1. Athanasios T. Giannitsis
Dynamic Properties of Nanoparticle Magnetic Fluids
Measured by Impedance Spectroscopy
Magnetic fluids: measurements of the dynamic properties of magnetic nanoparticle fluids using impedance spectroscopy. Microscopic parameters of the magnetic nanoparticles such as size distribution, relaxation times, magnetic anisotropy and gyromagnetic ratio are calculated from the magnetic susceptibility spectrum. Magnetic relaxations and resonances are measured in the linear and non-linear region of the magnetization curve by applying appropriate magnetic fields. The measurements confirm theories of relaxation and resonance in magnetic fluids.
I explored properties of magnetic nanoparticles in the context of Brownian motion, by extensively studying the effects of relaxation (both Brownian and Neel) and magnetic resonance of those particulate systems. The magnetic nanoparticles I tested were single domain particles with their core consisting of alloys of ferrite (manganese-ferrite, nickel, zinc etc) or maghemite. Many of the nanoparticles I studied were commercial ferrofluids (3-10 nm) and many were designed for medical applications including beads (100 nm). I used impedance analysers, sweeping at frequency range Hz-GHz , in order to detect the response of the nanoparticles.

2. What is a magnetic fluid?
Colloidal suspension of magnetic nanoparticles in carrier liquid
Paramagnetism: All known ferromagnetic materials have a Curie point well below their melting temperature. Thus, they lose their ferromagnetic properties before becoming liquid. Ferrofluid suspensions show liquid behavior coupled with superparamagnetic properties. That means that moderate magnetic fields can exhibit magnetic forces to the liquid, which are comparable to gravitational forces. To control flow of ferrofluids you need magnetic fields of mT. The alignment is counteracted by thermal motion yielding a paramagnetic magnetization behavior with an initial susceptibility.
Ferroparticle: the magnetic core of the ferroparticle may consist of materials such as magnetite, cobalt ferrite, manganese ferrite, zinc ferrite, nickel-zinc ferrite , or cobalt-zinc ferrite. The particles are considered to be single domain. Single domain means that the volume of the ferromagnet which consists of the particle, the so called magnetic volume, is composed of one only domain. Domains are areas that are formed in a ferromagnet as a result of energy minimization. Raikher and Shliomis calculated the critical size for mono-domainness for a typical ferromagnet and obtained a critical diameter of 30 nm. Magnetic moment of magnetic nanoparticle is 104 –105 μB. Strong exchange forces hold the adjacent spins parallel to each other, and the direction of these spins defines the direction of μm. These exchange forces fall off very rapidly with distance and only affect nearest neighbour pairs.
Anisotropy: A ferromagnetic crystal is not isotropic since it contains crystallographic axes, usually in uniaxial disposition. The electron orbits are linked to these axes and by their interaction with the spins (spin-orbit interactions) makes the spins align along these crystallographic axes. The resulting anisotropy is called magnetocrystalline anisotropy. The uniaxial anisotropy is expressed by the anisotropy constant K. For magnetite K has a value between ,000 J/m3, for manganese ferrite 4,000-6,000 J/m3, for cobalt ferrite 5,000 J/m3 and for nickel-zinc ferrite 3,000 J/m3.
1. Origin of magnetism is due to the orbital and spin motion of the electrons.
2. A ferroparticle consists of single domain ferromagnetic or ferrimagnetic material having anisotropy.
3. Bulk magnetization, Ms , of a magnetite ferroparticle is 0.4 T.
4. Average radius of magnetic nanoparticles spans between 2-10 nm. Surfactant is 2nm.
5. The carrier liquid can be either water, or oil such as kerosene, isoparaffin, diester, or toluene.

3. Applications Engineering - stepper motors - electromagnets
- transformers
- loudspeakers
- ink jet printing
- memories
- lubrication
- sealing
Science
- theoretical mechanics
- magnetism
- material science
- statistical physics
spectroscopy
chemical physics
Medicine
- bioelectronics
separation
targeting cells
Space technology:
- controllable flows
thermal agent
microgravity
Ferrofluids are an important category of physical systems, where microscopic magneto-mechanical effects can be manifested. Ferrofluids have a wide diversity of applications. Ferrofluids are ideal physical systems for basic research.

4. Impedance Spectroscopy Methods
Autobalancing bridge
Audio frequency measurements
0.1 Hz MHz
Radio frequency vector analyzer
Radio frequency measurements
1 MHz -1 GHz
Auto-balancing bridge method: is based on the elimination of the current through Zx. Measures impedances between 1mΩ-100MΩ at 5Hz-100 MHz.
Vector analyzer, I-V method: is based on the measurement of the source voltage and the calculation of the current through R by measuring the voltage across R. R has well known value. The output impedance matches the impedance of port extension. There are two arrangements, one for high impedance DUT and one for low impedance DUT. Measures impedances between 100mΩ-1MΩ at 1MHz-1GHz.
Networking technique: is based on the measurement of the reflected coefficient SΓ, that is the ratio of the reflected to the incident wave. For the case when the line is open SΓ=1, and for shorted case SΓ=-1. For the case when a 50 Ω load is connected SΓ=0; this means no reflection since the impedance of the 50 Ω load is equal to the characteristic impedance of the line. It is evident that –1<SΓ<1 . Measures impedances between 10Ω-100Ω at 50MHz-20GHz.
Network analyzer
Microwave frequency measurements
50 MHz – 20 GHz

5. Measurement setup
The dynamic magnetic susceptibility χ(ω) = χ΄(ω) - i χ΄΄(ω) can be measured by impedance spectroscopy, using alternating magnetic field, h(t), in combination with a static magnetic field, Hs .
Autobalancing bridge method for low frequency measurements
ΔL(ω)/L(ω) ,
ΔR(ω)/ωL, L=inductance of coil
Coils: Toroid coils were implemented with cores of mu-metal that are high permeability alloys and have minimum remain magnetization . A slit of 0.3 mm was cut on the core using a diamond saw. The winding was made by wounding copper wire around the metal core. The number of turns of the winding depends on the size of the coil and varies between turns. The DC resistance of the wire is 0.05Ω. The core has circumference 150 mm, cross-sectional area 9.52x9.52 mm2=90.7 mm2 and the length of the slit is 0.3 mm. The inductance of such coil spans between mH-μH. Above 1 MHz measurement is prohibited because the toroids resonate. For measurements below 50 Hz a large toroid (N=80) with bigger inductance is more suitable, since the autobalancing bridge cannot observe small changes in the impedance.

6. Coaxial line method for radio and microwave frequency measurements
Coaxial line: The cylindrical cell, in which the ferrofluid is placed, is the end section of the coaxial line. The diameter of the inner conductor is 3mm and the diameter of the outer conductor is 7mm. There are three different cells, of 9.5 mm, 1.8 mm and 1mm depth, to be used for three measurement frequency ranges, namely, 1 MHz-1 GHz, 100 MHz-6 GHz and 1 GHz-20 GHz. The cell behaves as an inductor when is shorted and it behaves as a capacitor when is open. Hewlett Packard supports short and open adaptors, for the 7 mm connector, operating sufficiently up to 40 GHz. Only the cell behaves as a pure inductor or a pure capacitor, depending on the termination; the rest of the coaxial line is both inductive and capacitive.
Equations: The equations above for χ(ω) are valid only under the assumption that the wavelength is much greater than the sample thickness (λ>>x). This assumption is not usually true for high frequencies, thus abnormal behaviour occurs. Dimensional resonance takes place when the depth is an odd multiple of quarter wavelengths. For the case of water-based ferrofluid, resonance will occur at 1, 3, 5 … GHz approximately. Fortunately for non-water based ferrofluids, such as these based on paraffin or kerosene, ε is small thus the propagation velocity is large and the resonance frequency one order higher than water-based fluids.
Electromagnet: the electromagnet can provide magnetic field up to 200 kA/m.

7. Magnetostatic properties
Magnetic fluids demonstrate:
1. Paramagnetic behavior
2. No hysteresis
linear region
nonlinear region
Ferrofluid: magnetite in water with saturation magnetization 400 G.
Ms: is the magnetization of each individual ferroparticle (bulk magnetization). Ms=0.4 T for the magnetite nanoparticle and Ms=1 T for the cobalt nanoparticle.
Curie law: states that the static susceptibility, χ0, defined as the magnetization divided by the applied static field, M(ξ)/ξ, is proportional to the inverse of the absolute temperature T. χ0 may be derived by expanding the Langevin function in a Taylor’s series in ξ. When ξ<<1 (valid for normal room temperatures) the expression for Μ(ξ) can be approximated by the first term, Μ(ξ)=Μsξ/3 = Μs μmHs/3kBT and thus the static susceptibility, χ0=Μsμm/3kBT=C/T. The Curie Law, together with the Langevin function, shows that the theory of a paramagnetic gas also holds for ferrofluids.
Figure: Data has been fitted with the Langevin equation modified to include the distribution of the particle’s sizes. This is realised by distributing the magnetic radius rm of the particles by a distribution function such as Normal (Gaussian) or Log-Normal. Most investigators prefer to use the Normal, Log-Normal or the Nakagami distribution (Normal is a special case of Nakagami distribution, whilst Nakagami is related to Gamma distribution).
Fitting parameters: Ms=0.4 T (magnetization of one individual ferroparticle), rm=5.5 nm (magnetic radius of a ferroparticle), μm=2.78x10-25 Wb m (magnetic moment of a ferroparticle), and σr is 0.6 and 0.45 for Normal and Log-Normal respectively.
Units: 1 Tesla = 10,000 Gauss = 800,000 A/m . If Ms in Tesla, H in A/m and μm in Wb m , then ξ=μmΗ/kT. If Μs in A/m, H in A/m then ξ= μ0 μmΗ/kT.
Langevin law of magnetization:
Curie law of static susceptibility:

8. Magnetodynamic properties
Magnetic relaxation is due to the rotation of the magnetic moment vector
- Brownian relaxation is due to Brownian motion.
- Néel relaxation is due to the rotation of in the magnetic core, due to
thermal activity.
Resonance is due to precession of about the axis of the effective field
(internal anisotropy field + external fields) and occurs when
the frequency of the precession of become equal to the frequency of a
probing alternating field which is used to excite the system.
Relaxation: is the phenomenon of returning from the polarised state to the original chaotic state. Brownian relaxation is related with the rotational Brownian motion of the whole ferroparticle, due to thermal agitation whilst the Néel relaxation is related only with the physical rotation of the moment itself within the magnetic core of the ferroparticle which is, again, due to thermal activity.
Resonance: Ferromagnetic resonance (or gyromagnetic resonance) occurs when the frequency of the alternating field matches the frequency of precession of μm , that is when the atomic magnetic moments precess about an externally applied field with frequency equal to the Larmor frequency ωL . However if ω is the same as ωres then moment and field can remain in phase. At this critical point the susceptibility changes sign denoting the change of phase.
Alternating field : frequency varies between Hz-GHz and strength varies between 0-13 kA/m
Static field : strength varies between 0-100kA/m
Figure: Precession of μm under the influence of a static field, Hs , in conjunction with an alternating field, ha(t) . On the graph is shown the orientation of the alternating field, ha(t) , at an arbitrary instant and its components along the three axes of space. The moment rotates anticlockwise. When the frequency of the alternating field equals the frequency of moment’s precession, the moment jumps to an other energy state, i.e. μm jumps under the x-y plane. The rotation of μm is in the clockwise direction, i.e. μm changes its rotational phase.

9. Dynamic magnetic susceptibility
and
L(ξ) is the Langevin function
Dynamic magnetic susceptibility χ(ω)=χ΄(ω)-i χ΄΄(ω) is frequency and field dependent.
The magnetic susceptibility χ(ω)=χ΄(ω)-i χ΄΄(ω) can be decomposed in
The components follow Debye and Landau-Lifshitz dispersions.
Debye: , , ,
Landau-Lifshitz: ,
The field, , dependence of follows Langevin profile:
The relaxation time is field, ξ , dependent:
Complex magnetic susceptibility: provides information on the microscopic parameters of a ferroparticle, namely the particle radius, the magnetic radius, the anisotropy constant, the internal anisotropy field and the gyromagnetic ratio.
Susceptibility: At low frequencies the moments can respond to the applied alternating field so that magnetisation is almost completely in phase with this field. Therefore the real part of susceptibility is large and the imaginary is practically zero. However as the frequency increases the magnetisation increasingly lags ha(t) which means that the real part decreases and the imaginary part increases and a point is reached when the imaginary component is a maximum. At higher frequencies, the system cannot respond to the applied alternating field and both real and imaginary susceptibility components decrease to zero. This type of response is characterized as a Debye type response and can accurately describe the behaviour of the system at low frequencies.
Landau-Lifshitz: Raikher and Shliomis have treated the resonance phenomena in terms of equations proposed by Landau-Lifshitz modified to include stochastic terms and derived expressions for the transverse susceptibility, which is equivalent to the resonance susceptibility component. Their expressions were subsequently simplified by Coffey.
Langevin function: L(ξ)=coth ξ-1/ξ. Accounts for paramagnetic behavior.

10. Theoretical spectrum of magnetic susceptibility
Susceptibility according to Debye and Landau-Lifshitz dispersions.
relaxation
resonance
Debye term :
(relaxation term)
Landau-Lifshitz:
(resonance term)

11. Magnetic relaxation Magnetite Fe3O4 magnetic fluid
Figure : Magnetite in water carrier 100 G, measured in frequency range 50 Hz-1 MHz using the toroid coil technique. The susceptibility of this fluid has an absorption Brownian peak at low frequency, which means the relaxation time of the particle is large. Large relaxation times denote agglomeration, since aggregated particles rotate slowly. The peak of χ’’ at frequency fmax=4.68 kHz corresponds to a relaxation time τ =3.4 x10-5 s from which we are able to calculate the mean particle radius, rp : for room temperature 300oK and viscosity 10-4 N/m2, rp =48 nm, which denotes aggregation.
Estimation of the average particle size
from the relaxation time
Estimation of the particle size
distribution from the fit

12. Field dependent susceptibility
Susceptibility: Cobalt-Ferrite in water (300 G).
Relaxation time: Coffey gave analytical closed form expressions for the relaxation times, which are dependent on the relative orientation of the fields. They derived the relaxation times by means of a method, which combines linear response theory and the effective eigenvalue method.
Field dependence of the relaxation time

13. Cole-Cole Cole-Cole parameter α accounts for size distribution
The fits are obtained by choosing optimum values for ξ and α for each individual curve. The Cole-Cole parameter α used here is 0.36 and it doesn’t change significantly for several values of biasing field. This is demonstrated by the Cole-Cole diagram in which we plot χ’ versus χ’’ . In a Cole-Cole diagram the data appear to form a semicircle with the centre under the horizontal axis. Each semicircle corresponds to a different field and each data point to a different frequency. If we connect all the data-points which correspond to the same frequency, but to different polarizing fields, we get the Cole-Cole diagram for constant frequency. Figure shows where the constant frequency curves (solid line) intersect the constant field curves (dots). The angle, ψ, formed between the radius and the horizontal axis is equal to ψ=απ/2. For the case of mono-dispersion α=0 (ψ=0), and the Cole-Cole diagram is a complete semicircle. For our data we found that for each individual data curve, ψ=0.565 rad. Thus, the radii of all the circles are parallel and the circle’s centres lie on the same line.
Cole-Cole parameter α accounts for size distribution

14. Nonlinear increment of susceptibility
Susceptibility difference between polarized and non polarized.
Coffey have shown theoretically that there are higher order terms of the external field which should be added in the Debye expression. He shows, for the dielectric case, a relaxation due to the nonlinear behaviour of the dielectric. The magnetic analogue of the latter theory is well suited for the case of the ferrofluids, in which we are able to observe the nonlinear effects. Coffey derived a theory for nonlinear relaxation in dielectric systems, the magnetic analogue of which, as shown here, can be easily applied for the case of ferrofluids. Experimental evidence of dielectric nonlinear relaxation has been given recently for several polar dielectric liquids, namely hexyl-cyanobiphenyl in benzene , solutions of pentyl-cyanobiphenyl in cyclohexane and in squalane, polyvinil acetate and trifluoroethylene-rich copolymers. There are experimental verifications of the validity of Coffey model for nonlinear susceptibility in dielectrics. It is our aim to present experimental evidence for the validity of this model in ferrofluids. The main differentiation between Coffey consideration and Debye’s derivation was the notion that there is no longer the connection between the aftereffect and the alternating-field solutions seen in the linear approximations. The ratio of the response to stimulus (transfer function or impulse response) now depends on the form of the applied field, and is not independent of it, as in the linear case. The term Δχ is the susceptibility difference between a polarised and an unpolarised measurement; it may also be represented in its complex form of Δχ=Δχ’ - i Δχ’’ . It is an important factor and can be represented graphically by subtracting the unpolarised data from the polarised data.
Anomalous rotational diffusion

15. Magnetic resonance
oil-based magnetic fluid consisting of magnetite Fe3O4 nanoparticles
relaxation
Ferrofluid: isoparaffin-based ferrofluid.
gyromagnetic ratio : is the ratio of magnetic moment to angular momentum.
Interaction field: fmax is written in terms of the interaction field, HR , where the interaction field can be written in terms of the interaction parameter I(rp) and a dimensionless parameter b, which is a function of the orientation of the anisotropy axis, i.e. HR = b HA +I.
Spectrum of χ΄(ω) and χ΄΄(ω) for frequency range 1 MHz-6 GHz

16. Field dependence Polarizing field 0-100 kA/m
Ferrofluid: 900 G (0.09 T) magnetite in isoparaffin with mean particle hydrodynamic radius of 5 nm. The prominent peaks in χ΄΄ between 10 MHz-100MHz in Fig.(b) are purely due to Néel relaxation while above 100 MHz resonance is dominant. It is important to notice that Néel relaxation may appear throughout the whole spectrum, however in the majority of ferrofluids it is dominant only between 1 MHz-100 MHz. Neel relaxation progressively dissapears because the effect of the external field is to effectively increase the value of the anisotropy field HA, and the barrier to rotation of the magnetic moments now becomes higher (VA=Kvm+μmHs).
Figure: (a) χ΄ for Hs = 0 – 50 kA/m, (b) χ΄΄ for the same values of Hs, (c) fres versus Hs. <HA> is estimated from the point where the line cuts the horizontal axis. The value of 40 kA/m is typical for magnetite. The slope of the line defines γ, (d) fmax versus Hs.
The slops define the gyromagnetic ratio γ

17. Characterization of various magnetic fluids
Ferrofluid type
gyromagnetic ratio
γ (m/A sec)
anisotropy field
(A/m)
Anisotropy constant
(J/m3)
Magnetite in water or isoparaffin
220, ,000
40,000
7,500-11,000
Cobalt ferrite in isoparaffin **
5,000
Cobalt in diester
200,000
126,000
63,000
Manganese ferrite in isoparaffin
230, ,000
20,000-40,000
4,000-6,000
Nickel-zinc ferrite in isoparaffin
250,000
3,000-3,500
** Have not been measured because its resonant frequency is outside the
experimental frequency range.
The wide range of values of γ and <K> for a particular fluid is due to the different proportion (stoichiometry) between the iron and the transition metal. For example, for manganese ferrite Mn0.1Fe0.9Fe2O4 we have measured γ=230,000 m/A s and <K>=6,000 J/m3, but for Mn0.7Fe0.3Fe2O4 we found γ=252,000 m/A s and <K>=2,400 J/m3.
Anisotropy constant K: HA=2K/Ms.

18. susceptibility fits as function of frequency
Fits obtained using Landau-Lifshitz model
unpolarized
polarized (60 kA/m)
Ferrofluid: Fits (dotted lines) of the susceptibility data (solid lines) of the 900 G magnetite in isoparaffin sample.
Figure: The fitting equation for χ(ω) used here is the Landau-Lifshitz equation. When the polarising field is zero or small, such as in case (a), the effective σ is small so the Néel relaxation operates and the total susceptibility is given by 1/3(χ||(ω,Hs)+2χ(ω,Hs)). χ(ω,Hs) is the resonance component (Landau-Lifshitz) and χ||(ω,Hs) the relaxation component (Debye type). The inclination of the χ΄ curve below 1 GHz is due to the appearance of Néel relaxation at the radio frequency range. When the polarising field becomes strong σ is very large and in this event the Néel relaxation is blocked (case (f)). The total susceptibility in this case equals χ(ω,Hs).
Fitting parameters: the normal distribution was used for rm and K. For all the fits we performed here: Ms=0.4 T, <K> =9,000 J/m3 with a standard deviation σK =0.7, <rm>=6 nm with standard deviation σr= 0.55, γ =230,000 m/A s and αd=0.25. The damping parameter, αd , exceeded the usual values ( ) which αd typically takes. The resonance frequency, ωres=γ(Hs+K/2Ms) , is directly dependent on Hs and by increasing the latter we have a shift in the resonance point of the fit. The limited range of values that αd can take ( ) was treated by Raikher and Shliomis and applies only to their equation. For the Landau-Lifshitz equation, αd may be lie between 0-1.
Scaife studied extensively the Landau-Lifshitz equation. He compared the Landau-Lifshitz type of absorption (also called ferromagnetic resonance absorption) with the Debye type and the Lorentz type of absorption. The difference between the Landau-Lifshitz absorption type and the Debye type is that the latter does not take account of the inertia which produces Larmor precession (is the undamped precession) about the magnetic axis. The Landau-Lifshitz absorption has many similarities, at low frequencies, with the Lorentz type of absorption, which is the resonance of a damped harmonic oscillator. Scaife also estimated the value of τ0.
Distributions: The choice of a distribution is arbitrary but most investigators prefer to use the Normal, Log-Normal or the Nakagami distribution (Normal is a special case of Nakagami distribution, whilst Nakagami is related to Gamma distribution).
The polydispersion of the radius and anisotropy constant of the nanoparticles is
accounted using distributions such as Normal (Gaussian), log-normal and Nakagami.

19. Cole-Cole χ΄ versus χ΄΄ for various fields
χ΄ versus χ΄΄ for various frequencies
Figure: (a) Cole-Cole diagram (type I) of the resonant curve of the 900 G magnetite in isoparaffin sample. Each curve corresponds to a constant polarising field. The tail appearing on the right hand side of the first three curves is due to the relaxation contribution, (b) Cole-Cole diagram (type II) of the same data where each curve corresponds to a constant frequency (i.e. 1, 2, 3, 4, 5 GHz). Each data point of each individual curve corresponds to a different polarising field, (c) Real susceptibility component as a function of the field and of the frequency, (d) Imaginary susceptibility component as a function of the field and of the frequency.
Explanation: The type I Cole-Cole diagram is presented in Fig. (a), in which each data curve corresponds to a different field. There are 17 curves, as many as the polarising fields which were used in this experiment. The values of the polarising fields used, are kA/m. At lower fields (<16.1 kA/m) the effective σ is small and the relaxation mechanism is active. The relaxation is responsible for the appearance of the tails on the right hand side of the first three curves. The rest of the curves are normal circles and do not have tails. In Fig. b) we present the type II Cole-Cole plot, in which each curve is associated with a constant frequency and each point of the curve with a particular field. Each curve consists of 17 points, as many as the polarising fields which were used in this experiment. The values of the 17 χ΄ points are presented in Fig. (c) for each individual frequency and field. The 17 values of χ΄΄ are presented in Fig. (d). We are now able to fit the type II Cole-Cole curves of Fig. (b) with the Landau-Lifshitz equation for each selected frequency (see next transparency).
χ΄ versus field and frequency
χ΄΄ versus field and frequency

20. susceptibility fits as function of field
Fits obtained using Landau-Lifshitz model
Figure: (g)-(h) Fits of the Cole-Cole curves of figure (b) by use of the Landau-Lifshitz equation. In all the four cases the fitting parameters used were the same. The parameters used were: Ms=0.4 T, K=9,000 J/m3, γ =230,000 m/A sec, τ0 =2x10-10 sec. In each fit the frequency took its correspondent value, i.e. (g) 4 GHz, (h) 5 GHz.
Each curve corresponds to a specific frequency and each point
of the curve corresponds to a particular polarizing field.
Cole-Cole diagrams ease the fitting
The parameters gained from the fit are γ, K and τ0.

21. Time domain
Decay of the magnetization of a magnetic fluid obtained from
the susceptibility spectrum by Inverse Fourier Transform.
Figure: The decay of magnetisation (after effect) for 900 G (0.09 T) magnetite in isoparaffin. Shows oscillation behavior as the polarizing field increases . Each curve refers to a separate Hs. Such a representation is called, the after-effect function.
After effect: It is possible by applying the inverse Fourier transform to the susceptibility data to derive the decay of the magnetisation, as a function of the time, after the switching off of the biasing field (after-effect). This is an easy method by which one can estimate where the time interval of interest is of the order of nanoseconds. As Hs increases, the profile of MsF shows oscillatory behaviour. This should be expected since an increase in Hs results in an effective increase of σ , or in other words to an elimination of the thermal agitation. The thermal agitation interrupts the oscillatory motion of the magnetic moment, μm , of the ferroparticles. When Hs is removed the decay type is exponential or oscillatory, depending on the magnitude of the field strength. Now, the value of the time which corresponds to the 1/e of the decay of M(t) (horizontal line drawn) is the Néel relaxation time.
Time interval is of the order of nanoseconds

22. Other resonance models
Coffey’s model ,
Figure: Comparison between Landau-Lifshitz equation fit (dashed line) and Coffey’s equation fit appropriately modified to include the parallel susceptibility component.
Coffey’s model: Coffey derived an equation for χ based on the analysis of Gilbert’s equation, which includes the thermal agitation effects (for all values of σ). Debye equation and that of Landau-Lifshitz are the two limits for very low and very high σ respectively, since Coffey’s equation covers the whole range of σ values. D(x) represents Dawson’s integral. Coffey derived his model based on the effective eigenvalue technique.
Landau-Lifshitz versus Coffey model: both phenomenological expressions, Landau-Lifshitz and Coffey’s, can be used. The first is suitable to describe the field dependence of χ, while the second is more appropriate to describe the frequency dependence of χ. Raikher and Shliomis have treated the resonance phenomena in terms of equations proposed by Landau-Lifshitz modified to include stochastic terms and derived expressions for the transverse susceptibility, which is equivalent to the resonance susceptibility component. Their expressions were subsequently simplified by Coffey. The Landau-Lifshitz equation is suitable for describing the system only for large values of σ, which means that thermal agitation does not greatly affect the movement of the moment, or equivalently the internal barrier is very large (Kvm>>kBT), so that the rotation of the moment with respect to the particle (Néel relaxation) does not change for long periods of time (i.e. we assume that μm is blocked) and the only change of the direction of μm is due to the precession within the particle. Hence all changes in orientation of μm are determined by the rotational motion of the particle. For the kind of experiments we perform (i.e. polarised measurements where the effective σ becomes large) and for the frequencies of our interest (>1 GHz), this approximation is tolerable. The simplicity of the Landau-Lifshitz equation, in contrast to the equation of Coffey,makes the use of this equation appropriate.
Landau-Lifshitz model versus Coffey’s model:
Coffey’s equation covers the whole range of σ values. Debye equation and
Landau-Lifshitz are the two limits for very low and very high σ respectively.

23. Application Estimation of losses in magnetic fluids Loss factor:
Definition: The loss factor tanδ is a quantity measuring losses in a magnetic material. The definistion comes from the circuits theory.
Figure: (a) The loss factor tanδ, of a 300 G cobalt-ferrite in water ferrofluid, as a function of the frequency for various polarizing fields, (b) The loss factor tanδ as a function of polarizing field for various frequencies, (c) A triangular representation of tanδ.
Explanation: tanδ increases gradually with the frequency up to a maximum and then decreases. Its magnitude is a decaying function of the biasing field. The fluid becomes less-lossy with increase of Hs . This is expected since χ, which defines tanδ, is a decaying function of the polarizing field. In the context of the application of magnetic fluids in the area of low-loss devices, this result is of significance as it indicates a possible technique of controlling such losses. It is possible to plot the tanδ data in a Cole-Cole diagram in which the curves correspond to constant frequencies. In this diagram (Fig. (c)) the horizontal axis is extended back to –1, so each point’s value represents the denominator of tanδ. Each data-point corresponds to a triangle from which tanδ is easily calculated from the lengths of the triangle’s sides.

24. Microwave absorption in magnetic fluids - Energy loss
Figure: 900 G magnetite isoparaffin-based ferrofluid. (a) Representation of tanδ in a type II Cole-Cole diagram. The horizontal edge of the drawn triangles equals 1+χ’ while the vertical edge equals to χ’’ , (b) tanδ as a function of frequency and polarising field.
Explanation: In the same manner we use the tan parameter to describe microwave absorption (losses) in magnetic systems and nanoparticle-based devices. Tanδ is both frequency and field dependent. As measurements show, tanδ decreases with increasing bias field and the frequency of the microwave signal. From this plot we can see that, for the same Hs , tanδ decreases as the frequency increases. At fields below 20 kA/m the biggest absorption appears for f=2 GHz. As the external magnetic strength increases, tanδ increases up to a maximum and then decreases rapidly down to zero. This is opposite to what we discovered in the low frequency case. There, tanδ decreased uniformly with increase in Hs and the frequency was the main factor which affects the losses. In the resonance region, the opposite happens. The losses are mainly affected by the biasing field, rather than the frequency. A fuller explanation of this may be found in the nature of the relaxation and resonance absorption mechanisms. By polarising the sample, in the relaxation case the absorption peak, fres , is not shifted as much as its counterpart is in the resonance case. In terms of absorption, in the relaxation case the changes of the frequency have more effect than the changes of the field, whilst in the resonance case the opposite applies.
It is obvious that Fig. (b) consists of a chart which one can use to control the losses in magneto-electronic devices, by a simple change to the amplitude or the frequency of the intrinsic wave (alternating current).
Also possible to estimate energy losses in Joule/m3 by applying the definitions:
(frequency dependent)
(time average)

25. Application Signal to noise ratio in magnetic nanoparticle systems
Definition : SNR=
Figure : (a) Signal-to-noise ratio of the magnetite in isoparaffin sample (unpolarised), (b) Signal-to-noise ratio of the cobalt in diester sample (unpolarised).
Explanation: The quality of a single-domain nanoparticle as a signal processing device is estimated with the aid of the signal-to-noise ratio (SNR) characteristic. The imposition of an external alternating field together with random noise, influences the magnetic switching within the magnetic particle and the SNR can be found in the framework of micromagnetic theory. The main material parameters essentially involved in the resulting expression are, the Néel relaxation time, τN, the precessional decay time, τ0, and also the magnetic viscosity, ηm. The description of the signal-to-noise ratio of a field-driven magnetisation of a fine ferroparticle is delivered by the theory of the superparamagnetic stochastic resonance*. The SNR is defined through the power spectral density function S().
* Fox defines the stochastic resonance as “noise-induced signal-to-noise ratio enhancement”. The main model in the stochastic resonance theory is an overdamped bistable oscillator in a thermal bath.
Magnetite Fe3O4 in isoparaffin
Cobalt ferrite CoFe2O4 in diester
The imposition of an external alternating field together with random noise,
influences the switching of the magnetic moment within the nanoparticle.

26. Application Potential use of alloys of ferrite-based magnetic
fluids, as adjustable magnetic resonators
Ni0.5Zn0.5Fe2O4
Ni0.3Zn0.7Fe2O4
ratio1:1
Ferrofluid: 900G.
Mixing: Either mix the two ferrofluids together, or devide the cell in two compartments that each contains a different ferrofluid.
Nickel-zinc ferrites of different stoichiometries
Ni0.3Zn0.7Fe2O4 and Ni0.5Zn0.5Fe2O4 mixed

27. Application Use of magnetic fluids in microgravity
Ferrofluid
Thermomagnetic convection:
A flow driven by magnetic forces in a fluid under influence of thermal gradients.
Gravity free conditions enhance the thermomagnetic convection effect.
Influence of magnetic fields on transport phenomena when a temperature gradient is present: microgravity research is the only possible tool to investigate these thermal transport phenomena especially the so called thermomagnetic convection in a quantitative way. A temperature gradient always induces a density gradient in the fluid which will yield disturbing buoyancy driven convection in terrestrial investigations. Thermomagnetic convection is a tool of heat and mass transfer control in nanosize material (like protein solutions and other forms of colloids) under microgravity. On the Earth magnetic forces for ordinary fluids exerted by a typical magnet are too insignificant, that is why colloids of monodomain particles (ferrofluid) with thousand times higher magnetic susceptibility are very convenient for ground-based modeling of magnetoconvection. Study of thermal and concentration ferrofluid magnetoconvection, as well as the measurements of the translation diffusion coefficient and thermal diffusion ratio, are complicated at laboratory conditions due to ubiquitous gravity sedimentation effects. Therefore, the investigations of pure magnetoconvection and the transport factors measurings are very useful under microgravity.
Magnetic force: The possibility to exert strong forces on ferrofluids is due to the high initial susceptibility which is in the order of χ=1 compared to χ= 10-3 for paramagnetic salt solutions. This means, that the magnetization of the fluid is about three orders of magnitude higher at weak magnetic fields, than it is known from usual paramagnetic liquids. Thus one can easily calculate, that the magnetic force density Fmag=μ0Μ gradH is comparable to the gravitational force for a standard ferrofluid in moderate magnetic field gradients. For example a magnetic field of about H= 20kA/m with a gradient of about gradH= A/m2 as it is typically present some 5cm from a pole of an electromagnet will exert a force density of about 14 kN/m3 to the standard fluid, while the gravitational force density on the same fluid is approximately 13 kN/m3. Thus the magnetic field is able to produce a force strong enough to lift the fluid out of the pool towards the pole of the magnet. This magnetic force enables the control of the flow of magnetic fluids.
ESA has scientific interest to study the effect of thermomagnetic convection
in the International Space Station and in parabolic flights. ZARM drop tower
in the University of Bremen was also employed.
Magnetization of a magnetic fluid saturates earlier in a gravity free
environment than on ground.

28. Application Potential use of magnetic fluids in microfluidics
Pipe diameter m
Magnetic strength Gauss
Ferrofluid type oil based
Surfactant hydrophobic
coils
Fmagn.= force acting on a single magnetic nanoparticle.
u= velocity of the nanoparticles.

29. Thanks are due to
Professor Paul Fannin, Trinity College Dublin (TCD), Ireland
Professor William Coffey, Trinity College Dublin (TCD), Ireland
Professor Brendan Scaife, Trinity College Dublin (TCD), Ireland
Professor Yury Kalmykov, University Perpignan (UPDV), France