Dilation or Compression

Consider the relative change in interfacial area AA/A brought about by an external stress. Dilation leads to dilution of the monolayer which, in turn, results in a rise of the interfacial tension y. Conversely, reducing the interfacial area causes y to decrease. The response of the monolayer to the imposed deformation may be more or less elastic or viscous. Elastic behavior is expected for monolayers in which the amphiphilic molecules are interconnected forming a two-dimensional gel. Also, when the rate of deformation is too high to allow for relaxation back to equilibrium by, for example, adsorption or desorption of amphiphiles to or from the interface, or by reorientation and/or reconformation of the molecules in the monolayer (especially in the case of polymers and proteins), the monolayer responds partly elastically. 

FIGURE 17.13 Interfacial deformations (a) dilation or compression and (b) shear. 

Analogous to the three-dimensional situation (cf. Section 17.1.1), the interfacial dilation (or compression) modulus KP is defined through 

If the monolayer is purely viscous, the change in the interfacial tension would be related to the rate at which the interfacial area is changed, as 

Monolayers seldom, if ever, show an entirely viscous behavior. They always present an elastic contribution. Interfacial dilation or compression causes a change in the interfacial tension which, after releasing the stress, relaxes with a characteristic time toward equilibrium. Thus, the interfacial tension change induced by changing the interfacial area is determined by an elastic and a viscous contribution that are likely to be additive  

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It is not necessary for the physicist to know how to compute the Coulomb functions. They are found in subroutine libraries, for example Barnett et al. (1974). A sufficient idea of their form is obtained by putting j = L = 0 in (4.62), when they are seen to be sinp and cosp respectively. The potential terms dilate or compress the sine and cosine waves, resulting in an overall phase shift at long range. 

If the viscoelastic material is under the effect of an isotropic deformation (dilatation or compression), the diagonal components of both the stress and strain tensors differ from zero. In analogy with Eq. (4.92), the relationship between the excitation and the response is given by... 

The deformations in the smectic A phase liquid crystals are the bending of the smectic layer (accordingly to the splay of the directors) and the dilation or compression of the layers. The energy is thus... 

From this figure we learn that pure deformation without any change in size (pure shear) and change in size without any change in shape of the body (dilatation or compression) have to be considered. For the latter case the tensor of strain becomes Uj,j 8 (Landau Lifschitz 1953). Any deformation can be given as the sum of pure shear and dilatation deformations. Therefore, we get... 

We now discuss the fundamental question of fluctuations in the columnar phase. Let us suppose that the liquid-like columns are along the z axis and that the two-dimensional lattice (assumed to be hexagonal) is parallel to the xy plane. The two basic deformations in such a structure are (i) the curvature deformation (or bending) of the columns without distortion of the lattice and (ii) lattice dilatation (or compression) without columnar curvature. There can also be coupling between the two types of distortion but, as shown by Kleman and Oswald, " the coupling term merely rescales the bend elastic constant of the columns. We shall consider only the vibrations of the lattice in its own plane. The free energy may be written... 

Two types of interfacial deformation are considered, dilation or compression, and shear, as shown in Figure 17.13. 

In a similar way as the Young modulus E and the Poisson ratio V are connected to the uniaxial extension test, the shear modulus G and the bulk modulus K are connected to simple shear and isotropic deformation (i.e. dilatation or compression). Note that, accidentally, it turns out that the shear modulus G equals the second Lame constant ju. Since for isotropic materials only two of the elastic constants are independent, the knowledge of any pair of them is sufficient to calculate the other constants and thus to describe the elastic behavior of isotropic materials completely. For easy reference in this chapter we list the most important interrelations between the elastic constants ... 

SmA layers is a very rare event (except for large K). A possible solution is that the nucleation starts from a nonuniform state. The main idea is simple a bulk dust particle or surface irregularity dilates (or compresses) the smectic layers and thus the initial state is characterized by some nonzero dilation energy. The nucleation of the TFCD means the substitution of the dilation by curvature deformations which generally have less energy. The... 

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