Lennard-Jones potential

class=skin-invert-image|alt=|thumb|320x320px|Graph of the Lennard-Jones potential function: Intermolecular potential energy as a function of the distance of a pair of particles. The potential minimum is at .

In computational chemistry, molecular physics, and physical chemistry, the Lennard-Jones potential (also termed the LJ potential or 12-6 potential; named for John Lennard-Jones) is an intermolecular pair potential. Out of all the intermolecular potentials, the Lennard-Jones potential is probably the one that has been the most extensively studied. It is considered an archetype model for simple yet realistic intermolecular interactions. The Lennard-Jones potential is often used as a building block in molecular models (a.k.a. force fields) for more complex substances. Many studies of the idealized "Lennard-Jones substance" use the potential to understand the physical nature of matter.

Overview

The Lennard-Jones potential is a simple model that still manages to describe the essential features of interactions between simple atoms and molecules: two interacting particles repel each other at very close distances, attract each other at moderate distances, and eventually stop interacting at infinite distance, as shown in the figure. The Lennard-Jones potential is a pair potential, i.e. no three- or multi-body interactions are covered by the potential.

The general Lennard-Jones potential combines a repulsive potential, <math>1/r^{n}</math>, with an attractive potential, <math>-1/r^{m}</math>, using empirically determined coefficients <math>A_n</math> and <math>B_m</math>: Lennard-Jones suggested using

<math>m=6</math> to match the London dispersion force and <math>n=12</math> based on experimental data. Setting <math>A_n = 4\varepsilon \sigma^{12}</math> and <math>B_m = 4\varepsilon \sigma^6</math> gives the widely used Lennard-Jones 12-6 potential:

<math display="block">V_\text{LJ}(r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6 \right] ,</math>

where is the distance between two interacting particles, is the depth of the potential well, and is the distance at which the particle-particle potential energy is zero. The Lennard-Jones 12-6 potential has its minimum at a distance of <math>r=r_{\rm min}= 2^{1/6}\sigma,</math> where the potential energy has the value <math>V=-\varepsilon.</math>

The Lennard-Jones potential is usually the standard choice for the development of theories for matter (especially soft-matter) as well as for the development and testing of computational methods and algorithms.

Numerous intermolecular potentials have been proposed in the past for the modeling of simple soft repulsive and attractive interactions between spherically symmetric particles, i.e. the general shape shown in the Figure. Examples for other potentials are the Morse potential, the Mie potential, the Buckingham potential and the Tang-Tönnies potential. While some of these may be more suited to modelling real fluids, the simplicity of the Lennard-Jones potential, as well as its often surprising ability to accurately capture real fluid behavior, has historically made it the pair-potential of greatest general importance. a series of landmark papers on the pair potentials that would ultimately be named for him. Hence, the Mie potential was actually proposed before the Lennard-Jones potential. The Mie potential is named after Gustav Mie. The Stockmayer potential is a combination of a Lennard-Jones potential superimposed by a dipole. Hence, Stockmayer particles are not spherically symmetric, but rather have an important orientational structure.

Two center Lennard-Jones potential The two center Lennard-Jones potential consists of two identical Lennard-Jones interaction sites (same <math>\varepsilon</math>, <math>\sigma</math>, <math>m</math>) that are bonded as a rigid body. It is often abbreviated as 2CLJ. Usually, the elongation (distance between the Lennard-Jones sites) is significantly smaller than the size parameter <math>\sigma</math>. Hence, the two interaction sites are significantly fused.
Lennard-Jones truncated & splined potential The Lennard-Jones truncated & splined potential is a rarely used yet useful potential. Similar to the more popular LJTS potential, it is sturdily truncated at a certain 'end' distance <math>r_\mathrm{end}</math> and no long-range interactions are considered beyond. While the LJTS potential is shifted such that the potential is continuous but the force is discontinuous, the Lennard-Jones truncated & splined potential is made continuous by using a spline function that ensures a continuous force.

Lennard-Jones truncated & shifted (LJTS) potential

class=skin-invert-image|thumb|770x770px|Comparison of the vapor–liquid equilibrium of the 'full' Lennard-Jones potential (black) and the 'Lennard-Jones truncated & shifted' potential (blue). The symbols indicate molecular simulation results; the lines indicate results from equations of state.

The Lennard-Jones truncated & shifted (LJTS) potential is an often used alternative to the 'full' Lennard-Jones potential (see Eq. (1)). The 'full' and the 'truncated & shifted' Lennard-Jones potential have to be kept strictly separate. They are simply two different intermolecular potentials yielding different thermophysical properties. The Lennard-Jones truncated & shifted potential is defined as

V_{\text{LJTS(r) = \begin{cases}

V_\text{LJ} (r)

V_\text{LJ} (r_\text{end})

~~~~ r \le r_\text{end}

~~~~ r > r_\text{end},

\end{cases}</math>

with

V_\text{LJ}(r)

4 \varepsilon

\left[

\left( \frac{\sigma}{r} \right)^{12}

\left( \frac{\sigma}{r} \right)^6

\right].

</math>

Hence, the LJTS potential is truncated at <math> r_\mathrm{end} </math> and shifted by the corresponding energy value <math> V_\mathrm{LJ}(r_\mathrm{end})</math>. The latter is applied to avoid a discontinuity jump of the potential at <math> r_\mathrm{end} </math>. For the LJTS potential, no long-range interactions beyond <math> r_\mathrm{end} </math> are required – neither explicitly nor implicitly. The most frequently used version of the Lennard-Jones truncated & shifted potential is the one with <math> r_\mathrm{end} = 2.5\,\sigma </math>. Nevertheless, different <math> r_\mathrm{end} </math> values have been used in the literature. Each LJTS potential with a given truncation radius <math> r_\mathrm{end} </math> has to be considered as a potential and accordingly a substance of its own.

The LJTS potential is computationally significantly cheaper than the 'full' Lennard-Jones potential, but still covers the essential physical features of matter (the presence of a critical and a triple point, soft repulsive and attractive interactions, phase equilibria etc.). Therefore, the LJTS potential is used for the testing of new algorithms, simulation methods, and new physical theories.

Interestingly, for homogeneous systems, the intermolecular forces that are calculated from the LJ and the LJTS potential at a given distance are the same (since <math> \text{d}V/\text{d}r </math> is the same), whereas the potential energy and the pressure are affected by the shifting. Also, the properties of the LJTS substance may furthermore be affected by the chosen simulation algorithm, i.e. MD or MC sampling (this is in general not the case for the 'full' Lennard-Jones potential).

For the LJTS potential with <math> r_\mathrm{end} = 2.5\,\sigma </math>, the potential energy shift is approximately 1/60 of the dispersion energy at the potential well: <math> V_\mathrm{LJ}(r_\mathrm{end} = 2.5\,\sigma) = -0.0163\,\varepsilon </math>. The Figure on the right shows the comparison of the vapor–liquid equilibrium of the 'full' Lennard-Jones potential and the 'Lennard-Jones truncated & shifted' potential. The 'full' Lennard-Jones potential results prevail a significantly higher critical temperature and pressure compared to the LJTS potential results, but the critical density is very similar. (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

[[File:LJ PhaseDiagram.png|class=skin-invert-image|thumb|347x347px|Phase diagram of the Lennard-Jones substance. Correlations and numeric values for the critical point and triple point(s) are taken from Refs. The solid lines indicate coexistence lines of two phases. The dashed lines indicate the vapor–liquid spinodal. The virial coefficients can for example be computed directly from the Lennard-potential using algebraic expressions and reported data has therefore no uncertainty. Molecular simulation results, e.g. the pressure at a given temperature and density has both statistical and systematic uncertainties.

Since the Lennard-Jonesium is the archetype for the modeling of simple yet realistic intermolecular interactions, a large number of thermophysical properties were studied and reported in the literature. and Wood and Parker after molecular simulations on "fast computing machines" 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

<math>T_\mathrm{tr} = 0.69 \pm 0.005 \,\varepsilon k_\mathrm{B}^{-1}</math>
<math>\rho_\mathrm{tr,gas} = 0.0017 \pm 0.004 \,\sigma^{-3}</math>
<math>\rho_\mathrm{tr,liq} = 0.845 \pm 0.009 \,\sigma^{-3}</math>
<math>\rho_\mathrm{tr,sol} = 0.961 \pm 0.007 \,\sigma^{-3}</math>
<math>p_\mathrm{tr} = 0.0012 \pm 0.0007 \,\varepsilon\sigma^{-3}</math>

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, <math>Z</math> and its derivatives can match the values of the ideal gas for special <math>T</math>, <math>\rho</math> 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 <math>\rho \rightarrow 0</math>. 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 <math>Z=p/(\rho T)=1</math>. The Zeno curve originates at the Boyle temperature <math>T_\mathrm{B} = 3.417927982\,\varepsilon k_\mathrm{B}^{-1}</math>, 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 <math>pvT</math> – 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 (<math>\varepsilon</math> or <math>\sigma</math>) 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 <math>\varepsilon_\mathrm{12}</math> or <math>\sigma_\mathrm{12}</math> from <math>\varepsilon_\mathrm{11}</math>, <math>\sigma_\mathrm{11}</math> and <math>\varepsilon_\mathrm{22}</math>, <math>\sigma_\mathrm{22}</math>. 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

<math display="block">\sigma_{12} = \eta_{12} \frac{\sigma_{11} + \sigma_{22{2} </math>

<math display="block">\varepsilon_{12} = \xi_{12} \sqrt{\varepsilon_{11} \varepsilon_{22 </math>

The parameter <math>\xi_{12} </math> is an additional state-independent interaction parameter for the mixture. The parameter <math>\eta_{12} </math> is usually set to unity since the arithmetic mean can be considered physically plausible for the cross-interaction size parameter. The parameter <math>\xi_{12} </math> 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 <math>k_{12} = 1- \xi_{12} </math>. For <math>\xi_{12} > 1 </math>, 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 <math>\xi_{12} < 1 </math>.

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 <math>\xi_{12}=1

</math>. For <math>\xi_{12}>1

</math> 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 <math>\xi_{12}<1

</math> 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 <math>\xi_{12}

</math> 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 <math>V_\mathrm{LJ}(r)</math> 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.