Fuzzy cylinder

This article reviews the following solution properties of liquid-crystalline stiff-chain polymers (1) osmotic pressure and osmotic compressibility, (2) phase behavior involving liquid crystal phasefs), (3) orientational order parameter, (4) translational and rotational diffusion coefficients, (5) zero-shear viscosity, and (6) rheological behavior in the liquid crystal state. Among the related theories, the scaled particle theory is chosen to compare with experimental results for properties (1H3), the fuzzy cylinder model theory for properties (4) and (5), and Doi s theory for property (6). In most cases the agreement between experiment and theory is satisfactory, enabling one to predict solution properties from basic molecular parameters. Procedures for data analysis are described in detail. 

The dynamic behavior of liquid-crystalline polymers in concentrated solution is strongly affected by the collision of polymer chains. We treat the interchain collision effect by modelling the stiff polymer chain by what we refer to as the fuzzy cylinder [19]. This model allows the translational and rotational (self-)diffusion coefficients as well as the stress of the solution to be formulated without resort to the hypothetical tube model (Sect. 6). The results of formulation are compared with experimental data in Sects. 7-9. 

Before proceeding to a review of both scaled particle theory and fuzzy cylinder model theory, it would be useful to mention briefly the unperturbed wormlike (sphero)cylinder model which is the basis of these theories. Usually the intramolecular excluded volume effect can be ignored in stiff-chain polymers even in good solvents, because the distant segments of such polymers have little chance of collision. Therefore, in the subsequent reference to wormlike chains, we always mean that they are unperturbed . 

In the rod limit, Le = L, and de = d. On the other hand, in the coil limit, Le - V6de, if we assume the chain to be in the unperturbed state. It is to be noted that the axial ratio of the fuzzy cylinder is greater than unity even in the coil limit. At intermediate N, L and the axial ratio Le/de may be calculated, respectively, from the Kratky-Porod equation [102,103] for the (unperturbed)... 

Fig. 14. The Kuhn segment number N dependence of the axial ratio Le/d for fuzzy cylinders with different d [104]...

We may have to consider that the segment distribution fluctuates in the cylindrical domain in order to formulate the effects of entanglement and jamming in a solution as illustrated in Fig. 13b. In other words, we may no longer be permitted to consider the fuzzy cylinder a hard-core cylinder of the geometry specified by Eq. (43), but have to make its periphery fluctuate. 

The diffusion coefficients at infinite dilution (D]0, D 0, and Dr0) for the fuzzy cylinder reduce to those for the wormlike cylinder, which can be calculated as explained in Appendix B. On the other hand, these diffusion coefficients, D, Dx, and Dr, for the fuzzy cylinder at finite concentrations can be formulated by use of the mean-field Green function method and the hole theory, as detailed below. 

To explain the Green function method for the formulation of Dx, D and D, of the fuzzy cylinder [19], we first consider the transverse diffusion process of a test fuzzy cylinder in the solution. As in the case of rodlike polymers [107], we imagine two hypothetical planes which are perpendicular to the axis of the cylinder and touch the bases of the cylinder (see Fig. 15a). The two planes move and rotate as the cylinder moves longitudinally and rotationally. Thus, we can consider the motion of the cylinder to be restricted to transverse diffusion inside the laminar region between the two planes. When some other fuzzy cylinders enter this laminar region, they may hinder the transverse diffusion of the test cylinder. When the test fuzzy cylinder and the portions of such other cylinders are projected onto one of the hypothetical planes, the transverse diffusion process of the test cylinder appears as a two-dimensional translational diffusion of a circle (the projection of the test cylinder) hindered by ribbon-like obstacles (cf. Fig. 15a). 

Fig. 15a-c. Projection procedures for the transverse, rotational and longitudinal diffusion processes of a fuzzy cylinder in concentrated solution... 

Since the fuzzy cylinders act as obstacles when entering the laminar region in Fig. 15a, the probability of their appearance (and disappearance) on the projection plane should be governed essentially by the longitudinal diffusion of the hindering or test fuzzy cylinder, and the lifetime t should obey the relation... 

The quantity pBX can be calculated from vex, c / x, where vex. x is the excluded volume between one hindering fuzzy cylinder and the laminar region of unit surface area, and is approximately equal to Le[l + fde/Le)]. Furthermore, the average length b L is proportional to Le for the longer sides of ribbon-like obstacles and to de for the shorter sides of the obstacles. 

Before inserting t, pBX, and bx obtained above into Eq. (44), we have to mention the fluctuation effect of the fuzzy cylinder. When a hindering fuzzy cylinder is entering the periphery of the laminar region, the hindrance that it exerts may be released by fluctuation of the segment distributions in the... 

The rotational diffusion coefficient of the fuzzy cylinder can be formulated in a similar way. For the rotational diffusion process, it is convenient to imagine a hypothetical sphere which has the diameter equal to Lc, just encloses the test fuzzy cylinder, and moves with the translation of the fuzzy cylinder. If the test cylinder and the portions of surrounding fuzzy cylinders entering the sphere are projected onto the spherical surface as depicted in Fig. 15b (cf. [108]), the rotational diffusion process of the test cylinder can be treated as the translational diffusion process of a circle on the hypothetical spherical surface with ribbon-like obstacles. 

In this model, the geometry of the critical hole must be specified. Here we simply assume that the critical hole for a semifiexible polymer chain is similar in shape to the fuzzy cylinder [19]. The similarity ratio X between the critical hole and the fuzzy cylinder is left as an adjustable parameter. The mutual excluded volume V x between the critical hole and one surrounding semifiexible chain can... 

With increasing polymer concentration, we may expect that the polymer global motion changes from the fuzzy cylinder model mechanism to the repta-tion model mechanism. The onset of the crossover should depend on the degree in which the lateral motion of a polymer chain is suppressed by entanglement with its surrounding chains, but it is difficult to estimate this degree. There are some disputes over it in the case of flexible polymers [20]. 

If we neglect the distortion of the segment distribution in the fuzzy cylinder by the shear flow, we can apply Doi s stress expression, Eq. (61), to fuzzy cylinder systems as it stands. The neglect of the distortion may be justified when the shear-rate is low. Equation (61) expresses the contribution of the end-over-end rotation of the chain to asegment distribution is not distorted, the orientational entropy term Sor in the static free energy expression contains only the orientational entropy loss of the entire chain, but not the conformational entropy loss cf. Sect. 2.3. 

Values of B calculated from the ordinate intercepts are shown in Fig. 23 as a plot of B/(2q)3 against the number of the Kuhn segments N. For N<4, the data points for the indicated systems almost fall on the solid curve which is calculated by Eq. (78) along with Eqs. (43), (51), (52), and Cr = 0. A few points around N 1 slightly deviate downward from the curve. Marked deviations of data points from the dotted lines for the thin rod limit, obtained from Eq. (78) with Le = L and de = 0, are due to chain flexibility the effect is appreciable even at N as small as 0.5. The good lit of the solid curve to the data points (at N 4) proves that the effect of chain flexibility on r 0 has been properly taken into account by the fuzzy cylinder model. 

Figure 24 shows the N dependence of VJ q)3 for aqueous xanthan and schizophyllan. For both systems, V increases monotonically with N. As mentioned in Sect. 6.3.2, if the critical hole for longtitudinal diffusion has similarity ratio X to the fuzzy cylinder, V x is given by Eq. (56), and if X and d are taken to be 0.11 and 2.2 nm for xanthan and 0.13 and 2.6 nm for schizophyllan (cf. Tables 2 and 6), this equation gives the solid curves shown in Fig. 24. They fit closely the data points for the two systems over the entire range of N examined.

In this article, we have surveyed typical properties of isotropic and liquid crystal solutions of liquid-crystalline stiff-chain polymers. It had already been shown that dilute solution properties of these polymers can be successfully described by the wormlike chain (or wormlike cylinder) model. We have here concerned ourselves with the properties of their concentrated solutions, with the main interest in the applicability of two molecular theories to them. They are the scaled particle theory for static properties and the fuzzy cylinder model theory for dynamical properties, both formulated on the wormlike cylinder model. In most cases, the calculated results were shown to describe representative experimental data successfully in terms of the parameters equal or close to those derived from dilute solution data. 

The fuzzy cylinders are assumed to interact dynamically through a hard-core potential. The rotational diffusion coefficient of the rods is computed as... 

Figure 22 Relative viscosities ti/rjo plotted against the molar concentration

Figure 23 Comparison of experimental >) (circles) for toluene solutions of PHIC at 25 °C with theoretical values (eqn [86]) based on the fuzzy cylinder model.

Figure 27 Concentration dependence of >s (filled symbols) and that of D (unfilled symbols) for CTC samples of the indicated molecular weights in THF at 25 °C. Curves, theoretical >s for the fuzzy cylinder model (eqns [105]-[107]).

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