Isotropic aggregate

For a transversely isotropic aggregate these Equations reduce to... 

For this situation of a transversely isotropic aggregate of transversely isotropic units, the Legendre addition theorem gives... 

For an isotropic aggregate, the stiffness averaging procedure had been proposed by Voigt, ° and the compliance averaging procedure by Reuss, many years previously. Each had been used to compare the elastic constants of single crystals with those of an isotropic aggregate of single crystals (see for example Ref. 12). 

The equations which predict the elastic constants of a partially oriented polymer involve orientation functions to define the orientation of the aggregate units. For example, the average extensional compliance S33 for a transversely isotropic aggregate of transversely isotropic structural units is given by S 33 = Su sin 0+S33Cos" 0-l-(2Si3-l-544) sin 0cos 9. 

In the following, we will discuss the behavior of concentrated solutions of several of the polymers described above in terms of these relations. It is well known that concentrated solutions of rodlike chains will undergo a phase transition from a disordered to an ordered state at some molecular weight dependence concentration.29 in the ordered state the rods form parallel arrays, usually leading to optical anisotropy. The first part of our discussion will be confined to optically isotropic, disordered solutions. In the final parts of this section we will turn to some properties of ordered optically anisotropic solutions and optically isotropic, aggregated solutions. 

Voigt has shown how, starting with the monocrystalline elastic stiffiiess moduli, Cy, it is possible to derive expressions for tne macroscopic shear modulus, Gy, and the bulk modulus, Ky. For a mac-roscopically isotropic aggregate of cubic monocrystais, the VoiGT approach yields ... 

Averaging the compliance constants defines the elastic properties of the isotropic aggregate in terms of 33 and s. This is called the Reuss average [76]. Averaging the stiffness constants defines the elastic properties of the aggregate in terms of C33 and C44. This is... 

It is interesting to apply the aggregate model to these data, calculating bounds for the elastic constants of an equivalent fibre by averaging the sheet constants in the plane normal to the sheet draw direction. This requires an extension of the mathematical treatment of Section 8.6.2 to deal with the case of a transversely isotropic aggregate of orthorhombic units. The basic equations have been given in detail elsewhere [99] so only the key results will be summarised here. If the orthorhombic unit constants are n, 13,..., 66 (Section... 

We assume that at the grain boundaries aggregates, that are responsible of the radiative emission, are randomly oriented and therefore giving rise to isotropic emission. 

It is also comparatively straightforward to-calculate P200, P220, P420 and P o for a biaxially oriented aggregate of transversely isotropic units in terms of the principal extension ratios Xx, X2 and (with X,X2 3 = 1). 

This species has been used as precursor of nanogold particles by electrodeposition. When the electrodeposition is induced from the isotropic state at 117 °C, the nanoparticles obtained are nanodots aggregated in a spherical-like shape. In contrast, the morphology of the nano particles prepared from the SmA mesophase at 111 °C consist of leaf-like forms interlocked in rosettes. 

Overview. Considerable research activities in the fields of isotropic SAXS and small-angle neutron scattering (SANS) are devoted to the investigation of ensembles of uncorrelated but identical or almost identical complex particles. Frequently these particles are studied in solution. Samples for such investigations must be supplied in a solution in which the particles do not aggregate. 

Figure 7.1 Illustration of different aggregation states obtained (from left to right) by increasing temperature crystal (K), smectic C (SmC), nematic (N) and isotropic (I). Row a shows macroscopic appearance of samples in row b, short-range microscopic ordering is represented (each bar represents a molecule) thermotropic phase diagram of row c illustrates relevant transition temperatures (Tm melting temperature Tsmc-N transition temperature between SmC and N Tc clearing temperature) row d shows different texture of different states as seen through polarizing microscope (with crossed polars, isotropic phase appears black).

The term microemulsion is applied in a wide sense to different types of liquid liquid systems. In this chapter, it refers to a liquid-liquid dispersion of droplets in the size range of about 10-200 nm that is both thermodynamically stable and optically isotropic. Thus, despite being two phase systems, microemulsions look like single phases to the naked eye. There are two types of microemulsions oil in water (O/W) and water in oil (W/O). The simplest system consists of oil, water, and an amphiphilic component that aggregates in either phase, or in both, entrapping the other phase to form... 

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