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The role of surface forces in heterogeneous nucleation
349
ColloidsandSurfaces, 61(1991)349-351 Elsevier Science Publishers B.V., Amsterdam
Brief Note
The role of surface forces in heterogeneous A-1. Rusanov
nucleation
and F.M. Kuni
Lrningrad State University, Leningrad
199034, USSR
(Received 14 June 1991;accepted 18 June 1991)
As an example, we consider the case of condensation of a gas on a small, wettable, spherical solid particle. A thin adsorbed film forms on the particle under the influence of surface forces. Usually, the vapor pressure and the chemical potential, pa, of an adsorbed film r.re lower than those of the bulk phase Q. pa increases while the film thickens. However, thickening is accompanied by an increase in the curvature radius, r, of the outer surface of the film. When the internal state of the film approaches the state of the bulk phase, the role of the curvature of the outer surface of the film becomes significant; the chemical potential p” should decrease with increasing r in accordance with the Kelvin equation. Thus, the superposition of the laws of behavior of thin films and curved surfaces leads to the result that the chemical potential of the film passes through a maximum when the film thickness is increased. The maximum corresponds to the limiting supersaturation which appears to be smaller for the case under consideration than for homogeneous nucleation. A quantitative theory may be constructed as follows. For a onecomponent phase, o(, at a given temperature, the chemical potential pa is determined only by the pressure pa, i.e. dpa = v” dp”
0)
where v is the molecular volume. We may set ua x const. for a condensed phase, so that the integration of Eqn (1)yields /.ia-ji~wla(pa-p~)
(3
where the subscript 03 refers to a flat boundary and p where r = h = 00 (h is the film thickness). In the case of a curved thin film, the pressure 0166-6622/91/$03.50
0
1991 Elsevier
between pa
Science Publishers
the phases CL
is often represented B.V. All rights reserved.
350
as the sum of the capillary p”
=pp
+
(20,/r)
and disjoining
pressures,
i.e.
- n(h)
(3)
where 0 is the surface tension and n(h) Substituting Eqn (3) into Eqn (2) gives
is the disjoining
pressure.
/P(h) Z /:‘, + u” [20, /(r, + h) - n(h)]
(4,
where rn = r - h is a given radius of the particle. Equation (4) rela.tes the dependence of the chemical potential of a curved film (as a function of the film thickness) to the disjoining pressure isotherm, II(h). I7 is greater than zero for stable films and, as is seen from Eqn (4), the disjoining pressure lowers the chemical potential whereas the capillary pressure increases the chemical potential. Introducing supersaturation 5 as defined by the expression =kTln(lt<)
P-P%
(5)
where k is Boltzmann’s rewrite Eqn (4) as ln(l
constant
+ s) z (u”/kT)[2cr,/(r,
and T is the temperature,
+ h) - n(h)]
(6)
The limiting supersaturation for the condensation particle is characterized by the f:ondition
dp”ldhlh+,, When
=
process
on the (7)
0
combined
ll’(ho)(r,
we may
with Eqn (4), this leads to the equation
+ ho)’ z5 - 20,
where R’= dll/dh, from which the thickness ho corressf,nding to the limiting supersaturation can be calculated. Several kinds of disjoining pressure isotherms are ’ 3wn from theory and experiment. Let us consider, as an example, th_ condensation of water on quartz particles. For flat films of water on quartz, experiment shows the following dependence [l,Zj: I7(h)=Kexp(-I+)
(9)
with constants K = 9.94 lo6 N rnB2 and Eqn (9) into Eqn (8) yields the equation l
r, e (21a,/IC)li2
exp (h,/21)
- ho
2 = 2.33 nm.
Substitution
of (10)
from which the limiting thickness, ho, can be calculated for a given particle radius. r,. Setting c’m = 75.6 mN m-l at 273 K, we obtain h ,~4nmforr,=10nmandh0~ 8 nm for r, = 20 nm. Respectively, we find the limiting supersaturation values so x 0.1 and go w 0.05 from
351
Eqn (6). Such small supersaturation values explain the fundamentally important fact that the water phase transitions in the atmosphere occur at very small supersaturations. REFERENCES 1 2
B.V. Derjaguin and Z.M. Zorin, Zh. Fiz. Khim., 29 (1955) 1755. N.V. Churaev and V.M. Muller, Surface Forces, B.V. Derjaguin, Bureau, New York, 1987.
Consultant
ColloidsandSurfaces, 61(1991)349-351 Elsevier Science Publishers B.V., Amsterdam
Brief Note
The role of surface forces in heterogeneous A-1. Rusanov
nucleation
and F.M. Kuni
Lrningrad State University, Leningrad
199034, USSR
(Received 14 June 1991;accepted 18 June 1991)
As an example, we consider the case of condensation of a gas on a small, wettable, spherical solid particle. A thin adsorbed film forms on the particle under the influence of surface forces. Usually, the vapor pressure and the chemical potential, pa, of an adsorbed film r.re lower than those of the bulk phase Q. pa increases while the film thickens. However, thickening is accompanied by an increase in the curvature radius, r, of the outer surface of the film. When the internal state of the film approaches the state of the bulk phase, the role of the curvature of the outer surface of the film becomes significant; the chemical potential p” should decrease with increasing r in accordance with the Kelvin equation. Thus, the superposition of the laws of behavior of thin films and curved surfaces leads to the result that the chemical potential of the film passes through a maximum when the film thickness is increased. The maximum corresponds to the limiting supersaturation which appears to be smaller for the case under consideration than for homogeneous nucleation. A quantitative theory may be constructed as follows. For a onecomponent phase, o(, at a given temperature, the chemical potential pa is determined only by the pressure pa, i.e. dpa = v” dp”
0)
where v is the molecular volume. We may set ua x const. for a condensed phase, so that the integration of Eqn (1)yields /.ia-ji~wla(pa-p~)
(3
where the subscript 03 refers to a flat boundary and p where r = h = 00 (h is the film thickness). In the case of a curved thin film, the pressure 0166-6622/91/$03.50
0
1991 Elsevier
between pa
Science Publishers
the phases CL
is often represented B.V. All rights reserved.
350
as the sum of the capillary p”
=pp
+
(20,/r)
and disjoining
pressures,
i.e.
- n(h)
(3)
where 0 is the surface tension and n(h) Substituting Eqn (3) into Eqn (2) gives
is the disjoining
pressure.
/P(h) Z /:‘, + u” [20, /(r, + h) - n(h)]
(4,
where rn = r - h is a given radius of the particle. Equation (4) rela.tes the dependence of the chemical potential of a curved film (as a function of the film thickness) to the disjoining pressure isotherm, II(h). I7 is greater than zero for stable films and, as is seen from Eqn (4), the disjoining pressure lowers the chemical potential whereas the capillary pressure increases the chemical potential. Introducing supersaturation 5 as defined by the expression =kTln(lt<)
P-P%
(5)
where k is Boltzmann’s rewrite Eqn (4) as ln(l
constant
+ s) z (u”/kT)[2cr,/(r,
and T is the temperature,
+ h) - n(h)]
(6)
The limiting supersaturation for the condensation particle is characterized by the f:ondition
dp”ldhlh+,, When
=
process
on the (7)
0
combined
ll’(ho)(r,
we may
with Eqn (4), this leads to the equation
+ ho)’ z5 - 20,
where R’= dll/dh, from which the thickness ho corressf,nding to the limiting supersaturation can be calculated. Several kinds of disjoining pressure isotherms are ’ 3wn from theory and experiment. Let us consider, as an example, th_ condensation of water on quartz particles. For flat films of water on quartz, experiment shows the following dependence [l,Zj: I7(h)=Kexp(-I+)
(9)
with constants K = 9.94 lo6 N rnB2 and Eqn (9) into Eqn (8) yields the equation l
r, e (21a,/IC)li2
exp (h,/21)
- ho
2 = 2.33 nm.
Substitution
of (10)
from which the limiting thickness, ho, can be calculated for a given particle radius. r,. Setting c’m = 75.6 mN m-l at 273 K, we obtain h ,~4nmforr,=10nmandh0~ 8 nm for r, = 20 nm. Respectively, we find the limiting supersaturation values so x 0.1 and go w 0.05 from
351
Eqn (6). Such small supersaturation values explain the fundamentally important fact that the water phase transitions in the atmosphere occur at very small supersaturations. REFERENCES 1 2
B.V. Derjaguin and Z.M. Zorin, Zh. Fiz. Khim., 29 (1955) 1755. N.V. Churaev and V.M. Muller, Surface Forces, B.V. Derjaguin, Bureau, New York, 1987.
Consultant