Primitive path, definition

In the course of investigation of polymer dynamics, all topological states of chains are equivalent due to the presence of their free ends and it is not necessary to construct the topological invariant. However, for every instantaneous chain configuration, its topological state is definite and is determined by the topological invariant - the primitive path. 

The primitive path is defined as the shortest line with the same topology as the original chain. Provide a precise definition of what is meant by the same topology . 

It is also assumed that the ends of fragments of the primitive path with the length 1 are fixed in a definite manner therefore, the Np and Lp values can be found fairly easily using /pj. The value can be expressed as follows [42] ... 

The justification for the above definition comes from the argument that the disentanglement rate represents a characteristic length for a disentangling chain divided by a characteristic time. We have chosen this length to be the primitive path and the time to be the reptation time. Using the scaling laws developed in the previous section, we can write... 

These g t) data do not favor the reptation model, but are not definitive enough to rule out the possibility of reptation motion. Thus, Kolinski et al. [57] stepped further by calculating the longitudinal and lateral displacements of the primitive path used in the Doi-Edwards theory. Such calculations should give us direct information on local chain motions. 

To provide a microscopic definition of entanglement. By the microscopic definition, we mean a definition that uses bead coordinates of chains around a particular chain as a function of time. Alternatively, one can try to define the tube or a primitive path. The challenge is not to come up with a reasonable proposal (several of them already exist ), but... 

In the following section we calculate these long-time properties from the simplest assumptions about the short-time dynamics of unentangled loops. These assumptions are based on the definition of unentangled loops given below, that provides a unique assignment of the loops to the steps of the primitive path. 

Figure 21 Definition of the primitive path and the imagination of the tube...

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