M. STEINZIG, F. H. HARLOW*. Group T-3, Los Alamos National .... the inverse of the square root of cooling rate, as shown in Fig. 3. For lack of conclusive ...
CHARACTERIZATION OF CAST METALS WITH PROBABILITY DISTRIBUTION FUNCTIONS M. STEINZIG, F. H. HARLOW* Group T-3, Los Alamos National Laboratory, Los Alamos, NM 87545 ABSTRACT Characterization of microstructure using a probability distribution function (PDF) provides a means for extracting useful information about material properties. In the extension of classical PDF methods developed in our research, material characteristics are evolved by propagating an initial PDF through time, using growth laws derived from consideration of heat flow and species diffusion, constrained by the Gibbs-Thomson law. A model is described here that allows for nucleation, followed by growth of nominally spherical grains according to a stable or unstable growth law. Results are presented for the final average grain size as a function of cooling rate for various nucleation parameters. In particular we show that the model describes linear variation of final grain size with the inverse cube root of cooling rate. Within a subset of casting parameters, the stable-to-unstable manifests itself as a bimodal distribution of final grain size. Calculations with the model are described for the liquid to epsilon phase transition in a plutonium 1 weight percent gallium alloy. INTRODUCTION Many of the physical processes that govern solidification are understood, and direct numerical simulations (DNS) such as those of Rappaz and Gandin [1] or Juric and Tryggvason [2] are being used to generate accurate characterizations of interesting features such as dendritic growth and coring. These DNS calculations are somewhat limited by the magnitude of scales being covered, which range from the sub-micron scale at nucleation to final grain sizes that might be as large as a centimeter. The computing power necessary for such a simulation is prohibitively large for most problems of practical interest (such as metal casting) and will probably remain so for the foreseeable future. In addition, the extraction of bulk material properties from the results of a DNS simulation are difficult. The use of a probability distribution function (PDF) to describe the stochastics of phase change may prove to be more practical in terms of today’s computer hardware, and eventually be more useful for extracting such bulk material properties as average elastic modulus, thereby allowing prediction of the material response to a variety of insults. In this paper, we discuss the formulation of a simple PDF model for the nucleation of grains in a liquid, and use the results of the nucleation to infer the variations in final grain size. We are interested in the solidification of plutonium, which can undergo a series of phase transitions while cooling to room temperature. The α phase is often undesirable, and so small amounts of gallium are used to alloy the material and limit the transition, as can be seen in Fig. 1. In this work, we present simulations of the liquid to ε transition, in an attempt to determine the final size of ε grains. FORMULATION The example described here assumes a problem homogeneous in space, and the only variable considered is R, the grain radius. The PDF is defined such that P (R)dR
(1)
Figure 1. Plutonium-gallium phase diagram for low gallium concentrations [3] is the probable number of grains per unit volume of size R in an interval of size space dR. The PDF is evolved using a Liouville equation, such that the total number of grains in any arbitrary interval of R space is conserved. ∂P ∂P R˙ + =0 (2) ∂t ∂R In addition to the Liouville equation, we have an energy equation L ∂fs ∂ Tb = T˙ + , ∂t ρcp ∂t
(3)
in which ρ is density, cp is specific heat, Tb is the bulk temperature, L is the latent heat, fs is the fraction solidified and T˙ is a specified term representing heat diffusion. The probable number of grains multiplied by the volumetric change rate can be integrated over all size space to give the rate of change of fraction solidified, as dfs = dt
Z
∞
˙ dR . 4πR2 RP
(4)
0
We have used two growth rate equations K 1 K2 R˙ s = R and
µ
1 1 − Rc R
¶
K1 K2 , R˙ u = β 4Rc2
with constants defined as K1 =
k , L
K2 =
2σTs , L
(5)
(6)
in which σ is the surface tension, k is the thermal conductivity of the liquid, Ts is the equilibrium solidification temperature, and Rc is the critical radius defined by the Gibbs-Thompson equation, so that for a sphere K2 Rc = . ∆T The undercooling is defined as ∆T = Ts − Tb + mℓ C+ , where mℓ is the slope of the liquidus line and C+ is the concentration of gallium just outside of the grains. Equation (6) is used for grains that exceed the Mullins and Sekerka [4] stability criterion of R/Rc ≥ 7, while Eq. (5) is used for those grains with R/Rc ≤ 7. The β parameter in the unstable growth rate equation is the ratio of the equivalent grain radius to the distance from the center of a grain to the longest tip. This formulation is meant to account for the dendritic growth phase, where growth is strongly tip-led, and the growing grain is no longer spherical, so that R represents an equivalent radius, i.e., the spherical radius with the same volume as the actual grain. No experimental results about the shape of growing ε grains has been located, and we have chosen β as 0.8 for these simulations. Because of the large gallium diffusivity in both liquid and ε-phase plutonium, the solidification is thermally controlled, and the lever rule is invoked for relating gallium concentration on the liquid side of the grain boundary to fs . The liquidus line is fit with a constant-slope approximation for use in the Gibbs-Thomson equation. The derivation of these equations is covered in more detail by Steinzig [5]. For this exercise, the equations are solved using a Lagrangian finite-difference code. The initial PDF represents a flat distribution, P0 , of nucleation sites, from the smallest at Rmin to the largest at Rm . Each simulation begins with liquid plutonium at a bulk temperature equal to the critical temperature of the largest nucleation site in the melt, Tb = Ts −
K2 + mℓ C+ . Rm
As heat is extracted, (through the T˙ term in the global energy equation) the temperature drops and progressively smaller sites are nucleated from the initial distribution in a manner similar to that described by Thevoz et al. [6]. Those nucleation sites that are activated become growing grains, and release latent heat that is included in the energy equation and works to counter the effect of T˙ . In some cases, Tb reaches a minimum and begins to increase toward Ts . This recalescence is shown in the curves plotted in Fig. 2 for various cooling rates; the higher the cooling rate, the more nucleation sites become activated. 670.00
Temperature (˚C)
669.95
669.90
669.85
669.80
669.75 0.0
5.0
10
15
Time (sec)
Figure 2. Calculated cooling curves for, from left to right, T˙ = 0.249, 0.062, and 0.028◦ C/sec.
It is expected that the number of nucleation sites activated (Na ), has a direct impact on the final grain size. Experimental evidence of this effect is given by the plots of Gardner [7], who shows a linear relationship when final grain size for δ-phase plutonium is plotted as a function of the inverse of the square root of cooling rate, as shown in Fig. 3. For lack of conclusive evidence regarding the structure of the ε-phase grains, we postulate a close correlation with δ-phase grain size when the cooling rate is low and the number of nucleation sites for the ε → δ transition is likely to be controlled by the geometry of the ε-phase grains. At high cooling rates (e.g., quenching) the ε → δ nucleation occurs at numerous sites throughout the ε-phase grains, resulting in patchworks of elongate δ-phase grains with directionality controlled by a relationship similar to that described by Kurdjumov and Sachs for ferrite growing into austenite. This structure, as described by Johnson [8], is shown is Fig. 4, which allows tentative inference regarding the ǫ-phase grain sizes, but more extensive data will be required for incisive comparison with our calculated results. Thus our calculations for this paper focus on two issues: — The variations of final ε-phase grain size with cooling rate for various initial nucleation densities. — The consequences of grain-growth rate switch from stable, Eq. (5), to unstable, Eq. (6).
0.006
Grain Radius (cm)
0.005 0.004 0.003 0.002 0.001 0 0.0
1.0
2.0
3.0
4.0
5.0 -1/2
[Cooling Rate]
6.0
(K/sec)
7.0
8.0
-1/2
Figure 3. Variations of δ-phase plutonium grain size with cooling rate [7].
Figure 4. Patchwork structure of δ-phase grains [8]. Width of picture represents approximately 250 µm of actual surface.
Figure 5 shows the variations in ε-phase grain size plotted as a function of the inverse cube root of the cooling rate, T˙ −1/3 . With P0 = 108 cm−4 (top curve), the grain growth is predominantly unstable, (R > 7 Rc ), the switch occurring quite soon after nucleation. With P0 = 1010 cm−4 (bottom curve), the growth becomes unstable (R > 7 Rc ) for a brief period in the early stages of growth, and then reverts to stable growth (R < 7 Rc ) for the rest of the solidification. To show the contrast between stable and unstable growth, we used values of P0 that are too small to produce the very small ǫ grain size (∼ 0.005 cm) inferred from the δ-grain patchwork signature data for ε-phase grain size. Despite the simple structure of the initial PDF, a linear variation of final grain size with T˙ −1/3 is well described even for the rather narrow parameter range in which the switch from stable to unstable growth occurs during the middle stages of the growth process. In this middle range, with P0 = 109 cm−4 , some of the grains grow unstably while others remain stable throughout their growth. The result is a bimodal distribution of final grain size, shown in Fig. 5 by the two middle curves. 0.16
Grain Radius (cm)
0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 0.0
1.0
2.0
3.0 -1/3
[Cooling Rate]
(K/sec)
4.0
5.0
-1/3
Figure 5. Variations of ε-phase plutonium grain size with cooling rate.
CONCLUSIONS AND DISCUSSION The variation of final grain size with cooling rate is calculated to be linear when plotted as a function of T˙ −1/3 . Three datum points given by Ferrera, et al. [9] for δ-phase plutonium are consistent with this same T˙ −1/3 variation, whereas the four datum points given by Gardner suggest T˙ −1/2 . The effects of changing from unstable to stable grain growth are manifested principally by the development of a bimodal distribution of final sizes over a small range of nucleation-site parameters. The PDF approach appears to be a useful and efficient way to characterize gain-size distributions in numerical metal-casting codes. ACKNOWLEDGEMENT We are pleased to acknowledge valuable discussions with Thomas Zocco, Frank Gibbs, Damir Juric, Douglas Kothe, Dana Knoll, Christian Charbon, Professor C. Beckermann, and Professor M. Rappaz. This work was performed under the auspices of the United States Department of Energy.
REFERENCES 1. M. Rappaz and Ch.-A. Gandin, Acta metall. mater. Vol 41, #2, p. 345, (1993) 2. D. Juric and G. Tryggvason, J. of Computational Physics 123, 127 (1996). 3. F. H. Ellinger, C. C. Land, and V. O. Struebing, J. Nucl. Mater. 12, 228 (1964). 4. W. W. Mullins and R. F. Sekerka, Journal of Applied Physics 34 323–329 (1965). 5. M. Steinzig, Doctoral Dissertation, New Mexico State University, expected in May 1999. 6. Ph. Thevoz, J. L. Desbiolles, M. Rappaz, Metalurgical Transactions A 20A, 311 (1989). 7. H. R. Gardner, Report BNWL-13, Battelle Northwest Laboratory (1965). 8. K. A. Johnson, Los Alamos National Laboratory report LA-2989 (1964). 9. D. W. Fererra, J. H. Doyle, M. R. Harvey, Report RFP-1800, Dow Chemical USA, (1972).
