Molecular between crosslinks

A cross-linked polymer has a density of 0.94 g cm" at 25°C and a molecular weight between crosslinks of 28,000. The conformation of one bond in the middle of the molecule changes from trans to gauche, and the molecule opens up by 120°. In w-butane, the trans to gauche transformation requires about 3.3 kJ mol". Estimate a value for AH of stretching based on this model, and use the law of cosines to estimate the magnitude of the opening up that results. 

The next step in the development of a model is to postulate a perfect network. By definition, a perfect network has no free chain ends. An actual network will contain dangling ends, but it is easier to begin with the perfect case and subsequently correct it to a more realistic picture. We define v as the number of subchains contained in this perfect network, a subchain being the portion of chain between the crosslink points. The molecular weight and degree of polymerization of the chain between crosslinks are defined to be Mj, and n, respectively. Note that these same symbols were used in the last chapter with different definitions. 

For thermosets with molecular weight between crosslinks M, the crosslink density Px, is described by Px N /Mx- As Px increases, the nets become tighter and stiffer, and thus require more stress to break via... 

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. 

Analysis of data pertaining to the modulus of PEO gels obtained by the polyaddition reaction [90] shows that even in this simplified case the network structure substantially deviates from the ideal one. For all samples studied, the molecular weight between crosslinks (M p) exceeds the molecular weight of the precursor (MJ. With decreasing precursor concentration the M xp/Mn ratio increases. Thus, at Mn = 5650 a decrease in precursor concentration from 50 to 20% increases the ratio from 2.3 to 12 most probably due to intramolecular cycle formation. 

Mr averaged molecular mass between crosslinks as calculated from... 

N number of molecular chains between crosslinks (network strands) in... 

The polymers of the present study were all based on the molecule bisphenol A (2,2-di(4-hydroxy-phenyl)propan). The molecular chains between crosslinks were built up as uniformly as possible in order to eliminate effects imposed by variations of the chemical composition. 

All of the studied polymers are described by the network scheme in Fig. 2.1. Consequently, the composition of the polymers is uniform but the molecular chains between crosslinks differ in length. The molecular mass between crosslinks is therefore a dominant parameter for the characterization of the networks. 

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 ... 

Mc averaged molecular mass of strands between crosslinks, f functionality of junction (f = 3),... 

Table 3.1. Young s modulus E at T = 500 K and average molecular mass between crosslinks...

The results of the tests are plotted in Fig. 4.1 according to Fox and Flory [39] against 1> the inverse molecular mass between crosslinks, which was determined from the rubbery modulus. The two sets of results agree basically, although the DSC results are consistently 8 K lower than the temperatures TXmax, derived from... 

Inverse molecular mass (effective) between crosslinks... 

Fig. 4.1. Glass transition temperatures of the polymers are plotted against l/IVlc, that is the inverse molecular mass between crosslinks.

Tgoa glass transition temperature of the uncrosslinked polymer Mc average molecular mass between crosslinks tj> empirical factor... 

Young s moduli are given in Table 5.1 as well as in Fig. 5.2, plotted against Me1, the inverse molecular mass between crosslinks. Obviously, the moduli of the polymers increased as the number of crosslinks multiplied. Such an effect could be caused by the additional number of rigid covalent bonds or, just as well, by the increased density of the crosslinked polymers (Sect. 5.1). 

Yield strength as determined in tensile tests [53] at ambient temperature was plotted in Fig. 6.1 against M 1, the inverse molecular mass between crosslinks. All the samples of polymer A (the most crosslinked polymer) failed before the polymer started to yield. Therefore, load-extension-curves were extrapolated up to a hypothetical yield strain in this case. The extrapolated tensile is marked by brackets (Table 6.1). 

Effective molecular mass between crosslinks tvtc/kg mol 1 Tensile yield strength cry/MPa Energy release rate Gic/Jm 2 Half crack opening displacement w = 6/2 = Gic/2cry w/pm Chain contour length (Eq. 7.9) lc/nm... 

Fig. 6.1. Yield strengths of the five polymers are plotted against 1/MC that is the inverse molecular mass between crosslinks. The diamond represents polymer E. Test temperature 23 °C. a and b represent results of flexural tests on small samples (thickness 1.3 mm) a annealed, b quenched,...

Fig. 6.6. Comparison of activation volume and average volume of the strands between crosslinks. The effective molecular mass Mc between crosslinks is varied from 0.1 to 10 kg/mol. The cubes represent the activation volume of 2 nm3.

Fig. 7.4. Half the crack opening displacement 8C, is plotted against the effective molecular mass N4C between crosslinks. 5C = GIC/uy was calculated from the results in Table 6.1, measured at 23 °C. Mc was determined from the moduli of the polymers in the rubbery state...

Broutman and McGarry [98] examined the effects of crosslinking on toughness as early as 1965. Bell [99] observed a threefold increase in notched impact strength as the molecular mass between crosslinks was increased. Schmid et al. [100] and Lohse et al. [101] pointed out the dominating effect of molecular strand length on the ultimate properties and the toughness of crosslinked polymers. Later, Batzer et al. [46], Schmid [44], and Fischer et al. [45] compared the behavior of various networks composed of epoxy resins. 

Both sets of experiments seem to support the proportionality of crack opening displacement 5C = 2w and molecular mass Mc between crosslinks as indicated by the slope 1 in the double logarithmic plot (Fig. 7.5). Even if Mc had to be adjusted due to doubts about the front factor in Eq. (4.3), the proportionality would stay unaffected. Consequently, the size of the deformation zone ahead of the crack is determined by the length of the molecular strands in the chemical network. 

Fig. 7.5. Half the crack opening displacement 6C is plotted against the effective molecular mass Mc between crosslinks.

Fig. 8.1. Effects of crosslinking on various properties (X) of the polymer. The deviation [X — XcoJ/Xa, from the thermoplastic property (X ) is plotted against Me1, the inverse molecular mass between crosslinks.

Figure 9. SANS measurements of R /Rt° and RL/R ° for stretched radiation cross-linked polystyrene. is determined by measurements in which the neutron is parallel (iso) and perpendicular (aniso) to the stretching direction. Mc is molecular weight between crosslinks. Theoretical curves 2 and 3 are drawn for tetrafunctional networks. Data from Ref. 21.

C. C. Han, H. Yu and their colleagues (23) have presented some new SANS data on end-linked trifunctional isoprene networks. These are shown in Figure 10. Those materials of low molecular weight between crosslinks exhibit greater chain deformation consistent with the thesis that the junction points are fixed. This is the reverse of that found by Beltzung et al. for siloxane networks. 

Inhomogeneities in a real network may occur either because of a continuous distribution of molecular weight between crosslinks or due to the regions of different average molecular weights (as may be the case in randomly crosslinked networks). 

Peppas and Merrill (1977) modified the original Flory-Rehner theory for hydrogels prepared in the presence of water. The presence of water effectively modifies the change of chemical potential due to the elastic forces. This term must now account for the volume fraction density of the chains during crosslinking. Equation (4) predicts the molecular weight between crosslinks in a neutral hydrogel prepared in the presence of water. 

Here, r is the stress applied to the polymer sample, p is the density of the polymer, R is the universal gas constant, T is the absolute experimental temperature, and Mc is the desired molecular weight between crosslinks. 

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