BCC晶体中<100>{010}和1/2<111>{110}位错的芯结构及滑动机制研究
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摘要
体心立方晶体中存在各种各样的位错,这些位错对晶体的力学性质具有非常重要的影响。研究位错的中心问题就是确定位错的芯结构问题。经典的Peierls-Nabarro (P-N)模型虽然能定量地给出位错的芯结构,但是,P-N模型建立在弹性连续介质近似的基础上,忽略了晶格离散效应对位错性质的影响,并且P-N模型对失配面间的非线性相互作用取了正弦近似,而此近似对于实际晶体来说过于粗糙。随着失配相互作用和广义层错能(γ-面)关系的揭示,P-N模型得到了很大的改善,但是,晶格离散效应问题的进展仍不尽人意。目前,基于点阵静力学的全离散位错晶格理论有了很大发展,并且已经给出了用于讨论位错芯结构的普适位错方程。本文的主要任务是用位错晶格理论具体讨论体心立方晶体材料中位错的芯结构以及Peierls应力,并进一步研究位错的运动机制问题,具体包括各向同性近似下Fe中<100>{010}刃位错的芯结构、Fe、Mo和W中1/2<111>{110}刃位错的芯结构、Ta中1/2<111>{110}螺位错的运动机制、各向异性近似下Fe中<100>{010}和1/2<111>{110}刃位错的芯结构、Mo中1/2<111>{110}刃位错的运动机制以及Mo中1/2<111>{110}混合位错的芯结构。主要内容如下:
(1)各向同性简立方晶格中的位错方程—一个可解模型
虽然具有任意晶格结构的晶体中的位错方程都可以根据位错晶格理论给出,但是位错方程中积分项的系数只能通过连续极限下与弹性理论的结果对比得到,而二阶微分项的系数也只能用去耦合表面(去掉与内部原子面耦合的表面)的声波波速近似表达。这些结果需要可解模型的验证和确认。这里讨论了简立方晶格是一个可解模型。根据晶格格林函数方法导出长波近似下的约化动力学矩阵,据此导出了简立方晶格中的位错方程。约化动力学矩阵由两部分组成:表面项动力学矩阵和半无限晶体(除去表面项)动力学矩阵。我们得到的半无限晶体动力学矩阵的矩阵元之间的关系与文献中根据晶格静力学和对称性原理给出的结果一致,而表面项动力学矩阵则与文献中的结果不相符。这是因为,文献中的表面项是在假定表面即{010}面为各向同性的条件下得到的,而对于我们考虑的模型,尽管简立方晶格是各向同性的,但其中的{010}面却为各向异性。通过与位错晶格理论给出的位错方程对比可知,我们得到的简立方晶格位错方程中二阶微分项的系数与位错晶格理论给出的结果不一致,而这不一致主要来源于表面项约化动力学矩阵。简立方晶格可解模型第一次明确给出了约化动力学矩阵中的所有常数,这对于使用位错方程研究诸如扭折(kink)之类的弯曲位错很有帮助。
(2) Fe中<100>{010}刃位错
Fe是典型的体心立方晶体而且被广泛应用,并且Fe中的位错也被广泛研究。<100>{010}型位错是bcc晶体中最为简单的一种位错。在数值模拟方面,Bullough和Perrin以及Gehlen等人采用Johnson原子势研究了Fe中<100>{010}刃位错的芯结构,他们的结果显示<100>{010}位错的位错芯非常窄而且芯半径在1.25b(b为伯格斯矢量)到1.65b之间。Chen(陈立群)等人用分子动力学方法模拟了Fe中<100>{010}刃位错的芯结构,他们得到的位错芯半径为1.65b。在理论研究方面,Yan(严家安)等人在考虑了广义层错能(γ-面)后,基于P-N模型研究了这种位错的芯结构。他们得到的有效芯半径(位错半宽)在0.85b到0.93b之间。很明显,理论计算结果比数值计算结果小很多。两者不一致的原因可能是没有考虑晶格离散效应对位错芯结构的影响。因此,用全离散的位错晶格理论研究此种位错的芯结构对于用位错晶格理论来解决bcc晶体中的位错问题来说具有理论代表意义。通过研究Fe中<100>{010}刃位错的芯结构我们发现,P-N模型(正弦力律)给出的位错半宽为0.70b ,但是考虑了晶格离散效应的修正后,位错半宽变宽近一倍,宽度变成1.30b,由此说明,对于Fe中的<100>{010}刃位错,晶格离散效应对位错芯结构非常重要,它可以使位错宽度加倍。通过比较P-N模型(正弦力律)和P-N模型(γ-面)给出的位错宽度,可以看出,在用γ-面代替P-N模型(正弦力律)中的正弦力律来表示失配面间的相互作用后,位错宽度并没有太大变化,只是由0.70b变到了0.88b-0.94b ,这说明Fe中的<100>{010}刃位错对γ-面不是很敏感。我们用P-N模型(γ-面)得出的位错宽度和文献中用γ-面代替正弦力律的P-N模型给出的结果非常接近,这说明我们的计算结果是准确的。最为重要的是,用位错晶格理论即修正的P-N模型给出的位错宽度为1.51 ? 1.57b,这和数值计算得到的结果1.25 ? 1.65b和1.67b符合得很好,这意味着位错晶格理论是合理可用的,而且它大大改善了理论预言和数值计算之间的符合程度。
(3) Fe、Mo和W中1/2<111>{110}刃位错
Mo和W也是典型的体心立方晶体,在工业生产中也被广泛应用。在bcc晶体中,1/2<111>{110}刃位错是最常见的刃位错,因此,用全离散的位错晶格理论研究这三种材料中的1/2<111>{110}刃位错的芯结构具有十分重要的意义。在拟合文献中给出的γ-面时,我们引进了三个拟合参数Δ1、Δ2和Δ3。含有这三个参数的γ-面的表达式可以很精确地描述文献中给出的γ-面,通过求解位错方程,可以给出位错芯结构参数与γ-面中三个拟合参数之间的关系。我们发现,实际上,表征位错的芯结构的位错宽度主要依赖于这三个参数的和Δ=Δ1 +Δ2 +Δ3,而对γ-面的其它细节不敏感,并且Δ与不稳定层错能存在线性关系,因此可以得出结论,位错宽度主要取决于不稳定层错能,即γ-面的能量极大值。这个结论有助于理解晶体中位错的结构和行为。对同种材料,不稳定层错能越大,位错宽度越窄,离散效应对位错宽度的修正效果则越明显。基于位错晶格理论,我们计算了三种材料中不同的不稳定层错能对应的位错半宽,计算得出的Fe中1/2<111>{110}刃位错的位错半宽在2 ? 4b之间,Mo的在2.7 ? 4.8b之间,W中的则在2.9 ? 4.6b之间,这些结果与数值计算的结果一致。这说明,位错晶格理论可以准确地描述位错的芯结构。
(4) Ta中1/2<111>{110}螺位错
1/2<111>{110}螺位错是体心立方晶体中最常见也最重要的位错,因为这种位错决定着体心立方晶体的塑性形变。体心立方晶体的高Peierls应力被认为是由螺位错的非平面芯结构造成的。静态时的螺位错的芯结构在三个{110}面上分解,但是,当螺位错运动时芯结构是否还在三个{110}面上分解还不确定。对bcc晶体中1/2<111>{110}螺位错的芯结构展开的数值计算和原子模拟结果显示,在外力作用下,螺位错如果要发生运动,它的芯结构首先要发生剧烈的变化。晶体Ta是在工业生产中广泛应用的一种材料,Ta中的1/2<111>{110}螺位错的Peierls应力也被广泛研究。但是,关于Ta中1/2<111>{110}螺位错的数值计算结果和实验值之间存在很大的差异。早期对Ta中螺位错的Peierls应力的模拟结果大约为1.5GPa,但是实验值只有260MPa。后来,虽然根据新的广义赝势理论和qEAM2力场得到的Peiers应力分别为660MPa和440MPa ,但是这些数值仍比实验值大很多。数值计算和实验值之所以会存在分歧,一个可能的原因就是没有正确理解Ta中1/2<111>{110}螺位错的运动机制,因为当螺位错运动时,位错的芯结构对Peiers应力非常重要。本文在位错晶格理论的基础上研究了Ta中具有平面且非分解芯结构的1/2<111>{110}螺位错的Peierls应力。在计算Peierls应力时考虑了P-N模型中忽略的但是与晶格离散效应有密切关系的弹性应变能。通过分析弹性应变能、晶格离散效应修正和正弦力律修正对Peierls应力的影响,可以得出,弹性应变能对Peierls应力的影响最大。在综合考虑了弹性应变能、晶格离散效应修正以及正弦力律修正对Peierls应力的影响后,计算得到的平面螺位错的Peierls应力大约为200MPa ,这个数值与实验观测值260MPa非常接近。由于具有平面芯结构的螺位错要比具有非平面芯结构的螺位错容易运动得多,因此,通过与实验值以及数值模拟结果比较可得,Ta中1/2<111>{110}螺位错在运动时的芯结构不再是非平面的,而可能是平面的。这个结果与数值计算的预想一样,即螺位错在受到外力作用时会由非平面的芯结构变成一个容易运动的芯结构,比如平面芯结构。
(5)各向异性近似下,<100>{010}和1/2<111>{110}刃位错的芯结构和运动机制
在研究位错的理论中,大多是在各向同性近似下进行的。由于实际的晶体都是各向异性的,因此,为了更精确地研究晶体中的位错芯结构需要考虑弹性各向异性的影响。基于位错晶格理论,各向异性在位错方程中主要表现在三个方面,晶格离散效应修正、各向异性能量因子以及滑移方向上的切变模量。通过分析单位长度上的位错能量可以求得不同材料中不同类型位错的各向异性能量因子,通过分析滑移方向上的受力情况可以求得滑移方向上的切变模量,而各向异性近似下的刃位错的晶格离散效应修正系数可以根据滑移面上的纵波波速和横波波速求得。通过比较各向同性以及各向异性近似下三个因子的取值可得,相比于各向同性近似,各向异性近似下的三个因子都有较大的变化。但是,综合考虑这三个因素后,各向异性对位错芯结构的影响却不是很明显,这是因为三个因子对位错芯结构的修正效果不一致,而且对于同一种晶体,如果位错类型不同,各向异性的修正效果也可能不同。考虑了各向异性影响后,相比于各向同性下的结果,Fe中<100>{010}刃位错的位错宽度变窄,而1/2<111>{110}刃位错的位错宽度却变宽了。此外,在各向异性近似下,还研究了Mo中1/2<111>{110}刃位错的运动机制。到目前为止,Mo中1/2<111>{110}刃位错的运动机制还是一个不确定的问题。在对此位错展开的原子模拟中,采用Finnis-Sinclair(F-S)势模拟得到的Peierls应力大约为50MPa,而且刃位错被认为是以kink机制运动的,但是后来采用同样的原子势模拟得到的Peierls应力只有25MPa ,而且人们认为kink机制在Mo中1/2<111>{110}刃位错的运动机制中并不占主导作用。此外,Liu等人采用F-S势模拟得到的Peierls应力大约为20 ? 39MPa。为了研究Mo中1/2<111>{110}刃位错的运动机制,我们在各向异性近似下计算了Mo中1/2<111>{110}刃位错的Peierls应力,并与数值计算的结果进行了比较。我们计算得到的Peierls应力大部分值为十几兆帕,最大值也只有51MPa。我们的结果与20 ? 39MPa的数值结果很符合。由于刃位错以刚性机制运动时所需要的Peierls应力要比以kink机制运动时所需要的Peierls应力小,因此,通过与数值模拟的结果相比较可以得出结论,Mo中1/2<111>{110}刃位错运动时是以刚性机制运动,而不是以kink机制运动。
(6) Mo中1/2<111>{110}混合位错
在体心立方晶体中,除了刃位错和螺位错外,还存在各种各样的混合位错。目前对混合位错的理论研究和数值模拟非常少,因此,用位错晶格理论研究体心立方晶体中的混合位错也非常重要。通过研究晶格离散效应对Mo中1/2<111>{110}刃位错、混合位错以及螺位错的修正可以看出,晶格离散效应对刃位错的修正最显著,因此,位错中的刃分量越多,晶格离散效应对此位错的修正就越明显。此外,对于同一种位错,位错宽度越窄,晶格离散效应的修正越明显,而且在刃位错、混合位错以及螺位错中,刃位错的位错宽度是最宽的,也因此,位错中的刃分量越多,位错宽度越宽。
There are various dislocations in body-centered-cubic (bcc) metals, and the dislocations are very important to the mechanical properties of the crystals. The crucial problem of the dislocation is the core structure. The classical Peierls-Nabarro (P-N) model can determine the core structure and Peierls stress quantitatively, however, because it is based on the elastic continuous medinum approximation, it neglects the influence of the lattice discrete effect to the dislocation core structure. Besides, in the classical P-N model, the nonlinear interaction of the misfit planes is described by the sinusoidal force law, whereas, it is inaccurate for the real materials. With the revealment of the relationship between the misfit interaction and the generalized stacking fault energy (γ-surface), the classical P-N model has been improved greatly, however, the problem of the lattice discrete effect is still unsatisfactory and needs further investigations. Up to now, the full discrete lattice theory of dislocation based on the lattice statics has been developped greatly, and the unified dislocation equation to investigate the dislocation core structure has also been provided. In this paper, the dislocation core structures and the Peierls stress of the bcc crystals have been investigated by using the dislocation lattice theory (the modified P-N model), furthermore, the movement mechanism of the dislocations have also been considered. The contents include the core structure of the <100>{010} edge dislocation in Fe, the core structures of the 1/2<111>{110} edge dislocations in Fe, Mo and W as well as the core structure of the moving 1/2<111>{110} screw dislocation in Ta in the isotropic approximation, the core structures of the <100>{010} and 1/2<111>{110} edge dislocations in Fe, the movement mechanism of the 1/2<111>{110} edge dislocation in Mo as well as the core structures of the 1/2<111>{110} mixed dislocations in Mo in anisotropic approximation. The main work and results involve:
(1) The dislocation equation of the simple cubic (SC) crystal in the isotropic approximation-a solvable model
Though the dislocation equations of the crystals with arbitral lattice structures can be obtained according to the lattice theory of dislocation, the coefficient of the intergral term can only be determined by comparing the result in continuous limit with that in the elastic theory, and the coefficient of the second-order differential term can also only be expressed approximately by the acoustic velocities of the uncoupled surface (a plane without the coupling with the internal atomic planes). It is needed for these results to be verified and validated by the solvable models. We have discussed the SC lattice which is a solvable model. The reduced dynamical matrix (RDM) in the long-wavelength approximation can be obtained according to the method of the lattice Green’s function, and then the dislocation equation of the SC crystal can be obtained. The RDM includes two parts, the surface term and the half-infinite crystal term (except for the surface term). The half-infinite crystal term we obtain is in according to the result according to the lattice statics and the symmetry principle, whereas, the surface term we obtain disagrees with the result in the reference. The reason is that the surface (videlicet the {010} plane) in reference is assumed to be isotropic, however, for the model we discussed, the {010} plane is anisotropic though the SC lattice is isotropic. Having compared with the dislocation equations obtained from the dislocation lattice theory, it can be found that the coefficients of the secone-order differential term is not in accordance with the results in the dislocation lattice theory, and this is mainly resulted from the surface term of the RDM. All the constants of the RDM are explicitly given by the solveable model of SC lattice firstly, and it is helpful to investigate the curved dislocations, such as the kink, by using the dislocation equations.
(2) The <100> {010} edge dislocation in Fe
Fe is a typical metal with bcc structure and is widely used in industry. The dislocations in Fe is investigated extensively and the <100> {010} edge dislocation is the simplest edge dislocation in bcc metals. Bullough and Perrin as well as Gehlen et al have studied the <100> {010} edge dislocation in Fe based on the atomic simulations using the Johson potential. Both their results indicate that the core of the <100> {010} edge dislocation in Fe is very narrow and the core radius they obtained is between 1.25b and 1.65b ( b is the Burgers vector). Chen et al have investigated the core structure of the <100> {010} edge dislocation in Fe by using the molecular dynamics simulation and the core radius they obtained is1.67b . In theory, Yan et al studied the dislocation based on the P-N model considering theγ-surface and the core half width (the core radius) they obtained is between0.85band0.93b . It is obvious that the core width they obtained is much smaller than those numerical results. The reason may be that the lattice discrete effect was ignored incorrectly. Therefore, it is meaningful in theory to investigate the <100> {010} edge dislocation in Fe by using the modified P-N model. The core width obtained from the classical P-N model (sinusoidal force law), in which the lattice discrete effect is not considered and the interaction between the misfit planes is described by the sinusoidal force law, is0.70b . However, having considered the lattice discrete effect, the core width is1.30b. It is indicated that the lattice discrete effect can make the core width nearly doubled. If the sinusoidal force law is replaced by theγ-surface, then the core width becomes0.88band0.94b , and there are no much changes in the core width relative to that in classical P-N model (sinusoidal force law). It is indicated that the <100> {010} edge dislocation in Fe is not sensitive to theγ-surface. The core width we obtain from the P-N model (γ-surface) is very close to the results in reference, and it indicates that our results are accurate. More important, the core width obtained from the modified P-N model is1.51 ? 1.57b, and it is in accordance with the numerical results 1.25 ? 1.65b and1.67b . It is indicated that the dislocation lattice theory is reasonable in investigating the dislocation core structures and the agreement between the theoretical prediction and the numerical simulation is improved remarkably by the modified P-N model.
(3) The 1/2<111> {110} edge dislocations in Fe, Mo and W
Mo and W are also the typical metals with bcc structures and they are also widely used in industry. The 1/2<111> {110} edge dislocation is the most commen edge dislocation in bcc metals, therefore, it is meaningful to investigate the core structures of the 1/2<111> {110} edge dislocations in Fe, Mo and W by using the modified P-N model. When fitting theγ-surface given by some references, the three fitting parametersΔ1 ,Δ2andΔ3 are introduced. The expression of theγ-surface with the three fitting parameters can describe theγ-surface appeared in the references accurately, and the relationship between the structure parameter of the dislocation core and the three parameters of theγ-surface can be obtained by solving the dislocation equation. It is found that the dislocation core width, which characterizes the dislocation, mainly depends upon the sum of the three fitting parametersΔ=Δ1 +Δ2 +Δ3, whereas, the core width is not sensitive to the other details of theγ-surface. Because the relationship betweenΔand the unstable fault energy is linear, therefore, it can be concluded that the dislocation core width mainly depends upon the unstable fault energy, videlicit the maximum energy of theγ-surface. This conclusion is helpful to understand the core structures and behaviours of the dislocations. For one material, if the unstable fault energy is larger, then the corresponding core width will be narrower, and the the correction of the lattice discrete effect to the dislocation core structure will be more significant. Based on the dislocation lattice theory, we calculate the core width of the 1/2<111> {110} edge dislocations in Fe, Mo and W. The core width of dislocation in Fe is between2b and4b , the core width in Mo is between2.7b and4.8b , and the core width in W is between2.9b and4.4b . These results are all in accordance with the numerical results. It is indicated that the dislocation lattice theory can describe the dislocation core structure accurately.
(4) The 1/2<111> {110} screw dislocation in Ta
The 1/2<111> {110} screw dislocation is the most important dislocation in bcc metals, because it is considered to be of primary importance in controlling the plastic deformation of bcc metals. It is suggested that the non-planar core structure can interpret the high Peierls stress of bcc metals. Though the idea of the famous three-way dissociation into three equivalent {110} planes is widely accepted, that is, the core structure of the static screw dislocation is dissociated into three equivalent {110} planes, however, it is speculated whether the core structure of the screw dislocation is still dissociated into three equivalent {110} planes when the screw dislocation moves. The core structures of the screw dislocations are investigated by some numerical calculations and atomistic simulations, and it is indicated that the core structures of the screw dislocations always undergo significant changes under the explicit stress before the screw dislocation moves. The metal Ta is widely used in industry, and the screw dislocation in Ta is widely investigated by some numerical simulations. However, there is still a discrepancy between the numerical simulations and the experimental data. The discrepancy is that the Peierls stress predicted by the numerical simulations is much higher than that observed at low temperature. The previous simulations of the screw dislocation in Ta are about1.5GPa , whereas the experimental data is260MPa . Later, the Peierls stress obtained from the new model generalized pseudopotential theory (MGPT) and the qEAM2 force field are about 660MPa and 440MPa , respectively, however, these results are still much larger than the experimental data. The reason of the discrepancy may be the incomplete understanding of the movement mechanism of the screw dislocation, and the configuration of the core structure is very crucial to the Peierls stress for the moving screw dislocation. In this paper, based on the dislocation lattice theory, the Peierls stress of the 1/2<111> {110} screw dislocation with planar and non-dissociated core structure in Ta has been calculated. The elastic strain energy which is associated with the lattice discrete effect and is ignored in classical P-N model is taken into account in calculating the Peierls stress. Through analylizing the effect of the elastic strain energy, the lattice discrete effect and the modification of the sinusoidal force law to the Peierls stress, it can be found that the elastic strain energy is the most important to the Peierls stress. The Peierls stress we obtain is about200MPa , and this result is very close to the experimental data260MPa . Because the Peierls stress of the screw dislocation with the planar core structure is smaller than that with the non-planar core structure, therefore, it is speculated that the core structure of the moving 1/2<111> {110} screw dislocation in Ta may be planar. Just as the prediction of the numerical results, the core structure of the screw dislocation in Ta undergoes significant changes, furthermore, under the external stress, the non-planar core structure of the screw dislocation changes into a planar core structure which moves more easily.
(5) The core structures of the <100> {010} and 1/2<111> {110} edge dislocations in anisotropic approximation
Among the theories investigating the dislocations, the most work is done under the isotropic approximation. However, the real materials are anisotropic, therefore, in order to investigate the dislocation core structures more accurately, it is necessary to consider the effect of the elastical anisotropy to the dislocation core structures. Based on the dislocation lattice theory, the elastical anisotropy is reflected in three factors in the modified P-N dislocation equation, the lattice discrete effect, the elastically anisotropic energy coefficient and the shear modulus. The energy coefficient can be obtained by analyzing the dislocation energies of unit length of the edge and screw dislocations, and the shear modulus can be obtained by analyzing the stress of the glide direction. Besides, the lattice discrete effect in anisotropic approximation can be obtained according to the velocities of the longitudinal wave and the transverse wave along the glide direction. Though the three factors in anisotropic case are different from those in isotropic case, having considered the corrections of the three factors together, there are no much changes between the results in anisotropic case and those in isotropic case, this is because the corrections of the three factors to the dislocation core structures are not always the same. For one material, if the dislocation is different, then the effect of the anisotropy to the core structure will be different. The elastical anisotropy can make the core width of the <100> {010} edge dislocation in Fe narrowed, whereas, it can make the core width of the 1/2<111> {110} edge dislocation in Fe broadened. Besides, the movement mechanism of the 1/2<111> {110} edge dislocation in Mo has been investigated in the anisotropic approximation. Up to now, the movement mechanism of the 1/2<111> {110} edge dislocation in Mo is still undetermined. The Peierls stress of the edge dislocation obtained by using the Finnis-Sinclair (F-S) potential is about50MPa , and it is assumed that the mobility of the edge dislocation is controlled by the kink mechanism at the non-zero temperature. However, the Peierls stress of the edge dislocation obtained by using the same potential is just25MPa , and it is indicated that the kink mechanism does not control the movement of the edge dislocation in Mo. Furthermore, the Peierls stress has also been calculated by Liu et al by using the same F-S potential, and the result they obtained is20 ? 39MPa. In order to investigate the movement mechanism of the 1/2<111> {110} edge dislocation in Mo, we calculate the Peierls stress of the 1/2<111> {110} edge dislocation in Mo, and make a comparison with the numerical simulations. The most Peierls stress we obtain is about dozen MPa and the maximum of the Peierls stress is just51MPa . Our results agree well with the atomistic simulation20 ? 39MPa. Because the Peierls stress of the edge dislocation moving with the rigid mechanism is smaller than that with the kink mechanism, therefore, it can be speculated that the 1/2<111> {110} edge dislocation in Mo may move in the rigid mechanism rather than the kink mechanism or other mechanisms when the external stress is loaded on the dislocation.
(6) The 1/2<111> {110} mixed dislocations in Mo
There are various mixed dislocations in bcc metals except for the edge dislocation and screw dislocation. However, there are little studies on the core structures of the mixed dislocations though they are also important to the plastic deformations of bcc metals. Through investigating the correction of the lattice discrete effect to the core structures of edge, screw as well as the mixed dislocations, it can be found that the correction of discrete effect to the edge dislocation is the most significant, therefore, more larger the edge components is, more significant the correction of discrete effect to the core structure is. Besides, for one dislocation, the correction of discrete effect to the core structure will be more significant if the dislocation core width is narrower. Among the edge, mixed and screw dislocations, the core width of the edge dislocation is the widest, therefore, if the edge component of dislocation is larger, then the dislocation core width is wider.
(1)各向同性简立方晶格中的位错方程—一个可解模型
虽然具有任意晶格结构的晶体中的位错方程都可以根据位错晶格理论给出,但是位错方程中积分项的系数只能通过连续极限下与弹性理论的结果对比得到,而二阶微分项的系数也只能用去耦合表面(去掉与内部原子面耦合的表面)的声波波速近似表达。这些结果需要可解模型的验证和确认。这里讨论了简立方晶格是一个可解模型。根据晶格格林函数方法导出长波近似下的约化动力学矩阵,据此导出了简立方晶格中的位错方程。约化动力学矩阵由两部分组成:表面项动力学矩阵和半无限晶体(除去表面项)动力学矩阵。我们得到的半无限晶体动力学矩阵的矩阵元之间的关系与文献中根据晶格静力学和对称性原理给出的结果一致,而表面项动力学矩阵则与文献中的结果不相符。这是因为,文献中的表面项是在假定表面即{010}面为各向同性的条件下得到的,而对于我们考虑的模型,尽管简立方晶格是各向同性的,但其中的{010}面却为各向异性。通过与位错晶格理论给出的位错方程对比可知,我们得到的简立方晶格位错方程中二阶微分项的系数与位错晶格理论给出的结果不一致,而这不一致主要来源于表面项约化动力学矩阵。简立方晶格可解模型第一次明确给出了约化动力学矩阵中的所有常数,这对于使用位错方程研究诸如扭折(kink)之类的弯曲位错很有帮助。
(2) Fe中<100>{010}刃位错
Fe是典型的体心立方晶体而且被广泛应用,并且Fe中的位错也被广泛研究。<100>{010}型位错是bcc晶体中最为简单的一种位错。在数值模拟方面,Bullough和Perrin以及Gehlen等人采用Johnson原子势研究了Fe中<100>{010}刃位错的芯结构,他们的结果显示<100>{010}位错的位错芯非常窄而且芯半径在1.25b(b为伯格斯矢量)到1.65b之间。Chen(陈立群)等人用分子动力学方法模拟了Fe中<100>{010}刃位错的芯结构,他们得到的位错芯半径为1.65b。在理论研究方面,Yan(严家安)等人在考虑了广义层错能(γ-面)后,基于P-N模型研究了这种位错的芯结构。他们得到的有效芯半径(位错半宽)在0.85b到0.93b之间。很明显,理论计算结果比数值计算结果小很多。两者不一致的原因可能是没有考虑晶格离散效应对位错芯结构的影响。因此,用全离散的位错晶格理论研究此种位错的芯结构对于用位错晶格理论来解决bcc晶体中的位错问题来说具有理论代表意义。通过研究Fe中<100>{010}刃位错的芯结构我们发现,P-N模型(正弦力律)给出的位错半宽为0.70b ,但是考虑了晶格离散效应的修正后,位错半宽变宽近一倍,宽度变成1.30b,由此说明,对于Fe中的<100>{010}刃位错,晶格离散效应对位错芯结构非常重要,它可以使位错宽度加倍。通过比较P-N模型(正弦力律)和P-N模型(γ-面)给出的位错宽度,可以看出,在用γ-面代替P-N模型(正弦力律)中的正弦力律来表示失配面间的相互作用后,位错宽度并没有太大变化,只是由0.70b变到了0.88b-0.94b ,这说明Fe中的<100>{010}刃位错对γ-面不是很敏感。我们用P-N模型(γ-面)得出的位错宽度和文献中用γ-面代替正弦力律的P-N模型给出的结果非常接近,这说明我们的计算结果是准确的。最为重要的是,用位错晶格理论即修正的P-N模型给出的位错宽度为1.51 ? 1.57b,这和数值计算得到的结果1.25 ? 1.65b和1.67b符合得很好,这意味着位错晶格理论是合理可用的,而且它大大改善了理论预言和数值计算之间的符合程度。
(3) Fe、Mo和W中1/2<111>{110}刃位错
Mo和W也是典型的体心立方晶体,在工业生产中也被广泛应用。在bcc晶体中,1/2<111>{110}刃位错是最常见的刃位错,因此,用全离散的位错晶格理论研究这三种材料中的1/2<111>{110}刃位错的芯结构具有十分重要的意义。在拟合文献中给出的γ-面时,我们引进了三个拟合参数Δ1、Δ2和Δ3。含有这三个参数的γ-面的表达式可以很精确地描述文献中给出的γ-面,通过求解位错方程,可以给出位错芯结构参数与γ-面中三个拟合参数之间的关系。我们发现,实际上,表征位错的芯结构的位错宽度主要依赖于这三个参数的和Δ=Δ1 +Δ2 +Δ3,而对γ-面的其它细节不敏感,并且Δ与不稳定层错能存在线性关系,因此可以得出结论,位错宽度主要取决于不稳定层错能,即γ-面的能量极大值。这个结论有助于理解晶体中位错的结构和行为。对同种材料,不稳定层错能越大,位错宽度越窄,离散效应对位错宽度的修正效果则越明显。基于位错晶格理论,我们计算了三种材料中不同的不稳定层错能对应的位错半宽,计算得出的Fe中1/2<111>{110}刃位错的位错半宽在2 ? 4b之间,Mo的在2.7 ? 4.8b之间,W中的则在2.9 ? 4.6b之间,这些结果与数值计算的结果一致。这说明,位错晶格理论可以准确地描述位错的芯结构。
(4) Ta中1/2<111>{110}螺位错
1/2<111>{110}螺位错是体心立方晶体中最常见也最重要的位错,因为这种位错决定着体心立方晶体的塑性形变。体心立方晶体的高Peierls应力被认为是由螺位错的非平面芯结构造成的。静态时的螺位错的芯结构在三个{110}面上分解,但是,当螺位错运动时芯结构是否还在三个{110}面上分解还不确定。对bcc晶体中1/2<111>{110}螺位错的芯结构展开的数值计算和原子模拟结果显示,在外力作用下,螺位错如果要发生运动,它的芯结构首先要发生剧烈的变化。晶体Ta是在工业生产中广泛应用的一种材料,Ta中的1/2<111>{110}螺位错的Peierls应力也被广泛研究。但是,关于Ta中1/2<111>{110}螺位错的数值计算结果和实验值之间存在很大的差异。早期对Ta中螺位错的Peierls应力的模拟结果大约为1.5GPa,但是实验值只有260MPa。后来,虽然根据新的广义赝势理论和qEAM2力场得到的Peiers应力分别为660MPa和440MPa ,但是这些数值仍比实验值大很多。数值计算和实验值之所以会存在分歧,一个可能的原因就是没有正确理解Ta中1/2<111>{110}螺位错的运动机制,因为当螺位错运动时,位错的芯结构对Peiers应力非常重要。本文在位错晶格理论的基础上研究了Ta中具有平面且非分解芯结构的1/2<111>{110}螺位错的Peierls应力。在计算Peierls应力时考虑了P-N模型中忽略的但是与晶格离散效应有密切关系的弹性应变能。通过分析弹性应变能、晶格离散效应修正和正弦力律修正对Peierls应力的影响,可以得出,弹性应变能对Peierls应力的影响最大。在综合考虑了弹性应变能、晶格离散效应修正以及正弦力律修正对Peierls应力的影响后,计算得到的平面螺位错的Peierls应力大约为200MPa ,这个数值与实验观测值260MPa非常接近。由于具有平面芯结构的螺位错要比具有非平面芯结构的螺位错容易运动得多,因此,通过与实验值以及数值模拟结果比较可得,Ta中1/2<111>{110}螺位错在运动时的芯结构不再是非平面的,而可能是平面的。这个结果与数值计算的预想一样,即螺位错在受到外力作用时会由非平面的芯结构变成一个容易运动的芯结构,比如平面芯结构。
(5)各向异性近似下,<100>{010}和1/2<111>{110}刃位错的芯结构和运动机制
在研究位错的理论中,大多是在各向同性近似下进行的。由于实际的晶体都是各向异性的,因此,为了更精确地研究晶体中的位错芯结构需要考虑弹性各向异性的影响。基于位错晶格理论,各向异性在位错方程中主要表现在三个方面,晶格离散效应修正、各向异性能量因子以及滑移方向上的切变模量。通过分析单位长度上的位错能量可以求得不同材料中不同类型位错的各向异性能量因子,通过分析滑移方向上的受力情况可以求得滑移方向上的切变模量,而各向异性近似下的刃位错的晶格离散效应修正系数可以根据滑移面上的纵波波速和横波波速求得。通过比较各向同性以及各向异性近似下三个因子的取值可得,相比于各向同性近似,各向异性近似下的三个因子都有较大的变化。但是,综合考虑这三个因素后,各向异性对位错芯结构的影响却不是很明显,这是因为三个因子对位错芯结构的修正效果不一致,而且对于同一种晶体,如果位错类型不同,各向异性的修正效果也可能不同。考虑了各向异性影响后,相比于各向同性下的结果,Fe中<100>{010}刃位错的位错宽度变窄,而1/2<111>{110}刃位错的位错宽度却变宽了。此外,在各向异性近似下,还研究了Mo中1/2<111>{110}刃位错的运动机制。到目前为止,Mo中1/2<111>{110}刃位错的运动机制还是一个不确定的问题。在对此位错展开的原子模拟中,采用Finnis-Sinclair(F-S)势模拟得到的Peierls应力大约为50MPa,而且刃位错被认为是以kink机制运动的,但是后来采用同样的原子势模拟得到的Peierls应力只有25MPa ,而且人们认为kink机制在Mo中1/2<111>{110}刃位错的运动机制中并不占主导作用。此外,Liu等人采用F-S势模拟得到的Peierls应力大约为20 ? 39MPa。为了研究Mo中1/2<111>{110}刃位错的运动机制,我们在各向异性近似下计算了Mo中1/2<111>{110}刃位错的Peierls应力,并与数值计算的结果进行了比较。我们计算得到的Peierls应力大部分值为十几兆帕,最大值也只有51MPa。我们的结果与20 ? 39MPa的数值结果很符合。由于刃位错以刚性机制运动时所需要的Peierls应力要比以kink机制运动时所需要的Peierls应力小,因此,通过与数值模拟的结果相比较可以得出结论,Mo中1/2<111>{110}刃位错运动时是以刚性机制运动,而不是以kink机制运动。
(6) Mo中1/2<111>{110}混合位错
在体心立方晶体中,除了刃位错和螺位错外,还存在各种各样的混合位错。目前对混合位错的理论研究和数值模拟非常少,因此,用位错晶格理论研究体心立方晶体中的混合位错也非常重要。通过研究晶格离散效应对Mo中1/2<111>{110}刃位错、混合位错以及螺位错的修正可以看出,晶格离散效应对刃位错的修正最显著,因此,位错中的刃分量越多,晶格离散效应对此位错的修正就越明显。此外,对于同一种位错,位错宽度越窄,晶格离散效应的修正越明显,而且在刃位错、混合位错以及螺位错中,刃位错的位错宽度是最宽的,也因此,位错中的刃分量越多,位错宽度越宽。
There are various dislocations in body-centered-cubic (bcc) metals, and the dislocations are very important to the mechanical properties of the crystals. The crucial problem of the dislocation is the core structure. The classical Peierls-Nabarro (P-N) model can determine the core structure and Peierls stress quantitatively, however, because it is based on the elastic continuous medinum approximation, it neglects the influence of the lattice discrete effect to the dislocation core structure. Besides, in the classical P-N model, the nonlinear interaction of the misfit planes is described by the sinusoidal force law, whereas, it is inaccurate for the real materials. With the revealment of the relationship between the misfit interaction and the generalized stacking fault energy (γ-surface), the classical P-N model has been improved greatly, however, the problem of the lattice discrete effect is still unsatisfactory and needs further investigations. Up to now, the full discrete lattice theory of dislocation based on the lattice statics has been developped greatly, and the unified dislocation equation to investigate the dislocation core structure has also been provided. In this paper, the dislocation core structures and the Peierls stress of the bcc crystals have been investigated by using the dislocation lattice theory (the modified P-N model), furthermore, the movement mechanism of the dislocations have also been considered. The contents include the core structure of the <100>{010} edge dislocation in Fe, the core structures of the 1/2<111>{110} edge dislocations in Fe, Mo and W as well as the core structure of the moving 1/2<111>{110} screw dislocation in Ta in the isotropic approximation, the core structures of the <100>{010} and 1/2<111>{110} edge dislocations in Fe, the movement mechanism of the 1/2<111>{110} edge dislocation in Mo as well as the core structures of the 1/2<111>{110} mixed dislocations in Mo in anisotropic approximation. The main work and results involve:
(1) The dislocation equation of the simple cubic (SC) crystal in the isotropic approximation-a solvable model
Though the dislocation equations of the crystals with arbitral lattice structures can be obtained according to the lattice theory of dislocation, the coefficient of the intergral term can only be determined by comparing the result in continuous limit with that in the elastic theory, and the coefficient of the second-order differential term can also only be expressed approximately by the acoustic velocities of the uncoupled surface (a plane without the coupling with the internal atomic planes). It is needed for these results to be verified and validated by the solvable models. We have discussed the SC lattice which is a solvable model. The reduced dynamical matrix (RDM) in the long-wavelength approximation can be obtained according to the method of the lattice Green’s function, and then the dislocation equation of the SC crystal can be obtained. The RDM includes two parts, the surface term and the half-infinite crystal term (except for the surface term). The half-infinite crystal term we obtain is in according to the result according to the lattice statics and the symmetry principle, whereas, the surface term we obtain disagrees with the result in the reference. The reason is that the surface (videlicet the {010} plane) in reference is assumed to be isotropic, however, for the model we discussed, the {010} plane is anisotropic though the SC lattice is isotropic. Having compared with the dislocation equations obtained from the dislocation lattice theory, it can be found that the coefficients of the secone-order differential term is not in accordance with the results in the dislocation lattice theory, and this is mainly resulted from the surface term of the RDM. All the constants of the RDM are explicitly given by the solveable model of SC lattice firstly, and it is helpful to investigate the curved dislocations, such as the kink, by using the dislocation equations.
(2) The <100> {010} edge dislocation in Fe
Fe is a typical metal with bcc structure and is widely used in industry. The dislocations in Fe is investigated extensively and the <100> {010} edge dislocation is the simplest edge dislocation in bcc metals. Bullough and Perrin as well as Gehlen et al have studied the <100> {010} edge dislocation in Fe based on the atomic simulations using the Johson potential. Both their results indicate that the core of the <100> {010} edge dislocation in Fe is very narrow and the core radius they obtained is between 1.25b and 1.65b ( b is the Burgers vector). Chen et al have investigated the core structure of the <100> {010} edge dislocation in Fe by using the molecular dynamics simulation and the core radius they obtained is1.67b . In theory, Yan et al studied the dislocation based on the P-N model considering theγ-surface and the core half width (the core radius) they obtained is between0.85band0.93b . It is obvious that the core width they obtained is much smaller than those numerical results. The reason may be that the lattice discrete effect was ignored incorrectly. Therefore, it is meaningful in theory to investigate the <100> {010} edge dislocation in Fe by using the modified P-N model. The core width obtained from the classical P-N model (sinusoidal force law), in which the lattice discrete effect is not considered and the interaction between the misfit planes is described by the sinusoidal force law, is0.70b . However, having considered the lattice discrete effect, the core width is1.30b. It is indicated that the lattice discrete effect can make the core width nearly doubled. If the sinusoidal force law is replaced by theγ-surface, then the core width becomes0.88band0.94b , and there are no much changes in the core width relative to that in classical P-N model (sinusoidal force law). It is indicated that the <100> {010} edge dislocation in Fe is not sensitive to theγ-surface. The core width we obtain from the P-N model (γ-surface) is very close to the results in reference, and it indicates that our results are accurate. More important, the core width obtained from the modified P-N model is1.51 ? 1.57b, and it is in accordance with the numerical results 1.25 ? 1.65b and1.67b . It is indicated that the dislocation lattice theory is reasonable in investigating the dislocation core structures and the agreement between the theoretical prediction and the numerical simulation is improved remarkably by the modified P-N model.
(3) The 1/2<111> {110} edge dislocations in Fe, Mo and W
Mo and W are also the typical metals with bcc structures and they are also widely used in industry. The 1/2<111> {110} edge dislocation is the most commen edge dislocation in bcc metals, therefore, it is meaningful to investigate the core structures of the 1/2<111> {110} edge dislocations in Fe, Mo and W by using the modified P-N model. When fitting theγ-surface given by some references, the three fitting parametersΔ1 ,Δ2andΔ3 are introduced. The expression of theγ-surface with the three fitting parameters can describe theγ-surface appeared in the references accurately, and the relationship between the structure parameter of the dislocation core and the three parameters of theγ-surface can be obtained by solving the dislocation equation. It is found that the dislocation core width, which characterizes the dislocation, mainly depends upon the sum of the three fitting parametersΔ=Δ1 +Δ2 +Δ3, whereas, the core width is not sensitive to the other details of theγ-surface. Because the relationship betweenΔand the unstable fault energy is linear, therefore, it can be concluded that the dislocation core width mainly depends upon the unstable fault energy, videlicit the maximum energy of theγ-surface. This conclusion is helpful to understand the core structures and behaviours of the dislocations. For one material, if the unstable fault energy is larger, then the corresponding core width will be narrower, and the the correction of the lattice discrete effect to the dislocation core structure will be more significant. Based on the dislocation lattice theory, we calculate the core width of the 1/2<111> {110} edge dislocations in Fe, Mo and W. The core width of dislocation in Fe is between2b and4b , the core width in Mo is between2.7b and4.8b , and the core width in W is between2.9b and4.4b . These results are all in accordance with the numerical results. It is indicated that the dislocation lattice theory can describe the dislocation core structure accurately.
(4) The 1/2<111> {110} screw dislocation in Ta
The 1/2<111> {110} screw dislocation is the most important dislocation in bcc metals, because it is considered to be of primary importance in controlling the plastic deformation of bcc metals. It is suggested that the non-planar core structure can interpret the high Peierls stress of bcc metals. Though the idea of the famous three-way dissociation into three equivalent {110} planes is widely accepted, that is, the core structure of the static screw dislocation is dissociated into three equivalent {110} planes, however, it is speculated whether the core structure of the screw dislocation is still dissociated into three equivalent {110} planes when the screw dislocation moves. The core structures of the screw dislocations are investigated by some numerical calculations and atomistic simulations, and it is indicated that the core structures of the screw dislocations always undergo significant changes under the explicit stress before the screw dislocation moves. The metal Ta is widely used in industry, and the screw dislocation in Ta is widely investigated by some numerical simulations. However, there is still a discrepancy between the numerical simulations and the experimental data. The discrepancy is that the Peierls stress predicted by the numerical simulations is much higher than that observed at low temperature. The previous simulations of the screw dislocation in Ta are about1.5GPa , whereas the experimental data is260MPa . Later, the Peierls stress obtained from the new model generalized pseudopotential theory (MGPT) and the qEAM2 force field are about 660MPa and 440MPa , respectively, however, these results are still much larger than the experimental data. The reason of the discrepancy may be the incomplete understanding of the movement mechanism of the screw dislocation, and the configuration of the core structure is very crucial to the Peierls stress for the moving screw dislocation. In this paper, based on the dislocation lattice theory, the Peierls stress of the 1/2<111> {110} screw dislocation with planar and non-dissociated core structure in Ta has been calculated. The elastic strain energy which is associated with the lattice discrete effect and is ignored in classical P-N model is taken into account in calculating the Peierls stress. Through analylizing the effect of the elastic strain energy, the lattice discrete effect and the modification of the sinusoidal force law to the Peierls stress, it can be found that the elastic strain energy is the most important to the Peierls stress. The Peierls stress we obtain is about200MPa , and this result is very close to the experimental data260MPa . Because the Peierls stress of the screw dislocation with the planar core structure is smaller than that with the non-planar core structure, therefore, it is speculated that the core structure of the moving 1/2<111> {110} screw dislocation in Ta may be planar. Just as the prediction of the numerical results, the core structure of the screw dislocation in Ta undergoes significant changes, furthermore, under the external stress, the non-planar core structure of the screw dislocation changes into a planar core structure which moves more easily.
(5) The core structures of the <100> {010} and 1/2<111> {110} edge dislocations in anisotropic approximation
Among the theories investigating the dislocations, the most work is done under the isotropic approximation. However, the real materials are anisotropic, therefore, in order to investigate the dislocation core structures more accurately, it is necessary to consider the effect of the elastical anisotropy to the dislocation core structures. Based on the dislocation lattice theory, the elastical anisotropy is reflected in three factors in the modified P-N dislocation equation, the lattice discrete effect, the elastically anisotropic energy coefficient and the shear modulus. The energy coefficient can be obtained by analyzing the dislocation energies of unit length of the edge and screw dislocations, and the shear modulus can be obtained by analyzing the stress of the glide direction. Besides, the lattice discrete effect in anisotropic approximation can be obtained according to the velocities of the longitudinal wave and the transverse wave along the glide direction. Though the three factors in anisotropic case are different from those in isotropic case, having considered the corrections of the three factors together, there are no much changes between the results in anisotropic case and those in isotropic case, this is because the corrections of the three factors to the dislocation core structures are not always the same. For one material, if the dislocation is different, then the effect of the anisotropy to the core structure will be different. The elastical anisotropy can make the core width of the <100> {010} edge dislocation in Fe narrowed, whereas, it can make the core width of the 1/2<111> {110} edge dislocation in Fe broadened. Besides, the movement mechanism of the 1/2<111> {110} edge dislocation in Mo has been investigated in the anisotropic approximation. Up to now, the movement mechanism of the 1/2<111> {110} edge dislocation in Mo is still undetermined. The Peierls stress of the edge dislocation obtained by using the Finnis-Sinclair (F-S) potential is about50MPa , and it is assumed that the mobility of the edge dislocation is controlled by the kink mechanism at the non-zero temperature. However, the Peierls stress of the edge dislocation obtained by using the same potential is just25MPa , and it is indicated that the kink mechanism does not control the movement of the edge dislocation in Mo. Furthermore, the Peierls stress has also been calculated by Liu et al by using the same F-S potential, and the result they obtained is20 ? 39MPa. In order to investigate the movement mechanism of the 1/2<111> {110} edge dislocation in Mo, we calculate the Peierls stress of the 1/2<111> {110} edge dislocation in Mo, and make a comparison with the numerical simulations. The most Peierls stress we obtain is about dozen MPa and the maximum of the Peierls stress is just51MPa . Our results agree well with the atomistic simulation20 ? 39MPa. Because the Peierls stress of the edge dislocation moving with the rigid mechanism is smaller than that with the kink mechanism, therefore, it can be speculated that the 1/2<111> {110} edge dislocation in Mo may move in the rigid mechanism rather than the kink mechanism or other mechanisms when the external stress is loaded on the dislocation.
(6) The 1/2<111> {110} mixed dislocations in Mo
There are various mixed dislocations in bcc metals except for the edge dislocation and screw dislocation. However, there are little studies on the core structures of the mixed dislocations though they are also important to the plastic deformations of bcc metals. Through investigating the correction of the lattice discrete effect to the core structures of edge, screw as well as the mixed dislocations, it can be found that the correction of discrete effect to the edge dislocation is the most significant, therefore, more larger the edge components is, more significant the correction of discrete effect to the core structure is. Besides, for one dislocation, the correction of discrete effect to the core structure will be more significant if the dislocation core width is narrower. Among the edge, mixed and screw dislocations, the core width of the edge dislocation is the widest, therefore, if the edge component of dislocation is larger, then the dislocation core width is wider.
引文
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