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A study of curie-weiss behaviour in ferrofluids
Journal of Magnetism North-Holland
A STUDY
and Magnetic
Materials
OF CURIE-WEISS
M. HOLMES,
K. O’GRADY
Magnetism Group, SEES,
47
85 (1990) 47-50
BEHAVIOUR
IN FERROFLUIDS
and J. POPPLEWELL
Dean St, Bangor, Gwynedd, LL57 I UT, UK
In this paper we have investigated concentration effects on the temperature variation of ac susceptibility for a Fe,O, ferrofluid. Our measurements show deviation from classical Langevin theory and a non-linear Curie-Weiss behaviour. This is attributed to the effects of interparticle interactions.
1. Introduction
Ferrofluids are stable colloidal suspensions of single-domain ferromagnetic particles in a nonmagnetic carrier liquid. Magnetic characterisation has shown that ferrofluids are superparamagnetic and can therefore be described by the classical Langevin function M=coth(a)
- l/a,
(1)
where a = pH/kT. If H is small so that pH/kT e 1 then M = L( a)/3. This gives us an expression for the initial susceptibility of a single particle of moment p X, = ,u2/3kT.
(2)
If we have n similar particles per unit volume then in the absence of particle interaction effects the volume susceptibility is given by X, = np2/3kT.
(3)
Evdokimov [l] has calculated the magnetic field due to a 30 A Fe,O, particle at a distance of 20 A to be approximately 1000 Oe. Since the particles in a ferrofluid are in constant motion by Brownian agitation it is probable that particles will come into such close proximity that dipolar interactions will be substantial. If this interaction is dominant over the thermal energy then particle aggregation will result. However, in a less severe regime it is thought that these interactions modify the behaviour of ferrofluids to cause deviations from Langevin theory. 0304-8853/90/$03.50
0 1990 - Elsevier Science Publishers
Interaction effects in ferrofluids are most apparent at low fields and hence the initial susceptibility (dM/d HH _ ,,) can be used as a sensitive measure of these. Many authors have considered a mean-field theory and attempted to represent the interactions by an internal field which is uniform and acting throughout the volume of magnetic material. Hence, the total field acting within the material is given by Ht”, = Harp + Hint.
(4)
Hi,, is assumed
zation
to be proportional to the magnetiand this leads to the Curie-Weiss law
X, = C/(T-
To).
(5)
The Curie-Weiss law predicts a linear dependence of reciprocal initial susceptibility on temperature with a non-zero temperature intercept. This intercept has been termed the Ordering Temperature [2] and thought to be indicative of the interaction strength. Curie-Weiss behaviour has been measured previously in ferrofluids by Soffge et al. [3], O’Grady et al. [4] and Popplewell et al. [5]. The data of O’Grady et al. shows a positive temperature intercept which indicates the initial susceptibility being enhanced by the interactions. This contrasts with the data of Popplewell et al. which shows negative ordering temperatures. This latter observation was attributed to the interactions being strong and causing aggregation with antiferromagnetic-like ordering of the particles. There is some doubt as to the applicability of the mean-field theory to ferrofluids since they are
B.V. (North-Holland)
4x
M. Holmes
et crl. / Currr-
dynamic systems and not a system of moments arranged on a regular lattice [6]. Several authors have attempted to model the interactions of an assembly of fine particles, e.g. Brown (Pair of Spheres Model, [7]), Kneller (Effective Volume Model [8] and Concentration Profile model, [9]). Mairiov (Mean Spherical model [lo]) with varying degrees of success. Interactions in ferrofluids have also been modelled numerically by using Monte Carlo techniques, e.g. Menear [2].
Weiss heharwur
tn ferrofluid.~
Xi larb I X 12
X A
E
= z
230 Kelwn 270
/
2. Experimental L
0
We have used a mutual inductance technique to measure ac susceptibility of a magnetite/paraffin oil/oleic acid ferrofluid at 210 Hz with an alternating magnetic field of 1 Oe peak-to-peak. The superposition of a coaxial dc field enabled the susceptibility to be measured at any field between zero and 500 Oe. Temperature variation (between 220-380 K) was achieved by insertion of the coil assembly into an Oxford Instruments CF1200 Continuous Flow Cryostat. Temperature measurement was by a gold-iron/chrome1 thermocouple with the measurement junction attached to the coil former. Measurements were made on cylindrical samples with an aspect ratio of 10
16
20
2L Is
28 lemu/g
I
shows that the effect of interactions is greater at lower temperatures when the thermal disordering is least. The near linearity at higher temperatures indicates that thermal disordering is so effective in offsetting the interaction enhancement that the system behaves as if it were almost non-interacting. Figure 2 shows X, (= X,/M,) plotted against M, for the same five temperatures. 2, is a powerful tool when comparing different concentrations
larbl
6 :
X A
= =
230 K 270
v
=
340
0
=
380
-A-
A-A
T
V-V o-o-
and discussion
Figure 1 shows the variation of Initial Susceptibility with concentration for five temperatures in the range 230-380 K. This graph shows departure from Langevin behaviour (i.e. X, linearly dependent on M,) with the susceptibility becoming enhanced. It can be seen that the departure from linearity is greatest at the lower temperatures. This
12
Fig. I. Plot of X, against concentration (I,) for 5 temperatures in the range 230-380 K.
3
Results
8
-0-
X-
V-V
-V-
-V-
o_o-o-o-
c
Fig.
2. Plot
of reduced initial susceptibility susceptibility (Is).
against
Initial
M. Holmes et al. / Cune- Werss hehavrour in ferrofluids
of a stable ferrofluid since any change in the ratio of X, to MS must be due solely to interaction effects on the initial susceptibility. From fig. 2 it can be seen that x, increases for each concentration as the temperature is lowered. These increases become greater at higher concentrations which suggests that the interaction strength is dependent upon concentration and inversely dependent upon temperature. The linearity of the relationship. between 2, and A4, deserves further investigation. Figure 3 shows reciprocal initial susceptibility plotted against temperature for five ferrofluid concentrations. Since preparation of the abstact for this paper we have extended the range of temperature measurements from 230-300 to 230-380 K. Subsequent data taken over this wider temperature range, given in fig. 3, shows the behaviour to be, in fact, non-linear. This indicates that the ferrofluids are not behaving according to classical Curie-Weiss theory since there is no straight line region for any of these curves from which an Ordering Temperature can be extrapolated. Departure from classical Curie-Weiss behaviour has been predicted by Brown [7] in his “Pair of Spheres Model” in which the l/T term for the variation in reciprocal susceptibility with temperature vanishes leaving the dependence on l/T’. Bradbury [ll] has shown that the linear behaviour predicted by Menear [2] is an artifact of two-dimensional mod-
l/Xi
12
(arb)
-
r rX
10
-
=
A
8-
6
X
= =
, -
J
2.8 emu/g X’
4.9
a.4
v
=
13.9
0
:
28.6
*r”
,x
.a
x’ .x’
8
L-
A’
_MfiA-A-A’
A, A-
A’
A,
A’
l
l/Xi
larbl
14
dc
-
field
.
=
200 Oe
_
V
:
160
_
x
=
120
-
A
=
80
0
=
40
,2
10
-
8-
A = 0
6 -A%@@
4-
2 0 I 0
50
100
150
1
I
200
250
300
350
temperature
Fig. 4. Graph temperature
IKI
showing the effect of an applied dc field on the dependence of reciprocal initial susceptibility.
elling and that the l/T term also disappears when modelling in three dimensions. An important consequence of recipriocal susceptibility depending on l/T2 rather than l/T is that a phase transition is no longer predicted. The approximation that pH -=c kT implies that the susceptibility is constant at low fields. Our ac Susceptometer was able to detect a change in the susceptibility with dc applied fields greater than 20 Oe. This means that the approximation becomes invalid for fields greater than this. In fig. 4 we show the effect of dc applied fields between zero and 200 Oe on the temperature variation of reciprocal susceptibility. It can be seen that tangents drawn to the data for higher fields would lead to negative ordering temperature. We think that this may explain the data of Popplewell et al. who used fields of up to 75 Oe, to measure initial susceptibility.
4. Conclusions
/0
49
and future work
_e2-
_*a-*-
v_v-v-
0-@
~oos~~;~~~~~~-_oo~-
0
I 0
so
100
150
200
250
I
300 350 temperature
Fig. 3. Plot of reciprocal initial susceptibility against ture for five ferrofluid concentrations.
I 400 K
t I
tempera-
In this paper we have used the Initial Susceptibility X, and the Reduced Initial Susceptibility X, to experimentally study concentration effects on interactions in ferrofluids. We have found that interactions enhance the susceptibility above the non-interacting value. This enhancement increases
50
M. Holmes er al. / Curre- Weiss hehnvmur rn ferrofluids
with ferrofluid concentration and decreases with increasing temperature. The variation in reciprocal initial susceptibility with temperature was found to be non-linear, thus showing departure from Curie behaviour. For future work we are planning to model the interactions in ferrofluids by developing a three-dimensional Monte Carlo Model.
References [I] V.B. Evdokimov. Russ. J. Phys. Chem. 37 (1963) 1018. [2] S. Menear, A. Bradbury and R.W Chantrell, J. Magn. Magn. Mat. 39 (1983) 17. [3] F. Soffge and E. Schmidbauer, J. Magn. Magn. Mat. 24 (1981) 54.
14) K. O’Grady. A. Bradbury, SW. Charles, S. Menear, J. Popplewell and R.W. Chantrell, J. Magn. Magn. Mat. 31-34 (1983) 958. [5] J. Popplewell, B. Abu, Aisheh and N.Y. Ayoub. J. Appl. Phys. 64 (19Xx) 5852. (61 W.F. Brown, in: Handbuch der Physik. Vol. 17 (Springer. Berlin. 1956) p. 47. [7] W.F. Brown. J. Appl. Phys. 38 (1967) 1017. [X] E. Kneller, Proc. Int. Conf. on Magn. (Nottingham, 1964) (Physical Society. London. 1965) p. 174. [9] E. Kneller and G. Trippel, J. Appl. Phya. 3X (1967) 993. [lOI K.I. Morozov. A.F. Pshenichnikov. Yu. L. Raikher and M.I. Schliomis. J. Magn. Magn. Mat. 65 (19X7) 269. [Ill A. Bradbury, G.A.R. Martin and R.W. Chantrell. J. Magn. Magn. Mat. 69 (1987) 5.
A STUDY
and Magnetic
Materials
OF CURIE-WEISS
M. HOLMES,
K. O’GRADY
Magnetism Group, SEES,
47
85 (1990) 47-50
BEHAVIOUR
IN FERROFLUIDS
and J. POPPLEWELL
Dean St, Bangor, Gwynedd, LL57 I UT, UK
In this paper we have investigated concentration effects on the temperature variation of ac susceptibility for a Fe,O, ferrofluid. Our measurements show deviation from classical Langevin theory and a non-linear Curie-Weiss behaviour. This is attributed to the effects of interparticle interactions.
1. Introduction
Ferrofluids are stable colloidal suspensions of single-domain ferromagnetic particles in a nonmagnetic carrier liquid. Magnetic characterisation has shown that ferrofluids are superparamagnetic and can therefore be described by the classical Langevin function M=coth(a)
- l/a,
(1)
where a = pH/kT. If H is small so that pH/kT e 1 then M = L( a)/3. This gives us an expression for the initial susceptibility of a single particle of moment p X, = ,u2/3kT.
(2)
If we have n similar particles per unit volume then in the absence of particle interaction effects the volume susceptibility is given by X, = np2/3kT.
(3)
Evdokimov [l] has calculated the magnetic field due to a 30 A Fe,O, particle at a distance of 20 A to be approximately 1000 Oe. Since the particles in a ferrofluid are in constant motion by Brownian agitation it is probable that particles will come into such close proximity that dipolar interactions will be substantial. If this interaction is dominant over the thermal energy then particle aggregation will result. However, in a less severe regime it is thought that these interactions modify the behaviour of ferrofluids to cause deviations from Langevin theory. 0304-8853/90/$03.50
0 1990 - Elsevier Science Publishers
Interaction effects in ferrofluids are most apparent at low fields and hence the initial susceptibility (dM/d HH _ ,,) can be used as a sensitive measure of these. Many authors have considered a mean-field theory and attempted to represent the interactions by an internal field which is uniform and acting throughout the volume of magnetic material. Hence, the total field acting within the material is given by Ht”, = Harp + Hint.
(4)
Hi,, is assumed
zation
to be proportional to the magnetiand this leads to the Curie-Weiss law
X, = C/(T-
To).
(5)
The Curie-Weiss law predicts a linear dependence of reciprocal initial susceptibility on temperature with a non-zero temperature intercept. This intercept has been termed the Ordering Temperature [2] and thought to be indicative of the interaction strength. Curie-Weiss behaviour has been measured previously in ferrofluids by Soffge et al. [3], O’Grady et al. [4] and Popplewell et al. [5]. The data of O’Grady et al. shows a positive temperature intercept which indicates the initial susceptibility being enhanced by the interactions. This contrasts with the data of Popplewell et al. which shows negative ordering temperatures. This latter observation was attributed to the interactions being strong and causing aggregation with antiferromagnetic-like ordering of the particles. There is some doubt as to the applicability of the mean-field theory to ferrofluids since they are
B.V. (North-Holland)
4x
M. Holmes
et crl. / Currr-
dynamic systems and not a system of moments arranged on a regular lattice [6]. Several authors have attempted to model the interactions of an assembly of fine particles, e.g. Brown (Pair of Spheres Model, [7]), Kneller (Effective Volume Model [8] and Concentration Profile model, [9]). Mairiov (Mean Spherical model [lo]) with varying degrees of success. Interactions in ferrofluids have also been modelled numerically by using Monte Carlo techniques, e.g. Menear [2].
Weiss heharwur
tn ferrofluid.~
Xi larb I X 12
X A
E
= z
230 Kelwn 270
/
2. Experimental L
0
We have used a mutual inductance technique to measure ac susceptibility of a magnetite/paraffin oil/oleic acid ferrofluid at 210 Hz with an alternating magnetic field of 1 Oe peak-to-peak. The superposition of a coaxial dc field enabled the susceptibility to be measured at any field between zero and 500 Oe. Temperature variation (between 220-380 K) was achieved by insertion of the coil assembly into an Oxford Instruments CF1200 Continuous Flow Cryostat. Temperature measurement was by a gold-iron/chrome1 thermocouple with the measurement junction attached to the coil former. Measurements were made on cylindrical samples with an aspect ratio of 10
16
20
2L Is
28 lemu/g
I
shows that the effect of interactions is greater at lower temperatures when the thermal disordering is least. The near linearity at higher temperatures indicates that thermal disordering is so effective in offsetting the interaction enhancement that the system behaves as if it were almost non-interacting. Figure 2 shows X, (= X,/M,) plotted against M, for the same five temperatures. 2, is a powerful tool when comparing different concentrations
larbl
6 :
X A
= =
230 K 270
v
=
340
0
=
380
-A-
A-A
T
V-V o-o-
and discussion
Figure 1 shows the variation of Initial Susceptibility with concentration for five temperatures in the range 230-380 K. This graph shows departure from Langevin behaviour (i.e. X, linearly dependent on M,) with the susceptibility becoming enhanced. It can be seen that the departure from linearity is greatest at the lower temperatures. This
12
Fig. I. Plot of X, against concentration (I,) for 5 temperatures in the range 230-380 K.
3
Results
8
-0-
X-
V-V
-V-
-V-
o_o-o-o-
c
Fig.
2. Plot
of reduced initial susceptibility susceptibility (Is).
against
Initial
M. Holmes et al. / Cune- Werss hehavrour in ferrofluids
of a stable ferrofluid since any change in the ratio of X, to MS must be due solely to interaction effects on the initial susceptibility. From fig. 2 it can be seen that x, increases for each concentration as the temperature is lowered. These increases become greater at higher concentrations which suggests that the interaction strength is dependent upon concentration and inversely dependent upon temperature. The linearity of the relationship. between 2, and A4, deserves further investigation. Figure 3 shows reciprocal initial susceptibility plotted against temperature for five ferrofluid concentrations. Since preparation of the abstact for this paper we have extended the range of temperature measurements from 230-300 to 230-380 K. Subsequent data taken over this wider temperature range, given in fig. 3, shows the behaviour to be, in fact, non-linear. This indicates that the ferrofluids are not behaving according to classical Curie-Weiss theory since there is no straight line region for any of these curves from which an Ordering Temperature can be extrapolated. Departure from classical Curie-Weiss behaviour has been predicted by Brown [7] in his “Pair of Spheres Model” in which the l/T term for the variation in reciprocal susceptibility with temperature vanishes leaving the dependence on l/T’. Bradbury [ll] has shown that the linear behaviour predicted by Menear [2] is an artifact of two-dimensional mod-
l/Xi
12
(arb)
-
r rX
10
-
=
A
8-
6
X
= =
, -
J
2.8 emu/g X’
4.9
a.4
v
=
13.9
0
:
28.6
*r”
,x
.a
x’ .x’
8
L-
A’
_MfiA-A-A’
A, A-
A’
A,
A’
l
l/Xi
larbl
14
dc
-
field
.
=
200 Oe
_
V
:
160
_
x
=
120
-
A
=
80
0
=
40
,2
10
-
8-
A = 0
6 -A%@@
4-
2 0 I 0
50
100
150
1
I
200
250
300
350
temperature
Fig. 4. Graph temperature
IKI
showing the effect of an applied dc field on the dependence of reciprocal initial susceptibility.
elling and that the l/T term also disappears when modelling in three dimensions. An important consequence of recipriocal susceptibility depending on l/T2 rather than l/T is that a phase transition is no longer predicted. The approximation that pH -=c kT implies that the susceptibility is constant at low fields. Our ac Susceptometer was able to detect a change in the susceptibility with dc applied fields greater than 20 Oe. This means that the approximation becomes invalid for fields greater than this. In fig. 4 we show the effect of dc applied fields between zero and 200 Oe on the temperature variation of reciprocal susceptibility. It can be seen that tangents drawn to the data for higher fields would lead to negative ordering temperature. We think that this may explain the data of Popplewell et al. who used fields of up to 75 Oe, to measure initial susceptibility.
4. Conclusions
/0
49
and future work
_e2-
_*a-*-
v_v-v-
0-@
~oos~~;~~~~~~-_oo~-
0
I 0
so
100
150
200
250
I
300 350 temperature
Fig. 3. Plot of reciprocal initial susceptibility against ture for five ferrofluid concentrations.
I 400 K
t I
tempera-
In this paper we have used the Initial Susceptibility X, and the Reduced Initial Susceptibility X, to experimentally study concentration effects on interactions in ferrofluids. We have found that interactions enhance the susceptibility above the non-interacting value. This enhancement increases
50
M. Holmes er al. / Curre- Weiss hehnvmur rn ferrofluids
with ferrofluid concentration and decreases with increasing temperature. The variation in reciprocal initial susceptibility with temperature was found to be non-linear, thus showing departure from Curie behaviour. For future work we are planning to model the interactions in ferrofluids by developing a three-dimensional Monte Carlo Model.
References [I] V.B. Evdokimov. Russ. J. Phys. Chem. 37 (1963) 1018. [2] S. Menear, A. Bradbury and R.W Chantrell, J. Magn. Magn. Mat. 39 (1983) 17. [3] F. Soffge and E. Schmidbauer, J. Magn. Magn. Mat. 24 (1981) 54.
14) K. O’Grady. A. Bradbury, SW. Charles, S. Menear, J. Popplewell and R.W. Chantrell, J. Magn. Magn. Mat. 31-34 (1983) 958. [5] J. Popplewell, B. Abu, Aisheh and N.Y. Ayoub. J. Appl. Phys. 64 (19Xx) 5852. (61 W.F. Brown, in: Handbuch der Physik. Vol. 17 (Springer. Berlin. 1956) p. 47. [7] W.F. Brown. J. Appl. Phys. 38 (1967) 1017. [X] E. Kneller, Proc. Int. Conf. on Magn. (Nottingham, 1964) (Physical Society. London. 1965) p. 174. [9] E. Kneller and G. Trippel, J. Appl. Phya. 3X (1967) 993. [lOI K.I. Morozov. A.F. Pshenichnikov. Yu. L. Raikher and M.I. Schliomis. J. Magn. Magn. Mat. 65 (19X7) 269. [Ill A. Bradbury, G.A.R. Martin and R.W. Chantrell. J. Magn. Magn. Mat. 69 (1987) 5.