Uncrossability constraints

Fig. 2A-C. Various representations of a polymer chain and its surroundings. The chain and segments of neighboring chains (A), the chain in a tube of uncrossable constraints provided by its neighbors (B), and the primitive path of a chain among the surrounding constraints provided by neighbors (a step length of the primitive path R = end-to-end vector of the chain) (C)...

Shaffer s bond fluctuation model (Shaffer, J. S., 1994. Effects of chain topology on polymer dynamics—Bulk melts, J. Chem. Phys., 101 4205 13). Polymers are grown as random walks on a simple cubic lattice, subjected to the excluded volume, chain connectivity, and chain uncrossability constraints described in the text. 

The Doi-Edwards theory assumes that reptation is the dominant mechanism for conformational relaxation of highly entangled linear chains. Each molecule has the dynamics of a Rouse chain, but its motions are now restricted spatially by a tube of uncrossable constraints, illustrated by the sketch in Fig. 3.38. The tube has a diameter corresponding to the mesh size, and each chain diffuses along its own tube at a rate that is governed by the Rouse diffusion coefficient (Eq. (3.37)). If the liquid is deformed, the tubes are distorted as in Fig. 3.39, and the resulting distortion of chain conformations produces a stress. The subsequent relaxation of stress with time corresponds precisely to the progressive movement of chains out of the distorted tubes and into random conformations by reptation. The theory contains two experimental parameters, the unattached mer diffusion coefficient T>o... 

Fig. 3.38. A depiction of a chain with the uncrossability constraint represented by a tube [56].

Padding, J.T. and Briels, W.J. (2001) Uncrossability constraints in mesoscopic polymer melt simulations non-Rouse behavior of C120H242./. Chem. Phys.,... 

How do we describe the entanglements and how do we calculate the relaxation times of a chain and other dynamical properties of the entangled system The statistical mechanics of such a disordered system with the uncrossability constraints is very difficult to formulate. Nevertheless there have been three kinds of approaches. 

At the same time, the above mentioned chain-like structure leads to the fact that different parts of polymer molecules fluctuating in space cannot go through each other without chain rupture. For the system of non-phantom closed chains, this fact means that only those space conformations that can be transformed continuously into one another are available (see Fig. 1). The adequate mathematical language for description of those physical effects is elaborated in the mathematical discipline called topology. That is why we also call the effects connected with chain uncrossability the topological constraints. 

Figure 45a-c shows an adaptation of the developed model to uniaxial stress-strain data of a pre-conditioned S-SBR-sample filled with 40 phr N220. The fits are obtained for the third stretching cycles at various prestrains by referring to Eqs. (38), (44), and (47) with different but constant strain amplification factors X=Xmax for every pre-strain. For illustrating the fitting procedure, the adaptation is performed in three steps. Since the evaluation of the nominal stress contribution of the strained filler clusters by the integral in Eq. (47) requires the nominal stress aR>1 of the rubber matrix, this quantity is developed in the first step shown in Fig. 45a. It is obtained by demanding an intersection of the simulated curves according to Eqs. (38) and (44) with the measured ones at maximum strain of each strain cycle, where all fragile filler clusters are broken and hence the stress contribution of the strained filler clusters vanishes. The adapted polymer parameters are Gc=0.176 MPa and neITe= 100, independent of pre-strain. According to the considerations at the end of Sect. 5.2.2, the tube constraint modulus is kept fixed at the value Ge=0.2 MPa, which is determined by the plateau modulus Gn° 0.4 MPa [174, 175] of the uncross-linked S-SBR-melt (Ge=l/2GN°). The adapted amplification factors Xmax for the different pre-strains ( max=l> 1-5, 2, 2.5, 3) are listed in the insert of Fig. 45a.

The dielectric relaxation frequency (/,) of 5CB at room temperature is in the order of 10 Hz [43], whereas/c of the neat LCE (without solvent) corresponding to SNE-7 (unpublished results from this laboratory) and/c of an uncross-linked side-chain EC polymer with the similar mesogen [44] are in the order of 10 Hz at 70°C (above the glass transition temperatures of ca. 50°C). The significantly lower values of /c for the neat LCE and the side-chain EC polymers are due to the constraint effect of the network and polymer backbone on the motion of the dangling... 

If the long polymer chains are interconnected by widely separated cross-links, the portions of chains between the cross-links can still assume the three states mentioned. In the case of melt state, a loosely cross-linked melt is called a rubber. The extent to which the original molecules can assume new conformations is limited because of the topological constraints of the cross-links. If the number of cross-links is increased, the portions of chains between cross-links become shorter. Finally, these sections of chains may be so short that rotations around single bonds and segmental diffusion are no longer possible and the system resembles a permanent glass even at temperatures at which an uncross-linked polymer would be a melt. 

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