Front-factor

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. 

The front factor of x can be treated as the spring constant of mbbery elasticity, which obeys Hooke s law. 

Only two of the exponents (a and n, for instance) are sufficient to describe the rheology of nearly critical gels. The front factor is more difficult to estimate, but it most likely differs on both sides. 

Time-temperature superposition [10] increases the accessible frequency window of the linear viscoelastic experiments. It applies to stable material states where the extent of reaction is fixed ( stopped samples ). Winter and Chambon [6] and Izuka et al. [121] showed that the relaxation exponent n is independent of temperature and that the front factor (gel stiffness) shifts with temperature... 

The dimensionless relaxation exponent n is allowed to take the values between 0 and 1. The front factor H0, with the dimension Pa and the characteristic time X0, depends on the specific choice of material. Various values have been assigned in the literature. The spectrum has only two independent parameters, since several constants are lumped into (H0Xo"). For certain materials (the special case of LST), the upper limit of the power law spectrum may diverge to infinity, Xu -> oo, without becoming inconsistent [18]. 

Comparison of the expressions for the elastic free energies for the affine and phantom network models shows that they differ only in the front factor. Expressions for the elastic free energy of more realistic models than the affine and phantom network models are given in the following section. 

For imperfect epoxy-amine or polyoxypropylene-urethane networks (Mc=103-10 ), the front factor, A, in the rubber elasticity theories was always higher than the phantom value which may be due to a contribution by trapped entanglements. The crosslinking density of the networks was controlled by excess amine or hydroxyl groups, respectively, or by addition of monoepoxide. The reduced equilibrium moduli (equal to the concentration of elastically active network chains) of epoxy networks were the same in dry and swollen states and fitted equally well the theory with chemical contribution and A 1 or the phantom network value of A and a trapped entanglement contribution due to the similar shape of both contributions. For polyurethane networks from polyoxypro-pylene triol (M=2700), A 2 if only the chemical contribution was considered which could be explained by a trapped entanglement contribution. 

According to the theory (1,2), for phantom imperfect (cf. also (17)) network the front factor A assumes the value... 

Figure 2. Theoretical curves for superimposed reduced moduli Gd /RT(mol/ cm 3) of epoxy-amine networks versus the gel fraction ws. Fraction of epoxy groups in monoepoxide is O, 0 , 0.2 A, 0.33 V, 0.5. Key O, , A, V, dry networks , A, , swollen networks value of front factor A indicated.

Ronca and Allegra (12) and Flory ( 1, 2) assume explicitly in their new rubber elasticity theory that trapped entanglements make no contribution to the equilibrium elastic modulus. It is proposed that chain entangling merely serves to suppress junction fluctuations at small deformations, thereby making the network deform affinely at small deformations. This means that the limiting value of the front factor is one for complete suppression of junction fluctuations. 

These expressions are those for a viscoelastic solid and assume the front factor is unity. The relaxation times, ip, are modified from simple Rouse values to allow for the motion of a strand rather than a complete chain. 

The first term having a front factor which scales with the square of contrast is related to the shape function (see Fig. 5) ... 

Figure 9. Reduced equilibrium modulus of polyurethane networks from POP trlols and MDI in dependence on the sol fraction. networks from POP triol Mjj - 708, o networks from POP triol Mjj = 2630. C-) calculated dependence using Flory junction fluctuation theory for the value of the front factor A indicated. (Reproduced from Ref. 57. Copyright 1982 American Chemical Society.)...

Figure 10. Dependence of the reduced equilibrium modulus of POP triol - MDI networks prepared in the presence of diluent. POP triol Mu= 708 stress-strain measurements in the presence of diluent (o) and after evaporation of the diluent ( ). Flory theory for the values of the front factor A indicated, theoretical dependence including trapped interchain constraints Numbers at curves Indicate the value of ry.

Figure 1, Ratio of molar mass between elastically effective junctions to front factor (M(-/A) relative to molar mass between junctions of the perfect network (M ) versus extent of intramolecular reaction at gelation (pj- (.) Polyurethane networks from hexamethylene diisocyanate (HDI) reacted with polyoxpropylene (POP) triols at 80°C in bulk and in nitrobenzene solution(5-7,12). Systems 1 and 2 HDI/POP triols >i= 33, V2= 61. Systems 3-6 ...

Malkin s autocatalytic model is an extension of the first-order reaction to account for the rapid rise in reaction rate with conversion. Equation 1.3 does not obey any mechanistic model because it was derived by an empirical approach of fitting the calorimetric data to the rate equation such that the deviations between the experimental data and the predicted data are minimized. The model, however, both gives a good fit to the experimental data and yields a single pre-exponential factor (also called the front factor [64]), k, activation energy, U, and autocatalytic term, b. The value of the front factor k allows a comparison of the efficiency of various initiators in the initial polymerization of caprolactam [62]. On the other hand, the value of the autocatalytic term, b, describes the intensity of the self-acceleration effect during chain growth [62]. 

The role of the isothermal and pseudo-first-order reaction assumptions on the observed value of activation energy was assessed to allow comparison of our data to previous work by modifying Malkin s autocatalytic equation so that the autocatalytic term b is equal to zero. The values of the activation energy and front factor were calculated using short-time, low-conversion data. By making the autocatalytic term equal to zero, the modified Malkin autocatalytic model becomes a first-order rate reaction. Table 1.2 shows that by assuming a... 

The terminal spectrum is furnished by cooperative motions which extend beyond slow points on chain in the equivalent system. The modulus associated with the terminal relaxations is vEkT, which is smaller by a factor of two than the value from a shifted Rouse spectrum. It is consistent with a front factor g = j given by some recent theories of rubber elasticity (Part 7). The terminal spectrum for E 1 has the Rouse spacings for all practical purposes, shifted along the time axis by an undetermined multiplying factor (essentially the slow point friction coefficient). Thus, the model does not predict the terminal spectrum narrowing which is observed experimentally. 

The front factor g as defined above5 is unity in all the earlier theories (17). Recently Duiser and Staverman (233) have obtained g = j and Imai and Gordon (259) g — 0.54 with Rouse model theories which make no a priori assumptions about the junction point locations after deformation. Edwards (260) also arrives at and Freed (261) deduces that g= 1 is an upper bound by similar approaches. The front factor usually assumed in the shifted relaxation theory of the plateau modulus is g = 1, although Chompff and Duiser (232) obtain g = j through their extension of the Duiser-Staverman result to entanglement networks. The physical reasons for the different values of g in different treatments are not clear at present. 

Note that this definition of front factor differs from that given earlier by Tobolsky (162), since the term /0 has been separated out. 

The Bueche-Mullins method has been applied in the separation of the modulus contributions of crosslinks and entanglements in several elastomers. A front factor of g = 1 was then used to determine Me. The Langley method has also been applied in a few cases, resulting in values of both g and Me. These works are summarized below results are collected in Table 7.2. 

Front factor in modulus equation from rubber elasticity theory (Part 7). Fraction of configurations of free chains which are consistent with specified end-to-end coordinates (Part 7). 

In the current statistical theory of rubber elasticity, it is suggested that the front-factor molecular forces. They have proposed a semiempirical equation of state taking into account the dependence... 

N is the number of statistical segments in a chain, 1 is their length and m the number of submolecules. This model includes the front factor only in C, but the parameter K0 is also temperature dependent it depends on the thermal expansion. Thus, the Priss tube model predicts different temperature dependences of C, and C.. It means that the entropy and energy contribution have to be dependent on X and include a considerable intermolecular part. 

In this connection, it is very interesting that the volume and intrachain changes obtained by various experimental methods 24,29,85) [Eq. (101)] agree well with Eq. (56) following from the Tobolsky-Shen semiempirical equation of state or the related phenomenological Eq. (76). The values of y determined from the data are rather small (0.1-0.3). As has been mentioned above, according to the semiempirical approach by Tobolsky and Shen one can formally suggest that the front-factor in Eq. (28) is pressure dependent. If it is really so, then the parameter y for rubbers can be considered as an experimental coefficient similar to the coefficient of thermal expansion and compressibility 29). 

On the other hand, Kilian 50) having analysed the strain-induced volume dilation 24 91 using the van der Waals equation of state (Fig. 6) emphasized that only pressure dependence of the interchain parameter, a, is required for a full explanation of the relative volume changes. He arrived at a conclusion that non-crystalline rubbers are anisotropic equilibrium liquids and a higher compressibility of NR was only necessary for fitting the extension data. Hence, on using the van der Waals approach, there is no need of postulating volume dependence of the front factor as proposed by Tobolsky and Shen. 

The front factor in Eq. (III-ll), introduced for the first time by Tobolsky (768,169) has subsequently been inserted by all other authors (57, 76). 

At smaller strains this result reduces to the Gaussian Eq. (HI-2), provided we remember that in the derivation of Eq. (IV-9), It was assumed that in the undeformed network the network chains are indeed in the unstrained reference state, so that the front factor (y2 l(r2 is lacking. 

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