Interfacial coupling

As mentioned earlier, the contact-mechanics-based experimental studies of interfacial adhesion primarily include (1) direct measurements of surface and interfacial energies of polymers and self-assembled monolayers (2) quantitative studies on the role of interfacial coupling agents in the adhesion of elastomers (3) adhesion of microparticles on surfaces and (4) adhesion of viscoelastic polymer particles. In these studies, a variety of experimental tools have been employed by different researchers. Each one of these tools offers certain advantages over the others. These experimental studies are reviewed in Section 4. 

Fig. 15. Schematic of the problem of adhesion of elastomers in the presence of interfacial coupling chains. Reproduced from ref. [98].

Calvert et al. [12] studied the interfacial coupling of alkoxy titanium and zirconium tricarboxyls with metal oxides. They showed that isopropoxytitanium and zirconium tristearates interact with silica and alumina by exclusive loss of isopropanol, and that the titanate and zirconate were resistant to desorption by treatment with hot water. 

Calvert, P. D., et al., Interfacial Coupling by Alkoxytitanium and Zirconium Tricarboxylate, in Adhesion Aspects of Polymer Coatings, K. L. Mittal, ed., Plenum Press, New York, 1983, p. 457. 

In order to carry out computations with the volume averaged equations on the form (3.138) or (3.139), we need to relate the average of products to products of averages and derive constitutive equations for the interfacial coupling terms. 

In the interfacial coupling terms the turbulent contributions are commonly taken into account through the empirical model parameters. 

In particular, referring to the introduction of the external forces as presented in sect 4.1.3 there are still no complete consensus in the literature regarding the treatment of the interfacial coupling terms like the steady drag-, added mass- and lift forces. In one view it is considered convenient to split the net force exerted by the interstitial fluid on the particle into two different contributions One virtual force applied by an undisturbed flow on a imaginary fluid particle which coincides with the solid particle in volume and shape, and a second contribution that represents the forces due to the perturbations in the flow. These flow disturbances are created by the presence of the particles. The phrase undisturbed flow thus refers to the flow that would be observed if the particle was not present. Neglecting the effects of the perturbations in the flow, the net force exerted on a particle (4.57) might be approximated by ... 

For comparison we briefly display the standard macroscopic transfer functions resulting from the averaging process, before we introduce the particular constitutive equations normally applied in chemical reaction engineering. This might help to elucidate the connection between the conventional fluid mechanics modeling framework and the customary chemical reaction engineering interfacial coupling terms. 

The interfacial and turbulence closures suggested in the literature also differ considering the anticipated importance of the bubble size distributions. It thus seemed obvious for many researchers that further progress on the flow pattern description was difficult to obtain without a proper description of the interfacial coupling terms, and especially on the contact area or projected area for the drag forces. The bubble column research thus turned towards the development of a dynamic multi-fluid model that is extended with a population balance module for the bubble size distribution. However, the existing models are still restricted in some way or another due to the large cpu demands required by 3D multi-fluid simulations. 

Several strategies have been proposed to treat these interfacial coupling terms in the design of appropriate numerical solution methods. A simple way to implement these interphase coupling terms is to apply an explicit discretization scheme. Alternatively, an alternating direction implicit (ADI)-like method can be applied in which the current phase variable in the coupling term can... 

Intermediate velocity values were then obtained by solving the momentum balances without the interfacial coupling terms using explicit discretization schemes for all variables except for the volume fractions. The intermediate volume fraction estimates obtained in step 1 were used in all terms except from the convective terms ... 

V diagonal matrix containing interfacial coupling terms... 

F interfacial coupling term in two-phase k-e turbulence model N/mP ) k,s)Ai surface average combined species mass transfer flux kg/ m s)... 

The interfacial coupling term S, in Eqs. (13) and (15) is a linear combination of several forces. Politis et al. [19] have compared the order of magnitude of the various forces and concluded that only the drag force and the virtual mass force need to be considered. 

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