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Properties of clusters
Chapter 6
Properties of clusters
We have seen in Chapter 3 that the main difficulty in the determination of the Gibbs free energy G(n) and, thereby, of the work W(n) for formation of an nsized cluster arises from the uncertainty concerning the cluster excess energy Gex(n ). The reason for this uncertainty is that the cluster is, in fact, a phase of finite size. The properties of such a phase differ from those of the corresponding bulk (i.e. infinitely large) phase. That is why, in eq. (3.9), the pressure Pn inside the n-sized cluster and the chemical potential//new, n of the molecules in the cluster are not equal to the outside pressure p and the chemical potential ~new o f the bulk new phase under otherwise the same conditions. In addition, the presence of interface boundary between the cluster and the old phase leads to the appearance of the cluster surface energy ~ n ) in the energy balance. Let us now consider the effect of the cluster size on Pn, /ffnew,n and some other thermodynamic characteristics of homogeneously formed clusters. Unless especially noted, the considerations will be confined to onecomponent clusters defined by the EDS.
6.1 Inside pressure For the pressure p~ inside a cluster of n molecules we have the generalized Laplace equation (3.10) so that for a concrete dependence ofp~ on n we need a model presentation of q~. In the scope of the capillarity approximation ~ is given by eq. (3.17) which, upon being used in (3.10), yields the following formula for EDS-defined clusters of both condensed and gaseous phasesPn =
P + 2c,,~,,/3 V~/3 + (1/3)d(cnG,,)/d V~/3 .
(6.1)
For clusters of condensed phases V~ and n are related in a simple way by eq. (3.13) and this allows expressing p~ from (6.1) as an explicit function of n" Pn = P + 2CnGn]3 0113nl/3 + (113 V~/3)d(cnO'n)/dn 1/3.
(6.2)
The representation of p, from (6.1) in explicit dependence of n for clusters of gas phases is complicated because of the non-linear relation, eq. (3.14), between V, and n in this case. For spherical clusters of both condensed and gas phases c, = (36z01/3 = constant and V~ is conveniently expressed by eq. (3.22) as a function of the cluster radius R. Equation (6.1) then becomes p,, = p + 2Gn/R + dGn/dR
(6.3)
Properties of clusters 71 which is the known formula [Ono and Kondo 1960; Toschev 1973a; Abraham 1974a] for pn of EDS-defined spherical clusters. In the classical nucleation theory cnand (r~ are regarded as n-independent quantities whose values c and (r characterize, respectively, the specified cluster shape and the specific surface energy of the planar interface (G is independent of the choice of the dividing surface and is the limiting value of (r, for n --4 oo). In this approximation eqs (6.1) and (6.2) take the simple form Pn =
P + 2cG/3 V2/3
(6.4)
for clusters of both condensed and gas phases and Pn = P + 2cG/3 U~/3n1/3
(6.5)
for clusters of condensed phases. Accordingly, eq. (6.3) reduces to the familiar Laplace equation (3.24). Equations (6.4), (6.5) and (3.24) say that the molecules inside any n-sized cluster are under pressure Pn >-P and that Pn ~ P when n (and, hence, V n o r R) tends to infinity. This is seen also in Fig. 6.1 a which depicts the dependence of Pn on n from eq. (6.4) (with Vn related to n in accordance with (3.21)) and from eq. (6.5) for steam bubbles in water at T = 583 K and for water droplets in vapours at T = 293 K under the atmospheric pressure p = 0.1 MPa. The calculation is done with c = (36z01/3 (spheres) and the parameter values listed in Tables 3.1 and 3.2. In the range of n < 100 molecules the dependences shown should be regarded as more or less qualitative, since employing the usual thermodynamic methods for the description of EDSdefined clusters of such small size is questionable.
6.2 Chemical potential The chemical potential/lnew, n of the molecules in an n-sized cluster is given by eq. (3.11). It is different from the chemical potential/1new(P) of the molecules in the bulk new phase at pressure p, because the pressure Pn inside the cluster is higher than the pressure p outside it. To find the dependence of ,t/new, n o n n we use again the capillarity approximation and consider EDS-defined clusters first of condensed phases. For them Vn can be approximated by (3.13) if they are treated as incompressible. Performing the integration in eq. (3.11) with Vn from (3.13) and using p~ from (6.2) then results in ]2new, n "- [ l n e w ( P )
+
2CnGn v2/3 /3nl/3 + ( U2/3/3)d(cnGn)/dn1/3.
(6.6)
For spherical clusters cn = (36zr) 1/3, n = (4~3v0)R 3 and (6.6) gives conveniently ]-/n as a function of the cluster radius R: ,t/new, n = [ l n e w ( P )
+
2GnUo[R + vodGn/dR.
(6.7)
Turning now to EDS-defined gaseous clusters, we can employ eq. (3.14) to perform the integration in (3.11). Doing that and using (6.1) yields
72
Nucleation: Basic Theory with Applications
1500
(a) 1000 t--
500 bubble I
I
I
I
9
,
I
!
...i
.
,
!.
.
.
.
I
,
,
.
O.
i---
(b)
6
bubble
4 ir
~
2
e--
:=k
~..~
droplet
0
bubble
1.0
t3
(c)
droplet
t--
0.5
1
200
400
600
800
1000
n Size dependence of (a) pressure inside cluster, (b) cluster chemical potential, and (c) cluster specific surface energy for water droplets in vapours at T = 293 K and steam bubbles in water at T = 583 K. F i g . 6.1
]-/new,,, -- ]-/new(P) -I-
kT In [1 + 2c,(r~/3p
Vf~13 -I-
(ll3p)d(c, Crn)ld Vt]/3 ] (6.8)
where Vn is a complicated function of n obtainable from eq. (3.14) with p,, from (6.1). For spherical clusters, however, it is much simpler to express V~ through the cluster radius R rather than through n. Using (3.22) in (6.8) and recalling that for this cluster shape c = (36Jr) 1/3 then leads to ~new,n --/-/new(P) +
kT In [1 + 2Crn/pR + (1/p)dcr,/dR].
(6.9)
In the scope of the classical nucleation theory with n-independent c,, and an, from eqs (6.6) and (6.8) it follows that /-/new, n -- ]-/new(P) +
2ccr 02/3 ]3n 1/3
(6.10)
Properties of clusters
73
for clusters of condensed phases [Zettlemoyer 1969] and that [/new, n = ]-/new(P) q" kT In (1 +
2cG/3p V2/3)
(6.11)
for gas-phase clusters. Accordingly, from (6.7) and (6.9) we find that if the clusters are spherical, ]Jnew, n = I/new(P) 4" 2 O v 0 / R
(6.12)
when they are liquid or solid [Zettlemoyer 1969] and [lnew,n "-[lnew(P) +
kT In (1 + 2G/pR)
(6.13)
when they are gaseous [Kaischew and Mutaftschiev 1962]. Equations (6.6)-(6.13) show that ,t/new,n(Pn ) ---+ ]anew(P) when n (and, hence, Vn or R) tends to infinity. This is understandable, since when the cluster is sufficiently large, the pressure Pn inside it is practically equal to the outside pressure p. The dependence of ]Jnew, n o n n for spherical water droplets in vapours at T = 293 K and for spherical steam bubbles in water at T = 583 K and p = 0.1 MPa is depicted in Fig. 6.1b. The calculation is done with the help of eqs (6.10) and (6.11) (in the latter Vn is expressed via n as required by eq. (3.21)). The parameter values used are those in Tables 3.1 and 3.2. As already noted, in the range of n < 100 molecules the dependences in Fig. 6.1 b may have only a qualitative character because of the questionable applicability of usual thermodynamics to EDS-defined clusters of such small size. Equations (6.10) and (6.11) provide direct evidence that the chemical potential [lnew, n* of the molecules in the nucleus equals the chemical potential Pold of the molecules in the ambient old phase. Indeed, setting n = n* in (6.10) and (6.11), inserting n* from (4.7) in (6.10) and V* from (4.15) in (6.11) and accounting for Ap from (2.1) and for p* and Ap from (4.14) and (2.10) leads again to eq. (4.4). In the same way it is easy to verify that [/new, n >/'/old for n < n* and that [gnew,n < I/old for n > n*. Physically, the former inequality implies that the subnuclei cannot form spontaneously, but only by chance: transfer of molecules from a spatial region (the old phase) where the chemical potential is lower to a spatial region (the subnucleus) where it is higher is thermodynamically unfavoured, i.e. it is not a 'natural process' [Guggenheim 1957].
6.3 Vapour pressure A bulk condensed phase can coexist with its vapours at a given temperature T if the vapour pressure is equal to the equilibrium (or saturation) pressure pe(T). For that reason Pe is often called vapour pressure of the condensed phase. Likewise, a vapour pressure Pe,n(T) is necessary for a given n-sized cluster of the condensed phase to coexist with the vapours around it. Since Pe and Pe,n are related to it/e and ,t/new,n, and Pe and/-/new,n are not equal because of the finite size of the cluster, the vapour pressure Pe,n of the n-sized cluster will also differ from Pe.
74 Nucleation: Basic Theory with Applications
To find Pe,n w e use the ideal-gas approximation for the vapours and assume that the cluster is incompressible. Then, from eqs (2.5) and (6.10) we have /-/old(Pe,n) "-]-/e d- kT In (Pe,n[Pe)
(6.14)
~new,n(Pn) ----/-/new(fie,n) q- 2cry v 2/3/3n 1/3
(6.15)
and for the chemical potentials of the molecules in the vapours around the nsized cluster and of the molecules inside the cluster, respectively. Equation (6.15) expresses/-/new,n in the scope of the classical nucleation theory and in it ~n~w(Pe,~) is given by eq. (2.6). Coexistence between the cluster and its vapours is possible only when//new,n(Pn) "-/t/old(fie,n) SO that from (2.6), (6.14) and (6.15) it follows that Pe,n = Pe exp [ 2 c t 7 0 2 / 3 / 3 k T n 1/3 + Uo(Pe,n- pe)/kT]
(6.16)
or, to a good approximation, Pe,n = Pe exp (2ccr v 2/3/3kTnl/3).
(6.17)
For spherical clusters (6.17) takes the form of the well-known Thomson (Lord Kelvin) equation [Thomson 1870, 1871 ] Pe,n = Pe exp (2crvo/kTR).
(6.18)
The approximation of the classical theory for constant tYn can be relaxed by using eq. (6.6) in lieu of (6.10) throughout the above derivation. Equation (6.17) is then replaced by Pe,n = Pe exp {( u~/3/3kT)[2CntYn/n 1/3 + d(cntYn)/dnl/3]},
(6.19)
and eq. (6.18) becomes [Ono and Kondo 1960; Abraham 1974a] Pe.,, = Pe exp [(vo/kT)(2cr,,/R + dcr,,/dR)].
(6.20)
The above formulae apply to EDS-defined condensed-phase clusters surrounded by their vapours. Similarly, it is possible to speak of the vapour p r e s s u r e Pe,n of gas-phase clusters embedded into bulk condensed (liquid or solid) phases. Physically, Pe,n is again that pressure of the vapours within the n-sized gaseous cluster, which ensures the coexistence (i.e. the chemical equilibrium) between the cluster and the condensed phase around it. Hence, Pe,n should not be confused with Pn which is the pressure inside the gaseous cluster necessary only for the cluster's mechanical equilibrium. However, despite their different physical nature, Pe,n and p~ can have the same value for a cluster with particular size. This means that such a gaseous cluster has the important property of being simultaneously in mechanical and chemical (i.e. complete thermodynamic) equilibrium. According to eqs (4.3) and (4.4), the nucleus is a cluster which has just this property. In the scope of the c~, cr~ = constant approximation, the other clusters (the sub- and the supernuclei) are only in partial thermodynamic equilibrium: though usually assumed to be in mechanical, they are not in chemical equilibrium, for they cannot coexist with the old phase.
Properties of clusters 75
To determine the vapour pressure Pea, of spherical gaseous clusters in the c,,cr, = constant approximation of the classical theory we use eq. (6.11) in the form ]-/new, n(Pe.n) = ]-/new(Pe.n -- 2C~[3 V113)
+ kT In [1 + 2ccr/3(pe,n - 2ccr/3 V2/3) V2/3].
(6.21)
This expression takes into account that when the pressure inside the cluster is Pe,n, in conformity with (6.4), the outside pressure is Pe,n - 2ccr/3 V2/3. With this outside pressure, from (2.6) it follows that we can use the formula flold(Pe,n
--
2ccr/3 V2/3) = Ire + vo(Pe,,- 2ccr/3 V,u3
-Pe)
(6.22)
for the chemical potential of the molecules in the condensed phase around the gaseous cluster (this phase is now the old one). Since coexistence between the cluster and the old phase requires equality of the above two chemical potentials, Pe,n can be found by setting equal the fight-hand sides of (6.21) and (6.22) noting that due to (2.5) ]-/new(Pe,n- 2ccr/3 V~/3) = Ire + kT In [(Pe,~ - 2ccr/3 V~/3)/pe],
since now the gas phase is the new one. The result is Pe,n = P e
exp [ - (vo/kT)(2ccr/3
V 113 +
Pe
-
Pe,n)]
(6.23)
or, to a good accuracy, Pe,,, = Pe exp (- 2ctrvd3kTV1/3).
(6.24)
In these formulae Vn is a complicated function of n obtainable from (3.21) with p replaced by Pe,n - 2ccr/3 V2/3. For spherical clusters it is more convenient to express Vn through the cluster radius R with the help of (3.22). Equation (6.24) then becomes [Ono and Kondo 1960] Pe,n = Pe exp (- 2Ovo/kTR).
(6.25)
If we want to avoid the approximation of the classical nucleation theory for n-independent Cn and on, we should repeat the above derivation, but with the aid of eqs (6.1) and (6.8) instead of (6.4) and (6.11), respectively. Equations (6.24) and (6.25) then take the form Pe,n = P e Pe,n =
exp {- (Vo/3kT)[2cnCrn/ V1/3 + d(cn~n)]d V1/3 ] }
Pe exp [- (vo/kT)(2tYn/R + dan~dR)]
(6.26) (6.27)
for EDS-defined gaseous clusters with arbitrary and with spherical [Ono and Kondo 1960] shape, respectively. The dependence of p~,,, on n from eqs (6.17) and (6.24) is illustrated in Fig. 6.2a for spherical water droplets at T = 293 K and steam bubbles at T = 583 K (the parameter values used are those listed in Tables 3.1 and 3.2). For the smallest clusters (e.g. of n < 100 molecules) these dependences are more or less only qualitative, since usual thermodynamics may fail in this range of cluster sizes. As seen, Pe,n ---)Pe for n --> ,~, but while Pe,n >-Pe for the droplets,
76
Nucleation: Basic Theory with Applications
c-
0 15
o
lo l--
0
g
5 0 1.0
-
(c)
f e-.
0.5
h-0
1
200
400
600
800
1000
Fig. 6.2 Size dependence of (a) vapour pressure of water droplets in vapours at T = 293 K and of steam bubbles in water at T = 583 K, (b) solubility of crystallites of sparingly soluble salts in aqueous solutions at room temperature, and (c) melting point of ice crystallites in water at atmospheric pressure.
Pe,n <- Pe for the bubbles. The physical reason for this distinction is that the
molecules at the convex surface of the droplet and at the concave surface of the bubble are bound, respectively, more weakly and more strongly than the molecules at the planar liquid/gas interface. Evaporation of the molecules is then, respectively, less and more difficult from the curved cluster surface than from the planar surface and this is reflected by the above inequalities between Pe (which is the vapour pressure over the planar surface) a n d Pe,n.
Properties of clusters
77
6.4 Solubility Let us consider a condensed-phase cluster of n molecules which is in a solution. Clearly, the increased chemical potential of the molecules in the cluster will affect the cluster solubility, i.e. the solute concentration at which the n-sized cluster can coexist with the solution. In other words, if C~,, is this concentration, we should expect it to differ from the equilibrium concentration (or the solubility) Ce of the bulk condensed phase of solute. The effect is completely analogous to the effect of the cluster size on the cluster vapour pressure Pe,n, because the chemical potential of the molecules in dilute vapours and solutions depends logarithmically on pressure and solute concentration, respectively (see eqs (2.5) and (2.11)). This means that for the solubility Ce,n of an n-sized cluster of condensed phase we can rewrite eqs (6.17) and (6.19) with Pe,n and Pe replaced by Ce, n and Ce, respectively: Ce, n = C e
exp (2cGv~/3/3kTn 1/3)
(6.28)
Ce,,~ = Ce exp {( U 2/3 ]3kT)[2CntTn/n 1/3 + d(cntTn)/dn 1/3] }.
(6.29)
For spherically shaped clusters these dependences become (cf. eqs (6.18) and (6.20)) Ce,n = Ce exp (2OVo/kTR)
(6.30)
Ce,,, = Ce exp [(vo/kT)(2Gn/R + dG,/dR)].
(6.31)
Equations (6.29) and (6.31) account for the dependence of the cluster specific surface energy cr~ on the size of the EDS-defined cluster and, in the c,,cr, = constant approximation of the classical theory, simplify to eqs (6.28) and (6.30), the latter being the known formula of Ostwald [1900] for the solubility of small crystals in dilute solutions. According to (6.28) and (6.30), the solubility of the condensed-phase clusters increases with decreasing cluster size. This is seen in Fig. 6.2b which represents the dependence of Ce,,, on n for spherical clusters, resulting from eq. (6.28) with the parameter values listed in Table 6.1 (these values are typical for crystallites of sparingly soluble salts in aqueous solutions at room temperature). We note again that in the small-size region (e.g. for n < 100 molecules) the application of eqs (6.28)(6.31) is questionable. Also, for more concentrated solutions, in these equations Ce, n and Ce must be replaced by the corresponding activities (see Chapter 2). Table 6.1 Values of various quantities used for calculation of different dependences for nucleation of crystallites of sparingly soluble salts in aqueous solutions at T = 293 K Vo (nm3)
do (nm)~
Ce (m-3)
O" (mJ/m2)
D (~tm2/s)
0.05
0.46
1023
1 O0
1000
acalculated from do = (6vo/~)1/3
78 Nucleation: Basic Theory with Applications
6.5 Melting point The dependence of the chemical potential of the molecules in a given cluster on the cluster size has an impact also on the melting point of the crystalline clusters when they are small enough. The melting point is the absolute temperature Te,n at which an n-sized crystalline cluster can coexist with the melt around it. This quantity is analogous to the melting point Te of the corresponding bulk crystal, i.e. to the temperature at which the bulk crystal and the melt are in two-phase equilibrium. Similar to Pe,n and Ce,,, Te,~ can be found from the condition for equality of the chemical potentials it/new, n and ~told of the molecules in the n-sized cluster (the crystallite) and the old phase (the melt) in which the cluster is formed. According to eqs (6.10) and (2.19), for/~new,~ and/~oJd we have the expressions ]Jnew,n(Te,n) = ~e(P) at- Snew(Te)(Ze- Te,n) at- 2cGv2/a/3nl/3 ~old(Te,n) = ]Je(P) + Sold(Te)(Te- Te,n)
(6.32) (6.33)
which are restricted by the approximation of the classical theory for nindependent c~ and tr, and by the approximations Snew, n(T) = constant = Snew(Te)and Sold(T) = constant = So]d(Te) for the molecular entropies Snew,n and Sold of the crystallite and the melt in the temperature range between Te,n and Te. Setting equal the fight-hand sides of the above equations we thus find that Te,n = Te - 2cG v 2/3/3Ase nil3
(6.34)
where Ase- So]d(Te) - Snew(Te) is the melting entropy per molecule. If the crystalline cluster is regarded as spherically shaped, (6.34) takes the form of the known formula of Thomson [ 1886] T~,, = T e - 2ov0/As~.
(6.35)
The approximation for constant Cn and or, can be relaxed by employing eq. (6.6) instead of (6.10) for the above derivation. The resulting generalization of (6.34) and (6.35) for EDS-defined clusters is Te,n = Te - ( 02/3/3 Ase)[2CnGn/n 1/3 + d(cnGn)/dn 1/3]
(6.36)
when the clusters are arbitrarily shaped and Te, n -" T e -(vo/Ase)(2Crn/R + dGn/dR)
(6.37)
when they are spherical. Equations (6.34) and (6.35) show that the melting point of the crystalline clusters is below that of the corresponding bulk crystal, the depression being more pronounced for the smaller clusters. Physically, this is a consequence of the weaker (on average) binding of the molecules at the finite-area surface of the crystalline cluster compared with their binding at the infinitely large crystal surface. The magnitude of the effect for spherical ice clusters in water at atmospheric pressure is illustrated in Fig. 6.2c which depicts the dependence of Te,n on n, calculated from eq. (6.34) with c = (367r)1/3 and the parameter
Properties of clusters
79
values given in Table 6.2. It is worth reiterating that the applicability of eqs (6.34)-(6.37) to the smallest clusters (e.g. of n < 100 molecules) is questionable. Table 6.2 Values of various quantities used for calculation of different dependences for nucleation of ice in water under atmospheric pressure Vo (nm 3)
do (nm) a
Te (K)
Ase/k
cr (mJ/m 2)
0(230 K) (mPa.s) b
77(240K) (mPa.s) b
7n
0.033
0.4
273.15
2.6
26
43
16
1
acalculated from do = (6Vo/tr)1/3 bcalculated from eq. (13.47)
6.6 Specific surface energy The specific surface energy is an important parameter in the physical chemistry of surfaces in general and in the classical nucleation theory in particular. For instance, eqs (6.1), (6.6), (6.8), (6.19), (6.26), (6.29) and (6.36) tell us that Pn, bin, Pe,n, Ce,n and Te,n for a cluster of specified shape (Cn = constant) depend on n not only directly, but also through the n-dependence of the cluster specific surface energy G,,. If the effect of the cluster size on crn manifests itself more strongly for smaller values of n, it might be expected that the approximation Gn = constant = cr of the classical theory may introduce significant inaccuracy in the thermodynamic description of the smaller clusters even in the size range in which such a description is unquestionable. Clearly, finding the n-dependence of ty, is an important problem and this may explain the interest in this problem shown first by Gibbs [1928] and then by many others [Tolman 1949; Kirkwood and Buff 1949; Defay and Prigogine 1966; Nishioka 1977, 1987, 1992; Rasmussen 1982a; Rasmussen et al. 1983; Schmelzer and Mahnke 1986; Larson and Garside 1986; S6hnel and Garside 1988; Nishioka et al. 1989; Dillmann and Meier 1989, 1991; Hadjiagapiou 1994; Laaksonen and McGraw 1996; McGraw and Laaksonen 1997]. Despite the great effort put in solving this problem, there is no generally agreed opinion on the dependence of o-n on n. We shall now outline briefly only the best known solutions of the problem. For further reading we refer to the detailed considerations of Ono and Kondo [1960], Rusanov [1967, 1978] and Baidakov [1994] as well as to a recent paper by Schmelzer et al. [ 1996]. When solving the problem of the dependence of tyn on n the choice of the dividing surface is of prime importance. Gibbs [1928] considered clusters defined by the so-called surface of tension (ST) which is chosen in such a way that the value of or, is minimal with respect to the values resulting from any other choice of the dividing surface. He derived a differential equation for the dependence of the specific surface energy on the curvature of the ST, which for ST-defined spherical clusters reads [Gibbs 1928] d(ln O'sT,n)/d(ln RST) = (280/RsT)/(1 + 280/RsT).
(6.38)
80 Nucleation: Basic Theory with Applications
Here the subscript 'ST' indicates that crn is referred to the ST, and S0 is defined by 60 = R - RST
(6.39)
where R and RST are the radii of the EDS- and the ST-defined clusters, respectively. The Gibbs parameter ~ is thus the distance between the EDS and the ST. Gibbs [1928] solved eq. (6.38) by assuming (i) that ~0 is constant and (ii) that RST >> 2~0. He thereby found that O'ST,n = Cr exp (- 26o/RsT)
(6.40)
where cr (independent of the choice of the dividing surface) is the specific surface energy of the planar interface (i.e. the value of CrST,,,at RST = ~). We may note, however, that Gibbs could have solved eq. (6.38) without using the second of the above assumptions. Indeed, the solution of the full eq. (6.38) with constant 80 is represented as CrsT,n = cr/(1 + 260/RsT)
(6.41)
which is the formula of Tolman [ 1949] and Kirkwood and Buff [1949]. Equation (6.38) was derived by Gibbs for the specific surface energy corresponding to the ST. However, it can be shown [Kondo 1956; Ono and Kondo 1960] that d(ln O'sT,n)/d(ln RST) = d(ln O'n)/d(ln R) so that eq. (6.38) remains in force also when the specific surface energy is referred to the EDS (we keep the notation o'n in this case). This means that with the help of (6.39) eq. (6.38) can be rewritten as d(ln Crn)/d(ln R ) = (260/R)/(1 + ~ / R ) .
(6.42)
If we now adopt Gibbs's assumptions, the second in the less restrictive form R >> S0, we arrive again at Gibbs's equation (6.40), but for EDSdefined clusters: O "n " -
O"
exp (- 2S0/R).
(6.43)
Similar to the derivation of (6.41), we can integrate the full equation (6.42), i.e. without utilizing the condition R >> ~o. The result is an= cr/(1 + ~0/R)2
(6.44)
which is the analogue of (6.41) in the case of EDS-defined clusters. This equation, which does not seem to have been reported hitherto, can be generalized to hold for arbitrarily shaped clusters of both condensed and gaseous phases if with the aid of (3.22) R is replaced by the volume V, of the EDS-defined cluster: cr~ = or~(1 + C6o/3 V~/3 )2.
(6.45)
Here c is the cluster shape factor equal to (36t01/3 for spheres, 6 for cubes, etc.
Properties of clusters 81
In accordance with (3.13), for condensed-phase clusters the explicit ndependence of cr,, from (6.45) is given by ty,, = or~(1 + Ct~o/3O~/3 n 1/3) 2.
(6.46)
For gaseous clusters, however, the dependence of o'n on n remains only implicitly expressed by (6.45). Indeed, in this case V,, and n are related in a complicated way by the equality n = (p/kT)V,, + (2ccr/3kT) V1/3[1 + (Ct~o/3V, l,/3)/(1 + Ct~o/3Vnl/3) 3] (6.47)
which follows from (3.14) upon using Pn from (6.1) with c,, = constant = c and dcr,,/d V,~/3 calculated from (6.45). The above results show that the effect of the cluster size on the cluster specific surface energy is controlled by the Gibbs parameter do: tS0 > 0 leads to a lower specific surface energy for the smaller clusters, and d;0 < 0 has the opposite effect (see, e.g. eqs (6.44)-(6.46)). It is, therefore, very important to know the sign and the absolute value of 60. Unfortunately, such a knowledge cannot be obtained from experiment, since neither the ST nor the EDS are real physical objects and it is thus impossible in principle to measure the distance 60 between them. Under such circumstances it remains only to resort to theoretical arguments and/or to determinations of ~ with the aid of numerical calculations. On such grounds it is believed that, typically, I~lis of the order of the molecular diameter do, but the question about the sign of is still open [Ono and Kondo 1960; Defay and Prigogine 1966; Nishioka 1977, 1992; Hadjiagapiou 1994; Baidakov 1994, 1995]. The argument concerning the magnitude of] S01is that since the EDS is positioned inside the surface layer (the spatial zone with varying molecular density), the ST can be expected to be close to the EDS and, hence, also inside this layer. Ergo, d;0 should not exceed the thickness of the surface layer which at low enough temperatures comprises one to a few molecular layers. Another unclear point is whether it is possible to treat ~ as an R, Rsz-independent quantity which may be approximated [Ono and Kondo 1960] by the distance between the EDS and the ST for the planar interface, i.e. by the limiting value of for both R ~ oo and RST ~ co. Recently, Hadjiagapiou [ 1994] has shown that 60 may decrease linearly with R, and Schmelzer et al. [ 1996] have demonstrated that different tS0(R,RsT) functions can affect essentially the size dependence of the cluster specific surface energy. Figure 6.1c illustrates the n-dependence of o'~ from eqs (6.45) and (6.46) for spherical EDS-defined steam bubbles and water droplets, respectively. For the bubbles T = 583 K and p = 0.1 MPa, and n is determined with the help of (6.47). The parameter values used are c = (36/r) 1/3, S 0 = 0.1 nm and those listed in Tables 3.1 and 3.2. As seen, ty~ does not differ more than 10% from its limiting value ty for n = oo (i.e. for the planar interface) when n > 1000 molecules. In the range of n < 100 molecules the change of ty, with respect to ty is already greater. However, in this range of n values the predicted effect cannot be regarded without reservation, because then the ~0(R, RST) = constant approximation is uncertain and, more importantly, the very
82
Nucleation: Basic Theory with Applications
applicability of usual thermodynamics is questionable. From this point of view, the numerous attempts in the nucleation literature to take account of the n-dependence of o'n, e.g. by means of eq. (6.41), appear more or less speculative. In the absence of a firm knowledge about the actual change of O'n with n we shall confine all further considerations within the approximation an = constant = cr of the classical nucleation theory. In Chapter 8 we shall see that an alternative determination of the cluster specific surface energy is possible in the scope of quasi-thermodynamics.
Properties of clusters
We have seen in Chapter 3 that the main difficulty in the determination of the Gibbs free energy G(n) and, thereby, of the work W(n) for formation of an nsized cluster arises from the uncertainty concerning the cluster excess energy Gex(n ). The reason for this uncertainty is that the cluster is, in fact, a phase of finite size. The properties of such a phase differ from those of the corresponding bulk (i.e. infinitely large) phase. That is why, in eq. (3.9), the pressure Pn inside the n-sized cluster and the chemical potential//new, n of the molecules in the cluster are not equal to the outside pressure p and the chemical potential ~new o f the bulk new phase under otherwise the same conditions. In addition, the presence of interface boundary between the cluster and the old phase leads to the appearance of the cluster surface energy ~ n ) in the energy balance. Let us now consider the effect of the cluster size on Pn, /ffnew,n and some other thermodynamic characteristics of homogeneously formed clusters. Unless especially noted, the considerations will be confined to onecomponent clusters defined by the EDS.
6.1 Inside pressure For the pressure p~ inside a cluster of n molecules we have the generalized Laplace equation (3.10) so that for a concrete dependence ofp~ on n we need a model presentation of q~. In the scope of the capillarity approximation ~ is given by eq. (3.17) which, upon being used in (3.10), yields the following formula for EDS-defined clusters of both condensed and gaseous phasesPn =
P + 2c,,~,,/3 V~/3 + (1/3)d(cnG,,)/d V~/3 .
(6.1)
For clusters of condensed phases V~ and n are related in a simple way by eq. (3.13) and this allows expressing p~ from (6.1) as an explicit function of n" Pn = P + 2CnGn]3 0113nl/3 + (113 V~/3)d(cnO'n)/dn 1/3.
(6.2)
The representation of p, from (6.1) in explicit dependence of n for clusters of gas phases is complicated because of the non-linear relation, eq. (3.14), between V, and n in this case. For spherical clusters of both condensed and gas phases c, = (36z01/3 = constant and V~ is conveniently expressed by eq. (3.22) as a function of the cluster radius R. Equation (6.1) then becomes p,, = p + 2Gn/R + dGn/dR
(6.3)
Properties of clusters 71 which is the known formula [Ono and Kondo 1960; Toschev 1973a; Abraham 1974a] for pn of EDS-defined spherical clusters. In the classical nucleation theory cnand (r~ are regarded as n-independent quantities whose values c and (r characterize, respectively, the specified cluster shape and the specific surface energy of the planar interface (G is independent of the choice of the dividing surface and is the limiting value of (r, for n --4 oo). In this approximation eqs (6.1) and (6.2) take the simple form Pn =
P + 2cG/3 V2/3
(6.4)
for clusters of both condensed and gas phases and Pn = P + 2cG/3 U~/3n1/3
(6.5)
for clusters of condensed phases. Accordingly, eq. (6.3) reduces to the familiar Laplace equation (3.24). Equations (6.4), (6.5) and (3.24) say that the molecules inside any n-sized cluster are under pressure Pn >-P and that Pn ~ P when n (and, hence, V n o r R) tends to infinity. This is seen also in Fig. 6.1 a which depicts the dependence of Pn on n from eq. (6.4) (with Vn related to n in accordance with (3.21)) and from eq. (6.5) for steam bubbles in water at T = 583 K and for water droplets in vapours at T = 293 K under the atmospheric pressure p = 0.1 MPa. The calculation is done with c = (36z01/3 (spheres) and the parameter values listed in Tables 3.1 and 3.2. In the range of n < 100 molecules the dependences shown should be regarded as more or less qualitative, since employing the usual thermodynamic methods for the description of EDSdefined clusters of such small size is questionable.
6.2 Chemical potential The chemical potential/lnew, n of the molecules in an n-sized cluster is given by eq. (3.11). It is different from the chemical potential/1new(P) of the molecules in the bulk new phase at pressure p, because the pressure Pn inside the cluster is higher than the pressure p outside it. To find the dependence of ,t/new, n o n n we use again the capillarity approximation and consider EDS-defined clusters first of condensed phases. For them Vn can be approximated by (3.13) if they are treated as incompressible. Performing the integration in eq. (3.11) with Vn from (3.13) and using p~ from (6.2) then results in ]2new, n "- [ l n e w ( P )
+
2CnGn v2/3 /3nl/3 + ( U2/3/3)d(cnGn)/dn1/3.
(6.6)
For spherical clusters cn = (36zr) 1/3, n = (4~3v0)R 3 and (6.6) gives conveniently ]-/n as a function of the cluster radius R: ,t/new, n = [ l n e w ( P )
+
2GnUo[R + vodGn/dR.
(6.7)
Turning now to EDS-defined gaseous clusters, we can employ eq. (3.14) to perform the integration in (3.11). Doing that and using (6.1) yields
72
Nucleation: Basic Theory with Applications
1500
(a) 1000 t--
500 bubble I
I
I
I
9
,
I
!
...i
.
,
!.
.
.
.
I
,
,
.
O.
i---
(b)
6
bubble
4 ir
~
2
e--
:=k
~..~
droplet
0
bubble
1.0
t3
(c)
droplet
t--
0.5
1
200
400
600
800
1000
n Size dependence of (a) pressure inside cluster, (b) cluster chemical potential, and (c) cluster specific surface energy for water droplets in vapours at T = 293 K and steam bubbles in water at T = 583 K. F i g . 6.1
]-/new,,, -- ]-/new(P) -I-
kT In [1 + 2c,(r~/3p
Vf~13 -I-
(ll3p)d(c, Crn)ld Vt]/3 ] (6.8)
where Vn is a complicated function of n obtainable from eq. (3.14) with p,, from (6.1). For spherical clusters, however, it is much simpler to express V~ through the cluster radius R rather than through n. Using (3.22) in (6.8) and recalling that for this cluster shape c = (36Jr) 1/3 then leads to ~new,n --/-/new(P) +
kT In [1 + 2Crn/pR + (1/p)dcr,/dR].
(6.9)
In the scope of the classical nucleation theory with n-independent c,, and an, from eqs (6.6) and (6.8) it follows that /-/new, n -- ]-/new(P) +
2ccr 02/3 ]3n 1/3
(6.10)
Properties of clusters
73
for clusters of condensed phases [Zettlemoyer 1969] and that [/new, n = ]-/new(P) q" kT In (1 +
2cG/3p V2/3)
(6.11)
for gas-phase clusters. Accordingly, from (6.7) and (6.9) we find that if the clusters are spherical, ]Jnew, n = I/new(P) 4" 2 O v 0 / R
(6.12)
when they are liquid or solid [Zettlemoyer 1969] and [lnew,n "-[lnew(P) +
kT In (1 + 2G/pR)
(6.13)
when they are gaseous [Kaischew and Mutaftschiev 1962]. Equations (6.6)-(6.13) show that ,t/new,n(Pn ) ---+ ]anew(P) when n (and, hence, Vn or R) tends to infinity. This is understandable, since when the cluster is sufficiently large, the pressure Pn inside it is practically equal to the outside pressure p. The dependence of ]Jnew, n o n n for spherical water droplets in vapours at T = 293 K and for spherical steam bubbles in water at T = 583 K and p = 0.1 MPa is depicted in Fig. 6.1b. The calculation is done with the help of eqs (6.10) and (6.11) (in the latter Vn is expressed via n as required by eq. (3.21)). The parameter values used are those in Tables 3.1 and 3.2. As already noted, in the range of n < 100 molecules the dependences in Fig. 6.1 b may have only a qualitative character because of the questionable applicability of usual thermodynamics to EDS-defined clusters of such small size. Equations (6.10) and (6.11) provide direct evidence that the chemical potential [lnew, n* of the molecules in the nucleus equals the chemical potential Pold of the molecules in the ambient old phase. Indeed, setting n = n* in (6.10) and (6.11), inserting n* from (4.7) in (6.10) and V* from (4.15) in (6.11) and accounting for Ap from (2.1) and for p* and Ap from (4.14) and (2.10) leads again to eq. (4.4). In the same way it is easy to verify that [/new, n >/'/old for n < n* and that [gnew,n < I/old for n > n*. Physically, the former inequality implies that the subnuclei cannot form spontaneously, but only by chance: transfer of molecules from a spatial region (the old phase) where the chemical potential is lower to a spatial region (the subnucleus) where it is higher is thermodynamically unfavoured, i.e. it is not a 'natural process' [Guggenheim 1957].
6.3 Vapour pressure A bulk condensed phase can coexist with its vapours at a given temperature T if the vapour pressure is equal to the equilibrium (or saturation) pressure pe(T). For that reason Pe is often called vapour pressure of the condensed phase. Likewise, a vapour pressure Pe,n(T) is necessary for a given n-sized cluster of the condensed phase to coexist with the vapours around it. Since Pe and Pe,n are related to it/e and ,t/new,n, and Pe and/-/new,n are not equal because of the finite size of the cluster, the vapour pressure Pe,n of the n-sized cluster will also differ from Pe.
74 Nucleation: Basic Theory with Applications
To find Pe,n w e use the ideal-gas approximation for the vapours and assume that the cluster is incompressible. Then, from eqs (2.5) and (6.10) we have /-/old(Pe,n) "-]-/e d- kT In (Pe,n[Pe)
(6.14)
~new,n(Pn) ----/-/new(fie,n) q- 2cry v 2/3/3n 1/3
(6.15)
and for the chemical potentials of the molecules in the vapours around the nsized cluster and of the molecules inside the cluster, respectively. Equation (6.15) expresses/-/new,n in the scope of the classical nucleation theory and in it ~n~w(Pe,~) is given by eq. (2.6). Coexistence between the cluster and its vapours is possible only when//new,n(Pn) "-/t/old(fie,n) SO that from (2.6), (6.14) and (6.15) it follows that Pe,n = Pe exp [ 2 c t 7 0 2 / 3 / 3 k T n 1/3 + Uo(Pe,n- pe)/kT]
(6.16)
or, to a good approximation, Pe,n = Pe exp (2ccr v 2/3/3kTnl/3).
(6.17)
For spherical clusters (6.17) takes the form of the well-known Thomson (Lord Kelvin) equation [Thomson 1870, 1871 ] Pe,n = Pe exp (2crvo/kTR).
(6.18)
The approximation of the classical theory for constant tYn can be relaxed by using eq. (6.6) in lieu of (6.10) throughout the above derivation. Equation (6.17) is then replaced by Pe,n = Pe exp {( u~/3/3kT)[2CntYn/n 1/3 + d(cntYn)/dnl/3]},
(6.19)
and eq. (6.18) becomes [Ono and Kondo 1960; Abraham 1974a] Pe.,, = Pe exp [(vo/kT)(2cr,,/R + dcr,,/dR)].
(6.20)
The above formulae apply to EDS-defined condensed-phase clusters surrounded by their vapours. Similarly, it is possible to speak of the vapour p r e s s u r e Pe,n of gas-phase clusters embedded into bulk condensed (liquid or solid) phases. Physically, Pe,n is again that pressure of the vapours within the n-sized gaseous cluster, which ensures the coexistence (i.e. the chemical equilibrium) between the cluster and the condensed phase around it. Hence, Pe,n should not be confused with Pn which is the pressure inside the gaseous cluster necessary only for the cluster's mechanical equilibrium. However, despite their different physical nature, Pe,n and p~ can have the same value for a cluster with particular size. This means that such a gaseous cluster has the important property of being simultaneously in mechanical and chemical (i.e. complete thermodynamic) equilibrium. According to eqs (4.3) and (4.4), the nucleus is a cluster which has just this property. In the scope of the c~, cr~ = constant approximation, the other clusters (the sub- and the supernuclei) are only in partial thermodynamic equilibrium: though usually assumed to be in mechanical, they are not in chemical equilibrium, for they cannot coexist with the old phase.
Properties of clusters 75
To determine the vapour pressure Pea, of spherical gaseous clusters in the c,,cr, = constant approximation of the classical theory we use eq. (6.11) in the form ]-/new, n(Pe.n) = ]-/new(Pe.n -- 2C~[3 V113)
+ kT In [1 + 2ccr/3(pe,n - 2ccr/3 V2/3) V2/3].
(6.21)
This expression takes into account that when the pressure inside the cluster is Pe,n, in conformity with (6.4), the outside pressure is Pe,n - 2ccr/3 V2/3. With this outside pressure, from (2.6) it follows that we can use the formula flold(Pe,n
--
2ccr/3 V2/3) = Ire + vo(Pe,,- 2ccr/3 V,u3
-Pe)
(6.22)
for the chemical potential of the molecules in the condensed phase around the gaseous cluster (this phase is now the old one). Since coexistence between the cluster and the old phase requires equality of the above two chemical potentials, Pe,n can be found by setting equal the fight-hand sides of (6.21) and (6.22) noting that due to (2.5) ]-/new(Pe,n- 2ccr/3 V~/3) = Ire + kT In [(Pe,~ - 2ccr/3 V~/3)/pe],
since now the gas phase is the new one. The result is Pe,n = P e
exp [ - (vo/kT)(2ccr/3
V 113 +
Pe
-
Pe,n)]
(6.23)
or, to a good accuracy, Pe,,, = Pe exp (- 2ctrvd3kTV1/3).
(6.24)
In these formulae Vn is a complicated function of n obtainable from (3.21) with p replaced by Pe,n - 2ccr/3 V2/3. For spherical clusters it is more convenient to express Vn through the cluster radius R with the help of (3.22). Equation (6.24) then becomes [Ono and Kondo 1960] Pe,n = Pe exp (- 2Ovo/kTR).
(6.25)
If we want to avoid the approximation of the classical nucleation theory for n-independent Cn and on, we should repeat the above derivation, but with the aid of eqs (6.1) and (6.8) instead of (6.4) and (6.11), respectively. Equations (6.24) and (6.25) then take the form Pe,n = P e Pe,n =
exp {- (Vo/3kT)[2cnCrn/ V1/3 + d(cn~n)]d V1/3 ] }
Pe exp [- (vo/kT)(2tYn/R + dan~dR)]
(6.26) (6.27)
for EDS-defined gaseous clusters with arbitrary and with spherical [Ono and Kondo 1960] shape, respectively. The dependence of p~,,, on n from eqs (6.17) and (6.24) is illustrated in Fig. 6.2a for spherical water droplets at T = 293 K and steam bubbles at T = 583 K (the parameter values used are those listed in Tables 3.1 and 3.2). For the smallest clusters (e.g. of n < 100 molecules) these dependences are more or less only qualitative, since usual thermodynamics may fail in this range of cluster sizes. As seen, Pe,n ---)Pe for n --> ,~, but while Pe,n >-Pe for the droplets,
76
Nucleation: Basic Theory with Applications
c-
0 15
o
lo l--
0
g
5 0 1.0
-
(c)
f e-.
0.5
h-0
1
200
400
600
800
1000
Fig. 6.2 Size dependence of (a) vapour pressure of water droplets in vapours at T = 293 K and of steam bubbles in water at T = 583 K, (b) solubility of crystallites of sparingly soluble salts in aqueous solutions at room temperature, and (c) melting point of ice crystallites in water at atmospheric pressure.
Pe,n <- Pe for the bubbles. The physical reason for this distinction is that the
molecules at the convex surface of the droplet and at the concave surface of the bubble are bound, respectively, more weakly and more strongly than the molecules at the planar liquid/gas interface. Evaporation of the molecules is then, respectively, less and more difficult from the curved cluster surface than from the planar surface and this is reflected by the above inequalities between Pe (which is the vapour pressure over the planar surface) a n d Pe,n.
Properties of clusters
77
6.4 Solubility Let us consider a condensed-phase cluster of n molecules which is in a solution. Clearly, the increased chemical potential of the molecules in the cluster will affect the cluster solubility, i.e. the solute concentration at which the n-sized cluster can coexist with the solution. In other words, if C~,, is this concentration, we should expect it to differ from the equilibrium concentration (or the solubility) Ce of the bulk condensed phase of solute. The effect is completely analogous to the effect of the cluster size on the cluster vapour pressure Pe,n, because the chemical potential of the molecules in dilute vapours and solutions depends logarithmically on pressure and solute concentration, respectively (see eqs (2.5) and (2.11)). This means that for the solubility Ce,n of an n-sized cluster of condensed phase we can rewrite eqs (6.17) and (6.19) with Pe,n and Pe replaced by Ce, n and Ce, respectively: Ce, n = C e
exp (2cGv~/3/3kTn 1/3)
(6.28)
Ce,,~ = Ce exp {( U 2/3 ]3kT)[2CntTn/n 1/3 + d(cntTn)/dn 1/3] }.
(6.29)
For spherically shaped clusters these dependences become (cf. eqs (6.18) and (6.20)) Ce,n = Ce exp (2OVo/kTR)
(6.30)
Ce,,, = Ce exp [(vo/kT)(2Gn/R + dG,/dR)].
(6.31)
Equations (6.29) and (6.31) account for the dependence of the cluster specific surface energy cr~ on the size of the EDS-defined cluster and, in the c,,cr, = constant approximation of the classical theory, simplify to eqs (6.28) and (6.30), the latter being the known formula of Ostwald [1900] for the solubility of small crystals in dilute solutions. According to (6.28) and (6.30), the solubility of the condensed-phase clusters increases with decreasing cluster size. This is seen in Fig. 6.2b which represents the dependence of Ce,,, on n for spherical clusters, resulting from eq. (6.28) with the parameter values listed in Table 6.1 (these values are typical for crystallites of sparingly soluble salts in aqueous solutions at room temperature). We note again that in the small-size region (e.g. for n < 100 molecules) the application of eqs (6.28)(6.31) is questionable. Also, for more concentrated solutions, in these equations Ce, n and Ce must be replaced by the corresponding activities (see Chapter 2). Table 6.1 Values of various quantities used for calculation of different dependences for nucleation of crystallites of sparingly soluble salts in aqueous solutions at T = 293 K Vo (nm3)
do (nm)~
Ce (m-3)
O" (mJ/m2)
D (~tm2/s)
0.05
0.46
1023
1 O0
1000
acalculated from do = (6vo/~)1/3
78 Nucleation: Basic Theory with Applications
6.5 Melting point The dependence of the chemical potential of the molecules in a given cluster on the cluster size has an impact also on the melting point of the crystalline clusters when they are small enough. The melting point is the absolute temperature Te,n at which an n-sized crystalline cluster can coexist with the melt around it. This quantity is analogous to the melting point Te of the corresponding bulk crystal, i.e. to the temperature at which the bulk crystal and the melt are in two-phase equilibrium. Similar to Pe,n and Ce,,, Te,~ can be found from the condition for equality of the chemical potentials it/new, n and ~told of the molecules in the n-sized cluster (the crystallite) and the old phase (the melt) in which the cluster is formed. According to eqs (6.10) and (2.19), for/~new,~ and/~oJd we have the expressions ]Jnew,n(Te,n) = ~e(P) at- Snew(Te)(Ze- Te,n) at- 2cGv2/a/3nl/3 ~old(Te,n) = ]Je(P) + Sold(Te)(Te- Te,n)
(6.32) (6.33)
which are restricted by the approximation of the classical theory for nindependent c~ and tr, and by the approximations Snew, n(T) = constant = Snew(Te)and Sold(T) = constant = So]d(Te) for the molecular entropies Snew,n and Sold of the crystallite and the melt in the temperature range between Te,n and Te. Setting equal the fight-hand sides of the above equations we thus find that Te,n = Te - 2cG v 2/3/3Ase nil3
(6.34)
where Ase- So]d(Te) - Snew(Te) is the melting entropy per molecule. If the crystalline cluster is regarded as spherically shaped, (6.34) takes the form of the known formula of Thomson [ 1886] T~,, = T e - 2ov0/As~.
(6.35)
The approximation for constant Cn and or, can be relaxed by employing eq. (6.6) instead of (6.10) for the above derivation. The resulting generalization of (6.34) and (6.35) for EDS-defined clusters is Te,n = Te - ( 02/3/3 Ase)[2CnGn/n 1/3 + d(cnGn)/dn 1/3]
(6.36)
when the clusters are arbitrarily shaped and Te, n -" T e -(vo/Ase)(2Crn/R + dGn/dR)
(6.37)
when they are spherical. Equations (6.34) and (6.35) show that the melting point of the crystalline clusters is below that of the corresponding bulk crystal, the depression being more pronounced for the smaller clusters. Physically, this is a consequence of the weaker (on average) binding of the molecules at the finite-area surface of the crystalline cluster compared with their binding at the infinitely large crystal surface. The magnitude of the effect for spherical ice clusters in water at atmospheric pressure is illustrated in Fig. 6.2c which depicts the dependence of Te,n on n, calculated from eq. (6.34) with c = (367r)1/3 and the parameter
Properties of clusters
79
values given in Table 6.2. It is worth reiterating that the applicability of eqs (6.34)-(6.37) to the smallest clusters (e.g. of n < 100 molecules) is questionable. Table 6.2 Values of various quantities used for calculation of different dependences for nucleation of ice in water under atmospheric pressure Vo (nm 3)
do (nm) a
Te (K)
Ase/k
cr (mJ/m 2)
0(230 K) (mPa.s) b
77(240K) (mPa.s) b
7n
0.033
0.4
273.15
2.6
26
43
16
1
acalculated from do = (6Vo/tr)1/3 bcalculated from eq. (13.47)
6.6 Specific surface energy The specific surface energy is an important parameter in the physical chemistry of surfaces in general and in the classical nucleation theory in particular. For instance, eqs (6.1), (6.6), (6.8), (6.19), (6.26), (6.29) and (6.36) tell us that Pn, bin, Pe,n, Ce,n and Te,n for a cluster of specified shape (Cn = constant) depend on n not only directly, but also through the n-dependence of the cluster specific surface energy G,,. If the effect of the cluster size on crn manifests itself more strongly for smaller values of n, it might be expected that the approximation Gn = constant = cr of the classical theory may introduce significant inaccuracy in the thermodynamic description of the smaller clusters even in the size range in which such a description is unquestionable. Clearly, finding the n-dependence of ty, is an important problem and this may explain the interest in this problem shown first by Gibbs [1928] and then by many others [Tolman 1949; Kirkwood and Buff 1949; Defay and Prigogine 1966; Nishioka 1977, 1987, 1992; Rasmussen 1982a; Rasmussen et al. 1983; Schmelzer and Mahnke 1986; Larson and Garside 1986; S6hnel and Garside 1988; Nishioka et al. 1989; Dillmann and Meier 1989, 1991; Hadjiagapiou 1994; Laaksonen and McGraw 1996; McGraw and Laaksonen 1997]. Despite the great effort put in solving this problem, there is no generally agreed opinion on the dependence of o-n on n. We shall now outline briefly only the best known solutions of the problem. For further reading we refer to the detailed considerations of Ono and Kondo [1960], Rusanov [1967, 1978] and Baidakov [1994] as well as to a recent paper by Schmelzer et al. [ 1996]. When solving the problem of the dependence of tyn on n the choice of the dividing surface is of prime importance. Gibbs [1928] considered clusters defined by the so-called surface of tension (ST) which is chosen in such a way that the value of or, is minimal with respect to the values resulting from any other choice of the dividing surface. He derived a differential equation for the dependence of the specific surface energy on the curvature of the ST, which for ST-defined spherical clusters reads [Gibbs 1928] d(ln O'sT,n)/d(ln RST) = (280/RsT)/(1 + 280/RsT).
(6.38)
80 Nucleation: Basic Theory with Applications
Here the subscript 'ST' indicates that crn is referred to the ST, and S0 is defined by 60 = R - RST
(6.39)
where R and RST are the radii of the EDS- and the ST-defined clusters, respectively. The Gibbs parameter ~ is thus the distance between the EDS and the ST. Gibbs [1928] solved eq. (6.38) by assuming (i) that ~0 is constant and (ii) that RST >> 2~0. He thereby found that O'ST,n = Cr exp (- 26o/RsT)
(6.40)
where cr (independent of the choice of the dividing surface) is the specific surface energy of the planar interface (i.e. the value of CrST,,,at RST = ~). We may note, however, that Gibbs could have solved eq. (6.38) without using the second of the above assumptions. Indeed, the solution of the full eq. (6.38) with constant 80 is represented as CrsT,n = cr/(1 + 260/RsT)
(6.41)
which is the formula of Tolman [ 1949] and Kirkwood and Buff [1949]. Equation (6.38) was derived by Gibbs for the specific surface energy corresponding to the ST. However, it can be shown [Kondo 1956; Ono and Kondo 1960] that d(ln O'sT,n)/d(ln RST) = d(ln O'n)/d(ln R) so that eq. (6.38) remains in force also when the specific surface energy is referred to the EDS (we keep the notation o'n in this case). This means that with the help of (6.39) eq. (6.38) can be rewritten as d(ln Crn)/d(ln R ) = (260/R)/(1 + ~ / R ) .
(6.42)
If we now adopt Gibbs's assumptions, the second in the less restrictive form R >> S0, we arrive again at Gibbs's equation (6.40), but for EDSdefined clusters: O "n " -
O"
exp (- 2S0/R).
(6.43)
Similar to the derivation of (6.41), we can integrate the full equation (6.42), i.e. without utilizing the condition R >> ~o. The result is an= cr/(1 + ~0/R)2
(6.44)
which is the analogue of (6.41) in the case of EDS-defined clusters. This equation, which does not seem to have been reported hitherto, can be generalized to hold for arbitrarily shaped clusters of both condensed and gaseous phases if with the aid of (3.22) R is replaced by the volume V, of the EDS-defined cluster: cr~ = or~(1 + C6o/3 V~/3 )2.
(6.45)
Here c is the cluster shape factor equal to (36t01/3 for spheres, 6 for cubes, etc.
Properties of clusters 81
In accordance with (3.13), for condensed-phase clusters the explicit ndependence of cr,, from (6.45) is given by ty,, = or~(1 + Ct~o/3O~/3 n 1/3) 2.
(6.46)
For gaseous clusters, however, the dependence of o'n on n remains only implicitly expressed by (6.45). Indeed, in this case V,, and n are related in a complicated way by the equality n = (p/kT)V,, + (2ccr/3kT) V1/3[1 + (Ct~o/3V, l,/3)/(1 + Ct~o/3Vnl/3) 3] (6.47)
which follows from (3.14) upon using Pn from (6.1) with c,, = constant = c and dcr,,/d V,~/3 calculated from (6.45). The above results show that the effect of the cluster size on the cluster specific surface energy is controlled by the Gibbs parameter do: tS0 > 0 leads to a lower specific surface energy for the smaller clusters, and d;0 < 0 has the opposite effect (see, e.g. eqs (6.44)-(6.46)). It is, therefore, very important to know the sign and the absolute value of 60. Unfortunately, such a knowledge cannot be obtained from experiment, since neither the ST nor the EDS are real physical objects and it is thus impossible in principle to measure the distance 60 between them. Under such circumstances it remains only to resort to theoretical arguments and/or to determinations of ~ with the aid of numerical calculations. On such grounds it is believed that, typically, I~lis of the order of the molecular diameter do, but the question about the sign of is still open [Ono and Kondo 1960; Defay and Prigogine 1966; Nishioka 1977, 1992; Hadjiagapiou 1994; Baidakov 1994, 1995]. The argument concerning the magnitude of] S01is that since the EDS is positioned inside the surface layer (the spatial zone with varying molecular density), the ST can be expected to be close to the EDS and, hence, also inside this layer. Ergo, d;0 should not exceed the thickness of the surface layer which at low enough temperatures comprises one to a few molecular layers. Another unclear point is whether it is possible to treat ~ as an R, Rsz-independent quantity which may be approximated [Ono and Kondo 1960] by the distance between the EDS and the ST for the planar interface, i.e. by the limiting value of for both R ~ oo and RST ~ co. Recently, Hadjiagapiou [ 1994] has shown that 60 may decrease linearly with R, and Schmelzer et al. [ 1996] have demonstrated that different tS0(R,RsT) functions can affect essentially the size dependence of the cluster specific surface energy. Figure 6.1c illustrates the n-dependence of o'~ from eqs (6.45) and (6.46) for spherical EDS-defined steam bubbles and water droplets, respectively. For the bubbles T = 583 K and p = 0.1 MPa, and n is determined with the help of (6.47). The parameter values used are c = (36/r) 1/3, S 0 = 0.1 nm and those listed in Tables 3.1 and 3.2. As seen, ty~ does not differ more than 10% from its limiting value ty for n = oo (i.e. for the planar interface) when n > 1000 molecules. In the range of n < 100 molecules the change of ty, with respect to ty is already greater. However, in this range of n values the predicted effect cannot be regarded without reservation, because then the ~0(R, RST) = constant approximation is uncertain and, more importantly, the very
82
Nucleation: Basic Theory with Applications
applicability of usual thermodynamics is questionable. From this point of view, the numerous attempts in the nucleation literature to take account of the n-dependence of o'n, e.g. by means of eq. (6.41), appear more or less speculative. In the absence of a firm knowledge about the actual change of O'n with n we shall confine all further considerations within the approximation an = constant = cr of the classical nucleation theory. In Chapter 8 we shall see that an alternative determination of the cluster specific surface energy is possible in the scope of quasi-thermodynamics.