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# Kinetic chain length

In polymer chemistry the kinetic chain length of a polymer, ν , is the average number of monomers added to each polymerizing radical during polymerization. During this process, a polymer chain is formed when units called monomers are bonded together to form longer chains known as polymers. Kinetic chain length is defined as the average number of monomer units consumed for each radical initiator that begins the polymerization of a chain and is a more general development of the average degree of polymerization. The kinetic chain length can be calculated several ways, and its value can describe certain characteristics of the material, including chain mobility, glass-transition temperature, and modulus of elasticity.

## Calculating chain length

For step-growth polymerization, the average kinetic chain length is defined as the ratio of the number of propagation steps to the number of initiation steps:

$v = \frac{R_p}{R_i} = \frac{R_p}{R_t}$

where Rp is the rate of propagation, Ri is the rate of initiation of polymerization, and Rt is the rate of termination of the polymer chain. The second form of the equation is valid at steady-state polymerization, as the chains are being initiated at the same rate they are being terminated (Ri = Rt). [1]

An analogous equation can be written for living polymerization, a type of addition polymerization, and is usually written as:

$\ v = \frac{[M]_0-[M]}{[I]_0}$

where [M]0-[M] represents the number of monomer units consumed, and [I]0 the number of radicals that initiate polymerization. When the reaction goes to completion, [M]=0, and then the kinetic chain length is equal to the number average degree of polymerization of the polymer.

### Notes

• Kinetic chain length is an average quantity, as not all polymer chains are identical in length.
• The value of ν depends on the nature and concentration of both the monomer and initiator involved.
• Kinetic chain length can be calculated with or without chain transfer being considered.[2]

## Kinetic chain length without transfer

### Termination by disproportionation

Termination by disproportionation occurs when an atom is transferred from one polymer free radical to another. The atom is usually hydrogen, and this results in two polymer chains.

In this situation, the average kinetic chain length is equal to the number average degree of polymerization (DPn):

$\ v = DP_n$

### Termination by combination

With combination, two radicals are joined together, destroying the radicals on each of the two chains and forming one polymeric chain. Here, the average kinetic chain length is defined as:

$\ v = \frac{DP_n}{2}$

## Kinetic chain length with chain transfer

In the case of chain transfer, another atom (often hydrogen) is transferred from a molecule in the system to the polymer radical. The original polymer chain is terminated and a new one is initiated.[3] As a result, the kinetic chain length is shortened. Thus, the kinetic chain length is redefined as:

$\ v_{tr} = \frac{R_p}{R_t + R_{tr}}$

where Rtr is the rate of transfer. The greater Rtr is, the shorter the kinetic chain length.

## Significance

The chain length of the polymer is important in many aspects of its properties.

• Viscosity - Chain entanglements are very important in viscous flow behavior (viscosity) of polymers. As the chain becomes longer, chain mobility decreases; that is, the chains become more entangled with each other.
• Glass-transition temperature - An increase in chain length often leads to an increase in the glass-transition temperature, Tg. The increased chain length causes the chains to become more entangled at a given temperature. Therefore, a temperature does not need to be as low for the material to act as a solid.
• Modulus of Elasticity - A longer chain length is also associated with a material tends to be tougher and has a higher modulus of elasticity, E, also known as the Young's modulus. The interaction of the chains causes the polymer to become stiffer.

## References

1. ^ Hiemenz, Paul C., and Timothy P. Lodge. Polymer Chemistry. 2nd ed. Boca Raton, FL: CRC Press, 2007. 94-96.
2. ^ Professor Paula Hammond, 10.569 Synthesis of Polymers Fall 2006 materials, MIT OpenCourseWare (http://ocw.mit.edu/index.html), Massachusetts Institute of Technology, 7 Dec 2007.
3. ^ "Chain Transfer." IUPAC Compendium of Chemical Terminology. 1997. IUPAC. 6 Dec. 2007 .