Network structure imperfections

To determine the crosslinking density from the equilibrium elastic modulus, Eq. (3.5) or some of its modifications are used. For example, this analysis has been performed for the PA Am-based hydrogels, both neutral [18] and polyelectrolyte [19,22,42,120,121]. For gels obtained by free-radical copolymerization, the network densities determined experimentally have been correlated with values calculated from the initial concentration of crosslinker. Figure 1 shows that the experimental molecular weight between crosslinks considerably exceeds the expected value in a wide range of monomer and crosslinker concentrations. These results as well as other data [19, 22, 42] point to various imperfections of the PAAm network structure. 

The properties of a polymer network depend not only on the molar masses, functionalities, chain structures, and proportions of reactants used to prepare the network but also on the conditions (concentration and temperature) of preparation. In the Gaussian sense, the perfect network can never be obtained in practice, but, through random or condensation polymerisations(T) of polyfunctional monomers and prepolymers, networks with imperfections which are to some extent quantifiable can be prepared, and the importance of such imperfections on network properties can be ascertained. In this context, the use of well-characterised random polymerisations for network preparation may be contrasted with the more traditional method of cross-linking polymer chains. With the latter, uncertainties can exist with regard to the... 

Looking first at the tan 6 curves. Figures 2 and 3, it should be noted that the breadth of the alpha peak is influenced very little by the presence of the alkyl substituents. Imperfections in the network structure are reported to lead to a broadening of the alpha peak ( ). As mentioned earlier, the peak temperature of the alpha peak, a measure of Tg, follows exactly the trend in Tg s as measured by DSC. 

In addition to terminal chains and entanglements, there are other types of network imperfections. Figure 6-2 shows that if a short chain were crosslinked only once, the crosslink is a wasted one because the chain cannot support elastic stress. Also, if a crosslink forms an intrachain loop, it is again an ineffective crosslink. Unfortunately, owing to its very complexity, it is at present impossible to completely characterize the network structure of an elastomer. 

The cycle rank of a network is defined as the number of cuts required to reduce the network to a tree ( ). It is a structural factor characteristic of the perfection of a network ( ) for a perfect network, the cycle rsuik per chain ( is given by C>l-2/f. The ( values for various networks are listed in Table 3. The gels produced at very high extents of reaction still eidiibit various kinds of structural imperfections I for example, the cycle rank of the... 

Equation (22) holds for phantom networks of any functionality, irrespective of their structural imperfections. In case b), fluctuations of junctions are assumed to be suppressed fully. The junctions themselves are considered to be firmly embedded in the medium and their position is transformed affinely with the macroscopic strain. This leads to the free energy expression for an f-functional network possibly containing free chain ends... 

An ideal amorphous semiconductor is thought of as a uniform three-dimensional random network structure in which the valence requirements of each atom are satisfied. Any real material contains, however, imperfections and deviations from the ideal disordered network in the form of dangling bonds, voids, or density fluctuations. Furthermore, some vitreous semiconductors contain structural units which are bonded together by weak van der Waals type forces. This section deals in particular with structural and compositional inhomogeneities whose own size or whose effect (by forming a... 

There is a diversity of theroetical models used to elucidate the relationships between the molecular parameters of the network and the various experimental results [33-57]. Hence, the resulting deduction of the molecular structure of the network can depend on the model chosen for data analysis. Additionally, the structure of the networks at the supermolecular level is a function of the preparation conditions (temperature, concentration at crosslinking, chemical nature of the crosslinker, etc.). During network formation imperfections in the structure may also develop. In many cases the crosslinking process leads to fixation of otherwise nonequilibrium states. A wide variety of molecular superstructures may be produced within networks prepared from the same starting materials. This makes comparisons of experimental results from different literature sources extremely complicated. Consequently, a simple tabulation of previously published data is not particularly useful. 

Since dangling chains are imperfections in a network structure, their presence should have a detrimental effect on ultimate properties such as the tensile strength, as gauged by the nominal stress, f, at rupture, f. ... 

The proton-jump model for diffusion is based upon the premise that the structure of liquid water derives from an icelike hydrogen-bonded network in addition to the two normal OH bonds in water each oxygen atom is linked to neighboring ones via hydrogen bonds. The linkage is imperfect and there are defects in the network. These imperfections, known... 

Since dangling chains constitute imperfections in a network structure, one would expect their presence to have a detrimental effect on the ultimate properties (//A )r and Qfr of an elastomer. This expectation is confirmed by an extensive series of results obtained on PDMS networks that had been tetrafunctionally cross-linked using a variety of techniques [130]. The largest values of the ultimate strength... 

The elastic free energy of a phantom network of Gaussian chains was obtained rigorously by Flory and is valid for networks of any functionality, irrespective of their structural imperfections. It is given in equation (101). The elastic equation of state for phantom networks may then be expressed by equation (102). Equation (84) is then recovered, as expected, because of the relationship shown in equation (103). 

Some polymer solutions show the phenomenon of gelation. The polymeric chains form association complexes at widely separated points at a certain reduced solvent power of the medium. This leads to the formation of a continuous physical network structure extending throughout the volume of the system. The assoeiation producing quasi-crosslinkages is a reversible pioeess, so that the gel may be liquified and reset many times. The nature of the linkages is rather imperfectly understood, but the phenomenon is usually encountered with more or less crystalline polymers. Sometimes, sharp X-ray diffraction patterns are observed which disappear at the melting point of the gel. 

Another type of network imperfection, resulting from cross-linking of two units not distantly related structurally, is indicated in Fig. 94. Cross-linkages such as B are wasted (except insofar as the loop may be involved in entanglements not otherwise operative). The proportion of these short path cross-linkages should be small ordinarily but could become very large if the cross-linking process were carried out in a dilute solution of the polymer. 

Note 3 In addition to loose ends, model networks usually contain ring structures as network imperfections. 

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