Evidence for a chemical clock in oscillatory formation of UiO-66

Chemical clocks are often used as exciting classroom experiments, where an induction time is followed by rapidly changing colours that expose oscillating concentration patterns. This type of reaction belongs to a class of nonlinear chemical kinetics also linked to chaos, wave propagation and Turing patterns. Despite its vastness in occurrence and applicability, the clock reaction is only well understood for liquid-state processes. Here we report a chemical clock reaction, in which a solidifying entity, metal–organic framework UiO-66, displays oscillations in crystal dimension and number, as shown by X-ray scattering. In rationalizing this result, we introduce a computational approach, the metal–organic molecular orbital methodology, to pinpoint interaction between the tectonic building blocks that construct the metal–organic framework material. In this way, we show that hydrochloric acid plays the role of autocatalyst, bridging separate processes of condensation and crystallization.


Supplementary Figure 3
In-situ WAXS data obtained for synthesis of UiO-66(Zr) in different solutions. "Eq." represents equivalent (with 1 eq. representing the typical amount denoted above). The colours are arbitrary.

Integral SAXS parameters
The definition of scattering vector q is q=(4π/λ) sinθ (dimension in nm -1 ), where θ is the Bragg angle that is usually used in XRD. The physical meaning of q is that all electrons in a plane perpendicular to q are in phase with respect to incoming light. The location of a particular electron with respect to an electron in the origin (spherical coordinates (0,0,0)) is defined by r. Therefore the phase of the electron is qr and the outcoming phase factor as recorded is given by e -iqr , which is the complex representation that is convenient for Fourier Transformation (FT). The magnitude of q is representative for the size of the scattering entities; the typical scale of the scatterer is of the order 2π/q. Now, the recorded amplitude is the sum of all recorded waves, which in our case is defined using an electron density ρ(r), integrated over all three dimensions: This is a Fourier Integral. Mathematically speaking, we need to square the amplitude by multiplying it by its complex conjugate to yield intensity. Physically speaking, only a pair of electrons can be responsible for interference, and therefore the diffraction pattern. The result of both interpretations is the same and yields a six-fold integral over two volume elements and two local spaces, with the distance between the electron pair defined as r 1 -r 2 . Rather than evaluating this integral, every pair of electrons is defined by one single, fictitious point r =r 1 -r 2 . This transformation is a FT in itself, and yields a new relation for I(q): Here, the angular brackets denote the auto-correlation, the summarizing of all electron pairs.
An additional advantage to mathematical simplification is that this -new-electron density is averaged over all electrons in the structure, and is highly characteristic for the morphology of the scatterer. When we take into account the isotropy in space, the problem becomes one-dimensional in r, and Debye formulated an orientationally averaged term for the phase factor. [31]

Eq. S.3
Then, making use of the absence of long-range order in a solute or dispersed system of scatterers, one measures only fluctuation relative to the background: For which: that has defined an auto-correlation by: γ(r) is therefore the average of the two fluctuations occurring at a distance r (the distance between the electron pair. This auto-correlation function is very distinct for the geometry of scattering entity. In our case we don't have very well defined crystals and we use the pairdistance distribution p(r) instead, which is derived below. First, one must summarize the simplifications and formulations of Eq. S.3 to Eq. S.6, in which the equation for intensity is made one-dimensional in spherical coordinates, the phase factor is simplified by the Debye equation, and electron density is transformed into an autocorrelation which is a deviation from the background and is directly characteristic for the geometry of the scatterer.
One can now write Eq. S.2 as:

Eq. S.7
This functional describes the shape of the intensity decay obtained in the SAXS experiment. The Inverse Fourier Transform of I(q) yields the function for γ(r), which by multiplication with r 2 yields p(r), carrying dimension [nm]: ! !

Eq. S.8
The p(r) function describes the paired-set of all distances between points within an object, and is used here to detect conformational changes in time within the growing MOF particles.
Before continuing, we must first consider that in order to perform FT analysis, a smooth dataset spanning from 0 to ∞ is needed. Experimental SAXS data have a cut-off at the beam stop at a small q-value (the value q = 0 can't be measured), plus a nonzero background value at large q, resulting in infinite values for the Fourier integral. The first problem can be solved by extrapolating SAXS data to q = 0 by using an approximation derived by Guinier in 1939 for single-particle scattering.
[32] The derivation is briefly discussed here. Two major criteria -which are not always met in literature -but which are required for the Guinier approximation to be valid are the following: (1) the particles must be well-separated in solution (2) there scatterer must have a centre of symmetry. In other words, the solution in which the MOF crystallises ought to be (at least relatively) dilute, and the MOF crystals and agglomerates should possess an axis of symmetry. The first criterium is met since -even though not all 2-aminoterephthalic acid is dissolved at the start of synthesis -crystallisation occurs from clear solutions. The second criterium also holds for the NH 2 -MIL-53(Al) and NH 2 -MIL-101(Al) frameworks, which on average, and to reasonable extent, possess a centre of symmetry. With condition (1), one can focus on the scattering by a single particle, and condition (2) simplifies the phase factor e -iqr , which has now become real for any orientation, and can be replaced by cos(qr). Now, Eq. S.1, for the amplitude, becomes:

Eq. S.10
Which leads to the Guinier Approximation

Eq. S.11
Where Δn e = (ΔρV) 2 and R g is the Radius of gyration.
In the data treatment R g is used to extrapolate data to q = 0. The second problem, the subtraction of the background and infinite integration can be treated using the approximation for the final slope as derived by Porod.
[32] For this, a scaled version of the correlation function is defined, such that:

Eq. S.12
This function was called the characteristic by Porod and is analytically derivable for most welldefined, constant morphologies, which opens the door to modeling This is done to obtain a parameter, γ 0 (r), which is related to the electronic structure (read: morphology) of the scatterer, and scaled by the electron density difference, which is usually assumed constant. γ 0 (r) plays an important role in modeling with SAXS, by defining a correlation length, but this is not considered in our data treatment, and will not be further considered (the interested reader is referred to the standard works of Glatter & Kratky and Feigin & Svergun). [32,33] For the Porod regime of the decay in SAXS (the higher q-values), γ 0 (r) can be expanded into a power series Now, it is possible to define γ 0 (r) as the volumetic overlap of a particle and a new particle moved by a distance r. For small r, it is obvious that γ 0 (r) is determined by the surface of the particle. This leads to the derivation of the fourth power law (see again: Glatter and Kratky), defined as:

Eq. S.14
Here S/V is the Porod surface-to-volume ratio of the scatterer, and Q inv is the Porod Invariant.
The latter needs consideration, as it is a fundamental parameter in SAXS. Setting r = 0 in γ(r) leads to: The final statement is easily derived from the definition of the auto-correlation (Eq. S.6), for which: Eq. S.16 follows from the fact that, at large electron-pair distances, 'correlation is lost', i.e. the average local scattering is equal to the scattering of the background. It must be stated that, although here the limit of r to infinity is taken, in reality, γ(r) goes to zero for very finite values that lie well within colloidal dimensions. Eq. S.17 follows directly from the definition of the correlation function in Eq. S.6. The result is important as it means that the integral factor in Eq.S.15 is only directly proportional to the mean square fluctuation of electron density, meaning that it is fully independent of temperature, crystallinity and/or crystal morphology, and is therefore called the Porod Invariant Q inv :

Eq. S.18
Using this definition, Eq. S.11 and Eq. S.14 are used to extrapolate the pattern obtained by synchrotron SAXS data (figure 3B.1). Here the power law in Eq. S.14 is used to extrapolate the q-space to infinity, using I(q) = x 1 q -4 + x 2 . Here, x 1 represents SQ/πV while x 2 corrects for incomplete subtraction of the background from experimental data. This data treatment yields a smooth dataset without Bragg peaks suitable for the application of eqs 3B.11 and 3B.14. Functions and parameters that can thus be calculated in this time-resolved experiment are: p(r), R g , V, and S/V. The first and the third are of most importance in this chapter, and therefore leaves us with the definition of V, the Porod Volume. Using the condition that for a single scatterer: