Foams hydrostatic equilibrium

When a foam is placed over a liquid phase and is in hydrostatic equilibrium, the pressure pL in Plateau borders at distance z from the liquid level decreases by a certain value ApL with regard to pressure Pl.o on the liquid surface... 

The pressure difference Ap causes an acceleration of foam drainage which runs until it is equalised by the capillary pressure. In order to reach capillary pressures higher than atmospheric pressure fine porous filter should be used and the pressure difference should be created by increasing pressure in the space above the foam. Under these conditions when hydrostatic equilibrium is reached the pressure of the liquid in Plateau borders and the capillary pressure will be, respectively, equal to... 

The analysis in [2] indicates that for a foam at hydrostatic equilibrium that is in contact with the foaming solution, Eqs. (4.15) and (4.16) cannot be employed to calculate the average expansion ratio or the critical foam height, which gives the boundary between the ability of a foam either to drain or to suck in liquid. That is so because the maximum volume of the liquid in a foam is a function of the 5-layer and actually does not depend on the whole foam column height. 

The rate of foam drainage is determined not only by the hydrodynamic characteristics of the foam (border shape and size, liquid phase viscosity, pressure gradient, mobility of the Iiquid/air interface, etc.) but also by the rate of internal foam (foam films and borders) collapse and the breakdown of the foam column. The decrease in the average foam dispersity (respectively the volume) leads the liberation of excess liquid which delays the establishment of hydrostatic equilibrium. However, liquid drainage causes an increase in the capillary and disjoining pressure, both of which accelerate further bubble coalescence and foam column breakdown. 

Fig. 5.12 depicts the lg Wit dependence for NaDoS foams with thin liquid films, h 16 nm, with CBF, h 8 nm and with NBF, h 4.2 nm. The differences between curve 1, 2 and 3, corresponding to the different foam film types, is clearly expressed and is valid not only for the curve slopes but also for t at which a plateau is reached, that itself corresponds to hydrostatic equilibrium. Fig. 5.12,a and 5.12,b plots the initial linear parts of the experimental Wit dependences where the black circles are for NBF, and the black squares are for CBF. It can be seen that the drainage rate is different for the different types of foam films. 

H0 a monotonous decrease in pressure with time is observed, being sharper at higher levels. Data about the rate of pressure change indicate that in a foam which is away from its hydrostatic equilibrium, a linear Ap(H) dependence is realised, the pressure remaining constant and corresponding to the level H0 (Ap is the difference between atmospheric and border pressure). At levels below H0 the pressure increases during the initial period and after a certain period of time begins to decrease. 

Pertsov et al. [19] and Kann [20] have proposed a logarithmic time dependence of the foam expansion ratio in order to determine the rate of bubble expansion. This approach is reasoned by the fact that at hydrostatic equilibrium further increase in foam expansion ratio occurs only when excess liquid is released with decreasing foam dispersity. It was experimentally established that at the final stage of internal foam collapse this increase in foam expansion ratio can be expressed by an exponential function [19,21]... 

An intrinsic property of a polyhedral foam is the reduced pressure in its Plateau borders. At the moment of foam formation the pressure in the borders depends mainly on the foam expansion ratio, dispersity and surface tension (see Eqs. (4.9) and (4.10)). At hydrostatic equilibrium the border pressure is expressed by Eqs. (1.37) and (1.38). 

In order to collect information about the influence of the pressure drop on the lifetime of foams with different types of foam films, foam columns of small heights (2-3 cm) were studied. It was found that the time needed to reach hydrostatic equilibrium pressure (the outflow of the excess liquid ceases) was considerably smaller (5-6 times) than the foam lifetime. This is realised at small pressure drops (up to 5-10 kPa). For that reason it is believed that in these experiments the foam column destruction runs mainly under equilibrium conditions (referring to hydrostatic pressure and drainage). 

Individual structural elements of the foam, such as films and borders, can be under hydrostatic equilibrium and can correspond to a true metastable state. Therefore, when there is no diffusion expansion of bubbles in a monodisperse foam, its state can be regarded as metastable in the whole disperse system. Krotov [5-7] has performed a detailed analysis of the real hydrodynamic stability of polyhedral foam by solving two problems determination of... 

Along with the diffusion foam collapse and the structural rearrangement, the hydrostatic equilibrium of borders is disturbed. This leads to the drainage of the released excess liquid. At low surfactant concentrations (lower than CMC) this liquid is surfactant enriched, compared to the initial solution. The degree of its concentrating depends on the dispersity of the initial foam and on the type of foam films. That is why a stable foam (of a lifetime longer than several minutes) can be formed even at initial surfactant concentration lower than the concentration Cw at which black spots are observed [53,54]. 

It is necessary to bear in mind that although Eqs. (4.9) and (4.10) are rigorously fulfilled at any hydrostatically equilibrium state of the foam, the capillary pressure exerts a strong influence on the drainage and foam stability. At a certain value of the capillary pressure, depending of foam dispersity and the foam film type, the foam lifetime becomes very short and the foam breaks down instantaneously. 

In the general case it is difficult to predict quantitatively the liquid carry-away with a foam. As mentioned in Chapter 5, analytical equations, describing the liquid distribution along the height of the foam column, have been derived only for a foam that is at hydrostatic equilibrium [57-59], Calculation of the liquid content in a non-equilibrium static foam, performed on the basis of the drainage model of Desai and Kumar [60], are given in [61]. 

The rapid coalescence can proceed differently, depending on the foam structure parameters (see Fig. 6.5). For foams whose initial expansion is higher than the equilibrium expansion, the decrease in dispersity does not lead to a local expansion change. Those foams the expansion of which is lower than the equilibrium expansion are characterized by a simultaneous sharp increase in the mean equivalent cell radius and the local expansion. For foams whose expansion is higher than the equilibrium expansion obtained experimental data indicate that the rapid decrease in foam dispersity corresponds to the time before the establishment of hydrostatic equilibrium when there is a rapid outflow of the dispersion medium from a low-expansion foams and its redistribution through the foam column height. 

The capillary pressure that compresses the oil drops in the GPBs gradually increases with the drainage of liquid from the foam. The condition for a hydrostatic equilibrium in the foam column requires the appearance of a vertical gradient of the capillary pressure that opposes the gravity [2,38]. At equilibrium, the capillary pressure at the top of the foam column should be approximately equal to the hydrostatic pressure [13,38] ... 

An important condition which has to be fulfilled when using this method for foam dispersity determination is the absence of an excess hydrostatic pressure in the foam liquid phase. This pressure is equalized to a considerable extent when an equilibrium distribution the foam expansion ratio and the border pressure along the height of the foam column is established. This can be controlled by measuring the pressure in the Plateau borders at a certain level of the foam column by means of a micromanometer. However, if this condition is overlooked, the hydrostatic pressure can introduce a considerable error in the results of bubble size measurements, especially in low expansion ratio foams. Probably, it is the influence of the unrecorded hydrostatic pressure that can explain the lack of correspondence between the bubble size in the foam and the excess pressure in them, observed by Aleynikov[49]. The... 

Probably, the first condition can be realised at the moment of foam formation, but then the foam becomes hydrostatically non-equilibrium and h(t) ho. For very stable foams this condition can be realised at high capillary pressures (pc 104 Pa). The second condition can be fulfilled in a foam with thin films at not very high capillary pressures. In order to have the condition r = ro = const conformed with it is necessary that... 

Experimental values of aggregative foam stability characteristics are presented in Table 6.1. The rate constants of rapid and slow coalescence can be assigned, in accordance with the above considerations, to hydrostatically non-equilibrium and equilibrium conditions, respectively. Table 6.1 Aggregative foam stability characteristics... 

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