脉冲电流作用下Pb-Al合金的液-固分相过程

图1 实验装置及合金凝固过程示意图

Fig.1 Schematic of the Bridgman-type solidification setup equipped with the ECPs producing device and the L-S phase separation process in a continuous solidifying Pb-Al alloy (rOz—the coordinate system, ECP—electric current pulse, L-S—liquid-solid)

在不施加脉冲电流的条件下，将直径0.1 mm的W-Re热电偶置于合金熔体的中心位置，使用XSR90-04V0型温度记录仪测定合金冷却凝固过程中熔体中心处的冷却曲线。使用电容储能式脉冲电源产生脉冲电流：储能电容器的电容为210 μF，放电频率为100 Hz。实验时通过调节储能电容器的充电电压来改变脉冲电流的峰值电流密度(j_max)。

2 实验结果

图2给出了不同峰值电流密度条件下Pb-0.15Al合金的凝固组织。图中黑色相和白色相分别为Al粒子和Pb基体。可见，在拉速为10 mm/s的连续凝固条件下，不施加脉冲电流时即可得到Al粒子均匀分布于Pb基体的复合凝固组织。施加脉冲电流以后，当峰值电流密度较小(j_max ≤ 4 × 10⁸ A/m²)时，Al粒子的尺寸及分布与不施加脉冲电流时基本相同；当峰值电流密度较大(j_max > 4 × 10⁸ A/m²)时，富Al粒子显著细化，且随j_max增大，细化效果增强，Al粒子的分布更加均匀弥散。图3所示为Al粒子的二维尺寸分布图。可知，Al粒子尺寸为单峰分布，且随着电流密度的增大，Al粒子尺寸分布的峰值提高、宽度下降且向左偏移。凝固组织中Al粒子的二维平均半径随j_max的变化情况在图4中给出。图5为试样表面处的凝固组织。可见，与不施加脉冲电流时相比，施加脉冲电流以后，试样表面出现了贫Al层。

图2

图2 不同峰值电流密度(j_max)条件下Pb-0.15Al合金凝固组织的背散射电子(BSE)像

Fig.2 Backscattered electron (BSE) images of the microstructures of the Pb-0.15Al alloys continuously solidified under the effect of ECPs with peak current densities (j_max) being 0 A/m² (a), 4 × 10⁸ A/m² (b), 8 × 10⁸ A/m² (c), and 1.2 × 10⁹ A/m² (d), respectively

图3

图3 不同j_max条件下Pb-0.15Al合金凝固组织中Al粒子的二维尺寸分布

Fig.3 2D size distributions of the Al-rich particles in the Pb-0.15Al alloys continuously solidified under the effect of ECPs with j_max being 0 A/m² (a), 4 × 10⁸ A/m² (b), 8 × 10⁸ A/m² (c), and 1.2 × 10⁹ A/m² (d), respectively

图4

图4 Pb-0.15Al合金凝固组织中Al粒子的二维平均半径( ${\bar{R}}_{2 D}$ )随j_max的变化

Fig.4 Average 2D radius ( ${\bar{R}}_{2 D}$ ) of the Al-rich particles in Pb-0.15Al alloys vsj_max

图5

图5 有/无脉冲电流下Pb-0.15Al合金试样表面处凝固组织的BSE像

Fig.5 BSE images of the surface microstructures of the Pb-0.15Al alloys solidified without ECPs (a) and with the ECPs of j_max = 8 × 10⁸ A/m² (b)

3 分析讨论

3.1 模型与模拟方法

定义Al粒子的半径分布函数f(r, z, t, R)。其中，(r, z)为rOz坐标系(图1，O为稳态凝固条件下凝固界面最低点)下的位置坐标； $t$ 为时间； $R$ 为粒子半径。f(r, z, t, R)dR表示位置(r, z)处、 $t$ 时刻，单位体积合金熔体中半径介于 $R$ 至( $R + d R$ )之间的Al粒子的数量。在考虑粒子的形核、长大及空间迁移的条件下， $f$ 满足如下控制方程^[22]:

\frac{\partial f}{\partial t} + \nabla \cdot [(u + V) f] + \frac{\partial}{\partial R} (v f) = \frac{\partial I}{\partial R} |_{R = R_{c}}

(1)

式中， $u = u_{S} + u_{e}$ (其中， $u_{S}$ 为密度差引起的粒子Stokes运动速度， $u_{e}$ 为电致粒子运动速度)，为Al粒子相对于基体熔体的运动速度； $V = V_{0} + V_{c}$ (其中， $V_{0}$ 为试样的提拉速度， $V_{c}$ 为熔体的对流运动速度)，为基体熔体的运动速度； $v$ 为粒子长大速率； $I$ 为粒子形核率； $R_{c}$ 为临界晶核半径。

$u_{S}$ 、 $u_{e}$ 分别满足^[23,24]：

u_{S} = \frac{2 (ρ^{β} - ρ^{m}) R^{2}}{9 η^{m}} g

(2)

u_{e} = \frac{2 μ r (σ_{e}^{m} - σ_{e}^{β}) R^{2} j^{2}}{9 η^{m} (2 σ_{e}^{m} + σ_{e}^{β})} i

(3)

式中， $ρ^{β}$ 、 $ρ^{m}$ 分别为Al粒子和基体熔体的密度； $η^{m}$ 为基体熔体的黏度； $g$ 为重力加速度；μ = 4π × 10^-7 N/A²，为磁导率； $σ_{e}^{β}$ 、 $σ_{e}^{m}$ 分别为Al粒子和基体熔体的电导率； $j$ 为电流密度； $i$ 为Or方向的单位向量。

$V$ 满足如下连续性方程和动量传输方程^[25]：

\nabla \cdot (ρ V) = 0

(4)

\frac{\partial (ρ V)}{\partial t} + ρ^{m} (V \cdot \nabla) V = - \nabla p + \nabla^{2} (η V) + ρ g - \frac{μ r j^{2}}{2} i - \frac{4 π}{3} \int_{0}^{\infty} \{f ρ^{β} R^{3} [(u + V) \cdot \nabla] (u + V)\} d R

(5)

式中， $ρ$ 为合金熔体(Al粒子与基体熔体的两相混合物)的密度；p为压力； $η = η^{m} (1 + 2.5 ϕ)$ (其中， $ϕ$ 为合金熔体内Al粒子的体积分数)，为合金熔体的黏度^[26]。

$v$ 满足下式^[27]：

v = D \frac{C^{m} - C_{I}^{m}}{C^{β} - C_{I}^{m}} \frac{1}{R} + \frac{R}{3} \frac{\frac{d C^{β}}{d T} \frac{d T}{d t}}{C^{β} - C_{I}^{m}}

(6)

式中， $D = k_{B} T /$ (4πη^mr_Al) (其中， $k_{B}$ 为Boltzmann常数； $T$ 为热力学温度； $r_{A l} = 0.136 n m$ 为Al的原子半径)，为Al在Pb中的扩散系数^[28]； $C^{β}$ 、 $C^{m}$ 分别为Al粒子和基体熔体中Al的摩尔浓度； $C_{I}^{m} = C_{e}^{m}$ exp(α_s / R)(其中， $C_{e}^{m}$ 为基体熔体中溶质Al的平衡摩尔浓度； $α_{s}$ 为毛细长度， $α_{s} = (2 σ Ω^{β}) / (k_{B} T)$ (其中， $σ = 0.135 J / m^{2}$ 为界面能^[29]， $Ω^{β}$ 为Al粒子的平均原子体积))，代表半径为 $R$ 的Al粒子与基体熔体界面处熔体中溶质Al的摩尔浓度。

$I$ 由下式计算^[30]：

I = N_{0} \cdot 4 n_{c}^{2 / 3} \cdot \frac{6 D}{λ^{2}} \cdot {(\frac{Δ G_{c}}{3 π k_{B} T n_{c}^{2}})}^{1 / 2} \cdot e x p (- \frac{Δ G_{c}}{k_{B} T})

(7)

式中， $N_{0}$ 为单位体积基体熔体中的原子数； $n_{c}$ 为具有临界晶核半径 $R_{c}$ 的Al粒子中的原子数； $λ$ 为Al原子的平均跳跃距离； $Δ G_{c} = 4 π R_{c}^{2} σ + 4 π R_{c}^{3} Δ G_{V} / 3$ (其中， $Δ G_{V} = Δ G_{V, 0} + Δ G_{V, e}$ ， $Δ G_{V, 0}$ 为无电流条件下固、液两相单位体积Gibbs自由能之差， $Δ G_{V, e}$ 为电流导致的自由能变化)，为Al粒子的形核能垒。 $Δ G_{V, e}$ 满足下式^[17,31]：

Δ G_{V, e} = μ [\frac{3}{2} l n (\frac{R_{s}}{R_{c}}) - \frac{65}{48} - \frac{5}{48} \frac{σ_{e}^{m} - σ_{e}^{β}}{σ_{e}^{β} + 2 σ_{e}^{m}}] R_{s}^{2} \frac{σ_{e}^{m} - σ_{e}^{β}}{σ_{e}^{β} + 2 σ_{e}^{m}} j^{2}

(8)

式中， $R_{s}$ 为试样半径。

连续凝固过程中，合金试样的温度场满足如下控制方程^[32]：

\frac{\partial (ρ c_{p} T)}{\partial t} + \nabla \cdot (V ρ^{m} c_{p}^{m} T) = \nabla^{2} (k T) + Q_{s / l} +

Q_{e} - \frac{4 π}{3} \nabla \cdot [\int_{0}^{\infty} (u + V) f (ρ^{β} c_{p}^{β} - ρ^{m} c_{p}^{m}) T R^{3} d R]

(9)

式中， $c_{p}$ 、 $c_{p}^{β}$ 、 $c_{p}^{m}$ 分别为合金熔体、Al粒子和基体熔体的比热容； $k$ 为合金熔体的热导率； $Q_{s / l} = ρ^{m} L V_{0} (1 - ϕ)$ (其中， $L$ 为凝固潜热)，为凝固界面处合金凝固潜热释放速率； $Q_{e} = j^{2} / σ_{e}$ (其中， $σ_{e}$ 为合金电导率)，为单位时间单位体积内由电流产生的Joule热。

合金熔体的浓度场满足如下控制方程^[33]：

\frac{\partial C}{\partial t} + \nabla \cdot (C^{m} V) = \nabla^{2} [D (C^{m} - C_{e}^{m}) (1 - ϕ)] -

\frac{4 π}{3} \nabla \cdot [\int_{0}^{\infty} (u + V) (C^{β} - C^{m}) f R^{3} d R]

(10)

式中， $C = (1 - ϕ) C^{m} + ϕ C^{β}$ ，为合金熔体内Al的摩尔浓度。

对于本实验条件，由于脉冲电流的趋肤深度^[16]δ = (πσ_eμF)^-0.5 ≈ 49 mm (其中， $F$ 为频率)，远大于试样半径，因此可以认为电流在试样径向上均匀分布。在一个脉冲周期内， $t$ 时刻的电流密度为^[17]：

j (t) = \frac{U}{R_{e} A} e x p (- \frac{t}{R_{e} C_{e}})

(11)

式中， $U$ 为储能电容器的充电电压； $R_{e}$ 为放电回路的总电阻； $A$ 为圆柱形试样的横截面面积； $C_{e}$ 为电容。

采用有限体积法将以上温度场、流场、浓度场和Al粒子半径分布函数控制方程离散并与合金相图^[4]进行耦合求解，即可模拟脉冲电流作用下Pb-Al合金连续凝固过程中的液-固分相凝固组织形成过程。模拟过程中使用的热物性参数见表1^[28]。

表1 Pb-Al体系的热物性参数^[28]

Table 1 Thermophysical property parameters of Pb-Al system^[28]

Parameter	Value	Unit
Dynamic viscosity of Pb $η^{m}$	0.0004636exp(1036.7 / T)	Pa·s
Thermal conductivity of liquid Pb $k^{m}$	15.88	W·m^-1·K^-1
Density of liquid Pb $ρ^{m}$	10670 - 1.32(T - 600.4)	kg·m^-3
Density of solid Al $ρ^{β}$	2700	kg·m^-3
Specific heat of liquid Pb $c_{p}^{m}$	127.61	J·kg^-1·K^-1
Latent heat of solidification of Pb $L$	24700	J·kg^-1
Electrical conductivity of liquid Pb $σ_{e}^{m}$	(0.0479T + 66.6)^-1	10^-8 S·m^-1
Electrical conductivity of solid Al $σ_{e}^{β}$	[15(exp(0.00057T)–1)]^-1	10^-8 S·m^-1

Note:T—thermodynamic temperature

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3.2 Pb-Al液-固分相合金凝固组织演变过程

当Pb-Al液-固分相合金的连续凝固过程进入稳定凝固状态以后，凝固界面前沿的合金熔体可以划分为如图1所示的2个区域，即均一熔体区和液-固分相区。在液-固分相区内，合金熔体的过冷度( $Δ T = T_{s} - T$ ，为平衡组元互溶温度 $T_{s}$ 与熔体温度 $T$ 的差值)、过饱和度( $S = X^{m} - X_{e}^{m}$ ，为基体熔体中Al的摩尔分数 $X^{m}$ 与平衡摩尔分数 $X_{e}^{m}$ 的差值)、Al粒子的形核率 $I$ 、数量密度 $N$ 及平均半径 $\bar{R}$ 沿试样中心轴 $z$ 的变化曲线如图6所示。当均一的合金熔体冷却进入液-固分相区时，随着温度持续降低，Al在Pb基体中溶解度下降，熔体的过饱和度持续升高；当温度降至某一值时，Al粒子开始从基体中形核析出；形核初始时，形核率较小、粒子数量较少，形核及长大消耗的溶质也较少，因而过饱和度持续升高；随着过饱和度的升高，形核率及粒子数量迅速增大，当形核及长大消耗溶质的速率与温度降低导致的基体熔体中溶质溶解度减小的速率相等时，熔体过饱和度和Al粒子形核率达到最大值；此后，由于大量的小粒子在高过饱和熔体中迅速长大，消耗溶质速率很高，熔体过饱和度和Al粒子形核率迅速降低；形核结束后，粒子在基体熔体中不断长大，并在熔体对流、相间密度差等作用下发生空间迁移，直至被凝固界面捕获。

图6

图6 不施加脉冲电流时Pb-0.15Al合金凝固界面前沿中心轴线处基体熔体过冷度( $Δ T$ )、过饱和度( $S$ )，Al粒子形核率( $I$ )、数量密度( $N$ )及平均半径( $\bar{R}$ )沿z轴变化曲线

Fig.6 Supercooling ( $Δ T$ ) and supersaturation ( $S$ ) of the matrix melt, and nucleation rate ( $I$ ), number density ( $N$ ), and average radius ( $\bar{R}$ ) of the Al-rich particles in front of the solidification interface along the central $z$ axis for Pb-0.15Al alloys solidified without ECPs (a), and the enlargement of the microstructure evolution in the nucleation region of the Al-rich particles (b)

3.3 Joule热效应的影响

首先，电流的Joule热效应使熔体的温度场发生变化。图7给出了不同峰值电流密度条件下凝固界面前沿试样中心轴线处的温度分布曲线。可见，随着j_max的增大，熔体的温度逐渐升高，相当于提高了熔体的保温温度，使熔体在冷却凝固过程中的温度梯度增大。

图7

图7 不同峰值电流密度条件下，Pb-0.15Al合金凝固界面前沿试样中心轴线处的温度分布曲线

Fig.7 Calculated temperature profile (solid lines) along the central z axis in front of the solidification interface when solidifying Pb-0.15Al alloys at the rate of 10 mm/s with the ECPs of different j_max together with the experimental results (symbols) without ECPs (T_s(C₀)—solubility line temperature of initial composition alloy, T_eu—eutectic reaction temperature of Al-Pb alloy)

其次，Joule热效应对温度场的改变使熔体对流也发生变化。图8a和b分别给出了常规凝固条件下和脉冲电流作用下Pb-0.15Al合金连续凝固时凝固界面前沿合金熔体内的温度场和流场。可见，试样表面附近熔体温度低、密度大，向下流动；试样中心熔体温度高、密度小，向上流动。最大对流速率出现在试样的中心轴线处。施加脉冲电流会增强合金熔体内的对流运动，最大对流速率随j_max的变化如图8c所示。

图8

图8 凝固界面前沿Pb-0.15Al合金熔体内的温度场和流场，最大对流速率(V_c,max)随j_max变化曲线，及凝固界面前沿中心轴线处，熔体对流运动速度的z分量(V_cz)沿z轴变化曲线

Fig.8 Temperature and flow fields of the melt in front of the solidification interface when solidifying Pb-0.15Al alloys without ECPs (a) and with the ECPs of j_max = 8 × 10⁸ A/m² (b), the maximum of the melt convection rate (V_c,max) vsj_max (c), and z-component of the melt convection velocity (V_cz) along the central z-axis (d) (Q_e—Joule heat of ECPs, f_e—electromagnetic force of ECPs)

脉冲电流Joule热效应对熔体温度场、流场的改变会影响合金的凝固组织。研究^[29,34]表明，逆重力连续凝固条件下，试样中心处熔体的对流运动速度向上，冷却速率较低，弥散相粒子形核率和数量密度较低、平均半径较大；试样近表面处熔体的对流运动速度向下，冷却速率较高，弥散相粒子形核率和数量密度较高、平均半径较小。电流的Joule热效应强化了熔体对流，使得弥散相粒子形核率、数量密度及平均半径沿试样径向分布的上述特征更加明显(如图9所示)，尤其是当j_max超过8 × 10⁸ A/m²后，Joule热导致熔体对流强度迅速增加(见图8c)，使合金凝固组织中易于出现个别大尺寸Al粒子，如图3c和d所示。

图9

图9 Pb-0.15Al合金凝固组织中Al粒子的数量密度、平均半径和体积分数沿试样径向的分布

Fig.9 Variations of $N$ and $\bar{R}$ (a) and the volume fraction ( $ϕ$ ) (b) of the Al-rich particles in the solidified Pb-0.15Al alloys along radial direction ( $Δ G_{V, e}$ —electromagnetic energy of ECPs)

3.4 电磁力的影响

首先，脉冲电流对熔体施加电磁力(f_e)可能会影响熔体的对流运动。图8d给出了凝固界面前沿中心轴线处熔体对流运动速度的z分量( $V_{c z}$ )沿z轴变化曲线。可见，单独考虑电流的电磁力时， $V_{c z}$ 与不施加脉冲电流时的基本相同；同时考虑电磁力与Joule热效应时， $V_{c z}$ 与单独考虑Joule热效应时的基本一致。即电磁力基本不改变熔体的对流运动。熔体对流增强是电流的Joule热效应增大了熔体内温度梯度导致的。

其次，脉冲电流对Al粒子施加电磁力会影响粒子的空间迁移。电致粒子迁移沿试样径向，速率u_e与(jR)²正相关(式3)。图10给出了试样表面附近平均尺寸粒子迁移速度的r分量( $u_{r} (\bar{R}, r = 2 m m)$ )沿z轴变化曲线。可见，随着j_max的增大， $| u_{r} (\bar{R}, r = 2 m m) |$ 逐渐增大。

图10

图10 试样表面附近平均尺寸粒子迁移速度的r分量( $u_{r} (\bar{R}, r = 2 m m)$ )沿z轴变化曲线

Fig.10 r-component of the average size particles' move-ment velocity at the position of r = 2 mm ( $u_{r} (\bar{R}, r = 2 m m)$ ) along the central z axis when solidifying Pb-0.15Al alloy with the ECPs of different j_max

脉冲电流的电磁力通过影响弥散相粒子迁移从而影响合金的凝固组织。由图9可知，单独考虑电磁力的影响时，凝固组织中Al粒子的数量密度、平均半径沿试样径向分布与常规条件下凝固时的基本一致，即电磁力对粒子数量密度及平均半径沿试样径向分布的影响很弱。但凝固组织中试样中心区域处Al粒子的体积分数增大，而试样表面处粒子的体积分数减小，形成贫Al层。这是Al粒子在电磁力的作用下由试样表面向心部迁移造成的，模拟结果与实验结果(图5)相符。利用脉冲电流的这一作用效果，可以制备出表面贫Al，而内部被Al粒子强化的具有壳/核型结构的特种铅基合金材料。

3.5 电磁能促进Al粒子形核的影响

在Pb-Al合金的液-固分相过程中施加脉冲电流会改变Al粒子的形核率。由于Al粒子的电导率大于Pb基体，脉冲电流的施加可以为Al粒子的形核提供额外的驱动力^[17]，提高Al粒子的形核率和数量密度，降低粒子平均半径，如图9a所示。

施加脉冲电流时，电流的作用是不连续的，可以将一个脉冲周期分成2个阶段，即有电流作用的阶段(I)和无电流作用的阶段(II)。阶段(I)持续的时间( $t_{I}$ )很短，阶段(II)持续的时间( $t_{I I}$ )较长，当脉冲频率为100 Hz时， $t_{I}$ 、 $t_{I I}$ 分别约为3和10⁴ μs。凝固组织中的Al粒子也可分成2部分，即阶段(I)中形核的Al粒子和阶段(II)中形核的Al粒子，它们的数量密度(N_I、N_II)分别为：

N_{I} = \frac{1}{V_{0} (t_{I} + t_{I I})} \int_{0}^{\infty} \int_{0}^{t_{I}} I (t) d t d z

(12)

N_{I I} = \frac{1}{V_{0} (t_{I} + t_{I I})} \int_{0}^{\infty} \int_{0}^{t_{I} + t_{I I}} I (t) d t d z

(13)

图11给出了凝固组织中 $N_{I}$ 、 $N_{I I}$ 和 ${\bar{R}}_{2 D}$ 随 $j_{m a x}$ 的变化关系，图12给出了不同 $j_{m a x}$ 条件下熔体温度、过饱和度和Al粒子形核率沿 $z$ 轴的分布。可见， $j_{m a x}$ 存在2个临界值：j_c1 ≈ 4 × 10⁸ A/m²，j_c2 ≈ 6 × 10⁸ A/m²。当 $j_{m a x} \leq j_{c 1}$ 时，虽然电流作用期间Al粒子的形核率有所增加，但增加幅度不够、且电流作用时间很短， $N_{I}$ 很小(见图11)，熔体的峰值过饱和度和Al粒子峰值形核率位置与不施加脉冲电流时的基本相同(见图12b)，进入阶段(II)后Al粒子仍然形核， $N_{I} ≪ N_{I I}$ ，即Al粒子的形核主要发生在无电流作用的阶段(II)，脉冲电流的影响很弱，凝固组织中Al粒子的平均半径与不施加脉冲电流时的基本相同(见图11)。当 $j_{c 1} < j_{m a x} < j_{c 2}$ 时，随着 $j_{m a x}$ 的增大，电流作用期间Al粒子的形核率和 $N_{I}$ 迅速增大，Al粒子在更高的温度、更低的过饱和度下开始形核并达到峰值，熔体的峰值过饱和度下降，阶段(II)中粒子的形核率减小(见图12c)， $N_{I}$ 逐渐追赶并超过 $N_{I I}$ ，凝固组织中Al粒子的数量密度( $N_{I} + N_{I I}$ )迅速增大，平均半径快速下降(图11)。当 $j_{m a x} \geq j_{c 2}$ 时，阶段(II)中粒子的形核率很小， $N_{I} ≫ N_{I I}$ ，即Al粒子的形核主要发生在有电流作用的阶段(I)。随着 $j_{m a x}$ 的进一步增大，凝固组织中Al粒子的数量密度持续增加，平均半径进一步减小(图11)。

图11

图11 凝固组织中Al粒子的N及 ${\bar{R}}_{2 D}$ 随j_max的变化

Fig.11 Curves of $N_{I}$ , $N_{I I}$ , $N_{I} + N_{I I}$ , and ${\bar{R}}_{2 D}$ vs $j_{m a x}$ ( $N_{I}$ —number density of the Al-rich particles nucleated during the period of the current density j > 0; $N_{I I}$ —number density of the Al-rich particles nucleated during the period of j = 0; j_c1 and j_c2—critical values of j_max)

图12

图12 不同j_max条件下Pb-0.15Al合金中心轴线处熔体温度( $T$ )、 $S$ 及 $I$ 沿 $z$ 轴变化曲线

Fig.12 Curves of $T$ , $S$ and $I$ vs $z$ with j_max values being 0 A/m² (a), 4 × 10⁸ A/m² (b), 6 × 10⁸ A/m² (c), and 8 × 10⁸ A/m² (d), respectively ( $I_{I}$ —Al-rich particles' nucleation rate at the moment of $j = j_{m a x}$ ; $I_{I I}$ —Al-rich particles' nucleation rate during the period of $j = 0$ . The parameter with subscript “max” refers to the maximum value for the corresponding parameter)

图13a和b给出了j_max = 8 × 10⁸ A/m²时凝固界面前沿中心轴线处ΔT、S、I、N及 $\bar{R}$ 沿 $z$ 轴的变化曲线。与图6对比可见，脉冲电流作用下凝固时，在凝固界面至峰值形核率区间内， $S$ 、 $N$ 和 $\bar{R}$ 沿 $z$ 轴呈现波动变化，波动幅度随 $z$ 的下降而逐渐衰减。这是电流非连续作用的结果。在一个脉冲周期内的电流作用期间，Al粒子在峰值过饱和度位置附近区域大量形核，使 $N$ 曲线上出现一个“峰”。形成的大量粒子在随后的凝固过程中扩散长大并消耗溶质，使当地熔体的过饱和度和Al粒子长大速率迅速下降，导致对应位置处 $S$ 、 $\bar{R}$ 曲线很快形成一个“谷”。在电脉冲结束后的无电流作用期间，粒子形核率很低， $N$ 曲线上形成一个“谷”，粒子扩散长大消耗溶质较少，导致对应位置处 $S$ 、 $\bar{R}$ 曲线形成一个“峰”。每次电脉冲作用下Al粒子的高形核率都会使 $N$ 曲线上出现一个“峰”，而使 $S$ 曲线和 $\bar{R}$ 曲线出现一个“谷”。随着试样的连续下拉凝固，曲线上的“峰”与“谷”逐渐向凝固界面移动，形成“谷”“峰”交替的波动状曲线。在溶质扩散及Al粒子长大、空间迁移的作用下， $S$ 、 $N$ 和 $\bar{R}$ 沿 $z$ 轴的波动变化幅度会在形核结束后的凝固过程中逐渐减小、消失，最终形成Al粒子均匀分布的凝固组织。图13c给出了图13a和b所示位置(No.1~6)处Al粒子的半径分布函数。可见，对应于形核峰值位置(No.1)的Al粒子半径分布曲线呈现多峰，最大的2个峰分别位于 $R \approx 0$ 及 $R \approx 0.4 μ m$ 处，前者对应于当前电脉冲作用下新形核的Al粒子，后者则对应于前一个电脉冲作用时形核的粒子。随着凝固过程的进行(对应于No.1→No.6)，两峰逐渐融合，最终变成单峰。凝固组织中Al粒子的二维尺寸分布见图3，模拟结果与实验结果相吻合。

图13

图13 峰值电流密度为8 × 10⁸ A/m²时凝固界面前沿中心轴线处， $Δ T$ 、 $S$ 、 $I$ 、 $N$ 、 $\bar{R}$ 沿 $z$ 轴变化曲线，及位置No.1~6处Al粒子的半径分布函数

Fig.13 Curves of $Δ T$ , $S$ , $I$ , $N$ , and $\bar{R}$ vs $z$ for Pb-0.15Al alloys solidified under the effect of ECPs (a, b), and the radius distribution function ( $f$ ) of the Al-rich particles at the positions of Nos.1~6 as marked in Figs.13a and b (c) with j_max = 8 × 10⁸ A/m² (Fig.13b is the local enlargement of Fig.13a)

4 结论

(1) 脉冲电流显著影响Pb-Al合金的液-固分相凝固过程和凝固组织。一方面，脉冲电流能有效提高Pb-Al合金液-固分相过程中Al粒子的形核驱动力，降低其形核能垒，提高粒子形核率，降低粒子尺寸，促进Pb-Al合金形成原位微/纳米Al粒子复合凝固组织；另一方面，脉冲电流使熔体中的Al粒子在电磁力的作用下向试样心部迁移，导致试样表面形成贫Al层，可用于控制制备由表面Pb壳和原位Al粒子铅基合金内核组成的壳/核型特种复合材料。

(2) 用脉冲电流细化Pb-Al合金液-固分相凝固过程中原位析出的Al粒子时 $j_{m a x}$ 存在2个临界值j_c1 ≈ 4 × 10⁸ A/m²，j_c2 ≈ 6 × 10⁸ A/m²。当j_max ≤ j_c1时，脉冲电流几乎不影响Al粒子的沉淀析出过程；当j_c1 < j_max < j_c2时，Al粒子的形核率和数量密度随着j_max的增大而快速升高；当j_max ≥ j_c2时，Al粒子的形核率和数量密度随着j_max的增大继续升高，但升高速度较慢。

(3) Pb-Al液-固分相合金在脉冲电流作用下凝固时，电流的非连续性导致Al粒子的形核率随时间发生变化，使形核位置附近粒子的数量密度、平均半径以及熔体的过饱和度沿试样轴线呈现波动变化。在溶质扩散及Al粒子长大和空间迁移的作用下，这种波动变化的幅度会在形核结束后的凝固过程中迅速减小、消失，最终形成Al粒子均匀分布的凝固组织。

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[D]. Shenyang: University of Chinese Academy of Sciences, 2014

江鸿翔

电流作用下偏晶合金连续凝固过程研究

[D]. 沈阳: 中国科学院大学, 2014

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J Z

Convective effect on the microstructure evolution during a liquid-liquid decomposition

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, Ratke

Repeated nucleation of minority phase droplets induced by drop motion

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, Guthrie

R I L

, translated by Xian

A P

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L W

. The Physical Properties of Liquid Metals [M]. Beijing: Science Press, 2006: 256

[本文引用: 4]

Iida

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著, 冼爱平, 王连文译. 液态金属的物理性能 [M]. 北京: 科学出版社, 2006: 256

[本文引用: 4]

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, et al.

Microstructure formation in directionally solidified Pb-Al alloy

[J]. Acta Metall. Sin., 2022, 58: 1072

DOI [本文引用: 2]

Pb is widely used as grid material for lead-acid batteries, an electrowinning electrode and a nuclear radiation shield. To improve the performance of these materials, alloying elements such as Ag, Sb, and Ca are commonly added. Pb's conductivity and strength can be improved using Al as an alloying element. However, the phase diagram of the Pb-Al alloy is characterized by the large liquid-liquid and liquid-solid miscibility gaps. When a homogeneous single-phase Pb-Al liquid is cooled into the miscibility gaps, Al-rich droplets/particles precipitate first from the melt, causing the Pb-Al alloy to form a microstructure with coarse Al-rich particles or serious phase segregation. Understanding the evolution of microstructure in the liquid-solid phase separation has remained a scientific challenge thus far. The solidification of the Pb-Al alloy is investigated using directional solidification experiments in this work. A numerical model is developed to describe the microstructure formation in a directionally solidified liquid-solid phase separation alloy using the population dynamics method. The evolution of the microstructure is simulated. The simulation results agree well with the experimental results. They show that a supercooling zone appears in front of the solidification interface, where the liquid-solid phase separation of the Pb-Al alloy occurs. In this zone, Al-rich particles (dispersed phase) form and grow by solute diffusing as they move toward the solidification interface. The nucleation rate and the number density of Al-rich particles increase as the solidification rate increases, whereas the average radius of the particles decreases. The Al-rich particles' Stokes movement velocity has the same direction as the melt's solidification velocity, resulting in an enrichment of Al-rich particles in front of the solidification interface. Because of the convective flow of the melt in front of the solidification interface, the cooling rate of the melt is unevenly distributed along the radial direction, resulting in an uneven distribution of nucleation rate, number density, and average radius of Al-rich particles. The formation of a solidification microstructure with the dispersive distribution of Al-rich particles is dependent on the solidification rate being fast enough to ensure that all size particles in the liquid-solid phase separation region move toward the solidification interface under the effect of the Stokes movement of Al-rich particles and the convective flow of melt.

李彦强, 赵九洲, 江鸿翔

等.

Pb-Al合金定向凝固组织形成过程

[J]. 金属学报, 2022, 58: 1072

DOI [本文引用: 2]

实验考察了Pb-Al液-固分相合金的定向凝固行为,建立了Pb-Al合金定向凝固模型,结合实验模拟分析了凝固组织形成过程。研究表明,在Pb-Al合金液-固分相过程中,凝固界面前沿存在一过冷区,富Al (弥散相)粒子在此区间内形核,并在向凝固界面移动过程中进行扩散长大,随着凝固速率的提高,弥散相粒子形核率升高、数量密度增大、平均半径减小。富Al粒子Stokes运动的方向与合金凝固方向相同,导致粒子在凝固界面前富集；熔体的对流运动导致弥散相粒子的形核率及数量密度沿试样径向不均匀分布。在弥散相粒子Stokes运动和熔体对流作用下,形成弥散型凝固组织的必要条件为合金的凝固速率足够高,能保证凝固界面前沿液-固分相区间内所有尺寸粒子均向凝固界面迁移。

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Gránásy

, Ratke

Homogeneous nucleation within the liquid miscibility gap of Zn-Pb alloys

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R S

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Deng

C K

, Jiang

H X

, Zhao

J Z

, et al.

Study on the solidification of Ag-Ni monotectic alloy

[J]. Acta Metall. Sin., 2020, 56: 212

The Ag-Ni alloy has high electrical conductivity, good thermal conductivity, high specific heat capacity, and excellent electrical wear resistance if the Ni-rich phase is dispersedly distributed in the Ag-based matrix. It has been widely used in the medium load contactors, magnetic starters, relays, etc. However, Ag-Ni alloy is a typical monotectic system. Generally, the liquid-liquid phase transformation leads to the formation of a solidification microstructure with serious phase segregation. So far, there have been few studies on the solidification process of Ag-Ni alloys and powder-metallurgical techniques are commonly used to prepare Ag-Ni alloys in industry. In this work, casting experiments and microhardness test were carried out with the Ag-Ni monotectic alloy. The samples with composite microstructure, in which the Ni-rich particles dispersed homogeneously in Ag matrix, were obtained. The microhardness of Ag-Ni alloy increases with the increase of nickel content and the cooling rate of the sample during solidification. When the cooling rate during the liquid-liquid phase transition of the Ag-4.0%Ni alloy reaches 1800 K/s, the microhardness of the Ag-4.0%Ni alloy is close to that of the Ag-10.0%Ni sheet electrical contacts produced by powder metallurgy. A model describing the microstructure evolution during cooling Ag-Ni monotectic alloy melt has been proposed. The process of microstructure formation has been simulated and discussed in details. The results indicate that the cooling rate during the nucleation of the Ni-rich droplets/particles has a dominant influence on the solidification microstructure. The average radius of the Ni-rich particles increases with the increase of nickel content, while it decreases with the increase of the cooling rate during solidification. The average radius of the Ni-rich particles shows an inverse square root dependence on the cooling rate during the nucleation of the Ni-rich droplets/particles. The Ostwald coarsening of the Ni-rich droplets/particles is very weak during cooling Ag-Ni monotectic alloy melt. Rapid/sub-rapid solidification has a good application prospect in the preparation of the high-performance Ag-Ni contact materials.

邓聪坤, 江鸿翔, 赵九洲

等.

Ag-Ni偏晶合金凝固过程研究

[J]. 金属学报, 2020, 56: 212

对Ag-Ni偏晶合金开展了快速/亚快速凝固实验，获得了富Ni相粒子均匀弥散分布于Ag基体的合金样品，Ag-Ni合金显微硬度随着合金Ni含量增加和试样凝固过程冷却速率升高而增大，当Ag-4.0%Ni合金液-液相变开始阶段熔体冷却速率达1800 K/s时，其显微硬度接近粉末冶金生产的Ag-10.0%Ni片状电触头的硬度。建立了描述Ag-Ni合金凝固组织演变的动力学模型，模拟计算了Ag-Ni合金凝固组织形成过程，分析讨论了合金成分和试样直径(冷却速率)对Ag-Ni合金凝固组织形成过程的影响。结果表明：富Ni相液滴/粒子形核阶段熔体的冷却速率对合金凝固组织弥散度具有决定性影响；合金的Ni含量越高、试样冷却速率越低，凝固组织中富Ni相粒子平均尺寸越大；Ag-Ni合金熔体冷却凝固时，富Ni相液滴/粒子的尺寸主要受形核和长大控制，Ostwald粗化作用很弱。

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J Z

, Ratke

Solidification microstructure and dynamics of metastable phase transformation in undercooled liquid Cu-Fe alloys

[J]. Acta Mater., 2006, 54: 1749

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Microstructure formation in a directionally solidified immiscible alloy

[J]. Metall. Mater. Trans., 2008, 39A: 3308