Dependence of Nanofluid Viscosity on Nanoparticle Size and Material

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1 4th Mco and ano Flows Confeence UCL London UK 7- Setembe 4 Deendence of anoflud Vscosty on anoatcle Sze and Mateal Valey Ya. RUDYAK * Segey L. KRASOLUTSKII Dens A. IVAOV * Coesondng autho: Tel.: ; Fax: 7(383)66483; Emal: Valey.Rudyak@mal.u The ovosbsk State Unvesty of Achtectue and Cvl Engneeng Russan Fedeaton Abstact The vscosty of nanofluds s studed by molecula dynamcs method. It s consdeed the model nanofluds based on lqud agon wth alumnum and lthum nanoatcles. The nanoatcle sze anges fom to 4 nm. The volume concentaton of nanoatcles s vaed fom % to %. It s shown that the vscosty of the nanoflud nceases wth deceasng nanoatcle sze. Potental contbuton to vscosty of nanoflud consdeably exceeds the knetc one. Theefoe aametes of molecule-molecule and nanoatcle-molecule otentals make the consdeable mact on vscosty of nanofluds whch thus deends on mateal of nanoatcles. Keywods: anofluds; anoatcles; Vscosty; Molecula Dynamcs; Effect of anoatcles Sze. Intoducton Snce nealy all alcatons of nanofluds ae elated to channel flows vscosty s a detemnng facto n the use. It has been well establshed that the vscosty of nanofluds s much hghe than the vscosty of conventonal dsesons at equal volume concentatons of dsesed atcles and s not descbed by any classcal theoy of the vscosty of these fluds (Ensten atchelo Mooney Kege etc.) (see evews [ 3]). It has been found that the effectve vscosty of nanofluds deends not only on the concentaton of nanoatcles but also on the sze. It s shown that the vscosty of the nanoflud nceases wth deceasng nanoatcle sze. Evdence fo ths has been ovded by molecula dynamcs smulatons fo the had-shee otental [4 5] as well as by exements [3 6 8]. Howeve ooste data have also been eoted [9 ]. In [3] t s noted that the vscosty of nanofluds may also deend on the nanoatcle mateal. Ths coelates wth the data of the knetc theoy of gas nanosusensons whee ths deendence has been well establshed [ ]. The fst MD calculatons have confmed the exstence of ths deendence [3]. Obtanng systematc data that would gve an unambguous answe to the queston of the deendence of the vscosty of nanofluds on the nanoatcle sze and the mateal s the obectve of ths ae. We studed the vscosty of model nanofluds n whch the cae flud was agon of densty ρ =.77 and temeatue T = 3 K. The dsesed comonent was lthum and alumnum nanoatcles wth a damete fom one to fou nanometes. The volume concentaton of nanoatcles was vaed fom % to %.. Smulaton technque The standad molecula dynamcs method was used (see e.g. [4 5]). A cubc cell and eodc bounday condtons wee used. Inteacton between the molecules of the cae flud was descbed by the Lennad- Jones otental Φ ( ) 4 [( / ) ( / ) ] 6 LJ = ε σ σ () whee σ s the effectve damete of the flud molecules ε s the deth of the otental well and = s the dstance between molecules and. Inteacton between cae flud molecules and a nanoatcle was descbed by the Rudyak Kasnolutsk (RK) otental [6 7] Ψ ) = Ψ ( ) Ψ ( ) () Ψ ( 9 3 = C ( ( - -

2 4th Mco and ano Flows Confeence UCL London UK 7- Setembe 4 ( ) ( ) R R a whee = 9 3 a = 9 /8 9 a = 3/ 3 V = ρ / m C 6 = (πε σ ) /3V C = (4πε σ ) / 45V 3 9 V = ρ / m. Hee ρ s the densty of the nanoatcle mateal m s the mass of the atom (molecule) of the nanoatcle R s the adus of the nanoatcle σ ε ae the aametes of the nteacton otental () between a cae flud molecule and a molecule of the nanoatcle. As the nanoatcle nteacton otental we use a secally constucted otental [8] whch fo monodsese nanoatcles has the fom U( = U 7 ( U( (3) whee π ~ ε ~ σ R U7( = 7 35 V ( ( ) R R 3 ( ( 5 5 ( ( π ~~ εσ 4R U = ( ln 3 V R. 4R Hee ~ R s the adus of the nanoatcles ε and ~ σ ae the aametes of the Lennad- Jones () nteacton otental of the nanoatcle atoms. Potentals () and (3) wee constucted unde the assumton that the nteacton between the cae-flud molecules and nanoatcle atoms and the nteacton of the nanoatcle atoms ae descbed by otental () esectvely wth the aametes σ ε and ~ σ ~ ε. The smulaton was efomed as follows. Caeflud molecules and nanoatcles wee unfomly laced n the smulaton cell accodng to the secfed values of the densty of the cae flud and the volume concentaton of nanoatcles. The densty of the ue flud was detemned by the aametes: ρ = σ 3 V and = V / V hee V s the volume of the system (cell) 3 V = σ / s the volume of close ackng of molecules. The volume of the smulaton cell fo the nanoflud was gven by the elaton 3 3 V = σ / 4π kv R / 3 whee and ae the numbe of molecules and nanoatcles esectvely. Wth the addton of nanoatcles the coeffcent k V was chosen so that the nanoflud essue was equal to the essue of ue agon of the gven densty. The ntal veloctes of molecules wee secfed accodng to the Maxwell dstbuton at the gven temeatue and those of nanoatcles wee set equal to zeo. The calculaton was stated afte an ntal elaxaton eod dung whch the whole system had come to equlbum. The ewton's equatons wee ntegated usng Schofeld s scheme [9]. Snce the otentals used hee have an nfnte ange they should be tuncated dung the smulaton. Fo otental () the tuncaton adus was.5σ and fo otental () t was ( R bσ ). The value of b was chosen so that at the tuncaton adus fo otental () the foce exeted on a molecule by a nanoatcle was equal to the foce of nteacton between two molecules at the tuncaton adus fo otental (). The quantty b deends on the nanoatcle adus mateal and the oetes of the cae-flud molecules. In atcula fo L nanoatcles (fo the damete of the nanoatcles was nm) n A b was equal to 3.78 and fo Al nanoatcles t was 4.. The nanoatcle nteacton otental was tuncated at a dstance at whch the nteacton foce between nanoatcles was zeo.e. the so-called soft-shee otental was used. It should be noted that the smulaton of the evoluton of nanofluds moses moe sevee estctons on the sze of the smulaton cell as comaed wth homogeneous fluds. Snce a nanoatcle should not nteact wth tself the sde length of the cubc cell L should be much geate than the ange of the otental - -

3 4th Mco and ano Flows Confeence UCL London UK 7- Setembe 4 ~ L >> R. In actce the dstance should be at least seveal tmes geate to avod the coesondng coelatons. The aametes of the nteacton otental of agon molecules wee as follows: σ = 3.45 Å and ε/k = 9.8 K []. The aametes of otentals () and (3) wee calculated usng the followng aametes of otental (): fo lthum σ = 3.95 Å and ε / k = 5 К [] and fo alumnum σ =.55 Å and ε / k = К. The latte aametes wee obtaned fom data on Young s modulus and the stuctue of the cystal lattce (face-centeed cubc) usng a method smla to that descbed n []. The aametes σ and ε wee detemned usng the smle combnatonal elatons: σ σ ~ = σ and ε εε~ =. 3 ehavo of the coelaton functons Shea vscosty coeffcent was calculated fom the Geen Kubo fomula [3] V η = J xy () t J xy ( t t ) J yz () t J yz ( t t ) 3k T () t J zx ( t t ) dt J zx (4) whee the xy comonent of the stess tenso fo a bnay mxtue has the fom J xy = m x y Fx y V = = = Fx y Fx y. (5) = = = Hee subsct = efes to molecules subsct to nanoatcles m s the mass of a atcle (molecule o nanoatcle) V s the volume of the system T s the temeatue of the medum s the tme to each a lateau value [4] F s the foce actng on a molecule o nanoatcle x and y ae the coodnates of the molecule o nanoatcle and and ae the numbes of molecules and nanoatcles esectvely. The angula backets n (4) demote an ensemble aveage. Equatons (4) wee obtaned n [3] fo molecula systems but n a book of one of the authos [5] t s shown that they have the same fom fo dsese systems n atcula fo nanofluds. Fluctuaton-dssatve theoem (4) gves the connecton between the vscosty coeffcent of the nanoflud and velocty coelaton functons (the functons n the angula backets). Theefoe the value of the vscosty coeffcent wll be detemned by the evoluton of the aoate coelaton functons. Accodng to (4) (5) the nanoflud vscosty coeffcent contans seveal dffeent contbutons η = ηk η ηk (6) whee the fst tem descbes the knetc contbuton ηk= k TV β () t m x y ( t t ) m x y dt (7) β= = = β the second tem descbes the otental contbuton η 4k TV = Fx y k TV k TV β= k l β () t F y ( t t ) dt x kl kl () t F y ( t t ) Fx y x kl kl dt = = k = l= k = l = () t F y ( t t ) Fx y x kl kl k = l= x kl ykl F () t Fx y ( t t ) dt and the thd descbes the coss effects η = m x y k () t ktv β = = β F k l x kl = k = l = y kl ( t t ) dt (8) () ( ) = m x y t F y t t dt (9) x kl kl In the fg. the tycal examle of coelaton functons χ enteng nto exessons (7)- (9) and defnng coeffcent of vscosty of the nanoflud A-Al wth atcles of damete nm s gven. Volume concentaton of nanoatcles equals % n ths examle. Pessue of the nanoflud equals to essue of - 3 -

4 4th Mco and ano Flows Confeence UCL London UK 7- Setembe 4 ue A at densty ρ =.77 and temeatue T = 3 K. Coelaton functons χ ae nomalzed on ( ) kt V tme s nomalzed on σ v T whee v T themal velocty of molecules of cayng meda. Functon χ on the fg. (dotted lne) coesonds to the knetc contbuton to coeffcent of vscosty (see (7)) functon χ (dash-dotted lne) coesonds to the otental one (the fomula (8)) functon χ 3 (dashed lne) coesonds to the otental-knetc one (the fomula (9)). At last the sold lne coesonds to the total coelaton functon χ = χ χ χ3. Fg.. The nomalzed knetc otental otentalknetc and total coelaton functons fo coeffcent of vscosty of nanoflud Analyss of the fg. shows that functon χ s manly detemned by the functon χ defnng the otental contbuton to tansot of mulse. It consdeably exceeds the knetc χ and otental-knetc χ 3 functons. The consequence of ths s that aametes of otentals () () make the consdeable mact on value of coeffcent of shft vscosty of nanofluds. Theefoe t should deend on mateal of nanoatcles. Ths ont of vew s confmed wth the data of modelng too. Table. Contbutons of dffeent comonents to the vscosty of nanofluds (μpa s) A Al-A L-A η η k η η k η η (η η ). 5.9 Ths table shows a comason of each of the contbutons fo agon and the two nanofluds consdeed. In both cases the concentaton of nanoatcles was equal to φ =.97 and the damete of the nanoatcles was nm. The essue of all thee fluds was the same and equal to the essue of ue A at a densty ρ =.77 and a temeatue T = 3 K. The sxth ow of the table shows the otental contbuton to the vscosty due to the nteacton of agon molecules the seventh ow shows the otental contbuton due to the coelaton between the ntemolecula foces and the nteacton foces of the molecules wth the atcle (the thd tem n fomula (8)). Fnally the last ow shows the contbuton due to the last tem n (8). Fom an analyss of the Table s data t aeas that fst although the oveall vscosty of nanofluds s sgnfcantly hghe than the vscosty of the cae flud the knetc contbutons η k ae lowe. Ths decease s due to a educton n the numbe densty of molecules n the nanoflud comaed to the cae flud. On the othe hand the knetc contbuton of nanoatcles s small snce the volume (and mass) of a nanoatcle s much hghe than the volume (mass) of the molecule wth the consevaton of the aveage knetc enegy of the molecules and nanoatcles. It should also be emhaszed that snce the flud s dense the contbuton of the knetc ocesses to the tansfe of the momentum (vscosty) s small: fo agon t s about % and fo the nanofluds 6 7%. The otental contbuton of the - 4 -

5 4th Mco and ano Flows Confeence UCL London UK 7- Setembe 4 molecules η (sxth ow of n the Table) n the nanofluds s also slghtly educed. The ole of the coss tems η k (ffth ow n the table) s small and s about ten ecent. In all ou calculatons the otental contbuton of the atcle nteacton was almost equal to zeo. Thus the ncease n the vscosty s due manly to nanoatcle molecule nteactons and the coelatons between molecule molecule and molecule nanoatcle nteactons. Deendence of value of these contbutons on a sze of the dsesed atcles s easy fo qualtatve estmatng. Let thee ae atcles wth aduses R and R fo defnteness we wll consde that R > R. At the gven volume concentaton numbes of those and othe atcles n a unt volume ae n nvese ooton to cubes of the 3 3 aduses: / = R / R. On the othe hand total coss-sectons of scatteng of molecules of a cayng flud wth those and wth othe atcles ae accodngly equal S = π R and S = π R. Theefoe the ato of these coss-sectons whch detemne value of consdeed otental contbutons aeas to be nvesely ootonal to the aduses: S / S = R / R. These smle estmatons show that wth gowth of atcle szes the elatve value of these contbutons wll decease. data fo the nanoflud wth L atcles and tangles eesent data fo the nanoflud wth Al atcles. In all cases the damete of the nanoatcles was nm. As a esult fo the nanoflud wth lthum nanoatcles n fomula (9) we obtaned a = 3. and a = 5.38 (dash-dot lne numbe n Fg. ) and fo the nanoflud wth alumnum nanoatcles a = 3.5 and a = 3.6 (dotted lne numbe n Fg..) In all cases the vscosty of two consdeed nanofluds s substantally hghe than that fo conventonal dsesons. Thus even at low atcle concentatons the effectve vscosty of nanofluds s not descbed by Ensten's theoy whch gves a =.5. The dashed lne numbe 3 n Fg. shows the deendence obtaned by atchelo [6] takng nto account the hydodynamc nteacton of dsesed atcles and the ownan moton. The atchelo fomula ovdes a good ft to vaous exemental data fo conventonal dsese systems but the values gven by ths fomula ae sgnfcantly smalle than those fo nanofluds (see Fg.). 4 Smulaton esults Snce all the nanofluds studed had a elatvely low concentaton of nanoatcles t mght be exected that the esultng vscosty would be descbed by a quadatc deendence on the volume concentaton of nanoatcles ϕ η / η η a ϕ a ϕ = =. (9) whee η s the vscosty of ue agon. Ou calculatons have shown that t s ndeed so. The deendence of the dmensonless vscosty η on the volume concentaton of nanoatcles s shown n Fg.. Hee the densty of the cae flud was ρ =.77 and the temeatue T = 3 K; ccles eesent Fg.. Deendence of the dmensonless vscosty of a nanoflud on a volume concentaton of nanoatcles whch ae made of dffeent mateals The second motant ont s that the vscostes of the nanofluds wth alumnum and lthum nanoatcles ae sgnfcantly dffeent. Thus the vscosty of nanofluds ndeed deends on the nanoatcle mateal. To answe the queston of how nanoflud - 5 -

6 4th Mco and ano Flows Confeence UCL London UK 7- Setembe 4 vscosty deends on the nanoatcle sze we efomed calculatons fo nanofluds wth lthum nanoatcles of damete and 4 nm at volume concentatons of nanoatcles φ = % and 4%. The esults can be aoxmated e.g. by the followng exonental deendence on the nanoatcle damete wth ecson about 5% η = η ( 5.5ϕ 4.94ϕ ) ex(.8d d) () whee D s the damete of the nanoatcles and d s the damete of the molecules. In the lmt D d >> the fomula () concdes wth atchelo fomula. 5 Dscusson of Results The data esented n the evous secton suggest that the effectve vscosty of nanofluds nceases wth nceasng concentaton of nanoatcles. At modeate volume concentatons of atcles the deendence of the vscosty on the concentaton of nanoatcles can be descbed by the quadatc functon (9) as fo conventonal dsesed fluds. At the same tme the vscosty of nanofluds at a gven atcle concentaton s substantally hghe than that fo conventonal dsesed fluds. Ths s consstent wth vtually all exstng exemental data (see [ 3]). In addton the vscosty of a nanoflud cannot be descbed by exstng classcal theoes because t deends not only on the concentaton of nanoatcles but also on the sze. Fo the model nanofluds consdeed n ths ae ths deendence s gven by elaton () whch s n qualtatve ageement wth the data of [3 6 8]. The vscosty of nanofluds deceases wth nceasng atcle sze. Physcally ths s a easonable esult snce as the atcle sze tends to the sze of conventonal macoscoc atcles elaton () should educe to the classcal fomulas fo the vscosty of nanofluds e.g. n atchelo fomula. Anothe motant featue that dstngushes the vscosty of nanofluds fom that of conventonal dsesons s ts deendence on the nanoatcle mateal. Ths deendence should dsaea wth nceasng nanoatcle sze. Howeve f the atcles ae suffcently small accountng fo ths deendence s motant and one cannot hoe to obtan a unvesal fomula fo the vscosty. It can be exected that ths deendence can be neglected even fo nanofluds wth a chaactestc atcle sze of moe than ten nanometes. Indect evdence fo ths s ovded by knetc theoy [ ] and molecula dynamcs smulatons of the dffuson of nanoatcles [7]. It should also be noted that the aeaance of the deendence on the nanoatcle mateal s not due to the choce of a atcula nteacton otental; t has also been ecoded fo the had-shee otental [4 5]. The aeaance of ths deendence s due to takng nto account the ndvdual chaactestcs of nanoatcles (n atcula the masses and nteacton otental aametes) n the nteacton wth cae flud molecules and wth each othe as s the case n conventonal molecula fluds. To see ths t s necessay to analyze the smulaton ocess of nanofluds fom a molecula-knetc ont of vew. The last nteestng queston s followng why do the vscosty coeffcents of nanoflud and couse dsesed flud wth low atcle concentaton dffe? A smle exlanaton of ths may be found n the followng consdeaton. In a susenson wth macoscoc atcles at volume concentatons less than o of the ode of 3 ϕ ~ the dstances between the atcles ae lage enough so that the nteacton can be neglected. Indeed suose that the 4 dsesed atcle sze s D ~ cm. Then at the secfed volume concentaton of these atcles the aveage dstance between them s 3 ~ 3 l m n whee n = 6ϕ /( π D ) s the numbe densty of the atcles. Fo these data n 9 ~ and l m ~ 3 cm. In ths case the mutual nfluence of atcles on the etubatons nduced by these atcles n the cae flud velocty feld can be neglected. Theefoe the vscosty coeffcent of the susenson can be descbed by Ensten's fomula η =.5ϕ. In nanofluds the atcle sze s of the ode - 6 -

7 4th Mco and ano Flows Confeence UCL London UK 7- Setembe 4 of β 7 cm whee the aamete β takes values of to. At the same volume 3 concentaton ( ϕ ~ ) the aveage dstance between nanoatcles n the flud s n 3 l ~ m nn. The numbe densty of nanoatcles 9 3 n ths case s of the ode of n n ~ β 9 3 and vaes fom to deendng on the sze of the atcles. Accodngly the dstance between the atcles vaes fom to 4 cm. In all cases the dstance between the nanoatcles ae of the ode of the sze. Theefoe we need to take nto account the hydodynamc nteacton of the atcles even at these low concentatons. 6 Concluson In concluson two thngs must be onted out. In ths ae we consdeed nanofluds based on a vey dense gas. Agon becomes a lqud at much lowe temeatues and then the elatve vscosty s even much hghe. Ths n atcula s due to the fact that n lquds the ole of the otental contbutons s even geate. In atcula n nanoflud A-Al at cayng meda densty ρ =.77 volume concentaton φ = 9.9% of alumnum atcles of damete nm at temeatue T = 4 K the elatve coeffcent of vscosty s η =.9 whle at T = 3 K t s η =.45. And second the choce of essue s a necessay condton fo coect comason of the tansot coeffcents of the cae flud and nanoflud. Ths s consstent wth exemental actce. 6 Achnoledgements Ths wok was atally suoted by the Russan Foundaton fo asc Reseach (Gant o. 3--5). Refeences. Mahbubul I.M. Sadu R. Amalna M.A. Latest develoments on the vscosty of nanofluds. Intenatonal J. Heat and Mass Tansfe Hossen S.Sh. Shahed A. and Vazfeshenas Y. A evew of elatons fo hyscal oetes of nanofluds. Austalan J. asc and Aled Scences 5() Rudyak V.Ya. Vscosty of nanofluds. Why t s not descbed by the classcal theoes. Advances n anoatcles Rudyak V.Ya. elkn A.A. Tomlna E.A. Egoov V.V. anoatcle fcton foce and effectve vscosty of nanofluds. Defect and Dffuson Foum Rudyak V.Ya. elkn A.A. and Egoov V.V. On the effectve vscosty of nanosusensons. Techncal Physcs 9 54(8) Rudyak V.Ya. Dmov S.V. Kuznetsov V.V. and adakhanov S.P. Measuement of the vscosty coeffcent of an ethylene glycol-based nanoflud wth slcon doxde atcles. Doklady Physcs 3 58(5) Rudyak V.Ya. Dmov S.V. and Kuznetsov V.V. About deendence of the nanoflud vscosty coeffcent on the temeatue and sze of the atcles. Techncal Physcs Lettes 3 39(7) Tmofeeva E.V. Smth D.S. Yu W. Fance D.M. Sngh D. Routbo J.L. Patcle sze and ntefacal effects on themo-hyscal and heat tansfe chaactestcs of wate-based -SC nanofluds. anotechnology () 573. do:.88/ /// He Y. Jn Y. Chen H. Dng Y. Cang D. and Lu H. Heat tansfe and flow behavo of aqueous susensons of TO nanoatcles (nanofluds) flowng uwad though a vetcal e. Int. J. Heat Mass Tansfe 7 5(-) guyen C.T. Desganges F. Roy G. Galans. Maeґ T. ouche S. and Mntsa H. Temeatue and atcle-sze deendent vscosty data fo wate-based nanofluds Hysteess henomenon. Int

8 4th Mco and ano Flows Confeence UCL London UK 7- Setembe 4 J. Heat and Flud Flow Rudyak V.Ya. Kasnolutsk S.L. About vscosty of aefed gas susensons wth nanoatcles. Doklady Physcs 3 48() Rudyak V.Ya. Kasnolutsky S.L. Effectve vscosty coeffcent fo aefed gas nanosusensons. Atmoshee and Ocean Otcs 4 7(5) Rudyak V. Dmov S. Kasnolutsk S. Ivanov D. Effectve vscosty of the nanofluds: Exemental studyng and molecula dynamcs smulaton. Techncal Poceedngs of the 3 STI anotechnology Confeence and Exo. STI-anotech Raaot D.C. The At of Molecula Dynamcs Smulaton. Cambdge Unvesty Pess oman G.E. Stegalov V.V. Stochastc Theoy of the Classcal Molecula Dynamcs Method. Mathematcal Models and Comute Smulatons 3 5(4) Rudyak V.Ya. Kasnolutsk S.L. The nteacton otental of dsesed atcles wth cae gas molecules. Raefed Gas Dynamcs XXI. Poc. st Int. Sym. on RGD. Toulouse Géadués- Édtons Rudyak V.Ya. Kasnolutsk S.L. Knetc descton of nanoatcle dffuson n aefed gas. Doklady Physcs 46() Rudyak V.Ya. Kasnolutsk S.L. and Ivanov D.A. The Inteacton Potental of anoatcles. Doklady Physcs 57() do:.34/s Schofeld P. Comute smulaton studes of the lqud state. Comut. Phys. Comm () Hschfelde J.O. Cutss Ch.F. and d R.. Molecula Theoy of Gases and Lquds. Wley ew Yok Aefyev K.M. Tansot ocesses n gases and lasma. Lenngad Enegoatomzdat Henz H. Vaa R.A. Fame.L. ak R.R. Accuate Smulaton of Sufaces and Intefaces of Face-Centeed Cubc Metals Usng 6 and 9 6 Lennad- Jones Potentals. J. Phys. Chem. C 8 (44) DOI:./893d 3. M.P. Allen D.J. Tldesley Comute Smulaton of Lquds Unvesty Pess Oxfod Rudyak V.Ya. elkn A.A. Ivanov D.A. Egoov V.V. The smulaton of tansot ocesses usng the method of molecula dynamcs. Self-dffuson coeffcent. Hgh Temeatue 8 46() Rudyak V.Ya. Statstcal Aeohydomechancs of Homogeneous and Heteogeneous Meda. Vol. : Hydomechancs. ovosbsk ovosbsk State Unvesty of Achtectue and Cvl Engneeng atchelo G.K. The effect of ownan moton on the bulk stess n a susenson of shecal atcles. J. Flud Mech () Rudyak V.Ya. Kasnolutsk S.L. Ivanov D.A. Molecula dynamcs smulaton of nanoatcle dffuson n dense fluds. Mcofludcs and anofludcs (4) DOI:.7/s