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Generalization of the Gibbs-Thomson equation
SURFACE
SCIENCE 3 (1965) 429-444 o North-Holland
GENERALIZATION
Publishing Co., Amsterdam
OF THE GIBBS-THOMSON CHARLES
EQUATION
A. JOHNSON
Edgar C. Bain Laboratory for Fundamental Research, United States Steel Corporation Research Center, MonroeviNe, Pennsylvania, U.S.A. Received 24 May 1965 The Gibbs-Thomson equation relates the chemical potential of the vapor in equilibrium with a spherical drop to the radius and isotropic surface free energy of the drop. It is shown that this equation has a simple generalization to the case of arbitrary anisotropic surface free energy. This general form of the Gibbs-Thomson equation provides a simple and direct connection between the size and shape of a crystal in equilibrium with its vapor, the chemical potential of the vapor in equilibrium with the crystal and the detailed behavior of the specific surface free energy, y, as a function of surface orientation. HerringI) has given two equations which relate the chemical potential of the vapor in equilibrium with a smoothly curved surface element or a facet on a given body of arbitrary (i.e., not necessarily equilibrium) shape to the detailed behavior of y as a function of surface orientation and to the size and geometry of the given body. It is shown that Herring’s equation for a smoothly curved surface element can be made to take a considerably simpler form by comparing the geometry of the given body to that of an equilibrium body; if the given body does in fact have the equilibrium shape the still simpler general Gibbs-Thomson equation is obtained. Herring’s equation for a facet is also shown to yield the generalized Gibbs-Thomson equation when the given body is an equilibrium body.
1. Introduction For a one component system the Gibbs-Thomson equation relates the chemical potential, p”(r), of the vapor in equilibrium with a spherical droplet of radius r to the chemical potential, &co), of the vapor in equilibrium with a flat surface of the same substance at the same temperature through
where y is the (isotropic) specific surface free energy of the droplet and LI,, is the molecular volume in the condensed phase. This equation is readily derived by considering the transport of material from the flat surface to the droplet so as to increase the radius of the droplet.2) The chemical potential of the vapor in equilibrium with a small body will be uniform if and only if the body is in equilibrium with respect to changes 429
430
CHARLES
A. JOHNSON
of shape. Equation (1) applies only to the case of isotropic surface free energy and spherical drops. It is clear, for example, that one cannot apply equation (1) to the rounded edge of a plate-like precipitate: for, if the surface free energy is assumed to be isotropic then the particle is not in shape equilibrium, while if it is supposed that the plate-like shape is the equilibrium shape, then the surface free energy is certainly not isotropic. A crystalline solid will, in general, have a specific surface free energy which varies with the orientation of the surface relative to the crystal axes: if surface orientation is denoted by the unit normal to the surface, A, and y is the specific surface free energy, then y=y (A). The question of the equilibrium shape of a small crystal was posed by Gibbss) and solved under some restrictions by Wulff4); Wulff’s solution was first proved rigorously by Dinghas5), and both solution and proof were shown to apply to the most general variation of y(A) by Herring.6) One can expect that a generalization of the Gibbs-Thomson equation will hold for equilibrium-shaped particles no matter what the behavior of y(A). Such a generalization, valid for equilibrium shapes which are (ideal) polyhedra, has been given by Volmer.7) As Herring6) has pointed out, polyhedra from only a rather special subclass of the possible equilibrium shapes of crystallites, and recent experimental work has clearly shown that a variety of other forms 0ccur.s) The principal result to be presented here is a generalization of the GibbsThomson equation which is valid for any variation of y(ii) whatever. A simple derivation, based on Wulff’s construction, will be given first. It will also be shown that the equation can be derived from the equations which Herringl) has given for the chemical potential of the vapor in equilibrium with a body of arbitrary shape and arbitrary y(h). In this connection Herring’s general equations will be put into forms which depend only on the geometry of the given body and that of an equilibrium-shape body of the same substance at the same temperature. The term vapor, as used here, is intended to denote the phase surrounding the small (condensed) body of interest, whether this is a vapor phase or a condensed phase. In applying the Gibbs-Thomson equation, as derived here, to solid state problems, it is necessary that: (1) the precipitate particle be in shape equilibrium ;* and (2), there be no significant strains in the system. (The further generalization to multi-component systems follows directly along the lines set down by Volmer.)T) With these conditions satisfied, the generalized Gibbs-Thomson equation relates the enhanced solubility of a * For a crystalline particle contained in a crystalline matrix, different precipitate-matrix orientation relationships will generally lead to differing metastable equilibrium shapes (and, hence, to differing y-plots). The results derived in this paper are valid for any particular orientation relationship which is stationary.
GENERALIZATION
OF THE GIBBS-THOMSON
EQUATION
431
small particle and, equivalently, the boundary values of the chemical potential for the diffusion problem involved in the coarsening of precipitates (Ostwald ripening)s), directly to y(A). Recent work on the coarsening of small precipitate particles has shown that, for suitable systems, values of y(A) for precipitate-matrix interfaces can be obtained in this way is). In the following two sections the connection between y(A) and the equilibrium shape (Wulff’s theorem) and the derivation of the generalized GibbsThomson equation are given for crystals possessing a center of symmetry. In section 4 the same program is carried out without the restriction to crystals having central symmetry. In section 5, Herring’s equations are introduced and interpreted in terms of the geometry of an equilibrium body; it is shown that both these equations reduce to the generalized GibbsThomson equation when applied to an equilibrium body. The results are summarized in section 4. 2. Wulff’s construction The construction given by Wulff4) determines the shape which, for given y(A) and fixed volume, has the least surface free energy; this shape can be referred to as the equilibrium shape (or body) corresponding to y(A).
Fig. 1. The y-plot and WullT’s construction of the equilibriumshape
432
CHARLES
A. JOHNSON
The behavior of y as a function of surface orientation, fi, can be described as follows: from a fixed origin, construct a vector p(A) along the direction I? such that the length of p(A) is proportional to y(A) (fig. 1). These vectors map out a closed surface which will be referred to as a “y-plot”. Following Wulff*~e), construct the plane which passes through the end of, and is normal to, p(a), for every A. The body which is contained within this set of planes is then geometrically similar to the equilibrium shape (fig. 1). From a practical point of view one ordinarily has the inverse of Wulff’s problem: given the equilibrium shape, what is y(h)? It is convenient to split the process of determining y(A) into two stages. The first is the determi-
Fig. 2. Construction of the relative y-plot from the equilibrium shape (full inversion symmetry).
nation of relative values of y(A); the second step is to place these values on an absolute scale. The Gibbs-Thomson equation, together with py from, say, coarsening experiments, can provide this absolute scale. But the first step, that of going from the equilibrium body to a relative y-plot, is not unique unless the crystal structure under consideration has full (3-dimensional) inversion symmetry. For the sake of simplicity the general case is treated later; in this and the next section it is assumed that the crystal (and, hence, the y-plot) has full inversion symmetry. From an origin at the center of inversion of the equilibrium body, we erect vectors r(A) which trace out the surface of the body (fig. 2). Wulff’s theorem states that the normal distance, measured from a fixed origin, of a
GENERALIZATION
OF THE GIBBS-THOMSON
433
EQUATION
surface element on an equilibrium body is proportional to the specific surface free energy of that element. (For the present case, symmetry determines the origin unambiguously.) Thus, defining 1(R) = r(A)*A we have r(h)= const. *L(A). To fix this constant, known. Then, with I(&,)=&, y(A) = FL(r)
suppose
that
r(fi,)=
= +)+
0
for all orientations
(2) yO is
(3)
0
A which occur on the equilibrium
body.
3. Derivation of the generalized Gibbs-Thomson equation Consider a crystallite of a pure substance which is in equilibrium with its vapor, and therefore has the equilibrium shape. The size of the crystallite is specified by the value of A, relevant in equation (3). If the chemical potentials of a molecule in the solid and vapor are respectively pLs(Ao)and &A,), then equilibrium implies ,u,(n,)=&n,).* Similarly, for a flat surface (that is, the surface of an infinitely large body) of the same substance at the same temperature, p,( CO)=~~( co). Now, following the treatment given by Lewis and Randallz), we consider the changes in free energy consequent upon evaporating dn molecules from the flat surface and condensing them on the equilibrium body in such a way as to increase the size of the equilibrium body without changing its shape. The volume free energy change in this process is dFV = [cl,@,) - ~~(a)]
dn T
(4)
which must be equal to the change in surface free energy of the crystallite, dF,. Because of the equality of solid and vapor chemical potentials, equation (4) is equivalent to dF, = [I*&~) The volume
(5)
- ~&o)]dn.
of the crystal may be written
as
v = Kn; where K is a shape factor. invariant, and thus
In the process
(6) under
consideration
the shape is
dV = 3K$d(1,), * As Herring I) has shown, the possibility of point defects in the crystal requires a generalization of this equation. This has no effect on the equation for the chemical potential of the vapor as a function of crystal size.
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CHARLES
A. JOHNSON
whence dn = z
= 3KG d &) QO
QO
’
(7)
where Q, is the molecular volume in the crystal. In order to evaluate the change in surface free energy, we shall use equation (3) to determine the total surface free energy of an equilibrium body. Consider an equilibrium body, as in the previous section, whose surface is mapped out by Y(A), r(d) being taken from the center of inversion of the body. Equation (3) holds, with A,, being the normal distance of the surface element of normal ii, from our chosen origin, and y(ii,)=y,. The total surface free energy of the body is y(ri)dS
F, =
(8)
s where the integral is taken over the surface of the body. Using equation equation (8) becomes
(3),
r(A)*I?dS.
F, = f 0 s s
The divergence
theorem,
which states that
s
f(x,y,z)-AdS
=
S
VfdI’* s ”
transforms the surface integral of equation (9) into the integral of V*r(fi) over the volume, I’, of the equilibrium body. Since V *r(A) = 3, we obtain FSzz3+ A0
d&$ s
(II) 0
V
Using
equation
(6) in (1 l), F, = 3Ky,i;,
(12)
and dF, = 6Ky,A, d (A,). Hence, from (7) and (12), with dF,=dF,,
2Y(A) P,(I,)-P”(oo)=~R,= I(A 0
(13)
* The divergence theorem holds provided that (i) f is differentiable and has continuous partial derivatives, and (ii) j f. S dS exists. Both conditions are fulfilled in the present problem.
GENERALIZATION
which is the generalization
OF THE GIBBS-THOMSON
of the Gibbs-Thomson
435
EQUATION
equation
for crystals
in
equilibrium with their vapor (for crystals having full inversion symmetry). According to equation (13), the chemical potential of the vapor in equilibrium with a small crystal is directly proportional to the specific surface free energy of any given surface orientation, I?, and inversely proportional to the width of the crystal (the distance between the parallel planes tangent to the body at opposite sides) for the same orientation fi. (According to Wulff’s theorem this ratio is the same for all surface orientations on an equilibrium body.) The proof of equation (13) contains no restrictive assumptions about the “smoothness” of the equilibrium body: the proof is valid for polyhedral equilibrium bodies as well as for smoothly curved equilibrium bodies. It does not, of course, apply to orientations (such as those between two facets of a polyhedron) which do not occur on the equilibrium body. It is apparent that for a spherical equilibrium body (isotropic surface free energy) equation (13) reduces to equation (1). 4. The Gibbs-Thomson equation for crystals of less than full inversion symmetry For crystals having less than full inversion symmetry the process of constructing a y-plot from the equilibrium body is not uniquely determined; this is because we no longer have physical reasons which uniquely determine the origin for the construction. It is none the less possible to obtain a GibbsThomson equation which, in this case, relates pV to pairs of values of the specific surface free energy for opposed surface orientations. We choose an origin anywhere within the equilibrium body and erect vectors r(li) which trace out the surface of the body (fig. 3). Let us suppose that we know the absolute values of the specific surface free energy, yi = y(A,) and y; = y( - A,), for a pair of opposed surface orientations + A,. Then, defining the normal distances of the surface elements f A, from the chosen origin to be 12; = r(A,)*A, (14) I, = r( - A,)-( - A,) we have a y-plot, based on the chosen
origin, r(ri)*ii.
(15)
This is not, in general, the “physical” y-plot, but it does have the two invariance properties required for the derivation of the Gibbs-Thomson equation :
436
CHARLES
A. JOHNSON
(1) The sum y(A) + y( - A) is the same no matter what the choice of origin. of the choice of origin. (2) The total surface free energy is also independent Equation (15) is the expression of Wulff’s theorem in the form: the sum of the surface free energies for opposed orientations rtr A is proportional to the width of the equilibrium body in the direction A. Using (15) the total surface free energy is F, = 1 y(R)dS
s
S ;r
Fig. 3.
Construction
The divergence
(16)
= (~)jr(A)*AdS.
Equilibrium
shape
of the relative y-plot from the equilibrium
theorem,
when applied
shape (general case).
to (16), yields
T,=(~)SV.r(A)dV=3(~)V
(17)
V
independent of the choice of origin. On writing the volume of the body as V = K (A,+ + A,)”
(18)
and considering changes of size d(Ai +A,), the remainder is the same as in the preceding section, leading to [A@,’
+ A,> - A (41
= 2
which reduces to equation (13) for any orientation are crystallographically equivalent.
of the derivation
(19) ri for which + A and - i?
GENERALIZATION
OF THE GIBBS-THOMSON
EQUATION
437
5. Herring’s equations Herringi) has given two equations which determine the chemical potential of the vapor in equilibrium with any surface element of a crystal of arbitrary shape for which y(A) is known. These equations determine, respectively, the value of pV above a smoothly curved surface element and the average value of pu, above a flat facet. The equations are only valid when equilibrium with respect to transfer of matter between the vapor and solid exists; their application to non-equilibrium shapes is therefore restricted to situations in which transport through the vapor phase is much slower than transport across the smface (e.g., solid-state reactions). For a non-equilibrium shape pu, will vary over the surface, so that p”=p,(A). These two equations will, in turn, be shown to reduce to equation (19) (the Gibbo-Thomson equation) when the crystal under consideration has the equilibrium shape. The derivations are carried out along the lines followed in the previous section: no assumption as to crystal symmetry is made.This involves an arbitrary choice of origin for constructing y(R) from the equilibrium shape. The invariance properties mentioned in the previous section are sufficient to make all final results independent of the choice of origin. 5.1. HERRING'S EQUATION FOR A SMOOTHLY CURVED SURFACE
Consider a surface element of orientation A, on a given body of arbitrary shape. It is assumed that the principle radii of curvature, R,(R,) and R,(ri,), of the body at A, are both non-zero and finite (the body is smoothly curved and not “flat” in any direction), and it is also assumed that the second derivatives of y with respect to changes of surface orientation are finite at A, (so that the equilibrium shape is not “flat” at A,). Then, according to Herringi), the difference between the chemical potential of the vapor in equilibrium with the surface element A, on the given body, &R,), and the chemical potential of the vapor in equilibrium with a very large body, c(“(co), is given by
where R,(A,) and &(A,) are the principal radii of curvature of the body at R,; these are the radii of the circles of curvature in the principal planes Pr and PZ. The angle a1 is defined as follows : let A, be a variable unit normal lying in the plane P1; then cos 8r E 4, -A,. Similarly, for A, lying in Pz, cos 8, =A, .A,. Q, is again the molecular volume in the condensed phase. The size of the body enters equation (20) through the values of R, and R,.
438
CHARLES
A. JOHNSON
The quantities in the square brackets of equation (20) can be expressed in terms of the geometry of an equilibrium body of the same substance. To this end we introduce an equilibrium body, of size determined by I(&-,)+ A( - &,)=A: + A,, with its crystal axes aligned parallel with those of the given body. At A, on the equilibrium body we construct planes parallel to P, and Pz ; these will not, in general, be the principal planes of the equilibrium body at fi,. To evaluate the quantity
1
a2y fi, [ y+ig we need to consider
the behavior
of am
as ii varies, with li constrained
to
(b)
Fig. 4.
a) Projection
of an equilibrium body onto one of the principal planes of the given body. b) Geometry of the curve Cl.
lie in PI. The surface elements, fi, of interest are those which are tangent to the cylinder, circumscribed about the equilibrium body, whose generators are normal to P, (fig. 4a)*. The intersection of this cylinder with the plane P, is the shadow of the equilibrium body when illuminated normal to P,. The curve, C,, which bounds this shadow determines the behavior of y(A) for li lying in PI. We choose an arbitrary origin, 0, within the equilibrium body and erect the vectors r(R) which trace out the surface of the equilibrium body. Then, * G. D. Chakerian, private communication.
GENERALIZATION
according
to equation
OF THE
(15),
YP)=
GIBBS-THOMSON
439
EQUATION
_ yo+ A,++ n,
( >
r(fi).A.
P(A).A=J(A) is the support function, based on the origin 0, of the equilibrium body. And, if we project 0 onto the plane Pi, thus defining an origin 0, for the curve C,, then, for orientations fi which lie in PI,A(h) is also the support function of the curve C,. Thus
(21) where A and d2A/de2, are to be measured along the curve C,. Let p1 (B,) be the radius of curvature of C, at A,, and 6 be the vector from 0, to the center of this circle of curvature (fig. 4b). Then, for orientations A sufficiently close to A,, r(A)=6+pi(A,,)ri, whence n(h) = A.6 + p,(A,).
(22)
On expressing the difference between neighboring surface unit normals ii and A, in terms of 8, =cos-‘(A.&,) and evaluating the appropriate limit, it can readily be shown that (23) so that, from equations
(21), (22) and (23)
Y(h,) In like manner,
+(&g&= (gi)Pl (fro).
if C, is the curve bounding
body when illuminated normal curvature of C, at A,, then y(A,) +
to plane
(24)
the shadow of the equilibrium P2, and p,(ri,)
is the radius
(gg&= (gi)P2@o).
of
(25)
Using (24) and (25) we can now rewrite equation (20) in a form which depends on the geometry of an equilibrium body, a pair of values of r(li) and the geometry of the given body, and does not require any particular crystal symmetry :
($!)[$$
P”(fiO)- P”(W)= J-20 where
R,(R,)and R,(A,)are the principal
+$$]
radii of curvature
(26) of the given
440
CHARLES
A. JOHNSON
bodyat4. ~,(fid and ~~(4,)are
the radii of curvature, at R,, of the shadows of an equilibrium body on the principal planes, P, and Pz, of the given body. These “shadow” curvatures are connected with the principal radii of curvature of the equilibrium body at A, in the following way 11): let CIbe the angle between PI and one of the principal planes of the equilibrium body at A, ; then, if the principal radii of the equilibrium body are R’l”(fi,) and Rsq(R,), p1 (ii,) = cos’c( Relq(i2,,)+ sin’cl Riq (A,) p2 (A,) = sin2a R”,‘(A,) + cos2a Rzq(f?,) .
(27)
For the special case a=0 (that is, the principal planes PI and P2 of the given body at A, are also principal planes of the equilibrium body at R,) Equations (27) reduce to ~1 (fi,) = R;‘(&,) ~2 (fir,>
=
(28)
R’,4(&,).
Suppose that the given body does in fact have the equilibrium shape; the given body differs from the equilibrium body which has been used to evaluate y(Ei) only by a scale factor. We adjust the size* of the equilibrium body to make this scale factor unity: the given body and the equilibrium body are now identical, so that RFq(li)=Ri(fi), i= 1, 2, and x=0 for all A. From equations (26) and (28)
(29) which is equation
(19), the generalized
Gibbs-Thomson
equation.
5.2. HERRING’S EQUATIONFOR A FLAT FACET Equations (20) or (26) apply only to the surface of a body at a point where it is continuously curved. Neither form of this equation can be applied to surface elements of a given body which are flat, i.e., facets, Herringl) has given an equation by which the average chemical potential of the vapor in equilibrium with a flat facet can be determined. In this section it will be shown that this equation reduces to the Gibbs-Thomson equation if the given body does in fact have the equilibrium shape. Consider a body having a flat facet, of area A, and surface normal vector A n,, which is bounded by an edge, so that the surface normal changes discontinuously on crossing this edge. Let s be distance measured along the edge (from some arbitrary starting point); further, let ri(s) be the normal to * The dimensions of the equilibrium body enter equation (26) only through the ratios pl (&)/(A,+ + LO-), i= 1, 2. These ratios are independent of the size of the equilibrium body.
GENERALIZATION
the element
of surface adjacent
OF THE GIBBS-THOMSON
441
EQUATION
to the facet at s, so that 19(s)~cos-~(A(s).A,)
is the dihedral angle across the edge at s (see fig. 5). Finally, let yl=y(A,) and r(s)=y(A(s)) be the specific surface free energies of the facet and of the orientations Herringl) the average R(A,), and large body,
bordering the facet, respectively. has shown that, using these definitions, the difference between chemical potential of the vapor in equilibrium with this facet, the chemical potential of the vapor in equilibrium with a very &(co), is given by
[Y(S)CSC~(S) - y,ctne(s>]ds
,E,(PI~)- P,(CO)=: 1
(30)
where the integral is to be taken around the edge. Q,, is the molecular volume in the condensed phase. We introduce an equilibrium body, of size (At + A,),
Fig. 5.
Geometry of a facet and its adjacent surface elements.
oriented so that its crystal axes are parallel to those of the given body. Then according to equation (15)
r(fi>=
Yo+ + YO
( 1A@) n,’
where IZ(A)=r(A). R, r(A) being the tracing vector of the surface, measured from an arbitrarily chosen origin, 0. Defining A,=A(A,) and A(s)=A(A(s)), (30) becomes jj,(fi,)
- p,(a)
=520d +y0 AI
csce kf=i-)+$ [A(s)
(s) - 1, ctn0 (s)] ds .
(31)
We suppose that the given body is, in fact, identical with the equilibrium body used to define A(A); the reduction of (31) to equation (19) consists in showing that * A, = + [q+sce(s) - 1, ctne(s)]ds. P
CHARSES
442
Referring
to fig. 5, the vector equation &ii,
obviously
A. JOHNSON
holds. Taking
that ii(s)*T(s)=O,
+ T,(s) = IL(s)A(s) + T(s)
the scalar product
A(s ) .li,=cosB(s)
of this equation
and ii(s). T,(s)=IT,(s)J
i(s) = A1 costi
with h(s), noting sin B(s),
+ ITI (s)l sine(s)
or ITI (s)l = L(s)cs&(s)
- 1, ctno(s).
But A, = 3
ITI (s)/ds
whence A, = f
[A(s)cscB(s)
- &ctn0(s)]ds.
(32)
Using (32) in (31) we have
the Gibbs-Thomson equation. (The bar, denoting an average value, has been removed from &fir) since a body which is everywhere in equilibrium with its vapor must have ,LL” the same everywhere.) 6. Summary It has been shown that, for a crystal that is everywhere in equilibrium with its vapor (and thus is in equilibrium with respect to changes of shape), the chemical potential of the vapor is related to the size and shape of the crystal and to its (anisotropic) specific surface free energy in a very simple way (equations (13) or (19)). This relationship, which may be called the generalized Gibbs-Thomson equation, bears a very strong resemblance to the classic Gibbs-Thomson equation for the vapor pressure of a fluid drop. This result is, on the one hand, an extension of Volmer’s7) result for crystals which are polyhedra and, on the other hand, it can also be obtained by applying Herring’s general equations to an equilibrium shape. The particular advantage of the generalized Gibbs-Thomson equation is that of simplicity, not only in form but in application. As an illustration of the application of the generalized Gibbs-Thomson equation, let us suppose that we have a small crystal possessing central symmetry, which is known to be in equilibrium (as far as changes of shape are concerned) with its environment. (This environment, which has been
GENERALIZATION
referred
OF THE GIBBS-THOMSON
to as a vapor phase throughout
EQUATION
this paper,
can equally
443
well be a
crystalline matrix in which the substance of the crystal is soluble.) Further, suppose that the chemical potential of the vapor, relative to that for an infinitely large body of the same substance at the same temperature has been determined, either directly by a measurement of vapor pressure or enhanced solubility (concentration gradient) or indirectly from data on the coarsening of a distribution of such crystals in a solid matrixs*ra). Then, from equation (13), the specific surface free energy of any particular surface orientation is given by
Y(fi)=
P”(iO> - AC4
(
2520
)
n(a)
(33)
where~~(2,)and~4co > are the chemical potentials
of the vapor in equilibrium with the crystal under consideration and with an infinitely large crystal, Q, is the molecular volume in the crystal, and ,4(A) is the normal distance from the center of symmetry of the crystal to the surface element of orientation A (A(b)) is half the width of the crystal in the direction R; see figs. 2 and 3). In situations where equilibrium obtains across every element of the surface, but not between widely separated elements of the surface (e.g., the approach to the equilibrium shape of a particle in a solid matrix or the advance of a curved interface in a diffusion-controlled solid state transformation reaction) then one or the other of Herring’s general equations must be used. Both of these equations require detailed knowledge of y(A). It has been shown that, in Herring’s equation for a smoothly curved surface element, this detailed knowledge of y(A) can equally well be replaced by a knowledge of the geometry of the equilibrium shape (equation 26). Acknowledgements The comments and suggestions of Professor G. D. Chakerian of the Mathematics Department of the University of California (Davis) were essential to the development of much of the subject matter of this paper, and are most gratefully acknowledged. Thanks are also due to Drs. J. C. M. Li, N. Louat and R. A. Oriani, all of this Laboratory, for very helpful discussions. References 1) C. Herring, The Physics of Powder Metallurgy (McGraw-Hill
Book Company, New York, 1951) p. 143f. 2) G. N. Lewis and M. Randall, Thermodynamics (2nd edition) (McGraw-Hill Book Company, New York, 1961) p. 482.
444
CHARLES
A. JOHNSON
3) J. W. Gibbs, Collected Works, Vol. I (Longmans, Green and Co., New York, p. 321. 4) G. Wulff, Zeit. f. Krist. 34 (1901) 449. 5) A. Dinghas, Zeit. f. Krist. 105 (1944) 304. 6) C. Herring, Structure and Properties of Solid Surfaces (Univ. of Chicago Chicago, 1952) p. 5. 7) M. Volmer, Kinetik der Phsenbildung (T. Steinkopf Verlag, Leipzig, 1939) p. 8) B. E. Sundquist, Acta Met. 12 (1964) 67, 585. 9) C. Wagner, Zeit. f. Elektrochemie 65 (1961) 581. 10) G. R. Speich and R. A. Oriani, to be published in Trans. Met. Sot. AIME. 11) W. Blaschke, Kreis und Kugel (2nd edition) (Walter de Gruyter and Co., Berlin, p. 115f.
1928)
Press, 87f.
1956)
SCIENCE 3 (1965) 429-444 o North-Holland
GENERALIZATION
Publishing Co., Amsterdam
OF THE GIBBS-THOMSON CHARLES
EQUATION
A. JOHNSON
Edgar C. Bain Laboratory for Fundamental Research, United States Steel Corporation Research Center, MonroeviNe, Pennsylvania, U.S.A. Received 24 May 1965 The Gibbs-Thomson equation relates the chemical potential of the vapor in equilibrium with a spherical drop to the radius and isotropic surface free energy of the drop. It is shown that this equation has a simple generalization to the case of arbitrary anisotropic surface free energy. This general form of the Gibbs-Thomson equation provides a simple and direct connection between the size and shape of a crystal in equilibrium with its vapor, the chemical potential of the vapor in equilibrium with the crystal and the detailed behavior of the specific surface free energy, y, as a function of surface orientation. HerringI) has given two equations which relate the chemical potential of the vapor in equilibrium with a smoothly curved surface element or a facet on a given body of arbitrary (i.e., not necessarily equilibrium) shape to the detailed behavior of y as a function of surface orientation and to the size and geometry of the given body. It is shown that Herring’s equation for a smoothly curved surface element can be made to take a considerably simpler form by comparing the geometry of the given body to that of an equilibrium body; if the given body does in fact have the equilibrium shape the still simpler general Gibbs-Thomson equation is obtained. Herring’s equation for a facet is also shown to yield the generalized Gibbs-Thomson equation when the given body is an equilibrium body.
1. Introduction For a one component system the Gibbs-Thomson equation relates the chemical potential, p”(r), of the vapor in equilibrium with a spherical droplet of radius r to the chemical potential, &co), of the vapor in equilibrium with a flat surface of the same substance at the same temperature through
where y is the (isotropic) specific surface free energy of the droplet and LI,, is the molecular volume in the condensed phase. This equation is readily derived by considering the transport of material from the flat surface to the droplet so as to increase the radius of the droplet.2) The chemical potential of the vapor in equilibrium with a small body will be uniform if and only if the body is in equilibrium with respect to changes 429
430
CHARLES
A. JOHNSON
of shape. Equation (1) applies only to the case of isotropic surface free energy and spherical drops. It is clear, for example, that one cannot apply equation (1) to the rounded edge of a plate-like precipitate: for, if the surface free energy is assumed to be isotropic then the particle is not in shape equilibrium, while if it is supposed that the plate-like shape is the equilibrium shape, then the surface free energy is certainly not isotropic. A crystalline solid will, in general, have a specific surface free energy which varies with the orientation of the surface relative to the crystal axes: if surface orientation is denoted by the unit normal to the surface, A, and y is the specific surface free energy, then y=y (A). The question of the equilibrium shape of a small crystal was posed by Gibbss) and solved under some restrictions by Wulff4); Wulff’s solution was first proved rigorously by Dinghas5), and both solution and proof were shown to apply to the most general variation of y(A) by Herring.6) One can expect that a generalization of the Gibbs-Thomson equation will hold for equilibrium-shaped particles no matter what the behavior of y(A). Such a generalization, valid for equilibrium shapes which are (ideal) polyhedra, has been given by Volmer.7) As Herring6) has pointed out, polyhedra from only a rather special subclass of the possible equilibrium shapes of crystallites, and recent experimental work has clearly shown that a variety of other forms 0ccur.s) The principal result to be presented here is a generalization of the GibbsThomson equation which is valid for any variation of y(ii) whatever. A simple derivation, based on Wulff’s construction, will be given first. It will also be shown that the equation can be derived from the equations which Herringl) has given for the chemical potential of the vapor in equilibrium with a body of arbitrary shape and arbitrary y(h). In this connection Herring’s general equations will be put into forms which depend only on the geometry of the given body and that of an equilibrium-shape body of the same substance at the same temperature. The term vapor, as used here, is intended to denote the phase surrounding the small (condensed) body of interest, whether this is a vapor phase or a condensed phase. In applying the Gibbs-Thomson equation, as derived here, to solid state problems, it is necessary that: (1) the precipitate particle be in shape equilibrium ;* and (2), there be no significant strains in the system. (The further generalization to multi-component systems follows directly along the lines set down by Volmer.)T) With these conditions satisfied, the generalized Gibbs-Thomson equation relates the enhanced solubility of a * For a crystalline particle contained in a crystalline matrix, different precipitate-matrix orientation relationships will generally lead to differing metastable equilibrium shapes (and, hence, to differing y-plots). The results derived in this paper are valid for any particular orientation relationship which is stationary.
GENERALIZATION
OF THE GIBBS-THOMSON
EQUATION
431
small particle and, equivalently, the boundary values of the chemical potential for the diffusion problem involved in the coarsening of precipitates (Ostwald ripening)s), directly to y(A). Recent work on the coarsening of small precipitate particles has shown that, for suitable systems, values of y(A) for precipitate-matrix interfaces can be obtained in this way is). In the following two sections the connection between y(A) and the equilibrium shape (Wulff’s theorem) and the derivation of the generalized GibbsThomson equation are given for crystals possessing a center of symmetry. In section 4 the same program is carried out without the restriction to crystals having central symmetry. In section 5, Herring’s equations are introduced and interpreted in terms of the geometry of an equilibrium body; it is shown that both these equations reduce to the generalized GibbsThomson equation when applied to an equilibrium body. The results are summarized in section 4. 2. Wulff’s construction The construction given by Wulff4) determines the shape which, for given y(A) and fixed volume, has the least surface free energy; this shape can be referred to as the equilibrium shape (or body) corresponding to y(A).
Fig. 1. The y-plot and WullT’s construction of the equilibriumshape
432
CHARLES
A. JOHNSON
The behavior of y as a function of surface orientation, fi, can be described as follows: from a fixed origin, construct a vector p(A) along the direction I? such that the length of p(A) is proportional to y(A) (fig. 1). These vectors map out a closed surface which will be referred to as a “y-plot”. Following Wulff*~e), construct the plane which passes through the end of, and is normal to, p(a), for every A. The body which is contained within this set of planes is then geometrically similar to the equilibrium shape (fig. 1). From a practical point of view one ordinarily has the inverse of Wulff’s problem: given the equilibrium shape, what is y(h)? It is convenient to split the process of determining y(A) into two stages. The first is the determi-
Fig. 2. Construction of the relative y-plot from the equilibrium shape (full inversion symmetry).
nation of relative values of y(A); the second step is to place these values on an absolute scale. The Gibbs-Thomson equation, together with py from, say, coarsening experiments, can provide this absolute scale. But the first step, that of going from the equilibrium body to a relative y-plot, is not unique unless the crystal structure under consideration has full (3-dimensional) inversion symmetry. For the sake of simplicity the general case is treated later; in this and the next section it is assumed that the crystal (and, hence, the y-plot) has full inversion symmetry. From an origin at the center of inversion of the equilibrium body, we erect vectors r(A) which trace out the surface of the body (fig. 2). Wulff’s theorem states that the normal distance, measured from a fixed origin, of a
GENERALIZATION
OF THE GIBBS-THOMSON
433
EQUATION
surface element on an equilibrium body is proportional to the specific surface free energy of that element. (For the present case, symmetry determines the origin unambiguously.) Thus, defining 1(R) = r(A)*A we have r(h)= const. *L(A). To fix this constant, known. Then, with I(&,)=&, y(A) = FL(r)
suppose
that
r(fi,)=
= +)+
0
for all orientations
(2) yO is
(3)
0
A which occur on the equilibrium
body.
3. Derivation of the generalized Gibbs-Thomson equation Consider a crystallite of a pure substance which is in equilibrium with its vapor, and therefore has the equilibrium shape. The size of the crystallite is specified by the value of A, relevant in equation (3). If the chemical potentials of a molecule in the solid and vapor are respectively pLs(Ao)and &A,), then equilibrium implies ,u,(n,)=&n,).* Similarly, for a flat surface (that is, the surface of an infinitely large body) of the same substance at the same temperature, p,( CO)=~~( co). Now, following the treatment given by Lewis and Randallz), we consider the changes in free energy consequent upon evaporating dn molecules from the flat surface and condensing them on the equilibrium body in such a way as to increase the size of the equilibrium body without changing its shape. The volume free energy change in this process is dFV = [cl,@,) - ~~(a)]
dn T
(4)
which must be equal to the change in surface free energy of the crystallite, dF,. Because of the equality of solid and vapor chemical potentials, equation (4) is equivalent to dF, = [I*&~) The volume
(5)
- ~&o)]dn.
of the crystal may be written
as
v = Kn; where K is a shape factor. invariant, and thus
In the process
(6) under
consideration
the shape is
dV = 3K$d(1,), * As Herring I) has shown, the possibility of point defects in the crystal requires a generalization of this equation. This has no effect on the equation for the chemical potential of the vapor as a function of crystal size.
434
CHARLES
A. JOHNSON
whence dn = z
= 3KG d &) QO
QO
’
(7)
where Q, is the molecular volume in the crystal. In order to evaluate the change in surface free energy, we shall use equation (3) to determine the total surface free energy of an equilibrium body. Consider an equilibrium body, as in the previous section, whose surface is mapped out by Y(A), r(d) being taken from the center of inversion of the body. Equation (3) holds, with A,, being the normal distance of the surface element of normal ii, from our chosen origin, and y(ii,)=y,. The total surface free energy of the body is y(ri)dS
F, =
(8)
s where the integral is taken over the surface of the body. Using equation equation (8) becomes
(3),
r(A)*I?dS.
F, = f 0 s s
The divergence
theorem,
which states that
s
f(x,y,z)-AdS
=
S
VfdI’* s ”
transforms the surface integral of equation (9) into the integral of V*r(fi) over the volume, I’, of the equilibrium body. Since V *r(A) = 3, we obtain FSzz3+ A0
d&$ s
(II) 0
V
Using
equation
(6) in (1 l), F, = 3Ky,i;,
(12)
and dF, = 6Ky,A, d (A,). Hence, from (7) and (12), with dF,=dF,,
2Y(A) P,(I,)-P”(oo)=~R,= I(A 0
(13)
* The divergence theorem holds provided that (i) f is differentiable and has continuous partial derivatives, and (ii) j f. S dS exists. Both conditions are fulfilled in the present problem.
GENERALIZATION
which is the generalization
OF THE GIBBS-THOMSON
of the Gibbs-Thomson
435
EQUATION
equation
for crystals
in
equilibrium with their vapor (for crystals having full inversion symmetry). According to equation (13), the chemical potential of the vapor in equilibrium with a small crystal is directly proportional to the specific surface free energy of any given surface orientation, I?, and inversely proportional to the width of the crystal (the distance between the parallel planes tangent to the body at opposite sides) for the same orientation fi. (According to Wulff’s theorem this ratio is the same for all surface orientations on an equilibrium body.) The proof of equation (13) contains no restrictive assumptions about the “smoothness” of the equilibrium body: the proof is valid for polyhedral equilibrium bodies as well as for smoothly curved equilibrium bodies. It does not, of course, apply to orientations (such as those between two facets of a polyhedron) which do not occur on the equilibrium body. It is apparent that for a spherical equilibrium body (isotropic surface free energy) equation (13) reduces to equation (1). 4. The Gibbs-Thomson equation for crystals of less than full inversion symmetry For crystals having less than full inversion symmetry the process of constructing a y-plot from the equilibrium body is not uniquely determined; this is because we no longer have physical reasons which uniquely determine the origin for the construction. It is none the less possible to obtain a GibbsThomson equation which, in this case, relates pV to pairs of values of the specific surface free energy for opposed surface orientations. We choose an origin anywhere within the equilibrium body and erect vectors r(li) which trace out the surface of the body (fig. 3). Let us suppose that we know the absolute values of the specific surface free energy, yi = y(A,) and y; = y( - A,), for a pair of opposed surface orientations + A,. Then, defining the normal distances of the surface elements f A, from the chosen origin to be 12; = r(A,)*A, (14) I, = r( - A,)-( - A,) we have a y-plot, based on the chosen
origin, r(ri)*ii.
(15)
This is not, in general, the “physical” y-plot, but it does have the two invariance properties required for the derivation of the Gibbs-Thomson equation :
436
CHARLES
A. JOHNSON
(1) The sum y(A) + y( - A) is the same no matter what the choice of origin. of the choice of origin. (2) The total surface free energy is also independent Equation (15) is the expression of Wulff’s theorem in the form: the sum of the surface free energies for opposed orientations rtr A is proportional to the width of the equilibrium body in the direction A. Using (15) the total surface free energy is F, = 1 y(R)dS
s
S ;r
Fig. 3.
Construction
The divergence
(16)
= (~)jr(A)*AdS.
Equilibrium
shape
of the relative y-plot from the equilibrium
theorem,
when applied
shape (general case).
to (16), yields
T,=(~)SV.r(A)dV=3(~)V
(17)
V
independent of the choice of origin. On writing the volume of the body as V = K (A,+ + A,)”
(18)
and considering changes of size d(Ai +A,), the remainder is the same as in the preceding section, leading to [A@,’
+ A,> - A (41
= 2
which reduces to equation (13) for any orientation are crystallographically equivalent.
of the derivation
(19) ri for which + A and - i?
GENERALIZATION
OF THE GIBBS-THOMSON
EQUATION
437
5. Herring’s equations Herringi) has given two equations which determine the chemical potential of the vapor in equilibrium with any surface element of a crystal of arbitrary shape for which y(A) is known. These equations determine, respectively, the value of pV above a smoothly curved surface element and the average value of pu, above a flat facet. The equations are only valid when equilibrium with respect to transfer of matter between the vapor and solid exists; their application to non-equilibrium shapes is therefore restricted to situations in which transport through the vapor phase is much slower than transport across the smface (e.g., solid-state reactions). For a non-equilibrium shape pu, will vary over the surface, so that p”=p,(A). These two equations will, in turn, be shown to reduce to equation (19) (the Gibbo-Thomson equation) when the crystal under consideration has the equilibrium shape. The derivations are carried out along the lines followed in the previous section: no assumption as to crystal symmetry is made.This involves an arbitrary choice of origin for constructing y(R) from the equilibrium shape. The invariance properties mentioned in the previous section are sufficient to make all final results independent of the choice of origin. 5.1. HERRING'S EQUATION FOR A SMOOTHLY CURVED SURFACE
Consider a surface element of orientation A, on a given body of arbitrary shape. It is assumed that the principle radii of curvature, R,(R,) and R,(ri,), of the body at A, are both non-zero and finite (the body is smoothly curved and not “flat” in any direction), and it is also assumed that the second derivatives of y with respect to changes of surface orientation are finite at A, (so that the equilibrium shape is not “flat” at A,). Then, according to Herringi), the difference between the chemical potential of the vapor in equilibrium with the surface element A, on the given body, &R,), and the chemical potential of the vapor in equilibrium with a very large body, c(“(co), is given by
where R,(A,) and &(A,) are the principal radii of curvature of the body at R,; these are the radii of the circles of curvature in the principal planes Pr and PZ. The angle a1 is defined as follows : let A, be a variable unit normal lying in the plane P1; then cos 8r E 4, -A,. Similarly, for A, lying in Pz, cos 8, =A, .A,. Q, is again the molecular volume in the condensed phase. The size of the body enters equation (20) through the values of R, and R,.
438
CHARLES
A. JOHNSON
The quantities in the square brackets of equation (20) can be expressed in terms of the geometry of an equilibrium body of the same substance. To this end we introduce an equilibrium body, of size determined by I(&-,)+ A( - &,)=A: + A,, with its crystal axes aligned parallel with those of the given body. At A, on the equilibrium body we construct planes parallel to P, and Pz ; these will not, in general, be the principal planes of the equilibrium body at fi,. To evaluate the quantity
1
a2y fi, [ y+ig we need to consider
the behavior
of am
as ii varies, with li constrained
to
(b)
Fig. 4.
a) Projection
of an equilibrium body onto one of the principal planes of the given body. b) Geometry of the curve Cl.
lie in PI. The surface elements, fi, of interest are those which are tangent to the cylinder, circumscribed about the equilibrium body, whose generators are normal to P, (fig. 4a)*. The intersection of this cylinder with the plane P, is the shadow of the equilibrium body when illuminated normal to P,. The curve, C,, which bounds this shadow determines the behavior of y(A) for li lying in PI. We choose an arbitrary origin, 0, within the equilibrium body and erect the vectors r(R) which trace out the surface of the equilibrium body. Then, * G. D. Chakerian, private communication.
GENERALIZATION
according
to equation
OF THE
(15),
YP)=
GIBBS-THOMSON
439
EQUATION
_ yo+ A,++ n,
( >
r(fi).A.
P(A).A=J(A) is the support function, based on the origin 0, of the equilibrium body. And, if we project 0 onto the plane Pi, thus defining an origin 0, for the curve C,, then, for orientations fi which lie in PI,A(h) is also the support function of the curve C,. Thus
(21) where A and d2A/de2, are to be measured along the curve C,. Let p1 (B,) be the radius of curvature of C, at A,, and 6 be the vector from 0, to the center of this circle of curvature (fig. 4b). Then, for orientations A sufficiently close to A,, r(A)=6+pi(A,,)ri, whence n(h) = A.6 + p,(A,).
(22)
On expressing the difference between neighboring surface unit normals ii and A, in terms of 8, =cos-‘(A.&,) and evaluating the appropriate limit, it can readily be shown that (23) so that, from equations
(21), (22) and (23)
Y(h,) In like manner,
+(&g&= (gi)Pl (fro).
if C, is the curve bounding
body when illuminated normal curvature of C, at A,, then y(A,) +
to plane
(24)
the shadow of the equilibrium P2, and p,(ri,)
is the radius
(gg&= (gi)P2@o).
of
(25)
Using (24) and (25) we can now rewrite equation (20) in a form which depends on the geometry of an equilibrium body, a pair of values of r(li) and the geometry of the given body, and does not require any particular crystal symmetry :
($!)[$$
P”(fiO)- P”(W)= J-20 where
R,(R,)and R,(A,)are the principal
+$$]
radii of curvature
(26) of the given
440
CHARLES
A. JOHNSON
bodyat4. ~,(fid and ~~(4,)are
the radii of curvature, at R,, of the shadows of an equilibrium body on the principal planes, P, and Pz, of the given body. These “shadow” curvatures are connected with the principal radii of curvature of the equilibrium body at A, in the following way 11): let CIbe the angle between PI and one of the principal planes of the equilibrium body at A, ; then, if the principal radii of the equilibrium body are R’l”(fi,) and Rsq(R,), p1 (ii,) = cos’c( Relq(i2,,)+ sin’cl Riq (A,) p2 (A,) = sin2a R”,‘(A,) + cos2a Rzq(f?,) .
(27)
For the special case a=0 (that is, the principal planes PI and P2 of the given body at A, are also principal planes of the equilibrium body at R,) Equations (27) reduce to ~1 (fi,) = R;‘(&,) ~2 (fir,>
=
(28)
R’,4(&,).
Suppose that the given body does in fact have the equilibrium shape; the given body differs from the equilibrium body which has been used to evaluate y(Ei) only by a scale factor. We adjust the size* of the equilibrium body to make this scale factor unity: the given body and the equilibrium body are now identical, so that RFq(li)=Ri(fi), i= 1, 2, and x=0 for all A. From equations (26) and (28)
(29) which is equation
(19), the generalized
Gibbs-Thomson
equation.
5.2. HERRING’S EQUATIONFOR A FLAT FACET Equations (20) or (26) apply only to the surface of a body at a point where it is continuously curved. Neither form of this equation can be applied to surface elements of a given body which are flat, i.e., facets, Herringl) has given an equation by which the average chemical potential of the vapor in equilibrium with a flat facet can be determined. In this section it will be shown that this equation reduces to the Gibbs-Thomson equation if the given body does in fact have the equilibrium shape. Consider a body having a flat facet, of area A, and surface normal vector A n,, which is bounded by an edge, so that the surface normal changes discontinuously on crossing this edge. Let s be distance measured along the edge (from some arbitrary starting point); further, let ri(s) be the normal to * The dimensions of the equilibrium body enter equation (26) only through the ratios pl (&)/(A,+ + LO-), i= 1, 2. These ratios are independent of the size of the equilibrium body.
GENERALIZATION
the element
of surface adjacent
OF THE GIBBS-THOMSON
441
EQUATION
to the facet at s, so that 19(s)~cos-~(A(s).A,)
is the dihedral angle across the edge at s (see fig. 5). Finally, let yl=y(A,) and r(s)=y(A(s)) be the specific surface free energies of the facet and of the orientations Herringl) the average R(A,), and large body,
bordering the facet, respectively. has shown that, using these definitions, the difference between chemical potential of the vapor in equilibrium with this facet, the chemical potential of the vapor in equilibrium with a very &(co), is given by
[Y(S)CSC~(S) - y,ctne(s>]ds
,E,(PI~)- P,(CO)=: 1
(30)
where the integral is to be taken around the edge. Q,, is the molecular volume in the condensed phase. We introduce an equilibrium body, of size (At + A,),
Fig. 5.
Geometry of a facet and its adjacent surface elements.
oriented so that its crystal axes are parallel to those of the given body. Then according to equation (15)
r(fi>=
Yo+ + YO
( 1A@) n,’
where IZ(A)=r(A). R, r(A) being the tracing vector of the surface, measured from an arbitrarily chosen origin, 0. Defining A,=A(A,) and A(s)=A(A(s)), (30) becomes jj,(fi,)
- p,(a)
=520d +y0 AI
csce kf=i-)+$ [A(s)
(s) - 1, ctn0 (s)] ds .
(31)
We suppose that the given body is, in fact, identical with the equilibrium body used to define A(A); the reduction of (31) to equation (19) consists in showing that * A, = + [q+sce(s) - 1, ctne(s)]ds. P
CHARSES
442
Referring
to fig. 5, the vector equation &ii,
obviously
A. JOHNSON
holds. Taking
that ii(s)*T(s)=O,
+ T,(s) = IL(s)A(s) + T(s)
the scalar product
A(s ) .li,=cosB(s)
of this equation
and ii(s). T,(s)=IT,(s)J
i(s) = A1 costi
with h(s), noting sin B(s),
+ ITI (s)l sine(s)
or ITI (s)l = L(s)cs&(s)
- 1, ctno(s).
But A, = 3
ITI (s)/ds
whence A, = f
[A(s)cscB(s)
- &ctn0(s)]ds.
(32)
Using (32) in (31) we have
the Gibbs-Thomson equation. (The bar, denoting an average value, has been removed from &fir) since a body which is everywhere in equilibrium with its vapor must have ,LL” the same everywhere.) 6. Summary It has been shown that, for a crystal that is everywhere in equilibrium with its vapor (and thus is in equilibrium with respect to changes of shape), the chemical potential of the vapor is related to the size and shape of the crystal and to its (anisotropic) specific surface free energy in a very simple way (equations (13) or (19)). This relationship, which may be called the generalized Gibbs-Thomson equation, bears a very strong resemblance to the classic Gibbs-Thomson equation for the vapor pressure of a fluid drop. This result is, on the one hand, an extension of Volmer’s7) result for crystals which are polyhedra and, on the other hand, it can also be obtained by applying Herring’s general equations to an equilibrium shape. The particular advantage of the generalized Gibbs-Thomson equation is that of simplicity, not only in form but in application. As an illustration of the application of the generalized Gibbs-Thomson equation, let us suppose that we have a small crystal possessing central symmetry, which is known to be in equilibrium (as far as changes of shape are concerned) with its environment. (This environment, which has been
GENERALIZATION
referred
OF THE GIBBS-THOMSON
to as a vapor phase throughout
EQUATION
this paper,
can equally
443
well be a
crystalline matrix in which the substance of the crystal is soluble.) Further, suppose that the chemical potential of the vapor, relative to that for an infinitely large body of the same substance at the same temperature has been determined, either directly by a measurement of vapor pressure or enhanced solubility (concentration gradient) or indirectly from data on the coarsening of a distribution of such crystals in a solid matrixs*ra). Then, from equation (13), the specific surface free energy of any particular surface orientation is given by
Y(fi)=
P”(iO> - AC4
(
2520
)
n(a)
(33)
where~~(2,)and~4co > are the chemical potentials
of the vapor in equilibrium with the crystal under consideration and with an infinitely large crystal, Q, is the molecular volume in the crystal, and ,4(A) is the normal distance from the center of symmetry of the crystal to the surface element of orientation A (A(b)) is half the width of the crystal in the direction R; see figs. 2 and 3). In situations where equilibrium obtains across every element of the surface, but not between widely separated elements of the surface (e.g., the approach to the equilibrium shape of a particle in a solid matrix or the advance of a curved interface in a diffusion-controlled solid state transformation reaction) then one or the other of Herring’s general equations must be used. Both of these equations require detailed knowledge of y(A). It has been shown that, in Herring’s equation for a smoothly curved surface element, this detailed knowledge of y(A) can equally well be replaced by a knowledge of the geometry of the equilibrium shape (equation 26). Acknowledgements The comments and suggestions of Professor G. D. Chakerian of the Mathematics Department of the University of California (Davis) were essential to the development of much of the subject matter of this paper, and are most gratefully acknowledged. Thanks are also due to Drs. J. C. M. Li, N. Louat and R. A. Oriani, all of this Laboratory, for very helpful discussions. References 1) C. Herring, The Physics of Powder Metallurgy (McGraw-Hill
Book Company, New York, 1951) p. 143f. 2) G. N. Lewis and M. Randall, Thermodynamics (2nd edition) (McGraw-Hill Book Company, New York, 1961) p. 482.
444
CHARLES
A. JOHNSON
3) J. W. Gibbs, Collected Works, Vol. I (Longmans, Green and Co., New York, p. 321. 4) G. Wulff, Zeit. f. Krist. 34 (1901) 449. 5) A. Dinghas, Zeit. f. Krist. 105 (1944) 304. 6) C. Herring, Structure and Properties of Solid Surfaces (Univ. of Chicago Chicago, 1952) p. 5. 7) M. Volmer, Kinetik der Phsenbildung (T. Steinkopf Verlag, Leipzig, 1939) p. 8) B. E. Sundquist, Acta Met. 12 (1964) 67, 585. 9) C. Wagner, Zeit. f. Elektrochemie 65 (1961) 581. 10) G. R. Speich and R. A. Oriani, to be published in Trans. Met. Sot. AIME. 11) W. Blaschke, Kreis und Kugel (2nd edition) (Walter de Gruyter and Co., Berlin, p. 115f.
1928)
Press, 87f.
1956)