This simulation applies Flory–Huggins lattice theory to a binary mixture of A and B, where A is usually a polymer with NA segments and B is a solvent (NB = 1) or a second polymer (NB > 1). Each segment and each solvent molecule occupies one lattice site; φ is the volume fraction of A.
The left plot shows the Gibbs free energy of mixing per lattice site, divided by RT, at the selected temperature. Each curve is labeled directly on the plot. The curve is blue where a single phase is stable, orange where it is metastable, and red where it is unstable. When two phases form, the green dashed line is the common tangent, and its points of tangency (open circles) are the compositions of the coexisting phases. The red diamonds mark the spinodal points, the inflection points where the curvature of the free-energy curve changes sign; between them the curve is concave down and a single phase is unstable.
The right plot is the phase diagram. The binodal (blue) encloses the two-phase region; the spinodal (red dashed) encloses the unstable region. The bright green point is the critical point, where the binodal and spinodal meet. When two phases form, the green horizontal tie line connects the compositions of the coexisting phases (open circles). The black point is the overall composition φ0 at temperature T; drag it, or drag along the free-energy curve, to change φ0 and T. Use the buttons above the phase diagram to plot temperature or the interaction parameter on the vertical axis.
The interaction parameter depends on temperature as = α + β/T, where α is the temperature-independent (entropic) part and β is the enthalpic coefficient. With β > 0 the mixture has an upper critical solution temperature (UCST); with β < 0 it has a lower critical solution temperature (LCST). The NA and NB sliders (lattice sites per molecule) are logarithmic, 1 to 1000.
In the Flory–Huggins lattice model, A molecules occupy NA sites and B molecules occupy NB sites. The Gibbs free energy of mixing per lattice site, divided by RT, is the sum of a combinatorial entropy term and an interaction term:
where is the volume fraction of A. The entropy of mixing is reduced by a factor for each component because chain segments are connected; this is why polymers mix much less readily than small molecules. The Flory–Huggins interaction parameter is taken to depend on temperature as
A single phase is locally stable where the free-energy curve is concave up. The spinodal is where the curvature is zero:
Inside the spinodal () any small composition fluctuation lowers the free energy, and the mixture separates spontaneously by spinodal decomposition.
The binodal and spinodal meet at the critical point, where the second and third derivatives are both zero:
For a polymer in a solvent () with long chains, and , so the critical point shifts to very low polymer concentration. The critical temperature follows from .
Two phases with compositions and coexist when both components have equal chemical potentials in the two phases. Graphically, this is the common tangent to the curve, which requires equal slopes and equal intercepts:
where
These two nonlinear equations are solved numerically by Newton's method in the variables , which keeps the solution accurate when one phase is almost pure (for long chains the polymer-poor phase can have far below ). Between the binodal and the spinodal a single phase is metastable: it is stable to small fluctuations, but two phases have lower free energy, so separation requires nucleation.
When the overall composition lies inside the binodal, the fraction of lattice sites (the volume fraction of the mixture) in the A-rich phase is
The free energy of the two-phase mixture lies on the common tangent at (open square on the left plot), below the single-phase curve.
| fractions of the mixture (by volume) in the A-poor and A-rich phases | |
| Gibbs free energy of mixing (J) | |
| dimensionless Gibbs free energy of mixing per lattice site, | |
| number of lattice sites occupied by one molecule of A or B (≈ degree of polymerization for a polymer; 1 for a small-molecule solvent) | |
| total number of lattice sites (mol) | |
| ideal gas constant, 8.314 J/(mol K) | |
| temperature (K) | |
| critical solution temperature (K) | |
| temperature-independent (entropic) part of the interaction parameter (dimensionless) | |
| enthalpic coefficient of the interaction parameter (K) | |
| volume fraction of A | |
| overall volume fraction of A in the mixture | |
| volume fractions of A in the coexisting A-poor and A-rich phases | |
| volume fraction of A at the critical point | |
| Flory–Huggins interaction parameter (dimensionless) | |
| interaction parameter at the critical point |
This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.