#### 2.1. Theory

The magnetic remanence of composite microspheres with embedded hard-magnetic nanoparticles depends on their magnetization history. This is discussed here theoretically, on the basis of embedded magnetic nanoparticles that are monodisperse and exhibit no Néel relaxation.

After treatment in a saturating magnetic field, the maximum remanence of composite microparticles is limited by the orientations and magnetic anisotropy energy of the embedded nanoparticles. In theory, full remanence could be attained if all the nanocrystals were identically oriented and if the anisotropy energy were sufficiently high compared to the thermal energy, so that it would prevent any orientational relaxation of the dipole moments in zero field. In that case, all the nanoparticle dipoles would point in the same direction along the same crystalline easy axis of magnetization. The microparticle dipole moment μ would then be equal to the number

N of embedded nanoparticles times the dipole moment μ

_{1} of a single nanoparticle: μ =

Nμ

_{1} (

Figure 2a).

A more realistic case is that the magnetic nanocrystals inside a microparticle are physically oriented at random. In zero field, the magnetization is then the vector sum of the nanoparticle dipoles with their various directions. Assuming that the easy axes of the nanoparticles are randomly oriented and that the dipoles have relaxed to the direction that is the closest to the previously applied field direction, the theoretical remanence in the field direction should on average be ½ of saturation magnetization (

Figure 2b). Taking the magnetic field treatment to be along the

x-axis, ½ is the average

x-component when the

x-components of randomly oriented unit vectors are suddenly all given a positive sign. Note that the direction of the net dipole of the microparticle in zero field does not have to be along the

x-axis. When the frame of reference is taken to be the direction of the net dipole of each microparticle (

Figure 2b), the average remanence typically tends to ⅔ (which we calculated by numerical simulation of the average vector sum of

N randomly oriented unit vectors with a positive

x-coordinate).

After a successful demagnetization treatment, the magnetization of the macroscopic sample as a whole should be completely negligible. On the scale of the individual microparticles, however, the situation is expected to be different. Even when all the nanoparticle dipoles inside one microparticle have random orientations, their vector sum is nonzero (

Figure 2c). Mathematically, the sum of

N randomly oriented unit vectors has a magnitude that is on average equal to the square root of

N. In other words, the ensemble of nanoparticle dipole vectors describes a random walk. Therefore, the minimal remanence is nonzero. The same effect applies to the individual microparticles and to the entire macroscopic sample, but the scales are different. Relative to the total number

N of nanoparticles present, the square root of

N is much, much smaller in the case of the macroscopic sample (with its 10

^{12} microparticles per 3 mL) than in the case of an individual microparticle (with only a few hundred nanoparticles). In this theory, the relative effect of demagnetization is inversely proportional to the square root of

N.

#### 2.2. Magnetic Content of the Composite Microspheres and Remanence on the Macroscopic Scale

The magnetization

M as a function of the applied magnetic field

H was measured for cobalt ferrite nanoparticles in dilute dispersion and for dry composite microparticles made from such cobalt ferrite nanoparticles (

Figure 3).

The average dipole moment and polydispersity of the nanoparticles can be calculated from the magnetization curve of the nanoparticles in dilute dispersion. The measured curve does not show hysteresis, allowing a polydisperse fit on the basis of the Langevin equation and a lognormal distribution [

16]. The Langevin equation describes how the magnetization

M depends on the magnetic field:

where M_{s} is the saturation magnetization,

and μ_{0} is the magnetic permeability of vacuum, μ is the magnetic dipole moment of a nanoparticle, H is the magnetic field, and k_{B}T is the thermal energy. For nanoparticles with a lognormal distribution and where interactions are negligible, the magnetization M at field H is given by

where

P(

μ) is the probability density function for μ.

Equation 4 was used for a theoretical fit of the magnetization curve in

Figure 3a to obtain the magnetic dipole moment of the individual nanoparticles. The magnetic dipole moment can be converted into an effective magnetic diameter

d_{m} via

where

m_{s} is the material-dependent saturation magnetization per unit volume, 240 kA/m for cobalt ferrite, which was obtained from the magnetization curve in

Figure 3a. This value for

m_{s} is much lower than the bulk value of 425 kA/m reported in the literature, probably due to the presence of a non-magnetic iron oxide layer [

17]. We assume that both the magnetic dipole moment and the magnetic diameter have a lognormal distribution:

where

P is the probability density function, μ

^{*} and

d_{m}^{*} are respectively the dipole moment and magnetic diameter at the maximum of the distribution, and the width of the distribution is described by

σ_{μ} = 3

σ_{d}. A fit of the data in

Figure 3a yields a mean magnetic dipole moment μ = 1.61 ×·10

^{−19} Am

^{2} and σ

_{μ} = 1.53, corresponding to

d_{m}= 11 nm and σ

_{d} = 0.51. The positions of the maxima of the distributions are calculated using

$\langle \mu \rangle ={\mu}^{*}\hspace{0.17em}exp[{\sigma}_{\mu}^{2}/2]$: μ

_{m}^{*} = 5.0·× 10

^{−2} Am

^{2} and

d_{m}^{*} = 9.7 nm. The lognormal distributions of the magnetic diameter and the probability density function of the magnetic moment are plotted in

Figure 4.

The magnetic content of the microspheres was estimated from the magnetization curve of the dry microspheres (

Figure 3b) and the bulk magnetization of cobalt ferrite: 42.1 mg of cobalt ferrite per gram of microspheres. Per silica microsphere, this corresponds to about 440 cobalt ferrite nanoparticles with a dipole moment of 1.61·× 10

^{−19} Am

^{2}. The likelihood that the nanoparticles are present in small clusters due to magnetic interactions can be estimated from the distribution in

Figure 4. The dimensionless dipolar contact interaction is given by

where

d is the particle diameter. About 13% of the nanoparticles is larger than 17 nm, corresponding to λ = 2, which is sufficient for nanoparticle aggregation [

18,

19]. This agrees with the presence of clusters in aqueous dispersions of cobalt ferrite nanoparticles of the type studied here as revealed by AC magnetic susceptibility measurements [

20].

Figure 3b indicates that the magnetic remanence of a dry sample of our microparticles is on the order of 30% after saturation magnetization treatment. This is significantly lower than the theoretically expected value of 50% (

Figure 2b). The reason is that part of the embedded nanoparticles shows relatively rapid Néel relaxation of the dipolar orientation inside the nanoparticles. An indication for the Néel relaxation rate of the nanoparticles is given by

where τ

_{0} is on the order of 10

^{−9} s,

K is the anisotropy constant, and

V_{m} is the magnetic volume of the nanoparticles. The precise value of the anisotropy constant

K for cobalt ferrite is not well known, with values of 120 kJ/m

^{3} [

21], 200 kJ/m

^{3} [

13,

22], 180–300 kJ/m

^{3} [

23] and 3150 kJ/m

^{3} [

22,

24] being quoted by different authors. For

K = 120 kJ/m

^{3}, cobalt ferrite particles smaller than 9.5 nm exhibit Néel relaxation with τ

_{N} < 100 s (see

Figure 5), so that they do not contribute to the remanence on the time scale of our measurements of the magnetization curves. Comparing this to the magnetic size distribution in

Figure 4 indicates that only about 60% of nanoparticles should contribute to the magnetic remanence in the magnetization curves of dry particles, in good agreement with 50% remanence expected without Néel relaxation and 30% remanence actually observed.

#### 2.4. AC Susceptibility Measurements

To study the magnetic remanence on the scale of the microparticles, the AC magnetic susceptibility was measured as a function of frequency while the particles were dispersed in ethanol. The susceptibility χ = χ′ − jχ″ consists of an in-phase “real” component χ′ and an out-of-phase “imaginary” component χ″, both of which are plotted in

Figure 8. Their dependence on the frequency

f is given by

where χ

_{0} is the low-frequency limit and

f_{char} is the characteristic frequency [

25]. Both components were numerically fitted jointly as a function of

f, assuming a lognormal distribution of

f_{char} [

12,

14]. The average characteristic frequency of 2.5 Hz is of the order expected for the Brownian rotation of particles with a diameter of 380 nm in ethanol (see

Equation 1, with η = 1.074 mPa s, the viscosity of ethanol at 25 °C [

12]). This indicates that magnetic relaxation of the sample requires rotation of the entire microparticles. In other words, the magnetic susceptibility is determined by the number of microparticles and their net permanent dipole moments, as opposed to being due to the Néel relaxation of the dipoles inside individual embedded nanoparticles [

12]. From the polydispersity of

f_{char}, a polydispersity of about 15% was calculated for the hydrodynamic radius, in agreement with a polydispersity of 18% from electron microscopy [

12]. The different spectra were obtained by magnetizing the same particles at different fields after an initial demagnetization treatment. The characteristic frequency is practically the same in all the spectra, indicating that the rotation remains that of single microparticles, as opposed to magnetic assemblies of particles, which would relax at much lower frequencies [

26].

To demagnetize the magnetic microparticles in colloidal dispersion, we froze the solvent of the dispersion using liquid nitrogen and rotated the frozen dispersion with respect to a magnetic field of fixed orientation but decreasing magnitude (see Section 3.2). During the demagnetization treatment, the nanoparticles respond individually to the magnetic field whereas the microparticles are unable to physically rotate in the frozen solvent. Whether or not the nanoparticles are present in microparticles does not affect the response of the nanoparticles. The susceptibility at 1 Hz after consecutive demagnetization or remagnetization treatments is shown in

Figure 9. The maximum susceptibility after magnetization treatment was a factor 7 higher than the minimum after demagnetization treatment, meaning that the dipole moment was higher by a factor of 2.7, since magnetic susceptibility scales with the square of the dipole moment. Intermediate values were obtained by magnetizing in fields lower than 400 kA/m, when the in-field magnetization does not yet saturate the sample at 77 K.

In principle, the microparticle dipole moment μ can be calculated from the low-frequency limit of the magnetic susceptibility, χ_{0}, and the number N of microparticles present per unit volume V, because

where μ

_{0} = 4π·× 10

^{−7} J A

^{−2}m

^{−1}. However, the dipole moment can be determined more reliably from the low-frequency limit of the magnetic susceptibility as a function of the amplitude

H of the applied alternating magnetic field, since this does not require precise knowledge of the concentration

N/

V [

12]:

where

L(α) and α are given by

Equations 2 and

3. Such data is presented in

Figure 10. The observation that the susceptibility does not increase but only decreases at increasing field amplitude is direct evidence of a permanent rather than an induced dipole moment of the microparticles [

12,

27]. The fits and the calculated dipole moments assume that the average alignment of the microparticle dipoles in an external magnetic field is given by the Langevin function [

12]. Assuming a lognormal distribution, the fitted polydispersity of the dipole moment was on the order of 30% after saturation magnetization treatment. This agrees with the 15% polydispersity in the microparticle radius and the fact that the nanoparticles are located in a spherical monolayer shell.

The AC susceptibility measurements are relatively insensitive to nanoparticles with rapid Néel relaxation, since χ depends on the square of the dipole moment (

Equation 12). The contribution due to Brownian rotation of a microparticle with a dipole moment μ

= N × μ

_{1}is proportional to

N^{2}, whereas the contribution of

N nanoparticles with a dipole moment μ

_{1} that respond individually by Néel relaxation is proportional to

N. Nevertheless, the AC susceptibility measurements do show evidence of Néel relaxation, be it on time scales of minutes to weeks: a slow decrease of the remanent magnetization (

Figure 11). The decrease in remanence with the logarithm of time is as expected for frozen ferrofluid spin glasses [

28,

29]. Two factors that affect the rate of decrease are the polydispersity in the nanoparticle dipole moments and the interactions between the dipoles.