Elasticity Gibbs

The stabihty of a single foam film can be explained by the Gibbs elasticity E which results from the reduction ia equiUbrium surface concentration of adsorbed surfactant molecules when the film is extended (15). This produces an iacrease ia equiUbrium surface tension that acts as a restoring force. The Gibbs elasticity is given by equation 1 where O is surface tension and is surface area of the film. 

Such nonequilihrium surface tension effects ate best described ia terms of dilatational moduh thanks to developments ia the theory and measurement of surface dilatational behavior. The complex dilatational modulus of a single surface is defined ia the same way as the Gibbs elasticity as ia equation 2 (the factor 2 is halved as only one surface is considered). 

A is the area of the surface. In a foam, where the surfaces are interconnected, the time-dependent Marangoni effect is important. A restoring force corresponding to the Gibbs elasticity will appear, because only a finite rate of absorption of the surface-active agent, which decreases the surface tension, can take place on the expansion and contraction of a foam. Thus the Marangoni effect is a kinetic effect. 

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... 

Other experiments performed by Bergeron [34] on air foams stabilized with ionic surfactants reveal that the so-called Gibbs or dilatational elasticity e may play an important role in the coalescence process. The Gibbs elasticity measures the variation of surface tension yi t associated to the variation of the surfactant surface concentration F ... 

It is probable that numerous interfacial parameters are involved (surface tension, spontaneous curvature, Gibbs elasticity, surface forces) and differ from one system to the other, according the nature of the surfactants and of the dispersed phase. Only systematic measurements of > will allow going beyond empirics. Besides the numerous fundamental questions, it is also necessary to measure practical reason, which is predicting the emulsion lifetime. This remains a serious challenge for anyone working in the field of emulsions because of the polydisperse and complex evolution of the droplet size distribution. Finally, it is clear that the mean-field approaches adopted to measure > are acceptable as long as the droplet polydispersity remains quite low (P < 50%) and that more elaborate models are required for very polydisperse systems to account for the spatial fiuctuations in the droplet distribution. 

Gibbs elasticity - [ENGINEERING, CHEMICALDATA CORRELATION] (Vol 9) - [THERMODYNAMICS] (Vol23) -and foam stability [FOAMS] (Vol 11)... 

Surfactants also reduce the coalescence of emulsion droplets. The latter process occurs as a result of thinning and disruption of the liquid film between the droplets on their close approach. The latter causes surface fluctuations, which may increase in amplitude and the film may collapse at the thinnest part. This process is prevented by the presence of surfactants at the O/W interface, which reduce the fluctuations as a result of the Gibbs elasticity and/or interfacial viscosity. In addition, the strong repulsion between the surfactant layers (which could be electrostatic and/or steric) prevents close approach of the droplets, and this reduces any film fluctuations. In addition, surfactants may form multilayers at the O/W interface (lamellar liquid crystalline structures), and this prevents coalescence of the droplets. 

The elasticity depends on the rate of film expansion. Under quasistatic equilibrium conditions its values are very low and in such a case it is called Gibbs elasticity. When there is no equilibrium it is called Maiangoni elasticity. The largest value of the elasticity modulus, acquired when the adsorption layer behaves as insoluble one, is called Marangoni dilatation modulus Em). 

Bianko and Marmur [99] have developed a new technique for the measurement of Gibbs elasticity of foam films. In order to exclude the effect of the mass transfer of the surfactant, the stretching of an isolated soap bubble is used. The surface tension needed for the calculation of the elasticity modulus is determined by the pressure in the bubble and the radius of curvature. The modulus obtained are considerably lower than those derived by the technique of Prins et al. [95]. 

Another indirect method for the estimation of Gibbs elasticity modulus is based on the determination of the surface dilatation modulus E in experiments in which the surfaces of the surfactant solutions undergo small amplitude deformations of oscillatory nature [100-102], It is shown [100, see also Chapter 7] that the concentration dependence of a Gibbs elasticity modulus at constant film thickness should be nearly the same as the concentration dependence of (twice) the surface elastic modulus E when film thickness and frequency are related by... 

Two types of elasticity could be distinguished equilibrium (Gibbs elasticity) and dynamic (Marangoni elasticity). According to Gibbs the modulus of elasticity of the film is... 

The rate of change in interfacial tension with surface area from its equilibrium value is termed as the Gibbs elasticity E = 2dy/d(ln A) (9). The factors which control E are the rate of transport toward or from the interface and the structure of surfactant as well as the... 

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. 

Assuming that the W/O emulsion behaves as a near hard-sphere dispersion, it is possible to apply the Dougherty-Krieger equation [7, 8] to obtain the effective volume fraction, 4>. The assumption that the W/O emulsion behaves as a near hard sphere dispersion is reasonable as the water droplets are stabihsed with a block copolymer with relatively short PHS chains (of the order of lOnm and less). Any lateral displacement of the polymer will be opposed by the high Gibbs elasticity of the adsorbed polymer layer, and the droplets will maintain their spherical shape up to high volume fractions. 

The partition of surfactant molecules between the oil and aqueous phases. With higher surfactant concentrations, the molecules with shorter EO chains (i.e., lower HLB number) may accumulate preferentially at the O/W interface. This may result in a reduction of the Gibbs elasticity, which in turn would cause an increase in the Ostwald ripening rate. 

Laplace pressure than the larger bubbles, but as the gas solubility increases with pressure the gas molecules will difTuse from the smaller to the larger bubbles. This process only occurs with spherical foam bubbles, and may be opposed by the Gibbs elasticity effect. Alternatively, rigid films produced using polymers may resist Ostwald ripening as a result of their high surface viscosity. 

The main deficiency of the early studies on Gibbs elasticity was that they were applied to thin films and diffusion from the bulk solution was neglected. In other words, the Gibbs theory applies to the case where there is insufficient surfactant molecules in the film to diffuse to the surface and lower the surface tension. This... 

Equation 5.73 visualizes the very strong dependence of the relaxation time to on the surfactant concentration c, in general, can vary with many orders of magnitude as a function of Cj. Equation 5.73 shows also that high Gibbs elasticity corresponds to short relaxation time, and vice versa. 

To directly measure the Gibbs elasticity Eo, or to precisely investigate the dynamics of surface tension, we need an experimental method, whose characteristic time is smaller compared with x. Equation 5.73 and the latter numerical example show that when the surfactant concentration is higher, the experimental method should be faster. 

FIGURE 5.6 (a) Plot of the slope coefficient 5, vs. the surfactant (DDBS) concentration the points are the values of 5, for the curves in Figure 5.5 the fine is the theoretical curve obtained using the procedure described after Equation 5.85 (no adjustable parameters), (b) Plots of the relaxation time and the Gibbs elasticity vs. the DDBS concentration is computed from the equilibrium surface tension isotherm = n (S, /Eq) is calculated using the above values of 5,. 

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... 

The Gibbs elasticity, Eq, favors the formation of emulsion 1, because it slows the film thinning. On the other hand, increased surface diffusivity, D, decreases this effect, because it helps the interfacial tension gradients to relax, thus facilitating the formation... 

The temperature strongly affects the solubility and surface activity of nonionic surfactants. It is well known that at higher temperature nonionic surfactants become more oil soluble, which favors the W/O emulsion. Thus, solubility may change the type of emulsion formed at the PIT. The surface activity has numerous implications the most important is the change of the Gibbs elasticity, Eq, and the interfacial tension, a. 

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