Random walks ideal

Polymer chains at low concentrations in good solvents adopt more expanded confonnations tlian ideal Gaussian chains because of tire excluded-volume effects. A suitable description of expanded chains in a good solvent is provided by tire self-avoiding random walk model. Flory 1151 showed, using a mean field approximation, that tire root mean square of tire end-to-end distance of an expanded chain scales as... 

We close these introductory remarks with a few comments on the methods which are actually used to study these models. They will for the most part be mentioned only very briefly. In the rest of this chapter, we shall focus mainly on computer simulations. Even those will not be explained in detail, for the simple reason that the models are too different and the simulation methods too many. Rather, we refer the reader to the available textbooks on simulation methods, e.g.. Ref. 32-35, and discuss only a few technical aspects here. In the case of atomistically realistic models, simulations are indeed the only possible way to approach these systems. Idealized microscopic models have usually been explored extensively by mean field methods. Even those can become quite involved for complex models, especially for chain models. One particularly popular and successful method to deal with chain molecules has been the self-consistent field theory. In a nutshell, it treats chains as random walks in a position-dependent chemical potential, which depends in turn on the conformational distributions of the chains in... 

This expression accounts for the configurational entropy of an ideal binary mixture with identical molecular sizes, but not for that of a polymer solution, since polymer chains are large and flexible. For that case, more contributions arise from the chain conformational entropy, first considered by Meyer [19] and then derived by Huggins [20] and Flory [21]. In analogy with a nonreversing random walk on a lattice, the conformational contribution of polymer chains to the partition function is given by... 

Self-avoiding random walks (SARW) statistics has been proposed [1] for single that is for non-interacting between themselves ideal polymeric chains (free-articulated Kuhn s chains [2]) into ideal solvents, in which the all-possible configurations of the polymeric chain are energetically equal. From this statistics follows, that under the absence of external forces the conformation of a polymeric chain takes the shape of the Flory ball, the most verisimilar radius Rf of which is described by known expression [3, 4]... 

The calculated root-mean-square displacement for a general sequence of jumps has two terms in Eq. 7.31. The first term, NT(r2), corresponds to an ideal random walk (see Eq. 7.47) and the second term arises from possible correlation effects when successive jumps do not occur completely at random. 

For a random walk, f = 1 because the double sum in Eq. 7.49 is zero and Eq. 7.50 reduces to the form of Eq. 7.47. In principle, f can have a wide range of values corresponding to physical processes relating to specific diffusion mechanisms. This is readily apparent in extreme cases of perfectly correlated one-dimensional diffusion on a lattice via nearest-neighbor jumps. When each jump is identical to its predecessor, Eq. 7.49 shows that the correlation factor f equals NT.6 Another extreme is the case of f = 0, which occurs if each individual jump is exactly opposite the previous jump. However, there are many real diffusion processes that are nearly ideal random walks and have values of f 1, which are described in more detail in Chapter 8. 

Gaussian coils are characterized by a gaussian probability distribution [2] for the monomers and describe adequately flexible polymer blocks. Ideal chains follow random walk statistics, i.e.,... 

In the following two sections we first consider colls In solutions which are so dilute that coll-coll Interaction does not play a role. Section 5.2a deals with random-walk chains where the Interaction between the units within one chain may be neglected. We denote these as ideal chains. Section 5.2b treats swollen coils In which the monomeric units repel each other due to so-called excluded volume effects. 

Though the ideal gas assumption would cause some error in predicting result, the reasonableness of the above suggested models can be explained by HIO (Higashi, Ito, Oishi) model (Higashi et al., 1963) which is based on the random walk of molecules. The HIO model was same with the model 4 in this paper when the single layer adsorption was assumed. 

Fig. 3. Scaling of round-trip times for a random walk in energy space sampling a flat histogram open squares) and the optimized histogram solid circles) for the two-dimensional fully frustrated Ising model. While for the multicanonical simulation a power-law slowdown of the round-trip times 0 N L ) is observed, the round-trip times for the optimized ensemble scale like 0([A ln A ] ) thereby approaching the ideal 0(A )-scaling of an unbiased Markovian random walk up to a logarithmic correction...

Figure 17. Simulations of confined polymer chains as ideal random walks between two hard impenetrable interfaces. Two populations exist free (nonimmobilized) chains, which contribute to the normal mode, and immobilized chains, which contribute to the confinement-induced mode via fluctuations of their terminal subchains.

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. 

Consider a particular random walk on a lattice with each step having independent Cartesian coordinates of either -fl or —1. The projection of this three-dimensional random walk onto each of the Cartesian coordinate axes is an independent one-dimensional random walk of unit step length (see Fig. 2,8 for an example of a two-dimensional projection). The fact that the one-dimensional components are independent of e ch other is an important property of any random walk (as well as any ideal polymer chain). 

The stretching along the x axis, shown in Fig. 2.13, makes the stretched conformation of an ideal chain a directed random walk of tension blobs. This conformation is sequential in the x direction, but the y and z directions... 

Consider a monomer of an ideal polymer trying to reach fellow monomers of the same chain via a CB radio (see Fig. 2.17). The number of monomers it can call depends on the range r of its transmitter. It can contact any monomer within the sphere of radius r of itself The number of monomers m that can be reached via a CB radio with range r is given by random walk statistics ... 

While the ideal chain discussed in Chapter 2 has a random walk... 

The length of a tube 7 occupied by an ideal chain can be estimated as a random walk of Njg compression blobs along the contour of the tube ... 

As expected, the size of the ideal chain along the contour of the tube is not affected by the confinement. This is an important property of an ideal chain. Deformation of the ideal chain in one direction does not affect its properties in the other directions because each coordinate s random walk is independent. 

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