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Nucleation kinetics of polar compounds. Effect of curvature and dipole—dipole interaction on surface energy
Volume 206, number 5,6
CHEMICAL PHYSICSLETTERS
I May 1993
Nucleation kinetics of polar compounds. Effect of curvature and dipole-dipole interaction on surface energy F. Joseph Kumar,
D. Jayaraman,
C. Subraraanian
and P. Ramasamy
CrystalGrowthCentre, Anna University,Madras, India
Received 2 I January 1993
Based on thermodynamical concepts, a new expression for the surface energy of a microcluster of a polar material has been derived incorporating the effect of curvature and the dipole-dipole interactions. Using the expression, the nucleation parameters of the polar materials chloroform and acetonitrile have been calculated. A study of the dependence of critical supersaturation on the temperature has also been made and compared with the available experimental results.
1. Introduction The study of homogeneous nucleation of liquid clusters from a vapour has evoked considerable interest among scientists since it forms the basis for a wide range of more complex phenomena such as nucleation of water droplets in the atmosphere, crystal growth, thermal polymerization and so on. It has been widely accepted that the Becker-Doring-Zeldovich theory or the socalled classical nucleation theory [ l31, in spite of its crude assumptions and possible fundamental errors [ 4,5 1, predicts the correct value of critical supersaturation required for the onset of homogeneous nucleation in good agreement with experiment in the case of most of the weakly to moderately polar compounds. The apparent experimental agreement of the predictions of the classical theory has been attributed to the large extent of cancellation of errors involved in the theory. Recently Wright et al. [ 61 reported that the predictions of the classical theory are in serious disagreement with experiment in the case of homogeneous nucleation of the highly polar acetonitrile vapour system (dipole moment= 3.9 D), wherein the dipole-dipole interaction is expected to play a major role in modifying the surface free energy of microclusters at the critical stage. The classical nucleation theory is based on the capillarity approximation according to which the Elsevier Science Publishers B.V.
macroscopic thermodynamic properties apply to the supersaturated vapour and the liquid phases and the interface separating the two phases can be described by the surface between bulk quantities of both the phases. The validity of this approximation has long been questioned since it creates a discontinuity in the chemical potential from the vapour to the liquid phase. One way of solving this problem is to use a surface free energy with a functional dependence on the cluster size in place of the bulk value as used in the classical theory. Various attempts have been made in this respect to determine the curvature dependence of surface tension [ 7-121, Nucleation kinetics of non-polar materials CC&, C2H2C14, SnCl, and TiC14 have been studied using the curvaturedependent surface tension in our early works [ 11,121. But since in the case of weakly to moderately polar materials the use of curvature-dependent surface tension in the nucleation theory ruins the close agreement with experiment, the effect of dipole-dipole interactions on the surface free energy of these materials also should be considered along with the curvature dependence of surface tension. In the present work we have derived a new expression for the curvature-dependent surface tension including the effect of dipole-dipole interaction on the surface energy. The contribution towards the work of formation of a droplet of a particular size “8 is considered to be due to (i) planar interfacial ten415
Volume 206, number 5,6
CHEMICAL PHYSICSLETTERS
sion, (ii) the surface energy gradient and (iii) dipole-dipole interaction. Also the truncated Tolman expression S=&,,( 1 +&Jr) has been used instead of assuming the Tolman length 6 to be a constant [ 8, lo]. With these modifications a new expression for the surface energy of microclusters is derived and incorporated into the classical nucleation theory and the nucleation parameters are calculated for the moderately polar material chloroform with a dipole moment of 1.02 D and the highly polar material acetonitrile with a dipole moment of 3.9 D. The results are compared with the experimental values of Katz et al. [ 131 and Wright et al. [6] obtained using an upward thermal diffusion cloud chamber.
7 May 1993
orientation involves an increase of the potential energy of a surface molecule by an amount 28 W=
s 0
pEc sin 2ede,
(1)
where p is the dipole moment of each molecular dipole, E is the electric field, c is the number of surface nearest neighbours, r is the radius of the droplet and x is the distance between nearest neighbours. For very small 0 taking sin 28=20=2x/r the above equation simplifies to W=,u2c/4xr2. The total increase of surface potential energy with respect to a planar surface is then
2. Effect of dipole-dipole interaction and curvature on surface energy In order to determine the effect of dipole-dipole interaction on the surface free energy, the model proposed by Abraham [ 31 is followed. According to this model there is an increase in the dipole-dipole interaction energy at the microcluster stage which gives a positive contribution to the surface energy. For a polar liquid the surface dipoles are oriented perpendicular to the liquid surface and parallel to one another. For a spherical droplet of nucleus size this parallel orientation is destroyed due to the surface curvature and each dipole is oriented at an angle with respect to its nearest neighbours, the angle being proportional to the radius of the droplet (fig. 1). Each dipole of a dipole pair is oppositely rotated by an angle 8 with respect to a line bisecting the line joining the dipoles. This deviation from the parallel
vopour
where j, is the total number of molecules in the surface of the droplet and the prefactor { arises because each dipole of a dipole pair shares one half of the interaction energy. From solid angle considerations we get the value of j, as j, = 16r2/x2 . Hence the dipole-dipole interaction results in an increase of the total surface energy far highly curved surfaces by an amount u,=2~2cfx3.
Let us now consider a one-component two-phase system consisting of a liquid microcluster in a supersaturated vapour. Let the state of the system be determined by its entropy S, mass m of the component constituting the two phases, volumes tP and ve of the homogeneous vapour and liquid phases and by area A and radius r of the spherical microcluster. Following Gibbs [ 14 1, for equilibrium of the system ($-pa)
Liquid
(aI
(bl
Fig. 1. Illustration of dipole alignment of polar molecules at (a) a flat interface, (b) a spherical interface.
416
(2)
dtP=cr.. d,4tgCwdr.
(3)
Here a, is defined as the surface free energy per unit area of a plane interface and u, as the surface free energy per unit radius per unit solid angle [ 1O-121. Integrating the above equation between the limits zero and r one can obtain the work of formation of a droplet of radius r and solid angle 47r
7May 1993
CHEMICAL PHYSICS LETTERS
Volume 206, number $6
3. Nucleation kinetics
I
(f-pal)
dve=u,A+uc4w.
(4)
0
To determine the curvature-dependent surface tension, the work of formation is equated to the product of the surface area of the droplet and the sum of the planar interfacial tension, change in surface energy corresponding to that radius and the energy contribution due to the dipole-dipole interaction,
=
(,,yr+-&. i-* -
dr
bF= $r3AG, +4xr2u( r, p) ,
P2C
(6)
2nx3r3 ’
Here a, is a constant and is defined as
where a( r, p) and 6 are strong functions of radius of the droplet. 6 is the distance between the surface of tension and the equimolar dividing surface. Using the Tolman expression 6=6,( 1 +&Jr) and substituting the value of a, in eq. (6),
xexp
”
(7)
Here S, is the radius of a monomer whose value is taken to be 1 nm. The above differential equation can be solved to get the expression for the curvature dependence of surface free energy including the correction due to dipole-dipole interaction as r*c[ l+ (26J3r) 4Kx3r=
xex
P(
- % (1+&/2r)
>
.
+ (&,/2r)*] >
(8)
.
(1+6,/2r) >
(10)
The value of f is calculated by the method of iteration, taking the classical value P= -2u,/AG, as the first approximation. The critical free energy change AF* is calculated by using the value off in eq. (9). The rate of homogeneous nucleation I is defined as the number of clusters nucleated per second in a unit volume of vapour, and is expressed as AF+ - kT (
P2C
(w~)21
[2+&/r+
- +
(
I=Kexp
=-w*
(9)
where AG,= &In (S) /v is the change in volume free energy, u(r, p) is the surface tension, S is the supersaturation ratio, v is the specific volume of the liquid and k is the Boltzmann constant. For Sz 1, AF passes through a maximum value at a particular radius called the critical radius. Differentiating eq. (9) and equating to zero the critical radius is obtained as P=$
Comparing eqs. (4 )’ and (5) we obtain
ddr,pu) uc ~____-
The free energy change associated with the formation of a droplet of size r in a supersaturated system at a pressure P and temperature T is
, >
(11)
where K is a factor which varies more slowly with P and T than the exponential term. K is usually expressed as
where Z is the Zeldovich factor whose value is typically of the order of lo-=, P is the actual pressure of the vapour, k is the Boltzmann constant, T is the temperature of the vapour, a( r, 1) is the surface free energy and m is the mass of a single vapour molecule. The supersaturation value at which I= 1 is called the critical supersaturation SC which can be computed for different temperatures. 417
Volume 206, number 5,6
CHEMICALPHYSICS LETTERS
7 May 1993
the size increases, the surface tension first falls rapidly below the bulk value and then rises exponentially lo the bulk value. In the case of the highly polar acetonitrile system the surface energy drops exponentially to the bulk value indicating the strong effect of dipole-dipole interaction of surface energy. The thermophysical data employed in the present calculations are presented in table 1 for the two systems. The present values of the critical radius and
4. Results and discussion Figs. 2 and 3 show the variation of surface free energy as a function of cluster size for chloroform and acetonitrile at T=230 and 260 K, respectively. It is found that the surface free energy for very small clusters of a size of a few angstroms (containing 1O-20 molecules) is higher than that of the bulk value. In the case of the weakly polar chloroform system, as
50
RADIUS
A
Fig. 2. Variation of surfax free energy with radius for chloroform, (0 ) radial dependence, and ( 0 ) total dependence.
0
5
10
15
20
25 RADIUS
30
(A
35
) effect due to dipole-dipole interaction
40
L5
5
A
Fig. 3. Variation of surface free energy with radius for acetonitrile, (0 ) radial dependence, (A ) effect due to dipole-dipole interaction and ( 0 ) total dependence.
418
Volume 206, number 5,6
CHEMICAL PHYSICS LETTERS
I May 1993
Table 1
Thennophysicaldata employed in the present calculations [ 13,6] chloroform
M=ll9.38kg/kmol, lnP,=
-4 :[
p=l.O2D, T-r (’
‘,I
c=3,
x=3x10-10m
+Arrln;+fAB(T-&,)+aAy(%r;)+&A&%T:)
>I
+ (&+ fA#?&t f AyT: + aAD;) 2 - 1 +ln 760Torr ( ACY=-0.14504, A/&1.259x10-‘, Ay=7.2079xlO-‘, A6=2.0001~10-’ L,=59.9 k&/kg, I&.=334.35K u=29.91-O.l295(T-273.15) ml/m’ &1.526~10’-1.845T-6.017~10-*T~-1.436x acetonitrile
(TinK) lo-‘T’kg/m3
M=41.053kg/kmol, ~=3.9D, c=3, x=3.5x10-‘“m logP.=7.9386621785.844/T (Tin K) u=31.82-O.i263(T-273.15) mJ/m’ (TinK) d=0.8073~10~-1.07(T-273.15)kg/m” (TinK)
Table 2 Variation of critical free energy change with and without the effect of dipole-dipole
s
(Tin “C)
Classical fl (A)
chloroform, 6 8 10 12
T=230 K, a,= 15.14 13.28 11.99 11.11
s
Classical f(A)
interaction
Present work p=O
Present workp=
1.02 D
AF*/kT
f (4
AF*/kT
f (A)
hF*fkT
35.49 mJ/m’ 111.31 82.64 67.40 51.87
14.89 12.76 11.47 10.58
90.04 64.38 50.95 42.66
14.89 12.76 11.47 10.58
93.60 67.92 54.48 46.18
Present work JGO AF*/kT
acetonitrile, T=260 K, u-=33.48 ml/m* 2 22.35 195.17 3 14.09 77.69 4 11.17 48.79 5 9.62 36.20
Present worky=3.92
D
r* (A)
AF*/kT
f (4
AF*/kT
21.29 13.02 10.07 8.50
168.97 61.16 35.69 24.92
21.29 13.02 10.07 8.50
198.03 90.03 64.31 53.34
the critical free energy change for the chloroform and acetonitrile vapour systems are given in table 2 along with the classical values for different supersaturations. From table 2 it can be noted that incorporation of the dipole-dipole interaction increases the critical energy barrier to nucleation. Compared with the classical values the energy barrier is decreased for the nucleation of weakly polar chloroform vapour and increased for the nucleation of strongly acetonitrile vapour in the present work. The plots of the critical supersaturation versus temperature as calculated using the present theory including the effect of dipole-dipole interaction in
the case of chloroform and acetonitrile are shown in figs. 4 and 5 along with the experimental values of Katz et al. [ 131 and Wright et al. [ 61, respectively. It is seen that the curve incorporating the effect of dipole-dipole interaction is closer to the experimental curve in the case of the highly polar acetonitrile system. Incorporation of the dipole-dipole interaction decreases the concentration of critical size clusters and consequently the nucleation rate. Hence the critical supersaturation required at a particular temperature is increased and there is better agreement with experiment. Also for the weakly polar material chloroform the incorporation of the dipole-dipole 419
Volume 206, number $6
I May 1993
CHEMICALPHYSICSLETTERS
J
210
215
220
225
230
235
TEMPERATURE
260
2b5
250
K
Fig. 4. Variation of critical supersaturation with temperature for condensation of chloroform vapour: (0 ) classical, ( 0 ) experimental, (m) present work including dipole-dipole interaction (p= 1.02D), and (A ) present work excluding dipole-dipole interaction (p= 0).
21 230
I 2&O
1 250
1
I 260 TEMPERATURE
270
I 280
29
K
Fig. 5. Variation of critical supersaturation with temperature for condensation of acetonitrile vapour: ( 0 ) classical, ( A ) experimental,
and (0 ) presentworkincludingdipole-dipoleinteraction (IL=3.9D ). interaction with the effect due to curvature yields a result which is closer to the experiment than when
the curvature dependence alone is considered.
420
References [l] J. Frenkel, Kinetic theory of liquids (Dover, New York, 1955). [2] A.C. Zettlemoyer, ed., Nucleation (Dekker, New York, 1969). [ 3) F.F. Abraham, Homogeneousnucleation theory (Academic Press, New York, 1974).
Volume 206, number 5,6
CHEMICALPHYSICSLETTERS
[4] J. Fedder, KC. Russell, J. Lothe.and G.M. Pound, Advan. Phys. 15 (1969) 111. [5] V. Ruth, J.P. Hirth and G.M. Pound, J. Chem. Phys. 88 (1988) 7079. [6] D. Wright, R. Caldwell and MS. El-Shall, Chem. Phys. Letters 176 (1991) 46. [ 71 R.C. Tolman, J. Chem. Phys. 17 ( 1949) 333. [8] D.H. Rasmussen, J. Cryst. Growth 56 (1982) 45. [91 K Nishioka, Phys. Rev A 39 ( 1989) 772. [ lo] D. Jayaraman, C. Subramauian and P. Ramasamy, J. Cryst. Growth 79 (1986) 997.
I May 1993
[ 111F. Joseph Kumar, D. Jayaraman, C. Subramanian and P. Ratnasamy, J. Mater. Sci. Lt. 10 (1991) 608. [ 121F. Joseph Ktmuu, D. Jayaraman, C. Subramanian and P. Ramasamy, in: Nucleation and atmospheric aerosols, eds. N. Fukuta and P.E. Wagner (A Deepak Publishing, Hampton, 1992) p. 59. [ 131J.L. Katz, P. Mirabel, C.J. Scoppa II and T.L. Virkler, J. Cl-rem.Phys. 65 (1976) 382. [ 141J.W. Gibbs, Collected works, Vol. 1 (Longmans and Green, London, 1932).
421
CHEMICAL PHYSICSLETTERS
I May 1993
Nucleation kinetics of polar compounds. Effect of curvature and dipole-dipole interaction on surface energy F. Joseph Kumar,
D. Jayaraman,
C. Subraraanian
and P. Ramasamy
CrystalGrowthCentre, Anna University,Madras, India
Received 2 I January 1993
Based on thermodynamical concepts, a new expression for the surface energy of a microcluster of a polar material has been derived incorporating the effect of curvature and the dipole-dipole interactions. Using the expression, the nucleation parameters of the polar materials chloroform and acetonitrile have been calculated. A study of the dependence of critical supersaturation on the temperature has also been made and compared with the available experimental results.
1. Introduction The study of homogeneous nucleation of liquid clusters from a vapour has evoked considerable interest among scientists since it forms the basis for a wide range of more complex phenomena such as nucleation of water droplets in the atmosphere, crystal growth, thermal polymerization and so on. It has been widely accepted that the Becker-Doring-Zeldovich theory or the socalled classical nucleation theory [ l31, in spite of its crude assumptions and possible fundamental errors [ 4,5 1, predicts the correct value of critical supersaturation required for the onset of homogeneous nucleation in good agreement with experiment in the case of most of the weakly to moderately polar compounds. The apparent experimental agreement of the predictions of the classical theory has been attributed to the large extent of cancellation of errors involved in the theory. Recently Wright et al. [ 61 reported that the predictions of the classical theory are in serious disagreement with experiment in the case of homogeneous nucleation of the highly polar acetonitrile vapour system (dipole moment= 3.9 D), wherein the dipole-dipole interaction is expected to play a major role in modifying the surface free energy of microclusters at the critical stage. The classical nucleation theory is based on the capillarity approximation according to which the Elsevier Science Publishers B.V.
macroscopic thermodynamic properties apply to the supersaturated vapour and the liquid phases and the interface separating the two phases can be described by the surface between bulk quantities of both the phases. The validity of this approximation has long been questioned since it creates a discontinuity in the chemical potential from the vapour to the liquid phase. One way of solving this problem is to use a surface free energy with a functional dependence on the cluster size in place of the bulk value as used in the classical theory. Various attempts have been made in this respect to determine the curvature dependence of surface tension [ 7-121, Nucleation kinetics of non-polar materials CC&, C2H2C14, SnCl, and TiC14 have been studied using the curvaturedependent surface tension in our early works [ 11,121. But since in the case of weakly to moderately polar materials the use of curvature-dependent surface tension in the nucleation theory ruins the close agreement with experiment, the effect of dipole-dipole interactions on the surface free energy of these materials also should be considered along with the curvature dependence of surface tension. In the present work we have derived a new expression for the curvature-dependent surface tension including the effect of dipole-dipole interaction on the surface energy. The contribution towards the work of formation of a droplet of a particular size “8 is considered to be due to (i) planar interfacial ten415
Volume 206, number 5,6
CHEMICAL PHYSICSLETTERS
sion, (ii) the surface energy gradient and (iii) dipole-dipole interaction. Also the truncated Tolman expression S=&,,( 1 +&Jr) has been used instead of assuming the Tolman length 6 to be a constant [ 8, lo]. With these modifications a new expression for the surface energy of microclusters is derived and incorporated into the classical nucleation theory and the nucleation parameters are calculated for the moderately polar material chloroform with a dipole moment of 1.02 D and the highly polar material acetonitrile with a dipole moment of 3.9 D. The results are compared with the experimental values of Katz et al. [ 131 and Wright et al. [6] obtained using an upward thermal diffusion cloud chamber.
7 May 1993
orientation involves an increase of the potential energy of a surface molecule by an amount 28 W=
s 0
pEc sin 2ede,
(1)
where p is the dipole moment of each molecular dipole, E is the electric field, c is the number of surface nearest neighbours, r is the radius of the droplet and x is the distance between nearest neighbours. For very small 0 taking sin 28=20=2x/r the above equation simplifies to W=,u2c/4xr2. The total increase of surface potential energy with respect to a planar surface is then
2. Effect of dipole-dipole interaction and curvature on surface energy In order to determine the effect of dipole-dipole interaction on the surface free energy, the model proposed by Abraham [ 31 is followed. According to this model there is an increase in the dipole-dipole interaction energy at the microcluster stage which gives a positive contribution to the surface energy. For a polar liquid the surface dipoles are oriented perpendicular to the liquid surface and parallel to one another. For a spherical droplet of nucleus size this parallel orientation is destroyed due to the surface curvature and each dipole is oriented at an angle with respect to its nearest neighbours, the angle being proportional to the radius of the droplet (fig. 1). Each dipole of a dipole pair is oppositely rotated by an angle 8 with respect to a line bisecting the line joining the dipoles. This deviation from the parallel
vopour
where j, is the total number of molecules in the surface of the droplet and the prefactor { arises because each dipole of a dipole pair shares one half of the interaction energy. From solid angle considerations we get the value of j, as j, = 16r2/x2 . Hence the dipole-dipole interaction results in an increase of the total surface energy far highly curved surfaces by an amount u,=2~2cfx3.
Let us now consider a one-component two-phase system consisting of a liquid microcluster in a supersaturated vapour. Let the state of the system be determined by its entropy S, mass m of the component constituting the two phases, volumes tP and ve of the homogeneous vapour and liquid phases and by area A and radius r of the spherical microcluster. Following Gibbs [ 14 1, for equilibrium of the system ($-pa)
Liquid
(aI
(bl
Fig. 1. Illustration of dipole alignment of polar molecules at (a) a flat interface, (b) a spherical interface.
416
(2)
dtP=cr.. d,4tgCwdr.
(3)
Here a, is defined as the surface free energy per unit area of a plane interface and u, as the surface free energy per unit radius per unit solid angle [ 1O-121. Integrating the above equation between the limits zero and r one can obtain the work of formation of a droplet of radius r and solid angle 47r
7May 1993
CHEMICAL PHYSICS LETTERS
Volume 206, number $6
3. Nucleation kinetics
I
(f-pal)
dve=u,A+uc4w.
(4)
0
To determine the curvature-dependent surface tension, the work of formation is equated to the product of the surface area of the droplet and the sum of the planar interfacial tension, change in surface energy corresponding to that radius and the energy contribution due to the dipole-dipole interaction,
=
(,,yr+-&. i-* -
dr
bF= $r3AG, +4xr2u( r, p) ,
P2C
(6)
2nx3r3 ’
Here a, is a constant and is defined as
where a( r, p) and 6 are strong functions of radius of the droplet. 6 is the distance between the surface of tension and the equimolar dividing surface. Using the Tolman expression 6=6,( 1 +&Jr) and substituting the value of a, in eq. (6),
xexp
”
(7)
Here S, is the radius of a monomer whose value is taken to be 1 nm. The above differential equation can be solved to get the expression for the curvature dependence of surface free energy including the correction due to dipole-dipole interaction as r*c[ l+ (26J3r) 4Kx3r=
xex
P(
- % (1+&/2r)
>
.
+ (&,/2r)*] >
(8)
.
(1+6,/2r) >
(10)
The value of f is calculated by the method of iteration, taking the classical value P= -2u,/AG, as the first approximation. The critical free energy change AF* is calculated by using the value off in eq. (9). The rate of homogeneous nucleation I is defined as the number of clusters nucleated per second in a unit volume of vapour, and is expressed as AF+ - kT (
P2C
(w~)21
[2+&/r+
- +
(
I=Kexp
=-w*
(9)
where AG,= &In (S) /v is the change in volume free energy, u(r, p) is the surface tension, S is the supersaturation ratio, v is the specific volume of the liquid and k is the Boltzmann constant. For Sz 1, AF passes through a maximum value at a particular radius called the critical radius. Differentiating eq. (9) and equating to zero the critical radius is obtained as P=$
Comparing eqs. (4 )’ and (5) we obtain
ddr,pu) uc ~____-
The free energy change associated with the formation of a droplet of size r in a supersaturated system at a pressure P and temperature T is
, >
(11)
where K is a factor which varies more slowly with P and T than the exponential term. K is usually expressed as
where Z is the Zeldovich factor whose value is typically of the order of lo-=, P is the actual pressure of the vapour, k is the Boltzmann constant, T is the temperature of the vapour, a( r, 1) is the surface free energy and m is the mass of a single vapour molecule. The supersaturation value at which I= 1 is called the critical supersaturation SC which can be computed for different temperatures. 417
Volume 206, number 5,6
CHEMICALPHYSICS LETTERS
7 May 1993
the size increases, the surface tension first falls rapidly below the bulk value and then rises exponentially lo the bulk value. In the case of the highly polar acetonitrile system the surface energy drops exponentially to the bulk value indicating the strong effect of dipole-dipole interaction of surface energy. The thermophysical data employed in the present calculations are presented in table 1 for the two systems. The present values of the critical radius and
4. Results and discussion Figs. 2 and 3 show the variation of surface free energy as a function of cluster size for chloroform and acetonitrile at T=230 and 260 K, respectively. It is found that the surface free energy for very small clusters of a size of a few angstroms (containing 1O-20 molecules) is higher than that of the bulk value. In the case of the weakly polar chloroform system, as
50
RADIUS
A
Fig. 2. Variation of surfax free energy with radius for chloroform, (0 ) radial dependence, and ( 0 ) total dependence.
0
5
10
15
20
25 RADIUS
30
(A
35
) effect due to dipole-dipole interaction
40
L5
5
A
Fig. 3. Variation of surface free energy with radius for acetonitrile, (0 ) radial dependence, (A ) effect due to dipole-dipole interaction and ( 0 ) total dependence.
418
Volume 206, number 5,6
CHEMICAL PHYSICS LETTERS
I May 1993
Table 1
Thennophysicaldata employed in the present calculations [ 13,6] chloroform
M=ll9.38kg/kmol, lnP,=
-4 :[
p=l.O2D, T-r (’
‘,I
c=3,
x=3x10-10m
+Arrln;+fAB(T-&,)+aAy(%r;)+&A&%T:)
>I
+ (&+ fA#?&t f AyT: + aAD;) 2 - 1 +ln 760Torr ( ACY=-0.14504, A/&1.259x10-‘, Ay=7.2079xlO-‘, A6=2.0001~10-’ L,=59.9 k&/kg, I&.=334.35K u=29.91-O.l295(T-273.15) ml/m’ &1.526~10’-1.845T-6.017~10-*T~-1.436x acetonitrile
(TinK) lo-‘T’kg/m3
M=41.053kg/kmol, ~=3.9D, c=3, x=3.5x10-‘“m logP.=7.9386621785.844/T (Tin K) u=31.82-O.i263(T-273.15) mJ/m’ (TinK) d=0.8073~10~-1.07(T-273.15)kg/m” (TinK)
Table 2 Variation of critical free energy change with and without the effect of dipole-dipole
s
(Tin “C)
Classical fl (A)
chloroform, 6 8 10 12
T=230 K, a,= 15.14 13.28 11.99 11.11
s
Classical f(A)
interaction
Present work p=O
Present workp=
1.02 D
AF*/kT
f (4
AF*/kT
f (A)
hF*fkT
35.49 mJ/m’ 111.31 82.64 67.40 51.87
14.89 12.76 11.47 10.58
90.04 64.38 50.95 42.66
14.89 12.76 11.47 10.58
93.60 67.92 54.48 46.18
Present work JGO AF*/kT
acetonitrile, T=260 K, u-=33.48 ml/m* 2 22.35 195.17 3 14.09 77.69 4 11.17 48.79 5 9.62 36.20
Present worky=3.92
D
r* (A)
AF*/kT
f (4
AF*/kT
21.29 13.02 10.07 8.50
168.97 61.16 35.69 24.92
21.29 13.02 10.07 8.50
198.03 90.03 64.31 53.34
the critical free energy change for the chloroform and acetonitrile vapour systems are given in table 2 along with the classical values for different supersaturations. From table 2 it can be noted that incorporation of the dipole-dipole interaction increases the critical energy barrier to nucleation. Compared with the classical values the energy barrier is decreased for the nucleation of weakly polar chloroform vapour and increased for the nucleation of strongly acetonitrile vapour in the present work. The plots of the critical supersaturation versus temperature as calculated using the present theory including the effect of dipole-dipole interaction in
the case of chloroform and acetonitrile are shown in figs. 4 and 5 along with the experimental values of Katz et al. [ 131 and Wright et al. [ 61, respectively. It is seen that the curve incorporating the effect of dipole-dipole interaction is closer to the experimental curve in the case of the highly polar acetonitrile system. Incorporation of the dipole-dipole interaction decreases the concentration of critical size clusters and consequently the nucleation rate. Hence the critical supersaturation required at a particular temperature is increased and there is better agreement with experiment. Also for the weakly polar material chloroform the incorporation of the dipole-dipole 419
Volume 206, number $6
I May 1993
CHEMICALPHYSICSLETTERS
J
210
215
220
225
230
235
TEMPERATURE
260
2b5
250
K
Fig. 4. Variation of critical supersaturation with temperature for condensation of chloroform vapour: (0 ) classical, ( 0 ) experimental, (m) present work including dipole-dipole interaction (p= 1.02D), and (A ) present work excluding dipole-dipole interaction (p= 0).
21 230
I 2&O
1 250
1
I 260 TEMPERATURE
270
I 280
29
K
Fig. 5. Variation of critical supersaturation with temperature for condensation of acetonitrile vapour: ( 0 ) classical, ( A ) experimental,
and (0 ) presentworkincludingdipole-dipoleinteraction (IL=3.9D ). interaction with the effect due to curvature yields a result which is closer to the experiment than when
the curvature dependence alone is considered.
420
References [l] J. Frenkel, Kinetic theory of liquids (Dover, New York, 1955). [2] A.C. Zettlemoyer, ed., Nucleation (Dekker, New York, 1969). [ 3) F.F. Abraham, Homogeneousnucleation theory (Academic Press, New York, 1974).
Volume 206, number 5,6
CHEMICALPHYSICSLETTERS
[4] J. Fedder, KC. Russell, J. Lothe.and G.M. Pound, Advan. Phys. 15 (1969) 111. [5] V. Ruth, J.P. Hirth and G.M. Pound, J. Chem. Phys. 88 (1988) 7079. [6] D. Wright, R. Caldwell and MS. El-Shall, Chem. Phys. Letters 176 (1991) 46. [ 71 R.C. Tolman, J. Chem. Phys. 17 ( 1949) 333. [8] D.H. Rasmussen, J. Cryst. Growth 56 (1982) 45. [91 K Nishioka, Phys. Rev A 39 ( 1989) 772. [ lo] D. Jayaraman, C. Subramauian and P. Ramasamy, J. Cryst. Growth 79 (1986) 997.
I May 1993
[ 111F. Joseph Kumar, D. Jayaraman, C. Subramanian and P. Ratnasamy, J. Mater. Sci. Lt. 10 (1991) 608. [ 121F. Joseph Ktmuu, D. Jayaraman, C. Subramanian and P. Ramasamy, in: Nucleation and atmospheric aerosols, eds. N. Fukuta and P.E. Wagner (A Deepak Publishing, Hampton, 1992) p. 59. [ 131J.L. Katz, P. Mirabel, C.J. Scoppa II and T.L. Virkler, J. Cl-rem.Phys. 65 (1976) 382. [ 141J.W. Gibbs, Collected works, Vol. 1 (Longmans and Green, London, 1932).
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