Kinetic study on preparation of substoichiometric titanium oxide via carbothermal process

Abstract

Gibbs free energy was applied to analyze the carbothermal reduction process of TiO₂. The phase stable diagram of Ti–C–CO was drawn, and it can be found that TiO₂ will be reduced to Ti–O Magnéli phase, Ti_nO_2n–1 (3 < n < 10) or Ti₃O₅ first and then further reduced to TiC. However, in the present study, it was found that Ti₄O₇ can be reduced to TiC directly when the temperature was lower than 800 K and lg(p _co/p ^θ) < −1.1. The reduction route was proved by the results of non-isothermal TG–DTA combined XRD. The isothermal analysis of carbothermal reduction process of TiO₂ via TG–DTA under the temperature of 1363, 1413, 1463 and 1513 K was performed. The whole reduction process was divided into two sections: Initially it was rate controlled by an interfacial reaction, and later was rate controlled by three-dimensional diffusion. The modified mathematical expressions were deduced. They fit the experimental curves well. Their activation energy of steps was 241.1 and 323.4 kJ mol⁻¹, respectively.

Introduction

Ti–O Magnéli phase compounds were a group of special substoichiometric Ti–O compounds; their general formula was Ti_nO_2n−1 (3 < n<10) [1–4]. Among these compounds, Ti₄O₇ attracted more attention due to its high electrical conductivity (>10³ S cm⁻¹ at room temperature [5–7]), thermal stability and corrosion resistance in acid medium [8, 9]. Because of these excellent properties, Ti₄O₇ or Ebonex™ was applied in the field of waste water treatment, especially in removing the sulfur in the high concentrated salt water [10, 11]. Also, it could be used as a catalyst support for precious metals; the supported catalysts exhibited high catalytic activity and stability in fuel cell applications [12–15]. Therefore, the applications of Ti₄O₇ in various kinds of batteries had been become more and more popular recently, such as the Ni–Zn secondary battery [16], metal–air battery [17, 18] and Li–S battery [19, 20].

Ti₄O₇ was reported to be synthesized by thermal reduction from TiO₂ by hydrogen- or carbon-containing raw materials [21, 22]. However, the reported conditions of reduction were completely different. The temperature was at the range of 1073–1573 K, and the reduction time was from 2 to 36 h [10, 13]. Therefore, the synthesis of substoichiometric Ti–O compounds needed to be studied systematically; meanwhile, 24 types of compounds in Ti–O system have been reported. The thermodynamic data were also reported in Merrit’s work [23]; Yang et al. [24] checked and reevaluated these data at 1304 K. In the present paper, the phase stable diagrams of Ti–O system and Ti–C–CO system were drawn based on the serial data at different temperatures from FactSage™ database [25]. The possible condition of reaction was selected. Systematical investigations of carbothermal reduction process were carried out by a thermogravity (TG) instrument, which was a common method for study carbothermal reduction [26–30]. The mechanism of reduction and mathematic expression was introduced. The experimental kinetic curves of reduction were described by the expression satisfactorily.

Thermodynamic consideration

For any pure substance at any temperature, its Gibbs free energy could be expressed as follows:

$$ G_{\text {T}}^{\theta } = H_{298}^{\theta } - TS_{298}^{\theta } + \int_{298}^{T} {C_{\text {p}} {\text{d}}T} - T\int_{298}^{T} {(C_{\text p} /T){\text{d}}T} $$

(1)

where $G_{\text {T}}^{\theta }$ was the standard Gibbs free energy of a pure substance at temperature T, $H_{298}^{\theta }$ was the standard formation enthalpy of the substance at 298 K, $S_{298}^{\theta }$ was the standard entropy of the substance at 298 K, C _p was the isobaric heat capacity of the substance.

For any reaction, the standard Gibbs free energy could be calculated by Eq. (2).

$$ \Delta_{\text{r}} G_{\text {T}}^{\theta } = \sum\limits_{\rm i} {\nu_{\rm i} G_{\text {i,T}}^{\theta } } $$

(2)

Here, i presented the component i, ν _i was the stoichiometric number of i in reaction, $G_{\text {i,T}}^{\theta }$ was the Gibbs free energy of i component at T temperature. Normally, the standard Gibbs free energy of a reaction could be written in the form of polynomial expression, such as

$$ \Delta_{\text{r}} G_{\text {T}}^{\theta } = {\text{A}} + {\text{B}}T + {\text{C}}T\ln T + {\text{D}}T^{2} + {\text{E}}T^{ - 1} $$

(3)

Here, A, B, C, D and E were different constants for a special reaction.

There could be a serial of reactions among Ti–O system compounds.

$$ n{\text{Ti}}_{\text {n} - 1} {\text{O}}_{2 \text {n} - 3(s)} + \frac{1}{2}{\text{O}}_{2({\rm g})} = (\text {n} - 1){\text{Ti}}_{\text {n}} {\text{O}}_{2\text{n} - 1(s)} $$

(4)

where Ti_n–1O_2n–3 and Ti_nO_2n–1 were the closest compounds in Ti–O binary phase diagram, n was the variable parameter. In these reactions, only oxygen was gas phase, and others were pure solid. Then, the Gibbs free energy could be written as:

$$ \Delta_{\text{r}} G = \Delta_{\text{r}} G^{\theta } + \frac{1}{2}RT\ln \frac{{p_{{O_{2} }} }}{{p^{\theta } }} $$

(5)

When Δ_r G = 0, i.e., at the equilibrium,

$$ \Delta_{\text{r}} G^{\theta } = - \frac{1}{2}RT\ln \frac{{p_{{{\text{O}}_{ 2} }} }}{{p^{\theta } }} $$

(6)

Combining polynomial expression of Δ_r G ^θ and Eq. (6), the relationship between temperature T and $\ln \frac{{p_{{{\text{O}}_{2} }} }}{{p^{\theta } }}$ was obtained:

$$ \Delta_{\text{r}} G_{\text {T},4}^{\theta } = {\text{A}} + {\text{B}}T + {\text{C}}T\ln T + {\text{D}}T^{2} + {\text{E}}T^{ - 1} = - \frac{1}{2}RT\ln \frac{{p_{{{\text{O}}_{ 2} }} }}{{p^{\theta } }} $$

(7)

Based on the data of Ti–O system in FactSage, the diagram of this relationship is shown in Fig. 1, called Ti–O system phase stable diagram.

Fig. 1

Phase stable diagram of Ti–O₂ system

Phase transformations of one compound were considered also.

$$ {\text{Ti}}_{\text n} {\text{O}}_{{2\text {n} - 1({\text{s, }}j)}} = {\text{Ti}}_{\text n} {\text{O}}_{{2\text {n} - 1({\text{s, }}\,j{ + 1})}} $$

(8)

Here, s represented solid phase, j was the variable number. Ti_nO_{2n–1 (s, j)} and Ti_nO_{2n–1 (s, j +1)} were two solid phases of compound Ti_nO_2n–1, and phase transformation occurred between them. Because the phase transformation does not affect by the oxygen partial pressure, it is normally a vertical line in Ti–O₂ phase stable diagram. At the temperature range in Fig. 1, there is no phase transformation.

From Fig. 1, the Ti₄O₇ could synthesize from TiO₂ if the oxygen partial pressure and temperature were accurately controlled. It can be seen from Fig. 1 that the lower the temperature was, and the wider the range of oxygen partial pressure for Ti₄O₇ existed. When the temperature was adjacent to 1273 K and $\lg \frac{{p_{{{\text{O}}_{ 2} }} }}{{p^{\theta } }} < 15.2$, the domain of Ti₄O₇ was expanded.

In the present work, carbon was used as the reducing agent and kept excess amount during process, so the activity of carbon is equal to unity, a _c = 1. The reduction reaction can be written as follows.

$$ (n - 1){\text{Ti}}_{\text n} {\text{O}}_{{2\text {n} - 1({\text{s}})}} + {\text{C}}_{{({\text{s}})}} = n{\text{Ti}}_{\text {n} - 1} {\text{O}}_{2\text {n} - 3(s)} + {\text{CO}}_{{ ( {\text{g)}}}} ,\,\quad \Delta_{\text{r}} G^{\theta } = - RT\ln \frac{{p_{\text{CO}} }}{{p^{\theta } }} $$

(9)

Then,

$$ \Delta_{\text{r}} G_{\text{T},9}^{\theta } = {\text{A}}^{{\prime }} + {\text{B}}^{{\prime }} T + {\text{C}}^{{\prime }} T\ln T + {\text{D}}^{{\prime }} T^{2} + {\text{E}}^{{\prime }} T^{ - 1} = - RT\ln \frac{{p_{\text{CO}} }}{{p^{\theta } }} $$

(10)

Here, A′, B′, C′, D′ and E′ were constants. The phase stable diagram of Ti–C–CO system is shown in Fig. 2.

Fig. 2

Phase stable diagram of Ti–C–CO system

Figure 2 implies that TiC would form at the low CO partial pressure and high temperature. Comparing Figs. 2 and 1, Ti₃O₅ can be reduced to TiC directly. When temperature was lower than 800 K and lg(p _co/p ^θ) < −1.1, Ti₄O₇ would reduce to TiC directly. Only at the non-equilibrium condition, TiC_xO_1−x can be synthesized [22, 24]. If the condition was lg(p _co/p ^θ) < −2.2 and T > 1000 K, the final product of reduction would be TiC at equilibrium. It could be concluded that controlling the CO partial pressure and temperature is key to produce the substoichiometric titanium oxide through carbothermal reduction process. Ti₄O₇ could be synthesized when temperature was lower than 1400 K and CO pressure was kept in the appropriate range. The equilibrium temperature at the standard state of TiO₂ phase transformation and carbothermal reduction in different titanium oxide are listed in Table 1. It showed that anatase TiO₂ prefers to rutile at any temperature thermodynamically. When temperature was higher than 860 K, synthesis reaction of the substoichiometric titanium oxide had begun. If the CO pressure kept at 101,325 Pa and the temperature was lower than 1055 K, the TiC product could not form during the reduction process.

Table 1 Equilibrium temperatures of possible reactions in Ti–C–CO system at standard state

Experimental

Carbothermal reduction of TiO₂

Anatase TiO₂ (mean particle size <2 μm, from Sinopharm Company, China) and carbon (acetylene black, Beijing Chemical Reagent Company) mass ratio of TiO₂ and C were 2:1 (C was in large excess amount); the mixture was put into mill pot and added some ethanol ball milling for 12 h. TiO₂ and C were contacted each other; normally the relative large TiO₂ particles were embraced by the tiny carbon particles, expressed by TiO₂@C. The TiO₂@C precursor was put into the Al₂O₃ crucible in a thermogravimeter (SEIKO 6300 TG/DTA). The non-isothermal reduction process was carried from ambient to 1673 K with the heating rate of 40 K min⁻¹ and the gas flow of 200 mL min⁻¹ Ar (99.99 %). The isothermal reduction processes were carried at 1323, 1373, 1423 and 1473 K for 2 h, respectively; they were also in 200 mL min⁻¹ Ar (99.99 %) gas flow; the heating rate was 40 K min⁻¹ to target temperature also. In order to understand the exact extent of reaction at the beginning of the isothermal processes, the heating rates of both non-isothermal and isothermal reductions were fixed 40 K min⁻¹.

Characterization

The crystalline phases of TiO₂@C precursor and the products after reducing were characterized by X-ray powder diffraction (XRD) analysis (Bruker™ D8 ADVANCE, Germany). The measurement was taken under the following conditions, Cu K_α1 irradiation at 40 kV and 40 mA, 2θ scanning range from 10° to 80° with 0.02° step, scanning speed of 6° min⁻¹. Morphologies were observed using a scanning electron microscope (SEM, Hitachi™ S-4300, Japan). Accelerating voltage was 15 kV, the filament current was 10 μA, and the secondary electron images were adopted to observe the changes of morphology.

Results and discussion

Analysis of the non-isothermal reduction process

TiO₂@C precursor was in Ar gas flow and raised temperature from ambient to 1673 K at the rate of 40 K min⁻¹. The TG and differential thermal analysis (DTA) curves are shown in Fig. 3. There were three peaks on the DTA curve. Their temperatures were around 1273, 1523 and 1623 K, respectively. Then, TiO₂@C precursor was heated to 1273, 1523 and 1623 K in Ar gas flow, respectively, and quenched in cooling air. The produced samples and the TiO₂@C precursor were examined by XRD. Their diffraction results are shown in Fig. 4. Diffraction pattern of TiO₂@C precursor only showed the anatase phase (JCPDS:21-1272), and no crystalline phase of carbon was found. It inferred that acetylene black carbon may be amorphous. Figure 3 shows an inconspicuous peak around 1273 K, but no mass change. A phase transformation occurred at this temperature. Some rutile (JCPDS:21-1276) was found in the sample which quenched at 1273 K. There was still no graphite phase found. When temperature reached around 1523 K, there was also an endothermic peak on DTA curve, and a mass loss peak appeared on the TG curve. The Ti₄O₇ (JCPDS:50-0787), Ti–O Magnéli phase, was found. Carbothermal reduction occurred, and some gaseous products emission caused the mass loss. As the temperature reached 1623 K, there was a massive endothermic peak in DTA signal and also mass loss accompanied. The quenched sample exposed TiC (JCPDS:65-0971) phase at XRD pattern. It might be some rest carbon there but not indexed in the diffraction peaks. The vigorous reduction occurred, large amount of oxygen from titanium oxide was deprived, and produced gas might be the reason of mass loss.

Fig. 3

TG–DTA curves of TiO₂@C from ambient to 1673 K at 40 K min⁻¹ rate, in 200 mL min⁻¹ Ar gas flow

Fig. 4

XRD results of TiO₂@C precursor and intermediate products at different reducing temperature. a Quenched from 1273 K, b quenched from 1523 K and c quenched from 1623 K

The TG–DTA combined with the XRD analysis of the quenched samples at different annealing processes showed that at the relative low temperature, part anatase first changed to rutile structure; then, TiO₂ was reduced to substoichiometric titanium oxide and finally reduced to TiC, with the increasing in the temperature. This process is coinciding with the thermodynamic consideration before.

Isothermal reduction

In order to investigate the kinetics and mechanism of the reduction process, especially the process from TiO₂ to Ti_nO_2n−1, isothermal reduction was running in the range of 1363–1513 K; 50 K is the interval for each process.

Since the temperature was in the range of 1323–1523 K during the isothermal reduction process, the product of reduction was substoichiometric titanium oxide, expressed by Ti_nO_2n−1. The carbothermal reduction was believed as a solid–solid reaction.

$$ n{\text{TiO}}_{2} + {\text{C}} = {\text{Ti}}_{\text {n}} {\text{O}}_{2\text {n} - 1} + {\text{CO}} $$

(11)

According to the non-isothermal reduction, at the setting temperature, the TiC was not formed. From Fig. 2, it can be seen that Ti₃O₅ was the nearest Ti–O compound to TiC in the phase stable diagram. So the limit of the reduction reaction at the setting temperature was assumed as:

$$ 3{\text{TiO}}_{2} + {\text{C}} = {\text{Ti}}_{ 3} {\text{O}}_{ 5} + {\text{CO}} $$

(12)

The conversion fraction X was defined as:

$$ X = \frac{{w_{0} - w_{\text {t}} }}{{w_{0} - w_{\infty } }} $$

(13)

where w _t was the mass of reaction system at the time t, whereas w ₀ was the original mass of reactants before the reduction reaction; w _∞ was the mass of reaction system reaches the limit when all the TiO₂ in reactants reduced to Ti₃O₅. X was the conversion fraction at the time t.

The isothermal conversion of TiO₂@C at 1363, 1413, 1463 and 1513 K is shown in Fig. 5. Temperature impacts the reduction seriously. High temperature brought the high reaction rate. Most conversion curves had two parts: At the initial stage, the reaction rate was relative fast; at the later part, the rate was slow. With the decrease in the temperature, the first part was prolonging and the conversion was reducing. The curve at 1363 K cannot be divided into two parts.

Fig. 5

Conversion X of the different temperature

Some basic assumptions

At the initial stage, TiO₂ was embraced by the carbon. The reaction (11), the direct reduction, started at first. With the reaction continued, the product layer formed at the surface of TiO₂ and isolated the contact of reactants. The CO gas played as a reducing medium, which can diffuse and penetrate through the product layer, reduced the TiO₂ to Ti_nO_2n−1 and produced CO₂.

$$ n{\text{TiO}}_{2} + {\text{CO}} = {\text{Ti}}_{\text{n}} {\text{O}}_{\text{2n - 1}}+ {\text{CO}}_{ 2} $$

(14)

CO₂ diffused through the layer of product and reacted with C to produced CO again. This was an indirectly reducing.

$$ {\text{CO}}_{2} + {\text{C}} = 2{\text{CO}} $$

(15)

Total reaction was reaction (14) plus reaction (15), the same as reaction (11). These two stages of reduction could be described by Fig. 6.

Fig. 6

Reaction models for two stages

Kinetic model

Three-dimensional interfacial reaction might be well for describing the first stage, and the three-dimensional diffusion might be fit for the second stage. So the conversion of the first stage was set as X _1, and the conversion of second stage was set as X ₂. Then, the relation among total conversion X, X ₁ and X ₂ could express as,

$$ X{ = }\xi_{1} X_{1} + \xi_{2} X_{2} $$

(16)

where ξ ₁ denoted the ratio of extent of interfacial reaction to that of ideal total reduction (12); in another word, it was the maximum amount of X which was contributed by the interfacial reaction. ξ ₂ was the theoretical ratio of extent of gas diffusion reduction to that of ideal total reduction (12). Here, there are just two reduction processes, so:

$$ \xi_{1} + \xi_{2} { = 1} $$

(17)

ξ ₁ and ξ ₂ were affected by the temperature, rate of gas diffusion, the thickness of the product layer, and so on.

At the initial stage, the carbon and TiO₂ contacted with each other at the spherical interfacial; the three-dimensional shrinkage model could describe this process. The expression could be written as:

$$ X_{1} = 1 - (1 - k_{1} t)^{3} $$

(18)

Here, X ₁ was the conversion fraction of interfacial reaction, k ₁ was the rate constant of interfacial reaction. Considering the TG was not stable at the start of setting temperature, the reaction had already begun before it was stable. The actual reaction time needed to modify is as follows:

$$ X_{1} = 1 - [1 - k_{1} (t + \Delta t)]^{3} $$

(19)

A corrected factor b ₀ was introduced to replace k ₁Δt. The reaction (18) could be rewritten as:

$$ X_{1} = 1 - [1 - (k_{1} t + b_{0} )]^{3} $$

(20)

At the second stage, gas diffusion was the rate-controlled step. If r and y are set as the original radius of titanium oxide particles and the thickness of the products layer, respectively, the original volume V _O could be described as:

$$ V_{\text{O}} = \frac{4}{3}\pi r_{{}}^{3} $$

(21)

the volume of unreactive part V _U could be described as:

$$ V_{\text{U}} = \frac{4}{3}\pi (r - y)^{3} $$

(22)

so the volume of the products layer V _P could be described as:

$$ V_{\text{P}} = V_{\text{U}} - V_{\text{O}} = \frac{4}{3}\pi [r^{3} - (r -y)^{3} ] $$

(23)

The conversion fraction of titanium oxide reduction reaction X could be written as:

$$ X = \frac{{r^{3} - (r - y)^{3} }}{{r^{3} }} = 1- \left( {1- \frac{y}{r}} \right)^{3} $$

(24)

$$ \frac{y}{r} = 1 - \left( {1 - X} \right)^{\frac{1}{3}} $$

(25)

There was a hypothesis that contact surface could be treated as large flat, thus, taken the parabolic equation y ² = kt in Eq. (24):

$$ y^{2} = r^{2} \left[ {1 - (1 - X)^{\frac{1}{3}} } \right]^{2} = kt $$

(26)

or

$$ \left[ {1 - (1 - X)^{\frac{1}{3}} } \right]^{2} = kt $$

(27)

three-dimensional diffusion, Jander model could provide an equation to portrait this.

$$ X_{2} = 1 - \left[ {1 - (k_{2} t)^{\frac{1}{2}} } \right]^{3} $$

(28)

where X ₂ was the conversion fraction of diffusion-controlled reduction step, k ₂ was the rate constant of diffusion-controlled reaction. The Jander model assumed the reaction was diffusion-controlled from the beginning, but there had been a product layer on the surface when the diffusion-controlled reduction started. Assuming the thickness of the product layer as r′, the volume of the products layer had to change to another equation, instead of Eq. (23), which is:

$$ V = \frac{4}{3}\pi \{ (r - r^{{\prime }} )^{3} - [(r - r^{{\prime }} ) - y]^{3} \} $$

(29)

So the conversion fraction of diffusion-controlled reduction could be written as follows:

$$ X_{2} = \left( {1 - \frac{r}{{r^{{\prime }} }}} \right)^{3} - \left[ {\left( {1 - \frac{r}{{r^{{\prime }} }}} \right) - (k_{2} t)^{\frac{1}{2}} } \right]^{3} $$

(30)

Another corrected factor r ₀ was introduced to replace $\frac{r}{{r^{{\prime }} }}$, r ₀ would be the relative thickness of product to the radius of initial particle. Consider the time increment, in the same time, the expression was

$$ X_{2} = (1 - r_{0} )^{3} - [(1 - r_{0} ) - (k_{2} t + b_{0} )^{\frac{1}{2}} ]^{3} $$

(31)

Model application

TiO₂@C was reduced in Ar gas flow, at 1413 K. After annealing for 60, 120 and 240 min, respectively, each sample was quenched to room temperature and observed by SEM. The morphologies of these samples and TiO₂@C precursor are shown in Fig. 7. It could be seen from Fig. 7a that carbon and TiO₂ mixed the particle size of TiO₂ was at a range of 0.3–1.2 μm. Carbon was much smaller than TiO₂, so it was believed that carbon was adsorbed on the surface of TiO₂ as that described in Fig. 6 (left part). It provided the conditions for interfacial chemical reaction. As the time of reaction passed by, it could be seen that some particles were in bright contrast in Fig. 7b–d and became more and more. These particles had smooth surfaces, which attribute to the carbon consuming and product forming. The particle size of reduced titanium oxide was at a range of 0.2–1.4 μm, which was very close to the size before. As Fig. 6 shows unreacted core model, when reaction went on, the outer diameter of particles was kept constant. It was coincide with the observed result.

Fig. 7

SEM of TiO₂@C precursor (a) and products that reduced in Ar gas flow, at 1413 K. After annealing for 60 min (b), 120 min (c) and 240 min (d), all the scar bars were 5 μm

Reaction curves at different temperatures were fitted by equations combined (16), (20) and (31). The fitting results are shown in Fig. 8. Normally the rate of interfacial reaction was faster than that of the gas diffusion process. The curve showed two stages, first was fast and second was relative slow. The curve was just fitted by ξ ₁ X ₁ at 1363 K (Fig. 8a), which meant only the interfacial reaction between carbon and TiO₂ occurred during the test time. Figure 8b–d is fitted by Eq. (16); the insets were the fitting process of ξ ₁ X ₁ in the first stage and the simulating of ξ ₂ X ₂ in the second part. All the experimental point and fitting results are merged in Fig. 8e. Temperature increasing accelerated the reaction rate. And the fraction of interfacial reaction was increasing, but the period of interfacial reaction decreased. The analysis result is listed in Table 2. When the interfacial reaction closed to the end, judging from the conversion X ₁ closed to 1, the conversion rate of this point was ξ ₁. The value of ξ ₁ was increasing with the raising of temperature. Based on the model in Fig. 6, the gas diffusion-controlled reduction should be after the interfacial reaction. Maybe at a short period of time, both gas diffusion-controlled reduction and solid interfacial reduction occurred simultaneously. So the starting time of the second part of reduction may not be after X ₁ = 1. But the product layer should accumulate to a certain thickness; the gas diffusion could be the rate-controlling step. The simulating results of times, when X ₁ = 0.95 and X ₁ = 0.99 under different temperatures, are listed in Table 2. The starting times of the second stage were between the time of X ₁ = 0.95 and X ₁ = 0.99, except at 1363 K. In a very short period of time, the reduction may be influenced by both solid interfacial reaction and gas diffusion process. The starting time of second stage became shorter with the increase in temperature.

Fig. 8

Simulation and experiment data of conversion X at 1363 K (a), 1413 K (b),1463 K (c), 1513 K (d), and 4 in 1 (e). Insets were simulation of ξ ₁ X ₁ and ξ ₂ X ₂, respectively

Table 2 Influences of temperature on the reaction mechanism

When the reduction temperature is at 1363 K, the curve is fitted by the first-stage model ξ ₁ X ₁, as shown in Fig. 8a. It meant at this temperature, during the first 100 min, reduction was only controlled by the solid interfacial reaction. At the other three temperatures, the reduction curves, Fig. 8b–d, could portrait by the two-stage model.

The fitting results are listed in Table 3.

Table 3 Fitting parameters of reduction curves

The parameter b ₀ is related to the reduction starting time before the TG apparatus reached the stable temperature. It was affected by the temperature. At 1363 K, it was a minus number, which means the starting time was later than the TG reached the temperature. With the temperature rising, the number became larger, which meant the reaction already started before TG reached the target temperature.

The parameter r ₀ is related to the accumulated thickness of reducing product during first stage. It may not be affected by the change of temperature much, so it was from about 0.13 to 0.16 when the temperature changed from 1413 to 1513 K.

The relationship between rate constant k and the temperature T obeyed Arrhenius equation (Eq. 32).

$$ \ln k = - \frac{{E_{\text{a}} }}{RT} + \ln A $$

(32)

The activation energy E _a and pre-exponential factor A were obtained from the slop and the intercept of Arrhenius plot, respectively (Fig. 9). The correlation coefficient of line fitting for the first stage and the second stage was 0.974 and 0.989, respectively.

Fig. 9

Arrhenius plots of the first-step (a) and the second-step (b) reaction

The kinetic expression of the first stage was:

$$ \xi_{1} X_{1} = \xi_{1} \{ 1 - [1 - (1.638 \times 10^{7} t{\text{e}}^{{{ - }{{ 2 4 1 1 0 6} \mathord{\left/ {\vphantom {{ 2 4 1 1 0 6} {\text{RT}}}} \right. \kern-0pt} {\text {RT}}}}} + b_{0} )]^{3} \} $$

(33)

and the second stage was:

$$ \xi_{2} X_{2} = (1 - \xi_{1} ) \cdot \left\{ {(1 - r_{0} )^{3} - \left[ {1 - r_{0} - (1.727 \times 10^{8} t{\text{e}}^{{{ - 323390}/{\text{RT}}}} + b_{ 0} )^{\frac{1}{2}} } \right]^{3} } \right\} $$

(34)

Summary

Based on the thermodynamic consideration, the temperature and the partial pressure of CO were the key impact factors impacting on preparing Ti–O Magnéli phase via carbothermal reduction process. The non-isothermal carbon reduction process showed that the anatase TiO₂ first changes to rutile structure at relative low temperature and then reduced to Ti–O Magnéli phase. When temperature was higher than 1523 K, it would reduce to TiC.

The isothermal carbon reduction process showed that there was a one stage of solid interfacial reaction stage at the 1363 K. But when temperature was rising, the reduction curves would show a solid interfacial reaction stage at initial, and a gas diffusion-controlled slow reduction at second. The expressions of these two stages were deduced. The stimulating curves were close to the experimental conversion curves. Activation energies E _a and pre-exponential factors A of two stages were obtained from the fitting results. The total expression of conversion curves was

$$ \begin{aligned} X = &\, \xi_{1} X_{1} + \xi_{2} X_{2} \\ = & \,\xi_{1} \{ 1 - [1 - (1.638 \times 10^{7} t{\text{e}}^{{{ - 241106}/{\text {RT}}}} + b_{ 0} )]^{3} \} \\ & + (1 - \xi_{1} ) \cdot \left\{ {(1 - r_{0} )^{3} - \left[ {1 - r_{0} - (1.727 \times 10^{8} t{\text{e}}^{{{ - 323390}/{\text {RT}}}} + b_{ 0} )^{\frac{1}{2}} } \right]^{3} } \right\} \\ \end{aligned} $$

(35)