Function average

An example of a time averaging function -4(a ) is the formula termed LongAverage in [7] ... 

Appendix B Calculation of the Jacobian Matrix of an Averaging Function... 

Determine the average functionality of the mixture. The total number of functional groups is 6.00 mol, but the total number of molecules initially present... 

The most commonly used isocyanate is a modified form of MDI. Such polymeric forms may be prepared, for example, by reacting phosgene with formaldehyde-aniline condensates which have average functionalities of between 2 and 7 and may be represented by the formula given in Figure 27.10. 

From the point of view of principles, it is interesting to note that the method based on the generalized form of Eq. III. 129 seems to be very closely connected both with Wigner s classical theory described in Section III.B and with Bohm and Pines plasma model (Krisement 1957). Following Krisement, we will replace the various trial functions flt /2,. . ., fn in Eq. III.9 by a single average function /, and Wigner s basic wave function (Eq. II1.7) takes then the simple form... 

The Carothers equation relates the number-average degree of polymerization to the extent of reaction and average functionality of a step-growth polymer. In the Carothers equation, the number-average degree of polymerization, X , relates to the extent of reaction, p, and average functionality, /avg, of the polymer system ... 

The molecular weight of a polymer will be reduced if either die extent of conversion or the average functionality is decreased. At 95% conversion of difunctional monomers, for example, Xn is only 20.25 The molecular weight is also related to a stoichiometric imbalance, r, which is normally defined to be less than 1.0 ... 

As mentioned previously, the use of multifunctional monomers results in branching. The introduction of branching and the formation of networks are typically accomplished using trifunctional monomers, and the average functionality of the polymerization process will exceed 2.0. As the average functionality increases, the extent of conversion for network formation decreases. In... 

Carbodiimide functionality can be produced by reacting isocyanates at elevated temperature with proper catalysis (Scheme 4.15). Although carbodiimides undergo a variety of reactions,23 most commonly as dehydrating agents, in the presence of excess isocyanate they will form uretone imines. This not only increases the average functionality of the isocyanate product but also lowers its freezing point. For example, a liquefied (or modified) version of 4,4,-MDI can... 

The constraints due to filler particles are somewhat more complicated to mimic analytically. A crude model is to assume solid spheres of an average functionality fp (—4) [15]. If... 

In POLYM the output data of KINREL are used with compositional information to calculate the number and mass average molecular masses (Rn and Rm, respectively) and number and end-group average functionalities (fp and fg> respectively) in the pre-gel region in all stages. In addition, the network characteristics such as sol fraction, mj, and the number of elastically active network chains per monomer (5), Ng, are calculated in the post-gel regime of stage 3. 

In all calculations the molar masses given in the top of Table I were used. First of all, the effects of variations in the concentration of trifunctional monomers were determined, as exemplified by the nine formulations of Table I and the resulting prepolymer characteristics after full conversion given in Table II. Formulations FIO to F40 result in branched prepolymers, which are cured in the third stage by difunctional monomers. On the other hand, formulations FOO to F04 result in the same linear prepolymer, which is subsequently cured with various mixtures of di- and trifunctional monomers. The number average functionalities of PI (and P2) and of the mixtures of E and F monomers are varied systematically between 2.0 and 2.4. Therefore, the only difference between formulations FjO and FOj is the stage in which the branching units are added. 

The averaged function e(y/dc) can be well represented by a relationship established by Martin [7] consisting of a parabolic and an attenuated cosine part. The parameters for Eqs. (3.2.14) and (3.2.15) given by Martin were newly determined based on MRI results of the type shown in Figure 3.2.9. The differing curves shown in Figure 3.2.9 can be taken as proof of the difficulty of measuring e with conventional means. Data... 

One system studied by the authors(3) used LG56 triol which had been characterised by molar-mass as well as end-group determinations. The data for the samples used are given in Table III, indicating a true number-average functionality (fn) of between 2.95 and 2.99. The triol was reacted with HDI in bulk (see Figure 1) and at various dilutions in benzene, and values of Nr and ac determined. To estimate functionality from ac, equations... 

Polyurethane networks were prepared from polyoxypropylene (POP) triols(Union Carbide Niax Polyols) after removal of water by azeotropic distillation with benzene. For Niax LHT 240, the number-average molecular weight determined by VPO was 710 and the number-average functionality fn, calculated from Mjj and the content of OH groupSj determined by using excess phenyl isocyanate and titration of unreacted phenyl isocyanate with dibutylamine, was 2.78 the content of residual water was 0.02 wt.-%. For the Niax LG-56, 1 =2630, fn=2.78, and the content of H2O was 0.02wt.-%. The triols were reacted with recrystallized 4,4"-diphenylmethane diisocyanate in the presence of 0.002 wt.-% dibutyltin dilaurate under exclusion of moisture at 80 C for 7 days. The molar ratio r0H = [OH]/ [NCO] varied between 1.0 and 1.8. For dry samples, the stress-strain dependences were measured at 60 C in nitrogen atmosphere. The relaxation was sufficiently fast and no extrapolation to infinite time was necessary. 

Polyurethane Networks. Again, for Tqjj. 1 the sol fraction fits well the theoretical curves, if the measured values of the average functionality fn of the polyoxypropylene (POP) triol, the final conversion of isocyanate groups, and the formation °f... 

After prolonged degassing of a large batch of PPG, analyses showed that its hydroxyl, unsaturation, and water contents were 0.97 meq/g, 0.033 meq/g, and 0.0035%, respectively. The hydroxyl content was determined by an acetylation method, carried out with acetic anhydride (10). The amounts of unsaturation and water were determined by the mecuric acetate and Karl Fischer methods (10), respectively. The obtained analytical results indicate that the number-average molecular weight of the dihydroxy material is 2062, provided its molecular weight is arbitrarily assumed to be twice that of the monohydroxy material, and that the mole fraction of the monohydroxy poly (propylene oxide) is 0.066. This value corresponds to a number-average functionality of 1.93 for the PPG. 

Triols. One triol used was the propylene oxide adduct of 1,2,6-hexanetriol, designated LHT-240 and supplied by Union Carbide Chemicals Company. After degassing, its hydroxyl, unsaturation, and water contents were 4.31 meq/g, <0.01 meq/g, and 0.004%, respectively. Thus, the material was essentially trifunctional, and its number average molecular weight was 696. If the unsaturation was exactly 0.01 meq/g, then the number-average functionality was 2.99. 

The average function E = f ipHipdr/ f ip2 dr. All the integrals in this expression contain the standard form... 

For the application of hyperbranched polymers as precursors in network formation, the functionality averages are important. The number-average functionality is given by the equation... 

This is also the observation in experiments with salt addition to enzyme systems. The data points do not always form strictly linear curves, but it seems fair enough in most cases to describe the averaged functional form by... 

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