Molecular-level derivation

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.^[6] 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,^[7] 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_g 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_g (it does not really drop to zero but to the much lower value of the rubber elasticity plateau). By setting ${\displaystyle G(T_{g})=0}$ at the point where the G drops abruptly and solving for T_g, one obtains the following relation:^[6]

${\displaystyle {T_{g}}={\frac {1}{\alpha _{T}}}(1-C-\phi _{c}^{*}+2\Lambda )-{\frac {2\Lambda M_{0}}{\alpha _{T}M_{n}}}}$

In this equation, ${\displaystyle \phi _{c}^{*}}$ 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 ${\displaystyle z_{\rm {co}}=0}$. If the monomers can be approximated as soft spheres, then ${\displaystyle \phi _{c}^{*}\approx 0.64}$ as in the jamming of soft frictionless spheres.^[8] In the presence of covalent bonds between monomers, as is the case in the polymer, the packing fraction is lowered, hence ${\displaystyle \phi _{c}=\phi _{c}^{*}-\Lambda \cdot {z_{\rm {co}}}}$, where ${\displaystyle \Lambda }$ 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 ${\displaystyle T_{g}}$ via thermal expansion, according to ${\displaystyle \phi _{c}=\exp(-\alpha _{T}T_{g}-C)}$, which comes from integrating the thermodynamic relation between thermal expansion coefficient ${\displaystyle \alpha _{T}}$ and volume V, ${\displaystyle \alpha _{T}={\frac {1}{V}}\,\left(\partial V/\partial T\right)}$, where ${\displaystyle \alpha _{T}}$ is the coefficient of thermal expansion of the polymer in the glassy state. Note the relation between packing fraction ${\displaystyle \phi }$ and total volume given by ${\displaystyle \phi =vN/V}$, where ${\displaystyle N}$ is the total number of monomers, with molecular volume ${\displaystyle v}$, contained in the total volume ${\displaystyle V}$ of the material, which has been used above. Hence ${\displaystyle C}$ is the integration constant in ${\displaystyle \ln(1/\phi )=\alpha _{T}T+C}$, and it was found that ${\displaystyle C\approx 0.48}$ for the case of polystyrene. Also, ${\displaystyle M_{0}}$ 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 ${\displaystyle M_{n}}$, and provides a molecular-level meaning to the empirical parameters present in the Fox-Flory equation. Furthermore, it predicts that ${\displaystyle T_{g}\sim 1/\alpha _{T}}$, i.e. that the glass transition temperature is inversely proportional to the thermal expansion coefficient in the glass state.

Alternative equations

While the Flory–Fox equation describes many polymers very well, it is more reliable for large values of M_n ^[9] and samples of narrow weight distribution. As a result, other equations have been proposed to provide better accuracy for certain polymers. For example:

${\displaystyle T_{g}=T_{g,\infty }-{\frac {K}{(M_{n}M_{w})^{\frac {1}{2}}}}.}$

This minor modification of the Flory–Fox equation, proposed by Ogawa,^[10] replaces the inverse dependence on M_n  with the square of the product of the number-average molecular weight, M_n , and weight-average molecular weight, M_w . Additionally, the equation:

${\displaystyle {\frac {1}{T_{g}}}={\frac {1}{T_{g,\infty }}}+{\frac {K}{T_{g,\infty }^{2}}}{\frac {1}{M_{n}}}}$

was proposed by Fox and Loshaek,^[11] and has been applied to polystyrene, polymethylmethacrylate, and polyisobutylene, among others.

However, it is important to note that despite the dependence of T_g  on molecular weight that the Flory-Fox and related equations describe, molecular weight is not necessarily a practical design parameter for controlling T_g  because the range over which it can be changed without altering the physical properties of the polymer due to molecular weight change is small.^[9]