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Driving force for nucleation
Chapter 2
Driving force for nucleation
In the preceding Chapter we have seen that when p' ~ Pe (but is limited between P~;s and Pg.s), the VDW fluid can be in two states with differently deep minima of G. Obviously, the transition from metastable (e.g. gas) to truly stable (e.g. liquid) state occurs because of the necessity for the fluid to occupy a lower-energy state. The same holds true for any other metastable phase and for this reason, in general, the thermodynamic driving force for the first-order phase transition and, hence, for the nucleation process, is the quantity p
A/I - (Gold - Gnew)/M =/-/old
-- ,//new
(2.1)
known as supersaturation. Physically, the supersaturation Ap is the gain in free energy per molecule (or atom) associated with the passage of the phase from the minimum with higher Gibbs free energy Gol d t o the minimum with lower Gibbs free energy Gnew (Pold and Ftneware, respectively, the chemical potentials of the old and the new phases at the corresponding minima). Equation (2.1) allows expressing A/I through experimentally controllable parameters in various particular cases. As an example, let us consider the transition of the VDW fluid from gas (old phase) to liquid (new phase), which occurs at reduced pressure p' satisfying the condition Pe < P' < Pg.s (this corresponds to transition from point h to point a in Fig. 1.1). Using eq. (1.4) yields p
Go~d = Gref (T') + MkTcr[3p'Vg/8 - 9/8Vg- T ' l n (3Vg- 1)] Gnew -" Gref (T') + MkTcr[3p'Vl"/8 - 9/8V[- T'ln (3Vl'- 1)]. Here Vg (p', T') and VI' (p', T') are, respectively, the reduced volumes of the gas and the liquid, at which G has a minimum (i.e. the abscissae of points h and a in Fig. 1.1). Substitution of Gold and Gnew from above into (2.1) leads to the exact formula
A,tt = kTcr {T'ln [(3Vl'- 1)/(3Vg- 1)] + (9/8)(1/V1' - l/Vg) + (3/8) p'(Vg - VI')1.
(2.2)
With the help of (1.6) this formula can be given the equivalent form
At.t = kTcr{T'ln [(3Vg(e- 1)(3Vl'- 1)/(3Vg- 1)(3Vl,e- 1)] + (9/8)(1/V~'- 1/VLe+ 1/Vg~e- 1/Vg)
+ (3/8)(p'Vg - p~Vg~ + p[VI,'~ - p'V[)}
(2.3)
10 Nucleation: Basic Theory. with Applications
which shows that A/d = 0 at p' = Pe. Indeed, at phase equilibrium there exists no driving force for first-order phase transition and, hence, for nucleation and then it is said that the old phase is saturated. Obviously, nucleation is impossible as well when A/d < 0: the old phase is then undersaturated (its G minimum is lower than that of the new phase and/~old > 1, 3Vg.e>> 1, VI' ~ Vl,'e = Moo/Vcr (the liquid is almost incompressible) and Vgp ' - VgPep~ (8/3) T' (the gas is nearly ideal). In view of the relation 3MkTcr = 8PcrVcr, it then follows from (2.3) that, approximately, P
t
A/d = k T In (P/Pe) - vo(P - Pe) = k T In
(Vg,e/Vg) -kT(Vl,elVg,e)(Vg,elVg- l)
(2.4)
where v0 is the molecular volume. Let us now see how Ay can be determined in some important cases of nucleation during first-order phase transitions in real systems (see also [Bohm 1981]). In finding the respective formulae for A/d we shall use eq. (2.1) and known expressions for the chemical potentials/dold and/dnew of the old and the new phases. (a) Condensation of vapours (p > Pe) In this case the dependence of the chemical potentials of the old phase (vapours) and the new phase (liquid or solid) on the actual pressure p of the vapours is given by [Guggenheim 1957] /dold(P) =/de -t- k T In (P[Pe) /dnew(P) =/de + Oo(P- Pe)
(2.5) (2.6)
where pe(T) is the phase-equilibrium pressure (i.e. the pressure of the saturated vapours of the liquid or the solid), and/de =/dold(Pe) =/dnew(Pe) is the chemical potential of the vapours and of the liquid or the solid at phase equilibrium. The above two expressions hold true provided the vapours behave as ideal gas, and the liquid or the solid is incompressible. Combining (2.1), (2.5) and (2.6) leads to (e.g. [Toschev 1973a]) A/d(p, T) = k T In (P]Pe) - Oo(P - P e )
(2.7)
which in most cases can be approximated sufficiently accurately by [Zettlemoyer 1969] Al~(p, T ) = k T In [p/pe(T)],
(2.8)
since usually V0Pe << k T and the second summand in (2.7) is negligible.
Driving force for nucleation
11
Worth noting is the coincidence of eqs (2.4) and (2.7). It is not accidental, since these two equations are derived under the same assumptions for the gas and the liquid phases and since in many respects the VDW fluid is an acceptable model for real fluids. That is why, if for the determination of Ap during condensation of vapours into a liquid it is necessary to account for the effects of gas non-ideality and/or liquid compressibility, instead of (2.7) or (2.8) one can use the full equations (2.2) or (2.3), which allow for these effects in the scope of the VDW model. The vapour non-ideality can be taken into account [Guggenheim 1957] also by replacing the P/Pe ratio in the logarithm of eqs (2.7) and (2.8) with the ratio fv/fv, e of the actual and the equilibrium fugacities fv and fv, e of the vapours. In condensation of molecular beams onto a substrate the experimentally controllable parameter usually is the impingement rate I of molecules (per second per m 2) rather than the pressure p. As I = p/(2xmokT) 1/2, in this case Ap is given approximately by (2.8) in the form [Sigsbee 1969; Lewis and Anderson 1978]
Ap(l, T)= kT In [I/le(T)]
(2.9)
where le = pe/(27cmokT) 1/2 is the equilibrium value of the impingement rate, m0 and T being, respectively, the mass of a molecule and the substrate temperature. This formula is valid when thermal equilibration of the deposited molecules with the substrate takes a sufficiently short time [Sigsbee 1969]. (b) Boiling, evaporation or sublimation (0 < p < Pe) In this case the old phase is the liquid or the solid, and the new phase is the vapour. Accordingly, eqs (2.5) and (2.6) are to be used in (2.1) with exchanged subscripts 'old' and 'new'. This leads again to Ap from eq. (2.7), but with changed sign of its r.h.s.:
Ap(p, T) = kT In
(pe/p)
-
OO(Pe-- p),
(2.1 O)
p being the actual pressure of the liquid in boiling or of the vapour in evaporation or sublimation. When v0 Pe << kT, Al~ can be approximated by
Ap(p, T)= kT In [pe(T)/p]. (c) Condensation of solute (a > ae) In this case the old phase is constituted of dissolved molecules which along with a solvent (treated as an inert medium) represent a liquid or solid solution, and the new phase is the liquid or solid condensate of the dissolved molecules. Rather than the pressure, the experimentally controllable parameter usually is the actual solute activity a, and the dependence of Pold on a is of the form [Guggenheim 1957] /-/old(a) = / / e 4- kT In (a/ae)
(2.11)
where ae(T) is the equilibrium activity, i.e. the activity at which the solute and the condensate are in phase equilibrium. Since Pnew is practically aindependent, approximately,
12 Nucleation: Basic Theory with Applications
Pnew(a) =/~new(ae) =/~e"
(2.12)
Combining the above two expressions with eq. (2.1) yields [Walton 1969a; S6hnel and Garside 1992] A/~(a, T) - k T l n [a/a~(T)].
(2.13)
For sufficiently dilute solutions a and ae can be replaced, respectively, with the actual and the equilibrium concentrations C and Ce(T) of the solute (Ce is also known as solubility) and (2.13) becomes [Nielsen 1964; S6hnel and Garside 1992] A/~(C, T) = kT In [C/Ce(T)].
(2.14)
It is worth noting that eqs (2.13) and (2.14) cover also the case of decay of solids by 'condensation' of atomic vacancies (the old 'phase') 'dissolved' in them [Hirth and Pound 1963]. This 'condensation' leads to the appearance of macroscopic cavities (the new 'phase') in the solid. Then a, ae and C, Ce are the actual and the equilibrium activities and concentrations of the vacancies in the solid. Equations (2.13) and (2.14) are applicable when the solute molecules in the solution are not dissociated into ions. In many cases, however, this is not the case and then, according to the condition for chemical equilibrium [Guggenheim 1957; Landau and Lifshitz 1976], ,t/old = V I ] . / 1 + V2,t/2 + . . .
+
(2.15)
Vk,tl k
where vi is the number of ith species in the molecule (i = 1, 2 . . . . . k), and pi is the chemical potential of the ith species in the solution. Using (2.11) in the form ].li(ai)- ltli,e + kT In (ai/ai,e)
leads to the following expression for/~old: ~told = ~t~ + k T l n [(al/al,e) v~ ( a z ] a z , e ) V 2
. . . (ak]ak,e)
vk ] .
Here a i is the actual activity of the ith species in the solution, a i , e and Pi,e are the equilibrium activity and chemical potential of the ith species, and ,t/e = Vl,t/1,e + V2,t/2,e + 9 9 9 +
Vk,Uk,e.
Thus, with the help of eq. (2.12) and the above expression for/~ola, it follows from (2.1) that A/~(H, T)= k T l n [H/He(T)]
(2.16)
v2 Vk where H = a~ ~a~2 .. a~ k and Fie = al,v1ea2,e.., akx are, respectively, the actual and equilibrium activity products of the solute [Nielsen 1964; S6hnel and Garside 1992]. As seen, eq. (2.16) is a generalization ofeqs (2.13) and (2.14) (for sufficiently dilute solutions H and I'I e a r e the products of the corresponding concentrations of ith species, II e being the so-called solubility product). An interesting point concerning eq. (2.16) is that even when the solution is supersaturated with 9
Driving f o r c e f o r nucleation
13
respect to only one of the species in the solution, but is saturated with respect to the others (e.g. a l / a l , e > 1, a2/a2,e = a3/a3,e = 9 9 9 = ak/ak,e = 1), there exists a driving force for condensation of the solute, since then A/j > 0. Moreover, sufficiently high supersaturation of the solution with regard to one (e.g. a l / a l,e >> 1) or more species can create a driving force (A/~ > 0) even if the solution is undersaturated with regard to one (e.g. a2/a2, e < 1) or more of the rest of the species. Detailed considerations concerning A/.t for crystallization from solutions can be found elsewhere [van Leeuwen 1979; van Leeuwen and Blomen 1979; S6hnel and Garside 1992]. (d) Dissolution (a < ae) Analogously to the case of boiling, evaporation or sublimation, the driving force for dissolution is that for condensation of solute, but taken with opposite sign, as now the old and the new phases are reversed. That is why, according to eqs (2.13), (2.14) and (2.16), for dissolution A~t is given by A/~(a, T) = k T In [ a e ( T ) / a ] ~ k T In [ C e ( T ) / C ]
(2.17)
and, more generally, by A/x(H, T) = k T In [FIe(T)/H].
(2.18)
(e) Crystallization of melt or polymorphic transformation by cooling (T < Te) In this case the old phase is a melt or a crystal with a given modification, and the new phase is a crystal (which has another modification in polymorphic transformation). The commonly used parameter to control Voidexperimentally is the temperature rather than the pressure and, for that reason, it is convenient to express Ap as a function of T. Isobarically, from thermodynamics [Guggenheim 1957], ~ ( T ) = ~e +
(2.19)
s(r') dr'
where s is the entropy per molecule (or atom), and Te is the absolute phaseequilibrium temperature (the melting point in melt crystallization). Using (2.19) twice, for/-told and ~new, and recalling (2.1) yields [Volmer 1939] AI.t(T) =
A s ( T ' ) dT'
(2.20)
where As(T) = Sold(T) - Snew(T). Obviously, the main problem in finding Abt is to know how the entropies Sold and Snew of the old and the new phases depend on T. In the absence of such a knowledge, following Bohm [ 1981 ], it is convenient to use the Taylor expansion AH(T) = _ Ase(T - Te) - (1/2)(dAs/d~e(T - Te)2 - (1/6)(d2As/dT2)e(T-
Te)3 - . . .
of A/I from (2.20) in the vicinity of T = Te where
mse, (dAs/dT)e
and (deAs/
14 Nucleation: Basic Theory with Applications
dT2)e are the values of As and its derivatives at T = Te. Recalling the known relationship [Guggenheim 1957]
s(r) =
fo
[Cp(
)/r']dr'
between the entropy and the heat capacity (per molecule) Cp at constant pressure, we can employ it twice (for the old and the new phase) to find that d A s / d T = Acp(T)/T p
d2As/dT 2 = A c p ( T ) / T - Acp(T)/T 2
where Acp - Cp,old- Cp,new, A Cp =--dAcp/dT, and Cp,oldand Cp,neware the molecular heat capacities of the old and the new phase, respectively. With these expressions for the As derivatives, a truncation of the above expansion of Ap results in All(T) = A s e A T - (Acp,e/2Te)AT 2 + [ ( Z e A c p , e - Acp,e)/6T 2] A T 3. ( 2 . 2 1 )
Here AT = Te - T
(2.22)
is the undercooling, Ase = Sola(Te) - Snew(Te) = ~,/Te is the difference in the molecular entropies of the old and the new phase at T = Te, ~, is the latent heat (per molecule) of crystallization or polymorphic transformation, ACp,e- Cp,old(Te) -- Cp,new(Te) and Acp,e- dAcp/dT at T = Te. Equation (2.21) reduces to the widely used formula [Volmer 1939] Ap(T) = AseAT
(2.23)
when the undercooling satisfies the condition AT << 2TeAse/Acp,e - 2A/Acp,e, which is usually met in crystallization of metal melts or in polymorphic transformation. For glass-forming, polymer and other more complex melts eq. (2.23) is not always sufficiently correct, but for them it often suffices to use (2.21) in the simpler form [Bohm 1981] Al.t (T) = A s e A T - (Acp,e/2Te)A T 2.
(2.24)
This equation is very similar to the one given by Jones and Chadwick [ 1971 ]. A number of other Ap(T) dependencies are also known in the literature (see, e.g., [Hoffman 1958; Gutzow et al. 1985; Toner et al. 1990; Kelton 1991; Gutzow and Schmelzer 1995]). Figure 2.1 shows the temperature dependence of Ap for Li20.2SIO2 melt for which Te = 1306 K, As e = 5.26k and Acp,e=l.56k [Kelton 1991]. The circles are the exact values computed by Kelton [1991] with the help of measured Acp(T) data, and the dashed and solid lines are drawn, respectively, according to eqs (2.23) and (2.24). It is seen that in this case (2.24) describes considerably better than (2.23) the actual dependence of Ap on AT. (f) Melting or polymorphic transformation by heating (T > Te) In this case all equations from case (e) remain in force taking into account that now the old phase is the crystal and the new phase is the melt or the
Driving f o r c e f o r nucleation
'
iHi
i
i
1
'~
'
'1
15
~
'
~
I
'
'
'
I
'
~
~
I
'
~
J'
/,
/ t
4 p
3
p tpp
s "
,p"//I
2
."
o
0
O
9
2
1t
0
0.2
,~
0.4
0.6
0.8
1.0
AT/T e
Fig. 2.1 Dependence of the supersaturation on the undercooling f o r Li20.2Si02 melt: circles - experimental data [Kelton 19911; circular c o n t o u r - eq. (2.23); solid line - eq. (2.24). crystal with another modification. This means that everywhere in these equations As~, Acp,~ and A cp,~ have to be taken with opposite sign. Since melting and polymorphic transformation often occur at T - Te, in many cases the approximation (2.23) is good enough: A/~(T) =-AseAT = Ase(T- Te).
(2.25)
(g) Electrochemical deposition (tp < tpe) In this case the old phase is ionic solute which along with a liquid or solid solvent forms electrolytic solution, and the new phase is a liquid or solid with a given Galvani potential q~. The role of the chemical potentials of the old and new phases is now played by the respective electrochemical potentials ~old and ~new [Lange and Nagel 1935; Guggenheim 1957] so that rather than by eq. (2.1), Ap is determined by AlL ~- ~old -- ~new.
(2.26)
The dependence of ~n~w on tp has the form [Lange and Nagel 1935; Guggenheim 1957] ~new ((~) -" ~e q" Zieo (tp - q)e)
where zi is the valency of the solute ions, e0 is the electronic charge, and tPe and ~e (T) are the phase-equilibrium Galvani and electrochemical potentials, respectively. Since ~ola is practically ~independent, i.e.
16 Nucleation: Basic Theory with Applications
~o~d (~o) = ~old(~0o) =
~o,
using the last two equations in (2.26) leads to the approximate formula [Volmer 1939; Vetter 1967] Ap(cp) - zieoACp
(2.27)
where Aq9 - r
r
(2.28)
is the overvoltage. (h) Electrochemical dissolution (q9 > q~e) Analogously to non-electrochemical dissolution (case (d)), in this case ~old and ~n~w have to be exchanged in eq. (2.26) and for that reason the supersaturation is again given by eq. (2.27), but with opposite sign of its r.h.s."
A/l(tp) = zie0(tp- tpe).
(2.29)
Summarizing, we see that Ap is determined always by eq. (2.1) or its appropriate generalization (e.g. eq. (2.26)). The difficulties in the determination of Ap arise from incomplete knowledge of the exact dependences of Polo and ]-/new on the respective experimentally controllable parameters. It must be emphasized that the correct determination of Ap in each concrete case is of great importance for the reliable confrontation of theory with experiment. For instance, in the case of melt crystallization the use of different approximate formulae for Ap can lead to considerable differences in the calculated values of some nucleation parameters [Kelton 1991].
Driving force for nucleation
In the preceding Chapter we have seen that when p' ~ Pe (but is limited between P~;s and Pg.s), the VDW fluid can be in two states with differently deep minima of G. Obviously, the transition from metastable (e.g. gas) to truly stable (e.g. liquid) state occurs because of the necessity for the fluid to occupy a lower-energy state. The same holds true for any other metastable phase and for this reason, in general, the thermodynamic driving force for the first-order phase transition and, hence, for the nucleation process, is the quantity p
A/I - (Gold - Gnew)/M =/-/old
-- ,//new
(2.1)
known as supersaturation. Physically, the supersaturation Ap is the gain in free energy per molecule (or atom) associated with the passage of the phase from the minimum with higher Gibbs free energy Gol d t o the minimum with lower Gibbs free energy Gnew (Pold and Ftneware, respectively, the chemical potentials of the old and the new phases at the corresponding minima). Equation (2.1) allows expressing A/I through experimentally controllable parameters in various particular cases. As an example, let us consider the transition of the VDW fluid from gas (old phase) to liquid (new phase), which occurs at reduced pressure p' satisfying the condition Pe < P' < Pg.s (this corresponds to transition from point h to point a in Fig. 1.1). Using eq. (1.4) yields p
Go~d = Gref (T') + MkTcr[3p'Vg/8 - 9/8Vg- T ' l n (3Vg- 1)] Gnew -" Gref (T') + MkTcr[3p'Vl"/8 - 9/8V[- T'ln (3Vl'- 1)]. Here Vg (p', T') and VI' (p', T') are, respectively, the reduced volumes of the gas and the liquid, at which G has a minimum (i.e. the abscissae of points h and a in Fig. 1.1). Substitution of Gold and Gnew from above into (2.1) leads to the exact formula
A,tt = kTcr {T'ln [(3Vl'- 1)/(3Vg- 1)] + (9/8)(1/V1' - l/Vg) + (3/8) p'(Vg - VI')1.
(2.2)
With the help of (1.6) this formula can be given the equivalent form
At.t = kTcr{T'ln [(3Vg(e- 1)(3Vl'- 1)/(3Vg- 1)(3Vl,e- 1)] + (9/8)(1/V~'- 1/VLe+ 1/Vg~e- 1/Vg)
+ (3/8)(p'Vg - p~Vg~ + p[VI,'~ - p'V[)}
(2.3)
10 Nucleation: Basic Theory. with Applications
which shows that A/d = 0 at p' = Pe. Indeed, at phase equilibrium there exists no driving force for first-order phase transition and, hence, for nucleation and then it is said that the old phase is saturated. Obviously, nucleation is impossible as well when A/d < 0: the old phase is then undersaturated (its G minimum is lower than that of the new phase and/~old > 1, 3Vg.e>> 1, VI' ~ Vl,'e = Moo/Vcr (the liquid is almost incompressible) and Vgp ' - VgPep~ (8/3) T' (the gas is nearly ideal). In view of the relation 3MkTcr = 8PcrVcr, it then follows from (2.3) that, approximately, P
t
A/d = k T In (P/Pe) - vo(P - Pe) = k T In
(Vg,e/Vg) -kT(Vl,elVg,e)(Vg,elVg- l)
(2.4)
where v0 is the molecular volume. Let us now see how Ay can be determined in some important cases of nucleation during first-order phase transitions in real systems (see also [Bohm 1981]). In finding the respective formulae for A/d we shall use eq. (2.1) and known expressions for the chemical potentials/dold and/dnew of the old and the new phases. (a) Condensation of vapours (p > Pe) In this case the dependence of the chemical potentials of the old phase (vapours) and the new phase (liquid or solid) on the actual pressure p of the vapours is given by [Guggenheim 1957] /dold(P) =/de -t- k T In (P[Pe) /dnew(P) =/de + Oo(P- Pe)
(2.5) (2.6)
where pe(T) is the phase-equilibrium pressure (i.e. the pressure of the saturated vapours of the liquid or the solid), and/de =/dold(Pe) =/dnew(Pe) is the chemical potential of the vapours and of the liquid or the solid at phase equilibrium. The above two expressions hold true provided the vapours behave as ideal gas, and the liquid or the solid is incompressible. Combining (2.1), (2.5) and (2.6) leads to (e.g. [Toschev 1973a]) A/d(p, T) = k T In (P]Pe) - Oo(P - P e )
(2.7)
which in most cases can be approximated sufficiently accurately by [Zettlemoyer 1969] Al~(p, T ) = k T In [p/pe(T)],
(2.8)
since usually V0Pe << k T and the second summand in (2.7) is negligible.
Driving force for nucleation
11
Worth noting is the coincidence of eqs (2.4) and (2.7). It is not accidental, since these two equations are derived under the same assumptions for the gas and the liquid phases and since in many respects the VDW fluid is an acceptable model for real fluids. That is why, if for the determination of Ap during condensation of vapours into a liquid it is necessary to account for the effects of gas non-ideality and/or liquid compressibility, instead of (2.7) or (2.8) one can use the full equations (2.2) or (2.3), which allow for these effects in the scope of the VDW model. The vapour non-ideality can be taken into account [Guggenheim 1957] also by replacing the P/Pe ratio in the logarithm of eqs (2.7) and (2.8) with the ratio fv/fv, e of the actual and the equilibrium fugacities fv and fv, e of the vapours. In condensation of molecular beams onto a substrate the experimentally controllable parameter usually is the impingement rate I of molecules (per second per m 2) rather than the pressure p. As I = p/(2xmokT) 1/2, in this case Ap is given approximately by (2.8) in the form [Sigsbee 1969; Lewis and Anderson 1978]
Ap(l, T)= kT In [I/le(T)]
(2.9)
where le = pe/(27cmokT) 1/2 is the equilibrium value of the impingement rate, m0 and T being, respectively, the mass of a molecule and the substrate temperature. This formula is valid when thermal equilibration of the deposited molecules with the substrate takes a sufficiently short time [Sigsbee 1969]. (b) Boiling, evaporation or sublimation (0 < p < Pe) In this case the old phase is the liquid or the solid, and the new phase is the vapour. Accordingly, eqs (2.5) and (2.6) are to be used in (2.1) with exchanged subscripts 'old' and 'new'. This leads again to Ap from eq. (2.7), but with changed sign of its r.h.s.:
Ap(p, T) = kT In
(pe/p)
-
OO(Pe-- p),
(2.1 O)
p being the actual pressure of the liquid in boiling or of the vapour in evaporation or sublimation. When v0 Pe << kT, Al~ can be approximated by
Ap(p, T)= kT In [pe(T)/p]. (c) Condensation of solute (a > ae) In this case the old phase is constituted of dissolved molecules which along with a solvent (treated as an inert medium) represent a liquid or solid solution, and the new phase is the liquid or solid condensate of the dissolved molecules. Rather than the pressure, the experimentally controllable parameter usually is the actual solute activity a, and the dependence of Pold on a is of the form [Guggenheim 1957] /-/old(a) = / / e 4- kT In (a/ae)
(2.11)
where ae(T) is the equilibrium activity, i.e. the activity at which the solute and the condensate are in phase equilibrium. Since Pnew is practically aindependent, approximately,
12 Nucleation: Basic Theory with Applications
Pnew(a) =/~new(ae) =/~e"
(2.12)
Combining the above two expressions with eq. (2.1) yields [Walton 1969a; S6hnel and Garside 1992] A/~(a, T) - k T l n [a/a~(T)].
(2.13)
For sufficiently dilute solutions a and ae can be replaced, respectively, with the actual and the equilibrium concentrations C and Ce(T) of the solute (Ce is also known as solubility) and (2.13) becomes [Nielsen 1964; S6hnel and Garside 1992] A/~(C, T) = kT In [C/Ce(T)].
(2.14)
It is worth noting that eqs (2.13) and (2.14) cover also the case of decay of solids by 'condensation' of atomic vacancies (the old 'phase') 'dissolved' in them [Hirth and Pound 1963]. This 'condensation' leads to the appearance of macroscopic cavities (the new 'phase') in the solid. Then a, ae and C, Ce are the actual and the equilibrium activities and concentrations of the vacancies in the solid. Equations (2.13) and (2.14) are applicable when the solute molecules in the solution are not dissociated into ions. In many cases, however, this is not the case and then, according to the condition for chemical equilibrium [Guggenheim 1957; Landau and Lifshitz 1976], ,t/old = V I ] . / 1 + V2,t/2 + . . .
+
(2.15)
Vk,tl k
where vi is the number of ith species in the molecule (i = 1, 2 . . . . . k), and pi is the chemical potential of the ith species in the solution. Using (2.11) in the form ].li(ai)- ltli,e + kT In (ai/ai,e)
leads to the following expression for/~old: ~told = ~t~ + k T l n [(al/al,e) v~ ( a z ] a z , e ) V 2
. . . (ak]ak,e)
vk ] .
Here a i is the actual activity of the ith species in the solution, a i , e and Pi,e are the equilibrium activity and chemical potential of the ith species, and ,t/e = Vl,t/1,e + V2,t/2,e + 9 9 9 +
Vk,Uk,e.
Thus, with the help of eq. (2.12) and the above expression for/~ola, it follows from (2.1) that A/~(H, T)= k T l n [H/He(T)]
(2.16)
v2 Vk where H = a~ ~a~2 .. a~ k and Fie = al,v1ea2,e.., akx are, respectively, the actual and equilibrium activity products of the solute [Nielsen 1964; S6hnel and Garside 1992]. As seen, eq. (2.16) is a generalization ofeqs (2.13) and (2.14) (for sufficiently dilute solutions H and I'I e a r e the products of the corresponding concentrations of ith species, II e being the so-called solubility product). An interesting point concerning eq. (2.16) is that even when the solution is supersaturated with 9
Driving f o r c e f o r nucleation
13
respect to only one of the species in the solution, but is saturated with respect to the others (e.g. a l / a l , e > 1, a2/a2,e = a3/a3,e = 9 9 9 = ak/ak,e = 1), there exists a driving force for condensation of the solute, since then A/j > 0. Moreover, sufficiently high supersaturation of the solution with regard to one (e.g. a l / a l,e >> 1) or more species can create a driving force (A/~ > 0) even if the solution is undersaturated with regard to one (e.g. a2/a2, e < 1) or more of the rest of the species. Detailed considerations concerning A/.t for crystallization from solutions can be found elsewhere [van Leeuwen 1979; van Leeuwen and Blomen 1979; S6hnel and Garside 1992]. (d) Dissolution (a < ae) Analogously to the case of boiling, evaporation or sublimation, the driving force for dissolution is that for condensation of solute, but taken with opposite sign, as now the old and the new phases are reversed. That is why, according to eqs (2.13), (2.14) and (2.16), for dissolution A~t is given by A/~(a, T) = k T In [ a e ( T ) / a ] ~ k T In [ C e ( T ) / C ]
(2.17)
and, more generally, by A/x(H, T) = k T In [FIe(T)/H].
(2.18)
(e) Crystallization of melt or polymorphic transformation by cooling (T < Te) In this case the old phase is a melt or a crystal with a given modification, and the new phase is a crystal (which has another modification in polymorphic transformation). The commonly used parameter to control Voidexperimentally is the temperature rather than the pressure and, for that reason, it is convenient to express Ap as a function of T. Isobarically, from thermodynamics [Guggenheim 1957], ~ ( T ) = ~e +
(2.19)
s(r') dr'
where s is the entropy per molecule (or atom), and Te is the absolute phaseequilibrium temperature (the melting point in melt crystallization). Using (2.19) twice, for/-told and ~new, and recalling (2.1) yields [Volmer 1939] AI.t(T) =
A s ( T ' ) dT'
(2.20)
where As(T) = Sold(T) - Snew(T). Obviously, the main problem in finding Abt is to know how the entropies Sold and Snew of the old and the new phases depend on T. In the absence of such a knowledge, following Bohm [ 1981 ], it is convenient to use the Taylor expansion AH(T) = _ Ase(T - Te) - (1/2)(dAs/d~e(T - Te)2 - (1/6)(d2As/dT2)e(T-
Te)3 - . . .
of A/I from (2.20) in the vicinity of T = Te where
mse, (dAs/dT)e
and (deAs/
14 Nucleation: Basic Theory with Applications
dT2)e are the values of As and its derivatives at T = Te. Recalling the known relationship [Guggenheim 1957]
s(r) =
fo
[Cp(
)/r']dr'
between the entropy and the heat capacity (per molecule) Cp at constant pressure, we can employ it twice (for the old and the new phase) to find that d A s / d T = Acp(T)/T p
d2As/dT 2 = A c p ( T ) / T - Acp(T)/T 2
where Acp - Cp,old- Cp,new, A Cp =--dAcp/dT, and Cp,oldand Cp,neware the molecular heat capacities of the old and the new phase, respectively. With these expressions for the As derivatives, a truncation of the above expansion of Ap results in All(T) = A s e A T - (Acp,e/2Te)AT 2 + [ ( Z e A c p , e - Acp,e)/6T 2] A T 3. ( 2 . 2 1 )
Here AT = Te - T
(2.22)
is the undercooling, Ase = Sola(Te) - Snew(Te) = ~,/Te is the difference in the molecular entropies of the old and the new phase at T = Te, ~, is the latent heat (per molecule) of crystallization or polymorphic transformation, ACp,e- Cp,old(Te) -- Cp,new(Te) and Acp,e- dAcp/dT at T = Te. Equation (2.21) reduces to the widely used formula [Volmer 1939] Ap(T) = AseAT
(2.23)
when the undercooling satisfies the condition AT << 2TeAse/Acp,e - 2A/Acp,e, which is usually met in crystallization of metal melts or in polymorphic transformation. For glass-forming, polymer and other more complex melts eq. (2.23) is not always sufficiently correct, but for them it often suffices to use (2.21) in the simpler form [Bohm 1981] Al.t (T) = A s e A T - (Acp,e/2Te)A T 2.
(2.24)
This equation is very similar to the one given by Jones and Chadwick [ 1971 ]. A number of other Ap(T) dependencies are also known in the literature (see, e.g., [Hoffman 1958; Gutzow et al. 1985; Toner et al. 1990; Kelton 1991; Gutzow and Schmelzer 1995]). Figure 2.1 shows the temperature dependence of Ap for Li20.2SIO2 melt for which Te = 1306 K, As e = 5.26k and Acp,e=l.56k [Kelton 1991]. The circles are the exact values computed by Kelton [1991] with the help of measured Acp(T) data, and the dashed and solid lines are drawn, respectively, according to eqs (2.23) and (2.24). It is seen that in this case (2.24) describes considerably better than (2.23) the actual dependence of Ap on AT. (f) Melting or polymorphic transformation by heating (T > Te) In this case all equations from case (e) remain in force taking into account that now the old phase is the crystal and the new phase is the melt or the
Driving f o r c e f o r nucleation
'
iHi
i
i
1
'~
'
'1
15
~
'
~
I
'
'
'
I
'
~
~
I
'
~
J'
/,
/ t
4 p
3
p tpp
s "
,p"//I
2
."
o
0
O
9
2
1t
0
0.2
,~
0.4
0.6
0.8
1.0
AT/T e
Fig. 2.1 Dependence of the supersaturation on the undercooling f o r Li20.2Si02 melt: circles - experimental data [Kelton 19911; circular c o n t o u r - eq. (2.23); solid line - eq. (2.24). crystal with another modification. This means that everywhere in these equations As~, Acp,~ and A cp,~ have to be taken with opposite sign. Since melting and polymorphic transformation often occur at T - Te, in many cases the approximation (2.23) is good enough: A/~(T) =-AseAT = Ase(T- Te).
(2.25)
(g) Electrochemical deposition (tp < tpe) In this case the old phase is ionic solute which along with a liquid or solid solvent forms electrolytic solution, and the new phase is a liquid or solid with a given Galvani potential q~. The role of the chemical potentials of the old and new phases is now played by the respective electrochemical potentials ~old and ~new [Lange and Nagel 1935; Guggenheim 1957] so that rather than by eq. (2.1), Ap is determined by AlL ~- ~old -- ~new.
(2.26)
The dependence of ~n~w on tp has the form [Lange and Nagel 1935; Guggenheim 1957] ~new ((~) -" ~e q" Zieo (tp - q)e)
where zi is the valency of the solute ions, e0 is the electronic charge, and tPe and ~e (T) are the phase-equilibrium Galvani and electrochemical potentials, respectively. Since ~ola is practically ~independent, i.e.
16 Nucleation: Basic Theory with Applications
~o~d (~o) = ~old(~0o) =
~o,
using the last two equations in (2.26) leads to the approximate formula [Volmer 1939; Vetter 1967] Ap(cp) - zieoACp
(2.27)
where Aq9 - r
r
(2.28)
is the overvoltage. (h) Electrochemical dissolution (q9 > q~e) Analogously to non-electrochemical dissolution (case (d)), in this case ~old and ~n~w have to be exchanged in eq. (2.26) and for that reason the supersaturation is again given by eq. (2.27), but with opposite sign of its r.h.s."
A/l(tp) = zie0(tp- tpe).
(2.29)
Summarizing, we see that Ap is determined always by eq. (2.1) or its appropriate generalization (e.g. eq. (2.26)). The difficulties in the determination of Ap arise from incomplete knowledge of the exact dependences of Polo and ]-/new on the respective experimentally controllable parameters. It must be emphasized that the correct determination of Ap in each concrete case is of great importance for the reliable confrontation of theory with experiment. For instance, in the case of melt crystallization the use of different approximate formulae for Ap can lead to considerable differences in the calculated values of some nucleation parameters [Kelton 1991].