Orientational distribution function equation

This can be inserted in equation (02.2.3) to give tlie orientational distribution function, and tlius into equation (02.2.6) to deteniiine the orientational order parameters. These are deteniiined self-consistently by variation of tlie interaction strength iin equation (c2.2.7). As pointed out by de Gemies and Frost [20] it is possible to obtain tlie Maier-Saupe potential from a simple variational, maximum entropy metliod based on tlie lowest-order anisotropic distribution function consistent witli a nematic phase. 

The averages in equation (5.7) represent averages over the orientation distribution function characterizing the angles 0, [Pg.83]

If the orientation distribution function is uniaxial, and symmetric about the x axis, the averages in equation (5.44) take on the following simple forms,... 

An exact solution for the orientation distribution function can be found in the absence of Brownian motion since the motion of the particles is deterministic and given by equation (7.107). In this case the orbit of each particle is uniquely determined from its initial conditions specified by the orbit constants, C and k. Indeed, the orientation distribu-... 

A consistent study of the linear and lowest nonlinear (quadratic) susceptibilities of a superparamagnetic system subjected to a constant (bias) field is presented. The particles forming the assembly are assumed to be uniaxial and identical. The method of study is mainly the numerical solution (which may be carried out with any given accuracy) of the Fokker-Planck equation for the orientational distribution function of the particle magnetic moment. Besides that, a simple heuristic expression for the quadratic response based on the effective relaxation... 

Taking thermal fluctuations into account, the motion of the particle magnetic moment is described by the orientational distribution function W(e,t) that obeys the Fokker-Planck equation (4.90). For the case considered here, the energy function is time-dependent ... 

The Fokker-Planck equation governing the evolution of the orientational distribution function W(e, n. t) for arbitrary i/ and i/j, that is, when the particle... 

Cooke BJ, Matheson AJ (1976) Dynamic viscosity of dilute polymer solutions at high frequencies of alternating shear stress. J Chem Soc Faraday Trans II 72(3) 679-685 Curtiss CF, Bird RB (1981a) A kinetic theory for polymer melts. I The equation for the single-link orientational distribution function. J Chem Phys 74 2016—2025 Curtiss CF, Bird RB (1981b) A kinetic theory for polymer melts. II The stress tensor and the rheological equation of state. J Chem Phys 74(3) 2026—2033 Daoud M, de Gennes PG (1979) Some remarks on the dynamics of polymer melts. J Polym Sci Polym Phys Ed 17 1971-1981... 

From the potential Vnem(vt). one can obtain the rod orientation distribution function 1/ (u), and hence the order parameter S, by a self-consistent calculation. At equilibrium, the distribution function V (u) is related to the potential Fnemfu) by Boltzmann s equation ... 

From slow-shear-rate solutions of the Smoluchowski equation, Eq. (11-3), with the Onsager potential, Semenov (1987) and Kuzuu and Doi (1983, 1984) computed the theoretical Leslie-Ericksen viscosities. They predicted that ai/a2 < 0 (i.e., tumbling behavior) for all concentrations in the nematic state. The ratio jai is directly related to the tumbling parameter X by X = (1 -h a3/a2)/(l — aj/aa). Note the tumbling parameter X is not to be confused with the persistence length Xp.) Thus, X < I whenever ai/a2 < 0. As discussed in Section 10.2.4.1, an approximate solution of Eq. (11-3) predicts that for long, thin, stiff molecules, X is related to the second and fourth moments Sa and S4 of the molecular orientational distribution function (Stepanov 1983 Kroger and Sellers 1995 Archer and Larson 1995) ... 

A complete description of the texture (or preferred orientation) is formulated as a probability for finding a particular crystallite orientation within the sample this is the orientation distribution function (ODF). For an ideally random powder the ODF is the same everywhere (ODF=l) while for a textured sample the ODF will have positive values both less and greater than unity. This ODF can be used to formulate a correction to the Bragg intensities via a fourdimensional surface general axis equation) that depends on both the direction in reciprocal space and the direction in sample coordinates ... 

Stein (67) has used the basic theory of Nishijima for developing equations for predicting the values of various ratios of the fluorescence intensities of differing polarization expected from a stretched ideal rubber. Stein s theoretical efforts made use of Roe and Krigbaum s (55) expression for the complete orientation distribution function derived from the Kuhn and Griin (58) statistical segment model. Stein s equations have yet to be verified by experiment. 

In order to obtain the maximum Z( Na ), we minimize Equation 2.19 with respect to the orientation distribution function while retaining the normalization condition 2.20. The Euler-Lagrange equation is... 

It has been shown that Onsager s approximation (Equation 2.25) on the orientational distribution function works well (error is within 7%). 

Hence, transformations of Maxwell equations, the change of orientational distribution functions to the form, close to the Boitzmann distribution with account of classical equations of motions yield the complex susceptibility /(co) determined by unperturbed collision-free motion of an individual particle in a given static potential well. In our approach, the complex permittivity e(co) is found as a simple rational function of this susceptibility /(co). 

Some methods, as described in the following section, can give the complete orientation distribution function, N 9). It is often usefnl to calculate from this the values of / 2(cos0)) and (P4(cos0)) for comparison with values obtained by other methods. They can be obtained from the equation... 

Extension the solution of two-dimensional wave equation to a three dimensional wave spectrum expression can get the directional spectrum it can be decomposed into a frequency spectrum and an orientation distribution function, such as... 

The orientation distribution function (ODF) can be formed as given in equation 1 from information of Pj(cos0) and P (cos0) [56],... 

Instead of formal minimization of the free energy leading to an integral equation for the orientation distribution function / we will first use the Gaussian distribution function which simplifies the calculations considerably, while leading to reasonably good results. This is illustrated in Fig. 6.6, where we plot the isotropic-nematic phase coexistence curve for L/D = 10 and q= On the ordinate the relative reservoir concentration of penetrable hard spheres is plotted versus the... 

In injection molding CAE, fiber orientation analysis is carried out combining with mold filling analysis. Using velocity distributions obtained from the mold filling analysis, the fiber orientation equation is numerically solved by a finite-difference method for predicting fiber orientation. The orientation distribution function is calculated and is applied to the estimation of mechanical properties for predicting warp age of molded parts. 

The parallel cross-linked pore model has been developed by Johnson and Stewart [1965] and Feng and Stewart [1973]. Equation (3.4-7) is considered to apply to a single pore of radius r in the solid, and the diffusivities are interpreted as the actual values, rather than effective diffusivities corrected for porosity and tortuosity. A pore size and orientation distribution function f r, Q) is defined. Then, f r, fi)d/dfi is the fraction open area of pores with radius rand a direction that forms an angle Q with the pellet axis. The total porosity is then... 

Kumar and Dattagupta studied the linear response of an assembly of noninteracting particles to a small oscillating field. An appropriate Fokker-Planck equation for the orientational distribution function of the particles was written down and solved approximately under the condition that KV kT. Particle size distribution was not taken into account, but random orientation of easy axes with respect to the magnetic field is... 

Now Eq. (3.23) is an approximation since, in reality, the fi2 has cylindrical symmetry in nematics. This point is deferred [see Eq. (3.57)], but note that additional terms [3.10] are needed in the above equation for (U12). The average over the orientations of molecule 2 only influences and requires an orientation distribution function /( 2) molecule 2. Since there is no (f) dependence in nematics because of uniaxial symmetry, the integral over (f) vanishes unless m is zero and L is even. Thus,... 

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