Nucleation rates in water—carbon dioxide solutions: The spherical cavity case

NOTES Nucleation Rates in Water-Carbon Dioxide Solutions: The Spherical Cavity Case INTRODUCTION Recently (1) one of us applied classical nucleation theory to predict homogeneous and some heterogeneous nucleation rates for H20/CO2 solutions in contact with surfaces containing defects of different geometries. The supersaturation ratios were appropriate for a freshly opened carbonated beverage. Even using double-precision arithmetic, problems involving numerical precision were encountered in the case of the spherical cavity. These difficulties precluded calculations for spherical cavities whose radii were smaller than 10-6 cm. Thus the above-cited work was inconclusive as to whether or not nucleation theory predicted that spherical cavities would be nucleation sites. We have reinvestigated this spherical cavity case, identified and eliminated the source of numerical imprecision, and extended the calculations to smaller cavities. In this note we report the method and results of tbese calculations.

Each term in the sum of Eq. [1] was determined iteratively by computer (IBM PC, double precision Microsoft BASIC). For a given cavity radius rs, an initial value of bubble radius r was increased incrementally until an integral number g of molecules in the bubble was determined according to the ideal gas law

g(r) = PcV(r)/kT,

[3]

where Ref. (1) explains how to calculate the gas pressure Pc in the bubble of volume V(r). Then the term [A(g)n(g)]-I was calculated, and r again incrementally changed until the next successife integer g was determined by [3]. This process was continued for all values o f g which contributed significantly to Eq. [1]. The numerical values necessary for this calculation are unchanged from those of Figs. 6 and 7 o f ( l ) . The area A(g) is found, since g = g(r), from A(g) = 27rr2{1 - cos c~(r)}.

DETAILS OF THE THEORETICAL MODEL Rather than use the integral approximation often employed in nucleation late calculations, we directly evaluated the nucleation rate J from the summation following from the unbalanced steady-state analysis (2):

J= ~ ' / ~ n ( g ; ( g ) "

where the symbols are identified in (1). As g increases and the bubble grows, it passes through phases as depicted in Fig, 1. In Fig. 1A we define the parameters describing bubble growth. With increasing g the bubble growth, progressing at constant contact angle 0, passes into the situation shown in Fig. 1B. Here g is finite but the bubble radius r is infinite. Sometime before reaching condition (B) the bubble will have attained critical size. As gincreases beyond condition (B), the bubble geometry changes to that shown in Fig. 1C. For relatively large spherical cavity radii rs we have found that g values resulting from the geometry in Fig. 1C add completely negligible contributions to the sum in Eq. [ 1] and may be ignored. However, for cavities whose radii are of the order of or smaller than the bubble critical radius, all growth phases shown in Fig. 1 may contribute to Eq. [1].

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For the growth geometry of Fig. 1A, a(r),/3(r), and W(g(r)) are given by Eqs. [17a], [17b], and [16] of Ref. (1), respectively. For the growth phase of Fig. 1C, we find

[1]

We follow the notation used in (1); specifically, n(g) is the number of bubbles, per unit area, containing g molecules and A(g) is the corresponding surface area of the bubble through which molecules may pass. The distribution n(g) is given by n(g) = N e x p [ - W(g)/kT], [2]

[4]

_l ( r~sin0 l c~(r) = tan l ~ J '

[5]

~f - r sin 0 l e(r) = tan- It, + r cos 01 '

[6]

7r

V(r) = ~ {r~(2 - 3 cos ¢~ + cos3/3) - r3(2 - 3 cos a + cos3a)},

[7]

and

W(g(r)) = 21ra{ri(l - cos c0 + r~cos 0(1 - cos/3)} - (Pc -

PL)V(r).

[81

The numerical precision problems mentioned earlier were found to be associated with computer evaluation of the expressions (1 - cos a) arid (2 - 3 cos a + cos3a) when a i s small. For example, double-precision MS-DOS BASIC library functions evaluate (2 - 3- cos a + cos3~) as exactly 0 for a = 0.01 lad; the correct answer is 7.5 × 10-9. To eliminate these difficulties we evaluated these expressions by series expansions when c~ < 0.06 lad. The more complicated expansion is (2 - 3 cos a + cos3~) = 3a4/4 - a6/4 + 91as/2240 + • • ..

[9]

296 Journal of Colloid and Interface Science, Vol. 123, No. 1, May 1988

NOTES

297

I

8 = 94 °

= 94 ° Liquid'~

I

#~

e-- 94° I

Liqluid X

I Liquid%-

r

"~/ar

Solid

I I I

r = ~o ot=O

Solid

Solid

(C)

(B)

(A)

FIG. 1. Phases of bubble growth in a spherical cavity. Symbols are identified in Ref. (1) and in the text, where further details of the growth process may be found.

'

I

''

'

I

'

I

'

i

i

I

7/////////////////////////////////////////////////////J

Nucleation Range -iOtz~-

0-=65 erg/cm z ....... 0-=75 erg/cm2

~.~ 0

-'%L- \'\

J

%:

\\ "x~

-I0 6

ILL,, 5 235 10-7

Figure 2 shows the results of our calculations for a = 65 and 75 erg/cm 2 and 0 = 94 ° and 130 °. For rs > 10-6 cm our present results agree with those calculated in the integral approximation of Ref. (1). Using the present approach, we have extended the curves to spherical cavities as small as 2 X 10 -7 c m . Note that curves for 0 = 130 ° intersect the nucleation range. We did not extend the curves for 0 = 94 ° below 2 X 10-7 cm for reasons described below.

CONCLUSIONS

-I0~a - -

-I@

RESULTS

-'~ .......... J'150"

,I

5

,I

5

,I 5

10-6 10-5 10-4 10-3 10-2 IO-L R, (cm)

RG. 2. The calculated dependence of nucleation rate J on spherical cavity radius rs for H20/CO2 solutions in contact with spherical cavities. The supersaturation ratio is 5.08, while the numerical values of other parameters are shown in the figure or are the same as those in Figs. 6 and 7 of Ref. (1). See text for a discussion of those curves which intersect the nucleation range.

We have also calculated the number of molecules in the critical bubble, ge, for various points on the curves of Fig. 2. For example, for the 0 = 94 ° curves and rs = 2 X 10 -7 c m we find gc = 2, even though the curves have not reached the nucleation range. For 0 = 130 ° we find that gc is less than 1 for the rs range where these curves intersect the nucleation range. Even for 0 = 140 ° (an extremely large value) and cr in the range of 65 to 75 erg/ c m 2, w e find that gc < 2 ifrs is small enough ( ~ 4 X 10 -7 cm) to place log~oJ in the range of 2 to 6. (These last results are not plotted in Fig. 2.) Obviously there are conceptual problems in assuming that the bulk parameters of nucleation theory such as O and a have validity when applied to situations involving one or two molecules (3). It seems clear that one cannot apply classical nucleation theory to spherical cavities small

Journal of Colloid and Interface Science, Vol. 123,No. 1, May 1988

298

NOTES

enough to become nucleation sites for the conditions typical of carbonated beverages. This note does give geometrical details for the spherical cavity calculation and identifies and corrects computational sources of numerical imprecision. One could thus follow this approach if one were interested in much higher supersaturation ratios where nucleation theory would be expected to predict that some spherical cavities could be nucleation sites. ACKNOWLEDGMENT We gratefully acknowledge financial support from the Centre College Faculty Development Committee in support of this work.

Journalof Colloidand InterfaceScience,Vol. 123,No. 1, May 1988

REFERENCES 1. Wilt, P. M., J. Colloid Interface Sci. 112, 530 (1986). 2. McDonald, J. E.,Amer. J. Phys. 31, 31 (1963). 3. McDonald, J. E., Amer. J. Phys. 30, 870 (1962). P. A. CIHOLAS

P. M. WILT

Division of Science and Mathematics Centre College Danville, Kentucky 40422 Received April 13, 1987; accepted June 29, 1987