# softmatter:Flory-Huggins Chi Parameter

The Flory-Huggins lattice model gives a relation for qualitative comparison regarding the effects of polymer length, concentration, and temperature on the phase behavior of a mixture of two flexible molecules. The free energy of mixing ΔF is given as[1]:

where φ_{Y} refers to the volume fraction and N_{Y} is the length (number of lattice sites) occupied by each polymer *Y*. The expression χ*k*_{B}*T* is the energy cost per lattice site of moving an **A** type bead from a medium composed completely of **A** into a medium composed completely of **B**. The parameter χ is therefore related to the contact energy between lattice sites, or the energy cost of intermingling two species. For polymer systems, χ is typically related to temperature as follows:

where *a* and *b* are fitting parameters; this is not necessarily valid over all temperature ranges[1].

For a two component mixture of symmetric polymers, we have N_{A} = N_{B} and φ_{A} = (1 - φ_{B} ), and thus the free energy of of mixing is minimized when[2]:

.

This can be used to determine the relationship between, for instance, χ and ε / *k*_{B}*T*. The general procedure for determining such relationships can be found in references [2] and [3]. These papers determined the relationship between χ and ε / *k*_{B}*T* for systems using Lennard-Jones interactions[2] and χ and Δ*a* for systems using linear forces like those used in Dissipative Particle Dynamics[3].

## χ mapping for Lennard-Jones

For systems using Lennard-Jones interactions, the following χ mapping was found[2]:

χ = (9.48 + / − 0.11)ε / *k*_{B}*T* − 0.09) for Number density = 0.85.

For Lennard-Jones polymers systems with small polymer chains, the effective χ*N*, where N is the length of the polymers, was determined to be[2]:

for Number density = 0.85.

## χ mapping for Dissipative Particle Dynamics

For systems using linear forces like those used in DPD, the following χ mapping was found[3]:

χ = (0.286 + / − 0.002)Δ*a*, for Number Density = 3,

χ = (0.689 + / − 0.002)Δ*a*, for Number Density = 5.

## References

- [1] R.G. Larson The Structure and Rheology of Complex Fluids
- [2] M.A. Horsch, Z-L Zhang, C.R. Iacovella, S.C. Glotzer.
*Hydrodynamics and microphase ordering in blockcopolymers: Are hydrodynamics required for ordered phases with periodicity in more than one dimension?*Journal of Chemical Physics, 121(22): 11455-11462, (2004) - [3] R. D. Groot and P. B. Warren,
*Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic simulation*, Journal of Chemical Physics, 107, 4423-4435, (1997)