Degree of Polymerization of Polymers

The Degree of polymerization (DP or Xn) is defined as number of repeating monomer units in the polymers.

The Degree of a polymerization is calculated by taking ratio of molecular weight of the polymer and molecular weight of the repeat unit.

Degree of Polymerization Formula

What is Polymerization?

Polymers derived from word “poly” means many and “mers” mean parts, are long chain molecules formed by repetition of many small simple chemical repeating units.

The repeating units are termed as monomers. These monomers can either repeat linearly or in the form of chains which are connected as branches to form a three-dimensional network. The monomers are linked with one-another via chemical reactions by a process called polymerization.

Types of Polymerization

In 1953, Paul Flory classified polymerization reactions in two groups; step growth polymerization (also called condensation reactions) and chain polymerization (also known as addition reactions).

Types of Polymerization
Types of Polymerization

Step Polymerization

In Step Growth Polymerization, bi-functional or multifunctional monomers react to first form dimers, then trimers, longer oligomers and then long chain polymers.

The mechanism of step polymerization is just like a group of people reaching out each other with their hands to form a human chain, in case of polymers the reactive sites act as two hands which join with one another to form a polymer chain.

As there can be more than one reactive site on a monomer so there is a high extent of polymerization resulting in a high molecular chain. Polyesters, polyamides, polyurethanes etc. are few common examples of step polymerization.

Difference between Step growth and Chain Growth Polymerization
Difference between Step growth and Chain Growth Polymerization

Chain Polymerization

In Chain polymerization, the reaction takes place by an active center which is a free radical or ion to form a macromolecule. The polymer growth takes place at the end points of the chains hence it is also termed as addition polymerization.

The mechanism of reaction involves three steps, initiation, propagation and termination.

Many common polymers are formed by chain polymerization such as polyethylene (PE), polyvinyl chloride (PVC), polypropylene (PP), polyimide (PI), polymethyl methacrylate, polyvinyl acetate.

How to Calculate Degree of Polymerization

Degree of polymerization can be calculated by dividing polymer’s molecular weight to repeating unit’s molecular weight.

Carothers Equation for Degree of Polymerization

Classification of Degree of Polymerization

The Degree of Polymerization is classified in two forms, the number-average degree of polymerization and weight-average degree of polymerization.

The number-average degree of polymerization is the weighted mean of degrees of polymerization, weighted by the mole fractions (or the number of molecules) of the polymer. It is experimentally determined by measurements of the osmotic pressure of the polymer.

The weight-average degree of polymerization is the weighted mean of degrees of polymerization, weighted by the weight fractions (or the overall weight of the molecules) of polymer. It is experimentally determined by measurements of Rayleigh light scattering by the polymer.

Degree of polymerization (DP) of Step-growth polymerization

In step-growth polymerization, the DP is calculated by Carothers’ equation  which is defined for linear and branched molecules.  For linear polymer formed by the reaction (usually by condensation) of two monomers in equimolar quantities, the equation of DP is:

Carothers Equation for Degree of Polymerization

In Linear polymerization where one monomer is in excess quality the equation is modified as:

Degree of Polymerization for linear polymerization

Where, r is the stoichiometric ratio of reactants, as the excess reactant is generally placed in denominator so r < 1 always.

The excess reactant is generally added to control the DP of polymers for a given value of p. When the limiting reagent monomer is added in limit of complete conversion, p → 1 and

Limiting Reagent monomer and DOP

When a monomer has functionality greater than one, it results in branching of polymer. In this case, the DP of polymers will depend on the average functionality of the monomer fav. The fav for a given reaction can be calculated as:

Formula for Functionality of Monomer

The modified Carothers equation for branched step polymerization will be:

Modified Carothers Equation for DOP

Degree of polymerization (DP) of Chain-growth polymerization

The degree of polymerization of chain-growth polymerization, cannot be calculated by Carothers.

At the beginning of reaction long chains are formed and longer reaction times have little effect on the average molecular weight it only increases the polymer yield. Hence, the DP of polymers given by formula below, is associated with the kinetic chain length, which is the average number of monomers polymerized for each chain initiated.