Interfacial deformation

Fracture Stress and Strain. Yielding and plastic deformation in the schematic representation of tensile deformation were associated with microfibrillation at the interface and stretching of the microfibrils. Because this representation was assumed to apply to both the core-shell and interconnected-interface models of compatibilization, the constrained-yielding approach was used without specific reference to the microstructure of the interface. In extending the discussion to fracture, however, it is useful to consider the interfacial-deformation mechanisms. Tensile deformation culminated in catastrophic fracture when the microfibrillated interface failed. This was inferred from the quasi-brittle fracture behavior of the uncompatibilized blend with VPS of 0.5, which indicated that the reduced load-bearing cross section after interfacial debonding could not support plastic deformation. Accordingly, the ultimate properties of the compatibilized blend depended on interfacial char-... 

The inadequacy of referencing a bulk rupture parameter, / Yade, to an interfacial deformation is obvious the structure, chemistry, quasi-hydrostatic stresses, temperatures and rates of strain are certainly not comparable in the two cases. A well quoted example of the quality of the Ratner-Lancaster relationship is shown in Figure... 

Let US consider two spherical emulsion drops approaching each other, which interact through the van der Waals attractive surface force. Sooner or later interfacial deformation will occur in the zone of drop-drop contact. The calculations (138) show that, if the drop radius a is greater than 80 jm, the drop interfaces bend inwards (under the action of the hydrodynamic pressure) and a dimple is formed in the contact zone soon the dimple transforms into an almost plane-parallel film (Fig. 2D). In contrast, if the drop radius... 

H. Tang, D.C. Martin, Microstiuctural smdies of interfacial deformation in painted thermoplastic polyolefins (TPOs). J. Mater. Sci. 37, 4783-4791 (2(K)2)... 

It is clear from Equations 1.1 and 1.2 that surface excess quantities do take into account the variation of composition and propalies across an interfacial region of finite thickness. As we shall see shortly, they can be used to define interfacial tension. Moreover, since all surface excess properties are assigned to the reference surface S, the area and curvature of S can be identified as the corresponding properties of the interface and used, for example, to describe interfacial deformation. 

The underlying idea behind the stability analysis is that all possible initial perturbations z(x) consistent with system boundary conditions may occur. If all of these perturbations diminish in amplitude with time, the system is stable. But if any permrbation grows, instability occurs and the interface never remms to its initial flat configuration. It is clear from Equation 5.2 that if the system is stable with respect to all initial interfacial deformations having the forms Zia and Z2a of individual Fourier components, it is stable with respect to a general deformation. But if it is unstable with respect to even one Fourier component, interfacial deformation can be expected to increase continuously with time, and there is no return to the initial state. 

Let us suppose that has the same periodic behavior as the interfacial deformation itself, so that... 

Note that these are four linear, homogeneous equations in the four unknowns D[ through D4. An obvious solution, but an uninteresting one because it implies no interfacial deformation and no flow, is the trivial solution Z)i = A = A = A = 0. According to the theory of linear equations, the condition for a nontrivial solution to exist is that the determinant of coefficients vanishes for Equations 5.43 through 5.46. This relationship, which of course does not contain any of the D S, may be solved to obtain the time factor Pas a function of the wavenumber a and the physical properties of the fluids. If the real part of pis positive, the... 

A general conelusion is also reached that the presence of surfactants makes the interface more rigid and provides resistance to lateral interfacial deformation, the measure of the latter being the interfacial velocities. It should be noted that the normal stress balance is satisfied with this solution, the same result as was found in Section 4 in the absence of surfactants. 

Two types of interfacial deformation are considered, dilation or compression, and shear, as shown in Figure 17.13. 

FIGURE 17.13 Interfacial deformations (a) dilation or compression and (b) shear. 

The role of interfacial deformation is considered in the stability analysis of fluid layers heated from below or above when there is an open interface to ambient air, and double diffusive transport of heat and solute thus leading to variations of interfacial tension that compete or cooperate with buoyancy phenomena. The onset of both oscillatory convection and steady patterns is described. 

Suppose the quantity of interfacial deformation is x, t), the thickness of the liquid layer is changed from d to d then we have the following ... 

In normal direction, the force causing interfacial deformation is equal to the force acting by the bulk liquid to the interface [37], that is,... 

Substituting to the mass balance equation and considering the disturbance of concentration and velocity as well as the interfacial deformation, the following equation is obtained after neglecting the high-order infinitesimal terms ... 

Finally, another experimentally relevant situation is that of a thin fluid film. The particles lie on a solid substrate that also sustains a thin fluid film or, alternatively, are trapped inside a liquid bridge (Figure 2.3). If the average thickness of the film is less than the particle diameter, the particles can get in touch with the fluid interface and capillary forces can play a role, hi this case, the wettability properties of the particle s surface are not so critical and the interfacial deformation can be brought about also by geometrical constraints of the setup, for example, conservation of volume of the liquid film. 

The capillary force as usually measured in experiments and as relevant in theoretical calculations calling for an effective interparticle force is actually a so-called mean force, that is, the result after all the degrees of freedom other than the particle position and orientation have been integrated out. Consider, for example, the simplest case of two identical, spherical particles a distance d apart. Let E pid) denote the parametric d-dependence of the free energy stemming from the terms affected by interfacial deformation (see, e.g., Equation 2.11). Then, E id) is a potential of mean force and the capillary force is defined as the derivative = -E fd). In this respect, two observations are in order concerning particularly the case n(r) 0 ... 

The first approach to the understanding of the capillary forces in equilibrium states involves the approximation of small interfacial deformations about a reference flat state. It will be understood under this approximation that both the departures from the flat interface and the spatial derivatives of the departure are as small as necessary to retain only the lowest-order terms in an expansion in those small quantities. This approximation simplifies considerably the analytical calculations and it also seems to describe well many of the experiments performed so far Because of the typically large values of y, considerable large forces already come into play upon tiny interfacial deformations additionally, the geometrical configurations usually considered are simple enough that no large derivatives appear. 

For completeness, we mention briefly the eqnivalent energy approach. In the small-deformation limit, the interfacial deformation is derived from variations at fixed pressure field ll(r) of the energy functional... 

Consider now a collection of particles. The electrostatic analogy establishes that the interfacial deformation can be written as a superposition of expansions of the form of Equation 2.12, one centered at each particle, but with certain multipole charges Qf + 5Qf (i =1,2, N), which do not have to be identical with the charges as if each particle were isolated ... 

The discussion concerning the capillary force, that is, the effective force between particles due to the interfacial deformation that they induce, proceeds in an analogous manner. We consider two particles in a fixed configuration determined on the XT-plane by their separation d and the orientation cp of the joining line with respect to the coordinates axes. In this configuration, each particle is characterized by the (permanent plus induced) capillary multipoles and Q , respectively, which depend on their full 3D position and orientation. One introduces an effective interaction potential of the form (see Appendix)... 

In the substrate-liquid film configuration of Figure 2.3, the interfacial deformation at a distance r from an isolated spherical particle in the small-deformation limit is Mjso,(r) = qJ2ny)KQ rl K), with... 

The interfacial deformation by an isolated quadrupole is U2(r, tp) = q lAizyr ) cos2(tp - Oj), which yields a maximum deformation at the contact line of the order of H= q lAiz R for a particle characterized by a typical size R it is useful to parametrize q by the length scale H. The capillary interaction between these particles will be described asymptotically by the quadrupole-quadrupole term of Equation 2.14, which can be rewritten as [9]... 

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