Final Project: Phase Separation in Binary Lennard-Jones Mixtures

Background¶

Many soft materials—such as polymer blends, lipid membranes, and even biological condensates—undergo phase separation when composed of two or more types of molecules. Whether a mixture remains homogeneous or separates into distinct domains depends on interaction strengths, temperature, and composition.

In this project, you will simulate a 2D (or 3D) binary mixture of LJ particles—types A and B—and investigate how varying parameters leads to mixing or phase separation.

You’ll explore:

• How particle identity affects interaction potentials

• How composition and temperature influence morphology

• How to construct a phase diagram from simulation data

Project Goals¶

• Implement binary Lennard-Jones simulations of A/B particles

• Study how tuning interspecies interaction ($\varepsilon_{AB}$) affects mixing

• Identify when and how phase separation occurs

• Estimate a phase diagram in the temperature–composition plane

Tasks¶

1. Modify MD Code for Binary System¶

• Introduce two particle types: A and B.

• Define three interaction strengths:

  • $\varepsilon_{AA} = 1.0$

  • $\varepsilon_{BB} = 1.0$

  • $\varepsilon_{AB} =$ tunable parameter (e.g., 0.3–1.2)

• Use the same $\sigma$ for simplicity, or explore asymmetric sizes.

Tip: Maintain an array of particle types, and loop over all particle pairs to assign the correct LJ parameters.

2. Initialize Mixtures¶

• Start with a random mixture of A and B particles (e.g., 50/50 composition).

• Use periodic boundary conditions and equilibrate for a long time.

3. Visualize and Detect Phase Separation¶

• Visualize particle positions, coloring A and B differently.

• At low $T$ and low $\varepsilon_{AB}$, you should observe domain formation (e.g., A-rich and B-rich regions).

• At high $\varepsilon_{AB}$ or high $T$, the system should remain mixed.

4. Scan Parameters and Construct a Phase Diagram¶

• Run simulations at various:

  • Temperatures (e.g., from 0.5 to 2.0 in LJ units)

  • Compositions (e.g., from 20:80 to 80:20 A:B)

  • $\varepsilon_{AB}$ values (e.g., 0.3 to 1.2)

• For each combination, record whether the system appears mixed or phase separated (by visual inspection or density profiles).

• Plot a phase diagram in the temperature–composition plane for a fixed $\varepsilon_{AB}$, showing the boundary between mixed and demixed regions.

5. Challenges (Optional)¶

• Measure radial distribution functions $g_{AA}(r)$, $g_{AB}(r)$, $g_{BB}(r)$

• Extract structure factor $S(k)$ to quantify ordering length scale

• Use clustering algorithms to identify domain size distributions

• Simulate in 3D or use anisotropic particles (e.g., dumbbells)

Learning Outcomes¶

• Understand microscopic origins of demixing in multi-component systems

• Learn how intermolecular forces determine macroscopic phase behavior

• Develop skills in parameter scanning and phase diagram construction

• Explore techniques for visualization and qualitative assessment of structure

Suggested Parameters¶