Interacting Molecules in a Dense Fluid

Reduced Spatial Probability Density Functions Let us start from the configuration integral for a system of N identical atoms:

Reduced Spatial Probability Density Functions As in the evaluation of virial coefficients, we can choose any convenient reference for the origin of the coordinate system, for example the position of molecule 1. The configuration integral becomes: Additional simplification is not possible because we can choose the reference system only once.

Reduced Spatial Probability Density Functions We will take a different approach. Consider a system of N identical (but distinguishable) atoms in a volume V at temperature T. The goal is to find the probability of finding molecules at certain positions near one another. We will use a probability density function to determine the probability of finding a molecule in a volume element around a location .

Reduced Spatial Probability Density Functions The probability that molecule A is in a small volume element around a location is equal to: The probability that molecule B is in a small volume element around a location is equal to:

Reduced Spatial Probability Density Functions The probability that molecules A or B are in this small volume element is The probability that any of the N molecules are in this small volume element is:

Reduced Spatial Probability Density Functions Now consider the probability that molecule A is in a small volume element around a location and that molecule B is in a small volume around a location and so forth for particles 3, 4, 5... This is more complicated than the previous situation because of the interactions between molecules in different volume elements which will affect this probability. This probability will be the Boltzmann factor of the interaction energy of this configuration divided by the sum of probabilities of all configurations.

Reduced Spatial Probability Density Functions This probability will be the Boltzmann factor of the interaction energy of this configuration divided by the sum of probabilities of all configurations.

Reduced Spatial Probability Density Functions Let us now consider a different probability: the probability that molecule A is in a small volume element around a location and that molecule B is in a small volume around a location and so forth up to a certain molecule n, regardless of the locations of molecules n+1, n+2,…N.

Reduced Spatial Probability Density Functions Let us reintroduce the idea that identical molecules are indistinguishable. There are N possible molecules for the first position and volume element; then there are N-1 molecules for the second position and volume element, etc…

Reduced Spatial Probability Density Functions The previous equation introduced the n-body correlation function, :

Reduced Spatial Probability Density Functions The previous equation introduced the n-body correlation function, : In most cases, we are interested in 2 or 3-body correlation functions. The 2-body correlation function is known as pair correlation function or radial distribution function.

Reduced Spatial Probability Density Functions The pair correlation function is:

Reduced Spatial Probability Density Functions Physical interpretation of the pair correlation function The probability of finding a molecule in the volume element around a location and simultaneously a second molecule in a volume element around a position is:

Reduced Spatial Probability Density Functions The pair correlation function is a function of the two position vectors and , temperature, and density. This should not be forgotten, even though it is common to find the notation For non-interacting molecules, the pair correlation function is:

Reduced Spatial Probability Density Functions Let us now find the probable number of molecules in a volume element given that there is a molecule in volume element This is computed as the likelihood of molecules being simultaneously in both volume elements divided by the likelihood of any of the N molecules being in volume element The likelihood of any of the N molecules being in volume element is:

Reduced Spatial Probability Density Functions

Thermodynamic Properties from the Pair Correlation Function Let us compute the average value of the configuration energy. Its value is:

Thermodynamic Properties from the Pair Correlation Function Let us assume the potential is pairwise additive, i.e.:

Thermodynamic Properties from the Pair Correlation Function Then:

Thermodynamic Properties from the Pair Correlation Function Then: Assuming the molecules are spherically symmetrical:

Thermodynamic Properties from the Pair Correlation Function Pressure: In rectangular coordinates:

Thermodynamic Properties from the Pair Correlation Function In rectangular coordinates: Let us now define: Then:

Thermodynamic Properties from the Pair Correlation Function Using pairwise additivity and rectangular coordinates, a long series of steps (please refer to the textbook, pages 192-3) leads to the pressure equation:

Pair Correlation Function at Low Density We can evaluate thermodynamic properties from the pair correlation function, but we need to find out its expression. This section is devoted to finding it for low density fluids. Let us “borrow” some slides from Chapter 7 to see how to evaluate the denominator. We will work on the numerator later on.

Pairwise additivity Assuming pairwise additivity, the Boltzmann factor in the configuration integral can be written as: The configuration integral is then:

Mayer cluster function and irreducible integrals The next point is to set up a framework that will allow the evaluation of the configuration integral. To do that, we begin by defining the Mayer cluster function as follows: Limiting behaviors of the Mayer cluster function:

Mayer cluster function and irreducible integrals We now use the Mayer cluster function in the configuration integral: We obtain:

Mayer cluster function and irreducible integrals Expanding as a series of products, we get: We can now break in a summation of integrals, the first of which is:

Mayer cluster function and irreducible integrals After some algebra, we proved in Chapter 7 that when we only keep the first two terms of the expansion: Then:

Pair Correlation Function at Low Density The development of the previous slides covers how to compute the denominator. We will now tackle the numerator, by expressing it in terms of Mayer cluster functions:

Pair Correlation Function at Low Density

Pair Correlation Function at Low Density We now combine the expressions for the numerator and denominator:

Pair Correlation Function at Low Density As an approximation, let us only keep the density-independent part (reasonable for low densities): For spherical particles:

LJ 12-6 potential and rdf

Low density and higher density rdf of the hard-sphere potential as a function of r/s

Pair Correlation Function at High Density Three methods: Experimental: X-ray, neutron scattering measurements. Statistical thermodynamics theory: next chapter Molecular simulations: molecular dynamics or Monte Carlo

Fluctuations and the Compressibility Equation Let us consider the grand canonical (temperature, volume, and chemical potential are specified) applied to a pure substance. The average number of molecules is:

Fluctuations and the Compressibility Equation The average of the squared number of molecules is:

Fluctuations and the Compressibility Equation Assuming the fluctuations follow a Gaussian distribution, the following difference is useful to compute the standard deviation: However: Therefore:

Fluctuations and the Compressibility Equation From classical thermodynamics: Then:

Fluctuations and the Compressibility Equation Again, from classical thermodynamics: Then: The isothermal compressibility is:

Fluctuations and the Compressibility Equation Then:

Fluctuations and the Compressibility Equation Remember now the definition of radial distribution function in the canonical ensemble: In the grand canonical ensemble, it is calculated in similar fashion using the average value of N(N-1), i.e.:

Fluctuations and the Compressibility Equation

Fluctuations and the Compressibility Equation Combining the two equations from the previous slide: Dividing by the average number of molecules, we obtain: Note that:

Fluctuations and the Compressibility Equation Combining the equations from the previous slide: In summary:

Coordination Number from the Pair Correlation Function Take as reference the center of a molecule at the origin of the coordinate system: Very close to it, there is an excluded volume in which the center of no other molecule is found. Then, there is a first coordination shell of atoms in which the local density will be higher than the bulk density. g(r) has a peak, greater than unity. Beyond this first coordination shell, g(r) exhibits a periodic pattern of peaks and valleys of decreasing amplitude. Typically, after 3 to 5 molecules diameters from the central molecule, this periodic pattern is barely visible.

Coordination Number from the Pair Correlation Function Source: http://cbp.tnw.utwente.nl/PolymeerDictaat/node14.html

Coordination Number from the Pair Correlation Function The first coordination shell is taken as the region from where g(r) first departs from zero to the point of minimum after the first peak of g(r). Assuming spherical symmetry, the number of molecules in the first coordination shell is: In a solid, this result depends on the type of crystal structure. In liquids, the maximum coordination number is commonly assumed to be equal to 10.

Coordination Number from the Pair Correlation Function The second coordination shell is taken as the region between the first and second points of minimum after the first peak of g(r). Assuming spherical symmetry, the number of molecules in the first coordination shell is:

Coordination Number from the Pair Correlation Function van der Waals atomic radii The radial distribution function of atoms modeled as hard spheres can be used to assign radii. Chemically bonded atoms will be separated by a distance smaller than the summation of their van der Waals radii. Non-chemically bonded atoms will generally be separated by a distance bigger than the summation of their van der Waals radii. If the atoms form hydrogen bonds, their separation distance will be smaller than the summation of their van der Waals radii. The modeling of substances and mixtures with hydrogen bonds is a major challenge in thermodynamics.