Brazilian Journal of Physics ISSN: Sociedade Brasileira de Física Brasil

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1 Brazilian Journal of Physics ISSN: Sociedade Brasileira de Física Brasil Scherer, Claudio Sochasic molecular dynamics of colloidal paricles Brazilian Journal of Physics, vol. 34, núm. A, june, 4, pp Sociedade Brasileira de Física Sâo Paulo, Brasil Available in: hp:// How o cie Complee issue More informaion abou his aricle Journal's homepage in redalyc.org Scienific Informaion Sysem Nework of Scienific Journals from Lain America, he Caribbean, Spain and Porugal Non-profi academic projec, developed under he open access iniiaive

2 44 Brazilian Journal of Physics, vol. 34, no. A, June, 4 Sochasic Molecular Dynamics of Colloidal Paricles Claudio Scherer Insiuo de Física, Universidade Federal do Rio Grande do Sul, CEP , Poro Alegre, RS, Brazil Received on Sepember, 3 Colloidal paricles move in he carrier liquid under he acion of several forces and orques. When he paricles carry a dipole momen, elecric or magneic, as in ferrofluids, he roaional and ranslaional moions are coupled because he field on a paricle depends on he spaial and direcional disribuion of he ohers and he force and orque on i depends on he field. Moreover, here is Brownian, as well as dissipaive forces and orques on each paricle. Consequenly, he numerical soluion of he equaions of moion requires, besides he echniques of Classical Molecular Dynamics, hose of Sochasic Dynamics. The algorihm is explained in some deail and applied on a ypical ferrofluid. For differen values of he emperaure, he possibiliy of he formaion of srucures is examined. 1 Inroducion Colloidal paricles have ranslaional and roaional moion, due o several forces and orques. Besides forces and orques due o applied fields and iner-paricle ineracions, he molecules of he carrier liquid collide incessanly wih he paricle, causing ranslaional and roaional Brownian moion. Their movemen inside he liquid is opposed by viscous, dissipaive, forces and orques. For simpliciy, we will consider only spherical paricles. We also assume ha each paricle carries a permanen dipole momen, which may be elecric or magneic. By using he appropriae Langevin equaions we simulae realizaions of he sochasic process which is he coupled moion of a sample of paricles during a ime which is sufficienly long for he hermodynamic equilibrium o be esablished. We use in he simulaion ypical values for he parameers, which correspond o realisic ferrofluids[1]. For recen references on ferrofluids, see he book Ferrofluids: Magneically Conrollable Fluids and Their Applicaions,ediedbySefanOdenbach[].Our procedure and resuls of he simulaions are compared wih hose of Wang, Holm and Mller[3], recenly published. The Equaions of Moion The equaions of moion for each paricle can be wrien as one vecor equaion for is roaion around is cener of mass and one vecor equaion for he ranslaion of he cener of mass. There is formally only one difference in he roaional equaions of moion of colloidal paricles having magneic or elecric dipole momens. This difference is in he exisence of an inrinsic angular momenum, S, associaed o he magneic momen µ, µ = gs (1) where g is he gyromagneic facor. There is no inrinsic angular momenum associaed wih elecric dipoles. The roaional equaion of moion for a solid paricle is given by Classical Mechanics, dj d = N () where J is he oal angular momenum and N is he oal orque. The magneic paricles of ferrofluids may be superparamagneic, when he magneic momen is free o roae wih respec o he paricle, or blocked, when he magneic momen is fixed in he paricle. The case of superparamagneic paricles in ferrofluids has a more complicaed dynamics because he magneic momen and he paricle s Euler angles are independen, bu ineracing, variables[4, 5, 6]. In his work we consider only blocked magneic paricles. For spherical magneic paricles he angular momenum may be wrien as J = Iω + S (3) where I is he momen of ineria and ω is he angular velociy. For elecric dipolar paricle he relaion is he same, excep ha S =. The ime derivaive of µ is relaed o ω, for blocked magneic or elecric dipoles, by dµ d = ω µ (4) The orque N has several origins. If here is a field F, which may be he magneic inducion B or he elecric field E, a he paricle s posiion, here is a orque N F = µ F. The field F is he sum of he applied field wih he fields due o he oher paricles. We assume ha he carrier liquid has no magneic or elecric momens. The hermal molecular moion of he liquid causes a sochasic orque, σ ξ(), on he paricle, where σ is a consan and ξ() we simulae

3 Claudio Scherer 443 by normalized whie noise, wih average zero and dela ype correlaion funcion, ξ() = (5) ξ i () ξ j ( ) = δ i,j δ( ) (6) where i,j indicae he Caresian componens. The roaional moion is opposed by a dissipaive orque, λ ω, due o he liquid s viscosiy η, hrough he Sokes relaion for roaion, λ = πη d 3 where d is he diameer of he spherical paricle. Einsein s relaion for he consans σ and λ and he emperaure T reads σ = λkt (7) where k is Bolzmann s consan. Summing up, we come o he following equaion for he roaional moion, I dω d + 1 dµ = µ F λω + σξ (8) g d which, ogeher wih Eq.(4), form, if F is known, a complee se of equaions for he vecors ω and µ. The same equaions describe also elecric dipolar paricles, excep ha he second erm a he LHS of Eq.(8) is absen. However, when paricle-paricle ineracion is imporan, as we wan o consider in his work, F is dependen on he paricle posiion r as well as on he oher paricles posiions and momens. Consequenly, we have o solve simulaneously he roaional and ranslaional equaions of moion for all paricles. The ranslaional moion is described by Newon s equaions, dr d = v (9) m dv d = f (1) for he posiion vecor r and velociy v. The force f on he paricle, like he orque, has several origins. If here is a field gradien F a he paricle s posiion, hen he force due o his field is f F = F µ. (11) The paricle-paricle ineracions are parly due o heir conribuion o he field F, bu we also assume a hard core ineracion which avoids wo paricles o come closer han a disance d beween heir ceners. This is no conained in he expression for f, bu will be inroduced direcly in he inegraion procedure. The sochasic force, due o he collisions of he liquid s molecules wih he paricle, has he form α Γ(),whereαisaconsan and Γ() is also modeled by normalized whie noise, like in Eqs. (5) and (6). There is also a dissipaive force γv opposing he ranslaional moion. The dissipaive consan γ is relaed o he viscosiy by he following Sokes relaion, γ =3πηd. (1) Einsein s relaion for he consans α and γ and he emperaure T reads α = γkt (13) Summing up, we arrive a he following equaion for he ranslaional moion: m dv = F µ γv + αγ() (14) d which, ogeher wih Eq.(9), form a complee se of equaions for he variables r and v if F (r,) and µ are known. However he paricle-paricle ineracions make F o depend on he magneic momens and posiions of all paricles, so ha simulaneous soluions of all equaions of moion is necessary. 3 Numerical Soluion of Langevin Equaions Before discussing he procedures o solve he equaions of moion of secion, we presen a brief inroducion o numerical soluions of sochasic differenial equaions wih whie noise erms, known as Langevin equaions. Consider an n-dimensional sochasic process X(), described by he following general Langevin equaion, dx = A(X,)+B(X,) ξ() (15) d where A(X,) is a well behaved n-dimensional vecor funcion, B is an n m marix and m is he number of independen componens of he normalized whie noise ξ(). If B does no depend on X we say he noise is addiive, oherwise i is called muliplicaive. For our presen purpose i is no necessary o allow for explici dependencies of A and B on,sohaa = A(X) and B = B(X) I is impossible o simulae, in he compuer, realizaions of he whie noise, since is correlaion ime iero. Therefore we inegrae formally Eq.(15) beween and +, = + X() X( + ) X() = A(X( ))d + + B(X( )) ξ( )d (16) Since A is a coninuous funcion of X, which is a coninuous funcion of, we know, from he mean value heorem, ha here is a leas one value of such ha he firs inegral above is + A(X( ))d = A(X( )) (17)

4 444 Brazilian Journal of Physics, vol. 34, no. A, June, 4 For he second inegral a he RHS of Eq.(16) we consider firs he case of addiive noise. Then B may be aken ouside he inegral, so ha he equaion becomes X() =A(X( )) + +B + ξ( )d (18) The inegral above is known as Wiener incremen, where W () =W ( + ) W () (19) W () = ξ( )d () is he Wiener process. The componens of he Wiener incremen W () are Gaussian sochasic processes wih zero averages and sandard deviaions equal. These properies of W are fundamenal for simulaing realizaions of he soluion X() of Eq.(15). Eq.(18) has hen a form very appropriae for numerical simulaion, X() =A(X( )) + B W () (1) In numerical simulaion, in Eq.(1), is he lengh of he ime sep for inegraion. If we choose a very small, X( ) may be subsiued simply by X(), or we may prefer a more precise algorihm, like Runge-Kua second order. The componens of W are generaed, a each ime sep, as he produc of by random Gaussian numbers wih zero average and uni variance. The case of muliplicaive noise is more complex. The rigorous reamens of Io and of Sraonovich give us he prescripions o follow, which are equivalen in he limi, bu for finie hey have differen speed of convergence o he exac resul. According o my own experience, by reaing several examples, he Sraonovich procedure converges more rapidly han ha of Io, bu, in many cases, i is more difficul o implemen. We use here he Sraonovich procedure, according o which he equaion equivalen o Eq.(1) for muliplicaive noise should be wrien as ( X() =A(X( )) + B X()+ X() ) W () () When we can isolae X() in Eq.(), very good, oherwise we may, for example, use simply B(X()) o obain a firs approximaionfor X() and hen subsiue his value in he RHS of he Equaion. 4 Numerical Simulaion on Ferrofluids Iniially we consider he orders of magniude of he erms in Eqs. (8) and (14). We sar wih he simples one, which is Eq.(14). Applying he procedure described in secion 3 o he sysem of Eqs. (9) and (14), follows and r( + ) =r()+ v() (3) v( + ) =v()+ 1 m ( F µ γ v()) + α W () m (4) An alernaive approximaion o hose equaions consiss in neglecing he ineria erm in Eq.(14), which leads o r( + ) =r()+ 1 γ F µ + α W () (5) γ The numerical simulaion of he sysem of Eqs. (3) and (4) and of Eq.(5) was made for he same realizaion of W (), wih F =. As parameers we used hose of a magneie spherical paricle of diameer d=1 nm in a liquid whose viscosiy is ha of waer, a T =3 K. The resuls for one componen of r() are shown in Fig. 1. We see ha he effec of aking ino accoun he mass is, indeed, negligible. This conclusion becomes even more eviden in Fig., where we used for he mass a value 1 imes bigger, keeping he oher parameers unchanged magneie spherical paricle d=1 nm T=3 K η of waer x() wih mass x() wihou mass ime(ns) Figure 1. Translaional moion of a magneie paricle, wih he sandard parameers given on he ex. The advanage of neglecing he mass is ha he convergence o he limi is much faser han when he mass erm is presen, which makes an imporan difference in CPU ime when we simulae a sysem of many ineracing paricles. Now we consider he equaion of roaional moion, Eq.(8), using hree alernaive approximaions: 1) neglecing he momen of ineria, I; ) neglecing he inrinsic angular momenum, S = µ/g; 3) neglecing boh, I and S.

5 Claudio Scherer x() wih mass 1 bigger x() wihou mass 3) Neglecing boh, I and S: In his case Eq.(8) becomes µ F λω + σξ = (3) I akes again some vecor algebra and he use of equaion ds d = ω s (33) o ransform Eq.(3) o he form where ds d = µ λ F + σ λ ξ s (34) ime(ns) Figure. Translaional moion of a paricle; he doed line corresponds o a paricle wih a ficiious mass 1 imes bigger han ha of our sandard magneie paricle 1) Neglecing I: Le us define he following symbols: s = µ/µ,whereµ = µ is a consan for blocked magneic momen; W = W s (s W ) is he componen of W perpendicular o s. Afer some vecor algebra we ge, neglecing erms of order higher han, where s = A 1 (g (s F ) + σ W ) (6) A = 1 λs z λsy λ 1 λs x (7) λs y λsx 1 being s x,s y, he Caresian componens of s, λ = gλ/µ and σ = gσ/µ. Finally, s( + ) =s + s. (8) We improve he qualiy of he simulaion by renormalizing s afer each inegraion sep, making s =1. F = F s(s F ) (35) The corresponding discreized Langevin equaion is s = µ λ F σ s W (36) λ In Fig. 3 we show () obained wih he hree approaches jus described, for he same realizaion of W (), using he parameers of a realisic magneic paricle in a ferrofluid: a spherical magneie paricle wih diameer of 1 nm in a liquid wih he viscosiy of waer a room emperaure and in presence of a weak magneic field of 1 G. We see ha aking ino accoun he momen of ineria, I, or he inrinsic angular momenum, S, has pracically no effec on he resul of he simulaion. To enlarge he effec of hose erms we repea he simulaion wih ficiious values of I and S 1 imes bigger han hose of he sandard magneie paricle of 1 nm in diameer, keeping he oher parameers unalered. The resul is shown in Fig. 4. In his case S has a considerable effec bu I is sill absoluely irrelevan, so ha, if one of hose erms should be aken ino accoun, ha should be S and no I. The opposie procedure was made in reference [3], as menioned in our inroducion. 1. ) Neglecing he inrinsic angular momenum S = µ/g : Again, afer some algebra and neglecing erms of order higher han, we come o he se of equaions ω = µ (s F ) λω + σ W (9) I(1 + λ /) ω = ω + ω (3) and s =( ω s) (31) The procedure is hen simple: for given ω(), s(), F () and W we calculae ω, ω and s in his order, and hen ω( + ) = ω() + ω and s( + ) =s()+ s..5 B z =1 G T = 3 K s, wih S z, wih I s, wihou I S z ime(ns) Figure 3. z-componen of he uni vecor parallel o µ; he doed line corresponds o aking S ino accoun; he solid line, which corresponds o aking I, bu no S, ino accoun, is indisinguishable from he do-marks, which corresponds o neglecing boh, I and S.

6 446 Brazilian Journal of Physics, vol. 34, no. A, June, Periodic Boundary Condiions p B z = 1 G T = 3 K I and S 1 x bigger ime(ns), wih S, wih I, wihou I S Figure 4. Same as Fig.3, bu for a ficiious paricle wih I and S 1 imes bigger han ha of our sandard magneie paricle. 5 The Iner-paricle Forces and Torques The forces beween he magneic paricles in a ferrofluid are of wo differen origins: 1) A shor range repulsion, which avoids wo paricles o overlap in space and ) magneic dipole-dipole force. The firs we simulae by a hard spherical core wih he size of he paricle; he dipolar force on a given paricle is calculaed in he following way: we calculae he field gradien B, whereb is he magneic inducion on he paricle s posiion due o he oher paricles in he sample, hen f p = B µ p (37) where µ p is he paricle s magneic momen. For a paricle a posiion r p, he magneic inducion due o he oher paricles is B(r p )= j p 3n(µ j n) µ j r p r j 3 (38) where n is a uni vecor in he direcion of r p r j. The orque on µ p is N p = µ p B(r p ) (39) Figure 5. A wo-dimensional skech of our procedure for periodic boundary condiion. The numerical simulaions on a random homogeneous disribuion of magneic dipoles have convinced us ha he conribuion o B(r p ) due o he dipoles which are more han a few iner-paricle disances apar from r p is oally negligible. This is so because when he spacial disribuion of he momens is precisely homogeneous, even if here is a preferenial direcion for heir orienaion, he field a he cener of a cube, due o he oher paricles in he cube, is zero. Therefore a non zero field comes only from deviaions from homogeneiy, which is really imporan only in shor disances. Our simulaion is done on a cubic box of side L of a ferrofluid conaining 1 magneic paricles. We use periodic boundary condiions. To calculae he inducion B a r p due o he paricle a r j, if he x-coordinae x p x j >L/ we subsiue x j by x j + L in Eq.(38), and similarly for he oher coordinaes; if x p x j < L/ we subsiue x j by x j L. Wih his procedure, he field on each paricles considers all oher paricles which are in he cube of side L of which he considered paricle is a he cener, as is indicaed in a wo-dimensional projecion in Fig. 5. To calculae he field on paricle p in Fig. 5, due o all oher paricles inside he simulaion box, in solid line, we use, in he case of paricle 1, is acual coordinaes bu in he cases of paricles, 3, 4 and 5, he coordinaes of heir periodic images, which are inside he box in doed line, of which p is he cener. Afer each inegraion sep we ranslae each paricle which wen ou of he simulaion box like paricle in he figure, o inside he box, a he posiion of is periodic image. We sar he simulaion by placing he magneic paricles a he sies of a cubic laice inside he simulaion box, he componens of he magneic momens are chosen a random and he modulus normalized o µ. The fields and field gradiens on he paricles are calculaed as explained above. For he inegraion we use he approach of he equaions wihou mass, momen of ineria and inrinsic angular momenum, as explained in secion 4. For realisic values of he parameers, as hose used for Figs. 1 and 3, room emperaure and zero applied field, afer sufficienly long simulaion ime, when he disribuion seem no o change signifi- 4

7 Claudio Scherer 447 canly anymore, he resuling disribuion is shown in Fig. 6 for a slice of he simulaion box, parallel o he x-y plane and of widh equal o he average iner-paricle disance. In Fig. 6 he lines indicae he projecion of he magneic momens on he x-y plane. We see ha nohing of he lines of aligned dipoles, as repored in reference [3], can be seen. On he oher hand, simulaion made on a ficiious ferrofluid a T=3K (no 3K) lead o Fig. 7, where lines similar o hose repored in reference [3] are seen. This resul makes us suspec ha an error in he generaion of he noise was made by he auhors of reference [3], resuling in noise erms much weaker han wha should be he case a 3 K Figure 7. Same as in Fig.6, bu for a ficiious ferrofluid a T=3 K Figure 6. A slice of he simulaion box a he end of he simulaion for T=3 K, showing he magneic paricles and heir magneic momens(lines) normalized o one and projeced on he x-y plane. References [1] R.E. Rosensweig, Ferrohydrodynamics, Dover Publicaions, Inc., New York, [] S. Odenbach (Ed.), Ferrofluids: Magneically Conrollable Fluids and Their Applicaions, Springer-Verlag, Berlin (). [3] Z. Wang, C. Holm, and H.W. Müller, Phys. Rev. E66, 145 (). [4] M.I. Shliomis and V.I. Sepanov, Adv. Chem. Phys. Series 87, 1 (1994). [5] C. Scherer and G. Mauis, Phys. Rev. E63, 1154 (1). [6] C. Scherer and T. Ricci, Braz. J. Phys. 31, 38 (1).