Monodisperse foam

Mathes, H., Plath, P. J., Generation of monodisperse foams using a micro-structured static mixer, in Proceedings of the Tunisian-German Conference of Smart Systems and Devices, submitted for publication (27-30 March 2001), Hammamet, Tunisia. 

In a monodisperse foam the deformation of spherical bubbles and formation of films at the places of their contact starts when the gas content in the system reaches - 50% (vol.) for simple cubic bubble packing or 74% for close (face-centred) cubic or hexagonal packing (foam expansion ratio - 4). In a polydisperse foam the transition to polyhedral structure starts at expansion ratio n - 10-20, according to [ 10] but, as reported in [51], this can occur at n < 4, the latter being more probable. The structure which corresponds to the transition of bubbles from spherical to polyhedral shape is called occasionally honeycomb structure. 

The volume and shape of Plateau borders depend on the expansion ratio of the foam. In a spherical monodisperse foam with close packing of bubbles all air/liquid interfaces are spherical and the liquid volume which belongs to one cell can be derived from the difference between the volumes of the corresponding polyhedron (for example, a dodecahedron) and the inscribed in it sphere, having in mind the co-ordination number of the foam cell. 

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... 

Polydispersity. If a liquid is injected into polydisperse cellular foam (a process which decreases the foam multiplicity) until the foam becomes spherical with the same size distribution of bubbles, then the obtained spherical foam is said to be equivalent. The equivalent foam is characterized by the multiplicity called the minimum multiplicity Kmjn. Obviously, the minimum multiplicity of polydisperse foam is larger than that of monodisperse foam, since in gaps between densely packed spheres of the same largest size, spheres of smaller sizes can be located. Thus the value K a can be used as a quantitative measure of polydispersity [214], While K a = 3.86 for monodisperse foam, for actual polydisperse foam we have 10 to 15 in practice Kma never exceeds 20 [480],... 

According to (7.1.15), the capillary rarefaction depends on the mean curvature s of the internal foam surface at the nodes of the foam structure. This variable was calculated in [378] for a monodisperse foam with cells modeled by pentagonal dodecahedra ... 

In the case of monodisperse foam, another approximation formula for the interface mean curvature was obtained in [156] by using the deformation theory [430] and the rounded dodecahedron model namely,... 

For monodisperse foam, the area 5b of the cross-section of the Plateau border is... 

In these formulas, K = 1/V. The coefficient 4//fmin is introduced for approximate consideration of polydispersity, since it is well known [481] that the hydroconductivity of polydisperse polyhedral foam is 1.5 to 2 times less than that of monodisperse foam. At the same time, substantiation of formula (7.4.3) for polydisperse foams is doubtful, since polydisperse foam can hardly be polyhedral. 

If we estimate the specific interfacial area e for a monodisperse foam by formula (7.1.4), then we can rewrite this relation as... 

One may also obtain a monodisperse foam by bubbling gas through a single capillary. This foam will become polydisperse upon further film rupture... 

Foam is a disperse system in which the dispersed phase is a gas (most commonly air) and the dispersion medium is a liquid (for aqueous foams, it is water). Foam structure and foam properties have been a subject of a number of comprehensive reviews [6, 17, 18]. From the viewpoint of practical applications, aqueous foams can be, provisionally, divided into two big classes dynamic (bubble) foams which are stable only when gas is constantly being dispersed in the liquid 2) medium and high-expansion foams capable of maintaining the volume during several hours or even days. In general, the basic surface science rules are established in foam models foam films, monodisperse foams in which the dispersed phase is in the form of spheres (bubble foams) or polyhedral (high-expansion foams). Meanwhile, real foams are considerably different from these models. First of all, the main foam structure parameters (dispersity, expansion, foam film thickness, pressure in the Plateau-Gibbs boarders) depend... 

Morrison and Ross (70) have indicated that, while Eqs (56-57) are undoubtedly correct for monodisperse foams, a rigorous proof of their validity for polydisperse systems was lacking. Such proof has since been provided by Hollinger (71), Crowley (72), and Crowley and Hall (73). 

The simplest way to form a nearly ideal foam is to introduce the gas into the liquid through a capillary tube. In that way individual bubbles of equal (almost) size will break off from the capillary tip under the action of surface tension. The process, however, must be slow in order to ensure that interfadal equihbrium is achieved for each bubble otherwise a monodisperse foam will not be produced. A much more rapid, but less controllable, procedure is to bubble gas into the system through a porous plug. In that process a highly polydisperse foam will result since many small bubbles will have the opportunity to coalesce while still attached to the plug. Even less consistent results will be obtained for foams produced by agitation. 

Foams are dispersions of bubbles where neighboring bubbles touch each other and form a jammed solid-like closed packing [62]. They are characterized by polyhedral bubbles and a high gas-phase fraction. When the gas fraction is relatively low, the bubbles retain their spherical shape (unless they are severely confined) and bubble suspensions are obtained. Monodisperse foams are advantageous, since coalescence, driven by the difference of Laplace pressure between neighboring bubbles, is reduced. Due to the high interfacial tension between gases and liquids, surfactants are usually introduced in the liquid phase to facilitate bubble formation and reduce coalescence. 

The structure of a foam depends on the relative proportions of the gas and liquid. Bubbles are spherical in foams containing a large amount of liquid (>25%). As the percentage of liquid decreases, the bubbles become less spherical. Foams with less than 2% liquid are almost completely polyhedral. For monodispersed foams (composed of identical bubbles), a gas volume fraction of 0.74 is considered to be the limit beyond which the bubbles cease to be spherical. This is because the highest volume fraction that identical... 

Because coalescence never occurs at the foam/liquid interface, = Ro, where Rq is the radius of a bubble in the freshly formed monodisperse foam. 

In an initially monodispersed foam, the bubble size is uniform within an element (i.e., all bubbles within the element have the same radius (i ) and... 

As mentioned earlier, coalescence gives rise to a bubble size distribution in an initially monodispersed foam. The number of bubbles decreases and the mean bubble volume increases. Owing to the complexity involved in computing the evolution of the bubble size distribution, we restrict our treatment to the calculation of the mean bubble volume F, which is required in the formulation of the bulk conservation equations. This is considered next. 

The mean film radius was taken to be 1 fo = 0.606i o> where Rq is the bubble radius (taken to be 0.2 mm) in the monodispersed foam before coalescence starts. The standard deviation 5 was taken to be 0.51 fo, and i Fmino and i Fc were taken to be 0.99/ fo and 1.01i FO. respectively, except when their effects on the results were examined. This represents a deviation of about 1% from the mean value. These values are arbitrary as we have no way of iden-... 

The upper limit for R beyond which the distribution is assumed to be Dirac delta Largest value of the film radius Smallest value of the film radius at a given level Molar gas constant A mean radius used in models for interbubble gas diifusion Bubble radius in the initially monodispersed foam Mean bubble radius Standard deviation Time... 

Various methods of foam generation that have been used in this context are first described, together with an indication of their respective advantages, disadvantages, and limitations. Although rarely considered in studies of antifoam action, we briefly include the issue of measurement of bubble size distributions because few methods of foam generation conveniently produce monodisperse foam. 

Comparison of this theory with experiment is at present largely limited to the prediction of Equation 5.31, which is of course no advance on the simple phenomenological approach. However, in principle, some of the quantities in Equation 5.26 are either directly accessible by experiment or could be calculated. Clearly the effect of antifoam on bubble size distribution P(r ) could be measured but rarely is— no reliable data are apparently to be found in the published literature. Alternatively the effect of antifoam concentration on monodisperse foam volumes would be instructive. In this case. Equation 5.27 reduces to... 

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