Langmuir 2004, 20, 1721-1723
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Correction for the Aerodynamic Resistance and Viscosity in Maximum Bubble Pressure Tensiometry V. B. Fainerman,† V. D. Mys,† A. V. Makievski,‡,§ and R. Miller*,‡ Medical Physicochemical Centre, Donetsk Medical University, 16 Ilych Avenue, Donetsk 83003, Ukraine, Max-Planck-Institut fu¨ r Kolloid- und Grenzfla¨ chenforschung, 14424 Golm, Germany, and SINTERFACE Technologies, Volmerstr. 5-7, 12489 Berlin, Germany Received July 2, 2003. In Final Form: November 5, 2003 The application of the maximum bubble pressure tensiometry at short adsorption times (below 0.5 s) is complicated by the two effects aerodynamic resistance of the capillary and viscosity of the studied liquid. A procedure is shown that demonstrates how the two effects can be taken into consideration to obtain correct dynamic surface tensions. The parameters are determined for a commercial steel capillary of 0.25-mm diameter and 60-mm length. The correction procedure, in turn, can even be used to estimate the viscosity of the liquid under investigation in case it is unknown.
1. Introduction For the measurement of dynamic surface tensions of surfactant solutions in the short time range, the maximum bubble pressure technique is widely used.1,2 The principle of this technique consists of the determination of the pressure in a bubble during its growth. The maximum pressure then correlates to the hemispherical bubble size.3 This is correct only for narrow capillaries, while for those with a capillary radius rcap g 0.1 mm, a correction of the bubble deformation due to gravity is needed. A comprehensive description is given in review articles and books, for example, in ref 4. Another important fact has to be considered as well: the pressure is not measured in the bubble directly but in the system connected to the capillary. Therefore, the measured pressure exceeds the pressure in the bubble, especially at short bubble lifetimes.4 This excess pressure is caused by the aerodynamic resistance of the capillary, by the inertia of the gas, and also by the viscosity of the studied liquid. The excessive pressure also depends on the ratio of the volume of the separating bubble to the volume of the measuring system, as discussed recently.5 A detailed analysis of these and many other effects observed in the maximum bubble pressure tensiometry was made by Kovalchuk and Dukhin.6 It was shown, in particular, that when the capillary is wide and short enough, the separation of bubbles is followed by an oscillation process, which leads quite rapidly to the elimination of the pressure difference between the capillary ends. Therefore, for such capillaries the excessive pressure in the system is quite small.7 However, it is impossible to account for all relevant contributions in the * Corresponding author. † Donetsk Medical University. ‡ Max-Planck-Institut fu ¨ r Kolloid- und Grenzfla¨chenforschung. § SINTERFACE Technologies. (1) Eastoe, J.; Dalton, J. S. Adv. Colloid Interface Sci. 2000, 85, 103. (2) Noskov, B. A. Adv. Colloid Interface Sci. 1996, 69, 63. (3) Bendure, R. L. J. Colloid Interface Sci. 1971, 35, 238. (4) Fainerman, V. B.; Miller, R. The maximum bubble pressure technique. In Drops and Bubbles in Interfacial Science; Mo¨bius, D., Miller, R., Eds.; Studies of Interface Science; Elsevier: Amsterdam, 1998; Vol. 6, pp 279-326. (5) Fainerman, V. B.; Miller, R.; Makievski, A. V. Accurate analysis of the bubble formation process in maximum bubble pressure tensiometry. Rev. Sci. Instrum., in press. (6) Kovalchuk, V. I.; Dukhin, S. S. Colloids Surf., A 2001, 192, 131.
framework of the existing theories. In this context, the effect of the viscosity of the studied liquid is especially difficult to access when the bubble growth conditions are nonstationary. Therefore, to correct for the aerodynamic resistance of the capillary, first a calibration of the capillary with a pure liquid of low viscosity (e.g., water) is performed. To make corrections for the viscosity, systematic studies are needed resulting in a dependency that allows correction of the obtained raw data.8 In the present paper, a procedure is described that is implemented in the BPA tensiometer (SINTERFACE, Berlin) for the correction of the aerodynamic resistance and viscosity effects. The advantage of this tensiometer is its ability to measure the lifetime and dead time directly. Because the dead time (the time during which the rapid growth and separation of the bubble takes place after the maximum pressure is attained) depends on the viscosity of the studied liquid, it is possible to make corrections for the viscosity of higher viscous liquids even when the viscosity is not known. 2. Results and Discussion Figure 1 illustrates the surface tension data obtained for water when a steel capillary of 0.25-mm diameter and 60-mm length was used. For the bubble surface lifetime tl < 1 s, the surface tension calculated from the pressure in the measuring system exceeds the actual value by up to 3 mN/m (at a lifetime of 0.01 s). To eliminate the effect caused by the aerodynamic resistance, the dependence of excess surface tension ∆γa on time in the time range below 1 s is approximated by a quadratic polynomial, where the polynomial coefficients are a function of the capillary length and radius. This value has to be subtracted from the measured surface tension γm to get the correct surface tension γc ) γm - ∆γa. γc is the surface tension of pure water measured at a time t > 1 s. In agreement with ref 9, the surface tension of pure liquids does not depend on time. For adsorption times down to 1 ms, this fact was demonstrated by the method of the oscillating jet with a (7) Lylyk, S. V.; Makievski, A. V.; Kovalchuk, V. I.; Schano, K.-H.; Fainerman, V. B.; Miller, R. Colloids Surf., A 1998, 135, 27. (8) Fainerman, V. B.; Makievski, A. V.; Miller, R. Colloids Surf., A 1993, 75, 229. (9) Defay, R.; Prigogine, I. Surface Tension and Adsorption; Longmans-Green: London, 1966.
10.1021/la030271y CCC: $27.50 © 2004 American Chemical Society Published on Web 02/06/2004
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Fainerman et al.
Figure 1. Dependence of the surface tension for pure water. Two experimental series.
Figure 3. Dead time as a function of the surface tension γm for the Triton ×100 (2 mmol/L) and C12EO6 (0.3 mmol/L) aqueous solutions (curves 1 and 2, respectively), the Triton X100 (2 mmol/L) solutions in 50% and 70% solutions of glycerine in water (curves 3 and 4, respectively), and 80% glycerine aqueous solution (curve 5); the viscosity values were 1 mm2/s (solutions 1 and 2) and 5, 23, and 52 mm2/s (solutions 3, 4, and 5).
of water at a given temperature. Therefore, the value of the dynamic surface tension, corrected with respect to the resistance and viscosity, is
γc ) γm - ∆γa - ∆γv
Figure 2. Dependence of the surface tension for aqueous glycerine solutions at 60% (1) and 70% (2) glycerine, where the viscosities are 11 and 23 mm2/s, respectively; two experimental series for each solution.
narrow elliptic orifice (0.35 × 0.2 mm)10 and by the maximum bubble pressure using short and wide capillaries.4 For solutions of higher viscosity, the excess pressure in the measuring system is even higher, as is illustrated in Figure 2. These higher values are due to the fact that, in addition to the capillary aerodynamic resistance ∆γa, the viscous hydrodynamic resistance of the liquid in the process of bubble growth ∆γv also becomes relevant. This extra correction ∆γv depends on the viscosity and surface tension of the studied liquid. A detailed theoretical analysis of such effects was given elsewhere.4,8 For a capillary of 0.25-mm diameter, this correction was determined via measurements performed for various liquids in the viscosity range of 1-100 mm2/s. As a result, the following relationships are obtained:
for 0.1 e tl e 1 s ∆γv ) -0.13 ln tl (P/Pcal) ln η
(1)
for 0.001 e tl e 0.1 s ∆γv ) (-0.32 - 0.25 ln tl)(P/Pcal) ln η
(2)
Here, tl is the measured lifetime in s, P is the actual pressure, Pcal is the calibration pressure for water, and η is the viscosity of the liquid in mm2/s. The coefficients given in these equations depend on the capillary radius, and their values decrease with increasing radius. In eqs 1 and 2, the ratio P/Pcal can be replaced by the ratio of surface tension of the studied liquid to the surface tension (10) Thomas, W. D. E.; Potter, L. J. Colloid Interface Sci. 1975, 50, 397.
(3)
As shown experimentally, the dead time td measured here by the BPA tensiometer decreases almost linearly with the noncorrected dynamic surface tension γm (or the pressure in the measuring system). This fact can be readily understood: the pressure increase leads to a decrease of the time necessary for the formation of the bubble of a certain size.4 It is important that the slope of the td versus surface tension curve depends on the viscosity of the liquid. Some examples of such dependencies are shown in Figure 3. The higher the liquid viscosity, the larger the slope of the dependence td(γ). This fact can be explained by an increasing bubble volume separating from the capillary, because the increase in the viscosity leads to a retardation of the separation process. It is important to note that for liquids of a low viscosity (for example water) the slope of the td(γ) curve does not depend on the concentration of the solution studied or on the type of the surfactant used. It is seen from Figure 3 that for the Triton X100 and C12EO6, the slopes are the same. Therefore, from the data illustrated in Figure 3 one can conclude that
td ) A - Rγ
(4)
where A and R are constants characteristic of the specific liquid. The dependencies of R on the viscosity of the liquid for solutions of various surfactants (water and aqueous glycerine solutions used as solvents) are shown in Figure 4. For viscosities η > 2 mm2/s, that is, for R > 0.0016 s/(mN/m), the dependence of R on η is quite linear. Therefore, we obtain
R ) B + βη
(5)
where B and β are constants. For a capillary of diameter 0.25 mm, one obtains B ) 0.0015 s/(mN/m) and β ) 0.000 092 s2/[mN/(m‚mm2)]. As in the case of eqs 1 and 2, the coefficients in eq 5 depend on the capillary radius. Therefore, the viscosity coefficient η for the studied solution, for which the viscosity is unknown, can be determined from eq 5, using the value of the constant R calculated from eq 4. Equations 1 and 2 can be used to
Maximum Bubble Pressure Tensiometry
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Figure 4. Dependence of the coefficient R on the liquid viscosity. Points represent experimental data; the straight line corresponds to eq 5.
Figure 5. Noncorrected (open symbols) and corrected (closed symbols) dependence of the surface tension for 2 mmol/L Triton X100 in a 50% glycerine solution.
calculate the correction for the viscosity of the liquid, and via eq 3 one can correct the surface tension. This procedure is implemented in the BPA tensiometer, where it is performed automatically. The results obtained from studies of the dynamic surface tension for 2 mmol/L Triton X100 in an aqueous 50% glycerine solution are shown as an example in Figure 5. For the dependence given by eq 4, the values of the parameters in eq 4 were obtained: A ) 0.172 19 s and R ) 0.002 058 s/(mN/m). The η value obtained with this R from eq 4 is η ) 6.1 mm2/s, instead of the 6.3 mm2/s measured directly for the water/glycerine mixture. The surface tension values, corrected for both the effects according to eq 3, are considerably lower (about 3-4 mN/m at short lifetimes) than those measured directly. Both effects, ∆γa and ∆γv, fade out at lifetimes tl > 0.5 s.
dynamic resistance of the capillary and the viscosity of the liquid under study. These corrections cannot be obtained via calibrations because all characteristic parameters of bubble formation are effected by these two effects. The obtained parameters are valid for the capillary used here (rcap ) 0.125 mm, length 60 mm). The procedure can be considered as a guide for determining the corrections for any experimental geometry. It is even possible to use the characteristic parameters of the viscous correction to determine the viscosity of the measured liquid in case it is unknown prior to the experiment. This option makes bubble pressure tensiometry an easy-to-use technique for dynamic surface tension measurements in the range of very short adsorption times.
3. Conclusions At short adsorption times, two important corrections have to be made to obtain unaffected surface tension data from maximum bubble pressure experiments: the aero-
Acknowledgment. The work was financially supported by a project of the European Space Agency (FASES MAP AO-99-052), the Ukrainian SFFR (Project No. 03.07/ 00227), and the DFG (Mi418/11-2). LA030271Y
