Number capillarity

When discussing the morphology it is useful to use the microrheology as a guide. At low stresses in a steady uniform shear flow, the deformation can be expressed by means of three dimensionless parameters the viscosity ratio, the capillarity number, and the reduced time, respectively ... 

The structure and morphology of immiscible blends depends on many factors among which the flow history and the interfacial properties are the most important. At high dilution, and at low flow rates the morphology of polymer blends is controlled by three dimensionless microrheologi-cal parameters (i) the viscosity ratio, where r j is the viscosity of the dispersed liquid and r 2 that of the matrix (ii) the capillarity number, k = d / Vj2, where d... 

Since, for = 0 to the quantity in the square bracket ranges from 1.00 to 1.18, the drop deformability D = 0.55k. Thus, a small deformation of Newtonian drops in Newtonian matrix varies linearly with the capillarity number. This proportionality was indeed demonstrated in Couette-type rheometer for a series of com symp/silicon oil emulsions [Elemans, 1989]. 

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 ... 

From the point of view of the drop deformation and breakup there are four regions of the reduced capillarity numbers xf, both in shear and elongation ... 

When values of the capillarity number and the reduced time are within the region of drop breakup, the mechanism of breakup depends on the viscosity ratio, X. In shear, four regions have been identified [Goldsmith and Mason, 1967] ... 

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... 

Since derivation of this relation considered only the drop-splitting mechanism and neglected coalescence, its validity may be limited to small capillarity numbers, K = 1-2, and low concentrations. 

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).

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

For K > 2 drops deform affinely with the matrix into long fibers. When subsequently the deforming stress decreases, causing the reduced capillarity number to fall below two, K < 2, the fibers disintegrate under the influence of the interfacial... 

At high dilution the morphology of an immiscible blend is controlled by the viscosity ratio, f, the capillarity number, K, and the reduced time, t, as defined in Eq 9.8. The interfacial and rheological properties enter into K, and t. As the concentration increases, the coalescence becomes increasingly important. This process is also controlled by the interphasial properties. 

For K > 2 the drops deform into stable filaments, which only upon reduction of k disintegrate by the capillarity forces into mini-droplets. The deformation and breakup processes require time - in shear flows the reduced time to break is tb > 100- When values of the capillarity number and the reduced time are within the region of drop breakup, the mechanism of breakup depends on the viscosity ratio, A, - in shear flow, when X > 3.8, the drops may deform, but they cannot break. Dispersing in extensional flow field is not subjected to this limitation. Furthermore, for this deformation mode Kcr (being proportional to drop diameter) is significantly smaller than that in shear (Grace 1982). 

At low concentration of a second polymer, blends have dispersed-phase morphology of a matrix and discrete second phase. As the concentration increases, at the percolation threshold volume fraction of the dispersed phase, (f>c 0.16, the blends structure changes into co-continuous. Maximum co-continuity is achieved at the phase inversion concentration, (py. The morphology as well as the level of stress leads to different viscosity-composition dependencies. The deformation and dispersion processes are best described by microrheology, using the three dimensionless parameters the viscosity ratio (2), the capillarity number (k), and the reduced time (f ), respectively (Taylor 1932) ... 

Fig. 18.20 Contrasted defoimability of Newtonian and viscoelastic drops - 2D drop length-to-width versus the capillarity number (Mighri and Huneault 2001)...

Starting with cell model of creeping flow, Choi and Schowalter [113] derived a constitutive equation for an emulsion of deformable Newtonian drops in a Newtonian matrix. The authors characterized the interphase with an ill-defined interfacial tension coefficient, Vu, affecting the capillarity number, k = (Judfvu. The analysis indicated that depending on magnitude of /cy the emulsion may be elastic, characterized by two relaxation times. For the steady-state shearing, the authors expressed the relative viscosity of emulsions and the first normal stress difference as ... 

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