Onsager method

In his paper7) Onsager has considered the liquid-crystalline transition in the system of rigid rods using two main assumptions a) the interaction of rods was assumed to be due to the pure steric repulsion (no attraction) b) the virial expansion method was used (for the details of the Onsager method see Sect. 2.4). Thus, the Onsager results... 

We now begin the analysis of the phase diagram by studying its high-temperature and low-temperature behaviors. Beforehand, in the following section, we will recall the main steps of the Onsager method in the form which is most convenient for further generalizations. 

Thus the fundamental approximation of the Onsager method is the second virial approximation. This method is therefore valid only at low polymer concentrations in the solution. The estimations for the second (B) and the third (C) virial coefficients of the interaction of rods give B p2d3 and C p3d6 (see25)) hence, the second virial approximation (Cc2 < Be) is valid if c < 1/pd3, or 0 < I5. 

The following step of the Onsager method involves the determination of the equilibrium function f(u) which minimizes functional (2.2). Unfortunately, the direct minimization leads to an integral equation, which cannot be solved in a standard way (see, however, Refs.29,30)), So, in Ref.75 the variational method was used, the trial function being chosen in the form... 

It should be emphasized that the application of the variational procedure in Ref.7) is the way to simplify the numerical calculations thus, the only fundamental physical limitation of the Onsager method is connected with the second virial approximation, i.e. with the condition 0 <1 1. 

Equations (2.6) as well as Eqs. (2.7) were obtained by use of some approximations. The approximations of the Flory method are connected with the lattice character of his model it is difficult to estimate the degree of their accuracy. The approximations of the Onsager method are due to (a) the application of the second virial approximation and (b) the application of the variational procedure. It is rather easy to eliminate the latter approximation by solving numerically with high degree of accuracy the integral equation which appears as a result of the exact minimization of expression (2.2). This has been done in Ref.30 the results are... 

To summarize we can say that for the athermal case the fundamental advatange of the Onsager method over the Flory method is due to the fact that the use of the former method permits to obtain the results which are asymptotically exact at p P- 1. 

We will apply the Onsager method (see Eq. (2.3)), taking into account both the attractive and the repulsive parts of the virial coefficient B(y). It is necessary to bear in mind that this method is valid only if i) [Pg.68]

It should be recalled that the qualitative form of the phase diagram for the solution of disconnected rods was obtained in Sect. 2 using the Onsager method, which was generalized to take into account the attraction of the rods. The same approach can be applied to the solution of semiflexibie macromolecules. 

Figure 18 Determination of the electron lifetime in TMP by the Onsager method. (Redrawn from the data of Gonidec, A., Rubbia, C., Schinzel, D., and Schmidt, W.F., Ionization Chambers with Room-Temperature Liquids for Calorimetry, CERN, 1988.)...

NEMATIC ORDERING IN AN ATHERMAL SOLUTION OF RIGID RODS (ONSAGER METHOD)... 

The basic propositions of the Onsager method consist of the following. Let N rods be positioned in volume V so that their concentration is c = N/V, and the... 

The fundamental approximation of the Onsager method thus consists of the fact that the interaction of the rods is taken into consideration in the second virial approximation as a consequence, this method is only applicable with a relatively low concentration of the solution of rods. 

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]... 

We emphasize that the only fundamental physical restriction of the Onsager method is due to the second virial tq)proximation, i.e, the condition d> 1. The use of the variation procedure is simply a way of simplifying the calculation. The integral equation which arises in precise minimization of functional (1.1) can be solved numerically with a high degree of precision this was done in [11,12]. As a result, the following was obtained ... 

The Onsager method is thus applicable with low concentrations of a solution of rods (high concentrations (O 1) [8, 12-16]. Parsons approach [13] is distinguished by the greatest simplicity and generality with respect to application to solutions of rigid-chain macromolecules. The central approximation in [13] consists of the hypothesis that the pair correlation... 

Equations (1.26)-(1.29) completely detomine the free aiCTgy of a solution of semiflexible macromolecules for the models illustrated in Fig. 1.2. The subsequent calculations performed in [32-35] in complete analogy with the Onsager method (cf. 1.1) resulted in the following conclusions. 

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