Gel-point predictions

Table 2.4. Gel-point predictions from percolation theory ...

Figure 4 shows the relationship between prepolymer content and viscosity for a xylene solution of BCB-2 prepolymer slightly below the gel point. Predictable coating thicknesses useful for multilayer structures were obtained from solutions with viscosities in the range of 10 cP to 150 cP. 

For the gel point prediction which is ap>phcable to the experimental data, it is necessary to introduce solvent molecules to the lattice model shown above. The introduction of solvent is the process to lay both monomer and solvent molecules randomly to lattice dots (that is, sites), which is known as the site percolation. In the lattice model, the site percolation assumes that, whenever a monomer is laid in the neighbour, the mutual bond is necessarily produced between neighbours. The ptrobabiUty of the site occupetion by the monomer is the dominant factor of the gelation In the experimental system, the probability of occupation correspends with the monomer concentration The cluster of infinite size appears above the threshold of the probability of occupjation... 

Furthermore, the molecular scheme for the gel point prediction and viscoelasticity calculation in the course of the network formation were described in Section 3 and 4, respectively. Although some simpler models are in demand, the frameworks currently used are too complicated to use conventionally. However, the effect of unequal reactivity on the delay of gel point could be derived by drawing the detailed molecular scheme. Conversely, it is necessary to set the model up to details to meet with the realistic experimental data. Such molecular parameters allows us to prepare materials near the gel point with a wide range of properties for applications, like adhesives, absorbents, vibration dampers, sealants, membranes etc. With suitable design, it will be possible to control network structures, relaxation character, and then mechanical properties to the requirements. 

Statistical theories which predict the gel point as a function of cross-linking have been developed. While the theoretical analysis of gelation caused by the... 

The classical theory predicts values for the dynamic exponents of s = 0 and z = 3. Since s = 0, the viscosity diverges at most logarithmically at the gel point. Using Eq. 1-14, a relaxation exponent of n = 1 can be attributed to classical theory [34], Dynamic scaling based on percolation theory [34,40] does not yield unique results for the dynamic exponents as it does for the static exponents. Several models can be found that result in different values for n, s and z. These models use either Rouse and Zimm limits of hydrodynamic interactions or Electrical Network analogies. The following values were reported [34,39] (Rouse, no hydrodynamic interactions) n = 0.66, s = 1.35, and z = 2.7, (Zimm, hydrodynamic interactions accounted for) n = 1, s = 0, and z = 2.7, and (Electrical Network) n = 0.71, s = 0.75 and z = 1.94. 

De Gennes [41] predicted that percolation theory should hold for crosslink-ing of small molecule precursors. However, he argued that for vulcanizing polymers (high Mw), only a very narrow regime near the gel point exists for which percolation is valid, i.e. these polymers should exhibit more mean fieldlike behavior. 

Predictions using the observed relaxation time spectrum at the gel point are consistent with further experimental observations. Such predictions require a constitutive equation, which now is available. Insertion of the CW spectrum, Eq. 1-5, into the equation for the stress, Eq. 3-1, results in the linear viscoelastic constitutive equation of critical gels, called the critical gel equation ... 

Fig. 13. Shear stress t12 and first normal stress difference N1 during start-up of shear flow at constant rate, y0 = 0.5 s 1, for PDMS near the gel point [71]. The broken line with a slope of one is predicted by the gel equation for finite strain. The critical strain for network rupture is reached at the point at which the shear stress attains its maximum value...

During our early experiments on chemical gels, when first observing the intermediate state with the self-similar spectrum, Eq. 1-5, we simply called it viscoelastic transition . Then, numerous solvent extraction and swelling experiments on crosslinking samples showed that the viscoelastic transition marks the transition from a completely soluble state to an insoluble state. The sol-gel transition and the viscoelastic transition were found to be indistinguishable within the detection limit of our experiments. The most simple explanation for this observation was that both phenomena coincide, and that Eqs. 1-1 and 1-5 are indeed expressions of the LST. Modeling calculations of Winter and Cham-bon [6] also showed that Eq. 1-1 predicts an infinite viscosity (see Sect. 4) and a zero equilibrium modulus. This is consistent with what one would expect for a material at the gel point. 

The factors which influence pre-gel intramolecular reaction in random polymerisations are shown to influence strongly the moduli of the networks formed at complete reaction. For the polyurethane and polyester networks studied, the moduli are always lower than those expected for no pre-gel intramolecular reaction, indicating the importance of such reaction in determining the number of elastically ineffective loops in the networks. In the limit of the ideal gel point, perfect networks are predicted to be formed. Perfect networks are not realised with bulk reaction systems. At a given extent of pre-gel intramolecular... 

The power-law variation of the dynamic moduli at the gel point has led to theories suggesting that the cross-linking clusters at the gel point are self-similar or fractal in nature (22). Percolation models have predicted that at the percolation threshold, where a cluster expands through the whole sample (i.e. gel point), this infinite cluster is self-similar (22). The cluster is characterized by a fractal dimension, df, which relates the molecular weight of the polymer to its spatial size R, such that... 

The two approaches to the problem of predicting the extent of reaction at the onset of gelation differ appreciably in their predictions of pc for the same system of reactants. The Carothers equation predicts the extent of reaction at which the number-average degree of polymerization becomes infinite. This must obviously yield a value of pc that is too large because polymer molecules larger than Xn are present and will reach the gel point earlier than those of size Xn. The statistical treatment theoretically overcomes this error, since it predicts the extent of reaction at which the polymer size distribution curve first extends into the region of infinite size. 

The gel point is usually determined experimentally as that point in the reaction at which the reacting mixture loses fluidity as indicated by the failure of bubbles to rise in it. Experimental observations of the gel point in a number of systems have confirmed the general utility of the Carothers and statistical approaches. Thus in the reactions of glycerol (a triol) with equivalent amounts of several diacids, the gel point was observed at an extent of reaction of 0.765 [Kienle and Petke, 1940, 1941], The predicted values of pc, are 0.709 and 0.833 from Eqs. 148 (statistical) and 2-139 (Carothers), respectively. Flory [1941] studied several systems composed of diethylene glycol (/ = 2), 1,2,3-propanetricarboxylic acid (/ = 3), and either succinic or adipic acid (/ = 2) with both stoichiometric and nonstoichiometric amounts of hydroxyl and carboxyl groups. Some of the experimentally observed pc values are shown in Table 2-9 along with the corresponding theoretical values calculated by both the Carothers and statistical equations. 

Percolation simulations on a computer on the other hand demonstrate nicely the formation of a well defined surface43"45. The prediction of the gel point still remains poor probably the extent of ring formation is overestimated in the presently used simplest models for percolation. 

Scaling predictions for the steady-state viscoelastic properties have also been developed. The growth of the equilibrium modulus after the gel point can be described as a function of x by... 

But it is difficult to apply zero shear predictions to measurements that have been performed at low, but nonzero, shear rates. Neither s nor u can be decisively determined at a fixed frequency near the gel point the only way to truly obtain s and u exponents from experiments is through the use of a theory capable of predicting the entire frequency dependence of the viscoelastic response. 

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