Question

What is the maximum and minimum values for average end-to-end distance in polymers? What is the probablity distribution function for the chain length between these range? Drive the equation for root-mean square end to end distance for N bonds, each with the length of 1 and no steric hindrance and all the chains are freely rotated?

Answer 1

the end to end distance represents the average distance between the first and the last segment of the polymer, and ranges between a maximum value and a minimum value. The maximum value appears when chains are fully extended, in a planar, zigzag configuration known as "all-trans", where the contour length can be calculated easily. The minimum value corresponds to the sum of Van der Waals radii in each end.

maximum value : minimum value :

The size of the polymer is given, in statistical terms, by the mean-square end-to-end distance, r². Other authors express the root mean-square end-to-end distance, that is to say, r^2½. The magnitude r² is defined according to:

Where W is a probability distribution function. let's consider the simplest model of a polymer chain, i.e. an ideal polymer, consisting of a series of N segments of length L. Let's assume that the chain segments are bonded according to a linear sequence, without any restriction regarding bonding angles w and internal angles of rotation l , so that the atoms are separated each other at fixed distances but located in any direction. Thus, the calculation of r2 can be made by means of a procedure known as random flight. According to this procedure and following a mathematical reasoning equation can be re-written as follows: ( if we take L as1 then r2f  = N ) Where subscript f indicates that a random flight approximation is being considered, originating a model known as freely jointed chain. Actually Factors like solvent type, chain type, and the groups attached to the polymer backbone, do cause interactions, generating deviations from the freely jointed chain model. For this reason r2  is higher than that obtained by calculation of the random flight. These interactions can be divided in: short range interactions and long range interactions. Short range interactions are related to the structural characteristics of the macromolecule, considering bond types and the interactions between segments or neighboring atoms. These factors originate steric repulsions, which limit the values of the internal angles of rotation since in such a case, they are not all equally probable. The magnitude of this effect is related to the size of the substituent groups. Hence the random coil will expand itself, in order to avoid such repulsions. This model is known as unperturbed dimension, since neither interactions between non-neighboring chain segments nor solvent interactions (long range interactions, to be discussed later) are being considered. To this end, the mean-square end-to-end distance of the unperturbed dimension,r2o, is expressed as follows: Where r2fr represents the mean-square end-to-end distance of the free rotation chain, that is to say, under the condition that the bonding angles w remain fixed, independently of the presence of substituent groups. The s factor, referred to as conformation factor, is a parameter related to the impediments to rotation that real chains show, compared to that with a free chain rotation. The s factor depends on temperature and sometimes on the solvent, and offers interesting information about the conformation of a certain macromolecule and usually, it is increased in the presence of bulky groups. Although the corrections introduced by the short range interactions offer a more approximate description of real macromolecules in dilute solutions, such interactions do not contemplate the behavior of non-neighboring chain segments, each one occupying a certain volume from which all the other segments are excluded. Such effect, known as excluded volume and its influence on the dimensions of the macromolecular chains, has been the subject of numerous studies for a long time. Its theoretical calculation has been carried out by means of statistic and the aid of computer simulation. To this end, in order to make possible the calculation of the end-to-end distance considering the excluded volume effects, the long range interactions have been introduced, which consider both solvent interactions and interactions between atoms or non-neighboring segments. One may speculate that the long range interactions can produce a bigger chain expansion over its unperturbed dimensions, since now, due to the excluded volume effect, such conformations where two remote segments can occupy the same space at the same time should be eliminated. Hence, the end-to end distance is given by: Where a is the linear expansion factor, and r2o the end-to-end distance of the unperturbed dimension. the r21/2 is the R.m.s. end to end distance.