Bulk fractal dimension

The adsorption of large polar molecules is expected to produce large unfolding of the protein, possibly terminating with its denaturation. Denaturation can be considered as a phase transition, characterized by a jump discontinuity of the adsorption isotherm, between native-like state (with bulk fractal dimension D = 3 and surface fractal dimension D = 2.1 - 2.2) and the denatured state (in which the protein can be seen as an excluded-volume polymer with D =5/3 [57]). Denaturation is therefore associated with a sudden variation of the fractal dimension of the protein. 

In two dimensions (2D), the exact value for the hull fractal dimension, dj =, differs from the bulk fractal dimension. Although percolation in 3D is of greater practicd interest, no exact result is known in this case. The aim of the present paper is to review briefly the numerical results obtained so far for the hulls of 3D percolation clusters and to discuss the possibility that the hull and bulk critical behaviors are the same in this case. 

The fractal dimension can be estimated by several techniques, including structured walk (Richardson s plot), bulk density-particle diameter relation, sorption behavior of gases, pore size distribution, and viscoelastic behavior. The fiactal dimension obtained by each method has its own physical meaning (Rahman, 1997). 

The relationships of surface fractal dimension, Ds, with flowability and bulk density have been reported. ... 

A number of methods have been proposed for particle shape analysis these include verbal description, various shape coefficients and shape factors, curvature signatures, moment invariants, solid shape descriptors, the octal chain code and mathematical functions like Fourier series expansion or fractal dimensions. As in particle size analysis, here one can also detect intense preoccupation with very detailed and accurate description of particle shape, and yet efforts to relate the shape-describing parameters to powder bulk behaviour are relatively scarce.10... 

Zeng, Y., C.J. Gantzer, and S.H. Anderson. 1996. Fractal dimension and lacunarity of bulk density determined with x-ray computed tomography. Soil Sci. Soc. Am. J. 60 1718... 

Equation (16) also holds for transfer across fractal interfaces. In the latter case, the fractal dimension dj refers to the (Euclidean) bulk in which particles diffuse, and is given hy d] = d = dx + I, d = 2, while dj equals the fractal dimension d of the interface itself. 

Sample Bulk density (g cmO Porosity (%) Pore density (tnm ) Average cross-sectional area (pm ) Roundness of pores Fractal dimension... 

It is worth noting that at the minimum value of d = 2, value A becomes negative. That means, in accordance with Equation (12.1), that tan < tan 8m- In other words, at small d (smooth surfaces of the filler particles) the packing of the polymer molecules at the interface may be more dense compared with the bulk. This fact leads to the diminishing molecular mobility in the interfacial layer [1, 37, 38] Extrapolation of the dependence of A on d to maximum value of d = 3 gives the limiting value of A 4.5. This value meets the value A = 4.2, that was derived from the extrapolation of the volume of the interfacial layer on d to d = 3 [39]. Thus, this analysis allows an estimation of the structural factors influencing formation of the adhesion joints. The main factors are fractal dimensions of the particle surface, d and of the polymer df, which determine adhesion at the polymer-filler particle interface. 

Up to now we considered pol5meric fiiactals behavior in Euclidean spaces only (for the most often realized in practice case fractals structure formation can occur in fractal spaces as well (fractal lattices in case of computer simulation), that influences essentially on polymeric fractals dimension value. This problem represents not only purely theoretical interest, but gives important practical applications. So, in case of polymer composites it has been shown [45] that particles (aggregates of particles) of filler form bulk network, having fractal dimension, changing within the wide enough limits. In its turn, this network defines composite polymer matrix structure, characterized by its fractal dimension polymer material properties. And on the contrary, the absence in particulate-filled polymer nanocomposites of such network results in polymer matrix structure invariability at nanofiller contents variation and its fractal dimension remains constant and equal to this parameter for matrix polymer [46]. 

As it has been shown above, a macromolecular coil in solution fractal dimension is defined by two groups of factors polymer-solvent interactions and macromolecular coil elements between themselves. As an approximation, the first from the indicated factors can be characterized by the difference of polymer 6 and solvent 6 solubility parameters A8= 6 6j [16]. As to the second group of factors, then a parameters number exists, influencing on value in some way chain rigidity, bulk side groups availability, hydrogen bond and so on. Since at present the strict theory of polymer regular solutions has not still developed, then analytical relationships... 

As it has been noted above, the calculation of solvent fractal dimension 8 according to the Eq. (18) of Chapter 1 shows, that the value 6 =0 (a separate molecule of low-molecular solvent is a point or zero-dimensional object [25]) is obtained for only D=5I1> or a macromolecular coil in good solvent [232]. Since such molecule always remains a zero-dimensional object, then it should be supposed, that the dimension 8 characterizes the structure of low-molecular solvent molecules totality ( swarm ), which participates in polymers dissolution process. The bulk interactions parameter e of a macromolecular coil is connected with dimension D hy the Eq. (39), from which it follows, that D=2 (the coil in 0-solvent, attractive and repulsive interactions are balanced) e=0, at D >2 e<0 characterizes attractive interactions of coil elements between themselves and at Dj<2 e>0—repulsive interactions. 

Kozlov, G. V. Afaunov, V. V Temiraev, K. B. Fractal dimension of biopolymers macromolecular coil in solution as a bulk interactions measure. Manuscript deposited to VINIU RAS, Moscow, 08.01.199SfP-V98. 

Particle property any property of a particle that is not (only) a bulk property of the dispersed phase there are several, not necessarily excluding categories (geometric properties, dynamic properties, interactions with electromagnetic or sound fields) these properties are related to particle size or, more generally, to the particle morphology, which also includes the particle shape (e.g. sphericity) and structural composition (e.g. fractal dimension). 

Even for a network whose connectivity can be described using a fractal dimension, d is the dimensionality of the space in which the network is represented i.e., d=2 for surface diffusion and 3 for bulk diffusion. 

Figure 14-12. Evolution of structural features (a) fractal dimension, (b) cluster size, (c) particle size, as a function of the bulk density for sintered and compressed gels.

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