Single polymer coil

The above derivation can also be applied to colloidal or polymer-based liquids and is then used to calculate the so-called form factors of soft matter samples. The major difference between a monoatomic liquid and a polymer chain in the melt or in solution is that the total structure factor consists of two parts. The first is the inter-particle structure factor and the second the intra-particle structure factor. This second part is also often called the particle form factor P(q). Using Eq. (2.38) it is straightforward to calculate P(q) for a given soft matter sample. A good example is the form factor of a single polymer coil in a melt [88, 92]. The pair correlation function of such a coil is given by... 

Quite naturally, a free single polymer coil is often called a Gaussian coil, after the distribution (6.16). 

In dilute solutions, the single polymer coil expands in the athermal solvent. In a good solvent, the coil will expand more significantly. In contrast, in a poor solvent, the chain units and the solvent undergo a phase separatiOTi under a proper thermodynamic condition. Consequently, the single chain will collapse drastically into a condensed sphere. Therefore, the internal concentration reaches... 

If the solution is too concentrated, macromolecular entanglements will scatter radiation at different time scales than single polymer coils. To prevent this, DLS is done on dilute solutions and results are extrapolated to zero concentration conditions. 

However, the reduced viscosity is also not totally independent of the concentration. Even though viscosimetric measurements are performed in the range of dilute solutions (below the critical concentration c where the single polymer coils start to interpenetrate), small polymer interactions (that are decreasing with a decreasing concentration) have to be considered. The true viscosity enhancing properties of a polymer is therefore the reduced viscosity extrapolated to c—>0 ... 

The intrinsic viscosity [q] has the same unit, [ml g" ], as the reduced viscosity rjred a better understanding, the intrinsic viscosity can be considered as a measure for the volume demand of the single polymer coil in ideally diluted solution. The intrinsic viscosity is proportional to the reciprocal density of the polymer coil in solution according to Eqs. (4.6) and (4.7). 

The intrinsic viscosity can be determined experimentally with the first two terms in Eq. (4.1). In a real polymer solvent system the state of an ideally dilute solution is never reached, even very dilute solutions have a finite amount of polymer. The second term in Eq. (4.1) captures these interactions between the single polymer coils in a not ideally dilute solution. When Eqs. (4.3)-(4.7) are inserted in Eq. (4.1), a linear dependency of the reduced viscosity / ed from the concentration c is obtained ... 

The diameter d of a single polymer coil can be obtained from the density the... 

The knowledge of the dimensions of a single polymer coil allows for the calculation of the solution volume filled with polymer. A matter of particular interest is the polymer concentration where the solution is completely filled with polymer coils and the coils start to interpenetrate as shown in Fig. 7.1. 

Equivalent volume of a single polymer coil in solution Volume of polymer in solution (equivalent fraction)... 

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