Second virial approximation

Recently, Chen [47] proposed to determine the equilibrium distribution function directly from a numerical analysis. Thusjie determined f(a) by requiring that AF be minimum for all variations of f(a) under the normalization condition. Using the second virial approximation for Sex in AF, he obtained... 

Abe et al. pointed out that the experimental S of PBLG [93] and PYPt [33] solutions at the phase boundary concentration cA largely departed from the prediction of Khokhlov and Semenov s second virial approximation theory [7,44]. Similar deviations of the scaled particle theory from experiment are seen in Fig. 12a,c, where the left ends of the theoretical curves and the experimental data points at the lowest c correspond to cA. Sato et al. [17] showed theoretically that ternary solutions containing two polymer species with different... 

In addition to the above effects, the intermolecular interaction may affect polymer dynamics through the thermodynamic force. This force makes chains align parallel with each other, and retards the chain rotational diffusion. This slowing down in the isotropic solution is referred to as the pretransition effect. The thermodynamic force also governs the unique rheological behavior of liquid-crystalline solutions as will be explained in Sect. 9. For rodlike polymer solutions, Doi [100] treated the thermodynamic force effects by adding a self-consistent mean field or a molecular field Vscf (a) to the external field potential h in Eq. (40b). Using the second virial approximation (cf. Sect. 2), he formulated Vscf(a), as follows [4] ... 

Larson s results [154] are divided into the three shear rate regimes - tumbling, wagging, and steady-state - as explained below. He chose the strength of mean-field potential 2L2dc in Eq. (41) to be 10.67, which corresponds to the concentration cA of the nematic phase coexisting with the isotropic phase (in the second virial approximation), and expressed the shear rate in terms of T defined by... 

Since Chen used the second virial approximation in his phase boundary calculation, it is adequate to compare his results with the phase boundary concentrations calculated from Onsager s expression of S (cf. Sect. 2.5) and cj(N) of Eq. (18) together with the OTF. Chen expressed the phase boundary concentrations in terms of reduced quantities Cj = L2dc j and cA = L2dc A, which depend only on N. As shown in Fig. Al, the relative differences in both Q and cA between the two procedures are less than 13% over the whole range of N. This confirms the relevance of the OTF for semiflexible polymer systems. 

Odijk [6] calculated ct(N) by using a Gaussian trial function for f(a). This trial function, as well as the second virial approximation for use in calculating the phase boundary concentrations, however, leads to c, and cA which differ more than 70% at N l from Chen s calculations. In addition, cj from the Gaussian trial function appears beyond the critical concentration at which the isotropic phase becomes unstable for large N [9]. Therefore, the Gaussian trial function for f(a) is inadequate. 

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. 

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... 

The free energy of the solution of chains in the second virial approximation differs from the free energy of the corresponding solution of disconnected segments (Eq. (2.3)) only in two respects. 

The independence of and of T must be expected in those regions of the phase diagram, where the attraction between the rods is negligible in comparison with their mutual repulsion. Since the high-temperature corridor of the phase diagram is situated at i 1/p I, i.e. in the region of validity of the second virial approximation, this statement can be reformulated as follows the values of and do not depend on temperature if the contribution of the attraction to the second virial coefficient, Ba, is much less than that of the repulsion, B,. 

Flory presented a more detailed analysis of the molecular theory of the hquid crystalline state of polymeric systems. This theory is discussed in the first chapter of this book, so we here shall only describe the principle of calculating the equilibrium phase composition without going into details. Note that Flory s theory embraces a wide range of concentrations, while Onsager s analysis is valid only for low concentrations, since it is based on the second virial approximation. 

The nematic to isotropic transition of rigid rods in solution is of the first order. If the axial ratio L/D is great, the concentration of rods 4> D/L 1, even at the N — I transition, meets the condition of the second virial approximation. 

It should be pointed out that to meet the second virial approximation, molecules must have a large L/D so that at the transition the solution is dilute. For molecules of axial ratio less than 10 the theory does not work well. In addition, the Onsager value of the density difference at the nematic — isotropic transition is greater than the experimental data. 

It is shown that the predictions of the Onsager theory are smaller than that of the experimental data. This discrepancy is due to the second virial approximation. Adopting higher terms of virial coefficients may improve the predictions. 

For example, in the second virial approximation this yields... 

The solution x = corresponding to the isotropic distribution is trivially satisfied at all densities and in any approximation. Below a certain critical density, (2.2.7) has only the isotropic root. Above this density two new roots appear the one which corresponds to low x gives t and the others are discarded. Using (2.2.5), the pressure can then be evaluated as a function of density or volume. The transition predicted by Onsager s theory in the second virial approximation is confirmed by this model even when all virial coefficients up to the seventh are included but the properties of the anisotropic phase depend rather sensitively on the order of the approximation. Fig. 2.2.2 gives the isotherm obtained in the sixth approximation, and it can be seen that the shape of the curve is characteristic of a first order transition. 

Generally speaking, the virial expansion model, asymptotically exact for very large aspect ratios, is applicable only when the volume fraction of rods is very small (second virial approximation), whereas the lattice model may be more reliable at rod concentrations in typical lyotropic phases. The implications of both models are still being explored today. Inclusion of a distribution of rod... 

The crowding approach, which has been based upon McMillan-Mayer solution theory, has anployed Equation 11.7 as a starting point (Davis-Searles et al. 2001 Shimizu and Boon 2004). This is based upon a second virial approximation. FST can even provide the condition upon which this approximation is accurate. Since Equation 11.7 holds under the condition that AN21 is negligibly small, and that this quantity is related to the partial molar volume via Equation 11.5, the proposed condition is... 

Show that in the limit L/D —> oo and low concentrations the above expression for the free energy reduces to the free energy in the second virial approximation (6.1)... 

Figure 4 compares the ideal behavior, that is, (s) - solute result when excluded volume interactions are included. We can understand the shape of the excluded volume curve in terms of our simple model for linear aggregates. Using In (ais a2)- soiute from Ref. 28 (second virial approximation neglecting polydispersity) and = s(xy exp[0 ]) / (cf. Eq. [20]) yields (s) (xsoiute exp[d>o + 22- soiutel) and a (s) for large s), that is, we obtain the same excluded volume induced growth behavior as shown in Figure 4. Note that excluded volume affects the size through (22, which means that the aggregate ends drive the growth.

Most theories of the polymeric nematic phase are based upon the model in which the liquid crystal is considered as a packed system interacting through its hard-core diameters. Historically, attention was first focused on lyotropic liquid crystals. The binary collision or second virial approximation is taken into account at low volume factions. A slightly different approach is to use a scaling law to describe the fluctuation of nematic order parameters. In this section we select three different methods to represent the three different treatments. 

The first term is the free energy of the translational motion of the rods the second term is the loss of orientational entropy, and the last term the fi ee energy of interaction of the rods in the second virial approximation. The term B y) is the... 

Note that the second virial approximation thus always holds in the vicinity of the transition provided... 

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. 

The simplest estimations of the virial coefficients of the steric interaction of the rods, second B L d and third C In (L/d) [8], show that the second virial approximation (Be C

liquid-crystalline transition in a solution of rods will take place just when O 1, and for this reason, the Onsager metlKxl is precise for studying this transition and the properties of the anisotropic phase which arises in the limit of L d (which is the most interesting for rigid-chain macromolecules) [9,10]. 

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