Interface deformation

At a given flow condition, different flow patterns were observed which can be classified into five distinct patterns depending on the interfacial configuration liquid alone (or liquid slug), gas core with a smooth thin liquid film, gas core with a smooth thick liquid film, gas core with a ring-shaped liquid film, and gas core with a deformed interface. 

Electrostatic interactions between a spherical charged protein particle and an oppositely charged, deformable interface can be estimated by evaluating the electrostatic force on a small segment of the interface as that produced by an adjacent flaf section on the protein surface. The strength of this interaction is dependent on the separation distance (b) between those two segments, and so will be a function of the position of the interfacial segment ... 

In the paragraphs below, we first examine the simple, analytical results that can be derived from the linear Poission-Boltzmann equation for a single particle interacting with a flat surface. Next, more complicated physical situations are considered, including interactions between many particles and a wall between a particle and a deformable interface between a protein and a wall and between a moving particle and a wall. In Sec. Ill, solutions to the nonlinear Poisson-Boltzmann equation are considered, and comparisons are made between the linear and nonlinear versions and also with more... 

C. Multiparticle Interactions, Deformable Interfaces, and Protein-Surface Interactions... 

Dungan and Hatton [12] solved Eq. (6) together with Eq. (48) for the problem depicted in Fig. 6, where a spherical particle is interacting with an oppositely charged deformable interface. To obtain their solution they used a boundary-integral method, in which the surfaces of the interface and sphere are discretized and assigned constant surface charge density boundary con-... 

FIG. 6 A sphere interacting with a deformable interface. (From Ref. 12.)... 

A hierarchy of approximations now exists for calculating interactions between a charged particle and a charged, planar interface in electrolyte solutions. At moderate surface potentials less than approximately 2(kT/e the linear Poisson-Boltzmann equation provides a good approximation in many circumstances, provided the solution is a 1 1 electrolyte at low to moderate ionic strength. The relative simplicity of the linear equation makes it particularly useful for examining problems that are complicated in other ways, such as interactions involving many particles, interactions with deformable interfaces, and interactions where the detailed structure and properties of the particle (or macromolecule) play an important role. 

Dispersion behaviour in systems with liquid/liquid or liquid/gas interfaces (i.e. droplet or bubbles) has traditionally been described in terms of rheological properties, wetting properties, including contact angle and interfacial tensions, or phase behaviour and stability measurements. Direct force measurements provide a means to fundamentally probe the interactions between deformable interfaces that significantly impact the dispersion (or emulsion) behaviour. 

The study of forces between deformable interfaces can be broken into two categories, the interactions between two sets of deformable interfaces (e.g., two oil drops in water), or a rigid particle and a single deformable interface. Study of the forces in these systems is motivated by the prevalence of both types of systems (drop-drop or drop-rigid particle) in industrial problems. For example, wetting and adhesion of oil emulsions in porous media are concerns in the petroleum industry for both liquid/liquid separations and oil recovery [1]. An understanding of the interaction forces between... 

Although many physical processes of interest to chemical reaction engineers involve absorption, heterogeneous reaction, surface mass transport, and interfacial mass transfer at moving and deforming interfaces, their main focus is concerned with the phenomena occurring at two particular types of interface systems. These are (1) the adsorption and reaction processes taking... 

Combining the general equation of films with deformable interfaces (Equation 5.255), the mass balance (Equations 5.276 and 5.277), and the boundary condition for the interfacial stresses (Equation 5.281), we can derive ... 

Figure 2-13. A sketch of a flat and deformed interface, showing the definition of the coordinates and the interface shape function as discussed in the text.

H. A. Stone, A simple derivation of the time-dependent convective-diffusion equation for surfactant transport along a deforming interface, Phys. Fluids A 2, 111-12 (1990). 

Finally, we have to apply the boundary conditions at the interface. These conditions are strictly applied at the deformed interface z = eh. However, the domain perturbation argument from the Rayleigh-Taylor section shows that the boundary conditions for the linearized disturbance problem can equally well be applied at the unperturbed surface, z = 0. 

The initial distance Hq is large compared with h, the thickness of the film at time t. The change in time, At, is the time it takes to reach a critical thickness for film rupture. Several versions of this equation exist that include internal circulation within the drop, rigid yet deformable interfaces, and complete interface mobility [64, 65]. 

A more precise approach is to introduce a coordinate frame aligned with a weakly deformed interface. Given the interface one can find unit... 

However, all these advantages raise a new set of fluid dynamical problems related to the deformable interface of the droplets the need to take into account interfacial tension and its variations and the complexity of singular events such as merging or splitting of drops. 

Since the first measurements of the electrostatic double-layer force with the AFM not even 10 years ago, the instrument has become a versatile tool to measure surface forces in aqueous electrolyte. Force measurements with the AFM confirmed that with continuum theory based on the Poisson-Boltzmann equation and appKed by Debye, Hiickel, Gouy, and Chapman, the electrostatic double layer can be adequately described for distances larger than 1 to 5 nm. It is valid for all materials investigated so far without exception. It also holds for deformable interfaces such as the air-water interface and the interface between two immiscible liquids. Even the behavior at high surface potentials seems to be described by continuum theory, although some questions still have to be clarified. For close distances, often the hydration force between hydrophilic surfaces influences the interaction. Between hydrophobic surfaces with contact angles above 80°, often the hydrophobic attraction dominates the total force. 

Williams and Janssen (20) studied the behavior of droplets in a simple shear flow in the presence of a protein emulsifier. The effect of two structurally diverse protein emulsifiers, P-lactoglobulin and P-casein, upon the breakup behavior of a single aqueous droplet in a Couette flow field has been studied over a wide range of protein concentrations. It was found that P-casein and low concentrations of P-lactoglobulin cause the droplets to be at least as stable as expected from conventional theories based on the equilibrium interfacial tension. In such cases the presence of the emulsifier at the deforming interface is thought to enhance the interfacial elasticity. This effect can be characterized by... 

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