A New Model for Nano-TiO2 Crystal Birth and Growth in Hydrothermal

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A new model for nano-TiO2 crystals birth and growth in hydrothermal treatment using oriented attachment approach Vasile Lavric, Raluca Isopescu, Valter Maurino, Francesco Pellegrino, Letizia Pellutie, Erik Ortel, and Vasile-Dan Hodoroaba Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.7b00302 • Publication Date (Web): 25 Sep 2017 Downloaded from http://pubs.acs.org on October 1, 2017

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Crystal Growth & Design

A new model for nano-TiO2 crystals birth and growth in hydrothermal treatment using oriented attachment approach Vasile Lavrica,b, Raluca Isopescu*a,b, Valter Maurinoc, Francesco Pellegrinoc, Letizia Pellutièc, Erik Orteld, Vasile-Dan Hodoroabad a – R&D Consultanta si Servicii, 266 -268, Calea Rahovei, 050912 Bucharest, Romania b - University “Politehnica” of Bucharest, G. Polizu Street 1-7, 011061 Bucharest, Romania c – Dipartimento di Chimica and NIS, Università di Torino, via P. Giuria 5, 10125 Torino, Italy d - Bundesanstalt für Materialforschung und –prüfung (BAM), Unter den Eichen 87, 12205 Berlin, Germany ABSTRACT The synthesis of TiO2 was studied in an original hydrothermal process that uses triethanolamine titanium complex Ti(TeoaH)2 as Ti precursor and triethanolamine (TeoaH3) as shape controller to obtain bipyramidal anatase nanoparticles. Backed-up by experimental evidence, i.e. time profiles for Ti(IV) species concentrations together with crystal shape and particle size distributions measured by dynamic light scattering and electron microscopy, a mathematical model was built. The model includes chemical reactions responsible for TiO2 generation in solution and the subsequent anatase nucleation and crystal growth. The oriented attachment mechanism was adopted to explain the builtup of crystals with equilibrium anatase structure (Wulff structure) and time-varying shape factor. This complex mathematical model was solved writing and validating an in-house software using the Matlab™ (Natick, MA, USA) environment. The process was simulated for a batch time of 50 hours

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and the results, in terms of main species concentration and crystal size distributions are in rather good agreement with the experimental measurements.

INTRODUCTION Nano-crystalline TiO2 is extensively studied due to numerous applications that may depend on its particular polymorphic phase, size and shape. The applications of nano-TiO2 generally refer to its photocatalytic1, 2 and photovoltaic properties3. Practical applications that use nano-TiO2 to attenuate UV light are sunscreens or similar cosmetics, various plastic-based products and containers, and clothing. Emerging applications include solar cells or environmental remediation of pollutants4, 5. TiO2 may be obtained in three polymorphic phases: rutile, anatase or brookite. Experimental and theoretical studies proved that the three crystalline phases have different characteristics that imply also different uses and applications6-8. There are several synthesis routes for nano-TiO2: hydrothermal9-12, solvothermal3, sol-gel13 or precipitation14. Hydrothermal synthesis is a method for obtaining single crystals, technique that depends on the solubility of minerals in water-based mineralizers under high pressure and temperature. The reaction implies the solubilization / re-precipitation of a suitable feeder. Advantages of hydrothermal synthesis are crystallization without extensive particle growth and a reduced agglomeration11. NanoTiO2 can be obtained by hydrothermal treatment of peptized precipitates of a titanium precursor with water in form of nanoparticles10, nanowires15, or nanotubes16. The evolution of crystal shape during growth is related to the species and their concentration in the synthesis environment, as they modulate the crystal surfaces energy. Moreover, surface adsorbed additives can substantially influence the final shape of the crystals, acting as shape controllers. The solvent effect on the crystal morphology is generated via the different interactions with different crystal planes, stabilizing preferentially one of them and, hence, the final shape of the crystal17.


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There are several experimental and theoretical studies that suggest various mechanisms of TiO2 crystal growth for hydrothermal processes9,

18-22

. For the hydrothermal TiO2 synthesis in alkaline

conditions, at 200°C and 2-16 h, a growth model based on oriented attachment (OA) along the [001] direction followed by increasing size by Ostwald ripening is proposed by Cho et al18. They noticed that the size and shape of the crystals depend on pH and process duration. OA was first noticed by Penn and Banfield19 in the synthesis of nanoanatase crystals in hydrothermal conditions. This mechanism was extensively studied by molecular dynamics simulation by Fichthorn and co-workers, based upon the intrinsic nanocrystal forces that act in vacuum and facilitate alignment and aggregation20, 21. Supplementary, they also introduced in their model the interactions between water molecules and crystal surface, when the process occurs in humid environment, where the growth is mediated by adsorbed water and hydroxyls22. In the present study, a new mathematical model for the simulation of nano-TiO2 hydrothermal synthesis and prediction of the product shape and particle sizes is proposed. The model is based on both theoretical considerations and experimental results, with the latter being used in generating the simplified hypotheses and in the validation step. The experiments are carried out in an original hydrothermal synthesis for bipyramidal anatase nanoparticles process that uses triethanolamine titanium complex Ti(TeoaH)2 as Ti precursor and triethanolamine (TeoaH3) as shape controller23. A mathematical model including nucleation and growth mechanisms together with the kinetics of the involved chemical reactions was developed, based upon first principles approach. This model aims to predict the behavior of the discontinuous crystallization process of TiO2 truncated bi-pyramidal anatase, considering the generally accepted chemical reactions implied in the pathway from Ti(TeoaH)2 to

TiO2, aq. In order to develop a reliable mathematical model, the liquid phase

chemical reactions were scrutinized, so that only the main accepted ones describing the thermodynamically possible generation of chemical compounds by Ti4+ in water system24,

25

are

retained. Then, the kinetics of this process was built, considering two types of processes, which are



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inter-related, although their time-scales are different. The first process is the continuous variation of the species concentrations in the solution, with a time scale in the order of magnitude of the reaction time. The second is represented by nucleation and growth of the crystals, seen like discrete events which change the crystals distribution and having a time scale several orders of magnitude smaller than the first process. Here, growth is understood as the increase of crystal dimensions through the addition of the minimum possible number of units. In fact, with respect to the time scale of species concentration variation, the nucleation and growth could be considered as instantaneous events.

EXPERIMENTAL Materials and Methods The precursor used for the hydrothermal synthesis of TiO2 nanoparticles is a complex of Ti (IV) with triethanolamine with molar ratio1:2. The synthesis is described elsewhere23. Briefly, 0.1 mol of Ti(IV) isopropoxide (Aldrich reagent grade 98%) are mixed with 2 mol of triethanolamine (Aldrich Reagent grade 98%) under nitrogen atmosphere. After reaction, the isopropyl alcohol formed is distilled off under vacuum. The synthesis product is a pale yellow viscous liquid, used without further purification. Other reagents used are at least analytical grade. Ultrapure water is produced with a MilliQ gradient apparatus (Millipore). The TiO2 NPs were synthesized by forced hydrolysis of an aqueous solution of the TiIV(triethanolamineH)2 complex, Ti(TeoaH)2, in a Teflon lined stainless steel reactor. A typical TiO2 synthesis batch time considered in our study is 50 hours. In order to estimate the variation of compositions during the process, several runs were performed, starting from the same initial conditions, and stopping the process after several treatment times, as presented in Table 1.


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Table 1. Conditions adopted for TiO2 NPs hydrothermal treatment preparation TiO2 synthesis Ti(TeoaH) set tag 2 (mM) UT_2-20_02h

59.2

Added TeoaH3 (mM) 40.0

Initial pH

Temperature (°C)

Crystallographic phase

170±2

Treatment time (hours) 2.00

10.0

UT_2-20_05h

58.8

40.0

10.0

170±2

4.67

Anatase

UT_2-20_15h

58.6

40.7

10.0

170±2

15.0

Anatase

UT_2-20_25h

56.4

40.0

10.0

170±2

25.0

Anatase

UT_2-20_35h

60.5

40.3

10.0

170±2

35.0

Anatase

UT_2-20_50h

59.2

40.0

10.0

170±2

50.0

Anatase

Anatase

The hydrothermal reaction was stopped after the reported treatment time by quenching the stainlesssteel reactor in water. Initial pHs were measured using a pH meter (Metrohm mod 720) equipped with a combined electrode (Metrohm mod 6.0232.100) calibrated against NIST traceable pH buffers (pH 3.00, 7.00, 10.00). The pH reached after the treatment time was measured after cooling the reaction mixture at ambient temperature. Total soluble Ti(IV) species were measured using a ThermoScientific ICAP-Qs ICP-MS, equipped with a quadrupolar mass analyzer and a quadrupolar collision/reaction cell with flat electrodes (flatpole). The instrument is calibrated with external standards in HCl/HNO3 10 g/L for Ti prepared by dilution of certified reference standards at 1000 mg/L, using 45Sc (100 ppb) as an internal standard. The 48Ti isotope was used. The calibration curve must contain at least five calibration points and the linear correlation coefficient, R2, should be > 0.99. The raw suspensions were filtered and diluted with HCl/HNO3 solution (10 g/L). The ESIHRMS analysis of a freshly prepared 40 mM Ti(TeoaH)2 water solution was performed diluting 1:100 the previously mentioned solution in methanol by using a LTQ- Orbitrap (Thermo Finnigan) hybrid HRMS equipped with an atmospheric pressure ionization (API) source, operated in electrospray ionization (ESI) mode (high voltage 3 kV, sample flow 10 µL /min).



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The particle size distribution of TiO2 obtained in each sample (Table 1) was measured by dynamic light scattering (DLS) and scanning electron microscopy in transmission mode (T-SEM). DLS measurements were carried out with an ALV NIBS (not invasive backscattering) apparatus (ALV, Langen Germany), equipped with a correlator mod ALV5000. Data analysis on the scattered light autocorrelation function was carried out with the Contin regularization algorithm, to obtain decay time and translational diffusion coefficient distributions. Conversion of translational diffusion coefficients to hydrodynamic radius, RH of the NPs is done with the Stokes-Einstein equation. DLS data were obtained analyzing the raw suspensions (if necessary diluting it using NH3 200 mM as a dispersant solvent) after sonication for 10-30 minutes in an ultrasound bath (95 W, 37 kHz) and in a closed vial to avoid NH3 evaporation. Recommended concentration range is 10-500 mg/L, depending on the size of the NPs. A standard operation procedure for screening characterization of morphology (size and shape) of TiO2 NPs by electron microscopy26 was also applied. A high-resolution scanning electron microscope (SEM) Supra 40 (Zeiss, Oberkochen, Germany) having a Schottky field emitter has been used. A dedicated sample holder27-28 has been employed to enable the operation of the SEM in the transmission mode (T-SEM) which is more sensitive to NP dimensional measurements. All the samples have been measured after preparation from suspension by dropping on typical TEM grids. The suitability of T-SEM as a metrological tool for NP size measurements has been demonstrated recently in several reports in the literature 27-29. The shape factor was calculated by evaluating the major and minor axes of ellipses fitted to the NP’s boundary by using the freeware software package ImageJ (https://imagej.nih.gov/ij/) and dividing them to each other for each identified NP. To compare the results of size measurements by DLS with T-SEM, the geometrical data determined by T-SEM were converted to hydrodynamic radii (assuming the NPs as prolate ellipsoids) with the Perrin formula 30-31


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p2 − 1 RH , p = a / b = 1, prolate = RS p 1/3 ln p + p 2 − 1

)

(

p2 − 1 RH = , p = b / a > 1, oblate RS p2/3 arctan p2 − 1

)

(

where RS is the radius of the sphere with the same volume as the NP. This way, a direct comparison between DLS measurements (for which the characteristic length is the RH as measured in NP suspension) and T-SEM images (which give geometric size parameters of NPs deposited on a substrate) is possible.

RESULT AND DISCUSSION 1 - Modeling the treatment processes 1.1 The stoichiometry Starting from experimental evidence concerning the Ti species present in the reaction medium (see section 2), and considering all the chemical reactions that may take place in aqueous solution, it is assumed that Ti4+ is generated from Ti(TeoaH)2 through the consecutive chemical reactions:  → Ti ( TeoaH )2 +2H 2 O ←  kfTiTEOA 2

krTiTEOA 2

Ti ( TeoaH ) +TeoaH 3 +2HO 2+

 → Ti ( TeoaH ) +2H 2 O ← 

(1) −

kfTiTEOA 2 p

2+

krTiTEOA 2 p

4+

Ti +TeoaH3 +2HO

−

(2)

Ti4+, that is assumed to form by equation (2), disappears from the reaction medium by a very fast stepwise Ti4+ hydrolysis, which eventually leads to TiO2 generation32. This assumption is in good agreement with experimental results expressed in total soluble Ti concentration (Table 3) during the hydrothermal treatment. NaOH is added in a quantity that allows the initial pH to be higher than 9 as required for subsequent anatase bi-pyramids to crystalize. As the phase analysis (Table 1 and



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XRD patterns in Supporting Information, Figure SI_10) proves that anatase is the only detected phase, the rutile and brookite phases were disregarded in the mathematical model.

Ti4+ hydrolysis generates H+, thus compensating the hydroxyl ions formed in the preceding steps (1) and (2) and limiting the pH increase:  → Ti ( OH ) +H + Ti 4+ +H 2 O ←  kfTi4p

3+

(3)

krTi4p

 → Ti ( OH ) +H + (4) Ti ( OH ) +H 2O ←  2 kfTiOH13p

3+

2+

krTiOH13p

 → Ti ( OH ) +H + (5) Ti ( OH )2 +H 2 O ←  3 kfTiOH22p

2+

+

krTiOH22p

 → Ti ( OH ) +H + Ti ( OH )3 +H 2 O ←  4 kfTiOH31p

+

krTiOH31p

(6)

In turn, Ti(OH)4 either releases two molecules of water to form TiO2 or reacts with the hydroxyl ion further (abundant if the reaction medium is alkaline) to give Ti ( OH )5− . kfTiOH 4  → TiO2 +2H 2 O Ti ( OH )4 ← 

(7)

kfTiOH 45  → Ti ( OH )Ti ( OH )4 +HO − ←  5

(8)

krTiOH 4

krTiOH 45

The formation of bi-pyramidal anatase crystals is realized at high pH, and, therefore, the initial pH of the Ti(TeoaH)2 solution is adjusted by adding NaOH, which is assumed totally hydrolyzed. Ti ( OH )5 interacts with Na , present in the solution, to form TiO 2 as well. +

-

kfTiOH 5 m  → TiO 2 +2H 2 O+NaOH Ti ( OH )5 +Na + ←  krTiOH 5 m

(9)

Although NaOH is practically totally dissociated, the dissociation equation is included just for closing the mass balance; to account for the total dissociation, a very low value was assumed for the reversible reaction rate krNaOH:  → Na + +HONaOH ←  kfNaOH

krNaOH

(10)


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Water dissociation was also included in the kinetic model for the same reason of closing the mass and ion balance: kfH 2 O  → H + +HOH 2 O ← 

(11)

krH 2O

The triethanolamine added as a shape controller, as well as the triethanolamine formed by the hydrolysis of Ti(TeoaH)2 has an alkaline character, dissociating in water: kfTEOA → TeoaH 3 +H 2O ← TeoaH 4 + +HO-

(12)

krTEOA

1.2 The chemical reactions kinetics Taking into account the hypothesis that all the aforementioned chemical reactions are elementary, the mass action law applies to get the equivalent reaction rates. Considering the concentration of water as nearly constant, since it is available in large excess (several orders of magnitude higher than of all other species implied in the chemical process), the first equivalent reaction rate (relation 13) becomes the relation (14), where kfTiTeoaH includes water concentration (55.56 mol·L-1, according to the initial reagents ratios). Whenever a chemical reaction implies water as reagent, its concentration will be included, by default, in the corresponding rate constant. Relation (15) represents the Ti4+ generation, while relations (16)-(19) describe the steps with Ti(IV) hydroxides formation. The rate of TiO2 formation in the solution is described by relation (20) while its interference with Na+ and HO- ions present in the alkaline environment, represented by relations (8) and (9) are described by equations (21) and (22). Even though, in most cases, any dissociation reaction is considered fast enough to be at equilibrium at any moment; in the present kinetics, we write the reaction rates for the three reversible dissociation processes (water, NaOH and triethanolamine), since nucleation and growth events are instantaneous and the ions resulted from dissociations are implicated in the rest of the chemical steps.

r1 = kfTiTEOA2* ⋅  Ti ( TeoaH )2  ⋅ [ H 2 O ] - krTiTEOA2 ⋅  TiTeoaH 2+  ⋅ [ TeoaH 3 ] ⋅  HO −  2

2

r1 = kfTiTEOA2 ⋅ Ti ( TeoaH ) 2  − krTiTEOA2 ⋅  TiteoaH 2+  ⋅ [ TeoaH 3 ] ⋅  HO −  2


2

2

(13) (14)


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2 2+ r2 = kfTiTEOA2 p ⋅ Ti ( TeoaH )  − krTiTEOA2 p ⋅ Ti 4+  ⋅ [ TeoaH3 ] ⋅  HO−   

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2

(15)

3+ r3 = kfTi 4 p ⋅ Ti 4+  − krTi 4 p ⋅ Ti ( OH )  ⋅  H +   

(16)

3+ 2+ r4 = kfTiOH 13 p ⋅ Ti ( OH )  - krTiOH 13 p ⋅ Ti ( OH )2  ⋅  H +     

(17)

2+ + r5 = kfTiOH 22 p ⋅ Ti ( OH )2  − krTiOH 22 p ⋅ Ti ( OH )3  ⋅  H +     

(18)

r6 = kfTiOH 31 p ⋅ Ti ( OH )3  − krTiOH 31 p ⋅ Ti ( OH )4  ⋅  H +   

(19)

r7 = kfTiOH 4 ⋅ Ti ( OH )4  − krTiOH 4 ⋅ [ TiO 2 ]

(20)

+

r8 = kfTiOH 45 ⋅ Ti ( OH )4  ⋅ HO−  − krTiOH 45 ⋅ Ti ( OH )5   

(21)

r9 = kfTiOH 5m ⋅ Ti ( OH )5  ⋅  Na +  − krTiOH5m ⋅ [ TiO2 ] ⋅ [ NaOH ]  

(22)

r10 = kfNaOH ⋅ [ NaOH ] -krNaOH ⋅  Na +  ⋅  HO − 

(23)

r11 = kfH 2O - krH 2O ⋅  H +  ⋅  HO − 

(24)

r12 = kfTEOA ⋅ [ TeoaH 3 ] − krTEOA ⋅ TeoaH 4 +  ⋅  HO − 

(25)

-

-

1.3 Crystallization mechanism and kinetic considerations Accepting that TiO2 crystals appear/form and grow through the OA mechanism, we considered that the anatase nanocrystals with a bi-pyramidal shape, (the Wulff shape, as it was observed experimentally19 and theoretically predicted33) are present in the system. The rest of the shapes/phases current in other processes were disregarded; this decision was backed up by the experimental results in the investigated working conditions. The Wulff anatase crystal has both facets [101] and [001]. Our kinetic considerations are based on the equilibrium anatase structure as derived by atomic scale simulation21.


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Figure 1. Bi-pyramidal TiO2 nanocrystal:

a) Wulff topology (adapted from

22

), b) schematic

representation of TiO2 units in the longitudinal section with the parameters of the model.

According to Fichthorn22, this structure is defined by several superimposed squared layers creating a symmetrical truncated bipyramid (Figure1a), characterized by height (as distance between the two (001) faces), denoted as a, and width, representing the length of the symmetry square side, denoted as b. The ratio a/b defines the shape factor, ϕ. One main hypothesis related to the crystal growth is that, no matter how big the crystal is, it has a central bi-layer square (the two layers observe a point symmetry). This bi-layer is the symmetry boundary between two mirrored square frusta (truncated square pyramid). According to experimental



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evidence (see paragraph 2 for a thorough discussion), a birth mechanism of the primary anatase crystals should include i) cuboids, followed by a rearrangement into ii) an elongated anatase form, which can be regarded as the seeds from which iii) the crystals will grow. We hypothesize that this elongated primary anatase crystal will have, on each side of the symmetry bi-layer, at least three double layers with the same sides as the symmetry layer, being characterized by shape factors higher than 2.2. Moreover, as experimental evidence suggests (see the sudden decrease of the shape factor from over 2.2 to around 1.6 in Table 3), the elongated anatase crystals of shape factors in the vicinity of 2.2-2.3 are energetically unstable and rearrange internally to form an anatase crystal with a shape factor in the vicinity of 1.6-1.7, increasing its width much more than its height. The timescale of this internal rearrangement is very small, of the same magnitude as the timescale of nucleation and growth. Therefore, the internal rearrangement will be regarded as an instantaneous process, as well. The OA growth mechanisms imply two discrete events: i) the attachment along an energetically favorable facet of two distinct crystals, followed by ii) the rearrangement of the parents into a newborn crystal. The frustum is formed by successions of pairs of layers, the units from one layer being rotated with 1800 with respect to the units from the other layer. Each successive pair of layers has one unit less and this holds up to the upper pair of layers forming the small square basis of the frustum (Figure 1a). An elongated anatase crystal has supplemental pairs of layers on each side of the symmetry bi-layer (Figure 1b), with the same number of units as the latter. These layers of equal squares form a cuboid between the two frusta. Accordingly, such an elongated crystal will have three parameters with which its geometry can be fully described (Figure 1b): m, the number of units/molecules of the side of the small square base of frustum; k, the number of pairs of (mirror) layers in a frustum (i.e. not


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belonging to the cuboid part); l, the number of pairs of (mirror) layers in the cuboid part, (i.e. in between the frusta and the symmetry bi-layer). Using the aforementioned parameters, we can approximate the width and the height of this crystal:

a = 2 ⋅ ( k + l + 1) and b = m + k −1. Considering that the projection of the crystal onto the plane ZX (see Figure 1b) could be fairly approximated by an ellipse, the equivalent dimensionless radius would be R = a ⋅ b 2 . The structure of the elongated anatase crystal implies that even for the smallest variations, with one unit/molecule only for m and one pair of layers only for k or l, the overall number of units associated with these crystals are considerably different (see Table 2 for NPs possible configurations for some small values of k, m and l).

Table 2 - The relationship between the parameters characterizing the geometry of the crystal and its number of molecules shape factor and dimensionless equivalent radius. Configurations of (Elongated, l >0) Anatase Crystals k m l Number of TiO2 ϕ R units/monomers 2 0 70 2 2.12 2 1 106 2.67 2.45 3 0 132 1.5 2.45 2 3 1 196 2 2.83 4 0 214 1.2 2.74 4 1 314 1.6 3.16 2 0 148 2 2.83 2 1 212 2.5 3.16 3 0 250 1.6 3.16 3 3 1 350 2 3.54 4 0 380 1.33 3.46 4 1 524 1.67 3.87 2 0 266 2 3.54 2 1 366 2.4 3.87 3 0 416 1.67 3.87 4 3 1 560 2 4.24 4 0 602 1.43 4.18 4 1 798 1.41 4.58 5 2 0 432 2 4.24



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2 3 3 4 4 5 5

1 0 1 0 1 0 1

576 638 834 888 1144 1182 1506

2.33 1.71 2 1.5 1.75 1.33 1.56

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4.58 4.58 4.95 4.9 5.29 5.2 5.61

The first observation, when analyzing the NPs from Table 2, is that the increase with only one TiO2 unit: − of the side of the upper square of the frustum, m, fattens the crystals, decreasing its aspect ratio; − of the number of pair layers in a frustum, k, slims the crystal, decreasing its aspect ratio; − of the height of the cuboid part of the crystal, l, increases its aspect ratio significantly. These important consequences of the variation of the crystal geometry with respect to its parameters will be fully used in choosing the crystal growth strategy. The second observation is related to the total number of units in a crystal, which changes discontinuously, each small increase in one of the parameters implying a significantly higher number of TiO2 units.

1.3.1 Birth mechanism According to the experimental evidences (see the discussion in paragraph 2), the birth mechanism of the primary elongated anatase crystals should include, as first stages, cuboids formation and agglomeration. The biggest cuboids would undergo a rearrangement into an elongated anatase form, which can be regarded as the seeds from which the TiO2 crystals will grow. We postulate that the backbone of the anatase crystal is the symmetry bi-layer. Therefore, this symmetry bi-layer will appear during the nucleation of the smallest cuboids, and be kept during the agglomeration of cuboids, the rearrangement of the biggest cuboids into the elongated anatase crystal and the further crystal growth processes. As such, the smallest cuboids would be cubes having a symmetry bi-layer


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and two double layers with the same number of units as in the symmetry bi-layer on each side of the latter. It should be emphasized that, since the layers are in symmetrical pairs, there could be no other cubes with a lesser number of units/monomers, still respecting the symmetry principles of the anatase crystal. We also postulate that the agglomeration process takes place only between cubes, the interactions between cubes and cuboids being less favorable – a cube and a cuboid would give birth to a crystal lacking symmetry. Accordingly, these primary cuboids will agglomerate two by two, forming larger and unstable cuboids with 12x12x6 dimensions, with a total of 864 TiO2 units (see Figure 2 for details). These unstable cuboids could, on one hand, split into the original precursors/cubes, or, on the other, suffer a rearrangement process, irreversible, to result in an elongated anatase crystal with the characteristics (k=5, m=2, l=3), which has the exact same number of units, 864, as the unstable biggest cuboid, and a shape factor of 3. These are the primary elongated anatase crystals, which will enter the process of growth through OA, eventually giving a quite narrow anatase crystals size distribution.

Figure 2. - Formation of the precursor cuboid of elongated anatase crystal from the primary cubes, appeared through nucleation (6x6x6 units, having a total of 216 monomers). They agglomerate two by two preserving the symmetry layers, into cuboids with 12x12x6 dimensions, eventually.



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1.3.2 Growth mechanism To define the growth mechanism, four hypotheses were assumed. The first hypothesis refers to the ability of two particles to collide and eventually generate a new anatase crystal. We postulate that the mechanism is OA, meaning that two crystals will join facets according to their electrostatic potential. During the OA, through which the anatase crystals grow, there will be no increase in l, the number of double layers situated on each side of the symmetry bilayer, since this is energetically unfavorable – this is true since any increase of the primary elongated anatase crystal should have as effect an increase of the aspect ratio over 3; this was not found experimentally. For example, an increase of one unit of l for the crystal (5,2,3) to (5,2,4) would imply a raise of the shape factor to 3.33. More, adding a unit to the small square of frustum, m, for the latter, (5,3,4), would decrease the shape factor to 2.86, while an increase of the cuboid part of this elongated anatase crystal with one unit to (5,3,5) would give a crystal with the shape factor of 3.14. We postulate, also, that there is no gradual decrease of the shape factor, the crystals losing their cuboid part suddenly, when the aspect ratio drops below 2.2-2.4. Till then, there will be a gradual decrease of the shape factor, due to the preferential increase in width of crystals, driven by the OA growth. After this key rearrangement of the biggest precursor cuboids, the anatase crystals will slowly drop their shape factor towards 1.4 and even less, since the growth in height is less favorable. We hypothesize that in order to successfully collide and aggregate, the two anatase crystals should have sufficiently close masses. If we take as metric k, then two crystals are close enough to aggregate if k1 − k2 ≤ 3 . This means that all the crystals from Table 2 can aggregate following an OA between themselves, but the smallest crystals (k=2) cannot aggregate with higher crystals, having k>5. The second hypothesis is related to the number of TiO2 units needed by the resulted crystal: if the latter would have more units than both parents, the deficit is covered, during aggregation, harvesting the necessary supplemental units from the solution (ripening off). On the other hand, if both parents


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have, together, a surplus of TiO2 units, aggregation would imply that a part of these units already crystallized to be solvated again (re-solvation). Although there are some experimental evidences that this phenomenon is possible (not in the case of anatase TiO2 crystals, but generally), we disregard it, for simplicity reasons. Thus, in the case of surplus, the collision is unsuccessful in our model (if future experimental evidence will prove the existence of re-solvation in the case of this process, implementing it in the software is an easy task). The third hypothesis related to the growth of anatase crystals is that it happens step-wise, through unity increments. This means that a crystal can grow with a single pair of layers on height, with a single monomer on the smallest square of frustum, or in both directions, the latter with a smaller rate of success, due to the increase of energetic demands during the rearrangement. If the operating conditions favor slim crystals, it is possible to envisage an exceptional increase with two increments of k. The forth hypothesis concerns the difference between the colliding parents and the direction of growth of the new crystal. If the parents have the same height, the new crystal will be higher with one increment k new = k old + 1 , whilst in all other cases, the latter will fatten with one increment m new = m old + 1 , keeping the height of the highest parent. Under selected operating conditions, the new

crystal could grow in height with two increments, k new = k old + 2 , but this should be a rare event. It should be emphasized that not all the collisions are effective in producing bigger crystals, but only a fraction, f growth , which is higher for the smallest crystals, and lower and lower as the involved parents have bigger and bigger sizes. We hypothesized that f growth = 0.5 for k=2, m=2, then starts decaying exponentially to fgrowth = 10−4 for k=70, m=25, which has more than 1.1 ⋅ 10 6 TiO2 units. This function resulted from the analysis of the experimental data, which showed the tendency of NPs to decrease their growth rate exponentially, as their sizes increase.



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1.4 The mathematical model The mathematical model of the crystallization process of TiO2 to anatase is based upon the mass and charge balance equations of all participating species. For all the active species, the time variation is given by the dichotomic simultaneous action of two main processes: formation and disappearance; there are situations in which one of the two might be absent. d  Ti ( TeoaH ) 2  dt

= − r1

(26)

2+ d Ti ( Teoa )    = r −r 1 2 dt

d [ TeoaH3] dt

(27)

= r1 + r2 − r12

(28)

d [H 2O]

= −2 ⋅ r1 − 2 ⋅ r2 − r3 − r4 − r5 − r6 dt + r7 + 2 ⋅ r9 − r11 − r12 d  HO -  dt d  Ti 4+  dt

= 2 ⋅ r1 + 2 ⋅ r2 − r8 + r10 + r11 + r12

(30)

= r2 − r3

(31)

3+ d  Ti ( OH )    =r −r 3 4 dt

d  H +  dt

= r3 + r4 + r5 + r6 + r11 + r12'

2+ d Ti ( OH )2    = r −r 4 5 dt

+ d  Ti ( OH )3    = r −r 5 6 dt

d  Ti ( OH )4  dt

(29)

= r6 − r7 − r8

(32)

(33)

(34)

(35)

(36)


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d Ti ( OH )5    = r −r 8 9 dt -

d [ NaOH ] dt

d  Na +  dt

= −r10

= − r9 + r10

+ d ( TeoaH4 )    =r 12 dt

(37)

(38)

(39)

(40)

The solvation of H+ and Na+ ions does not alter the model since the kinetic parameters are obtained by regression analysis, and their values include implicitly this effect. The crystallization processes including with nucleation, dissolution of small crystals and growth are described by equations (41)-(44). The equation (42) stands for the number variation of the smallest cube appeared from the TiO2 solution, when the latter’s concentration becomes greater than the critical concentration. In equations (41) and (42), kf1st represents the nucleation rate constant, considering first order kinetic with respect to TiO2 concentration in the liquid phase. In equations (41) and (42), NA stands for Avogadro’s constant (216 is the number of units of the primary cube) and k f 2 nd is the fraction of primary cubes agglomerating to form cuboids 12x6x6. N ⋅ 216  kf1st ⋅ [ TiO 2 ] + kr1st ⋅ cube , [ TiO 2 ] ³ [ TiO 2 ]crit − 14 4244 3 NA 3  generation of cube 6x6x6 144244 d [ TiO 2 ]  solvation of cube 6x6x6 = r7 + r9 +  N ⋅ 216 dt  kr1st ⋅ cube , [ TiO 2 ] < [ TiO 2 ]crit  NA 14 4 244 3  solvation of cube 6x6x6

dNcube dt

 [TiO2 ] ⋅ NA − kr1st ⋅ Ncube − kf2nd ⋅ Ncube + 2⋅ kr2nd ⋅ Ncuboid2 , [ TiO2 ] ³[ TiO2 ]crit kf1st 1424 3 144244 3 216 = agglomeration cuboid 12x6x6 from cuboid 12x6x6  2⋅ kr2nd ⋅ Ncuboid2 − kr1st ⋅ Ncube − kf2nd ⋅ Ncube , [ TiO2 ] kf2nd>kf3rd), while fGrowthUp represents the fraction of the cuboids 12x12x6 which enter the rearrangement process to

form anatase elongated crystal (5,2,3). In order to consider the probability of collisions of crystals having matching sizes, which drops significantly for bigger crystals, the parameter fGrowthLow is introduced, its value corresponding to the anatase crystal with the dimensions kmax and mmax (aspect ratio=1.51, dimensionless equivalent radius=78.1). As it might be seen from Figure SI_5 (bottom right field ‘fGrowth decrease’), we considered, beside the exponential decrease, the possibility of a linear decrease. When using the latter, the resulted NPs distribution was not satisfactory at all, indirectly confirming that the probability of successful collisions decays exponentially with the mass of the anatase crystal. Another parameter is the “Critical Aspect” – when the shape factor is in the former’s vicinity, in the interval [0.9, 1.1] multiplied with the ”Critical ratio”, the elongated anatase crystal attained the limit of stability and suffers a spontaneous process of passing to the anatase form. This is done throughout an increase on both dimensions, k and m. According to the kinetic model, the concentrations of Ti(IV) hydroxides vanish after about 3 h (See Supporting Information, Figure SI_4) and similarly the concentration of TiO2 in solution, which is consistent with experimental values. Figure 9 presents the time evolution of TiO2 concentration in solution, during a full batch. After 2 h, its concentration drops till the critical one considered to be 0.015 mol/m3. TiO2 forms during the chemical process and is consumed to produce the precursors of


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crystals, the primary cubes. It should be mentioned that, according to equation (42), some primary cubes could solvate back the TiO2 molecules in solution, as the hydrothermal treatment ends up with anatase formation meaning that the dissolution of primary cuboids (CuboidOne) is slower than its formation. This gives a slightly sinuous line (10-3-10-4 mol/m3) around the critical concentration for dissolved TiO2. One of the novelties of the present approach is the introduction of three cuboids, considered as intermediaries between the solvated TiO2 and the anatase crystal. According to the experimental evidences regarding the dimensions, shape factors and time of existence, these precursor cuboids are (see Figure 2): a primary cube (CuboidOne), of 6x6x6 units, which can solvate, bringing back to the solution the TiO2 molecules, a cuboid of 12x6x6 units (CuboidTwo), seen as the link between the former unstable cube and the final precursor, and this latter one, a cuboid of 12x12x6 units (CuboidFour), highly unstable, which generates the primary elongated anatase crystals through rearrangement. The primary cuboid profile (see Figure SI_6a in Supporting Information for details) follows closely the TiO2 profile in aqueous solution, as expected, since the former appears by agglomeration of the TiO2 , when the concentrations of the latter exceeds the critical one. The role of the primary cube is two-fold: start the simplest symmetric arrangement around the bi-layer and provide the building blocks for the second cuboid, the intermediate step. The concentration profile vs. time of this latter cuboid is presented in Figure SI_6b. Its peak is delayed with respect to the peak of its precursor, and, as such, it is consumed a little bit later, too. This delay is caused by the one order of magnitude difference between kf1st and kf2nd. Another interesting aspect is that, due to this one order of magnitude difference between kf2nd and kf3rd, the intermediary cuboid has time to reach the biggest concentration of all cuboids. Figure

SI_6(c) presents the time profile of the cuboid 12x12x6, which is the source of the primary elongated



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anatase crystal, with dimensions (5,2,3). It has the lowest concentration amongst all cuboids due to the very high ratio of passing into the more stable elongated anatase crystal, given by fGrowthUp. According to indirect experimental evidence (the decrease in the shape factor after two hours of process, and even more drastic after 4.67 hours- Table 4 and Figure 8), we might say that the model copes well with these findings, since the simulated process of generating the primary elongated anatase crystals ends after about 3-4 hours. During the rest of the batch time, there will be a continuous growth of the anatase crystals, till the final distribution will be attained, according to the growth model considered. By several test-runs, we observed that, as the parameters kDelta and mDelta are higher than one, the CPU time increases dramatically. This is due to the fact that these

parameters decide how different from each other the parent crystals could be, in order to interact through OA mechanism and give raise to a newborn crystal. In the case depicted in Figure SI_7 (see Supporting Information), both parameters have the same value, one, meaning that only crystals differing by only one double layer on height (k) or one unit on the small frustum square (m) can participate in the growth process. Even with this drastic restriction, the solution of the mathematical model took more than 9 hours. The values of kDelta=1 and mDelta=1 force the crystal distribution to have only 13 different crystals, which group in 13 classes of crystals – during the rest of the batch time, this distribution remains unchanged, while the concentrations of the crystals belonging to each class will do. The smaller crystals will have their concentration diminished during growth, while the biggest crystal will have its concentration increasing on time (Figure 9).


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Figure 9. Mass fraction distribution on classes of 5 nm, in time (kDelta=1 and mDelta=1). The legend stands for the time elapsed from the beginning of the batch, (1) 10 min, (2) 1h, (3) 2h, (4) 5h, (5) 50 h.

Analyzing the time profiles from Figure 9, we observe that, during the first two hours of the process, the mass fraction of all classes grows rather steady, except the larger crystals class (last class), of which the growth is slightly slower – the biggest crystals class has a mass fraction only a little bit higher after one hour, compared to the value after 10 min, contrary to the rest of the classes. As the time goes, after two hours, the increase of the mass fraction of the biggest crystals class starts accelerating. This collective increase in the mass fraction is due to the presence of the cuboids precursors in solution, which feed the chain through the primary elongated anatase crystal with dimensions (5,2,3). Once the cuboids vanished from the solution, the growth process starts reversing the aforementioned trend, the smaller crystals diminishing their concentration due to the coupling through OA, to give



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birth to new, larger NPs. On the other hand, the mass fraction of the biggest crystals class gets higher and higher, eventually the distribution narrowing to this single class (see cyan curve in Figure 9). For values of kDelta=2 and mDelta=2, the growth process becomes more complex, since the parent crystals could have heights differing by two double layers, or frustum smallest base being different also by two TiO2 units. For more than 4 and half hours of batch time, the CPU needed longer than 50 hours; we, then, decided to interrupt the computations. In order to illustrate the complexity of the computations, we present Figure 10, in which the mass fraction distribution on classes of 5 nm is traced in time.

Figure 10 Mass fraction distribution on classes of 5 nm, in time (kDelta=2 and mDelta=2). The legend stands for the time elapsed from the beginning of the batch: (1) 10 min, (2) 1 h, (3) 2 h, (4) 4.5 h.

As expected, there are significantly more different crystals in the distribution, in fact, 123, which give 55 classes – note the difference to only 13 different crystals in the previous case. This complicates the computations with at least one order of magnitude. It must be pointed out that


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grouping in classes of 5 nm is done to smooth the distribution curves, but the computations are done involving all 123 different NPs. As expected, the time behavior remains the same: as long as the cuboids are in solution, the mass fraction of the smaller crystals grows faster than the mass fraction of the bigger crystals. When the cuboids vanish, this process is reversed, the smaller crystals being consumed in the growing process, towards bigger crystals (magenta), when the mode at small sizes (about 2 nm) diminishes its importance. The disappearance of the mode at small sizes is in good agreement with experimental data (Figure 4). This behavior becomes clearer when the class length is increased, from 5 nm to 10 nm (see Supporting Information, Figure SI_8). Analyzing the time profiles, we conclude that, for the big crystal classes (larger than 10.5 nm) the increase in mass fraction from 10 min to 1 hour is insignificant, while the biggest jump is done between one hour and two hours (green vs. blue curves). After cuboids depletion, the bigger crystal mass fractions increase rapidly, at the expense of the smaller crystals. The reconstruction of number based distribution of TiO2 NPs from simulated results are in good agreement with experimental data which can also prove the predictive capability of the mathematical model (see Supporting Information, Figure SI_9). Since the mathematical model does not incorporate several time scales (we considered instantaneous the steps at very low time scales), its validity is general. Unfortunately, for short reaction times (order of seconds up to few minutes), we lack experimental results to calibrate the model accordingly.

CONCLUSIONS The experimental evidences reported allow to propose a formation and growth mechanism of the anatase NPs with the adopted hydrothermal method. Specifically:



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1) the complex chemical reactions undergone by the soluble Ti(IV) species in the Ti(TeoaH)2 water solutions, with the formation at ambient temperature of polynuclear species; 2) the formation of small diameter cuboids (0.5-2 nm) in the early stages of the hydrothermal treatment of Ti(TeoaH)2 solutions, which disappear during the treatment; 3) the decrease of the shape factor and the related NPs volume increase during the hydrothermal treatment, with the transition from elongated to bipyramidal particles. Backed-up by the experimental study, a mathematical model was developed to encompass the chemical reaction kinetics for TiO2 generation starting from titanium trimethylamine complex, and its subsequent crystallization process. The OA mechanism proved to capture the crystal growth with shape factor evolution in time. The complex mathematical model describing the chemical reactions, nucleation and growth of TiO2 nanocrystals was solved using the Matlab™ (Natick, MA, USA) environment. The simulation results, in terms of main species concentration and crystal size distribution, are in rather good agreement with the experimental measurements.

ACKNOWLEDGMENT The authors acknowledge the financial support from SETNanoMetro project in the frame of FP7NMP-2013_LARGE-7, project no 604577.

SUPPORTING INFORMATION ESI-HRMS spectra for Ti compounds, XRD patterns for synthetized materials, the software graphical interface, additional simulation results.

REFERENCES (1) Nakata, K.; Akira Fujishima A., J. Photochem. Photobiol. C: Photochem. Rev., 2012, 13, 169189.


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For Table of Contents Use Only

A new model for nano-TiO2 crystals birth and growth in hydrothermal treatment using oriented attachment approach Vasile Lavric, Raluca Isopescu, Valter Maurino, Francesco Pellegrino, Letizia Pellutiè, Erik Ortel, Vasile-Dan Hodoroaba

A new model for anatase nanoparticles synthesis in hydrothermal process starting from triethanolamine titanium complex includes chemical reactions for TiO2 generation and subsequent anatase nucleation and growth. The built-up of crystals is explained accepting the equilibrium structure (Wulff ). The model is validated using experimental data for titanium species concentration and time evolution of crystal size and shape.


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A New Model for Nano-TiO2 Crystal Birth and Growth in Hydrothermal

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