A unified model of microsegregation and coarsening

Using a coordinate transform in a standard microsegregation model it is shown that the effect of coarsening on the dilution of the liquid phase can be accounted for by adding a dimensionless parameter α ^c to the back-diffusion Fourier number. Through theory, for the case of parabolic growth, and numerical experiments for the constant cooling rate, it is shown that a constant value of α ^c =0.1 is a sound choice across a wide range of casting and alloy conditions. This finding allows for the development of a unified model for microsegregation and coarsening. In particular, previous approximate microsegregation models can be readily modified to account for coarsening.     

On the influence of coarsening on microsegregation

An analytical treatment of the influence of coarsening by ripening on microsegregation is presented. Comparison of results from the model with data on Al-4.5 wt pct Cu and Sn-Bi alloys indicates that an overestimation of the reduction in microsegregation results if it is assumed that ripening proceeds throughout the solidification range. This is attributed to the changing nature of coarsening processes, in which coalescence processes predominate over ripening at higher volume fractions solid.     

Quantification of microsegregation during rapid solidification of Al-Cu powders

A new technique is introduced to quantify microsegregation during rapid solidification. The quantification involves calculation of the average solute solubility in the primary phase during solidification of an Al-Cu binary alloy. The calculation is based on using volume percent eutectic and weight percent of second phase (in the eutectic), which were obtained experimentally. Neutron diffraction experiments and stereology calculation on scanning electron microscope images were done on impulse atomized Al-Cu alloys of three compositions (nominal), 5 wt pct Cu, 10 wt pct Cu, and 17 wt pct Cu, atomized under N₂ and He gas. Neutron diffraction experiments yielded weight percent CuAl₂ data and stereology yielded volume percent eutectic data. These two data were first used to determine the weight percent eutectic. Using the weight percent eutectic and weight percent CuAl₂ in mass and volume balance equations, the average solute solubility in the primary phase could be calculated. The experimental results of the amount of eutectic, tomography results from previous work, and results from the calculations suggest that the atomized droplets are in metastable state during the nucleation undercooling of the primary phase, and the effect of metastability propagates through to the eutectic formation stage. The metastable effect is more pronounced in alloys with higher solute composition.     

Microsegregation in Al-Cu alloys

A comparison has been made between the amount of microsegregation predicted by a numerical model and that found experimentally in Al-Cu alloys varying in composition between 1 and 8 wt pct Cu. A depleted region was predicted and observed experimentally near the Al₂Cu. The depleted region was formed below the eutectic temperature and had a significant effect on the ordered composition-fraction plots, particularly for high alloy compositions. Although the fit between experiment and theory was reasonably good, it was concluded that it was necessary to propose that local equilibrium was not maintained between the phases in the solid-state reactions.     

A study of microsegregation in Al-Cu using a novel single-pan scanning calorimeter

A single-pan scanning calorimeter has been developed that eliminates the smearing of latent heat that occurs in a conventional two-pan heat-flux differential scanning calorimeter (DSC). In the new calorimeter, accurate enthalpy/temperature data was obtained in pure Al without smearing, and excellent sensitivity to new phases was obtained in a multicomponent Al alloy (LM25). The calorimeter has been used to investigate microsegregation in an Al-4.45 wt pct Cu alloy. The enthalpy/temperature data fell between that calculated, assuming no mixing in the solid (Scheil) and complete mixing in the solid (equilibrium solidification). The amount of segregation agreed well with that calculated using a diffusion-based model of microsegregation. The difficulty of getting the fraction solid from the enthalpy data is discussed, and it is concluded that it is not possible to do so without using a microsegregation model. In addition, it is concluded that it is wrong to assume that the enthalpy of an alloy can be given by a specific heat term and a constant latent heat term that depend on fraction liquid as is assumed in most casting models. This article is based on a presentation given in the symposium “Fundamentals of Solidification,” which occurred at the TMS Fall meeting in Indianapolis, Indiana, November 4–8, 2001, under the auspices of the TMS Solidification Committee.     

Dendrite growth directions in aluminum-zinc alloys

The dendrite growth directions in fcc aluminum-zinc alloys have been measured by electron backscattered diffraction (EBSD) as a function of the zinc concentration,c _o . In specimens produced by directional or Bridgman solidification, 〈100〉 dendrites were observed up to 25 wt pct Zn, whereas, above 60 wt pct, the dendrite growth direction was clearly 〈110〉. In between these two concentrations, the angle, φ(c _o ), between 〈100〉 and the 〈hk0〉 dendrite growth direction, varied continuously between 0 and 45 deg asc _o increased. Following an analysis of growth directions suggested by Karma but restricted to two dimensions in a (001) plane,^[1] this angle was fitted with a function ¼ arcos (?η₄/4η₈), where η₄(c _o ) cos (4φ) and η₈(c _o ) cos (8φ) are the first two contributions to the solid-liquid interfacial stiffness anisotropy in a (001) plane. This analysis also gives a qualitative explanation to textured seaweed structures observed aroundc _o ≈ 25 and 60 wt pct. These findings are discussed in light of recent measurements of the weak interfacial solid-liquid energy anisotropy of aluminum^[2,3] and of dendrite growth directions in hcp Zn-Al alloys.^[4]     

Mathematical modeling of microsegregation in binary metallic alloys

A mathematical model was developed to calculate microsegregation in binary metallic alloys. This model utilized the mathematical techniques of the method of lines combined with invariant imbedding (MOL/II) to solve the problem of combined heat and mass transfer during and after solidification. Model predictions were compared to experimental measurements in the Al-Cu system and to other microsegregation models. The MOL/II model predicted nonequilibrium second-phase contents within ±3 pct at low and intermediate cooling rates, when dendrite-arm coarsening was included in the model. It also was able to reproduce concentration profiles reasonably well. The analytical models commonly used (equilibrium cooling, Scheil equation, Brody/ Flemings model, Clyne/Kurz model, Solari/Biloni model, and Basaran equation) in micro-segregation calculations were shown to be considerably less accurate than the numerical models (MOL/II and the Ogilvy/Kirkwood model). Formerly Graduate Research Assistant, The University of Michigan     

Discontinuous precipitation and coarsening in Al-Zn alloys

The morphology and kinetics of discontinuous precipitation (DP) and discontinuous coarsening (DC) in solution treated and isothermally aged Al-Zn alloys containing 39.3 and 59.3 at.% Zn have been investigated at temperatures ranging from 323 to 523 K by light microscopy, electron microscopy and X-ray diffraction. At all aging temperatures the supersaturated α solid solution was observed to decompose rapidly by DP into a lamellar mixture of solute depleted α phase and β phase precipitate. DP occurred so rapidly in the 59.5 at.% Zn alloy that the heat of transformation raised the temperature of the alloy significantly. With further aging a slower DC reaction transformed the lamellar DP into a coarser lamellar structure of the same two phases; however, the composition of the α phase of the DC was closer to the equilibrium solvus composition than that of the DP. With still further aging a second, much slower DC reaction was observed to decompose the lamellar product of the first DC reaction in the 59.5 at.% Zn alloy into a still coarser lamellar structure. Analysis of the kinetics of both the DP and DC reactions showed them to be controlled by boundary diffusion in the advancing reaction interface. Reaction front migration rates for both DP and DC increased markedly with increasing Zn content. This increase seems to be associated partially with an increase in boundary diffusivity with increasing Zn content.     

Dendrite coherency of Al-Si-Cu alloys

The dendrite coherency point of Al-Si-Cu alloys was determined by thermal analysis and rheological measurement methods by performing parallel measurements at two cooling rates for aluminum alloys across a wide range of silicon and copper contents. Contrary to previous findings, the two methods yield significantly different values for the fraction solid at the dendrite coherency point. This disparity is greatest for alloys of low solute concentration. The results from this study also contradict previously reported trends in the effect of cooling rate on the dendritic coherency point. Consideration of the results shows that thermal analysis is not a valid technique for the measurement of coherency. Analysis of the results from rheological testing indicates that silicon concentration has a dominant effect on grain size and dendritic morphology, independent of cooling rate and copper content, and thus is the factor that determines the fraction solid at dendrite coherency for Al-Si-Cu alloys.     

Precipitation effects on thermopower in Al-Cu alloys

The precipitation influence on the thermoelectric power (TEP) of Al-Cu alloys (with from 2 up to 5 wt% Cu) has been observed. It has been shown that incoherent θ precipitates have no influence on TEP, in agreement with previous observations in various metallic alloys where similar precipitates form. On the opposite metastable plate-like precipitates (GP zones, θ″, θ′) induce a contribution to TEP, which is the larger the higher the measuring temperature; furthermore a correlation between the sign of this contribution and the nature of the stresses induced by the particles is observed.     

Investigation of microstructural coarsening in Sn-Pb alloys

The central theme of this work is to investigate the kinetics of microstructural evolution at high volume fractions of the dispersed phase in a solid-liquid mixture. Until recently, the kinetics of coarsening in the high volume fraction range was not clearly established. A recent study focused on high volume fractions (V _v >0.90) revealed that the temporal scaling laws that describe phase coarsening change from the conventional cube root of time behavior to a fourth-power relationship. This work probes the variation of the temporal exponent with volume fraction of the dispersed phase (V _v >-0.60). An overview of the fundamentals of the physics involved in diffusion-limited coarsening is presented. Also explained is the relevance of phase coarsening in various applications. A succinct review of the attempts to understand the various parameters involved in coarsening is provided, with the Sn-Pb system chosen for this study for reasons apart from its importance as a commercial solder alloy system. Details of the experimental procedures are described, and, following this, the results are outlined and the underlying mechanisms discussed. The findings reveal that the temporal exponent changes as the volume fraction of the dispersed phase changes.     

Isotropic fiber coarsening in unidirectionally solidified eutectic alloys

A theory of coarsening has been developed for the case of fibers of circular cross section in unidirectionally solidified eutectic alloys. The fibers are assumed to be randomly distributed throughout the unoccupied space in the matrix (uniformly distributed), and diffusion of solute is assumed to occur under steady-state conditions with cylindrical symmetry. The theory predicts that the cube of the average fiber radius increases linearly with time for all fiber volume fractions. As the volume fraction increases, the coarsening rate increases and the theoretical distribution of fiber radii becomes broader. It is shown that isotropic fiber coarsening occurs at approximately half the rate at which the same phase would coarsen if it were in the form of spherical precipitates for volume fractions up to 0.6. It is also demonstrated that the fibers will coarsen at a faster rate when they are uniformly distributed than when they are arranged on an hexagonal lattice.     

Discontinuous coarsening of spinodally decomposed Cu-Ni-Fe alloys

The coarsening behavior of two Cu-Ni-Fe alloys has been found to include a discontinuous reaction which originates at larger misorientation grain boundaries. Segments of these boundaries migrate at a rate depending on temperature and time and consume the semicoherent matrix spinodal, leaving behind larger irregularly shaped semicoherent particles in a cellular morphology. The combined influence of a dominant driving force (reduction in total interphase interfacial energy) and sluggish boundary migration rate promotes additional coarsening among the boundary-fed particles, wherein their effective interparticle spacing (λ₂) varies as λ₂ = k₂t^0.7, compared to that of the matrix, λ₁ = k₁t^0.34. In both cases the rate of coarsening is greater in the alloy of asymmetrical composition than in the symmetrical alloy. No satisfactory account of Cu-Ni-Fe discontinuous coarsening reaction kinetics can be made based upon current theories of thermally activated growth processes.