Hydrodynamic processes in dynamic bubble pressure experiments

Colloids and Surfaces A: Physicochemical and Engineering Aspects 192 (2001) 157– 175 www.elsevier.com/locate/colsurfa

Hydrodynamic processes in dynamic bubble pressure experiments Part 5. The adsorption at the surface of a growing bubble N.A. Mishchuk a, S.S. Dukhin a, V.B. Fainerman b, V.I. Kovalchuk c, R. Miller d,* a

Institute of Colloid Chemistry and Chemistry of Water, 42 Vernadsky a6enue, 03680 Kie6, Ukraine b Institute of Technical Ecology, 25 She6chenko bl6d., 83017 Donetsk, Ukraine c Institute of Biocolloid Chemistry, 42 Vernadsky a6enue, 03680 Kie6, Ukraine d Max-Planck-Institut fu¨r Kolloid- und Grenzfla¨chenforschung, Forschungscampus Golm, 14476 Golm, Germany

Abstract The dynamics of the adsorption process at the surface of a growing bubble is described for a diffusion-controlled adsorption mechanism. The model is linked with the measuring procedure in maximum bubble pressure experiments as it is programmed in standard tensiometers. As a result the pressure– gas flow rate dependence is obtained which is the basis of the Lauda tensiometer MPT2. The theoretical results allow to define the necessary corrections for calculation procedures used in bubble pressure tensiometry, and to estimate possible experimental errors. Moreover, the importance of particular features of commercial instruments is demonstrated, such as the requirement of a large gas reservoir and a maximum length for capillaries needed to reach short adsorption times of microseconds and below. The influence of the surfactant concentration as studied for decyl dimethyl phosphine oxide on the particular shape of all characteristic dependencies is quantitatively reproduced by the theory. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Adsorption; Bubble pressure; Hydrodynamic processes

1. Introduction The maximum bubble pressure method is widely used to measure dynamic surface tension (see reviews [1 – 3]). For this method, commercial devices exist which allow measurements even in the millisecond time range. The theoretical background of the method was summarised in Ref. [2], * Corresponding author.

and the results of specific theoretical studies are reported by Dukhin and co-workers [4–11]. When the method is to be used in the millisecond and sub-millisecond time range, a number of fundamental problems have to be solved. As the excess pressure (from which the surface tension is calculated via the Laplace equation) is measured in the measuring system connected with the bubble through the capillary, rather than in the bubble itself, it becomes important either to minimise the

0927-7757/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 ( 0 1 ) 0 0 7 2 2 - 1

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pressure difference along the capillary, or to accurately account for this pressure difference. On the one hand, by considering inertia and viscosity of gas and liquid, and the non-stationarity of the flow processes [4– 9], one can introduce corresponding corrections into the calculations and optimise the geometry of the capillary and measuring system. On the other hand, the experimental procedure implemented in modern tensiometers, e.g. the MPT2 from Lauda (Germany), provide conditions to minimise the influence of viscosity, inertia and non-stationarity effects. Yet more complicated problems are connected with the subdivision of the physical bubble time between two detaching bubbles into the so-called lifetime and dead time. This can be done by simultaneous determination of the critical point in the pressure – gas flow rate dependence [2] and an optimisation of the geometry and properties of the capillary. It is important here to provide minimum penetration of the liquid into the capillary after the detachment of the bubble [10]. The most complicated question in maximum bubble pressure tensiometry deals with the relation of the pressure measured by the device with the surface tension and the respective surface lifetime. Here the effective lifetime is needed, which means that the expansion of the bubble during the measuring procedure has also to be taken into account. For this it is assumed that at the moment of bubble detachment the adsorption at the new bubble surface is almost zero, while the maximum adsorption corresponds to the moment when the growing bubble reaches a hemisphere with a radius of curvature equal to the capillary radius. This assumption was confirmed by comparison of MPT2-data with those obtained by other experimental methods, and with theoretical calculations for surfactant solutions using a diffusional adsorption mechanism [2]. However, it was shown in Ref. [11] that for concentrated solutions of surfactants this assumption is invalid. It seems important, therefore, to perform an analysis of the actual capacity of the maximum bubble pressure tensiometry by a direct comparison of the theoretical and experimental dependence of the pressure in the measuring system as a function of the gas flow rate through the capillary.

The theoretical model which describes the gradual variation of the bubble shape — from a flat meniscus to the hemisphere, and than to the final spherical bubble, was proposed in Ref. [11]. This model provided more insight into the relevant processes, in particular, the interrelation between adsorption, pressure inside the bubble, and rate of bubble growth. To simplify the model, the reservoir volume was assumed to be infinitely large. It will be shown below that this assumption is a good approximation in the sub-millisecond time range only, where pressure variations in the reservoir and the effect produced by this variation on the gas flow along the capillary can be neglected. Moreover, the formation of a single bubble was considered in Ref. [11], governed by certain initial conditions. However, to enable the thorough comparison between the theoretical results and the experimental data, the theory should account for the stationarity of the bubble formation process, i.e. at the time point when the bubble detaches, the pressure in the reservoir is equal to the value at the onset of bubble formation. The adsorption at the bubble surface should also comply with this requirement. In the proposed study, we present a solution of this self-consistent problem, an analysis of the main features of the process, and a comparison between theory and experimental results.

2. Theory The physical model considered in this work corresponds to the actual conditions implemented in the tensiometer MPT2. The continuous sequence of bubbles is formed at the tip of the capillary immersed into the solution. The opposite end of the capillary is connected to the reservoir into which the gas is supplied at a certain constant rate. The capillary length significantly exceeds the capillary radius. Under these conditions, the growth of a bubble is determined primarily by the aerodynamic resistance of the capillary, rather than the hydraulic resistance of the solution. The growing bubble surface shape is part of a regular sphere (Fig. 1), which can be confirmed by monitoring with a high-speed video system [12].

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As the capillary radius is small, and the hydraulic resistance of the solution is low, the non-sphericity of the bubble shape can be neglected, so that the growth of the bubble is described solely by the time dependence of the bubble radius R(t) or the characteristic angle q(t), cf. Fig. 1. The special processing of the capillary surface (hydrophobisation of the internal surface and the hydrophilisation of the frontal surface) and the sharpening of the capillary edge hinder penetration of the solution into the capillary after bubble detachment. This ensures that the wetting perimeter (points A, Fig. 1) remains virtually motionless during bubble growth. If the internal surface of the capillary is hydrophilic, then penetration of the solution into the capillary is possible; however, the penetration depth can be made negligible, if the capillary is wide and short enough [10,12]. The gas pressure in the capillary does not remain constant: the pressure decreases during the stage of fast bubble growth, while when the bubble growth becomes slower (when the bubble shape approaches the hemisphere) the pressure approaches its initial value. The pressure oscillations in the reservoir are very small, because the reservoir volume is much higher than the volume of the detaching bubble, however, these oscillations significantly affect the bubble growth process. The rate of bubble growth is usually determined by three factors: “ the aerodynamic relaxation, i.e. the rate of the pressure equilibration between the bubble and the reservoir;

Fig. 1. System of coordinates with characteristic angle q; the value q is y for a flat meniscus and y/2 for a hemisphere; q =30° for R= 2a0 and q: 5.5° for R =10a0.

“

159

the pressure relaxation in the reservoir after detachment of a bubble, which depends on the gas feed rate; “ the relaxation of surface tension caused by the adsorption of surfactants. The role played by each of these factors depends essentially on the stage of the bubble growth process and the device operation regime. At the initial stage the bubble growth is always determined by the aerodynamic relaxation along the capillary. In the initial time moment, the radius of curvature of the meniscus is very high (the value of q is close to y), the capillary pressure in the bubble is low, and the pressure difference along the capillary is high. This results in a fast inflow of gas through the capillary into the bubble. However, with bubble growth, the radius of curvature decreases rapidly, as does the pressure difference. If the bubble formation frequency is low enough such that the aerodynamic relaxation time is much smaller than the time interval between two consequent bubbles, then the equilibration of the pressure within the bubble and reservoir takes place before the reservoir pressure becomes equal to the maximum bubble pressure necessary for the detachment of the bubble. In the case of pure water, the rate of further bubble growth is determined by the rate of pressure restoration in the reservoir [9]. In the case of surfactant solution, the bubble growth is also promoted by the surface tension decrease in time, and if the adsorption is fast, then this factor could be more significant than the pressure restoration in the reservoir. However, in any case, to ensure the stationary regime of bubble formation, the pressure restoration process should take place. In the rapid bubble formation regime, the aerodynamic relaxation time is larger than the time interval between two bubbles, and therefore pressure equilibration cannot be completed, and the bubble detaches when the pressure in the reservoir is different from the pressure within the bubble. In the stopped gas flow regime, the pressure in the reservoir is not restored, and the detachment of bubbles is caused only by the decrease of surface tension. Hence, each subsequent bubble detaches at a smaller pressure in the reservoir. The time intervals between two bubbles become

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gradually larger. This regime is applied for surface tension measurements at large surface lifetimes and works only for surfactant solutions of sufficiently large concentration [2]. In the MPT2 device, similar to its earlier version MPT1, a constant size of the detaching bubbles is ensured. This is arranged by a pin located opposite to the capillary tip with some axial displacement, which leads to the detachment of the bubbles when they grow up to the defined size. After some transition time, a constant predefined gas flow leads to a stationary regime of bubble formation, as all bubbles are of the same size. Then the kinetics of bubble growth becomes the same for each bubble. When a bubble detaches from the capillary, all system parameters are restored to their initial values. Therefore, the whole measuring process is characterised by periodic initial conditions. In the present study, all features characteristic to the rapid regimes of bubble formation are disregarded. This enables one to neglect both the inertial effects and the processes caused by the gas compressibility in the capillary [4– 6,9]. Similarly to Ref. [11], we consider a non-ionic surfactant with a diffusion-controlled adsorption mechanism. For this case, the Langmuir isotherm is valid: KHCs G= KHCs 1+ Gmax

(1)

where G is the concentration of adsorbed molecules at the bubble surface (the adsorption value); KH is the adsorption constant; Cs is the surfactant concentration close to the bubble surface, i.e. at the internal boundary of the diffusion layer; and Gmax is the maximum adsorption. The adsorption process at the surface of a bubble is governed by an equation which describes the diffusion of molecules towards the surface and by an adsorption equation (which is just the boundary condition at the bubble surface), where the expansion of the surface is taken into account. In the locally flat model approximation, these equations read [13]:

(C(z, t) 1 dS(t) (C(z, t) ( 2C(z, t) = z +D (t S(t) dt (z (z 2

(2)

dG(t) 1 dS(t) (C(z, t) + G(t)= D dt S(t) dt (z Z = 0

(3)

)

where C(z, t) is the concentration of molecules in the solution, D is the diffusion coefficient, t is the time expired from the bubble detachment moment, z is the coordinate normal to the surface, S(t) is the surface area, and dS(t)/dt is the variation of this area with time. The solution of Eq. (2) based on the model developed in Ref. [11] allows to express the surfactant concentration gradient in Eq. (3) by

)

C0 − CS(t) (C = (z Z = 0

yDtef

(4)

' 

and, together with Eqs. (1) and (3) we get D Á ytef dG(t) Ã 1 dS(t) +Ã + dt G(t) ÃS(t) dt KH 1− Ä Gmax D = C ytef 0

'

 à ÃG(t) à Å



(5)

Here C0 is the surfactant concentration in the solution, tef(t)=

&

t



S 2(t) dt /S 2(t)

(6)

0

is the effective time which reflects the variation of the surfactant diffusion layer thickness at the bubble surface [11], expressed via the time-dependent variation of the bubble surface area S(t) S(t)=

2ya 20 1− cos q(t)

(7)

q(t) is the characteristic angle which gives the relation between the bubble radius R and the capillary radius a0 as: R(t)= a0/sin q(t). To obtain the solution of this problem, it is convenient to present the effective time tef(t), Eq. (6), and the characteristic angle q(t) in the differential form: dtef(t) 2t (t) dS(t) = 1− ef dt S(t) dt

(8)

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dq(t) V(t) =− (1 − cos q(t))2 dt a0

(9)

where the flow rate is determined from the Poiseuille equation via the pressure difference between bubble and reservoir DP*(t) = DP(t)− P| (t) [11] V(t)=

DP(t)a 20 DP(t)a 20 = (1 − k(t) sin q) 8lp 8lp

(10)

where DP(t) is the pressure difference between the reservoir and the liquid at the bubble surface, including the pressure constituent caused by the surface curvature (the Laplace pressure) P| (t)=

2|(t) R(t)

(11)

a0 and l are the radius and length of the capillary, respectively, and p is the gas viscosity. Here the auxiliary function k(t)=

2|(t) a0DP(t)

(12)

was introduced to account for the dependence of the pressure within the bubble on the dynamic surface tension |(t). The dynamic surface tension is related to the surface tension of the solvent |0 and the adsorption G(t) via the Szyszkowski– Langmuir equation



|(t)= |0 + RTGmax ln 1 −



G(t) Gmax

(13)

dDP DP + Patm =k (L−ya 20V(t)) dt Vres

G(t) t = 0 = G(t) t = T = G0 “

P (t) dPres(t) = k res (L − ya 20V(t)) dt Vres

“

(16)

the initial meniscus is planar q(t) t = 0 = y

“

where k is the adiabatic constant, Vres is the reservoir volume, and L is the gas flow rate per unit time at the reservoir inlet. The pressure variation in the reservoir leads to the corresponding variation of the pressure difference between reservoir and bubble DP, thus one can rewrite Eq. (14) as:

(15)

where Patm is atmospheric pressure. Therefore, the set of four first-order differential equations, Eqs. (5), (8), (9) and (15) is defined, which describe the interrelation between the adsorption G(t); the bubble size expressed via the angle q(t); the effective time tef(t) which accounts for the variations in the properties of diffusion layer at the vicinity of the growing bubble; and the pressure in the reservoir Pres(t), for predefined instrument parameters Vres, a0, l, L; the characteristics of the surfactant solution Gmax, KH, C0, D; and the gas viscosity p. To calculate the solution for the stationary regime of separate bubble formation, the set of differential equations should obey the following initial conditions: “ the adsorption at the onset of bubble formation (t=0) is equal to the adsorption at the final moment before bubble detachment (t= T); this problem was shown in Ref. [11] to be very complicated, and here the simplest assumption is employed, namely that the density of the adsorbed substances after the bubble detachment remains approximately equal to that immediately before the detachment

The above equations should be complemented with the equation describing the pressure variation in the reservoir. This can be derived from the balance of the gas input into the reservoir and the gas outflow from the reservoir through the capillary [9] (14)

161

(17)

the effective time at the bubble existence onset is zero tef(t) t = 0 = 0

(18)

the pressure in the reservoir at the initial moment of the bubble existence is equal to that at the final moment: Pres(t) t = 0 = Pres(t) t = T = Pres,0

(19)

In contrast to the model presented in Ref. [11], where the initial values of adsorption G0 and pressure DP0 for a single bubble were predefined,

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162

here we impose the requirement on the solution to satisfy the periodic conditions Eqs. (16) and (19): this requirement corresponds to the stationary regime of bubble sequence formation. The characteristic times usually employed in the maximum pressure method could be calculated from additional expressions: “ the lifetime tl q(tl)= y/2 or R(tl) = a0 “

(20)

total bubble existence time T =tl +td (including the lifetime tl and dead time td) R(t) t = T =R0

(21)

where R0 is the characteristic size of the bubble at which the detachment takes place. The experimentally measured gas flow rate L can be expressed in terms of the volume of the detaching bubble Vdet via the equation: L= Vdet/T

(22)

where for the characteristic size of the detaching bubble R0  10a0 the volume of the detaching bubble can be calculated from the simplified formula Vdet =

4yR 30 3

(23)

As the value of R0 is usually approximately equal to 10a0, the volume of the detaching bubble is roughly equal to the volume of the corresponding sphere 4yR 30/3. This value was used to simplify the calculations. The error due to non-sphericity of the bubble in the vicinity of the region of the capillary tip does not exceed 0.5%. The equations above are quite cumbersome. Therefore, only numerical solution of the problem is available. The results of the theoretical analysis are presented below in graphical form, and the role of each of the important parameters is illustrated.

3. Material and methods The surface tension measurements presented here were performed using the maximum bubble

pressure tensiometer MPT2, manufactured by Lauda, Germany. The device and method were described in detail elsewhere [2,13,14]. The time range reached in our experiments is 0.2 ms to 50 s (2–50 s in the stopped flow regime [2]). Two capillaries were used in the study: a hydrophilic capillary of radius a0 = 74.7 mm, and a capillary with hydrophobized internal surface, a0 =66.8 mm. The length of the narrow section for both capillaries was l= 10 mm. The gas reservoir volume of the instrument was 35 cm3. The decyl dimethyl phosphine oxide (C10DMPO) is a specially purified material prepared in our laboratory particularly for studies of surface phenomena. Bidistilled and deionised water was used to prepare the solutions. All measurements were performed at 25°C. The parameters characteristic for the adsorption of decyl dimethyl phosphine oxide are: KH = 8.29× 10 − 6 m, Gmax = 3.64× 10 − 6 mol m − 2, D= 4.8×10 − 10 m2 s − 1, according to Ref. [15]. 4. Results and discussion

4.1. Analysis of numerical calculations The most interesting characteristics to be analysed are the interrelation between the measured pressure and the gas flow through the capillary, the values of bubble lifetime (i.e. the time necessary for the bubble to become a hemisphere), the dead time (the time during which the bubble is expanding from the hemisphere to the detachment), and the surface pressure. It is also important to analyse the dependencies of adsorption and pressure difference between reservoir and bubble as functions of time and bubble size. As different conditions of the process studied lead to different values of the characteristic times, and the time dependence of the bubble radius is nonmonotonous, the decrease of this radius from infinity (at the bubble formation moment) to the capillary radius (at the dead time end) is followed by the increase of the bubble radius up to a certain value R0. The most demonstrative form to represent the results is the angle dependence. For some functions, the dependencies are presented in two forms, angle and time dependence.

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163

Fig. 2. Calculation of the adsorption value G(q) (a) and G(t) (b); pressure difference between the bubble and reservoir DP*(q) (c); time t(q) necessary for the bubble to achieve certain size expressed via the characteristic angle q (d); pressure oscillations in the reservoir Pres(q) (e) and surface tension |(q) (f) for the values of parameters: a0 =75 mm; l=1 cm; D =5 ×10 − 6 m2 s − 1; K H =10 − 6 m; C0 = 1 mol m − 3; Vres = 5 cm3; curve 1 corresponds to the gas flow rate L =120 mm3 s − 1; curve 2, L =105 mm3 s − 1; curve 3, L= 10 mm3 s − 1.

4.1.1. Analysis of adsorption 6ariation during bubble growth Some of the results are illustrated in Fig. 2(a– f). It is seen from Fig. 2(a,b) that two types of

curves can exist depending on the gas flow rate: a monotonous decrease of adsorption throughout the lifetime and some portion of dead time, followed by an adsorption increase at the dead time

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end (Fig. 2(a), curve 1), or an adsorption maximum at values of angle corresponding to the hemispherical bubble, i.e. at the lifetime end (Fig. 2(a,b), curves 2, 3). The existence of these two types of dependence is related to the ratio between the aerodynamic relaxation time and the bubble formation time. For low gas flow rates (Fig. 2(a,b), curve 3) the bubble formation time is much longer than the aerodynamic relaxation time (which is almost independent of the gas flow rate). The bubble growth becomes slower before the bubble reaches the hemisphere, because the pressure difference vanishes, and it takes some time for the pressure drop to arise either due to the pressure increase in the reservoir, or due to the adsorption process. During this time, the adsorption at the bubble surface increases significantly, because the surface area increase is very slow (for more details see below). In contrast, for high flow rates (curve 1) the time of bubble formation is very short, and the process is not delayed when the bubble shape becomes hemispherical. Hence, the expansion of the bubble surface is rapid over the lifetime stage, thus a continuous decrease of the adsorption is observed.

For surfactant solutions, additional processes, which are governed by the ratio of surface expansion rate and the adsorption kinetics, are quite significant. This becomes evident from the comparison of the rates of bubble surface expansion 1 dS(t) S(t) dt during the lifetime and the dead time, respectively. It is seen from Eqs. (7) and (9) that these changes strongly depend on the angle q (cf. Fig. 1), 1 dS(t) V(t) = sin q(t)(1− cos q(t)) S(t) dt a0

(24)

In particular, at the lifetime end, i.e. when q= y/2, the relative variation of the bubble surface area approaches a certain value, which actually depends only on V(t)





y V(t) 1 dS(t) V(t) 1+ − q : “ 2 a0 S(t) dt a0

(25)

At the dead time end, i.e. for q“ 0, the relative change of S is proportional not only to the velocity of the gas flow V(t), but also to the function q 3(t), which decreases rapidly

Fig. 2. (Continued)

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1 dS(t) V(t) q 3(t) “ S(t) dt a0 2

(26)

If the gas flow rate is large, the pressure DP is high, and therefore the function k(t) in Eq. (10) is significantly smaller than 1. In this case, the small factor q 3(t)/2 in Eq. (26) plays a much more significant role than the dependence of the gas flow velocity V(t) on angle and time in Eqs. (25) and (26). It follows then that the relative rate of area variation at the dead time end is essentially lower than at the lifetime end. For rapid surface area variation the adsorption from solution cannot compensate the expansion effect, and therefore the amount of surfactant molecules adsorbed at the bubble surface decreases during almost the whole period of bubble growth. In contrast, at the dead time end the surface expansion rate becomes much lower, and an increase of surfactant adsorption is observed, cf. curve 1 in Fig. 2(a). The more complicated shapes of curves 2 and 3 shown in Fig. 2(a) reflect some peculiarities in the variation of the bubble surface deformation rate at pre-critical (bubble formation regime) gas flow rates. At first, an aerodynamic relaxation takes place during a very short time, when both the pressure difference in the capillary and the gas flow velocity are high, which leads to a rapid growth of the bubble (decrease of q from 180 to 110°, curve 3, and to 100°, curve 2). The surface deformation rate is high, and therefore the adsorption becomes slower. For low gas flow rates, and therefore, for a relatively small initial pressure in the reservoir, the pressure difference in the capillary can vanish before the bubble reaches the hemisphere (cf. Fig. 2(c) for DP*(q), curves 2 and 3), i.e. DP*(t)=DP(t) −

2|(t) sin q(t) : 0 a0

for angle q(t) \ y/2. In this regime, one obtains sin q(t):

a0DP(t) 2|(t)

thus the bubble grows further (q decreases to y/2) both due to the increasing pressure in the reservoir and to the surface tension decrease. Correspondingly, the gas flow velocity

V(t)DP*(t) DP(t)(1− k(t) sin q)

165

(27)

approaches zero at angles close to y/2. This means that the rate of surface area variation also approaches zero: 1 dS(t) V(t) “ “0 S(t) dt a0 cf. Eq. (25). Therefore, the process of bubble expansion becomes extremely slow, and the adsorption process prevails over the surface expansion. The adsorption increases at an almost constant angle q or the variation of this angle is very slow (cf. Fig. 2(a), curves 2, 3). For very slow gas flow rates, the change in adsorption is almost synchronous with the bubble growth: the adsorption increases, however the pressure difference between the reservoir and bubble DP* remains constant because of the simultaneous decrease of the curvature radius, and therefore the capillary pressure in the bubble remains almost constant — for small flow rates we have 2|(t) sin q(t):DP(t):const a0 For angles close to y/2, the curvature radius almost does not change, because these values correspond to the minimum of the radius of curvature. In this case the bubble grows with almost constant adsorption and surface tension, i.e. with q: y/2 and low rates 2|(t) : DP(t): const a0 a small increase in the adsorption results in an increase of the pressure difference between the reservoir and the bubble DP* and hence to a small increase in the bubble size. This in turn results in a decrease of the adsorption, pressure difference, and so on. As a matter of fact, in this case the surface tension slowly increases, accompanied by a slow decrease of the adsorption (cf. curve 3 in Fig. 2(a), due to the slow increase of pressure in the reservoir DP(t). The excess pressure DP* is almost zero not only when the bubble is close to the hemisphere, but also after this time point, because the expansion

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of the bubble leads to a decrease of the surfactant surface concentration, which results in an increased surface tension and Laplace pressure, thus sustaining the pressure difference value close to zero (with an almost unchanged radius of curvature, as its values is close to the minimum). The particular features of the pressure variation determine the behaviour of the adsorption curves (Fig. 2(a) in particular, this leads to the fact that the adsorption maximum does not coincide with the end of the lifetime period (cf. curve 3 of Fig. 2(a). At the end of the total bubble time, similar to the onset of this process, the gas flow velocity is high, because the pressure difference is large (due to the relation sinq “ 0 and, therefore, (1− k(t) sin q)“ 1, cf. Eq. (27)). In this case, small pressure oscillations in the reservoir do not play any role. The surface expansion process at low and high flow rates is similar, therefore the curves describing the angular dependence of the adsorption approach each other at large and small values of q. The above analysis of the surface expansion rate corresponds to the dependence t(q), as given in Fig. 2(d). For high gas flow rates, when the variations in the flow velocity through the capillary are insignificant, the lifetime tl is essentially lower than the dead time td or total time of the existence of the bubble T =tl +td. For example, in the limiting case of very high pressure, i.e. when the gas flow velocity is almost constant and the detaching bubble has a radius equal to 10a0, the ratio of these two times can be estimated from the ratio of the bubble volumes corresponding to the lifetime and dead time end as tl 2ya 30/3 1 : : “0 T 4y(10a0)3/3 2000 cf. curve 1 in Fig. 2(d). For low gas flow rates, the lifetime becomes much higher because the flow velocity is almost zero. In this case the lifetime can be much higher than the dead time, therefore tl “ 1, T see curve 3 in Fig. 2(d). Fig. 2(e) illustrates the calculated pressure in the reservoir. During the short initial lifetime

stage, the pressure in the reservoir remains virtually constant. Only when the aerodynamic relaxation process is completed and the bubble growth becomes slower, this pressure increases. At the end of the dead time period the pressure acquires its initial value. The higher the imposed pressure is, the lower is k(t), and, therefore, the lower are the oscillations of the gas flow velocity through the capillary, which in turn leads to a decrease of the pressure oscillations in the reservoir. Finally, Fig. 2(f) illustrates the variations of surface tension: the minimum of surface tension corresponds to the maximum of adsorption, and vice versa, cf. Fig. 2(a). These dependencies substantiate the concept of separation of the bubble time into the lifetime and dead time, employed in the maximum bubble pressure method, and the assumption that surface tension at the end of lifetime acquires its minimum value [2]. It is instructive to emphasise some features of the calculated theoretical curves. It is seen from Fig. 2(a) that the adsorptions of a surfactant at the onset and at the end of the bubble time for all three illustrated gas flow rates are almost the same. It is also interesting to note that not only the adsorption maximum corresponds to the characteristic angle q: y/2, but also the minimum is localised at approximately one and the same angle q: 30°. This feature can be explained by the fact that the pressure differences between reservoir and bubble, and, therefore, the gas flow velocities in the capillary and the rates of bubble expansion for q‡y/2 for different curves are close to each other. At the same time, the relation between the minimum depths for curves 2 and 3 in Fig. 2(a) seems quite unexpected, at least at a first glance. The decrease in the gas flow rate would seem to make the adsorption factor more significant; however, the minimum at the adsorption curve is shifted towards the larger bubble size region (larger q values) and becomes deeper. For curve 3, the maximum, which is higher than that in curve 2, is followed by a deeper minimum. This less significant adsorption factor for curve 3 compared to curve 2 is caused by the thickness of the adsorption layer for the two cases. In the non-stationary regime the adsorption layer thickness is given by

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167

Fig. 3. (a) The dependence of the adsorption at the onset and the end of the bubble existence G0 =G(t) t = 0 =G(t) t = T (curves 1 –3) and of maximum adsorption value G* (curves 1%– 3%) on the gas flow rate L for Vres =50 cm3; all other parameters for curves 1, 1% are the same as in Fig. 2; curves 2, 2%; KH = 10 − 5 m, C0 =1 mol m − 3; curves 3, 3%, KH =10 − 6 m, C0 =10 mol m − 3. (b) The dependence of lifetime tl(L) (curves 1, 1%, 2, 3) and total time of bubble existence T(L) (curves 4, 4%) on the gas flow rate L; curves 1 – 4, Vres = 50 cm3, curves 1%–4%, Vres = 5 cm3; all other parameters for curves 1, 4, 1%, 4% are the same as in Fig. 2; curve 2, KH = 10 − 5 m, C0 = 1 mol m − 3; curve 3, KH = 10 − 6 m, C0 =10 mol m − 3.

l= 2Dt

(28)

and therefore, for curve 3 the diffusion layer thickness is higher than that for curve 2, because the bubble growth rate during the lifetime period is lower (cf. Fig. 2(d). Thus, the surfactant concentration gradient in the solution is lower, which results in a slower adsorption process. The qualitative differences between the adsorption curves described above can be related to the transition from the single-bubble regime to a gas jet regime [2]. The difference between curves 1 and 2 in Fig. 2(a) are qualitative in respect to the adsorption process. At the same time, curves 2 and 3 exhibit only quantitative differences between the characteristics of the process studied. As the difference in the gas flow rate between curves 1 and 2 is only 15 mm3 s − 1, while for curves 2 and 3 this difference is much higher (95 mm3 s − 1), it can be argued that the qualitative

change of the process takes place at a certain critical gas flow rate. This is just the change of the regime, which is well-known from the experiment: for small gas flow rates separate bubbles are slowly formed, governed by the retardation of the bubble growth at the lifetime end, see curves 2, 3. For high gas flow rates, a rapid generation of consecutive bubbles is observed, without any delay during the formation of hemispheres (lifetime is almost zero), i.e. the so-called jet regime [2] takes place, see curve 1.

4.1.2. Effect of gas flow rate The characteristic features of the change of adsorption at the bubble surface dependent on the gas flow rate L are illustrated in Fig. 3(a,b). These figures show the initial adsorption G0 = G(t) t = 0= G(t) t = T, the maximum adsorption G* at the lifetime end, the lifetime tl and total bubble time T. The increase in the adsorption constant or

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surfactant concentration, results quite naturally in higher adsorption values over the whole adsorption curves, including the initial adsorption G0 (curves 1–3) and the adsorption in the maximum region G* (curves 1% – 3%). At the same time, the adsorption behaviour at the onset and end point of bubble existence is different from the adsorption behaviour in the maximum region. Curves 1– 3 in Fig. 3(a) demonstrate that the initial adsorption G0 is almost independent of L. The maximum adsorption value G* slowly decreases in the range of low and medium gas flow rates, which is followed by a rapid decrease for rates of the order of L 90 – 110 mm3 s − 1, i.e. in the range where the transition from single bubbles to the jet regime happens. A qualitatively similar change in the behaviour is observed in Fig. 3(b), where almost vertical portions in the lifetime curves 1–4 and total bubble time curves 1% and 2% exist at small flow rates and are followed by almost horizontal portions for high flow rates. The dependencies 1%, 2%, and 3% of G* which correspond to the jet regime are shown as dashed lines. In this gas flow rate range, smooth expansions of the bubbles are observed, and no retardation takes place at the lifetime end. In this case the adsorption curves do not exhibit any maximum, and the values of G* show the adsorption value at the lifetime end at q = y/2. It should be noted that for a given gas flow rate the gas reservoir volume only slightly affects the adsorption value, and therefore the dependencies shown in Fig. 3(a) correspond to the same reservoir volume. However, under certain conditions the characteristic times depend essentially on the reservoir volume. While for medium and high gas flow rates this effect is insignificant, for low flow rates, when a pronounced maximum exists on the adsorption curve, the dependence of the characteristic times on the reservoir volume becomes considerable, as demonstrated in Fig. 3(b), curves 1, 2, and 1%, 2%. The total bubble time for the adsorption constants and surfactant concentrations shown in Fig. 3(b) are quite similar, and therefore only the two curves 4 and 4% were plotted. However, the lifetime values in this case are quite different. These time values are inversely proportional to the adsorption values: the higher

the adsorption, ceteris paribus, the more rapid is the bubble expansion, and the shorter is the bubble lifetime.

4.1.3. Effect of gas reser6oir 6olume Let us consider now in detail the influence of the reservoir volume V on the behaviour of adsorption curves (Fig. 4(a), the reservoir pressure (Fig. 4b), the pressure difference between the reservoir and the bubble (Fig. 4(c), the characteristic time (Fig. 4(d) and the surface tension (Fig. 4(e) for a certain medium gas flow rate L=20 mm3 s − 1. It is seen from Fig. 4(a) that the maximum adsorption for a high reservoir volume is observed when the bubble becomes a hemisphere (at the lifetime end), while for small reservoir volumes this maximum is located at much shorter times. This is caused by the differences in the time dependence of pressure for different reservoir volumes. In fact, for the same gas flow rate, the initial pressure in the reservoir is lower for a smaller reservoir volume (Fig. 4(b). Therefore, during the expansion of the bubble, the pressure difference between reservoir and bubble becomes zero earlier (Fig. 4(c), curves 1–3 and 1% –3% shown in the inset). A further growth of the bubble takes place due to increasing surfactant adsorption which leads to a decrease of the capillary pressure in the bubble, and due to the increase of the reservoir pressure. Naturally, if a retardation of the bubble growth occurs earlier, the adsorption also increases earlier. This, in turn, leads to a lower Laplace pressure and, therefore, to a larger pressure difference between reservoir and bubble, i.e. to a rapid expansion of the bubble and to a decreased adsorption. In addition, the variation of the reservoir pressure also plays its role. For smaller reservoir volumes, the increase of the pressure within the reservoir during the retarded growth of the bubble is considerably higher. The increase of the reservoir pressure promotes the growth of the bubble, and therefore accelerates the adsorption decrease. The processes which govern the expansion of the bubble also affect the relation between the time expired from bubble formation onset t and the bubble size expressed via the angle q, Fig. 4(d). For example, if q= 95°, then for a small

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reservoir we have t(q =95°) =0.067 s, while for a large reservoir this value is only t = 0.005 s. At the same time, an earlier increase in the adsorption

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accompanied by a simultaneous increase of the reservoir pressure indicates that a further bubble expansion happens more rapidly, so that the total

Fig. 4. Calculation of the adsorption value G(q) (a), reservoir pressure Pres(q) (b); pressure difference between the bubble and reservoir DP*(q) (c); time t(q) necessary for the bubble to achieve certain size expressed via the characteristic angle q (d). For curves 1 – 3 Vres = 5 cm3; for curves 1% –3% Vres = 50 cm3; gas flow rate L= 20 mm3 s − 1; curves 1, 1%, KH =10 − 6 m, C0 =1 mol m − 3; curves 2, 2%, KH =10 − 5 m, C0 = 1 mol m − 3; curves 3, 3%, KH =10 − 6 m, C0 =10 mol m − 3; all other parameters are the same as in Fig. 2.

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bubble time for both cases is roughly the same (for qB 75° the curves in Fig. 4(d) almost coincide). The increase of the adsorption constant or surfactant concentration results in higher adsorption values (cf. curves 2 and 3 in Fig. 4(a) and lower surface tensions (cf. curves 2 and 3 in Fig. 4(e), which in turn leads to a faster decrease of the pressure difference (Fig. 4(c). Therefore the curves 2 and 3 (Fig. 4(a) exhibit not only higher adsorption values, but also larger relative depths of the two minima. In spite of the fact that the initial adsorptions G0 for curves 2 and 3 differ from each other almost twice, the heights of the maximums at the lifetime end G* are almost the same. This is due to the fact that for a high enough surface coverage by surfactant, i.e. when the concentration gradient along the diffusion layer cross-section is small, any further adsorption process is governed by the adsorption constant rather than by the surfactant bulk concentration. It can be concluded from the data presented in Fig. 4 that a more convenient numerical analysis of experimental results requires large reservoir volumes. In this case the reservoir pressure remains almost constant, and the adsorption maximum is localised at the lifetime end. This enables one to decrease significantly any errors in the calculation of surface pressure and adsorption. Therefore, the choice of a large reservoir volume in the MPT2 tensiometer is appropriate [2].

4.1.4. Effect of capillary characteristics, adsorption constant and surfactant concentration As the experimental studies of the adsorption with the MPT2 tensiometer are based on measurements of the reservoir pressure, we present here some theoretical curves which illustrate the trends of the dependencies of measured pressure as a function of the gas flow rate (Fig. 5(a,b) for various values of the capillary radius and length, and various values of the adsorption constant and surfactant concentration. Each of the curves shown exhibit a sharp kink at a certain gas flow rate. In this range, the gas jet regime arises instead of a smooth variation of pressure corresponding to the formation of single bubbles.

Fig. 5. (a) The dependence of the reservoir pressure on the gas flow rate for various values of capillary radius: curves 1 – 3, a0 =75 mm; curves 1%– 3%, a0 =85 mm; curves 1¦ – 3¦, a0 = 95 mm; and adsorption constants: curves 1, 1%, 1¦, KH =10 − 6 m; curves 2, 2%, 2¦, KH =5 × 10 − 6 m; curves 3, 3%, 3¦, KH = 50× 10 − 6 m; parameters taken for the calculations are l=1 cm, Gmax =5 × 10 − 6 mol m − 2; D =5 × 10 − 10 m2 s − 1; C0 = 1 mol m − 3; Vres =5 cm3. (b) The dependence of the reservoir pressure on the gas flow rate for various values of surfactant concentration: curves 1 – 4, C0 =1 mol m − 3 and curves 1%–4%, C0 =10 mol m − 3 and capillary length: curves 1, 1%, l =0.7 cm, curves 2, 2%, l= 1 cm, curves 3, 3%, l =2 cm, curves 4, 4%, l= 3 cm; parameters taken for the calculations are a0 =75 mm; KH =10 − 6 m; Gmax =5 × 10 − 6 mol m − 2; D =5 × 10 − 10 m2 s − 1; Vres =5 cm3.

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It is seen from Fig. 5(a) that the increase of the capillary radius leads to a decrease of the pressure necessary to maintain the same gas flow rate. This can be explained by the dependence of the bubble growth rate on the bulk flow velocity, i.e. on the capillary cross-section area ya 2 and on the linear velocity of the gas flow V, which is proportional to the pressure and to the square of the capillary radius, as obtained from Eq. (10). Therefore, the larger the capillary radius, the lower the pressure necessary to produce the same effect. It is seen from Eq. (10) that the increase of the capillary length leads to a decrease of the gas flow velocity, that is, the same gas flow rate requires higher pressures (cf. Fig. 5(b). The pressure decrease with increasing values of the adsorption constant (Fig. 5(a) or surfactant concentration (Fig. 5(b) is quite natural. In these cases, a faster increase of adsorption and decrease of surface tension is obtained. Therefore, both the Laplace constituent of the pressure, Eq. (11), and the related factor k(t) in Eq. (10) become lower. Hence, the same gas flow velocity in the capillary is maintained by a lower pressure DP(t), and therefore, by a lower gas pressure in the reservoir. The factor k(t) in Eq. (10) is inversely proportional to the pressure DP(t), and becomes insignificant in the jet regime. This results in a low dependence of gas flow rate on adsorption, and the curves plotted for different adsorption constants (Fig. 5(a) and surfactant concentrations (Fig. 5(b) almost coincide. In contrast, for lower pressure (and, consequently, lower gas flow rates) the value k(t) sin q approaches 1, and the role played by the surface tension, i.e. by adsorption constant and surfactant concentration, becomes more important.

4.1.5. Estimation of effects caused by initial adsorption The numerical analysis presented above was based on the initial condition of Eq. (16). It was implied, in fact, that the detachment of a bubble does not affect the adsorption layer. However, the bubble detachment and the formation of a fresh meniscus is accompanied by processes which can both increase or decrease the initial adsorption at the new bubble surface. On the one hand, the

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bubble surface expansion process is not uniform. The region where the bubble is attached to the capillary undergoes only a slight expansion during the dead time period. Therefore the surfactant concentration in this region can be higher than the concentration averaged over the whole bubble surface. This can result in a situation where the initial surfactant concentration at the meniscus can be higher than it was at the bubble surface immediately before detachment. On the other hand, when the bubble is detached, rapid expansion of its base takes place, leading to a local decrease of the surfactant concentration. In addition, if the surface coverage by surfactant molecules is not very large, the surface layer at the bubble surface remains mobile and can be washed off by the liquid flow. It follows that, in principle, deviations from the conditions of Eq. (16) in either direction are possible. It can be shown that the solution obtained above is sufficiently stable with respect to small deviations from the conditions of Eq. (16). Fig. 6 illustrates three cases of the adsorption depen-

Fig. 6. The dependence of the adsorption on the angle for various initial conditions: curve 1, G0/Gmax =GT/Gmax; curve 2, G0/Gmax =0.7GT/Gmax; curve 3, G0/Gmax =1.3GT/Gmax; parameters taken for the calculations are a0 =75 mm; l=1 cm; KH =5 × 10 − 6 m; Gmax =5 × 10 − 6 mol m − 2; D =5 × 10 − 10 m2 s − 1; L =50 mm3 s − 1; Vres =50 cm3.

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dence on time for a given gas flow rate: the self-consistent case, G0/Gmax =GT/Gmax, curve 1; the case when the initial concentration is 30% lower than the concentration corresponding to bubble detachment, G0/Gmax =0.7GT/Gmax, curve 2; and the case when the initial concentration is 30% higher than the concentration corresponding to the bubble detachment, G0/Gmax =1.3GT/Gmax, curve 3. One can see that the difference between the curves is quite significant only in the initial time moment, while at the lifetime end this difference becomes rather small, and vanishes completely at bubble detachment. In both ‘perturbed’ cases (curves 2 and 3) the deviation of the adsorption at the lifetime end (q =y/2) from the selfconsistent value does not exceed 3%, that is, 10 times lower that the initial perturbation introduced at the meniscus formation onset. This fact can be explained by the effect of two competitive processes: adsorption decrease due to bubble expansion, and adsorption increase caused by the diffusion of new surfactant molecules towards the bubble surface. On the one hand, the concentration decrease due to bubble expansion is proportional to the initial concentration, i.e. the higher the initial values, the larger is the decrease. On the other hand, the higher is the initial surface concentration, the weaker is the diffusion flux towards the surface. These factors act together and lead to the convergence of the curves at the lifetime end, and their coincidence at the dead time end. It should be noted that the reservoir pressure in all three cases differs from each other by less than 0.5%. Therefore, it can be concluded that the error caused by small violations of the conditions of Eq. (16) are insignificant, which is very advantageous for the experiment.

4.2. Comparison between experimental results and theoretical calculations The theoretical curves presented above are in qualitative agreement with experimental results reported earlier [14]. A quantitative analysis of the theoretical values versus experimental data obtained for the non-ionic surfactant C10DMPO will be presented here to demonstrate the validity of the theory developed for a diffusion adsorption

mechanism. The adsorption and diffusion characteristics of the surfactant C10DMPO were reported elsewhere [15]. The results of our measurements and corresponding theoretical calculations are presented in Fig. 7(a) for a usual hydrophilic capillary and in Fig. 7(b) for a hydrophobised capillary. In spite of different surface properties of the two capillaries, the differences in the experimental data are caused only by the differences in the capillary radius (due to the hydrophobisation of the internal surface of the second capillary). This is obviously due to the fact that short capillaries were used, for which videomonitoring has shown that the detachment of bubbles and formation of the new meniscus does not involve any processes characteristic for long capillaries [12]. Although the solution of the problem of nonstationary adsorption at the surface of a growing bubble is quite cumbersome, the theoretical results obtained for both cases are in a good agreement with experimental data. The best correspondence is observed for low gas flow rates, while when the system approaches the jet regime the coincidence is noticeably worse. This disagreement is possibly caused by various hydrodynamic factors, which become more significant with increasing gas flow velocity through the capillary. These include deviations of the flow profile within the capillary from the Poiseuille profile, turbularisation, gas compressibility and inertial properties of gas and liquid [4–7]. All these factors lead to an increase in the resistance against the gas flow through the capillary and the resistance against the expansion of the bubble in the liquid, resulting therefore in an increase in the measured pressure at a fixed gas flow rate. The differences between the calculated and measured values are especially significant in the transition region to the jet regime with pure water (cf. curves 1 and 1%). This can possibly be ascribed to the differences in the state of the bubble surface. In contrast to bubbles covered by an adsorption layer, which are stabilised due to structural forces, for pure water, consecutive bubbles rapidly following each other can merge together; this can distort the ideal transition to the jet regime. In addition, for pure water, the formation of a series of bubbles is

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Fig. 7. (a, b) The dependence of measured pressure on the gas flow rate for the decyl dimethyl phosphine oxide adsorption with parameters: KH = 8.29× 10 − 6 m; Gmax = 3.64× 10 − 6 mol m − 3; D = 4.8 ×10 − 10 m2 s − 1 [15]; curves 1, 1%, pure water; curves 2, 2%, C0 = 0.1 mol m − 3; curves 3, 3%, C0 = 0.2 mol m − 3; curves 4, 4%, C0 =0.5 mol m − 3; curves 5, 5%, C0 =1 mol m − 3; curves 1 –5, experiment; curves 1%– 5%, theory; capillaries used: a0 =74.7 mm (a); 66.8 mm (b); l= 1 cm; Vres =35 cm3.

possible, separated by long time intervals between two series [9], which is ignored in the presented theory. It is seen from the results in Fig. 7(a,b) that in the studied surfactant concentration range the theoretical curves do not only approach the experimental points in a quantitative way: the curves also perfectly reproduce the shape of the experimental dependencies (the same slope of curves, similar position of the inflection points). Fig. 8 illustrates the experimental dynamic surface tensions for C10DMPO solutions at different concentrations as a function of surface lifetime, obtained for the hydrophilic capillary and the capillary with a hydrophobised internal surface. The difference between the two sets of data are small, being confined within the experimental error limits of the maximum bubble pressure method. These results clearly illustrate the applicability of the method: the lifetime range available for dynamic studies extends over five orders of magnitude.

5. Conclusions The dynamics of bubble growth at the capillary tip is analysed under the mechanism of diffusion controlled adsorption in the framework of the Szyszkowski–Langmuir adsorption model. The general features of the dependencies of adsorption, surface tension, bubble pressure and gas reservoir pressure at all stages of bubble growth on the capillary parameters, gas flow rate, reservoir volume, surfactant concentration and surface activity are studied. The results obtained indicate the approximate character of some assumptions the maximum bubble pressure is based on. It is shown, for example, that for certain conditions the point of maximum bubble pressure does not correspond to the adsorption maximum, and the adsorption minimum does not correspond to the lifetime onset. These theoretical results allow to define the necessary corrections for the calculation procedures used in bubble pressure tensiometry, and to estimate possible experimental errors.

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Fig. 8. The dependence of surface pressure for C10DMPO solutions on the surface lifetimes for the concentrations: curve 1, pure water data (the same for the hydrophilic and hydrophobic capillaries); curves 2, 2%, 0.1 mol m − 3; curves 3, 3%, 0.2 mol m − 3; curves 4, 4%, 1 mol m − 3; curves 5, 5%, 2 mol m − 3. Curves 2 – 5, hydrophilic capillary; 2%– 5%, hydrophobized capillary.

Moreover, the calculations show the validity of particular features as part of the measuring equipment and measurement procedures, e.g. the requirement of a large volume of the gas reservoir and short capillaries as used in the bubble pressure tensiometer MPT2 from Lauda. The essential achievement of the theory is the almost complete agreement between the experimental and theoretical dependencies of reservoir pressure on the gas flow rate for non-ionic surfactants characterised by a diffusion-controlled adsorption mechanism, i.e. the exact localisation of the critical point for the transition from a jet regime to the regime of individual bubbles. Also the influence of the surfactant concentration on the shape of all characteristic dependencies is reproduced in detail by the theory. Therefore, the present study summarises to a certain extent the earlier works [2,4– 14] and demonstrates that the maximum bubble pressure method can be regarded as one of the few experimental techniques which provides a correct interpretation of the actual dynamic adsorption characteristics of surfactants at the solution/gas interface.

Acknowledgements The work was financially supported by research fellowships of the Max Planck Society, an ESA project (FASES), and the Fonds der Chemischen Industrie (RM 400429).

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