Al-Li合金中 δ′/θ′/δ′复合沉淀相结构演化及稳定性的第一性原理探究

图1 δ'/θ'/δ'复合沉淀相中反相和同相结构示意图以及2种同相的界面结构

Fig.1 Schematics of anti-phase (a) and in-phase (b) in the δ'/θ'/δ' composite precipitate, and two kinds of in-phase interfacial structures (c) (The arrows are used to assist in showing the relationship for the opposite δ' phases)

为了选取稳定的同相界面结构,构建了包含不同Cu层数的复合沉淀相的结构模型,计算ΔH,计算公式如下^[18]：

Δ H = \frac{E (A l_{n} C u_{b} L i_{c}) - n E (A l) - b E (C u) - c E (L i)}{b + c}

(1)

式中,E(Al _n Cu _b Li _c )是复合沉淀相0 K下的总能；E(Al)、E(Cu)和E(Li)分别为组成元素Al、Cu和Li在平衡晶体结构下单个原子的能量；n、b和c分别为组成复合沉淀相Al、Cu和Li的原子个数。δ'/θ'/δ'复合沉淀相包含不同层Cu时,采用同相#1和同相#2界面结构计算所得的ΔH如图2所示。可以看出,不论是奇数还是偶数层Cu原子,#2都是能量上更稳定的同相界面结构。

图2

图2 δ'/θ'/δ'复合沉淀相包含不同层Cu时采用同相#1和同相#2界面结构计算所得的形成焓(ΔH)

Fig.2 Calculated formation enthalpies (ΔH) of the composite precipitate δ'/θ'/δ' containing different Cu layers with in-phase #1 and in-phase #2 two interfacial structures

以包含2层Cu的δ'/θ'/δ'复合沉淀相为例,讨论由稳定的同相#2结构转变为反相结构的情况。对于给定的同相结构,其反相结构包含2种,如图3b和c所示。图3b所示的反相结构,可以认为它是由同相#2结构沿[010]方向滑移a /2所得,定义为反相a /2[010] (其中a为基体晶格常数)。对于图3c所示的反相结构,姑且认为它的形成是在同相#2沿[010]方向滑移a /2的基础上,继续沿[100]方向滑移a /2得到,这里定义为反相a /2[110]。当然也可以直接由同相#2结构沿[110]方向滑移a /2得到。随着同相#2转变为反相结构,界面δ'端的原子结构变化如图3d所示。

图3

图3 δ'/θ'/δ'复合沉淀相由同相#2转变为反相a /2[010]和a /2[110]结构以及转变过程中界面δ'端的原子结构示意图

Fig.3 Schematics showing the transformation of the composite precipitate δ'/θ'/δ' from in-phase #2 (a) to anti-phase a /2[010] (b) and to anti-phase a /2[110] (c), and the atomic structure of the δ' phase during the transformation (d)

为了给出由同相#2结构转变为这2种反相结构的转变路径,计算了不同转变方式对应的滑移势能面以及沿不同滑移方向转变需要的激活能(E_activation),只考虑相界面的刚性运动,分别如图4a和b所示。可以发现,相比反相a /2[010]界面结构,反相a /2[110]结构的形成焓更低,表明对于包含偶数层Cu原子的δ'/θ'/δ'复合沉淀相,其两侧的δ'相应采用反相a /2[110]位相关系。观察不同滑移路径构成的势能面,可知由初始的同相#2得到稳定的反相a /2[110]结构,沿[110]方向滑移 $\sqrt[]{2}$ a /2是最优的转变路径。反之,由同相#2结构先沿着[100]方向滑移a /2得到反相a /2[100],再继续沿着[010]方向滑移a /2,需要克服的滑移势垒最高,其E_activation为26.835 kJ/mol,如图4b所示。此外,由初始的同相#2结构,不论是先沿着[010]方向朝着反相关系滑移,还是先沿着[100]方向朝着同相#1关系滑移,随着最终转变为反相a /2[110]结构,它们的E_activation都为26.835 kJ/mol。这是由同相#2界面结构的对称性决定的,显然这2种滑移方式都是热力学上更难实现的。而沿[110]方向滑移 $\sqrt[]{2}$ a /2,完成同相#2到反相a /2[110]结构的转变所需的E_activation仅为6.534 kJ/mol。因而,对于δ'/θ'/δ'复合沉淀相中的2种位相关系之间的转变,将依靠此方式实现。

图4

图4 δ'/θ'/δ'复合沉淀相由同相#2结构转变为反相结构的滑移势能面以及相应的滑移激活能

Fig.4 Surface free energy evolution of the δ'/θ'/δ' composite precipitate slipping from in-phase #2 to anti-phase (a), and the corresponding activ-ation energy (E_activation) along slipping path (b)

以包含2、3层Cu的δ'/θ'/δ'复合沉淀结构为例,计算了δ'/θ'/δ'复合沉淀相包含偶数和奇数层Cu原子时,采用同相#2和2种反相结构的形成焓,如图5所示。可以得知,当Cu原子为偶数层时,δ'/θ'/δ'复合沉淀相其两侧δ'将采取反相a / 2[110]结构；当Cu原子为奇数层时,则采取同相#2结构。这一理论计算结果与实验观测^[17]一致。同时,随着内层θ'相逐层长大(设从单层Cu开始),δ'/θ'/δ'复合沉淀两侧的δ'相将以同相#2→反相a / 2[110] →同相#2方式交替转换。

图5

图5 包含2、3层Cu的δ'/θ'/δ'复合沉淀相采取不同位相关系时的ΔH

Fig.5 ΔH of δ'/θ'/δ' composite precipitate containing 2 and 3 Cu-layers with different relationships for the opposite δ'

2.2 不同界面结构的界面能

为了从形核热力学角度阐述不同位相结构导致的δ'相在θ'相上形核、生长的差异,计算了上述3种位相关系对应的界面结构的界面能。然而,在求解θ'/δ'两相界面能时,采用传统的线性拟合法(linear fitting)^[25]、直接法(direct calculation)^[26]都难以满足计算要求。如,采用线性拟合法时,界面能的计算结果会是界面过渡区的平均值；而直接法将造成能量与化学计量比的不一致性^[19]。为了精确求解θ'/δ'界面能,构建了如图6所示的界面模型。图6a是θ'/δ'界面结构,将其按相种类划分出θ'和δ'相,分别得到图6b和c。需要说明的是,两相界面距离是通过构建不加真空层的块体结构,弛豫晶格参数、原子位置得到的。界面能(γ_interface)采取如下公式计算^[19]：

γ_{i n t e r f a c e} = (E_{s u r f a c e}^{θ^{'} / δ^{'}} - E_{s u r f a c e}^{θ^{'}} - E_{s u r f a c e}^{δ^{'}} + S γ_{s u r f a c e}^{θ^{'}} + S γ_{s u r f a c e}^{δ^{'}}) / S

(2)

式中, $E_{s u r f a c e}^{θ^{'} / δ^{'}}$ 为由θ'相和δ'相组成界面结构的能量； $E_{s u r f a c e}^{θ^{'}}$ 和 $E_{s u r f a c e}^{δ^{'}}$ 分别为将界面切片所得到包含真空层θ'和δ'相的能量,γ $_{s u r f a c e}^{θ^{'}}$ 和γ $_{s u r f a c e}^{δ^{'}}$ 分别为对应的表面能；S为界面面积。将图6a界面结构的总能与分开各相的总能做差后,将多减去每个相的表面能(γ_surface),因此需要加上。γ_surface计算公式如下^[27]：

γ_{s u r f a c e} = \frac{1}{2 A} [E_{s u r f a c e} - E_{b u l k} - \sum_{i}^{s p e c i e s} Δ n_{i} μ_{i}]

(3)

式中,A为表面面积,E_surface为表面结构的能量,E_bulk为独立相结构的能量。考虑到切开各相不符合化学计量比,n_i 为多出的原子个数,μ_i 为多出原子对应的化学势。不同位相结构的界面能结果如表1所示。界面能由高到低依次为：反相a /2[010] > 同相#2 > 反相a /2[110]。可见,当θ'和δ'相采取不同的界面结合时,将导致δ'相在先析出θ'相上的形核表现为明显的差异。值得注意的是,反相a /2[110]界面结构的界面能为负值(这里采取图3包含2层Cu时的3种位相关系定义),表明对于包含偶数层Cu的复合沉淀相,δ'相将在θ'相上采取反相a /2[110]的原子排列方式自发形核。当时效温度、Li原子浓度达到L1₂-δ′相有序化形核时,δ'相将总是保持这种界面结合方式在θ'相两端异质形核。当θ'相两端被δ'相包裹形成δ'/θ'/δ'三明治结构时,θ'相的生长、粗化受到限制。因此实验上往往观察到纳米级、窄而长的δ'/θ'/δ'复合结构。

图6

图6 求解θ'/δ'界面结构界面能的示意图

Fig.6 Schematics how solving for interfacial energy of θ'/δ' interfacial structures, the interfacial structures of θ'/δ' containing vacuum (a), separated θ' (b) and δ' (c) phases with surface

表1 3种界面结构的界面面积、界面能以及解理面各相的表面能

Table 1 Calculated interface area (S), interfacial energy (γ_interface), and relative surface energy (γ_surface) of three cleavage planes with different phase relationships

Structure	S / nm²	$γ_{s u r f a c e}^{θ'}$ / (J·m^-2)	$γ_{s u r f a c e}^{δ'}$ / (J·m^-2)	$γ_{i n t e r f a c e}$ / (J·m^-2)
In-phase #2	0.165	1.401	0.894	0.335
Anti-phase a / 2[010]	0.163	1.418	0.894	0.964
Anti-phase a / 2[110]	0.163	1.445	0.846	-0.015

2.3 不同界面结构的理想断裂强度

从界面断裂的理想强度出发可以得到δ'/θ'/δ'复合沉淀相采取不同位相结合的强度和稳定性。基于Rose等^[20]提出的关于金属材料界面断裂普遍存在的键能-位移(binding energy-displacement)关系理论,通过拟合键能(E_b)与位移(d)关系曲线,做键能关于拉伸位移的一阶导数,得到界面解理的应力(σ)随d的变化关系,具体公式如下^[20,28]：

E_{b} (d) = G_{c} [1 - (1 + \frac{d}{d_{i c}}) e x p (- \frac{d}{d_{i c}})]

(4)

σ (d) = \partial E_{b} / \partial d

(5)

式中,G_c为界面的解理能,等于界面断裂的分离功；d_ic为达到最大解理应力时的临界位移。如图7a所示,3种界面结构的键能-位移关系曲线与Rose定律基本吻合。图7b展示了解理应力随位移变化的关系。表2总结了3种界面结构在Rose断裂模型下的G_c、理想解理应力(σ_ic)以及对应的d_ic。G_c从高到低依次为反相a /2[110] > 同相#2 > 反相a /2[010],表明反相a /2[110]在3种界面结构中,其黏合强度最大；而反相a /2[010]界面的黏合强度最低。理想解理强度的计算结果与分离功一致,表明反相a /2[110]是强度最高、最稳定的δ'/θ'界面结合方式。

图7

图7 3种θ'/δ'界面结构的键能-位移曲线以及对应的解理应力变化曲线

Fig.7 Bonding energy (E_b)-displacement (d) curves for three θ'/δ' interface structures (a) and corres-ponding cleavage stress variation curves (b)

表2 基于Rose断裂模型拟合得到的3种界面结构的临界位移(d_ic),解理功(G_c)以及理想解理应力(σ_ic)

Table 2 Critical spacing (d_ic), cleavage energy (G_c), and the ideal cleavage stress (σ_ic) for three interfacial structures with different phase relationships evaluated by RGS model under tensile deformation

Structure	d_ic / nm	G_c / (J·m^-2)	σ_ic / GPa
In-phase #2	0.059	1.948	12.356
Anti-phase a / 2[010]	0.050	1.341	9.796
Anti-phase a / 2[110]	0.061	2.097	12.659

2.4 3种界面结构的成键分析

以包含2层Cu的δ'/θ'/δ'复合沉淀为例,进一步从化学键的成键角度分析界面间原子对成键对复合沉淀相结构稳定性的影响。图8展示了3种θ'/δ'界面结构,界面间不同原子对的成键示意图。可知,在这3种界面上主要存在2类Li—Al键,包括Li¹—Al²和Li¹—Al³；以及2类Al—Al键,包括Al¹—Al²和Al¹—Al³。表3和4分别总结了这些原子对的键长(b')以及相应的基于Hamilton矩阵的晶体轨道布居理论的原子对成键的积分值(-ICOHP)。对比不同原子对成键的-ICOHP可以得知,Al—Al键的成键强度要远大于Li—Al键,表明Al—Al键对界面结合、稳定界面结构起主导贡献。分析不同原子对的b'值得知,与同相#2和反相a / 2[110]不同的是,在反相a / 2[010]结构中,不在同一水平层(沿着δ'/θ'/δ'复合沉淀的(010)面)的原子对,包括Li¹—Al²和Al¹—Al³,其b'明显增加,增长约0.1 nm。而对于处于同一水平层的Li¹—Al³和Al¹—Al²,b'略有缩短,约0.020 nm。结合这些原子对的-ICOHP可以发现,对于键长拉伸的Li¹—Al²和Al¹—Al³,成键态积分值大幅降低,且只有其在同相#2和反相a /2[110]结构中的10%~20%；而对于键长略有缩短的Li¹—Al³和Al¹—Al²,成键态积分值大幅增加约200%。这直接导致在反相a /2[010]结构中,界面间的结合仅依靠位于同一水平层的原子,不同层原子并没有形成稳定的化学键。因此可以推测,以反相a /2[010]连接的δ'/θ'相界面使得δ'/θ'/δ'复合沉淀相的稳定性最差,导致其理想解理强度最低,黏合强度最低。

图8

图8 位于3种θ'/δ'界面处不同原子间的成键示意图

Fig.8 Schematics of different atomic bonding in three types of θ'/δ' interfaces, including in-phase #2 (a), anti-phase a /2[010] (b), and anti-phase a /2[010] (c)

表3 位于3种θ′/δ′界面处,不同原子对化学键的键长 (nm)

Table 3 Summaries of the bond lengths of different atomic interactions in three types of θ′/δ′ interfaces

Structure	Li¹—Al²	Li¹—Al³	Al¹—Al²	Al¹—Al³
In-phase #2	0.288	0.288	0.288	0.288
Anti-phase a / 2[010]	0.386	0.271	0.262	0.397
Anti-phase a / 2[110]	0.280	0.280	0.279	0.279

表4 基于Hamilton矩阵的晶体轨道布局理论,位于3种θ'/δ'界面结构中,不同原子对的积分值(-ICOHP) (eV)

Table 4 Summaries of the integral vales of different atomic interactions based on the crystal orbital Hamilton population (-ΙCOHP) analyses for three types of θ'/δ' interfaces

Structure	Li¹—Al²	Li¹—Al³	Al¹—Al²	Al¹—Al³
In-phase #2	0.256	0.256	1.655	1.655
Anti-phase a / 2[010]	0.052	0.515	3.265	0.178
Anti-phase a / 2[110]	0.264	0.265	2.031	2.031

进一步分析同相#2和反相a /2[110]界面间原子对的b'以及对应的-ICOHP结果可知,这些界面原子除了与位于同一水平层的原子形成稳定的化学键外,与非同一水平的邻近层原子也稳定成键,且成键强度相同。因而有效地将δ'和θ'相连接为一个整体。对于反相a /2[110]结构,相应原子对的键长略有缩短,键合增强。因而,以反相a /2[110]界面结构组装成的δ'/θ'/δ'复合沉淀相,其相结构最稳定。

将Al原子投影到3s、3p轨道,Li原子投影到1s、2s轨道,进一步分析稳定反相a /2[110]界面结构中不同轨道对Al—Al和Li—Al原子对键强的贡献。本工作采用的Li的赝势为Li_sv,由于其1s轨道能级对成键贡献相比2s轨道基本可以忽略,所以只讨论Li的2s轨道贡献。以Al¹—Al²、Li¹—Al³原子对为代表,由图9a可知,对于起主要强化贡献的Al—Al键,来自Al¹3p—Al²3p的贡献最大,占总成键态的50%。相反,来自Al¹3s—Al²3s的贡献可以忽略。其余的50%贡献来自Al¹—Al²键的3s—3p、3p—3s轨道贡献,且贡献基本相等。对于Li-Al原子对的不同轨道投影贡献,如图9b所示,其主要贡献来自Li¹2s—Al³2p成键态贡献,约占总成键态的57.3%,且主要位于低能级处。Li¹2s—Al³2s轨道贡献为34.6%,其余为Li¹1s轨道与Al³2s和2p轨道成键贡献,占到总成键态的8%。

图 9

图 9 基于Hamilton举证的晶体轨道布局理论,在反相a /2[110]界面结构中不同分子轨道对Al¹—Al²和Li¹—Al³原子对成键的贡献值

Fig.9 Different orbital-pair contributions to the total Al¹—Al² (a) and Li¹—Al³(b) interactions based on the crystal orbital Hamilton population analyses for the anti-phase a /2[110] interfacial structure (-pCOHP—project crystal orbital Hamilton population, E_f—Fermi level)

3 结论

(1) 对于复合沉淀相δ'/θ'/δ',当θ'相包含奇数Cu层时,δ'/θ'采取反相a /2[110]界面结构；当包含偶数Cu层时,δ'/θ'采取同相界面结构。通过滑移势能面计算,这2种位相随着θ'逐层生长,将通过沿界面 [110]方向滑移 $\sqrt[]{2}$ a /2实现。

(2) 当δ'/θ'/δ'相包含奇数(偶数)层Cu原子时,δ'/θ'采取同相(反相a /2[110])的界面结构,此时δ'相将在先析出的θ'相上自发形核。一旦δ'/θ'/δ'复合沉淀粒子形成,将有效阻碍基体中θ'相的粗化。

(3) 基于Rose界面断裂模型,对于包含奇数(偶数)层Cu原子的δ'/θ'/δ'相,同相#2 (反相a /2[110])结构的解理能与理想解理应力都是最大的,表明该界面结构是最稳定的。

(4) 界面处Al原子与邻近层Al和Li原子间高的成键强度决定了反相a /2[110]结构优异的稳定性。相比Li—Al键,Al—Al键对界面结构的稳定结合起主导作用。且主要来自Al原子对的3p—3p轨道贡献。其次是3s—3p和3p—3s轨道贡献,大约各占25%。

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Quantum-chemical computations of solids benefit enormously from numerically efficient plane-wave (PW) basis sets, and together with the projector augmented-wave (PAW) method, the latter have risen to one of the predominant standards in computational solid-state sciences. Despite their advantages, plane waves lack local information, which makes the interpretation of local densities-of-states (DOS) difficult and precludes the direct use of atom-resolved chemical bonding indicators such as the crystal orbital overlap population (COOP) and the crystal orbital Hamilton population (COHP) techniques. Recently, a number of methods have been proposed to overcome this fundamental issue, built around the concept of basis-set projection onto a local auxiliary basis. In this work, we propose a novel computational technique toward this goal by transferring the PW/PAW wavefunctions to a properly chosen local basis using analytically derived expressions. In particular, we describe a general approach to project both PW and PAW eigenstates onto given custom orbitals, which we then exemplify at the hand of contracted multiple-ζ Slater-type orbitals. The validity of the method presented here is illustrated by applications to chemical textbook examples-diamond, gallium arsenide, the transition-metal titanium-as well as nanoscale allotropes of carbon: a nanotube and the C60 fullerene. Remarkably, the analytical approach not only recovers the total and projected electronic DOS with a high degree of confidence, but it also yields a realistic chemical-bonding picture in the framework of the projected COHP method.Copyright © 2013 Wiley Periodicals, Inc.

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