CTE Analysis (Thermal Expansion)¶

The coefficient of thermal expansion (CTE) measures how a material's volume changes with temperature. In amorphouspy, CTE is computed from enthalpy-volume cross-correlations in NPT molecular dynamics simulations.

Theory¶

Statistical Mechanical Definition¶

The volumetric CTE \(\alpha_V\) can be computed from NPT ensemble fluctuations using the cross-correlation between enthalpy (\(H\)) and volume (\(V\)):

\[ \alpha_V = \frac{\langle \delta H \cdot \delta V \rangle}{k_B \, T^2 \, \langle V \rangle} \]

where:

Symbol	Description
\(\delta H = H - \langle H \rangle\)	Instantaneous enthalpy fluctuation
\(\delta V = V - \langle V \rangle\)	Instantaneous volume fluctuation
\(k_B\)	Boltzmann constant
\(T\)	Temperature
\(\langle V \rangle\)	Time-averaged volume

The linear CTE is approximately:

\[ \alpha_L \approx \frac{\alpha_V}{3} \]

Why H-V Cross-Correlation?¶

This approach is statistically more efficient than directly fitting volume vs. temperature curves, because: - It requires only a single NPT simulation at each temperature - The cross-correlation converges faster than volume fluctuations alone - It avoids the need for multiple simulations at slightly different temperatures

Usage¶

`cte_from_npt_fluctuations(temperature, enthalpy, volume, ...)`¶

This is the analysis function that computes CTE from pre-collected time series data.

from amorphouspy.analysis.cte import cte_from_npt_fluctuations
import numpy as np

# enthalpy and volume are 1D arrays from NPT simulation output
# (e.g., from LAMMPS thermo output)

cte = cte_from_npt_fluctuations(
    temperature=300.0,            # K
    enthalpy=enthalpy_array,      # Shape: (n_steps,), units: eV
    volume=volume_array,          # Shape: (n_steps,), units: ų
)

print(f"Volumetric CTE: {cte:.2e} K⁻¹")
print(f"Linear CTE: {cte/3:.2e} K⁻¹")

Parameters:

Parameter	Type	Description
`temperature`	`float`	Simulation temperature (K)
`enthalpy`	`np.ndarray`	Time series of enthalpy values (eV)
`volume`	`np.ndarray`	Time series of volume values (ų)

Returns:

Type	Description
`float`	Volumetric CTE (\(K^{-1}\))

Typical CTE Values for Oxide Glasses¶

Glass system	Linear CTE (\(\times 10^{-6}\) K⁻¹)
Vitreous SiO₂	0.5–0.6
Borosilicate (Pyrex-type)	3.2–3.5
Soda-lime silicate	8.5–9.5
Lead silicate	8–10
Aluminosilicate	4–6

Note: MD-computed CTE values may differ from experimental values due to the high cooling rates used in simulation and the limitations of classical potentials. Trends across compositions are generally well-reproduced.

Convergence¶

The H-V cross-correlation requires sufficiently long NPT trajectories for convergence. Guidelines:

Minimum: 100 ps of production data (after equilibration)
Recommended: 500 ps – 1 ns for reliable values
Check: Plot the running average of \(\langle \delta H \cdot \delta V \rangle\) to verify convergence

The CTE workflow module (amorphouspy.workflows.cte) includes built-in convergence checking.