Prediction of the gel point

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. 

X. Prediction of the Gel Point XL Morphology of Cross-Linked Polymers Xll. Problems References... 

Prediction of the gel point for XAf + YBgC is also presented in Table 3.3. 

Before dealing with theoretical predictions of the gel-point, some important network-forming step polymerizations will be described. 

The discrepancy between the experimental observation and these predictions led several authors to formulate a better prediction of the gel point, by using a reasoning similar to that previously described for (X- -X +Y- -Y) bivalent systems. Let us consider the particular case of a system constituted of bi- and trivalent monomers,... 

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. 

Fonnulas of this type are used especially for practical purposes since they make it possible to predict, with a sufficient accuracy, the viscosity of reactive systems [35-39]. A limitation of such formulas is that formally they do not make allowance for the existence of the gel point for which -> oo according to Eq. (1) at any finite time t, the viscosity is also finite. This has formal character since we can take for a gelpoint a certain level of value of the viscosity, for example, t] = 10 or lO" Pa and then t is the time for which this level is achieved. 

The Isaacson-Lubensky prediction [32] was confirmed by means of the enumeration of bond animals by Gaunt [33]. The shift of the excluded volume molecule to the ideal molecule is really a phase transition and not an asymptotic phenomenon [26,30-35]. This aspect is again encountered in Sect. 6 and 7 in the context of the estimation of the gel point. 

These pioneering simulation studies clearly demonstrate the presence of the percolation threshold for water under ordinary conditions. Although some polygonal closures can exist, the critical percolation threshold is apparently well predicted by Flory s theory of the gel point. 

Effective use of a thermosetting system requires prediction of the cure kinetics of the system [17] to consistently obtain the maximum and also to predict the flow behaviour of the curing resin, in particular to precisely locate when the sol-gel transition occurs. This is because the polymer can be easily shaped or processed only before the gel point, where it can still flow and can be easily formed with stresses applied relaxed to zero thereafter. Accurate knowledge of the gel point would therefore allow estimation of the optimum temperature and time for which the sample should be heated before being allowed to set the mould. The gel point can also be used to determine the activation energy for the cure reaction of the system [18]. 

We have discussed the experimental analysis of viscoelastic behavior accompanying the sol-gel transition. It has been widely known that the temperature dependence of relaxation time can be expressed by the WLF equation when the glass transition phenomenon is treated as a relaxation phenomenon. Similarly, in the sol-gel transition, it will be possible to predict the wide range of viscoelastic behavior if the temperature dependence of relaxation is known. In addition, viscoelastic behavior in the vicinity of the gel point will continue to be an interesting problem. 

One important type of reaction is the step reaction polymerization, which belongs to the class of gelation models where small higher functional units are added to themselves to form clusters. An early approach for equivalent numbers of different reactive groups predicts for the gel point... 

We repeat that the position of the gel point is not a universal quantity. Therefore, the phase diagrams shown in Figs. 5-8 should not be regarded as quantitative predictions from which one can judge the validity of the classical or percolation theory. It is the exponents defined at or near these phase transition lines which are universal and which allow a clear distinction and classification of competing theories. However, a complete and correct theory must predict both the correct exponents and the correct phase diagram. 

The weight fraction of gel calculated for this case from Eq. (45) is plotted against a in Fig. 70. It will be observed that the formation of the infinite network is indicated to commence suddenly at the critical point. This prediction of the theory is abundantly confirmed by the characteristic abruptness with which gel appears in polyfunctional condensations. Also shown in Fig. 70 are the weight fractions Wx of various species as functions of a, calculated according to Eq. (36). 

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