Simple Polymer Fractals

The first polymer example is obvious. Indeed, we have already discussed how a polymer chain is bent and entangled, just like a Brownian particle s path. So it is bound to be a fractal  

So what is the fractal dimensionality of an ideal isolated polymer chain The number of particles, that is, monomer units, is obviously proportional 

We shall give some more examples below. However, even now we can conclude that the self-similarity and the fractal structure are not an excep -tion but rather a rule in polymers and other complex systems. 

We were talking in Chapter 8 about the swelling of a real (not ideal) pol Tner coil — due to the fact that every monomer is not an infinitesimal point, but a body of maybe small, yet still finite, size. We have seen that the size of a swelling coil is R N /. A swollen coil is therefore also a fractal, with a fractional dimensionality df 5/3. 

Walter Stockmayer (1914-2004), at Dartmouth College in New Hampshire, showed that the size of such a tree containing N monomers is proportional to R Hence, a tree of this kind is a fractal, and its dimensionality 

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P(r) can be transformed into a distribution of the particle size as defined by the hydrodynamic radius Rh. But only for TDFRS, and not for PCS, a particle size distribution in terms of weight fractions can be obtained without any prior knowledge of the fractal dimension of the polymer molecule or colloid, which is expressed by the scaling relation of Eq. (39). This can be seen from the following simple arguments ... 

For Q<0, this distribution function is peaked around a maximum cluster size (2Q/(2Q-1))< >, where < > is the mean cluster size. 2Q=a+df1 is a parameter describing details of the aggregation mechanism, where a1 is an exponent considering the dependency of the diffusion constant A of the clusters on its particle number, i.e., A NAa. This exponent is in general not very well known. In a simple approach, the particles in the cluster can assumed to diffusion independent from each other, as, e.g., in the Rouse model of linear polymer chains. Then, the diffusion constant varies inversely with the number of particles in the cluster (A Na-1), implying 2Q=-0.44 for CCA-clusters with characteristic fractal dimension d =l.8. 

The r value is supposed to vary in the range 1/d < r < 1. It is believed that the r-fractals thus introduced provide a simple description for real polymers r = 1 for linear chains... 

Let us consider further simple technique of determination of fractal dimension -D of macromolecular coil in diluted polymer solution, within the framework of which the Eq. (11) was obtained. The determination of value is the first stage of macromolecular coils study within the framework of fractal analysis (see chapter 1) and the similar estimations are performed by measurement of the exponents in Mark-Kuhn-Hou-wink t5q)e equations, linking intrinsic viscosity [r ] (the Eq. (1)), translational diffrisivity or rate sedimentation coefficient with pol5mers molecular weight MM [3] ... 

As it was noted above, at present it becomes clear, that polymers in all their states and on different structural levels are fractals [16, 17]. This fundamental notion in principle changed the views on kinetics of processes, proceeding in polymers. In case of fractal reactions, that is, fractal objects reactions or reactions in fractal spaces, their rate fr with time t reduction is observed, that is expressed analytically by the Eq. (106) of Chapter 2. In its turn, the heterogeneity exponent h in the Eq. (106) of Chapter 2 is linked to the effective spectral dimension d according to the following simple equation [18] ... 

In the end the conclusion can be made, that reaction cessation in low-temperature polycondensation process is limited by pnrely physical factor, namely, by macromolecular coil density reaching of reactive medium density (mixture monomer — polymer solution) [121], that is possible for fractal objects only. Such conclusion follows from the simple analysis, adduced below. As it is known [122], the fractal object density pfr can be calculated according to the Eq. (73) ... 

Hence, the fractal analysis, that is a purely pltysical (structural) conception, and the irreversible aggregation models, closely connected with it, provide a simple quantitative description of both environment and time, whereas a reaction mechanism change also influences the reaction course of high-molecular systems. This is possible just owing to introduction of the polymer stmcture in its different states. 

There have been several attempts to extend the above Flory theory to correct for its failure for the i/g estimate (in d = 2) and for estimating i/, and vg on fractals. We give here a simple and elegant one (cf. [7]) which starts with a functional estimate of the radius of gyration distribution P(R) exp[—F(iJ)], instead of that for the free energy F(R). The form of the distribution function P R) of the polymer radius of gyration is given by... 

Finally, one can treat the fractal lattice as a simple model for a disordered substrate on which the polymer is adsorbed, and use the exact results found for polymers on fractals to develop some understanding about real experimental systems. But for this, this article... 

We have discussed only the equilibrium properties of polymers. Of course, in many real systems, the time scales for equilibriation can be very large. It is thus of interest to study non-equilibrium properties of statistical mechanical systems on fractals. A simple prototype is the study of kinetic Ising model on fractals. Closer to our interests here, one can study, say, the reptation motion of a polymer on the fractal substrate. This seems to be a rather good first model of motion of a polymer in gels. 

Hierarchy can be described in analogy to rope (stretched polymer molecules in domains that make up nanofibers, combined to microwhiskers, bundled into fibers that are spun into yarn that is twined to make up the rope). Wood and tendon are biological examples that have six or more hierarchical levels. Compared to these, fiber-reinforced matrix composites made up of simple massive fibers embedded in a metallic, ceramic, or polymer matrix are primitive. Hierarchical inorganic materials, as discussed in Chapter 7, can be made with processes for fractal-like solid products spinodal decomposition, diffusion-limited growth, particle precipitation from the vapor, and percolation. Fractal-like solids have holes and clusters of all sizes and are therefore hierarchical if the interactions... 

Let us stress that this is the fractal dimension of the polymers in the reaction bath. We assume that all polymers that constitute the sol have this same fractal dimension. This was calculated by renormalization group techniques and computer simulations [31,32,33,34]. We will give a simple Flory derivation [35] that is close to the former results for all space dimensions. The polydispersity exponent t can be shown to be related to the fractal dimension. 

The fractal exponent is thus significantly smaller than that of a simple random walk, and the structure of the coil is considerably more open. This result is in good agreement with experiment, as shown by the example of polystyrene dissolved in benzene, at sufficiently low concentrations that the polymers cannot penetrate each other (which would totally change the type of solution). 

The above examples show that it is a subtle task to gauge the strength of osmotic interactions of our fractal polymers. Even simple rearrangements of the particles can have a big effect on the interaction strength. Happily, though, the intersections of fractals (without any rearrangements) can be readily analyzed. Thus we consider the g r) of two fractals with dimensions Di and D2, both made from particles of radius a. One way to infer this g r) is by analyzing its effect on the number of intersections I r). In fractals, unlike clouds, these intersections are not statistically independent. If one intersection occurs at some point, many others must occur near it. We denote the number of extra intersections by M. That is, M is the number of intersections for those objects which have at least one intersection. 

In this subsection we will consider (distinct from the dendrimers of Sect. 8) another class of regular hyperbranched polymers. We recall that the quest for simpUcity in the study of complex systems has led to fruitful ideas. In polymers such an idea is seating, as forcefully pointed out by de Gennes [4j. Now, the price to be paid in going from linear chains to star polymers [33,194[, dendrimers [13,33,194,205] and general hyperbranched structures [216[ is that scaling (at least in its classical form) is not expected to hold anymore (at least not in a simple form, which implies power-law dependences on the frequency CO or on the time f). One of the reasons for this is that while several material classes (such as the Rouse chains) are fractal, more general structures do not necessarily behave as fractals. 

Abstract. The square and cubic lattice percolation problem and the selfavoiding random walk model were simulated by Monte Carlo method in order to obtain new understanding of the fractal properties of branched and hnear polymer molecules. The central point of this work refers to the comparison between the cluster properties as they emerge from the percolation problem on one hand and the random walk properties on the other hand. It is shown that in both models there is a drastic difference between two and three dimensional systems. In three dimensions it is possible to find a regime where the properties converge towards simple non-avoided random walk, while in two dimensions the topological reasons prevent a smooth transition of the properties pertaining to avoided and non-avoided random walks. 

In summary, we have discussed how interfacial effects can influence the viscoelastic properties of polymer coatings, polymer melts and solutes, and even simple nonpolar liquids within an interfacial boundary regime. In high molecular weight polymer systems, the interfacial boundary regime can reach up to hundreds of nanometers. The interfacial boundary layer of simple nonpolar fluids is restricted to a few nanometers. While outside the critical interfacial boundary layer interfacial effects on properties can be approached with phenomenological theories, modified or new theories are in demand within the structurally - or entropically cooled interfacial boundary layer. The modern theoretical approach of the fractal dimensionality will be discussed next. 

Thus, the stated above results demonstrated the fractal analysis possibilities at polymers local deformation description. In each from the described cases fractal dimension of either element has simple and clear physical significance that allows to obtain both empirical and analytic correlations between different structural levels in polymers and also describe their evolution in polymers deformation and failure processes. 

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