Determination of the blocking temperature of magnetic nanoparticles: The good, the bad and the ugly

In a magnetization vs. temperature (M vs. T) experiment, the blocking region of a magnetic nanoparticle (MNP) assembly is the interval of T values were the system begins to respond to an applied magnetic field H when heating the sample from the lower reachable temperature. The location of this region is determined by the anisotropy energy barrier depending on the applied field H, the volume V, the magnetic anisotropy constant K of the MNPs and the observing time of the technique. In the general case of a polysized sample, a representative blocking temperature value $T_B$ can be estimated from ZFC-FC experiments as a way to determine the effective anisotropy constant. In this work, a numerical solved Stoner-Wolfharth two level model with thermal agitation is used to simulate ZFC-FC curves of monosized and polysized samples and to determine the best method for obtaining a representative $T_B$ value of polysized samples. The results corroborate a technique based on the T derivative of the difference between ZFC and FC curves proposed by Micha et al(the good) and demonstrate its relation with two alternative methods: the ZFC maximum (the bad) and inflection point (the ugly). The derivative method is then applied to experimental data, obtaining the $T_B$ distribution of a polysized $Fe_3O_4$ MNP sample suspended in hexane with an excellent agreement with TEM characterization.


I. INTRODUCTION
Magnetic nanoparticles (MNPs) are been extensively studied due to their multiple applications in technology 1 and biomedicine 2,3 . Particles with sizes in the range [5,100]nm 4 present a magnetic behaviour determined by its volume, shape and composition, matrix viscosity and temperature, among other factors. In the simplest (however very useful) model, the MNPs of volume V and saturation magnetization M s are considered as almost spherical ellipsoids with a permanent moment m = M s V and a preferential magnetization axis (easy axis) in which the anisotropy energy E K = KV sin 2 [δ] is minimum, being K the effective anisotropy density constant and δ the angle between m and the easy axis. If the MNPs are fixed in the matrix and separated one from each other by a distance d > 3V 1/3 , dipolar interactions can be neglected 5 and the energy of the system can be expressed as the sum of the anisotropy energy and the Zeeman energy E H = −mH cos[θ]: with θ the angle between m and H ( fig. 1). This configuration is usually called Stoner-Wolfharth system since the publication of a work 6   In order to calculate the temperature dependence of the magnetic response for MNPs systems, it is necessary to consider the effect of thermal fluctuations that arXiv:1508.04413v1 [cond-mat.mes-hall] 18 Aug 2015 allow transitions between stable configurations. Doing so, it is possible to simulate M vs. T experiments as the extensively performed Zero Field Cooling-Field Cooling (ZFC-FC) routine. In this kind of experiments, a sample is cooled from a temperature where all particles show superparamagnetic behaviour to the lowest reachable temperature (usually around 3K), then, a small constant field usually lower than 8kA/m is applied, and the sample is heated to a temperature high enough to observe an initial growth and subsequent decrease in magnetization, i.e. were the sample show again superparamagnetic behaviour .The sample is then cooled again to the lowest temperature with the constant field still applied.
In the ideal case of a monosized, non interacting MNPs sample; a narrow temperature region should exist in which the system performs a transition between irreversible and reversible regimes. When heating with applied field, the thermal energy kT is initially much smaller than the anisotropy barrier KV so the magnetization remains null. Due to the exponential dependence of the Néel relaxation time with temperature 7 , when kT ∼ KV , the magnetization grows rapidly until its thermodynamic equilibrium value, defining the aforementioned transition region. The Blocking Temperature T B can be considered as the inflection point of this growing and its experimental determination is an important goal of the MNPs characterization.
Real samples always present a size dispersion, usually reasonably well described by a log-normal distribution. Different particle size implies different anisotropy barrier KV and therefore a different T B for each size fraction, so, in real ZFC-FC experiments, the blocking region is wide and the representative T B value is not well defined. There are several different criteria used to define a representative T B from ZFC-FC data of polysized samples. Some authors maintain the inflexion point (IP) criterion 8 while others report the maximum ZFC magnetization temperature (MAX) 910 , being all this criteria still in discussion 11 . In an alternative approach, Micha et al. 12 propose a method in which the T B distribution is obtained from the T derivative of the difference between ZFC and FC curves. An approximated theoretical justification for this method was presented by Mamiya et al 13 .
In this work, a SW model with thermal agitation is applied to obtain the temporal dependence of the magnetization M (t) for an ordered system of identical MNPs in a similar way to previous works of Lu 14 , Usov 15 and Carrey 16 . Temperature dependence dM (T )/dT is then obtained in order to numerically simulate the ZFC-FC curves. In contrast to the method implemented by Usov 17 where a stair-step approximation for the time evolution of the temperature was used, we consider a continuous time evolution. Finally, an ordered polysize system response is simulated by linear combination of the monosize curves weighted by a discrete log-normal distribution.
The validity of the method proposed by Micha et al was verified by comparing the T derivative of this ZFC-FC curve with the T B distribution obtained from the inflection points of each volume of the distribution. The resultant mean blocking temperature value < T B > is then compared, for several volume distributions, with the commonly used criteria for a representative T B : the inflection point temperature IP and the maximum MAX of the ZFC curve. Additionally, Micha's method is tested with experimental data of a magnetite MNPs frozen ferrofluid suspended in hexane comparing the obtained T B distribution with the one calculated from the TEM size information. In order to obtain an ordered system, the ferrofluid was frozen while a large constant was field applied.

II. MODEL
A SW-like model with thermal agitation and zero width energy minima approximation was developed in order to obtain ZFC-FC curves of fixed MNPs with size dispersion. Only the simplest case of an ordered system was considered, with all the MNPs oriented (easy axis orientation) in the direction of the field. This situation can be achieved experimentally by freezing a ferrofluid sample under a sufficiently strong applied field (∼ 7T ).

A. Magnetization vs. time equation
For a system of identical, fixed, non interacting MNPs of volume V , anisotropy constant K and saturation magnetization M s , with their anisotropy axes parallel to an external field H, the energy can be expressed as the sum of the anisotropy energy E k and the Zeeman energy E h 6 : being h = H/H k and H k = µ 0 M s /2K. In the range θ = [0, 2π], this energy landscape presents two minima, of The frequency of thermal inversions between minima i and j is the inverse of the Nèel relaxation time 1819 ,: with f 0 the "intrinsic frequency", times the Boltzman "success probability" depending on the ratio between thermal energy kT and barrier height D ij . The barrier between minima is symmetric for h = 0 with D du = D ud = KV (naming u and d to θ = 0 and θ = π directions respectively) and smaller for inversion to the field Energy landscapes as a function of theta for a fixed MNP under a magnetic field at θ = 0. Each color stands for a different value of the reduced field factor 2h. Blue: 2h = 0.1, purple: 2h = 0.5, gold: 2h = 2.
direction otherwise: It is a good approximation to consider the same f 0 value for both frequencies 20 .
Sample magnetization M in the direction of the applied field can be expressed in terms of saturation magnetization M s and the number of particles per unit volume magnetized in each direction N u and N d : with N the total number of particles per unit volume. So the time derivative of the magnetization can be written in terms of the population variation which is equal to the actual population times the inversion probability to each direction so the time derivative of the relative magnetization m is determined by the transcendental equation where C = KV /kT . The system is first cooled with zero field applied from a high temperature where all particles show superparamagnetic behaviour (ZFC, no showed), then, the field is turned on and the system is heated beyond the blocking region (ZFCW). Maintaining the applied field, the system is cooled (FC). The final heating can be performed with (FCW) or without applied field (TRM).

B. Magnetization vs. Temperature equation: ZFC-FC simulation
Temperature dependence of the magnetization can be obtained from 8 via the equation For a linear temperature variation T (t) = Bt + T 0 , the magnetization derivative is During the warming after zero field cooling (ZFCW for this chapter, usually called just ZFC), the exponential dependence of the inversion frequency with temperature in equation 3 determines a narrow "blocking region" wherein the MNPs, which were "blocked" at low temperature begin to respond to the field. Magnetization grows with temperature since the applied field has decreased the energy barrier for θ = π to θ = 0 inversion. The magnetization increasing reverts when thermal energy is much higher than the barrier, so the difference between inversion frequencies in each direction tends to disappear. The blocking temperature T B of the system is then defined as the inflection point of the magnetization growing when heating.
When the system is cooled again (FC), magnetization grows monotonically while the barrier height difference between inversions becomes increasingly significant against thermal energy. This growth stops when thermal energy becomes too low for inversions to occur within the experimental window time. If the system is then heated maintaining the applied field (FCW), magnetization values are the same than FC except for the blocking region where there is a small increase due to the assembly getting closer to the equilibrium state. If the final warming is done with no applied field (Thermal Remanent Magnetism, TRM), magnetization drops to zero in the blocking region when thermal energy is enough for the wells populations to equilibrate.
The magnetization values M p for a polysized sample are obtained by linear addition of the M V i values for each contemplated size V i , weighted by the corresponding volume and log-normal distribution LnN (V i ) value: The V i LnN (V i ) product stands for the relative volume distribution. Figure 4 shows the comparison between ZFC-FC simulations for samples with different size dispersion expressed as the scale parameter σ of the log-normal number distribution. A much wider transition region can be seen for the bigger dispersion so the different aforementioned criteria would define very separated values for a representative T B .

A. Micha's method verification
In order to verify Micha's method, several polysize ZFC-FC experiments were simulated using differents parameter sets varying σ and the mean radius. For each one of the used sets, the T derivative of the ZFC-FC difference was calculated. Then, the T B distribution was obtained from the monosize curves that were added to construct the polyzise simulation in equation 11: a ZFC curve was calculated for each class of the size distribution so each T B class comes from a volume class, maintaining the same relative height. Also IP and MAX values of the polysize ZFC curve were calculated and compared with the mean value < T B > of the distribution in each simulation ( fig. 5).
In all cases, the T B distribution and the ZFC-FC derivative are identical. Figure 6 shows the results for the simulation with 4.5 nm mean radius, σ = 0.5, 16kJ/m 3 anisotropy constant and a 4K/min heating rate.
Also, for a set of ZFC-FC curves calculated with the same mean volume, saturation magnetization, heating rate and anisotropy constant, by increasing scale parameter σ, < T B > stays constant while the polysize curve IP shifts to smaller temperatures and MAX shifts in opposite direction. Figure 7 shows the results for 4.5nm mean radius, 16kJ/m 3 anisotropy constant, 4K/min heating rate and σ = [0.1, 0.6].
This behaviour is the same in the hole studied size range. By normalizing IP values by the T B mean, all points fall in the same curve as shown in figure 8 while the variation for MAX is small.
Varying the heating rate and K does not affect the relation IP/< T B >. Meanwhile, the MAX/< T B > ratio changes strongly in the range [0.04; 400]K/min and noticeably in the range [1; 10]K/min and also depends on the K value. Figure 9 shows the results of varying the heating rate for R m = 10nm, K = 16kJ/m 3 and M s = 281kA/m. Figure 10 shows a parabolic fit over the IP/ < T B > values obtained for all the simulations. The curve is universal with small fluctuations due to numeric resolution. The obtained polynomial with fitting errors is taken in order to determine the size distribution of the particles (fig 11). A narrow log-normal number diameter distribution (LnN (x)) was obtained with a 9.54nm mean and a 1.73nm standard deviation. The relative TEM volume distribution was obtained from this results and fitted with a xLnN (x) function obtaining a scale parameter σ = 0.55 (2).
The ZFC-FC routine was carried with a 2.4K/min rate and a 8kA/m field on an encapsulated FF sample frozen under a 7T field in order to obtain an ordered system with all the MNP easy axes oriented parallel to the field. The ZFC-FC derivative was calculated and fitted with a xLnN (x) distribution using the TEM σ as a fixed parameter with a very good correspondence (figure 12).
where τ 0 = 1/f 0 is the inverse of the intrinsic inversion frequency. For a known volume distribution, this comparison can be used to determine the effective K value as the one that maximizes the coincidence between TEM and ZFC-FC distributions. In this case, a value of 34(2)kJ/m 3 was obtained with a very good correspondence between TEM and ZFC-FC data. This calculation implies some approximations: M s is considered independent from the temperature in the region of interest, and the relaxation time expression used for the blocking condition 12 considerate only the inversions in the direction of the field. While the first approximation is very reasonably, the blocking condition expression is accurate only in experiments with high µH/(kT ) ratios, where the reversal frequency are much smaller for the inversions to the antiparallel state.
Additionally, the IP/< T B > ratio was calculated obtaining a value of 0.7(1), compatible with polynomial expression obtained from the simulations.

IV. DISCUSSION AND CONCLUSIONS
The validity of the Micha's method to determine the T B distribution of non interacting MNPs assembly was demonstrated by numerical simulations and experimental data analysis.
A Stoner-Wolfarth model with thermal agitation was developed in order to simulate the ZFC-FC curves of polysized MNPs assembles. From this simulation it was clearly demonstrated that the temperature derivative of the ZFC-FC difference is in full coincidence with the T B distribution of the sample, calculated as the inflection points of each size ZFC curve. Additionally, it came clear from the results that the maximum (MAX) and the inflection point (IP) of the polysized ZFC curve are affected not only by the mean size of the particles, but by the size dispersion. Thus neither IP or MAX are direct estimators of the mean T B . This is an interesting result since these values are commonly used in magnetic characterizations and can lead to estimate T B values far from the mean. As an example, for a sample with σ = 0.5 and a heating rate of 4K/min, IP= 0.7 < T B > and MAX= 2.12 < T B >. Nevertheless, it was found that the IP/ < T B > ratio depends exclusively on σ, while MAX/ < T B > changes with K and the heating-cooling rate. This behavior reveals a connection between IP, one of the most commonly used T B criteria and the actual mean value of the blocking temperature. For a sample with known σ, the mean blocking temperature could be obtained from the universal curve presented in this work. This approach of obtaining size distribution information from an universal curve was presented before by Hansen et al 22 using a rougher model.
In the present development status, the ZFC-FC simulation algorithm does not include a "measurement time" parameter. Just the heating rate is used, so the authors assume that the simulated data points represent the values for "instantaneous" measurements. Therefore, the magnetization values obtained from the simulation won't be equivalent to the ones obtained in an experiment with the same parameters.
The Micha's method was then applied to characterize a sample of magnetite nanoparticles coated with oleic acid and suspended in hexane. The volume distribution of the sample was obtained from TEM analysis showing a narrow log-normal shape with a mean diameter of 9.54nm and a standard deviation of 1.73nm. In order to obtain an ordered system, the ferrofluid was frozen under a 7T magnetic field. Then, a ZFC-FC routine was carried and a T B distribution was obtained from the data. This distribution was fitted with a xLnN [x] function using the scale parameter σ obtained from the TEM data as a fixed fitting parameter. The high goodness of the fitting supports the validity of Micha's method and the low influence of magnetic interaction between particles which is consistent with the particle to particle distance imposed by the FF concentration. Additionally, the resultant IP/< T B > values is consistent with the universal curve obtained from the simulations.
Finally, the effective anisotropy constant of the particles was estimated as the value which gives the maximum coincidence between the ZFC-FC T B distribution and the one obtained from the TEM volume.
The results obtained in this work constitute just a first example of the potential of the presented model in combination with experimental characterization. There is work in progress to enhance the simulation algorithm in order to include the measurement time as a parameter and to considerate both inversions processes in the blocking condition.