Lennard–Jones Force and Well

The Lennard-Jones potential models intermolecular interactions, featuring a steep repulsive core and a longer-ranged attractive tail. The balance of these two terms creates a potential well that determines the equilibrium separation of two molecules.

Potential and Force

The potential energy V(r) depends on the distance r between particles, an energy scale epsilon, and a length scale sigma:

V(r) = 4 * epsilon * [(sigma/r)^12 - (sigma/r)^6]

The associated outward radial force is the negative derivative of the potential with respect to r:

F(r) = -dV/dr = 24 * epsilon / r * [2 * (sigma/r)^12 - (sigma/r)^6]

The potential goes to zero at r = sigma. The potential reaches a minimum of -epsilon at an equilibrium distance r_min = 2^(1/6) * sigma. At this minimum, the force is exactly zero. The curvature of the well near the minimum relates to the effective spring constant for small vibrations.

Finding the minimum

Consider two particles with Energy scale 1 and Length scale 1. Setting r to the minimum distance (r = 2^(1/6) * 1 ≈ 1.122):

The potential is evaluated as V = 4 * 1 * [(1/1.122)^12 - (1/1.122)^6] ≈ -1.0.
The outward radial force is F = 24 * 1 / 1.122 * [2 * (1/1.122)^12 - (1/1.122)^6] = 0.
The well depth is exactly the energy scale: 1.
The curvature at this minimum is 72 * 1 / (1.122^2) ≈ 57.146.

If the particles are pushed closer together to the zero crossing (r = 1), the potential becomes exactly 0.0, but the force becomes strongly positive (repulsive) at 24.0. Potential equal to zero does not imply force equal to zero.

Exploring the interaction

You can use the interactive module to see how the separation distance and scale parameters affect the potential energy and radial force.

Open the Lennard–Jones Force and Well interactive module. The simulation starts at the Minimum separation.
Notice the interacting pair shown at the top, and the corresponding point on the potential and force curves.
Change the Separation to "Zero crossing". Observe that the potential becomes zero, but the force shoots up to a positive value, indicating a strong repulsion.
Change the Separation to "Attractive". The potential rises slightly, and the force becomes negative, indicating an attractive pull trying to restore the particles to their minimum separation.
Adjust the Energy scale and Length scale to see how the well's depth and position change, and verify how the computed outputs respond.

Open Lennard–Jones Force and Well

Limitations of the Model

The Lennard-Jones potential is a simplified pairwise model. In reality, intermolecular forces arise from complex quantum mechanical interactions, including electron exchange and dispersion forces. It is not a complete multi-particle fluid simulation.

The r^-12 term is largely an empirical choice to model repulsion due to Pauli exclusion, chosen for mathematical convenience (as the square of the r^-6 dispersion term) rather than fundamental physical rigor. Additionally, as separation r approaches 0, the potential and force strictly diverge to infinity; they are never evaluated at exactly r = 0.

References

For more detailed information on intermolecular potentials and equations of state:

NIST TEQP Documentation: Model Potentials: Provides an overview of various pairwise potentials including the Lennard-Jones model used for thermophysical property modeling.

Lennard–Jones Force and Well