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Diffusion fields associated with prolate spheroids in size and shape coarsening
Pergamon PII S1359-6454(%)00201-7
.4cta marer. Vol. 45. No. 2. pp. 823-835. 1997 Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved 1359-6454,97 $17.00 + 7.00
DIFFUSION FIELDS ASSOCIATED WITH PROLATE SPHEROIDS IN SIZE AND SHAPE COARSENING Y. MOUt Department
and J. M. HOWE
of Materials Science and Engineering. University Charlottesville. VA 22903. U.S.A.
of Virginia.
(Receioed 22 March 1995; accepted 6 March 1996)
Abstract-The
diffusion field or solute concentration distributed around a prolate spheroidal particle simulating a rod- or needle-shaped precipitate has been solved for varying aspect ratios of the precipitate and varying concentrations along the precipitate surface owing to the curvature effect. With the prolate
spheroidal coordinates, the principal curvatures of the prolate spheroidal surface are derived as functions of the angular variable, and the Laplace field equation is separated into two Legendre equations either on the radial variable or on the angular variable. The analytical solution fitting the present boundary conditions is secured as the sum of a Legendre function of the second kind of order zero and a Legendre series. The Legendre function of the second kind gives the concentration distribution when the effect can be ignored, whereas the Legendre series represents the concentration contributed curvature effect. Numerical results of normalized concentrations are presented as functions of and angular variables for selected aspect ratios. The tangent component of the concentration contributed from the curvature effect, may cause complicated mass transfer. which is responsible
curvature from the the radial gradients. for shape
coarsening.
1. INTRODUCTION
non-equilibrium kinetic factors. Overall, agreement between the theory and experiment often requires serious modifications [5-181. In developing coarsening theory for non-spherical precipitates. some investigations [ 19-271 considered the formation and lateral movement of growth ledges [28, 291 on precipitate facets in coarsening kinetics. Quantitative analyses by Aikin and Plichta [26] and Aikin et al. [27] applied Jones and Trivedi’s results [30, 311 for the concentration gradient in front of a single ledge, obtained by assuming no solute penetration across the terrace and an equal moving speed for the whole ledge. Based on the ledge mechanism of precipitate growth or dissolution [32, 331, these analyses [26, 271 predict coarsening behaviour of disc-shaped particles, which is much closer to experiments than that predicted with the LSW approach modified but still holding the assumption of isotropic atomic processes. However, as a measure of the anisotropy of atomic processes, ledge densities or interledge spacings cannot be evaluated or estimated in these treatments; instead, experimentally measured interledge spacings were used as inputs to the analyses. Needle (or rod) and disc-shaped precipitates can be mathematically modelled as prolate and oblate spheroids, or elongated and ffattened spheres at the poles, respectively. Ham [34, 351 and Horvay and Cahn [36] developed kinetic theories for ellipsoidal and spheroidal growth, respectively. Although growth and coarsening are governed by similar diffusion processes, there exist some obvious differences. which prevent these theories from applying to
The coarsening behaviour of precipitate particles is one of the most important problems in the thermal stability of precipitation-hardened alloys at elevated ambient temperatures. In metallic alloys, small and densely distributed precipitates act as obstacles to dislocation motion. When coarsening occurs at elevated temperatures, it increases the size and decreases the number density of the precipitates, thereby reducing their effectiveness as strengtheners. Understanding the coarsening kinetics is an important step in improving the long-term high-temperature performance of these alloys. Following Greenwood’s first treatment [ 11, systematic coarsening theory, now referred to as the LSW theory, was developed in 1961 independently by Lifshitz and Slyozov [2] and by Wagner [3]. Although one of the assumptions in the LSW theory, i.e. steady-state diffusion, is valid for small supersaturations and is well satisfied for metallic coarsening processes [4], the assumption that precipitates are spheres and that the atomic attachment or detachment is isotropic 12. 31 mostly restricts application of the theory. Non-spherical particles frequently form as a result of anisotropy in interfacial energy in solid-solid phase transformations. Furthermore, many precipitates are needle or disc-shaped owing to high-level anisotropy in interfacial energy and
tTo whom all correspondence should be addressed at AFE Technologies, 368A Greenbrier Dr., Charlottesville. VA 22901, U.S.A. 823
824
MOU and HOWE:
DIFFUSION
FIELDS
ASSOCIATED
coarsening processes. First, needle- or disc-shaped particles are, in general, involved in concurrent size and shape coarsening (with varying aspect ratios) [26], whereas these growth theories only treat shape-conservative kinetics (with constant aspect ratios). Secondly, the effect of surface curvature on solute concentration (the Gibbs-Thompson effect [37]) is much more important in coarsening than in growth processes because growth or shrinking of particles in coarsening is determined by their curvatures in comparison with some mean curvature, whereas the driving force in growth is the decrease in volume free energy. If the curvature effect is considered, solute concentrations will vary along the particle surface. The growth theories mentioned above, however, assume a constant concentration along the surface. Any coarsening kinetics theory of spheroidal particles must be much more complex than the LSW theory for spherical particles because two independent variables, i.e. two principal axes or one of them and the aspect ratio, are required to describe the particle geometry. The present paper deals with the first problem in prolate spheroidal coarsening, namely, the solute concentration field distributed around a prolate spheroidal particle. The governing diffusion field equation will be solved under a boundary condition for varying solute concentrations reflecting the curvature effect. A companion paper [38] is devoted to the parallel problem for oblate spheroidal coarsening.
WITH
PROLATE
SPHEROIDS
diffusion field around any individual particle.) For emphasizing the needle morphology in practical coarsening problems, the half-length I and radius r of the particle are used interchangeably with the half major and minor axes, a and b of the generating ellipsis, respectively. As shape coarsening is assumed, the aspect ratio A, or a/b, is no longer constant. Figure 1 also shows three sets of schematic concentration curves starting from different surface positions with angles tll, ~12and c(~from the I axis. For each set of concentration curves, C, is the precipitate concentration (illustrated for solute-rich cases), and CO, is the matrix concentration at the particle surface with the curvature effect ignored. As CI decreases from n/2 to 0, the concentration increment C, from the curvature effect increases because the principal curvatures get greater. As a result, the matrix concentration at the precipitate surface, C, , is a function of the angle tl and the surface shape defined by the aspect ratio A. If the particle is very small compared with the neighbouring matrix area free of other precipitate particles, a constant matrix concentration C* [26, 271 can be assumed at infinitely remote distances. It is noticed that this assumption aims to avoid unnecessarily complicated mathematics at the beginning phase of the present development. In subsequent investigations in progress, this assumption will be released, so that the effect of precipitate volume fraction can be incorporated into spheroidal coarsening kinetics. 2.2. Prolate spheroidal coordinates
2. FIELD EQUATION AND ITS SOLUTION
2.1. Assumptions and the governing field equation
Similar to the LSW spherical coarsening [2, 31, the spheroidal coarsening treated hereafter is assumed to be controlled by a steady-state diffusion process. Accordingly, the time-dependent diffusion equation [39] becomes the Laplace equation v2c = 0
Equation (1) can be solved only if prolate spheroidal coordinates [40] are adopted because the boundary conditions of solute concentration at the precipitate surface and at a remote boundary can be separated only in these coordinates. The prolate spheroidal coordinates are related to the Cartesian
(1)
where C is the solute concentration, and V2 is the Laplacian operator, i.e. V2 = V. V. In the Cartesian coordinate system V=ii+$j+$k where i, j and k are the Cartesian unit vectors. The approach that assumes the concentration distribution satisfies equation (1) instead of the time-dependent diffusion equation is known as the invariable field approximation in growth kinetics literature [4]. Equation (1) has been used almost exclusively in the literature since it simplifies the mathematics that might otherwise be intractable. An isolated prolate spheroidal particle (Fig. 1) is assumed to be surrounded by a strain-free matrix. (The particle size distribution is not considered here since the purpose of this work is only to solve the
Y
a, < a, c
ator
a3
Pole b 1
0
Distance
Fig. 1. Schematic curves showing solute concentrations in the matrix as functions of distance from the precipitate centre. Inset is a prolate spheroid simulating a rod- and needle-shaped precipitate.
MOU and HOWE: DIFFUSION FIELDS ASSOCIATED WITH PROLATE SPHEROIDS coordinates
through the transformation X = c&’ y = c
2
equations
I
- l)(l - Y/2)cos 4.
(3a)
(t2 - l)(l - $) sin 4,
(3b)
Z = c&J,
825
e=o
(3c)
where c is the half interfocal distance of the prolate spheroidal particle c=Jz_bz
(4)
and 5, q and $I are the three prolate spheroidal coordinates. It is convenient to explain the coordinate geometrically if intermediate transformation parameters p and 8 are introduced so that c = cash p,
@a)
q = cos 0,
(5b)
where p and 0 are called the radial and angular variables, respectively, since they are analogous to their counterparts in the spherical coordinates [41]. A curvilinear coordinate system is characterized by its coordinate constant surfaces and coordinate curves [41]. In the Cartesian system, for example x. y and =-constant surfaces are respectively planes parallel to the Y-Z, Z-x and x-y planes inclusive, and coordinate curves are lines parallel to the x, y and z coordinates inclusive. Combining equations (3) and (5) yields
y + g
I
2 sin’ Q
-2 c2 co? 6
1.
(6b)
Equation (6a) represents an infinite array of prolate spheriods with half-major axes of c cash p and half-minor axes of c sinh p. The precipitate surface is a special (@)-constant surface, defined by 4 = 5, or p = Pi, so that a = c cash pp = ~5,.
(7a)
b=csinhp,=cfi.
(7b)
&=S=&&
(7c)
Equation (7~) results from equations (4) and (7a) and the definition of the aspect ratio. Since &, depends only on A, it may be called the shape parameter. The r/(0)-constant surfaces, represented by equation (6b), are hyperboloids of revolution of two sheets. The q-constant hyperboloids and l-constant prolate spheroids form a confocal system (with the same half focal distance c). It is seen from equation (3) that x/y = tan d, and that the : coordinate is independent of 4. Accordingly, $-constant surfaces are planes including the z axis and having an angle of 4 with the positive x axis. Figure 2 is a three-dimensional view of the prolate spheroidal coordinate system, shown as related to the Cartesian coordinates. The point P has coordinates
5 coordinate
\
Fig. 2. A three-dimensional view of the prolate spheroidal coordinates. shown as related to the Cartesian coordinates.
of (0.6, k/6, n/4) in the (p, 0, 4) format or (1.185, -0.866. 7c/4) in the (5, q, 4) format. At point P, the t coordinate curve is a hyperbola, which is the intersection curve between the &constant hyperboloid and &constant plane both including P. Similarly, the 0 and 4 coordinates are an ellipsis and a circle, respectively, resulting from intersection of the $-constant plane and <-constant spheroid and of the {-constant spheroid and B-constant hyperboloid, all crossing the point P. The prolate spheroidal coordinates are orthogonal since the three coordinate curves are mutually perpendicular everywhere. Figure 3 shows a d-constant section of the prolate spheroidal coordinate system with a half-focal distance of c. The X-JJ axis in the figure is any ray in the x-y plane, starting from the origin. The prolate spheroidal coordinate system is indeed built through a rotation of the 4-constant section around the Cartesian z axis. The scale factors are introduced into the application of curvilinear coordinates to build their scalar relationships to the Cartesian system [41]. For the prolate spheroidal system, these factors can now be calculated from equation (3)
h@ =
II dr
@
= c&
-
l)(l
-
$)
(SC)
826
MOU and HOWE:
DIFFUSION
where r is the Cartesian position r = xi + yj + zk. The gradient operator Laplacian V2 [41] in the prolate spheroidal then secured through the scale factors.
FIELDS ASSOCIATED WITH PROLATE SPHEROIDS vector of V and the system are
P (E
(9)
v*=v.v =
c2(t2L ‘i’){$[(r’ -4 Fig. 4. Principal curvatures of a prolate spheroidal surface B at the point P, defined as the curvatures of the curves Kq and K$ at P.
(10) These equations will be used later to derive the diffusion field equation appropriate to prolate spheroidal coarsening. 2.3. Principal curvatures and surface concentrations Since matrix concentrations at the precipitate surface are affected by the surface curvatures, it is essential that expressions for the surface curvatures of a prolate spheroid be evaluated in the prolate
spheroidal system. At a surface point P, shown in Fig. 4, the unit vector [ is normal to the precipitate surface Z with 5 = 5,. If a plane is drawn (then removed in the figure for clarity) through the unit vectors f and d, it intersects the precipitate surface along the ellipsis Kq. Similarly, the plane Tq is drawn through f and $, intersecting the precipitate surface along the curve K+. The curvatures of K, and K4 at P, known as the principal curvatures [41] and denoted as K, and Q, can be determined through the scale factors given by equation (8)
1 ah@ Icd= -h;h,ag
5, SC{ =
-452
P
_
I)(52
P
_
f)’
P
(lib)
Y x
These curvatures can also be expressed in terms of the half-major and half-minor axes of the generating ellipsis through equation (7) lc, = -
ab (a’ - c* cos* 0)3’
(124
Wb) --2
Fig. 3. A +-constant coordinate surface in the prolate spheroidal coordinates with a half focal distance of c, showing curves of intersection of the &constant surface with <-constant and q-constant surfaces.
Two special cases are worth noticing: 1IC,1= Iq 1= a/ b* at the two poles where f3 is 0 or z, and IIC,I = b/a* and IK+1= l/b along the equator where 0 is n/2. The Gibbs-Thompson equation [37], expanded only to the linear terms, gives the matrix concen-
DIFFUSION
MOU and HOWE: tration
at the prolate
spheroidal
surface
c,, = Co,[l + r(lJG!I +
are thus Legendre functions of the second kind en([). Solutions to equation (14) then have the form
5 = i,
IJ%Ill
(13)
concentration at the where C,,,,, is the matrix precipitate surface when both curvatures are 0, and r = oVm/[(Cp - Co,)W
(13a)
where 0 is the interfacial free energy of the precipitate surface, V,,,the partial molar volume of solvent in the precipitate, C, the precipitate concentration, R the gas constant, and T the absolute temperature. Although generally a function of r~ and 4, 0 is a function of only r~ here since the geometry of the problem possesses an infinity-fold rotation symmetry around the z axis.
where F,, are arbitrary constants. The specific solution to equation (14) that satisfies the present boundary conditions can be written as the sum of two contributions
where
Cd<, u) = ~Pd;rl)QdS)/Qdi’,) or equivalently
2.4. Solution qf the governing boundary conditions
equation
and
the
It is not surprising that the curvatures and, the matrix concentrations at the accordingly, precipitate surface are independent of 4 (equations (11)) because of the rotation symmetry. Substituting equation (10) with the $-dependence eliminated into equation (1) obtains the governing field equation
+-
1)$]+$1
-$)$]=O.
(14)
with M and N to be determined,
c,((,
-5~~~]+v(v+
l)E=O,
(15)
$1
-+]+v(v+
l)H=O
(16)
Fe,,(z)
(17)
where PI,(z) and Q”(Z) are Legendre functions of the first and second kind, E and Pare arbitrary constants, and u stands for B or H and z and 5 or 11. Solutions to the present problem are restricted by the bounding and periodic properties of the Legendre functions. It is noted that the domain of the radial variable is 1 < t < cc or 0 < p < co while that of the angular variable is - 1 < q < 1 or 0 < 0 < 27~. For the angular solutions H(u), P”(q) are bounded at ye= + 1 only if v values are restricted to non-negative integers n. Q.(q) must be rejected because they are unbounded [42]. The angular solutions to the present problem are thus Legendre polynomials P,,(q) or P,(cos t3). For the radial solutions Z(t), on the other hand, Pn(<) must be rejected because they are unbounded when 5 *co. The radial solutions Z(c)
WI
and
r/) = t F”P.(vl)Q.(S)/QfIC~,). n=O
(21)
The constants Q.(&,) in equations (20) and (21) are introduced to simplify expressions for M, N, and F’. The boundary conditions at the precipitate surface 4 = &, are specified as Co(&) = c,,,
(22a)
GJ(l~~I + IQI).
As assumed in Section 2.1, the boundary at 5 -+co can be stated as
where v is the separation parameter. Equations (15) and (16) are both Legendre’s equations, which have general solutions of the form [40]
GW
(note that PO(v) = 1 [43])
C,(&, r) = a[il
+ N
co(i”) = kfQo(5)/Qo(t,) + N
Let C({, r~) = E(e)H(y). Equation (14) can then be separated into two ordinary differential equations
u = EP,,(z) +
827
FIELDS ASSOCIATED WITH PROLATE SPHEROIDS
GQb) conditions
c*,
(23a)
C,(co, q) = 0.
Wb)
Co(m) =
Substitution of equation (22) into (19) verifies that these conditions are equivalent to equation (13). It is obvious that C(co, PI) = C* from equations (19) and (23). It is now clear that C,(l, n) is the concentration contributed from the curvature effect, and that C,({, q) is the concentration distribution when the curvature effect can be ignored. 2.5. Evaluation
of the coeficients
The evaluation of A4 and N is rather straight forward. Substitution of the boundary conditions for C,(t). equations (22a) and (23a), into equations (20b) gives M=Cnm--C*
and
N=C*.
(24)
The boundary condition, equation (23b), is satisfied since Legendre functions of the second kind Q.(t) approach 0 when t-0 [42]. Combining equations (21) and (22b) (substituting equation (11)
828
MOU and HOWE:
DIFFUSION FIELDS ASSOCIATED WITH PROLATE SPHEROIDS
for K, and icr) yields
where the two definite integrals are
= f
F,P”(r/).
(25)
n=O
In order to seek coefficients dependent precipitate shape, let
only on the
6 = F”Z/(COnlZ)
(26)
where I is the half-length of the precipitate needle, and I = a = cl, (equation (7b)). This simplification is based on the assumption that Z, on the interfacial free energy 0 (related to the equilibrium aspect ratio Ao), is a weak function of q compared with the surface curvature (the bracketed terms in equation (25)). When the actual (non-equilibrium) aspect ratio A is large (A is usually much greater than AC,), as is frequently the case in the coarsening of needle-shaped particles, the bracketed terms are a very strong function of rl because the dramatic changes in curvature from the equator where the curvature is 0, to polar positions where the curvature approaches infinity. Furthermore, this treatment has another convenience that only an average value of e is needed since the function a(q) is unknown in most cases. For theoretical completeness and more accurate treatments when the function a(q) is available, further development is necessary and currently in progress. This will be addressed in a separate publication because of its complexity. Substituting equation (26) into (25) yields
It is noted that the denominators of both integrands in equation (30) are even functions and that Legendre polynomials of odd orders are odd functions [39]. Hence, the two integrals with n odd equal 0, so that fn vanishes when n is odd. For even n or 2m, I,+(n) can be evaluated by a formula in Ref. [42] (p. 202, problem 16) Z+(2m) = 2iP2,(O)Q&Jt~
- 1)
(31)
where i is the imaginary unit. Since even-order Legendre functions of the second kind with pure imaginary arguments have pure imaginary values [38], Zm(2m) is real, as the integral expression itself may tell us in equation (30b). For evaluating Z,,(2m), a relation between the two integrals needs to be derived by taking the derivative of Z4(2m) with respect to 5,
= f
:I&_Cl;-
W’2f’n(v) drl = - 5,4. (32)
1,,(2m) can then be evaluated using equations (31) and (32) (33)
(27)
where Qim(i&$?) are the first derivatives of even-order Legendre functions of the second kind with a pure imaginary argument, taking real values 1381.
The coefficients fn can now be evaluated as follows: multiply both sides by Pm(q) (m are non-negative integers), and then integrate both sides of equation (27) from - 1 to 1 by applying the orthogonality properties of Legendre polynomials [39] 0 if m # n; Pm(v)f’n(v)drl = 2/(2n + 1) if m = n. f -I It is then obtained for non-negative
(28)
integers n that
h = (n + 1/2)%G - l)-“*[(t; - l)Z,(n) + Z&r)] (29)
In summary, subject to the boundary i.e. equations (22) and (23), or C(&, rl) = Co, + CC%
COJ(IK, ?I
=
I c*,
+
conditions,
h
I>,
(344 Wb)
the solution of the governing diffusion field equation, equation (14), is given by combining equations (19), (20b), (21) (24), (26), (2% (31) and (33) as C(5, q) = (C,, -
c*j gg$+ c*
829
MOU and HOWE: DIFFUSION FIELDS ASSOCIATED WITH PROLATE SPHEROIDS non-negative
where
integer in [39] (42) P?,,+ , (0) = 0.
I iQzm(iJ
Jm
(36j
1
3. NUMERICAL RESULTS AND DISCUSSION 3.1. Legendre polynomials second kind
and functions
of the
Legendre polynomials P,(s) and Legendre functions of the second kind Q.(X) satisfy the following recurrence formula [39] (written only for P,(X)) P,(x)
= (2 - l/?Z).XP~_,(.X)- (1 - Iln)P,+,(.u).
(37)
This recurrence relation makes the evaluation of P,,(x) and Qn(x) very efficient beginning with the first two Legendre polynomials and Legendre functions of the second kind [43] P&x) = 1,
1
x+1
PO(x) = - In .u_l, 2
P,(x)
= 5;
(38)
Ql(x)=~ln~-1; IX > 1.
(39)
This approach works fine for the Pfl(q) evaluation with - 1 < q < 1 or 0 > (3> 271 (actually, calculations are needed only for 0 < q < 1 or 0 > 0 > n/2 because of the symmetry of the diffusion field). It was found, however, that Qn(<) cannot be correctly evaluated for 5 > 1 with n increasing because this algorithm fails to tolerate the inevitable truncation error in the computer. An alternative was then taken with n decreasing; equation (37) thus takes the form (written for Q+,(X)) Q.-I(X) = (2 + I/n).yQ.(_x) - (1 + l/n)Q,+,(.u). (40) The two functions of the highest and second highest order then must be evaluated first through [39]
Q,z(x)
=
5 r=o
x”
2,
I
2”(n + r)!(n + 2r)! ; r!(2n + 2r + l)!
IX > 1 (41) where lOG5000 terms in the infinite series are required dependent on the 5 values and the desired accuracy. It is also feasible to use equation (41) to evaluate all en(t) involved. The values of Pn(0), needed in determining the coefficients fn, can be evaluated through equations (37) and (38). However. note that for anvd
(43)
Equation (42) is thus more convenient than equations (37) and (38) for the P>,(O) evaluation. According to equation (43), on the other hand, the f?,,, formula, equation (36), also holds for odd n, i.e. all fim _,are 0. In fact, equations (31) and (33) are also valid for n odd. Therefore, if P”(O) are evaluated through equations (37) and (38),f, for either even or odd n can be evaluated through equation (36) substituting n for 2m. This algorithm takes a slightly longer time to run but is simpler in logic because all iterations are performed consistently on consecutive integers. Different approaches may be used to check the calculational results. A detailed discussion of Legendre functions of the second kind with an imaginary argument is postponed to a companion paper [38] on oblate spheroidal coarsening in which these functions are solutions to the field equation. It suffices here to notice that equation (39) is also valid for an imaginary argument owing to the analytic continuation of Legendre functions of the second kind [42]. The quantity im is thus substituted for .Y in equation (38); the calculation of QZ,,,(i,/m) is carried out through the complex variable package implemented in many languages such as FORTRAN and C. It is found that the recurrence formula (37) with II increasing is suited to this evaluation. The first derivative of QO(:) can be obtained from equation (39) as Q;(Z) = l/(1 - ?)
(44)
where z is used instead of x to emphasize that it can have complex values. The first derivatives of Legendre functions of the second kind, of order n 3 2, are related to Legendre functions through the following recurrence formula [39] (written for Qn(:)) Q:(Z) = [n/(1 - =‘)I@-
I(=) - zQ,,(z)].
3.2. Solution with an insigr$cant
(45)
curvature efjrect
When the precipitate half-length I is very large for limited aspect ratios, the factor in front of the series in equation (35) becomes very small, so that the concentration contributed from the curvature effect may practically vanish. Alternatively, if the interfacial energy 0 is very small. the curvature effect also becomes insignificant. These limiting cases suggest that with insignificant curvature effects, the solution to the field equation, equation (35) reduces to C”(t) = (Co, - C*) a
+ C*
(46)
where C(t) is used instead of C(t, 9) since the solute concentration has no angular dependence.
830
MOU and HOWE:
Substituting
DIFFUSION
FIELDS ASSOCIATED WITH PROLATE SPHEROIDS
equations (7~) and (39) gives
c (5) _ C(5) - C* II co, - c* 1
ln5+l 5-1
=2ln(A+J’A’-I)
(47)
where the concentration constants are collected to form the normalized concentration c,,(g), which depends only on the radial variable 5 and aspect ratio A. It is further noticed that the effect of A on the normalized concentration is a separated factor. In other words, z’,(g) curves are different only in the “magnification” of the concentration axis. The plots in Fig. 5 show the relations between the normalized concentration c(t) and radial variable 4 for various aspect ratios from 1.25 to 20. For each of the curves, z’,(r) decreases from unity, the normalized concentration at the precipitate surface with 5 = 5,. The l, values are listed in the figure, corresponding to the selected aspect ratios. It is seen that z;,(t) decreases with increasing 5 more rapidly for larger aspect ratios. In fact, this conclusion can be drawn from equation (47) since the factor containing only A decreases with increasing A. Concentration gradients, however, cannot be simply compared because the radial variable 5 is not equivalent to a distance variable. Figure 6 shows a three-dimensional 2’,(c) plot for an aspect ratio of 5. The position coordinates correspond to the z and x-y axes in Fig. 3. Because the diffusion field has an infinity-fold rotational symmetry around the z axis, the two position coordinates can represent all positions in the diffusion field. Equipotentials, or surfaces of equal concentrations, are represented by confocal ellipses positioned at different concentration levels (only the first quadrant is drawn owing to the reflection symmetries). The ellipsis at the highest level
Curve
5,
c
Fig. 6. A thr_ee-dimensional plot of the normalized concentration C,,(t) as a function of distances along the z and x-y axes (see Fig. 3) for an aspect ratio of 5. The coordinate scales are normalized with the half focal distance c.
represents the precipitate surface, an equipotential with Co(&) = Co, or z’,(&) = 1. It is noticed that although solute concentrations are constant on individual prolate spheroidal surfaces, concentration gradients vary with the angular variable 8. This becomes clear after observing that equal concentration decrements are established over different distance intervals along different &constant concentration curves. In particular, the concentration gradient at the poles (0 = 0) is the maximum, and that along the equator (0 = 7c/2) is the minimum. 3.3. Contributions from the curvature effect The last term in equation (35) is contributed from the curvature effect. The factor in front of the series is a function of the particle size I and system thermodynamic parameters Co, and r. The series, on the other hand, depends on only precipitate shape, which may be represented by either the shape parameter 4, or aspect ratio A. The curvature contribution to the concentration distribution can thus be normalized as
A
0.8
0.6
0.4
0.2
1 0
2
4
6
8
10 5
Fig. 5. Functional relations between the normalized concentration c(t) and radial variable c for various aspect ratios from 1.25 to 20.
where f2,,, are given by equation (36). According to the convergence theorem of the Legendre series [43], the series in the left-hand side of equation (27) is convergent to the right-hand function for all - 1 < rl < 1 or 0 < 0 < 27~.Since the quotient of two Legendre functions of the second kind in equation (48) is no greater than unity [42], the series in equation (48) converges for all 5 and q in the
MOU and HOWE:
DIFFUSION
present problem. The speed of convergence, however, relies on the 4, values. In general, the smaller is the 5, value (the greater is the aspect ratio), the larger is the number of series terms required for a preset accuracy. It is noticed that the slowest convergence for a given 5, value is achieved when < = t,, under which condition the quotient factors in the series are all unity. Particularly, when [ = [, and VI= 1 (noting that PZm(l) = 1 [43]), substituting equation (27) into equation (48) obtains
C,(;“,. I)
=
i
.fim= &
n,= II
=24’.
P
831
FIELDS ASSOCIATED WITH PROLATE SPHEROIDS
A=5 Curve
0 (“)
::
11.25 0
: 5
22.5 45 90
1.3
1.4
(49)
This states that the sum of all fimcoefficients is 2A’, which is also the maximum of ?,(t, q) because PZm(q) Q 1 [43] (see equation (48)). This simple relation provides a very useful means to check the calculation of the f&, coefficients. Furthermore, it can be employed to determine the number of terms required for a preset accuracy in the sense that the infinite series has to be truncated in practical evaluations. In the numerical results presented later, for example, the maximum of m is selected as 20. 40 and 80 for an aspect ratio of 5, 10 and 20, respectively, for a relative error of less than 0.002. Figure 7 shows the numerical results of the normalized concentration c, or C,l/(C,,r) distributed around a prolate spheroidal precipitate with an aspect ratio of 5. There are nine curves in the figure, displaying c;, as functions of the angular variable 8 for selected 5 values beginning with {, (1.021 for A = 5) with increments doubled consecutively. It is observed that (?h reaches the maximum value of 2A’ along curve 1 (5 = r,) at q = 1 (0 = 0’). This represents the situation at the poles of prolate
1.1
1.2
1.5 5
(a) 3.5 A=5
2.5
I\\\
Curve
u (9
:
1 I!25 22.5
3
2-
1.5
l-
0.5 ’ 1.5
2
2.5
3
3.5
4
4.5 5
50
(b) A=5
G
40 n
1
curve
5
:. 3 4
1.021 1.041 1.082 1.165
i I 8 9
1.660 1.330 2.320 3.639 6.279
30 2 20
Fig. 8. The normalized concentration c, as a function of the radial variable ; for selected 0 values for an aspe$ ratio of 5: (a) 5 ranging from 0 to 1.5; and (b) magnified C, scale for t beyond 1.5.
~
I~~~, 15
30
45
60
75
90
e (“) Fig. 7. The normalized concentration ck contributed from the curvature effect as a function of the angular variable 0 for selected 5 values for an aspect ratio of 5.
spheroidal precipitates, where the two principal curvatures are the same and maximum. For all the curves, SK decrease with 0 increasing from 0 ’to 90.. However, c, changes slowly in an area around each pole, which expands with increasing [. The most rapid decrease of cK occurs in a 0 interval of about 15” beyond the polar area. The curvature effect on concentration is approximately uniform in a broad Q band some 45” up and down from the equator (see Fig. 1). At a distance of about 6.3 times the precipitate half-length (curve 9) from the precipitate centre, c, values are less than 1 and almost independent of 0. It is worth emphasizing that the actual concentration C, from the curvature effect has the same monotonicity as does the normalized
832
MOU and HOWE:
DIFFUSION
FIELDS ASSOCIATED WITH PROLATE SPHEROIDS
concentration cK for solute-rich precipitation (positive r). In the solute-poor case the actual concentration will increase with 0 increasing and reach its minimum at the poles. ?, can also be drawn as a function oft for selected 0 values, as shown in Fig. 8, again for A = 5. The five curves all begin with 5 = &, i.e. the precipitate surface, running to 5 = 1.5 in Fig. 8(a) and being continued in Fig. 8(b) in a magnified concentration scale for clarity. It is observed that the c?~ values starting from the precipitate surface are quite different along different &constant surfaces, though they arrive at almost the same value when 5 exceeds 3.5. Figures 9 and 10 show that the CC values decrease with increasing 0 along selected g-constant surfaces
200 , A = 10
2;, 160.
Curve
9 (“)
:.
11’25
: 5
22.5 45 90
120.
80
0
1.0
1.1
1.2
1.3
1.4
1.5 5
(a) A = 10
Curve
5 A =
1.005 1.010 1.020 1.040
: :
80
10
Curve
6 (“)
1 :
122.5 lo25
4 5
45 90
I 0
15
30
45
60
75 6 (“)
90
1’ 1.5
2
2.5
3
3.5
4 5
(a) (b) 20
h
E 16 x t\
A = 10
Curve
5
5
1.081 1.161 1.322
5
!
Fig. 10. The normalized concentration cx as a function of the radial variable 5 for selected 0 values for an aspect ratio of 10: (a) 5 ranging from 0 to 1.5; and (b) magnified C, scale for 5 beyond 1.5.
I 0
15
30
45
60
75 9 (“)
90
(b) Fig. 9. The normalized concentration of c as a function of the angular variable 6 for selected 5 values for an aspect ratio of 10: (a) four r values from 1.005 to 1.040; and (b) magnified & scale for five 5 values from 1.081 to 2.290.
and with increasing 5 along selected &constant surfaces, respectively, for an aspect ratio of 10. The maximum of &, 200, is much greater than that in Figs 7 and 8 for A = 5 since the maximum is proportional to the square of the aspect ratio. For an aspect ratio of 20, as shown in Figs 11 and 12, the maximum of c% becomes as great as 800 (Figs 1 l(a) and 12(a)). In general, the actual or total solute concentration around a prolate spheroidal particle, is the sum of the concentration with the curvature effect ignored and the concentration contributed from the curvature effect. However, the results shown in Figs 7-12 cannot simply be added to get the total concentration. The reason is that the two normalized
833
MOU and HOWE: DIFFUSION FIELDS ASSOCIATED WITH PROLATE SPHEROIDS concentrations, c0 and c,, are normalized with respect to different thermodynamic parameters. This is why they have very different scales; in particular, ?, can range from zero to infinity. The actual concentration can be obtained with the following relation given by substituting equations (47) and (48) into (35) C(5,?1) = (C”, - c*)G(5)
+ c* + (C”J/Y)C,(S,
80 70
G a50 4c-
a)
30
(50) where (?, and cK are normalized concentrations defined by equations (47) and (48) respectively. When a coarsening system is specified, the plotted results can be substituted into equation (50) to get the actual concentration for aspect ratios of 5, 10 or 20. Concentration gradients at the precipitate surface are essential for the analysis of coarsening kinetics. When the curvature effect is ignored, the solute concentration depends only on the radial variable (equation (46)). The concentration gradients are directed normal to the precipitate surface because the tangential component of the concentration gradient is zero, as can be seen from equation (9) (the coefficient of the unit vector Q). However, the concentration contributed from the curvature effect depends on both the radial and angular variable, so that the concentration gradient has both normal and tangential components. The normal components of the concentration can be qualitatively observed from the c,-< curves (Q is constant) in Figs 8, 10 and 12 at the slopes of the tangents to the curves. As shown in equation (9), however, their exact values are modified by the factors before the < partial derivative. The tangent components are qualitatively observed from 800 I\
A = 20
700
I \
1 z
1
4
1.001 1.003 1 .oos 1.010
0
45
(b)
80 A = 20
70 CK
5
\
60
Curve
5
5
1.020 1.040 1.080 1.160
50
76 8
40 30 20
;
10
9 L_
0
15
30
45
60
75
90
9 (“) (c) Fig. 11. The normalized concentration c, as a function of the angular variable B for selected 5 values for an aspect ratio of 20: four 5 values from 1.001 to 1.010 in 0 intervals of (a) O’-15” and (b) 15‘-90” with magnified r, scale for clarity; and (c) magnified i;, scale for five 5 values from 1.020 to 1.320.
the Sk-Q curves in Figs 7,9 and 11; they are non-zero except at positions with f3= 0’ or 6 = 90”, particularly at the poles and along the equator of the precipitate surface. The existence of the tangent components implies that complicated mass transfer occurs and is responsible for shape coarsening.
4. SUMMARY
Fig. 11
The diffusion field, or solute concentration as a function of position, around a prolate spheroidal precipitate has been solved under a boundary condition of varying concentrations along the precipitate surface owing to the curvature effect [37]. The physical assumptions made here are otherwise similar to those of the LSW theory addressing spherical coarsening [2, 31.
834
MOU and HOWE:
DIFFUSION
FIELDS ASSOCIATED WITH PROLATE SPHEROIDS
The prolate spheroidal coordinates [40] are adopted to ensure that the boundary conditions at the precipitate surface and at some remote surface are separable. Introduction of these coordinates also makes it easier to formulate the principal curvatures [40] of the precipitate surface. Under these coordinates the diffusion field equation, the Laplace equation [39], is separated into two Legendre equations on the radial and angular variables [39,42], whose solutions are Legendre polynomials and Legendre functions of the second kind [40,42,43]. The boundary conditions are fitted by forming a solution as the sum of two parts: the first is a Legendre function of the second kind of order zero, and the second is an infinite Legendre series [39,43]. 800 A = 20
I 640
Curve
0 (“)
1
480 80
1 lo25
z
2i.5 45 90
i
320
60
The series coefficients are expressed analytically with Legendre polynomials and Legendre functions. The first part of the solution displays the concentration distribution when the curvature effect is ignored, whereas the second part represents the concentration change contributed from the curvature effect. Numerical results are presented for normalized concentrations, defined to filter out the dependence of actual concentrations on the particle size and system thermodynamic parameters. Normalized concentrations with an ignored curvature effect are drawn as a function of the radial variable. For various aspect ratios, normalized concentrations contributed from the curvature effect are displayed as a function of the radial variable for selected angular values and of the angular variable for selected radial values. Concentration gradients contributed from the curvature effect have tangential components, which may produce complicated mass transfer and thus be responsible for shape coarsening. Acknowledgements-Appreciation is expressed to NASALaRC for support of this research under Grant NAG-l-745. One of the authors (J.M.H.) is also grateful to NSF for support under Grant DMR-9302493.
160. 1 REFERENCES
40 1.0
1.1
1.2
1.3
1.4
1.5
20
OL
1.0
1.1
1.2
1.3
1.4
1.5 5
(a) 8 A = 20 I c*
Curve
0 (“)
6
1. G. W. Greenwood, Acta metall. 4, 243 (1956). 2. I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids 19, 35 (1961). 3. C. Wagner, Z. Elektrochem. 65, 581 (1961). 4. H. B. Aaron, D. Fainstein and G. R. Kotler, J. Appl. Phys. 41, 4404 (1970). 5. R. A. Oriani, Acta metall. 12, 1399 (1964). 6. G. R. Speich and R. A. Oriani, Trans. TMS-AZME 233, 623 (1965). I. A. J. Ardell and R. B. Nicholson, Acta metall. 14, 1295 (1966). 8. A. J. Ardell and R. B. Nicholson, J. Phys. Chem. Solids 27, 1793 (1966). 9. G. P. Airey, T. A. Hughes and R. F. Mehl, Trans. TMS-AZME 242. 1853 (1968). 10. A. J. Ardell, in The M&ha&ms of Phase Transformations in Crystalline Solids, p. 111, The Metals Society,
London, (1969). 5
4
3
2 1.5 1.5
2
2.5
3
3.5
4
4.5 z 5
(b) Fig. 12. The normalized concentration c as a function of the radial variable 5 for selected 0 values for an aspect ratio of 20: (a) 5 ranging from 0 to 1.5 with an inset showing the whole ?K scale for curve 1; and (b) magnified ?& scale for 5 beyond 1.5.
11. G. Sauthoff and M. Kahlweit, Acta metall. 17, 1501 (1969). 12. A. J. Ardell, Metall. Trans. 1, 525 (1970). 13. P. K. Rastogi and A. J. Ardell, Acta metall. 19, 321 (1971). 14. J. D. Boyd and R. B. Nicholson, Acta metall. 19, 1379 (1971). 15. D. J. Chellman and A. J. Ardell, Acta metall. 22, 577 (1974). 16. F. Ferrante and R. D. Doherty, Acta metall. 27, 1603 (1979). K. Oki and T. Eguchi, Trans. 17. Y. Seno, Y. Tomokiyo, Jpn. Inst. Met. 24, 491 (1983). 18 H. Calderon and M. E. Fine. Mater. Sci. Enp. _ 63. 197 (1984). 19 G. J. Shiflet, H. I. Aaronson and T. H. Courtney, Acta metall. 27, 377 (1979). 20 P. Merle and F. Fouquet, Acta metall. 29, 1919 (1981). 21. P. Merle and J. Merlin, Acta metall. 29, 1929 (1981). 22. T. G. Zocco and M. R. Plichta, Scripta metall. 18, 555 (1984).
MOU
and HOWE:
DIFFUSION
FIELDS
ASSOCIATED
23. S. Elangovan. Ph.D. Dissertation, University of Utah, Salt Lake City. UT (1984). 24. K. E. Rajab and R. D. Doherty, Acta metall. 31, 2709 (1989). 25. R. D. Doherty and K. E. Rajab. Acta metall. 37, 2723 (1989). 26. R. M. Aikin, Jr and M. R. Plichta, Acta metall. 38, II (1990). 27. R. M. Aikin, Jr, S. Elangovan, T. G. Zocco and M. R. Plichta. Metall. Trans. A 22A, 1381 (1991). 28. H. I. Aaronson, in The Decomposition of Austenite b? DQ@ional Processes (edited by V. F. Zackay and H. I. Aaronson), p. 387. Interscience, New York (1962). 29. H. I. Aaronson, J. Microsr. 102, 275 (1974). 30. G. J. Jones and R. K. Trivedi, J. Appl. Ph~s. 42, 4299 (1971). 31. G. J. Jones and R. K. Trivedi, J. Cryst. Growth 29, 155 (1975). 32. H. 1. Aaronson. T. Furuhara, J. M. Rigsbee, W. T. Reynolds, Jr and J. M. Howe, Metal1 Trans. .4 21A, 2369 (1990). 33. H. I. Aaronson, Metall. Trans. 24A, 241 (1993).
34. 35. 36. 37. 38. 39.
40.
41.
42.
43.
WITH
PROLATE
SPHEROIDS
835
F. S. Ham. J. Phys. Chem. Solids 6, 335 (19.58). F. S. Ham, Quart. Appl. Math. 17, 137 (1959). G. Horvay and J. W. Cahn, Acta metall. 9, 695 (1961). E. A. Guggenheim, Thermodvnamics, pp. 160, 198, Interscience. New York (1957). Yiwen Mou and J. M. Howe, Metall. Mater. Trans. A. in press. L. A. Pipes and L. R. Harvill, Applied Mathematicsfor Engineers and Phvsicists. vv. 412434. 7999807. MlGraw-Hill. New’York (19%). P. M. Morse and H. Feshbach, Methods of Theoretical Physics, vol. I, pp. 21-31. 492-675. McGraw-Hill. New York (1953). P. E. Lewis and J. P. Ward, ktctor .4na[vsis .for Engineers and Scientists, pp. 2499278. Addison-Wesley, New York (1989). N. N. Lebedev. Special Functions and Their Applications, pp. 161-203. Prentice-Hall. Englewood Cliffs, NJ (1965). L. C. Andrews, Special Functions for Engineers and Applied Mathematicians, pp. 116165. MacMillan New York (1985).
.4cta marer. Vol. 45. No. 2. pp. 823-835. 1997 Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved 1359-6454,97 $17.00 + 7.00
DIFFUSION FIELDS ASSOCIATED WITH PROLATE SPHEROIDS IN SIZE AND SHAPE COARSENING Y. MOUt Department
and J. M. HOWE
of Materials Science and Engineering. University Charlottesville. VA 22903. U.S.A.
of Virginia.
(Receioed 22 March 1995; accepted 6 March 1996)
Abstract-The
diffusion field or solute concentration distributed around a prolate spheroidal particle simulating a rod- or needle-shaped precipitate has been solved for varying aspect ratios of the precipitate and varying concentrations along the precipitate surface owing to the curvature effect. With the prolate
spheroidal coordinates, the principal curvatures of the prolate spheroidal surface are derived as functions of the angular variable, and the Laplace field equation is separated into two Legendre equations either on the radial variable or on the angular variable. The analytical solution fitting the present boundary conditions is secured as the sum of a Legendre function of the second kind of order zero and a Legendre series. The Legendre function of the second kind gives the concentration distribution when the effect can be ignored, whereas the Legendre series represents the concentration contributed curvature effect. Numerical results of normalized concentrations are presented as functions of and angular variables for selected aspect ratios. The tangent component of the concentration contributed from the curvature effect, may cause complicated mass transfer. which is responsible
curvature from the the radial gradients. for shape
coarsening.
1. INTRODUCTION
non-equilibrium kinetic factors. Overall, agreement between the theory and experiment often requires serious modifications [5-181. In developing coarsening theory for non-spherical precipitates. some investigations [ 19-271 considered the formation and lateral movement of growth ledges [28, 291 on precipitate facets in coarsening kinetics. Quantitative analyses by Aikin and Plichta [26] and Aikin et al. [27] applied Jones and Trivedi’s results [30, 311 for the concentration gradient in front of a single ledge, obtained by assuming no solute penetration across the terrace and an equal moving speed for the whole ledge. Based on the ledge mechanism of precipitate growth or dissolution [32, 331, these analyses [26, 271 predict coarsening behaviour of disc-shaped particles, which is much closer to experiments than that predicted with the LSW approach modified but still holding the assumption of isotropic atomic processes. However, as a measure of the anisotropy of atomic processes, ledge densities or interledge spacings cannot be evaluated or estimated in these treatments; instead, experimentally measured interledge spacings were used as inputs to the analyses. Needle (or rod) and disc-shaped precipitates can be mathematically modelled as prolate and oblate spheroids, or elongated and ffattened spheres at the poles, respectively. Ham [34, 351 and Horvay and Cahn [36] developed kinetic theories for ellipsoidal and spheroidal growth, respectively. Although growth and coarsening are governed by similar diffusion processes, there exist some obvious differences. which prevent these theories from applying to
The coarsening behaviour of precipitate particles is one of the most important problems in the thermal stability of precipitation-hardened alloys at elevated ambient temperatures. In metallic alloys, small and densely distributed precipitates act as obstacles to dislocation motion. When coarsening occurs at elevated temperatures, it increases the size and decreases the number density of the precipitates, thereby reducing their effectiveness as strengtheners. Understanding the coarsening kinetics is an important step in improving the long-term high-temperature performance of these alloys. Following Greenwood’s first treatment [ 11, systematic coarsening theory, now referred to as the LSW theory, was developed in 1961 independently by Lifshitz and Slyozov [2] and by Wagner [3]. Although one of the assumptions in the LSW theory, i.e. steady-state diffusion, is valid for small supersaturations and is well satisfied for metallic coarsening processes [4], the assumption that precipitates are spheres and that the atomic attachment or detachment is isotropic 12. 31 mostly restricts application of the theory. Non-spherical particles frequently form as a result of anisotropy in interfacial energy in solid-solid phase transformations. Furthermore, many precipitates are needle or disc-shaped owing to high-level anisotropy in interfacial energy and
tTo whom all correspondence should be addressed at AFE Technologies, 368A Greenbrier Dr., Charlottesville. VA 22901, U.S.A. 823
824
MOU and HOWE:
DIFFUSION
FIELDS
ASSOCIATED
coarsening processes. First, needle- or disc-shaped particles are, in general, involved in concurrent size and shape coarsening (with varying aspect ratios) [26], whereas these growth theories only treat shape-conservative kinetics (with constant aspect ratios). Secondly, the effect of surface curvature on solute concentration (the Gibbs-Thompson effect [37]) is much more important in coarsening than in growth processes because growth or shrinking of particles in coarsening is determined by their curvatures in comparison with some mean curvature, whereas the driving force in growth is the decrease in volume free energy. If the curvature effect is considered, solute concentrations will vary along the particle surface. The growth theories mentioned above, however, assume a constant concentration along the surface. Any coarsening kinetics theory of spheroidal particles must be much more complex than the LSW theory for spherical particles because two independent variables, i.e. two principal axes or one of them and the aspect ratio, are required to describe the particle geometry. The present paper deals with the first problem in prolate spheroidal coarsening, namely, the solute concentration field distributed around a prolate spheroidal particle. The governing diffusion field equation will be solved under a boundary condition for varying solute concentrations reflecting the curvature effect. A companion paper [38] is devoted to the parallel problem for oblate spheroidal coarsening.
WITH
PROLATE
SPHEROIDS
diffusion field around any individual particle.) For emphasizing the needle morphology in practical coarsening problems, the half-length I and radius r of the particle are used interchangeably with the half major and minor axes, a and b of the generating ellipsis, respectively. As shape coarsening is assumed, the aspect ratio A, or a/b, is no longer constant. Figure 1 also shows three sets of schematic concentration curves starting from different surface positions with angles tll, ~12and c(~from the I axis. For each set of concentration curves, C, is the precipitate concentration (illustrated for solute-rich cases), and CO, is the matrix concentration at the particle surface with the curvature effect ignored. As CI decreases from n/2 to 0, the concentration increment C, from the curvature effect increases because the principal curvatures get greater. As a result, the matrix concentration at the precipitate surface, C, , is a function of the angle tl and the surface shape defined by the aspect ratio A. If the particle is very small compared with the neighbouring matrix area free of other precipitate particles, a constant matrix concentration C* [26, 271 can be assumed at infinitely remote distances. It is noticed that this assumption aims to avoid unnecessarily complicated mathematics at the beginning phase of the present development. In subsequent investigations in progress, this assumption will be released, so that the effect of precipitate volume fraction can be incorporated into spheroidal coarsening kinetics. 2.2. Prolate spheroidal coordinates
2. FIELD EQUATION AND ITS SOLUTION
2.1. Assumptions and the governing field equation
Similar to the LSW spherical coarsening [2, 31, the spheroidal coarsening treated hereafter is assumed to be controlled by a steady-state diffusion process. Accordingly, the time-dependent diffusion equation [39] becomes the Laplace equation v2c = 0
Equation (1) can be solved only if prolate spheroidal coordinates [40] are adopted because the boundary conditions of solute concentration at the precipitate surface and at a remote boundary can be separated only in these coordinates. The prolate spheroidal coordinates are related to the Cartesian
(1)
where C is the solute concentration, and V2 is the Laplacian operator, i.e. V2 = V. V. In the Cartesian coordinate system V=ii+$j+$k where i, j and k are the Cartesian unit vectors. The approach that assumes the concentration distribution satisfies equation (1) instead of the time-dependent diffusion equation is known as the invariable field approximation in growth kinetics literature [4]. Equation (1) has been used almost exclusively in the literature since it simplifies the mathematics that might otherwise be intractable. An isolated prolate spheroidal particle (Fig. 1) is assumed to be surrounded by a strain-free matrix. (The particle size distribution is not considered here since the purpose of this work is only to solve the
Y
a, < a, c
ator
a3
Pole b 1
0
Distance
Fig. 1. Schematic curves showing solute concentrations in the matrix as functions of distance from the precipitate centre. Inset is a prolate spheroid simulating a rod- and needle-shaped precipitate.
MOU and HOWE: DIFFUSION FIELDS ASSOCIATED WITH PROLATE SPHEROIDS coordinates
through the transformation X = c&’ y = c
2
equations
I
- l)(l - Y/2)cos 4.
(3a)
(t2 - l)(l - $) sin 4,
(3b)
Z = c&J,
825
e=o
(3c)
where c is the half interfocal distance of the prolate spheroidal particle c=Jz_bz
(4)
and 5, q and $I are the three prolate spheroidal coordinates. It is convenient to explain the coordinate geometrically if intermediate transformation parameters p and 8 are introduced so that c = cash p,
@a)
q = cos 0,
(5b)
where p and 0 are called the radial and angular variables, respectively, since they are analogous to their counterparts in the spherical coordinates [41]. A curvilinear coordinate system is characterized by its coordinate constant surfaces and coordinate curves [41]. In the Cartesian system, for example x. y and =-constant surfaces are respectively planes parallel to the Y-Z, Z-x and x-y planes inclusive, and coordinate curves are lines parallel to the x, y and z coordinates inclusive. Combining equations (3) and (5) yields
y + g
I
2 sin’ Q
-2 c2 co? 6
1.
(6b)
Equation (6a) represents an infinite array of prolate spheriods with half-major axes of c cash p and half-minor axes of c sinh p. The precipitate surface is a special (@)-constant surface, defined by 4 = 5, or p = Pi, so that a = c cash pp = ~5,.
(7a)
b=csinhp,=cfi.
(7b)
&=S=&&
(7c)
Equation (7~) results from equations (4) and (7a) and the definition of the aspect ratio. Since &, depends only on A, it may be called the shape parameter. The r/(0)-constant surfaces, represented by equation (6b), are hyperboloids of revolution of two sheets. The q-constant hyperboloids and l-constant prolate spheroids form a confocal system (with the same half focal distance c). It is seen from equation (3) that x/y = tan d, and that the : coordinate is independent of 4. Accordingly, $-constant surfaces are planes including the z axis and having an angle of 4 with the positive x axis. Figure 2 is a three-dimensional view of the prolate spheroidal coordinate system, shown as related to the Cartesian coordinates. The point P has coordinates
5 coordinate
\
Fig. 2. A three-dimensional view of the prolate spheroidal coordinates. shown as related to the Cartesian coordinates.
of (0.6, k/6, n/4) in the (p, 0, 4) format or (1.185, -0.866. 7c/4) in the (5, q, 4) format. At point P, the t coordinate curve is a hyperbola, which is the intersection curve between the &constant hyperboloid and &constant plane both including P. Similarly, the 0 and 4 coordinates are an ellipsis and a circle, respectively, resulting from intersection of the $-constant plane and <-constant spheroid and of the {-constant spheroid and B-constant hyperboloid, all crossing the point P. The prolate spheroidal coordinates are orthogonal since the three coordinate curves are mutually perpendicular everywhere. Figure 3 shows a d-constant section of the prolate spheroidal coordinate system with a half-focal distance of c. The X-JJ axis in the figure is any ray in the x-y plane, starting from the origin. The prolate spheroidal coordinate system is indeed built through a rotation of the 4-constant section around the Cartesian z axis. The scale factors are introduced into the application of curvilinear coordinates to build their scalar relationships to the Cartesian system [41]. For the prolate spheroidal system, these factors can now be calculated from equation (3)
h@ =
II dr
@
= c&
-
l)(l
-
$)
(SC)
826
MOU and HOWE:
DIFFUSION
where r is the Cartesian position r = xi + yj + zk. The gradient operator Laplacian V2 [41] in the prolate spheroidal then secured through the scale factors.
FIELDS ASSOCIATED WITH PROLATE SPHEROIDS vector of V and the system are
P (E
(9)
v*=v.v =
c2(t2L ‘i’){$[(r’ -4 Fig. 4. Principal curvatures of a prolate spheroidal surface B at the point P, defined as the curvatures of the curves Kq and K$ at P.
(10) These equations will be used later to derive the diffusion field equation appropriate to prolate spheroidal coarsening. 2.3. Principal curvatures and surface concentrations Since matrix concentrations at the precipitate surface are affected by the surface curvatures, it is essential that expressions for the surface curvatures of a prolate spheroid be evaluated in the prolate
spheroidal system. At a surface point P, shown in Fig. 4, the unit vector [ is normal to the precipitate surface Z with 5 = 5,. If a plane is drawn (then removed in the figure for clarity) through the unit vectors f and d, it intersects the precipitate surface along the ellipsis Kq. Similarly, the plane Tq is drawn through f and $, intersecting the precipitate surface along the curve K+. The curvatures of K, and K4 at P, known as the principal curvatures [41] and denoted as K, and Q, can be determined through the scale factors given by equation (8)
1 ah@ Icd= -h;h,ag
5, SC{ =
-452
P
_
I)(52
P
_
f)’
P
(lib)
Y x
These curvatures can also be expressed in terms of the half-major and half-minor axes of the generating ellipsis through equation (7) lc, = -
ab (a’ - c* cos* 0)3’
(124
Wb) --2
Fig. 3. A +-constant coordinate surface in the prolate spheroidal coordinates with a half focal distance of c, showing curves of intersection of the &constant surface with <-constant and q-constant surfaces.
Two special cases are worth noticing: 1IC,1= Iq 1= a/ b* at the two poles where f3 is 0 or z, and IIC,I = b/a* and IK+1= l/b along the equator where 0 is n/2. The Gibbs-Thompson equation [37], expanded only to the linear terms, gives the matrix concen-
DIFFUSION
MOU and HOWE: tration
at the prolate
spheroidal
surface
c,, = Co,[l + r(lJG!I +
are thus Legendre functions of the second kind en([). Solutions to equation (14) then have the form
5 = i,
IJ%Ill
(13)
concentration at the where C,,,,, is the matrix precipitate surface when both curvatures are 0, and r = oVm/[(Cp - Co,)W
(13a)
where 0 is the interfacial free energy of the precipitate surface, V,,,the partial molar volume of solvent in the precipitate, C, the precipitate concentration, R the gas constant, and T the absolute temperature. Although generally a function of r~ and 4, 0 is a function of only r~ here since the geometry of the problem possesses an infinity-fold rotation symmetry around the z axis.
where F,, are arbitrary constants. The specific solution to equation (14) that satisfies the present boundary conditions can be written as the sum of two contributions
where
Cd<, u) = ~Pd;rl)QdS)/Qdi’,) or equivalently
2.4. Solution qf the governing boundary conditions
equation
and
the
It is not surprising that the curvatures and, the matrix concentrations at the accordingly, precipitate surface are independent of 4 (equations (11)) because of the rotation symmetry. Substituting equation (10) with the $-dependence eliminated into equation (1) obtains the governing field equation
+-
1)$]+$1
-$)$]=O.
(14)
with M and N to be determined,
c,((,
-5~~~]+v(v+
l)E=O,
(15)
$1
-+]+v(v+
l)H=O
(16)
Fe,,(z)
(17)
where PI,(z) and Q”(Z) are Legendre functions of the first and second kind, E and Pare arbitrary constants, and u stands for B or H and z and 5 or 11. Solutions to the present problem are restricted by the bounding and periodic properties of the Legendre functions. It is noted that the domain of the radial variable is 1 < t < cc or 0 < p < co while that of the angular variable is - 1 < q < 1 or 0 < 0 < 27~. For the angular solutions H(u), P”(q) are bounded at ye= + 1 only if v values are restricted to non-negative integers n. Q.(q) must be rejected because they are unbounded [42]. The angular solutions to the present problem are thus Legendre polynomials P,,(q) or P,(cos t3). For the radial solutions Z(t), on the other hand, Pn(<) must be rejected because they are unbounded when 5 *co. The radial solutions Z(c)
WI
and
r/) = t F”P.(vl)Q.(S)/QfIC~,). n=O
(21)
The constants Q.(&,) in equations (20) and (21) are introduced to simplify expressions for M, N, and F’. The boundary conditions at the precipitate surface 4 = &, are specified as Co(&) = c,,,
(22a)
GJ(l~~I + IQI).
As assumed in Section 2.1, the boundary at 5 -+co can be stated as
where v is the separation parameter. Equations (15) and (16) are both Legendre’s equations, which have general solutions of the form [40]
GW
(note that PO(v) = 1 [43])
C,(&, r) = a[il
+ N
co(i”) = kfQo(5)/Qo(t,) + N
Let C({, r~) = E(e)H(y). Equation (14) can then be separated into two ordinary differential equations
u = EP,,(z) +
827
FIELDS ASSOCIATED WITH PROLATE SPHEROIDS
GQb) conditions
c*,
(23a)
C,(co, q) = 0.
Wb)
Co(m) =
Substitution of equation (22) into (19) verifies that these conditions are equivalent to equation (13). It is obvious that C(co, PI) = C* from equations (19) and (23). It is now clear that C,(l, n) is the concentration contributed from the curvature effect, and that C,({, q) is the concentration distribution when the curvature effect can be ignored. 2.5. Evaluation
of the coeficients
The evaluation of A4 and N is rather straight forward. Substitution of the boundary conditions for C,(t). equations (22a) and (23a), into equations (20b) gives M=Cnm--C*
and
N=C*.
(24)
The boundary condition, equation (23b), is satisfied since Legendre functions of the second kind Q.(t) approach 0 when t-0 [42]. Combining equations (21) and (22b) (substituting equation (11)
828
MOU and HOWE:
DIFFUSION FIELDS ASSOCIATED WITH PROLATE SPHEROIDS
for K, and icr) yields
where the two definite integrals are
= f
F,P”(r/).
(25)
n=O
In order to seek coefficients dependent precipitate shape, let
only on the
6 = F”Z/(COnlZ)
(26)
where I is the half-length of the precipitate needle, and I = a = cl, (equation (7b)). This simplification is based on the assumption that Z, on the interfacial free energy 0 (related to the equilibrium aspect ratio Ao), is a weak function of q compared with the surface curvature (the bracketed terms in equation (25)). When the actual (non-equilibrium) aspect ratio A is large (A is usually much greater than AC,), as is frequently the case in the coarsening of needle-shaped particles, the bracketed terms are a very strong function of rl because the dramatic changes in curvature from the equator where the curvature is 0, to polar positions where the curvature approaches infinity. Furthermore, this treatment has another convenience that only an average value of e is needed since the function a(q) is unknown in most cases. For theoretical completeness and more accurate treatments when the function a(q) is available, further development is necessary and currently in progress. This will be addressed in a separate publication because of its complexity. Substituting equation (26) into (25) yields
It is noted that the denominators of both integrands in equation (30) are even functions and that Legendre polynomials of odd orders are odd functions [39]. Hence, the two integrals with n odd equal 0, so that fn vanishes when n is odd. For even n or 2m, I,+(n) can be evaluated by a formula in Ref. [42] (p. 202, problem 16) Z+(2m) = 2iP2,(O)Q&Jt~
- 1)
(31)
where i is the imaginary unit. Since even-order Legendre functions of the second kind with pure imaginary arguments have pure imaginary values [38], Zm(2m) is real, as the integral expression itself may tell us in equation (30b). For evaluating Z,,(2m), a relation between the two integrals needs to be derived by taking the derivative of Z4(2m) with respect to 5,
= f
:I&_Cl;-
W’2f’n(v) drl = - 5,4. (32)
1,,(2m) can then be evaluated using equations (31) and (32) (33)
(27)
where Qim(i&$?) are the first derivatives of even-order Legendre functions of the second kind with a pure imaginary argument, taking real values 1381.
The coefficients fn can now be evaluated as follows: multiply both sides by Pm(q) (m are non-negative integers), and then integrate both sides of equation (27) from - 1 to 1 by applying the orthogonality properties of Legendre polynomials [39] 0 if m # n; Pm(v)f’n(v)drl = 2/(2n + 1) if m = n. f -I It is then obtained for non-negative
(28)
integers n that
h = (n + 1/2)%G - l)-“*[(t; - l)Z,(n) + Z&r)] (29)
In summary, subject to the boundary i.e. equations (22) and (23), or C(&, rl) = Co, + CC%
COJ(IK, ?I
=
I c*,
+
conditions,
h
I>,
(344 Wb)
the solution of the governing diffusion field equation, equation (14), is given by combining equations (19), (20b), (21) (24), (26), (2% (31) and (33) as C(5, q) = (C,, -
c*j gg$+ c*
829
MOU and HOWE: DIFFUSION FIELDS ASSOCIATED WITH PROLATE SPHEROIDS non-negative
where
integer in [39] (42) P?,,+ , (0) = 0.
I iQzm(iJ
Jm
(36j
1
3. NUMERICAL RESULTS AND DISCUSSION 3.1. Legendre polynomials second kind
and functions
of the
Legendre polynomials P,(s) and Legendre functions of the second kind Q.(X) satisfy the following recurrence formula [39] (written only for P,(X)) P,(x)
= (2 - l/?Z).XP~_,(.X)- (1 - Iln)P,+,(.u).
(37)
This recurrence relation makes the evaluation of P,,(x) and Qn(x) very efficient beginning with the first two Legendre polynomials and Legendre functions of the second kind [43] P&x) = 1,
1
x+1
PO(x) = - In .u_l, 2
P,(x)
= 5;
(38)
Ql(x)=~ln~-1; IX > 1.
(39)
This approach works fine for the Pfl(q) evaluation with - 1 < q < 1 or 0 > (3> 271 (actually, calculations are needed only for 0 < q < 1 or 0 > 0 > n/2 because of the symmetry of the diffusion field). It was found, however, that Qn(<) cannot be correctly evaluated for 5 > 1 with n increasing because this algorithm fails to tolerate the inevitable truncation error in the computer. An alternative was then taken with n decreasing; equation (37) thus takes the form (written for Q+,(X)) Q.-I(X) = (2 + I/n).yQ.(_x) - (1 + l/n)Q,+,(.u). (40) The two functions of the highest and second highest order then must be evaluated first through [39]
Q,z(x)
=
5 r=o
x”
2,
I
2”(n + r)!(n + 2r)! ; r!(2n + 2r + l)!
IX > 1 (41) where lOG5000 terms in the infinite series are required dependent on the 5 values and the desired accuracy. It is also feasible to use equation (41) to evaluate all en(t) involved. The values of Pn(0), needed in determining the coefficients fn, can be evaluated through equations (37) and (38). However. note that for anvd
(43)
Equation (42) is thus more convenient than equations (37) and (38) for the P>,(O) evaluation. According to equation (43), on the other hand, the f?,,, formula, equation (36), also holds for odd n, i.e. all fim _,are 0. In fact, equations (31) and (33) are also valid for n odd. Therefore, if P”(O) are evaluated through equations (37) and (38),f, for either even or odd n can be evaluated through equation (36) substituting n for 2m. This algorithm takes a slightly longer time to run but is simpler in logic because all iterations are performed consistently on consecutive integers. Different approaches may be used to check the calculational results. A detailed discussion of Legendre functions of the second kind with an imaginary argument is postponed to a companion paper [38] on oblate spheroidal coarsening in which these functions are solutions to the field equation. It suffices here to notice that equation (39) is also valid for an imaginary argument owing to the analytic continuation of Legendre functions of the second kind [42]. The quantity im is thus substituted for .Y in equation (38); the calculation of QZ,,,(i,/m) is carried out through the complex variable package implemented in many languages such as FORTRAN and C. It is found that the recurrence formula (37) with II increasing is suited to this evaluation. The first derivative of QO(:) can be obtained from equation (39) as Q;(Z) = l/(1 - ?)
(44)
where z is used instead of x to emphasize that it can have complex values. The first derivatives of Legendre functions of the second kind, of order n 3 2, are related to Legendre functions through the following recurrence formula [39] (written for Qn(:)) Q:(Z) = [n/(1 - =‘)I@-
I(=) - zQ,,(z)].
3.2. Solution with an insigr$cant
(45)
curvature efjrect
When the precipitate half-length I is very large for limited aspect ratios, the factor in front of the series in equation (35) becomes very small, so that the concentration contributed from the curvature effect may practically vanish. Alternatively, if the interfacial energy 0 is very small. the curvature effect also becomes insignificant. These limiting cases suggest that with insignificant curvature effects, the solution to the field equation, equation (35) reduces to C”(t) = (Co, - C*) a
+ C*
(46)
where C(t) is used instead of C(t, 9) since the solute concentration has no angular dependence.
830
MOU and HOWE:
Substituting
DIFFUSION
FIELDS ASSOCIATED WITH PROLATE SPHEROIDS
equations (7~) and (39) gives
c (5) _ C(5) - C* II co, - c* 1
ln5+l 5-1
=2ln(A+J’A’-I)
(47)
where the concentration constants are collected to form the normalized concentration c,,(g), which depends only on the radial variable 5 and aspect ratio A. It is further noticed that the effect of A on the normalized concentration is a separated factor. In other words, z’,(g) curves are different only in the “magnification” of the concentration axis. The plots in Fig. 5 show the relations between the normalized concentration c(t) and radial variable 4 for various aspect ratios from 1.25 to 20. For each of the curves, z’,(r) decreases from unity, the normalized concentration at the precipitate surface with 5 = 5,. The l, values are listed in the figure, corresponding to the selected aspect ratios. It is seen that z;,(t) decreases with increasing 5 more rapidly for larger aspect ratios. In fact, this conclusion can be drawn from equation (47) since the factor containing only A decreases with increasing A. Concentration gradients, however, cannot be simply compared because the radial variable 5 is not equivalent to a distance variable. Figure 6 shows a three-dimensional 2’,(c) plot for an aspect ratio of 5. The position coordinates correspond to the z and x-y axes in Fig. 3. Because the diffusion field has an infinity-fold rotational symmetry around the z axis, the two position coordinates can represent all positions in the diffusion field. Equipotentials, or surfaces of equal concentrations, are represented by confocal ellipses positioned at different concentration levels (only the first quadrant is drawn owing to the reflection symmetries). The ellipsis at the highest level
Curve
5,
c
Fig. 6. A thr_ee-dimensional plot of the normalized concentration C,,(t) as a function of distances along the z and x-y axes (see Fig. 3) for an aspect ratio of 5. The coordinate scales are normalized with the half focal distance c.
represents the precipitate surface, an equipotential with Co(&) = Co, or z’,(&) = 1. It is noticed that although solute concentrations are constant on individual prolate spheroidal surfaces, concentration gradients vary with the angular variable 8. This becomes clear after observing that equal concentration decrements are established over different distance intervals along different &constant concentration curves. In particular, the concentration gradient at the poles (0 = 0) is the maximum, and that along the equator (0 = 7c/2) is the minimum. 3.3. Contributions from the curvature effect The last term in equation (35) is contributed from the curvature effect. The factor in front of the series is a function of the particle size I and system thermodynamic parameters Co, and r. The series, on the other hand, depends on only precipitate shape, which may be represented by either the shape parameter 4, or aspect ratio A. The curvature contribution to the concentration distribution can thus be normalized as
A
0.8
0.6
0.4
0.2
1 0
2
4
6
8
10 5
Fig. 5. Functional relations between the normalized concentration c(t) and radial variable c for various aspect ratios from 1.25 to 20.
where f2,,, are given by equation (36). According to the convergence theorem of the Legendre series [43], the series in the left-hand side of equation (27) is convergent to the right-hand function for all - 1 < rl < 1 or 0 < 0 < 27~.Since the quotient of two Legendre functions of the second kind in equation (48) is no greater than unity [42], the series in equation (48) converges for all 5 and q in the
MOU and HOWE:
DIFFUSION
present problem. The speed of convergence, however, relies on the 4, values. In general, the smaller is the 5, value (the greater is the aspect ratio), the larger is the number of series terms required for a preset accuracy. It is noticed that the slowest convergence for a given 5, value is achieved when < = t,, under which condition the quotient factors in the series are all unity. Particularly, when [ = [, and VI= 1 (noting that PZm(l) = 1 [43]), substituting equation (27) into equation (48) obtains
C,(;“,. I)
=
i
.fim= &
n,= II
=24’.
P
831
FIELDS ASSOCIATED WITH PROLATE SPHEROIDS
A=5 Curve
0 (“)
::
11.25 0
: 5
22.5 45 90
1.3
1.4
(49)
This states that the sum of all fimcoefficients is 2A’, which is also the maximum of ?,(t, q) because PZm(q) Q 1 [43] (see equation (48)). This simple relation provides a very useful means to check the calculation of the f&, coefficients. Furthermore, it can be employed to determine the number of terms required for a preset accuracy in the sense that the infinite series has to be truncated in practical evaluations. In the numerical results presented later, for example, the maximum of m is selected as 20. 40 and 80 for an aspect ratio of 5, 10 and 20, respectively, for a relative error of less than 0.002. Figure 7 shows the numerical results of the normalized concentration c, or C,l/(C,,r) distributed around a prolate spheroidal precipitate with an aspect ratio of 5. There are nine curves in the figure, displaying c;, as functions of the angular variable 8 for selected 5 values beginning with {, (1.021 for A = 5) with increments doubled consecutively. It is observed that (?h reaches the maximum value of 2A’ along curve 1 (5 = r,) at q = 1 (0 = 0’). This represents the situation at the poles of prolate
1.1
1.2
1.5 5
(a) 3.5 A=5
2.5
I\\\
Curve
u (9
:
1 I!25 22.5
3
2-
1.5
l-
0.5 ’ 1.5
2
2.5
3
3.5
4
4.5 5
50
(b) A=5
G
40 n
1
curve
5
:. 3 4
1.021 1.041 1.082 1.165
i I 8 9
1.660 1.330 2.320 3.639 6.279
30 2 20
Fig. 8. The normalized concentration c, as a function of the radial variable ; for selected 0 values for an aspe$ ratio of 5: (a) 5 ranging from 0 to 1.5; and (b) magnified C, scale for t beyond 1.5.
~
I~~~, 15
30
45
60
75
90
e (“) Fig. 7. The normalized concentration ck contributed from the curvature effect as a function of the angular variable 0 for selected 5 values for an aspect ratio of 5.
spheroidal precipitates, where the two principal curvatures are the same and maximum. For all the curves, SK decrease with 0 increasing from 0 ’to 90.. However, c, changes slowly in an area around each pole, which expands with increasing [. The most rapid decrease of cK occurs in a 0 interval of about 15” beyond the polar area. The curvature effect on concentration is approximately uniform in a broad Q band some 45” up and down from the equator (see Fig. 1). At a distance of about 6.3 times the precipitate half-length (curve 9) from the precipitate centre, c, values are less than 1 and almost independent of 0. It is worth emphasizing that the actual concentration C, from the curvature effect has the same monotonicity as does the normalized
832
MOU and HOWE:
DIFFUSION
FIELDS ASSOCIATED WITH PROLATE SPHEROIDS
concentration cK for solute-rich precipitation (positive r). In the solute-poor case the actual concentration will increase with 0 increasing and reach its minimum at the poles. ?, can also be drawn as a function oft for selected 0 values, as shown in Fig. 8, again for A = 5. The five curves all begin with 5 = &, i.e. the precipitate surface, running to 5 = 1.5 in Fig. 8(a) and being continued in Fig. 8(b) in a magnified concentration scale for clarity. It is observed that the c?~ values starting from the precipitate surface are quite different along different &constant surfaces, though they arrive at almost the same value when 5 exceeds 3.5. Figures 9 and 10 show that the CC values decrease with increasing 0 along selected g-constant surfaces
200 , A = 10
2;, 160.
Curve
9 (“)
:.
11’25
: 5
22.5 45 90
120.
80
0
1.0
1.1
1.2
1.3
1.4
1.5 5
(a) A = 10
Curve
5 A =
1.005 1.010 1.020 1.040
: :
80
10
Curve
6 (“)
1 :
122.5 lo25
4 5
45 90
I 0
15
30
45
60
75 6 (“)
90
1’ 1.5
2
2.5
3
3.5
4 5
(a) (b) 20
h
E 16 x t\
A = 10
Curve
5
5
1.081 1.161 1.322
5
!
Fig. 10. The normalized concentration cx as a function of the radial variable 5 for selected 0 values for an aspect ratio of 10: (a) 5 ranging from 0 to 1.5; and (b) magnified C, scale for 5 beyond 1.5.
I 0
15
30
45
60
75 9 (“)
90
(b) Fig. 9. The normalized concentration of c as a function of the angular variable 6 for selected 5 values for an aspect ratio of 10: (a) four r values from 1.005 to 1.040; and (b) magnified & scale for five 5 values from 1.081 to 2.290.
and with increasing 5 along selected &constant surfaces, respectively, for an aspect ratio of 10. The maximum of &, 200, is much greater than that in Figs 7 and 8 for A = 5 since the maximum is proportional to the square of the aspect ratio. For an aspect ratio of 20, as shown in Figs 11 and 12, the maximum of c% becomes as great as 800 (Figs 1 l(a) and 12(a)). In general, the actual or total solute concentration around a prolate spheroidal particle, is the sum of the concentration with the curvature effect ignored and the concentration contributed from the curvature effect. However, the results shown in Figs 7-12 cannot simply be added to get the total concentration. The reason is that the two normalized
833
MOU and HOWE: DIFFUSION FIELDS ASSOCIATED WITH PROLATE SPHEROIDS concentrations, c0 and c,, are normalized with respect to different thermodynamic parameters. This is why they have very different scales; in particular, ?, can range from zero to infinity. The actual concentration can be obtained with the following relation given by substituting equations (47) and (48) into (35) C(5,?1) = (C”, - c*)G(5)
+ c* + (C”J/Y)C,(S,
80 70
G a50 4c-
a)
30
(50) where (?, and cK are normalized concentrations defined by equations (47) and (48) respectively. When a coarsening system is specified, the plotted results can be substituted into equation (50) to get the actual concentration for aspect ratios of 5, 10 or 20. Concentration gradients at the precipitate surface are essential for the analysis of coarsening kinetics. When the curvature effect is ignored, the solute concentration depends only on the radial variable (equation (46)). The concentration gradients are directed normal to the precipitate surface because the tangential component of the concentration gradient is zero, as can be seen from equation (9) (the coefficient of the unit vector Q). However, the concentration contributed from the curvature effect depends on both the radial and angular variable, so that the concentration gradient has both normal and tangential components. The normal components of the concentration can be qualitatively observed from the c,-< curves (Q is constant) in Figs 8, 10 and 12 at the slopes of the tangents to the curves. As shown in equation (9), however, their exact values are modified by the factors before the < partial derivative. The tangent components are qualitatively observed from 800 I\
A = 20
700
I \
1 z
1
4
1.001 1.003 1 .oos 1.010
0
45
(b)
80 A = 20
70 CK
5
\
60
Curve
5
5
1.020 1.040 1.080 1.160
50
76 8
40 30 20
;
10
9 L_
0
15
30
45
60
75
90
9 (“) (c) Fig. 11. The normalized concentration c, as a function of the angular variable B for selected 5 values for an aspect ratio of 20: four 5 values from 1.001 to 1.010 in 0 intervals of (a) O’-15” and (b) 15‘-90” with magnified r, scale for clarity; and (c) magnified i;, scale for five 5 values from 1.020 to 1.320.
the Sk-Q curves in Figs 7,9 and 11; they are non-zero except at positions with f3= 0’ or 6 = 90”, particularly at the poles and along the equator of the precipitate surface. The existence of the tangent components implies that complicated mass transfer occurs and is responsible for shape coarsening.
4. SUMMARY
Fig. 11
The diffusion field, or solute concentration as a function of position, around a prolate spheroidal precipitate has been solved under a boundary condition of varying concentrations along the precipitate surface owing to the curvature effect [37]. The physical assumptions made here are otherwise similar to those of the LSW theory addressing spherical coarsening [2, 31.
834
MOU and HOWE:
DIFFUSION
FIELDS ASSOCIATED WITH PROLATE SPHEROIDS
The prolate spheroidal coordinates [40] are adopted to ensure that the boundary conditions at the precipitate surface and at some remote surface are separable. Introduction of these coordinates also makes it easier to formulate the principal curvatures [40] of the precipitate surface. Under these coordinates the diffusion field equation, the Laplace equation [39], is separated into two Legendre equations on the radial and angular variables [39,42], whose solutions are Legendre polynomials and Legendre functions of the second kind [40,42,43]. The boundary conditions are fitted by forming a solution as the sum of two parts: the first is a Legendre function of the second kind of order zero, and the second is an infinite Legendre series [39,43]. 800 A = 20
I 640
Curve
0 (“)
1
480 80
1 lo25
z
2i.5 45 90
i
320
60
The series coefficients are expressed analytically with Legendre polynomials and Legendre functions. The first part of the solution displays the concentration distribution when the curvature effect is ignored, whereas the second part represents the concentration change contributed from the curvature effect. Numerical results are presented for normalized concentrations, defined to filter out the dependence of actual concentrations on the particle size and system thermodynamic parameters. Normalized concentrations with an ignored curvature effect are drawn as a function of the radial variable. For various aspect ratios, normalized concentrations contributed from the curvature effect are displayed as a function of the radial variable for selected angular values and of the angular variable for selected radial values. Concentration gradients contributed from the curvature effect have tangential components, which may produce complicated mass transfer and thus be responsible for shape coarsening. Acknowledgements-Appreciation is expressed to NASALaRC for support of this research under Grant NAG-l-745. One of the authors (J.M.H.) is also grateful to NSF for support under Grant DMR-9302493.
160. 1 REFERENCES
40 1.0
1.1
1.2
1.3
1.4
1.5
20
OL
1.0
1.1
1.2
1.3
1.4
1.5 5
(a) 8 A = 20 I c*
Curve
0 (“)
6
1. G. W. Greenwood, Acta metall. 4, 243 (1956). 2. I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids 19, 35 (1961). 3. C. Wagner, Z. Elektrochem. 65, 581 (1961). 4. H. B. Aaron, D. Fainstein and G. R. Kotler, J. Appl. Phys. 41, 4404 (1970). 5. R. A. Oriani, Acta metall. 12, 1399 (1964). 6. G. R. Speich and R. A. Oriani, Trans. TMS-AZME 233, 623 (1965). I. A. J. Ardell and R. B. Nicholson, Acta metall. 14, 1295 (1966). 8. A. J. Ardell and R. B. Nicholson, J. Phys. Chem. Solids 27, 1793 (1966). 9. G. P. Airey, T. A. Hughes and R. F. Mehl, Trans. TMS-AZME 242. 1853 (1968). 10. A. J. Ardell, in The M&ha&ms of Phase Transformations in Crystalline Solids, p. 111, The Metals Society,
London, (1969). 5
4
3
2 1.5 1.5
2
2.5
3
3.5
4
4.5 z 5
(b) Fig. 12. The normalized concentration c as a function of the radial variable 5 for selected 0 values for an aspect ratio of 20: (a) 5 ranging from 0 to 1.5 with an inset showing the whole ?K scale for curve 1; and (b) magnified ?& scale for 5 beyond 1.5.
11. G. Sauthoff and M. Kahlweit, Acta metall. 17, 1501 (1969). 12. A. J. Ardell, Metall. Trans. 1, 525 (1970). 13. P. K. Rastogi and A. J. Ardell, Acta metall. 19, 321 (1971). 14. J. D. Boyd and R. B. Nicholson, Acta metall. 19, 1379 (1971). 15. D. J. Chellman and A. J. Ardell, Acta metall. 22, 577 (1974). 16. F. Ferrante and R. D. Doherty, Acta metall. 27, 1603 (1979). K. Oki and T. Eguchi, Trans. 17. Y. Seno, Y. Tomokiyo, Jpn. Inst. Met. 24, 491 (1983). 18 H. Calderon and M. E. Fine. Mater. Sci. Enp. _ 63. 197 (1984). 19 G. J. Shiflet, H. I. Aaronson and T. H. Courtney, Acta metall. 27, 377 (1979). 20 P. Merle and F. Fouquet, Acta metall. 29, 1919 (1981). 21. P. Merle and J. Merlin, Acta metall. 29, 1929 (1981). 22. T. G. Zocco and M. R. Plichta, Scripta metall. 18, 555 (1984).
MOU
and HOWE:
DIFFUSION
FIELDS
ASSOCIATED
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