Curvilinear diffusion constant,

Here, we shall relate the translational diffusion motion of the center of mass of a polymer chain to the curvilinear diffusion constant. Experimentally, the translational diffusion constant can readily be measured. 

Then R — Rn-i = o,. Assume that in a time interval At, the primitive chain jumps forward or backward with equal probability one step of length a. Then the curvilinear diffusion constant can be defined by... 

The curvilinear diffusion constant of the chain in its tube is given by D= g7V5, where =(L/6)5q is the friction coefficient of the whole chain and 6=3.4 A is the contour length of one bp. The time it takes to disengage completely from a tube of length Na by this diffusion process is2 ... 

The Einstein formula relates the tube mobility to a curvilinear diffusion constant... 

As in Sect. 2.1, Dj is the curvilinear centre-of-mass diffusion constant of the chain, and is given in terms of the monomeric friction constant by the Einstein relation Dj =kT/Nl. L is as before the length of the primitive path, or tube length of the chain, which is Finally, we need the initial condition on p(s,t), which... 

The type of motion considered here is shown in Fig. 3. We follow one labeled chain in the layer. The chain reptates on the self-similar grid. It is subdivided into subunits (i) with a spatial size Zi equal to the distance to the wall, and a number of monomers gi - gizi) Instead of computing directly the diffusion constant Du, we use as an intermediate the mobility p = D /kT of the chain (where kT is the thermal energy). When the chain reptates, the subunit (i) has a certain curvilinear velocity Ud It is important to realize that Ud is different for different subunits what is conserved is the tube current J , i.e., the number of monomers, crossing one given point on the tube, per second. 

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