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Ferromagnetic particles suspended in a liquid carrier are generally referred to as ferroflu-ids [19]. The dynamic behavior of these ferrofluids is governed by two very different magnetization relaxation pathways. Either the magnetization can change within each magnetic particle (Neel relaxation) or the whole particle can rotate in the liquid (Brow-nian relaxation). We will first discuss each mechanism individually and then turn to systems with size dispersions, where both relaxation mechanisms can be present simultaneously.

In the following we will neglect magnetic interactions between magnetic particles, which may arise from dipolar coupling. This assumption is nontrivial, but in the case of biologically functionalized magnetic nanoparticles, the ligand coating helps to separate the individual particles independent of their concentration. The important parameter for assessing the importance of interactions is the ratio of interaction to thermal energies X = EiJkBT. Eint depends on both the saturation magnetization, particle volume, and in-terparticle distance (concentration of suspension). As an example, for magnetite (Fe3O4) particles with 10-nm diameter, X becomes negligibly small (X < 0.1) for ligand coatings with a thickness exceeding 8 nm [20]. In any case, interactions are more important for Neel (internal) relaxation and less so for Brownian relaxation [21].

The magnetization in each particle is coupled to the orientation of the particle through magnetic anisotropies, which may arise from the crystal structure or the overall shape of the particles. In nanoparticles the magnetic anisotropy is typically increased compared to bulk materials due to additional surface contributions [22]. These anisotropies create energy barriers AE between different orientations of the magnetization relative to the particle. In the simplest case of a uniaxial anisotropy (characterized by an anisotropy constant K) this energy barrier is given by AE = KVm, where Vm is the magnetic volume of the particle. In this case the Neel relaxation follows a simple Arrhenius law [23]:

The pre-exponential factor to is generally assumed to be between 10-8 and 10-12, see Ref. 24. Since the Neel relaxation only depends on parameters specific for the given magnetic materials (K, Vm, and to), it is not modified by targets bound to the ligand coating of the magnetic particles.

When magnetic particles are suspended in a liquid, the magnetization can also relax via rotational Brownian motion in addition to Neel relaxation [19]. The relaxation time associated with rotational motion is identical with the Brownian relaxation time tb in Eq. (4.1) and thus is independent of magnetic properties. As is shown in Figure 4.1, the two relaxation times tn and t b depend very differently on particle size. Notice that the particle diameter d in Figure 4.1 refers in one case (tn) to the magnetic diameter, while in the other case (t b) it indicates the hydrodynamic diameter. In general the two are not the same due to the surfactant coating of the magnetic particles.

Since for most particle diameters the timescales for the two relaxation mechanisms are very different, the two processes are generally independent of each other [25,26].

Figure 4.1. Neel tn (dashed line) and Brownian tb (solid line) relaxation time as a function of particle diameter D. D indicates the magnetic diameter for twhile it indicates the hydrodynamic diameter for tb. For the calculations based on Eqs. (4.1) and (4.2) we used bulk magnetic properties of magnetite (Fe3O4) and the viscosity of water at room temperature.

Figure 4.1. Neel tn (dashed line) and Brownian tb (solid line) relaxation time as a function of particle diameter D. D indicates the magnetic diameter for twhile it indicates the hydrodynamic diameter for tb. For the calculations based on Eqs. (4.1) and (4.2) we used bulk magnetic properties of magnetite (Fe3O4) and the viscosity of water at room temperature.

In this case, one can define an effective relaxation time t-1 = tn-1 + tb-1. From this expression it is clear that the shorter of the two relaxation times determines the dominant relaxation mechanism. As can be verified from Figure 4.1, below a critical particle size (determined by tn = t b) the relaxation is mainly Neel type, while above a critical particle size, it is mainly Brownian. Magnetic particles that are small enough, such that their relaxation is dominated by Neel relaxation, are also referred to as superparamagnetic or unblocked particles.

In principle the relaxation time can be determined directly by measuring the time-dependent response of the magnetization to a change in magnetic fields [27]. However, it is difficult to measure the time-dependent magnetization of ferrofluids with a time resolution below 1 msec. This means that the Brownian relaxation of submicron ferromagnet particles in aqueous solution cannot be determined by time-resolved measurements (see Table 4.1).

In order to get information about the relaxation mechanisms in ferrofluids it is often more convenient to determine the frequency-dependent susceptibility, instead of measuring the direct time dependence of the magnetization [28]. The complex susceptibility as a function of frequency w can be expressed as where t is the effective relaxation time and x o is the static susceptibility. As a function of frequency, the real part x' decreases monotonically with a step at the inverse relaxation time, while the imaginary part x'' has a peak with its maximum at the inverse relaxation time [29]. Generally, the peak in x'' occurs at low frequencies (up to 100 kHz) for Brownian relaxation [see Eq. (4.1)], while Neel relaxation (see Eq. (4.2) typically gives rise to a peak at higher frequencies (i.e., megahertz to gigahertz range).

In order to determine pure Brownian motion from magnetic susceptibility measurements, we have seen from Figure 4.1 that the particles need to have a minimum size to avoid Neel relaxation. However, if the magnetic particles become too large, the magnetization in each particle ceases to be homogeneous (single domain) and an inho-mogeneous magnetization structure, such as multiple magnetic domains, can develop [30]. In this case the magnetic response of each particle may no longer be dominated by Brownian relaxation, but instead the magnetization can change internally—that is, via domain-wall motion. Therefore, besides a lower limit there is also an upper limit for the suitable size of ferromagnetic particles for which their Brownian rotational motion can readily be detected via susceptibility measurements [31]. The maximum single-domain size depends on materials-specific parameters, such as exchange stiffness, magnetic anisotropy, and saturation magnetization [30]. We show in Figure 4.2 suitable particle sizes for stable single-domain magnetic particles (large enough to avoid Neel relaxation,

0 20 40 60 80 100 Particle Diameter (nm)

Figure 4.2. Particle diameters for stable single domain magnetic nanoparticles. (Adapted from Ref. 31.)

0 20 40 60 80 100 Particle Diameter (nm)

Figure 4.2. Particle diameters for stable single domain magnetic nanoparticles. (Adapted from Ref. 31.)

small enough to avoid multiple domains) for various magnetic materials. For almost all materials the useful particle size is of the order of tens of nanometers.

The frequency dependence of the complex magnetic susceptibility can be experimentally determined several ways. A small ac magnetic field applied to the sample results in a time-varying magnetic moment in accordance with Eq. (4.4) (x = dM/dH, where H is the applied magnetic field and M is the magnetic moment). By placing the sample into a pick-up coil, the time-dependent moment induces a current in the pick-up coil, thus allowing measurement of x without sample motion. Alternatively, the pick-up coil itself can be used as the source of the alternating magnetic field and the susceptibility can be determined directly from the impedance of the coil. For measurements at very high frequencies (>0.1 GHz) a transmission line can be used instead of a pick-up coil [32].

In Figure 4.3 we show an example of Brownian rotational relaxation measured via the frequency dependence x"■ The solid symbols show x" at 300 K for avidin-coated Fe3O4 particles suspended in a phosphate buffered saline (PBS, pH = 7) solution with a concentration « 2 x 1015 particles/ml. For these particles, the magnetic diameter is «10 nm and the hydrodynamic diameter is « 50 nm. A peak is observed at 210 Hz. The measurement was repeated at 250 K (open symbols in Figure 4.3), which is below the freezing point of the PBS solution. In this case the peak in x " disappears, verifying that the peak at 210 Hz in the 300 K data is indeed due to rotational Brownian motion [31]. The solid line in Figure 4.3 shows a fit of the data to Eqs. (4.1) (4.6) and (4.7) using a size dispersion of ±12% (see also Section 4.3.3).

Any real ferrofluid system will generally have a distribution of sizes for the ferromagnetic particles, which may influence the experimentally observed susceptibility. In fact, it has been shown that in some cases the Neel relaxation peak cannot be understood completely in terms of a size distribution alone, but a distribution of effective magnetic anisotropies is also required [33]. However, since in this review we are mostly concerned with

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