On the models for interparticle interactions in nanoparticle assemblies: comparison with experimental results

Transcription

1 Journal of Magnetism and Magnetic Materials 202 (1999) 251}267 On the models for interparticle interactions in nanoparticle assemblies: comparison with experimental results J.L. Dormann, D. Fiorani, E. Tronc* LMOV, UMR 8634 CNRS-Universite& de Versailles-Saint Quentin, Versailles Cedex, France IC-MAT, CNR, CP 10, Monterotondo Stazione, Italy Laboratoire de Chimie de la Matie% re Condense& e UMR 7574, T54-E5, Universite& Pierre et Marie Curie, 4 Place Jussieu, Paris Cedex 05, France Received 8 June 1998; received in revised form 9 November 1998 Abstract The models proposed to account for the variation of the blocking temperature of a magnetic particle in an assembly of interacting particles are re-discussed. Experimental results on the thermal variation of the relaxation time are reported in support of the discussion. Except few cases related to a collective state, all data are well explained by the Dormann}Bessais}Fiorani model accounting for the interparticle interactions, in the framework of the superparamagnetic NeH el}brown model. The energy barrier unambiguously increases with increasing interactions, even when they are very weak, in disagreement with the M+rup}Tronc model Published by Elsevier Science B.V. All rights reserved. PACS: G; J; G Keywords: Fine particles; Superparamagnetism; Dipolar interactions; Magnetic ordering 1. Introduction In a recent report [1], Hansen and M+rup (H&M) analyse two con#icting models [2,3] for superparamagnetic relaxation [4], which aim at determining the average relaxation time of the magnetic moment of a particle in a particle assembly and, in particular, the change of the energy barrier due to magnetic interparticle interactions. These models predict opposite e!ects. In the M+rup}Tronc (M+rup) model [3], valid for weak * Corresponding author. Tel.: # ; fax: # address: elt@ccr.jussieu.fr (E. Tronc) Deceased 6 June interaction strength, the interactions lead to a decrease of the energy barrier, whereas in the Dormann}Bessais}Fiorani (DBF) model [2,4], valid for weak and medium interaction strengths, they lead to an increase. Note that the Shtrikman}Wohlfarth model [5], "rst published on the subject, and its re"nement [6] lead to the same conclusions as the DBF model. H&M severely criticise the DBF model, basing on arguments that we "nd mostly invalid and partially wrong. This will be discussed in Section 2 after recalling the main points of these two models. Our objections [4] to the M+rup model, ignored in Ref. [1], will also be developed. According to Hansen and M+rup [1], when the interactions are not weak enough to satisfy the M+rup model, the dynamics is governed by the /99/$ - see front matter 1999 Published by Elsevier Science B.V. All rights reserved. PII: S ( 9 8 )

2 252 J.L. Dormann et al. / Journal of Magnetism and Magnetic Materials 202 (1999) 251}267 approach to a transition to an ordered state. H&M discuss the existence of such a state, propose an estimate of the transition temperature and compare their evaluation to some data reported in the literature. We do agree [4] that for su$ciently strong interactions, the properties are no longer relevant to superparamagnetism and that a change of magnetic behaviour occurs. However, the properties of the corresponding magnetic state are not yet established. Therefore, in our opinion, the prediction of an ordering temperature is problematic. This point will be discussed in Section 3. No model can cover the extreme complexity of real nanoparticle assemblies. Therefore, any model includes more or less drastic simpli"cations which can always be discussed a priori, and some of them contested. Therefore, the comparison between the predictions and experimental data is the only objective criterion for validating, or invalidating, a model. Accordingly, in Section 4, we shall review all the results [2,7}26] that we know concerning the dynamics in particle assemblies and providing reliable information on the temperature variation of the relaxation time. It will be shown that most of the data can be well explained using the DBF model, quantitatively when the sample characteristics are well known. Few data, relative to strongly interacting particles, suggest the approach to a collective magnetic state. It is unambiguous that no result shows a decrease of the energy barrier with increasing interparticle interactions. 2. On the models 2.1. The DBF model First of all, let us recall the Shtrikman}Wohlfarth model [5]. The energy barrier E of a particle with volume < and non-relaxing (intrinsic) magnetisation M is estimated to E "E #H M <th[h M </(k¹)], (1) where E is the energy barrier for the noninteracting particle, H a phenomenological inter- action "eld, ¹ the temperature and k the Boltzmann constant. Using the DBF model [2,4], if we limit the interactions to nearest neighbours, which represents ca. 90% of the total e!ect, we obtain E "E #n a M <C[a M </(k¹)], (2) where n is the average number of nearest neighbours, a +C /2 where C is the volume concentration of the particles in the sample, and C[ ) ] is the Langevin function. Eqs. (1) and (2) are similar. In fact, the DBF model follows the approach of Shtrikman and Wohlfarth and quanti"es the e!ect of the interactions. If the DBF model has no physical meaning, the Shtrikman}Wohlfarth model [5] and its re"nement [6] should be meaningless, too. Hansen and M+rup [1] addressed themselves to analyse in detail each assumption and approximation in the DBF model [2,4], which led them to establish an impressive list of &unrealistic' or &incorrect' items and to conclude to &unreliable' predictions. The raised criticisms can mainly be classi"ed by types of argument into three groups: (i) A series is relevant to the relaxation process. H&M put foward a &#ip'-process model, in which they expect the magnetic moment, m, of the particle to spend most of the time with its direction close to an easy direction. This is equivalent to the two-level model. This model is reasonable at low temperature, in the blocked state, because the Boltzmann statistics shows that the probability of presence of m is negligible outside the directions close to the energy minima. It does not hold at high temperature, in the unblocked state, because the probability of presence outside the minima becomes appreciable [4]. We concern ourselves with the estimation of the energy barrier, for evaluating the blocking temperature, ¹, which, in a simple view, separates the blocked state from the unblocked state in the experimental time window. Therefore, it is not certain at all that the &#ip'-process model is applicable. Accordingly, it can hardly be used as a solid basis for arguing against another approach. (ii) Another group of criticisms is based on microscopic calculations applying to particular particle arrangements, suggesting some misunderstanding of the aim of the DBF model. Its goal is to give an estimation, on the average, of the modi"cation of the energy barrier due to interactions in the case of a disordered assembly of particles with easy axes distributed at random. This estimation applies

3 J.L. Dormann et al. / Journal of Magnetism and Magnetic Materials 202 (1999) 251} in the temperature region of the average ¹ and at higher temperature but a priori not at low temperature, in the blocked state. Because it is impossible to establish an exact analytical expression (to be averaged) due to the complexity of the problem, we have proposed a kind of mean-"eld approach. The energy of particle i due to interaction with particle j is estimated considering all possible relative directions of m and m, and results from an average over all pairs of orientation. This is valid for any pair of particles in the sample and accounts for the #uctuations of the relative orientation of the two moments due to randomly orientated easy axes, di!erent relaxation rates and all interaction e!ects in the sample. Next, the e!ect is summed over j, which is correct because the fact that two neighbours interact does not modify the calculation assumptions. Obviously, our estimation does not apply to particular particle arrangements. The relevant criticisms and the two-particle example are beside the point. (iii) Other criticisms are relevant to a certain approximation in the calculation. They denote some misinterpretation of the determination of the relaxation time. Using the same notations as in Ref. [4] we write the interaction energy for particle i as E "!M < cos θ M a C[M M < a /(k¹)]cos θ (3) approximated to E "!E cos θ (4) with E "M < M a C[M M < a /(k¹)], (5) where M is the non-relaxing magnetisation of particle i or j, < is the volume of particle i, a is a parameter depending on the volume and the relative location of particle j, and θ is the angle between the magnetic moment and the easy direction of particle i. H&M note that this approximation is valid only for weak interactions, as a result of the low-argument approximation of the Langevin function. We agree with that, but the Langevin function approximation is not necessary. The used expression of the relaxation time, τ, equal to the inverse transition probability per unit time, is deduced from the NeH el}brown model [4,27}30] through the Kramers escape rate theory [31], which shows that E "(;!; ), (6) where ; and ; are the maximum and minimum values of the potential (in uniaxial symmetry), respectively. Considering Eq. (3), it is easy to see that ; and ; correspond to cos θ "0 and 1, respectively, so the di!erence is exactly E as given by Eq. (5). The DBF model does not yield the average energy barrier, as is claimed by H&M. It estimates the energy (potential) by averaging over all possible particle arrangements. The result (Eq. (3)) satis"es the uniaxial symmetry. Then Eq. (6) is applied for calculating τ, in agreement with the NeH el}brown model. The used τ expression, valid for uniaxial symmetry and in the absence of an applied "eld, and satisfactory for E /(k¹)*2.5 [30], is given by τ"τ exp[e /(k¹)] (7) with τ " π 4 m(0) E γ 1 #η η M (¹) M (0) k¹ E 1#k¹ E, (8) where m(0)"m(0)"m (0)<, γ is the electron gyromagnetic ratio and η is a dimensionless constant such that η "ηγ M (0), η being the damping constant. Concerning the τ expression, we remark that using oversimpli"ed expressions in quantitative "ts or modellings is not justi"ed: (i) It is incorrect to take τ constant, "xed at an arbitrary value. Unlike a pure Arrhenius law (τ "constant), Eq. (8) leads to a small but appreciable curvature in the log τ versus1/¹ plot, which cannot be neglected. The τ expression (Eq. (8)) is now well established, so there is no reason for not using it. It indeed contains an unknown parameter, namely η. Some recent data [7}10,32] give an approach to its order of magnitude, with "tted

4 254 J.L. Dormann et al. / Journal of Magnetism and Magnetic Materials 202 (1999) 251}267 values ranging between and 1. This is contested by H&M. (ii) Under a weak applied "eld, the potential of a uniaxial particle exhibits two unequivalent minima [4]. The two relaxation times, τ and τ, related to the jump from the lower minimum to the upper minimum and vice versa, respectively, must be taken into account. Good asymptotic expressions, including the τ formulation, are available for the "eld parallel to the easy axis [33,34] and for an oblique "eld with large and small values of η [35}38]. The only case unresolved exactly so far is that of a very weak oblique "eld but, in a "rst approximation, we can use Eqs. (7) and (8) with the correct E values. Note that the Kramers escape rate theory is still valid, so E is still given by Eq. (6), where ; now corresponds to the saddle point and ; to the appropriate minimum The M+rup model Discussing the M+rup model [3], "rst of all, we point out the name we use, &M+rup' instead of &M+rup}Tronc', since the coauthor (E.T.) has clearly withdrawn [4,7}10,39}41] since Second, we remark that our objections [4] have not been taken in consideration in Ref. [1]. They concern two basic aspects of the model: (i) The use of a unidirectional "eld to account for the interactions does not satisfy the symmetry of dipolar interactions, which are unchanged by inversion through a centre of symmetry. (ii) The treatment of the relaxation under "eld is not correct. Moreover, the oversimpli"ed expression of the relaxation time, based on a single relaxation time and an arbitrary constant τ value leads to signi"cant imprecisions. Let us give some explanations. As mentioned above, in the case of a very weak oblique "eld, in a "rst approximation we can use Eqs. (7) and (8) with the correct E values, deduced from Eq. (6) and given by E "K<[1#2h($cos υ!sin υ)#h], (9) where K is the anisotropy energy constant, υ the angle between the "eld B and the easy direction, h"bm /(2K);1, and $corresponds to the two energy barriers leading to the two relaxation times τ. In contrast to Eq. (9), the formula (Eq. (17) in Ref. [1]) giving E in the M+rup model contains the angle φ (the azimuthal angle of the magnetic moment). This is in contradiction with Eq. (6) and, therefore, incorrect. The transition probability per unit time is given by τ"τ #τ with (10) τ "2τ exp[e /(k¹)]. (11) Applying Eqs. (8) and (9), we obtain τ" τ π/4 α 1#β sin υ#β 2!5#3α 1#α (h sin υ#βh) exp[!α(1#h)] (12) with τ " <M γ 1 η #η M (¹) M (0) 1 k¹ (13) and α"k</(k¹)<1 and β"m <B/(k¹);1. This is substantially di!erent from the corresponding expression (Eq. (26) in Ref. [1]) in the M+rup model. Due to the lack of precise theoretical formulation for very weak h values, we cannot ensure the exactness of our calculation but it is likely to be much more realistic [42]. In Eq. (12), the terms 5#3α (h sin υ#βh) (14) 1#α come from the expression of τ. The pre-exponential factor up to "rst order in B depends on β"2hα compared to 3h. At the average blocking temperature, α is typically in the range of 14}20 for AC susceptibility (χ ) experiments and 2}7 for MoK s- sbauer spectroscopy. Hence, the 3h term is negligible in the former case but not in the latter. Thus, it is reasonable to neglect the h dependence of τ for χ data, but not for MoK ssbauer spectroscopy ones, the only type of data to which the M+rup model was applied [3]. It is clear that more rigorous calculations following the same approach as the M+rup model cannot

5 J.L. Dormann et al. / Journal of Magnetism and Magnetic Materials 202 (1999) 251} change the general trend that is, a "eld always leads to a decrease of τ, but seriously change the subsequent estimations. 3. Magnetic state of an assembly of particle magnetic moments According to Hansen and M+rup [1], if τ increases because of interactions the dynamics is no longer relevant to superparamagnetism but governed by the approach to a transition to an ordered state resembling the spin-glass (SG) phase. H&M discuss the existence of such an ordering, report an expression [43] for the ordering temperature and compare the estimations to available data taken as ordering temperatures. No ordered state has yet been proved to exist in disordered nanoparticle assemblies, although there are some experimental clues in favour of its possible occurrence for strongly interacting particles. Some observed properties present analogies with those of spin glasses and it is reasonable to conjecture a low-temperature state similar to the SG phase, since the SG ingredients, i.e. disorder and frustration, are also present due to the disordered arrangement of particles, random distribution of easy axes and volume distribution, and the intrinsic con#icting character of dipole}dipole interactions, which can be both ferromagnetic and antiferromagnetic. However no evidence of a &spin-glass transition' has been reported and further experiments are needed. Thus, in our opinion it is highly problematic to predict [43] an ordering temperature for a magnetic state which is not yet known, and rather illogical to base one's critique on mere speculations. Let us summarise the known and expected features relevant to the magnetic state of an assembly of particle moments Known and expected features If the interparticle interactions are negligible every particle has its own E. If they are appreciable they give rise to metastable states with additional energy minima and then modify the energy barriers, which become interdependent. The extra minima are all very shallow at high temperature due to the e!ect of thermal disorder and become gradually deeper as the temperature is lowered. Some of them can become very deep, depending on the topology and the relative magnitudes of the individual anisotropy energy and the interaction energy. If the interactions are negligible the magnetic state is superparamagnetic. The m #uctuation modes are localised to individual particles. As the temperature is lowered, a collection of independent blocking processes occurs governed by the individual E 's and described by the NeH el}brown model [4]. In the presence of interactions, two situations can be distinguished depending on whether the #uctuation modes are quasi-localised or whether they are extended. In the "rst case, the relaxation in each particle remains governed by its own E, although modi"ed by interactions, and the magnetic state is still superparamagnetic. In the second case, it is not possible to identify the individual E 's, even in average, only the energy of the particle assembly is signi"cative and a so-called collective state is present. This leads to two scenarios upon cooling: (i) a progressive inhomogeneous blocking, which could also lead, depending on the interaction strength, to a collective state at lower temperature; and (ii) a homogeneous freezing, as in spin glasses (SG) showing a thermodynamic phase transition. When the m #uctuations are very slow or not present, if the interactions are negligible the magnetic state is speromagnetic. If they are appreciable, no long-range order should exist without applied "eld. Numerical simulations [44] of hysteresis loops of "ne particles with random anisotropy directions and dipolar interactions were made at zero temperature. For zero anisotropy, the results show the existence of coercivity due to interactions. For high particle concentrations, a short-range local order is revealed. Upon increasing the anisotropy, a crossover between two regimes is observed. For low anisotropy, the static properties are roughly the same as for zero anisotropy. For high anisotropy, a single-particle-like behaviour in agreement with the Stoner}Wohlfarth model [45] is observed. No precise information concerning the magnetic state was given, which could in principle be SGlike.

6 256 J.L. Dormann et al. / Journal of Magnetism and Magnetic Materials 202 (1999) 251}267 Many disordered materials show properties resembling those of true SGs. The distinctive properties [46}48] of the latter are related to the existence of an order parameter, in agreement with the general theory of phase transitions [49], implying the critical power-law divergence of the non-linear susceptibility, χ, on approaching the phase transition temperature, ¹. It also results in a critical dynamical behaviour, i.e. a slowing down of the relaxation time of the spins on approaching ¹ from above, according to a power law. The frozen state is nonergodic with a hierarchical and ultrametric structure of the energy valleys in the phase space, as revealed by the time variation of the magnetisation, non-logarithmic and depending on the ageing time. However, the ageing phenomenon also occurs in bulk disordered magnetic materials that do not show a phase transition, due to relaxation between metastable states. Any disordered system of interacting particles exhibits numerous metastable con"gurations with a spectrum of energy di!erences, in which the system may become trapped upon cooling. Its energy always has a multi-valley structure in the phase space. This will always give rise to ageing e!ects on the magnetisation relaxation, observable if the relaxation times associated with secondary minima are inside the time window of the experiment. Therefore, for the same reasons as in bulk materials, ageing e!ects cannot, in any way, be a proof of an ordered state. We can only say that their absence is a likely proof of the absence of such a state. Any unfrozen system of interacting particles exhibits correlations in the dynamics of the moments. These correlations will always give rise to nonlinear e!ects on the DC susceptibility. Therefore, non-vanishing χ cannot by itself be a proof of a collective state and, a fortiori, of the approach to a phase transition. Non-vanishing χ s have been reported [10,20,21] but, as far as we know, not yet any critical behaviour, even in systems which show a critical slowing down of the relaxation time (see Section 4). Indeed, di!erences in critical behaviours should exist between particle magnetic moments and spins, for many reasons, but these di!erences are still to be established. Accordingly, at present the existence of a phase transition may still be questioned. H&M take τ versus ¹ variations that can be modelled by the Vogel}Fulcher (VF) law τ"τ exp[e /k(¹!¹ )] (15) as an evidence of the existence of a phase transition. Actually, this law is senseless near ¹ and so unable to prove a singularity. The VF law was introduced [50,51] in the "eld of spin glasses and disordered magnetism in order to account, within the superparamagnetic-particle approach, for interaction e!ects on the frequency dependence of the freezing temperature measured by AC susceptibility. It has been stated clearly that this law is phenomenological and meaningless in the vicinity of ¹, and that ¹ cannot be considered a transition temperature but simply an estimate of the interaction strength, only useful for a comparison between di!erent samples. For "ne-particle systems, the VF law can match practically any experimental τ versus ¹ variation. For non-interacting particles, it yields a better "t (with a small ¹ value) than an Arrhenius law, because of the variation of τ with ¹ (Eq. (8)). Obviously, it is not allowed to take ¹ for a transition temperature. For weak interactions, the energy barrier in the Shtrikman}Wohlfarth model (Eq. (1)) is given by E "E #(H M <)/(3k¹) (16) leading to E /(k¹)"e /[k(¹!¹ )] (17) with ¹ "(H M <)/(3kE ). (18) Eq. (17) is valid only if ¹ ;¹. Then ¹ cannot correspond to a temperature where τ diverges. It is a phenomenological temperature, which cannot, in any case, be assimilated or related to a transition temperature. It is clear that by simply checking a VF law, one cannot prove the existence of a phase transition. Therefore, we believe that the use of this law is non-informative. This will be illustrated in Section 4.

7 J.L. Dormann et al. / Journal of Magnetism and Magnetic Materials 202 (1999) 251} Dynamical regimes Homogeneous freezing In SG, the power-law divergence of the critical relaxation time is usually expressed [46}48] as τ"τ [¹ /(¹!¹ )], (19) where ¹ is the freezing temperature, frequency dependent, and ¹ the glass-transition temperature. The exponent depends on the law governing the phase transition [49], τ being related to the correlation length ξ as τ&ξ, and ξ diverging with temperature as ξ&[¹ /(¹!¹ )]. The exponent zν is in the range of 7}8. For the collective state of particle moments, we can expect a similar power law, as already observed in some samples (see Section 4). It is meaningful only in the vicinity of the critical temperature. At present it is di$cult to predict the value of the exponent, because the kind of phase corresponding to the collective state is not really known Inhomogeneous blocking In the superparamagnetic regime, where the individual behaviour dominates, the DBF model is applicable in the framework of the NeH el}brown model, with the relaxation-time expression being derived through the Kramers escape theory (Eq. (6)). The DBF model is a priori not valid in the blocked state and obviously not when a collective regime sets in. Let us recall its predictions using the "rst-order approximation. We label the interactions as weak or medium according to the purpose, by contrast with the so-called strong interactions leading to the collective state. The e!ective energy barrier (Eq. (2)) can be approximated in two di!erent ways according to the value of a M </(k¹), where ¹ is the blocking temperature related to the volume <. For a M </(k¹))1, which corresponds to weak interactions or high ¹ values E +E #n (a M <)/(3k¹). (20) For a M </(k¹)*2, which corresponds to medium interactions or low ¹ values E +E!n k¹#n a M <. (21) In this case, the relaxation time (Eq. (7)) becomes τ"τ exp(!n ) exp[(e #n a M <)/(k¹)]. (22) Therefore, in the temperature region where the model and the medium-interaction approximation (Eq. (21)) are valid, neglecting the temperature variation of E in τ (Eq. (8)) we obtain the same τ expression as for non-interacting particles, with an increased E value, and a τ value decreased by a factor of exp(!n ). Hence, in the log τ versus 1/¹ plot we obtain a quasi-straight line, with increased slope and intercept of the log τ axis shifted by!n /2.30 (neglecting E variation) with respect to the case without interactions. These predictions are in excellent agreement with experimental results (Section 4). 4. Comparison with experimental data All available results [2,7}26], so far to our knowledge, on interacting-particle systems with clear indications on the dynamical behaviour, are reviewed below. Other recent works [52}60] on the subject will not be discussed, because they are less related to the present questions, namely the magnetic state of the particle moments and the validity of the models proposed to account for the e!ect of interparticle interactions on the energy barrier and the average blocking temperature ¹. The data of interest deal mainly with the τ versus ¹ variation deduced from χ measurements. The temperature, ¹, at the maximum of the real part, χ, of the susceptibility can be a freezing temperature, ¹,or a blocking temperature, ¹, relative to the measuring time τ "1/ν, where ν is the frequency. In the superparamagnetic model [4,61], ¹ is related to the volume < "R</<, where the R factor is mostly determined by the width of the < distribution and usually of the order of 1}2. So the values of ¹ as a function of ν are directly comparable for a given sample, and between samples with the same distribution of particle size and shape. It is worth emphasizing that the results below, still partial, concern a variety of samples which are often not comparable directly, even if the chemical compound is the same. The shape of the particles and that of the sample, the presence of aggregates, the particle surface properties depend strongly on the preparation method. All these factors in#uence the properties and lead to more or less weak e!ects

8 258 J.L. Dormann et al. / Journal of Magnetism and Magnetic Materials 202 (1999) 251}267 superimposed on the main ones due to superparamagnetic (or collective) properties. In the absence of perfect samples, since the main e!ects are not de"nitively stated, it is di$cult to ascertain whether weak deviations from the properties expected from the superparamagnetic (or collectivestate) model are due to the model itself or whether they are due to extra phenomena determined by the above-cited factors Fe particles embedded in alumina [2,4] The "rst detailed investigation [2] of the e!ect of interparticle interactions on the superparamagnetic relaxation was performed by us on Fe particles embedded in alumina. Three samples with di!erent mean particle volumes (+40, 60 and 200 nm) and similar volume concentration (C +0.22) were studied by numerous techniques. The τ versus ¹ variation was determined by χ measurements (210}10 Hz) and MoK ssbauer spectroscopy experiments. The results are fairly well described using the DBF model. In the log τ versus 1/¹ plot, the χ data yield quasi-straight lines with an attempt time (+10 s) much lower than τ,asis predicted by the medium-interaction approximation (Eq. (21)). The MoK ssbauer data deviate slightly from the straight lines. This is due to the higher temperatures, as a result of the shorter measuring time than for χ measurements, which make the condition a M </(k¹)*2 and Eq. (21) no longer valid. Fits were performed using Brown's τ expression [27] and Eq. (5) for the interaction e!ect assuming average close packing (n "12). < and a M being "xed as were deduced experimentally, the E value was determined yielding the individual anisotropy energy constant. A very good scaling of the three sets of log τ versus 1/¹ data was obtained, fully consistent with scaling in volume. Neutron inelastic scattering measurements [62] con"rmed the high-temperature E value as deduced from Eq. (20) γ-fe 2 O 3 particles [7}10] Our second study, still running, deals with γ- Fe O particles prepared by a chemical route. Many samples were prepared with di!erent average particle volumes and di!erent topologies: quasiisolated particles dispersed in a polymer with average centre-to-centre distance between neighbouring particles varying between ca. 5D (C +0.6%) and 1.55D (C +20%), where D"(6</π), chain-like clusters of varying length, large agglomerates made up of entangled chains, also dispersed in the polymer, and dried powders. The degree of aggregation (average number of particles per cluster and average number of nearest neighbours within clusters) was determined by electron microscopy. Four series of investigation by χ and MoK ssbauer spectroscopy measurements have been reported [7}10]. The τ versus ¹ variation is in accordance with the NeH el}brown model for all samples with very weak interactions, and consistent with the DBF model for all the other samples but one. In the "rst case, least-squares "ts using Eqs. (7) and (8) with <"< and M being "xed at experimental values, yielded the values of E /k and η.in the second case, to determine the interaction e!ect we expressed the e!ective energy barrier as E "E # n E C[E /(k¹)] (23) where n and E can be expressed as a function of n and E, respectively, assuming a regular average arrangement of particles. With < and E /k being "xed as were found for the corresponding sample with negligible (very weak) interactions, n "xed according to the particle arrangement, M kept at experimental values and the thermal variation of E taken the same as that of M, least-squares "ts using Eqs. (7), (8) and (23) yielded the values of E (0)/k and η. The data of interest are listed in Tables 1 and Dispersions } concentration and agglomeration ewects [7] Figs. 1 and 2 show the log τ versus 1/¹ variation obtained for dispersions of quasi-isolated particles at low (sample IF) and high concentration (sample IN), and large agglomerates (sample Floc) with the same < distribution (series 33A, Table 1). Fig. 1 is relative to the diluted sample. The data are in accordance with the NeH el}brown model. Fig. 2 shows the data for the two samples of interacting

9 J.L. Dormann et al. / Journal of Magnetism and Magnetic Materials 202 (1999) 251} Table 1 Characteristic data for quasi-isolated γ-fe O particles at varying concentration in a polymer, and large agglomerates (Floc) (see text) Sample D (nm) C (%) E /k (K) η E (0)/k (K) Ref. 33A IF [7] IN } Floc } D C/ [8,9] C/25 } C/5 } C/1 } A IF [10] IN } Floc } Table 2 Characteristic data for chain-like clusters of γ-fe O particles (see text). D stands for the average particle diameter, and n for the average number of nearest-neighbouring particles inside a cluster, as deduced by electron microscopy Sample D (nm) n Attempt time (s) 18A A CH APV APV } Fig. 2. Thermal variation of the relaxation time for interacting γ-fe O particles in a polymer, quasi-isolated at high concentration (sample IN), strongly agglomerated (sample Floc) (from Ref. [7]). particles. The χ data yield quasi-straight lines with very low attempt time, about 10 s, and the MoK ssbauer data deviate slightly, as it was found for the Fe/Al O particles (Section 4.1). The interaction e!ect is larger than expected for dipole}dipole interactions. This was attributed to some implication of the surface magnetic state, as supported by several other features. Fig. 1. Thermal variation of the relaxation time for very weakly interacting γ-fe O particles in a polymer (from Ref. [7]) Dispersions } chaining ewect [7] Figs. 3 and 4 show the log τ versus 1/¹ variations obtained for chain-like clusters of

10 260 J.L. Dormann et al. / Journal of Magnetism and Magnetic Materials 202 (1999) 251}267 Fig. 3. Thermal variation of the relaxation time for small chains of γ-fe O particles in a polymer. Particle size e!ect (from Ref. [7]) Dispersions } concentration ewect [8,9] Fig. 5 shows the data obtained for quasi-isolated particles with the same < distribution (series 4D, Table 1) and di!erent volume fractions. For C "0.75% (sample C/50), the χ data yield a quasi-straight line in accordance with the NeH el}brown model. A weak curvature is observed for C "1.2% (sample C/25) and 4.7% (sample C/5). A quasi-straight line is obtained again for C "19.5% (sample C/1) in agreement with Eq. (21). The weak curvature for the intermediate concentrations results from the fact that E /(k¹)(2, so Eq. (21) does not apply. Three points may be emphasized in the present context: (i) For C "1.2%, the particles interact weakly. A collective state is unconceivable due to the large interparticle distance. ¹ as measured by χ experiments is systematically larger than for C "0.75%, where the interactions are negligible. This is in contradiction with the M+rup model. (ii) The "ts to the χ data give η values increasing with interactions, from 0.06 to 1.0. This leads to curve crossings at some τ value below which Fig. 4. Thermal variation of the relaxation time for chains of ca. 9.5-nm-γ-Fe O particles. Small chains (sample APV6), long chains with branchings (sample APV12) (from Ref. [7]). particles belonging to di!erent series (Table 2). The χ data still yield quasi-straight lines. The slope increases with < and n, the attempt time decreases with increasing n, with the shift between samples following exp(!n ), as is predicted by Eq. (21). Fig. 5. Thermal variation of the relaxation time for γ-fe O particles at increasing concentration in a polymer (from Refs. [8,9]).

11 J.L. Dormann et al. / Journal of Magnetism and Magnetic Materials 202 (1999) 251} ¹ decreases with an increase in interactions, as observed by MoK ssbauer spectroscopy. To attribute the decrease of ¹ to a decrease of E [3] is clearly incorrect. Indeed, the MoK ssbauer data show slight deviations from the "tted curves. This is due [4,7}9,39}41] to the lack of an accurate model describing the line shape as a function of the relaxation time, so only approximate ¹ values can be obtained. Moreover, as focussed on above, several energy barriers probably exist for interacting particles. χ measurements are mainly sensitive to the highest one, whereas MoK ssbauer spectroscopy experiences all of them. Hence, the ¹ values deduced from both techniques may not always be strictly comparable. (iii) The increase in interaction (concentration) within the present series of samples does not induce any drastic change in the τ versus ¹ variation. Only gradual modi"cations are observed, well explained using the DBF model in the framework of superparamagnetic relaxation Dispersions and powder [10] Fig. 6 shows the data obtained for another series of samples (36A, Table 1) with the same < distribution. Three samples consist in dispersions in the polymer: quasi-isolated particles at low (sample IF) and high (sample IN) concentration, and large agglomerates (sample Floc). The fourth sample consists in powdered particles. For sample IF, the χ data give a quasi-straight line in accordance with the NeH el}brown model. For samples IN and Floc, they yield quasi-straight lines with very low attempt time, about 10 s, and the MoK ssbauer data deviate slightly. These features are similar to those observed for the other series (Figs. 1, 2 and 5), so the same analysis was done. At a given concentration, low (C +0.7%) or high (C +20%), the di!erences in the various energy barriers with respect to the other sample series are mainly due to the di!erence in D. For the powder, ¹ is much higher than for the other samples. Its variation with τ is much weaker and cannot be described by the NeH el}brown model including any interaction term. A good "t was obtained with a power law according to Eq. (19) with ¹ "129.2$0.3 K, τ "10 s and (zν)"7.0$0.3. The τ value is practically the same as for samples IN and Floc. The exponent is similar to the SG one [46}48]. This demonstrates that τ presents a critical slowing down. ¹ is no longer a blocking temperature, it should be a freezing temperature. Studies of the relaxation of the zero-"eld-cooled magnetisation (M ) and its ageing-time dependence, and of the non-linear e!ects on the DC susceptibility showed that properties similar to the spin-glass ones were absent in sample IF, very weak in sample IN, and very clear in sample powder, but no critical behaviour near ¹ was observed. Thus, the observed properties indicate the existence of a collective glassy state, but without the characteristics of a &spin-glass transition' Permalloy particles in alumina [11] Fig. 7 shows the log τ versus 1/¹ variation reported for permalloy particles in alumina. A quasi-straight line with very low attempt time, about 10 s, is observed once more. The data were analysed using the DBF model on the basis of dipolar interactions. This yielded an individual anisotropy energy constant in agreement with other evaluations. Studies of M relaxation revealed waiting-time e!ects, and no critical behaviour. Fig. 6. Thermal variation of the relaxation time for γ-fe O particles with interparticle interactions of varying strength. Dispersions in a polymer (samples IF, IN and Floc) and powdered particles (sample Powder) (from Ref. [10]) Magnetite particles in a silicate glass [12] This study deals with 6 nm-particles of Fe O }γ-fe O solid solution, dispersed in a

12 262 J.L. Dormann et al. / Journal of Magnetism and Magnetic Materials 202 (1999) 251}267 Fig. 7. Thermal variation of the relaxation time for permalloy particles in alumina (reproduced from Ref. [11] with permission). Fig. 8. Thermal variation of the relaxation time for magnetite particles at varying volume concentration in a frozen liquid (plotted from data in Ref. [13]). calcium silicate glass. The volume concentration is seemingly of the same order as for our IN samples. The log τ versus 1/¹ variation determined from χ measurements and MoK ssbauer spectroscopy is again a straight line with low attempt time, of 10 s. The data were interpreted by means of the DBF model. Studies of the decay of the thermoremanent magnetisation (TRM) showed a temperature dependence that seemed insensitive to the interparticle interactions, in accordance with our preliminary TRM measurements on γ-fe O particles [39}41]. However, no conclusion can be drawn without an accurate analysis of the whole experimental process, which determines the e!ective "eld seen by the particle Magnetite ferroyuids [13] This study deals with magnetite particles at different concentrations (C "0.12%, 0.6%, 3% and 6%) in a frozen liquid. The properties were studied mainly by χ measurements and discussed using the VF law (Eq. (15)). As shown below, they can be well described by the DBF model. Fig. 8 shows the log τ versus 1/¹ variations plotted from the published data, relative to the temperature at the peak of the imaginary part χ of the susceptibility. For the most diluted sample (C "0.12%), where the interactions are negligible, the variation is fully consistent with the NeH el}brown model. For C "0.6%, where the interactions are very weak, a straight line is still observed, but a slight curvature cannot be excluded. The ¹ variation, the increase of slope and the small decrease of the attempt time with respect to the (0.12%) sample are consistent with the DBF model. For so low a concentration and in the temperature range of the χ maximum, a collective state is physically impossible. The (3%) and (6%) samples yield quasi-straight lines with higher slope and much lower attempt time (+10 s), once more. We can conclude that the results are coherent with the DBF model and give a further clear evidence of an increase of the energy barrier with increasing interactions, including very weak interactions. We note some inconsistency in Ref. [1]. H&M report an estimated ordering temperature, ¹, equal to 12, 50 and 82 K for the (0.6%), (3%) and (6%) samples, respectively. For each sample, for τ "10 s the χ -peak temperature is ca. 28 K above ¹, very far from ¹. Therefore, these peaks cannot correspond to the approach to a transition towards an ordered state. They can only correspond to a blocking process, for which E does increase with interactions Frozen suspensions of γ-fe 2 O 3 particles [14,15] γ-fe O particles dispersed in a solidi"ed hydrocarbon oil have been studied extensively, especially

13 J.L. Dormann et al. / Journal of Magnetism and Magnetic Materials 202 (1999) 251} for C "0.03% and 17%. For the diluted sample, the interactions are negligible and the properties are in agreement with the superparamagnetic model. For the concentrated sample, ageing e!ects on M relaxation, noise and χ experiments showed features similar to those observed in SGtype materials. However, as discussed above, they are not a proof of a collective state with a phase transition. They may be caused by metastability as well. From the published data [15], we estimated the log τ versus 1/¹ variations. Within the accuracy of our estimation, the variation in the (0.03%) sample is in full agreement with the NeH el}brown model, and a quasi-straight line with low attempt time, about 10 s, is obtained, once more, for the (17%) sample. χ data [14] for ν"125 Hz have also been reported for samples with C "0.3% and 3%. The comparison between the four samples clearly shows that the temperature of the χ (125 Hz) maximum increases with C, even for 0.3%, which is too low to give rise to a collective state. This is contradictory again with the M+rup model Fe-C ferroyuids [16] Amorphous Fe}C particles in a frozen liquid, with C "5% and 0.6%, were studied mainly by χ measurements. For the concentrated sample, the properties showed a critical slowing of the dynamics indicating a collective state at "nite temperature. Surprisingly, the strongly diluted sample did not exhibit the superparamagnetic properties expected for non-interacting particles, so the presence of small agglomerates of particles was inferred. Fig. 9 shows the log τ ("1/(2πν)) versus 1/¹ plots, drawn from the reported data. The plot relative to the concentrated sample presents a steep straight line on the low-temperature side, which indicates that the DBF model does not apply. But it applies very well to the diluted sample. Qualitatively, the low-temperature χ data yield a straight line with attempt time about 10 s, which corresponds to interactions coming from 10 to 12 near neighbours, on average. Departure from the straight line is observed as the temperature increases, as expected, since a M </(k¹) becomes Fig. 9. Thermal variation of the relaxation time for Fe}C particles at di!erent volume concentrations in a frozen liquid (plotted from data in Ref. [16]). less than 2. Hence, we can conclude that the sample is likely to be made up of strongly diluted agglomerates of a few tens of particles. It is worth noting that the τ versus ¹ variation was originally modelled by a VF law (Eq. (15)), with ¹ "10 K. H&M did not retain this result as being related to an ordering temperature, contrary to the data for the magnetite ferro#uids [13], although the τ versus ¹ variations look similar. Particle agglomeration in the diluted sample throws some doubt upon the homogeneity of the concentrated sample, thus limiting a possible comparison with other samples showing collective properties. Critical slowing down has also been found in metallic granular systems consisting in nanoparticles, such as Fe Ag granular "lms [17] Iron nitride magnetic yuids [18}21] Studies [18,19] of ε-fe N particles embedded in solidi"ed kerosene at eight di!erent dilutions have been reported. The samples are labelled d1}d8 in order of decreasing particle concentration. The three most diluted samples seem to exhibit superparamagnetic properties of non-interacting particles. The most concentrated sample presents a collective magnetic behaviour [20,21] as shown by the critical slowing down of the relaxation time. Few data [18,19] have been reported for the

14 264 J.L. Dormann et al. / Journal of Magnetism and Magnetic Materials 202 (1999) 251}267 intermediate samples (d5}d2) presented as containing weakly interacting particles. A progressive change in the properties is reported. The ¹ values estimated from M and M measurements are listed in Table 3. An increase of the energy barrier with increasing interactions is observed once more Numerical simulations [22}26] Attempts at simulating the interparticle-interaction e!ects by Monte-Carlo methods have been reported [22}26]. In Ref. [25], the calculations were performed for an assembly of particles with equal size, which reduces the degree of disorder and is not representative, in our opinion, of the full complexity of real particle assemblies. Nevertheless, an increase of the χ -peak temperature, ¹, with concentration (interactions) is observed. The concentrated system seems to show a collective behaviour at low temperature, but it is not indicated whether a critical slowing down occurs at ¹. Furthermore, only very few results are reported for the intermediate concentration. Most recently [26], calculations of the initial susceptibility were performed for a system of Co particles with a lognormal distribution of particle sizes and a packing density varying from 0 to 0.2. The results clearly show an increase of the peak temperature and, hence, of the energy barriers with packing density. This occurs from the lowest densities, which indicates increased E (¹ ) due to weak interaction e!ects, in agreement with the DBF model and in contradiction, once more, with the M+rup one. Table 3 Blocking temperature of iron-nitride particles with di!erent volume fractions in frozen kerosene. Data from Refs. [18,19] Sample C (%) ¹ (K) d }40 d }42 d4 23}44 d }48 d } Comparison with spin glasses Two criteria have often been used [2,4,46}48] to compare the frequency sensitivity of ¹ "¹ in di!erent samples and to distinguish between canonical spin glasses, other disordered magnetic compounds, where the freezing is progressive, and "ne particles. They are given by C " ¹ /(¹ log ν), independent of any model, and C "(¹!¹ )/¹, where ¹ is the characteristic temperature of the VF law (Eq. (15)). The criterion values evaluated near ν"10 Hz for representative sample series are indicated in Table 4. The ranges [2,4,46}48] characteristic of true spin glasses, of bulk disordered materials where the freezing is progressive, and of a noninteracting particle (from theory, C being deduced from the NeH el}brown model and C from a pure Arrhenius law, i.e. ¹ "0) have been included for comparison. In general, the three parameters in the VF law are correlated and there is not a unique good set of values. Therefore, for comparing C values we referred to a suitable common τ value. In each of the three series of iron-oxide particle samples, for the most diluted one the values of C and C are very close to those of the noninteracting particle. With increasing concentration (interactions) both C and C decrease, as expected, and reach, for the most concentrated samples, the typical range of inhomogeneous freezing (blocking) in bulk materials. C and C lie in this range for the Fe/Al O samples as well. For the γ-fe O }36A Powder sample, C and C lie in the characteristic range of homogeneous freezing. The three types of dynamical behaviour which can be distinguished for the above "ne-particle systems (i.e. similar to the non-interacting-particle one, to inhomogeneous or homogeneous freezing in disordered materials) correspond to the distinction we made from our analysis of the τ versus ¹ variation in the same samples, i.e. purely superparamagnetic, superparamagnetic modi"ed by interactions, and collective, respectively. This clearly supports our analysis. Finally, we point out that the DBF model, more or less modi"ed, has been used [63,64] successfully for modelling other "ne-particle properties.

15 J.L. Dormann et al. / Journal of Magnetism and Magnetic Materials 202 (1999) 251} Table 4 Frequency sensitivity of the AC susceptibility-peak temperature, ¹, in various particle samples evaluated near ν"10 Hz, and in bulk spin-glass-type materials (in italics). C stands for the volume concentration of particles in dispersions, ¹ and τ for the parameters of the Vogel}Fulcher law τ"1/ν"τ exp[e /(¹!¹ )] Sample Ref. C ¹ /¹ log ν (¹!¹ )/¹ τ (s) Fe/Al O [4] γ-fe O 4D C/50 [8,9] C/ C/ C/ A IF [10] IN Floc Powder Magnetite ferro#uids [13] Homogeneous freezing [2,4,46}48] 0.005} } Inhomogeneous freezing 0.03} } Non-interacting particle (theory) 0.10} Original data reported for ν+100 Hz. ν+50 Hz. 5. Conclusions The determination of the e!ects of interparticle interactions in an actual particle assembly, usually characterised by three degrees of disorder, i.e. topological disorder, volume distribution and random distribution of easy axes, is an extremely complex task. Therefore, modelling necessarily implies some approximations. We discussed the models proposed to account for the ewect of interparticle interactions on the variation of the average blocking temperature, ¹. Their adequacy to other parameters or to other ranges of temperature remains to be established. We showed that the criticisms raised by Hansen and M+rup concerning the DBF model are mostly irrelevant and partially wrong. Moreover, we pointed out serious objections to the M+rup model. However, we think that such a debate is not really interesting when many experimental data provide a good support to check the predictions, at least qualitatively and sometimes quantitatively. To the best of our knowledge, regarding the ¹ versus τ variation, nearly all the numerous data, obtained mainly from χ experiments, are consistent with the superparamagnetic model, namely the NeH el}brown model, and with the DBF model for the interparticle interaction e!ect. In a few cases where a critical slowing down of the dynamics is e!ective, revealing a transition towards a collective state, the DBF model is evidently not valid, because it assumes that one can de"ne the energy, at least in average, for a given particle, which is impossible in a collective state. In all cases, an increase of the energy barrier with increasing interactions is clearly observed, even if the interactions are intrinsically weak, in complete contradiction with the M+rup model. With growing interactions, progressive changes from the properties of non-interacting particles