Recovery, Recrystallization and Grain Growth
复原、再结晶和晶粒长大

Abstract  摘要

Strong plastic deformation of a polycrystalline material followed by annealing can lead to a new microstructure of the whole specimen which then composed of grains of new size, shape and crystal orientation distribution. This transformation is misleadingly called recrystallization. The process can be preceded by recovery, principally involving a change of the dislocation microstructure. Recrystallization can be followed by growth of the grains and then constituting the specimen. Recovery processes are of homogeneous nature. Recrystallization is of heterogeneous nature and thus exhibits something called “nucleation,” but which in fact implies growth of an existing subgrain of relatively low stored strain energy into surrounding material of relatively high stored strain energy. Various possible mechanisms of recovery and recrystallization are discussed. Remarks on (the limitations of) the kinetic analyses of these processes are given. The treatment of grain growth is preceded by a consideration of the notions grain-boundary energy and grain-boundary tension and of mechanical equilibrium at locations where grain boundaries meet. Grain-boundary curvature is identified as the driving force for grain growth, which leads to a description of the kinetics of grain growth, including effects of pinning forces.
多晶材料发生强烈塑性变形后进行退火，可导致整个试样产生新的微观结构，由新尺寸、形状和晶体取向分布的晶粒组成。这种转变被误认为是再结晶。在这一过程之前可能会发生回复，主要涉及位错微观结构的变化。再结晶之后可能是晶粒的生长，然后构成试样。复原过程具有均匀性。再结晶是异质的，因此表现出一种称为 "成核 "的现象，但实际上这意味着存储应变能相对较低的现有子晶粒向存储应变能相对较高的周围材料生长。本文讨论了各种可能的恢复和再结晶机制。对这些过程的动力学分析（局限性）进行了评论。在处理晶粒生长之前，首先要考虑晶界能量和晶界张力的概念，以及晶界交汇处的机械平衡。晶界曲率被确定为晶粒生长的驱动力，这导致了对晶粒生长动力学的描述，包括销钉力的影响。

Recrystallization has been identified as a process in metallic solids since the “old days” (last part of the nineteenth century), when it was supposed that cold working of a metallic workpiece destroyed its crystallinity and that subsequent heating restored the crystalline nature by a process then naturally coined with the name “recrystallization”. Nowadays, we would define recrystallization as a process that leads to a change of the crystal orientation (distribution) for the whole polycrystalline specimen, in association with a release of the stored strain energy as could have been induced by preceding cold work: a new microstructure results (Fig. 11.1). Recrystallization restores the properties as they were before the cold deformation. Recrystallization (and recovery and grain growth) occurs in all types of crystalline materials, so not only in metals. However, metals are the only important class of materials capable of experiencing pronounced plastic deformation at relatively low temperatures (i.e. low with respect to the melting temperatures), which explains that most of the corresponding research has been and is performed on metallic materials.
早在 "过去"（十九世纪后半期），人们就已将再结晶确定为金属固体中的一个过程，当时认为金属工件的冷加工会破坏其结晶性，而随后的加热会恢复其结晶性，这一过程自然而然地被命名为 "再结晶"。如今，我们将再结晶定义为一个导致整个多晶试样晶体取向（分布）发生变化的过程，同时释放出之前冷加工可能诱发的应变能：形成新的微观结构（图 11.1）。再结晶恢复了冷变形前的特性。再结晶（以及复原和晶粒长大）发生在所有类型的晶体材料中，因此不仅仅发生在金属中。不过，金属是唯一一类能够在相对较低的温度下（即相对于熔化温度较低）发生明显塑性变形的重要材料，这也是大部分相关研究一直以来都是针对金属材料进行的原因。

Fig. 11.1 图 11.1

Optical micrographs showing the microstructure of an Fe-4.65at%Al alloy after cold rolling to a degree of deformation of 90% (i.e. a reduction of sheet thickness of 90%), and after subsequent anneals at temperatures and for times as indicated in the micrographs. Upon progressive annealing the elongated grain morphology resulting after cold rolling is gradually replaced by a more or less equiaxed grain morphology as the result of recrystallization, involving the nucleation and growth of new grains in the deformed microstructure (micrographs made by S. Meka, Max Planck Institute for Metals Research)
光学显微照片显示了一种 Fe-4.65at%Al 合金在冷轧至 90% 变形程度（即薄板厚度减少 90%）后，以及随后在显微照片所示温度和时间下退火后的显微结构。在逐步退火过程中，冷轧后产生的细长晶粒形态逐渐被等轴晶粒形态所取代，这是再结晶的结果，包括在变形的微观结构中新晶粒的成核和生长（显微照片由马克斯-普朗克金属研究所的 S. Meka 制作）。

The industrial need for understanding the effects of deformation in material-forming production steps and of subsequent annealing processes is obvious. Then it may come as a surprise that even about 150 years of research in this area have not led to comprehensive models describing these processes on the basis of fundamental insight such that reliable application for technological purposes can be guaranteed. One of the main reasons for this deficiency is undoubtedly our still limited understanding of the plastically deformed state (cf. Chap. 12).
工业界显然需要了解材料成型生产步骤中的变形效应以及随后的退火过程。但令人惊讶的是，即使在这一领域进行了约 150 年的研究，也未能在基本洞察力的基础上建立描述这些过程的综合模型，从而保证可靠的技术应用。造成这一缺陷的主要原因之一无疑是我们对塑性变形状态的理解仍然有限（参见第 12 章）。

Recovery, implying a decrease of the density and a redistribution of defects in the deformed solid, precedes recrystallization. Grain growth can occur in the recrystallized microstructure. Thereby, the sense of a treatment of recovery, recrystallization and grain growth, in this order in this chapter, has been validated. Yet, it is recognized that overlapping of these processes can occur in a significant way.
再结晶之前会出现恢复，这意味着变形固体中的密度降低和缺陷重新分布。再结晶的微观结构中会出现晶粒长大。因此，本章按此顺序处理复原、再结晶和晶粒长大的观点已得到验证。不过，我们也认识到，这些过程可能会出现严重的重叠。

11.1 Recovery  11.1 恢复

The defects introduced by plastic deformation processes, as cold rolling, and of importance in subsequent recovery and recrystallization processes, are predominantly dislocations. Point defects, as vacancies, are also introduced upon plastic deformation, but these are usually already annealed out at low temperatures (e.g. in copper at temperatures below room temperature). In particular if the stacking fault energy is low (as holds for silver; copper has a stacking fault energy about three times larger than that of silver and aluminium has a high stacking fault energy about eight times larger than that of silver), dissociation of the dislocations occurs, cross-slip is hindered (cf. Sect. 5.2.8) and twinning becomes a preferred mode of plastic deformation. Also, if not enough slip systems are available, as can occur with hexagonal metals (cf. Sect. 5.2.8), the initial plastic deformation can occur by slip (dislocation glide), but deformation twinning can become important upon progressing plastic deformation.
塑性变形过程（如冷轧）引入的缺陷以及在随后的复原和再结晶过程中产生的重要缺陷主要是位错。塑性变形时也会产生点缺陷，如空位，但这些缺陷通常在低温下已经退火（如铜在室温以下）。特别是如果堆积断层能较低（银也是如此；铜的堆积断层能约为银的三倍，而铝的堆积断层能较高，约为银的八倍），位错就会发生解离，交叉滑移就会受到阻碍（参见第 5.2.8 节），孪晶就会成为首选的塑性变形模式。此外，如果没有足够的滑移系统，如六方金属（参见第 5.2.8 节），最初的塑性变形可通过滑移（位错滑行）发生，但随着塑性变形的进展，变形孪晶会变得重要。

Recovery, as induced by annealing after plastic deformation, leads to a change of the dislocation microstructure and thereby a partial restoration of the material properties as before the plastic deformation is realized. It should be remarked that recovery processes can also operate in materials containing dislocations and non-equilibrium amounts of point defects (as vacancies) and which have not been subjected to pronounced plastic deformation by the exertion of external mechanical loads: for example, irradiation (bombardment) by accelerated particles (e.g. ions) induces such a defect structure. In this last case recovery can restore fully the original material properties. In this section the discussion is confined to recovery in materials deformed plastically such that distinct permanent shape changes have resulted (as by cold rolling).
塑性变形后通过退火引起的恢复会导致位错微观结构的改变，从而部分恢复塑性变形前的材料特性。应该指出的是，恢复过程也可在含有位错和非平衡点缺陷（如空位）的材料中进行，这些材料并未因施加外部机械负荷而发生明显的塑性变形：例如，加速粒子（如离子）的辐照（轰击）可诱发这种缺陷结构。在最后一种情况下，恢复可以完全恢复材料的原始特性。在本节中，讨论的仅限于发生塑性变形的材料的恢复，这种变形会导致明显的永久性形状变化（如冷轧）。

During the rearrangement/partial annihilation of the dislocations in the process of recovery, the grain boundaries in the material do not move; the recovery process occurs more or less homogeneously throughout the material, in flagrant contrast with recrystallization, characterized by the sweeping of high-angle grain boundaries through the deformed matrix, which process thus takes place explicitly heterogeneously (see Sect. 11.2 and the discussion on homogeneous and heterogeneous transformations in Sect. 9.2).
在回复过程中位错的重新排列/部分湮灭期间，材料中的晶界不会移动；回复过程在整个材料中或多或少是均匀发生的，这与再结晶形成鲜明对比，再结晶的特点是高角度晶界穿过变形基体，因此这一过程明显是异质发生的（见第 11.2 节和第 9.2 节中关于同质和异质转变的讨论）。

The above discussion could be conceived as that recovery is induced, after the plastic deformation (by cold work), by annealing at an appropriate, elevated temperature (say, distinctly below half of the melting temperature in Kelvin). However, if the plastic deformation occurs at elevated temperature (as by hot rolling), recovery processes already run while the material is still deforming; one then speaks of dynamic recovery (similarly, one recognizes dynamic recrystallization).
上述讨论可以理解为，在塑性变形（通过冷加工）之后，通过在适当的高温（例如明显低于熔化温度一半的开尔文）下退火来诱导恢复。然而，如果塑性变形是在高温下发生的（如热轧），那么当材料仍在变形时，恢复过程就已经开始了；这就是动态恢复（类似于动态再结晶）。

11.1.1 Dislocation Annihilation and Rearrangement
11.1.1 位错湮灭和重排

The driving force for the migration of the dislocations leading to a different dislocation configuration and/or to a partial annihilation of dislocations is a reduction of the strain energy incorporated in the strain fields of the dislocations. This decrease of the stored energy in the material obviously decreases the driving force for the (largely) subsequent recrystallization (recovery and recrystallization may overlap; see later).
位错迁移导致不同位错配置和/或部分位错湮灭的驱动力是位错应变场中的应变能的降低。材料中存储能量的减少显然降低了（大部分）随后再结晶的驱动力（复原和再结晶可能重叠；见下文）。

The annihilation of dislocations can occur by various mechanisms. Dislocations can migrate by glide along a single slip plane, by cross-slip and by climb (see Sects. 5.2.5, 5.2.6 and 5.2.7).
位错的湮灭有多种机制。位错可以通过沿单一滑移面滑行、交叉滑移和爬升的方式迁移（见第 5.2.5、5.2.6 和 5.2.7 节）。

Evidently (edge) dislocations of opposite sign (cf. Sect. 5.2.3) on the same slip plane can become annihilated by gliding to contact (Fig. 11.2).
显然，同一滑移面上符号相反的（边缘）位错（参见第 5.2.3 节）可以通过滑行接触而湮灭（图 11.2）。

Fig. 11.2 图 11.2

Annihilation of two (edge) dislocations of opposite sign (cf. Sect. 5.2.3) by glide
两个符号相反的（边缘）位错（参见第 5.2.3 节）通过滑行湮灭

Fig. 11.3 图 11.3

Annihilation of two edge dislocations of opposite sign (cf. Sect. 5.2.3) by climb and glide
两个符号相反的边缘位错（参见第 5.2.3 节）通过爬行和滑行湮灭

If these two initial dislocations of opposite sign are of edge type and on two different glide planes, their possible annihilation requires a combination of climb and glide processes (Fig. 11.3). The climb step is outspokenly thermally activated (cf. Sect. 5.2.7), implying that, according to the mechanism considered here, dislocation annihilation can only occur at elevated temperatures. If the two initial dislocations on two different glide planes are of screw type, their annihilation can be established by cross-slip.
如果这两个符号相反的初始位错是边缘型的，并且位于两个不同的滑行面上，那么它们可能的湮灭需要爬升和滑行过程的结合（图 11.3）。爬升步骤明显是热激活的（参见第 5.2.7 节），这意味着根据本文所考虑的机制，差排湮灭只能在高温下发生。如果位于两个不同滑动面上的两个初始位错是螺钉型的，它们的湮灭可以通过交叉滑动来实现。

Dislocations may also glide along a slip plane and upon “colliding” with a grain boundary be incorporated into the grain-boundary structure. Thereby the dislocation as an isolated defect may loose its identity by local atomic shuffles in the grain boundary, in association with the loss of strain energy and, in this sense, annihilation of the dislocation has occurred as well.
差排也可能沿着滑移面滑行，并在与晶界 "碰撞 "后融入晶界结构中。因此，作为孤立缺陷的差排可能会因晶界中的局部原子变位而失去其特性，同时伴随应变能的损失，从这个意义上说，差排也发生了湮灭。

Release of strain energy can also be realized by rearrangement of the dislocations in a single grain of the material. Evidently, if the numbers of dislocations of opposite sign are unequal, complete dislocation annihilation by any of the first two processes mentioned above is impossible. The presence of unequal numbers of dislocations of opposite sign can be the result of bending of a single grain experiencing glide along a single slip plane: a curved grain results by an excess of edge dislocations of the same type (cf. the discussion on “geometrically necessary dislocations” in Sect. 12.14.2); see Fig. 11.4a. Upon annealing, these edge dislocations can strive for arrangements in “walls” and thus form low-angle tilt boundaries (cf. Sect. 5.3.1). This rearrangement is realized by climb and short-range glide (see Fig. 11.4b). The overlapping of “tensile” and “compressive” parts of the long-range strain fields of neighbouring dislocations in the dislocation wall provides the release of strain energy that is the driving force for this process; see the discussion of Eq. (5.15) in Sect. 5.3.1. As a result of the formation of these dislocation walls/low-angle tilt boundaries, the originally (i.e. after the plastic deformation) curved crystal-structure planes of the grain considered become the sides of a polygon: a series of subgrains has formed which are slightly differently oriented with respect to each other (with a view to the configuration shown in Fig. 11.4b: the subgrains are slightly rotated with respect to each other around an axis perpendicular to the plane of the drawing). One therefore names this phenomenon: polygonization. The process is revealed in X-ray diffraction patterns by the replacement of strongly broadened reflections, observed after the plastic deformation, by a series of neighbouring discrete spots, observed upon subsequent annealing (Cahn 1949).
应变能的释放也可以通过材料单个晶粒中位错的重新排列来实现。显然，如果符号相反的位错数量不相等，上述前两个过程中的任何一个都不可能完全湮灭位错。符号相反的差排数量不相等可能是单个晶粒沿单个滑移面滑行时发生弯曲的结果：相同类型的边缘差排过多导致晶粒弯曲（参见第 12.14.2 节中关于 "几何上必要的差排 "的讨论）；见图 11.4a。退火时，这些边缘位错可努力排列成 "壁"，从而形成低角度倾斜边界（参见第 5.3.1 节）。这种重新排列是通过爬升和短程滑行实现的（见图 11.4b）。位错壁中相邻位错长程应变场的 "拉伸 "和 "压缩 "部分的重叠提供了应变能的释放，而应变能正是这一过程的驱动力；参见第 5.3.1 节中公式 ( 5.15) 的讨论。由于这些位错壁/低角度倾斜边界的形成，晶粒原本（即塑性变形后）弯曲的晶体结构平面变成了多边形的两侧：形成了一系列彼此方向略有不同的子晶粒（如图 11.4b 所示：子晶粒绕着垂直于图平面的轴线彼此略微旋转）。因此，我们将这种现象命名为：多边形化。 在 X 射线衍射图样中，塑性变形后观察到的强烈加宽反射被随后退火后观察到的一系列相邻离散光斑所取代，从而揭示了这一过程（Cahn，1949 年）。

Fig. 11.4 图 11.4

a Bending of a single grain experiencing glide along a single slip plane: a curved grain results by an excess of edge dislocations of the same type. b Upon annealing, these edge dislocations can strive for arrangements in “walls”, by climb and short-range glide, and thus form low-angle tilt boundaries: Polygonization of a bended grain by rearrangement of edge dislocations
a 单个晶粒沿单个滑移面滑行发生弯曲：同类边缘位错过多导致晶粒弯曲。 b 退火后，这些边缘位错可通过爬升和短程滑行努力排列成 "墙"，从而形成低角度倾斜边界：边缘位错的重新排列使弯曲晶粒多边形化

The simple calculation for the energy per unit area of a low-angle tilt boundary, as given by Eq. (5.15), holds for an infinitely long dislocation wall. In practice, the dislocation walls (segments of low-angle tilt boundaries) in the polygonized microstructure may comprise ten dislocations and less. The process of aligning of edge dislocations of the same sign has also been observed for the misfit dislocations originally present in the interface of an A/B bicrystal (see Sect. 5.3.4). In that case diffusion annealing (specific observations were made for a thin Cu/Ni bicrystalline film) leads to the formation of dislocation walls initially comprising even only two edge dislocations (Fig. 11.5). This can be conceived as an extreme case of polygonization, where the driving force also is the release of dislocation-strain energy, albeit the dislocations were not induced by external mechanical action. Although the process is always driven by the release of dislocation-strain energy, the energy gain per dislocation for dislocation walls of (such) limited length cannot be assessed by application of Eq. (5.15); a numerical approach is required (Beers and Mittemeijer 1978).
低角度倾斜边界单位面积能量的简单计算，如公式 ( 5.15) 所示，对于无限长的差排壁是成立的。实际上，多边形微结构中的位错墙（低角度倾斜边界段）可能由十个或更少的位错组成。对于原本存在于 A/B 双晶界面中的错位位错，也观察到了同符号边缘位错的排列过程（见第 5.3.4 节）。在这种情况下，扩散退火（对铜/镍双晶薄膜进行了具体观察）会导致位错壁的形成，最初甚至只有两个边缘位错（图 11.5）。这可以看作是多边形化的极端情况，其驱动力也是位错应变能的释放，尽管位错不是由外部机械作用诱发的。虽然这一过程总是由位错应变能的释放所驱动，但对于（这种）长度有限的位错壁，每个位错的能量增益无法通过应用公式（5.15）来评估；需要采用数值方法（Beers 和 Mittemeijer，1978 年）。

Fig. 11.5 图 11.5

(Edge) Misfit dislocations originally located in the interface of an A/B bicrystal (top part of left part of the figure; see “I”) upon annealing can move away from the interface by climb (see “II”) and, subsequently, by glide can become aligned on top of each other (see “III”). A schematic depiction of this alignment process is shown in the right part of the figure: the dislocations align part by part. Thus dislocation walls can be formed initially comprising only two dislocations. Such observations have been made for Cu/Ni bicrystals (Beers and Mittemeijer 1978)
(边缘）原本位于 A/B 双晶界面上的错位位错（图中左侧上半部分，见 "I"）在退火后可通过爬升远离界面（见 "II"），随后通过滑行相互对齐（见 "III"）。图中右侧显示了这一排列过程的示意图：位错逐个排列。因此，最初形成的差排壁仅由两个差排组成。在铜/镍双晶中也观察到这种现象（Beers 和 Mittemeijer，1978 年）。

As discussed qualitatively with respect to Eq. (5.15) already, the energy of the dislocation wall per unit area increases with increasing dislocation density of the wall (increase of θ, decrease of D; cf. Eq. (5.14)), but the energy per dislocation in the small-angle boundary decreases with increasing dislocation density (increase of θ, decrease of D). Hence, after the polygonization has started, a driving force exists for enhancing the size of the subgrains (cf. Fig. 11.4b) by merging of adjacent dislocation walls/small-angle tilt boundaries.
正如公式( 5.15) 所定性讨论的，单位面积差排壁的能量随着差排壁密度的增加而增加（θ增加，D减小；参见公式( 5.14)），但小角度边界中每个差排的能量随着差排密度的增加而减小（θ增加，D减小）。因此，在多边形化开始后，相邻差排壁/小角度倾斜边界的合并会产生一种驱动力，从而增大子晶粒的尺寸（参见图 11.4b）。

The obvious mechanism to cause such subgrain coarsening is based on the migration and merging of low-angle boundaries. The migration rate of low-angle boundaries, as symmetrical tilt boundaries, by glide of the edge dislocations, composing the boundary, on their parallel slip planes, is relatively high.
导致这种亚晶粒粗化的明显机制是基于低角度边界的迁移和合并。低角度边界作为对称的倾斜边界，通过组成边界的边缘位错在其平行滑移面上的滑移，其迁移率相对较高。

An alternative mechanism leading to subgrain coarsening is the coalescence of adjacent subgrains preceded by subgrain rotation. The driving force for subgrain rotation is understood on the basis of, again, Eq. (5.15). Consider subgrain 1 with its surrounding neighbouring subgrains 2, 3, … (cf. Fig. 11.6). The decrease, by rotation of subgrain 1 with respect to its surrounding, static neighbours, of the misorientation along the boundary 1/2, separating subgrain 1 from subgrain 2, at the same time, will be associated with decreases or increases of the misorientations along the other boundaries of subgrain 1 with its neighbouring subgrains. Now, for the same change of misorientation, as described by the change of the angle θ, the change of energy according to Eq. (5.15) is the larger the smaller the misorientation, θ. Hence, there is a driving force for making the misorientation of the lowest-angle boundaries (even) smaller, as the cost for making, unavoidably and simultaneously, the misorientation of other, larger-angle boundaries larger, is smaller, because the energy gain (release) for the lowest-angle boundaries is larger per unit area boundary than the energy cost (absorption) for the other, larger-angle boundaries. So, provided the ratio of total amount lowest-angle boundary area and of total amount of larger-angle boundary area is not too small, the subgrain 1 can release energy by rotation such that the lowest-angle boundaries decrease their misorientation and the larger-angle boundaries increase their misorientation (Li 1962). Eventually, the misorientation along the boundary 1/2 vanishes, i.e. coalescence of subgrains 1 and 2 has effectively been realized. This subgrain rotation can be achieved by emittance of dislocations from the lowest-angle boundaries and their migration, as by climb and glide, to the larger-angle boundaries. Additionally, local atomic shuffles in the boundary regions can occur. This intellectually appealing mechanism for subgrain coarsening, by subgrain rotation and coalescence, is a feasible one from an energy point of view. However, conclusive experimental evidence for its importance for the subgrain coarsening occurring in reality lacks, and it has been concluded that subgrain coarsening is dominated by the above first discussed migration of low-angle boundaries (Humphreys and Hatherly 2004).
导致亚晶粒粗化的另一种机制是亚晶粒旋转之前相邻亚晶粒的凝聚。亚晶粒旋转的驱动力也是根据公式 ( 5.15) 理解的。考虑亚晶粒 1 及其周围的邻近亚晶粒 2、3、...（参见图 11.6）。子晶粒 1 相对于其周围的静态邻晶粒旋转时，沿分隔子晶粒 1 和子晶粒 2 的边界 1/2 的方向偏差减小，同时，沿子晶粒 1 与其邻晶粒的其他边界的方向偏差也会减小或增大。现在，对于由角度 θ 的变化所描述的相同的偏差变化，根据公式 ( 5.15) ，偏差 θ 越小，能量变化越大。因此，有一种使最小角度边界的不定向（甚至）变小的驱动力，因为使其他较大角度边界的不定向不可避免地同时变大的代价较小，因为最小角度边界的单位面积边界的能量增益（释放）大于其他较大角度边界的能量代价（吸收）。因此，只要最低角度边界的总面积与较大角度边界的总面积之比不是太小，亚晶粒 1 就能通过旋转释放能量，使最低角度边界的错误取向减小，而较大角度边界的错误取向增大（Li 1962）。最终，沿边界 1/2 的错向消失，即子晶粒 1 和 2 有效地实现了凝聚。 这种亚晶粒旋转可以通过位错从最低角度的边界逸出，并通过爬升和滑行迁移到较大角度的边界来实现。此外，边界区域还可能发生局部原子洗牌。从能量的角度来看，这种通过亚晶粒旋转和凝聚实现亚晶粒粗化的机制是可行的。然而，对于现实中发生的亚晶粒粗化，缺乏确凿的实验证据来证明其重要性，因此得出的结论是，亚晶粒粗化主要是由上述首次讨论的低角度边界迁移引起的（Humphreys 和 Hatherly，2004 年）。

Fig. 11.6 图 11.6

Coalescence of adjacent subgrains by subgrain rotation: By climb and glide of dislocations from the lower-(misorientation) angle grain boundary 1/2 to the higher-(misorientation) angle grain boundary 1/3, associated with rotation of grain 1, the grain boundary 1/2 is eliminated, while the misorientation angle of the grain boundary 1/3 is enlarged. This leads to a lowering of the total dislocation-strain energy of the system (see text)
相邻子晶粒因子晶粒旋转而聚合：位错从低取向角晶界1/2爬升和滑行到高取向角晶界1/3，与晶粒1的旋转有关，晶界1/2被消除，而晶界1/3的取向角被扩大。这导致系统的总位错应变能降低（见正文）

The simple picture sketched above provides a basis for understanding complex phenomena occurring in complicated dislocation microstructures which result from severe plastic deformation. In a pronounced stage of deformation of a ductile material (as a metal) the dislocations gather in regions of high dislocation density and a dislocation-cell structure develops within the grains, with a high dislocation density in the cell walls and a small dislocation density in-between (cf. Sect. 12.14.1). Annealing-induced recovery in such a microstructure replaces the tangled configuration of the dislocations in the cell walls into more regular arrangements as in low-angle boundaries and distinctly reduces the dislocation density within the cells. In the sense discussed above, one can say that the deformation cells have become subgrains.
以上的简单描述为理解严重塑性变形产生的复杂位错微结构中出现的复杂现象提供了基础。在韧性材料（如金属）的明显变形阶段，位错聚集在位错密度高的区域，晶粒内部形成位错晶胞结构，晶胞壁的位错密度高，晶胞之间的位错密度小（参见第 12.14.1 节）。在这种微观结构中，退火引起的恢复会将晶胞壁上纠结的位错配置转变为更规则的排列，就像低角度边界一样，并明显降低晶胞内的位错密度。在上述意义上，我们可以说变形晶胞已经变成了亚晶粒。

The formation of subgrains should not be considered as a recrystallization process: the orientation (distribution) does not change significantly by the above described processes of subgrain formation. But the subgrains discussed here can play a role in the initiation of recrystallization (see Sect. 11.2).
亚晶粒的形成不应被视为再结晶过程：上述亚晶粒形成过程不会显著改变取向（分布）。但本文讨论的亚晶粒可能在再结晶的启动过程中发挥作用（见第 11.2 节）。

11.1.2 Kinetics of Recovery
11.1.2 回收动力学

The recovery process occurs more or less homogeneously throughout the material. Consequently, the theory of heterogeneous transformations as dealt with in Sects. 10.4−10.15 has no direct relevance for recovery (to a large extent; but see the remark on the determination of the effective, overall activation energy below).
整个材料的恢复过程或多或少都是均匀的。因此，第 10.4- 10.15 节所述的异质转化理论与回收没有直接关系（在很大程度上；但请参 见关于确定有效的整体活化的注释）。10.4- 10.15 节所述的异质转化理论与回收没有直接关系（在很大程度上；但请参阅下文关于确定有效总活化能的注释）。

For homogeneous reactions, the probability for the transformation to occur is the same for all locations in the virginal system considered. As a result, the transformation rate decreases monotonically from t = 0 onwards. The prescription for the degree of transformation, f, according to Eqs. (10.2) and (10.3), implying dependence only on the “path variable”, β, is also fully compatible with the well-known result for homogeneous reactions (cf. Mittemeijer 1992):
对于均相反应，处女膜系统中所有位置发生转化的概率都是相同的。因此，转化率从 t = 0 开始单调递减。根据公式 ( 10.2) 和 ( 10.3) 对转化程度 f 的规定，意味着只依赖于 "路径变量 "β，这也完全符合众所周知的均相反应结果（参见 Mittemeijer 1992）：

$${\left(1-f\right)}^{1-m}=1-\beta \left(1-m\right)\text{for}m>1$$

(11.1a)

$$\ln \left(1-f\right)=-\beta \text{for}m=1$$

(11.1b)

where m is the so-called order of reaction (known from chemical reaction kinetics). The degree of transformation (here: degree of recovery) can be defined as indicated by Eq. (10.1), where p can be a physical parameter as the electrical resistivity, the hardness, the enthalpy (heat released), the yield limit (cf. Sect. 12.9), etc.
其中 m 是所谓的反应阶次（从化学反应动力学中得知）。转化程度（此处指复原程度）可按公式 ( 10.1) 进行定义，其中 p 可以是电阻率、硬度、焓（释放的热量）、屈服极限（参见第 12.9 节）等物理参数。

Recipes for the determination of the effective, overall activation energy of the homogeneously occurring recovery, described by Eq. (11.1), are the same as described for heterogeneous transformations in Sects. 10.15.1 and 10.15.2.
公式 ( 11.1) 所描述的确定均相恢复的有效总活化能的方法与第 10.15.1 和 10.15.2 节中异相转化的方法相同。10.15.1 和 10.15.2 节中对异质转化的描述相同。

Values for kinetic parameters, as the effective, overall activation energy, obtained by fitting expressions as Eq. (11.1) to experimental data, for a parameter p varying upon recovery, may be difficult to interpret. Recovery can be a composite process where various subprocesses may contribute simultaneously (cf. the discussion on and the unravelling of the effects of nucleation, growth (and impingement) modes on the overall kinetics of heterogeneous transformations in Chap. 10). Also, subprocesses may occur consecutively, prohibiting a direct application of Eq. (11.1).
在参数 p 随恢复而变化的情况下，通过公式 ( 11.1) 与实验数据拟合得到的有效总活化能等动力学参数值可能难以解释。复原可能是一个复合过程，其中各种子过程可能同时起作用（参见第 10 章中关于成核、生长（和撞击）模式对异质转化整体动力学影响的讨论和解释）。此外，子过程可能会连续发生，因此不能直接应用公式 ( 11.1)。

11.2 Recrystallization  11.2 再结晶

The heterogeneous formation of new, strain-free grains growing, by a migrating high-angle grain boundary, into the deformed matrix typifies the recrystallization process. This immediately indicates the driving force for recrystallization: the complete release of the strain energy induced by the preceding process of cold work and as remaining after the subsequent recrystallization-foregoing recovery. Hence, the driving force, −∆G_recryst, is given by (cf. Sect. 5.2.4 and Eq. (5.8)):
通过高角度晶界的迁移，新的无应变晶粒在变形基体中异质形成，这是再结晶过程的典型特征。这立即表明了再结晶的驱动力：完全释放前一冷加工过程引起的应变能，以及随后的再结晶-前向恢复后剩余的应变能。因此，驱动力 -∆G _recryst 由（参见第 5.2.4 节和式 ( 5.8)）给出：

$$-\mathrm{\Delta }{G}_{\text{recryst}}=E_{\text{elastic}}=\text{const}\cdot {\rho }_{\text{d}}G{b}^2$$

(11.2)

with the “const” having a value between 0.5 and 1.0 (see below Eq. (5.8)) and ρ_d as the dislocation density removed by the recrystallization. Strongly deformed, cold rolled metals exhibit dislocation densities as large as 5 × 10¹⁵ m⁻² (cf. Sect. 5.2.3). Taking G and b as for b.c.c. iron (ferrite) and the “const” equal to 1.0 it follows: −∆G_recryst equals about 2.6 × 10⁷ Pa = 2.6 × 10⁷ Nm/m³ = 2.6 × 10⁷ J/m³, which corresponds to about 0.18 kJ/mol, which should be considered as an upper estimate. This can be compared with the driving force of phase transformations as considered in Chap. 9. Obviously, in principle the driving force for a phase transformation, e.g. the transformation of phase α into phase β, can be very small: at the equilibrium temperature, the driving force, −∆G = G_α − G_β, equals zero. However, many phase transformations are induced remote from the state of equilibrium: for example, the decomposition of a supersaturated solid solution (retained by quenching), ${\alpha }^{\mathrm{\prime }}$, into the equilibrium phases α and β (see Fig. 9.1 and its discussion in the introduction of Chap. 9), for which the driving force is given by −∆G = −{(G_α + G_β) − $G_{{\alpha }^{\mathrm{\prime }}}$}. This last driving force can be of the order of 1 kJ/mole. It can thus be concluded that the driving force for recrystallization is rather small as compared to that of the last category of phase transformations.
const "的值介于 0.5 和 1.0 之间（见下文公式 ( 5.8)），ρ _d 是再结晶去除的位错密度。强变形冷轧金属的位错密度高达 5 × 10 ¹⁵ m ⁻² （参见第 5.2.3 节）。将 G 和 b 设为 b.c.c.铁（铁素体），且 "const "等于 1.0，可以得出-∆G _recryst 约等于 2.6 × 10 ⁷ Pa = 2.6 × 10 ⁷ Nm/m ³ = 2.6 × 10 ⁷ J/m ³ ，相当于约 0.18 kJ/mol，这应视为上限估计值。这可以与第 9 章中考虑的相变驱动力进行比较。显然，原则上相变的驱动力，例如相 α 向相 β 的转变，可以非常小：在平衡温度下，驱动力 -∆G = G _α - G _β 等于零。然而，许多相变都是在远离平衡状态的情况下发生的：例如，过饱和固溶体（通过淬火保留） ${\alpha }^{\mathrm{\prime }}$ 分解为平衡相 α 和 β（见图 9.1 及其在第 9 章引言中的讨论），其驱动力为 -∆G = -{(G _α + G _β ) - $G_{{\alpha }^{\mathrm{\prime }}}$ }。最后一个驱动力可能为 1 kJ/mole。因此可以得出结论，与上一类相变相比，再结晶的驱动力相当小。

Recrystallization phenomena have also been observed upon interdiffusion, as in the diffusion zone of A/B diffusion couples (e.g. in thin Cu/Ni bicrystalline films; Mittemeijer and Beers 1980); see the discussion in Sect. 8.9.2. (Note that for the example of the thin Cu/Ni bicrystalline films, also a special variant of polygonization was observed upon annealing; see Fig. 11.5 and its discussion in Sect. 11.1.1).
在 A/B 扩散耦合的扩散区，也观察到了相互扩散时的再结晶现象（例如在铜/镍双晶薄膜中；Mittemeijer 和 Beers，1980 年）；见第 8.9.2 节的讨论。(请注意，以铜/镍双晶薄膜为例，在退火时也观察到了一种特殊的多边形化变体；见图 11.5 及其在第 11.1.1 节中的讨论）。

11.2.1 “Nucleation” of Recrystallization
11.2.1 再结晶的 "成核"

Recrystallization was formerly conceived as a heterogeneous phase transformation in the sense of the treatment in Sect. 9.2. However, this can be considered a problematic point of view: nucleation as discussed in Sect. 9.2 does not occur in recrystallization. Thermally induced fluctuations in the deformed microstructure do not lead to the formation of a strain-free nucleus (particle of supercritical size; cf. Sect. 9.2) separated by a high-angle grain boundary from the matrix.
按照第 9.2 节的处理方法，再结晶以前被认为是一种异质相变。然而，这种观点是有问题的：第 9.2 节中讨论的成核现象并不发生在再结晶中。变形微观结构中的热诱导波动不会导致形成无应变核（超临界尺寸的颗粒；参见第 9.2 节），该核与基体之间存在高角度晶界。

The above statement can be illustrated by straightforward application of the treatment in Sect. 9.2. Consider Eq. (9.3). Replace $\mathrm{\Delta }{G}_{\text{chem}}^v$ by ∆G_recryst according to Eq. (11.2)_, take the interfacial energy, γ, equal to that for a high-angle grain boundary (i.e. of the order 1 J/m²), and recognize that for recrystallization $\mathrm{\Delta }{G}_{\text{strain}}^v$ is nil. Then it can be calculated from Eq. (9.3) that the critical Gibbs energy of nucleus formation, ∆G^* (cf. Eq. (9.5)), is very large, in association with a large value of the size for the particle of critical size (cf. Eq. (9.4)). Obviously, this is due to the relatively small driving force (cf. Eq. (11.2) discussed above) and the relatively large value for the interfacial energy. Hence, the nucleation rate, as given by Eq. (10.6), becomes very small. This consideration makes likely that initiation of recrystallization is not a nucleation process according to the theory for heterogeneous phase transformations dealt with in Chap. 9. What then are viable mechanisms for initiating recrystallization?
直接应用第 9.2 节中的处理方法即可说明上述论述。考虑公式 ( 9.3)。将 $\mathrm{\Delta }{G}_{\text{chem}}^v$ 替换为 ∆G _recryst 根据公式 ( 11.2) _, 取界面能 γ 等于高角度晶界的界面能（即 1 J/m 的数量级 ² ），并确认再结晶 $\mathrm{\Delta }{G}_{\text{strain}}^v$ 为零。根据公式 ( 9.3) 可以计算出，晶核形成的临界吉布斯能 ∆G ^* （参见公式 ( 9.5)）非常大，这与临界尺寸颗粒的尺寸值较大有关（参见公式 ( 9.4)）。显然，这是由于相对较小的驱动力（参见上文讨论的公式 ( 11.2)）和相对较大的界面能值造成的。因此，由公式 ( 10.6) 得出的成核率变得非常小。因此，根据第 9 章所述的异质相变理论，再结晶的引发可能不是一个成核过程。那么什么是启动再结晶的可行机制呢？

If genuine nucleation of a strain-free grain, separated by a mobile high-angle grain boundary from the deformed matrix, is impossible, it appears natural to look for regions in the deformed microstructure the growth of which would lead to a reduction of the stored energy in the specimen. In other words, the heterogeneity of the deformed microstructure may provide the key to the initiation of the recrystallization process.
如果不可能真正形成无应变晶粒（由变形基体的可移动高角度晶界分隔），那么自然要在变形微结构中寻找可导致试样中存储能量减少的生长区域。换句话说，变形微结构的异质性可能是启动再结晶过程的关键。

Strain-induced grain-boundary migration is thought to be initiated at a high-angle grain boundary in the deformed microstructure where the dislocation density at both sides of the boundary is significantly different due to the previous (cold) work, which can be a consequence of the dependence on crystal orientation of a grain to applied external loading. The situation can be as sketched in Fig. 11.7, where a (high-angle¹) grain boundary separates crystals A (relatively low value of stored energy per unit volume, E_A) and B (relatively high value of stored energy per unit volume, E_B). A part of the grain boundary can bulge out, from A into B under simultaneous elimination of a surplus stored energy per unit volume, ∆E_d, in the range given by the possible extremes, E_B − E_A, and if the bulging volume approximates a dislocation-free crystal, E_B. Thus, the gain in energy (energy released) is:
应变诱导的晶界迁移被认为是在变形微结构中的高角度晶界处开始的，由于之前的（冷）加工，晶界两侧的位错密度有很大差异，这可能是晶粒的晶体取向依赖于施加的外部荷载的结果。如图 11.7 所示，一个（高角度 ¹ ）晶界将晶体 A（单位体积储能值相对较低，E _A ）和晶体 B（单位体积储能值相对较高，E _B ）分隔开来。晶界的一部分可以凸起，从 A 进入 B，同时消除单位体积的剩余存储能量 ∆E _d ，范围在可能的极端值 E _B - E _A 之间，如果凸起的体积近似于无位错晶体，则 E _B 。因此，能量增益（释放的能量）为

$$\mathrm{\Delta }{E}_{\text{strain}}=\mathrm{\Delta }V\cdot \mathrm{\Delta }{E}_{\text{d}}$$

(11.3)

Fig. 11.7  图 11.7

Schematic depiction of strain-induced grain-boundary migration. The inhomogeneity of the deformed microstructure can bring about that a grain A of relatively low stored energy (dislocation poor) is adjacent to a grain B of relatively high stored energy (dislocation rich). Bulging out of the (high-angle) A/B grain boundary into grain B under simultaneous elimination of the surplus stored energy (annihilation of dislocations by the advancing grain boundary) releases stored (deformation) energy and thereby provides a possible mechanism for the initiation of recrystallization. Note: the dislocations as indicated in the figure are identical in grain A (same b and l; cf. Sect. 5.2.3) and identical in grain B. This has only been done to suggest that grain A and grain B have different crystallographic orientations; of course, in reality, dislocations of varying orientation of l and different orientations of b can occur in both grains
应变诱导的晶界迁移示意图。变形微结构的不均匀性会导致存储能相对较低（差排贫乏）的晶粒 A 与存储能相对较高（差排丰富）的晶粒 B 相邻。在多余储能被同时消除（位错被前进的晶界湮灭）的情况下，A/B 晶界（高角度）向 B 晶界凸出，释放了储能（变形），从而提供了启动再结晶的可能机制。注意：图中所示的位错在晶粒 A 中是相同的（相同的 b 和 l；参见第 5.2.3 节），在晶粒 B 中也是相同的。这样做只是为了表明晶粒 A 和晶粒 B 具有不同的晶体学取向；当然，在现实中，不同取向的 l 位错和不同取向的 b 位错可能出现在两个晶粒中

where ∆V is the volume of the “bulge”. However, the extension of the grain-boundary area by the “bulging” costs interfacial energy per unit interface area, γ. Thus, the cost in energy (energy absorbed) is:
其中 ∆V 是 "隆起 "的体积。然而，"隆起 "对晶粒边界区域的扩展会耗费单位界面面积的界面能量 γ。 因此，能量成本（吸收的能量）为：

$$\mathrm{\Delta }{E}_{\text{gb}}=\mathrm{\Delta }A\cdot \gamma$$

(11.4)

where ∆A is the increase in grain-boundary area due to the “bulging”. In order that grain-boundary bulging can occur, the condition
其中 ∆A 是由于 "隆起 "而增加的晶界面积。要发生晶界膨胀，条件是

$$\mathrm{\Delta }{E}_{\text{strain}}>\mathrm{\Delta }{E}_{\text{gb}}$$

(11.5)

must be fulfilled, and thus
必须实现，因此

$$\mathrm{\Delta }{E}_{\text{d}}>\gamma \cdot \frac{\mathrm{\Delta }A}{\mathrm{\Delta }V}$$

(11.6)

Now the “bulge” will be approximated as a spherical gap with radius R (see Fig. 11.7). Then, for constant L (2L is the diameter of the initially flat part of the grain boundary that bulges out) and variable R, it follows by straightforward calculus:
现在，"凸起 "将近似为半径为 R 的球形间隙（见图 11.7）。然后，对于常数 L（2L 是凸起的晶界最初平坦部分的直径）和变量 R，可以通过简单的微积分计算得出：

$$\frac{\mathrm{\Delta }A/\mathrm{\Delta }R}{\mathrm{\Delta }V/\mathrm{\Delta }R}=\frac{2}{R}$$

(11.7)

By substitution of the result given by Eq. (11.7) into the condition (11.6) it is finally obtained:
将公式 ( 11.7) 所示结果代入条件 ( 11.6) 即可得出：

$$R>\frac{2\gamma }{\mathrm{\Delta }{E}_{\text{d}}}$$

(11.8)

The smallest possible value of R equals L (cf. Fig. 11.7); then the “bulge” is a hemisphere (cf. the derivation of the largest minimal shear stress for the bowing out of a dislocation pinned at two pinning points, which occurs if the dislocation between the two pinning points is a half-circle; cf. Eq. (5.10) in Sect. 5.2.6). Hence, it follows that strain-induced grain-boundary migration can take place if
R 的最小值等于 L（参阅图 11.7）；那么 "凸起 "就是一个半球（参阅在两个销钉点销钉的差排弯曲时最大最小剪切应力的推导，如果两个销钉点之间的差排是一个半圆，就会发生这种情况；参阅第 5.2.6 节中的公式 ( 5.10)）。因此，应变诱导的晶界迁移可能发生在下列情况下

$$L>\frac{2\gamma }{\mathrm{\Delta }{E}_{\text{d}}}$$

(11.9)

This condition has first been formulated by Bailey (1960).
贝利（1960 年）首次提出了这一条件。

In the above discussion strain-induced grain-boundary migration was thought to occur along a grain boundary at one or more places, subject to the condition (11.9). Recognizing the microstructural inhomogeneity (even) within deformed grains, the above reasoning suggests that a single, large enough subgrain (cell) in a polygonized, dislocation-cell structured (cf. Sect. 11.1) microstructure, located at a A/B (high-angle) grain boundary, can act as the region initiating recrystallization (Fig. 11.8). The condition (11.8) can then be formulated as:
在上述讨论中，应变诱导的晶界迁移被认为是沿着晶界的一处或多处发生的，但必须符合条件 ( 11.9)。由于认识到变形晶粒内部微观结构的不均匀性（甚至），上述推理表明，位于 A/B （高角度）晶粒边界的多边形位错晶胞结构（参见第 11.1 节）微观结构中的单个足够大的亚晶粒（晶胞）可作为再结晶的起始区域（图 11.8）。条件 ( 11.8) 可以表述为

$$R_{\text{subgrain}}>\frac{2\gamma }{\mathrm{\Delta }{E}_{\text{d}}}$$

(11.10)

Fig. 11.8 图 11.8

Grains A and B exhibit a polygonized dislocation-cell (subgrain) structure. If a subgrain, located at the A/B (high-angle) grain boundary, is large enough, it can act as a location for the initiation of recrystallization, in accordance with the principle illustrated in Fig. 11.7 (see text)
晶粒 A 和 B 显示出多边形差排细胞（亚晶粒）结构。如果位于 A/B（高角度）晶粒边界的亚晶粒足够大，根据图 11.7 所示的原理（见正文），它可以作为再结晶的起始位置

where the subgrain shape has been taken (approximated) as a sphere and R_subgrain is the subgrain radius. In this case ∆E_d is given by the difference of (i) the strain energy of the polygonized, dislocation-cell structured grain B, into which the large subgrain of the polygonized, dislocation-cell structured grain A grows, and (ii) the strain energy of the large subgrain in grain A, which can be taken as nil (marginal dislocation density within the subgrain; cf. Sect. 11.1). Hence, ∆E_d = E_B. For the dislocation-cell structured grain B, the strain energy is governed by the amount of subgrain boundaries. The energy per unit area subgrain boundary in grain B is γ_B. The amount of subgrain boundary per unit volume in grain B is roughly 3/(2 <R_B>), with 2 <R_B> as the average subgrain diameter of grain B.² Consequently, ∆E_d = E_B = 3/(2 <R_B> ) ⋅ γ_B. Substitution of this result into condition (11.10) finally gives:
其中，亚晶粒形状（近似）为球形，R _subgrain 为亚晶粒半径。在这种情况下，ΔE _d 由以下两项之差给出：(i) 多角化、位错胞结构晶粒 B 的应变能，多角化、位错胞结构晶粒 A 的大亚晶粒长入其中；(ii) 晶粒 A 中大亚晶粒的应变能，可将其视为零（亚晶粒内的边际位错密度；参见第 11.1 节）。因此， ∆E _d = E _B 。对于位错单元结构晶粒 B，应变能受亚晶粒边界量的影响。晶粒 B 中单位面积亚晶粒边界的能量为 γ _B 。晶粒 B 中单位体积的亚晶粒边界量大致为 3/(2 <R _B >)，2 <R _B > 为晶粒 B 的平均亚晶粒直径。 ² 因此，∆E _d = E _B = 3/(2 <R _B > ) ⋅ γ _B 。将这一结果代入条件 ( 11.10) 最后得到：

$$R_{\text{subgrain}}>\left(\frac{4}{3}<R_B>\right)\cdot \left(\frac{\gamma }{{\gamma }_B}\right)$$

(11.11)

Thereby the condition for recrystallization to be initiated is not expressed as a condition for the difference in strain energy of adjacent grains (cf. conditions (11.8) and (11.9)): merely the size of a subgrain adjacent to the grain boundary is decisive for the mechanism considered here. Even if the stored, strain energies in both grains, A and B, are similar (same average subgrain/dislocation-cell size), the mechanism considered here can operate provided the size distribution of the subgrains is sufficiently wide.
因此，再结晶开始的条件并不是相邻晶粒应变能差异的条件（参见条件 ( 11.8) 和 ( 11.9)）：对于本文所考虑的机制而言，仅仅是晶界附近亚晶粒的尺寸起决定性作用。即使 A 和 B 晶粒中存储的应变能相似（子晶粒/错位小室的平均尺寸相同），只要子晶粒的尺寸分布足够宽，本文所考虑的机制也能发挥作用。

Finally it is remarked that subgrain coarsening, in the bulk of a polygonized/dislocation-cell structured grain, can be a precursor for the initiation of recrystallization. Such subgrain coarsening is dominated by the migration of low-angle boundaries (the boundaries of the subgrains), which by itself is no recrystallization (see discussion in Sect. 11.1; note that the subgrain growth discussed in the preceding paragraph involved a subgrain at a high-angle grain boundary that grows by migration of this high-angle boundary, thereby changing the orientation of the deformed material into which this subgrains grows: recrystallization). Two cases can be considered: (i) Upon traversing a grain, the orientation variation of the subgrains passed may be random, i.e. the (minor) variation experienced by subsequent subgrain-boundary passages is at random positive and negative. (ii) Alternatively, upon traversing the grain, there may be a systematic trend in the orientation variation of the subgrains³: while maintaining the minor magnitude of the variation of the orientation at each subgrain boundary, the systematic (minor) change of orientation can occur in the same direction and as a result the difference in orientation of the “first” subgrain and the “last” subgrain met along the passage can be relatively large. Now, for case (ii), suppose that subgrain coarsening starts at distant locations along the passage considered. Evidently, the growing subgrains will meet at some stage, thereby creating a higher-angle boundary than found before between adjacent subgrains along the passage (Fig. 11.9). On this basis recrystallization can be initiated as a consequence of subgrain coarsening in the presence of a gradient in the subgrain orientation.
最后要指出的是，在多角化/异位晶胞结构晶粒的主体中，亚晶粒粗化可能是再结晶开始的前兆。这种亚晶粒粗化主要是低角度边界（亚晶粒的边界）的迁移，其本身并不是再结晶（见第 11.1 节的讨论；请注意，前一段讨论的亚晶粒生长涉及高角度晶粒边界的亚晶粒，该亚晶粒通过迁移该高角度边界而生长，从而改变了该亚晶粒所生长的变形材料的取向：再结晶）。可以考虑两种情况：(i) 在穿越晶粒时，所经过的子晶粒的取向变化可能是随机的，即随后的子晶粒边界穿越所经历的（微小）变化是随机的正负变化。(ii) 另一种情况是，在穿越晶粒时，子晶粒的方位变化可能有系统的趋势： ³ ：在保持每个子晶粒边界的方位变化幅度较小的同时，方位的系统（微小）变化可能发生在同一方向上，因此，沿通道遇到的 "第一个 "子晶粒和 "最后一个 "子晶粒的方位差异可能相对较大。现在，对于情况 (ii)，假设亚晶粒粗化开始于所考虑的通道的远处。显然，不断长大的亚晶粒会在某个阶段相遇，从而在通道上相邻的亚晶粒之间形成比以前更高的角度边界（图 11.9）。在此基础上，由于亚晶粒取向存在梯度，亚晶粒粗化可以导致再结晶。

Fig. 11.9 图 11.9

Subgrain coarsening in an orientiation gradient. In the case considered, upon traversing a grain in the polygonized microstructure, there may be a systematic trend in the orientation variation of the subgrains: while maintaining the minor magnitude of the variation of the orientation at each subgrain boundary, the systematic (minor) change of orientation can occur in the same direction, and as a result, the difference in orientation of the “first” subgrain and the “last” subgrain met along the passage can be relatively large. If subgrain coarsening starts at distant locations along the passage considered, the formation of a higher-angle grain boundaries is possible as sketched in the figure
取向梯度中的子晶粒粗化。在所考虑的情况中，在多边形微观结构中穿越晶粒时，亚晶粒的取向变化可能会出现系统性趋势：在保持每个亚晶粒边界的取向变化幅度较小的同时，取向的系统性（较小）变化可能会发生在同一方向上，因此，在穿越过程中遇到的 "第一个 "亚晶粒和 "最后一个 "亚晶粒的取向差异可能会相对较大。如果亚晶粒粗化开始于所考虑的通道的远处，则可能形成较高角度的晶界，如图所示

The recrystallization mechanisms discussed above all imply that the orientations of the recrystallized material must have been present already in the deformed/recovered material. Yet, observations have been made where the orientations of new, recrystallized grains did not resemble those of the apparent parent grains. It may be speculated that in these cases local, relatively pronounced orientation variations occur in the immediate vicinity of grain boundaries (and grain-boundary junctions!) in the deformed microstructure, as a consequence of the incompatibilities of the intrinsic deformation behaviours of adjacent grains in a massive specimen (cf. the “Intermezzo: Grain interaction” at the end of Chap. 6 and see Mishra et al. 2009). If this is so, a mechanism as discussed above could operate, but this can be difficult to observe. Clearly, a similar discussion can be given for the observation of initiation of recrystallization at the interface with second-phase particles.⁴
以上讨论的再结晶机制都意味着，再结晶物质的取向必须已经存在于变形/复原的物质中。然而，在观察中发现，新的再结晶晶粒的取向与表观母晶粒的取向并不相似。可以推测，在这些情况下，由于大块试样中相邻晶粒的内在变形行为不相容，在变形微结构中紧邻晶粒边界（和晶粒边界交界处！）的地方发生了相对明显的取向变化（参见第 6 章末尾的 "插曲：晶粒相互作用"，以及 Mishra 等人，2009 年）。如果是这样，上文讨论的机制就可能起作用，但这很难观察到。显然，对于在第二相颗粒界面上观察再结晶的启动也可以进行类似的讨论。 ⁴

Intermezzo: The History of an Idea; the Subgrain as Origin of Recrystallization
插曲：一个想法的历史；亚晶粒是再结晶的起源

Burgers (W.G.; see also the “Intermezzo: A historical note about the Burgers vector” in Sect. 5.2.3) wrote the first, extended monograph on recrystallization: W.G. Burgers, “Rekristallisation, verformter Zustand und Erholung”, Handbuch der Metallphysik, vol. 3, p. 2, Akademischer Verlaggesellschaft Becker & Erler Kom.-Ges., Leipzig, 1941 (in German). In this book a remarkable discussion about the origin of recrystallization is given (Sects. 106–109 at pp. 233–262). The deformed microstructure is conceived as an assembly of more or less homogeneously strained “blocks” (“Gitterblöcke”) separated by highly deformed transition regions/layers. Then two different concepts for the initiation of recrystallization are considered⁵:
伯格斯（W.G.；另见第 5.2.3 节中的 "插曲：关于伯格斯矢量的历史说明"）撰写了第一部关于再结晶的扩展专著：W.G. Burgers，"Rekristallisation, verformter Zustand und Erholung"，Handbuch der Metallphysik，第 3 卷，第 2 页，Akademischer Verlaggesellschaft Becker & Erler Kom.-Ges.，莱比锡，1941 年（德文）。该书对再结晶的起源进行了深入探讨（第 106-109 节，第 233-262 页）。变形的微观结构被认为是由或多或少均匀应变的 "块"（"Gitterblöcke"）和高度变形的过渡区/层分隔而成。然后，考虑了两种不同的再结晶启动概念 ⁵ ：

• (i) Genuine nucleation of recrystallization nuclei at/in the highly deformed transition regions/layers;
  在高度变形的过渡区/层中真正形成再结晶核；

• (ii) Growth of “blocks” of, as compared to the surrounding “blocks”, relatively low strain energy, pre-existing (Burgers speaks of: “präformiert”) in the deformed microstructure and which are able to grow upon annealing, driven by the release of energy stored in the deformed surroundings of these “blocks” (cf. Fig. 110 at p. 246 of Burgers’ book).
  与周围的 "块体 "相比，应变能相对较低的 "块体 "在变形的微观结构中预先存在（伯格斯称之为："präformiert"），在退火时，由于这些 "块体 "周围变形环境中储存的能量释放，这些 "块体 "得以生长（参见伯格斯著作第 246 页，图 110）。

This second hypothesis, as formulated by Burgers, and, by the way, tributary to ideas earlier presented a.o. by Masing in 1920 and Dehlinger in 1933, sounds surprisingly modern: one is immediately tempted to identify the “low-energy block” with the cell/subgrain in a dislocation-cell structured or polygonized grain, presented above as the crucial structural entity to initiate recrystallization. Burgers presented this concept in 1941, which is long before polygonization was first described and its potential importance for the initiation of recrystallization was recognized (Cahn 1949; Beck 1949). Moreover, transmission electron microscopy, capable of revealing the presence of polygonized/dislocation-cell microstructures, emerged as an important technique for microstructural analysis not before the “fifties” of the past century: the first observations by TEM of dislocations were made in 1956.
布尔格斯提出的第二个假说，顺便说一下，与马辛（Masing）在1920年和德林格（Dehlinger）在1933年提出的观点相辅相成，听起来出奇地现代：人们会立即将 "低能块 "与位错晶胞结构或多边形化晶粒中的晶胞/子晶粒相提并论，认为上述晶胞/子晶粒是启动再结晶的关键结构实体。Burgers 在 1941 年提出了这一概念，这比人们首次描述多边形化及其对再结晶启动的潜在重要性要早得多（Cahn 1949；Beck 1949）。此外，透射电子显微镜能够揭示多边形化/位错电池微结构的存在，在上世纪 "五十年代 "之前就已成为微结构分析的重要技术：1956 年首次通过透射电子显微镜观察到位错。

Whereas Burgers in his evaluation, on the basis of the available experimental information at the time, could eventually not decide between the above extremes for the initialization of recrystallisation (Sect. 110, at pp. 260–262 in his book), research until now has established with certainty that pre-existing, i.e. after deformation/recovery, low-energy “blocks”, i.e. the dislocation cell or the subgrain, are the origins of recrystallization (Humphreys and Hatherly 2004).
Burgers在他的评估中，根据当时可用的实验信息，最终无法在上述两种极端情况中确定再结晶的初始化（见书中第110节，第260-262页），而迄今为止的研究已经确定，预先存在的，即变形/恢复后的低能 "块"，即位错单元或亚晶粒，是再结晶的起源（Humphreys 和 Hatherly，2004年）。

11.2.2 Kinetics of Recrystallization
11.2.2 再结晶动力学

The majority of the kinetic analyses performed of recrystallization adopt an approach as indicated for heterogeneous phase-transformation kinetics; see Chap. 10. “Nucleation”, growth and impingement are distinguished as three generally overlapping mechanisms. As shown in Sect. 10.8, this framework can lead to the classical Johnson–Mehl–Avrami equation, describing the degree of transformation (here: fraction recrystallized) as function of time at constant temperature (Eq. (10.20)). To emphasize the restricted validity of the classical JMA equation, the basis assumptions made in its derivation are listed here (again; see Sect. 10.11): isothermal transformation, either pure site saturation at t = 0 or pure continuous nucleation, high driving force in order that Arrhenius-type temperature dependences for the nucleation and growth rates are assured and randomly dispersed nuclei which grow isotropically.
对再结晶进行的动力学分析大多采用异相转变动力学的方法；见第 10 章。"成核"、生长和撞击被区分为三种通常相互重叠的机制。如第 10.8 节所示，这一框架可以引出经典的约翰逊-梅尔-阿夫拉米方程，描述恒温条件下转变程度（此处：再结晶部分）与时间的函数关系（式 ( 10.20)）。为了强调经典 JMA 方程的有限有效性，这里列出了推导该方程时所作的基本假设（再次参见第 10.11 节）：等温转变、t = 0 时的纯位点饱和或纯连续成核、高驱动力以确保成核和生长率的阿伦尼乌斯型温度相关性，以及随机分散的各向同性生长的晶核。

Particularly problematic with a view to application of the classical JMA equation to recrystallization is the assumption of a large driving force: as indicated at the start of Sect. 11.2, recrystallization is characterized by a small driving force. Further, a random dispersion of “nucleation” sites is unlikely (e.g. strain-induced boundary migration initiating at high-angle grain boundaries, implying a more “regular/periodic” “nucleation”; cf. discussion at the end of Sect. 10.9).
将经典的 JMA 方程应用于再结晶尤其存在问题的是假定存在较大的驱动力：如第 11.2 节开头所述，再结晶的特点是驱动力较小。此外，"成核 "点不可能随机分散（例如，应变诱导的边界迁移始于高角度晶界，这意味着 "成核 "更 "规则/周期"；参见第 10.9 节末尾的讨论）。

Yet, many applications of classical JMA analysis to recrystallization kinetics have been made. Especially use of inaccurate data and insensitive fitting may have led to seemingly successful fitting of the classical JMA equation (cf. Sect. 10.11).
然而，经典 JMA 分析在再结晶动力学中也有许多应用。特别是使用不准确的数据和不敏感的拟合可能导致经典 JMA 方程的拟合看似成功（参见第 10.11 节）。

Also, application of the generalized JMA equation (Eq. (10.23)) cannot be advised, as this equation, although compatible with a range of nucleation and growth modes, is still based on a random distribution of the “nuclei” to describe the effect of impingement. A more promising approach may therefore be adopting the generalized description of the extended volume (Eqs. (10.14)–(10.16)) and combine this with an appropriate impingement mode (e.g. see Eqs. (10.18b), (10.21) and (10.22)) and evaluate the degree of recrystallization (fraction recrystallized) numerically on the basis of the recipe described in Sect. 10.10.
此外，我们也不建议应用广义的 JMA 方程（式 ( 10.23)），因为该方程虽然与一系列成核和生长模式兼容，但仍然是基于 "核 "的随机分布来描述撞击效果的。因此，更有前途的方法可能是采用扩展体积的广义描述（式 ( 10.14)-( 10.16)），并将其与适当的撞击模式相结合（例如，见式 ( 10.18b)、( 10.21) 和 ( 10.22)），然后根据第 10.10 节中描述的配方对再结晶程度（再结晶部分）进行数值评估。

However, even then, one still is subject to the assumption of thermally activated “nucleation” and growth according to Arrhenius-type temperature dependencies (large driving force; see above). For example, in case of a small driving force for growth the recrystallization-front velocity, v, can be written as (cf. Sect. 10.6.1):
然而，即便如此，我们仍然需要假设热激活的 "成核 "和生长是根据阿伦尼乌斯类型的温度依赖性进行的（大驱动力；见上文）。例如，在生长驱动力较小的情况下，再结晶前沿速度 v 可以写成（参见第 10.6.1 节）：

$$v\left(T\left(t\right)\right)=M\left(-\mathrm{\Delta }G\right)=M_0\exp \left(-\frac{Q_G}{\text{RT}(t)}\right)\left(-\mathrm{\Delta }G\left(T\left(t\right)\right)\right)$$

(10.11)

The driving force, −∆G, can change with time and temperature, for example, due to ongoing recovery processes, in the not yet recrystallized matrix, while recrystallization runs. Then, an Arrhenius-type temperature dependence for growth generally does not hold. Of course, even in this case a numerical approach remains possible (the volume of the recrystallized particle nucleated at time τ must be calculated now by numerical integration according to Eq. (10.8)).
驱动力 -∆G 可能会随着时间和温度的变化而变化，例如，在再结晶过程中，由于尚未再结晶的基体中正在进行的恢复过程，驱动力 -∆G 可能会随着时间和温度的变化而变化。因此，阿伦尼乌斯式的生长温度依赖关系一般不成立。当然，即使在这种情况下，仍然可以采用数值方法（现在必须根据公式 ( 10.8) 通过数值积分计算在时间 τ 成核的再结晶颗粒的体积）。

Finally, it is remarked that interpretation of the value possibly determined for the effective, overall activation energy of recrystallization, is difficult without more ado. The effective activation energy incorporates contributions of “nucleation” and growth (e.g. see Sect. 10.12 and Eq. (10.24)). Procedures for unravelling the activation energies of “nucleation” and growth are possible (e.g. see Sect. 10.15.6).
最后，需要指出的是，如果不详细说明，很难解释可能确定的再结晶有效总活化能值。有效活化能包括 "成核 "和生长的贡献（例如，见第 10.12 节和式 ( 10.24)）。解开 "成核 "和生长活化能的程序是可行的（如见第 10.15.6 节）。

11.3 Grain Growth  11.3 谷物生长

After completion of the recrystallization process as discussed in Sect. 11.2, a coarsening of the microstructure can occur, driven by the release of grain-boundary energy: the larger grains grow at the expense of the smaller grains. As the driving force for this process is (even; cf. discussion at the beginning of Sect. 11.2) distinctly smaller than for recrystallization, the velocity of the migrating grain boundaries is smaller than in the case of recrystallization (cf. (again) Eq. (10.11) given directly above). Two cases of grain growth can be discerned:
如第 11.2 节所述，再结晶过程完成后，在晶界能量释放的驱动下，微观结构会发生粗化：大晶粒的生长以牺牲小晶粒为代价。由于这一过程的驱动力（甚至；参见第 11.2 节开头的讨论）明显小于再结晶的驱动力，因此迁移晶界的速度也小于再结晶的情况（再次参见上文直接给出的公式 ( 10.11)）。晶粒生长有两种情况：

• normal grain growth, characterized by an approximately uniform velocity for the migrating grain boundaries throughout the specimen, with the consequence that the grain size remains more or less uniform throughout the specimen, but increases during the process;
  正常晶粒生长，其特点是整个试样的晶界迁移速度大致均匀，因此整个试样的晶粒大小基本保持一致，但在加工过程中会增大；

• abnormal grain growth, characterized by mobile grain boundaries for only a few grains, with the result that these few grains become very large as compared to the remaining majority of the grains. This last process has, confusingly, also been called secondary recrystallization, as compared to the primary recrystallization discussed in Sect. 11.2 where the driving force is the decrease of stored strain energy.
  异常晶粒长大，其特点是只有少数晶粒的晶界可以移动，结果是这些少数晶粒与其余大多数晶粒相比变得非常大。与第 11.2 节中讨论的一次再结晶相比，最后一个过程也被称为二次再结晶，而后者的驱动力是存储应变能的减少。

11.3.1 The Grain-Boundary Network; on Grain-Boundary/Interfacial Energy and Tension
11.3.1 晶界网络；关于晶界/界面能量和张力

Obviously, thermodynamic equilibrium requires elimination of all grain boundaries in the specimen. Normally this ultimate, stable state is not reached. Instead, the arrangement of grain boundaries in a specimen can be such that metastable states occur.
显然，热力学平衡要求消除试样中的所有晶界。通常情况下，这种最终的稳定状态是无法达到的。相反，试样中的晶界排列可能会导致出现蜕变状态。

Changes in the arrangement and density of the grain boundaries/interfaces in a material can occur under the constraints of (i) preservation of the massive nature of the specimen (the grains must be space filling), and (ii) establishment of local mechanical equilibrium of grain-boundary/interface tensions at locations where grain boundaries meet, the so-called vertices.
材料中晶界/界面的排列和密度变化可在以下限制条件下发生：(i) 保持试样的块状性质（晶粒必须充满空间）；(ii) 在晶界交汇处（即所谓的顶点）建立晶界/界面张力的局部机械平衡。

Before proceeding, at this place, some digression on the concepts grain-boundary/interface energy and tension is necessary. The following discussion pertains to interfaces in general, i.e. including surfaces, grain boundaries and interphase boundaries, but only the notion grain boundary will be used, as “pars pro toto”.
在继续讨论之前，有必要先对晶界/界面能量和张力的概念做一些介绍。下面的讨论涉及一般界面，即包括表面、晶界和相间边界，但作为 "本体"，只使用晶界概念。

The atoms at a grain boundary generally possess a higher energy than the atoms in the bulk (of the grain considered), because of their less ideal or incomplete state of chemical bonding. The amount of energy the atoms at the grain boundary have, more than they would have as bulk atoms, is an “excess energy” and, per unit area grain boundary, is called the grain-boundary energy, γ_GB. The grain then strives for making the grain-boundary area as small as possible. Hence it costs energy to enlarge the grain-boundary area. Or, in other words, a force has to be applied, in the plane of the grain boundary and acting along a line in the grain-boundary area, in order to extend the grain-boundary area in the direction of the force (cf. Fig. 11.10). This force per unit length, i.e. tension/stress, along the line mentioned is σ_GB. On the basis of this reasoning it would follow: σ_GB dA (work done) = γ_GB dA (energy change), with dA as the increase of grain-boundary area per unit length along the line in the grain-boundary area considered. Consequently, the grain-boundary tension, σ_GB, has the same numerical value as the grain-boundary energy, γ_GB:
晶界上的原子由于其化学键的不理想或不完全状态，通常比主体（晶粒）上的原子具有更高的能量。晶界原子所具有的能量，比它们作为主体原子所具有的能量要多，这就是 "过剩能量"，单位面积晶界称为晶界能量，γ _GB 。晶粒会努力使晶界面积尽可能小。因此，扩大晶界面积需要耗费能量。或者换句话说，必须在晶界平面上沿晶界区域的一条线施加一个力，才能沿力的方向扩大晶界区域（参见图 11.10）。沿所述线的单位长度力（即拉力/应力）为 σ _GB 。根据这一推理可以得出： σ _GB dA（做功）= γ _GB dA（能量变化），其中 dA 是在所考虑的晶界区域内沿线每单位长度上增加的晶界面积。因此，晶界张力 σ _GB 与晶界能量 γ _GB 的数值相同：

$${\text{σ}}_{\text{GB}}={\gamma }_{\text{GB}}$$

(11.12a)

Fig. 11.10 图 11.10

Schematic depiction of the increase in grain-boundary area by moving a grain boundary: a force has to be applied, in the plane of the grain boundary and acting along a line in the grain-boundary area, in order to extend the grain-boundary area in the direction of the force. This is the origin of the notion grain-boundary tension/stress
移动晶界以增加晶界面积的示意图：必须在晶界平面上沿晶界面积的一条线施加一个力，才能使晶界面积沿力的方向扩展。这就是晶界张力/应力概念的起源

Note that σ_GB is expressed in Nm⁻¹ and γ_GB is expressed in Jm⁻² (1 J (energy) = 1 Nm (work)).
请注意，σ _GB 的单位是 Nm ⁻¹ ，γ _GB 的单位是 Jm ⁻² （1 J（能量）= 1 Nm（功））。

However, the discussion in the above paragraph has tacitly assumed that γ_GB does not depend on (the extension of) A. In order that this is true, it would be necessary that the density and the arrangement (structure) of the atoms in the grain boundary are unchanged upon change of A. This can be true for the surface of liquids, where atoms can rapidly, freely, move from the bulk to the surface, and vice versa, to accommodate imposed shape changes and thereby maintain the overall, equilibrium surface structure. For solids similar phenomena are less likely: a serious straining of the arrangement of grain-boundary atoms may occur (see next paragraph) without relaxation by the transfer of atoms from the bulk, or vice versa: solids are much more viscous than liquids and, in contrast with liquids, can support shear (see Sects. 12.7 and 12.16). Then the numerical values of σ_GB and γ_GB are not identical. For this case, one can proceed as follows. The change in Gibbs energy upon change of grain-boundary area dA is given by dG = d(γ_GB A) = γ_GB dA + A dγ_GB. From σ_GB dA (work done) = dG (energy change) = γ_GB dA + A dγ_GB, it then follows:
然而，上段的讨论默认 γ _GB 不依赖于（A 的延伸）。要做到这一点，就必须在 A 变化时，晶界原子的密度和排列（结构）保持不变。这对于液体表面来说是真实的，因为原子可以快速、自由地从主体移到表面，反之亦然，以适应外加的形状变化，从而保持整体的、平衡的表面结构。对于固体来说，类似的现象不太可能发生：晶界原子的排列可能会发生严重的应变（见下段），而不会因原子从体液转移而松弛，反之亦然：固体的粘度比液体高得多，与液体相反，固体可以承受剪切力（见第 12.7 和 12.16 节）。那么 σ _GB 和 γ _GB 的数值并不相同。在这种情况下，我们可以按以下步骤进行计算。晶界面积 dA 变化时的吉布斯能变化由 dG = d(γ _GB A) = γ _GB dA + A dγ _GB 给出。根据 σ _GB dA（做功）= dG（能量变化）= γ _GB dA + A dγ _GB, ，可以得出：

$${\text{σ}}_{\text{GB}}={\gamma }_{\text{GB}}+A\cdot \frac{\text{d}{\gamma }_{\text{GB}}}{\text{d}A}$$

(11.12b)

which reduces to Eq. (11.12a) if γ_GB does not depend on A. It should further be realized that for the results given by Eqs. (11.12a and 11.12b), the grain-boundary tension/stress is taken as isotropic, i.e. σ_GB does not depend on direction in the grain-boundary area. In view of the elastic anisotropy (cf. Sect. 12.3), this will generally not be true, but corresponding experimental data are extremely rare.
如果 γ _GB 並不取決於 A，則可還原為公式 ( 11.12a)。應進一步了解，對於公式 ( 11.12a 及 11.12b) 所給出的結果，晶界拉力/應力被視為各向同性，即 σ _GB 並不取決於晶界區域的方向。鉴于弹性各向异性（参阅第 12.3 节），这通常是不正确的，但相应的实验数据极为罕见。

The above discussion suggests that the differences between σ_GB and γ_GB are less pronounced for high-angle (more irregular atomic arrangement) than for low-angle (more regular atomic arrangement) grain boundaries.
上述讨论表明，对于高角度（原子排列更不规则）晶界，σ _GB 和 γ _GB 之间的差异没有低角度（原子排列更规则）晶界那么明显。

The origin of the straining in the grain-boundary area of a solid can be discussed as follows. Due to the lack of neighbours or having partly different neighbours, than as for the atoms in the bulk, the atoms in the peripheral grain boundary can have a coordination and bonding different from the bulk atoms, with the result that their strived for atomic volumes (nearest neighbour distances) and strived for arrangement can be different from those of the bulk atoms. However, the atoms at the periphery are constrained to remain in registry with the underlying atomic layers. Hence, the grain boundary experiences a grain-boundary strain/stress with respect to the preferred, strived for atomic positions (cf. Sutton and Balluffi 1995).
固体晶界区域应变的起源可以讨论如下。由于缺乏相邻原子或相邻原子部分不同于主体原子，外围晶界原子的配位和成键可能不同于主体原子，因此它们所争取的原子体积（近邻距离）和争取的排列可能不同于主体原子。然而，外围原子受限于与底层原子层保持一致。因此，相对于首选的、力争的原子位置，晶界会产生晶界应变/应力（参见 Sutton 和 Balluffi，1995 年）。

The concept of grain-boundary tension now allows defining a local mechanical equilibrium at common grain-boundary edges/“vertices”.
晶界张力的概念现在允许在共同晶界边缘/"顶点 "处定义局部机械平衡。

Consider Fig. 11.11: three grains, A, B and C, meet at a common edge, perpendicular to the plane of drawing. Given a sufficiently high atomic mobility, the grain boundaries will orient themselves at the edge/triple junction at O such that the grain-boundary tensions σ_A/B, σ_B/C and σ_A/C comply with a local mechanical equilibrium at O given by balance of the three grain-boundary tensions. Thus, vectorial equilibrium of the grain-boundary tension components along the A/B boundary plane leads to:
请看图 11.11：三个晶粒 A、B 和 C 在垂直于绘图平面的共同边缘相遇。如果原子流动性足够高，晶界将在 O 处的边缘/三重交界处定向，使晶界张力 σ _A/B 、σ _B/C 和 σ _A/C 符合三个晶界张力平衡后在 O 处的局部机械平衡。因此，沿 A/B 边界平面的晶粒边界张力分量的矢量平衡导致：

$${\text{σ}}_{\text{A/B}}+{\text{σ}}_{\text{B/C}}\cos {\theta }_{\text{B}}+{\text{σ}}_{\text{A/C}}\cos {\theta }_{\text{A}}=0$$

(11.13a)

Fig. 11.11 图 11.11

Illustration of local mechanical equilibrium of grain-boundary tensions at a grain-boundary triple junction (edge) of grains A, B and C
晶粒 A、B 和 C 的晶界三重交界处（边缘）的晶界张力局部机械平衡示意图

Equivalent expressions result considering vectorial equilibrium of grain-boundary tension components along the B/C and A/C boundary planes. Or, by vectorial equilibrium of the grain-boundary tension components perpendicular to the A/B, B/C and A/C boundary planes, a well-known relation is obtained:
考虑到沿 B/C 和 A/C 边界平面的晶界张力分量的矢量平衡，可得出等效表达式。或者，通过垂直于 A/B、B/C 和 A/C 边界平面的晶界拉力分量的矢量平衡，可以得到众所周知的关系式：

$$\frac{{\text{σ}}_{\text{A/B}}}{\sin {\theta }_{\text{C}}}=\frac{{\text{σ}}_{\text{B/C}}}{\sin {\theta }_{\text{A}}}=\frac{{\text{σ}}_{\text{A/C}}}{\sin {\theta }_{\text{B}}}$$

(11.13b)

If the three grain-boundary tensions involved have the same value, it follows that the so-called dihedral angles, θ_A, θ_B and θ_C, are given by 120º. Hence, in case of a single phase material with an isotropic grain-boundary tension, for a two-dimensional, massive arrangement of two-dimensional grains, or, in three dimensions for a massive arrangement of columnar, parallel grains, a microstructure of grains of hexagonal morphology would exhibit metastable (see above and begin of Sect. 11.3.2) equilibrium.
如果涉及的三个晶界张力值相同，则所谓的二面角 θ _A 、θ _B 和 θ _C 都是 120º。因此，如果是具有各向同性晶界张力的单相材料，对于二维晶粒的大量排列，或三维晶粒的大量平行排列，六方形态晶粒的微观结构将表现出可移动平衡（见上文和第 11.3.2 节的开头）。

A similar argument as above leads to the conclusion that, for the case of isotropic grain-boundary tension and four grains meeting at a corner (point/vertex), the balancing of the grain-boundary tensions involves that the angles between the grain edges at the corner will be 109º 28′, i.e. as pertains to the edges of the regular tetrahedron. Within the present context, it further holds for the three-dimensional grain-boundary network that a configuration of more than four grains (edges) at a corner is unstable, i.e. a balancing of grain-boundary tensions is impossible. The analogous statement for a two-dimensional network is that a configuration of more than three grains (edges) at a corner is unstable. Such an unstable configuration strives for decomposition in metastable configurations in each of which the grain-boundary tensions are balanced.
通过上述类似的论证可以得出结论：在各向同性的晶界张力和四个晶粒在一个角（点/顶点）处相遇的情况下，晶界张力的平衡涉及角处晶粒边缘之间的夹角将为 109º 28′，即与正四面体的边缘有关。在目前的情况下，三维晶界网络进一步认为，在一个角上超过四个晶粒（边）的配置是不稳定的，即晶界张力不可能平衡。二维网络的类似说法是，一个角上超过三个晶粒（边）的构型是不稳定的。这种不稳定构型会分解成可转移构型，在每个可转移构型中，晶粒-边界张力都是平衡的。

A special, important case follows if, for the case of three grains meeting at an edge, grains A and B are identical ($A^{\mathrm{\prime }}$  = A = B; see Fig. 11.12) and the grain-boundary tensions are isotropic. It follows from the balance of grain-boundary tensions in the plane of the $A^{\mathrm{\prime }}$/$A^{\mathrm{\prime }}$ boundary (perpendicular to the plane of drawing)⁶:
如果三个晶粒在边缘相遇，晶粒 A 和 B 相同 ( $A^{\mathrm{\prime }}$ = A = B；见图 11.12)，并且晶界张力是各向同性的，那么就会出现一种特殊的重要情况。根据 $A^{\mathrm{\prime }}$ / $A^{\mathrm{\prime }}$ 边界平面（垂直于拉伸平面）上的晶界张力平衡可得出 ⁶ ：

$${\text{σ}}_{{\text{A}}^{\mathrm{\prime }}/{\text{A}}^{\mathrm{\prime }}}=-2{\text{σ}}_{{\text{A}}^{\mathrm{\prime }}/\text{C}}\cdot \cos \left({\theta }_{{\text{A}}^{\mathrm{\prime }}}\right)=2{\text{σ}}_{{\text{A}}^{\mathrm{\prime }}/\text{C}}\cdot \cos \left({\theta }_{\text{C}}/2\right)$$

(11.14)

Fig. 11.12 图 11.12

Illustration of local mechanical equilibrium of grain-boundary tensions at a grain-boundary triple junction (edge) of grains, $A^{\mathrm{\prime }},A^{\mathrm{\prime }}$ and C for the case that the grain-boundary tensions are isotropic (the $A^{\mathrm{\prime }}$/$A^{\mathrm{\prime }}$ plane is a mirror plane)
在晶粒边界张力为各向同性的情况下（ $A^{\mathrm{\prime }}$ / $A^{\mathrm{\prime }}$ 平面为镜面），晶粒边界张力在晶粒、 $A^{\mathrm{\prime }},A^{\mathrm{\prime }}$ 和 C 的三重交界处（边缘）的局部机械平衡示意图

This is the same equation as Eq. (9.13) in Sect. 9.4.5, where the morphology of a second-phase particle developing on a grain boundary of the matrix was discussed in dependence on the strived for contact angle (${\text{σ}}_{{\text{A}}^{\mathrm{\prime }}/{\text{A}}^{\mathrm{\prime }}}$ being smaller or larger than 2 ${\text{σ}}_{{\text{A}}^{\mathrm{\prime }}/\text{C}}$).
这与第 9.4.5 节中的公式 ( 9.13) 相同，在第 9.4.5 节中讨论了在基体晶界上形成的第二相颗粒的形态与接触角 ( ${\text{σ}}_{{\text{A}}^{\mathrm{\prime }}/{\text{A}}^{\mathrm{\prime }}}$ 小于或大于 2 ${\text{σ}}_{{\text{A}}^{\mathrm{\prime }}/\text{C}}$ ) 的关系。

Absolute values for grain-boundary tensions may be difficult to determine; relative determinations, i.e. with reference to a specific grain-boundary tension, are more easily possible by application of Eqs. (11.13) and (11.14). For example, $A^{\mathrm{\prime }}$/$A^{\mathrm{\prime }}$ may stand for a grain boundary of the specimen, composed of $A^{\mathrm{\prime }}$ grains, intersecting the surface. Then, Eq. (11.14) predicts that, for local mechanical equilibrium of the surface and grain-boundary tensions at the point of intersection, a groove must develop at the point of intersection such that a contact (dihedral) angle θ_V occurs. “Grain C” here then should be interpreted as vacuum or the vapour phase in contact with $A^{\mathrm{\prime }}$ (see Fig. 11.13). On this basis $A^{\mathrm{\prime }}$/$A^{\mathrm{\prime }}$ grain-boundary tensions can be determined with respect to the same surface tension, supposed to be isotropic, i.e. independent of crystal orientation. To this end precise determination of the contact angle from the profile of the groove is a prerequisite. This is difficult, because the contact angle is established at the deepest position of the groove where it is very small (see Fig. 11.13). Accurate determination of the contact angle is possible applying atomic force microscopy (see Schöllhammer et al. 1999; cf. the description of scanning probe microscopy in the “Intermezzo: Combined nanoindentation and scanning probe microscopy” in Sect. 12.13). An experimental example is shown in Fig. 11.14.
晶界张力的绝对值可能难以确定；相对确定，即参照特定晶界张力，通过应用公式 ( 11.13) 和 ( 11.14) 更容易实现。例如， $A^{\mathrm{\prime }}$ / $A^{\mathrm{\prime }}$ 可能代表试样的晶界，由 $A^{\mathrm{\prime }}$ 晶粒组成，与表面相交。那么，根据公式 ( 11.14) 预测，为了使表面和晶界张力在交点处达到局部机械平衡，必须在交点处形成凹槽，从而产生接触（二面）角 θ _V 。因此，此处的 "晶粒 C "应解释为真空或与 $A^{\mathrm{\prime }}$ 接触的气相（见图 11.13）。在此基础上， $A^{\mathrm{\prime }}$ / $A^{\mathrm{\prime }}$ 晶界张力可以根据相同的表面张力确定，表面张力应该是各向同性的，即与晶体取向无关。为此，根据凹槽轮廓精确确定接触角是一个先决条件。这很困难，因为接触角是在沟槽最深的位置确定的，而那里的接触角非常小（见图 11.13）。使用原子力显微镜可以精确测定接触角（见 Schöllhammer 等人，1999 年；参见第 12.13 节 "间奏：纳米压痕和扫描探针显微镜的结合 "中对扫描探针显微镜的描述）。实验示例见图 11.14。

Fig. 11.13 图 11.13

Illustration of the local mechanical equilibrium of grain-boundary and surface tensions at the intersection of a grain boundary with the surface of the specimen, requiring the formation of a groove at the intersection of the grain-boundary with the surface
说明在晶界与试样表面的交点处晶界和表面张力的局部机械平衡，要求在晶界与表面的交点处形成凹槽

Fig. 11.14 图 11.14

Depth profile of a Σ19a grain-boundary groove (for the meaning of the symbol “Σ”, see the discussion on the coincidence site lattice (CSL) in Sect. 5.3) at the surface of a Cu-50at.ppmBi bicrystal annealed at 110 h for 1123 K, as measured by atomic force microscopy. The measured contact angle is 140.3°; note the difference in scales along abscissa and ordinate (taken from Schöllhammer et al. 1999)
原子力显微镜测量的 1123 K 退火 110 h 的铜-50at.ppm铋双晶表面的 Σ19a 晶界凹槽深度剖面图（符号 "Σ "的含义请参见第 5.3 节中关于重合位点晶格 (CSL) 的讨论）。测得的接触角为 140.3°；请注意横座标和纵座标上的刻度差异（摘自 Schöllhammer 等人，1999 年）。

Intermezzo: Interface Stabilized Microstructures
插曲：界面稳定微结构

The atoms at an interface of a solid phase with another solid phase, or with a liquid or vapour phase, or with the vacuum, generally possess a higher energy than the atoms in the bulk of that solid phase, because of their less ideal or incomplete state of chemical bonding (cf. the way the concept grain-boundary energy was introduced in the above text). The presence of specific interfaces can cause thermodynamic (energetic) stabilization of phases, which are metastable or unstable according to bulk thermodynamics (energetics). Obviously, corresponding observations can be made especially in thin films and thin film systems, characterized by a high interface density.
在固相与另一固相、液相或汽相、或真空的界面上的原子，由于其化学键的不理想或不完全状态，通常比该固相主体中的原子具有更高的能量（参见上文中引入晶界能概念的方式）。特定界面的存在可导致相的热力学（能量）稳定，而根据体热力学（能量学），这些界面是可转移或不稳定的。显然，在以高界面密度为特征的薄膜和薄膜系统中尤其可以观察到相应的现象。

An amorphous, solid phase, ${\alpha }^{\mathrm{\prime }}$, has a higher bulk energy (Gibbs energy; cf. Sect. 7.3) than the corresponding crystalline, solid phase, α. Now consider the situation of this amorphous phase, ${\alpha }^{\mathrm{\prime }}$, in contact with a crystalline phase, β (both phases of different composition). It can be shown that the energy of the interface between the amorphous phase ${\alpha }^{\mathrm{\prime }}$ and the crystalline phase β generally is smaller than between the crystalline phase α and the crystalline phase β (Jeurgens et al. 2009). Consequently, considering a layered structure of ${\alpha }^{\mathrm{\prime }}$ and β, the lower energy of the ${\alpha }^{\mathrm{\prime }}$/β interface, as compared to the energy of the α/β interface, can overcompensate the difference in bulk energy of the ${\alpha }^{\mathrm{\prime }}$ and α phases. Upon increasing thickness of the amorphous layer (phase), the relative contribution of the interface energy (proportional with the interface area), as compared to the contribution of the bulk energy (proportional with the product of interface area and thickness of the layer), decreases. Hence, up to a certain, critical thickness, the layer of the amorphous phase ${\alpha }^{\mathrm{\prime }}$ is energetically preferred over a layer of the crystalline phase α. In other words: the amorphous phase is the stable phase for a thickness smaller than the critical thickness.
无定形固相 ${\alpha }^{\mathrm{\prime }}$ 的体积能（吉布斯能；参见第 7.3 节）高于相应的晶体固相 α。现在考虑无定形相 ${\alpha }^{\mathrm{\prime }}$ 与晶体相 β（两相成分不同）接触的情况。可以看出，非晶相 ${\alpha }^{\mathrm{\prime }}$ 与结晶相 β 之间的界面能量一般小于结晶相 α 与结晶相 β 之间的界面能量（Jeurgens 等人，2009 年）。因此，考虑到 ${\alpha }^{\mathrm{\prime }}$ 和 β 的层状结构，与 α/β 界面的能量相比， ${\alpha }^{\mathrm{\prime }}$ /β 界面的能量较低，可以弥补 ${\alpha }^{\mathrm{\prime }}$ 和 α 相的体能差异。随着无定形层（相）厚度的增加，界面能（与界面面积成正比）与体能（与界面面积和层厚度的乘积成正比）的相对贡献会减小。因此，在达到一定临界厚度时，非晶相层 ${\alpha }^{\mathrm{\prime }}$ 在能量上比晶相层 α 更受青睐。换句话说：在厚度小于临界厚度时，非晶相是稳定的相。

The above reasoning has provided the explanation for the emergence of amorphous phases, instead of the expected, corresponding crystalline phases, at the interface of crystalline A/B couples upon diffusion annealing, whereas it was thought before that the presence of such amorphous phases was due to kinetic obstacles for the formation of the crystalline compound (Benedictus et al. 1996; Fig. 11.15a).
上述推理解释了在扩散退火过程中，结晶 A/B 对偶界面上出现无定形相而不是预期的相应结晶相的原因，而之前人们认为出现这种无定形相是由于结晶化合物形成的动力学障碍（Benedictus 等人，1996 年；图 11.15a）。

Fig. 11.15 图 11.15

a Formation of a compound phase AB at an interface between the two crystalline phases A and B. Dependent on interface energy values, the compound, product phase can be amorphous up to a certain, critical thickness beyond which the crystalline modification, with the lower “bulk” Gibbs energy, is stable. b Formation of an oxide phase at the surface of the crystalline (metal) phase Me upon oxidation in (gaseous) O₂. Dependent on interface- and surface energy values, the oxide phase can be amorphous up to a certain, critical thickness beyond which the crystalline modification, with the lower “bulk” Gibbs energy, is stable (cf. Fig. 11.16)
a 在两个结晶相 A 和 B 之间的界面上形成复合相 AB。根据界面能值的不同，复合相、产物相可以是无定形的，直至达到一定的临界厚度，超过该厚度，具有较低 "体积 "吉布斯能的结晶修饰就会稳定。 b 在（气态）O ₂ 中氧化时，在结晶（金属）相 Me 的表面形成氧化物相。根据界面能和表面能值的不同，氧化物相在一定临界厚度内可以是无定形的，超过该厚度后，具有较低 "体 "吉布斯能的晶体修饰就会稳定（参见图 11.16）。

Similarly, considering the oxidation of metals, it was shown that the amorphous state for the developing oxide layer can be the energetically stable configuration up to a certain critical thickness of the oxide layer (Reichel et al. 2008). The critical oxide-film thicknesses up to which the amorphous state is preferred energetically, because of its lower sum of interface and surface energies (Fig. 11.15b), as compared to the corresponding crystalline state, are shown for various metals as function of temperature in Fig. 11.16. Thus, also the well-known occurrence of an amorphous oxide film on aluminium in ambient at room temperature represents a state of equilibrium and is not the consequence of a kinetically obstructed crystallization, as has often been suggested (Fig. 11.16).
同样，考虑到金属的氧化，研究表明，在氧化层达到一定临界厚度时，正在形成的氧化层的无定形态可能是能量稳定的构型（Reichel 等人，2008 年）。图 11.16 显示了各种金属的临界氧化膜厚度，与相应的结晶态相比，无定形态的界面能和表面能总和较低（图 11.15b），因此在能量上更受青睐。因此，铝在室温环境中出现无定形氧化膜也是一种众所周知的平衡状态，而不是通常所说的结晶受阻的结果（图 11.16）。

Fig. 11.16 图 11.16

Critical oxide-film thickness, below which oxide films on specific substrates are amorphous (cf. Fig. 11.15b), as a function of oxide-growth temperature, for surfaces of indicated crystallographic orientation of selected materials. A negative value for the critical thickness implies that the oxide film is crystalline from the beginning of oxide-film growth (taken from Reichel et al. 2008)
临界氧化膜厚度，在此厚度以下，特定基底上的氧化膜为无定形膜（参见图 11.15b），与氧化物生长温度有关，适用于选定材料的指定晶向表面。临界厚度的负值意味着氧化膜从开始生长时就是晶体状的（摘自 Reichel 等人，2008 年）

11.3.2 Grain-Boundary Curvature-Driven Growth
11.3.2 晶界曲率驱动的增长

The discussion in the above Sect. 11.3.1 described conditions for mechanical equilibrium at locations where grain boundaries meet. Thereby a prescription for a complete state of metastable equilibrium for the entire grain-boundary network has not yet been established.
上文第 11.3.1 节的讨论描述了晶界交汇处的机械平衡条件。因此，对于整个晶界网络的完全蜕变平衡状态的规定尚未建立。

Obviously, a curved grain-boundary area between two parallel edges has a larger energy than the possible planar grain-boundary area between these two edges. Hence, energy is reduced if the curved grain-boundary area is replaced by the corresponding planar grain-boundary area. This can also be expressed as follows.
显然，两个平行边缘之间的弯曲晶界区比这两个边缘之间可能的平面晶界区具有更大的能量。因此，如果用相应的平面晶界区域代替曲面晶界区域，能量就会减少。这也可以表示为

A curved surface in three-dimensional space is characterized by two principal radii of curvature, r₁ and r₂ (which generally depend on location at the surface). Therefore, the force per unit area grain boundary, i.e. pressure, acting within the grain at the concave side (see footnote 9) of the curved grain boundary with grain-boundary tension σ_GB, at the location with radii of curvature r₁ and r₂, equals σ_GB/r₁ + σ_GB/r₂. If the boundary is part of a sphere, r = r₁ = r₂ and the pressure is given by the well-known result 2σ_GB/r.⁷ This (extra) pressure enhances the energy of the grain at the concave part of the grain boundary. The pressure becomes nil if r₁ and r₂ (or r) become(s) nil. Then a planar grain-boundary area results.⁸
三维空间中的曲面具有两个主要曲率半径，即 r ₁ 和 r ₂ （通常取决于曲面的位置）。因此，在曲率半径为 r ₁ 和 r ₂ 的位置，在具有晶界张力 σ _GB 的弯曲晶界凹面（见脚注 9）上，单位面积晶界力（即压力）等于 σ _GB /r ₁ + σ _GB /r ₂ 。如果边界是球体的一部分，则 r = r ₁ = r ₂ ，压力由众所周知的结果 2σ _GB /r 给出。 ⁷ 这种（额外的）压力会增强晶粒边界凹面部分的能量。如果 r ₁ 和 r ₂ （或 r）变为零，则压力变为零。那么就会产生一个平面晶界区。 ⁸

Fig. 11.17 图 11.17

Curved grain-boundary segment with its two principal radii of curvature r₁ and r₂. At the position shown r₁ = −r₂, and, consequently, the corresponding, local pressure on the grain interior is nil, a situation which always occurs in case of flat grain boundaries
具有两个主要曲率半径 r ₁ 和 r ₂ 的弯曲晶界段。在所示位置，r ₁ = -r ₂ ，因此，相应的，晶粒内部的局部压力为零，这种情况总是发生在平坦的晶界上

For isotropic σ_GB, it is possible to fill two-dimensional space (a plane) with polygons having planar faces compatible with the requirement of mechanical equilibrium at the junctions, i.e. the plane is filled with hexagons (see Eq. (11.13)) and thereby a fully (in the sense of the first paragraph of this section) metastable state for the grain-boundary network in two-dimensional space has been realized. However, a similar situation cannot be established in three-dimensional space: no regular polyhedron with planar faces can fill space under the requirement of local mechanical equilibrium of the grain-boundary tensions at the grain-boundary edges. As a consequence (part of) the grain boundaries are curved and a complete metastable equilibrium for the three-dimensional grain-boundary network can never be achieved: grain growth in the three-dimensional grain-boundary network is unavoidable.
對於各向同性的 σ _GB ，有可能在二維空間（一個平面）中填入符合交界處機械平衡要求的平面多面體，也就是在平面中填入六角形（見公式 ( 11.13)），因而在二維空間中實現了晶界網路的完全（在本節第一段的意義上）遷移狀態。然而，类似的情况在三维空间中并不存在：在晶界边缘的晶界张力局部机械平衡的要求下，没有一个平面正多面体可以填充空间。因此，（部分）晶界是弯曲的，三维晶界网络永远无法达到完全的可移动平衡：三维晶界网络中的晶粒生长是不可避免的。

The above discussion can be summarized by stating that the force due to the grain-boundary tension acting on curved grain boundaries induces grain-boundary migration in order to minimize this force, i.e. the curved grain boundaries tend to migrate towards their centre of curvature. Thus, as considered from the point of observation, a concave⁹ grain boundary moves inwardly and a convex grain boundary moves outwardly (Fig. 11.18).¹⁰
上述讨论可归纳为：作用在弯曲晶界上的晶界拉力会诱发晶界迁移，以尽量减小该拉力，即弯曲晶界倾向于向其曲率中心迁移。因此，从观察点来看，凹 ⁹ 晶界向内移动，凸晶界向外移动（图 11.18）。 ¹⁰

Fig. 11.18 图 11.18

Motions of grain boundaries as driven by grain-boundary tension. Curved grain-boundary segments tend to migrate to their centres of curvatures: a concave grain-boundary segments move inwardly, whereas b convex grain-boundary segments move outwardly
由晶界张力驱动的晶界运动。弯曲的晶界片段倾向于向其曲率中心移动：a 凹面晶界片段向内移动，而 b 凸面晶界片段向外移动

Consider a massive arrangement of parallel, columnar grains with isotropic grain-boundary tension. The system strives for local mechanical equilibrium at locations (edges/“junctions”) where grain boundaries meet. This implies that the system attempts to establish dihedral angles of 120º at the junctions/“vertices” (see below Eq. (11.13)). As a consequence, a grain with more than six sides in the planar arrangement will have convex grain boundaries and tend to grow, and a grain with less than six sides will have concave grain boundaries and tend to shrink (cf. Fig. 11.18a and b). Otherwise said: grain boundaries move into the material on their concave side; i.e. the material with the highest energy (subjected to the pressure σ_GB/r₁ + σ_GB/r₂).
考虑由平行柱状晶粒组成的具有各向同性晶界张力的大规模排列。系统会在晶界交汇的位置（边缘/"交界处"）努力实现局部机械平衡。这意味着系统试图在交界处/"顶点 "建立 120º 的二面角（见下文公式 ( 11.13)）。因此，在平面布置中，多于六个面的晶粒将具有凸晶界并趋于增长，而少于六个面的晶粒将具有凹晶界并趋于收缩（参见图 11.18a 和 b）。换句话说：晶界移动到其凹面的材料中；即具有最高能量的材料（受到压力 σ _GB /r ₁ + σ _GB /r ₂ ）。

At this place it is appropriate to indicate the difference in the direction of grain-boundary migration between the cases of recrystallization and of grain growth. Recrystallization can proceed by outward migration of concave grain boundaries: e.g. by strain-induced grain boundary migration at a high-angle grain boundary or by subgrain coarsening in the presence of a gradient in the subgrain orientation, as discussed in Sect. 11.2.1. This contrasts with grain growth, where grain-boundary segments of concave nature move inwardly. In the process of recrystallization the (sub)grain on the concave part of the boundary is strain-free, and (yet) this (sub)grain grows into the deformed matrix, i.e. in the direction opposite to that for grain growth, as indicated by the centre of curvature of the boundary. The process is driven by the difference in strain energy of the surrounding, deformed matrix and the growing, recrystallized grain, which suffices to overcompensate the unfavourable extension of grain-boundary length/area by the recrystallization processes indicated. Grain growth, in contrast with recrystallization, occurs in a strain-free matrix and therefore is driven by the decrease of grain-boundary density and thus grain-boundary energy, only.
这里应该指出再结晶和晶粒长大两种情况下晶界迁移方向的不同。再结晶可通过凹晶界的外移进行：例如，在高角度晶界上通过应变引起的晶界迁移，或在亚晶粒取向存在梯度的情况下通过亚晶粒粗化（如第 11.2.1 节所述）。这与晶粒生长形成鲜明对比，在晶粒生长过程中，具有凹面性质的晶界段向内移动。在再结晶过程中，边界凹面上的（亚）晶粒是无应变的，（然而）该（亚）晶粒向变形基体生长，即与晶粒生长方向相反，如边界曲率中心所示。这一过程由周围变形基体和生长的再结晶晶粒的应变能差驱动，足以弥补再结晶过程中晶粒边界长度/面积的不利扩展。与再结晶相反，晶粒生长发生在无应变基体中，因此只受晶界密度下降的驱动，因而也受晶界能量下降的驱动。

11.3.3 Kinetics of Grain Growth; Inhibition of Grain Growth
11.3.3 谷粒生长动力学；抑制谷粒生长

The extra pressure prevailing within the grain at the concave side of a curved grain boundary with grain-boundary tension σ_GB equals σ_GB(1/r₁ + 1/ r₂), with r₁ and r₂ as the principal radii of curvature (cf. Sect. 11.3.2). The driving force for grain growth per mole material swept by the moving grain boundary (moving to its centre of curvature; see what follows), −∆G (i.e. the release of Gibbs energy upon grain-boundary migration), is given by the product of the extra pressure, as indicated above, and the molar volume, V_m, and thus
在具有晶界张力 σ _GB 的弯曲晶界凹面上，晶粒内部的额外压力等于 σ _GB (1/r ₁ + 1/ r ₂ )，其中 r ₁ 和 r ₂ 为主要曲率半径（参见第 11.3.2 节）。移动晶界（向其曲率中心移动；见下文）所扫过的每摩尔材料的晶粒生长驱动力 -ΔG（即晶界迁移时释放的吉布斯能）由上述额外压力与摩尔体积 V _m 的乘积给出，因此

$$-\mathrm{\Delta }G={\text{σ}}_{\text{GB}}\left(\frac{1}{r_1}+\frac{1}{r_2}\right)\cdot V_{\text{m}}$$

(11.15)

Now consider a migrating grain-boundary segment with an average radius of curvature r. Then Eq. (11.15) can be written as:
现在考虑平均曲率半径为 r 的迁移晶界段：

$$-\mathrm{\Delta }G=\frac{\left(c{V}_{\text{m}}\right){\text{σ}}_{\text{GB}}}{r}$$

(11.16)

where the geometrical constant c generally depends on the shape of the moving part of the grain boundary: e.g. c equals 2 for the grain-boundary segment being part of a sphere.
其中几何常数 c 通常取决于晶界移动部分的形状：例如，如果晶界部分是球体的一部分，则 c 等于 2。

The grain-boundary velocity for small driving forces, as holds for grain growth, is given by Eq. (10.11) and thus, at constant temperature, is proportional with −∆G. The grain-boundary velocity also equals dr/dt. Hence, from Eqs. (10.11) and (11.16), it is obtained:
小驱动力下的晶界速度，如晶粒生长一样，由式 ( 10.11) 给出，因此，在恒温条件下，与 -∆G 成正比。因此，根据公式 ( 10.11) 和 ( 11.16) 可以得到

$$v=\frac{dr}{dt}=\frac{M\left(c{V}_{\text{m}}\right){\text{σ}}_{\text{GB}}}{r}$$

(11.17)

with M as the mobility of the grain boundary. Upon integration (at constant temperature) with respect to r with r = r₀ at t = 0, and assuming that σ_GB is isotropic, it follows:
M 为晶界的流动性。在 t = 0 时，对 r 进行积分（恒温），r = r ₀ ，并假设 σ _GB 是各向同性的：

$$r^2-r_0^2=2M\left(c{V}_{\text{m}}\right){\text{σ}}_{\text{GB}}t$$

(11.18)

The above treatment concerns a grain-boundary segment that, taking r₁ and r₂, or r, as positive values, moves to its centre of curvature with a radius of curvature that increases with time (Fig. 11.19). Now, to relate the change of r of an individual grain-boundary segment with a change of grain size, a bold step must be made: the (average) radius of curvature of the moving grain-boundary segment in Eq. (11.17) is equated with the average grain size (equivalent grain radius) of the specimen at each moment of time (at constant temperature) and thereby Eq. (11.18) describes the (average) grain growth occurring in the specimen, if r and r₀ are replaced by <r> and <r₀>,¹¹ respectively:
上述处理涉及一个晶界段，该晶界段以 r ₁ 和 r ₂ 或 r 为正值，向其曲率中心移动，曲率半径随时间而增加（图 11.19）。现在，要将单个晶界段的 r 变化与晶粒大小的变化联系起来，必须采取一个大胆的步骤：将式 ( 11.17) 中移动晶界段的（平均）曲率半径等同于平均曲率半径。17）中移动晶界段的（平均）曲率半径等同于试样在每个时刻（恒定温度下）的平均晶粒尺寸（等效晶粒半径），因此，如果将 r 和 r ₀ 分别替换为 和 <r ₀ >， ¹¹ ，则式 ( 11.18) 描述了试样中发生的（平均）晶粒增长：

$$\text{<}r{\text{>}}^2-\text{<}{r}_0{\text{>}}^2=2M\left(c{V}_{\text{m}}\right){\text{σ}}_{\text{GB}}t$$

(11.18a)

Fig. 11.19 图 11.19

Increase of the radius of curvature of a grain-boundary segment upon its migration. The (average) radius of curvature of the moving grain-boundary segment is equated with the average grain size (equivalent grain radius) of the specimen at each moment of time
晶界段移动时曲率半径的增加。移动晶界段的（平均）曲率半径等同于试样在每个时刻的平均晶粒尺寸（等效晶粒半径

The basis for a treatment like the above one was given in the middle of the previous century (e.g. see Burke and Turnbull 1952). The result is often written in general form:
类似上述处理方法的基础早在上世纪中叶就已给出（例如，见 Burke 和 Turnbull，1952 年）。其结果通常被写成一般形式：

$$\text{<}r{\text{>}}^{n^{\mathrm{\prime }}}-\text{<}{r}_0{\text{>}}^{n^{\mathrm{\prime }}}=\text{const}\cdot t$$

(11.19)

with $n^{\mathrm{\prime }}$ as so-called grain-growth exponent. (cf. the (unrelated) growth exponent introduced in Sect. 10.8 to describe phase-transformation kinetics).
$n^{\mathrm{\prime }}$ 作为所谓的晶粒生长指数。(参见第 10.8 节中为描述相变动力学而引入的（不相关的）生长指数）。

The parabolic relationship indicated by Eq. (11.18a) has often been questioned, as a considerable body of experimental work, after the fifties of the previous century, has provided values of $n^{\mathrm{\prime }}$ (cf. Eq. (11.19)) larger than 2 (up to 4). Recent theoretical analyses and computer simulations have only confirmed the validity of the parabolic relationship. It appears that much experimental work may have been imprecise (a similar remark was made regarding the application of the JMA equation for phase-transformation kinetics in Sect. 10.11). Further, in particular the ideal situation assumed for the derivation of Eq. (11.18a) can be incompatible with practical situations where small amounts of grain-boundary pinning, second-phase particles are present (see further below).
公式 ( 11.18a) 所表示的抛物线关系经常受到质疑，因为在上世纪五十年代之后，大量的实验工作都提供了大于 2（最多为 4）的 $n^{\mathrm{\prime }}$ 值（参见公式 ( 11.19)）。最近的理论分析和计算机模拟只是证实了抛物线关系的有效性。看来许多实验工作可能并不精确（第 10.11 节中对相变动力学中 JMA 方程的应用也有类似的评论）。此外，特别是推导公式 ( 11.18a) 时所假设的理想情况可能与存在少量晶界钉合、第二相颗粒的实际情况不符（详见下文）。

The driving force for grain growth depends on the value of the grain-boundary tension σ_GB (cf. Eqs. (11.15) and (11.16)) and thus depends of the structure of the grain boundary: a low-energy (low-angle) grain boundary experiences a smaller driving force and thus shows a smaller grain-boundary velocity (cf. Eq. (11.17)) than a high energy (high-angle) grain boundary. Hence, the distribution of the type of grain boundaries in the specimen,¹² and thus the crystallographic texture (for the notion texture, see Sect. 4.7) influences the rate of (average) grain growth. The distribution of the type of grain boundaries and the texture can change during grain growth, which by itself (i.e. apart from the decrease of grain-boundary density due to grain growth) will lead to a change of the grain-growth rate. The distribution of the type of grain boundaries in the specimen and the texture are a pair of most important parameters characterizing the microstructure, and this recognition has led to a field of activity called “grain-boundary engineering”. (Deformation and) Recrystallization and (subsequent) grain-growth procedures are devised in order to arrive at microstructures with optimal properties. Even today, it has to be admitted that most of this work is performed in practice on an empirical basis and that a great need exists for fundamental research in this area. The most significant gap in our knowledge concerns the atomic structure of, in particular moving, (high-angle) grain boundaries (cf. the discussion on “diffusion induced grain-boundary migration (DIGM)” in Sect. 8.9.2).
晶粒生長的驅動力取決於晶界張力 σ _GB 的值 (參閱公式 ( 11.15) 和 ( 11.16))，因此也取決於晶界的結構：低能量 (低角度) 的晶界比高能量 (高角度) 的晶界承受較小的驅動力，因此顯示較小的晶界速度 (參閱公式 ( 11.17))。因此，试样中晶界类型的分布、 ¹² 以及晶体学纹理（关于概念纹理，请参见第 4.7 节）都会影响（平均）晶粒生长速度。晶界类型和纹理的分布会在晶粒生长过程中发生变化，这本身（即除了晶粒生长导致晶界密度降低之外）会导致晶粒生长速率的变化。试样中晶界类型的分布和纹理是表征微观结构的一对最重要参数，这种认识导致了一个名为 "晶界工程 "的活动领域。(为了获得具有最佳性能的微观结构，人们设计了（变形和）再结晶以及（随后的）晶粒生长程序。即使到了今天，我们也不得不承认，大部分工作都是根据经验在实践中进行的，这一领域的基础研究仍有很大的需求。我们的知识中最重要的空白涉及原子结构，特别是移动的（高角度）晶界（参见第 8.9.2 节中关于 "扩散诱导晶界迁移 (DIGM) "的讨论）。

Obviously, upon continuation of normal grain growth as described above, the driving force for grain growth decreases as the grains become, rather uniformly, larger: see Eq. (11.16). It then becomes conceivable that upon prolonged annealing, the thermal activation, which expresses itself through the grain-boundary mobility, M (cf. Eq. (11.17)), is too small, in view of the strongly decreased value of the driving force, −∆G, in order to sustain a measurable grain-boundary migration rate, v. Consequently, the process of normal grain growth comes effectively to a halt.
显然，在继续上述正常晶粒生长时，随着晶粒均匀变大，晶粒生长的驱动力会减小：见公式 ( 11.16)。因此可以想象，在长时间退火后，由于驱动力 -∆G 的值大幅下降，通过晶界迁移率 M（参见公式 ( 11.17)）表现的热活化作用变得太小，无法维持可测量的晶界迁移率 v。

Or, at some prolonged stage of grain growth, the driving force has become that small that a possible grain-boundary pinning force becomes significant in view of the diminished value of the driving force as expressed by Eq. (11.16):
或者，在晶粒生长的某个较长阶段，驱动力已经变得很小，考虑到驱动力的减小值，可能的晶界夹紧力变得很重要，如公式 ( 11.16) 所示：

• (i) Effect of second-phase particles. The pinning of a grain boundary by a second-phase particle in a matrix may qualitatively be understood as follows. Upon intersection of the particle by the grain boundary, a part of the grain boundary, as large as the area of intersection, has been removed. Thereby grain-boundary energy has been released (see also the discussion at the end of Sect. 9.2): one could say that the grain boundary is attracted to the particle, or otherwise said: it costs energy to remove the grain boundary from the particle. It can be shown that this energy needed to disconnect particle and grain boundary is proportional with the grain-boundary tension and the size of the particle:
  第二相颗粒的影响。基体中的第二相颗粒对晶界的钉扎作用可定性地理解如下。当粒子与晶界相交时，晶界的一部分（与相交面积一样大）被移除。因此，晶界能量被释放了出来（另见第 9.2 节末尾的讨论）：可以说晶界被吸引到了粒子上，或者也可以说：从粒子上移除晶界需要耗费能量。可以证明，断开粒子和晶界所需的能量与晶界张力和粒子大小成正比：

  Consider a grain boundary intersecting a spherical particle. In order that the grain boundary, experiencing a driving force to move (e.g. see Eq. (11.16)), loses itself from the particle, it bows out (Fig. 11.20), because it thereby exerts a net force on the particle (which is, at the moment of loosening from the particle, equal to the opposite of the drag force exerted by the particle on the grain boundary) due to the grain-boundary tension σ_GB: the moving grain boundary exerts a force, per unit length junction grain-boundary/particle surface, in the positive, vertical y-direction, proportional to σ_GB (i.e. σ_GB cosψ). The total force is obtained by multiplying with the length of the (circular) junction of grain boundary and particle surface (i.e. 2πr_p cosθ). It is concluded that the total force to be exerted by the boundary to free itself from the particle is proportional with σ_GBr_p.
  考虑一个与球形粒子相交的晶界。晶界在移动过程中受到驱动力（例如，见公式 ( 11.16)），为了使晶界脱离质点，晶界会向外弯曲（图 11.20）。20），因为它因此对粒子施加了一个净力（在从粒子松开的瞬间，等于粒子对晶界施加的阻力的反方向），这是由于晶界张力 σ _GB ：移动的晶界在垂直 y 方向的正方向上施加了一个单位长度交界处晶界/粒子表面的力，与 σ _GB 成比例（即 σ _GB cosψ）。将总力与晶界和颗粒表面（圆形）交界处的长度相乘即可得到（即 2πr _p cosθ）。由此可以得出结论，晶界要从粒子中释放出来所要施加的总力与 σ _GB r _p 成正比。

  Fig. 11.20 图 11.20

  

  Illustration of grain-boundary pinning by a spherical, second-phase particle. The grain boundary experiences a driving force to move. It looses itself from the particle by bowing out, because it thereby exerts a net force on the particle. The force exerted on the particle equals the product of the component of σ_GB acting in the direction of the y-axis, σ_GBcosψ, multiplied with the length of the (circular) junction of grain boundary and particle surface, 2πr_pcosθ
  球形第二相颗粒对晶界钉扎的说明。晶界受到驱动力而移动。由于它对粒子施加了一个净力，因此会弓起而脱离粒子。施加在粒子上的力等于沿 y 轴方向作用的 σ _GB 的分量 σ _GB cosψ 乘以晶界和粒子表面（圆形）交界处的长度 2πr _p cosθ 的乘积。

  If the volume fraction of second-phase particles equals φ_p, it follows for the number of (spherical) particles per unit of volume, N_p: N_p = φ_p/((4/3)πr${}_{\mathrm{p}}^3$). The particles intersecting the (macroscopically planar) grain-boundary are located in a volume defined by planes parallel to the grain boundary and located at distances r_p above and below it. Hence, per unit area grain boundary, there are 2r_pN_p = 2φ_p/((4/3)πr${}_{\mathrm{p}}^2$) intersecting particles.
  如果第二相颗粒的体积分数等于 φ _p ，则单位体积的（球形）颗粒数 N _p ：N _p = φ _p /((4/3)πr ${}_{\mathrm{p}}^3$ )。与（宏观平面）晶界相交的粒子位于一个由平行于晶界的平面所定义的体积中，该体积位于晶界上下距离 r _p 。因此，单位面积晶界上有 2r _p N _p = 2φ _p /((4/3)πr ${}_{\mathrm{p}}^2$ ) 相交的粒子。

  Thus, it follows from the above treatment that the force (pressure) to be exerted by a grain boundary per unit area, to free itself from the pinning particles, is proportional with σ_GBφ_p/r_p. This corresponds with an energy barrier to overcome per mole material swept by the moving boundary, ∆G_pin, given by:
  因此，根据上述处理方法可以得出，单位面积上的晶界为摆脱钉销颗粒而施加的力（压力）与 σ _GB φ _p /r _p 成正比。这相当于移动边界每摩尔材料要克服的能量障碍 ∆G _pin ，其公式为：

  $$\mathrm{\Delta }{G}_{\text{pin}}=\frac{\left(c^{\mathrm{\prime }}{V}_{\text{m}}\right){\text{σ}}_{\text{GB}}{\text{φ}}_{\text{p}}}{r_{\text{p}}}$$

  (11.20)

  with $c^{\mathrm{\prime }}$ as a constant (e.g. of value 3/2). A consideration of this type is originally due to Zener; one also speaks of “Zener drag” or “Zener pinning” (cf. Nes et al. 1985).
  $c^{\mathrm{\prime }}$ 作为常数（例如值为 3/2）。这种类型的考虑最初是由齐纳提出的；也有人称之为 "齐纳阻力 "或 "齐纳引脚"（参见 Nes 等人，1985 年）。

  The net driving force for grain growth now follows from Eqs. (11.16) and (11.20):
  根据公式 ( 11.16) 和 ( 11.20) 可以得出谷物生长的净驱动力：

  $$-\mathrm{\Delta }G=\frac{\left({cV}_{\text{m}}\right){\text{σ}}_{\text{GB}}}{\text{<}r\text{>}}-\frac{\left(c^{\mathrm{\prime }}{V}_{\text{m}}\right){\text{σ}}_{\text{GB}}{\text{φ}}_{\text{p}}}{r_{\text{p}}}$$

  (11.21)

  and (cf. Eq. (11.17)):
  和（参见公式 ( 11.17)）：

  $$v=\frac{d\text{<}r\text{>}}{d\text{t}}={MV}_m(\frac{c{\text{σ}}_{\text{GB}}}{\text{<}r\text{>}}-\frac{c^{\mathrm{\prime }}{\text{σ}}_{\text{GB}}{\text{φ}}_{\text{p}}}{r_{\text{p}}})$$

  (11.22)

  It follows from this equation that at the start of grain growth, a parabolic growth law is obeyed (cf. Eq. (11.18)), but upon continued growth, the growth rate diminishes and growth is no longer possible when <r> has become that large that −∆G according to Eq. (11.21) has become nil. The corresponding limiting value of <r>, <r>_final, follows from −∆G = 0 and thus (cf. Eq. (11.21)):
  从该方程可以看出，在晶粒开始生长时，会遵循抛物线生长规律（参见公式 ( 11.18)），但在继续生长时，生长速率会减小，当 大到公式 ( 11.21) 中的 -∆G 为零时，就不再可能生长了。 的相应极限值为 _final ，由-∆G = 0 得出，因此（参见公式 ( 11.21)）：

  $$\text{<}r{\text{>}}_{\text{final }}=\left(\frac{c}{c^{\mathrm{\prime }}}\right).\left(\frac{r_{\text{p}}}{{\text{φ}}_{\text{p}}}\right)$$

  (11.23)

  where again (cf. discussion Eq. (11.18)), the (average) radius of curvature r_final is equated with the average grain size (equivalent grain radius). Practical values of c/$c^{\mathrm{\prime }}$ are in the range 1/3–1/2.
  其中（参见公式 ( 11.18) 的讨论），（平均）曲率半径 r _final 等同于平均晶粒尺寸（等效晶粒半径）。c/ $c^{\mathrm{\prime }}$ 的实际值范围为 1/3-1/2。

• (ii) Effect of surfaces. For a thin layer (or a fibre), the thickness of the layer (the diameter of the fibre) can be that small that the grain size becomes of the order of the layer thickness (fibre diameter). The tendency to reduce the grain-boundary energy in the system provides an explanation for the tendency for grain boundaries in thin layers (fibres) to be oriented perpendicular to the surface. Moreover, thermal grooving for grain boundaries intersecting the surface, as discussed below Eq. (11.14), in order to establish a balancing of the surface tension and the grain-boundary tension at the junction of surface and grain boundary, is compatible with a perpendicular orienting of the grain boundary with respect to the surface (cf. the symmetry of the geometry of the case discussed in Eq. (11.14) and see Fig. 11.13). Such surface grooves are a barrier for grain-boundary migration, i.e. a force has to be exerted by the grain boundary that is about to migrate. This parallels the discussion on “Zener drag”, i.e. the pinning effect of second-phase particles (see above). As a result, a limiting lateral grain size for the grains at the surface occurs, which follows from the net driving force being nil (cf. derivation of Eq. (11.23)). In practice, the (limiting) lateral grain size in thin films, composed of columnar grains traversing the thin film, is about two to three times the layer thickness.
  表面的影响。对于薄层（或纤维）来说，薄层的厚度（纤维直径）可以非常小，以至于晶粒大小与薄层厚度（纤维直径）相当。这种降低系统中晶界能量的趋势解释了为什么薄层（纤维）中的晶界倾向于垂直于表面。此外，为了在表面和晶界的交界处建立表面张力和晶界张力的平衡，在与表面相交的晶界上进行热开槽（如下面公式 ( 11.14) 所讨论的那样）与晶界垂直于表面的取向是一致的（参见公式 ( 11.14) 所讨论的情况的几何对称性和图 11.13）。这种表面凹槽是晶界迁移的障碍，也就是说，即将迁移的晶界必须施加一种力。这与关于 "齐纳阻力 "的讨论相似，即第二相颗粒的钉扎效应（见上文）。因此，表面晶粒的横向尺寸会达到极限，这是因为净驱动力为零（参见公式 ( 11.23) 的推导）。实际上，薄膜中由柱状晶粒组成的（极限）横向晶粒尺寸大约是层厚度的两到三倍。

• (iii) Effect of solute atoms. Solute atoms can influence the mobility of grain boundaries. The energy of a solute atom at the grain boundary is generally different from the energy of the solute atom in the bulk of the grain, as a direct consequence of the difference in the state of bonding (difference in the local atomic arrangement). A solute may thus be attracted to the boundary (and thereby energy is released) or it may be repelled from the boundary (it costs energy to move the solute atom from the bulk to the boundary). If the solute is attracted to the boundary, the solute concentration at the boundary is larger than in the bulk and one speaks of “solute segregation”. In this case, the solute atoms can induce a “solute drag” force on the moving boundary. As a result a limiting grain size occurs when the net driving force becomes nil (cf. derivation of Eq. (11.23)).
  溶质原子的影响。溶质原子会影响晶界的流动性。由于键合状态的不同（局部原子排列的不同），溶质原子在晶界的能量通常不同于溶质原子在晶粒主体的能量。因此，溶质可能被边界吸引（从而释放能量），也可能被边界排斥（将溶质原子从主体移至边界需要耗费能量）。如果溶质被吸引到边界上，边界上的溶质浓度就会大于体积中的浓度，这就是 "溶质分离"。在这种情况下，溶质原子会在移动的边界上产生 "溶质阻力"。因此，当净驱动力为零时，就会出现极限晶粒大小（参见公式 ( 11.23) 的推导）。

11.3.4 Abnormal Grain Growth
11.3.4 谷物生长异常

Restriction of the mobility of grain boundaries to only a small number of grains causes these grains to grow, by consuming the other, surrounding grains with virtually immobile grain boundaries, and become very large (in case of metals the size of these grains can easily become of the order of a centimetre). The inhomogeneous nature of the process and the (attempted) description of its kinetics in a way analogous to recrystallization (cf. Sect. 11.2.2) has led to the name “secondary recrystallization”, but, as indicated at the beginning of Sect. 11.3, the driving force for abnormal grain growth is of different origin and much smaller than the (already modest; see discussion of Eq. (11.2)) driving force of recrystalllization. Other names used, which speak for themselves, are “exaggerated grain growth” and “discontinuous grain growth”.
由于晶界的流动性只局限于少数晶粒，这些晶粒通过吞噬周围其他晶界几乎不流动的晶粒而长大，并变得非常大（在金属中，这些晶粒的大小很容易达到一厘米）。这一过程的非均质性以及以类似再结晶的方式对其动力学的（尝试性）描述（参见第 11.2.2 节）导致了 "二次再结晶 "这一名称的出现，但正如第 11.3 节开头所指出的，异常晶粒生长的驱动力来源不同，而且比再结晶的驱动力（已经很小，参见公式 ( 11.2) 的讨论）小得多。其他名称不言自明，如 "夸张晶粒长大 "和 "不连续晶粒长大"。

The normal sequence of events upon annealing a deformed material is: recovery, recrystallization, normal grain growth and abnormal grain growth, but overlapping of these processes can occur (cf. the introduction of this Chap. 11).
变形材料退火后的正常顺序是：恢复、再结晶、正常晶粒生长和异常晶粒生长，但这些过程也可能发生重叠（参见本章导言 11）。

The kinetic Eqs. (11.18a) and (11.19) pertain to the change of the average grain radius (grain size) during normal grain growth, i.e. taking place rather uniformly throughout the specimen. If the treatment is focussed on the growth behaviour of only a single grain, in the assembly of grains constituting the specimen, growth of this single grain is governed by the release of energy due to the elimination of the grain boundaries of the surrounding grains, which are consumed, and the counteracting cost of energy due to the increase of grain-boundary area (and thus energy) of the growing grain. As a result, it can be shown that the grain-boundary velocity of the single grain growing into its (static) surroundings is given by:
动力学公式 ( 11.18a) 和 ( 11.19) 涉及正常晶粒生长过程中平均晶粒半径（晶粒尺寸）的变化，即在整个试样中均匀发生。如果只处理构成试样的晶粒集合体中单个晶粒的生长行为，则该单个晶粒的生长受以下因素的支配：由于周围晶粒的晶界消除而消耗的能量释放，以及由于生长晶粒的晶界面积增加（从而消耗能量）而产生的能量消耗。因此，单个晶粒向其周围（静态）生长时的晶界速度由以下公式给出：

$$v=\frac{dr}{dt}=M{V}_m(\frac{c^{\mathrm{\prime }\mathrm{\prime }}<{\text{σ}}_{\mathrm{G}\mathrm{B}}>}{<r>}-\frac{c{\text{σ}}_{\mathrm{G}\mathrm{B}}}{r})$$

(11.24)

where <r> and <σ_GB> represent the average grain size and average grain-boundary tension of the static grains and r and and σ_GB indicate the grain size and grain-boundary tension of the growing grain, and where for spherical grains¹³ c = 2 and $c^{\mathrm{\prime }\mathrm{\prime }}$  = 3/2. Hence, growth of this single grain can occur if
其中 和 <σ _GB > 表示静态晶粒的平均晶粒大小和平均晶界张力，r 和 σ _GB 表示生长晶粒的晶粒大小和晶界张力，对于球形晶粒， ¹³ c = 2 和 $c^{\mathrm{\prime }\mathrm{\prime }}$ = 3/2。因此，单个晶粒的生长可以在以下条件下发生

$$\frac{r}{<r>}>\left(\frac{c}{c^{\mathrm{\prime }\mathrm{\prime }}}\right)\cdot \left(\frac{{\text{σ}}_{GB}}{<{\text{σ}}_{GB}>}\right)$$

(11.25a)

which for spherical grains¹³ and σ_GB =  <σ_GB> leads to
对于球形晶粒 ¹³ 和 σ _GB = <σ _GB >，可得出

$$\frac{r}{<r>}>\frac{4}{3}$$

(11.25b)

Equation (11.25) provides the criterion to be fulfilled in order that a single grain of (effective) radius r can grow into a static surrounding assembly of grains of average (effective) radius <r> . A well-known consequence of the result indicated by Eq. (11.25b) is that large grains grow at the expense of small grains.
公式 ( 11.25) 提供了单个（有效）半径为 r 的晶粒可以长成由平均（有效）半径为<r> 的晶粒组成的静态周围集合体所必须满足的标准。等式 ( 11.25b) 所示结果的一个众所周知的后果是，大颗粒的生长以牺牲小颗粒为代价。

So far, the occurrence of abnormal grain growth has not been dealt with by the above treatment. After normal grain growth a more or less uniform grain size occurs in the specimen. Thus, to explain abnormal grain growth, additional effects have to be considered.
迄今为止，上述处理方法尚未解决晶粒异常生长的问题。正常晶粒生长后，试样中会出现或多或少均匀的晶粒大小。因此，要解释异常晶粒长大，必须考虑额外的影响。

• (i) Effect of second-phase particles. In the presence of a volume fraction φ_p of (spherical) second-phase particles of radius r_p, the grain-boundary velocity of a single grain growing into its static surrounding grains is given by (cf. Eqs. (11.24) and (11.22)):
  第二相颗粒的影响。在半径为 r _p 的（球形）第二相颗粒的体积分数 φ _p 存在的情况下，单个晶粒长入其周围静止晶粒的晶界速度为（参见公式 ( 11.24) 和 ( 11.22)）：

  $$v=\frac{dr}{dt}=M{V}_m(\frac{c^{\mathrm{\prime }\mathrm{\prime }}<{\text{σ}}_{GB}>}{<r>}-\frac{c{\text{σ}}_{GB}}{r}-\frac{c^{\mathrm{\prime }}{\text{σ}}_{GB}{\text{φ}}_p}{r_p})$$

  (11.26)

  The consequence of Eq. (11.26) for individual grains in the specimen, upon consideration of the inequality $c^{\mathrm{\prime }\mathrm{\prime }}$  <σ_GB> / <r> − cσ_GB/r − $c^{\mathrm{\prime }}$ σ_GB φ_p/r_p > 0, is that some of them (the larger ones) can grow and others (the smaller ones) cannot. Hence, the grain-size distribution becomes wider during normal grain growth. Normal grain growth in the presence of second-phase particles is inhibited at a final average grain size given by Eq. (11.23). In the end only the largest grains fulfil the inequality indicated, and thus the occurrence of abnormal grain growth in the presence of second-phase particles may be understood.
  考虑到不等式 $c^{\mathrm{\prime }\mathrm{\prime }}$ <σ _GB > / - cσ _GB /r - $c^{\mathrm{\prime }}$ σ _GB φ _p /r _p > 0，公式 ( 11.26) 对试样中单个晶粒的结果是，其中一些（较大的晶粒）可以生长，而另一些（较小的晶粒）则不能生长。因此，在正常晶粒生长过程中，晶粒尺寸分布变得更宽。在存在第二相颗粒的情况下，正常晶粒生长会在公式 ( 11.23) 给出的最终平均晶粒尺寸处受到抑制。最终，只有最大的晶粒才符合所显示的不等式，因此可以理解在存在第二相颗粒的情况下发生的异常晶粒生长。

• (ii) Effect of surfaces. Evidently, for grains adjacent to the surface, a driving force for (lateral) growth occurs if the surface energy of the grain is lower than those of the surrounding grains at the surface. Clearly, this can be a very important effect for in particular thin films. For this case of abnormal grain growth three contributions to the driving force can be indicated: the decrease of surface energy and the decrease of grain-boundary energy (area) are two contributions driving abnormal growth, whereas the third contribution due to the pinning by surface grooves (see under (ii) at the end of Sect. 11.3.3) opposes the lateral growth of the growing grain. Such surface energy-driven abnormal grain growth obviously is associated with the development of specific crystallographic textures: for example, in case of f.c.c. metals abnormal growth is in particular observed for grains with {111} or {100} planes at the surface. Surface energy-driven abnormal grain growth can lead to grain growth with grain-boundary movement in directions opposite to those which would be expected on the basis of grain-boundary curvature-driven grain growth (cf. Sect. 11.3.2). As a special feature the effect of the outer atmosphere on the surface energy should be mentioned as a means to influence the developing crystallographic texture. Evidently, the occurrence and control of abnormal grain growth in thin films is of great importance to the microelectronic industry.
  表面的影响。显而易见，对于邻近表面的晶粒来说，如果晶粒的表面能低于表面周围晶粒的表面能，就会产生（横向）生长的驱动力。很明显，这对某些薄膜来说是非常重要的影响。对于这种异常晶粒生长的情况，可以指出驱动力的三个贡献：表面能的降低和晶粒边界能（面积）的降低是驱动异常生长的两个贡献，而由于表面沟槽的针刺作用（见第 11.3.3 节末尾的 (ii) 小节）而产生的第三个贡献则会阻碍生长晶粒的横向生长。这种表面能驱动的异常晶粒生长显然与特定晶体学纹理的发展有关：例如，在 f.c.c.金属中，表面具有{111}或{100}平面的晶粒尤其会出现异常生长。表面能驱动的异常晶粒生长可导致晶粒生长，其晶界运动方向与基于晶界曲率驱动晶粒生长的预期方向相反（参见第 11.3.2 节）。外层大气对表面能量的影响是影响晶体纹理发展的一个重要因素。显而易见，薄膜中异常晶粒生长的发生和控制对微电子工业非常重要。

• (iii) Effect of texture. As indicated in Sect. 11.3.3, the distribution of the type of grain boundaries in the specimen, and thus the crystallographic texture, influences the rate of (average) normal grain growth. Neighbouring grains of similar orientation are separated by low-angle grain boundaries of low grain-boundary tension (energy) which corresponds with a relatively low driving force for (normal) grain growth. Now, consider the presence of a grain of crystal orientation strikingly different from that pertaining to the crystallographic texture (dominating) component. This grain will generally have high-angle, high energy grain boundaries with neighbours compatible with the crystallographic texture component. Consequently, this grain may grow into its neighbours at a stage where the majority of the grains, belonging to the crystallographic texture component, have stopped their (normal) growth (note the similarity of the here discussed mechanism for abnormal grain growth with the growth of subgrains at a high-angle grain boundary as a mechanism for initiating primary recrystallization; cf. Sect. 11.2.1). This mechanism for abnormal grain growth will be the more prominent the stronger and sharper the crystallographic texture is, since the misorientation of the grains belonging to the crystallographic texture component is the smaller the more outspoken the crystallographic texture is.
  纹理的影响。如第 11.3.3 节所述，试样中晶界类型的分布以及晶体学纹理会影响（平均）正常晶粒的生长速度。取向相似的相邻晶粒由低角度晶界分隔，晶界张力（能量）较低，这相当于（正常）晶粒生长的驱动力相对较低。现在，考虑存在与晶体纹理（主导）成分相关的晶体取向截然不同的晶粒。这种晶粒通常具有高角度、高能量的晶界，其相邻晶界与晶体学纹理成分兼容。因此，这种晶粒可能会在属于结晶纹理成分的大多数晶粒停止（正常）生长的阶段向其相邻晶粒生长（注意此处讨论的异常晶粒生长机制与高角度晶粒边界上的亚晶粒生长作为启动原生再结晶机制的相似性；参见第 11.2.1 节）。晶粒异常生长的这一机制在结晶纹理越强、越锋利的情况下就越突出，因为结晶纹理越明显，属于结晶纹理成分的晶粒的错向就越小。

11.3.5 Particle Coarsening; Ostwald Ripening
11.3.5 颗粒粗化；奥斯特瓦尔德熟化

Consider a two-component (A and B) system (A-rich), which, at the temperature, pressure and composition considered, in equilibrium is constituted of two phases: the α phase (A-rich), which is the matrix, and the β phase (B-rich), which is finely dispersed as particles in the matrix, upon precipitation from the supersaturated solid solution (see Fig. 9.1 and its discussion). Even if the compositions of matrix and precipitate particles would satisfy the prescription given by the phase diagram for the “bulk” materials (but see below), genuine equilibrium has not been attained: the occurrence of many α/β interfaces (interphase boundaries) of variable curvatures (e.g. the β phase consists of a dispersion of spheres of variable size) provides the possibility of decrease of energy by letting the larger β phase particles (of larger radii of curvature) grow at the expense of the smaller β phase particles (of smaller radii of curvature) which thereby dissolve. This process of particle coarsening is often denoted as “Ostwald ripening”. Because of the similarity in origin of the driving forces of the particle coarsening process and of the process of grain growth of a homogenous material, dealt with above in this Sect. 11.3, i.e. decrease of interfacial area/interface energy, particle coarsening is considered here as well.
考虑一个双组分（A 和 B）体系（富 A），在所考虑的温度、压力和成分条件下，该体系在平衡状态下由两相构成：α 相（富 A），即基体；β 相（富 B），从过饱和固溶体中析出后，以微粒形式分散在基体中（见图 9.1 及其讨论）。即使基体和沉淀颗粒的成分符合 "块状 "材料相图的规定（但见下文），也没有达到真正的平衡：出现了许多曲率可变的 α/β 界面（相间边界）（如 β 相）。例如，β 相由大小不一的球体分散而成），这就为能量的降低提供了可能，即让较大的 β 相颗粒（曲率半径较大）生长，而让较小的β 相颗粒（曲率半径较小）溶解。这种颗粒粗化过程通常被称为 "奥斯特瓦尔德熟化"。由于颗粒粗化过程的驱动力与上文第 11.3 节所述的均质材料晶粒生长过程的驱动力（即界面面积/界面能的减少）的起源相似，因此这里也考虑颗粒粗化。

The pressure induced on a β phase particle in the α matrix by a curved α/β interface (of concave nature from the point of view of the β phase particle; cf. footnote 9) raises the Gibbs energy of the β phase particle, by an amount 2σ_α/β/r, with σ_α/β as the interface tension and r as the radius of the β phase particle taken as a sphere (cf. Eq. (11.16)). This increase of the energy of the β phase particle, due to the curvature of the α/β interface of specific interfacial tension, is called the “Gibbs–Thomson effect” or “capillary effect”.
弯曲的 α/β 界面（从 β 相粒子的角度看具有凹面性质；参见脚注 9）对 α 矩阵中的β 相粒子产生的压力使 β 相粒子的吉布斯能增加了 2σ _α/β /r，其中 σ _α/β 为界面张力，r 为作为球体的 β 相粒子的半径（参见公式 ( 11.16)）。由于具有特定界面张力的 α/β 界面的曲率而导致的 β 相粒子能量的增加被称为 "吉布斯-汤姆森效应 "或 "毛细管效应"。

The Gibbs–Thomson effect has an important consequence: the local solubility of B in the surrounding α matrix depends on the radius of curvature of the α/β interface and thus directly on the size of the β phase particle if the β phase particle is a sphere: the local solubility of B in the surrounding α matrix is the larger, the smaller r. It can be shown that this effect becomes important for nanosized β phase particles (i.e. r < 100 nm; cf. Sect. 12.14.2). As a consequence, concentration gradients occur in the α matrix containing a dispersion of β phase particles (of spherical shape and) of different sizes. The energy of the system thereby is decreased by a solute (B) flux in the matrix from the small β phase particles to the large β phase particles, i.e. from larger to smaller B concentration in the α matrix (Fig. 11.21). As a result, the small β phase particles become smaller and disappear eventually and the larger β phase particles grow.
吉布斯-汤姆森效应有一个重要的结果：B 在周围 α 基质中的局部溶解度取决于 α/β 界面的曲率半径，因此如果 β 相粒子是球体，则直接取决于 β 相粒子的大小：B 在周围 α 基质中的局部溶解度越大，r 越小。因此，在含有不同大小的 β 相颗粒（球形）的 α 矩阵中会出现浓度梯度。由于基质中的溶质（B）从小粒 β 相颗粒流向大粒 β 相颗粒，即 α 基质中的 B 浓度从大到小（图 11.21），从而降低了系统的能量。因此，小的β相颗粒变小并最终消失，而大的β相颗粒变大。

Fig. 11.21 图 11.21

(Precipitate-)Particle coarsening of second-phase, β particles (B-rich) in the matrix of parent phase α (A-rich) of the two-component system A-B. The Gibbs–Thomson effect causes the local solubility of B in the matrix (α) to be larger at the α/β interface for a small β phase particle (small radius of curvature of the particle/matrix interface) than at the α/β interface for a large β phase particle (large radius of curvature of the particle/matrix interface). As a consequence a net flux of B occurs in the matrix from small β phase particles to large β phase particles: the larger β phase particles will grow at the expense of the smaller β phase particles, i.e. coarsening, also called Ostwald ripening, takes place
(A-B 双组分体系母相 α（富含 A）基质中第二相 β 颗粒（富含 B）的（沉淀-）颗粒粗化。Gibbs-Thomson 效应使 B 在基质（α）中的局部溶解度在小 β 相粒子的 α/β 界面（粒子/基质界面曲率半径小）比在大 β 相粒子的 α/β 界面（粒子/基质界面曲率半径大）大。因此，基质中会出现从小 β 相颗粒到大β 相颗粒的 B 净流量：大β 相颗粒的增长将以牺牲小β 相颗粒为代价，即发生粗化（也称为奥斯特瓦尔德熟化）。

Adopting volume diffusion of B in the α matrix as rate determining process, the kinetics of β phase particle coarsening is often described by the Lifshitz–Slyozov–Wagner (1961) equation:
如果把 B 在 α 矩阵中的体积扩散作为速率决定过程，β 相颗粒粗化的动力学通常用 Lifshitz-Slyozov-Wagner （1961）方程来描述：

$$\text{<}r{\text{>}}^3-\text{<}{r}_0{>}^3=\text{const}\cdot t$$

(11.27)

with <r> and <r₀> as the β phase particle radii at t and t = 0, respectively (cf. Eqs. (11.18a) and (11.19)). The constant in this equation contains the product of the volume diffusion coefficient of B in the α matrix and the solubility of B in the α matrix for r infinitely large. Therefore the β phase particle coarsening can be strongly temperature dependent, as both the volume diffusion coefficient and the solubility of B in the α matrix generally strongly increase with temperature.
和 <r ₀ > 分别为 t 和 t = 0 时的β相粒子半径（参见式 ( 11.18a) 和 ( 11.19)）。该方程中的常数包含 B 在 α 矩阵中的体积扩散系数与 B 在 α 矩阵中的溶解度的乘积，当 r 无穷大时。因此，由于 B 在 α 矩阵中的体积扩散系数和溶解度通常会随着温度的升高而增大，β 相粒子的粗化可能与温度密切相关。