非平衡界面动力学理论

图1 不同相变类型中的界面动力学过程示意图

Fig.1 Schematics for the interface kinetic processes in different phase transformation types (L—liquid, S—solid, I—interface, A—solvent atom A, B—solute atom B, α—solid phase α, β—solid phase β, the same below)

(a) liquid/solid interface during melting

(b) solid/liquid interface during solidification

从字面上看，界面动力学状态关注的是界面所处的动力学状态，即平衡或非平衡^[5]。在实际的科学问题中，平衡或非平衡界面对凝固过程的影响十分广泛。例如，非平衡成分梯度可对凝固过程的温度梯度和浓度梯度产生重要影响，其不仅影响凝固过程中的温度分布，还影响晶体的生长方式和生长形态。通过控制凝固过程中的成分梯度，可有效改变组织形貌和相分布，进而影响材料的力学性能。从凝固理论的角度来看，界面状态与体积相的扩散动力学紧密相关，故在对界面动力学状态描述时，需同时考虑界面和体积相的动力学状态。同样以二元合金凝固为例，依据界面迁移速率的大小，界面状态可分为如下4类。

(1) 完全平衡。当界面速率趋近于零时，界面处固相和液相成分均为平衡成分，且液相和固相成分均一^[6]。此时，界面处于完全平衡状态，即：

$C S * = C S = C S e q, C L * = C L = C L e q$ (1)

式中， $C_{S}^{e q}$ 和 $C_{L}^{e q}$ 分别为固相和液相的平衡成分，C_S和C_L分别为固相和液相成分。

(2) 局域平衡。当界面迁移速率较小时，界面处固相和液相成分均为平衡成分，但液相和固相中存在成分梯度。此时，界面迁移速率仅由界面处固相和液相溶质扩散决定(也即扩散控制生长机制^[6,7])：

$J B L * - J B S * = V V m C L * - C S * = - D L V m ∂ C L * ∂ Z + D S V m ∂ C S * ∂ Z$

( $C S * = C S e q$ , $C L * = C L e q$ ) (2)

式中， $J_{B}^{S *}$ 和 $J_{B}^{L *}$ 分别为界面处固相和液相的溶质通量，V_m是摩尔体积(本文假设A和B 2种组成元素分别为溶剂和溶质中的置换组元，且其摩尔体积相等)，D_S和D_L分别为固相和液相中的溶质扩散系数，Z是坐标轴。需要说明的是，经典Zener扩散控制生长机制^[6,7]忽略了生长相中的溶质扩散(即式(2)中 $J_{B}^{S *}$ = 0且 $\partial C_{S}^{*} / \partial Z$ = 0)^[6,7]。

(3) 局域非平衡。当界面迁移速率较大但仍远小于液相中溶质扩散速率( $V_{D}^{L}$ )时，界面处固相和液相成分偏离平衡成分，且液相和固相中存在成分梯度。此时，界面迁移速率由界面处固相和液相溶质扩散以及界面迁移驱动力共同决定：

$J B L * - J B S * = V V m C L * - C S * = - D L V m ∂ C L * ∂ Z + D S V m ∂ C S * ∂ Z$ (3)

$V = - M Δ G m$ (4)

式中， $Δ$ G_m为界面迁移驱动力，M为界面移动性系数。式(4)即为经典的Christian界面控制生长机制^[8]。需要说明的是，此时固相和液相的溶质通量均正比于成分梯度，也即遵循经典Fick扩散定律。此外，由于界面迁移速率由扩散控制机制(式(3))和界面控制机制(式(4))共同决定，因此这种情况也常被称为混合控制生长机制^[9]。

(4) 完全非平衡。当界面迁移速率与液相中溶质扩散速率相当时，界面迁移速率同样由界面处固相和液相扩散以及界面迁移驱动力共同决定，但是固相和液相中的溶质通量不再仅正比于成分梯度，而且与扩散历史相关^[10]，也即：

$J B L * - J B S * = V V m C L * - C S * = - D L V m ∂ C L * ∂ Z - τ D L ∂ J B L * ∂ t +$

$D S V m ∂ C S * ∂ Z + τ D S ∂ J B S * ∂ t$ (5)

式中，t为时间， $τ_{D}^{S}$ = $D_{S} / (V_{D}^{S})^{2}$ ( $V_{D}^{S}$ 是固相中的溶质扩散速率)和 $τ_{D}^{L}$ = $D_{L} / (V_{D}^{L})^{2}$ ( $V_{D}^{L}$ 是液相中的溶质扩散速率)分别为固相和液相的溶质扩散弛豫参数。在此情形下，不仅界面处于非平衡状态，固相和液相的溶质扩散也偏离经典的Fick扩散定律而“远离平衡”，故本文将其定义为“完全非平衡”。需要特别说明的是，当界面迁移速率达到甚至超过液相中溶质扩散速率时，稳态时界面处固相成分与液相成分相等，且固相和液相中均无成分梯度^[11]。此时，发生无偏析凝固^[12,13] (即完全溶质截留，详见1.2节)，合金的界面迁移过程与纯物质相同，仅伴随着液相原子的有序化但无溶质原子在界面处的再分配。

界面动力学通常包括2类过程(图1)：界面迁移和溶质跨界面扩散。前者对应着界面迁移速率与界面迁移驱动力之间的关系(式(4))。以纯物质为例，通常情况下界面温度偏离平衡熔点温度越大，则驱动力越大、界面迁移速率也越大。类似地，可认为合金界面迁移动力学过程使得界面温度偏离平衡温度(或者说初始成分对应的液相线温度)^[14]。而后者对应着界面成分与界面迁移速率之间关系，或者说非平衡溶质分配系数( $k$ = $C_{S}^{*} / C_{L}^{*}$ )随界面迁移速率的变化关系。换言之，溶质跨界面扩散过程使得界面成分偏离平衡成分^[14]。

事实上，随着界面迁移速率的不断提高，k不断偏离平衡溶质分配系数( $k_{e}$ = $C_{S}^{e q} / C_{L}^{e q}$ )，直到k = 1，此现象被称为溶质截留效应^[15]。从物理机制的角度而言，溶质截留效应是指当界面迁移速率较大时，更多的溶质原子会被包裹入新生成的固相之中(图1)。尽管从热力学的角度而言，这些多余的溶质原子具有通过溶质跨界面扩散重新回到界面处液相中的驱动力，但由于界面迁移速率很快，新的固相会迅速生成，进而阻碍了上述溶质跨界面扩散过程的发生，故相比平衡成分，更多的溶质原子最终被保留到固相之中。这一非平衡效应最早可追溯到1969年Baker和Cahn^[16]的工作，他们发现在急冷条件下，Cd在Zn-Cd合金中的固溶度甚至可以超过其在平衡相图中的最大固溶度极限。

同样，与界面迁移动力学过程相对应，还存在另一非平衡效应，即溶质拖曳效应。从定义上讲，溶质拖曳效应是指溶质原子和界面的相互作用(即溶质跨界面扩散过程)会消耗界面迁移的驱动力，从而使界面迁移速率降低的现象。该平衡效应可追溯到1957年Lücke和Detert的工作^[17]，他们发现在高纯Al中加入0.01% (质量分数)的Mn和Fe后，再结晶速率均会大幅度降低。对于相界面，纯物质的界面驱动力全部用于驱动界面迁移，而合金中的跨界面扩散过程会消耗部分界面驱动力，进而使得界面迁移速率相比纯物质而言大幅降低。但当完全溶质截留发生后，溶质跨界面扩散过程被完全抑制，溶质拖曳效应也会随之消失。

需要特别指出的是，在固态相变领域，溶质拖曳效应在非平衡界面动力学理论建立之初便被广泛接受。但在凝固领域，凝固过程中是否存在溶质拖曳效应？溶质拖曳效应程度如何？这些问题一直伴随着非平衡凝固界面动力学理论的发展，且至今也未得到完全解决。然而，抛开液/固相变、固/液相变和固态相变的不同之处，如应力问题、对流问题等，3者从不可逆热力学的角度完全相同，故所建立的非平衡界面动力学理论也应完全相同。

本文从非平衡界面动力学的共性问题出发，详细论述基于尖锐界面理论假设的非平衡界面动力学理论的发展历史与现状。在总结经典理论模型和最新理论研究成果的基础上，分析这一领域的发展趋势并进行展望，以期推动人们对非平衡界面动力学的认识与理解，并为组织调控提供借鉴。需要说明的是，下文将以二元合金凝固为论述对象，但相关理论论述也同样适用于熔化和固态相变。

1 局域非平衡条件下的二元合金界面动力学理论

界面动力学理论是通过建立界面动力学通量与对应驱动力之间的关系来确定界面参变量之间的关系，进而得到界面温度和界面成分随界面速率的变化规律(稳定态情况)或者界面温度、界面成分和界面速率随时间的变化规律(非稳态情况)。因此，本节将首先介绍界面动力学过程，而后依次对稳态和非稳态界面动力学理论的发展进行论述。

1.1 界面动力学过程

Baker^[18]在1965年首次对非平衡界面动力学过程进行了描述，认为二元合金的凝固过程首先可以看作是溶剂A从液相原子变成固相原子和溶质B从液相原子变成固相原子2个过程(图2)。

图2

图2 液相直接凝固出目标固相成分的固/液界面动力学过程示意图

Fig.2 Schematic for the solid/liquid interface kinetic processes when the solid with the aimed composition is solidified directly from the liquid

对于迁移速率为V的平界面，单位时间、单位面积通过固/液界面的A原子和B原子数目，也即A原子和B原子的通量，分别为：

J_{A}^{S *} = \frac{V}{V_{m}} (1 - C_{S}^{*}), J_{B}^{S *} = \frac{V}{V_{m}} C_{S}^{*}

(6)

式中， $J_{A}^{S *}$ 为界面处固相中溶剂A的通量。 $J_{A}^{S *}$ 和 $J_{B}^{S *}$ 对应的共轭驱动力(X_A和X_B)为固相原子与液相原子的化学势之差( $Δ μ_{A}^{*}$ 和 $Δ μ_{B}^{*}$ )，也即：

X_{A} = Δ μ_{A}^{*} = μ_{A}^{S *} - μ_{A}^{L *}, X_{B} = Δ μ_{B}^{*} = μ_{B}^{S *} - μ_{B}^{L *}

(7)

式中， $μ_{A}^{S *}$ 和 $μ_{A}^{L *}$ 分别为界面处溶剂A在固相和液相中的化学势， $μ_{B}^{S *}$ 和 $μ_{B}^{L *}$ 分别为界面处溶质B在固相和液相中的化学势。因此，单位面积固/液界面迁移所引起的Gibbs自由能变化率( $\dot{G}$ _)为：

\dot{G} = \frac{V}{V_{m}} Δ G_{t o t a l} = \frac{V}{V_{m}} [(1 - C_{S}^{*}) Δ μ_{A}^{*} + C_{S}^{*} Δ μ_{B}^{*}]

(8)

式中， $Δ$ G_total为凝固1 mol原子时整个界面所耗散的Gibbs自由能。

其次，如前所述界面动力学可以看作是界面迁移(即液相原子转变为固相原子，但是成分仍为液相成分)和跨界面扩散(即新生成固相中多余溶质原子从固相跨界面扩散到液相) 2个过程。2者的通量(J_C和J_D)分别为：

J_{C} = \frac{V}{V_{m}}, J_{D} = \frac{V}{V_{m}} (C_{L}^{*} - C_{S}^{*})

(9)

式中，C和D分别代表界面迁移和界面扩散。由于上述2种情况下，初态和终态均相同，因此界面耗散的Gibbs自由能也应相同。故式(8)可改写为：

\dot{G} = \frac{V}{V_{m}} Δ G_{t o t a l} = \frac{V}{V_{m}} [(1 - C_{L}^{*}) Δ μ_{A}^{*} + C_{L}^{*} Δ μ_{B}^{*}] +

\frac{V}{V_{m}} (C_{L}^{*} - C_{S}^{*}) (Δ μ_{A}^{*} - Δ μ_{B}^{*})

(10)

上式表明界面迁移和跨界面扩散对应的共轭驱动力(X_C和X_D)分别为：

X_{C} = (1 - C_{L}^{*}) Δ μ_{A}^{*} + C_{L}^{*} Δ μ_{B}^{*}, X_{D} = Δ μ_{A}^{*} - Δ μ_{B}^{*}

(11)

前者是1 mol液相直接转变为固相所引起的Gibbs自由能变化；后者是溶质原子B由固相跨界面扩散到液相，而相应溶剂原子A由液相跨界面扩散到固相并占据之前溶质原子B位置所引起的Gibbs自由能变化(图3)。如图3所示，1 mol液相发生凝固时整个界面所耗散的Gibbs自由能为 $Δ G_{t o t a l}$ = $(1 - C_{S}^{*}) Δ μ_{A}^{*} + C_{S}^{*} Δ μ_{B}^{*}$ 。将固相Gibbs自由能曲线在成分 $C_{S}^{*}$ 的公切线向上平移到液相Gibbs自由能曲线的成分 $C_{L}^{*}$ 位置时， $Δ$ G_total被分为2部分：上部分为溶质跨界面扩散耗散过程的Gibbs自由能 $Δ G_{D}$ = $(C_{L}^{*} - C_{S}^{*}) (Δ μ_{A}^{*} - Δ μ_{B}^{*})$ ；下部分为界面迁移耗散过程的Gibbs自由能 $Δ G_{m}$ = $(1 - C_{L}^{*}) Δ μ_{A}^{*} + C_{L}^{*} Δ μ_{B}^{*}$ 。

图3

图3 稳态固/液界面动力学过程的摩尔Gibbs自由能示意图

Fig.3 Mole Gibbs free energy diagram for the solid/liquid interface kinetic processes under a steady-state condition (Green and red curves represent the Gibbs free energy curve of the solid phase and liquid phase, respectively, the same below. Total Gibbs free energy dissipated by the interface after solidification of 1 mol liquid is $Δ G_{t o t a l}$ = $(1 - C_{S}^{*}) Δ μ_{A}^{*} + C_{S}^{*} Δ μ_{B}^{*}$ . By translating the tangent of Gibbs free energy curve of solid at $C_{S}^{*}$ to the Gibbs free energy curve of liquid at $C_{L}^{*}$ , $Δ G_{t o t a l}$ is divided into two parts: The upper one for trans-interface diffusion is $Δ G_{D}$ = $(C_{L}^{*} - C_{S}^{*}) (Δ μ_{A}^{*} - Δ μ_{B}^{*})$ and latter part for interface migration is $Δ G_{m}$ = $(1 - C_{L}^{*}) Δ μ_{A}^{*} + C_{L}^{*} Δ μ_{B}^{*}$ . $Δ G_{m}$ —Gibbs free energy of interface migration, $Δ G_{D}$ —Gibbs free energy of trans-interface diffusion, $C_{S}^{*}$ —the solid composition at the interface, $C_{L}^{*}$ —the liquid composition at the interface, $Δ μ_{A}^{*}$ —the chemical potential of A across the interface, $Δ μ_{B}^{*}$ —the chemical potential of B across the interface, $μ_{A}^{L *}$ and $μ_{A}^{S *}$ —the chemical potential at the interface for liquid and solid phase B, $μ_{B}^{L *}$ and $μ_{B}^{S *}$ —the chemical potential at the interface for liquid and solid phase B, respectively, $g_{m}^{S}$ —the mole Gibbs free energy of the solid, $g_{m}^{L}$ —the mole Gibbs free energy of the liquid)

需要指出的是，Baker^[18]实际上从不可逆热力学的角度出发，建立了界面动力学的理论框架，该理论框架持续影响着后续理论工作的开展。然而，该理论实际上假设固相成分恒定(即为 $C_{S}$ = $C_{S}^{*}$ )，换而言之，其忽略了生长相中溶质扩散的影响：对于液/固相变而言，液相中溶质扩散系数通常比固相大几个数量级，该假设成立；对固/固相变而言，生长相和母相的扩散速率相当，甚至生长相中可能更大，因此生长相中的扩散通常不可忽略；对于固/液相变而言，液相中溶质扩散系数比固相大几个数量级，该假设同样不成立。Hillert等^[19,20]最早关注了上述问题，并从界面处的守恒方程出发进行了对比分析，界面处守恒方程可表示为：

J_{B}^{L *} - J_{B}^{S *} = \frac{V}{V_{m}} (C_{L}^{*} - C_{S}^{*})

(12)

当固相中溶质扩散可以忽略时，即 $J_{B}^{S *} = 0$ ，式(12)可简化为：

J_{B}^{L *} = \frac{V}{V_{m}} (C_{L}^{*} - C_{S}^{*})

(13)

比较式(9)和(13)可知： $J_{D}$ = $J_{B}^{L *}$ 。可见，跨界面扩散通量与界面处液相的溶质通量相等。此时，最终穿越界面进入固相的溶质成分 $C_{t r a n s}$ = $C_{S}^{*}$ ，整个界面耗散的摩尔Gibbs自由能 $Δ G_{t o t a l}$ = $(1 - C_{S}^{*}) Δ μ_{A}^{*} + C_{S}^{*} Δ μ_{B}^{*}$ 。而当固相中的溶质扩散不可忽略时，他们认为对于跨界面扩散通量， $J_{D}$ = $J_{B}^{L *}$ 仍然成立，此时 $C_{t r a n s}$ 可由式(12)得到：

J_{D} = J_{B}^{L *} = \frac{V}{V_{m}} [C_{L}^{*} - (C_{S}^{*} - \frac{V_{m}}{V} J_{B}^{S *})] =

\frac{V}{V_{m}} (C_{L}^{*} - C_{t r a n s})

(14)

即 $C_{t r a n s}$ = $C_{S}^{*} - J_{B}^{S *} V_{m} / V$ 。此时，C_trans是与界面处固相扩散有关的量，整个界面耗散的摩尔Gibbs自由能为 $Δ G_{t o t a l}$ = $(1 - C_{t r a n s}) Δ μ_{A}^{*} + C_{t r a n s} Δ μ_{B}^{*}$ ，且：

\dot{G} = \frac{V}{V_{m}} Δ G_{t o t a l} = \frac{V}{V_{m}} [(1 - C_{L}^{*}) Δ μ_{A}^{*} + C_{L}^{*} Δ μ_{B}^{*}] +

\frac{V}{V_{m}} (C_{L}^{*} - C_{t r a n s}) (Δ μ_{A}^{*} - Δ μ_{B}^{*})

(15)

Hillert等^[19,20]把Baker^[18]不考虑生长相内扩散的界面称为稳态界面，而把考虑生长相内扩散的界面称为非稳态界面(如图4)，在图中1 mol液相原子发生凝固时整个界面所耗散的Gibbs自由能为 $Δ G_{t o t a l}$ = $(1 - C_{t r a n s}) Δ μ_{A}^{*} + C_{t r a n s} Δ μ_{B}^{*}$ 。将固相Gibbs自由能曲线在成分 $C_{S}^{*}$ 的公切线向上平移到液相Gibbs自由能曲线的成分 $C_{L}^{*}$ 位置时， $Δ$ G_total被分为2部分：上部分为溶质跨界面扩散耗散过程的Gibbs自由能 $Δ G_{D}$ = $(C_{L}^{*} - C_{t r a n s}) (Δ μ_{A}^{*} - Δ μ_{B}^{*})$ ；下部分为界面迁移耗散过程的Gibbs自由能 $Δ$ G_m = $(1 - C_{L}^{*}) Δ μ_{A}^{*} + C_{L}^{*} Δ μ_{B}^{*}$ 。稳态和非稳态情况 $Δ$ G_total之差为 $Δ G^{t r a n s ‐ S}$ = $(C_{t r a n s} - C_{S}^{*}) (Δ μ_{A}^{*} - Δ μ_{B}^{*})$ 。此能量为将实际穿过界面成分从 $C_{S}^{*}$ 调整到C_trans所耗散的Gibbs自由能。

图4

图4 非稳态固/液界面动力学过程的摩尔Gibbs自由能示意

Fig.4 Mole Gibbs free energy diagram for the solid/liquid interface kinetic processes under a non-steady-state condition (Total Gibbs free energy dissipated by the interface after solidification of 1 mol liquid is $Δ G_{t o t a l}$ = $(1 - C_{t r a n s}) Δ μ_{A}^{*} + C_{t r a n s} Δ μ_{B}^{*}$ . By translating the tangent of Gibbs free energy curve of solid at $C_{S}^{*}$ to the Gibbs free energy curve of liquid at $C_{L}^{*}$ , $Δ$ G_total is divided into two parts: The upper one for trans-interface diffusion is $Δ G_{D}$ = $(C_{L}^{*} - C_{t r a n s}) (Δ μ_{A}^{*} - Δ μ_{B}^{*})$ and latter part for interface migration is $Δ G_{m}$ = $(1 - C_{L}^{*}) Δ μ_{A}^{*} + C_{L}^{*} Δ μ_{B}^{*}$ . The difference in $Δ$ G_total between the steady-state condition and the non-steady-state condition is $Δ G^{t r a n s ‐ S}$ = $(C_{t r a n s} - C_{S}^{*}) (Δ μ_{A}^{*} - Δ μ_{B}^{*})$ , which is the Gibbs free energy dissipated to adjust the actual composition transferred across the interface from $C_{S}^{*}$ to C_trans ( $C_{t r a n s}$ —the solute component that finally crosses the interface and enters the solid phase)

需要特别说明的是，上述结果是通过稳态和非稳态情况间的相互对比得到的，而值得质疑是为何2种情况下跨界面溶质扩散通量始终等于界面处母相的溶质扩散通量。针对这一问题，Wang等^[21]通过构造体系的Gibbs自由能，并对其进行时间求导运算，从而得出了Gibbs自由能的变化率。所得到的单位面积液/固界面迁移所引起的Gibbs自由能变化率为：

\dot{G} = \frac{V}{V_{m}} Δ G_{t o t a l} = \frac{V}{V_{m}} \{[C_{S}^{*} μ_{B}^{S *} + (1 - C_{S}^{*}) μ_{A}^{S *}] -

[C_{L}^{*} μ_{B}^{L *} + (1 - C_{L}^{*}) μ_{A}^{L *}]\} -

J_{B}^{S *} (μ_{B}^{S *} - μ_{A}^{S *}) + J_{B}^{L *} (μ_{B}^{L *} - μ_{A}^{L *})

(16)

根据式(16)，在非稳态条件下界面的动力学过程分为界面迁移、界面处固相扩散和界面处液相扩散3个过程，其对应的驱动力分别为固、液2相的摩尔Gibbs自由能差( $g_{m}^{S *} - g_{m}^{L *}$ )、界面处固相扩散势( ${\tilde{μ}}^{S *}$ = $μ_{B}^{S *} - μ_{A}^{S *}$ )和界面处液相的扩散势( ${\tilde{μ}}^{L *}$ = $μ_{B}^{L *} - μ_{A}^{L *}$ )。但由于界面溶质守恒条件(式(12))，这3个过程实际上相互依赖。如若将式(12)代入式(16)，即可发现式(16)与(15)等价，且C_trans有2种表达式：

C_{t r a n s} = C_{S}^{*} - \frac{V_{m}}{V} J_{B}^{S *} = C_{L}^{*} - \frac{V_{m}}{V} J_{B}^{L *}

(17)

进一步地，Kuang等^[22,23]认为在稳态条件下跨界面溶质扩散为一步完成，即从固相跳跃到液相，跨界面扩散通量 $J_{D}$ = $J_{B}^{L *}$ = $(C_{L}^{*} - C_{S}^{*}) V / V_{m}$ (图5a)。

图5

图5 跨界面溶质扩散示意图

Fig.5 Schematics for solute trans-interface diffusion in one step (a) and two steps (b) (For the former, there is only one flux for solute trans-interface diffusion, i.e., $J_{D}$ = $J_{B}^{L *}$ = $(C_{L}^{*} - C_{S}^{*}) V / V_{m}$ , whereas for the latter, there are two fluxes for solute trans-interface diffusion, i.e., $J_{D}^{S \to I}$ = $(C_{t r a n s}^{*} - C_{S}^{*}) V / V_{m}$ = $- J_{B}^{S *}$ and $J_{D}^{I \to L}$ = $(C_{L}^{*} - C_{t r a n s}^{*}) V / V_{m}$ = $J_{B}^{L *}$ . J_D—the flux of solute trans-interface diffusion, $J_{D}^{S \to I}$ —the flux of trans-interface diffusion from solid phase to interface, $J_{D}^{I \to L}$ —the flux of trans-interface diffusion from interface to liquid phase, $C_{t r a n s}^{*}$ —the solute component at the interface that finally crosses the interface and enters the solid phase, V—the velocity of the migrating interface, V_m—mole volume)

在非稳态条件下，跨界面溶质扩散为2步完成(图5b)：首先，从固相跳跃到界面，其中界面成分为 $C_{t r a n s}^{*}$ ，相应的跨界面扩散通量为 $J_{D}^{S \to I}$ = $(C_{t r a n s}^{*} - C_{S}^{*}) V / V_{m}$ = $- J_{B}^{S *}$ ；其次，从界面跳跃到液相，相应的跨界面扩散通量为 $J_{D}^{I \to L}$ = $(C_{L}^{*} - C_{t r a n s}^{*}) V / V_{m}$ = $J_{B}^{L *}$ 。由此可见，无论是界面处液相还是固相中的溶质扩散通量，均为跨界面溶质扩散通量。但Hillert等^[19,20]的工作仅选取了一种独立的动力学过程。试想一不断加热熔化和降温凝固的循环过程，对于Hillert等^[19,20]的工作，熔化过程需选择 $J_{B}^{S *}$ 为跨界面溶质通量，而在凝固过程中需选择 $J_{B}^{L *}$ 为跨界面溶质通量。而在Wang等^[21]和Kuang等^[22,23]工作中， $J_{B}^{S *}$ 和 $J_{B}^{L *}$ 均为跨界面通量，因此无论是熔化过程还是凝固过程，上述理论分析结果均适用。

1.2 稳态界面动力学理论

根据经典的Onsager唯象关系^[24,25]，Baker^[18]定义了通量和驱动力之间的关系，建立了经典的稳态动力学理论。针对上述2种通量和驱动力的选择，建立了2种模型：

J_{A}^{S *} = L_{A A} X_{A} + L_{A B} X_{B}, J_{B}^{S *} = L_{B A} X_{A} + L_{B B} X_{B}

(18)

J_{C} = L_{C C} X_{C} + L_{C D} X_{D}, J_{D} = L_{D C} X_{C} + L_{D D} X_{D}

(19)

式中，L_AA、L_AB、L_BA、L_BB、L_CC、L_CD、L_DC和L_DD均为动力学系数。且根据Onsager倒易关系^[24,25]，有L_AB = L_BA和L_CD = L_DC。由于2种通量和驱动力的选择是描述同一动力学问题，因此Baker^[18]首先对2种模型的等价性进行了研究。结果发现式(18)和(19)并不能同时满足Onsager的经典倒易关系，如当L_AB = L_BA成立时，L_CD = L_DC并不成立。因此在很长一段时间内人们都在关注经典Onsager倒易关系的普适性问题。Hillert^[26]在2006年完全解决了这一问题，发现2种通量和驱动力的选择完全等价，所建立的界面动力学模型也完全等价，且L_AB = L_BA和L_CD = L_DC也同时成立。但Baker^[18]这一开创性理论工作仅关注了动力学系数之间的关系以及应满足的条件，并未给出具体的表达式。

Aziz和Kaplan^[27]在1988年建立了另一经典的界面动力学模型。基于化学反应速率理论，该模型中通量和驱动力之间为幂指数关系而非如式(18)和(19)中的线性关系，并且还给定了动力学系数的表达式。相应的溶质截留模型，也即跨界面溶质扩散动力学过程为：

k = \frac{C_{S}^{*}}{C_{L}^{*}} = \frac{\frac{V}{V_{D I}} + κ_{e}}{\frac{V}{V_{D I}} + 1 - (1 - κ_{e}) C_{L}^{*}}

(20)

κ_{e} = \frac{C_{S}^{*} (1 - C_{L}^{*})}{C_{L}^{*} (1 - C_{S}^{*})} e x p (\frac{Δ μ_{A}^{*} - Δ μ_{B}^{*}}{R T})

(21)

式中，V_DI为界面溶质扩散速率，κ_e为溶液条件下的分配因子，R为气体常数，T为温度。相应的界面迁移动力学模型为：

V = V_{0} [1 - e x p (\frac{Δ G_{m}}{R T})]

(22)

式中，V₀为界面迁移速率上限， $Δ$ G_m为界面迁移耗散过程的Gibbs自由能。但关于 $Δ$ G_m，Aziz和Kaplan^[27]却给出了2种表达式：一种是 $Δ$ G_total (式(10))，一种是X_C (式(11))。前者为未考虑溶质拖曳模型，后者为考虑溶质拖曳模型(详见第4节论述)。对式(20)和(22)进行线性近似可得：

J_{D} = - \frac{V_{D I} C_{S}^{*} (1 - C_{L}^{*})}{V_{m} R T} X_{D}

(23)

V = - \frac{V_{0}}{R T} Δ G_{m}

(24)

由此可见，Aziz和Kaplan^[27]的工作仍在Baker^[18]的理论框架之内：首先，在动力学过程方面，Aziz和Kaplan^[27]选择了界面迁移和跨界面扩散；其次，在通量和驱动力关系方面，Aziz和Kaplan^[27]仅考虑了通量与其共轭驱动力间关系，并未涉及通量与非共轭驱动力间关系，即未考虑不同动力学过程之间的相互影响；最后，Aziz和Kaplan^[27]给出了相应动力学系数的表达式。

需要特别说明的是，目前界面动力学理论普遍采用界面迁移和跨界面扩散2类动力学过程，但也有少数工作仍采用实际A原子和B原子通过界面的通量，即 $J_{A}^{S *}$ 和 $J_{B}^{S *}$ (式(6))，来描述界面动力学过程，相应的动力学系数也是通过化学反应速率理论给定^[28]。此外，界面迁移的动力学系数来源于纯物质的界面动力学方程，故已被广泛认同。但有关跨界面扩散的动力学系数，不同学者给出了不同的表达式，如Buchmann和Rettenmayr^[29]认为其应为与成分无关的常数。近期，Hareland等^[30]认为该系数应满足成分的对称性(如式(23)中的 $C_{S}^{*} (1 - C_{L}^{*})$ 应为 $C_{L}^{*} (1 - C_{L}^{*})$ )。但这些质疑均不影响Aziz和Kaplan工作^[27]在这一领域的地位，其也是到目前为止最被认可的模型。

1.3 非稳态界面动力学理论

Hillert等^[19,20]仅讨论了非稳态条件下的界面动力学过程，并未明确给出界面动力学理论。随后，Buchmann和Rettenmayr^[29]在讨论凝固过程中的初始瞬态非平衡问题时，给出了通量和驱动力之间的关系：

J_{C} = - \frac{V_{0}}{V_{m} R T} [(1 - C_{L}^{*}) Δ μ_{A}^{*} + C_{L}^{*} Δ μ_{B}^{*}]

(25)

J_{D} = - \frac{V_{D I}}{V_{m} R T} [(C_{L}^{*} - C_{t r a n s}) (Δ μ_{A}^{*} - Δ μ_{B}^{*})]

(26)

式(25)与Aziz和Kaplan^[27]考虑溶质拖曳效应的模型在线性近似条件下(式(24))完全相同；式(26)相较于Aziz和Kaplan^[27]在线性条件下的溶质截留模型，跨界面扩散变成与界面处固相溶质扩散 $J_{B}^{S *}$ 相关的量(式(14))，对应的移动性系数变成了与穿越界面进入固相成分C_trans有关的量。

Wang等^[21]在Aziz和Kaplan^[27]工作的基础上考虑了界面处固相的溶质扩散。在假设界面处固相溶质扩散不耗散Gibbs自由能的前提下，推导出的界面动力学方程为：

V = V_{0} \{1 - e x p [\frac{C_{L}^{*} Δ μ_{B} + (1 - C_{L}^{*}) Δ μ_{A}}{R T}]\}

(27)

k = \frac{\frac{V}{V_{D I}} + κ_{e} + \frac{V_{m} J_{B}^{S *}}{C_{L}^{*} V_{D I}}}{\frac{V}{V_{D I}} + 1 - (1 - κ_{e}) C_{L}^{*}}

(28)

可见，界面处固相的溶质扩散仅影响跨界面溶质扩散方程(式(28))。需要说明的是，在分析界面动力学过程时，Wang等^[21]采用了热力学极值原理，故界面迁移的动力学遵循考虑溶质拖曳效应的动力学方程(式(27))，即与Aziz和Kaplan^[27]考虑溶质拖曳效应的方程一致。

基于两步跨界面扩散概念和热力学极值原理，Kuang等^[22,23]构建了多元置换固溶体合金和含间隙组元的多元固溶体合金的界面动力学模型。该模型不仅考虑了界面处母相的耗散过程，同时也考虑了界面处生长相中的耗散过程。对于二元合金凝固过程，相应的界面动力学模型为：

- \frac{J_{C}}{M_{J_{C}}} - (C_{S}^{*} - C_{L}^{*}) \frac{J_{B}^{S *}}{M_{J_{B}^{S *}}} = C_{L}^{*} Δ μ_{B}^{*} + (1 - C_{L}^{*}) Δ μ_{A}^{*}

(29)

- \frac{J_{B}^{L *}}{M_{J_{B}^{L *}}} - \frac{J_{B}^{S *}}{M_{J_{B}^{S *}}} = Δ μ_{B}^{*} - Δ μ_{A}^{*}

(30)

式中， $M_{J_{B}^{S *}}$ 、 $M_{J_{B}^{L *}}$ 和 $M_{J_{C}}$ 分别为界面处固相溶质扩散、界面处液相溶质扩散和界面迁移的动力学系数，表达式分别为：

M_{J_{B}^{S *}} = \frac{D_{S}}{a_{0} V_{m}} {[\frac{\partial (μ_{B}^{S *} - μ_{A}^{S *})}{\partial C_{S}^{*}}]}^{- 1},

M_{J_{B}^{L *}} = \frac{D_{L}}{a_{0} V_{m}} {[\frac{\partial (μ_{B}^{L *} - μ_{A}^{L *})}{\partial C_{L}^{*}}]}^{- 1},

M_{J_{C}} = \frac{V_{0}}{V_{m} R T}

(31)

式中， $a_{0}$ 为原子间面间距。需要说明的是，Buchmann和Rettenmayr^[29]的工作选择是独立的动力学过程，即界面迁移J_C和一步跨界面扩散 $J_{D}$ = $J_{B}^{L *}$ ，这与稳态界面动力学理论完全对应。Kuang等^[22,23]选取了界面迁移J_C、固相到界面的跨界面扩散 $J_{D}^{S \to I}$ = $- J_{B}^{S *}$ 和界面到液相的跨界面扩散 $J_{D}^{I \to L}$ = $J_{B}^{L *}$ 3个相互依赖的动力学过程，并认为这些过程均会耗散能量，且各自具有特定的动力学系数。一般而言，从3个相互依赖的动力学过程中选取2个独立的动力学过程，共存在3种选择，那么为何选择J_C和 $J_{D}$ = $J_{B}^{L *}$ 呢？这种选择本身就存在疑问。此外，如果将式(29)、(30)与式(12)联立并写成J_C、 $J_{B}^{S *}$ 和 $J_{B}^{L *}$ 的表达式，则可发现所得的界面动力学模型同时满足Onsager唯象关系和Onsager倒易关系^[24,25]，这也进一步佐证了模型的正确性。

2 完全非平衡条件下的二元合金界面动力学理论

根据本文定义，在完全非平衡条件下，不仅界面处于非平衡状态，而且体积相中的溶质扩散也不再遵循经典的Fick扩散定律^[10]：

J_{B}^{i *} = - \frac{D_{i}}{V_{m}} \nabla C_{i} - τ_{D}^{i} \frac{\partial J_{B}^{i}}{\partial t} (i = S, L)

(32)

式中，∇C_i 为浓度梯度。事实上，存在2种方法可将体积相扩散的非平衡效应耦合入界面动力学之中^[31]。

其一是动力学能量方法，即固定溶质扩散的移动性，改变溶质扩散的驱动力。此时式(30)可改写为：

J_{B}^{i} = - \frac{D_{i}}{V_{m}} {(\frac{\partial {\tilde{μ}}^{i}}{\partial C_{i}})}^{- 1} (\nabla {\tilde{μ}}^{i} + \frac{α_{i}}{V_{m}} \frac{\partial J_{B}^{i}}{\partial t}) (i = S, L)

(33)

式中， ${\tilde{μ}}^{i}$ = $μ_{B}^{i} - μ_{A}^{i}$ 为溶剂A和溶质B之间的相互扩散势， $\nabla {\tilde{μ}}^{i}$ 为化学式梯度，动力学系数 $α_{i}$ = $(\partial {\tilde{μ}}^{i} / \partial C_{i}) (V_{m}^{2} / (V_{D}^{i})^{2})$ 。这种方法可直接将不考虑溶质扩散非平衡效应模型中的 $\nabla {\tilde{μ}}^{i}$ 项替换成 $\nabla {\tilde{μ}}^{i}$ + $(α_{i} / V_{m}) \partial J_{B}^{i} / \partial t$ 或者将体系中的摩尔Gibbs自由能 $g_{m}^{i}$ 替换成 $g_{m}^{i} + 0.5 α_{i} (J_{B}^{i})^{2}$ ^[10]。

其二是有效移动性方法，即固定溶质扩散的驱动力，改变溶质扩散的移动性。此时式(33)可改写为：

J_{B}^{i} = - \frac{D_{i}}{V_{m}} {(\frac{\partial {\tilde{μ}}^{i}}{\partial C_{i}})}^{- 1} (1 + \frac{V_{m}}{{\begin{matrix} (V_{D}^{i}) \end{matrix}}^{2}} \frac{1}{\nabla C_{i}} \frac{\partial J_{B}^{i}}{\partial t}) \nabla {\tilde{μ}}^{i} =

- \frac{D_{i}}{V_{m}} (1 + \frac{V_{m}}{\begin{matrix} (V_{D}^{i})^{2} \end{matrix}} \frac{1}{\nabla C_{i}} \frac{\partial J_{B}^{i}}{\partial t}) \nabla C_{i} (i = S, L)

(34)

这种方法可直接将不考虑溶质扩散非平衡效应模型中的D_i 项替换成 $D_{i} / V_{m} [1 + (V_{m} / \begin{matrix} (V_{D}^{i})^{2} \end{matrix}) (1 / \nabla C_{i}) (\partial J_{B}^{i} / \partial t)]$ 或者在稳态条件下直接将 $D_{L}$ 项替换成 $D_{L} (1 - V^{2} / (V_{D}^{L})^{2})$ 。

在非平衡界面动力学理论领域，目前主要采用动力学能量方法对局域平衡条件下的理论模型进行改进。在稳态界面动力学理论方面，Galenko^[32,33]对Aziz和Kaplan^[27]的工作进行了改进，相应的理论模型为：

k = \frac{C_{S}^{*}}{C_{L}^{*}} = \frac{\frac{V}{V_{D I}} + ψ κ_{e}}{\frac{V}{V_{D I}} + ψ [1 - (1 - κ_{e}) C_{L}^{*}]}

(35)

V = V_{0} [1 - e x p (\frac{Δ G_{m}}{R T})]

(36)

式中，弛豫因子(ψ)可表示为：

ψ = \{\begin{matrix} 1 - \frac{V^{2}}{(V_{D}^{L})^{2}} & (V < V_{D}^{L}) \\ 0 & (V \geq V_{D}^{L}) \end{matrix}

(37)

考虑溶质拖曳效应的模型(式(36))中 $Δ$ G_m的驱动力仍为X_C (式(11))。对于不考虑溶质拖曳效应的模型：

Δ G_{C} = \{\begin{matrix} (1 - C_{S}^{*}) Δ μ_{A} + C_{S}^{*} Δ μ_{B} + (C_{L}^{*} - C_{S}^{*}) V_{m} α_{L} \frac{\partial J_{B}^{L}}{\partial t} & (V < V_{D}^{L}) \\ (1 - C_{L}^{*}) Δ μ_{A} + C_{L}^{*} Δ μ_{B} & (V \geq V_{D}^{L}) \end{matrix}

(38)

式中， $Δ$ G_C为不考虑溶质拖曳效应的Gibbs自由能。在稳态条件下，固相中的溶质扩散可忽略，因此本工作中仅考虑了液相中非平衡溶质扩散效应。当V = $V_{D}^{L}$ 时，完全溶质截留(即k = 1)会立刻发生，且溶质拖曳效应会立刻消失(( $C_{L}^{*} - C_{S}^{*}) (Δ μ_{A} - Δ μ_{B})$ = 0)。但需要特别说明的是，动力学能量本身并不能将非平衡溶质扩散效应耦合入界面动力学效应之中，如式(38)中的非平衡溶质扩散效应项 $(C_{L}^{*} - C_{S}^{*}) V_{m} α_{L} \partial J_{B}^{L} / \partial t$ ，从单位的角度其很明显与化学势不一致。

Wang等^[21]首次在非稳态的条件下对Aziz和Kaplan^[27]的工作进行了改进，并且同时考虑了固相和液相中的非平衡溶质扩散效应，但未考虑界面处固相溶质通量对界面处能量的耗散，相应的界面动力学模型为：

k = \frac{C_{S}^{*}}{C_{L}^{*}} = \frac{\frac{V}{V_{D I}} + ψ κ_{e} + \frac{V_{m} J_{B}^{S *}}{C_{L}^{*} V_{D I}}}{\frac{V}{V_{D I}} + ψ [1 - (1 - κ_{e}) C_{L}^{*}]}

(39)

V = V_{0} \{1 - e x p [\frac{1}{R T} (C_{L}^{*} Δ μ_{B} + (1 - C_{L}^{*}) Δ μ_{A} -

\frac{1}{2} α_{L} (J_{B}^{L *})^{2} + \frac{1}{2} α_{S} (J_{B}^{S *})^{2})]\}

(40)

式(39)表明界面处固相的溶质扩散会在非稳态条件下影响界面处的溶质分配，而式(40)表明界面处的液相和固相中非平衡溶质扩散均会影响界面迁移的驱动力。需要特别说明的是，在溶质截留方面，Wang等^[21]采用的是有效移动性方法耦合非平衡溶质扩散效应，而在界面迁移驱动力方面则采用的是摩尔Gibbs自由能示意图方法。这说明Wang等^[21]的工作也有待商榷。需要特别说明的是，在引入非平衡溶质扩散效应时，动力学能量方法同样在描述相场理论中无法耦合入短程扩散，且这种方法本身与热力学极值原理不相容，故采用有效移动性方法直接对局域平衡条件下得到的界面动力学理论进行拓展是较好的选择^[31]。此外，Wang等^[21]的工作实际上只选择了考虑溶质拖曳的理论模型，其原因在于热力学极值原理仅支持该类模型(如式(27)和当 $J_{B}^{S *}$ = 0时的式(29))。

3 二元合金部分溶质拖曳界面动力学理论

如前所述，Aziz和Kaplan^[27]在界面迁移的驱动力方面给出了考虑溶质拖曳效应和不考虑溶质拖曳效应2种表达式。在脉冲激光熔化Si-As合金的非平衡凝固^[34]、过冷Ni-B合金的非平衡枝晶生长^[35]等实验中，均发现不考虑溶质拖曳的模型与实验结果吻合较好。因此，Kittl等^[34]和Eckler等^[35]认为非平衡凝固过程遵循不考虑溶质拖曳的界面动力学模型。但是这一观点被Hillert^[36]质疑，其原因在于固态相变遵循考虑溶质拖曳效应的模型。如前所述，液/固相变与固态相变从不可逆热力学的角度并无本质区别，故所遵循的模型应完全相同。特别地，近期Wang等^[21,31]和kuang等^[22,23]工作表明，热力学极值原理只支持考虑溶质拖曳效应的理论模型，这也与Hillert^[36]的观点一致。此外，Wang等^[21]发现Kittl等^[34]分析所采用的Si-As合金相图与非平衡凝固实验条件并不一致。在此基础上可得出结论，液/固相变和固态相变均遵循考虑溶质拖曳效应的理论模型。

但事实上，本文所关注的所有理论模型均建立在尖锐界面理论假设之上，而实际的界面是有一定厚度的。对于厚界面或者说弥散界面而言，实际驱动界面迁移的驱动力是介于上述考虑溶质效应和不考虑溶质拖曳效应之间的，即部分溶质拖曳效应^[31]。这一结论也被Yang等^[37]有关分子动力学模拟的工作所证实。需要说明的是，Aziz和Boettinger^[38]首先提出了部分溶质拖曳的概念，并定义了介于液相成分和固相成分之间的有效成分(C_eff)：

C_{e f f} = γ C_{L}^{*} + (1 - γ) C_{S}^{*}

(41)

式中，γ是介于0和1之间的拖曳系数。基于此，界面迁移的动力学方程(如式(22))可改写为：

V = V_{0} \{1 - e x p [\frac{C_{e f f} Δ μ_{B} + (1 - C_{e f f}) Δ μ_{A}}{R T}]\}

(42)

由此可见，当γ = 0时，式(42)为不考虑溶质拖曳效应的理论模型，当γ = 1时为考虑溶质拖曳效应的理论模型，而当0 < γ < 1为部分溶质拖曳理论模型。在实际应用中，γ通常为拟合值，且对于不同合金及不同成分其值也不同。原则上，可以通过这种方法将上述所有理论模型中的界面迁移动力学方程中的相应成分进行替换，从而得到相应的部分溶质拖曳模型。然而，需要注意的是，在部分溶质拖曳的情况下，不仅界面迁移的驱动力发生变化，跨界面扩散的驱动力也会相应改变。因此，部分溶质拖曳不仅会影响界面迁移动力学过程，还会影响跨界面扩散过程。后者目前在相关研究中鲜有关注。

近期，受Kuang等^[22,23]的两步跨界面扩散概念启发，Hareland等^[30]对部分溶质拖曳理论模型进行了进一步改进。需要说明的是，该工作聚焦于多元合金体系，并旨在描述一般的相变过程(包括液/固相变和固/固相变)。为更清晰地展示其物理内涵，本文将其简化至如下二元合金的凝固情形。首先，Hareland等^[30]假设界面内成分为C_eff，相应的通量为J_D。则界面处固相一侧和液相一侧的成分守恒条件分别为：

J_{B}^{S *} - J_{D} = \frac{V}{V_{m}} (C_{S}^{*} - C_{e f f})

(43)

J_{D} - J_{B}^{L *} = \frac{V}{V_{m}} (C_{e f f} - C_{L}^{*})

(44)

其次，Hareland等^[30]假设界面处的动力学过程为界面迁移和跨界面扩散2个独立的动力学过程，其对应的通量分别为J_C和J_D。则将式(43)和(44)代入式(16)可得：

\dot{G} = \frac{V}{V_{m}} Δ G_{m} = J_{C} [C_{e f f} Δ μ_{B} + (1 - C_{e f f}) Δ μ_{A}] +

J_{D} (Δ μ_{A} - Δ μ_{B})

(45)

相应的界面动力学方程为：

J_{C} = - \frac{V_{0}}{V_{m} R T} [(1 - C_{e f f}) Δ μ_{A} + C_{e f f} Δ μ_{B}]

(46)

J_{D} = - \frac{V_{D I}}{V_{m} R T} C_{e f f} (1 - C_{e f f}) (Δ μ_{A} - Δ μ_{B})

(47)

相较于前人工作，Hareland等^[30]的模型从式(16)出发，实际上也考虑了固相溶质扩散的影响，故隶属于非稳态的界面动力学方程(图6)。如图6所示，1 mol液相发生凝固时整个界面所耗散的Gibbs自由能为 $Δ G_{t o t a l}$ = $(1 - C_{t r a n s}) Δ μ_{A}^{*} + C_{t r a n s} Δ μ_{B}^{*}$ 。将固相Gibbs自由能曲线在成分 $C_{S}^{*}$ 的公切线向上平移到液相Gibbs自由能曲线的成分C_eff位置时， $Δ$ G_total被分为2部分：上部分为溶质跨界面扩散耗散过程的Gibbs自由能 $Δ G_{D}$ = $(C_{e f f} - C_{t r a n s}) (Δ μ_{A}^{*} - Δ μ_{B}^{*})$ ；下部分为界面迁移耗散过程的Gibbs自由能 $Δ G_{m}$ = $(1 - C_{e f f}) Δ μ_{A}^{*} + C_{e f f} Δ μ_{B}^{*}$ 。稳态和非稳态情况 $Δ G_{t o t a l}$ 之差为 $Δ G^{t r a n s ‐ S}$ = $(C_{t r a n s} - C_{S}^{*}) (Δ μ_{A}^{*} - Δ μ_{B}^{*})$ 。此能量为将实际穿过界面进入固相成分从 $C_{S}^{*}$ 调整到 $C_{t r a n s}$ 所耗散的Gibbs自由能。部分溶质拖曳和完全溶质拖曳 $Δ$ G_m之差为 $Δ G^{L ‐ e f f}$ = $(C_{L}^{*} - C_{e f f}) (Δ μ_{A}^{*} - Δ μ_{B}^{*})$ 。此能量为将固/液界面前沿液相成分从 $C_{L}^{*}$ 调整到C_eff以发生界面迁移所耗散的能量。需要说明的是，Hareland等^[30]的工作混淆了非稳态的C_trans和C_eff，2者概念完全不同。另外在非稳态下式(41)C_eff的定义是否仍然适用也存在质疑，如当 $C_{S}^{*}$ < C_eff < C_trans就会存在问题。此外，由于J_D (式(43)和(44))是与C_eff有关的物理量，因此同样考虑了部分溶质拖曳对跨界面扩散能量耗散的影响：当γ = 0时， $C^{e f f}$ = $C_{S}^{*}$ (式(41))、 $J_{D}$ = $J_{B}^{S *}$ (式(43))；当γ = 1时， $C_{e f f}$ = $C_{L}^{*}$ (式(41))、 $J_{D}$ = $J_{B}^{L *}$ (式(44))；当0 < γ < 1时，根据式(41)、(43)和(44)，J_D始终位于 $J_{B}^{L *}$ 和 $J_{B}^{S *}$ 之间。最后，该模型中的动力学系数中采用了 $C_{e f f} (1 - C_{e f f})$ 这种完全对称的成分表达式，不存在Aziz和Kaplan^[27]模型中预测结果不合理的问题。但也要特别注意的是，在此工作中界面处实际上有4种动力学过程( $J_{B}^{S *}$ 、J_D、 $J_{B}^{L *}$ 和J_C)，2个限制条件(即式(43)和(44))，因此有2个独立的动力学过程。Hareland等^[30]选择了常用的J_C和J_D，而实际的选择可能有6种，这本身也值得商榷。其次，该工作未考虑体积相的非平衡溶质扩散效应。但该工作对后续界面动力学理论的发展具有重要启发作用，且在一定程度上会进一步推动部分溶质拖曳界面动力学理论模型的发展。

图6

图6 非稳态部分拖曳固/液界面动力学过程的摩尔Gibbs自由能示意图

Fig.6 Mole Gibbs free energy diagram for the solid/liquid interface kinetic processes with a partial solute-drag effect under a non-steady-state condition (The total Gibbs free energy dissipated by the interface after solidification of 1 mol liquid is $Δ G_{t o t a l}$ = $(1 - C_{t r a n s}) Δ μ_{A}^{*} + C_{t r a n s} Δ μ_{B}^{*}$ . By translating the tangent of Gibbs free energy curve of solid at $C_{S}^{*}$ to the Gibbs free energy curve of liquid at C_eff, $Δ$ G_total is divided into two parts. The upper one for trans-interface diffusion is $Δ G_{D}$ = $(C_{e f f} - C_{t r a n s}) (Δ μ_{A}^{*} - Δ μ_{B}^{*})$ and latter part for interface migration is $Δ$ G_m = $(1 - C_{e f f}) Δ μ_{A}^{*} + C_{e f f} Δ μ_{B}^{*}$ . The difference in $Δ$ G_D between the steady-state condition and the non-steady-state condition is $Δ G^{t r a n s ‐ S}$ = $(C_{t r a n s} - C_{S}^{*}) (Δ μ_{A}^{*} - Δ μ_{B}^{*})$ , which is the Gibbs free energy dissipated to adjust the actual composition transferred across the interface from $C_{S}^{*}$ to C_trans for trans-interface diffusion. The difference in $Δ$ G_m between the two cases with a full solute-drag effect and with a partial solute-drag effect is $Δ G^{L ‐ e f f}$ = $(C_{L}^{*} - C_{e f f}) (Δ μ_{A}^{*} - Δ μ_{B}^{*})$ , which is the Gibbs free energy dissipated to adjust the liquid composition ahead of the solid/liquid interface from $C_{L}^{*}$ to C_eff for interface migration. It should be pointed out that Hareland et al. ^[30] confused the concepts of C_trans under non-steady-state and C_eff. Furthermore, it is also queried that the definition of C_eff can be stilled be used for a non-steady state condition. For example, there will be a problem when $C_{S}^{*}$ < C_eff < C_trans (C_eff—effective concentration)

4 结论与展望

本文以二元合金凝固为例，分析总结了非平衡界面动力学理论的发展现状和发展趋势。主要结论如下。

(1) 针对局域平衡界面条件，目前已分别建立了稳态和非稳态条件下的非平衡界面动力学理论。采用的理论方法包括不可逆热力学、化学反应速率理论和热力学极值原理。

(2) 针对完全非平衡界面条件，同样建立了稳态和非稳态条件下的非平衡界面动力学理论。耦合体积相非平衡溶质扩散效应的方法主要包括动力学能量方法和有效移动性方法。

(3) 针对尖锐界面动力学理论假设的不足，通过引入界面有效成分的概念，建立了半拖曳非平衡界面动力学理论。

尽管该领域已取得诸多进展，但在理论研究方法和理论概念等方面仍存在诸多亟待解决的关键科学问题。

(1) 目前主要采用动力学能量方法引入体积相非平衡溶质扩散效应，但该方法实际上只能将该效应引入到体积相而不是同时引入到体积相和界面处。

(2) 目前半拖曳模型的界面有效成分仍然采用稳态条件下的定义，在非稳态条件下，该定义存在 $C_{S}^{*}$ < C_eff < C_trans的情况，故仍需新的定义。

(3) 目前的半拖曳模型采用的耗散过程为独立过程，而基于热力学极值原理的工作表明采用非独立耗散过程具有理论优势。独立过程与非独立过程的理论联系仍需深入研究。

(4) 化学反应速率理论为非线性热力学，而热力学极值原理和不可逆热力学为线性热力学，那么线性热力学到非线性热力学转变的临界条件也需进一步研究。

(5) 本文是以二元合金凝固为例进行分析讨论。尽管已构建了针对多元合金的相应理论模型^[39~41]，然而，对于当前研究领域的热点(多主元合金^[42~47])而言，由于其独特的成分构成特点，即不存在溶质和溶剂之分，其界面动力学理论仍需进一步研究与探索。

(6) 理论模型的验证问题。目前主要采用过冷枝晶生长实验^[48~54]和分子动力学模拟^[55~57]进行验证，前者与界面动力学理论不完全对应，而后者虽然具有无任何拟合参数拟合的潜力，但尚未发挥出来。故仍需开展相关理论、实验和模拟方法研究。

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Rapid solidification is a relevant physical phenomenon in material sciences, whose theoretical analysis requires going beyond the limits of local equilibrium statistical physics and thermodynamics and, in particular, taking account of ergodicity breaking and of generalized formulation of thermodynamics. The ergodicity breaking is related to the time symmetry breaking and to the presence of some kinds of fluxes and gradient flows making that an average of microscopic variables along time is different than an average over some chosen statistical ensemble. In fast processes, this is due, for instance, to the fact that the system has no time enough to explore the whole region of possible microscopic states in the phase space. Similarly to this, systems submitted to strong fluxes may have no time for reaching the whole phase space in local bulks during observable macroscopic time. Rapid solidification, ergodicity breaking and extended thermodynamics actually make a conceptually novel combination in the present overview: ergodicity breaking is expressed in general terms and then extended thermodynamics is formulated as a particular phenomenological expression and applied to describe the dynamics of the phenomenon. Using the formalism of micro- and meso-scopic dynamics we introduce a general view on non-ergodic fast transitions and provide a simplest description of a continuum theory based on the system of hyperbolic equations applicable to rapid solidification. Analysis of non-equilibrium effects, including interface kinetics, solute trapping and solute drag, is presented with their effect on the rapidly moving solid-liquid interface. Special attention is paid to the theory predictions compared with the kinetics obtained in experiments on samples processed by electromagnetic levitation facility and in molecular dynamics simulation. (C) 2019 Elsevier B.V.

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