Effect of Mechanical Stresses on Metastability of Heterophase Superparamagnetic Nanoparticles

Afremov Leonid1, a and Il`yushin Ilia2, b

School of Natural Science, Far-Easter Federal University, Vladivostok, Russia

aafremovl@mail.dvgu.ru, bfutted@gmail.com

Keywords: Heterophase particles, mechanical stress, magnetic states, critical field, metastable magnetic states, interfacial exchange interaction.

Abstract. Under the model of two-phase multiaxial nanoparticles the dependence of magnetic states

and critical fields of the phases magnetic reversal on mechanical stresses has been defined. By means

of phase diagram the effect of stresses on the basic and metastable magnetic states of heterophase

particles has been studied.

Introduction

Heterophase nature of magnetic materials can be determined by different factors. Thus,

titanomagnetites, being the main carrier of magnetic properties of natural materials, can undergo the

processes of decay [1] or oxidation [2, 3]. Multiphase artificial magnetic materials can be exemplified

by the nanoparticles systems utilized in magnetic storage devices. The ferric particles exposed to

superficial oxidation [4, 5] or magnetic nanoparticles coated with other materials, for example, the

particles of Fe2O3 coated with cobalt [6-7] can be used as carriers of magnetic properties of

magnetic memory elements. Amorphous magnets derived in a special way can be also related to

multiphase magnetic materials. The process of phases forming in these magnets is associated with

transition of amorphous material from metastable state to balanced state (see, for example, [9]). In

this connection, nanocristalline alloys with grains of another ferromagnetic material distributed in its

magnetic matrix are of special interest. A rather detailed theoretical investigation of magnetic states

and magnetization processes of heterophase particles system is presented in publications [9 – 11]. It

must be noted, in the publication [11] we have extended the model [10] to multiaxial two-phase

particles, dimension of which exceeds the dimension of superparamagnetic transition, and have

investigated the effect of mechanical stresses on magnetic states of such particles. And in the present

article we develop a method allowing to estimate the effect of mechanical stresses on metastability of

heterophase superparamagnetic nanoparticles.

Model

Let us use a model of two-phase nanoparticle (see Figure 1), the detailed description of which has

been presented in publication [11],

Figure 1: Illustration of the model of two-phase particle

excluding the thesis about the limiting of particle volume from below by the volume of

superparamagnetic transition.

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According to the publication [11], with neglect of thermal fluctuations in the absence of external

magnetic field two-phase particle can be in one of the following four states:

In the first «(↑↑)-state» the magnetic moments of the two phase are parallel and oriented along

the axis Oz;

In the second «(↑↓)-state» the phases are magnetized in antiparallel way, and the magnetic

moment of the first phase is oriented along the axis Oz;

The third «(↓↓)-state» differs from the first state in antiparallel orientation of the phases

magnetization regarding to the axis Oz;

In the fourth «(↓↑)-state» the magnetic moment of the second phase is oriented along and the

magnetic moment of the first phase – against the axis Oz.

If it’s impossible to neglect thermal fluctuations, then the magnetic moments of the phases can

transit from one state to another. According to [12], probability of transition in unit time is

determined by the height of potential barrier of transition from i-state to k-state:

, (1)

where is typical frequency of “attempts” of crossing the barrier, - Boltzmann

constant, - absolute temperature, - the maximal value of energy between the i-state and

k-state, - energy of equilibrium i-state. The detailed calculation of potential barriers is presented

in the Appendix B of publication [10]. The height of potential barrier is determined by critical

field of transition from -state to в -state , that can be evaluated out of condition of minimum

of total energy density characteristic to two-phase nanoparticle. This energy is composed of

Energy density of magnetocrystalline anisotropy of cubic symmetry crystal, which according

to [13] depending on the sign of anisotropy constant can be given by the following:

(2)

Energy density of grain in the field of mechanic stresses:

(3)

Energy density of inter-phase exchange interaction, which according to [9], can be of the

following form:

. (4)

The energy density of magnetic moment of grain in external magnetic field:

(5)

202 Progress in Materials and Processes

Energy density of magnetostatic interaction:

(6)

In the formulas (2) – (6) values of is a relative volume of the second phase, and are

dimensionless constants of magnetocrystalline anisotropy of the first or the second order of the 1-st or

the 2-nd phase, where and where ,

and are the phases magnetostriction constants, shape anisotropy constant

is stated through demagnetization coefficient along the long axis , depending

only on the elongation of the ellipsoid : , is the

constant of inter-phase exchange interaction, is the width of transitional area, having the order of

grating constant, and are the surface area and the volume of the second phase, respectively.

Minimization of density of nanoparticle total energy results in the following

expressions of critical fields. Thus, the critical field of transition from the first to the fourth state

(from the fourth to the first state) and from the third to the second (from the second to the third),

symmetrical regarding the sense of the magnetic field , is equal to:

(7)

In the same way we can define the critical fields of transition from the first to the second state (from

the second to the first) and from the third to the fourth (from the fourth to the third) state:

, (8)

from the third to the first state ( from the second to the fourth state) and vice versa:

. (9)

Where , , , if

and where , .

Let us introduce normalized to unity population density vector of a two-phase particle states

. If the initial state is

non-equilibrium state, then the transition to equilibrium state can be regarded as a Markovian process

with discrete states, described by the following system of four simultaneous equations:

, (10)

with initial conditions , where . Using the conditions of normalizing

and excluding , let us set out the equations (10) in matrix form:

Advanced Materials Research Vols. 602-604 203

, (11)

where

. (12)

Solutions of the equations (12) is convenient to present using the matrix exponent:

. (13)

Formulas (12), (13) totally define the population density of magnetic states of two-phase

superparamagnetic particles.

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