Lattice Parameters and Site Occupancy Factors of Magnetite-Maghemite

research papers

Journal of Applied Lattice parameters and site occupancy factors of Crystallography magnetite–maghemite core–shell nanoparticles. ISSN 1600-5767 A critical study

Received 3 June 2014 a b c Accepted 2 September 2014 Antonio Cervellino, Ruggero Frison, Giuseppe Cernuto, Antonietta Guagliardib,c* and Norberto Masciocchic

aSwiss Light Source, Paul Scherrer Institut, 5232 Villigen, Switzerland, bIstituto di Cristallografia and To.Sca.Lab, Italian National Council of Research, via Lucini 3, Como 22100, Italy, and cDipartimento Scienza e Alta Tecnologia, Universita` dell’Insubria, via Valleggio 11, Como 22100, Italy. Correspondence e-mail: [email protected]

The size-driven expansion and oxidation-driven contraction phenomena of nonstoichiometric magnetite–maghemite core–shell nanoparticles have been investigated by the total scattering Debye function approach. Results from a large set of samples are discussed in terms of signiﬁcant effects on the sample average lattice parameter and on the possibility of deriving the sample average oxidation level from accurate, diffraction-based, cell values. Controlling subtle experimental effects affecting the measurement of diffraction angles and correcting for extra-sample scattering contributions to the pattern intensity are crucial issues for accurately estimating lattice parameters and cation vacancies. The average nanoparticle stoichiometry appears to be controlled mainly by iron depletion of octahedral sites. A simple law with a single adjustable parameter, well correlating lattice parameter, stoichiometry and size effects of all the # 2014 International Union of Crystallography nanoparticles present in the whole set of samples used in this study, is proposed.

1. Introduction ments, a partial or complete oxidation in which iron ions Magnetic iron oxide nanoparticles (NPs) are materials of great diffuse outwards and generate vacant sites in the crystal interest in many technological fields (high-density storage lattice. Oxidation usually results in nonstoichiometric media, ferrofluids, electronic devices, catalysis; see, for magnetite–maghemite core–shell NPs (Fe3O4, < 1/3), or example, Reddy et al., 2012; Tang & Lo, 2013); owing to their can be driven to completion ( = 1/3) forming pure maghemite, biocompatibility, they are also attractive materials for with iron vacancies either randomly distributed (space group different biomedical applications (magnetic resonance Fd3m) or partially (cubic P4132; Shmakov et al., 1995) or imaging, hyperthermia, targeted drug delivery; see, for totally ordered (tetragonal P41212; Greaves, 1983; Jørgensen et example, Laurent et al., 2008; Xu & Sun, 2013). Here we focus al., 2007). Even when vacancy ordering phenomena can be on magnetite (Fe3O4) NPs and on their oxidized counterpart, neglected, assessing the degree of oxidation or, in other words, maghemite (-Fe2O3 or, equivalently, Fe2.67O4). In the form of the core–shell magnetite–maghemite ratio (see Salazar et al., NPs, these two iron oxide materials exhibit super-para- 2011; Frison, Cernuto, Cervellino, Zaharko et al., 2013) in NPs magnetic properties as a rather complex function of compo- of a few nanometres only is not an easy task. Accordingly, sition, structure and surface modifications, which depend on different experimental techniques are usually combined to both NP size and oxidation state. Despite the wide literature derive details about (i) the crystal structure, sample purity and available on this topic, such complexity makes some funda- lattice parameters (X-ray or neutron powder diffraction), (ii) mental (structural and functional) aspects still unclear. Among the NP size distribution (transmission electron microscopy), these falls the reliable evaluation of size-dependent oxidation and (iii) the sample oxidation state (Mo¨ssbauer or other degree and surface effects, particularly in NPs well below 20– spectroscopies). A deeper survey of the pertinent literature 30 nm and, intimately related, the dependency of the lattice shows a relatively large, and unexplained, spread of the lattice parameter and the extent of cation vacancies on both effects. parameters, in particular for maghemite NPs, that, in our Magnetite crystallizes in the so-called inverse spinel struc- opinion, required further investigation. Moreover, differently ture (Fd3m), with O atoms forming a closed-packed cubic from the bulk, where iron vacancies are mainly located at the lattice (Wyckoff position 32e), Fe3+ ions located in the tetra- octahedral site (Armstrong et al., 1966; Haneda & Morrish, hedral sites (Wyckoff position 8a), and a 1:1 mix of Fe2+ and 1977), vacancies occurring in both the octahedral and the Fe3+ ions filling the octahedral sites (Wyckoff position 16d). tetrahedral sites of cubic NPs are reported on the basis of The high instability of Fe2+ in air causes the NPs to undergo, as Rietveld-based (Jørgensen et al., 2007) and pair distribution a function of the particle size and the post-synthesis treat- function (Petkov et al., 2009) analyses.

J. Appl. Cryst. (2014). 47, 1755–1761 doi:10.1107/S1600576714019840 1755 research papers

In a recent paper on iron oxide NPs (Frison, Cernuto, NPs from any standard modern powder diffractometer) might Cervellino, Zaharko et al., 2013) we proposed a new total be a serious limitation to the possibility of deriving the sample scattering approach (Cervellino et al., 2010) relying on the stoichiometry from the lattice parameter in a reliable way. Debye (1915) equation. We demonstrated that, despite the Indeed, a cell parameter variation Áa =0.01A˚ corresponds to very close similarity of magnetite and cubic maghemite in an absolute Á(h3 i) value of 0.07, turning into a relative terms of crystal structures and lattice parameters [aMagnetite = stoichiometry error larger than 20%. Note that, for our set of ˚ ˚ 8.3967 (3) A (Bosi et al., 2009) and aMaghemite = 8.3457 A (s.u. samples, we derived stoichiometry values with estimated not available; Shin, 1988)] and the peak broadening due to the standard deviations in the range 0.004–0.02 (Frison, Cernuto, very small crystal domains, the size-dependent lattice para- Cervellino, Zaharko et al., 2013). In view of these results, meter and stoichiometry of core–shell magnetite–maghemite understanding whether relatively large fluctuations may NPs can be simultaneously determined from their synchrotron derive exclusively from sample features rather than from X-ray diffraction patterns and show a very good linear mutual (additional) inaccurate estimates of the lattice parameters correlation. Moreover, the number- and mass-based average (which might easily originate when peaks are broad and crystal domain size and the size distributions were also esti- necessary angular corrections are not applied) appears an mated. Additionally, performing the same analysis on important issue to be thoroughly investigated. laboratory data (Frison, Cernuto, Cervellino, Guagliardi & Accordingly, in this paper, we focus on size-driven expan- Masciocchi, 2013) collected in reflection geometry on a subset sion and oxidation-driven contraction phenomena and discuss of the available samples, comparable lattice parameters (as the quantitative evaluation of their influence on the average long as data were corrected for sample displacement effects) – lattice parameters of nonstoichiometric magnetite–maghemite but incompatible sample stoichiometry values – were core–shell NPs. Moreover, we also investigated the iron obtained, though fluorescence contributions were limited at vacancy distributions between the octahedral and tetrahedral the best of the detector’s capability. sites, determined for the first time within the total scattering According to these findings, can we use the (diffraction- Debye-equation-based approach. Owing to the important role based) accurate average lattice parameter of small iron oxide played in the DFA, details about the data pre-processing and NPs to derive a reliable estimation of their average stoichio- the DFA modelling are also presented, in xx2 and 3, respec- metry? While this is very likely to hold for single particles (or tively. Finally, we propose a simple law with a single adjustable monodisperse samples with constant stoichiometry), for parameter, which is well correlated with the lattice value, samples with relatively broad size distribution the answer is stoichiometry and size effects of all the NPs present in the not straightforward. Indeed, within our study (Frison, whole set of 30 samples used in this study. Cernuto, Cervellino, Zaharko et al., 2013), we found that the smallest NPs (typical sizes <5 nm) of the investigated samples 2+ 3+ (obtained by co-precipitation of Fe and Fe chlorides in 2. Angular and intensity corrections of magnetite– aqueous media) showed a clearly measurable size-dependent maghemite nanoparticles lattice expansion. Such an effect acts against the lattice Iron oxide NPs were prepared according to suitable modifi- contraction driven by size-dependent oxidation phenomena cations of the co-precipitation literature protocol (Massart, and – to the best of our knowledge – has never been detected 1981) as reported elsewhere (Frison, Cernuto, Cervellino, before on this kind of NP. We believe that this might have Zaharko et al., 2013). The synthesized samples were grouped caused the wide spread of the lattice parameters reported in with labels A–G, depending on the synthesis details (synop- the literature, where, for -Fe O , values range from 2 3 tically reported in Table S1 of the supporting information1). 8.334 (2) A˚ (Petkov et al., 2009) to 8.3457 (–) A˚ (Shin, 1988), Diffraction data were collected at the MS-X04SA beamline and the much narrower 8.394 (4)–8.400 (–) A˚ interval (– (Willmott et al., 2013) of the Swiss Light Source synchrotron indicates that s.u. values are not available; Gorski & Scherer, facility at the Paul Scherrer Institut, from samples loaded in 2010; Gotic et al., 2009) is attributed to Fe O NPs. While at 3 4 0.3 mm glass capillaries, using a Debye–Scherrer geometry the microscale such small variations might be neglected, at the and 15 keV (or 16 keV) radiation, a partial He beam path, and nanoscale they become highly important since they act as a Mythen II detector covering 120 with 0.0036 resolution possible indicators of surface effects (Palosz et al., 2003, 2010; (Bergamaschi et al., 2010). Before performing the DFA, Qi & Wang, 2005), as also disclosed by our Debye function following an approach similar to that proposed by Ritter et al. analysis (DFA) model on iron oxide NPs. Diehm et al. (2012) (1951) and Paalman & Pings (1962), angle-dependent intensity have recently reviewed this topic and found that, in a nearly corrections were applied to the raw data to account for sample systematic manner, small metal oxide NPs tend to slightly absorption effects, and (absorption-corrected) air and capil- expand (typically, but not exclusively, in the 0.1–0.4% range) lary scattering contributions, measured independently, were because of negative surface stress (within the frame of the subtracted (see Fig. 1a). Absorption correction curves were capillary pressure model). Therefore, for compounds like estimated by measuring the transmitted beam intensities from those here investigated, absolute lattice inaccuracies as large the filled capillary, while for the empty capillary the X-ray as 0.01 A˚ (i.e. smaller than the difference between Petkov’s and Shin’s reported parameters and about one order of 1 Supporting information for this paper is available from the IUCr electronic magnitude larger than the error presently achievable on small archives (Reference: AJ5238).

1756 Antonio Cervellino et al. Magnetite–maghemite core–shell nanoparticles J. Appl. Cryst. (2014). 47, 1755–1761 research papers attenuation coefficients (http://physics.nist.gov/PhysRefData/ where Q =2q, q =2sin/ is the length of the reciprocal FFast/html/form.html) computed from the nominal glass scattering vector, is the radiation wavelength, fj is the atomic composition (Hilgenberg GmbH 0140 and 0500) were used. form factor, Tj and oj are (adjustable) atomic Debye–Waller As an example, the curves for sample B4 within the capillary and site occupancy factors (SOFs), dij is the interatomic and for the glass capillary in filled and empty conditions are distance between atoms i and j, and N is the number of atoms shown in Fig. 1(b). In addition, using a certified silicon powder in the NP.

(NIST 640c) as an external standard, zero angle and x, y Atomistic models of pure Fe3O4 nanocrystals of spherical capillary offsets to the 2 values were also applied. Fig. 1(c) shape were generated following a concentric shells model to shows the angular and intensity-corrected pattern of sample build up a discrete population of NPs of increasing size, Ár,of B4 superimposed on the corresponding raw data; the angular subsequent spheres being equal to 0.328 nm [details are given shift of the 111 and 220 reflections of silicon 640c is apparent by Cervellino et al. (2010)]. The long-known problem of in the inset. In-house-developed (unpublished) numerical extremely heavy computing time required by the Debye algorithms and programs were used for all steps of these data equation was solved by sampling the interatomic distances pre-processing treatments. according to the algorithm described by Cervellino et al. (2006). After calculating and storing the set of sampled distances of each nanocrystal, the Debye pattern model of 3. The Debye function analysis of magnetite– each sample was calculated and further adjusted to the maghemite nanoparticles experimental one by optimizing the following: 3.1. The DFA modelling (i) The nanocrystals’ size distribution, assuming a lognormal In order to investigate the variability of the lattice para- function, the average (hDi) and standard deviation ()of meters and of the cation vacancies in magnetite–maghemite which were varied. The adoption of this function is justified by NPs with their size-dependent oxidation we used a modelling experimental observations (as reviewed by Granqvist & approach relying on the Debye (1915) equation in the Buhrman, 1976) and several theoretical models developed for formulation recently proposed by Cervellino et al. (2010): single-process-driven NP syntheses (Smoluchowsky, 1917; So¨derlund et al., 1998; Kiss et al., 1999, and references therein; PN 2 2 Ali-zade, 2008). IðQÞ¼ fjðQÞ oj j¼1 (ii) The size-dependent lattice parameter using an inverse PN linear function with two refined coefficients: þ 2 fjðQÞfiðQÞTjðQÞTiðQÞojoi sinðQdijÞ=ðQdijÞ; ð1Þ j>i¼1

Figure 1 (a) Raw synchrotron data collected on sample B4 (red line) and independently measured scattering patterns of the empty glass capillary (green line) and air (blue line). (b) Calculated absorption corrections for the iron oxide powder of sample B4 within the capillary (red line) and for the glass capillary in ﬁlled (green line) and empty (blue line) conditions. (c) Diffraction pattern of sample B4 (blue line), after 2/absorption corrections and glass/air contribution subtraction, with the original raw pattern (red line) superimposed. The inset shows the effects of the angular correction on the 111 and 220 peaks of the standard reference NIST silicon 640c. (d) Best ﬁt (continuous red line) of the synchrotron data (blue dots) of sample B4 as provided by the Debye function approach described in the text. The green trace describes the amorphous-like component (two-line ferrihydrite) modelled by scaling the experimental pattern of a sample synthesized for the purpose and collected under the same experimental conditions. Additional phenomenological contributions have not been added to the model.

J. Appl. Cryst. (2014). 47, 1755–1761 Antonio Cervellino et al. Magnetite–maghemite core–shell nanoparticles 1757 research papers

XsðnÞ¼! þð! Þðn0 þ 1=2Þ½1 ðn0 þ 3=2Þ=ðn þ 1=2Þ; distribution (solid blue line) or the size fraction >10 nm only ð2Þ (dotted line), to exclude the contribution of the surface relaxation effects. We then compared our results with those where n and n0 indicate cluster shell numbers and with Xs(n0)= collected in a recent paper (Gorski & Scherer, 2010), corre- and Xs(n0 +1) = !. lating the cell values (from laboratory powder diffraction (iii) The SOFs of Fe in the octahedral site [Fe(oct)], data) to the sample stoichiometry derived by independent according to the following law: Mo¨ssbauer spectroscopy of partially oxidized magnetite NPs of D ’ 20 nm (red points and solid red regression line). OðDÞ¼O1 þðO0 O1Þ expðD=OLÞ; ð3Þ Following the original paper, additional points from other where O0, O1 describe the SOF at the smallest and largest studies were also inserted (Feitknecht et al., 1962; Gotic et al., sizes, respectively, and OL the SOF variation length. 2009; Schmidbauer & Keller, 2006; Annersten & Hafner, 1973; (iv) The size-dependent isotropic Debye–Waller factor of Yang et al., 2004; Volenı´k et al., 1975); the dotted red regres- each atom, according to a three-coefficient law similar to the sion line was then obtained by considering the whole set of one used for SOF. points, although most of the newly added points refer to Parameter optimization was performed through the simplex significantly larger particles (micrometre-sized samples or NPs method (Nelder & Mead, 1965). Using a single lattice para- 150–200 nm large). The blue and red lines clearly indicate that meter for a core–shell NP is here justified by the close simi- meaningful cell deviations are obtained for the two sets of larity of either the end member ao values or their bulk moduli (see x4), as confirmed by the lack of evident peak broadening attributable to strain. Thanks to the high quality of ‘clean’ data, an additional component (a disordered iron oxy- hydroxide), with a scattering trace very similar to that of the two-line ferrihydrite (Drits et al., 1993), was found in variable amounts (15–30% in weight) in all samples. The experimental diffraction pattern collected on a synthetic sample of two-line ferrihydrite (G1) well described such a component and therefore was used as a blank curve added to the total pattern model and scaled by linear least squares. Results about the average NP size and width, lattice parameter, and stoichiometry of all samples are synoptically collected in Table S1 of the supporting information. For one of the investigated samples, the best fit is shown in Fig. 1(d). Additional information about the core–shell magnetite–maghemite ratio and the magnetic properties of the same samples can be found elsewhere (Frison, Cernuto, Cervellino, Zaharko et al., 2013). In addition, a few of the investigated samples are here further processed to study the iron vacancy distribution by simultaneously refining the SOFs of both the tetrahedral and the octahedral sites (see x3.3).

3.2. About the lattice parameters Despite the fact that the opposite trend would be expected on sample oxidation, a lattice expansion phenomenon was observed over the whole set of samples and particularly in Figure 2 those showing (mass-based) average sizes of around 10 nm (a) Average lattice parameter versus average sample stoichiometry h3 (about 80% of the total number); this behaviour was inter- i (blue dots); horizontal and vertical bars describe the corresponding preted as the effect of a surface relaxation, which is particu- error. The abscissa is here defined as h3 i, which represents the iron content in Feh3iO4 (h3 i = 2.667 for pure maghemite and h3 i =3 larly significant below 5 nm (see Fig. S1 of the supporting for pure magnetite). Regression lines were obtained by least-squares information). In view of these findings we wondered whether methods using the full size distribution (continuous blue line, orthogonal such a systematic size effect on the NPs’ average lattice least squares) or the NP fraction with sizes D > 10 nm only (dotted blue parameter might explain the large variability of values line). Red points and the associated regression line refer to NPs of similar size from Gorski & Scherer (2010); according to the original paper, the reported. Weighing the (size-dependent) cell of each nano- dotted red line is obtained by considering new points from other work. crystal according to the sample NPs’ (mass-based) size (b) Regression (solid) and 1 (dotted) lines computed using the full size distribution, we extracted the average lattice parameter and distribution of this work (Fit 1, blue lines) and that of Gorski and Scherer the average oxidation level h3 i of each investigated sample (Fit 2, red lines); the abscissa is here defined as x =3(h3 i) 8(x =0 for pure maghemite and x = 1 for pure magnetite) or, in other words, in (blue dots in Fig. 2a) and evaluated the least-squares line by the y = mx + q formulation, q is the 100% maghemite cell and (m + q)is orthogonal regression while considering either the full size NP the 100% magnetite one.

1758 Antonio Cervellino et al. Magnetite–maghemite core–shell nanoparticles J. Appl. Cryst. (2014). 47, 1755–1761 research papers nanosized samples (this work and Gorski’s), Table 1 particularly at low and medium sample oxida- Refined SOFs of Fe ions at the octahedral [Fe(oct)] and tetrahedral [Fe(tet)] sites (Wyckoff positions are 8a and 16d, respectively), average lattice parameter (hai ), and tion degrees (Áa ’ 0.01 A˚ is measured at h3 M stoichiometry (h3 iM) of samples A1 and A5 (case a) are compared with the values i = 2.75), while the tendency to convergence at obtained by taking the [Fe(tet)] sites fully occupied (SOFFe(tet) = 1) (case b). h i the largest 3 values is observed, probably Isotropic Debye–Waller thermal factors in case a were kept fixed to the values refined in case b. thanks to the larger size of slightly (or not at all) oxidized samples (D > 15 nm within our speci- Sample SOFFe(tet) SOFFe(oct) BFe(tet) BFe(oct) BO haiM h3 iM mens), whose NP fraction (<5 nm) is small or A1 (a) 0.985 (7) 0.882 (8) 0.452 0.936 0.365 8.362 (4) 2.75 (2) totally absent. A1 (b) 1.0 0.893 (6) 0.4 (1) 0.94 (3) 0.36 (7) 8.362 (1) 2.79 (1) A5 (a) 0.976 (7) 0.887 (6) 0.550 0.938 0.464 8.364 (3) 2.75 (1) As mentioned before, the estimated standard A5 (b) 1.0 0.905 (5) 0.550 (7) 0.94 (1) 0.46 (3) 8.364 (1) 2.81 (1) errors on lattice parameters play an important -Fe2O3 † 0.865 (1) 0.901 (1) 1.14 (2) 1.14 (2) 1.9 (5) 8.4053 (1) 2.667 role and are hereafter discussed in deeper -Fe2O3 ‡ 0.90 (5) 0.80 (5) – – – 8.370 (2) 2.5 detail. Regression (1) lines from this work † From Jørgensen et al. (2007). ‡ From Petkov et al. (2009). (Fit 1) and from Gorski’s and Scherer’s paper (Gorski & Scherer, 2010) (Fit 2) are shown in Fig. 2(b); here the abscissa is defined as x = 3(h3 i) 8(x = 0 for pure maghemite and x = 1 for pure to vacancies in the octahedral site. In particular, in situ X-ray magnetite) or, in other words, in the y=mx+ q formulation, q synchrotron data of -Fe2O3 NPs grown from an amorphous is the 100% maghemite cell and (m + q) is the 100% magnetite precursor (Jørgensen et al., 2007), analysed by the Rietveld one. Fit 1 refers to the whole population of NPs, the dashed (1969) method and with the cubic Fd3m structure, indicated (1) lines highlighting the significance of the small q value that the majority of the vacancies occur at the tetrahedral site changes. Considering also the q changes corresponding to the [SOFFe(tet) = 0.865 (1)] rather than at the octahedral one NP fraction >10 nm (not shown here for the sake of clarity), [SOFFe(oct) = 0.901 (1)]. An unexpected (hyper-magnetite- the extrapolated ‘maghemitic’ cells for the samples here like) lattice parameter of 8.4053 (1) A˚ was also reported for investigated range between 8.3479 (6) and 8.3449 (2) A˚ , this structure. Another example is given by Petkov et al. respectively, well in agreement with the reference value from (2009), in which the pair distribution function analysis of both Shin (1988) [8.3457 (–) A˚ ]; analogously, the extrapolated spherical and tetrapod-shaped iron oxide NPs (prepared ‘magnetitic’ cell of 8.394 (1) A˚ closely matches that from Bosi through thermal decomposition of organometallic precursors et al. (2009) [8.3967 (3) A˚ ]. Our results seem to indicate a clear in the presence of surfactants) provided even smaller SOFs of linear trend of the lattice parameter versus the sample both cation sites (well below the = 1/3 limit). Stimulated by oxidation degree within the range defined by the two these results, we carried out a new DFA for some of our mentioned reference cell values. Surface relaxation samples by simultaneously refining both Fe sites’ SOFs along phenomena can determine small deviations from the refer- with the lattice parameter. In order to limit the total number ence maghemite cell; however, they become relevant when the of refined parameters, the isotropic Debye–Waller thermal fraction of NPs <10 nm is significant (ca 40% in weight), and in factors were kept fixed to the values obtained by the previous such a case they should be properly accounted for. Similar refinement for all atoms. The results, reported in Table 1, are effects on the magnetite cell are probably not to be expected compared with the cases in which only SOFFe(oct) was refined. since very small NPs having a pure magnetite composition are Values from Jørgensen et al. (2007) and Petkov et al. (2009) rather unstable and get easily oxidized. (spherical case) are also given. Interestingly, our model indi- Much larger q value changes are obtained in Fit 2 (red cates a rather modest number of vacancies at the Fe(tet) site, lines), well beyond the parameter precision achievable with which (if present at all) never exceeds 3–4% of the total value. modern diffractometers. Moreover, Fig. 2(b) shows that Therefore, the DFA modelling of randomly distributed differences larger than those originating from surface relaxa- vacancies seems to confirm that, when small magnetite NPs tion effects are found for magnetite–maghemite NPs with are oxidized, (most of) the cation diffusion comes from iron similar oxidation degrees and sizes. Such large deviations, if removal from the octahedral site. not resulting from uncorrected experimental effects, in the case of Gorski’s NPs might reflect (yet undisclosed) surface NP contraction. Actually, we do not expect significant contributions to lattice variations from diverse synthetic 4. Predicting size effects on the lattice parameters routes. In this study we analysed a large number of samples, each constituted of magnetite–maghemite nanocrystals with a variably broad size distribution and in different average 3.3. About the Fe vacancies oxidation states, covering almost the full range of x [as

Recent literature reports on -Fe2O3 NPs have pointed out anticipated, x =3(h3 i) 8 = 0 for maghemite and x = 1 for the formation of cation vacancies at the tetrahedral Fe site, magnetite]. For each sample, the full size distribution has been while older reports, based on diffraction and spectroscopic determined, together with the size-dependent stoichiometry methods, systematically attributed the stoichiometric changes and lattice parameter. Given the common aqueous-based

J. Appl. Cryst. (2014). 47, 1755–1761 Antonio Cervellino et al. Magnetite–maghemite core–shell nanoparticles 1759 research papers synthesis method and the ubiquitous presence of a fraction of These two effects have then been combined in equation (9), amorphous ferrihydrite, which we inferred to be present as a which includes equation (4), giving the lattice parameter for surface layer (less than 1 nm thick) covering the nanocrys- D0 !1, and equation (8b), addressing the relative variations talline oxide (Frison, Cernuto, Cervellino, Zaharko et al., due to the small particle diameter. After extracting the lattice 2013), we can reliably state that the surface conditions of all parameter and composition versus particle diameter curves of particles in all samples are nearly identical. In these cases, all samples and evaluating pointwise standard deviations, we lattice parameter variations due to stoichiometry and to fitted the whole ensemble of data to a simple global law with a particle size, whose origin is in the interplay between bulk single refinable parameter, : elastic energy and surface tension (Diehm et al., 2012), can be combined to give an overall general law. Two contributions a ¼ ðÞ1 x aMaghemite þ xaMaghemite ð1 =DÞ: ð9Þ can be easily envisaged: 3 (a) For an ideal solid solution (i.e. with ÁHmix =0,ÁHmix The excellent fit (GoF ’ 1) gave = 2.05 (21) 10 nm, being the mixing enthalpy), for which molar volumes can be with a rather uniform residual distribution. A graphical considered additive (Retger’s law), and with small differences representation is given in Fig. 3 for NP diameters of up to between the physical properties of the end members, a linear 10 nm. It is worthy of note that this law predicts an inflation of dependency of the lattice parameters on stoichiometry, x, can the cell parameter upon decreasing the NP size while keeping be assumed: constant the oxidation degree. According to Diehm et al. a ¼ð1 xÞ a þ xa ð4Þ (2012), this is a common behaviour for oxides, meaning that Maghemite Magnetite the surface tension must be negative. (as in Vegard’s law). As for the constancy of with respect to x, this assumption (b) Slightly less known is the dependency of the lattice is largely justified by the similarity of the bulk moduli (B): parameter on particle size, which follows a D1 law. A thor- 186 GPa for magnetite (Finger et al., 1986) and 203 GPa for ough and complex derivation of this law (for faceted crystals) maghemite (Jiang et al., 1998). In fact, the relative error on our is given by Diehm et al. (2012); in the following, we provide a estimate of (about 10%) agrees with the relative difference simplified derivation, based on the assumption that each of the two B values; for the sake of completeness, we have also nanocrystal (of spherical shape) behaves as an isotropic performed a similar fit assuming a linear dependence of medium. In such an ideal case, the uniform isotropic strain is a with x, which, however, did not yield better results. Therefore, scalar parameter " = Áa/a. Thus, the NP diameter, the surface we believe that the data quality must be improved by one area and the volume depend on the strain as D = D0(1 + "), S = order of magnitude to be able to detect this subtle effect. 2 3 S0(1 + ") , V = V0(1 + ") , respectively, where D0, S0, V0 are the Significantly, the similarity of the two bulk moduli explains 2 unstrained quantities and, for a sphere, S0 = D0 and V0 = why strain gradients are negligible, even if in many samples 3 D0/6. Furthermore, the surface energy change due to strain the surface is (much) more oxidized than the core. As a can be easily derived as corollary, the magnitude of the surface tension for wet- synthesized (and probably ferrihydrite-terminated) magne- ÁE ¼ ½ð1 þ "Þ2 1S ¼ ð2" þ "2ÞD2; ð5Þ s 0 0 tite–maghemite NPs can be estimated to be 1.8 to where is the surface tension, defined as the quantity of 2.0 eV nm2 using either value of the bulk modulus. energy needed to create a unit of new surface. The bulk elastic strain energy change is

2 2 3 ÁEb ¼ð9=2Þ B " V0 ¼ð3=4Þ B " D0; ð6Þ the bulk modulus B (Ashcroft & Mermin, 1976) being the only elastic constant to be considered in isotropic media. There- fore, the total energy ÁEb + ÁEs is minimal when

" ¼½4=ð3BÞ=fD0 þ½4=ð3BÞg =ðD0 þ Þ: ð7Þ As =4/(3B) (which we have assumed to be independent of x, see below) has to be very small compared to D0, we can safely state that

" ’ =D0 ð8aÞ or, equivalently,

D ¼ D0ð1 =D0Þ: ð8bÞ During this derivation, we also assumed that the bulk modulus does not change with the (very small) volume change asso- Figure 3 Two-dimensional map of the predicted lattice parameters on varying the ciated with strain. Note also that the zero-strain condition is NP stoichiometry and diameter D. Only the portion with D <10nmis satisﬁed for D0 !1, that is, for an inﬁnite crystal. shown.

1760 Antonio Cervellino et al. Magnetite–maghemite core–shell nanoparticles J. Appl. Cryst. (2014). 47, 1755–1761 research papers

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J. Appl. Cryst. (2014). 47, 1755–1761 Antonio Cervellino et al. Magnetite–maghemite core–shell nanoparticles 1761