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14410

J. Phys. Chem. 1996, 100, 14410-14416

Control of the Size of Cobalt Ferrite Magnetic Fluids: Mo1 ssbauer Spectroscopy N. Moumen,† P. Bonville,*,‡ and M. P. Pileni*,†,§ Laboratoire SRSI, URA CNRS 1662, UniVersite Pierre et Marie Curie, (Paris VI), BP 52, 4 Place Jussieu, 75231 Paris cedex 05, France, C.E.A.-C.E. Saclay, DRECAM-S.P.E.C, 91191 Gif sur YVette, Cedex, France, and C.E.A.-C.E. Saclay, DRECAM-S.C.M, 91191 Gif sur YVette, Cedex, France ReceiVed: NoVember 10, 1995; In Final Form: February 12, 1996X

57Fe

Mo¨ssbauer absorption spectroscopy studies have been performed in zero magnetic field in the temperature range 4.2-200 K, in cobalt ferrite (CoFe2O4) magnetic fluids that differ in the mean size of their particles. For particles with an average diameter of 2 and 3 nm, a quadrupolar doublet progressively replaces the sixline hyperfine field pattern as the temperature increases. This is attributed to a superparamagnetic regime with respect to the time scale τM ) 10-8 s of 57Fe Mo¨ssbauer spectroscopy. For particles with an average diameter of 5 nm, no such superparamagnetic behavior is observed; from the thermal variation of the mean hyperfine field, an axial anisotropy constant KA ) (8 ( 2) × 106 erg cm-3 is deduced. The whole set of data, together with magnetic measurements, strongly suggests that magnetic anisotropy in these particles has an axial rather than cubic character and that the anisotropy constant varies as the inverse of the square of the particle diameter.

I. Introduction Increasing interest in recent years1,2 has been focused on the size dependent evolution of solid state properties in systems where chemical binding and electron delocalization are fairly strong. The physical properties of very small clusters generally differ from those of the free atoms or molecules themselves as well as from the properties of the bulk solids. Techniques3-9 have recently been developed which enable the synthesis of novel metal and semiconductor clusters and their physical characterization with an unprecedented precision. A challenge now exists to explore a wide array of new cluster-based materials. It is anticipated that their properties will be different from, and often superior to, those of conventional materials that have phase or grain structures on a coarser size scale. Small magnetic particles have surface properties that may differ from those at the interior. The elucidation of these differences, however, is particularly difficult because the surface morphology, which depends on the preparation technique, may vary. A magnetic fluid consists of kinetically stabilized ferro- or ferrimagnetic particles coated with a monomolecular layer of surfactant and colloidally dispersed in a nonmagnetic liquid matrix. In zero magnetic field each particle may be considered independent, and its magnetization direction is randomly oriented.10 The particles are sufficiently small so that thermal energy maintains them in a stable dispersion. Many phenomena in magnetic fluids have been interpreted under the assumption that the colloidal particles are well dispersed, i.e., that aggregates are absent. Many investigations have been performed on magnetic particles having a size in the range of 10-30 nm.11,12 The transition metal oxides having a spinel structure attract a great interest because of their remarkable electrical and magnetic properties. These properties depend upon the type of magnetic cations in the tetrahedral A site and octahedral B site of the spinel lattice and the relative strength of intra- and * Author to whom correspondence should be addressed. † Universite Pierre et Marie Curie. ‡ DRECAM-S.P.E.C. § DRECAM-S.C.M. X Abstract published in AdVance ACS Abstracts, July 15, 1996.

S0022-3654(95)03324-7 CCC: $12.00

inter-sublattice interactions. The cation distribution is affected by many factors such as temperature, pressure, composition, and method of preparation. In addition, the physical properties of these materials exhibit substantial changes from their “bulk” values when the crystallite size is sufficiently reduced.13,14 Cobalt ferrite, CoFe2O4, is usually assumed to have a collinear ferrimagnetic spin structure along the [100] direction of the cubic cell. It is an inverse spinel. The ratio Fe(A)/Fe(B) depends on the preparation process and is found to be between 0.6 and 0.9.15,16 In the range of 10-100 nm, the decrease in the saturation magnetization with decreasing size has been explained in terms of noncollinear spin arrangement at or near the surface of the particles. Such a noncollinear structure, mostly attributed to a surface effect, is most pronounced for the smallest particles.17 Relatively little work has been carried out on magnetic materials having sizes smaller than 10 nm. Nanocrystallites having an average size of 5-10 nm have been synthesized.18 Recently, Davies et al.19 produced cobalt ferrite particles characterized by an average size equal to 3 nm. In all these cases, syntheses have been performed by using very high concentrations of metallic salts and bases. The solutions were heated at relatively high temperature (close to 100 °C). The control of the size was achieved by changing the experimental conditions such as pH, ionic strength, complexation, etc. To our knowledge, very little work has been done on magnetic particles made in the presence of surfactant and at low reactant concentrations. Recently, reverse micelles have been used to synthesize metallic or boride cobalt nanoparticles having an average size equal to 3 and 4 nm.20-22 In previous papers,23,24 we have demonstrated that by using oil in water micelles it has been possible to control the size of the particles from 2 to 5 nm continuously. The size distribution is close to 30%. In these syntheses the concentration of the reactants are lower by 2 or 3 orders of magnitude compared to other preparations of similar materials. Furthermore, the reaction takes place at room temperature. These particles, when dispersed in a fluid, are characterized by a superparamagnetic behavior with respect to magnetic measurements. The saturation magnetization decreases with the decrease in the particle size. The anisotropy constant strongly increases with the decrease in the particle size. © 1996 American Chemical Society

Size of CoFe2O4 Magnetic Fluids

J. Phys. Chem., Vol. 100, No. 34, 1996 14411

By contrast, for dry powder made of nanosized particles,25 the magnetic behavior strongly differs; in a first approximation, it can be assumed to be a superparamagnetic behavior only for the smaller particles (2 and 3 nm diameter). The reduced remanence depends on the particle size and not on the preparation mode, whereas coercivity depends on the preparation mode. In the present work, we report on 57Fe Mo¨ssbauer absorption spectroscopy experiments, in zero magnetic field in the temperature range 4.2-200 K, for three ensembles of nanosized CoFe2O4 particles with average diameters 2, 3, and 5 nm. II. Experimental Section II.1. Materials. Sodium dodecyl sulfate (SDS) was bought from BDH, and iron chloride, FeCl2, and cobalt acetate, Co(CH3CO2)2, were procured from Fluka. Methylamine, CH3NH3OH, was purchased from Merck. All the compounds were sold as 99.99% purity Cobalt(II) and iron(II) dodecyl sulfate, Co(DS)2 and Fe(DS)2, were made as described elsewhere.24 II.2. Experiment. The morphology and particle size were investigated using a Phillips transmission electron microscope (model CM 20, 200 kV). The 57Fe Mo¨ssbauer absorption spectra were recorded using 57 a Co*:Rh γ-ray source mounted on a triangular velocity electromagnetic drive. To minimize interaction effects between particles, they were diluted in gelatin; the volume of the sample was 1.57 cm3, yielding an average distance of 10, 16, and 30 nm, respectively, for the 2, 3, and 5 nm particles. Above 250 K, the Mo¨ssbauer absorption vanishes, probably due to the fact that the solidification point of gelatin is around 250 K and that the very small particle size (the largest particle contains about 104 atoms) yields an important recoil energy. Indeed, for a 5 nm particle, the ratio of the recoil energy to the natural width of the Mo¨ssbauer line is ER/Γ ≈ 10, which prevents recoil-less absorption from being observed for the “free” particle. II.3. Histogram Treatment. Histograms are obtained by measuring the diameter Di of all the particles from different parts of the grid (×100000) for an average number of particles close to 500. The standard deviation, σ, is calculated from the following equation

σ ) {∑[ni(Di - D)2]/[N - 1]}1/2 where D and N are the average diameter and the number of particles, respectively. III. Treatment of Magnetic and Mo1 ssbauer Data. Small ferromagnetic and ferrimagnetic particles are characterized by superparamagnetism.26 The relaxation time of the magnetization strongly decreases with decreasing size of the particles. The magnetization follows the direction of the external field when present, and the coercive force appears to be negligible in contrast to that of permanent magnets. The first theory published by Ne´el27 considers the relaxation as a random hopping of the magnetization vector between the two easy directions in a particle. This process implies the crossing of an anistropy energy barrier, and thus the thermal variation of the relaxation time τ has an activation-like dependence. Following Ne´el, most authors have assumed a simple exponential temperature dependence for τ. (i) For an axial magnetocrystalline anisotropy27

( )

KAV τ ) τ0 exp kT

(1)

(ii) For a cubic magnetocrystalline anisotropy26 assuming a [100] easy magnetization axis and rotation restricted to (100) planes

( )

τ ) τ0 exp

KAV 4kT

(2)

where KA, V, k, and T are the anisotropy constant, the volume of the particle, Boltzmann’s constant, and the temperature, respectively, and τ0 ≈ 10-10 s is a microscopic relaxation time.28 The blocking temperature, TB, is defined (ideally for particles with a unique volume) as the temperature for which the relaxation time, τ, is equal to the characteristic time of the technique. For magnetic and 57Fe Mo¨ssbauer measurement, τχ and τM are 100 and 10-8 s, respectively. So we have for the case of axial anisotropy

() ()

ln

ln

KAV τχ ≈ 27 ) τ0 kT χ

(3)

KAV τM ≈5) τ0 kT M

(4)

B

B

This yields the theoretical ratio

TBM TBχ

≈5

(5)

The measure of the blocking temperature allows the determination of the anisotropy constant, KA. In the presence of a distribution of particle volumes, the volume V has to be replaced by the average volume 〈V〉 in eqs 3 and 4 in order to estimate KA . The blocking temperature TBχ is often identified with the temperature Tg for which the zfc curve shows a maximum.26 For very small particles, the width of the volume distribution has to be taken into account in the determination of the blocking temperature, and the ratio Tg/TBχ can significantly depart from unity. Simulations assuming a log-normal volume distribution29 have shown that Tg/TBχ is roughly equal to 3, 2.5, and 1 for the present particles with average diameter of 2, 3, and 5 nm, respectively. The Mo¨ssbauer blocking temperature TBM can be estimated when the average energy barrier KA〈V〉 of the particles is not much larger than the experimentally accessible temperature range. At a temperature T ) TBM defined in eq 4, the particles can be separated into two roughly equal populations: (i) those whose magnetization fluctuates “slowly” (τ > τM), which yield a six-line hyperfine field pattern; (ii) those fluctuating “rapidly” (τ < τM), showing a singleline or a two-line quadrupolar pattern for Fe ions in a noncubic site. From many studies of small particles, it has been observed that the hyperfine field of the slowly fluctuating particles decreases with increasing temperature more rapidly than in the bulk material.30 This has been explained by oscillations of the magnetization around an energy minimum.31,32 If θ is the angle between the magnetization vector and the easy direction, then the hyperfine field for a particle of volume V at a temperature T is given by

Hhf(V,T) ) Hhf(bulk,T)〈cos θ〉T

(6)

where Hhf(V,T) and Hhf(bulk, T) are the hyperfine fields of the particles and of the bulk phase, respectively, and 〈cos θ〉T is a Boltzmann average.

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Moumen et al.

For particles with uniaxial symmetry, the anisotropy energy per unit volume is26

E(θ) ) KA sin2 θ

(7)

where KA > 0. Then31

〈cos θ〉T )

xx e1/x - 1 2 1xxjeV2 dy ∫0

(8)

where x is defined as x ) kT/KAV. In the limit x , 1, one obtains

〈cos θ〉Tax ) 1 -

kT 2KAV

(9)

For particles with cubic symmetry, the anistropy energy per unit volume is26

E(θ) )

KA 2 [sin 2θ + (sin4 θ sin2 2φ)] 4

(10)

If KA > 0, the easy axis is a [100] axis, and assuming rotation of the magnetization between these [100] axes (φ ) 0 or π/2), one gets

〈cos θ〉Tcub )

1 1/2 (y2-y)/x dy ∫ e 2 0

∫1/1x2e(y -y )/x dy 4

2

Figure 1. Histograms of magnetic fluid made at various surfactant concentrations with [Co(DS)2]/[Fe(DS)2] ) 0.325, [Co(DS)2]/[NH2CH3] ) 1.3 × 10-2, [Fe(DS)2] ) 6.5 × 10-3 M (A), [Fe(DS)2] ) 1.3 × 10-2 M (B), and [Fe(DS)2] ) 2.6 × 10-2 M (C).

(11)

In the limit x , 1, we have shown that

〈cos θ〉Tcub ) 1 -

kT 1.56KAV

(12)

IV. Synthesis of CoFe2O4 Nanosized Particles. Recently, syntheses of cobalt ferrite particles differing by their average size from 2 to 5 nm have been described.24 Methylamine, CH3NH3OH, is added to a mixed micellar solution formed by Co(DS)2 and Fe(DS)2 surfactants. The solution is stirred for 2 h, and a magnetic precipitate appears. The supernatant is removed and replaced by pure bulk solvent. A brown magnetic solution called magnetic fluid is obtained. When the Fe(DS)2 concentration is increased from 6.5 × 10-3 to 2.6 × 10-2 M, with constant ratios of [Co(DS)2]/[Fe(DS)2] ) 0.325 and [Fe(DS)2]/[CH3NH3OH] ) 1.3 × 10-2, the particle size increases from 2 to 5 nm. This can be observed from the TEM pattern, and the size distribution is deduced from the histograms (Figure 1). From electron and X-ray diffraction spectra, it is shown that the particles keep the invert spinel crystalline structure and their stoichiometry as in the bulk phase. V. Results V.1. Particles with Average Diameter of 5 nm. The Mo¨ssbauer spectra of particles having 5 nm as a mean diameter recorded at various temperatures are characterized by a welldefined sextuplet (Figure 2). At 4.2 K, the spectrum is similar to that observed for larger particles (30-350 nm) by Haneda and Morrish17 and shows a sixth B-site line apparently less intense than the sixth A-site line as observed for bulk cobalt ferrite. A good fit of the two overlapping six-line hyperfine patterns corresponding to the 57Fe in B and A sites can be obtained with a ratio of A to B site patterns equal to 0.67. The hyperfine fields of A and B sites

Figure 2. 57Fe Mo¨ssbauer absorption spectra of the 5 nm diameter CoFe2O4 particles at various temperatures in zero magnetic field. Solid lines are fits as explained in the text.

are equal to 510 and 533 kOe, respectively. The line width (hwhm) of the A site pattern in 0.22 mm/s, whereas the B site spectrum has to be fitted with different widths for the three pairs of lines, i.e., 0.23, 0.27, and 0.40 mm/s with increasing splitting. This suggests that for small particles, as found by Haneda and Morrish17 the A to B site ratio is close to that of the “quenched” bulk sample (0.615,16). When the temperature is increased, no drastic changes in the Mo¨ssbauer spectra are observed. No doublet due to a superparamagnetic behavior is obtained. An increase in the line broadening with an increase in the temperature is observed. Above 77 K, the shoulder at the sixth line disappears due to a decrease in hyperfine field at the B site more rapidly than at the A site with increasing temperature. From the data fitted with one six-line pattern (parameters are summarized in Table 1), a decrease in the average hyperfine field, Hhf(T), with increasing temperature is observed (Table

Size of CoFe2O4 Magnetic Fluids

J. Phys. Chem., Vol. 100, No. 34, 1996 14413

TABLE 1: Parameters Deduced from Mo1 ssbauer Spectra of 5 nM Diameter Particles at Various Temperatures Hhf (kOe)

G (mm/s)

T(K)

A

B

A

B

4.2 77 120 150 200

510 516 516 503 491

533 516 509 503 491

0.22 0.26, 0.36, 0.43 0.26, 0.34, 0.40 0.33, 0.40, 0.48 0.26, 0.38, 0.48

0.23, 0.36, 0.43 0.26, 0.36, 0.43 0.26, 0.34, 0.40 0.33, 0.40, 0.48 0.26, 0.38, 0.48

TABLE 2: Thermal Variation of Ratio T/TN, rB, rA, r(T), and 〈cos θ〉T T(K) T

77

120

150

200

T/TN rB rA r(T) 〈cos θ〉T

0.1 0.989 1 0.993 0.992

0.15 0.984 1 0.990 0.981

0.1875 0.973 1 0.984 0.976

0.25 0.950 0.986 0.964 0.972

Figure 3. Thermal variation of 〈cos θ〉T obtained in the 5 nm diameter CoFe2O4 particles derived from the hyperfine field thermal variation. The dashed line corresponds to a linear decrease according to eq 9, with KA ) 7.5 × 106 erg cm-3.

1). Similar behavior has been observed previously for larger particles.31,32 The decrease in the hyperfine field with increasing temperature can be attributed to a thermal effect (TC for CoFe2O4 is close to 800 K15) and to oscillations of the magnetization around an energy minimum31,32 as described by eq 6. Then

Hhf(T) ) Hhf(bulk,T) 〈cos θ〉 ) r(T) Hhf(bulk,T)0)〈cos θ〉 (13) where r(T) is the bulk thermal reduction factor. This thermal factor was obtained using the hyperfine field curves for sites A and B given in refs 15 and 16 and by averaging over sites: r(T) ) 0.4rA(T) + 0.6rB(T). Table 2 gives the thermal variation of rA, rB, and r. The saturated mean value Hhf(bulk,T)0) estimated from refs 15 and 16 is Hhf(bulk,T)0) ) 525 kOe; the experimental value in the 5 nm particles is 524 kOe, showing that there is no reduction with respect to the bulk. From the value of the hyperfine fields given in Table 1, 〈cosθ〉T can be obtained using eq 13

(14)

Figure 4. 57Fe Mo¨ssbauer absorption spectra of the 3 nm diameter CoFe2O4 particles at various temperatures in zero magnetic field. Solid lines are fits as explained in the text.

Figure 3 shows the decrease of 〈cosθ〉T with temperature. The slope is found to equal 1.4 × 10-4 K-1. Assuming axial and cubic anisotropy, the anisotropy constants, KA, are deduced from eqs 9 and 12, respectively. In these equations the volume V is replaced by the average volume, 〈V〉. For an axial and a cubic anisotropy, respectively, the KA values are close and found to be KA ) (8 ( 2) × 106 and (10 ( 2) × 106 erg cm-3. Fits of the spectra at all temperatures were also performed with two six-line patterns, keeping the ratio of A to B site equal to 0.67 and introducing a distribution of hyperfine fields due to the volume distribution according to eqs 9 and 13 (solid lines in Figure 2). The line broadenings are very well reproduced with the value of the anisotropy constant determined above. V.2. Particles with Average Diameters of 2 and 3 nm. At 4.2 K, the Mo¨ssbauer spectra obtained for 2 and 3 nm particles (Figures 4 and 5) show a six-line hyperfine pattern as expected. However, the sixth line is not resolved as observed for 5 nm particles (Figure 2) and in the bulk phase.17 The different line widths show the presence of a distribution of hyperfine fields. Assuming a Gaussian distribution, the average hyperfine field for particles with an average diameter of 2 and

3 nm is 501 and 498 kOe, respectively (with a root-mean-square deviation ∆Hhf ) 26 kOe). These values are 5% smaller than those determined for 5 nm particles (525 kOe). This can be attributed to the increase in the total surface with the decrease in the particle size. This tends to frustrate the exchange interactions among cations. Figures 4 and 5 show the Mo¨ssbauer spectra recorded at various temperatures. In both cases, when temperature increases, a strong asymmetrical broadening of the lines and the appearance of a doublet whose intensity grows at the expense of the sextuplet can be observed. The maximum hyperfine field slightly decreases as temperature increases. This thermal behavior is characteristic of superparamagnetism and can be interpreted by assuming two populations: the particles which fluctuate slowly (τ > τM) give rise to the hyperfine sextuplet, whereas those having a rapid relaxation yield a quadrupolar doublet. A first approach to fit these spectra consists of a superposition of a hyperfine field distribution (taken here as a histogram with free weights) and a quadrupolar doublet. Such fits were performed for the 3 nm particles spectra and yield the thermal

〈cos θ〉T )

Hhf(T) r(T) Hhf(bulk,T)0)

14414 J. Phys. Chem., Vol. 100, No. 34, 1996

Moumen et al.

Figure 6. Thermal variation of the superparamagnetic fraction in the 3 nm diameter CoFe2O4 particles derived from the Mo¨ssbauer spectra.

Figure 5. 57Fe Mo¨ssbauer absorption spectra of the 2 nm diameter CoFe2O4 particles at various temperatures in zero magnetic field. Solid lines are fits as explained in the text.

variation of the superparamagnetic fraction (the doublet) shown in Figure 6. The blocking temperature TBM is found to equal 160 K. For the 2 nm particles, the rather large statistical dispersion precludes the use of such a fit, especially at intermediate temperatures. However, the blocking temperature can be estimated by inspection: TBM ) 100 K. We then proceeded to a more fundamental type of fit. Starting from the size distribution determined by TEM (Figure 1), the theoretical spectra are calculated by introducing a cutoff ratio (xc < 1) of the variable x ) kT/KAV. The cutoff value xc is defined as

τM ) τ0 exp(1/xc)

(15)

To fit the experimental data, the following are assumed: (i) For x < xc, the spectrum is a sextuplet with a hyperfine field given by eq 6, and for x > xc the spectrum is a quadrupolar doublet. This procedure is justified since the relaxation time τ, within the Ne´el approach, depends only on x. (ii) Between 4.2 and 180 K, the anisotropy constant, KA, is assumed to be temperature independent (see section VI). The axial anisotropy formula given in eq 8 is used for 〈cos θ〉T. (iii) The value Hhf(T)0) was kept at the value of 500 kOe, and the individual line widths are taken equal to the values found by fitting the 4.2 K spectrum with three pairs of Lorentzian lines. The quadruplet splitting is kept at the value ∆EQ ) 0.67 mm/s fitted on the highest temperature spectrum. The finding of a nonzero quadrupolar splitting indicates that the Fe A and B sites are distorted from cubic symmetry. By contrast, the quadrupolar splitting for the sextuplets is found to be very small. This is often encountered in small particle Mo¨ssbauer studies and probably arises from the fact that the distortion axes are randomly oriented with respect to the direction of the hyperfine field, i.e. the direction of the Fe magnetization.

Assuming the anisotropy constant KA is independent of the particle volume (i.e. particle size), reasonable fits are obtained. However, the KA values deduced from these fits depend on the size of the particles. They are found equal to 107 and 4 × 107 erg cm-3 for particles having an average diameter equal to 3 and 2 nm, respectively. Moreover, the cutoff value xc decreases as temperature increases, whereas coherence of the model implies a temperature independent xc value. Increasing the energy barriers for the smallest particles should restore a temperature independent xc, which can be done by assuming a law of variation of KA with diameter, d, in 1/d or 1/d2. The fits show that this assumption is correct. However, for the 3 nm particles, it turns out that it is not possible to distinguish between these variations. For the 2 nm particles, the smallest sizes are quite sensitive to the difference between a 1/d law and a 1/d2 law, and we found that the latter yields more coherent results. Finally, the following law yields reasonably good fits for both the 2 and 3 nm particles at all temperatures (solid lines in Figures 4 and 5)

KA ) (A + (B/d2)) × 106 erg cm-3

(16)

where A ) 4.5 ( 5 and B ) 55 ( 5. For the 2 and 3 nm particles, the xc value is found equal to 0.25 and 0.15, respectively. In the frame of the axial anisotropy model, these values are quite coherent. As matter of fact, by taking τ0 ) 10-10 s and τM ) 10-8 s, eq 15 yields xc ) 0.22. The above-derived law yields a mean value 〈KA〉 ≈ 8 × 106 erg cm-3 for the diameter range corresponding to the 5 nm particles. This is in good agreement with the value previously obtained from the thermal decrease of 〈cos θ〉T. V.3. Comparison of Anisotropy Constants Determined from Mo1 ssbauer and Magnetic Measurements. The magnetization curves24 have been obtained at 10 K for particles having an average size equal to 2, 3, and 5 nm, respectively. The presence of hysteresis with an increase in the coercivity with the particle size is observed. The magnetic field needed to reach the saturation magnetization depends on the size of the particles. For particles having an average size equal to 2 and 3 nm, the saturation is not reached even for a magnetic field equal to 40 kOe, whereas it is observed with 5 nm particles. The saturation magnetization, deduced from zero extrapolation of M vs 1/H, decreases with a decrease in the particle size. The ratio of the remanence to saturation magnetizations, Mr/Ms, deduced from the magnetization curve decreases with the decrease in particle size (Table 3). The samples are cooled in zero field to 10 K, and the magnetization is measured as a function of temperature in a 100 Oe field. The temperature Tg, corresponding to the maximum of the zfc curve, is given Table 3 for the various particles. Such a Tg temperature is usually assumed to be the blocking temperature. According to calcula-

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J. Phys. Chem., Vol. 100, No. 34, 1996 14415

TABLE 3: Variation with Particle Size of the Polydispersity, σ, reducut remanance, Mr/Ms, glass temperature, Tg, blocking temperature determined from magnetic measurement, TBχ, and from 57Fe Mo1 ssbauer spectroscopy, TBM, anisotropy constant determined from magnetic measurements, KAχ d (nm)

σ (%)

2 3 5

37 36 23

a

Mr/Ms

Tg (K)

TBχ (K)

TBM (K)

0.31 0.43 0.74

70 120 180

23 48 180

100 160 a

TBM/TBχ 4.3 3.3

(10-6

KAχ erg cm-3) 22.7 14 11.3

Not observable.

Figure 7. Variation of anisotropy constant KA with the inverse square diameter. The squares are the mean values obtained from magnetic measurements; the hashed area corresponds to law given by eq 16 obtained from the Mo¨ssbauer data.

tions made with small particles,29 the volume distribution of the particles has to be taken into account. From section III, the blocking temperature, TBχ is similar to Tg for 5 nm particles and strongly changed for the smaller ones. Table 3 gives the blocking temperature deduced from the maximum of the zfc curves taking into account the distribution of the volume of the particles. According to eq 3, the anisotropy constant, KAχ, determined from magnetic measurements is deduced (Table 3). A relatively good agreement of the anisotropy constants determined from Mo¨ssbauer and magnetic measurements is obtained. Figure 7 shows a similar dependence on 1/d2 of the anisotropy constant determined from both techniques. The comparison is, however, somewhat difficult since the KA value determined from the zfc curve is a mean value, implying an average of both KA and V in eq 3. VI. Discussion Bulk, CoFe2O4 is characterized by a strong cubic anisotropy of magnetocrystalline origin,33 mainly due to the Co2+ ions (3d7) which have a nonzero orbital momentum, while the Fe3+ ions have L ) 0. The anisotropy energy is given by eq 10 with KA > 0, which means that [100] is the easy magnetization axis. In the model described above, it is assumed that the anisotropy constant does not vary with temperature. By magnetic torque measurements in single crystals of a nonstoichiometric crystal (CoFe2O3.62), apparently having a Curie temperature equal to 400 K, Shenker34 measured a decrease by a factor of 3 of the anisotropy constant with the increase in temperature from 4 to 200 K. It is found equal to 17.5 106 and 3.5 × 106 erg cm-3 at 77 and 300 K, respectively. In the case of an stochiometric crystal (CoFe2O4), having a Curie temperature equal to 800 K, as in the bulk phase, Iizuki and Iida35 measured smaller values than those given by Shenker et al.34 They found a KA value equal to 10 × 106 and 2 × 106 ergs cm-3 at 77 and 300 K, respectively. At a given temperature, the KA values determined by these two groups strongly differ. From the data presented above, it has been demonstrated that the particles are characterized by a stoichiometric composition. Because of that we have

chosen to consider the data presented by Iizuki and Iida.35 The anisotropy constants determined in our experiments for the smallest particles are higher than those obtained by Iizuki and Iida35 for the bulk phase. Hence, even if the anisotropy determined in the present work would change with temperature, this would not explain the high value of the anisotropy constant and its dependence on 1/d2. For cubic anisotropy, the energy barrier to be crossed by the magnetization for KA > 0 is Eb ) KAV/4. This yields a cutoff value xc equal to 0.05. Assuming the anisotropy has a cubic character and the data above described are due to thermal variation of KA, the Mo¨ssbauer blocking temperatures are expected to be much lower than those experimentally observed. For the 5 nm particles with a cutoff value xc ) 0.05, the blocking temperature is reached at T ) 210 and 150 K from data in refs 34 and 32, respectively. This means a sizeable fraction of the 5 nm particles should undergo fast magnetization fluctuations and therefore show a Mo¨ssbauer quadrupolar doublet, which is not observed experimentally. This can be due to (i) A higher value of the anisotropy constant KA, that is to say, a value much higher than 107 erg cm-3; this value compared to the bulk phase is very large and such hypothesis seems rather unlikely. (ii) Interaction effects between particles; this leads to a higher blocking temperature than expected for independent particles. This objection is not easy to dismiss and would require a more thorough study of interaction effects as a function of particle dilution in gelatin. However, for the present dilution levels, the dipolar interactions between the 2 nm particles is of the order of 0.2 K, and it is about 10 times larger for the 5 nm particles, i.e. 2 K. These are at least 2 orders of magnitude smaller than the typical blocking temperatures. The measurements have been performed with 1% volume fraction of particles dispersed in gelatin with such a small value that the interaction between particles can be assumed to be rather small. Table 3 shows that the Mr/Ms ratio is 0.74 (see Table 3), close to the bulk cubic value. However, from the measured thermal decrease of 〈cosθ〉T, the anisotropy constants determined by assuming either an axial or cubic anisotropy are found to be KA ) (8 ( 2) × 106 and (10 ( 2) × 106 erg cm-3, respectively. These values, obtained in the temperature range 77-200 K, are close to that determined at low temperature for the bulk crystal.35 However, in such a temperature range, the thermal variation of KA cannot be taken to equal that of the bulk. Alternatively, there could be an extra (axial) contribution that would enhance the KA values so that they are compatible with our data. We have checked that by adding a constant contribution of 3 × 106 erg cm-3 to the KA(T) data of ref 35 we obtain a good agreement with our 〈cos θ〉T thermal variation within the error bars. So, it is possible that in the 5 nm particles the cubic anisotropy dominates and that a smaller axial contribution is present. For 2 and 3 nm particles, the dominant anisotropy could be not of magnetocrystalline origin. It can arise from surface effects which are expected to be important for such small particles. Diameter of 2 and 3 nm correspond to 3 and 4 unit cells, respectively, and more than 50% of the atoms at the surface. Lattice distortions could also play a role, as suggested by the observation of a sizeable electric field gradient on the 57Fe nucleus in the “superparamagnetic phase”. Then the K A value can differ, and its thermal variation can be less pronounced than for the bulk material if it is not directly linked with the magnetocrystalline energy. Whatever the origin of the magnetic anisotropy, the whole set of our Mo¨ssbauer data can be coherently interpreted in terms of an axial-like anisotropy, with

14416 J. Phys. Chem., Vol. 100, No. 34, 1996 energy barriers Eb ) KAV and with a temperature independent but size dependent anisotropy constant. Introducing a size dependent contribution to KA varying as 1/d2 yields a temperature independent cutoff value xc ≈ 0.2, which is characteristic for axial anisotropy. The value of xc is somewhat sensitive to the τ0 value (eq 15): assuming τ0 ) 10-12 s (instead of 10-10 s), one gets for the axial case xc ) 0.11 and for the cubic case xc ) 0.027. The experimental results are closer to the axial values. The presence of a dominant axial contribution to the anisotropy in the 2 and 3 nm particles is also strongly suggested by the value of the remanence to saturation magnetization ratio Mr/Ms (see Table 3 and ref 24), which is greatly reduced with respect to the bulk cubic value 0.83.36 Recently, a magnetization study of 3.5 nm mean diameter CoFe2O4 particles has been reported.18b From the analysis of the remanence magnetization curve, the authors conclude that the anisotropy in their particles is of cubic multiaxial type. This conclusion is at odds with the interpretation presented here for a similar system, prepared, however, by a somewhat different method (see Introduction). The problem consists in estimating whether there exists a “spin-pinning” shell at the surface of the particles and, if any, in measuring its size. A direct proof of the existence of such a shell can be provided by Mo¨ssbauer experiments with an applied magnetic field and also by very high-field magnetization measurements. Such experiments are planned for our particle systems, together with efforts to reduce the polydispersity of the size distribution. VII. Conclusion In this paper 57Fe Mo¨ssbauer spectroscopy has been performed in the temperature range 4.2-200 K in CoFe2O4 magnetic fluids differing by their sizes, prepared in micellar media. For 2 and 3 nm diameter particles, we observe a superparamagnetic behavior with respect to the 57Fe Mo¨ssbauer time scale, with blocking temperatures equal to 100 and 160 K, respectively. The fits of the spectra with a “two-population model” strongly suggest that the magnetic anisotropy is of axial rather than cubic type and that the anisotropy constant varies as 1/d2, where d is the particle diameter. For the 5 nm diameter particles, we observe no superparamagnetic behavior. From the thermal decrease of the hyperfine field, we derive an anisotropy constant KA ) (8 ( 2) × 106 erg cm-3, close to the cubic bulk low-temperature value. These results show that, for the present small particles, decreasing the particle size leads to a crossover from cubic to axial anisotropy. This behavior is confirmed by magnetic measurements. References and Notes (1) Ozin, G. A. AdV. Mater. 1992, 4, 612 and references therein.

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