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The microstructural development of binary alloys during directional solidification is studied. Cellular growth data for the Al-Cu and Pb-Sn binary alloy systems are analyzed in order evaluate the criteria of Kurz and Fisher (1981) and Trivedi (1984) for cellular-dendritic transition. It is observed that the experimental growth values do not correlate with the Kurz and Fisher or Trivedi data.
The development of theoretical models which characterize the planar-cellular and cell-dendrite transitions is described. The transitions are analyzed in terms of the Chalmers number, the solute Peclet number, and the tip stability parameter, which correlate microstructural features and processing conditions. The planar-cellular transition is examined using the constitutional supercooling theory of Chalmers et al., (1953) and it is observed that the Chalmers number is between 0 and 1 during dendritic and cellular growth. Analysis of cell-dendrite transition data reveal that the transition occurs when the solute Peclet number goes through a minimum, the primary arm spacings go through a maximum, and the Chalmers number is equal to 1/2. The relation between the tip stability parameter and the solute Peclet number is investigated and it is noted that the tip stability parameter is useful for studying dendritic growth in alloys.
A simple model for the constrained growth of an array of cells or dendrites in a binary alloy in the presence of an imposed positive temperature gradient in the liquid is proposed, with the dendritic or cell tip radius calculated using the marginal stability criterion of Langer and Muller-Krumbhaar (1977). This approach, an approach adopting the ad hoc assumption of minimum undercooling at the cell or dendrite tip, and an approach based on the stability criterion of Trivedi (1980) all predict tip radii to within 30 percent of each other, and yield a simple relationship between the tip radius and the growth conditions. Good agreement is found between predictions and data obtained in a succinonitrile-acetone system, and under the present experimental conditions, the dendritic tip stability parameter value is found to be twice that obtained previously, possibly due to a transition in morphology from a cellular structure with just a few side branches, to a more fully developed dendritic structure.
Since significantly different solidification structures of a given alloy can be obtained by varying experimental growth rates, it is desirable to understand the basic factors which control the formation and stability of these microstructures when conditions are altered. Directional solidification experiments are described and the results obtained in metallic and transparent organic systems are presented. Emphasis is on the characteristics of dendritic structures obtained under different solidification conditions. Specifically, the effect of the growth rate on the primary dendritic spacing, the secondary dendrite spacing, and the dendrite tip radius is discussed. It is shown that significant changes in the primary spacing are observed when a dendrite to cellular transition takes place at lower velocities. It is found that the primary cellular spacing is much smaller than the primary dendrite spacing so that a maximum in the primary spacing occurs as a function of velocity at the dendrite-cellular transition. A theoretical model is also described which quantitatively explains various microstructural features of dendritic and cellular structures.
The central focus of this research is to establish key scientific concepts that govern the selection of cellular and dendritic patterns during the directional solidification of alloys. Ground-based studies have established that the conditions under which cellular and dendritic microstructures form are precisely where convection effects are dominant in bulk samples. Thus, experimental data can not be obtained terrestrially under pure diffusive regime. Furthermore, reliable theoretical models are not yet possible which can quantitatively incorporate fluid flow in the pattern selection criterion. Consequently, microgravity experiments on cellular and dendritic growth are designed to obtain benchmark data under diffusive growth conditions that can be quantitatively analyzed and compared with the rigorous theoretical model to establish the fundamental principles that govern the selection of specific microstructure and its length scales. In the cellular structure, different cells in an array are strongly coupled so that the cellular pattern evolution is controlled by complex interactions between thermal diffusion, solute diffusion and interface effects. These interactions give infinity of solutions, and the system selects only a narrow band of solutions. The aim of this investigation is to obtain benchmark data and develop a rigorous theoretical model that will allow us to quantitatively establish the physics of this selection process.
The fundamental mechanisms responsible for both micro scale and macro scale compositional variations during cellular and dendritic solidification are discussed. It is shown that presently available models for cellular and dendritic growth in alloys predict only the elimination of macrosegregation during rapid solidification but that microsegregation still persists even in this regime. A simple model is proposed here to describe the complete elimination of microsegregation in the rapid solidification regime. A solid of ideally uniform composition can be produced with no micro or macro segregation at both very low large solidifation rates.
The dependence of primary arm spacings on growth speed was investigated for cellular and dendritic arrays in Pb-10 wt percent Sn samples directionally solidified under a constant positive thermal gradient in the melt. The gradient of constitutional supercooling was varied from almost zero (near the break-down of the planar liquid-solid interface at small growth speeds, cellular morphology) to near unity (large growth speeds, dendritic morphology). The spatial arrangements of cells and dendrites, as given by their coordination number, are not very different from each other. It appears that primary arm spacing maxima and the cell to dendrite transition are strongly influenced by the magnitude of the solute partition coefficient. The planar to cellular bifurcation is supercritical in Pb-Sn which has a high partition coefficient, as compared to the subcritical behavior reported in Al-Cu and succinonitrile-acetone, both of which have low partition coefficients. The primary arm spacing model due to Hunt agrees with the experimentally observed trend for the whole growth regime. There is a good quantitative agreement at higher gradients of supercooling. However, the model overpredicts the primary arm spacings at low gradients of constitutional supercooling.
The dependence of primary arm spacings on growth speed was investigated for cellular and dendritic arrays in Pb-10 wt percent Sn samples directionally solidified under a constant positive thermal gradient in the melt. The gradient of constitutional supercooling was varied from almost zero (near the break-down of the planar liquid-solid interface at small growth speeds, cellular morphology) to near unity (large growth speeds, dendritic morphology). The spatial arrangements of cells and dendrites, as given by their coordination number, are not very different from each other. It appears that primary arm spacing maxima and the cell to dendrite transition are strongly influenced by the magnitude of the solute partition coefficient. The planar to cellular bifurction is supercritical in Pb-Sn which has a high partition coefficient, as compared to the subcritical behavior reported in Al-Cu and succinonitrile-acetone, both of which have low partition coefficients. The primary arm spacing model due to Hunt agrees with the experimentally observed trend for the whole growth regime. There is a good quantitative agreement at higher grdients of supercooling. However, the model overpredicts the primary arm spacings at low gradients of constitutional supercooling.
Thermosolutal convection caused by the solute build up ahead of the growing arrays of cells and dendrites results in macrosegregation along the length of the Pb-Sn alloy (10 to 58 wt pct Sn) specimens when they are directionally solidified in a positive thermal gradient (melt on top, solid below, and gravity pointing down). At a constant thermal gradient, the extent of macrosegregation increases with decreasing growth speed as the microstructure changes from dendritic, to cellular and to planar. An empirical parameter, effective partition coefficient, obtained from the dependence of the longitudinal macrosegregation on fraction distance solidified can be used to represent the extent of macrosegregation.
The dependence of primary arm spacings on growth speed has been investigated for cellular and dendritic arrays in directionally solidified Pb-10 wt pct Sn. The spatial arrangements of cells and dendrites, as given by their coordination number, are not very different from each other. The primary arm spacing maxima and the cell-to-dendrite transition appear to be strongly influenced by the magnitude of the solute partition coefficient (k). The planar-to-cellular transition in Pb-Sn (k = 0.50) is supercritical as compared to the subcritical behavior reported in Al-Cu (k = 0.14) and succinonitrile-acetone (k = 0.1).
During the directional solidification of alloys, solute inhomogeneities transverse to the growth direction arise due to morphological instabilities (leading to cellular or dendritic growth) and/or due to convection in the melt. In the absence of convection, the conditions for the onset of morphological instability are given by the linear stability analysis of Mullins and Sekerka. For ordinary solidification rates, the predictions of linear stability analysis are similar to the constitutional supercooling criterion. However, at very rapid solidification rates, linear stability analysis predicts a vast increase in stabilization in comparison to constitutional supercooling.
Measurements of convective flow fields and solid-liquid interface shapes during the solidification of a pure and a slightly alloyed transparent material reveal that the convective transport of solute can cause a macroscopic depression to develop in the solid-liquid interface. This effect occurs under conditions close to those which are predicted to produce morphological instability of a planar interface. A cellular or dendritic microstructure later develops within the interface depression. The convection is attributed to the effect of radial temperature gradients in the crystal growth apparatus.
The mechanical and electrical properties of crystals produced by a unidirectional process depend strongly on the temperature and flow fields since these control the concentration of solute at the melt-crystal interface. The solute gradient there drives morphological instabilities that lead to cellular or dendritic interfaces. In the presentation several features of flow-solidification interactions will be discussed. These will include the effects of convection driven by density changes and buoyancy and the imposition of forced flow.
Directionally solidified aluminum-indium alloys of the monotectic composition were found to form an in situ rod composite which obeys a lambda exp 2 R = constant relation. The experimental data shows good agreement with previously reported results. A theoretical boundary between cellular and dendritic growth conditions was derived and compared with experiments. The unique wetting characteristics of the monotectic alloys can be utilized to tailor the interface structure in metal matrix composites. Metal matrix composites with monotectic and hypermonotectic Al-In matrices were made by pressure infiltration, remelted and directionally solidified to observe the wetting characteristics of the alloys as well as the effect on structure of solidification in the constrained field of the fiber interstices. Models for monotectic growth are modified to take into account solidification in these constrained fields.
Succinonitrile-1.9 wt pct acetone has been directionally solidified in 0.7 X 0.7-cm-square cross section pyrex ampoules in order to observe the cell/dendrite tip morphologies, not influenced by the 'wall effects', which are present during growth in the generally used thin (about 200 gm) crucibles. The tips do not maintain a steady-state shape, as is generally assumed. Instead, they fluctuate within a shape envelope. The extent of fluctuation increases with decreasing growth speed, as the micro structure changes from the dendritic to cellular. The influence of natural convection has been examined by comparing these morphologies with those grown, without convection, in the thin ampoules.
The aim of this investigation is to establish key scientific concepts that govern the selection of cellular and dendritic patterns during the directional solidification of alloys. We shall first address scientific concepts that are crucial in the selection of interface patterns. Next, the results of ground-based experimental studies in the Al-4.0 wt % Cu system will be described. Both experimental studies and theoretical calculations will be presented to establish the need for microgravity experiments.
Artificial neural networks comprising spiking neurons of a novel type have been conceived as improved pattern-analysis and pattern-recognition computational systems. These neurons are represented by a mathematical model denoted the state-variable model (SVM), which among other things, exploits a computational parallelism inherent in spiking-neuron geometry. Networks of SVM neurons offer advantages of speed and computational efficiency, relative to traditional artificial neural networks. The SVM also overcomes some of the limitations of prior spiking-neuron models. There are numerous potential pattern-recognition, tracking, and data-reduction (data preprocessing) applications for these SVM neural networks on Earth and in exploration of remote planets. Spiking neurons imitate biological neurons more closely than do the neurons of traditional artificial neural networks. A spiking neuron includes a central cell body (soma) surrounded by a tree-like interconnection network (dendrites). Spiking neurons are so named because they generate trains of output pulses (spikes) in response to inputs received from sensors or from other neurons. They gain their speed advantage over traditional neural networks by using the timing of individual spikes for computation, whereas traditional artificial neurons use averages of activity levels over time. Moreover, spiking neurons use the delays inherent in dendritic processing in order to efficiently encode the information content of incoming signals. Because traditional artificial neurons fail to capture this encoding, they have less processing capability, and so it is necessary to use more gates when implementing traditional artificial neurons in electronic circuitry. Such higher-order functions as dynamic tasking are effected by use of pools (collections) of spiking neurons interconnected by spike-transmitting fibers. The SVM includes adaptive thresholds and submodels of transport of ions (in imitation of such transport in biological neurons). These features enable the neurons to adapt their responses to high-rate inputs from sensors, and to adapt their firing thresholds to mitigate noise or effects of potential sensor failure. The mathematical derivation of the SVM starts from a prior model, known in the art as the point soma model, which captures all of the salient properties of neuronal response while keeping the computational cost low. The point-soma latency time is modified to be an exponentially decaying function of the strength of the applied potential. Choosing computational efficiency over biological fidelity, the dendrites surrounding a neuron are represented by simplified compartmental submodels and there are no dendritic spines. Updates to the dendritic potential, calcium-ion concentrations and conductances, and potassium-ion conductances are done by use of equations similar to those of the point soma. Diffusion processes in dendrites are modeled by averaging among nearest-neighbor compartments. Inputs to each of the dendritic compartments come from sensors. Alternatively or in addition, when an affected neuron is part of a pool, inputs can come from other spiking neurons. At present, SVM neural networks are implemented by computational simulation, using algorithms that encode the SVM and its submodels. However, it should be possible to implement these neural networks in hardware: The differential equations for the dendritic and cellular processes in the SVM model of spiking neurons map to equivalent circuits that can be implemented directly in analog very-large-scale integrated (VLSI) circuits.
Technique yields better mixing of impurities and superior qualiity crystals. Accellerated motion stirs melt which reduces temperature gradients and decreases boundary layer for diffusion of impurities near growing surface. Results better mixing of impurities into melt, decrease in tendency for dendritic growth or cellular growth and crystals with low dislocation density. Applied with success to solution growth and Czochralski growth, resulting in large crystals of superior quality.