J. Phys. Chem. 1995, 99, 7837-7844
7837
The Kelvin Relation: Stability, Fluctuation, and Factors Involved in Measurement Howard Reiss*J and Ger J. M. Koper Department of Physical and Macromolecular Chemistry, Gorlaeus Laboratories, Leiden University, P.O. Box 9502, 2300 RA Leiden, The Netherlands Received: November 7, 1994; In Final Form: February 9, 1 9 9 9
We have studied the Kelvin relation, e.g., an equilibrium between a two-component drop (in which the solute is involatile) and the vapor of the solvent component. This equilibrium and its stability have been analyzed previously, but not in a thermodynamically disciplined manner that employs the appropriate thermodynamic potential. Using the theory of the thermodynamic potential, we verify the previous conclusion that regimes of both stable and unstable equilibrium exist. However, we are able to show in addition that, contrary to the conventional wisdom, it is possible to have unstable equilibrium even in undersaturated solvent vapor. Also, because our analysis is based on the thermodynamic potential, we are able to characterize the fluctuations in drop size and to emphasize the experimental importance of the critical-like behavior that occurs at the boundary between a stable and an unstable regime. Finally, we are able to provide a general analytical description of the surface representing the free energy of the system as a function of drop composition at fixed super(under)saturation. This allows us to establish the connection (and continuity) between studies of the Kelvin relation (micrometer range) and of nucleation (nanometer range).
1. Perspective The Kelvin relation was f i s t published by William Thomson (Lord Kelvin) in 1871'q2 and predicted the increase of vapor pressure that a small drop would experience, over that of the bulk liquid. This capillarity-driven phenomenon is manifest in other forms, e.g., increased solubility of small crystals, depression of melting point, nucleation, bubble formation, equilibrium shape of crystals, microemulsions, stability of crystal surfaces, rates of evaporation, capillary rise, etc. In this more general guise it is usually referred to as the Gibbs-Thomson relation, thereby marking the indelible influence of Gibbs on the entire field. In spite of its wide use in these areas, after more than 120 years there does not appear to have been an absolutely quantitative and direct experimental verification of the relation, although especially during the past 15 years a few heroic efforts have been made. The measurements have been close to theory, but no closer than several percent. They have involved vapor pressure in c a p i l l a r i e ~ ,evaporation ~.~ drops of binary solution with an involatile solute,8 molecular force measurem e n t ~ homogeneous ,~ nucleation,10 and melting point depression.",'* The sources of error in each case are fairly well understood. For example, in the case of molecular force measurement^,^ even a monolayer of impurity adsorbed on the mica cylinders between which the liquid is suspended could cause considerable error. It should also be noted that evaporation rate measurements are not equilibrium measurements while the Kelvin relation itself does refer to equilibrium. This opens the possibility that the relation will never be confirmed with accuracy beyond that already achieved. Part of the problem lies in the smallness of the effect coupled with the smallness of the system that must be studied to reveal it. Another problem is that the Kelvin relation for a single component drop corresponds to an unstable equilibrium, for which equilibrium measurements are almost impossible. Still
* To whom correspondence should be addressed.
' Permanent
address: Department of Chemistry and Biochemistry, University of California, Los Angeles, Los Angeles, CA 90024-1569. @Abstractpublished in Advance ACS Absrracts, May 1, 1995.
0022-3654/95/2099-7837$09.00/0
another problem is associated with the fact that as the system (e.g., the drop) is made increasingly small, so as to enhance the effect, the relation itself must be altered in order to take account of the dependence of surface tension on curvature and, indeed, in order to define the radius of the drop. A thorough study of the thermodynamics of such altered Kelvin relations does not seem to have been made, although the formalism for accomplishing this appears to be a ~ a i l a b l e . ' ~ - ' ~ Because of its great importance throughout many areas of science and technology, the Kelvin (or Gibbs-Thomson) relation has been used even in the absence of precise experimental verification and has usually been validated in a qualitative manner through the observation of the phenomena that depend on it, e.g., Ostwald ripening, nucleation, evaporation, etc. Given the modern state-of-the-art in the measurement of light scattering, it is likely that the study of the vapor pressure of a liquid drop still holds the most promise for increasing the accuracy of the measurement of the relation-or for that m a t t e r t h e altered relation that must be considered as the drop size is decreased. As we indicate below, even if an improvement in accuracy proves to be impossible, the experiments will reveal other interesting features (some of practical importance) that are of value in themselves. Unfortunately, as has already been indicated, the vapor pressure of a single-component drop involves a thermodynamically unstable equilibrium for which equilibrium measurements are almost impossible. Most other techniques, e.g., vapor pressure above a meniscus, molecular force measurements, and melting point depression of thin films, involve stable equilibria. We note, in passing, that the measurement of melting point depression requires knowledge of the surface tension of a solid or the contact angle of a melt on its own solid, which is almost impossible to measure. In contrast, the liquid drop holds the possibility that all the relevant parameters, surface tension, vapor pressure, density, etc., can be measured independently, and for this reason the drop remains the most promising system with which to attempt verification. The problem of the unstable equilibrium was solved to a degree by Lah4er and Gruen in 19528 through the ingenious device of working with a drop of a binary solution containing 1995 American Chemical Society
7838 J. Phys. Chem., Vol. 99, No. 19, 1995 an involatile solute. The solution consisted of toluene (volatile solvent) and dioctyl phthalate (involatile solute). Such a drop could be shown to be in stable equilibrium (see refs 17-19 and pp 262-268 of ref 20) with the vapor of volatile solvent, and yet the features of that equilibrium were still, in part, controlled by the Kelvin effect. LaMer and Gruen went further than this by focusing the measurement, not on the vapor pressure of the drop but rather by means of light scattering, on its radius under conditions such that the vapor pressure was the controlled variable. Not only did they therefore avoid the difficulty of measuring small changes of vapor pressure, but also they were able to magnify the effect in the sense that the radius bore a nonlinear relation to the vapor pressure. Still the measurements of LaMer and Gruen remained incomplete in many respects, and others have neither attempted to duplicate nor extend them. The fact that the unstable equilibrium could be converted to a stable one by the influence of the involatile solute seems to have been first realized by Kohler,17-19 who used this idea in a very practical way to explain the difference between the stabilities of droplets in mists on the one hand and in clouds on the other hand. Defay and Dufour (see pp 262-268 of ref 20) have extended Kohler's ideas. Although the question of stability was central to the work of these authors, none of their work seems to have focused on the relevant thermodynamic potential whose curvature as a function of drop size would by itself define stability. Instead, they examined the behavior of the drop vapor pressure itself. This method is perfectly acceptable, but among other things it provides a more limited picture of the phenomenon; e.g., it does not permit the study of fluctuations and other kinds of behavior. For example, the opinion seems to have developed that a drop in equilibrium with undersaturated solvent vapor can never, on thermodynamic grounds, be in unstable equilibrium. Although the conditions for this to occur might be difficult to achieve in any real system, such an unstable equilibrium is not proscribed by thermodynamics-as we show below. Indeed, the transition from stable to unstable equilibrium, in both undersaturated and supersaturated vapors, by a drop containing involatile solute may represent another method for assessing the Kelvin relation, in addition to possessing an intrinsic interest in itself. Also, fluctuations near these transitions and even well within the stable domain are of interest in these respects. We address these issues in the following sections where we focus on an approach based on a study of the thermodynamic potential for the system consisting of a binary drop containing involatile solute and in equilibrium with the vapor of its volatile solvent, that vapor being maintained at constant pressure (the control variable), Le., on the system accessible to light scattering experiments similar to those performed by LaMer and Gruen.8 2. Thermodynamics
Our analysis will deal with only the simplest model of the binary drop-vapor system. We take the vapor to be ideal and the drop to be incompressible. The interface between the drop and the surrounding vapor is considered to be abrupt and of zero thickness at the radius r, and the fluid on either side is assumed to be of uniform density. A more exact treatment is left to the future, since here we are only interested in the gross features of the phenomenon. The most conventional method for deriving the Kelvin relation for this system is simple and direct (see p 260 of ref 20 and pp 165-167 of ref 21). Since the solute (component 2) is involatile, the vapor consists only of the volatile solvent (component l), and its chemical potential may be written in
Reiss and Koper the form
where pol is the chemical potential of pure solvent liquid, p is the pressure of the vapor, and pow is the vapor pressure of the pure liquid in bulk, Le., of a drop of pure solvent having infinite radius. k is the Boltzmann constant and T the temperature. In denoting the pressure of the vapor, we drop the subscript 1 since we never have to consider the partial pressure of the involatile solute, component 2. The chemical potential of the solvent in the drop is given by
pl(T,P)= p",(T,P)
+ kTln a ,
+ v , ( P - p ) + kTln a , = po,(T,p) + 2av,/r + kT In a , ( 2 . 2 ) = po,(T,p)
where P is the pressure inside the drop, IJ is the surface tension, V I is the partial molecular volume of the solvent in the drop, and a1 is its thermodynamic activity. In the last step, in eq 2.2, we have used the Laplace relation
P -p = 2 d r
(2.3)
and in the second step we have assumed the drop to be incompressible. Equating A1 to pl at equilibrium, we obtain from eqs 2.1 and 2.2
where the first exponential factor has been approximated as unity, since for a condensed system p"l(T,p) - p01(T,pom)in the exponent is exceedingly small. Equation 2.4 is the Kelvin relation for the binary drop in question. To demonstrate that the equilibrium to which eq 2.4 applies can indeed be stable; previous author^'^-^^ have simply plotted ln(plpo,) versus r, according to the equation, and argued (in terms of the response of the vapor pressure of the drop to a change in radius while the pressure of the vapor was maintained constant) that, under conditions such that the slope of this plot was positive, the equilibrium was stable. This is a perfectly valid argument, but the more usual (and more disciplined) approach to the question of stability in thermodynamics is to evaluate the appropriate thermodynamic potential and to show that the equilibrium corresponds to a minimum of that potential. The evaluation of the thermodynamic potential away from this minimum has additional uses. For example, it can be used to study fluctuations about that minimum. These fluctuations could be significant for small enough drops. For these various reasons we now proceed to evaluate and apply the appropriate thermodynamic potential. Since we are interested in a system at constant pressure, the most convenient potential is the Gibbs free energy which takes the form
where nl and 122 are the numbers of molecules of solvent and solute in the drop, ,MI and pz are the respective chemical potentials at the pressure outside of the drop (i.e., at the pressure p ) , A is the surface area of the drop, and N I is the number of solvent molecules in the vapor. Both 11and 0 have been defined previously. The system is closed so that nl N1 is constant. We will be interested in the variation of G, namely ( ~ G ) T , ~ ,
+
J. Phys. Chem., Vol. 99, No. 19, 1995 7839
The Kelvin Relation with respect to nl, so we write
(”)
=A(%)
T,p
n
t
+N(%)
+pl+p(3) T,p
T,p
T,p
(%)TJJ + n (%)T p +cJ(”)T,p +A(*)
T,p
(2.6)
Now, because the system is closed,
(2.7)
not correspond to the equilibrium of our interest unless both sides are set to zero. Nevertheless, eq 2.15 remains valid even when (aGlanl)T,p f 0 since it is necessary to introduce a constraint (virtual variation) to maintain the system in equilibrium (an equilibrium having an additional variable of state) along the paths of the variation so that T and p (equilibrium quantities) can be defined. This is the usual situation in thermodynamics (see pp 122-125 of ref 21), and for this reason, eqs 2.12 and 2.13, which are necessary for the derivation of eq 2.15, will still hold even when (aGlanl)T,pf 0. Assuming a spherical shape for the incompressible droplet its radius r is a unique function of nl and n2, namely
T,p
(5)=(&)(5)=-(&)= O T,p
aNl
T,p
T,p
3(n1v, (2.8)
(3)= o
(2.9)
T,p
since we do not allow the involatile solute (in the closed system) to enter the vapor phase. To investigate the remaining derivatives in eq 2.6, we recognize that both nl and n 2 are the sums of contributions from the bulk and the surface of the drop. We write
+ ni n2 = n2 + ni nl = nlb
(2.10)
b
(2.11)
where the superindex “b” refers to the bulk and the “s” to the surface. The Gibbs-Duhem relation requires
(2.12) while the Gibbs adsorption equation demands
+ns(%) T,p
=-A(*) T,p
4n
T,p
aNl
Equation 2.8 is true since, according to eq 2.1, ill depends only on T and p . We also have
ns(%)
r=[
(2.13) T,p
+ n2v2)
1
(2.16)
and its surface area is
where v2 is the partial molecular volume of the solute. Therefore
where we have used the thermodynamic relation
(2.19) which can be shown to be true even for the drop in which P changes with nl, the proof resting on the assumption of incompressibility. Substitution of eq 2.18 into eq 2.15 and setting ( a G l a n l ) T , p = 0 leads to eq 2.4; Le., we have recovered the Kelvin relation for the binary drop, this time, however, by working with the thermodynamic potential. To determine whether the extremum of G to which eq 2.4 corresponds is a maximum or a minimum, Le., whether the equilibrium is unstable or stable, we must see whether (a2G/ an12)T,pis negative or positive. Returning to eq 2.15, and ignoring pol(T,p) - p0l(T,p,), we find, by differentiating with respect to nl once more
Adding these equations yields
n
(2.20)
(*)
From eq 2.18 we have, by differentiation
= - A ( E ) T , p (2.14)
T,p
Substitution of eqs 2.1, 2.2, 2.7, 2.8, 2.9, and 2.14 into eq 2.6 yields
T,p
= - - V12 2zr4
= p o , ( ~ , p-) po,(~,po,)
+ kTln a, - kTln e + CJ PO-
(2.21)
where we have, once again, used eq 2.19. Substituting eqs 2.14 and 2.21 into eq 2.20 yields
(Z)T,p
(2.15) It is worth noting that eqs 2.12 and 2.13 apply to systems that remain in equilibrium during the variation, i.e., to systems undergoing reversible changes. At the same time eq 2.15 does
-
-
Now the standard state for the solvent has been chosen such that Q I XI as XI 1 where XI, the mole fraction of the
7840 J. Phys. Chem., Vol. 99, No. 19, 1995
Reiss and Koper
TABLE 1: Relevant Parameters in the LaMer and Gwen Experiment* (e.g., First Row of Their Table 6)a V I = 1.78 x m3 (toluene) v2 = 4.97 x m3 (dioctyl phthalate) u = 27 mN/m (almost independent of composition) ni = 1.13 x l o 8 n2 = 8.4 x lo6 t* = 1.8 x lo-”
T=300K
II
In this list n2 was obtained from the authors’ estimate of the radius of the initial dioctyl phthalate drop. a
solvent, is
(2.23) Also, in an ideal solution, a1 = XI for all compositions. Thus, for dilute or ideal solutions, eqs 2.4 and 2.22 become 2avllrkT
PIPo- = xle
(2.24)
r
Figure 1. Supersaturation S = In pip", as a function of droplet radius r for a two-component droplet. The point A marks the undersaturation regime from which droplet nuclei can grow to the stability point marked B.
to maintain an invariant Mie scattering pattern for as long as 30 min, which certainly suggests such stability.
3. Range of Stability
and
For the special cases of ideal or dilute solution, a1 in eq 2.4 may be set equal to XI = nl/(nl n2) and then for fixed 112
+
If the drop consists of a single component (only solvent) so that X I = 1 , then both eqs 2.22 and 2.25 yield
($), 2 n l 2
=-
x1 =
1 - 3n2v2/4nr3 1
+ 3n2(v1- v2)/4nr3
Thus, with al = XI and expressed as
0
(2.28)
T,p
which confirms that G is at a minimum and that the equilibrium was stable. In their experiments, LaMer and Gruen were able
Thus, for fixed n2, ln(p/po,) may be plotted versus r, in the manner of Kohler to obtain a curve of the form exhibited in Figure 1. Kohler (see also refs 17-20) identified the region of stable equilibrium as that given by the section of the curve in Figure 1 having a positive slope, i.e., the section to the left of the maximum indicated by B. In this region, addition of solvent molecules to the drop, so as to increase r, increases its vapor pressure so that the added molecules will evaporate and retum the drop to its original size. On the other hand, if molecules initially escape from the drop, its radius will be decreased and its vapor pressure reduced. Thus, molecules will condense on the drop and retum it to its original size. Thus, fluctuations will always regress so that the equilibrium will be stable. A similar argument (in which fluctuations do not regress) indicates that the equilibrium is unstable on the branch of the curve where the slope is negative, Le., to the right of the maximum marked with a B. At the maximum, which (significantly) lies in the region where the vapor is supersaturated @/PO- > 1),the system makes a transition from stable to unstable equilibrium. The value of r corresponding to the maximum can be determined by setting the derivative, with respect to r, of ln(plpo,) in eq 3.2 equal to zero, and the resulting equation may be shown to be identical with that obtained by equating the right side of eq 2.25 to zero. Consider a monodisperse assembly of drops initially in the stable region to the left of the maximum in Figure 1. If the supersaturation is increased, by any one of several methods, the drops may be brought to the maximum. If the increase in supersaturation was accomplished by an isothermal increase of
. I Phys. . Chem., Vol. 99, No. 19, 1995 7841
The Kelvin Relation
0'01
1
4. Stability Limits in Undersaturated Vapors The focus on ideal or dilute solutions, and on curves such as those in Figure 1, seems to have generated the idea that the binary drop-vapor equilibrium can never be unstable when p/p"- < 1. In fact, the study of stability by means of the thermodynamic potential shows that there is no purely thennodynamic reason why an unstable equilibrium should not be possible in an undersaturated vapor. However, the conditions under which such an unstable equilibrium would occur might be very difficult to achieve in a real system. To understand this better, consider eq 2.4 specialized to the case of an ideal solution. We obtain
P = xlPo,e Figure 2. Supersaturation S = In plp", as a function of droplet radius r for a two-component droplet for n2 = 2 x lo6, 5 x lo6, and 10'.
the pressure of the vapor, the drops would reach the maximum by growing. On the other hand, the most convenient means for bringing them to the maximum would involve lowering the temperature (and therefore p",) through an adiabatic expansion. The drops could play the role of preexisting heterogeneous nuclei, since, once in the unstable region, they could g o w . At the same time, if they remained on the unstable branch of the curve in Figure 1, they could still decrease in size so as to return to the maximum. The exact behavior would depend on the rate of cooling. For large enough values of n2, the level of supersaturation at the maximum, Le., the value of plp",, could be quite low. Figure 2 shows several plots of ln@/po,) for different values of 112. These plots show that the supersaturationat the maximum could be brought close to unity. Such low supersaturations could be achieved by a slow very controllable expansion, starting at the left position, marked A in Figure 1, Le., starting with easily obtainable undersaturated vapors. If the slow expansion were coupled with light scattering studies, then one would expect to see dramatic changes in the pattem of scattered light when the maximum was reached and large fluctuations became possible. It might then be possible to locate the maximum and to perform another experimental check on the validity of the Kelvin relation. We address the question of fluctuations in a later section. Since nucleation was mentioned above, it is worth indicating that an aspect of the stability phenomenon that forms the subject of this paper is, to a degree, contained in the phenomenon and theory of nucleation in binary systems.22 In supersaturated binary vapors, the free energy surface, plotted as a function of the numbers of molecules of its component species, and over which a growing cluster moves, usually contains a valley leading to a saddle point which marks the location of the nucleus. If one moves in the direction across the valley or the saddle, the free energy will pass through a minimum. If one of the components is almost involatile, e.g., sulfuric acid in a drop of aqueous sulfuric a ~ i d ,the ~ ~ cluster, , ~ ~modeled as a drop, will achieve quasistable equilibrium at this minimum with respect to the gain or loss of water molecules between the times of acquisition of sulfuric acid molecules. The minimum in question is exactly the minimum to which eq 2.4 applies, given that the second derivative in eq 2.22 in positive. Even if one of the components is not involatile, the existences of both the valley and the saddle point are due to phenomena very closely associated with the subject matter of this paper. We discuss the connection to nucleation theory more thoroughly in section 5.
2ovllrkT
(4.1)
Suppose the system is within the regime of stable equilibrium, and a fluctuation occurs such that a few solvent molecules evaporate. Then r will be reduced, but XI will also be reduced. The first effect tends to increase p, while the second will decrease it, and the ultimate direction of the change in p will depend on which effect dominates. The outcome is almost apparent. The Kelvin effect, Le., the first effect, is so small, for any reasonable values of (T and V I ,that it will surely lose the competition and p will be decreased. But then, if the ambient vapor pressure is fixed, the evaporated molecules will recondense so that the drop will be restored to its original condition. A similar argument can be used in dealing with a fluctuation involving the condensation of a few molecules. The vapor pressure of the drop will be increased, and the drop will retum to its original condition. Thus the equilibrium is stable. An examination of the argument reveals that the stability is a result of the smallness of the Kelvin effect and has no fundamental thermodynamic basis! Thus, discounting the Kelvin effect, it may still be possible for the system to become unstable at supersaturations below unity for other reasons. One such reason could involve nonideal behavior of the solution. Thus, we are motivated to examine eq 2.4 and 2.22 without approximating a1 by XI. The simplest model of al for a real solution is that derived from the mean field Bragg-Williams lattice approximation (see pp 345-348 of ref 25). In this approximation al may be expressed as
cw(1 - XI) "P{
2kT
'}
where
w = 2€*2- E l l - E2*
(4.3)
in which €11 and €22 are nearest-neighbor "bond" energies between pairs of solvent and solute molecules, respectively, while 612 is the "bond" energy of a nearest-neighbor solutesolvent pair. The quantity c is the coordination number of the lattice, i.e., the number of nearest neighbors possessed by a lattice site. Substitution of eq 4.2 into eq 2.4 gives l n P = Pom
cw(1 - XJ2 2kT
2-1 + In x1 + rkT
(4.4)
Now ignore the last term, on the right of eq 4.4, that represents the small Kelvin effect, and consider only the first two terms on the right. The second term is responsible for the negative slope of the curve in Figure 1, in the case w = 0, where the solution is ideal. As r decreases at fixed n2 so must XI and
Reiss and Koper
7842 J. Phys. Chem., Vol. 99, No. 19, 1995 g1 1.0 1
m
c
i
3
I Figure 3. Supersaturation S = In p/p"- as a function of droplet radius r for a two-componentdroplet of nonideal constituents. The nonideality is represented by the coefficient cwl2kT (see eq 4.4) having values 4, 4.5, and 5, the lowest curve referring to the lowest value for the coefficient. The C marks the instability region in the curve due to
nonideality. therefore In X I . But now consider the situation when w is large and positive. For values of r in the region where PIP"- < 1, XI certainly decreases as r decreases. However, because w is large and positive, the first term on the right of eq 4.4 will increase as r and XI decrease. If w is large enough, it will outweigh the decrease of In XI. At this point ln(p/p"-) will rise as r decreases so that a curve like that in Figure 1 will exhibit a negative slope even though PIP"- < 1 . A negative slope signifies an unstable equilibrium, so that in principle it should be possible to have an unstable regime under conditions such that the supersaturation p/p"- is less than unity, Le., when the system is undersaturated! Substitution of eq 3.1 into eq 4.4 yields
1 - 3n2v,/4xr3 1n[ 1
+ 3n2(v, - v2)/4nr
3]
+2
(4.5)
Figure 3 exhibits plots of In plp", versus r, derived from this equation for v2 = V I and several values of w. The occurrence of a negative slope with pip", < 1 is evident. The physical origin of a large positive w is the following. Suppose, € 1 2 , €11, and € 2 2 are all negative. Then, for w to be positive, eq 4.3 shows that (€11 c22)/2 must be more negative than €12; Le., solvent must like solvent and solute like solute better than they like each other. Such behavior will eventually lead to a miscibility gap and phase separation. Such critical behavior can be analyzed within the crude model, even in connection with eq 4.5, but we do not attempt this here. It is also possible for one of the separating phases to be a solid, e.g. sodium chloride precipitating from an aqueous solution. Indeed, Tang and Munkelwitz3' have, in essence, observed this phenomenon in levitated drops of aqueous ammonium sulfate. These authors observed the deliquescence of crystals of ammonium sulfate; i.e., they observed the opposite of eflorescence. Efflorescence is the phenomenon that corresponds to the process of interest to us, but these authors could not observe it directly due to the formation of supersaturated solutions of ammonium sulfate. Nevertheless, deliquescence implies efflorescence via the principle of reversible equilibrium. Again, we do not expand the analysis in the present paper. The existence of negative slopes at both small and large values of r gives rise to a stability regime bounded at both ends.
+
-1.0
I
Figure 4. Typical graph of g, (cf. eq 5.2) versus nl for some values of n2. From right to left the value of n2 increases.
At each end we may expect large fluctuations, and we develop the theory for these in the next section. 5. Fluctuations
In the present section we will simplify matters by taking
VI
= v2 = v, and because of incompressibility this volume will be independent of n1
+ n2 = n.
Thus, we can write
where the first factor on the right is constant. We shall continue to be interested in a system consisting of a vapor (ideal) within which there is a single drop at one particular position within the vapor. Since the pressure of the system is maintained constant, the system corresponds in statistical mechanics to the constant pressure ensemble (see pp 102-104 of ref 25), and we can ask the question, what is the chance that the drop will be observed to consist of nl n2 molecules when n2 is fixed, Le., what is the chance that the drop will contain nl solvent molecules? In order to answer this question, we shall first determine the range of values of n2 for which stable droplets exist. To this purpose, eq 2.15 may be written, using eq 2.18 and ignoring p0l(T,p)- po~(T,pom), as
+
where we have also set ai = XI. Also, assuming as we have been doing that o is constant, eq 2.25 becomes
---
A typical graph of gl as a function of nl for some values of n2 is given in Figure 4 . For n 00 we have gl = -In PIP"-, and on the other hand for ni 0 we have gl -00. Because gI has a negative slope for nl there must be a maximum nl = nl, for each value of n2. We shall assume that the conditions are such that at least for some values of n2 and PIP"- there are zeros for gl or, in other words, that gl(nl,) > 0. Then, because gl(nl,) is a decreasing function of n2, there is a value n*2 such that for n2 > n*2 there are no zeros for gl or, in other words, that gl(n1,) -= 0. First consider the case n2 < n*2, where there are two zeros, 1110 < rill. Careful analysis of g2 (eq 5.3) shows that the zero nl I refers to unstable equilibrium whereas the zero nlo refers to stable equilibrium. We now consider fluctuations around this value nlo. In the constant-pressure ensemble, the 00,
-
The Kelvin Relation
J. Phys. Chem., Vol. 99, No. 19, 1995 7843
chance that the drop will contain exactly nl molecules is given by
(5.4)
We note that the sum in the denominator in eq 5.4 has a maximum term that corresponds to the free energy to which eq 2.4 corresponds. The value of nl at this maximum is nlo, and we multiply numerator and denominator in eq 5.4 by exp(G(nl0)l kT). Then we get
For simplicity, we will deal with a drop that behaves as an ideal solution so that al in eq 2.4 may be replaced by XI as in eq 2.24. Then, as long as we are reasonably far to the left of point B in Figure 1, Le., in the region of thermodynamic stability, we expect the sum in the denominator of eq 5.2 to be dominated by the terms in the neighborhood of nl’ = nlo. For this reason we may expand the exponents in both numerator and denominator in powers of nl - nlo and retain terms out to the quadratic. Because eq 2.24 is based on the vanishing of the first derivative, Le., gl = 0, at B the quadratic term becomes the largest nonvanishing term in the expansion of G(nl) - G(nl0). With the use of eqs 5.2, 5.3 and 5.1 the above equation becomes
P(n1) =
Figure 5. Typical plot of the Gibbs free energy G(nl,nz) versus
nl
and n2. At the smaller values of n2 there is a “valley” whose floor slopes upward toward a saddle point. Beyond the saddle point, the valley floor slopes downward until the valley disappears, and the surface only slopes (steeply) downward toward increasing values of nl. Not shown, as one continues to larger values of nl, is a region (depending on conditions) where there is no valley, and the surface simply slopes downward toward increasing values of nl.
and h e n 8 experiment (see Table 1) and calculated the relative fluctuations to be extremely small, A(n1 - nlo) = 4 x low4. For 122 approaching n*2 the second derivative g2 of the Gibbs free energy (cf. eq 5.3) vanishes, and the fluctuations increase dramatically, diverging for n2 = n*2. At that point, the two zeros nlo and rill for stable and unstable equilibrium merge, and there is no equilibrium, neither stable nor unstable, for n2
=- n*2.
=K eXP[
-[
+
3kT(n10 n2)’I3n2- 2 0 ~ ( 4 d 3 v ) ” ~ n , ~
6kTn,o(nIo
+ nJ4I3 (5.6)
Because of the rapid decay of P(nl) as nl departs from nlo, we may integrate P(nl) dnl between the limits nl - nlo = -00 and in order to evaluate K by setting the integral equal to unity. The result is
+-
Using this value of K in eq 5.6, we can evaluate the relative fluctuation
we find, since (nl - n ~ o = ) 0,
- nlo> =
As a typical example, we have taken the data from the LaMer
The foregoing discussion is of great value in advancing a continuity of ideas between the field of nucleation in binary vapors (where drops or clusters are usually of mesoscopic dimensions (nanometers) and the study of the Kelvin relation in the micrometer range. Figure 5 exhibits a typical plot of G(n1,n2) based on the analysis just presented. At the smaller values of 112 we see a “valley” whose floor slopes upward toward a saddle point. Beyond the saddle point, the valley floor slopes downward until the valley disappears, and the surface only slopes downward toward increasing values of nl. This marks the point where the curves of Figure 2 no longer intersect the gl = 0 axis. Now, if the solute is completely involatile, the system cannot, without deliberate additions of component 2 to the drop, pass from one cross-sectional profile (at constant n2) in Figure 5 to another. Suppose then that the system is limited to a profile at a low value of n2 and finds itself in the valley on that profile. Because of fluctuations (that we have calculated to be very small), there is an infinitesimal but nonzero probability that the drop could pass over the hill to the right of the valley by adding molecules of component 1. Once over the hill, the drop will grow. This is actually a nucleation process, but one that involves only component 1. If the involatile component is not entirely involatile and the vapor contains it at small partial pressures, then fluctuations can occur which deliver the drop to the saddle point by the addition of molecules of component 2. Because of the small fraction of component 2 in the vapor, the drop will then continue to grow slowly (slow because of the slow addition of molecules of component 2). However, once it grows to the point where the valley has disappeared, it can grow rapidly through the addition of component 1, a process that carries it downhill on
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7844 J. Phys. Chem., Vol. 99, No. 19, 1995
the surface. The process involving component 2 is also a nucleation process. Indeed, the conventional wisdom has been that binary nucleation proceeds by means of a flux through the saddle. In this case, however, once the drop has passed the saddle, it may still grow slowly until it reaches the point where it can grow by the addition of component 1 only. Thus, this latter point is a second critical growth point beyond the “thermodynamic” one represented by the saddle point. It is in fact a “kinetic” critical point. Retuming to the first profile, if the supersaturation of component 1 is high enough, the barrier represented by the hill on the profile will be reduced to the point where nucleation along the profile (Le., at constant n2) will be rapid and will dominate the flux passing through the saddle. Thus, contrary to the conventional wisdom, the nucleation process in the binary system will not involve the saddle. Recently, Vehkamiiki et aLZ6and McGraw2’ have performed extensive numerical analyses on nucleation in HzS04-H20 vapor mixtures where HzS04 plays the role of the involatile component. Both of these studies indicate that the saddle point can be bypassed at high enough H20 supersaturations. 6. Discussion In conclusion, we have developed the appropriate thermodynamic potential for the treatment of equilibrium between a drop of binary solution, containing an involatile solute, and the vapor of its volatile component. This potential was used to reproduce earlier results obtained in another way by Kohler’7-’9 and to show that there was a regime in which the equilibrium, in question, was thermodynamically stable. However, we have also shown that for a nonideal solution there could be a regime, even when the vapor is undersaturated, within which the equilibrium is unstable. Furthermore, with the help of the thermodynamic potential, we have been able to discuss fluctuations in size of the drop within the stable regime and also at the point where transition to an unstable regime occurs. Light scattering studies in the neighborhood of this point might provide another check on the validity of the Kelvin relation. The transition at large drop sizes always occurs when the vapor is supersaturated, but the supersaturation at that point is also small. Thus, it could be produced in a controlled manner by a slow adiabatic expansion. However, this approach would involve the continuous change of temperature as well as pressure. The variation of temperature should lead to no additional experimental difficulty, but it would have to be taken account of. Since fluctuations become pronounced at the point of transition from stability to unstability, one would expect
dramatic effects in the light scattering. However, since the supersaturation is small, growth of the drops could be slow enough to inhibit the realization of these effects. The theory for the assessment of fluctuations (this paper) and growth28is available so that the ease of observation of the effects should be calculable. We leave this issue to a later paper. Finally, we leave this discussion by pointing out the close relation of this analysis to the theory of droplike microemulsions29.30and the many features of phase stability that must be considered in that field.
Acknowledgment. We thank D. Bedeaux and J. Groenewold for fruitful discussions. This work was supported by Brookhaven National Laboratory through Contract CW28/4-443835-HR5743 1. References and Notes (1) Thomson, W. Proc. R. Soc. Edinburgh 1870, 7, 63. (2) Thomson, W. Philos. Mag. 1871, 42, 448. (3) Thoma, M. Z . Phys. 1930, 62, 224. (4) Cohen, L. H.; Meyer, G. E. J . Am. Chem. SOC. 1940, 62, 2715. (5) Monchick, L.; Reiss, H. J . Chem. Phys. 1954, 22, 831. (6) Sambles, J. R.; Skinner, L. M.; Lisgarten, N. D. Proc. R. Soc. London A 1970, 318, 507. (7) Sambles, J. R. Proc. R. SOC. London A 1971, 324, 339. (8) LaMer, V. K.; Gruen, R. Trans. Faraday SOC. 1952, 48, 410. (9) Fisher, L. R.; Israelachvilli, J. N. J . Colloid Interface Sci. 1981, 80, 528. (10) Strey, R.; Wagner, P. E. J . Phys. Chem. 1994, 98, 7748. (1 1) Boiko, B. T.; Pugachev, A. T.; Bratsykhin, V. M. Sov. Phys.-Solid State (Engl. Transl.) 1969, 10, 2832. (12) Reiss, H.; Wilson, I. B. J . Colloid Sci. 1948, 3, 551. (13) Tolman, R. C. J . Chem. Phys. 1949, 17, 333. (14) Kirkwood, J. G.; Buff, F. P. J . Chem. Phys. 1949, 17, 338. (15) Blokhuis, E. M.; Bedeaux, D. J . Chem. Phys. 1991, 95, 6986. (16) Blokhuis, E. M.; Bedeaux, D. Physica A 1992, 284, 42. (17) Kohler, H. Geofys. Pub. Krestiania 1921, I , 1. (18) Kohler, H. Geofys. Pub. Krestiania 1922, 2, 6. (19) Kohler, H. Nova Acta Regiae SOC.Sci. Ups. 1950, 14, 9. (20) Defay, R.; Prigogine, I.; Bellemans, A. Surjace Tension and Adsorption (translated by D. H. Everett); Longmans, Green & Co.: London, 1966. (21) Reiss, H. Methods of Thermodynamics; Blaisdell: New York, 1965. (22) Reiss, H. J. Chem. Phys. 1950, 18, 840. (23) Heist, R. H.; Reiss, H. J . Chem. Phys. 1974, 61, 573. (24) Shugard, W. J.; Heist, R. H.; Reiss, H. J . Chem. Phys. 1974, 61, 5298. (25) Hill, T. L. Statistical Mechanics; Dover: New York, 1987. (26) Vehkamiki, H.; Paatero, 0.;Kulmala, M.; Laaksonen, A. Private communication (to be published), 1994. (27) McGraw, R. Private communication (to be published), 1994. (28) Friedlander, S. K. Smoke, Dust, and Haze; Wiley: New York, 1977. (29) Borkovec, M. J . Chem. Phys. 1989, 91, 6268. (30) Borkovec, M. Adv. Colloid Interface Sci. 1992, 37, 195. (31) Tang, I. N.; Munkelwitz, H. R. J . Colloid Interface Sci. 1984, 98, 430. JP943015Q
