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Determination of the magnetic anisotropy of ferrofluids from torque magnetometry data
83
Journal of Magnetism and Magnetic Materials 38 (1983) 83-92 North-Holland
DETERMINATION OF THE MAGNETIC TORQUE MAGNETOMETRY DATA
ANISOTROPY
OF FERROFLUIDS
FROM
R.W. CHANTRELL Dept. of Physics and Astronomy,
Preston Polytechnic, Corporation St., Preston PRI 2TQ, UK
B.K. TANNER and S.R. HOON Dept. of Physics, Durham University, South Road, Durham DHI SLE, UK
Received 17 January 1983; in revised form 18 March 1983
A theoretical analysis is presented which enables the uniaxial magnetic anisotropy constant K, of particles in ferrofluids, frozen in an external field, to be obtained from torque magnetometry measurements. The two-fold symmetry of the torque curve, found experimentally, is correctly predicted. An asymptotic solution is found which enables K, to be determined without recourse to iterative numerical methods. In this limit, the torque amplitude varies linearly with the inverse of the freezing field for large freezing fields. For all cases, extraction of K, requires an accurate knowledge of the particle size distribution parameters.
Introduction
The magnetic anisotropy of the small, single domain, particles constituting the magnetic component of a ferrofluid is a fundamentally important parameter in determining the magnetization reversal process. Until now values have had to be extracted by indirect fitting procedures [l]. There is good agreement in the literature that the anisotropy is predominantly uniaxial, that it is determined by the particle shape and that the anisotropy energy per unit volume can be described by a single constant K,, However, the exact value deduced depends very much on the technique used for measurements. For example by a fitting a theoretical model to the initial susceptibility as a function of temperature a K value of 4.3 x lo5 erg/cm3 was deduced for an Fe,O, fluid [l]. Fitting to the isothermal remanence magnetization of the same fluid yielded K = 2.7 X lo5 erg/cm* [ 11. From Mossbauer measurements a value of K = 5.1 x lo4 erg/cm3 was obtained [2] for a similar fluid. Torque magnetometry [3] is a well-established technique for the measurement of magnetic anisotropy in single and textured polycrystalline material. In a magnetic field, the magnetization of the sample tends to align with the applied field, but is prevented from doing so by the anisotropy energy. The net effect is a torque on a freely suspended sample given by the derivative of the anisotropy energy with respect to angle. From measurement of this torque in various directions and appropriate Fourier analysis, the various phenomenological anisotropy constants can be derived [3]. We have recently performed torque magnetometry experiments on ferrofluids frozen in large aligning fields. Well defined torque curves showing systematic variations with respect to both measuring and magnetizing field were obtained [4]. In this paper we present the theoretical analysis required to extract the uniaxial particle anisotropy constant K from the torque data. 03048853/83/0000-0000/$03.00
0 1983 North-Holland
R. W. Chantrell et al. / Magnetic anisotropy of ferrofruids
84
2. Torque on a system of equal diameter particles As is later proven, if the particles are randomly oriented no macroscopic torque will be detected in high measurement fields. In order to introduce a preferred alignment the ferrofluids are frozen in a strong magnetic field. This introduces a partial alignment with respect to the direction of this freezing field but a random distribution about this axis remains. Torque magnetometry is performed by measuring the torque induced as a measuring field B is rotated in any plane of the sample containing the freezing field direction. Consider then a sample constrained to the shape of a disc, frozen in a field B, applied in the plane of the disc. In the geometry shown in fig. 1 the disc plane is concomitant with the plane of the page. The measuring field is rotated in this plane. We define \k as the angle between the directions of the measuring field B,,, Q, as the angle between the easy axis c of a particle and B, and 8 as the angle between c and 1%. n is the angle which the plane containing c and B, makes with the disc plane. It is straightforward to show that, 8, !P, @ and n are related by cos@=cos8cos
?P+sin@sin
!Pcosn.
(1)
Let us consider initially the torque from a system of particles of equal diameter. Suppose that the measuring field is sufficiently large that, during measuring of the torque the moments are all pulled from the easy directions into the measurement field direction. Then the anisotropy energy per unit volume for a particle of diameter D and moment ~1is FK = K, sin2@,
(2)
where we assume that the particle anisotropy is uniaxial and fully characterized by the first anisotropy constant per unit volume K, and that the particle has volume aD3/6. Thus from eq. (1) and (2), F,=K,(l-(cosBcos!P+sin8sin~Pcosn)~). Now there is a random volume is F, = K,{l
distribution
easy axes about B,, so the average anisotropy
- cos2B cos2’k - i sin 28 sin 29(cos
= K,{ 1 - cos2B The anisotropy
of particle
(3)
energy
cos’!P
-
+
averaged
Fig. 1. Coordinate system with respect disc-shaped sample.
energy per unit
n) - sin28 sin29(cos2n))
sin’@ sin29}. over the distribution
to the plane of the
(4) of orientation
Fig. 2. Coordinate
with respect
system referred
to the freezing
to the moment
field
direction
c.
85
R. W. Chantrell et al. / Magnetic anisotropy of ferrojluidv
direction is, therefore given by 4=K,{l The
-f
sin29+($
sin’+-
l)(C0S28)}.
(5)
measured torque per unit particle volume r, is given by
= ;K,
r, = -a&a*
[1-
3(c0s28)
]
sin 2*.
(6)
For particles of volume V, the averaged torque is r, = f IX, [ 1 - 3(cos28)]sin
29
(7)
and for N identical particles, the total torque becomes Nr, = fNIX,
[ 1 - 3(cos29)] sin 2\k.
(8)
If the volumetric fraction of particles in the fluid is E, these particles take up or comprise a total fluid volume NV/E. Thus, for equal diameter particles the torque per unit volume ofj7uid r is given by r = &K, [ 1 - 3(cos28)] sin 2*.
(9)
For a real fluid this equation must be integrated over the particle size distribution. The alignment function (cos28), which is a function of particle size, is evaluated in the next section.
3. The alignment function 3.1. An abrupt blocked-unblocked
transition
The alignment function (cos28) will be determined by the distribution of easy axis directions locked in during the freezing. As the field aligns the magnetic moments, the easy axis are only aligned indirectly because of particle anisotropy correlating moment and easy axis. It is clear that for particle diameters D very much greater than the superparamagnetic diameter [5] DP (at the freezing temperature), the moments are locked to the easy axis as the anisotropy energy of these large particles is much greater than kT. For particles with D a: D, the thermal energy is much greater than the anisotropy energy and hence particles will tumble about the moment direction. Such small particles will thus be frozen in with axes randomly oriented. As a first approximation we consider a model involving an abrupt transition from fully blocked to fully unblocked axes at D,. Here, we define D, in terms of the condition that thermal excitation over the anisotropy energy barrier should not occur within the time take to freeze the sample t,. At a freezing temperature T, we have 0; = (hkT,/rK,
) ln( tr/r,),
(10)
where ~a = 1 ns. The unblocked, randomly oriented small particles can (cos2f3) = l/3 and h ence the torque due to these particles to average cos28 over a Maxwell-Boltzmann distribution average over cos B which every undergraduate performs to lW27r sin 8 cos’8 exp( @a cos B/kT,)dB
(cos2e) = J0
*2?r sin 0 exp(&
cos @/kT,)dO
be easily seen to have an orientation function is zero. For the fully blocked particles we need in an exactly analogous manner to the usual obtain the Langevin function. Specifically
=
I_bb( kTF/&,)2v2
exp( v)dv
(11) ’ / -b
exp(v)dv
’
R. W. Chantrell et al. / Magneiic anisotropy of ferrofruids
86
where v = pB,, cos B/kT, and b = pB,,/kTF. On evaluation of the integrals we have,
(cos*e)= 1 -i
coth b+s=
1 +(b),
(12)
where L(b) is the Langevin function. We now introduce the particle diameter distribution function f(r). y = D/D, and JJ, = D,/D,, where D, is the median particle diameter. The torque per unit volume of fluid according to this model is r= EK, sin 29
(13)
The above abrupt blocking transition approximation is similar to that introduced by NCel [5] and used to describe the time dependence of the magnetization of ferrofluid systems [6,7]. In this latter context the boundary between superparamagnetic and blocked particles can be considered abrupt because of the very rapid variation of the NM relaxation time with particle size. Even so, we have recently shown that it is necessary to sum the relaxation times of individual particles if a good fit to new, high precision, data is to be obtained [8]. Although it provides clear physical insights into the problem, in the context of torque magnetometry, the approximation is not at all satisfactory. While only a very small fraction of the particle size distribution contributes to most time dependent magnetization experiments, the torque experiments measure the effects of all particles and partial alignment of particles with D 5 D, gives a major contribution to the torque. As a result, the torque contribution from up to half the particles in a typical fluid can be overlooked, with the consequence that anisotropy constants derived using this model are artificially high. In the following section we present a derivation of the alignment function which includes particles only partially aligned during freezing. 3.2. Partial alignment during freezing For the general case, we must assume that the easy axis direction and moment direction are no longer locked. Let @ be the angle between them. The free energy of the particle can then be written as E = K,V sin*+ - pB, cos 8,
(14
using the coordinate system of fig. 1. It is important to note that the energy is independent of the angle between easy axis and freezing field, since coupling between easy axis and field is indirect, via the moment. Because of this the value of anisotropy has no bearing on the magnetization in the fluid state. This result will be proved later. It is convenient to choose a new coordinate system (fig. 2) centred on the moment direction cc. Then cos
e = cos J, cos + + sin Ic,sin $I cos E
(15)
and E=K,Vsin2+-pBOco~~.
With a = K,V/kT,, E, = E/kT,
(16)
we have = a sin*+ - b cos 4.
(17)
The partition function, z, is given by z=
V nsin 4 sin + exp( -a lJ 0 0
sin*+ + b cos #)d+d#.
(18)
R. W. Chantrell et al. / Magnetic anisotropy of ferrofruiidr
87
The alignment function can thus be written as ‘II nsin 4 sin $I cos2t9 exp( -a sin’+ + b cos #)d+d#dt (cos%) = 12W;2$-, sin # sin + exp( - sin2+ + b cos $)d+d$d[ 0
0
(19) ’
0
with cos28 obtained from eq. (15). It is here that we see that the appropriate choice of coordinate system has retained a simple form for the free energy, compressing all the complexities into the pre-exponential factor, and hence considerably simplifying the evaluation of (~0~~8). The dominator of eq. (19) is just 2 B times the partition function z which can be simply evaluated. The partition function is z = I(u) eeo(eb - e-‘)/b,
(20)
r(a) =I_+,’ exp( ax2)dx
(21)
where
(x is a dummy variable). It is noteworthy that eq. (20) enables us to prove very simply that the magnetization of a ferrofluid is the Langevin function and is therefore independent of the particle anisotropy, a result first shown by Krueger [9] a few years ago. From eq. (20), the reduced magnetization @is given immediately by @= (az/ab)/z
= coth b - (l/b)
= L(b).
(22)
Evaluation of the numerator of the alignment function (eq. (19)) is straightforward.
From eq. (15)
cos28 = cos2# cos2+ + 2 cos # cos S#Jsin JI sin (p cos I + sin’+ sin2+ cos25.
(23)
In eq. (19), the integral over 4 is between limits of 0 and 21~ and hence the second term in eq. (23) averages to zero. Thus, (cos28) = (2nz)-‘(lW
sin + cos2J/ exp(b cos J’,dqiV
+ lW sin 4 sin24 exp( b cos #)d$iW
sin C#I COS’C#S exp( -a sin2+)dg12Wd5
sin $I sin2$ exp( -a sin2+)d+12” cos’[dt).
(24)
This can be rewritten as (cos28) = (27rz)-‘(2?r[~,x2 +rr ‘,(l /
-x2)
ebXdx ePa/_‘,x2 euX2dx ebXedx e-O/:,(1
-x2)
e”“‘dx).
(25)
After considerable manipulation we arrive at @x28)
=
(1+(b))(
--&-$-)
-y
(-&-$-
I).
(26)
Thus, using eq. (9), the torque per unit volume of fluid r is r = &K,/o”[
1 - 3(cos28)] f( y)dy sin 2!P,
(27)
R. W. Chantrell er al. / Magnetic anisotropy of ferrofruiidr
88
where (cos28) is given by eq. (26). These two equations permit exact numerical solution and hence the determination of the particle anisotropy constant K, from the torque data.
4. The asymptotic limit of (I B 1, the strong coupling limit We recall that u = K,L’/kT,. z(a)
14
(
It is shown in the appendix that for a > 1,
1+1+3+... 2a
(28)
. 4a2
i
Z(a) is defined in eq. (20) and (21). Thus, to a first order approximation eO/uZ(a) = 1 - 1/2a
(29)
and hence (cos%) = 1 - (2/b)L(b).
(30)
This result is identical to that of eq. (12) in which it was assumed that the magnetic moment lay in the easy direction. We note that the torque is determined principally by the first term in eq. (26). (eO/(aZ(a)) = 1 and hence, in comparison to the first, the second term is negligible.) Thus the strength of the coupling between the easy axis and magnetic moment is largely represented by the function
@(,)=ef_L
2a’
al(a)
Taking the first two terms of Z(a) from eq. (28) yields @(u)=l-(l/u).
(32)
Fig. 3 shows the approximate solution (eq. (32)) and numerically evaluated exact solution (eq. (3 1)) of @(a) u I Mlerp(a)
0.8 0.6 04 0.2 -
I , 0
2
a 4
6
8
10
12
14
16
18
Fig. 3. Approximate and numerical solutions of the function e(a) as a function of a. For a > 1.5 the two solutions are very close.
20
0I
2
4
6
8
10
Fig. 4. Approximate and numerical solutions of the alignment integral I(a) as a function a.
89
R. W. Chantrell et al. / Magnetic anisotropy of ferrofluids
as a function of a. Provided that ~12 1.5, the approximate solution is satisfactory. The minimum diameter Dmin for which the approximate solution is valid is thus Dmin = (9kTF/?rK,)“3.
(33)
For TF = 178 K and K, = 2 X lo6 erg/cm, corresponding to a cobalt fluid in toluene, we have Dmin = 33 A. For Fe,O, particles in water, TF = 273 K and K, = 5 X lo5 erg/cm3. Hence Dmin = 60 A. As ferrofluids usually have median particle sizes greater than these values the approximate solution of @(a) can often be applied. Where a significant fraction of particles have D < Dmin, a numerical solution must be sought. Using eq. (26) and (32), for large a (co&?) = 1 - (2/b)L(b)
- l/a.
(34)
Hence, from eq. (6), we have for rp, the torque per unit particle volume on particles of diameter D,
1
%L(b)-l+&
sin29.
We note that as b + co, r, + - K,( 1 - 3/2a) sin 2+. This differs from the bulk single crystal anisotropy constant K, by 3K,/2a or 3kTF/2V. This can be thought of as an “entropy term” representing the fact that thermal agitation ensures that even in infinite field the alignment of easy axes will not be perfect. For a fluid containing particles of a single diameter of volumetric packing fraction E, the torque per unit fluid volume r is iL(b)-1
+&
1
sin2q.
However, as ferrofluids have a range of particle sizes, then for a particle volume distribution f(v) y = V/V,, and VYis the median particle volume, we have F= rK, sin 29
- 1 +&j/(Y)
where
d.v.
(37)
5. Variation of torque amplitude with large freezing field
In the limit b ZB 1 (where b = pBO/kTF), f=
-rK,
sin 2q
eq. (37) becomes
1-$-;)f(~)dy.
(38)
As the distribution is normalized, / o%Y)dy
(39)
= 1
and hence
it4, is the saturation magnetization per unit volume of the material comprising the magnetic particles. Many ferroflmds exhibit a lognormal distribution of particle volume fraction, namely
f(v) = (GJ_$
exp( - (ln y)2/2cr2).
(41)
R. W. Chantrell et al. / Magnetic anisotropy of ferrofluicis
90
Therefore, with the substitution r=a(lny)-a
(42)
it is easy to prove that
Jaoy-lf(y)dy
= exp(a2/2).
0
Substitution of eq. (43) into (40) yields, for b > 1 and a lognormal particle distribution function f=
-cK,
sin 2!# 1 -s I
I”
exp( a2/2) - *
S”0
exp( u2/2)
1.
(44)
We note that the torque amplitude at large field varies inversely with the freezing field B,,, a feature which has been confirmed experimentally [4]. By plotting f as a function of B; ’ and extrapolating to B; ’ = 0 we obtain the saturation torque r,,, as c;,,, = --E sin 29[ K, - (3kT,/2Vv)
exp( u2/2)].
(45)
In terms of a lognormal diameter distribution parameter un c,, = -
E
sin 2*[ K, - (9kT,/&)
exp( 9uk/2)].
(46)
6. Conclusions
The above theoretical analysis has shown the following features: of partially aligned frozen ferrofluids may be used to extract a value of the particle anisotropy constant K,. b) The torque from particles exhibiting uniaxial particle anisotropy varies as sin 29. This is confirmed experimentally [4]. c) The torque amplitude varies linearly with the inverse of the freezing field for large fields. This is also confirmed experimentally [4]. 4 An asymptotic solution for large a = K,V/kT, can be used satisfactorily for many ferrofluid systems. e) If a is not large, iterative numerical methods can be used to extract K,. 0 Determination of K, requires an accurate knowledge of the particle size distribution function.
4 Torque magnetometry
Acknowledgements
The work was supported in part by the University of Durham Special Research Projects Fund. We gratefully acknowledge the care with which Mrs. V. Todd and Mrs. M. Bradley typed the manuscript.
Appendix. Asymptotic solutions for the alignment integral
We require asymptotic solutions of the integral exp( ux2)dx,
(A.11
R. W. Chantrell et al. / Magnetic anisotropy of ferrofhds
91
valid for small and large a. A. 1. Strong coupling limit (a >s-1) limit is for a >> 1. Because the function
By far the most important Z(a) = 2l’ Substituting
is even,
exp( ax*)dx.
z *=l+x*gives
Z(a) = 2e”i’z(l
-z*)-“*
eCdL2dz.
(A-3)
When a x=. 1, the integrand is dominated by the variation of exp( - az*). This means attention to small z. Using the binomial expansion of (1 - z)-‘/* we have Z(a) = Ze”~‘$z*‘+‘B(n) where B(n)
we can restrict
e-11z2dz,
is the coefficient
our
(A-4)
of x” in the expansion
of (1 - z*). Thus,
Z(a) =2eUe(-l)B(n)l(n), 1
(A.5)
Z(n) =~‘z*‘+’
(A-6)
where, e-az2dz,
Since a x- 1, most of the variation of the integrand takes place for z small. introducing appreciable error we can extend the upper limit of the integral to infinity. Z(n) =lmz2’+i Substituting
eq. (A.7) into (A.5) and writing
Z(a)=2e”C-Writing
e-ar2dz = n!/2a”+‘.
n
(2n)!
1 (nl)24”
n!
the first few terms explicitly
1+1+3+x+2a
4a2
8a3
without
(A.7) B(n)
=- “,“;:G!LL. 1 .
2a”+’
Therefore, Then
explicitly
gives .
64.8)
gives (A.9)
It must be noted that, generally eq. (A.9) represents a divergent series. The ratio of successive term R(n) = (n-3)a-’ and thus for finite a, R(n) B 1 for large n and the series diverges. This is a consequence of the approximation made in the evaluation of Z(n). In this process the upper limit of the integral was taken to infinity. This will be a reasonable approximation for the first few terms where n is small, but will be increasingly invalid as n increases, For large n the approximate value of Z(n) will be a drastic over-estimate, thus producing the divergence of the series in eq. (A.8). As long as a is large enough so that the first few terms give sufficient accuracy, eq. (A.8) and (A.9) will give accurate values of Z(a).
R. W. Chantrell et al. / Magnetic anisotropy of jerrojiuids
92
The validity of this approach is tested in fig. 4. Here, the approximate
I(a)=; ( 1+&
solution
1
is compared with a numerical very satisfactory.
evaluation
of the full integral. It is clear that for a 2 1.5, the agreement is
A.2. Weak coupling limit a CC I Here I(a)
= /_:’
exp( ax*)dx
=
/ _+jl
+ ax*)dx
= 2( 1 + a/3).
This is not so useful as the strong coupling limit, but is included
(A.lO) for completeness.
References [II A. Tari, J. Popplewell and SW. Charles, J. Magn. Magn. Mat. 15-18 (1980) 1125. PI A. Tari, R.W. Chantrell, SW. Charles and J. Popplewell, Physica 97B (1979) 57. (31 R. Pearson, in: Experimental Magnetism, vol. I, eds. G. Kalvius and R. Tebble (Wiley, New York, 1979) p. 138. 141 S.R. Hoon, B.K. Tanner and M. Kilner J. Magn. Magn. Mat. 39 (1983) special issue. VI L. N&l, Advan. Phys 4 (1955) 191. VI K. G’Grady, R.W. Chantrell, J. Popplewell and S.W. Charles, Trans. IEEE Magn. MAG-17 (1981) 2943. [71 A.T. Cayless, S.R. Hoon, B.K. Tanner, R.W. Chantrell and M. Kilner, J. Magn. Magn. Mat. 30 (1983) 303. PI R.W. Chantrell, S.R. Hoon and B.K. Tanner, J. Magn. Magn. Mat. 38 (2) (1983) in press. [91 D.A. Krueger, J. Appl. Phys. 50 (1979) 8169.
Journal of Magnetism and Magnetic Materials 38 (1983) 83-92 North-Holland
DETERMINATION OF THE MAGNETIC TORQUE MAGNETOMETRY DATA
ANISOTROPY
OF FERROFLUIDS
FROM
R.W. CHANTRELL Dept. of Physics and Astronomy,
Preston Polytechnic, Corporation St., Preston PRI 2TQ, UK
B.K. TANNER and S.R. HOON Dept. of Physics, Durham University, South Road, Durham DHI SLE, UK
Received 17 January 1983; in revised form 18 March 1983
A theoretical analysis is presented which enables the uniaxial magnetic anisotropy constant K, of particles in ferrofluids, frozen in an external field, to be obtained from torque magnetometry measurements. The two-fold symmetry of the torque curve, found experimentally, is correctly predicted. An asymptotic solution is found which enables K, to be determined without recourse to iterative numerical methods. In this limit, the torque amplitude varies linearly with the inverse of the freezing field for large freezing fields. For all cases, extraction of K, requires an accurate knowledge of the particle size distribution parameters.
Introduction
The magnetic anisotropy of the small, single domain, particles constituting the magnetic component of a ferrofluid is a fundamentally important parameter in determining the magnetization reversal process. Until now values have had to be extracted by indirect fitting procedures [l]. There is good agreement in the literature that the anisotropy is predominantly uniaxial, that it is determined by the particle shape and that the anisotropy energy per unit volume can be described by a single constant K,, However, the exact value deduced depends very much on the technique used for measurements. For example by a fitting a theoretical model to the initial susceptibility as a function of temperature a K value of 4.3 x lo5 erg/cm3 was deduced for an Fe,O, fluid [l]. Fitting to the isothermal remanence magnetization of the same fluid yielded K = 2.7 X lo5 erg/cm* [ 11. From Mossbauer measurements a value of K = 5.1 x lo4 erg/cm3 was obtained [2] for a similar fluid. Torque magnetometry [3] is a well-established technique for the measurement of magnetic anisotropy in single and textured polycrystalline material. In a magnetic field, the magnetization of the sample tends to align with the applied field, but is prevented from doing so by the anisotropy energy. The net effect is a torque on a freely suspended sample given by the derivative of the anisotropy energy with respect to angle. From measurement of this torque in various directions and appropriate Fourier analysis, the various phenomenological anisotropy constants can be derived [3]. We have recently performed torque magnetometry experiments on ferrofluids frozen in large aligning fields. Well defined torque curves showing systematic variations with respect to both measuring and magnetizing field were obtained [4]. In this paper we present the theoretical analysis required to extract the uniaxial particle anisotropy constant K from the torque data. 03048853/83/0000-0000/$03.00
0 1983 North-Holland
R. W. Chantrell et al. / Magnetic anisotropy of ferrofruids
84
2. Torque on a system of equal diameter particles As is later proven, if the particles are randomly oriented no macroscopic torque will be detected in high measurement fields. In order to introduce a preferred alignment the ferrofluids are frozen in a strong magnetic field. This introduces a partial alignment with respect to the direction of this freezing field but a random distribution about this axis remains. Torque magnetometry is performed by measuring the torque induced as a measuring field B is rotated in any plane of the sample containing the freezing field direction. Consider then a sample constrained to the shape of a disc, frozen in a field B, applied in the plane of the disc. In the geometry shown in fig. 1 the disc plane is concomitant with the plane of the page. The measuring field is rotated in this plane. We define \k as the angle between the directions of the measuring field B,,, Q, as the angle between the easy axis c of a particle and B, and 8 as the angle between c and 1%. n is the angle which the plane containing c and B, makes with the disc plane. It is straightforward to show that, 8, !P, @ and n are related by cos@=cos8cos
?P+sin@sin
!Pcosn.
(1)
Let us consider initially the torque from a system of particles of equal diameter. Suppose that the measuring field is sufficiently large that, during measuring of the torque the moments are all pulled from the easy directions into the measurement field direction. Then the anisotropy energy per unit volume for a particle of diameter D and moment ~1is FK = K, sin2@,
(2)
where we assume that the particle anisotropy is uniaxial and fully characterized by the first anisotropy constant per unit volume K, and that the particle has volume aD3/6. Thus from eq. (1) and (2), F,=K,(l-(cosBcos!P+sin8sin~Pcosn)~). Now there is a random volume is F, = K,{l
distribution
easy axes about B,, so the average anisotropy
- cos2B cos2’k - i sin 28 sin 29(cos
= K,{ 1 - cos2B The anisotropy
of particle
(3)
energy
cos’!P
-
+
averaged
Fig. 1. Coordinate system with respect disc-shaped sample.
energy per unit
n) - sin28 sin29(cos2n))
sin’@ sin29}. over the distribution
to the plane of the
(4) of orientation
Fig. 2. Coordinate
with respect
system referred
to the freezing
to the moment
field
direction
c.
85
R. W. Chantrell et al. / Magnetic anisotropy of ferrojluidv
direction is, therefore given by 4=K,{l The
-f
sin29+($
sin’+-
l)(C0S28)}.
(5)
measured torque per unit particle volume r, is given by
= ;K,
r, = -a&a*
[1-
3(c0s28)
]
sin 2*.
(6)
For particles of volume V, the averaged torque is r, = f IX, [ 1 - 3(cos28)]sin
29
(7)
and for N identical particles, the total torque becomes Nr, = fNIX,
[ 1 - 3(cos29)] sin 2\k.
(8)
If the volumetric fraction of particles in the fluid is E, these particles take up or comprise a total fluid volume NV/E. Thus, for equal diameter particles the torque per unit volume ofj7uid r is given by r = &K, [ 1 - 3(cos28)] sin 2*.
(9)
For a real fluid this equation must be integrated over the particle size distribution. The alignment function (cos28), which is a function of particle size, is evaluated in the next section.
3. The alignment function 3.1. An abrupt blocked-unblocked
transition
The alignment function (cos28) will be determined by the distribution of easy axis directions locked in during the freezing. As the field aligns the magnetic moments, the easy axis are only aligned indirectly because of particle anisotropy correlating moment and easy axis. It is clear that for particle diameters D very much greater than the superparamagnetic diameter [5] DP (at the freezing temperature), the moments are locked to the easy axis as the anisotropy energy of these large particles is much greater than kT. For particles with D a: D, the thermal energy is much greater than the anisotropy energy and hence particles will tumble about the moment direction. Such small particles will thus be frozen in with axes randomly oriented. As a first approximation we consider a model involving an abrupt transition from fully blocked to fully unblocked axes at D,. Here, we define D, in terms of the condition that thermal excitation over the anisotropy energy barrier should not occur within the time take to freeze the sample t,. At a freezing temperature T, we have 0; = (hkT,/rK,
) ln( tr/r,),
(10)
where ~a = 1 ns. The unblocked, randomly oriented small particles can (cos2f3) = l/3 and h ence the torque due to these particles to average cos28 over a Maxwell-Boltzmann distribution average over cos B which every undergraduate performs to lW27r sin 8 cos’8 exp( @a cos B/kT,)dB
(cos2e) = J0
*2?r sin 0 exp(&
cos @/kT,)dO
be easily seen to have an orientation function is zero. For the fully blocked particles we need in an exactly analogous manner to the usual obtain the Langevin function. Specifically
=
I_bb( kTF/&,)2v2
exp( v)dv
(11) ’ / -b
exp(v)dv
’
R. W. Chantrell et al. / Magneiic anisotropy of ferrofruids
86
where v = pB,, cos B/kT, and b = pB,,/kTF. On evaluation of the integrals we have,
(cos*e)= 1 -i
coth b+s=
1 +(b),
(12)
where L(b) is the Langevin function. We now introduce the particle diameter distribution function f(r). y = D/D, and JJ, = D,/D,, where D, is the median particle diameter. The torque per unit volume of fluid according to this model is r= EK, sin 29
(13)
The above abrupt blocking transition approximation is similar to that introduced by NCel [5] and used to describe the time dependence of the magnetization of ferrofluid systems [6,7]. In this latter context the boundary between superparamagnetic and blocked particles can be considered abrupt because of the very rapid variation of the NM relaxation time with particle size. Even so, we have recently shown that it is necessary to sum the relaxation times of individual particles if a good fit to new, high precision, data is to be obtained [8]. Although it provides clear physical insights into the problem, in the context of torque magnetometry, the approximation is not at all satisfactory. While only a very small fraction of the particle size distribution contributes to most time dependent magnetization experiments, the torque experiments measure the effects of all particles and partial alignment of particles with D 5 D, gives a major contribution to the torque. As a result, the torque contribution from up to half the particles in a typical fluid can be overlooked, with the consequence that anisotropy constants derived using this model are artificially high. In the following section we present a derivation of the alignment function which includes particles only partially aligned during freezing. 3.2. Partial alignment during freezing For the general case, we must assume that the easy axis direction and moment direction are no longer locked. Let @ be the angle between them. The free energy of the particle can then be written as E = K,V sin*+ - pB, cos 8,
(14
using the coordinate system of fig. 1. It is important to note that the energy is independent of the angle between easy axis and freezing field, since coupling between easy axis and field is indirect, via the moment. Because of this the value of anisotropy has no bearing on the magnetization in the fluid state. This result will be proved later. It is convenient to choose a new coordinate system (fig. 2) centred on the moment direction cc. Then cos
e = cos J, cos + + sin Ic,sin $I cos E
(15)
and E=K,Vsin2+-pBOco~~.
With a = K,V/kT,, E, = E/kT,
(16)
we have = a sin*+ - b cos 4.
(17)
The partition function, z, is given by z=
V nsin 4 sin + exp( -a lJ 0 0
sin*+ + b cos #)d+d#.
(18)
R. W. Chantrell et al. / Magnetic anisotropy of ferrofruiidr
87
The alignment function can thus be written as ‘II nsin 4 sin $I cos2t9 exp( -a sin’+ + b cos #)d+d#dt (cos%) = 12W;2$-, sin # sin + exp( - sin2+ + b cos $)d+d$d[ 0
0
(19) ’
0
with cos28 obtained from eq. (15). It is here that we see that the appropriate choice of coordinate system has retained a simple form for the free energy, compressing all the complexities into the pre-exponential factor, and hence considerably simplifying the evaluation of (~0~~8). The dominator of eq. (19) is just 2 B times the partition function z which can be simply evaluated. The partition function is z = I(u) eeo(eb - e-‘)/b,
(20)
r(a) =I_+,’ exp( ax2)dx
(21)
where
(x is a dummy variable). It is noteworthy that eq. (20) enables us to prove very simply that the magnetization of a ferrofluid is the Langevin function and is therefore independent of the particle anisotropy, a result first shown by Krueger [9] a few years ago. From eq. (20), the reduced magnetization @is given immediately by @= (az/ab)/z
= coth b - (l/b)
= L(b).
(22)
Evaluation of the numerator of the alignment function (eq. (19)) is straightforward.
From eq. (15)
cos28 = cos2# cos2+ + 2 cos # cos S#Jsin JI sin (p cos I + sin’+ sin2+ cos25.
(23)
In eq. (19), the integral over 4 is between limits of 0 and 21~ and hence the second term in eq. (23) averages to zero. Thus, (cos28) = (2nz)-‘(lW
sin + cos2J/ exp(b cos J’,dqiV
+ lW sin 4 sin24 exp( b cos #)d$iW
sin C#I COS’C#S exp( -a sin2+)dg12Wd5
sin $I sin2$ exp( -a sin2+)d+12” cos’[dt).
(24)
This can be rewritten as (cos28) = (27rz)-‘(2?r[~,x2 +rr ‘,(l /
-x2)
ebXdx ePa/_‘,x2 euX2dx ebXedx e-O/:,(1
-x2)
e”“‘dx).
(25)
After considerable manipulation we arrive at @x28)
=
(1+(b))(
--&-$-)
-y
(-&-$-
I).
(26)
Thus, using eq. (9), the torque per unit volume of fluid r is r = &K,/o”[
1 - 3(cos28)] f( y)dy sin 2!P,
(27)
R. W. Chantrell er al. / Magnetic anisotropy of ferrofruiidr
88
where (cos28) is given by eq. (26). These two equations permit exact numerical solution and hence the determination of the particle anisotropy constant K, from the torque data.
4. The asymptotic limit of (I B 1, the strong coupling limit We recall that u = K,L’/kT,. z(a)
14
(
It is shown in the appendix that for a > 1,
1+1+3+... 2a
(28)
. 4a2
i
Z(a) is defined in eq. (20) and (21). Thus, to a first order approximation eO/uZ(a) = 1 - 1/2a
(29)
and hence (cos%) = 1 - (2/b)L(b).
(30)
This result is identical to that of eq. (12) in which it was assumed that the magnetic moment lay in the easy direction. We note that the torque is determined principally by the first term in eq. (26). (eO/(aZ(a)) = 1 and hence, in comparison to the first, the second term is negligible.) Thus the strength of the coupling between the easy axis and magnetic moment is largely represented by the function
@(,)=ef_L
2a’
al(a)
Taking the first two terms of Z(a) from eq. (28) yields @(u)=l-(l/u).
(32)
Fig. 3 shows the approximate solution (eq. (32)) and numerically evaluated exact solution (eq. (3 1)) of @(a) u I Mlerp(a)
0.8 0.6 04 0.2 -
I , 0
2
a 4
6
8
10
12
14
16
18
Fig. 3. Approximate and numerical solutions of the function e(a) as a function of a. For a > 1.5 the two solutions are very close.
20
0I
2
4
6
8
10
Fig. 4. Approximate and numerical solutions of the alignment integral I(a) as a function a.
89
R. W. Chantrell et al. / Magnetic anisotropy of ferrofluids
as a function of a. Provided that ~12 1.5, the approximate solution is satisfactory. The minimum diameter Dmin for which the approximate solution is valid is thus Dmin = (9kTF/?rK,)“3.
(33)
For TF = 178 K and K, = 2 X lo6 erg/cm, corresponding to a cobalt fluid in toluene, we have Dmin = 33 A. For Fe,O, particles in water, TF = 273 K and K, = 5 X lo5 erg/cm3. Hence Dmin = 60 A. As ferrofluids usually have median particle sizes greater than these values the approximate solution of @(a) can often be applied. Where a significant fraction of particles have D < Dmin, a numerical solution must be sought. Using eq. (26) and (32), for large a (co&?) = 1 - (2/b)L(b)
- l/a.
(34)
Hence, from eq. (6), we have for rp, the torque per unit particle volume on particles of diameter D,
1
%L(b)-l+&
sin29.
We note that as b + co, r, + - K,( 1 - 3/2a) sin 2+. This differs from the bulk single crystal anisotropy constant K, by 3K,/2a or 3kTF/2V. This can be thought of as an “entropy term” representing the fact that thermal agitation ensures that even in infinite field the alignment of easy axes will not be perfect. For a fluid containing particles of a single diameter of volumetric packing fraction E, the torque per unit fluid volume r is iL(b)-1
+&
1
sin2q.
However, as ferrofluids have a range of particle sizes, then for a particle volume distribution f(v) y = V/V,, and VYis the median particle volume, we have F= rK, sin 29
- 1 +&j/(Y)
where
d.v.
(37)
5. Variation of torque amplitude with large freezing field
In the limit b ZB 1 (where b = pBO/kTF), f=
-rK,
sin 2q
eq. (37) becomes
1-$-;)f(~)dy.
(38)
As the distribution is normalized, / o%Y)dy
(39)
= 1
and hence
it4, is the saturation magnetization per unit volume of the material comprising the magnetic particles. Many ferroflmds exhibit a lognormal distribution of particle volume fraction, namely
f(v) = (GJ_$
exp( - (ln y)2/2cr2).
(41)
R. W. Chantrell et al. / Magnetic anisotropy of ferrofluicis
90
Therefore, with the substitution r=a(lny)-a
(42)
it is easy to prove that
Jaoy-lf(y)dy
= exp(a2/2).
0
Substitution of eq. (43) into (40) yields, for b > 1 and a lognormal particle distribution function f=
-cK,
sin 2!# 1 -s I
I”
exp( a2/2) - *
S”0
exp( u2/2)
1.
(44)
We note that the torque amplitude at large field varies inversely with the freezing field B,,, a feature which has been confirmed experimentally [4]. By plotting f as a function of B; ’ and extrapolating to B; ’ = 0 we obtain the saturation torque r,,, as c;,,, = --E sin 29[ K, - (3kT,/2Vv)
exp( u2/2)].
(45)
In terms of a lognormal diameter distribution parameter un c,, = -
E
sin 2*[ K, - (9kT,/&)
exp( 9uk/2)].
(46)
6. Conclusions
The above theoretical analysis has shown the following features: of partially aligned frozen ferrofluids may be used to extract a value of the particle anisotropy constant K,. b) The torque from particles exhibiting uniaxial particle anisotropy varies as sin 29. This is confirmed experimentally [4]. c) The torque amplitude varies linearly with the inverse of the freezing field for large fields. This is also confirmed experimentally [4]. 4 An asymptotic solution for large a = K,V/kT, can be used satisfactorily for many ferrofluid systems. e) If a is not large, iterative numerical methods can be used to extract K,. 0 Determination of K, requires an accurate knowledge of the particle size distribution function.
4 Torque magnetometry
Acknowledgements
The work was supported in part by the University of Durham Special Research Projects Fund. We gratefully acknowledge the care with which Mrs. V. Todd and Mrs. M. Bradley typed the manuscript.
Appendix. Asymptotic solutions for the alignment integral
We require asymptotic solutions of the integral exp( ux2)dx,
(A.11
R. W. Chantrell et al. / Magnetic anisotropy of ferrofhds
91
valid for small and large a. A. 1. Strong coupling limit (a >s-1) limit is for a >> 1. Because the function
By far the most important Z(a) = 2l’ Substituting
is even,
exp( ax*)dx.
z *=l+x*gives
Z(a) = 2e”i’z(l
-z*)-“*
eCdL2dz.
(A-3)
When a x=. 1, the integrand is dominated by the variation of exp( - az*). This means attention to small z. Using the binomial expansion of (1 - z)-‘/* we have Z(a) = Ze”~‘$z*‘+‘B(n) where B(n)
we can restrict
e-11z2dz,
is the coefficient
our
(A-4)
of x” in the expansion
of (1 - z*). Thus,
Z(a) =2eUe(-l)B(n)l(n), 1
(A.5)
Z(n) =~‘z*‘+’
(A-6)
where, e-az2dz,
Since a x- 1, most of the variation of the integrand takes place for z small. introducing appreciable error we can extend the upper limit of the integral to infinity. Z(n) =lmz2’+i Substituting
eq. (A.7) into (A.5) and writing
Z(a)=2e”C-Writing
e-ar2dz = n!/2a”+‘.
n
(2n)!
1 (nl)24”
n!
the first few terms explicitly
1+1+3+x+2a
4a2
8a3
without
(A.7) B(n)
=- “,“;:G!LL. 1 .
2a”+’
Therefore, Then
explicitly
gives .
64.8)
gives (A.9)
It must be noted that, generally eq. (A.9) represents a divergent series. The ratio of successive term R(n) = (n-3)a-’ and thus for finite a, R(n) B 1 for large n and the series diverges. This is a consequence of the approximation made in the evaluation of Z(n). In this process the upper limit of the integral was taken to infinity. This will be a reasonable approximation for the first few terms where n is small, but will be increasingly invalid as n increases, For large n the approximate value of Z(n) will be a drastic over-estimate, thus producing the divergence of the series in eq. (A.8). As long as a is large enough so that the first few terms give sufficient accuracy, eq. (A.8) and (A.9) will give accurate values of Z(a).
R. W. Chantrell et al. / Magnetic anisotropy of jerrojiuids
92
The validity of this approach is tested in fig. 4. Here, the approximate
I(a)=; ( 1+&
solution
1
is compared with a numerical very satisfactory.
evaluation
of the full integral. It is clear that for a 2 1.5, the agreement is
A.2. Weak coupling limit a CC I Here I(a)
= /_:’
exp( ax*)dx
=
/ _+jl
+ ax*)dx
= 2( 1 + a/3).
This is not so useful as the strong coupling limit, but is included
(A.lO) for completeness.
References [II A. Tari, J. Popplewell and SW. Charles, J. Magn. Magn. Mat. 15-18 (1980) 1125. PI A. Tari, R.W. Chantrell, SW. Charles and J. Popplewell, Physica 97B (1979) 57. (31 R. Pearson, in: Experimental Magnetism, vol. I, eds. G. Kalvius and R. Tebble (Wiley, New York, 1979) p. 138. 141 S.R. Hoon, B.K. Tanner and M. Kilner J. Magn. Magn. Mat. 39 (1983) special issue. VI L. N&l, Advan. Phys 4 (1955) 191. VI K. G’Grady, R.W. Chantrell, J. Popplewell and S.W. Charles, Trans. IEEE Magn. MAG-17 (1981) 2943. [71 A.T. Cayless, S.R. Hoon, B.K. Tanner, R.W. Chantrell and M. Kilner, J. Magn. Magn. Mat. 30 (1983) 303. PI R.W. Chantrell, S.R. Hoon and B.K. Tanner, J. Magn. Magn. Mat. 38 (2) (1983) in press. [91 D.A. Krueger, J. Appl. Phys. 50 (1979) 8169.