Langmuir 1997, 13, 5663-5668
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Remarks on the Interpretation of Data from the Dynamic Drop Volume Method R. Miller,*,† S. A. Zholob,‡ A. V. Makievski,†,‡ P. Joos,§ and V. B. Fainerman‡ Max-Planck-Institut fu¨ r Kolloid- und Grenzfla¨ chenforschung, 5 Rudower Chaussee, D-12489 Berlin-Adlershof, Germany, Institute of Technical Ecology, 25 Shevchenko Blvd., 340017, Donetsk, Ukraine, and Department Scheikunde, Universitaire Instelling Antwerpen, Universiteitsplein 1, B-2600 Antwerpen (Wilrijk), Belgium Received February 4, 1997. In Final Form: June 5, 1997X The maximum bubble pressure, drop volume, and ring methods were used to measure the dynamic surface tension for aqueous solutions of dodecyldimethylphosphine oxide in a wide concentration range. It was shown that for some concentrations the experimental data obtained from the drop volume method do not agree with the results from the maximum bubble pressure method. The drop volume data differ also from values predicted by theoretical models which account for surface expansion and convection transfer only. To achieve agreement between data from the drop volume method and theoretical results, the actual drop surface area and the state of this surface at the initial time moment have to be included into the boundary conditions of the theory. Conditions are defined at which a discrepancy can be expected between different experimental techniques.
Introduction The dynamic drop volume method is a rather simple experimental technique and therefore widely used for studies of dynamic interfacial tension.1-6 However, the interpretation of the measured results is rather cumbersome. Usually it is necessary to introduce the hydrodynamic corrections for short drop formation times.1,4,5,7 Also the surface expansion and the convection transfer of the liquid from the bulk towards the surface are to be accounted for.8-10 The results of the drop volume method, being expressed in terms of the effective (adsorption) time, usually agree well with the data obtained from other methods.11-14 However, it will be shown below that one cannot expect agreement in each case, because the existing theoretical models for the drop volume method do not regard the actual value and state of the surface at the initial time moment. This theoretical handicap was overcome recently for the growing drop technique.15-18 In * Corresponding author. † Max-Planck-Institut fur Kolloid- und Grenzfla ¨ chenforschung. ‡ Institute of Technical Ecology. § Universitaire Instelling Antwerpen. X Abstract published in Advance ACS Abstracts, September 15, 1997. (1) Brady, A. P.; . Brown, A. G. In Monomolecular Layers; Sobotka, H., Ed.; Symp. Philadelphia, 1951; American Association for the Advancement of Science: Washington, DC, 1954; p 33. (2) Babu, S. R. J. Colloid Interface Sci. 1987, 115, 551. (3) Miller, R.; Schano, K.-H. Colloid Polym. Sci. 1986, 264, 277. (4) Jho, C.; Burke, R. J. Colloid Interface Sci. 1983, 95, 61. (5) Van Hunsel, J.; Bleys, G.; Joos, P. J. Colloid Interface Sci. 1986, 114, 432. (6) Kloubek, J. Colloid Polym. Sci. 1975, 253, 754. (7) Miller, R.; Schano, K.-H.; Hofmann, A. Colloids Surf. A 1994, 92, 189. (8) Miller, R. Colloid Polym. Sci. 1980, 258, 179. (9) Joos, P.; Rillaerts, E. J. Colloid Interface Sci. 1981, 79, 96. (10) Kuz,V. A. Langmuir 1993, 9, 3724. (11) Bleys, G.; Joos, P. J. Phys. Chem. 1985, 89, 1027. (12) Miller, R.; Joos, P.; Fainerman, V. B. Adv. Colloid Interface Sci. 1994, 49, 249. (13) Van Hunsel, J.; Joos, P. Colloids Surf. 1987, 24, 139. (14) Henderson, D. C., Micale, F. J. J. Colloid Interface Sci. 1993, 158, 289. (15) MacLeod, C. A. Radke, C. J. J. Colloid Interface Sci. 1993, 160, 435. (16) Nagarajan, R.; Wasan, D. T. J. Colloid Interface Sci. 1993, 159, 164. (17) MacLeod, C. A.; Radke, C. J. J. Colloid Interface Sci. 1994, 166, 73. (18) Joos, P.; Van Uffelen, M. J. Colloid Interface Sci. 1995, 171, 297.
S0743-7463(97)00112-1 CCC: $14.00
this model the interfacial tension of the drop at the initial time moment was assumed to be identical to the equilibrium value. The increase of the volume due to the constant liquid influx into the drop yields an extreme in the time dependence of the interfacial tension due to the surface expansion. In this method the interfacial tension is determined by continuous measurements of the pressure in the growing drop. The aim of present study is to apply the theoretical model of growing bubbles or drops developed by Joos and Van Uffelen18 for the interpretation of the results of dynamic surface tension measurements with the drop volume method and to compare this method with the maximum bubble pressure method (MBPM) and the ring tensiometry. Theory The dynamic adsorption well-known equation of Ward and Tordai19
xDπ [c xt - ∫
Γ(t) ) 2
0
xt
0
c(0,t-τ) dxτ]
(1)
where D is the diffusion coefficient, c0 and c(0,t) are the bulk and subsurface concentrations, respectively, t is the time, and τ is a dummy integration variable. For the case of a spherical drop which grows from the central point, the following equation was derived assuming a constant liquid influx into the drop and a radial flow of liquid within the drop8
Γ(t) )
x[ x D 2c π 0
∫0(3/7)
3t - t-2/3 7
t7/3
c(0,(3/7)t7/3 -
]
τ) dxτ (2) Comparing these two equations one can easily see that eq 2 can be transformed into eq 1 if the effective time tef ) (3/7)t is introduced. It was just this substitution of the effective time instead of the physical one which was performed in refs 11-14 to compare various methods used for the measurements of dynamic surface tension. In their growing drop studies MacLeod and Radke17 introduced (19) Ward, A. F. H.; Tordai, L. J. Chem. Phys. 1946, 14, 453.
© 1997 American Chemical Society
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Miller et al.
the actual adsorption and the state of the drop surface at the initial time; however only a numerical solution was available under the specific experimental conditions. Significant progress was achieved subsequently by Joos and Van Uffelen18 who have derived an analytical solution under special boundary conditions. It was assumed that the surface area of the growing drop increases according to the power law
Ω(t) ) Ω0(1 + Rt)n
(3)
where Ω0 is the initial surface area of the drop, R and n are constants. For the case of a spherical drop which grows under a constant liquid flow, n ) 2/3. The initial surface area can vary from πr2 to 2πr2, where r is the capillary radius.20 The general expression for the dynamic adsorption has the form18
x
Γ(λ) f(t) ) Γ0 + 2
∫0
D π
xλ
[c0 - c(0,t-τ)] d(xτ)
(4)
where Γ0 is the initial adsorption, f(t) ) (1 + Rt)n, and λ ) ∫0t[f(t)]2 dt. In the growing drop experiment the initial adsorption Γo is taken to be equal to the equilibrium value Γe. For the dynamic drop volume method however we assume that Γo ) Γ, where Γ the adsorption value at the droplet lifetime t. This means that at the initial moment of a drop life is identical to that just before the drop detachment, which is true only if there is no significant deformation of the drop surface from the moment of drop instability until the break-off of the liquid bridge. This assumption is better fulfilled for narrow capillaries (cf. ref 7). A certain drop surface deformation would lead to a slightly higher initial surface tension. Equation 4 for the case of small deviations from the equilibrium was transformed into a simple analytical expression for the surface tension jump18
(1 + Rt)n - (1 - ζ)
γ - γ∞ ) ξAx(2n +1)R
x(1 + Rt)2n+1 - 1
(5)
where γ and γ∞ are the dynamic and equilibrium surface tension, respectively, ξ ) (Γ/Γe)2, ξ ) (Γe - Γ)/Γ, and
A)
RTΓ2e c0
x4Dπ
(6)
R)
[( ) ]
1 Ω(t) t Ω0
1/n
-1
(8)
The parameter A in eq 6 is equal to the derivative dγ/ dtef-1/2 for t f ∞ (Joos-Hansen approximation21). If the values of the equilibrium adsorption for t and equilibrium surface tension obey the Szyszkowski-Langmuir equations, then the parameters ξ and ζ can be expressed via the dynamic and equilibrium surface pressure, Π and Π∞, respectively:
ξ)
ζ)
[
1 - exp(-Π/RTΓ∞)
1 - exp(-Π∞/RTΓ∞)
]
2
exp(-Π/RTΓ∞) - exp(-Π∞/RTΓ∞) 1 - exp(-Π∞/RTΓ∞)
(9)
(10)
where Π∞, γ0 - γ∞, Π ) γ0 - γ, Γ∞ is the maximum adsorption, R is the gas constant, T is the absolute temperature, and γ0 is the surface tension of the solvent. Materials and Methods The dynamic surface tension measurements were performed using three different methods: MPT1 (maximum bubble pressure method), TVT1 (drop volume method), and TD1 (ring tensiometer), all manufactured by Lauda, Germany. The devices and methods were described elsewhere.12,22,23 The time range is 0.001-20 s for the MPT1 tensiometer, 1-300 s for TVT1, and 10020000 s for TD1; therefore the time ranges of the methods used partially overlap one another. The experimental results obtained by MPT1 and TVT1 devices were represented as the dependence on the effective (i.e., reduced to the non-deformed plane surface) time.12,22-24 For the drop volume method the hydrodynamic effects characteristic to short time range (cf. refs 25 and 26) were taken into account. The tip radius of the capillary employed in the drop volume method was 1.05 mm. To study the physical pattern of the drop formation processes, a high-speed video system (SPEEDCAM, Weinberger AG, Dietikon, Switzerland) was employed. The dimethyldodecylphosphine oxide (DMDPO) was supplied by Gamma-Service Dr. Schano, Berlin, Germany; special purification of the material was performed which makes it suitable for interfacial studies. Bidistilled and deionized water was used to prepare the solutions. All measurements were performed at 25 °C. Results and Discussion
From video experiments (see below) it follows that the drop area for the drop life time t is approximately equal to the sum of the surface area of the residual and detaching drops after the liquid bridge breaks. The smaller the capillary radius is, the more accurate is this approximation. Therefore, at moment of drop detachment its area is approximately equal to the value defined by the expression
In Figure 1 the basic stages of drop evolution at the tip of a capillary at a constant liquid flow rate is presented: the residual drop in the initial time moment a), intermediate stage during growing (b), in the moment, when the stability got lost (c), and just before the break of the liquid bridge, connecting the falling and residual drops. These high-speed video experiments were used to calculate the drop surface area as a function of time. This
Ω(t) ) (4π)1/3(3Vd)2/3 +Ω0 ≈ 4.83Vd2/3 +Ω0
(22) Fainerman, V. B.; Miller, R.; Joos, P. Colloid Polym. Sci. 1994, 272, 731. (23) Miller, R.; Hofmann, A.; Hartmann, R.; Schano, K.-H.; Halbig, A. Adv. Mater. 1992, 4, 370. (24) Rusanov, A. I.; Prokhorov, V. A. Interfacial Tensiometry. In Studies of Interface Science; Mo¨bius, D., Miller, R., Eds.; Elsevier: Amsterdam, 1996; Vol. 3. (25) Miller, R.; Schano, K.-H.; Hofmann, A. Colloids Surf., A 1994, 92, 189. (26) Dukhin, S. S.; Kretzschmar, G.; Miller, R. Dynamics of Adsorption at Liquid Interfaces: Theory, Experiment, Application, in Studies of Interface Science; Mo¨bius, D., Miller, R., Eds.; Elsevier: Amsterdam, 1995; Vol. 1.
(7)
where Vd is the volume of the falling drop. Therefore for drops with lifetime t the value of the constant R can be calculated from the expression (20) Pierson, F. W.; Whitaker, S. J. Colloid Interface Sci. 1976, 54, 203, 219. (21) Fainerman, V. B.; Makievski, A. V. Miller, R. Colloids Surf. A 1994, 87, 61.
Dynamic Drop Volume Method
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Figure 1. Images of the water drop formation process. Time interval between subsequent drops was 616 ms.
dependence Ω(t) for a water drop at a drop formation time of t ) 616 ms is presented in Figure 2. The time moments corresponding to the images in Figure 1 are indicated. The solid line in Figure 2 is calculated from eq 3 with n ) 2/3. As one can see, the change of drop surface area t ) 0 to the moment when the drop loses its stability (c) is satisfactorily described by this equation. Only during the stage of formation and breaking of the liquid bridge does the drop area grow faster than that predicted by eq 3. The time interval between moments (c) and (d) has been called drop detachment time td.25 The directly determined value of td from this graph coincides with that obtained from the analysis of the volumes of falling drops as a function of
time as described in refs 25 and 26. In particular, for the example presented in Figures 1 and 2, a value of td ) 0.075 s results, while for a similar capillary td ) 0.07 s was obtained in ref 256. In the theoretical model of eqs 7 and 8 used here we neglect the indicated deviations during the stage td and apply the extrapolation of eq 3 over the entire drops lifetime interval. This approximation is satisfactory in view of the relatively small difference of the measured value Ω(t) at time td in comparison with the calculations from eq 3. The results of the dynamic surface tension studies for DMDPO solutions at various concentrations are presented
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Figure 2. Surface area of the drop in time: (0) experimental data (see Figure 1); solid line, calculated from eq 3. ΩΣ is the total area of falling and residual drop; Ωf is the area of falling drop.
Figure 3. Dynamic surface tension of DMDPO solution for c0 ) 5 × 10-9 mol/cm3, measured using the MBPM (9), drop volume method (4) and ring method ([). Curve 1 was calculated from the models of Ward and Tordai and Miller; curve 2 was calculated from the Joos-Van Uffelen model.
in Figures 3-6. For high DMDPO concentrations, respectively, it is evident that the character of the experimental dependencies of γ on t obtained using the MBPM and the ring method differ drastically from those measured by the drop volume method. In particular, while for low concentrations all three methods lead to the results which agree satisfactorily with each other within the experimental error (see Figures 3 and 4), the values of dynamic surface tension for large concentrations measured using the drop volume method are significantly lower than the MBPM data (Figures 5 and 6). Let us compare these experimental data with the dependencies of γ(t) calculated according to the different existing models of Ward and Tordai, eq 1, and Miller, eq 2, and, alternatively, of Joos and Van Uffelen, eq 5. To do so, in any case we need first a model for the equilibrium state of adsorption. The surface tension and the adsorption at equilibrium for the DMDPO are satisfactorily described by the Szyszkowski and Langmuir equations
Table 1. Adsorption Characteristics of the DMDPO
Π∞ ) γ0 - γ∞ ) RTΓ∞ ln(1 + c0/a)
(11)
and
Γe ) Γ∞
c0 c0 + a
(12)
respectively, with parameters Γ∞ ) 4.1 × 10-10 mol/cm2 and a ) 5 × 10-9 mol/cm3.27,28 It is seen from Figure 7 that the values of γ∞ obtained in this study agree with those published by others.28 The diffusion coefficient calculated by the equation of Wilke and Chang,29 is found to be 4.4 × 10-6 cm2/s. The polynomial method described in refs 30 and 31 was employed for calculations of eq 1. Table 1 summarizes the adsorption characteristics of the (27) Fang, J. P.; Wantke, K.-D.; Lunkenheimer, K. J. Phys. Chem. 1995, 99, 4632. (28) Lunkenheimer, K.; Haage, K.; Miller, R. Colloids Surf. 1987, 22, 215. (29) Wilke, C. R.; Chang, P. AIChE J 1995, 1, 264. (30) Ziller, M.; Miller, R. Colloid Polym. Sci. 1986, 264, 611. (31) Miller, R.; Kretzschmar, G. Adv. Colloid Interface Sci. 1991, 37, 97.
c0, mol/cm3 5 × 10-9 10-8 2 × 10-8 5 × 10-8 10-7 2 × 10-7
(dγ/dt-1/2)tf∞, mN m-1 s1/2 γ∞, mN/m Γe, -10 2 experiment/ 10 mol/cm eq 1 eqs 5 (eq 11) (eq 12) experiment or 2 and 6 63.3/64.8 60.5/60.1 56/54.9 47.6/48.2 41.4/40.9 34.8/33.9
2.05 2.73 3.28 3.73 3.90 4.00
84 68 56.7 32.8 18.7 9.6
66 81 60 22.5 12.7 5.2
87 78 56 29 15.8 8.3
DMDPO solutions studied. The values of the derivative
(dγ/dt-1/2)tf∞ )
RTΓe2 2c0
xDπ
calculated from the experimental dependencies agree well with those both from eq 1, and from the approximate JoosHansen relationship of eq 6. This indicates a diffusioncontrolled adsorption mechanism for the DMDPO and shows that the simple approximation of eq 6 can be used without any significant error. The results of the calculations from the two theoretical models are presented in Figures 3 to 6. One can see that at low DMDPO concentrations the values calculated from eq 1, agree well with the experimental data from all three methods. Note that the experimental data obtained from MBPM and drop volume method are represented as dependencies on the effective time, which makes the Ward and Tordai model equivalent to that proposed by Miller, eq 2. For high concentrations, however, which can be seen from Figures 5 and 6, only the MBPM and the ring method show an agreement with the model described by eq 1. The results of the calculations performed according to the Joos-Van Uffelen approximate of eq 5 for the minimal concentration studied (see Figure 3) are almost identical to those from eq 1, while for large concentrations agreement is obtained only for large t values. Although the data of MBPM and drop volume method deviate from each other, the drop volume results are satisfactorily described by the calculations according to the model of eq 5. This agreement becomes better with
Dynamic Drop Volume Method
Figure 4. Same as in Figure 3 for c0 ) 10-8 mol/cm3.
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Figure 6. Same as in Figure 3 for c0 ) 2 × 10-7 mol/cm3.
Figure 7. Comparison of the surface tension isotherm of DMDPO: present study (2) and ref 28 (0).
Figure 5. Same as in Figure 3 for c0 ) 10-7 mol/cm3. Curve 3 was calculated from the Joos-Van Uffelen model for r ) 5 mm.
decreasing values of γ - γ∞, i.e. with increasing time. This is caused obviously by the approximate nature of eq 5, which was in fact derived under the assumption of small deviations from equilibrium. Thus, for large deviations from equilibrium or at small times the error made by using eq 5 is significant, and consequently one has to apply the exact eq 4. Alternatively, numerical calculations according to the model developed in ref 17 can be performed. Additional reasons for the discrepancies between the theoretical model eq 5 and the experimental drop volume data for short times are the assumption that Γ0 ) Γ and the approximate character of eq 7 for Ω(t). It is obvious that the fast expansion of the surface during the period td results to Γ0 < Γ. However, the agreement between the experimental results from drop volume experiments and the values calculated from the model of eq 5 is satisfactory. The main source for the discrepancy between the experimental and theoretical dependence of γ on t using the models of Ward and Tordai, eq 1 or Miller, eq 2, is the fact that the correct initial value of adsorption has not been used. This discrepancy is characteristic of a rather narrow range of the parameter A given in eq 6. From results for DMDPO at c0 > 5 × 10-7 mol/cm3 and the analysis of experimental results (presented here and in
previous studies11-13, 32) it follows that for A > 20 mN m-1 s1/2 (for low concentrations) and for A < 2 mN m-1 s1/2 (for high concentrations and γ f γ∞) the results of the experiments performed using the drop volume method agree well with the theory described by eqs 1 and 2. In the first case this agreement is due to small initial adsorption, while for the second case γ is close to equilibrium and therefore the results of the theoretical models eqs 1 and 2 do not differ significantly from those obtained from eq 5. The calculations according to eq 5 were performed for various capillary tip radii. For r < 1 mm the dependence γ(t) is rather insensitive to radius variation, as Ω0 amounts to less than 10% of Ω(t). For larger r the ratio of Ω0 to Ω(t) increases, and the effect of the radius becomes more significant. For example, for r ) 5 mm the ratio Ω0/Ω(t) ) 0.5. The results of the calculations performed for r ) 5 mm presented in Figure 5 show that the discrepancy between the models eqs 1 or 2 and the experimental data becomes significant. Conclusions If the initial adsorption at the surface of the residual drop in dynamic drop volume experiments is close to the value at lifetime t, the shape of the γ dependence on t can be significantly affected. In particular, for values of (dγ/ dt-1/2)tf∞ in the range from 2 to 20 mN m-1 s1/2 the discrepancy between experiment and the standard adsorption kinetics theory can be significant. Then a reasonable theoretical model has to account for the initial (32) Fang, J.-P.; Joos, P. Colloids Surf. 1992, 65, 121.
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adsorption state at the drop surface. If this is done, as shown by the relationship (5) a satisfactory agreement with experimental data is obtained. The results of the present study explain why data obtained from the drop volume method in certain cases disagree with results measured by other methods. The conditions are given under which agreement will be achieved. Apart from these conditions significant differences between methods have to be expected.
Miller et al.
Acknowledgment. The work was financially supported by projects of the European Community (INTAS 93-2463-ext and INCO IC15-CT96-0809) and the Fonds der Chemischen Industrie (RM 400429). Research grants from the Max-Planck-Institut fu¨r Kolloid- und Grenzfla¨chenforschung (V.B.F) and the Alexander von Humboldt Stiftung (A.V.M.) are also gratefully acknowledged. LA970112C
