Kratky-Porod chain model

Figure 5.6 Kratky plot of the scattering function (circles) calculated16 from an atomistic model of polyethylene. The curves shown are those calculated for a Kratky-Porod chain, a Gaussian chain, and an asymptotic thin rod. (After Kirste and Oberthur.15)...

Although the polymer chains must possess chemical specificity in order to express their unique functions in various macromolecular processes, they exhibit certain universal behavior at larger length scales. By parametrizing the chemical details at the monomeric level, we have described various coarsegrained models, namely the Kuhn chain, Gaussian chain, and the wormlike Kratky-Porod chain. Chain stiffness is captured by the persistence length parameter. 

A comparison is presented between the behavior of unperturbed stars of finite size whose configurational statistics are evaluated by R1S theory and the Kratky-Porod wormlike chain model. Emphasis Is placed on the initial slopes of the characteristic ratio, C, or g when plotted as a function of the reciprocal of the number of bonds, n. 

From their light-scattering measurements Holtzer, Benoit, and Doty (126) concluded that the short-range interactions control the dimensions of cellulose nitrate chains, and they discussed their results in terms of the worm-like chain model of Kratky and Porod (142), obtaining a persistence length of about 34.7 A. In Fig. 21 these data are shown as a plot of (S yjMw against Mw. The open circles are the experimental points and the broken curve is that calculated from the equations for the worm-like chain model. The theoretical curve is claimed to reproduce the data to within the probable experimental error in all but two cases. 

The worm-like chain model (sometimes called the Kratky-Porod model) is a special case of the freely rotating chain model for very small values of the bond angle. This is a good model for very stiff polymers, such as double-stranded DNA for which the flexibility is due to fluctuations of the contour of the chain from a straight line rather than to trans-gauche bond rotations. For small values of the bond angle ( < 1), the cos 9 in Eq. (2.23) can be expanded about its value of unity at = 0 ... 

The quantity a is called the persistence length and is a measure of chain stiffiiess. The wormlike chain model (sometimes called the Porod-Kratky chain) is a special continuous curvature limit of the freely rotating chain, such that the bond length I goes to zero and the number of bonds n goes to infinity, but the contour length of the chain L = nl and the persistance length a are kept constant. In this limit... 

We emphasize that these deviations from the Kratky-Porod model that occur for semiflexible polymers both in equilibrium and in their response to stretching forces, were not properly noticed in most of the experiments. However, in analyzing data one normally does not have strictly monodisperse chains, and neither p nor L are independently known both parameters are usually used as adjustable fitting parameters. Because fp depends on d, and is also affected by solvent conditions, and for strongly stretched real chains other effects (related to the local chemical structure of the effective monomeric units) come into play, this failure is not surprising. However, some of the confusion over the actual values of p that... 

In the limit, the conformation of the chain is not zigzag bnt rather a smooth curve in a three-dimensional space, as illustrated in Figure 1.43. This model is called a wormlike chain or a Kratky-Porod model. A continuous... 

It was shown that with increasing internal chain stiffness the effective exponent y for Le — crosses over from a value of one toward two as the internal stiffness of a chain increases. The quadratic dependence of the electrostatic persistence length on the Debye radius for the discrete Kratky Porod model of the polyelectrolyte chain was recently obtained in [65]. It seems that the concept of electrostatic persistence length works better for intrinsically stiff chains rather than for flexible ones. Further computer simulations are required to exactly pinpoint the reason for its failure for weakly charged flexible polyelectrolytes. 

As discussed in Chapter 2, xanthan has a structure that is not quite a rigid rod since it has some degree of flexibility. This type of structure was described by Porod and Kratky as the worm-like chain model (Richards, 1980, p. 88). Although this may be visualised intuitively to be rather like a semi-flexible string of plastic pop-in beads, it requires the definition of the persistence length, /p, in order to develop the idea in a more quantitative way. This quantity is defined for an infinite polymer chain as follows ... 

The Kratky-Porod wormlike chain model [20,21] is widely used for describing conformational characteristics of less flexible chains. The polymer is viewed as a semi-flexible string (or worm) of overall contour length L with a continuous curvature. The chain is subdivided into N segments of length AL, which are linked at a supplementary angle r. The persistence length q (Fig. 1) is defined as... 

In 1984, Tricot [266] summarized viscosity results of various vinylic polyacids reported in literature, analyzed the data by the Kratky-Porod worm-like chain model and compared the results with the OSF theory and the theory of Fixman [251] and Le Bret [19]. At low ionic strength, the electrostatic persistence length approximately followed a scaling relation Ip (cf) in complete... 

Many polymer chains are not completely flexible under the usual experimental conditions of interest. In order to incorporate the local chain stiffness, the Kuhn model is modified slightly by introducing a bond angle 180-0 between the consecutive Kuhn steps, as sketched in Figure 2.12a. Obviously, this angle is a parameter to capture the backbone stiffness of the chain. Further, let us assume that the Kuhn steps are freely rotating, and now the model is called the Kratky-Porod or wormlike chain model. 

The Kratky and Porod model is an extension of the freely rotating chain model in which the valence angles between two units are close to 180°. This confers to such chains a certain persistence which prevents them from behaving like a random coil such chains are semi-rigid and are also called worm-like. The persistence length (fl) is defined as... 

When the end-to-end distance of the chain corresponds to its contour length, this macromolecule can be identified with a rod-like chain. Thus, the Kratky-Porod model is appropriate to describe the transition from a rigid rod to a flexible chain. 

Chain with local stiffiiess Kratky-Porod model. 

The simplest model for free (or linker) DNA is the wormlike-chain or Kratky-Porod model [48], It is based on the assumption that changing the contour of a linear chain by bending costs energy. If we describe the contour of length I by introducing the contour parameter s e [0, /], an infinitesimal segment of the contour (arc length) can be expressed in local coordinates by... 

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