Lamellar growth of eutectic equiaxed grains
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K2 l
Fig. 1—The spherical coordinate system.
[1]
where K1 and K2 are material constants. Although any pair of V and l will satisfy Eq. [1], experimental studies demonstrated that only a narrow range of interphase spacings can be chosen by a given system. Following the principle of growth at extremum,[2,3] three interdependent relationships were derived from Eq. [1] corresponding to the minimum spacing, lex.[1] 2 l ex z V 5 K2 /K1
[2a]
DT /=V 5 2 =K1 z K2
[2b]
DT 5 2 z K2 /lex
[2c]
Fig. 2—The assumed geometry of the periphery of the eutectic grain in a section perpendicular to the z direction (N 5 N(c) and wav 5 wav(c)).
Also, a maximum spacing, lM, was proposed by JH, for which the slope of the interface goes to infinity (a groove is formed at the middle of a lamella). Even though the extremum criterion has no fundamental justification, many experiments performed on regular eutectic systems[4–8] have shown that measured values closely satisfied Eqs. [2a] through [2c]. A mathematical model which takes into account the spherical geometry of eutectic grains will be developed. The main assumptions of this model are as follows:
(4) there is an isothermal solid/liquid interface; and (5) the grain geometry at the periphery of the grain, in sections perpendicular on the z direction, is that presented in Figure 2. Because of the assumptions (1), (2), and (5), the composition field has to satisfy the following differential equation:
(1) there is a steady-state solute field around the eutectic grains; (2) diffusion occurs only along the w and r directions (Figure 1); (3) the lamellar spacing, l, is much smaller than the diffusion distance, D/V, that is, growth occurs at low Peclet numbers;
with the boundary conditions
ADRIAN V. CATALINA, Graduate Research Assistant, and DORU M. STEFANESCU, University Research Professor and Director of the Solidification Laboratory, are with the Department of Metallurgical and Materials Engineering, The University of Alabama, Tuscaloosa, AL 35487. Manuscript submitted August 24, 1994. METALLURGICAL AND MATERIALS TRANSACTIONS A
]2C 2 ]C 1 ]2C 1 z 1 z 50 ]r 2 r ]r r 2 z sin2 c ]w 2 ]C ]w
w 5 p z w
]C ]w
w 5 ~p 1 21! z w
]C ]r
av
r5R 50
(c )
50
av
(c )
[3]
[4a] 50
[4b] [5]
t
where wav (c) is the average angle between two lamellae of the same kind on the grain contour; Rt is the grain radius at the end of solidification (final grain radius); and p is a VOLUME 27A, DECEMBER 1996—4205
number between 0 and N 2 1, where N is the number of pairs of a/b lamellae on the periphery of a given section of the eutectic grain (N 5 N(c)). The number N 5 N(c) can be defined as
@ # 2zp wav (c )
N(c ) 5 int
[6a]
or, for the equatorial plane for which c 5
~p2 !, and N
0
~p2 !,
corrective factor which compensates for the difference in the liquidus slopes and curvature undercoolings of the two phases;[9] and qa and qb are the density factors of the a and b phases, respectively. They are defined as the ratio between the density of each phase and the average density, r , of the
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