Dilatational modulus

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). 

Comprehensive evaluation of the results shows that the adsorption and desorption of alkanesulfonates, and other surfactants too, is diffusion-controlled. The dilatation modulus increases with increasing number of carbons because of the enhanced intermolecular interaction. This information is particularly significant in, for instance, foams. 

FIG. 28 Dependence of the interface dilatation modulus e on the concentration c of homologous alkanesuifonates and decylbenzenesulfonate according to Ref. 48. O, Sodium tetradecanesulfonate x, sodium decylbenzenesulfonate , sodium dodecanesulfonate A, sodium decanesulfonate). 

The complex interfacial dilational modulus ( ) is a key fundamental property governing foam and emulsion stability. It is defined as the interfacial tension increment (da) per unit fractional interfacial area change (dA/A) i.e.,... 

The shape of the curves for the dilational modulus (Figures 7 and 8) suggests a single relaxation mechanism, probably the unfolding of the demulsifier molecules at the interface. The frequency peak in the e"(f) plot is a measure of the characteristic relaxation time. A shorter relaxation time, by inducing faster film drainage, increases demulsification efficiency. 

For effective demulsification of a water-in-oil emulsion, both shear viscosity as well as dynamic tension gradient of the water-oil interface have to be lowered. The interfacial dilational modulus data indicate that the interfacial relaxation process occurs faster with an effective demulsifier. The electron spin resonance with labeled demulsifiers suggests that demulsifiers form clusters in the bulk oil. The unclustering and rearrangement of the demulsifier at the interface may affect the interfacial relaxation process. 

Figure 20. (a) The (dimensionless) lateral compressibility (dilatational modulus, elastic area expansion modulus) (left ordinate) and the dimensionless area per molecule (right ordinate) as a function of the tail length (t) of the PC lipids in equilibrium bilayer membranes. The conversion to real compressibilities and areas per molecule is discussed in the text, (b) The (dimensionless) surface tension and the (dimensionless) lateral compressibility as a function of the relative expansion for the C PC lipid... 

The surface rheological properties of the /3-lg/Tween 20 system at the macroscopic a/w interface were examined by a third method, namely surface dilation [40]. Sample data obtained are presented in Figure 24. The surface dilational modulus, (E) of a liquid is the ratio between the small change in surface tension (Ay) and the small change in surface area (AlnA). The surface dilational modulus is a complex quantity. The real part of the modulus is the storage modulus, e (often referred to as the surface dilational elasticity, Ed). The imaginary part is the loss modulus, e , which is related to the product of the surface dilational viscosity and the radial frequency ( jdu). 

The dilational rheology behavior of polymer monolayers is a very interesting aspect. If a polymer film is viewed as a macroscopy continuum medium, several types of motion are possible [96], As it has been explained by Monroy et al. [59], it is possible to distinguish two main types capillary (or out of plane) and dilational (or in plane) [59,60,97], The first one is a shear deformation, while for the second one there are both a compression - dilatation motion and a shear motion. Since dissipative effects do exist within the film, each of the motions consists of elastic and viscous components. The elastic constant for the capillary motion is the surface tension y, while for the second it is the dilatation elasticity e. The latter modulus depends upon the stress applied to the monolayer. For a uniaxial stress (as it is the case for capillary waves or for compression in a single barrier Langmuir trough) the dilatational modulus is the sum of the compression and shear moduli [98]... 

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... 

In this table we largely follow the lUPAC recommendations (sec. 3.10a), but complete adherence is not feasible because of interference with symbols already used elsewhere In FIGS. For instance, we prefer for the interfacial dilational modulus over the lUPAC recommendation f. 

Introducing the modulus (= absolute value in complex number language) of the surface dilational modulus K° as... 

Figure 3.48. Dilational modulus of bovine serum albumin and ovalbumin monolayers. (Courtesy of J. Benjamins).

Figure 3.91. Surface shear viscosity (a) and dilational modulus (b) of DPPC (L-a-dipalm-itoylphosphatidylchollne) and DMPE (L-a-dimyristoyl phosphatidylethanolamine) mono-layers. Inset 7c[a ) curves. (Redrawn from Kragel et al.. loc clt.)...

The first is that under equilibrium conditions the dilational modulus K° can be related to the interfacial excess Helmholtz energy F° through... 

Figure 4.27. Absolute value of the surface dilational modulus obtained by the wave technique (closed symbols) and from oscillating bubbles (open symbols). Surfactant, tridecyldimethyl phosphine oxide A, A c = 2 xlQ-S M O. c = 5 x lO M. Drawn curves fit to [4.5.431. Temperature 22°C. (Redrawn from Wantke et al. (loc. cit.).)...

The d)mamics of adsorption of emulsifiers at fluid interfaces have been determined by tensiometry and surface rheology (Figure 14.3) that is, from the time dependence of surface pressure and surface dilatational modulus (E). We found that tt and E increase with time (9), which should be associated with emulsifier adsorption (Patino and Nino, 1999 Nino et al., 2003 Carrera et al., 2005). 

In Figure 14.3c a normalization in a single master curve of E vs. tt data reflects the interfacial behavior of emulsifier adsorbed films for different emulsifier concentrations, at different adsorption times, and under different processing conditions (Nino and Patino, 2002 Nino et al., 2003). The plot suggests that interactions between adsorbed emulsifier molecules (residues) increase with tt. In fact, at lower tt values the slope of the -tt plot was close to 1, which corresponds to the behavior of an ideal gas with low emulsifier interactions. However, at higher tt values the slope changes, which implies an important nonideal behavior with higher molecular interactions as the amount of emulsifier at the interface increases. These data indicate that the interfacial activity and the surface dilatational modulus of emulsifier films are mainly a result of the amount of adsorbed emulsifier. 

The results with biosurfactant (protein and LMWE) monolayers indicate that the dilatational modulus is not only determined by the interactions between spread biosurfactant molecules (which depend on the surface pressure or surface density), but that the structure of the spread molecule also plays an important role (Nino et al., 2003). 

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