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DOI: 10.1039/C9NR07425B
(Paper)
Nanoscale, 2019, Advance Article

Ekaterina A. Elfimova^{a},
Alexey O. Ivanov^{a} and
Philip J. Camp*^{ab}
^{a}Ural Federal University, 51 Lenin Avenue, 620000 Ekaterinburg, Russian Federation. E-mail: Ekaterina.Elfimova@urfu.ru; Alexey.Ivanov@urfu.ru; Fax: +7 343 389 9540; Tel: +7 343 389 9477 Tel: +7 343 389 9540
^{b}School of Chemistry, University of Edinburgh, David Brewster Road, Edinburgh EH9 3FJ, Scotland. E-mail: philip.camp@ed.ac.uk; Tel: +44 131 650 4763

Received
28th August 2019
, Accepted 30th October 2019

First published on 31st October 2019

The magnetization curve and initial susceptibility of immobilized superparamagnetic nanoparticles are studied using statistical–mechanical theory and Monte Carlo computer simulations. The nanoparticles are considered to be distributed randomly within an implicit solid matrix, but with the easy axes distributed according to particular textures: these are aligned parallel or perpendicular to an external magnetic field, or randomly distributed. The magnetic properties are calculated as functions of the magnetic crystallographic anisotropy barrier (measured with respect to the thermal energy by a parameter σ), and the Langevin susceptibility (related to the dipolar coupling constant and the volume fraction). It is shown that the initial susceptibility χ is independent of σ in the random case, an increasing function of σ in the parallel case, and a decreasing function of σ in the perpendicular case. Including particle–particle interactions enhances χ, and especially so in the parallel case. A first-order modified mean-field (MMF1) theory is accurate as compared to the simulation results, except in the parallel case with a large value of σ. These observations can be explained in terms of the range and strength of the (effective) interactions and correlations between particles, and the effects of the orientational degrees of freedom. The full magnetization curves show that a parallel texture enhances the magnetization, while a perpendicular texture suppresses it, with the effects growing with increasing σ. In the random case, while the initial response is independent of σ, the high-field magnetization decreases with increasing σ. These trends can be explained by the energy required to rotate the magnetic moments with respect to the easy axes.

The fundamental magnetic properties of single superparamagnetic and ferromagnetic nanoparticles have been studied in detail, including the composition and architecture of the particles, and the effects on the static and dynamic responses to applied magnetic fields.^{24–29} The effects of interactions between magnetic nanoparticles have been investigated experimentally^{30–32} and in computer simulations.^{33–37} The links between the basic magnetic properties – such as the dynamic magnetic susceptibility spectrum – and power dissipation^{38} have been explored in the context of medical applications, such as hyperthermia treatments.^{39–41} The effects of the carrier liquid on heat dissipation have also been investigated.^{42}

The effects of magnetic interactions on the bulk properties of magnetic liquids are well understood. In particular, the magnetization curve M(H) and the initial susceptibility χ = (∂M/∂H)_{H=0} of ferrofluids can be predicted accurately using statistical–mechanical theory,^{43–45} as tested against experimental measurements^{46} and computer simulations.^{47–49} In such systems, whether the particles are superparamagnetic or ferromagnetic is unimportant, as long as the particles are free to rotate.

In this work, the response of interacting superparamagnetic nanoparticles (SNPs) immobilized in a solid matrix to an applied magnetic field is studied using statistical–mechanical theory and computer simulations. Here, the SNPs are dispersed uniformly throughout the matrix, while the orientations of the easy axes are subjected to various types of texturing. The orientation of a nanoparticle's magnetic moment is assumed to display uniaxial anisotropy, meaning that there is only one easy axis of alignment. The magnetization curve and the initial susceptibility are therefore controlled by the energy barrier separating the two degenerate alignments of a nanoparticle's magnetic moment with respect to its easy axis, the interaction energy between dipoles and the field, and the interactions between dipoles on different particles. The latter two effects are strongly influenced by the direction and degree of alignment of the easy axes with respect to the applied magnetic field. Herein, parallel, perpendicular, unidirectional, and isotropic distributions are considered. The reason for these choices is that the easy axes can be aligned in a liquid precursor solution using a strong magnetic field before initiating a chemical reaction or physical process that solidifies the suspending medium. The probing field can then be applied at any angle with respect to the easy-axes. The isotropic distribution is, of course, the default situation without any field applied during synthesis. It will be shown theoretically that interactions and textures have huge effects on the magnetic response, and particularly on the magnitude of χ, which is of course anisotropic in the case of the easy axes being aligned. Interactions can only be treated in an approximate manner, and in this work, the first-order modified mean-field (MMF1) approach will be exploited.^{43–45} The role of magnetic interactions between particles will nonetheless be shown to be substantial, and the accuracy of this approach will be demonstrated by comparison with Monte Carlo (MC) simulations. This type of system has been studied before. Carrey et al. studied the dynamic response of immobilized SNPs with parallel and isotropic distributions of the easy axes, using Stoner-Wohlfarth models and linear-response theory.^{50} Elrefai et al. established empirical expressions for the magnetization curves of immobilized non-interacting SNPs by fitting to numerical simulations, and then compared the results to experimental data.^{51} The novelty of the current work is that the static magnetic properties of immobilized SNPs are expressed in analytical form, and with interactions taken into account according to systematic statistical–mechanical theory.

The rest of the article is organized as follows. The essential features of SNPs, and the particle model adopted in this work, are defined in section 2. The statistical–mechanical framework of the theory is outlined in section 3, and the application to immobilized and orientationally textured systems is detailed in section 4. The MC simulations are described in section 5. The results are presented in section 6, in the form of direct comparisons between theory and simulation for various cases of orientational texture. The conclusions are presented in section 7.

The next issue is the orientation of the magnetic moment m inside the body of a particle. The Brownian translations and rotations of immobilized particles are suppressed, and so the magnetic moment can vary only by superparamagnetic fluctuations (the Néel mechanism). In the simplest case, the crystalline structure of the magnetic material has only one axis of easy magnetization (uni-axis magnetization). Therefore, the orientation of the particle is defined by the direction of the magnetic easy axis, denoted by the vector n; see Fig. 1(c). The magnetic moment of a particle has two degenerate ground-state directions, these being parallel and anti-parallel to the easy axis. The potential energy U_{N} as a function of the angle between m and n is shown schematically in Fig. 2. The energy barrier is proportional to the volume of the magnetic core v_{m} = πx^{3}/6, and the magnetic crystallographic anisotropy constant K, a material property. For common nanosized particles, the barrier (Kv_{m}) may be comparable to the thermal energy, and so thermal fluctuations result in stochastic reorientations of the magnetic moment. The mean value of the particle magnetic moment, measured over a long time, will be equal to zero. This behaviour is known as Néel superparamagnetism, and it is a characteristic of nanosized particles only. Superparamagnetic fluctuations are commonly described as the thermally activated rotations of the magnetic moment inside the particle magnetic core. Importantly, this mechanism means that even if particle positions and orientations (easy axes) are frozen, the magnetic moments are still able to rotate, subject to the potential energy U_{N}, and the interactions with the field and other magnetic moments.

Putting all of this together, the total potential energy of a configuration of N identical SNPs can be written in the form

(1) |

U_{N}(i) = −Kv_{m}(_{i}·_{i})^{2}
| (2) |

U_{m}(i) = −μ_{0}(m_{i}·H) = −μ_{0}mH(_{i}·ĥ)
| (3) |

(4) |

(5) |

(6) |

(7) |

The essential point here is that the magnetic response of immobilized particles is dictated by the internal rotation of the magnetic moments within the particles, rather than by the Brownian rotation of the particles. In sections 3 and 4, the magnetization curve and initial susceptibility will be calculated for systems of particles with various types of orientational distributions of the easy axes (). These results will be compared with those for ferrofluids, which will highlight the effects of the textures. The sample geometries and textures studied in this work are illustrated in Fig. 3. Fig. 3(a) represents the case of a ferrofluid, where the particles translate and rotate under the influence of Brownian forces, and the particle–particle and particle–field interactions. Fig. 3(b) shows an immobilized system, where the easy axes are aligned, and the particle positions are random. Fig. 3(c) shows an immobilized system in which the particle positions and easy axes are distributed randomly.

In all cases, the sample container is taken to be a highly elongated cylinder aligned along the laboratory z axis, and the applied magnetic field H = H(0, 0, 1) is in the same direction. This means that demagnetization effects can be neglected, and the internal magnetic field can be taken to be the same as the external applied field H. The centre-of-mass position of a particle is the radius vector r_{i} = r_{i}_{i}, where _{i} = (sinθ_{i}cosϕ_{i}, sinθ_{i}sinϕ_{i}, cosθ_{i}), θ_{i} is the polar angle with respect to the laboratory z axis, and ϕ_{i} is the azimuthal angle with respect to the laboratory x axis. The orientation (easy axis) of a particle is the unit vector _{i} = (sinξ_{i}cosψ_{i}, sinξ_{i}sinψ_{i}, cosξ_{i}), where ξ_{i} and ψ_{i} are, respectively, the polar and azimuthal angles in the laboratory frame. The magnetic moment on a particle is m_{i} = m_{i}, where _{i} = (sinω_{i}cosζ_{i}, sinω_{i}sinζ_{i}, cosω_{i}), and ω_{i} and ζ_{i} are, respectively, the polar and azimuthal angles in the body-fixed frame of the particle. These vectors are shown in Fig. 1(c). Now the problem is to study the magnetic properties of a system of N particles in a container with volume V at temperature T. The particle concentration ρ = N/V can be expressed in the dimensionless form ρd^{3}, or converted into the hard-sphere volume fraction φ = πρd^{3}/6.

(8) |

The integration over the unit vector _{i} is defined as

(9) |

(10) |

(11) |

Q is the partition function, given by the integral of the Boltzmann factor exp(−U/k_{B}T) over the degrees of freedom for all N particles. Differentiating eqn (11) with respect to _{1} gives

(12) |

(13) |

It is only the last term in eqn (12) that describes the interparticle correlations. In the limit of low concentration ρ → 0, the system becomes an ideal paramagnetic gas of non-interacting particles. Omitting the correlation term, the ideal one-particle probability density W_{0}(1) is then the solution of

(14) |

(15) |

The next step is to identify the effects of interparticle correlations, represented by the second term in eqn (12). It contains factors of concentration ρ and U_{d}/k_{B}T ∼ λ, in addition to the dependence of g_{2}(1, 2) on those variables. The following development is limited to the regime of low concentration (ρd^{3}, φ ≪ 1), and weak-to-moderate interactions (λ ∼ 1). The leading-order correction to eqn (12) is of order φλ, and can be separated out by neglecting the concentration dependence of the pair correlation function, and writing it as a product of two one-particle distribution functions:

g_{2}(1, 2) = W(1)W_{0}(2)Θ(1, 2) + (φλ).
| (16) |

Θ(1, 2) = exp[−U_{HS}(1, 2)/k_{B}T] is the Heaviside step-function, describing the impenetrability of two particles. Combining eqn (12) and (16) gives

(17) |

(18) |

The solution of eqn (17) is then the one-particle distribution function

(19) |

Comparing this result with the corresponding equation for the ideal paramagnetic system (15) makes the meaning of −U_{eff}(1)k_{B}T absolutely clear: it represents the average interaction energy between particle 1 and the effective magnetic field produced by the N − 1 other particles in the system. As a result, this theoretical approach is called the first-order modified mean-field (MMF1) theory.^{43–45}

(20) |

Here the subscript z indicates the z components of the corresponding vectors. The last expression can be written in the succinct form

U_{eff}(1) = (_{1}·G)
| (21) |

(22a) |

(22b) |

(22c) |

Here χ_{L} is the Langevin initial susceptibility

(23) |

(24) |

(25) |

M = M_{∞}L(α_{eff}),
| (26a) |

(26b) |

These expressions are valid for infinitely soft magnetic nanoparticles irrespective of whether they are suspended in a liquid and may translate or rotate freely, or they are immobilized in some rigid matrix. The only requirement is that the spatial distribution of particles inside the sample is uniform, i.e., no extensive self-assembly induced by magnetic or other colloidal forces takes place. The expressions in eqn (26) are coincident with the MMF1 predictions developed earlier for fluids of spherical particles with central, fixed, point dipoles.^{44} This equivalence is discussed further in section 4.

(27) |

(28) |

(29) |

This is precisely the same as the result obtained for soft magnetic nanoparticles in section 3. Therefore, eqn (26) holds true for ferrofluids, and the static (equilibrium) magnetization of a ferrofluid is influenced only by _{i}. It means that the easy axes of the particles, at equilibrium, adopt a favourable orientational distribution for a given applied external field due to Brownian rotation. As a result, the static magnetic properties of a fluid suspension of SNPs are independent of the height of the Néel energy barrier σ.

The MMF1 prediction (26) and its second-order correction (MMF2) were obtained almost twenty years ago for dipolar fluids that correspond to magnetically hard ferroparticles, with σ ≫ 1.^{43} Nonetheless, the MMF approach describes the static magnetic properties of real ferrofluids containing SNPs rather accurately,^{46} because the Brownian rotation means that the easy axes cannot influence the equilibrium distribution of the magnetic moments. The same MMF1 results also apply to soft magnetic particles (σ → 0) because the Néel rotation of the magnetic moments is unhindered. The static magnetic properties of dipolar fluids have been well studied by means of computer simulations (both MC and molecular dynamics), and the high accuracy of the MMF1 expressions has been demonstrated over the range χ_{L} ≤ 3.^{46,48} Higher-order corrections for treating concentrated ferrofluids at low temperatures have also been derived.^{47,49}

(30) |

(31) |

Note that R_{1}(α, 0) = sinh(α)/α. By symmetry, G_{x} = G_{y} = 0, and the z component is

(32) |

(33) |

(34) |

Note that R_{2}(α, 0) = L(α)sinh(α)/α. Substituting these expressions into eqn (21) and (19), gives for the magnetization

(35) |

(36) |

(37) |

(38) |

Note that R_{1}(0, σ) = R(σ), and the function A_{‖}(σ) coincides with the corresponding value introduced by Raikher and Shliomis.^{54} For magnetically soft particles, A_{‖}(0) = 1, and then eqn (35) and (37) coincide with (26). The limit of magnetically hard particles (σ → ∞) gives A_{‖} → 3 and the largest value of the initial magnetic susceptibility, χ_{‖} → 3χ_{L}(1 + χ_{L}). This limit is worth mentioning because it corresponds to the case of Ising particles, the magnetic moments of which are quantized in only two states: _{i} = ±1. The magnetization (35) in this limit becomes

M_{‖} → M_{∞}tanh(α + χ_{L}tanhα)
| (39) |

Fig. 4 Static magnetization curves for immobilized particles with χ_{L} = 2, and with parallel (a) and perpendicular (b) textures of the magnetic easy axes. The results are plotted as the reduced magnetization M/M_{∞} as a function of the dimensionless magnetic field strength (Langevin parameter) α. The dashed lines are for non-interacting (NI) particles, and the solid lines are the theoretical predictions for interacting particles according to eqn (35) (a) and eqn (44) (b). The relative anisotropy energies are σ = 0 (black), 3 (red), and 10 (green). |

(40) |

(41) |

Note that R_{3}(α, 0) = R_{1}(α, 0) = sinh(α)/α. By symmetry, G_{x} = G_{y} = 0, and the z component is

(42) |

(43) |

Note that R_{4}(α, 0) = R_{2}(α, 0) = L(α)sinh(α)/α. Here I_{0}(z) and I_{1}(z) are the modified Bessel functions of zero and first orders, respectively. Following the same development as in the parallel-texture case, the magnetization in the perpendicular case is

(44) |

(45) |

(46) |

(47) |

For magnetically soft particles, A_{⊥}(0) = 1, and eqn (44) and (46) coincide with (26). For magnetically hard particles, A_{⊥}(∞) = 0, and χ_{⊥} = 0.

Similar to the parallel-texture case, the interparticle interactions lead to an increase in the magnetization, as shown in Fig. 4(b). But the growth of the anisotropy parameter results in the opposite effect, because for higher barriers, a stronger magnetic field is required to rotate the magnetic moment away from the easy axis. Hence, the magnetization is a decreasing function of σ in this case.

(48) |

Here

(49) |

with the special cases R_{5}(α, 0, ξ) = R_{1}(α, 0) = sinh(α)/α, R_{5}(α, σ, 0) = R_{1}(α, σ), and R_{5}(α, σ, π/2) = R_{3}(α, σ). The x and y components of G are in general non-zero, and complicated, but they do not affect the magnetization, which is in the z direction. The z component is

(50) |

(51) |

and R_{6}(α, 0, ξ) = R_{2}(α, 0) = L(α)sinh(α)/α. For the arbitrary angle ξ, the z component of the magnetization is given by

(52) |

(53) |

The initial susceptibility is

(54) |

(55) |

It is interesting that there is a magic angle at which the coefficient A_{ξ}(σ, ξ_{0}) = 1 and is hence independent of σ. At this angle, the initial susceptibility of immobilized SNPs is given by the soft magnetic particle/ferrofluid expression in eqn (26b).

(56) |

(57) |

To calculate the initial susceptibility, it is necessary to first linearize the effective Langevin parameter with respect to the bare Langevin parameter. For small values of α,

(58) |

Therefore, the effective field is independent of the anisotropy parameter σ, and the initial susceptibility is equal to the usual MMF1 expression for soft magnetic particles and ferrofluids:

(59) |

Typical magnetization curves are shown in Fig. 5 for both interacting and non-interacting particles with χ_{L} = 2. It is clear that the initial linear response of the magnetization is independent of the anisotropy parameter σ. But the approach to the saturation magnetization is much slower with a large value of σ, and as in all of the preceding cases, interactions increase the magnetization for a given field strength.

Fig. 5 Static magnetization curves for immobilized particles with χ_{L} = 2, and with random orientations of the magnetic easy axes. The results are plotted as the reduced magnetization M/M_{∞} as a function of the dimensionless magnetic field strength (Langevin parameter) α. The dashed lines are for non-interacting particles, and the solid lines are the theoretical predictions for interacting particles according to eqn (56). The relative anisotropy energies are σ = 0 (black), 5 (red), and 10 (green). |

The initial susceptibility in the x direction was computed using the fluctuation formula

(60) |

The magnetization curves of dilute systems with various textures are shown in Fig. 7. With σ = 0, the magnetization curves are of course coincident for all textures. With σ = 10, the magnetization for the parallel texture grows rapidly with the applied field, while the magnetization remains low for the perpendicular texture. This is obviously consistent with the initial-susceptibility results presented in Fig. 6. For the random distribution, the initial slope is the same as that for the σ = 0 case, since the initial susceptibility is independent of σ, but the high-field behaviour is different due to the energetic cost of rotating the magnetic moments with respect to the easy axes; this effect was demonstrated already in Fig. 5. In all cases, the effects of interactions are weak, but they are nonetheless described well by the MMF1 theory, eqn (35), (44), and (56). Note that results are shown for σ = 10, but the behaviour of the magnetization curves is typical. As demonstrated in Fig. 4 and 5, the magnetization increases with increasing σ in the parallel case, and decreases in the random and perpendicular cases.

The initial susceptibilities of systems with φ = 0.05 and λ = 1 are shown in Fig. 8. The qualitative behaviour is no different from that of the more-dilute system, but the effects of interactions are more pronounced in this case. For the random and perpendicular textures, the MMF1 theory gives an excellent account of the interactions, with practically no deviation from the MC data. But for the parallel texture, there is a surprising effect: for non-interacting particles, the MC data agree exactly with the theory over the whole range of σ, showing that the flip algorithm is working as intended; but at the same time, the MMF1 susceptibility of interacting particles (37) appears to be valid only for low-to-moderate values of σ. Here, the MC susceptibility increases with σ more rapidly than the prediction of the model (filled red squares and solid red line). Moreover, with high values of σ, the susceptibility of interacting particles is about forty percent larger than that of non-interacting particles, despite the system being only weakly magnetic, with a Langevin susceptibility χ_{L} = 0.40. Increasing the concentration further does not change these trends (data not shown): the model (37) agrees well with MC data with low values of σ, but it underestimates the simulated susceptibility with large values of σ.

The corresponding magnetization curves for systems with σ = 0 and σ = 10 are shown in Fig. 9. On the whole, the agreement between theory and simulation is good: the effects of texture and the interactions are captured well by the theory. Qualitatively, the trends are the same as those discussed in connection with Fig. 7, but with the increased interactions giving a greater enhancement of the magnetization for a given texture and magnetic-field strength.

The comparison with simulation shows that the MMF1 theory is accurate at least for χ_{L} ≤ 0.40. In many biomedical applications, the volume fractions of magnetic material may be an order of magnitude smaller than those considered here. For instance, if φ ∼ 10^{–3}, then with λ = 1, χ_{L} ∼ 10^{–2}. The effects of interactions can be assessed using the initial magnetic susceptibility (χ, with interactions) divided by the ideal susceptibility (χ_{ideal}, without interactions). For the random texture, this ratio is simply

(61) |

and it is independent of σ. Taking this texture as a guide, enhancements of around 10% are to be expected when χ_{L} is about 0.3. Fig. 10 shows the ratios for parallel and perpendicular textures, and with χ_{L} = 0.01, 0.10, 0.16, and 0.40. Fig. 10(a) shows that over the range 0 ≤ σ ≤ 20, interactions within the parallel texture enhance the initial magnetic susceptibility by less than 1% with χ_{L} = 0.01, 3.3–9.5% with χ_{L} = 0.10, 5.3–15% with χ_{L} = 0.16, and 13–38% with χ_{L} = 0.40. With the perpendicular texture, the enhancements for σ = 0 are the same as with the random and parallel textures, and they decrease with increasing σ. The effects of interactions on χ are obviously mirrored in the initial, linear portion of the magnetization curve (not shown), but the effects on the magnetization decrease with increasing field strength due to the field–particle interactions overcoming the particle–particle interactions.

Summing up this section, a comparison of theoretical and simulation results shows that the effects of interactions on the initial static magnetic response of immobilized SNPs are much stronger when the easy axes are aligned parallel with the external field direction, than when they are randomly distributed or perpendicular to the field. While the MMF1 theory (accurate to leading order in the Langevin susceptibility χ_{L}) gives excellent predictions in the random and perpendicular cases, it is only accurate in the parallel case when the magnetic crystallographic anisotropy barrier Kv_{m} is not too large compared to the thermal energy k_{B}T (σ ∼ 1). This last condition means that the magnetic moment is not ‘blocked’ inside the body of the particle.

With large energy barriers (σ ≫ 1), the magnetic moments in the parallel texture appear to be strongly correlated, which results in a strong enhancement of the magnetic response, and especially the initial susceptibility. This can be explained in terms of the effect of orientational averaging on the range and strength of the (effective) interactions and correlations between the particles. With low-to-moderate values of σ, the superparamagnetic rotation is not blocked, and the orientational averaging produces an effective interaction between particles that is short ranged (∼−1/r_{ij}^{6}). Hence, the resulting correlations are weak. With large values of σ, all of the particle magnetic moments are approximately (anti-)parallel to one another, and hence the dipolar interactions and the resulting correlations are long-ranged (∼1/r_{ij}^{3}). Here, the interactions between particles are evaluated on the basis of two-particle correlations (17), and the many-body contributions to the pair correlation function should be included to improve the accuracy of the theory.

Orientational averaging also explains the relatively weak effects of interactions on the initial susceptibility with random and perpendicular textures. In these cases, the effective interactions are short-ranged, due to the azimuthal rotations of the magnetic moments in the perpendicular case, and the isotropic distribution of easy axes in the random case. Hence, the orientational correlations and the enhancement of the initial susceptibility are weak.

The initial susceptibility χ was found to depend on σ in very different ways, depending on the texture. With a random distribution, χ is independent of σ. With a parallel texture, χ increases with increasing σ, while with a perpendicular texture, χ decreases. In all cases, including interactions between particles leads to an enhancement of χ, but the enhancement is much stronger for the parallel texture than for the random and perpendicular textures. The MMF1 theory is accurate for the random and perpendicular cases with all values of σ, but for the parallel case, it is only reliable with small values of σ. All of these effects can be explained in terms of the effective interactions between the particles, after taking into account orientational averaging of the magnetic moments. When the magnetic moments are blocked and aligned parallel with the external magnetic field, the correlations that control the initial susceptibility are strong and long-ranged. The susceptibility in the random and perpendicular cases remains relatively low because of the possibility of orientational averaging, which renders the correlations short-ranged. Qualitatively, the theory captures all of the main effects of textures and interactions on the initial susceptibility.

The magnetization curves show several interesting features. Although the initial susceptibility of the random texture does not depend on σ, the high-field behaviour does, with the magnetization decreasing with increasing σ. This is due to the increasing energetic cost of rotating the magnetic moments with respect to the easy axes. The magnetization is strongly enhanced by a parallel texture, due to the alignment of the magnetic moments with the easy axes and the field. In contrast, the magnetization is strongly suppressed by a perpendicular texture, as it is restrained by the easy axes. The agreement between the MMF1 theory and MC simulation data is generally good, as the particle–field interaction energy becomes at least as significant as the particle–particle interaction energy.

The basic magnetic properties of immobilized SNPs are becoming increasingly important, due to the development of magnetic gels, elastomers, rubbers, glasses, etc. This work represents a significant step towards a detailed quantitative description of this technologically important class of functional materials.

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