Crosslinking junctions

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. 

Obviously, universal constant, cj) depends on the number of chains that are linked at a crosslink (junction functionality f) and probably on the chemical composition as well. 

A. The vector connecting the ends of a chain is deformed affinely. The crosslink junctions are fixed in space. 

Figure 2. Scattering intensity versus azimuthal angle for a uniaxially oriented elastomer, X is 3, x is 0.2. Phantom network where , f is 3 A, f is 4 V, f is 10. Crosslink junctions fixed, X.

In this review, we have given our attention to Gaussian network theories by which chain deformation and elastic forces can be related to macroscopic deformation directly. The results depend on crosslink junction fluctuations. In these models, chain deformation is greatest when crosslinks do not move and least in the phantom network model where junction fluctuations are largest. Much of the experimental data is consistent with these theories, but in some cases, (19,20) chain deformation is less than any of the above predictions. The recognition that a rearrangement of network junctions can take place in which chain extension is less than calculated from an affine model provides an explanation for some of these experiments, but leaves many questions unanswered. 

Most, but not all of the terminal units are near the outside of the dendrimer at any given time. The SAXS studies [17] of the segment density distribution have shown that there is an abrupt transition region at the outside of large PAM AM dendrimers. The combination of these two factors suggests that the terminal functionalities of dendrimers are accessible from the outside and available for chemical reactions such as attachment to surfaces, mounting of a catalyst, or for use as a crosslink junction. 

Here, v is Poisson s ratio which is equal to 0.5 for elastic materials such as hydrogels. Rubber elasticity theory describes the shear modulus in terms of structural parameters such as the molecular weight between crosslinks. In the rubber elasticity theory, the crosslink junctions are considered fixed in space [19]. Also, the network is considered ideal in that it contained no structural defects. Known as the affine network theory, it describes the shear modulus as... 

A subsequent theory [6] allowed for movement of the crosslink junctions through rearrangement of the chains and also accounted for the presence of terminal chains in the network structure. Terminal chains are those that are bound at one end by a crosslink but the other end is free. These terminal chains will not contribute to the elastic recovery of the network. This phantom network theory describes the shear modulus as... 

K is a numerical factor which accounts essentially for chain flexibility. The 2H NMR spectrum might be simulated by superposing the contributions of all chains, considered independently [35]. If the macroscopic deformation is transmitted affinely to crosslink junctions, P(R) may be written in the deformed network (the strain being along z) ... 

Fig. 9a-d. Correlations of volume (S) of sorbed benzene per gram of (Sty)[ x (DVB), particles (80% by weight) enmeshed in PTFE microfibers with (a) the cross-link density, 1 /X (where X, equal to 1/x when x is <0.1, is the number of backbone carbon atoms between crosslink junctions), (b) the reciprocal of crosslink density (X), (c) the square root of X, and (d) the cube root of X. 

The average number (k) of backbone carbon atoms between crosslinked junctions in such macronet structures is given by the ratio of the total number of atoms in the backbones of the crosslinked network divided by the total number of atoms that form crosslink junctions. Thus, in this macronet system, k is given by ... 

Fig. 18. Correlation of S, in mL of sorbed liquid per gram of crosslinked macronet polyester (prepared by Takahashi [144] in 1983) with the corresponding cube root of the average number of carbon atoms between crosslink junctions...

The above results indicate that Eq. 14 may be general for liquid sorption by crosslinked polymer networks, i.e. the volume, S, of liquid sorbed per gram of crosslinked polymer increases linearly with the cube root of the average number, X, of backbone atoms in the segments between crosslink junctions. The constants... 

Fig. 27. Correlation of the volume S in mL of sorbed I(CH2)nH liquids per gram of polymer with the corresponing cube root of the number (X) of backbone carbon atoms between crosslink junctions in (Sty), (DVB), polymers as a function of n...

Since ordered structural orientation is believed to be a prerequisite for exhibiting alternation in physical properties in monolayer coverage, I have interpreted the alternation exhibited in the C data for the liquids that comprise Z(CH2)nH series to mean that such moleclules are fixed to the monomer unit of poly(styrene) segments in the liquid-saturated gel-state via the Z substituent, and that these adsorbed molecules are distributed around the adsorption sites (i.e, the monomer units) in a well-defined orientation with respect to each site, such that the established orientation relative to that monomer unit is maintained despite the freedom of rotation and serpentine movement of the polymer segments between crosslink junctions in the liquid-saturated gel domain. 

The main assumption of the affine network model is that the ends of network strands (the crosslink junctions) are fixed in space and are displaced... 

The deformation dependence of the stress in the Edwards tube model is the same as in the classical models [Eqs (7.32) and (7.33)] because each entanglement effectively acts as another crosslink junction in the network. Therefore, the Edwards tube model is unable to explain the stress softening at intermediate deformations, demonstrated in Fig. 7.8. The reason for the classical functional form of the stress strain dependence is that the confining potential is assumed to be independent of deformation. 

On swelling, each network strand is stretched as the crosslink junctions move further apart. The stretching of an ideal chain was treated in... 

Crosslink junctions in the network are fixed at their mean positions. Upon deformation, these junctions transform affinely, that is, in the same ratio as the macroscopic deformation ratio of the elastomer sample. 

Figure 2.30 Idealized network structure of a crosslinked polymer. indicates a crosslink (junction) and —> signifies continuation of the network structure. Wavy, lines between crosslinks are active network chain segments. (Note that for a tetrafunctional crosslink, as shown here, the number of crosslinks is one-half the number of active network chain segments.)...

With regard to the mechanical reactirai of a polymer network to a stress applied, it is important that loose ends of macromolecules in a network structure are as shmrt as possible and/or their concentration is low. As these ends mostly extend out of the lamellas of crystallites then, while crossUnking is taking place in an amorphous phase and with the simultaneous presence of crystallites, a network with small loose ends should be formed. The crosslink junctions stabilize the natural molecular network (entanglements and crystallites), and every chain in the system is potentially elastically operative and can contribute to the stress in a tensile experiment [33]. The stabilization effect of chemical crosslinks on entanglements and crystallites may be the direct cause of observed differences in the determination of the amount of chemical crosslinks from mechanical property measurements and sol-gel analysis of the cross-linked polymer. 

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