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Magnetic properties of fine maghemite particles in an electroconducting polymer matrix
Journalof mnalnetlsm magnetic ~ l ~ materials
•i• ELSEVIER
Journal of Magnetism and Magnetic Materials 137 (1994) 205-218
Magnetic properties of fine maghemite particles in an electroconducting polymer matrix O. Jarjayes a, P.H. Fries b,., G. Bidan
a
a CEA/D~partement de Recherche Fondamentale sur la Mati~re Condens~e/SESAM, Grenoble, France b Electrochimie Mol~culaire, Chimie de Coordination CEN-Grenoble, 17 rue des Martyrs, 38054 Grenoble Cedex 9, France
Received 20 December 1993; in revised form 5 March 1994
Abstract The magnetic properties of 7-Fe203 grains of a few nanometers in size, included in films of conducting polypyrrole by electrochemical synthesis, are investigated. A theoretical treatment of the combined effects of the possible non-uniform spontaneous magnetization of the grains and of their crystalline anisotropy energies is presented. The magnetization of samples of this composite material has been measured at several temperatures as a function of the field up to 30 kOe. The interpretation of these data shows that the grains in the polymer behave as independent monodomains, coexist with the polymer without any notable interference, and have the same size distribution as in the ferrofluid solution used for the electrochemical inclusion.
1. Introduction Over the last thirty years, the properties of fine magnetic particles have been widely studied both for theoretical purposes and for technological applications such as high density magnetic recording. New developments have appeared in the design and synthesis of mesoscopic superparamagnetic materials, i.e. particles of ferro- or ferrimagnetic monodomains of a few nanometers. Ziolo et al. [ 1 ] reported in situ synthesis of ferrimagnetic maghemite, "y-Fe203 in art ion exchange resin polymer matrix in order to make a new optically transparent magnetic material. Marchesault et al. [2] studied small ferrite particles of F e 3 0 4 trapped in lignocellulosic polymers. Recently, Armes et al. [3] described the preparation and characterization of superparamagnetic conductive polyester textile composites by a two step solution deposition process with a view to applications in electromagnetic interference shielding. In this paper the magnetic properties of 3'-Fe203 ferrimagnetic maghemite particles included in films of conducting polymer matrix of polypyrrole (PPy) are investigated. The inclusion is based on the use of an anionic ferrofluid (FF) [4] which is a liquid solution of magnetic crystalline cores of a few nanometers, coated with citrate anions. Because of their coating the magnetic cores can be included amidst the polymer chains as classical doping anions in the course of electrochemical polymerization. This method is quite different from the mechanical process [5,6] where a vigourous stirring is performed in order to insert micrometer sized particles of metal oxide like WO3 into a PPy matrix. Our goal is to look for the possible interactions between the magnetic grains and the surrounding polymer. Does the PPy matrix modify the magnetic properties of the included monodomains through the citrate coating? Does it * Corresponding author. 0304-8853/94/$07.00 © 1994 Elsevier Science B.V. All fights reserved S S D 1 0 3 0 4 - 8 8 5 3 ( 94 ) 0 0 3 2 7 - N
O. Jarjayes et aL / Journal of Magnetism and Magnetic Materials 137 (1994) 205-218
206
favour some aggregation of the grains? In order to answer these questions the magnetization curves of the T-Fe203 particles in the polymer will be analyzed as a function of the external field and temperature, and compared to the magnetization measurements of the ferrofluid. The magnetization results yield a distribution of the particle sizes which will be interpreted with respect to the particle sizes obtained by X-ray diffraction or Transmission Electron Micrographs (TEM). Such magnetic studies were pioneered in 1959 by Beans and Livingston [7] who showed that single domain ferromagnetic particles can be superparamagnetic at elevated temperature, i.e. behave in a manner analogous to Langevin paramagnetic atoms. In 1978 Chantrell et al. [8] presented a method for determining the particle mean diameter from room temperature magnetization curves using a lognormal distribution. Many studies were then done on the magnetic and thermal behaviour of fine grains [9-11]. Very recently, Williams et al. [ 12] considered the anisotropy energies of monodomains with respect to their easy magnetization axes and studied the influence of these anisotropy effects on the superparamagnetism of ferrofluids of magnetite particles coated with an oleic acid. The experimental processes for synthesizing the ferrofluid and for inserting the y-Fe203 monodomains are summarized in Section 2. The surfactant coating or the polymer matrix can modify the spontaneous magnetization of the y-Fe203 particles near the surface by distorting its crystalline structure. The theoretical expressions of the macroscopic magnetization of solid and liquid samples containing a distribution of ferromagnetic particles are given in Section 3 when both the effects of the non-uniform spontaneous magnetization of the particles and of their crystalline anisotropy energy are taken into account. Using this theory the magnetization data of our T-Fe203 ferrofluid and PPy composites are interpreted in Section 4.
2. Experimental The ferrofluid was synthesized according to Massart's procedure [ 13 ]. The y-Fe203 particles were coated with citrate - O O C - C H z - C ( O H ) ( C O O - ) - C H 2 C O O - ligands (Cit) to give the corresponding ferrofluid anions yFe203/Cit. The y-FezO3/Cit incorporated polypyrrole films (T-Fe203/Cit-PPy) were prepared by electrolysis in a one compartment cell containing a non-stirred aqueous solution of 0.5 mol/1 pyrrole and of y-Fe203/Cit anions at a 0.1 mol / 1Fe concentration. The films with embedded 'y-Fe203 grains were deposited onto an optically transparent electrode of area 2 cm X 3 cm at 0.7 V with respect to the Saturated Calomel Electrode (SCE). The films, which are typically 100/xm thick, were then washed with distilled water and acetonitrile. After one night's free standing, they were stripped from the electrode substrate and dried under vacuum for 12 hours at 60°C. The electrochemical device is made of a potentiostat PAR 173, a potentiostat/galvanostat PAR 175 and a coulometer PAR 179 from EG&G Princeton Applied Research (PAR) connected to a Sefram TGM 164 recorder. The density of the (y-Fe203/Cit-PPy) composite material is 1.85 g/cm 3. Elemental analysis shows that the mass percentages of the composite films are p=26.27% for y-Fe203, 6.81% for the citrate anions, 46.25% for the polypyrrole, and 20.65% for the residual water. During the electrochemical synthesis the particles of T-Fe203 in aqueous solution around pH = 7 are coated with negatively charged citrate ligands. There is a strong Coulomb repulsion between two particles caused by their total superficial charges [ 14], q --- - 400e, as discussed in Section 4. The particles are attracted by the positive charges on the heterocycles of the polypyrrole as it forms on the transparent electrode, and progressively incorporated. The composite structure can be imagined as a dispersion of repulsive magnetic macroanions embedded in a hydrated positively charged polymer matrix which ensures the electric neutrality of the material. The density of T-Fe203 oxide in the grains is taken to be p = 4.60 g/cm 3 [ 15]. According to a numerical treatment of a Transmission Electron Micrograph (TEM) the grain diameters are distribPPy uted about a maximum probability diameter Dmp (TEM) = 50/~. An estimate of the number density of the particles is given by n
PPY~ p × 1.85
ox
~'rr/5 3 -
1.6 × 1018 grains/cm 3,
O. Jarjayes et at / Journal of Magnetism and Magnetic Materials 137 (1994) 205-218
207
if the grains are assumed to have the same diameter value/5 = 50/~. The strength of the grain magnetic interactions is related to the mean distance between the nearest neighbours. Because of their strong Coulomb repulsion the grains are expected to be dispersed in the polypyrrole. They have difficulty forming aggregates. As a first approximation, assuming that they occupy the nodes of a face-centred cubic lattice of cube side a~'~r~Y,,the mean distance between the centres of two neighbouring grains is estimated to be /~'~PY=a~PY/v~= 100 A ([16], pp. 18-21). This intercentre distance, which is rather large with respect to the grain diameters, is compatible with a weak interparticle coupling in the composite. The magnetic measurements were carried out with a Quantum Design SQUID magnetometer operating between 2 and 300 K in magnetic fields up to 50 kOe.
3. Models of superparamagnetic particles Consider a sample containing a dispersion of nearly spherical ferromagnetic particles. Our goal is to determine the distribution of the particle sizes from magnetization measurements as a function of the external magnetic field H. This can be achieved when the particles can be approximated as ferromagnetic monodomains at thermal equilibrium, the sizes of which are distributed according to a simple probability law. It is well known (see Ref. [ 7] and [ 17], pp. 749-754) that a magnetic material has a critical size under which the lowest magnetic energy corresponds to a single domain arrangement where the magnetic moments are parallel. An estimate of this critical size is given by the thickness of the Bloch walls ( [ 16], pp. 566-569) which are the transition layers between the monodomains. It is clear that this thickness gives a limit under which the material forms a monodomain. Consider a lattice of spins S which are coupled by exchange interactions J. If aL denotes a typical lattice constant of the spin array and K an estimate of the magnetocrystalline anisotropy energy density, a first approximation of the Bloch walls thickness is given by ( [ 16], pp. 566-569)
=( ffr2JS2)1/2. NaL ~ KaL ]
(1)
Let Ms(T) be the spontaneous magnetization of the material at temperature T. The magnetic moment of a single domain of volume V and orientation a is
m=Ms(T)Va.
(2)
In order to be at thermal equilibrium during the magnetization measurements the particle moments should fluctuate with a characteristic time ~-shorter than the time of experiment. The random changes of the directions of the moments can originate from two mechanisms. The first one is always present. It was proposed by Nrel ( [ 17], pp. 761-766, and [ 18] ) and is associated with the thermal fluctuations of the crystal lattice which is coupled to the spins S. The magnetization vector Ms(T) a fluctuates between the easy directions with a characteristic Nrel time ~'N defined as
1 TN
KV \
KBI/
wherefo is a frequency factor, which is usually taken to be 1 0 9 S - 1 [7,12], but can be much larger [ 19]. A second mechanism of random reorientations of the moments occurs in liquids. It corresponds to the Brownian rotational diffusion of the particles and has a characteristic Debye time 3~V
ZB= kBT'
(4)
where 7/is the liquid viscosity, eB is the time constant of the exponential decay of the time correlation functions of the spherical harmonics of order 1 which define the moment orientation [ 20].
208
O. Jarjayes et al. / Journal of Magnetism and Magnetic Materials 137 (1994) 205-218
Electron micrographs [ 10] show particles which can be approximated as spheres of diameters D, distributed according to a log-normal probability law
p(D)
D
exp
[ ln2(D/D°)~ 20"2
(5)
J'
where In(Do) is the mean value of In(D) and 0"2 the standard deviation. The maximum of the probability density p(D) occurs for the diameter value Drop = Doexp( - 0"2). The applicability of the log-normal law was examined by O'Grady and Bradbury [21] for 100 ferrofluids. They showed that a choice between the Gaussian and log-normal distributions of the particle sizes is, in general, adequate and that the form of the distribution is very likely associated with the technique of particle preparation. In principle Eqs. (2) and (5) should be sufficient to calculate the superparamagnetic behaviour of ferromagnetic particles when their diameters D are small enough for the monodomain condition D < to be met, where the Bloch wall thickness is given by Eq. (1), and for the fluctuation times rN or ~'u to be shorter than the magnetization measurement time. However, three other effects may modify the sample magnetism. Firstly, even for spherical particles without shape anisotropic magnetic energy there remains an anisotropic energy density associated with the crystal structure ([17], pp. 333-355). Following Williams et al. [12] the effects of this magnetocrystalline interaction are estimated by assuming that it is uniaxial in symmetry and of the simplest form
NaL
Nat.
Ex = KVsin20,
(6)
where O is the angle between the easy axis and the particle moment. Secondly, Sato et al. [22] found a sharp decrease of the saturation magnetization of fine ferrite particles when their sizes are reduced below 100 ]k. They attributed this decrease to a magnetically inactive surface layer, the thickness of which increases with the crystalline anisotropy up to about 10/1,. Furthermore, Nunes et al. [23] examined the magnetization density distribution of finely divided CoFe204 particles by measuring the X-ray and polarized neutron scattered intensities. Their measurements can be interpreted by assuming a magnetically anomalous surface shell of inactive fixed pinned spins which increases from 5 ,~ for the naked particles to 20 /~ when they are coated with a chemisorbed oleate CH3 (CH2) 7CH = CH(CH2) 7COO- surfactant. In order to take possible surface effects for our material into account we assume that the spontaneous magnetization Ms(T,R) inside the particle may be non-uniform and is likely to depend on the radial distance R from the particle centre. Because a surface layer seems magnetically inactive the form of the magnetization is simply taken to be
Ms (T,R) =
(7) otherwise,
where D is the particle diameter and AR the layer thickness. Thirdly, particle interactions have been the subject of much discussion [24,25]. The subject is difficult. A Curie-Weiss behaviour of the sample magnetism has been frequently observed for small external magnetic fields with pseudo-ordering 'Curie' or 'N6el' temperatures To. However, as far as we know, there is no satisfactory theory of these many-body interactions. In what follows the particles are sufficiently dilute for neglecting their interactions in a first approximation as long as they do not form aggregates. The consistency of this hypothesis will be checked by comparing the calculated magnetism of independent particles with the measured values. Assume that the ferromagnetic particles are distributed in a solid sample located in an external magnetic field /-/= H/~L which points in the f% direction of the laboratory reference frame ( OL, fL,JL, ]~L)"Consider a single-domain ferromagnetic particle of diameter D and volume V(D). Let ke be the unit vector defining the orientation of the easy magnetization axis and a, fl its spherical coordinates. The magnetic moment of the particle is
O. Jarjayes et al. / Journal of Magnetism and Magnetic Materials 137 (1994) 205-218
m=m(D)~= fMs(T,R)d3R~,
209
(8)
v
where the unit vector fi has the spherical coordinates 0,(h. The thermal average (ran) of the projection of the magnetic moment along the field is given by the Boltzmann distribution
ffcosO exp [
1
tiBET(0,~b,Ct,fl) ] S ~ 0 dOd~b
(mn) = m(D) (cos0) ~,~ = re(D)
,
(9)
f fexp[ -13BE.r(O,4~,a,~)]sinOdOd4~ where the total particle energy ET(O,~b,a,[3) is the sum of the Zeeman and anisotropy contributions
ET( O,4a,a, fl) = -- m( D )HcosO + KV(D) sin20,
(10a)
cosO= cos/3cos0 + sin/3sin0cos( ~b- a).
(10b)
with
Now let N be the number of ferromagnetic particles in the sample. Consider the subset of the particles with diameters between D and D + dD. The orientations ke of their easy axes have an uniform angle distribution. The angle average ((ran)) of the thermal averages (mn) of the projections of the magnetic moments of these particles along the field is 1
( ( mn ) ) = m( D ) ( ( cosO) c,.a) = -~-~,rrm(D ) f f ( cosO) ,,.asin I3 d Oda.
(11)
The total magnetic moment rnff°t of the sample along the field is the sum of the contributions of the particles with diameters between D and D + dD. It reads
l" m~' =NJm(D) ((cosO),~.t~)p(D)
dD.
(12)
If V t°t-- NfV(D)p(D) dD is the total volume of the particles of individual volumes V(D) the sample magnetization along the field per unit volume of magnetic material can be written as
m~ t fm(O)((cosO),~.t~)p(O ) dD Mn -
(13)
VtOt -
f v(O)p(O) dD where m (D) is defined by Eq. (8). The sample magnetization per gram of magnetic material is simply Mn/p where p is the density in g/cm 3. Note that Eq. (8) generalizes the treatment of Williams et al. [12] to particles with a non-uniform spontaneous magnetization. The magnetic surface effects which may reveal an interesting interaction with the polymer matrix can be taken into account. Assume now that the ferromagnetic particles are dispersed in a carder fluid. Consider a particle of diameter D. The thermal average (mn) of the projection of the magnetic moment along the field is given by the Boltzmann distribution
O. Jarjayes et al. /Journal of Magnetism and Magnetic Materials 137 (1994) 205-218
210
.
"r, = o.s
........... i .......... %
o.4+
.
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.............. J................ i ............ ~,...........
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i
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~
0
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i ............ i ......
.......... * .............. i............ i............. !.............
"-" M 0.2
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.......
i ...........
! i i i ~o ............................................................
I
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40
80
120
160
200
240
280
320
Temperature (K) Fig. 1. Measured values of the susceptibility X of a sample of polypyrrolc with embedded y-Fe203/Cit anions as the temperature is raised in a field of 30 Oe. The Zero Field Cooled (ZFC) and Field Cooled (FC) curves correspond to the sample freezing without and with the field prior to the measurements.
ff cosO ffexp[
exp[ -/?'BET( 0,~b,O, q~) ] sin0 d0d~b sinO dOdqb
(14)
- /~ET( 0,~b,O,q~) ] sin0 d0d~b sinO dOdqb,
where (9, • are the spherical coordinatesof the orientationunit vector]~ewith respect to a particle referenceframe, the axis of which points in the moment direction ~. According to Eq. (10a) the expression (14) reduces to the usual Langevinequation
m(D)H
(mt.t>=m(D)(cosO)=m(D)L[~nT ], where the Langevin function is defined as L(a) = coth(a) unit volume of magnetic material is given by
m tfft
(15)
1/a. The ferrofluid
magnetization along the field per
(m(D)L[m(D)H]D(D) dD .] [_ kBT _r
Mn = VtOt -
(16)
I V(D)p(D) dD Note that Eq. (13), which applies to a distribution of particles in a solid matrix, simplifies to Eq. (16) if the anisotropy energy of each particle is much smaller than its Zeeman energy, i.e. KV(D) << m(D)H. Numerical details for computing the expressions (13) and (16) of Mn are given in Appendix A.
4. Results and discussion Fig. 1 shows the variation with temperature of the susceptibility X of the y-Fe203/Cit-PPy films in a small magnetic field of 30 Oe. The Zero Field Cooling (ZFC) curve is obtained by cooling the sample down to 20 K at
O. Jarjayes et al. / Journal of Magnetism and Magnetic Materials 137 (1994) 205-218
211
~'~~l 7075 '.~.%' ' I~ . . . . . . . . . . . . . . . . . . . . - . . . . . . .
•
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.............................................................. i.................... ?,0 ................................. •
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55
. . . . . . . . . . . .
0
i ....
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. . . . . . . .
1000 2000 3000 4000 5000 6000 (Temperature) 3/2 (K) 3/2
Fig.2. Saturationmagnetization(MH/p)sa t versusT 3/2 in a fieldof 20kOe. zero field. The Field Cooled (FC) curve corresponds to the freezing of the same sample in the presence of a magnetic field of 30 Oe. The susceptibility measurements are performed with increasing temperature in a field H = 30 Oe. The peak in the ZFC curve results from the competition between two effects related to the increase of temperature. In the ZFC process the isotropic random distribution of the magnetic moments m of the particles is frozen. When the temperature starts to rise from 20 K the moments of the smallest particles, which have the shortest Nrel times ~'N defined by Eq. (3), are freed from their frozen orientations by the thermal fluctuations of the crystal lattice and tend to align parallel to the field. This causes an initial increase of the susceptibility. As the temperature further rises, the moments of larger particles become parallel to the field, but at the same time the thermal agitation begins to average out the sample magnetization. This effect finally dominates and the susceptibility decreases. The presence of a susceptibility maximum is often interpreted in terms of a superparamagnetic blocking temperature TB,m~x [25,26] below which an irreversible behaviour of the sample magnetization is clearly observed. Here, TB,mx-- 80 K. However, in any real system the distribution of anisotropy energies associated to the various particle sizes give rise to a distribution of blocking temperatures [ 19]. This explains why the ZFC and FC curves move apart below 160 K in a progressive way, contrary to the spin glasses for which a sharp cusp is observed. Thus, the superparamagnetic approximation can be safely applied above 160 K. The temperature dependence of the saturation magnetization (Mn/p)sat in a field of 20 kOe is reported in Fig. 2. Over a large temperature range (Mn/p)s,t follows the usual Bloch law in T 3/2, which can be deduced from the spin wave excitation model ( [ 16], pp. 566-569). The magnetization curves of the pure ferrofluid at 300 K and of the polymer with included monodomains at various temperatures are plotted in Figs. 3 and 4, respectively. The theoretical expressions of the sample magnetization, which are based on the concepts developed in Section 3, are fitted to the experimental data. The density of "y-Fe203 oxide in the grains is taken to be p = 4.60 g/cm 3 [ 15 ]. Firstly, a uniform spontaneous magnetization of the particles and an anisotropy energy constant K = 0 are assumed. The y-FezO3/Cit anions of the ferrofluid in the liquid state at 300 K are dilute enough to neglect their interactions. Indeed, the density of y-Fe203 in the solution is 0.141 g/cm 3. An estimate of the number density of the grains can be easily calculated in a reference monodisperse solution. It is given by nF F
0.141 pX 1~/)3 = 4.7 × 1017 grains/cm 3,
in case all the grains have the same diameter,/) = 50/~. Hereafter, it will be shown that this value approximates to the diameter Dmmpat the maximum of the distribution of the particle diameters. Extensive solution studies lead by Bacri and Massart [ 14] have shown that in our aqueous solution around pH = 7 the colloidal citrated T-Fe203
212
O. Jarjayes et al. / Journal of Magnetism and Magnetic Materials 137 (1994) 205-218
J
80
L
I
*
I
J
I
I
I
I
,
g-l]
MH/P [ emu
~
I
FF
60
40
20
0
r
u
r
l
u
0
l
f
I
l
l
I0
t
u
I
f
l
2O
30
HEkOe] Fig. 3. Magnetization MH/pa s a function of the magnetic field at 300 K for the liquid ferrofluid of T-Fe203/Cit anions. The solid line corresponds to the best fit of the expression (16).
I
1 O0
I
J
I
I
I
MH/P E emu
I
]
J
I
t
n
i
i
I
g-1 ]
8013 .....
J3
60
40-
20
0
I0
20
30
HEkOe3 Fig. 4. Magnetization MH/p as a function of the magnetic field at three different temperatures ( + : 300 K, X : 240 K, . : 200 K) for the polypyrrole matrix with embedded T-Fe203/Cit anions. The lines correspond to the best simultaneous fit of the expression (16) to these experimental values and to the saturation values at three other temperatures (O: 160 K, A: 120 K, [~: 80 K).
O. Jarjayes et al. / Journal of Magnetism and Magnetic Materials 137 (1994) 205-218
213
particles are macroanions with a superficial density of charge ,~ -- 1.25 × 10 - 2 e/~2. For the reference monodisperse solution the macroanion charge is q = -400e. The ionic strength [27] of an electrolyte solution is i = ½E,ciz 2 , with the summation running over all ionic species i of concentrations ci ( expressed in mol / 1) and of charges qi = z,e. Here, i = 0.05 mol/l, and a single dilute phase is expected [ 14]. Furthermore, according to the Debye-Hiickel theory [27], the effective screened Coulomb potential between two repulsive macroanions is
~w(R)
q2 exp[--K(R--/9)] if R>/5, cR (1 + v,Jff))
(17a)
where the inverse of the Debye length K is defined in cgs units as
~1/2
[ 8,rre2iNA K=~IOOOkBT¢)
,
(17b)
NA being the Avogadro number and ¢ the dielectric constant of water. In the reference solution at 300 K, ¢ = 78 and rd)=4.1. The values of the repulsive Coulomb potential flw(R) are 4.5 × 10a, 75, and 1.2 at R = 1, 2, and 3/), respectively, so that the macroanions should be well separated. The strength of the grain interactions is related to the mean distance between the nearest neighbours. This distance can be estimated to b e / ~ = 150 ~ in case the grains do not form aggregates, but occupy the nodes of a face-centred cubic lattice of cube side a = x/2R~ ( [ 16], pp. 18-21). The rather large intercentre mean distance Rf~ -FF of two neighbouring grains with respect to their diameters is compatible with a weak interparticle magnetic coupling in the solution. This will be checked now. A given magnetic particle in a sample is submitted to a local field H~oc which results from the external field H and from the magnetic dipolar interactions of the surrounding particles, the magnetic moments of which tend to be parallel to the external field. The exact calculation of the local field is a very complicated self-consistent many-body problem. However, in an uninterrupted sample it can be written as ( [ 17], pp. 1-31) /-/loc = / ' / + H I '
+/-/1 ".
(18a)
The dipolar field H i ' originates from all the particles located in a Lorentz sphere centred at the studied particle and having a diameter which is large with respect to the particle dimension/5, but small with respect to the sample size. It verifies
m(/5)
H~' < /~f3 •
(18b)
The dipolar field H~" results from all the particles in the sample outside the Lorentz sphere. When the sample shape is smooth and symmetric enough, H1" is constant inside the sample and given by H
l " -- - -4~M ~ sample-NMs~mple,
(18c)
where Ms~p|e is the sample magnetization, which is very different from the particle magnetization for dilute solutions, and where N is the tensor of the demagnetizing field coefficients. An estimate of the intensity of H1" is rt 471" HI =-~-M~ampl~.
(18d)
In an external field H = 200 Oe, the dipolar contribution 1-11' in our solution verifies Hi' <-50e, since/9 = ~ = 50 and Rfc,:----Rfo:-~= 150 A. The measured sample magnetization is Ms~pl~ ~ 2 emu/cm 3 so that Ha"~ 8 O¢. The local f i e l d / / ~ can be safely approximated to the external field H with an accuracy of about 5%, and due to the magnetization saturation this approximation is all the better as the external field increases. The hypothesis of independent magnetic particles is justified. On the other hand, for small external fields H < 50 Oe, the many-body interparticle magnetic interactions become significant.
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Using Eq. (16) the best fit gives a log-normal distribution of the particle sizes, characterized by a diameter Dm pFF__-54 ]k at the probability maximum and by a standard deviation o-vF = 0.40. The theoretical magnetization of the ferrofluid is shown in Fig. 3. Similarly, the grains in the composite are assumed to be independent. Again this hypothesis can be easily checked. In an external field H = 2 0 0 Oe, the dipolar contribution Ha' in the composite verifies H1' <15 Oe, since P P y (TEM) = 50/~k and/~fcc = ~ ,~ArPPY D- = Omp ,'- fcc PPY ~ 100 A. The measured composite magnetization is ,,, sample~ 7 emu/cm 3 SO that Hi" = 30 Oe. The local field H~oc can be approximated to the external field H with an accuracy of at least 20%, but probably better since the pieces of composite material and the grains themselves are randomly distributed in the sample holder, which tends to average the dipolar interactions. Using Eq. (13) the log-normal distribution of the grains can be determined by a least-squares fitting to the experimental sample magnetization at three temperatures 300, 240, and 200 K for all the values of the field up to 30 kOe and at three other lower temperatures 160, 120, and 80 K for two saturating fields. Indeed, the saturation magnetization in high fields, (MH/p)sat, follows the T 3/2 law of Fig. 2 and seems independent of the particle blocking at low temperature. The theoretical magnetization curves are represented in Fig. 4 and the associated parameters /-')PPY=51.3 A and o-PPY=0.39 so that nearly all particles have a diameter obtained for the log-normal law are ~mp D < 150 ~. If the fitting procedure is restricted to the measurements at 300 K a 'perfect' fit is observed, giving a value o f OPm~y = 52.9 ~, which is consistent with the previous value 51.3 ~. The hypothesis of a superparamagnetic regime above 160 K is validated. Another important point to notice is that the sizes of the y-Fe203/Cit particles, which were made during the same synthesis, are very similar in the ferrofluid and in the composite. The Dmp values differ by less than 5% and such differences can be equally observed between two samples of ferrofluid or of the composite material. This means that during the electrochemical synthesis no aggregation between the monodomains occurred and the preferential inclusion of the smallest particles was not favoured. In the case of the mesoscopic particles considered here the electrochemical inclusion process differs from the mechanical one for which Beck and Dahlhaus [28] observed that under a strong convection the small particles of WO3 of sizes 0.1/xm are deposited more easily in the PPy matrix, while the very large particles of diameters 10 /xm cannot even be permanently incorporated. An anisotropy energy constant, K--~ 105 erg/cm 3, can also be very roughly estimated from the fitting procedure. The fits are very little improved by the presence of the additional adjustable parameter K because its influence on the sample magnetization is often modest as discussed in the Appendix. Thus, the K value strongly depends on small experimental variations from one sample to another, on the number of temperatures, and on the field range for which the theoretical magnetization expression is fitted. The possible K values are higher than the value 4.7 × 104 erg/cm 3 which is measured in the bulk y-Fe203 material [ 15]. However, they compare favourably to the values which have been obtained for similar spinel iron oxide particles by M6ssbauer spectroscopy [29] and which increase with decreasing particle size. Furthermore, preliminary M6ssbauer studies [30] show an anisotropy constant K ~ 3 × 105 erg/cm 3 for the T-Fe203 particles in our composite material. The hypotheses which are the basis for the theoretical magnetization expressions (13) and (16) can now be checked. Suppose an anisotropy constant K = 3 × 105 erg/cm 3. Since S = 5/2, aL = 8.33 A [ 15], the monodomain condition (1) is verified for particle sizes NaL < 200 ]k [ 10] as long as "y-Fe203 is assumed to have an effective unique exchange constant, close to the value J = 1.9 × 10-14 erg of iron which shows a Curie temperature of the same order of magnitude as "y-Fe203 ( [ 15] and [ 16], pp. 566-569). As already discussed, the random changes of the directions of the monodomain magnetic moments occurs either by Brownian rotation in the liquid state or by N6el fluctuation. Of course, for our solid composite Brownian rotation is ruled out. Following Bean and Livingston [ 7 ] let ~'= 100 s be a typical time for a magnetic measurement. A particle of volume V in a solid matrix is considered to be blocked when its N6el time given by Eq. (3) verifies T< ZN, i.e. 32kBT< KV, iff~ ~- 1012 s-~ [ 19,30]. Using this possible K value it is easily seen that all the particles are at thermal equilibrium above 160 K, but that the large ones of diameters D > 130/~ are blocked near 80 K. These results are consistent with the previous determination of the superparamagnetic regime of the composite above 160 K through the comparison of the susceptibility measurements after Zero Field Cooling and Field Cooling.
O. Jarjayes et al. / Journal of Magnetism and Magnetic Materials 137 (1994) 205-218 Illllllltlllll]llll
215
[1111
35 103xp(D) 30 25 20 15 10
0
'
0
'
I
I
50
'
'
III
'
1O0
'
'
I
I
I
150
'
I
J I
P I
250
200
DEA-I Fig. 5. Bar chart of the distributionof the sizes of the y-Fe203grains in the polypyrrolematrix as measuredfrom TransmissionElectron Micrographs.Particlesize log-normaldistributionobtainedfromthe interpretationof the magnetizationexperiments(dashedline). The particle size distribution p ( D ) deduced from the analysis of the magnetization curves is compared to that obtained by a numerical treatment of a Transmission Electron Micrograph (TEM) in Fig. 5. The shapes of these two distributions are very similar. The value of the maximum probability diameter given by the magnetization Pry(TEM) ~-50±5 1~. Also measurements, Dr~Y(mag) ~50___5/~, is in good agreement with the TEM value, Drop note that the average particle diameter Dx ~- 75/~, which has been estimated from X-ray diffraction line widths using the Scherrer method, is higher than Drop [ 29]. Furthermore, the polydispersity appears to be slightly larger in the case of the magnetization measurements, probably because the aggregates of a few grains are seen as large particles. As observed by Bacri et al. [ 10] the spontaneous magnetization Ms(T) of the grains is about 20% smaller than the bulk value 74 emu/g at room temperature if a uniform spontaneous magnetization is assumed in the total volumes of the grains. However, this apparent reduction of Ms(T) may be suppressed if the lack of magnetization is supposed to be completely localized in a nonmagnetic surface layer. The width AR ---3.2 ~ of this layer can be estimated from the magnetization data by using the expression (8) for Ms (T,R) and by assuming that the spontaneous magnetization Ms(T) has the bulk value 74 emu/g at 300 K. The size distribution p ( D ) related to this model is shown in Fig. 5. The parameters Drop and tr calculated with and without a magnetically inactive layer differ by less than 5%. Finally, because the TEM study shows no particle of diameter above 150/~, the log-normal distribution of Fig. 5 has been truncated at this diameter value and renormalized. This procedure has a negligible effect on the values of Drop and tr. On the other hand, although the particles of diameters near 140 ,~ are very few, they notably contribute to the magnetization.
5. Conclusion We have been able to include ferromagnetic y-Fe203 grains of a few nanometers into an electroconducting polymer matrix of polypyrrole by electrochemical synthesis. The composite material obtained is in the form of thin films which are 100/zm thick. The theoretical treatment of the magnetism of such a material in terms of the magnetic properties of the grains and of their possible modifications by the inclusion has been developed. A careful analysis
o. Jarjayeset al. / JournalofMagnetismandMagneticMaterials137(1994)205-218
216
of the magnetization of composite samples has shown that the distribution of the grain sizes obeys a log-normal law with as most probable diameter, DmpPPY(mag) - 52/~. All the grains of the aqueous ferrofluid solution are equally incorporated into the polymer matrix whatever their sizes. The inclusion of the mesoscopic T-Fe203 particles leads to a fair dispersion in the polymer, with no magnetic modification of the particles and no aggregation. Moreover, no major interaction between the particles has been detected. As far as the polymer is concerned, its electrical conductivity as a function of temperature remains unchanged in the presence of the magnetic particles. Thus, for this class of composites the conducting properties of the polymer and the magnetic properties of the grains do coexist without any interference. Potential applications in electromagnetic interference shielding can be tested for this material and similar ones.
Acknowledgements Valuable conversations with Drs. P. Molho and P.A. Petit of the CNRS-Grenoble are appreciated. The financial support of the Rdgion Rh6ne-Alpes is gratefully acknowledged. We also thank the company Arjo, Wiggins, and Appleton in Apprieu Is6re for their interest in this work.
Appendix A
Let /3BE-r( 0,4',a,/3) = -- acos0 + bsin20,
(A1)
be the reduced energy with a =/3Bm(D)H, b =/3BKV(D). The calculation of the expression (13) of the sample magnetization MH first requires the computation of the double average
IfcosO
exp [ - ~3nET(O,4',a,/3) ]sinO dOd4'
C(a,b) = ((cos O)~,~) = ~
-sin/3 d flda.
(A2)
f f exp[ - /3.Ev( O,4',a,/3) ]sinO dOd4' Because the total energy Er given by Eq. (10) depends on the angles ce and 4' through their difference 4 ' - a, all the partial three-dimensional integrals over 0, 4', and/3 are independent of a, and Eq. (A2) simplifies to
cosO ff C(a,b) = ~ f . .
exp[ - ~3BEy(0,4',0,/3) ] sin0 d0d4'
.
.
.
.
.
.
sin/3 d/3.
(A3)
f f exp[ - ~ E ~ ( O,4',O,/3)]sinO dOd4' This integral is computed by the Cartesian product [ 31 ] of two Gauss-l_~gendre quadratures ( [ 32], pp. 324-331 ) over the angles 0 and/3 in [0,'rr], times the Gaussian composite trapezoidal rule ([32], pp. 341-343) over the angle 4' in [0, 2~r]. Because of the symmetry relation cos4'= cos(2n-r - 4') the quadrature over 4' can be reduced to the interval [ 0,'rr ]. Another simplification occurs because the contributions to the integral (A3) associated to/3 and "rr-/3 are equal. The Gaussian quadratures are very accurate integration methods, even for small numbers no, n+, and nt3 of nodes and weights. Although the exponential exp( -/3BET) has a very steep angular variation for large a and b values, the function C(a,b) is stabilized to about 0.5% for a,b< 100 if no, n,b, no=20. From a careful analysis of the ratio of the double integrals of Eq. (A3) it can be shown that
O. Jarjayes et al. /Journal o fMagnetism and Magnetic Materials 137 (1994) 205-218
217
[0 .8
0 .8 6 .~
O"
Fo <¢
'~,
A o
b.C.x~~-~
Fig. 6. The average C(a,b) of the projections along the field of the unit vectors parallel to tile magnetic moments of the grains having the same diameter value D. The variables a and b are the reduced Zeeman and anisotropy energies, respectively. ~/2
C(a,b) ---- / tanh(a cosOa)cosOasinfl d/3 if 1 << b,
(A4)
0
where 0t3 is the solution in [0, fl] of the equation asin0+ b s i n 2 ( 0 - / 3 ) = 0, and gives the maximum value of - flaET. Furthermore, when a << b, 08---/3 and Eq. (A4) simplifies to "~/2
C(a,b) --- / tanh(a cos/3)cos/3 sin/3 d/3 if 1 << b and a << b,
(AS)
0
which is independent of b. For small values of a, Eq. (A5) becomes C(a,b) =-a/3 =L(a). In other words, the magnetization of a sample of high anisotropy energy is the same as that of a sample without anisotropy energy at low magnetic fields! The cosine average C(a,b) is plotted as a function of a and b in Fig. 6. For each fixed b value C(a,b) has a shape which is similar to that of the Langevin function L(a), with smaller values as a increases. For instance, C(20,20) ---0.75 is 20% smaller than L(20) ---0.95, and according to Eq. (A5), when 1 << a << b, C(a,b) ~ 0.5 as a--* o%while L(a) ~ 1. This discussion clearly shows that an anisotropy energy of the form (6) will notably modify the magnetization of a solid sample, only if there is a large proportion of particles with diameters D such as 1 << /3Bm(D)H << /3BKV(D). Finally, the integrals over the diameter D which occur in the expressions (13) and (16) of the magnetization Mn, are computed by taking the new variable u such as D = Doexp (o-u), and by integrating in the interval - 4 < u < 4.
References [1] R.F. Ziolo, E.P. Giannelis, B.A. Weinstein, M.P. O'Horo, B.N. Ganguly, V. Mehrotra, M.W. Russel and D.R. Huffman, Science 257 (1992) 219. [2] R.H. Marchessault, S. Ricard and P. Rioux, Carbohydr. Res. 224 (1992) 133. [3] C. Forder, S.P. Armes, A.W. Simpson, C. Maggiore and M. Hawley, J. Mater. Chem. 3 (1993) 563. [4] G. Bidan, O. Jarjayes, J.M. Fruchart and E. Hannecart, Adv. Mater. 6 (1994) 152. [5] F. Beck, M. Dahlhaus and N. Zahedi, Electrochim. Acta 37 (1992) 1265. [6] H. Yoneyama, Y. Shoji and K. Kawai, Chem. Lett. (1989) 1067. [7] C.P. Bean and J.D. Livingston, J. Appl. Phys. 30 (1959) 120S. [8] R.W. Chantrell, J. Popplewell and S.W. Charles, IEEE Trans. on Magn. 14 (1978) 975.
218
O. Jarjayes et al. / Journal of Magnetism and Magnetic Materials 137 (1994) 205-218
[9] S.M. Aharoni and M.H. Litt, J. Appl. Phys. 42 (1971) 352. [ 10] J.C. Bacri, R. Perzynski, D. Salin, V. Cabuil and R. Massart, J. Magn. Magn. Mater. 62 (1986) 36. [ 11] J.C. Bacri, R. Perzynski and D. Salin, J. Magn. Magn. Mater. 71 (1988) 246. [ 12] H.D. Williams, K. O'Grady, M. El Hilo and R.W. Chantrell, J. Magn. Magn. 122 (1993) 129. [ 13 ] R. Massart, IEEE Trans on Magn. 17 (1981) 1247; R. Massart et al., French Patent 9006484 (1990). [ 14] J.C. Bacri, R. Perzynski, D. Salin, V. Cabuil and R. Massart, J. Magn. Magn. Mater. 85 (1990) 27. [ 15] G. Bate, in : Magnetic Oxide, Vol. 2, ed. D.J. Craik (Wiley, London, 1975) p. 703. [ 16] C. Kittel, Introduction h la Physique de l'Etat Solide (Dunod, Paris, 1972). [17] A. Herpin, Th~orie du Magnrtisme (PUF, Paris, 1968). [18] L. Nrel, C. R. Acad. Sci. 228 (1949) 664; Ann. Grophys. 5 (1949) 99. [ 19] D.P.E. Dickson, N.M.K. Reid, C. Hunt, H.D. Williams, M. E1-Hilo, and K. O'Grady, J. Magn. Magn. Mater. 125 ( 1993 ) 345. [20] A. Abragam, Les Principes du Magnrtisme Nuclraire (PUF, Paris, 1961) pp. 29%301. [21 ] K. O'Grady and A. Bradbury, J. Magn. Magn. Mater. 39 (1983) 91. [22] T. Sato, T. Iijima, M. Seki and N. Inagaki, J. Magn. Magn. Mater. 65 (1987) 252. [23] A.C. Nunes, C.F. Majkrzak and A.E. Berkowitz, J. Magn. Magn. Mater. 39 (1983) 59. [24] J. Popplewell, B. Abu Aisheh and N.Y. Ayoub, J. Appl. Phys. 64 (1988) 5852 and references therein. [25] S. Koutani, G. Gavoille and R. Gerardin, J. Magn. Magn. Mater. 123 (1993) 175. [26] S.H. Liou and C.L. Chien, J. Appl. Phys. 63 (1988) 4240. [27] R.S. Berry, S.A. Rice and J. Ross, Physical Chemistry (Wiley, New York, 1980) pp. 997-1006. [28] F. Beck and M. Dahlhaus, J. Appl. Electrochem. 23 (1993) 781. [29] E. Tronc and D. Bonnin, J. Physique Lett. France 46 (1985) 437. [30] O. Jarjayes and P. Auric (private communication). [31] A.H. Stroud, in: Mathematical Methods for Digital Computers, eds. A. Ralston and H.S. Will (Wiley, London, 1967), Vol. II, pp. 145155. [32] L.W. Johnson and R.D. Riess, Numerical Analysis (Addison-Wesley, London, 1982).
•i• ELSEVIER
Journal of Magnetism and Magnetic Materials 137 (1994) 205-218
Magnetic properties of fine maghemite particles in an electroconducting polymer matrix O. Jarjayes a, P.H. Fries b,., G. Bidan
a
a CEA/D~partement de Recherche Fondamentale sur la Mati~re Condens~e/SESAM, Grenoble, France b Electrochimie Mol~culaire, Chimie de Coordination CEN-Grenoble, 17 rue des Martyrs, 38054 Grenoble Cedex 9, France
Received 20 December 1993; in revised form 5 March 1994
Abstract The magnetic properties of 7-Fe203 grains of a few nanometers in size, included in films of conducting polypyrrole by electrochemical synthesis, are investigated. A theoretical treatment of the combined effects of the possible non-uniform spontaneous magnetization of the grains and of their crystalline anisotropy energies is presented. The magnetization of samples of this composite material has been measured at several temperatures as a function of the field up to 30 kOe. The interpretation of these data shows that the grains in the polymer behave as independent monodomains, coexist with the polymer without any notable interference, and have the same size distribution as in the ferrofluid solution used for the electrochemical inclusion.
1. Introduction Over the last thirty years, the properties of fine magnetic particles have been widely studied both for theoretical purposes and for technological applications such as high density magnetic recording. New developments have appeared in the design and synthesis of mesoscopic superparamagnetic materials, i.e. particles of ferro- or ferrimagnetic monodomains of a few nanometers. Ziolo et al. [ 1 ] reported in situ synthesis of ferrimagnetic maghemite, "y-Fe203 in art ion exchange resin polymer matrix in order to make a new optically transparent magnetic material. Marchesault et al. [2] studied small ferrite particles of F e 3 0 4 trapped in lignocellulosic polymers. Recently, Armes et al. [3] described the preparation and characterization of superparamagnetic conductive polyester textile composites by a two step solution deposition process with a view to applications in electromagnetic interference shielding. In this paper the magnetic properties of 3'-Fe203 ferrimagnetic maghemite particles included in films of conducting polymer matrix of polypyrrole (PPy) are investigated. The inclusion is based on the use of an anionic ferrofluid (FF) [4] which is a liquid solution of magnetic crystalline cores of a few nanometers, coated with citrate anions. Because of their coating the magnetic cores can be included amidst the polymer chains as classical doping anions in the course of electrochemical polymerization. This method is quite different from the mechanical process [5,6] where a vigourous stirring is performed in order to insert micrometer sized particles of metal oxide like WO3 into a PPy matrix. Our goal is to look for the possible interactions between the magnetic grains and the surrounding polymer. Does the PPy matrix modify the magnetic properties of the included monodomains through the citrate coating? Does it * Corresponding author. 0304-8853/94/$07.00 © 1994 Elsevier Science B.V. All fights reserved S S D 1 0 3 0 4 - 8 8 5 3 ( 94 ) 0 0 3 2 7 - N
O. Jarjayes et aL / Journal of Magnetism and Magnetic Materials 137 (1994) 205-218
206
favour some aggregation of the grains? In order to answer these questions the magnetization curves of the T-Fe203 particles in the polymer will be analyzed as a function of the external field and temperature, and compared to the magnetization measurements of the ferrofluid. The magnetization results yield a distribution of the particle sizes which will be interpreted with respect to the particle sizes obtained by X-ray diffraction or Transmission Electron Micrographs (TEM). Such magnetic studies were pioneered in 1959 by Beans and Livingston [7] who showed that single domain ferromagnetic particles can be superparamagnetic at elevated temperature, i.e. behave in a manner analogous to Langevin paramagnetic atoms. In 1978 Chantrell et al. [8] presented a method for determining the particle mean diameter from room temperature magnetization curves using a lognormal distribution. Many studies were then done on the magnetic and thermal behaviour of fine grains [9-11]. Very recently, Williams et al. [ 12] considered the anisotropy energies of monodomains with respect to their easy magnetization axes and studied the influence of these anisotropy effects on the superparamagnetism of ferrofluids of magnetite particles coated with an oleic acid. The experimental processes for synthesizing the ferrofluid and for inserting the y-Fe203 monodomains are summarized in Section 2. The surfactant coating or the polymer matrix can modify the spontaneous magnetization of the y-Fe203 particles near the surface by distorting its crystalline structure. The theoretical expressions of the macroscopic magnetization of solid and liquid samples containing a distribution of ferromagnetic particles are given in Section 3 when both the effects of the non-uniform spontaneous magnetization of the particles and of their crystalline anisotropy energy are taken into account. Using this theory the magnetization data of our T-Fe203 ferrofluid and PPy composites are interpreted in Section 4.
2. Experimental The ferrofluid was synthesized according to Massart's procedure [ 13 ]. The y-Fe203 particles were coated with citrate - O O C - C H z - C ( O H ) ( C O O - ) - C H 2 C O O - ligands (Cit) to give the corresponding ferrofluid anions yFe203/Cit. The y-FezO3/Cit incorporated polypyrrole films (T-Fe203/Cit-PPy) were prepared by electrolysis in a one compartment cell containing a non-stirred aqueous solution of 0.5 mol/1 pyrrole and of y-Fe203/Cit anions at a 0.1 mol / 1Fe concentration. The films with embedded 'y-Fe203 grains were deposited onto an optically transparent electrode of area 2 cm X 3 cm at 0.7 V with respect to the Saturated Calomel Electrode (SCE). The films, which are typically 100/xm thick, were then washed with distilled water and acetonitrile. After one night's free standing, they were stripped from the electrode substrate and dried under vacuum for 12 hours at 60°C. The electrochemical device is made of a potentiostat PAR 173, a potentiostat/galvanostat PAR 175 and a coulometer PAR 179 from EG&G Princeton Applied Research (PAR) connected to a Sefram TGM 164 recorder. The density of the (y-Fe203/Cit-PPy) composite material is 1.85 g/cm 3. Elemental analysis shows that the mass percentages of the composite films are p=26.27% for y-Fe203, 6.81% for the citrate anions, 46.25% for the polypyrrole, and 20.65% for the residual water. During the electrochemical synthesis the particles of T-Fe203 in aqueous solution around pH = 7 are coated with negatively charged citrate ligands. There is a strong Coulomb repulsion between two particles caused by their total superficial charges [ 14], q --- - 400e, as discussed in Section 4. The particles are attracted by the positive charges on the heterocycles of the polypyrrole as it forms on the transparent electrode, and progressively incorporated. The composite structure can be imagined as a dispersion of repulsive magnetic macroanions embedded in a hydrated positively charged polymer matrix which ensures the electric neutrality of the material. The density of T-Fe203 oxide in the grains is taken to be p = 4.60 g/cm 3 [ 15]. According to a numerical treatment of a Transmission Electron Micrograph (TEM) the grain diameters are distribPPy uted about a maximum probability diameter Dmp (TEM) = 50/~. An estimate of the number density of the particles is given by n
PPY~ p × 1.85
ox
~'rr/5 3 -
1.6 × 1018 grains/cm 3,
O. Jarjayes et at / Journal of Magnetism and Magnetic Materials 137 (1994) 205-218
207
if the grains are assumed to have the same diameter value/5 = 50/~. The strength of the grain magnetic interactions is related to the mean distance between the nearest neighbours. Because of their strong Coulomb repulsion the grains are expected to be dispersed in the polypyrrole. They have difficulty forming aggregates. As a first approximation, assuming that they occupy the nodes of a face-centred cubic lattice of cube side a~'~r~Y,,the mean distance between the centres of two neighbouring grains is estimated to be /~'~PY=a~PY/v~= 100 A ([16], pp. 18-21). This intercentre distance, which is rather large with respect to the grain diameters, is compatible with a weak interparticle coupling in the composite. The magnetic measurements were carried out with a Quantum Design SQUID magnetometer operating between 2 and 300 K in magnetic fields up to 50 kOe.
3. Models of superparamagnetic particles Consider a sample containing a dispersion of nearly spherical ferromagnetic particles. Our goal is to determine the distribution of the particle sizes from magnetization measurements as a function of the external magnetic field H. This can be achieved when the particles can be approximated as ferromagnetic monodomains at thermal equilibrium, the sizes of which are distributed according to a simple probability law. It is well known (see Ref. [ 7] and [ 17], pp. 749-754) that a magnetic material has a critical size under which the lowest magnetic energy corresponds to a single domain arrangement where the magnetic moments are parallel. An estimate of this critical size is given by the thickness of the Bloch walls ( [ 16], pp. 566-569) which are the transition layers between the monodomains. It is clear that this thickness gives a limit under which the material forms a monodomain. Consider a lattice of spins S which are coupled by exchange interactions J. If aL denotes a typical lattice constant of the spin array and K an estimate of the magnetocrystalline anisotropy energy density, a first approximation of the Bloch walls thickness is given by ( [ 16], pp. 566-569)
=( ffr2JS2)1/2. NaL ~ KaL ]
(1)
Let Ms(T) be the spontaneous magnetization of the material at temperature T. The magnetic moment of a single domain of volume V and orientation a is
m=Ms(T)Va.
(2)
In order to be at thermal equilibrium during the magnetization measurements the particle moments should fluctuate with a characteristic time ~-shorter than the time of experiment. The random changes of the directions of the moments can originate from two mechanisms. The first one is always present. It was proposed by Nrel ( [ 17], pp. 761-766, and [ 18] ) and is associated with the thermal fluctuations of the crystal lattice which is coupled to the spins S. The magnetization vector Ms(T) a fluctuates between the easy directions with a characteristic Nrel time ~'N defined as
1 TN
KV \
KBI/
wherefo is a frequency factor, which is usually taken to be 1 0 9 S - 1 [7,12], but can be much larger [ 19]. A second mechanism of random reorientations of the moments occurs in liquids. It corresponds to the Brownian rotational diffusion of the particles and has a characteristic Debye time 3~V
ZB= kBT'
(4)
where 7/is the liquid viscosity, eB is the time constant of the exponential decay of the time correlation functions of the spherical harmonics of order 1 which define the moment orientation [ 20].
208
O. Jarjayes et al. / Journal of Magnetism and Magnetic Materials 137 (1994) 205-218
Electron micrographs [ 10] show particles which can be approximated as spheres of diameters D, distributed according to a log-normal probability law
p(D)
D
exp
[ ln2(D/D°)~ 20"2
(5)
J'
where In(Do) is the mean value of In(D) and 0"2 the standard deviation. The maximum of the probability density p(D) occurs for the diameter value Drop = Doexp( - 0"2). The applicability of the log-normal law was examined by O'Grady and Bradbury [21] for 100 ferrofluids. They showed that a choice between the Gaussian and log-normal distributions of the particle sizes is, in general, adequate and that the form of the distribution is very likely associated with the technique of particle preparation. In principle Eqs. (2) and (5) should be sufficient to calculate the superparamagnetic behaviour of ferromagnetic particles when their diameters D are small enough for the monodomain condition D < to be met, where the Bloch wall thickness is given by Eq. (1), and for the fluctuation times rN or ~'u to be shorter than the magnetization measurement time. However, three other effects may modify the sample magnetism. Firstly, even for spherical particles without shape anisotropic magnetic energy there remains an anisotropic energy density associated with the crystal structure ([17], pp. 333-355). Following Williams et al. [12] the effects of this magnetocrystalline interaction are estimated by assuming that it is uniaxial in symmetry and of the simplest form
NaL
Nat.
Ex = KVsin20,
(6)
where O is the angle between the easy axis and the particle moment. Secondly, Sato et al. [22] found a sharp decrease of the saturation magnetization of fine ferrite particles when their sizes are reduced below 100 ]k. They attributed this decrease to a magnetically inactive surface layer, the thickness of which increases with the crystalline anisotropy up to about 10/1,. Furthermore, Nunes et al. [23] examined the magnetization density distribution of finely divided CoFe204 particles by measuring the X-ray and polarized neutron scattered intensities. Their measurements can be interpreted by assuming a magnetically anomalous surface shell of inactive fixed pinned spins which increases from 5 ,~ for the naked particles to 20 /~ when they are coated with a chemisorbed oleate CH3 (CH2) 7CH = CH(CH2) 7COO- surfactant. In order to take possible surface effects for our material into account we assume that the spontaneous magnetization Ms(T,R) inside the particle may be non-uniform and is likely to depend on the radial distance R from the particle centre. Because a surface layer seems magnetically inactive the form of the magnetization is simply taken to be
Ms (T,R) =
(7) otherwise,
where D is the particle diameter and AR the layer thickness. Thirdly, particle interactions have been the subject of much discussion [24,25]. The subject is difficult. A Curie-Weiss behaviour of the sample magnetism has been frequently observed for small external magnetic fields with pseudo-ordering 'Curie' or 'N6el' temperatures To. However, as far as we know, there is no satisfactory theory of these many-body interactions. In what follows the particles are sufficiently dilute for neglecting their interactions in a first approximation as long as they do not form aggregates. The consistency of this hypothesis will be checked by comparing the calculated magnetism of independent particles with the measured values. Assume that the ferromagnetic particles are distributed in a solid sample located in an external magnetic field /-/= H/~L which points in the f% direction of the laboratory reference frame ( OL, fL,JL, ]~L)"Consider a single-domain ferromagnetic particle of diameter D and volume V(D). Let ke be the unit vector defining the orientation of the easy magnetization axis and a, fl its spherical coordinates. The magnetic moment of the particle is
O. Jarjayes et al. / Journal of Magnetism and Magnetic Materials 137 (1994) 205-218
m=m(D)~= fMs(T,R)d3R~,
209
(8)
v
where the unit vector fi has the spherical coordinates 0,(h. The thermal average (ran) of the projection of the magnetic moment along the field is given by the Boltzmann distribution
ffcosO exp [
1
tiBET(0,~b,Ct,fl) ] S ~ 0 dOd~b
(mn) = m(D) (cos0) ~,~ = re(D)
,
(9)
f fexp[ -13BE.r(O,4~,a,~)]sinOdOd4~ where the total particle energy ET(O,~b,a,[3) is the sum of the Zeeman and anisotropy contributions
ET( O,4a,a, fl) = -- m( D )HcosO + KV(D) sin20,
(10a)
cosO= cos/3cos0 + sin/3sin0cos( ~b- a).
(10b)
with
Now let N be the number of ferromagnetic particles in the sample. Consider the subset of the particles with diameters between D and D + dD. The orientations ke of their easy axes have an uniform angle distribution. The angle average ((ran)) of the thermal averages (mn) of the projections of the magnetic moments of these particles along the field is 1
( ( mn ) ) = m( D ) ( ( cosO) c,.a) = -~-~,rrm(D ) f f ( cosO) ,,.asin I3 d Oda.
(11)
The total magnetic moment rnff°t of the sample along the field is the sum of the contributions of the particles with diameters between D and D + dD. It reads
l" m~' =NJm(D) ((cosO),~.t~)p(D)
dD.
(12)
If V t°t-- NfV(D)p(D) dD is the total volume of the particles of individual volumes V(D) the sample magnetization along the field per unit volume of magnetic material can be written as
m~ t fm(O)((cosO),~.t~)p(O ) dD Mn -
(13)
VtOt -
f v(O)p(O) dD where m (D) is defined by Eq. (8). The sample magnetization per gram of magnetic material is simply Mn/p where p is the density in g/cm 3. Note that Eq. (8) generalizes the treatment of Williams et al. [12] to particles with a non-uniform spontaneous magnetization. The magnetic surface effects which may reveal an interesting interaction with the polymer matrix can be taken into account. Assume now that the ferromagnetic particles are dispersed in a carder fluid. Consider a particle of diameter D. The thermal average (mn) of the projection of the magnetic moment along the field is given by the Boltzmann distribution
O. Jarjayes et al. /Journal of Magnetism and Magnetic Materials 137 (1994) 205-218
210
.
"r, = o.s
........... i .......... %
o.4+
.
.
.
.
.
.
.
.
.
.............. J................ i ............ ~,...........
.
i
I
~
0
.
.
.
i ............ i ......
.......... * .............. i............ i............. !.............
"-" M 0.2
.
.......
i ...........
! i i i ~o ............................................................
I
I
I
I
I
I
I
40
80
120
160
200
240
280
320
Temperature (K) Fig. 1. Measured values of the susceptibility X of a sample of polypyrrolc with embedded y-Fe203/Cit anions as the temperature is raised in a field of 30 Oe. The Zero Field Cooled (ZFC) and Field Cooled (FC) curves correspond to the sample freezing without and with the field prior to the measurements.
ff cosO ffexp[
exp[ -/?'BET( 0,~b,O, q~) ] sin0 d0d~b sinO dOdqb
(14)
- /~ET( 0,~b,O,q~) ] sin0 d0d~b sinO dOdqb,
where (9, • are the spherical coordinatesof the orientationunit vector]~ewith respect to a particle referenceframe, the axis of which points in the moment direction ~. According to Eq. (10a) the expression (14) reduces to the usual Langevinequation
m(D)H
(mt.t>=m(D)(cosO)=m(D)L[~nT ], where the Langevin function is defined as L(a) = coth(a) unit volume of magnetic material is given by
m tfft
(15)
1/a. The ferrofluid
magnetization along the field per
(m(D)L[m(D)H]D(D) dD .] [_ kBT _r
Mn = VtOt -
(16)
I V(D)p(D) dD Note that Eq. (13), which applies to a distribution of particles in a solid matrix, simplifies to Eq. (16) if the anisotropy energy of each particle is much smaller than its Zeeman energy, i.e. KV(D) << m(D)H. Numerical details for computing the expressions (13) and (16) of Mn are given in Appendix A.
4. Results and discussion Fig. 1 shows the variation with temperature of the susceptibility X of the y-Fe203/Cit-PPy films in a small magnetic field of 30 Oe. The Zero Field Cooling (ZFC) curve is obtained by cooling the sample down to 20 K at
O. Jarjayes et al. / Journal of Magnetism and Magnetic Materials 137 (1994) 205-218
211
~'~~l 7075 '.~.%' ' I~ . . . . . . . . . . . . . . . . . . . . - . . . . . . .
•
..............................................................................................
65
......................................• ............... .... ~............................................................
60
.............................................................. i.................... ?,0 ................................. •
..~
55
. . . . . . . . . . . .
0
i ....
, I
•
. . . . . . . .
1000 2000 3000 4000 5000 6000 (Temperature) 3/2 (K) 3/2
Fig.2. Saturationmagnetization(MH/p)sa t versusT 3/2 in a fieldof 20kOe. zero field. The Field Cooled (FC) curve corresponds to the freezing of the same sample in the presence of a magnetic field of 30 Oe. The susceptibility measurements are performed with increasing temperature in a field H = 30 Oe. The peak in the ZFC curve results from the competition between two effects related to the increase of temperature. In the ZFC process the isotropic random distribution of the magnetic moments m of the particles is frozen. When the temperature starts to rise from 20 K the moments of the smallest particles, which have the shortest Nrel times ~'N defined by Eq. (3), are freed from their frozen orientations by the thermal fluctuations of the crystal lattice and tend to align parallel to the field. This causes an initial increase of the susceptibility. As the temperature further rises, the moments of larger particles become parallel to the field, but at the same time the thermal agitation begins to average out the sample magnetization. This effect finally dominates and the susceptibility decreases. The presence of a susceptibility maximum is often interpreted in terms of a superparamagnetic blocking temperature TB,m~x [25,26] below which an irreversible behaviour of the sample magnetization is clearly observed. Here, TB,mx-- 80 K. However, in any real system the distribution of anisotropy energies associated to the various particle sizes give rise to a distribution of blocking temperatures [ 19]. This explains why the ZFC and FC curves move apart below 160 K in a progressive way, contrary to the spin glasses for which a sharp cusp is observed. Thus, the superparamagnetic approximation can be safely applied above 160 K. The temperature dependence of the saturation magnetization (Mn/p)sat in a field of 20 kOe is reported in Fig. 2. Over a large temperature range (Mn/p)s,t follows the usual Bloch law in T 3/2, which can be deduced from the spin wave excitation model ( [ 16], pp. 566-569). The magnetization curves of the pure ferrofluid at 300 K and of the polymer with included monodomains at various temperatures are plotted in Figs. 3 and 4, respectively. The theoretical expressions of the sample magnetization, which are based on the concepts developed in Section 3, are fitted to the experimental data. The density of "y-Fe203 oxide in the grains is taken to be p = 4.60 g/cm 3 [ 15 ]. Firstly, a uniform spontaneous magnetization of the particles and an anisotropy energy constant K = 0 are assumed. The y-FezO3/Cit anions of the ferrofluid in the liquid state at 300 K are dilute enough to neglect their interactions. Indeed, the density of y-Fe203 in the solution is 0.141 g/cm 3. An estimate of the number density of the grains can be easily calculated in a reference monodisperse solution. It is given by nF F
0.141 pX 1~/)3 = 4.7 × 1017 grains/cm 3,
in case all the grains have the same diameter,/) = 50/~. Hereafter, it will be shown that this value approximates to the diameter Dmmpat the maximum of the distribution of the particle diameters. Extensive solution studies lead by Bacri and Massart [ 14] have shown that in our aqueous solution around pH = 7 the colloidal citrated T-Fe203
212
O. Jarjayes et al. / Journal of Magnetism and Magnetic Materials 137 (1994) 205-218
J
80
L
I
*
I
J
I
I
I
I
,
g-l]
MH/P [ emu
~
I
FF
60
40
20
0
r
u
r
l
u
0
l
f
I
l
l
I0
t
u
I
f
l
2O
30
HEkOe] Fig. 3. Magnetization MH/pa s a function of the magnetic field at 300 K for the liquid ferrofluid of T-Fe203/Cit anions. The solid line corresponds to the best fit of the expression (16).
I
1 O0
I
J
I
I
I
MH/P E emu
I
]
J
I
t
n
i
i
I
g-1 ]
8013 .....
J3
60
40-
20
0
I0
20
30
HEkOe3 Fig. 4. Magnetization MH/p as a function of the magnetic field at three different temperatures ( + : 300 K, X : 240 K, . : 200 K) for the polypyrrole matrix with embedded T-Fe203/Cit anions. The lines correspond to the best simultaneous fit of the expression (16) to these experimental values and to the saturation values at three other temperatures (O: 160 K, A: 120 K, [~: 80 K).
O. Jarjayes et al. / Journal of Magnetism and Magnetic Materials 137 (1994) 205-218
213
particles are macroanions with a superficial density of charge ,~ -- 1.25 × 10 - 2 e/~2. For the reference monodisperse solution the macroanion charge is q = -400e. The ionic strength [27] of an electrolyte solution is i = ½E,ciz 2 , with the summation running over all ionic species i of concentrations ci ( expressed in mol / 1) and of charges qi = z,e. Here, i = 0.05 mol/l, and a single dilute phase is expected [ 14]. Furthermore, according to the Debye-Hiickel theory [27], the effective screened Coulomb potential between two repulsive macroanions is
~w(R)
q2 exp[--K(R--/9)] if R>/5, cR (1 + v,Jff))
(17a)
where the inverse of the Debye length K is defined in cgs units as
~1/2
[ 8,rre2iNA K=~IOOOkBT¢)
,
(17b)
NA being the Avogadro number and ¢ the dielectric constant of water. In the reference solution at 300 K, ¢ = 78 and rd)=4.1. The values of the repulsive Coulomb potential flw(R) are 4.5 × 10a, 75, and 1.2 at R = 1, 2, and 3/), respectively, so that the macroanions should be well separated. The strength of the grain interactions is related to the mean distance between the nearest neighbours. This distance can be estimated to b e / ~ = 150 ~ in case the grains do not form aggregates, but occupy the nodes of a face-centred cubic lattice of cube side a = x/2R~ ( [ 16], pp. 18-21). The rather large intercentre mean distance Rf~ -FF of two neighbouring grains with respect to their diameters is compatible with a weak interparticle magnetic coupling in the solution. This will be checked now. A given magnetic particle in a sample is submitted to a local field H~oc which results from the external field H and from the magnetic dipolar interactions of the surrounding particles, the magnetic moments of which tend to be parallel to the external field. The exact calculation of the local field is a very complicated self-consistent many-body problem. However, in an uninterrupted sample it can be written as ( [ 17], pp. 1-31) /-/loc = / ' / + H I '
+/-/1 ".
(18a)
The dipolar field H i ' originates from all the particles located in a Lorentz sphere centred at the studied particle and having a diameter which is large with respect to the particle dimension/5, but small with respect to the sample size. It verifies
m(/5)
H~' < /~f3 •
(18b)
The dipolar field H~" results from all the particles in the sample outside the Lorentz sphere. When the sample shape is smooth and symmetric enough, H1" is constant inside the sample and given by H
l " -- - -4~M ~ sample-NMs~mple,
(18c)
where Ms~p|e is the sample magnetization, which is very different from the particle magnetization for dilute solutions, and where N is the tensor of the demagnetizing field coefficients. An estimate of the intensity of H1" is rt 471" HI =-~-M~ampl~.
(18d)
In an external field H = 200 Oe, the dipolar contribution 1-11' in our solution verifies Hi' <-50e, since/9 = ~ = 50 and Rfc,:----Rfo:-~= 150 A. The measured sample magnetization is Ms~pl~ ~ 2 emu/cm 3 so that Ha"~ 8 O¢. The local f i e l d / / ~ can be safely approximated to the external field H with an accuracy of about 5%, and due to the magnetization saturation this approximation is all the better as the external field increases. The hypothesis of independent magnetic particles is justified. On the other hand, for small external fields H < 50 Oe, the many-body interparticle magnetic interactions become significant.
214
O. Jarjayes et al. / Journal of Magnetism and Magnetic Materials 137 (1994) 205-218
Using Eq. (16) the best fit gives a log-normal distribution of the particle sizes, characterized by a diameter Dm pFF__-54 ]k at the probability maximum and by a standard deviation o-vF = 0.40. The theoretical magnetization of the ferrofluid is shown in Fig. 3. Similarly, the grains in the composite are assumed to be independent. Again this hypothesis can be easily checked. In an external field H = 2 0 0 Oe, the dipolar contribution Ha' in the composite verifies H1' <15 Oe, since P P y (TEM) = 50/~k and/~fcc = ~ ,~ArPPY D- = Omp ,'- fcc PPY ~ 100 A. The measured composite magnetization is ,,, sample~ 7 emu/cm 3 SO that Hi" = 30 Oe. The local field H~oc can be approximated to the external field H with an accuracy of at least 20%, but probably better since the pieces of composite material and the grains themselves are randomly distributed in the sample holder, which tends to average the dipolar interactions. Using Eq. (13) the log-normal distribution of the grains can be determined by a least-squares fitting to the experimental sample magnetization at three temperatures 300, 240, and 200 K for all the values of the field up to 30 kOe and at three other lower temperatures 160, 120, and 80 K for two saturating fields. Indeed, the saturation magnetization in high fields, (MH/p)sat, follows the T 3/2 law of Fig. 2 and seems independent of the particle blocking at low temperature. The theoretical magnetization curves are represented in Fig. 4 and the associated parameters /-')PPY=51.3 A and o-PPY=0.39 so that nearly all particles have a diameter obtained for the log-normal law are ~mp D < 150 ~. If the fitting procedure is restricted to the measurements at 300 K a 'perfect' fit is observed, giving a value o f OPm~y = 52.9 ~, which is consistent with the previous value 51.3 ~. The hypothesis of a superparamagnetic regime above 160 K is validated. Another important point to notice is that the sizes of the y-Fe203/Cit particles, which were made during the same synthesis, are very similar in the ferrofluid and in the composite. The Dmp values differ by less than 5% and such differences can be equally observed between two samples of ferrofluid or of the composite material. This means that during the electrochemical synthesis no aggregation between the monodomains occurred and the preferential inclusion of the smallest particles was not favoured. In the case of the mesoscopic particles considered here the electrochemical inclusion process differs from the mechanical one for which Beck and Dahlhaus [28] observed that under a strong convection the small particles of WO3 of sizes 0.1/xm are deposited more easily in the PPy matrix, while the very large particles of diameters 10 /xm cannot even be permanently incorporated. An anisotropy energy constant, K--~ 105 erg/cm 3, can also be very roughly estimated from the fitting procedure. The fits are very little improved by the presence of the additional adjustable parameter K because its influence on the sample magnetization is often modest as discussed in the Appendix. Thus, the K value strongly depends on small experimental variations from one sample to another, on the number of temperatures, and on the field range for which the theoretical magnetization expression is fitted. The possible K values are higher than the value 4.7 × 104 erg/cm 3 which is measured in the bulk y-Fe203 material [ 15]. However, they compare favourably to the values which have been obtained for similar spinel iron oxide particles by M6ssbauer spectroscopy [29] and which increase with decreasing particle size. Furthermore, preliminary M6ssbauer studies [30] show an anisotropy constant K ~ 3 × 105 erg/cm 3 for the T-Fe203 particles in our composite material. The hypotheses which are the basis for the theoretical magnetization expressions (13) and (16) can now be checked. Suppose an anisotropy constant K = 3 × 105 erg/cm 3. Since S = 5/2, aL = 8.33 A [ 15], the monodomain condition (1) is verified for particle sizes NaL < 200 ]k [ 10] as long as "y-Fe203 is assumed to have an effective unique exchange constant, close to the value J = 1.9 × 10-14 erg of iron which shows a Curie temperature of the same order of magnitude as "y-Fe203 ( [ 15] and [ 16], pp. 566-569). As already discussed, the random changes of the directions of the monodomain magnetic moments occurs either by Brownian rotation in the liquid state or by N6el fluctuation. Of course, for our solid composite Brownian rotation is ruled out. Following Bean and Livingston [ 7 ] let ~'= 100 s be a typical time for a magnetic measurement. A particle of volume V in a solid matrix is considered to be blocked when its N6el time given by Eq. (3) verifies T< ZN, i.e. 32kBT< KV, iff~ ~- 1012 s-~ [ 19,30]. Using this possible K value it is easily seen that all the particles are at thermal equilibrium above 160 K, but that the large ones of diameters D > 130/~ are blocked near 80 K. These results are consistent with the previous determination of the superparamagnetic regime of the composite above 160 K through the comparison of the susceptibility measurements after Zero Field Cooling and Field Cooling.
O. Jarjayes et al. / Journal of Magnetism and Magnetic Materials 137 (1994) 205-218 Illllllltlllll]llll
215
[1111
35 103xp(D) 30 25 20 15 10
0
'
0
'
I
I
50
'
'
III
'
1O0
'
'
I
I
I
150
'
I
J I
P I
250
200
DEA-I Fig. 5. Bar chart of the distributionof the sizes of the y-Fe203grains in the polypyrrolematrix as measuredfrom TransmissionElectron Micrographs.Particlesize log-normaldistributionobtainedfromthe interpretationof the magnetizationexperiments(dashedline). The particle size distribution p ( D ) deduced from the analysis of the magnetization curves is compared to that obtained by a numerical treatment of a Transmission Electron Micrograph (TEM) in Fig. 5. The shapes of these two distributions are very similar. The value of the maximum probability diameter given by the magnetization Pry(TEM) ~-50±5 1~. Also measurements, Dr~Y(mag) ~50___5/~, is in good agreement with the TEM value, Drop note that the average particle diameter Dx ~- 75/~, which has been estimated from X-ray diffraction line widths using the Scherrer method, is higher than Drop [ 29]. Furthermore, the polydispersity appears to be slightly larger in the case of the magnetization measurements, probably because the aggregates of a few grains are seen as large particles. As observed by Bacri et al. [ 10] the spontaneous magnetization Ms(T) of the grains is about 20% smaller than the bulk value 74 emu/g at room temperature if a uniform spontaneous magnetization is assumed in the total volumes of the grains. However, this apparent reduction of Ms(T) may be suppressed if the lack of magnetization is supposed to be completely localized in a nonmagnetic surface layer. The width AR ---3.2 ~ of this layer can be estimated from the magnetization data by using the expression (8) for Ms (T,R) and by assuming that the spontaneous magnetization Ms(T) has the bulk value 74 emu/g at 300 K. The size distribution p ( D ) related to this model is shown in Fig. 5. The parameters Drop and tr calculated with and without a magnetically inactive layer differ by less than 5%. Finally, because the TEM study shows no particle of diameter above 150/~, the log-normal distribution of Fig. 5 has been truncated at this diameter value and renormalized. This procedure has a negligible effect on the values of Drop and tr. On the other hand, although the particles of diameters near 140 ,~ are very few, they notably contribute to the magnetization.
5. Conclusion We have been able to include ferromagnetic y-Fe203 grains of a few nanometers into an electroconducting polymer matrix of polypyrrole by electrochemical synthesis. The composite material obtained is in the form of thin films which are 100/zm thick. The theoretical treatment of the magnetism of such a material in terms of the magnetic properties of the grains and of their possible modifications by the inclusion has been developed. A careful analysis
o. Jarjayeset al. / JournalofMagnetismandMagneticMaterials137(1994)205-218
216
of the magnetization of composite samples has shown that the distribution of the grain sizes obeys a log-normal law with as most probable diameter, DmpPPY(mag) - 52/~. All the grains of the aqueous ferrofluid solution are equally incorporated into the polymer matrix whatever their sizes. The inclusion of the mesoscopic T-Fe203 particles leads to a fair dispersion in the polymer, with no magnetic modification of the particles and no aggregation. Moreover, no major interaction between the particles has been detected. As far as the polymer is concerned, its electrical conductivity as a function of temperature remains unchanged in the presence of the magnetic particles. Thus, for this class of composites the conducting properties of the polymer and the magnetic properties of the grains do coexist without any interference. Potential applications in electromagnetic interference shielding can be tested for this material and similar ones.
Acknowledgements Valuable conversations with Drs. P. Molho and P.A. Petit of the CNRS-Grenoble are appreciated. The financial support of the Rdgion Rh6ne-Alpes is gratefully acknowledged. We also thank the company Arjo, Wiggins, and Appleton in Apprieu Is6re for their interest in this work.
Appendix A
Let /3BE-r( 0,4',a,/3) = -- acos0 + bsin20,
(A1)
be the reduced energy with a =/3Bm(D)H, b =/3BKV(D). The calculation of the expression (13) of the sample magnetization MH first requires the computation of the double average
IfcosO
exp [ - ~3nET(O,4',a,/3) ]sinO dOd4'
C(a,b) = ((cos O)~,~) = ~
-sin/3 d flda.
(A2)
f f exp[ - /3.Ev( O,4',a,/3) ]sinO dOd4' Because the total energy Er given by Eq. (10) depends on the angles ce and 4' through their difference 4 ' - a, all the partial three-dimensional integrals over 0, 4', and/3 are independent of a, and Eq. (A2) simplifies to
cosO ff C(a,b) = ~ f . .
exp[ - ~3BEy(0,4',0,/3) ] sin0 d0d4'
.
.
.
.
.
.
sin/3 d/3.
(A3)
f f exp[ - ~ E ~ ( O,4',O,/3)]sinO dOd4' This integral is computed by the Cartesian product [ 31 ] of two Gauss-l_~gendre quadratures ( [ 32], pp. 324-331 ) over the angles 0 and/3 in [0,'rr], times the Gaussian composite trapezoidal rule ([32], pp. 341-343) over the angle 4' in [0, 2~r]. Because of the symmetry relation cos4'= cos(2n-r - 4') the quadrature over 4' can be reduced to the interval [ 0,'rr ]. Another simplification occurs because the contributions to the integral (A3) associated to/3 and "rr-/3 are equal. The Gaussian quadratures are very accurate integration methods, even for small numbers no, n+, and nt3 of nodes and weights. Although the exponential exp( -/3BET) has a very steep angular variation for large a and b values, the function C(a,b) is stabilized to about 0.5% for a,b< 100 if no, n,b, no=20. From a careful analysis of the ratio of the double integrals of Eq. (A3) it can be shown that
O. Jarjayes et al. /Journal o fMagnetism and Magnetic Materials 137 (1994) 205-218
217
[0 .8
0 .8 6 .~
O"
Fo <¢
'~,
A o
b.C.x~~-~
Fig. 6. The average C(a,b) of the projections along the field of the unit vectors parallel to tile magnetic moments of the grains having the same diameter value D. The variables a and b are the reduced Zeeman and anisotropy energies, respectively. ~/2
C(a,b) ---- / tanh(a cosOa)cosOasinfl d/3 if 1 << b,
(A4)
0
where 0t3 is the solution in [0, fl] of the equation asin0+ b s i n 2 ( 0 - / 3 ) = 0, and gives the maximum value of - flaET. Furthermore, when a << b, 08---/3 and Eq. (A4) simplifies to "~/2
C(a,b) --- / tanh(a cos/3)cos/3 sin/3 d/3 if 1 << b and a << b,
(AS)
0
which is independent of b. For small values of a, Eq. (A5) becomes C(a,b) =-a/3 =L(a). In other words, the magnetization of a sample of high anisotropy energy is the same as that of a sample without anisotropy energy at low magnetic fields! The cosine average C(a,b) is plotted as a function of a and b in Fig. 6. For each fixed b value C(a,b) has a shape which is similar to that of the Langevin function L(a), with smaller values as a increases. For instance, C(20,20) ---0.75 is 20% smaller than L(20) ---0.95, and according to Eq. (A5), when 1 << a << b, C(a,b) ~ 0.5 as a--* o%while L(a) ~ 1. This discussion clearly shows that an anisotropy energy of the form (6) will notably modify the magnetization of a solid sample, only if there is a large proportion of particles with diameters D such as 1 << /3Bm(D)H << /3BKV(D). Finally, the integrals over the diameter D which occur in the expressions (13) and (16) of the magnetization Mn, are computed by taking the new variable u such as D = Doexp (o-u), and by integrating in the interval - 4 < u < 4.
References [1] R.F. Ziolo, E.P. Giannelis, B.A. Weinstein, M.P. O'Horo, B.N. Ganguly, V. Mehrotra, M.W. Russel and D.R. Huffman, Science 257 (1992) 219. [2] R.H. Marchessault, S. Ricard and P. Rioux, Carbohydr. Res. 224 (1992) 133. [3] C. Forder, S.P. Armes, A.W. Simpson, C. Maggiore and M. Hawley, J. Mater. Chem. 3 (1993) 563. [4] G. Bidan, O. Jarjayes, J.M. Fruchart and E. Hannecart, Adv. Mater. 6 (1994) 152. [5] F. Beck, M. Dahlhaus and N. Zahedi, Electrochim. Acta 37 (1992) 1265. [6] H. Yoneyama, Y. Shoji and K. Kawai, Chem. Lett. (1989) 1067. [7] C.P. Bean and J.D. Livingston, J. Appl. Phys. 30 (1959) 120S. [8] R.W. Chantrell, J. Popplewell and S.W. Charles, IEEE Trans. on Magn. 14 (1978) 975.
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O. Jarjayes et al. / Journal of Magnetism and Magnetic Materials 137 (1994) 205-218
[9] S.M. Aharoni and M.H. Litt, J. Appl. Phys. 42 (1971) 352. [ 10] J.C. Bacri, R. Perzynski, D. Salin, V. Cabuil and R. Massart, J. Magn. Magn. Mater. 62 (1986) 36. [ 11] J.C. Bacri, R. Perzynski and D. Salin, J. Magn. Magn. Mater. 71 (1988) 246. [ 12] H.D. Williams, K. O'Grady, M. El Hilo and R.W. Chantrell, J. Magn. Magn. 122 (1993) 129. [ 13 ] R. Massart, IEEE Trans on Magn. 17 (1981) 1247; R. Massart et al., French Patent 9006484 (1990). [ 14] J.C. Bacri, R. Perzynski, D. Salin, V. Cabuil and R. Massart, J. Magn. Magn. Mater. 85 (1990) 27. [ 15] G. Bate, in : Magnetic Oxide, Vol. 2, ed. D.J. Craik (Wiley, London, 1975) p. 703. [ 16] C. Kittel, Introduction h la Physique de l'Etat Solide (Dunod, Paris, 1972). [17] A. Herpin, Th~orie du Magnrtisme (PUF, Paris, 1968). [18] L. Nrel, C. R. Acad. Sci. 228 (1949) 664; Ann. Grophys. 5 (1949) 99. [ 19] D.P.E. Dickson, N.M.K. Reid, C. Hunt, H.D. Williams, M. E1-Hilo, and K. O'Grady, J. Magn. Magn. Mater. 125 ( 1993 ) 345. [20] A. Abragam, Les Principes du Magnrtisme Nuclraire (PUF, Paris, 1961) pp. 29%301. [21 ] K. O'Grady and A. Bradbury, J. Magn. Magn. Mater. 39 (1983) 91. [22] T. Sato, T. Iijima, M. Seki and N. Inagaki, J. Magn. Magn. Mater. 65 (1987) 252. [23] A.C. Nunes, C.F. Majkrzak and A.E. Berkowitz, J. Magn. Magn. Mater. 39 (1983) 59. [24] J. Popplewell, B. Abu Aisheh and N.Y. Ayoub, J. Appl. Phys. 64 (1988) 5852 and references therein. [25] S. Koutani, G. Gavoille and R. Gerardin, J. Magn. Magn. Mater. 123 (1993) 175. [26] S.H. Liou and C.L. Chien, J. Appl. Phys. 63 (1988) 4240. [27] R.S. Berry, S.A. Rice and J. Ross, Physical Chemistry (Wiley, New York, 1980) pp. 997-1006. [28] F. Beck and M. Dahlhaus, J. Appl. Electrochem. 23 (1993) 781. [29] E. Tronc and D. Bonnin, J. Physique Lett. France 46 (1985) 437. [30] O. Jarjayes and P. Auric (private communication). [31] A.H. Stroud, in: Mathematical Methods for Digital Computers, eds. A. Ralston and H.S. Will (Wiley, London, 1967), Vol. II, pp. 145155. [32] L.W. Johnson and R.D. Riess, Numerical Analysis (Addison-Wesley, London, 1982).