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Thermal Marangoni Effects on Gas Bubbles Are Generally Accompanied by Solutal Marangoni Effects Steven Lubetkin Eli Lilly & Co., 2001 West Main Street, Greenfield, Indiana 46140 Received October 1, 2003 The thermal Marangoni force on a bubble, as originally predicted and experimentally verified by Young et al., has been measured by at least three later investigators. In each of the later cases, the measurements were complicated by oscillations in the size of the bubbles, which produced vertical oscillatory motion of the bubble in the combined thermal and gravitational fields. The authors ascribed this motion to the variable solubility of air in the working liquid as the bubble moved in the temperature gradient. It is pointed out here that this variable solubility implies a second, solutal Marangoni effect, which reduces the thermal forces. Since the solubilities of all gases are a function of temperature, solutal forces will generally accompany thermal forces on gas bubbles.
Background Young et al.1 predicted that a bubble in a thermal gradient would experience a force as a result of the temperature variability of the surface tension and also reported at the same time the first experimental data, which were qualitatively in accord with the theory. There have been at least three later direct measurements of this effect, by Hardy,2 Merritt and Subramanian,3 and Morick and Woermann,4 and there has been a thorough review of these experiments by Subramanian and Balasubramanian.5 The original measurements made by Young et al. showed that it was possible to bring a bubble to a stationary position in which the thermal Marangoni force balanced the gravitational force. The buoyancy (directed vertically upward, against gravity) is opposed by the thermal Marangoni force (in these experiments, directed downward). By variation of the temperature gradient, the expectation was that there would generally be a point at which these opposed forces balance, and the bubble could be brought to rest. This is exactly how the original experiments were conducted by Young et al., who pointed out that bringing the bubble to a stationary position “..avoid[s] the effects of changing bubble solubility with temperature.” In the three more recent measurements, the experimenters found it impossible to attain this stationary point: instead the bubble oscillated about a central position, in an approximately sinusoidal fashion. Together with the expected thermal Marangoni effect there was a second, variable force. This force was ascribed to variation in the buoyancy (and also in the thermal Marangoni effect) as a result of the radius R of the bubble changing, and the variation was approximately periodic. As the air bubble rose, entering cooler regions, where the working liquid, PDMS (poly(dimethylsiloxane)), was undersaturated with respect to dissolved air, it shrunk. As it shrunk, the (1) Young, N. O.; Goldstein, J. S.; Block, M. J. J. Fluid Mech. 1959, 6, 350. (2) Hardy, S. C. J. Colloid Interface Sci. 1979, 69, 157. (3) Merritt, R. M.; Subramanian, R. Shankar. J. Colloid Interface Sci. 1988, 125, 333. (4) Morick, F.; Woermann, D. Ber. Bunsen-Ges. Phys. Chem. 1993, 97, 961. (5) Subramanian, R. Shankar; Balasubramanian, R. The Motion of Bubbles and Drops in Reduced Gravity; Cambridge University Press: Cambridge, 2001.
buoyancy force (which goes as R3) was reduced, and the thermal Marangoni force (which goes as R2) was great enough to begin to accelerate it downward. As the bubble entered the hotter regions, supersaturated with air, it expanded, and the increased buoyancy caused the bubble to cease its downward motion and to begin to rise again toward the cooler region, and thus the approximately cyclical motion was established. The fact that Young et al. were able to avoid such oscillation while the later authors could not is significant and needs explanation. Regardless of the detailed explanation, the fact that such an oscillation takes place at all shows that the initially uniform distribution of dissolved gas is being perturbed. There are at least three factors (ignoring the negligible change of hydrostatic pressure as the bubble depth changes) involved in this oscillatory motion, of which two have been previously identified.2-4 First, the gases of which the bubble was composed (in the present cases, nitrogen and oxygen) are soluble to some extent in the PDMS. In common with all gases in liquids, the solubility increases as the temperature falls and decreases as the temperature rises. So the first factor affecting the buoyancy force is the bubble size as governed by the changing solubility of air in PDMS. In these experiments, the radius R typically changes by about a factor of 400% between the undersaturated and supersaturated regions. This factor is expected to change with time. Second, the bubble will expand and contract according to its position in the temperature gradient, even in the absence of any gas exchange with the surrounding liquid: contraction is expected in cooler regions, expansion in the hotter. For typical temperatures used in the experiments, the thermal expansion of the bubble amounts to about 5% in the radius, R, from the coldest to the hottest regions of the cell. The third factor is that superimposed on the temperature gradient, there will be a concentration gradient. Initially the system is uniform and fully equilibrated at ambient temperature 〈T0〉, and with a uniform concentration of gas 〈c0〉 throughout, corresponding to the gas solubility at the temperature 〈T0〉. When the temperature gradient is applied at time t ) 0, the cooler regions, measured relative to 〈T0〉, are initially undersaturated. Similarly, the relatively warmer regions are initially supersaturated. Supersaturation and undersaturation are by definition nonequilibrium states, so as time passes,
10.1021/la0358365 CCC: $25.00 © 2003 American Chemical Society Published on Web 11/19/2003
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We now examine these three routes in a little more detail: (1) Direct Diffusion. Thermal diffusion of dissolved gas (the Soret effect6) will take place with a characteristic time θ, given by
θ ) d2/π2D
Figure 1. Showing the temperature (right-hand side) and steady-state concentration (left-hand side) gradients.
and assuming a route is available for mass transfer, the system will approach equilibrium: the warm, supersaturated regions will lose dissolved gas to the cool, undersaturated regions. Thus, the amount of dissolved gas in the supersaturated region will be reduced, and the amount of gas in the undersaturated regions will be increased, both quantities being measured relative to the initial concentration 〈c0〉. The overall effect of this transport is that a concentration gradient is established as the system approaches steady state, as shown in Figure 1, and this gradient has consequences for the motion of the bubble. The Steady-State Concentration Gradient The cool, undersaturated region will gain air from the hot, supersaturated region. There are three possible routes for transport of gas, these being: (1) Gas is transported directly, by diffusion in solution from supersaturated regions to undersaturated regions through the bulk liquid. (2) Gas is transported by the motion of the bubble taking in gas from the supersaturated region and thus expanding and delivering the gas to the undersaturated region, thus contracting. Overall, this amounts to transport by the bubble. (3). Gas is transported directly through the interface of the PDMS drop from the atmosphere, because the system is open to the air. This route is available in the case of the experiment of Young et al.1 but not in the other three cases,2-4 where the PDMS is completely enclosed in a sample cell.
(1)
where D is the (isothermal, mutual) diffusion coefficient and d is the depth of liquid, here about 3 mm. This formula assumes parallel plate geometry, appropriate to the present experimental setup,7,8 and also treats the air as a single gas. There do not appear to be any published numerical values for the diffusion coefficient of N2 and O2 in PDMS, but it is to be expected that they would be less than the diffusion coefficients of these gases in water, simply based on the assumption that D is inversely proportional to the viscosity. Bierlein’s law allows a calculation of the kinetics of approach to the steady state9
n)1-
4 -9t/θ 4 -25t/θ 4 -49t/θ 4 -t/θ e e e e + + π 3π 5π 7π 4 -81t/θ e + ... (2) 9π
and a typical plot of the concentration versus time is given in Figure 2. The plot of the concentration as a function of the position in the cell is perhaps more revealing, showing that the steady-state (t ) ∞) spatial distribution is essentially a linear gradient of concentration from top to bottom. Figure 3 shows the plots of the solution to the equation9
n ) n0[1 + p(1 - n0)(1/2 - ξ) - 4p(1 - n0)(cos πξ) exp (-t/θ)/π2] (3) for various times, t, between 0 and 10 × θ: this latter is assumed to correspond to t ) ∞. The reduced positional coordinate is ξ, and n/n0 is the reduced molar concentration, here expressed as a percentage of the initial concentration. The variable p is the reduced temperature gradient, p ) στ, where σ ) Soret coefficient (Ds/D) and τ ) temperature gradient, or ∂T/∂ξ. The initial uniform concentration (100%) is shown by the horizontal line AB. The ultimate (steady-state) gradient is shown by the diagonal line CD.
Figure 2. The approach of the mid-point concentration to steady state, as predicted by Bierlein’s law, eq 2.
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Figure 3. The spatial distribution of concentration as a function of time after the temperature gradient is imposed, from eq 3, here solved for values appropriate to water, together with an assumed value for the Soret coefficient. Table 1. Variation of the Relaxation Time, θ, with Viscosity liquid
η/cP
θ/s
water PDMS PDMS
1 20 200
103 2 × 104 2 × 105
It is assumed that the results of using PDMS are predictable based only on the change in viscosity, η, and that D is inversely proportional to η. With this simplifying assumption, the estimates for θ are shown in Table 1. (2) Transfer by the Bubble. When the bubble is in the supersaturated region, it grows. As a direct consequence of this gain of matter by the bubble, the supersaturated region (taken as a whole) now has a concentration somewhat below what it had before imposition of the temperature gradient, since material has been removed. When the bubble rises into the undersaturated region, material is lost from the bubble into the surrounding liquid, thus increasing the concentration there. In its oscillatory path, the bubble repetitively extracts gas from the supersaturated region and deposits it in the undersaturated region. The overall effect is that gas is being transported from the supersaturated to the undersaturated region. Since the initial state is one of uniform concentration, the critical point is that the observed oscillations prove that transport of gas is taking place, and this in turn proves that there is a developing gradient of concentration. The magnitude of the gradient will obviously depend on the length of time during which the oscillatory motion has been occurring and on the interfacial area, and thus the cumulative amount of gas transferred. During the early stages, little material will have been transferred, and the concentration gradient will be correspondingly small. Later, it will be greater, and finally, at steady state, it will be at a maximum. The presence of oscillations shows that a concentration gradient is developing, and when oscillations cease, the maximum gradient exists. (6) de Groot, S R. L’Effet Soret: Diffusion Thermique dans les Phases Condensees; Noord-Hollandsche: Amsterdam, 1945. (7) Grew, K E. Transport Phenomena in Fluids; Hanley, Howard J. M., Ed.; Marcel Dekker: New York, 1969; p 333. (8) Chanu, J. Adv. Chem. Phys. 1967, 13, 349. (9) Bierlein, J. A. J. Chem. Phys. 1955, 23, 10.
(3) Direct Atmospheric Contact. In the experiment of Young et al. where the PDMS is held by surface tension between the anvils of a machinist’s micrometer, the outer surface of the drop of PDMS is fully exposed to the atmosphere and so can readily exchange both N2 and O2 with the surrounding air. Supersaturated regions lose N2 and O2 to the atmosphere; undersaturated regions gain N2 and O2. The surface area available for diffusion is much greater than that of the bubble (by a factor of about 104), thus the equilibration (steady state) is achieved very much more quickly. On the basis of the estimates of relative surface area, and allowing for the different viscosity, this gives a predicted increase in rate of the order of 10-100 times compared to the bubble alone. The details of the kinetics in the case of the bubble will be somewhat different than those for the air/PDMS drop interface, so that this rate prediction is only an estimate. Overall in cases 1 and 2, where the equilibration is slow, and the supersaturation and undersaturation persist for significant lengths of time (of the order of hours), the oscillation of the bubble will continue to occur as long as it takes to eliminate the supersaturation and undersaturation. In case 3, the supersaturation and undersaturation are relieved relatively quickly, and oscillation is not then observed. This is the explanation for the puzzling fact that Young et al. did not see oscillations, whereas the other three observers did. Effects of the Concentration Gradient Transport by some combination of the three mechanisms mentioned above gives rise to a concentration gradient. As a result, there is a solutal Marangoni force (driven by concentration) superimposed on the thermal Marangoni force (driven by temperature) but opposed to it. When there are thermal Marangoni forces acting on a gas bubble, there will generally be corresponding solutal forces, too: gas bubbles can only survive if the liquid phase is saturated (or more precisely, slightly supersaturated, because of the Laplace pressure, see below) with the gases of which the bubble is formed. Only gases for which the concentration dependence of the surface tension is approximately zero (e.g., helium in water) will not show this effect. The magnitude of the effect will be determined by both the b coefficient of the gas/liquid combination considered (see
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Table 2. γ ) γo + bp + cp2 + dp3 (p in bar) for Aqueous Solutions of Various Gases at 25 °C Adapted from Massoudi and King11 gas
b
c
He H2 O2 N2 Ar CO CH4 C2H4 C2H6 C3H8 N2O CO2 n-C4H10
0.0000 -0.0247 -0.0769 -0.0824 -0.0829 -0.1027 -0.1527 -0.6270 -0.4319 -0.9554 -0.6150 -0.7687 -2.3045
0.000101 0.000189 0.000189 0.000233 0.000444 0.003078 -0.001529 -0.057370 0.002795 0.005289 -0.575644
(5)
Since ∂T/∂z is of opposite sign to ∂c/∂z, while ∂γ/∂T and ∂γ/∂c are of the same sign, the second term in the braces results in a diminution in the magnitude of the thermal Marangoni effect, represented by the first term. Discussion and Conclusion -3.845 × 10-5 -4.037 × 10-5
Theory The behavior of gases acting as surfactants is well established, although not particularly well-known. Most of the data relate to inorganic or simple organic gases adsorbing at the water/gas interface, but there is also evidence that the same effects operate at the organic gas/ organic liquid interface for example from Lubetkin and Akhtar.10 In the same paper, there is a reasonably complete set of references to the literature on the effects of various gases on the surface tension of water. Unfortunately, data for the magnitude of the effects of O2 and N2 on the surface tension of PDMS do not appear to be available. Table 2 has a representative set of data for the effects of various gases dissolved in water on the surface tension, expressed as γ ) γo + bp + cp2 + dp3, where γ is the interfacial tension and p is the partial pressure of the gas, which by Henry’s law is proportional to c, the concentration of the gas. The quantity, γo, is usually taken to be the air/water interfacial tension at 1 bar. A simple measure of the efficacy of a gas at reducing the surface tension of the gas/water interface as a function of the applied pressure is thus the coefficient, b ) ∂γ/∂p, and the linear approximation, taking only the leading term in the expansion, is satisfactory for the present purpose. Thus the two factors affecting the surface tension are (1) the direct effect of the temperature and (2) the indirect effect of the temperature coefficient of the solubility which determines the concentration, and which in turn changes the surface tension. In the absence of any coupling, overall the effect is a linear combination of these two factors. This is assumed to be the case here. Using the expression for the force Fthermal on a bubble of radius R, at a position z in the vertical plane, due to the thermal Marangoni (absolute temperature, T) from Morick and Woermann
{[∂T∂γ][∂T∂z ]}
{[∂T∂γ][∂T∂z ] + [∂γ∂c][∂z∂c]}
Fcombined ) 2πR2
d
below, where values of b are given in Table 2 for various gas/water interfaces) and the sensitivity of the solubility of the gas to changes in temperature (the temperature coefficient of the Henry’s law constant).
Fthermal ) 2πR2
combined force, Fcombined
(4)
Now include the effect of the change in surface tension with concentration (c) of dissolved gas. This gives for the (10) Lubetkin, S. D.; Akhtar, M. J. Colloid Interface Sci. 1996, 180, 43. (11) Massoudi, R.; King, A. D., Jr. J. Phys. Chem. 1974, 78, 2262.
During the original measurements of Young et al. no oscillatory motion was observed. However, Hardy, Merritt and Subramanian and Morick and Woermann all reported such oscillations.2-4 The experimental procedure of the later investigators actually resulted in prolongation of this motion. Young used an open system, so that air could be freely exchanged at the surface of the suspended liquid PDMS drop. As discussed above, this greatly reduces the effects of the solubility changes, since it allows the undersaturation or supersaturation to be rapidly relieved through the bulk interface, rather than solely through the bubble interface, or by bulk diffusion. A conservative estimate of the ratio of the interfacial area available is that in the open system, where no oscillation was seen, the area exceeded that in the sealed, oscillatory experiments, by about a factor of 104. Hardy, Merritt and Subramanian and Morick and Woermann all used sealed systems where the only available interface was that of the bubble. When there is a large interfacial area available, as in Young’s system, the result is that equilibrium is achieved quickly, and the drive for oscillation is rapidly removed. In the sealed systems used later, the bubble itself is the main agent transferring gas from the warm (supersaturated) regions to the cool (undersaturated) regions, and the much smaller interfacial area greatly slows down the mass transport and thus the equilibration. Persistent oscillations are then observed. In both cases, the concentration of dissolved gas is a function of position, although the full concentration gradient will be achieved more slowly in the case of the sealed systems. Typical values of the quantity [∂γ/∂T] are given in refs 2-5. The average value for PDMS is about -0.063 mN m-1 K-1 with a standard deviation of (0.005. The average temperature gradient [∂T/∂z] used for example by Morick and Woermann was 9 × 103 K m-1; thus a typical overall thermal Marangoni effect is proportional to [∂γ/∂T][∂T/∂z] or -0.570 N m-2. To estimate the size of the concentration Marangoni effect, which is proportional to [∂γ/∂c][∂c/∂z], the values of b quoted in the Table 2 appropriate to water as the liquid are used. It has not proved possible to find any of the necessary numerical values for air/PDMS, so the magnitude of the concentration Marangoni effect has been estimated on the basis of the relevant properties for water.12 Note that although the surface tension of PDMS is much lower than that of water (about a factor of 3 less), the solubility of both N2 and O2 is greater in PDMS than in water. Overall, these two changes roughly cancel out. The solubilities of simple gases in liquids at low pressure obey Henry’s law: a doubling of the partial pressure corresponding to a doubling of the amount of gas dissolved. Concentration and partial pressure are thus easily interchangeable. For the temperature gradients used by Hardy, Merritt, and Subramanian and Morick and Woermann, the average concentration difference, c, between (12) Lubetkin, S. D. Electrochim. Acta 2002, 48, 357.
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Table 3. Solubilities of N2 and O2 in Water N2 O2
c273/g cm-3
c303/g cm-3
c273/c303
10.7 × 10-3 21.9 × 10-3
6.16 × 10-3 11.6 × 10-3
1.73 1.88
top and bottom, is thus equivalent to a change in partial pressure, p, of approximately 0.82 bar over the height of the cell, or about 3 mm. This value is derived from the concentrations for a typical value of ∆T of about 30 °C, corresponding to the middle range of temperature gradients used, as given in Table 3. Overall
[∂γ∂c][∂z∂c] ) [∂p∂γ][∂p∂z] ) 0.08 mN m
-1
/bar ×
0.82 bar/3mm ) 0.022 N m-2 This is about a 4% correction to the value for the thermal Marangoni effect, and since it opposes the thermal effect, it results in a diminution. There are a further two corrections which should also be taken into account. These are as follows: (1) N2 and O2 have different solubilities, and different temperature coefficients of the solubility. So, as the air bubble expands and contracts, its internal gas composition changes. For a general pair of gases, 1 and 2 at a given pressure, this variable composition will result in differing adsorbed amounts, Γ1 and Γ2 of the two gases, and so differing surface tension as a result of the differing values of b1 and b2. Thus once more, the surface tension is a function of the position of the bubble. In the absence of experimental data for the solubility of either gas in PDMS, or of the respective temperature coefficients, it is not possible to reliably estimate the size of this effect. For the case of N2 and O2 in water as the liquid, where b1 ≈ b2 it is expected to be small. Note that if b1 ) b2 accurately (or of course, for the case of a single gas), then this effect falls to zero. The Soret effect is also somewhat dependent on the chemical nature of the diffusing species and will thus differ between nitrogen and oxygen. The difference is expected to be subtle and unimportant in the present case. (2) By use of experimental values of the radii of the bubbles as measured by Morick and Woermann, there is approximately a factor of 4 in the radius between the top and the bottom of the travel (23 µm to 108 µm). The Laplace excess pressure, ∆p, is thus about a factor of 4 greater in the cold (top) region than in the hot (bottom). Since the adsorbed amount is proportional to the total pressure (the sum of the Laplace and hydrostatic pressures and ignoring the negligible change of hydrostatic pressure as the bubble
depth changes), the reduction in surface tension is asymmetrical: there will be a greater reduction at the top than the corresponding increase at the bottom. However, the absolute values are small. Here, the Laplace pressure is calculated for the PDMS/air interface. ∆p ) 2γ/R when R ) 23 µm, then with γ ) 21 mN m-1, ∆p ) 42 × 10-3/23 × 10-6 ) 1.8 × 103 ≈ 1/55 bar. At the bottom, with R ) 108 µm, ∆p ) 42 × 10-3/108 × 10-6 ) 4 × 102 ) 1/260 bar. The corresponding hydrostatic pressures are 101322 + 1830 ) 103152 and 101322 + 388 ) 101710. This is about a 2% difference. On the basis of the properties of the air/water interface, overall the thermal Marangoni effect will be reduced by about 6%. This is made up of a 4% reduction due to the solutal Marangoni effect and a further 2% due to the Laplace term in the internal pressure of the bubble. Hardy evaluated the coefficient [∂γ/∂T] from a direct measurement of the surface tension as a function of temperature using the pendent drop method, which is thus independent of the Marangoni effect, and obtained a value of 0.058 mN m-1 K-1. Hardy did not report the precise conditions used for the pendent drop experiment. From the balance of the Marangoni force with the buoyancy, the same author obtained an experimental value of 0.055 mN m-1 K-1. If the discrepancy were entirely due to the effects described here, the calculated reduction of 6% of the 0.058 mN m-1 K-1 would give 0.055 mN m-1 K-1, in agreement with the independent experimental value. This calculated correction is however based on the properties of the air/water interface, not the air/PDMS interface, so that this good agreement may prove coincidental. Control over some of the variables may be improved in a closed cell, and this was the motive for the changes away from the simpler experimental design of Young et al. These improvements have increased the complexity both in terms of the conduct of the experiment and in terms of interpretation of what was an elegant demonstration of the physical principles involved. An improvement in the current experimental arrangements might involve the use of a different control variable. The use of large and variable thermal gradients (which entails producing large supersaturations and undersaturations), to alter the balance between gravity and the thermal Marangoni effect, brings with it oscillatory motion and a slow approach to equilibrium of the concentration. A better alternative would be to use a relatively modest, fixed thermal gradient, and vary the bubble size, R, by controlled, variable pressure applied to the liquid, thus altering the balance between the Marangoni forces (∝R2) and gravitational forces (∝R3). LA0358365
