Theoretical study of the magnetization dynamics of non-dilute ferrofluids

Transcription

1 Thortcal study of th magntzaton dynamcs of non-dlut frrofluds D.V. Brkov, L.Yu. Iskakova, A.Yu. Zubarv Innovnt Tchnology Dvlopmnt, Prussngstr. 7B. D-07745, Jna, Grmany Ural Stat Unvrsty, Lnna Av 5, 60083, Ekatrnburg, Russa ABSTRACT Th papr s dvotd to th thortcal nvstgaton of th magntodpolar ntrpartcl ntracton ffct on rmagntzaton dynamcs n modratly concntratd frrofluds. W consdr a homognous (wthout partcl aggrgats) frroflud consstng of dntcal sphrcal partcls and mploy a rgd dpol modl, whr magntc momnt of a partcl s fxd wth rspct to th partcl tslf. In partcular, for th magntzaton rlaxaton aftr th xtrnal fld s nstantly swtchd off, w show that th magntodpolar ntracton lads to th ncras of th ntal magntzaton rlaxaton tm. For th complx ac-suscptblty χ( ω) = χ ( ω) + χ ( ω) w fnd that th ths ntracton lads to an ovrall ncras of χ ( ω) and shfts th χ ( ω) - pak towards lowr frquncs. Comparng rsults obtand wth our analytcal approach (scond ordr vral xpanson) to numrcal smulaton data (Langvn dynamcs mthod), w dmonstrat that th mployd vral xpanson approxmaton gvs a good qualtatv dscrpton of th frroflud magntzaton dynamcs and provds a satsfactory quanttatv agrmnt wth numrcal smulatons for th dc magntzaton rlaxaton - up to th partcl volum fracton φ ~ 0% and for th ac-suscptblty - up to φ 5 %. PACS numbr: Mm; Nd; 8.70.Dd. I. INTRODUCTION In ths papr w study th rmagntzaton dynamcs n frrofluds - collodally stabl suspnsons of magntc sngl-doman partcls n a carrr lqud (n ordr to prvnt aggrgaton of partcls du to th magntodpolar attracton, frroflud partcls ar covrd wth th spcal surfactant layrs). Du to th possblty to chang physcal paramtrs and control th bhavor of a frroflud by an xtrnally appld fld, such systms ar of a larg ntrst both for fundamntal and appld physcs. Frrofluds ar usd n many xstng tchnologs and ar supposd to b hghly promsng for a varty of potntal tchncal and mdcal applcatons []. Exprmnts dmonstrat that magntodpolar ntrpartcl ntracton changs sgnfcantly both th qulbrum [] and dynamcal [3] proprts of frrofluds. Thortcal modls of dynamcal proprts of dlut frrofluds wth vanshng ntrpartcl ntractons hav bn proposd n [4, 5, 6, 7]. Ths modls lad to vry accurat rsults for vry dlut frrofluds but can not xplan proprts and bhavor of frrofluds whr th ntrpartcl ntracton s sgnfcant. Dpndng on th nrgy of ths magntodpolar ntracton, t can lad thr to an apparanc of homognous short- and long-rang ntrpartcl corrlatons, or to a formaton of chan-lk, drop-lk and othr htrognous ntrnal structurs [8]. At prsnt thr s no gnral thory allowng to prdct ntrnal structur n non-dlut frrofluds at gvn xprmntal condtons. Thrfor t s rasonabl to consdr ffcts of dffrnt ntrnal structurs on th dynamcal phnomna n frrofluds sparatly. Such dalzd modls can provd bttr nsghts nto th nflunc of varous structurs and factors on th macroscopcal proprts of frrofluds.

2 Combnaton of corrspondng dalzd modls can srv as a bass for constructng thors of ral magntc fluds wth typcal long-rang ntrpartcl corrlatons, whr also varous htrognous partcl aggrgats ar prsnt. Howvr, bfor consdrng dynamcal proprts of a frroflud wth varous partcl aggrgats, th bhavor of a homognous systm should b proprly undrstood. For ths rason, n ths papr w prsnt a modl of th rmagntzaton dynamcs of a homognous frroflud consstng of dntcal partcls. It s assumd that magntc momnt of ach partcl has a constant magntud and s frozn nto th partcl body. W ralz, that ths modl s obvously th ovrsmplfcaton of ral frrofluds. Th problm s not only a mor or lss broad dstrbuton of gomtrc and magntc partcl paramtrs of ral frrofluds (ths can b asly ncludd nto th approachs usd by us) and partcl aggrgats oftn prsnt n ral systms (such aggrgats obvously rqur a spcal tratmnt). A vry mportant physcal aspct also s, that th ntrnsc magntc ansotropy of an ndvdual frroflud partcl s fnt (and usually not vn larg compard to th thrmal nrgy and magntodpolar ntracton fld), so that th magntc momnt can rotat wth rspct to th partcl tslf. Stll, th analyss of a smpl modl studd hr s a mandatory frst stp for undrstandng dynamcal proprts of ral frrofluds. Rmagntzaton dynamcs of th modl outlnd abov s studd both analytcally and usng numrcal smulatons. To obtan analytcal rsults wth maxmal mathmatcal accuracy, w tak nto account th magntodpolar ntrpartcl ntracton usng th rgular mthod of vral xpanson ovr th partcl concntraton. W assum that magntodpolar ntracton nrgy s lss or of th sam ordr of magntud as th thrmal nrgy. Othrws partcl aggrgats must appar n a frroflud, whch tratmnt s out of th framwork of ths papr. Nxt, n ordr to focus on th ffcts of magntodpolar ntracton, w nglct hr ffcts of th hydrodynamcal ntracton btwn partcls. Effcts of ths ntracton wll b consdrd n a sparat publcaton. Th papr s organzd n th followng way. In th nxt Scton w xplan n dtal our analytcal approach, drvng th govrnng quaton for th macroscopcal magntzaton dynamcs. In Sc. III w prsnt th numrcal smulaton mthodology and justfy our th choc of th short-rang rpulsv potntal. In Sc. IV w study th ffct of th magntodpolar ntrpartcl ntracton frst, on th magntzaton rlaxaton aftr a stp-ws (nstant) chang of th xtrnal fld and scond, on th ac-suscptblty of a frroflud. Hr w calculat corrspondng dynamcal systm bhavor and compar rsults of th analytcal approach to numrcal smulaton studs, stablshng th concntraton rgon whr th analytcal thory provds a quanttatvly accurat dscrpton of th magntzaton dynamcs n frrofluds. II. ANALYTICAL APPROACH AND BASIC EQUATION FOR THE MAGNETIZATION DYNAMICS W consdr a frroflud wth volum V contanng N dntcal sphrcal frromagntc partcls wth th damtr d. Th absolut magntud p mag of th partcl magntc momnt p,mag s constant, th momnt s frozn nto th partcl body. W ntroduc th unt vctor m = p,mag / p mag of th magntc momnt of th -th partcl and dnot th partcl radus-vctor by r. To calculat th macroscopc charactrstcs of ths systm, w must dtrmn th N-partcl dstrbuton functon P N (m, m N,r, r N ). It can b found by solvng th approprat Fokkr- Planck quaton, whr w hav to tak nto account magntodpolar ntractons btwn all partcls. Ths corrspondng quaton s: PN D Dt = I I ( ) + + I + () t r UPN UPN Dr PN Dt PN

3 whr th summaton n () s prformd ovr partcls and w hav usd th standard notaton I = m, =. m r Th potntal nrgy of th systm U = ( m ) + w j contans th nrgy du to th xtrnal fld (frst trm), whr th rducd fld s dfnd va th vacuum prmablty µ 0, and th local magntc fld H. 3 j = µ Magntodpolar ntracton nrgy ( nd trm n th xprsson for U) contans par ntracton trms w j (ntracton nrgy of partcls and j) w j µ 0 = p 4π m m j rj mrj m jrj mag 5 rj ( ) 3( )( ) whr r j s th radus-vctor btwn th cntrs of ths partcls. Rotatonal and translatonal partcl dffuson coffcnts D r = t 6Vpη, D =, 3πηd ar dtrmnd by th hydrodynamcal (ncludng th non-magntc shll) partcl damtr d, th carrr flud vscosty and th partcl volum V p = πd 3 /6. In solvng Eq. () w must tak nto account th condton that th partcls can not ovrlap: r j d. Th Fokkr-Planck quaton () can not b solvd xactly for two rasons. Th frst on s th wll-known problm of statstcal physcs ntrpartcl ntracton n a many-partcl systm dos not allow (narly always) to solv th govrnng quaton for a many-partcl dstrbuton functon or to calculat th Gbbs statstcal ntgral. Th scond rason s th purly mathmatcal dffculty arsng by th soluton of th Fokkr-Planck quaton vn for th sngl partcl. In ordr to ovrcom th scond problm, w us th ffctv-fld approach, suggstd n [4], whch s a vrson of th tral functon mthod. Accordng to ths approach, w wrt th functon P N n th form of an qulbrum Gbbs functon n som ffctv magntc fld H, whch must b dtrmnd, nstad of th ral fld H. Wth othr words, w postulat th valdty of th followng quaton: whr D D I I + + I + = () t ( ) r UPN U PN Dr PN Dt PN 0 U = m + w, Combnng () and (), w obtan: ( ) j PN D = t j pmagh = µ 0 r I I ( δ PN ), 0 p mag δ = (3) Up to ths pont all transformatons wr xact: nstad of th unknown functon P N w hav ntroducd th unknown rducd fld κ lnkd to P N by th Gbbs formula H

4 P N U = Z xp, U Z = xp dm... dmndr... dr N (4) Th crucal assumpton of ths mthod s that th ffctv fld dos not dpnd on th vctors m and r and that th componnts of ths fld can b found from th quaton for th frst statstcal momnt of th functon P N. Not that for dlut frrofluds ths mthod lads to a vry good agrmnt wth xprmnts and rsults of computr smulatons (s,.g., [9]). Smlar das hav bn succssfully usd by analyzng rhologcal proprts of frrofluds wth chan-lk aggrgats [0]. As usual n statstcal physcs, n a gnral cas ntrpartcl ntractons do not allow to calculat xactly th avrag valus of physcal quantts usng Eq. (4). From now on w suppos that partcl concntraton n our systm s not hgh and us th vral xpanson mthod. In Sc. V w show that ths mthod rprsnts a rasonabl approxmaton whn dscrbng dynamcal proprts of low and modratly concntratd frrofluds. It s convnnt, frst, to avrag th dstrbuton functon P N ovr coordnats r of all partcls. Introducng th Mayr functon f j = xp(-w j /)- and avragng (4) ovr all r, w obtan th avragd N-partcl dstrbuton functon p N n th form pn = PN dr = Z xp l ( + fj ) d m r k (5) l > j k Expandng (5) n a powr srs n f j, kpng only th frst two trms and prformng standard transformatons, w obtan pn = k N G + Qj V k > j ψ φ ( ) (6) Hr φ = NVp / V s th hydrodynamcal (ncludng th non-magntc partcl shll) volum concntraton of partcls and xp( m ) ψ = ψ ( ) =, snhκ xp( m) m 4, m z = d = π z κ Qj = f jdr j, G = Q r V ψ ψ,... =... p... k dm l= j k l, k =,... N Calculatng th ntgral n th xprsson for Q j, w must kp n mnd that th rsult dpnds on th shap of th (nfnt) ntgraton volum [0]. Th rason for ths s th long-rang charactr of th dpolar ntracton. A physcally corrct way of ntgraton must provd for a systm n thrmodynamcal qulbrum th qualty of th magntc fld n th ntgraton volum to th local physcal fld H n th sampl rgon whr ntractng partcls ar stuatd. For ths rason w must us th ntgraton cav as an nfntly long cylndr, drctd along th fld H, wth th tral (frst) partcl on th axs of ths cylndr, ntgratng ovr all postons of th nd partcl. Tchncally ths mans that n th ntgral for Q j (s abov) w should us a cylndrcal coordnat systm (ρ,, z) wth th z-axs along H. Frst, w must ntgrat ovr th z-coordnat of th nd partcl (from to +, thn ovr othr coordnats. Ths ntgraton ordr has bn succssfully usd n [] to calculat th frroflud qulbrum magntzaton. Unfortunatly, th complcatd form of th Mayr functon maks th analytcal calculaton of Q j mpossbl. Hr w rstrct ourslvs to th stuaton whn th dpolar ntracton nrgy w j 4

5 btwn partcls s wj htroaggrgats n th systm. or smallr. Obvously ths assumpton mans that thr ar no Expandng th Mayr functon n a powr srs n w j,, kpng only th lnar trms and usng th mthod from [] to calculat of th ntgral for Q j, w obtan Q ( ) = 8γ m m, G( x) = 4λL ( x), j j L( x) = coth x, G = G( κ ) (7) x µ p 0 mag whr th ntracton paramtr λ = charactrzs th rato of th dpolar ntracton 3 4π d nrgy of two closly placd partcls to th thrmal nrgy. Avragng Eq. (3) ovr th partcl postons, w com to an quaton dntcal to (3) wth p N nstad of P N. Multplyng th rsultng quaton by m and avragng ovr all m, w obtan = D mi ([ δ m ] p ), wth... =... (8)... N t r N Hr = m pn s th avrag of th orntaton vctor m of th tral partcl. Th frroflud magntzaton s M = np mag, Usng Eqs. (6) and (7) w fnd (ntroducng L = L(κ )) N n = φ V = V (9) p N dg ( κ ) = µ h, µ = L + Vp, h = H, (0) V dκ H In th thrmodynamcal lmt (prm mans a drvatv wth rspct to ) L L 8 LL dκ dg µ = + φ = + φλ () Takng nto account that th angular momntum oprator I s anthrmtan and usng th approxmaton (6) for p N, w obtan: m I N ([ δ m ] p ) = p = ψ + v bψ G ψ () N N V whr = [ δ ] m m and ([ ] ) b = ψ Q, and n th thrmodynamcal lmt m I δ m p = p = ψ + φ bψ G ψ (3) N N Combnng Eqs.(6), (7) and (3), aftr smpl transformatons w hav whr th functons ar dfnd va ([ ] ) m I δ m pn = Aδ B( h δ) h (4) A =A( ), B =B( ) (5) A( x) = L( x) / x + 8 φλ ( L ( x) C( x)) L( x) / x 5

6 B( x) = C( x) + 8 ϕλ ( L ( x) C( x)) L( x) / x, C( x) = 3 L( x) / x Substtutng Eq. (4) nto (8) and (0), w obtan = Dr Aδ B ( h δ ) h t (6) W arrvd at a systm of quatons (0),() and (6) for th vctors and. For subsqunt calculatons t s convnnt to wrt ths systm n th form of a sngl quaton wth rspct to. To ths nd w mploy th fact that = µ h and wrt κ = h J h + µ, (7) t t t 6 ( ) dµ J L L L L = = + 8 ϕλ ( ) + dκ Substtutng (7) nto th frst quaton of (6) and wrtng th scalar product of th rsult and vctor h, w obtan dκ Dr = ( B A )( h δ ) (8) dt J Fnally, nsrtng (9) nto (7) and th rsult nto (6), w arrv at th quaton d dt h A = D ( δ h ( h δ )) (9) µ Equatons (8) and (9) form a systm of quatons for and h, whch can b asly rducd to a sngl quaton ( δ) d A B A = Dr ( δ ) + κ δ (0) dt κ J µ κ To fnd th macroscopcal magntzaton M = np mag, w hav to solv Eq. (0) and substtut th rsult nto (0). III. NUMERICAL SIMULATIONS METHODOLOGY Our numrcal smulatons ar basd on th Langvn dynamcs formalsm, whr th quatons of moton for th rlvant dgrs of frdom charactrzng our systm ar solvd takng nto account thrmal fluctuatons. For th frroflud modl consdrd n ths papr (partcls wth fxd magntc momnts, whch ar not allowd to mov wth rspct to th partcls tslf) th systm of quatons for th dscrpton of frroflud dynamcs ncluds two quatons - for th translatonal and rotatonal partcl motons. For th tm scal of ntrst (~ 0-6 sc) nrtal trms can b nglctd du to small partcl szs (~ 0 nm) and a substantal carrr flud vscosty (~ 0. Ps) typcal for standard frrofluds. In ths approxmaton th quaton for th translatonal partcl moton n a frroflud can b smply dducd from th balanc btwn th vscous forc b dr/dt and all othr forcs actng on th -th frroflud partcl: dr dp rp fl b (,mag ) U dt = p H + F () Hr b dnots th vscous frcton coffcnt, whch for a sphrcal partcl wth th hydrodynamcal radus R hyd n a flud wth th vscosty s b = 6R hyd. Th frst trm on th rght-hand

7 sd rprsnts th magntodpolar ntracton forc rp 7 dp dp dp = U = mag F ( p H ) and th scond on - th strc rpulson forc F = U. Ths lattr forc s du to th non-magntc shll surroundng th magntc partcl krnl. Th choc of th rpulsv potntal U rp wll b dscussd n dtal blow. Th thrd trm s a stochastc thrmal forc F fl rsponsbl for a translatonal Brownan moton. Ths forc has δ-functonal corrlaton proprts rp fl fl F, ξ (0) Fj, ψ ( t) = b δjδξψ δ ( t) n our modl, whr th hydrodynamc ntracton btwn partcls s nglctd. Employng th sam approxmatons, w can wrt th quaton for partcl rotatonal moton as th balanc btwn th vscous torqu and all th othr torqus: ζ dp,mag dp fl = p,mag,mag,mag dt p H p T () Hr = 8R 3 hyd s th rotatonal vscous frcton coffcnt. Th frst trm on th r.h.s. s th torqu xrtd on th magntc momnt by th magntodpolar ntracton fld H dp. Ths torqu s drctly transfrrd on th partcl tslf du to th fxd momnt approxmaton of our modl. Th random torqu T fl du to th thrmal bath fluctuatons lads n th Langvn dynamcs formalsm to th rotatonal Brownan moton of th partcl. If th hydrodynamc ntracton s b nglctd, th componnts of T fl hav th sam smpl corrlaton proprts as for th random forc F fl fl fl : T, ξ ( t) Tj, ψ ( t ) = ζ δjδξψδ ( t t ). Th systm of stochastc dffrntal quatons (SDE) ()-() s solvd by th optmzd Bulrsch-Stor mthod (s [] for th dscrpton of th basc da of ths algorthm), whch convrgs to th Stratonovch soluton of ths SDEs. Mthods for th numrcal valuaton of th long-rang magntodpolar fld H dp prsnt n both quatons () and () ar dscussd n dtal n our rvw [3]. For ths study, whr a formaton of partcl aggrgats was not xpctd (th absnc of such aggrgats was confrmd by smulatons), w hav chosn th modfd Lorntz cavty mthod. W hav usd th cut-off radus of th Lorntz sphr RL = r, whr r s th man ntrpartcl dstanc. It was chckd that furthr ncras of R L dos not affct smulaton rsults wthn statstcal rrors. All rsults prsntd blow hav bn obtand for a systm of partcls wth th magntc cor radus R mag = 6 nm, non-magntc shll thcknss h = nm and th partcl matral magntzaton M = 400 G, for whch th ntracton paramtr λ (dfnd aftr Eq. (7)) s λ = 0.8. Th nxt mportant mthodcal quston s th choc of th short-rang rpulsv potntal U rp prsnt n (). Th corrspondng ssu was dscussd n [3] from th physcal pont of vw,.., consdrng th plausblty of th choc for U rp as a rprsntatv for a strc rpulson forc actng btwn surfactant-coatd magntc partcls n ral frroflud. In ths partcular rsarch, howvr, w hav an addtonal mthodcal problm: takng nto account that on of th man goals of ths study s th comparson btwn analytcal thory and numrcal smulatons, w hav to choos th rpulsv potntal n such a way that t dos ntroduc an artfcal bas nto such a comparson. Th smplst choc whch would nabl a most straghtforward comparson btwn analytcal thory and numrcal smulatons would b th hard-cor potntal (U rp = 0 for r > R hyd and U rp = for r < R hyd ). Ths choc would xactly corrspond to th condton that partcls ar not allowd to ovrlap usd by th analytcal soluton of th basc Eq. (). Unfortunatly, th hard cor-potntal s not dffrntabl, so that th dynamc quaton contanng t can not b solvd n a standard way. Instad, th so calld collson-basd algorthms (s [4] for th rvw of ths mthods) should b mployd, whr th valuaton of th nxt collson tm s

8 usd to dtrmn th maxmal tm stp and th systm bhavor aftr th partcl collson. Such algorthm prforms qut wll whn th hard-cor potntal only xsts n th systm undr study. Howvr, n th prsnc of anothr potntal (lk th magntodpolar ntracton prsnt n our cas), th valuaton of th collson tm bcoms a dlcat mattr and th collsonbasd algorthms ar known to work vry slow. For ths rason w hav chosn svral knds of analytcal short-rang potntals and tstd whthr and how th smulaton rsults dpnd on th knd of U rp. Our frst choc was th purly xponntal (Yukawa-typ) potntal U ( r) = B xp( ( r Rmag ) / h). Th dcay radus of rp ths potntal s qual to th non-magntc shll thcknss h and th ampltud B s chosn to b much largr than th maxmal magntodpolar ntracton nrgy of partcls wth th ntrpartcl dstanc qual to th magntc cor damtr: E = ( π / 3) M V (hr M s th dp partcl magntzaton and V mag - th volum of th magntc partcl cor). Th scond potntal tstd by us was th potntal of th scrnd-coulomb typ max rp xp( s / q) r Rmag U ( r) = Aq, s = (3) s h Hr th constant q controls th scrnng radus r scr = hq, and th ampltud A q was chosn so that th rpulson forc du to th potntal (3) was qual to th maxmal magntodpolar attracton forc actng btwn partcls placd at th dstanc r = R hyd. Whn th constant q dcrass, th scrnng radus rscr 0, and th ampltud Aq prsrvng th proprty that th rpulson forc F rp ( r = R hyd ) s qual to th maxmal magntodpolar attracton forc. In ths sns w can say that ths rpulsv potntal convrgs to th hard-cor potntal wth R cor = R hyd whn q 0. Tst smulaton rsults for th xponntal potntal dp max rp mag rp ( ) xp( ( mag ) / ) U r = B r R h wth B = 0 E and scrnd-coulomb potntals U ( r ) wth two vry dffrnt valus of th constant q (q = 0.5 and q = 4.0) ar shown n Fg.. Potntal dpndncs on th ntrpartcl dstanc for all thr potntals ar shown n Fg. (a). Th magntzaton tm-dpndncs m(t), computd for ths thr typs of U rp aftr th ntally appld magntc fld H = 00 O s nstantly swtchd off, ar dsplayd n th part (b). On can s that wthn th statstcal smulaton rrors all th tm-dpndncs for all thr potntals fully concd, thus nsurng th ndpndnc of th smulaton rsults on th choc of th short-rang potntal for our systm. Ths provs that th dffrncs btwn th analytcal thory and numrcal smulatons obsrvd and dscussd blow ar not du to an mpropr choc of th short-rang rpulsv potntal n our smulatons. Concludng ths dscusson, w rmnd that th quston concrnng th dpndnc of th qulbrum frroflud bhavor on th xact form of short-rang rpulson potntal was studd analytcally n [5] (s also rfrncs thrn). Th man rsult of ths study was that for dlut and modratly concntratd frroflud whr th thr-partcls corrlatons ar not vry mportant, th qulbrum magntzaton of a homognous (wthout partcl aggrgats) frroflud dos not dpnd on th form of ths short-rang potntal. Our numrcal smulatons show that ths concluson rmans tru also for th dynamcal proprts of a frroflud. IV. RESULTS AND DISCUSSION IV.. Rmagntzaton dynamcs aftr an nstantanous chang of an appld fld Lt us assum that at th tm t = 0 th magntud of an appld fld changs nstantanously from th ntal valu H to th fnal on H, whrby drcton of th fld rmans th sam. 8

9 Th analytcal approach outlnd abov (Sc. II) lads n ths cas to th followng vrson of th Eq. (0): whr th ntal condton s wth κ, = µ 0 p mag H, /. d A B dt ( κ ) = Dr J (4) κ = κ at t = 0 (5) Th Cauchy problm (4,5) can b asly solvd wth any commrcally avalabl softwar packag capabl to handl ordnary dffrntal quatons. Numrcal smulatons of th rmagntzaton dynamcs ar prformd n th followng way. W start wth th systm of partcls whch magntc momnts ar algnd n th drcton of th xtrnal fld H. Th systm s qulbratd n ths fld untl th magntzaton dos not changd anymor (n frams of statstcal rrors); th annalng tm ntrval t ann = 5t Br (t Br s th Brownan rlaxaton tm) s usually long nough to achv ths qulbrum. Aftrwards, th xtrnal fld s nstantly changd to H and th magntzaton rlaxaton s rcordd. To achv a hgh accuracy rqurd, n partcular, to dtrmn th rlaxaton tm, w hav prformd th avragng ovr N att = 3 ndpndnt runs for a systm of N p = 000 partcls. Corrspondng analytcal and numrcal smulaton rsults ar compard for th stp-ws dcras and ncras of th appld fld n Fg. and 4. For th magntzaton dcay aftr th xtrnal fld s swtchd off (Fg. ), on can s that th analytcal modl agrs wth th smulaton rsults farly wll for frrofluds wth concntraton of th magntc phas up to φ 6 %, what rprsnts - from th appld pont of vw - a modratly concntratd frroflud. It s ntrstng to not that th substantal contrbuton to th dsagrmnt btwn analytcal thory and numrcal smulatons rsults from th corrspondng dsagrmnt btwn th ntal (qulbrum) magntzaton valus. Th lattr s du to th ovrstmaton of th qulbrum frroflud magntzaton by th scond ordr vral xpanson approach. Ths mportant ssu s llustratd n Fg. 3, whch shows th concntraton dpndnc of th rlaxaton tm dfnd as mz trl = (6) d mz / dt κ = κ aftr th nstant fld dcras from H = 00 O to H = 0, whch corrsponds to th ntal stag of th rlaxaton procss shown n Fg.. Ths plot dmonstrats, on th on hand, that th magntc ntracton btwn partcls ncrass th magntzaton rlaxaton tm (dcrass ts rlaxaton rat) at th ntal rlaxaton stag. In th studd concntraton rang th ncras of t rl s narly lnar wth concntraton. Ths ncras s causd by th formaton of short-rang corrlaton btwn partcl momnts; th corrspondng corrlaton dgr ncrass wth th partcl concntraton du to th magntodpolar ntrpartcl ntracton. On th othr hand, Fg. 3 shows that a good agrmnt btwn th analytcal modl and numrcal smulatons concrnng th ntal rlaxaton tm prssts up to th hghst partcl volum concntraton studd hr (φ = 4 %, whch from th xprmntal pont of vw mans a hghly concntratd frroflud), so that th analytcal thory prdcts ths dynamcal systm fatur much bttr than ts qulbrum magntzaton valu. 9

10 Th sam ln of argumnts allows to xplan why th agrmnt btwn thory and smulatons s much bttr (prssts up to hghr concntratons) whn th xtrnal fld s ntally absnt (H(t = 0) = 0) and thn s nstantly swtchd on (s Fg. 4). In ths cas, frst of all, th ntal (for t = 0) qulbrum magntzaton s, of cours, absnt (M(t = 0) = 0) both n analytcal thory and smulatons. Morovr, on can show analytcally that n th scond ordr vral xpanson th ntal slop of th magntzaton curv dm z (t)/dt dos not dpnd on th partcl concntraton. Usng numrcal smulatons w hav vrfd, that ths analytcal rsult s vald up to th hghst studd concntraton φ = 4%. So for th magntzaton ncras aftr th xtrnal fld s swtchd on, th dscrpancy btwn analytcal thory and smulaton rsults arss du to th dffrnt rat of th magntzaton chang whn th systm bcoms magntzd up to som xtnt, as t can b sn from Fg. 4. ( Concludng ths subscton, w would lk to consdr th magntzaton rlaxaton aftr an nstant chang of th xtrnal fld whn th ffctv fld κ s narly qual to th fnal fld κ κ κ / κ ). In ths cas th rlaxaton tm, whch charactrzs ths 'lnar'rmagntzaton dynamcs, xhbts a non-trval dpndnc on th fnal fld valu κ, as w show blow. In th lnar approxmaton wth rspct to δκ = κ κ quaton (4) can b wrttn as d A B dt ( κ ) = Dr J On can asly show that n th sam approxmaton Eq. (7) lads to 0, A, B, J = A, B, J ( κ ) (7) dµ µ µ J =, whr τ =, µ = µ ( κ ) dt τ Dr A B whch allows a straghtforward calculaton of th rlaxaton tm τ. Corrspondng rsults prsntng th rlaxaton tm as a functon of th fnal fld κ ar shown n Fg. 5. On can s that th ntracton btwn partcls ncrass τ (.. dcrass th rmagntzaton rat) whn th fld κ s rlatvly wak and dcrass τ (acclrats th rmagntzaton) whn κ s hgh. Such a non-trval dpndnc of τ on th fnal fld κ s a rsult of th comptton btwn two factors. Th frst on s th usual ffct of th ntrpartcl ntracton, whch dcrass th rlaxaton rat analogous to th rmagntzaton dynamcs aftr a larg chang of an xtrnal fld occurs (s abov). Th scond factor s th wll known ffct of th ncras of a man partcl magntc momnt du to th ntracton btwn partcls [9, ]. Th last factor ncrass th rmagntzaton rat. Whn th fnal fld s wak or modrat, th frst factor domnats, whn th fld s strong th scond on. IV.. Complx suscptblty In ths scton w study th frroflud rspons to a lnarly polarzd oscllatng fld Analytcal approach. Equaton (0) now rads (8) H z = H 0 snω t, H x = H y = 0 (9) d A B = Dr ( κ 0 sn ωt) ), dt J pmagh0 κ 0 = µ 0 (30) Ths quaton can b also asly solvd numrcally. Substtutng κ (t) obtand from (30) nto Eqs. (0) and (), w fnd th man z-projcton of th momnt unt vctor µ z ( t) mz ( t). Th Fourr transforms

11 µ ( Ω ) = µ z ( t )sn( Ω t ) dt 0 µ ( Ω ) = µ z ( t ) cos( Ω t ) dt (3) provd th ral µ ( Ω ) and magnary µ ( Ω ) parts of µ ω, rlatd to th corrspondng parts of th frroflud magntzaton Fourr transform as M ω =np mag µ ω. Th ndx ω mans hr that th appld fld oscllats wth th frquncy ω. W dfn th rducd complx suscptblty rd ( ) ω h 0 ( ) 0 µ ω χ ω =, (3) whr th rducd fld s dfnd va th saturaton magntzaton of th partcl matral M S as h 0 = H 0 /M S. Th rducd suscptblty (3) s proportonal to th standard suscptblty χ = M/H 0 whch dscrbs th racton of th frroflud at th sam frquncy ω as th frquncy of th appld fld. Howvr, th rducd quantty χ rd s mor convnnt to study th ffcts of th ntrpartcl ntracton, bcaus th trval proportonalty of th standard suscptblty χ = M/H 0 to th partcl concntraton s lmnatd (w rmnd that µ z (t) s th avrag z-projcton of th magntc momnt unt vctor). Numrcal masurmnts of th complx suscptblty ar straghtforward and dscrbd n dtal n our rvw [3]. In short, w start smulatons from th stat wth chaotcally orntd partcl magntc momnts and annal th systm durng t ann = t Br n th absnc of an xtrnal fld. A shortr annalng tm - compard to th smulatons of th magntzaton rlaxaton dscrbd abov - s possbl, bcaus th avrag magntzaton dos not chang durng th qulbraton procss, so that only short-rang corrlatons btwn th partcl momnts hav to b stablshd. Aftrwards, w swtch on th oscllatng fld H = H0 z sn( ωt) and comput th nphas and out-of-phas rsponss of th z-componnt of magntzaton (L s th numbr of th tm stps). Dvdng th rsults by th fld ampltud and by th saturaton magntzaton of th systm (n ordr to lmnat th proportonalty of χ = M/H to th partcl concntraton, as by th dfnton (3)) R( χrd ) χrd = mz ( tl ) sn( ωtl ) h L 0 0 l= Im( χrd ) χrd = mz ( tl ) cos( ωtl ) h L L L l= w obtan th complx suscptblty pr partcl χ rd. To obtan th frquncy dpndnc of th ac-suscptblty at a gvn tmpratur χ rd(ω), w prform th masurmnts (33) at a st of frquncs suffcntly dns to rsolv all faturs of ths dpndnc. To obtan th rsults wth a suffcntly good statstcs, w hav carrd out th smulatons durng N cyc = 5 fld cycls at ach frquncy (so that smulatons ar spcally tm-consumng n th low-frquncy rgon), and prformd th avragng ovr N att = 8 ndpndnt runs for a systm wth N p = 500 partcls ach. Fg. 6 dmonstrats th comparson of analytcal rsults obtand usng (30-3) and numrcal smulatons for th ral χ rd and magnary χ rd suscptblty parts. From th qualtatv pont of vw, both analytcal approach and numrcal smulatons prdct th shft of th pak on th magnary suscptblty part ( χ rd ( ω) -dpndnc) towards lowr frquncs wth ncrasng (33) partcl volum fracton φ. Ths s n a qualtatv agrmnt wth th ncras of th rlaxaton tm t rl wth th growng partcl concntraton dscussd abov. Quanttatvly, w not that th dsagrmnt btwn analytcal thory and numrcal smulatons s mor sgnfcant (for on

12 and sam partcl concntraton) than for th magntzaton rlaxaton study prformd n th prvous subscton. Th xplanaton of ths phnomnon can b as follows. For all concntratons, th dvaton btwn th analytcal approach and smulaton rsults for th magnary part of th ac-suscptblty has dffrnt sgns for low and hgh frquncs (s Fg. 6). Takng nto account, that th magntzaton rlaxaton aftr an nstantanous chang of an xtrnal fld contans contrbutons from all frquncs, th dffrnc btwn th analytcal and numrcal suscptblts may b partally avragd out for th magntzaton rlaxaton procss. CONCLUSION In ths papr w hav studd th nflunc of th magntodpolar ntrpartcl ntracton on th rmagntzaton dynamcs of a homognous frroflud usng an analytcal modl and numrcal smulatons. Th analytcal modl s basd on th rgular scond ordr vral approxmaton and dos not contan any adjustabl paramtrs or hurstc constructons. It lads to a good quanttatv agrmnt wth computr smulaton rsults (whch can b consdrd as xact for our frroflud modl) up to th volum concntraton of magntc phas φ ~ 5-0%, dpndng on th typ of th rmagntzaton dynamcs undr study. W not, that ths volum concntraton can b consdrd as bng rlatvly hgh from th pont of vw of modrn frroflud applcatons. Our rsults show, that th magntodpolar ntracton ncrass th charactrstc tm of th magntzaton dcay mmdatly aftr th appld fld s swtchd off. For th magntzaton rlaxaton for th cas whn th ntal fld s clos to th fnal on, th rlaxaton tm dmonstrats a mor complcatd bhavor, ncrasng wth th partcl concntraton f th fnal fld s wak and dcrasng f ths fld s strong. Th man ffct of th magntodpolar ntracton on th frquncy dpndnc of th frroflud ac-suscptblty s twofold: ths ntracton nhancs ts magnary part, and shfts th pak on th χ ( ω) -dpndnc towards lowr frquncs, n accordanc wth th ncras of th systm rlaxaton tm mntond abov. Our study of th frroflud dynamcs has bn prformd for th fxd dpol modl, whr th partcl magntc momnt s fxd wth rspct to th partcl tslf. Th undrstandng of ths smpl modl s th ncssary frst stp for th thortcal analyss of ths complx systm. Howvr, w pont out, that n ordr to proprly undrstand th bhavor of ral frrofluds, th ncluson of th hydrodynamcal ntrpartcl ntracton and th xtnson of th modl to allow for th ntrnal magntc dgrs of frdom (rotaton of th magntc momnt rlatv to th partcl du to th fnt valu of th sngl-partcl magntc ansotropy) s ncssary. Acknowldgmnts Ths work has bn don undr th fnancal support of RFFI, grants N , , Ural, Fund CRDF, PG

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14 Fgur captons Fg.. Magntzaton rlaxaton curvs aftr swtchng th appld fld H = 00 off (at t/t vsc = 00) for a frroflud wth th magntc partcl volum fracton φ = 0 % smulatd wth varous short-rang rpulson potntals U(r) as shown n th lgnds. It can b clarly sn that rsults for varous U(r) concd wthn statstcal rrors. Fg.. Comparson of th analytcal thory (opn crcls) and numrcal smulaton rsults (sold lns): magntzaton rlaxaton m z (t) aftr swtchng off th xtrnal fld H = 00 O at t = 0 for varous volum fractons of magntc partcls φ. Analytcal rsults agr rasonably wll wth numrcal smulaton up to th concntraton φ 6%. Not that th dsagrmnt btwn smulatons and analytcs s largly du to th dffrnc btwn th ntal (qulbrum) magntzaton valus m q (H = 00). Partcl paramtrs: magntc cor radus R p = 6 nm, shll thcknss h = nm, magntzaton of th cor matral M = 400 G. Fg. 3. Concntraton dpndnc of th ntal rlaxaton tm calculatd analytcally usng th dfnton (6) (sold ln) and computd numrcally from th smulatd rlaxaton curvs m z (t) as dscrbd n th papr txt (opn squars, dashd ln s a gud for an y). In contrast to rlaxaton curvs, 'analytcal'and 'numrcal'ntal rlaxaton tms narly agr (wthn statstcal rrors of numrcal smulatons) up to th hghst studd concntraton φ = 4 %. Fg. 4. Th sam as n Fg. for th dc-magntzaton, whn th fld H = 00 O s nstantly swtchd on at t = 0. Partcl paramtrs ar th sam as n Fg.. Not, that th agrmnt btwn analytcal rsults and numrcal smulatons s much bttr than for th m z (t)-rlaxaton aftr swtchng th xtrnal fld off (compar to th Fg. ). Fg. 5. Dpndnc of th magntzaton rlaxaton tm t rl aftr th nstant chang of th appld fld Hnt Hfn on ts fnal valu H fn whn th ntal fld H nt s only slghtly smallr than H fn for varous partcl concntratons as shown n th lgnd. Not that th rlaxaton tm t rl ncrass wth th partcl concntraton φ for small fnal flds H fn < 0, but dcrass wth φ for larg flds H fn > 0. Fg. 6. Ral (top srs) and magnary (bottom srs) parts of th complx suscptblty χ(ω) of a frroflud wth th sam partcl paramtrs as on prvous fgurs for thr partcl concntratons as ndcatd n th lgnd. Not, that th agrmnt btwn numrcal rsults (full squars) and analytcal valus (opn crcls) for th suscptblty s sgnfcantly wors thn for th magntzaton rlaxaton (compar curvs on.g., Fg. and on ths fgur for on and th sam partcl concntraton). 4

15 E/E dm Mag.-dp. nrgy U = B xp(-s/q), q =, B = 0 U = (A q /s) xp(-s/q), q = 0.5 U = (A q /s) xp(-s/q), q = 4.0 s = (r-r p )/h (a) r mn = (R p + h) r /R p Extrnal fld swtchd off (b) m z (t) φ mag = 0 vol. % t/t vsc Fg. φ = φ = 4 <mz> num. sm. anal. thory t x D r t x D r φ = 6 φ = 0.0 <mz> t x D r t x D r Fg. 5

16 t rl x D num. sm. anal. thory Fg. 3 φ, vol. % m z φ = m z φ = 4 t x D r num. sm. anal. thory t x D r m z φ = 6 m z φ = 0.0 t x D r t x D r Fg. 4 6

17 0.8 t rl x D r φ = φ = 8 φ = 0. h fn Fg R(χ) φ =.4 R(χ) φ = 6.4 R(χ) φ = ω / D r ω / D r ω / D r Im(χ) φ = 0.8 Im(χ) φ = Im(χ) φ = 0.0 num. sm. anal. thory num. sm. anal. thory num. sm. anal. thory ω / D r ω / D r Fg. 6 ω / D r