Big Chemical Encyclopedia

Chemical substances, components, reactions, process design ...

Articles Figures Tables About

Distribution functions Onsager trial function

These expressions contain two orientation-dependent parameters p and ct(N), which can be calculated from the equilibrium orientational distribution function T(a) minimizing AF. When the Onsager trial function (Eq. (17)) is used for f(a), p and ct(N) are given as functions of the unknown parameter a the functional form of ct(N) has already appeared in Eq. (18), while p is expressed asymptotically as [2]... [Pg.98]

Now we compare the isotropic-liquid crystal phase boundary concentrations for various polymer solution systems with the scaled particle theory for the wormlike spherocylinder. If the equilibrium orientational distribution function f(a) in the coexisting liquid crystal phase is approximated by the Onsager trial... [Pg.106]

The order parameter S is the orientational average of the second-order Legendre polynomial P2(a n) (n = the director), and if the orientational distribution function is approximated by the Onsager trial function, it can be related to the degree of orientation parameter ot by... [Pg.118]

More convenient calculations of the phase transition can be performed by choosing a trial function for the orientation distribution function / with one or more variational parameters. The free energy as a function of these parameters can then be minimized with respect to these parameters. Onsager [8] chose the following function... [Pg.201]

Comparison of these results with exact values (6.12) and (6.14) shows that the trial function chosen by Onsager works quite well. Odijk [15, 16] realized that for large values of k, Onsager s orientational distribution function can be approximated by a Gaussian distribution function... [Pg.201]

The next step is to find the equilibrium distribution function of/(n). Onsager used a standard approximation method, namely, the variation principle (well known in mathematical physics) to attack the problem. Taking h as a variational parameter, a new angle 9 is defined between n and the direction of the anisotropy axis. He then took as a trial function... [Pg.142]

Equations (53) and (54), supplemented by Eq. (52) for the orientational distribution function and Eq. (49) for the free energy, can be used to determine the coexisting densities and the value of the order parameter at the transition. Eq. (52) was first solved approximately by Onsager who used the trial function... [Pg.88]

Calculations based on the full size distribution, Xj(Q), are tedious. A number of authors therefore replace Xj(f2) by Xl(Q), where L is a mean aggregate length. Using this monodispersity assumption in conjunction with Onsager s trial function approach Odijk [58] has investigated the effect of micellar flexibility on the isotropic-to-nematic phase transition in solutions of linear aggregates. Oz. is modeled in terms of the simple d = 1-model discussed above. In addition... [Pg.77]

Finding the equilibrium distribution function f(it) by minimizing the form of (1.1) is the next step in the Onsager method. Direct minimization results in an integral equation which can only be solved numaically (cf. [11]). For this reason, an approximate variation method with the following trial function was used in [7]... [Pg.4]

To derive the orientational distribution function in an equilibrium state, Onsager used a trial function [12]. To derive the analytical expression of the free energy, we here expand sin

[Pg.48]


See other pages where Distribution functions Onsager trial function is mentioned: [Pg.64]    [Pg.202]    [Pg.126]   
See also in sourсe #XX -- [ Pg.96 , Pg.153 ]




SEARCH



Onsager

Onsager function

Trial functions

© 2024 chempedia.info