Self-entanglements

It is expected, however, that the Gaussian representation is inadequate in transient elongational flow, even if the chain is only weakly deformed. During a fast deformation, the presence of non-equilibrium effects, like internal viscosity , noncrossability and self-entanglements will stiffen the molecular coil which is now capable of storing a much larger amount of elastic energy than that predicted from Eq. (113). 

However, this is true only for good solvent conditions, where (c) is also the correlation length, beyond which both the excluded volume and the hydrodynamic interaction are screened and self-entanglements (intramolecular... 

Under -conditions the situation is more complex. On one side the excluded volume interactions are canceled and E,(c) is only related to the screening length of the hydrodynamic interactions. In addition, there is a finite probability for the occurrence of self-entanglements which are separated by the average distance E,i(c) = ( (c)/)1/2. As a consequence the single chain dynamics as typical for dilute -conditions will be restricted to length scales r < (c) [155,156],... 

Finally for Cfe(c) 1 the unperturbed (not self-entangled) single-chain re-laxationjust known from good solvent conditions, takes place. S(Q, t)/S(Q, 0) is a universal function of (Q(Q,t)2/3 with Q(Q) = QZ(Q) In Fig 58b a schematic plot of the crossover behavior of the segmental dynamics under 0-conditions is shown. 

Under 0-conditions the single-chain behavior itself shows, at decreasing Q-values, a crossover from ordinary Zimm relaxation (Q(Q) Q3) to an intramolecular microgel mode (Q(Q) Q), which is due to the occurrence of self-entanglements... 

A useful toy theoretical model which captures the essential features of self-entangled dendritic polymers is the monodisperse Cayley tree in which each chain segment branches with a fixed functionality z at each of its ends, except those at the extremity of the molecule (see Eig. 13). Smaller versions of these structures, too low in molecular weight to be entangled, have been synthesised and are usually referred-to as dendrimers [47]. 

The simplest example of topological classification is the knotting of individual strands, i.e., self-entanglement. However, the effects attributed to entanglement in non-crosslinked polymers are clearly intermolecular. The simplest such case is that of pair-wise classification without self-entanglement. Consider the following three examples of strand pairs ... 

In general, it appears that the fraction of configurations in the various topological classes can be determined for models in which one of the elements is a fixed curve and the other is a random coil. The detailed calculations are intricate and difficult, however, and some simple generalizations are needed which could be used as a step towards building classification effects into the network theories. Classification for the case of two random coils and for self-entanglement are unsolved problems at the present time. 

Figure 2.4.5 Unusual structural arrangements can be observed for coordination polymers including (a) interpenetration, or polycatenation, and (b) self-entanglement.

The possibility of exploiting engineered coordination networks for practical applications (e.g., absorption of molecules, reactions in cavities) very much depends on whether the networks contain large empty spaces (channels, cavities, etc.), or whether the network is close packed because of interpenetration and self-entanglement. The possibility of a sponge-like behavior by which the network can change its shape, well, and shrink... 

Molecular geometry and connectivity can be used to construct descriptors for other shape features. In this section, we introduce another aspect of macromolecular shape the complexity of self-entanglements in a polymer. The analysis is restricted to backbones that is, only main chain atoms are considered. 

Self-entanglements. The occurrence of twists, turns, and convolutions in space curves can be characterized by a number of descriptors using geometry and connectivity. These include the overcrossing probability distribution of a backbone of arbitrary architecture (and related parameters, such as the mean number of overcrossings N), the writhing number W of a curve, and the twist 9" of a ribbon model. 

K. Koniaris and M. Muthukumar, J. Chem. Phys., 95, 2873 (1991). Self-Entanglements in Ring Polymers. 

G. A. Arteca, Phys. Rev. E, 51, 2600 (1995). Scaling Regimes of Molecular Size and Self-Entanglements in Very Compact Proteins. 

Non-Self-Entangled Long Chains in a Short-Chain Matrix... 

A first hint of this complication is found in dilute solutions. The theoretical models (Zimm or Rouse) that we introduced both consider phantom chains, they ignore the topological constraints due to the connections between polymer chains, and thus the effects of entanglements this turns out to be a reasonable assumption if the number of entanglements or knots is small and is actually the case for an isolated polymer in a good solvent. However, in a Gaussian chain of N monomers (in a 0 solvent), the number of knots scales as in a 0 solvent, polymer chains are self-entangled. 

In a semi-dilute solution in a good solvent, only entanglements between different chains exist this is not the case in a 0 solvent where both interchain entanglements and self-entanglements are present. This leads to the existence of two different relaxation modes and thus two different diffusion coefficients. In this section, we briefly discuss these two modes in a 0 solvent and the crossover effects between good and 0 solvents. 

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