text: The main shortcoming related to the free volume concept is that it is not so well defined at the molecular level. A more precise, molecular-level derivation of the Flory–Fox equation has been developed by Alessio Zaccone and Eugene Terentjev. The derivation is based on a molecular-level model of the temperature-dependent shear modulus G of glassy polymers. The shear modulus of glasses has two main contributions, one which is related to affine displacements of the monomers in response to the macroscopic strain, which is proportional to the local bonding environment and also to the non-covalent van der Waals-type interactions, and a negative contribution that corresponds to random (nonaffine) monomer-level displacements due to the local disorder. Due to thermal expansion, the first (affine) term decreases abruptly near the glass transition temperature T because of the weakening of the non-covalent interactions, while the negative nonaffine term is less affected by temperature. Experimentally, it is observed indeed that G drops sharply by many orders of magnitude at or near T (it does not really drop to zero but to the much lower value of the rubber elasticity plateau). By setting  at the point where the G drops abruptly and solving for T, one obtains the following relation:
  In this equation,  is the maximum volume fraction, or packing fraction, occupied by the monomers at the glass transition if there were no covalent bonds, i.e. in the limit of average number of covalent bonds per monomer . 
  If the monomers can be approximated as soft spheres, then  as in the jamming of soft frictionless spheres.  
  In the presence of covalent bonds between monomers, as is the case in the polymer, the packing fraction is lowered, hence , where  is a parameter that expresses the effect of topological constraints due to covalent bonds on the total packing fraction occupied by the monomers in a given polymer.
  Finally, the packing fraction occupied by the monomers in the absence of covalent bonds is related to  via thermal expansion, according to , which comes from integrating the thermodynamic relation between thermal expansion coefficient  and volume V, , where  is the coefficient of thermal expansion of the polymer in the glassy state. Note the relation between packing fraction  and total volume given by , where  is the total number of monomers, with molecular volume , contained in the total volume  of the material, which has been used above. Hence  is the integration constant in , and it was found that  for the case of polystyrene. Also,  is the molecular weight of one monomer in the polymer chain.
  Hence the above equation clearly recovers the Flory–Fox equation with its dependence on the number average molecular weight , and provides a molecular-level meaning to the empirical parameters present in the Fox-Flory equation. Furthermore, it predicts that , i.e. that the glass transition temperature is inversely proportional to the thermal expansion coefficient in the glass state.