ARTICLE pubs.acs.org/JPCA
Kinetic Analysis of One-Step Solid-State Reaction for Li4Ti5O12/C Xuebu Hu,*,†,|| Ziji Lin,‡ Kerun Yang,§ and Zhenghua Deng*,§,|| †
College of Chemistry and Materials Science, Sichuan Normal University, Chengdu, Sichuan 610066, China China National Quality Supervision & Inspection Center for Alcoholic Beverage Products and Processed Food, Luzhou, Sichuan 646100, China § Chengdu Institute of Organic Chemistry, Chinese Academy of Sciences, Chengdu, Sichuan 610041, China Zhongke Laifang Power Science & Technology Co., Ltd., Chengdu, Sichuan 610041, China
)
‡
bS Supporting Information ABSTRACT: The kinetics of one-step solid-state reaction of Li4Ti5O12/C in a dynamic nitrogen atmosphere was first studied by means of thermogravimetricdifferential thermal analysis technique at five different heating rates. According to the double equal-double steps method, the Li4Ti5O12/C solid-state reaction mechanism could be properly described as the Jander equation, which was a threedimensional diffusion with spherical symmetry, and the reaction mechanism functions were listed as follows: f(α) = 3/2(1 α)2/3[1 (1 α)1/3]1, G(α) = [1 (1 α)1/3]2. In FWO method, average activation energy, frequency factor, and reaction order were 284.40 kJ mol1, 2.51 1018 min1, and 1.01, respectively. However, the corresponding values in FRL method were 271.70 kJ mol1, 1.00 1017 min1, and 0.96, respectively. Moreover, the values of enthalpy of activation, Gibbs free energy of activation, and entropy of activation at the peak temperature were 272.06 kJ mol1, 240.16 kJ mol1, and 44.24 J mol1 K1, respectively.
1. INTRODUCTION As a new anode material for lithium-ion battery, spinel Li4Ti5O12 has attracted widespread attention because of its unique properties.1,2 However, low electric conductivity restricts its industrial applications.3 Carbon deposited on the Li4Ti5O12 particle surface has been proved as an effective way to improve the electric conductivity of the material.48 Currently, major synthesis methods of Li4Ti5O12/C composites include the high temperature solid-state method and the solgel method. Compared with the solgel method, the high temperature solid-state method shows the advantages of a simple process, easily controlled reaction conditions, and large-scale industrial production. Therefore, the high temperature solid-state method is widely used today for the synthesis of Li4Ti5O12/C composites.912 Some features of a solid-state reaction such as whether or not reacting between solid-state reactants, the reaction degree and the reaction controlling step directly affect the structure and performances of the final material. Therefore, it is very important to study the kinetics and mechanism of the solid-state reaction for the synthesis of the material. Kinetic analysis of the solid-state reactions by the thermogravimetric-differential thermal analysis (TG-DTA) technique can provide a theoretical basis for the synthesis of the materials and be used as the important evaluation indicators of the optimum conditions in industrial production. Currently, the technique is used in the study of the solid-state synthesis of electrode materials.1318 In this article, spinel Li4Ti5O12/C composites were synthesized via one-step solid-state reaction. The kinetics of the r 2011 American Chemical Society
solid-state synthesis of Li4Ti5O12/C was first studied by means of TG-DTA technique with different heating rates. The reaction mechanism functions were obtained according to the double equaldouble steps method. The FlynnWallOzawa (FWO) and FriedmanReichLevi (FRL) models were employed to investigate the activation energy (E), the frequency factor (A) and the reaction order (n). Moreover, the enthalpy of activation (ΔH#), the Gibbs free energy of activation (ΔG#), and the entropy of activation (ΔS#) at the peak temperature were calculated. All the conclusions could be applied as an important theoretical principle for the preparation and performance optimization of Li4Ti5O12/C composites via high temperature solid-state reaction.
2. EXPERIMENTAL SECTION 2.1. Chemicals. A stoichiometric amount of LiOH 3 H2O was dissolved into aqueous solutions of poly(acrylic acid) (PAA) to get PAALi. Then powdered TiO2 (Li:Ti = 4:5) was added into the PAALi solutions with continuous stirring for 6 h to yield homogeneous PAALi + TiO2 slurries. Subsequently, the above mixtures were dried at 120 °C. The precursors were obtained by grinding. 2.2. Thermogravimetric Analysis (TG) and X-ray Diffraction Analysis (XRD). A Netzsch STA 409PC/PG thermal analyzer was used for thermogravimetry-derivative thermogravimetry Received: August 7, 2011 Revised: October 6, 2011 Published: October 10, 2011 13413
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Figure 1. TG-DTA curves of the precursors under different heating rates.
analysis (TG-DTG) and thermogravimetric-differential thermal analysis (TG-DTA) with different scanning rates (5, 10, 15, 20, 25 K min1) from 30 to 750 °C at a nitrogen flow rate of 100 mL min1. X-ray diffraction (XRD) was carried on a Rigaku D/ Max 2550 powder diffractometer with Cu Kα radiation of λ = 1.5418 Å in the range 10° < 2θ < 90°.
3. THEORY Assuming the reaction process of the material is determined by two independent parameters, i.e., conversion rate (α) and temperature (T), the kinetics of heterogeneous reaction will obey the following equations.1922 dα ¼ k 3 f ðαÞ dt
ð1Þ
j¼
ð3Þ
dα A ¼ expðE=RTÞf ðαÞ dT j
ð4Þ
According to eq 1, the Arrhenius law (eq 2), and heating rate eq 3, eq 4 could be deduced, where t is time, k is reaction rate constant, f(α) is reaction mechanism function, A is frequency factor, E is activation energy, and R is gas constant. The purpose of this paper is to obtain the three factors of the above equations, i.e., E, A, and f(α). 3.1. FlynnWallOzawa (FWO) Method.2325. By integration of eq 4 the following equation is obtained. Z α
k ¼ A expðE=RTÞ
dT dt
dα A Z T E=RT A ¼ GðαÞ ¼ e dT ¼ ΔðTÞ 0 f ðαÞ j 0 j
ð2Þ 13414
ð5Þ
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Figure 2. TG (a) and DTG (b) curves of the precursors under different heating rates.
u¼
E RT
GðαÞ ¼
ð6Þ A Z T E=RT AE Z e dT ¼ j 0 jR
Table 1. Basis Kinetic Data Derived from the TG Curvesa j (K 3 min1)
eu AE PðuÞ du ¼ ∞ u2 jR ð7Þ u
temp range (°C) 25250
where G(α) is integral mechanism function. According to eqs 5 and 6, eq 7 is obtained. P(u) is commonly used Doyle formula. E ð8Þ RT Substituting eq 8 into eq 7 obtains the integral formula (eq 9) of FWO method. In accordance with the FWO method, the slope of log j vs 1/T for the same value of α gives the value of E, and A can be calculated by intercept when f(α) is coincident. AE E ð9Þ log j ¼ log 2:315 0:4567 GðαÞR RT
280650
log PD ðuÞ ¼ 2:315 0:4567
3.2. FriedmanReichLevi (FRL) Method.26,27. Equation 10 is FRL method which follows from logarithmic form of eq 4. jdα E ð10Þ ln ¼ ln½Af ðαÞ dT RT
In terms of the FRL method, the slope of ln(j dα/dT) vs 1/T for the same value of α gives the value of E, and A can be calculated by intercept when f(α) is coincident.
4. RESULTS AND DISCUSSION 4.1. Thermal Analysis. Figure 1 shows the TG-DTA curves of the precursors under different heating rates. It can be seen that all the precursors show similar weight loss curves. All TG curves from 30 to 200 °C mainly correspond to the vaporization of water and the decomposition of low molecular polymers. When the temperature is above 650 °C, the TG curves show no obvious weight loss, indicating that completion of the reaction makes Li4Ti5O12 completely crystallized. The temperature ranges 400500 and 350650 °C correspond to the largest weight loss and the largest heat change, respectively. For example, at a heating rate of 10 K min1, a large endothermic peak from 350 to 550 °C is attributable to the thermal decomposition of PAALi
30650 a
5
10
15
20
25
Ti (°C)
94.7
Tp (°C) mass loss (%)
117.9 131.7 140.3 141.7 146.5 1.25 1.50 1.52 1.57 1.66
Ti (°C)
416.1 426.4 431.9 433.3 433.6
Tp (°C)
432.8 445.4 452.4 451.0 457.3
mass loss (%)
6.11
103.5 109.7 109.8 110.4
6.52
6.62
6.50
7.25
total mass loss (%) 35.32 36.74 35.82 35.83 35.86
Ti: initial mass loss temperature. Tp: maximum temperature of the peak.
and solid-state reaction. A strong endothermic peak from 550 to 650 °C is related to growth and crystalline perfection of Li4Ti5O12 crystal particles. These conclusions can be proved in the results of XRD patterns (see Figure S in the Supporting Information for details). Figure 2 shows the TG and DTG curves of the precursors under different heating rates. It is obvious that all the precursors show two weight loss processes related to two temperature ranges 30250 and 300650 °C. The heating rate directly affected thermal decomposition rate. At heating rate of 5 and 25 K min1, the weight loss rate is fastest and slowest, respectively. Therefore, with the increase of the heating rate, the weight loss rate of the precursors gradually decreases. Table 1 lists some kinetic data derived from the TG curves. With the increase of heating rate, the maximum temperature of the peak (Tp) shifts to higher temperature, but total mass loss does not change significantly. 4.2. Nonisothermal Kinetics. 4.2.1. Determination of f(α) and G(α). FWO and FRL methods have a huge advantage; i.e., the activation energy can be calculated even though the specific reaction mechanism is not known. However, the reaction mechanism functions must be determined to calculate the frequency factor. Zhang et al.28,29 proposed a double equal-double steps method to process the data of thermal analysis kinetics. Equation 11 is obtained by eq 9. AE E 2:315 0:4567 logj ð11Þ log GðαÞ ¼ log R RT On substituting the values of α at the same temperature on different TG curves, different G(α)23, and j, values for the linear 13415
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of α are much more significant at 400, 450, and 500 °C. Therefore, calculating for r, k, and b only at the three temperatures, a part of the results are shown in Table 3. As shown in the table, the values of r and k at 450 °C are better than those at 400 and 500 °C in the same mechanism functions, which indicates that the reaction at 450 °C is more representative of the actual solidstate reaction process. By comparison of different mechanism functions at 450 °C, it is obvious that k in the Jander equation (D3, spherical symmetry) is closest to 1 and r reaches 0.990 11. These results show that the Jander equation can properly describe the solid-state reaction of Li4Ti5O12/C. The reaction mechanism is a three-dimensional diffusion with spherical symmetry, and the reaction mechanism functions are listed as follows: f(α) = 3/2(1 α)2/3[1 (1 α)1/3]1, G(α) = [1 (1 α)1/3]2. 4.2.2. Calculation of E and A. By substituting the values of α, j, T, and the reaction mechanism functions into eqs 9 and 10, linear fitting curves using the FWO and FRL methods are shown in Figure 3. The corresponding values of E and A are shown in Table 4. The values of E and A show an increasing trend with the increase of the conversion rate by two methods. However, the
correlation coefficient r, the slope k, and the intercept b at different temperatures are obtained by the linear least-squares method with log G(α) vs log j. If r is the best and k approaches 1, the relevant function is the probable mechanism function of the solid-state reaction. The values of α at the same temperature on different TG curves are listed in Table 2. It is clear that the values Table 2. Values of α at the Same Temperature on Different TG Curves α (K 3 min1) T (°C)
5
10
15
20
25
300
0.16
0.16
0.15
0.15
0.15
350
0.19
0.18
0.18
0.18
0.17
400 450
0.23 0.64
0.22 0.50
0.21 0.42
0.21 0.39
0.21 0.38
500
0.93
0. 90
0.86
0.84
0.84
550
0.97
0.96
0.96
0.96
0.96
600
0.99
0.99
0.99
0.99
0.99
Table 3. Classification of Kinetic Mechanism Functions and Relevant Parameters Based on the Double Equal-Double Steps Method T (°C)
function
f(α)
mechanism
G(α) α2
k
b
r 0.95838
0.12306
1.19449
450
-0.67649
0.07385
-0.992
500
0.13912
0.03741
0.98101
400
400 450
Parabola rule
D1
1/(2α)
Valensi
D2
[ln(1α)]1
α + (1α) ln(1α)
Jander
D2
(1 α)1/2[1 (1 α)1/2]1
[1 (1 α)1/2]2
0.12835 1.45581 0.95839 -0.76478 -0.04597 -0.99118 0.20975
0.02183
0.98255
0.13117
1.73626
0.95839
450
-0.81838
-0.24203
-0.99062
500
0.27666
0.06989
0.98367 0.9584
500 400
0.03349
0.51701
450
-0.21758
-0.12296
-0.99011
500 400
0.08535 2.06803
0.05482 0.13395
0.98423 0.9584
400
Jander
Jander
D3
D3 spherical symmetry
6(1 α)2/3[1 (1 α)1/3]1/2
/2(1 α)2/3[1 (1 α)1/3]1
3
[1 (1 α)1/3]1/2
[1 (1 α)1/3]2
450
-0.87033
-0.49182
-0.99011
500
0.34141
0.21928
0.98423 0.95839
/2[(1 α)1/3 1]1
1 2α/3 (1 α)2/3 0.13022
2.09536
450
-0.79975
-0.63035
-0.99082
500
0.25117
0.49818
0.98329
0.03491
0.26811
0.95841
400
400
GinstlingBrounshtein
AvramiErofeev
D3
A2
3
2(1α)[ln(1α)]1/2
[ln(1α)]1/2
450 500
-0.24533 0.12595
0.16947 -0.98903 0.30108 0.98495
0.02327
0.17874
450
-0.16355
0.11298
-0.98903
500
0.08396
0.20072
0.98495 0.95838
400
AvramiErofeev
A3
3(1α)[ln(1α)]2/3
[ln(1α)]2/3
0.06153
0.59724
450
-0.33824
0.03692
-0.992
500
0.06956
0.0187
0.98101
400
400 450
Mampel power
R1
α
0.95841
1
phase boundary reaction contraction cylinder
2(1 α)1/2
1 (1 α)1/2
phase boundary reaction contraction sphere
3(1 α)2/3
1 (1 α)1/3
500 400 450
0.06558 0.86813 0.95839 -0.40919 -0.12102 -0.99062 0.13833
0.03495
0.98367
0.06698
1.03401
0.9584
-0.43516
-0.24591
-0.99011
0.10964
0.98423
0.1707
500 13416
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Figure 3. Linear curves using FWO (a) and FRL (b) methods. α are 0.98, 0.95, 0.90, 0.80, 0.70, 0.60, 0.50, 0.40, 0.30, 0.20, 0.10, 0.05, and 0.02 from left to right (black slashes). j are 25, 20, 15, 10, and 5 K 3 min1 from top to bottom (color curves and points).
Table 4. Activation Energies and Frequency Factors Based on FWO and FRL Methods FWO method α
FRL method
E (kJ 3 mol1) log A (min1) E (kJ 3 mol1) log A (min1)
0.02
82.88 ( 18.60
7.9
94.31 ( 14.07
9.4
0.05
81.39 ( 8.36
7.3
92.03 ( 6.84
8.5
0.10
70.92 ( 4.48
5.5
77.36 ( 7.74
5.8
0.20
200.98 ( 9.94
13.4
254.38 ( 24.84
16.5
0.30 0.40
324.40 ( 70.38 303.57 ( 40.01
21.7 20.0
282.90 ( 47.53 266.54 ( 28.83
18.2 17.0
0.50
289.81 ( 26.74
18.8
261.98 ( 24.12
16.5
0.60
286.69 ( 21.02
18.4
269.86 ( 21.84
16.9
0.70
291.26 ( 18.59
18.6
281.86 ( 21.33
17.5
0.80
295.41 ( 17.01
18.6
277.09 ( 20.48
16.8
0.90
283.08 ( 12.73
17.4
278.97 ( 20.08
16.5
0.95
383.93 ( 34.88
23.2
368.35 ( 44.70
21.7
18.4
271.70
17.0
average 284.40
Figure 4. Linear curve of ln j vs 1/Tp for reaction system.
Table 5. E, A, and n of the Whole Reaction E (kJ 3 mol1)
log A (min1)
n
FWO method
284.40
18.4
1.01
FRL method Average
271.70 278.05
17.0 17.7
0.96 0.98
values of E and A by the FWO method are slightly higher than those by the FRL method. The first weight loss stage appears within α scope of 0.020.20. It belongs to the vaporization of water and presents small activation energy. Between 0.20 and 0.90, there exists the second weight loss, i.e., the main reaction stage, showing that the pyrolysis of the PAALi carbon skeleton and the formation of Li4Ti5O12 by solid phase reaction of lithium and titanium. The third weight loss stage appears within α scope of 0.900.95, i.e., the crystal perfect period during which the system needs more energy to promote Li4Ti5O12 crystal grain growth and crystal structure. Therefore, this stage shows large activation energy. To average for all E and A between 0.20 and 0.90, in the FWO method, the average activation energy and the frequency factor are 284.40 kJ mol1 and 2.51 1018 min1, respectively. However, if in the FRL method, the corresponding values are
271.70 kJ mol1 and 1.00 1017 min1, respectively. The whole reaction E and A are determined by the average values of FWO and FRL methods, as listed in Table 5. The results show that the values of the average activation energy and the frequency factor obtained by these two methods are very similar, and it is reasonable. 4.2.3. Calculation of n. According to Crane formula,30,31 d ln j E ¼ 2Tp dð1=Tp Þ nR
ð12Þ
When E/nR . 2Tp, the integral of Crane formula can be described as eq 13. ln j ¼
E 1 þ C nR Tp
ð13Þ
In terms of eq 13, the plot of ln j vs 1/Tp is a line and the slope is E/nR in a temperature range of 300650 °C. The results of n are shown in Figure 4. As shown in the figure, ln j vs 1/Tp has a good linear curve with a correlation coefficient of r2 > 0.97 and a slope of 33.87. Calculated values of n are shown in Table 5. Nonintegral values of n are obtained, indicating that solid-state reaction mechanism of Li4Ti5O12/C is complex, which can be reflected by several obvious weight loss peaks in the DTG curves. Meanwhile, the values of n are very similar in these two methods and it is reasonable that they are obtained by FWO and FRL 13417
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Table 6. Thermodynamic Parameters for the Kinetics of Li4Ti5O12/C by One-Step Solid-State Reaction j
ΔH#
ΔG#
ΔS#
Tp
(K 3 min1)
(kJ 3 mol1)
(kJ 3 mol1)
(J 3 mol1 3 K1)
(°C)
5 10
272.18 272.08
240.83 240.27
44.41 44.27
432.8 445.4
15
272.02
239.96
44.19
452.4
20
272.03
240.02
44.20
451.0
25
271.98
239.74
44.14
457.3
average
272.06
240.16
44.24
methods. These results are consistent with the conclusions of the average activation energy and the frequency factor. 4.2.4. Calculation of Thermodynamic Parameters. The thermodynamic parameters of activation can be calculated from the following equations.32 A expðE=RTÞ ¼
kB T expðΔG# =RTÞ h
ð14Þ
ΔH # ¼ E RT
ð15Þ
ΔG# ¼ ΔH # TΔS#
ð16Þ
where ΔG is Gibbs free energy of activation, ΔH is enthalpy of activation, ΔS# is entropy of activation, kB is Boltzmann constant, h is Planck constant, E is the activation energy of the whole reaction, and A is the frequency factor of the whole reaction. The values of ΔG#, ΔH#, and ΔS# at the peak temperature are listed in Table 6. As shown in the table, the positive values of ΔH# and ΔG# illustrate that considerable energy is required for the solid-state reaction. The positive values of ΔS# indicate that the solid-state reaction leads to disorder through the formation of Li4Ti5O12/C composites. The thermodynamic parameters of activation show that the formation of Li4Ti5O12/C composites is the entropy-driven one-step solid-state reaction mechanism. #
#
5. CONCLUSIONS The Li4Ti5O12/C solid-state reaction process was studied by TG-DTA at different scanning rates in a dynamic nitrogen atmosphere. The results demonstrate that the temperature ranges 400500 and 350650 °C correspond to the largest weight loss and the largest heat change, respectively. There exist two obvious endothermic peaks. One corresponds to the thermal decomposition of PAALi and solid-state reaction, the other is related to Li4Ti5O12 crystal particle growth and crystalline perfection. Using FWO and FRL methods, the Li4Ti5O12/C solid-state reaction mechanism can be properly described as the Jander equation, which is a three-dimensional diffusion with spherical symmetry, and the reaction mechanism functions are listed as follows: f(α) = 3/2(1 α)2/3[1 (1 α)1/3]1, G(α) = [1 (1 α) 1/3 2 ] . In the FWO method, the average activation energy and the frequency factor are 284.40 kJ mol1 and 2.51 1018 min1, respectively. However, if in the FRL method, the corresponding values are 271.70 kJ mol1 and 1.00 1017 min1, respectively. Moverover, the values of ΔH#, ΔG#, and ΔS# at the peak temperature are 272.06 kJ mol1, 240.16 kJ mol1, and 44.24 J mol1 K1, respectively. The thermodynamic parameters of activation indicate that the formation of Li4Ti5O12/C composites is the entropy-driven one-step solid-state reaction mechanism.
’ ASSOCIATED CONTENT
bS
Supporting Information. Figure S provides the XRD patterns of as-prepared Li4Ti5O12/C with heating rate of 10 K min1 at different temperatures. This information is available free of charge via the Internet at http://pubs.acs.org.
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