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#

Parameter

Typical Value

\( N \)

100–500

\( \varepsilon_{AA}, \varepsilon_{BB} \)

1.0

\( \varepsilon_{AB} \)

0.3–1.2

\( \sigma \)

1.0

Time step

0.001–0.005

\( T \)

0.5–2.0

Composition

A:B from 20:80 to 80:20