Two families of analytic γ-plots and their influence upon homogeneous nucleation kinetics

Surface Science 62 (1977) 695-706 0 North-Holland Publishing Company

TWO FAMILIES OF ANALYTIC -y-PLOTS AND THEIR INFLUENCE HOMOGENEOUS NUCLEATION KINETICS

UPON

J.K. LEE, D.W. DOOLEY, D.E. GRAHAM *, S.P. CLOUGH, C.L. WHITE ** and H.I. AARONSON Department of Metallurgical Engineering, Michigan 49931, USA

Received

4 October

1976; manuscript

Michigan Technological

received

University, Houghton,

in final form 23 November

1976

The equilibrium shapes corresponding to two different families of r-plots are constructed. One y-plot family comprises continuous variations from a sphere to an oblate ellipsoid. This set of y-plots yields a sharp edge in the corresponding equilibrium shape when its aspect ratio is less than l/J2. The other family consists of nephroids of revolution, varied in cross-sectional form from two slightly overlapping near-circles to an ellipse-like morphology. This family exhibits a facet at one boundary orientation in the equilibrium shape. For the analytical expression of the equilibrium shape, the E-vector formalism of Cahn and Hoffman is used and found to give results identical to those from the Euler-Lagrange method. The effects of the variations in equilibrium shape within the two families treated upon the principal parameters in the general equation for the time-dependent rate of nucleation are assessed in order to ascertain their relative influence on nucleation kinetics.

1. Introduction In recent papers [l-3], equilibrium shapes have been presented for nuclei formed homogeneously and at planar, disordered grain boundaries. These nuclei are composed of spheres or portions thereof and in some cases incorporate planar facets. In the present investigation, these considerations are extended to two other classes of nuclei, both formed homogeneously. For each, considerations begin with a family of analytical y-plots which can be expressed as continuous functions of the surface orientation; the equilibrium shapes corresponding to these plots are derived and then the variations in the principal terms in the general equation for the timedependent rate of nucleation [4] and the nucleation rate itself with progression through the families are described. One family is developed from a sphere which is continuously flattened into a disc-shaped oblate ellipsoid. The other family com* Now at General Electric Company, Schenectady, New York 12345, USA. ** Now at Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA. 695

696

J. K. Lee et al. / TNYJ ,fatrlilies of‘arzalvtie

y-plots

prises nephroids of revolution, varied from a pair of slightly overlapped near-circles to extensively overlapped ellipse-like shapes in their cross-section view through the centers. This latter family is chosen because it yields a facet at one nucleus: matrix boundary orientation. General equations for the equilibrium shape as a function of variations in the y-plots are derived, for subsequent application to the two families, using the recently developed E-vector formalism of Cahn and Hoffman [5,6]. These equations are shown to be identical to those obtained by Landau [7] by means of the Euler--Lagrange method.

2. Analytical

expressions

of the equilibrium

shape for a y-plot of revolution

The equilibrium shape for a given y-plot is determined by the Wulff construction [8,9]. An analysis of this construction led Herring [lo] to develop the tangent sphere construction, which is useful in forming the equilibrium shape for a given y-plot. Fig. 1 illustrates the tangent sphere construction: a sphere is drawn which passes through the origin of the y-plot, 0, and tangentially touches the y-plot surface at the end point of a radius vector of this plot. If any other portion of the yplot does not cut this sphere, the other end of that diameter of the sphere which begins at 0 will be on the periphery of the equilibrium shape. Recently, Cahn and Hoffman [5,6] designated this diameter as the &vector in their vector thermodynamic description of anisotropic surfaces. Analytical expressions of a .$-vector can be obtained from definitions given by

Fig. 1. Tangent sphere construction. 5 is the diameter of the tangent Wulff point, 0, and touches the y-plot surface at (yP, yz).

sphere

which

passes the

J.K. Lee et al. / Two families of analytic y-plots

697

Cahn and Hoffman:

(1) (2) where n is the unit normal vector specifying the orientation of a surface element with interfacial free energy per unit area, y. For a y-plot of revolution, y is a function of only an angle 8, defined in fig. 1. Since ti represents a change in orientation by the angle df3 = I dn 1, the derivative dn/d0 is a unit vector in the direction of dn. Therefore, eq. (2) may be written for a y-plot of revolution: [ . &/de

= dr/de

(3)

In terms of Cartesian coordinate

components

in the p-z plane (fig. 1):

n = (COS 8, sin e) , dn/dB = (-sin

(4)

8, cos e) .

(5)

Substituting eqs. (4) and (5) into eqs. (1) and (3) and solving for the Cartesian components & and g, of& <,, = y cos e - (dr/dr3) sin e , gz = y sin f3 + (dr/de)

(6)

cos e .

(7)

By solving the Euler-Lagrange equation for a two-dimensional equilibrium shape, Landau [7] has previously derived the results of eqs. (6) and (7) but in terms of the usual Cartesian coordinates x and y rather than of .$ and & for an equilibrium shape. 2.1. Ellipsoidal y-p/o ts An ellipsoidal y-plot of revolution y = P/(sin’e t p2 c0s2e)1’2

is described by [ 1 l] :

,

03)

where 0 is the orientation angle and fl is the aspect ratio of the ellipsoid, i.e., the ratio of the semi-minor axis to the semi-major axis. Physically, the aspect ratio /3 means the ratio of the value of y at 8 = n/2 to that at 0 = 0. Differentiating y with respect to 8: dy/d0 = (0’ - 7’) y tan e/P2 . Substituting

(9)

eq. (9) into eqs. (6) and (7) and rearranging them:

ip = 7 cos e [i - tan20 + y2 tm2e/fi2] .g, = y sin e [2 - r2/P2] .

,

(10) (11)

J.K. Lee et al, / Two families of anal.ytic y-plots

698

Fig. 2. Schematic drawing for the derivation of the largest aspect which causes formation of a sharp edge on the equilibrium shape.

ratio

of an ellipsoidal

y-plot

Eqs. (10) and (11) are the analytic expressions for the equilibrium shapes, corresponding to the y-plot described by eq. (8) when fi is larger than the critical value, l/d/2. When B< l/d2, certain orientations become physically unstable, causing the equilibrium shape to develop a sharp edge in the direction 0 = 0. It occurs at 19 = 0 because at this orientation y has its maximum value. The condition for a missing orientation to occur is that a plane perpendicular to a radius vector of the y-plot cuts the y-plot itself [8,9]. Let the distance from the Wulff point, 0, to the intersection of the plane perpendicular to a radius vector with the B = 0 plane be 1. Then I is simply given by (see fig. 2): I=

y/cos e =p/cos

8(sin20 tp*

If the minimum value a sharp edge in the 0 and setting it to zero, of 1 to have a minimum

c0s2e)1/2.

of 1 is less than y(0 = 0) = 1, the equilibrium shape will have = 0 direction. By differentiating eq. (12) with respect to 0 one finds the following relation as a condition for the value which is less than unity:

1 -2/3*=tan20. Since the right hand side is always positive orientations on the equilibrium shape P
(14

(13) for 0 # 0 in order to have missing

(14)

Fig. 3 shows the equilibrium shapes corresponding to ellipsoidal y-plots calculated from eqs. (10) and (11) of representative values of p. The outer light solid curve delineates the y-plots (eq. (8)) while the inner heavy solid curve describes the equilibrium shapes. Note that, in fig. 3d, there are no missing orientations while figs. 3a, b and c do have some. The broken curves show (incomplete) traces of these missing orientations as determined by eqs. (10) and (11) when /3 < ljd2.

J.K. Lee et al. / Two families of analytic y-plots

\\ + ,’ (o)b=

,’ ~\

2

699

Cc>31

0.125

(c)J3:0.50

(b)b=0.25

(d)b-0.75

Fig. 3. Ellipsoidal y-plots of revolution and their corresponding equilibrium shapes. The outer light solid curve denotes the r-plots while the inner heavy solid curve indicates the equilibrium shapes. The broken curve is a trace of some missing orientations on the equilibrium shape. p is the ratio of the value of y at 0 = n/2 to that of-r at .9 = 0.

Increasing departure of the aspect ratio /I from unity means physically an increasingly oblate ellipsoidal y-plot of revolution. When fl approaches zero, the equilibrium shape can be described as consisting of two abutting paraboloidal caps. In this circumstance, the y-plot can be assumed to be two parallel planes separated by the distance 20. Hence, substituting y = p/sin 0 into eqs. (6) and (7) and eliminating 8, one obtains:

E, = P - $/4P

(15)

which is the equation

of a paraboloid.

2.2. Nephroidal y-plots The nephroidal

-y-plots are given in parametric

form as [l l] :

yp = sin3u ;

(16)

y* = h cos u - cos3u )

(17)

where 0 G u < 27r, 1 < h < 3 and the orientation, 6 = tan-‘(y,/y,)

.

0, is given by:

(18)

J. K. Lee et al. / Tww families of analytic y-plots

700

Substituting r;, = {-(2h

eqs. (16))( 18) into eqs. (6) and (7) and rearranging: - 3) s6 - (5X2 ~ 15x + 12) s4 ~ (X ~ l)(?? - 9h + 12) s2

+(h-1)2(X-3)}{(2X-3)s3-3(X-l)s}-’,

(19)

~z= {-(2X-3)s4t(h-l)(h-3)s~-3(h~-l)2}cosu

(20)

(2h - 3) s* - 3(h - I )

where s = sin U. The value of X-l is the ratio of $0 = rr/2) to ~(0 = 0). This family of y-plots has an inwardly pointing cusp at 0 = +n/2 and hence exhibits a facet at the 0 = +rr/2 orientation in the corresponding equilibrium shapes as shown in fig. 4, where the outer light solid curve delineates the r-plots while the inner heavy solid curve denotes the equilibrium shapes. Physically, a torque term (I dy/de I) must be finite, and therefore, at a pointed cusp, the first derivative of y must be finite but discontinuous in a real system [8]. Nephroidal y-plots are not realistic in this sence since their first derivative becomes infinite at u = 0 or 0 = rr/2. However, their mathematical form (eqs. (16))(18)) is represented as a continuous function of the orientation, 8, which is of primary interest in this study.

Cc) h-l=050

Fig. 4. Nephroidal y-plots of revolution ratio of the value of y at @= n/2 (facet

(d) h-l:0

and their corresponding equilibrium free energy) to that of y at 0 = 0.

75

shapes.

h - 1 is the

J.K. Lee et al. / Two families of analytic y-plots 3.

701

Nucleation kinetics with the equilibrium shapes

In this section, the effects of the morphological changes accompanying progression through the two families of equilibrium shapes examined on the principal parameters in the general equation for the time-dependent rate of homogeneous nucleation will be examined. The critical nucleus can be considered to achieve the equilibrium shape, in the absence of strain energy, simply because its size is very small and this shape minimizes the free energy of activation, AC*. According to classical nucleation theory, the time-dependent nucleation rate, J*, is given by [1,41: .I* = .ZP*N exp(-AG*/kZJ

exp(-T/t)

= P exp(-T/t)

,

(21)

where AC* = 4VJAG,2 Z = V,AGJ(

,

127rkTAG*)‘1* ,

(22)

(23)

p* = ws* ,

(24)

7 = l/2(3*22 .

(25)

Here, P is the steady-state nucleation rate, Z = the Zeldovich non-equilibrium factor, b* = impingement rate of single atoms on the critical nucleus, N = number of nucleation sites per unit volume, 7 = incubation time, V, = atomic volume, AG, = volume free energy change, w = impingement flux of atoms on the critical nucleus interface, S* = the interface area of the critical nuclues at which atoms are easily attached, and VW is the volume of the critical nucleus in “Wulff space”, wherein specific interface free energy y replaces the linear dimension r as a coordinate [ 121. When the matrix:nucleus interface free energy, yap, is isotropic, the critical nucleus shape is a sphere and its Wulff volume, VW, becomes 4ny,$3. Fig. 5 shows the variation of AG*/AG,*, 0*/p,* and T/T, as a function of aspect ratio, /3, in the ellipsoidal y-plots. The subscript, s, designates values associated with a spherical nucleus. The value of y at 0 = 0 is taken to be equal to the interfacial free energy of a spherical nucleus, and all of the surface area of the critical nuclei is assumed equally able to accomodate atoms from the matrix phase. Since the volume and surface area of the critical nucleus decrease smoothly with decreasing p, all three rates decrease smoothly. In fig. 6, the variation of AG*/AGz, /3*/p,* and T/T~ for nephroidal y-plots are plotted as a function of X - 1. The value of y at 0 = 0 is again taken to be equal to the interfacial free energy of a spherical nucleus. Since the volume of the critical nucleus decreases with X - 1, AG*/AG,* also shows a smooth decrease. However, because all the critical nuclei with a nephroidal y-plot have facets at one orientation (fig. 4), a modification must be made in S* and hence in /3* [l] . The facets on a critical nucleus are immobile since the availability of an appropriate growth mecha-

J.K. Lee et al. / Two families of aualytic y-plots

702

,e( aspectratlo) I:@. 5. The variation of AG‘*/AG‘z, p*/pg and r/rs shapes derived horn ellipsoidal -y-plots, The subscript,

as a function

of aspect

s, refers to a spherical

ratio

for nucleus

nuckus.

nism such as the ledge mechanism is highly unlikely on a very small facet. Nevertheless, Johnson et al. [l] demonstrated a plausible growth path through which an embryo can develop a facet and achieve equilibrium shape without displacing the facet in the direction normal to itself. Therefore, only the curved interfacial area is taken into account in computing /J’*and 7. As h - 1 decreases, the ratio of the facet area to the curved area increases. This is why the /3*/p,* curve shows a plateau at X ~~ 1 z 0.25, and the r/r, curve has an extremum point at h - 1 = 0.45. Fig. 7 shows the

,-- ------------12-

/

-----.___

,’

--_

/’ /j 0

lo-

-----

00

0.1

02

L 03

1 04

0.5

1 0.6

ps’/,fj:

1 0.7

I

/

08

0.9

1 0

h-1

Fig. 6. The variation of AG*/AC,*, from nephroidal r-plots.

p*/pt

and T/T~ as a function

of A - 1 for shapes

obtained

J.K. Lee et al. / Two families of analytic y-plots

703

24.0 20.0 Ellipsoidal N*

7-plot

16.0-

N 12.0 7

4.0 -

fi or

A-1

Fig. 7. The variation of Z/Z, as a function of aspect ratio for nucleus shapes based upon ellipsoidal r-plots and h - 1 for those obtained from nephroidal -y-plots.

variation of Z/Z, as a function of fi and X - 1 for ellipsoidal and nephroidal y-plots, respectively. Since the Zeldovich nonequilibrium factor is inversely proportional to the square-root of AC* (see eq. (23)), the value of Z/Z, increases sharply when fl or h - 1 approaches zero. In fig. 8, logIo(P/.Q is plotted as a function of fl and h - 1 for ellipsoidal and nephroidal y-plots, respectively, where 4 is the steady-state nucleation rate for a spherical nucleus. Since the maximum value of AC* for a perceptible nucleation rate, say, one nucleus/sec/cm3, is -60 kT [13], the value of AC: is taken to be 60 kT in the calculation of P/G. When the aspect ratio of the ellipsoidal y-plots is less than -0.2, the value of AC* is so small that its role in determining P is negligible compared with that of the impingement factor, p*, which is proportional to the interfacial area of the critical nucleus. But the ratio /3*/p,* also decreases rapidly with decreasing aspect ratio, /3, and thus a slight decrease in logIo(P/Q is obtained in fig. 8 when fl is less than 0.2 for the case of ellipsoidal y-plots. For these comparisons, the isotropic interface free energy, y, of a spherical nucleus was taken to be equal to the value of y at f3= 0. This choice is rather arbitrary in the sense that for strong anisotropy such as fig. 3a, this y does not even appear on the equilibrium nucleus. Hence, one might consider as a reference value, a different isotropic value, doff, for a spherical nucleus. In the case, however, the results are simply those in figs. 5-7 multiplied by appropriate factors: they are R3, RF312, 2 5 R and R for AG*/Ac&., Z/ZTeff, /3*//3qek,and r/rYeff, respectively, and R is the

J.K. Lee et al. / Tu*o families of analytic q-plots

704

as a function of aspect ratio Kg. 8. The variation of loglo(Js/~) soidal y-plots and A - 1 for nuclei derived from nephroidal y-plots.

for nuclei

based

upon

ellip-

ratio of y at 0 = 0 to Yefr. For logre(Sjljsreff), one has: logI,(Js/s”,,ff) + ; logI&

= 1ogtcrVl.Q + logrO(sjs/JsyefJ = logt0VIJ9 - (AG,*/2.303kT)(l

~ k-‘)

.

+

(26)

4. Discussion A recent study of the simultaneous minimization of interfacial and elastic strain energies in calculating AC* for an ellipsoidal critical nucleus has shown that the critical nucleus morphology is primarily dictated by interfacial energy minimization unless the ratio of the strain energy to the driving force, AC,, exceeds about 0.75 [14]. Since at usual undercoolings this ratio is normally much smaller, the present calculations, which have ignored elastic strain energy, should be realistic for nearly all solid-solid transformations encountered in practice. Miller and Chadwick [ 151 have reported experimental studies on the equilibrium

J. K. Lee et al. / Two families of analytic y-plots

105

shape of small liquid droplets entrained within solid crystals in the solid + liquid region, Their observed shapes should be close to the equilibrium shapes because diffusion in the liquid phase is so rapid. Lenticular disc-type droplets similar to figs. 3(a) and 3(b) were observed in the Zn-Mg system, while the Cd-Zn system showed oblate ellipsoidal liquid droplets like the shape in fig. 3(d). In the Zn-Pb system, they observed liquid droplets of rounded-edge disc shape such as the particles of figs. 4b and 4c. As Herring [8] has indicated, one cannot of course be entirely certain that these small droplet morphologies are derived from ellipsoidal or nephroidal y-plots, respectively. The results of the nucleation kinetics calculations show that, as the aspect ratio of the ellipsoidal y-plot or the ratio of the facet free energy to the maximum curved interfacial free energy in the nephroidal y-plot decreases, the steady-state nucleation rate increases by a factor of up to 10” relative to the rate for a spherical nucleus. These effects of anisotropic interfacial energy upon nucleation kinetics are thus analogous to those noted by Johnson et al. [l] for the simpler case of faceted spherical shapes. And also as in the spherical case, the predominant influence of such anisotropy upon the nucleation rate is found to be exerted through the diminution of AC*.

Acknowledgements This work was largely supported by the Division of Materials Research of the National Science Foundation under Grants GH-37103 and DMR73-07504-A01 for which much appreciation is expressed. J.K. Lee also wishes to acknowledge supplementary support from the Division of Materials Research of the National Science Foundation under Grant DMR-76-06855.

References [l] [2] [3] [4] [5] [6] [7] [S] [9] [lo]

W.C. Johnson, C.L. White, P.E. Marth, P.K. Ruf, S.M. Tuominen, K.D. Wade, K.C. Russell and HI. Aaronson, Met. Trans. 6A (1975) 911. J.K. Lee and H.I. Aaronson, Acta Met. 23 (1975) 799. J.K. Lee and HI. Aaronson, Acta Met. 23 (1975) 809. K.C. Russell, Phase Transformations, Ed. HI. Aaronson (ASM, Metals Park, Ohio, 1968) p. 219. D.W. Hoffman and J.W. Cahn, Surface Sci. 31 (1972) 368. J.W. Cahn and D.W. Hoffman, Acta Met. 22 (1974) 1205. L.D. Landau, Collected Papers (Pergamon, Oxford, 1965) p. 540. C. Herring, in: Structure and Properties of Solid Surfaces, Eds. R. Gomer and C.S. Smith (Univ. of Chicago Press, Chicago, 1952) p. 5. W.W. Mullins, in: Metal Surfaces, Structure, Energetics and Kinetics, Eds. W.D. Robertson and N.A. Gjostein (ASM, Metals Park, Ohio, 1963) p. 63. C. Herring, Phys. Rev. 82 (1951) 87.

706 [ 111 1121 1131 [14] [15]

J.K. Lee et al. / Two families of analytic y-plots S.M. J.K. K.C. J.K. W.A.

Selby, Standard Mathematical Tables (CRC, Cleveland, Ohio, 1972) p. 371. Lee and H.I. Aaronson, Scripta Met. 8 (1974) 1451. Russell, Acta Met. 16 (1968) 761. Lee, D.M. Barnett and H.I. Aaronson, Met. Trans. (A) submitted. Muller and G.A. Chadwick, Proc. Roy. Sot. (London) A312 (1969) 257.