page_title: Flory–Fox equation
section_title: Overview
hierarchical_section_title: Flory–Fox equation / Overview
caption_attribution_description: English: A cartoon showing a polymer (represented by the white circles) and the free volume associated with its different parts. Polymer ends have more free volume than units within the chain.
context_page_description: Flory function
  Synonym: viscosity function, SI unit: mol⁻¹
  Coefficient connecting the intrinsic viscosity, the mean-square radius of gyration, and the molar mass of a chain macromolecule, according to the equation
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  Kirkwood–Riseman theory
  Theory, based on the pearl-necklace model, that describes the translational diffusive and viscous flows of an isolated linear macromolecule in solution in the theta state and accounts for the gradual change from freedraining behaviour at lower molecular weights to impermeable behaviour at higher molecular weights.
  Flory–Fox assumption
  Assumption that the Kirkwood–Riseman theory can be applied to linear isolated macromolecules in solution, independent of whether they are in the theta state.
  In polymer chemistry and polymer physics, the Flory–Fox equation is a simple empirical formula that relates molecular weight to the glass transition temperature of a polymer system. The equation was first proposed in 1950 by Paul J. Flory and Thomas G. Fox while at Cornell University. Their work on the subject overturned the previously held theory that the glass transition temperature was the temperature at which viscosity reached a maximum.
context_section_description: 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.  It is this concept of “free volume” that is observed by the Flory–Fox equation.
  Free volume can be most easily understood as a polymer chain's “elbow room” in relation to the other polymer chains surrounding it.  The more elbow room a chain has, the easier it is for the chain to move and achieve different physical conformations.  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.  End chain units exhibit greater free volume than units within the chain because the covalent bonds that make up the polymer are shorter than the intermolecular nearest neighbor distances found at the end of the chain.  In other words, chain end units are less dense than the covalently bonded interchain units.  This means that a polymer sample with long chain lengths (high molecular weights) will have fewer chain ends per total units and less free volume than a polymer sample consisting of short chains.  In short, chain ends can be viewed as an “impurity” when considering the packing of chains, and more impurity results in a lower Tg. 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 the 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. In special cases, the free volume around hydrophilic main chain sites can exceed the free volume associated to the hydrophilic polymer ends.
  Thus, glass transition temperature is dependent on free volume, which in turn is dependent on the average molecular weight of the polymer sample.  This relationship is described by the Flory–Fox equation.  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 figure to the left clearly displays this relationship – as molecular weight increases, the glass transition temperature increases asymptotically to Tg,∞   (in this arbitrary case shown in the image, Tg,∞  = 365 K).
caption: A polymer chain (represented by the white circles) exhibits more free volume (represented by the gray shading) at the ends of the chain than from units within the chain.
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