Exploiting GISAXS for the Study of a 3D Ordered Superlattice of Self-Assembled Colloidal Iron Oxide NanocrystalsDavide Altamura,† Vaclav Holy,‡ Dritan Siliqi,† Indira Chaitanya Lekshmi,§ Concetta Nobile,§

Giuseppe Maruccio,§,∥ P. Davide Cozzoli,§,∥ Lixin Fan,⊥ Fabia Gozzo,# and Cinzia Giannini*,†

†Institute of Crystallography (CNR-IC), V. Amendola 122/O, 70126-Bari, Italy‡Faculty of Mathematics and Physics, Charles University, Ke Karlovu 5, 121 16 Prague, Czech Republic§National Nanotechnology Laboratory (NNL), CNR Istituto Nanoscienze , c/o Distretto Tecnologico, via per Arnesano km 5, 73100Lecce, Italy∥Dipartimento di Matematica e Fisica “E. De Giorgi”, Universita del Salento, via per Arnesano, 73100 Lecce⊥Rigaku Innovative Technologies (RIT), 1900 Taylor Road, Auburn Hills, Michigan 48326, United States#Paul Scherrer Institut, Swiss Light Source, 5232 Villigen PSI, Switzerland

*S Supporting Information

ABSTRACT: A three-dimensional (3D) ordered superlattice of colloidaliron oxide nanocrystals obtained by magnetic-field-assisted self-assemblyhas been studied by grazing incidence small-angle X-ray scattering(GISAXS). A new model to simulate and interpret GISAXS patterns ispresented, which returns the structural and morphological details of 3Dnanocrystal-built supercrystals. The model is applied to a sample with asuitable surface morphology, allowing the observation of “volumediffraction” even at extremely low grazing incidence angle. In thisparticular case, the average fcc-like stacking of the nanocrystals (buildingblocks), their spherical shape, and statistical information on their sizedistribution and positions within the superlattice have been safely deduced.The proposed model is expected to be amendable for the analysis of morecomplex structures and applicable to a large variety of nanocrystal-basedassemblies.

■ INTRODUCTION

Three-dimensional (3D) superlattices composed of self-assembled colloidal inorganic nanocrystals (NCs) are of greatfundamental and technological interest due to expectation ofnovel optoelectronic, magnetic, mechanical, and catalyticproperties stemming from strong interactions among close-packed nanoscale building units. Advances in the controlledfabrication of such ordered NC-made materials are expected tolead to significant progress in the fabrication of artificialmesoscopic solids and devices.1,2

The fabrication of NC superlattices, with tunable orderedarrangements and lattice parameters, primarily requires theavailability of highly size- and shape-monodisperse NCs byrefined chemical synthesis protocols3 and the development ofversatile techniques for promoting NC assembly into program-mable architectures over large scales.4,5 Parallel to thesechallenges, an essential step toward NC superlattices withrationally designed properties involves understanding how thecollective chemical−physical behavior depends on the geo-metric features of the individual NC component blocks andtheir spatial organization in the assemblies, in terms ofcorrelation distances, symmetry, and degree of order.

Scanning and transmission electron microscopy techniques(SEM, TEM) are usually employed to easily provide a directimage of nanostructured materials;6−10 on the other hand, X-ray based techniques have been widely acknowledged aspowerful tools for the study of layered nanomaterials as well as3D NC-made superlattice systems,11−21 because of the highpenetration depth and large volumes sampled by the X-rays,especially in the cases in which SEM/TEM are limited by highelectron absorption (e.g., thick NC ensembles), low contrastdue to charging effects (e.g., insulating NCs), and difficulty inretrieving the 3D structure of the assembly from 2D projectedimages via tomographic reconstruction.In the case of NC assemblies lying on a flat substrate, grazing

incidence small-angle X-ray scattering (GISAXS) can providecomplete structural characterization of the superlattice throughcollection of a 2D reciprocal-space map even in a singlemeasurement. A detailed analysis of the map can allow thesimultaneous study of the average NC morphology (size,

Received: July 27, 2012Revised: September 21, 2012Published: October 1, 2012

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shape) and the structure of the NC superlattice unit cell,provided that suitable software for data analysis is available.Although the study of 2D in-plane assemblies of particles is

already well established, and software taking into account all X-ray scattering features in the grazing incidence geometry isavailable,22,23 the modeling of X-ray scattering from 3Dassemblies has been mainly devoted and limited to thecalculation and indexing of the expected diffraction spotpositions, related to periodic structures,7,24,25 and thereforehas still to be improved if microstructural/morphological detailshave to be returned. A further step toward the fitting of thewhole GISAXS intensity is represented by the NANODIFTcode, which is based on a discrete elements calculationapproach, and has been successfully applied to nanoporousmaterials.26 It is however limited to samples with thin filmmorphology and flat interfaces.On the other hand, the theoretical modeling of (1D, 2D, or

3D) disordered nanoparticle superlattices (SLs) has beenaddressed in a comprehensive way by M. Buljan et al.27 Inparticular, the authors analyzed GISAXS patterns generated byvarious types of 3D lattices (fabricated by physical techniques,such as ion beam irradiation and magnetron sputteringdeposition), which differed for the type and degree of order.The proposed models were successfully applied to describe thestructure of sufficiently diluted assemblies, in which the sizes

and positions of the quantum dots could be assumed to bestatistically uncorrelated (decoupling approximation- DA).In the present work, we provide a methodological

contribution to the GISAXS-based investigation of dense-packed 3D assemblies, through the structural characterizationof free-standing millimeter-scale supercrystals of iron oxideNCs. In contrast to our previous work,27 here we investigatehighly ordered 3D systems (supercrystals) of nanoparticles,which yield many “diffraction maxima” in the reciprocal space.This makes it possible to test in detail the validity of the 3Dscattering model and to accurately determine the 3D structuralparameters of the investigated sample. The investigated NCswere induced to self-assemble onto a flat substrate from aconcentrated colloidal solution, upon controlled solventevaporation assisted by an external magnetic field.28 Thesuperlattice structure and the morphology of the NC buildingblocks have been derived through the analysis of GISAXS data,based on the 3D short-range order model (SRO).27 Such amodel has been here further generalized to the case of a 3Dsuperlattice of NCs deposited onto a reflecting flat substrate,and it has been improved to take into account, through thelocal monodisperse approximation,17 the possible correlationbetween size and position of the NCs, which is likely to occurin non-diluted systems. Our model is demonstrated to returnthe main morphological and structural parameters of 3Dsuperlattice domains of colloidal iron oxide NCs. The obtained

Figure 1. (a) TEM image of the 2D monolayer of NCs on a Cu-supported carbon film of a TEM grid, which had self-assembled upon solventevaporation; (b−d) Low-resolution SEM images at different magnifications, showing the island-like features on the surface of the NC-builtsuperlattice films; (e, f) High-resolution SEM images of regions within cracks of the islands, where the 3D ordered NC packing can be clearlyappreciated.

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results, in agreement with those obtained by other techniques(TEM, SEM), demonstrate the applicability of the model alsoto the GISAXS study of nondiluted 3D NC assemblies.

■ EXPERIMENTAL SECTIONSample Preparation. Surfactant-capped ferrimagnetic iron oxide

NCs with the inverse spinel cubic lattice of structurally similar Fe3O4(magnetite) and/or γ-Fe2O3 (maghemite) phases were synthesized bya modified two-step seeded-growth protocol based on high-temper-ature decomposition of Fe(CO)5 in a ternary mixture of oleic acid,oleyl amine, and hexadecane-1,2-diol in 1-octadecene solvent at 280°C, as described elsewhere.29 NC superlattices were produced using amagnetically assisted self-assembly approach, similar to that describedin ref 28 with minor modifications. Briefly, a SiO2/Si substrate wasplaced onto the bottom of a 100 mL glass beaker and immersed into300 μL of a highly concentrated toluene solution of ∼10 nm ironoxide NCs ([Fe] = 150 μM). A 0.5 T magnet was located underneaththe vessel, in correspondence of the SiO2/Si substrate while thesolvent was allowed to evaporate in a quasi-saturated toluene-vaporatmosphere over a period of 72 h at room temperature. At the end ofthis process, a compact solid NC-built film with protrudingmicrometer-scale islands resulted. These samples were investigatedby SEM and GISAXS techniques.Characterization Techniques. Transmission electron microscopy

(TEM) images of the as-prepared NCs were recorded by a JEOL JEM1011 microscope operating at an accelerating voltage of 100 kV. Thesamples were prepared by dropping a dilute solution of the NCsdissolved in toluene onto carbon-coated copper grids and thenallowing the solvent to evaporate. Statistical size analysis was carriedout by counting and examining at least 300 particles on several wide-field TEM images with the help of dedicated software (Axio Vision).Scanning electron microscopy (SEM) imaging of NC superlattices

assembled over SiO2/Si substrate was performed by a FEINOVAnanoSEM200 microscope. Images were typically acquired atan accelerating voltage of 10 kV.GISAXS measurements were performed on a Rigaku three-pinhole

SAXS camera S-MAX 3000 with a sealed tube microsource 002+,focusing confocal optics, and a 2D multiwire proportional countinggas-filled detector with a diameter of 200 mm. The sample−detectordistance was 1.5 m (covered Q range 0.007−0.28 Å−1). Beamdivergence (fwhm) was approximately 0.5 mrad. The sample wasplaced on an automated high-weight capacity GISAXS stage andmeasured in the vacuum. The GISAXS patterns were collected fordifferent sample (in plane) orientations, with respect to the incidentbeam, with the value of the rotation angle Φ being varied over thewhole range. At each rotation angle, GISAXS patterns were collectedfor incidence angles (αi varying from 0.05° to 0.55° with a step of0.1°).

■ RESULTS AND DISCUSSIONElectron Microscopy. Figure 1a shows a representative

bright-field TEM overview of ∼10 nm iron oxide NCs thatwere used as building blocks to construct the 3D superlatticefilms investigated in the present work. The image demonstratesthe high degree of monodispersity of the NCs (size variance<5−7%), as a consequence of which the NCs tended tospontaneously organize into 2D ordered superlattices on theTEM grid upon slow solvent evaporation.Millimeter-scale supercrystals made of 3D ordered NCs were

obtained by a guided self-assembly procedure. The magneticallyresponsive NCs were driven to organize during slow solventevaporation from corresponding colloidal solutions under theaction of an externally applied magnetic field.28 SEMexamination at variable magnifications in Figure 1b−d revealedthe formation of a supercrystal solid film made of a compactarrangement of NCs, on which conic-shaped islands with basesand heights of about 100 μm were randomly distributed and

separated by several hundred micrometers. The islands wereenclosed by relatively smooth surfaces extending over severalsquare micrometers or, most frequently, exposed cracks andcorrugated surfaces with terraces. The sharp ridges visible atsuch stepped regions and/or in correspondence of inducedcracks revealed the inner superlattice organization into regularlystacked NC layers forming highly organized micrometer-scaledomains (Figure 1e and f). In most cases, the fast Fouriertransform (FFT) patterns calculated from relatively largesample areas in correspondence of individual islands as wellwere compatible with the formation of NC superlattices withboth cubic- and hexagonal-type distorted structures vieweddown the ⟨111⟩ and ⟨001⟩ directions, respectively (occasion-ally, disordered regions were identified). Direct cross-sectionalinspection of the NC multilayer stacking across terraces alongislands and at exposed cracks within them revealed cubicordering (Figure 1e and f).

GISAXS. The 3D structure of the assembled NC films wasstudied by grazing incidence small-angle X-ray scattering.GISAXS patterns were collected at different angles of incidenceto the substrate (see image sequences in Figures S1 and S2 inthe Supporting Information), all of which clearly showed thepresence of 3D order in NC positions. The best condition todetect diffraction spots from the NC assembly was found forthe smallest incidence angle of 0.05°.In the case of large flat homogeneous 3D assemblies, the

grazing incidence geometry normally allows the tuning of thepenetration depth, which could even be limited to the topmostNC layer, in the case of a very small incidence angle.On the contrary, due to the island-like surface morphology of

our samples (see Figure 1b−d), the small incidence angle of0.05° allowed the scattering contribution from the underlyingmaterial layers (including the NC superlattice film below theislands and the SiO2/Si support) to be minimized while thebeam could traverse the islands protruding out of film surface.This justifies the use of the 3D scattering model, and itsapplicability to self-assembled supercrystals of colloidal NCs, asdemonstrated below. Indeed, for λ = 1.54 Å the attenuationlength in a γ-Fe2O3 bulk sample is approximately L = 6 μm. Inthe case of a large compact assembly, of NCs with diameter D,and a constant separation distance δ between neighboring NCs,the beam is expected to irradiate a number of particles N = L/D, and cross an effective attenuation length, Lef f, given by thefollowing:

δ δ≈ + = +⎜ ⎟⎛⎝

⎞⎠L L N L

D1eff (1)

(here the beam absorption in the organic surfactant cappingbound to the NC surface has been neglected). The penetrationdepth (p) of an X-ray beam, impinging at an incidence angle αion such a layered structure, will be

α=p L sineff i (2)

As to the present iron oxide NC nanocrystals, if D = 10 nm(estimated from TEM) and δ = 2 nm (the latter is anoverestimated value) are assumed, an effective attenuationlength Lef f ≈ 7.2 μm should be considered. Consequently, at0.05° incidence, the penetration depth of X-rays should be p ≈6.3 nm, which is comparable to (even smaller than) the NCdiameter. It derives that, if the incidence angle is as small as0.05°, in the case of assembled NC superlattices uniformlycovering a substrate, the scattering signal mainly comes from

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the outermost exposed NC layer, and information on the 2Din-plane symmetry would be mainly achievable. Such acondition can be expected to hold in most cases wherebylarge and compact 3D assemblies of NCs are concerned.Conversely, the conic shape of the islands on the sample

studied in this work (Figure 1b−d) allowed us to systematicallyvary the number of NC layers sampled by the X-ray beam.Scattering from portions of the NC 3D assembly in proximityof the island surface was easily collected, and therefore, the X-ray scattering from different 3D regions of the assemblycontributed to the observed signal. The sketch reported inFigure 2 illustrates this concept, showing the relationshipsbetween the sample morphology and the scattering geometry;in the inset, an example of the 3D ordered region sampled bythe X-ray beam within the islands is reported, according to thestructure model derived by our analysis, as detailed in thefollowing paragraphs.By scanning across smaller to larger incidence angles, the

same diffraction peaks produced by the assembly could betracked, despite the fact that they were progressively over-whelmed by the increasing scattering contribution from theunderlying film, at larger incidence angles (see the sequence ofimages in Figures S1 and S2 in the Supporting Information).Therefore, our GISAXS analysis was focused on the patterncollected at the smaller incidence angle (Figure 4). Thepersistence of the peaks at larger (αi) values suggested that theNCs in the film adopted a spatial organization similar to thatwithin the islands.

■ GISAXS SIMULATION

In the structure model used for the calculation of the GISAXSintensity, the 3D assembly was described by a stacking of 2DNC arrays parallel to the substrate surface, as sketched in Figure3. Two in plane basis vectors (a′, b′) were introduced to

describe the average NC positions within the 2D arrays,whereas the out of plane vector (c′) described the relationshipbetween the average positions of NCs lying in consecutivearrays.The disorder in the NC positions was accounted for through

the 3D short-range order (SRO) model30 and the paracrystaltheory,27 which basically assume that the same fluctuations inthe positions occur for all the NCs, with respect to their firstorder neighbors, so that disorder in the superlattice increaseswith the distance from a given lattice node.The following parameters described the degree of order in

the positions of the NCs (Figure 3):

• σLL, describing the disorder in the mutual lateralpositions of the NCs belonging to the same 2D array;

Figure 2. Sketch of the GISAXS measurement: scattering geometry, sample morphology, and (inset) structure of a NC superlattice within a surface-protruding island, as derived by our analysis (see text).

Figure 3. Sketch of a cross section of the three-dimensional array ofNCs. The NCs in the ideal lattice and in their actual positions arerepresented by blue and gray ellipses, respectively. The basis vectors(a′, b′, c′) of the average lattice, the actual interparticle distances L(i),and the fluctuations of NC positions parallel to the substrate plane,within the same 2D layer (σLL) and in correlation with the underlyinglayer (σLV), and perpendicular to the substrate (σVV), are indicated.

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• σLV, describing the disorder in the mutual lateralpositions of the NCs lying in the subsequent arrays;

• σVV, accounting for the disorder in the mutual verticalpositions of the NCs in the subsequent arrays;

• σz, expressing the disorder in the vertical position of thewhole multilayer stack (ordered domain) with respect tothe substrate.

In this model, each NC was identified by a triplet of indexesn ≡ n1,2,3, such that the actual position of the n-th NC in thefirst 2D array (close to the substrate surface) was27

≡

= + + ···+ + + + ···+

R

L L L L L L

X Y( , , 0)n n n

n n1(1)

2(1)

1(1)

1(2)

2(2)

2(2)

(3)

where Ln(j) are random vectors parallel to the surface.

The average Ln(j) vectors were therefore represented by (a′,

b′):

⟨ ⟩ ≡ ′ ′ =a b jL , 1, 2nj( )

(4)

The root-mean square (rms) deviation of the lengths(moduli) of L(1,2) was σLL.The positions of the NCs in a two-dimensional array with n3

> 1 were correlated with the positions in the array n3 − 1 by thefollowing:

= +−R R Ln n n n n n n, , , , 1(3)

1 2 3 1 2 3 (5)

where

⟨ ⟩ = ′L cn(3)

(6)

The rms deviations of the lateral and vertical components ofthe vectors L(3) were represented by σLV and σVV, respectively.The vertical distance of the first 2D array from the substratesurface was assumed to be random as well, with mean value z1and rms deviation σz.NC shape was accounted for, in the calculation of the

scattered intensity, through the NC form factor (see the“Theoretical model” section in the Supporting Information fordetails). NCs were described by uniaxial ellipsoids with therotation axis perpendicular to the sample surface. The size ofthe NCs was assumed to be randomly varying, and the meanlateral and vertical radii were denoted as RL and RV,respectively. For the sake of simplicity, we assumed that theaspect ratio RV/RL was identical for all NCs and that the sizevariation followed the gamma distribution of order mR. Thelatter was related to the root-mean square (rms) deviations ofthe radii by σR = R/(mR)

1/2. The crystalline phase, and hencethe electron density homogeneity of the sample, was checkedby synchrotron radiation X-ray powder diffraction, whichconfirmed γ-Fe2O3 as the dominant phase, possibly incombination with Fe3O4 (see the SR-XRPD section in theSupporting Information).Differently from the theory previously published by M.

Buljan et al.,27 the model here presented was generalized totake into account several scattering processes contributing tothe GISAXS intensity (scattering of specularly reflected wave,specular reflection of the wave scattered from NCs, andspecular reflection of the primary wave followed by a scatteringprocess from NCs and specular reflection of the scatteredwave). All these processes were exactly described within thedistorted-wave Born approximation (DWBA) in the SupportingInformation (“Theoretical model” section). The analysis of the

GISAXS map was carried out both in the kinematic BornApproximation (BA) and in the DWBA descriptions, wherebythe former considers only single scattering processes from theNCs, while the latter takes into account scattering from bothNCs and the substrate surface. The main structural andmorphological parameters of the NC assembly were derived bysimulating the 2D GISAXS map, according to the structuremodel described above, and checking it against theexperimental data. The experimental map, collected at αi =0.05°, and the corresponding best fitting simulation arereported in Figure 4a and b, respectively. The simulation wasperformed in the Born approximation, where the correlationfunction for the local monodisperse approximation was used(GLMA in eq s7).

In the simulations, superlattice domains with differentazimuthal (i.e., in the substrate plane) orientations wereassumed. This assumption was justified by observing theGISAXS patterns collected for different azimuthal rotations ofthe sample, which did not reveal significant differences in thepatterns (see for example Figures S1 and S2 in the SupportingInformation, where the sample was at 0° and 45° relative to thedirect beam, respectively).The complete simulation in Figure 4b was obtained through

both direct comparison between the experimental andcalculated 2D GISAXS maps, for different values of thestructural parameters in the model, and the fitting of the verticaland horizontal linear cuts (Figure 4c−e), taken along thedirections marked by the white ticks in Figure 4a,b. Furtherdetails on the fitting procedure are reported in the SupportingInformation.The simulation performed in the kinematic approximation

(Figure 4) matched very well to the experimental data, whereasthe DWBA approach (Figure S5) led to additional scatteredintensity between the 2θf = 0 direction and the first correlation

Figure 4. (a) Experimental and (b) simulated GISAXS maps for 0.05°incidence. Linear cuts along (c) the vertical and (d,e) horizontaldirections, through the lines indicated by the white ticks in the maps.Data are simulated using the Born approximation, in the LMAdescription.

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peaks at 2θf ≈ 0.9°. Furthermore, the DWBA simulation isclearly affected by the specularly reflected wave from theassumed flat substrate, producing a horizontal streak at αf =0.05°, which was absent in the experimental map. Thisconfirmed that sample surface played no role in our GISAXSstudy, despite the very low grazing incidence angle set in themeasurement. Indeed, the undisturbed system used in theDWBA simulations (see the Supporting Information) assumesan ideally flat substrate surface producing a specularly reflectedwave. This is not the case, as a compact solid iron oxide NC-built film is present, on top of which the nanoparticles areconcentrated in large hillocks. Therefore, no large flat areas ofthe actual surface exist, so that the DWBA approach is notapplicable in our case. This is the reason why the results of thesimple first order Born approximation (BA) better match themeasured data. Beam divergence was taken into account in thesimulations as a Gaussian contribution with fwhm = 0.5 mrad,which was verified to have a negligible effect on the GISAXSpattern.The best fit parameters, relevant to the simulation reported

in Figure 4b-e, are listed in Tab. 1 (length units in nm, angles indegrees).The extracted morphological and structural parameters,

reported in Table 1, are in good agreement with TEM andSEM observations. The results in Table 1 indicate that thesuperlattice building blocks were spherical shaped NCs with amean diameter of 9.0 nm, in excellent agreement with resultsfrom XRPD (9 nm; see the Supporting Information) andTEM/SEM (Figure 1) analyses. The parameter σR confirmedthat the NC size polydispersity (σR/R) was smaller or equal to7%, as inferred from TEM statistical analysis.From the GISAXS fit, the ordered (coherent) domains were

deduced to consist of 5−6 NC layers stacked along a′ and b′and 3−4 NC layers stacked along c′. Such domains had at least40 different (azimuthal) orientations in the substrate plane.In Figure S6a in the Supporting Information, the GISAXS fit

performed within the BA/DA approach is reported forcomparison. Due to the narrow size distribution, the LMAand DA descriptions17 (see also the Supporting Information)led to similar results, although the model was able to highlightthe differences between the two descriptions, as shown by thelarger diffuse scattering in the DA simulated map (Figure S6a)when compared to the LMA one (Figure 4).Since the best fit GISAXS map had been calculated within

the Born (or kinematic) approximation, the vertical distance ofthe first 2D array from the substrate surface (z1) and its rmsfluctuations (σz) did not have any significant effect on the X-rayscattering pattern, in this particular case.The superlattice defined by the parameters in Table 1 can

actually be considered as a fcc-like structure with (111) planesoriented parallel to the substrate (the GISAXS pattern is indeedvery similar to those reported by other authors and indexed asfcc12,18).As sketched in Figure 5, in a perfect (111)-oriented fcc lattice

the (a, b, c) basis vectors, taken along the cube edges, arerelated to the basis vectors (a′, b′, c′), with a′, b′ laying in the(111) planes, through the following expressions:

′ = − ′ = − ′ = +a

b ab

c bc

a c2 2

( )2fcc fcc fcc

(7)

In such an ideal fcc structure, the basis vectors moduli shouldbe |afcc′ | = |bfcc′ | = |cfcc′ |, and their mutual angles should be αfcc =60°, βfcc = 120°, and γfcc = 120°, respectively. In Figure 5, thebasis vectors of the ideal fcc and of the actual structure areindicated by light and red arrows, respectively.The value of γ, slightly different from γfcc = 120°, is

responsible for the “bending” of the first lateral maxima, clearlyvisible in the GISAXS map. This fact can be deduced bycomparing Figure 4 and the simulations in Figures S6b and S6cof the Supporting Information, related to a perfect fcc latticeand to a fcc lattice distorted along the diagonal parallel to thevertical distance d represented in Figure 5. In a perfect fcclattice, this distance is related to the basis vectors through dfcc =(|a + b + c|)/3; therefore, for d = 8.0 ± 0.4 nm, as derived bythe fit, |afcc′ | = |bfcc′ | = |cfcc′ | = 9.9 nm and |a| = 14 nm should beexpected. Hence, the NCs can be regarded as being arranged ina distorted (111)-oriented fcc unit cell.

■ CONCLUSIONS

GISAXS data from a 3D superlattice of self-assembled ironoxide NCs were collected and analyzed against a newtheoretical model. In the considered specimen, NCs wereassembled in a compact film on a substrate, from which severalwell-separated, distinct 3D islands with conic shape protrudedoutward. Such a particular morphology of the sample allowedmeasuring the X-ray intensity scattered from 3D regions of the

Table 1. Structural Parameter Values from GISAXS Simulationa

a′ b′ c′ α β γ σLL σLV σVV RL/V σR

10.5 ± 0.5 12.0 ± 0.5 10.5 ± 0.5 52 ± 4 119 ± 5 125 ± 5 0.9 ± 0.2 0.5 ± 0.2 0.7 ± 0.2 4.5 ± 0.4 ≤0.3 ± 0.1aLengths and angle units are in nanometers and degrees, respectively.

Figure 5. Scheme of the (111)-oriented fcc unit cell, where the twosets of basis vectors (a,b,c) and (a′,b′,c′) for the ideal fcc arerepresented, together with the (a′,b′,c′) vectors for the actual structure(red arrows). The 2D NC layers represent (111) planes, separated bythe distance d.

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islands, even with the beam impinging at a very grazingincidence angle on the substrate.The presented model for GISAXS data analysis was

demonstrated to reproduce the scattering pattern of a NCsuperlattice with fcc-like structure, together with the scatteringsignal related to the morphological and statistical properties ofthe assembled NCs. This model therefore appears to besuccessfully applicable also to close-packed nanostructures,such as self-assembled NC-built supercrystals. It can be ingeneral applied to systems with different degrees of order andlattice structures oriented such that the primitive unit cell canbe described by two of the three basis vectors lying in the planeof the sample surface. The proposed model could be expectedto be successfully extended to the analysis of several other typesof assemblies based on NCs with different morphologies

■ ASSOCIATED CONTENT

*S Supporting InformationExperimental GISAXS maps collected at different incidenceangles and sample azimuthal orientations; XRD analysis; detailsof the theoretical model and the fitting procedure used forGISAXS data analysis; additional GISAXS simulations. Thismaterial is available free of charge via the Internet at http://pubs.acs.org.

■ AUTHOR INFORMATION

Corresponding Author*Phone: 0039-080-592-9167. Fax: 0039-080-592-9170. E-mail:cinzia.giannini@ic.cnr.it. Web: http://www.ic.cnr.it/giannini.php.

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

This work has been partially supported by the SEED project“X-ray synchrotron class rotating anode microsource for thestructural micro imaging of nanomaterials and engineeredbiotissues (XMI-LAB)”Italian Institute of Technology (IIT)Protocol no. 21537 of 23/12/2009. Dr. M. Ladisa isacknowledged for the fruitful discussions concerning the fittingalgorithms. V.H. acknowledges the support of the CzechScience Foundation (Project P204-11-0785). C.N. and P.D.C.acknowledge partial financial support by the Italian Ministry ofEducation, University and Research through the projectAEROCOMP (Contract MIUR No. DM48391). The SwissLight Source synchrotron facility at the Paul Scherrer Instituteis acknowledged for use of beamtime at the Materials Sciencebeamline powder diffraction station.

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