Autocatalysis and Oriented Attachment Direct the Synthesis of a Metal-Organic Framework.

Synthesis of porous, covalent crystals such as zeolites and metal-organic frameworks (MOFs) cannot be described adequately using existing crystallization theories. Even with the development of state-of-the-art experimental and computational tools, the identification of primary mechanisms of nucleation and growth of MOFs remains elusive. Here, using time-resolved in-situ X-ray scattering coupled with a six-parameter microkinetic model consisting of ∼1 billion reactions and up to ∼100 000 metal nodes, we identify autocatalysis and oriented attachment as previously unrecognized mechanisms of nucleation and growth of the MOF UiO-66. The secondary building unit (SBU) formation follows an autocatalytic initiation reaction driven by a self-templating mechanism. The induction time of MOF nucleation is determined by the relative rate of SBU attachment (chain extension) and the initiation reaction, whereas the MOF growth is primarily driven by the oriented attachment of reactive MOF crystals. The average size and polydispersity of MOFs are controlled by surface stabilization. Finally, the microkinetic model developed here is generalizable to different MOFs and other multicomponent systems.

Introduction

Multicomponent porous crystals, such as zeolites and metal–organic frameworks (MOFs), have greatly impacted the fields of gas separation, catalysis, separations, and adsorption.[1−4] The fundamental processes behind the crystallization of these porous materials are not adequately described by the well-known theories of classical crystallization. Although the classical crystallization theories were developed for noncovalent solidification of single-component systems, it is widely used to describe and draw insight into covalent crystallization of MOFs.[5,6] However, recent research on zeolite and MOF crystallization has shown multiple deviations from classical crystallization theories.[7,8]

Two-step nucleation is one such deviation that has been invoked for certain MOFs such as ZIF-8 and zeolites.[9−11] In fact, a three-step crystallization mechanism has been proposed recently for ZIF-8.[7] Nonclassical growth mechanisms have also been recently proposed as a deviation from the classical monomer-by-monomer based crystal growth, including for composite systems such as protein-MOF composites.[7,12−14] Nonclassical growth can occur because of the attachment of oligomers or amorphous clusters or through the oriented attachment of nanocrystals.[10,15] Previous literature has already indicated that nonclassical growth may play a part in MOF synthesis, where Cu2(ndc)2(dabco) nanoparticles have been observed to grow by oriented attachment using ex-situ transmission electron microscopy (TEM).[16] However, a mathematical description of the nonclassical nucleation mechanism needs to be formulated.

Studies aimed at understanding MOF formation generally utilize a combination of in-situ techniques combined with empirical models such as the Johnson–Mehl–Avrami–Kolmogorov (JMAK) and Gualtieri models.[17,18] Although these models provide information on crystallization rate constants, growth dimensionality, and probability of nucleation, the inherent empiricism limits the fundamental insights into the crystallization mechanism. Thus, the development of a kinetic model capable of detailing formation mechanisms remains a significant hurdle in understanding MOF crystallization.

In this article, we have utilized a combination of in-situ wide-angle X-ray scattering (WAXS) and reaction kinetic modeling to describe the crystallization of the prototypical MOF UiO-66 (see Figure ).[19−24] The novel reaction kinetics model utilized here makes several predictions that are fundamentally different than previously enumerated for MOF crystallization. We first show that a formation of a “building unit” (termed SBU) occurs through an autocatalytic mechanism. Second, the attachment of these SBUs to each other can then proceed through either classical or nonclassical (oriented attachment) growth, analogous to chain or step extension in polymeric systems, respectively. Third, we propose that there is a limit to the size of MOF crystal that can undergo oriented attachment. Finally, we show that the reaction kinetics system of equations can be reduced to obtain parameters related to crystal nucleation and growth, such as the activation energetics and rates, that are commonly obtained using other crystallization theories.

Figure 1

(a) Schematic of governing processes occurring during the synthesis of the MOF UiO-66. The first phase of MOF synthesis involves the autocatalytic initiation, where a metal node and a linker adsorb on an existing SBU surface, followed by their reaction to form the SBU, followed by the SBU desorption. Following initiation, the free SBUs (SBU1) undergo a chain reaction by sequential attachment with MOF clusters (SBU). MOFs can also grow via step reaction where intermediate size clusters (SBUi) undergo oriented attachment to form a bigger cluster. Finally, after attaining a sufficiently large size of MOF (SBUnk), the oriented attachment stops because of the stabilization of the MOF surface. (b) In-situ data showing integral intensity from the (111) peak diffraction, representative of crystalline fraction, of UiO-66 as a function of time (red curve). Four different phases can be identified in the sigmoidal growth curve of the crystalline volume fraction. The induction phase (blue shaded region) represents the onset of crystal nucleation due to autocatalytic initiation and chain reaction; the exponential growth phase (green shaded region) represents the steady growth of MOFs due to oriented attachment or step reaction; the transition phase (yellow shaded region) is determined by the onset of the surface-stabilization-driven termination reaction; and the stationary phase (red shaded region) is attained when there is negligible change in the crystalline volume fraction.

Results and Discussion

In-situ WAXS was used to measure UiO-66 formation as a function of different reaction temperatures and initial concentrations of the Zr-oxo cluster and H2BDC linker. Briefly, a reactor containing Zr metal oxo-cluster solution (9.5 mL) was stirred and held at a constant temperature, and a small quantity of linker solution (0.5 mL) was injected quickly. The integral intensity of the dominant diffraction plane (111) relative to the steady-state value was used to estimate the volume fraction of UiO-66 crystals produced in the reactor (see Section S1 and Figures S1 and S2 for details).[17,25] The increase in crystal volume fraction (or crystalline yield) over time follows a sigmoid curve (Figure ), which typically consists of four phases, named induction, (or lag), exponential, transition, and stationary (Figure b). The features of the sigmoid curve are shown in Figure S3 discussed in Section S2. Such a sigmoidal growth curve has been reported for the synthesis of zeolites, MOFs, and other nanomaterials.[26−31] The onset of the induction phase (or induction time) decreases with increasing temperature (Figures a–c) and increases with decreasing the initial concentration of Zr (Figures d, f) and linker (Figure e). The onset of the transition phase also follows a similar trend as induction time. The slope of the exponential growth phase decreases with increasing temperature and increases with decreasing initial concentration of Zr and linker.

Figure 2

Normalized crystalline volume fraction for experimental (red) and theoretical (black dashes) results for (a)14.4 mM Zr, 16.3 mM H2BDC at 25 °C; (b) 14.4 mM Zr, 16.3 mM H2BDC at 35 °C; (c) 14.4 mM Zr, 16.3 mM H2BDC at 45 °C (theoretical results predicted using the Arrhenius relationship); (d) 7.2 mM Zr, 16.3 mM H2BDC at 25 °C; (e) 14.4 mM Zr, 8.15 mM H2BDC at 25 °C; and (f) 28.8 mM Zr, 8.15 mM H2BDC at 25 °C (theoretical results predicted using Arrhenius relationship). The black dotted line shows the calculated volume fraction due to the formation of nuclei for each condition. The background fill represents the relative rate contribution to crystal volume fraction from initiation (blue), chain growth (green), step growth (yellow), and termination (red). Panels enclosed with the black box represent the results used to validate the microkinetic model, and panels enclosed with the blue box represent the results predicted using the microkinetic model. The y-axis labels in panel a apply to panels b–f as well.

To understand the mechanistic effects of the temperature and reactant concentrations on the induction time and the evolution of crystallization rates, we developed a comprehensive microkinetic model[32] considering a full reaction network (see Section S3 for details) consisting of ∼1 billion reactions while producing UiO-66 with up to 100 000 SBUs, or 62.5 nm crystallite size assuming an octahedral morphology. The microkinetic model[33] considers an initiation reaction for the synthesis of secondary building unit (SBU) that has a Zr metal node connected with six linkers, chain reactions involving sequential attachment of SBUs to growing MOFs, step reactions allowing oriented attachment of MOFs, and a termination reaction, where the step reaction yields a stable MOF that does not further undergo oriented attachment to a larger particle. Briefly, the activation energy and rate constant for initiation reaction were estimated and validated using integral intensities as a function of temperature in Figure a–c, and the reaction orders were determined from Figure a, d, and e. Figure The rate constants estimated for the chain and step reactions followed Flory’s approximation[34] such that they were independent of the size of oligomers (see Sections S4–S6, Figures S4–S17, and Table S1).

Figure 3

Validation of autocatalysis mechanism: (a) Complete FT-IR spectra of UiO-66 with the carbon–oxygen double bond peak highlighted. (b) Comparison of normalized volumetric rates observed in in-situ WAXS and FT-IR studies with the expected normalized volumetric rate theoretically expected without an autocatalysis mechanism. (c) Activation energy of SBU formation on the existing building unit (SBU) and unit cell (lattice). The red line represents the energetic barrier to form an SBU on another SBU, and the blue line represents the energetic barrier to form an SBU on a crystal lattice composed of four SBUs. (d) Estimated crystalline fraction curves with and without autocatalysis. The red curve in panel d is the theoretically estimated crystalline fraction curve shown by the dashed line in Figure a.

Figure shows the estimated crystal volume fraction and the normalized relative rates of initiation, chain, step, and termination reactions obtained from the microkinetic model. The synthesis of UiO-66 proceeds with an initiation reaction to form SBUs, which are gradually consumed by chain reactions to yield nuclei (defined as the smallest crystal size observed by WAXS at the onset of crystal volume fraction, see Table S2). The dotted lines in Figure show the calculated nucleation rate for the various conditions. As the reaction temperature increases (Figure a–c), the ratio of chain reaction rate to initiation reaction rate increases, leading to the consumption of SBUs and to the formation and consumption of nuclei at a faster rate, leading to a shorter induction time but also a decrease in the total volume fraction of nuclei present in solution (see Figure S18). The theoretical results shown in Figure c, f are predicted using the error minimized values of the parameters involved in the model (see Section S7).

On the other hand, decreasing the concentration of metal node or linker decreases the rate of the initiation reaction and hence the concentration of SBUs available for the subsequent chain reaction. The lower number of SBUs reduces the rate of chain reaction more as compared to the initiation reaction (Figures d–f). Therefore, both the induction time and the volume fraction of nuclei increase with decreasing the initial concentration of metal node and linker, contradictory to conventional models.[17,18,35]

We focused on understanding the initiation reaction mechanism as it determines the induction time. In ideal reactions, the concentration of free metal node and linker decreases stoichiometrically with an increase in the crystal volume fraction. Therefore, decreasing reactant concentrations should cause a monotonic decrease in the SBU rate of formation, resulting in the MOF volumetric growth rate. However, Figure shows an increase in the volumetric growth rate in the induction phase even as the metal node and linker deplete in the solution (see Figure S19), which indicates an autocatalytic initiation reaction.

Here we hypothesize that the SBU and its aggregates act as templates for further formation of SBUs via an autocatalytic initiation reaction (see Section S8 and Figure S20–S22). In this reaction, we hypothesize that the metal node adsorbs to the MOF template and reacts with free linkers to form a surface SBU, which then desorbs into the solution. To validate this hypothesis, we analyzed the spectra of UiO-66 during synthesis at various times using Fourier transformed infrared spectroscopy (FT-IR). Figure a shows the complete FT-IR spectra with the carbon–oxygen double bond (C=O 1670 cm–1) peak highlighted. The C=O bond on the terephthalic acid must be broken for the formation of UiO-66. Hence, the absorbance intensity shows a negative value at various times of UiO-66 synthesis due to background subtraction. Furthermore, the rate of normalized intensity change of C=O peak shows a rate increase, followed by a decrease as shown in Figure b, which is similar to the rate increase and decrease seen in in-situ XRD (WAXS). The rate without autocatalysis does not match the rate profiles as seen in in-situ WAXS and FT-IR. The in-situ experiments validate the autocatalysis mechanism of UiO-66 formation. The energy profiles in Figure c are the backward barriers obtained from the potential of the mean force approach,[36] where the transition state (peak) is described by a metal node in close proximity to the template, which then reversibly attaches with linkers to produce SBU at reaction coordinate = 1. Figure c shows a drastic reduction in the activation energy of SBU formation from 278 to 92 kJ/mol when the size of the template increases from an SBU to the unit cell composed of 4 SBUs. In Figure d, the theoretical result without the autocatalytic assumption does not show the sigmoidal curve and does not achieve the crystalline fraction observed in experiments. To further validate the hypothesis that the autocatalysis of initiation reaction proceeds through the self-templating mechanism and not just heterogeneous nucleation, we added graphite particles to the reactor to triple the accessible surface area (see Figure S23). Table S3 shows a slight reduction in the induction time from 146 to 110 s by the addition of graphite, indicating that additional nontemplating nucleation sites have minimal influence on the induction mechanism of UiO-66.

Figure shows the dominant pathway of nuclei formation at various experimental conditions obtained from the microkinetic model and yields insight into the evolution of the reaction network at the onset of nucleation. The nodes in each panel of Figure represent the cluster sizes increasing from 1 SBU1 to 129 SBU1 from left to right, as shown on the number line below each panel. Hence, the rightmost node represents the crystal of critical size of 8.6 nm. The solid black lines connecting the nodes represent the dominant pathway of nuclei formation, and the thickness of the lines is relative to the highest volumetric rate theoretically obtained in all of the experimental conditions. Figure a shows the experimental condition with initial concentrations of 14.4 mM Zr and 16.3 mM H2BDC at 25 °C. For this experimental condition, the onset of nucleation is near 45 s. In this case, nucleation occurs due to the attachment of clusters with similar lattice structures. Hence, the cluster sizes increase in powers of two, representing the attachment of SBUs with similar lattice structures. However, at the higher temperature shown in Figure b, the cluster size increase is greater than powers of two due to high volumetric rates at the onset of nucleation of 6 s. The increase in cluster size in powers greater than two is also observed for the case of halved initial concentration metal oxo-cluster concentration at 25 °C as shown in Figure c. The size increase, in this case, depends on the concentration of the clusters. The clusters with limiting concentration do not significantly contribute to the volumetric rate, and hence the dominant pathway of critical size nuclei formation shows longer jumps regardless of very low volumetric rates of the reactions involved.

Figure 4

Evolution of subcritical clusters to the critical size of the nucleus. The title represents the experimental conditions: (a) 14.4 mM Zr, 16.3 mM H2BDC, 25 °C; (b) 14.4 mM Zr, 16.3 mM H2BDC, 45 °C; (c) 7.2 mM Zr, 16.3 mM H2BDC, 25 °C. The nodes represent the cluster size, and the size is shown based on the number of SBU1 in the cluster. Each node is a part of the reaction network where the leftmost node represents the crystal size of 1 SBU1, and the rightmost node represents the crystal size of 129 SBU1. The size increase from left to right is represented by the number line given below each panel. The rightmost node represents the critical size (8.6 nm) of nuclei observed in in-situ WAXS. The black lines between the two nodes represent the dominant pathway of formation of critical size of nuclei. The thickness of the black line is relative to the highest volumetric rate observed in all of the experiments. Higher thickness implies a higher volumetric rate. Diamond-shaped nodes are part of the dominant pathway of nuclei formation. The color of each node represents the total number of reactions emerging out from each node. The pathway is shown for the time near the onset of nucleation as observed in in-situ WAXS.

The relative rates of the initiation reaction and chain reaction determine the induction time and nucleation rate. Similarly, the relative rates of chain and step reactions (or oriented attachment) govern the average crystal size and polydispersity. From the microkinetic model, we see that the step reaction contributes more to decreasing the nuclei number density and increasing the overall growth of each MOF crystal compared to the chain reaction (Figure a–f). The exponential phase of the sigmoidal growth curve is primarily due to step reactions. The transition phase of the growth is due to the onset of termination reactions, where MOF crystals are large and stop undergoing further step reactions. Finally, the stationary phase begins when step reactions are almost completely terminated, and the remaining growth is primarily due to chain reactions (Figure ).

Figure 5

Grain size for experimental (red curve) and theoretical (dashed curve) results for (a) 14.4 mM Zr, 16.3 mM H2BDC at 25 °C; (b) 14.4 mM Zr, 16.3 mM H2BDC at 35 °C; (c) 14.4 mM Zr, 16.3 mM H2BDC at 45 °C; (d) 7.2 mM Zr, 16.3 mM H2BDC at 25 °C; (e) 14.4 mM Zr, 8.15 mM H2BDC at 25 °C; (f) 28.8 mM Zr, 16.3 mM H2BDC at 25 °C. The background fill represents the volumetric rate contribution to grain size from chain growth (green), step growth (yellow), and termination (red). The total volumetric rate contribution from step growth is at least 1.5 times higher than the chain growth for all the cases. (g) Nonclassical addition of various cluster sizes for step growth; (h) termination events such as lattice mismatch, linker covered surface sites, and surface stabilization; (i) theoretical size distribution of crystal sizes as a function of the synthesis temperature; (j) theoretical crystal size distributions as a function of the starting metal oxo cluster concentration; (k) dynamics of free linkers in the solution per surface SBU; and (l) scaling relationship between the number of free linkers per surface SBU at a steady-state and the number of initial linkers per surface SBU of MACS. In panels a–f, the dotted line represents the calculated contribution of the chain growth to the overall grain size. Panels enclosed with the black box represent the results used to validate the microkinetic model, and panels enclosed with the blue box represent the results predicted using the microkinetic model. The y-axis labels in panel a apply to panels b–f as well.

To obtain insights into UiO-66 growth, we used the Scherrer equation to relate the full-width half-maximum (FWHM) of the diffraction from the (111) plane to the average grain size of MOFs (see Section S9). Figure a–c shows a sigmoidal curve (red) of the experimentally obtained average grain size as a function of time, at reaction temperatures of 25, 35, and 45 °C, respectively (see Figure S24 for versions with linear x-axis). As the temperature increases, the time for the onset of growth (defined as the onset of WAXS signal acquisition above background) decreases, and the volumetric growth rate increases. In the generalized reaction model, the MOF can grow either by a chain reaction (see Figure S25), indicative of a classical growth mechanism or a through step reaction (Figure g), indicative of a nonclassical growth mechanism. The simulated growth curves with (dash) and without (dot) the step reaction contribution show that the growth contribution from the step reaction is significantly higher than that for the chain reaction.

Corroborating the observation that step growth significantly contributes to the grain size, the colormap of the volumetric rate of reaction (molar rate divided by the molar density) in Figure a–c confirms that step reaction dominates MOF growth. Here, the green background color shows the volumetric rate contribution due to chain growth, orange shows the volumetric rate contribution due to step growth, and red shows the volumetric rate contribution due to the onset of termination. Figure a–c also shows that the step reaction rate increases with increasing temperature. This increase in the step growth rate leads to an earlier onset of growth as well. Figure i shows the predicted size distribution of the UiO-66 at the three different reaction temperatures from Figure a–c at a steady state. Although the variance does not change significantly with temperature, the polydispersity index decreases with increasing temperature, 1.0686 at 25 °C, 1.0577 at 35 °C, and 1.0492 at 45 °C. These results agree with existing analytical models of chain and step polymerization, where the average length of polymer is directly proportional to the growth rate constant, and polydispersity is inversely dependent on the growth rate constant.[34,37]

Panels d and e in Figure show the growth dynamics when the initial concentrations are halved for metal node and linker, respectively. The average grain size increases with decreasing metal node and linker concentration, which is due to a higher degree of oriented attachment and step-growth rate (orange). The inverse is also true and can be seen in Figure f, where two times higher concentration of metal node reduces the step growth rate resulting in smaller grain size at the steady-state. Figure j shows the increase in the average size and variance of the size distribution when the metal node concentration is decreased from 14.4 to 7.2 mM and the decrease in the average size and variance when the metal node concentration is increased from 14.4 to 28.8 mM at a fixed linker concentration and temperature. Using the microkinetic model, we show that stabilization of the MOF crystal surface is a primary reason for the termination of the step reaction and is responsible for the steady-state MOF size.

There are two major factors that determine termination: lattice mismatch during oriented attachment of MOFs and the reactivity of MOF surface (Figure h).[38,39] We propose that the oriented attachment of two MOF crystals begins with the alignment of surface lattices followed by a reaction between the surface SBUs, as observed before for nanoparticles.[39] Although the lattice alignment is dependent on the MOF surface and rotational motion, the surface reactivity is primarily governed by the number of exposed linkers per SBU on the surface of the two attaching crystals. For instance, the stable (111) surface of UiO-66 MOFs cannot undergo oriented attachment when the number of exposed linkers per SBU in both crystals is either four (linker saturated) or zero (modulator saturated) (see Figure S26). Thus, although a defect-free oriented attachment will occur only with exactly one exposed linker per pair of attaching SBUs, a lower number of linkers per SBU will still keep the surface reactive but will induce a higher number of defects in the MOF. In contrast, a higher number of linkers per SBU will compete for the vacant site for attachment to the metal node, resulting in a lower volumetric rate and larger crystal sizes. Furthermore, as the reaction proceeds faster because of the high temperature (Figure c), crystals are more likely to have defects, resulting in a lower than expected grain size as probed by in-situ WAXS.

Using the size distribution in panels i and j in Figure , we calculate the amount of free SBUs on the octahedral UiO-66 crystal surface. Figure k shows the evolution of free linkers in the solution present per surface SBU for the six different conditions corresponding to Figure a– f. It can be seen that the ratio of free linker per surface SBU stays constant at a steady state for varying reaction temperatures because the amount of the linker concentration and the metal node is initially the same. Therefore, decreasing the metal node concentration increases the ratio of free linker per surface SBU because the linker is initially present at a stoichiometrically higher ratio. However, we find that decreasing the linker concentration also increases the ratio of free linkers per surface SBU. In this case, we have a lower number of UiO-66 crystals, which are larger (Figure e), which reduces the SBU surface area and increases the linker to surface SBU ratio.

Increasing the linker to surface SBU ratio increases the reactivity of the MOF surface, as the concentration of the linker in solution is proportional to the number of linkers on the MOF surface. Therefore, increasing the free linker to surface SBU ratio allows oriented attachment over a longer time duration (see Figure a, d, and e), thereby permitting larger MOF crystals to aggregate. The largest size of MOF that can undergo oriented attachment to a larger crystal is referred to as the maximum aggregating crystal size (MACS). The estimated MACS is 15.1, 25.5, 60.3, and 9.6 nm for the experimental conditions in panels a and d–f in Figure , respectively. Figure l shows a scaling relationship between the number of free linkers per surface SBU at a steady-state and the number of initial linkers per surface SBU of MACS. Here, we see that conditions that allow for a larger number of free linkers per surface of the SBU allow for oriented attachment of larger crystallites.

Figure summarizes the different processes and the governing mechanisms for the microkinetic model. However, these results are not as straightforwardly compared to the nucleation and growth rates obtained from other models. For example, the classical crystallization model arises from a reaction kinetic model that considers only the monomeric attachment of building units, which can then be used to obtain nucleation and growth rates. To increase the applicability of our microkinetic model, we use the model to predict nucleation and growth rate kernels (functions). We compare these kernels with those estimated from the widely used Gualtieri model.[17]Figure a shows the estimated probability distribution of nucleation utilizing the Gualtieri model for different reaction temperatures as a function of reaction extent. The estimated normalized probability distribution using the microkinetic model is given in Figure S27. Figure b shows the Arrhenius plot of the estimated nucleation (kn) and growth (kg) rate constants from the Gualtieri model with activation energies of 78 and 83 kJ/mol, respectively (see Gualtieri Model Fitting section, Figure S28 and Table S4). The close values suggest the nucleation and growth mechanisms, while not explicitly defined in the Gualtieri model, are indistinguishable with respect to the rate-limiting crystallization mode.

Figure 6

Summary of various growth phases with the respective governing process and the corresponding conceptual figure. The induction phase is governed by the initiation reaction, catalyzed by the SBUs formed in the solution and resulting in a rapid increase of the SBU formation rate and sigmoidal kinetics. A large number of SBUs formed in the solution activates a cascade of the aggregation reaction, resulting in the exponential growth phase governed by step reaction (oriented attachment). Although both chain and step reactions are active in the exponential phase, the step reaction dominates the volumetric rate increase. The formation of larger crystals due to step growth decreases the surface-to-volume ratio, stabilizing the surface and reducing surface reactivity, causing the onset of the termination resulting in the transition phase. The absence of oriented attachment due to stable crystals moves the crystal growth to the stationary phase. In the stationary phase, crystal growth is governed by chain elongation until reactants are consumed.

Figure 7

Comparison of the reaction-aggregation model with the Gualtieri model and the nucleation and growth kernels. (a) Nucleation probability obtained from the Gualtieri model experimental fitting. (b) Arrhenius plots for nucleation and growth rate constants obtained using the Gualtieri model (red lines) and theoretical model (black lines with filled symbols), and initiation and aggregation rate constants obtained from the theoretical model (black lines with hollow symbols). (c) Theoretical nucleation rate as a function of reaction extent and reaction temperature. (d) Theoretical growth rate as a function of the extent of the reaction and the reaction temperature. (e) Nucleation rate as a function of the extent of the reaction at varying initial concentrations of the Zr-oxo cluster and the organic linker. (f) Growth rate as a function of the extent of the reaction at varying initial concentrations of the Zr-oxo cluster and the organic linker.

The microkinetic model can also be used to obtain the nucleation rate, where nucleation is defined as the formation of crystals of size greater than (an arbitrary) critical size (e.g., 8.6 nm). Figure c shows the nucleation rate distribution obtained from the microkinetic model. While similar trends in nucleation rate can be observed compared to the Gualtieri model, with maxima between a reaction extent of 0.1 and 0.3, the microkinetic model offers more insight into this behavior. In addition, our model can relate nucleation rate to metal node concentration, linker concentration, and reaction temperature (Figure c, d). The estimated form of the nucleation kernel is given below.where Ṅ (s–1) is the nucleation rate, ξ is the reaction extent, T (K) is the temperature, k0,n (s–1) is the pre-exponential factor of the nucleation rate constant kn, An is the rate order with respect to the reaction extent, Bn is the rate order with respect to limiting reactant, ΔGn (J mol–1) is the activation barrier for nucleation, and R (J mol–1 K–1) is the gas constant. Thus, the autocatalytic effect due to self-templating is captured by ξA term in eq .

Similarly, the linear growth rate can be computed using the microkinetic model as the rate of change of grain size for each condition in Figure . Figure d shows a sharp increase in the growth rate due to oriented attachment of smaller nuclei generated from autocatalytic initiation, followed by a constant decrease in growth rate due to depletion of limiting reactant. The estimated growth rate kernel is given below.where G (m s–1) is the growth rate of stable facet, k0,g (m s–1) is the pre-exponential factor of the growth rate constant kg, Ag is the rate order with respect to reaction extent, Bg is the rate order with respect to limiting reactant, and ΔGg (J mol–1) is the activation energy for growth. Figure b also shows the Arrhenius plot for the nucleation and growth rate constants from the microkinetic model with the activation energies of 19.9 and 66.2 kJ/mol, respectively. Panels e and f in Figure show the effect of varying metal node and linker concentrations on the rates of nucleation and growth, respectively. The order and the activation energies were obtained by nonlinear least-squares fitting.

The microkinetic model also yields insight into the reaction order and activation energy of reactions involved in the UiO-66 synthesis. The estimated reaction order for autocatalytic initiation reaction with 1:1 node-to-linker stoichiometry is 1.2 and 1.8 with respect to metal node and linker, respectively. In contrast, the reaction order for chain and step reactions matches exactly the stoichiometry of the reactants. This suggests the initiation reaction is nonelementary, whereas the chain and step reactions can be considered elementary. The activation energy for autocatalytic initiation is 52 kJ/mol (Figure b). Interestingly, the activation energy for chain and step reactions are identical −118 kJ/mol, which follows Flory’s approximation[34] that rate constants of polymerization reaction are independent of polymer length.

Conclusions and Perspectives

In this article, we identify key physical processes that govern the induction, nucleation, growth, and stabilization of UiO-66 crystals using time-resolved in-situ WAXS and large-scale microkinetic modeling. These physical processes are the formation of SBUs (or initiation), attachment of SBUs to the growing MOF (or chain growth), oriented attachment of MOF crystals (or step-growth), and surface stabilization of MOFs (or termination) (Figure ). A simplified overview and limitations of the work presented in the manuscript are given in Section S11. The flowchart of the work is given in Figure S29.

The time-resolved WAXS study shows an increase in the volumetric rate of MOF crystal formation in the induction phase, indicating autocatalysis of SBU formation that becomes more favorable with a larger template size. We find the exponential growth phase is primarily due to step-growth or oriented attachment. The estimated rate constant for step-growth is similar to chain growth, which suggests Flory’s approximation is also applicable to oligomerization reactions in MOF synthesis. The termination of step-growth is determined by the reactivity of the MOF crystal surface, and we identify that the average number of linkers per SBU on the surface of MOF can be used as a reactivity descriptor.

The nucleation and growth insights from a conventional Gualtieri analysis are also compared with the detailed microkinetic model. Although the Gualtieri model assumes the nucleation and growth probability that influences the estimated barriers, the microkinetic model can deconvolute the contributions from autocatalytic initiation, chain growth, step-growth (oriented attachment), and termination to accurately determine the activation barriers for nucleation and growth. These fundamental insights on the mechanism of MOF nucleation and growth that are revealed from in-situ X-ray scattering and microkinetic modeling will provide molecular control toward the synthesis of a wide range of MOFs.