- Email: [email protected]
Phase equilibrium of a liquid droplet formed on a solid particle
Chemical Engineering Science 61 (2006) 5925 – 5933 www.elsevier.com/locate/ces
Phase equilibrium of a liquid droplet formed on a solid particle J. Mitrovic ∗ Institut für Energie- und Verfahrenstechnik, Thermische Verfahrenstechnik und Anlagentechnik, Universität Paderborn, 33095 Paderborn, Germany Received 2 January 2006; received in revised form 4 May 2006; accepted 9 May 2006 Available online 19 May 2006
Abstract An equation of state of a two-phase system, deduced from the principles of classical thermodynamics, is used to illustrate the effects of interface curvature, electric charge, a soluble substance, and of disjoining film pressure on the equilibrium of a liquid droplet formed on a solid particle. The former three effects have been widely discussed in the past, whereas the effect of the disjoining film pressure has not been explored. The focus of the paper is, therefore, placed on the effect of the disjoining film pressure on the droplet equilibrium. Opposing the action of surface tension, the disjoining pressure is shown to lower the equilibrium vapour pressure at constant temperature and to increase the equilibrium temperature at constant pressure in comparison to the equilibrium in the reference state. This interplay provides a way bringing experimental findings in a better agreement with theoretical models. 䉷 2006 Elsevier Ltd. All rights reserved. Keywords: Drop; Aerosol; Phase equilibrium; Phase change; Coated solid particle; Disjoining pressure
1. Introduction When a superheated vapour is cooled down to its saturation temperature, condensation will immediately start if liquid phase is present or nucleation of the new phase can take place without shifting the vapour state towards the metastable state. In the absence of foreign action upon the vapour, which may arise e.g. from container walls or particles suspended in the vapour, usually phase transition does not occur at the nominal saturation temperature but at some subcooling. Nucleation expected under such condition is termed homogeneous nucleation. As can be taken from numerous sources, e.g. McDonald (1962) and Abraham (1974), one relationship of the classical homogeneous nucleation theory is G = 4r 2 − 43 L (V − L )r 3
(1)
with V − L = RT ln
pV , pV ∞
(2)
where G is the difference of the Gibbs free energy of the actual system, consisting of vapour and a droplet of the radius r at the ∗ Tel.: +49 5251 60 2409; fax: +49 5251 60 3207.
E-mail address: [email protected]. 0009-2509/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2006.05.013
pressure pV , and the reference system, consisting of vapour at the saturation pressure pV ∞ in equilibrium with the liquid of plane interface, both systems being at the same temperature T ; is the chemical potential, the indices V and L referring to vapour and liquid, respectively. Since pV > pV ∞ , that is, V > L , the energy difference G has a maximum Gmax =
16 3 3 (L RT ln(pV /pV ∞ ))2
(3)
at r = r∗ , r∗ =
2 . L RT ln(pV /pV ∞ )
(4)
The radius r∗ is usually called the critical radius of the droplet. With this radius, Eq. (1) becomes 2 1 L RT r ∗ r r pV G 1− . (5) = 4 ln r∗2 r∗ 3 r∗ p V ∞ As follows from Eqs. (4) and (5), a liquid droplet cannot be formed in a saturated vapour at pV = pV ∞ . For pV = pV ∞ , the radius r∗ according to Eq. (4) would become infinite, which means a liquid layer (or semi-infinite liquid space) of infinite surface area would form, and the energy difference G
5626
J. Mitrovic / Chemical Engineering Science 61 (2006) 5925 – 5933
Fig. 2. Illustration of the effects of curvature, dissolved substance, electric charge, and disjoining film pressure on droplet equilibrium.
Fig. 1. Dimensionless Gibbs free energy difference G of a droplet relative to the reference state at L RT r∗ / = 500.
according to Eq. (5) would become unlimited. Obviously, the saturated vapour must be shifted away from the saturation state and the new phase must have a finite surface area. In case of an isothermal shifting, a pressure increase in the vapour would have to take place resulting in a corresponding increase of the free energy of the vapour, which becomes supersaturated. Although this vapour would be metastable, its state would not change jumpwise to stable, lower-energy states, because a jump can only proceed through formation of tiny drops by clustering of vapour molecules. The increase in the free energy G of the system, required for formation of a droplet having the radius r according to Eq. (5) at L RT r ∗ /=500, is illustrated in Fig. 1. As is obvious from the figure, the dimensionless energy difference G/r∗2 decreases as pV /pV ∞ increases. At a fixed pV /pV ∞ , the ratio G/r∗2 passes a maximum determined by Eq. (3). Suppose now a droplet of a radius r in supersaturated homogeneous vapour large enough to apply the relationships of the classical thermodynamics. Since the process of phase transition under homogeneous conditions is a spontaneous one, the fate of the droplet just formed is determined by the change of free energy of the system caused by a fluctuation of the droplet size. If by such a fluctuation the free energy decreases, the system state moves towards a more stable state. This means that, in a spontaneous process, a droplet of r < r∗ would evaporate because its growth would be accompanied by an increase in the free energy which would be in contradiction to the spontaneous behaviour of the system. On the contrary, a droplet of r > r∗ would spontaneously grow. However, in order for the droplet to reach such a size, the energy barrier Gmax in Eq. (3) must be surmounted. The classical capillarity theory of homogeneous nucleation does not seem to convincingly explain the route on which the energy hill could be passed.
The validity of the above equations is restricted to pure vapours that are protected from external influences. If the vapour contains foreign substances, such as solid particles or ions, the thermodynamic conditions of equilibrium alter, and condensation is expected to set in at a lower supersaturation in comparison with pure vapours. The nucleation under such conditions is called heterogeneous. Both homogeneous and heterogeneous nucleation has been the subject matter of numerous papers, see e.g. Laaksonen et al. (1995) for further references. The aim of the present paper is to describe the equilibrium of a droplet which contains a solid particle. The substance making the solid particle is taken to be non-volatile; it may be soluble or insoluble in the liquid forming the (droplet) film. The particle is completely covered by a liquid film that is surrounded by and in equilibrium with its own vapour. The phase equilibrium is assumed to be simultaneously affected by • • • •
the interface curvature, the dissolved substance, an electric charge and the thickness of condensate film covering the solid particle.
These effects are illustrated in Fig. 2 and are measured in terms of supersaturation of vapour with respect to the equilibrium of the reference system consisting of pure substance (e.g. water) with a plane interface. The effect of the interface curvature is described by the well-known Kelvin equation (Thomson, 1871), while the effect of a dissolved substance in conjunction with the interface curvature has been considered, apparently for the first time, by Köhler (1926, 1950). The investigation of the effect of the electric charge seems to go back to Thomson (1903), whereas the last one—the effect of the film thickness resulting in a disjoining film pressure when condensation is taking place on a solid particle in conjunction with the solution and/or curvature effect on the droplet equilibrium—has not been reported in the literature. The present paper aims at closing the gap. First, a simple equation for the equilibrium of a droplet surrounded by the vapour phase is deduced from the principles
J. Mitrovic / Chemical Engineering Science 61 (2006) 5925 – 5933
5627
of thermodynamics. The derivation path describes a reversible evolution of a system by changing its coordinates. A pressure jump is thereby assumed to occur across the phase interface. Specifying this pressure jump, one describes the corresponding effect on the droplet equilibrium. From this point of view, the paper summarizes the basic models and gives to some extent a brief review of the knowledge in this area. 2. Equilibrium of a spherical droplet To derive a general equation for phase equilibrium, we start the considerations by writing down an expression for the fugacity f of a homogeneous phase in intrinsic equilibrium.1 This quantity is given by (see e.g. Van Ness and Abbott, 1982) d ln f = −
h − h v dT + dp, 2 RT RT
where h and h are the specific enthalpies of the phase in the actual and in an ideal state, respectively; the symbols R, v, p, and T denote the universal gas constant, the partial molar volume, the pressure, and the temperature, respectively. Eq. (6) is also valid for a two-phase system. Indicating its phases by the indices V and L, one obtains d ln
fL hV − h vV hL − h = dT − dp − dTL V V fV RTV RTV2 RTL2 vL + dpL . RTL
(7)
A system in thermodynamic equilibrium requires fL = fV and TL = TV , the former equality expressing the chemical, the latter one the thermal equilibrium. The pressures pL and pV , governing the mechanical equilibrium, need not necessarily be the same, and a pressure jump is assumed at the phase interface, Fig. 3. Denoting by p this pressure jump, p = pV − pL ,
Fig. 4. Evolution of a system.
(6)
(8)
where p > 0, or p < 0, Eq. (7) becomes vV hV − h L vL vL dpV = dTV + dpV − d(p). 2 RTV RT RT RTV V V
Describing the equilibrium of a two-phase system that is inhomogeneous with respect to the pressures in the phases, this expression represents an augmented Clapeyron–Clausius equation (Clapeyron, 1834; Clausius, 1850). As it shows, the vapour pressure pV depends on the temperature TV and the pressure difference p. Physical systems that follow Eq. (9) do not satisfy the classical (Gibbs) phase rule. The pressure difference p defined in Eq. (8) is a property of the system that—in single component systems—must not vary along the interface. If for instance one of the phases is disperse, e.g. liquid droplets in vapour, or vapour bubbles in liquid, the pressure jump p must be the same everywhere at the interface. For different values of p, the system would not be in mechanical equilibrium thus violating the thermodynamic equilibrium requirement. Consequently, Eq. (9) would be invalid. To perform integration in Eq. (9), the vapour phase is taken to be ideal, RTV /vV = pV . For vL /TV = const(=vL∞ /TV ∞ ) and hV − hL = h = const, an integration from the initial state (pV ∞ , TV ∞ , (p)∞ ) to the final state (pV , TV , p) gives2 ln
Fig. 3. Illustration of pressure p jump across the interface.
1 The considerations could also have been started from the Gibbs –Duhem–Helmholtz expression d(G/RT ) = −(h/RT 2 ) dT + (v/RT ) dp.
(9)
pV h TV − TV ∞ vL∞ = + (pV − pV ∞ ) pV ∞ RTV TV ∞ RTV ∞ vL∞ − (p − (p)∞ ). RTV ∞
(10)
Physically, the integration of Eq. (9) leading to Eq. (10) describes an evolution of the system. Such an evolution is illustrated in Fig. 4, where, for instance, the interface may be initially plane (a), whereas in the final state it is convex (b) or concave (c). 2 Note that the assumption v /T = const adopted for integration purL V poses ignores the anomaly of water.
5628
J. Mitrovic / Chemical Engineering Science 61 (2006) 5925 – 5933
For p = (p)∞ , Eq. (10) reduces to the relationship wellknown from literature. In general, however, the pressure difference p contains specific information about the system which is not included into the classical vapour pressure expression. Specifying the quantity p, as illustrated below, one formulates particular systems. 3. Particular cases 3.1. Effect of interface curvature Taking the pressure difference p to arise from the curvature of the interface and an action of the external potential on the phases, , we may write p = pV − pL = + ,
(11)
where denotes the surface tension; the curvature is positive ( > 0) when its centre lies on the side of the V -phase, Fig. 3. The physical significance of the quantity has been examined in an earlier paper by the same authour (Mitrovic, 2004). For (p)∞ = 0, Eqs. (10) and (11) give ln
pV h TV − TV ∞ vL∞ = + (pV − pV ∞ ) pV ∞ RTV TV ∞ RTV ∞ vL∞ ( + ). − RTV ∞
3.3. Effect of an electric charge (12)
In the considerations to follow, the quantity will be specified for cases illustrated in Fig. 2 and the effects of a dissolved substance, an electric charge, and the film thickness on the equilibrium conditions described using Eq. (12). 3.2. Effect of a dissolved substance Assuming the liquid phase to contain a dissolved substance, thus representing a solution, the dissolved substance causes an osmotic pressure which may be thought to arise from an external potential . Taking the solution at the temperature TV ∞ to be ideal, one may write ≡ = −
RTV ∞ ln(1 − x), vL∞
(13)
where x denotes the mole fraction of the solute. For a droplet of the radius r, the curvature is obtained from = dA/dV = 2/r, and Eq. (12), together with Eq. (13), gives ( < 0) ln
pV h TV − TV ∞ vL∞ = + (pV − pV ∞ ) pV ∞ RTV TV ∞ RTV ∞ vL∞ +2 + ln(1 − x). RTV ∞ r
As is obvious from this equation, a droplet in equilibrium with pure vapour at pV ∞ and TV ∞ requires a larger mole fraction x at a smaller droplet radius r. This interplay is rooted in counteracting effects of the curvature (surface tension) and the osmotic pressure on the equilibrium of the solution droplet, and has been examined by Köhler (1926, 1950), Byers (1965), Dufour and Defay (1963), Reiss and Koper (1995), Talanquer and Oxtoby (2003), and many others. Eq. (14) assumes a spatially homogeneous concentration of the solute in the droplet. If the solute is surface active, its higher concentration in the surface region affects the surface tension and the osmotic pressure. The droplet equilibrium will consequently depend on the surface activity of the solute. Sorjamaa et al. (2004) seem to be the first authors to examine the effect of the surface activity of the solute on the vapour pressure lowering (Raoult’s law) of solutions. Their paper gives all the important details concerning the equilibrium of a solution droplet basing on the Köhler (1926, 1950) theory. A further effect of the surface activity—not mentioned earlier—is the osmotic pressure stress which could, in case of spherical asymmetry, induce an osmotic flow thus affecting droplet formation.
(14)
Setting in Eq. (14) pV =pV ∞ and TV =TV ∞ , thus requiring the equilibrium conditions of the solution droplet to be the same as for the pure solvent at r → ∞, gives vL∞ x = 1 − exp −2 . (15) RTV ∞ r
If the droplet is not electrically neutral but carries an electric charge q, and if it is assumed to be a conducting sphere, the pressure pq acting on the droplet surface arising from this charge has been obtained by Thomson (1903) to be 1 q2 . (16) 8 r 4 This equation does not account for the sign of the electric charge. Furthermore, the assumption of the droplet to be a conductor has been discussed by Tohmfor and Volmer (1938). Considering the droplet as spherical condenser, they obtained the following expression: 1 1 q2 pq = , (17) 1− 8 r4 pq = pL − pV =
which, like Eq. (16), is invariant with respect to the charge polarization. For a sufficiently large dielectric constant , the two equations become identical. For further references on this effect, see e.g. Laaksonen et al. (1995). The effect of the electric charge on the droplet equilibrium follows immediately from Eq. (12) by replacing by the pressure pq according to Eq. (16) or (17). Eqs. (12) and (16) show, therefore, that any variation of the electric charge q is compensated by a corresponding change at least of one of the following system parameters: the temperature TV , the vapour pressure pV , and the droplet radius r. Taking TV and pV as constant, Eq. (12) with ≡ pq gives ( = −2/r) 1 1 q2 = C, (18) 1− −2 + r 8 r4 which means that—at droplet equilibrium—any variation in the electrostatic pressure is balanced by an opposite action of the capillary pressure /r.
J. Mitrovic / Chemical Engineering Science 61 (2006) 5925 – 5933
For pV = pV ∞ and TV = TV ∞ in Eq. (12), the constant C in Eq. (18) becomes zero. In this case, the droplet in equilibrium with the vapour phase is also in equilibrium with the liquid of plane interface, and the droplet equilibrium radius req follows from Eq. (18): 1/3 1 q2 1 1− . (19) req = 16 Deriving Eq. (18) with respect to r gives the boundary (r = rmax ) between the stable and unstable equilibrium states, 1/3 1 1 q2 . (20) 1− rmax = 4 A droplet of r > rmax is unstable, whereas one of r < rmax is stable. 3.4. Effect of film thickness Since the studies on dust, fogs, and clouds by Aitken (1880), the effect of solid particles on vapour condensation plays an important role in modelling of clouds formation in the atmosphere (see e.g. Laaksonen et al., 1995; Latham, 1969; Best, 1955; Fletcher, 1962, 1970; Wagner and Vali, 1988; Mason, 1971; Wright, 1936; Amelin, 1967). Fletcher (1958) has shown that insoluble spherical particles lower considerably the energy barrier of droplet formation. This energy barrier depends on the contact angle and the size of the particle in comparison with the critical droplet radius deduced from the classical nucleation theory. However, at zero contact angle, that is, at complete wetting of the particle surface, the critical radius of the vapour–liquid interface becomes smaller than the particle radius. When the solid particle is soluble, as is mostly the case in the atmosphere (Martin, 2000), formation of liquid phase on the particle surface is preceded by adsorption of solvent molecules on the particle surface. In this case, it is not clear what a shape of the liquid–vapour interface may be expected to form. Assuming the dissolution of the solid particle to occur preferably in the region of the three-phase line, formed by intersection of the system interfaces, a continuous liquid film is likely to cover the whole particle surface. On the contrary, in the event that the dissolution mainly occurs across the solid–liquid interface, a liquid cluster could basically penetrate the solid particle. The latter idea seems energetically less probable than the former one. For a general discussion of dissolution, the deliquescence of solid particle, the reader is referred to literature, e.g. Mirabel et al. (2000) and Russell and Ming (2002). Hanel (1976) has investigated condensation on solid particles consisting of soluble and insoluble parts. The model he adopted for equilibrium treatment—an insoluble particle covered by a solution film—resulted in a considerable effect of the solid particle on the equilibrium vapour pressure. However, this model has been considered by Gorbunov and Hamiltion (1997) to be less realistic because incomplete wetting of insoluble aerosol particles was mostly expected in reality. The latter authors adopted, in principle, the Fletcher (1958) model and replaced the pure liquid by a liquid solution. The expression
5629
for the vapour pressure thus obtained reduces in case of perfect wetting basically to the Köhler equation (Köhler, 1926, 1950), and the solid particle affects the phase equilibrium only by its volume. None of the models mentioned so far takes into account the effect arising from interactions substrate–film when a thin liquid film covers the particle surface. As is well-known, a liquid film formed on a solid substrate is subject to an action of the substrate which may result in a significant displacement of phase equilibrium (Derjaguin et al., 1987). To describe the effect of a solid particle on the droplet equilibrium, a spherical particle covered by a liquid film at perfect wetting will be assumed. In case of a thin film, the action of particle on the vapour–liquid interface can be accounted for in terms of disjoining film pressure p : p =
A , m
(21)
where the quantities A and m depend on the physical properties of the liquid and solid phase and, additionally, on the film thickness . Eq. (21) has been proposed by Derjaguin et al. (1987) for planar films. For thin films, we may set m = 3, whereas the constant A = A0 /(6), A0 being the Hamaker constant, is A = 1.3 × 10−21 J, approximately, see e.g. Mitrovic (2004) for details and further references. Taking the disjoining film pressure p in Eq. (21) to arise from an external potential, in Eq. (12) may be replaced by p from Eq. (21). This step accounts for the simultaneous effects of the interface curvature and the film thickness on the droplet equilibrium. Details on the disjoining film pressure on droplet equilibrium are given further below. 4. Resulting equilibrium equation Taking the single effects described by Eqs. (13), (16), and (21) to occur simultaneously and to be independent of each other, we may write + = −
RTV ∞ 1 q2 A ln(1 − x) + + m. vL∞ 8 r 4
(22)
With this expression, Eq. (12) describes the droplet equilibrium in terms of measurable quantities. Setting TV = TV ∞ or pV = pV ∞ , one formulates an isothermal, or an isobaric, system evolution. If the droplet of a radius r contains a spherical solid particle of the radius r0 , r = r0 + , and if pV and TV are fixed, Eqs. (12) and (22) give −
RTV ∞ 1 q2 A ln(1 − x) + + m = C, 4 vL∞ 8 r
(23)
where = −2/(r0 + ), C being a constant. The mole fraction x of the dissolved substance is defined by x=
n , n + nS
(24)
where n and nS are the numbers of moles of the solute and the solvent, respectively. Taking the solution to be dilute, assuming
5630
J. Mitrovic / Chemical Engineering Science 61 (2006) 5925 – 5933
n < < nS , and denoting by cS the molar concentration of the solvent in the droplet give x≈
n 1 n n = ≈ , nS cS V 4 cS r02
(25)
where V = (4/3)((r0 + )3 − r03 ) ≈ 4r02 . Eq. (23) thus becomes 1 1 RTV ∞ r0 1 n 1 − − ln 1 − 1 + ( /r0 ) 2 vL∞ 4 r03 cS /r0 +
1 q2 1 A 1 1 + = C. 3 4 m−1 16 r0 (1 + ( /r0 )) 2 r0 ( /r0 )m
(26)
Depending on the ratio /r0 , the constant C may be positive, negative, or zero. For C = 0, the droplet is in equilibrium with the surrounding vapour at pV = pV ∞ and TV = TV ∞ . At TV = TV ∞ and C > 0, it is pV > pV ∞ ; similarly, at pV = pV ∞ , it is TV > TV ∞ . The droplet under such conditions can be stable or unstable, depending on the change of pV and TV when is varied. Keeping the temperature TV constant, deriving Eq. (12) with respect to , and setting dpV /d =0 result in d(+)/d =0, or with Eq. (26) in 3 1 q2 m+1 1+ − r0 r0 4 r03 n/(r 30 cS ) 5 1 RTV ∞ r0 1+ − 8 vL∞ 1−(1/4)(n/(r 30 cS ))(1/( /r0 )) r0 m−1 5 m A × − 1+ = 0. (27) r0 2 r0m−1 r0 If solved for /r0 , this equation gives the boundary between the stable and unstable droplet states. However, since expression (27) is somewhat cumbersome, and, in addition, the effects of solute and electric charge has been considered in several publications (see e.g. Dufour and Defay, 1963; Sorjamaa et al., 2004; Fletcher, 1970, 1962), we will omit these effects by setting q = 0 and n = 0, and concentrate on the disjoining film effect. 5. Some details on film effect The equilibrium of a liquid layer formed on a spherical particle of the radius r0 can be obtained from Eqs. (12) and (22). Setting q = 0 and x = 0 gives pV h TV − TV ∞ vL∞ pV ∞ pV = + −1 ln pV ∞ RTV TV ∞ RTV ∞ pV ∞ vL∞ pV ∞ −2 RTV ∞ r0 pV ∞ 1 A 1 1 , (28) × − 2 r02 ((r/r0 ) − 1)3 r/r0 where r/r0 = 1 + ( /r0 ), Fig. 2.
For TV = TV ∞ , this equation describes the isothermal pressure change caused by simultaneous actions of capillary pressure and disjoining film pressure, pV pV vL∞ pV ∞ ln = −1 pV ∞ RTV ∞ pV ∞ vL∞ pV ∞ −2 RTV ∞ r0 pV ∞ 1 A 1 1 . (29) × − 2 r02 ((r/r0 ) − 1)3 r/r0 At pV = pV ∞ , we get TV − TV ∞ vL∞ =2 TV hr 0
1 A 1 1 − 2 3 2 r0 ((r/r0 ) − 1) r/r0
.
(30)
With A = 0, the droplet becomes homogeneous and its equilibrium is governed only by surface tension. In this case, Eqs. (29) and (30) give pV pV vL∞ pV ∞ vL∞ pV ∞ ln = −1 +2 , (31) pV ∞ RTV ∞ pV ∞ RTV ∞ pV ∞ r TV − TV ∞ vL∞ = −2 . TV hr
(32)
Omitting the first term on the right in Eq. (31), one obtains an expression first reported by von Helmholtz (1886), whereas Eq. (32) has been given by Thomson (1886) for the first time. Note that these equations rest on the assumption vL /TV = vL∞ /TV ∞ = const.3 Fig. 5 illustrates Eq. (29). Shown is the pressure ratio pV /pV ∞ as function of the dimensionless droplet radius r/r0 , where r0 is the radius of the solid particle. The monotonically decreasing vapour pressure pV /pV ∞ (dashed line) corresponds to a homogeneous droplet, without solid particle. The equilibrium of such a droplet is governed only by the droplet radius r, all other parameters being the same. This equilibrium curve is included into the figure for comparison purposes, and the radius r0 plays only a scaling role. The curve corresponding to the droplet with a solid particle (r/r0 > 1) shows a maximum (j(pV /pV ∞ )/jr =0) at r =rmax , a r0 1/2 rmax − r0 = = , (33) 1+ 1+4 2 a 3 A 1/2 a= . (34) 2 A droplet of r > rmax is unstable, whereas for r < rmax , it is stable. A variation of the droplet radius r in the region r > rmax at constant pressure pV of the vapour phase results in condensation, if r increases e.g. by a fluctuation, and in evaporation, 3 For p = p V V ∞ , that is, dpV = 0 and vL = vL∞ = const, Eq. (9) gives (p = −2/r, (p)∞ = 0)
T V − TV ∞ v = −2 L∞ , TV ∞ hr
(32a)
which differs from Eq. (32) in the denominator on the left-hand side.
J. Mitrovic / Chemical Engineering Science 61 (2006) 5925 – 5933
5631
Fig. 6. Isobaric equilibrium of a droplet without and with a solid particle. Fig. 5. Isothermal equilibrium of a droplet without and with a solid particle. The region affected by disjoining film pressure extends from r/r0 > 1 to rmax /r0 , approximately, with rmax according to Eq. (33).
if r is decreased. In the latter case, evaporation would stop in the region r < rmax at the saturation pressure equal to the initial pressure of the vapour, see enlargement in Fig. 5. This interplay is rooted in the opposite actions of surface tension and disjoining film pressure, and is entirely analogous to the one discussed e.g. by Reiss and Koper (1995) regarding the effect of a solute on the droplet equilibrium. To quantify the effect of disjoining film pressure on droplet equilibrium let the particle radius r0 be 10−8 m. With the value of the constant A = 1.3 × 10−21 J in Eq. (21), a film thickness of 10−8 m approximately matches the vapour pressure elevation due to interface curvature and condensation takes place on the droplet as on the plane interface of the same temperature, Fig. 5. This interplay should be considered when dealing with the droplet formation in technical processes. In connection with droplet formation in the atmosphere, e.g. formation of cloud droplets, solid particles are expected to incept and to promote vapour condensation at lower air humidity in comparison to dust-free humid air. Fig. 6 illustrates the equilibrium states under isobaric conditions according to Eq. (30). The temperature of the droplet containing the solid particle (r/r0 > 1) is above the droplet temperature without the particle. It first decreases, passes a minimum and then increases, asymptotically approaching the temperature of the homogeneous droplet from above. The minimum of the curve occurs at the same droplet radius as the maximum of the
vapour pressure curve shown in Fig. 5 and can be calculated from Eq. (33). A droplet left to the minimum is stable, on the right it is unstable and would grow if its radius increased by a fluctuation. Fig. 7 shows the dimensionless vapour pressure and the dimensionless equilibrium temperature as functions of the reduced film thickness /r0 . The curves correspond to different values of the dimensionless quantity A/(r02 ) at parameters noted in Figs. 5 and 6. Larger values of the ratio A/(r02 ) mean a stronger interaction between the solid particle and the liquid film. This interaction results in a disjoining film pressure which requires a lower vapour pressure at constant temperature and a higher temperature at constant pressure in comparison to a particle-free droplet. Since the Hamaker constant specifies the interaction energy of the film with the particle, the quantity A/(r02 ) measures this energy in terms of the free surface energy. The above considerations are also valid for the case of a soluble spherical particle. In such a case Eq. (28) has to be supplemented by the osmotic pressure, giving pV pV h TV − TV ∞ vL∞ pV ∞ ln = + −1 pV ∞ RTV TV ∞ RTV ∞ pV ∞ vL∞ pV ∞ 1 A 1 −2 RTV ∞ r0 pV ∞ 2 r02 ((r/r0 ) − 1)3 1 + ln(1 − x). − r/r0
(35)
5632
J. Mitrovic / Chemical Engineering Science 61 (2006) 5925 – 5933
glutamate. They considered all the effects treated in the literature that could reduce the supersaturation, but not the disjoining film effect. Also Wagner et al. (2003) give a summary of experiments on heterogeneous nucleation including vapour mixtures and soluble solid particles which are not completely dissolved. The authors varied the contact angle to bring their own experiments in a closer agreement with the Fletcher (1958) theory, but did not consider the disjoining action of the condensate film covering the solid nucleus. Viewed in the light of these experiments, the present paper—focussing on the disjoining film effect—seems to provide a way of how to narrow the discrepancy between the measured and calculated supersaturation. The present consideration concerning the disjoining film effect is principally immediately applicable to multicomponent films. Regarding condensation and nucleation of binary vapours the reader may be referred to a paper by Konopka (1997) for further references. In this context, however, it should be noted that tiny droplets, considered as nano-particles, may possess several degrees of freedom that affect the equilibrium pressure in comparison to a stationary droplet usually adopted for theoretical considerations. A paper by Kuhrt (1952) gives details on this subject, see also Abraham (1974).
Fig. 7. Isothermal (above) and isobaric (below) equilibrium of a droplet without and with a solid particle.
The concentration c of the solute in the liquid film, covering the solid nucleus of a soluble substance, can be obtained from ln
vL∞ c =2 , c∞ RTV ∞ r
(36)
where c∞ denotes the solubility of the solution with the plane interface (Webster and Cates, 1998). Note that Eq. (36) accounts for the effect of the interface curvature on the solubility and not the effect of the solid particle itself. The latter might also be important and has to be considered in a detailed analysis. 6. Experimental evidence and conclusions Experiments on nucleation under heterogeneous conditions are extensive and very complex. Chen et al. (1998, 2000) give an overview in this area and provide new data sets. The main conclusion from their experiments should be given here as originally stated: “. . ., if the macroscopic theory of heterogeneous nucleation is valid, then the results suggest that both SiC and SiO2 aerosols behave as having an ability to induce heterogeneous nucleation better than perfectly wetted particles.” The authors (Chen et al., 1998, 2000) experimented with water vapour using submicrometer particles of SiC, SiO2 , and naphthalene, as well as with n-butanol vapour and submicrometer charged and neutral particles of lactose and monosodium
Notation a A A0 c C f G h h m n p pL pV p q r R R T TL TV v V VL VV x
constant constant Hamaker constant concentration constant fugacity Gibbs free energy specific enthalpy, h in ideal state enthalpy of phase change (hV − hL ) exponent number of moles (solute) pressure pressure in liquid, pL∞ in reference state vapour pressure, pV ∞ in reference state pressure jump across interface, (p)∞ in reference state electric charge radius specific gas constant universal gas constant temperature liquid temperature vapour temperature, TV ∞ in reference state partial molar volume volume of droplet specific liquid volume, vL∞ in reference state specific vapour volume, vV ∞ in reference state mole fraction of solute
J. Mitrovic / Chemical Engineering Science 61 (2006) 5925 – 5933
Greek letters
film thickness external potential dielectric constant curvature of interface chemical potential osmotic pressure density surface tension
Indices eq L q S V 0 ∞ ∗ max
equilibrium, at pV = pV ∞ , TV = TV ∞ liquid due to electric charge solvent vapour due to film thickness solid particle reference state, plane interface equilibrium at maximum of pV
References Abraham, F.F., 1974. Homogenous Nucleation Theory. Academic Press, New York. Aitken, J., 1880–1881. On dust, fogs, and clouds. Transactions of the Royal Scottish Society of Arts, Edinburgh 30. Also Collected Scientific Papers of John Aitken, Cambridge University Press, Cambridge, 1923, pp. 34–64. Amelin, A.G., 1967. Theory of Fog Condensation. IPST Jerusalem. Best, A.C., 1955. Atomic explosion and condensation nuclei. Meteorological Magazine 84, 201–204. Byers, H.R., 1965. Vapour pressure over solution droplets. Journal of Chemical Education 42, 338–339. Chen, C.-C., Guo, M.-S., Tsai, Y.-J., Huang, C.-C., 1998. Heterogeneous nucleation of water vapor on submicrometer particles of SiC, SiO2 , and naphthalene. Journal of Colloid and Interface Science 198, 354–367. Chen, C.-C., Tao, C.-J., Shu, H.-J., 2000. Heterogeneous nucleation on nbutanol vapor on submicrometer charged and neutral particles of lactose and monosodium glutamate. Journal of Colloid and Interface Science 224, 11–22. Clapeyron, E., 1834. Puissance motrice de la chaleur. Journal de l’Ecole Polytechnique 14, 153–191. Also: Über die bewegende Kraft der Wärme. Annalen der Physik 135, 451–446, 566–586. Clausius, R., 1850. Über die bewegende Kraft der Wärme und die Gesetze, die sich daraus für die Wärmelehre selbst ableiten lassen. Annalen der Physik 200, 368–397 500–524. Derjaguin, B.V., Churaev, N.V., Muller, V.M., 1987. Surface Forces. Consultants Bureau, New York. Dufour, L., Defay, R., 1963. Thermodynamics of Clouds. Academic Press, New York. Fletcher, N.H., 1958. Size effect in heterogeneous nucleation. Journal of Chemical Physics 29, 572–576. Fletcher, N.H., 1962. The Physics of Rainclouds. Cambridge University Press, Cambridge. Fletcher, N.H., 1970. The Chemical Physics of Ice. Cambridge University Press, Cambridge. Gorbunov, B., Hamiltion, R., 1997. Water nucleation on aerosol particles containing both soluble and insoluble substances. Journal of Aerosol Science 28, 239–248.
5633
Hanel, G., 1976. Properties of atmospheric aerosol particles as function of the relative humidity at thermodynamic equilibrium with the surrounding moist air. Advances in Geophysics 19, 73–188. Köhler, H., 1926. Zur Thermodynamik der Kondensation an hydroskopischen Kernen und Bemerkungen über das Zusammenfließen der Tropfen. Med. Statens Meteorol.-Hydrogr. Anstalt (Stockholm) 3 (8), Köhler, H., 1950. On the problem of condensation in the atmosphere. Nova Acta Regiae Societatis Scientarum Upsaliensis Serie 14 (9), Konopka, P., 1997. Binary condensation on soluble aerosols—On the activation and growth of the stratospheric aerosol from coupled HNO3 –H2 O uptake. Journal of Aerosols Science 28, 1411–1424. Kuhrt, F., 1952. Das Tropfenmodell übersättigter realer Gase. Zeitschrift für Physik 131, 205–214. Laaksonen, A., Talanquer, V., Oxtoby, D.W., 1995. Nucleation: measurements, theory, and atmospheric applications. Annual Review of Physical Chemistry 46, 489–524. Latham, J., 1969. Cloud physics. Reports on Progress in Physics 32, 69–134. Martin, S.T., 2000. Phase transition of aqueous atmospheric particles. Chemical Reviews 100, 3403–3453. Mason, B.J., 1971. The Physics of Clouds. Clarendon Press, Oxford. McDonald, Y.E., 1962. Homogeneous nucleation of vapour condensation I. American Journal of Physics 30, 870–877. Mirabel, P., Reiss, H., Bowles, R.K., 2000. A theory for the deliquescence of small particles. Journal of Chemical Physics 113, 8200–8205. Mitrovic, J., 2004. On the equilibrium conditions of curved interfaces. International Journal of Heat Mass Transfer 47, 809–818. Reiss, H., Koper, G.J.M., 1995. The Kelvin equation: stability, fluctuation, and factors involved in measurement. Journal of Physical Chemistry 99, 7837–7844. Russell, L.M., Ming, Y., 2002. Deliquescence of small particles. Journal of Chemical Physics 116, 311–321. Sorjamaa, R., Svennigsson, B., Raatikainen, T., Henning, S., Bilde, M., Laaksonen, A., 2004. The role of surfactants in Köhler theory reconsidered. Atmospheric Chemistry and Physics 4, 2107–2117. Talanquer, V., Oxtoby, D.W., 2003. Formation of droplets on non-volatile soluble particles. Journal of Chemical Physics 119, 9121–9128. Thomson, J.J., 1886. Applications of Dynamics to Physics and Chemistry. Macmillan, London. Thomson, J.J., 1903. Conduction of Electricity Through Gases. Cambridge. Thomson, W., 1871. On the equilibrium of vapour at a curved surface of liquid. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science Series 4 42, 448–453. Tohmfor, G., Volmer, M., 1938. Die Keimbildung unter dem Einfluss elektrischer Ladungen. Annalen der Physik, 5 Folge 33, 109–131. Van Ness, H.C., Abbott, M.M., 1982. Classical Thermodynamics of Nonelectrolyte Solutions. McGraw-Hill Book Company, New York. von Helmholtz, R., 1886. Untersuchungen über Dämpfe und Nebel besonders über solche von Lösungen. Annalen der Physik und Chemie N. F. 27, 508–543. Wagner, P.E., Vali, G. (Eds.), 1988. Atmospheric Aerosols and Nucleation. Springer, Berlin. Wagner, P.E., Kaller, D., Vrtala, A., Lauri, A., Kulmala, M., Laaksonen, A., 2003. Nucleation probability in binary heterogeneous nucleation of water—n-propanol mixtures on insoluble and soluble nanoparticles. Physical Review E 67, 021605. Webster, A.J., Cates, M.E., 1998. Stabilization of emulsions by trapped species. Langmuir 14, 2068–2079. Wright, H.L., 1936. The size of atmospheric nuclei: some deductions from measurements of the number of charged and uncharged nuclei at Kew Observatory. Proceedings of the Physical Society 48 (Pt. 5), 675–689.
Further reading McDonald, Y.E., 1963. Homogeneous nucleation of vapour condensation II. American Journal of Physics 31, 31–41.
Phase equilibrium of a liquid droplet formed on a solid particle J. Mitrovic ∗ Institut für Energie- und Verfahrenstechnik, Thermische Verfahrenstechnik und Anlagentechnik, Universität Paderborn, 33095 Paderborn, Germany Received 2 January 2006; received in revised form 4 May 2006; accepted 9 May 2006 Available online 19 May 2006
Abstract An equation of state of a two-phase system, deduced from the principles of classical thermodynamics, is used to illustrate the effects of interface curvature, electric charge, a soluble substance, and of disjoining film pressure on the equilibrium of a liquid droplet formed on a solid particle. The former three effects have been widely discussed in the past, whereas the effect of the disjoining film pressure has not been explored. The focus of the paper is, therefore, placed on the effect of the disjoining film pressure on the droplet equilibrium. Opposing the action of surface tension, the disjoining pressure is shown to lower the equilibrium vapour pressure at constant temperature and to increase the equilibrium temperature at constant pressure in comparison to the equilibrium in the reference state. This interplay provides a way bringing experimental findings in a better agreement with theoretical models. 䉷 2006 Elsevier Ltd. All rights reserved. Keywords: Drop; Aerosol; Phase equilibrium; Phase change; Coated solid particle; Disjoining pressure
1. Introduction When a superheated vapour is cooled down to its saturation temperature, condensation will immediately start if liquid phase is present or nucleation of the new phase can take place without shifting the vapour state towards the metastable state. In the absence of foreign action upon the vapour, which may arise e.g. from container walls or particles suspended in the vapour, usually phase transition does not occur at the nominal saturation temperature but at some subcooling. Nucleation expected under such condition is termed homogeneous nucleation. As can be taken from numerous sources, e.g. McDonald (1962) and Abraham (1974), one relationship of the classical homogeneous nucleation theory is G = 4r 2 − 43 L (V − L )r 3
(1)
with V − L = RT ln
pV , pV ∞
(2)
where G is the difference of the Gibbs free energy of the actual system, consisting of vapour and a droplet of the radius r at the ∗ Tel.: +49 5251 60 2409; fax: +49 5251 60 3207.
E-mail address: [email protected]. 0009-2509/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2006.05.013
pressure pV , and the reference system, consisting of vapour at the saturation pressure pV ∞ in equilibrium with the liquid of plane interface, both systems being at the same temperature T ; is the chemical potential, the indices V and L referring to vapour and liquid, respectively. Since pV > pV ∞ , that is, V > L , the energy difference G has a maximum Gmax =
16 3 3 (L RT ln(pV /pV ∞ ))2
(3)
at r = r∗ , r∗ =
2 . L RT ln(pV /pV ∞ )
(4)
The radius r∗ is usually called the critical radius of the droplet. With this radius, Eq. (1) becomes 2 1 L RT r ∗ r r pV G 1− . (5) = 4 ln r∗2 r∗ 3 r∗ p V ∞ As follows from Eqs. (4) and (5), a liquid droplet cannot be formed in a saturated vapour at pV = pV ∞ . For pV = pV ∞ , the radius r∗ according to Eq. (4) would become infinite, which means a liquid layer (or semi-infinite liquid space) of infinite surface area would form, and the energy difference G
5626
J. Mitrovic / Chemical Engineering Science 61 (2006) 5925 – 5933
Fig. 2. Illustration of the effects of curvature, dissolved substance, electric charge, and disjoining film pressure on droplet equilibrium.
Fig. 1. Dimensionless Gibbs free energy difference G of a droplet relative to the reference state at L RT r∗ / = 500.
according to Eq. (5) would become unlimited. Obviously, the saturated vapour must be shifted away from the saturation state and the new phase must have a finite surface area. In case of an isothermal shifting, a pressure increase in the vapour would have to take place resulting in a corresponding increase of the free energy of the vapour, which becomes supersaturated. Although this vapour would be metastable, its state would not change jumpwise to stable, lower-energy states, because a jump can only proceed through formation of tiny drops by clustering of vapour molecules. The increase in the free energy G of the system, required for formation of a droplet having the radius r according to Eq. (5) at L RT r ∗ /=500, is illustrated in Fig. 1. As is obvious from the figure, the dimensionless energy difference G/r∗2 decreases as pV /pV ∞ increases. At a fixed pV /pV ∞ , the ratio G/r∗2 passes a maximum determined by Eq. (3). Suppose now a droplet of a radius r in supersaturated homogeneous vapour large enough to apply the relationships of the classical thermodynamics. Since the process of phase transition under homogeneous conditions is a spontaneous one, the fate of the droplet just formed is determined by the change of free energy of the system caused by a fluctuation of the droplet size. If by such a fluctuation the free energy decreases, the system state moves towards a more stable state. This means that, in a spontaneous process, a droplet of r < r∗ would evaporate because its growth would be accompanied by an increase in the free energy which would be in contradiction to the spontaneous behaviour of the system. On the contrary, a droplet of r > r∗ would spontaneously grow. However, in order for the droplet to reach such a size, the energy barrier Gmax in Eq. (3) must be surmounted. The classical capillarity theory of homogeneous nucleation does not seem to convincingly explain the route on which the energy hill could be passed.
The validity of the above equations is restricted to pure vapours that are protected from external influences. If the vapour contains foreign substances, such as solid particles or ions, the thermodynamic conditions of equilibrium alter, and condensation is expected to set in at a lower supersaturation in comparison with pure vapours. The nucleation under such conditions is called heterogeneous. Both homogeneous and heterogeneous nucleation has been the subject matter of numerous papers, see e.g. Laaksonen et al. (1995) for further references. The aim of the present paper is to describe the equilibrium of a droplet which contains a solid particle. The substance making the solid particle is taken to be non-volatile; it may be soluble or insoluble in the liquid forming the (droplet) film. The particle is completely covered by a liquid film that is surrounded by and in equilibrium with its own vapour. The phase equilibrium is assumed to be simultaneously affected by • • • •
the interface curvature, the dissolved substance, an electric charge and the thickness of condensate film covering the solid particle.
These effects are illustrated in Fig. 2 and are measured in terms of supersaturation of vapour with respect to the equilibrium of the reference system consisting of pure substance (e.g. water) with a plane interface. The effect of the interface curvature is described by the well-known Kelvin equation (Thomson, 1871), while the effect of a dissolved substance in conjunction with the interface curvature has been considered, apparently for the first time, by Köhler (1926, 1950). The investigation of the effect of the electric charge seems to go back to Thomson (1903), whereas the last one—the effect of the film thickness resulting in a disjoining film pressure when condensation is taking place on a solid particle in conjunction with the solution and/or curvature effect on the droplet equilibrium—has not been reported in the literature. The present paper aims at closing the gap. First, a simple equation for the equilibrium of a droplet surrounded by the vapour phase is deduced from the principles
J. Mitrovic / Chemical Engineering Science 61 (2006) 5925 – 5933
5627
of thermodynamics. The derivation path describes a reversible evolution of a system by changing its coordinates. A pressure jump is thereby assumed to occur across the phase interface. Specifying this pressure jump, one describes the corresponding effect on the droplet equilibrium. From this point of view, the paper summarizes the basic models and gives to some extent a brief review of the knowledge in this area. 2. Equilibrium of a spherical droplet To derive a general equation for phase equilibrium, we start the considerations by writing down an expression for the fugacity f of a homogeneous phase in intrinsic equilibrium.1 This quantity is given by (see e.g. Van Ness and Abbott, 1982) d ln f = −
h − h v dT + dp, 2 RT RT
where h and h are the specific enthalpies of the phase in the actual and in an ideal state, respectively; the symbols R, v, p, and T denote the universal gas constant, the partial molar volume, the pressure, and the temperature, respectively. Eq. (6) is also valid for a two-phase system. Indicating its phases by the indices V and L, one obtains d ln
fL hV − h vV hL − h = dT − dp − dTL V V fV RTV RTV2 RTL2 vL + dpL . RTL
(7)
A system in thermodynamic equilibrium requires fL = fV and TL = TV , the former equality expressing the chemical, the latter one the thermal equilibrium. The pressures pL and pV , governing the mechanical equilibrium, need not necessarily be the same, and a pressure jump is assumed at the phase interface, Fig. 3. Denoting by p this pressure jump, p = pV − pL ,
Fig. 4. Evolution of a system.
(6)
(8)
where p > 0, or p < 0, Eq. (7) becomes vV hV − h L vL vL dpV = dTV + dpV − d(p). 2 RTV RT RT RTV V V
Describing the equilibrium of a two-phase system that is inhomogeneous with respect to the pressures in the phases, this expression represents an augmented Clapeyron–Clausius equation (Clapeyron, 1834; Clausius, 1850). As it shows, the vapour pressure pV depends on the temperature TV and the pressure difference p. Physical systems that follow Eq. (9) do not satisfy the classical (Gibbs) phase rule. The pressure difference p defined in Eq. (8) is a property of the system that—in single component systems—must not vary along the interface. If for instance one of the phases is disperse, e.g. liquid droplets in vapour, or vapour bubbles in liquid, the pressure jump p must be the same everywhere at the interface. For different values of p, the system would not be in mechanical equilibrium thus violating the thermodynamic equilibrium requirement. Consequently, Eq. (9) would be invalid. To perform integration in Eq. (9), the vapour phase is taken to be ideal, RTV /vV = pV . For vL /TV = const(=vL∞ /TV ∞ ) and hV − hL = h = const, an integration from the initial state (pV ∞ , TV ∞ , (p)∞ ) to the final state (pV , TV , p) gives2 ln
Fig. 3. Illustration of pressure p jump across the interface.
1 The considerations could also have been started from the Gibbs –Duhem–Helmholtz expression d(G/RT ) = −(h/RT 2 ) dT + (v/RT ) dp.
(9)
pV h TV − TV ∞ vL∞ = + (pV − pV ∞ ) pV ∞ RTV TV ∞ RTV ∞ vL∞ − (p − (p)∞ ). RTV ∞
(10)
Physically, the integration of Eq. (9) leading to Eq. (10) describes an evolution of the system. Such an evolution is illustrated in Fig. 4, where, for instance, the interface may be initially plane (a), whereas in the final state it is convex (b) or concave (c). 2 Note that the assumption v /T = const adopted for integration purL V poses ignores the anomaly of water.
5628
J. Mitrovic / Chemical Engineering Science 61 (2006) 5925 – 5933
For p = (p)∞ , Eq. (10) reduces to the relationship wellknown from literature. In general, however, the pressure difference p contains specific information about the system which is not included into the classical vapour pressure expression. Specifying the quantity p, as illustrated below, one formulates particular systems. 3. Particular cases 3.1. Effect of interface curvature Taking the pressure difference p to arise from the curvature of the interface and an action of the external potential on the phases, , we may write p = pV − pL = + ,
(11)
where denotes the surface tension; the curvature is positive ( > 0) when its centre lies on the side of the V -phase, Fig. 3. The physical significance of the quantity has been examined in an earlier paper by the same authour (Mitrovic, 2004). For (p)∞ = 0, Eqs. (10) and (11) give ln
pV h TV − TV ∞ vL∞ = + (pV − pV ∞ ) pV ∞ RTV TV ∞ RTV ∞ vL∞ ( + ). − RTV ∞
3.3. Effect of an electric charge (12)
In the considerations to follow, the quantity will be specified for cases illustrated in Fig. 2 and the effects of a dissolved substance, an electric charge, and the film thickness on the equilibrium conditions described using Eq. (12). 3.2. Effect of a dissolved substance Assuming the liquid phase to contain a dissolved substance, thus representing a solution, the dissolved substance causes an osmotic pressure which may be thought to arise from an external potential . Taking the solution at the temperature TV ∞ to be ideal, one may write ≡ = −
RTV ∞ ln(1 − x), vL∞
(13)
where x denotes the mole fraction of the solute. For a droplet of the radius r, the curvature is obtained from = dA/dV = 2/r, and Eq. (12), together with Eq. (13), gives ( < 0) ln
pV h TV − TV ∞ vL∞ = + (pV − pV ∞ ) pV ∞ RTV TV ∞ RTV ∞ vL∞ +2 + ln(1 − x). RTV ∞ r
As is obvious from this equation, a droplet in equilibrium with pure vapour at pV ∞ and TV ∞ requires a larger mole fraction x at a smaller droplet radius r. This interplay is rooted in counteracting effects of the curvature (surface tension) and the osmotic pressure on the equilibrium of the solution droplet, and has been examined by Köhler (1926, 1950), Byers (1965), Dufour and Defay (1963), Reiss and Koper (1995), Talanquer and Oxtoby (2003), and many others. Eq. (14) assumes a spatially homogeneous concentration of the solute in the droplet. If the solute is surface active, its higher concentration in the surface region affects the surface tension and the osmotic pressure. The droplet equilibrium will consequently depend on the surface activity of the solute. Sorjamaa et al. (2004) seem to be the first authors to examine the effect of the surface activity of the solute on the vapour pressure lowering (Raoult’s law) of solutions. Their paper gives all the important details concerning the equilibrium of a solution droplet basing on the Köhler (1926, 1950) theory. A further effect of the surface activity—not mentioned earlier—is the osmotic pressure stress which could, in case of spherical asymmetry, induce an osmotic flow thus affecting droplet formation.
(14)
Setting in Eq. (14) pV =pV ∞ and TV =TV ∞ , thus requiring the equilibrium conditions of the solution droplet to be the same as for the pure solvent at r → ∞, gives vL∞ x = 1 − exp −2 . (15) RTV ∞ r
If the droplet is not electrically neutral but carries an electric charge q, and if it is assumed to be a conducting sphere, the pressure pq acting on the droplet surface arising from this charge has been obtained by Thomson (1903) to be 1 q2 . (16) 8 r 4 This equation does not account for the sign of the electric charge. Furthermore, the assumption of the droplet to be a conductor has been discussed by Tohmfor and Volmer (1938). Considering the droplet as spherical condenser, they obtained the following expression: 1 1 q2 pq = , (17) 1− 8 r4 pq = pL − pV =
which, like Eq. (16), is invariant with respect to the charge polarization. For a sufficiently large dielectric constant , the two equations become identical. For further references on this effect, see e.g. Laaksonen et al. (1995). The effect of the electric charge on the droplet equilibrium follows immediately from Eq. (12) by replacing by the pressure pq according to Eq. (16) or (17). Eqs. (12) and (16) show, therefore, that any variation of the electric charge q is compensated by a corresponding change at least of one of the following system parameters: the temperature TV , the vapour pressure pV , and the droplet radius r. Taking TV and pV as constant, Eq. (12) with ≡ pq gives ( = −2/r) 1 1 q2 = C, (18) 1− −2 + r 8 r4 which means that—at droplet equilibrium—any variation in the electrostatic pressure is balanced by an opposite action of the capillary pressure /r.
J. Mitrovic / Chemical Engineering Science 61 (2006) 5925 – 5933
For pV = pV ∞ and TV = TV ∞ in Eq. (12), the constant C in Eq. (18) becomes zero. In this case, the droplet in equilibrium with the vapour phase is also in equilibrium with the liquid of plane interface, and the droplet equilibrium radius req follows from Eq. (18): 1/3 1 q2 1 1− . (19) req = 16 Deriving Eq. (18) with respect to r gives the boundary (r = rmax ) between the stable and unstable equilibrium states, 1/3 1 1 q2 . (20) 1− rmax = 4 A droplet of r > rmax is unstable, whereas one of r < rmax is stable. 3.4. Effect of film thickness Since the studies on dust, fogs, and clouds by Aitken (1880), the effect of solid particles on vapour condensation plays an important role in modelling of clouds formation in the atmosphere (see e.g. Laaksonen et al., 1995; Latham, 1969; Best, 1955; Fletcher, 1962, 1970; Wagner and Vali, 1988; Mason, 1971; Wright, 1936; Amelin, 1967). Fletcher (1958) has shown that insoluble spherical particles lower considerably the energy barrier of droplet formation. This energy barrier depends on the contact angle and the size of the particle in comparison with the critical droplet radius deduced from the classical nucleation theory. However, at zero contact angle, that is, at complete wetting of the particle surface, the critical radius of the vapour–liquid interface becomes smaller than the particle radius. When the solid particle is soluble, as is mostly the case in the atmosphere (Martin, 2000), formation of liquid phase on the particle surface is preceded by adsorption of solvent molecules on the particle surface. In this case, it is not clear what a shape of the liquid–vapour interface may be expected to form. Assuming the dissolution of the solid particle to occur preferably in the region of the three-phase line, formed by intersection of the system interfaces, a continuous liquid film is likely to cover the whole particle surface. On the contrary, in the event that the dissolution mainly occurs across the solid–liquid interface, a liquid cluster could basically penetrate the solid particle. The latter idea seems energetically less probable than the former one. For a general discussion of dissolution, the deliquescence of solid particle, the reader is referred to literature, e.g. Mirabel et al. (2000) and Russell and Ming (2002). Hanel (1976) has investigated condensation on solid particles consisting of soluble and insoluble parts. The model he adopted for equilibrium treatment—an insoluble particle covered by a solution film—resulted in a considerable effect of the solid particle on the equilibrium vapour pressure. However, this model has been considered by Gorbunov and Hamiltion (1997) to be less realistic because incomplete wetting of insoluble aerosol particles was mostly expected in reality. The latter authors adopted, in principle, the Fletcher (1958) model and replaced the pure liquid by a liquid solution. The expression
5629
for the vapour pressure thus obtained reduces in case of perfect wetting basically to the Köhler equation (Köhler, 1926, 1950), and the solid particle affects the phase equilibrium only by its volume. None of the models mentioned so far takes into account the effect arising from interactions substrate–film when a thin liquid film covers the particle surface. As is well-known, a liquid film formed on a solid substrate is subject to an action of the substrate which may result in a significant displacement of phase equilibrium (Derjaguin et al., 1987). To describe the effect of a solid particle on the droplet equilibrium, a spherical particle covered by a liquid film at perfect wetting will be assumed. In case of a thin film, the action of particle on the vapour–liquid interface can be accounted for in terms of disjoining film pressure p : p =
A , m
(21)
where the quantities A and m depend on the physical properties of the liquid and solid phase and, additionally, on the film thickness . Eq. (21) has been proposed by Derjaguin et al. (1987) for planar films. For thin films, we may set m = 3, whereas the constant A = A0 /(6), A0 being the Hamaker constant, is A = 1.3 × 10−21 J, approximately, see e.g. Mitrovic (2004) for details and further references. Taking the disjoining film pressure p in Eq. (21) to arise from an external potential, in Eq. (12) may be replaced by p from Eq. (21). This step accounts for the simultaneous effects of the interface curvature and the film thickness on the droplet equilibrium. Details on the disjoining film pressure on droplet equilibrium are given further below. 4. Resulting equilibrium equation Taking the single effects described by Eqs. (13), (16), and (21) to occur simultaneously and to be independent of each other, we may write + = −
RTV ∞ 1 q2 A ln(1 − x) + + m. vL∞ 8 r 4
(22)
With this expression, Eq. (12) describes the droplet equilibrium in terms of measurable quantities. Setting TV = TV ∞ or pV = pV ∞ , one formulates an isothermal, or an isobaric, system evolution. If the droplet of a radius r contains a spherical solid particle of the radius r0 , r = r0 + , and if pV and TV are fixed, Eqs. (12) and (22) give −
RTV ∞ 1 q2 A ln(1 − x) + + m = C, 4 vL∞ 8 r
(23)
where = −2/(r0 + ), C being a constant. The mole fraction x of the dissolved substance is defined by x=
n , n + nS
(24)
where n and nS are the numbers of moles of the solute and the solvent, respectively. Taking the solution to be dilute, assuming
5630
J. Mitrovic / Chemical Engineering Science 61 (2006) 5925 – 5933
n < < nS , and denoting by cS the molar concentration of the solvent in the droplet give x≈
n 1 n n = ≈ , nS cS V 4 cS r02
(25)
where V = (4/3)((r0 + )3 − r03 ) ≈ 4r02 . Eq. (23) thus becomes 1 1 RTV ∞ r0 1 n 1 − − ln 1 − 1 + ( /r0 ) 2 vL∞ 4 r03 cS /r0 +
1 q2 1 A 1 1 + = C. 3 4 m−1 16 r0 (1 + ( /r0 )) 2 r0 ( /r0 )m
(26)
Depending on the ratio /r0 , the constant C may be positive, negative, or zero. For C = 0, the droplet is in equilibrium with the surrounding vapour at pV = pV ∞ and TV = TV ∞ . At TV = TV ∞ and C > 0, it is pV > pV ∞ ; similarly, at pV = pV ∞ , it is TV > TV ∞ . The droplet under such conditions can be stable or unstable, depending on the change of pV and TV when is varied. Keeping the temperature TV constant, deriving Eq. (12) with respect to , and setting dpV /d =0 result in d(+)/d =0, or with Eq. (26) in 3 1 q2 m+1 1+ − r0 r0 4 r03 n/(r 30 cS ) 5 1 RTV ∞ r0 1+ − 8 vL∞ 1−(1/4)(n/(r 30 cS ))(1/( /r0 )) r0 m−1 5 m A × − 1+ = 0. (27) r0 2 r0m−1 r0 If solved for /r0 , this equation gives the boundary between the stable and unstable droplet states. However, since expression (27) is somewhat cumbersome, and, in addition, the effects of solute and electric charge has been considered in several publications (see e.g. Dufour and Defay, 1963; Sorjamaa et al., 2004; Fletcher, 1970, 1962), we will omit these effects by setting q = 0 and n = 0, and concentrate on the disjoining film effect. 5. Some details on film effect The equilibrium of a liquid layer formed on a spherical particle of the radius r0 can be obtained from Eqs. (12) and (22). Setting q = 0 and x = 0 gives pV h TV − TV ∞ vL∞ pV ∞ pV = + −1 ln pV ∞ RTV TV ∞ RTV ∞ pV ∞ vL∞ pV ∞ −2 RTV ∞ r0 pV ∞ 1 A 1 1 , (28) × − 2 r02 ((r/r0 ) − 1)3 r/r0 where r/r0 = 1 + ( /r0 ), Fig. 2.
For TV = TV ∞ , this equation describes the isothermal pressure change caused by simultaneous actions of capillary pressure and disjoining film pressure, pV pV vL∞ pV ∞ ln = −1 pV ∞ RTV ∞ pV ∞ vL∞ pV ∞ −2 RTV ∞ r0 pV ∞ 1 A 1 1 . (29) × − 2 r02 ((r/r0 ) − 1)3 r/r0 At pV = pV ∞ , we get TV − TV ∞ vL∞ =2 TV hr 0
1 A 1 1 − 2 3 2 r0 ((r/r0 ) − 1) r/r0
.
(30)
With A = 0, the droplet becomes homogeneous and its equilibrium is governed only by surface tension. In this case, Eqs. (29) and (30) give pV pV vL∞ pV ∞ vL∞ pV ∞ ln = −1 +2 , (31) pV ∞ RTV ∞ pV ∞ RTV ∞ pV ∞ r TV − TV ∞ vL∞ = −2 . TV hr
(32)
Omitting the first term on the right in Eq. (31), one obtains an expression first reported by von Helmholtz (1886), whereas Eq. (32) has been given by Thomson (1886) for the first time. Note that these equations rest on the assumption vL /TV = vL∞ /TV ∞ = const.3 Fig. 5 illustrates Eq. (29). Shown is the pressure ratio pV /pV ∞ as function of the dimensionless droplet radius r/r0 , where r0 is the radius of the solid particle. The monotonically decreasing vapour pressure pV /pV ∞ (dashed line) corresponds to a homogeneous droplet, without solid particle. The equilibrium of such a droplet is governed only by the droplet radius r, all other parameters being the same. This equilibrium curve is included into the figure for comparison purposes, and the radius r0 plays only a scaling role. The curve corresponding to the droplet with a solid particle (r/r0 > 1) shows a maximum (j(pV /pV ∞ )/jr =0) at r =rmax , a r0 1/2 rmax − r0 = = , (33) 1+ 1+4 2 a 3 A 1/2 a= . (34) 2 A droplet of r > rmax is unstable, whereas for r < rmax , it is stable. A variation of the droplet radius r in the region r > rmax at constant pressure pV of the vapour phase results in condensation, if r increases e.g. by a fluctuation, and in evaporation, 3 For p = p V V ∞ , that is, dpV = 0 and vL = vL∞ = const, Eq. (9) gives (p = −2/r, (p)∞ = 0)
T V − TV ∞ v = −2 L∞ , TV ∞ hr
(32a)
which differs from Eq. (32) in the denominator on the left-hand side.
J. Mitrovic / Chemical Engineering Science 61 (2006) 5925 – 5933
5631
Fig. 6. Isobaric equilibrium of a droplet without and with a solid particle. Fig. 5. Isothermal equilibrium of a droplet without and with a solid particle. The region affected by disjoining film pressure extends from r/r0 > 1 to rmax /r0 , approximately, with rmax according to Eq. (33).
if r is decreased. In the latter case, evaporation would stop in the region r < rmax at the saturation pressure equal to the initial pressure of the vapour, see enlargement in Fig. 5. This interplay is rooted in the opposite actions of surface tension and disjoining film pressure, and is entirely analogous to the one discussed e.g. by Reiss and Koper (1995) regarding the effect of a solute on the droplet equilibrium. To quantify the effect of disjoining film pressure on droplet equilibrium let the particle radius r0 be 10−8 m. With the value of the constant A = 1.3 × 10−21 J in Eq. (21), a film thickness of 10−8 m approximately matches the vapour pressure elevation due to interface curvature and condensation takes place on the droplet as on the plane interface of the same temperature, Fig. 5. This interplay should be considered when dealing with the droplet formation in technical processes. In connection with droplet formation in the atmosphere, e.g. formation of cloud droplets, solid particles are expected to incept and to promote vapour condensation at lower air humidity in comparison to dust-free humid air. Fig. 6 illustrates the equilibrium states under isobaric conditions according to Eq. (30). The temperature of the droplet containing the solid particle (r/r0 > 1) is above the droplet temperature without the particle. It first decreases, passes a minimum and then increases, asymptotically approaching the temperature of the homogeneous droplet from above. The minimum of the curve occurs at the same droplet radius as the maximum of the
vapour pressure curve shown in Fig. 5 and can be calculated from Eq. (33). A droplet left to the minimum is stable, on the right it is unstable and would grow if its radius increased by a fluctuation. Fig. 7 shows the dimensionless vapour pressure and the dimensionless equilibrium temperature as functions of the reduced film thickness /r0 . The curves correspond to different values of the dimensionless quantity A/(r02 ) at parameters noted in Figs. 5 and 6. Larger values of the ratio A/(r02 ) mean a stronger interaction between the solid particle and the liquid film. This interaction results in a disjoining film pressure which requires a lower vapour pressure at constant temperature and a higher temperature at constant pressure in comparison to a particle-free droplet. Since the Hamaker constant specifies the interaction energy of the film with the particle, the quantity A/(r02 ) measures this energy in terms of the free surface energy. The above considerations are also valid for the case of a soluble spherical particle. In such a case Eq. (28) has to be supplemented by the osmotic pressure, giving pV pV h TV − TV ∞ vL∞ pV ∞ ln = + −1 pV ∞ RTV TV ∞ RTV ∞ pV ∞ vL∞ pV ∞ 1 A 1 −2 RTV ∞ r0 pV ∞ 2 r02 ((r/r0 ) − 1)3 1 + ln(1 − x). − r/r0
(35)
5632
J. Mitrovic / Chemical Engineering Science 61 (2006) 5925 – 5933
glutamate. They considered all the effects treated in the literature that could reduce the supersaturation, but not the disjoining film effect. Also Wagner et al. (2003) give a summary of experiments on heterogeneous nucleation including vapour mixtures and soluble solid particles which are not completely dissolved. The authors varied the contact angle to bring their own experiments in a closer agreement with the Fletcher (1958) theory, but did not consider the disjoining action of the condensate film covering the solid nucleus. Viewed in the light of these experiments, the present paper—focussing on the disjoining film effect—seems to provide a way of how to narrow the discrepancy between the measured and calculated supersaturation. The present consideration concerning the disjoining film effect is principally immediately applicable to multicomponent films. Regarding condensation and nucleation of binary vapours the reader may be referred to a paper by Konopka (1997) for further references. In this context, however, it should be noted that tiny droplets, considered as nano-particles, may possess several degrees of freedom that affect the equilibrium pressure in comparison to a stationary droplet usually adopted for theoretical considerations. A paper by Kuhrt (1952) gives details on this subject, see also Abraham (1974).
Fig. 7. Isothermal (above) and isobaric (below) equilibrium of a droplet without and with a solid particle.
The concentration c of the solute in the liquid film, covering the solid nucleus of a soluble substance, can be obtained from ln
vL∞ c =2 , c∞ RTV ∞ r
(36)
where c∞ denotes the solubility of the solution with the plane interface (Webster and Cates, 1998). Note that Eq. (36) accounts for the effect of the interface curvature on the solubility and not the effect of the solid particle itself. The latter might also be important and has to be considered in a detailed analysis. 6. Experimental evidence and conclusions Experiments on nucleation under heterogeneous conditions are extensive and very complex. Chen et al. (1998, 2000) give an overview in this area and provide new data sets. The main conclusion from their experiments should be given here as originally stated: “. . ., if the macroscopic theory of heterogeneous nucleation is valid, then the results suggest that both SiC and SiO2 aerosols behave as having an ability to induce heterogeneous nucleation better than perfectly wetted particles.” The authors (Chen et al., 1998, 2000) experimented with water vapour using submicrometer particles of SiC, SiO2 , and naphthalene, as well as with n-butanol vapour and submicrometer charged and neutral particles of lactose and monosodium
Notation a A A0 c C f G h h m n p pL pV p q r R R T TL TV v V VL VV x
constant constant Hamaker constant concentration constant fugacity Gibbs free energy specific enthalpy, h in ideal state enthalpy of phase change (hV − hL ) exponent number of moles (solute) pressure pressure in liquid, pL∞ in reference state vapour pressure, pV ∞ in reference state pressure jump across interface, (p)∞ in reference state electric charge radius specific gas constant universal gas constant temperature liquid temperature vapour temperature, TV ∞ in reference state partial molar volume volume of droplet specific liquid volume, vL∞ in reference state specific vapour volume, vV ∞ in reference state mole fraction of solute
J. Mitrovic / Chemical Engineering Science 61 (2006) 5925 – 5933
Greek letters
film thickness external potential dielectric constant curvature of interface chemical potential osmotic pressure density surface tension
Indices eq L q S V 0 ∞ ∗ max
equilibrium, at pV = pV ∞ , TV = TV ∞ liquid due to electric charge solvent vapour due to film thickness solid particle reference state, plane interface equilibrium at maximum of pV
References Abraham, F.F., 1974. Homogenous Nucleation Theory. Academic Press, New York. Aitken, J., 1880–1881. On dust, fogs, and clouds. Transactions of the Royal Scottish Society of Arts, Edinburgh 30. Also Collected Scientific Papers of John Aitken, Cambridge University Press, Cambridge, 1923, pp. 34–64. Amelin, A.G., 1967. Theory of Fog Condensation. IPST Jerusalem. Best, A.C., 1955. Atomic explosion and condensation nuclei. Meteorological Magazine 84, 201–204. Byers, H.R., 1965. Vapour pressure over solution droplets. Journal of Chemical Education 42, 338–339. Chen, C.-C., Guo, M.-S., Tsai, Y.-J., Huang, C.-C., 1998. Heterogeneous nucleation of water vapor on submicrometer particles of SiC, SiO2 , and naphthalene. Journal of Colloid and Interface Science 198, 354–367. Chen, C.-C., Tao, C.-J., Shu, H.-J., 2000. Heterogeneous nucleation on nbutanol vapor on submicrometer charged and neutral particles of lactose and monosodium glutamate. Journal of Colloid and Interface Science 224, 11–22. Clapeyron, E., 1834. Puissance motrice de la chaleur. Journal de l’Ecole Polytechnique 14, 153–191. Also: Über die bewegende Kraft der Wärme. Annalen der Physik 135, 451–446, 566–586. Clausius, R., 1850. Über die bewegende Kraft der Wärme und die Gesetze, die sich daraus für die Wärmelehre selbst ableiten lassen. Annalen der Physik 200, 368–397 500–524. Derjaguin, B.V., Churaev, N.V., Muller, V.M., 1987. Surface Forces. Consultants Bureau, New York. Dufour, L., Defay, R., 1963. Thermodynamics of Clouds. Academic Press, New York. Fletcher, N.H., 1958. Size effect in heterogeneous nucleation. Journal of Chemical Physics 29, 572–576. Fletcher, N.H., 1962. The Physics of Rainclouds. Cambridge University Press, Cambridge. Fletcher, N.H., 1970. The Chemical Physics of Ice. Cambridge University Press, Cambridge. Gorbunov, B., Hamiltion, R., 1997. Water nucleation on aerosol particles containing both soluble and insoluble substances. Journal of Aerosol Science 28, 239–248.
5633
Hanel, G., 1976. Properties of atmospheric aerosol particles as function of the relative humidity at thermodynamic equilibrium with the surrounding moist air. Advances in Geophysics 19, 73–188. Köhler, H., 1926. Zur Thermodynamik der Kondensation an hydroskopischen Kernen und Bemerkungen über das Zusammenfließen der Tropfen. Med. Statens Meteorol.-Hydrogr. Anstalt (Stockholm) 3 (8), Köhler, H., 1950. On the problem of condensation in the atmosphere. Nova Acta Regiae Societatis Scientarum Upsaliensis Serie 14 (9), Konopka, P., 1997. Binary condensation on soluble aerosols—On the activation and growth of the stratospheric aerosol from coupled HNO3 –H2 O uptake. Journal of Aerosols Science 28, 1411–1424. Kuhrt, F., 1952. Das Tropfenmodell übersättigter realer Gase. Zeitschrift für Physik 131, 205–214. Laaksonen, A., Talanquer, V., Oxtoby, D.W., 1995. Nucleation: measurements, theory, and atmospheric applications. Annual Review of Physical Chemistry 46, 489–524. Latham, J., 1969. Cloud physics. Reports on Progress in Physics 32, 69–134. Martin, S.T., 2000. Phase transition of aqueous atmospheric particles. Chemical Reviews 100, 3403–3453. Mason, B.J., 1971. The Physics of Clouds. Clarendon Press, Oxford. McDonald, Y.E., 1962. Homogeneous nucleation of vapour condensation I. American Journal of Physics 30, 870–877. Mirabel, P., Reiss, H., Bowles, R.K., 2000. A theory for the deliquescence of small particles. Journal of Chemical Physics 113, 8200–8205. Mitrovic, J., 2004. On the equilibrium conditions of curved interfaces. International Journal of Heat Mass Transfer 47, 809–818. Reiss, H., Koper, G.J.M., 1995. The Kelvin equation: stability, fluctuation, and factors involved in measurement. Journal of Physical Chemistry 99, 7837–7844. Russell, L.M., Ming, Y., 2002. Deliquescence of small particles. Journal of Chemical Physics 116, 311–321. Sorjamaa, R., Svennigsson, B., Raatikainen, T., Henning, S., Bilde, M., Laaksonen, A., 2004. The role of surfactants in Köhler theory reconsidered. Atmospheric Chemistry and Physics 4, 2107–2117. Talanquer, V., Oxtoby, D.W., 2003. Formation of droplets on non-volatile soluble particles. Journal of Chemical Physics 119, 9121–9128. Thomson, J.J., 1886. Applications of Dynamics to Physics and Chemistry. Macmillan, London. Thomson, J.J., 1903. Conduction of Electricity Through Gases. Cambridge. Thomson, W., 1871. On the equilibrium of vapour at a curved surface of liquid. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science Series 4 42, 448–453. Tohmfor, G., Volmer, M., 1938. Die Keimbildung unter dem Einfluss elektrischer Ladungen. Annalen der Physik, 5 Folge 33, 109–131. Van Ness, H.C., Abbott, M.M., 1982. Classical Thermodynamics of Nonelectrolyte Solutions. McGraw-Hill Book Company, New York. von Helmholtz, R., 1886. Untersuchungen über Dämpfe und Nebel besonders über solche von Lösungen. Annalen der Physik und Chemie N. F. 27, 508–543. Wagner, P.E., Vali, G. (Eds.), 1988. Atmospheric Aerosols and Nucleation. Springer, Berlin. Wagner, P.E., Kaller, D., Vrtala, A., Lauri, A., Kulmala, M., Laaksonen, A., 2003. Nucleation probability in binary heterogeneous nucleation of water—n-propanol mixtures on insoluble and soluble nanoparticles. Physical Review E 67, 021605. Webster, A.J., Cates, M.E., 1998. Stabilization of emulsions by trapped species. Langmuir 14, 2068–2079. Wright, H.L., 1936. The size of atmospheric nuclei: some deductions from measurements of the number of charged and uncharged nuclei at Kew Observatory. Proceedings of the Physical Society 48 (Pt. 5), 675–689.
Further reading McDonald, Y.E., 1963. Homogeneous nucleation of vapour condensation II. American Journal of Physics 31, 31–41.