Porod-Kratky wormlike chain

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. 

Figure S Scattering function calculated with the Yoshizaki and Yamakawa method. Reduc coordinates are used k = qb and F(k L) = k (L/b) P(q). The chain contour length is L = 80b. KP is the Kratky-Porod or wormlike chain. 1 and 2 refer to the helical...

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. 

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. 

To remedy this deficiency, one must consider locally rigid chains. The simplest example of such chains is the Kratky-Porod chain (wormlike chain),16 which is a continuous chain, and which, unlike the Brownian chain, has a finite length and, at each point, a well-defined tangent. 

Up to this point we have confined ourselves to ideally flexible chains. Thus, the theories developed on the models of such chains (for example, the spring-bead chain) should no longer be adequate for polymers whose chemical stmcture suggests considerable stiffness of the chain backbone. Many cheiin models may be used to formulate a theory of stiff or semi-flexible polymers in solution, but the most frequently adopted is the wormlike chain mentioned in Section 1.3 of Chapter 1 it is sometimes called the KP chain. This physical model was introduced long ago by Kratky and Porod [1] to represent cellulosic polymers. However, significant progress in the study of its dilute solution properties, static and dynamic, has occurred in the last two decades. 

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... 

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... 

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 ... 

The wormlike chain of Kratky and Porod [49K1] is characterized by a contour length L and a persistence length a. The latter increases with increasing stiffness, but is (on the basis of the model) independent of L. The relation between the radius of gyration and L for worm-like linear coils without excluded volume is[53Bl] ... 

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... 

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... 

The wormlike chain of Kratky and Porod (the KP chain) is a continuous limit of a freely rotating chain derived by letting N (the number of bonds) —> bond length) —> 0, and 0f (the supplementary bond angle) —> 0 under the conditions that... 

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. 

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