Fractal dimension ideal branched

An ideal randomly branched polymer is a fractal object with fractal dimension V — 4. In Chapter 6, we will learn how this polymer can fit into three-dimensional space. What is the ratio of molar masses M1/M2 of two ideal randomly branched polymers if the ratio of their sizes is R1/R2 = 3 ... 

Consider a monodisperse melt of randomly branched polymers with N Kuhn monomers of length b. Randomly branched polymers in an ideal state (in the absence of excluded volume interactions) have fractal dimension D = 4. Do these randomly branched polymers overlap in a three-dimensional monodisperse melt ... 

The fractal dimension of an ideal randomly branched polymer is 27 = 4 (because its degree of polymerization is proportional to its size to the fourth power N 2 ). In spaces with dimension d

Below the gel point, the system is self-similar on length scales smaller than the correlation length with a power law distribution of molar masses with Fisher exponent r = 5/2 [Eq. (6.78)]. Each branched molecule is a self-similar fractal with fractal dimension 27 = 4 for ideal branched mole-cules in the mean-field theory. The lower limit of this critical behaviour is the average distance between branch points (= ). There are very few... 

In the evolution of solids from solution, a wide spectrum of structures can be formed. In Fig. 4, a simple schematic representation of the structural boundary condition for gel formation is presented. At one extreme of the conditions, linear or nearly linear polymeric networks are formed. For these systems, the functionality of polymerization /, is nearly 2. This means there is little branching or cross-linking. The degree of cross-linking is nearly 0. In silica, gels of this type can be readily formed by catalysis with HCl or HNO3 under conditions of low water content (less than 4 mol water to 1 mol silicon alkoxide). The ideal fractal dimension for such a linear chain structure is 1. The phe-... 

The fractal dimension of purely statistical models, i.e., models without the effect of excluded volume, can be determined accurately [see Equation (11.9a)]. For linear polymers, this model corresponds to phantom random-walk. In the case of branched statistical fractals, the corresponding model is a statistical branched cluster, whose branching obeys the random-walk statistics. Since the root-mean-square distance between the random-walk ends is proportional to the number of walk steps N, then D = 2 irrespective of the space dimension. These types of structures have been studied [61, 75-77]. The value D = 4 irrespective of d was obtained for a branched fractal. Unlike ideal statistical models, models with excluded volume, i.e., those involving correlations, cannot be accurately solved in the general case. The Df values for these systems are usually found either using numerical methods such as the Monte Carlo method or taking into account the spatial position of a renormalisation group. 

Thus, the typical size of an ideal randomly branched tree scales as Rg Its fractal dimension is therefore df=4 ... 

Fractal aggregate, fractal agglomerate aggregates or agglomerates with a non-uniform distribution of the constiment particles, which typically coincides with a very porous, branch-like morphology fractal aggregates are characterised by a power-law decrease of the pair-correlation density function g(v) (Eq. (4.8)) and a power-law relationship between mass and size (Eq. (4.9)), in which the exponent is less than the Euclidean dimension fractal aggregates are not ideal fractal objects, but rather obey the fractal relationships only in a statistical sense (cf. Sect. 4.2.1). 

---