Jordan Journal of Physics

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1 Vlume, Number 1, 009. pp REVIEW ARTICLE Jrdan Jurnal f Physics Magnet-Optical and Magnet-Dielectric Anistrpy Effects in Magnetic Fluids N.A. Yusuf and I. O. Abu-Aljarayesh Department f Physics, Yarmuk University, Irbid, Jrdan. Received n: 1/8/008; Accepted n: 10/3/009 Abstract: An verview f the magnet-ptical and magnet-dielectric effects in magnetic fluids is presented encmpassing experimental results and different phenmenlgical mdels used t explain these effects. The rle f the different parameters n these effects, are discussed. It is suggested that the field induced-mechanical anistrpy prduced by agglmeratin and chain frmatin in magnetic fluids is the main cause fr these magnetptical and magnet-dielectric effects in magnetic fluids. Keywrds: Magnet-ptics, Magnetic fluids, Faraday rtatin, Faraday ellipticity, Transmissin, Optical anistrpy, Birefringence, Dichrism, Degree f plarizatin, Plarizatin effects, Magnet-dielectric effects, Dielectric anistrpy, Chain frmatin. Intrductin Under the applicatin f external magnetic fields, magnetic fluids, MF's, exhibit a range f magnet-effects such as magnet-ptical and magnet-dielectric effects. Magnet-ptical effects in magnetic fluids, have been investigated by many researchers using different experimental techniques as well as different theretical appraches because f their prmising ptential fr technlgical and industrial applicatins as well as their academic imprtance [1-4] The wrk n magnet-ptical effects has started as early as the middle f nineteenth century. chlten has reviewed the early wrk in magnet-ptical effects in cllids and shwed the imprtance f these studies in btaining infrmatin abut the magnetic particles and their spatial relatins [5]. The magnet-ptical effects t be addressed in this review are: Birefringence, Dichrism, Faraday rtatin, Faraday ellipticity, Degree f plarizatin and the transmissin f light; as well as magnetdielectric anistrpy effect. The induced magnet-ptical and magnet-dielectric effects in magnetic fluids are generally believed t be due t either field-induced magnetic r mechanical anistrpies in the fluid. This review is rganized as fllws: In sectin ne the texture in MF's is addressed. In sectin tw the magnet-ptical effects in MF's and their dependence n the varius parameters (magnetic field, cncentratin f magnetic particles, wavelength f the incident electrmagnetic waves and temperature) are presented. In sectin three the magnetdielectric anistrpy effect in MF's is presented. In the last sectin, numerical calculatins f the magnet-ptical and magnet-dielectric effects are presented. 1- Texture in Magnetic Fluids Under the applicatin f external magnetic fields, magnetic fluids becme textured. This texture culd be a magnetic texture due t the alignment f the permanent diple mments in the field directin, r it culd be Crrespnding Authr: N. A. Yusuf. nihadyusuf@yu.edu.j

2 Review Article Yusuf and Abu-Aljarayesh mechanical ne due t the agglmeratin f particles and chain frmatin in fluids. Cllidal suspensins f ferrmagnetic particles in magnetically passive liquid carriers have been fund in different states f aggregatin depending upn temperature, cncentratin f the magnetic particles and external magnetic fields. These systems have been bserved t frm lng chains as well as large cmpact clusters. Aggregatin and chain frmatin are assumed t be the main cause fr the appearance f mst magnet-ptical effects in magnetic fluids [5-4]. Theretical studies f chain frmatin in magnetic fluids based n particle-particle interactin were first carried ut by De Gennes and Pincus [6]. The main findings f this study can be summarized fr the lw cncentratin f magnetic particles in magnetic fluids as fllws: fr large external magnetic fields the particles tend t frm lng chains alng the field directin. In the absence f external field shrt chains still exist and behave like rds but randmly riented. Mrever as a chain length becmes mre than 4 D (D is the diameter f the particle) the lwest energy cnfrmatin is a ring, leading t magnetic flux clsure, thus the lnger chains either break int shrter chains r clse n it self. Elfimva [33] has attempted t establish a theretical mdel fr the fractal-like clusters induced in magnetic fluids and cncluded that an analytical result is rather difficult t btain. Fr chain frmatin t be established in a magnetic fluid, the magnetic particles must relax thrugh the Brwnian relaxatin mechanism, i.e., they shuld rtate physically in the field directin with their magnetic mments fixed in their easy axis directin with a relaxatin time τ B [37]. Anther cmpeting mechanism is the N e el relaxatin mechanism thrugh which the magnetic mments f the particles rtate against the anistrpy field and align in the field directin in a time τ N [37]. The relaxatin mechanism with the shrtest relaxatin time will dminate the relaxatin prcess f the particles. Particles, relaxing via the N e el relaxatin mechanism, d nt cntribute t chain frmatin r t the mechanical anistrpy in the sample. When the sample is in the frzen state, its viscsity is very high and the Brwnian relaxatin time is very large, cnsequently, the particles relax via the N e el relaxatin mechanism. Hwever, when the sample is in the liquid state bth relaxatin mechanisms are available fr the particles, and sme particles relax via the Brwnian relaxatin mechanism while the thers relax via the N e el relaxatin mechanism depending n the vlume f the particles and the temperature f the sample. As will be shwn when discussing the temperature dependence f magnet-ptical effects the nset f the Brwnian relaxatin ccurs when the tw relaxatin times becme equal (starting frm the frzen state f the sample). This equality ccurs fr a given vlume f the particles knwn as the hlimis vlumev. The hlimis vlume depends in a rather cmplex manner n temperature, viscsity, applied magnetic field, cncentratin and the size and shape f the particles. Particles with vlumes V V relax via the Brwnian relaxatin mechanism and thus cntribute t the chain frmatin in the sample. Althugh the nset f the physical rientatin f the particles in the field directin, and the chain frmatin is determined by the cmpetitin between thse tw relaxatin mechanisms, the degree f rientatin f the particles, and chain length r number f chains in the sample depend n ther parameters. Althugh the temperature and the applied field play direct and indirect rles in the relaxatin times τ N andτ B, they play tw cmpeting rles nce rientatin f particles and chain frmatin have started. The field plays an rienting rle while the temperature plays a randmizing rle due t thermal agitatin. At lw temperatures the rienting rle f the field is dminant, but at higher temperatures the randmizing rle f temperature becmes dminant. Tw mre parameters are crucial in the chain frmatin in the samples. The first is the interactin f the magnetic mments with the field, i.e., the Zeeman interactin which favrs energy-wise mre chains and lnger chains, and the chain- chain interactin which energy-wise favrs less number f chains, shrter chains and larger separatin between chains. It is imprtant t mentin that this chain-chain interactin becmes much

3 Magnet-Optical and Magnet-Dielectric Anistrpy Effects in Magnetic Fluids strnger nce the cncentratin f the sample is increased and may lead t the curling r even the clsure f sme chains. It is als imprtant t mentin that mst experimental investigatin, based n ptical bservatin, n the chain frmatin in magnetic fluids are usually carried ut n samples in the lw cncentratin regime where chain-chain interactin is weak [14,6]. It is the interplay f all these parameters that determines the ptimum cnditins fr chain frmatin in magnetic fluids and, cnsequently all ther magnet-ptical and magnet-dielectric effects. It is the cmplex nature f these parameters and even the mre cmplex nature f their interplay, we believe, that rendered the develpment f a cmplete analytical thery fr chain frmatin in magnetic fluids a frmidable task. Therefre ne has t rely n the available experimental and f Mnte- Carl simulatin results. In mst f the experimental wrk n chain frmatin in magnetic fluids, the samples are usually cntained in a transparent cell placed in a magnetic field. The dimensin f the cell in the field directin plays an imprtant rle n the length and number f chains frmed in the sample. Fr lw dimensin cells (thin film case), a spatial limitatin n the chain length is impsed resulting in a premature saturatin in the length, well befre the ptimum cnditins determined by the abve discussed cmpeting parameters is reached, and the faster frmatin f new chains. me experimental bservatins f chain frmatin have been indirect; Haas and Adams [5] cncluded that chains had frmed frm the diffractin pattern, characteristic f a grating, fund when light is passed thrugh a sample f magnetite-based ferrfluids perpendicular t the applied field. Ppplewell et al [6] have investigated the chain frmatin in magnetic fluid cmpsite thin films and have shwn bth experimentally and by Mnte Carl simulatin that extensive chain frmatin ccurs in relatively weak magnetic fields. They have als shwn that the chain length varies with the cncentratin accrding t the empirical frmula l α C 3l. Detailed experimental study n the fieldinduced agglmeratin in thin films f waterbased magnetic fluids has been reprted by Jnes and Niedba [7]. An experimental investigatin f the chain frmatin in magnetic fluids was als carried ut by Yusuf [8]. Recently Fang et al. [31] carried ut an experimental study n the magnetic-fieldinduced chain-like assembly structures in magnetic fluids. In their wrk they have bserved extensive chain frmatin under the applicatin f magnetic fields. Fang et al. suggested that the structure takes place in a tw stage prcess, aggregatin amng particles t frm chains, and aggregatin amng the chains t frm linear clusters, which are als active in frming the bserved aligned structures. Furthermre, they have als suggested that the chain length is sme times determined by the physical dimensins f the sample [31]. Anther recent study n chain-like aggregates in magnetic fluids was undertaken by Pshenichnikva, and Fedrenkb [30] using crss-fields measurements. The bias field is a direct field while the measuring field is an alternating field. Pshenichnikva, and Fedrenkb cncluded that chain frmatin was absent in highly cncentrated samples and that shrt chains existed in diluted samples. They have attributed this t the fact that, in a cncentrated sample, the clsedring structure is mre stable than lng chains. They have als suggested that shielding f the magnet- diple interactins reduces the pssibility f chain frmatin. Diluting the sample reduces the rle f shielding and breaks the balance between the attractive interactins and the steric-repulsive interactin, thus increases the chances f chain frmatin. In their wrk, Jnes and Niedba [7] had a drp f undiluted fluid sandwiched between tw parallel glass cver slips and placed nrmal t the ptic axis f a micrscpe. The magnetic field was applied parallel t the axis f the micrscpe, i.e. perpendicular t the plane f the thin film sample. In this experimental arrangement Jnes and Niedba bserved the number f chains per unit vlume, n, and fund it t increase rapidly with the applied field. The experimental set up used by Jnes and Niedba put sever limitatin n the length f frmed chains and lead t premature saturatin in the chain lengths as was suggested by Fang et al [31]. 3

4 Review Article Yusuf and Abu-Aljarayesh In his wrk, Yusuf [8] has avided this premature saturatin in the chain length by increasing the sample length in the field directin. The field and cncentratin dependence f chain frmatin in Fe particle magnetic fluid was investigated. The saturatin magnetizatins f the samples used are 5, 10, 15, 0 and 5 ka/m (saturatin magnetizatin f bulk Fe 3 O 4 is 485 ka/m). Thus these samples are diluted samples. Magnetic fields up t 71.6 ka/m are used in this wrk. Fig.1 shws the diffractin pattern btained when light was incident nrmal t the field and the plane f the fluid. The diffractin pattern bserved is similar t the pattern prduced by a diffractin grating and thus prvided indirect evidence n the chain frmatin in the fluid. The experimental set-up fr bserving the chain frmatin in the magnetic fluid is shwn in Fig.. FIG.. Viewing arrangement f the chains frmed in magnetic fluids The chain length in magnetic fluid with saturatin magnetizatin f 15 ka/m sample as a functin f applied magnetic field is shwn in Fig.5. The results in the figure shw that the chain length increases sharply at lw fields reaching 0.5 mm at relatively weak fields. A gradual increase in the chain length is bserved fr intermediate fields and at high fields a tendency twards saturatin is bserved. FIG. 1. The diffractin pattern bserved by shining a laser beam nrmal t the field and plane f the fluid. Fr viewing the chains the magnetic field is applied parallel t the plane f the fluid and perpendicular t the ptical axis f the micrscpe, and fr viewing the crss-sectin f the chains the field is applied perpendicular t the plane f the fluid and parallel t the ptical axis f the micrscpe. In Fig.3, An ptical micrgraph fr 15 ka/m saturatin magnetizatin magnetic fluid samples under 33.4 ka/m magnetic field shwing chain frmatin is presented, and in Fig.4, an ptical micrgraph fr the same sample shwing the crss-sectin f the chains is presented. FIG. 3. An ptical micrgraph fr 15 ka/m saturatin magnetizatin magnetic fluid under 33.4 ka/m magnetic field parallel t the plane f the fluid. FIG. 4. An ptical micrgraph btained by viewing the sample parallel t an applied magnetic field H = ka/m 4

5 Magnet-Optical and Magnet-Dielectric Anistrpy Effects in Magnetic Fluids In Fig.6 the chain length under the applicatin f an intermediate magnetic field (33.4 ka/m) is shwn as a functin f cncentratin (lw cncentratin regime) f the magnetic fluid. texture and chain frmatin n magnetptical effects will be discussed in the fllwing sectins. The different relatinships between chain length and cncentratin btained by different wrkers may be attributed t the dependence f the chain length n ther parameters such as the field, temperature and spatial limitatins impsed by the cells used in cntaining the sample under investigatin[3, 60, 61]. - Magnet-Optical Effects in Magnetic Fluids In this sectin the magnet-ptical effects including transmissin, Faraday rtatin, Faraday ellipticity, birefringence and dichrism are presented. The rle f texture and the varius parameters n these magnetptical effects is discussed. FIG. 5. The average chain length in a 15 ka/m saturatin magnetizatin sample versus applied magnetic field FIG. 6. The average chain length at H = 33.4 ka/m versus the saturatin magnetizatin f the fluid. The results shw that the chain length at intermediate field, i.e., befre saturatin, the chains length, is linear with saturatin magnetizatin. It is wrth mentining that fr thin samples, the chain length appraches the thickness f the sample at very lw fields and saturates prematurely. Cnsequently, the chain length is nt expected t increase any further with the field but rather an increase in the number f chains is bserved. The rle f.1- Magnet-Optical Effects and Texture in the Fluid.1.1- Linear Birefringence and Dichrism Birefringence, n, is defined as the difference between tw indices f refractin ne is fr an electrmagnetic wave linearly plarized in a given directin parallel t magnetic field directin, ( n ), and the ther fr an electrmagnetic wave linearly plarized in directin perpendicular t the magnetic field directin, ( n ). Dichrism, A hwever is defined as the difference in selective absrptin between these tw rthgnal states. Birefringence and dichrism have been measured by many investigatrs using different techniques [1, 56] Recently Kij et al. [57] has measured dichrism and birefringence in magnetite magnetic fluids using transmissin ellipsmetry. Using this technique they have investigated the spectral dependence f birefringence and dichrism and gt gd agreement with previus studies [69]. Here, we present sme f the experimental results relating birefringence t the texture in magnetic fluids. Davies and Llewellen [11] have bserved birefringence in a thin film f magnetic fluid sandwiched between thin slide cvers when it 5

6 Review Article Yusuf and Abu-Aljarayesh was in the liquid state. Hwever, when the magnetic fluid was dried under zer fields, birefringence was nt bserved and by adding a drp f liquid carrier t the dried sample, birefringence was bserved nce again. Bacri et al. [13] has implanted Fe 3 O 4 magnetic particles in a gel matrix and applied a magnetic field but they bserved n birefringence. Yusuf and c-wrkers [19] have studied birefringence in the temperature range K. Tw experimental prcedures were fllwed, the first by cling the sample t the lw temperature with n field applied. Then by applying an external magnetic field, the birefringence was measured. The results in Fig.7 shw that birefringence is absent at lw temperatures and cntinues t be absent till a specific temperature is reached then a weak birefringence is bserved. At this temperature the fluid is in a slurry state. By further increase in the temperature a steep increase in the birefringence is bserved (the fluid has reached its melting temperature) reaching a maximum value. Beynd this maximum an increase in temperature results in the reductin f birefringence. The secnd apprach is dne by applying a magnetic field n the sample starting frm high temperature and then cling the sample while the field is applied. The results als (Fig.7) shw that the birefringence increases by decreasing the temperature reaching a maximum value. Further decrease f temperature des nt change the value f birefringence. Discussin f this behavir is presented in the sectin devted t the temperature dependence f birefringence and dichrism. Drying the sample, implanting the magnetic particles in a gel matrix r freezing the magnetic fluid sample, have the same cmmn effect f preventing the physical mtin f the particles and thus prhibiting agglmeratin and chain frmatin in the sample. Therefre, the belief that the magnet ptical anistrpy, i.e, birefringence and dichrism, in magnetic fluids are due t the field-induced mechanical anistrpy f the sample is well established..1.- Circular Birefringence and Transmissin f Light Faraday rtatin is defined as the rtatin f the directin f plarizatin f a plane plarized electrmagnetic wave arund its prpagatin directin. This is usually explained in terms f the difference in the indices f the tw circularly plarized cmpnents, R- state and L -state. Faraday ellipticity, hwever, is a measure f the difference between the amplitudes f these tw circularly plarized states. Faraday rtatin, Faraday ellipticity and the transmissin f light thrugh a magnetic fluid were investigated by sme wrkers in the field [54-56, 58, 9]. Taketmi [54] studied the absrptin f light in thin films f magnetic fluids. Taketmi used tw different cncentratins and tw different thicknesses. It was shwn that the absrptin increases with cncentratin and thickness. Davies et al. [55] has studied the transmissin f micrwaves thrugh magnetic fluids and shwed that the transmissin fllws a Langevin- type behavir with the field similar t the magnetizatin f the fluid. Davies and Llewellen [56] have measured Faraday rtatin in highly diluted samples f Fe 3 O 4 and C particle magnetic fluid. Again their results suggested a Langevin-type behavir similar t the magnetizatin f the samples. Frm their results they cncluded that Faraday rtatin is gverned by the magnetizatin f the samples. FIG. 7. Birefringence versus temperature, a) warming up and b) cling dwn. 6

7 Magnet-Optical and Magnet-Dielectric Anistrpy Effects in Magnetic Fluids Rusan et al. [58] has investigated the dependence f the transmissin f light n the applied field and the cncentratin f the sample. In this investigatin the prpagatin f light is parallel t the applied field. The transmissin cefficient R defined as the rati f increase in the intensity f the transmitted light at a magnetic field H t the intensity f the transmitted light at zer applied fields was measured fr samples with different cncentratin, and fr samples f the same cncentratin but with different surfactant cntents. Furthermre, the cefficient R was measured fr the same sample but at tw different thicknesses. Their results shw that R fllws a Langevin-type behavir i.e., increasing almst linearly with the field at lw fields then gradually at intermediate fields tending t saturate at high fields. Hwever wapna et al. [9] in their recent study n the transmissin f light have used a different cnfiguratin. In their cnfiguratin the prpagatin f light is nrmal t the applied magnetic field. wapna et al. [6] results shw that the transmissin f light decreases with the applied field steeply at lw fields and then gradually at intermediate fields reaching almst a cnstant value at high fields. It is imprtant t mentin that the results f wapna et al. [9] and thse f Rusan et al. [59] are almst mirrr image f each ther abut the field axis. Bth f these tw grups attributed the change in the transmissin t the agglmeratin and chain frmatin in the magnetic fluid. Rusan et al. suggested that due t the chain frmatin under the applicatin f magnetic fields, the scattering crss-sectinal area f the particles decreases and the sample is arranged mechanically allwing fr the channeling f light in a directin parallel t the applied field. Therefre, the transmissin increases with the field. Using similar ideas but increasing the scattering crss sectins and partial blcking instead f channeling the decrease in the transmissin bserved by wapna et al. [9] may be explained. In their wrk, Rusan et al [58] has measured the transmissin f light fr the same sample at tw different thicknesses 0. and 1 mm. The results in Fig.8 shw that the transmissin cefficient R fr the thick sample is much higher than that fr the thin sample. Furthermre, they shw that R reaches saturatin in the thin sample while it is still increasing appreciably fr the thick sample. FIG. 8. Transmissin cefficient versus applied field at tw different thicknesses. This behavir is attributed t the premature saturatin f the chains in the thin samples. The same wrkers have als measured the transmissin cefficient in three samples with the same particle cncentratin, same liquid carrier but with slightly different surfactant cntents. Their results presented in Fig.9 shw that the higher the surfactant cntent in the sample is, the lwer the transmissin cefficient is and the faster it appraches saturatin. These results may be explained as fllws: decreasing the surfactant cntent f the sample breaks the balance between the attractive interactins and the repulsive interactin in favr f the attractive nes. Thus the pssibility f chain frmatin is increased [30] and cnsequently, the transmissin is increased. The cncentratin and field dependence f Faraday rtatin was thrughly investigated by Yusuf and c-wrkers [58-61] by studying samples with saturatin magnetizatin (6-100 ka/m). 7

8 Review Article Yusuf and Abu-Aljarayesh FIG. 9. Transmissin cefficient (R) versus field fr different surfactant cntents. Fig.10 shws Faraday rtatin, β fr samples with saturatin magnetizatin (6-5 ka/m) and thickness f 1 mm. The results shw that β fllws a Langevin-type behavir that is increasing sharply with the field at lw fields and tends t saturate at high fields similar t the magnetizatin f the samples. Furthermre the results als shw that β increases with the cncentratin f the sample and tends t saturate at higher fields fr higher cncentratins. Faraday rtatin β was measured fr tw identical samples but with tw different thicknesses. The results are presented in Fig.11. The results shw that Faraday rtatin in the thick sample is larger than that in the thin sample and surprisingly it saturates at fields fr the thicker sample higher than thse needed t saturate it fr the thinner sample. Faraday rtatin β was als measured fr samples f the same cncentratin but with slightly different surfactant cntents. The results are shwn in Fig.1. The results shw that Faraday rtatin increases with decreasing surfactant cntents and saturates at higher fields when the surfactant cntent is decreased as was previusly explained. FIG. 10. Faraday rtatin versus field fr different saturatin magnetizatin. If Faraday rtatin is nly gverned by the magnetizatin f the sample then the saturatin will take place at practically the same applied field fr the same sample regardless f the thickness f the sample. The difference in the saturatin field is attributed t the chain frmatin in the sample. As was mentined abve, chain frmatin may prematurely saturate in thin samples due t the physical limitatin impsed n the sample. FIG. 11. Faraday rtatin versus applied field at tw different thicknesses. 8

9 Magnet-Optical and Magnet-Dielectric Anistrpy Effects in Magnetic Fluids FIG. 1. Faraday rtatin versus field fr different surfactant cntents. As will be shwn in the sectin n the temperature dependence f magnet-ptical effects, a weak Faraday rtatin is still bserved in the frzen magnetic fluid when chain frmatin is physically prhibited. It is therefre believed that bth magnetizatin and chain frmatin cntribute t the Faraday rtatin in magnetic fluids; and here we attempt t separate these tw cntributins. Usually, fr a hmgeneus cllidal suspensin f single dmain fine ferrmagnetic particles, the Faraday rtatin is given by: M ( H ) β M ( H ) = C (1) M where β M (H ) is the Faraday rtatin at magnetic field H, M (H ) is the magnetizatin f the sample at magnetic field H, M is the saturatin magnetizatin f the sample and C is a cnstant. In rder t explain the results presented in Figs. (8, 9, 11-1), Yusuf and c-wrkers [60], intrduced a new term t the Faraday rtatin, β by invking the Verdet law. This term which is due t the chain frmatin is given by: β ( H ) VHl( H ) () C = where β C (H ) is the Faraday rtatin due t the chain frmatin, H is the lcal field, l (H ) is the chain length at field H and V is the Verdet cnstant. The ttal Faraday rtatin β is then given by: M ( H ) β ( H ) = C + VHl( H ) (3) M ince bth the magnetizatin f the sample and the chain frmatin fr weakly interacting system fllw a Langevin- type behavir, Faraday rtatin β is assumed t fllw a Langevin- type behavir expressed as: β ( H ) = β L( ah ) (4) where β is the saturatin Faraday rtatin, and a is a parameter that is saturatin magnetizatin dependent. The values f β are graphically btained by pltting the inverse f β against the inverse f the magnetic field H. By pltting the reduced Faraday rtatin ( β β ) versus magnetic field, Yusuf and c-wrkers [60] shwed that fr a given reduced Faraday rtatin the magnetic field is higher fr higher saturatin magnetizatin which implies that the parameter a is lwer fr higher saturatin magnetizatin. Furthermre, they have btained the parameter, a frm the linear part f the reduced Faraday rtatin curves and shwed that it is inversely prprtinal t the saturatin magnetizatin. In the experimental wrk f Yusuf and cwrkers, Faraday rtatin was measured using electrmagnetic waves f wavelength (λ = 600 nm) which is at least three rders f magnitude less than the chain length. Cnsequently, the lcal field felt by the electrmagnetic waves is practically cnstant and thus the cntributin t Faraday rtatin due t chain frmatin particularly at high fields is cnstant. Accrdingly, Faraday rtatin β is written as: M ( H ) β = C + C (5) M where C is the maximum cntributin f the chain frmatin t the Faraday rtatin. Pltting β versus ( M ( H ) M ), C is btained fr different samples with different saturatin magnetizatin and it is fund t be linear with saturatin magnetizatin. Nting that C is a cnstant multiplied by the chain length, it is btained that the chain length is linearly prprtinal t the saturatin 9

10 Review Article Yusuf and Abu-Aljarayesh magnetizatin, in agreement with the experimental results n chain frmatin btained by Yusuf [8]..- Field and Cncentratin Dependence f Magnet-Optical Effects The results n the field dependences f magnet-ptical effects shw that a given magnet-ptical effect, being birefringence, dichrism, transmissin, Faraday rtatin r Faraday ellipticity increases sharply and almst linearly with the field at lw applied fields, then gradually at intermediate fields appraching saturatin at high fields. The degree f plarizatin, D, f a partially plarized light defined as the rati f the intensity f the plarized cmpnent t the ttal intensity f light als behaves with the field in a similar manner. This behavir is assumed t fllw a Langiven-type behavir at lw cncentratin (weak interactins) and a mdified Langiven behavir at higher cncentratin f the samples (strnger interactins). This behavir f magnet-ptical effects with the field is similar t the behavirs f bth magnetizatin and chain frmatin. This general trend is explained in terms f tw cmpeting effects, the rienting effect (magnetic rientatin r physical rientatin) f the applied field that tends t align the particles magnetically r physically in the field directin; and the randmizing effect due t thermal agitatin which tends t disrupt the alignment in the field directin. The cncentratin dependence f mst magnet-ptical effects was investigated by many wrkers in the field. All experimental results shw that magnet-ptical effects increase with the cncentratin. Furthermre, the results shw that samples with higher cncentratins saturate at higher fields. Yusuf [78] has investigated the cncentratin dependence f birefringence and dichrism; and fund that bth birefringence and dichrism are linear with cncentratin in the range f cncentratin (lw cncentratin) used. These results are cnsistent with the results btained n chain frmatin versus cncentratin. Furthermre they are als in agreement with the theretical predictin f Fredriq and Hussier [79] fr diluted samples. Fr Faraday rtatin, Faraday ellipticity, the transmissin f light and the degree f plarizatin, nthing is fund in the literature t indicate a linear dependence n the cncentratin f the samples. Furthermre, it is nt expected t have such a linear dependence because bth Faraday rtatin and Faraday ellipticity are affected by the magnetizatin f the sample as well as by the chain frmatin. Hwever, althugh the transmissin f light is nt affected by magnetizatin and is nly affected by the chain frmatin and the reductin f gemetrical shadwing f the particles; and because the reductin f the gemetrical shadwing may nt be linear with cncentratin, the transmissin is fund nt t be linear with cncentratin. Althugh fr lw cncentratin samples, ne expects the degree f plarizatin t increase at least linearly with cncentratin, the results [70] shw that the degree f plarizatin at a given applied field is nt linear with cncentratin, and mst f the values lie belw the straight line with slpe 1. This may be explained in terms f the deplarizatin caused by mre scattering in samples with higher cncentratins due t much larger numbers f particles..3- The Wavelength Dependence f Magnet-Optical Effects The wavelength dependence f mst magnet-ptical effects has been investigated by sme wrkers in the field. The wavelength dependence fr Faraday rtatin, Faraday ellipticity, birefringence, dichrism and the degree f plarizatin is presented in the next subsectins Faraday Rtatin and Faraday Ellipticity The wavelength dependence f Faraday rtatin and Faraday ellipticity was theretically studied by Hui and trud [6]. Hui and trud have presented a thery f Faraday rtatin and Faraday ellipticity f a dilute suspensin f small particles embedded in a hst carrier. They cnsidered the dielectric functin f the Faraday active particles t be a cmplex, frequencydependant tensr and cnsidered the dielectric functin f the hst liquid t be a scalar 10

11 Magnet-Optical and Magnet-Dielectric Anistrpy Effects in Magnetic Fluids functin. Hui and trud used the Maxwell- Garnet [63] apprximatin t calculate the effective dielectric tensr f the suspensin in terms f thse f the particles and the liquid, and the vlume fractin f the particles in the suspensin. Frm the calculated dielectric tensr, they have determined the frequency dependence f Faraday rtatin and Faraday ellipticity f the magnetic fluid. Their calculated results fr Faraday rtatin and Faraday ellipticity are presented in Figs.13 and 14, respectively. Experimentally, the wavelength dependence f Faraday rtatin and Faraday ellipticity was investigated by Yusuf and cwrkers [64, 65]. The results are presented in Figs.15 and 16, respectively. As can be seen frm the Figs a gd agreement between the calculated and measured results is fund. Hwever, the relative width f the resnance in the experimental results is wider than that in the calculated results. FIG. 13. Faraday rtatin versus frequency, nrmalized t plasma frequency, calculated in [6]. FIG. 14. Faraday ellipticity versus frequency, nrmalized t plasma frequency, calculated in [6]. FIG. 15. Faraday rtatin versus frequency υ (experimental). This may be attributed t the fact that in the calculatin, Hui and trud have nly cnsidered mn dispersed spheres f equal size. While in reality the dispersed particles have shape and size distributins. Furthermre, due t the applicatin f an external magnetic field agglmeratin and chain frmatin are bund t ccur in the sample. These results seemingly indicate that agglmeratin and chain frmatin play a rle in the Faraday rtatin and Faraday ellipticity in magnetic fluids..3.- Birefringence and Dichrism Llewellen [66] has studied the wavelength dependence f birefringence and dichrism and successfully explained his experimental results in terms f Wiener thery [67] nly after taking int accunt the absrptin f light by bth the particles and the liquid carrier which is equivalent t cnsidering the dielectric functins t be cmplex. 11

12 Review Article Yusuf and Abu-Aljarayesh thrugh a wire grid emerge partially plarized with a given degree f plarizatin D. FIG. 16. Faraday ellipticity versus frequency ν (experimental) Yusuf et al. [68] has als investigated the wavelength dependence f birefringence and dichrism in the range f wavelengths ( nm) and in a magnetic field range (0-500 ka/m). Figs. 17 and 18 shw birefringence and dichrism versus field measured at different wavelengths. Figs. 19 and 0 shw birefringence and dichrism versus the wavelength f the electrmagnetic waves. Kij et al. [57] using transmissin ellipsmetry has investigated the wavelength dependence f birefringence and dichrism but fr a narrwer range f wavelengths. Their results are in gd agreement with the results f Yusuf et al. fr the range f wavelengths cmmn in bth wrks Degree f Plarizatin Plarizatin effects in magnetic fluids are a cnsequence f the chain frmatin in the fluid. ince mst magnet-ptical effects are dependent n the plarizatin state f the electrmagnetic waves used, it is infrmative t investigate the plarizatin effects f magnetic fluids under the applicatin f magnetic fields. The ccurrence f chain frmatin in magnetic fluids, thugh the chains are irregular in length and thickness, allws fr thinking ff the sample in terms f a wire grid plarizer. Natural (un-plarized) electrmagnetic waves when transmitted FIG. 17. Birefringence versus measuring field fr different wavelengths. FIG. 18. Dichrism versus measuring field fr different wavelengths The wavelength dependence f the Degree f plarizatin in magnetic fluids has been investigated by Yusuf et al. [70]. 1

13 Magnet-Optical and Magnet-Dielectric Anistrpy Effects in Magnetic Fluids FIG. 19. Birefringence versus wavelength at tw measuring field FIG. 1. Degree f plarizatin versus measuring field fr different wavelengths. FIG.. Degree f plarizatin versus measuring field fr different wavelengths. FIG. 0. Dichrism versus wavelength at tw measuring field. Figs. 1 and, shw the degree f plarizatin, D, measured at different wavelength pltted versus magnetic field fr a magnetic fluid sample with φ = The wavelengths λ used in this wrk ranges between nm [68]. The results shw that D increases with decreasing wavelength fr λ 488 nm exhibiting a peak at λ = 488 nm and then decreases with further decrease f wavelength FIG. 3. Degree f plarizatin measured at varius fields versus frequency. 13

14 Review Article Yusuf and Abu-Aljarayesh In Fig.3 the degree f plarizatin is pltted versus frequency. The results shw that D increases with frequency fr frequencies ν < 6 x Hz exhibiting a peak at ν = Hz, then decreases with higher frequencies. The degree f plarizatin like ther magnet-ptical effects, exhibits a resnance peak with frequency. Yusuf and cwrkers [68] attributed this behavir t the variatin in the absrptin f light with electric field vectrs parallel and perpendicular t the applied magnetic field with the frequency f the light used; and t the difference in the absrptin f these tw cmpnents. Fr light prpagating nrmal t the applied magnetic field, the Zeeman interactin results in three lines: a nn-shifted (ν = ν ) line with its electric field parallel t the applied magnetic field and tw shifted lines with their electric fields nrmal t the applied magnetic field and with frequencies ν = ν ± ν where ν is the change in frequency due t the Zeeman effect. The absrptin f these lines peaks at their perspective frequencies, i.e. at ν and ν ± ν. Fig. 4 shws the three Zeeman lines and their resnance absrptins. Therefre, starting with a frequency ν < ν - ν and by increasing the frequency, the first absrptin line at ν - ν fr the first transverse state is reached and then the transverse cmpnent is absrbed mre than the lngitudinal ne resulting in partially plarized light. With further increase in frequency the secnd absrptin line at ν fr the lngitudinal cmpnent is reached and then the lngitudinal cmpnent will be absrbed mre than the transverse cmpnent, again resulting in partially plarized light. Hwever, it is imprtant t nte that the height f the absrptin line fr the lngitudinal cmpnent is twice that fr the transverse cmpnent. As a result, the degree f plarizatin increases with frequency in the range (ν - ν) < ν < ν reaching a peak at ν. Increasing the frequency abve ν mves the system away frm the absrptin line at ν = ν t third absrptin line at ν = ν + ν fr the secnd transverse cmpnent, and cnsequently the degree f plarizatin decreases. FIG. 4. Zeeman lines and their resnance absrptins. Althugh, the Zeeman effect fr ne single particle, prduces partially plarized light, but fr an ensemble f randmly riented particles ne expects t have zer degree f plarizatin as a result f the superpsitin f a very large number f incherent light waves partially plarized with their planes f plarizatin randmly riented. As the results shw, the degree f plarizatin is zer under zer applied magnetic fields and cnsequently it is suggested that agglmeratin and chain frmatin play an imprtant rle in the plarizatin effects in magnetic fluids..4- The Temperature Dependence f Magnet-Optical Effects The temperature dependence f magnetptical effects was investigated by many wrkers in the field. This dependence and its investigatin have prvided mre understanding f the rigin f magnetptical effects. Taketmi et al. [33-35] has studied the birefringence in the temperature range K, well abve the melting pint f the liquid carrier, and fund that birefringence fllws a generalized Curie- Weiss type behavir assciated with psitive rdering temperatures. Yusuf and c-wrkers [18-1, 41, 4] have als investigated the temperature dependence f mst magnet-ptical effects in the temperatures range ( K). This range started frm temperatures well belw 14

15 Magnet-Optical and Magnet-Dielectric Anistrpy Effects in Magnetic Fluids the melting pint f the liquid carrier t temperatures well abve the melting pint f the liquid carrier. The results f the temperature dependence f Faraday rtatin and the transmissin f light are presented in Figs. 5 and 6. The results shw that belw a given temperature (T = 15 K), the transmissin f light, dse nt change by applying a magnetic field frm its value at zer applied field. Hwever, the Faraday rtatin fr these temperatures increases linearly with the field. Nting that in this temperature range, the magnetic fluid is in the frzen state and hence the chain frmatin is physically prhibited, again suggests that the transmissin f light is cntrlled mainly by the chain frmatin while Faraday rtatin has tw cmpnents ne is cntrlled by the magnetizatin and the ther is due t the chain frmatin in the fluid. Furthermre, the results shw that arund 150 K the transmissin f light starts t increase with the applied field and that Faraday rtatin als starts t increase with the field but nt linearly. Abve this temperature, bth the transmissin and Faraday rtatin fllw Langiven-type behavir with the applied field with values depending n the temperature at which measurements are taken. The results n the temperature dependence f the transmissin f light and Faraday rtatin are presented in Figs. 7 and 8, respectively. The results shw that belw 150 K, the change in the transmissin f light is zer and Faraday rtatin is very weak. Fr temperatures just abve 150 K bth the transmissin and Faraday rtatin increases slwly with temperature and arund T = 00 K they bth start t increase sharply with temperature exhibiting a cusp-like peak at T = 5 K. Abve this temperature bth the transmissin and Faraday rtatin decrease with temperature fllwing a generalized Curie-Weiss type behavir with negative rdering temperatures. FIG. 6. Faraday rtatin versus applied magnetic field at different temperatures. FIG. 5. Transmissin versus applied magnetic field at different temperatures. FIG. 7. Transmissin versus temperature at different applied magnetic field. 15

16 Review Article Yusuf and Abu-Aljarayesh FIG. 8. Faraday rtatin versus temperature at different applied magnetic field. The bservatin f negative rdering temperatures is in agreement with ur magnetic measurements carried n the same samples. They are als in agreement with the results f Ppplewell et al. [71] btained fr magnetizatin n Fe 3 O 4 magnetic fluid. tudies by ffge and chmidbauer [7] indicated a Curie-Weiss type behavir with negative rdering temperatures and have shwn tw regins f linearity; ne at lw temperatures yielding a negative rdering temperatures and the ther at high temperatures yielding a psitive rdering temperatures. The general temperature dependence f magnet-ptical anistrpy effect, i.e., birefringence and dichrism was thrughly investigated by Yusuf and cwrkers [18, 19, 40, and 41] in the temperature range K. Bth birefringence and dichrism were absent fr temperatures lwer than ~ 150 K. Fr temperatures just abve 150 K bth birefringence and dichrism are detected and with further increase f temperature, sharp increase in bth was detected reaching a cusp like maximum at arund ~ 00 K. With further increase f temperature bth birefringence and dichrism decreases with temperature. Typical results fr the birefringence and dichrism are shwn in Figs. 9 and 30. The general temperature dependence f the magnet-ptical anistrpy effect will be addressed and discussed in the next sectin. Here the Curie-Weiss type behavir will be discussed in cnnectin with the chain frmatin in the magnetic fluids. T explain the Curie-Weiss behavir in the magnet-ptical effects, Yusuf and cwrkers [19] invked the dependence f these effects n the chain frmatin and prpsed a tw-dimensinal mdel based n the Bltzmann distributin. They have assumed that the prbability that the magnetic axis f a given particle makes an angle θ with the applied magnetic field fllws a Bltzmann distributin, i.e., U P( θ ) = c exp( ) (6) kt where U is the particle ptential energy which cnsists mainly f tw cntributins, namely that is due t the applied field and that is due t the diple-diple interactin, c is a cnstant, T is the abslute temperature and k is the Bltzmann cnstant. FIG. 9. Birefringence versus temperature at different measuring fields The number f particles aligned in a given directin is given by the prduct f the prbability P (θ ) and the ttal number f particles, N in the sample. Hence the effective ttal chain length, l in a given directin is expressed as fllws: 16

17 Magnet-Optical and Magnet-Dielectric Anistrpy Effects in Magnetic Fluids U l = Ncd exp( ) (7) kt where d is the mean particle diameter. The chain lengths, l parallel t the field directin and, l perpendicular t the field directin are f particular imprtance in regard t induced magnet-ptical anistrpy in magnetic fluids. These tw chain lengths fr the case f negligible diple- diple interactins i.e. in weak samples are given by: µ H l = Ncd exp( ) (8) kt And l = Ncd (9) Ncdµ H n = (1) kt It is seen frm Eq. 1 that the birefringence, n fr high temperature r lw fields, fllws a Curie type law. When the intrinsic diple-diple interactin is nt negligible, Yusuf and cwrkers have treated the prblem in an analgus way as is usually dne in magnetism [70]. It is assumed that due t the chain frmatin, an internal field, in additin t the externally applied field, is established. This field is assumed t be prprtinal t the chain length. Cnsequently the chain lengths parallel and perpendicular t the field directin are expressed as: µ H + αlιι l = Ncd exp( ) (13) kt and l = αl Ncd exp( kt ) (14) Therefre, l fr a weakly interacting system and µ H << kt, is given by: µ H + α l l = Ncd ( ) (15) kt Rearrangement f eq. (15) yields the fllwing expressin: Ncdµ H l = k( T T ) (16) FIG. 30. Dichrism versus temperature at different measuring fields. The difference in the chain length in the tw directins is given by: µ H l = Ncd{exp( ) 1} (10) kt and when µ H << kt, it is reduced t: µ H l = Ncd ( ) (11) kt Assuming that the induced birefringence n is prprtinal t l, then, the birefringence is, within a cnstant, given by: with T, referred t as rdering temperature, is equal t ( Ncdα k ), cnsequently the birefringence n,within a cnstant, is given by: Ncdµ H n = k( T T ) (17) and hence the birefringence fllws a Curie- Weiss type behavir. Therefre, it is believed that the Curie r Curie-Weiss behavir f the magnet-dielectric anistrpy effect in magnetic fluids may be explained in terms f chain frmatin in the fluid. Furthermre, birefringence n as seen frm Eq. 1 and Eq. 17 is linearly prprtinal t the ttal number f particles, N in the sample which is in turn prprtinal t the cncentratin f the sample. Cnsequently, birefringence is linear with cncentratin. The general behavir f the magnetptical anistrpy effect with temperature has revealed the fllwing basic features [16-19,9, 30]: As is shwn in Figs. 9 and 30, magnetic fluids d nt exhibit any degree f ptical anistrpy belw a given temperature 17

18 Review Article Yusuf and Abu-Aljarayesh T ; fr T > T, magnetic fluids start t shw sme ptical anistrpy and the degree f this ptical anistrpy increases with temperature reaching a maximum at a temperature T m ; and fr T > T m the degree f the ptical anistrpy decreases with temperature. In additin t thse basic features it was fund that fr a given cncentratin f the magnetic fluid bth T and T m decrease with the applied measuring field being the lwest fr the highest field as is shwn in Figs. 31 and 3 [41, 4]. FIG. 33. Birefringence versus temperature fr different vlumic fractins. FIG. 34. Dichrism versus temperature fr different vlumic fractins. FIG. 31. Birefringence versus temperature at different measuring fields. FIG. 3. Dichrism versus temperature at different measuring fields. It is als fund that fr a given measuring field bth T and T m increase with the cncentratin f the magnetic fluid being the highest fr the highest cncentratin as is presented in Figs 33 and 36 [41, 4]. Furthermre, it has als been shwn that T m depends n the carrier used in the magnetic fluid [0] such that the lwer the viscsity and melting pint f the carrier are, the lwer T m is as is seen in Fig. 35. FIG. 35. Birefringence versus temperature at measuring field f 7.16 ka/m fr different liquid carriers. These basic features are basically explained in terms f tw cmpeting effects, mainly an rienting effect due t the applicatin f the external field and the randmizing effect due t thermal agitatin. Furthermre, as will be shwn in the last sectins f this article, these basic features have been successfully btained using 18

19 Magnet-Optical and Magnet-Dielectric Anistrpy Effects in Magnetic Fluids numerical calculatins based n these tw cmpeting effects. The rienting effect is achieved thrugh either the rtatin f the magnetic mments f the particles within the particles while the particles remain fixed in their cnfiguratin (Neél relaxatin mechanism) r thrugh the physical rtatin f the particles while the magnetic mments are fixed in the easy directin f the particles (Brwnian relaxatin mechanism). The first mechanism will prduce magnetic anistrpy but n mechanical anistrpy in the sample while the secnd mechanism will prduce bth mechanical and magnetic anistrpy. As was stated earlier, the magnet-ptical anistrpy effect is mainly due t the induced mechanical anistrpy caused by the agglmeratin and chain frmatin in the sample. Cnsequently, the magnet-ptical anistrpy effect will appear nly when the induced mechanical anistrpy is present in the sample. Therefre, it is imprtant t review the mechanism thrugh which the particles will rient themselves with the applied field. The magnetic mments f the cllidal particles in magnetic fluid may reach thermal equilibrium via tw distinct mechanisms. The first is the N e el relaxatin in which the magnetic mments f the particles rtate against the existing anistrpy barrier E in a time τ N given by [36]: M sb k BT 1 KV τ N = [ ][ ] exp( ) (18) αγk KV k T where α is an attenuatin factr, γ is the gyrmagnetic rati, K is the effective anistrpy cnstant, V is the magnetic vlume f the particle, k B is the Bltzmann cnstant, M sb the saturatin magnetizatin f bulk material, and T is the abslute temperature. The secnd, mechanism is the Brwnian relaxatin in which the particles rtate physically s that their easy axes align with the field while the magnetic mments are held fixed in the directin f the easy axis in a time τ given by [36]: B B τ B 3 V s = η k T B (19) where η is the viscsity f the magnetic fluid, V is the hydrdynamic vlume f the particle and s is a gemetrical factr ( s = 1 fr spherical particles). The dminant relaxatin mechanism is the ne with the shrter relaxatin time. As can be seen frm Eq. 8 the N e el relaxatin time grws expnentially with the magnetic vlume, therefre, nly small particles may relax via the N e el relaxatin mechanism. In magnetic fluids, there is always a size distributin f the particles, cnsequently a vlume fr which τ N = τ B exists. This vlume is knwn as the hlimis vlume, V s [36, and 37]. Only particles with V > V s will relax via the Brwnian mechanism, thus cntributing t the ptical anistrpy. It is, therefre, necessary t determine the hlimis vlume, V s. This is accmplished by equating the tw relaxatin times and assuming V = V yielding the fllwing transcendental relatin: q 3 6αγη exp( q) = M sb (0) where q = ( KV k BT ). The relatin in Eq. 0 is numerically slved fr the hlimis vlume, V s. As seen frm this equatin, the viscsity,η, is the determining factr fr the hlimis vlume at a given temperature and therefre it is necessary t state sme relatins related t the viscsity f a magnetic fluid. It is well established that the viscsity f a magnetic fluid varies with the cncentratin f the magnetic particles, the applied field, temperature and inter-particle interactins. The viscsity f a nn-aggregating, highly diluted (φ << 0.1) cllidal suspensin is apprximated by the Einstein frmula [37]: η = η ( 1 +.5φ ) (1) where η is the viscsity f the liquid carrier and. φ is the cncentratin f the sample. Fr φ ~ 0.1, the viscsity starts t deviate 19

20 Review Article Yusuf and Abu-Aljarayesh frm linearity and fr φ = 0.3 the viscsity increases sharply with a prnunced nnlinear behavir. The exact functinal dependence f the viscsity n temperature is nt well defined, but fr a qualitative discussin the viscsity (may be expressed as [39, 40]: η B ln( ) = η T T T T () η B ln( ) = T > T η n (3) T where B and B are characteristic psitive cnstants, T is a temperature belw the melting pint f the cllid, and n is a cnstant, usually taken t be larger than ne. Fr a cllidal suspensin, Eq. 3 will be applied in the liquid regin. It is relevant t mentin that the viscsity f magnetic fluids depends nt nly n the applied magnetic field but als n its directin [36]. It is wrth mentining that at lw fields the transverse and lngitudinal cmpnents f the magnetizatin have the same relaxatin time which isτ B. Fr high fields, hwever, the relaxatin time fr the lngitudinal cmpnent is still equal tτ B, while that fr the transverse cmpnent τ is given by [38]: τ τ B = (4) p where p = µ M sbvh k B T The magnetic particles in the fluid may have intrinsic ptical anistrpy due t their anistrpic shapes, althugh the magnetic fluid under zer fields will have n ptical anistrpies due t the randm rientatin f the particles in the fluid. When an external magnetic field is applied t the fluid, induced ptical anistrpy results as a cnsequence f the rientatin f the particles and t the field induced chain frmatin. The ptical anistrpy f a uniaxial singledmain particle suspended in a nnmagnetic liquid carrier under the applicatin f an external magnetic field was treated by Hartmann and Mende [16, 17] and chlten [14, 15]. In a carrier liquid with istrpic, real refractive index n this ptical anistrpy is given by [14-17] ~ κε n( T, H ) = ( g ~ g ~ a b ) Φ ( p, q) (5) n where g ~ a and g ~ b are the cmplex plarizabilities f the magnetic particles, which are usually given as: ~ ~ ~ n 1 {( ~ p n g i = ~ ) ~ (6) + n n N n } p where n ~ p is the anistrpic refractive index f the particle, and N i is the demagnetizatin cefficient which depends nly n the axial rati ( a b ) f the particle. Φ ( p, q), is an rientatin functin usually expressed as a prduct f tw functins, i.e. Φ ( p, q) = ξ ( q) f ( p) (7) where ξ (q) represents the cupling between the magnetic mment f the particle and its easy axis and is usually expressed as: 1 3 q exp( q) I( q) 1 ξ ( q) = ( ) (8) 4q I ( q) where q I ( q) = exp( x ) dx The functin f (p) is given by: i (9) 3L( p) f ( p) = [1 ] (30) p where L ( p) is the Langevin functin, and p = M VH k T. µ sb B The birefringence, n and dichrism, A, are btained frm the real and imaginary parts f the ptical anistrpy, n ~, i.e., κε n = Re( g ~ ~ a gb ) Φ ( p, q) (31) n And κε A = Im( g ~ a n g ~ b ) Φ ( p, q) (3) 0

21 Magnet-Optical and Magnet-Dielectric Anistrpy Effects in Magnetic Fluids When the particle size distributin is taken int cnsideratin, the ttal birefringence, n T, and ttal dichrism A T will be given by: V s n = nf( V ) dv (33) and T V s A = AF( V ) dv (34) T where F (V ) is a suitable particle size distributin ften taken as a lg-nrmal distributin, and V s, is the hlimis vlume which is btained by slving Eq. 0 numerically. The basic features bserved fr the magnet-ptical anistrpy effect against temperature are explained as fllws: Fr temperatures belw T, the magnetic fluid is in the frzen state with infinite viscsity and hence the particles are blcked against the physical rtatin resulting in n mechanical anistrpy in the sample and cnsequently the absence f the magnet-ptical anistrpy. When the temperature reaches, T, (the slurry state temperature) sme f the particles are unblcked and rient themselves in the field directin prducing sme degree f mechanical anistrpy and as a result magnet-ptical anistrpy appears. Increasing the temperature near T results in a sharp decrease in the viscsity f the sample allwing mre particles, in large numbers, t be unblcked and rient themselves in the field directin and thus an increase nt nly in the number f chains but als in their lengths leading t sharp enhancement f the ptical anistrpy. Further increase f temperature still results in reducing the viscsity f the sample and thus increases the chances fr unblcking the particles. But increasing the temperature has als anther effect that is a randmizing effect which results in disrienting the particles and thus in reducing the ptical anistrpy. It is the cmpetitin between these tw effects that determines whether the ptical anistrpy will increase r decrease. Fr temperatures abve T and belw sme temperature Tm the unblcking effect is the dminant ne and the ptical anistrpy increases till it reaches a maximum att m. By increasing the temperature abve T m the balance is disturbed and the randmizing effect becmes mre effective thugh the unblcking effect still plays a rle but with smaller number f particles. As a result the ptical anistrpy starts t decrease with temperature fr T > T m. Increasing the measuring magnetic field increases the magnetic trque exerted n each particle in the sample and cnsequently the particle will be able t rtate against a higher viscus trque, i.e., at a lwer temperature than that fr a weaker applied field. Hence the unblcking f particles starts at a lwer temperature, and the ptimum cnditin (temperature) will be lwer. Therefre, the temperatures at which the nset and the maximum f the ptical anistrpy, ccur, decrease with increasing the applied magnetic field. Increasing the cncentratin f sample, results in increasing bth the viscsity and the melting pint f the sample. It is bvius that increasing the viscsity and melting pint f the sample results in increasing bth T and T m. Using liquid carriers having different initial viscsities leads t changing the psitin f bth T and T m such that the lwer the initial viscsity is the lwer bth these tw temperatures are. The arguments that wuld be used t explain the effect f cncentratin and viscsity n the psitin f T and T m are similar t thse used t explain the effect f the measuring field. 3-Magnet-Dielectric Anistrpy Effects in Magnetic Fluids The magnet-dielectric effect in magnetic fluids has been investigated by many wrkers bth experimentally and theretically [43-53]. The experimental investigatins were based n impedance measurement techniques where the magnetic fluid is placed in a capacitr. Measurements f the impedance parameters 1

22 Review Article Yusuf and Abu-Aljarayesh such as the mdulus and phase are carried ut using a bridge r an RLC meter [43-50]. It is knwn that impedance measurement techniques suffer sme serius disadvantages such as electrde effects, parasitic impedances, skin depth and accuracy-related prblems. On the theretical side, Mnte Carl simulatins were used t calculate the magnet-dielectric effect [51-53]. trud and c-wrkers [6] have calculated sme magnet-ptical effects (Faraday rtatin and Faraday ellipticity) by determining the dependence f the ffdiagnal elements f the permittivity tensr n field, temperature and frequency. Their findings were experimentally verified by Yusuf and cwrkers [64, 65]. Using the ppsite apprach Llewellyn [66] has calculated the elements f the skew symmetric permittivity tensr f Fe particles frm their simultaneus measurements f birefringence, dichrism, Faraday rtatin and Faraday ellipticity in Fe particle magnetic fluids. Yusuf and c-wrkers [68, 69] have calculated the magnet-dielectric anistrpy effect in magnetic fluids frm the ptical anistrpy (i.e. birefringence and dichrism) measurements. They have investigated the field, cncentratin, Temperature and wavelength dependence f the magnetdielectric anistrpy effect. Bth the real and imaginary parts f the dielectric cnstant seen by light plarized parallel r perpendicular t the external magnetic field are calculated. Determining the magnet-dielectric anistrpy effects using ptical anistrpy measurements has the advantage ver the cnventinal methd in that it is highly accurate and is nt affected by electrde effects r skin depth and parasitic impedances. The dielectric cnstant f a magnetic fluid in the absence f an external magnetic field exhibits n anistrpy due t the randm rientatin f the particles. Therefre, the dielectric cnstant seen by light with different states f plarizatin is the same. Hwever, when a magnetic field is applied, rientatin f particles and field-induced chain frmatin in the field directin take place leading t tw different average lengths l in the field directin and l^ perpendicular t the field directin; cnsequently the dielectric cnstant will exhibit sme degree f anistrpy. The magnet-dielectric anistrpy factr g ( H, ω) is defined as: ε ( H, ω) ε (0, ω) g( H, ω) = (35) ε (0, ω) ε ( H, ω) The value f this factr btained by different investigatrs was either 1 r depending n the sample investigated [45, 48, 49, 50]. Using their magnet-ptical measurements, Yusuf and cwrkers [68, 69] have shwn that g = 1 fr thin samples (- dimensinal case) and that g = fr thick samples (3-dimensinal case) Derivatin f Fundamental Equatins In general, the dielectric cnstant f the magnetic fluid is cmplex and is written as: ~ ε = ε iε (36) where ε andε are the real and imaginary parts f the dielectric cnstant, respectively. Furthermre, the index f refractin f the fluid is als cmplex and is written as: n ~ = n ik (37) where n and k are the real index f refractin and the extinctin cefficient, respectively. Cnventinally, the dielectric cnstant and the index f refractin are related by: ~ n ~ ε = = n k ink (38) When an external magnetic field is applied, the dielectric cnstant ~ ε becmes anistrpic, thus exhibiting tw different behavirs fr light plarized parallel t the magnetic field directin ~ε and t that plarized ~ perpendicular t the magnetic field ε. imilarly, the index f refractin exhibits tw indices ~n, n ~, and the extinctin cefficient als exhibits tw values k, k. Accrdingly, Eq. (38) will be mdified such that it incrprates this induced anistrpy. The mdified equatins are:

23 Magnet-Optical and Magnet-Dielectric Anistrpy Effects in Magnetic Fluids ~ ε in k (39) = ε iε = n k and ~ ε ε = n k ε = i in k (40) By equating the real and imaginary parts in equatin (39) and in equatin (40), the fllwing equatins are btained: = n k ε (41a) ε (41 b) = n k and ε = n k (4 a) ε n k (4 b) = The distinctin between a tw dimensinal (thin) system and a three dimensinal (thick) system is manifested in the relatin jining n, n and n ; k, k, and k ; where n and k are the index f refractin and extinctin cefficient f the sample, at zer fields, respectively. Fr a tw dimensinal system Yusuf and cwrkers [68] used the fllwing relatins: n + n (43a) = and n k + k (43b) = k Hwever fr a three dimensinal sample they [66] used the fllwing equatins: n + n 3 (44a) = and n k + k 3 (44b) = k And by using the definitin f birefringence and dichrism expressed as: = n n (45a) n and = k k (45b) k One gets the tw sets f equatins, the first set is applicable t the tw dimensinal sample and cnsists f the fur fllwing equatins: n k ε = ( n + ) ( k + ) (46a) n k ε = ( n + )( k + ) (46b) n k ε = ( n ) ( k ) (46c) n k ε = ( n )( k ) (46d) while the secnd set is applicable t the three dimensinal sample and cnsists f the fllwing equatins: n k ε = ( n + ) ( k + ) (47a) 3 3 n k ε = ( n + )( k + ) (47b) 3 3 n k ε = ( n ) ( k ) (47c) 3 3 n k ε = ( n )( k ) (47d) 3 3 The last tw sets f equatins are the fundamental equatins fr calculating the magnet-dielectric anistrpy effect. They shw that magnet-dielectric effects can be determined frm ptically measured physical quantities such as birefringence n, dichrism A, the real refractive index n, and the extinctin cefficient k (under zer applied fields). It is imprtant t mentin that these physical quantities can be measured with excellent accuracies, e.g. values f n and k ~ 10-6 can easily be measured. The frequency dependence f the index f refractin, r the dielectric cnstant f matter (dispersin), is explained in terms f the distrtin f the internal charge distributin under the influence f an applied external electric field. This distrtin results in induced electric diple mments leading t an induced- electric plarizatin. Cnsidering the charge carriers t be elastically bund, the electric field f the electrmagnetic wave prvides the driving frce f the harmnic scillatr. Fllwing standard prcedures fr slving the equatin f a frced damped scillatr, the dielectric cnstant is given by: 3

24 Review Article Yusuf and Abu-Aljarayesh ~ q N ε = ε + (48) m( ω ω + iγω) where ε is the dielectric cnstant f free space, q the charge f the charge carriers, N the number f free charge carriers per unit vlume, m the mass f the charge carrier, ω the natural frequency f the scillatr, ω the frequency f the electrmagnetic wave and γ a damping parameter. Fr an elngated particle, the charge carriers can scillate alng the easy directin much easier than alng ther directins and with larger amplitudes leading t mre electric plarizatin in the easy directin. Therefre, the cntributin t the dielectric cnstant f charge carriers scillating alng the easy axis is the dminant ne. When a magnetic field is applied n the sample, the number f charge carriers, N, cntributing t the dielectric cnstant becmes directin dependant, thus leading t an induced anistrpy in the dielectric cnstant f the sample. Cnsequently, the dielectric cnstant f the sample under an applied magnetic field is given by: ~ ε x q N x = ε + (49) m( ω ω + iγω) q N y ε y = ε + (50) m( ω ω + iγω) and, ~ ε z where q N z = ε + (51) m( ω ω + iγω) ~ ε x, ~ ε y, and ~ ε z are the cmplex dielectric cnstants seen by the electrmagnetic waves plarized in the x, y, N x, N y and z directins respectively; and and N z are the number f charge carriers per unit vlume scillating in the x, y, and z directins respectively. In the absence f applied magnetic fields the densities f charge carriers scillating in the three directins are equal and thus N x = N y = N z = (1/3) N, where N is the density f charge carriers in the sample. Therefre, ~ ε x, ~ ε y and ~ ε z are all equal t ~ ε ze (dielectric cnstant under zer magnetic fields) and given by: ~ ~ ~ q N ε x = ε y = ε z = ε + (5) 3m( ω ω + iγω) When a magnetic field H is applied in the z directin, the particles tend t align in the field directin with a prbability P ( T, H ) given by: 1+ Φ ( p, q) P( T, H ) = (53) 3 where Φ ( p, q) is the rientatin functin previusly defined. This expressin fr the prbability insures that it is (1/3) at zer field r very high temperatures and is 1 at very high fields r very lw temperatures. Furthermre, the prbability P ( T, H ) at a given field and given temperature is the same fr samples f different cncentratin in the lw cncentratin case where particleparticle interactin is neglected. The densities f charge carriers are then given by: N z = NP( T, H ) (54) 1 P( T, H ) N x = N y = N (55) By substituting N x and N z in Eq. (49) and Eq. (51), ne gets the fllwing expressins fr the dielectric cnstant: ~ q NP( T, H ) ε = ε + (56) m( ω ω + iγω) ~ q N{1 P( T, H )} ε = ε 0 + (57) m( ω ω + iγω) And by separating the real and imaginary parts in Eq. (56), Eq. (57) ne btains the fllwing relatins: ε = ε + f ω) NP( T, ) (58) 1 ( H f1( ω) N(1 P( T, H )) ε = ε + (59) ε = f ω) NP( T, ) (60) ( H f ( ω) N{1 P( T, H )} ε = (61) 4

25 Magnet-Optical and Magnet-Dielectric Anistrpy Effects in Magnetic Fluids Nf ( ω) ε ze = ε + (6) 3 ( ω) ε Nf ze = 3 (63) where f ( ) and f ( ) are given by: 1 ω ω γωq f 1( ω) = (64) m{( ω ω ) + ( γω) } ( ω ω ) q f ( ω) = (65) m{( ω ω ) + ( γω) } The field-induced change in the real dielectric cnstants and that in the imaginary dielectric cnstant are then given by: 1 ε = f1 ( ω) N{ P( T, H ) } (66) ε = f1( ω) N{ P( T, H )} (67) 3 1 ε = f ( ω) N{ P( T, H ) } (68) ε = f ( ω) N{ P( T, H )} (69) 3 At a given frequency,ω, bth f 1( ω) and f ( ω) are cnstant and cnsequently, the field induced change in the real and imaginary parts f dielectric cnstant (at a given measuring field and temperature) in the lw cncentratin regime is linear with the density number N and thus is linear with cncentratin. The slpe f this linear relatinship is psitive fr the change in the dielectric cnstant, (real and imaginary parts) fr waves plarized in the field directin and is negative fr the change in the dielectric cnstant, (real and imaginary parts) fr thse plarized nrmal t the field directin. Hwever, fr a given sample, the field induced change in the real and imaginary parts f dielectric cnstant is field dependent since P ( T, H ) is field dependent. The change in the dielectric cnstant, (real and imaginary parts) fr waves plarized in the field directin is psitive and increases with the field while this change fr thse plarized nrmal t the field directin negative and its abslute value increases with the field. 3.- Field Dependence f the Dielectric Anistrpy Using the first set f equatins, i.e. Eq.46 (tw-dimensinal case) and the data n birefringence and dichrism presented in reference [68] and the measured values n and k f the sample, the dielectric cnstants ε and ε bth the real and imaginary parts were calculated versus field. The results f these calculatins are presented in Figs. 36 and 37. The results in Figs. 36 and 37 shw that at zer fields bth ε and ε are equal. Applying magnetic fields n the sample results in a frking behavir, i.e., an increase in bth the real and imaginary parts f ε and a decrease in bth the real and imaginary parts f ε. This frking effect is attributed t the chain frmatin in the directin f applied field. It is imprtant t nte that at any given applied field the tw differences ( ε - ε ) and ( ε - ε ) are equal and thus yield the value f 1 fr the dielectric anistrpy factr g. These results are cnsistent with the thery presented in Eqs imilarly, the change in the dielectric cnstant frm the measured values f birefringence and dichrism fr different samples at rm temperature and wavelength f 633 nm, and the measured values f n and k, was calculated fr a three dimensinal case using Eq. 47. The results are presented in Figs. 38 and 39. FIG. 36. Real parts ε and ε f the dielectric cnstant versus applied magnetic field. 5

26 Review Article Yusuf and Abu-Aljarayesh The results shw similar behavir with the field t that fr the tw-dimensinal case, but with ne fundamental difference, that is the change in the dielectric cnstant fr waves plarized in the field directin is twice the change (in abslute value) fr waves plarized nrmal t the field directin. Therefre, the dielectric anistrpy factr, g, is fr the three dimensinal case. FIG. 39. Field induced changes ε and ε versus applied magnetic field fr different cncentratins and λ = 633 nm. FIG. 37. Imaginary parts ε and ε f the dielectric cnstant versus applied magnetic field. FIG. 38. Field induced changes ε and ε versus applied magnetic field fr different cncentratins and λ = 633 nm Cncentratin Dependence f the Dielectric Anistrpy The cncentratin dependence f the dielectric anistrpy is calculated frm the measured values f birefringence and dichrism fr different samples taken at a given field and rm temperature and wavelength f 633 nm; and the measured values f n and k f these samples. The calculatins were fr a three dimensinal sample, i.e. using Eq. 47. The results f these calculatins fr the change in the dielectric cnstant, bth real and imaginary parts are presented in Fig. 40 and Fig. 41. The results shw that the change in the dielectric cnstant (real and imaginary parts) fr waves plarized in the field directin increases linearly with cncentratin; while this change fr thse plarized nrmal t the field directin decreases linearly with cncentratin. Nting that the cncentratins f the samples used are in the lw cncentratin regime, this linearity is in agreement with the thery presented in Eqs Furthermre, the results als shw that, fr all 6

27 Magnet-Optical and Magnet-Dielectric Anistrpy Effects in Magnetic Fluids cncentratins, the dielectric anistrpy factr g is. FIG. 40. Field induced changes ε and ε versus cncentratins fr different applied magnetic field and λ = 633 nm FIG. 41. Field induced changes ε and ε versus cncentratins fr different applied magnetic field and λ = 633 nm 3.4- Temperature Dependence f the Dielectric Anistrpy The temperature dependence f the magnet dielectric anistrpy was calculated fr a tw dimensinal case using measurements f birefringence and dichrism as a functin f temperature and the measured values f n and k. The results f these calculatins are shwn in Figs. 4 and 43. FIG. 4. Real parts ε and ε f the dielectric cnstant versus temperature. The results shw that befre a given temperature frking is nt bserved. Abve such a temperature a separatin between ε and ε fr bth the real and imaginary parts is bserved. Furthermre, the results shw that bth the real and imaginary parts f ε increase with temperature reaching a maximum at a given temperature then they bth decrease fr higher temperatures. Hwever, bth the real and imaginary parts f ε decrease with temperature reaching a minimum at a given temperature then bth, increase fr higher temperatures. The results als shw that the temperature at which ε reaches it maximum is the same temperature at which ε reaches its minimum. Thus the separatin is enhanced by increasing the temperature till it reaches a maximum and by further increase f temperature the separatin starts t decrease. Furthermre, the results als shw that at any temperature after the separatin between ε and ε, the tw differences ( ε - ε ) and ( ε - ε ) are equal and thus yield the value f 1 fr the dielectric anistrpy factr g. This behavir f the dielectric anistrpy is explained in terms f the tw cmpeting effects, the rienting effect and the randmizing effect. At lw temperature when the sample is in the frzen state, mechanical anistrpy des nt exist in the sample resulting in the absence f the dielectric anistrpy. By increasing the temperature, 7

28 Review Article Yusuf and Abu-Aljarayesh particles are unblcked and are allwed t rtate and rient in the field directin. Therefre the average prjectin f these particles in the field directin becmes larger than in a directin nrmal t the field. Cnsequently, a srt f mechanical anistrpy is present in the sample leading t a difference between ε and ε. Increasing the temperature further results in a higher rate f unblcking f particles cntributing t the dielectric anistrpy. This prcess cntinues till the dielectric anistrpy reaches a maximum at a given temperature where the cnditin f alignment and chain frmatin in the field cnditin is ptimum. Fr higher temperature the rle f thermal agitatin, i.e. randmizing effect becmes dminant leading t a reductin in the mechanical anistrpy and thus t a decrease in the dielectric anistrpy. Again it is the balance between these tw cmpeting effects that determines the behavir f the dielectric anistrpy in magnetic fluids. cefficient k f the samples (fr all wavelengths) is determined by measuring the transmissin f light thrugh cells f different ptical paths filled with the sample. The cells are standard cells made f the same glass having the same thickness. The wavelength and cncentratin dependence f the real part f the index f refractin f the magnetic fluids, under zer fields, used are shwn in Fig. 44 and Fig.45. The wavelength and cncentratin dependence f the extinctin cefficient, under zer fields, f the samples used are shwn in Fig. 46 and Fig. 47. FIG. 44. Real index f refractin, n versus wavelength. FIG. 43. Imaginary parts ε and ε f the dielectric cnstant versus temperature Wavelength Dependence f the Dielectric Anistrpy Measurements f birefringence were undertaken using a quartz lamp, and narrw band interference filters. imilarly, dichrism was measured using the same light surce and the wavelengths were als selected by narrw band interference filters. The real part f the refractive index, n f the sample under zer fields (fr all wavelength used) is determined by measuring the Brewster angle. Mrever, the extinctin FIG. 45. Real index f refractin, n versus cncentratin 8

29 Magnet-Optical and Magnet-Dielectric Anistrpy Effects in Magnetic Fluids fields, increases with wavelength reaching a maximum at a given wavelength then starts decreasing fr lnger wavelengths. Furthermre, the results als shw that the change in the real part f the nrmal cmpnent, ε, fr all measuring fields, decreases with wavelength reaching a minimum at a given wavelength then starts increasing fr lnger wavelengths. FIG. 46. Extinctin cefficient, k versus wavelength FIG. 48. Changes in the real parts, ε and ε versus wavelength at different measuring fields. FIG. 47. Extinctin cefficient, cncentratin k versus The change in bth the real and imaginary parts f the dielectric cnstant was calculated frm birefringence, dichrism; the real part f the index f refractin at zer fields and the extinctin cefficient f the samples at zer fields. These calculatins were undertaken fr a three dimensinal case. The results f the calculatins, fr the real part are presented in Fig. 48; while thse fr the imaginary part are presented in Fig. 49. The results in Fig. 48 shw that the change in the real part f the parallel cmpnent, ε, fr all measuring FIG. 49. Changes in the imaginary parts, ε and ε versus wavelength at different measuring fields. 9