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Inertial effects in the complex magnetic susceptibility of a ferrofluid in the presence of a dc bias field
AA lournal of netism netic brlals ELSIZVIER
Journal of Magnetism
and Magnetic
Materials
I64
(I 996) 133- 142
Inertial effects in the complex magnetic susceptibility of a ferrofluid in the presence of a dc bias field W.T. Coffey ’ Depuvtment ’ Institute
of Radio
Engineering
of Electronic
and Electronics
a-*, Yu.P. Kalmykov
and Electrical
qf the
Received
Engineering,
Russirm Academy
19 February
Trinity
of Sciences,
’
College, Dublin Vvedenskii
2, Irelnnd
Sq. I, Fry&o,
Moscow
141120, Rus~io
1996; revised 7 May 1996
Abstract The Euler-Langevin equation for rotational Brownian motion of a particle in a form appropriate to magnetic relaxation of ferrofluids (suspensions of single domain ferromagnetic particles) in the presence of a dc bias external magnetic field is used to include the inertia of particles. It is shown, by averaging the Langevin equation corrected for inertia, how approximate analytic expressions for the transverse and longitudinal components of the complex magnetic susceptibility may
be calculated directly from that equation with the aid of linear response theory. The method allows one to bypass the Fokker-Planck equation entirely. Sum rules for integrated absorption of electromagnetic radiation in suspensions of ferromagnetic Krpvrds:
particles are also deduced.
Complex susceptibility;
Ferrofluid;
Inertia: Langevin
equation
1. Introduction The study of the dynamic magnetic susceptibility of a ferrofluid (consisting of a colloidal suspension of fine ferromagnetic particles) is usually undertaken by constructing the underlying Fokker-Planck equation from the Landau-Lifshitz-Gilbert (LLG) equation augmented by a random noise term which is regarded as the Langevin equation of the process [ 1,2]. This method (pioneered by Brown [ 1] for single domain ferromagnetic particles and adapted to the study of the dynamic susceptibility [3] of a ferrofluid subjected to a strong dc bias field H,, superimposed on which is a weak ac field H(t) by
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author.
Fax:
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Martsenyuk et al. [4], Coffey et al. [5] and Waldron et al. [6]) generally involves one in complicated mathematical manipulations. Further it does not include the inertia of the particles which is necessary in order to calculate the magnetic susceptibility of ferrofluids at high frequencies. It is worthwhile therefore to seek an alternative approach which simultaneously circumvents the need to use the Fokker-Planck equation and allows one to include inertia1 effects. A single domain ferromagnetic particle suspended in a fluid is subjected to two separate mechanisms which cause variations in its magnetization. The first is due to the thermal fluctuations inside the particle (spontaneous remagnetization). This is the solid state mechanism known as NCel relaxation [7]. The second is due to the rotation of the particle within the fluid,
0 1996 Elsevier Science B.V. All rights reserved
since changes in the orientation of the particle will effect changes in the orientation of its magnetization. This mechanism is similar to that governing the Debye relaxation of polar molecules and is consequently known as Debye relaxation with characteristic time 7o given by [8]
where p is the dynamic viscosity, v is the volume of the particle, k is the Boltzmann constant and T is the temperature. For the values of typical parameters [4] and room temperatures rn has order of magnitude about lop6 s. The effect of the inertia of Brownian particles on relaxation processes in ferrofluids is usually ignored completely. This can be justified if one restricts oneself to investigations of the low frequency relaxation processes only (up to 10 MHz). However if we extend the frequency range of interest up to 1 GHz the inertia of the particles may play an essential role. Recently the effect of the moment of inertia of single domain ferromagnetic particles on the frequency dependent complex susceptibility x(w) = x’(w) - i ,$‘( w) of an isotropic ferrofluid was reported [9]. It was demonstrated that particle inertial effects that arise from rotational Brownian motion give rise to a resonance behavior, which is indicated by the real component x’(w) becoming negative at high frequencies. In order to explain experimental data Fannin and Coffey [9] suggested a model which takes into account both the Debye and NCel relaxation mechanisms. The model [9] was suggested for an isotropic ferrofluid. However, there is another experimental situation of magnetic measurements, namely: when a ferrofluid is subjected to a strong dc bias field. In this case the medium becomes anisotropic and the model [9] can be no longer applied. The primary purpose of this paper is to evaluate inertial effects in the complex magnetic susceptibility of a ferrofluid subjected to a dc bias field. For simplicity we consider an ensemble of non-interacting single domain ferromagnetic particles, where the NCel relaxation mechanism is blocked. We show how the equation of motion of the average magnetic dipole moment may be written down directly from the inertial Langevin equation for the rotational Brownian motion of the ferrofluid particle. A knowledge of the dynamic behavior of the average mag-
netic dipole moment will allow us to determine the complex magnetic susceptibility of the ferrofluid both in the inertial and in the noninertial (or Debye) limit as a particular case. This approach is similar to that used previously by us [5-7,10,1 l] in the theory of dielectric, magnetic and Kerr effect relaxation. It allows us to obtain in simple fashion (with the aid of linear response theory) closed form expressions for the longitudinal and transverse components of the complex susceptibility and also to deduce sum rules for integrated absorption of electromagnetic radiation in a ferrofluid subjected to a dc bias field.
2. The inertial Langevin particle
equation
for a ferrofluid
In order to discuss the Langevin equation for a single domain ferrofluid particle we first consider Gilbert’s equation for the dynamic behavior of the particle magnetization vector M in the presence of thermal agitation which is [ 1,2] $4(t)
=y[M(r)
x [HT(t)
- VQU)]]~
(1)
where y is the gyromagnetic ratio and r] is a phenomenological damping parameter, H, is the total magnetic field which can consist of the applied field H, the demagnetizing field H,, and a random white noise field h [3]. Gilbert’s Eq. (1) may be rearranged explicitly (as shown. e.g., in Ref. [7]) using the properties of the triple product formula to yield that equation in the Landau-Lifshitz [I] form: ;JV)
= g’M.\[ M(t)
x&l
xWt)l XW)l~ (2)
+h’[PV)
where M, is the (constant) magnitude of the magnetization, only its direction varies with time, the constants s’ and h’ are given by Y
if = M,( 1 +
y%+M\?)
h’= ’
w2 1 + y%j’M,’ ’
(3)
The Larmor precession of the dipole moment vector d = uM of the particle decays after a time T() given by [4] TV= (2r]y2K)-‘, where K is the effec-
W.T. Coffey, Yu.P. Kalmykol:/Journal
qfMqnetism
tive magnetic anisotropy constant. The second (NCel) relaxation time rN characterises reorientation of the magnetic dipole moment, by surmounting a potential barrier vK between different directions of the easiest magnetization. The estimation for this time is given by Brown’s formula [3,7]: Q-N
=
rOu-‘/2e’r,
CT=
which is valid for a~
v
K/kT,
2. At a~
10 [4]
Under this condition the internal state of every Brownian particle in a ferrofluid may be considered as at equilibrium (in the time mu the precession of the dipole moment has time to decay and the NCel relaxation mechanism is ‘frozen’). In a ferrofluid particle, where the NCel relaxation mechanism is blocked, orientational changes in M will be due to the rotational Brownian motion of the particle only. The dipole moment vector d will be directed along the axis of easiest magnetization. Consequently the magnetization will behave like that of a bar magnet when placed in a magnetic field. Consequently the magnetization of the particle obeys the kinematic equation [ 121 (which has the same mathematical form as the Gilbert equation)
and Magnetic Materials
due to the Brownian movement following properties:
= [a(t)
On differentiating M(t)
= [b(l)
x&I(t)].
where k is the Boltzmann constant, T is the absolute temperature, s(t) is the Dirac delta-function, 6,; is Kronecker’s delta, i,j = 1,2,3, which correspond to the Cartesian axes x,y,z of the laboratory coordinate system. The overbar means the ‘statistical average’ over an ensemble of particles which have started at time t with the same magnetization and angular velocity. The drag coefficient 5 can be also expressed in terms of the dynamic viscosity /3 of the fluid as follows [ = 6vp [4]. We have supposed in writing down Eq. (7) that the particle is a sphere with moment of inertia I. The magnetic field H(t) acting on the ferrofluid particle consists of a small time dependent external field H,(t) and a large constant field H, representing the dc bias field: H(t)
=H,(t)
(4)
+ [0(l)
= [h(t) + P(t)
+sa(r)
Eq. (5) can
- JJ(t)P(t) -M(t)(M(t)
.M(t))) *Ho))
+-r(t),
(9)
where x&f(t)] x M(t)
xwN1~
(6)
The quantity 0(t) is determined by the EulerLangevin equation for the angular velocity [ 131: HI(t)
+F(r)
(5)
or, using Eq. (4), ti(t)
(8)
t)
+ V(tP’(t) = v(%YZ
xni(t)].
+H,.
On using Eqs. (4), (7) and (8), therefore be written as
Eq. (4) we obtain x&I(r)]
so that N(t) has the
1\1( t) = 0,
Zn;i( t) + lni( $2(t)
135
I64 (19%) 133-142
= v[M(t)
xH(t)]
+N(t), (7)
where the term v[ M( t) X H(t)] is the torque due to the external magnetic fields acting on the particle, la(t) is the systematic damping torque and N(r) is the white noise random torque acting on the particle,
F(t)
= v(H,(t)M.?
L(r)
= [N(r)
x&f(r)].
-M(t)(M(r)
.H,(t)),
(10)
Eq. (9) is the vector Langevin equation of the problem. This equation is now analogous to the equation of motion of a polar molecule under the influence of electric fields [ 10,13]. The quantity that directly corresponds to the dipole moment of the polar molecule /.L in the theory of dielectric relaxation is the magnetic moment vM of an individual ferrofluid particle, not the magnetization M which is the magnetic moment per unit volume. Eq. (9) also means that we are treating each particle as a rigid magnetic dipole.
136
of Mupmrn
W.T. Coffee?; Yu.P. Kalm~koi~/Journal
We can now write Eq. (9) in terms of the Euler angles 6,(p,$ and the angular velocity components [12]. In the laboratory xyz coordinate system Eq. (9) becomes Zti_( 1) + &G;(t) - ZM,o,(
+ ZM;( t)( w:( r) + w;(t))
t) wj( t)sin (p(t)
= VZZ,( MS2- M;2( t)) + F_(t) ZIG,(r) +@&(r)
+zM.,(t)(+)
+ L(t),
(11)
+ U;(t))
-~M,~,(~)%(~)COSP(f) + ZMS02( t) w3( t)cos 6( t)sin ‘p(t) = -~~,M,(~)M,(~)~,(~) Iti&)
+#&)
(‘2)
+Z,,(t>,
+zM,.(t)(w;(t)
-ZM,w,(t)o,(r)sin
+w;
4
cp(t)
-zM,w,(t)w,(t)cos6(t)cos = - VH”M,.( t)M;(
cp(t)
t) + FY( t) + L,.( t
(‘3)
)
where w, = 8, w?= @sina, w3 = +cos6+ 4, F, and Li are the projections of the vectors F and L onto the x, v and z-axes MI = M, cos 6, M, = M, sin 6 cos cp, and M,. = M, sin 6 sin cp are the projections of the vector M onto the z, x and y-axes, respectively, the constant magnetic field H, is assumed to be directed along the z-axis. On averaging Eqs. (1 l)-( 13) we obtain
md Mqnetic.
Mrueriuls 164 f IYYh) 133-142
equation where the multiplicative noise term L(t) contributes a noise induced drift term to the average.) We remark also that M, and We in Eqs. (14)-( 16) and M,(t) and w,(t) in Eqs. (1 l&(13) have different meanings, namely, M,(t) and w,(t) in Eqs. (1 I)-(13) are the stochastic variables while M, and We in Eqs. (14)-(16) are the sharp (definite) values at time t: M,(t) = Mk and am = wk. Instead of using different symbols for the two quantities we have distinguished the sharp values at time t from the stochastic variables by deleting the time argument as in Refs. [13,14]. The quantities ML and We in Eqs. (14)-( 16) are themselves random variables with the probability density function W such that WdM,dw, is the probability of finding M, and w,! in the intervals CM,, M, + dM,) and (wk, wk + dw,), respectively. Therefore in order to obtain equations for the moments which govern the relaxation dynamics of the system we must also average Eqs. (14)-( 16) over W [6,10,13] we have: Z(ti,)
+ Uhj;(
6.1: + 0~;)) - ZM,( 0+ u,sin
= vH,(M,‘Z(M,)
(M;‘))
+ &Vi,)
Z(ti,.)
+ &I&)
- ZM,( w, w,sin
= vH,(M<‘Z&i, + @iv
M;2) +F_,
+ ZM,.( OJ; + co;)
+ ZM, co2 03cos
(14) - ZM, u, 03cos
p
6 sin cp= - vH,, M, M; + F, , (‘5)
Zti,.+
of
-ZM,w,w,cos6coscp=
+ co:) - ZM, w, u,sin
cp
-vHoM,M,+F,.. (16)
We have formed the average of Eqs. (l l)-(13) as in Ref. [ 131 noting that the averages of the noise terms L,(t) will vanish throughout because in the inertial Langevin equation M(t) is statistically independent of the white noise torque N(t). (This would not be however true of the noninertial Langevin
+Z(M,(w;
- ZM,( w, w3cos = - vff,(M,,M;)
= -vH,(M,.M;)
+ (F;).
8) ( 144
+ 0;))
cp) + ZM,( w? wicos + (F,>,
1.9sin cp) ( 15a)
+ Z(M,.( co; + co;)>
cp) - ZM,( w2 w3cos 6 cos q> + (F,>.
(*6a)
where the symbol ( ) designates an ensemble average with respect to the orientations and angular velocities at the moment t (see for details Ref. [ 131). The system of Eqs. (14a)-(16a) is very complicated, nevertheless it can in principle be solved by the method of Coffey et al. [IO,1 11 which has been suggested for similar problems in dielectric and Kerr-effect relaxation. However as is known from the theory of rotational diffusion [ 151 if the dimensionless parameter y=,
ZkT
i
’
which is always true for ferrofluid particles, the inertial effects can be evaluated with high accuracy
W.T. Coffee,: Yu.P. Kalmykou/
Journal of Magnetism and Magnetic Materials 164 (1996) 133-142
in the context of an approximate approach originally introduced in the theory of dielectric relaxation by Rocard [ 161, Dmitriev and Gurevich [ 171 and Powles
[lf31. In order to proceed we first consider Eqs. (14a)(16a) in the noninertial limit where I -+ 0. In this limit the Maxwell distribution for the angular velocities has set in so that orientation and angular velocity variables are decoupled from each other as far as the time behavior of the particle orientations is concerned. Thus on setting I = 0 in Eqs. (14a)-(16a) we have [(n;r,>
+ 2kT04;)
= VH,(M,2 - 04:))
+ (F;), (17)
5(&)
+ 2kT(M,)
= - vH,(M,M,)
+ (F,), (18)
[(f$)
+2kT(M,,)
= -vHo(MyM;)
+ (F,.). (19)
In writing these we have used the fact that in the diffusion limit angular velocity components about different axes are statistically independent and have taken into account the decoupling between functions of angular orientations and angular velocities [lo]: (wiojf(M))
=
(w;~,)(f(M))
=6;~T(f(M)).
(20)
Eqs. (17)-(19) govern the relaxational dynamics of the magnetization of a ferrofluid particle in the noninertial limit. Just as in the theory of dielectric relaxation [ 16- 181 we can approximately account for the inertia of the particles simply by adding the inertial terms to the right hand side of Eqs. (17)-( 19) Z(til)
+ f(ni,>
= vH,,(M,‘I( ii,)
+ l( if,)
= - vH,,(M,M;) I(kJ)
+ 5&,,)
= - vH,(M&)
+ 2kT(Mz) CM;))
+ (F;),
+ 2kT( M,) + (F,),
[ 171 and Powles [18] in order to take into account approximately the inertia of polar molecules in the theory of dielectric relaxation. However if H, # 0, Eqs. (21)-(23) become complicated. The problem is that Eqs. (21)-(23) are the first terms in an infinite hierarchy of differential-recurrence equations [ 131. It is obvious that in order to solve these equations, we must also obtain equations for CM;),
(M,M,),
(M!M,)
and so on. However, if we merely wish to evaluate the qualitative behavior of the complex susceptibility this difficulty may be circumvented by means of the effective eigenvalue method [3,13] which has been used by Martsenyuk et al. [4] in connection with the calculation of relaxation times of ferrofluids from the Fokker-Planck equation. This method was also used by Coffey et al. [5] in conjunction with the Langevin equation and linear response theory in the study of dielectric relaxation of a polar fluid under the influence of a constant electric field. It may also be used for the present problem.
3. Inertia corrected
magnetic
relaxation
The magnetization decay of a ferrofluid, which is under the influence of a constant field H,, a small constant external field H, (v(M . H, )/kT +X 1) having been switched off at time t = 0, is from linear response theory [7] (M,,(t))
- (M,,)o = x,,H,C,,(~)
(for the case of the longitudinal and
(24) relaxation,
(M.(t))-(M.)o=xrH,C,(t) (21)
137
(for the case of the transverse where
(22)
H,]IH,)
(25) relaxation,
H, I H,),
(26)
+ 2kT(M>) + (F,.).
In the case H, = 0, Eqs. (21)-(23) those used by Rocard [ 161, Dmitriev
(23)
are similar to and Gurevich
and
(27)
are the components bility tensor,
of the static magnetic
0
suscepti-
- (M;(O)% (28)
-
and
As was also shown in Refs. [4.6] the relaxation times 7v can be calculated by the effective eigenvalue method [ 131. In order to calculate the effective relaxation time r,, we have to set H,(t) = 0 (the switch off case), neglect the inertial term in Eqs. (2 l&(23), and finally use the formula [7,13]
(M,(O)) -
-_
7=-
Y
=
(ni,(O>>
C,(O) d,(O)
(34)
Thus using in Eqs. (21)-(23) we get the longitudinal effective relaxation time T,, [7]: (M;(O))
r,, = -
- (M,)o
(M,(O)) ev(M “
271 rif(B,P) // = a ;, ~ ev(M.“,,)IkTsin 8 dfidp I0 / 0
(00
(M;(O)) -’ vM,‘H,
density of magnetic particles. According to linear response theory [ 191 the corresponding complex magnetic susceptibilities x,,(o) and x1 (w) are given by = x,(w)
-
- v( M,‘(0))Ho
- 2kT( M;(O))
(cos’l9 >” - (cos 6 12,
N is the number
x,(w)
- CM,>”
ix;( 0)
rD (cos219)o - [(cos 6 - COS,6)“/2 (cos’6)o -
‘1)
- (cos6);
where r,, = [/2kT
(35)
’
1 - (cos’9)o and
x
e +‘C,(t)dt
t=_ VM, Ho
(+l,l). I
(30) As was shown in Refs. [4,6,13] in the noninertial limit the behavior of the magnetization decay (M,,(t)) - (M,,)” and the autocorrelation function C,,(r) may be approximated by a pure exponential. In this limit both (M,,(t)) - (M,, >O and C,(t) obey the equations
;[‘““” -
(M,)o]
+ r;‘[(M$))
kT
The same procedure relaxation yields [7] rl=
(M,,(O))
-
applied
to the transverse
- (M,,(O))0
(K(O)) 1 - (COS%)[, 7
(36)
D 1 + (cos’6)o
Now [7] -
=O
(t>O),
(31)
(cos~~),~ =L( <) and ( COS~IY), = 1 - 2L( t>/s, where
;qt,
+ T;‘c,(
t) = 0,
(32)
where rr (y = ]I,I) are the relaxation times. Equivalently in the limit of low frequencies the complex magnetic susceptibilities can be described in accordance with Eq. (30) by the Debye equation:
(33)
L(t)
=coth[-
is the Langevin F*L(5)
I,‘[ function. -5W5)
Hence
(37) (38)
W.T. Cqfey.
Yu.P. Kalmykm
/Journnl
qf Magnetism
Eqs. (37) and (38) coincide precisely with the formulae obtained by Martsenyuk et al. [4] and Coffey et al. [5,7]. We have shown in Ref. [61 that there is good agreement between the approximate analytical expressions for the effective relaxation times Eqs. (37), (38), and the exact relaxation times obtained by analytical solution of Eqs. (17)-c 19). The difference between the results predicted by these approximate and exact expressions does not exceed IO’% for all values of 6 [61. The Debye Eq. (33) is valid only in the limit of low frequencies, viz. rv w < 1 (for this condition we can neglect the contributions of the inertial terms). In the time domain the Debye equation implies that an abrupt change in the applied magnetic field would produce an instantaneous finite alteration of the rate of change of the magnetic dipole moment which is impossible in view of the finite rotational inertia of the particles. Just as for dielectric relaxation the consequence of this is the Debye plateau for magnetic absorption, viz. w$(w) + const at o + 00, while in general the product w&‘(w) must tend to zero in the limit of high frequencies [15]. The noninertial limit is determined by the ratio Z/c< lo-“IO- ‘” s for typical values of parameters [4]. For such values of I/[ one can neglect the inertial terms in Eqs. (14a)-( 16a) at low frequencies only (up to 10 MHz). Moreover. it shows clearly that at frequencies above 1 GHz the inertial effects become important. In the next section these effects will be incorporated in the theory. In the context of the effective eigenvalue approach [4,13] having determined the effective relaxation times, we reduce Eqs. (21)-(23) to a set of linear differential equations:
I(ii,) + i(tiy,) + +
und Magnetic
164 (156%) 133-142
139
plied parallel to the z and x-axes scopic longitudinal and transverse magnetization
induces macrocomponents of
(M;(t))
Mnterials
- (MC)” = (M;)exp(iwt)
and
= (M,)exp(iwt),
(42)
respectively. The corresponding complex magnetic susceptibility
components of the are then defined as
and x.(w)=
CM,)
lim -. H,-0 H,
(43) Substituting Eqs. (42) into (39) and (40) we then obtain from Eq. (43) the longitudinal and transverse
-0.21 -2
I -1
I 0 h$~
I 1
1 2
'J
- (MJO) = (FL (39)
An ac magnetic
field
H,(t) = H, exp(i ot)
ap-
Fig. I. The real x;i (a) and imaginary x;; (b) parts of the normalized longitudinal complex susceptibility as a function of log,,,(w~,). Curves I. 2 and 3 are plotted for 5 = 0.01, 2 and 5, respectively, and l/i = O.~T,,.
W.T. Coffe4\; Yu.P. Kalm~ko~~/Journal
140
components namely
of the complex
x,,(w> =1+
magnetic
of Magnetism
and Magnetic
Materials
164 (1996)
133-142
susceptibility
XII
(44)
i wr,, - ZW*T,,/{
and
xL(w)= 1 +iW+rl
XL
_------------
(45)
-Zw2r I /l’
where the relaxation times r,, and rL Eqs. (37) and (38) respectively,
are given by
XII= “2;TMSZ [ 1 - P( [) - 2L( [)/[I
(46)
and V2&MS2 L(S) XL= kT 5 are the components of the static susceptibility
(47)
Fig. 3. The o$(w)/G as a function of log10(w70). The inertia corrected Eq. (44) (dots) compared with w7o x;(w)/G as calculated from Debye’s Eq. (33) (dashed lines). Curves I, 2 and 3 are plotted for ( = 0.01, 2 and 5, respectively, and f/i = 0.17,.
tensor.
1.0 -
In Figs. 1 and 2 we plot the real and imaginary parts of the longitudinal component of the normalized complex susceptibility x,,(w)/G, where
0.6 :
0.6 -
V2NoM2 G=
0.4 -
0.2 -
0.0 I
02 -2
3W‘
.
These figures demonstrate clearly that particle inertial effects that arise from rotational Brownian motion give rise to a resonance behavior, which is indicated by the real component of the complex
-1
0 b,pJ
1
2
7.1
‘Or Oar
2. The
real x;, (a) and imaginary xi (b) parts of the normalized longitudinal complex susceptibility as a function of Iog,,,(wTo). Curves 1, 2 and 3 are plotted for 1/{=0.01~,, 0.17” and 0.5ro, respectively, and 5 = I.
Fig. 4. The w~;[(o)/G as a function of log,,( u-o). The inertia corrected Eq. (44) (dots) compared with Woo x;(w)/ G as calculated from Debye’s Eq. (33) (dashed line). Curves 1, 2 and 3 are plotted for I/[=O.OIT,. 0.17, and 0.57,, respectively and .$= I.
W.T. Coffey, Yu.P. Kalmykor/Journal
ofMagnetism
susceptibility x’(w) becoming negative at high frequencies. Moreover the inertia correction embodied in Eqs. (44)-(45) does remove the plateau in the high-frequency absorption spectrum (see Figs. 3 and 4). The qualitative behavior of the transverse component is similar to that of the longitudinal one and therefore we do not plot it.
t = 0 as follows:
the averaging (M,,(O)~,,(O))o = (M,(O)[
iti,
- MJO)(
+
?(,
-M’(O))])
C,,( W) = /,-eei
transform
-ii
=_- kT;2 [1 -
C,(O)+ i qo)
w
w3[p,(0)qO))o
kTw2
(48) of time). In
w;(O))
+
+ 0( Cl-‘).
-pwP,(O)
(49) = - 5
[2kT(sin’(O)co&(O))o
+ ~H,M,~((cos6(0)sin~8(O)cos~~(O)>o)]
asymptotic =_-
[ 1 + (c0&(0)>0]
kTM,’ (54)
I
---$i=
w’$‘( o’) dw’ + 0( w?).
(5’)
On taking account of Eqs. (53) and (54) the sum rules (52) become
Eqs. (49) and (51) we obtain the sum
LOX;
~(0)
q(O) 0
= - z
/0
w,(O)cos
I)
VHO -
we can easily derive the high-frequency expansion
M,q(O)
-M,u,(O)u,cos6(0)sin
On the other hand, on using the Kramers-Kronig relation [ 191
?z
(53)
-b%(O)-K(O)(w:(O)
+
V2N,oqO)~~(0))”
Comparing rule:
(cos*~(o))o]
=IM,(O)[
-
(since C,(t) is a real and even function this limit Eq. (30) predicts
x$4 -
bs3~(0))o)]
2 kTM,2 = ----q5)> IE
i
+ O( w-“)
x,( w> -
-
(M,(O)& (O>>o
-+0(&) 0X3
II
[2kT(cos28(0))0
$
- vH,,((cos6(0))0
,‘C?( t) dt
w
-
of the correlation
has the following asymptotic expansion [ 191 in the limit of high frequencies (that is w + 0)
C(w)
0
absorption =
The Fourier-Laplace function C,< t) namely
w?(O) + w;(O))
+M,w,(O)w,(O)sin9(0)
( 4. Sum rules for integrated
141
und Magnetic Materials 164 (1996) 133-142
/0
av2N,oqO)q(0))” 2kT
’
(52)
The equilibrium average (M,(O)M,(O))o in Eq. (52) may now be evaluated from the Eqs. (20) and (2 1) taking note of the decoupling between angular velocity and orientational functions at the moment of
Xwx;;(w)dw=
[wx;(m)dw=
;~~(x;(o)+2x’+))dw=
m2MS2No 15 L(5)? pyzNo
(55)
(1 -y),
(56)
“v’;‘No.
(57)
142
W.T. Cqffey, Yu.P. Kalm,vkoor~/Jourtml of Mugnrtism
The sum rules (55)~(57) are quite general and are valid for all models of inertial rotation of particles in a ferrofluid. In particular xi(w) and x’i (w) from Eqs. (44) and (45) satisfy these sum rules. Therefore it would be very interesting to check them experimentally. It should be noted that Eqs. (55)~(57) are the analogies of the sum rules for polar fluids in the presence of a constant external electric field [20].
5. Discussion
and conclusions
We have shown how analytic expressions may be obtained for the inertia corrected complex magnetic susceptibility of an assembly of noninteracting ferrofluid particles subject to a strong constant magnetic field H, superimposed on which is an alternating field H,(t) which is so weak as to cause only linear behavior in the response to that field. We have treated the relaxation behavior by computing the desired averages directly from the inertial Langevin equation. The use of the effective eigenvalue method allows us to treat in a simple manner the relaxation effects caused by the coupling between the constant field and the weak alternating field. The method also has the advantage that closed form expressions (44) and (45) are available for the complex susceptibility for all values of 5. Just as for the dielectric case [ 16- 181 the inertia correction removes the plateau in the high frequency absorption. The results obtained for the longitudinal and for the transverse components of the complex susceptibility can be also applied to a polar fluid in the presence of a constant electric field E,. As shown in Refs. [5,6] in this case equations for the effective dielectric relaxation times r,, and r1 are also given by Eqs. (37) and (38) where {= pE,/kT, p is the dipole moment of a polar molecule. Thus, Eqs. (44) and (45) predict the same field and frequency dependencies of magnetic and dielectric susceptibilities of a ferrofluid and a polar fluid respectively. The sum rules (55)-(57) are also similar to those for a polar fluid under the influence of a constant external field [20]. This is not unexpected because from a physical
irnd Mqwetic
Materids
164 f 1996) 133-142
point of view the behavior of a suspension of fine ferromagnetic particles in a constant magnetic field H,, is similar to that of a system of electric dipoles (polar molecules) in a constant electric field E,,.
Acknowledgements The support of this work by the British Council, the International Science Foundation, and Minnauki (Ministry of S cience) of the Russian Federation is gratefully acknowledged.
References ill W.F. Brown, Jr., Phys. Rev. 130 (19631 1677; J. Appl. Phys. Suppl. 30 (1959) 130s.
Dl T.L. Gilbert, Phya. Rev. 100 (1956) 1243. [31 Yu.L. Raikher and M.I. Shliomis, Adv. Chem. Phys. 87 (1994) 595. [41 M.A. Martsenyuk, Yu.L. Raikher and M.I. Shliomis, Sov. Phys. JETP 38 (1974) 413. [51 W.T. Coffey, Yu.P. Kalmykov and K.P. Quinn, J. Chem. Phys. 96 (1992) 5471. [61 J.T. Waldron, Yu.P. Kalmykov and W.T. Coffey, Phys. Rev. E 49 (19941 3976. [71 W.T. Coffey, P.J. Cregg and Yu.P. Kalmykov. Adv. Chem. Phys. 83 (19941 263. @I P. Debye, Polar Molecules (Chemical Catalog, New York, 1929). [91 P. Fannin and W.T. Coffey, Phys. Rev. E 52 (1995) 6129. [lOI W.T. Coffey, J. Chem. Phys. 95 (1991) 2026. [Ill W.T. Coffey, J.L. Dejardin, Yu.P. Kalmykov and K.P. Quinn, Chem. Phys. 164 (1991) 357. I121 L.D. Landau and E.M. Lifshitz, Mechanics, A Course of Theoretical Physics, Vol. I, 3rd edn. (Pergamon, Oxford, 19761. and J.T. Waldron, The [131 W.T. Coffey, Yu.P. Kalmykov Langevin Equation (World Scientific, New York, 19961. Equation (Springer, Berlin, iI41 H. Risken, The Fokker-Planck 1984). [I51 M.W. Evans, G.J. Evans, W.T. Coffey and P. Grigolini, Molecular Dynamics (Wiley, New York, 19821. ll61 Y. Rocard, J. Phys. Rad. 4 (1933) 247. [171 V.A. Dmitriev and C.B. Gurevich, Zh. Eksp. Teoret. Fir. 16 (1946) 937. [I81 J.G. Powles, Trans. Faraday Sot. 44 (I 9481 802. Fluctuations, Broken Symmetry [191 D. Forster, Hydrodynamic and Correlation Functions (Benjamin, Reading, MA, 1975). DO1 Yu.P. Kalmykov, Phys. Rev. A 45 (19921 7184.
Journal of Magnetism
and Magnetic
Materials
I64
(I 996) 133- 142
Inertial effects in the complex magnetic susceptibility of a ferrofluid in the presence of a dc bias field W.T. Coffey ’ Depuvtment ’ Institute
of Radio
Engineering
of Electronic
and Electronics
a-*, Yu.P. Kalmykov
and Electrical
qf the
Received
Engineering,
Russirm Academy
19 February
Trinity
of Sciences,
’
College, Dublin Vvedenskii
2, Irelnnd
Sq. I, Fry&o,
Moscow
141120, Rus~io
1996; revised 7 May 1996
Abstract The Euler-Langevin equation for rotational Brownian motion of a particle in a form appropriate to magnetic relaxation of ferrofluids (suspensions of single domain ferromagnetic particles) in the presence of a dc bias external magnetic field is used to include the inertia of particles. It is shown, by averaging the Langevin equation corrected for inertia, how approximate analytic expressions for the transverse and longitudinal components of the complex magnetic susceptibility may
be calculated directly from that equation with the aid of linear response theory. The method allows one to bypass the Fokker-Planck equation entirely. Sum rules for integrated absorption of electromagnetic radiation in suspensions of ferromagnetic Krpvrds:
particles are also deduced.
Complex susceptibility;
Ferrofluid;
Inertia: Langevin
equation
1. Introduction The study of the dynamic magnetic susceptibility of a ferrofluid (consisting of a colloidal suspension of fine ferromagnetic particles) is usually undertaken by constructing the underlying Fokker-Planck equation from the Landau-Lifshitz-Gilbert (LLG) equation augmented by a random noise term which is regarded as the Langevin equation of the process [ 1,2]. This method (pioneered by Brown [ 1] for single domain ferromagnetic particles and adapted to the study of the dynamic susceptibility [3] of a ferrofluid subjected to a strong dc bias field H,, superimposed on which is a weak ac field H(t) by
_ Corresponding [email protected]. 0304-8853/96/$15.00 PII
author.
Fax:
Copyright
SO304-8853(96)00390-3
+ 353-l-677-2442;
email:
Martsenyuk et al. [4], Coffey et al. [5] and Waldron et al. [6]) generally involves one in complicated mathematical manipulations. Further it does not include the inertia of the particles which is necessary in order to calculate the magnetic susceptibility of ferrofluids at high frequencies. It is worthwhile therefore to seek an alternative approach which simultaneously circumvents the need to use the Fokker-Planck equation and allows one to include inertia1 effects. A single domain ferromagnetic particle suspended in a fluid is subjected to two separate mechanisms which cause variations in its magnetization. The first is due to the thermal fluctuations inside the particle (spontaneous remagnetization). This is the solid state mechanism known as NCel relaxation [7]. The second is due to the rotation of the particle within the fluid,
0 1996 Elsevier Science B.V. All rights reserved
since changes in the orientation of the particle will effect changes in the orientation of its magnetization. This mechanism is similar to that governing the Debye relaxation of polar molecules and is consequently known as Debye relaxation with characteristic time 7o given by [8]
where p is the dynamic viscosity, v is the volume of the particle, k is the Boltzmann constant and T is the temperature. For the values of typical parameters [4] and room temperatures rn has order of magnitude about lop6 s. The effect of the inertia of Brownian particles on relaxation processes in ferrofluids is usually ignored completely. This can be justified if one restricts oneself to investigations of the low frequency relaxation processes only (up to 10 MHz). However if we extend the frequency range of interest up to 1 GHz the inertia of the particles may play an essential role. Recently the effect of the moment of inertia of single domain ferromagnetic particles on the frequency dependent complex susceptibility x(w) = x’(w) - i ,$‘( w) of an isotropic ferrofluid was reported [9]. It was demonstrated that particle inertial effects that arise from rotational Brownian motion give rise to a resonance behavior, which is indicated by the real component x’(w) becoming negative at high frequencies. In order to explain experimental data Fannin and Coffey [9] suggested a model which takes into account both the Debye and NCel relaxation mechanisms. The model [9] was suggested for an isotropic ferrofluid. However, there is another experimental situation of magnetic measurements, namely: when a ferrofluid is subjected to a strong dc bias field. In this case the medium becomes anisotropic and the model [9] can be no longer applied. The primary purpose of this paper is to evaluate inertial effects in the complex magnetic susceptibility of a ferrofluid subjected to a dc bias field. For simplicity we consider an ensemble of non-interacting single domain ferromagnetic particles, where the NCel relaxation mechanism is blocked. We show how the equation of motion of the average magnetic dipole moment may be written down directly from the inertial Langevin equation for the rotational Brownian motion of the ferrofluid particle. A knowledge of the dynamic behavior of the average mag-
netic dipole moment will allow us to determine the complex magnetic susceptibility of the ferrofluid both in the inertial and in the noninertial (or Debye) limit as a particular case. This approach is similar to that used previously by us [5-7,10,1 l] in the theory of dielectric, magnetic and Kerr effect relaxation. It allows us to obtain in simple fashion (with the aid of linear response theory) closed form expressions for the longitudinal and transverse components of the complex susceptibility and also to deduce sum rules for integrated absorption of electromagnetic radiation in a ferrofluid subjected to a dc bias field.
2. The inertial Langevin particle
equation
for a ferrofluid
In order to discuss the Langevin equation for a single domain ferrofluid particle we first consider Gilbert’s equation for the dynamic behavior of the particle magnetization vector M in the presence of thermal agitation which is [ 1,2] $4(t)
=y[M(r)
x [HT(t)
- VQU)]]~
(1)
where y is the gyromagnetic ratio and r] is a phenomenological damping parameter, H, is the total magnetic field which can consist of the applied field H, the demagnetizing field H,, and a random white noise field h [3]. Gilbert’s Eq. (1) may be rearranged explicitly (as shown. e.g., in Ref. [7]) using the properties of the triple product formula to yield that equation in the Landau-Lifshitz [I] form: ;JV)
= g’M.\[ M(t)
x&l
xWt)l XW)l~ (2)
+h’[PV)
where M, is the (constant) magnitude of the magnetization, only its direction varies with time, the constants s’ and h’ are given by Y
if = M,( 1 +
y%+M\?)
h’= ’
w2 1 + y%j’M,’ ’
(3)
The Larmor precession of the dipole moment vector d = uM of the particle decays after a time T() given by [4] TV= (2r]y2K)-‘, where K is the effec-
W.T. Coffey, Yu.P. Kalmykol:/Journal
qfMqnetism
tive magnetic anisotropy constant. The second (NCel) relaxation time rN characterises reorientation of the magnetic dipole moment, by surmounting a potential barrier vK between different directions of the easiest magnetization. The estimation for this time is given by Brown’s formula [3,7]: Q-N
=
rOu-‘/2e’r,
CT=
which is valid for a~
v
K/kT,
2. At a~
10 [4]
Under this condition the internal state of every Brownian particle in a ferrofluid may be considered as at equilibrium (in the time mu the precession of the dipole moment has time to decay and the NCel relaxation mechanism is ‘frozen’). In a ferrofluid particle, where the NCel relaxation mechanism is blocked, orientational changes in M will be due to the rotational Brownian motion of the particle only. The dipole moment vector d will be directed along the axis of easiest magnetization. Consequently the magnetization will behave like that of a bar magnet when placed in a magnetic field. Consequently the magnetization of the particle obeys the kinematic equation [ 121 (which has the same mathematical form as the Gilbert equation)
and Magnetic Materials
due to the Brownian movement following properties:
= [a(t)
On differentiating M(t)
= [b(l)
x&I(t)].
where k is the Boltzmann constant, T is the absolute temperature, s(t) is the Dirac delta-function, 6,; is Kronecker’s delta, i,j = 1,2,3, which correspond to the Cartesian axes x,y,z of the laboratory coordinate system. The overbar means the ‘statistical average’ over an ensemble of particles which have started at time t with the same magnetization and angular velocity. The drag coefficient 5 can be also expressed in terms of the dynamic viscosity /3 of the fluid as follows [ = 6vp [4]. We have supposed in writing down Eq. (7) that the particle is a sphere with moment of inertia I. The magnetic field H(t) acting on the ferrofluid particle consists of a small time dependent external field H,(t) and a large constant field H, representing the dc bias field: H(t)
=H,(t)
(4)
+ [0(l)
= [h(t) + P(t)
+sa(r)
Eq. (5) can
- JJ(t)P(t) -M(t)(M(t)
.M(t))) *Ho))
+-r(t),
(9)
where x&f(t)] x M(t)
xwN1~
(6)
The quantity 0(t) is determined by the EulerLangevin equation for the angular velocity [ 131: HI(t)
+F(r)
(5)
or, using Eq. (4), ti(t)
(8)
t)
+ V(tP’(t) = v(%YZ
xni(t)].
+H,.
On using Eqs. (4), (7) and (8), therefore be written as
Eq. (4) we obtain x&I(r)]
so that N(t) has the
1\1( t) = 0,
Zn;i( t) + lni( $2(t)
135
I64 (19%) 133-142
= v[M(t)
xH(t)]
+N(t), (7)
where the term v[ M( t) X H(t)] is the torque due to the external magnetic fields acting on the particle, la(t) is the systematic damping torque and N(r) is the white noise random torque acting on the particle,
F(t)
= v(H,(t)M.?
L(r)
= [N(r)
x&f(r)].
-M(t)(M(r)
.H,(t)),
(10)
Eq. (9) is the vector Langevin equation of the problem. This equation is now analogous to the equation of motion of a polar molecule under the influence of electric fields [ 10,13]. The quantity that directly corresponds to the dipole moment of the polar molecule /.L in the theory of dielectric relaxation is the magnetic moment vM of an individual ferrofluid particle, not the magnetization M which is the magnetic moment per unit volume. Eq. (9) also means that we are treating each particle as a rigid magnetic dipole.
136
of Mupmrn
W.T. Coffee?; Yu.P. Kalm~koi~/Journal
We can now write Eq. (9) in terms of the Euler angles 6,(p,$ and the angular velocity components [12]. In the laboratory xyz coordinate system Eq. (9) becomes Zti_( 1) + &G;(t) - ZM,o,(
+ ZM;( t)( w:( r) + w;(t))
t) wj( t)sin (p(t)
= VZZ,( MS2- M;2( t)) + F_(t) ZIG,(r) +@&(r)
+zM.,(t)(+)
+ L(t),
(11)
+ U;(t))
-~M,~,(~)%(~)COSP(f) + ZMS02( t) w3( t)cos 6( t)sin ‘p(t) = -~~,M,(~)M,(~)~,(~) Iti&)
+#&)
(‘2)
+Z,,(t>,
+zM,.(t)(w;(t)
-ZM,w,(t)o,(r)sin
+w;
4
cp(t)
-zM,w,(t)w,(t)cos6(t)cos = - VH”M,.( t)M;(
cp(t)
t) + FY( t) + L,.( t
(‘3)
)
where w, = 8, w?= @sina, w3 = +cos6+ 4, F, and Li are the projections of the vectors F and L onto the x, v and z-axes MI = M, cos 6, M, = M, sin 6 cos cp, and M,. = M, sin 6 sin cp are the projections of the vector M onto the z, x and y-axes, respectively, the constant magnetic field H, is assumed to be directed along the z-axis. On averaging Eqs. (1 l)-( 13) we obtain
md Mqnetic.
Mrueriuls 164 f IYYh) 133-142
equation where the multiplicative noise term L(t) contributes a noise induced drift term to the average.) We remark also that M, and We in Eqs. (14)-( 16) and M,(t) and w,(t) in Eqs. (1 l&(13) have different meanings, namely, M,(t) and w,(t) in Eqs. (1 I)-(13) are the stochastic variables while M, and We in Eqs. (14)-(16) are the sharp (definite) values at time t: M,(t) = Mk and am = wk. Instead of using different symbols for the two quantities we have distinguished the sharp values at time t from the stochastic variables by deleting the time argument as in Refs. [13,14]. The quantities ML and We in Eqs. (14)-( 16) are themselves random variables with the probability density function W such that WdM,dw, is the probability of finding M, and w,! in the intervals CM,, M, + dM,) and (wk, wk + dw,), respectively. Therefore in order to obtain equations for the moments which govern the relaxation dynamics of the system we must also average Eqs. (14)-( 16) over W [6,10,13] we have: Z(ti,)
+ Uhj;(
6.1: + 0~;)) - ZM,( 0+ u,sin
= vH,(M,‘Z(M,)
(M;‘))
+ &Vi,)
Z(ti,.)
+ &I&)
- ZM,( w, w,sin
= vH,(M<‘Z&i, + @iv
M;2) +F_,
+ ZM,.( OJ; + co;)
+ ZM, co2 03cos
(14) - ZM, u, 03cos
p
6 sin cp= - vH,, M, M; + F, , (‘5)
Zti,.+
of
-ZM,w,w,cos6coscp=
+ co:) - ZM, w, u,sin
cp
-vHoM,M,+F,.. (16)
We have formed the average of Eqs. (l l)-(13) as in Ref. [ 131 noting that the averages of the noise terms L,(t) will vanish throughout because in the inertial Langevin equation M(t) is statistically independent of the white noise torque N(t). (This would not be however true of the noninertial Langevin
+Z(M,(w;
- ZM,( w, w3cos = - vff,(M,,M;)
= -vH,(M,.M;)
+ (F;).
8) ( 144
+ 0;))
cp) + ZM,( w? wicos + (F,>,
1.9sin cp) ( 15a)
+ Z(M,.( co; + co;)>
cp) - ZM,( w2 w3cos 6 cos q> + (F,>.
(*6a)
where the symbol ( ) designates an ensemble average with respect to the orientations and angular velocities at the moment t (see for details Ref. [ 131). The system of Eqs. (14a)-(16a) is very complicated, nevertheless it can in principle be solved by the method of Coffey et al. [IO,1 11 which has been suggested for similar problems in dielectric and Kerr-effect relaxation. However as is known from the theory of rotational diffusion [ 151 if the dimensionless parameter y=,
ZkT
i
’
which is always true for ferrofluid particles, the inertial effects can be evaluated with high accuracy
W.T. Coffee,: Yu.P. Kalmykou/
Journal of Magnetism and Magnetic Materials 164 (1996) 133-142
in the context of an approximate approach originally introduced in the theory of dielectric relaxation by Rocard [ 161, Dmitriev and Gurevich [ 171 and Powles
[lf31. In order to proceed we first consider Eqs. (14a)(16a) in the noninertial limit where I -+ 0. In this limit the Maxwell distribution for the angular velocities has set in so that orientation and angular velocity variables are decoupled from each other as far as the time behavior of the particle orientations is concerned. Thus on setting I = 0 in Eqs. (14a)-(16a) we have [(n;r,>
+ 2kT04;)
= VH,(M,2 - 04:))
+ (F;), (17)
5(&)
+ 2kT(M,)
= - vH,(M,M,)
+ (F,), (18)
[(f$)
+2kT(M,,)
= -vHo(MyM;)
+ (F,.). (19)
In writing these we have used the fact that in the diffusion limit angular velocity components about different axes are statistically independent and have taken into account the decoupling between functions of angular orientations and angular velocities [lo]: (wiojf(M))
=
(w;~,)(f(M))
=6;~T(f(M)).
(20)
Eqs. (17)-(19) govern the relaxational dynamics of the magnetization of a ferrofluid particle in the noninertial limit. Just as in the theory of dielectric relaxation [ 16- 181 we can approximately account for the inertia of the particles simply by adding the inertial terms to the right hand side of Eqs. (17)-( 19) Z(til)
+ f(ni,>
= vH,,(M,‘I( ii,)
+ l( if,)
= - vH,,(M,M;) I(kJ)
+ 5&,,)
= - vH,(M&)
+ 2kT(Mz) CM;))
+ (F;),
+ 2kT( M,) + (F,),
[ 171 and Powles [18] in order to take into account approximately the inertia of polar molecules in the theory of dielectric relaxation. However if H, # 0, Eqs. (21)-(23) become complicated. The problem is that Eqs. (21)-(23) are the first terms in an infinite hierarchy of differential-recurrence equations [ 131. It is obvious that in order to solve these equations, we must also obtain equations for CM;),
(M,M,),
(M!M,)
and so on. However, if we merely wish to evaluate the qualitative behavior of the complex susceptibility this difficulty may be circumvented by means of the effective eigenvalue method [3,13] which has been used by Martsenyuk et al. [4] in connection with the calculation of relaxation times of ferrofluids from the Fokker-Planck equation. This method was also used by Coffey et al. [5] in conjunction with the Langevin equation and linear response theory in the study of dielectric relaxation of a polar fluid under the influence of a constant electric field. It may also be used for the present problem.
3. Inertia corrected
magnetic
relaxation
The magnetization decay of a ferrofluid, which is under the influence of a constant field H,, a small constant external field H, (v(M . H, )/kT +X 1) having been switched off at time t = 0, is from linear response theory [7] (M,,(t))
- (M,,)o = x,,H,C,,(~)
(for the case of the longitudinal and
(24) relaxation,
(M.(t))-(M.)o=xrH,C,(t) (21)
137
(for the case of the transverse where
(22)
H,]IH,)
(25) relaxation,
H, I H,),
(26)
+ 2kT(M>) + (F,.).
In the case H, = 0, Eqs. (21)-(23) those used by Rocard [ 161, Dmitriev
(23)
are similar to and Gurevich
and
(27)
are the components bility tensor,
of the static magnetic
suscepti-
- (M;(O)% (28)
-
and
As was also shown in Refs. [4.6] the relaxation times 7v can be calculated by the effective eigenvalue method [ 131. In order to calculate the effective relaxation time r,, we have to set H,(t) = 0 (the switch off case), neglect the inertial term in Eqs. (2 l&(23), and finally use the formula [7,13]
(M,(O)) -
-_
7=-
Y
=
(ni,(O>>
C,(O) d,(O)
(34)
Thus using in Eqs. (21)-(23) we get the longitudinal effective relaxation time T,, [7]: (M;(O))
r,, = -
- (M,)o
(M,(O)) ev(M “
271 rif(B,P) // = a ;, ~ ev(M.“,,)IkTsin 8 dfidp I0 / 0
(00
(M;(O)) -’ vM,‘H,
density of magnetic particles. According to linear response theory [ 191 the corresponding complex magnetic susceptibilities x,,(o) and x1 (w) are given by = x,(w)
-
- v( M,‘(0))Ho
- 2kT( M;(O))
(cos’l9 >” - (cos 6 12,
N is the number
x,(w)
- CM,>”
ix;( 0)
rD (cos219)o - [(cos 6 - COS,6)“/2 (cos’6)o -
‘1)
- (cos6);
where r,, = [/2kT
(35)
’
1 - (cos’9)o and
x
e +‘C,(t)dt
t=_ VM, Ho
(+l,l). I
(30) As was shown in Refs. [4,6,13] in the noninertial limit the behavior of the magnetization decay (M,,(t)) - (M,,)” and the autocorrelation function C,,(r) may be approximated by a pure exponential. In this limit both (M,,(t)) - (M,, >O and C,(t) obey the equations
;[‘““” -
(M,)o]
+ r;‘[(M$))
kT
The same procedure relaxation yields [7] rl=
(M,,(O))
-
applied
to the transverse
- (M,,(O))0
(K(O)) 1 - (COS%)[, 7
(36)
D 1 + (cos’6)o
Now [7] -
=O
(t>O),
(31)
(cos~~),~ =L( <) and ( COS~IY), = 1 - 2L( t>/s, where
;qt,
+ T;‘c,(
t) = 0,
(32)
where rr (y = ]I,I) are the relaxation times. Equivalently in the limit of low frequencies the complex magnetic susceptibilities can be described in accordance with Eq. (30) by the Debye equation:
(33)
L(t)
=coth[-
is the Langevin F*L(5)
I,‘[ function. -5W5)
Hence
(37) (38)
W.T. Cqfey.
Yu.P. Kalmykm
/Journnl
qf Magnetism
Eqs. (37) and (38) coincide precisely with the formulae obtained by Martsenyuk et al. [4] and Coffey et al. [5,7]. We have shown in Ref. [61 that there is good agreement between the approximate analytical expressions for the effective relaxation times Eqs. (37), (38), and the exact relaxation times obtained by analytical solution of Eqs. (17)-c 19). The difference between the results predicted by these approximate and exact expressions does not exceed IO’% for all values of 6 [61. The Debye Eq. (33) is valid only in the limit of low frequencies, viz. rv w < 1 (for this condition we can neglect the contributions of the inertial terms). In the time domain the Debye equation implies that an abrupt change in the applied magnetic field would produce an instantaneous finite alteration of the rate of change of the magnetic dipole moment which is impossible in view of the finite rotational inertia of the particles. Just as for dielectric relaxation the consequence of this is the Debye plateau for magnetic absorption, viz. w$(w) + const at o + 00, while in general the product w&‘(w) must tend to zero in the limit of high frequencies [15]. The noninertial limit is determined by the ratio Z/c< lo-“IO- ‘” s for typical values of parameters [4]. For such values of I/[ one can neglect the inertial terms in Eqs. (14a)-( 16a) at low frequencies only (up to 10 MHz). Moreover. it shows clearly that at frequencies above 1 GHz the inertial effects become important. In the next section these effects will be incorporated in the theory. In the context of the effective eigenvalue approach [4,13] having determined the effective relaxation times, we reduce Eqs. (21)-(23) to a set of linear differential equations:
I(ii,) + i(tiy,) + +
und Magnetic
164 (156%) 133-142
139
plied parallel to the z and x-axes scopic longitudinal and transverse magnetization
induces macrocomponents of
(M;(t))
Mnterials
- (MC)” = (M;)exp(iwt)
and
= (M,)exp(iwt),
(42)
respectively. The corresponding complex magnetic susceptibility
components of the are then defined as
and x.(w)=
CM,)
lim -. H,-0 H,
(43) Substituting Eqs. (42) into (39) and (40) we then obtain from Eq. (43) the longitudinal and transverse
-0.21 -2
I -1
I 0 h$~
I 1
1 2
'J
- (MJO) = (FL (39)
An ac magnetic
field
H,(t) = H, exp(i ot)
ap-
Fig. I. The real x;i (a) and imaginary x;; (b) parts of the normalized longitudinal complex susceptibility as a function of log,,,(w~,). Curves I. 2 and 3 are plotted for 5 = 0.01, 2 and 5, respectively, and l/i = O.~T,,.
W.T. Coffe4\; Yu.P. Kalm~ko~~/Journal
140
components namely
of the complex
x,,(w> =1+
magnetic
of Magnetism
and Magnetic
Materials
164 (1996)
133-142
susceptibility
XII
(44)
i wr,, - ZW*T,,/{
and
xL(w)= 1 +iW+rl
XL
_------------
(45)
-Zw2r I /l’
where the relaxation times r,, and rL Eqs. (37) and (38) respectively,
are given by
XII= “2;TMSZ [ 1 - P( [) - 2L( [)/[I
(46)
and V2&MS2 L(S) XL= kT 5 are the components of the static susceptibility
(47)
Fig. 3. The o$(w)/G as a function of log10(w70). The inertia corrected Eq. (44) (dots) compared with w7o x;(w)/G as calculated from Debye’s Eq. (33) (dashed lines). Curves I, 2 and 3 are plotted for ( = 0.01, 2 and 5, respectively, and f/i = 0.17,.
tensor.
1.0 -
In Figs. 1 and 2 we plot the real and imaginary parts of the longitudinal component of the normalized complex susceptibility x,,(w)/G, where
0.6 :
0.6 -
V2NoM2 G=
0.4 -
0.2 -
0.0 I
02 -2
3W‘
.
These figures demonstrate clearly that particle inertial effects that arise from rotational Brownian motion give rise to a resonance behavior, which is indicated by the real component of the complex
-1
0 b,pJ
1
2
7.1
‘Or Oar
2. The
real x;, (a) and imaginary xi (b) parts of the normalized longitudinal complex susceptibility as a function of Iog,,,(wTo). Curves 1, 2 and 3 are plotted for 1/{=0.01~,, 0.17” and 0.5ro, respectively, and 5 = I.
Fig. 4. The w~;[(o)/G as a function of log,,( u-o). The inertia corrected Eq. (44) (dots) compared with Woo x;(w)/ G as calculated from Debye’s Eq. (33) (dashed line). Curves 1, 2 and 3 are plotted for I/[=O.OIT,. 0.17, and 0.57,, respectively and .$= I.
W.T. Coffey, Yu.P. Kalmykor/Journal
ofMagnetism
susceptibility x’(w) becoming negative at high frequencies. Moreover the inertia correction embodied in Eqs. (44)-(45) does remove the plateau in the high-frequency absorption spectrum (see Figs. 3 and 4). The qualitative behavior of the transverse component is similar to that of the longitudinal one and therefore we do not plot it.
t = 0 as follows:
the averaging (M,,(O)~,,(O))o = (M,(O)[
iti,
- MJO)(
+
?(,
-M’(O))])
C,,( W) = /,-eei
transform
-ii
=_- kT;2 [1 -
C,(O)+ i qo)
w
w3[p,(0)qO))o
kTw2
(48) of time). In
w;(O))
+
+ 0( Cl-‘).
-pwP,(O)
(49) = - 5
[2kT(sin’(O)co&(O))o
+ ~H,M,~((cos6(0)sin~8(O)cos~~(O)>o)]
asymptotic =_-
[ 1 + (c0&(0)>0]
kTM,’ (54)
I
---$i=
w’$‘( o’) dw’ + 0( w?).
(5’)
On taking account of Eqs. (53) and (54) the sum rules (52) become
Eqs. (49) and (51) we obtain the sum
LOX;
~(0)
q(O) 0
= - z
/0
w,(O)cos
I)
VHO -
we can easily derive the high-frequency expansion
M,q(O)
-M,u,(O)u,cos6(0)sin
On the other hand, on using the Kramers-Kronig relation [ 191
?z
(53)
-b%(O)-K(O)(w:(O)
+
V2N,oqO)~~(0))”
Comparing rule:
(cos*~(o))o]
=IM,(O)[
-
(since C,(t) is a real and even function this limit Eq. (30) predicts
x$4 -
bs3~(0))o)]
2 kTM,2 = ----q5)> IE
i
+ O( w-“)
x,( w> -
-
(M,(O)& (O>>o
-+0(&) 0X3
II
[2kT(cos28(0))0
$
- vH,,((cos6(0))0
,‘C?( t) dt
w
-
of the correlation
has the following asymptotic expansion [ 191 in the limit of high frequencies (that is w + 0)
C(w)
0
absorption =
The Fourier-Laplace function C,< t) namely
w?(O) + w;(O))
+M,w,(O)w,(O)sin9(0)
( 4. Sum rules for integrated
141
und Magnetic Materials 164 (1996) 133-142
/0
av2N,oqO)q(0))” 2kT
’
(52)
The equilibrium average (M,(O)M,(O))o in Eq. (52) may now be evaluated from the Eqs. (20) and (2 1) taking note of the decoupling between angular velocity and orientational functions at the moment of
Xwx;;(w)dw=
[wx;(m)dw=
;~~(x;(o)+2x’+))dw=
m2MS2No 15 L(5)? pyzNo
(55)
(1 -y),
(56)
“v’;‘No.
(57)
142
W.T. Cqffey, Yu.P. Kalm,vkoor~/Jourtml of Mugnrtism
The sum rules (55)~(57) are quite general and are valid for all models of inertial rotation of particles in a ferrofluid. In particular xi(w) and x’i (w) from Eqs. (44) and (45) satisfy these sum rules. Therefore it would be very interesting to check them experimentally. It should be noted that Eqs. (55)~(57) are the analogies of the sum rules for polar fluids in the presence of a constant external electric field [20].
5. Discussion
and conclusions
We have shown how analytic expressions may be obtained for the inertia corrected complex magnetic susceptibility of an assembly of noninteracting ferrofluid particles subject to a strong constant magnetic field H, superimposed on which is an alternating field H,(t) which is so weak as to cause only linear behavior in the response to that field. We have treated the relaxation behavior by computing the desired averages directly from the inertial Langevin equation. The use of the effective eigenvalue method allows us to treat in a simple manner the relaxation effects caused by the coupling between the constant field and the weak alternating field. The method also has the advantage that closed form expressions (44) and (45) are available for the complex susceptibility for all values of 5. Just as for the dielectric case [ 16- 181 the inertia correction removes the plateau in the high frequency absorption. The results obtained for the longitudinal and for the transverse components of the complex susceptibility can be also applied to a polar fluid in the presence of a constant electric field E,. As shown in Refs. [5,6] in this case equations for the effective dielectric relaxation times r,, and r1 are also given by Eqs. (37) and (38) where {= pE,/kT, p is the dipole moment of a polar molecule. Thus, Eqs. (44) and (45) predict the same field and frequency dependencies of magnetic and dielectric susceptibilities of a ferrofluid and a polar fluid respectively. The sum rules (55)-(57) are also similar to those for a polar fluid under the influence of a constant external field [20]. This is not unexpected because from a physical
irnd Mqwetic
Materids
164 f 1996) 133-142
point of view the behavior of a suspension of fine ferromagnetic particles in a constant magnetic field H,, is similar to that of a system of electric dipoles (polar molecules) in a constant electric field E,,.
Acknowledgements The support of this work by the British Council, the International Science Foundation, and Minnauki (Ministry of S cience) of the Russian Federation is gratefully acknowledged.
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