This python code was developed by Yamanaka research group of Tokyo University of Agriculture and Technology in August 2019.
Let us consider an example: growth of phase B in phase A. This notebook shows one-dimensional phase-field simulation of the growth of phase B using Allen-Cahn equation.
For details of the model, please see reference (S. M. Allen and J. W. Cahn, "A Microscopic Theory for Antiphase Boundary Motion and Its Application to Antiphase Domain Coarsening," Acta Metallurgica, Vol. 27 (1979), pp. 1085–1095.).
An non-conserved order parameter $\phi$ (hereafter, we call it as "phase-field variable".) is used. The phase-field variable $\phi$ is defined as $\phi = 0$ in phase A and $\phi = 1$ in phase B. The phase-field variable $\phi$ smoothly changes from 0 to 1 in an "diffuse" interfacial region.
The total Gibbs free energy of the system is defined by $$ G = \int_{V} (g_{chem}(\phi) + g_{doub}(\phi) + g_{grad}(\nabla\phi)) dV $$ where, $g_{chem}$, $g_{doub}$, and $g_{grad}$ are the chemical free energy, the double-well potential energy and the gradient energy densities, respectively. In this source code, these energy densities are defined as follows: $$ g_{chem} = p(\phi)g_{B} + (1-p(\phi))g_{A} $$ $$ g_{doub} = Wq(\phi) $$ $$ g_{grad} = \frac{a^{2}}{2}|\nabla\phi|^{2} $$ where $g_{A}$ and $g_{B}$ are Gibbs free energy densities of pure phase A and pure phase B, respectively. Here, $g_{A}$ and $g_{B}$ are assumed to be constant.
$p(\phi)$ and $q(\phi)$ are an interpolation function and the double-well potential function. These functions are often defined as follows: $$ p(\phi) = \phi^{2}(3-2\phi) $$ $$ q(\phi) = \phi^{2}(1-\phi)^{2} $$
$W$ and $a$ are the height of double-well potential energy and the gradient energy coefficient, respectively, which are given by $$ W = \frac{6\sigma\delta}{b} $$ $$ a = \sqrt{\frac{3\sigma b}{\delta}} $$ where $\sigma$ is the interfacial energy of an interface between phase A and B. $\delta$ is the thickness (length in one-dimension) of the diffuse interface. $b$ is given as $b = 2\tanh^{-1}(1-2\lambda)$ where $\lambda$ is a constant which is used for determining the thickness of interfacial region. In this source code, we use $\lambda$ = 0.1 and then $b$ = 2.2.
The time evolution of the phase-field variable (that describes the migration of the interface) is given by Allen-Cahn equation by assuming that the total free energy of the system, $G$, decreases monotonically with time (i.e. The second law of thermodynamics): $$ \frac{\partial \phi}{\partial t} = -M_{\phi}\frac{\delta G}{\delta \phi}=-M_{\phi}\left(\frac{\partial p(\phi)}{\partial \phi}(g_{B}-g_{A})+W\frac{\partial q(\phi)}{\partial\phi}-a^{2}\nabla^{2}\phi\right) $$ where $M_{\phi}$ is the mobility of phase-field variable and is given by: $$ M_{\phi} = \frac{\sqrt{2W}}{6a}M $$ Here, $M$ is a "physical" mobility of interface between phase A and B. The term $g_{B}-g_{A}$ in the Allen-Cahn equation corresponds to the driving force of the growth of phase B.
Note that the Euler-Lagrange equation given by the following equation is used to calculate the functional derivative of $G$ with respect to the phase-field variable as: $$ \frac{\delta G}{\delta \phi}=\frac{\partial g}{\partial \phi}-\nabla\cdot\frac{\partial g}{\partial (\nabla \phi)} $$
In this notebook, you use NumPy, Matplotlib and Math libraries.
Numpy is a fundamental package for scientific computing in Python that provides a multidimensional array object.
Matplotlib is a Python library used for plotting the beautiful and attractive graphs and animations.
Math is a library for accessing to some common math functions and constants in Python
%matplotlib nbagg
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import animation
import math
nx = 64 # number of computational grids
dx = 0.5e-6 # spacing of computational grid [m]
eee = 5.e+5 # magnitude of driving force of growth of phase B: g_A - g_B [J/m3] = [Pa]
sigma = 1.0 # ineterfacial energy [J/m2]
delta = 4.*dx # interfacial thickness [m]
amobi = 4.e-14 # interfacial mobilitiy [m4/(Js)]
ram = 0.1 # paraneter which deternines the interfacial area
bbb = 2.*np.log((1.+(1.-2.*ram))/(1.-(1.-2.*ram)))/2. # The constant b = 2.1972 (please see the handout)
aaa = np.sqrt(3.*delta*sigma/bbb) # gradient energy coefficient "a"[(J/m)^(1/2)]
www = 6.*sigma*bbb/delta # potential height W [J/m3]
pmobi = amobi*math.sqrt(2.*www)/(6.*aaa) # mobility of phase-field [m3/(Js)]
dt = dx*dx/(5.*pmobi*aaa*aaa)/2 # time increment for a time step [s]
nsteps = 500 # total number of time step
p = np.zeros((nsteps,nx)) # phase-field variable for all time steps
driv = np.zeros((nsteps,nx)) # array for saving driving force term (only for visualization)
grad = np.zeros((nsteps,nx)) # array for saving gradient force term (only for visualization)
The initial nuclei of the phase B (region of $\phi = 1$) is located at the origin of the computational domain.
The initial profile of the phase-field variable along $x$-direction is calculated using the equilibrium profile (see Appendix in the handout for the derivation): $$ \phi = \frac{1}{2}\left(1-\tanh \frac{\sqrt{2W}}{2a}(x-r_{0}) \right) $$ where $r_{0}$ denotes the initial position of the interface.
r_nuclei = 10.*dx # length of the initial B phase
for i in range(0,nx):
r = i*dx - r_nuclei
p[0,i] = 0.5*(1.-np.tanh(np.sqrt(2.*www)/(2.*aaa)*r))
Allen-Cahn equation is discretized by simple finite difference method in this program.
The 1st-order foward Euler method is used for the time-integration. The 2nd-order central finite difference method is used for the spatial derivatives. The discretized Allen-Cahn equation is given as: $$ \phi^{t+\Delta t}_{i} = \phi^{t}_{i} + M_{\phi} \left[ 4W\phi^{t}_{i}\left(1-\phi^{t}_{i}\right)\left(\phi^{t}_{i}-\frac{1}{2}+\frac{3}{2W}(g_{A}-g_{B})\right) +a^{2}\left(\frac{\phi^{t}_{i+1}-2\phi^{t}_{i}+\phi^{t}_{i-1}}{(\Delta x)^{2}}\right) \right]\Delta t $$
For visuallization, arrays "driv" and "grad" are saved.
def do_timestep(p,driv,grad):
for t in range(nsteps-1):
for i in range(nx):
ip = i + 1
im = i - 1
if ip > nx - 1:
ip = nx -1
if im < 0:
im = 0
p[t+1,i] = p[t,i] + pmobi * ( 4.*www*p[t,i]*(1.-p[t,i])*(p[t,i]-0.5+3./(2.*www)*eee) + aaa*aaa*((p[t,ip] - 2*p[t,i] + p[t,im])/dx/dx) ) * dt
driv[t+1,i] = pmobi*(4.*www*p[t,i]*(1.-p[t,i])*(p[t,i]-0.5+3./(2.*www)*eee))
grad[t+1,i] = pmobi*(aaa*aaa*(p[t,ip] - 2*p[t,i] + p[t,im])/dx/dx)
Check the role of driving force and gradient terms in Allen-Cahn equation.
The driving force tem: $$ M_{\phi}\left[ 4W\phi^{t}_{i}\left(1-\phi^{t}_{i}\right)\left(\phi^{t}_{i}-\frac{1}{2}+\frac{3}{2W}(g_{A}-g_{B})\right) \right] $$ The gradient (or diffusion) term: $$ M_{\phi}\left[ a^{2}\left(\frac{\phi^{t}_{i+1}-2\phi^{t}_{i}+\phi^{t}_{i-1}}{(\Delta x)^{2}}\right) \right] $$
# calculate Allen-Cahn equation for "nsteps" times
do_timestep(p,driv,grad)
# visualize the result
fig = plt.figure(figsize=(7,9))
fig.set_dpi(100)
plt.subplots_adjust(hspace=0.5)
ax1 = fig.add_subplot(4,1,1)
ax2 = fig.add_subplot(4,1,2)
ax3 = fig.add_subplot(4,1,3)
ax4 = fig.add_subplot(4,1,4)
def animate(i):
ax1.clear()
plt.subplot(4, 1, 1)
plt.ylim([0,1])
plt.xlim([0,nx])
plt.xlabel('Grid number')
plt.ylabel('Phase-field variable')
plt.grid(True)
plt.plot(p[i,:],color="b",marker="o")
ax2.clear()
plt.subplot(4, 1, 2)
plt.ylim([-0.03,0.03])
plt.xlim([0,nx])
plt.xlabel('Grid number')
plt.ylabel('Driving force term')
plt.grid(True)
plt.plot(driv[i,:],color="r",marker="o")
ax3.clear()
plt.subplot(4, 1, 3)
plt.ylim([-0.03,0.03])
plt.xlim([0,nx])
plt.xlabel('Grid number')
plt.ylabel('Gradient term')
plt.grid(True)
plt.plot(grad[i,:],color="y",marker="o")
ax4.clear()
plt.subplot(4, 1, 4)
plt.xlim([0,nx])
plt.xlabel('Grid number')
plt.ylabel('Driving + Gradient')
plt.grid(True)
plt.plot(grad[i,:]+driv[i,:],marker="o")
anim = animation.FuncAnimation(fig,animate,frames=nsteps-1,interval=100,repeat=False)
plt.show()
