Физика полимеров: основы, модели и экспериментальные методы.

Polymer physics is the field of physics that studies polymers, their fluctuations, mechanical properties, as well as the kinetics of reactions involving degradation and polymerisation of polymers and monomers respectively. While it focuses on the perspective of condensed matter physics, polymer physics is originally a branch of statistical physics. Polymer physics and polymer chemistry are also related with the field of polymer science, where this is considered the applicative part of polymers. Polymers are large molecules and thus are very complicated for solving using a deterministic method. Yet, statistical approaches can yield results and are often pertinent, since large polymers (i. e., polymers with many monomers) are describable efficiently in the thermodynamic limit of infinitely many monomers (although the actual size is clearly finite). Thermal fluctuations continuously affect the shape of polymers in liquid solutions, and modeling their effect requires using principles from statistical mechanics and dynamics. As a corollary, temperature strongly affects the physical behavior of polymers in solution, causing phase transitions, melts, and so on. The statistical approach for polymer physics is based on an analogy between a polymer and either a Brownian motion, or other type of a random walk, the self avoiding walk. The simplest possible polymer model is presented by the ideal chain, corresponding to a simple random walk. Experimental approaches for characterizing polymers are also common, using polymer characterization methods, such as size exclusion chromatography, viscometry, dynamic light scattering, and Automatic Continuous Online Monitoring of Polymerization Reactions (ACOMP) for determining the chemical, physical, and material properties of polymers. These experimental methods also helped the mathematical modeling of polymers and even for a better understanding of the properties of polymers. Flory is considered the first scientist establishing the field of polymer physics. Doi and Edwards wrote a very famous book in polymer physics.

Models of polymer chains are split into two types: "ideal" models, and "real" models. Ideal chain models assume that there are no interactions between chain monomers. This assumption is valid for certain polymeric systems, where the positive and negative interactions between the monomer effectively cancel out. Ideal chain models provide a good starting point for the investigation of more complex systems and are better suited for equations with more parameters.

The freely jointed chain is the simplest model of a polymer. In this model, fixed length polymer segments are linearly connected, and all bond and torsion angles are equiprobable. The polymer can therefore be described by a simple random walk and ideal chain. The model can be extended to include extensible segments in order to represent bond stretching. The freely rotating chain improves the freely jointed chain model by taking into account that polymer segments make a fixed bond angle to neighbouring units because of specific chemical bonding. Under this fixed angle, the segments are still free to rotate and all torsion angles are equally likely. The hindered rotation model assumes that the torsion angle is hindered by a potential energy. This makes the probability of each torsion angle proportional to a Boltzmann factor:

, where is the potential determining the probability of each value of
In the rotational isomeric state model, the allowed torsion angles are determined by the positions of the minima in the rotational potential energy. Bond lengths and bond angles are constant. The Worm like chain is a more complex model. It takes the persistence length into account. Polymers are not completely flexible; bending them requires energy. At the length scale below persistence length, the polymer behaves more or less like a rigid rod.

Interactions between chain monomers can be modelled as excluded volume. This causes a reduction in the conformational possibilities of the chain, and leads to a self avoiding random walk. Self avoiding random walks have different statistics to simple random walks.

The ideal chain model assumes that polymer segments can overlap with each other as if the chain were a phantom chain. In reality, two segments cannot occupy the same space at the same time. This interaction between segments is called the excluded volume interaction. The simplest formulation of excluded volume is the self avoiding random walk, a random walk that cannot repeat its previous path. A path of this walk of N steps in three dimensions represents a conformation of a polymer with excluded volume interaction. Because of the self avoiding nature of this model, the number of possible conformations is significantly reduced. The radius of gyration is generally larger than that of the ideal chain.

The study of long chain polymers has been a source of problems within the realms of statistical mechanics since about the 1950s. One of the reasons however that scientists were interested in their study is that the equations governing the behavior of a polymer chain were independent of the chain chemistry. What is more, the governing equation turns out to be a random walk, or diffusive walk, in space. Indeed, the Schrödinger equation is itself a diffusion equation in imaginary time, t' = it.