Reptation monomer diffusion

Fig. 4. The number of monomers on the diffusion front of a simulated 2-d diffusion of reptating random coil chains versus the average monomer diffusion depth is shown for molecular weights 60-100 times the entanglement molecular weight.

Various factors govern autohesive tack, such as relaxation times (x) and monomer friction coefficient (Co) and have been estimated from the different crossover frequencies in the DMA frequency sweep master curves (as shown in Fig. 22a, b). The self-diffusion coefficient (D) of the samples has been calculated from the terminal relaxation time, xte, which is also called as the reptation time, xrep The D value has been calculated using the following equation ... 

Figure 3 Diffusion coefficient versus the polymerisation index of the labelled chains, N, in the case N P (filled circles, fixed matrix) and N=P (open squares) at T = 23°C. When the matrix is frozen, the power law D N 2 pO. typical of the reptation process, is observed down to very low values of N, leading to an evaluation of the minimum number of monomers to create an entanglement for PDMS Ng = 100. For comparison, data from reference 51, with P = N and T = 60°C, are reported as open circles.

An important question is to decide how far one can believe that a self-diffusion coefficient varying like N is characteristic of reptation. It has been argued that additional molecular weight dependences could exist and compensate for departures from the N 2 law [48 to 52]. Such an effect can come from the local monomer-monomer friction coefficient w hich appears as a prefactor in equation 8, hidden in the diffusion coefficient D. Several processes can combine and lead to a local friction which is molecular weight dependent, and W hich decreases when the polymer molecular weight is decreased. This is, for example, the... 

The diffusion coefficient ot the chain is controlled by the reptation time [Eq. (9.12)]. The linear polybutadiene chain with M= 130000gmol has A =1240 Kuhn monomers, with Kuhn length /)=10A and coil size R — hy/N = 350 A. Since linear polymers move a distance of order their own size in their reptation time, the reptation time of ri-ep = 0.2s at 25 °C enables estimation of the diffusion coefficient 6x... 

This curvilinear motion continues up to the reptation time trep where the chain has curvilinearly diffused the complete length of the tube, of order aNjN. At times longer than the reptation time (/ > Trep) the mean-square displacement of a monomer is approximately the same as the centre of mass of the chain and is a simple diffusion with diffusion coefficient D [Eq. (9.12)]. 

Consider the dynamics of an entangled ring polymer in an array of fixed obstacles [Fig. 9.40(a)]. The ring is not permanently trapped by the obstacles, but is able to diffuse. The ring does not have free ends and, therefore, classical snake-like reptation is not expected for it. An ideal untrapped ring polymer in an array of fixed topological obstacles is an unentangled loop formed by double-folded strands of Ne monomers each, similar to an arm of a star at the moment of complete retraction. 

Thus, the reptation model predicts that Dg decreases as N as the number N of monomers in the chain grows. When N is quite large, the diffusion coefficient is very low. As a result, if you bring two polymer melts together, they will tend to intermingle very slowly, even if the thermodynamics suggests that the mixed state is the most favorable one (i.e., if the two polymers are miscible). 

Suppose we have added some initiator to the solution of not-yet-polymerized monomers, and the reaction has begun. At first, the growing chains appear in a kind of dilute solution, in which the monomer molecules play the part of a solvent. With time, more and more molecules of the monomer become involved in the reaction. The concentration of the chains grows, and they begin to overlap. This is when the solution becomes semi-dilute. FYom this moment on, the character of the chains motion changes — they start moving by reptation. As we have already shown, this means that diffusion of polymer chains slows down substantially. 

Medium-to-high conversion The diffusion mechanism of the propagating radicals becomes complex after the onset of the gel effect. Large chains become immobile however, the chain ends may move by reptation or reaction diffusion. Monomer and short species may still be highly mobile in the polymerizing system. Translational center-of-mass diffusion may become the rate-determining step for radical-radical termination. 

On the other hand, the reptation theory proposed by de Gennes<4) assumes that a flexible chain is diffusing in a fixed three-dimensional mesh of obstacles that the chain cannot cross (Figure 2). Thus, the chain would be topologically constrained to move by a curvilinear, or snake-like, motion alone. This motion has been termed reptation (from reptile ). One can visualize that the flexible chain is reptating by a Brownian diffusion within a tube surrounded by obstacles, but motions proceed perpendicular to the axis if the tube is blocked. For a chain made from N monomers of size a, the coefficient of the curvilinear diffusion, along the tube is... 

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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