Temperature-Dependent Phase Formation

• The topic quantifies the likelihood of forming equilibrium and metastable phases by integrating free-energy landscapes with temperature-dependent kinetics.
• It utilizes statistical mechanics, first-principles energetics, and Bayesian inference to compute and rank phase probabilities with uncertainty measures.
• This approach informs synthesis and processing strategies in alloys, oxides, and functional materials by predicting optimal phase selection windows.

The temperature-dependent probability of phase formation quantifies, for a given thermodynamic system, the likelihood that a specified crystalline, amorphous, or metastable phase is realized at equilibrium or in a kinetically trapped state as a function of temperature. This probabilistic perspective integrates free-energy landscapes, statistical mechanics, and kinetic constraints, enabling rigorous predictions of phase selection, competition, and persistence across diverse materials classes—including alloys, oxides, intermetallic compounds, magnetics, and functional materials. The explicit temperature dependence emerges from the interplay of enthalpic stability, configurational and vibrational entropy, nucleation barriers, and external kinetic protocols (e.g., quenching, annealing, or constrained synthesis). Recent advances leverage first-principles energetics, canonical/semigrand ensemble Boltzmann statistics, and Bayesian uncertainty quantification to render quantitative, uncertainty-ranked phase-formation probabilities as core elements in computational materials design.

1. Statistical Thermodynamic Foundations

The unifying formalism for temperature-dependent phase probabilities is anchored in Boltzmann–Gibbs statistics. For a system whose microstates $i$ have energies $E_i$, and for which phases correspond to groupings of microstates (e.g., all arrangements sharing a given symmetry or order parameter), the probability of occupying phase $\alpha$ at temperature $T$ is given by the canonical partition function:

$$P_\alpha(T) = \frac{\sum_{i\in\alpha}\exp(-E_i / k_{\mathrm B}T)}{Z(T)}\,,\quad Z(T) = \sum_{j}\exp(-E_j / k_{\mathrm B}T)$$

If all microstates in $\alpha$ share a free energy $G_\alpha(T)$, this simplifies to a Boltzmann-weighted ratio:

$$P_\alpha(T) = \frac{\exp(-G_\alpha(T)/k_{\mathrm B}T)}{\sum_\beta \exp(-G_\beta(T)/k_{\mathrm B}T)}$$

This framework is central to phase competition in alloys, crystalline polymorphs, and magnetic/electronic systems. It forms the foundation for explicit calculations across a broad range of physical systems, such as the Ising model (Ding, 2023), multicomponent solid solutions, and ferroelectric transitions (Wang et al., 2022).

2. Free Energy Landscapes and Phase Formation

Quantitative prediction of $G_\alpha(T)$ incorporates enthalpic and entropic contributions:

$G_\alpha(T) = H_\alpha - T S_\alpha$

where $H_\alpha$ is the formation enthalpy (typically from DFT + convex hull calculations), and $S_\alpha$ includes configurational mixing, vibrational, magnetic, or electronic contributions, as relevant to the phase’s structure and disorder (Evans et al., 2020). For multistable systems, the full free energy landscape in order parameter space can be constructed—e.g., as a sum of parabolic wells (multi-well models) with degeneracies and curvatures parameterized by symmetry, energetics, and first-principles phonon data (Wang et al., 2022):

$$P_i(T) = \frac{g_i (2\pi k_B T/\kappa_i)^{N/2} \exp(-E_i/k_BT)}{ \sum_j g_j (2\pi k_B T/\kappa_j)^{N/2} \exp(-E_j/k_BT) }$$

Here $g_i$ is the degeneracy, $\kappa_i$ is the curvature, and $E_i$ is the zero-temperature energy of well $i$.

3. Kinetic Constraints and Metastable Phase Formation

In practical materials synthesis, many observed phases are not strictly equilibrium (lowest-$G$) states but kinetically stabilized intermediates. The temperature-dependent probability must then include both thermodynamic driving forces and kinetic barriers, as formalized in classical nucleation theory:

$$I(T) = I_0 \exp\left[ -\frac{\Delta G^* + Q_{\mathrm{diff}}}{k_B T} \right]$$

where $\Delta G^*$ is the critical nucleus formation barrier and $Q_{\mathrm{diff}}$ is a (possibly composition-dependent) diffusion activation energy. The net probability of phase $\alpha$ formation in a process with characteristic time $t$ is modeled as:

$$P_{\mathrm{form}}(T; x) \approx 1 - \exp\left[ - \int_0^t I(T') dt' \right ]$$

For systems subject to decomposition above a critical $T$, a survival probability under competing decomposition kinetics is included:

$$P_{\mathrm{survive}}(T; x) \approx \exp\left[ - \int_{T_{\mathrm{cryst}}(x)}^T k_{\mathrm{decomp}}(T'; x)\,dT' \right ]$$

The total probability is then $$P(T; x) = P_{\mathrm{form}}(T; x) \cdot P_{\mathrm{survive}}(T; x)$$. Experiments on Y-doped (Cr,Y)$_2$AlC MAX phases demonstrate, for example, a narrow $\sim$50 °C window for phase-pure growth, with strong temperature and yttrium-concentration dependence driven by both thermodynamic and kinetic factors (Azina et al., 2022).

4. Methods of Computation and Visualization

For realistic multicomponent alloys and complex chemistries, the calculation of $P_\alpha(T)$ proceeds through workflows involving:

• DFT evaluation of $H_\alpha$ for all candidate phases.
• Estimation of $S_\alpha$ (ideally via phonon calculations, configurational counting, or Monte Carlo).
• Construction of temperature grids and calculation of $G_\alpha(T)$ for each phase.
• Statistical-thermodynamic evaluation of phase probabilities:

$$P_\alpha(T) = \frac{\exp[-\Delta G_\alpha(T)/(k_BT)]}{\sum_\beta \exp[-\Delta G_\beta(T)/(k_BT)]}$$

with $\Delta G_\alpha(T)$ referenced to the lowest free energy at each $(T, x)$ (Evans et al., 2020, Miryashkin et al., 2023).

• Inverse Hull Web and similar graphical tools, mapping phases by reaction and formation energies to visualize stability/metastability boundaries as a function of $T$ and identify critical temperatures for phase transitions or metastable persistence (Evans et al., 2020).

For systems with finite simulation data and uncertainty, Bayesian inference is used to propagate errors from MD, melting-point, or phonon calculations to the posterior distribution of $G_\alpha(T)$, and hence to $P_\alpha(T)$, with uncertainty bands extracted via Monte Carlo sampling (Miryashkin et al., 2023).

5. Representative Models and Case Studies

A wide range of models illustrate the practical computation and measurement of temperature-dependent phase probabilities:

• Order–Disorder Alloys: In h.c.p.–Ti–Al, statistical thermodynamics and kinetic ordering models with temperature-dependent interaction parameters $W_n(T)$ yield equilibrium LRO parameters $\eta_{\mathrm{eq}}(T, c)$, interpreted as the probability of observing D0$_{19}$ ordering at $(T, c)$, and kinetics $d\eta/dt$ that accelerate predictably with $T$ (Radchenko et al., 2014).
• Multiphase Landscapes: In ferroelectric PbTiO$_3$, the occupation probability of each polarization well $P_i(T)$ follows the Boltzmann-mixing prescription, with a crossover from a polarized state to uniform occupation at $T_c$ (Wang et al., 2022).
• High Entropy Alloys: For solid solutions versus intermetallics in HEAs, the equilibrium persistence probability $P_{\mathrm{sol}}(T)$ is mapped in an Inverse Hull Web, with specific $T_{\mathrm{cs}}$ and $T_{\mathrm{ca}}$ scales correlating with experimental phase persistence (Evans et al., 2020).
• Metastable Thin Films: Experiments and DFT calculations on (Cr,Y)$_2$AlC demonstrate a non-monotonic $P(T; x)$, with strong $x$-dependence of crystallization and decomposition temperatures and a pronounced, narrow temperature window for single-phase formation (Azina et al., 2022).
• Hysteretic Transitions: For Pd–H, the occupation probability of the hydride phase is explicitly obtained from a constrained partition function, with thermodynamic hysteresis shifting the transition regime and altering $P_{\mathrm{hydride}}(T)$ (Rahm et al., 2021).

6. Experimental and Computational Determination

Experimental measurement of phase probabilities employs calorimetry (DSC, identifying $dP_{\mathrm{form}}/dT$ via exothermic peaks), in-situ diffraction (onset and extinction of phase-specific Bragg reflections), and compositional/microstructural analysis (EDS, TEM). These observations are compared with computational predictions based on statistical mechanics, ab initio energetics, and process-specific kinetic models.

Reported methods for computational determination include:

• Self-consistent field and kinetic equations for ordering process and LRO evolution (Radchenko et al., 2014).
• Thermodynamic integration of constrained-ensemble MC for coherent versus incoherent transitions, extracting free-energy plateaus and concentration-dependent probabilities (Rahm et al., 2021).
• Bayesian GP regression with error propagation from MD, yielding posterior $P_i(T, x)$ with confidence intervals (Miryashkin et al., 2023).

7. Implications and Generalizations

The explicit temperature-dependent probability of phase formation is central for materials design, describing both equilibrium phase stability and metastable persistence under fast synthesis, alloying, or constrained processing. The formalism is universal, applicable to systems with competing wells (structural, magnetic, electronic), symmetry-breaking order parameters, or complex free-energy landscapes. It also enables quantitative linking of calculated and measured phase diagrams, transition temperatures, hysteresis widths, and kinetic windows for synthesis (Evans et al., 2020, Wang et al., 2022, Azina et al., 2022).

A plausible implication is that integrating rigorous probabilistic models into high-throughput and machine-learning–guided materials exploration could enable uncertainty-ranked, process-aware design maps, directly expressing phase-survival, transformation, or loss as explicit functions of temperature and composition. This perspective is becoming standard for phase selection in complex multicomponent spaces and for designing functional or metastable materials with tailored thermodynamic and kinetic landscapes.