Conformation semi-flexible chains

The polymer we consider here is a semi-flexible chain which has some bending stiffness (Eq. 3). We first estimated the chain conformation in the melt. The calculated mean-square end-to-end distance R2n between atoms n-bond apart has shown that the chains have an ideal Gaussian conformation R2 is a linear function of n (see Fig. 35 given later). The value of R2 for n = 100... 

The conformational states of semi-flexible chains can be represented by the disorder parameter/, defined as... 

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. 

Fig. 8 General structure of semi-flexible oligobenzoate Hekates 34—36. (a) These stars may consist of three identical arms (34), two (35) or three (36) different arms. Thereby the length (n,m,l), the peripheral chains (R, R", R "), the linking groups X (OOC, COO, CONH) to the core and the substituents Y (H, I) can he adjusted, (b) Borderline conformers - star-shaped, f.-shaped and cone-shaped conformers - for a non-symmetric oligobenzoate scaffold (X = OOC). They can be created by rotation about the C-O single bond within the carboxy linking group to the core...

The conformation parameter a (=A/Af, where Af is A of a hypothetical chain with free internal rotation) for cellulose and its derivatives lies between 2.8-7.5 2 119,120) and the characteristic ratio ( = A2Mb//2, where Ax is the asymptotic value of A at infinite molecular weight, Mb is the mean molecular weight per skeletal bond, and / the mean bond length) is in the range 19-115. These unexpectedly large values of a and Cffi suggest that the molecules of cellulose and its derivatives behave as semi-flexible or even inflexible chains. For inflexible polymers, analysis of dilute solution properties by the pearl necklace model becomes theoretically inadequate. Thus, the applicability of this model to cellulose and its derivatives in solution should be carefully examined. 

A model presented by Wang Warner (1987) quantitatively describes the statistical mechanics of side chain nematic polymers with a semi-flexible backbone and stiff mesogenic side chains. In general, side chains and backbones have different orders, the former denoted by Sa and the latter denoted by Sb, respectively. The phase types, phase diagrams and backbone conformations in each phase is dependent on the competition of side chains of various lengths, x, and backbone of various stiffness, g. Mesogenic... 

Many other semi-flexible polymers are known which can be reasonably well modeled on wormlike chains. The Ml values obtained for most of such polymers are close to those calculated from crystallographic data, suggesting that stiff polymers in dilute solution maintain their crystalline conformations at least locally. 

Table 4.2 Equilibrated conformations from Monte Carlo simulations of complexes composed of a semi-flexible polyelectroly te and a single colloidal particle as a function of the solution ionic strength I and polyelectrolyte intrinsic rigidity k ng- By increasing the chain stiffness, solenoid conformations are progressively achieved at the particle surface, whereas an increase in ionic strength leads to the desorption of the polyelectrolyte.

The evolution from highly oriented polymers to in situ composites through PLCs has a common feature in that they all have negative thermal expansivity (as defined in section 8.1.1) in the orientation direction of the order of —10 K in a wide range below and around room temperature. As in oriented flexible polymers, these negative values are attributed to a decrease of the fully extended conformation of the chains induced by thermal vibration [19-21]. Various models and theories have been proposed semi-empirically as well as based on fundamental principles. In the following, a detailed discussion of the subject is presented. 

One can describe polymer samples in a similar way. First of aU, a polymer chain possesses semi-flexibility, that characterizes the intra-chain interactions for the most stable conformation persisting along the chain axis. Secondly, a polymer chain also holds complex inter-chain interactions. These two intrinsic characteristic factors dictate the basic physical behaviors of the same species of polymers. Besides these two intrinsic factors, each individual polymer sample possesses certain extrinsic characteristic factors, i.e., molecular weights and their distributions, topological architectures, and sequence irregularities. These extrinsic characteristic factors are also important in determining the physical behaviors of the polymer samples. 

The static flexibility of a semi-flexible polymer chain is related not only to the potential energy difference Ae, but also to the temperature T. The following quantities are often used to characterize the conformational states of semi-flexible polymer chains. 

However, the conformation statistics in Flory s treatment gives the conformational free energy, rather than the conformational entropy adapted in the Gibbs-DiMarzio theory. In addition, W was calculated with respect to the fully ordered state therefore. In W = 0 simply implies the return to the fully ordered state, rather than frozen in a disordered state. Furthermore, reflects the static semi-flexibility, while the glass transition should be related with the d3mamic semi-flexibility of polymer chains. Therefore, fundamental assumptions of the Gibbs-DiMarzio thermodynamic theory are misleading. 

In 1956, Hory introduced the semi-flexibility into the classical lattice statistical thermodynamic theory of polymer solutions (Flory 1956). From the classical lattice statistics of flexible polymers, we have derived the total number of ways to arrange polymer chains in a lattice space, as given by (8.15). The first two terms on the right-hand side of that equation are the combinational entropy between polymers and solvent molecules, and the last three terms belong to polymer conformational entropy. Thus the contribution of polymer conformation in the total partition function is... 

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