Polymer chains in MdM flow fields

The conformational properties and the dynamics of polymers in solutions under various types of flows have been a subject of considerable interest within the last decades. Much progress has been gained in the explanation of experimental data for systems in which the flow velocities are given functions in space and time, see Refs. [16-19]. On the other hand, the behaviour of polymers in random flows is less understood. In recent works [11] we (Oshanin and Blumen) succeeded in establishing analytically the behaviour of Rouse polymers [20] in MdM flow fields. The presentation here follows closely Ref [11]. 

In the Rouse model N monomers (beads) are coupled to each other via harmonic springs [16, 17, 20], As is well-known, the forces are of entropic origin. It is customary to revert to a continuous picture in which n, the bead s running number, takes real values. For a detailed discussion see Doi and Edwards [17]. The Langevin equations of motion for such a polymer in the MdM flow field are 

As before, the fluctuating forces on the rhs. of Eqs. 42-44 are Gaussian and also delta-correlated with respect to the running index [11, 17]. For the X and Z components, which are not subject to the flow, the procedure is standard [17]. Say, for the averaged X component of the end-to-end vector one has 

For isotropic situations (in the absence of flow fields) the end-to-end vector iV and the radius of gyration Rg of the polymer are related to Px and Xg through P = 3Px and Rg = 3Xg. Because of the anisotropy one has to consider in the MdM model the different components separately. 

The dynamics of a flexible polymer chain is richer than that of a single bead. In the Rouse model the dynamics of X (t) depends essentially on the time of observation t and on the Rouse time xr. 

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