Crossover From Individual to Collective Magnetism in Dense Nanoparticle Systems: Local Anisotropy Versus Dipolar Interactions

Dense systems of magnetic nanoparticles may exhibit dipolar collective behavior. However, two fundamental questions remain unsolved: i) whether the transition temperature may be affected by the particle anisotropy or it is essentially determined by the intensity of the interparticle dipolar interactions, and ii) what is the minimum ratio of dipole-dipole interaction ($E_\text{dd}$) to nanoparticle anisotropy ($K_{\text{ef}}V$, anisotropy $\times$ volume) energies necessary to crossover from individual to collective behavior. A series of particle assemblies with similarly intense dipolar interactions but widely varying anisotropy is studied. The $K_\text{ef}$ is tuned through different degrees of cobalt-doping in maghemite nanoparticles, resulting in a variation of nearly an order of magnitude. All the bare particle compacts display collective behavior, except the one made with the highest anisotropy particles, which presents ``marginal'' features. Thus, a threshold of $K_{\text{ef}} V/E_{\text{dd}} \approx 130$ to suppress collective behavior is derived, in good agreement with Monte Carlo simulations. This translates into a crossover value of $\approx 1.7$ for the easily accessible parameter $T_\text{MAX}$(interacting)$/T_\text{MAX}$(non-interacting) (ratio of the peak temperatures of the zero-field-cooled magnetization curves of interacting and dilute particle systems), which is successfully tested against the literature to predict the individual-like$/$collective behavior of any given interacting particle assembly comprising relatively uniform particles.


Introduction
Dense assemblies of magnetic nanoparticles continue to attract high interest, both from the technological [1][2][3][4] and basic science (e.g., superspin-glass dynamics, collective vs individual behavior) [5][6][7][8][9][10][11] viewpoints.The magnetic properties of the assembly depend markedly on the concentration.While dilute systems show magnetic properties similar to those of the individual particles, sufficiently concentrated assemblies may exhibit collective behavior, with properties (e.g., transition temperature or coercivity) distinctly different from those of their constituent particles.[16] In addition, in the recently discovered "liquid permanent magnets" dipolar interactions may play a role in the magnetic stabilization of the particles jammed at the interface of immiscible liquids. [17]Yet, the fundamental question as to the relative importance of the local anisotropy and the interparticle interactions in the blocking/freezing temperature and other magnetic properties of nanoparticle assemblies remains an open question, despite the relevance of this issue in many applications and the fundamental incentive provided by the contrasting theoretical descriptions of the effect.1] It is important to emphasize that while the behavior of particle systems with weak dipolar interactions can be simply accounted for by a modified Néel-Brown model, where the anisotropy of the particles still plays the main role in determining the blocking temperature, [22] dense random systems of dipolarly-interacting nanoparticles have far more complex characteristics.24][25][26] However, a fundamental difference between superspin-and classical spin-glass systems is the existence in the former of random (nanoparticle) anisotropy barriers (KefV) yielding strongly temperature-dependent local relaxation times.Their effect on the overall magnetic properties of dense assemblies has rarely been studied experimentally. [11]The abundant experimental literature on strongly interacting particle systems has delved into the effects of dipolar interactions by varying the concentration of the magnetic nanoparticles, first in frozen ferrofluids [21,[27][28][29] and nanogranular alloys in thin film or powder form, [30][31][32] and more recently by controlling the thickness of non-magnetic spacers (such as silica or dendrimer coatings) [20,22,24,[33][34][35] or the degree of powder compaction. [36]re, our experimental design takes on the opposite approach; namely, the packing fraction has been fixed (~ 60%, close to the theoretical maximum of random-close-packing of hard spheres, [37] in order to procure the strongest possible dipolar interactions), and the particle anisotropy systematically varied across an order of magnitude (by using maghemite nanoparticles doped with different amounts of Co). [38,39]Consequently, this shifts the focus from interparticle interactions to the importance of nanoparticle magnetic anisotropy.This strategy has allowed us to tackle the unsolved issue of the role of the local anisotropy barrier on the collective characteristics of dense assemblies of nanoparticles.The results, backed by Monte Carlo simulations and a literature review, show that sufficiently large local anisotropies will suppress the collective behavior of the assembly.In addition, they offer an estimate of the ratio of the relevant energies (anisotropy barrier and dipole-dipole interaction) that yields a crossover from individual to collective dynamics.

Preparation of dense nanoparticle systems
Four batches of highly monodisperse nanoparticles were synthesized through thermal decomposition using oleic acid (OA) as surfactant: [8,40] pure maghemite (γ-Fe2O3) nanoparticles with a mean diameter of 6.9 ± 0.6 nm and three types of cobalt-doped maghemite particles with similar sizes (6.7 ± 0.6 nm) and different fCo = Co/(Co+Fe) ratio, namely fCo = 0.11, 0.19 and 0.23. Figure 1 shows the transmission electron microscopy (TEM) images and particle size histograms, confirming the almost identical nanoparticle diameter independently of the Codoping.The narrow size distribution shown is representative of all the particles, with a polydispersity lower than 4%.Electron energy-loss spectroscopy (EELS) mapping analysis was performed to discard cation segregation during the synthesis.The images (see Figure S1) reveal a homogeneous distribution of the cobalt and iron ions across the whole nanoparticle, ruling out phase separation.After the chemical synthesis, a fraction of the particles was washed with acetone several times to remove the oleic acid surfactant bound to the particles, producing essentially bare particles.Figure S2 shows the thermogravimetric curves measured to quantify the content of oleic acid before (≈ 20%, very close to the expected value for a monolayer of oleic acid) and after (≈ 5%) the described washing.

Magnetic characterization
Figure 2 depicts the zero-field-cooled (ZFC) magnetization curves for the three series of samples described above: (a) silica-coated nanoparticles, (b) OA-coated nanoparticles and (c) bare nanoparticles.All the samples exhibit a ZFC maximum (TMAX) that increases, in all three series, with the content of Co [see also Figure 3(a), and Table 1 for numerical values].
The silica-coated particle assemblies [Figure 2(a)] provide the single particle behavior, as the thick diamagnetic shell makes magnetostatic interactions negligible. [8,22]In these systems, the relaxation time of the nanoparticle macrospin is simply governed by the ratio of the thermal (kBT) to the anisotropy barrier (KefV) energies, where Kef is the effective uniaxial anisotropy constant of the particle, as described by the Néel-Brown model  =  0 exp �      ⁄ �.When this relaxation time is smaller (greater) than the characteristic time of the measurement technique (  ), the magnetic response of the assembly is described as "superparamagnetic" ("blocked").The crossover temperature between the two regimes (for which  =   ) is known as the blocking temperature (TB) and is directly proportional to the product of the nanoparticle volume and its effective anisotropy constant (KefV).Therefore, given the constant volume across the silica-coated series the increase in TMAX (often taken as TB in these kind of dilute systems) across the series clearly indicates an anisotropy (Kef) enhancement due to the introduction of cobalt cations in the spinel structure. [38]Note that the same trends are observed when using the more accurate mean blocking temperature TBm (defined as the peak temperature in the -d(MFC-MZFC)/dT curve, [44][45][46] see Figure S3).The effective anisotropy constant can be readily calculated from the Néel-Brown equation using the TBm values, the measurement time   = 100 s (typical for dc SQUID magnetometry) and the attempt time  0 = 10 -13 s, yielding Kef ≈ 32, 228, 282 and 317 kJ m -3 for the pure maghemite, 11%, 19% and 23% Co-doped particles, respectively.Note that this attempt time value has been previously found to systematically fit the data of similar isolated particles better than the customary [22,23,26]  0 = 10 −10 s (in any case, this choice introduces a factor of only 1.25 in Kef), although faster attempt times have also been reported in both metallic [47] and spinel cobalt ferrite particles. [48]Comparable Kef values have been reported in the literature, e.g., 140 and 250 kJ m -3 (using  0 = 10 -10 s) for Co-doped iron oxide particles with similar size (5 and 6 nm, respectively) and cobalt content (15 and 19%, respectively). [38,49]n the series of pressed OA-coated nanoparticles [Figure 2(b)], the increase of TMAX with respect the corresponding isolated nanoparticles is stronger for the pure maghemite particles (∆T = 45 K) than for the three Co-doped particle systems (∆T ≈ 20 K).Interestingly, this offers a first indication that the TMAX shift is not determined solely by interparticle interactions, but it also depends on the nanoparticle anisotropy barrier.The OA-coated Co-doped samples are identified as "weakly/moderately interacting" systems with dynamics still described by a modified Arrhenius law, [50]  =  0 where the dipolar interactions simply enhance by an amount ∆ the single-particle anisotropy.
This "perturbation" description is suitable only for relatively small values of the ∆/KefV ratio, which explains why the lowest anisotropy particles (pure maghemite) of the OA-coated series showed collective, rather than modified single particle, behavior. [22]In a first, qualitative, approach, the collective behavior of this sample can be identified from the flat shape of the FC magnetization curve below TMAX with a magnetization close to MZFC(TMAX) [see grey line in Figure 2(d)].Note that a flat FC curve at low temperature is not indicative on its own of collective behavior; in fact, it is customarily observed in systems of isolated particles with a narrow size distribution and high enough TMAX, as is the case of our silica-coated Co-doped nanoparticles.On the other hand, in the three pellets comprising high-anisotropy OA-coated Co-doped nanoparticles, similarly strong dipolar interactions (in fact considerably stronger than those in traditional "dense" frozen ferrofluids or granular solids) [5,25,30,[51][52][53] were not enough to produce collective behavior [see, e.g., the FC curves (light-colored) of the 19% and 23% doping samples in Figure 2(e) and (f), comparable to the FC curves of the corresponding isolated nanoparticles (dashed lines)].
Table 1.Peak temperatures (TMAX) measured in the ZFC magnetization curves of the 12 samples (pressed discs) studied in this work.The error is less than 1 K in all cases.The Codoping atomic ratio is defined as fCo = Co/(Co + Fe).The estimated particle concentration (or packing fraction, C) for each type of particle coating is also indicated. [40,43]AX (K)  Note that in the OA-coated series, the pressed oleic acid introduces a uniform separation of about 1.3 nm between nanoparticles (estimated from the "magnetic" packing fraction), [43] ruling out the possibility of interparticle exchange coupling.Importantly, even in the bare nanoparticle series direct exchange coupling is also expected to play only a marginal role. [22]rstly, it is rather counterintuitive that the delicate indirect exchange at play in ferrites can propagate between metal ions belonging to different particles, unless there exists an exceptional crystalline coherence between aligned particles. [54,55]In addition, that the dominating type of interaction in the bare series is dipolar coupling is hinted by the very similar exchange bias fields (HE) measured at 5 K in the discs made of Co-doped nanoparticles with or without coating (see Figure S5 and the corresponding discussion).In short, HE in these samples is ascribed to the presence of surface spin disorder, [56][57][58][59][60][61][62] which would be affected by interparticle exchange interactions should there be present in the bare nanoparticle systems.The residual oleic acid bound to the nanoparticles will further hamper interparticle exchange.For a more detailed discussion, see the Supporting Information in Ref. [8].
As displayed in Figure 2(c), the strong interactions in the bare particle compacts enhance the ZFC peak temperatures well beyond the values measured in the OA-coated series.
Moreover, the flat FC curves, combined with the more quantitative features that will be presented later (i.e., ZFC memory effect and critical slowing down of the relaxation in SI), indicate a crossover from single-particle to collective (superspin glass like) dynamics with increasing packing fraction of the magnetic Particles.Therefore, in this series, the measured peak temperature (TMAX) reflects the collective freezing of the system upon cooling. [23,26]Yet, despite this collective character (see discussion below), TMAX is still not only controlled by the dipolar interactions, but it is also largely affected by the local anisotropy (nanoparticle energy barrier).This result is at odds with the conceptual phase diagram proposed by Mørup for dipolarly interacting particle systems, where the collective freezing temperature is determined exclusively by the strength of the dipolar interactions. [18]In fact, Figure 3(a) shows that TMAX increases with nanoparticle anisotropy (i.e., fCo) despite the concomitant reduction in dipolar interaction strength (green arrow in the figure) stemming from the small loss in saturation magnetization upon Co-doping (see Figure S6 and the corresponding discussion). [63,64]This small variation of dipolar interactions across the fCo series is an undesired effect of our Codoping strategy, yet it lends support to the argument.For  =   , T = TMAX, and solving for the latter: where TBm is the mean blocking temperature measured by magnetometry in isolated particles (KefV is taken here as the mean anisotropy barrier of the nanoparticles), and bs is a proportionality constant relating the Vogel-Fulcher divergence temperature Tint, customarily ascribed to interparticle interactions, [19][20][21] and the dipole-dipole interaction temperature   defined above.Although bs in Equation 3 will be systematically smaller than b0 in Mørup's simpler model for strongly interacting systems, namely   =  0   (where it was estimated to be of the order of 10) [18,65] , both parameters have essentially the same meaning, except that the influence of local anisotropy is overlooked in the latter model.Since Tdd (dipole-dipole interaction) simply sets the interaction energy scale, bS gathers everything else affecting the interaction energy, namely, as detailed in Ref. [18], the summation over many pairs of magnetic dipoles (long-range character of the dipolar interaction), the particle size distribution, as well as a factor of the order of unity relating the root-mean-square dipolar interaction energy and the (super)spin-glass ordering temperature.It must be emphasized that the Vogel-Fulcher law [from which Equation 3 is derived] describes the divergence of the relaxation time at a finite temperature Tint, and therefore Equation 3 is a priori only suitable for the analysis of strongly interacting systems.For non-or weakly-interacting systems, which do not yield such finite-T divergence, the modified Arrhenius expression in Equation 3 should be used instead.In short, the proposed "TMAX -TBm = bsTdd" relation is one way to introduce the influence of the particle anisotropy in strongly interacting particle systems, where, assuming that TMAX mimics the glass transition Tg, it suggests Tg = TBm + bsTdd, to be compared with Mørup's expression Tg = b0Tdd.
To test our data against Equation 3, Figure 3(c) plots TMAX for both the bare and OAcoated series as a function of the TBm measured in the corresponding dilute systems [in turn plotted vs fCo in Figure 3(b) from the data in Figure S2].It is important to underline the originality of this plot, whose abscissa is not some interaction proxy, as customarily found in the literature, [18,20,22,24,42] but the local anisotropy quantified through TBm.The clearly linear dependence and the fitted slope (close to 1 in both series) are in good agreement with Equation 3, unequivocally evidencing the important role of the single-particle anisotropy in the freezing temperature of strongly interacting assemblies.The intercept of the fits gives the average interaction temperature in the two series (  ����� ).In the bare series,   ����� = 157 K, which, considering an average Tdd ≈ 40 K, yields bs ≈ 3.9 (the range of individual bs values in the bare series is 2.7 -5.5).This value is indeed smaller than the b0 = TMAX/Tdd estimates for several concentrated nanoparticle systems reviewed by Mørup in Reference [18], in the range 4 -8.
Moreover, the smaller bs ≈ 2.1 of the OA-coated series appears consistent with a different qualitative prediction suggested by Mørup's diagram, namely the steeper variation of TMAX vs Tdd in systems with strong interactions (providing superspin glass behavior) compared with those with weaker interactions (presenting modified single-particle dynamics).Equation 3, thus, provides a method to estimate the interaction temperature simply from the ZFC curves of isolated and dense systems made of the same nanoparticles.In this regard, note that in dipolarlyinteracting systems, as in other systems with similar amounts of ferro-and antiferro-like interactions, [66,67] Curie-Weiss fits cannot determine the magnitude of the interactions.
This leads to the question as to how high the anisotropy barrier must raise (relative to the interparticle interaction) to suppress the collective spin glass-like behavior produced by dipolar interactions.First, note that the naïve notion that dipolar interactions, of the order of magnitude of the dipole-dipole interaction energy (Edd = kBTdd,), need to overcome the anisotropy energy barrier (KefV) to produce collective behavior must be dismissed, at least in particle assemblies with a random orientation of the uniaxial anisotropy axes.Note that the first three bare nanoparticle systems studied here, with energy ratios KefV/Edd ≈ 35•TBm/Tdd ranging from ≈ 6.6 (pure maghemite) to ≈ 118 (fCo = 0.19), show FC-ZFC curves typical of superspin glasses.This is not the case for the bare fCo = 0.23 sample (with the highest anisotropy), with a FC curve still increasing below TMAX and a broader ZFC maximum.This, together with relaxation features presented below, justifies labelling this sample as "marginal", marking the crossover from collective to single-particle dynamics with increasing KefV.Given the unavoidable distribution in nanoparticle anisotropy barrier, the broader ZFC peak in this sample would reflect the particularly wide distribution of relaxation times arising from a mixture of Néel (higher KefV particles) and critical (collective) dynamics.Moreover, the single-particle dynamics could operate in few-nanoparticles clusters (dimers, trimers, etc) of dipolarly-coupled high anisotropy particles, yielding long relaxation times.It is this heterogeneity in dynamics that defines the mentioned "crossover".Consequently, our experimental data allows us to estimate a crossover energy ratio of (KefV/Edd)c ≈ 130. Interestingly, this result is not far from the ratio implied in Mørup's phase diagram [(KefV/Edd)c ≈ 150; see Supporting Information], based on the experimental data available at the time, separating (modified) single-particle-like behavior from collective freezing in both Mössbauer and magnetometry experiments. [18]fortunately, the ratio KefV/Edd is rather inconvenient to experimentally predict the behavior of dense assemblies of nanoparticles.A more suitable parameter can be found rewriting Equation 3, using TBm = cTB (with TB defined as the ZFC peak temperature in the reference system of isolated particles), as which suggests the ratio TMAX/TB, more experimentally accessible than KefV/Edd, as a simple parameter to predict the single-particle/collective character of a given nanoparticle system and whether a superspin glass behavior can be expected in relatively dense samples.For example, pressed bare nanoparticles 8 nm in diameter, with TMAX/TB ≈ 4, have been previously described as a model superspin glass system. [42]To determine the crossover value we have used the data for fCo = 0. the interacting systems studied here (both the bare and OA-coated series), as well as other dense assemblies characterized by us or other groups. [5,20,23,24,33,40,42,68,69]Although this crossover ratio for collective behavior is far from being a universal [inasmuch as the parameters c and bs in Equation 4are not], it can be seen that the threshold value suggested above agrees very well with the reviewed studies (all of them dealing with relatively narrow particle size distributions and reporting TMAX, TB and whether the denser system presents collective or single-particle dynamics).It is particularly noteworthy that another "marginal" sample, reported by Hansen et al. in a 5 %vol concentrated FeC ferrofluid) lies also on the same crossover ratio. [5]Moreover, the dashed line in Figure 4 is a fit to Equation 4 of the bare nanoparticle systems studied here with the addition of another disc made of smaller (6.2 nm) maghemite particles. [40]As expected, the resulting value for the fitting parameter     is the same (within the error bars) as that obtained for the intercept     =   ����� in Figure 3(c) using the equivalent Equation 3. Overall, the data points distribution visually emphasizes the intuitive fact that it is easier to obtain collective behavior in systems comprising nanoparticles with lower anisotropy energy.The solid symbols correspond to systems with reported collective/superspin glass behavior, whereas hollow symbols are used for systems where the lack of collective behavior has been proven.Half-solid symbols correspond to systems with reported marginal behavior.The pink line, therefore, marks an experimental TMAX/TB threshold for superspin glass behavior.Larger symbols are used for compacts of essentially bare nanoparticles. [22,40,42]Grey symbols are used for pressed OAcoated or dendrimer-coated (down-triangle) nanoparticles. [20,22,24,33,69]Dark blue symbols are for frozen ferrofluids. [5]The nanoparticles are always maghemite [20,22,24,33,40,42,68] unless otherwise noted. [5,69]Importantly, all samples included in the graph were prepared using nanoparticles with a relatively narrow size distribution.The dashed line is a fit of the green datapoints to Equation 4, equivalent to the fit to Equation 3 in Figure 3.
From the crossover ratio (KefV/Edd)c ≈ 130 one may also calculate the minimum volume concentration/packing fraction of nanoparticles necessary to reach collective dipolar behavior,   , by simply inserting   =  0 4   2   , [65] which yields a particle-size independent   ≈ 0.1   0   2 .This leads to a realistic φc ≈ 0.58 for our fCo = 0.23 sample.An interesting corollary is that dipolar collective magnetism is not possible (i.e., φc ≳ 0.64 in random close-packed assemblies) for nanoparticulate materials with    2 ≳ 8.3 J kA −2 m −1 .This will be the case for hard particles with a modest saturation magnetization, such as ε-Fe2O3 nanoparticles (which yields an impossible φc ≈ 4 when using the K and MS values reported in Ref. 69) or CoPt, for which φc ≈ 0.60 using bulk values [55] (higher K and lower Ms values are customarily found in nanoparticles, thus pushing the critical concentration up to unphysical values).
In the discussion of Figure 3 above we claimed the relevance of the particle anisotropy even in systems showing collective dynamics.In this regard, to demonstrate the existence or the lack of such collective behavior, we have performed ZFC memory experiments (encompassing the phenomena of ageing and rejuvenation in spin glass-like systems) probing the system relaxation over several hours.This experiment, together with the previous ZFC/FC magnetization curves, and ac magnetic susceptibility (see SI), cover three of the four "key bulk measurements that determine a magnetic material to be a canonical spin glass" (the fourth one being specific heat), according to a recent review by J. A. Mydosh. [71]The ZFC memory effect is a remarkable phenomenon characteristic of the chaotic non-equilibrium (super) spin glass phase. [51,52,72,73]Figure 5 shows ZFC magnetization curves recorded after cooling without (reference curves) and with (memory curves) a 4 hour halt at  ℎ = (2 3 ⁄ ) MAX .A dip in the magnetization of the latter curve can be appreciated in the vicinity of the halt.The stop performed during cooling allows the system to "age", slowing down its dynamics towards the particular equilibrium state at the halt temperature; a state that is recovered later upon heating, thus the memory curve magnetization falls below the reference curve.The corresponding difference curves are shown in the lower inset of Figure 5, where ∆M is defined as the difference between the memory and reference curves �   () −    ()� normalized by   =    ( =  halt ).All samples display visible memory dips; however, their magnitude is strongly dependent on the doping ratio (see upper inset).While the pure maghemite sample exhibits a sharp memory effect (6 %), the Co-doped samples show weaker memory, barely noticeable in the 23% doped sample.This trend can be ascribed to the net increase of the magnetic anisotropy with Co-doping, in turn slowing down the individual nanoparticle relaxation and, thus, the overall collective effect, as have been previously described in both superspin [11,74] and atomic spin glasses. [75]A similar trend (from sharp collective features in the non-doped sample to marginal/anomalous behavior in the 23% doped sample) is observed in the frequency and temperature dependence of the ac susceptibility (see SI).

Monte Carlo simulations
[78][79][80][81][82][83] The Monte Carlo model is designed so that, although it is related to the experiment in terms of nanoparticle size, MS or Keff, it circumvents some of the experimental drawbacks (e.g., variations in the size, MS or surface disorder between constituents in the series).In this way, we avoid experimental issues that might complicate the interpretation of the data while making the results more general.The Hamiltonian includes only three terms, namely uniaxial anisotropy, dipolar interactions and Zeeman energy.FC and ZFC (with and without a halt to test memory) curves were simulated for ensembles corresponding to the silica-coated (dilute) and bare nanoparticles (60% packing fraction) compacts, where the uniaxial anisotropy was varied systematically.remove the oleic acid coating, obtaining what we call here bare particles (≈ 5%w oleic acid), a second fraction was separated to subsequently grow a silica shell following the method described elsewhere, [41] and a third fraction was left as it obtained (oleic acid-coated).All the different types of resulting nanoparticles were dried and the powder was pressed uniaxially under approximately 0.8 GPa to form dense discs.
Transmission electron microscopy (TEM) images have been acquired in a FEI Tecnai G2 F20 microscope operated at 200 kV.The size of the particles was determined by manually measuring the diameters of the particles from the TEM images.
A Quantum Design EverCool MPMS SQUID magnetometer was used for the magnetic characterization of the pellets with the applied field parallel to the disc plane.Zero-field-cooled (ZFC) and field-cooled (FC) magnetization curves were measured using a magnetic field strength of 0.5 mT (5 Oe) and a sweeping rate of 2.5 K/min.In the ZFC protocol the sample was cooled in zero-field from room temperature to 5 K, at which the magnetic field was applied, and the magnetization was recorded upon heating.Then, in the FC protocol, the sample was cooled in constant field and the magnetization recorded again upon heating.ZFC memory experiments were carried out using a similar protocol to that described in Reference [84] .In the "memory curve", the sample was cooled down to 5 K in zero field with a 4 h halt at  ℎ = 2  3 ⁄ .

Figure 1 .
Figure 1.Transmission electron microscopy (TEM) images of the pure maghemite (a, b) and Co-doped 19% maghemite particles (c, d).The silica-coated particles are shown on the right side, and the OA-coated particles on the left side (scale bar = 20 nm).The (mostly overlapping) size distributions of both samples (pure -blue-and doped -red-) are shown in the inset.Typical high-resolution scanning electrons microscopy (HRSEM) images of the surface of discs prepared with bare (e) and OA-coated (f) nanoparticles.

Figure 2 .
Figure 2. The left column shows the ZFC magnetization curves normalized by MZFC(TMAX) measured in an applied field of µ0H = 0.5 mT in pressed discs of (a) nanoparticles coated with a thick silica shell, (b) OA-coated nanoparticles, and (c) bare nanoparticles.The right column shows the ZFC and FC magnetization curves (measured in the same conditions) of (d) pure maghemite, (e) 19% and (f) 23% Co-doping particles with different coatings: silica (dashed lines), oleic acid (solid light-colored lines) and bare (solid dark-colored lines).
Remarkably, for the assemblies prepared from the uncoated (bare) particles, all samples exhibited flat FC curves at low temperatures indicating collective behavior, although clearly modulated by the varying nanoparticle anisotropy, except possibly the pellets made with the 23%-doped nanoparticles [see FC curves in Figure 2(e-f)].To better illustrate the above FC-ZFC irreversibility qualitative argument, Figure S4 plots a parameter expressing the rise of the FC plateau above the ZFC peak, FCrise = [Mplateau -MZFC(TMAX)]/MZFC(TMAX) for the different particle systems.

Figure 3 ( 2 )Figure 3 .
Figure3(a), therefore, shows that TMAX in any dipolar interacting nanoparticle system is determined by additive contributions from the dipolar interactions (characterized by the parameter   =  0   2  2 /  4 3 , where r is the mean distance between particles) as well as from the nanoparticle anisotropy energy barrier (KefV, here estimated from TBm, measured in the silica-coated nanoparticles).However, modified single-particle dynamics (Equation1),where the relaxation time (τ) still diverges only at T = 0 K, does not hold for more strongly interacting nanoparticle systems where τ is found to diverge at a finite (glass) temperature, signaling a phase transition to a superspin glass state.A phenomenological expression satisfying both requisites [namely, (i) TMAX determined by both Tdd and TBm, and (ii) critical divergence of the relaxation time at non-zero temperature] is the well-known Vogel-Fulcher law:[19]

Figure 4 .
Figure 4. Phase diagram of dipolarly interacting nanoparticle systems.The solid symbols correspond to systems with reported collective/superspin glass behavior, whereas hollow symbols are used for systems where the lack of collective behavior has been proven.Half-solid symbols correspond to systems with reported marginal behavior.The pink line, therefore, marks an experimental TMAX/TB threshold for superspin glass behavior.Larger symbols are used for compacts of essentially bare nanoparticles.[22,40,42]Grey symbols are used for pressed OAcoated or dendrimer-coated (down-triangle) nanoparticles.[20,22,24,33,69]Dark blue symbols are for frozen ferrofluids.[5]The nanoparticles are always maghemite[20,22,24,33,40,42,68] unless otherwise noted.[5,69]Importantly, all samples included in the graph were prepared using nanoparticles with a relatively narrow size distribution.The dashed line is a fit of the green datapoints to Equation4, equivalent to the fit to Equation 3 in Figure3.

Figure 5 .
Figure 5. ZFC memory experiments.The purple dotted lines are the reference curves measured without a halt.The solid lines are the ZFC magnetization curves measured after 4 hours stops at  ℎ = 2  3 ⁄ .The lower inset shows the difference curves, ∆M(T)/Mref(T = Thalt) ("memory dips"), with the depth plotted in the upper inset.The vertical dashed lines of the lower inset marks Thalt for each sample.

Figure 6
shows these curves for the end members of the simulated dense series (bare nanoparticles compacts).The qualitative agreement with the experimental data is remarkable, with the highest Kef dense sample showing lower memory effect.Yet more significant, the memory dip progressively fades with increasing anisotropy [see inset in Figure 6(b)], as observed experimentally.At the quantitative front, the dependence of TMAX (from simulations of the nanoparticle compacts) with the mean blocking temperature obtained from the corresponding dilute systems [in analogy to Figure 3(c)] is observed to be linear with a slope very close to unity [see inset in Figure 6(a)], in excellent agreement with the experimental data, thus corroborating the model in Equation 3. Finally, the crossover value (KefV/Edd)c_MC ≈ 140 found to transition from collective to individual behavior (corresponding to the anisotropy reducing the memory effect by a factor of three, as suggested by the experimental results) is remarkably similar to our experimental estimate.

Figure 6 .
Figure 6.(a) FC and ZFC magnetization curves simulated for the end members (lowest and highest Kef) of the series of dense systems (same filling factor, varying anisotropy).Solid/dash lines are used to denote ZFC curves obtained without/with a halt, respectively.The inset shows a plot analogous to Figure 3(c), with TBm calculated from d(MFC-MZFC)/dT as in the experiment.(b) "Memory curves" obtained as the difference between the ZFC curves described above normalized by Mref (T = Thalt).The vertical lines are the respective halt temperatures.The inset shows the dip depth as a function of the anisotropy constant (defined as k = KefV/20kB) of the nanoparticles across the simulated series.