J.D. Hunt

Cellular and dendritic growth. I

Abstract A theoretical model is developed to explain the variation in cellular or dendritic tip temperatures with velocity and temperature gradient. It is shown that the undercooling can be considered to arise as a result of build up of average solute concentration ahead of the growth front depending mainly on the imposed temperature gradient, together with an additional radial term necessary to satisfy mass balance at the interface. It is found that a good correlation is obtained between theory and experiment.

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.

A numerical analysis of dendritic and cellular array growth : the spacing adjustment mechanisms

Abstract A numerical model has been used to describe the steady and the non-steady growth of array cells and dendrites in a moving linear temperature field. It was found that the four parameter numerical results could be approximated by a single parameter for a wide range of growth conditions. The practical limitation on array spacing is discussed and the upper and lower spacing limits have been modelled numerically. A new easily formulated criterion for the lower array spacing limit is deduced. The models predict a small range of stable spacings for cells and a separate range for dendrites. Cells are found to be present at low and very high velocities with dendrites at intermediate velocities. The numerical model correctly predicts the absolute stability limit and the onset of constitutional undercooling. The agreement with experiment is generally good provided anisotropy in surface energy is included. The calculated tip radii for dendrites were found to be close to that predicted by the “marginal stability condition” over the complete range of steady state array spacings and so cannot be used to select the operating array spacing. For cells the calculated tip radii was different from that predicted by the “marginal stability condition”, but it is shown that the condition could be used to select a very approximate growth spacing.

The measurement of Al-Cu dendrite tip and eutectic interface temperatures and their use for predicting the extent of the eutectic range

Dendrite and eutectic interface temperature have been measured in the Al-CuAl2 system near the eutectic composition. The results are compared with theory. In contrast to previous work a good correlation was obtained with eutectic theory using the minimum undercooling condition. Less good agreement was obtained for the dendrite results. The extent of the eutectic range was predicted by the theoretical model but not by the experimental dendrite values. It is suggested that the discrepancies arise because of fluid flow in the vicinity of the dendrite tips.

Columnar and equiaxed growth: I. A model of a columnar front with a temperature dependent velocity

Abstract The columnar growth of an alloy is modelled using a finite difference method. The velocity of the columnar front is described by a parabolic undercooling dependency and the undercooling at the front is calculated dynamically during the analysis. The model is the necessary precursor to the modelling of equiaxed growth ahead of a columnar front. Results are presented and discussed.

Macroscopic stability of a planar, cellular or dendritic interface during directional freezing

Abstract Studies were made of the macroscopic interface shape of unidirectionally grown binary alloy rods. The interface shape was correlated with growth rate, and an explanation is proposed in terms of solute flow at the interface causing deviation of the interface shape from that of the isotherm. Curvature of the interface has been observed under conditions which have been thought to give a planar front.

The structures expected in a simple ternary eutectic system: Part 1. Theory

The structures which are to be expected during the steady state directional growth of alloys in a simple ternary eutectic system are discussed and it is predicted that for a particular velocity and temperature gradient five different structural regions can be observed. The structural regions of a two component system may be explained using a stability analysis and a competitive growth criterion, and it is suggested that these ideas may be extended to a three component system. Additional models have been proposed for the growth of a planar front two phase eutectic and single phase dendrites in a three component system.

Columnar and equiaxed growth: II. Equiaxed growth ahead of a columnar front

Abstract The growth of equiaxed grains ahead of a columnar front is modelled numerically using a finite difference approach. The essential dependency on undercooling of the velocity of the columnar front and the rate of growth of the equiaxed grains is included. Both conduction and convection in the bulk are treated. The effects of the casting variables on the columnar-equiaxed transition are noted and discussed.

The extent of the eutectic range

Abstract It is shown that the extent of the eutectic range both at high and low temperature gradients can be explained in terms of a competitive growth model, provided allowance is made for the change in dendrite growth temperature with temperature gradient. The extent of the eutectic range predicted for the Pb-Sn system has been compared with experimental results.

The structures expected in a simple ternary eutectic system: Part II. The Al-Ag-Cu ternary system

Ternary alloys of various compositions from the aluminum rich corner of the Al-Ag-Cu system were directionally solidified at several different growth rates ranging from 6.4 × 10−1 mm·S−1 to 5.6 × 10−3 mm· s−1. The region of two phase coupled growth between α-Al and CuAl2 was determined at a growth rate of 6.4 × 10−1 mm· s−1. The composition range over which a fully ternary eutectic structure formed was investigated for several different growth rates. The results are found to be consistent with the predictions of the competitive growth model set out in Part I,1 and it would seem that the ternary eutectic composition of the published phase diagram may be incorrect. Scanning electron microscopy, using the backscattered electron signal, was used, together with optical microscopy, to study the microstructures formed. The ternary eutectic between α-Al, Ag2Al, and CuAl2 was found to be semiregular, and the unusual morphology of the two phase dendrites between α-Al and Ag2Al is explained.