Ideal freely jointed segments chain

Figure 6.14 Picture of a linear polymer in the ideal freely jointed segments chain model.

One such model is the ideal freely jointed segments chain (Fig. 6.14). In this model the polymer is considered to consist of a chain of n links. We call each chain link a subunit . Each subunit has a length l. This parameter / can correspond to the length of a monomer but it can also be shorter or longer. The angle between adjacent chain links is taken to be arbitrary. The chain forms a random coil. To characterize the size and volume of such a coil we use the mean square of the end-to-end distance R2. The square-root of this value — we call it the size of a polymer chain — is given by... 

The factor a is 1 for the unperturbed coil defined in this way, and is larger (or smaller) when a polymer molecule is expanded (or compressed) due to polymer-solvent interactions. Thus, for an idealized freely jointed chain in theta solvent, we have = nl, and the corresponding mean square radius of gyration = nf-/G. When one accounts for the steric effects that prevent distant chain segments from overlapping (excluded volume effect), the dependence of (r ) is predicted to be (closer to experimental observation), rather than [Eq. (5)]. This polymer coil size is much larger than that based on polymer density, and hence markedly influences the viscosity behavior of polymers. 

Let us now at first treat in more detail the ideal chains, on the basis of the introduced freely jointed segments model. We choose numbers, from 0 to iVg,... 

For ease of calculation, we make a number of simplifying assumptions. These are relaxed in advanced treatments of the subject. First, rather than requiring tetrahedral bonds at each vertex of the chain, we allow all bond angles and assume that these are randomly distributed. Second, we ignore any excluded volumes or interactions between the segments of the chain. In this sense, our calculation is similar to the Bernoulli model of the ideal gas, which neglects intermolecular interactions. Our approximation is called the freely jointed chain model. 

Discuss the determination of the statistical segment for polyethylene whose structural unit is (—CH2 — CH2—)x knowing that C = (r )o/ / = T.6, where (r )o is the mean square end-to-end distance of a chain of n skeletal bonds each of length / (= 1.53 A), and nf- is the mean square end-to-end distance of the chain in the idealization that the skeletal bonds are freely jointed. 

Note that in the limit q 0, (2.62) yields, as expected, the flat plate result 6s/Rg = 2/y/n. The result in (2.62) holds for Gaussian ideal chains, implying the segment size b is smaller than all other length scales, Rg and R. For freely-jointed ideal chains the depletion thickness also depends on the size ratio b/R for R<50b [38]. 

For ideal polymer coils, the most significant and distinctive property is the Gaussian distribution of the end-to-end distances. By considering the different segments of the freely jointed chain which are statistically independent and can be represented by a Markov chain, one can derive the correction to the Gaussian distribution of the end-to end distances (see Reference 99). 

The freely-jointed chain model (Fig. 2a) is the simplest it has been described as the ideal gas of polymer physics, as all interactions between the chain segments, except the chemical bonds connecting them, are neglected. The model represents a chain as a sequence of steps, or rectilinear statistical (Kuhn) segments, connected together at their ends. Each statistical segment represents a sequence of several chemical bonds. 

Matsuoka and Cowman [32] proposed the consideration of two simple models to study the hydrodynamic properties of hyaluronan on the basis of inherent viscosity. The first model of the not freely-jointed or non-ideal coil is based on the statistical conformations of polymer chains. Under this model, the intrinsic viscosity [ /] is directly proportional to the volume occupied by mass units of the polymer segments. The volume is filled mainly with solvent and has low density of polymer segments. The second model is the freely-jointed chain model. It is assumed that the chain is more extended and its viscosity corresponds to that of the worm-like chains. The intrinsic viscosity changes as a square of the mean-squared end-to-end distance. Experimental studies of intrinsic viscosity of hyaluronan indicate that short chains act as freely-jointed chains and long chain of hyaluronan behave like the not freely-jointed chains. The molecular weight, at which these two types of behaviour co-exist, is approximately 3.75xl(yDa. The size of the polymer corresponds to a smallest coil of hyaluronan (Figure 4.6). 

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