing the symmetry difference between phases.

• In first-order transitions, the order parameter jumps discontinuously at the transition point. For example, the density difference $\rho_l - \rho_g$ drops abruptly to zero at a first-order liquid-gas transition (below the critical point).
• In second-order transitions, the order parameter goes to zero continuously as the transition is approached.
• Other examples: magnetization $M$ in ferromagnetic transitions, the fraction of atoms on preferred sublattice sites in order-disorder transitions in alloys.

Examples of first-order transitions

Solid-liquid transition

Melting and freezing are the most everyday first-order transitions. At the melting point, solid and liquid coexist in equilibrium, and the system absorbs (melting) or releases (freezing) the latent heat of fusion. The density changes discontinuously. In metal casting, controlling the solidification front is critical for determining grain structure and mechanical properties.

Liquid-gas transition

Vaporization and condensation involve large volume changes and correspondingly large latent heats. Water boiling at 100°C under 1 atm is the classic example. The bubbles you see during boiling are regions of gas phase coexisting with the surrounding liquid. This transition only exists as a true first-order transition below the critical point; above it, liquid and gas become indistinguishable.

Magnetic systems

Some magnetic materials exhibit first-order transitions. Metamagnetic transitions involve an abrupt jump in magnetization when an external field is applied to certain antiferromagnetic materials, flipping them into a ferromagnetic-like state. These transitions often display hysteresis: the field at which the transition occurs on increasing the field differs from the field on decreasing it, a signature of metastable states separated by energy barriers.

Clausius-Clapeyron equation

The Clausius-Clapeyron equation connects the slope of a coexistence curve in the $P$-$T$ plane to measurable thermodynamic quantities.

Derivation and significance

The derivation starts from the condition that chemical potentials of the two phases must be equal along the coexistence curve: $\mu_1(P,T) = \mu_2(P,T)$. Taking the total differential of both sides and using the thermodynamic identities $d\mu = -SdT + VdP$ (per mole) yields:

$\frac{dP}{dT} = \frac{L}{T\Delta V}$

where $L$ is the latent heat and $\Delta V = V_2 - V_1$ is the volume change. This tells you that if you know the latent heat and volume change, you can predict how the transition temperature shifts with pressure (and vice versa).

Applications to phase diagrams

• The positive slope of most solid-liquid coexistence curves follows from the fact that $L > 0$ and $\Delta V > 0$ (the liquid is less dense than the solid). Water is anomalous: ice is less dense than liquid water, so $\Delta V < 0$ and the melting curve has a negative slope.
• For liquid-gas transitions, $\Delta V$ is large and positive, giving a relatively gentle positive slope that steepens as you approach the critical point (where $\Delta V \to 0$ and $L \to 0$).
• The equation breaks down exactly at the critical point, where the first-order transition terminates and the coexistence curve ends.

Nucleation theory

Nucleation theory explains how a new phase first appears within a metastable parent phase. A system doesn't instantly transform the moment conditions favor the new phase; instead, small embryos of the new phase must form and grow.

Homogeneous nucleation

This occurs spontaneously in a pure, uniform system with no foreign surfaces or impurities. Thermal fluctuations randomly assemble small clusters of the new phase. These clusters face a competition:

• Volume term: the interior of the cluster gains free energy by being in the more stable phase (favorable, scales as $r^3$).
• Surface term: creating an interface costs energy (unfavorable, scales as $r^2$).

For small clusters, the surface cost dominates and the cluster tends to dissolve. Only clusters that reach the critical size can grow spontaneously.

Heterogeneous nucleation

In practice, nucleation almost always happens at impurities, container walls, scratches, or dust particles. These sites lower the effective surface energy, reducing the energy barrier compared to homogeneous nucleation. That's why boiling chips work in a chemistry lab: they provide nucleation sites so vapor bubbles form steadily rather than in sudden, dangerous bursts. Cloud formation similarly relies on dust and aerosol particles as nucleation sites for water droplets.

Critical nucleus size

The critical radius $r^*$ is the size at which the free energy cost of the surface exactly balances the free energy gain from the bulk. Nuclei smaller than $r^*$ tend to shrink and disappear; nuclei larger than $r^*$ grow spontaneously.

The critical radius depends on:

• Interfacial energy $\gamma$ (higher $\gamma$ means larger $r^*$)
• Degree of supersaturation or undercooling (greater driving force means smaller $r^*$, making nucleation easier)
• Temperature (affects both the driving force and the interfacial energy)

For a spherical nucleus, the free energy barrier scales as $\Delta G^* \propto \gamma^3 / (\Delta g)^2$, where $\Delta g$ is the bulk free energy difference per unit volume between the two phases.

Metastable states

Metastable states sit in local minima of the free energy landscape, separated from the true equilibrium state by energy barriers. The system can persist in a metastable state for a long time if the barrier is large compared to $k_BT$.

Superheating and supercooling

• Supercooling: a liquid cooled below its freezing point without solidifying. Water can be supercooled to about $-40°C$ under clean conditions. A sudden disturbance or introduction of a nucleation site triggers rapid crystallization.
• Superheating: a liquid heated above its boiling point without forming vapor bubbles. This happens in very clean, smooth containers (a common hazard when heating water in a microwave). When nucleation finally occurs, the result can be explosive boiling.

Both phenomena arise because the system remains trapped in the metastable phase due to the nucleation barrier.

Hysteresis in phase transitions

Hysteresis means the transition doesn't occur at the same value of the control parameter (temperature, field, pressure) on heating vs. cooling, or on increasing vs. decreasing the field. The system's state depends on its history.

This is a direct consequence of metastable states and energy barriers. In magnetic systems, the magnetization curve on increasing field differs from the curve on decreasing field, enclosing a hysteresis loop. Shape memory alloys like nitinol also show thermal hysteresis between their austenite and martensite phases, which is central to their functionality.

Landau theory

Landau theory provides a phenomenological description of phase transitions by expanding the free energy as a polynomial in the order parameter $\phi$, without requiring a microscopic model.

Free energy expansion

The general Landau expansion for a system that can exhibit a first-order transition is:

$F = F_0 + a\phi^2 + b\phi^3 + c\phi^4 + \cdots$

• The coefficient $a$ typically depends on temperature: $a = a_0(T - T_0)$, changing sign at some reference temperature $T_0$.
• The cubic term ($b\phi^3$) is the signature of a first-order transition. When $b \neq 0$, the free energy landscape can develop two unequal minima, and the order parameter jumps discontinuously as the system switches between them.
• If symmetry forbids odd powers ($b = 0$), you need to go to sixth order ($\phi^6$) with a negative $\phi^4$ coefficient to get a first-order transition. Otherwise, with $b = 0$ and $c > 0$, the transition is second-order.

Order parameter dynamics

The equilibrium value of $\phi$ is found by minimizing $F$ with respect to $\phi$: setting $\partial F / \partial \phi = 0$. For a first-order transition, this equation has multiple solutions, and the system jumps between them as temperature crosses the transition point.

The time-dependent behavior can be modeled by the time-dependent Ginzburg-Landau (TDGL) equation:

$\frac{\partial \phi}{\partial t} = -\Gamma \frac{\delta F}{\delta \phi}$

where $\Gamma$ is a kinetic coefficient. This describes how the order parameter relaxes toward the free energy minimum, including the dynamics of nucleation and domain growth during a first-order transition.

Coexistence curves

Coexistence curves map out the conditions (pressure, temperature, composition) where two phases exist in thermodynamic equilibrium.

Phase diagrams

A phase diagram is a graphical summary of which phase is stable under given conditions. The most common representations are $P$-$T$ diagrams and $T$-composition diagrams. Coexistence curves are the boundaries between single-phase regions. Along these curves, the chemical potentials of the two phases are equal, and both phases can coexist.

The $P$-$T$ diagram for water, for instance, shows three coexistence curves (solid-liquid, liquid-gas, solid-gas) meeting at the triple point.

Triple point and critical point

• Triple point: the unique $(P, T)$ where three phases coexist simultaneously. For water, this occurs at $T = 0.01°C$ and $P = 611.7 \text{ Pa}$. The triple point is an invariant point: by the Gibbs phase rule ($F = C - P + 2$), a single-component system with three phases has zero degrees of freedom.
• Critical point: the terminus of the liquid-gas coexistence curve. Beyond the critical temperature and pressure, there is no distinction between liquid and gas. At the critical point itself, $\Delta V \to 0$, $L \to 0$, and quantities like compressibility diverge. For water, the critical point is at $T_c = 374°C$ and $P_c = 22.1 \text{ MPa}$.

Experimental techniques

Calorimetry methods

Calorimetry directly measures heat flow during phase transitions, giving access to latent heat and heat capacity.

• Differential scanning calorimetry (DSC) heats a sample and a reference at the same rate, measuring the difference in heat flow. Peaks in the DSC signal correspond to latent heat release or absorption, making it straightforward to identify first-order transitions and measure $L$.
• Adiabatic calorimetry isolates the sample thermally and measures temperature changes precisely, yielding accurate heat capacities and enthalpy changes.
• Isothermal titration calorimetry (ITC) measures heat changes during mixing or reaction in solution, useful for transitions in soft matter and biological systems.

X-ray diffraction

X-ray diffraction probes the atomic-scale structure of materials, making it ideal for detecting structural changes during solid-state phase transitions.

• Powder X-ray diffraction identifies crystalline phases by their characteristic diffraction patterns. New peaks appearing or old peaks disappearing signals a phase transition.
• Single-crystal X-ray diffraction provides detailed atomic positions and bond lengths, revealing exactly how the structure rearranges.
• Temperature-dependent X-ray measurements can track lattice parameter changes through a transition in real time.

Neutron scattering

Neutrons complement X-rays because they interact with nuclei rather than electrons, making them sensitive to light elements (like hydrogen) and magnetic ordering.

• Elastic neutron scattering reveals structural changes, similar to X-ray diffraction but with different contrast.
• Inelastic neutron scattering probes excitations like phonons and magnons, showing how the dynamics of the system change across a transition.
• Small-angle neutron scattering (SANS) studies large-scale structures and is particularly useful for observing phase separation and domain formation.

Computational approaches

Monte Carlo simulations

Monte Carlo methods use random sampling to explore the configuration space of a many-particle system. The Metropolis algorithm generates configurations weighted by their Boltzmann factor $e^{-E/k_BT}$, allowing efficient sampling of equilibrium properties.

For first-order transitions, standard Monte Carlo can struggle because the system must cross a free energy barrier between phases. Techniques like multicanonical sampling and Wang-Landau algorithms help by flattening the energy histogram, enabling the simulation to visit both phases and accurately locate the transition.

Common applications include the Ising model (magnetic transitions) and lattice gas models (liquid-gas transitions).

Molecular dynamics

Molecular dynamics (MD) integrates Newton's equations of motion for each particle, producing trajectories that reveal both equilibrium and dynamic properties.

• NVE ensemble: constant energy, useful for checking energy conservation.
• NVT ensemble: constant temperature (via a thermostat), commonly used for equilibrium sampling.
• NPT ensemble: constant temperature and pressure, most directly comparable to experimental conditions for studying phase transitions.

MD is particularly valuable for studying the kinetics of nucleation and growth, the motion of phase boundaries, and non-equilibrium aspects of first-order transitions that Monte Carlo cannot easily capture.

Applications in materials science

Crystal growth processes

Controlled first-order transitions are the basis of crystal growth:

• Czochralski method: a seed crystal is slowly pulled from a melt, growing a large single crystal. This is how most silicon wafers for semiconductors are produced.
• Vapor phase epitaxy: gaseous precursors deposit crystalline layers on a substrate, used for III-V semiconductors (GaAs, InP) in optoelectronics.
• Solution growth: crystals form from supersaturated solutions. Protein crystallography relies on this to produce crystals suitable for X-ray diffraction.

In each case, controlling the nucleation rate and growth conditions determines crystal quality, defect density, and ultimately device performance.

Metallurgy and alloy formation

Phase transitions govern the microstructure and mechanical properties of metals and alloys.

• Heat treatment (annealing, quenching, tempering) manipulates first-order transitions to control grain size, phase composition, and hardness. Quenching steel, for example, traps carbon in a metastable martensite phase, producing a hard but brittle material.
• Eutectic reactions involve simultaneous solidification of two phases from a single liquid composition, producing fine-grained microstructures with useful properties.
• Shape memory alloys like nitinol undergo a reversible first-order transition between martensite and austenite phases, enabling applications in medical stents and actuators.