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Kinetic process of homogeneous nucleation incorporating the effect of curvature
Journal of Crystal Growth 79 (1986) 997-1000 North-Holland, A m s t e r d a m
997
KINETIC PROCESS OF HOMOGENEOUS NUCLEATION INCORPORATING THE EFFECT-OF CURVATURE D. J A Y A R A M A N , C. S U B R A M A N I A N , R. D H A N A S E K A R A N and P. R A M A S A M Y Crystal Growth Centre, Anna University, Madras-600 025, India
The dependence of interfacial tension on the size of the droplet in vapour phase homogeneous nucleation is studied. The work done by a droplet surrounded by the vapour is related to surface energy per unit solid angle and the surface energy per unit area of the plane interface. The expression for the dependence of surface energy on the size is derived. This permits the Jth potential to be a continuous function and the expression for the excess free energy required for the formation of the cluster is derived. In continuation of the above, the kinetic equation of the Frenkel distribution is modified and the dependence of cluster concentration on cluster size is studied.
1. Introduction
Models have been proposed [1-3] for the calculation of the size of nucleus based on m a n y different assumptions. Recently, Rasmussen et al. [4] developed a theory for the nucleation kinetic equation incorporating the dependence of interracial energy on the size of the cluster. According to him, the condensation of the first monomer is assumed to be in the same mother phase. It leads to the conclusion that ith cluster contains only ( i - 1) condensed monomers in addition to the first monomer in the mother phase. According to the classical nucleation theory, when a nucleus is born, the density of the two phases are assumed to be the same and then by monomer addition the nucleus grows. These concepts do not seem to be the real situation. In the present work we have made an attempt to find a relationship between the interfacial tension and the size of the droplet by a different approach which will provide a continuous pathway of transformation. Another aspect of the problem is that the Gibbs potential per m o n o m e r in the nucleus so far considered as constant is changing as the embryo grows. The change in J t h potential for the formation of a nucleus is worked out considering the two views above. Also assuming that the ith cluster should contain only i condensed monomers, the nucleation kinetic equation is deduced. The concentration of the critical
cluster is also obtained from the Frenkel distribution curve.
2. Dependence of interfacial energy on the size of the cluster
The resultant work done by a droplet of pressure P2 surrounding a supersaturated vapour at pressure P1 is ( P 2 - Pa)dV2, where dV2 is the increase in the volume of the droplet. This resultant work done is utilized for the increase of surface energy of the droplet associated with increase of surface area and the decrease the surface energy associated with increase in radius of curvature: (P2 - P1) dV2 = ao dA - occo d r ,
(1)
where o0 is the surface energy per unit area of a plane interface and oc is defined as the surface energy per unit solid angle per unit radius. For the homogeneous nucleation of a spherical shape, the solid angle o~ is 4~r. For the heterogeneous cap shaped nucleus, the value of ~o is equal to ~r [ 2 ( 1 - c o s 0 ) -
sin20 cos 0],
where 0 is the wetting angle. Expressing the change in volume for a spherical shape nucleus as dV2 = 4~rrZdr and the change in area as dA = 8~rrdr, eq.
0022-0248/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
998
D. Jayaraman et al. / Kinetic process of homogeneous nucleation
(1) becomes
ergy for monomer addition is assumed to vary with i as
200 oc /'2 - P1 . . . . + --. r r2
(2)
The value of % is related to o0 by 3, the radius of the monomer. Eq. (2) becomes 2%/ P 2 - P, = 7 / 1
Ill),
AqS, = Aq50(1 -
3 - ~r).
(3)
The work of formation of the droplet of size r from monomer radius 3 is the product of curvature dependent surface tension and surface area
(7)
where AO0 is the constant value of decrement in the Gibbs potential. For the formation of a dimer,
J 2 = ( N - 2 ) G I + 2(Ga-Ad&)+c~(r2)A2 .
(8)
For the formation of an i-mer, J / = ( N - i)~1 + i(~-2
_ Aep,) + o(ri)Ai,
(9)
[5]:
and before condensation
fsr( p2 -- P1) 4~rr2 d r = o( r )A.
(4)
J = N G ~.
(10)
The solution of eq. (4) yields a relation between o and r:
The change in the J t h potential for the formation of an i-mer is J, - J :
o(r) = O o ( 1 - 3 / r ),
AJ=J;-J
(5)
The above relationship is true in the case of heterogeneous nucleation of different shapes, where o~ assumes different values. For a cap shaped nucleus inserting the values of A, V2 and o~ in eq. (1), eq. (5) can be deduced. The final expression for the interfacial tension depends only on the size and not on the solid angle.
= i[(G2-
G1)-
AOi] + o ( r ) A
i.
(11)
Inserting eqs. (5) and (7) gives
A J = %r[r3AGv- A~v(r3 - 33)] + 4~ro0( r 2 - r3),
(12)
where 3. Expression for the change in free energy for the formation of a cluster The J t h potential for the formation of a cluster containing i molecules or atoms in a supersaturated system is written as J/= (N - i)G' + i(~2
_ Adpi) + o(r)Ai '
(6)
where N is the number of atoms in the mother phase, ~-1 is the Gibbs potential per monomer in the mother phase, ~-2 is the Gibbs potential of the first monomer in the condensed phase, and the third term is the surface energy. As the cluster grows, the Gibbs potential per monomer in the condensed phase is no longer constant due to the change of internal pressure in the duster. Let Aq~i denote the decrement in the Gibbs potential for the formation of an i-met. The decrement in en-
_ ~1
aGv=--
u
Ag, °
,
A~bv= - -v
A~v is the correction to be applied for AGv given in Rasmussen et al.'s treatment. Although the ith cluster contains i number of condensed monomers, it is to be noted that the correction to be applied for AGv is only due to the addition of (i - 1)mers with the first original monomer. Maximizing eq. (12), the expression for the critical radius of the nucleus is obtained: r* = - % -
%[1 + (3/Oo)(AGv--Aq, v)] AGv -- Aq5v
(13)
The substitution of r* in eq. (12) will yield the value of free energy change for the formation of critical nucleus.
D. Jayaraman et al. / Kinetic process of hornogeneous nucleation
where a = 47r2o0. Both exponential term and preexponential term affect the shape of the curve between N] and i.
4. Frenkel distribution
Following Frenkel's [6] considerations, the change in overall potential 4~ on forming all the clusters in the system from monomers can be written as
Aq~= E A J i + K T E X i
In Xi,
5. Results and discussion
(14)
The theory of nucleation presented in this paper is different from other authors for the following reasons: (i) the solid angle concept is introduced to arrive at the relation between the interfacial energy and the size of the cluster; (ii) the Gibbs potential per monomer of the nucleus is assumed to vary as the cluster grows; (iii) the shape of the Frenkel curve is affected substantially due to the above considerations. The derived nucleation kinetic equation, though altogether different in form, is not basically different from classical theory. The thermodynamic potential makes the pathway of phase transformation a continuous one. The variation of 0 / % with r/~ has been indicated in fig. 1,along with other authors' [4,7] works. The work of formation required for a very small cluster having dimensions of the order of 5 •~ is larger than that reported by Rasmussen et al.
where X~ is the mole fraction of clusters which contain i atoms per cluster. The second term in the right hand side of the equation corresponds to mixing of clusters with monomers under ideal conditions. Under stable condition ,/, is a minimum and Acp = 0:
A J,_ t - A J + k T ( X i _
1 In X~_ 1 - X , In X i ) = 0 .
(15) Making use of eq. (5), the condition F = 32Ni and the principle of multiple product, the above equation is written in the Frenkel form as
N,
F =
1
exp - kT,(AG - A~o)(i - 1)
+ a ( i 2 / 3 - 1)],
999
(16)
10
l O"
06
02
0
I
I
I
I
I
4
8
12
16
20
8 Fig. 1. Variation of surface free energy with the size of the nucleus and comparison with that of Gibbs and Rasmussen.
D. Jayaraman et al. / Kinetic process of homogeneous nucleation
1(313(3
and the values are: r * = 18.51 ,~ (r* = 29.78 A) and AG*/kT = 81.44 [(AG*/kT)c = 627.96]. The decrement in the Gibbs potential Aq~0 is fixed assuming the interfacial region has a thickness of 15 A and the jumping frequency is equal to 1.3 × 1011 vibrations/s. Assuming the interracial energy o to be 50 e r g / c m 2 and the supersaturation S = 1.5, the Frenkel curve is drawn as shown in fig. 2. It is to be mentioned here that the shape of the curve and concentration of the clusters are very much affected for higher values of S and o. Both the exponential term and the pre-exponential term are important for the shape of the curve. The exponential term contributes to the classical Boltzm a n n parabola with its minimum concentration at the classical critical size. Work is in progress to probe into the limitations of the kinetic nucleation equation with different values of o and S for different systems.
I°2°I 1018
1016 1014
I 1012 1010
10 s 10 6
10 4
References 10 2 10 0
0
10
20
30
40
50
60
70
Fig. 2. Frenkel distribution curve between the cluster concentration and duster size.
[4]. To show the extent to which the values of radius and free energy change as the critical point is altered, an illustrative calculation is carried out
BO
[1] J.P. Hirth and G.M. Pound, Condensation and Evaporation (Pergamon, Oxford, 1963). [2] R.A. Sigsbee, in: Nucleation, Ed. A.C. Zettlemoyer (Dekker, New York, 1969) p. 151. [3] D. Kashchiev, J. Chem. Phys. 10 (1982) 76. [4] D.H. Rasmussen, M.R. Appleby, G.L. Leedom and S.V. Babu, J. Crystal Growth 64 (1983) 229. [5] H. Reiss, Methods of Thermodynamics (Blaisdell, New York, 1965) p. 164. [6] J. Frenkel, Kinetic Theory of Liquids (Dover, New York, 1955) p. 366. [7] J.W. Gibbs, in: The Collected works of J. Wilfiard Gibbs, Vol. 1, Thermodynamics (Longman-Green, London, 1932) p. 232.
997
KINETIC PROCESS OF HOMOGENEOUS NUCLEATION INCORPORATING THE EFFECT-OF CURVATURE D. J A Y A R A M A N , C. S U B R A M A N I A N , R. D H A N A S E K A R A N and P. R A M A S A M Y Crystal Growth Centre, Anna University, Madras-600 025, India
The dependence of interfacial tension on the size of the droplet in vapour phase homogeneous nucleation is studied. The work done by a droplet surrounded by the vapour is related to surface energy per unit solid angle and the surface energy per unit area of the plane interface. The expression for the dependence of surface energy on the size is derived. This permits the Jth potential to be a continuous function and the expression for the excess free energy required for the formation of the cluster is derived. In continuation of the above, the kinetic equation of the Frenkel distribution is modified and the dependence of cluster concentration on cluster size is studied.
1. Introduction
Models have been proposed [1-3] for the calculation of the size of nucleus based on m a n y different assumptions. Recently, Rasmussen et al. [4] developed a theory for the nucleation kinetic equation incorporating the dependence of interracial energy on the size of the cluster. According to him, the condensation of the first monomer is assumed to be in the same mother phase. It leads to the conclusion that ith cluster contains only ( i - 1) condensed monomers in addition to the first monomer in the mother phase. According to the classical nucleation theory, when a nucleus is born, the density of the two phases are assumed to be the same and then by monomer addition the nucleus grows. These concepts do not seem to be the real situation. In the present work we have made an attempt to find a relationship between the interfacial tension and the size of the droplet by a different approach which will provide a continuous pathway of transformation. Another aspect of the problem is that the Gibbs potential per m o n o m e r in the nucleus so far considered as constant is changing as the embryo grows. The change in J t h potential for the formation of a nucleus is worked out considering the two views above. Also assuming that the ith cluster should contain only i condensed monomers, the nucleation kinetic equation is deduced. The concentration of the critical
cluster is also obtained from the Frenkel distribution curve.
2. Dependence of interfacial energy on the size of the cluster
The resultant work done by a droplet of pressure P2 surrounding a supersaturated vapour at pressure P1 is ( P 2 - Pa)dV2, where dV2 is the increase in the volume of the droplet. This resultant work done is utilized for the increase of surface energy of the droplet associated with increase of surface area and the decrease the surface energy associated with increase in radius of curvature: (P2 - P1) dV2 = ao dA - occo d r ,
(1)
where o0 is the surface energy per unit area of a plane interface and oc is defined as the surface energy per unit solid angle per unit radius. For the homogeneous nucleation of a spherical shape, the solid angle o~ is 4~r. For the heterogeneous cap shaped nucleus, the value of ~o is equal to ~r [ 2 ( 1 - c o s 0 ) -
sin20 cos 0],
where 0 is the wetting angle. Expressing the change in volume for a spherical shape nucleus as dV2 = 4~rrZdr and the change in area as dA = 8~rrdr, eq.
0022-0248/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
998
D. Jayaraman et al. / Kinetic process of homogeneous nucleation
(1) becomes
ergy for monomer addition is assumed to vary with i as
200 oc /'2 - P1 . . . . + --. r r2
(2)
The value of % is related to o0 by 3, the radius of the monomer. Eq. (2) becomes 2%/ P 2 - P, = 7 / 1
Ill),
AqS, = Aq50(1 -
3 - ~r).
(3)
The work of formation of the droplet of size r from monomer radius 3 is the product of curvature dependent surface tension and surface area
(7)
where AO0 is the constant value of decrement in the Gibbs potential. For the formation of a dimer,
J 2 = ( N - 2 ) G I + 2(Ga-Ad&)+c~(r2)A2 .
(8)
For the formation of an i-mer, J / = ( N - i)~1 + i(~-2
_ Aep,) + o(ri)Ai,
(9)
[5]:
and before condensation
fsr( p2 -- P1) 4~rr2 d r = o( r )A.
(4)
J = N G ~.
(10)
The solution of eq. (4) yields a relation between o and r:
The change in the J t h potential for the formation of an i-mer is J, - J :
o(r) = O o ( 1 - 3 / r ),
AJ=J;-J
(5)
The above relationship is true in the case of heterogeneous nucleation of different shapes, where o~ assumes different values. For a cap shaped nucleus inserting the values of A, V2 and o~ in eq. (1), eq. (5) can be deduced. The final expression for the interfacial tension depends only on the size and not on the solid angle.
= i[(G2-
G1)-
AOi] + o ( r ) A
i.
(11)
Inserting eqs. (5) and (7) gives
A J = %r[r3AGv- A~v(r3 - 33)] + 4~ro0( r 2 - r3),
(12)
where 3. Expression for the change in free energy for the formation of a cluster The J t h potential for the formation of a cluster containing i molecules or atoms in a supersaturated system is written as J/= (N - i)G' + i(~2
_ Adpi) + o(r)Ai '
(6)
where N is the number of atoms in the mother phase, ~-1 is the Gibbs potential per monomer in the mother phase, ~-2 is the Gibbs potential of the first monomer in the condensed phase, and the third term is the surface energy. As the cluster grows, the Gibbs potential per monomer in the condensed phase is no longer constant due to the change of internal pressure in the duster. Let Aq~i denote the decrement in the Gibbs potential for the formation of an i-met. The decrement in en-
_ ~1
aGv=--
u
Ag, °
,
A~bv= - -v
A~v is the correction to be applied for AGv given in Rasmussen et al.'s treatment. Although the ith cluster contains i number of condensed monomers, it is to be noted that the correction to be applied for AGv is only due to the addition of (i - 1)mers with the first original monomer. Maximizing eq. (12), the expression for the critical radius of the nucleus is obtained: r* = - % -
%[1 + (3/Oo)(AGv--Aq, v)] AGv -- Aq5v
(13)
The substitution of r* in eq. (12) will yield the value of free energy change for the formation of critical nucleus.
D. Jayaraman et al. / Kinetic process of hornogeneous nucleation
where a = 47r2o0. Both exponential term and preexponential term affect the shape of the curve between N] and i.
4. Frenkel distribution
Following Frenkel's [6] considerations, the change in overall potential 4~ on forming all the clusters in the system from monomers can be written as
Aq~= E A J i + K T E X i
In Xi,
5. Results and discussion
(14)
The theory of nucleation presented in this paper is different from other authors for the following reasons: (i) the solid angle concept is introduced to arrive at the relation between the interfacial energy and the size of the cluster; (ii) the Gibbs potential per monomer of the nucleus is assumed to vary as the cluster grows; (iii) the shape of the Frenkel curve is affected substantially due to the above considerations. The derived nucleation kinetic equation, though altogether different in form, is not basically different from classical theory. The thermodynamic potential makes the pathway of phase transformation a continuous one. The variation of 0 / % with r/~ has been indicated in fig. 1,along with other authors' [4,7] works. The work of formation required for a very small cluster having dimensions of the order of 5 •~ is larger than that reported by Rasmussen et al.
where X~ is the mole fraction of clusters which contain i atoms per cluster. The second term in the right hand side of the equation corresponds to mixing of clusters with monomers under ideal conditions. Under stable condition ,/, is a minimum and Acp = 0:
A J,_ t - A J + k T ( X i _
1 In X~_ 1 - X , In X i ) = 0 .
(15) Making use of eq. (5), the condition F = 32Ni and the principle of multiple product, the above equation is written in the Frenkel form as
N,
F =
1
exp - kT,(AG - A~o)(i - 1)
+ a ( i 2 / 3 - 1)],
999
(16)
10
l O"
06
02
0
I
I
I
I
I
4
8
12
16
20
8 Fig. 1. Variation of surface free energy with the size of the nucleus and comparison with that of Gibbs and Rasmussen.
D. Jayaraman et al. / Kinetic process of homogeneous nucleation
1(313(3
and the values are: r * = 18.51 ,~ (r* = 29.78 A) and AG*/kT = 81.44 [(AG*/kT)c = 627.96]. The decrement in the Gibbs potential Aq~0 is fixed assuming the interfacial region has a thickness of 15 A and the jumping frequency is equal to 1.3 × 1011 vibrations/s. Assuming the interracial energy o to be 50 e r g / c m 2 and the supersaturation S = 1.5, the Frenkel curve is drawn as shown in fig. 2. It is to be mentioned here that the shape of the curve and concentration of the clusters are very much affected for higher values of S and o. Both the exponential term and the pre-exponential term are important for the shape of the curve. The exponential term contributes to the classical Boltzm a n n parabola with its minimum concentration at the classical critical size. Work is in progress to probe into the limitations of the kinetic nucleation equation with different values of o and S for different systems.
I°2°I 1018
1016 1014
I 1012 1010
10 s 10 6
10 4
References 10 2 10 0
0
10
20
30
40
50
60
70
Fig. 2. Frenkel distribution curve between the cluster concentration and duster size.
[4]. To show the extent to which the values of radius and free energy change as the critical point is altered, an illustrative calculation is carried out
BO
[1] J.P. Hirth and G.M. Pound, Condensation and Evaporation (Pergamon, Oxford, 1963). [2] R.A. Sigsbee, in: Nucleation, Ed. A.C. Zettlemoyer (Dekker, New York, 1969) p. 151. [3] D. Kashchiev, J. Chem. Phys. 10 (1982) 76. [4] D.H. Rasmussen, M.R. Appleby, G.L. Leedom and S.V. Babu, J. Crystal Growth 64 (1983) 229. [5] H. Reiss, Methods of Thermodynamics (Blaisdell, New York, 1965) p. 164. [6] J. Frenkel, Kinetic Theory of Liquids (Dover, New York, 1955) p. 366. [7] J.W. Gibbs, in: The Collected works of J. Wilfiard Gibbs, Vol. 1, Thermodynamics (Longman-Green, London, 1932) p. 232.