Measurements of the Surface Elasticity in Medium Frequency Range Using the Oscillating Bubble Method

JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.

208, 34 – 48 (1998)

CS985799

Measurements of the Surface Elasticity in Medium Frequency Range Using the Oscillating Bubble Method Klaus-Dieter Wantke,1 Horst Fruhner, Jiping Fang, and Klaus Lunkenheimer Max-Planck-Institut fu¨r Kolloid- und Grenzfla¨chenforschung, Rudower Chaussee 5, D-12489 Berlin, Germany Received October 27, 1997; accepted August 11, 1998

deformation or by heating, leads not only to a new tension state but also to relaxation processes as a result of the disturbed thermodynamic equilibrium. This means the tension state is a function of its history, and the surface exhibits viscoelastic flow properties. A model describing these processes must take into account different rheological properties. However, for sinusoidal homogeneous dilatations only one modulus, the modulus of elasticity e ( f, c), can be introduced by the equation

Various experimental techniques are available for the investigation of dynamic surface tension, and a generally accepted theoretical model of these dynamics has been established. However, reliable rheological parameters of a fluid surface are very scarce. Therefore, comparisons of rheological parameters resulting from slow and faster processes or from theoretical calculations are required. In particular, a comprehensive experimental verification of the complex surface elasticity modulus which characterizes the dynamic behavior of a fluid surface in an appropriate manner is desirable. For this reason a new version of oscillating bubble method was developed which allows exact measurements of the complex elasticity modulus in the frequency range 3–500 Hz. With this method the assumptions of the theory of dynamic surface tension can be verified for medium frequencies. The new experimental results, in particular the experimental determination of the Gibbs elasticity, reveal that these assumptions are only approximately valid for faster processes. However, with a slight modification of the established model the experimental results can be explained. These experiments were carried out with solutions of tridecyl dimethyl phosphine oxide, fatty acids, n-alkanols, and triton X-100 at different surfactant concentrations. © 1998 Academic Press Key Words: adsorption layers; oscillating bubble method; oscillating barrier method; surface dilational properties; Gibbs elasticity; surface rheology.

D g 5 e ~ f, c!

The surface elasticity of surfactant solutions influences many technological processes such as foam stability, flotation, and stability of emulsions. The understanding of these processes is an important topic of colloid and surface science. This requires well-defined and reliable measuring techniques and a suitable rheological model. In the model used here, we assume that the bulk phase is an incompressible three-dimensional homogeneous fluid and that the surface can be considered as a two-dimensional compressible phase of variable composition. In a static state, the composition and the force balance are determined by the thermodynamic equilibrium between the bulk and the surface phase. An external disturbance, e.g., by To whom correspondence should be addressed.

0021-9797/98 $25.00 Copyright © 1998 by Academic Press All rights of reproduction in any form reserved.

[1]

where Dg 5 g 2 g# represents the change in the surface tension and DA/A 5 uDA/Auexp(i v t) represents the relative change in the surface area. The modulus of surface elasticity e ( f, c) is a complex function of the frequency with the bulk concentration c as a parameter. It involves elastic, viscous, and transport properties, which must be determined with the aid of a model by measurements as functions of frequency at different surfactant concentrations. Different types of experimental setups are used, and it is reasonable to ask for a comparison of their results. In addition, a generally accepted theory has been formulated by Ward and Tordai (1), as well as one for oscillating deformation by Lucassen et al. (2– 4) which allows the calculation of the surface elasticity modulus using measurements of the surface tension equilibrium isotherm. Both theories are equivalent (5, 6) from a mathematical point of view, and many theoretical calculations of the dynamic surface tension are based on these models. However, the comparison of rheological properties, like the surface elasticity, measured with different experimental setups or calculated with the aid of the isotherm equation are only discussed in depth for low frequencies (7). The justification of the extrapolation of these results to faster processes is unexplored. In addition, not many comparable values of surface rheological parameters valid for medium and higher frequencies are available in the literature (7–12), and the comparison of the results of different experimental and theoretical methods of determining surface rheological properties is required for evaluation of these methods. In particular, the as-

INTRODUCTION

1

DA , A

34

THE OSCILLATING BUBBLE METHOD

sumptions of established models also must be checked for faster processes. Therefore, the device of the oscillating bubble method was improved to provide reliable rheological parameters for medium frequencies (13). The advantages of this method are the well-defined surface adsorption states and a simple bulk flux behavior, which are important for the separation of the bulk influence. Using the new setup, different effects can be investigated such as the Gibbs elasticity, the bulk diffusion, the kinetic effects, and the intrinsic dilational viscosity of the surface. The method is very sensitive to different effects, and the frequency range allows a separation of the related parameters in most cases. For this aim the evaluation of many measurements is necessary, and not all results and aspects of such an investigation can be presented in one paper. Additionally, for a reliable quantitative description of some details, further improvement of the experimental setup is desirable. Therefore, the important effects of the dilational surface viscosity will be comprehensively discussed in another paper. The focus of the present article is the verification of the Lucassen/van den Tempel modulus (Lucassen/v. d. T. modulus) in the frequency range up to 500 Hz. Because of the great importance of the parameters of this model as inputs for numerous theoretical calculations, such independent experimental verifications in a broader frequency range are required. A simple way to realize this aim should be demonstrated. For this investigation, mainly surfactant solutions are chosen with surface elasticities which exhibit the typical slope of the Lucassen/v. d. T. modulus. This means the dynamic surface tension is only a function of the surface concentration. With the results of these measurements, experimental values of the Gibbs elasticity and the diffusion relaxation time can be determined. EXPERIMENTAL METHODS AND MATERIALS

35

difficult for a complex dynamic process due to the ambiguous interpretation of fit results in many cases. Therefore, the results of such experiments, e.g., of bubble pressure or oscillating jet experiments, are quite often only characterized by the dynamic surface tension and not by a modulus. However, a modulus e ( f, c) describes the response behavior of the local surface area on deformation in a general form and, although this modulus does not only include local properties (32), it represents a suitable input function for the theories of complex dynamic processes at the interface. The most frequently used techniques in this context are the oscillating barrier method for low frequencies (2, 3, 14, 15, 33–39) and the devices based on excitation of transversal (4, 10 –12, 15, 16, 40 – 45) and thermally excited waves for higher frequencies (17–19, 46). The oscillating barrier method gives reliable results; however, the frequency range is not sufficient for a complete verification of the assumptions of the theoretical models. The main problem in using transversal waves is the elimination of the influence of the bulk phase which requires a complicated model and high accuracy of measurements. Nevertheless, they give reliable results for low concentrated solutions (10 –12, 15). However, for more highly concentrated solutions the oscillating bubble method leads to much better results in the medium-frequency range because the simple flux behavior allows an easy elimination of the bulk influence. For this reason, some groups have improved the oscillating bubble method (13, 20, 21, 47–55). This method enables the determination of e ( f, c) in a broader frequency range with little effort. In addition, the method is sensitive to the changing of frequency and concentration, and therefore one can investigate special details of the dynamics at a fluid surface. In a first series of experiments we compare the bubble measurements with theoretical calculations using isotherm equations and with results of the classical oscillating barrier method.

Description of the Problems

The Oscillating Bubble Method

Only experiments with a harmonic change of area and no relaxation experiments are discussed in this paper. The modulus e ( f, c) defined by Eq. [1] is the appropriate function to characterize the results of such measurements. It includes different effects, and the various influences can be easily separated if this modulus is known in a broad frequency range. The complete knowledge of the response function of a sinusoidal surface deformation in such a range enables a simple, reliable verification of the established models of the dynamic surface tension. In the literature one finds only selective experimental data (in most cases for a single frequency) about the function e ( f, c) in the frequency range . 1 Hz (7–25). Nevertheless, much theoretical work was done to explain complex dynamic processes at an interface on the basis of this function using special assumptions (7, 26 –31). The results of these models were verified by experiments. However, the separation of the parameters related to the various effects remains

As mentioned previously, the aim of the development of the new version of the oscillating bubble method was the direct measurement of the modulus e ( f, c) in a broader frequency range. A device based on the oscillation of an incomplete bubble is easily put into practice. Two different principles of this method are known: first, the excitation via an air chamber and second, the excitation via a fluid chamber. The first version was built by Lunkenheimer (56 –59) according to the theoretical outlines proposed by Wantke (56 –59), and later by Johnson and Stebe (20, 47, 48), as well by Karapantsios and Palmer (55). Examples for the second version are the method of Enhorning (49, 60) and the setup of Fruhner (13, 54). New setups of the method of Enhorning were used by Chang and Franses (21, 50, 51) and Notter et al. (61, 62). Similar experiments are described by MacLeod and Radke (52) and Nagarajan and Wasan (53). Comparative considerations to these methods are given in (7, 54).

36

WANTKE ET AL.

Lu and Apfel (22, 23) and Tian et al. (24, 25) investigated the damping oscillations of a complete droplet built by a surfactant solution. The absence of a liquid–solid contact line is the main advantage of all methods using complete drops or bubbles. The disadvantage results from the difficulties in obtaining correct information about the pressure within the bulk phases. According to our experiments a sensitive pressure measurement is very important for reliable results. Using oscillating droplets, medium frequencies can be reached (200 Hz in (25)); however, the values of the measured Gibbs elasticity are very small. These and other examples demonstrate that the experimental determination of the complex surface elasticity represents an actual problem. For such measurements a broad frequency range and a small amplitude of oscillation are desirable. This was realized by Lunkenheimer (frequency range: 3 Hz , f , 150 Hz) and Fruhner (frequency range: 3 Hz , f , 500 Hz). In both cases calibration measurements were used for the elimination of the influence of the bulk phase. Therefore, the final equations for evaluating the measurements have a simple form. Johnson and Stebe replaced the calibration measurement by a theoretical calculation which was possible for their setup because they used bubbles of an approximately complete spherical shape and a low frequency range (0.001 Hz , f , 4 Hz). The method of Enhorning was developed for the evaluation of lung surfactant and requires a small frequency range (0.01 Hz , f , 3 Hz) and a greater magnitude of oscillation (49 –51, 60). This leads to higher harmonic terms of the basic oscillation frequency, and therefore, the comparison of these results with the results of other methods is difficult. Using this method, Chang and Franses (21, 50, 51) systematically investigated the response of the dynamic surface tension on a harmonic deformation. The method of Fruhner (13) was specially developed for the investigation of dilational properties of fluid surfaces in a medium-frequency range. It allows a simple determination of the complex elasticity modulus and a systematic study of the influence of surfactants on this modulus. Therefore, a comprehensive experimental verification of the theoretical model is possible using this apparatus. The principle of this newly designed oscillating bubble experiment is shown in Fig. 1. Within a closed measuring chamber a small hemispherical bubble is produced at the tip of a capillary. A piezoelectric driver generates sinusoidal oscillations of the bubble volume and consequently produces changes in the surface area and the radius. Changes in the bubble radius and surface area produce sinusoidal changes of the pressure in the measuring chamber, and these pressure changes are monitored by a sensitive pressure transducer which is mounted at the bottom of the chamber. The capillary used has a diameter of about 0.05 or 0.035 cm. A sharp edge at the tip of the capillary and special wetting properties are very important for the stability of the three-phase contact line. With this system

FIG. 1. Oscillating bubble setup for measuring the dilational properties of adsorption layers as a function of frequency: 1, cap with a capillary; 2, measuring chamber; 3, solution; 4, piezoelectric driver; 5, low-pressure quartz transducer; 6, amplifier and measuring instruments for signal amplitudes and phase angles; 7, frequency generator.

one can obtain information on the pumped volume, the change in area, the change in pressure, and the phase angles between these oscillations. Using these measurements, one can easily determine the surface elasticity values, as is demonstrated in Appendix A. For frequencies f . 400 Hz ( f . 500 Hz for a capillary diameter of d 5 0.035 cm) the higher harmonic oscillation of our system must be taken into account. In the low-frequency range the mechanical behavior of the system is stable until 0.1 Hz, although at the moment our electronic equipment gives useful results only for frequencies . 3 Hz. This situation will be improved in the future by using another electronic device which enables measurements down to 0.1 Hz. The influence of the kinetic effects on the complex elasticity values can then be studied in detail (47, 48, 54). The theoretical model describing the response of sinusoidal deformation of a plane fluid surface is well known. It leads to the Lucassen/v. d. T. modulus for the quotient of the change in surface tension and the relative expansion rate (2, 3). The formula is verified for low frequencies by experiments (2, 3, 7). In the theory of the oscillating bubble experiments, this modulus e ( f, c) is not really established (20, 21, 47–52). Various final equations are used for the evaluation of the experimental results; however, only some simple mathematical steps are necessary to transform the different equations into the standard form. Therefore, we complete our experimental consideration in a theoretical appendix (Appendix A), where the influence of the Lucassen/v. d. T. modulus and its modifications on the pressure balance at a spherical surface are demonstrated. Although most individual steps of this consideration are known, the present combination is yet missing. The understanding of Appendix A is necessary for the practical application of the method and the interpretation of the results. It allows a simple evaluation of the oscillating bubble measurements through the introduction of the complex surface elasticity modulus e ( f, c) in the pressure balance equation. The modifications of the

37

THE OSCILLATING BUBBLE METHOD

standard form of the elasticity modulus (Lucassen/v. d. T. modulus) are required for the interpretation of various observed experimental effects which will be comprehensively discussed in the future. With the oscillating bubble method, the amplitude, DA, of the deformation of the bubble surface caused by the motion of the piston is obtained, as well as the amplitude of the change in pressure Dp and the phase shift w between them. The influence of the bulk phase on the pressure measurements must be eliminated for the study of rheological properties of the surface. This is a standard problem of the surface rheology as long as the surface tension state is not measured directly. The formulation of the pressure balance equation and the elimination of the bulk influence are very simple for the oscillation of an ideal spherical surface (Appendix A). This is the main advantage of this method, compared to other experimental setups where the elimination procedure can lead to a complicated theoretical problem due to complicated flow behavior (7, 26 –31, 40 – 46). Then, the experimental verification of the theoretical model becomes uncertain, in particular for faster deformations of higher concentrated solutions. In the case of the oscillating bubble method, this problem is reduced upon determination of the calibration function F(t) and correction function g( f, c) (Appendix A). For pure water ( e ( f, c) [ 0, g( f, c) [ 0, and g 5 g0 5 const.) Eq. [A12] leads to Dp 1 5 F( f )DA/A. This means the calibration function F( f ) results from the measurement, Dp 1 , of dynamic pressure using pure water. The correction function g( f, c) has to be determined by independent measurements of the static surface tension and the bulk viscosity. The assumption of an ideal spherical geometry with a fixed center cannot be fulfilled by real experimental equipment. For an incomplete sphere Eq. [A1] is not exactly valid, and the shape of the bubble deviates slightly from an oscillating sphere. However, the force balance must be fulfilled at all surface areas, and the surface of the bubble at the bottom has the shape of an incomplete sphere with a mobile center. At varying surface areas of the bubble the contributions of the bulk and of the surface to the force balance are changed; however, these two changes compensate each other, and we must only consider the area at the bottom. The Navier–Stokes equation describing such flux behavior does not have the simple form of Eq. [A4], but the difference between the pressure at the bottom of the bubble and at the pressure transducer is given by an integral over the pressure gradient using any path. The influence of the bulk flow, represented by this integration function, can be eliminated by a calibration measurement as long as the flow within the bulk remains the same. This is fulfilled when the change in surface tension only causes an additional homogeneous bulk pressure and no change of the flow behavior by the boundary condition. One can determine the conditions for exact experimental results through systematic investigation of the calibration measurements and by optical control of the shape of the bubble.

FIG. 2. Calculated dynamic surface tension of tridecyl dimethyl phosphine oxide solutions: (■) c 5 5 3 1026 M, (F) c 5 1 3 1025 M, (Œ) c 5 2 3 1025 M, (}) c 5 5 3 1025 M.

The Oscillating Barrier Method As long as our method fails for frequencies , 3 Hz, we have to use the oscillating barrier method (2, 3, 33–39) to measure the surface elasticity in low-frequency range. This is a wellestablished technique. Basically, it is a modified Langmuir trough with two symmetrically oscillating barriers and a Wilhelmy plate for measuring the surface tension. The Langmuir trough is filled with the surfactant solution. The adsorption layer in its equilibrium state is expanded and compressed sinusoidally with a small amplitude in the frequency range 0.005– 0.5 Hz. Therefore, a constant level of the elasticity can be reached only for very low concentrated solutions, and the method is of limited applicability for verification of the value of the Gibbs elasticity. For reliable results, the measurements must commence from the adsorption equilibrium. As Fig. 2 indicates, different adsorption times are needed to reach the equilibrium of tridecyl dimethyl phosphine oxide solutions at different concentrations. Similar procedures were carried out with the other solutions. Surface active impurities of the materials used were carefully removed by an automatic purification apparatus. The trough and barriers which were made of Teflon were cleaned by sulfochromic acid and next rinsed in tap and distilled water. All experiments were carried out at 22 6 1°C. RESULTS

Some General Remarks on the Experimental Results If the experimental values of DA/A and Dp are known, then Eq. [A12] yields the complex elasticity modulus e ( f, c). In general, this modulus includes elastic and viscous effects of the surface, bulk diffusion, and kinetic effects. The particular form of the function e ( f, c) depends on the molecular exchange dynamics at the surface, which is demonstrated in Appendix A.

38

WANTKE ET AL.

The experimental e ( f, c) curve of a solution exhibits in most cases the dominant mechanism directly. According to the aim of this paper, we mainly consider examples without kinetic and viscous effects (h( f, c) [ 1 and k9 [ 0). Then, Eq. [A30] is reduced to the Lucassen/v. d. T. modulus. In this model it is assumed that after deformation, thermodynamic equilibrium is instantaneously established between surface and subsurface, and dynamic surface tension is only a function of the current surface concentration. The surface elasticity of many solutions also exhibits such behavior in the frequency range 3 Hz , f , 500 Hz. Two other examples are mentioned here. Their physical interpretation requires a comprehensive analysis which will be given in another paper; however, neglecting them completely leads to a wrong impression of the results. As mentioned above, with h( f, c) [ 1 and k9 [ 0, Eq. [A30] obtains the form of the Lucassen/v. d. T. modulus (2, 3)

e ~ f, c! 5 E~ f, c!exp~i w ~ f, c!! 5 e 0

1 1 z 1 iz , 1 1 2z 1 2z2

z5

Î

v0 , 2v

[2]

with the magnitude E~ f, c! 5

e0

Î1 1 2 z 1 2 z 2

.

[3]

The parameters which determine the curves are the Gibbs elasticity

e0 5 2

dg d g dc 5 2G d ln G dc dG

[4]

and the molecular exchange parameter

v 0 5 D~dc/dG! 2

[5]

(or the diffusion relaxation time t0 5 1/v0). These two parameters have in a direct or modified form (as derivative of a isotherm equation) a decisive influence on the results of many theoretical calculations of dynamic surface tension. Both parameters represent contact points between the mathematical hypothesis and the physical reality. Therefore, their independent experimental verification is important for the theory of many dynamic processes at surfaces. In particular, the exactness of the parameter dc/dG is a very critical point in all calculations (54), which shall be demonstrated in the following. From a purely mathematical point of view, Eqs. [2]–[5] are completely determined by the equilibrium isotherm equation. However, the derivation includes some assumptions and nu-

merical problems. Verification of these preconditions of the theory of dynamic surface tension requires the knowledge of the experimental values of e ( f, c) until their constant highfrequency limit e m (c). With the frequency range 3 Hz , f , 500 Hz, this limit can be reached in many cases. The possibilities for verification of the critical problems by experiments in low-frequency range are limited. For 2 p f ! v 0 , Eq. [2] leads to the approximation (2, 36)

e ~ f, c! 5 e 0~1 1 i!

dG dc

Î

pf G 2RT 5 ~1 1 i! D c

Î

pf . D

[6]

This corresponds to the formula of Hansen and Joos describing dynamic surface tension in the long time range (7, 63, 64). Here the critical parameter dc/dG is eliminated. Therefore, in this frequency range the experimental determination of e0 and v0 is impossible even if the experimental results show good agreement with the theoretical calculations. It is also difficult to separate the influence of e0 from the influence of v0 by a fit procedure as long as the constant level of the elasticity values with a vanishing phase angle is not reached. This is the main drawback of methods using only slow deformation rates. One needs measurements up to the high-frequency limit to verify the critical parameter dc/dG (or e0) of the theory of dynamic surface tension. The high-frequency limit directly gives the experimental value of the Gibbs elasticity e m , because there the function z becomes very small and loses its influence in Eqs. [2] and [3]. With this result the parameter v0 can be determined using the lower part of the elasticity curve. It is an advantage for the fit procedure of the elasticity functions using a broader frequency range that the single parameters of the theory have a different influence on the experimental results in various ranges. Comparison of the high-frequency limit e m (c) with the Gibbs elasticities e 0 (c) determined with the aid of the isotherm equation exhibits unacceptably large differences. Comparison of v0 and v m gives similar results. Such discrepancies have been found by other authors, too (4, 10 –12, 24). For the explanation of this effect we must consider some details of the theory (Appendix A). Both parameters, e 0 (c) and v0, include the differential quotient dc/dG. This differential quotient is a scale of the disturbance of the thermodynamic equilibrium within the bulk phase by a change in surface concentration according to the relation Dc 5 const.*DG. Using the Gibbs adsorption equation G52

c dg , RT dc

[7]

we obtain with Eq. [4] for the theoretical Gibbs elasticity

e 0 5 2G

S D YS

dg dc d g 5 2c dG dc dc

2

D

dg d 2g 1c 2 . dc dc

[8]

39

THE OSCILLATING BUBBLE METHOD

diffusion wave [A24]. The concentrations G v and G are different; however, a relation G( f, t) 5 G(G v ( f, t)) is established for a harmonic deformation independent of the thermodynamic state of the surface phase (equilibrium or nonequilibrium state). If DG v , and not DG, determines the bulk diffusion process, then, in the boundary condition Eq. [A18], the relation between the isotherm equation and the change of concentration, DG v , can be described with the aid of the function q~ f, c! 5

d ln G , d ln G v

[9]

and Eqs. [4] and [A17] must be substituted by

e ~ f, c! 5 FIG. 3. Isotherm of the equilibrium surface tension of tridecyl dimethyl phosphine oxide solutions: (■) experimental values; (—) fit results of the Langmuir model.

This means e0 and v0 are determined by the first and second derivative of the isotherm equation resulting from a fit procedure. In general, the difference between the mathematical idealization and the real physical process increases with further differentiation of such a function, and so, after the second derivation, the error in Eq. [8] is large. In addition, in the linear range of the g-log(c) isotherm near the CMC, the surface concentration G is approximately constant, according to Eq. [7]. Therefore, the value of dc/dG calculated with the help of the isotherm equation increases unrealistically. These remarks can be easily verified by determinations of e0 and v0 using different mathematical approximations of the same isotherm measurements. This is possibly the reason for the discrepancy between experimental and theoretical values. Additionally, in order to compare the experiments with the theory, the differential quotient in Eq. [4] must be replaced by the quotient of differences Dg/DG, where the value of DG/G remains smaller than 0.1 in our cases. Another possible explanation for the differences between the experimental values e m and v m and the theoretical values e0 and v0 is the following: All calculations are based on the assumption that surface tension is only a function of surface concentration and this concentration is determined by the Gibbs adsorption equation. Additionally, the boundary condition Eq. [A18] requires that the surface molecules be located within a monolayer. This means that only a monolayer has an influence on the surface tension. If these assumptions are not fulfilled, in particular for dynamic conditions, the model must be modified. The simplest way is to introduce a volume model with a fixed thickness of the surface phase like the Guggenheim model (65). The surface concentration G v of such a model can be defined by the ratio G v 5 N/A (N, number of molecules in the surface volume). The thickness of the surface volume should be small compared to the wavelength 2 p /k of the

d g d ln G d ln G v d ln G d ln G v d ln A

[10]

and

e 0 5 2q~ f 3 `, c!

dg . d ln G

[11]

All final equations must be invariant with respect to a variation of the thickness of the surface phase. For the determination of the correction functions, h( f, c) and q( f, c), by fit procedures we need very exact bubble or similar experiments in different frequency ranges. At the moment insufficient experiments are available for a detailed discussion, and it is not yet clear whether the differences between the theoretical and the experimental values of the Gibbs elasticity near the CMC are an effect of the mathematical procedure as discussed above or an effect of the physical model which neglects the influence of the thickness of the interface on the dynamic surface tension. This problem is not significant for slow processes as a consequence of the asymptotic Eq. [6]. However, it is an experimental fact resulting from measurements in a medium-frequency range, and an explanation is required. Discussion of the Experimental Results To demonstrate the ability of the oscillating bubble method we have compared the experimental and theoretical elasticity values of the following surfactant solutions: tridecyl dimethyl phosphine oxide, n-decanoic acid, n-heptanoic acid, n-nonanol, n-octanol, n-heptanol, n-hexanol, and triton X-100. These are typical examples of soluble surfactants where the constant level of the surface elasticity cannot be reached with the oscillating barrier method for higher concentrated solutions. However, the frequency range of the oscillating bubble method is large enough to get this limit. The ascent of the curves (Figs. 5–9) depends on the diffusion relaxation time t0 5 1/v0 which describes the influence of the bulk diffusion. A relaxation process at the surface becomes faster with increasing

40

WANTKE ET AL.

TABLE 1 Parameters of Elasticity Curves of Selected Surfactant Solutions

Subs. Tridecyl dimethyl phosphine oxide

Decanoic acid Heptanoic acid

Nonanol

Octanol

Heptanol

Hexanol

Triton X-100

c (mol dm23)

e0 (mN/m)

em (mN/m)

5 3 1026 1 3 1025 2 3 1025 5 3 1025 5 3 1025 1 3 1024 2 3 1024 5 3 1024 1 3 1023 2 3 1023 5 3 1025 1 3 1024 2 3 1024 3 3 1024 5 3 1024 1 3 1024 3 3 1024 6 3 1024 1 3 1023 2 3 1023 4 3 1024 7 3 1024 1 3 1023 2 3 1023 4 3 1023 6 3 1023 1 3 1023 2 3 1023 5 3 1023 1 3 1022 1 3 1025 5 3 1025 1.5 3 1024 2.5 3 1024

29.7 59.3 118.7 296.7 31.1 113.3 0.4 2.8 7.7 19.2 5.9 46.2 201 355.1 664 1.1 9.0 35.6 84.1 217 2.9 6.6 10.7 25.7 57.4 88.5 2.2 6.0 24.2 63.2 67.1 335 1005 1645

32.5 51 53 53 37.5 47.5 4.5 6.5 10.0 11.1 8.5 16.3 27 30.7 31.3 4.4 13 16.8 19.8 21.5 6.5 10.1 11.0 14.5 17.7 20.1 2.4 4.0 5.9 6.2 39.7 48 51.1 55.1

concentration. The dynamic behavior of the chosen solutions is known in principle, but the quantitative values of the parameters are uncertain. Measurements using the oscillating bubble method can help improve the situation. The statistical error of the method is small (uDEu ; 1 mN/m and uDwu ; 3° for 10 Hz , f , 400 Hz) as long as the mean bubble radius is constant. However, the need to adjust a new radius after a solution exchange caused a greater systematic error (uDE9u , 5 mN/m). For tridecyl dimethyl phosphine oxide the results are demonstrated with all details (Figs. 2– 6). In the other cases only essential information is shown (in Fig. 7 for decanoic acid, in Figs. 8 and 10 for n-alkanols, and in Fig. 9 for triton X-100). The equilibrium surface tension isotherm g (c) of the phosphine oxide ~ g ~c! 5 72.4 2 RT G `ln~1 1 c/a! mN/m, G ` 5 4.7 3 10 210 mol cm22, a 5 1.93 3 10 26 mol dm23)

v0 (s21) 0.00178 0.094 0.27 1.46 0.46 13.89 112 143 194 668 0.56 2.7 115.8 744 6796 1.6 4.6 46.2 421 806 106 119 167 627 4901 61720 659 721 3630 42140 0.50 219 16664 126960

vm (s21)

k9 (mN s/m)

0.0036 0.009 1.40 44.2 0.3 1.11 57.2 45.5 74.4 59.4 2.2 1.4 1.3 2.1 3.7 2.5 10 25 70 90 7 5.5 30 35 112 141 43 21 23 53 0.23 11 210 865

0 0 0 0.0001 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0015 0.0055 — — — — 0 0 0.00052 0.0024

is illustrated in Fig. 3. The comparisons between the experimental and the theoretical Gibbs elasticity ( e m 5 E( f 3 `, c) and e0) are demonstrated in Table 1, Fig. 4 (tridecyl dimethyl phosphine oxide), and Fig. 10 (n-alkanols). The frequency dependence of the elasticity modulus of n-heptanol and triton X-100 is plotted in Figs. 8 and 9. The missing frequency functions show the normal behavior according to Eq. [3] with the parameters e m , v m of Table 1. The isotherm equations of the fatty acids and the n-alkanols can be found in (66), and those of triton X-100 can be found in (67). The difference between the parameters e m , v m , resulting from the bubble measurements by fitting, and e0, v0, calculated with the aid of the equilibrium isotherm equation is obvious and cannot be ignored, in particular for higher concentrations. This corresponds to measurements using the transversal waves techniques (4, 10 –12), where the very high elasticity values resulting from the calculations near the CMC were never

THE OSCILLATING BUBBLE METHOD

FIG. 4. Comparision between Gibbs elasticity e0 (—, differentiation of the isotherm equation) and high-frequency limit of experimental elasticity values e m (F) of tridecyl dimethyl phosphine oxide solutions.

found. Measurements with the older version of the oscillating bubble method showed similar effects with a slightly higher magnitude. However, a systematic investigation of this problem was missing there (56 –59). The possible reasons for the difference between the elasticity measurement, e m 5 E( f 3 `, c), and the calculated Gibbs elasticity e0 in a higher concentration range are discussed above. The errors of the numerical procedure and the assumptions of the model are critical points in the theory. At the moment not enough measurements are available for a detailed interpretation of these effects, and we have to consider the

FIG. 5. Magnitude E( f ) of the surface elasticity of tridecyl dimethyl phosphine oxide solutions: 1, c 5 5 3 1026 M. (h) oscillating bubble experiments; (■) oscillating barrier experiments; (—) according to Eq. [3] with parameters of Table 1, columns 4 and 6 ( e 0 5 e m , e 0 5 v m ). 2, c 5 1 3 1025 M. (h) oscillating bubble experiments; (Œ) oscillating barrier experiments; (—) according to Eq. [3] with parameters of Table 1, columns 4 and 6 ( e 0 5 e m , v 0 5 v m ).

41

FIG. 6. Magnitude E( f ) of the surface elasticity of tridecyl dimethyl phophine oxide solutions: 1, c 5 2 3 1025 M. (‚) oscillating bubble experiments; (Œ) oscillating barrier experiments; (—) according to Eq. [3] with parameters of Table 1, columns 4 and 6 ( e 0 5 e m , v 0 5 v m ). 2, c 5 5 3 1025 M. (E) oscillating bubble experiments; (F) oscillating barrier experiments; (—) according to Eq. [3] with the parameters of Table 1, columns 4 and 6 ( e 0 5 e m , v 0 5 v m ).

parameters e m , v m for faster processes to be adjustable. The influence of a realistic variation of the diffusion coeffient (between D 5 2 3 1026 and D 5 9 3 1026 cm2 s21) is also too small for an explanation of the discrepancy. Therefore, we have chosen the value D 5 5 3 1026 cm2 s21 for the theoretical calculations and have interpreted the real, unknown diffusion coefficient as part of the adjustable parameter v m . The basic equation of the model then remains valid, as Figs.

FIG. 7. Magnitude E( f ) of the surface elasticity of decanoic acid solutions in 5 3 1023 M HCl: 1, c 5 5 3 1025 M. (E) oscillating bubble experiments; (F) oscillating barrier experiments; (—) according to Eq. [3] with parameters of Table 2, columns 4 and 6 ( e 0 5 e m , v 0 5 v m ). 2, c 5 1 3 1024 M. (‚) oscillating bubble experiments; (Œ) oscillating barrier experiments; (—) according to Eq. [3] with parameters of Table 1, columns 4 and 6 ( e 0 5 e m , v 0 5 v m ).

42

WANTKE ET AL.

TABLE 2 Calculated and Experimental Phase Angles (w 5 wt and w 5 wm) of the Elasticity Modulus of Tridecyl Dimethyl Phosphine Oxide Solutions in Degree f (s21):

0.005

0.01

0.025

0.05

0.1

10

w t of 5 3 1026 mol dm23: w m of 5 3 1026 mol dm23: w t of 1 3 1025 mol dm23: w m of 1 3 1025 mol dm23: w t of 2 3 1025 mol dm23: w m of 2 3 1025 mol dm23: w t of 5 3 1025 mol dm23: w m of 5 3 1025 mol dm23:

15.92 21 34.45 29.5 38.4 33.4 39.63 37.0

12.41 21.5 31.26 27.5 36.05 31.0 37.70 29.0

8.61 6.5 26.30 15.5 32.15 25.0 34.33 28.0

6.39 — 22.23 — 28.59 — 31.12 —

5.34 — 19.83 — 24.63 — 27.00 —

,1 ,5 2.7 ,5 4.46 ,5 5.55 11

5–7 demonstrate. Here, the magnitude of the elasticity E( f, c) measured with the oscillating bubble method and the oscillating barrier method is compared with fit results using Eq. [3]. These figures illustrate that the Lucassen/v. d. T. modulus also describes the frequency behavior for higher frequencies. The magnitude of the complex surface elasticity E( f, c) has the theoretical slope given by Eq. [3] with a constant level for high frequencies and a vanishing limit for f 3 0. However, we have used the parameters e m , v m of the fit procedure for these calculations instead e0, v0. For higher frequencies (100 Hz , f , 500 Hz), no change of the elasticity level was found with these solutions. Figures 5–7 also show that by using the oscillating barrier method the high-frequency limit can be reached only for very low concentrations, and there is better agreement between measured and calculated values of elasticity. The replacement of the bubble measurements by barrier measurements at low frequencies is only an interim step. By the formal extrapolation to lower frequencies using the bubble measurements within the small frequency range 3 , f , 5 Hz, we do not obtain the results of the barrier method. The shape of the curve of the bubble measurements within this range is a hint at the increasing error of our electric equipment for low

frequencies ( f 3 0). The present examples demonstrate only that bubble measurements are consistent with the results of the barrier technique if we exclude this critical range. A comprehensive comparison of both methods requires more exact measurements in the low-frequency range. At the moment, measurements of the elasticity values with an error of uDEu , 5 mN/m are not possible in the frequency range 0.1 Hz , f , 5 Hz using our bubble equipment. However, this range is important for the separation of the diffusion effects from the kinetic effects. The barrier method alone is not a sufficient experimental basis for the solution of this problem and, therefore, we will improve the bubble device. In particular, an exact determination of the phase angle is required, for low as well as for higher frequencies. The phase angles were measured in all cases as discussed in this paper. The determination of phase angles using our barrier equipment is not exact enough for the verification of a model. They exhibit only the expected tendency (Table 2). In the higherfrequency range ( f . 10 Hz) the phase angles measured with the bubble devices were small (,3°) for the presented solutions of decanoic acid, heptanoic acid, n-nonanol, and noctanol. This agrees with the theory because the phase angle

FIG. 8. Magnitude E( f ) of the surface elasticity of n-heptanol solutions using the oscillating bubble experiments: (3) c 5 4 3 1024 M, (}) c 5 1 3 1023 M, (Œ) c 5 4 3 1023 M, (■) c 5 6 3 1023 M; (—) according to Eq. [3] with parameters of Table 1, columns 4 and 6 ( e 0 5 e m , v 0 5 v m ).

FIG. 9. Magnitude E( f ) of the surface elasticity of triton X-100 solutions using the oscillating bubble experiments: (3) c 5 1 3 1025 M, (}) c 5 5 3 1025 M, (Œ) c 5 1.5 3 1024 M, (■) c 5 2.5 3 1024 M; (—) according to Eq. [3] with parameters of Table 1, columns 4 and 6 ( e 0 5 e m , v 0 5 v m ).

43

THE OSCILLATING BUBBLE METHOD

TABLE 3 Calculated and Experimental Phase Angles (w 5 wt and w 5 wm) of the Elasticity Modulus of n-Heptanol and Triton X-100 Solutions in Degree f (s21):

10

40

80

150

250

350

8.9 17 16.8 24 24.1 30

6.8 13 16.5 23 28.7 33

5.4 9 17.8 22 35.2 33

3.8 8 19.5 23 41.0 35

0.9 1 5.4 6 17.7 21 27.8 36

0.6 0 4.0 3 14.8 18 25.7 30

0.5 0 3.2 2 12.8 15 24.7 27

0.4 0 2.7 0 11.7 15 24.6 27

n-Heptanol 23

23

25

23

w t of 2 3 10 mol dm : w m of 2 3 1023 mol dm23: w t of 4 3 1023 mol dm23: w m of 4 3 1023 mol dm23: w t of 6 3 1023 mol dm23: w m of 6 3 1023 mol dm23:

19 16 26 22 27.7 24

11.9 19 18.9 25 22.8 27 Triton X-100

w t of 1 3 10 mol dm : w m of 1 3 1025 mol dm23: w t of 5 3 1025 mol dm23: w m of 5 3 1025 mol dm23: w t of 1.5 3 1024 mol dm23: w m of 1.5 3 1024 mol dm23: w t of 2.5 3 1024 mol dm23: w m of 2.5 3 1024 mol dm23:

2.4 2 12.9 12 29.4 26 36.0 46

1.2 1.5 7.3 9 21.6 24 30.5 42

must vanish in the level area. The higher concentrated solutions of n-heptanol and triton X-100 are examples for a more complex behavior of the surface. They are only mentioned here to demonstrate the limitations of the model. Figures 8 and 9 show the magnitude, and Table 3 shows the phase angle of the elasticity moduli for these examples. The behavior of higher concentrated hexanol solutions (c . 1022 mol dm23) is similar. These solutions do not have an extended level area with vanishing phase angle, but they show an increase of the magnitude and sometimes of the phase angles for higher frequencies above a small level range. Such characteristic behavior of e ( f, c) is an indication of the influence of an intrinsic dilational surface viscosity, and Eq. [A32] must be used for the evaluation of the results. For this reason Table 1 includes a column with the dilational surface viscosity k9. The interpretation of the viscous behavior of a fluid surface is a controversial topic. The problem is the separation of the intrinsic effects from the influence of bulk diffusion (54, 68). Both processes cause a phase angle; however, their frequency dependence is different (Appendix A). In the framework of a pure rheological formalism the different interpretation is not relevant, and some authors avoid discussion of this problem (49, 69). However, the dynamics of more complex systems like foams depend on the character of the surface processes, and a systematical investigation of the surface viscous effects is required. The oscillating bubble method is recommended for these studies because it is very sensitive to parameter changes. It allows the investigation of various molecular exchange mechanism and their influence on the dynamic surface tension by experiments on the basis of the general equation [A32].

SUMMARY

The surface elasticity modulus was investigated using a new version of the oscillating bubble method in the frequency range 3–500 Hz. This apparatus allows the experimental verification of the theory of dynamic surface tension for faster processes. Such comparisons are required for the determination of reliable surface rheological parameters which are needed for the investigation of complex fluid systems like foams. The newly developed oscillating bubble apparatus allows the experimental determination of the high-frequency limit of the surface elasticity for the complete concentration range of many surfactant solutions. The results exhibit that for faster processes the Gibbs elasticity e0 and the

FIG. 10. Comparison between Gibbs elasticity e0 (lines) and high-frequency limit of experimental elasticity values e m (symbols): 1, n-nonanol solutions (—, ■); 2, n-octanol solutions (—, F); 3, n-heptanol solutions (—, ); 4, n-hexanol solution (—, 3).

44

WANTKE ET AL.

diffusion exchange parameter v0 have to be replaced by adjustable values in the Lucassen/v. d. T. model. In some cases higher concentrated solutions exhibit atypical behavior. To investigate this, more measurements in a medium-frequency range are required. In particular, with an exact knowledge of the phase angle of the elasticity more details of dynamic processes at fluid interfaces can be explained. APPENDIX A

For the determination of the elasticity modulus using an oscillating bubble setup we have to consider the pressure balance at a the surface during an oscillation. This is simple for a complete ideal sphere. In such a case the flux velocity has only the radial component (49 –54) v r 5 v~r, t! 5

SD

of the inertial term. The magnitude of the displacement, represented by

us~r!u 5

SD

r0 2 uDru, r

is independent of the frequency as a consequence of the incompressibility of the bulk phase, and for a small magnitude of oscillations the second inertial term is negligible (v­v/­r ' 0). p s describes the pressure near the bubble surface. If we take into account only small deviations from an ideal sphere, the stress balance at the surface is given by

s grr 2 s srr 5 p s 2 p g 2 2 h 2

r0 v 0~t!, r

v 0~t! 5 i v uDruexp~i v t!,

­v 2 5 2 g, ­r rb

[A8]

[A1] which can be split into a static equation of the form

where r is the radial coordinate, r0 the mean radius of the bubble, v0 the velocity of the bubble surface, and Dr 5 uDruexp(ivt) the harmonic change of the bubble radius with the amplitude uDru. Equation [2] is valid for uDru ! r0. The general pressure tensor of a frictional flow within a bulk phase has the following components in spherical coordinates (61, 70): ­v s rr 5 2p 1 2 h , ­r

s qq 5 s ww 5 2p 1 2 h

p# s 2 p g 5 p# 0s 1 r gh9 2 p g 5 2

v . r

[A3]

D

­p ­ ­v 2 ­p ­v ­v 2 1 2h 1 v 52 5 1v . ­r ­r ­r r ­r ­t ­r

2 g# r9~h9!

[A9]

and in a dynamic one ( p g 5 const.) Dp s 2 2 h

[A2]

Neglecting the influence of gravity, the divergence of this tensor completed by the inertial terms leads to the special form of the Navier–Stokes equation (using Eq. [A1])

S

[A7]

­v 2 2 g# Dr, 5 2 Dg 1 ­r r0 ~r 0! 2

[A10]

where r b 5 r90 (h9) 1 Dr > r 0 1 Dr and h9 is the dip distance of the bubble area. Equation [A9] is the Young–Laplace relationship. Because the influence of gravity is negligible in the dynamic equation [A10] we can introduce a mean radius r 0 . Using Eqs. [A5] and [A6], this leads to

Dp 1 2 v 2B

@A4]

DA 2 g# 4i hv 2 Dr 2 Dr 5 2 Dg 1 A r0 ~r 0! 2 r0

[A11]

or This means we obtain the pressure at a measurement point r 1 within the bulk phase by r-integration in the form p 1 5 p# 1 1 uDp 1uexp~i w 9 1 i v t! 5 p s 1 v 2B

DA A

S

Dp 1 5 2

E

r0

r1

S DU U

r0 us~r!udr 5 r Ar 0 1 2 r1

Dr DA

[A12]

@A5] with

with the constant

rA B52 DA

D

DA 2 e ~ f, c! 1 F~ f ! 1 g~ f, c! r0 A

e ~ f, c! 5 E~ f, c!exp~i w ~ f, c!! 5 A [A6]

g~ f, c! 5

S

Dg , DA

[A13]

D

2 Dr 4i v ~ g# 2 g 0! 2 ~ h 2 h 0! A, ~r 0! 2 r0 DA

[A14]

45

THE OSCILLATING BUBBLE METHOD

and the calibration function F~ f ! 5 v 2B 2

4i vh 0 Dr 2 g 0 Dr A1 A. r 0 DA ~r 0! 2 DA

[A15]

For pure water ( e ( f, c) [ 0, g( f, c) [ 0, and g 5 g0 5 const.), Eq. [A12] is reduced to Dp 1 5 F( f )DA/A. This means the calibration function F( f ) results from the measurement, Dp 1 , of dynamic pressure using a pure water. The correction function g( f, c) has to be determined by independent measurements of static surface tension and bulk viscosity. The calibration and correction procedure is independent of the assumption of an ideal spherical geometry and is particularly important for real experimental systems. The justification of this is discussed above. For the interpretation of Eq. [A13] the dynamic surface tension must be calculated with the aid of a model. Most authors who have dealt with dynamic surface tension assume that this tension is only a function of the instantaneous surface concentrations G, and equilibrium is established at all times between the surface and the sublayer (7). These assumptions are used in the Lucassen/v. d. T. model, as well as in the Ward–Tordai equation. Other authors replace the equilibrium condition with a model describing the molecular exchange between surface and subsurface (20, 47, 48, 71–73). This includes the first case, although the relation to the Lucassen/ v.d.T. formulas is missing there. For the demonstration of the equivalence of both models and for a uniform description of the final equations, we repeat a known derivation of the Lucassen/v.d.T modulus introducing some complementations which take into account the kinetic effects (2, 3, 7, 47, 48, 71–73). We consider only solutions with one surfactant. The dynamic behavior of solutions which contain several surfactants can be described by similar equations (74). If the tension state of the surface is only a function of the surface concentration G, a change in surface tension can be represented by (G 5 n/A)

g 5 g#~G# ! 1

S

D

dg dg DA Dn 2G , [A16] DG 5 g#~G# ! 1 1 dG dG A A

This change must be equal to the difference in the adsorption and desorption rate which can be described with the standard assumption by the equation (20, 47, 48, 71–73) 1 dn 5 j r 5 P~c s, G! 2 Q~c s, G! A dt

[A19]

P~c s, G! 5 a exp~2E a /RT!c s~G ` 2 G!

[A20]

Q~c s, G! 5 a 9exp~2E b /RT!G,

[A21]

with

and

where Ea and Eb are energies of activation of adsorption and desorption, respectively. Equation [A19] represents an additional condition which is necessary for the solution of the diffusion problem. This equation includes the permanent equilibrium state between surface and subsurface as a special case. In the equilibrium case P(cs 5 c*, G) 5 Q(cs 5 c*, G) and jr 5 0, where the subsurface concentration cs results from the equilibrium isotherm equation. The permanent validity of this relation, cs 5 c*(G(t)), between the adsorption G(t) and the subsurface concentration cs 5 c*(t) is the additional condition in the models of Lucassen/v. d. T. and Ward and Tordai. This means the exchange of molecules between the subsurface and the surface is fast, compared to the time resolution of area changes, and the process is diffusion controlled. In addition, the thickness of the subsurface must be small in comparison to the wavelength of the related diffusion wave (Eq. [A24]). Then the thickness of the sublayer has no influence on the bulk diffusion, and the permanent equilibrium state between subsurface and surface can be considered as a limit case of the general condition [A19] which describes the exchange of molecules between surface and subsurface. If this exchange is not fast enough for the neglecting of its influence on the bulk diffusion process, the development of Eq. [A19] close to the equilibrium is useful, and this reads

and according to Eq. [1], the elasticity by d g d ln G e ~ f, c! 5 . d ln G d ln A

[A17]

n describes the number of molecules of the surfactant at the surface, G( A(t)) the surface concentration, and A the surface area. The change in number of molecules n is given by the law of mass conservation which reads at the surface 1 dn G d~ A~t!! dG ­c 5 1 5 2D A dt A dt dt ­y

U

. y50

[A18]

1 dn ­~P 2 Q! ­~P 2 Q! dG 5 ~c s 2 c# ! 1 ~c* 2 c# ! A dt ­c ­G dc* >

­~P 2 Q! ­c

U

~c s 2 c*!

[A22]

c5c*

because ­~P 2 Q! ­c

U

~c s 2 c# ! 1 c5c#

­~P 2 Q! dG ­G dc*

U

~c* 2 c# ! > 0. c*5c#

46

WANTKE ET AL.

Here c s (t) is the real concentration within the subsurface, responsible for the bulk diffusion process, and c* is a fictive subsurface concentration which is determined by the adsorption isotherms equation c*(G(t)). For the limit value d 5 ­(P 2 Q)/­c 3 ` the concentrations c s and c* become equal, and we obtain the equilibrium condition. The kinetic effects can be described easily with this development by the difference of the values c and c*. With this formal step the final equations have in all cases a form which is similar to the Lucassen/v. d. T. modulus, and the derivation needs only a small modification. This is an advantage for practical applications, and we repeat the derivation to demonstrate that (2, 3, 54, 58). In the case of an ideal sphere there is no lateral diffusion and the surface tension is homogeneous. In addition, the flux velocity has only a radial component, and, if the radius r 0 of the bubble is not too small ( =v / 2Dr 0 @ 1), the solution of the diffusion equation

dG dc* ­c s dc* dc s ­t

U

1 y50

d ln A dG dc* ­c s d ln G dc* dc s ­t

Dc 5 c 2 c# 5 uDcuexp@i b 1 ~1 1 i!ky 1 i v t#.

dDc* 5 D~1 1 i!kDc s 1 dDc s

y50

U

­c s ­y

S D Y S DJ

H

dg dc* dc s ­c s e ~ f, c! 5 2 11D d ln G dG dc* ­ y

. y50

­c s ­t

21

,

[A29] and with Eqs. [1] and [A26] we obtain

H

Î

D dc* h~ f, c! 2v dG

J

21

@A30#

for the surface elasticity where the parameter e0 represents the Gibbs elasticity

e0 5 2

[A24]

For the evaluation of the kinetic effects we can assume a special relation DA 5 DA(Dc*(Dc s (t))) between the change in surface area DA, the change in the real subsurface concentration Dc s 5 Dc(0, t), and the change in the fictive equilibrium concentration Dc* as a result of the oscillation. If the real subsurface concentration c s is different from the fictive concentration c*(G), a retardation must be taken into account. Using Eqs. [A18] and [A24] we obtain

5 2D

This allows replacement of Eq. [A17] by

[A23]

has for sinusoidal oscillations and one surfactant component the form ( y # 0)

U

[A28]

e ~ f, c! 5 e 0 1 1 ~1 2 i!

­ c ­c 5 ­ y 2 ­t 2

D

c) > 1; for a kinetic-controlled process, d 2 ! v D, and d 2 ; v D leads to a mix-controlled process. With this function Eq. [A18] reads

dg . d ln G

[A31]

As mentioned previously, in the general case the modulus e( f, c) must also include kinetic and viscous effects. The kinetic effects are taken into account by h( f, c) and the influence of the intrinsic dilational surface viscosity surface viscosity k9 can be described by an additional term in Eq. [A30] (20, 47, 54). This means the dynamic surface tension is a function not only of the surface concentration G but also of its time derivative. Then Eq. [A30] must be replaced by

[A25]

e ~ f, c!9 5 e ~ f, c! 1 i vk 9~1 2 u~ f !!.

[A32]

or dc s h~ f, c! 5 5 d/~d 1 ~1 1 i! Îv D/ 2! dc* 5 uh~ f, c!uexp~2i b 9~ f, c!!

[A26]

The experimental verification of this complete formula requires comprehensive measurements using higher concentrated solutions and various frequency ranges. With h( f, c) [ 1, and k9 5 0 (without kinetic and viscous effect) Eqs. [30] and [A32] lead to the known Lucassen/v. d. T. modulus (2, 3).

with

APPENDIX B: NOTATION

d5

­~P 2 Q! ­c

U

.

[A27]

c5c#

The function h( f, c) determines the kind of molecular exchange. For a diffusion-controlled process, d 2 @ v D or h( f,

a A DA B c 5 c# 1 Dc c# Dc 5 Dc(y, t )

constant of the Langmuir isotherm surface area change in surface area integration constant bulk concentrations (static and dynamic term) static term of bulk concentration dynamic term of bulk concentration

47

THE OSCILLATING BUBBLE METHOD c s 5 c# s 1 Dc s c* 5 c# * 1 Dc* d D E( f, c) E a, E b f F( f ) g( f, c) g h9 h( f, c) jr k n, N p 5 p# 1 Dp p s 5 p# s 1 Dp s p# s 5 p# 0s 1 r gh9 p 1 5 p# 1 1 Dp 1 pg P(c s , G) q( f, c) Q(c s , G) r, q , w r b 5 r90 (h9) 1 Dr r0 r1 Dr R s(r) t T u( f ) v 5 v r 5 v(r, t) v 0 (t) y 5 r0 2 r a, a9 b, b9

g 5 g# 1 Dg g0 G 5 G# 1 DG Gv G` e ( f ) 5 e ( f, c) e0 em z h h0 k9 r s rr , s qq , s ww s g s rr , s rr

subsurface concentration (static and dynamic term) fictive subsurface concentrations (static and dynamic term) sorption constant bulk diffusion coefficient amount of the complex surface elasticity activation energies frequency calibration function correction function gravitation constant dip distance of the bubble areas kinetic function diffusion current wave number of the diffusion wave number of molecules at the surface pressure within the bulk phase (static and dynamic term) bulk pressure near the bubble surface (static and dynamic term) static term of p s pressure at the pressure transducer (static and dynamic term) gas pressure adsorption rate relation between different defined surface concentrations desorption rate spherical coordinate system radius of curvature of a local bubble area (static and dynamic term) mean value of the bubble radius radial coordinate of the point of the pressure measurement dynamic term of the bubble radius gas constant radial displacement of the fluid time temperature weight function radial velocity of the flux radial velocity of the flux at the bubble surface coordinate perpendicular to the surface adsorption and desorption parameter phase angles of the oscillation of the bulk concentration surface tension (static and dynamic term) surface tension of the calibration solution surface concentrations (static and dynamic term) surface concentration of a volume model saturation adsorption complex surface elasticity Gibbs elasticity resulting from the equilibrium isotherm equation experimental high frequency limit of the elasticity molecular exchange function bulk viscosity bulk viscosity of the calibration fluid surface dilational viscosity density of the bulk solution components of the stress tensor in a spherical coordinate system radial components of the pressure tensor within the solution or gas volume

w ( f, c) w9 v 5 2pf v0 vm

phase angle of the elasticity modulus phase angle of the oscillation of the bulk pressure angular frequency theoretical value of the molecular exchange parameter experimental value of the molecular exchange parameter

ACKNOWLEDGMENTS The authors thank K. J. Stebe for helpful discussions on the boundary conditions and the Max-Planck-Gesellschaft and Professor H. Mo¨hwald for their support of the work.

REFERENCES 1. Ward, A. F. H., and Tordai, J., J. Phys. Chem. 14, 453 (1946). 2. Lucassen, J., and van den Tempel, M., Chem. Eng. Sci. 27, 1283 (1972). 3. Lucassen, J., and van den Tempel, M., J. Colloid Interface Sci. 41, 491 (1972). 4. Lucassen, J., and Hansen, R. S., J. Colloid Interface Sci. 23, 319 (1967). 5. Loglio, G., Tesei, U., and Cini, R., J. Colloid Interface Sci. 71, 316 (1976). 6. Loglio, G., Tesei, U., and Cini, R., Colloid Polymer. Sci. 264, 712 (1986). 7. Dukhin, S. S., Kretzschmar, G., and Miller, R., in “Studies in Interface Science” (D. Mo¨bius and R. Miller, Eds.), Vol. 1. Elsevier, Amsterdam, 1995. 8. Hempt, C., Lunkenheimer, K., and Miller, R., Z. Phys. Chem. (Leipzig) 266, 713 (1985). 9. Miller, R., Wu¨stneck, R., Kra¨gel, J., and Kretzschmar, G., Colloids Surf. A 111, 75 (1996). 10. Bonfillon, A., and Langevin, D., Langmuir 9, 2172 (1993). 11. Bonfillon, A., and Langevin, D., Langmuir 11, 2965 (1994). 12. Jayalakshmi, Y., and Langevin, D., Langmuir 11, 2965 (1995). 13. Fruhner, H., and Wantke, K.-D., Colloids Surf. A 114, 53 (1996). 14. van den Tempel, and Lucassen-Reynders, E. H., Adv. Colloid Interface Sci. 18, 218 (1983). 15. Noskov, B. A., Adv. Colloid Interface Sci. 69, 63 (1996). 16. Noskov, B. A., and Zubkova, T. U., J. Colloid Interface Sci. 69, 63 (1995). 17. Langevin, D., J. Colloid Interface Sci. 80, 412 (1981). 18. Earnshaw, J. C., and Hughes, C. J., Langmuir 7, 2419 (1991). 19. Peace, S. K., and Richards, R. W., Langmuir 14, 667 (1998). 20. Johnson, D. O., and Stebe, K. J., J. Colloid Interface Sci. 168, 21 (1994). 21. Chang, C.-H., and Franses, E. I., J. Colloid Interface Sci. 164, 107 (1994). 22. Lu, H. L., and Apfel, R. E., J. Colloid Interface Sci. 134, 245 (1990). 23. Lu, H. L., and Apfel, R. E., J. Fluid Mech. 222, 351 (1991). 24. Tian Y., Holt, R. G., and Apfel, R. E, Phys. Fluids 7, 2938 (1995). 25. Tian, Y., Holt, R. G., and Apfel, R., J. Colloid Interface Sci. 187, 1 (1997). 26. Liggieri, L., Ravera, F., and Passerone, A., Colloids Surf. A 114, 351 (1996). 27. Fainerman, V. B., and Miller, R., in “Studies in Interface Science” (D. Mo¨bius and R. Miller, Eds.), Vol. 6, p. 279. Elsevier, Amsterdam, 1998. 28. Fainerman, V. B., and Miller, R., Langmuir 13, 409 (1997). 29. Makievski, A. V., Fainerman, V. B., Miller, R., Bree, M., Liggieri, L., and Ravera, F., Colloids Surf. A 122 (1997). 30. Dukhin, S. S., Miller, R., and Loglio, G., in “Studies in Interface Science” (D. Mo¨bius and R. Miller, Eds.), Vol. 6, p. 367. Elsevier, Amsterdam, 1998. 31. Liggieri, L., and Ravera, F., in “Studies in Interface Science” (D. Mo¨bius and R. Miller, Eds.), Vol. 6, p. 239. Elsevier, Amsterdam, 1998. 32. Wantke, K.-D., J. Colloid Interface Sci. 144, 293 (1991). 33. Lucassen, J., Faraday Discuss. Chem. Soc. 59, 76 (1976). 34. Lucassen, J., Trans. Faraday Soc. 64, 2221 (1968).

48

WANTKE ET AL.

35. Lucassen, J., Trans. Faraday Soc. 64, 2230 (1968). 36. Lucassen, E. H., “Physical Chemistry of Surfactant Action,” p. 172, Surfactant Science Series, Vol. 11. Dekker, New York, 1981. 37. van den Tempel, M., Chem. Ing. Techn. 43, 1260 (1971). 38. Kitching, S, Johnson, G. D. W., Midmore, B. R., and T. M. Herrington, J. Colloid Interface Sci. 177, 58 (1996). 39. Fang, J., Wantke, K.-D., and Lunkenheimer, K., J. Colloid Interface Sci. 155, 31 (1996). 40. Levich, V. G., Acta Physicochem. URSS 14, 307 (1941). 41. Levich, V. G., Acta Physicochem. URSS 14, 322 (1941). 42. Levich, H. G., “Physicochemical Hydrodynamics.” Prentice Hall, Englewood Cliffs, NJ, 1962. 43. Hansen, R. S., J. Colloid Sci. 16, 549 (1961). 44. Hansen, R. S., and Mann J. A., J. Appl. Phys. 35, 152 (1964). 45. van den Tempel, M., and van de Riet, J. Chem. Phys. 42, 2765 (1964). 46. Earnshaw, J. C., and Hughes, C. J., Langmuir 7, 2419 (1991). 47. Johnson, D. O., and Stebe, K. J., J. Colloid Interface Sci. 182, 21 (1996). 48. Johnson, D. O., and Stebe, K. J., Colloids Surf. A 114, 41 (1994). 49. Hall, S., Bermel, M., Ko, Y., Palmer, H., Enhorning, G., and Notter, R., J. Appl. Physiol. 75, 467 (1993). 50. Chang, C.-H., and Franses, E. I., Chem. Eng. Sci. 49, 313 (1994). 51. Chang, C.-H., Coltharp, K. A., Park, S. Y., and Franses, E. I., Colloids Surf. A 114, 185 (1996). 52. MacLeod, C. A., and Radke, C. J., J. Colloid Interface Sci. 160, 435 (1993). 53. Nagarajan, R., and Wasan, D. T., J. Colloid Interface Sci. 159, 164 (1993). 54. Wantke, K.-D., and Fruhner, H., in “Drops and Bubbles in the Interface Research” (D. Mo¨bius and R. Miller, Eds.), Studies in Interface Science, Vol. 6, p. 327. Elsevier, Amsterdam, 1998.

55. Karapantsios, T. D., and Palmer, H. J., in “63rd Annual Colloid and Surface Science Symposium of The American Chemical Society,” Seattle, WA, p. 143. 1989. 56. Kretzschmar, G., and Lunkenheimer, K., Ber. Bunsenges. Phys. Chem. 74, 1064 (1970). 57. Lunkenheimer, K., Hartenstein, C., Miller, R., and Wantke, K.-D., Colloids Surf. 8, 271 (1984). 58. Wantke, K.-D., Miller, R., and Lunkenheimer, K., Z. Phys. Chem. (Leipzig) 261, 1177 (1980). 59. Wantke, K.-D., Lunkenheimer, K., and Hempt, C., J. Colloid Interface Sci. 159, 28 (1993). 60. Enhorning, G., J. Appl. Physiol. 43, 198 (1977). 61. Hall, S. B., Notter, R. H., Smith, R. J., and Hyde, R. W., J. Appl. Physiology 69, 1143 (1990). 62. Wang, Z., Hall, S., and Notter, R., J. Lipid Res. 36, 1283 (1995). 63. Hansen, R. S., J. Phys. Chem. 64, 637 (1960). 64. Rillaerts, E., and Joos, P., J. Phys. Chem. 86, 347 (1982). 65. Guggenheim, E. A., “Thermodynamics,” North-Holland, Amsterdam, 1967. 66. Malysa, K., Miller, R., and Lunkenheimer, K., Colloids Surf. 53, 47 (1991). 67. Fainermann, V. B., and Miller, R., Colloids Surf. A 97, 65 (1995). 68. Lucassen-Reynders, and Lucassen, J., Colloids Surf. A 85, 211 (1994). 69. Hirsa, A., Korenowski, G., Logory, L., and Judd, C., Langmuir 13, 3813 (1997). 70. Landau, L., and Lifschitz, E. M., “Fluid Mechanics,” Vol. 6, Pergamon Press, New York, 1959. 71. Baret, J. F., J. Colloid Interface Sci. 30, 1 (1969). 72. Adamczyk, Z., J. Colloid Interface Sci. 120, 477 (1987). 73. Gottier, G. N., Amundson, N. R., and Flumerfelt, R. W., J. Colloid Interface Sci. 114, 106 (1986). 74. Jiang, Q., Valentini, J. E., and Chiew, Y. C., J. Colloid Interface Sci. 174, 268 (1995).