Oseen theory

The Oseen theory embraces smectic mesophases, but is not really required for this case. The interpretation of the equilibrium structures assumed by smectic substances under a particular system of external influences may be carried out by essentially geometric arguments alone. The structures are conditioned by the existence of layers of uniform thickness, which may be freely curved, but in ways which do not require a breach of the layering in regions of greater extension than lines. These conditions automatically require the layers to be Dupin cyclides and the singular lines to be focal conics. Nothing, essentially, has been added... 

The LE theory is rather complex since it contains both viscous and elastic stresses. It can best be understood by considering viscous and elastic effects separately. If elastic effects are neglected, the LE equations reduce to Ericksens transversely isotropic fluidy while in the absence of flow the elastic stresses are just those of the Frank-Oseen theory (discussed below in Section 10.2.2). ... 

A further recent innovation is due to Ericksen [17] who proposes an extension to the Frank-Oseen theory in order to improve solutions modelling defects. To this end he incorporates some variation in the degree of alignment or the order parameter, and therefore proposes an energy of the form... 

Oseen C W 1929 The theory of liquid crystals Trans. Faraday See. 29 883-99... 

Note 3 The names of Oseen, Zocher, and Frank are associated with the development of the theory for the elastic behaviour of nematics and so the elastic constants may also be described as the Oseen-Zocher-Frank constants, although the term Frank constants is frequently used. 

There are also more limited treatments of scattering. McCartney (1976, Chaps. 4-6) confines his attention to scattering by atmospheric particles. This is also discussed by Twomey (1977, Chaps. 9-10) in his treatise on atmospheric aerosols. In Goody (1964, Chap. 7) there are discussions of absorption by gases and, in less detail, extinction by molecules and by droplets. Parts of books on electromagnetic theory or optics include the theory of scattering by a sphere, most notably Stratton (1941, pp. 563-573) and Born and Wolf (1965, pp. 633-664). The latter also derive the Ewald-Oseen extinction theorem and apply it to reflection and refraction at a plane interface (pp. 98-104). 

A general theory of curvature-elasticity in the molecularly uniaxial liquid crystals, similar to that of Oseen, is established on a revised basis. There are certain significant differences in particular one of his coefficients is shown to be zero in the classical liquid crystals. Another, which he did not recognize, does not interfere with the determination of the three principal coefficients. The way is therefore open for exact experimental determination of these coefficients, giving unusually direct information regarding the mutual orienting effect of molecules. 

Oseen likewise proceeded by setting up an expression for energy density, in terms of chosen measures of curvature. However, he based his argument on the postulate that the energy is expressible as a sum of energies between molecules taken in pairs. This is analogous to the way in which Cauchy set up the theory of elasticity for solids, and in that case it is known that the theory predicted fewer independent elastic constants than actually exist, and we may anticipate a similar consequence with Oseen s theory. 

It is worth remarking that the controversial conflict between the swarm theory and the continuum theory of liquid crystals is illusory. The swarm theory was a particular hypothetical and approximative approach to the statistical mechanical problem of interpreting properties which can be well defined in terms of a continuum theory. This point is seen less clearly from Oseen s point of departure than from that of the present paper. 

Comparison with oseen s theory According to Oseen, the energy is expressed by... 

We are not concerned with the first integral, which is not related to the dependence of energy on curvature (and which plays only a minor role in Oseen s theory). It is the integrand of the second integral which should be compared with our free-energy density g. Noting that... 

Our theory may be imderstood better if compared with the KR theory. Their theory has been developed along the observations discussed in Section 1. We note that Ff of Eq. (1.4) which depends on all the segments is replaced in their theory by a one body force determined by the ordering number of a segment irrespective of its location. For this reason it was necessary to replace the Oseen hydrodynamical interaction tensor by its average. 

The idea of Kirkwood (25) is combined with the Rouse model by Pyun and Fixman (14). The theory allows a uniform expansion of the bond length by a factor a such as introduced by Flory. The nondiagonal term of the Oseen tensor is considered but only to the first order by a perturbation method. Otherwise, their theory is identical to Zimm s theory in Hearst s version in the treatment of the integral equation (14). 

The application of the Lorentz-Lorenz equation gives a convincing demonstration of the general similarity of the linear response in gas and liquid but its application in the liquid introduces an approximation which has not yet been quantified. A more precise objective for the theory would be to calculate the frequency dependent susceptibility or refractive index directly. For a continuum model this may lead to a polarizability rigorously defined through the Lorentz-Lorenz equation as shown in treatments of the Ewald-Oseen theorem (see, for example Born and Wolf, plOO),59 but the polarizability defined in this way need not refer to one molecule and would not be precisely related to the gas parameters. 

The tendency of LCs to resist and recover from distortion to their orientation field bears clear analogy to the tendency of elastic solids to resist and recover from distortion of their shape (strain). Based on this idea, Oseen, Zocher, and Frank established a linear theory for the distortional elasticity of LCs. Ericksen incorporated this into hydrostatic and hydrodynamic theories for nematics, which were further augmented by Leslie with constitutive equations. The Leslie-Ericksen theory has been the most widely used LC flow theory to date. 

Kirkwood and Riseman 41) were the first to introduce this kind of perturbation into polymer kinetic theory, based on earlier hydro-dynamic studies by Burgers (/3) and Oseen (6/). The perturbation velocity v is given by ... 

Calculation of Complex Modulus. In the Zimm theory, the Oseen tensor is approximated by its average value over the equilibrium configuration ... 

The elasticity theory of liquid crystals was proposed by Oseen (1933) and Zocher (1933), and then modified by Frank into the form that has since... 

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