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# Polymer physics

Polymer physics is the field of physics associated to the study of 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 an aspect of the study of condensed matter physics, the field of polymer physics has developed as a branch of statistical physics. Polymer physics and polymer chemistry are part of the wider field of polymer science.

Disordered polymers are too complex to be described using a deterministic method. However statistical approaches can yield results and are often pertinent since large polymers (that is to say, polymers which contain a large number of monomers) can be described efficiently as systems at the thermodynamic limit.

Thermal fluctuations continuously affect the shape of polymers in liquid solutions, and modelling their effect requires a recourse to the principles of statistical mechanics. As a corollary temperature strongly affects the physical behavior of polymers in solution.

The statistical approach to polymer physics is based on an analogy between a polymer and either a brownian motion, or some other type of random walk. The simplest possible polymer model is presented by the ideal chain, corresponds to homogeneous random walk.

The Russian and Soviet schools of physics have been particularly active in the development of polymer physics.

## Models

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 canceled out . Ideal chain models provide a good starting point for investigation of more complex systems and is better suited for equations with more parameters.

### Ideal Chains

The freely-joined 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 freely-rotating chain improves the freely-jointed chain model by taking into account that polymer segments make a fixed 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:

$P(\theta)\propto{}\exp\left(-U(\theta)/kT\right)$

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 it causes bending energy. At the length scale below persistence length, the polymer behaves more or less like a rigid rod.

### Real Chains

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.

## Solvent and temperature effect

The statistics of a single polymer chain depends on the solvent. For good solvent the chain is more expanded while for bad solvent the chain segments stay close to each other. In the limit of a very bad solvent the polymer chain merely collapses to form a hard sphere, while in good solvent the chain swells in order to maximize the number of polymer-fluid contacts. For this case the radius of gyration is approximated using Flory's mean field approach which yields a scaling for the radius of gyration of:

Rg˜Nν,

where Rg is the radius of gyration of the polymer, N is the number of bond segments (N, which is the degree of polymerization) of the chain.

For good solvent, ν = 3 / 5; for bad solvent, ν = 1 / 3. Therefore polymer in good solvent has larger size and behaves like a fractal object. In bad solvent it behaves like a solid sphere.

In the so called θ solvent, ν = 1 / 2, which is the result of simple random walk. The chain behaves as if an ideal chain.

The quality of solvent depends also on temperature. For a flexible polymer, low temperature may correspond to poor quality and high temperature makes the same solvent good. At a particular temperature called theta (θ) temperature, the solvent behaves as if an ideal chain.

## Excluded volume interaction

Ideal chain model assumes that polymer segments can be overlapped with each other as if it is a phantom chain. In reality, two segments cannot occupy the same space at the same time. This interaction between segments is called 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, the number of possible conformation is significantly reduced. The radius of gyration is generally larger than that of ideal chain.

## Flexibility

Whether a polymer is flexible or not depends on the scale of interest. For example, the persistence length of double-stranded DNA is about 50nm. Looking at length scale smaller than 50nm, it behaves more or less like a rigid rod. At length scale much larger than 50nm, it behaves like a flexible chain.