Other Entanglements

Polythreaded systems contain motifs that are interweaved via rotaxane-like mechanical links and this implies the presence of closed loops as well as of elements that can thread the loops (Fig. 1.3.2). These two types of moieties may belong to the same unit or may be separately supplied by motifs having different structures. The constituent motifs could be, in principle, OD species, ID polymers or arrays of higher dimensionality. The resulting array can show the same or an increased dimensionality with respect to that of the polythreaded units. 

Known examples involving OD motifs, i.e. molecular beads (cucurbituril molecules), threaded by coordination polymers of different topology have been described by Kim and coworkers [80], 

Many examples of extended systems that cannot be disentangled have been reported very recently. The ID chain motif comprised of alternating rings and rods 

Numerous entangled arrays containing separable motifs (poly-pseudo-rotax-anes), like those shown in Fig. 1.3.18(c) and (d) have also been reported [see Ref. [3]]. 

We have attempted to rationalize here the structures of extended frames at different levels of complexity in terms of the network topology. With the aim of establishing useful relationships between structures and properties [84] a careful analysis of the topology is unavoidable, particularly for the complicated species that are increasingly being discovered. Indeed, phenomena like interpenetration are not only structural curiosities, with some esthetical appeal, but can play an effective role in the control of the properties of materials, as shown, inter alia, by their influence on network porosity [55 a, 65a] and by their capability to produce anomalous magnetic properties [85]. 

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The key point of our method is to entangle the initial state of the ancilla with the state of the system in such a way that the detection of the ancilla in its initial state implies that the system is also in its initial state. This is achieved through the coding procedure which has to ensure that after the exposition of the system to the action of errors, the initial state of the ancilla remains entangled only with the initial state of the system, whereas any other entanglement remains small during the time interval between consecutive measurements. 

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. 

If we assume the existence of a tube, we can proceed as follows [4]. For very small t the beads of each chain can move only very short distances and do not feel effects from other chains. Hence, the motion of the chain is essentially the same as that of a Rouse chain in dilute solution. This situation is sustained until t increases to n at which g i) becomes comparable to dt, i.e., the beads begin to feel the effect of other entangling chains. Here, as before, dt denotes the diameter of the tube. Using the known expression for g t) of an isolated Rouse chain, we find... 

In the case of star polymers as test chains, the more the number of arms, the smaller the possibility for stars to be detached from the matrix once they are attached to the matrix over much longer periods. Even though star test chains are smaller, their motions are still hindered by other entangled polymer coils. If most star test chains participate in the entanglement, the amount of "free" stars is too small to be detected. So the CONTIN analysis did not show two modes for LPS84, but we do not want to deny the possible existence of self-diffusion of star test chains because I I of LPS84 evidently deviated from the single exponential decay. 

There are a number of important concepts which emerge in our discussion of viscosity. Most of these will come up again in subsequent chapters as we discuss other mechanical states of polymers. The important concepts include free volume, relaxation time, spectrum of relaxation times, entanglement, the friction factor, and reptation. Special attention should be paid to these terms as they are introduced. 

In a similar fashion, a fraction of the velocity of the molecules with first-order coupling is transmitted to other molecules entangled with the latter. This is called second-order coupling (subscript 2). Still higher orders of effect radiate from the original molecule in the manner suggested by Fig. 2.13. Because of the... 

Equation (2.61) predicts a 3.5-power dependence of viscosity on molecular weight, amazingly close to the observed 3.4-power dependence. In this respect the model is a success. Unfortunately, there are other mechanical properties of highly entangled molecules in which the agreement between the Bueche theory and experiment are less satisfactory. Since we have not established the basis for these other criteria, we shall not go into specific details. It is informative to recognize that Eq. (2.61) contains many of the same factors as Eq. (2.56), the Debye expression for viscosity, which we symbolize t . If we factor the Bueche expression so as to separate the Debye terms, we obtain... 

Whether the beads representing subchains are imbedded in an array of small molecules or one of other polymer chains changes the friction factor in Eq. (2.47), but otherwise makes no difference in the model. This excludes chain entanglement effects and limits applicability to M < M., the threshold molecular weight for entanglements. 

An analogy to sHp dislocation is the movement of a caterpillar where a hump started at one end moves toward the other end until the entire caterpillar moves forward. Another analogy is the displacement of a mg by forming a hump at one end and moving it toward the other end. Strain hardening occurs because the dislocation density increases from about 10 dislocations/cm to as high as 10 /cm. This makes dislocation motion more difficult because dislocations interact with each other and become entangled. SHp tends to occur on more closely packed planes in close-packed directions. 

The two main amphibole asbestos fibers are amosite and crocidoHte, and both are hydrated siHcates of iron, magnesium, and sodium. The appearance of these fibers and of the corresponding nonfibrous amphiboles is shown in Figure 1. Although the macroscopic visual aspect of clusters of various types of asbestos fibers is similar, significant differences between chrysotile and amphiboles appear at the microscopic level. Under the electron microscope, chrysotile fibers are seen as clusters of fibrils, often entangled, suggesting loosely bonded, flexible fibrils (Fig. 2a). Amphibole fibers, on the other hand, usually appear as individual needles with a crystalline aspect (Fig. 2b). 

The alcohol swells the poly (ethyl methacrylate) beads, rapidly promoting diffusion of the plasticizer into the polymer. As a result of the polymer-chain entanglement, a gel is formed. The conditioner is applied to the denture and provides a cushioning effect alcohol and plasticizer are slowly leached out, and the material becomes rigid. To ensure resiliency, the conditioner must be replaced after a few days. Some materials exhibit high flow over a short period compared with others with low initial flow the latter remain active longer. 

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