JMAK Crystallization Kinetics

Introduction

The Johnson-Mehl-Avrami-Kolmogorov (JMAK) model is the cornerstone of crystallization kinetics in phase change materials. This tutorial will teach you how to model, simulate, and analyze crystallization processes using the JMAK framework, essential for understanding SET operations and retention behavior in PCM devices.

Learning Objectives

• Understand the JMAK equation and its parameters
• Determine Avrami exponent (n) from experimental data
• Extract activation energy using Arrhenius analysis
• Simulate isothermal and non-isothermal crystallization
• Apply JMAK to PCM device modeling

The JMAK Model

Practical Implementation

# JMAK Crystallization Simulation
import numpy as np
import matplotlib.pyplot as plt

class JMAKModel:
    def __init__(self, Ea=1.8, n=3.0, k0=1e16):
        self.Ea = Ea  # eV
        self.n = n    # Avrami exponent
        self.k0 = k0  # Pre-exponential factor
        self.kb = 8.617e-5  # eV/K
    
    def rate_constant(self, T):
        """Calculate temperature-dependent rate constant"""
        return self.k0 * np.exp(-self.Ea / (self.kb * T))
    
    def crystalline_fraction(self, t, T):
        """Calculate crystalline fraction at time t and temperature T"""
        k = self.rate_constant(T)
        return 1 - np.exp(-(k * t)**self.n)
    
    def time_to_fraction(self, X, T):
        """Calculate time to reach fraction X at temperature T"""
        k = self.rate_constant(T)
        return (-np.log(1 - X))**(1/self.n) / k

# Example usage
model = JMAKModel(Ea=1.8, n=3.0)
T = 500  # K
time = np.linspace(0, 100e-9, 1000)  # 0 to 100 ns

X = model.crystalline_fraction(time, T)
t50 = model.time_to_fraction(0.5, T)

print(f"Time to 50% crystallization: {t50*1e9:.2f} ns")

Avrami Exponent Interpretation

Advanced Exercise: Multi-Temperature Analysis