Sep 6, 2026 - 5 MIN READ

相变驱动力计算

基于 Gibbs-Duhem 关系与扩散势，推导多元两相体系中相变驱动力的两种表达形式及其等价关系，并附 Pandat 计算数据的验算示例。

Jing Zhong

在多元合金体系中，相变驱动力是判断新相能否析出以及析出倾向大小的关键热力学量。本文整理了基于 Gibbs-Duhem 关系的两种驱动力表达形式，以及二者的等价关系。

Reference: Hillert, M. Phase Equilibria, Phase Diagrams and Phase Transformations: Their Thermodynamic Basis, 2nd edition.; Cambridge University Press: Cambridge; New York, 2007.

相变驱动力计算

相变驱动力

Note 1: $\{\mu_i^{\alpha, 0}\}$ are fixed values for $\alpha$ phase, where $\{x_i\} = \{x_i^{\alpha, 0}\}$ .

Note 2: $\{\mu_i^{\beta, \textrm{new}}\}$ are fixed values for $\beta$ phase, where $\{x_i\} = \{x_i^{\beta, \textrm{new}}\}$ .

Note 3: $\{x_i^{\alpha, 0}\}$ and $\{x_i^{\beta, \textrm{new}}\}$ are taken as known value, and so are $\{\mu_i^{\alpha, 0}\}$ and $\{\mu_i^{\beta, \textrm{new}}\}$ .

表达形式一

对于多元两相体系，包含 $n$ 个组元，以及 $\alpha$ 和 $\beta$ 两个相。假设 $\alpha$ 相的成分为 $x_i^{\alpha, 0}$ ，而新相的成分为 $x_i^{\beta, \textrm{new}}$ 。根据Gibbs-Duhem关系， $\alpha$ 相相成分的能量切面可以表示为

f^\alpha=\sum_{i=1}^n x_i \mu_i^{\alpha, 0}

同理， $\beta$ 相相成分的能量切面可以表示为

f^\beta=\sum_{i=1}^n x_i \mu_i^{\beta, \textrm{new}}

根据驱动力的定义

\left.-\Delta G_m^{\alpha \rightarrow \beta}\right|_{\mathbf{x}=\mathbf{x}^{\beta \text { new }}}=f^\alpha-f^\beta

即

\left.-\Delta G_m^{\alpha \rightarrow \beta}\right|_{\mathbf{x}=\mathbf{x}^{\beta \text { new }}}=\sum_{i=1}^n x_i^{\beta, \text { new }} \mu_i^{\alpha, 0}-\sum_{i=1}^n x_i^{\beta, \text { new }} \mu_i^{\beta, \text { new }}=\sum_{i=1}^n x_i^{\beta, \text { new }}\left(\mu_i^{\alpha, 0}-\mu_i^{\beta, 0}\right)

表达形式二

采用扩散势也可以表示能量切面，如 $\alpha$ 相和 $\beta$ 相相成分的能量切面可以表示为

\begin{aligned} & f^\alpha=G_m^{\alpha, 0}+\sum_{i=1}^{n-1}\left(x_i-x_i^{\alpha, 0}\right) \tilde{\mu}_i^{\alpha, 0} \\ & f^\beta=G_m^{\beta, 0}+\sum_{i=1}^{n-1}\left(x_i-x_i^{\beta, 0}\right) \tilde{\mu}_i^{\beta, 0} \end{aligned}

驱动力可以表示为

\begin{aligned} & \left.-\Delta G_m^{\alpha \rightarrow \beta}\right|_{\mathbf{x}=\mathbf{x}^{\beta, \text { new }}}=G_m^{\alpha, 0}+\sum_{i=1}^{n-1}\left(x_i^{\beta, \text { new }}-x_i^{\alpha, 0}\right) \tilde{\mu}_i^{\alpha, 0}-G_m^{\beta, 0}-\sum_{i=1}^{n-1}\left(x_i^{\beta, \text { new }}-x_i^{\beta, \text { new }}\right) \tilde{\mu}_i^{\beta, 0} \\ & =G_m^{\alpha, 0}-G_m^{\beta, 0}+\sum_{i=1}^{n-1}\left(x_i^{\beta, \text { new }}-x_i^{\alpha, 0}\right) \tilde{\mu}_i^{\alpha, 0} \end{aligned}

两种表达形式的关系

\begin{aligned} & \left.\Delta G_m^{\alpha \rightarrow \beta}\right|_{\mathbf{x}=\mathbf{x}^{\beta, \text { new }}}=\sum_{i=1}^n x_i^{\beta, \text { new }} \mu_i^{\alpha, 0}-G_m^{\beta, \text { new }} \\ & =\sum_{i=1}^n\left(x_i^{\beta, \text { new }}-x_i^{\alpha, 0}\right) \mu_i^{\alpha, 0}-G_m^{\beta, \text { new }}+\sum_{i=1}^n x_i^{\alpha, 0} \mu_i^{\alpha, 0} \\ & =\sum_{i=1}^n\left(x_i^{\beta, \text { new }}-x_i^{\alpha, 0}\right) \mu_i^{\alpha, 0}-G_m^{\beta, \text { new }}+G_m^{\alpha, 0} \\ & =G_m^{\alpha, 0}-G_m^{\beta, \text { new }}+\sum_{i=1}^n\left(x_i^{\beta, \text { new }}-x_i^{\alpha, 0}\right)\left(\mu_i^{\alpha, 0}-\mu_n^{\alpha, 0}\right)+\mu_n^{\alpha, 0} \sum_{i=1}^n\left(x_i^{\beta, \text { new }}-x_i^{\alpha, 0}\right) \\ & =G_m^{\alpha, 0}-G_m^{\beta, \text { new }}+\sum_{i=1}^{n-1}\left(x_i^{\beta, \text { new }}-x_i^{\alpha, 0}\right) \tilde{\mu}_i^{\alpha, 0} \end{aligned}

以下为一个演算示例

# 能量数据为Pandat软件计算结果，Ni-Al体系的fcc与L12_fcc的计算结果，数据库为Pandat的镍基高温合金数据库。
x_matrix_al = 0.2
x_matrix_ni = 0.8
mu_matrix_al = -157793.0
mu_matrix_ni = -51148.8
gm_matrix = -72477.6

print("gm: ", gm_matrix, x_matrix_al * mu_matrix_al + x_matrix_ni * mu_matrix_ni)

x_l12_al = 0.24
x_l12_ni = 0.76
gm_l12 = -79069.11
mu_l12_al = -182955.13
mu_l12_ni = -46263.00
print("gm: ", gm_l12, x_l12_al * mu_l12_al + x_l12_ni * mu_l12_ni)
print("dp l12: ", mu_l12_al - mu_l12_ni)

driving_force_v1 = x_l12_al * (mu_matrix_al - mu_l12_al) + x_l12_ni * (mu_matrix_ni - mu_l12_ni)
print("driving_force_v1: ", driving_force_v1)

driving_force_v2 = gm_matrix - gm_l12 + (x_l12_al - x_matrix_al) * (mu_matrix_al - mu_matrix_ni)
print("driving_force_v2: ", driving_force_v2)

验算结果为

gm:  -72477.6 -72477.64000000001
gm:  -79069.11 -79069.1112
dp l12:  -136692.13
driving_force_v1:  2325.703199999998
driving_force_v2:  2325.7419999999975

参考说明

Hillert, M.

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