# softmatter:Square-Well Potential

The Square-Well potential (solid) with the The Lennard-Jones Potential (dotted).

The square-well potential is identical to the Hard Sphere Potential with the addition of an attractive well of depth − ε. Mathematically, this is expressed as follows:

$U_{HS}(r) = \begin{cases} \infty,&r<\sigma\\ -\epsilon, & \sigma < r < r_c\\ 0, & r \geq r_c \end{cases}$

where σ is the diameter of the particle and rc is the cutoff.

The square-well potential is often thought of as a coarse-grained treatment of The Lennard-Jones Potential because it also takes into account both excluded volume interaction and attractive interactions. Similar to the Hard Sphere Potential, the square-well potential is not numerically integrateable and cannot be used in Molecular Dynamics Simulation (MD), Brownian Dynamics Simulation (BD), or Dissipative Particle Dynamics Simulation (DPD). Potentials such as this are typically used in Monte Carlo Simulation (MC) methods or in event-based, collision dynamics methods (also called discontinuous MD).