Dynamic Properties of Magnetic Colloidal Particles and Holes

Dynamic Properties of Magnetic Colloidal Particles and Holes María del Carmen Miguel López Aquesta tesi doctoral està subjecta a la llicència ReconeixementNoComercial 4.0. Espanya de Creative Commons. Esta tesis doctoral está sujeta a la licencia Reconocimiento - NoComercial 4.0. España de Creative Commons. This doctoral thesis is licensed under the Creative Commons Attribution-NonCommercial 4.0. Spain License. Universitat de Barcelona Facultat de Física Dynamic properties of magnetic colloidal particles and holes Memoria presentada per María del Carmen Miguel López per optar al Grau de Doctor en Ciencies Físiques. Barcelona, 8 de setembre de 1995. Universitat de Barcelona Facultat de Física Dynamic properties of magnetic colloidal particles and holes Certifico que la present tesi doctoral ha estat realitzada sota la meya direcció Barcelona, 8 de setembre de 1995. Dr. J. M. Rubí Capaceti, Catedrátic de Física de la Materia Condensada de la Universitat de Barcelona Propiedades dinámicas de partículas y huecos magnéticos coloidales Introducción Las partículas magnéticas de pequeño tamaño son de gran importancia en multitud de áreas de la ciencia y la tecnología. No obstante, hay escasos ejemplos de sistemas formados por estas partículas suspendidas en un medio líquido de manera que las interacciones magnéticas desempeñen un papel principal, tanto en su comportamiento dinámico como en la formación de estructuras. Entre estos pocos casos figuran los ferrofluidos y las partículas de látex dispersas en un ferroftuido (huecos magnéticos). Un fluido magnético o ferrofluido es una suspensión coloidal estable de partículas mono dominio de un material magnético en un líquido común, como agua u otro compuesto orgánico. Inicialmente fueron concebidos como la primera muestra de un líquido artificial en el que las propiedades magnéticas y reológicas se viesen incremen tadas hasta ser comparables a las de un sólido ferromagnético, y se manufacturaron debido al hecho de que la existencia de monofases ferromagnéticas líquidas reales era bastante improbable. Aunque la es tabilidad de la suspensión está asegurada esencialmente por el movimiento browniano de las partículas, usualmente éstas se recubren de una capa de surfactante a fin de inhibir su coalescencia a distancias cortas. Los ferroftuidos tienen muchas aplicaciones tecnológicas, incluyéndose entre ellas los precintos de exclusión para las disqueteras de ordenador, amortiguadores en al tavoces, impresoras, instrumentos para tranferencia de masa y calor, lubricantes, etc. Por ejemplo, resulta complicado mantener un lubricante dentro de la zona de contacto de las parte móviles de un determinado mecanismo ya que las fuerzas centrífugas o la geometría tienden a expulsarlo. Sin embargo, un campo magnético puede ayudar a retener un lubricante magnético en la zona de fricción donde es más necesario. Más aún, la eficacia de un lubricante depende principalmente de su viscosidad y de su capacidad para formar una capa adsorbida. Los lubricantes magnéticos son más efec tivos que los comunes incluso en ausencia de campos magnéticos porque las partículas magnéticas que lo componen se ven atraídas por las superficies de acero y aumentan así tanto la viscosidad como el grosor de esta capa límite. Los ferrofluidos encuen tran también aplicaciones en ecología y medicina. Por ejemplo, sirven para eliminar residuos oleosos de los vertidos industriales. Los fluidos magnéticos con un hidrocar buro como base se pueden disolver en los productos oleosos para formar un ferrofluido más diluido. Un flujo de agua que contiene esta suspensión circula a través de un separador magnético donde un campo magnético que varía fuertemente en el espacio separa los residuos magnéticos del agua. Pero son los múltiples y cualitativamente diversos efectos procedentes de la interacción entre ferrofluidos y campos magnéticos los que han abierto la posibilidad de nuevas y prometedoras aplicaciones tecnológicas. Además de sus propiedades magnéticas, ópticas y reológicas, que discutiremos a lo largo de esta monografía, los ferrofluidos exhiben otros fenómenos poco comunes de gran interés científico, como las inestabiliades superficiales ycuriosas estructuras de no equilibrio. Hace algunos años se descubrió que las esferas mono dispersas de poliestireno, dis persas en un ferrofluido, constituyen un modelo conveniente para el estudio de varios tipos de fenómenos que muestran transiciones orden-desorden. La base de ello es que las esferas desplazan al ferrofluido y se comportan como huecos magnéticos con un momento magnético efectivo igual al momento total del ferrofluido desplazado. Las esferas (entre 1 y 100¡,tm) son mucho mayores que las partículas magnéticas del fer refluido (100.4) y, por tanto, se comportan como si se hallasen en un medio magnético uniforme. En general, las microesferas mono dispersas presentan una gran variedad de apli caciones industriales, científicas y médicas, debido a la singularidad de sus muchas propiedades como el poseer una distribución de tamaños extremadamente uniforme, una forma esférica casi perfecta, con un diámetro determinado con gran precisión, la gran diversidad de tipos de partículas (sólidas, porosas, magnéticas, etc.) y la posibilidad de manufacturar sus propiedades superficiales. Las microesferas de poli estireno con inclusiones magnéticas se han empleado satisfactoriamente para separar diversos materiales orgánicos (células, virus, micelas, etc.) El principio de esta apli cación se basa en una interacción selectiva entre los antígenos de la membrana de las células y los anticuerpos situados sobre las esferas. Las esferas con las células adheri das pueden eliminarse posteriormente por medio de un campo magnético. Es posible limpiar así la médula espinal de pacientes afectados de neuroblastoma o de ciertas formas de leucemia, limitando los tratamientos con radioterapia y/o quimioterapia, que puede afectar perjudicialmente al resto de células sanas del organismo. Mediante esta técnica, es posible limpiar hasta una décima parte de la médula de los pacientes, antes de recurrir a otras terapias más radicales. Las partículas magnéticas se pueden usar también como marcadores si se las recubre con ciertos componentes químcicos o con anticuerpos. Durante la migración de las partículas dispersas, éstas se adhieren a blancos específicos y facilitan su localización. Otra prometedora aplicación es su uso como portadores; se sitúan ciertos productos químicos sobre la superficie de las partículas y se transporta a éstas a los lugares donde las medicinas serán efectivas. Los bioquímicos están valorando actualmente las potencialidades de este método para tratar el cáncer. Conclusiones A lo largo de esta monografía nos hemos ocupado del estudio de sistemas fluidos, tanto con monodominios magnéticos como con dos tipos distintos de partículas, magnéticas y no magnéticas, en dispersión en un líquido newtoniano en situaciones fuera del equi librio. El comportamiento de estos sistemas se ve influenciado en gran medida por la presencia de un campo magnético externo, lo que da lugar a nuevos fenómenos que han sido el fundamento de muchas aplicaciones prácticas. Sin embargo, esta influencia depende de los diferentes procesos de relajación que tienen lugar dentro las partículas, con respecto a su ejes cristalinos, así como fuera de ellas, con respecto al fluido porta dor. Hemos descrito cuáles son estos procesos y obtenido la dependencia con respecto de los parámetros que describen las partículas y el fluido, de algunos coeficienes que caracterizan las propiedades reológicas, magéticas y ópticas de la suspensión coloidal. Para ser más precisos, en la primera parte del Capítulo 1 hemos analizado la dinámica de una esfera ferromagnética en la que el momento magnético se encuentra fijado rígidamente a su cuerpo, así como también las viscosidades de una suspensión diluida de este tipo de partículas. La base teórica fundamental es la ecuación de Navier-Stokes, en la que se ha incluído una fuerza inducida que representa las per turbaciones introducidas en la dinámica del fluido por el movimiento de la partícula, y una fuente de ruido tipo Langevin, procedente de las fluctuaciones de los campos hidrodinámicos. Esta ecuación da cuenta del acoplamiento de las dinámicas del flu ido y la partícula. Una expansión multipolar de las cantidades que aparecen en la solución formal de la ecuación de Navier-Stokes nos permite calcular las expresiones para la fuerza y el momento de las fuerzas ejercidos sobre la partícula, los cuales con tienen contribuciones aleatorias cuyas propiedades estadísticas están dictadas por la teoría de fluctuaciones hidrodinámicas. Hemos empleado dos métodos para obtener las viscosidades de cizalladura y rotacional. Uno de ellos es determinista, y se basa en la fórmula de Kirkwood para el tensor de presiones viscosas, el cual a su vez está relacionado con el multipolo de segundo orden de la fuerza inducida. El otro método se basa en la teoría de respuesta lineal, que da los coeficientes de transporte en términos de funciones de correlación dependientes del tiempo. La presencia del campo magnético provoca la aparición de contribuciones antisimétricas al tensor de presiones y da lugar a la viscosidad rotacional que aparece como un nuevo coeficiente de transporte. Esta parte antisimétrica del tensor de presiones proviene del balance que se establece entre el momento de las fuerzas ejercidas por el campo magnético sobre un dipolo y el momento de las fuerzas hidrodinámicas. Como consecuencia, el momento angular de las partículas puede diferir del valor de la vorticidad del fluido en el punto que aquella ocupa. Este formalismo es bastante general y se emplea en la segunda parte del Capítulo para calcular la dependencia de la viscosidad rotacional con respecto de la energía magnética de anisotropía del material. Considerando otro límite específico, en el cual los momentos magnéticos ya han relajado y apuntan en la dirección del campo, obtenemos que la viscosidad aumenta al aumentar el parámetro de anisotropía, alcanzando un límite de saturación. Hemos comparado nuestros resul tados con los de otros autores y con los datos experimentales disponibles para dipolos rígidos. Los resultados procedentes de otra solución aproximada de la ecuación esta cionaria de Smoluchowski sobreestiman los nuestros, mientras que los basados en ecuaciones de relajación fenomenológicas propuestas para el momento angular interno de la suspensión son muy próximos a los que hemos obtenido nosotros, y reproducen bastante bien los datos experimentales. Siguiendo un procedimiento análogo, en el Capítulo 11 hemos presentado un for malismo general para estudiar la dinámica de relajación de partículas ferromagnéticas, con el propósito principal de proporcionar expresiones explícitas tanto para la viscosi dad como para algunos tiempos de relajación que caracterizan diferentes propiedades del material (birrefringencia, susceptibilidad magnética, ... ). Los resultados cubren todo el rango de situaciones experimentales posibles. Hemos obtenido la ecuación de Smoluchowski que describe la evolución de la distribución de probabilidad de los gra dos de libertad relevantes de las partículas y que nos permite obtener una jerarquía de ecuaciones dinámicas para las diferentes funciones de correlación. Esta jerarquía se puede cerrar utilizando las conocidas aproximaciones de desacoplo apropiadas. A partir de las ecuaciones dinámicas para las correlaciones, uno puede encontrar expre siones para los tiempos de relajación característicos y éstos constituyen el punto de partida para determinar los coeficientes de transporte por medio de las fórmulas de Green-Kubo. En particular, hemos visto que la viscosidad rotacional alcanza de nuevo un límite de saturación, pero ahora depende los dos parámetros que comparan la en ergía magnética y la energía de anisotropía con la energía térmica. Para contrastar la validez de nuestro formalismo, hemos comparado nuestros resultados para el tiempo de relajación de las partículas con los experimentos de birrefringencia llevados a cabo con dos tipos de materiales ferromagnéticos. En ambos casos, nuestros resultados concuerdan muy bien con los datos experimentales. También hemos proporcionado una expresión general para la susceptibilidad magnética compleja del fluido magnético bajo la acción simultánea de un campo magnético polarizante y de un pequeño campo alterno, perpendiculares entre sí, lo que abre posibilidades para nuevas medidas ex perimentales. En el Capítulo 111 hemos estudiado la dinámica de partículas no magnéticas, o hue cos magnéticos, suspendidos en un ferrofluido en presencia de un campo magnético rotativo. Calculamos las fuerzas hidrodinámicas y el momento de las fuerzas ejer cidas sobre el hueco, de donde podemos identificar los tensores de fricción trasla cional yrotacional. Estas cantidades dependen de la fracción volúmica de partículas magnéticas y del campo magnético, el cual otorga al sistema un carácter anisótropo. El conocimiento de la fuerza y del momento de las fuerzas que actúan sobre el hueco nos permite estudiar las dinámicas de traslación y de rotación de las partículas cuando aplicamos un campo magnético rotativo. Nos hemos concentrado particularmente en el caso en que el ferrofluido está en reposo y el hueco puede girar bajo la influen cia del campo magnético. Bajo estas condiciones, hemos probado que la velocidad angular del hueco es proporcional a la frecuencia del campo, pero que tiene una di rección opuesta. El coeficiente de proporcionalidad es lineal con la fracción volúmica de las partículas de ferrofluido, y depende de una función de la intensidad del campo magnético que muestra un comportamiento de saturación. Hemos comparado nue stros resultados con experimentos realizados con partículas de poliestireno dispersas en diferentes ferrofluidos. En el rango de frecuencias consideradas, hemos reproducido los resultados experimentales concernientes a la velocidad de rotación del hueco como función del campo magnético externo. También hemos realizado un estudio preliminar de las interacciones hidrodinámicas entre los huecos en el ferrofluido. Esencialmente, hemos obtenido expresiones para los tensores de Oseen y Rotne-Prager de un fer rofluido como el primer paso en el estudio del importante papel que las interacciones hidrodinámicas pueden desempeñar en las propiedades físicas de estos materiales a bajas concentraciones. Para mostrar la estructura de estos tensores y su influencia en la dinámica de los huecos magnéticos, hemos dado también expresiones para las valocidad de una pareja de partículas depositándose bajo la acción de la gravedad en el seno de un ferrofluido. Como esperábamos, para diferentes configuraciones iniciales, la velocidad depende de los parámetros que caracterizan el ferrofluido. Resulta útil señalar que, aunque hemos realizado el estudio para un ferrofluido compuesto por dipolos rígidos, para el que la energía de anisotropía es mucho mayor que la energía asociada con la interacción de los momentos magnéticos con el campo externo, se puee hacer una análisis similar en una situación más general, en la que estas dos energías de las partículas magnéticas presenten valores arbitrarios. El Capítulo IV se presenta como un estudio preliminar de los fenómenos de agre gación que tienen lugar en sistemas de partículas magnéticas en suspensión, así como de las estructuras resultantes. Nuestro primer propósito ha sido elucidar la influencia de las interacciones hidrodinámicas en la cinética del proceso de agregación. Hemos extendido la teoría clásica de la coagulación de Smoluchowski para dar cuenta de la presencia de estas interacciones, importantes cuando se traspasa el régimen de con centraciones diluidas. Tales interacciones actúan antes de que las diferentes partículas lleguen a la esfera de influencia de una partícula dada. Hemos obtenido las ecuaciones cinéticas para el proceso de agregación y, a partir de ellas, hemos analizado la for mación de agregados. Nuestra principal conclusión es que la presencia de interacciones hidrodinámicas ralentiza el proceso de agregación. Como un segundo problema, hemos estudiado la dinámica de una cadena de partículas magnéticas bajo la influencia de una flujo externo elongacional. En particular, hemos calculado las contribuciones de la cadena al tensor de presiones del sistema a partir de la ecuación reológica propuesta por Kramers. De esta cantidad hemos obtenido la corrección a la viscosidad debida a la presencia de interacciones dipolares. Estos resultados preliminares constituyen el objeto de futuros trabajos. Diferentes líneas de investigación pueden surgir de los contenidos de esta tesis. En tre ellas, podemos citar la necesidad de incorporar en el análisis efectos inerciales y de la polidispersidad de las partículas. Como comentamos brevemente en la introducción de la tesis, los efectos inerciales darían lugar a un régimen oscilatorio adicional en los procesos de relajación para valores muy altos del campo magnético. Sin embargo, resultan ser esenciales para la descripción de la dinámica rotacional de las partículas a frecuencias muy altas. Con respecto a la polidispersidad, debido a que los difer entes procesos de relajación dependen de forma distinta del volumen de las partículas, sus contribuciones a las magnitudes calculadas no son las mismas en un medio po lidisperso. Por ejemplo, el espectro de los tiempos de relajación puede ensancharse. Otra posible línea de investigación consistiría en ir más allá de la respuesta lineal del sistema en condiciones alejadas del equilibrio. 1. The [erromagnetic particles 3 Figure .1: Scanning eler tron micrograph of magnelic rnic rospheres with anti-H2 mon oclonal antiboclies att.ached lo liepatocyte. 1 The ferromagnetic particles We will disregarcl for t.he moment the liquid in which tlie particles are dispersed and concentrate in the descr ipt.ion of t.he fine ferromagnetic par tic les. Ferromagnet.ic par t.icles in a ferrofluid are monoclomain and possess almost con st ant magnet.ic moments. It is well known that ferromagnet.ic crystals consist of re gions where the magnetizat.ion is orientecl clilTerently. This regions are called magnetic domains. The shape and size of these magnetic dorriains in thermodynarnic equilib rium are determined by the condition that t.he total free energy of tlie material should be a minimum. Moreover, the magnetic dorriains are separated from each other by domain walls, i.e., a transition layer in which the dist.ribution of the magnetization orientation is nonuniform but changes continuously. Frenkel and Dorfman [7] pre dicted that by reducing the size of a ferromagnetic sarnple, one would reach a point at which domain boundaries would no longer be energetically favorable so that the whole sample would become a single dornain. On one hancl, every domain wall means an additional arnount of surface tension energy which limits the number of possible dornains. On the other h and , the energy associalecl to t.he rnagnetic field surrounding the sample favors their forrnation. Based on this simple argumenl of competition one can estimate the critica! size (de) below which the sarnple can be considered a mag netic monodomain. In the esti rn at.ion one should distinguish between strong and weak magnetocrystalline anisotropy ber ause it also hine!ers tlie cr eat ion of a domain wall 4 INTRODUCTION favoring the in crease of the critical monodomain size. But, even for the less favored cases of weak magnetocrystalline anisotropy, the estimations give de "" 300..1 which is well aboye the mean size of the ferromagnetic particles used in ferrofluids whose typical diameter does not usually exceed 150..1. Thus, even allowing for a possible polydispersity, the colloidal dispersed particles are definitely monodomain. Although we are interested in enhancing for instance the magnetic properties of the magnetic liquid, and these are clearIy determined by the suspended particles, there is a restriction in their size to essentially avoid aggregation phenomena which are mainly due to dipolar magnetic energy. Ifwe define a dimensionless parameter A = m2 jd3kBT comparing dipolar magnetic and thermal energies, where m and d are the magnetic moment strength and the diameter of one particle, the stability condition imposes that A ::; 1 which can only be achieved if the particle diameter is small enough (for a ferromagnetic material whose saturation magnetization M. ::: 500G this condition holds if d ::; 100..1). An intermediate solution of the problem can be thought consisting in reducing the linear size of the particles but keeping the same volume fraction (¡fJ = n Vm), where n is the particle number density and Vm the volume of one monodomain. However, this strategy also manifests sorne problems. By pursuing this method one arrives at highly dispersed colloidal suspension but no longer magnetic. It is clear that the electron spins responsible for the magnetic ordering are affected by the presence of the boundary of the volume in which they are enclosed. The absence of partners for exchange interaction manifest itself on the formation of a thin demagnetized layer whose thickness can be comparable to that ofthe whole particle. This fact is verified by experiments [8, 9]. Moreover, there exists an alternative point of view which explains this demagnetized layers by chemical modification of the particles surfaces due, for instance, to oxidation or to the interaction with the surfactant which is usually added to stabilize colloidal suspensions. This question has been the subject of many studies [10, 11, 12] and it remains open. For our purposes in this thesis, this problem it is not crucial because we will always assume that m = M. Vm with Vm the volume of the magnetic core, which due to the aboye considerations can be less than the geometric volume. When looking at the particles, another aspect that should be kept in mind is that even though it might seem that a monodomain magnetic particle behaves like a permanent magnet, this is not always the case. The orientation of the poles in a magnet is fixed by the magnetic anisotropy energy. At room temperature, the dimensionless parameter (J' = Ka VmjkBT, comparing anisotropy and thermal energies 1. The ferromagnetíc partícles 5 with Ka the effective anisotropy constant, for a monodomain particle is not large (d", 100A, Ka '" 104-105 J 1m3). Consequently, the probability ofthermal rotational fluctuations of the magnetic moment inside the particle becomes important. At u ::; 1, the magnetic moment are not practically affected by the potential barrier and its motion is similar to the Brownian rotation of a colloidal particle in a viscous liquido Néel was the first to indicate the possibility of fluctuational remagnetization of small particles, that is why this process is usually referred to as the N éel relaxation. The characteristic time of this spontaneous reorientation, Tq depends on the parameter U. For time intervals shorter than T the particle behaves like a permanent magnet, whereas for measuring times greater than T the particle's magnetic moment can be considered zero. These differences can be corroborated, for instan ce, by means of Mossbauer spectroscopy. The alignment tends to be disrupted by thermal agitation, and of course, beyond the Curie temperature the domain possesses no magnetization any longer. In the above discussion we have considered the relaxation of the magnetic moment in the absence of an ezternal magnetic field. But, if a magnetic field is applied the energy of a magnetically uniaxial particle is the following u = -m' H - Ka Vm(71· kl, (1.1 ) where m = mR is the magnetic moment of the particle, H is the external magnetic field, and 71 is the unit vector along the direction of the axis of easy magnetization, or the anisotropy axis. The effective constant Ka contains, in general, contributions coming from the crystalline anisotropy of the particle as well as the shape anisotropy. From this expression it will be easy to see that the dynamics of the two degrees of freedom for a particle in a liquid, R and 71, are coupled. It is also worth mentioning a this point that there will be essentially two relevant dimensionless parameters in the analysis. These parameters are the above introduced u = Ka VmlkBT and the parameter p. = mH IkBT, comparing the magnetic energy of interaction with the magnetic field and therrnal energy. The derivative of this magnetic energy with respect to m determines the value and orientation of the effective magnetic field: H- - et: -H2KaVmA(A RA) eJJ - --a - - + n n . , m m (1.2) which includes the external field H and the anisotropy field Ha directed along the anisotropy axis. In the absence of the external field, the magnetic moment is just under the action of the anisotropy field and there are two equivalent equilibrium 6 INTRODUCTION orientations R = 71 and R = -71 between which the relaxation can take place. Now, the equilibrium condition is given by the absence of magnetic torques acting upon the particle, so that in the absence of thermal fluctuations the magnetic moment is parallel to Helf. When this is the case there are two different orientational relaxation processes of the magnetic moment of the particle relative to its crystallographic axes. The intrinsec motion of the magnetic moment consist of a regular precession around the effective field and of chaotic reorientations due to thermal fluctuations. The regular motion relative to the crystalline axes of the particle is described by the classical Landau-Gilbert equation, which we introduce in Chapter JI. This equation represents the precession of the magnetic moment with the Larmor frequency WL as well as the decay of this motion due to collisions, magnetoelastic interaction, ... Associated to this decay there is a characteristic time To = CiWL., where Ci is a dimensionless damping constant. Another characteristic time TD = (2Dm)-1 is connected with the rotational diffusion of the magnetic moment inside the particle Dm = kBTh, where, as we will see in Chapter JI, h plays the role of a rotational mobility of the magnetic momento 2 Motion of the magnetic particle in the liquid Besides the internal motion relative to the particle body, the magnetic moment also undergoes an external rotational diffusion as a consequence of the motion of the mag netic particles in the liquid they are suspended. The ferromagnetic particles suspended in a nonmagnetic fluid experience the action of the carrier liquid through viscous frie tion. We will now consider the rotational Brownian motion of a colloidal particle. Its deterministic motion is described by the equation for the rotational dynamics of a solid body suspended in a viscous liquid (2.1 ) where 1 is the moment of inertia of the particle, fi its angular velocity, �r the rotaional friction coefficient of the particle in a viscous liquid, and f the external torques acting on the particle. Jn the absence of external torques, the rotational motion of the particle decays wi th the characteristic time TI = 1 / �r. For a spherical particle in the Stokes approximation �r = 81f110a3, where 110 is the viscosity of the liquid and a is the radius of the particle. 3. The rotational diffusion equation 7 If we consider r¡ = 1O-2ps and d -- 100Á, it follows that TI -- lO-lIS. This value is so small that in all cases of practical interest the inertial term in Eq. (2.1) may be neglected in comparison with the viscous one. Indeed, this term may be relevant only for characteristic frequencies ofexternal excitation � 100GHz. Compared to the characteristic times of hydrodynamic processes TI is almost zero, Besides TI there exists a much larger characteristic time of the rotational motion of the particle. This time is determined by the rotational Brownian motion of the axis n representing the particle. This Brownian time is given by TB = (2Dr )-1, which taking into account the Stokes-Einstein relation it can be rewritten as TB = f.r/2kBT. For the same values given aboye at room temperature, TB - 10-6. During this time, the particle under the influence of thermal fluctuations rotates by a finite angle. The higher the viscosity ofthe carrier liquid the slower the rotation. Conversely, the higher the viscosity the bigger the time TI. Nevertheless, TB and TI will be comparable just for 770 - 1O-4ps, and such a viscosity would just correspond to aerosols or cryogenic liquids. In real ferrofluids, depending on the particle volume and on the magnetic anisotropy constant the relation among the different characteristic times, and of the parameters J1. and (7 may be arbitrary. Thus the relaxation time of the magnetization will be in general a combination of the different times already introduced. The limiting case of an infinitely strong coupling for (7 � 1, i.e. when the magnetic moment of the particle is rigidly coupled to the easy axis of magnetization, is known as the rigid dipole modelo Despite its simplicity, this approximation is widely used in the theory of magnetic fluids, and allows one to explain a wide range of magnetic and hydrodynamic phenomena experimentally observed. 3 The rotational diffusion equation As a single ferromagnetic particle experiences both a systematic damping and random thermal fluctuations of its magnetization and itself, the study of its dynamics can be performed by following two different methods: Langevirr's approach to the theory of Brownian motion, and by Brown's intuitive method [9] which is an adaptation of the arguments developed by Einstein in 1905, or in other words, by Fokker-Planck or Smoluchowski diffusion equations for the probability density. In order to accom plish this, Brown followed the procedure of Wang and Uhlenbeck [14] together with the Stratonovich definition of the derivative of a stochastic variable [15]. He also proposed an alternative and simple approach to writing down the Fokker-Planck or 8 INTRODUCTION Smoluchowski equation using a continuity equation argument as Einstein did in his treatment about the translational Brownian movement. Although Brown's approach was developed for a ferromagnetic particle in a solid matrix, his formalism can be further developed to describe the dynamic behavior of suspensions of these particles in fluids. Shliomis and co-workers [10] first obtained this equation for a suspension of rigid dipoles, and then for the general case with arbitrary values of the ratio ¡.JIu, they also deduced the appropriate Smoluchowski equation from a model similar to the itinerant oscillator model, [12]-[14]. The normalized stationary solution of the Smoluchowski equation enables one to obtain any equilibrium orientational characteristics of an assembly of magnetic par ticles. Moreover, in most of the cases one has to deal with the moments of the distribution function. Particularly interesting among these moments is the first one, which determines the equilibrium magnetization. To find out the solution of the kinetic equation of rotational diffusion different techniques have been proposed, especially for the rigid dipole model. For instance, for an assembly of rigid dipoles under constant external conditions any deviation of the distribution function from its equilibrium value may be expanded into a series of normal modes, each of which decays according to a simple exponential law. The relaxation time spectrum of the distribution function comes from the eigenvalues of the resulting equation. In fact, the lifetime of any departure from equilibrium is determined by the relaxation time of the most long-living modes. The effective field method and the decoupling approximations constitute other approachs to the problem. From the diffusion equation one can easily obtain an infinite set of coupled equa tions for the moments of the distribution. In order to solve the set, one can truncate it somewhere. But , if one wants to study the field dependence of the relaxation process, the number of calculations and the difficulty in understanding the results obtained grows drastically with the external field. To avoid these inconveniences related to the numerical solution, it would help to have a nontrivial scheme of closure of the moment equations that is capable of giving a compact analytical description of the orientational processes in a ferrofluid. Such a closure prescription will be better the smaller the number of equations. Ideally, for a suspension of rigid dipoles one should have only one equation, since only the first moment has a direct physical meaning. In such a way one would be able to give a correct description of the averaged dynamics of the magnet.ic particles for a wide range of values of the field, anisotropy, temperature, etc. The first idea meeting all these requirements for a suspension of rigid dipoles is ihe effective jield method [10], considered by Leontovitch in his book [21]. Thus, in 3. The rotationaJ diffusion equation 9 the nonequilibrium state one may consider any arbitrary value of the magnetízatíon as an equilibrium one in a certaín effective field. During the relaxatíon process, the effective field tends to the true field, so that the magnetízation relaxes via a sequence of quasi-equilibriurn states. Furthermore, one should represent the nonequilíbrium distribution function in the same form as the stationary distribution with the effec tive field. This technique also yields an expression for the field dependence of the relaxation times of a suspension of rigid dipoles under nonstationary condítíons but in a quiescent fluid. In this monograph we propose another alternative and simple procedure to solve the set of infinite and coupled equations for the rnoments of the distribution functíon which appears to províde good results not only for the rigid dipole model but in any arbitrary situatíon includíng a moving suspension or the case of finite anisotropy energy. The method is based on the decoupling approximalion of sorne of the quantities involved in the analysis, to be precise, of the quantities which vanish at equilibrium, when averaged, and those which are different from zero. Thís approximation may be justified from the fact that in equilibrium both quantities are not correlated, so, in non-equilibriurn conditions but in a linear regime, for instan ce when there is small but non-vanishing value of the vorticity of the fluid flow, we will assume that these components remain uncorrelated. In addition, we ensure the main characteristics of the decoupled quantities such that after performing the decouplings they are stíll proportional to the vorticity, or the invariance under reflections of the easy axis of magnetizatíon, ñ, and so on. We really expect that the decouplings are more accurate for quite small and high values of the external field, because in the former case hígher order moments will be neglígíble and in the latter the dynamícs of the particles ís mainly determined by the field rather than being influenced by Brownian motion. There are many possible ways of carrying out the decouplings. Such approxirnations are broadly used in the context of stochastic processes, and particularly, we will see throughout the thesis the most convenient decouplings to be performed in our system. In this system the stability of the stationary distribution in a constant external field appears as a natural fact. Furthermore, from these approximations one usu ally obtain a single characteristic time of the relaxation process. In the absence of an external field the relaxation of the initial disordered distribution takes place by means of free orientational diffusion which is a monotonic process with just one char acteristic time. In the strong field limit the dynamics of the magnetic moments is mainly deterministic, and they approach the field direction monotonically according to an exponential law with a characteristic time. Thus, although the external field 10 INTRODUCTION wilI reduce the relaxation time, it does not change the character of the decay process. In principIe, the formal cause of this behavior is that we are neglecting the inertial ter m due to their smalIness. Keeping this term would lead to an oscillatory regime of the relaxation process for enormous and almost unreachable values of the external field. However, inertial effects are essential if one considers the high-frequency be havior of rotating objects of molecular size. Here the treatment becomes much more complicated. A review of the problem and particular resulta is given by Coffey el al. [20]. 4 Macroscopic hydrodynamic theory of magnetic fluids Having characterized the different relaxation mechanism studied, we proceed to ana lyze their implications in the macroscopic behavior of the system. Ferrofluids can be treated as a homogeneous one-component monophase fluids. This approximation implies that the processes that should be considered are such that their characteristic dimensions are much larger than the size of the constituent magnetic particles. Thus, the magnetization ofthe system is assumed to be distributed throughout any elementary fluid volume. Establishing the equations of motion and heat transfer in a ferroftuid constitute one of the most important problems to study in these systems. Moreover, their peculiar properties manifest themselves most evidently in their hydrodynamic behavior. The set of hydrodynamic equations for an uncharged, magnetizable, and elec tricalIy nonpolarizable fluid is based on the balance equations of mass, linear and angular momentum, and heat transfer for a homogeneous monophase medium. As regards the momentum balance equation one should take into account the MaxwelI tensor, which in this case is antisymmetric unless the magnetization density vector and the field strength are paralIel (due to the finite value of the relaxation time of the transverse component of the magnetization, this condition does not hold under the influence of hydrodynamic torques acting upon the particles either in magnetic fields that change their direction or in a fluid having been set in motion). The asymmetry of the stress tensor demands a generalization of classical hydrodynamics in the sense that one should take into account the rotational degrees of freedom of the particles. Thus one ought to consider that the angular momentum density is made off two parts, the orbital associated with the translational motion of the suspended particles and 4. Macroscopic hydrodynamic theory of magnetic fluids 11 molecules of the solvent, and the spin momentum caused by the rotation of the par ticles. For simple fluids this spin is zero and the symmetry of the stress tensor is a necessary and sufficient condition for the conservation of the angular momentum. The situation is different if there exists interna! rotation because the stress tensor should not be symmetric any longer. The different fluid elements move with different veloc ities relative to the angular velocity of the liquid suspending them giving rise to an internal friction. In the absence of external torques acting on the particles, the differ ence between the averaged angular velocity of the particles and the local value of the vorticity of the fluid flow disappears as fast as inertial effects do. On the other hand, this difference results in an irreversible process (with dissipation of kinetic energy) of transfer of angular momentum between solid and liquid phases of the suspension due to the viscosity of the solvento Related to the presence of antisymmetric stresses in a ferrofluid there appears a new transport coefficient: the rotational viscosity, that in general contributes to the effective viscosity of the suspension. This contribution tends to an asymptotic limit at high field strength, it is maximum when the magnetic field and the vorticity are perpendicular and is zero when both vectors are parallel since in this case, if we neglect inertial effects, the angular velocity of the particle is equal to the vorticity of the fluid at the point occupied by the particle and only the contribution due to symmetric stresses will remain. In other words, the ferrofluid motion modifies the value of its magnetization, so if we consider a magnetic fluid with nonzero vorticity uniform shear (Couette flow) in a constant magnetic field perpendicular to the plane of shear, the field tends to align the particle magnetic moments and, therefore, the particles themselves. Moreover, the vorticity flow simultaneously attempts to rotate the particles destroying the magnetization created by the field. By means of the Smoluchowski equation for the rotational motion of the particles and after considering all their possible mechanism of orientation, we will be able to provide a solution of the macroscopic equation for the magnetization in a moving fluid. This magnetization, in turn, enters the equations of hydrodynamic motion of a magnetic fluid and thus affects the state of motion of the fluid. The rotational viscosity comes quite easily from this analysis using a rheologica! equation of state for the antisymmetric part of the pressure tensor or by using the corresponding Green-Kubo formula. To correct the hydrodynamics for fluctuations, one must add to the total fluid pressure tensor a thermal fluctuating pressure tensor whose average, but not square average, vanishes. This fluctuation reflects the fact that the first principles from which the complete macroscopic theories must follow, are time reversible. AIso the 12 INTRODUCTION fluctuation are seen to be related to the dissipation through the fluctuation-dissipation relation, showing indeed that a dissipative theory is not complete until fluctuations are included. In particular, if one consider the quiescent equilibrium states of a magnetic fluid in a nonuniform stationary magnetic field, one observes that such a field is not able to induce the ferrofluid motion, the magnetic pressure is balanced by the hydrostatic pressure. This fact provides the basis for numerous practical applications [2]. Once we have characterized the macroscopic behavior of a ferrofluid, giving ex pressions for the viscosities, as well as for the average of its magnetization, one can go further and analyze the dynamics of a nonmagnetic particle (magnetic hole) suspended in a ferrofluid. Recently, an increasing interest in the study of the dynamic properties of this magnetic holes has arisen [5, 3, 4]. Although the particles are not magnetic, when they are suspended in a carrier ferrofluid they acquire an induced magnet.ic moment equal to the magnetic moment of the ferrofluid volume they displace. The interaction of these induced magnetic moments of the holes causes a number of pecu liar phenomena, such as the order-disorder transition in magnetic hole lattices and the non-linear phenomena observed in assemblies of holes [7]. Furthermore, knowledge of the dynamics of such particles may constitute a way of characterizing the transport properties of the ferrofluid. For example, the friction coefficient of the particle gives us information about the viscosity of the carrier fluid. The rotational dynamics of the particle is strongly influenced by the presence of a rotating magnetic field. We have found that the hole rotates in the opposite direction to that of the field at low and moderate frequencies of the field rotation, and this result is corroborated by recent experiments, obtaining quite a good agreement in the frequency range we are considering. As we have commented previously, for high frequencies one ought to account for inertial effects, and for the ferromagnetic resonance of the ferrofluid, which is the Larmor precession of the magnetic moment inside the particle excited by an alternating magnetic field. The system holes-ferrofluid can be modeled as a suspension of particles (holes) m a carrier fluid (ferrofluid) because different length scales exist for the ferrofluid and the holes. Consequently, the ferrofluid can be viewed as a continuous medium through which the holes may move. The dynamics of the ferrofluid is governed, at the continuum level, by the aforementioned generalized equations. BIBLIOGRAPHY 19 [33] E. Lemaire, Y. Grasselli and G. Bossis, J. Phys. JI France 2, 359 (1992); J. Phys. JI France 4, 253 (1994). [34] D.J. Klingenberg, C.F. Zukoski, and J.C. Hill, J. Appl. Phys. 73,4644 (1993). [35] S. Miyazima, P. Meakin, and F. Family, Phys. Rev. A 36, 1421 (1987). [36] J.J.M. Janssen, J.J.M. Baltussen, A.P. van Gelder, and J.A.A.J. Perenboom, J. Phys. D 23, 1447 (1990). [37] M.E. van Leeuwen and B. Smit, Phys. Rev. Lett. 71,3991 (1993). [38] R. Zhang and M. Widom, Phys. Rev. E 49, R3591 (1994). [39] G.S. Rushbrooke, G. Stell, and J.S. Roye, Mol. Phys. 26,1199 (1973). [40] J.J. Weis and D. Levesque, Phys. Rev. Lett. 71, 2729 (1993). [41] M.J. Stevens and G.S. Grest, Phys. Rev. Lett. 72,3686 (1994). CHAPTERI DYNAMICS OF FERROMAGNETIC PARTICLES IN SUSPENSION: LIMIT CASES In this chapter we study the dynamics of a ferromagnetic particle and compute the transport coefficients of a dilute suspension constituted by these particles under the action of a constant magnetic field, We will develop a formalism which makes it possible to construct a general scheme to analyze the dynamics of the systern. In par ticular, we carry out an explicit calculation of the viscosities for a dilute suspension of spherical particles. The viscosities are essentially computed by means of Green-Kubo formulas in the linear response theory framework. Moreover, the Smoluchowski equa tion for the orientational degrees of freedom of the particles enable us to obtain the averaged values and the correlation functions involved in the calculations. Depend ing on the ratio Ji/u, comparing both magnetic and anisotropy energies, two simple experimentally reachable regimes show up, and their analysis constitutes the scope of this chapter. The first part of it deals with the limit u � Ji, that will be referred to as suspension of rigid dipoles. In the second part, we have analyzed the implications of the sphere's anisotropy energy on its dynamics at high magnetic field (Ji � u). The rotational viscosity has also been calculated as a function of the anisotropy parameter. For a given geometry, this magnitude is responsible for the in crease of the effective viscosity of the system due to the presence of an external magnetic field. Our results are compared with that obtained by other authors based on different approaches. 20 Part 1 RIGID DIPOLES 21 22 CHAPTER l. LIMIT CASES 1 Introduction Transport phenomena in colloidal suspensions and polymer solutions have played a prominent role in the study of these systems due to their potential applications to different are as of physics, physico-chemistry and biophysics. In dealing with transport phenomena, one is mainly con cerned with two levels of description. In the macroscopic level, one considers the system as a whole and establishes evolution equations for the relevant quantities in the framework of theories of continuum media and non equilibrium thermodynamics. In the other level, one distinguishes the suspended objects from the carrier fluid and uses Langevin or Smoluchowski descriptions which should be complemented with the knowledge of the dynamics of the individual objects, i.e., we have to know the force and the torque exerted by the fluid on the part.icle moving through it. Furthermore, one of the main problems about transport phenomena in different systems is the calculation of transport coefficients, which are characteristic of the response of the system to external forces. This task has been accornplished exten sively for particles of different shape and there exist many well-founded results in the literature [1, 2]. In this part, we will deal with suspensions of dipolar particles cornposed of a carrier fluid and particles having dipolar moments rigidly attached. In particular we will discuss the case of ferromagnetic particles. Thus, one may consider that the magnetic moments are only oriented by the magnetic field, the rotational Brownian motion of the ferromagnetic particles and by the external flow. This last mechanism depends on the shape of the particles. As an example, when the particles are spheres they are oriented by the vorticity of the flow, whereas elongated particles are oriented by the elongational flow as well. The possibility of the orientation of the particles by the magnetic field is responsible for the peculiar behavior of the viscosities of ferrofluids. Particularly, it has been shown that in the case of a suspension of ferromagnetic particles, there appears a new transport coefficient: the rotational viscosity, related to the presence of antisymmetric stresses [3]. The rotational viscosity of the suspension was calculated by Shliomis [4] using a continuum medium approach in which the difference between the vorticity of the fluid and the averaged angular velocity of the particles (spin) leads to the introduction of an internal angular momentum of the fluid elements. For this reason, it beco mes necessary to formulate a new balance equation for this quantity, which is coupled with the momentum balance equation. Our purpose in this part of the thesis is to present a unified formalism able to 1. Introduction n=R. n=R H • n=R e:=R e-n=R • 23 Figure 1.1: In suspension of rigid dipoles the magnetic moments are rigidly coupled to the easy axis of magnetization of the particles and relax together. describe the dynamics of the particles and to compute the transport coefficients. Here we present two versions: one where the viscosities are calculated from a rheological equation of state, giving the expression of the pressure tensor, and the other based on the linear response theory, which assumes the existence of fluctuations in the fl uid and requires the knowledge of the fluctuating dynamics, To this end, we have organized this part of the chapter in the following way. In Section 2, we analyze the dynamics of a ferromagnetic sphere by using an induced force in the Navier-Stokes equation, accounting for the perturbation caused by the sphere due to its motion. We find the expression for the force and torque exerted on the particle which split up into systematic and random contributions, and discuss the origin of antisymmetric stresses. Likewise, in Section 3 the expressions for the shear and rotational viscosities are obtained using a rheological equation of state for the contribution to the pressure tensor due to the presence of particles in suspension. We introduce the Smoluchowski equation and we get the expression for the average of the magnetic moment to obtain the rotational viscosity. Section 4 is devoted to the calculation of the viscosities by using Green-Kubo formulas. For this purpose, we need to know the fluctuation dynamics which follows from fluctuating hydrodynamics [5], and from the analysis of the corresponding Smoluchowski equation for the orientation of the particles [6]-[8]. Finally, in the last section we summarize our main results. 24 CHAPTER 1. LIMIT CASES 2 Dynamics of a ferromagnetic sphere Let us consider a dilute suspension of ferromagnetic spherical particles of radius a immersed in a nonpolar incompressible solvent, under the influence of an external magnetic field. Our starting point for analyzing the stationary motion of the particle will be the linearized stationary Navier-Stokes equation (2.1 ) in which p(r, t) is the pressure field, 1/0 the viscosity of the solvent and ver, t) is the velocity field. Owing to the incompressible nature of the carrier fluid the velocity field also satisfies V'. v= O. (2.2) In Eq, (2.1) we have assumed that the perturbation caused by the motíon of the par ticle may be taken into account by íntroducíng an induced force field ¡i(r,t) [10,11]. Furthermore, we have considered the possíbilíty of fluctuations in the fluid by means of the Langevin-Iike fluctuating source -V'. IlR, coming from the decomposition of the viscous pressure tensor in its systematic and random IlR parts [5]. To be consístent, the induced force should be defined in such a way that stick boundary conditions at the surface of the particle, ver, t) = ü(t) + fi(t) x aii, for Ir - Rcm(t)1 = a, (2.3) are satisfied. In this equation ü and ñ are the translational and rotational velocities of the particle, respectively, Rcm(t) is the position of the center of mass of the sphere, and n == (r - Rcm(t»/W - Rcm(t)l. Since inertial and memory effects relative to the motion of the particle are neglected in the stationary case, the induced force is a surface force and can be expressed as [10, 11] (2.4) where 1(n) is an induced force density per unít area. The velocity field given by Eq. (2.1) ís therefore valíd ín the whole space, even ínside the spheres. To compute the mobility tensor we need, first of all, to know the formal solution for the velocíty field. Thís solution follows from (2.1) by Fouríer transforming in f. In fact, after elíminatíon of the pressure, by applying the transversal projector (1 - kk), wíth k == {, one obtains 2. Dynamics oE a Eerromagnetic sphere 25 (2.5) where we have used the incompressibility condition which, in Fourier space, reads k . v = O and we have introduced the propagator - 1 "" T(/c) = -/c2 (1 - H), 'lo (2.6) with 1 as the unit matrix and Vo the unperturbed velocity field in the absence of the particle. The propagator T(k) is the Fourier transform of the Oseen tensor given by T(rj = _1_(1 + rr), 811'770 r where r == f. This formal solution can be rewritten in real space in the form (2.7) v(rj = vo(rj + J di 'T(i - i') . fi(i') + vR(rj, where the random velocity field is given by the inverse Fourier transform of (2.8) (2.9) The average and correlation of this quantity then follows from the stochastic prop erties of the randorn part of the viscous pressure tensor. According to fluctuating hy drodynamics, llR introduces a Gaussian white noise stochastic process of zero mean and fluctuation-dissipation theorem [5] R - R -, 3 - -, , ( Ilij(k, t)IlI:1(k ,t») = 2kBT '1ot:.ijl:l (211') 8(k + k) 8(t - t ), (2.10) where we have defined t:.ijl:1 = 8ik8j1 + 8il8jk - �8ij8kl. The formal solution (2.8) gives the velocity field at any point. Therefore, for a given point at the surface of the sphere, in view of the stick boundary condition, one has (2.11) where we have introduced the response function p(n, n') p(n, n') = (2a:)3 J if exp (iak. (n - n'» T(k). (2.12) 26 CHAPTER 1. LIMIT CASES To proceed further, we will expand the velocities and the induced force of (2.11) in multipoles [10, 11]. For an unspecified quantity �(a71) of that equation, one has �( A) = � (21 + 1)!! Al. !li'+1 an L..J I! n 0 . '=0 Bere n' is an irreducible tensor of rank 1, i.e. the tensor of rank 1 traceless and symmetric in any pair of its indexes, constructed with the vector ñ. The symbol 0 stands for full contraction and !li'+1 is the multipole of order 1 + 1, which is a tensor (2.13) of rank 1 + 1. These quantities are given by !li'+1 = _!_ J dñ n'�(ñ). 411' Inserting these expressions into the formal solution (2.11), one gets (2.14) v(l+I) = � (2/' + 1)!! 1'(/+1,1'+1) 0 p"+I) + v(l+I) + v(l+I) L..J I'! o R' 1'=0 which relates the multipoles of the velocity and of the induced force density. In this (2.15 ) expression the matrix elements 1'(1+1,1'+1) have been defined as 1'(1+1,1'+1) = (417r) J dñ J dñ' n' 1'(71, ñ') ñ/'. (2.16) The representation of the mobility kernel in terms of irreducible multipoles is diagonal in the indexes 1 and I' [10, 11]. As an example, we have I'C1,1) = .?:!:.I. 37]0 According to equation (2.15) for 1 = O, we get (2.17) (2.18) The multipole v(l) can be computed through its definition outlined in equation (2.14). One has v(l) = _!_Jdñ (ü+ñ x añ) = ü. 47r (2.19) In the same way, by assuming that vo( r') = �. (Rcm +añ) at the surface of the particle, we obtain v�l) = _!_ J dñ �. (Rcm + añ) = � . Rcm. 411' (2.20) 2. Dynamics of a ferromagnetic sphere 27 Since p(ii.) is the force per unit area exerted by the particle on the fluid, the drag force exerted by the fluid on the particle is given by (2.21) according to Eq. (2.14). Joining the results given through Eq. (2.19)-(2.21) and using Eq. (2.18) and the expression for the rnatrix elernent given in Eq. (2.17), we get the hydrodynarnic force -H -- ::(l) F =-67r7]oa(u-f3·Rem-vR), (2.22) frorn which one identifies the friction coefficient 67r7]oa corresponding to the inverse of the rnobili ty, as well as the Brownian force FB = -67r7]oatf¡i). We can also proceed by considering the case 1 = 1 in Eq. (2.12). One rnay show that this equation gives rise to V(2) = 31l(2,2) 0 :¡::{2) + v�2) + v�), where the rnobility 1l(2,2) follows frorn Eq. (2.12) and is given by (2.23) 1l(2,2) = _a_S + �A. 157]0 97]0 In this expression S and A have been defined as the isotropic tensors (2.24 ) (2.25) and 1 Aijkl = "2 (6il6jk - 6ik6jl), syrnrnetric and antisyrnrnetric in any pair of its indexes. The rnobility (2.24), together with the results (2.26) V(2) = _!_ J dñ il [11 + ñ x aii.] = � E . ñ 47r 3 (2.27) and (2.28) obtained by rneans of Eq. (2.14), with E being the Levi-Civita tensor and T standing for the transposed rnatrix, can be used in (2.23). One finally arrives at the expression 28 CHAPTER l. LIMIT CASES a - T (2) a 1 1 .-1'2) -(E·n-� )-VR =-(-S+-A)0.r' . 3 '70 5 3 (2.29) The induced force multipole :¡:(2) may be obtained from this equation. Multiplying both sides of Eq. (2.29) by S, one has (2.30) where, for an arbitrary tensor ofsecond rank T, T(') stands for its symmetric traceless part T��) = �(T." + T" - �TLL6 .. ) '} - 2 '} }' 3 ."" '} . On the other hand, multiplying both sides of Eq. (2.29) by A, one obtains (2.31) :¡:(2a) = '7o(E' ñ + �(a) _ 3'70 v�a), a where, again for an arbitrary tensor of second rank, T(a) is defined as (2.32) (2.33) Moreover, from Eq. (2.29) we can obtain the equation for the hydrodynamic torque since, as in Eq. (2.21), we have (2.34 ) so that only the antisymmetric part of :¡:(2) is involved. Making use of the fact that E : E = -21, in Eq. (2.29), we arrive at (2.35) where we have defined the vorticity Wo == �E : �(a), and one can identify the Brownian torque TB = -127r'7oa2E: v�). As follows from (2.22) and (2.35), the expressions for the force and torque split up into systematic and random contributions. The latter originates from the presence of fluctuations in the fluid and their statistical properties follow from fluctuating hydrodynamics according to Eq. (2.9) and (2.10). Once we have obtained the hydrodynamic force and torque, we can proceed to obtain the equations for the translational and rotational motion of the partícle. Since ferromagnetic particles are very small (radius about 100 Á) inertial effects can be 3. Shear and rotational viscosities of the ferrofluid 35 3.3 Rotational viscosity and the external magnetic field The quantity between brakets in Eq. (3.5) can be transformed by using the equation for the rotational motion of a ferromagnetic particle, which follows from the total angular momentum equation dñ -1 drñ - _ _ - I-;¡¡+"Yo Tt=-{r(O-wo)+mxH (3.3.1) where 1 is its moment of inertia, and "Yo the electron gyromagnetic factor. We will neglect the term coming from the angular momentum of the electrons determining the magnetic moment of the particles. Moreover, if the particle is small enough, as occurs for ferrofluids, inertial effects can also be neglected. Therefore the hydrodynamical and magnetic torques balance each other out and Eq. (3.5) transforms into (3.3.2) In view of (3.3.2), we conclude that the antisymmetric part of the pressure tensor depends on the relative orientation of the magnetic moment and the field. To compute the corresponding transport coefficient namely, the rotational viscosity, T¡r, we have to average this part over all possible orientations of k We then obtain (3.3.3) where the average on the right-hand side follows from the expression for the first moment of R computed by the Smoluchowski equation and given by (3.2.6) along with (3.2.8). We finally obtain (3.3.4) This equation yields the rotational viscosity 3 �-tanh� T¡r = "2T¡0l/J � + tanh s ' This last quantity behaves for � «: 1 as (3.3.5) 1 2 T¡r == ¡T¡ol/J� and is then quadratic in the field. For � � 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 Figure 1.2: The quantity r¡r/r¡. versus the parameter p. The solid line corresponds to our result given in Eq. (3.3.5) and with the result coming from a phenomenological theory proposed in Ref.[4], whereas the dashed line is obtained from the solution of the Smoluchowski equation given in Ref.[6]. The dots represent experimental data from Ref. [13]. 4. Calculatíon of the viscosities using Green-Kubo formulas 37 Consequently, the rotational viscosity tends towards the asymptotic limit �1104>. In Fig. 1.2 we have plotted the reduced rotational viscosity 11r/11., with 11. == �1104>, as a function of the parameter ¡J. The presence of the rotational viscosity leads to the introduction of an effective viscosity of the ferrofluid. For the case of aplanar Couette flow where the applied magnetic field is choosen perpendicular to the vorticity, this quantity gives the increase of the viscosity of the suspensión due to the action of the external field. These resulta agree with the ones obtained in Ref. [4] by means of a phenomenological theory and with the experiments performed by Mc Tague [17]. In our former calculation of the first moment of R, we have neglected a term of second order in the vorticity. This approximation can be removed by computing the distribution function up to second order in this quantity at low magnetic field. The resulting averages can be given, for example, when the vorticity is parallel to fr. For the average of the y-component, one has (3.3.8) which is in good agreement with the corresponding result obtained in Ref. [15] by means of Brownian dynamics. Note, however, that since the ferromagnetic particles are very small, second order corrections in the vorticity are only important at very high shear rates. 4 Calculation of the viscosities using Green-Kubo formulas Our purpose in this section is to present an alternative derivation of the shear and ro tational viscosities. These quantities wiJl be computed from the corresponding Green Kubo formulas arising from linear response theory, which have been widely used to calculate transport coefficients. The contribution of the particles to the shear viscosity, introduced in Eq. (3.4), is then given by (4.1 ) In order to compute the correlation of the symmetric part of the pressure tensor, we will use the relationship between the pressure tensor and the second order multipole of the induced force, given through eqs. (3.2) and (2.30). Provided that our reference state should be in equilibrium conditions (f3 = O), this relationship is the foJlowing 38 CHAPTER I. LIMIT CASES TI (.) ( ) - 20 2 (2.) ( ) p,:Cz t - - 1ra 7]oV R,:c' t , (4.2) which used in (4.1), yields (4.3) The correlation in this expression can be computed from fluctuating hydrodynam ics. In fact, from (2.9) and (2.10) we get In real space this expression is given by (4.5) Using now the multipole expansion, for points at the surface of the particle, we can obtain the corresponding correlation of the second multipole of the random velocity. One arrives at (4.6) where the matrix elernent 1-'(2,2) has been given in (2.24). For the symmetric part of this correlation, one has (4.7) Employing this expression in (3.3) and performing the integral, we then come to the result (4.8) which corresponds to the contribution to the viscosity of the suspension owing to the presence of particles. The rotational viscosity can be computed by rneans of the same procedure. The Green-Kubo formula is now T/r = Vk1BT 100 dt(TI��lz(t)TI��lz(O)) or in terms of the corresponding axial vectors (4.9) 4. Calculation of the víscosirfes using Green-Kubo formulas 39 n- = Vk�T 100 dt(n��J(t)n��J(o»). Making use of equations (3.2) and (2.32) and the definition (3.1.3), we get (4.10) (4.11) where we have employed again the requirement f3 = O, for the reference state. The correlation in Eq. (4.10) then follows from this last expression. One has (II��J(t)n��J(o») = (4'11"a3)2r¡�([Oy(t) + 23a (f : v�")(t»Yl [01/(0) + 23a (f : v�")(O»!ll) (4.12) or alternatively 2 (n��J(t)n��J(O») = (41Ta3)2�� (m x H)y(t)(m x H}y(O»), (4.13) with �r = 81Ta3r¡o being the rotation friction coefficient. To obtain this equality we have used the equation of rotational motion of the particle (4.14) where the expression for the hydrodynamic torque is given in (2.35) and fM is the magnetic torque. In (4.14) we have neglected inertial and gyromagnetic effects, thus the torques balance each other out. This fact provides the equation to derive (4.13) from (4.12). The rotational viscosity can be finally expressed as 3 2 roo ,, TJr = 2¡fJTJoDrJl Jo dt(R.,(t)R.,(O»), (4.15) where Jl is the Langevin parameter n;;�, comparing magnetic and thermal energies, and we have taken the magnetic field pointing towards the z-direction. To perform the integral in (4.15) we have to know the correlation function for the x-component of the vector Él. The evolution equation for this correlation comes from the Smoluchowski equation (4.16) valid in the absence of vorticity. In fact, starting from (4.16) one may derive the evolution equation for the correlation 40 CHAPTER l. LlMlT CASES d(R(�/l(O») = Dr{ -2(R(t)R(0») + ¡d! (R(O)} - IJ(R(t)R(O)R(t») . Ji}, ( 4.17) which for the perpendicular component yields (4.18) To salve this equation we will introduce a decoupling approximation as in the previous section. This approximation consists of decoupling the perpendicular and parallel components of the vector R, due to the different nature of the dynamics of both components, as we already pointed out, and can be formulated in the following way: (4.19) In addition, in the linear regime, we may approximate in (4.19) (RII(t») by its equi librium value (RII(O»)eq = C(IJ). This equilibrium average is computed with the equilibrium distribution function IJ A A t/!eq = . h exp(IJR· H). 411'SIn IJ Consequently, Eq.( 4.18) transforms into the relaxation equation ( 4.20) (4.21) from which we may identify the relaxation time TJ. (4.22) Thus we expect that the correlation decays exponentially according to (4.23) The rotational viscosity then follows from the Green-Kubo formula (4.15) together with (4.23). Performing the integral, one arrives at (4.24) In view of the result (R.:(0)2)eq = �, which can also be obtained by computing the equilibrium average with the equilibrium distribution function (4.20), one obtains 5. Conclusions 41 3 Iltanh Il TJr = "2 TJo4J Il + tanh Il ' (4.25) which, as we expected, also coincides with the corresponding expression obtained by Shliomis using a continuum theory [4] and with the result of [7]. 5 Conclusions In this part we have analyzed the dynamics of a ferromagnetic spherical particle, as well as the viscosities of a dilute suspension constituted by these particles. Our theoretical framework is based on the Navier-Stokes equation in which an induced force, resulting from the perturbation introduced in the dynamics of the fluid by the particle, and a stochastic Langevin source, coming from the fluctuations of the hydrodynamical fields, have been included. This equation accounts for the coupled dynamics of the fluid and the particle. A multipolar expansion of the quantities appearing in the formal solution of the Navier-Stokes-Langevin equation leads to the expressions for the force and torque exerted on the particle which contain random contributions whose statistical properties are dictated by fluctuating hydrodynamics [5]. We use two methods to obtain the shear and rotational viscosities. One is deter ministic and is based on the Kirkwood formula for the viscous pressure tensor which is shown to be related to the second order multipole of the induced force. The other is based on the linear response theory giving the transport coefficients in terms of time-deperident correlation functions. The presence of the field is responsible for the appearance of an antisymmetric contribution to the pressure tensor and introduces the rotational viscosity as a new transport coefficient. The existence of antisymmetric stresses comes from the fact that the torque exerted by the magnetic field on a dipole and the hydrodynamic torque balance each other out. Consequently, the angular ve locity of the particle may differ from the value of the vorticity of the fluid at the point it occupies. We have studied here a simple case in which the transport coefficients have already been calculated by other authors by means of different methods. Our main purpose has been to develop a rather general formalism that can be useful in the study of transport phenomena in ferrofluids in different and more intricate situations, which constitute the subject of sorne of the following parts of the monograph. Furthermore, the methods developed here can also be applied to study the dependence of the viscos ity on frequency [16], or for higher concentrations, in which case dipolar interactions 42 CHAPTER J. LJMJT CASES play an essential role. Work is being done in the last case in order to find further theoretical results. Part II 80FT MAGNETIC MATERIAL8 IN HIGH MAGNETIC FIELD8 43 44 CHAPTER l. LIMIT CASES n __ n n H Figure 1.3: For soft magnetic materials in sufficiently high magnetic fields the magnetic moments orient themselves in the direction of the field very quickly. Then it takes place the mechanical rotation of the particles to the stationary orientation. 6 Introduction In this part of the chapter we apply the formalism developed in the previous part to the case in which after perturbing the system, the relaxation towards the magnetic field takes place in two steps: first a quick relaxation of the magnetic moment, then a mechanical rotation of the particle towards the equilibrium orientation, with the magnetic moment also parallel to the easy axis of magnetization [19]. This situation occurs for high magnetic fields and the energy of the magnetic particles reduces to the energy of anisotropy. Experimentally, it is found that the magnetization tends to lie along certain crystallographic axes; this effect is known as crystalline anisotropy. It is additional to the directional effects that occur when the samples' shape lacks spherical or cubic symmetry. The existence of crystalline anisotropy may be dernon strated by the magnetization curves. It is clear that much smaller fields are required to magnetize the crystals to saturation along certain directions than along others. The crystallographic axes along which the magnetization tends to lie are called easy directions; the axes along which it is most diffícult to produce saturation are called hard directions. With this goal in mind, this part has been distributed as follows. Section 7 is intended as a short review of the rotational dynamics which was previously analyzed. 9. Conclusions 51 Ref. [18] using a phenomenological approach, based on the formulation of a relaxation equation for the internal angular momentum of the system in a continuum description. However, we have observed again sorne discrepancies in our results with respect to the ones obtained by those authors from an approximated solution of the Smoluchowski equation [18], especially for intermediate values of the anisotropy parameter. 9 Conclusions By means of the formalism developed in the previous part, we have presented in this part a calculation of the rotational viscosity based on the Green-Kubo formula for this transport coefficient. Our results show that the viscosity increases when increas ing the anisotropy parameter and reaches a saturation limito We have compared our curve with the ones obtained in Ref.[18]. Their result coming from a solution of the stationary Smoluchowski equation overestimates ours, whereas the one based on a phenomenological relaxation equation they propose for the internal angular momen tum of the suspension, is closer to ours, It would be interesting to gather experimental data in order to decide which of the three approaches is the most convincing. We have also computed the rotational viscosity by means of the rheological equa tion of state proposed by Kirkwood, giving the pressure tensor in terms of the hy drodynamic force exerted by the fluid on the magnetic partide. The average of the pressure tensor has been calculated from the stationary solution of the Smoluchowski equation (9.1 ) which now contains a contribution due to the vorticity of the flow. Following the steps indicated in [7] we get the same result (8.22). In the same way, we could solve the more general case in which both, the relaxation of the magnetic moment towards the field and towards the easy axis of magnetization, take place. Sorne related theoretical results have been obtained in [1] and it will be the subject of the following chapter. Bibliography [1] J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (Kluwer Aca demic Publishers, Dordrecht, The Netherlands, 1991). [2] H.L. Frisch and R. Simha, in: Rheology, vol.Il, F. R. Eirich, ed. (Acad. Press, New York, 1956). [3] S.R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (Dover,New York,1984). [4] M. 1. Shliomis, Sov. Phys. JETP 34 (1972) 1291. [5] L. Landau and E. M. Lifshitz, Statistical Physics, Part 2 (Pergamon Press, Ox ford, 1981). [6] M. A. Martsenyuk, Yu. L. Raikher and M. 1. Shliomis, Sov. 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. Pérez-Madrid, in: Complex Fluids, L. Garrido, ed., Lecture Notes in Physies, vol. 415 (Springer-Verlag, Berlin). [9] M.C. Miguel, J. Bonet Avalos, A. Pérez-Madrid and J.M. Rubí, Physiea A 193 359 (1993). [10] P. Mazur and D. Bedeaux, Physica 76 (1974) 235. [11] P. Mazur and W. Van Saarloos, Physiea 115 A (1982) 21. [12] H. Brenner, J. Colloid. Interface Sci. 1 (1970) 141. [13] J. P. Me Tague, J. Chem. Phys. 51 (1969) 133. 52 BIBLIOGRAPHY 53 [14] C. Salueña, A. Pérez-Madrid and J. M. Rubí, J. Colloid. Interface Sci. 164, (1994) 263. [15] A. O. Cebers, Magnit. gidrodin.4 (1984) 17 (Magnetohydrodynamics,1984). [16J C. Salueña and J. M. Rubí, J. Chem. Phys. 102 (1995) 3812. [17] J.-C. Bacri, K. Djerfi, S. Neveu and R. Perzynski, J. Mag. Mag. Mater. 123 (1993) 67. [18] Yu. L. Raikher and M. 1. Shliomis, SOy. Phys. J. Appl. Mech. Tech. Phys. 15, (1974) 470. [19] M.I. Shliomis and V.1. Stepanov, J. Magn. Magn. Mater. 122 (1993) 196. CHAPTER II GENERAL DYNAMICS In the previous chapter we have analyzed the dynamics of a ferromagnetic particle and the transport coefficients of a dilute suspension constituted by them in two lim iting cases, namely, a suspension of rigid dipoles and a suspension of a soft magnetic material under the influence of a high magnetic field. The main goal of this chapter is to provide a general theory covering the whole range of experimental situations. Indeed, different measurements carried out for mag netic liquids show that there is finite coupling between the orientation of the mag netic moment of a ferromagnetic monodomain and the orientation of the particle itself (characterized by the orientation of its crystalline axes). Because of this coupling, the relaxation of the magnetic moments takes place in two different ways that proceed simultaneously: rotation within the particle and together with the particle with re spect to the carrier liquido Both processes are of rotational diffusion type. Thus, for all practical purposes, the Smoluchowski equation, describing the evolution of the probability density for the directions of both the magnetic moment and the axis of easy magnetization of the particles, has been used. In particular, we compute the rotational viscosity from a Green-Kubo formula and give an expression for different relaxation times. These characteristic times come from the dynamic equations for the correlation functions which, in the linear response theory framework, are involved in the calculation of sorne of the material's physical properties we are interested in (optical, magnetic, ... ). Our results agree quite well with experiments performed with different samples of ferromagnetic particles, which permit to distinguish the different relaxation regimes occurring when the size and the nature of the magnetic material of the grains are freely modified. 54 1. Introduction 55 1 Introduction Systems of single-domain ferromagnetic particles immersed in a solid or liquid phase exhibit a number of interesting relaxation phenomena which have been the subject matter of many experimental and theoretical analyses [1]-[4]. These phenomena are essential in the study of the dynamics of these particles, and, particularly, have a c1ear influence when determining the effective viscosity, the dynamic birefringence, and the magnetic susceptibility. One of the main peculiarities of these systems is that their properties are greatly influenced by the presence of an external magnetic field. It is precisely this fact which has been the basis of many practical applications [1]. The rotational dynamics of a ferromagnetic particle embedded in a liquid phase is the result of the competition of three orientational mechanisms related to the exter nal field, the axis of easy magnetization, and rotational Brownian motion. That is, whereas the magnetic moment of the ferromagnetic particle relaxes towards the direc tion of the magnetic field, the axis of easy magnetization tends to be aligned with the magnetic moment, thus giving rise to different coupled relaxation phenomena. Un til recently, the most frequent case that has been studied in the literature deals with rigid-dipoles [4]-[6], for which the anisotropy energy is dominant due to the large value of the anisotropy constant, and because the radius of the particle usually exceeds a critical value. What is more, when looking to the relaxation phenomena described by a rigid dipole, one disregards the precessional motion of the magnetic moment, and, consequently, the associated dissipation. Under these conditions, we cannot talk about the relaxation of the axis of easy magnetization towards the magnetic moment any longer, instead both vectors relax together. However, there are materials for which the anisotropy energy may be comparable to the energy associated with the interaction with the magnetic field, or even smaller. Therefore, a general theory en compassing such a wide variety of situations and accounting for experimental results should be developed. The presence of different relaxation mechanisms has implica tions in the form of the effective viscosity of the system, which exhibits significant corrections when compared to the viscosity of a suspension of non-magnetic particles of the same shape. Another point of interest is the appearance of relaxation times which are usually involved in the characterization of certain physical properties, and which are suitable for being measured by means of different experimental techniques. The purpose of this chapter is to present a theory capable of giving expressions for the relevant transport coefficients of the system and of the corresponding charac teristic relaxation times determining for instance, the effective viscosity, the dynamic 56 CHAPTER Il. GENERAL DYNAMICS n n H Figure 11.1: The magnetic moments are oriented along an intermediate direction be tween that of the magnetic field and the easy axis of magnetization when both the anisotropy energy and the magnetic energy of interaction with the field are compara ble. The relaxation of these two degrees of freedom is coupled. birefringence, and the magnetic susceptibility of the suspension. We will focus on the general situation in which the magnetic and anisotropy energies of the particles may have arbitrary values. The formalism we have developed is based on the linear response theory where the correlation dynamics comes from a Smoluchowski equa tion. As we will show in one of the sections, our result for the relaxation time of the rotation of the particle is compared to experimental data and agrees quite well with birefringence experiments. We have distributed the chapter in the following way: in Section 2, we establish basic equations describing the dynamics of the degrees of freedom. Of particular interest is the Smoluchowski equation for the probability density which is given in a general case for unspecified values of the magnetic and anisotropy energies. There are different ways of deriving such equation. Although we will not go through this question, the equation we propose can be compared to another one obtained previously from a different theoretical method. In Section 3, we deal with the calculation of the rotational viscosity using a Green-Kubo equation proposed from the linear response theory. This method leads to an expression for this transport coefficient which is studied in particular situations ofinterest. Section 4 is devoted to the calculation ofthe relaxation times of the particles and the transversal component of the magnetization 2. Coupled dynamícs oE the degrees oi Ereedom 57 when considering the different orientational mechanisms. We have compared our results to experiments done for two samples of very common ferromagnetic particles for which the size and the nature ofthe magnetic material clearly establish different values of the magnetic energy and the energy of anisotropy, obtaining a good agreement in both situations. Finally, in the last section we summarize our main results. 2 Coupled dynamics of the degrees of freedom The energy of a spherical single-domain ferromagnetic particle under the action of an external magnetic field is the sum of two contributions. These contributions origi nate from the externally imposed magnetic field and the presence of an axis of easy magnetization (for uniaxial crystals). Its expression is given by (2.1 ) where ñi = mR is the magnetic moment of the particles, H is the external magnetic field, Ka is the first anisotropy constant (assumed positive), Vm is the magnetic volume of one of these spheres, and n is the unit vector along the direction of the axis of easy magnetization for .uniaxial magnetic crystals. It is clear from Eq. (2.1) that in the general case where both contributions may take arbitrary values, the relaxation mechanisms of the degrees of freedom, R and ñ, of the ferromagnetic spheres in suspension are coupled. The deterministic dynamics of R is governed by the Landau-Gilbert equation [7], proposed to study the relaxation of the magnetic moments of magnetic particles ern bedded in a solid matrix dR 'Yo {)U • dR . - = ---_ x R0/- X R. dt m {)R dt (2.2) From this equation, one may identify the two mechanisms responsible for the varia tion of k the effective field HeJl ex: -�, which causes a Larmor precessional motion of R, and the mean field, H d ex: - 4ft, which introduces a damping due to the col lisions of the electrons deterrnining the magnetic moment of the domain in a metal, or in a semiconductor, due to magnetoelastic interactions. In Eq. (2.2), 'Yo is the gyromagnetic ratio for an electron, and the quantity O/ plays the role of a darnping coefficient. The Landau-Gilbert equation can be rewritten such that 58 CHAPTER n. GENERAL DYNAMICS dEl _ au _ _ - = -hR X -_ x R + WL X R dt aR ' (2.3) with WL = gDe" as the Larmor frequency of the precessional motion, and where 9 == 1'0(1 + 0'2)-1 and h == �(1 + 0'2)-1 [2]. This equation is valid in a stationary frame of reference. If the ferromagnetic particle is rotating itself with the angular velocity ñ, we must modify Eq.(2.3) by adding on its right hand si de the corresponding contribution coming from the rotation. One then has dH ( _ au) _ _ _ di = -h R x aH x R + (WL + n) x R. (2.4) Furthermore, the dynamics of ñ follows from the kinematic relation dñ - di = n x ñ, (2.5) This expression can be rewritten as dñ [_ 1 - -] _ -d = Wo + -m x H x n, t �r with �r = 811'7]oa3 being the rotational friction coefficient of the particles; 7]0 is the (2.6) viscosity of the carrier fluid, a is the hydrodynamic radius of the particles, and Wo the vorticity of the carrier fluid. One arrives at this expression after using the deterministic part of the balance equation for the total angular momentum (2.7) provided that we neglect the ter m accounting for the inertial effects and the ter m com ing from the angular momentum of the electrons determining the magnetic mornent of the particles. Here fB is the Brownian torque acting on the particle [8]. The evolution ofthe probability density, 1/;(¡,t), with "Y == (H,ñ), is governed by the Smoluchowski equation. When ñ = O, which corresponds to the case of particles embedded in a solid matrix, the Smoluchowski equation was deduced by Brown [9] from the Landau-Gilbert equation. In these last conditions, and in order to preserve the total angular momentum conservation, the small magnetic particle should be embedded in a large rigid solid matrix. Otherwise, there will appear an elastic twist in the matrix, and one should take into account the corresponding elastic torque in the equation for the total angular momentum of the particle. The manner in which Brown expressed his intuitive method of deriving the Smoluchowski equation was to consider 2. Coupled dynamics oi the degrees oi Ereedom 59 the effect of thermal fluctuations on the probability density. Brown suggested that thermal agitation causes tP to become more uniform so that, in an equation describing its time evolution, thermal agitation gives rise to a diffusion term in tP. Shliomis and co-workers [10] obtained this equation for a suspension of rigid dipoles. If we define the dimensionless parameters /J = �lf and (7 = K,,"aF, comparing magnetic and anisotropy energies to thermal energy, respectively, this last situation corresponds to the limit (7 � /J. Rhaiker and Shliomis [11] also proposed the Smoluchowski equation for the opposite limit (7 <€:: /J, in which the dipoles are rapidly oriented towards the field direction. As regards the general case for arbitrary values of the ratio /JI (7, Shliomis and co-workers also deduced the appropriate Smoluchowski equation from a model similar to the itinerant oscillator model, [12]-[14]. In such a general situation, the Smoluchowski equation can al so be obtained from the continuity equation in the space spanned by the degrees of freedom "'{ == (R, n) fJtPb,t) a « ) ( ).) at =-8"'(. J"'{,t +tP"'{,t"'{, where � == d"'{ldt and the current Jb,t) is given by the Fick's law (2.8) Jb, t) = -D . atPb, t), 8"'( (2.9) with D being a diffusion matrix. Combining (2.8) and (2.9) we then arrive at atP a a . 8t = 8"'( . (D . 8"'( tP - tP"'{). (2.10) The diffusion matrix is related to the mobility matrix b through the Einstein relation, D = kBTb. After inserting eqs. (2.4) and (2.6) into (2.10), we can rewrite Eq. (2.10) in the forrn (2.11) where we have a contribution coming from a non-potential current -Ynon-pot, and the mobilities b are found to be (for more details see Appendix A) - ;r (1 - nn), 1 '- - (h + {r)(1 - RR), TI,' " bRn == {r [(n . R)1 - nR]. (2.12) 60 CHAPTER II. GENERAL DYNAMICS Here the symbol T stands for transposition. These expressions can be employed in Eq.(2.11). After sorne mathematical transformations (see detailed calculations in Appendix A), one then arrives at the Smoluchowski equation fN &t = --- - U -- Dr(nR + nñ)· [tP(nR + nñ) kBT + (nR + nñ)tP] - - U - - DmnR· {,pnR kBT + nRtP} - nR· (WLtP) ('RR + ññ) . (wotP), (2.13) + h -ñ=' 8 d--R' 8 ·1 kT· w ere "-ñ n x 8ñ an n R = x 7iR are rotationa operators, D; == T IS the Brownian rotational diffusion coefficient, and Dm == kBTh can be interpreted as the diffusion coefficient of the magnetic moment inside the particles. These diffusion coefficients are related to two relaxation times involved in the Smoluchowski equation, namely TD = (2Dm)-1 related to the chaotic reorientations of ni inside the particle due to thermal fluctuations, and the Brownian time TB = (2Dr )-1. The Smoluchowski equation (2.13) agrees with the corresponding one obtained in Ref. [12] by using a model similar to the itinerant oscillator model and will be used in our subsequent analysis. 3 Green-Kubo formula for the rotational viscosity In this section, we we will focus on the determination of the rotational viscosity from the corresponding Green-Kubo formula. This formula gives this transport coefficient in terms of the correlation function of the axial vector, ñ�a), related to the antisyrn metric part of the contribution of the particles to the pressure tensor [8], (3.1 ) where V is the volume of the system. In Ref. [8], W! obtained a relationship between the particle contribution to the pressure tensor, ñp, and the external torque experienced by the particle during its motion. As a result, if we consider the magnetic field pointing towards the z-direction, it is found that the rotational viscosity can be finalIy rewritten as 3 [''''' T}r = "2 rPT}o ti. 1-'2 Jo dt(Rr(t)Rr(O»), with rP = 4/3;a' being the volume fraction of particles. (3.2) 3. Green-Kubo formula {or the rotational viscosity 67 1.0 �----...,.-----�-----�----� 0.8 DjD,=1 11=100 0.6 0.4 0.2 0.0 �----�----�------._----� O 5 10 15 20 o Figure 11.3: Rotational viscosity versus the parameter O" for different values of the ratio Dm/ D,. and of the parameter ¡J. 68 CHAPTER Il. GENERAL DYNAMICS respectively, which also coincide with the results given in Ref.[12]. Even in this case, with ¡t � 1, if u takes moderate or small values, the rotational viscosity does not reach its saturation value any longer. In particular, if u - O, this saturation value depends on the ratio Dm/Dr as we have pointed out in Eq. (3.22). That is, the dissipation not only depends on the solvent viscosity, but also on the damping constant O' and the gyromagnetic factor 10. Additionally, the rotational viscosity increases with u until it reaches its saturation value when u - oo. The behavior of '7r / '7. as a function of u, and for different values of ¡t is depicted in Fig. 11.3, from which we can corroborate the main features of our previous analysis. The ratio Dm/ D; has been obtained after considering the following values of the involved quantities: O' '" 10-2, 10 '" 107e-1 s-1, M. '" 103e, '70 '" 1O-2c.!J; and the fact that the magnetic volume is almost the same as the hydrodynamic volume of the particles, with a = 10-6cm. An alternative procedure used to calculate the viscosity involves a rheological equation of state for the pressure tensor. In this way, it is also possible to obtain exactly the same expression for the rotational viscosity (3.16), but now in the presence of a linear vorticity field in stationary conditions. 4 Relaxation times. Comparison with experiments 4.1 Transient birefringence in crossed fields The relaxation of the optical birefringence induced by magnetic grains dispersed m the analyzed medium, is among the simplest experimental tools available for rheologi cal studies of viscoelastic solutions. A viscosimeter based on the determination of the relaxation of small magnetic particles in suspension in the studied medium turns the viscosity determination into an optical birefringence measurement with many advan tages: i) it is a nondestructive method, ii) we just need to add a very small amount of particles (<p '" 10-4), iii) it does not need any mechanical system, iv) the use of a laser beam for the birefringence measurement enables us the determination of the viscosity in the volume of a few mm3. In addition, as the size of the magnetic probe is of the order of 100A, the measured viscosity is a local quantity. The main limitation of the usual transient birefringence devices is the polydispersity of the magnetic particles. Usually, a log-normal distribution of spherical grain diameters, d, as the one given through the expression 4. Relaxation times. Comparison with experimente 69 P(d) = �CTd exp ( - 2�2 In2 (�) ) , (4.1.1) where do and CT are the mean diameter and the variance of the distribution, respec tively, is suitable to describe the samples. Thus, in order to get an exponential relax ation with only one characteristic time a sample whose value of CT is about 0.3 or less must be used. Each particle is first characterized through a static birefringence measurement as a function of the magnetic field. This is due to the anisotropy of the electric susceptibility tensor relating the polarizability of the system to the incident electric field. The main causes of the anisotropy are the internal optical anisotropy of the magnetic material (crystalline anisotropy) and the shape anisotropy of the particles. Concerning the dynamic behavior of the birefringence, different experiments can be carried out: In a liquid, if the magnetic field is switched off abruptly, particles thermally relax towards random directions. Thus looking at the birefringence relaxation is comparing viscous to thermal energies. If a square pulse of magnetic field H is applied to the ferrofluid solution, magnetic particles first tend to align along the field leading to birefringence 6n(t), and, as the field is switched off, they therrnally relax to random directions. Birefringence decreases exponentially according to Perrin's law [18) 8n(t) = 8n(d, H)exp-t/T(d) (4.1.2) where the characteristic time is T(d) = (6Dr )-1, with D; the rotational diffusion coefficient. In the quantity 8n(t), the polydispersity of the samples modifies both 8n(d, H) and T(d). Owing to this fact, .6.n(t) is not a simple exponential function of time. But, in order to characterize Lln(t), one can use the shortest time deduced from the initial slope of the function In(.6.n(t)) versus t. In a viscoelastic medium, looking at the birefringence response to an alternating magnetic field allows to reach the viscosity and the elastic modulus of the medium at high frequencies. If a large magnetic field is superimposed, there is a magnetic restoring force and viscous energy is compared to that. Instead of relaxing at random, magnetic moments relax towards the direction of the superimposed field. In this case, the relaxation time associated with the orientation mechanism of the particles, TR, has been measured recently by Bacri et al. [19, 20). This quantity comes from the relaxation time of the light intensity collected in a photocell after crossing the sample in the presence 70 CHAPTER TI. GENERAL DYNAMICS of an external magnetic field, and when applying additional pulses of magnetic field to perturb the sample. These experiments permit us to distinguish the different relaxation regimes occurring when the size and the nature of the magnetic material of the particles are modified. These regimes are determined by the parameters 1-' and u, comparing magnetic and anisotropy energies with thermal energy, respectively. In the experiments, it was observed that ferrofluid particles, prevented from moving by being quenched in a tight gel network, do not exhibit birefringence although they still show magnetization. Consequently, the birefringence of the solution is closely related to a mechanical alignment of the particles along the equilibrium orientation. In Refs. [8, 15, 21], two opposite limits were considered, one where u � 1-' (rigid dipole approximation) and another for which u «: 1-'. For the forrner limit, the re laxation time for the perpendicular component of the magnetization is found to be [8, 15] ( 4.1.3) which, in the case when 1-' -+ 00, tends to -DI = 1!..a. In the latter case, relaxation rIJ IJ occurs in two steps, first a quick relaxation of R towards il, then a mechanical rotation of the particle to the equilibrium orientation with the easy axis of magnetization parallel to R and ñ. Under these conditions, the characteristic relaxation time could be obtained from the equation of motion of whatever component of the correlatíon function (ñl.nz)(t)(nl.nz)(O)} [21]. Performing the correspondíng decouplings in its evolution equation, we obtain the relaxation time ( )-1 ( 1 )-1 ( 1 )-1 T = 2Dr u + Q(u) = TB U + Q(u) , (4.1.4) which for u -+ 00, tends to 2D1 = zu , -o u In the present analysis, we consider the less stringent case for which the ratio 1-'/ u may take arbitrary values. Consequently, both pararneters 1-' and u, are expected to determine the relaxation time associated with the rotational relaxation of the particles. As for the limit u «: 1-' discussed previously, the appropriate quantity for describing this mechanical relaxational motion is again a component of the correlation function (nl.nz)(t)(ñl.nz)(O)}. From the Smoluchowski equation, we obtain its dynamic equa tion �r d(nrnz)(tltrnz)(O)) = I-'(nr(n· R))(t)(nrnz)(O)) -1-'(n;Rr)(t)(nrnz)(O)} -1-'(nrnzRz)(t)(nrnz)(O)} - 6(n",nz)(t)(nrnz)(0)}. (4.1.5) 4. Relaxation times. Comparison with experiments 71 Proceeding along the same lines as in the previous section, we arrive at a closed set of three differential equations for the correlation functions (Rz(t)(nznz)(O»), «n ... (ñ. R))(t)(nznz)(O)), and «nznz)(t)(nznz)(O»). It is worthwhile pointing out that the quantities at time t appearing in the three independent correlation functions are the same as in the previous section. For the sake of simplicity, we wiIl introduce the vector Q = (Q 1, Q2, Q3), w hose com ponents are the Laplace transforms of (Rz (t)( nz n z )( O»), {(nz(ñ.R»(t)(nznz)(O»), and {(nznz)(t)(nznz)(O)}, respectively; and the vector QO = (Q¡, Q:í, Q3) representing the initial values of these correlation functions. These initial values can be calculated with the equilibrium probability density at t = O. The system of differential equations can be written in matrix notation as (4.1.6) with Á the coefficients matrix (3.13). We are particularly interested in the relaxation dynamics of «nrnz)(t)(nznz)(O»). From Eq. (4.1.6), we obtain its Laplace transform from which we can identify the relaxation time we are interested in - D-1 l' Q3(S) TR - r Im-Qo .-0 3 ( 4.1.8) In figure H.4, we have represented the relaxation time TR versus jJ for different ferromagnetic samples, but, in order to compare with experimental data from Ref. [19], we have also represented the relaxation time TR versus H-1 in figure 11.5. The data correspond to two samples of magnetic particles of the same mean size but made of different magnetic materials, namely CoFe204 and I - Fe203. The Co ferrite sample has an anisotropy constant K = 2 . 105�, and the saturation value of the magnetization is M. :::: 250�A. For the maghemite, the anisotropy constant is K = 4 . 103� and M. :::: 270�A. With these values, the Co-ferrite sample can be considered as a rigid dipole (O" � jJ). On the other hand, the maghemite particles are such that jJ � 0". Regarding the values of the anisotropy constant, 0", and the ratio m/kBT = jJoM. Vm/kBT , we have taken 0""'" 15, m/kBT,..., 2.8· 1O-4:¡ for the maghemite and 0""'" 565, m/kBT,..., 1.8· 1O-4:¡ for the Co-ferrite. For both samples, the values of the remaining quantities are: TB ,..., 4.5ms and Dm/ D; ,..., 1. As it was observed in the experiments for the Co-ferrite sample, TR tends to zero when H - oo. Both R and ñ, quickly relax towards the field direction due to the 72 CHAPTER II. GENERAL DYNAMICS 1.5 l' (ms) 1.0 R Figure H.4: Relaxation time of the particles as a function of J.l. for the Co-ferrite and maghemite samples. 4. ReJaxation times. Comparison with experimente 73 rigidity of the dipoles. For the maghemite sample TR tends to a fixed, non-vanishing value ( ..... 0.3), which can also be obtained from Eq. (4.1.4). Under these particular conditions, the magnetic moments rapidly relax towards the field direction, but due to the moderate value of (T, the relaxation of the easy axis of magnetization, ñ, or in other words, the mechanical relaxation of the particles, takes place in a finite period of time. Concerning the behavior of TR when H - O, we observe that TR - 6b. = Zf independently of the value of the parameter (T. This time corresponds to the well known characteristic relaxation time of the correlations of the components of the second order tensor (fin - f) for a purely diffusive process [27]. In figure 11.5, we have also represented the extrapolation of the relaxation time coming from rather simple considerations made in Ref. [22] for the rigid dipole limit under the action of a very large external magnetic field. Our results agree with the asymptotic behavior in its validity range, but at the same time, they show the deviations at intermediate and low magnetic field. These simple arguments can also be proposed for the opposite case p � (T reproducing the asyrnptotic value TR ..... 0.3ms for the maghemite sample, but they are not able to explain the p-dependence of the relaxation time for this material. 4.2 Transverse complex susceptibility In this section, we are interested in the linear response of the dispersion to an ae magnetic field of small amplitude. From an experimental point of view, it is much easier to observe different effects such as the saturation of the magnetization or the frequency dependen ce of its relaxation for a ferrofluid than for a polar dielectric fluid, because both the field strength and the frequencies required are much lower. Moreover, as we will see below, in a magnetic fluid the variation of the complex susceptibility with frequency depends on the relaxation time of the magnetic momento In the absence of an external de field, Raikher and Shliomis [23] derived expres sions for the complex ae susceptibility, X(w) = X'(w) - iX"(w), of a single domain uniaxial particle. They calculated both the parallel and perpendicular susceptibilities with respect to the easy axis of magnetization. For that particular situation, they showed that the frequency dependen ce of the susceptibility was at the sarne time a function of the parameter (T, comparing anisotropy energy to the thermal energy. The applied ac field orientates both the magnetic moments and the particle axes, but in a linear approximation, i.e. for small values of the applied ac field, they neglected 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 Figure 11.5: Relaxation time of the particles as a function of H-1 for the Co-ferrite and maghemite samples. Experimental data of Ref. [19] correspond to the dots. The dashed line corresponds to the limit J.l � 1 for the Co-ferrite, Ref.[22]. 4. Relaxation times. Comparison with experiments 75 the field-induced partiele orientation. Thus, for a weak measuring field, the partí ele anisotropy axes were just oriented in a random fashion due to Brownian motion. Moreover, in Ref. [4] the authors obtained the expressions for the longitudinal and transverse components of the complex dielectric susceptibility tensor for a system of noninteracting polar molecules under the simultaneous action of a constant external electric field and a small ae electric field. This situation would be equivalent to the previously mentioned rigid dipole approximation. In this case, the susceptibilities depend on the bias field. For experimental measures in a liquid, it is much simpler to determine the parallel and perpendicular susceptibilities with respect to a fixed direction like that of the de field than with respect to the axis of easy magnetization of the partieles. Thus, in this subsection, we obtain the expression for the perpendic ular susceptibility with respect to the bias field ii, i.e. the perturbing external field j{' .1 j{, when not only the magnetic moments but also the partieles themselves are oriented by the external field, such as we have described throughout the chapter. We obtain that the susceptibility depends on the bias field through the parameter /-l. From linear response theory, the decay of the magnetization perpendicular com ponent under the influence of a constant field H, another small constant external field H' (H' .1 H and such that mH'JkBT < 1) having been switched off at time t = O, is (MJ.(t)) - (MJ.(O)) = Xl.H'CJ.(t), (4.2.1) where (4.2.2) is the perpendicular component of the magnetization, 2 xl. = �:T ({Ri(O)) - (RJ.(0))2) is the perpendicular component of the static magnetic susceptibility, and (4.2.3) ( 4.2.4) is the autocorrelation function of any perpendicular component of the magnetization. The corresponding complex magnetic susceptibility XJ.(w) is (4.2.5) 76 CHAPTER JI. GENERAL DYNAMICS In the limit oflow frequencies, Eq. (4.2.5) may be written as X.L(w) � x�(1iWTol), (4.2.6) where (4.2.7) is the relaxation time. Moreover, Col (s) is the Laplace transform ofthe autocorrelation function. Eq. (4.2.6) can be written down, up to the same order of accuracy, in the form of the Debye equation (4.2.8) This is the rotational diffusion limit where the behavior of Col(t) and, consequently, of (Mol (t)) - (Mol(O)) may be approximated by the exponential (4.2.9) The relaxation time follows from (4.2.7) together with (4.2.4), and (3.8-3.14). In fact, when the perturbing external field is pointing towards the e", axis, H' = H' É"" and the constant polarizing magnetic field coincides with the ez axis, H = H «., our autocorrelation function reduces to C (t) = (R",(t)R",(O)) '" (R;(O)) (4.2.10) Consequently, D-1 l' R1(s) Tol = r 1m--;:;o . • -0 "'1 In Fig. 11.6, we have represented Tol versus JJ for the different values of the parameter (4.2.11) (J' corresponding to different available materials. The Debye spectra (single relaxation time approximation) is given by ( 4.2.12) where we have defined XO = nm2/3kBT as the static value of the susceptibility in the absence of the constant field. In Figs. 11.7 and 11.8 we plot the real X� (w) and imaginary x1 (w) parts of the transverse component of the normalized complex susceptibility for different values of APPENDIX A 83 Analogously, the potential terms (A3), (A6), and (A8) can be written in the form 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/;(Rfl + Rñ)U x R = -,' '1/;-(1 - RR)· -, + 'I/;-{(R· ñ)1 - Rñ}· -, , {r oR oR {r oR {r on (AlO) _!_�. ('I/;(1lfl + 1lñ)U x ñ) = �. ('I/;_!_(1 - ññ) . o� + 'I/;_!_ {(Rñ)1 _ nR} . o�) , {r on on {r on {r a R (All) where the underlined factors may be identified to the mobility matrices defined in Eq, (2.12). Moreover, the diffusion matrix is given by D = kBTb, so that we can also rewrite the diffusive factors in terms of the rotational operators (A13) a ( 0'1/; ) a "' ,0'1/; -- _. DR .- =D -·{(R·n)I-Rn}·-=DR· ·R··I. oR n oil r oR oñ r R n 'P, (A14) :n . (DnR. !�) = DrRñ ·1lfl'l/;· (A15) Finally, substituting eqs. (A2)-(A8) and (A12)-(A15) in Eq. (A 1), we arrive at equation (2.13). Appendix B Decoupling approximations In Section 3, we have introduced decouplings of sorne correlation functions appearing in the evolution equations for the correlations. The purpose of this appendix is to give more details about the procedure followed to carry out such decouplings. In Eq. (3.4), the approximated quantities are (Rr n, )(t)Rr(O)) '" (Rr( t)Rr(O)) en, )eq = .c(¡¡ )(Rr(t)Rr(O)), (Bl) and where linearizations in time have been performed. The approximation (Bl) was al ready discussed in the text (see Eq. (3.5) and comments below). For the correlation (B2), we decouple the quantities Rr and (ñ· R)2 because they are not coupled in equilibrium conditions, (Rr(ñ· k)2)eq = O, (Rr)eq = O, and (n· R)2)eq # O. Thus, in a situation not far frorn equilibrium, we will assume that both quantities remain also decoupled. By similar arguments, in Eq. (3.6) we have al so performed the following approxi mations (Rr(nz(n· R)))(t)Rr(O))""" (R",(t)Rr(O))«nz(n· R)))eq = .c(¡¡)Q(u)(Rr(t)Rr(O)), (B3) (nr(Rz(n· R)))(t)Rr(O)) '" «nr(n· R))(t)Rr(O))(Rz)eq = .c(¡¡)(nr(n· R))(t)Rr(O)), (B4) and «nr(n . R)(l - (n· R)2))(t)Rr(0)) ....., (nr(n . R))(t)Rr(O))(l _ (�. �)4)eq). (B5) «n . R)2)eq In Eq. (3.7), we have used 84 APPENDIX B 85 (n�R.,)(t)R.,(O)) - (R.,(t)R.,(O))(n;)eq = CC(JJ) (1 - 3Q(u)) + Q(u))(Rr(t)Rr(O)), JJ (B6) (nrnzRz)(t)R.,(O)) - (nrnz)(t)Rr(O))(Rz)eq = C(JJ)(n.,nz)(t)Rr(O))o (B7) Notice that the decoupling in the correlation (n.,(ñ o R)3)(t)Rr(O)) of Eq. (B5) is (nr(ñ o R))(t)Rr(O)) ¡�:::�:l:: and not (nr(ñ o R))(t)Rr(O))(ñ o R)2)eq, which leads to divergencies of the rotational viscosity at srnall values of JJo As regards this fact, it is worthwhile to emphasize that these approximations are more accurate for moderate and higher values ofparameters JJ and a . This type oftruncation was already proposed by Stratonovich in the context of stochastic processes, Bibliography [1] Proceedings 01 the Sixth International Conference on Magnetic Fluids, edited by V. Cabuil, J.-C. Bacri, and R. Perzynski (North-Holland, Arnsterdarn, 1993). [2] W.T. Coffey, P.J. Cregg and Yu.P. Kalrnykov, On the Theory of Debye and Neel Relaxation of Single Domain Ferromagnetic Pariicles, edited by 1. Prigogine and S.A. Rice, Adv. in Chern. Phys., Vol. 83, (Wiley Interscience, New York, 1992), p.263. [3] P.C. Fannin and S.W. Charles, J. Phys. D: Appl. Phys. 22, 187 (1989); J. Phys. D: Appl. Phys. 24, 76 (1991). [4] J.T. Waldron, Yu.P. Kalrnykov and W.T. Coffey, Phys. Rev. E 49,3976 (1994). [5] Yu. L. Raikher and M. I. Shliornis, Relaxation phenomena in Condensed M alter, edited by Williarn Coffey, Adv. in Chern. Phys., Vol. 87 (Wiley Interscience, New York, 1994) p. 595. [6] V.G. Bashtovoy, B.M. Berkovsky and A. N. Vislovich, Introduction to Thermo mechanics of Magnetic Fluids (Springer-Verlag, Berlin, 1988). [7] T. L. Gilbert, Phys. Rev. 100, 1243 (1955). [8] M.C. Miguel, J. Bonet A valos , A. Pérez-Madrid and J.M. Rubí, Physica A 193 359 (1993). [9] W. F. Brown, Phys. Rev. 130, 1667 (1963). [10] M. A. Martsenyuk, Yu. L. Raikher and M. I. Shliornis, SOy. Phys. JETP 38,413 (1974). [11] Yu. L. Raikher and M. J. Shliornis, SOy. Phys. J. Appl. Mech. Tech. Phys, 15, 470 (1974). 86 BIBLIOGRAPHY 87 [12] M.I. Shliomis and V.I. Stepanov, J. Magn. Magn. Mater. 122 196 (1993). [13] J.R. Calderwood and W.T. Coffey, Proc. Roy. Soco A 356 269 (1977). [14] N.G. van Kampen, Stochastic Processes in Physics and Chemistry, (North Holland, Amsterdam, 1992). [15] M.1. Shliomis, SOy. Phys. JETP 34 1291 (1972). [16] M. Doi and S.F. Edwards, The Theory 01 Po/ymer Dynamics, (Clarendon Press, Oxford, 1986). [17] J.P. McTague, J. Chem. Phys. 51133 (1969). [18] F. Perrin, J. Phys. Radium 533 (1934). [19] J.-C. Bacri, K. Djerfi, S. Neveu and R. Perzynski, J. Magn. Magn. Mater. 123 67 (1993). [20] J.-C. Bacri, J. Dumas, D. Gorse, R. Perzynski and D. Salin, J. Physique Lett. 46 L-1l99 (1985). [21] M.C. Miguel, J.M. Rubí and A. Pérez-Madrid, Physica A 20324 (1994). [22] J.-C. Bacri and R. Perzynski,Proceedings 01 the XII Silges Conference, edited by L. Garrido (SpringerVerlag, Berlin, 1993). [23] Yu.L. Raikher and M.I. Shliomis, SOy. Phys. JETP (Engl. transl.) 40 526 (1974). [24] P.C. Fannin, B.K.P. Scaife and S.W. Charles, J. Magn. Magn. Mater. 122 159 (1993). [25] M.1. Shliomis and V.I. Stepanov, J. Magn. Magn. Mater. 122 176 (1993); Relax ation phenomena in Condensed Maiter, edited by William Coffey, Adv. in Chem. Phys., Vol. 87 (Wiley Interscience, New York, 1994) p.I. CHAPTER III DYNAMICS OF MAGNETIC HOLES DISPERSED IN A FERROFLUID Once we have characterized the macroscopic behavior of a ferrofluid, giving expressions for the viscosities and, correspondingly, for both the symmetric and antisymmetric parts of the pressure tensor, as well as for the average of its magnetization, in this chapter we analyze the dynamics of a nonmagnetic particle (magnetic hole) suspended in a ferrofluid. The rotational dynamics of the particle is strongly influenced by the presence of a rotating magnetic field. As a result, we have found that the hole rotates in the opposite direction to that of the field. Our analysis is valid at low and moderate frequencies of the field rotation and may be compared to recent experiments obtaining quite a good agreement in the frequency range we are considering, in which a linear relationship between both the rotation frequency of the field and the angular velocity of the particle is found. In addition, the dependence of the frequency of the particle on the magnetic field strength is also compared to the experiments. For a slightly greater concentration of holes, we also determine the Rotne-Prager and Oseen equivalent tensors, as the first steps in the characterization of hydrodynamic interactions between the nonmagnetic particles in the ferrofluid. 88 1. Introduction 89 1 Introduction Recently, an increasing interest in the study of the dynamic properties of the so called magnetic holes, which are colloidal nonmagnetic particles dispersed in a carrier magnetic fluid, has arisen [1]-[4]. Although the particles are not magnetic, when they are suspended in a carrier ferrofluid they acquire an induced magnetic moment equal to the magnetic moment of the ferrofluid volume they displace. The interaction of these induced magnetic moments of the holes causes a number of peculiar phenomena, such as the order-disorder transition in magnetic hole lattices [5, 6] and the non linear phenomena observed in assemblies of holes [7]. Furthermore, knowledge of the dynamics of such particles may constitute a way of characterizing the transport properties of the ferrofluid. For example, the friction coefficient of the particle gives us information about the viscosity of the carrier fluid. The system holes-ferrofluid can be modeled as a suspension of particles (holes) in a carrier fluid (ferrofluid). This simplification can be accomplished when different length scales exist for the ferrofluid and the holes. Consequently, the ferrofluid can be viewed as a continuous medium through which the holes may move. The dynamics of the ferrofluid is governed, at the continuum level, by a generalized Navier-Stokes equation. The rotations of the ferromagnetic particles lead to the presence of an antisymmetric part of the pressure tensor giving rise to the appearance of a new transport coefficient: the rotational viscosity [8, 9]. This coefficient enters the expression of the effective viscosity of the suspension [10]. The purpose of this chapter is to analyze the translational and rotational dynamics of a magnetic hole immersed in a ferrofluid under the influence of a rotating magnetic field. In particular, we are interested in explaining the phenomenon of the rotation of the hole in the opposite direction to that of the rotating field, as has been observed in recent experiments. As we will see, the expression for the angular velocity of the hole comes from its corresponding rotational equation of motion involving the hydrodynamic torque exerted by the fluid on the particle, which is calculated in a nonstationary situation. Furthermore, we intend to characterize the hydrodynamic interactions between the non-magnetic particles in a ferrofluid by giving expressions for the correspond ing Oseen and Rotne-Prager equivalent tensors. In particular, the Oseen tensor for a simple fluid is well known for being the simplest approximation when computing hydrodynamic interactions, i.e. when the suspension is dilute enough that the par ticles are located at a relative large distan ce from each other. This means that the 90 CHAPTER III. DYNAMICS OF MAGNETIC HOLES suspension of non-magnetic particles under consideration is dilute enough to avoid aggregation phenomena among the holes, but at the same time, there is a sufficient number of particles that makes it necessary to take into account hydrodynamic in teractions between pairs of spheres. As an example to understand the influence of hydrodynamic interactions on the dynamics of the magnetic holes, we will also study their sedimentation in the presence of an external magnetic field. With this goal in mind, we have distributed the chapter in the following way. In Section 2, we formulate the basic equations for the whole system in the continuum approximation. Under this approximation, the magnetic hole may be viewed as a macroscopic particle moving through a continuum medium with internal degrees of freedom. In Section 3, we analyze the motion of the hole until obtaining explicit expressions for the force and torque exerted on the particle. From these equations, we then derive the translational and rotational friction tensors. Moreover, from the analysis of the rotational motion, we infer a linear law relating the angular velocity of the hole and the rotation frequency of the field. These results are compared to ex periments in Section 4. In Section 5 we study the hydrodynamic interactions between pairs of magnetic holes. We derive the expressions of the Oseen and Rotne-Prager equivalent tensors and we study the sedimentation of two holes in the presence of an external magnetic field for different configurations in order to show the influence of such interaction. Finally, in the last section, we summarize our main results. 2 Basic equations in the continuum approximation Let us consider a dilute suspension ofspherical magnetic holes ofradius a in a ferrofluid under the influence of an external magnetic field, which rotates with the constant frequency WO' In the continuum approach, the ferrofluid (understood as a suspension of small magnetic particles in a non-polar solvent) is assumed to be a continuum medium with a new hydrodynamic field: the spin or mean angular velocity of the volume elements of the continuum [8]. This approximation is justified under the mild requirement that the size of the holes (tipically about 10-3 -1O-4cm) be much larger than that of the ferromagnetic particles (l0-6cm). The magnetic field rotation frequencies are taken low enough to consider the qua sistationary limit in which the derivatives of the hydrodynamic fields, although time dependent, may be neglected. Under this approximation, the equations of motion are (2.1) 2. Basie equations in the eontinuum approximation 91 0= 2ña + M x ¡¡ + T', (2.2) where p(r, t), v(r, t), and ñ(a)(r, t) == -1/2E : II(a)(r, t) are the pressure, velocity, and the axial vector related to the antisymmetric part of the pressure tensor, II(a), re spectively. Moreover, E is the Levi-Civita tensor, M(r, t) is the average magnetization density, and ¡¡ = H f¡ is the external magnetic field. The appearance of antisym metric stresses in the ferrofluid comes from the difference between the mean angular velocity and the vorticity of the flow, 1/2"V x v. In addition, we have introduced the induced force and torque densities, fi(r, t) and :¡:i(r, t), which originate from the perturbation caused by the motion of the holeo The shear viscosity '1, according to Einstein's law is 5 T/ = T/o(l + '24». (2.3) This expression holds up to linear order in the volume fraction of magnetic particles 4> = 411V IV, where b is the radius of one ferromagnetic monodomain, and V is the volume occupied by the system. We have also defined T/o as the viscosity of the car rier fluid. Furthermore, these equations are complemented with the incompressibility condition "V . v = O . The induced force and torque density fields, fi(r, t) and :¡:i(r, t) are introduced so that the equations of motion of the fluid, given by eqs. (2.1) and (2.2), are also valid for the points inside the sphere [11, 12]. This consideration imposes the requirement fi(r, t) = T'(r, t) = O, for ¡r - Rcm(t)1 > a. (2.4) AdditionalIy, they must be chosen in such a way for the velocity and the pressure fields to satisfy v(r, t) = ti(t) + ñ(t) x añ, for ¡r - Rcm(t)1 = a, (2.5) p(r, t) = O, for ¡r - Rcm(t)1 < a, (2.6) and the magnetization of the hole M(r, t) = O, for ¡r - Rcm(t)1 � a. (2.7) In these equations, we have introduced ti and ñ as the translational and rotational velocities of the particle, respectively; Rcm(t) as the position of the center of mass of the sphere, and ñ == (r - Rcm(t))/¡r - Rcm(t)l. 92 CHAPTER IIl. DYNAMICS OF MAGNETIC HaLES After considering the conditions (2.4) - (2.7), and for the quasistationary case, ft(ü(t) + ñ(t) x añ) ::= O, we have fi(r,t) = f(ñ,t)8(JiRcm(t)la), (2.8) and fÍ(i, t) = fÍ(t)8(a - Ji - Rcm(t)l), (2.9) where 8 is the Heaviside step function. We will now proceed to derive a formal solution in terms of the induced force and torque densities with the purpose of obtaining expressions for the hydrodynamic force and torque on the nonmagnetic particle as a function of its velocity and the velocity fields in the absence of the sphere. The procedure we use does not require explicit knowledge of either fi(i, t), ?(r, t) or the real velocity field v(i, t). Equations (2.1) and (2.2) can be combined to eliminate the term proportional to the antisymmetric axial vector, ñ-, in Eq. (2.1). We then obtain (2.10) where we have defined ji as the combination of the induced force and torque fields (2.11) To solve for the velocity field from Eq. (2.10) , we need to know the expression for the magnetization density. The procedure used to obtain this quantity was introduced in Ref. [13] for the case of a constant magnetic field, and was based upon the solution of the corresponding Smoluchowski equation. In appendix A, we present an extension of the method for rotating magnetic fields. Our soIution is (2.12) where n is the number density of dipoles, (R) is the averaged orientation vector of each magnetic moment, p. = mH /kBT is the Langevin parameter, comparing magnetic and therrnal energies, with m the magnetic moment strength, assumed constant, kB the Boltzmann 's constant, and D; = krrT is the rotationaI diffusion coefficient; with �,. = 81r7Job3 the rotationaI friction coefficient of a ferromagnetic particle. Additionally, C(p.) = coth p. - 1/ u, is the Langevin function and F(p.) == 2+c;it¡J) is introduced and obtained in appendix A. 4. Comparison with experiments 99 This expression is also similar to the corresponding one obtained by Sellers and Bren ner for gravitational dipoles. It is worth mentioning that, for the rotational motion, the time dependen ce of the magnetic field gives rise to an extra term in the hydrodynamic torque proportional to dd1 x H for the quasistationary analysis. As we will see, for the field rotation frequencies under consideration, this term is responsible for the counter-rotation of the hole in respect to the field rotation, as observed by Helgesen and Skjeltorp [2] and by Popplewell ei al. [15]. The magnetic hole rotational equation of rnotion is given by (3.2.16) where 1 is the moment of inertia of the partícle. Again, since the inertial ter m is negligible, the result is that fH = O. For a ferrofiuid initially at rest, where ít = O, and up to the first order in the volume fraction, q¡, from Eq. (3.2.14) we obtain - dH - f2(t) = q¡JjF(Jj)( dt x H). (3.2.17) In particular, if we consider an external magnetic field rotating in the XY plane with angular frequency Wa, H (t) = cos water -í-sin wate", the magnetic hole angular velocity reduces to (3.2.18) From this expression, we then conclude that the direction of the rotation of the spheres is opposite the field rotation. Additionally, we obtain a linear relationship between the field frequency and the rotation frequency of the spheres for the range of field frequencies we are considering in our analysis, that is, for frequencies which enable us to perform a quasistationary treatment. Moreover, under these conditions, ñ is independent of the size of the spheres, it only depends on the volume fraction of magnetic particles and on the Langevin parameter. 4 Comparison with experiments The rotation ofmagnetic holes induced by a rotating magnetic field has been observed in recent experiments [2, 15]. The experimental setup consists of a thin layer of ferrofiuid, confined between two glass plates, in which spherical particles of polystyrene are dispersed. When applying a magnetic field rotating in the plane of the plates, 100 CHAPTER III. DYNAMICS OF MAGNETIC HOLES it was observed that the particles rotate in the opposite direction. The experiments were performed for kerosene-based [2] and water-based ferrofluids [2, 15]. The rotation frequency of the hole, ñ, for the kerosene-based ferrofluid, which showed good signs of homogeneity, was measured for different samples containing polystyrene spheres of various sizes. They saw that ñ was independent of the size of the spheres, for the size range they took into consideration. Furthermore, in the intermediate frequency range, they observed a linear relationship between the angular velocity of the sphere and the angular velocity of the magnetic field. In the case of a water-based ferrofluid, it has been clarified that it shows a weak tendency of sedimentation and aggregation phenomena under the action of the field [2]. In the same reference, the different response of the magnetic hole when suspended in both types of ferrofluids was also emphasized. At this point, it should be noted that there is not total agreement between the results of [2] and [15] concerning such a ferrofluid. In particular, Popplewell and coworkers conclude that particles of about 2a == 70J.lm rotate opposite the direction of the field, for any value of "-'o, whereas smaller particles, 2a == 20J.lm, rotate in the field direction at field frequencies below 70H z and in the opposite direction at higher frequencies. However, Helgesen and Skjeltorp were not able to reproduce these results. AdditionalIy, in both papers there are al so experimental data relating the holes rotation frequency and the external field strength at intermediate values of "-'o' In regards to the coupling between translational and rotational motion, the ex perimental observations corroborate our results. This can easily be concluded after inspection of the equations for the force and torque given byeqs. (3.1.7) and (3.2.14) . Moreover, our result given through Eq. (3.2.18) is in complete agreement with the observations carried out for the kerosene-based ferroftuid in the range of intermediate field frequencies, which, as we have stated before, enables us to perform a quasista tionary study of the system. This can be seen in Fig. 1Il.l for a ferroftuid with the same characteristic as that considered in Ref. [2], that is, for volume fraction of mag netic particles tP == 0.1 and saturation magnetization of 200G. On the other hand, as far as the theory is concerned, we can also indicate that we have obtained a rotation frequency which does not depend on the viscosity of the solvent (3.2.18) . Despite the possibility of the inhomogeneities in water-based ferrofluids, mentioned in Ref. [2], the dependence of n on the field strength seems to be represented by a saturation curve as the one corresponding to the quantity J.lF(J.l) = J.ltanhJ.l J.l + tanh u (4.1 ) 4. Comparíson wíth experíments 101 0.010 0.001 10 100 fH (Hz) 1000 Figure 111.1: Rotation frequency ofa magnetic hole in a kerosene-based ferrofluid layer versus the frequency fH of the rotating magnetic field for H = 140e. The direction of rotation of the spheres is opposite to that of the field. 102 CHAPTER IIl. DYNAMICS OF MAGNETIC HOLES 0.4 fs (Hz) 0.3 0.6 0.5 0.2 0.1 0.0 ...___ '-- __ ----''- __ --' ---I. --' O 20 40 60 80 100 H (G) Figure 111.2: Variation of particle rotation with the field strength. Rotation counter to the field rotation. (a = 35¡..¿m, lB = 60Hz) 5. Hydrodynamic interactions between pairs of magnetic hoJes 103 involved in our Eq. (3.2.18). In Fig. 11I.2, we have represented the frequency as a function of the imposed field for a ferrofluid with the same magnetization as that considered by Popplewell and coworkers. Finally, it is worth mentioning that in accordance with Ref. [2], we cannot reproduce the dependence of the frequency on the size of the particles as observed in Ref. [15]. 5 Hydrodynamic interactions between pairs of mag netic holes Hydrodynamic interactions among different objects moving in a viscous fluid have been extensively studied because of their effects on the dynamics and consequently on certain properties of the whole dispersion [17, 18]. The first studies of these inter actions were undertaken by Smoluchowski using the so-called methods of reflection. But, due to their complexity, they have only been applied to the case of a pair of particles. On the other hand, the method of induced forces, first introduced by Mazur [19] and generalized in Ref. [12] for a system of N spheres moving in a simple liquid, permits to calculate the mobility tensors to any desired order of approximation as an expansion in the inverse distan ce between the particles. In this section, we first develop the latter technique to study the hydrodynamic interactions between pairs of spherical nonmagnetic particles moving in an anisotropic magnetic liquido 5.1 Oseen and Rotne-Prager tensors for the ferroftuid We will now consider a suspension of N spherical magnetic holes of radii ai, (i = 1, ... , N), in an unbounded incompressible ferrofluid under the influence of an external magnetic field, which may rotate with constant angular velocity Wo. The suspension under consideration is dilute enough to avoid aggregation due to the interaction of the induced magnetic moments of the holes. On the other hand, it contains a sufficient number of particles so that hydrodynamic interactions between pairs of spheres must be considered. As in the previous sections, the ferrofluid is assumed to be a continuum medium based on the fact that the size of the holes is much larger than that of the ferromagnetic particles constituting such a ferrofluid. Our starting point for analyzing the quasistationary motion of the particles will be the linearized equations of conservation of the linear and angular momenta, which now read 104 CHAPTER IJI. DYNAMICS OF MAGNETIC HOLES N 0= -\7p+ r¡\72V - \7 x ü(a) + L fj"d, j=1 (5.1.1 ) N 0= 2ña + M x .i1 + L T1nd, i=1 (5.1.2) together with 0= \7 . V, (5.1.3) where we have introduced a set of induced forces and torques densities, fj"d(r, t) and T1nd(r, t), (j = 1, ... , N) which originate from the perturbation caused by the motion of the holes. As we have explained before, the extension of the fluid velocity field inside the particles imposes the requirement fj"d(r, t) = Tjnd(r, t) = O, for ¡r - Rj(t)1 > ai and (j = 1, ... , N). Ri(t) gives the position of the sphere center at time t. Furthermore, the set of equations including induced forces and torques must be completely equivalent with the original boundary value problem ver, t) = uj(t) + nj(t) x (r - Rj(t», for ¡r - Rj(t)1 � aj, (5.1.4) i.e., stick boundary conditions at the surfaces of the spheres. Similarly, for the hydro static pressure we impose the condition p(r, t) = O, for ¡r - Ri(t)1 < ai' (5.1.5 ) and for the magnetization (5.1.6) where we have introduced iij and ñi as the translational and rotational velocities of the particle i. respectively. Consequently, considering these conditions and for the quasistationary case, all the induced force and torque densities must be of the same form as in Section 2. If we want to consider a rotating magnetic field, its angular frequency should be low enough to consider the quasistationary limit in which the derivatives of the hydrodynamic fields, although time dependent, may be neglected. In appendix A we obtain the expression for the magnetization density. Substituting this result in Eq. (5.1.2), and combining eqs. (5.1.1) and (5.1.2) we can write the following equation 5. Hydrodynamic interactions between pairs oi magnetic holes 105 N 0= - \7p + (7] + 7]r )\72V' + 7]r \7 X h(Ji . \7 X V) + 'L .fjnd, i=l where 7]r is the rotational viscosity, and where we have defined :frd as the combination of the induced force and torque field densities (5.1.7) (5.1.8) In order to salve formally the equation of motion of the fluid (5.1.7), we introduce Fourier transforms of the velocity field V'. From now on, we wíll omit the explicit dependence on time of the different fields. We also define the Fourier transform of the induced force density rr: in a reference frame in which sphere j is at rest at the origin (5.1.9) After Fourier transforming this equation and applying the transversal projector (1 - H:), with k == k/k, one arrives at the formal solution for the transversal component of the velocity field N v(k) = T(k, t) . 'L exp( -ik . Ri ):fjnd(k), i=l where, as in Section 2, we have introduced the propagator (2.16). As we have indicated before, the spheres are allowed to move with arbitrary ve locity through the fluid, which may itself be in arbitrary stationary non-uniform un- (5.1.10) perturbed flow. For the sake of simplicity, we will consider that the unperturbed fluid is at rest, and we wíll study the hydrodynamic interactions which are set up between the spheres when they move. In particular our main goal is to calculate the transla tional mobility up to a given order of approximation in a series expansion in powers of inverse distances between the spheres, when only hydrodynamic interactions between two spheres contribute. The formal solution can be rewritten in real space N v(r,t)= 'Lj dr'T(r-r,).:Fjnd(r'). i=l In particular, for those points on the surface of the sphere i, and after taking into (5.1.11) consideration the properties of the generalized induced force, we have 106 CHAPTER JII. DYNAMICS OF MAGNETIC HOLES - N J - ñ· X ¡!nd ii(Ri +a¡ñ¡) = � dñj¡.J.(ñ¡,ñj)' [/td(ñj) - ) 2 J ], 3=1 where we have introduced the response function (5.1.12) (5.1.13) with the definition R;j = R; - Rj. For the purpose of evaluating hydrodynamic interactions, it is convenient to introduce irreducible induced force multipoles, defined in terms of the surface induced forces according to 1 J A =ind .....{1+1)=_ dñ, A�[/,!nd(A.)_n¡XT¡ ] :r i 471' nI nI I nI 2' (5.1.14) where ñ: is an irreducible tensor of rank 1, traceless and symmetric in any pair of its indices. The expansion in terms of irreducible force multipoles, which is written in a coordinate free way, is equivalent to an expansion in spherical harmonics, to which it can be reduced if polar coordinates are used. We also introduce irreducible surface moments of the fluid velocity field, and we obtain a relationship between the induced force multipoles and the surface moments of the fluid velocity field through a hierarchy of equations. It is this hierarchy which will then enable us to obtain expressions for the mobility tensors which relate the forces and torques on the spheres to their lineal and angular velocities. The general relationship between the (1 + 1) order multipoles of the quantities defined over the surface of the sphere is found to 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 involves the tensors (5.1.16) As we already pointed out in Section 3, the first order irreducible rnultipole will allow us to determine the translational mobility. Thus, our starting point will be the relationship between velocity and force multipoles which comes from Eq. (5.1.15) for 1 = O (1) _ � � (2/' + 1)!! (1,/'+1) �I'+l) V¡ - � L... I'! ¡.J.¡j 8 j , )=1/'=0 (5.1.17) 5. Hydrodynamíc ínteractíons between paírs of magnetíc hoJes 107 where the multipole vP) is vP) = 411f' J dn¡ (ü¡ + ñ¡ x a¡n¡) = Ü¡. (5.1.18) Moreover, using the fact that the drag force exerted by the fluid on the particle j (written in terms of the induced force) is (5.1.19) it follows that �1) = _!_ Jdn.(flnd _ nj x fJ) = _ FjH . J 41f' J J 2 41f'a� J (5.1.20) In Section 3, we already gave the exact form of this relationship for the case i = j. We obtained, 5 3 .. -61f'7]oa¡{[1 + 24> + ¡4>¡.tF(¡.t)](l - H H) 5 3 .. + [1 + 24> + 24>¡.tF(¡.t)]H H} . Ü¡, (5.1.21) Now, we willlook at the case i f j. In this case, the tensors ¡.tg+I,II+1) are no longer diagonal in their upper indices, so that there is a coupling between different multipoles. But, if we set l = O, we will restri-r ourselves to the lowest value of l' (1' = O), because beyond that, we will obtain contributions proportional to (al R)P with p 2:: 4 [12], negligible for the concentration of holes we are considering. The corresponding matrix is �U,l) = (;!2) J dk J dk exp(ik. R;j) h-1(k). (1 - kk)jo(ka¡)jo(kaj). (5.1.22) where jo(x) is the spherical Bessel function of zero order. In appendix e we obtain that (1,1 ) �¡j aJ 5 3 A• 27]oR;j ([1-"24> - "24>¡.tF(¡.t)](l + R;jR;j) 3 AA A A A A + 24>¡.tF(¡.t)[1 - H H - (R;j x H)(R;j x H)]} aJ(a¡+aJ) 5 3 l· A 27]oR�j {[1-"24> - "24>¡.tF(¡.t)](-31 - R;jR;j) 3 1 'A A A A A "24>¡.tF(¡.t)[3(1 - H H) - in; x H)(R;j x H)]), (5.1.23) + 108 CHAPTER lII. DYNAMICS OF MAGNETIC HaLES from which the translational mobility tensor comes up straightforwardly, after taking into account eqs. (5.1.17) - (5.1.20) RP 1 53 AA 1''' = {[1- -f/¡ - -f/¡pF(p)](1 + R;.R;.) IJ 81r7]0 R;j 22 J J 3 AA AA A A + 2f/¡pF(p)[1 - H H - (R;j x H)(R;j x H)]} (a; + aJ) 5 3 1 A' + 3 ([l- -f/¡ - -lPpF(p)](-1 - R;jR;j) 81r7]oRij 2 2 3 3 1 A A • •• • 2f/¡pF(p)[3(1 - H H) - (R;j x H)(R;j x H)]), (5.1.24) This mobility tensor represents the velocity of a sphere i with respect to the fluid velocity due to the motion of the particle i. and per unit of force exerted on sphere i. It is valid up to the first order in the volume fraction of magnetic particles, f/¡, and up to third order in the ratio (al R;j), for any value of the Langevin parameter p. From this expression, we can also obtain the corresponding Oseen tensor for the ferrofluid. The Oseen tensor is the first approximation when computing hydrodynamic effects. Nevertheless, it is a valid approximation when the particles are far apart enough, and we can assume that particle j generates the same perturbation as would be produced by a point force situated at its center. Thus, the Oseen tensor is obtained after neglecting the third power of (al R;j), o I'ij 1 {[1- �f/¡ - �f/¡pF(p)](1 + Ri·Ri'} 81r7]0 R;j 2 2 J } 3 . + 2f/¡pF(p)[1 - H H - (R;j X H)(Rij x Hm, (5.1.25) Obviously, if we set the volume fraction of magnetic particles equal to zero, f/¡ = O, this Oseen tensor reduces to the well-known expression 1 . . I'P)· = 8 R; (1 + R;jR;j), 1r7]0 'j which is the propagator of the perturbations in a simple fluid. (5.1.26) 5.2 Sedimentation of two magnetic holes in a ferrofluid To illustrate the influence of hydrodynamic interactions on the suspension of magnetic holes, we will study the sedimentation in the presence of an external magnetic field. Appendix B Explicit derivation of Eq. (5.1.24) Starting from Eq. (5.1.22) we can make an expansion such that 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 reference frame in which the ez-axis is parallel to the unitary vector �i (H = Rii�j), we may write dk = d�ijd<Pii, where �ij = Rij . k is the cosine of the polar angle between �j and k, and �ij is the azimutal 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 of the identity 1 ¡1 loo . ¡1 dq - dxxP dyyqe,:r:y = iq dxxP-8(x) = 8pqp!(-1)P, 211" -1 -00 -1 dxq (B4) it then follows that the terms of order k4 and higher of the expansion in the integrand of Eq. (B3) give vanishing contributions upon integration, since any element of the tensor h-1(k). (1 - kk) will be, after integration over �ij, a polynomial in �ij whose highest order is proportional to (ti [12], (B5) where 1 1 ( 5 3 ) el = -- '" - 1 - -� - -�¡.JF(¡.J) , 71 + r¡r 710 2 2 (B6) and 115 116 APPENDIX B 'Ir 1 3 ) C2 = ...., --l/JJJF(JJ . '1( '1 + 'Ir) '10 2 Thus, we can rewrite Eq. (B3) in the following form (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)+ 6Rtj {)2�ij6(�¡j) . (B8) For the sake of simplicity, we define (B9) with 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) After performing the integrals appearing in eqs. (B10) and (B11), they reduce to 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 of solving the integral in Eq. (B10) is the following where we have introduced the constants (91, ... ,94) which can be found by contracting the tensors in both si des of the equation, firstly with the identity matrix, 1, and sec ondly with the tensor R;j R;j. In doing so, and after calculating the simple scalar inte grals resulting from these contractions, we arrive at a determined system of equations for the aboye introduced constants. The solutions are 91 = 92 = 93 = -94 = aJ /2. APPENDIX B 117 The integral in Eq. (B11) can be carried out following exactly the same procedure. Finally, once we have substituted the constants Cl and C2 given through eqs. (B6) and (B7), Eq. (B9) turns out to be Eq. (5.1.23). Bibliography [1] H.S. Sellers and H. Brenner, PCH PhysicoChemical Hydrodynamics 11, (1989) 455. [2] G. Helgesen and A.T. Skjeltorp, J. Magn. Magn. Mat. 97, (1991) 25. [3] G. Helgesen and A.T. Skjeltorp, J. Appl. Phys. 69 (12), (1991) 8277. [4] P. Davies, J. Popplewell, G. Martin, A. Bradbury and R.W. Chantrell, J. Phys. D, Appl. Phys. 19, (1986) 469. [5] A.T. Skjeltorp, Phys. Rev. Lett. 51, (1983) 2306; J. Magn. Magn. Mat. 65, (1987) 195; J. Appl. Phys. 57 (1), (1985) 3285. [6] G. Helgesen and A.T. Skjeltorp, Physica A 170, (1991) 488; Physica A 176, (1991) 37. [7] G. Helgesen, P. Pieranski and A.T. Skjeltorp, Phys. Rev. Lett. 64, (1990) 1425; Phys. Rev. A 42, (1990) 7271. [8] H. Brenner, J. Colloid Interface Sci. 1, (1970) 141; Ann. Rev. Fluid Mech. 2, (1970) 137; Int. J. Engng. Sci. 22, (1984) 645. [9] V.G. Bashtovoy, B.M. Berkovsky and A. N. Vislovich, Introduction io Thermo mechanics 01 Magnetic Fluids (Springer-Verlag, Berlin, 1988). [10] J. P. Mc Tague, J. Chem. Phys. 51, (1969) 133. [11] P. Mazur and D. Bedeaux, Physica 76, (1974) 235. [12] P. Mazur and W. Van Saarloos, 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. Bonet Avales, A. Pérez-Madrid and J.M. Rubí, Physica A 193, (1993) 359. [15] J. Popplewell, R.E. Rosensweig and R.J. Johnston, IEEE Trans. Magn. 26, (1990) 1852. [16] M. A. Martsenyuk, Yu. L. Raikher and M. 1. Shliomis, SOy. Phys. JETP 38, (1974) 413. [17] W.B. Russel, n.A. Saville, W.R. Schowalter, Col/oidal Dispersions, (Cambridge University Press, Cambridge, 1989). [18] J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (Kluwer, The Netherlands, 1991) [19] P. Mazur, Physica HOA, (1982) 128. CHAPTERIV AGGREGATION PHENOMENA This chapter is intended to be an introductory analysis of the aggregation phenomena that arise for instance, as the size of the particles in a ferroftuid increases or at higher concentrations. In such conditions the dipole-dipole iteraction between the particles is enhanced and Brownian motion is no longer able to stabilize the suspension. Moreover, in the presence of a high magnetic field the particles show up a tendency to form chains. We study the kinetics of the formation of the aggregates by means of the Smoluchowski theory of coagulation in colloids accounting for hydrodynamic interactions. These interactions become relevant for the concentration of particles which give rise to these phenomena and slow down the aggregation process. In addition, the rheology of the chains that are usually observed in systems with dipolar interactions is studied for a rather simplified situation in order to elucidate the effects of the dipolar magnetic interactions in the contribution of the chains to the pressure tensor of the suspension. 120 1. Introduction 121 1 Introduction Colloidal dispersions play an important role in many natural phenomena as well as in various industrial processes. The stability of the suspension against aggregation of the particles is of essential importan ce for its behavior [1]. As long as one can neglect interparticle interactions, one does not observe aggregation phenomena and then the main problem is to understand how the physical properties of the fluid, for example its viscosity, are modified due to the presence of the particles. But when interactions between particles start to be important, the preceding problem becomes much more complicated. In this case, the aggregation mechanism in a colloidal solution is quite complex because the fluid does keep into an active role. The nature of inter-particle interactions depends on the properties of the fluid, and, concerning its driving motion, hydrodynamic effects cannot be neglected. The concepts of hard and soft particles are commonly used in the literature [2]. A hard particle is a rigid sphere on which only hydrodynamic forces are acting, they do not undergo other interaction forces except for an infinitely strong repulsion on contact to prevent interpenetration. In contrast, a soft particle is a sphere subject to interactions, other than hydrodynamic ones, felt at finite distances. In the last case, one can think of such spheres as having a surface boundary layer (sphere of influence) whose thickness is determined by a characteristic interaction length. Thus, for electro statically stabilized dispersions this thickness can be identified with the double-Iayer thickness, for van der Waals interactions with the thickness where the interaction en ergy becomes comparable to the thermal energy, and for sterically stabilized colloids with the length of the polymer tails adsorbed on the surface of the suspended particles. This idea was in a certain way exploited by von Smoluchowski [3], who first analized the problem of coagulation in the absence of any repulsive barrier (rapid coagulation). A sphere with a surface boundary layer can be considered soft because this layer can be deformed and penetrated during the approach of two particles. Moreover, in the literature the term coagulation is applied to aggregation that is induced by the van der Waals attraction between the colloidal particles. On the other hand, the term floccu lation is reserved for polymer-induced aggregation. Coagulation usually gives rise to compact aggregates whereas flocculation frequently produces more open structures. Particularly, the monodomain magnetic particles are usually sufficiently small that Brownian forces, along with short-range steric repulsion due to either natural or syn thetic polymers, dominate in the zero-field limit to guarantee the stability of the particles. However, when a non very dilute suspension of these particles is under 122 CHAPTER IV. AGGREGATION PHENOMENA the action of an applied magnetic field, particle aggregation occurs when the rnag netic attraction between the permanent moments is strong enough to outweigh the stabilizing forces. Thus the thickness of the boundary layer can be identified with the thickness for which the magnetic interaction energy is comparable to the thermal energy rather than with the length of the polymer, due to the long range character of such interaction. In the late 1930's, Winslow [4] observed interesting phenomena when dielectric particles suspended in oil were subject to an electric field. He saw the induced forma tion of chains of particles aligned with the electric field and, even of more practical importan ce, that the effective viscosity of the suspension could be varied by orders of magnitude by modifying the applied electric field. However, studies on electrorhe ological fluids are hindered by many problems related to surface charge, electro de polarization, adsorbed water, field inhomogeneities, etc. An analogous field-induced behavior is shown by magnetorheological fluids, e.g., in a suspension of magnetizable paramagnetic particles in a nonmagnetic fluid [5] or in a suspension of nonmagnetiz able spheres in a ferrofluid (magnetic holes, see previous chapter). In apure ferrofluid one observes similar phenomena but, due to fact that the particles are permanently magnetized, there are polarization forces even without an applied magnetic field. In all these cases, one has the advantage that they are not susceptible to the aboye mentioned problems. Thus, studying the kinetics of the formation of the aggregates and their rheology are topics of practical importance and has provided the bases of theoretical and experimental studies [6]-[8] as well as numerical simulations [9]-[11]. Having all these in mind, the structure of this chapter is as follows, In Section 2 we discuss the influence of hydrodynamic interaction in the kinetics of aggregates forma tion by using the Smoluchowski's theory of coagulation. We write the corresponding kinetic differential equations giving the time evolution of the dynamic cluster size distribution function and solve them for a rather simplified case. In this section we also discuss the scaling behavior of the cluster size distribution. Section 3 is intended to be an introductory analysis of the rheology of the chains which commonly appear when the aggregation process occurs under the action of an external field. (falta) The conclusions are summed up in the last section. 2. Hydrodynamic interactions in the Smoluchowski theory of coagulation 123 2 Hydrodynamic interactions in the Smoluchowski theory of coagulation The investigation of aggregation processes by means of computer simulations have been enhanced during the last 10 years after the pioneering work ofWitten and Sander where they proposed their diffusion-limited aggregation model referred to as DLA [12]. This work stimulated the development of the diffusion-limited cluster-cluster aggregation model by Meakin [13] and Kolb et al. [14], which, particularly, provides a better understanding of structural and kinetic aspects of colloid aggregation. The coagulation process can also be discussed in terms ofthe Smoluchowski's theory which, as we will see below, due to the nature of the approximation inherent in its derivation is a mean-field theory. Nevertheless, this theory is expected to hold for dimensions greater than the upper critical dimension aboye which fluctuations become irrelevant and that for this coagulation processes is 2. Smoluchowski first found out a very interesting application of the theory of Brow nian motion in the coagulation exhibited by colloidal particles when an electrolyte is added to the solution. His theory is based on the suggestion of Zsigmondy that coag ulation is a consequence of the existen ce of a sphere 01 infiuence of a certain radius R surrounding each colloidal particle such that its Brownian motion remains unaffected unless another particle enters within its sphere of influence. When this happens they stick to one another to form a single unit. For this particular problem, the spheres of influence are supposed to originate in the formation of electric double layers around each particle. The double particle continues moving randomly so long as it does not come within the spheres of influence of a single or another double particle. Then we will have the formation of a triple or a quadrupole particle, and so on. This pro cess wilI eventualIy lead to the total coagulation of alI the colloidal particles into one cluster. Recent real-time experiments performed with suspensions of superparamagnetic latex particles [15] corroborate Smoluchowski's hypotheses. In this case, the mag netic field induces chain formation, and one observes two aggregation time scales. When particles are far enough away from each other that their dipolar interaction is weak relative to kBT, Brownian motion dominates. When this random motion hap pens to bring two particles close enough that their dipolar interaction is of the same order as kBT, a rapid transition between random and ballistic motion occurs and the particles aggregate immediately. Moreover, numerical simulations incorporating induced-dipolar interaction forces and Brownian motion also corroborate that there is 124 CHAPTER IV. AGGREGATION PHENOMENA a citrical value of the interparticle separation within which particle motion ceases to be random. Thus, each particle can be thought of as having a capture volume defined by equating dipolar interaction energy to thermal energy. If one particle enters this volume, the two particles undergo ballistic aggregation. On the other hand, the appli cability of the Smoluchowski equation is limited to low cluster concentrations because it assumes that there are only binary collisions. We are interested in determining the concentrations nI, n2, ... , of single, double, triple, quadrupole, etc., particles at time t given that at time t = O there are no single particles. Moreover, once we know these concentrations we can write the scaling laws which show how these quantities behave for long times or for large radius of the spheres of influence. But, in this section we will introduce a modification to the simple theory proposed by Smoluchowski by considering hydrodynamic effects, For distan ces greater or of the same order of magnitude as the interparticle separation threshold (- 2a - 7a, with a the radius of one particle), hydrodynamic interactions should be taken into account when studying the diffusive motion of the particles. There are previous attemps to incorporate this effects in the final stages of the approach of two non-rnagnetic particles (i.e. when just short range interactions between the particles are considered) when, because it is difficult for the remaining film of Iiquid to escape, the process is clearly slowed down. Honig et al. [16] for instance examined the problem deriving an approximate relation for a modified diffusion coefficient: D(h) 1 + 2aj3h ------�- = --------�------- D(h -+ 00) 1 + 13aj6a + a2 j3h2 (2.1 ) where h = r - 2a is the distan ce of closest approach. This effect was also tested experimentally by Lichtenbelt el al. [17] who found that the rate of coagulation was reduced to less than half of the Smoluchowski value for those colloidal suspensions. However, when dealing with long range interactions, we believe that it would be enough to introduce hydrodynamic interactions represented by the Oseen and Rotne Prager tensors discussed in the previous chapter. This will enable us to find analytic expressions not only for the rate of coagulation but also for the different concentrations of clusters nI, n2, ... 2.1 Kinetic equation of irreversible aggregation For the time intervals we are interested in the motion of the particles is diffusive, i.e., local fluctuations in the concentration due to Brownian motion determine a global diffusion from the higher to the lower concentrated regions. We will start considering 2. Hydrodynamic interactions in the Smoluchowski theory of coagulation 131 - For large values of R, i.e., for a long range interaction, and provided that 3a/2R < 1 L will be also large so that we can rewrite Eq. (2.2.15) as follows no ( 1 )-(1:+1) no ( k )2 _ . nI: = (noLt)2 1 + noLt :::: k2 notL e;;;n, or, equivalently (2.3.1) nI: ex k-T f (;'1) , where we found out the characteristic exponents T = 2 and r = 1. - On the other hand, for sufficiently large times (2.3.2) nI: = no (1 + _1_) -(1:+1) ...., no (_k_) 2 e--;;!n (noLt)2 noLt - k2 noLt ' from which it is possible to write the size distribution function under the following (2.3.3) scaling forrn (2.3.4) whose characteristic exponents are T = 2 and z = 1. Vicsek and Family [21] and independently Kolb [22] introduced a dynamic scaling description for the cluster size distribution in the cluster-cluster aggregation model. Monte CarIo simulations showed that a dynamic scaling of the aboye introduced form (2.3.5) represented well the behavior of nl:(t). Here f(x) is a scaling function which depends on the dimension and on the cluster mobility. The exponent z has been measured both experimentally and by Monte CarIo simulations of diffusion-limited cluster-cluster aggregation. The experimental result are generally in agreement with the simulations and, at the same time, the Mote Carlo results agree with the Smoluchowski approach in d = 2,3, but disagree in one dimensiono That was consistent with the observation that Smoluchowski's theory, without the incorporation of spatial fluctuations, fails below a critical dimension de = 2. As an application of the scaling laws, let us look at the case of magnetic particles with dipolar interactions, when we neglect hydrodynamic effects, i.e. for large values of R. In this case, as we already pointed out in the introduction, we can estimate the value of R in the following way 132 CHAPTER IV. AGGREGATION PHENOMENA (2.3.6) which follows by equating both magnetic and therrnal energies. Consequently, (2.3.7) so that the concentration of clusters of k particles scales with the particle's rnagnetic moment (2.3.8) with characteristic exponents T = 2 y 'Y = 2/3. Janssen el al. [11] used a numerical approach to sol ve the basic equation for flocculation, considering a cylindrically sym metric interaction in order to model the dipolar interaction in paramagnetic particles. In particular they investigated the rate of initial flocculation J = 87r DnoR/W, where in their notation W stands for the stability factor which indicates how the interac tions modify the total flux in comparison with the case of rapid aggregation, in which no repulsive forces are present to slow down the process. They obtained that this stability factor depends on the dimensionless parameter characterizing the dipolar interaction, or in other words on the magnetic moment of the particles, Wex m-2/3, with the same characteristic exponent. This result was also verified experimentally at high magnetic fields. Moreover, recent experimental investigations on the magnetic-field-induced chain formation of superparamagnetic latex particles [15] show a power-law dependen ce on time as can also be obtained by the Smoluchowski equation and three-dimensional simulations of dipolar particles, But they also found that the value of the exponent z have a weak inverse dependen ce on the particle volume fraction and in the dirnen sionless constant characterizing the dipolar interaction strength. We plan to pursue work on this experimental observations in the near future. 3 Chains of magnetic particles in an elongational flow Colloidal particles may give rise to either fixed or free structure aggregates depending on the nature of the aggregation processes and the type of interparticle bonds estab lished [25]. We will consider the case in which the spheres are in close contact but are 3. Chains of magnetic particles in an elongational flow 133 Figure IV.1: Chain of magnetic particles in an elongational flow. All the magnetic moments towards the direction of the high external field applied. still able to translate relative to one another under the action of an elongational flow. If the particles are non-magnetic, this internal motion is resisted by hydrodynamic lubrication forces and, although the relative motion is very small, it has seen to have a big effect on the stress transmission. Now, the particles interact with each other because of the dipolar magnetic interaction and we want to know the effect of this interaction on the internal motion and the viscosity of such a system. We define the following dimensionless parameters ..\ == m2 j(d3kBT) and J.l = mH jkBT, where m and d are the magnetic moment and the diameter of each sphere, H is the external magnetic field strength, kB is the Boltzmann constant, and T the absolute temperature. When J.l - 00, all the magnetic dipoles align parallel to the ex ternal magnetic field. Under this condition, the energetically favourable arrengement of N grains is a linear chain. As we will see below, the analytic study of these par ticular aggregates requires considerable approximations, even in the limit case ..\ � 1 and J.l - oo. On the other hand, entropy considerations will suggest that other more complex structures may arise when the particle density is too large or ..\ is too small [26]. But for low density and large ..\, the most favourable phase is a linear chain. Direct calculations [26] show that despite there are long-range contributions to the energy of a chain containing N particles, the amplitude of the vibrations and oscillations of the spheres are almost entirely taken into account by just considering the effects due to first and second nearest-neighbour interactions. The long range effects still exist, but they are dominated by short range attractive contributions. As a matter of fact, the important regions correspond to relative distances among the 134 CHAPTER IV. AGGREGATION PHENOMENA particles r """ d and relative deviations (J """ O or 11'. In addition, for our magnetic colloids the interactions of one grain are essentially saturated when it has come into close contact with two others. Moreover, in the high field limit, the state of a sphere will be entirely described by the position of its centre. This will simplify considerably the equation of continuity that describes the conservation of system points in the configuration space. Finally, we will disregard LondonVan der Waals forces as well as steric effects. 3.1 Contribution of the chains to the pressure tensor of the system Apart from the contribution of the solvent to the total pressure tensor of a supension, there is another contribution coming from the direct interaction of the suspended particles [27, 28]. Thus, following Kramers' theory, the contribution of a chain to the pressure tensor of the whole system is 1 � (� 8Vmag ) N - 1 IIp = - V � qk� + -V-kBT1, k=1 s» where V is the volume of the system, q¡, are the relative position vectors, Vmag is the dipole-dipole magnetic potential, N is the number of particles of the chain, and 1 is (3.1.1 ) the unitary matrix. In order to do the averages appearing on the right hand side of Eq. (3.1.1), we need to solve the corresponding diffusion equation in the configurational space: al/; '" {8 '" TT ( 8l/; 8Vmag) 8 �} - = � -=-. �""'ij . kBT-� + -_-l/; - -_ . (f3. H¡)l/; 8t i en; j 8Rj 8Rj 8H¡ (3.1.2) where ti; is the postion vector of the í-sphere, ,..,.Tl is the relative translational rnobil ity, and f3 is the elongational rate (f3 = f3T) corresponding to the stationary hornoge neous external flow, Vo = f3 . r. We are also assuming that the spheres move through the solvent without disturbing the velocity field (free draining), so that TT 1 ,..,. .. = --IÓij. 1) 611'7]a where we consider that all the spheres in the chain have exactly the same radius (3.1.3) a. Moreover, we will disregard the rotational contribution to the diffusion equation 3. Chains oi magnetic particles in an eJongationaJ flow 135 because in the high field limit the state of a particle is fully represented by the position of its centre. Substituting Eq. (3.1.3) into Eq. (3.1.2), we arrive at ot/J = kBT I: {02_t/J + o_ . (oVmag!kBT t/J) - .!. . (f3 . R;)t/J} . &t 61!'1Ja i oR? aH; aH; aH; In addition, as we are considering the limit A ::» 1, J-l -> 00, we will assurne that (3.1.4 ) the potential can be expressed as a sum of the nearest neighbours interaction V. N-l mag "'"' k T = L....J �;,i+l B ;=1 As we have previously indicated, the amplitude of the vibrations and oscillations of (3.1.5) the spheres are well represented by such interactions. After taking into account that the most important contributions come from relative distances r ..... dand relative deviations (J "-' O, each term in the sum of Eq. (3.1.5) can be written as follows �;,;+1 "-' -A(2 - 3(J? - 6�;) (3.1.6) where we have introduced the relative position vector q; = Ri+1 - R; whose cornpo nents in spherical coordinates are q; == (q;, (Ji, <p). Moreover, we have carried out a development around the maximum contribution q; d(1 + �;) with �i "-' O (Ji O. (3.1.7) Now, we will write the diffusion equation in terms of the relative position vectors, q;. But before that, we introduce the transformation q; = I: Bil:RI: with s., = 6i+1,1: - 6i,1;, 1: from which one infers the relation (3.1.8) a a -_ = I: Bki-;;= aH; k os» (3.1.9) with if í = j if í = j ± 1 otherwise (3.1.10) 136 eHAPTER IV. AGGREGATION PHENOMENA Taking into account Eqs. (3.1.8)-(3.1.10), the diffusion equation reads The stationary solution for this homogeneous potential flow field is ./. (- -) e {-Vmag 311''1a(a "''''c - -)} 'I',t q1," ·,QN-1 = eep -¡¡-;¡;- + k T p : L...,;L...,; ij qi qj , B B i j (3.1.12) where e is a normalization constant and eij is the Kramers' matrix defined in the following way: e .. _ { i(N - j)/N ifi s j 1) - j(N _ i)/N if j � i (3.1.13) Once we know the stationary solution of the diffusion equation, we will compute the contribution of the chain to the pressure tensor, i.e., the average appearing on the right hand si de of equation (3.1.1) (fik O��ag) = J fl dif¡ (fik O��ag) 1/J.t(Q1,"" Vv-1)' s» 1=1 s» As we are interested in finding the Newtonian viscosity tensor, we can expand the exponential factor (37r'1a)/(kBT)(f3: Li Li eijq'¡ijj) in Eq. (3.1.12) up to first order in f3. If we were also interested in the first non-Newtonian contribution we would include the second order, ... At first we will use (3.1.14) or if we introduce the translational diffusion coefficient for a single particle D = kBT/(67r'1a) and Eq. (3.1.5) where e' is the new normalization constant that we compute in Appendix A 3. ehaíns of magnetic particJes in an elongational fIow 137 1 C' (3.1.17) + Consequently, Eq. (3.1.14) reduces to ( _ OVmag) q 1: ----;:;-:; os» e' ¡/if dq¡ exp( -�I,I+l)) (iJ: o�,:ag) �l Ü {l + 2�(�. ��C;;;;q;)}. (3.1.18) In order to evaluate this average, we will decompose it into different parts (detailed calculations have been collected in Appendix B). Up to first in {3 we have (iJ: O�'ikag ) = kBT {1 + � � Cil:{3· fzfz + 6�A (1 - l�A) el;k{3· (1 - fZfZ)} (3.1.19) Once we have been able to compute this average, the contribution of one chain to the pressure tensor (3.1.1) In view of Eq. (3.1.20), we can conclude that we obtain an expansion in powers of the inverse of the parameter A, comparing dipolar and thermal energies, starting from the assymptotic value which corresponds to the rigid chain limit and increasing or decreasing when A decreases, i.e. when the chain becomes more flexible, depending upon the structure of the flow rate. It worths pointing out that, as we are not taking into account the hydrodynamic interaction between the particles, the rigid chain limit pressure tensor grows simply as the third power of the chain length. This interactions are the responsible for the logarithmic term that should also appear when dealing with a long straight line of spheres [27], but which is less important for shorter chains. 138 CHAPTER IV. AGGREGATION PHENOMENA Moreover, from this expression we can also obtain the contribution of the chain to the viscosity tensor, 'Ip IIp = -2'1p : 13 (3.1.21) and (3.1.22) Here S is a symmetric tensor such that Sijkl = 1/2(6ik6jl + 6iI6jk). As the external magnetic field that we are applying is such that J1. --4 00, the chain will be always oriented in the direction of the field and smoothly vibrates and oscillates around this orientation. In this case, the symmetries of the fluid flow essentially determine the characteristics of both the pressure and the viscosity tensor. Thus, as the chain is oriented along the z-axis, if the flow field rate f3 is diagonal, the pressure tensor will also be diagonal, i.e. it will be a symmetric tensor. On the other hand, we will have, in general, both a symmetric and an antisymmetric contribution for a non-diagonal flow rateo Let us consider sorne particular cases: i) Flow through a pore oriented along the same direction as the chain. In thís case the flow rate 13 has the followíng form: (3.1.23) where {3 is the flow rate strength, and the systern has a rotational symmetry around the z-axis. The chain in these conditions is stretched by the elongational flow giving rise to the so-called elogational viscosity defined as follows _ IIzz - 1I.u = 11'flod3 (N2 _ 1) {2N � (1 _ _3_)} 2{3 8V + 3>' 18>.· (3.1.24) For a fluid flow with rotational syrnmetry around the x (or y) axis, we will have two different contributions coming from the differences _ 1I.u - IIzz = 1f7]od3 (N2 _ 1) {N � (1 _ _3_)} 2{3 8V + 3>' 18>" (3.1.25) 3. Chaíns of magnetic particles in an elongational flow 139 (3.1.26) Moreover, in both cases we have a contribution coming from the trace of the pressure tensor, which for the flow rate given by Eq. (3.1.23) reads (3.1.27) and for the flow with rotational symmetry around the x axis is TrTI = 1rr¡od3 (N2 _ 1) {N _ 2. (1 _ �)} 6/3 8V 3>' 18>.· (3.1.28) ii) Planar elongational flow. This flow can be generated by four rotating infinite cylinders. If we locate the cylinders such that the flow rate is again a diago nal matrix, the pressure tensor will be diagonal and symmetric in view of Eq. (3.1.20), i.e. for f3=/3(��� ) O O -1 (3.1.29) we will find a similar behavior of the system as the one described aboye. On the other hand if we rotate the four cylinders 450, i.e. for f3=/3(� � �) 1 O -1 (3.1.30) the pressure tensor has both a symmetric and an antisymmetric part. Related to these parts, we will find not only elongational viscosities but al so a shear viscosity and a rotational viscosity given by (.) _ _ _ _ 1rTlod3 2 _ { 2. ( _�)} TI.,z - 2r¡/3 - 2 16V (N 1) N + 3>' 1 18>' /3, (3.1.31) (a) _ _ _ _ 1rr¡od3 (N2 _ ) { _ _!_ ( _�)} TI.,z - 2r¡r/3 - 2 16V 1 N 3>' 1 18>' /3, (3.1.32) where we identify the shear and rotational viscosities 140 CHAPTER IV. AGGREGATION PHENOMENA (3.1.33) (3.1.34) 4 Conclusions This chapter is intended as a preliminary study of the aggregation phenomena taking place in systems of magnetic particles in suspension and of the resulting structures. Our first point has been to elucidate the influence of the hydrodynamic interac tions (HI) in the kinetics of the aggregation process. We have extended the classic Smoluchowski theory of coagulation to account for the presence of HI ocurring when one goes beyond the dilute regime. Such interactions act before the different parti eles arrive at the sphere of influence of a given particle. We have obtained the kinetic equations for the aggregation process and from them we have analyzed the cluster for mation. Our main conclusion is that the presence of HI slows down the aggregation process. As a second problem we have studied the dynamics of a chain of magnetic particles under the influence of an external elongational flow. In particular, we have computed the contribution of the chain to the pressure tensor of the system from the rheological equation of state proposed by Kramers. From this quantity we have obtained the correction to the viscosities due to the presence of dipolar interactions. These preliminary results will constitute the subject of future work.