Ideal chain models

As has probably become obvious already, the study of micelles has been one of the big topics in simulations of systems with surfactants. We have cited many of the related publications in the previous sections. Here, we shall discuss some special aspects of micelle simulations in order to illustrate the use of idealized chain models for this type of problem. 

There is no distinction between Xg and Xp in the Rouse description of linear chains. A third relaxation mechanism contemplates the disentangling of two or more intertwined arms. This relaxation is considerably slower and strongly dependent on f. Obviously, this feature cannot be described by the ideal chain model. 

This is the most successful ideal chain model used to calculate the details of conformations of different polymers. In this model, bond lengths I and bond angles d are fixed (constant). 

Table 2.2 summarizes the assumptions of the ideal chain models. The worm-like chain model is a special case of the freely rotating chain with a small value of the bond angle 6. Moving from left to right in Table 2.2, the models become progressively more specific (and more realistic). As more constraints are adopted, the chain becomes stiffer, reflected in larger Coo-... 

Table 2.2 Assumptions and predictions of ideal chain models FJC, freely jointed chain FRC, freely rotating chain HR, hindered rotation RIS, rotational isomeric state...

If the internal rotation surrounding each backbone bond contains three possible conformation states, the internal rotation of a long chain will generate an astronomical amount of possible conformation states. In such a case, we could not count them one-by-one, and thus have to make conformation statistics on the basis of a simplified ideal chain model. 

We have discussed the ideal-chain model in Sect. 2.2 by incorporating short-range restrictions into the freely-jointed-chain model first the fixed bond angles, then the hindered internal rotation. In this way, we reached the description of semiflexibility of the real polymer chains. The mean-square end-to-end distances of chains in different models are given below. 

This is the so-called Flory-Fisher scaling law (De Gennes 1979). The critical exponent v = 1 in (4.21) at the dimensionality d = 1 v = 3/4 at d = 2 v = 3/5 at d = 3 and v = 1/2 at d = 4. These critical exponents are consistent with that of self-avoiding walks obtained above from the computer simulations. The scaling law for the ideal chain model occurs only in 4D space of SAWs. In 3D space, the renormalization group theory yields the critical exponent as v = 0.588 0.001, which is in good consistency with the computer simulation results (Le Guillou and Zinn-Justin 1977). 

Due to the screening effect in the volume exclusion of polymer chains, singlechain conformation in the concentrated solutions will exhibit the size scaling similar to the ideal-chain model, as... 

Why do we say that the Rouse-chain model rests on the ideal-chain model ... 

The simplest model to describe polymers is the ideal-chain model. For books on polymer physics where all the relevant background material can be found see [11-19]. In this model the polymer consists of M subunits, each with a fixed bond length b, and their orientation is completely independent of the orientation and positions of previous monomers, even to the extent that two different monomers can occupy the same position in space there is no excluded volume. This model plays the same role in polymer physics as an ideal gas in molecular physics. It allows to describe the polymer chain as a (Gaussian) random walk of M steps, as depicted in Fig. 2.8. 

An exponent of more than 1/3 makes sense here for a three-dimensional (3D) polymer in solution because the polymer fills a volume space incompletely (a solid material fills space with v = 1/3). This ideal chain model for our polymer is modified, however, when we start to think of behavior of a real polymer chain, for which there can be monomer-monomer and monomer-solvent interactions. 

Freely jointed chain model See Ideal chain model... 

Ideal chain model In polymer science, a model for the polymer chain in which the monomers are freely joined with no interactions between monomers. 

In the heart of all ideal chain models is an efficient algorithm for the calculation of the partition function and concentrations fields. The algorithm is either called the matrix scheme (in ID lattice theories), and/or propagator... 

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