Planar external fields

Note that the external field vanishes in the normal surface tension calculation. In the fully local approximation there is no surface tension. Thus we can obtain the surface tension y associated with a planar interface of area A by the expression... 

In the first period, which ended with a review [18], the complex susceptibility x (0)) was expressed through the law of motion of the particles perturbed by a.c. external field E(t). The results of these calculations rigorously coincide with those obtained, for example, in Refs. 22 and 23, respectively, for the planar and spatial extended diffusion model (compare with our Ref. 18, pp. 65 and 68). The most important results of this period are (i) the planar confined rotator model [ 17, p. 70 20], which has found a number of applications in our and other [24—31] works (ii) the composite so-called confined rotator-extended diffusion model. However, this approach had no perspectives because of troublesome calculations of the susceptibility x ( )-... 

Fig. 24 Magnetic susceptibility for BABI at 10,000 Oe external field. A-D represent fits to the experimental data (A) square planar AFM system with J/k = — 1.6 K (B) square planar bilayer AFM system with J2olk = —1.4 K and interlayer Jik = —1.3 K (C) AFM spin pairing with Jjk = —3.8 K C is same model as C, with Jik = — 2.4 K D is same model as B, with J2T>lk = — 1.2K and interlayer Jik = — 1.9K (from calorimetric analysis).

We wish to study the effects of planar Couette flow on a system that is in the NPT (fully flexible box) ensemble. In this section, we consider the effects of the external field alone on the dynamics of the cell. The intrinsic cell dynamics arising out of the internal stress is assumed implicitly. The constant NPT ensemble can be employed in simulations of crystalline materials, so as to perform dynamics consistent with the cell geometry. In this section, we assume that the shear field is applied to anisotropic systems such as liquid crystals, or crystalline polytetrafluoroethylene. For an anisotropic solid, we assume that the shear field is oriented in such a way that different weakly interacting planes of atoms in the solid slide past each other. The methodology presented is quite general hence it is straightforward to apply for simulations of shear flow in liquids in a cubic box, as well. 

A difference between the perturbation considered here and that in the section on LRT considered earlier is the term involving qo, the position at which the drift velocity (i.e., the velocity contribution from the external field) of the fluid is zero. This term was chosen to be zero in the treatment for bulk fluids for simplicity it must be used here because confinement has broken the translational invariance of the system. The perturbation generates a planar Couette flow in the fluid between two surfaces ... 

In addition, the fluid is confined between two planar solid surfaces (slit-pore) exerting an external field on the fluid molecules. Specifically, these solid... 

Therefore, GMR angle sensors have a natural 360° measuring range, in contrast to the 180° covered by AMR sensors. The angle signal calculated from the arctangent does not repeat after 180° as occurs with AMR sensors, so that no additional sensor or planar coils are necessary to sense a complete rotation of an external field. 

The atomic charges as defined by Eq. [50] thus uniquely account for the force on nucleus A due to an external field in the z direction also exactly reproduce the molecular dipole moment, so that the electrostatic potential at large distances is also exactly reproduced by these atomic charges. (In fact, it is usually found that these charges reproduce the molecular electrostatic potential better than the total molecular dipole moment. ) Thus, for planar molecules it is possible to define a consistent set of atomic charges that account for both the molecular electrostatic potential and the intermolecular forces. Atomic charges as defined by other methods may reproduce the molecular electrostatic potential very well, but in general will not reproduce the perpendicular forces. 

Figure 5. Switching in a regular fiber array examples of director fields for different 77 a T = 1.0, R = 5a, and w = 1 yz-cioss sections through the fiber center). From left to right homogeneous (h), deformed (d), and saturated (s) structure. Anchoring easy axis is planar and z, while the external field E is directed along y. Note that the d-structure is twisted along the x-axis, while there is no twist in a simple nematic slab. This, however, does not affect the qualitative analogy of the two systems.

The optical-field-induced Freedericksz transition for a twist deformation by a normally incident laser beam in a planar-aligned nematic liquid crystal is studied. The Euler equation for the molecular director and the equations describing the evolution of the beam polarization in the birefringent medium are solved simultaneously in the small-perturbation limit. The stability of the undistorted state is investigated. An alternate series of stable and unstable bifurcations is found. This phenomenon has no analog in the Freedericksz transition induced by dc electric and magnetic external fields. 

The model is qualitatively as follows. At high frequencies an electro-convective instability with a periodic velocity distribution in the plane of the layer appears in all samples (in the isotropic phase, in the planar and homeotropically oriented nematics, in cholesterics [111], in nematics with smectic order, and even in smectic A [77]), because of the nonuniform distribution of the space charge along the direction of the external field Ez)-... 

In the general case, the classes of homotopic mappings of the line y threaded through a planar soliton form the relative homotopy group 7Ti(iR, 91), where 91 is the OP space far from the core of the soliton, shrunk (as compared to the complete OP space 91) by additional interactions (external field, boundary conditions, etc.). If 91 consists of a single point, as in Figure 5.18, 7Ti(9I, 91) coincides with the fundamental group 7Ti(91) [77], [78]. 

Just as a disclination in an external field can give rise to a planar soliton, a point defect can give rise to a linear soliton. Linear solitons are described by the classes of mappings of the surface a crossing the soliton into the OP spaces 91 and 91, i.e., by the elements of the second relative group 7T2(91, 91). 

The thermodynamics of curved surfaces is more subtle than that of planar, and we discuss only the most important case, the spherical surface, for which the two principal radii of curvature are equal. Hie extension of the argument to other curved surfaces is beset with difficulties into which we do not enter. The spherical surface, the bubble or the drop, is the only one that is stable in ffie absence of an external field. The original analysis of Gibbs was clarified and its consequences worked out by Tolman, whose work Koenig extended to multicomponent systems. Buff, Hill, and Kondo describe explicitly how the surface tension depends on the position of the dividing surface to which it is referred, or at which it is calculated. 

In 4.2 we set out the statistical mechanics of three-dimensional systems of arbitrary inhomogeneity in the presence of an arbitrary external field we now specialize these equations to a system with a square interface in the x,y-plane of area A = L, whose equimolar dividing surface ( (x, y) has its mean height at z = 0, and whose mean planarity is maintained by a gravitational potential o(z) = mgz, where m is the mass of a molecule. The positions of a pair of points r and f2 can, as before, be represented by their heights Zi and Z2 and the vectors Si and 82, which are the projections of Ci and t2 onto the x, y-plane. 

If we design the coupling of the external field to the system in such a way that the dissipative flux is equal to one of the Navier-Stokes fluxes (such as the shear stress in planar Couette flow or the heat flux in thermal conductivity), it can be shown - provided the system satisfies a number of fairly simple conditions (Evans Morriss 1990) - that the response is proportional to the Green-Kubo time integral for the corresponding Navier-Stokes transport coefficient. This means that the linear response of the system to the fictitious external field is exactly related to linear response of a real system to a real Navier-Stokes force, thereby enabling the calculation of the relevant transport coefficient. 

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