Zn-Pb偏晶合金微量界面活性元素选取

图1 添加不同微量元素前后Zn-4.0%Pb合金显微组织的背散射电子(BSE)像

Fig.1 BSE images of the Zn-4.0%Pb alloys without any trace element addition (a) and with additions of 0.057% (atomic fraction, the same below) trace elements of Sn (b), In (c), Cu (d), and Bi (e)

图2

图2 添加不同微量元素前后Zn-4.0%Pb合金中富Pb相粒子的二维平均直径

Fig.2 Two-dimensional (2D) average diameters of the Pb-rich particles in Zn-4.0%Pb alloys without and with additions of 0.057% trace elements (Dotted line represents 2D average diameters of the Pb-rich particles in Zn-4.0%Pb alloys without any trace element addition)

3 分析与讨论

3.1 微量元素种类对Zn-Pb合金相图的影响

微量元素i可能通过改变合金相图或作为界面活性元素影响Zn-Pb合金的凝固组织。因此，首先分析添加微量元素对Zn-Pb合金相图的影响。

Zn-Pb-i (i = Sn、In、Cu、Bi)三元体系中δ相(δ = L₁、L₂或Int，L₁、L₂和Int分别代表Zn基体、富Pb相液滴和液/液界面)的摩尔Gibbs自由能( $G_{m}^{δ}$ )可用亚规则模型^[22]描述：

G_{m}^{δ} = x_{Z n}^{δ} {}^{0}G_{Z n}^{δ} + x_{P b}^{δ} {}^{0}G_{P b}^{δ} + x_{i}^{δ} {}^{0}G_{i}^{δ} +

R_{g} T (x_{Z n}^{δ} l n x_{Z n}^{δ} + x_{P b}^{δ} l n x_{P b}^{δ} + x_{i}^{δ} l n x_{i}^{δ}) +

x_{Z n}^{δ} x_{P b}^{δ} L_{Z n - P b}^{} + x_{Z n}^{δ} x_{i}^{δ} L_{Z n - i}^{} + x_{P b}^{δ} x_{i}^{δ} L_{P b - i}^{} +

x_{Z n}^{δ} x_{P b}^{δ} x_{i}^{δ} L_{Z n - P b - i}^{}

(1)

式中， $x_{n}^{δ}$ 为n (n = Zn，Pb，i)组元在δ相中的摩尔分数； ${}^{0}_{n}^{δ}$ 为δ相中n纯组元的摩尔Gibbs自由能；L_i_-_j 为n-j二元合金在液相中的二元相互作用参数， $L_{n - j}^{} = \sum L_{n - j}^{k} {(x_{n}^{δ} - x_{j}^{δ})}^{k}$ ，组元j = Zn、Pb、i (n ≠ j)，k为常数， $L_{n - j}^{k}$ 为二元相互作用参数中k次项的系数， $L_{n - j}^{k} = A_{k} + B_{k} T$ ，A_k 和B_k 为常数，T为温度； $L_{Z n - P b - i}^{} = L_{Z n - P b - i}^{0} x_{Z n}^{δ} + L_{Z n - P b - i}^{1} x_{P b}^{δ} + L_{Z n - P b - i}^{2} x_{i}^{δ}$ 为Zn-Pb-i三元合金在液相中的三元相互作用参数， $L_{Z n - P b - i}^{k} = A_{k}^{'} + B_{k}^{'} T$ ， $L_{Z n - P b - i}^{k}$ 为三元相互作用参数中k次项的系数， $A_{k}^{'}$ 、 $B_{k}^{'}$ 为常数；R_g为气体常数，取值8.314 J/(mol·K)。

基于式(1)，使用最小自由能方法^[24]结合纯组元的摩尔Gibbs自由能^[23]和表1^[25~29]中的相互作用参数，计算不同微量元素作用下的Zn-Pb合金相图，得到了添加不同微量元素前后Zn-4.0%Pb合金的互溶温度和临界温度(图3)及互溶温度下合金熔体中两液相的平衡成分(表2)。可见，未添加微量元素时，Zn-4.0%Pb合金的互溶温度(T_b0)和临界温度(T_c0)分别为820.3和1066.3 K；添加0.057%的Sn、In、Cu元素后，互溶温度(T_b)和临界温度(T_c)及两液相成分变化均很小，基本不影响Zn-Pb合金相图；而添加0.057%的Bi元素后，T_b和T_c明显升高，弥散相液滴成分发生明显变化。因此，Bi元素并非通过作为界面活性元素来影响Zn-Pb合金的凝固组织。

表1 Zn-Pb-i (= Sn、In、Cu、Bi)体系在液相中的相互作用参数^[25~29]

Table 1 Interaction parameters in the liquid phase of the Zn-Pb-i (i = Sn, In, Cu, and Bi) system^[25-29]

System	Parameter / (J·mol^-1)	Ref.	System	Parameter / (J·mol^-1)	Ref.
Pb-Zn	$L_{P b - Z n}^{0}$ = 23259 - 3.6869T	[25]	Cu-Pb	$L_{C u - P b}^{0}$ = 31008 - 7.195T	[27]
	$L_{P b - Z n}^{1}$ = -44 - 2.9623T			$L_{C u - P b}^{1}$ = 15345 - 10.826T
	$L_{P b - Z n}^{2}$ = -4987 + 6.8175T			$L_{C u - P b}^{2}$ = -6493 + 5.947T
	$L_{P b - Z n}^{3}$ = -9978 + 8.2T			$L_{C u - P b}^{3}$ = -18416 + 13.16T
	$L_{P b - Z n}^{4}$ = 13275 - 10.2T		Cu-Zn	$L_{C u - Z n}^{0}$ = -40696 + 12.653T	[27]
Pb-Sn	$L_{P b - S n}^{0}$ = 6200 - 0.418T	[25]		$L_{C u - Z n}^{1}$ = 4403 - 6.554T
	$L_{P b - S n}^{1}$ = 790 - 1.914T			$L_{C u - Z n}^{2}$ = 7818 - 3.254T
Sn-Zn	$L_{S n - Z n}^{0}$ = 12558 - 8.7041T	[25]	Zn-Pb-Cu	$L_{Z n - P b - C u}^{0}$ = 95000 + 50T	[27]
	$L_{S n - Z n}^{1}$ = -5623 + 4.196T			$L_{Z n - P b - C u}^{1}$ = 40000 - 50T
	$L_{S n - Z n}^{2}$ = 4149 - 4.0910T			$L_{Z n - P b - C u}^{2}$ = 37000
Zn-Pb-Sn	$L_{Z n - P b - S n}^{0}$ = -42814 + 30.2878T	[25]	Zn-Pb-In	$L_{Z n - P b - I n}^{0}$ = -13.116T	[26]
	$L_{Z n - P b - S n}^{1}$ = 19983 - 32.5467T			$L_{Z n - P b - I n}^{1}$ = 2.421T
	$L_{Z n - P b - S n}^{2}$ = -23758 - 45.7394T			$L_{Z n - P b - I n}^{2}$ = -4.776T
In-Pb	$L_{I n - P b}^{0}$ = 3679 - 1.0797T	[26]	Bi-Zn	$L_{B i - P b}^{0}$ = 18265.09 - 8.67630T	[29]
	$L_{I n - P b}^{1}$ = 605 - 1.3688T			$L_{B i - Z n}^{1}$ = -6061.21 + 0.79581T
In-Zn	$L_{I n - Z n}^{0}$ = 12401 - 4.4498T	[26]		$L_{B i - Z n}^{2}$ = -6422.60 + 11.71966T
	$L_{I n - Z n}^{1}$ = -3186 + 1.7560T			$L_{B i - Z n}^{3}$ = 7227.44 - 9.29050T
	$L_{I n - Z n}^{2}$ = 679.0			$L_{B i - Z n}^{4}$ = 21123.07 - 27.14705T
Bi-Pb	$L_{B i - P b}^{0}$ = -4340.36 - 2.896T	[28]		$L_{B i - Z n}^{5}$ = -20747.56 + 22.01759T
	$L_{B i - P b}^{1}$ = -67.21			$L_{B i - Z n}^{6}$ = -7600.36 + 13.15957T

Note: $L n – j k$ —coefficient of the kth order term in the binary interaction parameter of the n-j system (n = Zn, Pb, i; j = Zn, Pb, i; n ≠ j); $L Z n – P b – i k$ —coefficient of the kth order term in the ternary interaction parameters of the Zn-Pb-i system; T—temperature

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图3

图3 添加0.057%不同微量元素前后Zn-4.0%Pb合金的互溶温度和临界温度

Fig.3 Mutual solubility temperature and critical temperature of the Zn-4.0%Pb alloys without and with additions of 0.057% trace elements (T_b0 and T_b are the mutual solubility temperature of Zn-4.0%Pb alloy without and with 0.057% trace element addition, respectively; T_c0 and T_c are the critical temperature of Zn-4.0%Pb alloy without and with 0.057% trace element addition, respectively)

表2 互溶温度下添加不同微量元素前后Zn-4.0%Pb合金的两液相平衡成分 (mole fraction / %)

Table 2 Phase compositions of the two liquid phases in Zn-4.0%Pb alloys without and with the addition of 0.057% trace elements at T_b

Element i	$x_{Z n}^{L_{1}}$	$x_{P b}^{L_{1}}$	$x_{i}^{L_{1}}$	$x_{Z n}^{L_{2}}$	$x_{P b}^{L_{2}}$	$x_{i}^{L_{2}}$
-	98.702	1.298	-	12.021	87.979	-
Sn	98.646	1.297	0.057	12.067	87.793	0.140
In	98.646	1.297	0.057	12.019	87.790	0.191
Cu	98.646	1.297	0.057	12.102	87.897	1.85 × 10^-4
Bi	98.646	1.297	0.057	12.042	84.081	3.877

Note: $x n L 1$ —atomic fraction of element n in Zn matrix (L₁) phase; $x n L 2$ —atomic fraction of element n in Pb-rich droplet (L₂) phase

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3.2 微量元素在液/液界面处的偏析行为

当由溶剂原子Zn、溶质原子Pb和微量元素i组成的均一熔体冷却至T_b时，富Pb相液滴形核，发生液-液相变。根据Gibbs吸附等温式，元素i作为界面活性元素的前提条件是元素i倾向于富集在液/液界面处^[30]。因此，构建了如下描述微量元素i在液/液界面处偏析行为的模型。

由于富Pb相液滴开始形核时L₂相中原子的数量密度极低，因此主要考虑由L₁相和液/液界面组成的体系。在常压(P)条件下，元素i在体系内的摩尔Gibbs自由能(G_i )可表示为^[22]：

G_{i} = E_{i} - T S_{i} + P V_{m} = x_{i}^{L_{1}} (E_{i}^{L_{1}} - T S_{i}^{L_{1}}) +

x_{i}^{I n t} (E_{i}^{I n t} - T S_{i}^{I n t}) - R_{g} T [x_{i}^{L_{1}} l n x_{i}^{L_{1}} +

(1 - x_{i}^{L_{1}}) l n (1 - x_{i}^{L_{1}}) + x_{i}^{I n t} l n x_{i}^{I n t} +

(1 - x_{i}^{I n t}) l n (1 - x_{i}^{I n t})] + P V_{m}

(2)

E_{i} = x_{i}^{L_{1}} E_{i}^{L_{1}} + x_{i}^{I n t} E_{i}^{I n t}

(3)

S_{i} = x_{i}^{L_{1}} S_{i}^{L_{1}} + x_{i}^{I n t} S_{i}^{I n t} + S^{″}

(4)

S^{″} = R_{g} [x_{i}^{L_{1}} l n x_{i}^{L_{1}} + (1 - x_{i}^{L_{1}}) l n (1 - x_{i}^{L_{1}}) +

x_{i}^{I n t} l n x_{i}^{I n t} + (1 - x_{i}^{I n t}) l n (1 - x_{i}^{I n t})]

(5)

式中，E_i 和S_i 分别为体系内元素i的内能和熵； $E_{i}^{L_{1}}$ 和 $E_{i}^{I n t}$ 分别为元素i在L₁相中和在液/液界面处的摩尔内能； $S_{i}^{L_{1}}$ 和 $S_{i}^{I n t}$ 分别为元素i在L₁相中和在液/液界面处的摩尔熵； $S^{″}$ 为摩尔混合熵；V_m为合金熔体的摩尔体积。因合金熔体不可压缩，V_m在给定温度下为常数。

当体系达到平衡状态时，应满足 $\partial G_{i} / \partial x_{i}^{I n t}$ = 0，可得：

l n [(\frac{x_{i}^{I n t}}{1 - x_{i}^{I n t}}) / (\frac{x_{i}^{L_{1}}}{1 - x_{i}^{L_{1}}})] =

- \frac{(E_{i}^{I n t} - E_{i}^{L_{1}}) - T (S_{i}^{I n t} - S_{i}^{L_{1}})}{R_{g} T}

(6)

采用准化学近似方法计算每摩尔i原子从L₁液相移动到液/液界面处的内能改变量 $(E_{i}^{I n t} - E_{i}^{L_{1}})$ 。只考虑最近邻的键， $E_{i}^{I n t} - E_{i}^{L_{1}}$ 可以通过i原子在界面单原子层与i原子在L₁液相中的键能差异计算。当L₁液相中1 mol溶质i原子与界面单原子层1 mol Zn原子交换位置时， $E_{i}^{I n t} - E_{i}^{L_{1}}$ 可表示为：

E_{i}^{I n t} - E_{i}^{L_{1}} = E_{i}^{L_{1} B} + E_{i}^{I n t B} + E_{Z n}^{I n t B} + E_{Z n}^{L_{1} B}

(7)

E_{i}^{L_{1} B} = - Z_{L} (x_{i}^{L_{1}} ε_{i i} + x_{P b}^{L_{1}} ε_{P b i} + x_{Z n}^{L_{1}} ε_{Z n i}) -

Z_{1} (x_{i}^{L_{1}} ε_{i i} + x_{P b}^{L_{1}} ε_{P b i} + x_{Z n}^{L_{1}} ε_{Z n i}) -

Z_{1} (x_{i}^{I n t} ε_{i i} + x_{P b}^{I n t} ε_{P b i} + x_{Z n}^{I n t} ε_{Z n i})

(8)

E_{i}^{I n t B} = Z_{L} (x_{i}^{I n t} ε_{i i} + x_{P b}^{I n t} ε_{P b i} + x_{Z n}^{I n t} ε_{Z n i}) +

Z_{1} (x_{i}^{L_{1}} ε_{i i} + x_{P b}^{L_{1}} ε_{P b i} + x_{Z n}^{L_{1}} ε_{Z n i}) +

Z_{1} (x_{i}^{L_{2}} ε_{i i}^{'} + x_{P b}^{L_{2}} ε_{P b i}^{'} + x_{Z n}^{L_{2}} ε_{Z n i}^{'})

(9)

E_{Z n}^{I n t B} = - Z_{L} (x_{Z n}^{I n t} ε_{Z n Z n} + x_{i}^{I n t} ε_{Z n i} + x_{P b}^{I n t} ε_{Z n P b}) -

Z_{1} (x_{Z n}^{L_{1}} ε_{Z n Z n} + x_{i}^{L_{1}} ε_{Z n i} + x_{P b}^{L_{1}} ε_{Z n P b}) -

Z_{1} (x_{Z n}^{L_{2}} ε_{Z n Z n}^{'} + x_{i}^{L_{2}} ε_{Z n i}^{'} + x_{P b}^{L_{2}} ε_{Z n P b}^{'})

(10)

E_{Z n}^{L_{1} B} = Z_{L} (x_{Z n}^{L_{1}} ε_{Z n Z n} + x_{i}^{L_{1}} ε_{Z n i} + x_{P b}^{L_{1}} ε_{Z n P b}) +

Z_{1} (x_{Z n}^{L_{1}} ε_{Z n Z n} + x_{i}^{L_{1}} ε_{Z n i} + x_{P b}^{L_{1}} ε_{Z n P b}) +

Z_{1} (x_{Z n}^{I n t} ε_{Z n Z n} + x_{i}^{I n t} ε_{Z n i} + x_{P b}^{I n t} ε_{Z n P b})

(11)

式中， $E_{i}^{L_{1} B}$ 和 $E_{i}^{I n t B}$ 分别为i原子在L₁相单原子层和界面单原子层断裂其最近邻键时所需能量； $E_{Z n}^{I n t B}$ 和 $E_{Z n}^{L_{1} B}$ 分别为Zn原子在界面单原子层和L₁相单原子层断裂其最近邻键时所需能量；ε_ZnZn、ε_Zn_i 、ε_Pb_i 、ε_ZnPb和ε_ii 分别为L₁相单原子层或界面单原子层中1 mol Zn—Zn、Zn—i、Pb—i、Zn—Pb、i—i键的键能；Z_L和Z_l分别界面单原子层内的原子与该层内原子和邻层原子的配位数，Z_L = 6、Z_l = 3； $ε_{i i}^{'}$ 、 $ε_{Z n i}^{'}$ 和 $ε_{P b i}^{'}$ 分别为界面处1 mol i原子与相邻L₁相单原子层中i原子、Zn原子、Pb原子间的键能， $ε_{i i}^{'} \approx ε_{i i}$ ； $ε_{Z n P b}^{'}$ 、 $ε_{Z n Z n}^{'}$ 分别为界面处1 mol Zn原子与相邻L₁相单原子层中Pb原子、Zn原子间的键能， $ε_{Z n Z n}^{'} \approx ε_{Z n Z n}$ 。

联立式(7)~(11)可得：

E_{i}^{I n t} - E_{i}^{L_{1}} = κ_{Z n - i} [x_{i}^{I n t} (2 Z_{1} - 2 Z_{L}) + 2 Z_{L} x_{i}^{L_{1}} - Z_{1}] +

ω x_{P b}^{L_{2}} (σ_{P b / i}^{0} - σ_{Z n / P b}^{0} - σ_{Z n / i}^{0}) +

(κ_{Z n - P b} + κ_{Z n - i} - κ_{P b - i}) \cdot

[Z_{L} (x_{P b}^{L_{1}} - x_{P b}^{I n t}) + Z_{1} x_{P b}^{I n t}] +

ω σ_{Z n / i}^{0} (1 - 2 x_{i}^{L_{2}})

(12)

式中，κ_n_-_j 为熔体中1 mol n原子与1 mol j原子间的平均键能， $κ_{n - j} = ε_{n j} - \frac{1}{2} (ε_{n n} + ε_{j j})$ ^[22]， $ε_{n n}$ 和 $ε_{j j}$ 分别为L₁相单原子层或界面单原子层中1 mol n—n和j—j键的键能； $σ_{n / j}^{0}$ 为n-j体系中富n熔体与富j熔体之间的界面能， $σ_{n / j}^{0} = Z_{1} [ε_{n j}^{'} - \frac{1}{2} (ε_{n n} + ε_{j j})] / ω$ ^[31]，如果Zn-i体系或Pb-i体系不属于偏晶合金体系，则 $σ_{n / j}^{0}$ = 0 J/m²；ω为液/液界面的摩尔面积， $ω = x_{Z n}^{I n t} ω_{Z n} + x_{i}^{I n t} ω_{i}$ ^[17]，由于溶质原子i的摩尔分数很低，因此在该模型中取 $ω \approx ω_{Z n} = 1.06 N_{a}^{1 / 3}$ · $V_{m Z n}^{2 / 3}$ ^[31]， $N_{a}^{}$ 为Avogadro常数， $ω_{Z n}$ 和 $ω_{i}$ 分别为纯Zn熔体和纯i熔体与气相间界面的摩尔面积，V_mZn为纯Zn熔体的摩尔体积。κ_n_–_j 可表示为^[32]：

κ_{n - j} = \frac{L_{n - j}}{Z}

(13)

式中，Z为纯Zn熔体中原子的配位数，Z = 12。

将式(13)代入式(12)，可得：

E_{i}^{I n t} - E_{i}^{L_{1}} = \frac{L_{Z n - i}}{Z} [x_{i}^{I n t} (2 Z_{1} - 2 Z_{L}) + 2 Z_{L} x_{i}^{L_{1}} - Z_{1}] +

ω x_{P b}^{L_{2}} (σ_{P b / i}^{0} - σ_{Z n / P b}^{0} - σ_{Z n / i}^{0}) +

\frac{(L_{Z n - P b} + L_{Z n - i} - L_{P b - i})}{Z} \cdot

[Z_{L} (x_{P b}^{L_{1}} - x_{P b}^{I n t}) + Z_{1} x_{P b}^{I n t}] +

ω σ_{Z n / i}^{0} (1 - 2 x_{i}^{L_{2}})

(14)

将式(14)代入式(6)，并利用 $(S_{i}^{L_{1}} - S_{i}^{I n t}) \approx 2 (- 3.5 \pm 1)$ J/(mol·K)^[32]，可得：

l n [(\frac{x_{i}^{I n t}}{1 - x_{i}^{I n t}}) / (\frac{x_{i}^{L_{1}}}{1 - x_{i}^{L_{1}}})] =

- \frac{1}{R_{g} T} \cdot \{\frac{L_{Z n - i}}{Z} [x_{i}^{I n t} (2 Z_{1} - 2 Z_{L}) + 2 Z_{L} x_{i}^{L_{1}} - Z_{1}] -

\frac{}{} 2 (3.5 \pm 1) T\} - \frac{1}{R_{g} T} \{\frac{(L_{Z n - P b} + L_{Z n - i} - L_{P b - i})}{Z} \cdot

[Z_{L} (x_{P b}^{L_{1}} - x_{P b}^{I n t}) + Z_{1} x_{P b}^{I n t}] + ω [x_{P b}^{L_{2}} (σ_{P b / i}^{0} -

σ_{Z n / P b}^{0} - σ_{Z n / i}^{0}) + σ_{Z n / i}^{0} (1 - 2 x_{i}^{L_{2}})]\}

(15)

仅当元素i在界面处浓度高于元素i在基体中浓度(即 $x_{i}^{I n t}$ / $x_{i}^{L_{1}}$ > 1)时，微量元素i才会在液/液界面处富集。

利用式(15)计算了互溶温度下Zn-4.0%Pb合金熔体中微量元素的偏析程度，计算中使用的摩尔体积和界面能如表3所示，计算结果如图4所示。结果表明，元素Cu满足 $x_{C u}^{I n t}$ / $x_{C u}^{L_{1}}$ = 0.85 < 1，不能富集于界面处，不可作为Zn-Pb合金的界面活性元素；虽然元素Bi满足 $x_{B i}^{I n t}$ / $x_{B i}^{L_{1}}$ = 9.58 > 1，但因其明显改变Zn-Pb合金相图，亦不能作为Zn-Pb合金的界面活性元素；元素Sn和In满足 $x_{S n}^{I n t}$ / $x_{S n}^{L_{1}}$ = 5.77 > 1和 $x_{I n}^{I n t}$ / $x_{I n}^{L_{1}}$ = 6.36 > 1，明显富集于液/液界面处，可作为Zn-Pb合金的界面活性元素。

表3 计算时使用的摩尔体积和界面能

Table 3 Molecular volume and interfacial energies used in calculations

Parameter	Value	Unit	Ref.
V_mZn	1.013 × 10^-5	m³·mol^-1	[33]
$σ_{Z n / P b}^{0}$	0.0915	J·m^-2	[34]
$σ_{C u / P b}^{0}$	0.110	J·m^-2	[35]
$σ_{Z n / B i}^{0}$	0.060	J·m^-2	[8]

Note:V_mZn—mole volume of Zn, $σ Z n / P b 0$ —interface energy between Zn-rich melt and Pb-rich melt in Zn-Pb alloy melt, $σ C u / P b 0$ —interface energy between Cu-rich melt and Pb-rich melt in Cu-Pb alloy melt, $σ Z n / B i 0$ —interface energy between Zn-rich melt and Bi-rich melt in Zn-Bi alloy melt

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图4

图4 互溶温度下Zn-4.0%Pb合金中不同微量元素的界面浓度( $x_{i}^{I n t}$ )与基体浓度( $x_{i}^{L_{1}}$ )的比值

Fig.4 Ratios of interface concentration to matrix concentration $x_{i}^{I n t} / x_{i}^{L_{1}}$ of 0.057% trace elements in the Zn-4.0%Pb alloy at T_b ( $x_{i}^{I n t}$ —atomic fraction of element i in the interface, $x_{i}^{L_{1}}$ —atomic fraction of element i in L₁ phase; dotted line represents $x_{i}^{I n t}$ / $x_{i}^{L_{1}}$ = 1, the same below)

3.3 微量元素在液/液界面处偏析行为的影响因素

式(15)表明，微量元素i在Zn-4.0%Pb合金液/液界面处的偏析行为取决于组元间的相互作用参数(L_Zn-Pb、L_Zn-_i 、L_Pb-_i )和二元合金体系中两液相间的界面能( $σ_{Z n / P b}^{0}$ 、 $σ_{Z n / i}^{0}$ 、 $σ_{P b / i}^{0}$ )。由于界面能取决于组元间的相互作用参数(式(16)^[31])，故微量元素i在液/液界面处的偏析行为主要取决于组元间的相互作用参数。

σ_{n / j}^{0} = \frac{2 α L_{n - j} + (7 \pm 2) T}{ω_{n / j}}

(16)

式中，α为界面断键数与体相断键数的比值，α = 0.182；ω_n_/_j 为富n液相与富j液相间界面的摩尔面积。

当合金成分确定时，L_Zn-Pb为常数，因此微量元素i在界面处的偏析行为主要由L_Zn-_i 和L_Pb-_i 决定。下面着重分析L_Zn-_i 和L_Pb-_i 对微量元素i在Zn-Pb合金中偏析行为的影响。

L_Zn-_i / L_Zn-Pb反映了Zn原子与i原子间的相互作用强度，图5为Zn-4.0%Pb合金中 $x_{i}^{I n t}$ / $x_{i}^{L_{1}}$ 随L_Zn-_i / L_Zn-Pb的变化。可见， $x_{i}^{I n t}$ / $x_{i}^{L_{1}}$ 随着L_Zn-_i / L_Zn-Pb的增大而增加。这说明，当Zn原子与i原子间吸引作用较强(L_Zn-_i $≪$ 0)时， $x_{i}^{I n t}$ / $x_{i}^{L_{1}}$ $≪$ 1，i元素倾向于分布在富Zn的L₁相中，本工作中，Cu元素属于该类元素(L_Zn-Cu / L_Zn-Pb = -0.79)；当Zn原子与i原子间排斥作用较强(L_Zn-_i $≫$ 0)时， $x_{i}^{I n t}$ / $x_{i}^{L_{1}}$ $≫$ 1，i元素倾向于分布在富Pb的L₂相中，本工作中，Bi属于该类元素(L_Zn-Bi / L_Zn-Pb= 0.81)；当Zn原子与i原子间的相互作用(吸引或排斥)较弱时，i元素倾向于富集到界面处，本工作中，Sn、In属于该类元素(L_Zn-Sn / L_Zn-Pb= 0.27，L_Zn-In / L_Zn-Pb= 0.36)。

图5

图5 互溶温度下Zn-4.0%Pb合金中 $x_{i}^{I n t}$ / $x_{i}^{L_{1}}$ 随Zn-i和Zn-Pb合金在液相中相互作用参数比值(L_Zn-_i / L_Zn-Pb)的变化

Fig.5 Variations of $x_{i}^{I n t}$ / $x_{i}^{L_{1}}$ with L_Zn-_i / L_Zn-Pb in Zn-4.0%Pb alloys with additions of 0.057% trace elements at T_b (L_Zn-_i —binary interaction parameter of the Zn-i system, L_Zn-Pb—binary interaction parameter of the Zn-Pb system; blue line represents the dependency of $x_{i}^{I n t}$ / $x_{i}^{L_{1}}$ on L_Zn-_i / L_Zn-Pb)

L_Pb-_i / L_Zn-Pb反映了Pb原子与i原子之间的相互作用强度，图6为Zn-4.0%合金中 $x_{i}^{I n t}$ / $x_{i}^{L_{1}}$ 随L_Pb-_i / L_Zn-Pb的变化。可见， $x_{i}^{I n t}$ / $x_{i}^{L_{1}}$ 随着L_Pb-_i / L_Zn-Pb的增大而减小，这说明，当Pb原子与i原子间吸引作用较强(L_Pb-_i $≪$ 0)时， $x_{i}^{I n t}$ / $x_{i}^{L_{1}}$ $≫$ 1，i元素倾向于分布在L₂相中，本工作中，Bi属于该类元素(L_Pb-Bi / L_Zn-Pb = -0.22)；当Pb原子与i原子间排斥作用较强(L_Pb-_i $≫ 0$ )时， $x_{i}^{I n t}$ / $x_{i}^{L_{1}}$ $≪$ 1，i元素倾向于分布在L₁相中，本工作中，Cu属于该类元素(L_Pb-Cu / L_Zn-Pb = 0.80)；当Pb原子与i原子间相互作用(吸引或排斥)较弱时，i元素倾向于富集在界面处，本工作中，Sn、In属于该类元素(L_Pb-Sn / L_Zn-Pb = 0.27，L_Pb-In / L_Zn-Pb = 0.11)。

图6

图6 互溶温度下Zn-4.0%Pb合金中 $x_{i}^{I n t}$ / $x_{i}^{L_{1}}$ 随Pb-i和Zn-Pb合金在液相中相互作用参数比值(L_Pb-_i / L_Zn-Pb)的变化

Fig.6 Variations of $x_{i}^{I n t}$ / $x_{i}^{L_{1}}$ with L_Pb-_i / L_Zn-Pb in Zn-4.0%Pb alloys with additions of 0.057% trace elements at T_b (L_Pb-_i —binary interaction parameter of the Pb-i system; blue line represents the dependency of $x_{i}^{I n t}$ / $x_{i}^{L_{1}}$ on L_Pb-_i / L_Zn-Pb)

综上所述，仅当L_Zn-_i 和L_Pb-_i 的值适中时，即Zn原子和Pb原子与i原子间的相互作用(吸引或排斥)均较弱时，元素i倾向于在界面处富集，可作为界面活性元素提高弥散相液滴的形核率、降低液滴的Marangoni迁移速率、细化弥散相液滴，进而促进弥散型凝固组织的形成。

4 结论

(1) Sn、In可作为微量添加元素调控Zn-Pb合金液-液相变过程，促进Zn-Pb合金凝固形成弥散型原位Pb粒子凝固组织；

(2) 建立了描述Zn-Pb合金液-液相变过程中微量元素在Zn基体与富Pb液滴间液/液界面处偏析行为的理论模型。计算结果表明，在Zn-Pb合金液-液相变过程中，微量元素Sn、In可作为界面活性元素富集于液/液界面处，降低液-液界面能，提高富Pb相液滴形核率，降低液滴的Marangoni迁移速率。

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