Influence of Crystallography on Ferrite Nucleation at Austenite Grain-Boundary Faces,Edges, and Corners in a Co-15Fe Alloy

The nucleation of ferrite precipitates at austenite grain faces, edges (triple lines), and corners (quadruple points) was studied in a Co-15Fe alloy in which the matrix phase was retained upon cooling to room temperature by serial sectioning coupled with electron backscatter diffraction analysis. Nearly half of the edges and corners were vacant at an undercooling of 60 K from the γ/(α + γ) boundary where the precipitation occurred significantly at grain faces. A significant proportion of precipitates had Kurdjumov–Sachs (K–S) and to a lesser extent Nishiyama–Wassermann (N–W) orientation relationships with more than one grain at all boundary sites. Vacant edges and corners were readily observed, of which the misorientations of matrix grain boundaries would permit a precipitate to have a specific orientation relationship with multiple grains. Small differences in the nucleation activation energy among the grain faces, edges, and corners may lend support to a view proposed from experiments of nucleation in Fe-C base alloys that ferrite nuclei are more or less surrounded by low-energy facets of α/γ phase boundary.     

Topological Characteristics of Two-Dimensional Grain Growth-Simulation and Analysis

A two-dimensional (2-D) grain growth simulation, using a curvature-driven vertex model applied to an initially Monte Carlo (MC)–generated microstructure, has been developed for analysis of the topological characteristics of the process, including the rates of different topological events and the effect of these event frequencies on the evolving grain structure. Findings include a constant ratio of the number fraction of disappearing grains to the area fraction swept by the grain boundary of 4/3; constant fractions of boundary sweeping due to grain disappearance (~2 pct), boundary-switching events (~8 pct), and simple boundary motion (~90 pct); and a constant ratio of 1.34 boundary-switching events per grain disappearance. An affinity term was developed to describe the tendency of grains of different edge classes to contact each other or to be involved in different topological events, relative to random behavior. The highest and lowest edge classes, 3 and 12, respectively, exhibited the highest affinity for mutual contact, a 16 times random occurrence, but an affinity of ~0 for contact with themselves. Intermediate edge classes showed an affinity of ~1, random contact, with other classes or with themselves. Few-edged grains showed ~0 affinity for contacting a disappearing trigon or gaining an edge in an edge-switching event, but had a high affinity, approaching 6, for losing an edge in a switching event. Many-edged grains showed the opposite trends and intermediate edge classes showed a random or less tendency for participation in any topological event. It was shown statistically that the growth rate of individual grains is controlled solely by the edge class, with essentially no direct effect from the grain size. The evolution of the grain area and edge class distributions were monitored throughout the transition from transient to steady-state grain growth, with a steady state achieved after loss of approximately 1/3 of the initial grains. The steady-state values of the coefficients of variation (CVs) of the edge class and grain area distributions were ~0.2 and 0.7, respectively.     

Measurement of average grain volume and certain topological parameters by serial section analysis

Serial section analysis is suitable for the measurement of such topological parameters of a polycrystalline aggregate as number of grains, of grain faces, of grain edges and of grain corners in unit volume of the material. The number of grains in unit volume is of particular interest, because its reciprocal is the average grain volume, which is the only available shape-insensitive measure of grain size. An experimental approach to the serial section analysis of polycrystalline structures is developed.     

The spherical image concept applied to grain structures

The spherical image of the boundary of a microstructural feature is a map of the directions of its surface normals on the unit sphere of orientation. For closed surfaces it is a topological invariant. For curved faced polyhedra, as are found in grain structures or other tessellations of three dimensional space, the corners, edges and faces all contribute to the spherical image. The geometries of the spherical images of corners, edges and faces are described in this paper. These geometric concepts are applied to individual cells, the cell set, and the triple lines and quadruple points in the corresponding tessellation. Equations are derived relating the contributions of these spherical images to the number of faces on a single cell and to the average number of faces per grain for the structure. Application of these relations to grain structures governed by local capillarity equilibrium demonstrate that it is unlikely that the average number of faces per grain will deviate significantly from 13.4.     

Measurement of average grain volume and certain topological parameters by serial section analysis

Serial section analysis is suitable for the measurement of such topological parameters of a polycrystalline aggregate as number of grains, of grain faces, of grain edges and of grain corners in unit volume of the material. The number of grains in unit volume is of particular interest, because its reciprocal is the average grain volume, which is the only available shape-insensitive measure of grain size. An experimental approach to the serial section analysis of polycrystalline structures is developed.

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Formerly Candidate for the Ph.D at the University of Florida

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Formerly Candidate for the MS at the University of Florida.     

Transient behavior of diffusion-induced creep and creep rupture

Steady state solutions to three types of diffusion problems: creep, grain boundary sliding and intergranular crack growth, have been published in the literature. This paper considers, in detail, the events which occur between the time when the external stress is applied and the time when the steady state is eventually reached. The time constant of the transient has been calculated. It is shown how the grain boundary tractions change with time from the initial “elastic” configuration (when sliding has been elastically accommodated) to the steady state “diffusional” configuration (when the sliding rate is diffusionally accommodated). This requires infusion of excess grain boundary dislocations; the distribution of these dislocations is calculated. The results are applied to problems of diffusional creep, grain boundary sliding and intergranular crack growth.     

Relationship between grain

The relationship between the average mean grain boundary curvature,H, and the mean linear grain intercept, λ, has been experimentally determined for aluminum. The ratioH/λ ^{- 1} = 0.31 was found to remain constant throughout grain growth. This value yields a driving pressure for grain growth approximately three times smaller than that derived by conventional modeling and yields realistic results in models of particle- or pore-inhibited grain growth.     

Geometry of grain boundaries

The first part of this paper deals with the problem of describing and measuring microstructure in exact terms. The Euclidean parameters: volume, area, length, and angle can be measured and expressed only in terms of the total of each in unit volume. Average properties, such as average grain diameter, are accessible only through the topological parameters, specifically number in unit volume, which can be measured only by serial sectioning. The parameters which have been used to represent the concept of grain size are analyzed and shown in most cases to represent a function of grain boundary area. In a second section of this paper the geometric problem of plastic slip through a grain boundary is analyzed. A method is proposed by which all of the components of the deformation, as well as the crystallographic directions, can be manipulated simultaneously through the use of a stereographic projection. The third section of this paper is concerned with the geometry of grain growth. The polycrystalline state is described as a grain boundary network, which must respond to the requirements of surface tension. The several topological changes in a network which can contribute to grain growth are described.     

Grain boundary mobility during recrystallization of copper

Average grain boundary migration rates during recrystallization of cold-deformed copper were estimated from stereological measurements. In the same material, the instantaneous driving forces for boundary migration during recrystallization were calculated from calorimetric measurements of the release of the stored energy of cold work. The migration rate dependence on driving force was analyzed in the context of grain boundary migration rate theory, and within experimental error, a linear dependence was observed. The average mobility of grain boundaries migrating during recrystallization of cold-worked copper at 121°C was calculated to be 6.31×10⁻¹⁰ (m⁴ s⁻¹ MJ⁻¹). This result was found to be consistent with single boundary, curvature-driven grain boundary mobilities measured in copper at higher temperatures. It was also demonstrated that the average grain boundary mobility was reasonably within the expectation (order of magnitude uncertainty) of the Turnbull single process model of boundary migration with a process akin to grain boundary self-diffusivity as the rate-controlling atomic mechanism.     

Grain boundary mobility during recrystallization of copper

Average grain boundary migration rates during recrystallization of cold-deformed copper were estimated from stereological measurements. In the same material, the instantaneous driving forces for boundary migration during recrystallization were calculated from calorimetric measurements of the release of the stored energy of cold work. The migration rate dependence on driving force was analyzed in the context of grain boundary migration rate theory, and within experimental error, a linear dependence was observed. The average mobility of grain boundaries migrating during recrystallization of cold-worked copper at 121 °C was calculated to be 6.31×10⁻¹⁰ (m⁴ s⁻¹ MJ⁻¹). This result was found to be consistent with single boundary, curvature-driven grain boundary mobilities measured in copper at higher temperatures. It was also demonstrated that the average grain boundary mobility was reasonably within the expectation (order of magnitude uncertainty) of the Turnbull single process model of boundary migration with a process akin to grain boundary self-diffusivity as the rate-controlling atomic mechanism.     

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.     

Interpretation of the creep behavior of nanocrystalline Ni in terms of dislocation accommodated boundary sliding

Creep behavior and deformation-induced grain growth in electrodeposited (ED) nanocrystalline (nc) Ni with a grain size of about 20 nm were studied over more than five orders of magnitude of strain rate (10⁻⁹ s⁻¹ to 2×10⁻⁴ s⁻¹) at 393 K (0.23 T _m, where T _mis the melting point). In addition, the activation energy for the creep in ED nc-Ni was determined by using the temperature change procedure. The results show that the creep behavior of the material is characterized by (a) a stress exponent that increases continuously from about 4.5 to about 30 with increasing applied stress; (b) an apparent activation energy for creep in the range of 126 to 141 kJ/mol; (c) an activation volume of about 20 b ³ where b is the Burgers vector; and (d) a grain size that upon loading, grows, attaining a constant value once steady-state creep is approached. The mechanical characteristics cannot be accounted for by current deformation processes. Analysis of the creep data along with consideration of available information leads to the suggestion that the creep behavior of nc-Ni arises from a deformation process that is based on the concept of dislocation-accommodated boundary sliding. By quantitatively developing this concept, a rate-controlling deformation process is formulated. It is shown that the predictions of this process, are in good agreement with experimental results and trends.     

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.     

The evolution of grain boundary character during superplastic deformation of an Al-6 pct Cu-0.4 pct Zr alloy

The evolution of microstructure, texture, and microtexture in an Al-6 pct Cu-0.4 pct Zr alloy was studied during mechanical testing at 480 °C and a strain rate of 5·10⁻⁴ s⁻¹. The as-processed material had an elongated, banded microstructure and a deformation texture with orientation distribution along the β-orientation fiber. The true strain vs true stress curve exhibited three stages: I, II, and III. Work hardening occurred in stages I and III, whereas nearly steady-state behavior was observed in stage II. A bimodal distribution of boundary disorientation angles was evident in as-processed material and persisted into stage I, with peaks at 5–15 deg in the low-angle boundary (LAB) regime and at 45–60 deg in the high-angle boundary (HAB) regime. An increase in strain rate sensitivity coefficient, m, in stage I was accompanied by fragmentation of the initial microstructure, leading to the formation of new grains. During stage II the strain rate sensitivity coefficient, m, attained a value of 0.5, which is consistent with the onset of grain boundary sliding. In stage III, the texture and the grain boundary disorientation distribution became randomized while the m value decreased. Grain elongation and cavity formation at second-phase particles and along grain boundaries were seen in samples deformed to failure. The as-processed microstructure is described in terms of deformation banding, and a model for the evolution of such a structure during superplastic deformation is proposed.     

Effect of dihedral angle on the morphology of grains in a matrix phase

Minimum interface energy configurations of a uniformly intermixed grain-matrix aggregate are determined for various dihedral angles and matrix contents by numerical analysis of a model which consists of a rhombic dodecahedron grain in contact with matrix at its curved surfaces along truncated edges and corners. For dihedral angles, Φ, greater than 90 deg, the total interface energy,E, increases monotonically with the matrix volume fraction,V _m . For Φ= 0 deg,E decreases withV _m until the grains become spherical atV _m = 26 pct. For 0 deg Φ ≤ 75 deg,E vs V _m curves show the minima which represent the stable configurations to be obtained whenV _m can be freely varied. For Φ ≤ 60 deg, the matrix is always continuous along the grain edges. For Φ 75 deg, the matrix becomes separated at the grain corners below certain critical values ofV _m . The contiguity decreases monotonically withV _m . The slope ofE vs V _m curve is shown to be an effective pressure on the specimen surface, which represents the driving force for changing the grain configuration with a corresponding change ofV _m while keeping the grain volume constant. The implications of these results on solid state sintering, liquid phase sintering, and the penetration of liquid into liquid phase sintered alloys are discussed. Finally, the results of a previous analysis by Beere are shown to disagree with the present work for systems with low dihedral angles apparently because of inaccuracy in his calculation.     

Effect of the Degree of Prior Cold Work on the Grain Volume Distribution and the Rate of Grain Growth of Recrystallized Aluminum

The grain volume distribution of recrystallized aluminum, determined by separating and weighing the individual grains, has been found to be log normal, its spread in size being expressed by the standard deviation of the distribution (In σ_v), which remains constant during steady state grain growth. The value of In σ_v is established by the degree of cold working that precedes annealing, being smaller the greater the degree of cold work. Grain growth proceeds the more rapidly the larger the value of In σ_v that is, the smaller the degree of prior cold work. The distributions of the numbers of faces per grain and of edges per face are also log normal and are proportional to the grain volume distribution. Thus, the relative number of three-edged faces increases with In σ_v and accounts for the observed increase in the rate of grain growth.     

Low stress and superplastic creep behavior of Zn-22 Pct Al eutectoid alloy

The temperature, grain size, and stress dependence of steady-state strain-rates were investigated for Zn-22 pct Al eutectoid alloy using double shear type specimens. Tests were performed on specimens of grain size from 1.3 m to 3.7 μm over a range of temperature from 450 to 525 K. The applied stresses were in the range between 10⁻¹ and 4.0 x 10¹ MPa with the resulting true strain-rates ranging from 10⁻⁷ to 10⁻²s⁻¹. The alloy exhibited two distinct regions of constant stress dependence with stress exponents 1 and 2.2. The transition stresses between these two regions were between 3 × 10⁻¹ and 7 x 10⁻¹ MPa. The true activation energies associated with these regions were found to be 95.9 ±2.1 and 69.9 ±2.1 kJ/mol, respectively. The grains remained equiaxed following large deformation, although there is evidence of excessive grain growth and “grain clustering.” Specimens deformed in both regions showed few or no dislocations within the grains. There was no evidence of subgrain formation or dislocation pile-up at the grain boundaries. It is possible that dislocations found during deformation were annihilated when stress was withdrawn or lost during thin foil preparation. There was clear evidence of primary stage in both regions. However, it is suggested that whereas the primary stage in Region II is due to dislocation multiplication process, at low stresses the primary stage in Region I may be due to elastic bowing of the existing dislocations in the structure. At low stresses Nabarro-Herring diffusional creep was found to be the rate controlling mechanism. At intermediate stresses, superplastic creep was found to be the dominant mechanism. The transition between these two mechanisms, as well as Coble creep and the ranges of manifestation for superplastic creep, are analyzed and the importance of the preexponential term A, in the dimensionless relation:γκT/DGb = A(b/d) ^p(Τ/G)ⁿ and the role of the concurrent grain growth are emphasized. A. K. S. YU, formerly with the Materials Science Section, Department of Mechanical Engineering, University of California. An erratum to this article is available at .     

Discussion of grain growth in aluminum

The data of Rhines and Craig (RC) (Met Trans., 1974, vol. 5, pp. 413-25) are partially reinterpreted. For the first regime of grain growth, average grain volume is proportional to time, in accordance with RC, but for the second, it is proportional to the three-halves power of time. The structural factor σ is found to be constant only in the first regime, and its value is redetermined from the data to be about 2.65. Theoretical derivations of this value are also given and are found to be related to grain form (e.g. a tetrakaidecahedron). Grain boundary motion in the second regime is controlled by drag due to lattice diffusion of solute atoms from the boundary, and an activation energy of about 27 kcal/mole (113 kJ/mole) for this process is calculated from the data. This work was performed while the author was on sabbatical leave at the School of Applied Sciences, University of Sussex, England.     

Two-dimensional grain growth in rapidly solidified succinonitrile films

The kinetics and topological mechanisms of normal grain growth have been examined throughin situ dynamic studies on rapidly solidified succinonitrile (SCN). Thein situ studies allowed for continuous monitoring of the evolution of individual grains during growth. We have assessed the Mullins—Von Neumann topological grain growth law and the Burke—Turnbull parabolic rate law and have determined rate constants that describe grain growth. This work demonstrates that both laws are both obeyed globally and consistently. Thesein situ studies permit one to follow the unit operations associated with grain growth kinetics. This article demonstrates the usefulness of succinonitrile as a model analog system for studying grain growth. This article is based on a presentation made in the symposium “Fine Grains And Their Growth in Rapidly Solidified Materials,” TMS Materials Week ’93, Pittsburgh, PA, October 18–21, 1993.     

Simulation of Nucleation of Proeutectoid Ferrite at Austenite Grain Boundaries during Continuous Cooling

The nucleation kinetics of proeutectoid ferrite during continuous cooling in three Fe-C-Mn-Si steels, measured in-situ by three-dimensional X-ray diffraction microscope, are compared with numerical simulation that takes into account differences in the activation energy of nucleation among grain boundary faces, edges, and corners. The essential feature of ferrite nucleation in the 0.21 pct C steel, i.e., nucleation occurred just below Ae₃ and ceased at a small undercooling, is reproduced taking into account the site consumption, primarily at grain corners and overlap of solute diffusion fields in the grain boundary region or the matrix and assuming a very small or almost null activation energy of nucleation. In the 0.35 and 0.45 pct C steels, small activation energy, as reported by Offerman et al., was not unequivocally obtained because ferrite nucleation occurred at considerably large undercoolings, even below the paraequilibrium Ae₃ in these steels. The increasing rate of the observed particle number with decreasing temperature is considerably smaller than calculation. This article is based on a presentation given in the symposium entitled “Solid-State Nucleation and Critical Nuclei during First Order Diffusional Phase Transformations,” which occurred October 15–19, 2006 during the MS&T meeting in Cincinnati, Ohio under the auspices of the TMS/ASMI Phase Transformations Committee.

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