Avrami Kinetics Calculator
Calculated Results
Understanding Avrami Kinetics in Phase Transformations
Avrami kinetics, also known as the Johnson-Mehl-Avrami-Kolmogorov (JMAK) model, describes how solids transform from one phase to another at a constant temperature. Whether it is the crystallization of a polymer melt, the growth of grains in a metal, or any nucleation and growth process, the Avrami equation is the primary tool for predicting the speed of the reaction.
The Avrami Equation
The core formula used in this calculator is:
X = 1 – exp(-k * tn)
Where:
- X: The fraction of the material transformed (ranging from 0 to 1).
- t: The time elapsed since the start of the transformation.
- k: The Avrami rate constant, which incorporates both nucleation and growth rates.
- n: The Avrami exponent, a dimensionless number reflecting the dimensionality of growth and nucleation mechanism.
Interpretation of the Avrami Exponent (n)
The value of 'n' provides deep insights into the physical nature of the reaction:
| Exponent (n) | Growth Geometry | Nucleation Type |
|---|---|---|
| 1 | Needle-like (1D) | Instantaneous / Site Saturation |
| 2 | Disk-like (2D) | Instantaneous / Site Saturation |
| 3 | Spherical (3D) | Instantaneous / Site Saturation |
| 4 | Spherical (3D) | Constant Rate Nucleation |
Practical Example
Imagine a polymer undergoing isothermal crystallization. If experimental data suggests an Avrami constant (k) of 0.005 min⁻ⁿ and an exponent (n) of 3.0, you might want to know how much has crystallized after 10 minutes.
1. Inputs: t = 10, k = 0.005, n = 3.0
2. Calculation: X = 1 – exp(-0.005 * 10³)
3. Calculation: X = 1 – exp(-0.005 * 1000) = 1 – exp(-5) ≈ 0.993
In this case, the reaction is 99.3% complete after 10 minutes.
Instantaneous Transformation Rate
The calculator also provides the dX/dt, which is the slope of the transformation curve at that specific moment. This is calculated using the derivative:
dX/dt = n * k * t(n-1) * exp(-k * tn)
This value is crucial for understanding when the reaction speed peaks, which typically occurs mid-way through the transformation before site impingement slows the process down.