Wormlike cylinder

Figure 20. Model of two consecutive growth stages for the polystyrene phase. (1) Early stages microgels of increasing size. (2) Intermediate to final stages Domains become wormlike cylinders and then interconnected cylinders that exhibit dualphase continuity 41).

In the present article, we focus on the scaled particle theory as the theoretical basis for interpreting the static solution properties of liquid-crystalline polymers. It is a statistical mechanical theory originally proposed to formulate the equation of state of hard sphere fluids [11], and has been applied to obtain approximate analytical expressions for the thermodynamic quantities of solutions of hard (sphero)cylinders [12-16] or wormlike hard spherocylinders [17, 18]. Its superiority to the Onsager theory lies in that it takes higher virial terms into account, and it is distinctive from the Flory theory in that it uses no artificial lattice model. We survey this theory for wormlike hard spherocylinders in Sect. 2, and compare its predictions with typical data of various static solution properties of liquid-crystalline polymers in Sects. 3-5. As is well known, the wormlike chain (or wormlike cylinder) is a simple yet adequate model for describing dilute solution properties of stiff or semiflexible polymers. 

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. 

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. 

These diffusion coefficients of a wormlike cylinder at infinite dilution are expressed by [19]. 

Wormlike chain model Wormlike touched beads model Wormlike cylinder model... 

Neglecting the draining term in the Kirkwood-Riseman integral equation1Z3), Yamakawa and Fujii12+) derived an expression of fq] for unperturbed wormlike cylinders (Fig. 37 c), which reads... 

The most important of recent theoretical studies on semi-flexible polymers is probably the formulation of Yamakawa and Fuji [2,3] for the steady transport coefficients of the wormlike cylinder. This hydrodynamic model, depicted in Figure 5-2, is a smooth cylinder whose centroid obeys the statistics of wormlike chains. In the figure, r denotes the normal radius vector drawn from a contour... 

Fig. 5-2. Wormlike cylinder. Dashed line, cyclinder axis whose shape obeys the statistics of wormlike chains. 

The Yamakawa-Fujii theory [2, 3] was developed by using the Kirkwood-Riseman formalism with the effect of chain thickness approximately taken into account. The following remarks may be in order. The Oseen interaction tensor was preaveraged. Force points were distributed along the centroid of the wormlike cylinder (not over the entire domain occupied by the cylinder). The no-slip hydrodynamic condition was approximated by equating the mean solvent velocity over each cross-section of the cylinder to the velocity of the cylinder at that cross-section (Burgers approximate boundary condition). 

The Yamakawa-Fujii theory neglects the friction on the end surfaces of the wormlike cylinder. Norisuye et al. [23] and Yoshizaki and Yeimakawa [25] estimated them by capping each end of a wormlike cylinder with a hemisphere and a straight spheroid, respectively. The results showed negligible effects on / unless the axial ratio p of the cylinder is smaller than 10, implying that the Yamakawa-Fujii theory of / is applicable down to vety low M. 

Importantly, Yoshizaki and Yamakawa [25] found that, in contrast to /, [77] of a wormlike cylinder undergoes significant end surface effects until the axial ratio p reaches about 50, on the basis of numerical solutions to the Navier-Stokes equation with the no-slip boundary condition for spheroid cylinders, spheres, and prolate and oblate ellipsoids of rotation. They constructed an empirical interpolation formula for [ y] of a spheroid cylinder which reduces to eq 2.37 for p > 1 and to the Einstein value at p = 1. Then, with its aid, Yamakawa and Yoshizaki [4] formulated a modified theory of [77] for wormlike cylinders which agrees with the Yamakawa-Fujii theory [3] for Lj lq > 2.278 and with the Einstein value at Ljd = 1, regardless of dj2q smaller than 0.1. However, no formulation has as yet been made for L/2q < 2.278 and d/2q > 0.1, i.e., for short flexible cylinders. In what follows, the Yamakawa-Yoshizaki modification is referred to as the Yamakawa-Fujii-Yoshizaki theoiy. 

Fig. 5-7. Molecular weight dependence of [77] and 50 (sedimentation coefficient) for schizophyllan triple helix in water [36]. Curve calculated for an unperturbed wormlike cylinder with q = 200 nm, Ml = 2150 nm, and d = 2.6 nm gives best fits to both [77] and 3o data.

A wormlike chain is specified by the persistence length A and the contour length Lp. However, it does not have a thickness. We need to give it a diameter b for the chain to have a finite diffusion coefficient. The model is called a wormlike cylinder (Fig. 3.62). The expressions for the center-of-mass diffusion coefficient and the intrinsic viscosity were derived by Yamakawa et al. in the rigid-rod asymptote and the flexible-chain asymptote in a series of h/A and A/A-... 

Figure 3.62. Wormlike cylinder has a finite thickness b in addition to the natnre of the wormhke chain.

For example, estimates for ItiIr given by Yamakawa and Fuji with a wormlike cylinder model and Elzner and Ptitsyn for a wormlike chain of beads are shown in Fig. 4. For a rodlike chain (small d/p) the relation for [r ] can be approximated by the equation... 

Finally, some rather recent devdopments must be noted. Several years ago, Yamakawa and co-workers [25-27] developed the wormlike continuous cylinder model. This approach models the polymer as a continuous cylinder of hydrodynamic diameter d, contour length L, and persistence length q (or Kuhn length / ). The axis of the cylinder conforms to wormlike chain statistics. More recently, Yamakawa and co-workers [28] have developed the helical wormlike chain model. This is a more complicated and detailed model, which requires a total of five chain parameters to be evaluated as compared to only two, q and L, for the wormlike chain model and three for a wormlike cylinder. Conversely, the helical wormlike chain model allows a more rigorous description of properties, and especially of local dynamics of semi-flexible chains. In large part due to the complexity of this model, it has not yet gained widespread use among experimentalists. Yamakawa and co-workers [29-31] have interpreted experimental data for several polymers in terms of this model. 

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... 

Figure 6 Bright-field TEM images of nanostructured Pt metal/ PI-/>-PDMAEMA block copolymer composites (a) spherical micellar morphology (b) wormlike cylinders morphology (c) lamellar morphology and (d) inverse hexagonal morphology. From Z. Li, H. Sai, S. C. Warren, M. Kamperman, H. Arora, S. M. Gruner, U. Wiesner, Metal Nanoparticle/Block Copolymer Composite Assembly and Disassembly, Chem. Mater. 21 (2009), 5578-5584, Figure 3.

For KP chains, Yamakawa and Fujii numerically obtained [//] values, but the result turned out to be incomplete for short wormlike cylinders for which effects from cylinder ends are significant. Yamakawa and YoshizaM later modified the theory so as to give the Einstein equation for rigid spheres in the limit of L/d = 1, that is,... 

Figure 24 Theoretical UHh plotted against L/d for wormlike cylinders with indicated id values. ...

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