Isothermal and non-isothermal sublimation kinetics of

Materials Chemistry and Physics 143 (2014) 1075e1081

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Isothermal and non-isothermal sublimation kinetics of zirconium tetrachloride (ZrCl4) for producing nuclear grade Zr Jae Hong Shin a, Mi Sun Choi b, Dong Joon Min c, Joo Hyun Park a, * a

Department of Materials Engineering, Hanyang University, Ansan 426-791, Republic of Korea Research Institute of Industrial Science and Technology (RIST), Pohang 790-330, Republic of Korea c Department of Materials Science and Engineering, Yonsei University, Seoul 120-749, Republic of Korea b

h i g h l i g h t s Sublimation kinetics of ZrCl4 was quantitatively analyzed using TGA method. Isothermal and non-isothermal sublimation kinetics were quantitatively evaluated. Activation energies of isothermal and non-isothermal kinetics were obtained. Sublimation mechanism was proposed from kinetic analyses and SEM observations. This kinetic information will be very useful in production of nuclear grade Zr.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 10 April 2013 Received in revised form 25 September 2013 Accepted 3 November 2013

Sublimation of ZrCl4 is important for the production of nuclear grade metallic Zr in Kroll’s process. The sublimation kinetics of ZrCl4 was investigated by thermogravimetric analysis under both isothermal and non-isothermal conditions. The sublimation rate of ZrCl4 increased with increasing temperature under isothermal conditions. ZrCl4 sublimation was confirmed to be a zero-order process under isothermal conditions, whereas it was first-order kinetics under non-isothermal conditions. The activation energy of ZrCl4 sublimation under isothermal conditions was 21.7 kJ mol1. The activation energy for nonisothermal sublimation was 101.4 kJ mol1 and 108.1 kJ mol1 with the Kissinger method and Flynn eWalleOzawa method, respectively. These non-isothermal activation energies were very close to the heat of sublimation (103.3 kJ mol1). Sublimation occurs by two elementary steps: surface reaction and desorption. Therefore, the overall activation energy of ZrCl4 sublimation is 104.8 (3.4) kJ mol1. The activation energy of the surface reaction and desorption steps are proposed to be 83.1 kJ mol1 and 21.7 kJ mol1, respectively. Ó 2013 Elsevier B.V. All rights reserved.

Keywords: Inorganic compounds Thermogravimetric analysis Phase transitions Thermal properties Thermodynamic properties Transport properties

1. Introduction Zirconium and its alloy are used extensively as nuclear fuel cladding materials in water-cooled nuclear power reactors [1e3]. However, only a few countries in the world have access to commercial-scale zirconium industries and have the capability to manufacture reactor-grade zirconium. Metallic zirconium (sponge) can be produced from a magnesiothermic reduction known as Kroll’s process [4,5], which is similar to that used for titanium production except that the raw material (TiCl4) is in the liquid state.

* Corresponding author. E-mail address: [email protected] (J.H. Park). 0254-0584/$ e see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.matchemphys.2013.11.007

In zirconium production, the raw material is zirconium tetrachloride (ZrCl4), which sublimates at about 330 C and melts at about 440 C (triple point). The rate of ZrCl4 sublimation may control the feeding rate of reactants in a vessel in a way that affects production efficiency. According to the reports by Reshetnikov and Oblomeev [6,7], the influence of the rate of ZrCl4 sublimation on the feeding rate is greater than the influence of the chemical reaction rate between ZrCl4 and liquid magnesium on the process rate. An understanding of the mechanism of ZrCl4 sublimation and the related kinetic data are required to optimize zirconium sponge production. There have been a few reports on the overall process of zirconium sponge production that include information on the distillation process coupled with thermodynamic data of ZrCl4 sublimation [4e10]. The vapor pressures of ZrCl4 from 207 to 416 C and from 437 to 463 C were originally reported by Palko et al. [8]

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They also determined the heats of vaporization and sublimation for ZrCl4. More recently, Liu et al. [9] measured the vapor pressure of ZrCl4 from room temperature to 340 C. To the best of our knowledge, only equilibrium thermodynamic data on ZrCl4 sublimation have been reported, whereas the kinetics of ZrCl4 sublimation has not yet been investigated. In the present study, we attempted to confirm the sublimation mechanism of ZrCl4 based on a kinetic study of its sublimation using thermogravimetic analysis (TGA). The activation energy of ZrCl4 sublimation was measured under both isothermal and nonisothermal conditions. 2. Experimental Thermogravimetric analysis (TGA) was employed to measure the rate of sublimation for reagent-grade zirconium tetrachloride (99.99% purity). An electronic balance set on a kanthal super electric furnace was connected to a computer to record the weight loss of the sample due to sublimation. The temperature was calibrated using a K-type thermocouple in an alumina sheath placed in the hot zone of the furnace. The experimental temperature ranged from 300 to 550 C. A fused alumina crucible (ID: 14 mm, OD: 18 mm, HT: 40 mm) filled with ZrCl4 powder (1.0 g) was connected to the electronic balance sensor by molybdenum wire and was hung in the hot zone of the furnace. ZrCl4 weight loss by sublimation was automatically recorded every second at each temperature (300, 330, 350, 400, 450, 500, and 550 C) for the isothermal experiments. ZrCl4 weight loss was also measured under non-isothermal conditions with heating rates of 17 C min1 and 8 C min1. After the experiments, the morphology of whole samples was examined by scanning electron microscopy coupled with an energy dispersive spectroscope. In the preliminary experiments, the effect of the flow rate of Ar carrier gas on the ZrCl4 sublimation rate by varying the Ar flow rate from 200 to 500 ml min1 at 330 C. It was founded that the sublimation behavior of ZrCl4 was not noticeably affected by Ar flow rate within analytical error limit. Hence, an inert atmosphere was maintained in the reaction chamber during the experiments by purging with purified Ar gas of which flow rate was 400 ml min1. Here, the Ar gas was purified by passing it through Mg turning furnace at 450 C. 3. Theoretical considerations

f ðaÞ ¼ am $ð1 aÞn $½ lnð1 aÞp

(3)

where n, m, and p are empirically obtained exponent factors, one of which is always equal to zero [11,12]. All functions covered in this work are listed in Table 1 in differential and integral form [11e16]. Eq. (3) can be represented in the integral form, function g(a), at constant temperature as follows:

gðaÞ ¼

Za 0

da ¼ kðTÞ$t f ðaÞ

(4)

The slope of a line g(a) against time t at a fixed temperature, i.e. Eq. (4), is used to calculate the rate constant k(T), which is given by the Arrhenius equation as follows:

Ea kðTÞ ¼ A$exp RT lnkðTÞ ¼ lnA

(5)

Ea 1 $ R T

(6)

where A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is the absolute temperature. Thus, the activation energy can be deduced from an Arrhenius plot. 3.2. Non-isothermal kinetics 3.2.1. Kissinger method For non-isothermal conditions, the heating rate b is given by

b¼

dT dt

(7)

Combining Eqs. (1) and (5) gives

da Ea ¼ A$exp $f ðaÞ dt RT

(8)

Thus, under non-isothermal conditions, the basic equation of solid-state decomposition kinetics is obtained from Eqs. (1) and (8) as follows.

da A Ea ¼ $exp $f ðaÞ b dT RT

(9)

3.1. Isothermal kinetics Solid-state decomposition kinetics is usually described by the following basic equation [11]:

da ¼ kðTÞ$f ðaÞ dt

ðWi Wt Þ Wi Wf

Model

f(a)

g(a)

Observations

F0, R1, P1

Constant

a

F1/2, R2

(1 a)1/2

2[1 (1 a)1/2]

F2/3, R3

(1 a)2/3

3[1 (1 a)2/3]

F1 F2 P2 A2

(1 a) (1 a)2

ln(1 a) a/(1 a) 2a1/2 2[ln(1 a)]1/2

Zero-order kinetics; one-dimensional advance of reaction interface One-half-order kinetics; two-dimensional advance of reaction interface Two-thirds-order kinetics; three-dimensional advance of reaction interface First-order kinetics Second-order kinetics Power law AvramieYerofeyev equation

a2/2 ln[a/(1 a)]

Parabolic law ProuteTompkins equation

(1)

where k(T) is the rate constant at absolute temperature T, t is the reaction time, and a is a degree of reaction, which is defined as follows:

a ¼

Table 1 Algebraic expressions of f(a) and g(a) for kinetic models considered in this study.

(2)

where Wi and Wf are the initial and final sample mass, respectively, and Wt is sample mass at any time t. According to Sesták [11], the kinetic function f(a) is related to the reaction mechanism, and can be written in the following general form:

D1 P-T

a1/2

(1 a)[ln (1 a)]1/2 1/a a (1 a)


The Kissinger method is based on the rate equation at the maximum reaction rate [17]. At this point, ðd2 a=dt 2 Þ or ðd2 a=dT 2 Þ is equal to zero. Thus, Eq. (10) is obtained from Eq. (8).

d a Ea b A Ea dT a $f ð ¼ $ $exp Þ$ dt RT dt 2 RT 2 b Ea da 0 þ A$f ðaÞ$exp ¼ 0 $ RTm dt

2

(10)

! Ea da 0 a ð Þ$exp ¼ 0 $ þ A$f RTm dt RTm 2

d2 a ¼ dt 2

Ea b

(11)

where Tm is the temperature at which the second derivatives of ðd2 a=dt 2 Þ or ðd2 a=dT 2 Þ equal zero. As follows from Eq. (11),

Ea b Ea 0 a ¼ A$f ð Þ$exp 2 RTm RTm

(12)

b

!

AR Ea ¼ lnðf 0 ðaÞÞ þ ln Ea RTm

2 Tm

3.2.3. CoatseRedfern method CoatseRedfern method is an integral method as likely as FlynneWalleOzawa method and it involves the function of reaction model. In the CoatseRedfern method, an asymptotic approximation is used for solution of the p(x) function as follows [22].

(18)

Therefore, Eq. (19) is obtained by combining Eqs. (14) and (18).

gðaÞ AR 2RT Ea ¼ ln ln $ 1 bEa Ea RT T2

(19)

Thus, the activation energy can be obtained by plotting the lefthand side (including the model, g(a)) versus T1. 4. Results and discussion 4.1. Thermogravimetric analysis

Eq. (12) can be rearranged after inserting logarithms into the Kissinger equation:

ln

from the slope of a plot of the natural logarithm of the heating rate lnbi versus 1=Ta i .

2 expð xÞ $ pðxÞy 1 x x2

From Eqs. (9) and (10), the following equation is deduced.

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The weight loss of ZrCl4 as a function of time at each isotherm in the temperature range of 300e550 C is shown in Fig. 1. Weight loss drastically increased with increasing temperature. Approximately

(13)

A plot of the left side of this equation as a function of reciprocal temperature gives a straight line with slope equal to the activation energy [17]. 3.2.2. FlynneWalleOzawa method The kinetics of sublimation was also studied using the isoconversional method of FlynneWalleOzawa [18,19]. In a nonisothermal system, f(a) can be expressed in integral form as the function g(a) as follows:

gðaÞ ¼

Za 0

da A ¼ $ b f ðaÞ

ZT 0

Ea AEa $dT ¼ $pðxÞ exp bR RT

(14)

where x is Ea =RT and the p(x) function is defined as follows

Zx pðxÞy N

expð xÞ $dx x2

(15)

Eq. (15) has no analytical solution but has many approximations [18e21]. The FlynneWalleOzawa method is one of the most popular one. The p(x) function was described by Flynn, Wall, and Ozawa using Doyle’s approximation as follows [18e20]:

pðxÞy0:0048$expð1:0516$xÞ

(16)

Eq. (14) is then rearranged for b,

Aa Ea Ea 5:331 1:0516 lnbi ¼ ln RgðaÞ RTa

(17)

where subscripts i and a denote a given heating rate and value of conversion, respectively. The activation energy is then calculated

Fig. 1. Weight loss (%) vs. time under (a) isothermal and (b) non-isothermal conditions.

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Table 2 Slope of g(a) vs. time for each kinetics model at 300 C. Slope of g(a) vs. time for each kinetics model at 300 C. Model

Slope

R-squared

F0, R1, P1 F1/2, R2 F2/3, R3 F1 F2 P2 A2 D1 P-T

0.0368 0.0559 0.0670 0.1067 1.7627 0.0838 0.1349 0.0152 0.0854

0.995 0.980 0.958 0.849 0.251 0.970 0.991 0.951 0.350

80% of the initial weight of ZrCl4 was sublimated when temperatures were lower than 350 C, while approximately 90% of the initial weight of ZrCl4 was sublimated at temperatures greater than 400 C. ZrCl4 weight loss with time under non-isothermal conditions with heating rates of 8 C min1 and 17 C min1 is also shown in Fig. 1. Weight loss drastically increased after about 330 C, the sublimation temperature of ZrCl4. Maximum sample loss was higher with a heating rate of 8 C min1 than with a heating rate 17 C min1. 4.2. Reaction model of sublimation of ZrCl4 compound under isothermal conditions According to Eq. (4), g(a) should have a linear relationship with time. The slope of the plot of g(a) versus time at 300 C is listed in Table 2. Slope was calculated by linear regression analysis, and Rsquared represents the coefficient of determination for each reaction model. The F0 and A2 models (R-squared is 0.995 and 0.991, respectively) show a good linear relationship between g(a) and time. However, the A2 model involves random nucleation and subsequent growth, which is not applicable to the sublimation mechanism. The F0 model for chemical decomposition is called ‘zero-order kinetics’. Dollimore suggested necessary assumptions for the application of a model based on zero-order kinetics as follows [23]: 1. The existence of a reaction interface. 2. The rate of reaction for a unit area of interface is constant at any given temperature. 3. For a small incremental change in temperature or time, the area of the reaction interface does not change.

Fig. 3. SEM images of samples at different degrees of reaction (a) at 550 C.

A schematic illustration of ZrCl4 sublimation in the present experimental conditions is presented in Fig. 2. The area of the interface of ZrCl4 samples is constant to the cross-sectional area of the crucible and the interface moves one dimensionally,

For zero-order kinetics, the movement of the sample interface under isothermal conditions should only be one-dimensional and the interface area should not change.

Fig. 2. Schematic illustration of ZrCl4 sublimation.

Fig. 4. g(a) vs. time under isothermal conditions at different temperatures.


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Table 3 Slope of g (a) vs. time at each temperature for zero-order process. Temp ( C)

k(T) (min1)

Intercept

R-squared

300 330 350 400 450 500 550

0.0368 0.0453 0.0582 0.0758 0.0889 0.1199 0.1562

0 0 0 0 0 0 0

0.995 0.993 0.993 0.992 0.992 0.991 0.995

representing zero-order kinetics. SEM analysis was performed to observe the morphology of all samples, and the results are shown in Fig. 3. The crucible was initially filled with pure ZrCl4 powder, which has widely variable particle diameters. From the particle size distribution during and after the sublimation process, each particle was initially sintered and then sublimated. In other words, ZrCl4 compact in the crucible behaved as a bulk solid. A number of previous studies have reported that the reaction mechanism for the evaporation or sublimation of the condensed phase follows zeroorder kinetics [12,23e27]. 4.3. Isothermal kinetics Since ZrCl4 sublimation is a zero-order process, the mass fraction sublimated is equal to the product of time and the rate constant, i.e. gðaÞ ¼ a ¼ kðTÞ$t. Fig. 4 shows g(a) vs. time curves at each temperature. According to Eq. (4), the rate constant k(T) was calculated from the slope of fitted lines at each temperature. The calculated rate constants, listed in Table 3, increase with increasing temperature. Based on Eq. (5), the activation energy of ZrCl4 sublimation was obtained from an Arrhenius plot, as shown in Fig. 5. The activation energy is 21.7 kJ mol1. 4.4. Non-isothermal kinetics

Fig. 6. da/dT vs. temperature under non-isothermal conditions at heating rate of (a) 17 C min1 and (b) 8 C min1.

To determine Tm at different heating rates, ðda=dTÞ was plotted against temperature at heating rates of 17 C min1 and 8 C min1, as shown in Fig. 6. At a heating rate of 17 C min1, ðd2 a=dT 2 Þ equaled zero at Tm ¼ 423.2 C, whereas Tm ¼ 397.6 C at a heating rate of 8 C min1. The Kissinger method was adopted to calculate the activation energy of ZrCl4 sublimation under non-isothermal 2 Þ versus ð1=TÞ, in which conditions. Fig. 7 shows a plot of ðlnb=Tm the slope equals activation energy. The activation energy was calculated to be 101.4 kJ mol1.

Alternatively, the isoconversional plots based on the FlynneWalle Ozawa method is appeared in Fig. 8. The calculated activation energies of ZrCl4 sublimation at each conversion rate are summarized in Table 4. The average activation energy of ZrCl4 sublimation obtained from the Flynn-Wall-Ozawa method is 108.1 kJ mol1, which is very close to that obtained with the Kissinger method.

-1.0 -1.5

lnk (T )

-2.0 -2.5 -3.0 -3.5 -4.0 1.00

-2.622

1.25

1.50

1.75

2.00

-1

1000/T (K ) Fig. 5. Arrhenius plot of rate constant against reciprocal temperature.

2 Þ vs. T1 curve. Fig. 7. ðlnb=Tm

1080


Fig. 9. Kinetic plot indicating F1 model by using CoatseRedfern method.

Fig. 8. ln (b) vs. T1 curve.

Table 4 Activation energy of ZrCl4 sublimation under non-isothermal conditions obtained by Kissinger and FlynneWalleOzawa methods. Activation energy (kJ mol1)

Method Kissinger

101.4

FlynneWalleOzawa

Degree of reaction (a)

Average

0.2 0.3 0.4 0.5 0.6 0.7

99.6 107.5 95.4 129.5 112.4 104.1 108.1

The calculated activation energies for different models at each heating rate by using CoatseRedfern method are listed in Table 5. The A2 model (R2 ¼ 0.981 and 0.994 for heating rate ¼ 8 C min1 and 17 C min1, respectively) show a good linear relationship between lngðaÞ=T 2 and 1=T. However, in view of the activation energy as shown in Table 5 and Fig. 9, the F1 model (R2 ¼ 0.969 and 0.965) is more consistent with the experimental results than A2 model. Consequently, the F1 model, viz. first-order kinetics, is acceptable for ZrCl4 sublimation kinetics under non-isothermal condition. 4.5. Multistep sublimation

these values are relatively close to the heat of ZrCl4 sublimation ZrCl4 ðDHsub ¼ 103:3 kJ mol1 Þ reported by Palko et al. [8] Nevertheless, the activation energy of ZrCl4 sublimation is inconsistent for non-isothermal and isothermal conditions. According to Somorjai [28], a sublimation reaction has two elementary steps. First, atoms break away from their neighbors in the crystal lattice, and second, atoms are removed into the gas phase. The first step, called the surface reaction step, involves atoms breaking away from a kink site and then diffusing on the surface until they are ready to vaporize. The second step, called the desorption step, involves the diffusion of atoms into gas phase after desorbing sublimation substance from the sublimation surface. Thus, sublimation probably has two activation energies. As discussed above, the activation energy of ZrCl4 sublimation measured under non-isothermal conditions in the present study (Ea ¼ 104.8 (3.4) kJ mol1) is close to the heat of ZrCl4 sublimation. This indicates that the activation energy of ZrCl4 sublimation under non-isothermal conditions represents overall activation energy that involves two energy barriers for two steps. The activation energy of ZrCl4 sublimation under isothermal conditions is low (Ea ¼ 21.7 kJ mol1) compared to the heat of ZrCl4 sublimation. This indicates that one of the two steps was skipped, because ZrCl4 was suddenly subjected to high temperatures when loaded into the furnace.

From Table 4, the Kissinger and Flynn-Wall-Ozawa methods afford similar activation energies for ZrCl4 sublimation. Moreover, Table 5 Activation energy of ZrCl4 sublimation under non-isothermal conditions by Coatse Redfern method. Activation energy of ZrCl4 sublimation under non-isothermal conditions by CoatseRedfern method. Model

F0, R1, P1 F1/2, R2 F2/3, R3 F1 F2 P2 A2 D1 P-T

8 C min1

17 C min1

Ea (kJ mol1)

R-squared

Ea (kJ mol1)

R-squared

25.1 45.6 37.2 99.2 243.7 6.7 37.1 61.8 132.7

0.738 0.921 0.869 0.969 0.970 0.434 0.981 0.814 0.820

54.7 64.1 67.5 95.0 121.6 21.5 31.6 121.2 252.4

0.973 0.988 0.991 0.965 0.997 0.954 0.994 0.978 0.780

Fig. 10. Schematic diagram of activation energy of ZrCl4 sublimation.


The activation energy for vacuum sublimation can be approximated by the heat of sublimation. Equilibrium can likely be established in all surface reaction steps, leading to vaporization. In addition, the desorption of substance atoms from the surface requires no extra activation energy [28]. Under atmospheric pressure, the desorption step may have an activation energy, although it has a relatively low value, as shown in Fig. 10. Thus, activation energy measured under isothermal conditions can be considered the energy barrier for the desorption step. There is a difference between non-isothermal activation energy, or the overall energy barrier of ZrCl4 sublimation, and isothermal activation energy, or the energy barrier for desorption. The activation energy for the surface reaction step is estimated to be about 83.1 kJ mol1. From these results one can conclude that the sublimation of ZrCl4 under isothermal condition is mainly controlled by desorption step. However, because the activation energy of surface reaction step is higher than that of desorption step, the sublimation of ZrCl4 is limited by surface reaction step under non-isothermal condition. 5. Conclusions The sublimation kinetics of ZrCl4 was investigated using the TGA method under isothermal and non-isothermal conditions. The sublimation rate of ZrCl4 increased with increasing temperature under isothermal conditions. ZrCl4 sublimation was confirmed to be a zero-order process under isothermal condition, and its activation energy under isothermal conditions was determined to be 21.7 kJ mol1. The activation energy of non-isothermal sublimation was determined to be 101.4 kJ mol1 and 108.1 kJ mol1 using the Kissinger and FlynneWalleOzawa methods, respectively. By employing the CoatseRedfern method, the sublimation of ZrCl4 was confirmed to be a first-order reaction under non-isothermal conditions. The non-isothermal activation energies were very close to the heat of sublimation (103.3 kJ mol1). Sublimation occurs by the two elementary steps of surface reaction and desorption. Therefore, the overall activation energy of ZrCl4 sublimation is 104.8 (3.4) kJ mol1. The activation energy of the surface reaction and desorption steps were proposed to 83.1 kJ mol1 and 21.7 kJ mol1, respectively. Thus, the sublimation of ZrCl4 under isothermal condition is mainly controlled by desorption step, whereas it is limited by surface reaction step under non-isothermal condition. References [1] M. Griffiths, R.A. Holt, Microstructural aspects of accelerated deformation of Zircaloy nuclear reactor components during service, J. Nucl. Mater. 225 (1995) 245e258.

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[2] N. Dupin, I. Ansara, C. Servant, C. Toffolon, C. Lemaignan, J.C. Brachet, A thermodynamic database for zirconium alloys, J. Nucl. Mater. 275 (1999) 287e295. [3] N. Stojilovic, E.T. Bender, R.D. Ramsier, Surface chemistry of zirconium, Prog. Surf. Sci. 78 (2005) 101e184. [4] W.J. Kroll, A.W. Schlechten, W.R. Carmody, L.A. Yerkes, H.P. Holmes, H.L. Gilbert, Recent progress in the metallurgy of malleable zirconium, Trans. Electrochem. Soc. 92 (1947) 99e113. [5] W.J. Kroll, C.T. Anderson, H.P. Holmes, L.A. Yerkes, H.L. Gilbert, Large-scale laboratory production of ductile zirconium, J. Electrochem. Soc. 94 (1948) 1e20. [6] F.G. Reshetnikov, E.N. Oblomeev, Mechanism of formation of zirconium sponge in zirconium production by the magnesiothermic process, Sov. J. At. Energy 2 (1957) 561e564. [7] F.G. Reshetnikov, E.N. Oblomeev, Study of the velocity of the magnesium reduction process of zirconium production, Sov. J. At. Energy 4 (1958) 459e463. [8] A.A. Palko, A.D. Ryon, D.W. Kuhn, The vapor pressure of zirconium tetrachloride and hafnium tetrachloride, J. Phys. Chem. 62 (1958) 319e322. [9] C. Liu, B. Liu, Y. Shao, Z. Li, C. Tang, Vapor pressure and thermochemical properties of ZrCl4 for ZrC coating of coated fuel particles, Trans. Nonferrous Met. Soc. China 18 (2008) 728e732. [10] K. Kapoor, C. Padmaprabu, D. Nandi, On the evolution of morphology of zirconium sponge during reduction and distillation, Mater. Charact. 59 (2008) 213e222. [11] J. Sesták, Study of the kinetics of the mechanism of solid-state reactions at increasing temperatures, Thermochim. Acta 3 (1971) 1e12. [12] L.T. Vlaev, V.G. Georgieva, G.G. Gospodinov, Kinetics of isothermal decomposition of ZnSeO3 and CdSeO3, J. Therm. Anal. Cal. 79 (2005) 163e168. [13] J.H. Sharp, G.W. Brindley, B.N.N. Achar, Numerical data for some commonly used solid state reaction equations, J. Am. Ceram. Soc. 49 (1966) 379e382. [14] J. Sesták, V. Satava, W.W. Wendlandt, The study of heterogeneous processes by thermal analysis, Thermochim. Acta 7 (1973) 333e556. [15] M.E. Brown, D. Dollimore, A.K. Galwey, in: C.H. Bamford, C.F.H. Tipper (Eds.), Comprehensive Chemical Kinetics (Reactions in the Solid State), Elsevier, Amsterdam, 1980. [16] J.P. Elder, A comment on mathematical expressions used in solid state reaction kinetics studies by thermal analysis, Thermochim. Acta 171 (1990) 77e85. [17] H.E. Kissinger, Reaction kinetics in differential thermal analysis, Anal. Chem. 29 (1957) 1702e1706. [18] J.H. Flynn, L.A. Wall, A quick, direct method for the determination of activation energy from thermogravimetric data, J. Appl. Polym. Sci. Part B Polym. Lett. 4 (1966) 323e328. [19] T. Ozawa, A new method of analyzing thermogravimetric data, Bull. Chem. Soc. Jpn. 38 (1965) 1881e1886. [20] C.D. Doyle, Kinetic analysis of Thermogravimetric data, J. Appl. Polym. Sci. 5 (1961) 285e292. [21] H.J. Chen, K.M. Lai, Methods for determining the kinetics parameters from non-isothermal thermogravimetry, J. Chem. Eng. Jpn. 37 (2004) 1172e1178. [22] A.W. Coats, J.P. Redfern, Kinetic parameters from thermogravimetric data, Nature 201 (1964) 68e69. [23] D. Dollimore, A breath of fresh air, Thermochim. Acta 340e41 (1999) 19e29. [24] S. Miyamoto, A theory of the rate of sublimation, Trans. Faraday Soc. 29 (1933) 794e797. [25] D.M. Price, M. Hawkins, Calorimetry of two disperse dyes using thermogravimetry, Thermochim. Acta 315 (1998) 19e24. [26] D.M. Price, Volatilization, evaporation and vapour pressure studies using a thermobalance, J. Therm. Anal. Cal. 64 (2001) 315e322. [27] M. Su ceska, M. Raji c, S. Mate ci c, S. Zeman, Z. Jalový, Kinetics and heats of sublimation and evaporation of 1,3,3-trinitroazetidine (TNAZ), J. Therm. Anal. Cal. 74 (2003) 853e866. [28] G.A. Somorjai, Mechanism of sublimation, Science 162 (1968) 755e760.