Random walk conformation

This model, however, is only valid for chain molecules at zero Kelvin. When 7V 0 the chain parts possess thermal energy vibration causes them to move in random directions, which always results in a contraction of the stretched chain. The chain tends to a state of higher probability, and eventually reaches a fully unoriented random conformation, as described in 2.4 ( random walk conformation ). To... 

One of the most important phenomena in the polymer solvation is the change in the overall size of the polymer chain upon solvation. In fact at equilibrium the average size of isolated polymer molecules in solution is a function of solvent quality and varies from expanded conformations in good solvents to random walk conformations in poor solvents. This is referred to as collapse transition and was first predicted by Stockmayer [82] more than 45 years ago. The phenomenon was observed by Nishio et al. [83] and Swislow et al. [84] more than 25 years ago and is still a subject of much experimental, computational, and theoretical research today. So far many investigators have tried to study the chain size with solvation using a variety of methods. 

In this section, we will consider the excluded volume interaction follow-ing a similar scaling approach, rhe main idea is that of a thermal length scale (the thermal blob). On length scales smaller than the thermal blob size the excluded volume interactions are weaker than the thermal energy kT and the conformations of these small sections of the chain are nearly ideal. The thermal blob contains gr monomers in a random walk conformation ... 

The dynamics of an entangled chain in an array of fixed obstacles can also be studied by Monte Carlo simulations. An initial unrestricted random walk conformation of a chain on a lattice (representing a chain in a melt) could be obtained using the method of section 9.6.2.2. The topological entanglement net of surrounding chains is represented by obstacles, sketched as solid circles in the middle of each elementary cell in Fig. 9.32. 

FIGURE 12.5 Random walk conformation of a polymer molecnle with chain elements of persistence length... 

It is stressed again that since is an average quantity, the coil is spherical on average. However, each individnal polymer molecule has a nonspherical asymmetrical shape. Furthermore, the monomer distribntion in the statistical coil is Ganssian that is, the density of the polymer segments decreases outwards from the center. It can be proven that for a random walk conformation, the segmental density distribution Pp(r) is given by... 

The distributions shown in Figure 12.6 are for random walk conformations, that is, at 9 conditions, for which % = 0.5. Note that in a better solvent, % < 0.5, the polymer coil swells, which implies even lower polymer volume fractions. The swelling may be acconnted for by introducing a linear expansion coefficient a, defined as... 

In the random walk conformation, the polymer molecule attains maximum rotational freedom along the chain resulting in a root mean square end-to-end distance given by Equation 12.9. The probability Q for a polymer chain to deviate from the random coil conformation and, therefore, having an end-to-end distance deviating from e is given by a Gauss equation. 

In a random walk conformation, the average segment density Pp g varies with whereas for a swollen coil, pp is proportional to At c the average segment concentration in solution equals that in an individual coil and, hence, Ce and c Ap° The formation of a nonconnected network is accompanied by sudden changes in physical-chemical properties of the system, for instance, an abrupt increase in the viscosity. 

A theoretical model has been proposed to treat the concentration dependence of the flexible polymer coil dimension in semidilute solutions by a perfectly reflecting spherical wall around each segment of the coil, which influences the random walk conformation. In a 0-solution the intersegmental interaction is expressed by a potential... 

Nucleic acids are worm-like molecules with a persistence length p of order 50 nmJ9,20 in a solution or a dilute gel, a DNA molecule has a random-walk conformation with a radius of gyration given by21 ... 

Equation (35) indicates that initially, the stretching grows exponentially with time, which can be seen in Fig. 5. Since (A (0)> = (2iVa /3ir) for a random-walk conformation, the average stretching rate is given by... 

In diluted gels, one has al>p, and DNA is expected to fill pores with a local random-walk conformation, giving In this case, the... 

Therefore, the Kratky-Porod model reduces to the rod conformation for ip L, and the freely-jointed random-walk conformation for ip <[Pg.33]

Fig. 7. Illustration of a polymer chain (the tagged chain) confined in the fictitious tube of diameter d formed by the matrix. The contour line of the tube is called the primitive path having a random-walk conformation with a step length a=d. The four characteristic types of dynamic processes (dotted arrow lines) and their time constants Zs, Zg, Zr, and za defined in the frame of the Doi/Edwards tube/reptation model are indicated...

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