Polymer Fractal

Polymer-polymer fractal interfaces may result from the interdiffusion of monomers or of polymers themselves. Koizumi et al. [31] annealed the interface between polystyrene and a styrene-isoprene diblock polymer at 150 C and showed extensive roughening of the interface by mutual interdiffusion on a micron scale (Fig. 8). 

Data on the fractal forms of macromolecules, the existence of which is predetermined by thermodynamic nonequilibrium and by the presence of deterministic order, are considered. The limitations of the concept of polymer fractal (macromolecular coil), of the Vilgis concept and of the possibility of modelling in terms of the percolation theory and diffusion-limited irreversible aggregation are discussed. It is noted that not only macromolecular coils but also the segments of macromolecules between topological fixing points (crosslinks, entanglements) are stochastic fractals this is confirmed by the model of structure formation in a network polymer. 

Cates [56] introduced the notion polymer fractal by replacing the rigid bonds in a percolation cluster by flexible (phantom) links. This model can be used to describe the gelation process (Figure 11.2) [57]. Within the framework of this approach, Cates [56] described variations of the structure and the dynamics of dilute solutions of polymers. 

Cates showed [56] that relationship (11.9) is also valid for structures containing loops. In the case of an ideal polymer fractal with Gaussian chains and the connectivity parameter (t/j), the fractal dimension df) of a percolation cluster with phantom bonds can be expressed as follows ... 

For t/j = 1 (linear chains). Equation (11.9a) provides the correct value, d = 2, corresponding to a macromolecular coil at the 0-point (see Table 11.2). As noted previously, d = 4/3 for a percolation cluster, irrespective of the dimension of the Euclidean space (see Table 11.1) therefore, from Equation (11.9a), we obtain df= 4, which is consistent with the Flory-Stockmayer theory [60] for phantom chains. For three-dimensional space, d > 3 has no physical meaning because the object cannot be packed more densely than an object having a Euclidean dimension. It is evident that this discrepancy is due to the phantom nature of the polymer chains postulated by Cates [56] it is therefore, necessary to take into account self-interactions of chains due to which the dimension of a polymer fractal assumes a value that has a physical meaning. 

In Equation (11.9a), df has the same meaning as the fractal dimension of phantom Gaussian chains therefore, the standard mean-field theory, taking account of the factor of excluded volume, can be applied to a polymer fractal [57]. 

In the case of percolation clusters. Equation (11.10a) gives D = 2 lor d = 4/3 and d=3 [57], while for linear polymers, D = 5/3 for d = 1 and d = 3 [61], which corresponds to an impermeable coil in a good solvent (see Table 11.2, state 3). The resulting dimensions are typical of macromolecular coils in monomeric solvents. Using the concept of polymer fractals, one can answer the question of what would happen if the monomeric solvent is replaced by a polymeric one, i.e., whether the polymer clusters and the clusters of a high-molecular-mass solvent would be separated from one another or entangled . This question can be answered by the equation [61] ... 

The final relationships for the dimensions of systems involving mixtures of polymer fractals (fractals in low- and high-molecular-mass solvents, melts of identical and arbitrary fractals) are given in Table 11.3 [61]. 

Table 11.3 Final relations for the dimensions of systems comprising mixtures of polymer fractals [61] ...

Some possible approximations have been considered by Cates [56], who concentrated attention on macromolecular entanglements, which play an important role in the description of the behaviour of block polymers [86-89]. Cates believes that the fact that the concept of polymer fractal neglects the effects of macromolecular entanglements is the main drawback of this theory. Nevertheless, Cates [56] introduced several simplifications that make it possible to ignore these effects for dilute solutions and relatively low molecular masses. However, in the opinion of Cates, even in the case of predominant influence of entanglements, theoretical interpretation of this phenomenon is impossible without preliminary investigation of the properties of the system in terms of Rouse-Zimm dynamics, which can serve as the basis for a more complex theory. It was assumed [56] that the effects of entanglement can be due to the substantially enhanced local friction of macromolecules. 

Now we attempt to estimate D within the framework of the theoretical views on polymer fractals [22, 36, 56, 61-64]. It is assumed [68] that the fractal dimensions d, d, and D are related to one another via an expression similar to Equation (11.8). The relationship between the parameters d and D has been considered [6]. It is believed that d is a dynamic value which responds to a change in the conditions of the interaction of a macromolecule with its surrounding and D is a static parameter. However, in our opinion, the opposite situation occurs in reality this is indicated by the following facts, some of which have been noted previously. It has been shown experimentally [72] that a two-fold increase in the crosslinking density does not change d, while, according to the plot shown in Figure 11.12b, has an appreciable influence on D. Moreover, the monomer and the crosslinked epoxy polymer have nearly identical d values (see Table 11.4). Helman... 

Where the morphology of a network polymer corresponds to the second variant, its structure can be represented as a mixture of arbitrary polymer fractals in the vicinity of the gelation point the spectral dimension of the globule is and that of the inter-globule area is 83. The 83 value can be estimated using the relationship [61] ... 

Fractals are opaque (nonleaking) one for another if increases at growth, that is, at u+ fi>d and transparent (leaking), if an intersections number decreases at enhancement [39]. In other words, for the case d=3 and two fractals with the same dimension ZT the Eq. (13) predicts transparent polymers fractal at Z) <1.5, that corresponds completely to the data of Ref [12],... 

For the case of dimensionally connected polymer fractals (good solvent) in a cylindrical pore the most important parameter is the minimum pore diameter through which a branched polymer still pass. This size is found from the conditions that for parallel direction R N is given by the equation as well [57] ... 

Dolbin, I. V Kozlov, G. V. Zaikov, G. E. The Structural Stabilization of POlymers Fractal Models. Moscow, Publishers Academy of Natural Sciences , 2007, 328 p. 

Metal-polymer fractal interfaces may result from processes such as vacuum deposition and chemical vapor deposition where metal atoms can diffuse considerable distances into the polymer. Mazur et al. [76,77] electrodeposited silver within a polyimide film. The Silver [I] solution was able to diffuse into the polymer film where it... 

Kozlov, G. V. Bejev, A. A. Autoacceleration (autostopping) in reactions of curing of cross-linked polymers fractal arralysis. In book Fractals and Local Order in Polymeric Materials. Ed. Kozlov, G. Zaikov, G. New York, Nova Science FTiblishers, Inc., 2001, 37-42. 

T. A. Vilgis, Polymer fractals and the unique treatment of polymers, J. Phys. France) 49, 1481 (1988). 

The development of new molecular closure schemes was guided by analysis of the nature of the failure of the MSA closure. In particular, the analytic predictions derived by Schweizer and Curro for the renormalized chi parameter and critical temperature of a binary symmetric blend of linear polymeric fractals of mass fractal dimension embedded in a spatial dimension D are especially revealing. The key aspect of the mass fractal model is the scaling relation or growth law between polymer size and degree of polymerization Ny cr. The non-mean-field scaling, or chi-parameter renormalization, was shown to be directly correlated with the average number of close contacts between a pair of polymer fractals in D space dimensions N /R if the polymer and/or... 

Kozlov, G. V., Ozden, S., Dolbin, I. V. (2002). Small Angle X-Ray Studies of the Amorphous Polymers Fractal Structure. Russian Polymer News, 7(2), 35-38. 

In the present section a number of modern physical concepts for the description of the structure of crosslinked polymers is used the thermodynamic concept, the cluster model of amorphous state structure of polymers, fractal analysis, irreversible aggregation models and the thermal cluster model. Within the frameworks of the thermodynamic approach the interconnection of structural and molecular characteristics of crosslinked polymers with disorder parameter 8 is considered [69]. According to the concept [69] the indicated parameter, connected with the thermal mobility of molecules near the melting temperature, is expressed by Formula 1.28. Since p. is given by Equation 1.29 then Relationship 1.30 can be received from combination of Equations 1.28 and 1.29. 

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