Hard spheres
Encyclopedia
Hard spheres are widely used as model particles in the statistical mechanical
theory of fluids and solids. They are defined simply as impenetrable spheres that cannot overlap in space. They mimic the extremely strong repulsion that atoms and spherical molecules experience at very close distances. Hard spheres systems are studied by analytical means, by molecular dynamics
simulations, and by the experimental study of certain colloid
al model systems.
are particles with the following pairwise interaction potential:
where
and 
are the positions of the two particles.
s for hard spheres can be determined analytically
Higher-order ones can be determined numerically using Monte Carlo integration. We list
A table of virial coefficients for up to eight dimensions can be found on the page Hard sphere: virial coefficients.
.
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
theory of fluids and solids. They are defined simply as impenetrable spheres that cannot overlap in space. They mimic the extremely strong repulsion that atoms and spherical molecules experience at very close distances. Hard spheres systems are studied by analytical means, by molecular dynamics
Molecular dynamics
Molecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...
simulations, and by the experimental study of certain colloid
Colloid
A colloid is a substance microscopically dispersed evenly throughout another substance.A colloidal system consists of two separate phases: a dispersed phase and a continuous phase . A colloidal system may be solid, liquid, or gaseous.Many familiar substances are colloids, as shown in the chart below...
al model systems.
Formal definition
Hard spheres of diameter are particles with the following pairwise interaction potential:where
and are the positions of the two particles.Hard-spheres gas
The first three virial coefficientVirial coefficient
Virial coefficients B_i appear as coefficients in the virial expansion of the pressure of a many-particle system in powers of the density, providing systematic corrections to the ideal gas law...
s for hard spheres can be determined analytically
 |
= |  |
 |
= |  |
 |
= |  |
Higher-order ones can be determined numerically using Monte Carlo integration. We list
 |
= |  |
 |
= |  |
 |
= |  |
A table of virial coefficients for up to eight dimensions can be found on the page Hard sphere: virial coefficients.
Hard-spheres liquid
The static structure factor of the hard-spheres liquid can be calculated using the Percus–Yevick approximationPercus–Yevick approximation
In statistical mechanics the Percus–Yevick approximation is a closure relation to solve the Ornstein–Zernike equation. It is also referred to as the Percus–Yevick equation. It is commonly used in fluid theory to obtain e.g...
.
Literature
- J. P. Hansen and I. R. McDonald Theory of Simple Liquids Academic Press, London (1986)
- Hard sphere model page on SklogWiki.