Euclidean space, dimension dependence

In the previous section we analyzed the random walk of molecules in Euclidean space and found that their mean square displacement is proportional to time, (2.5). Interestingly, this important finding is not true when diffusion is studied in fractals and disordered media. The difference arises from the fact that the nearest-neighbor sites visited by the walker are equivalent in spaces with integer dimensions but are not equivalent in fractals and disordered media. In these media the mean correlations between different steps (UjUk) are not equal to zero, in contrast to what happens in Euclidean space cf. derivation of (2.6). In reality, the anisotropic structure of fractals and disordered media makes the value of each of the correlations u-jui structurally and temporally dependent. In other words, the value of each pair u-ju-i-- depends on where the walker is at the successive times j and k, and the Brownian path on a fractal may be a fractal of a fractal [9]. Since the correlations u.juk do not average out, the final important result is (UjUk) / 0, which is the underlying cause of anomalous diffusion. In reality, the mean square displacement does not increase linearly with time in anomalous diffusion and (2.5) is no longer exact. 

The unlabeled triangle is the simplex in E (2-simplex) and the unlabeled tetrahedron is the simplex in (3-simplex) evidently, whether enantiomorphous -simplexes can be partitioned into homochirality classes depends on the dimension of E". Recall that an /j-simplex is a convex hull of + 1 points that do not lie in any (n - l)-dimensional subspace and that are linearly independent that is, whenever one of the points is fked, the n vectors that link it to the other n points form a basis for an n-dimensional Euclidean space An n-simplex may be visualized as an n-dimensional polytope (a geometrical figure in E" bounded by lines, planes, or hyperplanes) that has n + vertices, n n + )/2 edges, and is bounded by n + 1 (u — l)-dimensional subspaces. It has been shown that the homochirality problem for the simplex in E is shared by all -sim-... 

The spectral dimension d is a true property of a fractal and is determined only by its connectivity. It differs from the mass scaling index [see Equation (11.1)] or fractal dimension df and from the scaling index of the diffusion constant 8 by the fact that it does not depend on the way in which a fractal has been inserted into the Euclidean space with dimension d. The dependence of d and 8 on df is described by Equation (11.3). 

Equation (11.18) has two peculiar features firstly, D depends appreciably on d and, secondly, there exists a critical dimension of Euclidean space d = % for which D = 4 in accordance with the ideal statistical model, i.e., the model withont correlations. At d > 8, the correlations cansed by the effect of excluded volume are no longer significant and Df does not change. The value d = % was found in studies of branched polymers and lattice animals [79]. Calcnlations using formula (11.18) are in good agreement with the known results for lattice animals. ... 

It is considered that is determined by the properties of the supermolecular structure of epoxy polymers, namely, by the degree of local order in them. This assumption arises due to the well-known succession of relationships between the diffusion process and d [28], between the diffusion process and the fluctuation free volume [119], and between and the cluster structure [120]. Therefore, the fractal dimension of the cluster structure of crosslinked polymers [105, 110], estimated from Equation (11.27), is taken to be d. Since the D value is determined for an Euclidean space with d = 2, it was also assumed in Equation (11.27) that d = 2. The d value was found for a three-dimensional space therefore, a graph [121] for converting d from three- to two-dimensional space was used in subsequent calculations. The D values were further calculated from Equation (11.8) in terms of the AO and AS hypotheses. Figure 11.13 shows the dependences of D on calculated from Equations (11.8) and (11.39). The dependences of D on and the absolute magnitudes of D are in good agreement, especially when the AO hypothesis is used. 

Particles sizes were established using atomic power microscopy data (see Fig. 1.2). For each of the three nanocomposites studied not less than 200 particles were measured (the sizes of which were divided into 10 groups and the mean values of N and r were obtained). The dependences N(r) in double logarithmic coordinates were plotted, which proved to be linear and the values of were calculated according to their slope (see Fig. 1.5). It is obvious, that from such an approach that the fractal dimension is determined in two-dimensional Euclidean space, whereas real nanocomposites should be considered in three-dimensional Euchdean space. The following relationship can be used for recalculation for a three-dimensional space ... 

If we again adopt that / =y=0.6, then the value =1.67 accurately coincides with the dimensions of the fractal of the chain df=(d + 2)/3=5/3 and the value ) =V, coincides with the Flory index. In this case the dimensions of fractal of traps and characteristic times d according to equations (8.62) and (8.59), is equal to characteristic times set z, of macroradical relaxation is completely determined by the fractality of traps, i.e., by dimension di. In the case that the set of traps is not a fractal (for example, at the uniform distribution of traps in the reactive zone when the dimension of the f ractal of traps formally coincides with the dimension of the Euclidean space di=d), then, in accordance with equation (8.59) the set of characteristic times r, would be characterized by a fractal dimension equal to infinity and t would not depend on chain length and time of chain propagation, and would be presented by a constant value T, Tp p ,. Under this variant we have 1= 1, 1 and stretched exponential law transforms into simple exponential law. 

As it is known, autohesion strength (coupling of the identical material surfaces) depends on interactions between some groups of polymers and treats usually in purely chemical terms on a qualitative level [1, 2], In addition, the structure of neither polymer in volume nor its elements (for the example, macromolecular coil) is taken into consideration. The authors [3] showed that shear strength of autohesive joint depended on macromolecular coils contacts number A on the boundary of division polymer-polymer. This means, that value is defined by the macromolecular coil structure, which can be described within the frameworks of fiactal analysis with the help of three dimensions fractal (Hausdorff) spectral (fraction) J and the dimension of Euclidean space d, in which ifactal is considered [4]. As it is known [5], the dimension characterizes macromolecular coil connectivity degree and varies from 1.0 for linear chain up to 1.33 for very branched macromolecules. In connection with this the question arises, how the value influences on autohesive joint strength x or, in other words, what polymers are more preferable for the indicated joint formation - linear or branched ones. The purpose of the present communication is theoretical investigation of this elfect within the frameworks of fractal analysis. 

Attempts by the authors [21-23] to linearise the dependence of (1-a) on t for the system EPS-4/DDM by Equations 3.9 and 3.10 were not successful. Therefore the following assumption was made [21-23]. Equation 3.9 describes reaction kinetics of low-molecular substances at large density fluctuations in Euclidean space with dimension d, which is equal to 3 in the considered case. If we assume that the formation of fractal clusters (microgels) with dimension D defines the reaction curing course in a fractal space with dimension D, the dimension d in Equation 3.9 should be replaced by D. The dependence of In (1-a) on corresponding to Equation 3.9 with... 

In Figure 9.1 the comparison of dimensions and for the studied EP is adduced. Their good correspondence indicates unequivocally that their loosely packed matrix, which serves simultaneously as a natural nanocomposite matrix, is the fractal space where the nanocluster structure of epoxy polymers is formed. Since for linear amorphous polymers = 3 [9], i.e., their nanostructure formation is realised in three-dimensional Euclidean space, then the conclusion that chemical crosslinking network availability in the considered EP serves as the indicated distinction cause is obvious enough. In Figure 9.2 the dependence of on crosslinking density is... 

In Figure 9.14 the dependence K (D ) is adduced, which has shown linear decay with growth in and at = 3, i.e., at nanostructure formation in Euclidean space, K = 0 and the structure of epoxy polymers does not undergo changes (formation of nanoclusters) in its creation process. Let us note that such treatment is confirmed by the data for particulate-filled polymer nanocomposites, for which the structure formation proceeds in Euclidean space and the polymer matrix dimension of nanocomposites is constant and equal to this parameter for a matrix polymer [40]. The similar, but weaker, dependence K (D ) was found for a linear amorphous polymer (polycarbonate, a dashed line in Figure 9.14), which is due to the absence of such a powerful factor as chemical crosslinking nodes network. 

In Euclidean cZ-dimensional spaces Flory exponent depends only on d. A good (although not exact) estimation was given by Flory formula, Eq. (9) [7, 8], It is well known [47] that the critical phenomena depend by the decisive mode on various ftactal characteristics of basic stmcture. It becomes obvious, that excepting the ftactal (Hausdorflf) dimension Dj. physical phenomena on ft actals depend on many other dimensions, including skeleton fractal dimension [48], dimension of minimum (or chemical) distance [29] and so on. It also becomes clear, that regular random walks on fractals have anomalous fractal dimension [49] and that the vibrational excitations spectrum is characterized by spectral (fiac-ton) dimension d=2D d [41, 50],... 

In Fig. 21 the kinetic curves conversion degree—reaction duration Q-t for two polyols on the basis of ethyleneglycole (PO-1) and propylene-glycole (PO-2) are adduced. As it was to be expected, these curves had autodecelerated character, that is, reaction rate was decreased with time. Such type of kinetic curves is typical for fractal reactions, to which either fractal objects reactions or reactions in fractal spaces are attributed [85], In case of Euclidean reactions the linear kinetics (i> =const) is observed. The general Eq. (2.107) was used for the description of fractal reactions kinetics. From this relationship it follows, that the plot Q t) construction in double logarithmic coordinates allows to determine the exponent value in this relationship and, hence, the fractal dimension value. In Fig. 3.22 such dependence for PO-1 is adduced, from which it follows, that it consists of two linear sections, allowing to perform the indicated above estimation. For small t t 50 min) the linear section slope is higher and A =2.648 and for i>50 min A =2.693. Such A increase or macromolecular coil density enhancement in reaction course is predicted by the irreversible... 

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