Frequency dependence and long time relaxation of the susceptibility of the magnetic fluids

Journal of Magnetism and Magnetic Materials 122 (1993) 176-181 North-Holland

Frequency dependence and long time relaxation of the susceptibility of the magnetic fluids M.I. Shliomis a n d V.I. S t e p a n o v htstitute (if" Continuous Media Mechanics, Urals Braneh
1. Introduction

1. Presence of two competing mechanisms of the magnetization relaxation: the Brownian and

It is known that there is a definite similarity between the magnetic properties of ferrofluids and those of spin glasses [1-6]. Both former and latter demonstrate temperature maximum of the initial susceptibility X(W, T) at the ultralow frequencies o~ and extremely wide (up to 6 - 8 orders) spectrum of relaxation times. In our opinion, however, the analogy between magnetic fluids and spin glasses is superficial. As a matter of fact, in the spin glass the temperature T , of the maximum of x(T) is recognized as a temperaturc of a cooperative 'freezing' of the magnetic moments of the particles. At the same time in magnetic fluids the value of T , coincides with or is close to the solidification temperature and depends very weakly on the magnetic particle number density n. In the spin glass the temperature T , of a phase transition is proportional to the concentration n of the carriers of magnetic moments. We consider that interpretation of the observed dependencies X(w, T) in the magnetic fluids must be based on one-particle effects: interaction of particles is not very important. In the framework of one-particle theory one must only take into account three circumstances:

the N6el ones. The efficiency of the latter strongly depends on the particle size, but that of the former on the viscosity of liquid carrier. That is why two other factors are of a such importance. 2. Polydispersedness of ferrofIuids. This really results in an enormous extension of the N6el relaxation times spectrum. Averaging with the proper particle-size distribution function changes the low-frequency Debye susceptibility %(~o, T) unrecognizably.

Correspondence to." M.I. Shliomis, Institute of Continuous Media Mechanics, Urals Branch of the Russian Academy of Sciences, 614061 Perm, Russia. 11304-8853/93/$06.00 ~ 1993

3. Blocking of the rotational degrees of J?eedom of the particles on solidification of a carrier liquid. With freezing of the liquid the Brownian relaxation mechanism becomes ineffective which leads to the appearance of a maximum on the x(T) CUlWe.

Taking into account all these factors it is possible to describe properly the dynamic susceptibility of magnetic fluids in the framework of a single-particle theory.

2. Susceptibility of monodisperse ferrocolloids We are interested on the linear responsc on the ac magnetic field H = Hoc ~''' of small amplitude: ~ = m H 0 / k T < < l , where m = M ~ V is the value of the magnetic moment of a particle, M, is the saturation magnetization of the ferromagnet, and V is the volume of a particle. In a dc field (w = 0) the equilibrium distribution of orientations of the magnetic moment m

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M.L Shliomis, V.L Stepanov / Suseeptibility of magnetic fluids

and easy magnetization axis n of the particle is given by the function [7] I¥1, = C exp[o-(en) 2 +

(1)

Here e = m / m and h = H / H , and or = K V / k T , where K is the density of the effective magnetic anisotropy of a particle. The dimensionless parameter ~r characterizes a coupling between the orientation of the magnetic moment m and the position of a particle itself (orientation of its easy axis n). Due to this coupling the magnetization of the particle induced by the external field depends on the angle formed by the vectors h and n. Therefore one can distinguish between longitudinal XII and transversal X • susceptibilities. The magnetization of the assembly of particles with n II h is determined by the formula

M IL = n m ( c o s 0) = n m ~

1e~rX-'dx

(7 0

0

5

(2)

For the magnetization of the particle assembly with n 3_h one finds M± = nm(sin 0 cos 05)

1 ~r>> 1: R = ~

_

2 w£~e,~X2dx that is

kT

2R

(3)

_

_

_

) .

""

,

(4b)

we come to the conclusion that X II monotonously increases with or and grows from the 'Langevin' value n m 2 / 3 k T at cr << 1 to the 'Ising' n m 2 / k T at or >> 1 - thus strong uniaxial anisotropy causes the problem to be one-dimensional. Simultaneously X± decreases from Langevin susceptibility Xo = n m e / 3 k T to zero (see fig. 1). Eqs. (2) and (3) may be conveniently written down in the form

s = I(3R'/R

= ?lDl ~

X± -

""

3

1 + 2~r + 4~ 2 +

X u =X0(I+2S), ~2Wcos205 d05fol(1 - x z) e~X-'dx

nm 2 R - R '

1~0

Fig. 1. A static susceptibility as a function of o- in the cases of parallel or perpendicular orientations of the easy magnetization axis with respect to the measuring field.

(4a)

where cos 0 = (en) and the angular brackets denote averaging over the distribution function (1). One then gets for the susceptibility X II = M II/H £1e'~X:d x

1-

or<< 1: R = 1 + ~ r / 3 + ~ 2 / 1 0 + o ' 3 / 4 2 +

f0

R(o')=

2-

Using the asymptotic expansion of the function R ( ~ ) presented in Appendix of [8]:

fo x -~ e,rX2dx

nm 2 R'

xo

~( eh )]

= C e"""':[1 + ~(eh)].

)(Ib- kT R '

177

-

X.=X0(1-S),

1).

(5)

Here S(o-) is the average of the Legendre polynomial Pe(x) = (3x 2 - 1)/2. An applied field orientates both magnetic moments m and the particle axes n. But in linear approximation, i.e. to a term O(sC), the effect of field-induced particle orientation is absent. It follows from symmetry consideration. Since the di-

M. 1. Shliomis, ld L Stepanot, / SusceptibiliG of magnetic fluids

178

rections n and - n are equivalent, then interpreted physically they can only have bilinear combinations n,nk, which are invariant with respect to the sign change of n. T h e r e f o r e the degree of orientation is ((nh) 2) ,~:2. Thus, in the weak measuring field the particle anisotropy axes are oriented in a r a n d o m fashion. Consequently the susceptibility averaged over particle orientations is given by the formula X=~(Xll +2X~).

- I, % t =1"1 1 + 1"B

rs I =r/l+rBl

,

(7)

where

1"D

(1+2S)/(1-S)

at o - < 2 ;

( f v v / 2 ) o r 3/2e"

at ~r_> 2;

1-S 1", = 2 r D 2+S

7

J

/y

100-

10~

(6)

On substituting here )¢11 and X • from (5), one gets X =,V0: the initial susceptibility does not d e p e n d on the magnetic anisotropy of the particles. In a weak ac field the solution of the F o k k e r Planck equation [7] gives for the relaxation times of the parallel and perpendicular (to n) components of the vector m the following expressions:

1"/

1000 :

for all o-.

(8a) (8b)

H e r e r B = 3 r l V / k T and r o = 3txVm/kT, where r/ is the viscosity of the liquid and /z is the 'magnetic viscosity'. The latter characterize the process of intrinsic friction accompanied by the motion of the magnetic m o m e n t with respect to the body of the particle. The asymptotic (at er > 2) expression for r l (see (8)) got the name of N6el time r N. D e p e n d e n c e of longitudinal and transversal relaxation times on G is shown in fig. 2. Since we know the static susceptibility (5) and the relaxational times (7) we may now suggest an approximate formula for the dynamic susceptibility (compare with (6)):

X = ' [3 X l l ( l + i w r l l )

i +2x±(l+iwr±)

']

(9)

0.1

i 0

i

i

i 5

i (3-

110

Fig. 2. D e p e n d e n c e o f t h e p a r a l l e l ( l ) a n d p e r p e n d i c u l a r ( t ) to t h e e a s y m a g n e t i z a t i o n axis r e l a x a t i o n t i m e s o n ~r.

which is useful up to the L a r m o r frequency of the precession of vector m in the field of the anisotropy; this region of ferromagnetic resonance was studied in detail in [8]. For ultrafine particles (or _< 1) we have X II = X ~ = Xo and r; z, = r D. According to our estimation, the magnetic viscosity of magnetite is # = 10-4 g / c m • sec thus r D << r B. Therefore, as it follows from (7), r II = r i = rD and the formula (9) looks in this case especially simple: X =Xc~/( 1 + i~°rD). It describes very well experimental data [9] concerning the complex susceptibility X(~o)= X ' ( ~ o ) - i x " ( w ) of magnetic fluids containing particles with mcan diameter d nm and narrow size distribution. With increasing particle sizes the relaxation time r 1 decreases a s TI)/O" , SO up to very high frequencies ~o < G / r D = K/31~ the condition w r • << 1 is valid. Thus, at finite value of ~r and m o d e r a t e frequencies only the parallel c o m p o n e n t of thc susceptibility undergoes a dispersion. The latter is most noticeable in a narrow band of frequencies near 1

o0, =1-11 = ( r N + r B ) / r N 1 - B " The correlator q ( t ) = (~(t)s33~(0)) is a convenient function to describe the relaxation of magnetic m o m e n t ~ = M ~ of a sample of magnetic

M.L Shfiomis,

V. L

Stepanov / Susceptibifity of magnetic fluids

fluid (23 is a volume of the sample) after abrupt turn off of the field. According to the Fluctuation-dissipative theorem the Fourier transform of this correlator relates to the imaginary part of the susceptibility by the relation

q(w) = ( 2kT/w)23X"( w).

(10)

2.50

Z

2.00

2 ~

179

~

1.5O

Using (9) one finds in the t-presentation

q(t)=(Um2/3)[(l+2S)

+2(1-S)

e

e t/~,,

'/~~].

1.00

(11)

where N is the full number of magnetic particles in the ferrofluid sample.

3. Susceptibility of polydisperse colloids

V(/3 + 1). (12)

exp

This function is normalized on unit distribution density, F(x) is the G a m m a function, /3 and d 0

.

0.02

0.01 I

0.00

, ~

50

. . . . .I . . . . . . . 100 , 150

2 0.00

The distribution of particle sizes of real colloids may be described accurately by the G a m m a distribution [6]

~(d) = ~

0.50

200

d,A Fig. 3. Size distribution of magnetic particles according to formula (12) for d = 8 nm and /3 = 2.5 (curve 1), 8 (curve 2) and 15 (curve 3).

......

10 -2

~

.......

~

1

.......

~

' I I [111~ ....... ~

10 2

.......

~

......

10 4

~

. . . . Ill

...... ~

10 s

......

10 a CO Fig. 4. Frequency dependence of the complex susceptibility (X' - solid lines, X " - dashed lines) related to volume concentration of the magnetic phase of the colloid. Curve 1: T = 150 K, and curve 2:300 K. The calculations were carried out for next parameters values: K = 3 x 105 erg/cm 3, M, = 480 G,/~=4xl0 5g/cm.s,d=8nmand/3=8.

are the parameters of the distribution. The diameter, averaged with this function, is expressed via these parameters as d = (/3 + 1)d 0. A plot of the function (12) is shown in fig. 3. Below we discuss dependencies X(w) and x ( T ) of the polydisperse colloid obtained after averaging the 'monodisperse' susceptibility (9) over distribution (12). The temperature dependence of the viscosity of the liquid was also taken into account (r/ is contained in the expression %); we used the Vogel-Fulcher formula ~ = % exp[A/(T- T0)]. According to data of Pshenichnikov, this formula at rl0 = 5.77 × 10 3 p, A = 396 K and T 0 = 162 K accurately approximates the viscosity of kerosene-based ferrofluids in a range from room temperature up to the temperature of loss of fluidity ( -= 205-220 K). The frequency dependencies of the complex susceptibility for solidificated (T~ = 150 K) and liquid (T z = 300 K) ferrofluids are shown in figs. 4 and 5. At temperature T~ the Brownian relaxation mechanism is 'frozen' ( r B = oo), so at ~r > 2 we have r ll = r N. As frequency increases the blocking condition tor N >_ 1 is fulfilled for more

M.I. Shliomis. U..l. Stepanor / Susceptibility q# magnetic.lluids

180

and more fine particles. Therefore the magnetic m o m e n t s of a large n u m b e r of particles (to begin with the smallest) are late in their response to the low measuring field, that is they do not bring a contribution to the susceptibility. As one can see in the plots, X'(w, T l) decreases with the logarithm of w quasilincarly. Such d e p e n d e n c e of X' corresponds to constant imaginary part: X " = const. At room t e m p e r a t u r e T~ there are always some n u m b e r of large particles (with 7 N > ~-~) for which the Brownian mechanism of relaxation is more efficient then the Ndel one. To the contribution of the Brownian particles corresponds a horizontal plateau on the curve X'(co, T2). The magnetic m o m e n t s of the particles of size d~ is determined by the equation ~'N = ~'B- In the case of magnetite colloids we have d~ = 15 nm. Note that while the Ndel particles are blocked with increase of the frequency of a field gradually, the Brownian ones are ' t u r n e d off' almost instantaneously as if they were in a monodispersc colloid. This difference is explained by the strong dependence r N ( V ) and the weak d e p e n d e n c e ~-B(V). The same sets of parameters, excluding parameter /3 which determines the width of the distribution (12), were used for calculations of the plots in fig. 4 and 5. In the former /3 = 8 and in

5.00

X

4.00

2

3.00

2.00

21/ \\ /I \\

1

1.00

0.00 10 -~

1

10 2

10 "~

10 ~

09 F i g . 5. T h e s a m e as in fig. 4 b u t fl)r # = 2.5.

10 a

4

2

T ~o

2;o

2~o

Fig. 6. Temperature dependence of the initial susceplibility m dc (dashed line) and ac field (solid lines): w/2"rr = I0 : H z ( c u r v e 1), 10

i ( c u r v e 2) a n d I ( c u r v e 3).

the latter/3 = 2.5. The smaller /3 is the wider is particle sizes distribution sce fig. 3. Therefore as /3 decreases the n u m b e r of Brownian (i.c. large) particles in the colloid increases and their contribution the susceptibility grows as well (compare curves 2 in figs. 4 and 5). Fig. 6 demonstrates the temperature dependence of the initial susceptibility in dc and ac fields. The temperature maximum on the inffalow frequencies is explained by the rapkt growth of the viscosity of the liquid at cooling, which leads to blocking of the Brownian particlcs. Although for mean diameter ( l = 8 nm the number of such particles (with d > d~) is not great, their contribution to the susceptibility is very essential because they are large and Xo cx d". We have drawn attention above on the dispersion regions, where X'(o)) oc In ~o and, in accordance with the K r a m e r s - K r o n i g relation, X"c~ const. In this case, as seen from (10), corrclator q(w) c~ l / w , that is wc have a well-known expression for 1/w noise. On transforming from w- to the t-presentation, one finds q(t)O~ in t. All the formulae of this short paragraph are really equivalent and all of them correspond to the phen o m e n o n of the long-time logarithmic relaxation.

M.L Shliomis, V.L Stepanot, /Susceptibility of magneticfluids

181

The presence of a continuous spectrum of the r e l a x a t i o n times 7 N with even d i s t r i b u t i o n of t h e i r l o g a r i t h m is the r e a s o n of existence o f this p h e n o m e n o n in the m a g n e t i c colloids. T h e cont i n u o u s s p e c t r u m of t h e r e l a x a t i o n times t a k e s p l a c e in spin glasses too, b u t it arises t h e r e from a m u c h m o r e c o m p l e x physical p a t t e r n [10].

t o w a r d s e a c h other, which inevitably l e a d s to t h e b r e a k i n g up o f the spatial h o m o g e n e i t y of the ferrocolloid: stratification of it will o c c u r (transition of the first kind [11]) in regions with high a n d low c o n c e n t r a t i o n o f the m a g n e t i c particles. Such a stratified system is i m c o m p a t i b l e with the notion of a spin glass.

4. Conclusion

References

W e w o u l d like to n o t e that r e g a r d l e s s of the validity o f t h e t h e o r y s t a t e d above, t h e t r a n s i t i o n o f t h e m a g n e t i c fluids in the state o f the spin (or d i p o l e ) glass is in p r i n c i p l e not possible. F o r e x a m p l e a t r a n s i t i o n to the spin-glass state w o u l d m e a n collective ' f r e e z i n g ' of the o r i e n t a t i o n s o f p a r t i c l e m a g n e t i c m o m e n t s m, c a u s e d by t h e d i p o l e - d i p o l e i n t e r a c t i o n s b e t w e e n them. Suppose t h a t r is the m e a n d i s t a n c e b e t w e e n p a r t i cles, t h e n the i n t e r a c t i o n e n e r g y b e t w e e n two n e i g h b o r d i p o l e s is of o r d e r m 2 r - 3 ~ m2n; on e q u a t i n g this e n e r g y to kT, we g e t - a n e s t i m a t e o f t h e f r e e z i n g t e m p e r a t u r e n a m e l y T . = nm2/k. R e a l i z a t i o n o f t h e spin-glass state is impossible in m a g n e t i c fluids b e c a u s e the p a r t i c l e s have translational d e g r e e s of f r e e d o m . T h e strong (at T < T . ) m a g n e t o - d i p o l e a t t r a c t i o n invoked i m p o s e s a d i r e c t e d drift o f t h e p a r t i c l e s in the liquid matrix

[1] A. Tari, J. Popplewell and S.W. Charles, J. Magn. Magn. Mater. 15-18 (1980) 1125. [2] Yu.I. Dikanskii, Magn. Gydrodinamika N 3 (1982) 33. [3] K. O'Grady, J. Popplewell and S.W. Charles, J. Magn. Magn. Mater. 39 (1983) 56. [4] Yu.L. Raikher and A.F. Pshenichnikov, Sov. Phys. JETP Lett. 41 (3) (1985) 132. [5] A.F. Pshenichnikov and M.I. Shliomis, Bull. Acad. Sci. USSR Phys. Ser. 51 (6) (1987) 26. [6] A.F. Pshenichnikov and A.V. Lebedev, Sov. Phys. JETP 68 (1989) 498. [7] M.I. Shliomis and V.I. Stepanov, J. Magn. Magn. Mater. 122 (1993) 196. [8] Yu.L. Raikher and M.I. Shliomis, Sov. Phys. JETP 40 (1974) 526. [9] P.C. Fannin and S.W. Charles, J. Phys D: Appl. Phys. 24 (1991) 76. [10] S.L. Ginzburg, Irreversible Phenomena in Spin Glasses (Nauka, Moscow, 1989). [11] K.I. Morozov, Bull. Acad. Sci. USSR Phys. Ser. 51 (6) (1987) 32.