Flory–Fox equation
The equation was first proposed in 1950 by Paul J. Flory and Thomas G. Fox while at Cornell University.
[2] While its accuracy is usually limited to samples of narrow range molecular weight distributions, it serves as a good starting point for more complex structure-property relationships.
Recent molecular simulations have demonstrated that while the functional form of the Flory-Fox relation holds for a wide range of molecular architectures (linear chain, bottlebrush, star, and ring polymers), however, the central free-volume argument of the Flory-Fox relation does not hold since branched polymers, despite having more free ends, form materials of higher density and glass transition temperature increases.
[3] [4] The Flory–Fox equation relates the number-average molecular weight, Mn, to the glass transition temperature, Tg, as shown below: where Tg,∞ is the maximum glass transition temperature that can be achieved at a theoretical infinite molecular weight and K is an empirical parameter that is related to the free volume present in the polymer sample.
Free volume decreases upon cooling from the rubbery state until the glass transition temperature at which point it reaches some critical minimum value and molecular rearrangement is effectively “frozen” out, so the polymer chains lack sufficient free volume to achieve different physical conformations.
This ability to achieve different physical conformations is called segmental mobility.
Free volume not only depends on temperature, but also on the number of polymer chain ends present in the system.
Recent computer simulation study showed that the classical picture of mobility around polymer chain can differ in the presence of plasticizer, especially if molecules of plasticizer can create hydrogen bonds with specific sites of the polymer chain, such as hydrophilic or hydrophobic groups.
In such a case, polymer chain ends exhibit only a mere increase of the associated free volume as compared to the average associated free volume around main chain monomers.
Low molecular weight values result in lower glass transition temperatures whereas increasing values of molecular weight result in an asymptotic approach of the glass transition temperature to Tg,∞ .
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 Tg because of the weakening of the non-covalent interactions, while the negative nonaffine term is less affected by temperature.
at the point where the G drops abruptly and solving for Tg, one obtains the following relation:[6] 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
[8] In the presence of covalent bonds between monomers, as is the case in the polymer, the packing fraction is lowered, hence
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
, and provides a molecular-level meaning to the empirical parameters present in the Fox-Flory equation.
While the Flory–Fox equation describes many polymers very well, it is more reliable for large values of Mn [9] and samples of narrow weight distribution.
For example: This minor modification of the Flory–Fox equation, proposed by Ogawa,[10] replaces the inverse dependence on Mn with the square of the product of the number-average molecular weight, Mn , and weight-average molecular weight, Mw .
Additionally, the equation: 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 Tg on molecular weight that the Flory-Fox and related equations describe, molecular weight is not necessarily a practical design parameter for controlling Tg 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] The Flory–Fox equation serves the purpose of providing a model for how glass transition temperature changes over a given molecular weight range.
Another method to modify the glass transition temperature is to add a small amount of low molecular weight diluent, commonly known as a plasticizer, to the polymer.
The presence of a low molecular weight additive increases the free volume of the system and subsequently lowers Tg , thus allowing for rubbery properties at lower temperatures.
This effect is described by the Fox equation: Where w1 and w2 are weight fractions of components 1 and 2, respectively.
In general, the accuracy of the Fox equation is very good and it is commonly also applied to predict the glass transition temperature in (miscible) polymer blends and statistical copolymers.
