Scaling theories, polymer networks

Polymer networks (e.g., elastomers and gels) can be described on the microscopic scale by typical structural parameters like mesh size but also on the macroscopic scale by their bulk properties like swelling (uptake of liquids) or mechanical behavior. As the most important polymer networks used in bionanotechnology and biomedicine are hydrogel networks, we will focus on experimental methods and theories that are commonly used to analyze such hydrogel networks. 

Part II continues with a section on various approaches and transitions. Chapter 6 covers polymer networks and transitions from nano- to macroscale by Plavsic. The following chapter is on the atomic scale imaging of oscillation and chemical waves at catalytic surface reactions by Elokhin and Gorodetskii. Then next chapter relates the characterization of catalysts by means of an oscillatory reaction written by Kolar-Anic, Anic, and Cupic. Then Dugic, Rakovic, and Plavsic address polymer conformational stability and transitions based on a quantum decoherence theory approach. Chapter 10 of this section, by Jaric and Kuzmanovic, presents a perspective of the physics of interfaces from a standpoint of continuum physics. 

Key Words Crosslinked Rubber, Hory-Rehner Hypothesis, Gels, Networks, Polymer, Rubber Elasticity, Scaling Theory, Solution Thermodynamics, Swelling, Valanis-Landel Function. 

The osmotic response of swollen polymeric networks was studied on the basis of the scaling theory by Horkay et al. [17-19,22,23,133]. They measured both the swelling pressure, CO, and the shear modulus of gels, G, at different stages of dilution. The swelling pressure vs. polymer volume fraction data were analyzed according to the equation [22]... 

This chapter studies the local and global structures of polymer networks. For the local structure, we focus on the internal structure of cross-Unk junctions, and study how they affect the sol-gel transition. For the global structure, we focus on the topological connectivity of the network, such as cycle ranks, elastically effective chains, etc., and study how they affect the elastic properties of the networks. We then move to the self-similarity of the structures near the gel point, and derive some important scaling laws on the basis of percolation theory. Finally, we refer to the percolation in continuum media, focusing on the coexistence of gelation and phase separation in spherical coUoid particles interacting with the adhesive square well potential. 

The important physical properties of absorbent polymers are dependent on the precise structure of the polymer network. Of key importance for use in personal care applications are the equilibrium swelling capacity, the rate of swelling and the modulus of the swollen gel. Molecular theories of rubber elasticity describe the relationship between the molecular structure of a crosslinked polymer and the amount of swelling and elastic modulus which result. These theories are therefore more useful to the synthetic polymer chemist than are the scaling theories of rubber elasticity. 

In the percolation model the values for exponents are far from the mean field results. But essentially we have not yet introduced polymer chains directly, and we assumed that the networks are the result of a non-linear polymerization. Vulcanization starts from preformed polymers. Consider a melt of polymer chains with a (unique) length and add crosslinks which react with the polymer chains to link them together. One might expect a crossover from percolation (very short chains) to the vulcanization model, and we ask for the values of exponents here. Note that vulcanization describes a liquid-solid transition as well, as it is just another manifestation. Following ref. 80 the problem is considered by scaling theory, formulated in arbitrary Euclidian space dimension d. 

In uncrosslinked polymer melts it is argued that if the chains are long enough a topological network is established, and within an appropriate time scale the material behaves as a rubber. We do not discuss such temporary networks. An extended discussion can be found in the textbook by Doi and Edwards. We restrict ourselves to permanently crosslinked networks. Again theory has to be modest and only idealized cases can be solved. 

Since the prediction of the solute diffusion coefficient in a swollen matrix is complex and no quantitative theory is yet possible, Lustig and Peppas [74] made use of the scaling concept, arriving at a functional dependence of the solute diffusion coefficient on structural characteristics of the network. The resulting scaling law thus avoids a detailed description of the polymer structure and yet provides a dependence on the parameters involved. The final form of the scaling law for description of the solute diffusion in gels is... 

Although a network is not present in a concentrated solution, there exists a characteristic length, which had earlier been assumed the distance between neighbouring network sites. The characteristic length is a dynamic one. There are no temporary knots in a polymer system, though there is a characteristic time, which is the lifetime of the frozen large-scale conformation of a macromolecule in the system. So, the conceptions of intermediate length and characteristic time are based on deeper ideas and are reflected in the theory. 

Network polymers can also be made by chemically linking linear or branched polymers. The process whereby such a preformed polymer is converted to a network structure is called cross-linking. Vulcanization is an equivalent term that is used mainly for rubbers. The rubber in a tire is cross-linked to form a network. The molecular weight of the polymer is not really infinite even if all the rubber in the tire is part of a single molecule (this is possible, at least in theory), since the size of the tire is finite. Its molecular weight is infinite, however, on the scale applied in polymer measurements, which require the sample to be soluble in a solvent. 

Mikos, S.A. Smart, J.D. Scaling concepts and molecular theories of synthetic polymers to glycoproteinic networks. In Bioadhesive Drug Delivery Systems Lenaerts, V., Gurney, R., Eds. CRC Press Boca Raton, FL, 1990 25. 

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