Dimensions of a Real Polymer Coil

The assumption in Determination of the polymer coil dimensions from the intrinsic viscosity in Chap. 7 that the polymer coils in solution behave like hard spheres with a constant density inside the coil and a fixed boundary to the solvent is only a simple approximation. In reality, a polymer chain shows a dynamic behavior with fast and statistically changing conformations. 

However, it is possible to describe the dimensions even of dynamic polymer coils in solution with the radius of gyration Rq and the average end-to-end distance 1/2 chain. They can be calculated for a polymer coil in its unperturbed di- 

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At theta-conditions, the intramolecular interactions of the polymer chain are compensated by the solvation force of the solvent molecules and the polymer coil resumes its unperturbed dimensions ( Dimensions of a real polymer coil in Chap. 8). These theta-solvents correspond to thermodynamically poor solvents. The temperature at which the theta-conditions occur (theta-temperature) is normally close to the precipitation point of the polymer-solvent system. 

These calculations still assume that the polymer coil in solution is a hard sphere with an even density throughout the sphere and with a fixed boundary to the solvent. For a more realistic discussion of the dimensions of a real polymer coil in solution, the reader is referred to Chap. 8. In particular, the correlation of diameter dy molar mass M and intrinsic viscosity [rj] in Eq. (7.4) is discussed in detail in Chap. 8 in the form of the Fox-Flory equation that correlates the intrinsic viscosity with the radius of gyration Rq of a polymer coil and with the molar mass ... 

In a real polymer coil, not all angles r are possible. As one can see in Fig. 8.3, in a real polymer chain, the bond angle r is fixed and the rotation of the chain is restricted and reduced to the most probable torsion angle 0. Additional short-range interactions of the polymer chain segments can be captured with an additional factor In consideration of all these short-range interactions, the end-to-end distance of a polymer coil in its unperturbed dimensions, o, can be described with the following expression ... 

Nevertheless, even the critical concentration obtained from absolute polymer coil dimensions, is only a relative value since the radii measured with scattering experiments are not equal to the hydrodynamic radii of the same polymer coils in solution. A detailed discussion on how to calculate a hydrodynamic radius is given in The critical concentration of a real coil in Chap. 8. 

The methods of conformational statistics, discussed so far, had as starting point the real polymer chain. The aim was to relate the dimensions of the coiled polymer molecule statistically to the mutual displaceability of the chain atoms. Nearly exact relationships are obtained for a large number of freely jointed or freely rotating elements. Under conditions of restricted movability, however, the statistical equations can generally not be solved and empirical factors like s, a and a are introduced. 

The analysis of experimental data on light scattering from polymer solutions, as well as studies of the hydrodynamic properties of the solution, demonstrates that in a wide range of molecular weight, the apparent dimensions of the macromolecules grow proportionally to the square root of the number of the monomer units (molecular weight). These observations point to the random coil behavior of real macromolecules in solutions. 

In the models for polymer chain conformation that we have considered so far, the polymer chain is allowed to intersect itself, because each link is a vector that takes up no volume. This is clearly unrealistic for real polymer molecules, where the segments occupy a certain volume and the chain cannot cross itself. This leads to excluded volume, which cannot be occupied by other segments. Polymer coils which have excluded volume are said to be perturbed, whereas (r )J gives the unperturbed dimensions of the coil assuming volumeless links. The perturbed dimensions (r ) / are related to the unperturbed dimensions by the expansion factor, a ... 

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