Shedding Light on the Birth of Hybrid Perovskites

Our paper has been published in Chemistry of Materials! 

 

We combined synchrotron SAXS and environmental TEM to reveal the pathway leding from precursors to MAPI perovskites. This result was also featured on the cover.

 

Confinement Effects on the Structure of Entropy-Induced Supercrystals

Our paper has just been published in Small. Using LCTEM we were able to follow the desorption kinetics of gold nanoparticles from a 2D supercrystal deposited on a solid substrate.

The time and dose dependence of the process reveals that the desorption is due to loss of adhesion between the particles and the substrate, presumably due to charge buildup under the electron beam.

de Gennes narrowing

 In colloidal solutions, a widely-used relation connects the scale-dependent collective diffusion constant and the structure factor:
\begin{equation}
D_c(q) =\frac{D_0}{S(q)}
\label{eq:dGn}
\end{equation} and is generally known as de Gennes narrowing since its use by de Gennes in the context of quasi-elastic neutron scattering from liquids [1].

Power laws in small-angle scattering - part II

In the first part I showed that the SAXS intensity scattered by a platelet system goes like \( I(q) \sim q^{-2}\), at least in some intermediate (but as yet unspecified) q range. Here I will show that for thin rods this dependence becomes \( q^{-1}\), I will then derive the terminal (Porod) behaviour \( q^{-4}\) and briefly consider the transition between these two regimes. 

Power laws in small-angle scattering - part I

The small-angle X-ray scattering (SAXS) spectrum of particles with a well-defined shape (such as rods or platelets) is often characterized by a power-law dependence: \( I(q) \sim q^{-\alpha}\), where the exponent \( \alpha \) is directly related to the particle geometry. For "compact" particles, the large-\( q \) intensity scales as \( q^{-4}\) (Porod regime). Below, I'll give the most compact and yet -hopefully- understandable derivation I can think of for these power laws.

To simplify the derivation, we'll consider these objects as infinitely thin and infinitely large, meaning that we'll be looking at them on length scales much larger than their thickness and much smaller than their lateral extension. The approximation is legitimate, since it is in this range of length (or, conversely, scattering vector) that the power-law regimes are encountered.
As discussed above, the Patterson function is similar to the density and thus we will apply the same approximation to \(P(\mathbf{r})\), which is the natural descriptor of the system, due to its intimate relation with the intensity \(I(\mathbf{q}) = \left | \tilde{\rho}(\mathbf{q}) \right |^2\). 

Patterson functions

Fourier transforms

We will use the following convention for the Fourier transforms:\begin{equation} \begin{split} \rho(\mathbf{q}) = \mathcal{F} [\rho(\mathbf{r})](\mathbf{q}) & \triangleq \int_{\mathcal{V}} \rho(\mathbf{r}) \exp(-i \mathbf{q} \mathbf{r}) {\textrm d} \mathbf{r} \\ \rho(\mathbf{r}) = \mathcal{F}^{-1} [\tilde{\rho}(\mathbf{q})](\mathbf{r}) & \triangleq \dfrac{1}{(2\pi)^3}\int_{\mathbb{R}^3} \tilde{\rho}(\mathbf{q}) \exp(i \mathbf{q} \mathbf{r}) {\textrm d} \mathbf{q} \end{split} \label{eq:Fourierdef} \end{equation} where we integrate in real space over the (as yet unspecified) volume of interest \(\mathcal{V}\) and in reciprocal space over the entire \(\mathbb{R}^3\).

Wiener-Khinchine theorem

The autocorrelation of the real-space density function is \(\Gamma_{\rho \rho} = \int_{\mathcal{V}} \rho(\mathbf{r}') \rho(\mathbf{r}'+\mathbf{r}) {\textrm d} \mathbf{r}'\), which can be developed (using the second line of \eqref{eq:Fourierdef}) into:\begin{equation} \begin{split} \Gamma_{\rho \rho}(\mathbf{r}) & = \dfrac{1}{(2\pi)^6} \int_{\mathcal{V}} {\textrm d} \mathbf{r}' \rho(\mathbf{r}') \int_{\mathbb{R}^3} {\textrm d} \mathbf{q} \, \tilde{\rho}(\mathbf{q}) \exp(i \mathbf{q} \mathbf{r}') \int_{\mathbb{R}^3} {\textrm d} \mathbf{q}' \tilde{\rho}(\mathbf{q}') \exp[i \mathbf{q}' (\mathbf{r}' + \mathbf{r})] \\ & = \dfrac{1}{(2\pi)^6} \int_{\mathbb{R}^3} {\textrm d} \mathbf{q} \int_{\mathbb{R}^3} {\textrm d} \mathbf{q}' \tilde{\rho}(\mathbf{q}) \tilde{\rho}(\mathbf{q}') \exp(i \mathbf{q}' \mathbf{r}) \underbrace{\int_{\mathcal{V}} {\textrm d} \mathbf{r}' \exp[i (\mathbf{q} + \mathbf{q}') \mathbf{r} ]}_{(2\pi)^3 \delta (\mathbf{q} + \mathbf{q}')} \\ & = \dfrac{1}{(2\pi)^3} \int_{\mathbb{R}^3} {\textrm d} \mathbf{q}' \exp(i \mathbf{q}' \mathbf{r}) \tilde{\rho}(\mathbf{q}') \underbrace{\int_{\mathbb{R}^3} {\textrm d} \mathbf{q} \, \tilde{\rho}(\mathbf{q}) \delta (\mathbf{q} + \mathbf{q}')}_{\tilde{\rho}(-\mathbf{q}')} \end{split} \end{equation} where we assumed that everything converges, and thus we can interchange the integration order at will. Dropping the prime and noting that \(\tilde{\rho}(-\mathbf{q}) = \overline{\tilde{\rho}(\mathbf{q})}\) (Friedel's law) we finally prove the Wiener-Khinchine theorem: the autocorrelation function of the scattering length density is the inverse Fourier transform of its spectral density: \begin{equation} \Gamma_{\rho \rho}(\mathbf{r}) = \dfrac{1}{(2\pi)^3} \int_{\mathbb{R}^3} {\textrm d} \mathbf{q} \exp(i \mathbf{q} \mathbf{r}) \left | \tilde{\rho}(\mathbf{q}) \right |^2 = \mathcal{F}^{-1} [|\tilde{\rho}(\mathbf{q})|^2] \label{eq:WK} \end{equation}

The Patterson function

As discussed during the lecture, the scattered intensity is precisely the spectral density of \(\rho(\mathbf{r})\): \(I(\mathbf{q}) = \left | \tilde{\rho}(\mathbf{q}) \right |^2\). Unlike \(\rho(\mathbf{r})\) itself, its autocorrelation \(\Gamma_{\rho \rho}(\mathbf{r})\) is directly accessible via Fourier transform from the experimental data, provided their quality and \(q\)-range are sufficient. Since it is frequently used in crystallography, it has a specific name: the Patterson function, denoted by \(P(\mathbf{r})\).

Correlation and convolution

In reciprocal space, the signal recorded by the detector at position \(\mathbf{q}\) is characterized by the electric field amplitude \(E(\mathbf{q})\), but the experimentally accessible quantity is its modulus squared, the intensity \(I(\mathbf{q}) = |E(\mathbf{q})|^2\). In real space, the structure is described by the density function \(\rho(\mathbf{r})\) but, as we will see in the next post, it is useful to define two new types of functions "of the type of the square", but where the two instances of \(\rho\) are evaluated in different space points.

Polymorphous Packing of Pentagonal Nanoprisms

 Our paper has just been published in Nano Letters.

Pentagonal packing is a long-standing and rich mathematical topic: in two dimensions, the optimal (highest packing fraction η=0.921) packing of regular pentagons is a double-lattice arrangement, called the "pentagonal ice ray".

We pack pentagonal nanoprism into long-range mesocrystals by evaporation induced self-assembly and find the ice-ray structure, but also an arrangement devised by Albrecht Dürer in the 16^th century, with a slightly lower packing fraction (η=0.854), as well as intermediate polymorphs that can be obtained by a continuous sliding transformation between these two configurations.

We discuss the subtle relation between the orientational and positional order, as well as the presence of defects in the lattices.

Extracting the morphology of gold bipyramids from SAXS experiments

Our paper has just been published in Journal of Applied Crystallography.

 

We validate the use of the bicone model for extracting the form factor of gold bipyramids in solution from SAXS data.