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Comments on “the thermodynamics of cluster formation in nucleation theory”
Surface Science 104 (1981) L213-L216 North-Holland Publishing Company
SURFACE SCIENCE LETTERS
COMMENTSON“THETHERMODYNAMICSOFCLUSTERFORMATION INNUCLEATIONTHEORY" Kazumi NISHIOKA
*
Department of Metallurgical Engineering, Michigan Technological University, Houghton, Michigan 49931, USA
and Kenneth
C. RUSSELL
Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Received
4 August
1980; accepted
for publication
18 November
1980
(hereafter referred to as L-P) The controversy over the 1017 Lothe-Pound correction factor [l] to the rate equation in classical nucleation theory is still active [2,3]. Most authors attempted to disprove L-P, and in all cases but one the authors have resorted to statistical mechanical models for a droplet to advance their respective points. Blander and Katz [4] (hereafter referred to as B-K) utilized a rather novel approach in attacking this dilemma. They claimed that the Gibbs free energy of droplet formation had never been properly defined, and attempted a disproof of thr IL-P treatment by some rearrangements of the equations of the chemical thermcriynamics of nucleation. This note is intended to show that even where B-K’s ri~,?i.I!?i?!:.lisrnics are correct, their interpretations are in error, and that their methCX!‘:! -:$I*I’~i;sibly help resolve the controversy over the factor of 1017. I’ k. 1‘Ia :t;e law of mass action to write Ai
tlcip,,:‘p ‘) = -nRT
In S + [p,” -
np,],
(1)
wm rc p,, is the pressure of n-mer, and p” is the standard state pressure of n-mer (which B-K choose to take as the same as that for monomer). II,” is the chemical potential of the n-mer in the standard state, and pe is chemical potential of the bulk liquid phase under the saturation pressure pe. B-K note that because in classical nucleation theory the concentration of n-mer per unit volume N,, is given by N, = N1 exp(-AGIRT).
* On leave from Japan.
Department
(2)
of Precision
Mechanics,
Tokushima
University,
Tokushima
770,
L214
K. Nishioka, K.C. Russell / Thermodynamics of cluster formation
With an Nr prefactor, it is improper to identify the RHS of eq. (1) as AC. To do so would indeed violate the law of mass action. The consistent value of AG differs by a term RL ln(pI/po), B-K wrote the resulting function in the form AC=-(n-1)RTlnSt
[&&(n-l)~,,].
(3)
They then interpret the bracketed term as the “standard chemical potential change for one mole of monomer vapor + (n - 1) moles of liquid to form one mole of n-mer . . .‘I, and identify this term with the surface free energy, uA(n), in the classical nucleation theory. They then make the crucial assertion that since the difference pz - py appears in eq. (3) any contribution that free translation and rotation of the n-mer make to ~2 will be largely canceled by contributions of similar motions of the monomeric molecule to py. Let us rearrange eq. (3) in a rather more concise form: AG=-nRTlnS+
[&IIP,+RT~~O,~/~~)],
(3a)
or even more simply as: AG=-nRTlnS+
[&np,],
(3b)
where & is the chemical potential of the n-mer in a standard state of pr, and ~_l~is the chemical potential of the saturated vapor. We now consider the arguments of B-K from several points of view. Classical nucleation theory is ostensibly based upon the thermodynamics of Gibbs [5], who first calculated the reversible work (or free energy) of forming a droplet. Later authors have mainly been concerned with the prefactors to the exponential in eq. (2). Gibbs writes the reversible work as: LV=-VAp+uA(n),
(4)
where I/ denotes the volume enclosed by the dividing surface and Ap the pressure difference between the bulk liquid and the supersaturated vapor. The bulk liquid here possesses the same chemical potential as that of the vapor. Gibbs states that “W consists of two parts, one of which is always positive, and is expressed by the product of the superficial tension and the area of the surface of tension and the other is always negative, and is numerically equal to the product of the difference of the pressure by the volume of the interior mass”. Gibbs goes on to describe how the surface work could in principle be measured by extruding the droplet through an orifice in a body of pressurized bulk liquid. Although Gibbs employs the surface of tension, eq. (4) remains valid for any choices of the dividing surface. When the equimolecular dividing surface is chosen, the first term in eq. (4) becomes equivalent to -nRT In S in eq. (1) under the approximation that (pr - p,) is neglected in comparison with ($1 - pr), where pt is the pressure of the bulk liquid which possesses the same chemical potential as that of the supersaturated vapor. Note that the first term in eq. (4) is not an integral multiple of -RT In S under the other choices of the dividing surface. All this leads us to conclude that B-K’s definitions are inconsistent with those of
K. Nishioka,
K.C. Russell / Thermodynamics
of cluster formation
L215
Gibbs. Their -(n - I)RT In S cannot be identified with the first term in Gibbs’ equation (4). Their second term, which they interpret as the free energy change on placing (n - 1) liquid molecules onto a gaseous monomer to form an n-mer bears little resemblance to Gibbs’ description of the surface term in W. Let us consider the more physical arrangement of AC [eq. (3a)] and B-K’s main assertion, namely that the appearance of /_$ as the difference I*,”- &’ may be used to eliminate most of the 10” factor. As noted before, the term p” in eq. (3a) entered as the standard state for n-mer, so that /J: does not appear naturally in AC. Of course RTln(pr/p’) may be interpreted as the chemical potential difference between any gaseous species at pressure p1 and p”. The first two terms in the bracket in eq. (3a) are readily interpreted as the free energy change on creating an n-mer at pressure p” from n molecules of the bulk liquid with pe. The final term is the free energy change on expanding the n-mer gas from p” to pr . The form of AC in eq. (3b) illustrates this simple fact. The prefactor is determined by the standard state chosen for n-mer, so that all standard state terms in the expression for N, cancel, as they must. The standard state is a computational convenience and as such cannot affect the value of such physical quantities as concentrations and pressures. Several authors [6-81 have previously noted that in classical nucleation theory N,, is standard state dependent. B-K are correct in pointing this out again. Also, the translational and rotational entropy of all the monomer is fully accounted for when one takes
we= d’
+RT lnhhO),
(5)
in the derivation of AG. The only way one get np, in the exponent is by relating it to the chemical potential of n vapor molecules. The rotational and translational entropy of the gaseous monomer is essentially equated to the binding energy of the liquid molecule, and there is nothing left over to cancel with the y1-mer. Further, we fail to understand the basing of arguments on the final value of a physical quantity such as AG on the appearance of two (canceling) standard state terms. Standard state terms must appear in just this way for any treatment to be consistent with the very chemical thermodynamics invoked by B-K. Finally, B-K assert that if the RHS of eq. (1) is taken as the free energy to form an n-mer, one needs to eliminate a p” even in the L-P treatment. They misquote the L-P translational free energy as AGt = -RT ln[(2mnkT/n2)3’2
kT/p, 1,
(6)
where m is n-mer mass. kT/p, is the molecular volume in the gas phase at the pressure pn. The corresponding factor in the original L-P paper is 52, where “. . . !A is their molecular volume in the gaseous standard state of p and T . . .” *. So R = KT/ p” in the present terminology. * B-K apparently confuse the original paper, which is framed in Gibbs free energy and standard states with equivalent alternate treatments, which obtain the equilibrium number of n-mer from minimizing the Helmholtz free energy. See, for example, Feder et al. (71.
L216
K. Nishioka, K.C. Russell / Thermodynamics of cluster formation
Had L-P indeed written AC, as in eq. (6), their treatment would be erroneous and inconsistent with chemical thermodynamics. In fact, their treatment is consistent with the law of mass action and gives any desired prefactor, simply by taking that as the standard state of n-mer. The corresponding factor also appears in AG,, giving cancellation in any case. Lothe and Pound took the actual monomer pressure, pr, as their standard state so as to obtain Nr as a prefactor and allow easy comparison of their treatment with the classical theory. We would like to summarize our remarks as follows: (1) B-K’s thermodynamic manipulations are essentially correct. Their conclusion that the usual formulation of classical condensation theory is inconsistent with chemical thermodynamics is correct, but not original. Their treatment is incapable of in any way resolving the controversy over the L-P nucleation theory. (2) Their separation of the free energy of droplet formation into “surface” and “volume” terms is inconsistent with Gibbs. (3) Their argument that appearance of the chemical potential of n-mer as the difference p,O- 111implies a cancellation of much of the 10” factor is erroneous. Instead, the appearance of the standard potentials as a difference is a requirement of chemical thermodynamics. (4) B-K misquote the L-P theory in comparing the latter’s treatment to the requirements of chemical thermodynamics. The equations as originaZly written are consistent with these requirements. (5) This note takes no position on the correctness of the L-P statistical mechanics. We do assert, however, that qualitative or quantitative disproof or confirmation of their treatment must be based on a statistical mechanical model for the n-mer which would allow a calculation of pi that is convincing to those Involved in the controversy. The authors wish to thank Professor G.M. Pound of Stanford bringing the paper by Blander and Katz to their attention
Ciniversity for
References [l] J. Lothe and G.M. Pound, J. Chem. Phys. 36 (1962) 2080. [2] A.C. Zettlemoyer, Ed., Nucleation (Dekker, New York, 1969) pp. i 146 [3] A.C. Zettlemoyer, Ed., Advances in Colloid and Interface Science. \‘,:I. :’ f Flsevier, Amsterdam, 1977). [4] M. Blander and J.L. Katz, J. Statist. Phys. 4 (1972) 55. [5] J.W. Gibbs, The Collected Works, Vol. I, Thermodynamics (Yale Univ. Press, New Haven, 1957) pp. 2522258. [6] W.G. Courtney, J. Chem. Phys. 35 (1961) 2249. [7] J. Feder, K.C. Russell, J. Lothe and G.M. Pound, Advan. Phys. 15 (1966) 111. [8] K.C. Russell, in: Fundamental Processes in Materials Sciences, Vol. 3, Eds. P.L. de Bruyn and L.J. Bonis (Plenum, New York, 1966).
SURFACE SCIENCE LETTERS
COMMENTSON“THETHERMODYNAMICSOFCLUSTERFORMATION INNUCLEATIONTHEORY" Kazumi NISHIOKA
*
Department of Metallurgical Engineering, Michigan Technological University, Houghton, Michigan 49931, USA
and Kenneth
C. RUSSELL
Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Received
4 August
1980; accepted
for publication
18 November
1980
(hereafter referred to as L-P) The controversy over the 1017 Lothe-Pound correction factor [l] to the rate equation in classical nucleation theory is still active [2,3]. Most authors attempted to disprove L-P, and in all cases but one the authors have resorted to statistical mechanical models for a droplet to advance their respective points. Blander and Katz [4] (hereafter referred to as B-K) utilized a rather novel approach in attacking this dilemma. They claimed that the Gibbs free energy of droplet formation had never been properly defined, and attempted a disproof of thr IL-P treatment by some rearrangements of the equations of the chemical thermcriynamics of nucleation. This note is intended to show that even where B-K’s ri~,?i.I!?i?!:.lisrnics are correct, their interpretations are in error, and that their methCX!‘:! -:$I*I’~i;sibly help resolve the controversy over the factor of 1017. I’ k. 1‘Ia :t;e law of mass action to write Ai
tlcip,,:‘p ‘) = -nRT
In S + [p,” -
np,],
(1)
wm rc p,, is the pressure of n-mer, and p” is the standard state pressure of n-mer (which B-K choose to take as the same as that for monomer). II,” is the chemical potential of the n-mer in the standard state, and pe is chemical potential of the bulk liquid phase under the saturation pressure pe. B-K note that because in classical nucleation theory the concentration of n-mer per unit volume N,, is given by N, = N1 exp(-AGIRT).
* On leave from Japan.
Department
(2)
of Precision
Mechanics,
Tokushima
University,
Tokushima
770,
L214
K. Nishioka, K.C. Russell / Thermodynamics of cluster formation
With an Nr prefactor, it is improper to identify the RHS of eq. (1) as AC. To do so would indeed violate the law of mass action. The consistent value of AG differs by a term RL ln(pI/po), B-K wrote the resulting function in the form AC=-(n-1)RTlnSt
[&&(n-l)~,,].
(3)
They then interpret the bracketed term as the “standard chemical potential change for one mole of monomer vapor + (n - 1) moles of liquid to form one mole of n-mer . . .‘I, and identify this term with the surface free energy, uA(n), in the classical nucleation theory. They then make the crucial assertion that since the difference pz - py appears in eq. (3) any contribution that free translation and rotation of the n-mer make to ~2 will be largely canceled by contributions of similar motions of the monomeric molecule to py. Let us rearrange eq. (3) in a rather more concise form: AG=-nRTlnS+
[&IIP,+RT~~O,~/~~)],
(3a)
or even more simply as: AG=-nRTlnS+
[&np,],
(3b)
where & is the chemical potential of the n-mer in a standard state of pr, and ~_l~is the chemical potential of the saturated vapor. We now consider the arguments of B-K from several points of view. Classical nucleation theory is ostensibly based upon the thermodynamics of Gibbs [5], who first calculated the reversible work (or free energy) of forming a droplet. Later authors have mainly been concerned with the prefactors to the exponential in eq. (2). Gibbs writes the reversible work as: LV=-VAp+uA(n),
(4)
where I/ denotes the volume enclosed by the dividing surface and Ap the pressure difference between the bulk liquid and the supersaturated vapor. The bulk liquid here possesses the same chemical potential as that of the vapor. Gibbs states that “W consists of two parts, one of which is always positive, and is expressed by the product of the superficial tension and the area of the surface of tension and the other is always negative, and is numerically equal to the product of the difference of the pressure by the volume of the interior mass”. Gibbs goes on to describe how the surface work could in principle be measured by extruding the droplet through an orifice in a body of pressurized bulk liquid. Although Gibbs employs the surface of tension, eq. (4) remains valid for any choices of the dividing surface. When the equimolecular dividing surface is chosen, the first term in eq. (4) becomes equivalent to -nRT In S in eq. (1) under the approximation that (pr - p,) is neglected in comparison with ($1 - pr), where pt is the pressure of the bulk liquid which possesses the same chemical potential as that of the supersaturated vapor. Note that the first term in eq. (4) is not an integral multiple of -RT In S under the other choices of the dividing surface. All this leads us to conclude that B-K’s definitions are inconsistent with those of
K. Nishioka,
K.C. Russell / Thermodynamics
of cluster formation
L215
Gibbs. Their -(n - I)RT In S cannot be identified with the first term in Gibbs’ equation (4). Their second term, which they interpret as the free energy change on placing (n - 1) liquid molecules onto a gaseous monomer to form an n-mer bears little resemblance to Gibbs’ description of the surface term in W. Let us consider the more physical arrangement of AC [eq. (3a)] and B-K’s main assertion, namely that the appearance of /_$ as the difference I*,”- &’ may be used to eliminate most of the 10” factor. As noted before, the term p” in eq. (3a) entered as the standard state for n-mer, so that /J: does not appear naturally in AC. Of course RTln(pr/p’) may be interpreted as the chemical potential difference between any gaseous species at pressure p1 and p”. The first two terms in the bracket in eq. (3a) are readily interpreted as the free energy change on creating an n-mer at pressure p” from n molecules of the bulk liquid with pe. The final term is the free energy change on expanding the n-mer gas from p” to pr . The form of AC in eq. (3b) illustrates this simple fact. The prefactor is determined by the standard state chosen for n-mer, so that all standard state terms in the expression for N, cancel, as they must. The standard state is a computational convenience and as such cannot affect the value of such physical quantities as concentrations and pressures. Several authors [6-81 have previously noted that in classical nucleation theory N,, is standard state dependent. B-K are correct in pointing this out again. Also, the translational and rotational entropy of all the monomer is fully accounted for when one takes
we= d’
+RT lnhhO),
(5)
in the derivation of AG. The only way one get np, in the exponent is by relating it to the chemical potential of n vapor molecules. The rotational and translational entropy of the gaseous monomer is essentially equated to the binding energy of the liquid molecule, and there is nothing left over to cancel with the y1-mer. Further, we fail to understand the basing of arguments on the final value of a physical quantity such as AG on the appearance of two (canceling) standard state terms. Standard state terms must appear in just this way for any treatment to be consistent with the very chemical thermodynamics invoked by B-K. Finally, B-K assert that if the RHS of eq. (1) is taken as the free energy to form an n-mer, one needs to eliminate a p” even in the L-P treatment. They misquote the L-P translational free energy as AGt = -RT ln[(2mnkT/n2)3’2
kT/p, 1,
(6)
where m is n-mer mass. kT/p, is the molecular volume in the gas phase at the pressure pn. The corresponding factor in the original L-P paper is 52, where “. . . !A is their molecular volume in the gaseous standard state of p and T . . .” *. So R = KT/ p” in the present terminology. * B-K apparently confuse the original paper, which is framed in Gibbs free energy and standard states with equivalent alternate treatments, which obtain the equilibrium number of n-mer from minimizing the Helmholtz free energy. See, for example, Feder et al. (71.
L216
K. Nishioka, K.C. Russell / Thermodynamics of cluster formation
Had L-P indeed written AC, as in eq. (6), their treatment would be erroneous and inconsistent with chemical thermodynamics. In fact, their treatment is consistent with the law of mass action and gives any desired prefactor, simply by taking that as the standard state of n-mer. The corresponding factor also appears in AG,, giving cancellation in any case. Lothe and Pound took the actual monomer pressure, pr, as their standard state so as to obtain Nr as a prefactor and allow easy comparison of their treatment with the classical theory. We would like to summarize our remarks as follows: (1) B-K’s thermodynamic manipulations are essentially correct. Their conclusion that the usual formulation of classical condensation theory is inconsistent with chemical thermodynamics is correct, but not original. Their treatment is incapable of in any way resolving the controversy over the L-P nucleation theory. (2) Their separation of the free energy of droplet formation into “surface” and “volume” terms is inconsistent with Gibbs. (3) Their argument that appearance of the chemical potential of n-mer as the difference p,O- 111implies a cancellation of much of the 10” factor is erroneous. Instead, the appearance of the standard potentials as a difference is a requirement of chemical thermodynamics. (4) B-K misquote the L-P theory in comparing the latter’s treatment to the requirements of chemical thermodynamics. The equations as originaZly written are consistent with these requirements. (5) This note takes no position on the correctness of the L-P statistical mechanics. We do assert, however, that qualitative or quantitative disproof or confirmation of their treatment must be based on a statistical mechanical model for the n-mer which would allow a calculation of pi that is convincing to those Involved in the controversy. The authors wish to thank Professor G.M. Pound of Stanford bringing the paper by Blander and Katz to their attention
Ciniversity for
References [l] J. Lothe and G.M. Pound, J. Chem. Phys. 36 (1962) 2080. [2] A.C. Zettlemoyer, Ed., Nucleation (Dekker, New York, 1969) pp. i 146 [3] A.C. Zettlemoyer, Ed., Advances in Colloid and Interface Science. \‘,:I. :’ f Flsevier, Amsterdam, 1977). [4] M. Blander and J.L. Katz, J. Statist. Phys. 4 (1972) 55. [5] J.W. Gibbs, The Collected Works, Vol. I, Thermodynamics (Yale Univ. Press, New Haven, 1957) pp. 2522258. [6] W.G. Courtney, J. Chem. Phys. 35 (1961) 2249. [7] J. Feder, K.C. Russell, J. Lothe and G.M. Pound, Advan. Phys. 15 (1966) 111. [8] K.C. Russell, in: Fundamental Processes in Materials Sciences, Vol. 3, Eds. P.L. de Bruyn and L.J. Bonis (Plenum, New York, 1966).