Tailoring the Kinetic Behavior of Hydride Forming Materials for Hydrogen Storage DOI: http://dx.doi.org/10.5772/intechopen.82433

contracting models, (3) diffusion models, and (4) autocatalytic models. In Table 3, the integral form of the gas-solid kinetic models is described. There are more gas-solid models, but it is not the aim to perform exhaustive description of them.

Among the steps involved in the hydrogenation/dehydrogenation processes (Figure 5), it is usually found a slowest one, which limits the overall reaction rate of the process. Thus, the slowest step is commonly called "rate-limiting step." The determination of the rate-limiting step depends on the kind of hydride and the experimental conditions. Identifying the rate-limiting step requires the application of gas-solid models shown in Table 3. The determination of the rate-limiting step is carried out by measuring kinetic curves under constant temperature and pressure, as the ones shown in Figure 4. Once, the hydrogen uptake and release against time is expressed in terms of hydrogen fraction, α, the integral models can be applied to build a graph of g(α) as function of time.

A first approach to study which process limits the hydrogenation/dehydrogenation kinetic behavior is to find which model has the best linear fitting of the integral



dα

Steps involved in the hydrogenation and dehydrogenation processes under dynamic conditions.

sections, each dependence will be explained.

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material G(α)

for the evaluation of G(α):

expressed:

134

Figure 5.

where the overall reaction rate, dα/dt (α = hydrogen fraction and t = time) is function of the temperature, K(T), of the pressure, F(P), and of the intrinsic and morphological changes of the material occurring during the hydrogenation/dehydrogenation process, G(α), which is function of the hydrogen fraction (α). As shown by Eq. (5), the dependencies of the rate on the K(T), F(P), and G(α) can be investigated independently by keeping two of them constant. In the following

At constant temperature and pressure, the reaction rate depends on intrinsic factors and morphological changes of the solid (defects, crystalline structure, size and morphology of the particles, etc.), i.e., G(α) [34, 35]. Thus, Eq. (5) can be

k is the kinetic constant and it contains the temperature and pressure dependences as seen in Eq. (7). Reordering Eq. (6), it is possible to obtain an expression

dα

where g(α) is the integral form of the gas-solid kinetic models for the material's changes. The gas-solid kinetic models describe different physical phenomena. There are four main sets of models: (1) nucleation and growth models, (2) geometrical

3.2 Dependence on the intrinsic and morphological changes of the

dα

gð Þ¼ α

ðα 0

dt <sup>¼</sup> K Tð Þ� F Pð Þ� <sup>G</sup>ð Þ <sup>α</sup> , (5)

dt <sup>¼</sup> <sup>k</sup> � <sup>G</sup>ð Þ <sup>α</sup> (6)

<sup>G</sup>ð Þ <sup>α</sup> <sup>¼</sup> <sup>k</sup> � t, (8)

k ¼ K Tð Þ� F Pð Þ, (7)

form, g(α), of the solid-state models against time. First, the hydrogen fraction range considered for the fitting is usually from 0.1 to 0.9. On one hand, at the beginning of the process the initial points involve a high degree of uncertainty, mainly for fast rates in the range of seconds. On the other hand, the final stage of the process, reaching the saturation of the material, does not play any role in the determination of the rate-limiting step. As an example, Figure 6 exhibits g(α) generated from each model, Table 3, as a function of time in the range of α between 0.10 and 0.90 for the hydrogenation and dehydrogenation process shown in Figure 4. Then, a liner fitting was performed in each curve.

As seen, the simple linear fitting of the integral form of the solid state-models does not provide a clear clue about the rate-limiting step for the hydrogenation and dehydrogenation rates. There are several models with suitable fitting goodness (highlighted in bold). For these reason, it is possible to apply other procedure such as the reduced time method proposed by Sharp and Jones [37, 38]. By applying this method, a plot for the theoretical reduced time (t/t0.5)theoretical as a function of the experimental reduced time (t/t0.5)experimental is built as shown in Figure 7. The (t/t0.5)theoretical is obtained by expressing the equation of the integral form of solid-state models in terms of the time at α = 0.5, i.e., t0.5. As an example, for the case of the integral form of the solid-gas model JMA with n = 1, the (t/t0.5)theoretical is obtained as follows:

$$[-\ln\left(1-a\right)] = k \times t \tag{9}$$

the reduced time method is the one showing linear fitting goodness and slope equal

Reduced time method: Plots (t/t0.5)theoretical vs. (t/t0.5)experimental and fitting for (A) hydrogenation

Tailoring the Kinetic Behavior of Hydride Forming Materials for Hydrogen Storage

Figure 7 shows that for the case of the hydrogenation rate the model JMA, n = 1, Table 3, provides the best linear fitting (slope = 1, intercept = 0 and goodness ≈ 1). For this reason, the overall reaction rate is limited by one-dimensional growth with interface-controlled reaction rate. This rate-limiting step is related to the step (f) described in Figure 5. For the dehydrogenation rate, the rate-limiting step is the two-dimensional growth of contracting volume with constant interface rate (CA),

In fact, the kinetic analysis based on the solid-state models should be also based on additional experimental evidence obtained from the material science characterizations for morphological and microstructural changes before and after and upon

Under constant pressure, the hydrogenation/dehydrogenation reactions speed up as the temperature increases and this temperature dependence is described by the Arrhenius expression. The dependence of the hydrogenation/dehydrogenation

where A is the pre-exponential factor or frequency factor and represents the frequency of collisions between reactant molecules, Ea is the apparent activation energy, which is the energy required to start the reaction, R is the gas constant, and

The activation energy is a relevant kinetic parameter to evaluate the kinetic performance of a hydride forming material. Isothermal and non-isothermal measurements can be performed to obtain the activation energy. Two procedures are commonly used. The first consists in performing isothermal measurements at different temperature and constant pressure, such as the ones shown in Figure 4. Then, by the application of the gas-solid models, it is possible to calculate the kinetic constant "k," once identified the rate-limiting step as explained in Section 3.1. As expressed in Eq. (7), k depends on K(T) and F(P). Considering isobaric measure-

ments, k is only function of K(T) following the Arrhenius expression:

K Tð Þ¼ A � exp ð Þ �Ea=RT , (12)

k ¼ A � exp ð Þ �Ea=RT , (13)

to 1 and intercept equal to 0.

and (B) dehydrogenation rates.

Figure 7.

related to the step (a) shown in Figure 5.

DOI: http://dx.doi.org/10.5772/intechopen.82433

3.3 Temperature dependence K(T)

T is the absolute temperature.

137

hydrogenation and dehydrogenation kinetic processes.

reaction rates on the temperature follows Eq. (12):

$$[-\ln\left(\mathbf{1} - \mathbf{0.5}\right)] = k \times t\_{0.5} \tag{10}$$

The right term of Eq. (10) is equal to 0.69. Replacing 0.69 in Eq. (10) and dividing (9) over (10), Eq. (11) provides the (t/t0.5)theoretical

$$\frac{\left[-\ln\left(1-a\right)\right]}{0.69} = \left[\frac{t}{t\_{0.5}}\right]\_{theoretical} \tag{11}$$

The (t/t0.5)experimental is directly obtained from the experimental results, Figure 4, by dividing the time during the measurement (t) over the time at α = 0.5. The rate-limiting step is determined by three parameters, i.e., the fitting goodness, slope, and intercept of the linear fitting. This represents an advantage in comparison with the simple linear fitting of the integral form of the solid-state models, with just the fitting goodness as one decision parameter. The best fitting obtained from

#### Figure 6.

Application and fitting of the integral form of the solid-state models (Table 3) to the hydrogenation/ dehydrogenation rates shown in Figure 4.

Tailoring the Kinetic Behavior of Hydride Forming Materials for Hydrogen Storage DOI: http://dx.doi.org/10.5772/intechopen.82433

#### Figure 7.

form, g(α), of the solid-state models against time. First, the hydrogen fraction range considered for the fitting is usually from 0.1 to 0.9. On one hand, at the beginning of the process the initial points involve a high degree of uncertainty, mainly for fast rates in the range of seconds. On the other hand, the final stage of the process, reaching the saturation of the material, does not play any role in the determination of the rate-limiting step. As an example, Figure 6 exhibits g(α) generated from each model, Table 3, as a function of time in the range of α between 0.10 and 0.90 for the hydrogenation and dehydrogenation process shown in Figure 4. Then, a liner

As seen, the simple linear fitting of the integral form of the solid state-models does not provide a clear clue about the rate-limiting step for the hydrogenation and dehydrogenation rates. There are several models with suitable fitting goodness (highlighted in bold). For these reason, it is possible to apply other procedure such as the reduced time method proposed by Sharp and Jones [37, 38]. By applying this method, a plot for the theoretical reduced time (t/t0.5)theoretical as a function of the experimental reduced time (t/t0.5)experimental is built as shown in Figure 7. The (t/t0.5)theoretical is obtained by expressing the equation of the integral form of solid-state models in terms of the time at α = 0.5, i.e., t0.5. As an example, for the case of the integral form of the solid-gas model JMA with n = 1, the (t/t0.5)theoret-

The right term of Eq. (10) is equal to 0.69. Replacing 0.69 in Eq. (10) and

<sup>0</sup>:<sup>69</sup> <sup>¼</sup> <sup>t</sup>

Application and fitting of the integral form of the solid-state models (Table 3) to the hydrogenation/

The (t/t0.5)experimental is directly obtained from the experimental results, Figure 4, by dividing the time during the measurement (t) over the time at α = 0.5. The rate-limiting step is determined by three parameters, i.e., the fitting goodness, slope, and intercept of the linear fitting. This represents an advantage in comparison with the simple linear fitting of the integral form of the solid-state models, with just the fitting goodness as one decision parameter. The best fitting obtained from

t0:<sup>5</sup> 

theoretical

dividing (9) over (10), Eq. (11) provides the (t/t0.5)theoretical

½ � �ln 1ð Þ � α

½ �¼ �ln 1ð Þ � α k � t (9) ½�ln 1ð Þ � 0:5 � ¼ k � t0:<sup>5</sup> (10)

(11)

fitting was performed in each curve.

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ical is obtained as follows:

Figure 6.

136

dehydrogenation rates shown in Figure 4.

Reduced time method: Plots (t/t0.5)theoretical vs. (t/t0.5)experimental and fitting for (A) hydrogenation and (B) dehydrogenation rates.

the reduced time method is the one showing linear fitting goodness and slope equal to 1 and intercept equal to 0.

Figure 7 shows that for the case of the hydrogenation rate the model JMA, n = 1, Table 3, provides the best linear fitting (slope = 1, intercept = 0 and goodness ≈ 1). For this reason, the overall reaction rate is limited by one-dimensional growth with interface-controlled reaction rate. This rate-limiting step is related to the step (f) described in Figure 5. For the dehydrogenation rate, the rate-limiting step is the two-dimensional growth of contracting volume with constant interface rate (CA), related to the step (a) shown in Figure 5.

In fact, the kinetic analysis based on the solid-state models should be also based on additional experimental evidence obtained from the material science characterizations for morphological and microstructural changes before and after and upon hydrogenation and dehydrogenation kinetic processes.

#### 3.3 Temperature dependence K(T)

Under constant pressure, the hydrogenation/dehydrogenation reactions speed up as the temperature increases and this temperature dependence is described by the Arrhenius expression. The dependence of the hydrogenation/dehydrogenation reaction rates on the temperature follows Eq. (12):

$$K(T) = \mathbf{A} \times \exp\left(\mathrm{-Ea}/\_{\mathrm{RT}}\right),\tag{12}$$

where A is the pre-exponential factor or frequency factor and represents the frequency of collisions between reactant molecules, Ea is the apparent activation energy, which is the energy required to start the reaction, R is the gas constant, and T is the absolute temperature.

The activation energy is a relevant kinetic parameter to evaluate the kinetic performance of a hydride forming material. Isothermal and non-isothermal measurements can be performed to obtain the activation energy. Two procedures are commonly used. The first consists in performing isothermal measurements at different temperature and constant pressure, such as the ones shown in Figure 4. Then, by the application of the gas-solid models, it is possible to calculate the kinetic constant "k," once identified the rate-limiting step as explained in Section 3.1. As expressed in Eq. (7), k depends on K(T) and F(P). Considering isobaric measurements, k is only function of K(T) following the Arrhenius expression:

k ¼ A � exp ð Þ �Ea=RT , (13)

Applying natural logarithm to Eq. (13):

$$
\ln k = \ln A + \ln \left( {}^{-Ea} \!/ {}^{/}\!/ \!/ \!/ \!/ \!/ \!/ \!/ \!/ \!/ \!/ \!/ \!/ \!/ \!/ \!/ \!/ \!/ \!/ \!/} \tag{14}$$

Plotting lnk against 1/T, the slope of the linear fitting provides the Ea in kJ/mol H2 and the intercept the frequency factor A, as seen in Figure 8A. The second method involves non-isothermal measurements done via calorimetric methods such as scanning differential calorimetry (DSC) and differential temperature analysis (DTA) [39]. This method is the Kissinger one [40]. The method consists in determining temperature maxima (Tm) of the exothermal (hydrogenation)/ endothermal (dehydrogenation) events obtained from the DSC or DTA measurements at different constant heating rate (ø). The change of Tm with ø is directly related to the nature of the reaction.

At a constant heating rate (ø), constant pressure (F(P)) and considering k has an Arrhenius dependence on the temperature, Eq. (13), Eq. (5) can be expressed:

$$\frac{da}{dt} = \frac{A}{dt} \times \exp\left(\,^{-Ea}\!/\_{RT}\right) \times G(a) \tag{15}$$

Assuming that the fraction at the peak maximum and G(α) are independent on ø, reordering and applying natural logarithm to expression (15), it is possible to write:

$$\ln\left[\frac{\mathcal{Q}}{T\_m^2}\right] = -\frac{Ea}{RT\_m} + \ln\left[\frac{AR}{Ea}\right] + \mathcal{C} \tag{16}$$

interface and diffusion controlled processes are the most common ones for the

Tailoring the Kinetic Behavior of Hydride Forming Materials for Hydrogen Storage

DOI: http://dx.doi.org/10.5772/intechopen.82433

It is important to point out that the isothermal method for Ea requires the knowledge of the rate-limiting step, thus it depends on the solid-state model. On the contrary, the non-isothermal method is independent on the solid-state model, which represents an advantage over the first method. However, for the kinetic modeling of the reaction rates, as indicated by Eq. (5), the determination of G(α) is demanding. For this reason, at the time proceed with kinetic modeling, or just determine Ea via measurements of the hydrogenation or dehydrogenation rates at isothermal conditions, it is necessary to work far away from the equilibrium pressures to avoid the effects of the driving force, i.e., the F(P). In fact, rightly speaking, the K(T) must be corrected by the effects of the driving force by determining F(P), since the Ea values obtained after the correction are different from the one without; as an example we can mention the works published by Fenández et al. [51] and Jepsen et al. [52]. In the next subsection, then, the F(P) dependency is described.

hydride compounds formation/decomposition [41–51].

Activation energies, reaction mechanisms, and rate-limiting steps.

Table 4.

139

Figure 8B shows the DSC curves at different ø for a dehydrogenation process, from which the peak maxima are selected. Then, a plot of the ln(ø/Tm2) against 1/T is built and the Ea in kJ/mol H2 is obtained from the slope of the linear fitting, as shown in Figure 8C.

Table 4 shows examples for experimental activation energy (Ea) values, reaction mechanism, and rate-limiting steps for different materials. As seen, the lowest Ea values are related to surface controlled process such as physisorption and chemisorption (Figure 5: Hydrogenation process—Steps a and b and Dehydrogenation process—Steps d and e). On the contrary, bulk processes such as interface controlled and diffusion processes present higher Ea values (Figure 5: Hydrogenation process—Steps c, e, and f and Dehydrogenation process—Steps a and b). The

#### Figure 8.

Activation energy determination: (A) isothermal method, (B) DSC curves for the (C) non-isothermal method.


#### Table 4.

Applying natural logarithm to Eq. (13):

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related to the nature of the reaction.

write:

Figure 8.

138

shown in Figure 8C.

lnk ¼ lnA þ lnð Þ �Ea=RT (14)

dt � exp ð Þ� �Ea=RT <sup>G</sup>ð Þ <sup>α</sup> (15)

þ C (16)

Plotting lnk against 1/T, the slope of the linear fitting provides the Ea in kJ/mol H2 and the intercept the frequency factor A, as seen in Figure 8A. The second method involves non-isothermal measurements done via calorimetric methods such as scanning differential calorimetry (DSC) and differential temperature analysis (DTA) [39]. This method is the Kissinger one [40]. The method consists in deter-

mining temperature maxima (Tm) of the exothermal (hydrogenation)/

dα dt <sup>¼</sup> <sup>A</sup>

ln <sup>∅</sup> T2 m

" #

endothermal (dehydrogenation) events obtained from the DSC or DTA measurements at different constant heating rate (ø). The change of Tm with ø is directly

Arrhenius dependence on the temperature, Eq. (13), Eq. (5) can be expressed:

¼ � Ea RTm

At a constant heating rate (ø), constant pressure (F(P)) and considering k has an

Assuming that the fraction at the peak maximum and G(α) are independent on ø, reordering and applying natural logarithm to expression (15), it is possible to

Figure 8B shows the DSC curves at different ø for a dehydrogenation process, from which the peak maxima are selected. Then, a plot of the ln(ø/Tm2) against 1/T is built and the Ea in kJ/mol H2 is obtained from the slope of the linear fitting, as

Table 4 shows examples for experimental activation energy (Ea) values, reaction mechanism, and rate-limiting steps for different materials. As seen, the lowest Ea values are related to surface controlled process such as physisorption and chemisorption (Figure 5: Hydrogenation process—Steps a and b and Dehydrogenation process—Steps d and e). On the contrary, bulk processes such as interface controlled and diffusion processes present higher Ea values (Figure 5: Hydrogenation process—Steps c, e, and f and Dehydrogenation process—Steps a and b). The

Activation energy determination: (A) isothermal method, (B) DSC curves for the (C) non-isothermal method.

<sup>þ</sup> ln AR Ea � �

Activation energies, reaction mechanisms, and rate-limiting steps.

interface and diffusion controlled processes are the most common ones for the hydride compounds formation/decomposition [41–51].

It is important to point out that the isothermal method for Ea requires the knowledge of the rate-limiting step, thus it depends on the solid-state model. On the contrary, the non-isothermal method is independent on the solid-state model, which represents an advantage over the first method. However, for the kinetic modeling of the reaction rates, as indicated by Eq. (5), the determination of G(α) is demanding. For this reason, at the time proceed with kinetic modeling, or just determine Ea via measurements of the hydrogenation or dehydrogenation rates at isothermal conditions, it is necessary to work far away from the equilibrium pressures to avoid the effects of the driving force, i.e., the F(P). In fact, rightly speaking, the K(T) must be corrected by the effects of the driving force by determining F(P), since the Ea values obtained after the correction are different from the one without; as an example we can mention the works published by Fenández et al. [51] and Jepsen et al. [52]. In the next subsection, then, the F(P) dependency is described.
