Size dependence of the surface spin disorder and surface anisotropy constant in ferrite nanoparticles

The magnetic properties of nanoscale magnets are greatly influenced by surface anisotropy. So far, its quantification is based on the examination of the blocking temperature shift within a series of nanoparticles of varying sizes. In this scenario, the surface anisotropy is assumed to be a particle size-independent quantity. However, there is no solid experimental proof to support this simplified picture. On the contrary, our work unravels the size-dependent magnetic morphology and surface anisotropy in highly uniform magnetic nanoparticles using small-angle polarized neutron scattering. We observed that the surface anisotropy constant does not depend on the nanoparticle's size in the range of 3–9 nm. Furthermore, our results demonstrate that the surface spins are less prone to polarization with increasing nanoparticle size. Our study thus proves the size dependence of the surface spin disorder and the surface anisotropy constant in fine nanomagnets. These findings open new routes in materials based on a controlled surface spin disorder, which is essential for future applications of nanomagnets in biomedicine and magnonics.


Introduction
Disorder, the most favorable phenomenon for reducing a system's total energy, crucially alters materials' physical and chemical properties, whether in the form of chemical, structural, magnetic, or geometric disorder. 1,2 Therefore, in material science, controlling and tuning disorders is benecial. Decreasing the size of materials to the nanoscale, the effect of disorder becomes increasingly important. 3 Magnetic nanoparticles (NPs) promising technological and biomedical applications in for example, 4,5 magnetic recording, 6-10 magnetic uid hyperthermia, [11][12][13] magnetic manipulation for isolation of target biomolecules and rapid mixing, 14,15 and magnetic resonance imaging (MRI) 16,17 are due to their particular physical properties arising from their large surface-to-volume ratio. Different applications require distinct magnetic properties, for instance, data-storage applications demand high magnetic anisotropy to preserve thermal stability, 18 while magnetic hyperthermia and MRI necessitate superparamagnetic NPs. 19 The resulting magnetic properties and performance are affected not only by the disorder but also by various aspects, such as shape, [20][21][22] size, [23][24][25][26] chemical composition, 27 crystal phase, 28 surface coating 29 and effects. 30,31 However, surface disorder phenomena that arise from vacancies or the presence of antiphase boundaries, [32][33][34][35][36] such as spin disorder 37 and spin canting, 38,39 have the strongest inuence on the magnetic properties of NPs, 3 i.e., spontaneous magnetization, superparamagnetic behavior, coercivity, and exchange interaction. The effective magnetic anisotropy energy, E a = K eff $V, describes the alignment of the particle's moment in the eld direction, and it is an important parameter for many applications, for example, in data storage, it describes the efficiency of storing medium. E a depends on the effective anisotropy constant, K eff , and volume of material, V, and thus, studying their correlation with the particle size is important. 40 In small NPs, where a large surface-to-volume ratio drives the magnetic properties, the effective anisotropy constant depends not only on the bulk magnetocrystalline anisotropy, K b but also on the contribution from surface atoms. For spherical NPs, the effective anisotropy constant can be described as follows: where d and K s represent the particle diameter and surface anisotropy constants, respectively, and 6/d is the ratio of the surface area to the volume of a spherical nanoparticle. 41,42 Until now, the K eff was extracted from the shi of the blocking temperature using two different approaches, either from the real part of the susceptibility (AC susceptometry) or from IRM experiments, [43][44][45][46][47][48][49] and the obtained K s was attributed to the material. However, the assumption that the surface anisotropy constant is size-independent for samples prepared with the same synthesis method has never been proven experimentally. In contrast, magnetic small-angle polarized neutron scattering (SANS) studies of magnetic NPs uncovered that spin textures of NPs can be complicated, having uniform and different nonuniform, canted, or core-shell-type congurations. [50][51][52][53][54][55][56][57][58][59][60][61][62][63] Recently, a study by Zákutná et al. spatially resolved for the rst time the surface disorder energy within one NP-size batch using half-polarized SANS (SANSPOL). 64 The disorder energy, E dis is dened as where M z (H), V H mag max , and V mag (H min ) are the longitudinal magnetization at the applied eld, and magnetized volumes at H max and H min , respectively. Aerward, they accessed the effective anisotropy constant according to the following equation: where vE dis and vV mag are the derivatives of the disorder energy and magnetic volume, respectively. From the effective anisotropy constant, the spatially resolved surface anisotropy constant, K S , can be obtained: Thus, with the aim to validate experimentally eqn (1), according to which the surface anisotropy constant does not depend on the size of the material, using SANSPOL experiments and the analysis proposed by Zákutná et al., 64 we studied the surface spin disorder in spherical CoFe 2 O 4 NPs, prepared with the solvothermal method, having different coherent domain sizes of 3.1(1), 6.3(1), and 8.6(1) nm and narrow size distribution of s log < 20%. To isolate the effect of the size, we selected samples having the same chemical composition, crystalline and particle shape, and without interparticle interactions. For the rst time, we prove that albeit the energy needed for the polarization of the spins into the eld direction dramatically changes with NP size, the surface anisotropy constant remains the same in this size range. Moreover, we show that even applying the saturation eld, not all the spins at the NP surface are polarized along the eld direction, and the degree of polarization depends on the total crystalline and NP size. This study addresses, for the rst time experimentally, open fundamental questions about the surface anisotropy constant and progresses the design and characterization of magnetic NPs for their future technological and biomedical applications.  65 The metaloleates were prepared according to Repko et al. 66 An aqueous solution of Co(II) and Fe(III) nitrates in a ratio of 1 : 2 was added to a solution of sodium oleate, obtained by mixing sodium hydroxide with oleic acid in water and ethanol mixture (1 : 1). Aerwards, 20 mL of hexane was added, and the mixture was kept at the reux temperature for 1 h. Then, the reaction solution was cooled down to room temperature. The water phase was removed, and the organic phase was mixed with 20 mL of water, 5 mL of ethanol, and 5 mL of hexane and reuxed for another 30 min to remove the inorganic residuals. This step was repeated twice. Finally, the water phase was discarded from the mixture, and the remaining organic solvents were evaporated. The viscous metal-oleate product was then dissolved in 15 mL of pentanol. Cobalt ferrite NPs were synthesized by a solvothermal method of the metal-oleate complex at 220 or 180°C for 10 h in a stainless-steel autoclave (Berghof DAB-2). The details of the synthesis are reported in the Table S1. † Aer synthesis, the NPs were rst separated from the solvent with a permanent magnet. Then, NPs were washed twice using 10 mL of hexane and 10 mL of water in ultrasound bath with a magnet separation to remove the residue of oleic acid and organic solvents. Aerward, the NPs were dispersed in 5 mL of hexane and centrifuged at 3000 rpm for 5 min to remove agglomerates. Only the stable part of the NP dispersion was kept for further characterization. The sample names are labelled as S3, S6, and S9 sample, where the number indicates coherent domain size and letter S the spherical morphology.

Electron microscopy
Transmission electron microscopy (TEM) measurements were done at the Tecnai G2 Spirit microscope from FEI operating at 120 kV, and high-resolution scanning TEM (HRSTEM) measurements were carried out on JEOL NEOARM 200 F operating at 200 kV equipped with Schottky-FEG cathode and C S corrector, respectively. In both techniques, the toluene dispersion of NPs was dropped at the Cu grid with 400 mesh coated by carbon foil. The TEM micrographs were taken in bright-eld mode, and the size distribution was made by manually measuring at least 200 particle sizes from different TEM micrographs in ImageJ soware. 67 Acquisition of HRSTEM micrographs was made in annular bright (ABF) and dark-eld (ADF) mode.

Powder X-ray diffraction
Powder X-ray diffraction (PXRD) measurements were performed at the Panalytical X'pert Pro diffractometer equipped with Cu K a radiation and a secondary monochromator. The data were measured in powder form at the glass holder. The data analysis was done by Rietveld renement with implemented spherical harmonics function in the WinPLOTR soware within the FullProf soware package. 68 The instrumental broadening contribution of the diffractometer was extracted from the standard measurement of LaB 6 from NIST. Within the Rietveld analysis, the spherical harmonics function 69 describing the preferred orientation of crystallites was used to obtain averaged crystallite shape. The coherent domain size and shape were visualized by the GFourier program within the FullProf. 68

Small-angle X-ray scattering
Small-angle X-ray scattering (SAXS) experiments were performed at Xenocs Xeus 2.0 equipped with Cu and Mo K a microfocus X-ray sources, toroidal parallel beam producing Xray mirrors, two sets of beam collimating scatter-less slits and Dectris PILATUS 200K detector. The experiments were performed at a detector distance of 2.50 m using both Cu and Mo wavelengths to cover the maximum accessible Q-range. Measured 2D intensity proles were azimuthally integrated. Resulted 1D SAXS patterns were corrected to the capillary/ sample thickness, the transmission of samples, and properly scaled solvent and capillary signal was subtracted.

Small-angle neutron scattering
Half-polarized small-angle neutron scattering (SANSPOL) was carried out at the D33 instrument at the Institut Laue-Langevin, Grenoble, France. 70 The experiments were done in two instrument congurations with 2.8 and 7.8 m collimation and 2 and 7.8 m detector to sample distance, respectively. Due to the small size of the S3 sample, SANSPOL data were collected only at high Q-values. The neutron beam of the wavelength of 5 Å with the wavelength spread of Dl/l = 10% was used. The aperture of the rectangular shape and dimension of 5 × 7 mm was used to focus the neutron beam on the sample. At the sample, the magnetic eld up to 1.34 T was applied (other applied magnetic elds: 0.7, 0.1, 0.05, 0.03, 0.02, and 0.01 T) in the horizontal direction perpendicular to the neutron beam. The data were corrected to the background scattering, blocked beam, and polarization efficiency of the V-shaped polarizer (0.91) and ipper (0.99). The scattering intensity was transformed to the absolute scale by measuring the intensity of the empty beam. The pure nuclear scattering cross-section was extracted by sector averaging (with a total opening of 20°) in the magnetic eld direction (horizontal) at the saturating magnetic eld (1.34 T). The nuclear-magnetic scattering cross sections were obtained from sector averaging (with a total opening of 20°) perpendicular to the applied magnetic eld (vertical) for all used external magnetic elds.

Macroscopic magnetization
Magnetization measurements were performed in a physical property measurement system (PPMS) from Quantum Design (QD) equipped with Vibrating Sample Magnetometry (VSM) module. The measurements were done on powder samples enclosed in capsules with epoxy glue to prevent the physical rotation of the grains in the magnetic eld. Temperature dependence of magnetization was measured under zero-eld cooled (ZFC), and eld-cooled conditions to 2 K, and the magnetization response of the samples were collected while warming the samples from 2 K to 350 K with an applied eld of 50 mT. The isothermal magnetizations were collected at several temperatures (10 K, 50 K, 60 K, 100 K, 298 K, and 350 K depending on the sample) in the range of the applied magnetic eld of ±7 T.

Micromagnetic simulations
The micromagnetic simulations were performed using mumax3. 71 The simulations were based on single NPs of cubic computational cells of 1 nm, both uniaxial and cubic anisotropies were considered in separate simulations.

Thermogravimetric analysis
Thermogravimetric analysis (TGA) was carried out at the was performed on SETARAM SETSYS evolution 1750 in the nitrogen atmosphere (40 mL min −1 ). Powder samples (3-10 mg) were placed into a 100 mL alumina crucible and mounted on a Pt/Rh DSC rod. Samples were heated up to 800°C with a heating rate of 5°C min −1 .

Inductively coupled plasma optical emission spectroscopy
Inductively Coupled Plasma-Optical Emission Spectrometry (ICP-OES) was performed using an Agilent 5110 device. The calibration line was performed in the range 1-10 mgL −1 at wavelengths 228.615 nm for cobalt and 238.20 nm for iron. The samples were prepared for the analysis as follows: 1 mL of H 2 O 2 and milliQ water were added to approximately 6 mg of the powder sample under magnetic stirring to decompose the organic shell. When the formation of the foam was not observed anymore, we added 6 mL of HCl and 2 mL of HNO 3 . Then, the solution was heated at 80°C for 2 h and ltered in a volumetric ask.

Structure, morphology, and chemical composition
To reveal precisely the size dependence of the surface anisotropy constant, differently sized cobalt ferrite NPs of the same shape and reasonable size distribution were synthesized with the same protocol, according to Sanna et al. 65 The precise stoichiometry of Co 0.96 Fe 2.03 O 4 was conrmed by inductively coupled plasma optical emission spectroscopy for all samples. Furthermore, all samples consist of spherically shaped NPs with different mean particle sizes 4.4(1) (S3 sample), 8.1(1) (S6 sample), and 11.1(1) nm (S9 sample) having a reasonably narrow size distribution of s log < 20% determined from SAXS (Table S2 †), which is in good agreement with the TEM results ( Fig. S2 and Table S2 †). Moreover, using the Rietveld analysis with implemented spherical harmonic function, we obtained a spherical shape of the coherent domain size for all the samples, as reported in the inserts in Fig. S1 and Table S2. † The Rietveld analysis revealed coherent domain sizes of 3.1(1), 6.3(1), and 8.6(1) nm for S3, S6, and S9 samples, respectively, which are signicantly smaller than their physical particle size. This disagreement between crystallite size and particle size is due to the structural disorder or lack of crystallinity at the NP's surface conrmed also by HRSTEM (Fig. S3 †), leading to a possible presence of surface effects. It must be stressed that the exact stoichiometry, narrow size distribution, and crystalline and physical shape make the set of samples suitable for studying merely the effect of the physical size on the surface anisotropy constant. Furthermore, we extracted pure nuclear scattering cross-sections at the saturation eld of the sample from SANSPOL experiments to retrieve additional information on the NP's nuclear morphology. The pure nuclear scattering cross-sections ( Fig. 1) were described using the core-surfactant model, wherein the core represents the inorganic particle size, and the surfactant corresponds to the oleate (OA) ligand shell. The core particle sizes were xed according to the obtained SAXS results (r nuc ), and only the OA shell thickness (d OA ) was rened (with xed r OA = 0.78 × 10 −7 Å −2 ), leading to the thickness of d OA = 1.3(1) nm for all samples (as reported in Table S2 †), in agreement with previously published results. 64,72 Moreover, the presence of the Guinier plateau at the low Qranges in nuclear SANS cross-sections indicates that the dispersions consist of non-interacting NPs.

Averaged macroscopic magnetic properties
From zero-eld-cooled and eld-cooled curves of samples ( Fig. S4 †), the typical shi of the blocking temperature to higher values with increasing the physical size of NPs is obtained (see further description in the ESI, Table S6 †). From isothermal macroscopic averaged magnetization measurements, bimodal magnetic dipole distribution was obtained by numerical inversion method (Fig. S5 †), supporting the possible presence of spin disorder but also interparticle interactions that are commonly present in powder samples. However, one broad monomodal magnetic moment distribution can describe these bi-modal distributions. The averaged dipole magnetic moment of 3.35  (1730 m B ) was obtained for S3, S6, and S9 sample, respectively. It has to be mentioned that samples were measured in powder form, and thus results might be inuenced by interparticle interactions as well as spin disorder effects. Moreover, to support our experimental observations, we performed micromagnetic simulations using mumax3 (Fig. S6-S8 †). 71

Effective and surface anisotropy
To unveil surface phenomena in the prepared set of samples, the magnetic morphology of single NP has to be resolved with the spatial resolution. Therefore, magnetic SANS experiments were done on the NPs dispersion with low concentrations (1-3 mg mL −1 ) to avoid the effects of the interparticle interaction on the NP's magnetic properties. 73 The magnetic morphology of the samples was revealed by the core-shell-surfactant form factor renement of the vertical sectors (90°to the applied eld with 20°opening) of the magnetic-nuclear cross-sections (I − Q , I + Q ) at different applied magnetic eld strengths (Fig. 2, Table S3-S5 in the ESI †).
In this model, we dene the magnetized part of the NPs as the core (r mag ), while the shell is the non-magnetized part, named the disordered layer (d dis ), which allows us to unveil the presence of possible surface phenomena. At the maximal applied magnetic eld (H max = 1.34 T), all samples show a smaller magnetic radius than the physical size, conrming the presence of the layer without magnetization. Nevertheless, the amount of the magnetized volume decreases with increasing NP size. The smallest NPs (S3 sample) are almost fully magnetized with only 7.6(2)% of non-magnetized part, while for larger particles, the disordered volume fraction increases. Indeed, the largest NPs (S9 sample) have a non-magnetic volume of 25.9(8)% (Table 1). The magnetic scattering length density, r mag , increases with the applied magnetic eld due to the alignment of the magnetic moment of the NPs along the eld, supporting the previous work of Zákutná et al. 64 Moreover, a different degree of dependence of the magnetized volume and r mag with the applied magnetic eld is observed for different NPs sizes (Table 1). These results report the evidence of the size dependence of the surface spin disorder.
The obtained eld-dependence of r mag and d dis is visualized in Fig. 3. At the lowest applied eld (H min = 10 mT), the samples show the size of magnetized NPs close to the coherent domain size (r XRD ) with a signicant non-magnetic volume fraction (4 spin (H min )) between 40-50% depending on the sample (Table  1), which reduces gradually with an increasing magnetic eld. Nevertheless, while the smallest sample with a coherent domain size of 3.1(1) is almost fully magnetized at H max with only 7.6(2)%, the larger samples still have signicant nonmagnetic volume (up to 25.9(8)% for the S9 sample). This is understood as the smaller particles are easier to magnetize, which is directly related to the energy needed to polarize spins into the eld direction, known as disorder energy. These results are in excellent agreement with macroscopic magnetization measurements, whereby the non-regularized numerical inversion approach 74 of the magnetization curve at the 298 K, two populations of the magnetic moment were obtained corresponding to the core and shell (Fig. S5 †). Moreover, the averaged magnetized radii of 2.31(1), 3.96(1), and 4.91(2) nm were extracted for S3, S6, and S9 samples, respectively, representing the same total magnetized volume as received from SANSPOL at the highest applied magnetic eld. Furthermore, to obtain more in-depth information on the surface disorder, we calculated the disorder energy from SANSPOL results according to Zákutná et al., 64 using eqn (2). The longitudinal magnetization is directly proportional to the decrease of the r mag :    64 To support the phenomenological equation (eqn (1)), we spatially resolve the surface anisotropy constant according to Zákutná et al., 64 K S = K eff × r mag /3, independently for each NP's size. As reported in Table 2, we obtained the spatially resolved K S = 6 mJ m −2 for each sample, conrming that the surface anisotropy constant does not depend on the NPs size in the case of the ideal batch of samples, i.e. where the composition, coherent domain shape, and physical morphology is same. We also provide the volume averaged disorder anisotropy, hK s i, calculated as hK s i = E dis (-H max )/V nuc × r nuc /3, by considering the whole nuclear volume of NPs from the disorder energy at the highest applied magnetic eld (Table 2). Obtained hK s i values are in good agreement with previously reported values of surface anisotropies of different ferrite NPs. 46,48,75,76 These results experimentally support the theoretical description, according to which the surface-tovolume ratio does not inuence the surface anisotropy, however, the exchange interactions lead to reduced magnetic particle size compared with the structural size. 46,48,75,76

Conclusions
Within this work, we prove experimentally that the classical approach of extracting the averaged surface anisotropy constant, as interpolation of the blocking temperature shi from AC susceptometry data using different NP size batches, is entirely correct and reasonable. However, it is worth underlying that this approach should be applied only if the NPs have been prepared with the same synthesis method and possess the same chemical composition, polydispersity, and physical and crystallite shape. Furthermore, we demonstrate that NP's size increases the energy needed to polarize spins into the eld direction. These observations are crucial in designing the new materials with controlled surface disorder necessary for the improved magnetic NP applications in magnetic hyperthermia, drug delivery, data storage, and the development of rare element-free permanent magnets.