Wormlike polymer

Figure 3 Scheme of a wormlike polymer chain. The contour length is indicated by L. The points s and represent partial contour lengths from the origin 0. The tangents at s and s (with respect to the tangent at 0) are indicated by t s) and t s ), respectively. The end-to-end distance is h . The point 0 represents a second arbitrary origin (see text). 

Gans, C. Schnee, J. Scherf, U. Staikos, G. Pierri, E. Dondos, A. Viscosimetric determination of the statistical segment length of wormlike polymers. Polymer, 1998, iPfi/ , 4155 158. 

Here, we outline universal large-scale features that apply for ideal (freely jointed or wormlike) polymer chains irrespective of the mechanism of their flexibility, provided the number of... 

The preparation of networks from wormlike polymers such as polyesters and polyamides, and from stiifer polymers such as polyimides and poly-p-phenyl-enes, will be described in Sect. 3 below. In addition to the chemical nature of the stiff segments, attention will be paid to the rigidity or flexibility of the junction points, and to the sequence of reactions performed in order to obtain various rigid aromatic networks. 

B. Tinland, G. Maret, and M. Rinaudo. Reptation in semidilute solutions of wormlike polymers. Macromolecules, 23 (1990), 596-602. 

Ottinger, H.C. (2004) Coarse-graining of wormlike polymer chains for substantiating reptation. /. Non-Newtonian Fluid Mech., 120, 207-213. 

Living polymers and wormlike micelles suggest an interesting field for basic research in which the constant process of scission and recombination of the... 

In contrast to statics, the relaxational kinetics of living polymers and of giant wormlike micelles is unique (and different in both cases). It is entirely determined by the processes of scission/recombination and results in a nonlinear approach to equilibrium. A comparison of simulational results and laboratory observations in this respect is still missing and would be highly desirable. 

Subsequent work by Johansson and Lofroth [183] compared this result with those obtained from Brownian dynamics simulation of hard-sphere diffusion in polymer networks of wormlike chains. They concluded that their theory gave excellent agreement for small particles. For larger particles, the theory predicted a faster diffusion than was observed. They have also compared the diffusion coefficients from Eq. (73) to the experimental values [182] for diffusion of poly(ethylene glycol) in k-carrageenan gels and solutions. It was found that their theory can successfully predict the diffusion of solutes in both flexible and stiff polymer systems. Equation (73) is an example of the so-called stretched exponential function discussed further later. 

Further modification of the above nanostructures is useful for obtaining new functional materials. Thirdly, we apply the dopant-induced laser ablation technique to site-selectively doped thin diblock copolymer films with spheres (sea-island), cylinders (hole-network), and wormlike structures on the nanoscale [19, 20]. When the dye-doped component parts are ablated away by laser light, the films are modified selectively. Concerning the laser ablation of diblock copolymer films, Lengl et al. carried out the excimer laser ablation of diblock copolymer monolayer films, forming spherical micelles loaded with an Au salt to obtain metallic Au nanodots [21]. They used the laser ablation to remove the polymer matrix. In our experiment, however, the laser ablation is used to remove one component of block copolymers. Thereby, we can expect to obtain new functional materials with novel nanostmctures. 

The most austere representation of a polymer backbone considers continuous space curves with a persistence in their tangent direction. The Porod-Kratky model [99,100] for a chain molecule incorporates the concept of constant curvature c0 everywhere on the chain skeleton c0 being dependent on the chemical structure of the polymer. It is frequently referred to as the wormlike chain, and detailed studies of this model have already appeared in the literature [101-103], In his model, Santos accounts for the polymer-like behavior of stream lines by enforcing this property of constant curvature. 

In order to illustrate the potential applications of rheo-NMR five examples have been chosen. The first example deals with wormlike micelles [22] in which NMR velocim-etry is used to profile anomalous deformational flow and deuterium NMR spectroscopy is used to determine micellar ordering in the flow. The second example concerns flow in a soft glassy material comprising a solution of intermittently jammed star polymers [23], a system in which flow fluctuations are apparent. The third... 

This full set of self-consistent equations is clearly very difficult to solve, even numerically. However, good approximations of closed integral type have been proposed. These essentially ignore the s-dependence of the survival and orientation functions, which makes them a physically appeaUng approach in the case of wormlike surfactants [71,72]. For ordinary monodisperse polymers the following approximate integral constitutive equation results ... 

If the ideas of Marrucci [69] are correct and the non-monotonic predictions of the simple Doi-Edwards theory need to be modified in the case of polymer melts (for a recent development see [78]), then an explanation will be required for the apparent difference at high shear rates between melts and wormlike micelle solutions. There is also evidence that ordinary entangled polymer solutions do exhibit non-monotonic shear-stress behaviour [79]. As in the field of linear deformations, it may be that a study of the apparently more complex branched polymers in strong flows may shed light on their deceptively simple linear cous-... 

Thermotropic liquid crystalline polymers, like polyesters containing mesogenic units on the main chain, may not be described by the wormlike chain model (cf. Sect. 1.2). The present article does not consider this type of polymers. 

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. 

Liquid-Crystalline Polymers Viewed as the Wormlike Chain... 

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 . 

We begin by formulating the free energy of liquid-crystalline polymer solutions using the wormlike hard spherocylinder model, a cylinder with hemispheres at both ends. This model allows the intermolecular excluded volume to be expressed more simply than a hard cylinder. It is characterized by the length of the cylinder part Lc( 3 L - d), the Kuhn segment number N, and the hard-core diameter d. We assume that the interaction potential between them is given by... 

The orientation dependent parameter p defined by Eq. (11) becomes unity in the isotropic state, and decreases as the polymers are uniaxially oriented. Therefore, it follows from Eqs. (9) and (10) that the wormlike hard spherocylinder system has a smaller translational entropy loss from the ideal solution in the liquid crystal state than in the isotropic state. This difference drives the system to form a liquid crystal phase. However, in order to determine the equilibrium orientation of the system, the orientation dependence of Sor has to be formulated, and this is done in Sect. 2.3. 

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