# 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]:

$\frac{\Delta F}{k_BT} = \frac{\phi_A ln \phi_A}{N_A} + \frac{\phi_B ln \phi_B}{N_B} + \chi \phi_A \phi_B$

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:

$\chi = \frac{a}{T} + b$

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 NA = NB and φA = (1 - φB ), and thus the free energy of of mixing is minimized when[2]:

$\chi N_A = \frac{ln[(1-\phi_A)/\phi_A]}{1-2\phi_A}$.

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 [2] and [3]. These papers determined the relationship between χ and ε / kBT for systems using Lennard-Jones interactions[2] and χ and Δa 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 / kBT − 0.09) for Number density = 0.85.

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

$(\chi N)_{eff}= \frac{(9.48 +/- 0.11)\epsilon/k_BT-0.09)N}{1+3.9N^{-0.71}}$ 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)Δa, for Number Density = 3,

χ = (0.689 + / − 0.002)Δa, for Number Density = 5.