The structure of kinks at dislocation interphase boundaries and their role in boundary migration—II. Kinetic analyses including kink motion

A kinetic model was formulated to account for the influence of the density of orientation, thermal and intersection kinks on the velocity of dislocation ledges during precipitate growth. The relative contributions of each mechanism to the ledge velocity are characterized and the kinetic model is compared with existing experimental data on dislocation ledge motion in the Al-Ag system.     

Computer simulation of morphological changes of grain boundary precipitates growing by the ledge mechanism

A previously developed computer model was modified to simulate the growth of grain boundary precipitates which grow by the ledge mechanism. The ledges were assumed to be nucleated in the grain boundary region at constant, parabolically decreasing, and random rates and to grow under the control of volume diffusion of solute to or from the riser of ledges. At lower under coolings at which the motion of individual ledges is slow, late-nucleated ledges soon catch up with first-nucleated ones, and precipitates tend to extend along the grain boundary: the overall precipitate shape is essentially that of a grain boundary allotriomorph. At larger undercoolings, first-nucleated ledges move fast to form a protuberance similar to Widmanstätten sideplates, while late-nucleated ones stay near the grain boundary region. The transition of precipitate shape from one to the other occurs in a very narrow range of supersaturation. The results are compared with various characteristics of the growth of proeutectoid ferrite allotriomorphs and sideplates in Fe-C alloys documented in the literature.     

Atomic mechanisms of diffusional nucleation and growth and comparisons with their counterparts in shear transformations

An integrated overview is presented of a viewpoint on the present understanding of nucleation and growth mechanisms in both diffusional and shear (martensitic) transformations. Special emphasis is placed on the roles played by the anisotropy of interphase boundary structure and energy and also upon elastic shear strain energy in both types of transformation. Even though diffusional nucleation is based on random statistical fluctuations, use of the time reversal principle shows that interfacial energy anisotropy leads to accurately reproducible orientation relationships and hence to partially or fully coherent boundaries, even when nucleation at a grain boundary requires an irrational orientation relationship to obtain. Since the fully coherent boundary areas separating most linear misfit compensating defects are wholly immobile during diffusional growth because of the improbability of moving substitutional atoms even temporarily into interstitial sites under conditions normally encountered, partially and fully coherent interphase boundaries should be immovable without the intervention of growth ledges. These ledges, however, must be heavily kinked and usually irregular in both spacing and path if they, too, are not to be similarly trapped. On the other hand, the large shear strain energy usually associated with martensite requires that its formation be initiated through a process which avoids the activation barrier associated with nucleation, perhaps by the Olson-Cohen matrix dislocation rearrangement mechanism. During growth, certain ledges on martensite plates serve as transformation dislocations and perform the crystal structure change (Bain strain). However, the terraces between these ledges in martensite (unlike those present during diffusional growth) are also mobile during non-fcc/hcp transformations; glissile dislocations on these terraces perform the lattice invariant deformation. Growth ledges operative during both diffusional and shear growth probably migrate by means of kink mechanisms. However, diffusional kinks appear to be nonconservative and sessile (and therefore resist immediate transmission of elastic shear strain energy), whereas those associated with martensitic growth must be conservative and glissile (and fully transmit such strain energy). The broad faces of both diffusionally and martensitically formed plates contain an invariant line, as emphasized by Dahmen and Weatherly. However, in the diffusional case, minimization of growth ledge formation kinetics seems to be the main role thereby played, whereas in martensitic growth, the main purpose of such an interface is to minimize elastic shear strain energy. The latter minimization requires that martensite forms as plates (or perhaps as laths) enclosed by a pair of invariant line-containing interfaces. During diffusional transformations, on the other hand, other interfaces at which growth ledge formation kinetics are not too much faster than those at the invariant line interface can also comprise a significant portion of the interfacial area, thereby leading to the formation of other, quite different morphologies, such as intragranular idiomorphs and grain boundary allotriomorphs. Critical problems remaining unsolved in diffusional transformations include calculation of critical nucleus shapes when the crystal structures of the two phases are significantly different, highly accurate calculation of the energies of the interphase boundaries thus formed, and direct observation of atomic scale kinks on the risers of growth ledges by means of a yet-to-be-invented three-dimensional (3-D) atomic-resolution form of transmission electron microscopy. Experimental identification and characterization of transformation dislocations and experimental testing of “nucleation” mechanisms are now of special importance in fundamental studies of martensitic transformations.     

On a stochastic theory of grain growth

A theory of isothermal grain growth in polycrystalline solids, which treats grain growth as a statistical or stochastic process, is presented. In this treatment deterministic equation for the rate of grain growth is made stochastic by the addition of a “noise” term. The noise or fluctuations are used to model the effect of complex topologically connected structure of the specimen on grain boundary motion, in addition to such motion directed by surface tension forces. Such considerations lead to a second order partial differential equation (Fokker-Planck equation) for the grain size distribution. Many of the major attributes of grain growth are shown to be a natural consequence of this equation. The solution obtained for this equation is a modified form of Rayleigh distribution which in many respects is similar of log normal distribution. Grain size distribution is also obtained from independent statistical consideration and is shown to be approximately log normal. Extension of the mathematical analysis to the case of Ostwald ripening is indicated.     

Effect of size distribution on the kinetics of normal grain growth and of particle coarsening

A simple model is presented to describe the kinetics of normal grain growth. The final expression incorporates a factor which depends on the grain size distribution but no assumption is made on the distribution function. A similar expression is also derived for particle coarsening.     

Morphology and Growth Kinetics of Straight and Kinked Tin Whiskers

Time-lapse SEM studies of Sn whiskers were conducted to estimate growth kinetics and document whisker morphologies. For straight whiskers, growth rates of 3 to 4 microns per day were measured at room temperature. Two types of kinked whiskers were observed. For Type A kinks, the original growth segment spatial orientation remains unchanged, there are no other changes in morphology or diameter, and growth continues. For Type B kinks, the spatial orientation of the original segment changes and it appears that the whisker bends over. Whiskers with Type B kinks show changes in morphology and diameter at the base, indicating grain boundary motion in the film, which eliminates the conditions suitable for long-term whisker growth. To estimate the errors in the whisker growth measurements, a technique is presented to correct for SEM projection effects. With this technique, the actual growth angles and lengths of a large number of whiskers were collected. It was found that most whiskers grow at moderate or shallow angles with respect to the surface; few straight whiskers grow nearly normal to the surface. In addition, there is no simple correlation between growth angles and lengths for whiskers observed over an approximate 2-year period.     

A Three-Dimensional Cellular Automata Model Coupling Energy and Curvature-Driven Mechanisms for Austenitic Grain Growth

A 3D cellular automata model is used to simulate normal austenitic grain growth in this study. The proposed model considers both the curvature- and thermodynamics-driven mechanisms of growth. The 3D grain growth kinetics shows good agreement with the Beck equation. Moreover, the growth exponent and grain size distribution calculated by the proposed model coincides well with experimental and simulation results from other researchers. A linear relationship is found between the average relative grain size and the grain face number. More specifically, for average relative grain sizes exceeding ~0.5, the number of faces increases linearly with relative grain size. For average relative grain sizes <0.5, this relationship is changed. Results simulated by the proposed model are translated to physical meaning by adjusting the actual temperature, space, and time for austenitic grain growth. The calibrated results are found to be in agreement with the simulation results from other research as well as the experimental results. By means of calibration of the proposed model, we can reliably predict the grain size in actual grain growth.     

Junction disclinations in plastically deformed crystals

It is shown that defects of rotation type form in polycrystals during the plastic deformation process at junctions and kinks of grain boundaries. Their geometric properties are analysed and it is proved that in general these defects cannot be reduced to disclinations, but represent more complex linear-planar entities. The analysis shows that dipoles of defects of rotation type appear at ledges of boundaries. Such ledges are considered as dipole sources of the partial disclinations.     

The role of ledges in the proeutectoid ferrite and proeutectoid cementite reactions in steel

The influence of interphase boundary ledges on the growth and morphology of proeutectoid ferrite and proeutectoid cementite precipitates in steel is examined. After reviewing current theoretical treatments of growth by the ledge mechanism, investigations that clearly document the presence and motion of ledges with thermionic emission electron microscopy (THEEM) and transmission electron microscopy (TEM) are reviewed. A fundamental distinction is made between two types of ledges: (1) mobile growth ledges whose lateral migration displaces the inter-phase boundary and (2) misfit-compensating structural ledges. Both types of ledges strongly affect the apparent habit plane and aspect ratio of precipitate plates. Agreement between measured growth rates of proeutectoid ferrite and cementite (plates and allotriomorphs) and predicted growth kinetics assuming volume diffusion-controlled migration of ledge-free disordered boundaries is shown to be consistently poor. Physically realistic growth models should incorporate the ledge mechanism. More accurate comparisons of the growth models with experimental data will need to account for observed ledge heights, interledge spacings, and ledge velocities. In this vein, the sluggish growth kinetics of cementite allotriomorphs observed in an Fe-C alloy are shown to be quantitatively consistent with a strong increase in interledge spacing with time. This paper is based on a presentation made in the symposium “The Role of Ledges in Phase Transformations” presented as part of the 1989 Fall Meeting of TMS-MSD, October 1–5, 1989, in Indianapolis, IN, under the auspices of the Phase Transformations Committee of the Materials Science Division, ASM INTERNATIONAL.     

Numerical simulation of Zn coating solidification

A numerical model, which simulates nucleation and growth of Zn grains, has been developed in order to describe quantitatively the solidification of Zn coatings during the hot-dipping process. The inputs of the model are the nucleation distribution, which has been measured by electron backscattered diffraction (EBSD), and the dendritic growth kinetics, calculated with an analytical model of a parabolic dendrite tip modified to account for the interactions with the coating interfaces. The model predicts the shapes of the grain envelopes as a function of the grain orientation and the texture induced by growth. Three types of grain envelopes have been evidenced, depending on the angle between the c-axis and the normal to the coating plane. Moreover, it has been shown that growth reinforces the already existing {00.1} nucleation texture, in good agreement with experimental data. The model also predicts the cooling curve, including recalescence, and the grain size. Thus, it is used to describe the effects of Pb additions on solidification. In particular, it has been shown that Pb increases the nucleation undercooling and strongly decreases the density of active nuclei, thus resulting in a much larger grain size.     

Thickening of grain-boundary α allotriomorphs in a Ti-Cr alloy by multiple sets of ledges

A computer model is developed to simulate the growth of grain-boundary allotriomorphs having more than one set of growth ledges at their interfaces. The growth is controlled by the volume diffusion of solute to or from the riser of a ledge. The time dependence of the growth rate of two orthogonal sets of ledges is found to be somewhat different from that of a single set of ledges. However, the operation of multiple sets of ledges is unlikely to alter significantly the growth kinetics of grain-boundary allotriomorphs from those predicted from the disordered growth theory, except at small ledge spacings or at short reaction times. Faster growth kinetics of proeutectoid α allotriomorphs than those of either planar or ellipsoidal disordered boundaries which have been reported in a Ti-6.6 at. pet Cr alloy are not likely to be accounted for with the heights and spacings of double sets of ledges actually observed on the interfaces of allotriomorphs. Hence, the grain-and interphase-boundary diffusion-assisted growth of precipitates, (rejector plate mechanism, RPM) appears to be operative during the growth of a allotriomorphs, as previously proposed on the basis of growth-rate measurements. This paper is based on a presentation made in the symposium “The Role of Ledges in Phase Transformations” presented as part of the 1989 Fall Meeting of TMS-MSD, October 1–5, 1989, in Indianapolis, IN, under the auspices of the Phase Transformations Committee of the Materials Science Division, ASM INTERNATIONAL.     

正常晶粒长大数值模拟

从统计角度建立了正常晶粒长大的三维模型,该模型假定在奥氏体晶粒长大过程中,每个粒子体积的变化不仅与最近邻的晶粒有关,而且与系统中所有晶粒均有关,界面能的降低是晶粒长大的驱动力。通过模型计算可获得任意时刻的晶粒尺寸分布以及平均晶粒大小。第二相粒子的阻碍作用被引入到模型。温度升高,第二相粒子溶解,对正常晶粒长大的阻碍作用减小。初始晶粒尺寸分布采用对数正态分布。     

Modeling diffusional growth during austenite decomposition to ferrite in polycrystalline FeC alloys

A grain-boundary-nucleated, diffusional growth model of austenite decomposition to proeutectoid ferrite is developed for polycrystalline iron-carbon alloys. The diffusion equation is solved under restricted diffusion conditions using the quasi-static method and employing local thermodynamic equilibrium at the disordered austenite:ferrite interface. Decomposition kinetics for a model polycrystalline material consisting of a log-normal distribution of spherical grains are calculated numerically. Effects of temperature, overall carbon concentration, volume change, austenite grain size and carbon buildup in the centers of the austenite grains are included in the treatment. A scaling factor is deduced that enables the effect of austenite grain size on transformation kinetics to be characterized provided kinetic information is available for one grain size. Experiments carried out on a laboratory steel verified the applicability of the scaling factor, Also, partial I-T and C-T diagrams can be computed from the model and sample calculations are presented for an iron + 0.036 wt% carbon steel.     

Simulation of subgrain growth by subgrain rotation: A one-dimensional model

It has been known for many years that subgrain growth can occur by subgrain rotation. However, the kinetics of such a growth process have not been worked out. In this investigation, a simple model for such a process, namely, a single row of subgrains, has been studied by computer simulation. The rate of growth of the mean subgrain size and the variation in the distribution of the subgrain misorientations with time have been established. The differences between subgrain growth by rotation and normal grain growth are also discussed.     

Weibull尺寸分布晶粒组织演变的仿真研究

采用一种改进的Potts模型Monte Carlo算法,对具有Weibull尺寸分布(参数β=3.47)的晶粒组织进行了3D正常晶粒长大过程的仿真研究.仿真结果表明:整个晶粒长大过程遵循抛物线长大规律,晶粒生长指数为0.501,非常接近理论值0.5.晶粒长大过程可分为过渡阶段与准稳态长大两个阶段.Weibull尺寸分布参数β由过渡阶段的3.47逐渐演变为准稳态阶段的2.76,准稳态阶段晶粒尺寸分布参数保持β=2.76不变.晶粒的平均面数〈f〉随仿真时间的增加而增大,在准稳态阶段后期趋近于稳定数值.晶粒面数分布为Lognormal分布,最高频率面数f为10,个体晶粒面数范围为3~43.     

The thermal and metallurgical state of steel strip during hot rolling: Part III. Microstructural evolution

A mathematical model has been developed to compute the changes in the austenite grain size during rolling in a hot-strip mill. The heat-transfer model described in the first of this series of papers has been employed to calculate the temperature distribution through the thickness which serves as a basis for the microstructure model. Single-and double-hit compression tests have been conducted at temperatures of 900 °C, 850°C, 950 °C, and 875 °C on 0.34 and 0.05 pct carbon steels to determine the degree of recrystallization by metallographic evaluation of quenched samples and by measuring the magnitude of fractional softening. The Institut de Recherches de la Sidérurgie Francaise, (IRSID) Saint Germain-en-Laye, France equation has been found to yield the best characterization of the observed recrystallization kinetics. The equations representing static recrystallization kinetics, recrystallized grain size, and grain growth kinetics have been incorporated in the model. The principle of additivity has been invoked to permit application of the isothermal recrystallization data to the nonisothermal cooling conditions. The model has been validated by comparing predicted austenite grain sizes with measurements made on samples quenched after one to four passes of rolling on the CANMET pilot mill. The austenite grain size evolution during rolling of a 0.34 pct carbon steel on Stelco’s Lake Erie Works (LEW) hot-strip mill has been computed with the aid of the model. The grain size decreased from an initial value of 180μm to 35μm in the first pass due to the high reduction of 46 pct. The changes in austenite grain size in subsequent passes were found to be small in comparison because of the lower per pass reductions. It has been shown that the equation employed to represent grain growth kinetics in the interstand region has a significant influence on the computed final grain size. Altering the rolling schedule had a negligible influence on the final grain size for a given finished gage. A 200°C increase in entry temperature to the mill resulted in a 20μm increase in final grain size, which is significant. This can be attributed to increased grain growth at the higher temperature. Formerly Graduate Student, The Centre for Metallurgical Process Engineering, The University of British Columbia Metallurgical transactions a     

晶粒正常长大过程的元胞自动机模拟

 采用正六边形网格划分法,最近邻邻居关系以及概率性转换规则的二维元胞自动机（CA）法,对再结晶完成后的晶粒长大行为进行模拟。并利用正方形区域内圆形晶粒的收缩过程对模型的正确性进行验证。在此基础上,分析了微观结构的演化过程,给出了晶粒长大的动力学和界面能的变化,同时讨论了晶粒尺寸和晶粒边数的分布特点。结果表明：晶粒形态的演化过程遵循晶粒正常长大理论,晶粒尺寸和边数的分布具有时间不变性的特点,界面能的变化遵循能量降低原则。表明本模型能够较好的模拟晶粒的正常长大过程。     

Abnormal grain growth in materials containing particles

The behaviour of an abnormally large grain in a matrix containing stable particles in which normal grain growth is initially possible is examined by means of a theoretical model. It is found that an abnormally large grain will initially decrease its size relative to the critical radius of the distribution. As normal grain growth continues and the grain growth rate decreases due to the increasing action of the particles this situation is reversed and the abnormally large grain starts to increase its size relative to the critical radius. The model satisfactorily explains why the occurrence of severe abnormal grain growth in the presence of particles is often apparently “erratic”.     

Microstructural path concept applied to normal grain growth

Normal grain growth in solids is considered in the light of a microstructural path concept. The topological model first presented by Rhines and Craig is corrected and reformulated so as to include the Doherty sweep constant. The reformulated model predicts the experimentally observed linear time dependence for volumetric grain growth only when the average curvature per grain remains constant during growth. A geometrical path function for normal grain growth relating average microstructural properties on a per grain basis is proposed and compared with experimental observations on aluminum and titanium. By combining the geometrical path function with the topological model, further comprehension of normal grain growth kinetics results, specifically the role of the average grain shape.