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The structure of ferrofluids: A status report
Current Opinion in Colloid & Interface Science 10 (2005) 133 – 140 www.elsevier.com/locate/cocis
The structure of ferrofluids: A status report C. Holm a,b,*, J.-J. Weis c,1 a
b
Max-Planck-Institut fu¨r Polymerforschung, Ackermannweg 10, D-55128 Mainz, Germany Frankfurt Institute for Advanced Studies (FIAS), JW Goethe-Universita¨t Frankfurt, Max-von-Laue-Strasse 1, D-60438 Frankfurt am Main, Germany c Laboratoire de Physique The´orique, UMR 8627 Baˆtiment 210, Universite´ Paris-Sud, 91405 Orsay Cedex, France Available online 29 August 2005
Abstract We review the recent advances in our understanding of the behavior of ferrofluids that occurred during the last 5 years. Ferrofluids can be viewed as a particular interesting class of dipolar fluids, which have a wide range of potential application in biomedicine and technology that is responsible for the fast growing number of research papers in this area. We will concentrate on the issues of phase behavior and microstructure formation with and without an externally applied magnetic field. In addition we discuss the influence of polydispersity that is an almost unavoidable feature of any real ferrofluid. We will focus on the theoretical aspects of those issues, however, we tried also to discuss new results coming from experiments and relate them to our current theoretical understanding of ferrofluids. D 2005 Elsevier Ltd. All rights reserved. Keywords: Ferrofluids; Computer simulation; Phase behavior; Dipolar fluids
1. Introduction Ferrofluids (FF) are suspensions of small magnetic particles with a mean diameter of about 10 nm in appropriate carrier liquids. The particles contain only a single magnetic domain and can thus be treated as small thermally agitated permanent magnets in a carrier liquid. The special feature of FF is the combination of normal liquid behavior with superparamagnetic properties. Moreover, some properties like the viscosity, the phase behavior, or their optical birefringence properties, can be changed by applying an external magnetic field. FF possess a wide range of potential technical and biomedical applications [1&]. There is, of course, a close relationship to more general dipolar fluids, like magnetorheological or electrorheological fluids, or any fluid
* Corresponding author. Frankfurt Institute for Advanced Studies (FIAS), JW Goethe-Universita¨t Frankfurt, Max-von-Laue-Strasse 1, D-60438 Frankfurt am Main, Germany. Tel.: +49 69 79847505; fax: +49 69 79847611. E-mail addresses: [email protected] (C. Holm), [email protected] (J.-J. Weis). 1 Tel.: +33 1 69157786; fax: +33 1 69158287. 1359-0294/$ - see front matter D 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.cocis.2005.07.005
composed of particles interacting with permanent point dipolar interactions. For spherically symmetric dipolar particles the interaction between a particle i and j with point dipole moments m i and m j separated by a distance r ij is given " # 3ðmi Irij Þðmj Irij Þ 1 vðrij ; mi ; mj Þ¼vsr ðrij Þ þ 3 mi Imj ð1Þ r2ij rij where v sr (r) is an isotropic short-range potential mostly taken to be a hard sphere potential of diameter s, a soft sphere potential 4e(s / r)12 or a Lennard-Jones potential 4e[(s / r)12 (s / r)6]; r ij = r j r i is the vector joining the centers of mass of the particles. If the short range potential is a hard core, we talk about the dipolar hard sphere (DHS) model that is a popular toy model used for analytical calculations. If one uses instead a soft repulsive short range potential one calls it a dipolar soft sphere (DSS) model, or in the case of a full Lennard-Jones potential, it is called a Stockmayer fluid (SMF). Throughout the review we will use reduced units which differ according to whether the short range potential is strictly of a hard or soft core type with or without dispersion forces (soft core or LJ interaction). For hard core potentials a
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thermodynamic state is characterized by the reduced density q* = Nr 3 / V (N number of particles, V volume, r characteristic length of the hard core) and a reduced dipole moment l* = (l 2 / r 3kT)1 / 2 (l dipole moment, T temperature, k Boltzmann constant) or a reduced temperature T* = 1 / l*2. For soft core potentials, a state is defined by the reduced density q* = Nr 3 / V, the reduced temperature T* = kT / e and the reduced dipole moment l* = (l 2 / r 3e)1/2, where r and e are the parameters defining the range and the strength of the short range potential. It is also convenient to define the dimensionless coupling parameter k = l*2 which is half the ratio of the dipolar energy of two aligned dipoles at close contact to the thermal energy. The idea of this article is to review the recent progress made in investigating the phase behavior and the microstructure of FF during the last five years. For earlier results we refer to the review paper by Cabuil [2&&]. Since the dipolar interaction is anisotropic and can change from repulsive to attractive, the phase behavior is not easy to predict. An apparent simple question is whether DHS and DSS models can show a liquid –gas (l – g) coexistence. If one averages the dipole– dipole interaction weighted by its Boltzmann factor one obtains an attractive van der Waals (vdW) potential ( 1 / r 6) at large distances. Simple mean-field theories have predicted its occurrence, but systematic Monte Carlo (MC) studies of the DHS model have shown that the l– g transition was not present in the expected domain of densities and temperatures. MC simulations performed in the NpT and Gibbs ensembles for temperatures between T* = 0.222 and 0.18 and densities 0.1 q* 0.4 [3] and in the NVT ensemble, mainly along the isotherm T* = 0.0816 and densities 0.02 q* 0.3 [4&,5], did not find any evidence for a l– g transition. Another complex issue is connected to the microstructure of the fluid. Already early arguments by deGennes and Pincus [6] have pointed to the existence of clusters and chains, which are due to the attractive parts of the dipolar interactions. Small chains (dimers, trimers) start forming when the dipolar energy exceeds the thermal energy (k¨3). By increasing the dipole moment (or lowering the temperature) the chains grow and an equilibrium state establishes consisting of a distribution of chains, rings or more complex structures of different lengths, breaking and reforming, with relative concentrations depending on density and temperature. These have been observed by numerous researchers regardless if they are using DHS, DSS, or SMF. There have been suggestions that the presence of these clusters is actually responsible for the absence of the l– g transition [7&]. The application of an external field changes drastically the fluid structure, since it orients the dipoles along the applied field. This results in an increase in cluster or chain sizes, which in turn can also effect dramatically the phase behavior. The last issue addresses the influence of polydispersity, a factor which is inevitably present in all commercial FF, and which
recently have been taken into account via simulations and density functional calculations. The review is structured in the following way: first, we present the current knowledge of the equilibrium phase diagram in the absence of external fields. Then we discuss the influence of an applied external magnetic field on the phase diagram. Next we discuss our knowledge of the equilibrium microstructure in 3D systems, and we conclude with a discussion about the effects due to polydispersity.
2. Structure and phase behavior at zero external field In FF the phase diagram results essentially from the balance of three types of forces [2&&,8&]: repulsive forces due to the coating of the magnetic cores by either surfactant (surfacted FF) or charges (ionic FF) which prevent irreversible aggregation of the particles, attractive van der Waals forces and magnetic dipole –dipole interactions which can be either attractive or repulsive depending on the orientations of the dipole moments with respect to the interparticle distance vector. A pleasant feature of FF is that the relative strengths and ranges of the various interactions potentials can be controlled by the diameter of the magnetic cores, thickness of the surfactant layer or the ionic strength of the solution in the case of electrostatically stabilized FF. In conventional FF like maghemite (c-Fe2O3) or magnetite (Fe3O4), with typical particle size 5 –10 nm, the magnetic interaction is generally weak (k < 1), mainly reinforcing the vdW attraction and the phase behavior (in zero field) is expected to be similar to that of atomic fluids or nonmagnetic colloids. This picture has been confirmed by a series of experiments on citrate coated maghemite particles by varying the ionic strength of the solution [8&]. At high ionic strength (strong screening of the Coulomb interaction) the system is dominated by the attractive forces, and gas – liquid coexistence with a critical point and a (glassy) solid phase is observed [8& – 10]. In the opposite case (low screening) the repulsive forces dominate and only a fluid-(amorphous) solid transition occurs [8&]. When the dipolar interactions are much stronger than the thermal energies, the fluid develops clusters, like dimers, trimers, and longer chains. The microstructure of a particular fluid is hard to observe experimentally, although some techniques such as neutron scattering can probe the small scale structure of the fluid, such as the surfactant layer that is responsible for the steric stabilization [11]. The most clean results can, of course, be obtained with a computer experiment on a model system, through molecular dynamics (MD) or MC calculation, see [12&&] for an account of the history. By analyzing the configurations obtained in the simulations one can perform a cluster analysis of the ferrofluid systems based either on an energy
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the region T*c å 0.15(k å 2.6) and density q c * å 0.1. In quasi-2d DHS systems a phase transition has been conjectured to occur from a dilute phase characterized by a number of disconnected clusters to a condensed phase characterized by a network that includes most of the particles in the system [31]. An other rather unexpected feature of the phase diagram of 3d DHS is the existence of spontaneous ferromagnetic ordering at sufficiently high density and low temperature even in the absence of an applied magnetic field [12&&,13&&,14&&]. The critical line of the isotropic –ferromagnetic phase transition of DHS has recently been determined accurately in the high density range q* = 0.80 –0.88 by MC simulations in conjunction with finite size scaling analysis [32&]. If supplemented with earlier more approximate estimates of the critical temperature for DSS [33&,34&], k c is found to vary along the transition line from k c å 3.8 at q* = 0.88 to k c å 6.9 at q* = 0.7 [32&]. As can be noted these values of k are within the range of coupling strengths covered with the newly synthesized large size magnetite particles [23&]. Nevertheless, the rather high densities at which the transition occurs still precludes experimental 0.5
1.0 fluid of monomers
1.5
networks
rings
and chains
normal liquid
2.0
solid
ferroelectric
µ∗
criterion or on a proximity criterion. In the former, two particles are considered to be bound if their dipolar interaction energy is less than a predefined value of the contact energy of two perfectly coaligned dipolar particles, whereas is the latter a distance criterion of the order of the particle diameter is used to define members of a cluster. In a monodisperse system of small particles (k 2), the average cluster size is only slightly above 1 at zero field, and the magnetic susceptibility can be predicted accurately over a large density range by mean-field models or more elaborate statistical models [12&&,13&&,14&&,15&,16]. For larger values of the coupling parameter clusters or chains appear, which lead to a break down of the mean-field models. Some theoretical models have been developed which include the chain formation in a self-consistent way, which can improve considerably the predictions for larger values of the coupling constant [17,18&]. In zero field the influence of the dipole –dipole interaction on phase behavior of FF manifests itself significantly only for magnetic interactions stronger than k > 2 – 2.5. This range of magnetic strengths has been reached by synthesis of e-Co nanocrystals [19,20] and iron (Fe0.75Co0.25) dispersions (k å 2.5) [21,22&] which have larger bulk saturation magnetizations than, for instance, maghemite, and quite recently of magnetite particles of large size (k up to 7) [23&]. Recently Co particles have been synthesized with large magnetic moment (k up to 14) [24] which can be valuable model systems studying strong dipolar couplings. Magnetite colloids extracted from magnetotactic bacteria have even larger dipole strengths (k å 70) [25] but size control and purification are difficult, though some progress has been made in this direction [26,27]. Cryo-TEM images of vitrified thin films of 21 nm surfactant coated magnetite particles (k å 7) clearly display single chains, containing up to 20 particles with head-to-tail orientation of the dipole moments, branched chains and ring (flux-closure) structures [23&] in striking resemblance with the morphologies obtained in quasi-2d dipolar hard sphere systems at comparable couplings and densities [28&,29]. In contrast, in a dispersion of particles with smaller diameter (16 nm) (k å 2) the interaction is dominated by repulsive forces and no chaining is observed [23&]. In accord with simulation results of quasi-2d and 3d dipolar hard and soft spheres [14&&,12&&] the experimental results with iron [22&] and (large size) magnetite particles [23&] do not give evidence for a gas – liquid transition. For these large particles attractive vdW interactions (which increase with particle volume slower than the magnetic interactions) hardly contribute and the simple DHS model appears to be a fair representation of the system. It would be of interest if other characteristics of the phase diagram of DHS [12&&,14&&] could be reproduced by experiment. Thus from simulations [30&] and theoretical arguments [7&] it has been suggested that the 3d DHS undergo a first order fluid – fluid transition between chain phases of different morphologies (so far not characterized) with a critical point [30&] in
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2.5 ferroelectric liquid
3.0 0.0
0.2
0.4
0.6 ρ∗
0.8
? 1.0
Fig. 1. Sketch of the phase diagram of 3D DHS in the temperature-density plane. The reduced temperature is related to the reduced dipole moment by T* = 1 / l*2. At high temperature (l* 1.5) the system consists mainly of monomers. At lower temperatures and low density the DHS particles are associated in chains and rings. Upon increase of the density the average cluster sizes increase, the rings unbind in favour of chain-like segments and the structure changes to that of a network spanning the (finite) simulation box. At densities q* 0.2 each particle is coordinated with a whole shell of particles and the network structure breaks down: the system is more like a normal liquid [5,40]. Further compression gives rise to a ferroelectrically ordered state [5,33&,37]. Between the ferroelectric solid and the ferroelectric liquid a small region (dotted region) may exist with ferroelectric columnar ordering of the particles [33&,38]. The star marks the critical point of a condensation transition conjectured in [30&,40]. The triangles indicate the limits of mechanical stability of the bct (open triangle) and fcc (filled triangle) solid phases [37].
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confirmation of the possibility of ferromagnetic ordering in zero field. A large increase of ferromagnetic fluctuations in the static susceptibility upon lowering the temperature of iron nitride particles at density q* å 0.3 has been interpreted as an indication of a ferromagnetic transition [35]. Such a conclusion has, however, been refuted from susceptibility measurements in concentrated magnetite colloid solutions [36]. A number of theories, mostly of a mean field type, predict ferromagnetic ordering but the critical temperature is generally strongly overestimated; for a review see [32&]. For convenience, we also include here as Fig. 1 a sketch of the phase diagram, based on our current understanding.
3. The influence of external fields In contrast to the zero field case, even small magnetic interactions can have large effects on the structural organization of FF in finite fields. An external field induces anisotropic microstructures which can be investigated by a variety of experimental techniques including optical and electron microscopy, small-angle neutron (SANS) and X-ray (SAXS) scattering or, for model systems, by computer simulation. The precise structural arrangements of these microstructures depend on a variety of parameters like on magnetic interaction strength, field strength, orientation of the field with respect to the sample, rate of change of the field, sample thickness, polydispersity, volume fraction, temperature, amongst others. Understandingly, a systematic investigation of the influence on the microstructure and phase behavior of all these factors is neither available by experiment nor by computer simulations. Even the morphology of a single cluster depends on many parameters. It has been argued that from a certain length on, the equilibrium cluster shape changes from linear to drop like at zero field [39], but an applied external field will favor naturally extended chains. This can have severe implications for occurrence of the l–g transition. The appearance of clusters changes the microscopic magnetization behavior of the FF. Again, due to the neglect of correlations, mean-field and statistical models tend to break down at values of the coupling constant k 3, and at larger densities [15&,16]. The anisotropic structure factor and its characteristic features have been analyzed in simulational studies [40 – 42]. The structure factor is projected onto a certain geometric plane [41 – 43&] in order to get the information of the anisotropic structural properties of the model ferrofluid of interest. This is experimentally accessible via neutron scattering [43&,44], and is in good agreement with the simulations. On a qualitative level the morphologies of the structures observed either in thin fluid films of FF confined between glass plates (Hele-Shaw cell) or FF deposited on a substrate subjected to a perpendicular magnetic field have been characterized as needles, chains, columns, sheets, bent walls, lamellar, labyrinthine or worm-like structures. In a
weak magnetic field the dipole moments align with the field and the particles associate into chains with random positions and lengths oriented parallel to the field direction [45]. In larger fields the chains will span the sample and coalesce into thicker column-like structures which eventually organize into hexagonally ordered structures [45] in thin films (2 – 10 Am) [45] or into a glassy state in thicker films (125 Am) [46&]. Further increase of the field in thin films, at least for low sweep rates, has been shown to entail splitting of each column into two columns and formation of a second level hexagonal phase [45]. Distances between columns have been recorded as a function of magnetic field [47], film thickness [48], sweep rate [45,49] and temperature [50]. In cobalt nanocrystals that have been dispersed in a nonpolar solvent, deposited on a substrate while subjected to a strong magnetic field perpendicular to the substrate [51& –53], pattern formation has been found and related to the occurrence of a gas – liquid transition during the evaporation process. In the concentrated phase the columns are made of self-assembled nanocrystals in a face-centeredcubic (fcc) structure. Column and labyrinth formation is sensitive to the particle size distribution [52]. For particle sizes in the range 5 – 8 nm and a narrow size distribution ( 13%) the columns are isolated and well defined with closely packed nanoparticles inside the columns [53]. The average length and width of the particles remains unchanged upon increasing the field strength (0.25 – 0.59 T); no transition takes place from a columnar to a labyrinth pattern. In contrast, for larger size distributions (13 > %) [52] defects at the ends and the edges of the columns resulting from a decrease of cohesion of the nanoparticles inside the columns favor fusion of the columns into labyrinthine structures. Explanation of these pattern is generally based on minimization of a free energy comprising magnetic, interfacial and entropic contributions [54 – 56&]. A recently developed theory which takes into account the nonlinear relationship between magnetization and field at high field strength [56&] accurately reproduces the characteristics of hexagonal ordering in cobalt nanocrystals [51&], in particular the dependence of cylinder radius on applied field, volume fraction and cylinder height. The theory when applied to idealized stripe and hexagonal patterns predicts the existence of hexagonal patterns at low volume fractions and labyrinthine patterns at large volume fractions but no transition between hexagonal and striped structures by variation of the external field (at given volume fraction) [56&,57] in contrast to a theory by Lacoste and Lubensky [55] using a mean field approach for the magnetic free energy and the entropy of the hard sphere fluid, and to experimental findings for magnetite FF [46&]. The self-assembly into elongated clusters when a magnetic field is applied parallel to the sample has been investigated recently for maghemite nanocrystals [58], magnetite nanoparticles [59], cobalt nanocrystals [60] and iron FF [22&,61]. For the system of weakly dipolar maghemite nanocrystals deposited by evaporation it has
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been shown that the mesoscopic organization of the particles can be controlled by the thickness and nature of the surface coating. With thin coatings the particles form elongated structures in the direction of the field while they are randomly deposited with thicker coatings [58]. In the former case vdW forces dominate and the nanocrystals form spherical clusters in zero field. When the magnetic field is applied these clusters acquire a dipole moment much larger than that of the individual particles and attract each other to form larger elongated aggregates. When the thickness of the surface coating is increased (maintaining the average dipole moment and size of the particles unchanged) the vdW forces rapidly diminish and the observed structure results from the dominant repulsive forces [58]. Simulation studies varying the relative contributions of repulsive and attractive forces reproduce these results qualitatively [62]. Field induced pseudo-crystalline ordering has been inferred from SANS using polarized neutrons and synchrotron SAXS scattering data in concentrated Co-FF [63,64]. The particles are found to be arranged in hexagonal planes with the magnetic moments and the directions aligned along the magnetic field. The interplane distance varies with sample. Simulation studies addressing the formation of hexagonal or labyrinthe patterns in dipolar fluids subjected to an applied field present two complications related, on the one hand, to system size which has necessarily to be large if regular patterns are to be observed, on the other hand, to a correct treatment of the long range dipolar interactions in finite simulation cells which lack periodic boundary conditions in all space directions [12&&] as is the case in most experimental situations (FF confined between glass plates or adsorbed on a substrate). Not surprisingly, simulations are still often restricted to small system sizes, and sometimes a simple truncation scheme for the dipolar interactions is used despite for the fact that possible serious artefacts can be introduced in this way. A 3d simulation [65] study of dipolar particles has provided evidence for the formation of thick chain-like (columnar) aggregates at appropriate values of the volume fraction, magnetic coupling constant and field strength but the system size was not large enough to demonstrate hexagonal ordering of the columns. Regular ordering of columns (stripes) has been shown more convincingly in a large quasi-2d system of dipolar hard spheres in an external field parallel to the system layer [66]. The width of the stripes is shown to increase roughly linearly with density. A recent simulation study [67] of strongly interacting dipolar particles (k = 18) confined between two repulsive parallel walls has investigated the influence of wall separation on pattern formation in the presence of a magnetic field perpendicular to the plates. At short separations cyclic formation of chains and meandering walls are noted while at separations larger than about ten particle diameters the structure gradually changes to branched walls, then thicker ones with body-centered-
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tetragonal arrangement of the particles in the walls. The results may, however, be seriously flawed by the truncation of the dipolar potential which is potentially dangerous at the high coupling parameter considered.
4. Polydispersity Polydispersity is inherent in all colloidal dispersions and needs to be considered in the interpretation of experimental data as it affects magnetic properties and phase diagrams. In commonly studied FF like magnetite polydispersity may be as large as 40% but can be reduced by size sorting methods [68], magnetic fractionization methods [69,70], or fractionation by size-selective precipitation [71]. Metallic iron [21] or cobalt [72] particles, cobalt ferrite-silica [73] particles or cobalt nanocrystals [74] can be prepared with a narrower size distribution (10%). The particle size distribution is generally adequately described by a lognormal or a gamma distribution [75] whose parameters can be obtained, for instance, from magneto-granulometric analysis of magnetization curves [76] or, more directly, from electron microscope images. Due to the large number of degrees of freedom involved in polydiperse systems a theoretical determination of the phase behavior may be quite complex enabling coexistence of an arbitrary number of phases [77]. In particular, the coexisting phases may have different size distributions than the parent distribution if the latter is asymmetric. However general relations between the moments of parent and daughter distributions have been shown to exist [78]. Although this universal law of fractionation has been derived only for slightly polydisperse systems, confirmation of the law could be demonstrated for a highly polydisperse iron oxide system [79]. Insight in the influence of either bidispersity or polydispersity on phase diagrams, magnetization curves or aggregate microstructures can be gained from computer simulations by comparison with corresponding results for a monodisperse system of particles having, e.g., the same average diameter. Simulations by Kruse et al. [80] using potential parameters which reflect as close as possible magnetite FF supply analysis of size, composition and shape of the clusters formed in an external magnetic field. The finding that the mean particle radius in clusters roughly equals the mean radius of all the particles gives indication that particles of all sizes and not only the larger ones contribute to the composition of the clusters in close agreement with 2d-SAXS experiments [81&]. From this observation and the fact that in the considered magnetite system the average aggregation parameter k < 0.5 it is concluded that vdW forces play a significant role to achieve thermally stable clusters [80]. Simulations by Castro et al. [82] of a FF typical of magnetite analyze the distribution of small aggregates as a function of volume fraction and compare with results for a monodisperse system. Liquid-
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vapor coexistence curves for a polydisperse system have been contrasted in Ref. [83&] with its monodisperse equivalent: they are narrower in the polydisperse case, have a higher critical temperature and a lower critical density. With increasing field strength the coexistence curves broaden, the critical temperature increases but the critical density is little affected [83&]. At high density ferroelectric ordering observed in zero field is reduced by dipolar polydispersity [84]. The influence of polydispersity on the effect of magnetostriction has been investigated in [85]. Here it was found that the non-linear susceptibility due to magnetostriction was very sensitive to the polydispersity, and that measuring the harmonics could give information about the particle size distribution in the sample under investigation. The structure of polydispersed FF was discussed theoretically on the basis of a bidisperse model where the systems are supposed to consist of two fractions of magnetic particles with significant size differences [17,86 – 88]. The main structural character in these systems is proposed to be the chainlike aggregates formed by the large particles. Some small particles might be attached to the ends of these aggregates, but most of them remain in the non-aggregated state [86 –88]. A recent study also includes the effect of vdW and depletion interactions and studies the effects on the l– g transition [89]. A bidisperse solution was also studied by MD simulation [90&] whose size and dipolar couplings were chosen to mimic experimentally investigated ferrofluid [91&,17]. The initial susceptibility v and magnetization curve of the systems show a strong dependence on the volume fraction / L of the larger particles. With the increase of / L , the magnetization M of the system has a much faster increment at weak fields, and thus leads to a larger v. A cluster analysis indicated that this is due to the aggregation of the large particles in the systems. The average size of these clusters increases with increasing / L . It was found that the volume fraction of smaller particles had a significant effect on the structural properties of the system. A topological analysis of the structure reveals that the majority of the small particles remain non-aggregated. Only a small number of them are attracted to the ends of the chains formed by large particles. The influence of polydispersity on the structural behavior of monolayers of particles interacting by steric repulsive and dipolar potentials has been investigated by MC simulations for a Gaussian size distribution with standard deviations 0.2 and 0.35 and same average diameter [92,93]. Aggregation, in zero as well as finite fields, is favored compared to a monodisperse system with coupling strength equal to the average value due to association of the bigger particles. The particles most frequently found in the clusters are those having a diameter larger than the average diameter. In the absence of a field clusters have a more clump-like aspect (with internal network structure) than the necklace-like chain structures typical of monodisperse systems [92]. In low external fields the structure is sensitive to the size
distribution, in stronger fields the thickness of the straight chain-like aggregates depends on size distribution but not on field strength [93]. More complete though approximate phase diagrams have been obtained for binary mixtures of DHS [94 –97] and SMF [98] over a wide range of temperature, density, concentration and coupling strengths using a modified mean field theory. Focal topics are the interplay of ferromagnetic phases and demixing transitions for highly asymmetric mixtures, the temperature and density shifts of the isotropic-ferromagnetic transition compared to the onecomponent case, the modification of the phase diagram by application of an magnetic field. One should bear in mind, though, that mean fluid theories are generally at best qualitative due to neglect of correlations; comparison with more sophisticated integral equation results indeed reveals significant differences [95]. Detailed comparisons with simulation results are presently unavailable. Recent theoretical developments and simulation studies in areas of frozen FF or confined by disordered porous media have been reviewed in an article by Klapp [14&&] and we refer to this work for details.
5. Conclusions To conclude, the study of ferrofluids has enjoyed increasing attention during the last decade. We have tried to summarize the most important developments that happened during the last five years, focusing on the phase behavior and the microstructure of these interesting fluids. Novel insight has been gained through the combined effort of computer simulations, more elaborate experimental investigations, and refined theoretical approaches. Nevertheless, the detailed structure of the phase diagram even in the absence of any external field still needs to be resolved, and a complete understanding of the complex topology of the microstructure and its relation to the macroscopic phase behavior has still to be gained in future investigations.
Acknowledgment C.H. acknowledges funding by the DFG in the priority program SPP 1004 under Grant No. HO 1108/8-4.
References and recommended readings [1] S. Odenbach (Ed.), Ferrofluids, Magnetically Controllable Fluids and & Their Applications, Lect. Notes Phys., Springer, New York, 2002. [2] V. Cabuil, Phase behavior of magnetic nanoparticles dispersions in && bulk and confined geometries, Curr. Opin. Colloid Interface Sci. 5 (2000) 44 – 48. & of special interest. && of outstanding interest.
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The structure of ferrofluids: A status report C. Holm a,b,*, J.-J. Weis c,1 a
b
Max-Planck-Institut fu¨r Polymerforschung, Ackermannweg 10, D-55128 Mainz, Germany Frankfurt Institute for Advanced Studies (FIAS), JW Goethe-Universita¨t Frankfurt, Max-von-Laue-Strasse 1, D-60438 Frankfurt am Main, Germany c Laboratoire de Physique The´orique, UMR 8627 Baˆtiment 210, Universite´ Paris-Sud, 91405 Orsay Cedex, France Available online 29 August 2005
Abstract We review the recent advances in our understanding of the behavior of ferrofluids that occurred during the last 5 years. Ferrofluids can be viewed as a particular interesting class of dipolar fluids, which have a wide range of potential application in biomedicine and technology that is responsible for the fast growing number of research papers in this area. We will concentrate on the issues of phase behavior and microstructure formation with and without an externally applied magnetic field. In addition we discuss the influence of polydispersity that is an almost unavoidable feature of any real ferrofluid. We will focus on the theoretical aspects of those issues, however, we tried also to discuss new results coming from experiments and relate them to our current theoretical understanding of ferrofluids. D 2005 Elsevier Ltd. All rights reserved. Keywords: Ferrofluids; Computer simulation; Phase behavior; Dipolar fluids
1. Introduction Ferrofluids (FF) are suspensions of small magnetic particles with a mean diameter of about 10 nm in appropriate carrier liquids. The particles contain only a single magnetic domain and can thus be treated as small thermally agitated permanent magnets in a carrier liquid. The special feature of FF is the combination of normal liquid behavior with superparamagnetic properties. Moreover, some properties like the viscosity, the phase behavior, or their optical birefringence properties, can be changed by applying an external magnetic field. FF possess a wide range of potential technical and biomedical applications [1&]. There is, of course, a close relationship to more general dipolar fluids, like magnetorheological or electrorheological fluids, or any fluid
* Corresponding author. Frankfurt Institute for Advanced Studies (FIAS), JW Goethe-Universita¨t Frankfurt, Max-von-Laue-Strasse 1, D-60438 Frankfurt am Main, Germany. Tel.: +49 69 79847505; fax: +49 69 79847611. E-mail addresses: [email protected] (C. Holm), [email protected] (J.-J. Weis). 1 Tel.: +33 1 69157786; fax: +33 1 69158287. 1359-0294/$ - see front matter D 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.cocis.2005.07.005
composed of particles interacting with permanent point dipolar interactions. For spherically symmetric dipolar particles the interaction between a particle i and j with point dipole moments m i and m j separated by a distance r ij is given " # 3ðmi Irij Þðmj Irij Þ 1 vðrij ; mi ; mj Þ¼vsr ðrij Þ þ 3 mi Imj ð1Þ r2ij rij where v sr (r) is an isotropic short-range potential mostly taken to be a hard sphere potential of diameter s, a soft sphere potential 4e(s / r)12 or a Lennard-Jones potential 4e[(s / r)12 (s / r)6]; r ij = r j r i is the vector joining the centers of mass of the particles. If the short range potential is a hard core, we talk about the dipolar hard sphere (DHS) model that is a popular toy model used for analytical calculations. If one uses instead a soft repulsive short range potential one calls it a dipolar soft sphere (DSS) model, or in the case of a full Lennard-Jones potential, it is called a Stockmayer fluid (SMF). Throughout the review we will use reduced units which differ according to whether the short range potential is strictly of a hard or soft core type with or without dispersion forces (soft core or LJ interaction). For hard core potentials a
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thermodynamic state is characterized by the reduced density q* = Nr 3 / V (N number of particles, V volume, r characteristic length of the hard core) and a reduced dipole moment l* = (l 2 / r 3kT)1 / 2 (l dipole moment, T temperature, k Boltzmann constant) or a reduced temperature T* = 1 / l*2. For soft core potentials, a state is defined by the reduced density q* = Nr 3 / V, the reduced temperature T* = kT / e and the reduced dipole moment l* = (l 2 / r 3e)1/2, where r and e are the parameters defining the range and the strength of the short range potential. It is also convenient to define the dimensionless coupling parameter k = l*2 which is half the ratio of the dipolar energy of two aligned dipoles at close contact to the thermal energy. The idea of this article is to review the recent progress made in investigating the phase behavior and the microstructure of FF during the last five years. For earlier results we refer to the review paper by Cabuil [2&&]. Since the dipolar interaction is anisotropic and can change from repulsive to attractive, the phase behavior is not easy to predict. An apparent simple question is whether DHS and DSS models can show a liquid –gas (l – g) coexistence. If one averages the dipole– dipole interaction weighted by its Boltzmann factor one obtains an attractive van der Waals (vdW) potential ( 1 / r 6) at large distances. Simple mean-field theories have predicted its occurrence, but systematic Monte Carlo (MC) studies of the DHS model have shown that the l– g transition was not present in the expected domain of densities and temperatures. MC simulations performed in the NpT and Gibbs ensembles for temperatures between T* = 0.222 and 0.18 and densities 0.1 q* 0.4 [3] and in the NVT ensemble, mainly along the isotherm T* = 0.0816 and densities 0.02 q* 0.3 [4&,5], did not find any evidence for a l– g transition. Another complex issue is connected to the microstructure of the fluid. Already early arguments by deGennes and Pincus [6] have pointed to the existence of clusters and chains, which are due to the attractive parts of the dipolar interactions. Small chains (dimers, trimers) start forming when the dipolar energy exceeds the thermal energy (k¨3). By increasing the dipole moment (or lowering the temperature) the chains grow and an equilibrium state establishes consisting of a distribution of chains, rings or more complex structures of different lengths, breaking and reforming, with relative concentrations depending on density and temperature. These have been observed by numerous researchers regardless if they are using DHS, DSS, or SMF. There have been suggestions that the presence of these clusters is actually responsible for the absence of the l– g transition [7&]. The application of an external field changes drastically the fluid structure, since it orients the dipoles along the applied field. This results in an increase in cluster or chain sizes, which in turn can also effect dramatically the phase behavior. The last issue addresses the influence of polydispersity, a factor which is inevitably present in all commercial FF, and which
recently have been taken into account via simulations and density functional calculations. The review is structured in the following way: first, we present the current knowledge of the equilibrium phase diagram in the absence of external fields. Then we discuss the influence of an applied external magnetic field on the phase diagram. Next we discuss our knowledge of the equilibrium microstructure in 3D systems, and we conclude with a discussion about the effects due to polydispersity.
2. Structure and phase behavior at zero external field In FF the phase diagram results essentially from the balance of three types of forces [2&&,8&]: repulsive forces due to the coating of the magnetic cores by either surfactant (surfacted FF) or charges (ionic FF) which prevent irreversible aggregation of the particles, attractive van der Waals forces and magnetic dipole –dipole interactions which can be either attractive or repulsive depending on the orientations of the dipole moments with respect to the interparticle distance vector. A pleasant feature of FF is that the relative strengths and ranges of the various interactions potentials can be controlled by the diameter of the magnetic cores, thickness of the surfactant layer or the ionic strength of the solution in the case of electrostatically stabilized FF. In conventional FF like maghemite (c-Fe2O3) or magnetite (Fe3O4), with typical particle size 5 –10 nm, the magnetic interaction is generally weak (k < 1), mainly reinforcing the vdW attraction and the phase behavior (in zero field) is expected to be similar to that of atomic fluids or nonmagnetic colloids. This picture has been confirmed by a series of experiments on citrate coated maghemite particles by varying the ionic strength of the solution [8&]. At high ionic strength (strong screening of the Coulomb interaction) the system is dominated by the attractive forces, and gas – liquid coexistence with a critical point and a (glassy) solid phase is observed [8& – 10]. In the opposite case (low screening) the repulsive forces dominate and only a fluid-(amorphous) solid transition occurs [8&]. When the dipolar interactions are much stronger than the thermal energies, the fluid develops clusters, like dimers, trimers, and longer chains. The microstructure of a particular fluid is hard to observe experimentally, although some techniques such as neutron scattering can probe the small scale structure of the fluid, such as the surfactant layer that is responsible for the steric stabilization [11]. The most clean results can, of course, be obtained with a computer experiment on a model system, through molecular dynamics (MD) or MC calculation, see [12&&] for an account of the history. By analyzing the configurations obtained in the simulations one can perform a cluster analysis of the ferrofluid systems based either on an energy
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the region T*c å 0.15(k å 2.6) and density q c * å 0.1. In quasi-2d DHS systems a phase transition has been conjectured to occur from a dilute phase characterized by a number of disconnected clusters to a condensed phase characterized by a network that includes most of the particles in the system [31]. An other rather unexpected feature of the phase diagram of 3d DHS is the existence of spontaneous ferromagnetic ordering at sufficiently high density and low temperature even in the absence of an applied magnetic field [12&&,13&&,14&&]. The critical line of the isotropic –ferromagnetic phase transition of DHS has recently been determined accurately in the high density range q* = 0.80 –0.88 by MC simulations in conjunction with finite size scaling analysis [32&]. If supplemented with earlier more approximate estimates of the critical temperature for DSS [33&,34&], k c is found to vary along the transition line from k c å 3.8 at q* = 0.88 to k c å 6.9 at q* = 0.7 [32&]. As can be noted these values of k are within the range of coupling strengths covered with the newly synthesized large size magnetite particles [23&]. Nevertheless, the rather high densities at which the transition occurs still precludes experimental 0.5
1.0 fluid of monomers
1.5
networks
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and chains
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solid
ferroelectric
µ∗
criterion or on a proximity criterion. In the former, two particles are considered to be bound if their dipolar interaction energy is less than a predefined value of the contact energy of two perfectly coaligned dipolar particles, whereas is the latter a distance criterion of the order of the particle diameter is used to define members of a cluster. In a monodisperse system of small particles (k 2), the average cluster size is only slightly above 1 at zero field, and the magnetic susceptibility can be predicted accurately over a large density range by mean-field models or more elaborate statistical models [12&&,13&&,14&&,15&,16]. For larger values of the coupling parameter clusters or chains appear, which lead to a break down of the mean-field models. Some theoretical models have been developed which include the chain formation in a self-consistent way, which can improve considerably the predictions for larger values of the coupling constant [17,18&]. In zero field the influence of the dipole –dipole interaction on phase behavior of FF manifests itself significantly only for magnetic interactions stronger than k > 2 – 2.5. This range of magnetic strengths has been reached by synthesis of e-Co nanocrystals [19,20] and iron (Fe0.75Co0.25) dispersions (k å 2.5) [21,22&] which have larger bulk saturation magnetizations than, for instance, maghemite, and quite recently of magnetite particles of large size (k up to 7) [23&]. Recently Co particles have been synthesized with large magnetic moment (k up to 14) [24] which can be valuable model systems studying strong dipolar couplings. Magnetite colloids extracted from magnetotactic bacteria have even larger dipole strengths (k å 70) [25] but size control and purification are difficult, though some progress has been made in this direction [26,27]. Cryo-TEM images of vitrified thin films of 21 nm surfactant coated magnetite particles (k å 7) clearly display single chains, containing up to 20 particles with head-to-tail orientation of the dipole moments, branched chains and ring (flux-closure) structures [23&] in striking resemblance with the morphologies obtained in quasi-2d dipolar hard sphere systems at comparable couplings and densities [28&,29]. In contrast, in a dispersion of particles with smaller diameter (16 nm) (k å 2) the interaction is dominated by repulsive forces and no chaining is observed [23&]. In accord with simulation results of quasi-2d and 3d dipolar hard and soft spheres [14&&,12&&] the experimental results with iron [22&] and (large size) magnetite particles [23&] do not give evidence for a gas – liquid transition. For these large particles attractive vdW interactions (which increase with particle volume slower than the magnetic interactions) hardly contribute and the simple DHS model appears to be a fair representation of the system. It would be of interest if other characteristics of the phase diagram of DHS [12&&,14&&] could be reproduced by experiment. Thus from simulations [30&] and theoretical arguments [7&] it has been suggested that the 3d DHS undergo a first order fluid – fluid transition between chain phases of different morphologies (so far not characterized) with a critical point [30&] in
135
2.5 ferroelectric liquid
3.0 0.0
0.2
0.4
0.6 ρ∗
0.8
? 1.0
Fig. 1. Sketch of the phase diagram of 3D DHS in the temperature-density plane. The reduced temperature is related to the reduced dipole moment by T* = 1 / l*2. At high temperature (l* 1.5) the system consists mainly of monomers. At lower temperatures and low density the DHS particles are associated in chains and rings. Upon increase of the density the average cluster sizes increase, the rings unbind in favour of chain-like segments and the structure changes to that of a network spanning the (finite) simulation box. At densities q* 0.2 each particle is coordinated with a whole shell of particles and the network structure breaks down: the system is more like a normal liquid [5,40]. Further compression gives rise to a ferroelectrically ordered state [5,33&,37]. Between the ferroelectric solid and the ferroelectric liquid a small region (dotted region) may exist with ferroelectric columnar ordering of the particles [33&,38]. The star marks the critical point of a condensation transition conjectured in [30&,40]. The triangles indicate the limits of mechanical stability of the bct (open triangle) and fcc (filled triangle) solid phases [37].
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confirmation of the possibility of ferromagnetic ordering in zero field. A large increase of ferromagnetic fluctuations in the static susceptibility upon lowering the temperature of iron nitride particles at density q* å 0.3 has been interpreted as an indication of a ferromagnetic transition [35]. Such a conclusion has, however, been refuted from susceptibility measurements in concentrated magnetite colloid solutions [36]. A number of theories, mostly of a mean field type, predict ferromagnetic ordering but the critical temperature is generally strongly overestimated; for a review see [32&]. For convenience, we also include here as Fig. 1 a sketch of the phase diagram, based on our current understanding.
3. The influence of external fields In contrast to the zero field case, even small magnetic interactions can have large effects on the structural organization of FF in finite fields. An external field induces anisotropic microstructures which can be investigated by a variety of experimental techniques including optical and electron microscopy, small-angle neutron (SANS) and X-ray (SAXS) scattering or, for model systems, by computer simulation. The precise structural arrangements of these microstructures depend on a variety of parameters like on magnetic interaction strength, field strength, orientation of the field with respect to the sample, rate of change of the field, sample thickness, polydispersity, volume fraction, temperature, amongst others. Understandingly, a systematic investigation of the influence on the microstructure and phase behavior of all these factors is neither available by experiment nor by computer simulations. Even the morphology of a single cluster depends on many parameters. It has been argued that from a certain length on, the equilibrium cluster shape changes from linear to drop like at zero field [39], but an applied external field will favor naturally extended chains. This can have severe implications for occurrence of the l–g transition. The appearance of clusters changes the microscopic magnetization behavior of the FF. Again, due to the neglect of correlations, mean-field and statistical models tend to break down at values of the coupling constant k 3, and at larger densities [15&,16]. The anisotropic structure factor and its characteristic features have been analyzed in simulational studies [40 – 42]. The structure factor is projected onto a certain geometric plane [41 – 43&] in order to get the information of the anisotropic structural properties of the model ferrofluid of interest. This is experimentally accessible via neutron scattering [43&,44], and is in good agreement with the simulations. On a qualitative level the morphologies of the structures observed either in thin fluid films of FF confined between glass plates (Hele-Shaw cell) or FF deposited on a substrate subjected to a perpendicular magnetic field have been characterized as needles, chains, columns, sheets, bent walls, lamellar, labyrinthine or worm-like structures. In a
weak magnetic field the dipole moments align with the field and the particles associate into chains with random positions and lengths oriented parallel to the field direction [45]. In larger fields the chains will span the sample and coalesce into thicker column-like structures which eventually organize into hexagonally ordered structures [45] in thin films (2 – 10 Am) [45] or into a glassy state in thicker films (125 Am) [46&]. Further increase of the field in thin films, at least for low sweep rates, has been shown to entail splitting of each column into two columns and formation of a second level hexagonal phase [45]. Distances between columns have been recorded as a function of magnetic field [47], film thickness [48], sweep rate [45,49] and temperature [50]. In cobalt nanocrystals that have been dispersed in a nonpolar solvent, deposited on a substrate while subjected to a strong magnetic field perpendicular to the substrate [51& –53], pattern formation has been found and related to the occurrence of a gas – liquid transition during the evaporation process. In the concentrated phase the columns are made of self-assembled nanocrystals in a face-centeredcubic (fcc) structure. Column and labyrinth formation is sensitive to the particle size distribution [52]. For particle sizes in the range 5 – 8 nm and a narrow size distribution ( 13%) the columns are isolated and well defined with closely packed nanoparticles inside the columns [53]. The average length and width of the particles remains unchanged upon increasing the field strength (0.25 – 0.59 T); no transition takes place from a columnar to a labyrinth pattern. In contrast, for larger size distributions (13 > %) [52] defects at the ends and the edges of the columns resulting from a decrease of cohesion of the nanoparticles inside the columns favor fusion of the columns into labyrinthine structures. Explanation of these pattern is generally based on minimization of a free energy comprising magnetic, interfacial and entropic contributions [54 – 56&]. A recently developed theory which takes into account the nonlinear relationship between magnetization and field at high field strength [56&] accurately reproduces the characteristics of hexagonal ordering in cobalt nanocrystals [51&], in particular the dependence of cylinder radius on applied field, volume fraction and cylinder height. The theory when applied to idealized stripe and hexagonal patterns predicts the existence of hexagonal patterns at low volume fractions and labyrinthine patterns at large volume fractions but no transition between hexagonal and striped structures by variation of the external field (at given volume fraction) [56&,57] in contrast to a theory by Lacoste and Lubensky [55] using a mean field approach for the magnetic free energy and the entropy of the hard sphere fluid, and to experimental findings for magnetite FF [46&]. The self-assembly into elongated clusters when a magnetic field is applied parallel to the sample has been investigated recently for maghemite nanocrystals [58], magnetite nanoparticles [59], cobalt nanocrystals [60] and iron FF [22&,61]. For the system of weakly dipolar maghemite nanocrystals deposited by evaporation it has
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been shown that the mesoscopic organization of the particles can be controlled by the thickness and nature of the surface coating. With thin coatings the particles form elongated structures in the direction of the field while they are randomly deposited with thicker coatings [58]. In the former case vdW forces dominate and the nanocrystals form spherical clusters in zero field. When the magnetic field is applied these clusters acquire a dipole moment much larger than that of the individual particles and attract each other to form larger elongated aggregates. When the thickness of the surface coating is increased (maintaining the average dipole moment and size of the particles unchanged) the vdW forces rapidly diminish and the observed structure results from the dominant repulsive forces [58]. Simulation studies varying the relative contributions of repulsive and attractive forces reproduce these results qualitatively [62]. Field induced pseudo-crystalline ordering has been inferred from SANS using polarized neutrons and synchrotron SAXS scattering data in concentrated Co-FF [63,64]. The particles are found to be arranged in hexagonal planes with the magnetic moments and the directions aligned along the magnetic field. The interplane distance varies with sample. Simulation studies addressing the formation of hexagonal or labyrinthe patterns in dipolar fluids subjected to an applied field present two complications related, on the one hand, to system size which has necessarily to be large if regular patterns are to be observed, on the other hand, to a correct treatment of the long range dipolar interactions in finite simulation cells which lack periodic boundary conditions in all space directions [12&&] as is the case in most experimental situations (FF confined between glass plates or adsorbed on a substrate). Not surprisingly, simulations are still often restricted to small system sizes, and sometimes a simple truncation scheme for the dipolar interactions is used despite for the fact that possible serious artefacts can be introduced in this way. A 3d simulation [65] study of dipolar particles has provided evidence for the formation of thick chain-like (columnar) aggregates at appropriate values of the volume fraction, magnetic coupling constant and field strength but the system size was not large enough to demonstrate hexagonal ordering of the columns. Regular ordering of columns (stripes) has been shown more convincingly in a large quasi-2d system of dipolar hard spheres in an external field parallel to the system layer [66]. The width of the stripes is shown to increase roughly linearly with density. A recent simulation study [67] of strongly interacting dipolar particles (k = 18) confined between two repulsive parallel walls has investigated the influence of wall separation on pattern formation in the presence of a magnetic field perpendicular to the plates. At short separations cyclic formation of chains and meandering walls are noted while at separations larger than about ten particle diameters the structure gradually changes to branched walls, then thicker ones with body-centered-
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tetragonal arrangement of the particles in the walls. The results may, however, be seriously flawed by the truncation of the dipolar potential which is potentially dangerous at the high coupling parameter considered.
4. Polydispersity Polydispersity is inherent in all colloidal dispersions and needs to be considered in the interpretation of experimental data as it affects magnetic properties and phase diagrams. In commonly studied FF like magnetite polydispersity may be as large as 40% but can be reduced by size sorting methods [68], magnetic fractionization methods [69,70], or fractionation by size-selective precipitation [71]. Metallic iron [21] or cobalt [72] particles, cobalt ferrite-silica [73] particles or cobalt nanocrystals [74] can be prepared with a narrower size distribution (10%). The particle size distribution is generally adequately described by a lognormal or a gamma distribution [75] whose parameters can be obtained, for instance, from magneto-granulometric analysis of magnetization curves [76] or, more directly, from electron microscope images. Due to the large number of degrees of freedom involved in polydiperse systems a theoretical determination of the phase behavior may be quite complex enabling coexistence of an arbitrary number of phases [77]. In particular, the coexisting phases may have different size distributions than the parent distribution if the latter is asymmetric. However general relations between the moments of parent and daughter distributions have been shown to exist [78]. Although this universal law of fractionation has been derived only for slightly polydisperse systems, confirmation of the law could be demonstrated for a highly polydisperse iron oxide system [79]. Insight in the influence of either bidispersity or polydispersity on phase diagrams, magnetization curves or aggregate microstructures can be gained from computer simulations by comparison with corresponding results for a monodisperse system of particles having, e.g., the same average diameter. Simulations by Kruse et al. [80] using potential parameters which reflect as close as possible magnetite FF supply analysis of size, composition and shape of the clusters formed in an external magnetic field. The finding that the mean particle radius in clusters roughly equals the mean radius of all the particles gives indication that particles of all sizes and not only the larger ones contribute to the composition of the clusters in close agreement with 2d-SAXS experiments [81&]. From this observation and the fact that in the considered magnetite system the average aggregation parameter k < 0.5 it is concluded that vdW forces play a significant role to achieve thermally stable clusters [80]. Simulations by Castro et al. [82] of a FF typical of magnetite analyze the distribution of small aggregates as a function of volume fraction and compare with results for a monodisperse system. Liquid-
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vapor coexistence curves for a polydisperse system have been contrasted in Ref. [83&] with its monodisperse equivalent: they are narrower in the polydisperse case, have a higher critical temperature and a lower critical density. With increasing field strength the coexistence curves broaden, the critical temperature increases but the critical density is little affected [83&]. At high density ferroelectric ordering observed in zero field is reduced by dipolar polydispersity [84]. The influence of polydispersity on the effect of magnetostriction has been investigated in [85]. Here it was found that the non-linear susceptibility due to magnetostriction was very sensitive to the polydispersity, and that measuring the harmonics could give information about the particle size distribution in the sample under investigation. The structure of polydispersed FF was discussed theoretically on the basis of a bidisperse model where the systems are supposed to consist of two fractions of magnetic particles with significant size differences [17,86 – 88]. The main structural character in these systems is proposed to be the chainlike aggregates formed by the large particles. Some small particles might be attached to the ends of these aggregates, but most of them remain in the non-aggregated state [86 –88]. A recent study also includes the effect of vdW and depletion interactions and studies the effects on the l– g transition [89]. A bidisperse solution was also studied by MD simulation [90&] whose size and dipolar couplings were chosen to mimic experimentally investigated ferrofluid [91&,17]. The initial susceptibility v and magnetization curve of the systems show a strong dependence on the volume fraction / L of the larger particles. With the increase of / L , the magnetization M of the system has a much faster increment at weak fields, and thus leads to a larger v. A cluster analysis indicated that this is due to the aggregation of the large particles in the systems. The average size of these clusters increases with increasing / L . It was found that the volume fraction of smaller particles had a significant effect on the structural properties of the system. A topological analysis of the structure reveals that the majority of the small particles remain non-aggregated. Only a small number of them are attracted to the ends of the chains formed by large particles. The influence of polydispersity on the structural behavior of monolayers of particles interacting by steric repulsive and dipolar potentials has been investigated by MC simulations for a Gaussian size distribution with standard deviations 0.2 and 0.35 and same average diameter [92,93]. Aggregation, in zero as well as finite fields, is favored compared to a monodisperse system with coupling strength equal to the average value due to association of the bigger particles. The particles most frequently found in the clusters are those having a diameter larger than the average diameter. In the absence of a field clusters have a more clump-like aspect (with internal network structure) than the necklace-like chain structures typical of monodisperse systems [92]. In low external fields the structure is sensitive to the size
distribution, in stronger fields the thickness of the straight chain-like aggregates depends on size distribution but not on field strength [93]. More complete though approximate phase diagrams have been obtained for binary mixtures of DHS [94 –97] and SMF [98] over a wide range of temperature, density, concentration and coupling strengths using a modified mean field theory. Focal topics are the interplay of ferromagnetic phases and demixing transitions for highly asymmetric mixtures, the temperature and density shifts of the isotropic-ferromagnetic transition compared to the onecomponent case, the modification of the phase diagram by application of an magnetic field. One should bear in mind, though, that mean fluid theories are generally at best qualitative due to neglect of correlations; comparison with more sophisticated integral equation results indeed reveals significant differences [95]. Detailed comparisons with simulation results are presently unavailable. Recent theoretical developments and simulation studies in areas of frozen FF or confined by disordered porous media have been reviewed in an article by Klapp [14&&] and we refer to this work for details.
5. Conclusions To conclude, the study of ferrofluids has enjoyed increasing attention during the last decade. We have tried to summarize the most important developments that happened during the last five years, focusing on the phase behavior and the microstructure of these interesting fluids. Novel insight has been gained through the combined effort of computer simulations, more elaborate experimental investigations, and refined theoretical approaches. Nevertheless, the detailed structure of the phase diagram even in the absence of any external field still needs to be resolved, and a complete understanding of the complex topology of the microstructure and its relation to the macroscopic phase behavior has still to be gained in future investigations.
Acknowledgment C.H. acknowledges funding by the DFG in the priority program SPP 1004 under Grant No. HO 1108/8-4.
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