# Intermolecular forces

A keen understanding of intermolecular forces is imperative for discussing the different phases of matter. In general, the intermolecular forces of a solid are greater than those of a liquid. This trend can be observed when looking at the force it takes to separate a solid as compared to that required to separate a liquid. Also, the molecules in a solid are much more confined to their position in the solid’s structure as compared to the molecules of a liquid, thereby affecting their ability to move. Most solids and liquids are deemed incompressible. The underlying reason for their “incompressibility” is that the molecules repel each other when they are forced closer than their normal spacing; the closer they become, the greater the repelling force (Tien and Lienhard, 1979). A gas differs from both a solid and a liquid in that its kinetic energy is great enough to overcome the intermolecular forces, causing the molecules to separate without restraint. The intermolecular forces in a gas decrease as the distance between the molecules increases. Both gravitational and electrical forces contribute to intermolecular forces; for many solids and liquids, the electrical forces are on the order of 1029 times greater than the gravitational force. Therefore, the gravitational forces are typically ignored. To quantify the intermolecular forces, a potential function φ(r) is defined as the energy required to bring two molecules, which are initially separated by an infinite distance, to a finite separation distance r. The form of the function always depends on the nature of the forces between molecules, which can be either repulsive or attractive depending on intermolecular spacing. When the molecules are close together, a repulsive electrical force is dominant. The repulsive force is due to interference of the electron orbits between two molecules, and it increases rapidly as the distance between two molecules decreases. When the molecules are not very close to each other, the forces acting between molecules are attractive in nature and generally fall into one of three categories. The first category is electrostatic forces, which occur between molecules that have a finite dipole moment, such as water or alcohol. The second category is induction forces, which occur when a permanently-charged particle or dipole induces a dipole in a nearby neutral molecule. The third category is dispersion forces, which are caused by transient dipoles in nominally-neutral molecules or atoms. An accurate representation of the intermolecular potential function φ(r) should account for all of the forces discussed above. It should be able to reflect repulsive forces for small spacing and attractive forces in the intermediate distance. When the distance between molecules is very large, there should be no intermolecular forces. While the exact form of φ(r) is not known, the following Lennard-Jones 6-12 potential function provides a satisfactory empirical expression for nonpolar molecules:

$\phi (r) = 4\varepsilon \left[ {{{\left( {\frac{{{r_0}}}{r}} \right)}^{12}} - {{\left( {\frac{{{r_0}}}{r}} \right)}^6}} \right]\qquad \qquad(1)$

where $\varepsilon$ is a constant and r0 is a characteristic length. Both of them depend on the type of the molecules. Figure 1 shows the Lennard-Jones 6-12 potential as a function of distance between two molecules. When the distance between molecules is small, the Lennard-Jones potential decreases with increasing distance until a point is reached after which the repulsive force dominates, and it is necessary to add energy to the system in order to bring the molecules any closer. As the molecules separate, there is a distance, rmin at which the Lennard-Jones potential becomes minimum. As the molecules move further apart, the Lennard-Jones potential increases with increasing distance between molecules and the attractive force dominates. The Lennard-Jones potential approaches zero when the molecular

Lennard-Jones 6-12 potential vs. distance between two spherical, nonpolar molecules.

distance becomes very large. When the Lennard-Jones potential is at minimum, the following condition is satisfied:

$\frac{{d\phi ({r_{\min }})}}{{dr}} = 0\qquad \qquad(2)$

Substituting eq. (1) into eq. (2), one obtains,

${r_{\min }} = {2^{1/6}}{r_0} \approx 1.12{r_0}\qquad \qquad(3)$

For typical gas molecules, r0 ranges from 0.25 to 0.4 nm, which result a range from 0.28 to 0.45 nm for rmin.

The Lennard-Jones potential at this point is

$\phi ({r_{\min }}) = - \varepsilon\qquad \qquad(4)$

When looking at the three phases of matter in the context of the figure on the right, some relationships can be described. For the solid state, the atoms are limited to vibrating about the equilibrium position, because they do not have enough energy]] to overcome the attractive force. The molecules in a liquid are free to move because they have a higher level of vibrational energy]], but they have approximately the same molecular distance as the molecules of a solid. The energy required to overcome the attractive forces in a solid, thus allowing the molecules to move freely, corresponds to the latent heat of fusion. In the gaseous phase, on the other hand, the molecules are so far apart that they are virtually unaffected by intermolecular forces. The energy required to create vapor by separating closely-spaced molecules in a liquid corresponds to the latent heat of vaporization. Simulation of phase change at the molecular level is not necessary for many applications in macro spatial and time scales. For heat transfer at micro spatial and time scales, the continuum transport model breaks down and simulation at the molecular level becomes necessary. One example that requires molecular dynamics simulation is heat transfer and phase change during ultrashort pulsed laser materials processing (Wang and Xu, 2002). This process is very complex because it involves extremely high rates of heating (on the order of 1016 K/s) and high temperature gradients (on the order of 1011 K/m). The motion of each molecule i in the system is described by Newton’s second law, i.e.,

$\sum\limits_{j = 1(j \ne i)}^N {{\mathbf{F}}_{ij} } = m_i \frac{{d^2 {\mathbf{r}}_i }} {{dt^2 }}, i = 1,2,3...n$ $i = 1,2,3...n\qquad \qquad(5)$

where mi and ${{\mathbf{r}}_i}$ are the mass and position of the ith molecule in the system. In arriving at eq. (5), it is assumed that the molecules are monatomic and have only three degrees of freedom of motion. For molecules with more than one atom, it is also necessary to consider the effect of rotation. The Lennard-Jones potential between the ith and jth molecules is obtained by

${\phi _{ij}} = 4\varepsilon \left[ {{{\left( {\frac{{{r_0}}}{{{r_{ij}}}}} \right)}^{12}} - {{\left( {\frac{{{r_0}}}{{{r_{ij}}}}} \right)}^6}} \right]\qquad \qquad(6)$

The force between the ith and jth molecules can be obtained from

${{\mathbf{F}}_{ij}} = - \nabla {\phi _{ij}} = \frac{{24\varepsilon }}{{{r_o}}}\left[ {{{\left( {\frac{{{r_o}}}{{{r_{ij}}}}} \right)}^{13}} - {{\left( {\frac{{{r_o}}}{{{r_{ij}}}}} \right)}^7}} \right]\frac{{{{\mathbf{r}}_{ij}}}}{{{r_{ij}}}}\qquad \qquad(7)$

where rij is the distance between the ith and jth molecules.

The transport properties can be obtained by the kinetic theory in the form of very complicated multiple integrals that involve intermolecular forces. For a non-polar substance that Lennard-Jones potential is valid, these integrals can be evaluated numerically. For a pure gas, the self-diffusivity, D, viscosity, μ, and thermal conductivity, k, are (Bird et al., 2002)

$D = \frac{3}{{8}}\frac{{\sqrt {\pi m{k_b}T} }}{{\pi {\sigma ^2}{\Omega _D }}}\frac{1}{{\rho }}\qquad \qquad(8)$
$\mu = \frac{5}{{16}}\frac{{\sqrt {\pi m{k_b}T} }}{{\pi {\sigma ^2}{\Omega _\mu }}}\qquad \qquad(9)$
$k = \frac{{25}}{{32}}\frac{{\sqrt {\pi m{k_b}T} }}{{\pi {\sigma ^2}{\Omega _k}}}{\bar c_v}\qquad \qquad(10)$

where σ is collision diameter, and ${\bar c_v}$ in is the molar specific heat under constant volume. The dimensionless collision integrals are related by ${\Omega _\mu } = {\Omega _k} \approx 1.1{\Omega _D}$ and are slow varying functions of ${k_b}T/\varepsilon$ ($\varepsilon$ is a characteristic energy]] of molecular interaction). If all molecules can be assumed to be rigid balls, all collision integrals will become unity. It follows from eqs. (8) – (10) that the Prandtl and Schmidt numbers are, respectively, 0.66 and 0.75, which is a very good approximation for monatomic gases (e.g., helium in Table C.3). The transport properties at system length scales of less than 10λ will be different from the macroscopic properties, because the gas molecules are not free to move as they naturally would. While the transport properties discussed here are limited to low-density monatomic gases, the discussion can also be extended to polyatomic gases, and monatomic and polyatomic liquids.

## References

Bird, R.B., Stewart, W.E., and Lightfoot, E.N., 2002, Transport Phenomena, 2nd edition, John Wiley & Sons, New York.
Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA
Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.
Tien, C.L., and Lienhard, J.H., 1979, Statistical Thermodynamics, Hemisphere, Washington, D.C.
Wang, X., and Xu, X., 2002, “Molecular Dynamics Simulation of Heat Transfer and Phase Change During Laser Material Interaction,” ASME Journal of Heat Transfer, Vol. 124, pp. 265-274.