2 Interfacial Tension
2.5 Interfacial Tension Results
2.5.2 Interfacial Tension as Mass Transfer Proceeds
temperature. The turning point of this temperature dependence is at around the critical point of carbon dioxide where it changes from the gaseous to a supercritical one.
The value of the interfacial tension decreases first drastically with increasing pressure at every investigated temperature. Increasing pressure means also a higher solubility of carbon dioxide in the drop phase. This is in accordance with the Gibbs adsorption law (see Eq. 2-22) where it is stated, that an adsorption—characterized by an increase in the component concentration xi— causes a fall in the interfacial tension.
The fall of the interfacial tension in the first part is nearly linear. A further increase in the pressure behind this linear fall does not affect the interfacial tension significantly. From here on, the interfacial tension remains almost constant [81,194]. This is the second linear part of the interfacial tension curve. The linear dependency of the interfacial tension on the pressure at lower pressure conditions can be explained by means of the solubility of carbon dioxide in the drop phase. At lower pressures, the solubility of carbon dioxide in the liquid phase increases linearly with the pressure. However, the gradient of the solubility becomes smaller as the pressure increases until the curve reaches a nearly constant value where the solubility changes hardly with increasing pressure. This is the explanation for the behaviour of the interfacial tension dependency on the pressure. The interfacial tension is lowered with increasing pressure as long as the pressure still has an impact on the CO2 solubility. Once the drop phase has been saturated, a further increase in the pressure does not bring more carbon dioxide into the drop phase, and thus, it has hardly any influence on the interfacial tension.
The intersection of the first linear and the second horizontal line at 294 K is drawn in Fig. 2-10 and the turning point is approximately at the critical point of carbon dioxide where it turns from the gaseous to the supercritical one at 7.4 MPa.
The interfacial tension of ethanol is far lower than that of water. The same pressure effect can be seen here too. At ambient pressure, it has a value of 22 mN/m but at 5 MPa this value becomes 6.7 mN/m. At 373 K and 10 MPa ethanol has an interfacial tension of only 3.5 mN/m.
2.5.2 Interfacial Tension as Mass Transfer Proceeds
In Fig. 2-11 the interfacial tension of pure water against carbon dioxide at 294 K and 8 MPa, estimated by means of pure drop density as well as mixture density at saturated state, and water-ethanol mixture against presaturated carbon dioxide at 313 K and 10 MPa are shown. The interfacial tension of the two systems change in opposite directions with time.
An increase in the interfacial tension is seen in the water-ethanol mixture. Here, water is mixed with 10 wt% ethanol and the surrounded carbon dioxide phase is also presaturated with a small amount of this mixture. The mixture used for the presaturation of the embedding phase is approximately 5 g. The view cell used here has an inner volume of about 88.5 ml.
Apparently, the embedding phase is not saturated yet. The solubility of ethanol in the carbon dioxide is far higher than the amount provided for the presaturation. That is why ethanol is
2.5 Interfacial Tension Results
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transported from the aqueous drop phase to the continuous phase. The desorption of ethanol results in an increase of the drop interfacial tension from the initial value 18.5 mN/m to 21.3 mN/m. Although there are also mutual solubilities of water and carbon dioxide but compared to the amount of dissolved ethanol and its impact on the interfacial tension, they do not affect this property significantly. This desorption phenomenon which causes an increase in the interfacial tension is in good agreement with the Gibb’s adsorption law given in Ch. 2.1.
The required density data for the calculation of interfacial tension of water-ethanol mixture against carbon dioxide are taken from the literature. The density data of water-ethanol mixture reported in [152] is used. The density change of the continuous phase due to presaturation is assumed to be negligible and the density of pure carbon dioxide in [144] is taken.
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0 200 400 600 800 1000 1200
t [s]
Figure 2-11: Interfacial tension of water and water-ethanol mixture during the mass transfer.
As for the system pure water drop against carbon dioxide, carbon dioxide is brought into the drop phase and lowers the interfacial tension of the water phase according to the Gibbs adsorption law (Eq. 2-18). Here, considering the composition of the drop phase, the calculation of the liquid interfacial tension against carbon dioxide is done twice. First by means of pure water density, and later, by using the saturated density of water and carbon dioxide mixture. In both cases, it is assumed that the continuous phase contains only carbon dioxide. The solubility of water in carbon dioxide is taken as negligible.
Applying the pure densities of water and carbon dioxide, the interfacial tension falls from 34 mN/m to 28 mN/m in 800 s. If the saturated density of the mixture of water and carbon dioxide is taken for the drop phase, the interfacial tension is lowered from 37 mN/m to 30.6 mN/m. It is believed that after 800 s the drop phase is saturated by carbon dioxide so that after 800 s there is no significant change in the interfacial tension.
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The consideration about which density is to be used for the drop phase in estimating the interfacial tension—whether the pure density or the mixture density—goes back to the already aged-discussion about what role the bulk phase plays in the interfacial tension. There exist two possibilities. In the first one, the interfacial tension depends solely on the composition of the interface. In the second one, not only the interface but also the bulk phase determines the interfacial tension.
Assuming the first possibility (the interfacial tension is affected solely by the composition of the interface), the static interfacial tension should have been reached instantaneously after the drop is formed. During the formation of the drop, the forced convection around the drop surface governs the mass transfer between the phases. Thus, within a very short period of time, the concentration at the interface has reached saturation. But the fact is, the water drop shape in carbon dioxide still changes during the saturation time of the bulk phase, that is, in 800 s. The alteration in the shape of the drop signals that the static interfacial tension has not yet been obtained. As it can be seen in Fig. 2-11, the interfacial tension still changes with time regardless of whether the pure water density or the mixture density is taken into account in calculating the interfacial tension value. The saturation value is obtained after waiting for 800 s.
Morgner [136] tried to understand this fact by suggesting the idea that the pressure difference over the meniscus is responsible for the drop shape alteration during the measurement. The relaxation of this pressure difference takes some time and causes the change in the drop shape.
The alteration of the drop shape has no correlation with the static interfacial tension. To Morgner, the static interfacial tension should have been reached immediately (microseconds) after it hangs at the tip of the capillary.
According to him, the pressure difference &p perpendicular to an interface changes as much as 100 MPa/Å. That means, an interface of 1 Å thick brings a pressure difference of as high as a maximum of 100 MPa. Assuming a water drop diameter of 5 mm and letting carbon dioxide be transported across the interface, through a computer simulation Morgner stated that due to the transfer of carbon dioxide there is a pressure difference as much as 2 MPa over the interface [136].
To reexamine the validity of Morgner’s statement about the magnitude of the pressure difference across the interface, the Young-Laplace Equation (Eq. 2-32) is applied. In this equation, the pressure difference over the meniscus is correlated with the interfacial tension and the principal radii R1 and R2. Here, the radius R1 in the Young-Laplace Equation is set equal to R2 (Eq. 2-34). Both radii are the same:
If the drop geometry is assumed as spherical. In this case, the pressure difference &p over the meniscus is the same over the whole drop surface.
At apex, such as given in Eq. 2-34. This means, the pressure difference is calculated only at the point &pApex . The drop shape remains as it is (no geometrical simplification is made here).
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In the following, the radius of a drop is calculated in cases where the pressure differences over its meniscus are 100 MPa (maximal pressure difference possible) and 2 MPa. The system used here is water in a carbon dioxide environment at 293 K and pressures up to 25 MPa. The interfacial tension data are taken from the literature [61,182]. At 293 K and 0.1 MPa, the interfacial tension of water in a carbon dioxide environment is 70.32 mN/m. Following the suggestion of Morgner, the radius of the drop should be 14 Å (if the pressure difference is 100 MPa) or 7 Å if a pressure difference of 2 MPa is assumed. The theoretical radii obtained from the calculation can be seen in Table 2-2.
Table 2-2: Calculated drop radii assuming a pressure difference of 2 MPa and 100 MPa.
Temp. p &p R
[K] [MPa] [MPa] [Å] [mN/m] Source
293 0.1 2 7 70.32 [182]
293 2 2 6 57.27 [182]
293 4 2 5 46.69 [182]
293 6 2 3 34.38 [182]
293 8 2 3 25.92 [182]
293 10 2 2 24.53 [182]
293 15 2 2 23.1 [61]
293 20 2 2 20.3 [61]
293 25 2 1 13.6 [61]
293 0.1 100 14 70.32 [182]
293 2 100 11 57.27 [182]
293 4 100 9 46.69 [182]
293 6 100 7 34.38 [182]
293 8 100 5 25.92 [182]
293 10 100 5 24.53 [182]
293 15 100 5 23.1 [61]
293 20 100 4 20.3 [61]
293 25 100 3 13.6 [61]
According to the radii values obtained in Tab. 2-2, the radii lie in nanometer range.
Macroscopically seen, it is impossible to obtain such drops. Probably, the calculated pressure difference over the meniscus is overestimated. In the following, an attempt is made to calculate the pressure difference over the meniscus of a drop with a diameter of 5 mm. The same system as used in Tab. 2-2 is applied here, water drop in a carbon dioxide environment at 293 K and pressures up to 25 MPa.
It can be seen in Tab. 2-3 that the pressure difference over the meniscus is far lower. The pressure differences calculated here are 105- to 107-times lower than that forecast by Morgner.
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Schwuger [168] estimated the pressure difference over the meniscus of a water drop with a diameter of 2 mm. He took an interfacial tension of 73 MPa and found out that the pressure difference is 1.5 mbar (or 1.5*10-4 MPa) which is in the same order of magnitude as the value calculated in Tab. 2-3.
Table 2-3: Calculated pressure difference over the meniscus of a drop whose diameter is 5 mm.
Temp. p &p R
[K] [MPa] [MPa] [mm] [mN/m] Source
293 0.1 5.63E-05 2.5 70.32 [182]
293 2 4.58E-05 2.5 57.27 [182]
293 4 3.74E-05 2.5 46.69 [182]
293 6 2.75E-05 2.5 34.38 [182]
293 8 2.07E-05 2.5 25.92 [182]
293 10 1.96E-05 2.5 24.53 [182]
293 15 1.85E-05 2.5 23.1 [61]
293 20 1.62E-05 2.5 20.3 [61]
293 25 1.09E-05 2.5 13.6 [61]
The pressure difference as shown in Tab. 2-3 seems more reasonable than the one calculated by Morgner. This is due to the fact that a pressure difference—like a concentration and a temperature difference—will cause fluid flow in order to reach an (mechanical) equilibrium.
There is no reasonable argument which can explain why even though the pressure difference over the drop meniscus is as high as 2 MPa (or even 100 MPa), macroscopically seen, the drop remains in its static state.
Hence, the idea that it is the pressure difference which causes an alteration in the drop shape during the interfacial measurement is disproved. Later, in his report [136], Morgner also admitted that the pressure difference is not the property which is responsible for the geometry change of the drop. However, he still could not clearly explain what causes the changes in the drop shape during the measurement if the static interfacial tension has really been reached microseconds after the drop is formed.
Although the first possibility cannot be applied to explain the change in interfacial tension with time, Morgner’s argument about the instantaneous saturation of the interface is reasonable. It is believed that within a very short period of time the interface is saturated. This does not mean that the concentration equilibrium between the phases has been reached. Even though the interface is saturated, the bulk phase has not yet reached the equilibrium. A certain saturation time is still required to reach equilibrium between those bulk phases.
It is apparent that the composition of the bulk phase does play an important role in the interfacial tension. Even though the saturation of the interface is reached instantaneously as the drop hangs at the tip of the capillary, the static interfacial tension or the saturated interfacial
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tension has not yet been reached. The proof that the bulk composition affects the interfacial tension is that the drop shape changes as the mass transfer proceeds. And an alteration in the drop shape signals a change in the interfacial tension itself.
To be very precise, the drop density which should be applied in estimating the interfacial tension is the instantaneous density of the mixture in the drop phase. This density data can be obtained by hanging a drop at a capillary which is connected to a magnetic suspension balance (see Chapter 2.3.2). Using this method, the mass transfer into a drop can be tracked very well and the density change with time can be obtained. Here, the geometry of the interface is important. The diagram of density as a function of time is not the same whether a water drop is hung in the carbon dioxide environment or just a plane water surface is brought into touch with carbon dioxide. The final saturation density will be the same, but not the instantaneous one which is essential in estimating the precise instantaneous interfacial tension.
If the instantaneous drop density is applied, the degree of mass transfer is considered in the interfacial tension estimation. The interfacial tension diagram obtained will lie between both curves shown in Fig. 2-11. At the beginning, the drop density is closer to the pure water density.
Shortly before the equilibrium state is reached, it will be very similar to the saturation density.
Thus, the real interfacial tension will lie between these two diagrams. It starts from the interfacial tension diagram applying the pure drop phase density and ends in the diagram applying the saturation density of the drop phase.
The interfacial tension error appearing here due to the density used for the drop phase can be estimated by calculating the difference between the interfacial tension obtained using the pure and the saturated density. The mean difference between both interfacial tensions is 2.9 mN/m.
That means, applying the interfacial tension either calculated by means of the pure drop density or by the saturated drop density causes an error in the interfacial tension of less than 2.9 mN/m.
The degree of importance in using the instantaneous drop mixture density depends on the solubility of carbon dioxide into the drop phase. For systems where the solubility between the phases is not that high (for example water-carbon dioxide system), it is less important to apply the instantaneous drop mixture density. However, once the solubility of carbon dioxide in the drop phase is high (for example systems containing alcohols or oil), the change in the instantaneous drop mixture density becomes higher. It is therefore more essential to apply the instantaneous drop density for such systems in order to minimize the imprecision in the interfacial tension estimation.
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