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The nucleation of water on hexadecane The nuclei number determination by the reverse Wilson chamber (RWC) method
Colloids and Surfaces, 52 (1991) 175-184 Elsevier Science Publishers B.V., Amsterdam
175
The nucleation of water on hexadecane The nuclei number determination by the reverse Wilson chamber (RWC) method V.M. Chakarov’, A.D. AlexandroP, B.V. Toshevb and A.D. Scheludkoasb “Department of Nucleation, Central Laboratory of Mineral Processing, Bulgarian Academy of Sciences, 1 A. Ivanov Avenue, Sofia 1126 (Bulgaria) ‘Department of Physical Chemistry, Faculty of Chemistry, Sofia University, 1 A. Ivanov Avenue, Sofia 1126 (Bulgaria) (Received 25 January
1990; accepted 9 April 1990)
Abstract A modified version of the reverse Wilson chamber (RWC) suitable for direct observation of droplet growth and droplet (nuclei) determination is proposed. The observed independence of nuclei number per cm* as a function of the supersaturation is in contradiction with the classical (Volmer) theory of heterogeneous nucleation. This anomalous behaviour, however, supports our previous interpretation of heterogeneous nucleation as a barrierless process.
INTRODUCTION
The main purpose of the present study is to develop a new version of the reverse Wilson chamber (RWC) method, proposed [l-5] for heterogeneous nucleation investigation, in such a way, that direct observation of droplet growth and nuclei number determination will be possible. In our previous papers [l-5] we investigated the condensation from supersaturated water vapours on a liquid (hexadecane) substrate, using a light-scattering technique for detection of the nucleation and growth processes. As was described in detail, at supersaturations higher than the critical one, the amplitudes of light scattering impulses (I,,,) increase in an almost linear fashion with increasing supersaturation, which allows us to extrapolate the straight line to zero impulse height (I,,, = 0). The value of the supersaturation thus obtained is interpreted [3,4 ] as the onset of the barrierless nucleation and referred to as critical supersaturation (In S,). In the present investigation the last modification of RWC, described in Ref. [5], is used, but the photocell is replaced by photographic units, either for stationary (exposure time 0.5 s) or speed (50 pictures/s) photography. In all
0166-6622/91/$03.50
0 1991-
Elsevier Science Publishers
B.V.
176
experiments the compression time, controlled with an accuracy of 0.01 s, was 0.1 s. The initial substrate temperature was 25 -t 0.1’ C. The juxtaposition of these two RWC versions, by detection of light scattered by droplets condensed on the substrate and by photographic registration of the process, can be made by comparison of the results, obtained by both methods at a given supersaturation. It will become clear in the following that both RWC versions possess advantages, but they unavoidably exhibit some disadvantages, and only their appropriate combination will give more complete information about heterogeneous nucleation. RESULTS
AND DISCUSSION
A typical picture of photocurrent versus time dependence is presented in Fig. 1. This measurement was performed at initial supersaturation In S = 0.282. As can be seen from this curve, the photocurrent (I) sharply increases with time, passing through a maximum, and then gradually decreases up to I= 0. In Fig. 2 the droplet evolution in time is shown, the droplet diameter (D ) being determined from the moving pictures of the process at the same supersaturation as in Fig. 1.
L
‘* 1ime.s
*O
Fig. 1. Intensity of the light scattered by the droplets, formed on the liquid (hexadecane) substrate as a function of time. The measurement is performed at supersaturation In S = 0.282, initial substrate temperature 25°C and compression time 0.1 s.
177
1
Tlme.sec
2
Fig. 2. Droplet diameter (D) versus time. This dependence is obtained under the same conditions as in Fig. 1, but with the aid of speed photography (50 pictures/s). Open circles mark the mean values of droplet diameter at a given time, obtained from 15 parallel measurements of D.
Unfortunately, the poorer sensitivity of the photograph can provide data only for the region indicated by a horizontal dashed line in Fig. 1. However, the comparison of these two curves shows a qualitative agreement between them in the region accessible for investigation. It is known that there are two main reasons for the increase in the amplitudes of light scattering impulses, namely, increase in droplet concentration or increase in their size. It should be noted that simultaneous action of these two factors is expected, because, according to the Rayleigh theory, the intensity of light scattered by spherical nonmetal objects is proportional to Cv’, where v is the volume of the sphere and C is their number. However, from the curves, presented in Figs 1 and 2 we cannot answer the question what is the dominant factor in I,,, versus In S increase. This important question can be answered when investigating the dependence of the droplet number per cm* (2) on the supersaturation. The number of droplets, or nuclei/cm* (Z), was determined visually by counting them on photographs made at different supersaturations. Figure 3 is an arbitrarily chosen photograph of droplets, condensed on the hexadecane substrate at supersaturation In S = 0.256. The value of 2 is 3.58. lo4 drops/cm*, the observed area being 1.03 low3 cm*. The 2 versus In S curve is given in Fig. 4. As can be seen, at supersaturations under the critical one no condensation occurs, but at higher supersaturations the number of drops (nuclei) /cm* remains almost constant over a wide supersaturation range. We assumed that each macroscopic droplet, accessible for light microscope observation, corresponds to just one nucleus, e.g., the coalescence during the early stage of the process is neglected. The average value of 2 is 3.63.104 drops/cm* (for additional details see the Appendix). The results presented in Figs l-4 allow us to conclude that the linear relationship between l
178
O o”
0
_I’ Oo
0
0
0 Ooo 0
0
0
0
0
000
o
0
O
0
000
o
0
0
0
0
0
0
Z :35&104
Ins=0256
Fig. 3. Arbitrarily chosen light-microscope photo (after being worked up) of the droplets, deposited on the hexadecane substrate at In S = 0.256, initial substrate temperature 25’ C, compression time 0.1 s and magnification 500 x .
/
7 016
L
+ 020
026
028
032
0 36 Ins
Fig. 4. The number of droplets (nuclei)/cm’ as a function of the supersaturation. The average values of 2 within each series of measurements (In S = const ) are indicated by open circles. Vertical bars mark the standard deviation for each series of measurements. The mean value of In S, obtained in Ref. [5] is given with a bar, perpendicular to the abscissa. The solid line is obtained by the linear regression method (for details see the Appendix). Note that water droplets appear for the first time at supersaturation, which is in the error interval of In S (indicated by two horizontal arrows), obtained from Ref. [ 51.
the photocurrent amplitude (I,,,) and the supersaturation (In S) is due to an increase in droplet size, but not in droplet concentration. This important result supports our previous interpretation of the heterogeneous nucleation as a barrierless process [ 3,4] and is in complete contradiction to the classical (Volmer) theory of heterogeneous nucleation; the latter predicts an exponential increase in the nucleation rate with (In S)’ [see Eqns (l)-(3)]. The main result of the classical heterogeneous nucleation theory [ 14-161 is given by the well-known Volmer equation [ 171: J=Aexp
( - W/kT)
(1)
where A is a kinetic preexponential factor, estimated to be 10z5 (see e.g., Refs [4,X3] ); W is the work of nucleus formation; k is the Boltzmann constant; T
179
is the absolute temperature; and J is the nucleation rate (nuclei number/cm2 s-l). In the framework of the Volmer theory, the work of nucleus formation W is calculated according to the following equation: W=PoVc/2=
(47d3)
s)2 {0:(2-3~0s
@*)&
aI,
+cos3a,,)
+
a; (2 - 3cos cqco + cos3 a,,,} In Eqn (2), P,=2al/R1 =2a2/R2 is the capillary pressure, R, and R2 are the corresponding radii of curvature, V, is the volume of nucleus and Uis the molecular volume of water. The angles cy,, and 0~~~can be determined through the classical Neumann-triangle equations (x=0): 03
-
0,
cos
a!1,
-
(7.2 cos
cQco
o1sin cyI, - a, sin cy2a --0
-0 -
(3a) (3b)
In our previous studies [l-5], following the classical procedure of Volmer and Flood [ 191, we calculated the critical supersaturation for an arbitrarily chosen value of J, namely, J= lo2 (1 nucleus per 0.1 cm2 observable area per 0.1 s): In S,=O.755. If we now repeat these calculations for the experimentally determined value of 2 (Z=3.63.104 drops/cm2) and time 0.1 s, we get In S,=O.821, and the discrepancy between the Volmer theory and the experiment becomes even larger. In the following we shall give an explanation of the results, shown in Fig. 4, in terms of the theory of barrierless heterogeneous nucleation [ 6-81. For the critical supersaturation at the water condensation on hexadecane we assumed the average value obtained in Ref. [5]: In S,=O.204. The values of the three surface tensions are taken from the literature: a, = 72.00 dyn cm-’ [9,10] for the surface tension of water, a,=53.30 dyn cm-’ for the water/ hexadecane interfacial tension [ 111, and cr,=27.40 dyn cm-’ [ 12,13,21] for the surface tension of hexadecane. All these surface tensions are measured at 25°C. The molecular volume of water is 2.93*10-23 cm3 and kT=4.09~10-‘4 erg at the same temperature. For the line tension the value of - 1.9*10P5 dyn is assumed (for additional details see Ref. [ 51). With these data and the equations presented below we can calculate three important parameters: supersaturation (In S); the number of molecules in the droplet (N); and the work of equilibrium droplet formation ( W) as a function of the angle ay,. Expressions for the above quantities are derived by using Eqns (4)- (9) from Ref. [ 3 ] and the formula for the work of droplet formation [ 7,8] : In S= (2a,ti/7&T)f(cz1)
(5)
180
N= V/U= (K3/f “(a,)3${
(Z-3cos
cxl +cos%!,) + (a&Q3(2-3~0s
CY,+cos3cu,)}
(6) (7)
where I is the wetting perimeter f(cu,) =sin cul(a3 -U1cos
and the function f( cyl ) is defined by the equation
cy, -0,cos
cr,)
The angle cx2is determined as a function by making use of the condition [ 31: cr,sin cxl -azsin
(3) of cr, for the given values of CJ~and a2
cx2=O
(9)
For ran= 72.00 dyn cm-’ and a,= 53.30 dyn cm-‘, the values of (x2 run between 0 and 47.75’) aI being ranged between 0 and 123.04”. The calculation of In S, N, and Was a function of cx!,was performed with the aid of BASICA program, run on a 16-bit PC; the increment of cyl being lo. The relationship between the supersaturation and the number of molecules in the droplet (N) is graphically demonstrated in Fig. 5. This curve is discussed in detail elsewhere [ 6-81. It is important to note, that to each supersaturation correspond two droplets, different in size, which are in equilibrium with their surroundings. The droplet (St), formed on the left branch (smaller in size) is in a stable equilibrium with its surroundings, and, therefore, is not a nucleus of the new phase. The nucleus of the new phase is the droplet (unst), formed on the right branch of the curve (bigger in size ), which is in unstable equilibrium with its environment and can grow spontaneously.
0.20-
Fig. 5. Supersaturation
(In S) versus number of molecules in the equilibrium
droplet.
181
The work of formation vapours is given by W=A-& -Q,
of both equilibrium
= (P,v+rc1)/2
droplets
from supersaturated
(7a)
The Q-potential value Q2 corresponds to the equilibrium state of a small fluid drop, formed at the phase boundary between two bulk phases. These homogeneous phases with their interface are the initial thermodynamic state, characterized by the Sz, value. However, this initial state cannot be realized if Eqn (7a), applied for the smaller stable drops, gives s2,
(10)
The maximum of the curve In S versus N corresponds to the onset of barrierless condensation with W, - W, = 0. For the barrier-determined process W, - W, > 0. It holds even when W, is negative for some range of supersaturations. Then it would mean that
Fig. 6. Isothermal and reversible work w (in M’units) of stable (St) and unstable (unst) droplet formation as a function of the supersaturation (In S): curve 1 stable droplet; curve 2 unstable droplet.
182
(K= - 1.9*10P5 dyn) the barrierless condensation occurs at supersaturations above In S, = 0.204. There exists a narrow range of supersaturations where the process of condensation is barrier determined. At supersaturations below, say In S=O.16, the heterogeneous formation of a new bulk phase is practically impossible. This physical picture differs drastically from the classical GibbsVolmer description of the heterogeneous nucleation process. Obviously the independence of the droplet number 2 on In S (Fig. 4) is in accordance with the consideration of heterogeneous nucleation as a barrierless process. CONCLUSIONS
The final conclusion from the present investigation is that the classical (Volmer) theory of heterogeneous nucleation, not only fails to predict the critical supersaturation, but is not able to describe, even qualitatively, the overall dynamics of the process. This is an indication that heterogeneous nucleation has, probably, a completely different nature in comparison with homogeneous nucleation. It will be rather interesting to know the factors that govern the heterogeneous nucleation kinetics and to give a detailed explanation of the independence of 2 as a function of In S, based on an appropriate model. This will be the subject of our future theoretical and experimental investigations. APPENDIX
The most important on In S (Fig. 4
result obtained
in this study is the independence
[ 201, assuming
Z=a+bln
S the sum
-a-h
(2,
that (A.1 1
by minimizing
QZiil
of 2
W”
(A.21
which leads to the following set of equations:
aQ
da=-2
i
(Zi-U-blnSi)=O
i=l
JQ db
-_=-2
i
(A.3) (_q-
a- bln Si)ln Si =O
i=l
In the above equations
n denotes
the number
of (Zi, In Si) pairs; a and b are
183
regression parameters. The calculation was performed by simple BASICA program, run on a 16-bit PC, which gave a=3.63*104 and a practically negligible value for b. The use of polynomials of higher order gives no reasonable result, because of the large scattering of the data, presented in Fig. 4 (approximately 5 times). The second important feature, which is to be elucidated, concerns the mode of obtaining the data for 2 as a function of In S. The procedure consists of photographing the droplets, condensed on the liquid substrate at consecutively increased supersaturation. One can expect that the drops, formed during a given supersaturation pulse, are not completely evaporated, and are still present on the hexadecane surface (although not accessible for light microscope observation), promoting the nucleation during the next pulse. In order to avoid this objection we performed two series of measurements at different supersaturations, replacing the substrate (hexadecane film) with fresh hexadecane each time. With each substrate one experiment only is made. At supersaturation In S= 0.282 we obtained values of 2 ranged between 2.64~10~ and 4.62*104 drops/cm2 (average value of 2 was 3.64~10~) from 16 experiments, while at supersaturation In S= 0.452 2 was between 2.08. lo4 and 4.91. lo4 drops/cm2 (average value of 2 was 3.46. 104), the number of measurements being the same. This is an indication that the nucleation process is not promoted by water microdroplets which are not completely evaporated as far as the scattering of these data is comparable with those obtained by applying rising supersaturation pulses without replacement of the hexadecane film. ACKNOWLEDGEMENT
The assistance of Mr St. Christov is gratefully acknowledged.
from Bulgarian
TV in speed photography
REFERENCES 1 2 3 4 5 6 7 8 9
A. Scheludko, V. Chakarov and B. Toshev, J. Colloid Interface Sci., 82 (1981) 83. V. Chakarov, A. Scheludko and M. Zembala, J. Colloid Interface Sci., 92 (1983) 35. V. Chakarov, Colloid Polym. Sci., 261 (1983) 452. A. Scheludko and V. Chakarov, Colloid Polym. Sci., 261 (1983) 776. V. Chakarov, M. Zembala, 0. Novozhilova and A. Scheludko, Colloid Polym. Sci., 265 (1987) 347. A. Scheludko, Colloids Surfaces, 7 (1983) 81. B. Radoev, A. Scheludko and B. Toshev, J. Colloid Interface Sci., 113 (1986) 1. B. Toshev, D. Platikanov and A. Scheludko, Langmuir, 4 (1988) 489. Handbook of Chemistry and Physics, R.C. Weast (Ed.), 57th edn, CRC Press, Cleveland, OH, 1976, p. F-45.
184 10 11 12 13 14 15 16 17 18 19 20 21
Handbook of Physical and Chemical Constants, A.A. Ravdel and A.M. Ponomareva (Eds), Khimia, Leningrad, 1983, p. 20, in Russian. R. Aveyard and D.A. Haydon, Trans. Faraday Sot., 61 (1965) 2255. E. Have, E. Shafrin and M. Zisman, J. Phys. Chem., 58 (1954) 236. G. Kiirijsi and Sz. Kovatz, J. Chem. Eng. Data, 26 (1981) 323. M. Volmer, Kinetik der Phasenbildung, Steincopff, Leipzig, 1939. S. Kotake and 1.1. Glass, Prog. Aerospace Sci., 19 (1981) 129. J.P. Hirth and K.L. Moazed, in G. Haas and R.E. Thou (Eds), Physics of Thin Films, Vol. 4, Academic Press, New York, 1967, pp. 97-136. M. Volmer and A. Weber, Z. Phys. Chem. Abt. A, 119 (1926) 277. D.F. Mitchell, Ph.D. Thesis, Clarkson College of Technology, Potsdam, NY, 1976. M. Volmer and H. Flood, Z. Phys. Chem. Abt. A, 170 (1934) 273. J.H. Pollard, A. Handbook of Numerical and Statistical Techniques, Cambridge University Press, 1977. J.J. Jasper and E.V. Kring, J. Phys. Chem., 59 (1955) 1019.
175
The nucleation of water on hexadecane The nuclei number determination by the reverse Wilson chamber (RWC) method V.M. Chakarov’, A.D. AlexandroP, B.V. Toshevb and A.D. Scheludkoasb “Department of Nucleation, Central Laboratory of Mineral Processing, Bulgarian Academy of Sciences, 1 A. Ivanov Avenue, Sofia 1126 (Bulgaria) ‘Department of Physical Chemistry, Faculty of Chemistry, Sofia University, 1 A. Ivanov Avenue, Sofia 1126 (Bulgaria) (Received 25 January
1990; accepted 9 April 1990)
Abstract A modified version of the reverse Wilson chamber (RWC) suitable for direct observation of droplet growth and droplet (nuclei) determination is proposed. The observed independence of nuclei number per cm* as a function of the supersaturation is in contradiction with the classical (Volmer) theory of heterogeneous nucleation. This anomalous behaviour, however, supports our previous interpretation of heterogeneous nucleation as a barrierless process.
INTRODUCTION
The main purpose of the present study is to develop a new version of the reverse Wilson chamber (RWC) method, proposed [l-5] for heterogeneous nucleation investigation, in such a way, that direct observation of droplet growth and nuclei number determination will be possible. In our previous papers [l-5] we investigated the condensation from supersaturated water vapours on a liquid (hexadecane) substrate, using a light-scattering technique for detection of the nucleation and growth processes. As was described in detail, at supersaturations higher than the critical one, the amplitudes of light scattering impulses (I,,,) increase in an almost linear fashion with increasing supersaturation, which allows us to extrapolate the straight line to zero impulse height (I,,, = 0). The value of the supersaturation thus obtained is interpreted [3,4 ] as the onset of the barrierless nucleation and referred to as critical supersaturation (In S,). In the present investigation the last modification of RWC, described in Ref. [5], is used, but the photocell is replaced by photographic units, either for stationary (exposure time 0.5 s) or speed (50 pictures/s) photography. In all
0166-6622/91/$03.50
0 1991-
Elsevier Science Publishers
B.V.
176
experiments the compression time, controlled with an accuracy of 0.01 s, was 0.1 s. The initial substrate temperature was 25 -t 0.1’ C. The juxtaposition of these two RWC versions, by detection of light scattered by droplets condensed on the substrate and by photographic registration of the process, can be made by comparison of the results, obtained by both methods at a given supersaturation. It will become clear in the following that both RWC versions possess advantages, but they unavoidably exhibit some disadvantages, and only their appropriate combination will give more complete information about heterogeneous nucleation. RESULTS
AND DISCUSSION
A typical picture of photocurrent versus time dependence is presented in Fig. 1. This measurement was performed at initial supersaturation In S = 0.282. As can be seen from this curve, the photocurrent (I) sharply increases with time, passing through a maximum, and then gradually decreases up to I= 0. In Fig. 2 the droplet evolution in time is shown, the droplet diameter (D ) being determined from the moving pictures of the process at the same supersaturation as in Fig. 1.
L
‘* 1ime.s
*O
Fig. 1. Intensity of the light scattered by the droplets, formed on the liquid (hexadecane) substrate as a function of time. The measurement is performed at supersaturation In S = 0.282, initial substrate temperature 25°C and compression time 0.1 s.
177
1
Tlme.sec
2
Fig. 2. Droplet diameter (D) versus time. This dependence is obtained under the same conditions as in Fig. 1, but with the aid of speed photography (50 pictures/s). Open circles mark the mean values of droplet diameter at a given time, obtained from 15 parallel measurements of D.
Unfortunately, the poorer sensitivity of the photograph can provide data only for the region indicated by a horizontal dashed line in Fig. 1. However, the comparison of these two curves shows a qualitative agreement between them in the region accessible for investigation. It is known that there are two main reasons for the increase in the amplitudes of light scattering impulses, namely, increase in droplet concentration or increase in their size. It should be noted that simultaneous action of these two factors is expected, because, according to the Rayleigh theory, the intensity of light scattered by spherical nonmetal objects is proportional to Cv’, where v is the volume of the sphere and C is their number. However, from the curves, presented in Figs 1 and 2 we cannot answer the question what is the dominant factor in I,,, versus In S increase. This important question can be answered when investigating the dependence of the droplet number per cm* (2) on the supersaturation. The number of droplets, or nuclei/cm* (Z), was determined visually by counting them on photographs made at different supersaturations. Figure 3 is an arbitrarily chosen photograph of droplets, condensed on the hexadecane substrate at supersaturation In S = 0.256. The value of 2 is 3.58. lo4 drops/cm*, the observed area being 1.03 low3 cm*. The 2 versus In S curve is given in Fig. 4. As can be seen, at supersaturations under the critical one no condensation occurs, but at higher supersaturations the number of drops (nuclei) /cm* remains almost constant over a wide supersaturation range. We assumed that each macroscopic droplet, accessible for light microscope observation, corresponds to just one nucleus, e.g., the coalescence during the early stage of the process is neglected. The average value of 2 is 3.63.104 drops/cm* (for additional details see the Appendix). The results presented in Figs l-4 allow us to conclude that the linear relationship between l
178
O o”
0
_I’ Oo
0
0
0 Ooo 0
0
0
0
0
000
o
0
O
0
000
o
0
0
0
0
0
0
Z :35&104
Ins=0256
Fig. 3. Arbitrarily chosen light-microscope photo (after being worked up) of the droplets, deposited on the hexadecane substrate at In S = 0.256, initial substrate temperature 25’ C, compression time 0.1 s and magnification 500 x .
/
7 016
L
+ 020
026
028
032
0 36 Ins
Fig. 4. The number of droplets (nuclei)/cm’ as a function of the supersaturation. The average values of 2 within each series of measurements (In S = const ) are indicated by open circles. Vertical bars mark the standard deviation for each series of measurements. The mean value of In S, obtained in Ref. [5] is given with a bar, perpendicular to the abscissa. The solid line is obtained by the linear regression method (for details see the Appendix). Note that water droplets appear for the first time at supersaturation, which is in the error interval of In S (indicated by two horizontal arrows), obtained from Ref. [ 51.
the photocurrent amplitude (I,,,) and the supersaturation (In S) is due to an increase in droplet size, but not in droplet concentration. This important result supports our previous interpretation of the heterogeneous nucleation as a barrierless process [ 3,4] and is in complete contradiction to the classical (Volmer) theory of heterogeneous nucleation; the latter predicts an exponential increase in the nucleation rate with (In S)’ [see Eqns (l)-(3)]. The main result of the classical heterogeneous nucleation theory [ 14-161 is given by the well-known Volmer equation [ 171: J=Aexp
( - W/kT)
(1)
where A is a kinetic preexponential factor, estimated to be 10z5 (see e.g., Refs [4,X3] ); W is the work of nucleus formation; k is the Boltzmann constant; T
179
is the absolute temperature; and J is the nucleation rate (nuclei number/cm2 s-l). In the framework of the Volmer theory, the work of nucleus formation W is calculated according to the following equation: W=PoVc/2=
(47d3)
s)2 {0:(2-3~0s
@*)&
aI,
+cos3a,,)
+
a; (2 - 3cos cqco + cos3 a,,,} In Eqn (2), P,=2al/R1 =2a2/R2 is the capillary pressure, R, and R2 are the corresponding radii of curvature, V, is the volume of nucleus and Uis the molecular volume of water. The angles cy,, and 0~~~can be determined through the classical Neumann-triangle equations (x=0): 03
-
0,
cos
a!1,
-
(7.2 cos
cQco
o1sin cyI, - a, sin cy2a --0
-0 -
(3a) (3b)
In our previous studies [l-5], following the classical procedure of Volmer and Flood [ 191, we calculated the critical supersaturation for an arbitrarily chosen value of J, namely, J= lo2 (1 nucleus per 0.1 cm2 observable area per 0.1 s): In S,=O.755. If we now repeat these calculations for the experimentally determined value of 2 (Z=3.63.104 drops/cm2) and time 0.1 s, we get In S,=O.821, and the discrepancy between the Volmer theory and the experiment becomes even larger. In the following we shall give an explanation of the results, shown in Fig. 4, in terms of the theory of barrierless heterogeneous nucleation [ 6-81. For the critical supersaturation at the water condensation on hexadecane we assumed the average value obtained in Ref. [5]: In S,=O.204. The values of the three surface tensions are taken from the literature: a, = 72.00 dyn cm-’ [9,10] for the surface tension of water, a,=53.30 dyn cm-’ for the water/ hexadecane interfacial tension [ 111, and cr,=27.40 dyn cm-’ [ 12,13,21] for the surface tension of hexadecane. All these surface tensions are measured at 25°C. The molecular volume of water is 2.93*10-23 cm3 and kT=4.09~10-‘4 erg at the same temperature. For the line tension the value of - 1.9*10P5 dyn is assumed (for additional details see Ref. [ 51). With these data and the equations presented below we can calculate three important parameters: supersaturation (In S); the number of molecules in the droplet (N); and the work of equilibrium droplet formation ( W) as a function of the angle ay,. Expressions for the above quantities are derived by using Eqns (4)- (9) from Ref. [ 3 ] and the formula for the work of droplet formation [ 7,8] : In S= (2a,ti/7&T)f(cz1)
(5)
180
N= V/U= (K3/f “(a,)3${
(Z-3cos
cxl +cos%!,) + (a&Q3(2-3~0s
CY,+cos3cu,)}
(6) (7)
where I is the wetting perimeter f(cu,) =sin cul(a3 -U1cos
and the function f( cyl ) is defined by the equation
cy, -0,cos
cr,)
The angle cx2is determined as a function by making use of the condition [ 31: cr,sin cxl -azsin
(3) of cr, for the given values of CJ~and a2
cx2=O
(9)
For ran= 72.00 dyn cm-’ and a,= 53.30 dyn cm-‘, the values of (x2 run between 0 and 47.75’) aI being ranged between 0 and 123.04”. The calculation of In S, N, and Was a function of cx!,was performed with the aid of BASICA program, run on a 16-bit PC; the increment of cyl being lo. The relationship between the supersaturation and the number of molecules in the droplet (N) is graphically demonstrated in Fig. 5. This curve is discussed in detail elsewhere [ 6-81. It is important to note, that to each supersaturation correspond two droplets, different in size, which are in equilibrium with their surroundings. The droplet (St), formed on the left branch (smaller in size) is in a stable equilibrium with its surroundings, and, therefore, is not a nucleus of the new phase. The nucleus of the new phase is the droplet (unst), formed on the right branch of the curve (bigger in size ), which is in unstable equilibrium with its environment and can grow spontaneously.
0.20-
Fig. 5. Supersaturation
(In S) versus number of molecules in the equilibrium
droplet.
181
The work of formation vapours is given by W=A-& -Q,
of both equilibrium
= (P,v+rc1)/2
droplets
from supersaturated
(7a)
The Q-potential value Q2 corresponds to the equilibrium state of a small fluid drop, formed at the phase boundary between two bulk phases. These homogeneous phases with their interface are the initial thermodynamic state, characterized by the Sz, value. However, this initial state cannot be realized if Eqn (7a), applied for the smaller stable drops, gives s2,
(10)
The maximum of the curve In S versus N corresponds to the onset of barrierless condensation with W, - W, = 0. For the barrier-determined process W, - W, > 0. It holds even when W, is negative for some range of supersaturations. Then it would mean that
Fig. 6. Isothermal and reversible work w (in M’units) of stable (St) and unstable (unst) droplet formation as a function of the supersaturation (In S): curve 1 stable droplet; curve 2 unstable droplet.
182
(K= - 1.9*10P5 dyn) the barrierless condensation occurs at supersaturations above In S, = 0.204. There exists a narrow range of supersaturations where the process of condensation is barrier determined. At supersaturations below, say In S=O.16, the heterogeneous formation of a new bulk phase is practically impossible. This physical picture differs drastically from the classical GibbsVolmer description of the heterogeneous nucleation process. Obviously the independence of the droplet number 2 on In S (Fig. 4) is in accordance with the consideration of heterogeneous nucleation as a barrierless process. CONCLUSIONS
The final conclusion from the present investigation is that the classical (Volmer) theory of heterogeneous nucleation, not only fails to predict the critical supersaturation, but is not able to describe, even qualitatively, the overall dynamics of the process. This is an indication that heterogeneous nucleation has, probably, a completely different nature in comparison with homogeneous nucleation. It will be rather interesting to know the factors that govern the heterogeneous nucleation kinetics and to give a detailed explanation of the independence of 2 as a function of In S, based on an appropriate model. This will be the subject of our future theoretical and experimental investigations. APPENDIX
The most important on In S (Fig. 4
result obtained
in this study is the independence
[ 201, assuming
Z=a+bln
S the sum
-a-h
(2,
that (A.1 1
by minimizing
QZiil
of 2
W”
(A.21
which leads to the following set of equations:
aQ
da=-2
i
(Zi-U-blnSi)=O
i=l
JQ db
-_=-2
i
(A.3) (_q-
a- bln Si)ln Si =O
i=l
In the above equations
n denotes
the number
of (Zi, In Si) pairs; a and b are
183
regression parameters. The calculation was performed by simple BASICA program, run on a 16-bit PC, which gave a=3.63*104 and a practically negligible value for b. The use of polynomials of higher order gives no reasonable result, because of the large scattering of the data, presented in Fig. 4 (approximately 5 times). The second important feature, which is to be elucidated, concerns the mode of obtaining the data for 2 as a function of In S. The procedure consists of photographing the droplets, condensed on the liquid substrate at consecutively increased supersaturation. One can expect that the drops, formed during a given supersaturation pulse, are not completely evaporated, and are still present on the hexadecane surface (although not accessible for light microscope observation), promoting the nucleation during the next pulse. In order to avoid this objection we performed two series of measurements at different supersaturations, replacing the substrate (hexadecane film) with fresh hexadecane each time. With each substrate one experiment only is made. At supersaturation In S= 0.282 we obtained values of 2 ranged between 2.64~10~ and 4.62*104 drops/cm2 (average value of 2 was 3.64~10~) from 16 experiments, while at supersaturation In S= 0.452 2 was between 2.08. lo4 and 4.91. lo4 drops/cm2 (average value of 2 was 3.46. 104), the number of measurements being the same. This is an indication that the nucleation process is not promoted by water microdroplets which are not completely evaporated as far as the scattering of these data is comparable with those obtained by applying rising supersaturation pulses without replacement of the hexadecane film. ACKNOWLEDGEMENT
The assistance of Mr St. Christov is gratefully acknowledged.
from Bulgarian
TV in speed photography
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