The aim of this paper is to present an analytical calculation of the chemical
potential of a Lennard Jones fluid. The integration range is divided into two
regions. In the small distance region,which is $r\leq\sigma$ in the usual
notation,the integration range had to be cut off in order to avoid the
occurence of divergences.In the large distance region,the calculation is
technically simpler. The calculation reported here will be useful in all kinds
of studies concerning phase equilibrium in a $LJ$ fluid. Interesting kinds of
such systems are the giant planets and the icy satellites in various planetary
systems,but also the (so far) hypothetical quark stars

Celebonovic, V.

Serbian Astronomical Journal

English

arXiv

The aim of this paper is to present results of analytical calculation of chemical potential of a Lennard Jones (LJ) fluid performed in two ways: by using the thermodynamical formalism and the formalism of statistical mechanics. The integration range is divided into two regions. In the small distance region, which is $rleqsigma$ in the usual notation, the integration range had to be cut off in order to avoid the occurence of divergences. In the large distance region, the calculation is technically simpler. The calculation reported here will be useful in all kinds of studies concerning phase equilibrium in a LJ fluid. Interesting kinds of such systems are the giant planets and the icy satellites in various planetary systems, but also the (so far) hypothetical quark stars

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Serb. Astron. J. } 181 (2010), 51 - 55 UDC 52–335DOI: 10.2298/SAJ1081051C Preliminary reportCHEMICAL POTENTIAL OF A LENNARD JONES FLUIDV. ČelebonovićInstitute of Physics, Pregrevica 118, 11080 Zemun-Belgrade, SerbiaE–mail: vladan@ipb.ac.rs(Received: August 26, 2010; Accepted: November 18, 2010)SUMMARY: The aim of this paper is to present results of analytical calculationof chemical potential of a Lennard Jones (LJ) fluid performed in two ways: by usingthe thermodynamical formalism and the formalism of statistical mechanics. Theintegration range is divided into two regions. In the small distance region, whichis r ≤ σ in the usual notation, the integration range had to be cut off in order toavoid the occurence of divergences. In the large distance region, the calculationis technically simpler. The calculation reported here will be useful in all kinds ofstudies concerning phase equilibrium in a LJ fluid. Interesting kinds of such systemsare the giant planets and the icy satellites in various planetary systems, but alsothe (so far) hypothetical quark stars.Key words. Equation of state1. INTRODUCTIONThe aim of this paper is to present a calcu-lation of chemical potential of a fluid consisting ofneutral atoms or molecules. Interest in such systemshas considerably increased towards the end of the lastcentury, as a consequence of progress in planetaryscience. Until the end of October 2010, according todata at http://exoplanet.eu, as much as 493 planetsoutside the Solar system have been detected. It hasbeen shown that 423 of them have masses M ≤ 5MJwhere MJ is the mass of Jupiter. The major part(398) of the stars with planets have masses equal toor smaller than the solar mass, and 151 planet hasthe semi-major axis of the orbit between 1 and 3astronomical units.Judging by the experience from our Solar Sys-tem, it is expectable that this interval of distancesfrom a star corresponds to temperatures under whichfluids consisting of neutral atoms and molecules canexist. It is known that giant planets have huge at-mospheres and dense fluid interiors. Another classof planetologically interesting systems, in which flu-ids are important, are the icy satellites in our plan-etary system. For example, it has been concludedfrom data accumulated in the course of the Galileomission, that Jovian satellites Europa and possiblyCallisto almost certainly have fluid oceans beneaththeir surfaces.Calculations to be discussed in this paper canalso find applications in theoretical studies of quarkstars. These are (so far hypothetical) phases of ex-tremely dense matter, expected to occur in interiorsof neutron stars. In a recent study, aiming to con-strain the parameters of solid quark matter by usingdata on the binary pulsar PSRJ1614 − 2230, theLennard-Jones model was used to describe the coldquark matter in quark stars (Lai and Xu 2010). Itwas shown there that if the number of quarks in aquark cluster is Nq < 103 there is enough parameterspace for the existence of quark stars with masseshigher than 2 solar masses.Modelling theoretically the internal structureof celestial objects ranging from the icy satellitesand/or giant planets to quark stars, demands the51V. ČELEBONOVIĆa knowledge of chemical potential of the fluid in theirinteriors.A necessary preparatory step in such a studymust be the determination of interparticle interac-tion potential. Obviously, for a fluid or any otherkind of a system to be in equilibrium, the interparti-cle potential must be a combination of an attractiveand a repulsive term.It is known that there exists an attractive force- called the van der Waals (vdW) force (for exam-ple Margenau 1939 or Dzyaloshinskii, Lifchitz andPitaevskii 1961) between a pair of neutral atoms ormolecules at a mutual distance larger than their re-spective dimensions. The potential corresponding tothe vdW force is proportional to r−6, where r is theinterparticle distance. As shown by F. London, thephysical origin of the vdW force is the interaction ofinstantenous multipoles, while the repulsive contri-bution is of electrostatic origin.The vdW forces are anisotropic,which rendersthem additionally complicated (Dzyaloshinskii, Lif-chitz and Pitaevskii 1961). However, their isotropicpart is often approximated by the so called Lennard-Jones (LJ) potential. All the calculations in whatfollows will deal with this particular model poten-tial. The LJ model potential has the form:u(r) = 4ε[(σr)12 − (σr)6]. (1)The symbol ε denotes the depth of the potential well,while σ is the diameter of the molecular ”hard core”.Obviously, limr→0u(r) = ∞. It can simply be shownthat limr→σu(r) = 0 and that (∂u(r)/∂r) = 0 forrmin = 21/6σ. The depth of the potential well isu(rmin) = −ε.An example of the LJ potential drawn for theparticular case of CH4, with values of ε and σ from(Reichl 1988), is represented in Fig. 1. This particu-lar molecule is interesting in two research fields: theplanetary science, because it is present in the atmo-spheres of the giant planets, but also in studies ofthe interstellar medium. In the figure, the distanceis expressed in units of σ and the potential dividedby the Boltzmann constant kB given in units of K.0.8 1.0 1.2 1.4 1.6 1.8 2.0-20002004006008001000  u/k B [K]distance [ ]LJ potential for CH4Fig. 1. The LJ potential for methane (CH4).2. THE METHOD OF CALCULATION OFTHE CHEMICAL POTENTIALChemical potential of a fluid (or any other sys-tem) can be calculated in two different ways: by us-ing the general thermodynamical formalism, or bythe general formalism of statistical mechanics.2.1. The thermodynamical formalismIn this approach the calculation starts fromthe definition of the Gibbs potential:G = U − TS + PV (2)where all the symbols have their standard meanings.Using the virial expansion, the pressure can be ex-pressed as (Landau and Lifchitz 1976):P ∼= PID(1 + NVB) (3)where N is the number of particles in the systemand V the volume. The symbol B denotes the sec-ond virial coefficient, given by:B =12∫ ∞0(1− exp−u(r)/T )dV (4)and PID is the pressure of the ideal gas. The symbolu(r) denotes the interaction potential. Inserting Eq.(3) into Eq. (2), it follows thatG = U − TS + PV = GID + NPIDB . (5)Chemical potential is defined as µ =(∂G/∂N)P,T , which implies that:µ = (∂GID∂N)P,T + PIDB + NB(∂PID∂N)P,T (6)orµ = µID + PIDB + NB(∂PID∂N)P,T . (7)The equation of state of ideal gas is PIDV = NTwhich finally leads toµ = µID + 2PIDB . (8)For the particular case of the LJ potential, it can beshown that the second virial coefficient is given by(Reichl 1988):B(T ) = −(b0/2)∞∑n=01n!Γ(2n− 14)(εT)2n+14 (9)where b0 = 2πσ3/3 and Γ denotes the Gamma func-tion. Inserting Eq. (9) into Eq. (8) it follows that:52CHEMICAL POTENTIAL OF A LENNARD JONES FLUIDµ = µID − b0pID∞∑n=01n!Γ(2n− 14)(εT)2n+14 (10)which is the result for chemical potential. Limitingthe sum in this expression to terms up to and includ-ing n = 1, it follows that:µ ∼= µID + 2pIDb0[2.45083− 1.8128( εT)1/2]×( εT)1/4 . (11)2.2. The formalism of statistical mechanicsThe chemical potential of a fluid is given by(Hill 1987):µkBT= ln(ρλ3) +ρkBT∫ 10dγ∫ ∞0dr×4πr2u(r)g(r) (12)where γ denotes the ”coupling parameter” (Hill1987), ρ is the particle number density , u(r) theinteraction potential and g(r) is the radial distribu-tion function. The symbol h̄ is the Planck constantdivided by 2π, m is the particle mass and λ is thethermal wavelength given byλ = (2πh̄2mkBT)1/2 .Expression (10) is valid under the condition ρλ3 > 1,which leads to:ρ ≥(mkBT2πh̄2)3/2. (13)The radial distribution function is a ”bridge”relating macroscopic thermodynamic properties withinterparticle interactions in any kind of a substance.In the theory of liquids, g(r) can be determined fromfirst principles (Hill 1987) just by assuming a suit-able form of intermolecular potential (Morsali et al.2005). In what follows, the result for g(r) obtainedin Morsali et al. (2005) will be used. Changing thevariable from r to x = r/σ, and performing the inte-gration over γ, it follows that:µkBT= ln(ρλ3) + 4πσ3ρkBT∫ ∞0dxx2u(x)g(x) .(14)The domain of integration can be divided into twosubdomains x ∈ [0, 1] and x ∈ [1,∞]I = σ3∫ ∞0dxx2u(x)g(x) =σ3[∫ 10dxx2u(x)g1(x) +∫ ∞1dxx2u(x)× g2(x)]= σ3 × [I1 + I2] . (15)The divergence of the LJ potential which occurswhen x → 0 can be bypassed either by introducing asuitable change of the range of integration x ∈ [x0, 1]instead of x ∈ [0, 1] with x0 6= 0, or by changing theform of the potential in the domain x ∈ [0, 1]. Forx ∈ [0, 1] the function g(r) has the formg1(x) = s exp[−(mx + n)4] (16)and for x ∈ [1,∞] the radial distribution function isg2(x) = 1 +1x2exp[−(ax + b)] sin[(cx + d)] +1x2exp[−(gx + h)] cos[(kx + l)] (17)where a, b, c, d, g, h, k, l, m, n and s are functions ofpressure, temperature and density given in Morsaliet al. (2005).The appropriate boundary conditions, thatthe radial distribution function should tend to 1 inthe limits of zero density and infinite distance, andthe consequences of these conditions are also dis-cussed there. As a consequence, the functions b, d, hand l are functions of density only, n is the functionof temperature only and the other functions dependon the temperature and density (Morsali et al. 2005).3. THE CALCULATION3.1. The case x ∈ [0, 1]With the change of variables x = r/σ, the LJpotential gets the form:u(x) = 4ε[x−12 − x−6] . (18)Inserting Eqs. (16) and (18) into the expres-sion for I1 in Eq. (15), it follows that:I1 = 4sε∞∑l=0(−1l)(l!)∫ 1x=x0(1x10− 1x4)(mx + n)4ldx . (19)Performing the integrations, after some alge-bra, it finally follows that:I1 ∼= 89sε× [1 +275m4 −2710m8 − 9m1254+ .. +12x90− n42x90+ ..] . (20)53V. ČELEBONOVIĆ3.2. The case x ∈ [1,∞]In this case, the calculation of chemical po-tential is more straightforward. Inserting Eqs. (17)and (18) into the expression for I2 in Eq. (15), andperforming the integration, gives the following ap-proximate result for the integral I2:I2 ∼= −8ε9 + πε cos[d] cosh[b][a5120− a3c212+ac424− c5cos[c] cos[d] + . . .] . (21)3.3. The chemical potentialAccording to Eqs. (14) and (15) chemical po-tential is given by:µ = kBT ln(ρλ3) + 4πρσ3(I1 + I2) (22)where the first terms of I1 and I2 are given by Eqs.(20) and (21).Inserting Eqs. (20) and (21) into Eq. (22), onegets a simple analytical approximation for chemicalpotential of a LJ fluid.4. DISCUSSION AND CONCLUSIONSIn this paper, we have obtained an approxi-mate analytical expression for chemical potential ofa Lennard Jones fluid. Two ways in which such anexpression can be obtained have been presented, andboth of these approaches have been applied.The approach based on the general thermo-dynamic formalism gives a result, expressed as Eq.(10), which is both mathematically and physicallysimpler. It contains just two variables which char-acterize the material under consideration - these areσ - the diameter of the molecular ”hard core”, andε - the depth of the potential well. Note that thechemical potential obtained in this way for a certainvalue of the ratio ε/T reduces to the value µID.The formalism of statistical mechanics is bothmathematically and physically more complex. Thegeneral conclusion is that the chemical potential de-pends on thermodynamic parameters of the fluidthrough the functions a-s, which are in turn func-tions of the pressure and/or density and/or temper-ature (Morsali et al. 2005), but also on the inter-action parameters. The approximate expression forchemical potential of a LJ fluid is:µ ∼= kBT ln(ρλ3) + 329 πρεσ3s[1 +27m45+.. +12x90+ .. +a3scos[d] cosh[b](3a2320− 3c232)+..] . (23)All the symbols in this expression have their stan-dard meanings, or were introduced in Morsali et al.(2005). Mathematically, the symbol x0 denotes thecut off radius of the LJ potential introduced in cal-culations in order to avoid the occurence of diver-gences. Physically, this quantity represents the inter-particle distance at which the pressure ionisation oc-curs. Qualitatively speaking, the pressure excitationand/or ionisation occur because electronic energieschange under the influence of the external pressurefield. For details about this process see, for example,Kothari (1938).The calculation presented in this paper wasmotivated by recent advances in planetary science.As a consequence of numerous discoveries of giant ex-oplanets, modellisation of their internal structure hasregained importance. These planets consist mostlyof fluids, and accordingly an obvious need for a the-oretical ”preparation of the ground” for the modelli-sation of their interiors has occured. Studies of phaseequilibrium and phase transitions demand an explicitknowledge of chemical potential. Some preliminaryresults in that direction have recently been obtained(Celebonovic 2009) in the limit of small density andwithout taking into account chemical potential. An-other interesting problem, which becomes accessiblefor study with the results obtained in this paper isthe behaviour of chemical potential of a LJ fluid withchanges of its thermodynamical parameters. Someaspects of both of these problems will be discussedin a future work.Acknowledgements – This work was financed by theMinistry of Science Technology and Development ofSerbia under its project 141007. I am grateful tothe referee for helpful comments about the previousversion of the manuscript.REFERENCESCelebonovic, V.: 2009, Publ. Astron. Obs. Belgrade,86, 319.Dzyaloshinskii, I. E., Lifchitz, E. M. and Pitaevskii,L. P.: 1961, Sov. Physics-Uspekhi, 4, 153.Hill, T. L.: 1987, Statistical Mechanics: Principlesand Selected Applications, Dover PublicationsInc., New York.Lai, X. Y. and Xu, R. X.: 2010, arXiv:1011.0526 v2.Landau, L. D. and Lifchitz, E. M.: 1976, Statistich-eskaya Fizika, Nauka, Moscow.Kothari, D. S.: 1938, Proc. Roy. Soc., A165, 486.Margenau, H.: 1939, Rev. Mod. Phys., 11, 1.Morsali, A., Goharshadi, E. K., Ali Mansoori, G. andAbbaspour, M.: 2005, Chemical Physics, 310,11.Reichl, L. E.: 1988, A Modern Course in StatisticalPhysics, E. Arnold Publishers, Ltd., London.54CHEMICAL POTENTIAL OF A LENNARD JONES FLUIDHEMIJSKI POTENCIJAL LENARD-ONSOVOG FLUIDAV. ČelebonovićInstitute of Physics, Pregrevica 118, 11080 Zemun-Belgrade, SerbiaE–mail: vladan@ipb.ac.rsUDK 52–335Prethodno saopxteǌeU ovom radu izveden je na dva naqinaanalitiqki izraz za hemijski potencijalLenard-onsovog fluida. Izraqunavaǌe jeizvedeno koristei termodinamiqki for-malizam, kao i formalizam statistiqkemehanike. Oblast integracije je podeǉena nadve pod-oblasti. Dobijeni rezultati biekorisni u svim istraivaǌima vezanim zaravnoteu faza i fazne prelaze u Lenard-onsovom fluidu. Primeri takvih sistemaznaqajnih u astronomiji su velike planete unaxem planetarnom sistemu, ali i van ǌega,kao i veliki sateliti za koje se smatra dasadre okeane ispod povrxina. Nova istrai-vaǌa pokazuju da bi Lenard-onsov fluid mo-gao da ima primenu i u prouqavaǌima zvezdakoje se sastoje od kvarkova.55

Chemical Potential of a Lennard Jones Fluid

The aim of this paper is to present results of analytical calculation of
   chemical potential of a Lennard Jones (LJ) fluid performed in two ways: by
   using the thermodynamical formalism and the formalism of statistical
   mechanics. The integration range is divided into two regions. In the small
   distance region, which is r ? ? in the usual notation, the integration range
   had to be cut off in order to avoid the occurrence of divergences. In the
   large distance region, the calculation is technically simpler. The
   calculation reported here will be useful in all kinds of studies concerning
   phase equilibrium in a LJ fluid. Interesting kinds of such systems are the
   giant planets and the icy satellites in various planetary systems, but also
   the (so far) hypothetical quark stars.</jats:p

V. Celebonovic

Crossref

Chemical potential of a Lennard Jones fluid

The aim of this paper is to present results of analytical calculation of
   chemical potential of a Lennard Jones (LJ) fluid performed in two ways: by
   using the thermodynamical formalism and the formalism of statistical
   mechanics. The integration range is divided into two regions. In the small
   distance region, which is r ≤ σ in the usual notation, the integration range
   had to be cut off in order to avoid the occurrence of divergences. In the
   large distance region, the calculation is technically simpler. The
   calculation reported here will be useful in all kinds of studies concerning
   phase equilibrium in a LJ fluid. Interesting kinds of such systems are the
   giant planets and the icy satellites in various planetary systems, but also
   the (so far) hypothetical quark stars

The chemical potential of a Lennard Jones fluid

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