     Next: The Ornstein-Zernike equation and Up: Integral equations for polymer Previous: Flory's hypothesis

Consider a simple atomic liquid  or colloidal suspension . The average density at a distance r of a given particle is defined to be , i.e. (2.3)

Here is the average density in the fluid. Notice that is a conditional density; it is the density at , given a particle is present in the origin. A qualitative picture of g(r) is given in Fig. (2.1); it is called the radial distribution function .

It is clear that g(r) should go to 1 for large r. At very short rthe radial distribution function must be zero, because two particles cannot occupy the same space. Outside the van der Waals diameter, there is a peak because the remaining N-1 particles try to diffuse into the region occupied by the one at the origin. The surplus of particles in the first peak causes a lack of particles a little bit further on, explaining the minimum of g(r) around . Radial distribution functions may be measured by means of scattering experiments, for example by means of X-ray or neutron scattering   in the case of atomic liquids, or by means of SAXS, SANS or light scattering  in the case of colloidal suspensions. Generally the sample produces a scattering amplitude (2.4)

at the detector. Here is the scattering power at position , and is defined by (2.5)

where and are the wavevectors  of the outgoing and incoming wave respectively. The measured intensity is proportional to (2.6)

Writing the scattering power like (2.7)

we find after some algebra = (2.8) = (2.9)

The important quantity for us is = = = (2.10)

The -function only gives a contribution at , i.e. to the forward scattering. At all other k-values one measures the scattering factor (2.11)

i.e. essentially the Fourier transform of the total correlation function h(r)=g(r)-1.

Once we have determined the radial distribution function we may calculate several quantities of interest, like (2.12) (2.13)

where is the interaction potential  between two particles in the system, and Uand P are the thermodynamic energy and pressure  respectively. Another important relation is the compressibility equation (2.14)

For more information we refer to any book on liquid state theory, like for example J. -P. Hansen  and I. R. McDonald , Theory of simple liquids.     Next: The Ornstein-Zernike equation and Up: Integral equations for polymer Previous: Flory's hypothesis
W.J. Briels