Paper:Periodic lattices of arbitrary nano-objects: modeling and applications for self-assembled systems

This is a summary/discussion of the results from:

• Yager, K.G.; Zhang, Y.; Lu, F.; Gang, O. "Periodic lattices of arbitrary nano-objects: modeling and applications for self-assembled systems" Journal of Applied Crystallography 2014, 47, 118–129. doi: 10.1107/S160057671302832X

This paper describes the modeling of x-ray and neutron scattering data for lattices of nano-objects. The presented formalism enables both simulation or fitting of small-angle scattering data 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.

Math

Formalism

Randomly oriented crystals give scattering intensity:

${\begin{alignedat}{2}I_{0}(q)&=C\langle |F(\mathbf {q} )|^{2}S_{0}(\mathbf {q} )\rangle \\&=CP(q)\left\langle {\frac {|F(\mathbf {q} )|^{2}}{P(q)}}S_{0}(\mathbf {q} )\right\rangle \\&=CP(q)S_{0}(q)\end{alignedat}}$

Where the structure factor is defined by an orientational average (randomly oriented crystal(s)):

$S_{0}(q)\equiv \left\langle {\frac {|F(\mathbf {q} )|^{2}}{P(q)}}S_{0}(\mathbf {q} )\right\rangle$

We can compute a structure factor for a crystal-like material by considering an ideal lattice factor:

$Z_{0}(q)=c\sum _{\{hkl\}}^{m_{hkl}}{\frac {1}{q_{hkl}^{2}}}\left|\sum _{j=1}^{n_{c}}F_{j}(M_{j}\cdot \mathbf {q} _{hkl})e^{2\pi i(x_{j}h+y_{j}k+z_{j}l)}\right|^{2}e^{-\sigma _{D}^{2}q_{hkl}^{2}a^{2}}L_{hkl}(q-q_{hkl})$

Where c is a constant, and L is the peak shape and the $\sigma _{D}$ term is the Debye-Waller factor. So that the structure factor is:

$S_{0}(q)={\frac {Z_{0}(q)}{P(q)}}={\frac {c}{q^{2}P(q)}}\sum _{\{hkl\}}^{m_{hkl}}\left|\sum _{j=1}^{n_{c}}F_{j}(M_{j}\cdot \mathbf {q} _{hkl})e^{2\pi i(x_{j}h+y_{j}k+z_{j}l)}\right|^{2}e^{-\sigma _{D}^{2}q_{hkl}^{2}a^{2}}L_{hkl}(q-q_{hkl})$

And the intensity is then:

$I_{0}(q)={\frac {c}{q^{2}}}\sum _{\{hkl\}}^{m_{hkl}}\left|\sum _{j=1}^{n_{c}}F_{j}(M_{j}\cdot \mathbf {q} _{hkl})e^{2\pi i(x_{j}h+y_{j}k+z_{j}l)}\right|^{2}e^{-\sigma _{D}^{2}q_{hkl}^{2}a^{2}}L_{hkl}(q-q_{hkl})$

The form factor amplitude is computed via:

${\begin{alignedat}{2}F(\mathbf {q} )&=\int \limits \rho (\mathbf {r} )e^{i\mathbf {q} \cdot \mathbf {r} }\mathrm {d} V\\\end{alignedat}}$

Or, for an object of uniform density within the volume V:

${\begin{alignedat}{2}F(\mathbf {q} )&=\Delta \rho \int \limits _{V}e^{i\mathbf {q} \cdot \mathbf {r} }\mathrm {d} \mathbf {r} \\\end{alignedat}}$

The (isotropic) form factor intensity is an average over all possible particle orientations:

${\begin{alignedat}{2}P(q)&=\left\langle |F(\mathbf {q} )|^{2}\right\rangle \\&=\int \limits _{S}|F(\mathbf {q} )|^{2}\mathrm {d} \mathbf {s} \\&=\int _{\phi =0}^{2\pi }\int _{\theta =0}^{\pi }|F(-q\sin \theta \cos \phi ,q\sin \theta \sin \phi ,q\cos \theta )|^{2}\sin \theta \mathrm {d} \theta \mathrm {d} \phi \end{alignedat}}$

For a mixture of particles, the measured isotropic form factor intensity is a weighted sum of the various constituents:

${\begin{alignedat}{2}P_{\mathrm {total} }(q)&=\sum _{j=1}^{n_{c}}c_{j}P_{j}(q)\end{alignedat}}$

Where the $c_{j}$ are scaling factors (concentrations) for the constituents.

Background Scattering

In actual measurements, the intensity has a background:

${\begin{alignedat}{2}I_{\mathrm {meas} }(q)&=P(q)S_{\mathrm {ideal} }(q)+pq^{-\alpha }+c\end{alignedat}}$

Where c is a constant background, p is a constant prefactor, and $\alpha$ describes the scaling of the q-dependent background (which dominates at small q). If one experimentally obtains the structure factor by doing: $S_{\mathrm {meas} }(q)=I_{\mathrm {meas} }(q)/P(q)$ this implies:

${\begin{alignedat}{2}S_{\mathrm {meas} }(q)&={\frac {I_{\mathrm {meas} }(q)}{P(q)}}\\&={\frac {P(q)S_{\mathrm {ideal} }(q)+pq^{-\alpha }+c}{P(q)}}\\&=S_{\mathrm {ideal} }(q)+{\frac {pq^{-\alpha }+c}{P(q)}}\end{alignedat}}$

Background from Form Factor

In some cases, the background scattering comes from the (isotropic) form factor of the particles. For example if clusters are in solution alongside the free particles. In such cases the structure factor will tend towards 1 form large q:

$\lim _{q\rightarrow \infty }S_{\mathrm {meas} }(q)=S_{\mathrm {ideal} }(q)+{\frac {P(q)}{P(q)}}=0+1$

Note that most structure-factors decay as q⁶, so a background that takes this into account would be:

${\begin{alignedat}{2}\mathrm {background} =p_{1}q^{-\alpha }+p_{2}q^{-6}\end{alignedat}}$

Sources

The paper builds upon work presented in:

Basics

• Michael Kotlarchyk and Sow‐Hsin Chen Analysis of small angle neutron scattering spectra from polydisperse interacting colloids J. Chem. Phys. 1983, 79, 2461 doi:10.1063/1.446055
• Georg Pabst, Michael Rappolt, Heinz Amenitsch, and Peter Laggner Structural information from multilamellar liposomes at full hydration: Full q-range fitting with high quality x-ray data Phys. Rev. E 2000, 62, 4000–4009 doi: 10.1103/PhysRevE.62.4000

Formalism

• Supplementary Information of: Matthew R. Jones, Robert J. Macfarlane, Byeongdu Lee, Jian Zhang, Kaylie L. Young, Andrew J. Senesi, and Chad A. Mirkin DNA-nanoparticle superlattices formed from anisotropic building blocks Nature Materials 2010, 9, 913-917 doi: 10.1038/nmat2870
• S. Förster, A. Timmann, M. Konrad, C. Schellbach, A. Meyer, S.S. Funari, P. Mulvaney, R. Knott, J. Scattering Curves of Ordered Mesoscopic Materials Phys. Chem. B 2005, 109 (4), 1347–1360 DOI: 10.1021/jp0467494