Phantom modulus

The quantities B, and g, are given by Eq. (18a) and (18b). Fitting stress-strain data at different degrees of swelling by combination of Eq, (33), (34) and (35) enables one to determine x, and (presumed to be equal to Mq). Flory and Erman have performed the fitting by assuming = 1 and have determined x, C and the phantom modulus for randomly cross-linked poly(dimethylsiloxane) (PDMS) networks. These results were recently further exploited by Queslel and Mark. 

Flory has derived the elastic free energy of dilation of a network with account of restrictions of fluctuations of junctions. Quantitative agreement has been reported for vapor sorption measurements. Particularly impressive is reproduction of the observation that the product of the linear expansion ratiok and the elastic contribution (pi — p.i)e, to the chemkal potential of the dilumt in a swollen network exhibits a maximum with increase in k, which is contrary to previous theory It is convenient to compare the phantom modulus obtained by stress-str measurements to that obtained from swelling equilibrium studies... 

For a perfect end-linked network of functionality [Pg.143]

To account for the intermediate behavior between the affine and phantom modulus of a real network, Graessley has proposed an expression for the small strain modulus G as a function of an empirical paran ter having a value between 0 (afSne) and 1 (jdiantom). Specifically,... 

For imperfect networks, the comparison of the ratio 2Ci/vRT with 1 — 2/q> is no longer possible because the number of effective chains v is not known, and became not all of the junctions have the same functionality. Nevertheless, 2Ci can be compared with the value of the phantom modulus, which is (v, — pj RT [see Eq. (4)], and also with (v — p) RT. The variation of the ratio 2Ci/(v, — p ) RT with M is reported in Fig. 8 and 9 for trifunctional and tetrafunctional PDMS networks, resp Aively. Here M is different from because an active chain can be formed by two or more chains bound with difunctional junctions. The constant 2Ci is always hi er than the phantom modulus and the conclusions are similar to those readied in Section 4.2. 

Fig. 8. Dependence of the ratio between 2Ci and the phantom modulus on the number-average molecular wri t of primary chains for impofect trifunctional end-linked PDMS networks Mark >(0)...

We saw in the previous paragrajA that tl structure factor at zero deformation is less than unity for imperf t trifiiiKticHial nrtworks wten M < 10 gmol" . For these low molecular weights brtween cro links, the ntm-equilibrium effects are certainly avoided and it is then possible to apply the Flory theory. The value of the phantom modulus is given by the branching theory howev it is necessary to calculate... 

It is interesting to note that junction fluctuations increase in the direction of stretching but decrease in the direction perpendicular to it. Therefore the modulus decreases in the direction of stretching, but increases in the normal direction since the state of the network probed in this direction tends to be more nearly affine. The curve of [/" ] versus 1/a is sigmoidal. The parameters k and f of poly(dimethylsiloxane) networks are determined in Figure 17 (155) the intercept of the sigmoidal curves is the phantom modulus. This Flory-Erman theory has been compared successfully with such experiments in elongation and compression (155,162,162-166). It has not yet been extended to take account of limited chain extensibility or strain-induced crystallization (167). 

The modern theory of real networks now permits a more accurate determination of network structures through use of equations (84),(87), and (95) (187-204). Stress-strain measurements can he analyzed as shown in Figure 17. The phantom modulus thus determined leads to v and Me through equations (75) and (76) (189). Swelling equilibrium data are similarly analyzed through equation (94), with the parameter k given hy equation (87) (189). 

The deformation ratios t=l, 2, 3, characterizing the deformation of the swollen network relative to the unswollen, undeformed state are related to a by equations (92), (94) and (97). The ratio depends on the functionality through equation (53). In the limit of vanishing constraints, equations (122) apply. The reduced stress then l omes equal to the phantom modulus [/ ]ph-... 

In the case of perfect networks, combination of equations (47), (107), (112) and (113) yields equation (124). Thus, k depends on the inverse square-root of the phantom modulus and is independent of swelling. The factor is Avogadro s number, which appears in equation (124) since n in equation (112) is the number of junctions and not the number of moles of junctions. In the case of randomly cross-linked networks, use of equation (61) yields equation (125). In the case of networks formed by random cross-linking of star polymers, equation (63) is used instead of equation (61) to derive the expression for k. The other parameter C is the result of the relationship between k and network inhomogeneities and its magnitude is estimated by experiment. 

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