A new analytical approach to predict spacing selection in lamellar and rod eutectic systems

The Jackson and Hunt (JH) theory has been modified to relax the assumption of isothermal solid/liquid interface used in their treatment. Based on the predictions of this modified theory, the traditional definitions of regular and irregular eutectics are revised. For regular eutectics, the new model identifies a range of spacing within the limits defined by the minimum undercooling of the α and β phases. For the irregular Al-Si eutectic system, two different spacing selection mechanisms were identified: (1) for a particular growth rate, a nearly isothermal interface can be achieved at a unique minimum spacing λ _I ; (2) the average spacing (λ _av >λ _I ) is essentially dictated by the undercooling of the faceted phase. Based on the modified theoretical model, a semiempirical expression has been developed to account for the influence of the temperature gradient, which is dominant in the irregular Al-Si system. The behavior of the Fe-Fe₃C eutectic is also discussed. The theoretical calculations have been found to be in good agreement with the published experimental measurements.     

Eutectic Growth in Three Dimensions

Critical experimental studies have been carried out to examine the stability of eutectic morphology in three-dimensional (3-D) samples under diffusive growth conditions. By directionally solidifying capillary samples of the well-characterized Al-Cu eutectic alloy, it is shown that the observed minimum spacing agrees with the value predicted by the Jackson and Hunt (JH) model, but the range of stable spacing is reduced significantly in three dimensions. The ratio of the maximum to minimum eutectic spacing in three dimensions is found to be only 1.2 compared to the predicted value of 2.0 in two dimensions. The narrow range of stable spacing is shown to be due to the instabilities in the third dimension that forms when the local spacing becomes larger than some critical spacing value, which corresponds to the maximum stable spacing. A new mechanism of lamellar creation in the third dimension is observed in which lamella with a local spacing larger than the critical value becomes unstable and forms a sidewise perturbation that becomes enlarged at the leading front and then propagates parallel to the lamella to create a new lamella. Alternately, an array of sidewise perturbations form, which then coalesce at their leading fronts and then become detached from the parent lamella to form a new lamella. This article is based on a presentation made in the symposium entitled “Solidification Modeling and Microstructure Formation: In Honor of Prof. John Hunt,” which occurred March 13–15, 2006, during the TMS Spring Meeting in San Antonio, Texas, under the auspices of the TMS Materials Processing and Manufacturing Division, Solidification Committee.     

Interface structure of α-β brass two-phase bicrystals made by solid state diffusion couple method

The interface structure of the phase boundary in α-β brass two-phase bicrystals made by solid state diffusion was investigated by X-ray Laue method and electron microscopy. The macroscopic observation by X-ray Laue method snowed that the close-packed planes of both phases, (111)α and (110)β plane-matched at interface and the normal to the interface had no preferential orientation. Intrinsic interface dislocations were observed by electron microscopy as a set of regular fine lines with an equal spacing. The regular fine lines were interpreted by the plane-matching theory. The spacing and direction of a set of regular fine lines depended on the mismatch (rotation and spacing) between (111)α and (110)β.     

Behavior of Dilative Sand Interfaces in a Geotribology Framework

Frictional resistance along the exterior of an embedded structure or structural element develops through relative displacement at the interface. An understanding of how surface topography influences interface strength and deformation behavior is required to develop comprehensive interface models for soil-structure analyses, to develop interface design methods and for producing enhanced construction materials. This paper presents the results of an investigation to quantify the influence of surface topography on shear stress and volume change behavior of dilative granular material interface systems. The root spacing, asperity spacing, asperity height, and asperity angle of machined, idealized surfaces are systematically varied. Direct interface shear test results using Ottawa 20/30 sand and glass microbeads show that maximum interface efficiency for these materials is achieved for a asperity spacing to median grain diameter ratio between 1.0 and 3.0, and an asperity height to median grain diameter ratio greater than 0.9. An asperity angle of 50 degrees or greater yields maximum efficiency for any given asperity spacing or height. The results suggest that interface behavior is governed by predictable geometric and mechanical relationships that are applicable to more complex manufactured surfaces.     

A Numerical Finite Difference Model of Steady State Cellular and Dendritic Growth

A theoretical model is developed to treat the steady state growth of an array of cells and dendrites in a positive temperature gradient using finite difference techniques. The coupled three dimensional heat and solute flow equations have been solved in a ‘half-cell’ repeat unit with radial symmetry using the conservation boundary conditions at the solid-liquid interface. As, however, the problem is a free boundary one, there is an additional condition to be satisfied at the solid-liquid interface. Using this ‘equilibrium’ condition, self-consistent cell shapes have been obtained. Calculations have been carried out at several different growth velocities for fixed values of alloy content and liquid gradient. At each velocity self-consistent shapes have been found up to a maximum value of the cell spacing. The effect of surface energy and the kinetic coefficient on growth behavior has also been examined. Formerly Research Fellow, Department of Metallurgy, Oxford University This paper is based on a presentation made at the symposium “Establishment of Microstructural Spacing during Dendritic and Cooperative Growth” held at the annual meeting of the AIME in Atlanta, Georgia on March 7, 1983 under the joint sponsorship of the ASM-MSD Phase Transformations Committee and the TMS-AIME Solidification Committee.     

Numerical modeling of cellular/dendritic array growth: spacing and structure predictions

A numerical model of cellular and dendritic growth has been developed that can predict cellular and dendritic spacings, undercoolings, and the transition between structures. Fully self-consistent solutions are produced for axisymmetric interface shapes. An important feature of the model is that the spacing selection mechanism has been treated. A small, stable range of spacings is predicted for both cells and dendrites, and these agree well with experiment at both low and high velocities. By suitable nondimensionalization, relatively simple analytic expressions can be used to fit the numerical results. These expressions provide an insight into the cellular and dendritic growth processes and are useful for comparing theory with experiment. This article is based on a presentation made at the “Analysis and Modeling of Solidification” symposium as part of the 1994 Fall meeting of TMS in Rosemont, Illinois, October 2–6, 1994, under the auspices of the TMS Solidification Committee.     

高速线材吐丝机自平衡吐丝管设计

指出了高速线材吐丝管应满足的基本要求 ;提出了吐丝管曲线设计的理论和方法 ;构造了吐丝管曲线方程并说明了有关参数的确定思路 ,为吐丝机、吐丝管的国产化提供了理论依据 ,也为使用数控弯管机加工吐丝管提供了参考 ,具有重要的实际意义     

Eutectic interface configurations during melting

Experimental studies are presented on the melting of a lamellar eutectic structure in a transparent model alloy system. Steady-state morphologies have been characterized for different dissolution rates. Both coupled and noncoupled melting of eutectic are observed. A diffusive model of coupled eutectic melting has been developed and compared with the experimental results. Because the eutectic spacing is fixed, the system is shown to select the shape so that the average position of the interface for the two phases remains isothermal. A comparison of the predicted shapes with the experimental data shows that the higher melting point phase adjusts its curvature to equate the average temperature of the two phases when the experimental conditions are in the coupled melting regime. The conditions for a coupled melting to be unstable are described for which one of the phases extends further into the liquid. When this extension is significant, a spherodization of the leading phase is observed.     

Interlamellar Spacing in Directionally Solidified Eutectic Thin Films

Fault-free lamellar structures were grown in eutectic thin films of Pb-Sn, Cd-Pb, and Al-Al₂Cu. The 2 μ thick films were directionally solidified by either a scanning laser or a quartz iodine lamp. A thermal gradient at the solid-liquid interface was estimated to be 8000°C/cm for the laser heat source compared to 200°C/cm for the lamp. In each alloy the lamellar spacing was larger than values estimated from previous experiments with bulk material. Defects in the films were observed to generate V-shaped waves of bent plates that provide a mechanism for increasing the lamellar spacing. Computer-simulated microstructures for two dimensional lamellar growth were made by calculating the diffusion in the liquid ahead of an irregular lamellar structure assuming a planar interface, finding the shape of the solid-liquid interface, and finding the trajectories of the three phase junctions. The mechanism of two-dimensional eutectic growth was discussed. This paper is based on a presentation made at the symposium “Establishment of Microstructural Spacing during Dendritic and Cooperative Growth” held at the annual meeting of the AIME in Atlanta, Georgia on March 7, 1983 under the joint sponsorship of the ASM-MSD Phase Transformations Committee and the TMS-AIME Solidification Committee     

Modeling on Dendrite Growth of Medium Carbon Steel during Continuous Casting

Dendrite growth is an important phenomenon during steel solidification. In the current paper, a numerical method was used to analyse and calculate the dendrite tip radius, dendrite growth velocity, liquid concentration, temperature gradient, cooling rate, secondary dendrite arm spacing, and the dendrite tip temperature in front of the solid/liquid (S/L) interface for the solidification process of medium carbon steels during continuous casting. The current model was well validated by published models and measurement data. The results show that in the continuous casting process, the dendrite growth rate is dominated by the casting speed. Dendrite growth rate, liquid concentration at the S/L interface, temperature gradient and cooling rate decrease with proceeding solidification and solid shell thickness growth, while other parameters such as dendrite tip radius, secondary dendrite arm spacing, and dendrite tip temperature in front of the S/L interface become larger with solidification progress and solid shell thickness growth. Parametric investigations were carried out. The effects of the stability coefficient, temperature gradient and casting speed on the micro‐structural parameters were discussed. Under the same conditions, higher casting speed promotes coarser secondary dendrite arm spacing and enlarges the dendrite tip radius, while decreasing temperature gradient, reducing the dendrite growth rate and making the solute distribute more uniform.     

Modeling of rolling texture development in a ferritic chromium steel

The development of crystallographic texture during rolling of a ferritic chromium steel containing 11 pct Cr was examined experimentally as well as by polycrystal modeling at large strains (up to 90 pct thickness reduction). The initial shape of the grains was very much elongated in the direction of rolling. A strong rolling direction (RD) fiber (<110> parallel to the rolling direction) has been observed at large strains in the experiment. The Taylor viscoplastic model, the relaxed-constraints pancake model, and the self-consistent viscoplastic approach were employed to simulate the texture development. Strain hardening was accounted for by microscopic hardening laws, for which the parameters were obtained from uniaxial tensile tests. It has been found that among the three models considered, the self-consistent viscoplastic model (the version tuned to finite-element results) yielded the best agreement with the experimentally observed texture evolution. Strong effects of grain shape and hardening have been found. The pancake model was also able to reproduce the main characteristics of the texture because of the flattened initial grain shape.     

The mechanics of toughness development in ductile phase reinforced brittle matrix composites

This study is a numerical exercise to theoretically analyze toughening in brittle materials consisting of ductile reinforcements, on the basis of crack bridging by ductile particles. The effects of the particle constitute behavior, the shape of the stress-displacement law, the matrix fracture toughness and the overall elastic modulus of the composite on toughness have been illustrated by simple fracture mechanics calculations of self-consistent crack opening displacement profiles and crack bridging stress distributions. An approach to calculate the crack surface displacements, crack bridging stresses and stress intensity factors in any specimen geometry, using weight functions, is presented. Different types of idealized particle stress-displacement responses were used in the calculations. The opening characteristics of a bridge crack and the evolution of toughness for these types have been examined. The conditions for maximum bridging and toughness as well as the conditions at which a fully bridged crack transforms to a partially bridged one have been identified. The role of individual parameters used in constructing the stress-displacement law on composite toughness has been assessed. The toughness has been found to manifest from a complex interaction of the matrix fracture toughness, the composite modulus and the flow behavior of the ductile particle.     

A Review of the Data on the Interlamellar Spacing of Pearlite

After defining interlamellar spacing the various optical and electron optical methods for measuring spacing are outlined. It is clear for both isothermal and forced velocity transformation conditions that pearlite can grow at a constant velocity with a range of true spacings. The minimum true spacing and mean true spacing are not related by a constant factor, but this may vary from system to system and with temperature in a given system. The relationship between interlamellar spacing and temperature for isothermal growth conditions and between translation velocity and spacing for forced-velocity growth conditions is reviewed for a range of steels and nonferrous alloys. It is seen that the velocity-spacing relationship for the two modes of transformation is the same. For isothermal transformation a linear relationship between reciprocal spacing and temperature is generally observed, but for steels containing alloy additions there is little evidence of the predicted inflexion corresponding to a temperature at which alloy partitioning at the transformation front ceases. The lack of precise interfacial energy data makes it difficult to determine reliably the relationship between measured and critical spacings, although it seems likely to be in accord with the maximum growth rate or maximum rate of entropy production optimization criteria. This paper is based on a presentation made at the symposium “Establishment of Microstructural Spacing during Dendritic and Cooperative Growth” held at the annual meeting of the AIME in Atlanta, Georgia on March 7, 1983 under the joint sponsorship of the ASM-MSD Phase Transformations Committee and the TMS-AIME Solidification Committee.     

Overstability of lamellar eutectic growth below the minimum-undercooling spacing

We investigate the stability of lamellar eutectic growth by thin-sample directional solidification experiments and two-dimensional phase-field simulations. We find that lamellar patterns can be morphologically stable for spacings smaller than the minimum undercooling spacing λ _m . Key to this finding is the direct experimental measurement of the relationship between the front undercooling and spacing, which identifies λ _m independently of the Jackson and Hunt (JH) theory and of uncertainties of alloy parameters. This finding conflicts with the common belief that patterns with λ<λ _m should be unstable, which is based on the Jackson-Hunt-Cahn assumption that lamellae grow normal to the envelope of the front. Our simulation results reveal that lamellae also move parallel to this envelope to reduce spacing gradients, thereby weakly violating this assumption but strongly overstabilizing patterns for a range of spacing below λ _m that increases with G/V (temperature gradient to growth rate ratio). This range is much larger than predicted by previous stability analyses and can be significant for standard experimental conditions. An analytical expression is obtained phenomenologically, which predicts well the variation of the smallest stable spacing with G/V. We also present results that shed light on the history-dependent selection and long-time evolution of the experimentally observed range of spacings.     

The development of solidification microstructures in the presence of lateral constraints

The effect of a sudden change in cross section on the microstructural development has been investigated by the directional solidification technique. Experiments have been carried out in a transparent model system of succinonitrile-acetone so that the dynamical changes in the interface shapes can be monitoredin situ as the interface travels from a region of uniform cross section to a region of sharply reduced cross section. Different experimental conditions which give rise to initial steady-state planar, cellular, and dendritic interfaces have been investigated. Significant changes in microstructures have been observed as the interface approaches a sharply reduced cross section. The planar interface undergoes transitions to cellular and dendritic morphologies as the cross section is reduced, and reverse transitions are observed as the cross section is then increased gradually to its original value. An initial cellular interface is found to become dendritic as the cross section is reduced, and again it becomes cellular as the cross section is increased. When the experimental conditions are designed to give initial dendritic structures, the change in microstructure is found to occur only when the reduced cross section is of the order of primary dendrite spacing. When the reduced cross section is more than about 5 times the primary spacing, no appreciable change is observed in the dendritic array which travels across the cross-sectional change. The dynamical changes in the interface shape and the microstructural transitions that occur with the change in cross section have been examined quantitatively and discussed. Formerly with Ames Laboratory     

The development of solidification microstructures in the presence of lateral constraints

The effect of a sudden change in cross section on the microstructural development has been investigated by the directional solidification technique. Experiments have been carried out in a transparent model system of succinonitrile-acetone so that the dynamical changes in the interface shapes can be monitoredin situ as the interface travels from a region of uniform cross section to a region of sharply reduced cross section. Different experimental conditions which give rise to initial steady-state planar, cellular, and dendritic interfaces have been investigated. Significant changes in microstructures have been observed as the interface approaches a sharply reduced cross section. The planar interface undergoes transitions to cellular and dendritic morphologies as the cross section is reduced, and reverse transitions are observed as the cross section is then increased gradually to its original value. An initial cellular interface is found to become dendritic as the cross section is reduced, and again it becomes cellular as the cross section is increased. When the experimental conditions are designed to give initial dendritic structures, the change in microstructure is found to occur only when the reduced cross section is of the order of primary dendrite spacing. When the reduced cross section is more than about 5 times the primary spacing, no appreciable change is observed in the dendritic array which travels across the cross-sectional change. The dynamical changes in the interface shape and the microstructural transitions that occur with the change in cross section have been examined quantitatively and discussed. Formerly with Ames Laboratory     

Two-Zone Microstructures in Al-18Si Alloy Powders

Hypereutectic Al-18 wt pct Si alloy is widely used in automotive industry as a wear-resistant alloy for engine components. However, in the last few years, this traditional composition is being considered for processing by different rapid solidification methods. Positive points include its low thermal expansion and uniform distribution of surface oxides. Nevertheless, the microstructural aspects of Al-Si powders of 18 wt pct Si are still need to be addressed, such as, the eutectic Si morphology, size, and distribution generated by different process conditions during rapid solidification. Based on a detailed quantitative analysis of the microstructures of rapid solidified Al-18 wt pct Si in this work, solidification conditions that yield specific Si morphologies, Si spacing, and thermal cooling conditions are outlined. The focus is determining the solidification conditions that will yield a specified shape of eutectic Si. It is shown that Si morphology is dependent on a combination of growth velocity (based on modified JH model) and temperature gradient. Furthermore, the highest hardness is achieved with globular morphologies of Si. The processing conditions required to achieve these properties are outlined.     

Morphology of NiFe2O4 precipitation in NiO

The morphology of NiFe₂O₄ precipitates in a NiO matrix has been studied using TEM. The spinel precipitates had a dendritic morphology when a polycrystalline sample of Fe-doped NiO was cooled at a constant rate from above the solvus temperature. They nucleate as octahedra bounded by {111} interfaces which then grow by a ledge mechanism until they reach a critical size and this shape becomes unstable. The precipitates then adopt a dendritic morphology characterized by branches in the 〈001〉 directions. The branches are bounded by {111} and {011} facets. HREM showed that during the initial stages of growth the interface is composed of a series of closely spaced ledges which are typically one spinel lattice-spacing high. Later, the ledge spacing decreases resulting in large flat {111} and {011} facets. Secondary branches were observed to form under large supersaturations of Fe when the precipitate spacing was large. The shape of the precipitates during the early stages of dendritic growth are very similar to those predicted by a Monte Carlo simulation which incorporates diffusional growth and anisotropic interface kinetics. The dendritic morphology is associated with the small lattice misfit, the large diffusion rate in Fe-doped NiO and the large undercooling necessary for nucleation. The diffusion rate is particularly large due to the concentration of cation vacancies being approximately half of the iron concentration.     

Eutectic growth: Selection of interlamellar spacings

Critical experimental studies in a model transparent organic system have been carried out to establish the interlamellar spacing selection criterion for directionally solidified eutectic alloys. It is shown that the spacing selection is not very sharp and a finite distribution of spacing is observed for a given velocity. The width of the distribution is found to increase as the velocity is decreased. This band of spacings is shown to lie within the stable zone predicted by the lamellar stability theory, and the average spacing lies slightly above the value which corresponds to the minimum undercooling condition. A number of velocity change experiments were also carried out in which the initial spacing was either larger or smaller than the final spacing. For both these conditions, the average eutectic spacing is found to drift toward the spacing value which is slightly larger than the smallest stable spacing for the final velocity. All experimental results show that the stable range of eutectic spacings is much smaller than that discussed by Jackson and Hunt.