Avrami Method

avrami_method(time: np.ndarray, relative_crystallinity: np.ndarray) → Tuple[float, float, float]

Perform Avrami analysis for isothermal crystallization kinetics.

Parameters:

  • time (np.ndarray) – Time data.

  • relative_crystallinity (np.ndarray) – Relative crystallinity data.

Returns:

Tuple containing Avrami exponent (n), rate constant (k), and coefficient of determination (R^2).

Return type:

Tuple[float, float, float]

Theory

The Avrami equation describes the kinetics of phase transformations, particularly crystallization processes. It relates the fraction of transformed material to time:

\[X(t) = 1 - \exp(-(kt)^n)\]
where:

  • X(t) is the fraction of transformed material at time t

  • k is the rate constant

  • n is the Avrami exponent

The Avrami exponent (n) provides information about the nucleation and growth mechanisms:

  • n = 1: One-dimensional growth from instantaneous nuclei

  • n = 2: Two-dimensional growth from instantaneous nuclei

  • n = 3: Three-dimensional growth from instantaneous nuclei

  • n = 4: Three-dimensional growth from sporadic nuclei

Usage Example

import numpy as np
from pkynetics.model_fitting_methods import avrami_method

# Generate sample data
time = np.linspace(0, 100, 100)
relative_crystallinity = 1 - np.exp(-(0.01 * time) ** 2.5)

# Perform Avrami analysis
n, k, r_squared = avrami_method(time, relative_crystallinity)

print(f"Avrami exponent (n): {n:.2f}")
print(f"Rate constant (k): {k:.4e}")
print(f"R-squared: {r_squared:.4f}")

Parameters

  • time (np.ndarray): An array of time values. Must be in ascending order and start from a non-negative value.

  • relative_crystallinity (np.ndarray): An array of relative crystallinity values. Must be between 0 and 1.

Returns

A tuple containing:

  1. n (float): The Avrami exponent.

  2. k (float): The rate constant.

  3. r_squared (float): The coefficient of determination (R^2) of the fit.

Raises

  • ValueError: If input arrays have different lengths or contain invalid values.

Notes

  • The function uses non-linear least squares to fit the Avrami equation to the data.

  • Ensure that your time and relative crystallinity data are properly normalized and preprocessed before using this function.

  • The R-squared value provides a measure of how well the model fits the data. Values closer to 1 indicate a better fit.

See Also