Weibull Cumulative Distribution Function for Modeling the Isothermal

Article pubs.acs.org/IECR

Weibull Cumulative Distribution Function for Modeling the Isothermal Kinetics of the Titanium-oxo-alkoxy Cluster Growth Zorica Z. Baroš*,† and Borivoj K. Adnađević‡ †

Higher Education School of Professional StudiesBelgrade PolytechnicBrankova 17, 11000 Belgrade, Serbia Faculty of Physical Chemistry, University of Belgrade, Studentski trg 12-16, 11001 Belgrade, Serbia

‡

ABSTRACT: The isothermal kinetics of the titanium-oxo-alkoxy cluster (TOAC) growth obtained in the controlled hydrolysis−condensation reaction of Ti(OPri)4 with H2O in n-propanol solution was investigated. Kinetic curves of the TOAC growth were measured at five different temperatures: 298, 303, 308, 313, and 318 K. It was established that the kinetic curves of the TOAC growth could be mathematically modeled by the cumulative two-parameter Weibull probability distribution function of the growth time (t) in a wide range of the degree of growth (0.05 ≤ DG ≤ 0.95). It was found that the dependencies of the shape parameter (β) and scale parameter (η) on temperature (T) can be described by polynomial functions. The activation energy of TOAC growth (Ea,DG), as well as the pre-exponential factor (ln ADG) for different degrees of growth, were calculated using Friedman’s isoconversional method. On the basis of knowing Ea,DG(DG) dependence, the procedure for calculating isothermal dependence of Ea vs growth time was described. The isothermal probability distribution density functions of Ea for the TOAC growth process were calculated. It was established that the conversion kinetic curves of the TOAC growth process could be fully described by a model of the infinite number of parallel first-order growth reactions with time-varying value of the preexponential factor and activation energies which are distributed in accordance with a specific function of Ea.

1. INTRODUCTION

hydrolysis Ti(OR)(OH)3 species follows the stoichiometry of the corresponding hydrolysis reaction for their formation. Soloviev et al.5−7,10 studied kinetics of the initial stage of TiO2 nanoparticles formation using UV absorption and light scattering (LS) technique. The sol−gel process was carried out by using titanium tetraisopropoxide (Ti(OPri)4), water, and 2propanol solutions (c = 0.1 M and h = 0.1−3). It was found that the hydrolysis−condensation reaction takes place during the mixing of reagents and that it is complete at low values of h (h ≤ 1). The molar ratio h > 1 is needed for the particles to grow. The total nuclei mass, consisting of titanium-oxo-alkoxy clusters, is formed, almost instantaneously, during the initial hydrolysis and condensation reaction, while h ≤ 1, and it is conserved during the process. Therefore, further growth of nuclei was carried out by nucleus aggregation, or in other words, the particles were grown by nucleus agglomeration. The reaction limited aggregation (RLA) regime was suggested for the aggregation stage of the sol−gel process in mention. By comparing the intensity of light scattered by the suspension (I), with the mean hydrodynamic radius of particles (R), obtained from the granulometric measurements, they concluded that the particles are fractal in nature, i.e. the mass m of a single particle with the hydrodynamic radius R is proportional to RDf, where Df is a fractal dimension. Rivallin et al.11−13 observed the temperature dependence (T = 278−313 K) of the titanium oxide sols precipitation kinetics by the hydrolysis of Ti(OPri)4 as a metal organic precursor in 2propanol solution. Experimental studies were carried out by

One of the most recent nanomaterials that has attracted a great attention, due to its unique properties, is titanium dioxide. Monodisperse TiO2 powders possess interesting optical, dielectrical, and catalytical properties, which result in industrial applications such as pigments, fillers, catalyst supports, and photocatalysts. Control of size, shape, and structure of the colloidal precursor is an important factor in determining the properties of the final material which requires a detailed understanding of the mechanism of titanium oxide colloidal particle formation. The basic sol−gel chemistry of titanium metal alkoxides has been studied extensively during the last three decades and is now well-characterized. A significant progress has been achieved concerning the structure and reactivity of the participating unitsnanometric colloidal particles (sols).1−18 Although the underlying basic chemistry of the sol−gel process has been extensively studied, information concerning the kinetics of titanium oxide colloidal particle formation is much less prevalent. Barringer et al.1 studied the formation of spherical, monodispersed TiO2 powders with a controlled hydrolysis of titanium tetraethoxideTi(OC2H5)4in a diluted water solution of ethanol. Powders were synthesized at room temperature, by using tetraethoxide and water molar concentrations of c = 0.1−0.2 M and cw = 0.3−0.7 M, respectively. The molar ratio of water to tetraethoxide (h = cw/c) was always greater than 2.5. They derived the kinetic equation which represents the accumulation rate of the rate-controlling hydrolysis species having the formula Ti(OR)(OH)3, for which supersaturation was required for particle nucleation. On the basis of this, they concluded that the formation rate of © 2013 American Chemical Society

Received: Revised: Accepted: Published: 1836

August 8, 2012 January 10, 2013 January 11, 2013 January 11, 2013 dx.doi.org/10.1021/ie3021363 | Ind. Eng. Chem. Res. 2013, 52, 1836−1844

Industrial & Engineering Chemistry Research

Article

308, 313, and 318 K), for a fixed value of titanium tetraisopropoxide molar concentration c = 0.04 M, and a fixed value of the molar ratio h = 20. The TOAC formation was performed in the following way: The reaction vessel with volume of 250 mL was filled with 100 mL of the n-propanol. The reaction vessel was then placed in the thermostatic water bath for the purpose of heating the npropanol solution to a predefined temperature. A 1.067 mL of titanium tetraisopropoxide was added into a heated n-propanol solution in order to obtain an appropriate value of c. The formed reaction mixture was homogenized by stirring with a magnetic stirrer (450 rpm), and a 1.25 mL of distilled water was added dropwise into the reaction mixture in order to achieve a predetermined molar ratio (h = 20) of water to tetraisopropoxide. The reactive solution’s temperature, for each individual experiment, was kept constant, within ±1 K. 2.3. Using UV−vis Spectroscopy for Determining the Degree of Growth. In their study, Bityurin et al.19 developed a procedure for monitoring the growth of titanium oxide nanoparticles by applying the UV−vis spectroscopy, based on the increase of the colloid turbidity due to the particle scattering and the intraband transitions.6 Having that in mind, in this paper, the formation kinetics of the TOAC was monitored in situ by measuring the UV−vis absorption spectra of the reactive solutions within the 250−780 nm wavelength range, using the UV−visible Spectometer GBC Cintra 10e from Scientific Equipment, Ltd. An immersion quartz Suprasil probe (Hellma, 661.500-QX) in direct contact with the reactive solution was used to measure the transmitted light within an optical path of 0.1−2 cm. A UV−vis spectometer was connected to the optical probe through the standard fiber-optic cable (Hellma, 041.002-UV). Data collection was carried out in a continuous way with a scanning speed of 100 nm min−1 and with the scanning step of 0.427 nm. For characterization of the TOAC formation kinetics, absorbance changes versus time were recorded additionally, at predefined wavelength λ = 350 nm. Bearing in mind that the increase in absorbance is caused by the increase in dimensions of the particles and by a constant rate of their growth, it can be assumed that the degree of growth for the particle’s dimension (DG) is given with a relation:

using a sol−gel reactor with temperature and atmosphere control, ultrarapid reagent mixing and in situ measurements (LS method involving the optical fiber probe was used) of the precipitation. From these measurements the activation energy of the titanium oxide sol formation Ea = 0.33 ± 0.02 eV was obtained and related to the secondary hydrolysis−condensation reactions between surface Ti−OPri groups. They established an existence of almost linear decrease of the precipitate’s fractal dimension Df, as the temperature increases (Df = 1.53 at 278 K and Df = 1 at 313 K). By combining chemical and hydrodynamic influences on the particle aggregation mechanism, a kinetic model which describes the rate of sol nanoparticle aggregation, during the induction period of a sol−gel process, was developed. Azouani et al.14 investigated nucleation and growth of titanium-oxo-alkoxy TixOy(OiPr)z clusters which were created in the Ti(OiPr)4 sol−gel process in water solution of 2propanol, at the molar ratio, h, between 1.0 and 2.6. Four different domains of the cluster/nanoparticle growth kinetics were identified: h < 1.45 (I), 1.45 ≤ h ≤ 1.75 (II), 1.75 < h ≤ 2.0 (III), and h > 2.0 (IV). They derived the kinetic model that describes the induction kinetics in the fourth domain of the stable nuclei growth (h > 2.0), according to which the induction time depends on the inverse 6 and 5 powers for initial Ti-atom concentration (c) and the molar ratio (h) respectively. The hydrolysis of titanium(IV) tetraisopropoxide and the following condensation and aggregation to titania nanoparticles in isopropanol solution, under slow water addition into the solution, were investigated by in situ time-resolved X-ray and UV−vis spectroscopies.16 At the early stage of the Ti(OPri)4 hydrolysis−condensation reaction in isopropanol solution, a fast increase of absorption is observed. During this stage, polynuclear titanium species, identified as dodecatitanate species, are produced from the polycondensation reaction of the hydrolyzed isopropoxide precursor. In the intermediate stage, the absorption varies much more slowly. At this stage titania nanoparticles are formed by a cluster−cluster growth mechanism involving a very slow consumption of the dodecatitanate species following a pseudo first-order reaction. As a consequence, the kinetics of the cluster−cluster growth process of the titania nanoparticles can be described by a linear time dependency, as expected in the reaction limited regime proposed in previous studies.13,15 It is assumed that distinct increase of the absorption in the advanced stage is mainly governed by the changing size of the newly formed titania nanoparticles, which is related to their sudden aggregation. Taking into account the complexity of all processes mentioned above and kinetic models for their description, in this paper the possibility of applying the cumulative twoparameter (η,β) Weibull probability distribution function of one variable (the growth timet) to describe the growth kinetics of the TOAC, obtained in the controlled hydrolysis− condensation reaction of Ti(OPri)4 with H2O in n-propanol solution, was investigated.

DG =

A − A0 A max − A 0

(1)

where A is the current value of the absorbance, A0 is the value of the absorbance at the beginning of the increase in absorbance with time, and Amax is the maximum value of the absorbance at a given wavelength. These values are determined graphically (see Figure 1). 2.4. Isoconversional Friedman’s Method for Determination of Kinetic Parameters of the Process. The activation energy of the investigated growth process for various degrees of TOAC growth was established by Friedman’s method20 which is based on the following. The rate of the process in the condensed state is generally a function of temperature and the conversion process itself:

2. EXPERIMENTAL SECTION 2.1. Materials. For the TOAC synthesis, the following chemicals were used: titanium tetraisopropoxide, Ti(OPri)4, (98% purity by Acros Organics), n-propanol (provided by Lachner, 0.1% maximum water content), and distilled water. 2.2. Formation of the Reactive Solutions. Experiments were performed at each of five selected temperatures (298, 303,

dα = θ (T , α ) dt

(2)

i.e., 1837

dx.doi.org/10.1021/ie3021363 | Ind. Eng. Chem. Res. 2013, 52, 1836−1844


Article

independently of each other. In that case, we can asssume that the following is valid: α=1−

∞

∫0

Φ(Ea , T )m(Ea) dEa

(7)

where Φ(Ea,T) equals ⎛ A Φ(Ea , T ) = exp⎜ − ⎝ γ

∫0

T

⎛ E ⎞ ⎞ exp⎜ − a ⎟ dT ⎟ ⎝ RT ⎠ ⎠

(8)

where γ = dT/dt is the heating rate. By using a variable x = Ea/RT, eq 8 is rewritten as follows: ⎧ AE ⎛ e−x Φ(Ea , T ) = exp⎨− a ⎜ − ⎩ γR ⎝ x

(3)

⎡ ART 2 ⎛ E ⎞⎤ Φ(Ea , T ) = exp⎢ − exp⎜ − a ⎟⎥ ⎝ RT ⎠⎦ ⎣ γEa

Φ(Ea , T ) = U (Ea − Ea,s)

(4)

(10)

(11)

This approximation assumes that only single reactions, whose activation energy is Ea,s, occur at a given temperature T. Then, the eq 7 could be simplified in the following form: α=1−

∫E

∞

m(Ea) dEa

a,s

(12)

According to eq 11, we can write

(5)

α=

According to the isoconversional principle, the analytic form of the function f(α) does not change with α, and then eq 5 can easily be transformed to Ea, α ⎛ dα ⎞ = ln(ν)α = const = ln[A ·f (α)] − ln⎜ ⎟ ⎝ dt ⎠α = const RT

(9)

Having in mind that the Φ(Ea,T) function changes rather steeply with activation energy at a given temperature, it seems reasonable to represent Φ(Ea,T) by the step function U at an activation energy Ea = Ea,s:

where Ea is the activation energy, A is the pre-exponential factor, and R is the gas constant. Then, on substitution of eq 4 into eq 3, the following equation is obtained: ⎛ E ⎞ dα = A ·f (α) exp⎜ − a ⎟ ⎝ RT ⎠ dt

e − x ⎞⎫ d x ⎟⎬ ⎠⎭ x

where p(x) is the so-called “p-function”, well-known in the field of thermal analysis. By employing an approximation p(x) = e−x/ x2, we can write

where, dα/dt is the reaction rate, α is the conversion degree, k(T) is the temperature-dependent rate constant, t is the time, T is the temperature, and f(α) is the differential reaction model associated with a certain theoretical reaction mechanism. The dependence of the rate constant on temperature is usually described by the Arrhenius law:

⎛ E ⎞ k(T ) = A exp⎜ − a ⎟ ⎝ RT ⎠

∞

⎫ ⎧ AE = exp⎨− a p(x)⎬ ⎭ ⎩ γR

Figure 1. The kinetic changes of UV−vis spectra, during the TOAC formation and growth, for c = 0.04 M, h = 20, and temperature T = 298 K, measured at five different moments in time.

dα = k(T ) ·f (α) dt

∫x

∫0

Ea,s

m(Ea) dEa

(13)

therefore, m(Ea,s) is derived by differentiating eq 13 with respect to Ea,s as

m(Ea,s) =

(6)

where vα=const is the reaction rate for the defined α. From the slope and the y-intercept of the ln vα vs 1/T dependence, the values of the kinetic parameters, the activation energy Ea,α, and pre-exponential factor ln Aα for a particular degree of conversion, i.e. for a certain degree of TOAC growth in our case, are obtained. 2.5. Miura−Maki Integral Method for Determining the Probability Distribution Density Curve of the Activation Energies. Miura and Maki21 developed a simple method for determining the shape of the probability distribution density function of the activation energies m(Ea) for kinetically complex chemical reactions (or physicochemical processes). This method is based on the so-called distribution activation energy model (DAEM),22 according to which the complex chemical reaction (or physicochemical process) is treated as a set of several first-order parallel reactions with different activation energies, m(Ea), all occurring simultaneously but

dα dEa,s

(14)

Thus, the density distribution function of activation energies could be directly obtained by differentiating the experimentally determined relationship: α vs Ea, i.e. m(Ea) = |dα/dEa|.

3. RESULTS AND DISCUSSION The evolution of UV absorption spectra, during the TOAC formation for c = 0.04 M, h = 20, and temperature T = 298 K, is shown in Figure 1, by five curves representing A vs λ dependencies, measured at five different moments in time during the process (ttotal ≤ 50, = 73, = 83, = 91, and ≥ 102 min). Figure 1 shows that the UV−vis absorption spectra of reactive solutions were recorded within the wavelength range 250−780 nm. On the basis of the presented UV−vis spectra, it can be easily concluded that at all investigated times of TOAC formation, for wavelengths ranging from 250 to 340 nm, 1838



Article

absorbance of the reaction system has a constant value. The absorbance of the reaction system decreases with the increase of wavelengths in the spectral range between 340 and 380 nm. Finally, for wavelengths ranging from 380 to 780 nm value of absorbance is close to zero (without changing with the change of λ), due to the fact that the reaction system becomes completely transparent to vis radiation. Comparative analysis of the reaction system’s absorbance changes with time of TOAC formation, for the predefined wavelength (350 nm), leads to a clear conclusion that the absorbance value increases at fixed λ, during the time interval of 50 min ≤ ttotal ≤ 102 min. This increase is especially noticeable in the range of 345 nm ≤ λ ≤ 365 nm. Figure 2 shows kinetic curves of the isothermal TOAC growth. On isothermal growth curves, three characteristic

Table 1. Influence of Operating Temperature (T) on Nucleation Rate (vn)

a

T (K)

vn (min−1)

kinetic parameters

298 303 308 313 318

0.02041 0.02332 0.02670 0.03060 0.03522

Ea,n = (20.6 ± 0.5) kJ mol−1 ln An = 4.42 min−1 R2 a = 0.998 SDb = 0.00944

Squared linear correlation coefficient. bStandard deviation.

Ti(OR)4 to Ti(OR)3OH and the TOAC formation through the alcoxolation reaction.23 On the basis of the obtained results, it can be concluded that the TOAC dimensions increase with increasing time from the critical diameter to a certain maximum value at which the precipitation of particles occurs. Having that in mind it is possible to define the so-called degree of growth (DG) as a ratio of the particle’s size at a certain moment in time and the maximum value of the particle’s size. If we assume (a) that the different sized particles are being formed with different probabilities; (b) that the probability of forming particles with different sizes changes with time; (c) that the degree of growth is proportional to the probability of forming particles with different sizes; and (d) that the time-dependent probabilities of forming particles with different sizes can be described by the cumulative two-parameter Weibull probability distribution function of the growth timeP(t), then the following is valid: ⎡ ⎛ ⎞β⎤ t DG ≈ P(t ) = 1 − exp⎢ −⎜ ⎟ ⎥ ⎢⎣ ⎝ η ⎠ ⎥⎦

Figure 2. Isothermal TOAC growth curves at different operating temperatures: 298, 303, 308, 313, and 318 K.

(15)

where t is the growth time, β is the shape parameter, and η is the scale parameter. The growth time (t) was calculated as a difference between current time (ttotal) and the time (tind) necessary for the formation of the nuclei of critical dimensions. In order to determine the values of the Weibull function parametersη and β, eq 15 was linearized in the following form:

shapes of DG changes are clearly observed, which correspond to three different stages of TOAC formationforming of TOAC nuclei with critical dimensions, growth and, finally, agglomeration and precipitation. During the nucleation stage, the formation of TOAC nuclei with critical dimensions occurs, and because of that, the growth rate is close to zero and it does not change for a long period of time during TOAC formation. Sudden, almost linear increase in growth rate with the duration of TOAC formation, occurs in the second stage of TOAC formation, and is an immediate consequence of the formed TOAC nuclei growth. In the final stage of TOAC formation, agglomeration and precipitation, the value of DG does not change, due to the stop in further increase of TOAC dimensions. In order to completely determine the kinetic model of the TOAC formation process, the kinetics of nucleation was initially examined. Assuming that the nucleation rate (vn) is inversely proportional to the duration of the induction period (tind), the rate of TOAC nucleation was determined (see Figure 2). Table 1 shows the influence of temperature on the nucleation rate. The rate of nucleation exponentially increases with the increase of temperature (T). Having this in mind, by applying the Arrhenius equation, the apparent activation energy (Ea,n) and pre-exponential factor (ln An) of TOAC nucleation process was determined (Table 1). The calculated value of Ea,n = 20.6 kJ mol−1 is in accordance with the kinetically limiting stage of the suggested TOAC formation mechanismfast hydrolysis of

⎛1⎞ ln[− ln(1 − DG)] = β ln⎜ ⎟ + β ln t ⎝η⎠

(16)

Figure 3 shows the isothermal dependencies of ln[−ln(1 − DG)] on ln t. As it can be clearly seen in Figure 3, isothermal dependencies of ln[−ln(1 − DG)] on ln t, at all the investigated temperatures, in a wide range of the degree of growth (ΔDGthe range of applicability), are straight lines. The linearity of the ln[−ln(1 − DG)] vs ln t dependence confirms the possibility of fitting experimental results with the suggested TOAC growth model (eq 15) and enables the determination of η and β values by the use of this linear dependency’s slope and its interception with the y-axis, respectively. Table 2 shows changes with temperature for the values of the following parameters: η, β, ΔDG, squared linear correlation coefficients (R2), and standard deviations (SD). At all of the investigated temperatures the degree of growth takes values from an extremely wide range (ΔDG = 0.05− 0.95), and the quality of fitting the experimental results with the Weibull distribution function, quantified with R2, is satisfying (see Table 2). 1839



Article

Figure 4. Dependencies of ln vDG,Ti on inverse temperature (1/T) for different degrees of TOAC growth. Solid lines are linear fittings corresponding to different DG.

Figure 3. Weibull plots for TOAC growth process at different operating temperatures: 298, 303, 308, 313, and 318 K, for degrees of growth in the range of 0.05 ≤ DG ≤ 0.95.

As can be seen from the obtained results presented in Figure 4, there was a linear relationship between the ln vDG,Ti and the inverse temperature (1/T) for all of the degrees of TOAC growth. Each of the isoconversional lines for a chosen constant value of DG shows a high linear correlation coefficient (R) (in all cases R is higher than 0.992). From the slopes and yintercepts of these straight lines, the values of the kinetic parameters (Ea,DG and ln ADG, respectively), for each individual value of the degree of TOAC growth (0.05 ≤ DG ≤ 0.95), are obtained. The dependence of Ea,DG vs DG is shown in Figure 5. Each point in this figure is calculated by using the slope of the above linear relationship for different values of DG, within certain error limits, specified by bars.

Table 2. Range of Applicability for the Degree of Growth (ΔDG), the Parameters β and η, as well as the Corresponding Values of R2 and SD, Obtained by the Weibull Plot Method for the TOAC Growth Process at Five Operating Temperatures T (K)

β

η (min)

ΔDG

R2

SD

298 303 308 313 318

3.36 3.30 3.23 3.15 3.05

36.96 36.18 35.82 35.95 36.71

0.05−0.95 0.05−0.95 0.05−0.95 0.05−0.95 0.05−0.95

0.986 0.986 0.987 0.988 0.989

0.1339 0.1311 0.1278 0.1240 0.1193

As temperature increases, obtained values of the shape parameter (β) decrease. The values of the scale parameter (η) primarily decrease with the increasing of the operating temperature from 298 to 308 K, and then, they increase with further increase in the operating temperature from 308 to 318 K. The dependencies of shape parameter (β) and the scale parameter (η) on temperature (T) can be described by the polynomial functions: βT = a + bT

(17)

ηT = c + dT + eT 2

(18) −1

where a = 7.9083, b = −0.01522 K , c = 1007.9236 min, d = −6.2978 min K−1, and e = 0.0102 min K−2 are the corresponding fitting constants necessary to describe the relationship between the parameters mentioned above and the temperature. The possibility for a quality fitting of the TOAC growth experimental curves with the Weibull distribution function indicates a complex structure for the kinetic model of the growth process. Assuming that the complex kinetic nature of the TOAC growth process is associated with the reactivity distribution of the reaction species included in a process of calculating the activation energy (Ea,DG) for different degrees of TOAC growth, the Ea,DG was calculated using the Friedman’s isoconversional method. Figure 4 presents the dependencies ln vDG,Ti = f(1/T) for different degrees of TOAC growth (DG), where T is the temperature and vDG,Ti = 1/tDG,Ti is the isothermal growth rate at a certain degree of growth.

Figure 5. Dependence of the activation energy (Ea,DG), calculated by the isoconversional method using eq 6, on the degree of growth (DG) for TOAC growth process.

From Figure 5, it can be seen that the activation energy Ea,DG is a complex function of DG. The complex form of Ea,DG changes with DG is a direct proof of a complex kinetic model for TOAC growth. On the curve showing the Ea,DG dependence on DG, three characteristic shape changes of Ea,DG with the increase of DG can be observed. With DG ≤ 0.30, Ea,DG values concavely decrease as DG increases, from 16.2 to 12.5 kJ mol−1. In the range 0.30 ≤ DG ≤ 0.65, the increase of DG leads to an almost linear decrease of Ea,DG from 12.5 to 10.4 kJ mol−1. With DG ≥ 0.65, as DG increases, consequently, Ea,DG decreases convexly from 10.4 kJ mol−1 to approximately 8.3 kJ mol−1. 1840



Article

of activation energies. Due to the fact that, in this case, we also know the probability distribution density function of growth timesp(t)

Calculated values of the activation energy for the TOAC growth process (from 16.2 kJ mol−1 to approximately 8.3 kJ mol−1) are quite different from the activation energy Ea = 0.33 ± 0.02 eV (about 31.8 kJ mol−1) that has been obtained and assigned to the secondary hydrolysis−condensation reactions between surface Ti−OPri groups in the work of Rivallin et al.12 The established difference in the values of the activation energy is probably the result of determining the activation energy at different experimental conditions (c and h). The calculation of Ea in the work of Rivallin et al.12 was done for the reaction system with processing parameters c = 0.146 M and h = 2.46, which are considerably different from values of c and h in our measurements (c = 0.04 M and h = 20). The existence of Ea,DG dependence on DG and it is complex form, both suggest that the TOAC growth is a complex kinetic reaction. Having this in mind and also the previously proven possibility of describing the TOAC growth kinetics by using the Weibull probability distribution function of growth time, it seams logical to assume that kinetic complexity of the TOAC growth is a consequence of energetic heterogeneities of the reaction species, as is the existence of a certain probability distribution density function for activation energies. Using the Miura procedure, the temperature-invariant form of the probability distribution density function for activation energiesm(Ea)is calculated. The dependence of m(Ea) on Ea,DG is shown in Figure 6 by bold solid line.

p(t ) =

β⎛ t ⎞ dDG = ⎜ ⎟ η⎝η⎠ dt

β−1

⎡ ⎛ ⎞β⎤ t exp⎢ −⎜ ⎟ ⎥ ⎢⎣ ⎝ η ⎠ ⎥⎦

(19)

if we determine how time changes with energy Ea, we can calculate the above-mentioned isothermal forms of the probability distribution density function of activation energies. On the basis of the previously determined dependencies (eqs 17 and 18) and by applying the expression (derived from the eq 15) for isothermal growth’s duration time t T = ηT[−ln(1 − DG)]1/ βT

(20)

it is possible, at any chosen temperature (T), to calculate an exact value of tT corresponding to each value of DG. This functional relationship enables the transformation of the temperature-invariant dependence of Ea,DG on DG into isothermal dependencies Ea vs t. Figure 7 shows isothermal changes of Ea with time for TOAC growth.

Figure 7. Isothermal changes of Ea with time for the TOAC growth process.

Isothermal changes of Ea with time are similar in shape and suggest that Ea for TOAC growth indeed decreases with time, during the TOAC growth process. Isothermal changes of Ea with growth time can be described by the following equation:

Figure 6. Influence of operating temperature (T) on the shape of the probability distribution density function of activation energies (z(Ea), calculated from eq 28), for TOAC growth process. The experimental probability distribution density functionm(Ea), obtained by the Miura procedureis also presented in the same figure by a bold solid line.

Ea = φ(t ) =

1 ⎛t − ϕ⎞ ⎟ ln⎜ ε ⎝ τ ⎠

(21)

where τ, ε, and ϕ are the fitting coefficients. The coefficients τ and ϕ must have the dimension of time, whereas coefficient ε has the dimension of moles per kilojoule. The effect of the experimental temperature (T) on the values of the fitting coefficients τ, ε, and ϕ are shown in Table 3. The absolute values of fitting coefficients τ and ε increase with the increase in temperature. On the other hand, the absolute value of the fitting coefficient ϕ decreases with the increase of temperature. The change of Ea during the TOAC growth process leads to a change in the growth rate constant. If we assume (a) that, in accordance with Hamill et al.,24 the change with time of the rate constant can be described with an expression

From Figure 6, we can see that the curve showing the m(Ea) dependence on Ea,DG is relatively wide and bell-shaped, with left asymmetry and with the following characteristics: the most probable value of Ea (Ea,max = 10.22 kJ mol−1), the maximum intensity of the density distribution function m(Ea)max = 0.18449 mol kJ−1, the relative full width of the probability distribution density function at half-maximum for Ea (r-HW = 0.52980), and the shape factor or the asymmetry factor SF = 0.49863. Describing kinetics of the TOAC growth process with the Weibull probability distribution function, allows us to calculate isothermal forms of the probability distribution density function

k(t ) = Bt β− 1 1841

(22)



Article

almost the same, which means that they are practically independent of temperature and have the following characteristics: the most probable value of Ea (Ea,max = 10.54 kJ mol−1), the maximum intensity of the density distribution function z(Ea)max = 0.18850 mol kJ−1, the relative full width of the probability distribution density function at half-maximum for Ea (r-HW = 0.56352), and the shape factor SF = 0.66429. The calculated isothermal z(Ea) curves are shaped with wellexpressed maximum and with a relatively wide left-asymmetric peak. The most probable value of Ea (Ea,max = 10.22 kJ mol−1), on the experimental m(Ea) curve, obtained by the Miura procedure (Figure 6, bold solid line), and the other specified basic characteristics are a little lower than the corresponding values for distribution density curvesz(Ea), calculated by eq 28, which additionally confirms previously assumed energetic heterogeneity of the TOAC growth process. On the basis of the previously defined isothermal probability distribution density functions of Ea for TOAC growth processz(Ea) (eq 28)and the functional dependence Ea = φ(t) (eq 21), it becomes realistically possible to model the TOAC growth process by an infinite number of parallel firstorder growth reactions in the following forms:23

Table 3. Influence of Operating Temperature (T) on the Values of Fitting Coefficients τ, ε, and ϕ for the TOAC Growth Process T (K) 298 303 308 313 318

ε (mol kJ−1)

τ (min) 180 181 186 194 209

± ± ± ± ±

2 2 2 2 3

−0.147 −0.150 −0.153 −0.157 −0.162

± ± ± ± ±

0.002 0.002 0.002 0.002 0.002

ϕ (min) −6.8 −6.7 −6.5 −6.4 −6.2

± ± ± ± ±

0.1 0.1 0.1 0.1 0.1

where B is a constant and (b) that the growth rate (dDG/dt) can be expressed as −

dDG = k(t ) ·(1 − DG) dt

(23)

it is easy to derive the equation which describes the TOAC growth kinetics: ⎡ ⎛ Bt β ⎞⎤ DG = 1 − exp⎢ −⎜ ⎟⎥ ⎢⎣ ⎝ β ⎠⎥⎦

(24) β

If we assume that B = β/η , the expression above can be transformed into a form that corresponds to the cumulative Weibull probability distribution function of the growth time (eq 15), which was the basic presumption of this paper. Knowing the form of the pre-exponential factor’s dependence on growth time, we are able to derive the expression for the isothermal change of Ea: ⎡ ⎛ t ⎞⎤ Ea = E0 − RT ⎢(β − 1)ln⎜ ⎟⎥ ⎝ η ⎠⎦ ⎣

nTi(OR)4 + l[Ti(OR)3 OH] → [Ti n + lOl ](OR)4n + 2l + l ROH

[Ti n + lOl ](OR)4n + 2l + l1[Ti(OR)3 OH] → [Ti n + l + l1Ol + l1](OR)4n + 2l1 + l1ROH

z(Ea) =

DGT =

(31)

∫E

Ea

z(Ea) dEa

0

(32)

By substituting the functional dependence Ea = φ(t) (eq 21) in eq 32, we derive the expression:

(26)

DGT =

∫0

t

z[φ(t )]·φ′(t ) dt

(33)

or (27)

The distribution function z(Ea) is now given in the following form:

DGT =

β−1 βτε exp(εEa) ⎡ τ exp(εEa) + ϕ ⎤ z(Ea) = ⎥ ⎢ ⎦ ⎣ η η β⎫ ⎧ ⎪ ⎡ τ exp(εE ) + ϕ ⎤ ⎪ a × exp⎨ − ⎥ ⎬ ⎢ ⎪ ⎪ ⎦ ⎭ η ⎩ ⎣

dDG dEa

it follows that dDG = z(Ea) dEa and, therefore,

we are able to calculate the isothermal forms of the probability distribution density function of Ea for TOAC growth process z(Ea)by using the following relation: dψ z(Ea) = p(ψ (Ea)) · dEa

(30)

These reactions are defined with eq 14 and their activation energies are distributed according to z(Ea). In this case, since, by definition it applies that

(25)

where E0 is the value of Ea at β = 1. Since the values of β and η are greater than 1, it follows, from eq 25, that during the TOAC growth process, Ea values decrease in time as it was previously calculated based on the experimental results. If we know the probability distribution density function of growth time (eq 19), which is assumed to be the starting function, and the isothermal dependencies of growth time on Ea (from the eq 21): t = ψ (Ea) = τ exp(εEa) + ϕ

(29)

∫0

t

⎧ ⎛ ⎞ βT − 1 ⎡ ⎛ ⎞ βT ⎤⎫ ⎪ βT t ⎪ t ⎨ ⎜⎜ ⎟⎟ exp⎢ −⎜⎜ ⎟⎟ ⎥⎬ dt ⎢ ⎝ η ⎠ ⎥⎪ ⎪ ηT ⎝ ηT ⎠ T ⎣ ⎦⎭ ⎩

(34)

Equation 34 enables the calculation of isothermal TOAC growth curves, based on the knowledge of η, β, ε, τ, and ϕ values. Figure 8b shows the calculated conversion kinetic curves of the TOAC growth process at different temperatures: 298, 303, 308, 313, and 318 K together with the experimentally obtained kinetic curves (Figure 8a) in order to estimate the extent to which the investigated growth process can be described by the applied theory. By comparing the TOAC growth curves shown in Figure 8a and b, it can be concluded that the calculated conversion kinetic curves (Figure 8b) are almost identical with the experimental

(28)

The probability distribution density functions of Ea for TOAC growth process, calculated by eq 28 for different operating temperatures, are shown in Figure 6 by dashed lines. For all five operating temperatures the probability distribution density functions of apparent activation energies z(Ea) are 1842



■

ACKNOWLEDGMENTS

■

REFERENCES

Article

This investigation was supported by the Ministry of Science and Technological Development of the Republic of Serbia, through Project No. 172015 OI.

(1) Barringer, E. A.; Bowen, H. K. High-Purity, Monodisperse TiO2 Powders by Hydrolysis of Titanium Tetraethoxide. 1. Synthesis and Physical Properties. Langmuir 1985, 1, 414. (2) Hartel, R. W.; Berglund, K. A. Precipitation Kinetics of the Titanium Isopropoxide Hydrolysis Reaction. Mater. Res. Soc. Symp. Proc. 1986, 73, 633. (3) Jean, J. H.; Ring, T. A. Nucleation and Growth of Monosized TiO2 Powders from Alcohol Solution. Langmuir 1986, 2, 251. (4) Livage, J.; Henry, M.; Sanchez, C. Sol-Gel Chemistry of Transition Metal Oxides. Prog. Solid State Chem. 1988, 18, 259. (5) Soloviev, A. Procédé Sol−Gel: Étude par Diffusion de la Lumiére de la Cinétique de Croissance des Particules Pendant L’HydrolyseCondensation de L’Isopropoxyde de Titane (IV). Ph.D. Dissertation, University Paris-13, Villetaneuse, 2000. (6) Soloviev, A.; Tufeu, R.; Sanchez, C.; Kanaev, A. V. Nucleation Stage in the Ti(OPri)4 Sol−Gel Process. J. Phys. Chem. B 2001, 105, 4175. (7) Soloviev, A.; Ivanov, D.; Tufeu, R.; Kanaev, A. V. Nanoparticle Growth During the Induction Period of the Sol-Gel Process. J. Mater. Sci. Lett. 2001, 20, 905. (8) Golubko, N. V.; Yanovskaya, M. I.; Romm, I. P.; Ozerin, A. N. Hydrolysis of Titanium Alkoxides: Thermochemical, Electron Microscopy, Saxs Studies. J. Sol-Gel Sci. Technol. 2001, 20, 245. (9) Oskam, G.; Nellore, A.; Penn, R. L.; Searson, P. C. The Growth Kinetics of TiO2 Nanoparticles from Titanium(IV) Alkoxide at High Water/Titanium Ratio. J. Phys. Chem. B 2003, 107, 1734. (10) Soloviev, A.; Jensen, H.; Søgaard, G.; Kanaev, A. V. Aggregation Kinetics of Sol-Gel Process Based on Titanium Tetraisopropoxide. J. Mater. Sci. 2003, 38, 3315. (11) Rivallin, M. Evolution de Sols Nanométriques D’Oxyde de Titane Durant L’Induction d’une Précipitation de Type Sol-Gel en Réacteur à Micromélange Rapide: Mesures Granulométriques In-Situ et Modélisation. Ph.D. Dissertation, L’Ecole Nationale Supérieure des Mines de Paris−ENSMP, Paris, France, 2003. (12) Rivallin, M.; Benmami, M.; Gaunand, A.; Kanaev, A. Temperature Dependence of the Titanium Oxide Sols Precipitation Kinetics in the Sol−Gel Process. Chem. Phys. Lett. 2004, 398, 157. (13) Rivallin, M.; Benmami, M.; Kanaev, A.; Gaunand, A. Sol−Gel Reactor With Rapid Micromixing Modelling and Measurements of Titanium Oxide Nano-particle Growth. Chem. Eng. Res. Des. 2005, 83, 67. (14) Azouani, R.; Soloviev, A.; Benmami, M.; Chhor, K.; Bocquet, J.F.; Kanaev, A. Stability and Growth of Titanium-oxo-alkoxy TixOy(OiPr)z Clusters. J. Phys. Chem. C 2007, 111, 16243. (15) Marchisio, D. L.; Omegna, F.; Barresi, A. A. Production of TiO2 Nanoparticles with Controlled Characteristics by Means of a Vortex Reactor. Chem. Eng. J. 2009, 146, 456. (16) Stötzel, J.; Lützenkirchen-Hecht, D.; Frahm, R.; Santilli, C. V.; Pulcinelli, S. H.; Kaminski, R.; Fonda, E.; Villain, F.; Briois, V. QEXAFS and UV/Vis Simultaneous Monitoring of the TiO2Nanoparticles Formation by Hydrolytic Sol−Gel Route. J. Phys. Chem. C 2010, 114, 6228. (17) Mehranpour, H.; Askari, M.; Ghamsari, M. S.; Farzalibeik, H. Nanoparticle Synthesis & Applications: Application of Sugimoto Model on Particle Size Prediction of Colloidal TiO2 Nanoparticles. In Nanotechnology 2010: Advanced Materials, CNTs, Particles, Films and Composites; NSTI-Nanotech: Austin, TX, 2010; Vol. 1, Chapter 3, pp 436−439. (18) Mehranpour, H.; Askari, M.; Ghamsari, M. S. Nucleation and Growth of TiO2 Nanoparticles. In Nanomaterials; Rahman, M. M., Ed.; InTech: New York, December 2011; Chapter 1, pp 3−26.

Figure 8. TOAC growth curves (a) experimentally obtained and (b) calculated by the eq 34, at different operating temperatures: 298, 303, 308, 313, and 318 K.

ones (Figure 8a), which confirms validity of the proposed kinetic model for the description of the TOAC growth process.

3. CONCLUSIONS The conversion kinetic curves of the TOAC growth process can be mathematically modeled by the cumulative two-parameter Weibull probability distribution function of the growth time. The Weibull distribution function’s parameters: the shape parameter (β) and the scale parameter (η) are polynomial functions of temperature. As temperature increases, obtained values of the shape parameter (β) decrease. The values of the scale parameter (η) primarily decrease with the increasing of the operating temperature from 298 to 308 K and then increase with further increase in the operating temperature from 308 to 318 K. The activation energy of the TOAC growth process decreases with the increase of the degree of growth and the growth time, as well. The calculated isothermal probability distribution density functions of Ea for the TOAC growth process, z(Ea), are independent of the operating temperature (T). The conversion kinetic curves of the TOAC growth process can be fully described by a model of the infinite number of parallel first-order growth reactions with time-varying value of the pre-exponential factor and activation energies which are distributed in accordance with a specific function of Ea. The kinetic complexity of the TOAC growth process is a consequence of energetic heterogeneities of the reaction species, thus the existence of certain probability distribution density function of activation energies.

■

AUTHOR INFORMATION

Corresponding Author

*Tel.: +381 11 2633127. Fax: +381 11 2632341. E-mail addresses: [email protected]; zoricabar@open. telekom.rs. Notes

The authors declare no competing financial interest. 1843



Article

(19) Bityurin, N.; Znaidi, L.; Marteau, P.; Kanaev, A. UV Absorption of Titanium Oxide Based Gels. Chem. Phys. Lett. 2003, 367, 690. (20) Friedman, H. A. Kinetics of Thermal Degradation of CharForming Plastics from Thermogravimetry. Application to a Phenolic Plastic. J. Polym. Sci. C 1964, 6, 183. (21) Miura, K. A New and Simple Method to Estimate f(E) and k0(E) in the Distributed Activation Energy Model from Three Sets of Experimental Data. Energy Fuels 1995, 9, 302. (22) Burnham, A. K.; Braun, R. L. Global Kinetic Analysis of Complex Materials. Energy Fuels 1999, 13, 1. (23) Baroš, Z. Z.; Adnađević, B. K. The Influence of the Molar Ratio [H2O]/[Ti(OR)4] on the Kinetics of the Titanium-oxo-alkoxy Clusters Nucleation. Russ. J. Phys. Chem. A 2011, 85, 2295. (24) Hamill, W. H.; Funabashi, K. Kinetics of Electron Trapping Reactions in Amorphous Solids; a Non-Gaussian Diffusion Model. Phys. Rev. B 1977, 16, 5523.

1844


Weibull Cumulative Distribution Function for Modeling the Isothermal

Recommend Documents