Torques molecular field

The torque acting on the director is given by n x h, where h is called the molecular field, which can be derived from the Euler-Lagrange equation. The energy density corresponding to the flexoelectric polarization is given by — Pfl E and the molecular field can be expressed in the form ... 

Mathematically the molecular field vector h can be found using the Euler-Lagrange approach by a variation of the elastic and magnetic (or electric) parts of the free energy with respect to the director variable n(r) (with a constraint of = 1). For the elastic torque, in the absence of the external field, the splay, twist and bend terms of h are obtained [9] fi-om the Frank energy (8.16) ... 

As during relaxation the external field is switched off, the molecular field includes only the elastic torque. For the pure twist distortion and our geometry, the molecular field vector h is opposite to the magnetic field, i.e. directed opposite to X- axis. Its absolute value is (Section 8.3.3). The torque (n x h) is... 

In each particular situation, these torques may be balanced by the elastic, surface or viscous torques. The magnetic and electric field torques may be obtained differently. Using minimisation of the free energy with respect to the director one obtains the molecular field introduced earlier, see Eq. (8.27) and then finds the torques as vector products with the director. Let us show it. The magnetic free energy density is given by... 

By analogy, the molecular field coming from minimisation of (11.45) is given by (8.27) and the corresponding torque exerted by hg on the director is equal to (11.42) ... 

As we mentioned earlier, as a result of the significant elastic torque from molecules surrounding the laser beam (besides the boundary elastic torque), the field required to create finite molecular reorientation (which we shall denote 6th) is in general, larger than that associated with infinite-beam-size lasers. Figure 4 shows a plot of the value of 6th for which nonzero reorientation r) occurred. As a function of Wo/d, we note that 6th for d>... 

The potential of mean torque Uext( >, ) is responsible for the alignment of a conformer and originates from the molecular field of its neighbors. It depends on the conformational state n of the molecule and on the polar... 

The equilibrium condition for the elastic torques corresponds to the minimum of the elastic energy under the constraint of constant director length. It is eonvenient to introduce a molecular field h [36]... 

The seminal molecular field theory of nematics was developed by Maier and Saupe [20] who showed that the potential of mean torque for a uniaxial molecule in a nematic is given by... 

The first molecular field theory of biaxial nematics was presented by Preiser [14] indeed it was his prediction which stimulated the hunt for thermotropic biaxial nematics. An alternative vision of the theory was then given by Straley [17] and, although not primarily concerned with liquid crystals, Boccara et al [33] have presented a theory applicable to uniaxial and biaxial nematic phases. The key feature of these theories is the potential of mean torque which is written using a second rank interaction as... 

This result is not appropriate when the particles interact with a molecular field because the thermodynamic internal energy (not to be confused with the intramolecular energy) caused by molecular interactions is counted twice.This situation obtains because particles both generate and experience the molecular field. To correct this expression for the free energy we need to evaluate the contribution to the internal energy caused by the anisotropic interactions. The starting point is eqn (16) for the potential of mean torque for the nth conformer which can be rewritten in terms of the segmental interactions as... 

Electro-osmosis has been defined in the literature in many indirect ways, but the simplest definition comes from the Oxford English Dictionary, which defines it as the effect of an external electric held on a system undergoing osmosis or reverse osmosis. Electro-osmosis is not a well-understood phenomenon, and this especially apphes to polar non-ionic solutions. Recent hterature and many standard text and reference books present a rather confused picture, and some imply directly or indirectly that it cannot take place in uniform electric fields [31-35]. This assumption is perhaps based on the fact that the interaction of an external electric held on a polar molecule can produce only a net torque, but no net force. This therefore appears to be an ideal problem for molecular simulation to address, and we will describe here how molecular simulation has helped to understand this phenomenon [26]. Electro-osmosis has many important applications in both the hfe and physical sciences, including processes as diverse as water desahnation, soil purification, and drug delivery. 

Strong nonresonant laser fields affect molecular rotation by exerting an angle-dependent torque on the field-induced molecular dipole [15-20]. The interaction is described by the potential... 

Debye obtained his result by solving a forced diffusion equation Ci.e., with torque of the applied field included) for the distribution of dipole coordinate p - pcosS, with 6 the polar angle between the dipole axis and tSe field, and the same result for the model follows very simply from equation (3) using the time dependent distribution function in the absence of the field (5). The relaxation time is given by td = 1/2D, which for a molecular sphere of volume v rotating in fluid of viscosity n becomes... 

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