金属学报, 2024, 60(4): 537-547 DOI: 10.11900/0412.1961.2022.00096

研究论文

第一性原理研究钛合金中的沉淀强化

程坤1,2, 陈树明1,2, 曹烁1, 刘建荣1, 马英杰1, 范群波3, 程兴旺3, 杨锐1, 胡青苗,1

1 中国科学院金属研究所 师昌绪先进材料创新中心 沈阳 110016

2 中国科学技术大学 材料科学与工程学院 沈阳 110016

3 北京理工大学 材料学院 冲击环境材料技术重点实验室 北京 100081

Precipitation Strengthening in Titanium Alloys from First Principles Investigation

CHENG Kun1,2, CHEN Shuming1,2, CAO Shuo1, LIU Jianrong1, MA Yingjie1, FAN Qunbo3, CHENG Xingwang3, YANG Rui1, HU Qingmiao,1

1 Shi -changxu Innovation Center for Advanced Materials, Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, China

2 School of Materials Science and Engineering, University of Science and Technology of China, Shenyang 110016, China

3 National Key Laboratory of Science and Technology on Materials Under Shock and Impact, School of Materials Science and Technology, Beijing University of Technology, Beijing 100081, China

通讯作者: 胡青苗,qmhu@imr.ac.cn,主要从事工程合金计算设计方面的研究

责任编辑: 李海兰

收稿日期: 2022-03-07   修回日期: 2022-04-25  

基金资助: 国家自然科学基金(52071315)
国家自然科学基金(U2106215)
国家自然科学基金(52001307)
国家科技重大专项项目(J2019-VI-0012-0126)
国家博士后基金项目(2019M661149)

Corresponding authors: HU Qingmiao, professor, Tel:(024)23971813, E-mail:qmhu@imr.ac.cn

Received: 2022-03-07   Revised: 2022-04-25  

Fund supported: National Natural Science Foundation of China(52071315)
National Natural Science Foundation of China(U2106215)
National Natural Science Foundation of China(52001307)
National Science and Technology Major Project(J2019-VI-0012-0126)
China Postdoctoral Science Foundation(2019M661149)

作者简介 About authors

程 坤,男,1996年生,硕士生

摘要

为研究合金化对沉淀强化行为的影响,采用第一性原理方法计算了二元Ti-xM (M = Al、V、Cr、Mn、Fe、Co、Ni、Zr、Nb、Mo、Ta、W)合金弹性模量随成分的变化,提出了弹性模量Mo当量概念,以高效计算复杂成分钛合金(如Ti-Al-V合金以及Ti55521)的弹性模量;结合弹性模量及Russell-Brown沉淀强化模型,研究了二元Ti-xM (M = V、Cr、Mn、Fe、Co、Ni、Nb、Mo、Ta、W)合金以及Ti55521合金中的沉淀强化。结果显示,在体积分数及沉淀相颗粒尺寸相同的情况下,Co、Fe、W、Mo、Ni、Mn沉淀强化作用较强,Cr、Nb、Ta强化作用居中,V强化作用最弱。随合金元素含量x增加,沉淀强化作用均有所增强。热机械处理Ti55521合金经短时时效后,沉淀强化作用有所减弱,但长时时效后,沉淀强化效果增强。

关键词: 第一性原理; 钛合金; 弹性模量; 沉淀强化; 强度

Abstract

Titanium alloys have shown wide application potential in the areas such as aerospace and marine because of their comprehensive properties, including high specific strength, ductility, corrosion resistance, and damage tolerance. Given the rapid development of new-generation advanced military hardware toward large scale, high-speed, light-weight, and structure-complicated titanium alloys experience increasingly harsh application environments. Thus, developing novel high-strength and high-toughness titanium alloys is an important direction in the field of titanium research. To date, the compositional design of titanium alloys is performed within the framework of some empirical rules without involving strengthening and toughening mechanisms. This kind of approach can hardly achieve an accurate and efficient material design. Based on the abovementioned background, the effect of alloying on the precipitation strengthening of the α + β dual-phase titanium alloy was studied by using the first-principles exact muffin-tin orbital method in combination with a coherent potential approximation. High-strength and high-toughness titanium alloys obtain its high strength through precipitation strengthening in the β-phase matrix with α-phase precipitates. The influence of alloying on the precipitation strengthening is crucial to the understanding and prediction of alloy strength and rational alloy design. In the present work, the elastic moduli and lattice constants of a serial binary titanium alloy Ti-xM (M = Al, V, Cr, Mn, Fe, Co, Ni, Nb, Mo, Ta, W) against the composition x were calculated using the first-principles method. Based on which, the elastic moduli of the titanium alloy with a complex composition (such as Ti-Al-V and Ti55521) were evaluated using the concept of elastic Mo equivalency. Subsequently, the precipitation strengthening of binary titanium alloys and the Ti55521 alloy was evaluated by using the elastic modulus within the framework of the modulus strengthening model. Result shows that alloying elements, such as Co, Fe, W, Mo, Ni, and Mn, have the strongest precipitation strengthening effect for the same particle size and volume fraction of α precipitates, followed by Cr, Nb, and Ta, whereas V is the weakest. The strengthening effect increases with the content of alloying element. For the Ti55521 alloy prepared by using a thermal mechanical process, subsequent short-time aging weakens the precipitation strengthening effect compared with long-time aging.

Keywords: first principles; titanium alloy; elastic modulus; precipitation strengthening; strength

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本文引用格式

程坤, 陈树明, 曹烁, 刘建荣, 马英杰, 范群波, 程兴旺, 杨锐, 胡青苗. 第一性原理研究钛合金中的沉淀强化[J]. 金属学报, 2024, 60(4): 537-547 DOI:10.11900/0412.1961.2022.00096

CHENG Kun, CHEN Shuming, CAO Shuo, LIU Jianrong, MA Yingjie, FAN Qunbo, CHENG Xingwang, YANG Rui, HU Qingmiao. Precipitation Strengthening in Titanium Alloys from First Principles Investigation[J]. Acta Metallurgica Sinica, 2024, 60(4): 537-547 DOI:10.11900/0412.1961.2022.00096

钛合金具有较高的比强度、良好的韧性和耐腐蚀性、较高的损伤容限,在航空航天、海洋等重要装备领域有着广泛的应用[1~3]。随新一代装备向高速化、大型化、轻量化、结构复杂化方向发展,对钛合金结构材料性能的要求愈加苛刻。开发新型高强高韧钛合金是钛合金研究领域最为重要的方向之一[4,5]

目前,常用高强高韧钛合金的共同特点是:合金成分以Ti-Al-Mo-V-Cr为主,或辅以少量Nb、Fe、Zr等元素,经固溶时效处理,在β相中析出α沉淀相,通过沉淀强化提高合金强度[6~8]。因此,研究钛合金中的沉淀强化对理解、预测高强高韧钛合金的强度,进而进行合理的成分及工艺设计,具有重要的意义。

根据沉淀强化理论,若沉淀相与基体具有不同的剪切模量,位错接近或进入沉淀相时,引起位错自由能变化,产生强化效应。沉淀相与基体的模量差别越大,沉淀强化效应越强烈。基于此,Kelly[9]、Melander和Persson[10]、Russell和Brown[11]等发展了模量强化模型,并成功描述了Al-Zn、Al-Zn-Mg及Fe-Cu等合金中的沉淀强化行为。其中,Russell-Brown模型形式简洁优美,简单易用,且同时适于描述沉淀强化中的绕过及切过机制,应用较为广泛。

理论计算[12,13]表明,纯Ti α相的剪切模量约为62 GPa,β相的剪切模量约为19 GPa,差别相当大。因此,模量强化对α + β双相钛合金的强度可能会有较大影响。另外,合金化成分显著改变钛合金αβ相的剪切模量。因此,通过合理的成分优化,调节模量强化行为,有望成为优化高强高韧钛合金的有效手段之一。

基于上述研究背景,本工作采用第一性原理方法,利用Russell-Brown模量强化模型,研究了合金化及微观组织参数如沉淀相体积分数、颗粒尺寸等对α + β双相钛合金沉淀强化的影响,研究对象包括二元Ti-xM (M = Al、V、Cr、Mn、Fe、Co、Ni、Zr、Nb、Mo、Ta、W;x为原子分数,%,下同)以及高强高韧Ti55521合金。

1 计算方法与模型

1.1 Russell-Brown模型

根据Russell-Brown模型,沉淀强化引起的强度增量(Δσ)[11]

Δσ=0.8GbL1-E1E2212
(1)

式中,L为沉淀相粒子的间距;G为基体的剪切模量;b为基体的Burgers矢量模,对于bcc结构金属,b=3a / 2,其中,a为基体的晶格常数;E1E2分别为位错穿过相界面时界面两侧单位长度位错线的能量,当沉淀相一侧位错线能量Ep小于基体一侧能量Em时,E1 / E2=Ep / Em,反之,E1 / E2=Em / Ep

Ep / Em可近似为[11]

EpEm=EpEmlg(R / r0)lg(r / r0)+lg(r / R)lg(r / r0)
(2)

式中,R为沉淀相颗粒半径;rr0分别为计算位错能量时所用的外截半径和内截半径,一般情况下,r0选取为2.5br选取为1000r0EpEm分别表示无穷大沉淀相和基体中单位位错线的能量,且Ep / EmGp / Gm,其中,GpGm分别为沉淀相及基体的剪切模量。

式(2)可知,若Gp / Gm < 1,则Ep / Em < 1,且Ep / Em随沉淀相半径R的增加而减小,即,E1 / E2=Ep / Em减小。若Gp / Gm > 1,则Ep / Em > 1,且Ep / Em随着R增加而增加,E1 / E2=Em / Ep亦减小。当E1 / E2趋近0时,Δσ趋近Orowan绕过机制引起的强度增量ΔσOrowan,颗粒表现为强障碍。随着半径的减小,E1 / E2逐渐增加,沉淀相更容易被剪切。当E1/ E2=1时,模量强化对强度没有贡献。因此,Russell-Brown模型可以描述位错绕过或切过机制引起的强化效应[14]

1.2 微观组织参数对模量强化的影响

为考虑微观组织对沉淀强化的影响,本工作利用沉淀相尺寸R、体积分数f与沉淀相颗粒间距L的关系,在Russell-Brown模量强化模型中引入了这些微观组织参数。

设沉淀相为半径R、体积Vα=43πR3的球形颗粒,颗粒数量为N,且均匀分布在体积为Ω的合金内,则可将合金划分为N个正方体,每个正方体的体积为V=Ω / N=2R+L3,如图1所示,沉淀相体积分数可表达为:

f=NVαNV=43πR32R+L3 
(3)

图1

图1   基体内均匀分布的球形沉淀相示意图

Fig.1   Schematic representation of the spherical precipitation particle distributed homogenously in the matrix (R—radius of precipitate, L—spacing between precipitation phase particles)


即,

L=R4π3f1/3-2R
(4)

结合式(1)、(2)及(4)即可计算沉淀强化引起的强度增量Δσ。在上述模型中,fR及小正/四方体体积V的不同变化组合,对应着不同因素对强化的影响。

1.3 弹性常数及弹性模量

由上述可知,沉淀相及基体的弹性剪切模量是计算模量强化的关键参数。高强高韧钛合金中的bcc结构基体β相有3个独立的弹性常数C11C12C44。对bcc晶胞施加正交应变张量和单斜应变张量[15]

εo000-εo000εo21-εo2
(5)
0εm0εm0000εm21-εm2
(6)

式中,εoεm取值0~0.05,间隔0.01。立方晶体的剪切弹性模量C'C44通过如下能量-应变公式拟合求解:

Eεo=E0+2V0C'εo2
(7)
Eεm=E0+2V0C44εm2
(8)

式中,EεoEεm分别为正交和单斜应变下晶胞的能量,E0为未加应变时晶胞的能量,V0为晶胞的平衡体积。最终可获得C'=12C11-C12和剪切弹性模量C44。结合体模量B与弹性常数C11C12的关系:

B=13C11+2C12
(9)

可获得独立弹性常数C11C12B可通过在体积优化过程中得到的E~V数据拟合获得。

高强高韧钛合金中的强化相为hcp结构的α相,有5个独立的弹性常数,C11C12C13C33C44。当对其施加正交 式(5)和单斜变形 式(6)时,可以得到C66C44,其中,C66=12(C11-C12)。对于六角结构晶体,存在无量纲量P

P=-dln(c / a)0dlnV=C33-C11-C12+C13Cs
(10)

其中,

Cs=C11+C12+2C33-4C13=9(c / a)02V2E(V, c / a)(c / a)2
(11)

式中,ac为晶格常数,(c / a)0为体积为V的晶胞的平衡c / a。通过计算hcp晶胞的能量E随其Vc / a的变化,利用(c / a)0~VEV, c / a~(c / a)关系,可得到PCs。对于钛合金,c / aV的关系不大,可忽略不计,因此,B可表示为:

B2C11+C12+4C13+C3392C13+C333
(12)

利用上述关系式,可求出六方晶系的5个独立弹性常数。通过计算得到的单晶弹性常数,根据Hill平均[16]进一步计算多晶弹性模量(如体模量B、Young's模量Y、剪切模量G等)。

1.4 计算参数

本工作采用基于密度泛函理论(density functional theory,DFT)[17]的精确muffin-tin轨道(exact muffin-tin orbitals,EMTO)第一性原理方法[15,18]计算弹性常数及弹性模量。在电子自洽计算过程中,单电子Kohn-Sham方程采用Green函数技术求解,其中,有效势采用最优重叠muffin-tin近似处理,但在计算总能量时,引入全电荷密度修正[15]。单电子有效势中的电子交换关联泛函采用Perdew等[19]广义梯度近似描述。电子波函数采用精确muffin-tin轨道展开,基组中包含spdf轨道成分。将Al-3s23p1、Ti-3p64s23d2、V-3p64s23d3、Cr-3p64s13d5、Mn-3p64s23d5、Fe-3p64s23d6、Co-3p64s23d7、Ni-3p63d84s2、Zr-4p65s24d2、Nb-4p65s24d3、Mo-4p65s14d5、Ta-5s25p65d3、W-5s25p65d4作为价电子处理。对于αβ相,简约Brillouin区均匀k点取样网格分别选取21 × 21 × 21和15 × 15 × 13。总能量收敛标准为10-7 Ry。合金中合金的无序分布用相干势近似(coherent potential approximation,CPA)[15,18]描述,可在一个单胞内实现合金成分的任意变化。计算过程中,对bcc结构β相的体积(或晶格常数)、hcp结构α相的体积V和晶格参数c / a进行了优化。

2 结果与讨论

2.1 晶格常数

2.1.1 纯Ti

表1[20~24]中列出了纯Ti的αβ相的晶格参数。同时列出了第一性原理平面波赝势方法VASP计算[20~22]及实验测得[23,24]的晶格参数。EMTO计算得到的纯α-Ti的晶格常数a与VASP计算及实验测得的晶格参数符合良好,误差不到1%。EMTO计算得到的c / a比VASP及实验值大约1.3%。EMTO及VASP计算得到的β-Ti的晶格参数符合良好。纯β-Ti为高温相,因此,没有低温晶格参数实验值可比较。Lynch及Tanaka[24]利用实验测得的β-Ti-Mo及β-Ti-Re二元合金的晶格参数随成分的变化,外推得到室温纯β-Ti的晶格参数为0.3281 nm,与理论计算值非常接近。

表1   纯Ti αβ相的晶格常数(a、c)与实验值和其他理论值的比较[20~24]

Table 1  Lattice parameters (a, c) of the α and β phases of pure Ti in comparison with available values from experimental measurements and other theoretical calculations[20~24]

PhaseMethoda / nmc / a

α

EMTO0.29331.611
VASP[20]0.29241.595
VASP[21]0.29311.584
VASP[22]0.29461.584
Exp.[23]0.29511.585
βEMTO0.3261-
VASP[20]0.3252-
VASP[22]0.3264-
Exp.[24]0.3281-

Note: EMTO—exact muffin-tin orbitals, VASP—Vienna ab-initio simulation package

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2.1.2 二元钛合金

图2为Ti-xM (M = Al、V、Cr、Mn、Fe、Co、Ni、Zr、Nb、Mo、Ta、W)合金的晶格常数随成分的变化。如图2a所示,α相晶格常数a随着合金元素(除Zr外)含量增加而线性降低,其中Nb含量对a的影响很小。大多数合金元素均使α相的c / a增加,但在低浓度时,Fe使得c / a略有降低。β相的a随合金元素含量的变化与α相相似,与实验上观测到的趋势符合[25]图2ac中,αβ相的晶格常数a与合金成分x均呈线性关系,符合Vegard定律[26]。上述结果与文献报道[20~22,27]的无序二元钛合金α相及β相晶格参数第一性原理计算结果趋势一致。

图2

图2   二元钛合金Ti-xM (M = Al、V、Cr、Mn、Fe、Co、Ni、Zr、Nb、Mo、Ta、W)的晶格常数随成分x的变化

Fig.2   Lattice parameters of binary Ti-xM (M = Al, V, Cr, Mn, Fe, Co, Ni, Zr, Nb, Mo, Ta, W) against the composition x (atomic fraction)

(a) a of the α phase

(b) c / a of the α phase

(c) a of the β phase


2.2 弹性性质

2.2.1 纯Ti

表2[20~22,28~30]对比了EMTO、VASP[20~22]计算以及实验测得[28~30]的纯Ti αβ相的弹性常数。可见,EMTO计算得到的α相弹性常数与VASP及实验值相比有一定差距,其中,C11C33C66偏大,C12C13偏小。EMTO计算得到的βC11C44偏大。EMTO计算得到的弹性常数误差较大的可能原因是muffin-tin轨道球形近似对Ti来说不够精确。不过,研究[12]表明,EMTO-CPA方法能准确预测合金的弹性常数随成分的变化趋势。因此,EMTO-CPA在钛合金成分设计中,仍然能够发挥重要作用。

表2   纯Ti中αβ相的弹性常数与实验值和其他理论值的比较[20~22,28~30] (GPa)

Table 2  Elastic constants of the α and β phases of pure Ti in comparison with available values from experimental measurements and other theoretical calculations[20-22,28-30]

PhaseMethodC11C12C13C33C44C66
αEMTO201.644.254.4222.649.378.7
VASP[20]204.059.177.0192.945.972.5
VASP[21]176.684.577.0190.241.546.1
VASP[22]171.686.672.6190.641.142.5
Exp.[28]176.186.968.3190.550.844.6
Exp.[29]162.492.068.5180.746.635.2
Exp.[30]155.091.079.0173.065.032.0
βEMTO102.2103.6--65.9-
VASP[20]76.6121.9--30.0-
VASP[22]87.8112.2--39.8-

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2.2.2 二元钛合金

本工作计算了二元Ti-xM (M = Al、V、Cr、Mn、Fe、Co、Ni、Zr、Nb、Mo、Ta、W)合金αβ相弹性常数随成分x的变化,这里不作详细讨论。需要指出的是,部分二元Ti-xM合金的弹性常数已有第一性原理计算报道[12,20~22],本工作计算得到的弹性常数随合金成分变化的趋势与文献报道一致。

根据单晶弹性常数,本工作采用Hill平均[16]计算了αβ相的Young's模量、体模量以及剪切模量,如图34所示。除Ti-xZr合金外,αβ相的体模量均随着x的增加而增加。α相的Young's模量和剪切模量均降低,相反β相的Young's模量和剪切模量均升高。

图3

图3   二元钛合金Ti-xM (M = Al、V、Cr、Mn、Fe、Co、Ni、Zr、Nb、Mo、Ta、W)的α相多晶弹性模量随x的变化

Fig.3   Polycrystalline elastic moduli of the α phase of binary Ti-xM (M = Al, V, Cr, Mn, Fe, Co, Ni, Zr, Nb, Mo, Ta, W) against x

(a) bulk modulus (B)

(b) shear modulus (G)

(c) Young's modulus (Y)


图4

图4   二元钛合金Ti-xM (M = Al、V、Cr、Mn、Fe、Co、Ni、Zr、Nb、Mo、Ta、W)的β相多晶弹性模量随x的变化

Fig.4   Polycrystalline elastic moduli of the β phase of binary Ti-xM (M = Al, V, Cr, Mn, Fe, Co, Ni, Zr, Nb, Mo, Ta, W) against x

(a) bulk modulus

(b) shear modulus

(c) Young's modulus


在钛合金α相中,各合金元素的固溶度较低。由图3可见,在低合金元素M浓度(x < 15%)时,α弹性模量与x满足较好的线性关系,线性拟合系数见表3。大多数合金元素能在β-Ti中无限固溶。由图4可见,β相弹性模量与x可用二次曲线拟合,拟合系数也列于表3中。各拟合曲线的截距均为纯Ti的弹性模量。

表3   二元Ti-xM合金剪切模量GM含量x的拟合系数

Table 3  Parameters for the fitting of the shear modulus G and composition x relationship of binary Ti-xM alloy (For the α phase, G-x is fitted with G=62.49+mxx0.15. For the β phase, G-x is fitted with G=18.69+mx+nx2, x0.30. The intercepts in the fitting equations (62.49 GPa and 18.69 GPa) are respectively the shear moduli of the α and β phases of pure Ti)

Mαβ
mmn
Al0.87 ± 1.2938.84 ± 4.57107.37 ± 18.26
V-116.97 ± 0.42103.95 ± 3.96-142.15 ± 15.85
Cr-254.92 ± 11.39161.50 ± 6.75-255.07 ± 26.99
Mn-279.12 ± 3.47198.59 ± 9.37-270.81 ± 37.46
Fe-225.56 ± 8.13227.59 ± 9.73-288.98 ± 38.94
Co-167.85 ± 8.85240.66 ± 7.59-353.45 ± 30.36
Ni-110.81 ± 5.70216.68 ± 6.73-353.20 ± 26.90
Zr-19.07 ± 0.3538.28 ± 3.52-61.30 ± 14.07
Nb-90.08 ± 0.76148.00 ± 5.10-212.12 ± 20.40
Mo-200.39 ± 1.61207.58 ± 9.16-308.80 ± 36.65
Ta-48.13 ± 1.72145.14 ± 4.63-161.78 ± 18.53
W-162.35 ± 1.68215.07 ± 8.60-239.50 ± 34.42

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2.2.3 多元合金

钛合金一般为复杂成分多元合金。在钛合金设计过程中,需要对成分进行优化迭代,若每次迭代均进行第一性原理计算,成本较高。因此,需要发展弹性模量的高通量计算方法。本工作尝试利用二元钛合金的弹性模量估算多元钛合金的弹性模量。

表3中,弹性模量拟合公式中的一次项系数可以表征各元素对弹性模量的影响。据此,可定义弹性模量的Mo当量系数,即:

γM=mMmMo
(13)

其中,mMmMo分别为Ti-xM合金和Ti-xMo合金模量-成分x拟合一次项系数。表4列出了各合金元素的剪切模量Mo当量系数。这样,可以把多元合金中各元素的含量转化为相当的Mo含量,由Mo含量确定合金的弹性模量。

表4   不同合金元素在αβ相中的剪切模量Mo当量系数(γM)

Table 4  Shear modulus Mo-equivalencies (γM) of various alloying elements in the α and β phases of Ti

PhaseAlVCrMnFeCoNiZrNbTaW
α-0.0040.5841.2721.3931.1260.8380.5530.0950.4500.2400.810
β0.1870.5010.7780.9571.0961.1591.0440.1840.7130.6991.036

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弹性模量Mo当量意味着,含量为xM的合金元素M对钛合金模量的影响相当于含量为γMxM的Mo对模量的影响。例如,二元钛合金中V的α相剪切模量Mo当量系数为γVα=0.584β相剪切模量的Mo当量系数为γVβ=0.501。据此,V含量为xV=0.10α相钛合金剪切模量(50.59 GPa)相当于Mo含量为xVMo=0.10γVα=0.0584α相钛合金Young's模量(50.78 GPa);V含量为xV=0.10β相钛合金剪切模量(27.75 GPa)相当于Mo含量为xVMo=0.10γVβ=0.0501β相钛合金Young's模量(28.30 GPa)。

对于多元钛合金,可根据合金中各元素的含量及弹性模量Mo当量系数,确定合金的弹性模量Mo当量,再由二元Ti-Mo合金的弹性模量-成分关系,估算弹性模量。图5比较了用剪切模量Mo当量系数估算的Ti-Al-V合金的剪切模量以及采用第一性原理直接计算得到的结果。由图5可见,不同成分Ti-Al-V合金剪切模量的估算值与直接计算值之间的误差范围为0.38%~4.15%。第一性原理计算得到的纯金属的弹性模量与实验值的误差一般< 15%。因此,采用剪切Mo当量法估算剪切模量所产生的误差是可以接受的。

图5

图5   Ti-Al-V合金α相和β相剪切模量EMTO-CPA计算值(灰色方柱)以及该计算值与采用模量Mo当量方法的估算值的绝对误差(青色方柱)

Fig.5   Shear moduli of α phase (a) and β phase (b) of Ti-Al-V alloy from direct EMTO-CPA calculations (gray column) and the error of the shear modulus evaluated with modulus Mo-equivalency (cyan column)


直接计算与采用弹性模量Mo当量估算得到的弹性模量的差别主要来自2个方面:其一,没有考虑多元合金中不同合金元素原子间的相互作用对弹性模量的影响;其二,对于β相,弹性模量和成分间的关系为二次曲线,弹性模量Mo当量系数仅考虑一次项系数,忽略了二次项系数的贡献。在x较小时,二次项x2可以忽略,估算得到的弹性模量比较准确。但在x较高时,二次项对估算弹性模量会有一定的影响。

2.3 双相钛合金的模量强化

2.3.1 二元合金

利用2.2节中计算得到的αβ相的弹性模量,本工作首先研究了二元Ti-xM (M = V、Cr、Mn、Fe、Co、Ni、Nb、Mo、Ta、W)合金中的模量强化。由于以上元素为β相稳定元素,在β相中有较高固溶度,在α相中固溶度极低。经长期时效后,这些元素集中在β相中,在α相中的含量较少。因此,在计算过程中将α相视为纯Ti,合金元素都集中在β相中。

作为代表性合金,图6给出了β相成分为Ti-5%Mo合金模量强化强度增量(Δσ)α相体积分数f及颗粒尺寸R的变化。可以看出,各微观组织参数对Δσ的影响,分别讨论如下。

图6

图6   β相成分为5%Mo的Ti-Mo合金中,模量强化强度增量(Δσ)随沉淀相粒子半径(R)和体积分数(f)的变化

Fig.6   Strength increment (Δσ) induced by the modulus strengthening effect as a function of the radius (R) and volume fraction (f) of the precipitation particle in Ti-Mo alloy with β phase composition of 5%Mo


(i) 若沉淀相f不变,R增加,Δσ减小,如图6中平行于横坐标轴的直线(f=20%)所示。根据 式(4),f一定时,R增加,则粒子间距L增加,因此,合金划分成的每个小正/四方体的体积V=2R+L3增加(见图1)。在合金总体积Ω不变的情况下,V增加,意味着沉淀相颗粒数N=Ω / V减小。因此,f不变、R增加时,沉淀相颗粒数减少,导致Δσ减小。

(ii) 若V=2R+L3保持不变,即,N保持不变,则随R增加,f同时增加,Δσ随之增加,如图6中曲线(2R+L=200 nm)所示。由图1可知,在这种情况下,沉淀相颗粒间距L减小。

(iii) 若保持沉淀相R不变, f增加,则Δσ增加,如图6中平行于纵坐标轴的直线(R=100 nm)所示。R不变的情况下,f增加,则L减小,V=2R+L3减小,N=Ω / V增加。

其他二元钛合金沉淀强化强度增量随沉淀相Rf的变化趋势与β相成分为Ti-5%Mo合金相似,这里不再赘述。

图7给出了α沉淀相R=100 nm、f= 20%时,Ti-xM (M = V、Cr、Mn、Fe、Co、Ni、Nb、Mo、Ta、W)合金的模量强化强度增量Δσβx的变化。由图7可见,x = 5%时,合金元素大致可以分为3类,其中,I类元素包括Co、Fe、W、Mo、Ni、Mn,模量强化作用较强;II类元素包括Cr、Nb、Ta,强化作用居中;III类元素为V,强化作用最弱。随x增加,合金元素的沉淀强化作用均有所增强。但在x > 10%时,I类元素的沉淀强化作用随x增加趋缓,而II类及III类元素的沉淀强化作用持续增加。在x = 15%时,I类与II类元素的差别逐渐消失。

图7

图7   在沉淀相R = 100 nm、f = 20%时,二元α + β双相Ti-xM (M = V、Cr、Mn、Fe、Co、Ni、Nb、Mo、Ta、W)合金的模量强化强度增量Δσx的变化

Fig.7   Δσ induced by the precipitation strengthening against x of the β phase of the α + β dual phase Ti-xM alloy (M = V, Cr, Mn, Fe, Co, Ni, Nb, Mo, Ta, W) with R = 100 nm and f = 20%


2.3.2 高强高韧Ti55521合金中的模量强化

作为模量强化模型的应用示例,本工作研究了高强高韧Ti-5Al-5Mo-5V-2Cr-1Fe (Ti55521)合金中的模量强化。Ahmed等[31]系统研究了热处理工艺条件对Ti55521合金微观组织及力学性能的影响。他们对Ti55521合金进行了热机械处理(TMP),即,在950℃保温120 s,冷却到900℃,变形25%,再冷却到800℃,保温600 s,变形60%。然后,在650℃分别进行了3.6 ks (Aged I)及14.4 ks (Aged II)的时效处理。并对TMP、Aged I、Aged II 3种热处理状态下合金中的微观组织、元素分配等进行了表征,见表5

表5   不同热处理条件下,Ti55521合金中的元素分配、α相体积分数及颗粒尺寸[31],以及根据实验获得的αβ相成分计算得到的剪切模量

Table 5  Element partitions, f and R of α phase precipitate[31], and G calculated with the experimental compositions of the α and β phases of Ti55521 alloys under various heat treatments

StatePhaseElement partition / (atomic fraction, %)f / %R / nmG / GPa
TiAlMoVCrFeCal.Mo Eq.
TMPα86.99.21.32.20.30.122 ± 33560.0856.40
β81.28.43.25.21.10.9--34.9735.20
Aged Iα86.511.30.21.50.30.238 ± 622561.9959.20
β75.36.54.88.43.21.8--41.1542.52
Aged IIα86.47120.11.20.20.0350 ± 342562.9660.41
β76.15.84.37.84.51.5--40.9142.23

Note: TMP (thermal mechanical processed): held at 950oC for 120 s, then cooled to 900oC and deformed by 25%, then cooled to 800oC and held for 600 s, deformed by 60%; Aged I: held at 650oC for 3.6 ks after TMP; Aged II: held at 650oC for 14.4 ks after TMP

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利用Ahmed等[31]实验测量的αβ相成分,本工作采用EMTO-CPA直接计算了两相的剪切模量,同时采用Mo当量法估算了它们的剪切模量,列于表5中。直接计算及估算得到的β相剪切模量符合相当好,误差仅为1%左右。α相剪切模量误差相对较大,但也不超过5%。多元复杂Ti55521合金的剪切模量计算再次表明,采用弹性模量Mo当量法估算钛合金的弹性模量切实可行。

表5中可知,时效后,α相及β相剪切模量均有所增加,时效时间继续延长对剪切模量的影响减弱。剪切模量的变化与两相中的合金元素分配有密切关系。时效后,α相中α稳定元素Al含量增加,β稳定元素V及Mo等减少,使得α相剪切模量增加(见图3)。时效时间延长对合金元素再分配的影响减弱。类似的,时效使得β相中β稳定元素Mo、V、Cr、Fe含量增加,因此,β相剪切模量增加(见图4)。

利用表5中列出的相体积分数、沉淀相颗粒尺寸及剪切模量,本工作计算了Ti55521合金中的模量强化强度增量,结果列于表6。可见,短时时效后沉淀强化强度增量Δσ略有下降,但长时时效使得沉淀强化强度增量增加。沉淀强化强度增量Δσ随时效时间的非单调变化趋势可能是合理的。随着时效时间延长,沉淀相的fR均增加。由图6可见,fRΔσ的影响趋势是相反的,即,f增加,Δσ增加,但R增加Δσ减小,2者相互竞争,可能使得Δσ随时效时间非单调变化。

Ahmed等[31]的实验测试表明,随时效时间延长,Ti55521合金的屈服强度单调增加(见表6),与沉淀强化强度增量的变化趋势并不相同。需要指出的是,本工作重点考虑的是沉淀强化引起的强度变化。事实上,合金的屈服强度还包括基体的强度。从TMP状态到Aged I状态,基体β相中β稳定元素含量有较大幅度增加,基体强度也显著增加,因此,虽然沉淀强化强度增量Δσ有所下降,合金的整体强度也会增加。从Aged I到Aged II状态,基体β相中β稳定元素Cr含量略有增加,但其他β稳定元素含量下降,使得基体β相强度下降,因此,合金整体强度的增加应源自沉淀强化。本工作采用固溶强化理论计算了两相钛合金中固溶强化所导致的强度增量,证实了上述分析。

表6   不同热处理制度下沉淀强化强度增量Ti55521合金的Δσ计算值及屈服强度实验值(σ0.2) (MPa)

Table 6  Calculated strength increment Δσ induced by precipitation strengthening and experimental yield strength (σ0.2) of Ti55521 alloy after various heat treatments

StateΔσσ0.2 (exp.)
TMP237.8 ± 42.8980 ± 15
Aged I161.1 ± 80.31080 ± 18
Aged II364.3 ± 148.31200 ± 12

Note: The calculation error of Δσ in the table is calculated based on the measurement range of the volume fraction (see Table 5)

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3 结论

(1) 采用EMTO方法结合CPA,计算了二元钛合金Ti-xM (M = Al、V、Cr、Mn、Fe、Co、Ni、Zr、Nb、Mo、Ta、W)的弹性性质。结果显示,除Zr之外,αβ相的体模量均随着合金元素含量x的增加而增加。α相的Young's模量和剪切模量均降低,相反β相的Young's模量和剪切模量均升高。

(2) 提出用弹性模量Mo当量估算多元钛合金αβ的弹性模量。测试计算表明,Ti-Al-V及Ti55521合金的弹性模量估算值与第一性原理方法直接计算值误差均不超过5%。

(3) 采用Russell-Brown模量强化模型,研究了α + β双相钛合金中的沉淀强化。结果表明,根据β稳定元素对二元合金模量强化的贡献,钛合金中的β稳定元素可分为3类:I类元素包括Co、Fe、W、Mo、Ni、Mn,模量强化作用较强;II类元素包括Cr、Nb、Ta,强化作用居中;III类元素为V,强化作用最弱。随x增加,沉淀强化作用均有所增强,但I类与II类元素的沉淀强化效果差别逐渐消失;热机械处理高强高韧Ti55521合金经短时时效后,沉淀强化作用减弱,但长时时效后,沉淀强化效果增强。

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