Interfacial tension gradient model

The gradient model for interfacial tension described in Eqs. III-42 and III-43 is limited to interaction potentials that decay more rapidly than r. Thus it can be applied to the Lennard-Jones potential but not to a longer range interaction such as dipole-dipole interaction. Where does this limitation come from, and what does it imply for interfacial tensions of various liquids ... 

IHP) (the Helmholtz condenser formula is used in connection with it), located at the surface of the layer of Stem adsorbed ions, and an outer Helmholtz plane (OHP), located on the plane of centers of the next layer of ions marking the beginning of the diffuse layer. These planes, marked IHP and OHP in Fig. V-3 are merely planes of average electrical property the actual local potentials, if they could be measured, must vary wildly between locations where there is an adsorbed ion and places where only water resides on the surface. For liquid surfaces, discussed in Section V-7C, the interface will not be smooth due to thermal waves (Section IV-3). Sweeney and co-workers applied gradient theory (see Chapter III) to model the electric double layer and interfacial tension of a hydrocarbon-aqueous electrolyte interface [27]. 

Harrison and colleagues [40] measured the interfadal tension between SCCO2 and polyethylene glycol (MW 600) using a tandem variable-volume pendent drop tensiometer, as shown schematically in Fig. 10.3. At 45 °C the interfacial tension between the PEG-CO2 rich phase and the SCCO2 phase was reduced from 6.9 dyn cm at 100 bar to 3.08 dyn cm at 300 bar. Experimental observations were accurately predicted using a gradient model which utilized the lattice equation of state. In another piece of work, the effect of surfactants on poly-mer/C02 interfacial tension was addressed [42]. 

The first theoretical modeling of polymer blends was carried out by Helfand and his colleagues [25-27]. Their lattice theory approximated the polymer 1-polymer 2 interactions via the Huggins-Flory binary parameter, Xrt-The derivation provided expressions for the concentration gradient across the interface, as well as for the interfacial tension coefficient, Vi2, and the interphase thickness, Al ... 

In this section interfacial tensions are computed with the gradient theoiy of van der Waals in which the Peng Robinson (PR) equation of state [9] has been incorporated. The combination of these two models originally was presented by Carey et al. [8]. 

To model the properties of interfaces of mixtures with polar species, suitable equations of state need to be used in conjunction with the gradient theory. This is more important for mixtures than for pure fluids. For polar and even associating species the APACT has been applied, while for non-polar species like alkanes the Peng-Robinson equation of state has been selected. Calculations show that the interfacial tensions obtained with the former model are in good agreement with the experimental tensions. Even the interfacial tensions of mixtures containing water can be described accurately. 

Comelisse, P.M.W. (1997) The gradient theory applied, simultaneous modelling of interfacial tension and phase behaviour, PhD-thesis, Delft University of Thechnology, The Netherlands. 

The interfacial tension between CO2 and various condensed phases including water [63], styrene oligomers[64] and polyethylene glycol[8] has been measured with a tandem variable-volume tensiometer and modeled with gradient theory. Also surfactants were added to these systems to lower the interfacial tension and to determine the surfactant adsorption. A critical micelle concentration was observed for... 

The critical film thickness for rupture is of the order of 50 A. If the interaction time of the drops is too short to reach the critical film thickness, the drops will not coalesce. The drainage of the film is the rate-determining step in coalescence of deformable drops in polymer blends. Various models have been proposed to describe the film drainage. One model assumes fully mobile interfaces, another model assumes immobile interfaces, and a third model assumes partially mobile interfaces. The mobility of the interfaces is strongly dependent on the presence of impurities, such as surfactants. Surfactants reduce the mobility of the interfaces due to interfacial tension gradients [315]. 

In a series of papers [7,106,107], we have combined our EoS model with the density gradient approximation of inhomogeneous systans [99-105]. In Refs. [7,106,107], we have addressed in three alternative ways the problon of consistency and equivalence of the various methods of calculating the interfacial tension. In the first case [106], we have simulated the number density profile across the interface with the classical hyperbolic tangent expression [92] (Equation 2.138). In the second case [7], this profile was obtained from the free-energy minimization condition [103,105]. 

A conceptually different approach to the calculaticHi of interfacial tensions is the use of the generalized square-gradient approach as embodied in the work of Cahn and Hilliard [216]. The Cahn-Hilliard theory provides a means for relating a particular equation of state, based on a specific statistical mechanical model, to surface and interfacial properties. The local free energy, g, in a region of nonuniform composition will depend on the local composition as well as the composition of the immediate environment. Thus, g can be expressed in terms of an expansion in the local composition and the local composition derivatives. Use of an appropriate free energy expression derived from statistical mechanics permits calculation of the surface or interfacial tension. 

These factors were chosen as such that the small deformation model of Taylor is recovered for small deformations. The left-hand side of Equation 19.6 represents the change of S with time and its rotation with the overall flow field. The first term on the right-hand side represents the restoring action of the interfacial tension, where g(S) is introduced to preserve the droplet volume. The second term on the right-hand side captures the deformation of the droplet with the flow. This equation allows to accurately predict the droplet deformation in flows with an arbitrary but uniform velocity gradient. In case of shear flow, the following expression is obtained for the steady state deformation parameter [30] ... 

In Eq. 9, E is the interfacial tension, p the pressure, Vy the undisturbed velocity gradient tensor and Vy its transpose, tjm is the viscosity of the continuous phase, V is the total volume of the system, n is the unit vector orthogonal to the interface between the two phases, u is the velocity at the interface, dA is the area of an interfacial element and the integrals are evaluated over the whole interfadal area of the system, A. Since the constituents are assumed to be Newtonian all nonlinear contributions to the stress a(t) are caused entirely by the deformation of the droplet interface. The unit vectors n and u describe this deformation and can be computed using the Maffettone-Minale (MM) model for different frequencies and amplitudes. The MM model uses a second rank, symmetric and positive definite... 

The displacement flows can be miscible (brine after polymer solution, C02 after oil, steam after water) or immiscible (water after oil). In the former case, it is the mixing process itself which has to be understood and modeled steam recovery requires the thermal transport problem to be accurately modeled. In the latter case, the two fluid phases coexist within the porous medium their relative proportions are determined not only by flow and mixing processes, but equally by interfacial and surface tensions between the three phases (matrix material included). Local (capillary) variations in pressure between the two fluid phases become important. The overall flow field is determined by large-scale pressure gradients. 

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