Elastic constants layer compressibility

Here Fq is tire free energy of the isotropic phase. As usual, tire z direction is nonnal to tire layers. Thus, two elastic constants, B (compression) and (splay), are necessary to describe tire elasticity of a smectic phase [20,19, 86]. 

A more complete description of smectic A needs to take into account the compressibility of the layers, though, of course, the elastic constant for compression may be expected to be quite large. The basic ideas of this model were put forward by de Gennes. > We consider an idealized structure which has negligible positional correlation within each smectic layer and which is optically uniaxial and non-ferroelectric. For small displacements u of the layers normal to their planes, the free energy density in the presence of a magnetic field along z, the layer normal, takes the form... 

Close to the N-SmA transition, the vanishing of the elastic constant of compression of the layers amplifies the fluctuations of the phase whereas eritical fluctuations of the amplitude P(r) are expected to be important too. 

Ribotta and co-workers were the first to observe that a uniaxial pressure applied normal to the smectic A layers induces a transition to the smectic C phase when the stress exceeds a threshold value [130]. The behavior is much more pronounced when the stress is applied at a temperature close the SmA-SmB transition. The finite tilt angle induced can be directly related to B, the elastic constant of compression. Further studies have shown that the temperature dependence of the critical stress and strain re-... 

For small-molecule thermotropic smectic-A phases, typical values of two elastic constants are K 10 dyn and B 10 dyn/cm (Ostwald and Allain 1985). For lyotropic smectics, such as those made from surfactants in oil or water solvents, the layer compression modulus B can be much lower (see Chapter 12). From B and K, a length scale A. = ( 1 /B) 1 nm is defined it is called the permeation depth and its magnitude... 

Here the A terms describe curvature distortions of the smectic planes, the B terms the distortions of the director when the smectic planes are unperturbed, and the C terms the coupling between these two types of distortions. All the coefficients are approximately of the same order of magnitude as the nematic elastic constants. A term of the type B(du/dzY may also be included to allow for the compression of the layers, but we shall neglect it in the present discussion. 

The coefficient B is the elastic constant associated with the layer compressions. This represents the solid-like term along the layer normal. The second term (nematic-like) describes how much energy is required to bend the layers. In describing the SmA phase, both elastic constants K and B are very important. The former, which is higher order. 

Yet another technique, measuring ultrasonic velocity anisotropies in the vicinity of the NA transition, however, also find anisotropies consistent with crossover behaviour. Sonntag et al [52] studied the divergence of three elastic constants, the bulk compression constant A, the layer compression constant B and the bulk-layer coupling constant C. B and C have critical exponents that are unequal and are in between that of the 3DXY values and anisotropic scaling values. 

Wlrile quaternary layers and stmctures can be exactly lattice matched to tire InP substrate, strain is often used to alter tire gap or carrier transport properties. In Ga In s or Ga In Asj grown on InP, strain can be introduced by moving away from tire lattice-matched composition. In sufficiently tliin layers, strain is accommodated elastically, witliout any change in the in-plane lattice constant. In tliis material, strain can be eitlier compressive, witli tire lattice constant of tire layer trying to be larger tlian tliat of tire substrate, or tensile. 

If it concerns a monolayer of an amphiphile that is insoluble in the bordering phases, the modulus is purely elastic (although at strong compression, i.e., large AA/A, the surface layer may collapse), and SD is constant in time and independent of the dilatation rate. If the surfactant is soluble, exchange of surfactant between interface and bulk occurs, and Esr> will be time dependent. This means that also an apparent surface dilatational viscosity can be measured ... 

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