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Limits of oscillation frequencies in drop and bubble shape tensiometry
Colloids and Surfaces A: Physicochem. Eng. Aspects 261 (2005) 25–28
Limits of oscillation frequencies in drop and bubble shape tensiometry M.E. Lesera , S. Acquistapacea , A. Cagnab , A.V. Makievskic , R. Millerd,∗ a
d
Nestec Ltd., Nestl´e Research Centre, CH-1000 Lausanne 26, Switzerland b IT Concept, Longessaigne 69770, France c SINTERFACE Technologies, Berlin 12489, Germany MPI Kolloid- und Grenzfl¨achenforschung, Potsdam/Golm 14424, Germany Received 21 July 2004; accepted 2 November 2004 Available online 21 January 2005
Abstract To determine the dilational rheology of surface layers, the profile analysis tensiometry can be used with oscillating drops or bubbles. The methodology limits for these oscillations depend on the liquids’ properties, such as density, viscosity and surface tension. For the most frequently studied water/air interface, the maximum oscillation frequency is of the order of 1 Hz, although much higher frequencies are technically feasible by the existing profile analysis tensiometers. For f > 1 Hz, deviations of the drops/bubbles from the Laplacian shape mimic non-zero dilational elasticities for the pure water/air and ethanol/air interface. For liquids of higher viscosity, the critical frequency is much lower. © 2004 Elsevier B.V. All rights reserved. Keywords: Surface rheology; Dilational viscoelasticity; Profile analysis tensiometry; Oscillating drops and bubbles; Limiting frequency
1. Introduction Dilational rheology has much impact on quite a number of technological processes, such as foaming or emulsification [1–3]. The formation process and the mechanisms of stabilisation or destabilisation are linked to the dynamic interfacial properties, such as the interfacial dilational viscoelasticity. Only recently, commercial instruments became available for measuring the viscoelasticity of adsorption layers. This made the topic quite interesting to many research groups. The only methodology accessible via commercial instruments to measure the elasticity and viscosity of adsorption layers is the drop or bubble shape analysis as described in detail recently [4]. The main principle of drop and bubble profile tensiometers is to determine the interfacial tension from the shape of drops or bubbles via the Gauss–Laplace equation. In addition, by applying transient or harmonic ∗
Corresponding author. E-mail address: [email protected] (R. Miller).
0927-7757/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2004.11.043
perturbations, this method can also supply information on the dilational rheology of interfacial layers. The Gauss–Laplace equation is derived for static conditions [5]. If a drop or bubble is growing or shrinking during the experiment its shape can deviate from an equilibrium state. The same is true for oscillating systems, i.e. when a drop or bubble is forced on purpose to oscillate in order to determine the dilational elasticity or viscosity. The technical parameters of some instruments allow to perform oscillations of a drop or bubble at various frequencies, from very slow oscillations of about 10−3 Hz to frequencies of even 10 Hz and above. At high oscillation frequencies hydrodynamic effects cause deviations of drop and bubble profiles from a Laplacian shape. Thus, high-frequency studies require other techniques, such as the oscillating bubble method, which works with small spherical bubbles [6,7] and reaches frequencies of up to 150 Hz, or the capillary wave damping techniques [8], which can work even at oscillation frequencies of up to 1000 Hz. It is the purpose of this contribution to analyse the oscillation behaviour of drops and bubbles for the water/air
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M.E. Leser et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 261 (2005) 25–28
and oil/air systems to estimate the limiting frequency up to which this method yields correct values. For surfactant-free systems, both the elasticity and viscosity should be zero [9]. Any positive or negative values for the elasticity or viscosity would then refer to artefacts caused by a deformation rate larger than a critical value.
2. Materials and methods For the present studies two instruments were applied: the Tracker from IT Concept (France) and the PAT1 from SINTERFACE Technologies (Germany). The main elements of the method are CCD camera, frame grabber, light source, dosing system, cuvette with needle for drop/bubble formation. Both instruments are computer-controlled and use special software for fitting the Laplace equation to the drop or bubble shape coordinates. The computer-controlled dosing system was used to generate oscillations at frequencies up to about 0.5 Hz, while for higher frequencies a piezo-system was applied. The two rheological parameters elasticity and viscosity of the interfacial layer were obtained from a Fourier analysis of the measured signals. The volume of drops or bubbles was about 10 l. Although the two instruments used are based on quite different hardware and software, there were no significant differences between the results obtained for the same capillary diameter and the same samples. The studies were performed with pure water (γ = 72.0 mN/m, η = 1 cP), ethanol (γ = 22 mN/m, η = 1 cP), and their mixtures. In additional experiments drops of paraffin oil (γ = 31.4 mN/m, η = 122 cP, purchased from Sigma) and silicon oil (γ = 20.5 mN/m, η = 1000 cP, gift from Dow Corning) were studied. The volume of drops and bubbles were about 10 l, the applied volume amplitudes between 2 and 15%. At this point we want to note that the purity of the used oil samples were of minor importance as they have been studied at the liquid/air interface. Obviously, there are surfactants also adsorbing at such an interface (cf. [10]), however, blank tests have shown that there is no change in surface tension over a long period of time and hence there is no elasticity expected for these liquid surfaces.
Fig. 1. Surface tension and apparent dilational elasticity modulus E as a function of oscillation frequency for an air bubble in pure water.
Fig. 2. Surface tension and apparent dilational elasticity modulus E as a function of oscillation frequency for a drop of pure water in air.
above 1 Hz an apparent elasticity is obtained, which is mimicked by the non-ideal oscillation behaviour of the drop or bubble. Repeat experiments show that the obtained values have quite a large scattering, in contrast to low-frequency experiments. In any case, for frequencies above 1 Hz this apparent dilational elasticity is quite significant and of the order of 30 and 50 mN/m at 9 Hz. Such values typically correspond to packed surfactant adsorption layers. The generated oscillations at the studied frequencies are still harmonic, as one can see from the example shown in Fig. 3. The generated oscillation of the drop volume follows a sinusoidal function perfectly. However, the fitting of the Laplace equation to the dynamic drop or bubble shape coordinates shows that only up to a certain frequency the observed deviations along the shape are scattered normally (Fig. 4a). For higher oscillation frequencies, the deviation between the Laplace equation and the determined drop or bubble
3. Results and discussion The results of oscillation experiments with air bubbles in water and water drops in air are shown in Figs. 1 and 2, respectively. Up to a frequency of f = 1 Hz, the measured surface tension γ is independent of frequency and corresponds to the correct value, obtained for a drop in rest. For 3 and 6 Hz the value measured with the drop is still correct while that measured for the bubble is slightly increased. The surface elasticity, determined from the surface tension change during oscillations for bubbles and drops is also shown in Figs. 1 and 2. Here it becomes evident that for frequencies
Fig. 3. Volume oscillation of a water drop at 6 Hz.
M.E. Leser et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 261 (2005) 25–28
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Fig. 6. Apparent dilational elasticity modulus as a function of oscillation frequency for drops of water (), water/ethanol 86:14 (), ethanol (), amplitude of volume oscillations 8%.
already obtained. The much larger amplitude of 8% as compared to 2% in Fig. 5 does not change the critical frequency but increases the absolute values of the apparent elasticity modulus E. When we plot the results as a function of the dimensionless parameter Ca:Re we get linear dependencies (Fig. 7). Here Ca is the capillary number and Re the Reynolds number, defined by Fig. 4. Deviation of the Laplace equation from the drop shape as a function of the points around the drop profile: (a) 1 Hz and (b) 6 Hz.
coordinates shows a special pattern, which demonstrates the systematic deviation of the dynamic shapes from the Laplace equation (Fig. 4b). The higher the generated frequency is, the more pronounced are the patterns. This is observed for drops as well as for bubbles, and no significant difference appears for the studied water/air interface weather a drop or bubble is used. In order to analyse how the absolute surface tension of a liquid influences the critical frequency for oscillation experiments, measurements were performed with a 86:14 mixture of water and ethanol (γ = 46 mN/m) and of pure ethanol (γ = 22 mN/m). The results are shown in Figs. 5 and 6. As one can see, at 2 Hz a measurable dilational elasticity modulus is
Fig. 5. Apparent dilational elasticity modulus as a function of oscillation frequency for drops of water (), water/ethanol 86:14 (), ethanol (), amplitude of volume oscillations 2%.
Ca = ηvmax /γrγ
and
Re = ρvmax /rγ η
where η is the bulk viscosity, rγ the mean radius of curvature of the drop at the apex, ρ the density of the liquid, and vmax the maximum liquid flow rate estimated for a drop oscillation with an amplitude of about 5% with respect to the volume. The reason for the rather linear dependencies remains unclear and requires a deeper analysis. Another series of experiments was performed with viscous oils. Fig. 8 shows the apparent elasticities obtained from oscillations with drops of paraffin oil and silicon oil, respectively. While the paraffin oil yields elasticity moduli comparable with ethanol, however, the critical frequency at which measurable apparent elasticities were obtained, is much lower – of the order of 0.25 Hz. The apparent elasticity moduli for
Fig. 7. Apparent dilational elasticity modulus as a function of the dimensionless parameter Re:Ca for drops of water (), water/ethanol 86:14 (), ethanol (), amplitude of volume oscillations 8%.
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M.E. Leser et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 261 (2005) 25–28
A much lower critical frequency of about 0.1 Hz is obtained for drops of a very viscous silicon oil. For liquid/liquid interfaces a more complicated situation exists which needs to be analysed in detail before the rheological behaviour of any adsorption layers at such interfaces is studied. Acknowledgement This work was financially supported by a project of the European Space Agency (FASES MAP AO-99-052). Fig. 8. Apparent dilational elasticity modulus as a function of oscillation frequency for drops of silicon oil (䊉), paraffin oil (), amplitude of volume oscillations 2%.
the studied silicon oil are much larger as compared to paraffin oil, and the critical frequency is of the order of 0.1 Hz.
4. Conclusions The studies show that for oscillations with frequencies up to 1 Hz the “dynamic” drop and bubble shapes can be well described by the Gauss–Laplace equation, when the viscosity is η ≤ 1 cP. For higher frequencies (3 Hz) the results suggest already a certain dilational elasticity modulus, which cannot exist for a pure water/air or ethanol/air interface. This apparent modulus is mimicked by deviations of the drop/bubble shape from that given by the Laplace equation. Thus, the shape analysis tensiometry is applicable for dilational rheology studies up to a frequency of 1 Hz as maximum. This holds true for drops and bubbles. For higher frequencies alternative methods have to be applied in order to measure the elasticity and viscosity of interfacial layers.
References [1] A. Williams, J.J.M. Janssen, A. Prins, Colloid. Surf. A 125 (1997) 189. [2] E. Manev, R.J. Pugh, J. Colloid Interf. Sci. 186 (1997) 493. [3] D. Langevin, Curr. Opin. Colloid Interf. Sci. 3 (1998) 600. [4] G. Loglio, P. Pandolfini, R. Miller, A.V. Makievski, F. Ravera, M. Ferrari, L. Liggieri, Drop and bubble shape analysis as tool for dilational rheology studies of interfacial layers, in: D. M¨obius, R. Miller (Eds.), Novel Methods to Study Interfacial Layers, Studies in Interface Science, vol. 11, Elsevier, Amsterdam, 2001, p. 439. [5] J. Gaydos, The Gauss Laplace equation for liquid axisymmetric menisci, in: D. M¨obius, R. Miller (Eds.), Drops and Bubbles in Interfacial Research, Studies in Interface Science, vol. 6, Elsevier, Amsterdam, 1998, p. 1. [6] K.D. Wantke, H. Fruhner, J. Colloid Interf. Sci. 237 (2001) 185. [7] V.I. Kovalchuk, J. Kr¨agel, E.V. Aksenenko, G. Loglio, L. Liggieri, Oscillating bubble and drop techniques, in: D. M¨obius, R. Miller (Eds.), Novel Methods to Study Interfacial Layers, Studies in Interface Science, vol. 11, Elsevier, Amsterdam, 2001, p. 485. [8] B.A. Noskov, G. Loglio, Colloid. Surf. A 143 (1998) 167. [9] J. Lucassen, M. van den Tempel, Chem. Eng. Sci. 27 (1972) 1283. [10] M. Mellema, J. Benjamins, Colloid. Surf. A 237 (2004) 113.
Limits of oscillation frequencies in drop and bubble shape tensiometry M.E. Lesera , S. Acquistapacea , A. Cagnab , A.V. Makievskic , R. Millerd,∗ a
d
Nestec Ltd., Nestl´e Research Centre, CH-1000 Lausanne 26, Switzerland b IT Concept, Longessaigne 69770, France c SINTERFACE Technologies, Berlin 12489, Germany MPI Kolloid- und Grenzfl¨achenforschung, Potsdam/Golm 14424, Germany Received 21 July 2004; accepted 2 November 2004 Available online 21 January 2005
Abstract To determine the dilational rheology of surface layers, the profile analysis tensiometry can be used with oscillating drops or bubbles. The methodology limits for these oscillations depend on the liquids’ properties, such as density, viscosity and surface tension. For the most frequently studied water/air interface, the maximum oscillation frequency is of the order of 1 Hz, although much higher frequencies are technically feasible by the existing profile analysis tensiometers. For f > 1 Hz, deviations of the drops/bubbles from the Laplacian shape mimic non-zero dilational elasticities for the pure water/air and ethanol/air interface. For liquids of higher viscosity, the critical frequency is much lower. © 2004 Elsevier B.V. All rights reserved. Keywords: Surface rheology; Dilational viscoelasticity; Profile analysis tensiometry; Oscillating drops and bubbles; Limiting frequency
1. Introduction Dilational rheology has much impact on quite a number of technological processes, such as foaming or emulsification [1–3]. The formation process and the mechanisms of stabilisation or destabilisation are linked to the dynamic interfacial properties, such as the interfacial dilational viscoelasticity. Only recently, commercial instruments became available for measuring the viscoelasticity of adsorption layers. This made the topic quite interesting to many research groups. The only methodology accessible via commercial instruments to measure the elasticity and viscosity of adsorption layers is the drop or bubble shape analysis as described in detail recently [4]. The main principle of drop and bubble profile tensiometers is to determine the interfacial tension from the shape of drops or bubbles via the Gauss–Laplace equation. In addition, by applying transient or harmonic ∗
Corresponding author. E-mail address: [email protected] (R. Miller).
0927-7757/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2004.11.043
perturbations, this method can also supply information on the dilational rheology of interfacial layers. The Gauss–Laplace equation is derived for static conditions [5]. If a drop or bubble is growing or shrinking during the experiment its shape can deviate from an equilibrium state. The same is true for oscillating systems, i.e. when a drop or bubble is forced on purpose to oscillate in order to determine the dilational elasticity or viscosity. The technical parameters of some instruments allow to perform oscillations of a drop or bubble at various frequencies, from very slow oscillations of about 10−3 Hz to frequencies of even 10 Hz and above. At high oscillation frequencies hydrodynamic effects cause deviations of drop and bubble profiles from a Laplacian shape. Thus, high-frequency studies require other techniques, such as the oscillating bubble method, which works with small spherical bubbles [6,7] and reaches frequencies of up to 150 Hz, or the capillary wave damping techniques [8], which can work even at oscillation frequencies of up to 1000 Hz. It is the purpose of this contribution to analyse the oscillation behaviour of drops and bubbles for the water/air
26
M.E. Leser et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 261 (2005) 25–28
and oil/air systems to estimate the limiting frequency up to which this method yields correct values. For surfactant-free systems, both the elasticity and viscosity should be zero [9]. Any positive or negative values for the elasticity or viscosity would then refer to artefacts caused by a deformation rate larger than a critical value.
2. Materials and methods For the present studies two instruments were applied: the Tracker from IT Concept (France) and the PAT1 from SINTERFACE Technologies (Germany). The main elements of the method are CCD camera, frame grabber, light source, dosing system, cuvette with needle for drop/bubble formation. Both instruments are computer-controlled and use special software for fitting the Laplace equation to the drop or bubble shape coordinates. The computer-controlled dosing system was used to generate oscillations at frequencies up to about 0.5 Hz, while for higher frequencies a piezo-system was applied. The two rheological parameters elasticity and viscosity of the interfacial layer were obtained from a Fourier analysis of the measured signals. The volume of drops or bubbles was about 10 l. Although the two instruments used are based on quite different hardware and software, there were no significant differences between the results obtained for the same capillary diameter and the same samples. The studies were performed with pure water (γ = 72.0 mN/m, η = 1 cP), ethanol (γ = 22 mN/m, η = 1 cP), and their mixtures. In additional experiments drops of paraffin oil (γ = 31.4 mN/m, η = 122 cP, purchased from Sigma) and silicon oil (γ = 20.5 mN/m, η = 1000 cP, gift from Dow Corning) were studied. The volume of drops and bubbles were about 10 l, the applied volume amplitudes between 2 and 15%. At this point we want to note that the purity of the used oil samples were of minor importance as they have been studied at the liquid/air interface. Obviously, there are surfactants also adsorbing at such an interface (cf. [10]), however, blank tests have shown that there is no change in surface tension over a long period of time and hence there is no elasticity expected for these liquid surfaces.
Fig. 1. Surface tension and apparent dilational elasticity modulus E as a function of oscillation frequency for an air bubble in pure water.
Fig. 2. Surface tension and apparent dilational elasticity modulus E as a function of oscillation frequency for a drop of pure water in air.
above 1 Hz an apparent elasticity is obtained, which is mimicked by the non-ideal oscillation behaviour of the drop or bubble. Repeat experiments show that the obtained values have quite a large scattering, in contrast to low-frequency experiments. In any case, for frequencies above 1 Hz this apparent dilational elasticity is quite significant and of the order of 30 and 50 mN/m at 9 Hz. Such values typically correspond to packed surfactant adsorption layers. The generated oscillations at the studied frequencies are still harmonic, as one can see from the example shown in Fig. 3. The generated oscillation of the drop volume follows a sinusoidal function perfectly. However, the fitting of the Laplace equation to the dynamic drop or bubble shape coordinates shows that only up to a certain frequency the observed deviations along the shape are scattered normally (Fig. 4a). For higher oscillation frequencies, the deviation between the Laplace equation and the determined drop or bubble
3. Results and discussion The results of oscillation experiments with air bubbles in water and water drops in air are shown in Figs. 1 and 2, respectively. Up to a frequency of f = 1 Hz, the measured surface tension γ is independent of frequency and corresponds to the correct value, obtained for a drop in rest. For 3 and 6 Hz the value measured with the drop is still correct while that measured for the bubble is slightly increased. The surface elasticity, determined from the surface tension change during oscillations for bubbles and drops is also shown in Figs. 1 and 2. Here it becomes evident that for frequencies
Fig. 3. Volume oscillation of a water drop at 6 Hz.
M.E. Leser et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 261 (2005) 25–28
27
Fig. 6. Apparent dilational elasticity modulus as a function of oscillation frequency for drops of water (), water/ethanol 86:14 (), ethanol (), amplitude of volume oscillations 8%.
already obtained. The much larger amplitude of 8% as compared to 2% in Fig. 5 does not change the critical frequency but increases the absolute values of the apparent elasticity modulus E. When we plot the results as a function of the dimensionless parameter Ca:Re we get linear dependencies (Fig. 7). Here Ca is the capillary number and Re the Reynolds number, defined by Fig. 4. Deviation of the Laplace equation from the drop shape as a function of the points around the drop profile: (a) 1 Hz and (b) 6 Hz.
coordinates shows a special pattern, which demonstrates the systematic deviation of the dynamic shapes from the Laplace equation (Fig. 4b). The higher the generated frequency is, the more pronounced are the patterns. This is observed for drops as well as for bubbles, and no significant difference appears for the studied water/air interface weather a drop or bubble is used. In order to analyse how the absolute surface tension of a liquid influences the critical frequency for oscillation experiments, measurements were performed with a 86:14 mixture of water and ethanol (γ = 46 mN/m) and of pure ethanol (γ = 22 mN/m). The results are shown in Figs. 5 and 6. As one can see, at 2 Hz a measurable dilational elasticity modulus is
Fig. 5. Apparent dilational elasticity modulus as a function of oscillation frequency for drops of water (), water/ethanol 86:14 (), ethanol (), amplitude of volume oscillations 2%.
Ca = ηvmax /γrγ
and
Re = ρvmax /rγ η
where η is the bulk viscosity, rγ the mean radius of curvature of the drop at the apex, ρ the density of the liquid, and vmax the maximum liquid flow rate estimated for a drop oscillation with an amplitude of about 5% with respect to the volume. The reason for the rather linear dependencies remains unclear and requires a deeper analysis. Another series of experiments was performed with viscous oils. Fig. 8 shows the apparent elasticities obtained from oscillations with drops of paraffin oil and silicon oil, respectively. While the paraffin oil yields elasticity moduli comparable with ethanol, however, the critical frequency at which measurable apparent elasticities were obtained, is much lower – of the order of 0.25 Hz. The apparent elasticity moduli for
Fig. 7. Apparent dilational elasticity modulus as a function of the dimensionless parameter Re:Ca for drops of water (), water/ethanol 86:14 (), ethanol (), amplitude of volume oscillations 8%.
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M.E. Leser et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 261 (2005) 25–28
A much lower critical frequency of about 0.1 Hz is obtained for drops of a very viscous silicon oil. For liquid/liquid interfaces a more complicated situation exists which needs to be analysed in detail before the rheological behaviour of any adsorption layers at such interfaces is studied. Acknowledgement This work was financially supported by a project of the European Space Agency (FASES MAP AO-99-052). Fig. 8. Apparent dilational elasticity modulus as a function of oscillation frequency for drops of silicon oil (䊉), paraffin oil (), amplitude of volume oscillations 2%.
the studied silicon oil are much larger as compared to paraffin oil, and the critical frequency is of the order of 0.1 Hz.
4. Conclusions The studies show that for oscillations with frequencies up to 1 Hz the “dynamic” drop and bubble shapes can be well described by the Gauss–Laplace equation, when the viscosity is η ≤ 1 cP. For higher frequencies (3 Hz) the results suggest already a certain dilational elasticity modulus, which cannot exist for a pure water/air or ethanol/air interface. This apparent modulus is mimicked by deviations of the drop/bubble shape from that given by the Laplace equation. Thus, the shape analysis tensiometry is applicable for dilational rheology studies up to a frequency of 1 Hz as maximum. This holds true for drops and bubbles. For higher frequencies alternative methods have to be applied in order to measure the elasticity and viscosity of interfacial layers.
References [1] A. Williams, J.J.M. Janssen, A. Prins, Colloid. Surf. A 125 (1997) 189. [2] E. Manev, R.J. Pugh, J. Colloid Interf. Sci. 186 (1997) 493. [3] D. Langevin, Curr. Opin. Colloid Interf. Sci. 3 (1998) 600. [4] G. Loglio, P. Pandolfini, R. Miller, A.V. Makievski, F. Ravera, M. Ferrari, L. Liggieri, Drop and bubble shape analysis as tool for dilational rheology studies of interfacial layers, in: D. M¨obius, R. Miller (Eds.), Novel Methods to Study Interfacial Layers, Studies in Interface Science, vol. 11, Elsevier, Amsterdam, 2001, p. 439. [5] J. Gaydos, The Gauss Laplace equation for liquid axisymmetric menisci, in: D. M¨obius, R. Miller (Eds.), Drops and Bubbles in Interfacial Research, Studies in Interface Science, vol. 6, Elsevier, Amsterdam, 1998, p. 1. [6] K.D. Wantke, H. Fruhner, J. Colloid Interf. Sci. 237 (2001) 185. [7] V.I. Kovalchuk, J. Kr¨agel, E.V. Aksenenko, G. Loglio, L. Liggieri, Oscillating bubble and drop techniques, in: D. M¨obius, R. Miller (Eds.), Novel Methods to Study Interfacial Layers, Studies in Interface Science, vol. 11, Elsevier, Amsterdam, 2001, p. 485. [8] B.A. Noskov, G. Loglio, Colloid. Surf. A 143 (1998) 167. [9] J. Lucassen, M. van den Tempel, Chem. Eng. Sci. 27 (1972) 1283. [10] M. Mellema, J. Benjamins, Colloid. Surf. A 237 (2004) 113.