Crosslinks fluctuations

The experimental error was very large, with the apparent chain deformation greater than that expected for a phantom network, and closest to the curve anticipated where crosslink fluctuations are completely suppressed. 

This constitutes the formal basis of the theory. What is crucial is the potential function V(R jRc) and how it changes as the network is deformed. Po(R), N(Rc) and V(R Rc) contain important parameters. Chain geometry and crosslink fluctuations yield values of some parameters. It is at this point where experimental results provide considerable guidance. 

Small deformations of the polymers will not cause undue stretching of the randomly coiled chains between crosslinks. Therefore, the established theory of rubber elasticity [8, 23, 24, 25] is applicable if the strands are freely fluctuating. At temperatures well above their glass transition, the molecular strands are usually quite mobile. Under these premises the Young s modulus of the rubberlike polymer in thermal equilibrium is given by ... 

Cations can be seen as acting as ionic crosslinks between polyanion chains. Although this may appear a naive concept, crosslinking can be seen as equivalent to attractions between polyions resulting from the fluctuation of the counterion distribution (Section 4.2.13). Moreover, it relates to the classical theory of gelation associated with Flory (1953). Divalent cations (Zn and Ca +) have the potential to link two polyanion chains. Of course, unlike covalent crosslinks, ionic links are easily broken and re-formed under stress there could therefore be chain slipping and this may explain the plastic nature of zinc polycarboxylate cement. 

B. The mean vector connecting the chain ends is deformed affinely. The crosslink junctions fluctuate according to the theory of Brownian motion (2, 5, 7) ... 

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. 

The elastic free energy AFe causes difficulty because of its sensitivity to the crystallization model assumed. To estimate AFe for lamellar morphology, consider first an important property of a network, amorphous or crystalline. Network crosslinks are considerably restricted in their fluctuations. Fluctuations of crosslinks several chains removed from a particular chain are therefore inconsequential for that chain. A chain in the interior of a path traced through several sequentially connected chains behaves as if the path ends are securely anchored at fixed positions ( 7). If Gj chain vectors make up the path, then... 

If the fluctuations of crosslinks are considered, a functionality-dependent factor, 1 -2/f, has to be applied (28-31)... 

In order to enable these fluctuations to occur, the network chains are assumed to be "phantom" in nature i.e. their material properties are dismissed and they act only to exert forces on the junctions to which they are attached. With common networks having tetrafunctional junctions, the results of the two approaches differ by a factor of two. Identical results are only obtained from both theories, when the functionality is infinite. From a practical viewpoint, however, a value of about 20 for f can already be equated to infinity because crosslink densities can hardly be obtained with an accuracy better than 10%. 

Our results tend to approach the affine theory with increasing z. That means, that the fluctuationsof crosslinks are more and more restricted, the reason for this being the change in microstructure. (This is quite different from the strain dependent restriction of fluctuations as predicted by Flory s recent theory.)... 

Concluding, we can state that the absolute values of the small-strain moduli, which are greater for networks having comblike crosslinks, than for those with tetrafunctional junctions, are understandable, if we assume that the fluctuations of junctions are restricted by the very short chains. The strain dependent measurements do not agree quantitatively with the recent theory, although the trends are in accordance. An exact correspondence... 

The phantom network behaviour corresponding to volumeless chains which can freely interpenetrate one through the other and thus to unrestricted fluctuations of crosslinks should be approached in swollen systems or at high strains (proportionality to the Mooney-Rivlin constant C-j). For suppressed fluctuations of crosslinks, which then are displaced affinely with the strain, A for the small-strain modulus (equal to C1+C2) approaches unity. This situation should be characteristic of bulk systems. The constraints arising from interchain interactions important at low strains should be reflected in an increase of Aabove the phantom value and no extra Gee contribution to the modulus is necessary. The upper limit of the predicted equilibrium modulus corresponds therefore, A = 1. 

Star-shaped macromolecules have also been synthesized by using the monofunctional "living" precursor as an Initiator for the polymerization of a small amount of a divinyl monomer. A small crosslinked nodule is formed, which is connected with the p chains that have contributed to Its Initiation. It turns out that fluctuations on the value of p within a sample remain rather small, and consequently the star polymers obtained by this method can also be considered as tailor-made polymers. Recently star molecules with deuterium labeled central nodule have been synthesized according to the... 

The computer-assisted modelling of formation of the epoxy-amine networks indicates that the density fluctuations of the crosslinks does not exceed the value predicted by the statistical theory, which is much lower than the size of the globules in... 

Critical Behavior of Gels. In 1977, the critical phenomena were discovered in the light scattered from an acrylamide gel in water [18]. As the temperature was lowered, both the scattered intensity and the fluctuation time of the scattered light increased and appeared to diverge at —17 °C. The phenomenon was explained as the critical density fluctuations of polymer networks although the polymers were crosslinked [19, 20]. 

When crosslinks are introduced to these polymer solutions, the concentration fluctuations are perturbed due to the presence of crosslinks. The exact solution for the scattering function from gels has not been found yet because of the... 

If we exclude the case of pre-existing order, we have so far considered a network as a random, but completely homogeneous structure. It should now be mentioned that the crosslinking process itself may give rise to "aggregation of network elements and therefore, in the swollen state, to significant fluctuations in segment density. 

The exact result shown in Eq. (IV-4) indicates that appreciable errors are introduced if the Gaussian approximation is used below N = 6. The use of Eq. (IV-4) in a complete network theory (compare e.g. Eq. III-2) has not been undertaken because of mathematical difficulties. Instead, Treloar has shown that in a tetrahedral four chain arrangement around a central crosslink, the fluctuations in the crosslink... 

Wang and Goth (178) have applied the JG-type derivation (see Section III-l) using a series expansion of qt (r ) valid at not too large extensions. In this way the artificial splitting up in three sets of network chains, all having the same chain-end distances, is avoided. The assumption of affine deformation. however, still has to be made moreover, the fluctuations in crosslink position were assumed to be independent of the imposed strain. 

A significant difference between chemically crosslinked networks and hydrogen bonding (fluctuation) networks, is the absence of the plateau region in the high temperature region. 

It is important to note that in the inhomogeneous gel, the average crosslinking density is not a relevant parameter for determining the frictional pore size of the gel. It is the spatial correlation length of the density fluctuations that determines the bulk frictional behavior of water in the gel. 

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