On the coexistence of phases in a Lennard Jones liquid:first results
aa r X i v : . [ a s t r o - ph . E P ] F e b Publ. Astron. Obs. Belgrade No. 94 (2009), 1 - 4
Contributed paper
ON THE COEXISTENCE OF PHASES IN ALENNARD JONES FLUID:FIRST RESULTS
V. ˇCELEBONOVI ´C
Institute of Physics,Pregrevica 118, 11080 Zemun-Belgrade, SerbiaE–mail [email protected]
Abstract.
The aim of this paper is to investigate the conditions for the coexistence of phasesin a Lennard Jones fluid.The calculation has been performed within the virial developementmethod,and as a result,a simple approximate relation has been obtained between the numberdensities of two coexisting phases and the interparticle interaction potentials in them. Theresults of this work could have implications for modelling of giant planet interiors.This hasbecome important,due to the discovery of more than 300 extrasolar planets.
1. INTRODUCTION
In the last 5-6 years of the XX century,interest in planetology has drastically risen.This is due partially to results obtained within our planetary system,but even moreso to discoveries of extrasolar planets. Until the end of November 2008., accordingto data at http://exoplanet.eu , 329 exoplanets have been detected and 269 of themhave masses M ≤ M J where M J is the mass of Jupiter. This testifies about theinterest of modelling the internal structure of the giant planets. For a recent studysee,for example,(Vorberger et al.,2007 or Nettlemann et al.,2008).The aim of this contribution is to present preliminary results on the conditionsunder which two phases in a Lennard-Jones fluid can be in equilibrium. Phases aredefined as regions of the parameter space within which properties of a material areuniform.The condition for their equilibrium is the equality of pressures,temperaturesand chemical potentials. In this contribution only the equality of pressures and tem-peratures will be considered; accordingly the results will be only preliminary. Theequality of chemical potentials will be included in future work.
2. METHOD
The virial development of the equation of state is a method in which the equa-tion of state (
EOS ) of a fluid can be expressed as a power series in the density,andthe coefficients take into account the interactions present in the system,in which anincreasing number of particles takes part (e.g. Reichl 1988). Presented at the 15 th National Conf.of Astronomers of Serbia,Belgrade,October 2-5 . ,2008.Theproceedings will appear in Proc.Astron.Obs.Belgrade.Disregard the volume and page numbers. . ˇCELEBONOVI ´C The mathematical form of the
EOS of a fluid within the virial development is pvk B T = ∞ X l =1 a l ( T )( λ v ) l − (1)All the symbols on the left side of Eq.(1) have their standard meanings,while on theright hand side, a l are the so called virial coefficients, λ is the thermal wavelength and v is the inverse number density of the system v = V /N . The thermal wavelength isgiven by (for example Reichl,1988) λ = ( 2 π ¯ h mk B T ) / (2)where ¯ h is Planck’s constant and m the particle mass. The LJ model potential hasthe form u ( r ) = 4 ǫ h ( σr ) − ( σr ) i (3)The symbol ǫ denotes the depth of the potential,while σ is the diameter of the molec-ular ”hard core”. The first coefficient in Eq.(1) is a l = 1 ,while the second one isgiven by a ( T ) = − (2 / πN A β Z ∞ exp( − βu )( ∂u∂r ) r dr (4)(Maitland,G,C.,Rigby,M.,Smith.B.E.and Wakeham,W.A.,1987). In this and other ex-pressions β = 1 /k B T , T is the temperature and k B is the Boltzmann constant.It canbe shown that the second virial coefficient for the Lennard-Jones potential is given by a ( T ∗ ) = b ∞ X j =0 γ j (1 /T ∗ ) (2 j +1) / (5)where γ j = − ( j +1 / j ! Γ( 2 j −
14 ) (6) T ∗ = 1 βǫ (7)and b = (2 π/ N A σ ,where N A is Avogadro’s number. The chemical potential of afluid is given by (Hill,1987): µk B T = ln( nλ ) + nk B T Z dγ Z ∞ dr πr u ( r ) g ( r ) (8)where n is the particle number density and g ( r ) is the radial distribution function. Theproblem with the calculation of µ is the determination of g ( r ),which is a complicatedtask in statistical mechanics.In the future,it will be attempted to insert the form of g ( r ) proposed in (Morsali et.al.,2005) into Eq.(8).2 N THE COEXISTENCE OF PHASES IN A L-J FLUID
3. THE CALCULATION
In order to render the
EOS within the virial development physically applicable,the power series in Eq.(1) has to be convergent. This means that the results areapplicable only under the condition λ v = nλ ≺ n ≺ (cid:16) m π (cid:17) / (cid:20) ( k B T ) / ¯ h (cid:21) (10)Denote the two mutually non-interacting phases which make up the system by ”1”and ”2”. Applying Eq.(1),gives p = n k B T ∞ X l =1 a l ( T )( λ v ) l − (11)and p = n k B T ∞ X r =1 c r ( T )( λ v ) r − (12)Inserting the conditions of the equality of pressures and temperatures needed for thecoexistence in equilibrium of the two phases ,one gets the following expression for theratio of densities in them n n = P ∞ r =1 c r ( T )( n λ ) r − P ∞ l =1 a l ( T )( n λ ) l − (13)Taking into account that the equality of the temperatures implies the equality of thethermal wavelengths, this expression can be transformed into the following form ∞ X l =1 a l ( T ) n l λ l − = ∞ X r =1 c r ( T ) n r λ r − (14)Limiting the sums to the first two terms,it follows that a n + a n λ = c n + c n λ (15)Taking into account that the first virial coefficient is 1,and introducing n − n = x ,the last expression can be solved to give x = (1 / λa ) h − − λa n + p a n λ (1 + c n λ ) i (16)This result is mathematically simple,but physically interesting. It gives the differencebetween the number densities of two phases coexisting in equilibrium,expressed interms of the density of one of the phases ,the thermal wavelength and the second3 . ˇCELEBONOVI ´C virial coefficient in both of them. Expression (16) can further be transformed to givefinally n = − λa h − p a n λ (1 + c n λ ) i (17)which is positive for 4 a n (1 + c n λ ) ≻ any potential can be inserted in it.Turning to the case of a L − J fluid,the values of the first few coefficients γ j are: γ = 1 . γ = − . γ = − . a = 2 π N A σ (cid:20) . ǫk B T ) / − . ǫk B T ) / − . ǫk B T ) / + ... (cid:21) (18)Inserting Eq.(18) into Eq.(17),one would obtain an expression ”linking” the numberdensities of the two phases coexisting in equilibrium with the parameters of interpar-ticle potentials in them. How could this result be applied in studies of the interiors ofthe Jovian planets? Within any celestial body, the particle number density increaseswith increasing depth. Therefore, the parameters of the L − J potential would alsobe density dependent,which implies that the virial coefficients would also be depth(an density) dependent.This means that the possibility of phase coexistence would bedensity dependent.A closely related problem is the density dependence of the phasetransition pressure in a system with a L − J potential. These and other issues relatedto problems of the behaviour of a L − J fluid under high density will be studied infuture work.
4. Acknowledgement
This contribution has been prepared within the research project 141007 financed bythe Ministry of Science,Technology and Development of Serbia. I am grateful to thereferee for helpful comments.
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