17775
2008, 112, 17775–17777 Published on Web 10/25/2008
Dependence of the Properties of Magnetic Nanoparticles on the Interparticle Separation Rakesh Voggu,† N. Kumar,‡ and C. N. R. Rao*,† Chemistry and Physics of Materials Unit and CSIR unit of Excellence in Chemistry, Jawaharlal Nehru Centre for AdVanced Scientific Research, Jakkur P.O., Bangalore -560 064, India, and Raman Research Institute, Bangalore 560 080, India ReceiVed: September 12, 2008; ReVised Manuscript ReceiVed: October 15, 2008
Dependence of the magnetic properties of FePt nanoparticles with an average diameter, D, of 5 nm on the interparticle separation, d, has been investigated by employing different spacer molecules. The observed decrease in the blocking temperature and Curie-Weiss temperature with increase in d as well as the increase in saturation magnetization with increase in d have been explained on the basis of a simple model for the case d , D where the magnetic nanoparticles are treated as finite dipoles coupled through anisotropic multipolar interactions. Properties of magnetic nanoparticles have been investigated by several workers and some of the novel features exhibited by these materials have been known for some time.1 For example, nanoparticles of antiferromagnetic materials are expected to show a small net magnetic moment due to finite size effects.2 Such a behavior has been found in nanoparticles of oxides such as MnO, CoO, and NiO.3,4 A problem of considerable interest related to magnetic nanoparticles is the dependence of their properties on the interparticle separation. It is commonly believed that the variation in properties such as the blocking temperature, TB, with the interparticle separation, d, is governed by dipolar interaction. There have been a few studies on this aspect in the past few years. Vestal et al.5 have varied the interparticle distance in spinel ferrite nanoparticles by varying the concentration in solution, and found that TB increases with the increase in concentration, suggesting thereby that TB increases with a decrease in interparticle separation. A study of the magnetic interaction between iron oxide nanoparticles by Frankamp et al.6 through dendrimer-mediated selfassembly has shown that TB decreases with an increase in interparticle separation. These authors comment on how their results deviate from the dependence predicted on the basis of dipolar interaction between the magnetic nanoparticles. DNAassembled FePt nanoparticles exhibit a decrease in TB compared to the pristine ones.7 An increase in TB with increase in interparticle separation seems to have been observed in chains of 13 nm γ-Fe2O3 nanoparticles prepared by using 11-(10carboxy-decyldisulfanyl)-undecanoic acid.8 Clearly, the nature of the dependence of TB of magnetic nanoparticles on the interparticle separation, d, is not understood. We felt that this may be because one has not employed the correct model to interpret magnetic properties as a function of d. We would expect the interaction would be essentially dipolar if d is large relative to the particle diameter, D. The interaction would, however, be multipolar if d is small compared to D. We have * To whom correspondence should be addressed. E-mail: cnrrao@ jncasr.ac.in. Fax: (+91)80-22082766. † Jawaharlal Nehru Centre for Advanced Scientific Research. ‡ Raman Research Institute.
10.1021/jp808128r CCC: $40.75
carefully investigated the variation of the magnetic properties of FePt nanoparticles (D ) 5 nm) as a function of interparticle separation, by varying d by using alkanedithiols and other capping agents. The 5 nm FePt particles are superparamagnetic at 300 K and show magnetic hysteresis at low temperatures.9,10 Zero-field cooled magnetization data show a maximum in magnetization at the blocking temperature, TB. We have determined the variation of TB, the Curie-Weiss temperature, θ, and the saturation magnetization, M, with the interparticle separation, and discussed the observed trends in the light of a simple model. It may be noted that the variation of magnetization or the Curie-Weiss temperature with d has not been examined in any of the studies reported in the literature. FePt nanoparticles with an average diameter of 5 nm were prepared by the procedure described by Nandwana et al.10 Transmission electron microscopy (TEM) was employed to determine the average size of the particles. When the 5 nm FePt nanoparticles were prepared in the presence of the oleyl amine and oleic acid, the interparticle separation was 2.4 nm. Oleyl amine and oleic acid are readily removed on the addition of alkanedithiols. The dithiol binds two nanoparticles on either side, the interparticle separation being determined by the alkane chain length. To obtain alkanedithiol-linked particles, a known quantity of FePt nanoparticles was taken in a hexane solution of alkanedithiol of known concentration. The solid that precipitated out or obtained after evaporation of the solvent was used to measure the magnetic properties. Magnetic measurements were carried out by employing the vibrating sample magnetometer (VSM) option in the Physical Properties Measuring System (PPMS, Quantum Design, USA). The dependence of magnetic properties on the concentration of the dithiol was first examined. It was found that the value of the magnetization varies only slightly with the alkanedithiol concentration. The blocking temperature, TB, in the zero-field cooled data, and the Curie-Weiss temperature, θ, obtained from high-temperature inverse susceptibility data, were in the range of 87 ( 5 and 95 ( 5 K respectively over dithiol concentrations of 0.25 - 2 M. We have used a constant concentration of 0.5 mM of the alkanedithiols in the present study, since all of the particles would be dithiol-linked at this concentration. 2008 American Chemical Society
17776 J. Phys. Chem. C, Vol. 112, No. 46, 2008
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Figure 3. Variation of saturation magnetization at 5 K with the interparticle separation, d.
Figure 1. (a) Magnetic hysteresis at 5 K exhibited by 5 nm FePt nanoparticles at 5 K with different interparticle separations. (b) Temperature-variation of zero-field cooled magnetization. Figure 4. Plot of normalized magnetostatic energy U(d, D) for broadside on configuration (normalized) with respect to a point dipolar interaction at a distance (D + d) for the same µ value, against d/D.
Figure 2. Variation of TB and θ with the interparticle separation, d. Note that at the largest d, d/D ≈ 0.5.
With the different spacer molecules employed by us, the interparticle separation could be varied between 0.4 and 2.4 nm. In Figure 1, we show the results of magnetic measurements with two spacer molecules with d of 0.4 and 2.4 nm. From Figure 1a, we readily see that the nature of the hysteresis varies with d. Thus, the value of the saturation magnetization, Ms, increases with increase in d. From Figure 1b, we see that TB shifts to lower temperatures with increase in d. Since the average diameter of the FePt nanoparticles, D, is 5 nm, the spacer length, d, is always less than D (d , D). It is impractical to have a situation where d . D with alkanedithiols and such spacer molecules. Furthermore, nanoparticles with very small D do not possess the desirable magnetic properties. In Figure 2, panels a and b, we show the variation of the blocking temperature, TB, and the Curie-Weiss temperature, θ, respectively with the spacer length, d. The θ values were obtained from the hightemperature inverse susceptibility data. We see that TB decreases with the increase in d. The nature of variation of TB with d found here is comparable to that reported in the literature on the basis of concentration-dependence or of dendrimer-mediated
assemblies.5,6 The Curie-Weiss temperature, θ, varies with d in the same manner as TB. This is interesting since θ is a thermodynamic quantity unlike TB. In Figure 3, we show the variation of the saturation magnetization, MS, at 5 K with d, to demonstrate how Ms increases with d. The decrease in TB with increase in d can be understood on the basis of the model described by Dormann et al.11 which predicts a decrease in the energy barrier with increase in d. However, the exact maner in which TB varies with d is another matter. Theoretical calculations predict that TB of magnetic particles interacting through dipolar coupling should follow an inverse cubic dependence on the interparticle separation.12 Our data as well as the literature data6 do not follow the inverse cubic dependence. That is, the slope of the plot of log TB against log d deviates significantly from-3. The possibility of d6 dependence up to a spacing of 1.0 nm or so has been suggested.6 In other words, a modest increase in d has been considered sufficient to result in independent nanoparticles. The real situation, however, appears to be different. In order to understand the nature of variation of TB and θ with d, we have calculated the magnetostatic interaction energy between two identical, spherical nanoparticles as a function of the particle diameter, D, and the interparticle separation d, with D > d. We are dealing here with a multipolar finite magnets,13 with the purely dipolar behavior dominating only in the limit of large separation d . D. In deriving our expression for the normalized magnetostatic interaction energy, u, we take advantage of the fact that a uniformly magnetized spherical body of magnetic moment density µ is equivalent to a surface magnetic charge density σ(θ,φ) ) µ cos θ, where we have taken the direction of magnetization as the polar axis. The total magnetic moment is, given by, M ) µ(4π/3)(D/2)3 which is the magnetic moment of the spherical single-domain nanoparticles. The problem of finding the interaction energy between two uniformly magnetized spheres reduces to that of evaluating the interaction energy of these two surface charge densities. It is essentially
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J. Phys. Chem. C, Vol. 112, No. 46, 2008 17777
the product of these two densities divided by their relative distance apart, integrated over the respective spherical surfaces. For two antiparallel magnetized spheres in the broad-side on configuration, we obtain the following expression for u:
u)
U(d, D, µ) d 3 9 × ) 2 1+ Udipolar(d + D, 0, µ) 4π D
(
∫0π ∫0π ∫02π ∫02π
)
sin θ1 sin θ2 cos θ1 cos θ 2dθ1dθ2dφ1dφ2
√f(1, 2)
where
M(d) ) [2µ sinh(2µH/kT)]/[cosh(2µH/kT) +
2 d 1 1 f(1,2) ) 1 + + sin θ1 sin φ1 - sin θ2 sin φ2 + D 2 2 2 1 1 sin θ1 cos φ1 - sin θ2 cos φ2 + 2 2 2 1 1 cos θ1 - cos θ2 2 2
[(
exp(2U(d,D)µ2)]
)
(
Clearly, the induced magnetization M(d) increases monotonically as the interparticle spacing d increases. Recall that U(d,D) decreases monotonically as d increases (Figure 4). In order to interpret this more explicitly for the saturation magnetization, Ms(d), consider the high-field limit of 2 µH/kT . 1. Here,
)
)]
(
The above relation is exact. The normalization factor is the purely dipolar interaction energy for the two finite dipoles replaced by two point dipoles each of moment M placed at the centers of the respective spheres, i.e., at a distance d + D apart. It may be noted that the above expression for the normalized interaction tends to unity as d/D f ∞ as it should. This is readily verified by expanding 1/[f(1,2)]1/2 in powers of (1 + d/D)-1 and retaining the leading (nonzero) term obtained on performing the integrals of the trigonometric functions involved. The above expression can be simplified to a more tractable, but approximate form for numerical integration. In physical terms, here we essentially collapse the two uniformly magnetized finite-diameter spheres to two extended distributions of point dipoles along the line joining the two spheres in the broad-side on configuration. This gives the normalized interaction energy u as
u≡
(
U(d, D, µ) Udipolar(d + D, 0, µ
=
ferromagnetic system with its isotropic exchange coupling orders to saturation when a relatively small polarizing magnetic field is applied. In the present case, however, the anisotropic multipolar magnetic coupling between the nanoparticles is in conflict with the effect of a polarizing field. This effectively reduces the saturation magnetization. In the simple case of two nanomagnets each of magnetic moment µ, in the broad-side-on configuration with the magnetostatic coupling U(d,D), the magnetic polarization M(d) induced by a polarizing field H is given by
9 d 1+ 16 D
( )(
)
) ∫-1+1 ∫-1+1 dx1 dx2 3
((
(1 - x12)(1 - x22) 3 d 1 1 + - (x1 - x2) D 2
)
)
We see at once that u f 1 as d/D f ∞. That is, at large distances, the finite dipoles act as point dipoles as expected. In Figure 4, we show the variation of normalized u with d/D. For fixed D, as in our experiments, it represents the variation of u with d. The above treatment ignores nonmagnetic contributions to the binding. It is, however, gratifying, given the simplicity of this minimal model, that the plot in Figure 4 contains essential features of the experimental results in Figure 2, panels a and b. Thus, both TB and θ decrease with increase in d just as the magnetostatic interaction energy in Figure 4. The initial fall in both cases is fast and then tends to level off for large d. This is understandable since TB, is due to the energy barrier subtended by anisotropic interactions that render the arrested relaxation ultra slow for T < TB and must scale with u. Unlike the relaxational blocking temperature, the Curie-Weiss temperature is a thermodynamic quantity. It too must track u, this being the only coupling energy scale in the problem responsible for the cooperative phenomenon. The observed decrease in the Curie-Wiess temperature with increasing d is qualitatively similar to what is expected. The variation of the magnetization with d is rather subtle but can be understood as follows. First, note that a conventional
Ms(d) ∼ 2µ tanh(2µH/kT)[1 {exp(2U(d, D)µ2)/cosh(2µH/kT)}] Accordingly, Ms(d) increases with increasing spacing d (i.e., with decreasing U(d,D)). This is in qualitative agreement with the observed variation of saturation magnetization with d in Figure 3. In conclusion, the dependence of the magnetic properties of FePt nanoparticles on the interparticle separation has been studied by using of nonmagnetic spacers of d , D, the diameter of the nanoparticles. The observed variations of the blocking temperature and the Curie-Weiss temperature and also of the saturation magnetization with d, are found to be qualitatively consistent with a simple model of the nanomagnetic particles as finite dipoles coupled through anisotropic multipolar interactions. Basically, the magnetostatic interaction is the only relevant energy scale in the problem. Based on the present study, we are able to explain the results reported by other workers5-8 Almost all the data reported in the literature are for d , D. It is, therefore, understandable that they not only report a decrease in TB with increase in interparticle separation, but also a variation similar to that in Figure 2a. References and Notes (1) Rao, C. N. R.; Kulkarni, G. U.; Thomas P. J. Nanocrystals: Synthesis, Properties and Application; Springer: Berlin, 2007. (2) Neel, L. Low Temp. Phy., DeWitt, C. Eds.; Gordon and Breach: New York, 1962. (3) Ghosh, M.; Biswas, K.; Sundaresan, A.; Rao, C. N. R. J. Mater. Chem. 2006, 16, 106. (4) Ghosh, M.; Sampathkumaran, E. V.; Rao, C. N. R. Chem. Mater. 2005, 17, 2348. (5) Vestal, C. R.; Song, Q.; Zhang, Z. J. J. Phys. Chem. B 2005, 108, 18222. (6) Frankamp, B. L.; Boal, A. K.; Tuominen, M. T.; Rotello, V. M. J. Am. Chem. Soc. 2005, 127, 9731. (7) Srivastava, S.; Samanta, B.; Arumugan, P.; Han, G.; Rotello, V. M. J. Mater. Chem. 2007, 17, 52. (8) Nakata, K.; Hu, Y.; Uzun, O.; Stellacci, F. AdV. Mater. 2008, DOI: 0.1002/adma.200800022. (9) Sun, S. AdV. Mater. 2006, 18, 393. (10) Nandwana, V.; Elkins, K. E.; Poudyal, N.; Chaubey, G. S.; Yano, K.; Liu, J. P. J. Phys. Chem. C 2007, 111, 4185. (11) (a) Dormann, J. L.; Bessais, L.; Fiorani, D. J. Phys. C. Solid State Phys. 1988, 21, 2013. (b) Dormann, J. L.; Fiorani, D.; Tronc, E. AdV. Chem. Phys. 1997, 98, 283. (12) Kechrakos, D.; Trohidou, K. N. App. Phys. Lett. 2002, 81, 4574. (13) Jackson, J. D. Classical Electrodynamics; Wiley Eastern Ltd.: New Delhi, 1978.
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