Fixed obstacles

FIG. 26 A snapshot of a typical hoof-like form of a driven 32-bead chain (light beads) at overcritical bias B = 0.625 through a dilute medium of fixed obstacles (dark beads) of density Cobs = 0.125. The direction of flow is indicated by the arrow. [21]... 

Introduction of the reptation concept by De Gennes [43] led to further essential progress. Proceeding from the notion of a reptile-like motion of the polymer chains within a tube of fixed obstacles, De Gennes [43-45], Doi [46,47] and Edwards [48] were able to confirm Bueche s 3.4-power-law for polymer melts and concentrated polymer solution. This concept has the disadvantage that it is valid only for homogeneous solutions and no statements about flow behaviour at finite shear rates are analysed. 

The reptation model (225) also appears to produce a Rouse spectrum at long times. In order to renew its configurations a chain must diffuse out of the tunnel defined by the fixed obstacles along its length. De Gennes calculates the autocorrelation function for the end-separation vector, obtaining... 

DeGennes,P.G. Reptation of a polymer chain in the presence of fixed obstacles. J. Chem. Phys. 55,572-579 (1971). 

The basic models consider well-defined star-branched polymers. De Gennes [11] imagined in 1975 a simple relaxation mechanism of a branch based on the Brownian motion of an arm of a star-branched molecule in a network of fixed obstacles (Figure 13). From statistical considerations, the time necessary for a branch to renew its configuration is ... 

Other computer simulations, such as the Evans Edwards model of a chain in an array of fixed obstacles (described in detail in Section 9.6.2) exhibit fluctuations of the tube length and also find stronger molar mass dependences of relaxation time t and diffusion coefficient... 

The dynamics of an entangled chain in an array of fixed obstacles can also be studied by Monte Carlo simulations. An initial unrestricted random walk conformation of a chain on a lattice (representing a chain in a melt) could be obtained using the method of section 9.6.2.2. The topological entanglement net of surrounding chains is represented by obstacles, sketched as solid circles in the middle of each elementary cell in Fig. 9.32. 

Figure 9.34 displays the mapping of a chain in an array of fixed obstacles to the repton model. Topological obstacles form a lattice with cell size equal to the tube diameter. Roughly Ar monomers are in cell I, between the end of the chain A and the point B where the chain finally leaves cell I for... 

Consider a regular g-generation dendrimer entangled in an array of fixed obstacles. The functionality of each junction point is/and the number of... 

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. 

In the de Gennes approach, the polymer chain is assumed to be contained in a hypothetical tube [Fig. 2.32(a)] which is placed initially in a three-dimensional network formed from other entangled chains. Although for simplicity these network knots are shown in Fig. 2.31 as fixed obstacles around which the chain under consideration must wriggle during translation, in practice these obstacles would also be in motion. The contours of the tube are then defined by the position of the entanglement points in the network. 

Figure 2.31 A model for reptation. The chain P moves among fixed obstacles, O, but cannot cross any of them.

S. Goldstein, The steady flow of viscous fluid past a fixed obstacle at small Reynolds numbers, Proc. R. Soc. Ser. A, 123, 225-35 (1929). 

The reptation model, on the other hand, recognizes such impediments to chain motion. In Figure 3-24a a single chain is shown along with randomly placed dots that represent fixed obstacles to the chain motion. The reptation model suggests that the chain must move through this obstacle course in a worm-like fashion as relaxation occurs. 

Figure 2.28 (a) Schematic representation of a polymer chain confined in a hypothetical tube contoured by fixed obstacles, (b) Movement of a kink along the chain. (After Cowie, 1991.)... 

H4. A controUed airborne aircraft gets too close to a fixed obstacle other than a safe point of touchdown on assigned runway (known as controlled flight into terrain or CFIT). 

---