THEORY OF POLYMERS WITH FROZEN STRUCTURES (original) (raw)
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Theory of polymers with a frozen structure
A field theory describing polymers with a frozen topological structure is constructed. Branched polymers with a frozen molecular-structure distribution are studied in detail. It is shown that, because of the strong polydispersity, a semidilute solution of such polymers cannot be described by means of the concept ofblobs. It is also shown that the swelling of a molecule with a tree-type structure in a good solvent occurs in a nonaffine manner, and the dependence of its swelling coefficient on the spatial scale is calculated. The dependence of the swelling coefficient of a cyclic molecule on the quality of the solvent is obtained. The swelling of an infinite polymer gel network with a fixed topological structure in a good solvent is considered separately. The thermodynamic and correlation functions of the gel network are calculated, and it is shown that in the scaling region such a network can be described in the framework of the concept of blobs, within which there exist correlations characteristic of isolated branched molecules. It is also shown that the equilibrium density of the swollen gel is determined by a condition on the threshold for overlap of such molecules. The results obtained make it possible to use the results of numerous papers devoted to the investigation of "ordinary" blobs. Other problems that can be considered in the framework of the proposed field theory are also mentioned.
Scaling theory of star polymers and general polymer networks in bulk and semi-infinite good solvents
Journal de Physique, 1988
Nous établissons une théorie d'échelle des réseaux généraux de polymères, dans de bons solvants, en volume et en milieu semi-infini, en utilisant l'équivalence entre la fonction génératrice du nombre total de configurations et la fonction de corrélation à plusieurs spins du modèle de Heisenberg classique à n composantes dans la limite n ~ 0. Dans le cas de réseaux de polymères à topologie fixée G, composés de f chaînes linéaires de longueur f, le nombre total de configurations se comporte, pour f grand, comme Ng(l, l, ..., l) ~ l03B3g-103BClf. L'exposant 03B3g peut s'exprimer entièrement en terme des exposants y (f) et 03B3s(f) des polymères étoilés (libres et à centre fixé) à f branches. Quand les g des f branches du polymère étoilé sont attachés à une surface par leurs extrémités, les exposants 03B3g ~ 03B311 ... 1 (f) sont donnés en terme de ceux des polymères étoilés à f branches et des exposants des chaînes linéaires 03B311 ...1 (f) = 03B3 (f) + 03BD + g [03B3 11-03B31]. De plus, l'exposant 03B3g pour les polymères en forme de peigne (avec g unités trifonctionnelles) se réduit à une combinaison linéaire des polymères étoilés à 3 branches, y (3), et de l'exposant du nombre de configurations des chaines linéaires, 03B3 : 03B3comb (g) = 03B3 + g [03B3(3)-03B3 ]. Les exposants de polymères étoilés 03B3 (f), 03B3s(f) et 03B311...1 (f) sont calculés dans la théorie du champ moyen et dans le développement en 03B5.
Recent developments in the theory of polymer gels
Journal of Computer-Aided Materials Design, 1996
We review the results of a recently developed theory of randomly cross-linked polymer gels. The theory is based on the exact statistical mechanical solution of the Edwards model which takes into account both the frozen disorder of network structure and excluded-volume effects. Predictions are made for the behavior of individual network chains, the density correlation functions which are directly measured by neutron and light-scattering experiments and for the thermodynamics of gels. We now have a complete statistical description of polymer gels in good solvents, ranging from monomer length scales to the continuum limit.
Polymer Gels: Frozen Inhomogeneities and Density Fluctuations
Macromolecules, 1996
We present a phenomenological theory of randomly cross-linked polymer networks based on the separation of solid-like and liquid-like degrees of freedom and taking into account the frozen inhomogeneity of network structure. The complete solution of the statistical mechanics of this model is given, and the monomer density correlation functions are calculated for neutral gels in good and Θ solvents. The theoretical scattering curves are compared to the results of small angle neutron scattering and light scattering experiments, and new experimental tests of our theory are proposed. † Permanent address:
Lattice model for cold and warm swelling of polymers in water
Physical Review E, 2000
We define a lattice model for the interaction of a polymer with water. We solve the model in a suitable approximation. In the case of a non-polar homopolymer, for reasonable values of the parameters, the polymer is found in a non-compact conformation at low temperature; as the temperature grows, there is a sharp transition towards a compact state, then, at higher temperatures, the polymer swells again. This behaviour closely reminds that of proteins, that are unfolded at both low and high temperatures. PACS number(s): 05.20.-y; 05.40.Fb; 61.25.Hq; 87.10.+e
Paradoxes of Thermodynamics of Swelling Equilibria of Polymers in Liquids and Vapors
2011
An automatic registration of the changing size of a single spherical microbead of a cross-linked polymer was applied for studying the swelling process of the bead by the sorption of vapors and/or liquids. Many representatives of all three basic types of polymeric networks, gel-type, hypercrosslinked, and macroporous, were examined. Only the first two display large volume changes and prove suitable for following the kinetics and extent of swelling by the above dilatometric technique. The results unambiguously prove that swelling of all polymeric networks in liquids is always higher than in corresponding saturated vapors (Schroeder's paradox). The general nature of this phenomenon implies that the absolute activity of any sorbate in its liquid form is always larger than in the form of its saturated vapor. Surprisingly, gels with any solvent contents, which fall into the broad range between the vapor-equilibrated and liquidequilibrated extreme contents, retain their volumes constant in the saturated vapor atmosphere. This paradox of a wide range of gels swollen to a different extent and, nevertheless, standing in equilibrium with saturated vapor is explained by the specificity of the network polymers, namely, that the energy of the solventpolymer interactions is easily compensated by the energy of remaining between-chain interactions at any solvent content in the above range. Therefore, the strain-free swollen gels do not generate enhanced vapor pressure, but neither display the ability to take up more sorbate from its vapor.
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 1995
Models of linear polymers with competing interactions favoring rami6cation into thin branches on one hand, and compacti6cation on the other hand, are shown to possess a phase diagram with swollen linear, swollen branched, and compact regimes. Between each pair of these regimes, a multicritical transition occurs. The three transitions merge in a single, very unstable multicritical point. Evidence of such behavior is obtained by renormalization methods applied exactly to a model on a hierarchical lattice, and approximately to a system on a square lattice. The hierarchical model calculations, discussed in much detail, are particularly extensive and give the advantage of an almost complete characterization of the different scaling behaviors, exact at least within the limits of numerical accuracy. The strong qualitative evidence from renormalization group approaches is further con6rmed by an extensive exact enumeration analysis for a third model on the square lattice. The scalings at various multicritical transitions are studied in detail and universality issues are discussed.
Physical Review E, 2016
Mass fractal scaling, reflected in the mass fractal dimension d f , is independently impacted by topology, reflected in the connectivity dimension c, and by tortuosity, reflected in the minimum dimension d min. The mass fractal dimension is related to these other dimensions by d f = cd min. Branched fractal structures have a higher mass fractal dimension compared to linear structures due to a higher c, and extended structures have a lower dimension compared to convoluted self-avoiding and Gaussian walks due to a lower d min. It is found, in this work, that macromolecules in thermodynamic equilibrium display a fixed mass fractal dimension d f under good solvent conditions, regardless of chain topology. These equilibrium structures accommodate changes in chain topology such as branching c by a decrease in chain tortuosity d min. Symmetric star polymers are used to understand the structure of complex macromolecular topologies. A recently published hybrid Unified scattering function accounts for interarm correlations in symmetric star polymers along with polymer-solvent interaction for chains of arbitrary scaling dimension. Dilute solutions of linear, three-arm and six-arm polyisoprene stars are studied under good solvent conditions in deuterated p-xylene. Reduced chain tortuosity can be viewed as steric straightening of the arms. Steric effects for star topologies are quantified, and it is found that steric straightening of arms is more significant for lower-molecular-weight arms. The observation of constant d f is explained through a modification of Flory-Krigbaum theory for branched polymers.