CONDENSATION ON DIPOLAR MOLECULES LOUIS A. M A D O N N A Engineering Dicision, Pennsylaania Military College, Chester, Pa. S T E V E N L. P E A K E Guy Research and Deoelopment Co., Pittsburgh, Pa.
An attempt was made to correlate the supersaturation data of Volmer and Flood for several vapors by assuming heterogeneous nucleation due to dipolar molecules rather than by Volmer's assumption of homogeneous nucleation. The free energy contributed to an embryo by the presence of a dipole was found to b e negative, and therefore it is theoretically possible for dipolar molecules to cause heterogeneous nucleation. Using Volmer's observed critical supersaturation ratios, for each vapor the critical free energies of dipole-based embryos were calculated a t different assumed values of dielectric constant. From these, plots of kinetic coefficient vs. assumed dielectric constant were made.
HE condensation of a vapor into liquid droplets is a fairly Tcommon phenomenon in nature. Rain and the formation of clouds and mists are two familiar examples. T h e process is usually treated quantitatively by means of a kinetic rate equation analogous to that used for determining the rate of chemical reactions. T h e number of droplets (nucleation rate) appearing per unit time in a unit volume of vapor is given by :
J
=
Cexp
(-F )
kTln
If the free energy, AF, is plotted as a function of the radius, the resulting curve has the form shown in Figure 1. T h e curve maximum yields the critical free energy difference and the radius of the critical nucleus, which, because it possesses this maximum free energy, is the smallest embryonic droplet which is metastable Ivith respect to the vapor. T h e value of r* may be found by differentiating Equation 2 with respect to r , setting equal to zero, and solving for r : -2 M y = pAF'
T h e quantities r, and rk are found by solving for the tlvo values of r in Equation 5 :
AF*
T h e free energy difference between any nucleus or droplet and the vapor from which it is formed in the case of homogeneous nucleation is given by:
p*
Similarly, for nucleation on ions, one case of heterogeneous nucleation, the change in free energy, I F , is given by the Thompson equation :
*I:(
2 MY pRT In
(:)*
= 2 rV -
r
D"e2 4v 8 m
~
I n the pure carrier gas or as a result of the interaction of the carrier gas with the substance undergoing nucleation there might exist dipolar molecules. It could logically be assumed that dipoles exist in the supersaturated vapor, and heterogeneous rather than homogeneous nucleation occurred. For this reason, the investigation reported here \vas undertaken. Assumption of Dipoles
The free energy difference, AF, in Equation 1 for an embryonic droplet of given radius groivn from a dipolar molecule would differ from AF for an embryo of the same radius grown spontaneously-Le., by homogeneous growth. They Lvould differ by the quantity of electric work, rye,,involved in forming the dipole-based embryo. For such an embryo, Equation 2 becomes :
(24
T h e kinetic coefficient, C, for the case of homogeneous nucleation was derived by Becker and Doering ( I ) . T h e rate equation becomes :
J =
dP'
LYdTA%-312p)2
Table I .
2);(
p*/pm*
4;R2 - 16 r M 2 y 3 exP
(
3 p 2 k T RT In
-
P
9 (3)
I n Equation 3, setting J equal to 1, values of ( p / p -)* were calculated for the conditions a t which Volmer and Flood (2, 3 ) had obtained experimental supersaturation ratios. T h e results are compared in Table I. 80
l&EC FUNDAMENTALS
Observed and Calculated Supersaturation Ratios
Material IVater, 275.2 K. IVater, 261 . 0 K. Methanol, 270.0 e K. Ethanol, 273.0' K. 1-Propanol, 270.0" K. 2-Propanol, 265.0' K. 1-Butanol, 270.0' K. Nitromethane, 252' K. Ethyl acetate, 242.0 a K.
Obsd. 4.2 f 0.1 5.0 3.0 2.3 3 .O 2.8 4.6 6.0 8 . 6 to 1 2 . 3
Calrd. 4.2 5.0 1.8 2.3 3.2 2.9 4.5 6.2 10.4
2.0
-
D - 80 0
1.8
-
1.6
-
D a
15
1.4-
1.2
-
1.0
-
u)
?
-* (u
x
Figure 1.
q
Relation of free energy to radius
0.8 II
0.6
D = 1.85
D-
1.5
0.4-
Vacuum
0.2
-
I Figure i!. Model assumed for calculating electric work
Figure 3.
T o obtain an estimate of the value of TYe1, the spherical model shown in Figure 2 \$as used to represent a dipole-based embryo. I t is doubtful if the results obtained using a more complex model such as a n ellipsoid would differ sufficiently from those obtained from the spherical model to justify the great increase in the complexity of the mathematics entailed. Assuming such a model, the electric work involvrd in putting the shell of material around the previously exposed dipole is given by:
n=1,3,5
.....
[uo,~""(n
I ' I ' 1: I' T! 1
3
d = 1.5 x 10-8 cm. a = 3.0 X 10-8 cm.
T h e distance, b , to the cuter wall of the shell is the radius, r , of the embryo. T h e charge, e , on the dipole was assumed to be one electronic charge. Substituting values into Equation 7 and factoring 10-8 from under the summation sign yields:
1
l
1
1
1
7
8
9
5rad8s of embryo
1
1
1
1
10
II
12
13
Angstroms
1
1
14
15
Relation of electric work to radius of embryo
where r is now in Angstrom units and ?1,'1 in ergs. rI'@l is now assumed to be a function of only r, the radius of rhe embryo, and D,the dielectric constant of the material forming the shell around the dipole. Substitution of the correct value of D into Equation 8 would make Tl'el, and therefore AF in Equation 6, a function of r , and possibly could be maximized to determine I F as was done for the homogeneous case. By Equation 1, then, the value of the kinetic coefficient, C. could be found
+ nD + Dj(1 + nD + nj-
This equation will be derived in a separate publication. Even assuming that the dipoles exist, their dimensions are not known. As reasonable values, therefore. the following were assumed :
4
dnn2(1- D)%(n
+ 1)
(7)
for the case of condensation on dipoles uhich produced the observed critical supersaturation ratios. I t is unlikely. boa.ever, that the bulk phase values of the dielectric constant available for various materials are applicable to the materials when they exist as rather small aggregates of molecules such as those dealt with here. Volmer and Flood 13), for example, successfully correlated their ion nucleation data (Equations 3 and 4) for u'ater Tvith a dielectric constant of 1.85, whereas water in the bulk phase has a dielectric constant of 80. Equation l , then, actually contains tivo unknoivns : the kinetic coefficient, C. and the dielectric constant, D. Since, by letting J equal 1, Equation 1 may be easily rearranged to make C a n
VOL. 5
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81
D = 1.85
Table 11. Electric Work Required to Form a Hollow, Spherical Shell of Radius r Made of Material Having Dielectric Constant D Dielectric Constant, D r, A. 3.5 4 5 6
7 8 9 10 11 12 13 14
15
1.5
7.85
... ...
0.288 0.429
0.431 0.452 0.464 0.4715
0.6269
... ... ... ... ...
...
0.673
...
10 75 80 X lo", Ergs ... 1.057 ... ... , . . 1.3406 .., ... 1.5334 .., 1.'318 1.6005 _ ... 1.910 1.358 i.6314 1.733 1.9156 1.381 1.6479 1.7459 1.9185 1 . 3 9 4 1.657 1.7536 .,.
2.2
20 I .8
5
- W,i
~
/w/o Wel
I .6 El
P
!homogeneous)
1.4
~~~
0.689 0.690
1.408
0.6947
1.4143 1.672
1.7650
0.6974
1.41?7
1.7670 1.9232
1.668
1.675
1.?61
Y u)
I. 2
F
W
1.922
I
... ...
Q
1.0 0.8 0.6
0.4
explicit function of D, the calculations were made by assuming values of D and solving for C. Curves of C us. D plotted for the materials in Table I should reveal any value of C which would correlate their physical constants with their observed critical supersaturation. A common ordinate for all the curves either at their individual bulk phase D or at some fractional value of their bulk phase dielectric constant, the same fraction for each material, would be a n indication of correlation. Since r and the assumed value of D are independent of the material under consideration, the electric work, We,, may be calculated for a range of values of r and D and be equally applicable to all the materials under consideration. If it is recalled that AF* is determined by maximizing AF with respect to r , it can be seen that A F for the case of dipoles must be maximized graphically. The expression for W e l in Equation 6 makes impossible the maximization of the latter by analytical methods. In order to reduce the number of calculations necessary to obtain W e ,for a graphical determination of AF*, as well as to reveal the nature of the variation of Wei as a function of r and D , curves of W e , us. r were plotted for different values of the dielectric constant. The curves shown in Figure 3 are plots of the tabulated values in Table 11. As an example of the manner in which the values shown in Table I1 were found, the calculation of W e ,from Equation 8 is shown in Table 111. The electric work involved in forming a hollow shell of material of dielectric constant equal to 80 is calculated for embryos of 6-, 7-, and 8-A. radius, respectively.
W e , = (0.4558
x
10-10) (1
0.2
0.0
3
4
5
6
radius of embryo Figure 4. embryo
7
8
9
- Angstroms
10
Relation of free energy to radius of
a t D = 80 for all values of r , and the ( r z n T 1 ) factors calculated for one value of r obviously can be used for all values of D . Once the factors have become evaluated at n = 1 > 3, and 5 for all values of D and 7 that are of interest, it is then only necessary to insert them into the general term of the series under the summation sign, sum, and multiply by (0.4558 x 10-lO) (1 - D) to obtain JVe1. Values of the first two terms of Equation 6 must also be calculated for the graphical maximization of F.
Values of y, T , and ( p / p m ) *, as well as the bulk phase dielectric constant for the materials under consideration, are tabulated in Table V. All values except dielectric constants are from Volmer ( 2 ) . T h e graphical determination of AF+ for water at 275.2' K., assuming a dielectric constant of 1.85, was made by assembling and calculating values in the manner shown in Table I V .
(1.5)2n(r2n+1- 3.0Zn+l)
- D) n=1,3,5..
.
n ( l $- D,
3.04n*2(1 - D ) % ( n f 1) +
-
1
+ n ( D + 1)
D = 80
n=1,3,5,,
,
Y"*"5.U""'
The numbers tabulated under Z n = i in Table I11 are the values of the ith term of the series to be summed in Equation 8, determined by inserting the factors already calculated for n = 1, 3 , and 5. The increment contributed to the summation by the third term being only about 0.1% of the first term, the terms for the higher values of n were not determined. The sum of the first three terms multiplied by -36.008 X 10-10 gives the electric work, as shown. The factors independent of r may be used to evaluate W e l 82
l&EC FUNDAMENTALS
The plot of AF' is shown in Figure 4. I t was found in this and subsequent calculations that the curve of W e , (Figure 3) was so nearly horizontal near (7*) that W e lmerely lowered the curve of AFo us. r. T h e location of the maximum with respect to r was essentially independent of Wel. If the critical radius falls in the horizontal portion of the 147e1 curve for any material, then the critical radius is the same as for the case of homogeneous nucleation, and may be calculated by use of Equation 2a.
Table 111. ( 1.5p
Example Calculations 81 n
3.0 2 n
2.25 11.391 57.664
27.0 2.187 X lo3 1.7714 X l o 5 r=6A.
n 1 3 5 Total Wel, erg
+I
21 6 2.7993 X l o 5 3.6278 X l o 8
Table IV.
r = 8A.
r=7A.
+n+l
2,-i
343 8.2357 X loK 1.9774 X l o 9
5.1525 1.6123 0.6712 0.5320 -1.9156
X X X X X
-0.2064 X -0.490 -0.956 -1.652 -2.62 -3.92 -4.22 -4.69 -5.57 -7.65
0 -0.429 X -0.568 + -0.627 -0.660+ -0.673 -0,676' -0.680+ -0.681 -0.689
Table V. Molecular
T,
Material
Hz0
wt.
OK.
275.2 261 . O 270.0 273.0 270.0 265.0 270.0 252
Methanol Ethanol 1-Propanol 2-Propanol 1-Butanol Nitromethane
18 18 32.0 46 60.1 60.1 74.1 61
Data from Volmer (2, 3) Density, Y , Ergs G./Cc. sq. cm. 75.23 1 77.28 1 0.81 24.8 0.81 24.0 0.182 25.4 0.81 23.1 0.83 26.1 1.2 40.6
Values of Log,& as a Function of Assumed Dielectric Constant AF* X
T,
K.
D
275.2
Methanol Ethanol
1.85 2.1 2.8 261.0 1 . 8 5 2.1 2.8 270.0 1 . 1 1.2 273.0 1 . 1 1.85 2.8 8 m
1-Propanol
X 10-4 X X 10-6 X X
AF
AFw/oTl/',li(hom)
0 . 6 4 4 X 10-l2 0.591 0.739 1.12 1.345 1.45 1.454 1.445 1.40 1.10
0.644 X 10-lz 1.020 1.307 1 .75 2.00 2.12 2.13 2.132.08 1.79
270.0
1.85 w
2-Propanol
265.0
1.85
1-Butanol
270.0
1.85
Nitromethane
252
1.85 8
m a
m
ra, A . rki, A .
3.69 3.92 4.35 3.71 3.94 4.39 2.97 3.90 2.75 5.15 5.94 6.84 7.32 4.98 6.94 -5.16 7.21 5.02 7.14 4.40 6.00 6.53
7.73 7.64 7.46 7.40 7.31 7.03 7.67 7.35 14.03 13.67 13.42 13.08
12.89 14.49 13.90 14.69 14.03 13.15 12.38 10.46 9.82 9.54
7012 Ergs 0.463 0.365 0.220 0.386 0,292 0,142 0.2372
0.099 1.494 0.761 0.536 0.323 0.235 1.045 0.367 0.951 0.380 0.749 0.218 0.608 0.168 0.089
5.30 4.17 2.516 4.65 3.52 1.71 2.765 1.154 16.5 8.77 6.19 3.72 2.71 12.15 4.28 11.3 4.50 8.72 2.54 7.59 2.097 1.111
4.21 5.03 3.0 2.34 3.05 2.80 4.60 6.05
J
In J = 0 = In C logl0C
UB 80 80 3 7 . 5 (0"C.) 28.4 (O'C.) 24.8 ( 0 " C.) 15 .7 (20' C.) 7 . 8 ( 1 9 " C.) 6 (assumed value)
(P/$m)*obs.
By Equation 1, the kinetic coefficient was calculated from the values of AF*
(Calculated from 'Volmer's homogeneous data) Material
5.1605 1.6123 0.6712 0.5328 -1.9185
Free Energy Change for Homogeneous and Dipole-Based Nucleation as a Function of Embryo Radius
0 . 8 5 0 X 10-l2 1.510 2.263 3.40 4.625 6.04 6.35 6.815 7.65 9.44
Table VI.
2.n-i
512 2.0971 X 106 8.590 X l o 8
4 rr2y
r 3 4 5 6 7 8 8.2 8.5 9 10
+ 1)
1.1111 x 105 1.468 X l o 9 1.4471 X 1013
161 323 485
r z n +1
2,-i 5.1369 X 1.612 X 0.6712 X 0.5305 X -1.910 X
3 . 0 4 n + 2 ( -7Q)%(n 1+81n
+ 80
= ce - A F * / k T
- AF*/kT
when J = 1 droplet per second per cc. log10C =
AF* ~
2.303 k T
AF*, r*, and logloC are tabulated in Table VII. Conclusions
The fact that the W 6 , calculated here was negative, thus lowering the free energy of the critical embryo, shows that heterogeneous nucleation on dipoles is theoretically possible. T h e manner in which AF* varies with the assumed dielectric constant is shown implicitly by Figure 5, since by Equation l a log,& is proportional to AF*. Assuming the bulk phase dielectric constant for water, the critical free energy is lowered from about 430 cal. per mole of H20 for homogeneous nucleation to 42 cal. per mole of HnO for condensation on dipoles. One possible method of experimentally verifying the dipole assumption would be to inject the vapor of a substance known to consist of dipole molecules into a cloud chamber before an expansion was made. The vapor injected must necessarily be inert toward the other materials in the cloud chamber and VOL. 5
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83
30
(Data of Volmer and Flood) O
0
-1-
25
0
H20, 2 7 5 . 2 O K H20, 261.0 *K Methanol, 270.0.K Ethanol, 2 7 3 . O 0 K I-Propanol, 2 7 0 . 0 ° K Propanol, 265.0.K I-Butanol, 2 7 0 . 0 O K
+i-
20
x
I Nitromethane, 2 5 2 . 0 . K A $0 ions, 2 6 5 . 0 *K
.I5
u 0 m
- IO 0
5
0
-5
I 0.2
I
0.4
Figure 5.
I 0.6
I 0.8
I
1.0
I 1.2
I 1.4
I
1.6
d
1,8
Relation of kinetic coefficient to dielectric constant Data of Volmer and Flood (3) Assumption of dipoles
o
H20, 275.2.K
0
H20, 261.0.K
-I-
Methanol, 2 7 0.0 * K Ethanol, 2 7 3 . q d * K A I- Propanol, 2 7 0.0 K 2 Propanol, 2 6 5 . 0 0 ~ x I-Butanol, 2 7 0 . 0 . K I Nitromethane, 252,O.K A H20, ions, 2 6 5 . 0 . K
+
I
+ -
asymptote= OD
Figure 6.
84
I&EC FUNDAMENTALS
Relation of kinetic coefficient to dielectric constant Assumption of ions
280
Maximum Free Energy Change (AF* in Ergs) and Kinetic Coefficient (log& Constant for Condensation and Dipoles
Table VII.
D 1.o
1.5 1.85 5 10 15 80
D 1.0 1.85 5 10 15 80
Water, 275.2' K. r* AF* >< 7012 logl0C 8.2 2.13 24.35
Water, 267.0' K. r* AF* X 10l2 logloc 8.0 2.038 24.59
8.2 8.2 8.5 8.0 8.0
7.5 8.0 8.0 8.0 8.0
1.454 0,746 0.4'70 0.3'741 0.21
16.67 8.528 5.37 4.277 2.401
I-Propanol, 270.0" K. r* AF* >: 70'2 logl& 27.41 15.0 2.352 1.6546 19.28 15.0 10.89 15.0 0.9343 7.89 15.0 0.6'77 0.585 6.82 15.0 5.00 15.0 0.429
1.371 0.657 0.390 0,292 0.1195
Log,&
Ethanol, 273.0" K , AF* X r* 1012 log1& 14.0 2.006 23.12
Methanol, 270.0" K . r* AF* X logioc 8.0 0.6553 7.637 7.5 0.193 2.25 7.5 -0.010 -0.116
16.53 7.92 4.702 3.52 1.44
... ... ... ...
2-Propano1, 265.0" K. r* AF* X 10l2 log& 15.0 2.209 26.23 15.0 17.95 1.5116 15.0 0.7913 9.40 15.0 0.534 6.34 15.0 0.442 5.25 15.0 0.286 3.40
Table VIII.
as Functions of Assumed Dielectric
r* 13.5 13.5 13.5 13.5 13.5 13.5
...
14.0 14.0 14.0 14.0 14.0
...
... ..* ...
...
... ...
1.311 0.589 0.336 0.239 0.083
15.11 6.79 3.87 2.75 0.957
A'itromethane, 252' K. AF* X r* 1012 log1 oc 11.0 2.037 25.43 11.0 1.347 16.82 11.0 0.629 7.85 11.0 0.369 4.61 11.0 0.276 3.45 11.0 0.115 1.44
I-Butanol, 270.0" K. AF* X 10l2 logloC 2.022 23.56 1.327 15.46 0.607 7.07 0.349 4.07 0.257 2.99 0.099 1.15
as a Function of Assumed Dielectric Constant
[Calculated from Volmer's heterogeneous ion data (3)] T, K.
Material
HzO
( y = 77 ergs/sq.
($/pa)* = 4 . 1 D g =
80
cm.)
265
D 1.85 5 10 20 80
be present in a known con.centration. A lowering of the critical supersaturation ratio would indicate that nucleation had been produced by the dipoles.
ra, A.
a
=
b J C AF*
= = = =
d
= =
k
R N T r
D" iM AFB e
D V
(p/pJ
P
pm
= = = = = =
= = = = =
= =
=
distance from center of dipole to inner wall of shell distance from center of dipole to outer wall of shell number of droplets per second per cc. kinetic coefficient free energy change accompanying formation of a stable nucleus from vapor one-half length of the dipole Boltzmann constant, R/h' gas constant Avogadro number temperature in degrees K. radius of droplet or nucleus 1 - 1/D molecular weight molar free energy difference between phases in bulk (AFB = -RT lnp/pm) charge on ion ( e = 4.774 X 10-10 stat coulomb) dielectric constant of condensing phase volume of a mole of liquid critical supersaturation ratio vapor pressure of new phase (droplet) vapor pressure of plane surface (normal vapor pressure) electric work in forming depositing phase on dipole
AF* x 70'2 0.782 0.3055 0.2085 0.1620 0.1370
loglac 9.29 3.6274 2,4757 1.9235 1.6266
GREEKSYMBOLS condensation coefficient
a
=
7r
= 3.1416 = surface free energy per unit area of interface be-
Y Nomenclature
rk*, A. 8.52 8.14 8.0 7.91 7.85
3.55 4.60 4.90 5.05 5.18
6 P
tween two phases surface free energy per unit area of interface between two phases = density of liquid phase =
SUBSCRIPTS k = nucleus a = embryo in equilibrium with vapor
literature Cited (1) Becker, R., Doering, IV., Ann. Physik 24, 719 (1935). (2) Volmer, M., "Kinetik der Phasenbildung," Verlag von Theodor Steinkopff, Dresden and Leipzig, 1939, Edwards Brothers, Inc., Ann Arbor, Mich., 1945. (3) Volmer, M., Flood, H., Z. Physik Chem. 179 A, 273 (1934).
RECEIVED for review April 15, 1965 ACCEPTED September 28, 1965 Research sponsored by the Geophysics Division, Air Force Cambridge Research Center, Cambridge, Mass., under contract A. F. 19(122)-185.
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