Capillarity critical

A homogeneous metastable phase is always stable with respect to the fonnation of infinitesimal droplets, provided the surface tension a is positive. Between this extreme and the other thennodynamic equilibrium state, which is inhomogeneous and consists of two coexisting phases, a critical size droplet state exists, which is in unstable equilibrium. In the classical theory, one makes the capillarity approxunation the critical droplet is assumed homogeneous up to the boundary separating it from the metastable background and is assumed to be the same as the new phase in the bulk. Then the work of fonnation W R) of such a droplet of arbitrary radius R is the sum of the... 

The specific surface area of a ceramic powder can be measured by gas adsorption. Gas adsorption processes may be classified as physical or chemical, depending on the nature of atomic forces involved. Chemical adsorption (e.g., H2O and AI2O3) is caused by chemical reaction at the surface. Physical adsorption (e.g., N2 on AI2O3) is caused by molecular interaction forces and is important only at a temperature below the critical temperature of the gas. With physical adsorption the heat erf adsorption is on the same order of magnitude as that for liquefaction of the gas. Because the adsorption forces are weak and similar to liquefaction, the capillarity of the pore structure effects the adsorbed amount. The quantity of gas adsorbed in the monolayer allows the calculation of the specific surface area. The monolayer capacity (V ,) must be determined when a second layer is forming before the first layer is complete. Theories to describe the adsorption process are based on simplified models of gas adsorption and of the solid surface and pore structure. 

Figure 1, Composition of the critical cluster (bubble of critical size), Xgas, thermodynamic driving force of critical bubble formation, CJ,radius of the critical bubble, Rc, and work of critical bubble formation, [J Gc, computed for the case of boiling in binary liquid-gas solutions in dependence on supersturation here expressed via the density of the liquid puq (for the details see Ref 21). By the number (1), the results are shown computed via the classical Gibbs approach employing the capillarity approximation, number (2) refers to computations via the generalized Gibbs approach and number (3) to computations via the van der Waals square gradient density functional method.

It is evident that the question of determination of the size of the critical cluster is a rather non-trivial problem, it depends qualitatively on the definition of the size, i.e., for which of the dividing surfaces the parameter has to be computed and which assumptions are employed in its determination. Employing the classical Gibbs method and the capillarity approximation, we arrive at curve 1 in Fig.lc. 

Nearly all experimental coexistence curves, whether from liquid-gas equilibrium, liquid mixtures, order-disorder in alloys, or in ferromagnetic materials, are far from parabolic, and more nearly cubic, even far below the critical temperature. This was known for fluid systems, at least to some experimentalists, more than one hundred years ago. Verschaffelt (1900), from a careful analysis of data (pressure-volume and densities) on isopentane, concluded that the best fit was with p = 0.34 and 5 = 4.26, far from the classical values. Van Laar apparently rejected this conclusion, believing that, at least very close to the critical temperature, the coexistence curve must become parabolic. Even earlier, van der Waals, who had derived a classical theory of capillarity with a surface-tension exponent of 3/2, found (1893)... 

The principal limitation to the classical theory is seen as the capillarity approximation, the attribution of bulk properties to the critical cluster (see Figure 11.8). Most modifications to the classical theory retain the basic capillarity approximation but introduce correction factors to the model [e.g., see Hale (1986), Dillmann and Meier (1989, 1991), and Delale and Meier (1993)]. 

It is convenient to express the capillarity number in its reduced form K = K / K, where the critical capillary number, K., is defined as the minimum capillarity number sufficient to cause breakup of the deformed drop. Many experimental studies have been carried out to establish dependency of K on X. For simple shear and uniaxial extensional flow, De Bruijn [1989] found that droplets break most easily when 0.1 4 ... 

Table 7.5. Parameters of the critical capillarity number for drop burst in shear and extension in Newtonian systems [R. A. de Bruijn, 1989].

The microrheology makes it possible to expect that (i) The drop size is influenced by the following variables viscosity and elasticity ratios, dynamic interfacial tension coefficient, critical capillarity number, composition, flow field type, and flow field intensity (ii) In Newtonian liquid systems subjected to a simple shear field, the drop breaks the easiest when the viscosity ratio falls within the range 0.3 < A- < 1.5, while drops having A- > 3.8 can not be broken in shear (iii) The droplet breakup is easier in elongational flow fields than in shear flow fields the relative efficiency of the elongational field dramatically increases for large values of A, > 1 (iv) Drop deformation and breakup in viscoelastic systems seems to be more difficult than that observed for Newtonian systems (v) When the concentration of the minor phase exceeds a critical value, ( ) >( ) = 0.005, the effect of coalescence must be taken into account (vi) Even when the theoretical predictions of droplet deformation and breakup... 

Drop deformation in shear that leads to fibrillation was recently examined using microscopy, light scattering and fluorescence [Kim et al., 1997]. The authors selected to work with systems near the critical conditions of miscibility, thus where the flow affects miscibility and reduces the value of The drop aspect ratio, p, plotted as a function of the capillarity number, K, showed two distinct regimes. For K < K., p was directly proportional to K, whereas for K > K., p followed more complex behavior, with an asymptote that corresponds to flow-induced homogenization. 

Figure 7.34. Critical capillarity number vs. viscosity ratio is shear flow (sohd hues) and extension (dash line).

Similarly as also the critical time for drop breakup, varies with k. When values of the capillarity number and the reduced time are within the region of the drop breakup, the mechanism of... 

Figure 9.7. Critical capillarity number for drop breakup in shear and extensional flow.

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