Aggregation models, irreversible

TABLE 6.1 The parameters of irreversible aggregation model of nanofiUer particles aggregates growth... 

The adduced in Figs. 55 and 56 plots allow one to proceed to F-2 polycondensation mechanism analysis within the framework of irreversible aggregation models [ 129-131]. First of all it should be noted, that val-... 

Hence, the performed above analysis has shown that different solvents using in low-temperature nonequilibriiun polycondensation process can result not only in symthesized polymer quantitative characteristics change, but also in reaction mechanism and polymer chain structure change. This effect is comparable with the observed one at the same polymer receiving by methods of equilibrium and nonequilibrium polycondensation. Let us note, that the fractal analysis and irreversible aggregation models allow in principle to predict symthesized polymer properties as a function of a solvent, used in synthesis process. The stated above results confirm Al-exandrowicz s conclusion [134] about the fact that kinetics of branched polymers formation effects on their topological structures distribution and macromolecules mean shape. 

The obtained results allow one to perform the system EPS-l/DDM curing kinetic analysis within the framework of irreversible aggregation models [160]. In general case the relationship between and R can be written as follows [160] ... 

By its molecular weight PDMDAAC with A4M 4xlO is closer to oligomer than to polymer. Within the framework of irreversible aggregation model [160] it has been shown that at small N small rodlike... 

It is also obvious that the linear seale a introduction is important especially at different pol5nners flocculating ability comparison. Nevertheless, the authors [179,184] do not try to attain complete quantitative correspondence of theory and experiment, but want to demonstrate the principal possibility of such complex phenomenon as flocculation description within the framework of fractal analysis and irreversible aggregation models. 

Let us consider the parameter estimation within the framework of Witten-Sander irreversible aggregation model [ 194]. As it has been shown in Refs. [195, 196], the amorphous glassy polymers stmcture can be simulated as totality of Witten-Sander clusters (WS clusters) large munber, having radius R., which is determined as follows [197] ... 

In Fig. 98 the dependences In (S n) on Sf for fom PDMDAAC polymers, synthesized at different c, are shown. As it follows from the plots of Fig. 98, all four MWD curves are described by the sole curve. This is the most important result, confirming correctness of irreversible aggregation models usage for the description of pol5mierization process [222, 223]. 

Kozlov, G. V. Temitaev, K. B. Ovchatenko, E. N. Lipatov, Yu.S. The description of low-temperature polycondensation process within the framework of irreversible aggregation models. Reports of National Academy of Sciences of Ukraine, 1999, 12, 136-140. 

Kozlov, G. V. Yanovskii Yu.G. Kubica, S. Zaikov, G. E. A nanofiller particles aggregation in elastomeric nanocomposites the irreversible aggregation model. Przetworstwo Tworzyw, 2011, 5,413-416. 

Kozlov, G. V Malkanduev Yu.A. Zaikov, G. E. Molecular weight distribution of poly(dimethyl diallyl ammonium chloride) analysis within the framewoik of irreversible aggregation model. In book Fractal Analysis of Polymers From Synthesis to Composites. Ed. Kozlov, G. Zaikov, G. Novikov, V. New York, Nova Science Publishers, Inc. 2003, 131-139. 

Hence, the results stated above have shown that the structural analysis of polyesterification reaction in melt, using fiiactal analysis and irreversible aggregation models notions, allows to give precise enough description of this reaction even witliout applying purely chemical aspects. Let us note that fractal analysis is a more strict mathematical calculus than often used for synthesis kinetics description empirical equations. 

One of these situations shows the absence of frmdamental relationships structure properties for polymer solutions and melts. In the present monograph, this deficiency is removed by using modem physical conceptions—fractal analysis and irreversible aggregation models. 

THE DESCRIPTION OE POLYCONDENSATION KINETICS WITHIN THE FRAMEWORKS OF IRREVERSIBLE AGGREGATION MODELS... 

Let us note in conclusion, that polycondensation processes treatment within the frameworks of irreversible aggregation models does not exlude conclusions, made on the purely chemical grounds. Let us explain this on one example. In paper [51] it has been shown, that Ph-2 destraction is observed in N,N-dimeth-ylformamide. Within the frameworks of irreversible aggregation models this means, that two processes are realized in parallel aggregation and destmction. As it has been shown in work [103], such combination results to the value Df of final aggregates increasing. Actually, D, value for Ph-2 macromolecular coil in N,N-dimethylformamide is equal to 2.03, that corresponds well to D, theoretical value for such aggregation type [103]. 

For Euclidean object Dj.= d and pfr = const = pEnc. For fractal object increasing (or MM and N) in polycondensation process at a=const results to pfr reduction in virtne of the condition Dj,< d [52]. Therefore there are critical values MMc(Nc) and tc, above which synthesis reaction ceases [118, 119, 121]. This process is simulated within the frameworks of irreversible aggregation models according to the mechanism cluster-cluster. 

And at last, the third factor is defined completely by the macromolecular coil fractal properties. Earlier synthesized macromolecnle Ph-2 stmcture change because of reagents relation variation has not been supposed [48, 127], Nevertheless, irreversible aggregation models predict such possibility. It is obvious, that at deviation from equimolar relation reagent surplus cannot find partner for reaction realization, that will be resulted to p reduction and enhancement [123], This effect quantitative estimation can be obtained from the Eqs. (74) and (77) combination at the condition, that p reduces proportionally to any reagent surplus. The similar estimations have shown increasing from 1.55 at equimolar relation (CATA)/(PP) up to 1.70 at (CATA)/(PP) = 2 1. 

Hence, the stated above results have shown the principal possibihty of fcactal analysis and irreversible aggregation models for the description of reactive mass stirring intensity effects in interfacial nonequilibrium polycondensation process. It is obviorrs, that from the practical point of view the theoretically correct choice of coefficients in the Eqs. (26) and (28) for the substitution in them proportionahty sign on equahty sign is the most complex question [132],... 

Hence, the stated above results have shown, that fractal analysis and irreversible aggregation models application allows to obtain the clear physical interpretation of copolycondensation process and estimate its quantitative characteristics. The fractal dimension D. of macromolecular coil in solution is the main characteristic, controlling this process [142]. 

Kozlov, G. V Burya, A. I. Temiraev, K. B. Mikitaev, A. K. Chigvintseva, O. P. The description of low-temperature polycondensation kinetics within the frameworks of irreversible aggregation models and fractal analysis. Problems of Chemistry and Chemical Technology, 1998, (3), 26-29. 

At the synthesis beginning solution contains only monomer, which within the frameworks of irreversible aggregation models can be considered as particles, uniting later in a cluster (macromolecular coil). As it is known [21], within the frameworks of the indicated models such mechanism is called mechanism particle-cluster and aggregates with fractal dimension 2.5 is the result of its action. Besides, the value Cj was calculated according to the Eq. (18) with the following parameters using t = 0.5 min, i3r=4.8 mol/l s and Q=8.3><10-3. 

Let us note one more important aspect. As the experimental studies [1] have shown, the curves reaching of the saturation section occurs earlier, than the kinetic curves Tliis observation is also explained within the frameworks of irreversible aggregation models. As it is known [24], the critical (the greatest) macromolecular coil gyration radius is determined according to the Eq. (48) of... 

The second regime is characterized by approximately the same aggregate and reactive medium densities and for it scaling is described by the Eq. (31). The relation between N and Q within the frameworks of irreversible aggregation models is described as follows. The temporal evolution of Rg in the model DLA cluster-cluster can be described by the Eq. (70) of Chapter 1. For the considered case all parameters excepting c, t and which in the Eq. (70) of Chapter 1 right-hand part can be accepted as constant ones and then [44] ... 

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