#### Abstract

In terms of the two-phase nanoparticles model, the effect of mechanical stress on the magnetic state of both uniaxial and multiaxial heterophase magnetic is investigated. The spectrum of critical fields of reversal of phases' magnetic moments was calculated and phase diagrams were drawn to assess the effect of mechanical stress on the degree of metastability of two-phase nanoparticles' magnetic states. By the example of epitaxial cobalt-coated -Fe_{2}O_{3} particles, a theoretical analysis of the effect of uniaxial mechanical stress on the magnetization of a system of noninteracting heterophase nanoparticles is investigated. It was shown that tension reduced and compression increased coercive force , while the residual saturation magnetization was not changed under the influence of mechanical stress.

#### 1. Introduction

One of the factors that influence the magnetization is the heterophase of nanoparticles determining magnetic properties of nanodispersed materials. This heterophase may be specified by various factors. Thus, according to some authors (see, e.g., [1–5]), the formation of adjacent magnetic phases in natural magnetic materials is associated with oxidation or decomposition processes of a solid solution occurring in magnetic-ordered grains. For example, titanomagnetite being the main carriers of magnetic properties of natural materials can be exposed to decay [3] or oxidation [4, 5] processes. As a result of titanomagnetite collapse in nanoparticles, the extraction of both titanium depleted and enriched phases ultimately leads to a change in magnetic properties of a natural material. As an example of multiphase artificial magnetic materials can be nanoparticles with core/shell structure. Currently there are many different types of core/shell nanoparticles, such as metal/metal [6–11], nonmetal/metal [12–14], metal/polymer [15, 16], nonmetal/nonmetal, polymer/nonmetal, and polymer/polymer [17]. Development of technology of core/shell nanoparticles allowed for obtaining stable particles of nanometer scale [18–20], the particle size of 1–100 nm in high demand in many areas, particularly important to its application in medicine [21] and the production of memory storage [22]. Core/shell nanoparticles containing metals (Fe, Co, gold, platinum, palladium, Mn, Zn, etc.) are used in the manufacture of materials for storage (magnetic memory) and creating the design of microwave and electromagnetic devices [23–25] as well as creating medicines [26], drug delivery [27–29], oncology [30], and biomedicine [31, 32]. Polymer and nonmetallic core/shell nanoparticles are a new class of materials for electronics, for example, the organic light-emitting diodes (OLEDs), organic photovoltaics (OPVs), sensors, and organic field effect transistors (OFETs) [33–35]. Carriers of magnetic memory elements’ magnetic properties may be represented by iron particles [36–38] exposed to surface oxidation or magnetic nanoparticles coated with other material, such as cobalt-coated -Fe_{2}O_{3} particles [39–41]. According to Mössbauer spectroscopy [36] oxidized coating of iron particles is a mixture of -Fe_{2}O_{3} and Fe_{3}O_{4}, while the central part is composed of -Fe. The thickness of the coating may be around 10% of the radius of the particles, and the amount of phase composed of iron oxides is more than 50%, according to other reports [37]. These particles can be either cubic, spherical [42, 43], or very extended [37]. Note that magnetic properties of these nanoparticles can change significantly once they transit to heterophase state. Thus, an epitaxial cobalt coating of -Fe_{2}O_{3} particles increases coercive force [41] that is associated by the authors [44, 45] with the interfacial exchange interaction. Interfacial exchange interaction can change not only the coercivity but also other magnetic characteristics [9, 46]. Amorphous magnetic materials obtained in a special way may also refer to multiphase ones. The process of phase formation in such magnets associates with amorphous material transition from metastable to equilibrium state (see, e.g., [47–49]). Nanocrystalline alloys’ magnetic matrix that contains grains of other ferromagnets is of special interest here. Exchange interaction between the matrix and grains leads to collective phenomena, which causes a high performance of both magnetically soft and hard materials (e.g., Fe, Cu, Nb, Si, and B and Nd, Fe, and B, resp.). Along with the exchange interaction, there is mechanical stress at the interphase interface that is able to alter magnetic states of both phases due to size differences of neighboring phases’ lattices. For example, the transformation in iron and its alloys → is accompanied by a grain extension (1.5%) and, as a consequence, by significant deformations. However, the heterogeneity of the interphase interface can lead to a significant reduction of internal stress associated with the → transformation [50]. It should be noted that the mechanical stresses can have a significant effect on the magnetic properties of nanoparticles [51–55].

Rather detailed theoretical study of magnetic states and magnetization processes of the system of uniaxial two-phase particles is presented in the works [39, 46, 51, 54, 56–59]. In this paper, we attempt to expand the model [58, 59] to multiaxis dual-phase particles and studies of the effect of mechanical stress on the magnetic states and magnetic properties of systems of such particles.

#### 2. The Model of a Uniaxial Two-Phase Nanoparticle

In contrast to the model of two neighboring plane-parallel phases used in the works [56–59], we use a more plausible model of “inclusion.”(1) Homogeneously magnetized nanoparticle (phase ) with the volume = and the shape of an ellipsoid with the elongation = contains a uniformly magnetized ellipsoidal inclusion (phase ) with the volume = , whose long axis coincides with the long axis of the particle oriented along the axis (see Figure 1).(2) The axes of crystallographic anisotropy and of both uniaxial ferromagnets are considered to be parallel to the axis , and the vectors of phases’ spontaneous magnetization and are located in the plane containing long axes of magnetic phases and make angles and with and the axis , respectively.(3) The external magnetic field is placed along the axis as uniaxial mechanical stress .(4) Thermal fluctuations of phases’ magnetic moments may be neglected.

We exclude from consideration magnetoelastic interfacial interaction, which may be more or less justified only in case of highly disordered distribution of magnetic atoms in the boundary layer (see, e.g., [50]).

In the accepted approximation, free energy of nanoparticle located in an external field can be written as the sum of(i) crystallographic anisotropy energy: (ii) interaction energy of magnetic moment with its own magnetic field which in accordance with [66] can be represented as (iii) energy of magnetoelastic interaction with uniaxial mechanical stress : (iv) exchange interaction energy across the border, which according to [56] can be defined as (v) interaction energy of magnetic moment with an external magnetic field In (1)–(5) , , dimensionless constants of anisotropy and phases’ shape anisotropy are —the first anisotropy constant, and , , —the second and third magnetostriction constants of a uniaxial crystal; is the volume of a particle, is the surface area that separates the phases, is the ratio of the inclusion volume to the volume of the whole particle, is the constant of interfacial exchange interaction, and is the width of the transition region with the order of a constant lattice. Note that the shape anisotropy constant is expressed through the demagnetizing factor along the long axis depending only on the elongation of the ellipsoid :

Analysis of two-phase particle’s magnetic states is a procedure of minimization of the total free energy () of a nanoparticle: where the effective anisotropy constants and the position of are determined by the following formulas: The constants of an interfacial interaction and are expressed through the constants of magnetostatic and exchange interaction of the phases: Minimization of free energy (7) by and leads to a system of equations determining static states of phases’ magnetic moments: where . The system of (10) allows to determine the main and metastable states of nanoparticles’ magnetic moment.

#### 3. Static States of the Magnetic Moment of Uniaxial Nanoparticle

The system of equations, within the accuracy of notations, transfers into the system of equations obtained earlier in [58, 59], so neglecting thermal fluctuations of the magnetic moment and in the absence of an external magnetic field when and (an “easy axis”), two-phase particles can be in one of the following states:(i)the first “(↑↑)-state”: magnetic moments of both phases are parallel and directed along the axis ;(ii)the second “(↑*↓*)-state”: the phases are antiparallelly magnetized, and the magnetic moment of the first phase is directed along the axis ;(iii)the third “(*↓**↓*)-state”: it differs from the first one in the antiparallel orientation of the phases magnetization relative to the axis ;(iv)the fourth “(*↓*↑)-state”: the magnetic moment of the second phase is directed along the axis ; the magnetic moment of the first phase is directed against the axis .

Besides, if the magnetostatic interaction between the phases dominates the exchange interaction , the second and fourth states are stable, while the first and third are metastable, since the free energy of a particle in these states is less than in the first and third, where Otherwise , the first and third states are metastable.

The same states are realized in the case when the constants and have different signs. If and (an “easy plane”), a two-phase particle can be in one of the four states listed above with the phases’ magnetic moments perpendicular to the axis .

Following (10), if at least one of the constants or is positive, then when the external magnetic field obtains a critical value , the transition from one state to another will occur. For example, the critical field of the transition from the third state to the second or from the first state to the fourth is In much the same way we define the critical field of the transition from the third to the fourth (from the first to the second) state: from the forth to the first (from the second to the third): from the second to the first (from the forth to the third): from the third to the first and vice versa: and from the second to the fourth and vice versa:

In the same way we can calculate the spectrum of critical fields when and . In this case we can use the relations (11)–(16), having replaced by .

#### 4. Magnetic States of Multiaxis Two-Phase Nanoparticle

Let nanoparticle’s magnetic phases be represented by the crystals of cubic symmetry with the constants of crystalline anisotropy of the first and the second order, respectively. Let us perform a solution of the problem of the magnetic states of a multiphase particle in the framework of the following assumptions.(1)Crystallographic directions , , and of the phases coincide with the axes , , and , respectively (see Figure 1), if the first constants of the phases’ crystalline anisotropy are positive. Otherwise (), we combine the “easy axis” (direction ) with the axis .(2)We use the condition of magnetic uniaxiality of a multiaxis crystal’s grain [57, 67], the essence of which is that at a certain elongation its shape anisotropy prevails over the crystalline anisotropy. The process of magnetization of these particles is similar to magnetization of uniaxial particles with an effective constant determined by the total free energy. So for materials with (e.g., titanomagnetite or magnetite), the condition of magnetic uniaxiality is satisfied when the elongation .

As well as in Section 1, we build an expression for the total free energy, which includes(i) crystalline anisotropy energy, which, according to [62], depending on the sign of the constant of anisotropy has the following form:(ii)(iii) energy of a grain in the field of mechanical stress energy of magnetostatic interaction , interfacial exchange interaction , and energy of the magnetic moment of a particle in the external magnetic field defined by the relations (2), (4), and (5), respectively. In the relations (17) and (18) and are dimensionless constant of crystalline anisotropy of the first or second order of the first or second phase, respectively, and are the anisotropy constants of the first and second order of a cubic crystal, and , where and , where , and are the constants of phases’ magnetostriction.

According to the second condition stated above, magnetic states of a multiaxis two-phase particle are equivalent to states of a uniaxial two-phase particle (see Section 2), whose effective anisotropy constants and are expressed through the constants of a multiaxial crystal:

Expressions (11)–(16) and (19) allow us to study the effect of mechanical stress and geometric ( and ) and (, , , ) characteristics on the magnetic state of both single-axis and multiaxis crystallographic two-phase nanoparticles.

#### 5. Diagrams of the Magnetic States of Heterophase Nanoparticles

For comparison, let us consider two types of two-phase nanoparticles, whose magnetic characteristics are presented in Table 1: relatively low coercitive nanoparticles of magnetite (first phase) that include titanomagnetite [3, 60] (Fe_{3}O_{4}-Fe_{2.44}Ti_{0.56}O_{4}) and high coercitive nanoparticles -Fe_{2}O_{3} (second phase) epitaxially cobalt ferrite coated [39–41] (Co Fe_{2}O_{4}--Fe_{2}O_{3}).

Critical fields of magnetic reversal (11)–(16) depend essentially on the elongation and the relative volume of one of the particle’s phases. Moreover, for some and values the critical fields can be negative. Negativity should be comprehended as the inability to implement the metastable state controlled by this critical field. This statement may be interpreted in the following way. If an ensemble is made up of particles with different relative sizes and elongation of the second phase, the point A on the phase diagram (see Figure 2) can be associated with a particle that at (dark area) can be in either metastable or ground state and only in the ground state if (light area).

Note that natural limitation of the volume of introduction (of the second phase) and the relation between large semimajor axes of particles’ ellipsoids and implementation impose restrictions on the choice of and . Indeed, if we use expressions for the volume and the long axis of the elongated ellipsoid (, , , —minor axis of the ellipsoid), we can obtain the following relation . Obviously, the range of permissible values of and depends significantly on the elongation of nanoparticles (see Figure 2). The maximum value of and is limited with the value , since the demagnetizing factor decreases rapidly with () increasing.

In the study of the effect of mechanical stress on the magnetic states and magnetization processes of an ensemble of two-phase nanoparticles we use a dimensionless constant
As shown in Table 1, stress signs and match each other for the particles (Fe_{3}O_{4}-Fe_{2.44}Ti_{0.56}O_{4}), while negative values () correspond to tensile stress and positive values correspond to compression stress for cobalt-coated particles -Fe_{2}O_{3}.

Tables 2 and 3 show the basic diagram and the two-phase particles of the metastable states (CoFe_{2}O_{4}--Fe_{2}O_{3}) and (Fe_{3}O_{4}-Fe_{2.44}Ti_{0.56}O_{4}), respectively, differing magnitude interfacial exchange interaction compressive () or tensile () stresses. The particles, which were representative points in the phase diagram in the region highlighted in the dark color, can be both in the ground and metastable states (with a parallel or antiparallel orientation of the magnetic moments of the phases). The points in the light area correspond to particles in its ground part in Table 1 as footnote. Please check state in which depending on the size of the interfacial exchange interaction magnetic moments are parallel or antiparallel phase with each other. Figures in the second column 2 and the second row of Table 3 illustrate the effect on the metastability biphasic nanoparticles .

Let us define the degree of metastability of the nanoparticles system as the ratio of the areas: the dark side of the chart to the general—dark and light. Figure 3 shows that the metastable state is realized only at , and the degree of metastability reaches its maximum at a point . And, if the magnetic moments of the phases are antiparallel in the ground state at , then at they are parallel.

**(a)**

**(b)**

Table 2 and Figure 3(a) show that the stress does not affect the metastability of cobalt-coated particles -Fe_{2}O_{3}. Such a weak dependence of the magnetic states on the stress is associated with high values of anisotropy and shape anisotropy. The effect of mechanical stress on the degree of nanoparticles (Fe_{3}O_{4}-Fe_{2.44}Ti_{0.56}O_{4}) metastability is more significant (see Table 3 and Figure 3(b)). Tension increases both the degree of metastability and the range of the constants of interfacial exchange interaction where metastable states are implemented (see Figure 3(b)); compression leads to the opposite effect. Described dependence of metastability on mechanical stresses was expected. Since in accordance with the relations (19), an increase in tensile stress increases the effective constant of anisotropy and, therefore, to expansion of the area of metastable states, while compressive stress reduces metastability (see Figure 3(b)).

#### 6. Magnetization of the System of Two-Phase Nanoparticles

As shown in the previous section, a two-phase particle in one of the four magnetic states can be represented by a point on a diagram (see Figure 2). We assume that two-phase nanoparticles are uniformly distributed with respect to the relative volume of the second phase () and its elongation (). Then the points representing these nanoparticles are distributed on the diagram uniformly. Therefore, to calculate the dependence of a number of particles () located in each of the four magnetic states of the magnetic field we should integrate it with respect to the diagram area that satisfies conditions (see Figure 4).

So in the case of noninteracting nanoparticles the magnetic states vector population of the two-phase particles is defined as here is the density of distribution on the diagram of a number of particles in state and is the number of particles in one of the four equilibrium states at. and , and they are related as follows: can be defined from the condition that at in the state of zero magnetization the first and third as well as second and fourth states of nanoparticles are equivalent: where is the number of particles and is the total area of the diagram (see Figure 2).

Relations (21)–(23) defining the population vector allow for calculating the value of magnetization of the two-phase nanoparticles system: where is the volume concentration of the magnetic phase and is the volume that the system occupies.

#### 7. The Effect of Mechanical Stress on the Magnetization Curve and Hysteresis Characteristics of an Ensemble of Cobalt-Coated Nanoparticles -Fe_{2}O_{3}

Calculation of the magnetization curve and hysteresis loop held with the help of relations (11)–(16) and (21)–(24) is shown in Figure 5.

Dependence of the coercive force on the relative volume of cobalt coating is determined not only by mechanical stress but also by the interfacial exchange (see Figure 6).

It is easy to see that tension shifts magnetization curves to lower magnetic , but compression has the opposite effect—magnetization curves shift to larger fields relatively to the undeformed state. Thus, the mechanical stress does not affect the saturation magnetization , which coincides with the saturation remanence and decreases with increasing thickness of the membrane irritation. Recent experiments conducted on the core/shell nanoparticles of hematite/magnetite showed that is almost independent of stress [68]. We obtained that the dependence of the saturation magnetization of the relative amount of cobalt coating is in qualitative agreement with experimental data [69]. reduction system core/shell nanoparticles with increasing thickness of the shell due to the fact that with increasing is increased contribution less magnetic material and a magnetic role decline.

Dependence of the coercive force on the relative volume of cobalt coating is determined not only by mechanical stress but also by the interfacial exchange (see Figure 6). In addition, the negative exchange interaction leads to a reduction in general, and the positive increases relative to the state . We calculated the dependence of the coercive force of which is in qualitative agreement with experimental data [70]. Where shown, the dependence of the coercive force is not determined by the particle size and the relative thickness of coating core/shell nanoparticles.

We also note a qualitative agreement with experimental data [68]; we obtained depending from mechanical stresses. In the same paper was received experimental linear dependence of the effective anisotropy constant of stress which is in good agreement with the results obtained by us (see (19)). Dependence characteristics of the coercive force on the interfacial exchange interaction are shown in Figure 7. Obviously, the nonmonotonic behavior is typical for nanoparticles with a large () or low () cobalt coating thickness and implemented in both positive and negative values .

Since the coercive force is determined by the minimum of the range of critical fields, the nonmonotonic behavior is related to the difference in the dependence of these fields on the exchange interaction constant (see (10) and (11)–(16)). The above mentioned nonmonotonic behavior was not observed in [39, 65, 66, 71], which is obviously associated with a narrower range of critical fields of magnetization reversal of two-phase particles than that in this study. A qualitative comparison of the results with the similar calculations presented in the works [8, 54, 69, 71, 72] shows that coercivity increases up to saturation along with the increase in the amount of phase, just as it was mentioned in those works.

#### 8. Conclusions

In contrast to earlier models of two-phase nanoparticles [56–59, 67, 73] the model presented in this work allows for studying the effect of mechanical stress and magnetic field on the magnetic state of both uniaxis and multiaxis heterophase magnets. The estimated range of the critical fields of magnetization reversal of the phases’ magnetic moments allowed for building phase diagrams and evaluating the effect of mechanical stress on the degree of metastability of nanoparticles’ magnetic states. In the framework of our model, using an example of epitaxially cobalt-coated particles - we carried out a theoretical analysis of the effect of uniaxial stress on the magnetization process of the system of noninteracting heterophase nanoparticles. As it was shown tension leads to reduction and compression increases coercive force , while the residual saturation magnetization does not change under the influence of mechanical stress. The coercive force of the nanoparticles system changes in a nonmonotonic way as far as the interfacial exchange interaction increases and strongly depends on the thickness of cobalt coating.

The simulation of the effect of mechanical stresses on the magnetization of the system of two-phase single-domain noninteracting grains carried out in this study led to consistent results, which allows for using our model in analyzing magnetic properties of an ensemble of heterophase nanoparticles.