Network chain dimensions

Table 1.5 Network chain dimensions according to small angle neutron scattering...

The ratios of mean-squared dimensions appearing in Equation (13) are microscopic quantities. To express the elastic free energy of a network in terms of the macroscopic (laboratory) state of deformation, an assumption has to be made to relate microscopic chain dimensions to macroscopic deformation. Their relation to macroscopic deformations imposed on the network has been a main area of research in the area of rubber-like elasticity. Several models have been proposed for this purpose, which are discussed in the following sections. Before that, however, we describe the macroscopic deformation, stress, and the modulus of a network. 

IR dichroism has also been particularly helpful in this regard. Of predominant interest is the orientation factor S=( 1/2)(3—1) (see Chapter 8), which can be obtained experimentally from the ratio of absorbances of a chosen peak parallel and perpendicular to the direction in which an elastomer is stretched [5,249]. One representation of such results is the effect of network chain length on the reduced orientation factor [S]=S/(72—2 1), where X is the elongation. A comparison is made among typical theoretical results in which the affine model assumes the chain dimensions to change linearly with the imposed macroscopic strain, and the phantom model allows for junction fluctuations that make the relationship nonlinear. The experimental results were found to be close to the phantom relationship. Combined techniques, such as Fourier-transform infrared (FTIR) spectroscopy combined with rheometry (see Chapter 8), are also of increasing interest [250]. 

Fluctuations are larger in networks of low functionality and they are unaffected by sample deformation. The mean squared chain dimensions in the principal directions are less anisotropic than in the macroscopic sample. This is the phantom network model. 

If network unfolding takes place so that distances between junctions connecting the ends of a polymer chain deform less than that of a phantom network, molecular dimensions change less than by any other of the models considered. This is easily seen from the data presented for a not equal to zero. 

It is always easy to calculate idealized scattering curves for perfect networks. The experimental systems vary from the ideal to a greater or lesser degree. Accordingly, any estimate of the correctness of a theoretical analysis which is based on an interpretation of experiment must be put forth with caution since defects in the network may play a role in the physical properties being measured. This caveat applies to the SANS measurement of chain dimensions as well as to the more common determinations of stress-strain and swelling behavior. 

A second important characteristic is the value otj. of the elongation at which rupture occurs. The corresponding values of r/rm show that rupture generally occurred at approximately 80-90% of maximum chain extensibility (12). These quantitative results on chain dimensions are very important but may not apply directly to other networks, in which the chains could have very different configurational characteristics and in which the chain length distribution would presumably be quite different from the very unusual bimodal distribution intentionally produced in the present networks. 

Microgels are distinguished from linear and branched macromolecules by their fixed shape which limits the number of conformations of their network chains like in crosslinked polymers of macroscopic dimensions. The feature of microgels common with linear and branched macromolecules is their ability to form colloidal solutions. This property opens up a number of methods to analyze microgels such as viscometry and determination of molar mass which are not applicable to the characterization of other crosslinked polymers. 

To illustrate how the effect of the adsorption on the modulus of the filled gel may be modelled we consider the interaction of the same HEUR polymer as described above but in this case filled with poly(ethylmetha-crylate) latex particles. In this case the particle surface is not so hydrophobic but adsorption of the poly (ethylene oxide) backbone is possible. Note that if a terminal hydrophobe of a chain is detached from a micellar cluster and is adsorbed onto the surface, there is no net change in the number of network links and hence the only change in modulus would be due to the volume fraction of the filler. It is only if the backbone is adsorbed that an increase in the number density of network links is produced. As the particles are relatively large compared to the chain dimensions, each adsorption site leads to one additional link. The situation is shown schematically in Figure 2.13. If the number density of additional network links is JVL, we may now write the relative modulus Gr — G/Gf as... 

Sequential IPN. The preceding analysis does not apply to the case of a sequential IPN. The formation of this system originates with the synthesis of the network (1). Then, network (1) is swollen with monomer (2) which is subsequently polymerized in situ to form a second network. Due to perturbed chain dimensions, the modulus of the first network is higher than the corresponding modulus in the unswollen state by a factor equal to v [ ] ... 

Flory has recently summarized the experimental evidence pertaining to local correlation and their effects on chain dimensions (49). There is experimental support for local alignment from optical properties such as stress-optical coefficients in networks (both unswelled and swelled in solvents of varying asymmetry), and from the depolarization of scattered light in the undiluted state and at infinite dilution. The results for polymers however, turn out to be not greatly different from those for asymmetric small molecule liquids. The effect of... 

Curro and Mark 38) have proposed a new non-Gaussian theory of rubber elasticity based on rotational isomeric state simulations of network chain configurations. Specifically, Monte Carlo calculations were used to determine the distribution functions for end-to-end dimensions of the network chains. The utilization of these distribution functions instead of the Gaussian function yields a large decreases in the entropy of the network chains. 

This conclusion permits comparison of the thermomechanical and thermoelastic results for various networks. The most reliable data are summarized in Table 2. The temperature coefficients of the unperturbed dimensions of chains d In intermolecular interactions of the configuration of the network chains. 

In Eq. (III-9) the deformation ratios are defined with respect to a reference state in which the chain dimensions are such that they do not exert any elastic forces on the crosslinks (state of normal coiling). In general, the chains in a network may not actually be in this state at the beginning of a deformation experiment, because the ciosslinking process may quite well exert a, largely unknown, influence on the chain dimensions. 

Considering next swollen networks, the situation becomes somewhat more complicated. In the first place a free energy of mixing will be needed in addition to the network free energy. Furthermore, the reference state may, in principle, depend upon the nature of the diluent and the amount of it. Also, the effect of the crosslinks on the chain dimensions in the reference state is unknown, and may be a function of the diluent content. Since none of these finer adjustments in the reference state have been given a quantitative molecular basis, we will only formally introduce the... 

If qc is known, it is tempting to put q0 = qe, because this would eliminate one of the unknown network parameters. Eq. (III-18) shows that this may not be correct because the largely unknown influence of the crosslinks upon the chain dimensions is then neglected. 

As pointed out in Chapter III, Section 1 some specific diluent effects, or even remnants of the excluded volume effect on chain dimensions, may be present in swollen networks. Flory and Hoeve (88, 89) have stated never to have found such effects, but especially Rijke s experiments on highly swollen poly(methyl methacrylates) do point in this direction. Fig. 15 shows the relation between q0 in a series of diluents (Rijke assumed A = 1) and the second virial coefficient of the uncrosslinked polymer in those solvents. Apparently a relation, which could be interpreted as pointing to an excluded volume effect in q0, exists. A criticism which could be raised against Rijke s work lies in the fact that he determined % in a separate osmotic experiment on the polymer solutions. This introduces an uncertainty because % in the network may be different. More fundamentally incorrect is the use of the Flory-Huggins free enthalpy expression because it implies constant segment density in the swollen network. We have seen that this means that the reference dimensions excluded volume effect. 

The parameter rl r, sometimes referred to as the front factor, can be regarded as the average deviation of the network chains from the dimensions they would assume if they were isolated and free from all constraints. For an ideal elastomer network, the front factor is unity. 

Polymer precursor Af X 10 Dry network Grain dimension in dry network d, A Root square end-to-end distance of free chain, A Good solvent 0-Solvent ... 

As can be seen from Table 1.5, chain dimensions in the dry PDMS networks are found to be much smaller than those of the unperturbed macromolecules in the -solvent, toluene. 

Undoubtedly, in the preparation of model networks from rather long polymeric precurson by the end-hnking reactions, the strands between the junction points wid maintain the conformation of coils. In accordance with classical ideas, the cods in the dry model network should acquire unperturbed dimensions, simdar to cods in a -solvent. However, on contacting a model polystyrene network with cyclohexane at room temperature, which is far below the -point for linear polystyrene (34.5°C), its volume increases by a factor of 3. Thus, the solvent breaks some polymer-polymer interactions, stimulates swelling of the network, and may result in an increase of chain dimensions. Hence, even at room temperature, the polystyrene chains prefer replacing the alien chain segments by cyclohexane molecides. 

Figure 3.12 shows a shear strain y imposed on a rubber block containing a network chain with an end-to-end vector r. It is assumed that the crosslink deformation is affine with the rubber deformation, meaning that the components of r change in proportion to the rubber block dimensions. The components ty and are unchanged, but r becomes... 

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