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Keywords:

  • algorithms;
  • crystal growth;
  • nanoparticles;
  • organic–inorganic hybrid composites;
  • synthesis optimization

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results and Discussion
  6. Conclusion
  7. Experimental Section
  8. Acknowledgements
  9. Supporting Information

The local, deterministic optimisation algorithm BOBYQA (bound optimisation by quadratic approximation) was applied for the computer-assisted optimisation of inorganic–organic hybrid compounds. Four proof-of-concept studies were performed: 1) increasing the crystallinity and 2) inhibiting the crystallisation of [Ca(H2O)2(H2PMBC)2] (H2PMBC=HO3P-CH2-C4H6-COOH), 3) increasing the crystal size of [Bi(H2O)(BTC)] (BTC3−=benzene-1,3,5-tricarboxylate) and 4) tuning the particle sizes of [Al(OH)(CDC)]⋅x H2O (CDC2−=trans-1,4-cyclohexane dicarboxylate) to a desired value. The measurable quantities of crystallinity, crystal size and hydrodynamic diameter were used as the quality criteria for the optimisation, and the parameters of reaction time and temperature, molar ratios and overall concentration of the starting materials as well as the stirring rate were varied. The crystallinity of [Ca(H2O)2(H2PMBC)2] was increased in three optimisation steps by approximately 14 %, which was accompanied by an increase in crystal size by a factor of approximately 40. These crystals were suitable for structure determination from single-crystal X-ray diffraction data. The crystallisation of the same compound could be completely inhibited and clear solutions were obtained. The average crystal size of [Bi(H2O)(BTC)] was increased from (22.2±4.5) to (34.8±9.5) μm and the upper limit increased from 30.0 to 57.7 μm over the course of the optimisation. For the application in Bragg stacks, the variation of particle sizes of [Al(OH)(CDC)]⋅x H2O was studied. Although we aimed at a decrease to 100 nm, a lower limit of 460 nm and polydispersity index of 0.03 were accomplished. The convergence of the algorithm indicates that the optimisation progress is close to completion and the found value for the particle size is close to the lower limit in the system of the chosen parameters. These proofs-of-concept studies demonstrate the potential of optimisation algorithms like BOBYQA in synthesis optimisation experiments. At the same time, the convergence behaviour of the algorithm gives an indication of the progress of an optimisation.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results and Discussion
  6. Conclusion
  7. Experimental Section
  8. Acknowledgements
  9. Supporting Information

Inorganic–organic hybrid compounds have recently been the topic of intense investigations owing to their potential applications in gas storage and separation1, 2 and drug release3, 4 or as luminescence substances,5, 6 magnetic materials5, 6 or heterogeneous catalysts.7, 8 Such compounds are composed of metal cations and organic ligand molecules, which usually are polyfunctional carboxylates,8 phosphonates7 or amines.9 Often these compounds are obtained by solvothermal synthesis.10

For the characterisation and practical applications of inorganic–organic hybrid compounds the tuning of crystal sizes and particle morphology is of great importance. For example, the particle size in microporous materials is related to the sorption kinetics,11 which can influence the catalytic activity.12 In semiconductors the luminescence properties can be varied.13 Furthermore, a challenge in the crystal structure determination of new compounds is the growth of suitable single crystals. Whereas large crystals (>100 μm in one dimension) can be easily characterised by single-crystal X-ray diffraction, microcrystalline substances necessitate the use of more elaborate experimental methods such as electron diffraction18, 19 or X-ray microdiffraction.14 Alternatively, X-ray powder diffraction can be used, but in this case structure solution is not straightforward.17, 18 By taking all these points into account, it is apparent that the discovery of a compound must often be followed by optimisation of the synthesis conditions.

An optimisation process can mathematically be viewed as finding the optimum in a utility function, which relates a quality criterion to a set of n variables.15 This criterion can be any property (e.g., crystal lengths, average particle sizes, crystallinity, purity, porosity, or catalytic activity) or a combination of such properties.15 The variables that are varied in a synthesis are parameters such as concentration and molar ratio of the starting materials, solvent, reaction temperature, reaction time, stirring rate and so forth.

Generally, an optimisation can be carried out by systematically varying parameters (factorial design), by using chemical information (rational design) or the optimised parameters are obtained by computer simulations.10 Most often, combinations of these approaches are used. In a factorial design, a system of n parameters is investigated by choosing for each parameter m different values.16 The number of reactions to be carried out is mn—scaling exponentially with the number of parameters and multiplicatively with the number of values investigated for each parameter. Thus the number of reactions in a complicated system can be staggering. In contrast, rational design uses chemical information, for example, personal knowledge, database analysis17 or optimisation algorithms.18 This approach can be intricate since complex reaction systems such as solvothermal reactions are difficult or even impossible to predict.19 Computer simulations can also be implemented to predict reaction conditions that should lead to a desired product.15 These use statistical or quantum mechanical models for calculating the energy states and reaction pathways in phase space. This approach is time consuming and necessitates the use of large computing power. As a consequence, often simplified models are employed at the cost/expense of reducing the quality of the simulation, for example, DFT calculations instead of ab initio, or parameterised force-field methods instead of quantum mechanical approaches.20

The use of these three methods has been demonstrated in the optimisation of inorganic–organic hybrid materials, for example, factorial design has been applied in the high-throughput investigation of a copper phosphonoethanesulfonate,21 rational design has been utilised for the prediction of new metal–organic frameworks22 or the simulation of the sorption selectivity of MOFs.23

In an algorithm-based optimisation process, experimental data, that is, the quality criterion, is fed into the algorithms, which calculate the reaction parameters for the next optimisation step. In this way, a sequence of experiments can be used to optimise a property. Computer-assisted synthesis optimisations have five relevant characteristics: 1) they combine the power of algorithms with the scientific expertise of the chemist, 2) they quantify the synthetic parameters and thus permit the automation of an optimisation, 3) the amount of computing power and time is small compared to computer simulations and 4) the convergence of the quality criterion indicates the progress of the optimisation. Nevertheless, 5) a careful and precise choice of a quantifiable quality criterion is required. Most optimisation studies have been carried out using global indeterministic optimisation algorithms. Thus genetic algorithms have been applied in cases ranging from catalyst optimisation24, 25 to the prediction of stable alloys26 and the design of organic polymers.27 Since these global, indeterministic optimisation algorithms are time consuming and expensive owing to the number of optimisation steps necessary, local, deterministic optimisation algorithms should allow for a faster optimisation process, if there are only few optima in the parameter space. Recent studies suggest that this may indeed be true in most cases.15

Herein, we demonstrate the use of the deterministic and local optimisation algorithm BOBYQA (bound optimisation by quadratic approximation) in the synthesis of inorganic–organic hybrid compounds to increase the crystallinity or inhibit the crystallisation of a compound and to adjust the particle size towards a predefined value.

Methods

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results and Discussion
  6. Conclusion
  7. Experimental Section
  8. Acknowledgements
  9. Supporting Information

For the iterative optimisation, the algorithm BOBYQA28 was utilised. It is a highly efficient local optimisation algorithm developed by M. J. D. Powell and is described in more detail in the Supporting Information. As a general mathematical algorithm, it can be utilised for almost any kind of optimisation problem. Only two studies have been reported that apply BOBYQA. It was used for predicting well locations in fossil-fuel reservoirs29 and in combination with genetic algorithms for optimising force-field parameters for SiOH compounds.30 The closely related but older NEWUOA algorithm, which is almost identical to BOBYQA but does not set bounds (restrictions) on the parameters, has been used in various studies. For example, it was employed to tailor ultra-short laser pulses in a theoretical model to selectively break molecular bonds,31 to evaluate magnetic resonance imaging measurements of the human brain,32 or in image processing for estimating the amount of noise and blur in a distorted image.33

The BOBYQA algorithm requires only a small number x of data points to start an optimisation. This number depends on the number of parameters under investigation (x≤2n+1, n=number of parameters), that is, a starting point plus 2n initial data points. These additional points have to be arranged around the starting point, at a distance of the initial resolution ρ, in all n dimensions and directions. Therefore only 2n experiments are necessary to start the algorithm. The algorithm uses these x values of the quality criterion to set up a first approximation of the n-dimensional function and calculates the next set of parameters. Additional details on the operation of the algorithm are given in the Supporting Information.

BOBYQA is a mathematical algorithm implemented in Fortran. To apply the algorithm the program Chrysop (chemical/crystallinity optimisation) was written. Chrysop loads an input file containing the number of dimensions (parameters), the initial resolution ρ of BOBYQA (initial step size in parameter space), the final resolution at which the algorithm stops, the ranges of the parameters and the values of the x data points. These values encompass the parameters and the value of the quality criterion.

Over the course of the iterative optimisation, Chrysop utilises the input data to calculate a set of parameters for the next reaction. The experiment is performed at the calculated parameters and the quality criterion is determined. This new value is subsequently fed into Chrysop. If necessary, the resolution ρ is automatically adjusted and the next set of parameters is calculated. This procedure is repeated until the quality criterion fulfils the requirements or the optimisation has converged, that is, the step size has declined so far that no further improvement is to be expected. Since BOBYQA progresses in finite steps and since its internal quadratic approximation models are necessarily inexact, changes in the quality criterion value cannot be used as a reliable convergence criterion. However, the step size is progressively reduced based on accumulated history of the optimisation progress. Therefore, a lower threshold on the step size is a feasible convergence test.

Since BOBYQA uses the same resolution in all dimensions, the parameters of the optimisation have to be rescaled to be in similar orders of magnitude. Otherwise the algorithm would calculate steps that are either too small to cause a measurable difference in the experiment (e.g., a difference in reaction time of a few seconds) or too big to constitute a reasonable optimisation step (e.g., a temperature change of 300 °C).

Results and Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results and Discussion
  6. Conclusion
  7. Experimental Section
  8. Acknowledgements
  9. Supporting Information

Particle and crystal size as well as the crystallinity of materials are important factors that influence their properties. In a proof-of-concept study we have investigated the synthesis of inorganic–organic hybrid compounds towards the optimisation of crystal size, crystallinity and particle size. This correlates with the general targets of increasing or decreasing a property or tuning a property to a specific value.

To be more specific, the BOBYQA algorithm was used 1) to increase the crystallinity and phase purity of [Ca(H2O)2(H2PMBC)2] (H2PMBC=HO3P-CH2-C4H6-COOH),34 or 2) to inhibit its crystallisation, 3) to increase the crystal size of [Bi(H2O)(BTC)] (BTC3−=benzene-1,3,5-tricarboxylate) and 4) to tune the particle size of [Al(OH)(CDC)]⋅x H2O (CDC2−=trans-1,4-cyclohexane dicarboxylate)35 towards 100 nm (Table 1). Details concerning these products are given in the Supporting Information.

Table 1. List of optimisation studies, the synthesis parameters and the quality criterion used.
Form of optimisationProof of conceptParametersQuality criterion[a]
  1. [a]All quality criteria are defined with negative algebraic signs, since BOBYQA searches for a minimum (see Supporting Information for further details).

increasing1) increasing crystallinity of [Ca(H2O)2(H2PMBC)2]concentrations of metal source, ligand and base−(IproductIimpurities); I: intensity of 010 reflection
3) increasing crystal size of [Bi(H2O)(BTC)]concentration of the starting materials, reaction time and temperaturenegative averaged crystal length
decreasing2) inhibiting the crystallisation of [Ca(H2O)2(H2PMBC)2]concentrations of metal source, ligand and base−25 000+(IproductIimpurities); I: intensity of 001 reflection
tuning to a specific value4) particle size of [Al(OH)(CDC)]⋅x H2O to 100 nmconcentration of the starting materials, reaction time, temperature and stirring rate−107+(d−100)2; d: mean particle size

The systems were chosen to validate the approach for a property that can be easily measured and quantified (i.e., peak height in a powder X-ray diffraction (PXRD) pattern) as well as a property that is instrumental for the characterisation of an inorganic–organic hybrid material (crystal size) or is distinctly relevant for practical applications of such materials (particle sizes).

Increasing the crystallinity of [Ca(H2O)2(H2PMBC)2]

The crystallinity of a sample correlates with the intensity of the reflections in its PXRD pattern provided that identical sample amounts are used and that the experimental conditions are identical. To optimise the crystallinity of the desired phase and to simultaneously avoid crystalline impurities, the quality criterion was defined as −(IproductIimpurities), in which Iproduct is the height of the 010 Bragg reflection of [Ca(H2O)2(H2PMBC)2] above the base line and Iimpurities is the height of the most intense peak of any impurity. All experiments were repeated at least three times and a standard was used to normalise the measured intensities (see the Experimental Section).

The modus operandi in using Chrysop is exemplified in detail in the following paragraphs. The three parameters for the optimisation study were the concentrations of the three starting materials Ca(NO3)2⋅4 H2O, H3PMBC and KOH. Other parameters such as reaction temperature, reaction time or solvent were kept constant. The starting point, the parameter range and the initial resolution (ρ) are chosen and the program calculates the six initial data points that are needed to use the algorithm. After these reactions were carried out and the crystallinity was determined, all necessary data is available to start BOBYQA (Table 2). The corresponding input file for Chrysop is shown in Figure S5 in the Supporting Information.

Table 2. Chrysop input data for increasing the crystallinity of [Ca(H2O)2(H2PMBC)2]. Formulae in square brackets denote concentrations. A step of one in the parameter range corresponds to a change in concentration of 40 mmol L−1.
number of dimensions (=parameters)3
initial and final resolution0.5/10−5
range for x (=[Ca(NO3)2]⋅4 H2O)0 to 10
range for y (=[H3PMBC])0 to 10
range for z (=[KOH])0 to 10
  
 [Ca(NO3)2]⋅4 H2O[H3PMBC][KOH]Intensity/counts
starting points1115612±182
initial data points1.5116714±122
 0.5119468±223
 11.516746±860
 10.510 (clear solution)
 111.56985±145
 110.56557±188

The iterative optimisation proceeds as follows: Chrysop loads the input file, adjusts the resolution (ρ) and calculates the next set of parameters for the first optimisation step—in this case to ρ=0.1 and x:y:z=1.523:1.101:1.069 (Figure S6). After carrying out the corresponding experiments (again: for statistical reasons for each set of parameters the reaction was carried out six times) and measuring the quality criterion (=−6296.5±257), this value is typed into the program as the 7th data point and all points are subsequently used by BOBYQA to calculate the parameters for the next optimisation step (Figure S6). After three optimisation steps the optimisation was stopped since crystals suitable for single-crystal X-ray diffraction measurements were obtained (Figure 1).

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Figure 1. SEM images of [Ca(H2O)2(H2PMBC)2] obtained at the starting point in parameter space (left) and the refined ratios (0.43:1.056:0.766) (right).

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The course of the optimisation is depicted in Figure 2 (left). The individual PXRD patterns of the optimisation steps are depicted in the Supporting Information (Figure S12). The PXRD patterns of the reaction products obtained at the starting point and the optimised set of parameters are depicted in Figure 2 (right).

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Figure 2. Course of the three optimisation steps for increasing the crystallinity of [Ca(H2O)2(H2PMBC)2] (left) and the comparison between the PXRD pattern of product found at the point 0.43:1.056:0.766 (right, top), at the point 1:1:1 (right, middle) and the difference between the two (right, bottom). For clarity the ordinate offsets of the middle and the top pattern were set to 10 000 and 20 000 counts, respectively; the abscissa offsets were set to 1.275 and 2.55°, respectively.

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The optimised parameters (0.43:1.056:0.766) can also be correlated with common crystallisation trends. According to the chemical formula [Ca(H2O)2(H2PMBC)2] a molar ratio [Ca2+]/[H3PMBC] of 1:2 in the reaction mixture should lead to a phase-pure product. The decrease of the base concentration (from 1 to 0.766) should result in a lower concentration of the deprotonated ligand H2PMBC, which in turn should decrease the crystal-growth rate, giving rise to larger crystals.

This increase of measured crystallinity was accompanied by a drastic increase in crystal size. We attribute this enlargement as well as the fact that the main increase in the powder patterns is observed for the 010 reflection to textural effects, which could not be prevented by grinding the sample. Since the principal axis of the crystal corresponds to the a axis (determined by single-crystal diffraction), the increase of the 010 reflection intensity is probably due to the larger needles.

To demonstrate that the optimisation increased the overall crystallinity, the reflections up to 19.3° (2θ) were integrated, and the subtracted baseline was approximated in a simple linear model (Figure 3). The values confirm the trend observed by the peak height of the 010 reflection.

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Figure 3. Increase of the crystallinity of [Ca(H2O)2(H2PMBC)2] as determined by integration of the reflections up to 19.3° (2θ).

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Inhibiting the crystallisation of [Ca(H2O)2(H2PMBC)2]

To demonstrate that the algorithm can also be used to decrease a property, a second set of experiments on the crystallisation of [Ca(H2O)2(H2PMBC)2] was conducted. This time the optimisation experiment was performed to find a set of parameters at which [Ca(H2O)2(H2PMBC)2] is no longer observed, that is, no crystallisation occurs. Using the starting point 1:1:1 and a different set of six initial points (Table 3) the first calculated parameter set (1.01:0.503:0.963) in the optimisation immediately led to the complete inhibition of crystallisation. The PXRD patterns are depicted in Figure S13 in the Supporting Information.

Table 3. Chrysop input data for inhibiting the crystallisation of [Ca(H2O)2(H2PMBC)2]. Formulae in square brackets denote concentrations; the overall sum of the concentrations was set to 100 mmol L−1.
number of dimensions (=parameters)3
initial and final resolution0.25/10−5
range for x (=[Ca(NO3)2]⋅4 H2O)0 to 10
range for y (=[H3PMBC])0 to 10
range for z (=[KOH])0 to 10
  
 [Ca(NO3)2]⋅4 H2O[H3PMBC][KOH]Intensity/counts
starting points11112 670±2071
initial data points111.2513 922±973
 110.7517 558±692
 11.25115 965±1530
 10.7516727±851
 1.251116 620±258
 0.751118 395±1395

The optimised parameters (1.01:0.503:0.963) can also be correlated with common crystallisation trends. By decreasing the concentration of the ligand and simultaneously keeping the amount of base constant (ligand/base ≈1:2) the ligand remains in solution.

Increasing the crystal size of [Bi(H2O)(BTC)]

While the optimisation of the crystallinity and phase purity of a compound is important for the in-depth characterisation and possible applications, often the size of individual crystals is of minor importance. However, single crystals of sufficient size are necessary for a crystal-structure determination. The phase purity is in this case of minor importance. In the case of increasing the crystal size of [Bi(H2O)(BTC)], the average crystal length in μm as observed by SEM imaging was used as the quality criterion. For each sample, ten representative crystals were used and each reaction was performed six times for statistical reasons. The following parameters were optimised: reaction temperature, reaction time, and overall concentration of the starting materials (molar ratio Bi3+/H3BTC=1:1). The actual parameters were rescaled: temperature range from 0 to 10 represented a range from 35 to 185 °C (ΔT=1=15 °C), the range of reaction time from 0 to 20 represented a range from 0 to 200 min (Δt=1=10 min) and the concentration range from 1 to 20 represented a range from 0 to 284.5 mmol L−1c=1=15 mmol L−1). The rescaled values of the starting point and the six initial points are given in Table 4.

Table 4. Chrysop input data for increasing the crystal size of [Bi(H2O)(BTC)]. The parameter values are given in rescaled numbers. The values in common units (°C, min, mmo L−1) are given in Table S3.
number of dimensions (=parameters)3
initial and final resolution1/10−5
range for x (=temperature)0 to 10
range for y (=reaction time)0 to 20
range for z (=overall concentration)1 to 20
  
 TemperatureReaction timeOverall conc.Av. crystal size [μm]
starting point52522.2±4.5
initial data points52616.8±3.7
 52416.2±3.3
 62520.2±4.5
 42524.3±5.0
 53528.2±6.2
 51510.2±1.6

At the starting point, the crystal sizes varied in a relatively small range with an average length of (22.2±4.5) μm (Figure 4, left). After seven optimisation steps, the algorithm calculated a reaction temperature of 91.0 °C, a reaction time of 35.4 min and an overall concentration of the reactants of 53.5 mmol L−1. By using these parameters, the average crystal size increased to (34.8±9.5) μm and the size range drastically increased giving crystals ranging from 9.7 to 57.7 μm (Figure 4, left). The maximum change in the reaction parameters (denoted as the step size) for each optimisation step is depicted in Figure 4 (right). During optimisation, the step size decreased from 1.262 to 0.224, which indicated a convergence in the direction of the optimum (Figure 4, right).

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Figure 4. Crystal sizes with measured size ranges (left) and step sizes (right) of the data points proposed by the BOBYQA algorithm. The error bars depict the standard deviation of the averaged crystal size in the six samples measured for each optimisation step; the broad bars represent the entire scope of the distribution.

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In comparison to the starting parameters, larger crystals are obtained at a lower reaction temperature (90.4 °C versus 110 °C), a longer reaction time (38 min versus 20 min) and a lower overall concentration of the reactants (55.1 mmol L−1 versus 59.5 mmol L−1). Over the course of the optimisation experiment, BOBYQA reduced the reaction temperature. In the third step this parameter was calculated as 63 °C with a reaction time of 37 min, which led to significantly smaller crystals. Thus, in the fourth step the algorithm increased the temperature again to 105 °C and subsequently decreased it iteratively to a value of 90.4 °C.

As depicted in Figure 5, the size and the size distribution of the crystals obtained at the starting value (top) differ distinctly from those at optimised reaction conditions (bottom). The initial reaction product contains many small crystals with a length of around 22 μm on average. The optimised reaction product contains crystals with an average size of approximately 35 μm and also the size range is considerably larger, ranging from 9.7 to 57.7 μm.

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Figure 5. SEM micrographs of the initial (top) and the optimised product (bottom).

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Tuning the particle size of [Al(OH)(CDC)]⋅x H2O

The aluminium carboxylate [Al(OH)(CDC)]⋅x H2O (also known as CAU-13, CAU=Christian-Albrechts University) is a microporous inorganic–organic hybrid compound containing trans-1,4-cyclohexane dicarboxylate (CDC2−) ions.35 Such materials have recently been used for the construction of one-dimensional photonic crystals (Bragg stacks), which are investigated as sensor materials.11 For such applications, monodisperse particles with a size in the 100 nm range are advantageous. The hydrodynamic diameter (d, measured by dynamic light scattering) of CAU-13 was used to define the quality criterion (−107+(d−100)2) for the optimisation (Figure S11). The quadratic function was chosen to generate a minimum at 100 nm. When no product or a different product was obtained at a set of parameters, the corresponding data point was defined as zero.

The reactions were carried out under microwave-assisted heating and the following parameters were optimised: reaction temperature, reaction time, stirring rate and overall concentration of the starting materials (molar ratio Al3+/H2CDC=1:1). Since input into BOBYQA has to be rescaled to similar orders of magnitude, the parameters were rescaled in the following way: the concentration range from 0 to 15 represented a range from 0 to 1500 mmol L−1c=1=100 mmol L−1), the range of reaction time from 0 to 10 represented a range from 0 to 100 min (Δt=1=10 min), the temperature range from 0 to 13 represented a range from 50 to 180 °C (ΔT=1=10 °C) and the range of the stirring rate from 3 to 9 represented a range from 300 to 900 rpm (Δv=1=100 rpm). The values of the starting point and the six initial points are given in Table 5.

Table 5. Chrysop input data for tuning the particle sizes of [Al(OH)(CDC)]⋅x H2O. The parameter values are given in rescaled numbers; the values in common units are given in Table S4.
number of dimensions (=parameters)4
initial and final resolution1/10−5
range for w (=overall concentration)0 to 15
range for x (=reaction time)0 to 10
range for y (=temperature)0 to 13
range for z (=stirring rate)3 to 9
  
 TReaction tOverall conc.Stirring rateAv. particle size [nm]
starting points34.5861184±179
initial data points44.5861029±20
 24.586533±14
 35.5861113±110
 34.596906±31
 34.5871072±41
 33.586620±59

At the starting point the particle size had an average value of (1184±179) nm (Figure 6, left), which decreased over the course of the experiment to a value of (462±44) nm. The optimisation was stopped after six optimisation runs since the step size calculated by BOBYQA decreased from 1.96 to 0.1, which indicates that the parameters are very close to the optimum (Figure 6, right). Therefore the optimised particle size seems to be close to the smallest value possible by exploring these four parameters. Nevertheless, smaller particles could be achievable by exploring additional parameters such as other solvents, solvent mixtures or addition of modulators.

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Figure 6. Particle sizes with standard deviations (left) and step sizes proposed by BOBYQA (right) as functions of the step number. The particle size for step 1 was not determined since the product was X-ray amorphous and the quality criterion for this point was set to zero.

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The particle size mainly depends on the overall concentration. In comparison to the starting parameters, smaller particles are obtained at a lower concentration (300 versus 204 mmol L−1), and reaction time (45 versus 44 min), reaction temperature (130 °C in both parameter sets), and stirring rate (600 rpm in both parameter sets) deviate only slightly. The progress of the optimisation is demonstrated in Figure 7. The particle size distribution at the starting point and the two optimisation steps three and six are shown for one reaction product each. The sets of the six curves for the starting point and the third and sixth optimisation step are shown in Figures S14–S16. Whereas at the starting point a broad distribution (polydispersity index: 0.13) with maxima around 1178 nm was observed, the subsequent optimisation led to particles with maxima at 540 and 590 nm and a polydispersity index of 0.03 and 0.13 for steps three and six, respectively.

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Figure 7. Intensity distributions as functions of hydrodynamic diameters measured by dynamic light scattering for products obtained at the starting point and step three and six of the optimisation.

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Conclusion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results and Discussion
  6. Conclusion
  7. Experimental Section
  8. Acknowledgements
  9. Supporting Information

The study demonstrates that local, deterministic optimisation algorithms like BOBYQA can be utilised to optimise product properties of complex, multiparameter reaction systems. Three solvothermal reaction systems were chosen to optimise different inorganic–organic hybrid compounds towards 1) increasing crystallinity, 2) inhibiting crystallisation, 3) increasing crystal sizes and 4) tuning particle size. These examples represent the three fundamental targets in optimisation: increasing a property (1 and 3), decreasing a property (2) and tuning a property to a specific value (4). Since BOBYQA is a mathematical algorithm that is not limited to any specific field or research, we anticipate that it could be useful for optimising other complex chemical systems. Owing to its simplicity, BOBYQA should also be well suited for the incorporation into automated experimental setups.

Experimental Section

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results and Discussion
  6. Conclusion
  7. Experimental Section
  8. Acknowledgements
  9. Supporting Information

General

With the exception of 4-(phosphonomethyl)benzoic acid (H3O2P-CH2-C6H4-COOH, H3PMBC) all chemicals are commercially available and were used as received without further purification. The ligand H3PMBC was synthesised in an Arbuzov reaction starting from 4-(bromomethyl)benzoic acid and subsequent hydrolysis.36

High-throughput investigations under conventional electric heating were carried out in a custom-made high-throughput autoclave with a capacity of 24 Teflon-lined reactors with a maximum volume of 2 mL each.37 Reactions in glass reactors were performed in 2–5 mL glass vials (Biotage, item number 351521), which were heated in a conventional oven or a Biotage Initiator microwave oven for conventional solvothermal synthesis or microwave-assisted heating, respectively. All products were characterised by powder X-ray diffraction using a STOE Stadi P Kombi high-throughput diffractometer. SEM micrographs were recorded using a Philips XL30 or a JEOL JSM-6500F scanning electron microscope. Particle size distributions of dispersions were measured as hydrodynamic diameters by dynamic light scattering using a DelsaNano C apparatus (Beckman&Coulter).

Increasing the crystallinity or inhibiting the crystallisation of [Ca(H2O)2(H2PMBC)2]

The system Ca(NO3)2⋅4 H2O/H3PMBC/KOH/H2O was investigated by using a high-throughput autoclave containing 24 Teflon-lined reactors with a maximum volume of 2 mL each.37 The reactions were carried out using a total volume of 1.5 mL water. For each reaction the ligand was weighed in as a solid, whereas Ca(NO3)2⋅4 H2O and potassium hydroxide were added as aqueous solutions with concentrations of 423 mmol L−1 and 2 mol L−1, respectively. The reactor was heated to 130 °C over a duration of 2 h, the temperature was held for 12 h and then slowly lowered to room temperature over a period of 12 h.

For statistical reasons, the reactions at the starting point and at the data points in the optimisation reaction were each carried out six times. The reactions at the six initial data points equidistant around the starting point were carried out three times each, since the optimisation is less sensitive to errors in the initial data points compared to those further along in the optimisation. Reaction products were filtered off and the products of all reactions at a certain set of parameters were combined.

The phase identification and intensity measurement were carried out by PXRD. The products were thoroughly ground before the measurements. Three to six samples of each reaction product (5 mg) were used in the study to increase the crystallinity. To study the inhibition of the crystallisation, a sample holder with 8.3 mm3 volume was employed and three to six samples of each reaction product were measured. To normalise the intensity of the incident X-rays a Teflon foil (0.1 mm thickness) was used as an external standard for each measurement.

The only parameters investigated were the molar ratios and the overall concentration of the starting materials. The points in the parameter space are described by number triplets [Ca(NO3)2]⋅4 H2O/[H3PMBC]/[KOH]=x:y:z. For the crystallisation optimisation a value of 1 represents a concentration of 40 mmol L−1. The starting point is 1:1:1 and the initial step size was set to 0.5, which led to the initial data points 0.5:1:1, 1.5:1:1, 1:0.5:1, 1:1.5:1, 1:1:0.5 and 1:1:1.5. Details on the reaction conditions are given in Table S1. The quality criterion for increasing the crystallinity was defined as −(IproductIimpurities), in which Iproduct is the height of the 010 Bragg reflection of [Ca(H2O)2(H2PMBC)2] above the base line and Iimpurities is the height of the most intense peak of an impurity, in case an impurity was found. The explanation for the choice of the quality criterion is described in detail in the Supporting Information.

To inhibit the crystallisation of [Ca(H2O)2(H2PMBC)2] the sum of all the concentrations was set to 100 mmol L−1. The initial data point 1:1:1 was chosen and the initial step size was set to 0.25; this led to the initial data points 0.75:1:1, 1.25:1:1, 1:0.75:1, 1:1.25:1, 1:1:0.75 and 1:1:1.25. Details on the reaction conditions are given in Table S2. The quality criterion for inhibiting the crystallisation was defined as −25000+(IproductIimpurities), in which Iproduct is the height of the 001 Bragg reflection of [Ca(H2O)2(H2PMBC)2] above the base line and Iimpurities is the height of the most intense peak of an impurity, in case an impurity was found. The explanation for the choice of the quality criterion is described in detail in the Supporting Information.

Increasing the crystal sizes of [Bi(H2O)(BTC)]

The syntheses of [Bi(H2O)(BTC)] were carried out in 2–5 mL glass vials (Biotage item number 351521). In each reaction Bi(NO3)3⋅5 H2O and trimesic acid (H3BTC) were weighed in and methanol (2 mL) was added. The vial was closed by a seal and a metal cap and the reaction mixture was homogenised by shaking.

For the crystal size optimisation of the bismuth trimesate the three parameters reaction temperature, reaction time and overall concentration of the reactants (Bi(NO3)3⋅5 H2O, H3BTC) were chosen and their values were rescaled to have a step size of 1 equal to 15 °C, 10 min and 15 mmol L−1, respectively (for the need of rescaling, see the Supporting Information). As starting values, a reaction temperature of 110 °C, a reaction time of 20 min and an overall concentration of 59.5 mmol L−1 were chosen. The overall concentration corresponds to 59.5 μmol Bi salt and 59.5 μmol H3BTC in 2 mL methanol. The assigned ranges were 35 to 185 °C, 0 to 200 min and 0 to 284.5 mmol L−1, respectively, though the synthesis optimisation did not approach these boundaries. Details on the reaction conditions are given in Table S3.

The crystal size was measured by scanning electron microscopy and was averaged over ten representative single crystals. Each optimisation step was carried out six times and the results were averaged. The points in the parameter space are described by number triplets of overall concentration/reaction time/temperature=x:y:z. The averaged crystal length in micrometres was used as the quality criterion.

Tuning the particle sizes of [Al(OH)(CDC)]⋅x H2O

[Al(OH)(CDC)]⋅x H2O was synthesised under microwave-assisted heating. The four parameters reaction temperature, reaction time, stirring rate and overall concentration of the reactants (AlCl3⋅6 H2O and H2CDC) were chosen and their values were rescaled to have a step size of 1 equal to 10 °C, 10 min, 100 rpm and 100 mmol L−1, respectively. The starting point corresponds to a reaction temperature of 130 °C, a reaction time of 45 min, a stirring rate of 600 rpm and an overall concentration of 300 mmol L−1 (525 μmol Al salt and 525 μmol H2CDC in a mixture of 2.8 mL dimethylformamide and 0.7 mL water) and the assigned ranges were 50 to 180 °C, 0 to 100 min, 300 to 900 rpm and 0 to 1500 mmol L−1, respectively. Details on the reaction conditions are given in Table S4.

The syntheses of [Al(OH)(CDC)]⋅x H2O were carried out in 2–5 mL glass vials (Biotage item number 351521). AlCl3⋅6 H2O and H2CDC were weighed in and H2O (3.5 mL) was added. The synthesis was carried out under microwave-assisted heating (Biotage Initiator). The raw product was isolated by centrifugation at 10 000 rpm for 15 min and was successively treated with dimethylformamide (DMF; 3.5 mL) and ethanol (3.5 mL) at 130 °C for 45 min before it was finally dispersed in water. These steps are necessary since the framework of [Al(OH)(CDC)] is flexible and thus the structure varies with the type of guest species. The final product was identified by PXRD.

The particle size was measured as the median of the hydrodynamic diameter distribution observed by dynamic light scattering measured twice for each sample with a DelsaNano C apparatus (Beckman&Coulter). The calculation of the particle size distribution (numerical basis) was performed using the CONTIN particle size distribution analysis routines as implemented in the Delsa Nano 2.31 software. Each optimisation step was carried out three times and the results were averaged.

The points in the parameter space are described by the number quartets (overall concentration/reaction time/temperature/stirring rate=w:x:y:z). The quality criterion was defined as −107+(d−100)2 in which d is the particle size given as the hydrodynamic diameter in nanometre as measured by dynamic light scattering. The explanation for the choice of the quality criterion is described in detail in the Supporting Information.

Acknowledgements

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results and Discussion
  6. Conclusion
  7. Experimental Section
  8. Acknowledgements
  9. Supporting Information

Financial support by the DFG (SPP 1362, STO-643/5–2) is gratefully acknowledged.

Supporting Information

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results and Discussion
  6. Conclusion
  7. Experimental Section
  8. Acknowledgements
  9. Supporting Information

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