Wormlike chain, semiflexible

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. 

Garcia Molina, J.J., Lopez Martinez, M.C. and Garcia de la Torre, J. (1990) Computer simulation of hydrodynamic properties of semiflexible macromolecules Randomly broken chains, wormlike chains, and analysis of properties... 

Polymer molecules in this regime are called semiflexible or wormlike chains. 

Comparison of eqns [10] and [ 11 ] to eqns [2] and [3] points out that long wormlike chain acquires on the large scale the random coil conformation and its large-scale properties coincide with those of an equivalent freely jointed chain comprising Ni = L/2lp statistical segments each of length t = 2lp. Hence, both the freely jointed and the wormlike persistence chain models can be applied for desaiption of large-scale conformational properties of flexible and semiflexible chain polymers. 

In the part devoted to neutral polymers, we mentioned that semiflexible and stiff chains do not obey the behavior predicted by the Kuhn model. Restricted flexibility of the chain can be caused by the presence of stiff units with multiple bonds or bulky pendant groups, but it can be a result of external conditions or stimuli. In the preceding part, it was explained in detail that repulsive interactions together with entropic forces increase the stiffness of PE chains. Hence, a sudden pH change can be used as a stimulus affecting the stiffness of annealed PE chains. The properties of semiflexible polymers are usually treated at the level of the wormlike chain (WLC) model developed by Kratky and Porod [31]. The persistence length, /p, is an important parameter strongly related to the WLC model and has been used as the most common characteristic of chain flexibility—in both theoretical and experimental studies. It is used to describe orientational correlations between successive bond vectors in a polymer chain in terms of the normalized orientation correlation function, C(s) = (r,.r,+j). For the WRC model, this function decays exponentially ... 

Adopting a continuum wormlike chain model Wilhelm and Frey [27] have calculated analytically the end-to-end distance distribution function of a semiflexible polymer. They obtained the following expression ... 

If we fix the energy scale in a way that the entirely straight chain has zero bending energy, the discrete semiflexible, wormlike-chain polymer model can be written as... 

The conclusion that we draw from the discussion in this chapter is that the structural behavior of self-interacting semiflexible polymers cannot be adequately described by the wormlike-chain model that does not allow for the description of stmctural transitions. As we have seen, stmctural transitions also occur for semiflexible polymers and have to be taken into account. As in the case of purely flexible polymers, finite-size effects are responsible for a variety of solid, i.e., ordered, crystalline phases. This implies that the detailed stmctural behavior needs to be considered in the understanding of biomolecular processes on short length scales. It is therefore also relevant for the nanofabrication of molecular devices. 

For semiflexible chains characterized by contour length much larger than persistence length, Semenov and Kokhlov demonstrated that the values of the phase boundaries between the isotropic and nematic states, as well as the order parameter of the nematic phase could be predicted [291], Following Onsager [277], it was shown that the boundary Ci-m depends only on the ratio between the radius and the persistence length [277,291,292], For nematic wormlike micelles, this limit was found to vary between 20 and 45 wt. %, in fair agreement with the theoretical values [44],... 

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