A Kinetic Model for the Hydrolytic Nucleation and Growth of TiO2

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C: Physical Processes in Nanomaterials and Nanostructures 2

A Kinetic Model for the Hydrolytic Nucleation and Growth of TiO Nanoparticles Attila Forgács, Krisztián Moldován, Petra Herman, Edina Baranyai, István Fábián, Gabor Lente, and József Kalmár J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b04227 • Publication Date (Web): 30 Jul 2018 Downloaded from http://pubs.acs.org on July 30, 2018

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The Journal of Physical Chemistry

A Kinetic Model for the Hydrolytic Nucleation and Growth of TiO2 Nanoparticles Attila Forgácsa, Krisztián Moldována, Petra Hermana, Edina Baranyaia, István Fábiána,b, Gábor Lentea,c,* and József Kalmára,* a

Department of Inorganic and Analytical Chemistry, University of Debrecen, Egyetem tér 1, H-

4032 Hungary b

MTA-DE Redox and Homogeneous Catalytic Reaction Mechanisms Research Group, Egyetem

tér 1, H-4032 Hungary c

Department of General and Physical Chemistry, University of Pécs, Ifjúság útja 6, Pécs, H-7624

Hungary *Corresponding authors: e-mail: [email protected], [email protected]

Abstract. A simple kinetic model is derived to describe the formation of TiO2 particles up to the size of a few hundred nanometers in aqueous suspension. The model system for the kinetic experiments is the hydrolysis and condensation of titanium(IV)-bis(ammonium-lactato)dihydroxide (TIBALDH) under basic conditions. The formation of nanoparticles was followed by dynamic light scattering (DLS) and UV-vis spectrometric methods. The turbidity (i.e. the apparent absorbance at 500 nm) of a stable TiO2 suspension is shown to be proportional to the TiO2 concentration at constant particle size and proportional to the size at constant Ti concentration. The compilation of the DLS and UV-vis data yields characteristic sigmoid shaped kinetic curves for the evolution of particle size. A kinetic model with 3 reaction steps is postulated which provides an excellent fit to the experimental data. First, the rapid hydrolysis of the precursor takes place to give primary particles for the subsequent steps. The dimerization of 2 primary particles is slow and this is followed by the formation of larger particles in the step-bystep addition of subsequent primary units. An integrated rate equation was developed to predict the time-dependent mean particle size as the function of the initial precursor concentration. An important feature of the model is that a simple continuous function describes the temporal evolution of average particle size up to d = 600 nm. Previously published kinetic data representing various reaction systems were also successfully interpreted by the proposed model. –1–

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1. INTRODUCTION Semiconductor transition metal oxide nanoparticles are in the focus of materials science research owing to their widespread use in photocatalytic, electrocatalytic and solar energy related fields. Each specific application requires particles of different shape and size because these parameters determine the fundamental properties of the semiconductors. A versatile method for the synthesis of stable metal oxide aqueous sols is the controlled hydrolysis and condensation of a suitable precursor.1-3 Usually, the ultimate goal of the corresponding synthetic studies is to determine how the reaction conditions affect the quality of the final product, i.e. the size distribution, stability and shape of the nanoparticles. In order to achieve the required quality by design, the underlying mechanisms of nanoparticle formation need to be understood. On the one hand, the molecular mechanisms of these processes are thoroughly studied and well described.4,5 On the other hand, there have only been a handful of mechanistic-realistic kinetic models developed so far that successfully account for the nucleation and growth of nanoparticles under particular conditions.6-9 Furthermore, several of these models have fundamental limitations (e.g. the lack of mass balance, insufficient output information on size distribution, or requirement of a priori knowledge on product quality), as pointed out by Rempel et.al.10 Besides the mechanistic-realistic models, several empirical equations can be found in the literature that can be utilized with restrictions to achieve size control during synthesis.11 These were mainly adopted from phase-transition kinetic theories.12,13

Two, mechanism-based kinetic models are particularly powerful in describing nucleation and nanoparticle growth. The model published by Rivallin et.al.14-18 consists of 2 steps: 1) the fast hydrolysis of the precursor and 2) the aggregation of the particles. The hydrolysis is reversible and the primary particles formed in the reaction condensate into bigger particles. Condensation –2–


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becomes irreversible at a critical number of the oxygen bridges formed. This model considers that the total mass of the particles remains constant after the hydrolysis of the precursor, and only the particle size distribution changes due to aggregation. Growth takes place as a bimolecular reaction between 2 particles. There are no constraints for the sizes of the aggregating particles, but the bimolecular rate constant is proportional to the size. The mathematical solution derived for this kinetic model is able to describe the initial, accelerating size growth of nanoparticles, but does not give information on the final particle sizes. The other kinetic model was published by Rempel et.al.10 This model states that after the formation of primary particles (nuclei) in the hydrolysis of the precursor, growth takes place in reversible, step-by-step attachment of primary particles. The attachment rate of small particles is high, which quickly depletes the pool of the primary units. This kinetic region is characterized by size distribution focusing and yields small nanoparticles. Overall, the model distinguishes 5 kinetic stages and does not provide a continuous equation to describe the size evolution of particles in a broad size range. In the present study, our aim was to develop a universal kinetic model which yields a closed analytical solution in the form of a continuous function that describes the temporal evolution of particle size up to a few hundred nanometers. In order to provide appropriate experimental data for testing the kinetic model, we studied systematically the formation of TiO2 particles during the hydrolysis of titanium(IV)-bis(ammonium-lactato)-dihydroxide (TIBALDH) under basic conditions. This reaction yields stable, monodisperse TiO2 suspensions in the size range of d = 15 – 1000 nm depending on the conditions.19-21 The temporal size evolution of the nanoparticles was recorded by dynamic light scattering (DLS) and UV-vis spectrometric methods.

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2. EXPERIMENTAL 2.1. Materials and solutions Titanium(IV)-bis(ammonium-lactato)-dihydroxide (TIBALDH) 50wt.% aqueous solution was purchased from Sigma-Aldrich, and used as received. All aqueous solutions were prepared by ultra-filtered water (ρ = 18.2 MΩcm by Milli-Q from Millipore). Other chemicals (HCl, HNO3, NaOH, CH3COOH and HF) were ACS reagent grade (Sigma-Aldrich) and used without further purification. When necessary, pH potentiometric measurements were performed using a Metrohm 785 DMP Titrino automatic titrator unit equipped with a 6.0262.100 pH-electrode.22

2.2. Formation kinetics of TiO2 particles 2.2.1. Starting the reaction The kinetics of the formation of TiO2 nanoparticles was studied via the basic hydrolysis and condensation of TIBALDH. The reactant solutions were filtered by using a 0.20 µm pore size PTFE syringe filter before each experiment. The reaction was started by mixing a TIBALDH solution with a NaOH solution. All kinetic runs were started in a custom built installation in fixed geometry (Fig. S1 in the Supporting Information) in order to ensure identical mixing conditions. A 5.0 mL syringe contained the Ti-precursor solution. It was fixed above a 25 mL beaker (50 mm × 30 mm) containing the NaOH solution. The TIBALDH solution was driven directly into the NaOH solution through a steel needle (0.6 mm × 40 mm) in ca. 2 s under constant stirring realized by a 16 mm × 5 mm PTFE coated magnetic stirring bar at 1300 rpm. The needle was positioned at 5 mm from the wall of the beaker and 10 mm below water level. In each experiment, 2.00 mL of TIBALDH solution (c = 6.00 mM – 220 mM) was added to 10.00 mL of 0.10 M NaOH solution. After stirring the mixture for 30 s, identical samples were taken and the –5–



formation of TiO2 particles were followed simultaneously by dynamic light scattering (DLS) and UV-vis spectrometric methods. The fixed geometry ensured the high reproducibility of the kinetic runs. The changing TiO2 suspensions were stable in the sense that no sedimentation of the newly formed particles took place for 8–10 h under the applied experimental conditions.

2.2.2. Dynamic light scattering spectrometry An aliquot of the TIBALDH + NaOH reaction mixture was placed into a thermostated cuvette and DLS autocorrelation curves were recorded in every minute. Stirring was not applied. The DLS measurements were performed on a Brookhaven light scattering instrument equipped with BI-9000 digital correlator (Brookhaven Instruments Corp., USA). The light source was a solid-state, vertically polarized laser operating at 533.4 nm. The measurements were performed at 25 °C, the scattering angle (ϴ) was 90°. The autocorrelation function of the scattered light intensity was collected in homodyne mode.23 The evaluation of the normalized autocorrelation functions was performed by the instrument controlling software that implements a standardized formalism [ISO 13321 (1996), ISO 22412 (2017)] using cumulants analysis to calculate harmonic intensity averaged particle diameter (z-average size, dz) and polydispersity index (PDI). The temporal change of these calculated parameters are analyzed in this study. Neither the Rayleigh, nor the Mie theory is applied for calculating dz and PDI. In order to prove that the investigated reaction yields monodisperse TiO2 particles, and thus that the reported dz values are representative, the autocorrelation functions at 50 nm < dz < 400 nm were resolved by i) multi-exponential fitting using a non-negatively constrained least squares algorithm ii) inverse Laplace transformation. Both of these methods suggest the unimodal size distribution of the particles, and the calculated distribution by intensity is always a single symmetrical peak. Further –6–


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details on the DLS measurements, as well as representative as-recorded experimental data are given in the Supporting Information.

2.2.3. UV-vis spectrophotometry The TIBALDH + NaOH reaction mixture was placed into a quartz cuvette and UV-vis spectra were recorded in a conventional spectrophotometric setup in every minute for 5 h. The cell was thermostated at 25.0 °C, and stirring was not applied. The measurements were performed in a fiber-optic UV-vis spectrophotometer equipped with a fast CCD detector (Avantes).24 The absorbance change was followed in the 170–1100 nm wavelength range in 1 nm steps. TIBALDH and TiO2 absorb light in the UV range. Besides this effect, the light scattering of the TiO2 particles cause the extinction of the transient light in the visible range as well. The formation and aggregation of the TiO2 particles were followed by analyzing the turbidity (i.e. the measured apparent absorbance values at 500 nm). The calibration of the method is detailed in the Results and Discussion (Section 3.2.). The precipitation of the particles was checked experimentally at long reaction times. i) The cuvette was taken from the cell and stirred for 10 s before taking a spectrum again. No difference was found before and after stirring. ii) The cuvette was scanned horizontally, and the turbidity was found to be uniform along the horizontal axis. These observations strongly suggest that the spatial distribution of the particles is homogeneous in the cell even at long reaction times.

2.3. Preparation of stable TiO2 suspensions Stable, monodisperse TiO2 suspensions of different particle sizes were prepared from TIBALDH. The basic hydrolysis of the precursor was started similarly to the kinetic experiments, and after a given reaction time, the process was chemically frozen by the addition of a –7–



stoichiometric amount of acetic acid. The exact reaction time needed to reach the desired particle size distribution was approximated from the kinetic experiments. The TiO2 particles were collected by centrifugation in a Beckman J2-21 centrifuge (2 h at 20000 rpm). The pellets were decanted and resuspended in an acetic acid – acetate buffer up to the desired concentration. The size distribution of particles was measured by DLS, and the Ti concentration of the suspension was measured by ICP-OES (see Section 2.4).

2.4. Elemental analysis of Ti The Ti contents of TIBALDH solutions, as-prepared TiO2 samples and the corresponding supernatants were measured by inductively coupled plasma optical emission spectrometry (Agilent Technologies ICP-OES SVDV 5100). The TIBALDH solutions were directly introduced into the ICP instrument. When necessary, the samples were diluted by 0.01 M HNO3 immediately before the analysis. A special Teflon nebulizer (OneNeb) was applied for the sample introduction. An Ar humidifier was connected externally to the ICP-OES instrument. Four-point calibration was applied (0 –100 mgL–1), diluted from a mono-element Ti standard solution of 1000 mg L–1 (Scharlau). Intensity values were collected at three different wavelengths of Ti. The concentrations were calculated from the intensity of the 337.280 nm line which was found optimal considering the precision of the calibration curve and the signal/background ratio. The experimental parameters are given in Table S1 in the Supporting Information. In those experiments where the basic hydrolysis of TIBALDH was chemically frozen (Section 2.3), both the as-prepared TiO2 and the corresponding supernatant were analyzed. Solid TiO2 was solubilized by microwave-assisted digestion. Published methods were adapted for the analysis.25,26 TiO2 was recovered by centrifugation and the deposited solid particles were dried at –8–


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60 °C for 24 h in a Horo drying cabinet. The solid material was digested in a microwave system (Ethos Up, Milestone) in closed PTFE vessels in two steps. In the first step, the mixture of 9.0 mL 65wt.% HNO3, 4.0 mL 38wt.% HF and 3.0 mL 37wt.% HCl were used and the samples were heated at 180 °C for 9.5 min at 1600 W. Then the vessels were opened and allowed to cool back to room temperature. 20 mL 4.5% H3BO3 was added to remove the HF content. The addition of H3BO3 was necessary because the sample introduction system in the ICP spectrometer is made of glass and quartz. In the second step of the digestion, the PTFE vessels were placed again into the microwave system for another 10 min at 160 °C and 1600 W. After digestion, the samples were filled up to the final volume of 100 mL with 1.9% H3BO3 solution. The solubilized TiO2 and the centrifuged supernatants were analyzed using the same ICP-OES method as in the case of the TIBALDH samples.

2.5. Softwares and calculation methods The raw spectrometric data were processed by the instrument controlling softwares. Basic mathematical calculations were performed with MS Excel. Non-linear least squares fitting of the experimental data was done by using the Levenberg-Marquardt algorithm with Micromath Scientist v2.0 software package.

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3. RESULTS AND DISCUSSION 3.1. Stoichiometry of the hydrolysis The stoichiometry of the TIBALDH + NaOH hydrolytic reaction was determined in batch experiments. The initial molar ratio of NaOH/TIBALDH was varied from 1 to 10, and excess NaOH was quantified by pH potentiometric titration after the hydrolysis and condensation was complete (5 h). The amount of the excess NaOH was plotted versus the amount of the initial NaOH. This gave an estimate of 2.3 ± 0.2 for the stoichiometric ratio of NaOH/TIBALDH (Fig. 1).

Figure 1. The stoichiometry of the TIBALDH + NaOH hydrolytic reaction. The x-axis shows the initial reactant ratio and the y-axis shows the amount of the excess NaOH over the initial TIBALDH. NaOH concentration was determined by pH potentiometric titration after the hydrolysis and condensation was complete.

3.2. Kinetic experiments Following the hydrolysis of the precursor, the average size of the TiO2 particles increases in time according to a characteristic trend. Simultaneous kinetic curves recorded by DLS and UV-vis spectrometric methods are shown in Fig. 2. Two fundamental parameters of the

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suspension may change as the reaction progresses: 1) the size distribution of the particles, 2) the concentration of the produced TiO2. The DLS method is sensitive to both of these parameters, but only the former can be quantified as discussed in the Experimental (Section 2.2.2). In UV-vis spectrophotometry, the decrease of the intensity of the transient light is caused by the scattering of the TiO2 particles produced. Generally, we found that the apparent absorbance measured at 500 nm (i.e. the turbidity) increases with increasing TiO2 concertation and with increasing particle size. Stable, monodisperse TiO2 suspensions (see Section 2.3) were used for the calibration of the UV-vis method. It was established that the absorbance at 500 nm is directly proportional to the TiO2 concentration in a dilution series of a monodisperse suspension. The extinction coefficient is the slope of this linear calibration curve (Fig. 3 left panel). The above defined extinction coefficient increases in a parabolic manner with increasing particle size (Fig. 3 right panel). It is important to note that the right panel of Fig. 3 was obtained at constant Ti concentration and not at constant particle concentration. In this form, it can be regarded as an empirical calibration curve.

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Figure 2. The size evolution of TiO2 particles following the basic hydrolysis of TIBALDH. Two sets of experiments are shown. One column corresponds to simultaneously measured kinetic curves by DLS spectrometry (A, B) and UV-vis spectrophotometry (C, D). Panels E and F show the corresponding experimental points plotted versus each other. The initial TIBALDH concentration is different in the 2 experiments: c0(Ti) = 37.0 mM (A, C, E) and c0(Ti) = 5.00 mM (B, D, F). c0(NaOH) = 83.3 mM.

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Figure 3. Calibration of the UV-vis spectrophotometric method for the quantitative analysis of TiO2 suspensions. Left panel: absorbance (turbidity) measured at 500 nm versus the mass concentration of the monodisperse TiO2 suspension. Not all series are shown for clarity. Right panel: extinction coefficient versus TiO2 particle size. This panel represents constant Ti concentration and not constant particle concentration. The preparation of the standard suspensions is given in the Experimental (Section. 2.3).

In order to find correlation between the DLS and the UV-vis data, the temporal change of the total TiO2 concentration had to be followed by an independent method. This was achieved by chemically freezing the hydrolysis and condensation at different reaction times, and quantifying the solid TiO2 content together with the unreacted Ti in the supernatant by ICP-OES (see. Sections 2.3 and 2.4). The method was validated by checking the overall Ti recovery from the 2 sources, which was between 94% and 102% for all kinetic runs. The most important observation from these experiments is that the amount of TiO2 recovered after freezing the reaction was independent of the elapsed time as shown in Fig. S3 in the Supporting Information. This means that the total concentration of TiO2 does not change during the aging of the suspension, only the size of the particles increases. This finding is further supported by similar results published in connection with analogous hydrolytic reactions.14,15,18

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Thus, the parallel DLS and UV-vis kinetic curves can be compared to each other in a straightforward way by applying the following considerations. First, the TiO2 concentration of a suspension does not change during a kinetic run, meaning that the trend in the UV-vis data is exclusively caused by the increase of the average particle size. Second, the measured DLS and UV-vis data strongly correlate with each other in the simultaneous runs, as demonstrated in Fig. 2 panels E and F. Third, if there is no temporal change in TiO2 concentration, the time-dependent average particle size can be independently calculated from the UV-vis kinetic curves using the calibration based on the TiO2 standards. Thus, the most reliable approach to correlate DLS and UV-vis data is to convert the measured absorbances into particle size values. The empirical calibration curve in Fig. 3 (right panel) makes it possible to convert turbidity data directly to particle size when the total Ti concentration is constant in time. As seen in Fig. 4, the timedependent mean particle size values calculated from UV-vis data and measured by DLS are in good agreement with a maximum deviation of 10% between the corresponding points. The most important feature of this approach is that it allows the collection of high quality kinetic data with high time resolution up to ca. dZ = 600 nm. Overlaid kinetic curves illustrating the effect of the variation of the initial TIBALDH concentration on the size evolution of TiO2 particles are shown in Fig. S4 in the Supporting Information.

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Figure 4. The formation of TiO2 particles following the basic hydrolysis of TIBALDH. Time-dependent mean particle size values measured by DLS (red) and calculated from UV-vis data (blue). The initial TIBALDH concentrations are given on the panels. c0(NaOH) = 83.3 mM.

3.3. Kinetic model The kinetic model considered in the present work consists of the following 3 irreversible reaction steps: P fast → C1 kn 2C1 → C2 k

g, i C i + C1 → C i +1

(R1) i≥2

k g,i = ik g

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This is a typical nucleation-growth model for nanoparticles. A Ti precursor (P) is rapidly hydrolyzed into the TiO2 primary particle (C1). C1 is a solid nanoparticle with a well-defined size (d0). The dimerization of this primary particle is slow, and further size growth can only occur through the addition of a single unit to a larger nanoparticle. SEM pictures (Fig. S5 in the Supporting Information) show that the as-prepared larger particles are indeed built up from small primary particle units, supporting that growth takes place via primary particle attachment. In addition, the rate constant of the reaction between a larger particle (C2, C3, etc.) and a primary particle (C1) is proportional to the number of the primary units (i.e. the mass) of the larger particle. The proportionality of the rate constant to particle mass can be rationalized if the forming particles are regarded as mass fractals. As such, their reactive surface area is proportional to their mass.27 Importantly, it is not necessary to assume the formation of spherical particles in the model. The only point where particle shape may play some role is the calculation of particle size from particle mass. Here, the important aspect is that all the particles should have roughly identical shape, but it does not need to be spherical. As far as the individual reaction steps are concerned, this model is similar to that proposed by Rempel et.al.10 However, there are three important differences between the two models. In Rempel’s model i) reversible growth steps are considered; ii) kn = kg, i.e. the dimerization of the primary particle is kinetically not distinguished from particle growth; iii) the growth rate constant was proposed to be proportional to the surface area of the spherical nanoparticle (kg,i = i2/3kg) instead of its mass. Rempel’s model predicts too high attachment rates to smaller particles, thus the primary particle pool is quickly depleted. As a consequence, this leads to an early stop of the particle growth, which is in clear contradiction with the experimental data obtained in this work. Another family of kinetic models was developed by Rivallin et.al. and McKoy et.al.18,28,29 In their approach, a continuous distribution of possible particle sizes is considered, rather than the – 16 –


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discrete values limited to the integer multiples of the primary particle as in the model proposed here and in the work of Rempel et.al.10 A further difference is that the reactions between larger particles are allowed, i.e. the particle growth is not limited to the attachment of a primary unit only in Rivallin’s model.18 This model defines a kernel function (Ki,j) that gives the rate constant of the reaction between 2 nanoparticles containing i and j primary units. The model presented here is a discretized version of this very general class of models with the following specific kernel function: K1,1 = kn K1,i = K i ,1 = ikg Ki, j = 0

(1) if i ≥ 2 and j ≥ 2

The present model postulates that the Ti precursor is transformed into C1 primary particles on a time scale that is fast compared to the time resolution of the kinetic experiments. Therefore, it is sufficient to consider only the concentrations of the Ci nanoparticles as dependent variables. The simultaneous ordinary differential equations describing the time evolution of the concentrations of the different TiO2 particles take the following form: ∞ d [C1 ] = −2k n [C1 ] 2 − ∑ ik g [C1 ][C i ] dt i =2

d [C 2 ] = k n [C1 ]2 − 2k g [C1 ][C 2 ] dt

(2)

d [C i ] = (i − 1)k g [C1 ][C i −1 ] − ik g [C1 ][C i ] dt

i≥3

To simplify the formulas used in the mathematical derivations, it is useful to define dimensionless variables by scaling.30 This has the additional benefit of reducing the number of parameters as well. The following dimensionless quantities are introduced:

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ci =

[C i ] [P]0

τ = k g [P]0 t

α=

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(3)

kn kg

The ci quantities are scaled concentrations, [P]0 is the initial concentration of the Ti precursor, τ is the scaled time, whereas α is the ratio of the dimerization and particle growth rate constants (see R1). Using these new variables, the original set of differential equations can be transformed into a somewhat simpler one. First, all equations are divided by kg and the square of [P]0: d [C1 ] k g [P]02 dt d [C 2 ] k g [P]02 dt d [C i ] k g [P]02 dt

=−

=

∞

2k n [C1 ]2

− ∑i

k g [P]02

k n [C1 ] 2 k g [P] 02

= (i − 1)

−

[C1 ][C i ] [P] 02

i =2

2k g [C1 ][C 2 ]

(4)

k g [P] 02

k g [C1 ][C i −1 ] k g [P] 02

−i

k g [C1 ][C i ] k g [P]02

i≥3

In this form, once the equation dτ = kg[P]0dt is obtained through the differentiation of the equation defining the scaled time (eq. 3), it is easy to identify the new quantities and state the differential equations using only them: ∞ dc1 = −2αc12 − ∑ ic1ci dτ i =2

dc 2 = αc12 − 2c1c2 dτ dci = (i − 1)c1ci −1 − ic1ci dτ

(5)

i≥3

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The final goal of the derivation presented here is not to find each ci, but to give the average TiO2 nanoparticle size as a function of time. Average properties can be defined in a number of different ways. It is assumed here that the radii of the TiO2 particles are proportional to the cube root of the number of primary units in them, which is directly proportional to their mass. Let d0 be the reference size, i.e. the size of the primary particle. With this, the number-average size (dN) of the population of nanoparticles can be defined by the following formula: ∞

dN = d

∑i

1/ 3

ci

i =1 0 ∞

(6)

∑c

i

i =1

A mass-average size (dM) is defined like this: ∞

dM = d

∑i

4/3

ci

i =1 0 ∞

(7)

∑ ic

i

i =1

Theoretically, the method dynamic light scattering (DLS) gives a quantity that is called the Zaverage size (dZ) of the particles and can be approximated as follows (see eq. S6 in the Supporting Information): ∞

∑i c 2

i

dZ = d0

i =1 ∞

∑i

(8) 5/3

ci

i =1

Finally, the average for which the simplest analytical formulas can be derived could be called the cube-root number-average (dC): ∞

∑ ic

i

dC = d0 3

i =1 ∞

(9)

∑ ci i =1

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By mathematical properties dN ≤ dC ≤ dM ≤ dZ holds for the same population of nanoparticles. The polydispersities measured by DLS in a kinetic run were always below 0.3, which means that the different averages are reasonably close. In addition, the shape of the kinetic curves is also similar in all experiments. Therefore, a comparison between measured data and theoretically predicted dC values (the one for which analytical formulas are the simplest) seems entirely sufficient. To obtain dC values as a function of time, the method of moments was used similarly to some previous works.18,28,29 The qth moment of the ci variables is defined as: ∞

µ q = ∑ i q ci

(10)

i =1

It should be emphasized that q can be any real number, its values are not limited to positive integers. All of the average sizes defined earlier can be calculated from the moments. In particular, dC can be obtained from the ratio of the first and zeroth moments as follows:

dC = d0 3

µ1 µ0

(11)

The major advantage of this approach is that it is not necessary to find the analytical formulas for all ci scaled concentrations, using the moments only is sufficient. As shown in the Supporting Information, the differential equation giving the first moment is as follows:

dµ1 =0 dτ

(12)

This equation states the simple fact that the first moment (µ1) is independent of time. In chemical thinking, this result might be viewed as highly natural, as the physical information content of the first moment is the (scaled) sum of the primary units in all the TiO2 nanoparticles in the system, which cannot be changed during aggregation because of mass conservation. The fact, that the

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total amount of primary particles is conserved after the prompt hydrolysis of the precursor was directly shown by the ICP analysis of chemically frozen reaction mixtures (see Section 3.2 and Fig. S3 in the Supporting Information). As the concentrations were scaled by [P]0, the time independent value of µ1 will simply be 1. In a similar fashion, the physical information content of the zeroth moment is the sum of the scaled number of TiO2 nanoparticles per volume unit. Because the total amount of primary particles is conserved after the hydrolysis of the precursor, nanoparticle concentration decreases in time during the particle growth process due to aggregation. In any other way, mass conservation would be violated. This means that the first derivative of µ0 will always be negative. As detailed in the Supporting Information, the full expression to give the zeroth moment is as follows:

µ0 =

[

(

α 1− α α + + τ − ln 2αe τ + 1 − 2α τ 2 2α − 1 (1 − 2α ) 2αe + 1 − 2α (1 − 2α )

(

)

)]

(13)

This is an exact formula that can be used for any values of the parameters. However, the typical and practically useful parameter values make it possible to find further simplifications. First of all, nanoparticles of reasonable size can only be formed if the initial dimerization of 2 primary particles is a lot slower than later particle growth. Mathematically, this can be transformed into the inequality 0 < α 1 − lnα), then the following form is obtained:

µ0 ≅ −α +

( )

1 −τ 1 −τ  1  e + ατ − αln e τ − αln(2α ) = e + α ln − 1 2α 2α  2α 

(15)

Therefore, the time dependence of the cube-root number-average can be given from eq. 11 as:

dC = d0

3

1 1 −τ  1  e + α ln − 1 2α  2α 

(16)

A further simplification is possible when the time is short enough for the exponential term to dominate in the denominator under the cube root in this equation (τ < −2 lnα):

d C ≅ d0

1 3

1 −τ e 2α

= d 0 3 2αe τ = d 0 3 2α e τ / 3

(17)

Therefore, in this limited time range (1 − lnα < τ < −2 lnα), an exponential function describes the time dependence of the average particle size well. This might seem a narrow region at first, but it is exactly the time range where most of the increase in particle size detected by DLS occurs. Using the original parameters of the model instead of the scaled ones, eqs. 16 and 17 take the following final form: dC = d0 3

kg 2k n

dC ≅ d0 3

e

− k g [P] 0 t

1  k  k + n  ln g − 1 k g  2k n 

(18)

2kn k g [P]0 t / 3 e kg

(19)

Equation 18 gives and excellent fit to the experimental kinetic curves, as demonstrated in Fig. 5.

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Figure 5. The measured time-dependent mean particle size values (markers) are fitted by eq. 18, which is derived from the kinetic model of the R1 reaction equations (red line). The initial TIBALDH concentrations are given on the panels. c0(NaOH) = 83.3 mM.

The calculated kobs = kg[P]0 and d0 values are given versus c0(NaOH) over c0(TIBALDH) in Fig. 6. The kobs values deviate from the predicted invers proportionality under a ca. 5-fold stoichiometric (10-fold concentration) excess of NaOH. The explanation is that the hydrolysis of TIBALDH is not complete when NaOH is not in a high excess (see. Fig. 1), which is in contradiction with the postulated kinetic model.20 The observation that the time needed to reach ca. 600 nm particle size is longer for higher precursor concentration is in agreement with the

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model. A higher initial precursor concentration also means the higher concentration of primary particles, and the higher concentration of larger particles. For an increase in the average size of the particles, a much larger loss of primary particle concentration is needed under these conditions. Therefore, the rate in the increase of average size is a somewhat complicated function of the initial precursor concentration.

Figure 6. The kobs = kg[P]0 (left panel) and the d0 (right panel) values defined by eq. 18 versus the initial NaOH concentration over the initial Ti precursor concentration. The dashed line represents the 5-fold stoichiometric excess of NaOH.

Nevertheless, it is clearly demonstrated that eq. 18 can successfully be applied to model and predict the size evolution of hydrolytically produced nanoparticles in a broad time and size scale. However, it should be emphasized that the use of eq. 18 is valid only when α = kn/kg

A Kinetic Model for the Hydrolytic Nucleation and Growth of TiO2

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