Aggregate model application

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]. 

THEORETICAL PRINCIPLES OF IRREVERSIBLE AGGREGATION MODELS APPLICATION... 

Application of Aggregation Models in Reducing the Binary Dimension... 

In the next few pages we shall discuss the question of local interfacial structures bounding idealised aggregates, tiled by blocks of fixed dimensions. The model represents one extreme idealisation of the molecular constituents that form the aggregate, most applicable to small surfactant molecules. At the other extreme, the block dimensions are not set a priori, they must be determined as a function of the temperature, concentration, etc. This case will be dealt with later. The welding of two concepts, a fluid-like mixture of hydrocarbons, with that of an idealised block is at first sight contradictory. However it can be shown to be consistent in a first order theory [2]. 

For practical modeling applications, this multiactor breakdown of the GES system must still be strongly aggregated, as it is... 

Thus, the disperse nanofiller particle aggregation in elastomeric matrix can be described theoretically wilhin the frameworks of a modified model of irreversible aggregation particle-cluster. The obligatory consideration of nanofiller initial particle size is a feature of the indicated model application to real system description. The indicated particles diffusion in polymer matrix obeys classical laws of Newtonian liquids hydrod5mamics. The offered approach allows to predict nanoparticle aggregate final parameters as a function of the initial particles size, their contents, and other factors. 

In a somewhat later study of mechanical anisotropy, Hennig and Kausch-Blecken von Schmeling have both independently considered the application and possible development of the aggregate model. Kausch reviewed the applicability of compliance and stiffness averaging predictions for several polymers. He noted that the compliance averaging predictions with the pseudo-affine deformation scheme were close to the experimentally observed behaviour for nylon 66, Dacron and regenerated... 

A major part of this study was to examine the applicability of the Ward rotating element aggregate model to low density polyethylene. As discussed in Chapter 8 satisfactory agreement was obtained when account was taken of the non-affine processes which occur in the early stages of drawing." In addition, however, these experiments formed the basis of... 

Let us make two remaiks in conclusion. As the estimations have shown, the constant coefficient in square brackets of the Eq. (58) does not always give the exact MM estimation and, probably, is adjustable coefficient Secondly, as Kucha-nov pointed out in annotations to paper [96], in polymerization real processes intramolecular reactions resulted to cyclic fragments formation, that makes aggregation models, similar to the considered above, application scarcely probable. However, Kolb [97] demonstrated, that loops (cycles) availability did not infln-ences on the value D. Therefore the present model is applicable to polymers, forming cyclic fragments, which aromatic polyformals are [92]. 

Hence, the stated above results have shown that DMDAACh molecular weight distribution can be simulated and predicted within the frameworks of an irreversible aggregation model cluster-cluster. The shape and position of MWD curve are controlled by a number of factors, such as macromolecular coil stmcture, stochastic contribution of a coil enviromnent to a polymerization intensity, and the level of destraction of a coil during its synthesis process. These factors can be linked by simple relationships to technological characteristics of the polymerization process, for example, c and that is essentially important for practical applications of the considered theoretical model [89] [23]. 

Hence, the stated above results have shown that the change of microgels stmcture, characterized by its fractal dimension, in the system EPS-4/DDM curing reaction course influences on both steric factor value and curing reaction conversion degree. The irreversible aggregation models and fractal analysis application... 

Fig. 7 Application of aggregate models [Eqs. (3) and (4)] to the data plotted using filled symbols in Fig. 6. (From Ref. 67 with permission.)...

Northolt and Sikkema [62] also noted that yielding is most pronounced in lower modulus LLCP fibers and that this phenomenon is less apparent in samples that have increased modulus. Furthermore, they quantified yield strain values of 0.7% for poly(p-phenylene benzobisthiazole) (PBTZ), 0.8% for poly(p-phenylene benzobisoxazole)(PBO), and 0.5% for poly(p-phenylene terephthalamide) fibers. The similarity of these levels with that of HBA/HNA TPCP suggests that these materials are governed by a common deformation process. This supports the application of the aggregate model to describe elastic extension of this less-crystalline TLCP. 

The aggregate model would not appear to be generally applicable to high-density polyethylene and polypropylene. It appears that for polypropylene the aggregate model is applicable only at low draw ratios [78], As discussed above, there are simultaneous changes in morphology and molecular mobility at higher draw ratios. 

Very often the elasticity modulus (or reinforcement degree) of polymer composites (nanocomposites) is described within the frameworks of numerous micromechanical models, which proceed from elasticity modulus of matrix polymer and filler (nanofiller) and the latter volume contents [10]. Additionally it is supposed, that the indicated above characteristics of a filler are approximately equal to the corresponding parameters of compact material, fi om which a filler is prepared. This practice is inapplicable absolutely in case of polymer nanocomposites with fine-grained nanofiller, since in this case a polymer is reinforced by nanofiller fractal aggregates, whose elasticity modulus and density differ essentially from compact material characteristics (see the Eq. (10.3)) [5, 9]. Therefore, the microcomposites models application, as a rule. 

The applicability of irreversible aggregation models for theoretical desc ription of particulate nanofiller particles aggregation processes in polymer nanocomposites has been shown. Analysis within the framewoik of the indicated models allows to reveal either factors influence on aggregation degree. 

Application of fractal analysis and irreversible aggregation models for the description of crosslinked polymer curing processes allows it to be elucidated that macromolecular coil (microgel) structure, characterised by its fractal dimension, plays a larger role than purely chemical aspects. Such an approach allows a quantitative description of both curing process kinetics and its final results to be received. 

The fractals theory and its application to various physical and chemical processes have recently undergone a large amount of development [1-7]. For simplification of understanding of the results represented in subsequent chapters some main notions and definitions are briefly considered and reasons for the application of fractal analysis (and connected with it irreversible aggregation models) for description of the structure and properties of polymer materials and composites on this basis are shown. 

The irreversible aggregation models, which were prepared for such practically important process descriptions as flocculation, coagulation, polymerisation and so on, have recently been widespread in physics [1-6]. Several examples of successful application of these models for the description of a number of real processes were obtained [1-11 ]. Therefore the use of these models for the description of polymerisation processes, in particular curing processes of crosslinked polymers, is of undoubted interest. It must be noted that application of percolation and some other models for this goal did not give expected results [12]. 

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