A mathematical model for the solute drag effect on recrystallization

The model for the solute drag effect in phase transformations has been applied to recrystallization, i.e., moving grain boundaries. In this model, the total driving force is dissipated by the interfacial energy, the finite interfacial mobility, the solute drag in boundaries, and diffusion in the matrix ahead of the interface, of which all are taken into account consistently. The effects of the Gibbs energy of segregation and the diffusivity of impurity atoms in boundaries were investigated. The results show that the Gibbs energy of segregation mainly affects the critical composition at which the drastic change in the boundary velocity appears, and the diffusivity of impurity atoms in boundaries mainly affects the velocity reduced by the solute drag effect. In other words, the Gibbs energy of segregation and the diffusivity of impurity atoms in boundaries can be evaluated from experimental data by means of the present model. This model was applied to the Al-Mg system, and the Gibbs energy of segregation and the diffusivity of Mg in boundaries were evaluated from experimental data. The evaluated Gibbs energy of segregation agrees with the estimate based on elastic energy considerations. The diffusivity estimated from this model is smaller than that measured along the grain boundary.     

镁在高温合金中的晶界效应和作用机制

偏聚到晶界的镁原子可将原子错配度较小的其它溶质原子从晶界驱逐到晶内点阵或晶界相中，并可增强间隙原子向晶界的偏聚；镁原子可进入晶界相的单胞中，从而促使晶界相球化，并降低其稳定性。     

The non-equilibrium segregation of boron during the recrystalization of Nb-treated HSLA steels

The effect of boron segregation on recrystallization was investigated in two Nb and two Nb-free steels. The results indicate that the addition of B to a Nb steel retards the recrystallization of austenite after high temperature deformation to a significant degree. By contrast, the effect of B when added alone is not of commercial importance. At deformation temperatures above 950°C, the influence of boron is mainly due to grain boundary segregation and solute drag; below 900°C, the retardation results from the precipitation of Nb carbonitride. During isothermal holding, there are two types of non-equilibrium segregation on boundaries. Strain-induced segregation occurs on the original boundaries; it increases and attains a maximum rapidly, then decreases and disappears after 60 s of holding at 1000°C. A second type of non-equilibrium segregation occurs on newly formed boundaries during recrystallization. It develops quickly as the new boundaries are formed and begin to move, and can persist until grain impingement occurs and the motion is halted. The enhanced effect of B in the presence of Nb is attributed to the formation of (Nb, B) complexes, which appear to increase the solute drag force and decrease the boundary velocity.     

Sulfur segregation to grain boundaries in Ni3Al and Ni3(AI,Ti) alloys

The segregation of S to grain boundaries in Ni₃Al and Ni₃(Al, Ti) has been studied using Auger electron spectroscopy. The S concentration at the grain boundaries decreases more slowly with increasing temperature than would be predicted by segregation models based on a single solute binding energy to the grain boundaries. This behavior, which can be interpreted as an increase in the effective solute binding energy for a grain boundary as a function of temperature, is consistent with predictions of a model based on the existence of a spectrum of solute binding energies for grain boundaries. Formerly Professor and Head of the Department of Metallurgical Engineering, Michigan Technological University     

The transformation phenomenon in Fe-Mo-C alloys: A solute drag approach

A recently modified solute drag model, taking into account the interfacial segregation of alloy elements, is applied to the decomposition of austenite into ferrite in Fe-Mo-C alloys. The calculations are carried out with assessed thermodynamic and kinetic data stored in databases. The effect of the Gibbs energy of segregation is discussed in connection with published experimental observations of the bay in the time-temperature-transformation (TTT) curve for initiation of transformation. The solute draglike effect suggested in the literature is discussed in comparison with the present quantitative calculation.     

低作用应力引起的溶质在晶界非平衡偏聚或贫化

提出了关于应力时效（高温及作用应力）引起溶质原子的晶界非平衡偏聚或贫化机理的一个新模型。此模型基于如下假设：晶界相对于完整晶体而言在强度上是弱化区。当多晶体受到一个在弹性范围内的作用应力时,晶界优先变形。若压应力作用时,晶界作为激发射空位;若拉应力作用时,晶界作为阱吸收空位,应力引起的过饱和空位与溶质原子形成复合体,此复合体在基体中的扩散速率远高于溶质原子。作用应力对晶界溶质偏聚或贫化的影响,是由复合体的扩散和溶质原子的反向扩散之间的平衡所决定。根据此模型,应力时效过程中存在一个临界时间,在此时间,应力引起的晶界偏聚或贫化程度将达到一个极大值。此模型已由ShinodaT等所作的磷在钢中的应力时效实验和MisraR D K所作的硫在钢中的应力时效实验结果所证实。     

A Model for Static Recrystallization with Simultaneous Precipitation and Solute Drag

In the present work, we introduce a state parameter-based microstructure evolution model, which incorporates the effect of solute atoms and precipitates on recrystallization kinetics. The model accounts for local precipitate coarsening at grain boundaries, which promotes an average grain boundary movement even if the Zener pinning force exceeds the driving force for recrystallization. The impact of solute drag on the grain boundary mobility as well as simultaneous precipitation is discussed in detail. The model is validated on experimental data on recrystallization in V-micro-alloyed steel, where excellent agreement is achieved.     

Effect of Rare-Earth Additions on the Texture of Wrought Magnesium Alloys: The Role of Grain Boundary Segregation

Magnesium alloys that contain certain rare-earth (RE) additions are known to have improved formability and this can be partly attributed to the different texture they display after recrystallization. Previous experimental work has identified segregation of RE to grain boundaries and dislocations as being potentially important in producing this change in behavior. In the present paper, two classical models (Langmuir–McClean and Cahn–Lücke–Stüwe) are used to explore the likely effect of RE additions on grain boundary solute concentration and drag. It is demonstrated that a wide range of RE elements are predicted to segregate strongly to grain boundaries due to the large atomic size misfit with magnesium. The maximum level of segregation is produced for elements such as Y or Gd that combine a high misfit and high bulk solubility. Segregated Y is predicted to produce a solute drag pressure on migrating boundaries several orders of magnitude greater than that obtained by Al or Zn additions. It is demonstrated that while this drag is predicted to be insufficient to strongly retard static recrystallization under typical annealing conditions, it is expected to suppress dynamic recrystallization by any mechanism requiring boundary migration.     

Grain boundary damping in substitutional alloys

A study was made of the grain boundary damping characteristics of twelve Cu?Ni alloys, chosen to cover the entire range of solid solutions. The height, temperature, and activation energy of the solute peak was used to investigate solute segregation to grain boundaries. A criterion was developed to determine solute build up in the grain boundaries, and when applied to other alloy systems, it verified the well known equilibrium segregation concept due to Mclean.¹² The qualitative model for the low temperature peak in fcc pure metals¹⁰ was modified to incorporate segregation effects and thus explain the origin of the solute peak in dilute alloys. The model explained, without modification, the behavior of a concentrated alloy, as long as it was a complete solid solution.     

The Effect of Solute Atoms on Aluminum Grain Boundary Sliding at Elevated Temperature

Grain boundary sliding (GBS) is an important deformation mechanism for elevated temperature forming processes. Molecular dynamics simulations are used to investigate the effect of solute atoms in near grain boundaries (GBs) on the sliding of Al bicrystals at 750 K (477 °C). The threshold stress for GBS is computed for a variety of GBs with different structures and energies. Without solute atoms, low-energy GBs tend to exhibit significantly less sliding than high-energy GBs. Simulation results show that elements which tend to phase segregate from Al, such as Si, can enhance GBS in high-energy GBs by weakening Al bonds and by increasing atomic mobility. In comparison, intermetallic forming elements, such as Mg, will form immobile Mg-Al clusters, decrease diffusivity, and inhibit GBS.     

Hydrogen induced grain boundary fracture in high purity nickel and its alloys—Enhanced hydrogen diffusion along grain boundaries

Embrittlement of nickel by solute hydrogen is usually accompanied by a change in fracture mode from ductile rupture to an intergranular mode. When hydrogen is supplied at the external surface, the kinetics of this embrittlement is shown to be more rapid than can be accounted for by lattice diffusion of hydrogen. The embrittlement kinetics are shown to be consistent with hydrogen diffusion along grain boundaries and the grain boundary diffusivity is derived over the temperature range 274–314 K. The effects of carbon and sulfur grain boundary segregation on the kinetics of grain boundary embrittlement were also investigated. Segregation of carbon decreases the extent of embrittlement while sulfur segregation increases the amount of embrittlement relative to that observed in pure nickel. These effects are interpreted in terms of the effects of segregated solutes on hydrogen grain boundary diffusivity and on the critical hydrogen concentration for intergranular fracture.     

Segregation and wetting transition at dislocations

We investigate solute segregation and wetting transition at dislocations and the corresponding drag effect on dislocation glide using a continuum model developed previously for grain boundary and based on gradient thermodynamics. The dislocation core structure and stress field are described by the newly developed phase field model. This study differs from much previous work because it takes into account not only the long-range elastic interactions but the short-range chemical interactions between solute atoms and dislocation core as well as among solute atoms themselves. The latter leads to the prediction of a wetting transition at the dislocation core with respect to varying temperature, solute concentration, or dislocation velocity. The transition temperatures obtained during heating and cooling are different from each other, leading to a hysteresis loop in the solute concentration-temperature plot and the solute concentration-velocity plot. These predictions could provide new insights into the phenomena of sharp yield point drop and strain aging observed in metal alloys. This article is based on a presentation made in the “Hillert Symposium on Thermodynamics & Kinetics of Migrating Interfaces in Steels and Other Complex Alloys,” December 2–3, 2004, organized by The Royal Institute of Technology in Stockholm, Sweden.     

Role of grain boundary segregation in diffusional creep

The high temperature deformation of polycrystalline materials by the stress directed flow of vacancies is now a well established creep mechanism which operates in two temperature regimes: high temperature, or Nabarro-Herring creep, in which lattice diffusion is rate determining, and low temperature, or Coble creep, in which grain boundary diffusion predominates. Basic studies have been conducted mostly with pure metals for which there exists in general a good correspondence between predicted and observed behavior. Multicomponent engineering alloys will normally experience, as part of their processing history or service lives, the segregation enrichment of interfaces such as grain boundaries by species present in solid solution. The aim of this paper is to evaluate the experimental information and to explore the manner in which this segregation affects the principal forms of diffusional creep. Cases of retarded Herring-Nabarro creep are analyzed in terms of the efficacy of grain boundaries as sources and sinks for vacancies: strongly bound segregant atoms at grain boundaries affect the mobility of defects and hence control the operation of vacancy sources. Recently, observations have been made on the effect of strongly segregating solutes on grain boundary diffusivity. Such behavior influences Coble creep rates, producing in general a retardation. Here we assess the magnitude of the effect induced by various surface active species on grain boundary diffusivity and consequently on Coble creep; predictions show that in general, small amounts of highly surface active impurities induce a remarkable inhibition of this form of creep.     

Carbide Formation and Growth Kinetics Enhanced by Tensile Stress and Elevated Temperature Intergranular Cracking in 2.25Cr-1.5W Heat-Resistant Steels

The carbide growth kinetics enhanced in grain interiors is mainly due to the increased diffusivity in the direction normal to the tensile stress. The accelerated kinetics at the grain boundaries normal to the tensile stress is due to the increased grain boundary energy and the widened grain boundary path producing the increased diffusivity in the direction normal to the tensile stress. A strong segregation behavior of impurities to the grain boundary carbide interfaces follows the enhanced grain boundary carbide growth kinetics.     

Liquid- solid interface migration at grain boundary regions during transient liquid phase brazing

A two-dimensional (2-D) finite difference model has been used to analyze the effect of grain boundary regions on the migration of the liquid-solid interface during transient liquid phase (TLP) brazing of Ni with Ni-11 wt pct P filler metal. This work has been carried out to explain the differences observed between actual and calculated completion times for isothermal solid- ification during TLP brazing and the faster isothermal solidification rates when brazing fine- grained nickel-base material. Modeling considers the situation where the grain boundary intersects the liquid-solid interface at right angles. Four factors are considered in addition to solute diffusion in solid and liquid phases,viz., (1) high diffusivity at the grain boundary region, (2) the balance between the grain boundary energy and the liquid-solid interfacial energy, (3) the interfacial energy due to the curvature of the liquid-solid interface, and (4) diffusional flow along the liquid-solid interface (produced by the gradient of solute chemical potential resulting from factors (2) and (3)). Increased solute diffusivity at the grain boundary region has a neg- ligible effect on migration of the liquid-solid interface in the bulk region and shifts the interface at the grain boundary region in a direction opposite that observed in actual brazed samples. On the other hand, when factors (2) through (4) above are taken into account, the liquid-solid interface in the region of the grain boundary is displaced in the same direction as in the ex- perimental results and liquid penetration comparable with the experimental results occurs at the grain boundary region. Factors (2) through (4) accelerate the isothermal solidification process in the bulk region in accordance with actual experimental test results. Formerly Visiting Scientists Formerly Visiting Scientists     

On the Effect of Atoms in Solid Solution on Grain Growth Kinetics

The discrepancy between the classical grain growth law in high purity metals (grain size \( D \propto t^{1/2} \) ) and experimental measurements has long been a subject of debate. It is generally believed that a time growth exponent less than 1/2 is due to small amounts of impurity atoms in solid solution even in high purity metals. The present authors have recently developed a new approach to solute drag based on solute pinning of grain boundaries, which turns out to be mathematically simpler than the classic theory for solute drag. This new approach has been combined with a simple parametric law for the growth of the mean grain size to simulate the growth kinetics in dilute solid solution metals. Experimental grain growth curves in the cases of aluminum, iron, and lead containing small amounts of impurities have been well accounted for.     

Grain boundary structure and energetics

The subject of grain boundary structure and energetics is reviewed, with the progress over forty years measured in terms of describing boundaries with greater misorientation and with greater crystallographic complexity. Approaches based on continuum and lattice geometric concepts are discussed, with the emphasis placed on delineating the limitations inherent in each approach. Grain boundary energy measurements of relative and absolute energies are then described briefly. Experimental techniques stressed are those which provide accurate and incisive information on energy as a function of misorientation. The theory of heterophase dislocations is then discussed in terms of predicting equilibrium states at grain boundaries. New variational formulations of constrained and unconstrained equilibrium are presented, which yield differential equations for microscopic equilibrium in terms of the chemical and elastic energy density distributions along a grain boundary. The precise equilibrium form of the strain accommodation centers periodically distributed along a boundary is shown to be a unique eigenfunction of the constrained equilibrium equation. The influence of a second component on heterophase grain boundaries is presented along the lines first shown by Voronkov. Both the grain boundary " surface excess " and the stress-modified diffusion of the solute are considered. Finally, the influence of boundary structure on the segregation, Gibbs absorption, and diffusion of solutes is shown to be significant even in the case of relatively simple boundaries of modest (<15 deg) misorientation. This paper is based on a presentation made at a symposium on “Fundamentals of Grain Boundary Segregation” held at the Niagara Falls Meeting of The Metallurgical Society of AIME, September 21, 1976, under the sponsorship of the Physical Metallurgy Committee.     

Grain boundary migration and recrystallization in Ni induced by C diffusion

When Ni specimens annealed at 1200°C for 24 h are heat-treated at 900, 1000, and 1100°C in a carburizing atmosphere, some of the grain boundaries migrate and new grains form at the specimen edges and corners. Some of the migrating grain boundaries reverse their directions. The specimen heat treated in vacuum do not show any change of the grain boundary structure. The diffusion of C is thus shown to induce the grain boundary migration and recrystallization in Ni at three temperatures. The observed grain boundary migration behaviour does not appear to be consistent with the diffusional coherency strain theory because of the high lattice diffusivity of C. The new grains appear to nucleate because of the lattice parameter change induced by the diffusion of both substitutional and interstitial solute atoms.     

On the chemistry of grain boundary segregation and grain boundary fracture

This paper considers the problem of impurity segregation in metals and the effect of these impurities on grain boundary cohesion. The primary goal of this paper is to provide a physical model that will allow us to think about these two processes. We describe both of them in chemical terms. Segregation is treated as a distribution of a solute between two phases. In this way, it is a typical example of heterogeneous equilibrium. We also consider the various driving forces for solute segregation and find that the correlation between decreased solubility and increased segregation, first proposed by Hondros and Seah,^[9] is still an adequate one. We introduce the discussion of grain boundary fracture by pointing out that as the impurity enters the boundary, it establishes chemical bonds with the structural units of the boundary. The segregated boundary can then be thought of as a string of molecular units with bonds of different types. Some of these bonds will be weaker than others, and they will be the ones that eventually fracture when a stress is applied. We consider the cause of these weak bonds and suggest that the primary reason for them is the transfer of electronic charge from the metal atoms to the impurity, as proposed in previous work.^[3] However, some of the ideas in the earlier models should be amended based on new results obtained from the quantum mechanical analysis of bonding in metals presented by McAdon and Goddard.^[10,11] We also suggest that intergranular brittleness of intermetallic compounds such as Ni₃Al, which occurs in the absence of impurity segregation, can be explained by the charge distribution present at the grain boundary. Finally, we provide a critique of other models that have been used to describe grain boundary fracture and segregation. This paper is based on a presentation made in the symposium “Interface Science and Engineering” presented during the 1988 World Materials Congress and the TMS Fall Meeting, Chicago, IL, September 26–29, 1988, under the auspices of the ASM-MSD Surfaces and Interfaces Committee and the TMS Electronic Device Materials Committee.