Stiff chains

As with flexible chains, most studies of conformational behavior of stiff chains have involved light scattering and intrinsic viscosity studies of dilute polymer solutions. Excluded volume effects are of much diminished significance for stififer chains, so measured persistence lengths usually show only a mild dependence on the nature of the solvent. In fact, Norisuye and Fujita [72] have shown that excluded volume effects become measurable only when chains are very long (L 100 q). Thus, the choice of solvent is normally of less importance in studying the conformation of stiff chains than it is for flexible ones. Temperature, however, frequently has a pronounced impact on the value of q for stiff chain polymers [73]. 

measurement of Rg by light scattering leads directly to q. However, light scattering measurements on stiff polymers are often difficult, and conformational studies are most frequently conducted using intrinsic viscosity measurements. 

The Yamakawa-Fujii wormlike cylinder model [25,26] has been widely used for estimation of I (or q) of stiff chain materials from intrinsic viscosities. This approach is based on the equation 

Bohdanecky [83] has recently described a simple graohical procedure that allows easy evaluation of the wormlike cylinder parameters. He showed that d is related to by 

The subscript oo denotes the random coil value. From this approach unperturbed dimensions can readily be evaluated. The Kuhn length f may be computed as 

---

Salto N, Takahashi K and Yunoli Y 1967 The statistical mechanical theory of stiff chains J Phys. See. Japan 22 219... 

The various mechanical properties of polyamides may be traced in many instances to the possibility of intermolecular hydrogen bonding between the polymer molecules and to the relatively stiff chains these substances possess. The latter, in turn, may be understood by considering still another equilibrium, this one among resonance structures along the chain backbone ... 

Within these temperature ranges the polymers are, like the polycarbonates, tough. Because of the stiff chain and the resultant high Tg, processing temperatures need to be above 300°C. 

When this is carried out in suitable solvents at temperatures in the range 75-120°C, soluble products will be obtained. Polymeric MDI is usually used as the isocyanate component and this results in a stiff chain molecule. One such product is reported to have a of 200-220°C. 

The softening behaviour of a thermoplastic material depends to a large extent on the flexibility of the chain and the ability to crystallise. Significant cross-linking of a reasonably stiff-chained polymer will lead to material that is unlikely to soften below its decomposition temperature. Intermediate to the linear and cross-linked polymers are various ladder polymers in which the polymer molecule consists of a pair of more-or-less parallel chains bridged in a manner analogous to the rungs of a ladder. 

Fig. 9. Solid-state NMR spectra of stiff chain aromatic polyesters containing sulfur bonds and tentative assignements of their signals, 401. A contact time of 2 ms and a pulse repetition time of 10 s were used...

Both low molecular weight materials [145] and polymers [146,147] can show liquid crystallinity. In the case of polymers, it frequently occurs in very stiff chains such as the Kevlars and other aromatic polyamides. It can also occur with flexible chains, however, and it is these flexible chains in the elastomeric state that are the focus of the present discussion. One reason such liquid-crystalline elastomers are of particular interest is the fact that (i) they can be extensively deformed (as described for elastomers throughout this chapter), (ii) the deformation produces alignment of the chains, and (iii) alignment of the chains is central to the formation of liquid-crystalline phases. Because of fascinating properties related to their novel structures, liquid-crystalline elastomers have been the subject of numerous studies, as described in several detailed reviews [148-150]. The purpose here will be to mention some typical elastomers exhibiting liquid crystallinity, to describe some of their properties, and to provide interpretations of some of these properties in molecular terms. 

Polyisocyanates have attracted much attention owing to their liquid crystalline properties, stiff-chain solution characteristics, and induced optical activities associated with helical chain conformation (Scheme 8). Pattern and Novak [39]... 

Fig.1. Regular star macromolecules with/=3,4, and 8 arms of identical length. The arms or rays can consist of rather stiff chains, but are in most cases flexible chains. The global structure is determined by the overall shape of the whole macromolecule the internal structure is indicated by a domain that is much smaller than the overall dimension but still larger than a few Kuhn segments...

Figure 5.6 shows that the PDMS data perfectly match the prediction of the simple Rouse model up to the highest Q-values, whereas the PIB data show severe deviations from the Rouse model (Fig. 5.3) and the stiff chain model (Fig. 5.4). From the fact that two polymers with very similar structural parameters but strongly different torsional barriers display completely different relaxation behaviour the conclusion is compelling that there must be an addi-...

Linear polymers, polystyrene and cellulose triacetate exhibit differences in hydrodynamic behavior in solution. Cellulose and its derivatives are known to have highly extended and stiff chain molecules below a Dp of about 300, but as the Dp Increases above 300 the chain tends to assume the character of a random coll (27,28). The assumption that hydrodynamic volume control fractionation in GPC may not be true for polystyrene and cellulose triacetate, though it has been found satisfactory for non-polar polymers in good solvents (29). 

The positive intercepts in Figure 7 show that post-gel(inelastic) loop formation is influenced by the same factors as pre-gel intramolecular reaction but is not determined solely by them. The important conclusion is that imperfections still occur in the limit of infinite reactant molar masses or very stiff chains (vb - ). They are a demonstration of a law-of-mass-action effect. Because they are intercepts in the limit vb - >, spatial correlations between reacting groups are absent and random reaction occurs. Intramolecular reaction occurs post-gel simply because of the unlimited number of groups per molecule in the gel fraction. The present values of p , (0.06 for f=3 and 0.03 for f=4 are derived from modulus measure- ments, assuming two junction points per lost per inelastic loop in f=3 networks and one junction point lost per loop in f=4 networks. 

Note 1 The model describes the whole spectrum of chains with different degrees of chain stiffness from rigid rods to random coils, and is particularly useful for representing stiff chains. 

Lyotropic LCP s are processed from a solution, thermotropic ones from the melt. In both cases the flow pattern provides the necessary orientation of the stiff chains. 

The chain stiffness inflnences the height of the glass-rnbber transition temperatnre (and of the melting point), bnt not the stiffness of the polymer below Tg (in the glassy state). Extremely stiff chains show the effect of the formation of LCP s (liqnid-crystalline polymers), by which very high stiffness is reached, bnt only in the direction of the orientation. 

The free volume, i.e., the volume not occupied by the polymer molecules, is similar for polymers at the Tg and increases as the temperature is increased. More-mobile short chains have lower entropy values and hence lower Tm values, while less-mobile stiff chains have higher entropy values and higher Tm values. 

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. 

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 . 

Table 1. Persistence length q and molar mass per unit contour length ML for liquid-crystalline stiff-chain polymers...

The numerical coefficients in these equations as well as the numerical constants Av>i in Eq. (32) are given in Table 5. In fact, Eq. (32) approximates the results of direct numerical analysis to within 3% for 0.0015< d< 0.15, N> 0.05, and L/d > 5, the conditions which are fulfilled by most stiff-chain polymer solution systems studied so far. Equation (32) is more accurate at small N than our previous theory [18], in which slightly different empirical equations for c, and cA were proposed. 

Figures 7 and 8 display such plots for various lyotropic liquid-crystalline polymer systems, which range in q from 5.3 to 200 nm. As expected, most data points come close to the theoretical curve. This finding suggests that liquid crystallinity of stiff-chain or semiflexible polymer solutions has its main origin in the hard-core repulsion of the polymers.

A ternary system consisting of two polymer species of the same kind having different molecular weights and a solvent is the simplest case of polydisperse polymer solutions. Therefore, it is a prototype for investigating polydispersity effects on polymer solution properties. In 1978, Abe and Flory [74] studied theoretically the phase behavior in ternary solutions of rodlike polymers using the Flory lattice theory [3], Subsequently, ternary phase diagrams have been measured for several stiff-chain polymer solution systems, and work [6,17] has been done to improve the Abe-Flory theory. 

In the second half of this article, we discuss dynamic properties of stiff-chain liquid-crystalline polymers in solution. If the position and orientation of a stiff or semiflexible chain in a solution is specified by its center of mass and end-to-end vector, respectively, the translational and rotational motions of the whole chain can be described in terms of the time-dependent single-particle distribution function f(r, a t), where r and a are the position vector of the center of mass and the unit vector parallel to the end-to-end vector of the chain, respectively, and t is time, (a should be distinguished from the unit tangent vector to the chain contour appearing in the previous sections, except for rodlike polymers.) Since this distribution function cannot describe internal motions of the chain, our discussion below is restricted to such global chain dynamics as translational and rotational diffusion and zero-shear viscosity. 

---