Nonlinear susceptibilities of superparamagnetic fine particles

~ ELSEVIER

Journalof m:dgneusm magnetic materials

Jonrnal of Magnetism and Magnetic Materials 196-197 (1999) 88-90

Nonlinear susceptibilities of superparamagnetic fine particles Y u i . Raikher, V.I. Stepanov* Institute of Continuous Media Mechanics RAS. 1. Korolyov St., 614013 Perm, Russia

Abstract Derivation of a theory for linear and cubic dynamic susceptibilities of random assemblies of fine magnetic particles is outlined. The origin of differences between the low-frequency magnetic spectra of solid systems and magnetic suspensions (ferrofluids) is discussed. ::23;, 1999 Elsevier Science B.V. All rights reserved. Keywords. Magnetic fine particles; Superparamagnetism; Nonlinear dynamic susceptibility

When analyzing magnetic fine particle assemblies, solid or fluid (i.e., magnetic suspensions), a classical tool is the granulometry analysis based on the static magnetization curve. A more promising is a scheme, taken from the spin glass science, where linear and nonlinear dynamic susceptibilities are measured simultaneously. However, to realize it, one needs an adequate theory. Surprisingly, up to nowadays the superparamagnetic theory founded by N6el [1] and masterly developed by Brown [2] lacks a nonlinear extension. Hereby we outline the relevant features of the appropriate model. The starting point is an isolated nanoparticle embedded in a nonmagnetic matrix. The particle is single-domain so that its magnetization I is uniform over its volume v. Then the particle magnetic m o m e n t p = Iv may be described by the unit vector e = #/ic Implicitly, this means that the temperature is well below the Curie point. We assume that the magnetic anisotropy is uniaxial with the energy density K and denote the easy axis direction by the unit vector n. The orientation-dependent part of the particle energy and the equilibrium distribution function of e and n write, respectively,

where ~r = K c / k T and { = l v H / k T are dimensionless parameters, and h is the unit vector of the magnetic field. Basically, the difference between the susceptibilities of the same particle in a solid or fluid environment is due to the difference of the available configurational space. For a solid matrix it is produced by rotations of vector e, whereas for a fluid matrix it is e ® n . In the partition function Z of Eq. (2) the presence of mechanical degrees of freedom is indicated by integration over n. For an entrapped particle only integration over e is allowed in Z while n remains a parameter. Averaging over the latter, when necessary, is applied to the free energy F ~, In Z. In the framework of statistical thermodynamics, the set of initial (equilibrium) susceptibilities is obtained as the even derivatives (2nd, 4th . . . . J of the free energy F with respect to the field at H - , 0. However, see, for example, Refs. [3,4], the static linear susceptibility X is rather weakly sensitive to the state of the matrix. In particular, 7, for a random solid system and magnetic fluid are identical. Only cubic susceptibilities reveal the distinction:

U = - K v ( e ' n ) 2 - Iv(e'H),

(1)

Z~olia --

(2)

Here c is the particle number concentration and $2 is the internal magnetic order parameter generically defined as

C/4V 4 • (3)

45k3T3(l + 2S~),

CI4"c 4 Z f<3) luk

--

45k3TY

(3)

Wo = Z - 1 exp[a(e, n) 2 + ~(e" h)], Z = I i e x p [ a ( e " n) 2 + ~(e" h)] deed2n,

JJ

*Corresponding author. Fax: [email protected].

+ 7-3422-336957: e-mail:

SMr) = (Pl(e" n))o = fPl(e" n)Wo(e" n) d(e" n),

0304-8853/99/$ - see front matter :ii: 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 8 ) 0 0 6 7 2 - 6

i4)

Yu.L. Raikher, KI. Stepanov / Journal of Magnetism and Magnetic Materials 196-197 (1999) 88-90 with Pl being a Legendre polynomial and Wo the distribution function from Eq. (2) at H = 0. All S~ tend to unity for ~r --, ~ . The difference in configurational freedom all the more manifests itself in nonequilibrium properties. Let us first consider a mechanically fixed particle. Hereafter we restrict ourselves by the low-frequency range, that is ~o is sufficiently smaller than the intrinsic Larmor precession frequency (2K7//) with y being the gyromagnetic ratio. U p o n that, the motion of e is of a relaxational type. Its two principal modes are dubbed intrawell and interwell with regard to the orientational potential imposed by the magnetic anisotropy. Macroscopically, in terms of the Landau-Lifshitz equation (LLE), the response time for intrawell modes is of the order of the precession damping time Zo = 1/2ctTK and is weakly temperature dependent. Here ct is the spin-lattice relaxation parameter of LLE. For low potential barriers (or<<1) the interwell time reduces to ZD = aZO, i.e., that of the free orientational diffusion ofe. At a > 1 the interwell time approaches the N6el asymptotic form rN = ~0 exp(a). For arbitrary tr the motion of e is determined by the solution of the relevant F o k k e r - P l a n c k equation [2]

~W = ~J~W3dU/kT

+ In W),

(5)

for a time-dependent distribution function W(t). Here 3~ = (e x ~3/0e) is the operator of infinitesimal rotations of e. In Ref. [4] a procedure of numerically-exact solution of Eq. (5) is described and by means of it the linear and cubic susceptibilities for a solid system of uniaxial fine particles are obtained. With allowance for the polydispersity of real samples, the model yields a fairly good interpretation to the measurements [5] done with C o - C u composites. Also, using the numerical procedure as a testing tool, we show that the interpolation formula 1 (1 - i~oqo) Z~3~= 4 Zh3~°'id(1 + iegqo)(1 + 3ia~zlo)'

as rB = 3qv/kT, where t/ is the viscosity of the liquid matrix, one gets [cf. Eq. (5)] the F o k k e r - P l a n c k equation in the form [7] -~ W -]- 3e•LW =

JeW3 e

(U/kBT + In W),

(8)

where 3, = (n × O/On) is the operator of infinitesimal rotations over n, and the vector of Larmor frequency is defined as I~L = -- (7/IVm)(dU/~e). Eq. (8) is solved by expanding the distribution function W over the sets of 'external' (defined by n) and 'internal' (defined by e) spherical functions. It is much more cumbersome than that for Eq. (5). The obtained cubic susceptibilities for a random solid system and a magnetic fluid are compared in Fig. 1. Let us consider the effect of the matrix fluidity on the nonlinear magnetic spectra. In the low-frequency range, the susceptibility of a solid system, as given by formula (6), is governed by the effective interwell time (7). When decreasing temperature, Z~32 shows relaxational peaks at o9r10 ~ 1. On further cooling, the response of a solid system tends to zero. Indeed [6], at a >~ 3 the relaxation time rio is close to ry. The appearance of the exponent exp(a) makes Z~1), and all the more Z~31, to abruptly vanish at low temperatures. In other words, a rigid magnetic dipole fixed in a solid matrix has zero susceptibility. The low-temperature limit for a fluid matrix is completely different. Although the interwell transitions freeze with the temperature decrease, the emerging rigid magnetic dipole retains its mobility due to the 'external' degrees of freedom. The corresponding time is of the order of rB. When lowering temperature, its growth ( oc a) is much slower than the exponential ( oz exp(a)) change of ZN. Due to that, vector e reorients itself by rotating together with the particle body. Note that the present model does not take into account the possible freezing of the fluid. The appropriate extension is under preparation.

40

"

(7)

In a fluid system, the rotational freedom of the particles affects the susceptibilities in two ways. First, the applied field deforms the orientational distribution function of the easy axes. Second, if out of equilibrium, the orientational diffusion of the particle axes yields one more channel of magnetic relaxation - together with the particle body. Writing down the reference (Debye) time

20

20 0

+

(Je q- Jn)W(Je -I- Jn) -]-

(6)

works very well in the whole low-frequency ( ~ o < 1) range if to take (after Ref. [6]) the effective relaxation time as

rio = "cD 2----~-

89

-20

~/ il

' I ' 0.05 ~010 ! ~

0.15 1/o ~ .......L~2 -20

~--

-40

40

-60

-60

-80

Imx°)

0./S0.15020

V

Fig. 1. Comparison of temperature dependencies of the components of cubic magnetic susceptibilities for a solid random system (solid lines) and a magnetic fluid (dashed lines) at the probing field frequency ~oz0 = 10 5 and the ratio To/~n = 10 - 4 . Vertical axes are scaled in arbitrary units.

90

Yu.L. Raikher, EL Stepanov / Journal of Magnetism and Magnetic Materials 196-197 (1999) 88-90

The fact that at high-temperatures ~(3~ in a fluid system exceeds that in a solid one (see Fig. 1) is of qualitative importance. F r o m the magnetic viewpoint, in both systems one deals with a developed superparamagnetism, i.e., the particles polarize in the applied field with the reference time tO. However, due to uniaxial magnetic anisotropy, the particle polarizability along the easy axis n is greater than that in the transverse directions. Whereas in a solid system the distribution o f n is fixed, in a fluid matrix it adjusts to the applied field thus enhancing the magnetic response. The strength of the effect is ~ H a and the time scale ~ ra. The work was done under auspices of Grant 96-0663 from INTAS.

References [1] L. N6el, Ann. Geophys. 5 (19491 99. C.R. Acad. Sci. (Paris) 228 (1949) 664. [2] W.F. Brown Jr., Phys. Rev. 130 (1963) 1677. [3] J.L. Garcla-Palacios, F.G. Lb,zaro, Phys. Rev. B 55 (1997) 1006. [4] Yu.L. Raikher, V.I. Stepanov, Phys. Rev. B 55 (1997) 15005. [5] T. Bitoh, K. Ohba, M. Takamatsu, T. Shirane, S. Chikzawa, J. Phys. Soc. Japan 64 (1995) 1311. [6] W.T. Coffey, P.J. Cregg, D.S.F. Crothers, J.T. Waldron, A.W. Wickstead, J. Magn. Magn. Mater. 131 ~1994) L301. [7] M.I. Shliomis, V.1. Stepanov, Adv. Chem. Phys. 87 (1994) 1.