基于离散效应修正的位错芯结构及Peierls应力计算
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摘要
晶体材料中存在大量的位错缺陷,这些位错缺陷对晶体的电学、光学、磁学、特别是力学性质具有重要的影响。位错缺陷的中心问题是位错的芯结构问题。位错芯结构与表征位错滑移性的Peierls应力、位错之间相互作用细部特征之间的关系十分密切。而滑移性与位错相互作用和材料的范性及加工硬化等现象直接关联。可以说,位错芯结构的揭示是认识理解位错相关现象的第一步,也是最为重要的一步。经典的位错Peierls-Nabarro (P-N)模型虽然能够定量地给出位错的芯宽度和Peierls应力。但是P-N模型是建立在弹性近似的基础上,不能够反应晶格离散效应对位错性质的影响。目前,基于点阵静力学的全离散位错晶格理论已基本建立起来,用于讨论位错芯结构的位错方程也已经给出,能够弥补P-N模型的不足。本文的任务是考虑离散效应修正后讨论具体材料位错的芯结构及Peierls应力,具体包括新型B2结构金属间化合物YAg和YCu中<100>{010}位错和对称分解<111>{110}超位错、钙钛矿结构SrTiO_3中< 110>{100}混合位错以及面心立方晶体分解1/2<110>{111}位错。主要内容如下:
(1)位错方程的变分原理和计算Peierls应力的参数导数法
基于变分原理近似求解包含晶格离散效应修正的位错方程。首先将位错方程的求解转化为变分极值问题,验证了位错方程的求解和变分极值问题的等价性。试探解取成截断近似法中的位错解一致,包含一个位错芯结构参数,变分泛函能够清楚地表示为位错芯结构参数的函数。在忽略离散效应后,位错解和Peierls经典的位错解一致,说明此试探解是一个好的试探解。变分原理定出二维三角格子和简立方格子模型中的芯结构参数分别为0.71和0.68,截断近似法得到芯结构参数值分别为0.72和0.68,结果表明正弦力律的变分解和截断近似法给出的解极为吻合。此外,考虑了正弦力律修正对位错芯结构的影响,变分参数随力律修正因子的增大而减小。利用Foreman的参数导数法导出了包含弹性应变能贡献和力律修正项的Peierls能量和Peierls应力表达式。在正弦力律下,参数导数方法得到的结果和幂级数展开方法得到的结果完全一致,但参数导数方法能够极大地简化计算过程和计算结果。
(2)钇银和钇铜中<100>{010}位错
YAg和YCu是近年来发现的典型的新型B2结构金属间化合物,它们具有良好的力学性能。目前,仍缺乏对YAg和YCu中位错性质的研究。本文根据变分原理和参数导数法的理论结果,给出了YAg和YCu中<100>{010}位错的芯宽度和Peierls应力,位错芯宽度分别为2.12b和1.94b,Peierls应力为3.5×10 ?3μ和5.8×10 ~(-3)μ。作为比较还给出了传统B2结构NiAl中<100>{010}刃位错芯宽度和Peierls应力,结果分别为1.38b和5047MPa,NiAl的Peierls应力理论预言结果与Schroll等人的数值结果几千MPa在数量级上符合的很好。讨论了不稳定层错能和位错芯宽度以及Peierls应力之间的关系。YAg和YCu<100>{010}位错的不稳定层错能比NiAl <100>{010}位错的不稳定层错能小,因而位错芯宽度更宽,Peierls应力更小,位错更容易滑动。此外,还讨论了位错芯宽度和离散效应修正因子以及正弦力律修正系数之间的关系。
(3)钇银和钇铜中<111>{110}分解超位错
B2结构金属间化合物中最容易滑动的位错是<111>{110}分解超位错,但由于存在分解使得问题相对复杂。采用以分解宽度和超部分位错芯宽度为变分参数的试探解描述YAg和YCu中<111>{110}分解超位错的芯结构,变分泛函能够表示成分解宽度和芯宽度的函数,从而定出YAg和YCu中刃和螺分解超位错的芯结构。Xie等人实验上测定的YAg中刃超位错的分解宽度为0.6nm,本文的理论预言结果为0.572nm,两者十分接近。忽略离散效应后,得到的超位错分解宽度为0.431nm,和实验值差别很大,因此,在计算分解宽度时需要考虑离散效应修正。近似计算得到YAg和YCu中超部分位错的Peierls应力分别为1.80×10 ~(-3)μ和3.13×10 ~(-3)μ。虽然<111>{110}位错的不稳定层错能比<100>{010}位错的不稳定层错能大,但得到的Peierls应力反而更小,原因在于位错的分解效应使得<111>{110}位错的几何结构因子比<100>{010}位错的几何结构因子大,而Peierls应力随几何因子的增大而指数减小。
(4)钛酸锶中<110>{001}混合位错和面心立方晶体中分解位错
研究了钙钛矿结构SrTiO3中<110>{001}45混合位错的性质。由于混合位错具有刃分量和螺分量,因此需要采用二维位错方程才能确定两方向分量的位移场。本文验证了束缚路径近似的合理性,并在束缚路径近似下把二维位错方程退化为一维位错方程。通过一维位错方程得到混合位错的Peierls应力为0.22GPa,刃位错和螺位错的Peierls应力分别为0.17GPa和0.46GPa。混合位错的Peierls应力比刃位错的Peierls应力大,但比螺位错的Peierls应力小,这说明能够用一维位错方程估计混合位错的Peierls应力。还计算了面心立方晶体(Al、Cu、Ag和Au)中1/2<110>{111}刃位错的分解芯结构。由于此位错芯结构的复杂性,本文采用弹性理论的结果来确定分解宽度,通过变分原理确定刃分量和螺分量位错芯宽度。考虑离散效应修正得到部分位错刃分量和螺分量的芯宽度比P-N模型中得到的要宽。
Dislocations are the most abundant defects in materials that affect electronic, optic, magnetic especially the mechanical properties of materials. The crucial proplem of dislocation theory is the core structure, which is closely related with the Peierls stress and the interaction between dislocations that is correlated with the ductility and work hardening of materials. In a word, the determination of the core structure is the first and the most important step while disclosing the properties of dislocations. The classical Peierls-Nabarro (P-N) model can determine the core structure and Peierls stress quantitily. However, P-N model is based on the linear elastic theory that neglects the lattice discrete effect. Recently, lattice theory of dislocation based on the lattice statics has been constructed and dislocation equation to discuss the core sturucture is provided that can recover the defect of P-N model. In this paper, the core structure and Peierls stress of compact <100>{010} dislocation and collinear dissociated <111>{110} superdislocation in the novel B2 structure intermetallics YAg and YCu, <110>{100} mixed dislocation in SrTiO3 and 1/2<110>{111} dislocation in FCC crystals have been disscussed. The main work and results involve:
(1) Variational principle for dislocation equation and the method for Peierls stress
The variational principle is applied to solve the dislocation taking into account the discreteness effect. The dislocation equation is changed into the equivalent variational extreme problem. The trial solution is chosed to be the same as that in the truncate approximation method that is a good trial solution while the parameter is taken to be zero that is the same as the Peierls solution. The variational functional can be represented by parameter of core structure. Take the two-dimentional triangular lattice and the simple cubic lattice as an example, the value of parameters obtained by variational principle are 0.71 and 0.68 for sinusoidal force law,the value of parameters obtained by variational principle are 0.72 and 0.68 for sinusoidal force law. The results show that both variational principle and truncate approximation method are effective method to solve the dislocation equation. The modification of the sinusoidal force law show that variational parameter decreases as the increasing of the modification factor. The parametric derivation method of Foremanethod is applied to derive Peierls energy and Peierls stress considering the contribution of elastic strain energy that is neglected in P-N model and the term of modification of sinusoidal force law. The result for Peierls energy and Peierls stress are the same obtained by parametric derivation method and power series expansion method for the sinusoidal force law. But the expression obtained by parametric derivation method is much brief than that obtained by power series expansion method and is convenient to estimate the Peierls stress of specific materials.
(2) <100>{010} dislocation in YAg and YCu
YAg and YCu are the typical novel B2 structure intermetallics that possess the good mechanical properties. There is lack of the study of dislocation properties in literatures. The core structure and Peierls stress are calculated for the <100>{010} dislocation in novel B2 structure intermetallics YAg and YCu taking into account the modification of discrete effect. The core width of <100>{010} dislocation are 2.12b and 1.94b, the correspongding Peierls stress are 3.5×10 ~(-3)μand 5.8×10 ~(-3)μ, respectively. Furthermore, the core width and Peierls stress of <100>{010} dislocation in traditional B2 structure NiAl are also calculate for comparison, and the results are 1.38b and 5047MPa , that is in order agreement with the numerical result, the result is acceptable while evaluating the Peierls stress due to the approximation. The results show that the unstable stacking fault energy is the key parameter controlling the core width and Peierls stress. The unstable stacking fault energy of <100>{010} dislocations in YAg and YCu is smaller than that of <100>{010} dislocation in NiAl. Thus, core width of dislocation are wider and Peierls stress are smaller. Core width is wider as the increasing of discreteness effect factor and the decreasing of the modification factor of sinusoidal force law.
(3) <100>{010} dissociated superdislocations in YAg and YCu
Dissociated <111>{110} superdislocations are the most easiest slip dislocations in B2 structure intermetallics. The dissociated properties result in some complex. The trial solution for <111>{110} dissociated superdislocations in YAg and YCu are presented, dissociated width and core width of superpartials are two core structure paramenters. The variatiaonal functional can be represented by the two parameters, and the core structure of edge and screw <111>{110} superdislocations in YAg and YCu are determined. The experimental vaule for edge <111>{110} superdislocations in YAg provided by Xie is 0.6nm, the theoretical value predicted by this paper is 0.572nm, the dissociated width obtained in P-N is 0.431nm that is underestimate the dissociated width with the elastic continuum approximation. The Peierls stress of <111>{110} superpartials in YAg and YCu are 1.80×10 ~(-3)μand 3.13×10 ~(-3)μ, respectively. The Peierls stress of <111>{110} superpartials are smaller than that of <100>{010} dislocation, although the unstable stacking fault energy of <111>{110} superpartials are larger than that of <100>{010} dislocation due to the dissociated effect. The dissociated effect leads to the geometric structure factor of <111>{110} dislocation that is larger than of <100>{010} dislocationand and Peierls stress exponential decreases as the increasing of the geometric strucuture factor.
(4) <110>{001} mixed dislocation in SrTiO3 and dislocations in FCC crystal
The properties of <110>{001}45 mixed dislocation in perovskite structure SrTiO3 are discussed taking into account the discrete effect. It’s necessary to deal with the mixed dislocation with two-dimensional dislocation equation due to the edge and screw components. The rationality of the constrained path approximation is stated. In the constrained path approximation, the two dimensional dislocation reduced into one dimensional mixed dislocation equation. While the discrete effect is neglected, the new equation is the same as the mixed dislocation equation in P-N model. Peierls stress of the <110>{001} mixed dislocation obtained with reduced dislocation equation is 0.22GPa and Peierls stress of <110>{001} edge and screw dislocations are 0.17GPa and 0.46GPa, respectively. This shows that the one dimensional dislocation equation is a valid method to deal with the mobility of mixed dislocation due to the Peierls stress of mixed dislocation in the region of edge and screw dislocations. The dissociated core structure of edge 1/2<110>{111} dislocation in FCC crystals(Al、Cu、Ag and Au) are examined. The core structure of these dislocations is complex, the dissociated width is determined by the elastic theory, the width of edge and screw component of partials are determined by variational principle. While the modification of discrete effect is taken into account the width obtained are wider than that obtained in P-N model.
(1)位错方程的变分原理和计算Peierls应力的参数导数法
基于变分原理近似求解包含晶格离散效应修正的位错方程。首先将位错方程的求解转化为变分极值问题,验证了位错方程的求解和变分极值问题的等价性。试探解取成截断近似法中的位错解一致,包含一个位错芯结构参数,变分泛函能够清楚地表示为位错芯结构参数的函数。在忽略离散效应后,位错解和Peierls经典的位错解一致,说明此试探解是一个好的试探解。变分原理定出二维三角格子和简立方格子模型中的芯结构参数分别为0.71和0.68,截断近似法得到芯结构参数值分别为0.72和0.68,结果表明正弦力律的变分解和截断近似法给出的解极为吻合。此外,考虑了正弦力律修正对位错芯结构的影响,变分参数随力律修正因子的增大而减小。利用Foreman的参数导数法导出了包含弹性应变能贡献和力律修正项的Peierls能量和Peierls应力表达式。在正弦力律下,参数导数方法得到的结果和幂级数展开方法得到的结果完全一致,但参数导数方法能够极大地简化计算过程和计算结果。
(2)钇银和钇铜中<100>{010}位错
YAg和YCu是近年来发现的典型的新型B2结构金属间化合物,它们具有良好的力学性能。目前,仍缺乏对YAg和YCu中位错性质的研究。本文根据变分原理和参数导数法的理论结果,给出了YAg和YCu中<100>{010}位错的芯宽度和Peierls应力,位错芯宽度分别为2.12b和1.94b,Peierls应力为3.5×10 ?3μ和5.8×10 ~(-3)μ。作为比较还给出了传统B2结构NiAl中<100>{010}刃位错芯宽度和Peierls应力,结果分别为1.38b和5047MPa,NiAl的Peierls应力理论预言结果与Schroll等人的数值结果几千MPa在数量级上符合的很好。讨论了不稳定层错能和位错芯宽度以及Peierls应力之间的关系。YAg和YCu<100>{010}位错的不稳定层错能比NiAl <100>{010}位错的不稳定层错能小,因而位错芯宽度更宽,Peierls应力更小,位错更容易滑动。此外,还讨论了位错芯宽度和离散效应修正因子以及正弦力律修正系数之间的关系。
(3)钇银和钇铜中<111>{110}分解超位错
B2结构金属间化合物中最容易滑动的位错是<111>{110}分解超位错,但由于存在分解使得问题相对复杂。采用以分解宽度和超部分位错芯宽度为变分参数的试探解描述YAg和YCu中<111>{110}分解超位错的芯结构,变分泛函能够表示成分解宽度和芯宽度的函数,从而定出YAg和YCu中刃和螺分解超位错的芯结构。Xie等人实验上测定的YAg中刃超位错的分解宽度为0.6nm,本文的理论预言结果为0.572nm,两者十分接近。忽略离散效应后,得到的超位错分解宽度为0.431nm,和实验值差别很大,因此,在计算分解宽度时需要考虑离散效应修正。近似计算得到YAg和YCu中超部分位错的Peierls应力分别为1.80×10 ~(-3)μ和3.13×10 ~(-3)μ。虽然<111>{110}位错的不稳定层错能比<100>{010}位错的不稳定层错能大,但得到的Peierls应力反而更小,原因在于位错的分解效应使得<111>{110}位错的几何结构因子比<100>{010}位错的几何结构因子大,而Peierls应力随几何因子的增大而指数减小。
(4)钛酸锶中<110>{001}混合位错和面心立方晶体中分解位错
研究了钙钛矿结构SrTiO3中<110>{001}45混合位错的性质。由于混合位错具有刃分量和螺分量,因此需要采用二维位错方程才能确定两方向分量的位移场。本文验证了束缚路径近似的合理性,并在束缚路径近似下把二维位错方程退化为一维位错方程。通过一维位错方程得到混合位错的Peierls应力为0.22GPa,刃位错和螺位错的Peierls应力分别为0.17GPa和0.46GPa。混合位错的Peierls应力比刃位错的Peierls应力大,但比螺位错的Peierls应力小,这说明能够用一维位错方程估计混合位错的Peierls应力。还计算了面心立方晶体(Al、Cu、Ag和Au)中1/2<110>{111}刃位错的分解芯结构。由于此位错芯结构的复杂性,本文采用弹性理论的结果来确定分解宽度,通过变分原理确定刃分量和螺分量位错芯宽度。考虑离散效应修正得到部分位错刃分量和螺分量的芯宽度比P-N模型中得到的要宽。
Dislocations are the most abundant defects in materials that affect electronic, optic, magnetic especially the mechanical properties of materials. The crucial proplem of dislocation theory is the core structure, which is closely related with the Peierls stress and the interaction between dislocations that is correlated with the ductility and work hardening of materials. In a word, the determination of the core structure is the first and the most important step while disclosing the properties of dislocations. The classical Peierls-Nabarro (P-N) model can determine the core structure and Peierls stress quantitily. However, P-N model is based on the linear elastic theory that neglects the lattice discrete effect. Recently, lattice theory of dislocation based on the lattice statics has been constructed and dislocation equation to discuss the core sturucture is provided that can recover the defect of P-N model. In this paper, the core structure and Peierls stress of compact <100>{010} dislocation and collinear dissociated <111>{110} superdislocation in the novel B2 structure intermetallics YAg and YCu, <110>{100} mixed dislocation in SrTiO3 and 1/2<110>{111} dislocation in FCC crystals have been disscussed. The main work and results involve:
(1) Variational principle for dislocation equation and the method for Peierls stress
The variational principle is applied to solve the dislocation taking into account the discreteness effect. The dislocation equation is changed into the equivalent variational extreme problem. The trial solution is chosed to be the same as that in the truncate approximation method that is a good trial solution while the parameter is taken to be zero that is the same as the Peierls solution. The variational functional can be represented by parameter of core structure. Take the two-dimentional triangular lattice and the simple cubic lattice as an example, the value of parameters obtained by variational principle are 0.71 and 0.68 for sinusoidal force law,the value of parameters obtained by variational principle are 0.72 and 0.68 for sinusoidal force law. The results show that both variational principle and truncate approximation method are effective method to solve the dislocation equation. The modification of the sinusoidal force law show that variational parameter decreases as the increasing of the modification factor. The parametric derivation method of Foremanethod is applied to derive Peierls energy and Peierls stress considering the contribution of elastic strain energy that is neglected in P-N model and the term of modification of sinusoidal force law. The result for Peierls energy and Peierls stress are the same obtained by parametric derivation method and power series expansion method for the sinusoidal force law. But the expression obtained by parametric derivation method is much brief than that obtained by power series expansion method and is convenient to estimate the Peierls stress of specific materials.
(2) <100>{010} dislocation in YAg and YCu
YAg and YCu are the typical novel B2 structure intermetallics that possess the good mechanical properties. There is lack of the study of dislocation properties in literatures. The core structure and Peierls stress are calculated for the <100>{010} dislocation in novel B2 structure intermetallics YAg and YCu taking into account the modification of discrete effect. The core width of <100>{010} dislocation are 2.12b and 1.94b, the correspongding Peierls stress are 3.5×10 ~(-3)μand 5.8×10 ~(-3)μ, respectively. Furthermore, the core width and Peierls stress of <100>{010} dislocation in traditional B2 structure NiAl are also calculate for comparison, and the results are 1.38b and 5047MPa , that is in order agreement with the numerical result, the result is acceptable while evaluating the Peierls stress due to the approximation. The results show that the unstable stacking fault energy is the key parameter controlling the core width and Peierls stress. The unstable stacking fault energy of <100>{010} dislocations in YAg and YCu is smaller than that of <100>{010} dislocation in NiAl. Thus, core width of dislocation are wider and Peierls stress are smaller. Core width is wider as the increasing of discreteness effect factor and the decreasing of the modification factor of sinusoidal force law.
(3) <100>{010} dissociated superdislocations in YAg and YCu
Dissociated <111>{110} superdislocations are the most easiest slip dislocations in B2 structure intermetallics. The dissociated properties result in some complex. The trial solution for <111>{110} dissociated superdislocations in YAg and YCu are presented, dissociated width and core width of superpartials are two core structure paramenters. The variatiaonal functional can be represented by the two parameters, and the core structure of edge and screw <111>{110} superdislocations in YAg and YCu are determined. The experimental vaule for edge <111>{110} superdislocations in YAg provided by Xie is 0.6nm, the theoretical value predicted by this paper is 0.572nm, the dissociated width obtained in P-N is 0.431nm that is underestimate the dissociated width with the elastic continuum approximation. The Peierls stress of <111>{110} superpartials in YAg and YCu are 1.80×10 ~(-3)μand 3.13×10 ~(-3)μ, respectively. The Peierls stress of <111>{110} superpartials are smaller than that of <100>{010} dislocation, although the unstable stacking fault energy of <111>{110} superpartials are larger than that of <100>{010} dislocation due to the dissociated effect. The dissociated effect leads to the geometric structure factor of <111>{110} dislocation that is larger than of <100>{010} dislocationand and Peierls stress exponential decreases as the increasing of the geometric strucuture factor.
(4) <110>{001} mixed dislocation in SrTiO3 and dislocations in FCC crystal
The properties of <110>{001}45 mixed dislocation in perovskite structure SrTiO3 are discussed taking into account the discrete effect. It’s necessary to deal with the mixed dislocation with two-dimensional dislocation equation due to the edge and screw components. The rationality of the constrained path approximation is stated. In the constrained path approximation, the two dimensional dislocation reduced into one dimensional mixed dislocation equation. While the discrete effect is neglected, the new equation is the same as the mixed dislocation equation in P-N model. Peierls stress of the <110>{001} mixed dislocation obtained with reduced dislocation equation is 0.22GPa and Peierls stress of <110>{001} edge and screw dislocations are 0.17GPa and 0.46GPa, respectively. This shows that the one dimensional dislocation equation is a valid method to deal with the mobility of mixed dislocation due to the Peierls stress of mixed dislocation in the region of edge and screw dislocations. The dissociated core structure of edge 1/2<110>{111} dislocation in FCC crystals(Al、Cu、Ag and Au) are examined. The core structure of these dislocations is complex, the dissociated width is determined by the elastic theory, the width of edge and screw component of partials are determined by variational principle. While the modification of discrete effect is taken into account the width obtained are wider than that obtained in P-N model.
引文
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