Mobile interface

An attempt has been made by Tsouris and Tavlarides[5611 to improve previous models for breakup and coalescence of droplets in turbulent dispersions based on existing frameworks and recent advances. In both the breakup and coalescence models, two-step mecha-nisms were considered. A droplet breakup function was introduced as a product of droplet-eddy collision frequency and breakup efficiency that reflect the energetics of turbulent liquid-liquid dispersions. Similarly, a coalescencefunction was defined as a product of droplet-droplet collision frequency and coalescence efficiency. The existing coalescence efficiency model was modified to account for the effects of film drainage on droplets with partially mobile interfaces. A probability density function for secondary droplets was also proposed on the basis of the energy requirements for the formation of secondary droplets. These models eliminated several inconsistencies in previous studies, and are applicable to dense dispersions. 

Substitution of Equations (36) and (37) into Equation (35) generates a complicated differential equation with a solution that relates the shape of an axially symmetrical interface to y. In principle, then, Equation (35) permits us to understand the shapes assumed by mobile interfaces and suggests that y might be measurable through a study of these shapes. We do not pursue this any further at this point, but return to the question of the shape of deformable surfaces in Section 6.8b. In the next section we examine another consequence of the fact that curved surfaces experience an extra pressure because of the tension in the surface. We know from experience that many thermodynamic phenomena are pressure sensitive. Next we examine the effecl of the increment in pressure small particles experience due to surface curvature on their thermodynamic properties. 

A second way of classifying the material is on the basis of the experimental methods involved. For mobile interfaces, surface tension is easily measured. For these it is easiest to examine the surface tension-adsorption relationship starting with surface tension data. When insoluble surface films are involved, we shall see how the difference in y between a clean surface and one with an adsorbed film may be measured directly. For solid surfaces, surface tension is not readily available from experiments. In this case adsorption may be measurable directly, and the relationship between adsorption and surface tension may be examined from the reverse perspective. 

We have noted previously that measuring 7 as a function of concentration is a convenient means of determining the surface excess of a substance at a mobile interface. In view of the complications arising from charge considerations, the need for an independent method for measuring surface excess becomes apparent. Some elaborate techniques have been developed that involve skimming a thin layer off the surface of a solution and comparing its concentration with that of the bulk solution. 

GAINES, G.L., Insoluble Monolayers at Liquid-Gas Interfaces, Interscience (1966) ALEXANDER, A.E. and HIBBERD, G.E., Determination of properties of insoluble monolayers at mobile interfaces , in reference 17, Part 5, pp. 557-589... 

Direct measurement of kL is not possible but, given separate evaluations of k a and a, kL can be obtained. Such a procedure was first adopted by Calderbank (13) using extensive physical absorption measurements. He showed that agitation intensity and bubble size had no effect on the mass transfer coefficient, kL, but large bubbles with mobile interfaces have greater mass transfer coefficients than small rigid bubbles. 

Thus, Sh oc (P/F)1/4. For particles with a mobile interface (large particles), alternate relations need to be used. These are... 

For "thick films" where disjoining pressure effects are negligible, the constant B in Equation 6 ranges from 1.337 for the perfectly mobile interface case to 2.123 for the immobile interface case. 

Of direct interest in the study here is the pressure drop across the bubble in the liquid phase. For the mobile interface case this is given in dimensionless form by... 

It also turns out that for the mobile interface case the pressure in the gas phase across the lamella of Figure 5(b) is given by... 

Equation 16 indicates that for bubble trains with mobile interfaces, the pressure drop is related to the number of lamellae in the train (or number of bubbles per unit length). This important observation, first noted by Hirasaki and Lawson (1),... 

Viscous Pressure Drop. For continuous bubble trains with perfectly mobile interfaces moving through a given Dm channel, the dynamic pressure drop in the gas (foam) phase over a single tube segment D follows from Equation 16 ... 

Here, the dependence on displacement rate (through Ca) is stronger than that for the mobile interface case. Also, bubble size does not appear in this relation as in Equation 44. Further, it is noted that ks varies with(j)l/3, , and a 2/3, the latter being different from those found in the mobile interface case. 

This expression is valid for a partially mobile interface analogous expressions hold for fully mobile and immobile interfaces (Minale et al. 1997). Here, he is the critical thickness of the hquid gap between droplets at which coalescence occurs. Theoretically, one expects he (AHa/STrr), where Ah is the Hamaker constant (Chesters 1991). A value he = 0.2pm gives the solid line in Fig. 9-10, this value of he is an order of magnitude larger than predicted, no doubt because he is used as a fitting parameter to accomodate rough approximations used in the theory. 

When a protein adsorbs from a solution in which the pH is close to its isoelectric point, the rate of adsorption at mobile interfaces is controlled by the rate of diffusion to the interface and the interfacial pressure barrier. However, when the protein molecule takes on a net electrical charge, an additional barrier to adsorption appears, owing to the electrical potential set up at the interface by the adsorbed protein. 

The Gibbs equation contains three independent variables T, a, and p (defined either via concentration or pressure, c or p, respectively), and is a typical thermodynamic relationship. Therefore, it is not possible to retrieve any particular (quantitative) data without having additional information. In order to establish a direct relationship between any two of these three variables, it is necessary to have an independent expression relating them. The latter may be in a form of an empirical relationship, based on experimental studies of the interfacial phenomena (or the experimental data themselves). In such cases the Gibbs equation allows one to establish the dependencies that are difficult to obtain from experiments by using other experimentally determined relationships. For example, the surface tension is relatively easy to measure at mobile interfaces, such as liquid - gas and liquid - liquid ones (see Chapter I). For water soluble surfactants these measurements yield the surface tension as a function of concentration (i.e., the surface tension isotherm). The Gibbs equation allows one then to convert the surface tension isotherm to the adsorption isotherm, T (c), which is difficult to obtain experimentally. 

When adsorption takes place at the surface of a highly porous solid adsorbent, the surface excess can be readily measured, e.g. by measuring the increase in the adsorbent weight in the case of adsorption from vapor, or by following the decrease in the adsorbate concentration during adsorption from solutions. Studies of the adsorption dependence on vapor pressure (or solution concentration) reveal T(p) (or T(c)) adsorption isotherms. In both cases the two-dimensional pressure isotherm can be established from the Gibbs equation (see Chapter II, 2, and Chapter VII, 4). Therefore, it is as a rule possible to establish the dependence between the two of three variables present in the Gibbs equation the surface tension isotherm, a(c), for mobile interfaces and soluble surfactants, the two-dimensional pressure, tt(c), isotherm for insoluble... 

The necessary condition for spontaneous formation of disperse system and condition of its equilibrium with a macroscopic phase can be also obtained by utilizing the concepts of theory of fluctuations. This may be conveniently illustrated by the example of a highly mobile interface, such as liquid - liquid or liquid-vapor. The surface of liquid is not completely flat thermal fluctuations result in the appearance of capillary waves. It was shown by L. Mandelshtam (1914) that in the vicinity of critical point, e.g. around the temperature corresponding to a complete mixing of two liquids, the interface acquires substantial... 

In concentrated systems with highly mobile interfaces (foams and emulsions) capillary phenomena of the first kind, related to the surface curvature in regions of film - macroscopic phase contact or in the regions where three films come into contact, may play a significant role in the energy and dynamics of film thinning. As shown in Fig. VII-2, a concave surface is formed in these types of regions. Under this surface the pressure is lowered by the amount equal to capillary pressure (see Chapter I, 3),... 

In the case of highly mobile interface between dispersed phase and dispersion medium (as in foams and emulsions) the condition of zero fluid flow velocity at interface (non-slip condition), determining the validity of Reynolds equation, may not be obeyed. In this case the decrease in the film thickness occurs at a greater rate. However, in foam and emulsion films stabilized by surfactant adsorption layers the conditions of fluid outflow from an interlayer are close to those of outflow from a gap between solid surfaces even in cases when surfactant molecules do not form a continuous solid-like film. This is the case because at surfactant adsorption below Tmax the motion of fluid surface leads to the transfer of some portions of surfactant adsorption layer from central regions of film to peripherical ones, adjacent to the Gibbs-Plateau channels. As a result, the value of adsorption decreases in the center of film, but increases at the periphery, which stipulates the appearance of the surface... 

If the interface is stationary, or if it translates without accelerating, then a steady-state force balance given by equation (8-180) states that the sum of all surface-related forces acting on the interface must vanish. Body forces are not an issue because the system (i.e., the gas-liquid interface) exhibits negligible volume. The total mass flux vector of an adjacent phase relative to a mobile interface is... 

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