Carbonatation and Decarbonatation Kinetics in the La2O3-La2O2CO3 System under CO2 Gas Flows

Abstract

The carbonatation of La2O3 oxide and the decarbonatation of lanthanum carbonate phase La2O2CO3 are investigated using thermal and thermogravimetry analyses under CO2 gas flow. The initial phase La2O3 is first elaborated from pyrolysis of a LaOHCO3 precursor. Then, thermal and thermogravimetry analyses are carried out under CO2 flow, as temperature increases then decreases. The carbonatation kinetics of La2O3 is determined at three fixed temperatures. Electrical impedance spectroscopy is performed to determine the electrical responses associated with ionic mobilities and phase changes, in the temperature range 25 to ^{900[composite function]} C. The electrical conduction during heating under _CO2 gas flow should be linked to two regimes of ionic conduction of the carbonate ions. From these electrical measurements, the ionic mobility of carbonate ions ^{CO3 2-} is found to be close to 0.003 ·^{10-4 cm2 s-1 V-1} at ^{750[composite function]} C for the monoclinic La2O2CO3 phase.

Bahcine Bakiz 1, 2 and Frédéric Guinneton 1 and Madjid Arab 1 and Abdeljalil Benlhachemi 2 and Sylvie Villain 1 and Pierre Satre 1 and Jean-Raymond Gavarri 1

Recommended by Sridhar Komarneni

1, Institut Matériaux Microélectronique & Nanosciences de Provence, UMR CNRS 6242, Université du SUD Toulon-Var, BP 20132, 83957 La Garde Cedex, France
2, Laboratoire Matériaux et Environnement, Faculté des Sciences, Université Ibn Zohr, BP 8106, 80000 Agadir, Morocco

Received 26 May 2010; Revised 2 July 2010; Accepted 5 August 2010

1. Introduction

The lanthanum-based system La₂ O₃ -CO₂ -H₂ O is characterized by successive phases LaOHCO₃ , La₂ O₂ CO₃ , and La₂ O₃ , stable in various temperature ranges [1-9], depending on the partial pressures of CO₂ and H₂ O [10-15]. It has been established that the decomposition of the hydroxycarbonate LaOHCO₃ under air generally gives the La₂ O₂ CO₃ dioxycarbonate phase [6]. However, this last phase exists under three polymorphic structural varieties with tetragonal, hexagonal, and monoclinic crystal lattices [16-19]. The carbonatation of La₂ O₃ under pure CO₂ generally gives the main La₂ O₂ CO₃ phase; however, the obtained system can be complex, with presence of several polymorphic structures depending on the experimental synthesis conditions [10, 18, 20]. In their work concerning the TG analysis of the La₂ O₂ CO₃ phase under CO₂ gas, the authors of [2] showed that a small amount of La₂ O(CO₃ )₂ was probably formed as an additional phase. This phase was also studied by authors of [21].

In room conditions, the lanthanum oxide is highly sensitive to environmental water. In a previous study [22], we have established correlations between the thermal decomposition and the electrical responses of compacted pellets of this LaOHCO₃ phase, subjected to pyrolysis under air: we have shown that strong variations in conductances accompanied these phase changes. We have also established that these LaOHCO₃ , La₂ O₂ CO₃ , and La₂ O₃ phases have the capacity to convert carbon monoxide into CO₂ at relatively low temperature: at 200-^300° C, the L phase is a good catalyst converting CO into CO₂ , while it might be sensitive to CO₂ only above ^500° C.

In the present study, we focus our attention on phase changes during carbonatation and decarbonatation processes, respectively, of the La₂ O₃ phase and of the La₂ O₂ CO₃ phase. The main objective of this approach should reside in connecting the weight variations due to these phase transformations with electrical responses, in order to appreciate their potential efficiency in gas sensing devices. These correlations between mass losses and electrical responses are not known, and they could deliver interesting information on the electrical sensitivity of such systems.

2. Experimental Details

The LaOHCO₃ hydroxycarbonate was first prepared via a specific route [22, 23] based on a thermal treatment at ^80° C of three aqueous solutions of La(NO₃ )₃ · 6H₂ O, urea CO(NH₂ )₂ , and polyvinyl-pyrrolydine (PVP) polymer. The La₂ O₃ oxide was obtained by pyrolysis of this LaOHCO₃ precursor.

The various chemical steps can be summarized as follows:

(i) First initial decomposition processes under air as temperature increases (25-^1200° C): [figure omitted; refer to PDF] [figure omitted; refer to PDF]

(ii) Carbonatation and decarbonatation under pure CO₂ as temperature increases (25-^1200° C): [figure omitted; refer to PDF] [figure omitted; refer to PDF]

(iii): Recarbonatation under pure CO₂ as temperature decreases (1200 to ^25° C):

[figure omitted; refer to PDF]

In the previous equations, in bracket [1 to 3] we have designated phases obtained after a transformation process (decomposition, carbonatation, and decarbonatation). Theses phases have not the same characteristics (various morphologies and specific surfaces).

The polycrystalline samples were systematically analyzed by X-ray diffraction, using a D5000 Siemens-Bruker diffractometer, equipped with a copper X-ray source (wavelength λ=1.54 ^10-10 m), and with a monochromator eliminating Kβ radiation. The experiments were carried out using classical θ-2θ configuration.

Thermal and Thermogravimetric analyses (DTA-TG) were carried out using SETARAM DSC 92 equipment, with a thermal rate of ^10° C/minute, under CO₂ pure gas (rate of flow of 33 cm³ · s^-1 ).

Electrical measurements under CO₂ gas flow were performed using a Solartron electrical impedance spectrometer working with a maximal tension of 1 V, in the frequency range 100 to 10⁷ Hz. A reactive homemade cell was used to perform experiments under various gas flows (air, CO₂ ) at various temperatures ranging between 25 and ^900° C. The spectrometer delivers Nyquist representations of the resulting impedances recorded at fixed temperatures: the resistance value is classically obtained by extrapolation of the experimental Nyquist circles, and using electrical equivalent circuits (parallel R -C circuits) generated by the software. We have selected specific electrical circuits with a resistance (R ) parallel to a constant phase element CPE=^{(jC* ω)n} where the exponent n is comprised between 1 and 0, and ^C* is a term similar to a capacitance for n=1 (the unit of ^C* depends on n ).

To obtain electrical analyses of sample surfaces reacting with gas flows, the powder samples were first compacted under a pressure of 5 kbar in a cylindrical cell. Then, the obtained cylindrical pellet was cut in form of a rectangular plate, with platinum electrodes fixed on two parallel faces (dimensions 2.3 × 8 mm). The distance between the electrodes is 9 mm. This configuration (adapted to the reactive cell) allows a determination of the electrical properties of a significant material surface exposed to gas action. In a later step, these results might be used to test a hypothetical gas sensor sensitive to CO₂ .

3. Results

3.1. Carbonatation-Decarbonatation Processes

3.1.1. Heating Process under CO₂ Flow

The La₂ O₃ sample, initially obtained from thermal decomposition of LaOHCO₃ , has been subjected to thermal and thermogravimetry analyses under CO₂ gas flow, with temperature increasing from 25 to ^1200° C. The resulting TG-DTA curves are reported on Figure 1. A strong exothermic DTA peak is observed at ^525° C: it is related to the carbonatation of La₂ O₃ with formation of the La₂ O₂ CO₃ phase. Then, at ^960° C, we observe an endothermic feature corresponding with the decomposition of the carbonate phase. Above ^980° C the La₂ O₃ phase stabilizes. A small endothermic feature is observed at ^375° C: it might be associated with a partial dehydration of the sample due to the high sensitivity to environmental water of La₂ O₃ . The progressive mass evolution observed in the TG curve of Figure 1, as temperature increases, is directly associated with the classical buoyancy. A similar effect will be observed during the cooling process.

Figure 1: Weight gain associated with carbonatation of La₂ O₃ (first step): exothermic peak at ^520° C linked with weight gain due to formation of the monoclinic La₂ O₂ CO₃ phase; endothermic peak due to decarbonatation of La₂ O₂ CO₃ phase and formation of final La₂ O₃ (second step).

[figure omitted; refer to PDF]

3.1.2. Cooling Experiments under CO₂ Flow

Using cooling experiments, we have analyzed the carbonatation of La₂ O₃ from ^1200° C to ^25° C. The results are represented on Figure 2. The formation of La₂ O₂ CO₃ starts from ^820° C and is maximum at ^750° C. The exothermic peak associated with the crystallization of La₂ O₂ CO₃ carbonate is observed at ^790° C. This temperature of carbonatation is strongly different from the one obtained during the heating process.

Figure 2: Evolution of La₂ O₃ weight during cooling process, under CO₂ flow: formation of La₂ O₂ CO₃ phase (exothermic peak) then relative stabilization of this phase as temperature decreases.

[figure omitted; refer to PDF]

At each step involving a stabilized phase, we have carried out X-ray diffraction analyses to identify the obtained phases. We have confirmed that, in the case of thermal decomposition under air of LaOHCO₃ phase, two different tetragonal and hexagonal La₂ O₂ CO₃ structures are simultaneously observed. In the case of carbonatation of the La₂ O₃ phase in the temperature range 500 to ^700° C, we observe the formation of the La₂ O₂ CO₃ phase. The La₂ O(CO₃ )₂ phase was not observed in our experiments. This fact was previously reported by other authors [10, 18]. On Figure 3, we have reported the X-ray diffraction pattern characteristic of the monoclinic La₂ O₂ CO₃ phase heated at ^520° C under CO₂ flow, during 3 hours. The refined cell parameters are a=0.4073±0.0003 nm; b=1.3503±0.0008 nm; c=0.4079±0.0005 nm; β=^90.89° . In the pattern, a weak trace of the hexagonal phase (noted as *) is observed.

Figure 3: X-ray diffraction pattern of La₂ O₂ CO₃ (monoclinic) obtained by heating La₂ O₃ at ^520° C under CO₂ flow. Trace of hexagonal phase (noted *).

[figure omitted; refer to PDF]

3.2. Kinetics Study of Carbonatation of La₂ O₃ at Fixed Temperatures

We have performed a weight analysis of the La₂ O₃ powder, obtained from the thermal decomposition of the initial LaOHCO₃ phase, under CO₂ gas flow at three constant temperatures. The CO₂ gas flow rate was 33 cm³ · s^-1 . In the SETARAM equipment, a fast temperature increase is first applied to the sample, and then, the temperatures are successively fixed to 450, 480, and ^500° C. The three initial masses of La₂ O₃ are successively (at T=450 , 480, and ^500° C) _m0 =74.36 mg, 70.73 mg, and 39.26 mg. The data evolutions have been interpreted in terms of an elemental Avrami's model [24] (using a single mechanism approach): [figure omitted; refer to PDF]

(i) t is the reaction time;

(ii) Δ_m0 is the limit mass of CO₂ involved in the carbonate formation La₂ O₂ CO₃ from a mass m₀ of La₂ O₃ ;

(iii): Δm is the CO₂ mass having reacted with La₂ O₃ at the time t ;

(iv) k is a kinetics parameter depending of temperature;

(v) p is the exponent characteristic of the reaction mechanism (p>2 for complex mechanisms, p<1 , for example, for mechanisms involving diffusion barriers).

To test the degree of validity of this Avrami's model, we have reported the function Y versus ln (t) on Figure 4: [figure omitted; refer to PDF] For a single crystal growth mechanism, the variation of Y versus ln (t) should have been linear. Presently, the representation of Figure 4 is not linear: this should be mainly due to the existence of at least two different crystal growth mechanisms, with two periods of mass gain corresponding to two behaviors.

Figure 4: Test of Avrami's model validity: the representation of Y=ln [-ln (Δ_m0 -Δm)/Δ_m0 ] versus ln (t) (t in mn) shows that two main types of behaviors can be observed as time increases.

[figure omitted; refer to PDF]

In Table 1, we have reported the values of the kinetics parameters _k1 and _k2 and exponents _p1 and _p2 , corresponding with the two different behaviors in which a linear correlation might be observed. The parameters _k1 , _p1 are relative to the first period depending on temperature, and the parameters _k2 , _p2 are relative to the second period. The _k1 and _k2 are thermally activated with activation energies of, respectively, 7.6 and 2.8 eV. The _p1 exponent is quasi-constant, while the _p2 exponent is close to 1 at ^450° C and becomes very weak at higher temperatures.

Table 1: Parameters extracted from Avrami's model: _k1 and _k2 kinetics parameters and _p1 and _p2 exponents, respectively, associated with the fast and slow regimes (1st and 2nd periods).

	1st period	2nd period
T (°C)	Time range (min)	k1	p1	Time range (min)	k2	p2
450	45[arrow right]112	1.5 10-7	3.3	167[arrow right]320	0.02	0.9
480	6[arrow right]17	1.4 10-5	3.9	30[arrow right]90	0.54	0.3
500	3[arrow right]7	1.2 10-3	3.4	30[arrow right]90	1.10	0.2
	Activation energies E1 (k1 )	7.6 (eV)	Activation energies E2 (k2 )	2.8 (eV)

The first growth regime should be associated with a fast carbonatation of grain surfaces associated with complex diffusion mechanisms. During this period, a carbonate shell enveloping oxide grains probably should be formed. The second growth regime should be associated with reaction and diffusion in grain cores, with a decrease of the reaction rate due to the carbonate shell: the resulting slow diffusion regime could govern the global reaction speed.

3.3. Electrical Analyses under CO₂ Gas Flows

To correlate the phase modifications to electrical behaviors, we have analyzed compacted powder samples in the electrical cell. In this experiment, a rectangular compacted sample resulting from the total decomposition of the initial LaOHCO₃ sample has been subjected to a progressive heating, under pure CO₂ gas flow. Between 600 and ^700° C, carbonatation occurs, thus involving a strong increase in conductance mainly due to the ionic mobility of ^{_CO3 2-} carbonate ions. Then, above ^750° C decarbonation occurs, involving a decrease of conductance due to ^{_CO3 2-} carbonate ions elimination and formation of La₂ O₃ . This oxide should be formed at ^950° C.

On Figure 5, we have reported the ln (Σ) values versus temperature (total Σ values). We observe a strong decrease of ln (Σ) between ^450° C and ^750° C: the carbonatation of La₂ O₃ should start from ^450° C, with a first regime up to ^600° C and a second regime up to ^750° C. Two activation energies for the conduction behavior can be determined: 2.5 (first regime) and 1.4 eV (second regime). In this carbonatation domain, the ionic conduction plays a major role with mobile species ^{_CO3 2-} .

Figure 5: Evolution of ln (Σ) versus 10³ /T of initial La₂ O₃ sample during its carbonatation and decarbonatation, in the temperature range 300 to ^950° C, under a constant CO₂ gas flow: (a) starting step of carbonatation between 450 and ^750° C with two conduction regimes; (b) decarbonatation step above ^750° C.

[figure omitted; refer to PDF]

Above ^750° C, we observe a strong decrease in the ln (Σ) values: in this temperature range, decarbonatation occurs in a continuous way, with the elimination of ^{_CO3 2-} ions. As La₂ O₃ phase stabilizes, the resistance reaches a stabilized value.

Using the observed value Σ=1.7·^{10-4 Ω-1} at ^750° C (on Figure 5: maximum value of Σ just before decomposition), and considering as negligible the conductance of La₂ O₃ at the same temperature (close to 10^-8 to 10^{-9 Ω-1} at ^750° C), we have evaluated an ionic conductance due to ^{_CO3 2-} ions to ΔΣ=1.7·^{10-4 Ω-1} . From this evaluation of ΔΣ , we have determined the order of magnitude of the carbonate ion mobility u (^{_CO3 2-} ) at ^750° C. Other values could be derived from the data obtained in the temperature range 450 to ^700° C. The concentration of carbonate ions _Cion has been calculated from the effective density of the sample μ=5.1 g· cm^-3 (for a theoretical crystal density of 6.51 g.cm^-3 ) and using the sample volume V=0.1656 cm³ . A value of _Cion =0.0155 mol· cm^-3 has been obtained. To determine the mobility, we have used the classical relations: [figure omitted; refer to PDF] where σ is the conductivity, S and L are the surface and separation distance of the two electrodes, and where Q=193 000 C· mol^-1 . The relation giving the conductivity assumes an activity coefficient of 1: it only delivers an order of magnitude for the mobility.

We have obtained an order of magnitude of u (^{_CO3 2-} ) = (0.003±0.001 ) 10^-4 cm² s^-1 V^-1 for a carbonate ion moving at ^750° C mainly along grain boundaries (or grain surfaces), and partly in the grain cores. This relatively high mobility can be associated with the activation energy of 1.4 eV (in the temperature range 600 to ^750° C) as calculated above.

4. Discussion-Conclusions

The carbonatation kinetics of La₂ O₃ has been determined at various temperatures. In the case of mass gain analyses, an elemental Avrami's approach has allowed determining a complex two-step mechanism of growth: (i) a fast surface carbonatation with carbonate shell formation and (ii) a diffusion mechanism in grain cores with slower kinetics. The electrical analyses argue in favor of two different conduction mechanisms: during carbonatation at increasing temperature, the first activation energy (2.5 eV) should be associated with ionic conduction at grain surfaces, and the second activation energy (1.4 eV) should due to an increasing contribution of the conduction in the bulk. Correlatively, it should be remarked that, in thermal analyses, the stability range is observed from 500 to ^850° C, while in electrical analyses, this stability range is observed from 500 to ^750° C. This can be explained by the two different heating kinetics conditions used in the two experiments.

Finally, we observe a relatively high ionic mobility mainly due to the ^{_CO3 2-} ions in La₂ O₂ CO₃ at ^750° C. In our evaluation, we have neglected the ionic conduction of oxygen ions.

It should be concluded that these phase modifications associated with high ionic conduction might be used as electrical sensitive material to detect CO₂ , provide temperatures that could be fixed close to 400-^550° C (carbonatation of La₂ O₃ phase) and ^750° C to restore the initial La₂ O₃ phase.

Acknowledgments

The authors gratefully acknowledge the Provence-Alpes-Côte d'Azur Regional Council, the General Council of Var, and the agglomeration community of Toulon Provence Mediterranean for their helpful financial supports. This paper was developed in the general framework of ARCUS CERES project (2008-2010).

References

[1] E. Béchade, I. Julien, T. Iwata, O. Masson, P. Thomas, E. Champion, K. Fukuda, "Synthesis of lanthanum silicate oxyapatite materials as a solid oxide fuel cell electrolyte," Journal of the European Ceramic Society , vol. 28, no. 14, pp. 2717-2724, 2008.

[2] A. N. Shirsat, M. Ali, K. N. G. Kaimal, S. R. Bharadwaj, D. Das, "Thermochemistry of _La2O2CO3 decomposition," Thermochimica Acta , vol. 399, no. 1-2, pp. 167-170, 2003.

[3] S. Valange, A. Beauchaud, J. Barrault, Z. Gabelica, M. Daturi, F. Can, "Lanthanum oxides for the selective synthesis of phytosterol esters: correlation between catalytic and acid-base properties," Journal of Catalysis , vol. 251, no. 1, pp. 113-122, 2007.

[4] O. K. Nikol'skaya, L. N. Dem'yanets, "Hydrothermal crystallization in the systems _{La2(CO3 )3} ·6_H2 O -Ca_CO3 (Ba_CO3 )-R-_H2 O (R=_Na2 , _CO3K2CO3 , NaH_CO3 , KH_CO3 , NaCl, _NH4 Cl, CO_{(NH2 )2} )," Inorganic Materials , vol. 41, no. 11, pp. 1206-1212, 2005.

[5] M. L. Panchula, M. Akinc, "Morphology of lanthanum carbonate particles prepared by homogeneous precipitation," Journal of the European Ceramic Society , vol. 16, no. 8, pp. 833-841, 1996.

[6] H. Wakita, S. Kinoshita, "A synthetic study of the solid solutions in the systems and _{La2(CH3 )3} ·8_H2 O -_{Ce2(CO3 )3} ·8_H2 O and La(OH)_CO3 -Ce(OH)_CO3 ," Bulletin of the Chemical Society of Japan , vol. 52, pp. 428-432, 1979.

[7] G. A. M. Hussein, H. M. Ismail, "Characterization of lanthanum oxide formed as a final decomposition product of lanthanum acetylacetonate: thermoanalytical, spectroscopic and microscopic studies," Powder Technology , vol. 84, no. 2, pp. 185-190, 1995.

[8] B. Bakiz, K. Ouzaouit, A. Benlhachemi, "Multiphase lanthanum hydroxycarbonates and langasite ceramics for gas sensors," Physical and Chemical News , vol. 41, pp. 55-60, 2008.

[9] B. A. A. Balboul, A. M. El-Roudi, E. Samir, A. G. Othman, "Non-isothermal studies of the decomposition course of lanthanum oxalate decahydrate," Thermochimica Acta , vol. 387, no. 2, pp. 109-114, 2002.

[10] T. J. Toops, A. B. Walters, M. A. Vannice, "The effect of _CO2 and _H2 O on the kinetics of NO reduction by _CH4 over a _La2O3 /γ_-Al2O3 catalyst," Journal of Catalysis , vol. 214, no. 2, pp. 292-307, 2003.

[11] Y. Cui, H. Zhang, H. Xu, W. Li, "The _CO2 reforming of _CH4 over Ni/_La2O3 /α -_Al2O3 catalysts: the effect of _La2O3 contents on the kinetic performance," Applied Catalysis A , vol. 331, no. 1, pp. 60-69, 2007.

[12] J. Gao, Z. Hou, J. Guo, Y. Zhu, X. Zheng, "Catalytic conversion of methane and _CO2 to synthesis gas over a _La2O3 -modified _SiO2 supported Ni catalyst in fluidized-bed reactor," Catalysis Today , vol. 131, no. 1-4, pp. 278-284, 2008.

[13] T. Esaka, K. Motoike, " _CO2 absorption and desorption of _Bi2O3 -_La2O3 powders prepared by mechanical synthesis," Materials Research Bulletin , vol. 39, no. 11, pp. 1581-1587, 2004.

[14] T. Esaka, K. Motoike, "Absorption and desorption of carbon dioxide in the rare earth oxide-doped _Bi2O3 powder," Journal of Alloys and Compounds , vol. 408-412, pp. 480-483, 2006.

[15] H. Chen, H. Yu, F. Peng, H. Wang, J. Yang, M. Pan, "Efficient and stable oxidative steam reforming of ethanol for hydrogen production: effect of in situ dispersion of Ir over Ir/_La2O3 ," Journal of Catalysis , vol. 269, no. 2, pp. 281-290, 2010., [email protected]; [email protected]

[16] A. Olafsen, A.-K. Larsson, H. Fjellvåg, B. C. Hauback, "On the crystal structure of _Ln2O2CO3 II (Ln=La and Nd)," Journal of Solid State Chemistry , vol. 158, no. 1, pp. 14-24, 2001.

[17] K. Koyabu, T. Masui, S. Tamura, N. Imanaka, "Synthesis of a new phosphor based on rare earth oxycarbonate," Journal of Alloys and Compounds , vol. 408-412, pp. 867-870, 2006.

[18] H. Chen, H. Yu, F. Peng, H. Wang, J. Yang, M. Pan, "Efficient and stable oxidative steam reforming of ethanol for hydrogen production: effect of in situ dispersion of Ir over Ir/_La2O3 ," Journal of Catalysis , vol. 269, no. 2, pp. 281-290, 2010., [email protected]; [email protected]

[19] K. Efimov, M. Arnold, J. Martynczuk, A. Feldhoff, "Crystalline intermediate phases in the sol-gel-based synthesis of _La2NiO4+δ ," Journal of the American Ceramic Society , vol. 92, no. 4, pp. 876-880, 2009.

[20] T. J. Toops, A. B. Walters, M. A. Vannice, "The effect of _CO2 , _H2 O and _SO2 on the kinetics of NO reduction by _CH4 over _La2O3 ," Applied Catalysis B , vol. 38, no. 3, pp. 183-199, 2002.

[21] S. R. Dharwadkar, M. S. Kumbhar, M. S. Chandrasekharaiah, M. D. Karkhanavala, "Thermal decomposition of lanthanum formate," Journal of Inorganic and Nuclear Chemistry , vol. 42, no. 11, pp. 1621-1625, 1980.

[22] B. Bakiz, F. Guinneton, M. Arab, S. Villain, A. Benlhachemi, J. R. Gavarri, "Temperature dependent electrical properties and catalytic activities of _La2O3 -_CO2 -_H2 O phase system," Advances in Materials Science and Engineering , vol. 2009, 2009., [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]

[23] B. Bakiz, F. Guinneton, J.-P. Dallas, S. Villain, J.-R. Gavarri, "From cerium oxycarbonate to nanostructured ceria: relations between synthesis, thermal process and morphologies," Journal of Crystal Growth , vol. 310, no. 12, pp. 3055-3061, 2008.

[24] E. C. N. Lopes, F. S. C. Dos Anjos, E. F. S. Vieira, A. R. Cestari, "An alternative Avrami equation to evaluate kinetic parameters of the interaction of Hg(II) with thin chitosan membranes," Journal of Colloid and Interface Science , vol. 263, no. 2, pp. 542-547, 2003.