JOURNAL OF MATERIALS SCIENCE LETTERS 10 (1991) 608-610

Curvature dependence of surface free energy and nucleation kinetics of CCI4 and C2H2CI4 vapours F. JOSEPH KUMAR, D. JAYARAMAN, C. SUBRAMANIAN, P. RAMASAMY

Crystal Growth Centre, Anna University, Madras 600 025, India

Based on a quasi-thermodynamical model, Tolman [1] has derived an expression for the curvature dependence of interracial tension. Rasmussen [2] has derived a similar expression based on the thermodynamical treatment of the capillarity phenomena. Similar work has been reported by Vogelsberger et al. [3-5] and Nonnenmacher [6]. Nishioka [7] has formulated a theory for the curvature dependence of interracial tension by employing computer simulation results instead of the Tolman's length 6. Recently Jayaraman et al. [8] have derived a new expression for the curvature dependence by introducing the solid angle concept and assuming that the ith cluster contains i monomers. In the present treatment the work of formation of a droplet of a particular radius r is equated to the product of the area of the droplet and the sum of the planar interfacial tension and the change in the surface energy corresponding to that radius. Also the truncated Tolman's expression 6 = 6~ [1 + (6~/r)] has been used instead of assuming 6 to be a constant. With the above modifications a different expression for the curvature-dependent surface tension is derived. After incorporating the curvature-dependent surface tension in the classical nucleation theory, the predicted values of the critical supersaturations of CC14 and C2H2C14 vapours at different temperatures are compared with the experimental results obtained by Katz et al. [9] using the upward thermal diffusion cloud chamber. Consider a liquid microcluster in a supersaturated vapour. The microcluster is assumed to be spherical. Let P~ and P# be the pressures inside the homogeneous vapour and liquid phases, respectively. Following Gibbs [10] the change in free energy of the system between the two phases is d E i = T d S + g d m - P ~ d V ~ - P{JdV/3 + o ~ d A + o~codr

(1)

Here o-~ is the surface free energy per unit area of a plane interface and o-c is the surface free energy per unit radius per unit solid angle [8]. The change in the Gibbs free energy for a system without interface, i.e. consisting only of a bulk oi phase at a pressure P~ and volume V = V ~ + V/3, is dE2 = T d S + #drn - p ~ d V

(2)

The free energy of the interface can be obtained from Equations 1 and 2 as dEi = ( pc~ _ p # ) d V # _ o-~dA - o-co)dr

608

At equilibrium dEi = O; therefore (po~ _ p ~ ) d V f i = o-~dA + occodr

(3)

Integrating the above equation between the limits O and r one can obtain the work of formation of a droplet of radius r: f [ (P~ - P ~ ) d V ~ = o-~A + o-ccor

(4)

In the case of homogeneous nucleation of a spherical droplet, we have solid angle c0 = 4sT. Therefore r

fo ( P ~" - P f i ) d V f~ = o ~ A + Crc4JVr

(5)

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 and the change in surface energy corresponding to that radius: fo ( P : - P#)dV/J = c~ +

.r A

(6)

Comparing Equations 5 and 6, o-~A + 4sro-cr = (o-~ + ~ - - r r ) r ) A

do(r) dr

(7)

o-c r2

0

(8)

Though both o(r) and 6 are strong functions of radius, their product is assumed to be a constant and equated to o~:

o(r)6

=

=o-(r)6=(1 + @~, )

(9)

Here 6~ is the difference between the radius of the surface of tension and the equimolar dividing surface and taken to be 0.1nm. Substituting for Oc in Equation 8, dr

r2 6~ 1 +

=0

(10)

The solution of the above differential equation is o(r) = o-~ exp

1 + ~-r

(11)

The variation of o(r) with radius is shown in Fig. 1. It is found that the variation is more rapid at the beginning of nucleation. Beyond the radius 1 nm the variation is not appreciable and approaches the bulk value. 0261-8028/91 $03.00 + .12

© 1991 Chapman and Hall Ltd.

]°0

m

0.8

~0.6 b

0.4

//

0.2--

I

0

r

I

I

~

0~4

0

I

~

I

i

I

0,8

i

I

I

1.2

I

i

I

,

I;i

1.6

2.0

I

4.0

r Into)

Figure l Variation of surface tension with radius: (O) present work, (A) Tolman's theory, (X) Rasmussen's theory.

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 A F = 4srr 2 or(r) - 4 s r r 3 A G v

(12)

where AGv = k T l n S / v is the change in volume free energy, o ( r ) is the curvature-dependent surface free energy, S is the supersaturation ratio, v is the specific volume of the liquid, and k is the Boltzmann constant. Substituting o ( r ) from Equation 11 in Equation 12 A F = 4srr2cr~ exp

1 -}- 2 7

-- 42EF3AGv (13)

For S > 1, A F passes through a maximum value at a particular radius called the critical radius. Differenti-

ating Equation 13 and equating to zero, the critical radius is obtained as r* =

exp

1+

2r + 6~ +

F

(14) The value of r* is calculated by the method of iteration, taking the classical value r* = 2o~AGv as the first approximation. The critical free energy change AF* is calculated by using the value of r* in Equation 14. Tables I and I! show the present values of the critical radius and critical free energy change for the CC14 and CaH2C14 systems for various supersaturations, along with the classical values. The incorporation of the curvature correction to the

T A B L E I Critical radius and critical free energy change of CC14 vapour Classical values

Present values

System

P P*

r*(nm)

AF* k T

r*(nm)

o'(r*)

AF* k T

CC14: cr~ = 29.85

4.0 4.4 4.8 5.2 5.6 6.0 6.4 5.8

1.803 1.687 1.593 1.516 1.451 1.395 1.346 1.304

109.14 95.55 85.24 77.17 70.67 65.33 60.81 57.10

1.751 1.635 1.541 1.464 1.399 1.343 1.294 1.252

28.15 28.03 27.92 27.82 27.72 27.63 27.55 27.47

91.26 78.81 69.46 62.17 56.34 51.56 47.59 44.23

erg/cm-2 0.02985Jm-2), T = 270 K

T A B L E II Critical radius and critical free energy change of C2H2C14vapour Classical values

Present values

System

PP*

r*(nm)

AF*kT

r*(nm)

or(r*)

AF*kT

C2H 2C14: o~ = 40.42 erg cm-2 (0.04042Jm -2, T = 260 K

10 20 30 40 50 60

1.656 1.273 1.121 1.034 0.975 0.931

129.51 76.51 59.35 50.49 44.84 40.96

1.605 1.221 1.069 0.981 0.922 0.878

37.90 37.11 36.64 36.31 36.05 35.83

106.45 58.88 43.88 36.23 31.45 28.17

609

interfacial tension reduces the critical radius and consequently the critical energy barrier of nucleation. The rate of homogeneous nucleation I is defined as the n u m b e r of clusters nucleated per second in unit volume of vapour, and is expressed as I = K exp \ - ~ T - - ]

4.5

5.5

(15)

where K is a factor which varies m o r e slowly with P and T c o m p a r e d with the exponential term. K is usually expressed as

~25 c

1.5 K =

u ~mm

Z

(16)

where Z is the Zeldovich factor whose value is typically of the order of 10 -2 and m is the mass of a single vapour molecule. The supersaturation value at which I = 1 is called the critical supersaturation So, which can be computed for different temperatures. The plots of the critical supersaturation versus t e m p e r a t u r e for carbon tetrachloride (CC14) and tetrachloro ethane (C2H2C14) are shown in Figs 2 and 3 along with the experimental values of Katz et al. [9] and classical values. The plots show that the 10

8 -

2 o

0

I 24.0

I

I 260

I

I 280

I

I

[

300

Temperature, T [K) Figures Variation of critical supersaturation with temperature

for condensation of C2H_~C14vapour: (©) present work, (X) experimental, (~) classicaltheory. classical curve lies above the experimental curve. Incorporation of the curvative correction to the interracial tension gives rise to an increase in the concentration of critical size clusters aand hence the nucleation rate. Hence the critical supersaturation required at a particular t e m p e r a t u r e is lowered. For the CCl 4 system the evaluated critical supersaturations approach the experimental values at higher temperatures. However, in the C2H2C14 system our results are closer to the experimental values than the classical values. It is interesting to note that in both systems our results are close to the experimental values than the values predicted by T o l m a n ' s [1] and Rasmussen's [2] expressions.

P .=,7

References

c~

i, 2. 3.

~6

R . C . T O L M A N , J. Chem. Phys. 17 (1949) 333. D . H . R A S M U S S E N , J. Cryst. Growth 56 (1982) 45. w . V O G E L S B E R G E R et al., Z. Phys. Chem. (Leipzig)

257 (1976) 580.

o

4. Idem, ibid. 261 (1980) 1217. 5. Idem. ibid. 264 (i983) 265.

6. T. F. NONNENMACHER, Chem. Phys. Lett. 47 (1977) 507. 7. K. NISHIOKA, Phys. Rev. A 16 (1977) 2143.

5

8.

4 250

I

260

i

I

9.

i

270 280 T e m p e r a t u r % T (K)

290

Figure2 Variation of critical supersaturation with temperature for condensation of CC14 vapour: (O) present work, (X) experimental, (A) classical theory.

610

10.

D. J A Y A R A M A N , C. S U B R A M A N I A N and P. R A M A SAMY, J. Cryst. Growth 79 (1986) 997. J . L . K A T Z , P. M I R A B E L , C. J. S C O P P A II and T. L. V I R K L E R , J. Chem. Phys. 65 (1976) 382. GIBBS

R e c e i v e d 9 July and accepted 8 October 1990