Creator:David Turnbull and J.C. Fisher
Date Created:June 28, 1948
Place Created:Schenectady, New York
Keywords:nucleation in condensed systems
Context:article reprinted from The Journal of Chemical Physics
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Reprinted from The Journal of Chemical Physics, Vol. 17, No. 4, 429, April, 1949
Prinled in U. S A
Errata: Rate of Nucleation in Condensed Systems
(J Chem: Pliys. 17. 71 (1949)] J. C. Fisher
General Electric Company Research Laboratory, Schenectady. Srw York
'"jpHE first line of the second column, page 71, should end
i'~(2A/3£)\ and Eq. (15) should read
Reprinted from The Journal of Chemical Physics, Vol. 17, No. 1. 71-73, Jannary, 1940
Printed m u. s. A.
Rate of Nucleation in Condensed Systems
l). Fursbull* ANixjJ. C. Fisher* General Electric Company. Schenectady. Xnc York (Received June 28, 1948)
On the basis of the nucleation theory developed by Volmer, Becker, and co-workers, and the theory of absolute reaction rates, ail expression is derived fur the absolute rate of nucleation in condensed systems.
BECKER1 has proposed the following type of expression for the rate of nucleation in condensed systems (i.e., liquid-solid or solid-solid transformations)	,
r* = Kexpi-(AF.* + q)/kT], . (1)
where A/\* is the maximum free energy necessary for nucleus formation, q is the energy of activation for diffusion across the phase boundary (or within the solid solution when the transformation involves the separation of a phase having a different composition), and A" is an undetermined constant. Although Becker and Doring* were able to evaluate semiquantitative^' the factor corresponding to K for the rate of nucleation of a liquid from a supersaturated vapor, to the authors' knowledge no theory has been proposed for nucleation in condensed systems from which K can be specified.
It is the purpose of the present paper to derive an expression for r*, applying to condensed systems, on the basis of the theory of absolute reaction rates.
Nucleation theory frequently leads to an expression of the form
&F,/kT=Aii — Bi	(2)
for the local free energy change associated with the formation of a region of a new phase j3 in a parent phase «. In this equation i is the number of atoms or molecules in the transformed region. A is proportional to the interfacial free energy per unit area of a-/3-interface, and B is proportional to the bulk free energy difference between /3 and a in the absence of surfaces.
The curve of AFt/kT versus i passes through a
* Research Associate, General Electric Research Laboratory.
1 R. Recker, Ann. Phvsik, 32, 128 (1938).
' R. Becker and W. Daring, Ann. d. Physik [5] 24, 719 (1935).
maximum ^FS/kT^AA^/nB* at i* = {2A/3B). then decreases without limit as shown in Fig. 1* Subcritical nuclei containing fewer than i* atoms require free energy for further growth, while those containing more than i* grow freely with decreasing free energy. Since nuclei generally grow one atom at a time as the result of statistical thermal fluctuations, it is evident that small nuclei with fewer than i* atpms will usually disappear without reaching critical size. Only occasionally will a long chain of favorable energy fluctuations produce a nucleus exceeding the critical size.
The steady state rate of nucleation for a given transformation corresponds to constant equal net forward rates for the following set of reactions
An + ai^ftn+l,	^
+011^/3111+31
where an represents an atom of phase a, /?, a nucleus of phase /3 containing t atoms, and where 0m is the smallest nucleus of phase Further, the concentration of /3-nuclei must decrease to zero as the nucleus size increases,
lim [0,]=O,	(4)
i~* «>
since otherwise the percentage of untransformed a will not differ from zero.
Consider the reaction
iSi+ai^i+i	(5)
at steady state. Referred to phase a as the standard state, the free energy of 3,+ o<i is AFi while that of /3,+i is AFi+\. Assuming that the intermediate configurations between (3,+ai and /3<+i correspond to free energies greater than either AF, or A/\+l, each path by which and

72
D. TURNBULL AND J . C. FISHER
ai can be combined to give /3i+i will have an intermediate configuration of maximum free energy. For one of these paths there will be a particular intermediate configuration corresponding to the least maximum free energy. This configuration may be called an activated complex. The free energy difference between the activated complex and the mean of A/\ and A/\+i will be denoted by A/*.
Figure 2 indicates schematically the free energy variation associated with the combination of fi, and ai to give /Si+i (or the decomposition of /Si+i to give ot\) by a path having the least possible maximum free energy. If represents the steady state concentration of /3< nuclei, the rate of the forward reaction
0,+ai—»/8»+i
can be written as
r+ = n.aJikT/h) exp( - AfS/kT) (6)
according to the theory of absolute reaction rates.3 Here Oil' is the number of a-atoms in contact with each 0, nucleus, and n.Oji' is therefore the concentration of a-atoms available for the reaction. (kT/h) exp( — Af\*/kT) is the specific reaction rate for the formation of the activated complex with free energy increment A/i* as shown in Fig. 2.
The rate of the reverse reaction /3,fl—»/3,-fai can be written as
r- = ni+iaifl(kT/h) exp(-Aft*/kT), (7)
where is the number of /3-atoms in contact with a per /3,+i nucleus. The net forward rate of the reaction is therefore
r* = r —r~ = (*r/A)i«[»jai exp(-Afi*/kT)
-«i+Io2exp(-A/2V^r)]. (8)
Fig. 1. (aFi/KT)=Ai*-Bi versus i.
' S. Glasstone, K. J. Laidler, and H. F.yring, The Theory of Rate Processes (McGraw-Hill Book Company, Inc., New York, 1941).
Equation (8) can be solved approximately for w, by making the following simplifications:
(a)	The number of a-atoms in contact with a nucleus is not strictly equal to the number of
/5 atoms in contact with a at the surface of a nucleus. However, the difference is negligible for all but the smallest nuclei, and <ii = <72 = a will be assumed.
(b)	rt, and AFi will be assumed smooth functions of i although they have meaning only for integral values of i. The change in w, as i changes by one unit will be approximated by (drti/di)Ai — drn/di \ similarly the change in AFi will be approximated by d(AFt)/di. The approximation is good for all but the smallest nuclei.
(3) The quantities (l/kT)d(AF,)/di and (1 /«,) X(dtii/di) will both be assumed small in comparison with unity. Again the approximation is poor only for the smallest nuclei. Noting that
Ah* = Af* + \ (AFi+AFi+l) —AFi = A/* + i(AF,-+1-AF.) = Af*+$d(AFi)/di, AfS = Ar~\d{AF,)/di, r* = (-akT/h) exp(-Af*/kT)
X [»,(\A i -»- B) + dm/dt] t'». (9)
Defining
* = ,y(a-r)exp(-Ar/^ (10)
Eq. (9) reduces to the differential equation
dni/di+(lAi-*-B)n,=-Ri~i. (11) The solution to Eq. (11) is
J exp{Ail — Bi)i~ldi
+n,oexp(Aioi-Bio) , (12)
Fig. 2. .Schematic representation of free energy of activation.
NUCLEATION
73
where ni0 is"the steady state number of j9,0 nuclei.
The steady state concentration of subcritical nuclei containing say one-third the critical number of atoms does not differ appreciably from the equilibrium concentration assuming no nucleation. Taking it**i*/3,	t
w,0= w exp( — A/\0. kT) =n exp[— (.-hV —-6to)],
where n is the number of atoms of untransformed a. The expression for the steady state number of fit nuclei reduces to
tii = exp( — A /*',/kT)
x[-Rj exp(AFi/kTti M+nJ. (13)
Since
lim exp( — \Fi/kT) = i — «
and it is required that
lim w, = 0,
i—»oc
it must follow that
lim | -R £ exp(±F,/kT)i-»</?' +wj = 0, (14) and hence that
R = fl/[I	(15)
The exact location of the lower limit of integration in Eq. (15) is unimportant, since for to intermediate between zero and i* practically all the contribution to the integral comes from values of i very near i*. Replacing hF.'kT by a Taylor's expansion about i-i* and neglecting all
terms after the third, the integral can be
evaluated as
1 = (,9r/A)H*-i exp(AF*/kT). (16)
The steady state rateof nucleation is: therefore,
r* = n*(A/9ic)l(NkT/k)
XexP[-(A/M-AF*)/*7'] (17)
nuclei per second per mole of untransformed material, where n* is the number of surface atoms in the critical size nucleus. The quantity n*(A/9ir)l proves to be within a factor of ten of unity for most nucleation problems of interest, giving
r*—i NkT/h) exp[-(A/* + AF*)/*r] (18)
nuclei per mole per second to an order of magnitude.
Equation (18) was derive*! for nucleation processes that do not require long-range diffusion. A/* is the free energy of activation for the short-range diffusion of atoms or molecules moving a fraction of an atomic distance across an interface to join a new lattice. However, long-range diffusion is to be expected in phase transformations that involve more than one component, since the new phase and the old are generally of different composition. Equation (18) approximates the rate of nucleation in a two or more component system when A/* is taken as the activation energy for diffusion of the most slowly moving component and when the expression is multiplied by the mole fraction of the precipitating component.
ACKNOWLEDGMENT
The authors are pleased to acknowledge some helpful conversations on this problem with Dr. Harvey Brooks of the Knolls Atomic Power Laboratory.