Border radius of curvature

In order to compare the structural parameters of the foam model studied by Kruglyakov et al. [18] with the respective parameters of a real polydisperse foam (individual bubbles of different degree of polyherdisity) Kachalova et. al. [19] performed measurements of the average border radius of curvature of foams with variable expansion ratio. The foam studied, generated by the set-up shown in Fig. 1.4, was obtained from a nonionic surfactant solution of Triton-X-100 (a commercial product) to which NaCl (0.4 mol dm 3) was added. The expansion ratio was determined conductometrically with correction of the change in electrolyte concentration due to the internal foam destruction. The electrolyte concentration... 

Another technique employed in the evaluation of Plateau border pressure, introduced by Lalchev et al. [35] and Khristov et al. [36], involves measurement of border radius of curvature from microphotographs and further calculation from Eq. (1.40). 

For several values of the pressure drop the real profile of border radius of curvature does not correspond to the theoretical, calculated from Eq. (5.23). That is why instead of using Eq. (5.23) the experimental r t) dependence was used to calculate the number of borders. For example, for a NaDoS foam with 0.4 mol dm 3 NaCl the r(l) dependence is in good agreement with the next formula... 

Fig. 5.3. Profile of border radius of curvature when liquid flows through a foam under pressure drop (a)...

The foam expansion ratio can be characterised by the liquid volume fraction in the foam, which is the sum of the volume fractions of the films, plateau borders and vertexes. Alternatively, the foam density can be used as a measure of the foam expansion ratio. The reduced pressure in the foam plateau border can be measured using a capillary manometer [4], while the bubble size and shape distribution in a foam can be determined by microphotography of the foam. Information about the liquid distribution between films and plateau borders is obtained from the data on the border radius of curvature, the film thickness, and the film-to-plateau border number ratio obtained in an elementary foam cell. 

Fig. 3. Two-dimensional schematic illustrating the distribution of Hquid between the Plateau borders and the films separating three adjacent gas bubbles. The radius of curvature r of the interface at the Plateau border depends on the Hquid content and the competition between surface tension and interfacial forces, (a) Flat films and highly curved borders occur for dry foams with strong interfacial forces, (b) Nearly spherical bubbles occur for wet foams where...

The radius of curvature of plateau border can be related to the area of the plateau border Op and the film thickness xp for a regular dodecahedral arrangement from geometric considerations by (9),... 

From the above equation, the variation of equilibrium disjoining pressure and the radius of curvature of plateau border with position for a concentrated emulsion can be obtained. If the polarizabilities of the oil, water and the adsorbed protein layer (the effective Hamaker constants), the net charge of protein molecule, ionic strength, protein-solvent interaction and the thickness of the adsorbed protein layer are known, the disjoining pressure II(x/7) can be related to the film thickness using equations 9 -20. The variation of equilitnium film thickness with position in the emulsion can then be calculated. From the knowledge of r and Xp, the variation of cross sectional area of plateau border Qp and the continuous phase liquid holdup e with position can then be calculated using equations 7 and 21 respectively. The results of such calculations for different parameters are presented in the next session. 

In case of a fully polyhedral foam R/r 1 when the contribution of the vertex (by volume and length) can be neglected compared to the total volume of the liquid in one cell, the border along the whole length between the vertexes has identical area of the cross-section, and a single radius of curvature. The area of the cross-section of border A is determined by the radius of curvature of the border (r) and the film thickness (h) (Fig. 1.9)... 

If the condition for polyhedricity R/r 1 is not fulfilled, the radius of curvature and the area of the cross-section become dependent on the co-ordinates along the length of the Plateau border. Analytical dependence of the radius of curvature on the co-ordinates (the border profile) at different foam expansion ratio is not found. 

Based on the studies of border and film shape in the dodecahedral model Kruglyakov et al. [18] and Kachalova et al. [19] have proposed an expression for foam expansion ratio, using a cylindrical border model with the same cross-sectional radius of curvature. The volume of excess vertex parts was considered in order to estimate the effect of the longitudinal radius of curvature on the border shape... 

Analysis of other border profile models (linear r(l) dependence with constant or variable radius of curvature at border mouth, the relation (5.23), etc.) shows that the simplest analytical equation for the profiles of both pressure and radius of curvature is obtained if the function r(l) is given as a parabolic expression r2 = 2p l (where p is the parabolic parameter). This equation is in good agreement with the experimental data. The parabolic parameter can be determined from the experimental r [) dependence measuring the pressure at various levels in the foam. 

Fig. S.4. Parabolic model of border profile at variable (a) and constant (b) radius of curvature at border...

The second model of a parabolic profile (at constant radius of curvature at the border mouth) gives the following expression about the pressure gradient... 

Eqs. (5.42) and (5.44) can be used to calculate the time for establishing a given pressure (or for reaching a given radius of curvature) at the upper foam layer (border outset). Here, the parabolic parameter /0 values are not required for the calculation. However, these formulae, can be used only if during the whole drainage process the border profile is described by the parabolic function. 

The extrapolation of the lines in Fig. 5.9,a to a zero value of l shows that the minimum radius of curvature rmin (radius at border mouth) remains constant with time while the parabolic parameters p and lo change. This corresponds to a border model with a parabolic profile and constant radius of curvature at the border mouth. With the increase in foam... 

Equations connecting the accumulation ratio to parameters of foam structure (dispersity, expansion ratio, radius of curvature of Plateau borders) can be derived from the balance equations of the surfactant and the liquid phase, and the data on foam structure. 

Formulas for geometric parameters. The radius of curvature R of nodal menisci is one of the most important quantitative characteristics of foam. The other quantity connected with the former is the radius Rb of Plateau borders. 

Although, according to (7.1.15), the capillary rarefaction is a local variable, its values at nodes and in Plateau borders are equalized rather rapidly due to the flow of liquid, which allows one to assume that the capillary rarefaction at nodes and in Plateau borders is a unified integral characteristic of foam. This means that the mean curvature of menisci at nodes and borders is the same. However, since nodes possess a spherical curvature and Plateau borders a cylindrical curvature, the radius of curvature of the latter must be two times less than that of the nodal menisci,... 

Figure 3 shows an additional pressure variation between the Plateau borders (P0), where the radius of curvature is relatively small (P1B), and in the more laminar part of the lamella (PA), where the radius of curvature is relatively large (P1A). In the Figure 3 illustration, the principal radii of curvature have been assumed to be equal at a given location in the lamella... 

Figure 14.16 Plateau borders formed by latex particles as they deform prior to coming into contact, (a) A cross-section through a layer of nearly dry film showing deformed particles, (b) An expanded view of a plateau border with bilayers, showing the radius of curvature and the bilayer thickness, two parameters obtained from the wide-ai neutron-scattering e]q)eriments of ref. 31. (Reprinted with permission from ref. [31]. Copyright 1992 American Chemical Society.)...

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