Model freely-jointed

We briefly describe three models, namely, the Gaussian Chain model. Freely Jointed Chain (FJC) model, and Worm Like Chain (WLC) model, which have been extensively used to describe the force-extension curves of biopolymers. The advantages of these models come from their simplicity and allowing one to derive analytical expressions in a simple form. In these models a polymer chain consists of N beads (monomers) of contour length L. A point in d-dimensional space represents each monomer and the distance between two consecutive monomers is R, i - R, (see O Fig. 8-1). The energy with force / (along x-direction) in Gaussian model is expressed as (Doi and Edwards 1986)... 

Another simplified model is the freely jointed or random flight chain model. It assumes all bond and conformation angles can have any value with no energy penalty, and gives a simplified statistical description of elasticity and average end-to-end distance. 

Fig. 1. Freely jointed bead-rod model of a chain formed by(N + 1) beads and N rigid links of length Ip...

It is worth recalling that any of the molecular force laws given by Eqs. (13-16) are derived within the framework of the freely-jointed model which considers the polymer chain as completely limp except for the spring force which resists stretching thus f(r) is purely entropic in nature and comes from the flexibility of the joints which permits the existence of a large number of conformations. With rodlike polymers, the statistical number of conformations is reduced to one and f(r) actually vanishes when the chain is in a fully extended state. 

FIGURE 21.4 Nanofishing of a single polystyrene (PS) chain in cyclohexane. The solvent temperature was about 35°C (0 temperature). A cantilever with a 110 pN nm spring constant was used. The worm-like chain (WLC), solid line, and the freely jointed chain (FJC), dashed line models were used to obtain fitting curves. (From Nakajima, K., Watabe, H., and Nishi, T., Polymer, 47, 2505, 2006.)... 

X 10 N and the value obtained by the simplest form, 1.45 X 10 " N m n = 918 and a = 0.31 nm for Equation 21.1). These comparisons imphed that the measurements were consistent with the theoretical predictions. The deviation between the rupture length of 260.9 nm and the fitted-contour length indicated that the polymer chain was not fully stretched at the rupture event. The reason for this was that the rupture event was a stochastic process and was dependent on many factors such as pulling speed, bond strength, and temperature. The vahdity of the freely jointed (FJC) model (dashed fine) was also checked ... 

The foregoing derivation may appear artificial in view of the assumptions involved. The contribution of a given bond to x is by no means restricted to the two unique values, + as has been assumed. On the contrary, one may show that all values of h from 0 to Z occur with equal probability for freely jointed connections between links. A more detailed study of the problem shows that the final result is unaffected by this assumption so long as n is large. The freely jointed chain model under consideration is an artifice also, but the form of the results obtained will be shown to apply also to real polymer chains. 

First approaches at modeling the viscoelasticity of polymer solutions on the basis of a molecular theory can be traced back to Rouse [33], who derived the so-called bead-spring model for flexible coiled polymers. It is assumed that the macromolecules can be treated as threads consisting of N beads freely jointed by (N-l) springs. Furthermore, it is considered that the solution is ideally dilute, so that intermolecular interactions can be neglected. 

The elastic contribution to Eq. (5) is a restraining force which opposes tendencies to swell. This constraint is entropic in nature the number of configurations which can accommodate a given extension are reduced as the extension is increased the minimum entropy state would be a fully extended chain, which has only a single configuration. While this picture of rubber elasticity is well established, the best model for use with swollen gels is not. Perhaps the most familiar model is still Flory s model for a network of freely jointed, random-walk chains, cross-linked in the bulk state by connecting four chains at a point [47] ... 

The focus of this chapter is on an intermediate class of models, a picture of which is shown in Fig. 1. The polymer molecule is a string of beads that interact via simple site-site interaction potentials. The simplest model is the freely jointed hard-sphere chain model where each molecule consists of a pearl necklace of tangent hard spheres of diameter a. There are no additional bending or torsional potentials. The next level of complexity is when a stiffness is introduced that is a function of the bond angle. In the semiflexible chain model, each molecule consists of a string of hard spheres with an additional bending potential, EB = kBTe( 1 + cos 0), where kB is Boltzmann s constant, T is... 

The wall-PRISM equation has been implemented for a number of hard-chain models including freely jointed [94] and semiflexible [96] tangent hard-sphere chains, freely rotating fused-hard-sphere chains [97], and united atom models of alkanes, isotactic polypropylene, polyisobutylene, and polydimethyl siloxane [95]. In all implementations to date, to my knowledge, the theory has been used exclusively for the stmcture of hard-sphere chains at smooth structureless hard walls. 

The density functional theory is at least as accurate for other models of polymers as it is for freely jointed hard chains, but this is not the case with the... 

Cyclisation of long-chain molecules is a field where theory has far preceded experiment. In his pioneering treatment of flexible chains in terms of the freely-jointed chain model, Kuhn (1934) derived for the local concentration Ceff of one chain end in the neighbourhood of the other (see p. 7) expression (56) where Aa is Avogadro s number and Ceff is given in moles per... 

The main parameters used to describe a polymer chain are the polymerization index N, which counts the number of repeat units or monomers along the chain, and the size of one monomer or the distance between two neighboring monomers. The monomer size ranges from a few Angstroms for synthetic polymers to a few nanometers for biopolymers. The simplest theoretical description of flexible chain conformations is achieved with the so-called freely-jointed chain (FJC) model, where a polymer consisting of N + I monomers is represented by N bonds defined by bond vectors r/ with j= Each bond vector has a fixed length r,j = a corresponding to the... 

Fig. 2 Top Freely jointed chain (FJC) model, where N bonds of length a are connected to form a flexihle chain with a certain end-to-end distance R. Bottom in the simplified model, appropriate for more advanced theoretical calculations, a continuous line is governed hy some bending rigidity or line tension. This continuous model can be used when the relevant length scales are much larger than the monomer size...

The problem of conformational analysis of a chain is, therefore, that of calculating C. The approximation of the freely jointed chain, with no correlation between successive bonds, 69, gives a value of = 1. If one introduces into the model a constant value for the bond angles, but permits free rotation around the bonds (in the formula 70 all points of the base circumferences of the... 

A mean-square helical hydrophobic moment,

, is defined for polypeptides in analogy to the mean-square dipole moment, , for polymer chains. For a freely jointed polymer chain, is given by X rr , where mi denotes the dipole moment associated with bond /. In the absence of any correlations in the hydrophobic moments of individual amino acid residues In the helix,

is specified by X Wj2, where H denotes the hydrophobicity of residue /, Matrix-generation schemes are formulated that permit rapid evaluation of

and . The behaviour of

I

is illustrated by calculations performed for model sequential copolypeptides. 

As a contribution to the study of these problems, stochastic models are here developed for two cases a freely-jointed chain in any number of dimensions, and a one-dimensional chain with nearest-neighbor correlations. Our work has been directly inspired by two different sources the Monte Carlo studies by Verdier23,24 of the dynamics of chains confined to simple cubic lattices, and the analytical treatment by Glauber25 of the dynamics of linear Ising models. No attempt is made in this work to introduce the effects of excluded volume or hydrodynamic interactions. 

Furthermore, it may be seen that for all the normal modes of relaxation, including the most rapid, the freely jointed chain model and the Rouse model are identical if we set n = N + 1 that is, the relaxation time xp of the pth normal mode of a freely-jointed chain is the same as that of a Rouse marcromolecule composed of N + 1 subchains, each of mean square end-to-end length b2. Moreover, for the special choice a = 0, Eq. (10) is true for arbitrarily large departures from equilibrium. We thus seem to have confirmed analytically the discovery of Verdier24 that quite short chains executing a stochastic process described by Eqs. (1) and (3) on a simple cubic lattice display Rouse relaxation behavior. Of course, Verdier s Monte Carlo technique permits study of excluded volume effects, quite beyond the range of our present efforts. 

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