Polymer physics

• 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.[10] 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.[11]
• 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:

${\displaystyle P(\theta )\propto {}\exp \left(-U(\theta )/kT\right)}$, where ${\displaystyle U(\theta )}$ is the potential determining the probability of each value of ${\displaystyle \theta }$.

• 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 first example of a random walk is one in space, whereby a particle undergoes a random motion due to external forces in its surrounding medium. A typical example would be a pollen grain in a beaker of water. If one could somehow "dye" the path the pollen grain has taken, the path observed is defined as a random walk.

Consider a toy problem, of a train moving along a 1D track in the x-direction. Suppose that the train moves either a distance of +b or −b (b is the same for each step), depending on whether a coin lands heads or tails when flipped. Lets start by considering the statistics of the steps the toy train takes (where S_i is the ith step taken):

${\displaystyle \langle S_{i}\rangle =0}$ ; due to a priori equal probabilities

${\displaystyle \langle S_{i}S_{j}\rangle =b^{2}\delta _{ij}.}$

The second quantity is known as the correlation function. The delta is the kronecker delta which tells us that if the indices i and j are different, then the result is 0, but if i = j then the kronecker delta is 1, so the correlation function returns a value of b². This makes sense, because if i = j then we are considering the same step. Rather trivially then it can be shown that the average displacement of the train on the x-axis is 0;

${\displaystyle x=\sum _{i=1}^{N}S_{i}}$

${\displaystyle \langle x\rangle =\left\langle \sum _{i=1}^{N}S_{i}\right\rangle }$

${\displaystyle \langle x\rangle =\sum _{i=1}^{N}\langle S_{i}\rangle .}$

As stated ${\displaystyle \langle S_{i}\rangle =0}$, so the sum is still 0. It can also be shown, using the same method demonstrated above, to calculate the root mean square value of problem. The result of this calculation is given below

${\displaystyle x_{\mathrm {rms} }={\sqrt {\langle x^{2}\rangle }}=b{\sqrt {N}}.}$

From the diffusion equation it can be shown that the distance a diffusing particle moves in a medium is proportional to the root of the time the system has been diffusing for, where the proportionality constant is the root of the diffusion constant. The above relation, although cosmetically different reveals similar physics, where N is simply the number of steps moved (is loosely connected with time) and b is the characteristic step length. As a consequence we can consider diffusion as a random walk process.

Random walks in space can be thought of as snapshots of the path taken by a random walker in time. One such example is the spatial configuration of long chain polymers.

There are two types of random walk in space: self-avoiding random walks, where the links of the polymer chain interact and do not overlap in space, and pure random walks, where the links of the polymer chain are non-interacting and links are free to lie on top of one another. The former type is most applicable to physical systems, but their solutions are harder to get at from first principles.

By considering a freely jointed, non-interacting polymer chain, the end-to-end vector is

${\displaystyle \mathbf {R} =\sum _{i=1}^{N}\mathbf {r} _{i}}$

where r_i is the vector position of the i-th link in the chain. As a result of the central limit theorem, if N ≫ 1 then we expect a Gaussian distribution for the end-to-end vector. We can also make statements of the statistics of the links themselves;

• ${\displaystyle \langle \mathbf {r} _{i}\rangle =0}$ ; by the isotropy of space
• ${\displaystyle \langle \mathbf {r} _{i}\cdot \mathbf {r} _{j}\rangle =3b^{2}\delta _{ij}}$ ; all the links in the chain are uncorrelated with one another

Using the statistics of the individual links, it is easily shown that

${\displaystyle \langle \mathbf {R} \rangle =0}$

${\displaystyle \langle \mathbf {R} \cdot \mathbf {R} \rangle =3Nb^{2}}$.

Notice this last result is the same as that found for random walks in time.

Assuming, as stated, that that distribution of end-to-end vectors for a very large number of identical polymer chains is gaussian, the probability distribution has the following form

${\displaystyle P={\frac {1}{\left({\frac {2\pi Nb^{2}}{3}}\right)^{3/2}}}\exp \left({\frac {-3\mathbf {R} \cdot \mathbf {R} }{2Nb^{2}}}\right).}$

What use is this to us? Recall that according to the principle of equally likely a priori probabilities, the number of microstates, Ω, at some physical value is directly proportional to the probability distribution at that physical value, viz;

${\displaystyle \Omega \left(\mathbf {R} \right)=cP\left(\mathbf {R} \right)}$

where c is an arbitrary proportionality constant. Given our distribution function, there is a maxima corresponding to R = 0. Physically this amounts to there being more microstates which have an end-to-end vector of 0 than any other microstate. Now by considering

${\displaystyle S\left(\mathbf {R} \right)=k_{B}\ln \Omega {\left(\mathbf {R} \right)}}$

${\displaystyle \Delta S\left(\mathbf {R} \right)=S\left(\mathbf {R} \right)-S\left(0\right)}$

${\displaystyle \Delta F=-T\Delta S\left(\mathbf {R} \right)}$

where F is the Helmholtz free energy, and it can be shown that

${\displaystyle \Delta F=k_{B}T{\frac {3R^{2}}{2Nb^{2}}}={\frac {1}{2}}KR^{2}\quad ;K={\frac {3k_{B}T}{Nb^{2}}}.}$

which has the same form as the potential energy of a spring, obeying Hooke's law.

This result is known as the entropic spring result and amounts to saying that upon stretching a polymer chain you are doing work on the system to drag it away from its (preferred) equilibrium state. An example of this is a common elastic band, composed of long chain (rubber) polymers. By stretching the elastic band you are doing work on the system and the band behaves like a conventional spring, except that unlike the case with a metal spring, all of the work done appears immediately as thermal energy, much as in the thermodynamically similar case of compressing an ideal gas in a piston.

It might at first be astonishing that the work done in stretching the polymer chain can be related entirely to the change in entropy of the system as a result of the stretching. However, this is typical of systems that do not store any energy as potential energy, such as ideal gases. That such systems are entirely driven by entropy changes at a given temperature, can be seen whenever it is the case that are allowed to do work on the surroundings (such as when an elastic band does work on the environment by contracting, or an ideal gas does work on the environment by expanding). Because the free energy change in such cases derives entirely from entropy change rather than internal (potential) energy conversion, in both cases the work done can be drawn entirely from thermal energy in the polymer, with 100% efficiency of conversion of thermal energy to work. In both the ideal gas and the polymer, this is made possible by a material entropy increase from contraction that makes up for the loss of entropy from absorption of the thermal energy, and cooling of the material.