Backbone fractal dimension

According to this model, the clusters act as molecular springs with end to end distance , consisting of Nr backbone units of length d. The connectivity of the backbone units is characterized by the backbone fractal dimension dfjj. Due to the fractal nature of CCA-clusters it holds that... 

Backbone fractal dimension 54 Bending-twisting deformation 55 BET-surface area 13-14, 20 Brake system, frictional 145 Breakdown, filler cluster 64, 76... 

The fractal dimension D is used to quantify the micro structure of the fat crystal networks, where d is the Euclidean dimension, x is the backbone fractal dimension that is estimated between 1 and 1.3. The backbone fractal dimension describes the tortuosity of the effective chain of stress transduction within a cluster of particles yielding under an externally applied stress (Shih et al. 1990 Kantor and Webman 1984). 

In fact, if one measures the total number of bonds (sites) on the infinite cluster at the percolation threshold (pc) in a (large) box of linear size L, then this number or the mass of the infinite cluster will be seen to scale with L as where die (< d) is called the fractal dimension of the infinite cluster at the percolation threshold. Similar measurements for the backbone (excluding the dangling ends of the infinite cluster) give the backbone mass scaling as, de < die, where dfi is called the backbone (fractal) dimension. In fact, die can be very easily related to the embedding Euclidean dimension d of the cluster by... 

Dinsmore and Weitz (2002) [60] examined a model system of polymethylmethacrylate particles dyed with rhodamine in a density- and refractive-index-matched solution of decalin and cyclohexylbromide. They used CLSM to follow the (very slow) aggregation in real time and determined the particle positions in full three-dimensional detail. From this they performed a comprehensive structural analysis of the gels, including measurement of coordination numbers and backbone fractal dimension of the structures, as well as the much more commonly measured mass fractal dimension. 

The backbone fractal dimension is the mass fractal dimension of the backbone , i.e. that chain of particles which connects two arbitrary particles in the structure. 

Note that the fractal dimensions discussed here are the fractal dimensions of the excitation transfer paths connecting the hydration centers located on the inner surface of the pores. Due to the low humidity, all of the water molecules absorbed by the materials are bound to these centers. The paths of the excitation transfer span along the fractal pore surface and depict the backbone of clusters formed by the pores on a scale that is larger than the characteristic distance between the hydration centers on the pore surface. Thus the fractal dimension of the paths Dp approximates the real surface fractal dimension in the considered scale interval. For random porous structures, Dp can be also associated with the fractal dimension D, of the porous space Dp = Dr. Therefore, the fractal dimension Dp can be used for porosity calculations in the framework of the fractal models of the porosity. 

Isvoran, A. Licz, A. Unipan, L. Morariu, V.V. Determination of the fractal dimension of the lysozyme backbone of three different organism. Chaos, Solitons Fractals 2001, 12, 757-760. 

To assign physical meaning to D, we associate D with the mass fractal dimension of the protein, defined by M LD. Then D is the slope of log M versus log L, where M is the mass of all protein atoms enclosed by concentric spheres of radius L. For a completely collapsed, space-filling polymer, D is 3, but D may be generally less than 3 in proteins due to a combination of compact structures and spacing between them. Figure 14 shows log M versus log L for cytochrome c. The concentric spheres of radius L are centered on one of the Ca s of the protein backbone. We compute M for values of L from 2 A to 16 A. The plot appears linear with slope 2.15. Continuing in this way, we compute... 

Figure 14. Plot of log M versus log L for cytochrome c, where values of M are the masses of all protein atoms enclosed by concentric spheres of radius L centered at a backbone atom. The slope of the line fit to the data for L from 2 A to 16 A gives the mass fractal dimension, in this case 2.15.

N(r) can be related to the fractal dimension D of the protein-surfactant complex by the equation N r) = (r/R jP. For a freely diffusing micelle the fractal dimension would be 3, but in a complex the distribution of micelles is dictated by the topology of the polypeptide backbone, so that the fractal dimension is less than 3. At 1% by weight of lithium n-dodecylsulphate, D is 2.3 and decreases to 1.76 at 3%, consistent with a transition from a compact state to a more open random coil in which a string of constant-sized micelles are distributed along the hydrophobic patches of a denatured random coil, although the coil will be restrained by the 17 disulphide bonds in the BSA structure. 

After having discussed the behavior of SAWs on deterministic fractals, we move on to the second major topic of this chapter, namely the numerical study of SAWs on random fractals, the latter modelled by percolation. As non-trivial changes in the exponents characterizing the structure of SAWs on the incipient percolation cluster (and as a consequence on its backbone) axe only expected at criticality [5], i.e. for probability p of available sites being p = Pc, the following discussion is restricted to this case. A summary of exponents and fractal dimensions characterizing critical percolation is given in Table 1. 

Fat system Fractal dimension from image analysis Fractal dimension from rheology (weak-link regime) Percent deviation Fractal backbone dimension x... 

Under acid-catalyzed conditions with molar hydrolysis ratios r = 1 or 2.5, aging (at 0 C, room temperature, or under reflux) served to increase the concentration of cyclic tetrasiloxanes, which in turn influenced the intermediate-scale structure of polymers formed during subs uent acid- or base-catalyzed hydrolysis. Presumably incorporation of structurally rigid cyclic tetramers into the siloxane backbone leads to more extended structures characterized by larger size and lower mass fractal dimension. 

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