Average loop length

With the knowledge of /x(AT) we may derive all the bundle statistics. As an example, assuming that m is the smallest number of chain atoms in a loop, the average loop length is given by (see Fig. 1, m is a number of chain atoms)... 

From the maximum term of Eq. (B-43), taking into account, Eqs. (B-39) and (B-40) the average loop length PB, the average train length Ps, and the fraction p of adsorbed segments can be calculated as functions of ys, yB, as, aB, and AFS. The last quantity is related to by... 

Fig. 4. Average loop length PB (proportional to film thickness) and adsorbed amount dip as a function of...

Let us assume that the amorphous fraction forms a dense layer with an average loop length of gr1 segments on either side of the crystal, and that the surface is sufficiently extended to obtain a homogeneous density of segments ca in the amorphous fraction. Then, the height of the amorphous layers, see Fig. (2.2) is given by... 

In the following I consider only the case p > 1, which is the physical relevant solution for single chain crystals. The scaling variable can be related to the average loop length in the amorphous fraction... 

The opposite case of y 1 can only be realized if the average loop length is very large g p)- The latter must be excluded in order to avoid the brush... 

At this point a third intermediate approach deserves mentioning. It is due to Allegra [43] who proposed that polymer crystallization is controlled by a metastable equilibrium distribution of intramolecular clusters, the so-called bundles , forming in the liquid phase. These subsequently aggregate to the side surfaces of the crystals, driven by van der Waals interactions. The lamellar thickness is determined by the average contour length of the loops within the bundles. Although the model can... 

In view of this disagreement, as well as of evidence from polymer mesophases and MD simulations, we also propose an alternative model, based on the concept that the attractive interactions are so short-lived as to be effectively delocalized. As a consequence, bridges separating consecutive bundles are also taken into account in the evaluation of the average stem length of the growing crystal, in addition to the crystalline stems and to the loops... 

In this section, we present the two-mode models for loop and recycle reactors. In a loop reactor (Fig. 10) of loop length L, a flow rate of qm with an average velocity of u-m) enters and leaves the reactor at points a = 0 and x — l, respectively (where a is the length coordinate along the loop), and the total flow rate in the loop is Q + qin between points a = 0 and x — l, and is Q between points x — l and x — L, due to a recycle rate of Q. The recycle ratio A is the ratio of the volume of fluid returned to the reactor entrance per unit time to the volume of fluid leaving the system per unit time, and is given by A = Q/q-m. 

The average strand length is nearly proportional to the slab width. It is actually smaller than it, since loops axe always more probable than bridges (for a given strand length). 

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