はじめに
フェーズフィールド法は材料組織形態の時間変化を予測するシミュレーション手法です。今回はフェーズフィールド法で用いられる最もベーシックな方程式であるアレン-カーン方程式の導出についてまとめます。
アレン-カーン方程式
以下にアレン-カーン方程式を示します。
\begin{align}
\frac{\partial \phi}{\partial t}=-M_{\phi}\frac{\delta G}{\delta \phi} \tag{1}\\
\end{align}
\frac{\partial \phi}{\partial t}=-M_{\phi}\frac{\delta G}{\delta \phi} \tag{1}\\
\end{align}
ここで\(G\)は組織中のギブスの自由エネルギー、\(\phi\)は秩序変数、\(M_{\phi}\)はフェーズフィールドモビリティと呼ばれる界面の移動しやすさに関するパラメーターです。
\(G\)は秩序変数\(\phi, \nabla\phi,\nabla ^2\phi\)の関数\(G(\phi, \nabla\phi,\nabla ^2\phi)\)として書けます。このことはギブスの自由エネルギーが相界面の勾配、相界面の曲率に依存することを示しています。（この辺はまた別の記事で。）
\(\phi\)は時間・空間の関数\(\phi(x,t)\)なので、\(\phi\)を変数として持つ\(G\)は汎関数（関数の関数）と言えます。\(\delta\)は汎関数の微分を表す記号です。
アレン-カーン方程式は非保存量の秩序変数\(\phi\)（凝固、相変態、再結晶など）の時間発展を記述する方程式です。これを解くには材料組織中のギブスの自由エネルギー\(G\)をモデル化し、汎関数微分項\(\frac{\delta G}{\delta \phi}\)を計算する必要があります。
物理的背景と導出
アレン-カーン方程式\((1)\)はギブスの自由エネルギー最小化の概念から導くことが出来ます。少々長くなりますが、順に確認していきましょう。ギブスの自由エネルギーは次のように記述できます。
\begin{align}
G(\phi, \nabla\phi,\nabla ^2\phi)=\int_V g(\phi, \nabla\phi,\nabla ^2\phi)dV\tag{2}\\
\end{align}
G(\phi, \nabla\phi,\nabla ^2\phi)=\int_V g(\phi, \nabla\phi,\nabla ^2\phi)dV\tag{2}\\
\end{align}
ここで\(g\)はギブスの自由エネルギーの空間密度を表し、\(G\)と同様に秩序変数\(\phi, \nabla\phi,\nabla ^2\phi\)の関数\(g(\phi, \nabla\phi,\nabla ^2\phi)\)として書けます。
また、\(\nabla\phi\)は次のベクトル関数です。
\begin{align}
\nabla \phi=\left(\frac{\partial \phi}{\partial x},\frac{\partial\phi}{\partial y},\frac{\partial \phi}{\partial z}\right)\tag{3}\\
\end{align}
\nabla \phi=\left(\frac{\partial \phi}{\partial x},\frac{\partial\phi}{\partial y},\frac{\partial \phi}{\partial z}\right)\tag{3}\\
\end{align}
\(\nabla ^2\phi\)はラプラシアンを表し、次のスカラー関数として書けます。
\begin{align}
\nabla^2 \phi=\frac{\partial ^2 \phi}{\partial x ^2}+\frac{\partial^2 \phi}{\partial y^2}+\frac{\partial^2 \phi}{\partial z^2}\tag{4}\\
\end{align}
\nabla^2 \phi=\frac{\partial ^2 \phi}{\partial x ^2}+\frac{\partial^2 \phi}{\partial y^2}+\frac{\partial^2 \phi}{\partial z^2}\tag{4}\\
\end{align}
冒頭で述べたギブスの自由エネルギーの最小化とは「等温・等圧過程での自発変化はギブスの自由エネルギーGが減少する方向に生じる。」という熱力学上の性質です。このことは数式を用いると次のように表現できます。
\begin{align}
\frac{dG}{dt}\leq0 \tag{5}\\
\end{align}
\frac{dG}{dt}\leq0 \tag{5}\\
\end{align}
ここからは上式左辺\(\frac{dG}{dt}\)を計算することで、アレンカーン方程式\((1)\)を満たすとき、ギブスの自由エネルギー最小化（式\((5)\)）が成り立つことを確認していきます。
式\((2)\)より\(\frac{dG}{dt}\)を全微分で表現すると次式を得ます。
\begin{align}
\frac{dG(\phi, \nabla\phi,\nabla ^2\phi)}{dt}&=\int_V \frac{dg(\phi, \nabla\phi,\nabla ^2\phi)}{dt}dV\\
&=\int_V \frac{\partial g}{\partial \phi}\frac{\partial \phi}{\partial t}
+\frac{\partial g}{\partial (\nabla \phi)}\cdot \frac{\partial (\nabla\phi)}{\partial t}
+\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial (\nabla^2 \phi)}{\partial t}dV
\tag{6}
\end{align}
\frac{dG(\phi, \nabla\phi,\nabla ^2\phi)}{dt}&=\int_V \frac{dg(\phi, \nabla\phi,\nabla ^2\phi)}{dt}dV\\
&=\int_V \frac{\partial g}{\partial \phi}\frac{\partial \phi}{\partial t}
+\frac{\partial g}{\partial (\nabla \phi)}\cdot \frac{\partial (\nabla\phi)}{\partial t}
+\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial (\nabla^2 \phi)}{\partial t}dV
\tag{6}
\end{align}
ここで右辺第二項はベクトル関数同士の内積になっていることに注意しましょう。
まず右辺第二項について考えます。この項はベクトル関数による微分を含んでおり少々分かりづらいですが、成分で書き下すと次のように書けます。
\begin{align}
\int_V\frac{\partial g}{\partial (\nabla \phi)}\cdot \frac{\partial (\nabla\phi)}{\partial t}dV
&=
\int_V
\begin{pmatrix}
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial x}\right)}\\
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial y}\right)}\\
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial z}\right)}
\end{pmatrix}
\cdot
\begin{pmatrix}
\frac{\partial \left(\frac{\partial \phi}{\partial x}\right)}{\partial t}\\
\frac{\partial \left(\frac{\partial \phi}{\partial y}\right)}{\partial t}\\
\frac{\partial \left(\frac{\partial \phi}{\partial z}\right)}{\partial t}\\
\end{pmatrix}
dV\\
&= \int_V
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial x}\right)}
\frac{\partial \left(\frac{\partial \phi}{\partial x}\right)}{\partial t}
+
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial y}\right)}
\frac{\partial \left(\frac{\partial \phi}{\partial y}\right)}{\partial t}
+
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial z}\right)}
\frac{\partial \left(\frac{\partial \phi}{\partial z}\right)}{\partial t}dV
\tag{7}
\end{align}
\int_V\frac{\partial g}{\partial (\nabla \phi)}\cdot \frac{\partial (\nabla\phi)}{\partial t}dV
&=
\int_V
\begin{pmatrix}
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial x}\right)}\\
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial y}\right)}\\
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial z}\right)}
\end{pmatrix}
\cdot
\begin{pmatrix}
\frac{\partial \left(\frac{\partial \phi}{\partial x}\right)}{\partial t}\\
\frac{\partial \left(\frac{\partial \phi}{\partial y}\right)}{\partial t}\\
\frac{\partial \left(\frac{\partial \phi}{\partial z}\right)}{\partial t}\\
\end{pmatrix}
dV\\
&= \int_V
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial x}\right)}
\frac{\partial \left(\frac{\partial \phi}{\partial x}\right)}{\partial t}
+
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial y}\right)}
\frac{\partial \left(\frac{\partial \phi}{\partial y}\right)}{\partial t}
+
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial z}\right)}
\frac{\partial \left(\frac{\partial \phi}{\partial z}\right)}{\partial t}dV
\tag{7}
\end{align}
上式は\(g\)の全微分の形と一致しています。上式に部分積分（積の微分公式による変形）を適用すると次のよう変形できます。
\begin{align}
&=\int_V
\frac{\partial }{\partial x}
\left(
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial x}\right)}
\frac{\partial \phi}{\partial t}
\right)
–
\frac{\partial }{\partial x}
\left(
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial x}\right)}
\right)
\frac{\partial \phi}{\partial t}\\
&\qquad \qquad
+
\frac{\partial }{\partial y}
\left(
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial y}\right)}
\frac{\partial \phi}{\partial t}
\right)
–
\frac{\partial }{\partial y}
\left(
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial y}\right)}
\right)
\frac{\partial \phi}{\partial t}\\
&\qquad \qquad \qquad+
\frac{\partial }{\partial y}
\left(
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial z}\right)}
\frac{\partial \phi}{\partial t}
\right)
–
\frac{\partial }{\partial z}
\left(
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial z}\right)}
\right)
\frac{\partial \phi}{\partial t}
dV\\
&= \int_V
\nabla \cdot
\begin{pmatrix}
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial x}\right)}\frac{\partial \phi}{\partial t} \\
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial y}\right)}\frac{\partial \phi}{\partial t}\\
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial z}\right)}\frac{\partial \phi}{\partial t}
\end{pmatrix}
-\nabla \cdot
\begin{pmatrix}
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial x}\right)}\\
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial y}\right)}\\
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial z}\right)}
\end{pmatrix}
\frac{\partial \phi}{\partial t}
dV\\
&= \int_V
\nabla \cdot
\left(\frac{\partial g}{\partial \left(\nabla \phi\right)}\frac{\partial \phi}{\partial t}\right)
-\nabla \cdot
\left(\frac{\partial g}{\partial \left(\nabla \phi\right)}\right)
\frac{\partial \phi}{\partial t}
dV\\
\tag{8}
\end{align}
&=\int_V
\frac{\partial }{\partial x}
\left(
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial x}\right)}
\frac{\partial \phi}{\partial t}
\right)
–
\frac{\partial }{\partial x}
\left(
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial x}\right)}
\right)
\frac{\partial \phi}{\partial t}\\
&\qquad \qquad
+
\frac{\partial }{\partial y}
\left(
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial y}\right)}
\frac{\partial \phi}{\partial t}
\right)
–
\frac{\partial }{\partial y}
\left(
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial y}\right)}
\right)
\frac{\partial \phi}{\partial t}\\
&\qquad \qquad \qquad+
\frac{\partial }{\partial y}
\left(
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial z}\right)}
\frac{\partial \phi}{\partial t}
\right)
–
\frac{\partial }{\partial z}
\left(
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial z}\right)}
\right)
\frac{\partial \phi}{\partial t}
dV\\
&= \int_V
\nabla \cdot
\begin{pmatrix}
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial x}\right)}\frac{\partial \phi}{\partial t} \\
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial y}\right)}\frac{\partial \phi}{\partial t}\\
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial z}\right)}\frac{\partial \phi}{\partial t}
\end{pmatrix}
-\nabla \cdot
\begin{pmatrix}
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial x}\right)}\\
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial y}\right)}\\
\frac{\partial g}{\partial \left(\frac{\partial \phi}{\partial z}\right)}
\end{pmatrix}
\frac{\partial \phi}{\partial t}
dV\\
&= \int_V
\nabla \cdot
\left(\frac{\partial g}{\partial \left(\nabla \phi\right)}\frac{\partial \phi}{\partial t}\right)
-\nabla \cdot
\left(\frac{\partial g}{\partial \left(\nabla \phi\right)}\right)
\frac{\partial \phi}{\partial t}
dV\\
\tag{8}
\end{align}
第一項についてガウスの発散定理を用いると次のように書けます。
\begin{align}
= \int_S
\left(\frac{\partial g}{\partial \left(\nabla \phi\right)}\frac{\partial \phi}{\partial t}\right)\cdot \boldsymbol{n} dS
-\int_V \nabla \cdot
\left(\frac{\partial g}{\partial \left(\nabla \phi\right)}\right)
\frac{\partial \phi}{\partial t}
dV\tag{9}\\
\end{align}
= \int_S
\left(\frac{\partial g}{\partial \left(\nabla \phi\right)}\frac{\partial \phi}{\partial t}\right)\cdot \boldsymbol{n} dS
-\int_V \nabla \cdot
\left(\frac{\partial g}{\partial \left(\nabla \phi\right)}\right)
\frac{\partial \phi}{\partial t}
dV\tag{9}\\
\end{align}
上記の\(\boldsymbol{n}\)は境界に垂直な方向（法線方向）の単位ベクトルです。
ここで計算領域が十分広く、領域境界周辺は常に平衡状態であると仮定し、秩序変数は時間によらず一定、すなわち\(\frac{\partial\phi}{\partial t}=0\)が成立するものとします。（これはディリクレ条件に相当します。）すると式\((9)\)の第一項は0となり、最終的に式\((6)\)第二項は次のように書けます。
\begin{align}
\int_V\frac{\partial g}{\partial (\nabla \phi)}\cdot \frac{\partial (\nabla\phi)}{\partial t}dV
=-\int_V\nabla \cdot
\left(\frac{\partial g}{\partial \left(\nabla \phi\right)}\right)
\frac{\partial \phi}{\partial t}
dV\tag{10}\\
\end{align}
\int_V\frac{\partial g}{\partial (\nabla \phi)}\cdot \frac{\partial (\nabla\phi)}{\partial t}dV
=-\int_V\nabla \cdot
\left(\frac{\partial g}{\partial \left(\nabla \phi\right)}\right)
\frac{\partial \phi}{\partial t}
dV\tag{10}\\
\end{align}
同様に式\((6)\)第三項についても変形します。第二項はベクトル関数\(\nabla \phi\)の汎関数でしたが、第三項はラプラシアンによるスカラー関数\(\nabla^2 \phi\)の汎関数であることに注意します。第三項を成分表記で書き下すと以下のように書けます。
\begin{align}
\int_V \frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial (\nabla^2 \phi)}{\partial t}dV
&=
\int_V \frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial^2 \phi }{\partial x^2}+\frac{\partial^2 \phi }{\partial y^2}+\frac{\partial^2 \phi }{\partial z^2}\right)}{\partial t}dV\\
&=
\int_V \frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial^2 \phi }{\partial x^2}\right)}{\partial t}
+
\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial^2 \phi }{\partial y^2}\right)}{\partial t}
+
\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial^2 \phi }{\partial z^2}\right)}{\partial t}
dV\tag{11}
\end{align}
\int_V \frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial (\nabla^2 \phi)}{\partial t}dV
&=
\int_V \frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial^2 \phi }{\partial x^2}+\frac{\partial^2 \phi }{\partial y^2}+\frac{\partial^2 \phi }{\partial z^2}\right)}{\partial t}dV\\
&=
\int_V \frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial^2 \phi }{\partial x^2}\right)}{\partial t}
+
\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial^2 \phi }{\partial y^2}\right)}{\partial t}
+
\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial^2 \phi }{\partial z^2}\right)}{\partial t}
dV\tag{11}
\end{align}
上式各項に部分積分（積の微分公式による変形）を適用すると
\begin{align}
&=
\int_V \frac{\partial }{\partial x}\left(\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial \phi }{\partial x}\right)}{\partial t}\right)
–
\frac{\partial }{\partial x}\left(\frac{\partial g}{\partial (\nabla ^2 \phi)}\right)\frac{\partial \left(\frac{\partial \phi }{\partial x}\right)}{\partial t}\\
& \qquad \qquad +
\frac{\partial }{\partial y}\left(\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial \phi }{\partial y}\right)}{\partial t}\right)
–
\frac{\partial }{\partial y}\left(\frac{\partial g}{\partial (\nabla ^2 \phi)}\right)\frac{\partial \left(\frac{\partial \phi }{\partial y}\right)}{\partial t}\\
& \qquad \qquad \qquad \qquad +
\frac{\partial }{\partial z}\left(\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial \phi }{\partial z}\right)}{\partial t}\right)
–
\frac{\partial }{\partial z}\left(\frac{\partial g}{\partial (\nabla ^2 \phi)}\right)\frac{\partial \left(\frac{\partial \phi }{\partial z}\right)}{\partial t}dV\\
&= \int_V
\nabla \cdot \begin{pmatrix}
\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial \phi}{\partial x}\right)}{\partial t} \\
\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial \phi}{\partial y}\right)}{\partial t}\\
\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial \phi}{\partial z}\right)}{\partial t}
\end{pmatrix}dV\\
&\qquad \qquad -\int_V
\frac{\partial }{\partial x}
\left(
\frac{\partial g}{\partial (\nabla ^2 \phi)}
\right)
\frac{\partial \left(\frac{\partial \phi}{\partial x}\right)}{\partial t}
+
\frac{\partial }{\partial y}
\left(
\frac{\partial g}{\partial (\nabla ^2 \phi)}
\right)
\frac{\partial \left(\frac{\partial \phi}{\partial y}\right)}{\partial t}
+
\frac{\partial }{\partial z}
\left(
\frac{\partial g}{\partial (\nabla ^2 \phi)}
\right)
\frac{\partial \left(\frac{\partial \phi}{\partial z}\right)}{\partial t}
dV
\tag{12}
\end{align}
&=
\int_V \frac{\partial }{\partial x}\left(\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial \phi }{\partial x}\right)}{\partial t}\right)
–
\frac{\partial }{\partial x}\left(\frac{\partial g}{\partial (\nabla ^2 \phi)}\right)\frac{\partial \left(\frac{\partial \phi }{\partial x}\right)}{\partial t}\\
& \qquad \qquad +
\frac{\partial }{\partial y}\left(\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial \phi }{\partial y}\right)}{\partial t}\right)
–
\frac{\partial }{\partial y}\left(\frac{\partial g}{\partial (\nabla ^2 \phi)}\right)\frac{\partial \left(\frac{\partial \phi }{\partial y}\right)}{\partial t}\\
& \qquad \qquad \qquad \qquad +
\frac{\partial }{\partial z}\left(\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial \phi }{\partial z}\right)}{\partial t}\right)
–
\frac{\partial }{\partial z}\left(\frac{\partial g}{\partial (\nabla ^2 \phi)}\right)\frac{\partial \left(\frac{\partial \phi }{\partial z}\right)}{\partial t}dV\\
&= \int_V
\nabla \cdot \begin{pmatrix}
\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial \phi}{\partial x}\right)}{\partial t} \\
\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial \phi}{\partial y}\right)}{\partial t}\\
\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial \phi}{\partial z}\right)}{\partial t}
\end{pmatrix}dV\\
&\qquad \qquad -\int_V
\frac{\partial }{\partial x}
\left(
\frac{\partial g}{\partial (\nabla ^2 \phi)}
\right)
\frac{\partial \left(\frac{\partial \phi}{\partial x}\right)}{\partial t}
+
\frac{\partial }{\partial y}
\left(
\frac{\partial g}{\partial (\nabla ^2 \phi)}
\right)
\frac{\partial \left(\frac{\partial \phi}{\partial y}\right)}{\partial t}
+
\frac{\partial }{\partial z}
\left(
\frac{\partial g}{\partial (\nabla ^2 \phi)}
\right)
\frac{\partial \left(\frac{\partial \phi}{\partial z}\right)}{\partial t}
dV
\tag{12}
\end{align}
さらにガウスの発散定理を用いると次のように変形できます。
\begin{align}
&= \int_S
\begin{pmatrix}
\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial \phi}{\partial x}\right)}{\partial t} \\
\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial \phi}{\partial y}\right)}{\partial t}\\
\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial \phi}{\partial z}\right)}{\partial t}
\end{pmatrix}\cdot \boldsymbol{n} dS\\
&\qquad \qquad -\int_V
\frac{\partial }{\partial x}
\left(
\frac{\partial g}{\partial (\nabla ^2 \phi)}
\right)
\frac{\partial \left(\frac{\partial \phi}{\partial x}\right)}{\partial t}
+
\frac{\partial }{\partial y}
\left(
\frac{\partial g}{\partial (\nabla ^2 \phi)}
\right)
\frac{\partial \left(\frac{\partial \phi}{\partial y}\right)}{\partial t}
+
\frac{\partial }{\partial z}
\left(
\frac{\partial g}{\partial (\nabla ^2 \phi)}
\right)
\frac{\partial \left(\frac{\partial \phi}{\partial z}\right)}{\partial t}
dV\tag{13}
\end{align}
&= \int_S
\begin{pmatrix}
\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial \phi}{\partial x}\right)}{\partial t} \\
\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial \phi}{\partial y}\right)}{\partial t}\\
\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial \left(\frac{\partial \phi}{\partial z}\right)}{\partial t}
\end{pmatrix}\cdot \boldsymbol{n} dS\\
&\qquad \qquad -\int_V
\frac{\partial }{\partial x}
\left(
\frac{\partial g}{\partial (\nabla ^2 \phi)}
\right)
\frac{\partial \left(\frac{\partial \phi}{\partial x}\right)}{\partial t}
+
\frac{\partial }{\partial y}
\left(
\frac{\partial g}{\partial (\nabla ^2 \phi)}
\right)
\frac{\partial \left(\frac{\partial \phi}{\partial y}\right)}{\partial t}
+
\frac{\partial }{\partial z}
\left(
\frac{\partial g}{\partial (\nabla ^2 \phi)}
\right)
\frac{\partial \left(\frac{\partial \phi}{\partial z}\right)}{\partial t}
dV\tag{13}
\end{align}
式\((6)\)第二項の時と同様に計算領域が十分広く、領域境界周辺は常に平衡状態であると仮定し、境界において秩序変数の勾配は一定、すなわち\(\frac{\partial\phi}{\partial x}=\frac{\partial\phi}{\partial y}=\frac{\partial\phi}{\partial z}=0\)が成立するものとします。（これはノイマン条件に相当します。）すると式\((13)\)の第一項は0となり、次のように書けます。
\begin{align}
&=-\int_V
\frac{\partial }{\partial x}
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}
\right)
\frac{\partial \left(\frac{\partial \phi}{\partial x}\right)}{\partial t}
+
\frac{\partial }{\partial y}
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi \right)}
\right)
\frac{\partial \left(\frac{\partial \phi}{\partial y}\right)}{\partial t}
+
\frac{\partial }{\partial z}
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}
\right)
\frac{\partial \left(\frac{\partial \phi}{\partial z}\right)}{\partial t}
dV\tag{14}\\
\end{align}
&=-\int_V
\frac{\partial }{\partial x}
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}
\right)
\frac{\partial \left(\frac{\partial \phi}{\partial x}\right)}{\partial t}
+
\frac{\partial }{\partial y}
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi \right)}
\right)
\frac{\partial \left(\frac{\partial \phi}{\partial y}\right)}{\partial t}
+
\frac{\partial }{\partial z}
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}
\right)
\frac{\partial \left(\frac{\partial \phi}{\partial z}\right)}{\partial t}
dV\tag{14}\\
\end{align}
さて、しつこいですが上式はさらに部分積分できそうです。
\begin{align}
&=\int_V
\frac{\partial^2 }{\partial x^2}
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}
\frac{\partial \phi}{\partial t}
\right)
–
\frac{\partial^2 }{\partial x^2}
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}
\right)
\frac{\partial \phi}{\partial t}\\
&\qquad \qquad
+
\frac{\partial^2 }{\partial y^2}
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}
\frac{\partial \phi}{\partial t}
\right)
–
\frac{\partial ^2}{\partial y^2}
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}
\right)
\frac{\partial \phi}{\partial t}\\
&\qquad \qquad \qquad+
\frac{\partial^2 }{\partial z^2}
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}
\frac{\partial \phi}{\partial t}
\right)
–
\frac{\partial^2 }{\partial z^2}
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}
\right)
\frac{\partial \phi}{\partial t}
dV\\
&= \int_V
\nabla ^2
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}\frac{\partial \phi}{\partial t}
\right)
-\nabla ^2
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}\right)
\frac{\partial \phi}{\partial t}
dV
\tag{15}\\
\end{align}
&=\int_V
\frac{\partial^2 }{\partial x^2}
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}
\frac{\partial \phi}{\partial t}
\right)
–
\frac{\partial^2 }{\partial x^2}
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}
\right)
\frac{\partial \phi}{\partial t}\\
&\qquad \qquad
+
\frac{\partial^2 }{\partial y^2}
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}
\frac{\partial \phi}{\partial t}
\right)
–
\frac{\partial ^2}{\partial y^2}
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}
\right)
\frac{\partial \phi}{\partial t}\\
&\qquad \qquad \qquad+
\frac{\partial^2 }{\partial z^2}
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}
\frac{\partial \phi}{\partial t}
\right)
–
\frac{\partial^2 }{\partial z^2}
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}
\right)
\frac{\partial \phi}{\partial t}
dV\\
&= \int_V
\nabla ^2
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}\frac{\partial \phi}{\partial t}
\right)
-\nabla ^2
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}\right)
\frac{\partial \phi}{\partial t}
dV
\tag{15}\\
\end{align}
再び第一項についてガウスの発散定理を用いると
\begin{align}
&= \int_S
\nabla
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}\frac{\partial \phi}{\partial t}
\right)
\cdot \boldsymbol{n} dS
-\int_V\nabla ^2
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}\right)
\frac{\partial \phi}{\partial t}
dV
\tag{16}\\
\end{align}
&= \int_S
\nabla
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}\frac{\partial \phi}{\partial t}
\right)
\cdot \boldsymbol{n} dS
-\int_V\nabla ^2
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}\right)
\frac{\partial \phi}{\partial t}
dV
\tag{16}\\
\end{align}
上述の通り、領域境界において\(\frac{d\phi}{dt}=0\)を仮定すると面積分は消えて次式を得ます。
\begin{align}
&=
-\int_V\nabla ^2
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}\right)
\frac{\partial \phi}{\partial t}
dV
\tag{17}\\
\end{align}
&=
-\int_V\nabla ^2
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}\right)
\frac{\partial \phi}{\partial t}
dV
\tag{17}\\
\end{align}
以上より最終的に式(6)第三項は次のように書けます。
\begin{align}
\int_V
\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial (\nabla^2 \phi)}{\partial t}dV
=
-\int_V\nabla ^2
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}\right)
\frac{\partial \phi}{\partial t}
dV
\tag{18}
\end{align}
\int_V
\frac{\partial g}{\partial (\nabla ^2 \phi)}\frac{\partial (\nabla^2 \phi)}{\partial t}dV
=
-\int_V\nabla ^2
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}\right)
\frac{\partial \phi}{\partial t}
dV
\tag{18}
\end{align}
ここまでで式\((6)\)の各項の変形が完了しました。式\((10),(18)\)を式\((6)\)に代入すると
\begin{align}
\frac{dG(\phi, \nabla\phi,\nabla ^2\phi)}{dt}&=\int_V \frac{dg(\phi, \nabla\phi,\nabla ^2\phi)}{dt}dV\\
&=\int_V \frac{\partial g}{\partial \phi}\frac{\partial \phi}{\partial t}
-\nabla \cdot
\left(\frac{\partial g}{\partial \left(\nabla \phi\right)}\right)
\frac{\partial \phi}{\partial t}
-\nabla ^2
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}\right)
\frac{\partial \phi}{\partial t}
dV\\
&=\int_V \left[\frac{\partial g}{\partial \phi}
-\nabla \cdot
\left(\frac{\partial g}{\partial \left(\nabla \phi\right)}\right)
-\nabla ^2
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}\right)\right]
\frac{\partial \phi}{\partial t}dV
\tag{19}
\end{align}
\frac{dG(\phi, \nabla\phi,\nabla ^2\phi)}{dt}&=\int_V \frac{dg(\phi, \nabla\phi,\nabla ^2\phi)}{dt}dV\\
&=\int_V \frac{\partial g}{\partial \phi}\frac{\partial \phi}{\partial t}
-\nabla \cdot
\left(\frac{\partial g}{\partial \left(\nabla \phi\right)}\right)
\frac{\partial \phi}{\partial t}
-\nabla ^2
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}\right)
\frac{\partial \phi}{\partial t}
dV\\
&=\int_V \left[\frac{\partial g}{\partial \phi}
-\nabla \cdot
\left(\frac{\partial g}{\partial \left(\nabla \phi\right)}\right)
-\nabla ^2
\left(
\frac{\partial g}{\partial \left(\nabla ^2 \phi\right)}\right)\right]
\frac{\partial \phi}{\partial t}dV
\tag{19}
\end{align}
右辺の括弧の中は\(G\)の\(\phi\)に関する汎関数微分\(\frac{\delta G}{\delta \phi}\)の定義そのものなので次式のように書けます。
\begin{align}
\frac{dG}{dt}
=\int_V
\frac{\delta G}{\delta \phi}
\frac{\partial \phi}{\partial t}dV
\tag{20}
\end{align}
\frac{dG}{dt}
=\int_V
\frac{\delta G}{\delta \phi}
\frac{\partial \phi}{\partial t}dV
\tag{20}
\end{align}
上記がギブスの自由エネルギーの時間変動の汎関数微分による表現です。この式にアレンカーン方程式\((1)\)を代入します。
\begin{align}
\frac{dG}{dt}=-M_{\phi}\int_V \left(\frac{\delta G}{\delta \phi}\right)^2dV\tag{21}\\
\end{align}
\frac{dG}{dt}=-M_{\phi}\int_V \left(\frac{\delta G}{\delta \phi}\right)^2dV\tag{21}\\
\end{align}
積分の中身は2乗項なので積分自体は正の値を持ちます。また、フェーズフィールドモビリティ\(M_{\phi}\)も通常、正の値を持ちます。したがって右辺全体としてはマイナス符号により負の値を取ります。
\begin{align}
\frac{dG}{dt}=-M_{\phi}\int_V \left(\frac{\delta G}{\delta \phi}\right)^2dV\leq 0\tag{22}\\
\end{align}
\frac{dG}{dt}=-M_{\phi}\int_V \left(\frac{\delta G}{\delta \phi}\right)^2dV\leq 0\tag{22}\\
\end{align}
上式はギブスの自由エネルギーの最小化の式\((5)\)と一致しており、アレン-カーン方程式が成立するとき、ギブスの自由エネルギーは時間とともに減少することが分かります。
おわりに
今回はアレン-カーン方程式の導出についてまとめました。この辺は教科書では一行で飛ばされることが多いようですが、実際追ってみると中々ヘビーな式変形をしています。次はギブスの自由エネルギーの数式表現についてまとめたいと思います。
参考文献
[1] 高木 知弘ら, “フェーズフィールド法”, 養賢堂, 2012/3/2
[2] 山中 晃徳ら, “Pythonによるフェーズフィールド法入門 基礎理論からデータ同化の実装まで”, 丸善出版, 2023/12/15
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