Trapezoidal area change

There is an extra oscillation module, based on direct measurements of the capillary pressure, which operates from 1 to 150 Hz. There is also an additional accessory for the PAT1 for low-frequency oscillations. The range of surface and interfacial tension is 1 to 1000 mN/m with a resolution of 0.1 mN/m. The instrument allows for transient relaxation measurements, using perturbations such as ramp, square pulse, or trapezoidal area changes. 

Fig. 6.1. Schematic of different transient area changes (solid lines) and the corresponding interfacial tension responses (dotted lines) (a) - step type, (b) - ramp type, (c) - square pulse, (d) -trapezoidal change...

It was shown by Loglio et al. (1991a) that the most useful disturbance for interfacial relaxation experiments is the trapezoidal area change. For time regimes realised in most of the transient relaxation experiments the trapezoidal area change can be approximated adequately by a square pulse. For the square pulse area change we obtain ... 

The software driven apparatus allows different types of area changes step and ramp type, square pulse and trapezoidal as well as sinusoidal area deformations. The construction ensures that area changes are almost isotropic. Area changes used in transient and harmonic relaxation experiments are of the order of 1 to 5%. The surface tension response measured via the Wilhelmy balance has an accuracy of better than 0.1 mN/m. 

More convenient for relaxation studies are the square pulse (cf. Fig. 6.1, (c)) or its practical equivalence the trapezoidal area change (cf Fig. 6.1, (d)). The F-transforms of the respective area changes are as follows (cf Miller et al. 1991) ... 

The result for the trapezoidal area change is given in Chapter 6. 

The whole theoretical treatment of the derivation of interfacial response functions was discussed recently by Miller et al. [160]. It was shown by Loglio et al. [144, 161, 162] that exchange of matter functions derived for harmonic disturbances can be applied to transient ones. As the result for a diffusion-controlled exchange of matter, using the theory of Lucassen and van den Tempel [157], the following functions result for a trapezoidal area change [162]... 

In Fig. 17 some typieal interfacial tension changes are shown which have been obtained for a trapezoidal area change of an aqueous protein solution drop in tetradecane. The elasticity can be calculated from the initial jump of the y t) dependence immediately following an expansion or compression. 

As mentioned above the oscillating drop or bubble method, based on profile analysis tensiometry, is the most recently developed method to investigate the surface relaxation of soluble adsorption layers. By increasing/decreasing the volume of a pendent drop or bubble, a variety of area changes can be performed, such as step, square pulse, ramp type, trapezoidal, and of course harmonic area changes at low frequencies. 

Here (0 is the oscillation frequency, and the parameter cOb is the characteristic frequency, which is inverse proportinal to the diffusion relaxation time Xd given in Eq. (35). This characteristic frequency exists also for any transient relaxation processes. The interfacial response functions for a number of transient relaxations were discussed recently by Loglio et al. (2001). Among these, the trapezoidal area change is the most general perturbation which contains area changes such as the step or ramp type and the square pulse as particular cases. 

Drop and bubble shape tensiometry is a modem and very effective tool for measuring dynamic and static interfacial tensions. An automatic instrument with an accurate computer controlled dosing system is discussed in detail. Due to an active control loop experiments under various conditions can be performed constant drop/bubble volume, surface area, or height, trapezoidal, ramp type, step type and sinusoidal area changes. The theoretical basis of the method, the fitting procedure to the Gauss-Laplace equation and the key procedures for calibration of the instrument are analysed and described. 

Although no direct comparison with other commercial products are given we can state that the instrument PATl discussed here has the best features in respect to interfacial rheology studies. It provides a comfortable function generator for any type of transient and harmonic relaxation studies and also the theoretical tools to analyse trapezoidal and sinusoidal relaxation experiments. The on-line control of the interfacial area changes is very accurate and the oscillations performed in the range between 0.01 and 0.2 Hz are ideally smooth sinusoidal functions in contrast to experiments performed with other instruments. 

The first chore is to change all the units so they re the same — right now they re in inches and feet. You could change everything to inches, but a cubic foot is 12 x 12 x 12 cubic inches, and that makes the numbers a bit big to work with. Instead, you can change the 6 and 9 inches to feet and work with relatively nice fractions. The area of the trapezoidal end of the gutter is... 

Viscosity is directly related to structural levels. Comparing available values of viscosity rj with the density of the active faults of the Cenozoic era within the trapezoid area 5x5° and making an evaluation, it is not difficult for us to notice the inverse nonlinear tendency of mutual change between them (Fig. 1 (a)). 

All the systems were thixotropic. Both the ascending and the descending branches of the flow curves had plastic properties. The thixotropic area was determined from the area delimited by the two curves with the trapezoidal method. A certain super-molecular arrangement (structure) is postulated within the system by the theory of thixotropy, and this structure will change owing to the effect of a deforming force. 

---