Molecular dynamics simulations of momentum and thermal diffusion properties of near-critical argon along isobars.

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The

Journal

of

Supercritical

Fluids

jou rn al h om ep a g e :w w w . e l s e v i e r . c o m / l o c a t e / s u p f l u

Molecular

dynamics

simulations

of

momentum

and

thermal

diffusion

properties

of

near-critical

argon

along

isobars

Jakler

Nichele

a,b

_,

_Itamar

_Borges

_Jr.

a,b

_,

_Alan

_B.

_Oliveira

c

_,

_Leonardo

_S.

_Alves

a,d,∗ a_Defense_Engineering_Graduate_Program,_Military_Institute_of_Engineering,_22290-270_Rio_de_Janeiro,_Brazil

b_Chemical_Engineering_Department,_Military_Institute_of_Engineering,_22290-270_Rio_de_Janeiro,_Brazil c_Department_of_Physics,_Federal_University_of_Ouro_Preto,_35400-00_Ouro_Preto,_Brazil

d_Theoretical_and_Applied_Mechanics_Laboratory,_Mechanical_Engineering_Department,_Fluminense_Federal_University,_24210-240_Niterói,_Brazil

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received18October2015

Receivedinrevisedform11April2016 Accepted11April2016

Availableonline12April2016 Keywords:

Shearviscosity Bulkviscosity Thermalconductivity

Equilibriummoleculardynamics Stokeshypothesis

a

b

s

t

r

a

c

t

Threebasicdiffusionpropertiesofargon–shearviscosity,bulkviscosityandthermalconductivity–were studiedintheneighborhoodofthecriticalpointusingmoleculardynamics(MD)andtheLennard-Jones potentialenergyfunction.MDsimulationswereperformedalongthe1.0Pcand1.2Pcisobars.Green-Kubo

relationsandaLennard-Jonespairpotentialwereused.FourdifferentsetsofLennard-Jonesparameters wereused.Acomparisonofcomputedshearviscosityandthermalconductivityvalueswithdataavailable fromtheNationalInstituteofStandardsandTechnology(NIST)displayedagoodagreement.Resultsfor bulkviscosityindicatedthatvaluesofthispropertycannotbeneglectedinthisthermodynamicregion, aresultthatviolatesthetraditionalandmuch-assumedStokeshypothesisinclassicalﬂuidmechanics. Furthermore,itwasshownthatintheneighborhoodofthecriticalregionthebulkviscositycanhave largervaluesthantheshearviscosity.

1. Introduction

Fluiddynamicsandheattransfermodelsarestronglyrelated

toaccurate values for transport coefﬁcients. The Navier-Stokes

equation[1]andtheenergyequation[2]incorporateforagiven

substancethreetransport coefﬁcients thatconnect thephysical

dynamicsofmomentumandheattothreeproperties,namely,bulk

viscosity,shearviscosityandthermalconductivity.

Thereareintheliteratureseveraldatasetsthatcollect

exper-imental values of shear and thermal conductivity for simple

substances. Theelectronic tablesfrom theNational Instituteof

StandardsandTechnology(NIST)[3],forinstance,allowthe

solu-tionofagreatnumberoftypicalengineeringproblems.Inthese

cases,thebulkviscosityisgenerallyneglectedbasedonthevalidity

ofStokeshypothesis[4]evenwhentheﬂuidisneither

incompress-iblenoramonoatomicdilutegas[5].Hence,bulkviscosityvalues

arerarelyfoundintables,technicalworksandengineeringprojects,

eventhoughtheyareneededforthecompletespeciﬁcationofthe

Navier-Stokesstresstensor.

∗ Correspondingauthorat:DefenseEngineeringGraduateProgram,Military Insti-tuteofEngineering,22290-270RiodeJaneiro,Brazil.

E-mailaddress:leonardo.alves@mec.uff.br(L.S.Alves).

However,intheneighborhoodofthecriticalpoint,availability

oftransportcoefﬁcientdataisscarce.Moreover,theoryand

exper-imentdisplayanomaliesinthebehaviorofsomethermodynamic

andtransportproperties,usuallyappearsasadivergencetowards

inﬁniteorzerodependingontheproperty[6–10].Inparticular,

thehighcompressibilityofnear-criticalﬂuidsreducesthe

accu-racyofexperimentalmeasurements.Forthisclassofﬂuids,bulk

viscositydataareevenmorelimitedsincetheacousticabsorption

ofsoundwaves–themostusedexperimentaltechnique[11,12]

–issubject tohighlevelsoferror.Toovercomethisdrawback,

Brillouinscatteringwasdeveloped asa promising techniqueto

measurebulkviscosity,thoughitisstillnotwidelyknownandused

[13].

In order to overcome the aforementioned experimental

limitations, an interesting possibility is performing molecular

simulationsthatcansuitablymodelthenear-criticalstateand

com-putetransportcoefﬁcients.Oneofthesemethodsisequilibrium

moleculardynamics(MD)[14–16].Inthisapproach,fundamental

thermophysicalpropertiessuchasenergy,temperature,pressure

canbedeterminedfromspeciﬁcaveragesofthedynamic

evolu-tionofanensembleofmolecules.Otherpropertiesareobtained

using relationsfrom thermodynamics and statisticalmechanics

including time correlations, e.g., transport properties [17].This

methodisusefulintheinvestigationsofphasediagrams[18]and

chemical reactions [19]. In the framework of MD, particle

tra-http://dx.doi.org/10.1016/j.supﬂu.2016.04.004

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10

20

30

40

1.0

1.2

1.4

1.6

1.8

2.0 η

s

(μ

Pa

·s

)

T

r

10

20

30

40

1.0

1.2

1.4

1.6

1.8

2.0 η

s

(μ

Pa

·s

)

T

r

10

20

30

40

50

60

1.0

1.2

1.4

1.6

1.8

2.0 η

s

(μ

Pa

·s)

T

r

(d)

10

20

30

40

50

60

1.0

1.2

1.4

1.6

1.8

2.0 η

s

(μ

Pa

·s)

T

r

(c)

10

20

30

40

50

60

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1 η

s

(μ

Pa

·s

)

T

_r

(e)

Fig.1.Shearviscositycomputedusing104_argon_atoms_in_the_MD_simulation_for_P₌_1.0P

cinarangeofreducedtemperaturesfrom1.0to2.0usingtheLennard-Jonespotential

withdifferentparameterspairs–(a)[25],(b)[26],(c)[29],(d)[30]–and(e)theaveragevalueswitherrorbars.

jectoriesin phase spacedeﬁned bypositions andvelocities are

computedfromtheforcesresultingfromaninteractionpotential

energybetweenpairsofatomsandmolecules.Themostemployed

modelfornon-bondedparticlesistheLennard-Jones(L-J)

poten-tial[20]. It hasa simple mathematical expressionand leadsto

accurateresultsnotably inthestudyofnoblegases.TheL-Jhas

alreadybeenusedtocomputetransportpropertiesof

supercrit-ical ﬂuidsfar fromtheir critical points, transport properties of

near-criticalﬂuidsusingother approachesand otherproperties

[21–26].

In this work, we investigated transport properties of

near-critical argon using equilibrium MD. The present simulations

are based on Green-Kubo theory and use four different

ver-sionsofargonLennard-Jonespotentialsavailableintheliterature.

The importance of investigating argonin this workis twofold.

First, it is the most abundant noble gas in the atmosphere.

Second, it is an important substance employed for calibration

purposes in laboratoriesbecauseit is chemicallyinert and has

a low cost. The present resultswere compared with available

experimental data, which allows one to estimate their range

of applicability. Due to the scarcity of bulk viscosity

experi-mental values, especial emphasis was given to this transport

propertyinsuchawaythattheycanbeusedinsimulationsof

compressible ﬂuid ﬂow [27] aswell as heat transfer problems

[28].

2. Fundamentals

2.1. MoleculardynamicsandtheLennard-Jonespotential

ThebasisoftheMDtechniqueisthecomputationalsimulation

ofNewton’s equationofmotionfora setofinteractingmoving

particles.Forasystemconsistingofargonatomsinaregularbox,

onlyinteractionsbetweenpairsofnon-bondedatomswere

consid-ered.Inotherwords,many-bodyinteractionsarenotconsidered

explicitly.

Apairpotentialsuitablymodelstheinteractionsbetweenatom

pairs.ThesimplestrealisticmodelforthispurposeistheL-J12-6

potential.ItisatwoparametersmodelgivenbyEq.(1)

ULJ(rij)=

4␧

/rij12−/rij6

, rij≤rc 0, rij>rc (1)

whereε isthedepthofthepotentialwell,is thezero-energy

distance,r_ij isthedistancebetweenparticlesiandj,andr_c isa

cut-offdistancethatplaysanimportantroleinreducingthe

com-putationalcost.Tousethispotentialitisnecessarytocorrectthe

computedvalueofthepotentialinordertoincludethesmall

ener-giesresultingfromlong-rangeinteractionsfartherthanrc[20].The

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10

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1.0

1.2

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2.0 η

s

(μ

Pa·

s)

T

r

10

20

30

40

50

1.0

1.2

1.4

1.6

1.8

2.0 η

s

(μ

Pa·

s)

T

r

10

20

30

40

50

60

1.0

1.2

1.4

1.6

1.8

2.0 η

s

(μ

Pa

·s

)

T

r

(d)

10

20

30

40

50

60

1.0

1.2

1.4

1.6

1.8

2.0 η

s

(μ

Pa

·s

)

T

r

(c)

10

20

30

40

50

60

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1 η

s

(μ

Pa·

s)

T

r

(e)

Fig.2.Shearviscositycomputedusing104_argon_atoms_in_the_MD_simulation_for_P₌_1.2P

cinarangeofreducedtemperaturesfrom1.0to2.0usingLennard-Jonespotential

withdifferentparameterspairs–(a)[25],(b)[26],(c)[29],(d)[30]–and(e)theaveragevalueswitherrorbars.

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 0 10 20 30 40 Norm a liz ed J p mn co rr e la ti o n t(ps) P = 1.00 Pc Tr = 1.00 Tr = 1.10 Tr = 2.00 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 0 10 20 30 40 Norm a liz ed J p mn co rr e la ti o n t(ps) P = 1.20 Pc Tr = 1.00 Tr = 1.10 Tr = 2.00

Fig.3.Correlationsofoff-diagonalstresstensorelementsof104_argon_atoms_in_the_MD_simulation_of_1.0P

cand1.2Pcisobarsforthreetemperatures1.00Tc,1.10Tcand2.00Tc

usingLennard-Jonespotentialparametersfrom[25].

termdescribesattractioneffects.Eachi–jpairofparticlesinteracts

viaaforcefijthatcanbecalculatedfrom

fij=−

∇

ULJ(rij) (2)

Themotionof eachparticleisdeterminedfromitsresultant

force. An important feature of MD is that an accurate

poten-tialenergyexpressiongenerallyleadstoaccuratethermophysical

properties. When using the Lennard-Jones 12-6 expression, all

possiblemodelchoicesarerestrictedtoselectingandεvalues

inadditiontoaspecialcareconcerningthercvaluesinallcases.In

ordertoconsideruncertaintiesoftheLJpotentialparametrization

ofargon,fourdifferentpairsofandεandtheirrespectivecritical

constantswereusedandcomparedinthiswork.Theyareshown

inTable1.

InMDsimulationsitispossibletoadopttheconceptof

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0

20

40

1.0

1.2

1.4

1.6

1.8

2.0 κ

(mW/

m·

K

)

T

r

0

20

40

1.0

1.2

1.4

1.6

1.8

2.0 κ

(mW/

m·

K

)

T

r

0

20

40

60

80

1.0

1.2

1.4

1.6

1.8

2.0 κ

(m

W

/m·

K

)

T

r

(d)

0

20

40

60

80

1.0

1.2

1.4

1.6

1.8

2.0 κ

(m

W

/m·

K

)

T

r

(c)

0

20

40

60

80

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1 κ

(m

W

/m·

K

)

T

r

(e)

104.921

Fig.4.Thermalconductivityof104_argon_atoms_for_P₌_1.0P

cinarangeofreducedtemperaturesfrom1.0to2.0usingLennard-Jonespotentialwithdifferentparameters

pairs–(a)[25],(b)[26],(c)[29],(d)[30]–and(e)theaveragevalueswitherrorbars.

Table1

ParameterpairsfortheLennard-Jones12-6potentialargonatomsalongwiththeir respectivecriticalconstants.

ModelParameters CriticalConstants Ref.

␴(Å) ␧/kB(K−1₎ _Tc_(K) _Pc_(atm) _␳c_(kg/m3₎

3.405 119.8 150.86 48.34 535.397 Michelsetal.[25]

3.43 119.3 150.86 – 535.397 Levelt[26]

3.345 125.7 150.86 – 527.265 White[29]

3.3952 116.79 153.00 51.42 532.107 Vrabecetal.[30]

Tc:criticaltemperature;Pc:criticalpressure;␳c:criticaldensity;kB:Boltzmann

constant.

isolatedsystem(e.g.,microcanonicalorNVEensemble),aclosed

system (canonical, isochoric-isothermal or NVT ensemble), an

isobaric-isothermalsystem(NPTensemble),among other

possi-bilities,withoutusingpartitionfunctionsthatmaybedifﬁcultto

compute.Beforephysicalargumentsareconsidered,thechoiceof

anensembleisrelatedtotheuseofaspeciﬁcnumericalintegration

schemeoftheequationsofmotion.Eachalgorithmcaninclude

arti-ﬁcialthermostatsorbarostatstocontrolthebasicthermodynamic

variables—N,V,TandP[14].Themathematicalexpressionsusedin

MDandderivedfromstatisticalmechanicstheoryforcomputing

thethermophysicalpropertiesofinterestmustbeobtainedfrom

suitableensembles.

It is also important to avoid edge effects. For this

pur-pose,MDsimulationsofbulksystemsgenerallyemployperiodic

boundaryconditions.Theseartiﬁcialboundariesimposedonthe

systemrestrict thesimulation boxsize. Thisapproach is called

minimal-image convention and it states that the characteristic

dimensionofthesimulationbox–thelengthforacubicbox–must

begreaterthantwicethecutoffradiusrc.

2.2. Green-Kuboformulasfortransportproperties

Equilibrium MD employs Green-Kubo formulas [31] for the

computation of shear and bulk viscosity as well as of thermal

conductivity.Thethreecoefﬁcientscouldbeobtained

experimen-tally by perturbing the systemtowards non-equilibrium states

and observing its response that is associated with irreversible

processes. Therefore,in principle,to computethemone should

usenon-equilibriummoleculardynamics(NEMD)insteadof

equi-librium MD. NEMDisbased onthedirectapplication ofa heat

ormomentumﬂuxonthesimulation box[32,33].On theother

hand,theﬂuctuationdissipationtheoremallowsthecomputation

oftransportcoefﬁcientsfromthemicroscopicﬂuctuationsofthe

system.Thistheoremstatesthatsmallﬂuctuationsofthesystem

arerelatedtointernalrelaxationprocesses.Therefore,transport

coefﬁcientsthencanbecalculatedasintegralsoftime-correlation

functions of the microscopic stress tensor and internal energy

ﬂuxesinanequilibriumstate.

Shear viscositys is related totheresistanceof theﬂuid to

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0

20

40

60

1.0

1.2

1.4

1.6

1.8

2.0 κ

(mW

/m·

K

)

T

r

0

20

40

60

1.0

1.2

1.4

1.6

1.8

2.0 κ

(mW

/m·

K

)

T

r

0

20

40

60

80

1.0

1.2

1.4

1.6

1.8

2.0 κ

(mW

/m·

K

)

T

r

(d)

0

20

40

60

80

1.0

1.2

1.4

1.6

1.8

2.0 κ

(mW

/m·

K

)

T

r

(c)

0

20

40

60

80

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1 κ

(mW

/m

·K

)

T

r

Average Results for κ and NIST data at 1.00P

c

Fig.5.Thermalconductivityof104_argon_atoms_for_P₌_1.2P

cinarangeofreducedtemperaturesfrom1.0to2.0usingLennard-Jonespotentialwithdifferentparameterpairs

–(a)[25],(b)[26],(c)[29],(d)[30]–and(e)theaveragevalueswitherrorbars.

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80 100 Norm a lize d J Q co rr e la ti o n t(ps) P = 1.00 Pc Tr = 1.00 Tr = 1.10 Tr = 2.00 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80 100 Norm a lize d J Q co rr e la ti o n t(ps) P = 1.20 Pc Tr = 1.00 Tr = 1.10 Tr = 2.00

Fig.6.Heatﬂuxcorrelationsof104_argon_atoms_of_1.0P

cand1.2Pcisobarsforthreetemperatures1.00Tc,1.10Tcand2.00TcusingLennard-Jonespotentialparametersfrom

[25].

propertyisobtainedbyintegratingtheensembleaverageofthe

timecorrelationfunctionofthestresstensorJp_off-diagonal

ele-ments,whichrepresentthemomentumﬂuxes,

s= 1 3kBVT

m<n

∞ 0 Jp mn(0)Jmnp (t)dt (3)

wheremandnareindexesrepresentingtheCartesiancoordinates

x,yandzwithm /=n,k_BistheBoltzmannconstant,andVandTare

respectivelythevolumeandtemperatureofthesystem.

Thebulkviscositybdescribestheresistanceofaﬂuidto

dilata-tionassumingasmallvolumeelementofconstantshape.Inthe

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(μ

Pa·

s)

T

r

10

20

30

40

1.0

1.2

1.4

1.6

1.8

2.0 η

b

(μ

Pa·

s)

T

r

10

20

30

40

50

60

1.0

1.2

1.4

1.6

1.8

2.0 η

b

(μ

Pa·

s)

T

r

(d)

10

20

30

40

50

60

1.0

1.2

1.4

1.6

1.8

2.0 η

b

(μ

Pa·

s)

T

r

(c)

10

20

30

40

50

60

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1 η

b

(μ

Pa

·s

)

T

r

(e)

206.263 180.856 2777.02 798.348

Fig.7. Bulkviscosityof104_argon_atoms_for_1.0P

cisobarinarangeofreducedtemperaturesfrom1.0to2.0usingLennard-Jonesmodelwithdifferentparameterspairs–(a)

[25],(b)[26],(c)[29],(d)[30]–and(e)theaveragevaluesofbcomparedtotheaveragevaluesofsforthesameisobar.

theensembleaverageofthetimecorrelationfunctionofthestress

tensor_Jp_diagonal_elements_minus_the_product_PV._In_other_words,

b= 1 3kBVT

m

∞ 0 (Jp mm(0)−P(0)V)(Jmmp (t)−P(t)V)dt (4)

Thermalconductivity,,physicallyexpressestheabilityofthe

substanceintransferringthermalenergywhensubjectto

temper-aturegradients.Itiscomputedbyintegratingtheensembleaverage

oftimecorrelationfunctionofthemicroscopicheatﬂuxJQ_vector,

namely, = 1 3kBVT2

_∞ 0 JQ₍₀₎_·_JQ_(t)dt ₍₅₎

InEqs.(3)and(4),thestresstensorJp_can_be_computed_from

Jp₌ N

i=1 mivivi− N

i=1 N

j>i rij−

∇

Uij

(6)

wherem_iisthemassoftheparticlei,U_ij theinteraction

poten-tial between particles i and j, and the prime means matrix

transpose.Diagonalelementshavetheindexmequalton.InEq.

(5),theenergyﬂuxvectoriscalculatedfrom

JQ ₌ 1 V

⎡

⎣

N i=1 eivi+ 1 2 N

i<j1 (−

∇

Uij·(vi+vj))rij

⎤

⎦

(7)

wheree_iistheenergy,potentialandkinetic,ofparticleiwhereas

v

_iisthevelocityofparticlei.

2.3. Near-criticaleffects

Thethermophysicalandrheologicalpropertiesofasubstance

in the neighborhood of the critical point are characterized by

ananomalousbehavior.Thesecriticalanomaliesareobservedas

divergenceseithertozeroorinﬁnity.Theoryandexperimentshow

thatbulkandshearviscositiesandthermalconductivitydivergeto

inﬁnityinthevicinityofthecriticalpoint[34,23,35–39,21,40,41].

Oneimportantfeatureofthesedivergencesistheiruniversality,

i.e.,theyaredescribedbyuniversalparametersandexpressions

thatdonotdependonthenatureoftheﬂuid.Thesemodelsare

powerlawshavingasingleparameterthatindicatestheseparation

ofeachthermodynamicstatefromthecriticalpoint.Forthetwo

viscositiesandthermalconductivity,thisparameteristhe

temper-aturedifferenceT=T−Tc.However,itisalsoimportanttoidentify

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b

(μ

Pa·

s)

T

r

10

30

50

70

90

1.0

1.2

1.4

1.6

1.8

2.0 η

b

(μ

Pa·

s)

T

r

10

30

50

70

90

110

1.0

1.2

1.4

1.6

1.8

2.0 η

b

(μ

Pa·

s)

T

r

(d)

10

30

50

70

90

110

1.0

1.2

1.4

1.6

1.8

2.0 η

b

(μ

Pa·

s)

T

r

(c)

10

30

50

70

90

110

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1 η

b

(μ

Pa·

s)

T

r

(e)

296.885 (for 1.04Tc)

Fig.8.Bulkviscosityof104_argon_atoms_for_1.2P

cisobarinarangeofreducedtemperaturesfrom1.0to2.0usingLennard-Jonesmodelwithdifferentparameterspairs–(a)

[25],(b)[26],(c)[29],(d)[30]–and(e)theaveragevaluesofbcomparedtotheaveragevaluesofsforthesameisobar.

Fig.9. Correlationsofdiagonalstresstensorelementsof104_argon_atoms_of_1.0P

cand1.2Pcisobarsforthreetemperatures1.00Tc,1.10Tcand2.00TcusingLennard-Jones

potentialparametersfrom[26].

differentpathsleadtodifferentpowerlawcoefﬁcientsthatcould

berelatedthroughanequationofstate[6].Forinstance,alongthe

criticalisochore,onehas

≈◦T+− ₍₈₎

s≈s◦T− (9)

where=0.630±0.001,=0.031±0.004and =1.239±0.002are

universalexponentsobtainedfromrenormalizationgrouptheory

[42], with ◦ and s◦ depending on the substance. In

particu-lar,studiesseekingtounderstandthebehaviorofbulkviscosity

nearcriticalconditionsarerare.Fiveofthemostrepresentative

onesbasedeitheronMonteCarlosimulations,oronMDandeven

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importantsinceitcanhelpovercomingthesedifﬁcultiesasshown next.

3. Computationalmethods

The MDsimulations werecarried out usingtheopen source

codeLAMMPS[47].Shearviscosityandthermalconductivitywere

computedintheNVTensembleusinga Nosé-Hooverstyle

non-Hamiltonian equation of motion [48] in order to improve the

control ofthe simulation temperature.In order topreventany

inﬂuenceover pressure ﬂuctuations,bulk viscosityvalues were

obtainedintheNVEensembleusingtheVerletalgorithm.

Followingrecent papers focusing onnearcritical MD

simu-lations[9,49],weused104 _argon_atoms_in _all_simulations._This

numberofparticlesleadstoabettercontrolofthethermodynamic

stateinthesimulationboxandimprovedstatisticalaverages.The

numericalintegrationofthetimecorrelationfunctionsofthe

trans-portcoefﬁcients(Eqs.(3)–(5))usedthetrapezoidalruleoverall

timesteps.Eachsimulationwasequilibratedfor2ns(2×106_time

steps)beforeaverageswerecomputedforatimespanof10ns(107

timesteps).Atimestepof1fswasused.Correlationsofheatand

momentumﬂuxeswerecomputedforasamplinglengthof200ps

(2×105 _time_steps)._A _large_cut_off _radius_of₈ _was_chosen_to

improvetheaccuracyinthecalculationoftheinteractionenergies

andbetterdescribelong-rangecorrelationstypicalofthecritical

region.Thecorrectioninthepotentialforr_ij>rcwascomputed

inte-gratingaunitaryradialdistributionfunctionforthesedistances

[20].By employinga largevalueoftherc distance,simulations

weremorerealisticforadenseﬂuidanddescribedmoreaccurately

long-rangecorrections.Periodicboundaryconditionswereused.A

temperaturerangefrom1.0Tcupto2.0Tcalongtwoisobars–1.0Pc

and1.2Pc–wasconsidered.ThefourpairsofL-Jparametersshown

inTable1wereusedinthecomputations.Moredetailsaboutthe

simulationprocedurecanbefoundelsewhere[18].

4. Resultsanddiscussion

Figs.1and2showshearviscosityresultsofargonobtainedfrom

theMDsimulationsalongthe1.0Pc and1.2Pc isobarsusingfour

differentsetsofL-Jparameterstakenfromtheliterature(Table1).

Therearenosigniﬁcantdifferencesbetweentheresults

com-puted using all the L-J parameter pairs according to Table 1.

Therefore,computationof theshearviscosityisbarelysensitive

tosmallvariationsintheparametersoftheinteractionpotential.

Furthermore,MDsimulationswereabletoreproducethe

behav-ioroftheshearviscositycriticaldivergencequitewell.Figs.1eand

2eshowthatsimulationsareingoodagreementwithNISTdata.

TheresultsobtainedusingMichelsetal.[25]parametersfortheL-J

potentialexhibitabetterdescriptionofthecriticaldivergencein

theshearviscosity.

ToensuretheconvergenceoftheresultsinFigs.1and2,we

present in Fig.3 a setof normalizedJp

mn correlation functions

ofthreerepresentativetemperaturesforbothisobars.Theresults

agreewithdataalreadyreportedinliterature[50].

Figs.4–6showthecomputedthermalconductivityvaluesand

theheatﬂuxcorrelationfunctionsofargon,asdonefortheshear

viscosity.TheresultsarealsoingoodagreementwiththeNISTdata.

Allparameterspairsleadtosimilarresultswithsmall

discrepan-ciesascomparedtotheaveragevalues.Theparameterpairfrom

Michelsetal.[25]displaysasmootherbehaviorincomparisonto

NISTdataandbetterrepresentedthecriticaldivergence.Theerror

barsaresmallerascomparedtotheshearviscosityones.InFig.6,

thisplotshowscorrelationsupto100psandthesimulations

con-sideredthecorrelationsupto200ps.

Finally,Figs.7–9collectsMDbulkviscosityresultsfor argon

andthesetofcorrelationfunctionsusedtocomputethem.Since

thesedataarenotavailableinNISTtables,theycanbeespecially

useful.Onceagain,allfourpairsofL-Jparametersleadtosimilar

results,butinthiscasetheL-JparametersfromWhite[29]showed

aclearlysmootherbehaviorandabetterdescriptionofthe

criti-caldivergence.Largedeviationswerefoundintheaveragevalueof

bulkviscosityresults.Thesedeviationsareconsiderablylargerthan

deviationsreportedforshearviscosityandthermalconductivity.

Moreover,Fig.9showsthatthediagonalstresstensorcorrelations

exhibitabetterconvergenceifcomparedtothermalconductivity

correlations.Thecorrelationanalysisofthebulkviscositysuggests

thatthis propertyiseasiertocomputeascomparedtothermal

conductivityevennearthecriticalpoint.

AnotherimportantfeatureinFigs.7and8isthenon-zerovalue

ofthebulkviscosityfarfromthecriticaltemperature.AtTr=2.0

(_≈300K)thispropertyreachesvaluesaround30_␮Paswithan

aver-agedeviationof±5␮Pasconsideringtheresultsobtainedfromall

setofL-Jparameters.Thisresultshowsthateventhoughargonis

amonoatomicgas,theStokeshypothesisaswellasthekineticgas

theoryassumptions[52]donotholdandthuscannotbeused.

Fur-thermore,thispropertyhasthesameorderofmagnitudeofthe

shearviscosityvaluesand,hence,cannotbeneglectedin

simula-tionsofthemomentumconservationequations.

Figs.7eand8ecomparebothviscositiesobtainedusing

aver-agesovertheMDresultsandthefoursetsofL-Jparametersateach

T_rvalue.Theseresultsindicatethatbulkviscosityvaluesare

actu-allylargerthantheirshearcounterpartsforthethermodynamic

statesconsideredhere.Thisisanimportantﬁndingbecause

classi-calﬂuidmechanicsanalysisneglectsthebulkviscosity[53].From

theseﬁguresweproposeanempiricalrelationbetweenboth

vis-cositiesateachisobar:oneshoulduseb≈1.3satthe1.0Pcisobar

andb≈1.6satthe1.2Pcisobar.Therefore,thetraditional

assump-tionofneglectingthisdifferencecannotbeaprioriassumedinthe

neighborhoodofthecriticalpoint.

5. Conclusions

Inthisworkitwasshownthatmoleculardynamicssimulations

usingtheLennard-Jonespotentialtodescribethepotentialenergy

betweenargonatomsallowonetoobtainaccuratevaluesofboth

shearviscosityandthermalconductivityalongisobarsinthe

neigh-borhoodofthecriticalpoint.Thecomputedpropertieswereingood

agreementwithexperimentaldatacollectedintheNISTdatabase.

Thisgoodagreementleadsustocomputealsothebulkviscosity

inthesamethermodynamicregioneventhoughNISTdataisnot

availableinthiscase.

Intheneighborhoodofthecriticalpoint,theresultsforbulk

vis-cosityobtainedwithmoleculardynamicsshowthat:(a)shearand

bulkviscositiesdivergealongthesameisobarwithsimilar

behav-ior;(b)bulkviscositydoesnotgotozeroawayfromthecritical

pointinspiteofbeingclosetoit,thuscontradictingStokes

hypoth-esisandkinetictheoryevenforamonoatomicdilutegas;(c)bulk

viscosityisnotonlyofthesameorderofmagnitudeasshear

viscos-itynearthecriticalpoint,butitisactuallylargerthanitscounterpart

inthisregion.

The present approach of usingmolecular dynamicsto

com-pute diffusionproperties of a ﬂuid in the neighborhood of the

criticalpointisgeneralandwehaveshownthatcanbeusedto

(9)

complicatedandpracticalsystems(e.g.,N2).Investigationsalong

theselinesareunderwayinourgroup.

Thecompletesetofresultsforthetransportpropertiesis

avail-ableasanexcelﬁleintheSupplementarymaterialofthiswork.

Acknowledgements

L.S.deB.A.thanksBrazilianagenciesFAPERJ(throughgrants

E-26/110.846/2011and E-26/111.670/2012),CNPq (throughGrant

MCT/CNPq14-481072/2012-8)andCapes. I.B.and J.N.thankthe

sameagenciesforsupportofthisresearchaswelltheBrazilian

Army.L.S.deB.A.acknowledgestheﬁnancial supportfromthe

UnitedStatesAirForce(throughtheSOARDgrant

FA9550-13-1-0178). A.B.O. thanksCNPq(through Grants303820/2013-6 and

459852/2014-0)aswellaspartialﬁnancialsupportfrom

PROPP-UFOP(throughAuxílioaPesquisadores2015Grant).Wewouldlike

alsotothanktherefereesforthevaluablesuggestionsconcerning

dataanalysis.

AppendixA. Supplementarydata

Supplementarydataassociatedwiththisarticlecanbefound,

intheonlineversion,athttp://dx.doi.org/10.1016/j.supﬂu.2016.04.

004.

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