Coefficient of Thermal Expansion

Estimates the volumetric coefficient of thermal expansion from NPT molecular dynamics run at several temperatures.

Overview

The coefficient of thermal expansion (CTE) measures how much a material's volume changes with temperature at constant pressure. CTEAnalyzer gets there empirically rather than from a closed-form model: it runs an NPT MD simulation at each temperature in temperatures (default 300, 600, 900 K), lets the cell reach its equilibrium volume under the target pressure, and fits a line through the resulting volume-temperature points. This requires an MD-capable calculator (a BaseMDCalculator subclass); a plain BaseCalculator without a run() method raises. The MD ensemble (ensemble, default "npt_berendsen") and target pressure (default 1.0 atm) are applied identically at every temperature.

Theory

The volumetric CTE is the thermodynamic derivative

\[ \alpha_V = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P \]

CTEAnalyzer approximates this by running NPT-MD at each configured temperature \(T_i\), reading off the final cell volume \(V_i\) once the run completes, and fitting a straight line through the \((T_i, V_i)\) points:

\[ V(T) \approx V_{\text{ref}} + m\,(T - T_{\text{ref}}) \]

where \(m = dV/dT\) is the fitted slope and \(V_{\text{ref}}\) is the volume at the lowest sampled temperature \(T_{\text{ref}}\). The volumetric CTE is then the slope normalized by that reference volume:

\[ \alpha_V \approx \frac{m}{V_{\text{ref}}} \]

reported both in \(\text{K}^{-1}\) (cte) and in ppm/K (cte_ppm, \(\alpha_V \times 10^6\)). Because this is a linear fit, accuracy improves with more temperature points and longer runs (steps) that let the cell fully equilibrate at each temperature; at least two distinct temperatures are required.