Freely-jointed chains random walk

The results of the three-dimensional random walk, based on the freely-jointed chain, has permitted the derivation of the equilibrium statistical distribution function of the end-to-end vector of the chain (the underscript eq denotes the equilibrium configuration) [24] ... 

Fig. 75.—Vectorial representation in two dimensions of a freely jointed chain. A random walk of fifty steps.

Random-Walk Statistics The Freely Jointed Chain... 

At equilibrium, the distribution of conformations in a solvent at the theta temperature (see Section 2.3.1.2), or in a concentrated solution, is given by a set of random walks or, equivalently, by the conformations of a. freely jointed chain (see Section 2.2.3.2). If one end of the freely jointed chain with links, each of length bjc, lies at the origin, then-the probability, jrodR, that the other end lies at a position between R and R + dR is approximately a Gaussian function (Flory 1969 Larson 1988) ... 

Every possible conformation of an ideal chain can be mapped onto a random walk. A particle making random steps defines a random walk. If the length of each step is constant and the direction of each step is independent of all previous steps, the trajectory of this random walk is one conforma-tion of a freely jointed chain. Hence, random walk statistics and ideal chain statistics are similar. 

The previous problem showed that the equivalent freely jointed chain follows random walk statistics even if the effective monomer is renormalized to be larger than b. What is the smallest effective monomer size for which this renormalization works ... 

The analysis of this model is similar to that of the well-known random-walk model, which was first developed to describe the random movement of molecules in an ideal gas. The only difference now is that for the freely jointed chain, each step is of equal length 1. To analyze the model one end of the chain may be fixed at the origin O of a three-dimensional rectangular coordinate system, as shown in Fig. A2.1(b), and the probability, P(x,y,z), of finding the other end within a small volume element dx.dy.dz at a particular point with coordinates x,y,z) may be calculated. Such calculation leads to an equation of the form (Young and Lovell, 1990) ... 

The freely jointed chain is a model that treats the conformation of a polymer molecule as a mathematical random walk. This is a simple model for chain conformahon in which... 

The end-to-end vector, r, connects the first and last segments of a polymer chain. For the freely jointed chain, its mean squared value may be calculated within the assump-hons of the model from a consideration of a random walk of n vectors, each of length /. 

Continuous Space The random walks are not limited to those on a lattice. Here, we consider a random walker who jumps by a fixed distance b. The trajectory is shown in Figure 1.18 for a two-dimensional version of the continuous-space random walk. Starting at Tq, the walker moves by Ari, Ar2,..., Ar v to arrive at r v in a total N steps. When the direction is random in three dimensions, the trajectory represents a freely jointed chain (Table 1.1). Like a random walk on the lattice, the ith jump Ar, is not correlated with the yth jump Ar if i j. As long as Ar, satisfies Eq. 1.19, the displacement in a total N steps has the same statistical... 

Eignre 1.19 compares a freely jointed chain with a fixed bond length b (also called a segment length) and a bead-spring model with (Ar ) = b, both in two dimensions. Examples of a 100-step random walk are shown. The bead-spring model can have greater density flnctnations for the same Nb. ... 

Dimensions of Ideal Chains Now we obtain Rp and R for ideal chains whose conformations are given as trajectories of random walkers. They include a random walk on a lattice, a freely jointed chain, a bead-spring model, and any other model that satisfies the requirement of Markoffian property (Eq. 1.19). The bond vector r, - r, i of the ith bond is then the displacement vector Ar, of the ith step. We assume Eq. 1.19 only. Then the end-to-end distance is Nb. To calculate / g, we note that a part of the ideal chain is also ideal. The formula of the mean square end-to-end distance we obtained for a random walk applies to the mean square distance between the ith and/th monomers on the chain just by replacing N with i - j. ... 

In the simplest model for polymer coils, the chain is supposed to consist of n volume-less links of length / which can rotate freely in space. This model is then called the freely jointed chain model. Since each link can adopt any orientation, the polymer coil effectively executes a random walk, as sketched in Fig. 2.2. This is similar to the Brownian motion of microscopic particles suspended in a fluid (Section 1.4). The effect of the random walk statistics is that the chain coils back, and even crosses itself, many times, leading to a dense clumped up structure. The statistics of random walks were worked out for Brownian motion by Einstein (Section 1.4), and we can simply use the same result for random polymer coils. It turns out that the mean-square end-to-end distance is... 

Vectorial representation in two dimensions of a freely jointed chain. A random walk of 50 steps, (a) Two-dimensional random coil represented as a random walk, (b) Representation of a hindered chain in two dimensions. A random walk of SO steps with series between successive bonds limited to n/2. The scale in (b) is identical to that in (a) for an unrestricted random walk of the same number of steps. From Flory (1953). 

Analysis of the Random Walk Model. If the average dimensions of freely jointed coils can be determined using the previously described calculation, the latter cannot be used to evaluate the distributions corresponding to these average values. Such an information can be obtained by identifying a freely jointed chain with the random walk of a particle. 

Chains can rotate around dihedral angles. Inevitably then, the average length of polyethylene chains will be smaller than Lnax-The simplest model to determine the average length (L) of chains with M monomers is the freely jointed ideal chain. In a freely jointed chain, the dihedral angles can take on any value with equal probability. In an ideal chain there are no interactions between the monomers. In other words, the polymer is assumed to be a random walk in three dimensions (Fig. 11.2). Lord Rayleigh developed the model of unrestricted random walks in the early 19(X)s and Werner Kuhn first adopted this model in 1934 for polymer conformations. 

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