# Hard sphere

Final Project for AP225 Fall 2011, written by Hyerim Hwang

## Contents

## Definition

Most of matter is made up of atoms and molecules, and then phase of the condensed matter is solid or liquid. That gives us the fact that there is an attractive force between molecules. There is also a repulsive force, which prevents matter from completely collapsing. The short-range interaction is essentially quantum mechanical; the electron orbitals of neighboring molecules begin to interact. Much of the liquids and solids assumes that the repulsion is infinite when the molecules overlap, but there is a certain long-range interaction for larger separation which is known as a hard sphere potential. In shortly, hard spheres are thought to be model particles in the statistical mechanical theory of fluids and solids which are studied by analytically with simulations. They cannot overlap in space and mimic the strong repulsion that atoms and spherical molecules experience at very close distances. In comparison with soft spheres, internal structure of hard spheres does not change with concentration. There is a change in soft spheres at high concentration due to overlapping and deformation of the surfaces.

## Colloidal Interactions

Entropic interactions are exlained in the context of suspensions of hard spheres. Hard-sphere colloids lack attractive and long-range interactions, which compete with entropic effects to produce ordered phases. It was noted that in mixtures of different size spherical particles an ordered arrangement of large spheres can increase the total entropy of the system by increasing the entropy of the small spheres. The box in the Figure 1 contains a few large spheres and many small spheres. The entropy of a small sphere depends on the number of positions it can occupy in the box. More free volume means more entropy for the small spheres. Since the center of mass of the small sphere cannot penetrate within a/2 of the large sphere surface, a region of "excluded volume" surrounds each large sphere. Figure 1 illustrates the interactions of hard spheres. The small sphere centers are excluded from the shaded blud regions. The red regions correspond to the overlap of excluded volumes which means the increased volume for small spheres.

## References

1. Israelachvili, Jacob N. (2011). Intermolecular and Surface Forces. Academic Press. ISBN 9780123919274.

2. Jones, Richard A. N. (2002). Soft Condensed Matter. University Press. ISBN 9780198505891.

3. Eckert, T., Richtering, W., 2008. "Thermodynamic and Hydrodynamic Interaction in Concentrated Microgel Suspensions: Hard or Soft Sphere Behavior?" J. Chem. Phys. 129, 124902. �

## Additional Readings

1. Cloitre, M. (2010). High Solid Dispersions (Advances in Polymer Science). Springer 1st Edition. ISBN 9780521864299.

2. Fischer, Earl K. (2008) Colloidal Dispersions. Fischer Press. ISBN 9781443729345.