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Experimentally, it was found that in the case of ferro‐ or ferrimagnetic crystalline materials, there is a dependence of the magnetization of the single crystal on the crystallographic directions. The dependence of the magnetization of the crystalline magnetic material on the crystallographic axes determines the magnetocrystalline anisotropy (Kneller 1962; Caizer 2004a, 2019). Thus, the magnetization curves that are obtained in the same external magnetic field depend on the direction in which the crystalline material is magnetized. This type of magnetic anisotropy is characteristic of all bulk single crystalline (ferro‐ or ferromagnetic) magnetic materials (Fe, Co, Ni, Cd, their alloys, Fe oxides [Fe₃O₄, γ‐Fe₂O₃], etc.).

For example, in the case of the Ni bulk single crystal ferromagnetic material, which crystallizes in the cube system with centered volume (vcc), the magnetization curves have the shape shown in Figure 1.8 (Baberschke 2001). Thus, the magnetization is made easiest following the crystallographic direction [1,1,1], which represents the large diagonal of the cube, and its magnetization is made the hardest following the crystallographic direction [1,0,0], which is the edge of the cube. The direction [1,1,1] in this case is called the direction or axis of easy magnetization (e.m.a.), and the direction [1,0,0] is called the axis of hard magnetization (h.m.a).

Figure 1.8 (a) The crystallographic systems for Ni‐single crystal.

Source: Caizer (2016). Reprinted by permission from Springer Nature;

(b) Room temperature magnetization curves for Ni along the easy ([111]) and hard ([100]) direction.

Source: Based on Wijn (1986).

In the case of bulk ferromagnetic monocrystalline material with cubic symmetry, the energy of magnetocrystalline anisotropy can be determined with the following formula:

(1.15)

written as a series development of powers using the model proposed by Beker, Doring, Akulov, Mason (Kneller 1962; Herpin 1968), based on the symmetry properties of the crystal. In general, it was found that it is sufficient to use only the first two terms of development, in K₁ and K₂. In Eq. (1.15), K₁ and K₂ are the magnetocystalline anisotopy constants, and α₁, α₂ and α₃ are the cosine directors of the vector of spontaneous magnetization (M_s) in relation to the main crystallographic axes of the cube. In some cases, even the first term of development is sufficient. For example, in the case of the ferromagnetic Fe single crystal, K₁ = 4.8 × 10⁴ J m⁻³ and K₂ = 5 × 10³ J m⁻³ were found, where K₁ is in this case with approximately one order of magnitude larger than K₂.

In the case of uniaxial, hexagonal symmetry, as in the case of the ferromagnetic Co monocrystalline (Figure 1.9), the magnetocrystalline anisotropy energy is expressed as a function of the angle fi between the spontaneous magnetization vector M_s and the main axis of symmetry:

(1.16)

Figure 1.9 The crystallographic systems for Co‐single crystal.

Source: Caizer (2016). Reprinted by permission from Springer Nature.

Also, in this case, the first two terms are used (in K₁ and K₂) in the energy expression of uniaxial magnetocrystalline anisotropy.

In this case, the main axis of symmetry is the easy magnetization axis (e.m.a), and the direction perpendicular to it is the hard magnetization axis (h.m.a).

In the case of bulk magnetic material, there is another important form of magnetic anisotropy, which should not be neglected as it can become dominant in some cases. This is the anisotropy of the shape (Kneller 1962; Caizer 2004a), which shows that the magnetization of a sample depends on its shape.

As a general case, approximating the shape of the sample by the ellipsoid of revolution: a > b = c, where a, b, and c are the semiaxes of the ellipsoid (Figure 1.10), the anisotropy energy due to the shape of the sample is expressed by the following formula:

(1.17)

where N_a and N_b are the demagnetization factors along the a and b directions of the ellipsoid, and θ is the angle that the spontaneous magnetization vector M_s makes with the main axis (a) of the ellipsoid.

Figure 1.10 The crystal approximated by an ellipsoid (general case).

Source: Caizer (2019). Reprinted by permission of Taylor & Francis Ltd.

When the magnetization of the ellipsoid in the external magnetic field is done along the direction of the a‐axis, the shape anisotropy energy is minimum (or zero). In contrast, when magnetization is done along a direction perpendicular to the a‐axis (e.g. the b, c, or other directions), the shape anisotropy energy is maximum. In the latter case, taking into account the Eqs (1.17) and (1.16), the shape anisotropy constant can be expressed by the following formula:

(1.18)

in the approximation of the first order.

When the magnetic material is reduced to the nanoscale, these forms of magnetic anisotropy remain valid. In addition, in the case of magnetic nanoparticles, the shape anisotropy becomes very important, reaching in some cases even larger, or much larger than the magnetocrystalline anisotropy. Thus, the neglect of this first aspect leads to important errors from a magnetic point of view, incompatible with the physical reality. For example, if the nanoparticle is spherical in shape (Figure 1.5), the semiaxes a, b, c become equal (Figure 1.10), and equal to the radius of the sphere. According to Eqs. the shape anisotropy constant in this case is zero, as is the energy. So there is no shape anisotropy in the case of spherical nanoparticles. In contrast, in the case of elongated nanoparticles, when a ≫ b = c, and when they are soft magnetic (magnetocrystalline anisotropy is reduced), the shape anisotropy exceeds the magnetocrystalline anisotropy, or even becomes dominant. This is an important aspect in the case of magnetic nanoparticles that must be taken into account not only in practical applications, including biomedical ones, but also in theoretical calculations and models/experiments.

Moreover, in the case of nanoparticles, generally in the field of nanometers, when the volume‐to‐surface ratio of spherical nanoparticles increases: S/V ~ 1/D, D is the diameter of the nanoparticle, a new form of anisotopy appears which must be taken into account, namely: surface anisotropy (Caizer 2019). This is because in the case of small and very small nanoparticles, and for soft magnetic materials such as ferrites (e.g. Fe₃O₄, γ‐Fe₂O₃, Ni–ZnFe₂O₄, MnFe₂O₄), this form of anisotropy becomes dominant, sometimes much more larger than the magnetocrystalline (Caizer 2004b). This form of anisotropy results from the surface effects that occur in the case of small nanoparticles where the surface spines have an important contribution, the symmetry of the bonds with the nearest neighbors being different from that inside the core of nanoparticle. These aspects were first highlighted by Néel (1954) who showed that in the case of crystals with cubic symmetries and for surfaces of type (111) or (100) the surface anisotropy energy can be written as follows:

(1.19)

where B is the angle that the spontaneous magnetization vector M_s makes with the direction of the external normal to the considered surface (Figure 1.11), and K_s is the surface anisotropy constant (expressed in J m⁻²).

Figure 1.11 The orientation of spontaneous magnetization relative to normal on the surface (a) (100) for the monocrystal with cubic symmetry.

Source: Caizer (2019). Reprinted by permission of Taylor & Francis Ltd.

For example (Caizer 2004a), in the case of spherical nanoparticles with a diameter D of 10 nm, the value of ~6 × 10³ expressed in J m⁻³ is obtained for the surface anisotropy constant. This value is five times higher than the magnetocrystalline anisotropy of the Ni–Zn ferrite, which is 1.5 × 10³ J m⁻³ (Broese Van Groenou et al. 1967). Therefore, this form of magnetic anisotropy must be considered in the case of magnetic nanoparticles.

Moreover, in the process of abstaining from nanoparticles, as a result of physico‐chemical methods of preparation, or when the nanoparticles are surfacted or embedded in different solid matrices, elastic stresses can occur which induces an additional magnetic anisotropy (stress anisotropy) compared to those above. And this anisotropy, in the case of nanoparticles, can become large or high compared to magnetocrystalline anisotropy, and it must be taken into account when it appears. Example Coey (Coey and Khalafalla 1972) obtains a value of 1.2 × 10⁵ J m⁻³ for nanoparticles of 6.5 nm in diameter and Vassiliou et al. (1993) obtains the value of the anisotropy constant of 4.4 × 10⁵ J m⁻³, values that are approximately twice higher in magnitude than the magnetocrystalline anisotropy constant of the α‐Fe₂O₃ massive ferrite (K₁ = 4.6 × 10³ J m⁻³).

To conclude, in the case of magnetic nanoparticles, a magnetic anisotropy determined by magnetocrystalline anisotropy, shape anisotropy, surface anisotropy, and induced anisotropy must be considered:

(1.20)

and an effective magnetic anisotropy constant

(1.21)

respectively.

Typically, in the case of magnetic nanoparticles, this effective anisotropy constant increases when the magnetic nanoparticles become smaller, generally below 15–20 nm, depending on the nature of the material (Figure 1.12).

Figure 1.12 (a) Schematic view of the general spin canting geometry (the core@shell model). (b) Theoretical M_s (blue solid line) and K_eff (orange solid line) versus magnetite nanoparticle diameter D at 300 K. Horizontal dashed–dotted lines exhibit M_sb and K_b, respectively. Theoretical data are compared with experimental ones. Blue diamonds from Abbas et al. (2013), blue hexagons from Goya et al. (2003), orange triangle from Guardia et al. (2007), orange squares from Vargas et al. (2008), orange pentagon from Ferguson et al. (2011), and orange circle from Park et al. (2004). Note: 1 J (m⁻³T)⁻¹ = 1 A m⁻¹.

Source: Wu et al. (2017). Reprinted by permission of IOP Publishing.

The Figure 1.12b also shows the variation of the saturation magnetization (see Section 1.1.4) when the diameter of the nanoparticles decreases, this becoming smaller when the size of the nanoparticles decreases.

Moreover, in the case of magnetic core‐shell nanoparticles, the presence of a unidirectional anisotropy (Nogues et al. 2005; Caizer 2019) has recently been highlighted as a result of the coupling between neighboring layers (surface‐layer core) with different magnetic orders of magnetic moments in the network: ferromagnetic core (FM) and antiferomagnetic shell (AFM) (Figure 1.12). Also, in Ref. (Berkowitz and Kodama 2006) a review of the unidirectional (exchange) anisotropy for different FM‐AFM nanostructures may be found. In the case of CoFe₂O₄/NiO ferrimagnetic/antiferromagnetic nanocomposites, a similar behavior was found (Peddis et al. 2009) (Figure 1.13).

Figure 1.13 (a) Schematic drawing of a core‐shell structure and (b) transmission electron microscopy (TEM) image of an oxidized Co particle.

Source: Reprinted from Nogues et al. (2005), with permission from Elsevier.

Such situations may occur frequently in the case of different more complex magnetic nanostructures which are currently developed in nanotechnology and bionanotechnology for various applications.