Surfactant adsorption derivation

Surface heterogeneity may be inferred from emission studies such as those studies by de Schrijver and co-workers on P and on R adsorbed on clay minerals [197,198]. In the case of adsorbed pyrene and its derivatives, there is considerable evidence for surface mobility (on clays, metal oxides, sulfides), as from the work of Thomas [199], de Mayo and co-workers [200], Singer [201] and Stahlberg et al. [202]. There has also been evidence for ground-state bimolecular association of adsorbed pyrene [66,203]. The sensitivity of pyrene to the polarity of its environment allows its use as a probe of surface polarity [204,205]. Pyrene or ofter emitters may be used as probes to study the structure of an adsorbate film, as in the case of Triton X-100 on silica [206], sodium dodecyl sulfate at the alumina surface [207] and hexadecyltrimethylammonium chloride adsorbed onto silver electrodes from water and dimethylformamide [208]. In all cases progressive structural changes were concluded to occur with increasing surfactant adsorption. 

The objective in this section is to derive a mathematical model that can be used to extract the rate of adsorption from experimentally obtained dynamic surface tension data. Various investigators have speculated that the mechanism of surfactant adsorption involves two subsequent steps ... 

Correlation equations relating surfactant chemical structure to performance characteristics and physical properties have been established. One atmosphere foaming properties of alcohol ethoxyl-ates and alcohol ethoxylate derivatives have been related to surfactant hydrophobe carbon chain length, ethylene oxide content, aqueous phase salinity, and temperature. Similar correlations have been established for critical micelle concentration, surfactant cloud point, and surfactant adsorption. 

The nonpolar portion of surfactant ions has an important role in promoting the adsorption process because it increases the affinity of these organic ions to the interfacial region. The effect derives from mutual attraction between the hydrophobic tails as well as their tendency to escape from an aqueous environment. That mechanism is precisely the same one which causes the spontaneous formation of micelles in aqueous solution and is known as the hydrophobic effect [78]. In the case of surfactant adsorption, it is responsible for the formation of surface aggregates. However, it is not easy to accurately predict the shape and the size of such molecular associations in the same way that the structure of bulk aggregates can be determined from the geometry of the molecule. This is because the surface imposes different restrictions on the organization of the adsorbed layer. 

As this subject was covered in detail in Chapter 5, only a summary will be provided at this point. Surfactant adsorption is usually reversible, and hence thermodynamics can be applied for deriving the adsorption isotherm. Eor example, the adsorption of ionic surfactants onto hydrophobic surfaces may be represented by the Stern-Langmuir isotherm [13]. Consider a substrate containing sites (molm ) on which F molm of surfactant ions are adsorbed. The surface coverage 0 is (F/NJ and the fraction of uncovered surface is (1 — 0). The Stern-Langmuir... 

Each surfactant adsorption isotherm (that of Langmuir, Volmer, Frumkin, etc.), and the related expressions for the surface tension and surface chemical potential, can be derived from an expression for the surface free energy, F, which corresponds to a given physical model. This derivation helps us obtain (or identify) the self-consistent system of equations, referring to a given model, which is to be applied to interpret a set of experimental data. Combination of equations corresponding to different models (say, Langmuir adsorption isotherm with Frumkin surface tension isotherm) is incorrect and must be avoided. 

Substituting Q from Table 5.5 into Equation 5.86, and integrating, we can derive explicit expressions for the relaxation of surfactant adsorption ... 

The surfaces of fluid particles can be treated as tangentially immobile when they are covered by dense surfactant adsorption monolayers that can resist tangential stresses.In such a case, the bubbles or droplets behave as flexible balls with immobile surfaces. When the fluid particles are rather small (say, microemulsion droplets), they can behave like hard spheres therefore, some relations considered below, which were originally derived for solid particles, can be also applied to fluid particles. 

A comparison of the Ml nonlinear relationship (2-155) with the linear form (2-156) is shown in Fig. 2 22. Although (2-156) is often used in fluid dynamic calculations because of the simplicity that derives from the fact that it is linear, it can be seen from Fig. 2 22 that it requires T/ Too <strong reduction in the interfacial tension as F Too often has important consequences in flow systems, because the convection of surfactant on the interface often produces local regions of high concentration, but these will all be missed if the linear form (2-156) is used. Further discussion of surfactant adsorption at the fluid interface can be found either in classic textbooks48 or the research literature (e g., Pawar and Stebe)49... 

Theoretical modelling of surfactant adsorption uses as a starting point the theoretical isotherms derived from statistical and kinetic data for the gas-solid interface. The most common model is the one based on the Langmuir adsorption isotherm (see Chapter 11,2) ... 

Stabilization of emulsions by powders can be viewed as a simple example of structural- mechanical barrier, which is a strong factor of stabilization of colloid dispersions (see Chapter VIII, 5). The stabilization of relatively large droplets by microemulsions, which can be formed upon the transfer of surfactant molecules through the interface with low a (Fig. VII-10), is a phenomenon of similar nature. The surfactant adsorption layers, especially those of surface active polymers, are also capable of generating strong structural mechanical barrier at interfaces in emulsions. Many natural polymers, such as gelatin, proteins, saccharides and their derivatives, are all effective emulsifiers for direct emulsions. It was shown by Izmailova et al [49-52]. that the gel-alike structured layer that is formed by these substances at the surface of droplets may completely prevent coalescence of emulsion drops. 

The Stem-Martynov isotherm does not take into account the intermolecular interaction of adsorbed molecules, as it is considered in the Frumkin isotherm (cf Eq. (2.43)). The classical version of the Framkin isotherm was derived for nonionic surfactant adsorption layers. The incorporation of electrostatic interaction was proposed by Borwankar Wasan (1986, 1988). 

The conditions for the second and third regimes of formation of a dynamic adsorption layer of ionic surfactant will be presented without derivation. They are similar to the equivalent conditions derived in Section 8.6. The second regime, at which the value of surfactant adsorption only slightly deviates from equilibrium and the bubble surface is only slightly retarded, is realized under conditions... 

The equations which describe the reorientation of surfactant molecules in the surface layer can be derived from Eqs. (2.26) and (2.27). It is assumed that the reorientation results in a variation of the partial molar area coj. Note that for the derivation of Eq. (2.7) it was assumed that the tOj are independent of y. This requirement, however, does not contradict with the assumption of variable molar areas, because only the ratio of the surfactant adsorptions in different states, i.e., the states with different coj-values, depends on y. 

The characteristic time of surfactant adsorption at a fluid interface is an important parameter for suriaetant-stabilized dynamic systems sueh as emulsions. Sutherland (22) derived an expression deseribing flie relaxation of a small dilatation of an initially equilibrium adsorption monolayer... 

The fact that many surfactant systems cannot be adequately described by the Frumkin mode was the reason that other models have been derived. A comprehensive overview of such models was given recently elsewhere (Fainerman et al. 1998). We want to discuss two of the most recent models considering changes in orientation of adsorbed molecules and formation of two-dimensional aggregates (Fainerman et al. 2002). These new models are suitable to describe quite a number of surfactant adsorption layers much better than classical models do. 

Abstract We review a new theoretical approach to the kinetics of surfactant adsorption at fluid-fluid interfaces. It yields a more complete description of the kinetics both in the aqueous solution and at the interface, deriving all equations from a free-energy functional. It also provides a general method to calculate dynamic surface tensions. For non-ionic surfactants, the results coincide with previous models. Non-ionic surfactants are shown to usually undergo diffusion-limited adsorption, in agreement with the experiments. Strong electrostatic interactions in salt-free ionic surfactant solutions are found to... 

Surfactant adsorption theories are based on different physical and geometrical models of the adsorbed layer, resulting in a variety of surface equations of state or equivalently in several different adsorption isotherms. The usual approach in the theoretical description of the adsorption of ionic surfactants is the generalization of an adsorption isotherm (or equation of state) of nonionic surfactants by incorporating the electrostatic contribution in the adsorption free energy [4, 5, 6, 7, 8]. The validity of the ionic models derived is usually tested by applying the models for the description of the surface... 

The surfactant adsorption isotherm (Table 4.2) can be derived by setting the obtained expression for the surface chemical potential p,j equal to the bulk ch ical potential of the surfactant molecules in the subsurface layer (i.e., equilibrium between surface and subsurface is assumed) [11] ... 

It should be noted the result mentioned earlier holds only for the van der Waals (or Volmer) isotherm. Instead, if the Frumkin (or Langmuir) isotherm is used, the value of a obtained from the surface tension fits is about 33% greater than that obtained from molecular size [44], A possible explanation of this difference could be the fact that the Frumkin (and Langmuir) isotherm is statistically derived for localized adsorption and is more appropriate to describe adsorption at solid interfaces. In contrast, the van der Waals (and Volmer) isotherm is derived for nonlocalized adsorption, and they provide a more adequate theoretical desaiption of the surfactant adsorption at liqnid-flnid interfaces. This conclnsion refers also to the calculation of the surface (Gibbs) elasticity by means of the two types of isotherms [44]. 

In Ref [114], an approach to the dynamics of ionic surfactant adsorption was developed, which is simpler as both concept and application, but agrees very well with the experiment. Analytical asymptotic expressions for the dynamic surface tension of ionic surfactant solutions are derived in the general case of nonstationary interfacial expansion. Because the diffusion layer is much wider than the EDL, the equations contain a small parameter. The resulting perturbation problem is singular and it is solved by means of the method of matched asymptotic expansions [115]. The derived general expression for the dynamic surface tension is simplified for two important special cases, which are considered in the following section. 

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