Dynamic Properties of Magnetic Colloidal Particles and Holes

Dynamic P ope ies o Magne ic Colloidal Pa icles and Holes Ma ía del Ca men Miguel López Aques a esi doc o al es à subjec a a la llicència Reconeixemen - NoCome cial 4.0. Espanya de C ea i e Commons. Es a esis doc o al es á suje a a la licencia Reconocimien o - NoCome cial 4.0. España de C ea i e Commons. This doc o al hesis is licensed unde he C ea i e Commons A ibu ion-NonComme cial 4.0. Spain License. Uni e si a de Ba celona Facul a de Física Dynamic p ope ies o magne ic colloidal pa icles and holes Memo ia p esen ada pe Ma ía del Ca men Miguel López pe op a al G au de Doc o en Ciencies Físiques. Ba celona, 8 de se emb e de 1995. Uni e si a de Ba celona Facul a de Física Dynamic p ope ies o magne ic colloidal pa icles and holes Ce i ico que la p esen esi doc o al ha es a eali zada so a la meya di ecció Ba celona, 8 de se emb e de 1995. D . J. M. Rubí Capace i, Ca ed á ic de Física de la Ma e ia Condensada de la Uni e si a de Ba celona P opiedades dinámicas de pa ículas y huecos magné icos coloidales In oducción Las pa ículas magné icas de pequeño amaño son de g an impo ancia en mul i ud de á eas de la ciencia y la ecnología. No obs an e, hay escasos ejemplos de sis emas o mados po es as pa ículas suspendidas en un medio líquido de mane a que las in e acciones magné icas desempeñen un papel p incipal, an o en su compo amien o dinámico como en la o mación de es uc u as. En e es os pocos casos igu an los e o luidos y las pa ículas de lá ex dispe sas en un e o uido (huecos magné icos). Un luido magné ico o e o luido es una suspensión coloidal es able de pa ículas mono dominio de un ma e ial magné ico en un líquido común, como agua u o o compues o o gánico. Inicialmen e ue on concebidos como la p ime a mues a de un líquido a i icial en el que las p opiedades magné icas y eológicas se iesen inc emen adas has a se compa ables a las de un sólido e omagné ico, y se manu ac u a on debido al hecho de que la exis encia de mono ases e omagné icas líquidas eales e a bas an e imp obable. Aunque la es abilidad de la suspensión es á asegu ada esencialmen e po el mo imien o b owniano de las pa ículas, usualmen e és as se ecub en de una capa de su ac an e a in de inhibi su coalescencia a dis ancias co as. Los e o uidos ienen muchas aplicaciones ecnológicas, incluyéndose en e ellas los p ecin os de exclusión pa a las disque e as de o denado , amo iguado es en al a oces, imp eso as, ins umen os pa a an e encia de masa y calo , lub ican es, e c. Po ejemplo, esul a complicado man ene un lub ican e den o de la zona de con ac o de las pa e mó iles de un de e minado mecanismo ya que las ue zas cen í ugas o la geome ía ienden a expulsa lo. Sin emba go, un campo magné ico puede ayuda a e ene un lub ican e magné ico en la zona de icción donde es más necesa io. Más aún, la e icacia de un lub ican e depende p incipalmen e de su iscosidad y de su capacidad pa a o ma una capa adso bida. Los lub ican es magné icos son más e ec i os que los comunes incluso en ausencia de campos magné icos po que las pa ículas magné icas que lo componen se en a aídas po las supe icies de ace o y aumen an así an o la iscosidad como el g oso de es a capa lími e. Los e o luidos encuen an ambién aplicaciones en ecología y medicina. Po ejemplo, si en pa a elimina esiduos oleosos de los e idos indus iales. Los luidos magné icos con un hid oca  bu o como base se pueden disol e en los p oduc os oleosos pa a o ma un e o luido más diluido. Un lujo de agua que con iene es a suspensión ci cula a a és de un sepa ado magné ico donde un campo magné ico que a ía ue emen e en el espacio sepa a los esiduos magné icos del agua. Pe o son los múl iples y cuali a i amen e di e sos e ec os p oceden es de la in e acción en e e o luidos y campos magné icos los que han abie o la posibilidad de nue as y p ome edo as aplicaciones ecnológicas. Además de sus p opiedades magné icas, óp icas y eológicas, que discu i emos a lo la go de es a monog a ía, los e o luidos exhiben o os enómenos poco comunes de g an in e és cien í ico, como las ines abiliades supe iciales ycu iosas es uc u as de no equilib io. Hace algunos años se descub ió que las es e as mono dispe sas de polies i eno, dis pe sas en un e o luido, cons i uyen un modelo con enien e pa a el es udio de a ios ipos de enómenos que mues an ansiciones o den-deso den. La base de ello es que las es e as desplazan al e o luido y se compo an como huecos magné icos con un momen o magné ico e ec i o igual al momen o o al del e o luido desplazado. Las es e as (en e 1 y 100¡, m) son mucho mayo es que las pa ículas magné icas del e  e luido (100.4) y, po an o, se compo an como si se hallasen en un medio magné ico uni o me. En gene al, las mic oes e as mono dispe sas p esen an una g an a iedad de apli caciones indus iales, cien í icas y médicas, debido a la singula idad de sus muchas p opiedades como el posee una dis ibución de amaños ex emadamen e uni o me, una o ma es é ica casi pe ec a, con un diáme o de e minado con g an p ecisión, la g an di e sidad de ipos de pa ículas (sólidas, po osas, magné icas, e c.) y la posibilidad de manu ac u a sus p opiedades supe iciales. Las mic oes e as de poli es i eno con inclusiones magné icas se han empleado sa is ac o iamen e pa a sepa a di e sos ma e iales o gánicos (células, i us, micelas, e c.) El p incipio de es a apli cación se basa en una in e acción selec i a en e los an ígenos de la memb ana de las células y los an icue pos si uados sob e las es e as. Las es e as con las células adhe i das pueden elimina se pos e io men e po medio de un campo magné ico. Es posible limpia así la médula espinal de pacien es a ec ados de neu oblas oma o de cie as o mas de leucemia, limi ando los a amien os con adio e apia y/o quimio e apia, que puede a ec a pe judicialmen e al es o de células sanas del o ganismo. Median e es a écnica, es posible limpia has a una décima pa e de la médula de los pacien es, an es de ecu i a o as e apias más adicales. Las pa ículas magné icas se pueden usa ambién como ma cado es si se las ecub e con cie os componen es químcicos o con an icue pos. Du an e la mig ación de las pa ículas dispe sas, és as se adhie en a blancos especí icos y acili an su localización. O a p ome edo a aplicación es su uso como po ado es; se si úan cie os p oduc os químicos sob e la supe icie de las pa ículas y se anspo a a és as a los luga es donde las medicinas se án e ec i as. Los bioquímicos es án alo ando ac ualmen e las po encialidades de es e mé odo pa a a a el cánce . Conclusiones A lo la go de es a monog a ía nos hemos ocupado del es udio de sis emas luidos, an o con monodominios magné icos como con dos ipos dis in os de pa ículas, magné icas y no magné icas, en dispe sión en un líquido new oniano en si uaciones ue a del equi lib io. El compo amien o de es os sis emas se e in luenciado en g an medida po la p esencia de un campo magné ico ex e no, lo que da luga a nue os enómenos que han sido el undamen o de muchas aplicaciones p ác icas. Sin emba go, es a in luencia depende de los di e en es p ocesos de elajación que ienen luga den o las pa ículas, con espec o a su ejes c is alinos, así como ue a de ellas, con espec o al luido po a do . Hemos desc i o cuáles son es os p ocesos y ob enido la dependencia con espec o de los pa áme os que desc iben las pa ículas y el luido, de algunos coe icienes que ca ac e izan las p opiedades eológicas, magé icas y óp icas de la suspensión coloidal. Pa a se más p ecisos, en la p ime a pa e del Capí ulo 1 hemos analizado la dinámica de una es e a e omagné ica en la que el momen o magné ico se encuen a ijado ígidamen e a su cue po, así como ambién las iscosidades de una suspensión diluida de es e ipo de pa ículas. La base eó ica undamen al es la ecuación de Na ie -S okes, en la que se ha incluído una ue za inducida que ep esen a las pe  u baciones in oducidas en la dinámica del luido po el mo imien o de la pa ícula, y una uen e de uido ipo Lange in, p oceden e de las luc uaciones de los campos hid odinámicos. Es a ecuación da cuen a del acoplamien o de las dinámicas del lu ido y la pa ícula. Una expansión mul ipola de las can idades que apa ecen en la solución o mal de la ecuación de Na ie -S okes nos pe mi e calcula las exp esiones pa a la ue za y el momen o de las ue zas eje cidos sob e la pa ícula, los cuales con ienen con ibuciones alea o ias cuyas p opiedades es adís icas es án dic adas po la eo ía de luc uaciones hid odinámicas. Hemos empleado dos mé odos pa a ob ene las iscosidades de cizalladu a y o acional. Uno de ellos es de e minis a, y se basa en la ó mula de Ki kwood pa a el enso de p esiones iscosas, el cual a su ez es á elacionado con el mul ipolo de segundo o den de la ue za inducida. El o o mé odo se basa en la eo ía de espues a lineal, que da los coe icien es de anspo e en é minos de unciones de co elación dependien es del iempo. La p esencia del campo magné ico p o oca la apa ición de con ibuciones an isimé icas al enso de p esiones y da luga a la iscosidad o acional que apa ece como un nue o coe icien e de anspo e. Es a pa e an isimé ica del enso de p esiones p o iene del balance que se es ablece en e el momen o de las ue zas eje cidas po el campo magné ico sob e un dipolo y el momen o de las ue zas hid odinámicas. Como consecuencia, el momen o angula de las pa ículas puede di e i del alo de la o icidad del luido en el pun o que aquella ocupa. Es e o malismo es bas an e gene al y se emplea en la segunda pa e del Capí ulo pa a calcula la dependencia de la iscosidad o acional con espec o de la ene gía magné ica de aniso opía del ma e ial. Conside ando o o lími e especí ico, en el cual los momen os magné icos ya han elajado y apun an en la di ección del campo, ob enemos que la iscosidad aumen a al aumen a el pa áme o de aniso opía, alcanzando un lími e de sa u ación. Hemos compa ado nues os esul ados con los de o os au o es y con los da os expe imen ales disponibles pa a dipolos ígidos. Los esul ados p oceden es de o a solución ap oximada de la ecuación es a ciona ia de Smoluchowski sob ees iman los nues os, mien as que los basados en ecuaciones de elajación enomenológicas p opues as pa a el momen o angula in e no de la suspensión son muy p óximos a los que hemos ob enido noso os, y ep oducen bas an e bien los da os expe imen ales. Siguiendo un p ocedimien o análogo, en el Capí ulo 11 hemos p esen ado un o  malismo gene al pa a es udia la dinámica de elajación de pa ículas e omagné icas, con el p opósi o p incipal de p opo ciona exp esiones explíci as an o pa a la iscosi dad como pa a algunos iempos de elajación que ca ac e izan di e en es p opiedades del ma e ial (bi e ingencia, suscep ibilidad magné ica, ... ). Los esul ados cub en odo el ango de si uaciones expe imen ales posibles. Hemos ob enido la ecuación de Smoluchowski que desc ibe la e olución de la dis ibución de p obabilidad de los g a dos de libe ad ele an es de las pa ículas y que nos pe mi e ob ene una je a quía de ecuaciones dinámicas pa a las di e en es unciones de co elación. Es a je a quía se puede ce a u ilizando las conocidas ap oximaciones de desacoplo ap opiadas. A pa i de las ecuaciones dinámicas pa a las co elaciones, uno puede encon a exp e siones pa a los iempos de elajación ca ac e ís icos y és os cons i uyen el pun o de pa ida pa a de e mina los coe icien es de anspo e po medio de las ó mulas de G een-Kubo. En pa icula , hemos is o que la iscosidad o acional alcanza de nue o un lími e de sa u ación, pe o aho a depende los dos pa áme os que compa an la en e gía magné ica y la ene gía de aniso opía con la ene gía é mica. Pa a con as a la alidez de nues o o malismo, hemos compa ado nues os esul ados pa a el iempo de elajación de las pa ículas con los expe imen os de bi e ingencia lle ados a cabo con dos ipos de ma e iales e omagné icos. En ambos casos, nues os esul ados concue dan muy bien con los da os expe imen ales. También hemos p opo cionado una exp esión gene al pa a la suscep ibilidad magné ica compleja del luido magné ico bajo la acción simul ánea de un campo magné ico pola izan e y de un pequeño campo al e no, pe pendicula es en e sí, lo que ab e posibilidades pa a nue as medidas ex pe imen ales. En el Capí ulo 111 hemos es udiado la dinámica de pa ículas no magné icas, o hue cos magné icos, suspendidos en un e o luido en p esencia de un campo magné ico o a i o. Calculamos las ue zas hid odinámicas y el momen o de las ue zas eje  cidas sob e el hueco, de donde podemos iden i ica los enso es de icción asla cional y o acional. Es as can idades dependen de la acción olúmica de pa ículas magné icas y del campo magné ico, el cual o o ga al sis ema un ca ác e anisó opo. El conocimien o de la ue za y del momen o de las ue zas que ac úan sob e el hueco nos pe mi e es udia las dinámicas de aslación y de o ación de las pa ículas cuando aplicamos un campo magné ico o a i o. Nos hemos concen ado pa icula men e en el caso en que el e o luido es á en eposo y el hueco puede gi a bajo la in luen cia del campo magné ico. Bajo es as condiciones, hemos p obado que la elocidad angula del hueco es p opo cional a la ecuencia del campo, pe o que iene una di ección opues a. El coe icien e de p opo cionalidad es lineal con la acción olúmica de las pa ículas de e o luido, y depende de una unción de la in ensidad del campo magné ico que mues a un compo amien o de sa u ación. Hemos compa ado nue s os esul ados con expe imen os ealizados con pa ículas de polies i eno dispe sas en di e en es e o luidos. En el ango de ecuencias conside adas, hemos ep oducido los esul ados expe imen ales conce nien es a la elocidad de o ación del hueco como unción del campo magné ico ex e no. También hemos ealizado un es udio p elimina de las in e acciones hid odinámicas en e los huecos en el e o luido. Esencialmen e, hemos ob enido exp esiones pa a los enso es de Oseen y Ro ne-P age de un e  o luido como el p ime paso en el es udio del impo an e papel que las in e acciones hid odinámicas pueden desempeña en las p opiedades ísicas de es os ma e iales a bajas concen aciones. Pa a mos a la es uc u a de es os enso es y su in luencia en la dinámica de los huecos magné icos, hemos dado ambién exp esiones pa a las alocidad de una pa eja de pa ículas deposi ándose bajo la acción de la g a edad en el seno de un e o luido. Como espe ábamos, pa a di e en es con igu aciones iniciales, la elocidad depende de los pa áme os que ca ac e izan el e o luido. Resul a ú il señala que, aunque hemos ealizado el es udio pa a un e o luido compues o po dipolos ígidos, pa a el que la ene gía de aniso opía es mucho mayo que la ene gía asociada con la in e acción de los momen os magné icos con el campo ex e no, se puee hace una análisis simila en una si uación más gene al, en la que es as dos ene gías de las pa ículas magné icas p esen en alo es a bi a ios. El Capí ulo IV se p esen a como un es udio p elimina de los enómenos de ag e gación que ienen luga en sis emas de pa ículas magné icas en suspensión, así como de las es uc u as esul an es. Nues o p ime p opósi o ha sido elucida la in luencia de las in e acciones hid odinámicas en la ciné ica del p oceso de ag egación. Hemos ex endido la eo ía clásica de la coagulación de Smoluchowski pa a da cuen a de la p esencia de es as in e acciones, impo an es cuando se aspasa el égimen de con cen aciones diluidas. Tales in e acciones ac úan an es de que las di e en es pa ículas lleguen a la es e a de in luencia de una pa ícula dada. Hemos ob enido las ecuaciones ciné icas pa a el p oceso de ag egación y, a pa i de ellas, hemos analizado la o  mación de ag egados. Nues a p incipal conclusión es que la p esencia de in e acciones hid odinámicas alen iza el p oceso de ag egación. Como un segundo p oblema, hemos es udiado la dinámica de una cadena de pa ículas magné icas bajo la in luencia de una lujo ex e no elongacional. En pa icula , hemos calculado las con ibuciones de la cadena al enso de p esiones del sis ema a pa i de la ecuación eológica p opues a po K ame s. De es a can idad hemos ob enido la co ección a la iscosidad debida a la p esencia de in e acciones dipola es. Es os esul ados p elimina es cons i uyen el obje o de u u os abajos. Di e en es líneas de in es igación pueden su gi de los con enidos de es a esis. En e ellas, podemos ci a la necesidad de inco po a en el análisis e ec os ine ciales y de la polidispe sidad de las pa ículas. Como comen amos b e emen e en la in oducción de la esis, los e ec os ine ciales da ían luga a un égimen oscila o io adicional en los p ocesos de elajación pa a alo es muy al os del campo magné ico. Sin emba go, esul an se esenciales pa a la desc ipción de la dinámica o acional de las pa ículas a ecuencias muy al as. Con espec o a la polidispe sidad, debido a que los di e  en es p ocesos de elajación dependen de o ma dis in a del olumen de las pa ículas, sus con ibuciones a las magni udes calculadas no son las mismas en un medio po lidispe so. Po ejemplo, el espec o de los iempos de elajación puede ensancha se. O a posible línea de in es igación consis i ía en i más allá de la espues a lineal del sis ema en condiciones alejadas del equilib io. 1. The [e omagne ic pa icles 3 Figu e .1: Scanning ele on mic og aph o magnelic nic osphe es wi h an i-H2 mon oclonal an iboclies a .ached lo liepa ocy e. 1 The e omagne ic pa icles We will dis ega cl o .he momen he liquid in which lie pa icles a e dispe sed and concen a e in he desc ip .ion o .he ine e omagne ic pa ic les. Fe omagne .ic pa .icles in a e o luid a e monoclomain and possess almos con s an magne .ic momen s. I is well known ha e omagne .ic c ys als consis o e gions whe e he magne iza .ion is o ien ecl clilTe en ly. This egions a e called magne ic domains. The shape and size o hese magne ic do iains in he modyna nic equilib ium a e de e mined by he condi ion ha .he o al ee ene gy o lie ma e ial should be a minimum. Mo eo e , he magne ic do iains a e sepa a ed om each o he by domain walls, i.e., a ansi ion laye in which he dis . ibu ion o he magne iza ion o ien a ion is nonuni o m bu changes con inuously. F enkel and Do man [7] p e dic ed ha by educing he size o a e omagne ic sa nple, one would each a poin a which domain bounda ies would no longe be ene ge ically a o able so ha he whole sample would become a single do nain. On one hancl, e e y domain wall means an addi ional a noun o su ace ension ene gy which limi s he numbe o possible do nains. On he o he h and , he ene gy associalecl o .he nagne ic ield su ounding he sample a o s hei o na ion. Based on his simple a gumenl o compe i ion one can es ima e he c i ica! size (de) below which he sa nple can be conside ed a mag ne ic monodomain. In he es i n a .ion one should dis inguish be ween s ong and weak magne oc ys alline aniso opy be ause i also hine!e s lie c ea ion o a domain wall 4 INTRODUCTION a o ing he in c ease o he c i ical monodomain size. Bu , e en o he less a o ed cases o weak magne oc ys alline aniso opy, he es ima ions gi e de "" 300..1 which is well aboye he mean size o he e omagne ic pa icles used in e o luids whose ypical diame e does no usually exceed 150..1. Thus, e en allowing o a possible polydispe si y, he colloidal dispe sed pa icles a e de ini ely monodomain. Al hough we a e in e es ed in enhancing o ins ance he magne ic p ope ies o he magne ic liquid, and hese a e clea Iy de e mined by he suspended pa icles, he e is a es ic ion in hei size o essen ially a oid agg ega ion phenomena which a e mainly due o dipola magne ic ene gy. I we de ine a dimensionless pa ame e A = m2 jd3kBT compa ing dipola magne ic and he mal ene gies, whe e m and d a e he magne ic momen s eng h and he diame e o one pa icle, he s abili y condi ion imposes ha A ::; 1 which can only be achie ed i he pa icle diame e is small enough ( o a e omagne ic ma e ial whose sa u a ion magne iza ion M. ::: 500G his condi ion holds i d ::; 100..1). An in e media e solu ion o he p oblem can be hough consis ing in educing he linea size o he pa icles bu keeping he same olume ac ion (¡ J = n Vm), whe e n is he pa icle numbe densi y and Vm he olume o one monodomain. Howe e , his s a egy also mani es s so ne p oblems. By pu suing his me hod one a i es a highly dispe sed colloidal suspension bu no longe magne ic. I is clea ha he elec on spins esponsible o he magne ic o de ing a e a ec ed by he p esence o he bounda y o he olume in which hey a e enclosed. The absence o pa ne s o exchange in e ac ion mani es i sel on he o ma ion o a hin demagne ized laye whose hickness can be compa able o ha o he whole pa icle. This ac is e i ied by expe imen s [8, 9]. Mo eo e , he e exis s an al e na i e poin o iew which explains his demagne ized laye s by chemical modi ica ion o he pa icles su aces due, o ins ance, o oxida ion o o he in e ac ion wi h he su ac an which is usually added o s abilize colloidal suspensions. This ques ion has been he subjec o many s udies [10, 11, 12] and i emains open. Fo ou pu poses in his hesis, his p oblem i is no c ucial because we will always assume ha m = M. Vm wi h Vm he olume o he magne ic co e, which due o he aboye conside a ions can be less han he geome ic olume. When looking a he pa icles, ano he aspec ha should be kep in mind is ha e en hough i migh seem ha a monodomain magne ic pa icle beha es like a pe manen magne , his is no always he case. The o ien a ion o he poles in a magne is ixed by he magne ic aniso opy ene gy. A oom empe a u e, he dimensionless pa ame e (J' = Ka VmjkBT, compa ing aniso opy and he mal ene gies 1. The e omagne íc pa ícles 5 wi h Ka he e ec i e aniso opy cons an , o a monodomain pa icle is no la ge (d", 100A, Ka '" 104-105 J 1m3). Consequen ly, he p obabili y o he mal o a ional luc ua ions o he magne ic momen inside he pa icle becomes impo an . A u ::; 1, he magne ic momen a e no p ac ically a ec ed by he po en ial ba ie and i s mo ion is simila o he B ownian o a ion o a colloidal pa icle in a iscous liquido Néel was he i s o indica e he possibili y o luc ua ional emagne iza ion o small pa icles, ha is why his p ocess is usually e e ed o as he N éel elaxa ion. The cha ac e is ic ime o his spon aneous eo ien a ion, Tq depends on he pa ame e U. Fo ime in e als sho e han T he pa icle beha es like a pe manen magne , whe eas o measu ing imes g ea e han T he pa icle's magne ic momen can be conside ed ze o. These di e ences can be co obo a ed, o ins an ce, by means o Mossbaue spec oscopy. The alignmen ends o be dis up ed by he mal agi a ion, and o cou se, beyond he Cu ie empe a u e he domain possesses no magne iza ion any longe . In he abo e discussion we ha e conside ed he elaxa ion o he magne ic momen in he absence o an ez e nal magne ic ield. Bu , i a magne ic ield is applied he ene gy o a magne ically uniaxial pa icle is he ollowing u = -m' H - Ka Vm(71· kl, (1.1 ) whe e m = mR is he magne ic momen o he pa icle, H is he ex e nal magne ic ield, and 71 is he uni ec o along he di ec ion o he axis o easy magne iza ion, o he aniso opy axis. The e ec i e cons an Ka con ains, in gene al, con ibu ions coming om he c ys alline aniso opy o he pa icle as well as he shape aniso opy. F om his exp ession i will be easy o see ha he dynamics o he wo deg ees o eedom o a pa icle in a liquid, R and 71, a e coupled. I is also wo h men ioning a his poin ha he e will be essen ially wo ele an dimensionless pa ame e s in he analysis. These pa ame e s a e he abo e in oduced u = Ka VmlkBT and he pa ame e p. = mH IkBT, compa ing he magne ic ene gy o in e ac ion wi h he magne ic ield and he nal ene gy. The de i a i e o his magne ic ene gy wi h espec o m de e mines he alue and o ien a ion o he e ec i e magne ic ield: H- - e : -H- 2KaVmA(A RA) eJJ - --a - - + n n . , m m (1.2) which includes he ex e nal ield H and he aniso opy ield Ha di ec ed along he aniso opy axis. In he absence o he ex e nal ield, he magne ic momen is jus unde he ac ion o he aniso opy ield and he e a e wo equi alen equilib ium 6 INTRODUCTION o ien a ions R = 71 and R = -71 be ween which he elaxa ion can ake place. Now, he equilib ium condi ion is gi en by he absence o magne ic o ques ac ing upon he pa icle, so ha in he absence o he mal luc ua ions he magne ic momen is pa allel o Hel . When his is he case he e a e wo di e en o ien a ional elaxa ion p ocesses o he magne ic momen o he pa icle ela i e o i s c ys allog aphic axes. The in insec mo ion o he magne ic momen consis o a egula p ecession a ound he e ec i e ield and o chao ic eo ien a ions due o he mal luc ua ions. The egula mo ion ela i e o he c ys alline axes o he pa icle is desc ibed by he classical Landau-Gilbe equa ion, which we in oduce in Chap e JI. This equa ion ep esen s he p ecession o he magne ic momen wi h he La mo equency WL as well as he decay o his mo ion due o collisions, magne oelas ic in e ac ion, ... Associa ed o his decay he e is a cha ac e is ic ime To = CiWL., whe e Ci is a dimensionless damping cons an . Ano he cha ac e is ic ime TD = (2Dm)-1 is connec ed wi h he o a ional di usion o he magne ic momen inside he pa icle Dm = kBTh, whe e, as we will see in Chap e JI, h plays he ole o a o a ional mobili y o he magne ic momen o 2 Mo ion o he magne ic pa icle in he liquid Besides he in e nal mo ion ela i e o he pa icle body, he magne ic momen also unde goes an ex e nal o a ional di usion as a consequence o he mo ion o he mag ne ic pa icles in he liquid hey a e suspended. The e omagne ic pa icles suspended in a nonmagne ic luid expe ience he ac ion o he ca ie liquid h ough iscous ie ion. We will now conside he o a ional B ownian mo ion o a colloidal pa icle. I s de e minis ic mo ion is desc ibed by he equa ion o he o a ional dynamics o a solid body suspended in a iscous liquid (2.1 ) whe e 1 is he momen o ine ia o he pa icle, i i s angula eloci y, � he o aional ic ion coe icien o he pa icle in a iscous liquid, and he ex e nal o ques ac ing on he pa icle. Jn he absence o ex e nal o ques, he o a ional mo ion o he pa icle decays wi h he cha ac e is ic ime TI = 1 / � . Fo a sphe ical pa icle in he S okes app oxima ion � = 81 110a3, whe e 110 is he iscosi y o he liquid and a is he adius o he pa icle. 3. The o a ional di usion equa ion 7 I we conside ¡ = 1O-2ps and d -- 100Á, i ollows ha TI -- lO-lIS. This alue is so small ha in all cases o p ac ical in e es he ine ial e m in Eq. (2.1) may be neglec ed in compa ison wi h he iscous one. Indeed, his e m may be ele an only o cha ac e is ic equencies o ex e nal exci a ion � 100GHz. Compa ed o he cha ac e is ic imes o hyd odynamic p ocesses TI is almos ze o, Besides TI he e exis s a much la ge cha ac e is ic ime o he o a ional mo ion o he pa icle. This ime is de e mined by he o a ional B ownian mo ion o he axis n ep esen ing he pa icle. This B ownian ime is gi en by TB = (2D )-1, which aking in o accoun he S okes-Eins ein ela ion i can be ew i en as TB = . /2kBT. Fo he same alues gi en aboye a oom empe a u e, TB - 10-6. Du ing his ime, he pa icle unde he in luence o he mal luc ua ions o a es by a ini e angle. The highe he iscosi y o he ca ie liquid he slowe he o a ion. Con e sely, he highe he iscosi y he bigge he ime TI. Ne e heless, TB and TI will be compa able jus o 770 - 1O-4ps, and such a iscosi y would jus co espond o ae osols o c yogenic liquids. In eal e o luids, depending on he pa icle olume and on he magne ic aniso opy cons an he ela ion among he di e en cha ac e is ic imes, and o he pa ame e s J1. and (7 may be a bi a y. Thus he elaxa ion ime o he magne iza ion will be in gene al a combina ion o he di e en imes al eady in oduced. The limi ing case o an in ini ely s ong coupling o (7 � 1, i.e. when he magne ic momen o he pa icle is igidly coupled o he easy axis o magne iza ion, is known as he igid dipole modelo Despi e i s simplici y, his app oxima ion is widely used in he heo y o magne ic luids, and allows one o explain a wide ange o magne ic and hyd odynamic phenomena expe imen ally obse ed. 3 The o a ional di usion equa ion As a single e omagne ic pa icle expe iences bo h a sys ema ic damping and andom he mal luc ua ions o i s magne iza ion and i sel , he s udy o i s dynamics can be pe o med by ollowing wo di e en me hods: Lange i 's app oach o he heo y o B ownian mo ion, and by B own's in ui i e me hod [9] which is an adap a ion o he a gumen s de eloped by Eins ein in 1905, o in o he wo ds, by Fokke -Planck o Smoluchowski di usion equa ions o he p obabili y densi y. In o de o accom plish his, B own ollowed he p ocedu e o Wang and Uhlenbeck [14] oge he wi h he S a ono ich de ini ion o he de i a i e o a s ochas ic a iable [15]. He also p oposed an al e na i e and simple app oach o w i ing down he Fokke -Planck o 8 INTRODUCTION Smoluchowski equa ion using a con inui y equa ion a gumen as Eins ein did in his ea men abou he ansla ional B ownian mo emen . Al hough B own's app oach was de eloped o a e omagne ic pa icle in a solid ma ix, his o malism can be u he de eloped o desc ibe he dynamic beha io o suspensions o hese pa icles in luids. Shliomis and co-wo ke s [10] i s ob ained his equa ion o a suspension o igid dipoles, and hen o he gene al case wi h a bi a y alues o he a io ¡.JIu, hey also deduced he app op ia e Smoluchowski equa ion om a model simila o he i ine an oscilla o model, [12]-[14]. The no malized s a iona y solu ion o he Smoluchowski equa ion enables one o ob ain any equilib ium o ien a ional cha ac e is ics o an assembly o magne ic pa  icles. Mo eo e , in mos o he cases one has o deal wi h he momen s o he dis ibu ion unc ion. Pa icula ly in e es ing among hese momen s is he i s one, which de e mines he equilib ium magne iza ion. To ind ou he solu ion o he kine ic equa ion o o a ional di usion di e en echniques ha e been p oposed, especially o he igid dipole model. Fo ins ance, o an assembly o igid dipoles unde cons an ex e nal condi ions any de ia ion o he dis ibu ion unc ion om i s equilib ium alue may be expanded in o a se ies o no mal modes, each o which decays acco ding o a simple exponen ial law. The elaxa ion ime spec um o he dis ibu ion unc ion comes om he eigen alues o he esul ing equa ion. In ac , he li e ime o any depa u e om equilib ium is de e mined by he elaxa ion ime o he mos long-li ing modes. The e ec i e ield me hod and he decoupling app oxima ions cons i u e o he app oachs o he p oblem. F om he di usion equa ion one can easily ob ain an in ini e se o coupled equa ions o he momen s o he dis ibu ion. In o de o sol e he se , one can unca e i somewhe e. Bu , i one wan s o s udy he ield dependence o he elaxa ion p ocess, he numbe o calcula ions and he di icul y in unde s anding he esul s ob ained g ows d as ically wi h he ex e nal ield. To a oid hese incon eniences ela ed o he nume ical solu ion, i would help o ha e a non i ial scheme o closu e o he momen equa ions ha is capable o gi ing a compac analy ical desc ip ion o he o ien a ional p ocesses in a e o luid. Such a closu e p esc ip ion will be be e he smalle he numbe o equa ions. Ideally, o a suspension o igid dipoles one should ha e only one equa ion, since only he i s momen has a di ec physical meaning. In such a way one would be able o gi e a co ec desc ip ion o he a e aged dynamics o he magne .ic pa icles o a wide ange o alues o he ield, aniso opy, empe a u e, e c. The i s idea mee ing all hese equi emen s o a suspension o igid dipoles is ihe e ec i e jield me hod [10], conside ed by Leon o i ch in his book [21]. Thus, in 3. The o a ionaJ di usion equa ion 9 he nonequilib ium s a e one may conside any a bi a y alue o he magne íza íon as an equilib ium one in a ce aín e ec i e ield. Du ing he elaxa íon p ocess, he e ec i e ield ends o he ue ield, so ha he magne íza ion elaxes ia a sequence o quasi-equilib iu n s a es. Fu he mo e, one should ep esen he nonequilíb ium dis ibu ion unc ion in he same o m as he s a iona y dis ibu ion wi h he e ec i e ield. This echnique also yields an exp ession o he ield dependence o he elaxa ion imes o a suspension o igid dipoles unde nons a iona y condí íons bu in a quiescen luid. In his monog aph we p opose ano he al e na i e and simple p ocedu e o sol e he se o in ini e and coupled equa ions o he nomen s o he dis ibu ion unc íon which appea s o p o íde good esul s no only o he igid dipole model bu in any a bi a y si ua íon includíng a mo ing suspension o he case o ini e aniso opy ene gy. The me hod is based on he decoupling app oximalion o so ne o he quan i ies in ol ed in he analysis, o be p ecise, o he quan i ies which anish a equilib ium, when a e aged, and hose which a e di e en om ze o. Thís app oxima ion may be jus i ied om he ac ha in equilib ium bo h quan i ies a e no co ela ed, so, in non-equilib iu n condi ions bu in a linea egime, o ins an ce when he e is small bu non- anishing alue o he o ici y o he luid low, we will assume ha hese componen s emain unco ela ed. In addi ion, we ensu e he main cha ac e is ics o he decoupled quan i ies such ha a e pe o ming he decouplings hey a e s íll p opo ional o he o ici y, o he in a iance unde e lec ions o he easy axis o magne iza íon, ñ, and so on. We eally expec ha he decouplings a e mo e accu a e o qui e small and high alues o he ex e nal ield, because in he o me case híghe o de momen s will be neglígíble and in he la e he dynamícs o he pa icles ís mainly de e mined by he ield a he han being in luenced by B ownian mo ion. The e a e many possible ways o ca ying ou he decouplings. Such app oxi na ions a e b oadly used in he con ex o s ochas ic p ocesses, and pa icula ly, we will see h oughou he hesis he mos con enien decouplings o be pe o med in ou sys em. In his sys em he s abili y o he s a iona y dis ibu ion in a cons an ex e nal ield appea s as a na u al ac . Fu he mo e, om hese app oxima ions one usu ally ob ain a single cha ac e is ic ime o he elaxa ion p ocess. In he absence o an ex e nal ield he elaxa ion o he ini ial diso de ed dis ibu ion akes place by means o ee o ien a ional di usion which is a mono onic p ocess wi h jus one cha  ac e is ic ime. In he s ong ield limi he dynamics o he magne ic momen s is mainly de e minis ic, and hey app oach he ield di ec ion mono onically acco ding o an exponen ial law wi h a cha ac e is ic ime. Thus, al hough he ex e nal ield 10 INTRODUCTION wilI educe he elaxa ion ime, i does no change he cha ac e o he decay p ocess. In p incipIe, he o mal cause o his beha io is ha we a e neglec ing he ine ial e m due o hei smalIness. Keeping his e m would lead o an oscilla o y egime o he elaxa ion p ocess o eno mous and almos un eachable alues o he ex e nal ield. Howe e , ine ial e ec s a e essen ial i one conside s he high- equency be ha io o o a ing objec s o molecula size. He e he ea men becomes much mo e complica ed. A e iew o he p oblem and pa icula esul a is gi en by Co ey el al. [20]. 4 Mac oscopic hyd odynamic heo y o magne ic luids Ha ing cha ac e ized he di e en elaxa ion mechanism s udied, we p oceed o ana lyze hei implica ions in he mac oscopic beha io o he sys em. Fe o luids can be ea ed as a homogeneous one-componen monophase luids. This app oxima ion implies ha he p ocesses ha should be conside ed a e such ha hei cha ac e is ic dimensions a e much la ge han he size o he cons i uen magne ic pa icles. Thus, he magne iza ion o he sys em is assumed o be dis ibu ed h oughou any elemen a y luid olume. Es ablishing he equa ions o mo ion and hea ans e in a e o uid cons i u e one o he mos impo an p oblems o s udy in hese sys ems. Mo eo e , hei peculia p ope ies mani es hemsel es mos e iden ly in hei hyd odynamic beha io . The se o hyd odynamic equa ions o an uncha ged, magne izable, and elec icalIy nonpola izable luid is based on he balance equa ions o mass, linea and angula momen um, and hea ans e o a homogeneous monophase medium. As ega ds he momen um balance equa ion one should ake in o accoun he MaxwelI enso , which in his case is an isymme ic unless he magne iza ion densi y ec o and he ield s eng h a e pa alIel (due o he ini e alue o he elaxa ion ime o he ans e se componen o he magne iza ion, his condi ion does no hold unde he in luence o hyd odynamic o ques ac ing upon he pa icles ei he in magne ic ields ha change hei di ec ion o in a luid ha ing been se in mo ion). The asymme y o he s ess enso demands a gene aliza ion o classical hyd odynamics in he sense ha one should ake in o accoun he o a ional deg ees o eedom o he pa icles. Thus one ough o conside ha he angula momen um densi y is made o wo pa s, he o bi al associa ed wi h he ansla ional mo ion o he suspended pa icles and 4. Mac oscopic hyd odynamic heo y o magne ic luids 11 molecules o he sol en , and he spin momen um caused by he o a ion o he pa  icles. Fo simple luids his spin is ze o and he symme y o he s ess enso is a necessa y and su icien condi ion o he conse a ion o he angula momen um. The si ua ion is di e en i he e exis s in e na! o a ion because he s ess enso should no be symme ic any longe . The di e en luid elemen s mo e wi h di e en eloc i ies ela i e o he angula eloci y o he liquid suspending hem gi ing ise o an in e nal ic ion. In he absence o ex e nal o ques ac ing on he pa icles, he di e  ence be ween he a e aged angula eloci y o he pa icles and he local alue o he o ici y o he luid low disappea s as as as ine ial e ec s do. On he o he hand, his di e ence esul s in an i e e sible p ocess (wi h dissipa ion o kine ic ene gy) o ans e o angula momen um be ween solid and liquid phases o he suspension due o he iscosi y o he sol en o Rela ed o he p esence o an isymme ic s esses in a e o luid he e appea s a new anspo coe icien : he o a ional iscosi y, ha in gene al con ibu es o he e ec i e iscosi y o he suspension. This con ibu ion ends o an asymp o ic limi a high ield s eng h, i is maximum when he magne ic ield and he o ici y a e pe pendicula and is ze o when bo h ec o s a e pa allel since in his case, i we neglec ine ial e ec s, he angula eloci y o he pa icle is equal o he o ici y o he luid a he poin occupied by he pa icle and only he con ibu ion due o symme ic s esses will emain. In o he wo ds, he e o luid mo ion modi ies he alue o i s magne iza ion, so i we conside a magne ic luid wi h nonze o o ici y uni o m shea (Coue e low) in a cons an magne ic ield pe pendicula o he plane o shea , he ield ends o align he pa icle magne ic momen s and, he e o e, he pa icles hemsel es. Mo eo e , he o ici y low simul aneously a emp s o o a e he pa icles des oying he magne iza ion c ea ed by he ield. By means o he Smoluchowski equa ion o he o a ional mo ion o he pa icles and a e conside ing all hei possible mechanism o o ien a ion, we will be able o p o ide a solu ion o he mac oscopic equa ion o he magne iza ion in a mo ing luid. This magne iza ion, in u n, en e s he equa ions o hyd odynamic mo ion o a magne ic luid and hus a ec s he s a e o mo ion o he luid. The o a ional iscosi y comes qui e easily om his analysis using a heologica! equa ion o s a e o he an isymme ic pa o he p essu e enso o by using he co esponding G een-Kubo o mula. To co ec he hyd odynamics o luc ua ions, one mus add o he o al luid p essu e enso a he mal luc ua ing p essu e enso whose a e age, bu no squa e a e age, anishes. This luc ua ion e lec s he ac ha he i s p inciples om which he comple e mac oscopic heo ies mus ollow, a e ime e e sible. AIso he 12 INTRODUCTION luc ua ion a e seen o be ela ed o he dissipa ion h ough he luc ua ion-dissipa ion ela ion, showing indeed ha a dissipa i e heo y is no comple e un il luc ua ions a e included. In pa icula , i one conside he quiescen equilib ium s a es o a magne ic luid in a nonuni o m s a iona y magne ic ield, one obse es ha such a ield is no able o induce he e o luid mo ion, he magne ic p essu e is balanced by he hyd os a ic p essu e. This ac p o ides he basis o nume ous p ac ical applica ions [2]. Once we ha e cha ac e ized he mac oscopic beha io o a e o luid, gi ing ex p essions o he iscosi ies, as well as o he a e age o i s magne iza ion, one can go u he and analyze he dynamics o a nonmagne ic pa icle (magne ic hole) suspended in a e o luid. Recen ly, an inc easing in e es in he s udy o he dynamic p ope ies o his magne ic holes has a isen [5, 3, 4]. Al hough he pa icles a e no magne ic, when hey a e suspended in a ca ie e o luid hey acqui e an induced magne .ic momen equal o he magne ic momen o he e o luid olume hey displace. The in e ac ion o hese induced magne ic momen s o he holes causes a numbe o pecu lia phenomena, such as he o de -diso de ansi ion in magne ic hole la ices and he non-linea phenomena obse ed in assemblies o holes [7]. Fu he mo e, knowledge o he dynamics o such pa icles may cons i u e a way o cha ac e izing he anspo p ope ies o he e o luid. Fo example, he ic ion coe icien o he pa icle gi es us in o ma ion abou he iscosi y o he ca ie luid. The o a ional dynamics o he pa icle is s ongly in luenced by he p esence o a o a ing magne ic ield. We ha e ound ha he hole o a es in he opposi e di ec ion o ha o he ield a low and mode a e equencies o he ield o a ion, and his esul is co obo a ed by ecen expe imen s, ob aining qui e a good ag eemen in he equency ange we a e conside ing. As we ha e commen ed p e iously, o high equencies one ough o accoun o ine ial e ec s, and o he e omagne ic esonance o he e o luid, which is he La mo p ecession o he magne ic momen inside he pa icle exci ed by an al e na ing magne ic ield. The sys em holes- e o luid can be modeled as a suspension o pa icles (holes) m a ca ie luid ( e o luid) because di e en leng h scales exis o he e o luid and he holes. Consequen ly, he e o luid can be iewed as a con inuous medium h ough which he holes may mo e. The dynamics o he e o luid is go e ned, a he con inuum le el, by he a o emen ioned gene alized equa ions. BIBLIOGRAPHY 19 [33] E. Lemai e, Y. G asselli and G. Bossis, J. Phys. JI F ance 2, 359 (1992); J. Phys. JI F ance 4, 253 (1994). [34] D.J. Klingenbe g, C.F. Zukoski, and J.C. Hill, J. Appl. Phys. 73,4644 (1993). [35] S. Miyazima, P. Meakin, and F. Family, Phys. Re . A 36, 1421 (1987). [36] J.J.M. Janssen, J.J.M. Bal ussen, A.P. an Gelde , and J.A.A.J. Pe enboom, J. Phys. D 23, 1447 (1990). [37] M.E. an Leeuwen and B. Smi , Phys. Re . Le . 71,3991 (1993). [38] R. Zhang and M. Widom, Phys. Re . E 49, R3591 (1994). [39] G.S. Rushb ooke, G. S ell, and J.S. Roye, Mol. Phys. 26,1199 (1973). [40] J.J. Weis and D. Le esque, Phys. Re . Le . 71, 2729 (1993). [41] M.J. S e ens and G.S. G es , Phys. Re . Le . 72,3686 (1994). CHAPTERI DYNAMICS OF FERROMAGNETIC PARTICLES IN SUSPENSION: LIMIT CASES In his chap e we s udy he dynamics o a e omagne ic pa icle and compu e he anspo coe icien s o a dilu e suspension cons i u ed by hese pa icles unde he ac ion o a cons an magne ic ield, We will de elop a o malism which makes i possible o cons uc a gene al scheme o analyze he dynamics o he sys e n. In pa  icula , we ca y ou an explici calcula ion o he iscosi ies o a dilu e suspension o sphe ical pa icles. The iscosi ies a e essen ially compu ed by means o G een-Kubo o mulas in he linea esponse heo y amewo k. Mo eo e , he Smoluchowski equa ion o he o ien a ional deg ees o eedom o he pa icles enable us o ob ain he a e aged alues and he co ela ion unc ions in ol ed in he calcula ions. Depend ing on he a io Ji/u, compa ing bo h magne ic and aniso opy ene gies, wo simple expe imen ally eachable egimes show up, and hei analysis cons i u es he scope o his chap e . The i s pa o i deals wi h he limi u � Ji, ha will be e e ed o as suspension o igid dipoles. In he second pa , we ha e analyzed he implica ions o he sphe e's aniso opy ene gy on i s dynamics a high magne ic ield (Ji � u). The o a ional iscosi y has also been calcula ed as a unc ion o he aniso opy pa ame e . Fo a gi en geome y, his magni ude is esponsible o he in c ease o he e ec i e iscosi y o he sys em due o he p esence o an ex e nal magne ic ield. Ou esul s a e compa ed wi h ha ob ained by o he au ho s based on di e en app oaches. 20 Pa 1 RIGID DIPOLES 21 22 CHAPTER l. LIMIT CASES 1 In oduc ion T anspo phenomena in colloidal suspensions and polyme solu ions ha e played a p ominen ole in he s udy o hese sys ems due o hei po en ial applica ions o di e en a e as o physics, physico-chemis y and biophysics. In dealing wi h anspo phenomena, one is mainly con ce ned wi h wo le els o desc ip ion. In he mac oscopic le el, one conside s he sys em as a whole and es ablishes e olu ion equa ions o he ele an quan i ies in he amewo k o heo ies o con inuum media and non equilib ium he modynamics. In he o he le el, one dis inguishes he suspended objec s om he ca ie luid and uses Lange in o Smoluchowski desc ip ions which should be complemen ed wi h he knowledge o he dynamics o he indi idual objec s, i.e., we ha e o know he o ce and he o que exe ed by he luid on he pa .icle mo ing h ough i . Fu he mo e, one o he main p oblems abou anspo phenomena in di e en sys ems is he calcula ion o anspo coe icien s, which a e cha ac e is ic o he esponse o he sys em o ex e nal o ces. This ask has been acco nplished ex en si ely o pa icles o di e en shape and he e exis many well- ounded esul s in he li e a u e [1, 2]. In his pa , we will deal wi h suspensions o dipola pa icles co nposed o a ca ie luid and pa icles ha ing dipola momen s igidly a ached. In pa icula we will discuss he case o e omagne ic pa icles. Thus, one may conside ha he magne ic momen s a e only o ien ed by he magne ic ield, he o a ional B ownian mo ion o he e omagne ic pa icles and by he ex e nal low. This las mechanism depends on he shape o he pa icles. As an example, when he pa icles a e sphe es hey a e o ien ed by he o ici y o he low, whe eas elonga ed pa icles a e o ien ed by he elonga ional low as well. The possibili y o he o ien a ion o he pa icles by he magne ic ield is esponsible o he peculia beha io o he iscosi ies o e o luids. Pa icula ly, i has been shown ha in he case o a suspension o e omagne ic pa icles, he e appea s a new anspo coe icien : he o a ional iscosi y, ela ed o he p esence o an isymme ic s esses [3]. The o a ional iscosi y o he suspension was calcula ed by Shliomis [4] using a con inuum medium app oach in which he di e ence be ween he o ici y o he luid and he a e aged angula eloci y o he pa icles (spin) leads o he in oduc ion o an in e nal angula momen um o he luid elemen s. Fo his eason, i beco mes necessa y o o mula e a new balance equa ion o his quan i y, which is coupled wi h he momen um balance equa ion. Ou pu pose in his pa o he hesis is o p esen a uni ied o malism able o 1. In oduc ion n=R. n=R H • n=R e:=R e-n=R • 23 Figu e 1.1: In suspension o igid dipoles he magne ic momen s a e igidly coupled o he easy axis o magne iza ion o he pa icles and elax oge he . desc ibe he dynamics o he pa icles and o compu e he anspo coe icien s. He e we p esen wo e sions: one whe e he iscosi ies a e calcula ed om a heological equa ion o s a e, gi ing he exp ession o he p essu e enso , and he o he based on he linea esponse heo y, which assumes he exis ence o luc ua ions in he l uid and equi es he knowledge o he luc ua ing dynamics, To his end, we ha e o ganized his pa o he chap e in he ollowing way. In Sec ion 2, we analyze he dynamics o a e omagne ic sphe e by using an induced o ce in he Na ie -S okes equa ion, accoun ing o he pe u ba ion caused by he sphe e due o i s mo ion. We ind he exp ession o he o ce and o que exe ed on he pa icle which spli up in o sys ema ic and andom con ibu ions, and discuss he o igin o an isymme ic s esses. Likewise, in Sec ion 3 he exp essions o he shea and o a ional iscosi ies a e ob ained using a heological equa ion o s a e o he con ibu ion o he p essu e enso due o he p esence o pa icles in suspension. We in oduce he Smoluchowski equa ion and we ge he exp ession o he a e age o he magne ic momen o ob ain he o a ional iscosi y. Sec ion 4 is de o ed o he calcula ion o he iscosi ies by using G een-Kubo o mulas. Fo his pu pose, we need o know he luc ua ion dynamics which ollows om luc ua ing hyd odynamics [5], and om he analysis o he co esponding Smoluchowski equa ion o he o ien a ion o he pa icles [6]-[8]. Finally, in he las sec ion we summa ize ou main esul s. 24 CHAPTER 1. LIMIT CASES 2 Dynamics o a e omagne ic sphe e Le us conside a dilu e suspension o e omagne ic sphe ical pa icles o adius a imme sed in a nonpola incomp essible sol en , unde he in luence o an ex e nal magne ic ield. Ou s a ing poin o analyzing he s a iona y mo ion o he pa icle will be he linea ized s a iona y Na ie -S okes equa ion (2.1 ) in which p( , ) is he p essu e ield, 1/0 he iscosi y o he sol en and e , ) is he eloci y ield. Owing o he incomp essible na u e o he ca ie luid he eloci y ield also sa is ies V'. = O. (2.2) In Eq, (2.1) we ha e assumed ha he pe u ba ion caused by he mo íon o he pa  icle may be aken in o accoun by ín oducíng an induced o ce ield ¡i( , ) [10,11]. Fu he mo e, we ha e conside ed he possíbilí y o luc ua ions in he luid by means o he Lange in-Iike luc ua ing sou ce -V'. IlR, coming om he decomposi ion o he iscous p essu e enso in i s sys ema ic and andom IlR pa s [5]. To be consís en , he induced o ce should be de ined in such a way ha s ick bounda y condi ions a he su ace o he pa icle, e , ) = ü( ) + i( ) x aii, o I - Rcm( )1 = a, (2.3) a e sa is ied. In his equa ion ü and ñ a e he ansla ional and o a ional eloci ies o he pa icle, espec i ely, Rcm( ) is he posi ion o he cen e o mass o he sphe e, and n == ( - Rcm( »/W - Rcm( )l. Since ine ial and memo y e ec s ela i e o he mo ion o he pa icle a e neglec ed in he s a iona y case, he induced o ce is a su ace o ce and can be exp essed as [10, 11] (2.4) whe e 1(n) is an induced o ce densi y pe uní a ea. The eloci y ield gi en by Eq. (2.1) ís he e o e alíd ín he whole space, e en ínside he sphe es. To compu e he mobili y enso we need, i s o all, o know he o mal solu ion o he elocí y ield. Thís solu ion ollows om (2.1) by Fou íe ans o ming in . In ac , a e elímina íon o he p essu e, by applying he ans e sal p ojec o (1 - kk), wí h k == {, one ob ains 2. Dynamics oE a Ee omagne ic sphe e 25 (2.5) whe e we ha e used he incomp essibili y condi ion which, in Fou ie space, eads k . = O and we ha e in oduced he p opaga o - 1 "" T(/c) = -/c2 (1 - H), 'lo (2.6) wi h 1 as he uni ma ix and Vo he unpe u bed eloci y ield in he absence o he pa icle. The p opaga o T(k) is he Fou ie ans o m o he Oseen enso gi en by T( j = _1_(1 + ), 811'770 whe e == . This o mal solu ion can be ew i en in eal space in he o m (2.7) ( j = o( j + J di 'T(i - i') . i(i') + R( j, whe e he andom eloci y ield is gi en by he in e se Fou ie ans o m o (2.8) (2.9) The a e age and co ela ion o his quan i y hen ollows om he s ochas ic p op e ies o he ando n pa o he iscous p essu e enso . Acco ding o luc ua ing hy d odynamics, llR in oduces a Gaussian whi e noise s ochas ic p ocess o ze o mean and luc ua ion-dissipa ion heo em [5] R - R -, 3 - -, , ( Ilij(k, )IlI:1(k , ») = 2kBT '1o :.ijl:l (211') 8(k + k) 8( - ), (2.10) whe e we ha e de ined :.ijl:1 = 8ik8j1 + 8il8jk - �8ij8kl. The o mal solu ion (2.8) gi es he eloci y ield a any poin . The e o e, o a gi en poin a he su ace o he sphe e, in iew o he s ick bounda y condi ion, one has (2.11) whe e we ha e in oduced he esponse unc ion p(n, n') p(n, n') = (2a:)3 J i exp (iak. (n - n'» T(k). (2.12) 26 CHAPTER 1. LIMIT CASES To p oceed u he , we will expand he eloci ies and he induced o ce o (2.11) in mul ipoles [10, 11]. Fo an unspeci ied quan i y �(a71) o ha equa ion, one has �( A) = � (21 + 1)!! Al. !li'+1 an L..J I! n 0 . '=0 Be e n' is an i educible enso o ank 1, i.e. he enso o ank 1 aceless and symme ic in any pai o i s indexes, cons uc ed wi h he ec o ñ. The symbol 0 s ands o ull con ac ion and !li'+1 is he mul ipole o o de 1 + 1, which is a enso (2.13) o ank 1 + 1. These quan i ies a e gi en by !li'+1 = _!_ J dñ n'�(ñ). 411' Inse ing hese exp essions in o he o mal solu ion (2.11), one ge s (2.14) (l+I) = � (2/' + 1)!! 1'(/+1,1'+1) 0 p"+I) + (l+I) + (l+I) L..J I'! o R' 1'=0 which ela es he mul ipoles o he eloci y and o he induced o ce densi y. In his (2.15 ) exp ession he ma ix elemen s 1'(1+1,1'+1) ha e been de ined as 1'(1+1,1'+1) = (417 ) J dñ J dñ' n' 1'(71, ñ') ñ/'. (2.16) The ep esen a ion o he mobili y ke nel in e ms o i educible mul ipoles is diagonal in he indexes 1 and I' [10, 11]. As an example, we ha e I'C1,1) = .?:!:.I. 37]0 Acco ding o equa ion (2.15) o 1 = O, we ge (2.17) (2.18) The mul ipole (l) can be compu ed h ough i s de ini ion ou lined in equa ion (2.14). One has (l) = _!_Jdñ (ü+ñ x añ) = ü. 47 (2.19) In he same way, by assuming ha o( ') = �. (Rcm +añ) a he su ace o he pa icle, we ob ain �l) = _!_ J dñ �. (Rcm + añ) = � . Rcm. 411' (2.20) 2. Dynamics o a e omagne ic sphe e 27 Since p(ii.) is he o ce pe uni a ea exe ed by he pa icle on he luid, he d ag o ce exe ed by he luid on he pa icle is gi en by (2.21) acco ding o Eq. (2.14). Joining he esul s gi en h ough Eq. (2.19)-(2.21) and using Eq. (2.18) and he exp ession o he na ix ele nen gi en in Eq. (2.17), we ge he hyd odyna nic o ce -H -- ::(l) F =-67 7]oa(u- 3·Rem- R), (2.22) o n which one iden i ies he ic ion coe icien 67 7]oa co esponding o he in e se o he nobili y, as well as he B ownian o ce FB = -67 7]oa ¡i). We can also p oceed by conside ing he case 1 = 1 in Eq. (2.12). One nay show ha his equa ion gi es ise o V(2) = 31l(2,2) 0 :¡::{2) + �2) + �), whe e he nobili y 1l(2,2) ollows o n Eq. (2.12) and is gi en by (2.23) 1l(2,2) = _a_S + �A. 157]0 97]0 In his exp ession S and A ha e been de ined as he iso opic enso s (2.24 ) (2.25) and 1 Aijkl = "2 (6il6jk - 6ik6jl), sy n ne ic and an isy n ne ic in any pai o i s indexes. The nobili y (2.24), oge he wi h he esul s (2.26) V(2) = _!_ J dñ il [11 + ñ x aii.] = � E . ñ 47 3 (2.27) and (2.28) ob ained by neans o Eq. (2.14), wi h E being he Le i-Ci i a enso and T s anding o he ansposed na ix, can be used in (2.23). One inally a i es a he exp ession 28 CHAPTER l. LIMIT CASES a - T (2) a 1 1 .-1'2) -(E·n-� )-VR =-(-S+-A)0. ' . 3 '70 5 3 (2.29) The induced o ce mul ipole :¡:(2) may be ob ained om his equa ion. Mul iplying bo h sides o Eq. (2.29) by S, one has (2.30) whe e, o an a bi a y enso o second ank T, T(') s ands o i s symme ic aceless pa T��) = �(T." + T" - �TLL6 .. ) '} - 2 '} }' 3 ."" '} . On he o he hand, mul iplying bo h sides o Eq. (2.29) by A, one ob ains (2.31) :¡:(2a) = '7o(E' ñ + �(a) _ 3'70 �a), a whe e, again o an a bi a y enso o second ank, T(a) is de ined as (2.32) (2.33) Mo eo e , om Eq. (2.29) we can ob ain he equa ion o he hyd odynamic o que since, as in Eq. (2.21), we ha e (2.34 ) so ha only he an isymme ic pa o :¡:(2) is in ol ed. Making use o he ac ha E : E = -21, in Eq. (2.29), we a i e a (2.35) whe e we ha e de ined he o ici y Wo == �E : �(a), and one can iden i y he B ownian o que TB = -127 '7oa2E: �). As ollows om (2.22) and (2.35), he exp essions o he o ce and o que spli up in o sys ema ic and andom con ibu ions. The la e o igina es om he p esence o luc ua ions in he luid and hei s a is ical p ope ies ollow om luc ua ing hyd odynamics acco ding o Eq. (2.9) and (2.10). Once we ha e ob ained he hyd odynamic o ce and o que, we can p oceed o ob ain he equa ions o he ansla ional and o a ional mo ion o he pa ícle. Since e omagne ic pa icles a e e y small ( adius abou 100 Á) ine ial e ec s can be 3. Shea and o a ional iscosi ies o he e o luid 35 3.3 Ro a ional iscosi y and he ex e nal magne ic ield The quan i y be ween b ake s in Eq. (3.5) can be ans o med by using he equa ion o he o a ional mo ion o a e omagne ic pa icle, which ollows om he o al angula momen um equa ion dñ -1 d ñ - _ _ - I-;¡¡+"Yo T =-{ (O-wo)+mxH (3.3.1) whe e 1 is i s momen o ine ia, and "Yo he elec on gy omagne ic ac o . We will neglec he e m coming om he angula momen um o he elec ons de e mining he magne ic momen o he pa icles. Mo eo e , i he pa icle is small enough, as occu s o e o luids, ine ial e ec s can also be neglec ed. The e o e he hyd odynamical and magne ic o ques balance each o he ou and Eq. (3.5) ans o ms in o (3.3.2) In iew o (3.3.2), we conclude ha he an isymme ic pa o he p essu e enso depends on he ela i e o ien a ion o he magne ic momen and he ield. To compu e he co esponding anspo coe icien namely, he o a ional iscosi y, T¡ , we ha e o a e age his pa o e all possible o ien a ions o k We hen ob ain (3.3.3) whe e he a e age on he igh -hand side ollows om he exp ession o he i s momen o R compu ed by he Smoluchowski equa ion and gi en by (3.2.6) along wi h (3.2.8). We inally ob ain (3.3.4) This equa ion yields he o a ional iscosi y 3 �- anh� T¡ = "2T¡0l/J � + anh s ' This las quan i y beha es o � «: 1 as (3.3.5) 1 2 T¡ == ¡T¡ol/J� and is hen quad a ic in he ield. Fo � � 1, one has (3.3.6) (3.3.7) 36 CHAPTER l. LIMIT CASES 1.0 �--------�--------�----------�--------� 0.8 --- __ - 0.6 0.4 0.2 0.0 � � _' L- � O 5 10 15 20 Jl Figu e 1.2: The quan i y ¡ / ¡. e sus he pa ame e p. The solid line co esponds o ou esul gi en in Eq. (3.3.5) and wi h he esul coming om a phenomenological heo y p oposed in Re .[4], whe eas he dashed line is ob ained om he solu ion o he Smoluchowski equa ion gi en in Re .[6]. The do s ep esen expe imen al da a om Re . [13]. 4. Calcula íon o he iscosi ies using G een-Kubo o mulas 37 Consequen ly, he o a ional iscosi y ends owa ds he asymp o ic limi �1104>. In Fig. 1.2 we ha e plo ed he educed o a ional iscosi y 11 /11., wi h 11. == �1104>, as a unc ion o he pa ame e ¡J. The p esence o he o a ional iscosi y leads o he in oduc ion o an e ec i e iscosi y o he e o luid. Fo he case o aplana Coue e low whe e he applied magne ic ield is choosen pe pendicula o he o ici y, his quan i y gi es he inc ease o he iscosi y o he suspensión due o he ac ion o he ex e nal ield. These esul a ag ee wi h he ones ob ained in Re . [4] by means o a phenomenological heo y and wi h he expe imen s pe o med by Mc Tague [17]. In ou o me calcula ion o he i s momen o R, we ha e neglec ed a e m o second o de in he o ici y. This app oxima ion can be emo ed by compu ing he dis ibu ion unc ion up o second o de in his quan i y a low magne ic ield. The esul ing a e ages can be gi en, o example, when he o ici y is pa allel o . Fo he a e age o he y-componen , one has (3.3.8) which is in good ag eemen wi h he co esponding esul ob ained in Re . [15] by means o B ownian dynamics. No e, howe e , ha since he e omagne ic pa icles a e e y small, second o de co ec ions in he o ici y a e only impo an a e y high shea a es. 4 Calcula ion o he iscosi ies using G een-Kubo o mulas Ou pu pose in his sec ion is o p esen an al e na i e de i a ion o he shea and o a ional iscosi ies. These quan i ies wiJl be compu ed om he co esponding G een Kubo o mulas a ising om linea esponse heo y, which ha e been widely used o calcula e anspo coe icien s. The con ibu ion o he pa icles o he shea iscosi y, in oduced in Eq. (3.4), is hen gi en by (4.1 ) In o de o compu e he co ela ion o he symme ic pa o he p essu e enso , we will use he ela ionship be ween he p essu e enso and he second o de mul ipole o he induced o ce, gi en h ough eqs. (3.2) and (2.30). P o ided ha ou e e ence s a e should be in equilib ium condi ions ( 3 = O), his ela ionship is he oJlowing 38 CHAPTER I. LIMIT CASES TI (.) ( ) - 20 2 (2.) ( ) p,:Cz - - 1 a 7]oV R,:c' , (4.2) which used in (4.1), yields (4.3) The co ela ion in his exp ession can be compu ed om luc ua ing hyd odynam ics. In ac , om (2.9) and (2.10) we ge In eal space his exp ession is gi en by (4.5) Using now he mul ipole expansion, o poin s a he su ace o he pa icle, we can ob ain he co esponding co ela ion o he second mul ipole o he andom eloci y. One a i es a (4.6) whe e he ma ix ele nen 1-'(2,2) has been gi en in (2.24). Fo he symme ic pa o his co ela ion, one has (4.7) Employing his exp ession in (3.3) and pe o ming he in eg al, we hen come o he esul (4.8) which co esponds o he con ibu ion o he iscosi y o he suspension owing o he p esence o pa icles. The o a ional iscosi y can be compu ed by neans o he same p ocedu e. The G een-Kubo o mula is now T/ = Vk1BT 100 d (TI��lz( )TI��lz(O)) o in e ms o he co esponding axial ec o s (4.9) 4. Calcula ion o he íscosi es using G een-Kubo o mulas 39 n- = Vk�T 100 d (n��J( )n��J(o»). Making use o equa ions (3.2) and (2.32) and he de ini ion (3.1.3), we ge (4.10) (4.11) whe e we ha e employed again he equi emen 3 = O, o he e e ence s a e. The co ela ion in Eq. (4.10) hen ollows om his las exp ession. One has (II��J( )n��J(o») = (4'11"a3)2 ¡�([Oy( ) + 23a ( : �")( »Yl [01/(0) + 23a ( : �")(O»!ll) (4.12) o al e na i ely 2 (n��J( )n��J(O») = (41Ta3)2�� (m x H)y( )(m x H}y(O»), (4.13) wi h � = 81Ta3 ¡o being he o a ion ic ion coe icien . To ob ain his equali y we ha e used he equa ion o o a ional mo ion o he pa icle (4.14) whe e he exp ession o he hyd odynamic o que is gi en in (2.35) and M is he magne ic o que. In (4.14) we ha e neglec ed ine ial and gy omagne ic e ec s, hus he o ques balance each o he ou . This ac p o ides he equa ion o de i e (4.13) om (4.12). The o a ional iscosi y can be inally exp essed as 3 2 oo ,, TJ = 2¡ JTJoD Jl Jo d (R.,( )R.,(O»), (4.15) whe e Jl is he Lange in pa ame e n;;�, compa ing magne ic and he mal ene gies, and we ha e aken he magne ic ield poin ing owa ds he z-di ec ion. To pe o m he in eg al in (4.15) we ha e o know he co ela ion unc ion o he x-componen o he ec o Él. The e olu ion equa ion o his co ela ion comes om he Smoluchowski equa ion (4.16) alid in he absence o o ici y. In ac , s a ing om (4.16) one may de i e he e olu ion equa ion o he co ela ion 40 CHAPTER l. LlMlT CASES d(R(�/l(O») = D { -2(R( )R(0») + ¡d! (R(O)} - IJ(R( )R(O)R( ») . Ji}, ( 4.17) which o he pe pendicula componen yields (4.18) To sal e his equa ion we will in oduce a decoupling app oxima ion as in he p e ious sec ion. This app oxima ion consis s o decoupling he pe pendicula and pa allel componen s o he ec o R, due o he di e en na u e o he dynamics o bo h componen s, as we al eady poin ed ou , and can be o mula ed in he ollowing way: (4.19) In addi ion, in he linea egime, we may app oxima e in (4.19) (RII( ») by i s equi lib ium alue (RII(O»)eq = C(IJ). This equilib ium a e age is compu ed wi h he equilib ium dis ibu ion unc ion IJ A A /!eq = . h exp(IJR· H). 411'SIn IJ Consequen ly, Eq.( 4.18) ans o ms in o he elaxa ion equa ion ( 4.20) (4.21) om which we may iden i y he elaxa ion ime TJ. (4.22) Thus we expec ha he co ela ion decays exponen ially acco ding o (4.23) The o a ional iscosi y hen ollows om he G een-Kubo o mula (4.15) oge he wi h (4.23). Pe o ming he in eg al, one a i es a (4.24) In iew o he esul (R.:(0)2)eq = �, which can also be ob ained by compu ing he equilib ium a e age wi h he equilib ium dis ibu ion unc ion (4.20), one ob ains 5. Conclusions 41 3 Il- anh Il TJ = "2 TJo4J Il + anh Il ' (4.25) which, as we expec ed, also coincides wi h he co esponding exp ession ob ained by Shliomis using a con inuum heo y [4] and wi h he esul o [7]. 5 Conclusions In his pa we ha e analyzed he dynamics o a e omagne ic sphe ical pa icle, as well as he iscosi ies o a dilu e suspension cons i u ed by hese pa icles. Ou heo e ical amewo k is based on he Na ie -S okes equa ion in which an induced o ce, esul ing om he pe u ba ion in oduced in he dynamics o he luid by he pa icle, and a s ochas ic Lange in sou ce, coming om he luc ua ions o he hyd odynamical ields, ha e been included. This equa ion accoun s o he coupled dynamics o he luid and he pa icle. A mul ipola expansion o he quan i ies appea ing in he o mal solu ion o he Na ie -S okes-Lange in equa ion leads o he exp essions o he o ce and o que exe ed on he pa icle which con ain andom con ibu ions whose s a is ical p ope ies a e dic a ed by luc ua ing hyd odynamics [5]. We use wo me hods o ob ain he shea and o a ional iscosi ies. One is de e  minis ic and is based on he Ki kwood o mula o he iscous p essu e enso which is shown o be ela ed o he second o de mul ipole o he induced o ce. The o he is based on he linea esponse heo y gi ing he anspo coe icien s in e ms o ime-depe iden co ela ion unc ions. The p esence o he ield is esponsible o he appea ance o an an isymme ic con ibu ion o he p essu e enso and in oduces he o a ional iscosi y as a new anspo coe icien . The exis ence o an isymme ic s esses comes om he ac ha he o que exe ed by he magne ic ield on a dipole and he hyd odynamic o que balance each o he ou . Consequen ly, he angula e loci y o he pa icle may di e om he alue o he o ici y o he luid a he poin i occupies. We ha e s udied he e a simple case in which he anspo coe icien s ha e al eady been calcula ed by o he au ho s by means o di e en me hods. Ou main pu pose has been o de elop a a he gene al o malism ha can be use ul in he s udy o anspo phenomena in e o luids in di e en and mo e in ica e si ua ions, which cons i u e he subjec o so ne o he ollowing pa s o he monog aph. Fu he mo e, he me hods de eloped he e can also be applied o s udy he dependence o he iscos i y on equency [16], o o highe concen a ions, in which case dipola in e ac ions 42 CHAPTER J. LJMJT CASES play an essen ial ole. Wo k is being done in he las case in o de o ind u he heo e ical esul s. Pa II 80FT MAGNETIC MATERIAL8 IN HIGH MAGNETIC FIELD8 43 44 CHAPTER l. LIMIT CASES n __ n n H Figu e 1.3: Fo so magne ic ma e ials in su icien ly high magne ic ields he magne ic momen s o ien hemsel es in he di ec ion o he ield e y quickly. Then i akes place he mechanical o a ion o he pa icles o he s a iona y o ien a ion. 6 In oduc ion In his pa o he chap e we apply he o malism de eloped in he p e ious pa o he case in which a e pe u bing he sys em, he elaxa ion owa ds he magne ic ield akes place in wo s eps: i s a quick elaxa ion o he magne ic momen , hen a mechanical o a ion o he pa icle owa ds he equilib ium o ien a ion, wi h he magne ic momen also pa allel o he easy axis o magne iza ion [19]. This si ua ion occu s o high magne ic ields and he ene gy o he magne ic pa icles educes o he ene gy o aniso opy. Expe imen ally, i is ound ha he magne iza ion ends o lie along ce ain c ys allog aphic axes; his e ec is known as c ys alline aniso opy. I is addi ional o he di ec ional e ec s ha occu when he samples' shape lacks sphe ical o cubic symme y. The exis ence o c ys alline aniso opy may be de non s a ed by he magne iza ion cu es. I is clea ha much smalle ields a e equi ed o magne ize he c ys als o sa u a ion along ce ain di ec ions han along o he s. The c ys allog aphic axes along which he magne iza ion ends o lie a e called easy di ec ions; he axes along which i is mos di ícul o p oduce sa u a ion a e called ha d di ec ions. Wi h his goal in mind, his pa has been dis ibu ed as ollows. Sec ion 7 is in ended as a sho e iew o he o a ional dynamics which was p e iously analyzed. 9. Conclusions 51 Re . [18] using a phenomenological app oach, based on he o mula ion o a elaxa ion equa ion o he in e nal angula momen um o he sys em in a con inuum desc ip ion. Howe e , we ha e obse ed again so ne disc epancies in ou esul s wi h espec o he ones ob ained by hose au ho s om an app oxima ed solu ion o he Smoluchowski equa ion [18], especially o in e media e alues o he aniso opy pa ame e . 9 Conclusions By means o he o malism de eloped in he p e ious pa , we ha e p esen ed in his pa a calcula ion o he o a ional iscosi y based on he G een-Kubo o mula o his anspo coe icien . Ou esul s show ha he iscosi y inc eases when inc eas ing he aniso opy pa ame e and eaches a sa u a ion limi o We ha e compa ed ou cu e wi h he ones ob ained in Re .[18]. Thei esul coming om a solu ion o he s a iona y Smoluchowski equa ion o e es ima es ou s, whe eas he one based on a phenomenological elaxa ion equa ion hey p opose o he in e nal angula momen um o he suspension, is close o ou s, I would be in e es ing o ga he expe imen al da a in o de o decide which o he h ee app oaches is he mos con incing. We ha e also compu ed he o a ional iscosi y by means o he heological equa ion o s a e p oposed by Ki kwood, gi ing he p essu e enso in e ms o he hy d odynamic o ce exe ed by he luid on he magne ic pa ide. The a e age o he p essu e enso has been calcula ed om he s a iona y solu ion o he Smoluchowski equa ion (9.1 ) which now con ains a con ibu ion due o he o ici y o he low. Following he s eps indica ed in [7] we ge he same esul (8.22). In he same way, we could sol e he mo e gene al case in which bo h, he elaxa ion o he magne ic momen owa ds he ield and owa ds he easy axis o magne iza ion, ake place. So ne ela ed heo e ical esul s ha e been ob ained in [1] and i will be he subjec o he ollowing chap e . Bibliog aphy [1] J. Happel and H. B enne , Low Reynolds Numbe Hyd odynamics (Kluwe Aca demic Publishe s, Do d ech , The Ne he lands, 1991). [2] H.L. F isch and R. Simha, in: Rheology, ol.Il, F. R. Ei ich, ed. (Acad. P ess, New Yo k, 1956). [3] S.R. de G oo and P. Mazu , Non-Equilib ium The modynamics (Do e ,New Yo k,1984). [4] M. 1. Shliomis, So . Phys. JETP 34 (1972) 1291. [5] L. Landau and E. M. Li shi z, S a is ical Physics, Pa 2 (Pe gamon P ess, Ox o d, 1981). [6] M. A. Ma senyuk, Yu. L. Raikhe and M. 1. Shliomis, So . Phys. JETP 38 (1974) 413. [7] J.M. Rubí and M.C. Miguel, Physica A 194 (1993) 209. [8] J. M. Rubí, C. Salueña, and A. Pé ez-Mad id, in: Complex Fluids, L. Ga ido, ed., Lec u e No es in Physies, ol. 415 (Sp inge -Ve lag, Be lin). [9] M.C. Miguel, J. Bone A alos, A. Pé ez-Mad id and J.M. Rubí, Physiea A 193 359 (1993). [10] P. Mazu and D. Bedeaux, Physica 76 (1974) 235. [11] P. Mazu and W. Van Saa loos, Physiea 115 A (1982) 21. [12] H. B enne , J. Colloid. In e ace Sci. 1 (1970) 141. [13] J. P. Me Tague, J. Chem. Phys. 51 (1969) 133. 52 BIBLIOGRAPHY 53 [14] C. Salueña, A. Pé ez-Mad id and J. M. Rubí, J. Colloid. In e ace Sci. 164, (1994) 263. [15] A. O. Cebe s, Magni . gid odin.4 (1984) 17 (Magne ohyd odynamics,1984). [16J C. Salueña and J. M. Rubí, J. Chem. Phys. 102 (1995) 3812. [17] J.-C. Bac i, K. Dje i, S. Ne eu and R. Pe zynski, J. Mag. Mag. Ma e . 123 (1993) 67. [18] Yu. L. Raikhe and M. 1. Shliomis, SOy. Phys. J. Appl. Mech. Tech. Phys. 15, (1974) 470. [19] M.I. Shliomis and V.1. S epano , J. Magn. Magn. Ma e . 122 (1993) 196. CHAPTER II GENERAL DYNAMICS In he p e ious chap e we ha e analyzed he dynamics o a e omagne ic pa icle and he anspo coe icien s o a dilu e suspension cons i u ed by hem in wo lim i ing cases, namely, a suspension o igid dipoles and a suspension o a so magne ic ma e ial unde he in luence o a high magne ic ield. The main goal o his chap e is o p o ide a gene al heo y co e ing he whole ange o expe imen al si ua ions. Indeed, di e en measu emen s ca ied ou o mag ne ic liquids show ha he e is ini e coupling be ween he o ien a ion o he mag ne ic momen o a e omagne ic monodomain and he o ien a ion o he pa icle i sel (cha ac e ized by he o ien a ion o i s c ys alline axes). Because o his coupling, he elaxa ion o he magne ic momen s akes place in wo di e en ways ha p oceed simul aneously: o a ion wi hin he pa icle and oge he wi h he pa icle wi h e spec o he ca ie liquido Bo h p ocesses a e o o a ional di usion ype. Thus, o all p ac ical pu poses, he Smoluchowski equa ion, desc ibing he e olu ion o he p obabili y densi y o he di ec ions o bo h he magne ic momen and he axis o easy magne iza ion o he pa icles, has been used. In pa icula , we compu e he o a ional iscosi y om a G een-Kubo o mula and gi e an exp ession o di e en elaxa ion imes. These cha ac e is ic imes come om he dynamic equa ions o he co ela ion unc ions which, in he linea esponse heo y amewo k, a e in ol ed in he calcula ion o so ne o he ma e ial's physical p ope ies we a e in e es ed in (op ical, magne ic, ... ). Ou esul s ag ee qui e well wi h expe imen s pe o med wi h di e en samples o e omagne ic pa icles, which pe mi o dis inguish he di e en elaxa ion egimes occu ing when he size and he na u e o he magne ic ma e ial o he g ains a e eely modi ied. 54 1. In oduc ion 55 1 In oduc ion Sys ems o single-domain e omagne ic pa icles imme sed in a solid o liquid phase exhibi a numbe o in e es ing elaxa ion phenomena which ha e been he subjec ma e o many expe imen al and heo e ical analyses [1]-[4]. These phenomena a e essen ial in he s udy o he dynamics o hese pa icles, and, pa icula ly, ha e a c1ea in luence when de e mining he e ec i e iscosi y, he dynamic bi e ingence, and he magne ic suscep ibili y. One o he main peculia i ies o hese sys ems is ha hei p ope ies a e g ea ly in luenced by he p esence o an ex e nal magne ic ield. I is p ecisely his ac which has been he basis o many p ac ical applica ions [1]. The o a ional dynamics o a e omagne ic pa icle embedded in a liquid phase is he esul o he compe i ion o h ee o ien a ional mechanisms ela ed o he ex e  nal ield, he axis o easy magne iza ion, and o a ional B ownian mo ion. Tha is, whe eas he magne ic momen o he e omagne ic pa icle elaxes owa ds he di ec ion o he magne ic ield, he axis o easy magne iza ion ends o be aligned wi h he magne ic momen , hus gi ing ise o di e en coupled elaxa ion phenomena. Un il ecen ly, he mos equen case ha has been s udied in he li e a u e deals wi h igid-dipoles [4]-[6], o which he aniso opy ene gy is dominan due o he la ge alue o he aniso opy cons an , and because he adius o he pa icle usually exceeds a c i ical alue. Wha is mo e, when looking o he elaxa ion phenomena desc ibed by a igid dipole, one dis ega ds he p ecessional mo ion o he magne ic momen , and, consequen ly, he associa ed dissipa ion. Unde hese condi ions, we canno alk abou he elaxa ion o he axis o easy magne iza ion owa ds he magne ic momen any longe , ins ead bo h ec o s elax oge he . Howe e , he e a e ma e ials o which he aniso opy ene gy may be compa able o he ene gy associa ed wi h he in e ac ion wi h he magne ic ield, o e en smalle . The e o e, a gene al heo y en compassing such a wide a ie y o si ua ions and accoun ing o expe imen al esul s should be de eloped. The p esence o di e en elaxa ion mechanisms has implica ions in he o m o he e ec i e iscosi y o he sys em, which exhibi s signi ican co ec ions when compa ed o he iscosi y o a suspension o non-magne ic pa icles o he same shape. Ano he poin o in e es is he appea ance o elaxa ion imes which a e usually in ol ed in he cha ac e iza ion o ce ain physical p ope ies, and which a e sui able o being measu ed by means o di e en expe imen al echniques. The pu pose o his chap e is o p esen a heo y capable o gi ing exp essions o he ele an anspo coe icien s o he sys em and o he co esponding cha ac e is ic elaxa ion imes de e mining o ins ance, he e ec i e iscosi y, he dynamic 56 CHAPTER Il. GENERAL DYNAMICS n n H Figu e 11.1: The magne ic momen s a e o ien ed along an in e media e di ec ion be ween ha o he magne ic ield and he easy axis o magne iza ion when bo h he aniso opy ene gy and he magne ic ene gy o in e ac ion wi h he ield a e compa a ble. The elaxa ion o hese wo deg ees o eedom is coupled. bi e ingence, and he magne ic suscep ibili y o he suspension. We will ocus on he gene al si ua ion in which he magne ic and aniso opy ene gies o he pa icles may ha e a bi a y alues. The o malism we ha e de eloped is based on he linea esponse heo y whe e he co ela ion dynamics comes om a Smoluchowski equa ion. As we will show in one o he sec ions, ou esul o he elaxa ion ime o he o a ion o he pa icle is compa ed o expe imen al da a and ag ees qui e well wi h bi e ingence expe imen s. We ha e dis ibu ed he chap e in he ollowing way: in Sec ion 2, we es ablish basic equa ions desc ibing he dynamics o he deg ees o eedom. O pa icula in e es is he Smoluchowski equa ion o he p obabili y densi y which is gi en in a gene al case o unspeci ied alues o he magne ic and aniso opy ene gies. The e a e di e en ways o de i ing such equa ion. Al hough we will no go h ough his ques ion, he equa ion we p opose can be compa ed o ano he one ob ained p e iously om a di e en heo e ical me hod. In Sec ion 3, we deal wi h he calcula ion o he o a ional iscosi y using a G een-Kubo equa ion p oposed om he linea esponse heo y. This me hod leads o an exp ession o his anspo coe icien which is s udied in pa icula si ua ions o in e es . Sec ion 4 is de o ed o he calcula ion o he elaxa ion imes o he pa icles and he ans e sal componen o he magne iza ion 2. Coupled dynamícs oE he deg ees oi E eedom 57 when conside ing he di e en o ien a ional mechanisms. We ha e compa ed ou esul s o expe imen s done o wo samples o e y common e omagne ic pa icles o which he size and he na u e o he magne ic ma e ial clea ly es ablish di e en alues o he magne ic ene gy and he ene gy o aniso opy, ob aining a good ag eemen in bo h si ua ions. Finally, in he las sec ion we summa ize ou main esul s. 2 Coupled dynamics o he deg ees o eedom The ene gy o a sphe ical single-domain e omagne ic pa icle unde he ac ion o an ex e nal magne ic ield is he sum o wo con ibu ions. These con ibu ions o igi na e om he ex e nally imposed magne ic ield and he p esence o an axis o easy magne iza ion ( o uniaxial c ys als). I s exp ession is gi en by (2.1 ) whe e ñi = mR is he magne ic momen o he pa icles, H is he ex e nal magne ic ield, Ka is he i s aniso opy cons an (assumed posi i e), Vm is he magne ic olume o one o hese sphe es, and n is he uni ec o along he di ec ion o he axis o easy magne iza ion o .uniaxial magne ic c ys als. I is clea om Eq. (2.1) ha in he gene al case whe e bo h con ibu ions may ake a bi a y alues, he elaxa ion mechanisms o he deg ees o eedom, R and ñ, o he e omagne ic sphe es in suspension a e coupled. The de e minis ic dynamics o R is go e ned by he Landau-Gilbe equa ion [7], p oposed o s udy he elaxa ion o he magne ic momen s o magne ic pa icles e n bedded in a solid ma ix dR 'Yo {)U • dR . - = ---_ x R- 0/- X R. d m {)R d (2.2) F om his equa ion, one may iden i y he wo mechanisms esponsible o he a ia ion o k he e ec i e ield HeJl ex: -�, which causes a La mo p ecessional mo ion o R, and he mean ield, H d ex: - 4 , which in oduces a damping due o he col lisions o he elec ons de e nining he magne ic momen o he domain in a me al, o in a semiconduc o , due o magne oelas ic in e ac ions. In Eq. (2.2), 'Yo is he gy omagne ic a io o an elec on, and he quan i y O/ plays he ole o a da nping coe icien . The Landau-Gilbe equa ion can be ew i en such ha 58 CHAPTER n. GENERAL DYNAMICS dEl _ au _ _ - = -hR X -_ x R + WL X R d aR ' (2.3) wi h WL = gDe" as he La mo equency o he p ecessional mo ion, and whe e 9 == 1'0(1 + 0'2)-1 and h == �(1 + 0'2)-1 [2]. This equa ion is alid in a s a iona y ame o e e ence. I he e omagne ic pa icle is o a ing i sel wi h he angula eloci y ñ, we mus modi y Eq.(2.3) by adding on i s igh hand si de he co esponding con ibu ion coming om he o a ion. One hen has dH ( _ au) _ _ _ di = -h R x aH x R + (WL + n) x R. (2.4) Fu he mo e, he dynamics o ñ ollows om he kinema ic ela ion dñ - di = n x ñ, (2.5) This exp ession can be ew i en as dñ [_ 1 - -] _ -d = Wo + -m x H x n, � wi h � = 811'7]oa3 being he o a ional ic ion coe icien o he pa icles; 7]0 is he (2.6) iscosi y o he ca ie luid, a is he hyd odynamic adius o he pa icles, and Wo he o ici y o he ca ie luid. One a i es a his exp ession a e using he de e minis ic pa o he balance equa ion o he o al angula momen um (2.7) p o ided ha we neglec he e m accoun ing o he ine ial e ec s and he e m com ing om he angula momen um o he elec ons de e mining he magne ic mo nen o he pa icles. He e B is he B ownian o que ac ing on he pa icle [8]. The e olu ion o he p obabili y densi y, 1/;(¡, ), wi h "Y == (H,ñ), is go e ned by he Smoluchowski equa ion. When ñ = O, which co esponds o he case o pa icles embedded in a solid ma ix, he Smoluchowski equa ion was deduced by B own [9] om he Landau-Gilbe equa ion. In hese las condi ions, and in o de o p ese e he o al angula momen um conse a ion, he small magne ic pa icle should be embedded in a la ge igid solid ma ix. O he wise, he e will appea an elas ic wis in he ma ix, and one should ake in o accoun he co esponding elas ic o que in he equa ion o he o al angula momen um o he pa icle. The manne in which B own exp essed his in ui i e me hod o de i ing he Smoluchowski equa ion was o conside 2. Coupled dynamics oi he deg ees oi E eedom 59 he e ec o he mal luc ua ions on he p obabili y densi y. B own sugges ed ha he mal agi a ion causes P o become mo e uni o m so ha , in an equa ion desc ibing i s ime e olu ion, he mal agi a ion gi es ise o a di usion e m in P. Shliomis and co-wo ke s [10] ob ained his equa ion o a suspension o igid dipoles. I we de ine he dimensionless pa ame e s /J = �l and (7 = K,,"aF, compa ing magne ic and aniso opy ene gies o he mal ene gy, espec i ely, his las si ua ion co esponds o he limi (7 � /J. Rhaike and Shliomis [11] also p oposed he Smoluchowski equa ion o he opposi e limi (7 <€:: /J, in which he dipoles a e apidly o ien ed owa ds he ield di ec ion. As ega ds he gene al case o a bi a y alues o he a io /JI (7, Shliomis and co-wo ke s also deduced he app op ia e Smoluchowski equa ion om a model simila o he i ine an oscilla o model, [12]-[14]. In such a gene al si ua ion, he Smoluchowski equa ion can al so be ob ained om he con inui y equa ion in he space spanned by he deg ees o eedom "'{ == (R, n) J Pb, ) a « ) ( ).) a =-8"'(. J"'{, + P"'{, "'{, whe e � == d"'{ld and he cu en Jb, ) is gi en by he Fick's law (2.8) Jb, ) = -D . a Pb, ), 8"'( (2.9) wi h D being a di usion ma ix. Combining (2.8) and (2.9) we hen a i e a a P a a . 8 = 8"'( . (D . 8"'( P - P"'{). (2.10) The di usion ma ix is ela ed o he mobili y ma ix b h ough he Eins ein ela ion, D = kBTb. A e inse ing eqs. (2.4) and (2.6) in o (2.10), we can ew i e Eq. (2.10) in he o n (2.11) whe e we ha e a con ibu ion coming om a non-po en ial cu en -Ynon-po , and he mobili ies b a e ound o be ( o mo e de ails see Appendix A) - ; (1 - nn), 1 '- - (h + { )(1 - RR), TI,' " bRn == { [(n . R)1 - nR]. (2.12) 60 CHAPTER II. GENERAL DYNAMICS He e he symbol T s ands o ansposi ion. These exp essions can be employed in Eq.(2.11). A e so ne ma hema ical ans o ma ions (see de ailed calcula ions in Appendix A), one hen a i es a he Smoluchowski equa ion N & = --- - U -- D (nR + nñ)· [ P(nR + nñ) kBT + (nR + nñ) P] - - U - - DmnR· {,pnR kBT + nR P} - nR· (WL P) ('RR + ññ) . (wo P), (2.13) + h -ñ=' 8 d--R' 8 ·1 kT· w e e "-ñ n x 8ñ an n R = x 7iR a e o a iona ope a o s, D; == T IS he B ownian o a ional di usion coe icien , and Dm == kBTh can be in e p e ed as he di usion coe icien o he magne ic momen inside he pa icles. These di usion coe icien s a e ela ed o wo elaxa ion imes in ol ed in he Smoluchowski equa ion, namely TD = (2Dm)-1 ela ed o he chao ic eo ien a ions o ni inside he pa icle due o he mal luc ua ions, and he B ownian ime TB = (2D )-1. The Smoluchowski equa ion (2.13) ag ees wi h he co esponding one ob ained in Re . [12] by using a model simila o he i ine an oscilla o model and will be used in ou subsequen analysis. 3 G een-Kubo o mula o he o a ional iscosi y In his sec ion, we we will ocus on he de e mina ion o he o a ional iscosi y om he co esponding G een-Kubo o mula. This o mula gi es his anspo coe icien in e ms o he co ela ion unc ion o he axial ec o , ñ�a), ela ed o he an isy n me ic pa o he con ibu ion o he pa icles o he p essu e enso [8], (3.1 ) whe e V is he olume o he sys em. In Re . [8], W! ob ained a ela ionship be ween he pa icle con ibu ion o he p essu e enso , ñp, and he ex e nal o que expe ienced by he pa icle du ing i s mo ion. As a esul , i we conside he magne ic ield poin ing owa ds he z-di ec ion, i is ound ha he o a ional iscosi y can be inalIy ew i en as 3 [''''' T} = "2 PT}o i. 1-'2 Jo d (R ( )R (O»), wi h P = 4/3;a' being he olume ac ion o pa icles. (3.2) 3. G een-Kubo o mula {o he o a ional iscosi y 67 1.0 �----...,.-----�-----�----� 0.8 DjD,=1 11=100 0.6 0.4 0.2 0.0 �----�----�------._----� O 5 10 15 20 o Figu e 11.3: Ro a ional iscosi y e sus he pa ame e O" o di e en alues o he a io Dm/ D,. and o he pa ame e ¡J. 68 CHAPTER Il. GENERAL DYNAMICS espec i ely, which also coincide wi h he esul s gi en in Re .[12]. E en in his case, wi h ¡ � 1, i u akes mode a e o small alues, he o a ional iscosi y does no each i s sa u a ion alue any longe . In pa icula , i u - O, his sa u a ion alue depends on he a io Dm/D as we ha e poin ed ou in Eq. (3.22). Tha is, he dissipa ion no only depends on he sol en iscosi y, bu also on he damping cons an O' and he gy omagne ic ac o 10. Addi ionally, he o a ional iscosi y inc eases wi h u un il i eaches i s sa u a ion alue when u - oo. The beha io o '7 / '7. as a unc ion o u, and o di e en alues o ¡ is depic ed in Fig. 11.3, om which we can co obo a e he main ea u es o ou p e ious analysis. The a io Dm/ D; has been ob ained a e conside ing he ollowing alues o he in ol ed quan i ies: O' '" 10-2, 10 '" 107e-1 s-1, M. '" 103e, '70 '" 1O-2c.!J; and he ac ha he magne ic olume is almos he same as he hyd odynamic olume o he pa icles, wi h a = 10-6cm. An al e na i e p ocedu e used o calcula e he iscosi y in ol es a heological equa ion o s a e o he p essu e enso . In his way, i is also possible o ob ain exac ly he same exp ession o he o a ional iscosi y (3.16), bu now in he p esence o a linea o ici y ield in s a iona y condi ions. 4 Relaxa ion imes. Compa ison wi h expe imen s 4.1 T ansien bi e ingence in c ossed ields The elaxa ion o he op ical bi e ingence induced by magne ic g ains dispe sed m he analyzed medium, is among he simples expe imen al ools a ailable o heologi cal s udies o iscoelas ic solu ions. A iscosime e based on he de e mina ion o he elaxa ion o small magne ic pa icles in suspension in he s udied medium u ns he iscosi y de e mina ion in o an op ical bi e ingence measu emen wi h many ad an ages: i) i is a nondes uc i e me hod, ii) we jus need o add a e y small amoun o pa icles (<p '" 10-4), iii) i does no need any mechanical sys em, i ) he use o a lase beam o he bi e ingence measu emen enables us he de e mina ion o he iscosi y in he olume o a ew mm3. In addi ion, as he size o he magne ic p obe is o he o de o 100A, he measu ed iscosi y is a local quan i y. The main limi a ion o he usual ansien bi e ingence de ices is he polydispe si y o he magne ic pa icles. Usually, a log-no mal dis ibu ion o sphe ical g ain diame e s, d, as he one gi en h ough he exp ession 4. Relaxa ion imes. Compa ison wi h expe imen e 69 P(d) = �CTd exp ( - 2�2 In2 (�) ) , (4.1.1) whe e do and CT a e he mean diame e and he a iance o he dis ibu ion, espec i ely, is sui able o desc ibe he samples. Thus, in o de o ge an exponen ial elax a ion wi h only one cha ac e is ic ime a sample whose alue o CT is abou 0.3 o less mus be used. Each pa icle is i s cha ac e ized h ough a s a ic bi e ingence measu emen as a unc ion o he magne ic ield. This is due o he aniso opy o he elec ic suscep ibili y enso ela ing he pola izabili y o he sys em o he inciden elec ic ield. The main causes o he aniso opy a e he in e nal op ical aniso opy o he magne ic ma e ial (c ys alline aniso opy) and he shape aniso opy o he pa icles. Conce ning he dynamic beha io o he bi e ingence, di e en expe imen s can be ca ied ou : In a liquid, i he magne ic ield is swi ched o ab up ly, pa icles he mally elax owa ds andom di ec ions. Thus looking a he bi e ingence elaxa ion is compa ing iscous o he mal ene gies. I a squa e pulse o magne ic ield H is applied o he e o luid solu ion, magne ic pa icles i s end o align along he ield leading o bi e ingence 6n( ), and, as he ield is swi ched o , hey he nally elax o andom di ec ions. Bi e ingence dec eases exponen ially acco ding o Pe in's law [18) 8n( ) = 8n(d, H)exp- /T(d) (4.1.2) whe e he cha ac e is ic ime is T(d) = (6D )-1, wi h D; he o a ional di usion coe icien . In he quan i y 8n( ), he polydispe si y o he samples modi ies bo h 8n(d, H) and T(d). Owing o his ac , .6.n( ) is no a simple exponen ial unc ion o ime. Bu , in o de o cha ac e ize Lln( ), one can use he sho es ime deduced om he ini ial slope o he unc ion In(.6.n( )) e sus . In a iscoelas ic medium, looking a he bi e ingence esponse o an al e na ing magne ic ield allows o each he iscosi y and he elas ic modulus o he medium a high equencies. I a la ge magne ic ield is supe imposed, he e is a magne ic es o ing o ce and iscous ene gy is compa ed o ha . Ins ead o elaxing a andom, magne ic momen s elax owa ds he di ec ion o he supe imposed ield. In his case, he elaxa ion ime associa ed wi h he o ien a ion mechanism o he pa icles, TR, has been measu ed ecen ly by Bac i e al. [19, 20). This quan i y comes om he elaxa ion ime o he ligh in ensi y collec ed in a pho ocell a e c ossing he sample in he p esence 70 CHAPTER TI. GENERAL DYNAMICS o an ex e nal magne ic ield, and when applying addi ional pulses o magne ic ield o pe u b he sample. These expe imen s pe mi us o dis inguish he di e en elaxa ion egimes occu ing when he size and he na u e o he magne ic ma e ial o he pa icles a e modi ied. These egimes a e de e mined by he pa ame e s 1-' and u, compa ing magne ic and aniso opy ene gies wi h he mal ene gy, espec i ely. In he expe imen s, i was obse ed ha e o luid pa icles, p e en ed om mo ing by being quenched in a igh gel ne wo k, do no exhibi bi e ingence al hough hey s ill show magne iza ion. Consequen ly, he bi e ingence o he solu ion is closely ela ed o a mechanical alignmen o he pa icles along he equilib ium o ien a ion. In Re s. [8, 15, 21], wo opposi e limi s we e conside ed, one whe e u � 1-' ( igid dipole app oxima ion) and ano he o which u «: 1-'. Fo he o ne limi , he e laxa ion ime o he pe pendicula componen o he magne iza ion is ound o be [8, 15] ( 4.1.3) which, in he case when 1-' -+ 00, ends o -DI = 1!..a. In he la e case, elaxa ion IJ IJ occu s in wo s eps, i s a quick elaxa ion o R owa ds il, hen a mechanical o a ion o he pa icle o he equilib ium o ien a ion wi h he easy axis o magne iza ion pa allel o R and ñ. Unde hese condi ions, he cha ac e is ic elaxa ion ime could be ob ained om he equa ion o mo ion o wha e e componen o he co ela íon unc ion (ñl.nz)( )(nl.nz)(O)} [21]. Pe o ming he co espondíng decouplings in i s e olu ion equa ion, we ob ain he elaxa ion ime ( )-1 ( 1 )-1 ( 1 )-1 T = 2D u + Q(u) = TB U + Q(u) , (4.1.4) which o u -+ 00, ends o 2D1 = zu , -o u In he p esen analysis, we conside he less s ingen case o which he a io 1-'/ u may ake a bi a y alues. Consequen ly, bo h pa a ne e s 1-' and u, a e expec ed o de e mine he elaxa ion ime associa ed wi h he o a ional elaxa ion o he pa icles. As o he limi u «: 1-' discussed p e iously, he app op ia e quan i y o desc ibing his mechanical elaxa ional mo ion is again a componen o he co ela ion unc ion (nl.nz)( )(ñl.nz)(O)}. F om he Smoluchowski equa ion, we ob ain i s dynamic equa ion � d(n nz)( l nz)(O)) = I-'(n (n· R))( )(n nz)(O)) -1-'(n;R )( )(n nz)(O)} -1-'(n nzRz)( )(n nz)(O)} - 6(n",nz)( )(n nz)(0)}. (4.1.5) 4. Relaxa ion imes. Compa ison wi h expe imen s 71 P oceeding along he same lines as in he p e ious sec ion, we a i e a a closed se o h ee di e en ial equa ions o he co ela ion unc ions (Rz( )(nznz)(O»), «n ... (ñ. R))( )(nznz)(O)), and «nznz)( )(nznz)(O»). I is wo hwhile poin ing ou ha he quan i ies a ime appea ing in he h ee independen co ela ion unc ions a e he same as in he p e ious sec ion. Fo he sake o simplici y, we wiIl in oduce he ec o Q = (Q 1, Q2, Q3), w hose com ponen s a e he Laplace ans o ms o (Rz ( )( nz n z )( O»), {(nz(ñ.R»( )(nznz)(O»), and {(nznz)( )(nznz)(O)}, espec i ely; and he ec o QO = (Q¡, Q:í, Q3) ep esen ing he ini ial alues o hese co ela ion unc ions. These ini ial alues can be calcula ed wi h he equilib ium p obabili y densi y a = O. The sys em o di e en ial equa ions can be w i en in ma ix no a ion as (4.1.6) wi h Á he coe icien s ma ix (3.13). We a e pa icula ly in e es ed in he elaxa ion dynamics o «n nz)( )(nznz)(O»). F om Eq. (4.1.6), we ob ain i s Laplace ans o m om which we can iden i y he elaxa ion ime we a e in e es ed in - D-1 l' Q3(S) TR - Im-Qo .-0 3 ( 4.1.8) In igu e H.4, we ha e ep esen ed he elaxa ion ime TR e sus jJ o di e en e omagne ic samples, bu , in o de o compa e wi h expe imen al da a om Re . [19], we ha e also ep esen ed he elaxa ion ime TR e sus H-1 in igu e 11.5. The da a co espond o wo samples o magne ic pa icles o he same mean size bu made o di e en magne ic ma e ials, namely CoFe204 and I - Fe203. The Co e i e sample has an aniso opy cons an K = 2 . 105�, and he sa u a ion alue o he magne iza ion is M. :::: 250�A. Fo he maghemi e, he aniso opy cons an is K = 4 . 103� and M. :::: 270�A. Wi h hese alues, he Co- e i e sample can be conside ed as a igid dipole (O" � jJ). On he o he hand, he maghemi e pa icles a e such ha jJ � 0". Rega ding he alues o he aniso opy cons an , 0", and he a io m/kBT = jJoM. Vm/kBT , we ha e aken 0""'" 15, m/kBT,..., 2.8· 1O-4:¡ o he maghemi e and 0""'" 565, m/kBT,..., 1.8· 1O-4:¡ o he Co- e i e. Fo bo h samples, he alues o he emaining quan i ies a e: TB ,..., 4.5ms and Dm/ D; ,..., 1. As i was obse ed in he expe imen s o he Co- e i e sample, TR ends o ze o when H - oo. Bo h R and ñ, quickly elax owa ds he ield di ec ion due o he 72 CHAPTER II. GENERAL DYNAMICS 1.5 l' (ms) 1.0 R Figu e H.4: Relaxa ion ime o he pa icles as a unc ion o J.l. o he Co- e i e and maghemi e samples. 4. ReJaxa ion imes. Compa ison wi h expe imen e 73 igidi y o he dipoles. Fo he maghemi e sample TR ends o a ixed, non- anishing alue ( ..... 0.3), which can also be ob ained om Eq. (4.1.4). Unde hese pa icula condi ions, he magne ic momen s apidly elax owa ds he ield di ec ion, bu due o he mode a e alue o (T, he elaxa ion o he easy axis o magne iza ion, ñ, o in o he wo ds, he mechanical elaxa ion o he pa icles, akes place in a ini e pe iod o ime. Conce ning he beha io o TR when H - O, we obse e ha TR - 6b. = Z independen ly o he alue o he pa ame e (T. This ime co esponds o he well known cha ac e is ic elaxa ion ime o he co ela ions o he componen s o he second o de enso ( in - ) o a pu ely di usi e p ocess [27]. In igu e 11.5, we ha e also ep esen ed he ex apola ion o he elaxa ion ime coming om a he simple conside a ions made in Re . [22] o he igid dipole limi unde he ac ion o a e y la ge ex e nal magne ic ield. Ou esul s ag ee wi h he asymp o ic beha io in i s alidi y ange, bu a he same ime, hey show he de ia ions a in e media e and low magne ic ield. These simple a gumen s can also be p oposed o he opposi e case p � (T ep oducing he asy np o ic alue TR ..... 0.3ms o he maghemi e sample, bu hey a e no able o explain he p-dependence o he elaxa ion ime o his ma e ial. 4.2 T ans e se complex suscep ibili y In his sec ion, we a e in e es ed in he linea esponse o he dispe sion o an ae magne ic ield o small ampli ude. F om an expe imen al poin o iew, i is much easie o obse e di e en e ec s such as he sa u a ion o he magne iza ion o he equency dependen ce o i s elaxa ion o a e o luid han o a pola dielec ic luid, because bo h he ield s eng h and he equencies equi ed a e much lowe . Mo eo e , as we will see below, in a magne ic luid he a ia ion o he complex suscep ibili y wi h equency depends on he elaxa ion ime o he magne ic momen o In he absence o an ex e nal de ield, Raikhe and Shliomis [23] de i ed exp es sions o he complex ae suscep ibili y, X(w) = X'(w) - iX"(w), o a single domain uniaxial pa icle. They calcula ed bo h he pa allel and pe pendicula suscep ibili ies wi h espec o he easy axis o magne iza ion. Fo ha pa icula si ua ion, hey showed ha he equency dependen ce o he suscep ibili y was a he sa ne ime a unc ion o he pa ame e (T, compa ing aniso opy ene gy o he he mal ene gy. The applied ac ield o ien a es bo h he magne ic momen s and he pa icle axes, bu in a linea app oxima ion, i.e. o small alues o he applied ac ield, hey neglec ed 74 CHAPTER JI. GENERAL DYNAMICS / / / / / / / / / / / / / / / / 0.8 0.4 0.2 0.0 ...._ ...._ .....__ ...._ ......... --' 0.0 0.5 1.0 1.5 H-l(lO·sm/A) 2.0 2.5 Figu e 11.5: Relaxa ion ime o he pa icles as a unc ion o H-1 o he Co- e i e and maghemi e samples. Expe imen al da a o Re . [19] co espond o he do s. The dashed line co esponds o he limi J.l � 1 o he Co- e i e, Re .[22]. 4. Relaxa ion imes. Compa ison wi h expe imen s 75 he ield-induced pa iele o ien a ion. Thus, o a weak measu ing ield, he pa í ele aniso opy axes we e jus o ien ed in a andom ashion due o B ownian mo ion. Mo eo e , in Re . [4] he au ho s ob ained he exp essions o he longi udinal and ans e se componen s o he complex dielec ic suscep ibili y enso o a sys em o nonin e ac ing pola molecules unde he simul aneous ac ion o a cons an ex e nal elec ic ield and a small ae elec ic ield. This si ua ion would be equi alen o he p e iously men ioned igid dipole app oxima ion. In his case, he suscep ibili ies depend on he bias ield. Fo expe imen al measu es in a liquid, i is much simple o de e mine he pa allel and pe pendicula suscep ibili ies wi h espec o a ixed di ec ion like ha o he de ield han wi h espec o he axis o easy magne iza ion o he pa ieles. Thus, in his subsec ion, we ob ain he exp ession o he pe pendic ula suscep ibili y wi h espec o he bias ield ii, i.e. he pe u bing ex e nal ield j{' .1 j{, when no only he magne ic momen s bu also he pa ieles hemsel es a e o ien ed by he ex e nal ield, such as we ha e desc ibed h oughou he chap e . We ob ain ha he suscep ibili y depends on he bias ield h ough he pa ame e /-l. F om linea esponse heo y, he decay o he magne iza ion pe pendicula com ponen unde he in luence o a cons an ield H, ano he small cons an ex e nal ield H' (H' .1 H and such ha mH'JkBT < 1) ha ing been swi ched o a ime = O, is (MJ.( )) - (MJ.(O)) = Xl.H'CJ.( ), (4.2.1) whe e (4.2.2) is he pe pendicula componen o he magne iza ion, 2 xl. = �:T ({Ri(O)) - (RJ.(0))2) is he pe pendicula componen o he s a ic magne ic suscep ibili y, and (4.2.3) ( 4.2.4) is he au oco ela ion unc ion o any pe pendicula componen o he magne iza ion. The co esponding complex magne ic suscep ibili y XJ.(w) is (4.2.5) 76 CHAPTER JI. GENERAL DYNAMICS In he limi o low equencies, Eq. (4.2.5) may be w i en as X.L(w) � x�(1- iWTol), (4.2.6) whe e (4.2.7) is he elaxa ion ime. Mo eo e , Col (s) is he Laplace ans o m o he au oco ela ion unc ion. Eq. (4.2.6) can be w i en down, up o he same o de o accu acy, in he o m o he Debye equa ion (4.2.8) This is he o a ional di usion limi whe e he beha io o Col( ) and, consequen ly, o (Mol ( )) - (Mol(O)) may be app oxima ed by he exponen ial (4.2.9) The elaxa ion ime ollows om (4.2.7) oge he wi h (4.2.4), and (3.8-3.14). In ac , when he pe u bing ex e nal ield is poin ing owa ds he e", axis, H' = H' É"" and he cons an pola izing magne ic ield coincides wi h he ez axis, H = H «., ou au oco ela ion unc ion educes o C ( ) = (R",( )R",(O)) '" (R;(O)) (4.2.10) Consequen ly, D-1 l' R1(s) Tol = 1m--;:;o . • -0 "'1 In Fig. 11.6, we ha e ep esen ed Tol e sus JJ o he di e en alues o he pa ame e (4.2.11) (J' co esponding o di e en a ailable ma e ials. The Debye spec a (single elaxa ion ime app oxima ion) is gi en by ( 4.2.12) whe e we ha e de ined XO = nm2/3kBT as he s a ic alue o he suscep ibili y in he absence o he cons an ield. In Figs. 11.7 and 11.8 we plo he eal X� (w) and imagina y x1 (w) pa s o he ans e se componen o he no malized complex suscep ibili y o di e en alues o APPENDIX A 83 Analogously, he po en ial e ms (A3), (A6), and (A8) can be w i en in he o m a ( (, OU) ,) a ( "OU) h-,· '1/; R x -, x R = -, 'l/;h(1 - RR) . -. oR oR oR oR ' (A9) 1 a (- - ,) a (1 ,,0U 1, 'OU) --,' 'I/;(R l + Rñ)U x R = -,' '1/;-(1 - RR)· -, + 'I/;-{(R· ñ)1 - Rñ}· -, , { oR oR { oR { on (AlO) _!_�. ('I/;(1l l + 1lñ)U x ñ) = �. ('I/;_!_(1 - ññ) . o� + 'I/;_!_ {(R- ñ)1 _ nR} . o�) , { on on { on { a R (All) whe e he unde lined ac o s may be iden i ied o he mobili y ma ices de ined in Eq, (2.12). Mo eo e , he di usion ma ix is gi en by D = kBTb, so ha we can also ew i e he di usi e ac o s in e ms o he o a ional ope a o s (A13) a ( 0'1/; ) a "' ,0'1/; -- _. DR .- =D -·{(R·n)I-Rn}·-=DR· ·R··I. oR n oil oR oñ R n 'P, (A14) :n . (DnR. !�) = D Rñ ·1l l'l/;· (A15) Finally, subs i u ing eqs. (A2)-(A8) and (A12)-(A15) in Eq. (A 1), we a i e a equa ion (2.13). Appendix B Decoupling app oxima ions In Sec ion 3, we ha e in oduced decouplings o so ne co ela ion unc ions appea ing in he e olu ion equa ions o he co ela ions. The pu pose o his appendix is o gi e mo e de ails abou he p ocedu e ollowed o ca y ou such decouplings. In Eq. (3.4), he app oxima ed quan i ies a e (R n, )( )R (O)) '" (R ( )R (O)) en, )eq = .c(¡¡ )(R ( )R (O)), (Bl) and whe e linea iza ions in ime ha e been pe o med. The app oxima ion (Bl) was al eady discussed in he ex (see Eq. (3.5) and commen s below). Fo he co ela ion (B2), we decouple he quan i ies R and (ñ· R)2 because hey a e no coupled in equilib ium condi ions, (R (ñ· k)2)eq = O, (R )eq = O, and (n· R)2)eq # O. Thus, in a si ua ion no a o n equilib ium, we will assume ha bo h quan i ies emain also decoupled. By simila a gumen s, in Eq. (3.6) we ha e al so pe o med he ollowing app oxi ma ions (R (nz(n· R)))( )R (O))""" (R",( )R (O))«nz(n· R)))eq = .c(¡¡)Q(u)(R ( )R (O)), (B3) (n (Rz(n· R)))( )R (O)) '" «n (n· R))( )R (O))(Rz)eq = .c(¡¡)(n (n· R))( )R (O)), (B4) and «n (n . R)(l - (n· R)2))( )R (0)) ....., (n (n . R))( )R (O))(l _ (�. �)4)eq). (B5) «n . R)2)eq In Eq. (3.7), we ha e used 84 APPENDIX B 85 (n�R.,)( )R.,(O)) - (R.,( )R.,(O))(n;)eq = CC(JJ) (1 - 3Q(u)) + Q(u))(R ( )R (O)), JJ (B6) (n nzRz)( )R.,(O)) - (n nz)( )R (O))(Rz)eq = C(JJ)(n.,nz)( )R (O))o (B7) No ice ha he decoupling in he co ela ion (n.,(ñ o R)3)( )R (O)) o Eq. (B5) is (n (ñ o R))( )R (O)) ¡�:::�:l:: and no (n (ñ o R))( )R (O))(ñ o R)2)eq, which leads o di e gencies o he o a ional iscosi y a s nall alues o JJo As ega ds his ac , i is wo hwhile o emphasize ha hese app oxima ions a e mo e accu a e o mode a e and highe alues o pa ame e s JJ and a . This ype o unca ion was al eady p oposed by S a ono ich in he con ex o s ochas ic p ocesses, Bibliog aphy [1] P oceedings 01 he Six h In e na ional Con e ence on Magne ic Fluids, edi ed by V. Cabuil, J.-C. Bac i, and R. Pe zynski (No h-Holland, A ns e da n, 1993). [2] W.T. Co ey, P.J. C egg and Yu.P. Kal nyko , On he Theo y o Debye and Neel Relaxa ion o Single Domain Fe omagne ic Pa iicles, edi ed by 1. P igogine and S.A. Rice, Ad . in Che n. Phys., Vol. 83, (Wiley In e science, New Yo k, 1992), p.263. [3] P.C. Fannin and S.W. Cha les, J. Phys. D: Appl. Phys. 22, 187 (1989); J. Phys. D: Appl. Phys. 24, 76 (1991). [4] J.T. Wald on, Yu.P. Kal nyko and W.T. Co ey, Phys. Re . E 49,3976 (1994). [5] Yu. L. Raikhe and M. I. Shlio nis, Relaxa ion phenomena in Condensed M al e , edi ed by Willia n Co ey, Ad . in Che n. Phys., Vol. 87 (Wiley In e science, New Yo k, 1994) p. 595. [6] V.G. Bash o oy, B.M. Be ko sky and A. N. Vislo ich, In oduc ion o The mo mechanics o Magne ic Fluids (Sp inge -Ve lag, Be lin, 1988). [7] T. L. Gilbe , Phys. Re . 100, 1243 (1955). [8] M.C. Miguel, J. Bone A alos , A. Pé ez-Mad id and J.M. Rubí, Physica A 193 359 (1993). [9] W. F. B own, Phys. Re . 130, 1667 (1963). [10] M. A. Ma senyuk, Yu. L. Raikhe and M. I. Shlio nis, SOy. Phys. JETP 38,413 (1974). [11] Yu. L. Raikhe and M. J. Shlio nis, SOy. Phys. J. Appl. Mech. Tech. Phys, 15, 470 (1974). 86 BIBLIOGRAPHY 87 [12] M.I. Shliomis and V.I. S epano , J. Magn. Magn. Ma e . 122 196 (1993). [13] J.R. Calde wood and W.T. Co ey, P oc. Roy. Soco A 356 269 (1977). [14] N.G. an Kampen, S ochas ic P ocesses in Physics and Chemis y, (No h Holland, Ams e dam, 1992). [15] M.1. Shliomis, SOy. Phys. JETP 34 1291 (1972). [16] M. Doi and S.F. Edwa ds, The Theo y 01 Po/yme Dynamics, (Cla endon P ess, Ox o d, 1986). [17] J.P. McTague, J. Chem. Phys. 51133 (1969). [18] F. Pe in, J. Phys. Radium 533 (1934). [19] J.-C. Bac i, K. Dje i, S. Ne eu and R. Pe zynski, J. Magn. Magn. Ma e . 123 67 (1993). [20] J.-C. Bac i, J. Dumas, D. Go se, R. Pe zynski and D. Salin, J. Physique Le . 46 L-1l99 (1985). [21] M.C. Miguel, J.M. Rubí and A. Pé ez-Mad id, Physica A 20324 (1994). [22] J.-C. Bac i and R. Pe zynski,P oceedings 01 he XII Silges Con e ence, edi ed by L. Ga ido (Sp inge - Ve lag, Be lin, 1993). [23] Yu.L. Raikhe and M.I. Shliomis, SOy. Phys. JETP (Engl. ansl.) 40 526 (1974). [24] P.C. Fannin, B.K.P. Scai e and S.W. Cha les, J. Magn. Magn. Ma e . 122 159 (1993). [25] M.1. Shliomis and V.I. S epano , J. Magn. Magn. Ma e . 122 176 (1993); Relax a ion phenomena in Condensed Mai e , edi ed by William Co ey, Ad . in Chem. Phys., Vol. 87 (Wiley In e science, New Yo k, 1994) p.I. CHAPTER III DYNAMICS OF MAGNETIC HOLES DISPERSED IN A FERROFLUID Once we ha e cha ac e ized he mac oscopic beha io o a e o luid, gi ing exp essions o he iscosi ies and, co espondingly, o bo h he symme ic and an isymme ic pa s o he p essu e enso , as well as o he a e age o i s magne iza ion, in his chap e we analyze he dynamics o a nonmagne ic pa icle (magne ic hole) suspended in a e o luid. The o a ional dynamics o he pa icle is s ongly in luenced by he p esence o a o a ing magne ic ield. As a esul , we ha e ound ha he hole o a es in he opposi e di ec ion o ha o he ield. Ou analysis is alid a low and mode a e equencies o he ield o a ion and may be compa ed o ecen expe imen s ob aining qui e a good ag eemen in he equency ange we a e conside ing, in which a linea ela ionship be ween bo h he o a ion equency o he ield and he angula eloci y o he pa icle is ound. In addi ion, he dependence o he equency o he pa icle on he magne ic ield s eng h is also compa ed o he expe imen s. Fo a sligh ly g ea e concen a ion o holes, we also de e mine he Ro ne-P age and Oseen equi alen enso s, as he i s s eps in he cha ac e iza ion o hyd odynamic in e ac ions be ween he nonmagne ic pa icles in he e o luid. 88 1. In oduc ion 89 1 In oduc ion Recen ly, an inc easing in e es in he s udy o he dynamic p ope ies o he so called magne ic holes, which a e colloidal nonmagne ic pa icles dispe sed in a ca ie magne ic luid, has a isen [1]-[4]. Al hough he pa icles a e no magne ic, when hey a e suspended in a ca ie e o luid hey acqui e an induced magne ic momen equal o he magne ic momen o he e o luid olume hey displace. The in e ac ion o hese induced magne ic momen s o he holes causes a numbe o peculia phenomena, such as he o de -diso de ansi ion in magne ic hole la ices [5, 6] and he non linea phenomena obse ed in assemblies o holes [7]. Fu he mo e, knowledge o he dynamics o such pa icles may cons i u e a way o cha ac e izing he anspo p ope ies o he e o luid. Fo example, he ic ion coe icien o he pa icle gi es us in o ma ion abou he iscosi y o he ca ie luid. The sys em holes- e o luid can be modeled as a suspension o pa icles (holes) in a ca ie luid ( e o luid). This simpli ica ion can be accomplished when di e en leng h scales exis o he e o luid and he holes. Consequen ly, he e o luid can be iewed as a con inuous medium h ough which he holes may mo e. The dynamics o he e o luid is go e ned, a he con inuum le el, by a gene alized Na ie -S okes equa ion. The o a ions o he e omagne ic pa icles lead o he p esence o an an isymme ic pa o he p essu e enso gi ing ise o he appea ance o a new anspo coe icien : he o a ional iscosi y [8, 9]. This coe icien en e s he exp ession o he e ec i e iscosi y o he suspension [10]. The pu pose o his chap e is o analyze he ansla ional and o a ional dynamics o a magne ic hole imme sed in a e o luid unde he in luence o a o a ing magne ic ield. In pa icula , we a e in e es ed in explaining he phenomenon o he o a ion o he hole in he opposi e di ec ion o ha o he o a ing ield, as has been obse ed in ecen expe imen s. As we will see, he exp ession o he angula eloci y o he hole comes om i s co esponding o a ional equa ion o mo ion in ol ing he hyd odynamic o que exe ed by he luid on he pa icle, which is calcula ed in a nons a iona y si ua ion. Fu he mo e, we in end o cha ac e ize he hyd odynamic in e ac ions be ween he non-magne ic pa icles in a e o luid by gi ing exp essions o he co espond ing Oseen and Ro ne-P age equi alen enso s. In pa icula , he Oseen enso o a simple luid is well known o being he simples app oxima ion when compu ing hyd odynamic in e ac ions, i.e. when he suspension is dilu e enough ha he pa  icles a e loca ed a a ela i e la ge dis an ce om each o he . This means ha he 90 CHAPTER III. DYNAMICS OF MAGNETIC HOLES suspension o non-magne ic pa icles unde conside a ion is dilu e enough o a oid agg ega ion phenomena among he holes, bu a he same ime, he e is a su icien numbe o pa icles ha makes i necessa y o ake in o accoun hyd odynamic in e ac ions be ween pai s o sphe es. As an example o unde s and he in luence o hyd odynamic in e ac ions on he dynamics o he magne ic holes, we will also s udy hei sedimen a ion in he p esence o an ex e nal magne ic ield. Wi h his goal in mind, we ha e dis ibu ed he chap e in he ollowing way. In Sec ion 2, we o mula e he basic equa ions o he whole sys em in he con inuum app oxima ion. Unde his app oxima ion, he magne ic hole may be iewed as a mac oscopic pa icle mo ing h ough a con inuum medium wi h in e nal deg ees o eedom. In Sec ion 3, we analyze he mo ion o he hole un il ob aining explici exp essions o he o ce and o que exe ed on he pa icle. F om hese equa ions, we hen de i e he ansla ional and o a ional ic ion enso s. Mo eo e , om he analysis o he o a ional mo ion, we in e a linea law ela ing he angula eloci y o he hole and he o a ion equency o he ield. These esul s a e compa ed o ex pe imen s in Sec ion 4. In Sec ion 5 we s udy he hyd odynamic in e ac ions be ween pai s o magne ic holes. We de i e he exp essions o he Oseen and Ro ne-P age equi alen enso s and we s udy he sedimen a ion o wo holes in he p esence o an ex e nal magne ic ield o di e en con igu a ions in o de o show he in luence o such in e ac ion. Finally, in he las sec ion, we summa ize ou main esul s. 2 Basic equa ions in he con inuum app oxima ion Le us conside a dilu e suspension o sphe ical magne ic holes o adius a in a e o luid unde he in luence o an ex e nal magne ic ield, which o a es wi h he cons an equency WO' In he con inuum app oach, he e o luid (unde s ood as a suspension o small magne ic pa icles in a non-pola sol en ) is assumed o be a con inuum medium wi h a new hyd odynamic ield: he spin o mean angula eloci y o he olume elemen s o he con inuum [8]. This app oxima ion is jus i ied unde he mild equi emen ha he size o he holes ( ipically abou 10-3 -1O-4cm) be much la ge han ha o he e omagne ic pa icles (l0-6cm). The magne ic ield o a ion equencies a e aken low enough o conside he qua sis a iona y limi in which he de i a i es o he hyd odynamic ields, al hough ime dependen , may be neglec ed. Unde his app oxima ion, he equa ions o mo ion a e (2.1) 2. Basie equa ions in he eon inuum app oxima ion 91 0= 2ña + M x ¡¡ + T', (2.2) whe e p( , ), ( , ), and ñ(a)( , ) == -1/2E : II(a)( , ) a e he p essu e, eloci y, and he axial ec o ela ed o he an isymme ic pa o he p essu e enso , II(a), e spec i ely. Mo eo e , E is he Le i-Ci i a enso , M( , ) is he a e age magne iza ion densi y, and ¡¡ = H ¡ is he ex e nal magne ic ield. The appea ance o an isym me ic s esses in he e o luid comes om he di e ence be ween he mean angula eloci y and he o ici y o he low, 1/2"V x . In addi ion, we ha e in oduced he induced o ce and o que densi ies, i( , ) and :¡:i( , ), which o igina e om he pe u ba ion caused by he mo ion o he holeo The shea iscosi y '1, acco ding o Eins ein's law is 5 T/ = T/o(l + '24». (2.3) This exp ession holds up o linea o de in he olume ac ion o magne ic pa icles 4> = 411V IV, whe e b is he adius o one e omagne ic monodomain, and V is he olume occupied by he sys em. We ha e also de ined T/o as he iscosi y o he ca  ie luid. Fu he mo e, hese equa ions a e complemen ed wi h he incomp essibili y condi ion "V . = O . The induced o ce and o que densi y ields, i( , ) and :¡:i( , ) a e in oduced so ha he equa ions o mo ion o he luid, gi en by eqs. (2.1) and (2.2), a e also alid o he poin s inside he sphe e [11, 12]. This conside a ion imposes he equi emen i( , ) = T'( , ) = O, o ¡ - Rcm( )1 > a. (2.4) Addi ionalIy, hey mus be chosen in such a way o he eloci y and he p essu e ields o sa is y ( , ) = i( ) + ñ( ) x añ, o ¡ - Rcm( )1 = a, (2.5) p( , ) = O, o ¡ - Rcm( )1 < a, (2.6) and he magne iza ion o he hole M( , ) = O, o ¡ - Rcm( )1 � a. (2.7) In hese equa ions, we ha e in oduced i and ñ as he ansla ional and o a ional eloci ies o he pa icle, espec i ely; Rcm( ) as he posi ion o he cen e o mass o he sphe e, and ñ == ( - Rcm( ))/¡ - Rcm( )l. 92 CHAPTER IIl. DYNAMICS OF MAGNETIC HaLES A e conside ing he condi ions (2.4) - (2.7), and o he quasis a iona y case, (ü( ) + ñ( ) x añ) ::= O, we ha e i( , ) = (ñ, )8(Ji- Rcm( )l- a), (2.8) and Í(i, ) = Í( )8(a - Ji - Rcm( )l), (2.9) whe e 8 is he Hea iside s ep unc ion. We will now p oceed o de i e a o mal solu ion in e ms o he induced o ce and o que densi ies wi h he pu pose o ob aining exp essions o he hyd odynamic o ce and o que on he nonmagne ic pa icle as a unc ion o i s eloci y and he eloci y ields in he absence o he sphe e. The p ocedu e we use does no equi e explici knowledge o ei he i(i, ), ?( , ) o he eal eloci y ield (i, ). Equa ions (2.1) and (2.2) can be combined o elimina e he e m p opo ional o he an isymme ic axial ec o , ñ-, in Eq. (2.1). We hen ob ain (2.10) whe e we ha e de ined ji as he combina ion o he induced o ce and o que ields (2.11) To sol e o he eloci y ield om Eq. (2.10) , we need o know he exp ession o he magne iza ion densi y. The p ocedu e used o ob ain his quan i y was in oduced in Re . [13] o he case o a cons an magne ic ield, and was based upon he solu ion o he co esponding Smoluchowski equa ion. In appendix A, we p esen an ex ension o he me hod o o a ing magne ic ields. Ou soIu ion is (2.12) whe e n is he numbe densi y o dipoles, (R) is he a e aged o ien a ion ec o o each magne ic momen , p. = mH /kBT is he Lange in pa ame e , compa ing magne ic and he nal ene gies, wi h m he magne ic momen s eng h, assumed cons an , kB he Bol zmann 's cons an , and D; = k T is he o a ionaI di usion coe icien ; wi h �,. = 81 7Job3 he o a ionaI ic ion coe icien o a e omagne ic pa icle. Addi ionally, C(p.) = co h p. - 1/ u, is he Lange in unc ion and F(p.) == 2+c;i ¡J) is in oduced and ob ained in appendix A. 4. Compa ison wi h expe imen s 99 This exp ession is also simila o he co esponding one ob ained by Selle s and B en ne o g a i a ional dipoles. I is wo h men ioning ha , o he o a ional mo ion, he ime dependen ce o he magne ic ield gi es ise o an ex a e m in he hyd odynamic o que p opo ional o dd1 x H o he quasis a iona y analysis. As we will see, o he ield o a ion equencies unde conside a ion, his e m is esponsible o he coun e - o a ion o he hole in espec o he ield o a ion, as obse ed by Helgesen and Skjel o p [2] and by Popplewell ei al. [15]. The magne ic hole o a ional equa ion o no ion is gi en by (3.2.16) whe e 1 is he momen o ine ia o he pa ícle. Again, since he ine ial e m is negligible, he esul is ha H = O. Fo a e o iuid ini ially a es , whe e í = O, and up o he i s o de in he olume ac ion, q¡, om Eq. (3.2.14) we ob ain - dH - 2( ) = q¡JjF(Jj)( d x H). (3.2.17) In pa icula , i we conside an ex e nal magne ic ield o a ing in he XY plane wi h angula equency Wa, H ( ) = cos wa e -í-sin wa e", he magne ic hole angula eloci y educes o (3.2.18) F om his exp ession, we hen conclude ha he di ec ion o he o a ion o he sphe es is opposi e he ield o a ion. Addi ionally, we ob ain a linea ela ionship be ween he ield equency and he o a ion equency o he sphe es o he ange o ield equencies we a e conside ing in ou analysis, ha is, o equencies which enable us o pe o m a quasis a iona y ea men . Mo eo e , unde hese condi ions, ñ is independen o he size o he sphe es, i only depends on he olume ac ion o magne ic pa icles and on he Lange in pa ame e . 4 Compa ison wi h expe imen s The o a ion o magne ic holes induced by a o a ing magne ic ield has been obse ed in ecen expe imen s [2, 15]. The expe imen al se up consis s o a hin laye o e o iuid, con ined be ween wo glass pla es, in which sphe ical pa icles o polys y ene a e dispe sed. When applying a magne ic ield o a ing in he plane o he pla es, 100 CHAPTER III. DYNAMICS OF MAGNETIC HOLES i was obse ed ha he pa icles o a e in he opposi e di ec ion. The expe imen s we e pe o med o ke osene-based [2] and wa e -based e o luids [2, 15]. The o a ion equency o he hole, ñ, o he ke osene-based e o luid, which showed good signs o homogenei y, was measu ed o di e en samples con aining polys y ene sphe es o a ious sizes. They saw ha ñ was independen o he size o he sphe es, o he size ange hey ook in o conside a ion. Fu he mo e, in he in e media e equency ange, hey obse ed a linea ela ionship be ween he angula eloci y o he sphe e and he angula eloci y o he magne ic ield. In he case o a wa e -based e o luid, i has been cla i ied ha i shows a weak endency o sedimen a ion and agg ega ion phenomena unde he ac ion o he ield [2]. In he same e e ence, he di e en esponse o he magne ic hole when suspended in bo h ypes o e o luids was also emphasized. A his poin , i should be no ed ha he e is no o al ag eemen be ween he esul s o [2] and [15] conce ning such a e o luid. In pa icula , Popplewell and cowo ke s conclude ha pa icles o abou 2a == 70J.lm o a e opposi e he di ec ion o he ield, o any alue o "-'o, whe eas smalle pa icles, 2a == 20J.lm, o a e in he ield di ec ion a ield equencies below 70H z and in he opposi e di ec ion a highe equencies. Howe e , Helgesen and Skjel o p we e no able o ep oduce hese esul s. Addi ionalIy, in bo h pape s he e a e al so expe imen al da a ela ing he holes o a ion equency and he ex e nal ield s eng h a in e media e alues o "-'o' In ega ds o he coupling be ween ansla ional and o a ional mo ion, he ex pe imen al obse a ions co obo a e ou esul s. This can easily be concluded a e inspec ion o he equa ions o he o ce and o que gi en byeqs. (3.1.7) and (3.2.14) . Mo eo e , ou esul gi en h ough Eq. (3.2.18) is in comple e ag eemen wi h he obse a ions ca ied ou o he ke osene-based e o uid in he ange o in e media e ield equencies, which, as we ha e s a ed be o e, enables us o pe o m a quasis a iona y s udy o he sys em. This can be seen in Fig. 1Il.l o a e o uid wi h he same cha ac e is ic as ha conside ed in Re . [2], ha is, o olume ac ion o mag ne ic pa icles P == 0.1 and sa u a ion magne iza ion o 200G. On he o he hand, as a as he heo y is conce ned, we can also indica e ha we ha e ob ained a o a ion equency which does no depend on he iscosi y o he sol en (3.2.18) . Despi e he possibili y o he inhomogenei ies in wa e -based e o luids, men ioned in Re . [2], he dependence o n on he ield s eng h seems o be ep esen ed by a sa u a ion cu e as he one co esponding o he quan i y J.lF(J.l) = J.l- anhJ.l J.l + anh u (4.1 ) 4. Compa íson wí h expe ímen s 101 0.010 0.001 10 100 H (Hz) 1000 Figu e 111.1: Ro a ion equency o a magne ic hole in a ke osene-based e o luid laye e sus he equency H o he o a ing magne ic ield o H = 140e. The di ec ion o o a ion o he sphe es is opposi e o ha o he ield. 102 CHAPTER IIl. DYNAMICS OF MAGNETIC HOLES 0.4 s (Hz) 0.3 0.6 0.5 0.2 0.1 0.0 ...___ '-- __ ----''- __ --' ---I. --' O 20 40 60 80 100 H (G) Figu e 111.2: Va ia ion o pa icle o a ion wi h he ield s eng h. Ro a ion coun e o he ield o a ion. (a = 35¡..¿m, lB = 60Hz) 5. Hyd odynamic in e ac ions be ween pai s o magne ic hoJes 103 in ol ed in ou Eq. (3.2.18). In Fig. 11I.2, we ha e ep esen ed he equency as a unc ion o he imposed ield o a e o luid wi h he same magne iza ion as ha conside ed by Popplewell and cowo ke s. Finally, i is wo h men ioning ha in acco dance wi h Re . [2], we canno ep oduce he dependence o he equency on he size o he pa icles as obse ed in Re . [15]. 5 Hyd odynamic in e ac ions be ween pai s o mag ne ic holes Hyd odynamic in e ac ions among di e en objec s mo ing in a iscous luid ha e been ex ensi ely s udied because o hei e ec s on he dynamics and consequen ly on ce ain p ope ies o he whole dispe sion [17, 18]. The i s s udies o hese in e  ac ions we e unde aken by Smoluchowski using he so-called me hods o e lec ion. Bu , due o hei complexi y, hey ha e only been applied o he case o a pai o pa icles. On he o he hand, he me hod o induced o ces, i s in oduced by Mazu [19] and gene alized in Re . [12] o a sys em o N sphe es mo ing in a simple liquid, pe mi s o calcula e he mobili y enso s o any desi ed o de o app oxima ion as an expansion in he in e se dis an ce be ween he pa icles. In his sec ion, we i s de elop he la e echnique o s udy he hyd odynamic in e ac ions be ween pai s o sphe ical nonmagne ic pa icles mo ing in an aniso opic magne ic liquido 5.1 Oseen and Ro ne-P age enso s o he e o uid We will now conside a suspension o N sphe ical magne ic holes o adii ai, (i = 1, ... , N), in an unbounded incomp essible e o luid unde he in luence o an ex e nal magne ic ield, which may o a e wi h cons an angula eloci y Wo. The suspension unde conside a ion is dilu e enough o a oid agg ega ion due o he in e ac ion o he induced magne ic momen s o he holes. On he o he hand, i con ains a su icien numbe o pa icles so ha hyd odynamic in e ac ions be ween pai s o sphe es mus be conside ed. As in he p e ious sec ions, he e o luid is assumed o be a con inuum medium based on he ac ha he size o he holes is much la ge han ha o he e omagne ic pa icles cons i u ing such a e o luid. Ou s a ing poin o analyzing he quasis a iona y mo ion o he pa icles will be he linea ized equa ions o conse a ion o he linea and angula momen a, which now ead 104 CHAPTER IJI. DYNAMICS OF MAGNETIC HOLES N 0= - 7p+ ¡ 72V - 7 x ü(a) + L j"d, j=1 (5.1.1 ) N 0= 2ña + M x .i1 + L T1nd, i=1 (5.1.2) oge he wi h 0= 7 . V, (5.1.3) whe e we ha e in oduced a se o induced o ces and o ques densi ies, j"d( , ) and T1nd( , ), (j = 1, ... , N) which o igina e om he pe u ba ion caused by he mo ion o he holes. As we ha e explained be o e, he ex ension o he luid eloci y ield inside he pa icles imposes he equi emen j"d( , ) = Tjnd( , ) = O, o ¡ - Rj( )1 > ai and (j = 1, ... , N). Ri( ) gi es he posi ion o he sphe e cen e a ime . Fu he mo e, he se o equa ions including induced o ces and o ques mus be comple ely equi alen wi h he o iginal bounda y alue p oblem e , ) = uj( ) + nj( ) x ( - Rj( », o ¡ - Rj( )1 � aj, (5.1.4) i.e., s ick bounda y condi ions a he su aces o he sphe es. Simila ly, o he hyd o s a ic p essu e we impose he condi ion p( , ) = O, o ¡ - Ri( )1 < ai' (5.1.5 ) and o he magne iza ion (5.1.6) whe e we ha e in oduced iij and ñi as he ansla ional and o a ional eloci ies o he pa icle i. espec i ely. Consequen ly, conside ing hese condi ions and o he quasis a iona y case, all he induced o ce and o que densi ies mus be o he same o m as in Sec ion 2. I we wan o conside a o a ing magne ic ield, i s angula equency should be low enough o conside he quasis a iona y limi in which he de i a i es o he hyd odynamic ields, al hough ime dependen , may be neglec ed. In appendix A we ob ain he exp ession o he magne iza ion densi y. Subs i u ing his esul in Eq. (5.1.2), and combining eqs. (5.1.1) and (5.1.2) we can w i e he ollowing equa ion 5. Hyd odynamic in e ac ions be ween pai s oi magne ic holes 105 N 0= - 7p + (7] + 7] ) 72V' + 7] 7 X h(Ji . 7 X V) + 'L . jnd, i=l whe e 7] is he o a ional iscosi y, and whe e we ha e de ined : d as he combina ion o he induced o ce and o que ield densi ies (5.1.7) (5.1.8) In o de o sal e o mally he equa ion o mo ion o he luid (5.1.7), we in oduce Fou ie ans o ms o he eloci y ield V'. F om now on, we wíll omi he explici dependence on ime o he di e en ields. We also de ine he Fou ie ans o m o he induced o ce densi y : in a e e ence ame in which sphe e j is a es a he o igin (5.1.9) A e Fou ie ans o ming his equa ion and applying he ans e sal p ojec o (1 - H:), wi h k == k/k, one a i es a he o mal solu ion o he ans e sal componen o he eloci y ield N (k) = T(k, ) . 'L exp( -ik . Ri ): jnd(k), i=l whe e, as in Sec ion 2, we ha e in oduced he p opaga o (2.16). As we ha e indica ed be o e, he sphe es a e allowed o mo e wi h a bi a y e loci y h ough he luid, which may i sel be in a bi a y s a iona y non-uni o m un- (5.1.10) pe u bed low. Fo he sake o simplici y, we will conside ha he unpe u bed luid is a es , and we wíll s udy he hyd odynamic in e ac ions which a e se up be ween he sphe es when hey mo e. In pa icula ou main goal is o calcula e he ansla ional mobili y up o a gi en o de o app oxima ion in a se ies expansion in powe s o in e se dis ances be ween he sphe es, when only hyd odynamic in e ac ions be ween wo sphe es con ibu e. The o mal solu ion can be ew i en in eal space N ( , )= 'Lj d 'T( - ,).:Fjnd( '). i=l In pa icula , o hose poin s on he su ace o he sphe e i, and a e aking in o (5.1.11) conside a ion he p ope ies o he gene alized induced o ce, we ha e 106 CHAPTER JII. DYNAMICS OF MAGNETIC HOLES - N J - ñ· X ¡!nd ii(Ri +a¡ñ¡) = � dñj¡.J.(ñ¡,ñj)' [/ d(ñj) - ) 2 J ], 3=1 whe e we ha e in oduced he esponse unc ion (5.1.12) (5.1.13) wi h he de ini ion R;j = R; - Rj. Fo he pu pose o e alua ing hyd odynamic in e ac ions, i is con enien o in oduce i educible induced o ce mul ipoles, de ined in e ms o he su ace induced o ces acco ding o 1 J A =ind .....{1+1)=_ dñ, A�[/,!nd(A.)_n¡XT¡ ] : i 471' nI nI I nI 2' (5.1.14) whe e ñ: is an i educible enso o ank 1, aceless and symme ic in any pai o i s indices. The expansion in e ms o i educible o ce mul ipoles, which is w i en in a coo dina e ee way, is equi alen o an expansion in sphe ical ha monics, o which i can be educed i pola coo dina es a e used. We also in oduce i educible su ace momen s o he luid eloci y ield, and we ob ain a ela ionship be ween he induced o ce mul ipoles and he su ace momen s o he luid eloci y ield h ough a hie a chy o equa ions. I is his hie a chy which will hen enable us o ob ain exp essions o he mobili y enso s which ela e he o ces and o ques on he sphe es o hei lineal and angula eloci ies. The gene al ela ionship be ween he (1 + 1) o de mul ipoles o he quan i ies de ined o e he su ace o he sphe e is ound o be, (1+1) _ � � (2/' + 1)!! (1+1,1'+1) �I'+l) V¡ - L... L... 1" ¡.J.¡j 8 j , j=ll'=O . (5.1.15) which in ol es he enso s (5.1.16) As we al eady poin ed ou in Sec ion 3, he i s o de i educible nul ipole will allow us o de e mine he ansla ional mobili y. Thus, ou s a ing poin will be he ela ionship be ween eloci y and o ce mul ipoles which comes om Eq. (5.1.15) o 1 = O (1) _ � � (2/' + 1)!! (1,/'+1) �I'+l) V¡ - � L... I'! ¡.J.¡j 8 j , )=1/'=0 (5.1.17) 5. Hyd odynamíc ín e ac íons be ween paí s o magne íc hoJes 107 whe e he mul ipole P) is P) = 411 ' J dn¡ (ü¡ + ñ¡ x a¡n¡) = Ü¡. (5.1.18) Mo eo e , using he ac ha he d ag o ce exe ed by he luid on he pa icle j (w i en in e ms o he induced o ce) is (5.1.19) i ollows ha �1) = _!_ Jdn.( lnd _ nj x J) = _ FjH . J 41 ' J J 2 41 'a� J (5.1.20) In Sec ion 3, we al eady ga e he exac o m o his ela ionship o he case i = j. We ob ained, 5 3 .. -61 '7]oa¡{[1 + 24> + ¡4>¡. F(¡. )](l - H H) 5 3 .. + [1 + 24> + 24>¡. F(¡. )]H H} . Ü¡, (5.1.21) Now, we willlook a he case i j. In his case, he enso s ¡. g+I,II+1) a e no longe diagonal in hei uppe indices, so ha he e is a coupling be ween di e en mul ipoles. Bu , i we se l = O, we will es i- ou sel es o he lowes alue o l' (1' = O), because beyond ha , we will ob ain con ibu ions p opo ional o (al R)P wi h p 2:: 4 [12], negligible o he concen a ion o holes we a e conside ing. The co esponding ma ix is �U,l) = (;!2) J dk J dk exp(ik. R;j) h-1(k). (1 - kk)jo(ka¡)jo(kaj). (5.1.22) whe e jo(x) is he sphe ical Bessel unc ion o ze o o de . In appendix e we ob ain ha (1,1 ) �¡j aJ 5 3 A• 27]oR;j ([1-"24> - "24>¡. F(¡. )](l + R;jR;j) 3 AA A A A A + 24>¡. F(¡. )[1 - H H - (R;j x H)(R;j x H)]} aJ(a¡+aJ) 5 3 l· A 27]oR�j {[1-"24> - "24>¡. F(¡. )](-31 - R;jR;j) 3 1 'A A A A A "24>¡. F(¡. )[3(1 - H H) - in; x H)(R;j x H)]), (5.1.23) + 108 CHAPTER lII. DYNAMICS OF MAGNETIC HaLES om which he ansla ional mobili y enso comes up s aigh o wa dly, a e aking in o accoun eqs. (5.1.17) - (5.1.20) RP 1 53 AA 1''' = {[1- - /¡ - - /¡pF(p)](1 + R;.R;.) IJ 81 7]0 R;j 22 J J 3 AA AA A A + 2 /¡pF(p)[1 - H H - (R;j x H)(R;j x H)]} (a; + aJ) 5 3 1 A' + 3 ([l- - /¡ - -lPpF(p)](-1 - R;jR;j) 81 7]oRij 2 2 3 3 1 A A • •• • 2 /¡pF(p)[3(1 - H H) - (R;j x H)(R;j x H)]), (5.1.24) This mobili y enso ep esen s he eloci y o a sphe e i wi h espec o he luid eloci y due o he mo ion o he pa icle i. and pe uni o o ce exe ed on sphe e i. I is alid up o he i s o de in he olume ac ion o magne ic pa icles, /¡, and up o hi d o de in he a io (al R;j), o any alue o he Lange in pa ame e p. F om his exp ession, we can also ob ain he co esponding Oseen enso o he e o luid. The Oseen enso is he i s app oxima ion when compu ing hyd odynamic e ec s. Ne e heless, i is a alid app oxima ion when he pa icles a e a apa enough, and we can assume ha pa icle j gene a es he same pe u ba ion as would be p oduced by a poin o ce si ua ed a i s cen e . Thus, he Oseen enso is ob ained a e neglec ing he hi d powe o (al R;j), o I'ij 1 {[1- � /¡ - � /¡pF(p)](1 + Ri·Ri'} 81 7]0 R;j 2 2 J } 3 . + 2 /¡pF(p)[1 - H H - (R;j X H)(Rij x Hm, (5.1.25) Ob iously, i we se he olume ac ion o magne ic pa icles equal o ze o, /¡ = O, his Oseen enso educes o he well-known exp ession 1 . . I'P)· = 8 R; (1 + R;jR;j), 1 7]0 'j which is he p opaga o o he pe u ba ions in a simple luid. (5.1.26) 5.2 Sedimen a ion o wo magne ic holes in a e o luid To illus a e he in luence o hyd odynamic in e ac ions on he suspension o magne ic holes, we will s udy he sedimen a ion in he p esence o an ex e nal magne ic ield. Appendix B Explici de i a ion o Eq. (5.1.24) S a ing om Eq. (5.1.22) we can make an expansion such ha I-'g,l) = (;!2) J dk J dk exp(ik. Hij) h-1(k). (1 - kk) (1 _ (a? ; aJ) k2 + O(k4») . (B2) In a e e ence ame in which he ez-axis is pa allel o he uni a y ec o �i (H = Rii�j), we may w i e dk = d�ijd<Pii, whe e �ij = Rij . k is he cosine o he pola angle be ween �j and k, and �ij is he azimu al angle. (1,1) I-'ij (2�2) {11 d�ij 121( d�ij 100 dk exp(ikR;i�ii) h-1(k). (1 - kk) (1- (a?; aJ) e + O(k4») . (B3) Making use o he iden i y 1 ¡1 loo . ¡1 dq - dxxP dyyqe,: :y = iq dxxP-8(x) = 8pqp!(-1)P, 211" -1 -00 -1 dxq (B4) i hen ollows ha he e ms o o de k4 and highe o he expansion in he in eg and o Eq. (B3) gi e anishing con ibu ions upon in eg a ion, since any elemen o he enso h-1(k). (1 - kk) will be, a e in eg a ion o e �ij, a polynomial in �ij whose highes o de is p opo ional o ( i [12], (B5) whe e 1 1 ( 5 3 ) el = -- '" - 1 - -� - -�¡.JF(¡.J) , 71 + ¡ 710 2 2 (B6) and 115 116 APPENDIX B 'I 1 3 ) C2 = ...., --l/JJJF(JJ . '1( '1 + 'I ) '10 2 Thus, we can ew i e Eq. (B3) in he ollowing o m (B7) 2 JI 1211" ( 2 + 2) �2 ) (1,1) aj -1 '" ai aj u lJij =(21T)R.;j _ld{¡j o dl/Jijh (k)·(I-kk) 6(�ij)+ 6R j {)2�ij6(�¡j) . (B8) Fo he sake o simplici y, we de ine (B9) wi h GU,l) = (;!) ¡11 d{ij 1211" dl/Jij ( Cl (1 - H) + C2(k x il)(k x il») 6(�ij), and (BlO) 2( 2 + 2) JI 1211" J::l2 (1,1) aj ai aj (". " ') u Hij = (121T) -1 d�ij o d�ij C1(1 - kk) + C2(k x H)(k x H) {)2�ij 6(�ij). (B11) A e pe o ming he in eg als appea ing in eqs. (B10) and (B11), hey educe o 2 d;],l) = a� (Cl(l + R;Jl-ij) + C2[1 - il il - (R;j x il)(R;j x il)l) , (B12) The way o sol ing he in eg al in Eq. (B10) is he ollowing whe e we ha e in oduced he cons an s (91, ... ,94) which can be ound by con ac ing he enso s in bo h si des o he equa ion, i s ly wi h he iden i y ma ix, 1, and sec ondly wi h he enso R;j R;j. In doing so, and a e calcula ing he simple scala in e g als esul ing om hese con ac ions, we a i e a a de e mined sys em o equa ions o he aboye in oduced cons an s. The solu ions a e 91 = 92 = 93 = -94 = aJ /2. APPENDIX B 117 The in eg al in Eq. (B11) can be ca ied ou ollowing exac ly he same p ocedu e. Finally, once we ha e subs i u ed he cons an s Cl and C2 gi en h ough eqs. (B6) and (B7), Eq. (B9) u ns ou o be Eq. (5.1.23). Bibliog aphy [1] H.S. Selle s and H. B enne , PCH PhysicoChemical Hyd odynamics 11, (1989) 455. [2] G. Helgesen and A.T. Skjel o p, J. Magn. Magn. Ma . 97, (1991) 25. [3] G. Helgesen and A.T. Skjel o p, J. Appl. Phys. 69 (12), (1991) 8277. [4] P. Da ies, J. Popplewell, G. Ma in, A. B adbu y and R.W. Chan ell, J. Phys. D, Appl. Phys. 19, (1986) 469. [5] A.T. Skjel o p, Phys. Re . Le . 51, (1983) 2306; J. Magn. Magn. Ma . 65, (1987) 195; J. Appl. Phys. 57 (1), (1985) 3285. [6] G. Helgesen and A.T. Skjel o p, Physica A 170, (1991) 488; Physica A 176, (1991) 37. [7] G. Helgesen, P. Pie anski and A.T. Skjel o p, Phys. Re . Le . 64, (1990) 1425; Phys. Re . A 42, (1990) 7271. [8] H. B enne , J. Colloid In e ace Sci. 1, (1970) 141; Ann. Re . Fluid Mech. 2, (1970) 137; In . J. Engng. Sci. 22, (1984) 645. [9] V.G. Bash o oy, B.M. Be ko sky and A. N. Vislo ich, In oduc ion io The mo mechanics 01 Magne ic Fluids (Sp inge -Ve lag, Be lin, 1988). [10] J. P. Mc Tague, J. Chem. Phys. 51, (1969) 133. [11] P. Mazu and D. Bedeaux, Physica 76, (1974) 235. [12] P. Mazu and W. Van Saa loos, Physica 115 A, (1982) 21. [13] J.M. Rubí and M.C. Miguel, Physica A 194, (1993) 209. 118 BIBLIOGRAPHY 119 [14] M.C. Miguel, J. Bone A ales, A. Pé ez-Mad id and J.M. Rubí, Physica A 193, (1993) 359. [15] J. Popplewell, R.E. Rosensweig and R.J. Johns on, IEEE T ans. Magn. 26, (1990) 1852. [16] M. A. Ma senyuk, Yu. L. Raikhe and M. 1. Shliomis, SOy. Phys. JETP 38, (1974) 413. [17] W.B. Russel, n.A. Sa ille, W.R. Schowal e , Col/oidal Dispe sions, (Camb idge Uni e si y P ess, Camb idge, 1989). [18] J. Happel and H. B enne , Low Reynolds Numbe Hyd odynamics (Kluwe , The Ne he lands, 1991) [19] P. Mazu , Physica HOA, (1982) 128. CHAPTERIV AGGREGATION PHENOMENA This chap e is in ended o be an in oduc o y analysis o he agg ega ion phenomena ha a ise o ins ance, as he size o he pa icles in a e o uid inc eases o a highe concen a ions. In such condi ions he dipole-dipole i e ac ion be ween he pa icles is enhanced and B ownian mo ion is no longe able o s abilize he suspension. Mo eo e , in he p esence o a high magne ic ield he pa icles show up a endency o o m chains. We s udy he kine ics o he o ma ion o he agg ega es by means o he Smoluchowski heo y o coagula ion in colloids accoun ing o hyd odynamic in e ac ions. These in e ac ions become ele an o he concen a ion o pa icles which gi e ise o hese phenomena and slow down he agg ega ion p ocess. In addi ion, he heology o he chains ha a e usually obse ed in sys ems wi h dipola in e ac ions is s udied o a a he simpli ied si ua ion in o de o elucida e he e ec s o he dipola magne ic in e ac ions in he con ibu ion o he chains o he p essu e enso o he suspension. 120 1. In oduc ion 121 1 In oduc ion Colloidal dispe sions play an impo an ole in many na u al phenomena as well as in a ious indus ial p ocesses. The s abili y o he suspension agains agg ega ion o he pa icles is o essen ial impo an ce o i s beha io [1]. As long as one can neglec in e pa icle in e ac ions, one does no obse e agg ega ion phenomena and hen he main p oblem is o unde s and how he physical p ope ies o he luid, o example i s iscosi y, a e modi ied due o he p esence o he pa icles. Bu when in e ac ions be ween pa icles s a o be impo an , he p eceding p oblem becomes much mo e complica ed. In his case, he agg ega ion mechanism in a colloidal solu ion is qui e complex because he luid does keep in o an ac i e ole. The na u e o in e -pa icle in e ac ions depends on he p ope ies o he luid, and, conce ning i s d i ing mo ion, hyd odynamic e ec s canno be neglec ed. The concep s o ha d and so pa icles a e commonly used in he li e a u e [2]. A ha d pa icle is a igid sphe e on which only hyd odynamic o ces a e ac ing, hey do no unde go o he in e ac ion o ces excep o an in ini ely s ong epulsion on con ac o p e en in e pene a ion. In con as , a so pa icle is a sphe e subjec o in e ac ions, o he han hyd odynamic ones, el a ini e dis ances. In he las case, one can hink o such sphe es as ha ing a su ace bounda y laye (sphe e o in luence) whose hickness is de e mined by a cha ac e is ic in e ac ion leng h. Thus, o elec o s a ically s abilized dispe sions his hickness can be iden i ied wi h he double-Iaye hickness, o an de Waals in e ac ions wi h he hickness whe e he in e ac ion en e gy becomes compa able o he he mal ene gy, and o s e ically s abilized colloids wi h he leng h o he polyme ails adso bed on he su ace o he suspended pa icles. This idea was in a ce ain way exploi ed by on Smoluchowski [3], who i s analized he p oblem o coagula ion in he absence o any epulsi e ba ie ( apid coagula ion). A sphe e wi h a su ace bounda y laye can be conside ed so because his laye can be de o med and pene a ed du ing he app oach o wo pa icles. Mo eo e , in he li e a u e he e m coagula ion is applied o agg ega ion ha is induced by he an de Waals a ac ion be ween he colloidal pa icles. On he o he hand, he e m loccu la ion is ese ed o polyme -induced agg ega ion. Coagula ion usually gi es ise o compac agg ega es whe eas loccula ion equen ly p oduces mo e open s uc u es. Pa icula ly, he monodomain magne ic pa icles a e usually su icien ly small ha B ownian o ces, along wi h sho - ange s e ic epulsion due o ei he na u al o syn he ic polyme s, domina e in he ze o- ield limi o gua an ee he s abili y o he pa icles. Howe e , when a non e y dilu e suspension o hese pa icles is unde 122 CHAPTER IV. AGGREGATION PHENOMENA he ac ion o an applied magne ic ield, pa icle agg ega ion occu s when he nag ne ic a ac ion be ween he pe manen momen s is s ong enough o ou weigh he s abilizing o ces. Thus he hickness o he bounda y laye can be iden i ied wi h he hickness o which he magne ic in e ac ion ene gy is compa able o he he mal ene gy a he han wi h he leng h o he polyme , due o he long ange cha ac e o such in e ac ion. In he la e 1930's, Winslow [4] obse ed in e es ing phenomena when dielec ic pa icles suspended in oil we e subjec o an elec ic ield. He saw he induced o ma ion o chains o pa icles aligned wi h he elec ic ield and, e en o mo e p ac ical impo an ce, ha he e ec i e iscosi y o he suspension could be a ied by o de s o magni ude by modi ying he applied elec ic ield. Howe e , s udies on elec o he ological luids a e hinde ed by many p oblems ela ed o su ace cha ge, elec o de pola iza ion, adso bed wa e , ield inhomogenei ies, e c. An analogous ield-induced beha io is shown by magne o heological luids, e.g., in a suspension o magne izable pa amagne ic pa icles in a nonmagne ic luid [5] o in a suspension o nonmagne iz able sphe es in a e o luid (magne ic holes, see p e ious chap e ). In apu e e o luid one obse es simila phenomena bu , due o ac ha he pa icles a e pe manen ly magne ized, he e a e pola iza ion o ces e en wi hou an applied magne ic ield. In all hese cases, one has he ad an age ha hey a e no suscep ible o he aboye men ioned p oblems. Thus, s udying he kine ics o he o ma ion o he agg ega es and hei heology a e opics o p ac ical impo ance and has p o ided he bases o heo e ical and expe imen al s udies [6]-[8] as well as nume ical simula ions [9]-[11]. Ha ing all hese in mind, he s uc u e o his chap e is as ollows, In Sec ion 2 we discuss he in luence o hyd odynamic in e ac ion in he kine ics o agg ega es o ma ion by using he Smoluchowski's heo y o coagula ion. We w i e he co esponding kine ic di e en ial equa ions gi ing he ime e olu ion o he dynamic clus e size dis ibu ion unc ion and sol e hem o a a he simpli ied case. In his sec ion we also discuss he scaling beha io o he clus e size dis ibu ion. Sec ion 3 is in ended o be an in oduc o y analysis o he heology o he chains which commonly appea when he agg ega ion p ocess occu s unde he ac ion o an ex e nal ield. ( al a) The conclusions a e summed up in he las sec ion. 2. Hyd odynamic in e ac ions in he Smoluchowski heo y o coagula ion 123 2 Hyd odynamic in e ac ions in he Smoluchowski heo y o coagula ion The in es iga ion o agg ega ion p ocesses by means o compu e simula ions ha e been enhanced du ing he las 10 yea s a e he pionee ing wo k o Wi en and Sande whe e hey p oposed hei di usion-limi ed agg ega ion model e e ed o as DLA [12]. This wo k s imula ed he de elopmen o he di usion-limi ed clus e -clus e agg ega ion model by Meakin [13] and Kolb e al. [14], which, pa icula ly, p o ides a be e unde s anding o s uc u al and kine ic aspec s o colloid agg ega ion. The coagula ion p ocess can also be discussed in e ms o he Smoluchowski's heo y which, as we will see below, due o he na u e o he app oxima ion inhe en in i s de i a ion is a mean- ield heo y. Ne e heless, his heo y is expec ed o hold o dimensions g ea e han he uppe c i ical dimension aboye which luc ua ions become i ele an and ha o his coagula ion p ocesses is 2. Smoluchowski i s ound ou a e y in e es ing applica ion o he heo y o B ow nian mo ion in he coagula ion exhibi ed by colloidal pa icles when an elec oly e is added o he solu ion. His heo y is based on he sugges ion o Zsigmondy ha coag ula ion is a consequence o he exis en ce o a sphe e 01 in iuence o a ce ain adius R su ounding each colloidal pa icle such ha i s B ownian mo ion emains una ec ed unless ano he pa icle en e s wi hin i s sphe e o in luence. When his happens hey s ick o one ano he o o m a single uni . Fo his pa icula p oblem, he sphe es o in luence a e supposed o o igina e in he o ma ion o elec ic double laye s a ound each pa icle. The double pa icle con inues mo ing andomly so long as i does no come wi hin he sphe es o in luence o a single o ano he double pa icle. Then we will ha e he o ma ion o a iple o a quad upole pa icle, and so on. This p o cess wilI e en ualIy lead o he o al coagula ion o alI he colloidal pa icles in o one clus e . Recen eal- ime expe imen s pe o med wi h suspensions o supe pa amagne ic la ex pa icles [15] co obo a e Smoluchowski's hypo heses. In his case, he mag ne ic ield induces chain o ma ion, and one obse es wo agg ega ion ime scales. When pa icles a e a enough away om each o he ha hei dipola in e ac ion is weak ela i e o kBT, B ownian mo ion domina es. When his andom mo ion hap pens o b ing wo pa icles close enough ha hei dipola in e ac ion is o he same o de as kBT, a apid ansi ion be ween andom and ballis ic mo ion occu s and he pa icles agg ega e immedia ely. Mo eo e , nume ical simula ions inco po a ing induced-dipola in e ac ion o ces and B ownian mo ion also co obo a e ha he e is 124 CHAPTER IV. AGGREGATION PHENOMENA a ci ical alue o he in e pa icle sepa a ion wi hin which pa icle mo ion ceases o be andom. Thus, each pa icle can be hough o as ha ing a cap u e olume de ined by equa ing dipola in e ac ion ene gy o he mal ene gy. I one pa icle en e s his olume, he wo pa icles unde go ballis ic agg ega ion. On he o he hand, he appli cabili y o he Smoluchowski equa ion is limi ed o low clus e concen a ions because i assumes ha he e a e only bina y collisions. We a e in e es ed in de e mining he concen a ions nI, n2, ... , o single, double, iple, quad upole, e c., pa icles a ime gi en ha a ime = O he e a e no single pa icles. Mo eo e , once we know hese concen a ions we can w i e he scaling laws which show how hese quan i ies beha e o long imes o o la ge adius o he sphe es o in luence. Bu , in his sec ion we will in oduce a modi ica ion o he simple heo y p oposed by Smoluchowski by conside ing hyd odynamic e ec s, Fo dis an ces g ea e o o he same o de o magni ude as he in e pa icle sepa a ion h eshold (- 2a - 7a, wi h a he adius o one pa icle), hyd odynamic in e ac ions should be aken in o accoun when s udying he di usi e mo ion o he pa icles. The e a e p e ious a emps o inco po a e his e ec s in he inal s ages o he app oach o wo non- nagne ic pa icles (i.e. when jus sho ange in e ac ions be ween he pa icles a e conside ed) when, because i is di icul o he emaining ilm o Iiquid o escape, he p ocess is clea ly slowed down. Honig e al. [16] o ins ance examined he p oblem de i ing an app oxima e ela ion o a modi ied di usion coe icien : D(h) 1 + 2aj3h ------�- = --------�------- D(h -+ 00) 1 + 13aj6a + a2 j3h2 (2.1 ) whe e h = - 2a is he dis an ce o closes app oach. This e ec was also es ed expe imen ally by Lich enbel el al. [17] who ound ha he a e o coagula ion was educed o less han hal o he Smoluchowski alue o hose colloidal suspensions. Howe e , when dealing wi h long ange in e ac ions, we belie e ha i would be enough o in oduce hyd odynamic in e ac ions ep esen ed by he Oseen and Ro ne P age enso s discussed in he p e ious chap e . This will enable us o ind analy ic exp essions no only o he a e o coagula ion bu also o he di e en concen a ions o clus e s nI, n2, ... 2.1 Kine ic equa ion o i e e sible agg ega ion Fo he ime in e als we a e in e es ed in he mo ion o he pa icles is di usi e, i.e., local luc ua ions in he concen a ion due o B ownian mo ion de e mine a global di usion om he highe o he lowe concen a ed egions. We will s a conside ing 2. Hyd odynamic in e ac ions in he Smoluchowski heo y o coagula ion 131 - Fo la ge alues o R, i.e., o a long ange in e ac ion, and p o ided ha 3a/2R < 1 L will be also la ge so ha we can ew i e Eq. (2.2.15) as ollows no ( 1 )-(1:+1) no ( k )2 _ . nI: = (noL )2 1 + noL :::: k2 no L e;;;n, o , equi alen ly (2.3.1) nI: ex k-T (;'1) , whe e we ound ou he cha ac e is ic exponen s T = 2 and = 1. - On he o he hand, o su icien ly la ge imes (2.3.2) nI: = no (1 + _1_) -(1:+1) ...., no (_k_) 2 e--;;!n (noL )2 noL - k2 noL ' om which i is possible o w i e he size dis ibu ion unc ion unde he ollowing (2.3.3) scaling o n (2.3.4) whose cha ac e is ic exponen s a e T = 2 and z = 1. Vicsek and Family [21] and independen ly Kolb [22] in oduced a dynamic scaling desc ip ion o he clus e size dis ibu ion in he clus e -clus e agg ega ion model. Mon e Ca Io simula ions showed ha a dynamic scaling o he aboye in oduced o m (2.3.5) ep esen ed well he beha io o nl:( ). He e (x) is a scaling unc ion which depends on he dimension and on he clus e mobili y. The exponen z has been measu ed bo h expe imen ally and by Mon e Ca Io simula ions o di usion-limi ed clus e -clus e agg ega ion. The expe imen al esul a e gene ally in ag eemen wi h he simula ions and, a he same ime, he Mo e Ca lo esul s ag ee wi h he Smoluchowski app oach in d = 2,3, bu disag ee in one dimensiono Tha was consis en wi h he obse a ion ha Smoluchowski's heo y, wi hou he inco po a ion o spa ial luc ua ions, ails below a c i ical dimension de = 2. As an applica ion o he scaling laws, le us look a he case o magne ic pa icles wi h dipola in e ac ions, when we neglec hyd odynamic e ec s, i.e. o la ge alues o R. In his case, as we al eady poin ed ou in he in oduc ion, we can es ima e he alue o R in he ollowing way 132 CHAPTER IV. AGGREGATION PHENOMENA (2.3.6) which ollows by equa ing bo h magne ic and he nal ene gies. Consequen ly, (2.3.7) so ha he concen a ion o clus e s o k pa icles scales wi h he pa icle's nagne ic momen (2.3.8) wi h cha ac e is ic exponen s T = 2 y 'Y = 2/3. Janssen el al. [11] used a nume ical app oach o sol e he basic equa ion o loccula ion, conside ing a cylind ically sym me ic in e ac ion in o de o model he dipola in e ac ion in pa amagne ic pa icles. In pa icula hey in es iga ed he a e o ini ial loccula ion J = 87 DnoR/W, whe e in hei no a ion W s ands o he s abili y ac o which indica es how he in e ac ions modi y he o al lux in compa ison wi h he case o apid agg ega ion, in which no epulsi e o ces a e p esen o slow down he p ocess. They ob ained ha his s abili y ac o depends on he dimensionless pa ame e cha ac e izing he dipola in e ac ion, o in o he wo ds on he magne ic momen o he pa icles, Wex m-2/3, wi h he same cha ac e is ic exponen . This esul was also e i ied expe imen ally a high magne ic ields. Mo eo e , ecen expe imen al in es iga ions on he magne ic- ield-induced chain o ma ion o supe pa amagne ic la ex pa icles [15] show a powe -law dependen ce on ime as can also be ob ained by he Smoluchowski equa ion and h ee-dimensional simula ions o dipola pa icles, Bu hey also ound ha he alue o he exponen z ha e a weak in e se dependen ce on he pa icle olume ac ion and in he di nen sionless cons an cha ac e izing he dipola in e ac ion s eng h. We plan o pu sue wo k on his expe imen al obse a ions in he nea u u e. 3 Chains o magne ic pa icles in an elonga ional low Colloidal pa icles may gi e ise o ei he ixed o ee s uc u e agg ega es depending on he na u e o he agg ega ion p ocesses and he ype o in e pa icle bonds es ab lished [25]. We will conside he case in which he sphe es a e in close con ac bu a e 3. Chains o magne ic pa icles in an elonga ional low 133 Figu e IV.1: Chain o magne ic pa icles in an elonga ional low. All he magne ic momen s owa ds he di ec ion o he high ex e nal ield applied. s ill able o ansla e ela i e o one ano he unde he ac ion o an elonga ional low. I he pa icles a e non-magne ic, his in e nal mo ion is esis ed by hyd odynamic lub ica ion o ces and, al hough he ela i e mo ion is e y small, i has seen o ha e a big e ec on he s ess ansmission. Now, he pa icles in e ac wi h each o he because o he dipola magne ic in e ac ion and we wan o know he e ec o his in e ac ion on he in e nal mo ion and he iscosi y o such a sys em. We de ine he ollowing dimensionless pa ame e s .. == m2 j(d3kBT) and J.l = mH jkBT, whe e m and d a e he magne ic momen and he diame e o each sphe e, H is he ex e nal magne ic ield s eng h, kB is he Bol zmann cons an , and T he absolu e empe a u e. When J.l - 00, all he magne ic dipoles align pa allel o he ex e nal magne ic ield. Unde his condi ion, he ene ge ically a ou able a engemen o N g ains is a linea chain. As we will see below, he analy ic s udy o hese pa  icula agg ega es equi es conside able app oxima ions, e en in he limi case .. � 1 and J.l - oo. On he o he hand, en opy conside a ions will sugges ha o he mo e complex s uc u es may a ise when he pa icle densi y is oo la ge o .. is oo small [26]. Bu o low densi y and la ge .. , he mos a ou able phase is a linea chain. Di ec calcula ions [26] show ha despi e he e a e long- ange con ibu ions o he ene gy o a chain con aining N pa icles, he ampli ude o he ib a ions and oscilla ions o he sphe es a e almos en i ely aken in o accoun by jus conside ing he e ec s due o i s and second nea es -neighbou in e ac ions. The long ange e ec s s ill exis , bu hey a e domina ed by sho ange a ac i e con ibu ions. As a ma e o ac , he impo an egions co espond o ela i e dis ances among he 134 CHAPTER IV. AGGREGATION PHENOMENA pa icles """ d and ela i e de ia ions (J """ O o 11'. In addi ion, o ou magne ic colloids he in e ac ions o one g ain a e essen ially sa u a ed when i has come in o close con ac wi h wo o he s. Mo eo e , in he high ield limi , he s a e o a sphe e will be en i ely desc ibed by he posi ion o i s cen e. This will simpli y conside ably he equa ion o con inui y ha desc ibes he conse a ion o sys em poin s in he con igu a ion space. Finally, we will dis ega d London- Van de Waals o ces as well as s e ic e ec s. 3.1 Con ibu ion o he chains o he p essu e enso o he sys em Apa om he con ibu ion o he sol en o he o al p essu e enso o a supension, he e is ano he con ibu ion coming om he di ec in e ac ion o he suspended pa icles [27, 28]. Thus, ollowing K ame s' heo y, he con ibu ion o a chain o he p essu e enso o he whole sys em is 1 � (� 8Vmag ) N - 1 IIp = - V � qk� + -V-kBT1, k=1 s» whe e V is he olume o he sys em, q¡, a e he ela i e posi ion ec o s, Vmag is he dipole-dipole magne ic po en ial, N is he numbe o pa icles o he chain, and 1 is (3.1.1 ) he uni a y ma ix. In o de o do he a e ages appea ing on he igh hand side o Eq. (3.1.1), we need o sol e he co esponding di usion equa ion in he con igu a ional space: al/; '" {8 '" TT ( 8l/; 8Vmag) 8 �} - = � -=-. �""'ij . kBT-� + -_-l/; - -_ . ( 3. H¡)l/; 8 i en; j 8Rj 8Rj 8H¡ (3.1.2) whe e i; is he pos ion ec o o he í-sphe e, ,..,.Tl is he ela i e ansla ional nobil i y, and 3 is he elonga ional a e ( 3 = 3T) co esponding o he s a iona y ho noge neous ex e nal low, Vo = 3 . . We a e also assuming ha he sphe es mo e h ough he sol en wi hou dis u bing he eloci y ield ( ee d aining), so ha TT 1 ,..,. .. = --IÓij. 1) 611'7]a whe e we conside ha all he sphe es in he chain ha e exac ly he same adius (3.1.3) a. Mo eo e , we will dis ega d he o a ional con ibu ion o he di usion equa ion 3. Chains oi magne ic pa icles in an eJonga ionaJ low 135 because in he high ield limi he s a e o a pa icle is ully ep esen ed by he posi ion o i s cen e. Subs i u ing Eq. (3.1.3) in o Eq. (3.1.2), we a i e a o /J = kBT I: {02_ /J + o_ . (oVmag!kBT /J) - .!. . ( 3 . R;) /J} . & 61!'1Ja i oR? aH; aH; aH; In addi ion, as we a e conside ing he limi A ::» 1, J-l -> 00, we will assu ne ha (3.1.4 ) he po en ial can be exp essed as a sum o he nea es neighbou s in e ac ion V. N-l mag "'"' k T = L....J �;,i+l B ;=1 As we ha e p e iously indica ed, he ampli ude o he ib a ions and oscilla ions o (3.1.5) he sphe es a e well ep esen ed by such in e ac ions. A e aking in o accoun ha he mos impo an con ibu ions come om ela i e dis ances ..... dand ela i e de ia ions (J "-' O, each e m in he sum o Eq. (3.1.5) can be w i en as ollows �;,;+1 "-' -A(2 - 3(J? - 6�;) (3.1.6) whe e we ha e in oduced he ela i e posi ion ec o q; = Ri+1 - R; whose co npo nen s in sphe ical coo dina es a e q; == (q;, (Ji, <p). Mo eo e , we ha e ca ied ou a de elopmen a ound he maximum con ibu ion q; d(1 + �;) wi h �i "-' O (Ji O. (3.1.7) Now, we will w i e he di usion equa ion in e ms o he ela i e posi ion ec o s, q;. Bu be o e ha , we in oduce he ans o ma ion q; = I: Bil:RI: wi h s., = 6i+1,1: - 6i,1;, 1: om which one in e s he ela ion (3.1.8) a a -_ = I: Bki-;;= aH; k os» (3.1.9) wi h i í = j i í = j ± 1 o he wise (3.1.10) 136 eHAPTER IV. AGGREGATION PHENOMENA Taking in o accoun Eqs. (3.1.8)-(3.1.10), he di usion equa ion eads The s a iona y solu ion o his homogeneous po en ial low ield is ./. (- -) e {-Vmag 311''1a(a "''''c - -)} 'I', q1," ·,QN-1 = eep -¡¡-;¡;- + k T p : L...,;L...,; ij qi qj , B B i j (3.1.12) whe e e is a no maliza ion cons an and eij is he K ame s' ma ix de ined in he ollowing way: e .. _ { i(N - j)/N i i s j 1) - j(N _ i)/N i j � i (3.1.13) Once we know he s a iona y solu ion o he di usion equa ion, we will compu e he con ibu ion o he chain o he p essu e enso , i.e., he a e age appea ing on he igh hand si de o equa ion (3.1.1) ( ik O��ag) = J l di ¡ ( ik O��ag) 1/J. (Q1,"" V -1)' s» 1=1 s» As we a e in e es ed in inding he New onian iscosi y enso , we can expand he exponen ial ac o (37 '1a)/(kBT)( 3: Li Li eijq'¡ijj) in Eq. (3.1.12) up o i s o de in 3. I we we e also in e es ed in he i s non-New onian con ibu ion we would include he second o de , ... A i s we will use (3.1.14) o i we in oduce he ansla ional di usion coe icien o a single pa icle D = kBT/(67 '1a) and Eq. (3.1.5) whe e e' is he new no maliza ion cons an ha we compu e in Appendix A 3. ehaíns o magne ic pa icJes in an elonga ional Iow 137 1 C' (3.1.17) + Consequen ly, Eq. (3.1.14) educes o ( _ OVmag) q 1: ----;:;-:; os» e' ¡/i dq¡ exp( -�I,I+l)) (iJ: o�,:ag) �l Ü {l + 2�(�. ��C;;;;q;)}. (3.1.18) In o de o e alua e his a e age, we will decompose i in o di e en pa s (de ailed calcula ions ha e been collec ed in Appendix B). Up o i s in {3 we ha e (iJ: O�'ikag ) = kBT {1 + � � Cil:{3· z z + 6�A (1 - l�A) el;k{3· (1 - Z Z)} (3.1.19) Once we ha e been able o compu e his a e age, he con ibu ion o one chain o he p essu e enso (3.1.1) In iew o Eq. (3.1.20), we can conclude ha we ob ain an expansion in powe s o he in e se o he pa ame e A, compa ing dipola and he mal ene gies, s a ing om he assymp o ic alue which co esponds o he igid chain limi and inc easing o dec easing when A dec eases, i.e. when he chain becomes mo e lexible, depending upon he s uc u e o he low a e. I wo hs poin ing ou ha , as we a e no aking in o accoun he hyd odynamic in e ac ion be ween he pa icles, he igid chain limi p essu e enso g ows simply as he hi d powe o he chain leng h. This in e ac ions a e he esponsible o he loga i hmic e m ha should also appea when dealing wi h a long s aigh line o sphe es [27], bu which is less impo an o sho e chains. 138 CHAPTER IV. AGGREGATION PHENOMENA Mo eo e , om his exp ession we can also ob ain he con ibu ion o he chain o he iscosi y enso , 'Ip IIp = -2'1p : 13 (3.1.21) and (3.1.22) He e S is a symme ic enso such ha Sijkl = 1/2(6ik6jl + 6iI6jk). As he ex e nal magne ic ield ha we a e applying is such ha J1. --4 00, he chain will be always o ien ed in he di ec ion o he ield and smoo hly ib a es and oscilla es a ound his o ien a ion. In his case, he symme ies o he luid low essen ially de e mine he cha ac e is ics o bo h he p essu e and he iscosi y enso . Thus, as he chain is o ien ed along he z-axis, i he low ield a e 3 is diagonal, he p essu e enso will also be diagonal, i.e. i will be a symme ic enso . On he o he hand, we will ha e, in gene al, bo h a symme ic and an an isymme ic con ibu ion o a non-diagonal low a eo Le us conside so ne pa icula cases: i) Flow h ough a po e o ien ed along he same di ec ion as he chain. In hís case he low a e 13 has he ollowíng o m: (3.1.23) whe e {3 is he low a e s eng h, and he sys e n has a o a ional symme y a ound he z-axis. The chain in hese condi ions is s e ched by he elonga ional low gi ing ise o he so-called eloga ional iscosi y de ined as ollows _ IIzz - 1I.u = 11' lod3 (N2 _ 1) {2N � (1 _ _3_)} 2{3 8V + 3>' 18>.· (3.1.24) Fo a luid low wi h o a ional sy nme y a ound he x (o y) axis, we will ha e wo di e en con ibu ions coming om he di e ences _ 1I.u - IIzz = 1 7]od3 (N2 _ 1) {N � (1 _ _3_)} 2{3 8V + 3>' 18>" (3.1.25) 3. Chaíns o magne ic pa icles in an elonga ional low 139 (3.1.26) Mo eo e , in bo h cases we ha e a con ibu ion coming om he ace o he p essu e enso , which o he low a e gi en by Eq. (3.1.23) eads (3.1.27) and o he low wi h o a ional symme y a ound he x axis is T TI = 1 ¡od3 (N2 _ 1) {N _ 2. (1 _ �)} 6/3 8V 3>' 18>.· (3.1.28) ii) Plana elonga ional low. This low can be gene a ed by ou o a ing in ini e cylinde s. I we loca e he cylinde s such ha he low a e is again a diago nal ma ix, he p essu e enso will be diagonal and symme ic in iew o Eq. (3.1.20), i.e. o 3=/3(��� ) O O -1 (3.1.29) we will ind a simila beha io o he sys em as he one desc ibed aboye. On he o he hand i we o a e he ou cylinde s 450, i.e. o 3=/3(� � �) 1 O -1 (3.1.30) he p essu e enso has bo h a symme ic and an an isymme ic pa . Rela ed o hese pa s, we will ind no only elonga ional iscosi ies bu al so a shea iscosi y and a o a ional iscosi y gi en by (.) _ _ _ _ 1 Tlod3 2 _ { 2. ( _�)} TI.,z - 2 ¡/3 - 2 16V (N 1) N + 3>' 1 18>' /3, (3.1.31) (a) _ _ _ _ 1 ¡od3 (N2 _ ) { _ _!_ ( _�)} TI.,z - 2 ¡ /3 - 2 16V 1 N 3>' 1 18>' /3, (3.1.32) whe e we iden i y he shea and o a ional iscosi ies 140 CHAPTER IV. AGGREGATION PHENOMENA (3.1.33) (3.1.34) 4 Conclusions This chap e is in ended as a p elimina y s udy o he agg ega ion phenomena aking place in sys ems o magne ic pa icles in suspension and o he esul ing s uc u es. Ou i s poin has been o elucida e he in luence o he hyd odynamic in e ac ions (HI) in he kine ics o he agg ega ion p ocess. We ha e ex ended he classic Smoluchowski heo y o coagula ion o accoun o he p esence o HI ocu ing when one goes beyond he dilu e egime. Such in e ac ions ac be o e he di e en pa i eles a i e a he sphe e o in luence o a gi en pa icle. We ha e ob ained he kine ic equa ions o he agg ega ion p ocess and om hem we ha e analyzed he clus e o  ma ion. Ou main conclusion is ha he p esence o HI slows down he agg ega ion p ocess. As a second p oblem we ha e s udied he dynamics o a chain o magne ic pa icles unde he in luence o an ex e nal elonga ional low. In pa icula , we ha e compu ed he con ibu ion o he chain o he p essu e enso o he sys em om he heological equa ion o s a e p oposed by K ame s. F om his quan i y we ha e ob ained he co ec ion o he iscosi ies due o he p esence o dipola in e ac ions. These p elimina y esul s will cons i u e he subjec o u u e wo k.