Modulus bending

By virtue of their simple stnicture, some properties of continuum models can be solved analytically in a mean field approxunation. The phase behaviour interfacial properties and the wetting properties have been explored. The effect of fluctuations is hrvestigated in Monte Carlo simulations as well as non-equilibrium phenomena (e.g., phase separation kinetics). Extensions of this one-order-parameter model are described in the review by Gompper and Schick [76]. A very interesting feature of tiiese models is that effective quantities of the interface—like the interfacial tension and the bending moduli—can be expressed as a fiinctional of the order parameter profiles across an interface [78]. These quantities can then be used as input for an even more coarse-grained description. 

A thermodynamic treatment, similar to that used for microemulsions, as well as an approximate statistical mechanical one, are developed to explain the phase transition in monolayers of insoluble surfactants [3.8], A similar thermodynamic approach is applied to multilamellar liquid crystals, and it is shown that, for a given set of interactions and bending moduli, only narrow ranges of the thicknesses of the water and oil layers are allowed [3.9]. 

Since the model was developed for two bilayers in water and was compared to experimental results for lipid bilayer/ water multilayers, no complete accuracy is expected for the values obtained by fitting. Particularly, the values obtained for the bending moduli are somewhat larger than those provided by literature. This might be a consequence of the increased rigidity of a multilayer, when compared to that of two bilayers, due to the correlation between the fluctuations of adjacent bilayers. 

The bending moduli of the AOT films in the presence ofvarious electrolytes are not known. Experiment showed that19 at low AOT concentrations, the surface tension... 

Figure 3. The experimental data for the external pressure II vs average thickness (d) are compared to the results of the present theory, which accounts for the undulation of the interfaces combined with the traditional DLVO treatment of interactions. Different bending moduli affect not only the strength of the interactions but also its slope (traditionally related to the Debye screening length).

The lifetime of the film depends on pfei) hence it decreases for interfaces with low bending moduli (less rigid interfaces are likely to fluctuate with a larger amplitude and thus can reach easier the maximum ofthe enthalpy). Therefore, while the undulation repulsion increases the average thickness of the film at a given external pressure (Figure 6a), it decreases its stability (Figure 5). 

Bending moduli of the neat resin and the laminates in the fiber direction were measured using three point bending tests in RT-air and 77K-LN2- Th specimens used for neat resin were 4 mm thick, 5 mm wide... 

EpoxyH in 77K-LN2 was more than twice higher than that at RT. Change of the matrix modulus affected the shear moduli of laminates, G, calculated by the Halpin-1 equation. Since the shear deformation of laminates contributes to the bending deformation, the bending moduli in fiber direction, E i s often lower than the tensile moduli in the fiber direction, Ej. The estimated Ej based on E and the consider-... 

A variant is the micro-pipette method, which is also similar to the maximum bubble pressure technique. A drop of the liquid to be studied is drawn by suction into the tip of a micropipette. The inner diameter of the pipette must be smaller than the radius of the drop the minimum suction pressure needed to force the droplet into the capillary can be related to the surface tension of the liquid, using the Young-Laplace equation [1.1.212). This technique can also be used to obtain interfacial tensions, say of individual emulsion droplets. Experimental problems include accounting for the extent of wetting of the inner lumen of the capillary, rate problems because of the time-dependence of surfactant (if any) adsorption on the capillary and, for narrow capillaries accounting for the work needed to bend the interface. Indeed, this method has also been used to measure bending moduli (sec. 1.15). 

Here, as before, J is the first, or mean, curvature and K the second, or Gauss, curvature, see [1.1.4 and 5[. is the spontaneous mean curvature. For symmetrical interfaces = 0. Here, we shall neglect the spontaneous Gauss curvature. The interfacial tension for the unbent surface is y(0,0). The two bending moduli have dimensions of energy and are of the order of fcT. We shall call k and k the... 

Bending moduli can in principle be obtained for two types of systems (i) extended, flat surfaces or interfaces, the subject matter of this section, and (ii) surfaces that are already strongly curved, and for which y is zero or extremely low, such as in vesicles or micro-emulsions. For instance such moduli can be inferred from shape fluctuations, from the Kerr effect (sec. 1.7.14] or from polydispersity using some scattering technique. We repeat that this type of measurement is often ambiguous because the bending contributions to the Helmholtz energy can only be estimated when all other contributions are accurately known. 

Table 1.6. Bending moduli (fcj) for some flat monolayers.

Flooding of oilfields is another example where interfacial reactions lead to changes of the Interfaclal tension. The issue is essentially a wetting problem the oil in the pores of the shales should be displaced by the fluid applied, which may be (caustic) water, a polymer solution, steam, or still something else, depending on the demands of the system, including the relative water- or oil-wettability of the rock. Under conditions where the interfacial tension is, or becomes, very low not only does the tension itself become relevant but cdso the bending moduli. 

Henceforth we shall use the term capillary waves, or capillary ripples for waves that are so small that interfacial tension contributes significantly to their properties. Two types of such waves can be distinguished spontaneous, or thermal waves and those externally applied. The former type is always present they are caused by spontaneous fluctuations cind have a stochastic nature. In secs. 1.10 and 1.15 it was shown how from these fluctuations interfacial tensions and bending moduli could be obtained. Now the second type will be considered. Transverse or longitudinal perturbations can be applied to the interface, for example by bringing in a mechanically driven oscillator (see sec. 3.7). Such waves are damped, meaning that the amplitude Is attenuated. Damping takes place by viscous friction in the... 

In sec. 1.15 the foundations of such a treatment were already laid in connection with the aim of measuring the bending moduli. Now we shall make a further step by considering explicitly how the quantities characterizing the curvature are related to the composition of the solution, that is, to the various chemical... 

Equations ]4.7.9] are thermodynamic interpretations of the bending moduli for the case that the curvature and the chemical potential are coupled. This is the situation we wemted to describe. De-coupling can, for instance, take place in dilute micro-emulsions then the adsorption or desorption involved In bending the interface, or making more droplets, leads to a negligible change in concentration and hence to almost constant chemical potentials. In that limit. 

Holderer, O., Frielinghaus, H., Byelov, D., Monkenbusch, M., AUgaier, J. and Richter, D. (2005) Dynamic properties of microemulsins modified with homopolymers and diblock copolymers The determination of bending moduli and renormalization effects. /. Chem. Phys., 122, 094908/1-8. 

Under osmotic stress, the aqueous compartment shrinks and the mono-layer around the water cylinder bends. The osmotic work can be translated into monolayer- and bilayer-bending moduli (7) to yield a bilayer value of about 1.5 X 10 12 erg, which is in qualitative agreement with measurements from bilayer mechanics (38). It is possible to refine the analysis of Hn dehydration to include the change in area per lipid molecule and to extract both a bending modulus and a lateral compressibility moduli. This refinement gives moduli in close quantitative agreement with other measures of mechanical deformation (39). 

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