Zimm equation

Solutions of lightly sulfonated polystyrene (SPS) sodium salts with 0 and 19% perdeuterated chains were studied throughout a concentration range of 0. -4.0 g dL" in perdeuterated tetrahydrofuran (THF-ds). A 0.5 g dL solution of unmodified polystyrene (PS) was also measured for comparison. The scattered intensity I(q) can be interpreted using the well-known Zimm equation (12)... 

Scattering of light is a natural phenomenon arising from the interaction of light with the matter. There are several equations describing the concentration and angular dependence of the intensity of light scattered by a dilute polymer solution. One of them is called the Zimm equation ... 

In its most simple form for measurements made at low angle and infinite dilution, the Zimm equation reduces to ... 

Stein, R.S. and Hadziioannou, G. (1984) Generalization of the Zimm equation for scattering from concentrated... 

Equihbrium characterization of polymers usually takes place in dilute solution where 1 2AxM 3Ax M and over an angular range such that (f <1. In this case, the Zimm equation reduces to one of its most frequently used forms ... 

A convenient form of the Zimm equation for compolymer analysis, which is only slightly different than Equation 11.21, is ... 

The solid line incScates the calculated results from the generalized Zimm equation the volume fraction is 7.3x10-3... 

Substituting the preaveraged result of the Oseen tensor and performing the normal mode analysis of the Zimm equation for the various Rouse modes, we can calculate the mean square displacement of the center-of-mass of the chain, mean square displacement of a labeled monomer, translational friction coefficient of the chain, and the relaxation times of the various Rouse modes with the Zimm dynamics (Doi and Edwards 1986). The main results of these calculations are the following. 

It can also be shown from the preaveraged Kirkwood-Riseman-Zimm equation that the mean square displacement of a labeled monomer i is proportional to the 2/3 power of time for times shorter than rzimm. 

The results of Equations 7.46, 7.48, and 7.50 for the Zimm dynamics are entirely consistent with the universal laws expected in Section 7.2.1 and are fully supported by experimental data in dilute solutions. If the hydrodynamic interaction among segments is suppressed in the Kirkwood-Riseman-Zimm equation, then the problem reduces to the Rouse dynamics and all results of Section 7.2.2 are recovered. 

Plotting the intensity I(q) versus tf or log/( ) versus q yields the radius of gyration. Zimm representation is another way of processing the data, especially when several concentrations, C, are used [17]. Zimm equation reads ... 

The important difference between the Rouse and Zimm equations of motion comes from the nonlocality of the mobility tensor... 

Within this preaveraging approximation, the Zimm equation of motion becomes linear and can be studied by the same methods as the Rouse equation. The main results are summarized here. 

The second equation is very similar to the Rouse and Zimm equation of motion (Equations 12 and 22). The form of the friction force between monomers and solvent is not specified here, it will be determined by the use of the no-slip Stokes condition... 

---