Junction concentration network

Figure 11 Morphology diagram for PB-fc-PEO in water (concentration of 1 wt%) as a function of molecular size and composition. AfpB is the degree of polymerization of the PB block and wpEo is the weight fraction of the PEO blocks. Four basic morphologies were identified bilayers (B), Y junctions (Y), network, N cylinders (C), and spheres (S). (Adapted with permission from Ref. 40. American Association for the Advancement of Science, 2003.)...

If the concentration of junction points is high enough, even branches will contain branches. Eventually a point is reached at which the amount of branching is so extensive that the polymer molecule becomes a giant three-dimensional network. When this condition is achieved, the molecule is said to be cross-linked. In this case, an entire macroscopic object may be considered to consist of essentially one molecule. The forces which give cohesiveness to such a body are covalent bonds, not intermolecular forces. Accordingly, the mechanical behavior of cross-linked bodies is much different from those without cross-linking. 

The concentration of crosslink junctions in the network is also important if too low, flow will be possible if too high, the maximum attainable elongation will be decreased. From the point of view of theoretical analysis, the length of chain between crosslink points must be long enough to be described by random flight statistics. 

Gelatin stmctures have been studied with the aid of an electron microscope (23). The stmcture of the gel is a combination of fine and coarse interchain networks the ratio depends on the temperature during the polymer-polymer and polymer-solvent interaction lea ding to bond formation. The rigidity of the gel is approximately proportional to the square of the gelatin concentration. Crystallites, indicated by x-ray diffraction pattern, are beUeved to be at the junctions of the polypeptide chains (24). 

The equilibrium shear modulus of two similar polyurethane elastomers is shown to depend on both the concentration of elastically active chains, vc, and topological interactions between such chains (trapped entanglements). The elastomers were carefully prepared in different ways from the same amounts of toluene-2,4-diisocyanate, a polypropylene oxide) (PPO) triol, a dihydroxy-terminated PPO, and a monohydroxy PPO in small amount. Provided the network junctions do not fluctuate significantly, the modulus of both elastomers can be expressed as c( 1 + ve/vc)RT, the average value of vth>c being 0.61. The quantity vc equals TeG ax/RT, where TeG ax is the contribution of the topological interactions to the modulus. Both vc and Te were calculated from the sol fraction and the initial formulation. Discussed briefly is the dependence of the ultimate tensile properties on extension rate. 

Let us now assume momentarily that the elastomers are perfect, in the sense that all functional groups (originally present in equivalent amounts) have reacted in such a manner that no sol exists. Under such idealized conditions, each molecule of monohydroxy PPO will in effect produce one inactive junction. (Strictly speaking, i molecules of monohydroxy PPO can produce somewhat more than i inactive junctions.) Thus, the concentration of junctions in the network becomes about 0.073 mole/kg instead of 0.100 mole/kg (Table II). As each junction gives 3/2 chains, the modulus, after equating h to zero in eq 1, is given by... 

Let Pxi be the probability that a chain originating in a trifunctional moiety terminates in the gel. For a trifunctional moiety to be an active junction, all three emanating chains must terminate in the gel, the probability of which is P3j. Thus, the concentration of active junctions is CtriolD Pxi and the concentration of active chains is (3/2)CtriolDP3j, where CtriolD denotes the mole/cm3 of trifunctional moieties in the network. 

To a first approximation, which neglects changes in average chain structure, the loss in elastically active junction point concentration may be translated directly into loss in concentration of elastically active chains and increase in the value of M, . For a perfect network in the dry state, the concentration of elastically active chains is given by the equations... 

Systems such as the concentrated solution of the UHMWPE in paraffin oil (2-8% w/w) contain a three-dimensional molecular network in which the junction points are produced by secondary valence bonds which cause crystalline regions and by physical entanglements of different life times. Entanglements that are trapped between crystallites have, like the crystallites, essentially infinite life times. 

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