Diffusion in monodisperse ferrofluids

Transcription

1 Journal of Physcs: Conference Seres PAPE OPEN ACCESS Dffuson n monodsperse ferrofluds To ce hs arcle: Peredo-Oríz e al 7 J. Phys.: Conf. Ser Vew he arcle onlne for updaes and enhancemens. elaed conen - oaonal Brownan Moon on Sphere Surface and oaonalelaxaon Ekrem Aydner - New echnque for measurng he complex suscepbly of ferrofluds P C Fannn, B K P Scafe and W Charles - Langevn equaon mehod for he roaonal Brownan moon and orenaonal relaxaon n lquds W T Coffey, Yu P Kalmykov and S V Tov Ths conen was downloaded from IP address on /4/9 a 9:6

2 Inernaonal Conference on ecen Trends n Physcs 6 (ICTP6) Journal of Physcs: Conference Seres 755 (6) do:.88/ /755// Dffuson n monodsperse ferrofluds Peredo-Oríz, M Hernández-Conreras and Hernández Gómez Deparameno de Físca, Cenro de Invesgacón y Esudos Avanzados del Insuo Polécnco Naconal, Aparado Posal 4-74, Méxco Dsro Federal, Méxco. Deparameno de Compuacón, Cenro de Invesgacón y Esudos Avanzados del Insuo Polécnco Naconal, Aparado Posal 4-74, Méxco Dsro Federal, Méxco. E-mal: marher@fs.cnvesav.mx Absrac. The dffuson coeffcens characerzng he ranslaonal and roaonal Brownan moon of a parcle n a concenraed suspenson were deermned n he long me dffusve regme for a ferroflud suspenson wh he use of a Langevn equaon approach. These dynamcal properes depend on he equlbrum mcro-srucural nformaon of he suspenson and ake no accoun he effec of drec ansoropc ner-parcle's neracons on self-dffuson. The comparson of hs heory wh Brownan dynamc smulaons resuls s made n erms of collod densy and dpole neracon srengh.. Inroducon Nowdays, he effec of he drec ansoropc neracons among magnec nanoparcles and exernal magnec feld gradens, on her collecve dffuson s currenly nvesgaed wh forced aylegh scaerng echnques [] and dynamc lgh scaerng [] n ferroflud suspensons of up o % volume fracons. Ye, he non-drec long range hydrodynamc neracon (HI) n he sysem s expermenally measured by means of x-ray correlaon specroscopy [] and wh neuron spn-echo expermens [4,5,6]. Whereas, he descrpon of he collecve dffuson has been accomplshed wh graden dffuson approach on concenraed suspensons [7,8], and hrough he mean feld knec Smoluchowsk equaon for low and moderae volume fracons [9]. However, here s a clear absence of expermenal sudes of he roaonal and ranslaonal self- and racer dffuson coeffcens n mono- and poldsperse ferrofluds whch remans o be performed n he already well characerzed expermenal sysems. The above ced heores have been useful o descrbe he ranslaonal self-dffuson of Brownan dynamc smulaons were dsspave HI was gnored for model sysems of homogeneous moderaely concenraed suspensons [9]. Thus, due o he lack of HI here are no cross correlaon of dsnc parcle's veloces and herefore he collecve graden dffuson coeffcen concdes wh he self-dffuson expresson [,9]. The smulaon sudes of reference [9] confrmed good agreemen wh To whom any correspondence should be addressed. Conen from hs work may be used under he erms of he Creave Commons Arbuon. lcence. Any furher dsrbuon of hs work mus manan arbuon o he auhor(s) and he le of he work, journal caon and DOI. Publshed under lcence by Ld

3 graden dffuson heory. In he presen paper we show ha a Langevn equaon descrpon of he ranslaonal and roaonal Brownan dynamcs (BD) of a collodal parcle n concenraed ferroflud suspensons whou HI provdes, n he long me dffusve regme he ranslaonal self-dffuson coeffcen whch yelds good quanave predcon as compared o BD resuls. We provde expressons boh for he self- ranslaonal and roaonal coeffcens whch depend n he equlbrum mcrosrucural nformaon of he flud hrough he ansoropc par correlaon funcon. In ferrofluds he roaonal dffuson of he parcles can be measured wh ransen magnec brefrngence relaxaon [] and small-angle neuron-scaerng [6].. Langevn equaon of parcle veloces and of her local concenraon The collodal suspenson conaned n a volume V conss of a carrer flud plus he monodsperse sysem of N parcles of equal mass M and momen of nera I, where each parcle can be eher of sphercal or rod-lke shape havng an embedded opcal ansoropy or a permanen dpolar momen. The orenaon of a parcle's man axs of symmery s defned wh respec o racer's body fxed frame. Thus, anoher parcle's orenaon wll be denoed n hs frame by he polar angles of s orenaon Ω & = (θ &, φ & ) for =,, N. Then he oal poenal energy of he sysem s U = &, ψ & / where s assumed ha ψ & s he drec parwse neracon poenal beween parcles. Noce ha we wll no consder n hs paper he conrbuon of he non-addve HI among parcles. The equaon of moon governng he parcle ranslaonal v() and angular ω() velocy of s roaon can be wren as dv() =- z () + () + W[ y(, W)] d (, W,) d ò " () M V f drd r n r where he dynamcal flucuaon n concenraon abou s equlbrum value s eq eq dn(, r W,) = n(, r W,) -n (, r W) snce he equlbrum concenraon n does no conrbue o he ne force and orque componen. The hrd erm n he Langevn equaon above represens he drec forces and orques on he dffusng parcle by he hos collodal parcles around. We defned he generalzed velocy V() = ( V, w), and mass M j = Mdj (, j =,,), dji wh - I - he prncpal momens of nera of he parcle (, j = 4,5,6). The free parcle's frcon coeffcen s dagonal wh elemens V = V = V ^, V = V, V 44 = V 55 = V, and V 66 =, whch are exernal npus o he heory. The general random erm f() represens he force and orque mpared on he parcle by s collson wh he solven molecules on s surface. The Langevn equaon above s coupled o he flucuaon n concenraon whose evoluon equaon can be wren n s mos general form as d n(, r W,) eq = [ Ñn ( r, W )] V( ) - dd dw d dw L(, W;, W, - ) ò r r r r - s ( r, W ; r, W ) dn( r, W ; r, W, ) + h( r, W, ), ()

4 where s - s he nverse of he sac correlaon funcon s(, r W; r, W ): =ádn(, r W;) dn( r, W ;) ñ [] whch wll be denoed smply as s and smlarly for he oher quanes dn (), c(), n eq, y, L (), h (). Thus, fulflls he deny - ò d r W s(, r W; r, W ) s ( r, W ; r, W ) = d( r - r ) d( W-W ). () In equaon () L s an Onsager coeffcen ha sasfes he flucuaon-dsspaon relaon wh he saonary random erm of zero mean áh(, r W;)( h r, W ; ) ñ = L(, r W; r, W ; - ).. Effecve frcon funcon The formal soluon of equaon () s known and gven n references [,4] as eq dn( ) = c( ) dn() + d c( ) [ Ñ n ] V ( ) + d c( ) -h( ), ò ò (4) where = ò dd r W, and he propagaor c sasfes c() - =- d L( - ) s c( ), ò (5) wh nal condon c( = ) = d( r-r ) d( W-W ). Subsung equaon (4) n (), we oban he effecve Langevn equaon of he parcle dv() M =- z V() + f() - d Dz( - ) () + (). d ò V F (6) Where he dynamcal frcon funcon can be expressed n dfferen equvalen forms wh he use of eq he Werhem-Love equaon bñ y =-s - Ñ n [5], D V() = b[ Ñy] c() s [ Ñy] W = b[ Ñy] C ( ) [ Ñy] W kt B [ eq ] - ( ) [ eq ] = Ñn s c Ñn. (7) W Snce he observable D V () s relaed o he measured mean squared dsplacemen, s average over he sold W= 4p s performed. Here b = / ktwh k B B he Bolzmann consan, T he emperaure, and F () sasfyng he flucuaon-dsspaon relaonshp FF () () = ktd V (). The general expressons for he frcon funcon n (7) can also be used o sudy racer dffuson on rod-shaped parcles suspensons [6]. Such as fd vruses on whch he expermenal echnques of brefrngence, and forced aylegh scaerng measure he dffuson properes. On he oher hand, he propagaor C () = c() s : = dn () dn() governs he collecve relaxaon of he parcle's confguraon varables ( r,w ) due o hermal flucuaons, and has nal condon C() = s : = dn() dn(). The equaon of C () s obaned from (5) yeldng B

5 C(, r W, W, Wr;) =-r[ D Ñ + D Ñ ] dd d s - W W ( -, WW,, W - ) ò r r r r r C( r -r, W, W, W ; - ). (8) 4. Monodsperse ferroflud r -r For a hard ferroflud suspenson made of sphercal parcles wh permanen magnec momen of magnude, µ, we oban a hard sphere (HS) dameer d wh a Lennard-Jones shor range (sr) repulsve neracon. Thus, he oal par drec neracon s y = usr + fdd 6 d d usr 4 e æ æ ö æ ö = ö ç - ç, r.5 < d (9) ç è r ø è r ø è ø / be =.5, Plus he dpolar neracon f dd =-µ µ D( W, pw, W r )/4 r. µ s he magnec permeably of vacuum. D( W, W, W r ): = ( r u)( r u) -( u u ), Where r = r / r s a unary vecor along vecor r and wh orenaon W r. u= u ( qf, ) s a unary vecor of dpole orenaon. eq A equlbrum, he bulk profle concenraon of parcles around he racer n = rg(, r W, W) s proporonal o he par correlaon funcon whch for a dpolar flud s provded by s frs hree erms n a seres expanson of sphercal harmoncs. Here r = N / V s he number densy of parcles and g(, r W, W ) has he fxed space roaonal nvaran expanson of Werhem [7,8] g(, r W, W ) = g() r + h () r D + h () r D, () wh D= u : u. There s an equvalen roaonal nvaran expanson of () gven n [9] / f () r m n l f(, r æ ö WW, ) = (4 p ) å åç Ymµ ( W) Y n( W ) Yll( Wr), (l + ) µnl èµ n l () ø Wh he denfcaon g = g(), r h =- hd ()/ r, = h hd ()/ r []. We deermned hese mcro-srucural properes usng he reacon feld mehod as boundary condon [, ] n he Brownan dynamc smulaon [6]. The vecor posons were decomposed no s parallel ( ) and ^ perpendcular( ) projecons along he parcle s axs of symmery (n hs case u ), r= r + r where for parcle s r = [ u r] u, herefore [6] D D D r ( +D ) = r ( ) + F ( ) ( ) +Dr u z ^ ^ D ^ ^ ^ r ( +D ) = r ( ) + F ( ) +Dr e ( ) + r e z (). () Here F =-ÑU, r, r ^, r ^ are hree uncorrelaed Gaussan random numbers of varance and e (), e () wo orhogonal un vecors whch are perpendcular o u (), D / z 4

6 D u ( +D ) = u ( ) + T( ) u ( ) + xe () + xe (). () z T =-LU u. The angular operaor L u = u / u. x, x are wo uncorrelaed Gaussan random numbers of varance D / z. We used for z = phd, z = phd and N = 5 parcles. To -4 equlbrae he sysem 4 me seps of sze D : =D / = 5, = phd / kbt are performed. Afer hs, mllon more new parcle's confguraons were generaed and used confguraons for sascs o ge he mcro-srucural funcons of fgure (a), and dynamcal properes such as he long me and me-dependen ranslaonal and roaonal selfdffuson. Usng he defnons [] å á d ( r- r ) ñ j, j( ¹ j) () =, gr N4pr r á å d ( r- rj ) u u jñ, j( ¹ j) h () r =, N4pr r á å d ( r- rj )[ u rjuj rj - u uj ] ñ, j( ¹ j) hd () r =. N4pr We deermned ypcal correlaon funcons as dsplayed n fgure (a) for a dmensonless number densy r = nd =. and reduced dpolar srengh µ µ µ / 4.75 pk BTd g r s calculaed as he usual soropc par correlaon funcon []. By akng he Fourer-Bessel (4) = =. The erm ( ) ( ) 4 l = p ò ( ) ( ) [] wh j l l beng he sphercal Bessel funcon, and Laplace w Ca ( k, w) = ò de C ( k, ) of (8) we fnd a a nn mn wc, a k w C, a k w r p l D k D m m å C, a k w s -, a n f k r j kr f r dr ransform,, [- (, )- (, = )] =- 4 ( + )[ + ( + )] (-) (, )[ ], (5) 5

7 Fgure. Fgure (a) gves g( r ) (black dos ) hd ( r ) (open crcles ) and boom curve for hd () r (connuous lne) from BD smulaons of equaons (-) for he model ferroflud of sof Lennard- Jones shor range repulson and dpolar neracon wh r =. and µ =.75. Noce ha all funcons are dfferen from zero for r/ d â, whch fxes he lower lm n he radal dependence of equaons (-). Fgure (b) depcs he ranslaonal D/ D (symbol ) and roaonal D / D (symbol self-dffuson coeffcens calculaed from heory (-) for µ =.75 and as a funcon of reduced densy r. No HI's are ncluded n he calculaon on he properes of (b). m+ n æm n lö Where = (- ) and C, a ( k, w) = å ç C ( k, w) l= m-n èa -a []. Usng he above values of ø =,,, hen for dpolar lquds a =, ± [9], and usng he approxmaon D = D, D = D [4], we ge for equaon (5) C [ s ( k)] (, ) =. (6) , a, a k w a w r4 p( ) ( D k D )[ s - ( k)], a Moreover, he nverse relaon holds æm n lö C ( k, w) = (l+ ) ç C ( k, w). a èa a ø Thus C ( k, w) = C ( k, w), nf å ( m, n), a =-nf ( m, n) - (7) C ( k, w) = [ C, ( k, w) - C, ( k, w)] () 6

8 C ( k, w) = [ C, ( k, w) + C, ( k, w)]. (8) () The srucure facor S = s / r and oal correlaon funcon h are relaed by s ( k) = rs ( k) = r[ + ( ) h (- k)], mn mn a mn, a, a, a r S, ( k) = + h ( k), d r æ ö S, ( k) = - h ( k) h -, d ç () () è ø r æ ö S, ± ( k) = - +ç h ( k) h, d ç () () è ø (9) wh h = g -. Therefore from he Laplace ransform of he second deny of (7) we aan n he long me lm w = [4] for g =^,, where [4] HS D z ( w = ) =D z + z, () g HS z S, x - dxx 6pf + S, ( x) D z = g [ ( ) ] ò. () Noce ha n he above equaon () he frs erm s he hard sphere conrbuon [4] ha resuls from he hrd form of D z n equaon (7) whch depends on he graden of parcle concenraon eq Ñ n. Ths frs conrbuon was used here nsead of he Lennard-Jones shor range conrbuon o he frcon ha would s aaned from he second deny of (7) ha depends on Ñ u. The second sr par D z resuls from he las deny of (7). Here x: = ks. Whereas he conrbuon o he g ranslaonal frcon, n he drecon perpendcular o he man axs of symmery of he parcle, due o drec neracons s [4] 96 zfµ ( ) j( x) D z ^ = dx [6( S+, ( x)) 9( S, ( x)) ]. 5 W ò () æ D ö ç x + d è D ø 4 On he oher hand, he componen of he frcon parallel o he man body axs s D z = D. z^ Moreover, for he roaonal frcon we ge 96 zfµ ( ) j ( x) D z = dx [67( + S, ( x)) 8( S, ( x)) ]. 5 W ò () æ D ö ç x + d è D ø 7

9 Fgure. Fgure (a) s he ranslaonal selfdffuson coeffcen of he ferroflud as a funcon of parcle's volume fracon f. Comparson s made beween heory predcon of equaons (-) (symbol ) and BD smulaons aken from reference [9] (symbol  ) for dpole magnude µ =.58. Fgure (b) descrbes he roaonal dffuson coeffcen ha resuls from equaon (). There s no HI among parcles ncluded. Fgure. Fgure (a) s he calculaed normalzed ranslaonal dffuson coeffcen from equaons (-) (symbol ), and BD resuls are from reference [9] (symbol Â), as a funcon of dpole momen µ for fxed volume fracon f =.5 /.58. Correspondng roaonal dffuson from equaon () s depced n (b). Thus, he componens of he self-dffuson properes are ^ D / D = ( +D z = ( w )/ z ) -, g g g g g =, ^,, z = z = z, and he average oal ranslaonal and roaonal dffuson coeffcen are, respecvely, D= ( D + D ) /, D ^ / D and D = k T / z. Fgure (b) s he plo of he average B ranslaonal propery D/ D (black dos), and roaonal self-dffuson D / D (open crcles) obaned from he heory, equaons (-), as a funcon of r for µ =.75. In general, boh ranslaonal and roaonal dffuson of he parcle dmnsh by an ncrease n ferroflud densy. In fgure (a) we compared heory (-) for D/ D (symbol ) versus he Brownan dynamcs resuls of reference [9] denoed wh Â, as funcon of volume fracon and fxed dpolar srengh µ =.58. Noce ha n reference [9] he reduced dpolar srengh s denoed as l = µ µ p ktd wh d =.d, and s value s relaed o ours µ l /4 B c =.58. Thus, hey used he value l =. The fracon volume defnon c 8

10 used n reference [9] s r p /6 whch s relaed o our expresson for hs quany as d c f = ( rd c p / 6) /.58. Fgure (b) s he heorecal predcon of () for he roaonal dffuson. Fgure (a) provdes he comparson of heory (-) (symbol ) and Brownan dynamc smulaon (symbol Â) of reference [9] for ranslaonal dffuson a volume fracon f =.5 /.58 where he value of.5 was used by auhors of reference [9]. We ploed hs propery as funcon of dpolar srengh µ = l.58. Whereas fgure (b) s he correspondng heory, resuls for D / D. We consder now a ferroflud under he nfluence of a consan exernal magnec feld H. Therefore, he energy of he sysem s. Fgure 4 depcs he me-dependen self-dffuson coeffcens resuls from Brownan dynamc smulaon only a volume fracon.5 /.58 / and f = versus reduced me hree magnudes of parcle's dpole momen µ, and under appled magnec feld n dmensonless uns of H µµ H kbt = ( / ) = 4, where H = H. The feld orenaon s along he Z axs. For ranslaonal dffuson, we deermned he coeffcen as he componen D () D (), D () ^ = xx yy N ( ) = (/ ) á[ ( ) - ()] ñ/ 6 = D Nå r r, where [5] refers o dffuson parallel o axs X and Y bu perpendcular o he dffuson along he man parcle symmery axs whch s along Z [9], drecon abou whch we defned he componen D = D. For he roaonal moon, we used r D () =-ln[ á u () u ()]/ ñ [6]. Fgure 4(a) shows ha a he lowes dpole srengh = = µ =.4, and H = he shor me dffuson lm presens a plaeau n he dffuson coeffcen (upper connuous lne) where remans consan and hen drops connuously due o drec ner- parcle's neracons ha resran s movemen. For H = 4 he shor me secon of he dffuson, curve lowers (doed curve) and hen converges for longer mes o he zero feld case above. For a slghly larger dpole, srengh of µ =.59 eher wh feld (do-dash lne) or no (dash lne) boh curves almos concde n general. For he hghes dpolar case of µ = 4.64, n conras o he lowes dpole momen case before, he feld now ncreases, n general, he average ranslaonal dffuson D ()/ D a all mes (second curve from boom up connuous lne curve) as compared o he case of he zero feld homogeneous suspenson (boom curve hck black lne). Fgure 4(b) depcs he roaonal dffuson whou magnec feld a he hree dpole momens dscussed before. In hs plo and for he wo curves wh correspondng.4,.59 D s gven zz µ = values he dffuson coeffcen ( ) correcly only n he small me nerval / where he mean square dsplacemen s well defned whereas for larger mes s ncorrecly provded by he remanng par of hose wo curves. A he hghes dpole momen (boom lne) he roaonal dffuson lowers as 9

11 Fgure 4. Fgure (a) s he BD smulaon resuls from equaons (-) for he me-dependen ranslaonal self-dffuson coeffcen versus reduced me / a volume fracon f =.5 /.58 for fve values of reduced dpole momen µ wh and whou magnec feld H. Fgure (b) yelds he correspondng me-dependen roaonal dffuson of he parcle. A funcon of me due o he accumulave effec of ner- parcle's neracons ha lowers he roaonal movemen of he parcle. I should be noce ha smlar me-dependen ranslaonal dffuson properes have been measured n a super-paramagnec collod confned o a surface [6], and hrough BD smulaons n suspensons of hard spherocylnders [5]. 5. Conclusons A Langevn equaon heory allowed us he deermnaon of he frcon conrbuon due o drec nerparcle's neracons ha experence a collod parcle durng s dffuson n a concenraed monodsperse suspenson. In he long me dffusve lm he resulng expressons of he parcle selfdffuson coeffcen of s ranslaon and roaonal movemen shows a good comparson wh Brownan dynamc resuls of reference [9] as a funcon of collod concenraon and dpole srengh of neracon. The expresson we derved for he frcon funcon apply o collodal sysems made of rod-shaped parcles wh ansoropc parwse ner- parcle's neracons. Is generalzaon for collodal mxures s sraghforward. Fuure work ncorporaes he effec of exernal graden magnec feld and he dsspave HI. These dynamcal properes are feasble o be measured expermenally [].

12 Acknowledgmens The auhors acknowledge o he General Coordnaon of Informaon and Communcaons Technologes (CGSTIC) a CINVESTAV for provdng HPC resources on he Hybrd Supercompuer ʺ Xuhcoalʺ ha have conrbued o he research resuls repored whn hs paper. eferences [] Bacr C, Cebers A, Bourdon A, Demouchy G, Heegaard B M, Kashevsky B and Perzynsk 995 Phys. ev. E 5 96 [] Merelj A, Cmok L and Copc M 9 Phys. ev. E [] Wagner J, Fscher B, Auenreh T and Hempelmann 6 J. Phys.: Condens. Maer 8 S697 [4] Gazeau F, Boué F, Dubos E and Perzynsk J. Phys.: Condens. Maer 5 S5 [5] Gazeau F, Dubos E, Bacr J C, Boué F, Cebers A and Perzynsk Phys. ev. E 65 4 [6] Mérgue G, Dubos E, Jarda M, Bourdon A, Demouchy G, Dupos V, Farago B, Perzynsk and Turq P 6 J. Phys. Condens. Maer 8 S685 [7] Morozov K I 996 Phys. ev. E 5 84 [8] Pshenchnkov A F, Elfmova E A and Ivanov A O J. Chem. Phys [9] Ilg P and Kroger M 5 Phys. ev. E 7 54 [] Gomer 99 ep. Prog. Phys [] Bacr J C, Perzynsk, Saln D and Servas J 987 J. Phys. (Pars) [] Hansen J P and McDonald I 98 The Theory of Smple Lquds (London: Academc Press) [] Hernández-Conreras M, Medna-Noyola M and Vzcarra-endón A 996 Physca A 4 7 [4] Hernández-Conreras M and uíz-esrada H Phys. ev. E 68 [5] Gubbns K E 98 Chem. Phys. Le [6] Löwen H 994 Phys. ev. E 5 [7] Werhem M S 97 J. Chem. Phys [8] Levesque D, Paey G N and Wes J J 977 Mol. Phys [9] Blum L 97 J. Chem. Phys [] Gray C G, Gubbns K E and Josln C G Theory of Molecular Fluds (USA: Oxford Unversy Press) [] Allen M P and Tldesley D J 989 Compuer Smulaon of Lquds (Clarendon Press) [] Fres P H and Paey G N 985 J. Chem. Phys [] Wes J J and Levesque D 99 Phys. ev. E [4] Nägele G, Medna-Noyola M, Klen and Arauz-Lara J L 988 Physca A 49 [5] Krchhoff T, Löwen H and Klen 996 Phys. ev. E 5 5 [6] Kollmann M, Hund, nn B, Nägele G, Zahn K, Kröng H, Mare G, Klen and Dhon J K G Europhys. Le