softmatter:Flory-Huggins Chi Parameter
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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:
where φY refers to the volume fraction and NY is the length (number of lattice sites) occupied by each polymer Y. The expression χkBT 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.
For a two component mixture of symmetric polymers, we have NA = NB and φA = (1 - φB ), and thus the free energy of of mixing is minimized when:
This can be used to determine the relationship between, for instance, χ and ε / kBT. The general procedure for determining such relationships can be found in references  and . These papers determined the relationship between χ and ε / kBT for systems using Lennard-Jones interactions and χ and Δa for systems using linear forces like those used in Dissipative Particle Dynamics.
χ mapping for Lennard-Jones
For systems using Lennard-Jones interactions, the following χ mapping was found:
χ = (9.48 + / − 0.11)ε / kBT − 0.09) for Number density = 0.85.
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:
χ = (0.286 + / − 0.002)Δa, for Number Density = 3,
χ = (0.689 + / − 0.002)Δa, for Number Density = 5.
-  R.G. Larson The Structure and Rheology of Complex Fluids
-  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)
-  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)