Analyze miscibility limits with responsive inputs and clear outputs. Review equations, examples, and downloadable reports. Built for researchers studying coexistence curves in binary systems.
These example values are generated with N1 = 1 and N2 = 1 using χ(T) = 520/T + 0.10. They illustrate how the coexistence gap widens as χ rises.
| Temperature | χ | Binodal Lower | Binodal Upper |
|---|---|---|---|
| 266.666667 | 2.050000 | 0.366075 | 0.633925 |
| 253.658537 | 2.150000 | 0.277713 | 0.722287 |
| 236.363636 | 2.300000 | 0.203923 | 0.796077 |
| 208.000000 | 2.600000 | 0.123971 | 0.876029 |
The calculator uses the Flory-Huggins free-energy density for a binary mixture:
f(φ) = (φ/N1) ln(φ) + ((1-φ)/N2) ln(1-φ) + χφ(1-φ)
The binodal points φα and φβ satisfy a common tangent:
f′(φα) = f′(φβ) and f(φβ) - f(φα) = f′(φα)(φβ - φα)
The spinodal comes from the second derivative:
d²f/dφ² = 1/(N1φ) + 1/(N2(1-φ)) - 2χ = 0
The critical values are:
χc = 0.5(1/√N1 + 1/√N2)² and φc = √N2 / (√N1 + √N2)
A binodal curve marks coexistence compositions where a single mixture separates into two equilibrium phases. It is central in polymer physics, alloy thermodynamics, solution theory, and liquid-liquid phase separation studies. Comparing the binodal and spinodal helps identify metastable and unstable regions across composition and temperature space.
The binodal curve shows the two compositions that can coexist at equilibrium under one temperature or interaction parameter. Inside the gap, the mixture prefers separation into two phases rather than staying uniform.
The binodal marks equilibrium coexistence limits. The spinodal marks absolute instability. Between them, the mixture is metastable and usually needs nucleation. Inside the spinodal, fluctuations can grow spontaneously.
N1 and N2 control the entropy terms in the Flory-Huggins model. They often represent degree of polymerization or segment count. Changing them shifts the critical point and the shape of the coexistence boundary.
χ is the interaction parameter. Larger χ usually means less favorable mixing and stronger tendency toward phase separation. When χ rises above the critical value, a miscibility gap can appear.
Yes. The calculator supports χ(T)=A/T+B, which is a common approximation in solution thermodynamics. It lets you explore how changing temperature modifies the coexistence curve and critical behavior.
If the current χ is below the critical value, no binodal pair exists in this model. The mixture remains homogeneous for the chosen parameters, so the coexistence compositions are not returned.
No. They are numerical results from the selected theoretical model and your inputs. They are useful for analysis, estimation, and comparison, but experimental systems may require fitted parameters or richer thermodynamic descriptions.
Use a consistent unit system for temperature and the constants in χ(T). Volume fraction φ stays dimensionless. The model will only be meaningful when A, B, and temperature are supplied consistently.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.