Gibbs dilational elasticity

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 ... 

In the presence of liquid flow, the situation becomes more complicated due to the creation of surface tension gradients [17]. These gradients, described by the Gibbs dilational elasticity [17], e, initiate a flow of mass along the interface in direction of a higher surface or interfacial tension (the Marangoni effect), e is given by the... 

The effect of surface dilational elasticity was first formulated by Gibbs (1957) as... 

The surface dilational elasticity was firstly defined by Gibbs (1906)... 

In recent years, several theoretical and experimental attempts have been performed to develop methods based on oscillations of supported drops or bubbles. For example, Tian et al. used quadrupole shape oscillations in order to estimate the equilibrium surface tension, Gibbs elasticity, and surface dilational viscosity [203]. Pratt and Thoraval [204] used a pulsed drop rheometer for measurements of the interfacial tension relaxation process of some oil soluble surfactants. The pulsed drop rheometer is based on an instantaneous expansion of a pendant water drop formed at the tip of a capillary in oil. After perturbation an interfacial relaxation sets in. The interfacial pressure decay is followed as a function of time. The oscillating bubble system uses oscillations of a bubble formed at the tip of a capillary. The amplitudes of the bubble area and pressure oscillations are measured to determine the dilational elasticity while the frequency dependence of the phase shift yields the exchange of matter mechanism at the bubble surface [205,206]. 

Film Elasticity. The differential change in surface tension with relative change in area. Also termed surface elasticity, dilational elasticity, areal elasticity, compressional modulus, surface dilational modulus, or modulus of surface elasticity. For fluid films the surface tension of one surface is used. The Gibbs film (surface) elasticity is the equilibrium value. If the surface tension is dynamic (time-dependent) in character then, for nonequilibrium values, the term Marangoni film (surface) elasticity is used. The compressibility of a film is the inverse of the film elasticity. 

The interfacial dilational elasticity, , results from the presence of interfacial tension gradients due to inhomogeneous surfactant or polymer films. The regions that are depleted from the film have higher interfacial tension than those containing the adsorbed film. As a result, an interfacial tension gradient djIdA is set and the Gibbs dilational elasticity may be defined as... 

A useful method for the measurement of c is the pulsed drop method [18]. Rapid expansion of a droplet at the end of a capUlary horn a radius r to I2 is obtained by application of pressure. The pressure drop within the droplet is measured as a function of time using a sensitive pressure transducer. From the pressure drop one can obtain the interfacial tension as a function of time. The Gibbs dilational elasticity is determined from values of the time dependent interfacial tension. Measurement can be made as a function of frequency as is illustrated in Fig. 5.17 for stearic acid at the decane-water interface at pH = 2.5. 

A torsion pendulum apparatus was used to get information about the shear rheological properties of the surfactants at the interface. This sensitive instrument was not capable of detecting significant elastic or viscous properties except for Lecithine. The analysis provided data that indicated a low viscous resistance for Lecithine with an interfacial shear loss modulus of about 0.03 mN/m. Hence, we can conclude, that no network-like superstructures were formed, and that the large dilatational elastic response was mainly caused by Gibbs- and Marangoni-effects. 

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). 

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). 

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... 

Equations [3.6.16 and 17] define the interfacial viscous and elastic components if surfaces are deformed by shear. Their counterparts refer to deformation by dilation (extension), or compression. Now we are concerned with relative extensions AAI A, or, infinitesimally, d In A. As before, for purely elastic surfaces the following two options should be considered (a) there is a network-type elasticity, as in a two-dimensional gel and (b) such a skin is absent elasticity is of the Gibbs... 

Dilational moduli play an important part in a number of practical processes involving interfaclal area changes, where K° is a measure of the resistance a monolayer has against creating an interfaclal tension gradient Vy upon extension or compression. Emulsification and foam formation are representatives of such processes. On the other hand, once Vy has been created, the modulus controls the rate of relaxation. Historically, only static values have been considered. Recall that for elasticities of the Gibbs type, expressions for K° can be formulated on the basis of two-dimensional equations of state, see 13.4.30 and 30a]. In more modem developments the modulus is also studied under dynamic, or non-equilibrium... 

E.H. Lucassen-Reynders, Surface Elasticity and Viscosity in Compression/ Dilation, in Anionic Surfactants Physical Chemistry and Surfactant Action E.H. Lucassen-Reynders, Ed., Marcel Dekker (1981). (Review of dllatlonal rheology mode, emphasis on Gibbs monolayers includes discussion on 2D equations of state.)... 

The definition of Gibbs elasticity given by Eq. (19) corresponds to an instantaneous (Aft t ) dilatation of the adsorption layer (that contributes to o ) without affecting the diffuse layer and o. The dependence of o on Ty for nonionic surfactants is the same as the dependence of o on Ty for ionic surfactants, cf Eqs (7) and (19). Equations (8) and (20) then show that the expressions for Eq in Table 2 are valid for both nonionic and ionic siufactants. The effect of the surface electric potential on the Gibbs elasticity Eq of an ionic adsorption monolayer is implicit, through the equilibrium siufactant adsorption T y which depends on the electric properties of the interface. To illustrate this let us consider the case of Langmuir adsorption isotherm for an ionic surfactant (17) ... 

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