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On the application of thermodynamic potentials to the theory of heterogeneous nucleation
On the Application of Thermodynamic Potentials to the Theory of Heterogeneous Nucleation Y. ZIMMELS Mineral Engineering Department, Technion l i T , Haifa, Israel
Received August 15, 1975; accepted February 19, 1976 The application of thermodynamic potential to the nucleation theory is considered, and the Grand Canonical potential is being used as a suitable potential to describe the nucleation system. It is suggested that there is a thermodynamic path of heterogeneous nucleation whereby the free energy of formation of the nuclei is rendered a minimum value. This path depends on a particular degree of spreading of the spherical cap-shaped nucleus, which is determined in turn by the imposed conditions of minimum free energy of formation. INTRODUCTION
The purpose of this paper is to show that there is a thermodynamic path whereby the free energy of formation in heterogeneous nucleations is rendered a minimum value. As the spherical cap-shaped nucleus is capable of changing its curvature by spreading, as well as by increasing its volume, the free energy of formation should depend on the degree of spreading as well as on the volume of the nucleus. By use of the Grand Canonical potential it is shown that the preferable path of minimum free energy of formation is dependent on a particular degree of spreading of the nucleus. THEORY
As shown by the classical nucleation theory, the free energy of formation of a nucleus from its supersaturated vapors is given by (1) AGh .... = ~zrr~AG~ + 4rrr2"rLV
[11
2 -- 3 cos 0 + cos'~ 0 AGhet = ~rr3AG~
4
+2~rr~(1 -- cos O)3'LV +~,~ sin ~ o ( ~ aG~ = - ( R T / v ) In (Pr'/P~')
-
r~)
I-2~
~3-]
where AGhomis the free energy of formation of a spherical nucleus in homogeneous nucleation and AGhot is the free energy of formation of a spherical cap-shaped nucleus on a flat solid surface in heterogeneous nucleation. The first term on the left is a bulk term, the rest being due to interfacial effect. The latter can be regarded as a summation of a two-step nucleation process, as was suggested by Gibbs (1, 2). The process consists of the formation within a supersaturated vapor ( ' ) of pressure pr', of a critical nucleus of radius re, volume Vo, and containing c molecules. Once formed, the pressure within the nucleus will be pr" = 2~,v/rc. Imagine a bulk of the liquid ( " ) confined in a cylinder by a piston under the pressure p / ' . The chemical potential of the confined liquid will be the same as that of the external vapor. Now let c molecules condense into the cylinder by withdrawal of the piston. The system does work ( p 2 " - - p 2 ' ) V o , which accomplishes the first step. In the second step, a small droplet of radius ro is extruded into the vapor. The work done on the system is gained as interfacial energy, i.e., 41rro23,. In this way, Eqs. [-1] and [2"] are obtained. The two-step approach is helpful, since the nucleation process is generally described either by the Gibbs or by the
446 Journal of Colloid and Interface Science, Vol. 57, No. 3. December 1976
Copyright ~ 1976 by Academic Press, Inc. All rights of reproduction in any form reserved.
HETEROGENEOUS
447
NUCLEATION
tlchuholtz potentials. These potentials are the previously. This means that, when E(T, u) is Legendre transformation of the total energy used, the free energy of formation is obtaincd E=E(S,V,N) into E = E ( T , p , N ) and by simple integration. In homogeneous nucleation, the nucleus E = E ( T , V, N), respectively. The Gibbs and Helmholtz potential are not functions of the radius (rs) is assumed to be the only geometchemical potential that characterize nucleation rical variable, the value of which determines the curvature of the nucleus. Accordingly, systems. However, the Grand Canonical potential is tz = #(r,). Heterogeneous nucleation is a difa function of T, #, V, since it is the Legendre ferent case, since the nucleus is free to change transformation of the total energy into its curvature by spreading, as well as by inE = E ( T , u). This notation (3) does not in- creasing its volume (i.e., growing). Therefore, clude V and should be understood as an it is useful to define the parameters that can abbreviation. The application of the Grand be used to describe either the extent of growth Canonical potential should be well established or the degree of spreading. The geometrical once it is realized that the nucleation system is contact angle q, (not the equilibrium angle) at characterized by its chemical potential, tem- constant nucleus volume will measure the perature, and volume. Recently it was clearly degree of spreading (i.e., it is a shape stated by the "theory of small systems" (4, 5) parameter) whereas the nucleus radius r~ at that the environmental variables of systems of constant 4~ will measure the extent of growth the type discussed here are ~, T, V (or/~, T, p), (i.e., it is a size parameter), r, is in fact given depending upon whether the small system is by r, = [(3/4~-)V3~. The radius of curvature subject to volume or pressure variations. Sys- of the nucleus r is a flmction both of the tems like colloidal particles, micelles (6), volume V and the degree of spreading, i.e., macromolecules, and crystallite embryos, are r = r(r~, 4~) hence # = #(r8,4~) and E ( T , # ) also included in this category. Some concepts = E ( T , r,, 4~). The critical radius r,, can be relating the small-system thermodynamics to evaluated by the application of the Grand Canonical potential will be introduced later. ~r, -m, = 0 [6] The general application of E ( T , ~) to the theory of heterogeneous nucleation will be in the same way as in the homogeneous discussed first. The complete differential of E ( T , # ) (3), nucleation. However, since 4~ is independent of r,, Eq. [63 should be subjected to including the interfacial term, is given by IOE(T, u) ]
dE(T, ~) = - - S d T -- pdV -- Nd~
•
=
o.
[7]
+ Z .~,dA,. [ 4 ] i
Integration of Eq. [4] at constant temperature and volume yields E ( T , tz) -- E ( T , uo) = - - N ( u -- #o)
+ Z -~(a, - A,0).
[5]
i
Equation [5] is the general form of Eqs. [-1] and [2], and includes, by the very definition of E ( T , I~) and given thermodynamic process, each term of the two-step process described
Equation [7] is a simple but rather fundamental thermodynamic condition. Its meaning can be described as follows. The heterogeneous nucleation system should follow a thermodynamic path which renders the system minimum free energy of formation. If such a minimum exists, it should satisfy Eq. [7"]. Figure 1 shows an example of a typical isothermal three-dimensional surface E ( T , ~ ) = E (T, r,, ~). The figure shows that there is a definite path for which the nucleation system
Journal of Colloid and Interface Science, Vol. 57, No. 3, December 1976
448
Y. ZIMMELS
E(T,lu)
J
/
~J
/J FIG. 1. Schematicrepresentation of E(T, I~) = E(T, rs, ¢).
has the minimum free energy of formation. Obviously, this particular path is thermodynamically preferable. Furthermore, if equilibrium is to persist, the equilibrium contact angle (0) should satisfy Eq. [-7-]. We proceed to look at possible thermodynamic approaches suitable to solve Eq. [7-] via Eq. [-4-]. The small-system approach (4-6) is more detailed and takes into consideration the subdivision potential. The latter describes the tendency of the small system to subdivide further, and is a function of the thermodynamic state of the system. Although still a theoretical concept it is highly relevant to the understanding of nucleation systems. However, the classical approach, which will be discussed first, offers a simple and clear view, but should be regarded rather as an approximation.
the spherical cap-shaped nucleus. Since in Eq. [7], re = const (i.e., V = const), differentiations of Eq. [-4] with respect to q~ and setting the result to zero should yield the conditions characterizing the path of minimum free energy of formation. Thus we have OE(T, u)"l
J
O# Oep
OAi '
Oep
. Do-1
Combining Eqs. [9] and [10] and setting the result to zero yields 1 Or 2 V ' Y L V - - - - + ~ 7 i ( O A i / 0 4 ) = 0. r 2 0¢
[11]
The contact angle that satisfies Eq. [11] is the one with which the minimum free energy of formation of the heterogeneous nucleation system is obtained.
THE CLASSICALAPPROACH THE SMALL-SYSTEM APPROACH The classical approach employs basically macroscopic thermodynamics. The chemical potential of the supersaturated vapor phase is given by -- #o = R T l n ( p r ' / p . ' ) .
[-8]
Once the critical nuclei are formed, i.e., r~ > re, #-
#o = (2"YLv/r)v,
[-9]
Before the chemical potential term is evaluated, some introductory basic equations will be discussed. The notation used in this section, which is similar to that suggested by Hill (4, 5), differentiates between total and average values over a small system. The complete differential of the total energy E, is given by dE, = TdS, -- p d V , + Y~ # , d N . + d n
where r = r(rs, ¢) is the radius of curvature of Journal of Colloid and Interface Science, Vol. 57, No. 3, December 1976
i
[-12]
HETEROGENEOUS NUCLEATION where [,13-1
N~ = N ' n
is the subdivision potential (4), and n the number of small systems each containing an average of N' molecules. The following average quantities over a small system were defined : E = E,/n,
S = S,/n,
X,' = X . / n
04-I
and Eq. [,12-] takes the form dE = T d S -
PdV + ~ mdN/.
[-15-]
i
The extensive variables in Eq. [,15,] are now average quantities over a small system. The differentials of the familiar Legendre transformation of E (i.e., G and F) take the form (3) dG = - - S d r + V d P + Z m d N {
[-16]
i
dF = - S d T
- P d V + Y'. u , d N / .
[17]
i
For the present purpose it is easier to work with total measurable variables than to employ unknown average quantities. Therefore, it is suggested that Eq. [,,12-] be treated differently. Combining Eqs. [-12,] and [-13] yields dEt = T d S ~ - P d V t
+ zu~dy. + d x . / X L
[,18-]
We are interested in the case N / = const, since the spreading of the spherical cap-shaped nucleus involves neither volume changes nor exchange of matter. Equation [-18-] may be put in the form
449
chemical potential, the complete differential of the Grand Canonical potential takes the form d E ( T , ~) = - - S , d T -- P d V , -- Y] Nt,d¢~ -F Z 7,dA,. i
[-22]
i
Although the detailed functional dependence of ~/N' and/7 on q~is not yet known, Eq. [-22-] is valuable to the understanding of the mechanism and stages of heterogeneous nucleation. The larger is e and the smaller is N' (i.e., the size of the nucleus) the larger will be/7. This would result in a smaller minimum free energy of formation. An extreme example of a high value of e / N ' (and hence of/7) may be suggested as being within the category of adsorption. Each adsorption site can be regarded as a heterogeneous nucleation site. Also, the patchwise adsorption can be regarded as the first degenerate nucleation typified by a high value of the subdivision potential. As the adsorption builds up, the subdivision potential is expected to decrease (while N' increases) resulting in the formation of the cap-shaped nuclei. This simple model shows that the contact angle should be influenced by the state of subdivision, i.e., by the generalized chemical potential /7. As the nuclei increase in size, E/N'--~O, i.e., /7---~u and the macroscopial classical approach can be applied to the determination of the minimum free energy of formation of the already stable droplet. CONCLUSIONS
(a) Nucleation systems (at constant temperature and volume) are characterized by their chemical potential. In heterogeneous + E (m + ,/X,')dNt<. [-19-] nucleation, the chemical potential is a function i of the extent of growth, (rs) as well as the A new chemical potential is defined, namely: degree of spreading (~), i.e., # = #(r,, ~). Z~ = u~ + # N , ' . [-20-] (b) A suitable Legendre transformation of the total energy, that can be used in explorCombining Eqs. [-19-] and [-20-], we get ing the thermodynamics of nucleation systems ,lEt = TdS, -- P d V t + Z £ f l N , . [,21-] is the Grand Canonical potential E ( T , u). i (c) The application of E ( T , # ) to the In the light of the new definition of the heterogeneous nucleation shows that there is a
dE~ = TdSt -- P d V t
Journal of Colloid and Interface Science, Vol. 57, No. 3, December 1976
450
Y. ZIMMELS
preferable nucleation path whereby the free energy of formation is rendered a minimum value. (d) If the rate of change of E(T, ~) with 4~ (at a constant T, r,) is set to zero, the minimum free energy of formation is obtained. (e) The small-system approach indicates that a range of subdivision potentials m a y be related to a range in which adsorption turns into heterogeneous nucleation. APPENDIX : NOMENCLATURE
A c
= Interfacial area. = Number of molecules which form a nucleus. E = Total energy or the average over a small system of the total energy. E(T, u) = The Grand Canonical potential. F = The Helmholtz potential. AGv, AGh...... AGh~ = Gibbs free energy change per unit volume, in homogeneous and in heterogeneous nucleation, respectively. i = Subindex denoting the ith component, or ith type of interface. )v = Number of moles. N' = Number of unit (e.g., molecules) forming the small system. n = Number of small systems. p, p/, p~' = Pressure : general, above interface with radius of curvature r, and ~ , respectively. r = Radius of curvature.
r, re R S
t T v
V, Vc ~, ~LV, ~SL, 7sv
0 4~ #, #0
= Size paralneter. = Radius of the critical nuclei. = Gas constant. = Entropy, or the average over a small system of the total entropy. = Subindex denoting total amount. = Absolute temperature. = Molar volume. = Volume, and volume of the critical nucleus. = Interfacial tensions: general, liquid-vapor, solidliquid, and solid-vapor, respectively. = Contact angle. = The angle denoting the degree of spreading. = Chemical potential: general and of standards reference state. = Chemical potential including effect of subdivision potential. = Subdivision potential. REFERENCES
1. ZETTLE3KOYER, A. C., Ed., "Nucleation," Marcel Dekker, New York, 1960. 2. GIBBs, J. W., loc. cit. (1906), ref. 57 p. 252. 3. CALLEN, H. B., "Thermodynamics," Wiley, New York, 1960. 4. HILL, T. L., "Thermodynamics of Small Systems," Benjamin, New York, Part I, 1963 ; Part II, 1964. 5. HILL, T. L., J. Chem. Phys. 36, 12, 153 (1962). 6. HALL, D. G. AND PI~TmCA, B. A., in "Surfactant Science," Series 1, "Nonionic Surfactants," (M. J. Schick, Ed.), p. 516, Marcel Dekker, New York, 1967.
Journal of Colloid and Inlerface Science, Vol. 57, No. 3, December 1976
Received August 15, 1975; accepted February 19, 1976 The application of thermodynamic potential to the nucleation theory is considered, and the Grand Canonical potential is being used as a suitable potential to describe the nucleation system. It is suggested that there is a thermodynamic path of heterogeneous nucleation whereby the free energy of formation of the nuclei is rendered a minimum value. This path depends on a particular degree of spreading of the spherical cap-shaped nucleus, which is determined in turn by the imposed conditions of minimum free energy of formation. INTRODUCTION
The purpose of this paper is to show that there is a thermodynamic path whereby the free energy of formation in heterogeneous nucleations is rendered a minimum value. As the spherical cap-shaped nucleus is capable of changing its curvature by spreading, as well as by increasing its volume, the free energy of formation should depend on the degree of spreading as well as on the volume of the nucleus. By use of the Grand Canonical potential it is shown that the preferable path of minimum free energy of formation is dependent on a particular degree of spreading of the nucleus. THEORY
As shown by the classical nucleation theory, the free energy of formation of a nucleus from its supersaturated vapors is given by (1) AGh .... = ~zrr~AG~ + 4rrr2"rLV
[11
2 -- 3 cos 0 + cos'~ 0 AGhet = ~rr3AG~
4
+2~rr~(1 -- cos O)3'LV +~,~ sin ~ o ( ~ aG~ = - ( R T / v ) In (Pr'/P~')
-
r~)
I-2~
~3-]
where AGhomis the free energy of formation of a spherical nucleus in homogeneous nucleation and AGhot is the free energy of formation of a spherical cap-shaped nucleus on a flat solid surface in heterogeneous nucleation. The first term on the left is a bulk term, the rest being due to interfacial effect. The latter can be regarded as a summation of a two-step nucleation process, as was suggested by Gibbs (1, 2). The process consists of the formation within a supersaturated vapor ( ' ) of pressure pr', of a critical nucleus of radius re, volume Vo, and containing c molecules. Once formed, the pressure within the nucleus will be pr" = 2~,v/rc. Imagine a bulk of the liquid ( " ) confined in a cylinder by a piston under the pressure p / ' . The chemical potential of the confined liquid will be the same as that of the external vapor. Now let c molecules condense into the cylinder by withdrawal of the piston. The system does work ( p 2 " - - p 2 ' ) V o , which accomplishes the first step. In the second step, a small droplet of radius ro is extruded into the vapor. The work done on the system is gained as interfacial energy, i.e., 41rro23,. In this way, Eqs. [-1] and [2"] are obtained. The two-step approach is helpful, since the nucleation process is generally described either by the Gibbs or by the
446 Journal of Colloid and Interface Science, Vol. 57, No. 3. December 1976
Copyright ~ 1976 by Academic Press, Inc. All rights of reproduction in any form reserved.
HETEROGENEOUS
447
NUCLEATION
tlchuholtz potentials. These potentials are the previously. This means that, when E(T, u) is Legendre transformation of the total energy used, the free energy of formation is obtaincd E=E(S,V,N) into E = E ( T , p , N ) and by simple integration. In homogeneous nucleation, the nucleus E = E ( T , V, N), respectively. The Gibbs and Helmholtz potential are not functions of the radius (rs) is assumed to be the only geometchemical potential that characterize nucleation rical variable, the value of which determines the curvature of the nucleus. Accordingly, systems. However, the Grand Canonical potential is tz = #(r,). Heterogeneous nucleation is a difa function of T, #, V, since it is the Legendre ferent case, since the nucleus is free to change transformation of the total energy into its curvature by spreading, as well as by inE = E ( T , u). This notation (3) does not in- creasing its volume (i.e., growing). Therefore, clude V and should be understood as an it is useful to define the parameters that can abbreviation. The application of the Grand be used to describe either the extent of growth Canonical potential should be well established or the degree of spreading. The geometrical once it is realized that the nucleation system is contact angle q, (not the equilibrium angle) at characterized by its chemical potential, tem- constant nucleus volume will measure the perature, and volume. Recently it was clearly degree of spreading (i.e., it is a shape stated by the "theory of small systems" (4, 5) parameter) whereas the nucleus radius r~ at that the environmental variables of systems of constant 4~ will measure the extent of growth the type discussed here are ~, T, V (or/~, T, p), (i.e., it is a size parameter), r, is in fact given depending upon whether the small system is by r, = [(3/4~-)V3~. The radius of curvature subject to volume or pressure variations. Sys- of the nucleus r is a flmction both of the tems like colloidal particles, micelles (6), volume V and the degree of spreading, i.e., macromolecules, and crystallite embryos, are r = r(r~, 4~) hence # = #(r8,4~) and E ( T , # ) also included in this category. Some concepts = E ( T , r,, 4~). The critical radius r,, can be relating the small-system thermodynamics to evaluated by the application of the Grand Canonical potential will be introduced later. ~r, -m, = 0 [6] The general application of E ( T , ~) to the theory of heterogeneous nucleation will be in the same way as in the homogeneous discussed first. The complete differential of E ( T , # ) (3), nucleation. However, since 4~ is independent of r,, Eq. [63 should be subjected to including the interfacial term, is given by IOE(T, u) ]
dE(T, ~) = - - S d T -- pdV -- Nd~
•
=
o.
[7]
+ Z .~,dA,. [ 4 ] i
Integration of Eq. [4] at constant temperature and volume yields E ( T , tz) -- E ( T , uo) = - - N ( u -- #o)
+ Z -~(a, - A,0).
[5]
i
Equation [5] is the general form of Eqs. [-1] and [2], and includes, by the very definition of E ( T , I~) and given thermodynamic process, each term of the two-step process described
Equation [7] is a simple but rather fundamental thermodynamic condition. Its meaning can be described as follows. The heterogeneous nucleation system should follow a thermodynamic path which renders the system minimum free energy of formation. If such a minimum exists, it should satisfy Eq. [7"]. Figure 1 shows an example of a typical isothermal three-dimensional surface E ( T , ~ ) = E (T, r,, ~). The figure shows that there is a definite path for which the nucleation system
Journal of Colloid and Interface Science, Vol. 57, No. 3, December 1976
448
Y. ZIMMELS
E(T,lu)
J
/
~J
/J FIG. 1. Schematicrepresentation of E(T, I~) = E(T, rs, ¢).
has the minimum free energy of formation. Obviously, this particular path is thermodynamically preferable. Furthermore, if equilibrium is to persist, the equilibrium contact angle (0) should satisfy Eq. [-7-]. We proceed to look at possible thermodynamic approaches suitable to solve Eq. [7-] via Eq. [-4-]. The small-system approach (4-6) is more detailed and takes into consideration the subdivision potential. The latter describes the tendency of the small system to subdivide further, and is a function of the thermodynamic state of the system. Although still a theoretical concept it is highly relevant to the understanding of nucleation systems. However, the classical approach, which will be discussed first, offers a simple and clear view, but should be regarded rather as an approximation.
the spherical cap-shaped nucleus. Since in Eq. [7], re = const (i.e., V = const), differentiations of Eq. [-4] with respect to q~ and setting the result to zero should yield the conditions characterizing the path of minimum free energy of formation. Thus we have OE(T, u)"l
J
O# Oep
OAi '
Oep
. Do-1
Combining Eqs. [9] and [10] and setting the result to zero yields 1 Or 2 V ' Y L V - - - - + ~ 7 i ( O A i / 0 4 ) = 0. r 2 0¢
[11]
The contact angle that satisfies Eq. [11] is the one with which the minimum free energy of formation of the heterogeneous nucleation system is obtained.
THE CLASSICALAPPROACH THE SMALL-SYSTEM APPROACH The classical approach employs basically macroscopic thermodynamics. The chemical potential of the supersaturated vapor phase is given by -- #o = R T l n ( p r ' / p . ' ) .
[-8]
Once the critical nuclei are formed, i.e., r~ > re, #-
#o = (2"YLv/r)v,
[-9]
Before the chemical potential term is evaluated, some introductory basic equations will be discussed. The notation used in this section, which is similar to that suggested by Hill (4, 5), differentiates between total and average values over a small system. The complete differential of the total energy E, is given by dE, = TdS, -- p d V , + Y~ # , d N . + d n
where r = r(rs, ¢) is the radius of curvature of Journal of Colloid and Interface Science, Vol. 57, No. 3, December 1976
i
[-12]
HETEROGENEOUS NUCLEATION where [,13-1
N~ = N ' n
is the subdivision potential (4), and n the number of small systems each containing an average of N' molecules. The following average quantities over a small system were defined : E = E,/n,
S = S,/n,
X,' = X . / n
04-I
and Eq. [,12-] takes the form dE = T d S -
PdV + ~ mdN/.
[-15-]
i
The extensive variables in Eq. [,15,] are now average quantities over a small system. The differentials of the familiar Legendre transformation of E (i.e., G and F) take the form (3) dG = - - S d r + V d P + Z m d N {
[-16]
i
dF = - S d T
- P d V + Y'. u , d N / .
[17]
i
For the present purpose it is easier to work with total measurable variables than to employ unknown average quantities. Therefore, it is suggested that Eq. [,,12-] be treated differently. Combining Eqs. [-12,] and [-13] yields dEt = T d S ~ - P d V t
+ zu~dy. + d x . / X L
[,18-]
We are interested in the case N / = const, since the spreading of the spherical cap-shaped nucleus involves neither volume changes nor exchange of matter. Equation [-18-] may be put in the form
449
chemical potential, the complete differential of the Grand Canonical potential takes the form d E ( T , ~) = - - S , d T -- P d V , -- Y] Nt,d¢~ -F Z 7,dA,. i
[-22]
i
Although the detailed functional dependence of ~/N' and/7 on q~is not yet known, Eq. [-22-] is valuable to the understanding of the mechanism and stages of heterogeneous nucleation. The larger is e and the smaller is N' (i.e., the size of the nucleus) the larger will be/7. This would result in a smaller minimum free energy of formation. An extreme example of a high value of e / N ' (and hence of/7) may be suggested as being within the category of adsorption. Each adsorption site can be regarded as a heterogeneous nucleation site. Also, the patchwise adsorption can be regarded as the first degenerate nucleation typified by a high value of the subdivision potential. As the adsorption builds up, the subdivision potential is expected to decrease (while N' increases) resulting in the formation of the cap-shaped nuclei. This simple model shows that the contact angle should be influenced by the state of subdivision, i.e., by the generalized chemical potential /7. As the nuclei increase in size, E/N'--~O, i.e., /7---~u and the macroscopial classical approach can be applied to the determination of the minimum free energy of formation of the already stable droplet. CONCLUSIONS
(a) Nucleation systems (at constant temperature and volume) are characterized by their chemical potential. In heterogeneous + E (m + ,/X,')dNt<. [-19-] nucleation, the chemical potential is a function i of the extent of growth, (rs) as well as the A new chemical potential is defined, namely: degree of spreading (~), i.e., # = #(r,, ~). Z~ = u~ + # N , ' . [-20-] (b) A suitable Legendre transformation of the total energy, that can be used in explorCombining Eqs. [-19-] and [-20-], we get ing the thermodynamics of nucleation systems ,lEt = TdS, -- P d V t + Z £ f l N , . [,21-] is the Grand Canonical potential E ( T , u). i (c) The application of E ( T , # ) to the In the light of the new definition of the heterogeneous nucleation shows that there is a
dE~ = TdSt -- P d V t
Journal of Colloid and Interface Science, Vol. 57, No. 3, December 1976
450
Y. ZIMMELS
preferable nucleation path whereby the free energy of formation is rendered a minimum value. (d) If the rate of change of E(T, ~) with 4~ (at a constant T, r,) is set to zero, the minimum free energy of formation is obtained. (e) The small-system approach indicates that a range of subdivision potentials m a y be related to a range in which adsorption turns into heterogeneous nucleation. APPENDIX : NOMENCLATURE
A c
= Interfacial area. = Number of molecules which form a nucleus. E = Total energy or the average over a small system of the total energy. E(T, u) = The Grand Canonical potential. F = The Helmholtz potential. AGv, AGh...... AGh~ = Gibbs free energy change per unit volume, in homogeneous and in heterogeneous nucleation, respectively. i = Subindex denoting the ith component, or ith type of interface. )v = Number of moles. N' = Number of unit (e.g., molecules) forming the small system. n = Number of small systems. p, p/, p~' = Pressure : general, above interface with radius of curvature r, and ~ , respectively. r = Radius of curvature.
r, re R S
t T v
V, Vc ~, ~LV, ~SL, 7sv
0 4~ #, #0
= Size paralneter. = Radius of the critical nuclei. = Gas constant. = Entropy, or the average over a small system of the total entropy. = Subindex denoting total amount. = Absolute temperature. = Molar volume. = Volume, and volume of the critical nucleus. = Interfacial tensions: general, liquid-vapor, solidliquid, and solid-vapor, respectively. = Contact angle. = The angle denoting the degree of spreading. = Chemical potential: general and of standards reference state. = Chemical potential including effect of subdivision potential. = Subdivision potential. REFERENCES
1. ZETTLE3KOYER, A. C., Ed., "Nucleation," Marcel Dekker, New York, 1960. 2. GIBBs, J. W., loc. cit. (1906), ref. 57 p. 252. 3. CALLEN, H. B., "Thermodynamics," Wiley, New York, 1960. 4. HILL, T. L., "Thermodynamics of Small Systems," Benjamin, New York, Part I, 1963 ; Part II, 1964. 5. HILL, T. L., J. Chem. Phys. 36, 12, 153 (1962). 6. HALL, D. G. AND PI~TmCA, B. A., in "Surfactant Science," Series 1, "Nonionic Surfactants," (M. J. Schick, Ed.), p. 516, Marcel Dekker, New York, 1967.
Journal of Colloid and Inlerface Science, Vol. 57, No. 3, December 1976