Diffusion of surfactant

An evaluation of the retardation effects of surfactants on the steady velocity of a single drop (or bubble) under the influence of gravity has been made by Levich (L3) and extended recently by Newman (Nl). A further generalization to the domain of flow around an ensemble of many drops or bubbles in the presence of surfactants has been completed most recently by Waslo and Gal-Or (Wl). The terminal velocity of the ensemble is expressed in terms of the dispersed-phase holdup fraction and reduces to Levich s solution for a single particle when approaches zero. The basic theoretical principles governing these retardation effects will be demonstrated here for the case of a single drop or bubble. Thermodynamically, this is a case where coupling effects between the diffusion of surfactants (first-order tensorial transfer) and viscous flow (second-order tensorial transfer) takes place. Subject to the Curie principle, it demonstrates that this retardation effect occurs on a nonisotropic interface. Therefore, it is necessary to express the concentration of surfactants T, as it varies from point to point on the interface, in terms of the coordinates of the interface, i.e.,... 

The flux of surface-active agents from the surface into the bulk of the liquid may be controlled by the slower of the following processes 1) adsorption or desorption of surfactants at the surface or 2) diffusion of surfactants from the liquid bulk to the surface. Consequently, Levich evaluated the solution for a single drop... 

The overall set of partial differential equations that can be considered as a mathematical characterization of the processing system of gas-liquid dispersions should include such environmental parameters as composition, temperature, and velocity, in addition to the equations of bubble-size and residence-time distributions that describe the dependence of bubble nucleation and growth on the bubble environmental factors. A simultaneous solution of this set of differential equations with the appropriate initial and boundary conditions is needed to evaluate the behavior of the system. Subject to the Curie principle, this set of equations should include the possibilities of coupling effects among the various fluxes involved. In dispersions, the possibilities of couplings between fluxes that differ from each other by an odd tensorial rank exist. (An example is the coupling effect between diffusion of surfactants and the hydrodynamics of bubble velocity as treated in Section III.) As yet no analytical solution of the complete set of equations has been found because of the mathematical difficulties involved. To simplify matters, the pertinent transfer equation is usually solved independently, with some simplifying assumptions. 

If the supply of surfactant to and from the interface is very fast compared to surface convection, then adsorption equilibrium is attained along the entire bubble. In this case the bubble achieves a constant surface tension, and the formal results of Bretherton apply, only now for a bubble with an equilibrium surface excess concentration of surfactant. The net mass-transfer rate of surfactant to the interface is controlled by the slower of the adsorption-desorption kinetics and the diffusion of surfactant from the bulk solution. The characteris-... 

It may be expected that the rates of transfer of surfactant and alcohol, Dx and Dy, are affected by the negative feedback of Z. In other words, the diffusion rate, Dx, of surfactant from the aqueous phase to the interface may decrease with the net increase in the concentration of surfactant at the interface, X plus Z . A similar situation may hold for the diffusion rate, Dy, of alcohol from the aqueous phase to the interface. Hence, the system kinetics may be considered under the following assumptions (a) the concentration of surfactant and alcohol in the bulk aqueous phase, Xb and Yb, remain constant (b) the rates of diffusion of surfactant alcohol from the bulk aqueous phase to the interface are expressed as Dx(Xb - X ) and Dy(Yb - Yj), respectively (c) the negative feedback of Zi on the diffusion of X and Y are given Yb - kiZj and - k2Zj, respectively (d) the rate of step (iv) is expressed as a function, F(Xi( Yj), with the rate constant k3 and (e) the rate of step (iv) is expressed as a function, G(Zj), with the rate constant k4. 

Freshly prepared macroemulsions change their properties with time. The time scale can vary from seconds (then it might not even be appropriate to talk about an emulsion) to many years. To understand the evolution of emulsions we have to take different effects into account. First, any reduction of the surface tension reduces the driving force of coalescence and stabilizes emulsions. Second, repulsive interfacial film and interdroplet forces can prevent droplet coalescence and delay demulsification. Here, all those forces discussed in Section 6.5.3 are relevant. Third, dynamic effects such as the diffusion of surfactants into and out of the interface can have a drastic effect. 

An absence of the Gibbs-Marangoni effect is the main reason why pure liquids do not foam. It is also interesting, in this respect, to observe that foams from moderately concentrated solutions of soaps, detergents, etc., tend to be less stable than those formed from more dilute solutions. With the more concentrated solutions, the increase in surface tension which results from local thinning is more rapidly nullified by diffusion of surfactant from the bulk solution. The opposition to fluctuations in film thickness by corresponding fluctuations in surface tension is, therefore, less effective. 

Barrier breakdown leading to an inflammatory response due to diffusion of surfactant into epidermis Debonding of cells and inter-cellular mechanical movement... 

Figure 3.8 shows the dynamic surface tension of a pure anionic and a non-ionic surfactant dependent on the absorption time after the creation of new surface for different concentrations [9]. For both surfactants, the time dependence of the surface tension is greatly reduced when the concentration increases and this effect is especially pronounced when the critical micelle concentration is reached. The reason for this dependence is the diffusion of surfactant molecules and micellar aggregates to the surface which influences the surface tension on newly generated surfaces. This dynamic effect of surface tension can probably be attributed to the observation that an optimum of the washing efficiency usually occurs well above the critical micelle concentration. The effect is an important factor for cleaning and institutional washing where short process times are common. 

Constants Dv and Co are determined as free parameters in the non-linear regression of the experimental J(t) dependence along with the theoretical one calculated by the least square root method. The theoretical curve calculated at Co = 1. 7-10 4 mol dm 3 and >v = 4-1 O 6 cm2 s 1 is presented with solid line in the figure. The approximation of the lattice model of the amphiphile bilayer Dv is related to the coefficient of lateral diffusion of surfactant molecules building up the bilayer by the degree of filling 0 (respectively, of vacancies 6V)... 

The rates of diffusion of solutes and surfactants in and out of micelles have been measured using photophysical techniques. The most commonly used method is to measure the deactivation of excited states of the probe by added quenchers, which are only soluble in the aqueous phase. The measurement of either the decrease in emission intensity or a shortening of the emission lifetime of the probe can be employed to determine exit and entrance rates out of and into micelles 7d). The ability of an added quencher to deactivate an excited state is determined by the relative locations and rates of diffusion of the quenchers and excited states. Incorporation of either the quencher or excited state into a surfactant allows one to determine the rates of diffusion of surfactants. Because of the large dynamic range available with fluorescent and phosphorescent probes (Fig. 3), rates as fast as... 

Gerbacia and Rosano (46) have determined the interfacial tension at oil-water interface after alcohol injection into one of the phases. They observed that the interfacial tension could be temporarily lowered to zero due to the diffusion of alcohol through the interface. They concluded that the diffusion of surfactant molecules across the interface is an important requirement for reducing interfacial tension temporarily to zero as well as for the formation of microemulsions. They further claimed that the formation of microemulsions depend on the order in which components are added. 

Use of mixed surfactant films In many cases the used of mixed surfactants (e.g., anionic and nonionic or long chain alcohols) can reduce coalescence as a result of several effects a high Gibbs elasticity high surface viscosity and hindered diffusion of surfactant molecules from the film. 

Diffusion of surfactant molecules from the bulk solution to the subsurface layer... 

In the opposite case, when the surfactant is soluble in the continuous phase, the Marangoni effect becomes operative and the rate of film thinning becomes dependent on the surface (Gibbs) elasticity (see Equation 5.282). Moreover, the convection-driven local depletion of the surfactant monolayers in the central area of the film surfaces gives rise to fluxes of bulk and surface diffusion of surfactant molecules. The exact solution of the gives the following... 

Similar experiments were carried out in which drops that were mixtures of /i-decane and various alcohols were injected into dilute solutions of a zwitterionic (amine oxide) surfactant. Here, too, the lamellar phase was the first intermediate phase observed when the system was initially above the PIT. However, with alcohols of intermediate chain length such as /i-heptanol, it formed more rapidly than with oleyl alcohol, and the many, small myelinic figures that developed broke up quickly into tiny droplets in a process resembling an explosion.The high speed of the inversion to hydrophilic conditions was caused by diffusion of n-heptanol into the aqueous phase, which is faster than diffusion of surfactant into the drop. The alcohol also made the lamellar phase more fluid and thereby promoted the rapid breakup of myelinic figures into droplets. Further loss of alcohol caused both the lamellar phase and the remaining microemulsion to become supersaturated in oil, which produced spontaneous emulsification of oil. 

Some differences between the isotherms and the elasticity curves in Figures 3 and 4 are worth noting. In particular, the concentrations reflecting monomolecular coverage appear to be a little smaller than for the isotherms in Figure 3, the elasticity values in Figure 3 are also smaller than those in Figure 4. This can be connected with enhanced diffusion of surfactants dissolved in ethanol from the surface to the subsurface microlayer due to parametrically excited surface waves and container vibrations. 

In the spread of a surfactant film that is due to Marangoni-driven flow in the thin film, the surface diffusion of surfactant and gravitational and capillary contributions to the motion in the film are often assumed to be negligible. What conditions, in terms of... 

The role of adsorption kinetics and the diffusion of surfactants is especially important in controlling capillary impregnation. According to studies by N.N. Churaev, the solution impregnating the capillary quickly loses its dissolved surfactant due to adsorption of the latter on capillary walls, so the rate of impregnation may be limited by the diffusional transport of surfactant from the bulk of the solution to the menisci in the pores. 

Now, the system containing three ions is considered again. The rate of diffusion of surfactant anions must decrease with increasing electrolyte concentration and at... 

The non-steady diffusion of surfactant ions is a problem similar to the non-steady diffusion of non-ionic surfactant, which was described in Chapter 4. There is a specific distinction caused by the electrostatic retardation effect. The non-steady transport of ionic surfactants to the adsorption layer is a two-step process, consisting of the diffusion outside and inside the DL. 

Many surfactant solutions show dynamic surface tension behavior. That is, some time is required to establish the equilibrium surface tension. If the surface area of the solution is suddenly increased or decreased (locally), then the adsorbed surfactant layer at the interface would require some time to restore its equilibrium surface concentration by diffusion of surfactant from or to the bulk liquid. In the meantime, the original adsorbed surfactant layer is either expanded or contracted because surface tension gradients are now in effect, Gibbs—Marangoni forces arise and act in opposition to the initial disturbance. The dissipation of surface tension gradients to achieve equilibrium embodies the interface with a finite elasticity. This fact explains why some substances that lower surface tension do not stabilize foams (6) They do not have the required rate of approach to equilibrium after a surface expansion or contraction. In other words, they do not have the requisite surface elasticity. 

Anderson and Wennerstrom [33] calculated the geometrical obstruction factors of the self-diffusion of surfactant and solvent molecules in ordered bicontinuous microstructures, which serve as good approximations also for the disordered bicontinuous microemulsions and L3 (sponge) phases. The geometrical obstruction factor is defined as the relative diffusion coefficient DIDq, where D is the diffusion coefficient in the structured surfactant system and Z)q is the diffusion coefficient in the pure solvent. In a bicontinuous microemulsion the geometrical obstruction factor depends on the water/oil ratio. An expansion around the balanced (equal volumes of water and oil) state gives, to leading order. 

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