## Cohesive Energy and Lennard-Jones Potential

The cohesive energy is one of the most important properties of polymers. It can be derived from the depth of the molecular potential well. If no specific interactions are present, the well depth is the minimum of the Lennard-Jones potential:

*φ* = *φ _{0}* · [(

*r*/

_{0}*r*)

^{12}- 2(

*r*/

_{0}*r*)

^{6}]

*φ _{0}* is the energy responsible for the strength of a material. To be more specific, it is the energy that has to be overcome by mechanical and thermal energies to deform the material, and therefore, determines the magnitude of the bulk thermo-mechanical properties of a polymer.
The equivalent bulk parameter is the

*cohesive energy*,

*E*. It is defined as the energy per mole required to eliminate all intermolecular forces.

_{coh}^{2}

The volume of a repeat unit can be described in terms of its
molar van der Waals volume. This volume is connected to the intermolecular distance *r* by

*V* = *N _{avo}*

*r*

^{3}/

*q*

where *N _{avo}* is the Avogadro's number and

*q*is a constant that corrects for the geometry of the repeat units. On substituting the molecular distances with the volumes, one arrives at the following relationship between volume and intermolecular distance:

*V _{0}* /

*V*= (

*r*/

_{0}*r*)

^{3}

where *V _{0}* is the molar volume at the minimum in the potential well
which is set equal to the molar volume of the (glassy) polymer at 0 K. If we assume a hexagonal interaction cells, then there are six interactions per unit cell with two mer units per interaction, i.e. 3

*φ*is the total interaction energy per unit cell. The corresponding volume of this interaction is approximately four mer unit volumes, so that the energy density equals

_{0}*e _{0}* = 3

*N*/ (4

_{avo}φ_{0}*V*)

or

*φ _{0}* = 4

*E*/ 3

_{coh}*N*≈

_{avo}*E*/ 4.5 · 10

_{coh}^{23}

where *E _{coh}* is the cohesive energy in units of J/mol and

*φ*is in units of J/(mer-unit interaction). Substituting the expressions for

_{0}*φ*and

_{0}*r*/

_{0}*r*into the Lennard-Jones equation, the total potential energy per mole of a material reads

*W* ≈ *K E _{coh}* · [(

*V*/

_{0}*V*)

^{4}- 2(

*V*/

_{0}*V*)

^{2}]

where *K* is a numerical fitting parameter which has a value of about two
at room temperature.^{3} The cohesive energy has a considerable advantage over *φ _{0}* as an input parameter in models.
For example, it can be calculated with group contribution methods and for many compounds
experimental values have been reported whereas the literature on

*φ*is rather sparse.

_{0}##### References and Notes

- FD. Porter,
*Group Interaction Modelling of Polymer Properties*, Marcel Dekker, New York (1995). -
In the case of low molecular weight compounds, the cohesive energy is equal to the energy required to evaporate the material:

*E*≈ Δ_{coh}*H*-_{vap}*RT* - J.T. Seitz,
*J. of Appl. Poly. Sci.*, Vol. 49, 1331-1351 (1993)