Network knots

Fig. 24a and b. Schematic simplified representation of the basic difference between a regular a and irregular b network knots (1), chains connecting nearest-neighbor knots (2), pendant chains (3)... 

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. 

The value ofVjA is determined by the concentration of network knots. These knots usually have a functionality of 3 or 4. This functionality depends on the type of curing agent. Crosslinked polyurethanes cured by polyols with three OH-groups are examples of the three-functional network. Rubbers cured through double bond addition are examples of four-functional networks. 

Mj molar mass of a network chain between two network knots... 

Therefore, counterions are mostly localized around the network knots, as well as the polymer chains, due to the deep potential wells and valleys. The charge density of counterions decreased very sharply with an increase in the distance from the polymer chain. Counterions located in the deep potential valley ( 7 ) should bond strongly to the polyion. The amount of these bound counterions would increase with an increase in the cross-linking density. 

Block copolymers or graft copolymers made up of soft and rigid polymer sequences. Styrene block copolymers like polystyrene (PS) blocks [styrene-butadiene-styrene (SBS), styrene-isoprene-styrene (SIS), and styrene-ethylene-butylene-styrene (SEBS)], and polyester TPEs belong to this family. Structurally, the thermoplastic blocks form physical network knots within the polydiene. 

Thermoplastic elasto-ionomers, whose network knots are thermolabile ionic microdomains formed by associations between ionic groups (carboxylic or sulfonic acids) present in non-polar polymer chains. ... 

Kauffinan, L.H., Lomonaco, S. q-Deformed Spin Networks, Knot Polynomials and Anyonic Topological Quantum Computation, quant-ph/0606114 v2 Kauffman, L.H., Lomonaco, S.J. Braiding Operators are Universal QuantumGates. New Journal of Physics 6(134), 1-39 (2004)... 

The 4,4 -bipy and bpe derivatives display a similar crystal structure to that of Cd(4,4 -bipy)2[Ag(CN)2]2 reported by Iwamoto et al. [89]. It consists of the interpenetration of two identical 3D networks. The knots of the networks are defined by the iron(II) and silver(I) atoms. Each iron(II) atom located on an inversion centre defines an elongated octahedron whose axial positions are occupied by the nitrogen atoms of two 4,4 -bipy ligands. In addition, each 4,4 -bipy ligand binds a silver atom so that it is three-coordinated. This is the reason why the [Ag(CN)2] group is bent (see Fig. 18). 

Scheme 29. Unreported macromolecular architectures containing defined topological bonds polycatenane 9, linear poly[3]catenane 74, poly[2]catenane network 75, multicatenane network 76, rigid polymeric catenane 77, polymeric trefoil knot 78, and polyknot 79.

Upon cooling of solutions of long-chain polymers it may happen that only parts of the chain eventually crystallize together with parts of other polymer chains. When a number of such small crystallites are formed they operate as knots in a network of flexible polymer chains in solution and one obtains a gel. Processing of such gels into strong fibres and films is applied commercially and has been reviewed [13,14]. 

Every knot connects four chains by chemical bonds. At T > Tg, both chains and knots take part in thermal motion as Brownian particles, at T < Tg, the network is in a glassy state, large-scale conformations of the chains are frozen, the motion of the knots is negligible. 

Although a network is not present in a concentrated solution, there exists a characteristic length, which had earlier been assumed the distance between neighbouring network sites. The characteristic length is a dynamic one. There are no temporary knots in a polymer system, though there is a characteristic time, which is the lifetime of the frozen large-scale conformation of a macromolecule in the system. So, the conceptions of intermediate length and characteristic time are based on deeper ideas and are reflected in the theory. 

Network formation was generated on the lattice (hexagonal, tetragonal and cubic) with the total number of crosslinks amounting to 2-4 x 102. Tetrafunctional amine molecules were fixed in the lattice knots. This lattice was immersed in uniform liquid of diepoxide molecules which could penetrate the lattice freely. The network appeared as a result of an addition reaction between diepoxides and amine groups of crosslinks. 

The Continuous Plankton Recorder (CPR) survey consists of a dense network of transects across the North Atlantic and the North Sea. Plankton recorders are towed on a monthly basis at a depth of 8-10 m from ships-of-oppportunity that travel at 10-18 knots. The sampling mechanism inside the recorders consists of a narrow band of filtering silk (mesh 270 pm) that is driven by an impeller at the rear of the recorder at a speed adjusted according to the speed of the ship. The silk catches particles entering the 12 mm2 aperture while it passes (at a rate of 10 cm per 10 nautical miles, 18.5 km) through the end of a wide tunnel behind the narrow opening in front. About 3 m3... 

Surface wind speed and direction from six sites were available at four hour intervals during the 24 hour sample collection. These data indicate that surface winds in the Tacoma-Seattle area were consistently southwesterly at 10 knots on February 14-15. A rawindsonde confirmed southwesterly flow aloft. Frontal precipitation as rain occurred with the mean rainfall accumulation across the network of 0.5 cm. in a 3 hour period. 

An alternate approach to the analysis of the network geometry from the viewpoint of multiplicity and stability is due to Beretta and his coworkers (1979, 1981) the latter approach is based on knot theory. The Schlosser-Feinberg theory reproduces some of the Beretta-type theory results concerning stability from a somewhat different viewpoint. 

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