Periodic lattices of arbitrary nano-objects: modeling and applications for self-assembled systems
A formalism is described which enables the simulation or fitting of small-angle scattering data (X-ray or neutron) for periodic heterogeneous lattices of arbitrary nano-objects. Generality is maximized by allowing for particle mixtures, anisotropic nano-objects and definable orientations of nano-objects within the unit cell. The model is elaborated by including a variety of kinds of disorder relevant to self-assembling systems: finite grain size, polydispersity in particle properties, positional and orientation disorder of particles, and substitutional or vacancy defects within the lattice. The applicability of the approach is demonstrated by fitting experimental X-ray scattering data. In particular, the article provides examples of superlattices self-assembled from isotropic and anisotropic nanoparticles which interact through complementary DNA coronas.
Nanomaterials hold great promise for improvement of both conventional material properties (e.g. strength; Podsiadlo et al., 2011) and new properties and applications (e.g. self-cleaning surfaces; Zhang, Xia et al., 2012). A wide variety of ordered periodic nanomaterials are in particular being studied for the unique ability to tune properties, including optical (e.g. plasmonics, metamaterials) (Xiong et al., 2010), catalytic and transport (diffusivity, conductivity). Many novel fabrication strategies are being investigated, including interference lithography (Lu & Lipson, 2010; Xia et al., 2011; Xie et al., 2012), block-copolymer self-assembly (Fasolka & Mayes, 2001; Cheng et al., 2006; Kim & Hinsberg, 2008), dense nanoparticle packing (Henzie et al., 2012; Damasceno et al., 2012; Ni et al., 2012), evaporation-driven assembly (Narayanan et al., 2004; Bigioni et al., 2006), and nanoparticle interactions via electrostatics (Kalsin et al., 2006; Shevchenko et al., 2006), polymers or programmable DNA linkers (Park et al., 2008; Nykypanchuk et al., 2008). Design of these periodic nanomaterials is increasingly moving towards complicated lattices where the constituent objects may have complex and anisotropic structure (Quan & Fang, 2010; Jones et al., 2010); advances in nanoparticle synthesis (Murray et al., 2000; Glotzer & Solomon, 2007) have enabled production of a variety of nanoparticle shapes: spheres, cubes, octahedra, truncated octahedra, rods, plates, tetrapods etc.
X-ray scattering is a powerful tool for studying these nanostructured systems: powders and superlattices in solution can be studied using small-angle X-ray scattering (SAXS), whereas thin films can be probed using grazing incidence either in reflection mode (GISAXS) (Levine et al., 1989; Doshi et al., 2003; Renaud et al., 2009; Rauscher et al., 1999; Bian et al., 2011; Papadakis et al., 2008; Zhang, Luo et al., 2012) or via the recently described sub-horizon (GTSAXS) mode (Lu et al., 2013; Mahadevapuram et al., 2013). Assigning the peaks seen in such data sets is in principle straightforward: the nanoscale periodic order leads to well defined small-angle peaks analogous to the sharp diffraction peaks observed at wide angles for atomic crystals. However, quantitative data interpretation requires special considerations owing to the potential complexity of the scattering objects sitting on a nanoscale lattice. Conventional crystallographic scattering theory is derived under the assumption that lattice sites are occupied by atoms; this constrains the form factor associated with each lattice site (e.g. typically isotropic). This limiting case cannot be naively extrapolated to nanoscale analogs, since nano-objects may be highly anisotropic, and different nano-objects within the lattice may have vastly different relative sizes and shapes. Moreover, nanoscale systems rarely exhibit the perfection of atomic crystals, which complicates data interpretation. Various publications have gone beyond atomic crystallography and derived expressions for scattering from nanoscale lattices of particular symmetries (Ruland & Smarsly, 2004; Förster et al., 2005; Ruland & Smarsly, 2005; Ruland & Smarsly, 2007; Lazzari et al., 2007), which have proven extremely useful in analyzing the scattering of many nanoscale morphologies. The scattering for various nano- or mesoscale particles or aggregate shapes in solution or at surfaces has also been studied in some detail (Kline, 2006; Kohlbrecher, 2010; Förster et al., 2010; Babonneau, 2010; Favre-Nicolin et al., 2011), including the case of supramolecular assemblies (Ben-Nun et al., 2010; Szekely et al., 2010).
Here, we describe a generalized scattering formalism that can be used to predict, or quantitatively fit, X-ray or neutron scattering data for periodic lattices of arbitrary nano-objects (Fig. 1). We in particular consider generic unit cells that may have multiple different objects, and allow for the constituents to have arbitrary shape or internal structure, and arbitrary orientation within the lattice. The model is constructed such that one can use analytical expressions or numerical computations for the constituent form factors, thereby maintaining generality while allowing for computational efficiency when closed solutions are known. The experimental realization of nanoparticle superlattices typically involves the use of soft organic linkers, which generate relatively weak and broad effective interactions. This inherent sensitivity allows for subtle tuning of interparticle association, distances and orientations, leading to rich landscapes of possible structures. However, the lower barriers to defect formation associated with these weaker interactions frequently lead to structures with larger fluctuations and greater amounts of disorder. We note that the disorder observed in self-assembled nanosystems is not merely a matter of imperfect processing protocol: the disorder may be intrinsic, thermodynamic and unavoidable, as, for instance, in the case of liquid crystal and plastic crystal phases (Agarwal & Escobedo, 2011; de Graaf et al., 2012; Damasceno et al., 2012). Thus, we elaborate our model by explicitly including a variety of kinds of disorder, so as to enable quantitative comparison with experimental X-ray scattering data.
We revisit the standard derivation of scattering intensity for an ensemble of scatterers (Kotlarchyk & Chen, 1983; Warren, 1990; Guinier, 1994), recasting the expressions into a form useful for lattices of multiple nano-components. We begin with the general expression for scattering intensity, which is simply an ensemble average of intensities (square of the total scattering amplitude) where we explicitly sum over every scatterer in the sample:
is the scattering contribution of scatterer n, whose interpretation depends on the kind of scattering (e.g. nuclear scattering length in the case of neutron scattering), rn is the position vector and q is the scattering vector [q = |q| = (4π/λ)sinθ, where θ is half the scattering angle and λ is the wavelength of the incident radiation]. The ensemble average may correspond to different realizations of the system or to integrating the scattering data over a certain time frame. We denote this average with a subscript e so as to differentiate from other averages introduced later. We now split the summation into a triple summation by conceptually dividing the sample into Nn sub-cells, where each sub-cell contains Nj particles (nano-objects), and each particle contains Np elementary scatterers (e.g. electrons). The position vector is now denoted to emphasize that it points to scatterer njp (pth scatterer of particle j in sub-cell n):
Note that at this stage we have not sacrificed generality, since in principle each sub-cell could contain only a single scatterer. However, as we shall see shortly, this formulation is convenient as it simplifies considerably in cases where particular sub-cells reoccur throughout the sample. We decompose the position vector into a component that points to the sub-cell (), a component that points from the origin of the sub-cell to the center-of-mass of the particle j (), and a component that points from that center-of-mass to the final position (): . So
The third summation is a well known quantity: the per-particle form factor amplitude. We thus define
For convenience we also define
We now consider the form of the scattering intensity for a crystal-like lattice of particles. In such a case, is effectively the form factor of the unit cell, and the sum over n is a sum over the Nn identical unit cells in the sample. We also convert from to I(q) by including in the an average over grains at all possible orientations (powder-like sample), denoted by the subscript o.
For a perfect crystal, all are identical. As is well known (Warren, 1990; Guinier, 1994), the sum over identical unit cells serves to define a peak shape. A small number of unit cells interfere constructively to give a broad peak centered at the reciprocal-lattice spacing, whereas a progressively larger lattice produces a progressively sharper peak. The peak positions are defined by the symmetry of the lattice. We thus convert to a sum over Miller indices (see supporting information1 for details):
where mhkl is the multiplicity of the reflection hkl which appears in reciprocal space at . L is a peak-shape function. A variety of peak shapes are commonly used, including Gaussian, Lorentzian, Pearson and pseudo-Voigt (Young & Wiles, 1982). A peak shape that can vary continuously from Gaussian to Lorentzian (Micha, 1998; Förster et al., 2005) enables one to account for the varying contributions from grain size/shape and instrumental effects (refer to supplementary information for details).
is the Lorentz factor for a d-dimensional lattice, where Ω is a solid angle. For a three-dimensional lattice is the surface area, in reciprocal space, over which the intensity from the hkl reflection is uniformly spread owing to the orientational averaging. We let and note that in practice c is used as a scaling factor to account for a variety of effects. For instance, scattering intensity scales with scattering volume (intersection between the incident beam and the sample), or with concentration of scattering objects for solution scattering. When comparing with experimental data, corrections related to flux, polarization, energy dependence of scattering and the geometry of the experiment can also be included. We also define Z0(q) to be the lattice factor. We note, however, that traditional lattice factors consider only the positions of objects (e.g. atoms) in the unit cell and do not include the form factors. In our case, though, we cannot extract Fj from Z0 since we wish to allow for arbitrary and distinct form factors for each particle in the lattice.
xj, yj and zj are fractional coordinates within the unit cell, and we introduce Mj as a rotation matrix to account for the relative orientation of particle j within the unit cell.
The form factor given by equation (4) can be rewritten in terms of an arbitrary distribution of scattering density (electron density, when considering X-ray scattering), which is integrated over all space:
When considering organizations of nano-objects, we are typically not concerned with the arrangement of atoms within the particles. We can thus use a form of that averages over atomic length scales. Experimentally, this corresponds to ignoring the high-q data, which encode atomic positions; the resultant predictions are nevertheless valid in the small-angle regime, which encodes nanoscale order. If the nano-object of interest has a uniform internal density, the form factor can be simplified to an integral over the volume (V) of the object:
where is the scattering contrast of the nano-object (relative to the ambient medium). The spherically averaged form factor is a quantity that frequently appears in scattering expressions. We define Pj(q) to be the orientationally averaged (isotropic) form factor intensity:
In later sections we consider polydispersity in the nano-objects, in which case the above average would also include integrations over those distributions. Note that for , we obtain
As expected, scattering intensity scales with the square of the scattering contrast and the particle volume. For multi-component lattices, this has the effect of greatly emphasizing larger particles. For instance, a twofold increase in particle diameter results in a (23)2 = 64-fold increase in scattering intensity. In lattices where one particle is much larger, the smaller particles can frequently be neglected. Similarly, the large electron density of metallic and inorganic nanoparticles (Au, Ag, CdSe etc.) will tend to overwhelm any organic coatings, linkers or weakly scattering nano-objects (e.g. proteins).
The scattering intensity is frequently split into contributions from the form factor and the structure factor:
where S0(q) is the ideal structure factor (assuming no disorder):
This version of the structure factor is similar to that which has been used to describe anisotropic particle lattices (Jones et al., 2010), except we again emphasize that we allow for distinct form factors, and distinct spatial orientations, for every nano-object in the unit cell. The utility of S(q) is that it divides out contributions to the total scattering that arise from the particle volume. As we shall see, however, this does not mean that S(q) is unaffected by the particle properties (shape, orientation etc.). P(q) is the isotropic form factor intensity for the particles in the lattice. This quantity is affected by the size, shape and composition of particles, but not by their position or orientation within a lattice (in other words, it lacks structural information). As will be shown below, for multi-component systems this takes the form
where cj are weighting factors. We note that P(q) is what would be measured as the scattering intensity if the constituent particles were disassembled and became free floating (in which case cj correspond to concentrations). For systems with reversible self-assembly behavior, the structures can be eliminated by heating the solution above the order–disorder temperature, enabling convenient measurement of P(q) on the exact sample that I(q) is measured on.
Unlike atoms and small molecules, polymers and nanoparticles are synthesized with a certain polydispersity. In order to account for distributions in particle properties (size, shape, orientation etc.), we recast the scattering intensity into a form that highlights the variance of interparticle scattering. We assume that the per-particle properties are not correlated to their positions, so that we can write equation (6) (the summation over n) as (van Beurten & Vrij, 1981; Kotlarchyk & Chen, 1983)
Note that the inner angle brackets represent an average over the particle distributions (subscript d). This average can be written
where is the Kronecker delta function, and the term in square brackets is effectively a variance. This introduces a corresponding variance term into the scattered intensity:
Note that the orientational average of the outer angle brackets has been included in the averages for our new definitions:
Fluctuations (e.g. thermal) of the particle positions within the unit cell lead to lattice disorder, which decorrelates scatterers and thereby extinguishes scattering intensity (especially in higher-order peaks). We introduce
where is the relative r.m.s. displacement for a lattice of size a and is called the Debye–Waller factor. As has been demonstrated (Chipman & Paskin, 1959a,b; Inouye & Kirschner, 2003; Förster et al., 2005), inclusion of lattice disorder leads to an expression for scattering intensity of
The term in the square brackets is effectively a revised definition of the structure factor, S(q), which includes disorder from polydispersity and positional fluctuations. Disorder extinguishes intensity in structural peaks and shifts intensity instead to a diffuse scattering term: , which appears as a broad baseline in experimental data, appearing in S(q) as a baseline rising from 0 to 1 as q increases (Pabst et al., 2000). The diffuse scattering term can have some structure, however, owing to oscillations which appear in .
The average form factor intensity can be computed via
where we have again assumed a decoupling between the particle distributions, which are averaged over by the angular brackets, and the particle positions within the unit cell. This approximate form of P(q) corresponds to what would be measured experimentally: for a lattice of nano-objects that is dissociated, the measured scattering will be an incoherent sum of the isotropic form factor contributions of each particle type, weighted by the relative occurrence of that particle type. One can compute numerically using equation (22), or approximate by neglecting interparticle effects, which are expected to be negligible for relatively monodisperse systems (van Beurten & Vrij, 1981). The numerator of β becomes
For monodisperse particles, . For particle size distributions of finite width , it has been shown (Forster et al., 2005) that the ratio has an oscillating and a non-oscillating part, where one can approximate the non-oscillating scaling as for particles of radius R. Fig. 2 plots some example diffuse scattering contributions to S(q). As expected, increasing size polydispersity causes an increase in the diffuse scattering. For realistic nanoparticle polydispersities, the β contribution has substantial oscillations. However, inclusion of the Debye–Waller contribution tends to overwhelm these oscillations. In particular, for soft systems the Lindemann criterion provides an estimate of the r.m.s. particle displacement , at which point the lattice disorder dominates the diffuse scattering. The limit of the diffuse scattering can in principle be used as a measure of polydispersity. In practice, experimental data typically include many contributions to the low-q scattering, thereby complicating the analysis.
Summarizing the derivation results, we model the scattering intensity using
where c is a scaling constant, S(q) is the structure factor and P(q) is the average form factor intensity:
The diffuse scattering  includes an exponential Debye–Waller factor G(q) as well as the effect of particle polydispersity . The lattice factor, responsible for structural scattering, is computed as
where L(q-qhkl) is a peak-shape function, and are the form factors for the various particles in the unit cell. Each Fj can be distinct, allowing for heterogeneous lattices of arbitrary complexity. This formalism can take advantage of the numerous published form factors for a variety of particle shapes (Pedersen, 1997; Lazzari et al., 2007; Babonneau, 2010), including platonic solids (Li et al., 2011), ellipsoids of revolution (Sjöberg, 1999) and particle clusters (Ben-Nun et al., 2010; Szekely et al., 2010). It also allows for handling more complex shapes by solving equations (10) or (11) numerically. Similarly, Pj(q) can be either introduced as an analytical expression or handled numerically. In this article, we focus on the application of this formalism to nanoscale objects, in which case structural peaks are expected to appear in the small-angle regime. However, the generality of the approach means that it can be used for larger- or smaller-scale systems equally well, simply by reinterpreting the nature of the scattering objects.
A particularly powerful method for generating superlattices of nano-objects is to use particles coated with engineered DNA sequences. The strong yet reversible hybridization of complementary DNA sequences induces specific interactions between particles, leading to programmable higher-order assembly. Particle–particle interactions may be direct, wherein two particle types bear DNA sequences that are complementary, or indirect, wherein particles are complementary to the ends of a linker strand which is subsequently introduced. Spacer sequences can be introduced to regulate interparticle distances, while the ratio between single-stranded (ss) and double-stranded (ds) regions can tune stiffness. Overall this method provides for programmable and tunable interactions, which in turn provides control over the assembled superlattice. Simultaneously, this approach is amenable to any particle size or shape. We exploit these assembly motifs herein to generate a variety of model self-assembled structures.
Experimentally, small-angle scattering measurements of nano-object lattices yield I(q), which is typically converted into the corresponding S(q) by dividing by P(q). For DNA-assembled nanoparticle superlattices, P(q) can be conveniently measured by heating the solution above the DNA hybridization melt temperature (usually 313–353 K), which disassembles the superlattice. For other kinds of particle assembly, one can similarly measure the constituent particle mixtures in solution. Alternatively, one can compute P(q) theoretically from the known particle sizes, shapes and relative concentrations. Fig. 3 shows experimental data for a binary superlattice, where two nanoparticle types (Au and CdTe) bearing engineered DNA sequences are connected using complementary linker DNA strands. By using the described formalism, we are able to test any possible lattice arrangement for consistency with observed peak positions. We can further quantitatively fit the data in order to obtain structural parameters of interest.
We implemented our formalism [equations (34)–(37)] in the programming language Python, taking advantage of existing Python libraries for efficient numerical computations (numpy; Oliphant, 2007) and plotting (matplotlib; Hunter, 2007). In order to restrict the number of free parameters in the fit, we use independent means [scanning and transmission electron microscopy (SEM and TEM), dynamic light scattering and free-particle SAXS experiments] to measure nanoparticle size and polydispersity. Having manually selected the lattice type for consistency with peak positions, the lattice spacing can be fitted by itself based only on the position of the primary (highest intensity) peak. Furthermore, the lattice constant and symmetry can be evaluated for consistency with the physical assembly (stoichiometry and approximate particle–particle distance based on DNA length). Despite the generality and flexibility of our model, this then leaves relatively few free parameters to fit: the scaling factor (c), the thermal disorder Debye–Waller factor (), and the peak shape (ν) and width (δ). Each of these parameters has a distinct effect on the model. Fitting can be performed using any of a variety of established minimization routines. For the fits shown here, the simulated annealing package of the GNU Scientific Library (http://www.gnu.org/software/gsl/ ) was used. In simulated annealing, the fit error is treated as an energy to be minimized. Exploration of the energy landscape involves random motions that at first are large (analogous to high temperature), but which become progressively smaller (analogous to cooling). This produces a relatively robust exploration of the parameter space, avoiding immediate trapping in local minima.
The explicit inclusion of disorder gives rise to the diffuse scattering which underlies the structural peaks in S(q). However, the formalism as presented does not account for experimental background, which may arise from the empty cell, the solvent medium or detector noise. One can either carefully subtract such contributions from the experimental I(q) or add an ad hoc background to the theoretical scattering intensity described here. The appearance of substantial small-q scattering in the experimental data is not captured in the model. This scattering probably arises from the overall size/shape of the superlattice aggregates themselves (the mesoscale morphology). In principle, one could explicitly add such a contribution to the model in such a way that the average aggregate size was consistent with the correlation length embedded in the peak width.
We demonstrate here the dramatic effect that nanoparticle size and shape can have on scattering data. Any particle shape can be introduced into the formalism by using an analytic or numerical computation of the anisotropic form factor. For illustrative purposes we consider a particle shape known as a superball (Elkies et al., 1991; Jiao et al., 2009; Zhang et al., 2011; Ni et al., 2012), described by the equation
The parameter p allows one to continuously vary the particle shape. Specifically, p = 1 simply yields a sphere, whereas produces a cube. Thus produces a rounded cube where the curvature can be continuously varied. Also, p = 0.5 produces an octahedron, and yields a convex octahedron (bulging faces). Finally, produces a concave octahedron (the limit yields an object of zero volume). This diversity enables continuous exploration of the effect of particle shape; with respect to fitting experimental data, it provides a means to account for non-ideality in the shape of nanoparticles (which tend not to be ideal cubes, spheres or octahedra).
Fig. 4 shows S(q) curves for a variety of particle shapes and sizes, sitting on a simple cubic lattice with a = 85 nm. The particles are aligned with the unit cell with no orientational spread. The anisotropic form factor [equation (11)] is solved numerically by discretizing real space (for the distances explored here, a partitioning of 32 ×32 ×32 cells was found to be sufficient to avoid artifacts within the q range of interest); spherical averaging [equation (12)] was performed using angular increments of . The scattering varies tremendously based on the nano-object's structure. Particle shape has a non-uniform influence on the intensity of various peaks (Fig. 4b), emphasizing that one cannot simply use the atomic form factors of conventional crystallography when simulating nanoparticles. Another tempting approximation is to use a simple shape (e.g. sphere) to describe a nanoparticle's form factor. The diagonal elements in Fig. 4 (lower left to top right) are isovolumetric by design (thus the particles have the same total scattering power). The strong variation in the scattering curves demonstrates that isovolumetric approximations fail in the general case. Of course, in cases of high-symmetry (nearly isotropic) nanoparticles, or in cases of extreme orientational disorder (plastic crystals), isotropic form factors may yield a good approximation. The nano-object size interferes non-trivially with the structural scattering peaks (Fig. 4c), leading to non-monotonic modulation of peak intensities. This is especially apparent in cases where the particle size is comparable to the lattice spacing: the particle form factor's maxima and minima can overlap with the structural peaks, leading to beat patterns in peak intensities. Thus the particle form factor can effectively extinguish structural peaks. This can be seen most clearly for the 81.7 nm cube () scattering, where only a subset of peaks are visible. This emphasizes that, for nanoparticle superlattices, peak positions alone are insufficient to confirm lattice symmetry.
For anisotropic nano-objects, the orientation within the unit cell also plays a role. The particle form factors are anisotropic and modulate structural peaks differently depending on their orientation relative to the crystal axes. Fig. 5 explores the orientation effect for two representative systems: (a) a simple cubic arrangement of cubes and (b) a body-centered cubic arrangement of octahedra; particles are rotated in concert around an axis parallel to an edge of the unit cell (one of the three degenerate directions for the cubic cell). For both particle types, reorientation of the nano-object causes substantial changes in peak intensities. Moreover, the presence of isotropic particles (simulated by replacing the particle form factor by an isotropic orientational average) tends to smooth out higher-order peaks, making such states distinguishable from well aligned nanoparticle systems. For the experimental examples presented, the simple cubic arrangement of cubes (Fig. 5a) is best described by a cube orientation that maximizes face-to-face interactions, consistent with physical expectation that DNA-coated cubes will attempt to maximize the number of DNA contacts between neighbors. In the case of a body-centered cubic (b.c.c.) arrangement of octahedra (Fig. 5b), no model curve provides a perfect agreement with the experimental data. This implies the presence of some form of disorder (see next section). In particular, the larger interparticle spacing in this system is expected to provide the octahedra with considerable rotational freedom. Thus a plastic crystal (rotator phase), with substantial orientational disorder, should be considered. In general, these results suggest that one can measure the orientation of nano-objects within superlattices by careful fitting of experimental data.
Inclusion of disorder into scattering models is frequently necessary when comparing with experimental data. Nanoscale systems, in particular, can exhibit substantial amounts of disorder from both nano-object polydispersity and intrinsic defects arising from the softness of the interaction potentials used for assembly.
As with atomic systems, the finite grain size for crystalline domains is captured in the peak width: large grains yield sharp intense peaks, whereas small grains produce broad peaks. The peak width can be converted into a correlation length (ξ) via a Debye–Scherrer analysis (Langford & Wilson, 1978; Smilgies, 2009). Superlattices generated with soft linkers are likely to also exhibit distortion modes at the scale of the aggregate size (e.g. bend, splay, twist). To a first approximation, these distortions will simply decorrelate distant unit cells, which is captured by the peak width. This, however, points out that one must be careful in interpreting the Debye–Scherrer correlation distance as the aggregate size. Instead, the correlation distance establishes a minimum aggregate size. An avenue for future investigation may be to consider more rigorously the effect of large-scale lattice distortions: an appropriate theory accounting for soft-linker bonding networks could predict distortion modes and their effect on the scattering curve.
Thermal fluctuations in particle positions are captured by the Debye–Waller factor (), whose effect is to extinguish the intensity in structural peaks and instead generate diffuse scattering. Modeling could potentially provide estimates, based on interaction potentials, for the value of , thereby reducing the number of fit parameters. The present formalism assumes that the fluctuations of particle positions are isotropic and uncorrelated. This assumption is demonstrably effective in many cases; however, we note that solving for the modes of the nanoparticle motions (which, via their interactions, form a system of coupled oscillators) could provide a more rigorous form for G(q). In particular, an anisotropic Debye–Waller factor would be necessary to account for the anisotropic thermal fluctuations of particles connected via directional soft linkers. In principle, fitting data under the constraints of such models could provide a measure of the angle-dependent interparticle interaction potentials.
Distributions in the size and shape of nanoparticles are explicitly included in the model. These distributions alter the particle form factors, smearing out deep minima and generally broadening structural peaks. They also lead to diffuse scattering, which is captured in the present model. Fitting for the size and shape polydispersity from S(q) is in general difficult. Such parameters are better obtained from secondary measurements, such as microscopy (TEM, SEM or atomic force microscopy), or by fitting P(q) for free particles independently.
We explored the effect of substitutional (compositional) variation, a defect wherein the `wrong' particle type occasionally sits at a site in a heterogeneous lattice. We simulate this behavior by considering an expanded simulation cell which contains many unit cells, allowing us to introduce random defects into some fraction of unit cells. This introduces artifacts arising from the interference between the defects at the scale of the expanded cell size, which can be minimized by averaging over multiple realizations or by ignoring the q range of these effects. Fig. 6(a) shows simulation curves as a function of substitutional disorder. For the binary body-centered arrangement simulated, we observe peaks located at the simple cubic symmetry positions, since the two particle types are distinct. An increase in substitutional disorder extinguishes the odd-numbered peaks. In the limit of disorder (where any given lattice site has equal probability of housing either particle type), the scattering becomes that of pure b.c.c., since any given lattice site is statistically identical. More generally, this type of defect causes some peak intensities to vary strongly compared to others (Fig. 6a, lower panel), as dictated by how the defect alters the lattice symmetry. This in principle provides a way to measure disorder from the scattering data, by accounting for relative peak intensities.
We similarly explored the effect of vacancy defects, wherein certain lattice sites are missing particles (Fig. 6b). Such defects are particularly important given recent work suggesting that vacancies can improve positional order (Smallenburg et al., 2012). We observe that site occupation influences the relative peak heights measurably. When one species is entirely absent, the system reverts to canonical simple cubic scattering. A certain composition can negate the mismatch in scattering power of the two particles, yielding scattering akin to canonical (single particle type) b.c.c. The variation in peak intensities is thus non-monotonic (Fig. 6b, lower panel). As with substitutional disorder, one can in principle fit experimental data to determine the extent of vacancy defects in both particle types. We note, however, that substantial ambiguities exist: e.g. both substitutional disorder and vacancy defects can give rise to similar kinds of changes in relative peak heights. However, it is also worth noting that some of the presented kinds of disorder may not be physical: e.g. for binary assemblies of complementary particles, superlattice connectivity is not satisfied if a particular species is absent beyond a certain threshold. Physical considerations allow one to constrain the search space for a particular system. We note that input from theoretical models, able to delineate these boundaries, would be beneficial. In actual particle assemblies, vacancies may be accompanied by rearrangements of nearby particles, effectively delocalizing the defect (Smallenburg et al., 2012). Such defects would modulate peak heights, while simultaneously decorrelating particle positions, effectively increasing the Debye–Waller factor. Finally, we note that these occupation/vacancy results can also be used to interpret the scattering data of systems where small-particle inclusions populate interstitial holes of another lattice (Filion et al., 2011).
Fig. 7 shows the effect of orientational spread, again obtained using an expanded simulation cell where particle orientations randomly deviate from the ideal orientation (a uniform orientation distribution is assumed, with maximum angular deviation as denoted in the figure). Orientational disorder extinguishes higher-order peaks, similar to the Debye–Waller factor; however, this suppression is in principle distinguishable from positional disorder. Fig. 7(c) shows the intensity variation for selected peaks: orientational disorder has a relatively subtle influence on relative peak intensities, making it a more challenging parameter to measure by fitting experimental data.
Another kind of disorder which sometimes exists in experimental systems is the formation of a mixture of different lattice types. Generally such a situation can be handled by a simple weighted average of the scattering for the various possible lattice configurations. A more complicated case is where occasional lattice defects cause intimate mixing of different kinds of unit cells (at a scale smaller than the coherence length of the probing radiation). Such cases can be approximated by averaging many realizations of expanded unit cells which include the defective states.
A natural question is whether the disorder inherent to many soft systems makes structural identification impossible. Fig. 8 shows experimental data for a mixture of nanoparticles known to form a b.c.c.-like arrangement (as described in Fig. 3), but where the DNA linker length and annealing protocol lead to a relatively disordered structure. Simulations for three possible binary lattices, consistent with the 1:1 assembly stoichiometry, are shown. The Debye–Waller factor and peak width have been set to reproduce the experimentally observed disorder (, ). Although the experimental data contain only a few very broad peaks, it is nevertheless possible to exclude some lattices, while showing that other lattices are consistent with the experiment. Thus, even in highly disordered systems, quantitative modeling can be used to narrow down the possible classes of particle arrangements, even if quantitative fitting of the scattering data is not feasible.
We have demonstrated a formalism, applicable to X-ray and neutron small-angle scattering data, which can simulate data for extended periodic lattices of arbitrary nano-objects. We allow for heterogeneous lattices, where each particle can have an arbitrary size, shape, composition, internal structure and orientation. By explicitly including various kinds of disorder, we allow for quantitative fitting of experimental data.
A great many parameters of experimental interest have a measurable effect on the observed scattering. For instance, one can in principle fit the scattering data to obtain a measure of particle orientation within the lattice or the number of vacancy defects. However, in many cases simultaneously fitting for a variety of effects may be ill posed. In such cases, additional information from other measurement techniques, or constraints from theoretical models, can be highly valuable. This is of course a generic issue with scattering: one cannot unambiguously fit data, but must instead use physical insights to select an appropriate model, which can then be used to fit data and extract parameters of interest.
Many aspects of disorder are intrinsic to self-assembling systems. The disorder does not exist merely from imperfect processing, but is a thermodynamic aspect of the final structure, arising from competition between interactions, packing, entropy etc. For instance, assembly with large interparticle distances (relative to particle size) will in general provide rotational freedom, yielding plastic crystals (rotator phases). Although this intrinsic disorder represents a challenge in terms of fitting scattering data, we have demonstrated that explicit inclusion of relevant kinds of disorder can be used to analyze realistic experimental systems.
Our model demonstrates that the particle form factor can interfere substantially with the structural peaks, modulating peak intensities considerably. This effect is analogous to atomic crystallography, where electron density within the unit cell controls peak heights, while unit-cell symmetry dictates the array of peak positions. However, for nanoparticle systems, the relative size of the scattering entities and the unit cell can vary dramatically: from nearly space-filling particle packing to nearly point-like scatterers. In some cases, structural peaks can be effectively eliminated by minima in the particle form factor. It is also noteworthy that, for disordered systems, the broad peaks can overlap substantially, giving rise to convolved peaks which appear to shift in position as the relative contributions of various sub-peaks increase and decrease.
We have identified several areas for further elaboration of this model. (1) It would be useful to be able to extract the overall aggregate size/shape by fitting the very small angle regime in a manner consistent with the peak width (average correlation length). (2) Modeling of the lattice fluctuation modes could provide physical estimates for the (possibly anisotropic) Debye–Waller factor, reducing the number of free fit parameters, or inversely allowing for extraction of the particle interaction potentials from experimental data. (3) Similar models could provide insight into the likely large-scale fluctuations in the shapes of aggregates, which would also have implications for scattering data. (4) Theoretical estimates of the minimal connectivity necessary to maintain particular lattice types would help limit the search space of possibilities when considering defects. (5) Some of the approximations used in the present model may fail for some experimental systems. In particular, neglecting cross-correlations (e.g. between particle size and position), as is done in equations (28) and (32), may not be valid for certain kinds of particle assembly. It would be informative to test whether such correlations have measurable effects.
This research was carried out in whole at the Center for Functional Nanomaterials, Brookhaven National Laboratory, which is supported by the US Department of Energy, Office of Basic Energy Sciences, under contract No. DE-AC02-98CH10886.
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