Ideal Chain - The Model

The Model

N monomers form the polymer, whose total unfolded length is:

, where N is the number of monomers.

In this very simple approach where no interactions between monomers are considered, the energy of the polymer is taken to be independent of its shape, which means that at thermodynamic equilibrium, all of its shape configurations are equally likely to occur as the polymer fluctuates in time, according to the Maxwell-Boltzmann distribution.

Let us call the total end to end vector of an ideal chain and the vectors corresponding to individual monomers. Those random vectors have components in the three directions of space. Most of the expressions given in this article assume that the number of monomers N is large, so that the central limit theorem applies. The figure below shows a sketch of a (short) ideal chain.

The two ends of the chain are not coincident, but they fluctuate around each other, so that of course:

Throughout the article the brackets will be used to denote the mean (of values taken over time) of a random variable or a random vector, as above.

Since are independent, it follows from the Central limit theorem that is distributed according to a normal distribution (or gaussian distribution): precisely, in 3D, and are distributed according to a normal distribution of mean 0 and of variance:

So that . The end to end vector of the chain is distributed according to the following probability density function:

The average end-to-end distance of the polymer is:

A quantity frequently used in polymer physics is the radius of gyration:

It is worth noting that the above average end-to-end distance, which in the case of this simple model is also the typical amplitude of the system's fluctuations, becomes negligible compared to the total unfolded length of the polymer at the thermodynamic limit. This result is a general property of statistical systems.

Mathematical remark: the rigorous demonstration of the expression of the density of probability is not as direct as it appears above: from the application of the usual (1D) central limit theorem one can deduce that, and are distributed according to a centered normal distribution of variance . Then, the expression given above for is not the only one that is compatible with such distribution for, and . However, since the components of the vectors are uncorrelated for the random walk we are considering, it follows that, and are also uncorrelated. This additional condition can only be fulfilled if is distributed according to . Alternatively, this result can also be demonstrated by applying a multidimensional generalization of the central limit theorem, or through symmetry arguments.