Effective surface age

Fainerman et al. (1993c, 1994b) presented an analysis of the adsorption proeess at the bubble surface in order to derive a relation between the bubble life time and the effective surface age Tj. The basis of this analysis is the condition of constant pressure p at any moment of bubble life in the time interval 0 < t < t. 

At surface tensions y close to the one of the solvent y, the effective surface age is coincides with the bubble life time. The function 1 / (24 +1) changes from 1 down to 0.5 at low surface tensions, and therefore, the effective surface age z at small surface tensions amounts to about 50% of the bubble life time z. 

The design of a maximum bubble pressure method for high bubble formation frequencies must address three main problems the measurement of bubble pressure, the measurement of bubble formation frequency, and the estimation of surface lifetime and effective surface age. 

As one can see, Eq. (5.20) involves only experimentally available values. The critical point in the dependence p on L can easily be located. In the software the location is automatically calculated by an algorithm based on the Poiseuille law. The calculation of the effective surface age (effective adsorption time) can be made using Eq. 4.62, derived in Chapter 4. The derivation of the relative surface deformation rate is based on the eondition p = eonst. For values of y not too close to y (for example, for aqueous solutions below 60 mN/m), is approximately equal to 0.5, and consequently x, t/2. 

Most of the instruments, based on the principle of maximum bubble pressure, do not allow the effective surface age to be calculated because the conditions during the bubble formation are unknown. These methods yield only a dependence of surface tension on bubble frequency or bubble time x. The graph in Fig. 5.14 shows the remarkable differences of the three possible form of final data y(Xb), y(x), and y(Xg). 

Fig. 5.14 Dynamic surface tension y in dependence of bubble time Tj, (— ), bubble life time t (—), and effective surface age Tj, ( ) (schematically)...

The determination of the effective surface age is the key for comparison of results obtained by different experimental techniques. If for example the drop volume technique is used in its "classical" version, which is based on continuously growing drops, dynamic surface tensions are obtained as a function of drop formation time. It was shown in the previous chapter, that the process of adsorption at the surface of a growing drop is overlapped by a radial flow inside the drop, which changes the diffusion profile. In addition, the drop area increases and... 

Similar situations exist for other methods. If quantitative theories are applied to data interpretation, all peculiarities of a method have to be considered. However, this does not allow a direct comparison of experimental results from different methods, because the time scales are different and depend on the specific experimental conditions. A solution to this problem is the determination of the effective surface age, which then allows a direct comparison between experimental data independent of their origin. 

Fig. S.26 Schematic relations between specific experimental time scales and the effective surface age (adsorption time) for several methods drop volume (1), maximum bubble pressure (2), growing drop (3)...

The figure contains the original data as well as the recalculated results in form of surface tension as a function of the effective surface age x. The original data y(t) of the bubble pressure method are transferred into y(x3) by more or less a shift in the y/log t - plot, according to Eq. (4.62). The drop volume data were corrected first with respect to the hydrodynamic effect at drop formation times t < 30 s using Eq. (5.17) and then the effective surface age x was calculated from the drop formation time tj p using the approximate relation x = tdrop/3 (Cf. Section 5.9.1.). 

It becomes clear that the apparent surface tension is significantly increased by up to 1 mN/m at drop times up to about 10 s due to the hydrodynamic effect. Only the corrected dynamic surface tensions y as functions of the recalculated effective surface age are displayed in the following Figs 5.31 and Fig. 5.32. 

Summarising, the agreement of experimental data, obtained with methods of different physical principles, was shown to be excellent when displayed as functions of the effective surface age. 

In the book by Joos [16] as well as in original papers, some special cases of this general approach have been discussed. It was shown that such stress relaxation experiments are well suited for studying the dilational rheology of interfacial layers, which yield the dilational elasticity as a function of the effective surface age teff... 

There are a number of techniques available to measure the surface or interfacial tension of liquid systems, which together cover a wide range of time. In many cases, several methods are required in order to receive the complete surface tension time dependence of a surfactant system. One of the important points in this respect is that the data obtained from different experimental techniques have to be recalculated such that a common time scale results, i.e. one has to calculate the effective surface age from the experimental time, which is typically determined by the condition of the methods. For example, the maximum bubble pressure... 

The lifetime, the time period from the moment of bubble detachment and formation of a new bubble up to the moment of maximum pressure results as the difference t = tb - tj. From this lifetime t the effective surface age teff can be calculated via the relationship... 

The effective surface age has been determined on the basis of a diffusion controlled adsorption model. Similar to the maximum bubble pressure technique, where the bubble time is longer than teff, the drop formation time t is also significantly longer than the effective age teff and can be obtained as teff=3t/7 [185], This estimation assumes a radial flow inside and outside the growing drop and a homogeneous expansion of the drop surface. 

Dynamic surface tensions of different aqueous Triton X-n solutions plotted as function of effective surface age ten symbols are the same as in Fig. 4.21... 

The data shown in Fig. 21 are partially those of Fig. 20, however, plotted as a function of the effective surface age and restricted to the short time interval t < 5 ms. This demonstrate that the maximum bubble pressure technique is able to provide results in the sub-millisecond time range. 

Figure 12.8. Dynamic surface tension data of a dimethyldode-cylphosphine oxide solution, as measured by the drop volume method (TVT2, LAUDA, Germany) 0, dynamic mode , quasistatic mode x, recalculated to the effective surface age...

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