Radial Distribution Function for One-Dimensional Square-Well and Square-Shoulder Fluids

interaction potential

square well

square shoulder

square (shoulder) width

0.5

temperature

packing fraction

0.3

function

g(r)

S(k)

β P

= 1.149

∂P

∂ρ

= 1.422

= -0.338

In statistical mechanics, the radial distribution function represents the distribution of interparticle separations[1].

This Demonstration shows the results of exact statistical-mechanical computations of the radial distribution function

g(r)

and the structure factor

S(k)

[2] for a one-dimensional system of particles interacting via square-well and square-shoulder potentials[3, 4]. We also find the values of the ratio

βP/ρ

(where

β=1/kT

is the inverse temperature,

is the pressure, and

is the number density), the scaled inverse isothermal compressibility

β∂P/∂ρ

, and the excess internal energy per particle

/ϵ

. Sliders let you control the width of the square well or shoulder, the reduced temperature, and the packing fraction. The quantities

βP/ρ

and

/ϵ

are also plotted as functions of the packing fraction.

Details

In this one-dimensional system, the particles interact with an impenetrable hard core of length

plus either an attractive square-well or a repulsive square-shoulder potential of energy

and width

. This can be represented by a pairwise potential-energy function

V(r)

. The exact solution for the radial distribution function

g(r)

is worked out in references[3] and[4]. As a very rudimentary approximation,

g(r)≈

-V(r)/kT

. The structure factor is given by

S(k)=1+ρ∫(g(r)-1)

ikr

References

[1] Wikipedia. "Radial Distribution Function." (Apr 10, 2015) en.wikipedia.org/wiki/Radial_distribution_function.

[2] Wikipedia. "Structure Factor." (Apr 10, 2015) en.wikipedia.org/wiki/Structure_factor.

[3] Z. W. Salsburg, R. W. Zwanzig, and J. G. Kirkwood, "Molecular Distribution Functions in a One-Dimensional Fluid," The Journal of Chemical Physics, 21(6), 1953 pp. 1098–1107. doi:10.1063/1.1699116.

[4] S. B. Yuste and A. Santos, "A Model for the Structure of Square-Well Fluids," The Journal of Statistical Physics, 101(3), 1994 pp. 2355–2364. doi:10.1063/1.467676.

External Links

Radial Distribution Function for Sticky Hard Rods

Radial Distribution Function for Hard Spheres

Permanent Citation

Andrés Santos

"Radial Distribution Function for One-Dimensional Square-Well and Square-Shoulder Fluids"
http://demonstrations.wolfram.com/RadialDistributionFunctionForOneDimensionalSquareWellAndSqua/
Wolfram Demonstrations Project
Published: April 17, 2015