CTE Simulation¶

This workflow computes the coefficient of thermal expansion (CTE) from NPT molecular dynamics simulations using the enthalpy-volume cross-correlation method.

Method¶

NPT Fluctuation Approach¶

The volumetric CTE is computed from the cross-correlation of enthalpy and volume fluctuations in a single NPT simulation:

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

The linear CTE is then:

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

This approach requires only one NPT run per temperature point, making it computationally efficient.

Alternative: Direct Volume Method¶

The CTE can also be extracted from the slope of the volume-temperature curve:

\[ \alpha_L = \frac{1}{3V_0} \frac{dV}{dT} \]

This requires running NPT at multiple closely-spaced temperatures and fitting the \(V(T)\) curve.

The fluctuation method (default in amorphouspy) is generally preferred for single-point calculations, while the direct method is useful for identifying the glass transition temperature from the \(V(T)\) plot.

Usage¶

`cte_from_fluctuations_simulation(structure, potential, ...)`¶

from amorphouspy import cte_from_fluctuations_simulation

result = cte_from_fluctuations_simulation(
    structure=glass_structure,
    potential=potential,
    temperature=300.0,           # K
    production_steps=200_000,    # Default production steps
    equilibration_steps=100_000, # Default equilibration steps
    timestep=1.0,                # fs
    pressure=1e-4,               # 1 bar in GPa
)

# Extract results from nested summary
summary = result["summary"]
print(f"Volumetric CTE: {summary['CTE_V_mean']:.2e} K⁻¹")
print(f"Linear CTE: {summary['CTE_x_mean']:.2e} K⁻¹")

Parameters:

Parameter	Type	Default	Description
`structure`	`Atoms`	—	Equilibrated glass structure
`potential`	`str`	—	Path to LAMMPS potential file
`temperature`	`float`	`300.0`	Simulation temperature (K)
`pressure`	`float`	`1e-4`	Target pressure (GPa)
`production_steps`	`int`	`200_000`	Steps for individual production runs
`equilibration_steps`	`int`	`100_000`	Equilibration steps
`timestep`	`float`	`1.0`	MD timestep (fs)

Returns:

Returns a nested dictionary with "summary" and "data" keys. The "summary" contains averaged values and uncertainties.

`temperature_scan_simulation(structure, potential, ...)`¶

For a complete thermal expansion study across a temperature range, use temperature_scan_simulation:

import numpy as np
from amorphouspy import temperature_scan_simulation

temperatures = np.arange(100, 1200, 100).tolist()  # 100 to 1100 K

result = temperature_scan_simulation(
    glass_structure, 
    potential,
    temperature=temperatures,
    production_steps=200_000,
)

# Access data for each temperature
# Result structure: result["data"]["01_100K"]["run01"]...

The glass transition temperature (\(T_g\)) appears as a change in slope of the \(V(T)\) curve: - Below \(T_g\): lower CTE (glassy state)
- Above \(T_g\): higher CTE (liquid-like state)

Convergence¶

The H-V cross-correlation is sensitive to convergence. Guidelines:

Duration (ps)	Reliability
< 100	Poor — large fluctuations
100–500	Moderate — acceptable for trends
500–1000	Good — reliable single-point values
> 1000	Excellent — publication quality

The CTE workflow includes built-in convergence checking that monitors the running average of \(\langle \delta H \cdot \delta V \rangle\) and reports whether the simulation has converged.

Tip: If you see very noisy CTE values, increase the production run length. The H-V cross-correlation has higher statistical noise than simple volume averaging.