The Lennard-Jones potential is not only of fundamental importance in
computational chemistry and
soft-matter physics, but also for the modeling of real substances. The Lennard-Jones potential is used for fundamental studies on the behavior of matter and for elucidating atomistic phenomena. It is also often used for somewhat special use cases, e.g. for studying thermophysical properties of two- or four-dimensional substances (instead of the classical three spatial directions of our universe). There are two main applications of the Lennard-Jones potentials: (i) for studying the hypothetical Lennard-Jones substance
Statistical mechanics can be used to study the Lennard-Jones potential and to obtain thermophysical properties of the 'Lennard-Jones substance'. The Lennard-Jones substance is often referred to as 'Lennard-Jonesium,'
Thermophysical properties of the Lennard-Jones substance " became available in 1953. Transport property data of Lennard-Jones fluids have been compiled by Bell et al. and Lautenschaeger and Hasse. Figure on the right shows the
phase diagram of the Lennard-Jones fluid. Phase equilibria of the Lennard-Jones potential have been studied numerous times and are accordingly known today with good precision. The Figure shows results correlations derived from computer experiment results (hence, lines instead of data points are shown). The mean intermolecular interaction of a Lennard-Jones particle strongly depends on the
thermodynamic state, i.e., temperature and pressure (or density). For solid states, the attractive Lennard-Jones interaction plays a dominant role – especially at low temperatures. For liquid states, no ordered structure is present compared to solid states. The mean potential energy per particle is negative. For gaseous states, attractive interactions of the Lennard-Jones potential play a minor role – since they are far distanced. The main part of the internal energy is stored as
kinetic energy for gaseous states. At supercritical states, the attractive Lennard-Jones interaction plays a minor role. With increasing temperature, the mean kinetic energy of the particles increases and exceeds the energy well of the Lennard-Jones potential. Hence, the particles mainly interact by the potentials' soft repulsive interactions and the mean potential energy per particle is accordingly positive. Overall, due to the large timespan the Lennard-Jones potential has been studied and thermophysical property data has been reported in the literature and computational resources were insufficient for accurate simulations (to modern standards), a noticeable amount of data is known to be dubious.|alt= The triple point is presently assumed to be located at • T_\mathrm{tr} = 0.69 \pm 0.005 \,\varepsilon k_\mathrm{B}^{-1} • \rho_\mathrm{tr,gas} = 0.0017 \pm 0.004 \,\sigma^{-3} • \rho_\mathrm{tr,liq} = 0.845 \pm 0.009 \,\sigma^{-3} • \rho_\mathrm{tr,sol} = 0.961 \pm 0.007 \,\sigma^{-3} • p_\mathrm{tr} = 0.0012 \pm 0.0007 \,\varepsilon\sigma^{-3} The uncertainties represent the scattering of data from different authors. yield an illustrative description of essential features of the Lennard-Jones potential. Brown's characteristic curves are defined as curves on which a certain thermodynamic property of the substance matches that of an
ideal gas. For a real fluid, Z and its derivatives can match the values of the ideal gas for special T, \rho combinations only as a result of Gibbs' phase rule. The resulting points collectively constitute a characteristic curve. Four main characteristic curves are defined: One 0th-order (named
Zeno curve) and three 1st-order curves (named
Amagat,
Boyle, and
Charles curve). The characteristic curve are required to have a negative or zero curvature throughout and a single maximum in a double-logarithmic pressure-temperature diagram. Furthermore, Brown's characteristic curves and the virial coefficients are directly linked in the limit of the ideal gas and are therefore known exactly at \rho \rightarrow 0. Both computer simulation results and equation of state results have been reported in the literature for the Lennard-Jones potential. Points on the Zeno curve Z have a
compressibility factor of unity Z=p/(\rho T)=1. The Zeno curve originates at the
Boyle temperature T_\mathrm{B} = 3.417927982\,\varepsilon k_\mathrm{B}^{-1}, surrounds the critical point, and has a slope of unity in the low temperature limit. Also transport properties (viscosity, heat conductivity, and self diffusion coefficient) of the Lennard-Jones fluid have been studied, but the database is significantly less dense than for homogeneous equilibrium properties like pvT – or internal energy data. Moreover, a large number of analytical models (
equations of state) have been developed for the description of the Lennard-Jones fluid (see below for details).
Properties of the Lennard-Jones solid The database and knowledge for the Lennard-Jones solid is significantly poorer than for the fluid phases. It was realized early that the interactions in solid phases should not be approximated to be pair-wise additive – especially for metals. Nevertheless, the Lennard-Jones potential is used in solid-state physics due to its simplicity and computational efficiency. Hence, the basic properties of the solid phases and the solid–fluid phase equilibria have been investigated several times, e.g. Refs. and Leland and
Rowlinson and co-workers. Those are today the basis of most theories for mixtures. Mixtures of two or more Lennard-Jones components are set up by changing at least one potential interaction parameter (\varepsilon or \sigma) of one of the components with respect to the other. For a binary mixture, this yields three types of pair interactions that are all modeled by the Lennard-Jones potential: 1-1, 2-2, and 1-2 interactions. For the cross interactions 1–2, additional assumptions are required for the specification of parameters \varepsilon_\mathrm{12} or \sigma_\mathrm{12} from \varepsilon_\mathrm{11}, \sigma_\mathrm{11} and \varepsilon_\mathrm{22}, \sigma_\mathrm{22}. Various choices (all more or less empirical and not rigorously based on physical arguments) can be used for these so-called combination rules. The most widely used \sigma_{12} = \eta_{12} \frac{\sigma_{11} + \sigma_{22}}{2} \varepsilon_{12} = \xi_{12} \sqrt{\varepsilon_{11} \varepsilon_{22}} The parameter \xi_{12} is an additional state-independent interaction parameter for the mixture. The parameter \eta_{12} is usually set to unity since the
arithmetic mean can be considered physically plausible for the cross-interaction size parameter. The parameter \xi_{12} on the other hand is often used to adjust the
geometric mean so as to reproduce the phase behavior of the model mixture. For analytical models, e.g.
equations of state, the deviation parameter is usually written as k_{12} = 1- \xi_{12} . For \xi_{12} > 1 , the cross-interaction dispersion energy and accordingly the attractive force between unlike particles is intensified, and the attractive forces between unlike particles are diminished for \xi_{12} . For Lennard-Jones mixtures, both fluid and solid
phase equilibria can be studied, i.e.
vapor–liquid,
liquid–liquid, gas–gas, solid–vapor,
solid–liquid, and solid–solid. Accordingly, different types of
triple points (three-phase equilibria) and
critical points can exist as well as different
eutectic and
azeotropic points. have been studied more comprehensively then phase equilibria comprising solid phases. A large number of different Lennard-Jones mixtures have been studied in the literature. To date, no standard for such has been established. Usually, the binary interaction parameters and the two component parameters are chosen such that a mixture with properties convenient for a given task are obtained. Yet, this often makes comparisons tricky. For the fluid phase behavior, mixtures exhibit practically ideal behavior (in the sense of
Raoult's law) for \xi_{12}=1 . For \xi_{12}>1 attractive interactions prevail and the mixtures tend to form high-boiling azeotropes, i.e. a lower pressure than pure components' vapor pressures is required to stabilize the vapor–liquid equilibrium. For \xi_{12} repulsive interactions prevail and mixtures tend to form low-boiling azeotropes, i.e. a higher pressure than pure components' vapor pressures is required to stabilize the vapor–liquid equilibrium since the mean dispersive forces are decreased. Particularly low values of \xi_{12} furthermore will result in liquid–liquid miscibility gaps. Also various types of phase equilibria comprising solid phases have been studied in the literature, e.g. by
Carol and co-workers. Due to the fundamental importance of the Lennard-Jones potential, most currently available molecular-based EOS are built around the Lennard-Jones fluid. They have been comprehensively reviewed by Stephan et al. Equations of state for the Lennard-Jones fluid are of particular importance in
soft-matter physics and
physical chemistry, used as starting point for the development of EOS for complex fluids, e.g.
polymers and associating fluids. The monomer units of these models are usually directly adapted from Lennard-Jones EOS as a building block, e.g. the PHC EOS, the BACKONE EOS, and
SAFT type EOS. More than 30 Lennard-Jones EOS have been proposed in the literature. A comprehensive evaluation describe the Lennard-Jones potential with good and similar accuracy, but none of them is outstanding. Three of those EOS show an unacceptable unphysical behavior in some fluid region, e.g. multiple van der Waals loops, while being elsewise reasonably precise. Only the Lennard-Jones EOS of Kolafa and Nezbeda was found to be less precise for practically all available reference data the OPLS force field, and the MolMod force field (an overview of
molecular force fields is out of the scope of the present article). For the state-of-the-art modeling of solid-state materials, more elaborate multi-body potentials (e.g.
EAM potentials) are used. The Lennard-Jones potential yields a good approximation of intermolecular interactions for many applications: The macroscopic properties computed using the Lennard-Jones potential are in good agreement with experimental data for simple substances like argon on one side and the potential function V_\mathrm{LJ}(r) is in fair agreement with results from
quantum chemistry on the other side. The Lennard-Jones potential gives a good description of molecular interactions in
fluid phases, whereas molecular interactions in solid phases are only roughly well described. This is mainly due to the fact that multi-body interactions play a significant role in solid phases, which are not comprised in the Lennard-Jones potential. Therefore, the Lennard-Jones potential is extensively used in
soft-matter physics and associated fields, whereas it is less frequently used in
solid-state physics. Due to its simplicity, the Lennard-Jones potential is often used to describe the properties of gases and simple fluids and to model dispersive and repulsive interactions in
molecular models. It is especially accurate for
noble gas atoms and
methane. It is furthermore a good approximation for molecular interactions at long and short distances for neutral atoms and molecules. Therefore, the Lennard-Jones potential is very often used as a building block of
molecular models of complex molecules, e.g.
alkanes or
water. The Lennard-Jones potential can also be used to model the
adsorption interactions at solid–fluid interfaces, i.e.
physisorption or
chemisorption. It is well accepted, that the main limitations of the Lennard-Jones potential lie in the fact the potential is a
pair potential (does not cover multi-body interactions) and that the 1/r^{12} exponent term is used for the repulsion. Results from quantum chemistry suggest that a higher exponent than 12 has to be used, i.e. a steeper potential. Furthermore, the Lennard-Jones potential has a limited flexibility, i.e. only the two model parameters \varepsilon and \sigma can be used for the fitting to describe a real substance. ==See also==