Surfactant distribution

As shown above, the distribution coefficient K is the key parameter for the adsorption of surfactants at any liquid/liquid interface. There are only few attempts known from literature where K is determined. A straightforward procedure is the measurement of the bulk concentrations after equilibration of the two immiscible phases. This, however, only works for particular surfactants, and common analytic techniques usually fail at low surfactant concentration. An indirect method consists in the measurement of the surface tension of the aqueous phase. Via the y-c isotherm as a master curve, the concentration can be evaluated then and hence the distribution coefficient [140]. 

More recently, the adsorption dynamics of CnDMPO, C12DMPO and CioDMPO at freshly formed water-hexane interfaces has been investigated as a function of the initial partition conditions and as a function of the relative volumes between the two liquids [141]. The Table 4.1 summarises some K-values for the system water-hexane. 

As one can see these surfactants have large values of the partition coefficient, which enhances the influence of the transfer on the adsorption rate. 

Direct methods such as HPLC can be used to determine the amount of surfactant in both phases having been in contact before to reach the distribution equilibrium. For the non-ionics of the Triton type experiments have been described by Czichocki et al. [143]. It is shown that such analytical methods are very time consuming, however once available they provide very aecurate results. For ionic surfactants there are several other methods which can determine the amount of the surfactant at least in the aqueous phase, such as selective electrodes. 

Interfacial relaxation methods are typically based on a perturbation of the equilibrium state of an interfacial layer (equilibrium within the interfacial layer and with the adjacent bulk phases) by small changes of the interfacial area. The small relative change in area is defined by 

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Here we also consider sorption kinetics as the mass-transfer barrier to surfactant migration to and from the interface, and we follow the Levich framework. However, our analysis does not confine all surface-tension gradients to the constant thickness film. Rather, we treat the bubble shape and the surfactant distribution along the interface in a consistent fashion. 

The expected surfactant distribution is also portrayed qualitatively in Figure 2. At low Ca, recirculation eddies in the liquid phase lead to two stagnation rings around the bubble, as shown by the two pairs of heavy black dots on the interface (18>19). Near the bubble front, surfactant molecules are swept along the interface and away from the stagnation perimeter. They are not instantaneously replenished from the bulk solution. Accordingly, a surface stress, rg, develops along the interface... 

II provides a transition between the two asymptotic limits. Viscous stresses now scale by the local thickness of the film, h, and the bubble shape varies from the constant thickness film to the spherical segment. Here the surfactant distribution along the interface may be important. Fortunately, for small capillary numbers, dh/dx < 1 and the lubrication approximation may be used throughout. Region II is quantified below. 

The above expression and the quasistatic adsorption assumption provide the additional information necessary to establish both the bubble profile, h(x) from Equation 1, and the surfactant distribution, T(x) from Equation 3. 

It is based on equilibrium properties and is directly related to the Gibbs elasticity (17.). In the present context a gauges how strongly the surface tension depends on the surfactant distribution along the bubble interface. Second, captures the kinetics of the adsorption process and is defined by... 

When p approaches infinity, Equation 7 reveals that equals zero, which corresponds to infinitely fast sorption kinetics and to an equilibrium surfactant distribution. In this case Equation 6 becomes that of Bretherton for a constant-tension bubble. Equation 6 also reduces to Bretherton s case when a approaches zero. However, a - 0 means that the surface tension does not change its value with changes in surfactant adsorption, which is not highly likely. Typical values for a with aqueous surfactants near the critical micelle concentration are around unity (2JL) ... 

Figures 4 and depict the calculated surfactant distribution, expressed as 0, for the bubble front and rear, respectively.

Figure 4. The surfactant distribution at the bubble front expressed as a deviation from equilibrium.

Indirect techniques, such as conductivity measurements and the determination of the surface tension were carried out to get more information about the surfactant distribution during the polymerization and were applied to characterize the droplet or particle sizes before and after the polymerization without diluting the system [23]. As a powerful method small angle neutron scattering experiments were applied to characterize the droplet or particle sizes before and after the polymerization without diluting the system [23]. 

Fig. 3.11. Folding of T4 DNA by the addition of the gemini surfactant. Distributions of the long-axis length of T4 DNA at different concentrations [cs] of the surfactant. Coil, partially folded, and completely folded states are distinguished by the different colorings. Also shown are FM and AFM images with the corresponding schematic representation of the partially folded state ([cs] =0.2 pM) and completely folded state ([cs] = 1.0pM). The FM and AFM observations are of the same DNA molecules attached to a mica surface. A rings-on-a-string structure is clearly seen for the partially folded DNA, while the completely folded DNA assumes a network structure composed of many fused rings (see [19] for more details)...

The surfactant distribution between a foam and a solution is characterised by the accumulation ratio (//). There are a minimum (//min) and a maximum (//max) accumulation ratios [3,22-24,44]... 

R. Wagner, Y. Wu, L. Richter, J. Reiners, J. Weissmuller, A. De Montigny, Appl. Organometal. Chem., 1999, 13(1), 21-28. SUicon-modified carbohydrate surfactants. VIII. equilibrium wetting of perfluorinated solid surfaces by solutions of surfactants above and below the critical micelle concentration-surfactant distribution between liquid-vapor and solid-Uquid interfaces. ... 

Chan and Shah (26) proposed a unified theory to explain the ultralow interfacial tension minimum observed in dilute petroleum sulfonate solution/oil systems encountered in tertiary oil recovery processes. For several variables such as the salinity, the oil chain length and the surfactant concentration, the minimum in interfacial tension was found to occur when the equilibrated aqueous phase was at CMC. This interfacial minimum also corresponded to the partition coefficient near unity for surfactant distribution in oil and brine. It was observed that the minimum in ultralow interfacial tension occurs when the concentration of the surfactant monomers in aqueous phase is maximum. 

FIGURE 5.42 Damping of convection-driven surface tension gradients by influx of surfactant from the drop interior, (a) Since the mass transport is proportional to the perturbation, the larger the perturbation, the stronger the flux tending to eliminate it. (b) Uniform surfactant distribution is finally reached. 

Figure 7-19. A pictorial representation of the distribution of surfactant on the surface of a rising gas bubble. The small sticks at the bubble interface are intended to represent surfactant, which adsorbs preferentially at the gas-liquid interface. The fluid motion, from the top of the bubble toward the bottom, convects surfactant toward the rear of the bubble where it tends to accumulate. This tendency is counteracted to some extent by diffusion that tends toward a uniform surfactant distribution.

The latter is know as the interface Peclet number and provides a measure of the relative importance of convection and diffusion in determining the surfactant distribution on the interface. 

The balance of Marangoni and viscous stresses (8.153), reformulated in terms of T, is integrated to obtain the surfactant distribution and yields T as a function of ( ) and the dimensionless Marangoni number Ma. The surfactant distribution can be integrated over the cap region to obtain the total amount on the surface, M. The variable M is also computed independently from the surfactant conservation equations and equating the two expressions yields Once ( ) is specified, the drag coefficient and terminal velocity can be calculated. 

From the Frumkin equation (9.61) and the Marangoni stress balance, a differential equation for the surfactant distribution may be obtained... 

With surface-active material present the convection of the liquid, from right to left in the reference frame in which the bubble is stationary, results in a nonuniform surfactant distribution on the bubble surface. The surfactant is swept toward the rear of the bubble where it accumulates. As a consequence, the surface tension varies along the bubble with its lowest value at the rear end. The surface tension gradient exerts a tractive force on the bubble and increases its resistance to motion under a driving pressure gradient pi P -... 

Therefore, Eq. (2.15) which is valid for non-ionic surfactants, has to be replaced for ionic surfactants by Eq. (2.43). The constant b in the adsorption isotherm (2.45) is related to the constant of ionic surfactant distribution between surface and bulk as... 

The classical problem of adsorption dynamics is the prediction of the evolution of the interfacial adsorption T(t) for a "freshly" formed interface, i.e. with the initial condition of Eq. (4.10), and a homogeneous surfactant distribution in the two bulk phases... 

The transfer work can be calculated from the coefficient Kfof the surfactant distribution coefficient between the two phases. The values of transfer work for a number of substances are tabulated, for example, in Ref 262. For a ho-... 

Various expressions for HOR were proposed, expressed via the surfactant distribution coefficient between the phases, and the surfactant s adsorption activity (5). The advantage of HOR as compared to the HLB system is that, for a particular choice of the standard state, this index does not depend on the surfactant concentration, the type of the organic phase, or the presence of various additives soluble in water and oil. Methods were also proposed to determine the HOR for mixtures of surfactants (262). For these systems, however, this index is not additive anymore. The HOR values for mixtures are shifted towards that characteristic for the component which possesses the higher value of the distribution coefficient. Another deficiency of the HOR concept is its suggestiveness it was mentioned above that this value depends on the coordinate of the HLC. However, for the HOR values other than unity, the HLC position-dependent work of the introduction of a surfactant molecule into the surface layer is uniquely determined by the HOR. 

Kunieda intensified the studies on phase behavior and formation of microemulsions in mixed-surfactant systems [66-76], in order to understand the relationship between maximum solubilization of microemulsions and surfactant distribution of mixed surfactants at the water/oil interface in the microemulsion phase. He developed a method to calculate the net composition of each surfactant at the interface in the bicontinuous microemulsions assuming that the monomeric solubihty of each surfactant in oil is the same as in the oil microdomain of the microemulsions [69]. Using this approach, the distribution of surfactants in the different domains of bicontinuous microemulsions (Figure 9) could be quantified [70-75], even if the complete microstracture of these systems was not completely elucidated. 

Figure 5.15 Surfactant distribution with S = 10 and M = -10 fixed for several values of time t (shown in the upper right). The surface tension gradients driving the wave up...

The presence of surfactants, which adsorb at the liquid-vapor interface and reduce the surface tension, can also have a large impact on the evaporation-driven pattern formation. Inhomogeneities in the surfactant distribution create a surface tension gradient and a corresponding (additional) Marangoni flow. With respect to the Rayleigh-Benard convection, it has been shown that the surfactant-driven flow can favor the formation of convection cells and considerably alter the deposition patterns [7]. 

It should be emphasized that the entire study reported in this paper relates to the low surfactant concentration (< 0.5%) and does not involve the formation of middle phase microemulsions (23), etc. in this oil displacement process. At all times, the oil/brine/surfactant systems were composed of only two phases, oil and brine, with surfactant distributed in both phases. Also, this study is carried out at low salinity (< 2% NaCl) although we have reported elsewhere on high salinity formulations (24-26) which can produce ultralow IFT in millidynes/cm range at salt concentrations as high as 32%. 

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