Molecular Simulation/Solids

The Structure and Dynamics of Simple Solids
Solids have regular periodic structures where the atoms are held at fixed lattice points in the 3-D rigid framework. This regular repeating structure can take different forms and is usually represented by a unit cell, where the entire solid structure is the repeated translations of this cell. Since the atoms cannot move freely, this allows solids to have a fixed volume and shape, which is only distorted by an applied force. However, atoms in solids do still have motion. They vibrate around their lattice positions and therefore their position fluctuates slightly around the lattice point. This can be thought of as the atom being tethered to the point, but being able to slightly vibrate around it. The intermolecular interactions between the atoms keep them in their fixed positions. These forces depend on the composition of the solid and could include forces such as London dispersion, dipole-dipole, quadrupole-quadrupole, and hydrogen bonding. The temperatures at which these solids occur are low enough that the atoms do not have enough energy to overcome these forces and move away from their fixed position. This keeps the solid tightly packed and eliminates most of the space between atoms or molecules. This gives solids their dense property. Figure 1 shows a molecular dynamics simulation of solid argon at 50 K, where the atoms are all vibrating around their fixed lattice points. Argon atoms only have london dispersion intermolecular forces, which are described by the Lennard-Jones potential. The Lennard-Jones equation below accounts for the London dispersion attractive forces and the Pauli repulsion forces between atoms.

$$\mathcal{V}\left( r \right) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6} \right] = \varepsilon \left[ \left(\frac{R_{min}}{r}\right)^{12} - 2\left(\frac{R_{min}}{r}\right)^{6} \right]$$

The intermolecular distance between the argon atoms $$r$$ is equal to $$R_{\min}$$. This means that the atoms are at the same distance as the minima of this potential energy function. This maximizes the intermolecular forces by giving the most negative potential. The atoms in the solid argon are held together by these strong forces of attraction and are tightly packed to minimize empty space.

Differences in the Radial Distribution Function
The radial distribution function $$g(r)$$ relates the bulk density $$\rho$$ of a solid, liquid, or gas to the local density $$\rho$$$$(r)$$ at a distance $$r$$ from a certain molecule or atom. The equation that relates these parameters is found below.

$$\rho(r)=\rho^{bulk}g(r) $$

The radial distribution functions of solid, liquid, and gaseous argon can be seen in Figure 2. In a solid, particles are found at defined positions, which is shown by the discrete peaks at values of $$\sigma$$, $$\sqrt2$$$$\sigma$$, $$\sqrt3$$$$\sigma$$, $$2$$$$\sigma$$, etc. The peaks of this radial distribution function are also broadened due to the molecules fluctuating around their lattice positions and occupying slightly different positions in this range. The regions of the function with $$g(r)=0$$ are regions where there is a zero probability of finding another molecule or atom. There is a zero probability between the peaks in a solid radial distribution function because of the regular structure where all of the molecules are packed tightly to most efficiently fill the space. This leaves regular intervals of spaces where no atoms or molecules are present. Also, each peak in a radial distribution function represents a coordination sphere where there is a high probability of finding molecules. Each subsequent peak represents a coordination sphere that is farther from the origin molecule and therefore the nearest neighbours are in the first coordination sphere. It is also important to note that $$g(r) \approx 0$$ when $$r < \sigma$$. In this scenario, the electron density clouds of the two atoms are overlapping, causing the potential energy to be prohibitively high.

In contrast, the radial distribution function of a gas only has one peak/coordination sphere, which then decays to the bulk density, represented by $$g(r)=1$$. This simple radial distribution function is a consequence of a density that is so low that only the interactions of individual pairs of gas molecules affect the radial distribution function. The density is higher around the origin molecule due to strong london dispersion forces in this area, but the forces decay off quickly. The radial distribution function of liquids also differs from that of the solids. Molecules in a liquid have the ability to move around, but their positions are still correlated due to intermolecular forces between the molecules. This allows liquids to have periodic peaks in the radial distribution function, as shown in Figure 2. Thus, liquids also have coordination spheres where it is more likely to find molecules at these distances from an origin molecule, and thus there will be a greater local density at these positions. However, there is still a lower density than in solids due to the fluidity of liquids and the molecules being able to change positions. The radial distribution function of a liquid has its peaks at intervals of $$\sigma$$, which is due to the looser packing of the molecules in a liquid compared to in a solid. This looser packing is due to the coordination spheres not being bound to fixed positions. There is also a lower probability of finding molecules in the second coordination sphere due to Pauli repulsion interactions with the first sphere. Due to the disordered nature of liquids the radial distribution function eventually decays to one and returns to the bulk density at large $$r$$ values as the positions are no longer correlated to each other.

Simple liquids, such as liquid argon, are packed most efficiently to avoid repulsive interactions between the atoms, but there is still some spaces between them. Solids are packed very tightly so that the empty space between them is as little as possible and most crevices are filled. Their fixed positions allow them to maintain this tight packing of the atoms to minimize wasted space. Liquids also try to minimize this space, but are less tightly packed than solids because of their ability to move around and change positions. They have more energy to overcome the intermolecular forces correlating their positions. This difference in packing is seen in the radial distribution functions with the occurrence of the solid peaks at closer intervals than the liquid peaks.