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Influence of faceting upon the equilibrium shape of nuclei at grain boundaries—I. Two-dimensions
INFLUENCE
OF FACETING UPON THE AT GRAIN BOUNDARIES-I. JONG
K.
LEE?
and
H.
EQUILIBRIUM SHAPE TWO-DIMENSIONS* I.
OF NUCLEI
MRONSONt
The exact equilibrium sh8pe of two-dimensional nuclei (or of any other smell particles) at a grain boundany which is constrained to rem8in planar, both in the presence and the ebsence of a feoet at one orientatlon of the nucleus-matrix boundary. is found by minimizing the total interf8cial free energy subject to the constrclint of 8 constant volume. These sh8pes, as well as those obtiined by 8llowing the grain boundery to puoker in order t,o achieve force b818nces at its junctions with the nucleus, are 8160 derived graphically through 8 new generalization of the Wulff construction. The free energy of activation, A@+. for the formation of critical nuclei in these shapes is calculated 8s 8 function of the angle between the facet and the grain boundary under representative conditions of relative interfeaiel free energy; AC* is found to be significantly less at small values of this angle under most circumstances. cmd also to be smaller at puckered than at planar grain boundaries, though oft,en by negligible amounts. INFLUENCE
DES F.4CETTES SUR LA FORME A L’&QUILIBRE DES GERMES AUX JOIXTS DE GRAINS-I. GERMES BIDIMENSIOKNELS
SITUES
En minimis8nt 1’6nergie libre totale d’interface soumise il la contrainte d’un volume constant, on trouve la forme ex8cte it 1’Pquilibre d‘un germe bidimensionnel (ou de t.oute 8utre petite pa&cub) situ6 i\un joint de grains assujetti it rest.er plen en prCsence ou en l’absence d’une faxette, pour une orientation de l’interface germe-matrice. Grbe 6~une nouvelle g&+alisation de la construction de Wulff, on obtient Pgalement ces formes graphiquement. ainsi que celles qui se produisent si l’on permet au joint de grains de .se plisser de man&e B bquilibrer les forces au contact du germe. On calcule 1’6nergie libre d’8ctivstion AG* de formation de germes critiques de tee formes en fonction de l’angle entre la facet& et le joint de grains, pour des valeurs reprCsent,atives de I’bnergie libre interfaciale relative; on trouve que AG* est p6nCralement nettement plus faible pour les petites valeurs de cet ctngle. et egalement plus faible pour les joints de grains plis&s clue pour les joints plans, bien yue la diffbrence soit souoent n6gligeable. EISFLUD
DER
FACETTIERVXG
AL-F DIE GLEICHGEWICHTSGESTALT _4?i KORNGREXZEX-I. ZWEI DIMEXSIONES
VOX’
REIMEX
Die erakt,e Gleichgewichtsgestalt zweidimensionaler Keime (oder aller anderen kleinen Teilchen) an Korngrenzen wird untersucht. Die Korngrenzen sollen sowohl bei Gegenwarlrt 816 8uch bei Abwesenheit einer Facette an einer Orientierung der Keim-Mat,ris-Grenzflbche eben bleiben. Die Gleichgewichtegestalt wird durch Minimierung der gesamten GpnzflBchenergie bei konstantem Volumen gewonnen. Diese Keimformen und such die Formen, die entstehen. wenn Korngrenzen Felten zur Bildung einer Gleichgewichtslege mit den Keimen bilden. werden gr8phisch ebgeleitet mit Hilfe einer neuen Verallgemeinerung der W7ulff-Konstruktion. Die Aktivierungsenergie AG* der Bildung kritischer Keime mit diesen Formen wird 81s Funktion des Rinkels zwischen Facette und Korngrenze berechnet. Gnter Annahme repriisentativer Werte relativer Grenzfliichenenergien ergibt sich, daD AG* bei kleinen Werten dieses Wink& unter den meisten Umst.&nden betriichtlich kleiner ist und d8D AG* such an Korngrenzen mit F&en klciner 81s an ebenen Korngrenzen iat. wenn such manchmal nur urn einen vernachl&ssigbaren Betrag.
1. INTRODUCTION
The equilibrium shape of a precipitat,e crystal is of particular int,erest, because this is the morphology with the lowest free energy of activation for nucleation. In a companion paper,“) these shapes are calculated for three-dimensional nuclei formed at planar, disordered grain boundaries under the assumption t’hat an energy cusp facet appears at only one boundary orient,at,ion and in only one of the t.wo matrix grains forming the grain boundary. When the angle which the facet makes with the grain boundary exceeds a critical angle, however, rigorous minimization of the free energy is found to be not possible at t,he present time. It will be shown in this paper, however. that the equilibrium or minimum free energy morphology for the entire range of tnodimensional a,nalogues of these nuclei can be rigorousI> * Received May 4. 1974: revised October 24. 19i4. i Department of Metallurgical Engineering. Michigan Technological University. Houghton, Michigan 49931. n.S..4. ACT-4 2
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determined. These results will be used in the companion paper to evaluate more accurately the equilibrium shapes of the three-dimensional nuclei. In both papers, the simplifying assumption has been made that. the strain energy arising from area1 or volumetric misfit between the nucleus and matrix lattices can be ignored. This assumption is exact when the critical nucleus is an ellipsoid and Young’s modulus and Poisson’s ratio (in the isotropic elasticity approximation) are the same in both phases, for st)rain energy is independent, of morphology under t,hese circumstances.(*) Even when the elastic moduli are tinequal or anisotropic elasticity is inkoduced, however. the tacit assumption t.hat’ strain energy can be adequately accounted for simply by subtracting its value from t$hat for the area1 or volume free energy change wit~hout simultaneous minimization of interfacial and strain energies should be a good first approximation when the misfit, is small (as is often the case in solid-solid tSransformations). 799
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When some portion of the critical nucleus is part of a sphere, the free energy of activation for its formation is proportional to the third power of the inter-facial energy of the spherical surface (multiplied by a factor reflecting the other inter-facial energies operative13’) divided by the second power of the sum of the area1 or volume free energy and strain energy. Since strain energy is proportional to the square of the linear misfit between the two lattices, failure to minimize interfacial energy and strain energy simultaneously should thus not introduce a significant error in the activation free energy when the misfit is small. We shall first find the equilibrium shape of twodimensional nuclei developed upon a substrate or at a grain boundary by minimizing their total interfacial free energy under the constraint of a constant vo1ume.t Although the primary thrust of this paper is toward faceted nuclei, a useful introduction is provided by the consideration of their non-faceted counterparts. Then, based upon Winterbottom’s(4) concept of negative surface tension, a completely generalized Wulff construction will be developed and applied to the various classes of nucleus shape under consideration. Up to this point in the development, the assumption will be made that the grain boundary is constrained to remain planar during formation of the crit.ical nucleus. It will then be shown that under considerable ranges of conditions this assumption prevents the achievement of equilibrium shape. In order to achieve such a shape, the grain boundary must often be allowed to undergo deflections, or ‘puckering’(s) which enable it to join the nucleus at equilibrium angles. Finally, the variation of the free energy of formation of a critical nucleus, AG*, as a function of the angle, 4, c6) between the facet and the grain boundary formed by rotation of the facet about an axis lying parallel to and below the grain boundary plane (see Fig. I), will be calculated for both the planar and the puckered grain boundary nucleus configurations. Physically, 4 corresponds to the angle between the plane of the grain boundary and the closely matching conjugate habit planes which are responsible for the presence of the facet. In principle, $ can be taken as freely variable ; symmetry considerations, however, restrict the angular range which must be investigated to the range 0°-90”.
t In view of the analogy between two- and three-dimensional nuclei, u is denoted as surface rather than as line tension and wis termed volume rather than area. The term ‘surfaoe t,ension’ is understood to represent specifio interfacial free energy.
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2. DERIVATION
OF
EQUILIBRIUM
SHAPE
In this section we derive the equilibrium shape by minimizing the total surface free energy of the system with respect to the appropriate shape parameter, such as the cont,act angle or dihedral angle. (a) A liquid nucleus on a sub&rate (no facet) A liquid particle on a substrate is shown in Fig. la. The total surface free energy is given by: E =ja dl = 2R(&s,,
+ sin @I+, -
a,,)},
(I)
where 1 is the interface length, R is the radius of the basis circle from which the nucleus is taken, 0 is the contact angle, and ofi,, uBs and u,, are energies or tensions of the vapor(a)-liquid(@), liquid-substrate(s) and vapor-substrate boundaries. Using the constant-volume co&raint, v = 3RP(20-sin 20), by substituting the relationship for R obtained from it into equation (1) and setting aE/% = 0, one obtains the well-known Young equation, a 111-
as8 = oEp cos 8.
(b) -4 nucle26e at a grain boundary The tot,al interfacial
(2)
(no facet)
free energy (Fig. 1b) is given by
R = 2R(2&$
-
u,, Sin e),.
(3)
where a,, and a,, are the interfacial energies of a disordered nucleus-matrix and a disordered grain boundary, respectively, and both are assumed independent of boundary orientation. Again using the constant-volume constraint, v = Ra(2f3 - sin Se), and setting aE/&3 = 0, one obtains u Jlol= 2~,, ~08 e =
~~~(~08 e1 +
cos e,),
(4)
where in this situation 0, = 0, = 0. These relationships (equations 2 and 4) have been derived by many investigators.(4’7-10’ However, since they provide a useful introduction to more complicated cases we have rederived them. (c) A nucleue with a facet in contact with a grain boundary We assume that the facet, with energy cZgc, appears only the ‘upper’ portion of the nucleus (Fig. lc) because the necessary lattice orientation relationship exists only with respect to the upper matrix grain. In case of bEsc .> go.., a range of 4 exist,s in which the facet does not touch the grain boundary.‘@ The smallest t,ilt angle at which the facet meem the grain boundary is denoted as q&. When a,,” < +a,,, C& = 0 since a facet is always in contract with the -gain boundary in this situation.
LEE
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SHAPE
OF
NUCLEI-I
sin e2 = - cosbe csc +.
FIG. 1. Nucleus shapes: (a) on a substrate, (b) unfaceted grain boundary allotriomorph, (c) with one facet in contact with a grain boundary and (d) with two facets in contact with a grain boundary.
E = R{(Ze, + 26 - ~)o,~ - 2 sin e20,, - ZU,~~(COS 8, csc 4 - sin S)},
Atkention is therefore directed in this subsection to the case in which 4 2 &, but C#J is still sufficient,ly small so t,hat the facet appears at but one side of the nucleus. Since we have the same radius of curvature for each curved interface by assuming the same chemical potential all along it, we have for 8,, 2 sin e2 = sin 8, -
cos b csc (b -
cos 8, cot 4,
(5)
as shown in Appendix 1. Also as developed on the basis of the derivations in Appendix 1, the total int.erfacial free energy of the nucleus is given by: E = R((8, -+ 28, + 6 + c$ - 77)o,8 - 26,, . sin 8, - aa,' x (COS8, $ COS (6 + #))CSC(6}, (6) and the constant-volume
(10)
The expressions for the botal interfacial free energy and the constant volume can be obtained from equations (6 and 7) by replacing 8, with 6 - C#J,and adding contributions due to the parallelogram ABEC:
Cd)
Cc)
80 1
Fig. l(d). In this c&8e the shape parameters to be determined are b and x/R, where x is the perpendicular distance from the grain boundary to the junction of t,he disordered interface m-ith the shorter facet,, i.e. Al? sin C$as shown in Fig. l(d). Since 3: is described by X/R = cos (6 - 4) - cos e,, we can again use 6 and 8, as shape parameters. As in case (c), we have the same radius of curvature for each curved interface by assuming the same chemical potential. Hence we obtain for 8,,
(b)
(a)
THE
constraint is written as
2’ = -$R”{e, T tie, T b + + - 77- 2 sin e2 x .(cose1 -+ cos8,) + tcose, -i-cos (6 f- C#))cos bcsc C/J}. (7)
(11)
and r = @?{zf+ + 2b - x - 2 sin e,(cos8, -c cos 8,) + 2 cosb(cos8, csc6,- sin b)}. (i’) As before, we substitute equation (12) into (11) for R, set iYE/ae, = 0 and iTIE/&? = 0, and solve t,he resultant equations for a,, and cage (see Appendix 2). Thus we obtain for equilibrium conditions (13)
and uol== U,+OS Equation
e1 + cos e,).
(14)
(14) may be written in terms of x:
(X/R)equil = cog (6 - 4) - cos8, = cos (6 - 4) + cos e2 - u,,/u,,.(15) 3. A
GENERALIZED
WULFF
CONSTRUCTION*
A Wulff construction will now be presented for each of the classes of critical nucleus, i.e. of equilibSubst’ituting equation (7) into (6), setting aE/ae, = 0, rium particle shape, discussed in the preceding and aEli% = 0, and solving the final results simulAlthough several investigators’11e12) sugsection. taneously, one obtains for the equilibrium shape gested that the equilibrium shape could be found in (see Appendix 1) : t,his manner, Winterbottom”) was t,he first to do so O.c= -uap cos 6. (8) for a particle at an interface, developing such a aB construction for a liquid drop upon a substrate with oll = ump(c~~ 8, + cos e,). (5 (9) the aid of the concept of negative surface tension. We note that, from equation (5), e1 # 8,. (d) A ,sucleus with two parallel facets in contact with a grain boundary Once the angle C# is larger than a certain angle, denoted as tQe second critical angle, &, t.he equilibrium
shape
develops
two
parallel
facet,s
as shout
in
* After this paper had been submitted for review, the authors learned (through the courtesy of Professor K. C. Russell) that) J. IV’. Cahn and D. IV. Hoffman (Acta Net. 22, 1203, 1974) had devised essentially the same modifications of the Wulff construction. Since the objectives of the two investigations are quite different, however. there is comparatively little overlap in the applications which have been made of this construction.
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Another solution to this general problem is introduced here through incorporation of en ‘auxiliary y-plot’ in the Wulff construction. We define the auxiliary y-plot as a circle whose radius is proportional to the surface tension of a disordered nucleus-matrix boundary, baa, and whose center, 0’, is located et a distance proportional to the energy of the interface at which nucleation occurs, a,,, from the Wulff point, 0, in the direction perpendicular to the nucleating interface. An appropriate geometric operation will be used to develop the equilibrium shape in each case. (a) -4 nude248 on a substrate In this case the distance between points 0 and 0’ is proportional tot a,,, the surface tension between the a(vapor) phase and the substrate, as shown in Fig. 2. We draw a line parallel to the substrate, which is separated by a distance of aBs from bhe Wulff point 0. The portion of the auxiliary y-plot lying below the horizontal line is the equilibrium the distance shape. Note from Fig. 2 that O'B, between 0’ and the horizontal line, is equal to the product of oafi and sin (90’ - 0) and thus to olrBcos 8. Hence this construction yields directly cr,, = upa + aaS cos 0, i.e. the equilibrium condition given by equation (2). Note that if G,,~is less than o,, - oas, the equilibrium shape vanishes and hence complete wetting occurs. (b) d nucleus at a grain boundary (no facet) In order to equate 0,,/2 with a,, cos 8, and thus
Fm. 3. Wulff construotion for a nucleus at 8 grain bound. 8ry (no facet).
fulllll the equilibrium condition, equation (a), the horizontal line now lies at the intersection of the Wulff and the auxiliary Wulff circles and the equilibrium shape is represented by the entire area of overlap of the two circles (see Fig. 3). Again whenever there is no intersection between these circles, i.e. a,, r 2%,, complete wetting is achieved. As in case (a), we have drawn a horizontal line. Such lines: Ai in Fig. 2 and ,4T in Fig. 3, represent the location of the interface at which nucleation occurs. Therefore the essential point in the generalized Wulff construction is to locate this horizontal line. Only one such line lying between points 0 and 0’ can meet the equilibrium condition in each case. (c) 3 ltucleus with one facet id contact with the grain boudary
In this situation we have three conditions, equations (3, 8 and 9). Equation (8) can be easily fulfllled by
Fro. 2. WuB
construction
for 8 nucleus on a substrate.
t Hereeftm the proportionality equal to unity.
constant
will be essumed
drawing a facet (B& in Fig. 4) which is located at a distance of a,,” from the Wulff point, 0, but makes the angle 4 with a horizontal line. Equation (9) is of exactly the same form as equations (2 and 4), for the liquid nucleus on a substrate and for the unfaceted nucleus at a grain boundary, respectively. Similarly to equation (a), and especially to equation (4), equation (9) determines the position of the horizontal line, corresponding to the interface at which nucleation occurs, located between points S and T in Fig. 4. Let the horizontal lines be A P,the junction
LEE
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803
Two of the tilt angles pertinent to the one-facet nucleus morphology are : f$_ = cos-l (-u,&,~)
+ m-1
(u&u,& if
-
72,
Us; > 2
(16)
and +c, = --OS-l
0 cm
(--u&$/u,&
-+ cos-1 (u,,/2u,B)
+ n, if a,,” > 1.‘U,fi2 -
a
0’
Fro. 1. Wulff construction for a one-facet nucleus at planar grain boundary.
a
bet,ween AP and 00’ be the point D, the dihedral angle with the main y-plot circle be 8, and the dihedral angle with the auxiliary circle be 0,. Then we have uaa=-ti?=?i6;iii? cos e, -+ uaB cog e,? =(T i.e. equation (9). The other condition; equat,ion (5): enables locat,ion
u~~214~
(17)
The fist one, equation (16), which was previously ment.ioned in Section 2(c), is termed the first critical angle and the second one, equation (17), is denoted as the non-facet, critical angle, above which no faceting occurs. In terms of these two critical angles, t.he following results are obtained from the Wulff construction. ( 1) When u,gc > &u,, and 0 I $ I +,,. If the facet B& lies inside the auxiliary y-plot, i.e. does not intersect this circle, the facet’ does not touch the grain boundary. This particular case is shown in Fig. 5. Recently Aaronson and Aaronts) discussed this case extensively. Because t,he facet is not in contact with the grain boundary equations (6 and 7) should be replaced with the following, which are independent’ of 6:
aB
E, = R((48 -J- 26 -
27~)~~~-
2 sin Bu,,
i
2 sin ~3u,~~} of dP, in effect bhrough determinat.ion of 8, and oz. z1 = lpy2e + 6 - 71 - sin 28 - sin b cos 6), Equation (5) is derived in Appendix 1. This relationship arises from the equality of the distance, AC, where 0 = COS-l (0J2uaB), i.e. e1 = 8, = 8. between the t.ao junctions of the upper portion wit.11
(18) (19)
the grain boundary and the distance, A’C’, between the two junctions of the lower portion wit,h the grain boundary. obviously required for continuit,y. Hence we have only to find the horizontal line which satisfies this equality. Since sa’ = c’p for an>* horizontal line, bhe particular line needed is t’hat which makes cc’ = c’p and hence A,4’ = cc’. The following construct,ion is the only means found through which to locate C’. On a point by point basis, draw the curve which is equidistant from t’he facet, B& and the cr,&rcle segment, Bz; at each point,, t,he equidistant, position is determined along a line parallel to the grain boundary. This curve, B?, is shown dotted in Fig. 4. The point of intersection Ibetween the equidistant curve and t,he auxiliary y-plot is t.he point C’. The equilibrium shape of the upper portion is given by ABC while that of the lower portion is the circle segment, A’TC’.
_--L\____ Oi\
G,
“\
\\
8'
0
FIG. 5. Rulff construction for a one-facet nucleus at a grain boundary when r$ 5 & and u,ge > lo,,.
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ct,,8/4aud I$~, 5 4 I 90’.
This situation arises when the facet B&stays out of the auxiliary y-plot. In this case there is no faceting in the equilibrium shape. The equilibrium shape is the unfaceted nucleus at a grain boundary discussed in Sections 2(b) and 3(b). (3) When 4 2 0 and a,,” I; +cr,.. The entire analysis of this section and of Section 2(c) holds true even when u,~” < &r,, and 4 > 0. However, in order to prove this statement in a limiting case, i.e. in which the facet sinks into the grain boundary at 4 = 0, multiply both sides of equation (5) by sin +. Setting 4 = 0, we obtain cos 19~= - cos 6 = ~~~~/a.~. Substituting this result in equation (9) we recover equation (2) and hence Fig. 2. Note that when %S,bC < (%a - u,& and 0 < 4 < co& (~,~“/a,~) cos-1 (b,, - b,&, complete wetting occurs. (d) A two-facet nucleus at a grain boundary As the tilt angle + increases the opposite facet, BT
(see Fig. 4) in the Wulff plot, moves up toward
the grain boundary.
If the facet B,g’ meets the grain
boundary line AC, the equilibrium shape contains two parallel facets. In Section 5 it will be shown that under this condition the two-facet shape is always more stable than its one-facet counterpart. The Wulff construction for the t,wo-facet morphology is shown in Fig. 6. Since the dihedral angle, 0,, for the lower portion is already fixed through equations (10 and 13), we first locate the grain boundary line
Fro. 6. WuliY construction for a two-facet nucleus at a plmer grain boundary. (X/%lmII = 0, i.e. the angle at which a second facet, parallel to the first, begins to appear, as the second critical angle, &:
(20) When (x/R),,,~~ has a negative value, t,here is no two-facet equilibrium shape. 4. PUCKER
MECHANISM
There has been considerable debate about the Young equation, equation (2) because of an undC and then check whether the resultant shape balanced component of interfacial tension, aZ8, in the meets the equilibrium condition equation (15). vertical direction (Fig. la). Considering the force The point C’ (and thus the horizontal line AT) is balance at junctions A and C in Figs. 4 and lc, we given as the point of intersection between the auxiliary note that when the facet meets t,he grain boundary, y-circle and a vertical line (z’ in Fig. 6 which is 0, # 0,, a force balance exists only in the direction parallel to the grain boundary at junction d and in parallel to 00’ and is separated by c.lac csc 4 from the neither the parallel nor the perpendicular dire&on Wulff point,, 0. The equilibrium shape for the upper at junction C. To achieve complete force balances, portion is AB’BC while the counterpart for the lower the grain boundary must pucker such that one side portion is A’TC’. moves up or down relative to the other side as shown The proof of equation (15) for this morphology is in Figs. 7 and 8. In Fig. 7, where a single facet is given by: -present, t.he right side of the grain boundary (‘R’) is X/R = (F? - OD)/OB’ displaced downward with respect to the left side =(zL_ - o’il’ CO8e,))/E (‘L’). To prove a complete force balance at junction C in Fig. 7, we examine a force balance in the direction parallel to the facet B& since no torque term is -
alo + ‘TX8cos 0,
aZs
11 = co9 (6 - 4, + CO88, - ~&,s, where 0, is given by equation (10). We denote the particular tilt angle which satisfies
necessary in this direction. We note that uo’. cos 4 = OH + O’H’ = a,,” f
amB. co9 (4 +
< CO’O), where
00’ = o,,. It is clear that force balances in all other directions at the junction C require use of t,he torque
term pertaining
to
the facet,
B<.
The
LEE
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4ARONSOn’:
AXD
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805
formation of an embryo, AG”, is /
/
1’ !
\ ‘\
coe
AG” =
\/ ‘\,
. J
B dl + v +AG, = E + v - .&2,,.
where AG, is the volume free energy change of transformation, and E and v are the total surface free energy and the volume, respectively, for each shape. Eote that for each of the shapes considered E = k,R and c = k,R2, where E, and k, are constants involving only int,erfacial energies. Substituting these relationships into equation (21). the critical radius, RR*.is evaluated by setting a(AG”)/aR = 0. and thus AG* is found to be :
i
AG* = (-
FIG;. 7. Wulff construction for a one-facet nucleus at a puckered grain boundary.
n,
!
(21)
w
_r___+-_----__“~/
\
-&)E”it.
(22)
In Figs. 9 and 10, AG*/AG,* is plotted as a function of C#for several values of (~.~~/cr~~, where AG* pertains to faceted and AG,* to unfaceted nuclei (Fig. lb) formed at a grain boundary. The solid curves represent nucleus shapes at planar grain boundaries (Figs. 4 and 6) while the dotted curves represent, those with puckering (Figs. 5 and 8). In Fig. 9 the value of CJJ(J.~is equal to 1.05, while it is taken to be 0.5 in Fig. 10. When a,,’ > +a,,, AG*/AG,* is independent of 4 when C$5 #Q~.(~) The pucker mechanism is seen to yield a lower AC* than the planar boundary configuration, though over appreciable ranges of 4 and d..,c/~,8 the difference between
10
_ I___
FIG. 8. WuWf construction for a two-facet nucleus at a puckered grain boundary. 04
5. FREE
ENERGY CRITICAL
OF FORMATION NUCLEUS, AG*
FOR
/
0 300
puckered version of the two-facet, nucleus is shown in Fig. 8. In this case, both portions of the grain boundary- are displaced, in opposite directions.
7
O3i
::
:
i
A
In nucleation theory, one of the most important physical quantities is the free energy of activat,ion, AG*, for t,he formation of the critical nucleus. For a two-dimensional shape, the standard free energy of
9. Variation of AW/Aff,* with 4, where AQ,* is the the critical free energy of formation for an unfaceted grain boundary allotriomorph (a,,jn,+q = 1.05).
FIG.
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1976
because the diminution in the grain boundary area destroyed with rising 4 is more than offset, only in the case of Fig. 10, by the growing proportion of the nucleus-matrix interface which is faceted as a result of the relatively small value of a,,. We also note in Fig. 10 that there arises, in the pucker mechanism, a discontinuity at small values of ~.~“/a,, and I$ = 0. The discontinuity develops because a twofacet morphology is always possible in the pucker mechanism at small values of ~,~~/a,~ and cJa,, and 4 > 0. 6. SUMMARY
The equilibrium shape of t,wo-dimensional critical nuclei upon a substrate and at a grain boundary was derived in the absence and the presence of an energy cusp facet at one boundary orientation by minimizing the total interfacial free energy subject t.o the 0.j...... constraint of constant volume. One family of shapes is derived with an additional constraint, i.e. main0.1 6- aO.5 tenance of a planar grain boundary. A second & I family was derived by allowing the grain boundary 0.0 0 10 20 30 40 50 60 70 00 90 t’o pucker, with an infinite range of deflection, in 0 (degrees) order to achieve force balances at the junctions with Fxo. 10. Variation of AG*/AG,* with +, where AG,* iu the critical free energy of formation for 8n unfaceted grain the nucleus, These shapes were alsoderivedgraphically boundary allotriomorph (cr,Juap = 0.5). through a new generalization of the familiar Wulff The first family of shapes, i.e. with a the two AG*‘s is very small. This arises because in construction. planar grain boundary, will be seen, in the companion two dimensions t,he nucleus and the grain boundary paper,(i) to play an important role in determining the have point junctions and thus the grain boundary shapes of the counterpart three-dimensional nuclei, can pucker without an increase in its length as long whose equilibrium shapes cannot be developed as as the range of puckering is infinite.t However, rigorously and must therefore be constructed with this will be shown, in the companion paper,(r) not to t.he assistance of the results on two-dimensional be true in the three-dimensional case For small values of o,~~/u,~, a one-facet shape nuclei obtained in the present paper. The ratio of the free energy of activation for critical obtains when # < $,, and a two-facet shape develops nucleus formation at a grain boundary for a faceted at larger values of 4. The angular range over which the two-facet morphology occurs is seen to decrease nucleus to that for an unfaceted nucleus was calcuwith increasing a,gc/o,,r. As ~,,“/a,, approaches lated as a function of the tilt angle of the facet unity, the unfaceted morphology (Fig. lb) develops relative to the boundary. This ratio often varies instead of the two-faceted one at larger tilt angles. appreciably with t.ilt angle (4) ; when ads < u_, Hence $C,, the lowest tilt angle at which the unfaceted the ratio usually increases rapidly with 4 at small tilt shape has the smallest AG*, replaces de,. This result angles. AC* for critical nuclei at puckered grain is illustrated by the uppermost curves (r~,~~/a~~= boundaries is always less in the two-dimensional case than at planar boundaries. Over appreciable ranges 0.9) in Fig. 9, where +,, r 84”. Since o,, is usually greater than or equal to a,,, of d and a.scla=a, however, this difference is very the situation in which u,./u,~ = 0.5, used in Fig. 10, small. A finding of particular interest is that when will not be frequently encountered. We note, however, the facet energy is nearly equal to that of a disordered a drastic difference between Figs. 9 and 10. In Fig. interphase boundary, at high tilt angles an unfaceted 10, AG* is seen usually to diminish rather t,han to nucleus replaces one with two facets at parallel increase with +, as it does in Fig. 9. This occurs orientations as the minimum energy morphology.
L_____j
7 Consider a right triangle of height h, adjacent side R and hypotenuse S. The inorement AL may be estimated aa: AL = lim (S - R) N lim (R; $h*/R* - R) = 0. R-a,
R-m
ACKNOWLEDGEMENTS
This work was supported by National Science Foundation Grant GH-37103, for which much
ACT.4
808
METALLURGICA,
VOL.
23,
1975
or
(All)
~6 + wrl){x f Y(COSe1 f co9 e?) (Uf - 2 cos sj = {c + ~(~0s 8, f cos e2)
(A12) APPENDIX
Two-Facet
2
+ W(-cos
b)}(S f
Y5 7 Zy),
(B3)
where 8,’ = sin 6 csc 4 set e2,
Nucleus at a Grain Boundary
u = 2 f 2e,f,
As in Appendix 1, by substituting equation (12) into (11) for R, setting aEja0, = 0, and dividing and both sides by (2v)%B: 0 = (27 sin 8, csc f$}{2e, + 28 - 97 - 2kn e,lcos8, + co~ e,)+ 2 co8 d(co~e, x csc+ - sin 6)) - ${2e, + 28 - 77 - 2 sin 8,. [ - 27j(c0~8, csc 4 - sin 6)) x (2sin 8, esin e1 + 2( --co8 @ain 8, csc $1,
v=
-28,
co~ e9:
w = 2 co9 6.
Finally solving equations (B2 and B3) for 6 and q : u ESCC -a,,
co9 b,
034)
and uolDI = 8,8(~os e1 f cos e,).
(B5)
Pl)
where E = a,,Ja,, and r] = a,,“/~,,. equation (Bl): . 71{~ + Y(COS8, + cos e,) + z(-cos = (-cos
Rearranging 6))
6)(X + Yl + Zr)),
032)
where 9 = 2e2 + 26 - T, Y = -2 sin ea and Z= -2(cos er csc 4 - sin 6). By employing the s&me procedure for 6: + 28,’ - 2 co~ es . e,q + 2rl cos 8) (28, + 26 - T - 2 sin e,(c0s e1 f cos e,) 2 cos G(COS8, csc 4 - sin a)} - +{2e, 28 - T -2 sin e,t - ~~(COSe1 MC 4 sin 6)){2 + 28,’ - 28,’ cos e2 . (co9 8, + ~08 e,) - 2 sin G(cos 8, csc 4 - sin 6) - 2cosa 61,
0 = (2 x + + -
REFERENCES 1. J. K. LEE and H. I. AARONSON, dcfa -Mel. 23,809 (1975). 2. D. M. BARNETT, J. K. LEE, H. I. AUZOSSON and K. C. RUSSELL,Scripta Met. 8, 1447 (1974). 3. W. C. JOHNSON,C. WHITE, P. MARTE, P. RVF, S. Tuom-VEN, K. D. W-&DE, K. C. RUSSELLand H. I. .~_%Ro~~oN, Met. Trana. In press. 4. W. L. WLYTERBOTTOM, rlcta ivet. 15, 303 (1967). 5. K. N. Tn and D. T~NBULL, dcta iMe& 15, 1317 (1967). 6. H. I. AA~ONSON and H. B. doa, ,Met. Tram. 3, 2733
119721. 7. J. WILED GIBBS, The Collected Work8 of J. Willard Gibbs, Vo’olume I-Thermodvnamicu. D. 160. Yale Universitv Press, New Haven. C&m. (1926). 8. C. HERRWO, The Phyeics of Powder Metallurgy, edited by W. E. KINGSTON,p. 167. MeGraw-Hill, lu’ewYork (1951). 9. W. W. MULLINS.Metal &rfaeea: Structure, Energetic8 and Kinetics, edited by W. D. ROBERTSOXand N. A. GJO~TEI~, p. 63. AS&f, ;Metels Park, Ohio (1963). 10. R. E. JOW-SON, J. Phye. Chem. 811, 166.5 (1959). 11. R. KAISCHEW. drbeitatagung Featkoper Physik, p. 81. Dresden (1952). 12. E. BAC-ER,2. Kri8taUogr. 110, 372 (1958).
OF FACETING UPON THE AT GRAIN BOUNDARIES-I. JONG
K.
LEE?
and
H.
EQUILIBRIUM SHAPE TWO-DIMENSIONS* I.
OF NUCLEI
MRONSONt
The exact equilibrium sh8pe of two-dimensional nuclei (or of any other smell particles) at a grain boundany which is constrained to rem8in planar, both in the presence and the ebsence of a feoet at one orientatlon of the nucleus-matrix boundary. is found by minimizing the total interf8cial free energy subject to the constrclint of 8 constant volume. These sh8pes, as well as those obtiined by 8llowing the grain boundery to puoker in order t,o achieve force b818nces at its junctions with the nucleus, are 8160 derived graphically through 8 new generalization of the Wulff construction. The free energy of activation, A@+. for the formation of critical nuclei in these shapes is calculated 8s 8 function of the angle between the facet and the grain boundary under representative conditions of relative interfeaiel free energy; AC* is found to be significantly less at small values of this angle under most circumstances. cmd also to be smaller at puckered than at planar grain boundaries, though oft,en by negligible amounts. INFLUENCE
DES F.4CETTES SUR LA FORME A L’&QUILIBRE DES GERMES AUX JOIXTS DE GRAINS-I. GERMES BIDIMENSIOKNELS
SITUES
En minimis8nt 1’6nergie libre totale d’interface soumise il la contrainte d’un volume constant, on trouve la forme ex8cte it 1’Pquilibre d‘un germe bidimensionnel (ou de t.oute 8utre petite pa&cub) situ6 i\un joint de grains assujetti it rest.er plen en prCsence ou en l’absence d’une faxette, pour une orientation de l’interface germe-matrice. Grbe 6~une nouvelle g&+alisation de la construction de Wulff, on obtient Pgalement ces formes graphiquement. ainsi que celles qui se produisent si l’on permet au joint de grains de .se plisser de man&e B bquilibrer les forces au contact du germe. On calcule 1’6nergie libre d’8ctivstion AG* de formation de germes critiques de tee formes en fonction de l’angle entre la facet& et le joint de grains, pour des valeurs reprCsent,atives de I’bnergie libre interfaciale relative; on trouve que AG* est p6nCralement nettement plus faible pour les petites valeurs de cet ctngle. et egalement plus faible pour les joints de grains plis&s clue pour les joints plans, bien yue la diffbrence soit souoent n6gligeable. EISFLUD
DER
FACETTIERVXG
AL-F DIE GLEICHGEWICHTSGESTALT _4?i KORNGREXZEX-I. ZWEI DIMEXSIONES
VOX’
REIMEX
Die erakt,e Gleichgewichtsgestalt zweidimensionaler Keime (oder aller anderen kleinen Teilchen) an Korngrenzen wird untersucht. Die Korngrenzen sollen sowohl bei Gegenwarlrt 816 8uch bei Abwesenheit einer Facette an einer Orientierung der Keim-Mat,ris-Grenzflbche eben bleiben. Die Gleichgewichtegestalt wird durch Minimierung der gesamten GpnzflBchenergie bei konstantem Volumen gewonnen. Diese Keimformen und such die Formen, die entstehen. wenn Korngrenzen Felten zur Bildung einer Gleichgewichtslege mit den Keimen bilden. werden gr8phisch ebgeleitet mit Hilfe einer neuen Verallgemeinerung der W7ulff-Konstruktion. Die Aktivierungsenergie AG* der Bildung kritischer Keime mit diesen Formen wird 81s Funktion des Rinkels zwischen Facette und Korngrenze berechnet. Gnter Annahme repriisentativer Werte relativer Grenzfliichenenergien ergibt sich, daD AG* bei kleinen Werten dieses Wink& unter den meisten Umst.&nden betriichtlich kleiner ist und d8D AG* such an Korngrenzen mit F&en klciner 81s an ebenen Korngrenzen iat. wenn such manchmal nur urn einen vernachl&ssigbaren Betrag.
1. INTRODUCTION
The equilibrium shape of a precipitat,e crystal is of particular int,erest, because this is the morphology with the lowest free energy of activation for nucleation. In a companion paper,“) these shapes are calculated for three-dimensional nuclei formed at planar, disordered grain boundaries under the assumption t’hat an energy cusp facet appears at only one boundary orient,at,ion and in only one of the t.wo matrix grains forming the grain boundary. When the angle which the facet makes with the grain boundary exceeds a critical angle, however, rigorous minimization of the free energy is found to be not possible at t,he present time. It will be shown in this paper, however. that the equilibrium or minimum free energy morphology for the entire range of tnodimensional a,nalogues of these nuclei can be rigorousI> * Received May 4. 1974: revised October 24. 19i4. i Department of Metallurgical Engineering. Michigan Technological University. Houghton, Michigan 49931. n.S..4. ACT-4 2
METALLI:RGIC.&
VOL.
23. JULY
1955
determined. These results will be used in the companion paper to evaluate more accurately the equilibrium shapes of the three-dimensional nuclei. In both papers, the simplifying assumption has been made that. the strain energy arising from area1 or volumetric misfit between the nucleus and matrix lattices can be ignored. This assumption is exact when the critical nucleus is an ellipsoid and Young’s modulus and Poisson’s ratio (in the isotropic elasticity approximation) are the same in both phases, for st)rain energy is independent, of morphology under t,hese circumstances.(*) Even when the elastic moduli are tinequal or anisotropic elasticity is inkoduced, however. the tacit assumption t.hat’ strain energy can be adequately accounted for simply by subtracting its value from t$hat for the area1 or volume free energy change wit~hout simultaneous minimization of interfacial and strain energies should be a good first approximation when the misfit, is small (as is often the case in solid-solid tSransformations). 799
800
ACTA
METALLURGICA,
When some portion of the critical nucleus is part of a sphere, the free energy of activation for its formation is proportional to the third power of the inter-facial energy of the spherical surface (multiplied by a factor reflecting the other inter-facial energies operative13’) divided by the second power of the sum of the area1 or volume free energy and strain energy. Since strain energy is proportional to the square of the linear misfit between the two lattices, failure to minimize interfacial energy and strain energy simultaneously should thus not introduce a significant error in the activation free energy when the misfit is small. We shall first find the equilibrium shape of twodimensional nuclei developed upon a substrate or at a grain boundary by minimizing their total interfacial free energy under the constraint of a constant vo1ume.t Although the primary thrust of this paper is toward faceted nuclei, a useful introduction is provided by the consideration of their non-faceted counterparts. Then, based upon Winterbottom’s(4) concept of negative surface tension, a completely generalized Wulff construction will be developed and applied to the various classes of nucleus shape under consideration. Up to this point in the development, the assumption will be made that the grain boundary is constrained to remain planar during formation of the crit.ical nucleus. It will then be shown that under considerable ranges of conditions this assumption prevents the achievement of equilibrium shape. In order to achieve such a shape, the grain boundary must often be allowed to undergo deflections, or ‘puckering’(s) which enable it to join the nucleus at equilibrium angles. Finally, the variation of the free energy of formation of a critical nucleus, AG*, as a function of the angle, 4, c6) between the facet and the grain boundary formed by rotation of the facet about an axis lying parallel to and below the grain boundary plane (see Fig. I), will be calculated for both the planar and the puckered grain boundary nucleus configurations. Physically, 4 corresponds to the angle between the plane of the grain boundary and the closely matching conjugate habit planes which are responsible for the presence of the facet. In principle, $ can be taken as freely variable ; symmetry considerations, however, restrict the angular range which must be investigated to the range 0°-90”.
t In view of the analogy between two- and three-dimensional nuclei, u is denoted as surface rather than as line tension and wis termed volume rather than area. The term ‘surfaoe t,ension’ is understood to represent specifio interfacial free energy.
VOL.
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197.5
2. DERIVATION
OF
EQUILIBRIUM
SHAPE
In this section we derive the equilibrium shape by minimizing the total surface free energy of the system with respect to the appropriate shape parameter, such as the cont,act angle or dihedral angle. (a) A liquid nucleus on a sub&rate (no facet) A liquid particle on a substrate is shown in Fig. la. The total surface free energy is given by: E =ja dl = 2R(&s,,
+ sin @I+, -
a,,)},
(I)
where 1 is the interface length, R is the radius of the basis circle from which the nucleus is taken, 0 is the contact angle, and ofi,, uBs and u,, are energies or tensions of the vapor(a)-liquid(@), liquid-substrate(s) and vapor-substrate boundaries. Using the constant-volume co&raint, v = 3RP(20-sin 20), by substituting the relationship for R obtained from it into equation (1) and setting aE/% = 0, one obtains the well-known Young equation, a 111-
as8 = oEp cos 8.
(b) -4 nucle26e at a grain boundary The tot,al interfacial
(2)
(no facet)
free energy (Fig. 1b) is given by
R = 2R(2&$
-
u,, Sin e),.
(3)
where a,, and a,, are the interfacial energies of a disordered nucleus-matrix and a disordered grain boundary, respectively, and both are assumed independent of boundary orientation. Again using the constant-volume constraint, v = Ra(2f3 - sin Se), and setting aE/&3 = 0, one obtains u Jlol= 2~,, ~08 e =
~~~(~08 e1 +
cos e,),
(4)
where in this situation 0, = 0, = 0. These relationships (equations 2 and 4) have been derived by many investigators.(4’7-10’ However, since they provide a useful introduction to more complicated cases we have rederived them. (c) A nucleue with a facet in contact with a grain boundary We assume that the facet, with energy cZgc, appears only the ‘upper’ portion of the nucleus (Fig. lc) because the necessary lattice orientation relationship exists only with respect to the upper matrix grain. In case of bEsc .> go.., a range of 4 exist,s in which the facet does not touch the grain boundary.‘@ The smallest t,ilt angle at which the facet meem the grain boundary is denoted as q&. When a,,” < +a,,, C& = 0 since a facet is always in contract with the -gain boundary in this situation.
LEE
AND
FACETIKG
AARONSOK:
An’D
SHAPE
OF
NUCLEI-I
sin e2 = - cosbe csc +.
FIG. 1. Nucleus shapes: (a) on a substrate, (b) unfaceted grain boundary allotriomorph, (c) with one facet in contact with a grain boundary and (d) with two facets in contact with a grain boundary.
E = R{(Ze, + 26 - ~)o,~ - 2 sin e20,, - ZU,~~(COS 8, csc 4 - sin S)},
Atkention is therefore directed in this subsection to the case in which 4 2 &, but C#J is still sufficient,ly small so t,hat the facet appears at but one side of the nucleus. Since we have the same radius of curvature for each curved interface by assuming the same chemical potential all along it, we have for 8,, 2 sin e2 = sin 8, -
cos b csc (b -
cos 8, cot 4,
(5)
as shown in Appendix 1. Also as developed on the basis of the derivations in Appendix 1, the total int.erfacial free energy of the nucleus is given by: E = R((8, -+ 28, + 6 + c$ - 77)o,8 - 26,, . sin 8, - aa,' x (COS8, $ COS (6 + #))CSC(6}, (6) and the constant-volume
(10)
The expressions for the botal interfacial free energy and the constant volume can be obtained from equations (6 and 7) by replacing 8, with 6 - C#J,and adding contributions due to the parallelogram ABEC:
Cd)
Cc)
80 1
Fig. l(d). In this c&8e the shape parameters to be determined are b and x/R, where x is the perpendicular distance from the grain boundary to the junction of t,he disordered interface m-ith the shorter facet,, i.e. Al? sin C$as shown in Fig. l(d). Since 3: is described by X/R = cos (6 - 4) - cos e,, we can again use 6 and 8, as shape parameters. As in case (c), we have the same radius of curvature for each curved interface by assuming the same chemical potential. Hence we obtain for 8,,
(b)
(a)
THE
constraint is written as
2’ = -$R”{e, T tie, T b + + - 77- 2 sin e2 x .(cose1 -+ cos8,) + tcose, -i-cos (6 f- C#))cos bcsc C/J}. (7)
(11)
and r = @?{zf+ + 2b - x - 2 sin e,(cos8, -c cos 8,) + 2 cosb(cos8, csc6,- sin b)}. (i’) As before, we substitute equation (12) into (11) for R, set iYE/ae, = 0 and iTIE/&? = 0, and solve t,he resultant equations for a,, and cage (see Appendix 2). Thus we obtain for equilibrium conditions (13)
and uol== U,+OS Equation
e1 + cos e,).
(14)
(14) may be written in terms of x:
(X/R)equil = cog (6 - 4) - cos8, = cos (6 - 4) + cos e2 - u,,/u,,.(15) 3. A
GENERALIZED
WULFF
CONSTRUCTION*
A Wulff construction will now be presented for each of the classes of critical nucleus, i.e. of equilibSubst’ituting equation (7) into (6), setting aE/ae, = 0, rium particle shape, discussed in the preceding and aEli% = 0, and solving the final results simulAlthough several investigators’11e12) sugsection. taneously, one obtains for the equilibrium shape gested that the equilibrium shape could be found in (see Appendix 1) : t,his manner, Winterbottom”) was t,he first to do so O.c= -uap cos 6. (8) for a particle at an interface, developing such a aB construction for a liquid drop upon a substrate with oll = ump(c~~ 8, + cos e,). (5 (9) the aid of the concept of negative surface tension. We note that, from equation (5), e1 # 8,. (d) A ,sucleus with two parallel facets in contact with a grain boundary Once the angle C# is larger than a certain angle, denoted as tQe second critical angle, &, t.he equilibrium
shape
develops
two
parallel
facet,s
as shout
in
* After this paper had been submitted for review, the authors learned (through the courtesy of Professor K. C. Russell) that) J. IV’. Cahn and D. IV. Hoffman (Acta Net. 22, 1203, 1974) had devised essentially the same modifications of the Wulff construction. Since the objectives of the two investigations are quite different, however. there is comparatively little overlap in the applications which have been made of this construction.
ACTA
802
METALLURGICA,
VOL.
23,
1976
Another solution to this general problem is introduced here through incorporation of en ‘auxiliary y-plot’ in the Wulff construction. We define the auxiliary y-plot as a circle whose radius is proportional to the surface tension of a disordered nucleus-matrix boundary, baa, and whose center, 0’, is located et a distance proportional to the energy of the interface at which nucleation occurs, a,,, from the Wulff point, 0, in the direction perpendicular to the nucleating interface. An appropriate geometric operation will be used to develop the equilibrium shape in each case. (a) -4 nude248 on a substrate In this case the distance between points 0 and 0’ is proportional tot a,,, the surface tension between the a(vapor) phase and the substrate, as shown in Fig. 2. We draw a line parallel to the substrate, which is separated by a distance of aBs from bhe Wulff point 0. The portion of the auxiliary y-plot lying below the horizontal line is the equilibrium the distance shape. Note from Fig. 2 that O'B, between 0’ and the horizontal line, is equal to the product of oafi and sin (90’ - 0) and thus to olrBcos 8. Hence this construction yields directly cr,, = upa + aaS cos 0, i.e. the equilibrium condition given by equation (2). Note that if G,,~is less than o,, - oas, the equilibrium shape vanishes and hence complete wetting occurs. (b) d nucleus at a grain boundary (no facet) In order to equate 0,,/2 with a,, cos 8, and thus
Fm. 3. Wulff construotion for a nucleus at 8 grain bound. 8ry (no facet).
fulllll the equilibrium condition, equation (a), the horizontal line now lies at the intersection of the Wulff and the auxiliary Wulff circles and the equilibrium shape is represented by the entire area of overlap of the two circles (see Fig. 3). Again whenever there is no intersection between these circles, i.e. a,, r 2%,, complete wetting is achieved. As in case (a), we have drawn a horizontal line. Such lines: Ai in Fig. 2 and ,4T in Fig. 3, represent the location of the interface at which nucleation occurs. Therefore the essential point in the generalized Wulff construction is to locate this horizontal line. Only one such line lying between points 0 and 0’ can meet the equilibrium condition in each case. (c) 3 ltucleus with one facet id contact with the grain boudary
In this situation we have three conditions, equations (3, 8 and 9). Equation (8) can be easily fulfllled by
Fro. 2. WuB
construction
for 8 nucleus on a substrate.
t Hereeftm the proportionality equal to unity.
constant
will be essumed
drawing a facet (B& in Fig. 4) which is located at a distance of a,,” from the Wulff point, 0, but makes the angle 4 with a horizontal line. Equation (9) is of exactly the same form as equations (2 and 4), for the liquid nucleus on a substrate and for the unfaceted nucleus at a grain boundary, respectively. Similarly to equation (a), and especially to equation (4), equation (9) determines the position of the horizontal line, corresponding to the interface at which nucleation occurs, located between points S and T in Fig. 4. Let the horizontal lines be A P,the junction
LEE
AND
AARONSOK:
FACETING
AND
THE
SHAPE
OF
SUCLEI-I
803
Two of the tilt angles pertinent to the one-facet nucleus morphology are : f$_ = cos-l (-u,&,~)
+ m-1
(u&u,& if
-
72,
Us; > 2
(16)
and +c, = --OS-l
0 cm
(--u&$/u,&
-+ cos-1 (u,,/2u,B)
+ n, if a,,” > 1.‘U,fi2 -
a
0’
Fro. 1. Wulff construction for a one-facet nucleus at planar grain boundary.
a
bet,ween AP and 00’ be the point D, the dihedral angle with the main y-plot circle be 8, and the dihedral angle with the auxiliary circle be 0,. Then we have uaa=-ti?=?i6;iii? cos e, -+ uaB cog e,? =(T i.e. equation (9). The other condition; equat,ion (5): enables locat,ion
u~~214~
(17)
The fist one, equation (16), which was previously ment.ioned in Section 2(c), is termed the first critical angle and the second one, equation (17), is denoted as the non-facet, critical angle, above which no faceting occurs. In terms of these two critical angles, t.he following results are obtained from the Wulff construction. ( 1) When u,gc > &u,, and 0 I $ I +,,. If the facet B& lies inside the auxiliary y-plot, i.e. does not intersect this circle, the facet’ does not touch the grain boundary. This particular case is shown in Fig. 5. Recently Aaronson and Aaronts) discussed this case extensively. Because t,he facet is not in contact with the grain boundary equations (6 and 7) should be replaced with the following, which are independent’ of 6:
aB
E, = R((48 -J- 26 -
27~)~~~-
2 sin Bu,,
i
2 sin ~3u,~~} of dP, in effect bhrough determinat.ion of 8, and oz. z1 = lpy2e + 6 - 71 - sin 28 - sin b cos 6), Equation (5) is derived in Appendix 1. This relationship arises from the equality of the distance, AC, where 0 = COS-l (0J2uaB), i.e. e1 = 8, = 8. between the t.ao junctions of the upper portion wit.11
(18) (19)
the grain boundary and the distance, A’C’, between the two junctions of the lower portion wit,h the grain boundary. obviously required for continuit,y. Hence we have only to find the horizontal line which satisfies this equality. Since sa’ = c’p for an>* horizontal line, bhe particular line needed is t’hat which makes cc’ = c’p and hence A,4’ = cc’. The following construct,ion is the only means found through which to locate C’. On a point by point basis, draw the curve which is equidistant from t’he facet, B& and the cr,&rcle segment, Bz; at each point,, t,he equidistant, position is determined along a line parallel to the grain boundary. This curve, B?, is shown dotted in Fig. 4. The point of intersection Ibetween the equidistant curve and t,he auxiliary y-plot is t.he point C’. The equilibrium shape of the upper portion is given by ABC while that of the lower portion is the circle segment, A’TC’.
_--L\____ Oi\
G,
“\
\\
8'
0
FIG. 5. Rulff construction for a one-facet nucleus at a grain boundary when r$ 5 & and u,ge > lo,,.
ACTA
804
(2) When G,,~ > ~‘/a,~~ -
METALLURQICA,
VOL.
23,
1973
ct,,8/4aud I$~, 5 4 I 90’.
This situation arises when the facet B&stays out of the auxiliary y-plot. In this case there is no faceting in the equilibrium shape. The equilibrium shape is the unfaceted nucleus at a grain boundary discussed in Sections 2(b) and 3(b). (3) When 4 2 0 and a,,” I; +cr,.. The entire analysis of this section and of Section 2(c) holds true even when u,~” < &r,, and 4 > 0. However, in order to prove this statement in a limiting case, i.e. in which the facet sinks into the grain boundary at 4 = 0, multiply both sides of equation (5) by sin +. Setting 4 = 0, we obtain cos 19~= - cos 6 = ~~~~/a.~. Substituting this result in equation (9) we recover equation (2) and hence Fig. 2. Note that when %S,bC < (%a - u,& and 0 < 4 < co& (~,~“/a,~) cos-1 (b,, - b,&, complete wetting occurs. (d) A two-facet nucleus at a grain boundary As the tilt angle + increases the opposite facet, BT
(see Fig. 4) in the Wulff plot, moves up toward
the grain boundary.
If the facet B,g’ meets the grain
boundary line AC, the equilibrium shape contains two parallel facets. In Section 5 it will be shown that under this condition the two-facet shape is always more stable than its one-facet counterpart. The Wulff construction for the t,wo-facet morphology is shown in Fig. 6. Since the dihedral angle, 0,, for the lower portion is already fixed through equations (10 and 13), we first locate the grain boundary line
Fro. 6. WuliY construction for a two-facet nucleus at a plmer grain boundary. (X/%lmII = 0, i.e. the angle at which a second facet, parallel to the first, begins to appear, as the second critical angle, &:
(20) When (x/R),,,~~ has a negative value, t,here is no two-facet equilibrium shape. 4. PUCKER
MECHANISM
There has been considerable debate about the Young equation, equation (2) because of an undC and then check whether the resultant shape balanced component of interfacial tension, aZ8, in the meets the equilibrium condition equation (15). vertical direction (Fig. la). Considering the force The point C’ (and thus the horizontal line AT) is balance at junctions A and C in Figs. 4 and lc, we given as the point of intersection between the auxiliary note that when the facet meets t,he grain boundary, y-circle and a vertical line (z’ in Fig. 6 which is 0, # 0,, a force balance exists only in the direction parallel to the grain boundary at junction d and in parallel to 00’ and is separated by c.lac csc 4 from the neither the parallel nor the perpendicular dire&on Wulff point,, 0. The equilibrium shape for the upper at junction C. To achieve complete force balances, portion is AB’BC while the counterpart for the lower the grain boundary must pucker such that one side portion is A’TC’. moves up or down relative to the other side as shown The proof of equation (15) for this morphology is in Figs. 7 and 8. In Fig. 7, where a single facet is given by: -present, t.he right side of the grain boundary (‘R’) is X/R = (F? - OD)/OB’ displaced downward with respect to the left side =(zL_ - o’il’ CO8e,))/E (‘L’). To prove a complete force balance at junction C in Fig. 7, we examine a force balance in the direction parallel to the facet B& since no torque term is -
alo + ‘TX8cos 0,
aZs
11 = co9 (6 - 4, + CO88, - ~&,s, where 0, is given by equation (10). We denote the particular tilt angle which satisfies
necessary in this direction. We note that uo’. cos 4 = OH + O’H’ = a,,” f
amB. co9 (4 +
< CO’O), where
00’ = o,,. It is clear that force balances in all other directions at the junction C require use of t,he torque
term pertaining
to
the facet,
B<.
The
LEE
AND
FACETING
4ARONSOn’:
AXD
THE
SHAPE
OF
KCCLEI-I
805
formation of an embryo, AG”, is /
/
1’ !
\ ‘\
coe
AG” =
\/ ‘\,
. J
B dl + v +AG, = E + v - .&2,,.
where AG, is the volume free energy change of transformation, and E and v are the total surface free energy and the volume, respectively, for each shape. Eote that for each of the shapes considered E = k,R and c = k,R2, where E, and k, are constants involving only int,erfacial energies. Substituting these relationships into equation (21). the critical radius, RR*.is evaluated by setting a(AG”)/aR = 0. and thus AG* is found to be :
i
AG* = (-
FIG;. 7. Wulff construction for a one-facet nucleus at a puckered grain boundary.
n,
!
(21)
w
_r___+-_----__“~/
\
-&)E”it.
(22)
In Figs. 9 and 10, AG*/AG,* is plotted as a function of C#for several values of (~.~~/cr~~, where AG* pertains to faceted and AG,* to unfaceted nuclei (Fig. lb) formed at a grain boundary. The solid curves represent nucleus shapes at planar grain boundaries (Figs. 4 and 6) while the dotted curves represent, those with puckering (Figs. 5 and 8). In Fig. 9 the value of CJJ(J.~is equal to 1.05, while it is taken to be 0.5 in Fig. 10. When a,,’ > +a,,, AG*/AG,* is independent of 4 when C$5 #Q~.(~) The pucker mechanism is seen to yield a lower AC* than the planar boundary configuration, though over appreciable ranges of 4 and d..,c/~,8 the difference between
10
_ I___
FIG. 8. WuWf construction for a two-facet nucleus at a puckered grain boundary. 04
5. FREE
ENERGY CRITICAL
OF FORMATION NUCLEUS, AG*
FOR
/
0 300
puckered version of the two-facet, nucleus is shown in Fig. 8. In this case, both portions of the grain boundary- are displaced, in opposite directions.
7
O3i
::
:
i
A
In nucleation theory, one of the most important physical quantities is the free energy of activat,ion, AG*, for t,he formation of the critical nucleus. For a two-dimensional shape, the standard free energy of
9. Variation of AW/Aff,* with 4, where AQ,* is the the critical free energy of formation for an unfaceted grain boundary allotriomorph (a,,jn,+q = 1.05).
FIG.
ACTA
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1976
because the diminution in the grain boundary area destroyed with rising 4 is more than offset, only in the case of Fig. 10, by the growing proportion of the nucleus-matrix interface which is faceted as a result of the relatively small value of a,,. We also note in Fig. 10 that there arises, in the pucker mechanism, a discontinuity at small values of ~.~“/a,, and I$ = 0. The discontinuity develops because a twofacet morphology is always possible in the pucker mechanism at small values of ~,~~/a,~ and cJa,, and 4 > 0. 6. SUMMARY
The equilibrium shape of t,wo-dimensional critical nuclei upon a substrate and at a grain boundary was derived in the absence and the presence of an energy cusp facet at one boundary orientation by minimizing the total interfacial free energy subject t.o the 0.j...... constraint of constant volume. One family of shapes is derived with an additional constraint, i.e. main0.1 6- aO.5 tenance of a planar grain boundary. A second & I family was derived by allowing the grain boundary 0.0 0 10 20 30 40 50 60 70 00 90 t’o pucker, with an infinite range of deflection, in 0 (degrees) order to achieve force balances at the junctions with Fxo. 10. Variation of AG*/AG,* with +, where AG,* iu the critical free energy of formation for 8n unfaceted grain the nucleus, These shapes were alsoderivedgraphically boundary allotriomorph (cr,Juap = 0.5). through a new generalization of the familiar Wulff The first family of shapes, i.e. with a the two AG*‘s is very small. This arises because in construction. planar grain boundary, will be seen, in the companion two dimensions t,he nucleus and the grain boundary paper,(i) to play an important role in determining the have point junctions and thus the grain boundary shapes of the counterpart three-dimensional nuclei, can pucker without an increase in its length as long whose equilibrium shapes cannot be developed as as the range of puckering is infinite.t However, rigorously and must therefore be constructed with this will be shown, in the companion paper,(r) not to t.he assistance of the results on two-dimensional be true in the three-dimensional case For small values of o,~~/u,~, a one-facet shape nuclei obtained in the present paper. The ratio of the free energy of activation for critical obtains when # < $,, and a two-facet shape develops nucleus formation at a grain boundary for a faceted at larger values of 4. The angular range over which the two-facet morphology occurs is seen to decrease nucleus to that for an unfaceted nucleus was calcuwith increasing a,gc/o,,r. As ~,,“/a,, approaches lated as a function of the tilt angle of the facet unity, the unfaceted morphology (Fig. lb) develops relative to the boundary. This ratio often varies instead of the two-faceted one at larger tilt angles. appreciably with t.ilt angle (4) ; when ads < u_, Hence $C,, the lowest tilt angle at which the unfaceted the ratio usually increases rapidly with 4 at small tilt shape has the smallest AG*, replaces de,. This result angles. AC* for critical nuclei at puckered grain is illustrated by the uppermost curves (r~,~~/a~~= boundaries is always less in the two-dimensional case than at planar boundaries. Over appreciable ranges 0.9) in Fig. 9, where +,, r 84”. Since o,, is usually greater than or equal to a,,, of d and a.scla=a, however, this difference is very the situation in which u,./u,~ = 0.5, used in Fig. 10, small. A finding of particular interest is that when will not be frequently encountered. We note, however, the facet energy is nearly equal to that of a disordered a drastic difference between Figs. 9 and 10. In Fig. interphase boundary, at high tilt angles an unfaceted 10, AG* is seen usually to diminish rather t,han to nucleus replaces one with two facets at parallel increase with +, as it does in Fig. 9. This occurs orientations as the minimum energy morphology.
L_____j
7 Consider a right triangle of height h, adjacent side R and hypotenuse S. The inorement AL may be estimated aa: AL = lim (S - R) N lim (R; $h*/R* - R) = 0. R-a,
R-m
ACKNOWLEDGEMENTS
This work was supported by National Science Foundation Grant GH-37103, for which much
ACT.4
808
METALLURGICA,
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23,
1975
or
(All)
~6 + wrl){x f Y(COSe1 f co9 e?) (Uf - 2 cos sj = {c + ~(~0s 8, f cos e2)
(A12) APPENDIX
Two-Facet
2
+ W(-cos
b)}(S f
Y5 7 Zy),
(B3)
where 8,’ = sin 6 csc 4 set e2,
Nucleus at a Grain Boundary
u = 2 f 2e,f,
As in Appendix 1, by substituting equation (12) into (11) for R, setting aEja0, = 0, and dividing and both sides by (2v)%B: 0 = (27 sin 8, csc f$}{2e, + 28 - 97 - 2kn e,lcos8, + co~ e,)+ 2 co8 d(co~e, x csc+ - sin 6)) - ${2e, + 28 - 77 - 2 sin 8,. [ - 27j(c0~8, csc 4 - sin 6)) x (2sin 8, esin e1 + 2( --co8 @ain 8, csc $1,
v=
-28,
co~ e9:
w = 2 co9 6.
Finally solving equations (B2 and B3) for 6 and q : u ESCC -a,,
co9 b,
034)
and uolDI = 8,8(~os e1 f cos e,).
(B5)
Pl)
where E = a,,Ja,, and r] = a,,“/~,,. equation (Bl): . 71{~ + Y(COS8, + cos e,) + z(-cos = (-cos
Rearranging 6))
6)(X + Yl + Zr)),
032)
where 9 = 2e2 + 26 - T, Y = -2 sin ea and Z= -2(cos er csc 4 - sin 6). By employing the s&me procedure for 6: + 28,’ - 2 co~ es . e,q + 2rl cos 8) (28, + 26 - T - 2 sin e,(c0s e1 f cos e,) 2 cos G(COS8, csc 4 - sin a)} - +{2e, 28 - T -2 sin e,t - ~~(COSe1 MC 4 sin 6)){2 + 28,’ - 28,’ cos e2 . (co9 8, + ~08 e,) - 2 sin G(cos 8, csc 4 - sin 6) - 2cosa 61,
0 = (2 x + + -
REFERENCES 1. J. K. LEE and H. I. AARONSON, dcfa -Mel. 23,809 (1975). 2. D. M. BARNETT, J. K. LEE, H. I. AUZOSSON and K. C. RUSSELL,Scripta Met. 8, 1447 (1974). 3. W. C. JOHNSON,C. WHITE, P. MARTE, P. RVF, S. Tuom-VEN, K. D. W-&DE, K. C. RUSSELLand H. I. .~_%Ro~~oN, Met. Trana. In press. 4. W. L. WLYTERBOTTOM, rlcta ivet. 15, 303 (1967). 5. K. N. Tn and D. T~NBULL, dcta iMe& 15, 1317 (1967). 6. H. I. AA~ONSON and H. B. doa, ,Met. Tram. 3, 2733
119721. 7. J. WILED GIBBS, The Collected Work8 of J. Willard Gibbs, Vo’olume I-Thermodvnamicu. D. 160. Yale Universitv Press, New Haven. C&m. (1926). 8. C. HERRWO, The Phyeics of Powder Metallurgy, edited by W. E. KINGSTON,p. 167. MeGraw-Hill, lu’ewYork (1951). 9. W. W. MULLINS.Metal &rfaeea: Structure, Energetic8 and Kinetics, edited by W. D. ROBERTSOXand N. A. GJO~TEI~, p. 63. AS&f, ;Metels Park, Ohio (1963). 10. R. E. JOW-SON, J. Phye. Chem. 811, 166.5 (1959). 11. R. KAISCHEW. drbeitatagung Featkoper Physik, p. 81. Dresden (1952). 12. E. BAC-ER,2. Kri8taUogr. 110, 372 (1958).