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Fig.
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Framework diagram of this review.
| Cite this article as: |
Qian Li, Xi Lin, Qun Luo, Yuʼan Chen, Jingfeng Wang, Bin Jiang, and Fusheng Pan, Kinetics of the hydrogen absorption and desorption processes of hydrogen storage alloys: A review, Int. J. Miner. Metall. Mater., 29(2022), No. 1, pp.32-48. https://dx.doi.org/10.1007/s12613-021-2337-8
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High hydrogen absorption and desorption rates are two significant index parameters for the applications of hydrogen storage tanks. The analysis of the hydrogen absorption and desorption behavior using the isothermal kinetic models is an efficient way to investigate the kinetic mechanism. Multitudinous kinetic models have been developed to describe the kinetic process. However, these kinetic models were deduced based on some assumptions and only appropriate for specific kinetic measurement methods and rate-controlling steps (RCSs), which sometimes lead to confusion during application. The kinetic analysis procedures using these kinetic models, as well as the key kinetic parameters, are unclear for many researchers who are unfamiliar with this field. These problems will prevent the kinetic models and their analysis methods from revealing the kinetic mechanism of hydrogen storage alloys. Thus, this review mainly focuses on the summarization of kinetic models based on different kinetic measurement methods and RCSs for the chemisorption, surface penetration, diffusion of hydrogen, nucleation and growth, and chemical reaction processes. The analysis procedures of kinetic experimental data are expounded, as well as the effects of temperature, hydrogen pressure, and particle radius. The applications of the kinetic models for different hydrogen storage alloys are also introduced.
Hydrogen storage is one of the key technologies to realize the applications of hydrogen energy, which can reduce environmental pollution and contribute to carbon neutrality. Hydrogen storage alloys (HSAs), such as LaNi5, TiFe, TiMn2, and Mg2Ni, are a type of potential hydrogen storage material because of their high volumetric hydrogen density, high safety, and applicable working temperature and pressure [1−3]. HSA is usually employed as a material for hydrogen storage tanks (HSTs) used in specific applications, such as thermoelectric cogeneration systems [4] and fuel cell forklifts [5]. However, the low hydrogen charging and discharging rates of HST, which are caused by the heat of the hydriding/dehydriding reactions, are the main bottleneck for their applications [6−7]. The enhanced kinetics of the hydrogen absorption and desorption reactions of HSA and the appropriate design of HST are two ways to improve the rate of HST. To enhance the kinetics of the hydrogen absorption and desorption reactions of HSA, it needs to be modified based on an in-depth understanding of its kinetic mechanism. The HST can be designed effectively using numerical simulation. In the mathematical model, the kinetic equation is a key governing equation and determines the accuracy of the simulation [6]. Thus, how to analyze the kinetic mechanism of the hydrogen absorption and desorption processes of HSA is a core topic for the applications of hydrogen energy.
Generally, the reactions can be mainly classified into the solid–solid, gas–solid, and liquid–solid reactions [8−12]. The hydriding/dehydriding reactions are typical gas–solid reactions. The kinetic mechanism of the hydriding/dehydriding reactions can be determined through the analysis of the hydrogen absorption and desorption behavior using the isothermal kinetic models. Both external factors (temperature and pressure) and internal factors (elementary composition, particle radius, particle surface states, and hydrogen diffusion coefficient) can influence the kinetic process of hydrogenation/dehydrogenation [13]. Numerous isothermal kinetic models, such as the Jander [14−19], Chou [20−26], and Johnson–Mehl–Avrami–Kolmogorov (JMAK) [27−32] models, have been proposed and extensively applied in the investigation of hydrogenation/dehydrogenation. The effects of the external factors are expressed in the kinetic models as the activation energy and pressure terms. By contrast, the effects of the internal factors, except for the particle radius, are difficult to express in the kinetic models explicitly. Thus, the rate constant is always used to represent the effects of the internal factors on hydrogenation and dehydrogenation in many kinetic models. Moreover, the volume expansion and contraction of particles during the hydrogen absorption and desorption processes could influence the prediction accuracy of the kinetic models for metal hydrides (MH) with large volume changes. However, some models rarely consider this factor. Thus, the appropriate kinetic model is indispensable to the analysis of the hydrogen absorption and desorption behavior.
Many processes exist objectively in the hydrogen absorption and desorption reactions of HSA. The hydrogen absorption and desorption processes can be generally divided into [12,33−35] (1) chemisorption, (2) surface penetration, (3) diffusion of hydrogen, (4) nucleation and growth, and (5) chemical reaction. However, the aforementioned isothermal kinetic models were deduced based on some assumptions and only appropriate for specific rate-controlling steps (RCS) and kinetic measurement methods. Thus, researchers without an in-depth understanding of kinetic models need the appropriate model to analyze experimental data. Many published reviews focused on the introduction of HSA [36−37] or its kinetic models [35,38] rather than the scope of application and kinetic analysis method according to the RCSs and kinetic measurement methods. Thus, we summarize the kinetic models according to different RCSs and kinetic measurement methods, as well as the progress of the development of new kinetic models.
This review mainly focuses on the isothermal kinetic models, kinetic parameters, and general analysis procedures of hydrogenation and dehydrogenation reactions. The framework of this review is shown in Fig. 1, as follows: (1) introducing the general isothermal kinetic model, including the model assumptions at different kinetic measurement methods and RCSs (Section 2); (2) summarizing the different analysis procedures of kinetic models, and the effects of temperature, pressure, and particle radius on the kinetics (Section 3); (3) illustrating the applications of the kinetic models for different types of HSA (Section 4).
The general hydrogen absorption processes of metals or intermetallics are shown in Fig. 2. The hydrogen desorption of MH is the reverse process of hydrogen absorption. The hydrogen absorption processes can be described as follows: (a) The hydrogen molecules are physisorbed on the surface of the particle at a certain pressure. (b) The hydrogen molecules are dissociated to hydrogen atoms and chemisorbed on the surface of the metal (i.e., chemisorption). (c) The chemisorbed hydrogen atoms diffuse from the surface into the particle (i.e., surface penetration). (d) The MH nucleates and grows when the hydrogen concentration exceeds the solid solubility limit. Moreover, a chemical reaction occurs on the surface of MH in some HSAs with relatively low hydrogen solid solubility.
α represents the solid solution of hydrogen in alloys, which is formed first, as shown in Fig. 2(a). r0 and r are the radius of particle and unreacted core, respectively. When the diffusion rates of hydrogen atoms in the α phase are larger than the rates of chemisorption and surface penetration, the diffusion of hydrogen atoms in the α phase and physisorption are not regarded as the RCS. When the pressure of H2 is higher than the MH equilibrium pressure, the phase transition of α → β can occur. First, the β layer will be generated on the surface of the particle. Then, the β layer will grow along the radial direction of the particle. Thus, hydrogen must diffuse through the β layer to the front of the phase transition of α → β (i.e., diffusion of hydrogen atoms in the β phase). If the nucleation and growth rates of β are slow, then there will be an α + β two-phase region between α and β (as shown in Fig. 2(b)). If the nucleation and growth rates of β are fast, then the α + β two-phase region will decrease until it can hardly be observed. Thereby, the phase transition of α → β can be simplified by a chemical reaction (as shown in Fig. 2(c)). The interface between α and β can be simplified as a continuously moving interface, which has been used as the assumption in many kinetic models, such as the Jander and Chou models.
The main hydrogen absorption and desorption measurement methods are the Sievert and flowing volumetric methods [39], as shown in Fig. 3. The Sievert method is the most common measurement technique [39], which can be easily implemented using the system of known volume when the temperature and pressure are measured accurately. The hydrogen absorption or desorption capacity can be calculated by the modified state equation of ideal gas using pressure change. In most cases, the initial pressure of the hydrogen volume is higher than the platform pressure of the α–β equilibrium. The β layer will be generated rapidly, and the state of the particle shown in Fig. 2(a) can be ignored. Thus, kinetic models, such as the Jander, JMAK, and Chou models [14], can be applied in most cases.
In the flowing volumetric method, the change of hydrogen amount in the sample cell is controlled by the mass flow controller (MFC), shown in Fig. 3(b) [39]. When the pressure and void volume of the sample cell are known, the hydrogen absorption or desorption capacity of HSA can be easily calculated. If the volume of the sample cell is small, then the capacity of HSA has an approximately linear relationship with time. When the pressure is less than the MH equilibrium pressure, the phase transition of α → β cannot occur. In this situation, the hydrogen absorption process may be controlled by two RCSs at different pressure ranges [6]. Because of the continuous pressure change, the experiment using the flowing volumetric method is regarded as a suitable way to reveal the kinetic mechanism of hydrogenation or dehydrogenation at different pressure ranges. However, only a few studies using the flowing volumetric method and the corresponding kinetic model have been published.
The kinetic models applied to the hydrogen absorption and desorption processes using the Sievert method, which include chemisorption, surface penetration, diffusion of hydrogen, nucleation and growth, and chemical reaction, are given in this review.
The process of chemisorption can be expressed as follows:
| H2+2(ch)↔2H(ch) | (1) |
where (ch) is the empty site of chemisorption on the surface of the particle and H(ch) is the absorbed hydrogen. Generally, the process of chemisorption needs the activation energy E. With the increase of the percentage of the absorbed hydrogen on the particle surface (θch), the activation energy will increase, and the rate of chemisorption will decrease. The rate of chemisorption vch can be calculated by the Elovich equation [40], as follows:
| vch=kabch,0exp(−Eabch+aθchRT)(1−θch)2P+kdech,0exp(−Edech+bθchRT)θ2ch | (2) |
where
| vch=kabch,0exp(−Eabch+aθchRT)P | (3) |
When θch has a medium value, Eq. (2) can be simplified as follows:
| vch=kabch,0exp(−Eabch+aθchRT)P+kdech,0exp(−Edech+bθchRT) | (4) |
Aside from the Elovich equation, the chemisorption rate can be expressed as follows [12]:
| vch=kchr0(P−Peq) | (5) |
where r0 is the initial particle radius, Peq is the MH equilibrium pressure, kch is the rate constant of chemisorption. In Eq. (5), the effects of particle radius and equilibrium pressure on the chemisorption rate are considered. Then, combined with the Arrhenius equation, Eq. (5) can be expressed as [12]
| ξ=kch,0r0exp(−ERT)(P−Peq)t | (6) |
where ξ is the reaction fraction and t is the time, kch,0 is the factor for the rate constant of chemisorption.
The process of surface penetration can be expressed as follows:
| H(ch)↔[H]+(ch) | (7) |
where [H] is hydrogen in the α or β phase. The rate of surface penetration vsp can be expressed as follows [12,33]:
| vsp=kabspθch−kdespCβ | (8) |
where
| vsp=kspr0(P1/2−P1/2eq) | (9) |
where ksp is the rate constants of surface penetration. Then, Eq. (9) can be changed to the form of reaction fraction with time, as follows:
| ξ=ksp,0r0exp(−ERT)(P1/2−P1/2eq)t | (10) |
where ksp,0 is the factor for the rate constant of surface penetration.
For the RCSs of chemisorption and surface penetration, the reaction fraction has a linear relationship with time. The main difference is the pressure term in the kinetic equations expressed in Eqs. (6) and (10). Thus, the effect of pressure on these two RCSs needs to be investigated.
Many kinetic models, such as the Jander [18], Ginstling–Brounshtein (G-B) [41], Valensi–Carter (V-C) [42], and Chou [33] models, have been used to describe the diffusion of hydrogen. These kinetic models are all derived based on the diffusion of hydrogen as RSC, with the temperature and pressure assumed to be constant during the hydrogen absorption and desorption processes.
(1) Jander model.
When the RCS is the diffusion of hydrogen, the reaction rate decreases proportionally with the thickness of the β layer. The Jander model assumes that the area of the α/β phase interface and the particle volume are constant [18,43]. Then, the moving rate of the α/β phase interface can be expressed based on Fick’s law, as follows:
| ρ∂r∂t=−DΔCr0−r | (11) |
where ρ is the density, D is the diffusion coefficient, ∆C is the hydrogen concentration difference between the surface and the α/β phase interface, and r is the radius of the α/β phase interface. The reaction fraction can be calculated by the following equation [44]:
| ξ=1−(rr0)d | (12) |
where d is the dimensionality. For spherical, cylindrical, and plate particles, d is equal to 3, 2, and 1, respectively. Combining Eqs. (11) and (12), the Jander model can be expressed as follows:
| [1−(1−ξ)1/d]2=2DΔCr20t=kdit | (13) |
where kdi is the rate constant of hydrogen diffusion. The form of the Jander model is concise. However, as the area of the α/β phase interface changes with the hydrogen absorption and desorption reactions, a deviation between the Jander model and the actual situation is observed. Thus, the Jander model is applied to the hydrogen absorption and desorption processes of small particles because of the slight change of the area of the α/β phase interface in small particles.
(2) G-B model.
The G-B model is developed based on the Jander model without the assumption that the area of the α/β phase interface is constant [41,45]. For the cylindrical particle, the moving rate of the α/β phase interface can be derived as follows:
| ρ∂r∂t=−DΔCrln(r0/r) | (14) |
For the spherical particle, the moving rate of the α/β phase interface can be derived as follows:
| ρ∂r∂t=−DΔCr0(r0−r)r | (15) |
Then, the G-B models for the cylindrical and spherical particles can be expressed as Eqs. (16) and (17), respectively:
| (1−ξ)ln(1−ξ)+ξ=4DΔCr20ρt=kdit | (16) |
| 1−23ξ−(1−ξ)2/3=2DΔCr20ρt=kdit | (17) |
The G-B model is more complex than the Jander model, but it is more accurate than the Jander model because it considers the area variation of the α/β phase interface.
(3) V-C model.
The V-C model considers the volume variation of the spherical particle before and after the hydrogen absorption and desorption processes. Thus, Eq. (11) can be modified as follows [42]:
| ρ∂r∂t=−DΔCr−r2/[zr30+r3(1−z)]1/3 | (18) |
where z is the volumetric expansion ratio. Then, the V-C model can be expressed as follows:
| z−[1+(z−1)ξ]2/3−(z−1)(1−ξ)2/3z−1=2DΔCr20ρt=kdit | (19) |
Given the volume expansion and contraction of particles, the equation of the V-C model becomes complex. In most cases, the alloy particle is small because of pulverization during the activation procedures. The effect of the volume expansion and contraction of particles on the kinetic curve obtained using the V-C model is inapparent compared with that obtained using the Jander model.
(4) Chou model.
The Chou model focuses on the physical interpretation of the generalized rate constant kdi [14,20−26,33,44]. The effect of temperature on kdi is embodied by the Arrhenius equation. The effect of pressure is embodied by the pressure term, which indicates the chemical driving force. Then, kdi can be expressed as follows:
| kdi=kdi,0r20exp(−ERT)(P1/2−P1/2eq) | (20) |
where ksp,0 is the factor for the rate constant of hydrogen diffusion. Similar to that of the Jander model, the equation of the Chou model for the RCS of the diffusion of hydrogen can be expressed as follows [33]:
| [1−(1−ξ)1/d]2=kdi,0r20exp(−ERT)(P1/2−P1/2eq)t | (21) |
Further considering the volume expansion and contraction of different particles, Eq. (21) can be modified as follows [44]. For the spherical particle:
| ξ∫0[z−(z−1)(1−ξ)]1/3−(1−ξ)1/33(1−ξ)2/3dξ=kdi,0r20exp(−ERT)(P1/2−P1/2eq)t | (22) |
For the cylindrical particle:
| ξ∫0[z1−ξ−(z−1)]1/2dξ−ξ=kdi,0r20exp(−ERT)(P1/2−P1/2eq)t | (23) |
For the plate particle:
| ξ2=kdi,0r20exp(−ERT)(P1/2−P1/2eq)t | (24) |
The Chou model describes the functional relationship between rate and temperature, pressure, particle radius, and equilibrium pressure. Moreover, the Chou model proposes the concept of “characteristic time tc,” whose value is equal to 1/kdi, to describe the time required for hydrogen absorption and desorption reactions to reach a specific reaction fraction (such as 100%, 80%, or 50%) [33].
Generally, the nucleation and growth processes can be described by the JMAK model.
(1) Classical JMAK (C-JMAK) model.
The well-known C-JMAK model can be expressed as follows [14,46−47]:
| ξ=1−exp(−kngtn) | (25) |
where n is the Avrami exponent, and kng is the rate constant of nucleation and growth. Generally, the value of n is equal to or greater than 0.5. Eq. (25) can be transformed into:
| [−ln(1−ξ)]1/n=(kng)1/nt | (26) |
or
| ln[−ln(1−ξ)]=nlnt+lnkng | (27) |
The RCS in the nucleation and growth processes may be the diffusion or movement of the interface [38]. Notably, this diffusion represents the diffusion of the hydrogen and metal atoms during the nucleation and growth processes. In Section 2.2.3, the diffusion of hydrogen indicates the long-range diffusion of the hydrogen atom in the β layer. The C-JMAK model has been widely applied to describe the hydrogen absorption and desorption processes of HSA because of its high practicality and the simplified form of its equation. In the C-JMAK model, the RCS should be evaluated based on the value of n. However, as the Avrami exponent n is the fitting value, the RCS corresponding to n should be further investigated for a specific hydrogen storage material (or refer to published works).
(2) Nucleation-index-incorporated JMAK (NI-JMAK) model.
The NI-JMAK model is developed from the C-JMAK model to consider the effect of the self-catalysis of the nucleation process [14,28]. The NI-JMAK model can be expressed as follows:
| ξ=1−exp{−kc+d/mng,0Beta(c,dm+1)exp(−cEn−dmEgRT)tc+d/m} | (28) |
where c is the nucleation index, m is the growth mode parameter, d/m is the growth index, En is the activation energy of nucleation, and Eg is the activation energy of growth. The NI-JMAK model utilizes five independent variables to describe the nucleation and growth processes under the isothermal condition. Because of the complicated form of the equation, its application to the hydrogen absorption and desorption processes of HSA is limited. However, the C-JMAK model is recommended as the fitting kinetic model for the RCS of nucleation and growth.
The chemical reaction process can be expressed as follows:
| [H](α/β)+M(α/β)↔MH(α/β) | (29) |
where [H](α/β), M(α/β), and MH(α/β) are the hydrogen, hydrogen storage, and metal hydride at the α/β phase interface, respectively. Many kinetic models, such as the contracting volume (CV) and Chou models, have been used to describe the chemical reaction process.
(1) CV model.
The chemical reaction process is simplified as the process of the movement of the α/β phase interface with a constant rate [48] and can be expressed as follows:
| ∂r∂t=−kint | (30) |
where kint is the moving rate of the α/β phase interface. Combining Eqs. (12) and (30), the equation of the CV model can be expressed as follows:
| 1−(1−ξ)1/d=kintr0t=kcrt | (31) |
where kcr is the rate constant of chemical reaction.
(2) Chou model.
The equation of the Chou model for the RCS of the chemical reaction process can be expressed as follows [33]:
| 1−(1−ξ)1/d=kcr,0r0exp(−ERT)(P1/2−P1/2eq)t | (32) |
where kcr,0 is the factor for the rate constant of chemical reaction. The comparison between the CV and Chou models shows that the forms of their equations are similar. However, the Chou model expands the rate constant kcr into a function of temperature, particle radius, pressure, and equilibrium pressure, which endows the model with more physical significance.
Many kinetic models suitable for the processes using the Sievert method have been introduced in this section according to the difference in RCSs. Generally, all kinetic models should be applied to fit the experimental data tested by the Sievert method so that the RCS can be decided. However, to reduce the workload of researchers, the following suggestions are given: (1) When the hydrogen pressure is far from the hydrogen absorption or desorption platform pressure, the diffusion of hydrogen, nucleation and growth, and chemical reaction are likely to be the RCS. By contrast, when the hydrogen pressure is close to the hydrogen absorption or desorption platform pressure, chemisorption or surface penetration is likely to be the RCS. (2) For the HSA with a small particle radius and volumetric expansion ratio, the effect of the α/β phase interface area and the volume expansion and contraction of particles can be ignored. Thus, the Jander or Chou model is proposed for this type of HSA.
Only a few studies focusing on kinetic models appropriated for the hydrogen absorption and desorption processes using the flowing volumetric method have been published. Lin et al. [6,49] employed a kinetic model at constant flow to analyze the hydrogen absorption and desorption processes of HSA using the flowing volumetric method. As the hydrogen flows are constants, the curve of reaction fraction with time is nearly linear when the hydrogen concentration is less than the maximum concentration of HSA. Thus, the curve of pressure with time is analyzed. Because of the continuous pressure curve, the effect of pressure on the kinetic process can be easily revealed.
The kinetic equation for the chemisorption process can be expressed as follows [6]:
| P=±r0Fkch,0exp(−ERT)+Peq | (33) |
where F is the flow per unit mass. “+” and “−” in “
| Peq=f(H/M)exp[−ΔHR(1T−1Tref)] | (34) |
where H/M is the hydrogen concentration (defined as the mole ratio of hydrogen and alloy), Tref is the reference temperature of the PCT curve, f(H/M) is the polynomial equation of the PCT curve at Tref. The hydrogen in the void area of the sample cell can be ignored when the volume of the void area is small. Then, H/M can be calculated by the following equation:
| H/M=(H/M)0±2M0VmFt | (35) |
where Vm is the standard molar volume of gas, M0 is the relative molar mass, and (H/M)0 is the initial hydrogen concentration.
The kinetic equation for the surface penetration process can be expressed as follows [6]:
| P=[±r0Fksp,0exp(−ERT)+√Peq]2 | (36) |
The kinetic equation for the diffusion of hydrogen can be expressed as follows [6]:
| P={r20F[1−(1−ξ)1/3]kdi,0exp(−ERT)(1−ξ)1/3+√Peq}2 | (37) |
where ξ can be calculated by the following equation:
| ξ=H/MH/Mmax | (38) |
where H/Mmax is the maximum of H/M.
The kinetic equation for the nucleation and growth processes can be expressed as follows:
| P=Peqexp[±Fkng,0exp(−ERT)n[−ln(1−ξ)]n−1n(1−ξ)] | (39) |
The kinetic equation for the hydrogen absorption of the chemical reaction process can be expressed as follows:
| P=[r0Fkcr,0exp(−ERT)(1−ξ)2/3+√Peq]2 | (40) |
The kinetic equation for the hydrogen desorption of the chemical reaction process can be expressed as follows:
| P=[−r0Fkcr,0exp(−ERT)ξ2/3+√Peq]2 | (41) |
Substituting Eqs. (34) and (35) into Eqs. (33), (36), (37), and (39) to (41), the equations of pressure with time at a constant flow are established for different processes. The form of the equation of the kinetic model at constant flow is complicated. The key to the kinetic model at constant flow is the accuracy of the prediction of Peq.
The kinetic models can be used to fit the experimental data of the hydrogen absorption and desorption reactions of HSA according to different kinetic measurement methods. The RCS of hydrogenation or dehydrogenation at different pressures, temperatures, and particle radii can be analyzed using the fitted results, as well as the values of activation energy and rate constant. In this section, the analysis of the experimental kinetic data is summarized.
According to the experimental data from different kinetic measurement methods, the analysis procedures have slight differences. For the Sievert method, the experimental data of reaction fraction with time, as well as the rate with reaction fraction, can be applied to analyze the kinetic mechanism. For the flowing volumetric method, the experimental data of pressure with time is applied. In this section, these analysis procedures are all summarized.
The kinetic model can be used to analyze the experimental kinetic data of reaction fraction ξ with time t. The hydrogen absorption and desorption processes of HSA are mostly controlled by chemisorption, surface permeation, diffusion of hydrogen, nucleation and growth, and chemical reaction. The general formula of the kinetic equation can be written as follows:
| f(ξ)=kt | (42) |
where f(ξ) is the function of ξ which depends on the type of RCS. The rate constant k can be derived as follows:
| k=k0exp(−ERT)h(r0)g(P,Peq) | (43) |
where h(r0) and g(P, Peq) are the influence terms of the particle radius and pressure, respectively, and k0 is the factor of rate constant. The analysis procedures are shown in Fig. 4. According to the fitted results of the f(ξ)–t curve using the kinetic models for different RCSs, the RCS and rate constant are determined. Then, repeating the fitting process using the f(ξ)–t curves under different test conditions, including temperature, pressure, and particle radius, the activation energy E, k0, h(r0), and g(P, Peq) are all determined.
The RCS may change with the hydrogen concentration of some HSAs. Given the difference in f(ξ) for different RCSs, the multistage fitted results cannot be easily presented in one plot of f(ξ)–t visually. Thus, the plot of ξ–t is used to analyze the kinetic process, as shown in Fig. 5. Most kinetic models can be transformed into the form of ξ = f(t). A complete ξ–t curve can be processed in three stages. The RCS of Stages 1 and 3 can be determined by the corresponding kinetic model. Stage 2 is the transitional stage, and its RCS is mixed. Its hydrogen absorption and desorption rate v2 can be calculated by the following equation:
| v2=11/v1+1/v3 | (44) |
where v1 and v3 are the hydrogen absorption and desorption rates of Stages 1 and 3, respectively.
The plot of rate and reaction fraction can also be used to analyze the kinetic process tested by the Sievert method [12,50]. When the particle of HSA is spherical, regardless of particle volume change, ∂ξ/∂t and ξ can be plotted in Fig. 6 according to the kinetic models discussed in Section 2.2. For the chemisorption or surface penetration process, the rate does not change with the reaction fraction. Thus, the plot of ∂ξ/∂t–ξ is a straight line. For the diffusion of hydrogen, the rate decreases gradually with the increase of the reaction fraction. Thus, the plot of ∂ξ/∂t–ξ shows a semi-concave curve. For the nucleation and growth processes, the rate initially increases and subsequently decreases. Thus, the plot of ∂ξ/∂t–ξ shows a completely convex curve, which slightly differs with different n values. For the chemical reaction process, the rate also decreases gradually with the increase of the reaction fraction. However, the plot of ∂ξ/∂t–ξ shows a semi-convex curve. The comparison between the experimental data and the ∂ξ/∂t–ξ curves plotted in Fig. 6 shows that the rate is controlled by the nucleation and growth processes (n = 3) at ξ < 0.2, which then transform into the diffusion of hydrogen at ξ > 0.2. Thus, the plot of rate and reaction fraction can be applied to estimate the RCS at different reaction fraction ranges according to the line type.
The plot of pressure and time is used to analyze the kinetic process tested by the flowing volumetric method [6,49]. According to the kinetic model at constant flow discussed in Section 2.3, P and t, as well as the fitted results, can be plotted in Fig. 7. The precondition for the application of this kinetic analysis method is the equilibrium pressure equation, which can be determined from the experimental PCT curves [51−54]. When the pressure is lower than or close to the MH equilibrium pressure, the β phase cannot be generated, and the potential RCS is chemisorption or surface penetration. When the pressure is higher than the MH equilibrium pressure, the β phase layer can be generated rapidly, and the potential RCS is the diffusion of hydrogen, nucleation and growth, or chemical reaction. Thus, according to the fitted result, the RCS can be determined at different pressure ranges.
Given that the hydrogen absorption and desorption reactions of HSA are exothermic and endothermic processes, respectively, the temperature in the sample cell has a considerable effect on the reaction rate. The Arrhenius equation is a well-known equation used to embody the effect on the rate constant [55−56], which can be expressed as follows:
| k=Aexp(−ERT) | (45) |
where A is the pre-exponential factor. According to the fitted result of kinetic models, the values of rate constant k at different temperatures can be determined. Then, plotting the curve of lnk and 1/T, the value of E is evaluated. Table 1 shows the activation energy values of some typical HSAs. The activation energy values are compared in absorbed or desorbed one mol H2 in this work. Generally, k increases with temperature, and the value of E is positive. However, as k is related to the difference between pressure and equilibrium pressure, i.e., g (P, Peq) term in Eq. (43), the value of g (P, Peq) is considered to be close to the experimental data. Otherwise, the phenomenon that the value of k decreases with temperature may occur.
| No. | HSA | Test conditions | RCS | E / (kJ∙mol−1) | Ref. |
| 1 | LaNi5 | 293–333 K, P − Peq = 100 kPa | Chemisorption | 19.66 (ab) | [57] |
| 2 | LaNi4.5Al0.5 | 273–343 K, P − Peq = 32.4 kPa | Chemisorption | 39.77 (ab) | [58] |
| 3 | MmNi4.2Al0.8 | 288–308 K, 0.013 MPa | Diffusion of hydrogen | 48.09 (de) | [59] |
| 4 | La1.5Ni0.5Mg17 | 523–573 K, 0.755 MPa | Diffusion of hydrogen | 90 (ab) | [60] |
| 5 | ZrCo | 544–603 K, P/P0 = 0.9–1 MPa | Nucleation and growth | 120 (ab) | [61] |
| 6 | TiFe | 243–298 K, 0.02 MPa | Chemisorption | 3.35 (ab) | [62] |
| 7 | TiFe0.8Ni0.2 | 373–423 K, |Peq − P0.5|/Peqa | Diffusion of hydrogen | 53.13 (ab) | [63] |
| 8 | Zr0.2Ho0.8Co2 | 323–473 K, 0–5 MPa | ― | 15 (ab) | [64] |
| 9 | Ti33V20Cr47 | 293–353 K, 3.5 MPa | Diffusion of hydrogen | 25.7 (ab) | [65] |
| 10 | Mg2Ni | 528–623 K, 4 MPa | Diffusion of hydrogen | 44 (ab) | [22] |
| 11 | Mg1.9Ag0.1Ni | 523–573 K, 0.768 MPa | Diffusion of hydrogen | 52 (ab) | [66] |
| 12 | MgH2−5at%V | 453–573 K, 0.015 MPa | ― | 62 (de) | [67] |
| Note: aP0.5 is the pressure at the reaction fraction of 0.5. | |||||
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The driving force of the α → β phase transformation is the difference between hydrogen concentration and equilibrium hydrogen concentration, which is macroscopically represented by the difference between pressure and equilibrium pressure of HSA. Therefore, for the hydrogen absorption process, a high pressure leads to a fast hydrogen absorption rate. For the dehydrogenation process, a high pressure leads to a slow dehydrogenation rate. Notably, if the hydrogen absorption pressure is lower than the equilibrium pressure of α → β phase transformation, then only the hydrogen solid solution of the α phase will be generated. When the hydrogen absorption pressure is higher than the equilibrium pressure of the α → β phase transformation, the β phase layer will be generated on the surface of the particle rapidly. Therefore, the RCS will be different at different pressure ranges. The relationship between the rate constant and the pressure and equilibrium pressure can be expressed as follows:
| k=k0g(P,Peq) | (46) |
In the Chou model, g(P, Peq) is transformed into P1/2 −
| No. | HSA | Test conditions | RCS | g(P, Peq) | Ref. |
| 1 | LaNi5 | P − Peq = 50−200 kPa, 303 K | Chemisorption | P − Peq | [57] |
| 2 | LaNi4.5Al0.5 | P − Peq = 25.3−66.9 kPa, 298 K | Chemisorption | P − Peq | [58] |
| 3 | La1.5Ni0.5Mg17 | 0.256–0.992 MPa, 553 K | Diffusion of hydrogen | [(P − Peq)/Peq]2 | [60] |
| 4 | Mg2Ni | 0.275–1.133 MPa, 553 K | Diffusion of hydrogen | P1/2 − P1/2eq | [22] |
| 5 | MgH2−5at%V | 0.007–0.4 MPa, 533–673 K | ― | 1 − P/Peq | [68] |
| 6 | Mg−10wt%Ni | P − Peq = 20.1–54.6 kPa, 573 K | ― | [P − Peq]1.25 | [69] |
| 7 | ZrCo | <1 MPa, 300 K | Chemisorption | P2 | [70] |
| 8 | TiFe | 1.717–2.626 MPa, 294 K | Chemisorption | P − Peq | [62] |
| 9 | TiFe0.8Ni0.2 | 0.246–0.733 MPa, 408 K | ― | (Peq − P)/Peq | [71] |
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Generally, the specific surface area and diffusion cross section increase with the decrease of the particle radius [52,72−73]. Thus, decreasing the particle radius of HSA can enhance the hydrogen absorption and desorption rates. The relationship between rate constant and particle radius can be expressed as follows:
| k=k0h(r0) | (47) |
For different hydrogen absorption and desorption processes, the influence of particle radius is also different: for physical adsorption, chemical adsorption, surface penetration, and chemical reaction, h(r0) is generally expressed as 1/r0; for the diffusion of hydrogen, h(r0) is generally expressed as
The hydrogen absorption and desorption processes of HSA are complex and can be influenced by the preparation methods, powder particle state, and catalyst. Different kinds of HSA may have different RCSs; even the HSA with similar composition could have a different RCS. Notably, the kinetic model is semi-empirical. The completed hydrogen absorption and desorption kinetic mechanism should be further analyzed by conducting other experiments, such as scanning electron microscopy (SEM) and transmission electron microscopy (TEM) [74]. In this section, the kinetic studies of the hydrogen absorption and desorption processes of AB5-type, AB2-type, AB-type, and Mg-based HSAs are introduced.
LaNi5 is a typical AB5-type HSA, and its hydrogen concentration curve at different pressure differences dP of hydrogen absorption pressure and equilibrium pressure is shown in Fig. 8(a) [75]. The maximum hydrogen concentration increases with the increase of dP. Based on the CV model, the curves of 1 − (1 − ξ)1/3 and t are plotted in Fig. 8(b). Notably, 1 − (1 − ξ)1/3 has a linear relationship with t, and the values of R2 are higher than 0.98. The fitted slopes are the value of the rate constant. Thus, the RCS for the hydrogen absorption process of LaNi5 is the chemical reaction. With the decrease of dP from 200 to 50 kPa, the rate constants kcr are 0.01457, 0.00768, and 0.00461 s−1. The equilibrium pressure Peq of the alloy at 323 K is approximately 530 kPa. Assuming that the form of g(P, Peq) in Eq. (43) is ln(P/Peq), the curve of kcr and ln(P/Peq) can be plotted in Fig. 8(c). The linear fitted result of kcr and ln(P/Peq) indicates that the form of g(P, Peq) in Eq. (43) can be evaluated as ln(P/Peq). Thus, the kinetic equation of LaNi5 can be written as follows:
| 1 - {(1 - \xi )^{1/3}} = k_{{\text{cr}}}' \ln (P/{P_{{\text{eq}}}})t = {k_{{\text{cr,0}}}}\exp \Bigg( - \frac{E}{{RT}}\Bigg)\ln (P/{P_{{\text{eq}}}})t | (48) |
where
Then, according to the experimental kinetic curves at different temperatures, the values of kcr and
| 1 - {(1 - \xi )^{1/3}} = 507.75\exp \Bigg( - \frac{{3019.74}}{T}\Bigg)\ln (P/{P_{{\text{eq}}}})t | (49) |
However, different researchers have different estimations of the RCS of the hydrogen absorption process of LaNi5 alloy [24,75−80]. Table 3 shows some published results for the RCS of LaNi5 alloys. Although the composition of LaNi5 alloy is the same, the RCS and activation energy evaluated by different researchers are different. This finding indicates that the hydriding/dehydriding kinetic mechanism of HSA is complex, which is related not only to the composition but also to the test conditions, preparation method of the alloy, particle and its surface states, and impurity. Thus, researchers should consider these factors when analyzing the kinetic mechanism and refer to other published works. Other characterization methods, such as SEM and TEM, are necessary [74,81].
| No. | Test conditions | Kinetic model | RCS | E / (kJ·mol−1) | Ref. |
| 1 | 288–353 K, <1 MPa | CV | Chemical reaction | 32.2 | [75] |
| 2 | 273–363 K, 0.5 MPa | ― | Nucleation and growth | 31.8 | [76] |
| 3 | 333–338 K, 2 MPa | ― | Mixed (chemisorption and diffusion of hydrogen) | ― | [77] |
| 4 | 298–323 K, Pr = 2–5a | C-JMAK (n = 1) | Nucleation and growth (one-dimensional diffusion) | 27 | [78] |
| 5 | 298–313 K, Pr = 2–5 | C-JMAK (n = 1) | Nucleation and growth (one-dimensional diffusion) | 30–40 | [79] |
| 6 | 303–333 K, Pr = 2 | Jander | Diffusion of hydrogen | 27.7 | [80] |
| 7 | 303 K, 0.6–1 MPa | Chou | Diffusion of hydrogen | ― | [24] |
| Note: a Pr is the ratio of pressure to equilibrium pressure. | |||||
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The substitution of alloying elements can change the RCS and activation energy of the hydrogen absorption and desorption processes in AB5 alloys [49,82−84]. The LaNi5−xAlx alloy is a common ternary AB5 alloy [82]. Fig. 9 shows the hydrogen absorption and desorption pressure curves of LaNi4.25Al0.75 [49] at different temperatures and particle radii tested by the flowing volumetric method. The fitted results indicate that the RCSs for the hydrogen absorption process of LaNi4.25Al0.75 at different pressure ranges are surface permeation and nucleation and growth of MH, whereas the RCS for the hydrogen desorption process is only the nucleation and growth of MH.
Moreover, because of the high cost of the elementary substance La, the mixed rare earth Mm can be used to replace La [85−86]. The RCS of MmNi5 alloy can be determined based on the curves of ∂ξ/∂t and ξ. Notably, the RCS of the hydrogen absorption process changes with the pressure and reaction fraction [50], as shown in Fig. 10. The lines are re-plotted according to the kinetic equations (6)−(8) given in Ref. [50]. At the initial stage of the hydrogen absorption reaction (ξ < 0.35), when the hydrogen absorption pressure P is low and close to the equilibrium pressure of 1 MPa (P ≤ 1.5 MPa), the RCS is surface permeation. When the hydrogen absorption pressure continues to increase, the RCS will change to nucleation and growth (n ≈ 1.55). With the further increase of the reaction fraction (ξ > 0.4), the RCS changes from surface penetration (or nucleation and growth) to the diffusion of hydrogen.
The AB2-type alloy has a Laves phase structure [87], such as Zr-based [23,88−89] and Ti-based alloy [90−91]. Ti1.02Cr1.0Fe0.7−xMn0.3Alx (0 ≤ x ≤ 0.1) is a typical Zr-based alloy, and its hydrogen desorption process can be described by the C-JMAK model (n = 1). The RCS is the nucleation and growth of MH (one-dimensional diffusion), and the activation energy is within 7.4−9.9 kJ∙mol−1 [92]. Moreover, because of the complex phase composition of AB2-type alloy with multiple MH phases, the hydrogen absorption and desorption mechanism may change with the processes of the hydrogen absorption and desorption reactions. The hydrogen absorption process of Zr0.2Ho0.8Fe2 alloy has two hydrogen absorption stages [93]. Although both stages can be fitted by the C-JMAK model (n = 1), the rate constant and activation energy are different. The activation energies of both stages are 24 and 2 kJ∙mol−1 [93].
ZrCo-based alloy is one of the typical AB-type HSA, and the RCS of the hydrogen absorption reaction in the temperature range of 544−603 K is nucleation and growth with the activation energy of 120 kJ∙mol−1 [61]. The fluorinated and Ni-plated ZrCo alloys are controlled by chemisorption, but the hydrogen absorption kinetics of Ni-plated ZrCo alloys is significantly improved because of the easy dissociation of H2 on Ni at the particle surface [94]. The C-JMAK model is used to analyze the hydrogen absorption kinetics curves of the ZrCo0.95Cr0.05 alloy [95]. The fitted results show that the values of n for ZrCo0.95Cr0.05 at 373, 348, and 323 K are 2.83, 1.36, and 1.07, respectively. Notably, with the decrease of temperature, the RCS of the hydrogen absorption reaction of ZrCo0.95Cr0.05 changes from the three-dimensional diffusion to the one-dimensional diffusion of nucleation and growth.
The kinetic properties of hydrogen absorption and desorption of pure Mg are poor but can be enhanced by alloying or adding catalysts [96−100]. Table 4 shows the kinetic results of Mg-based HSAs. For pure Mg, the hydrogen absorption reaction is initially controlled by chemisorption and subsequently controlled by the diffusion of hydrogen [109]. For MgH2, the hydrogen absorption reaction is controlled by the diffusion of hydrogen [110]. Recently, Shriniwasan et al. [81,111] investigated the hydrogen absorption mechanism of Mg by combining SEM and kinetic analysis. They determined that the RCSs of Mg are chemisorption at the initial stage, followed by the movement of the interface during the nucleation and growth processes, and finally, the diffusion of hydrogen [81].
| No. | HSAs | Preparation methods | Test conditions | Kinetic model | RCS | E / (kJ∙mol−1) | Ref. |
| 1 | Mg−30wt% LaNi5 | MA; HC | 302−423 K, 1 MPa | Chou | Surface penetration; Diffusion of hydrogen | 28.0 (ab); 25.2 (ab) | [25] |
| 2 | La1.5Ni0.5Mg17 | HC | 523−573 K, 0.755 MPa | Jander | Diffusion of hydrogen | 90 (ab) | [60] |
| 3 | Nd4Mg80Ni8 | Melting | 373−623 K, 3.4 MPa; 564−620 K, Vacuum | Chou | Diffusion of hydrogen (ab); Surface penetration (de) | 82.3 (ab) 97.5 (de) | [101] |
| 4 | Mg18In1Ni3 | MA | 493−583 K, Vacuum | C-JMAK (n = 1) | Nucleation and growth (one-dimensional diffusion) | 107 (de) | [102] |
| 5 | LaNiMg17 | MA | 553−623 K, 0.755 MPa | Chou | Diffusion of hydrogen | 71 (ab) | [103] |
| 6 | Mg−Y2O3; Mg−Y | Arc melting | 473−573 K, 3 MPa | C-JMAK | ― | 79.9 (ab) 56.2 (ab) | [104] |
| 7 | Mg80Ce18Ni2 | Melting | 523−585 K, 3.5 MPa | C-JMAK (n = 0.53) | Nucleation and growth (one-dimensional diffusion) | 63 (de) | [105] |
| 8 | Mg12NiY | Melting | 523−623 K, Vacuum | C-JMAK (n = 0.6−0.8) | ― | 44.96 (de) | [106] |
| 9 | Mg2In0.1Ni | Melting | 467−507 K, Vacuum | C-JMAK (n = 0.6−0.8) | Nucleation and growth (one-dimensional diffusion) | 28.9 (de) | [107] |
| 10 | Mg−TiH1.971−TiH1.5 | HC | 298−673 K, 4 MPa; 473−673 K, Vacuum | C-JMAK (n = 0.3−1.1) | Nucleation and growth (one-dimensional diffusion) | 12.5 (ab); 46.2 (de) | [108] |
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The alloying method can introduce a new phase to Mg [101,112−113], which can change/increase the reaction path of the hydrogen absorption and desorption processes [114−117], significantly reduce the activation energy and hydrogen desorption temperature of Mg alloy, and change the RCS. Mg2Ni is a typical binary Mg-based alloy [118−120]. The addition of Ni can considerably increase the rate of the hydrogen absorption and desorption reactions, and the temperature of the hydrogen desorption reaction of Mg2Ni is significantly lower than that of pure Mg. The rate of the hydrogen absorption reaction of Mg2Ni is linearly related to pressure, and the RCS is chemisorption with the activation energy of 19.6 kJ∙mol−1 [121].
Generally, two or more elements, such as Nd, Ni, Ce, Y, La, In, and Ag, are added to Mg to improve the kinetic properties of the Mg alloy [55,66,98,117,122−123]. The Mg18In1Ni3 alloy prepared by ball milling can absorb 3.8wt% hydrogen in 20 min at 553 K and desorb 1wt% hydrogen in 10 min at 493 K. The C-JMAK model was used to fit the hydrogen desorption kinetic curves, and the values of n are close to 1, indicating that the RCS was the one-dimensional diffusion of nucleation and growth [102]. The calculated activation energy was 107 kJ∙mol−1, which was lower than the 160 kJ∙mol−1 of pure Mg [41]. The Chou, Jander, and C-JMAK models have been used to systematically analyze the hydrogen absorption and desorption kinetics of Mg−Ni−La [60,103], Mg−Ni−Ce [124], and Mg−Ni−Nd [101]. The kinetic properties of the LaNiMg17 alloy prepared by the mechanical alloying (MA) method are improved because of the introduction of surface defects. The RCS of the hydrogen absorption and desorption reaction is the diffusion of hydrogen. The activation energy of the hydrogen absorption reaction of the LaNiMg17 alloy prepared by the MA method is 71 kJ∙mol−1 [103], whereas that of the La1.5Ni0.5Mg17 alloy prepared by the hydriding combustion (HC) method is 90 kJ∙mol−1 [60]. The hydrogen absorption reaction of the Nd4Mg80Ni8 alloy is controlled by the diffusion of hydrogen with the activation energy of 82.3 kJ∙mol−1 , whereas the hydrogen desorption reaction of the Nd4Mg80Ni8 alloy is controlled by surface penetration with the activation energy of 97.5 kJ∙mol−1 [101].
The addition of a catalyst to Mg is a common method to improve the kinetic performance of the hydrogen absorption and desorption reactions [125]. The catalysts mainly include oxides, complex hydrides, and La and Ni compounds [104,126−128]. For Mg−Y2O3 and Mg−Y composite powders prepared by arc plasma melting, Mg−Y2O3 can adsorb 5wt% (573 K) hydrogen at 100 s, whereas Mg−Y only adsorbs 2.3wt% hydrogen under the same conditions [104]. The C-JMAK model was used to analyze the kinetics of the two powders, and the values of activation energy were 79.9 and 56.2 kJ∙mol−1. The high pre-exponential factor was the reason for the fast hydrogen absorption rate of Mg−Y2O3 [104]. The hydrogen absorption kinetics of Mg−xwt%LaNi5 (x = 10−50) composites prepared by the MA, microwave sintering, and traditional sintering methods were determined as follows [25]: (1) The RCS of Mg−xwt%LaNi5 was the diffusion of hydrogen, and the kinetic properties of Mg−xwt%LaNi5 were enhanced with the increase of LaNi5 content. (2) Mg−30wt%LaNi5 prepared by mechanical ball milling was controlled by surface penetration, whereas the alloys prepared by HC and microwave sintering were controlled by the diffusion of hydrogen atoms. The hydrogen absorption and desorption rates and the kinetic mechanism can be influenced by the amount of catalyst or catalytic phase and the preparation method.
Kinetics is significant for the development of HSA and the design of the HST using numerical simulation. In this paper, we review the kinetic models and their corresponding analysis procedures, as well as their applications to HSA. The development of theoretical kinetic models lagged behind experimental investigations. Many kinetic models only use the rate constant to express the effect of temperature or pressure on reaction rate, and only a few models can express the relationship between reaction fraction and multiple factors, such as temperature, pressure, and particle size. Most kinetic models are constructed based on the isothermal process, and only the NI-JMAK model can extend to the non-isothermal condition using the parameters obtained from the isothermal condition. The application of kinetic models faces the problem that some kinetic models are presented as differential equations of reaction rate, which cannot be conveniently applied in practice. Therefore, a model that has analytical formulas, is applicable under isothermal and non-isothermal conditions, and considers multiple factors is desirable for further hydrogenation and dehydrogenation kinetic analysis.
The kinetic mechanism of the hydrogen absorption and desorption reactions of AB5-type, AB2-type, AB-type, and Mg-based HSAs is not the same, which is related to the composition of the alloys, preparation methods, powder particle states, type and content of catalysts, and test conditions of hydrogen absorption and desorption. In general, alloying and adding catalysts can improve the hydrogen absorption and desorption kinetics of the alloy and reduce the activation energy but do not necessarily change the RCS. According to the current research results, most of the hydrogen absorption and desorption processes are controlled by the diffusion of hydrogen or the nucleation and growth of MH. Thus, we propose the use of the kinetic models for the diffusion of hydrogen or the nucleation and growth processes to analyze the experimental data and the use of other kinetic models for other RCSs. After obtaining the best-fit model, the kinetic parameters, such as the rate constant, activation energy, Avrami exponent, and forms of the pressure and particle radius terms, can be determined according to the fitted results. Notably, accurate and unambiguous experimental data are the prerequisite for kinetic analysis. Then, the accurate kinetic equation and its parameters can be used in the mathematical model to design the HST. In the future, a high-accuracy and concise kinetic model is necessary to describe the actual hydrogen absorption and desorption processes of HSA. The kinetic analysis may be combined with in situ experiments or molecular dynamics theory to explain the kinetic mechanism. We believe that this review is helpful for researchers focusing on the kinetic mechanism of HSA or the design of the HST by providing systematic fundamentals, analysis procedures, and application of kinetic models.
This work was financially supported by the Chongqing Special Key Project of Technology Innovation and Application Development, China (No. cstc2019jscx-dxwtB0029), the National Natural Science Foundation of China (Nos. 51871143 and U2102212), the Science and Technology Committee of Shanghai, China (No. 19010500400), and the Shanghai Rising-Star Program (No. 21QA1403200).
All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.
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Figures(10) / Tables(4)
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| No. | HSA | Test conditions | RCS | E / (kJ∙mol−1) | Ref. |
| 1 | LaNi5 | 293–333 K, P − Peq = 100 kPa | Chemisorption | 19.66 (ab) | [57] |
| 2 | LaNi4.5Al0.5 | 273–343 K, P − Peq = 32.4 kPa | Chemisorption | 39.77 (ab) | [58] |
| 3 | MmNi4.2Al0.8 | 288–308 K, 0.013 MPa | Diffusion of hydrogen | 48.09 (de) | [59] |
| 4 | La1.5Ni0.5Mg17 | 523–573 K, 0.755 MPa | Diffusion of hydrogen | 90 (ab) | [60] |
| 5 | ZrCo | 544–603 K, P/P0 = 0.9–1 MPa | Nucleation and growth | 120 (ab) | [61] |
| 6 | TiFe | 243–298 K, 0.02 MPa | Chemisorption | 3.35 (ab) | [62] |
| 7 | TiFe0.8Ni0.2 | 373–423 K, |Peq − P0.5|/Peqa | Diffusion of hydrogen | 53.13 (ab) | [63] |
| 8 | Zr0.2Ho0.8Co2 | 323–473 K, 0–5 MPa | ― | 15 (ab) | [64] |
| 9 | Ti33V20Cr47 | 293–353 K, 3.5 MPa | Diffusion of hydrogen | 25.7 (ab) | [65] |
| 10 | Mg2Ni | 528–623 K, 4 MPa | Diffusion of hydrogen | 44 (ab) | [22] |
| 11 | Mg1.9Ag0.1Ni | 523–573 K, 0.768 MPa | Diffusion of hydrogen | 52 (ab) | [66] |
| 12 | MgH2−5at%V | 453–573 K, 0.015 MPa | ― | 62 (de) | [67] |
| Note: aP0.5 is the pressure at the reaction fraction of 0.5. | |||||
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| No. | HSA | Test conditions | RCS | g(P, Peq) | Ref. |
| 1 | LaNi5 | P − Peq = 50−200 kPa, 303 K | Chemisorption | P − Peq | [57] |
| 2 | LaNi4.5Al0.5 | P − Peq = 25.3−66.9 kPa, 298 K | Chemisorption | P − Peq | [58] |
| 3 | La1.5Ni0.5Mg17 | 0.256–0.992 MPa, 553 K | Diffusion of hydrogen | [(P − Peq)/Peq]2 | [60] |
| 4 | Mg2Ni | 0.275–1.133 MPa, 553 K | Diffusion of hydrogen | P1/2 − P_{\rm eq}^{1/2} | [22] |
| 5 | MgH2−5at%V | 0.007–0.4 MPa, 533–673 K | ― | 1 − P/Peq | [68] |
| 6 | Mg−10wt%Ni | P − Peq = 20.1–54.6 kPa, 573 K | ― | [P − Peq]1.25 | [69] |
| 7 | ZrCo | <1 MPa, 300 K | Chemisorption | P2 | [70] |
| 8 | TiFe | 1.717–2.626 MPa, 294 K | Chemisorption | P − Peq | [62] |
| 9 | TiFe0.8Ni0.2 | 0.246–0.733 MPa, 408 K | ― | (Peq − P)/Peq | [71] |
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| No. | Test conditions | Kinetic model | RCS | E / (kJ·mol−1) | Ref. |
| 1 | 288–353 K, <1 MPa | CV | Chemical reaction | 32.2 | [75] |
| 2 | 273–363 K, 0.5 MPa | ― | Nucleation and growth | 31.8 | [76] |
| 3 | 333–338 K, 2 MPa | ― | Mixed (chemisorption and diffusion of hydrogen) | ― | [77] |
| 4 | 298–323 K, Pr = 2–5a | C-JMAK (n = 1) | Nucleation and growth (one-dimensional diffusion) | 27 | [78] |
| 5 | 298–313 K, Pr = 2–5 | C-JMAK (n = 1) | Nucleation and growth (one-dimensional diffusion) | 30–40 | [79] |
| 6 | 303–333 K, Pr = 2 | Jander | Diffusion of hydrogen | 27.7 | [80] |
| 7 | 303 K, 0.6–1 MPa | Chou | Diffusion of hydrogen | ― | [24] |
| Note: a Pr is the ratio of pressure to equilibrium pressure. | |||||
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| No. | HSAs | Preparation methods | Test conditions | Kinetic model | RCS | E / (kJ∙mol−1) | Ref. |
| 1 | Mg−30wt% LaNi5 | MA; HC | 302−423 K, 1 MPa | Chou | Surface penetration; Diffusion of hydrogen | 28.0 (ab); 25.2 (ab) | [25] |
| 2 | La1.5Ni0.5Mg17 | HC | 523−573 K, 0.755 MPa | Jander | Diffusion of hydrogen | 90 (ab) | [60] |
| 3 | Nd4Mg80Ni8 | Melting | 373−623 K, 3.4 MPa; 564−620 K, Vacuum | Chou | Diffusion of hydrogen (ab); Surface penetration (de) | 82.3 (ab) 97.5 (de) | [101] |
| 4 | Mg18In1Ni3 | MA | 493−583 K, Vacuum | C-JMAK (n = 1) | Nucleation and growth (one-dimensional diffusion) | 107 (de) | [102] |
| 5 | LaNiMg17 | MA | 553−623 K, 0.755 MPa | Chou | Diffusion of hydrogen | 71 (ab) | [103] |
| 6 | Mg−Y2O3; Mg−Y | Arc melting | 473−573 K, 3 MPa | C-JMAK | ― | 79.9 (ab) 56.2 (ab) | [104] |
| 7 | Mg80Ce18Ni2 | Melting | 523−585 K, 3.5 MPa | C-JMAK (n = 0.53) | Nucleation and growth (one-dimensional diffusion) | 63 (de) | [105] |
| 8 | Mg12NiY | Melting | 523−623 K, Vacuum | C-JMAK (n = 0.6−0.8) | ― | 44.96 (de) | [106] |
| 9 | Mg2In0.1Ni | Melting | 467−507 K, Vacuum | C-JMAK (n = 0.6−0.8) | Nucleation and growth (one-dimensional diffusion) | 28.9 (de) | [107] |
| 10 | Mg−TiH1.971−TiH1.5 | HC | 298−673 K, 4 MPa; 473−673 K, Vacuum | C-JMAK (n = 0.3−1.1) | Nucleation and growth (one-dimensional diffusion) | 12.5 (ab); 46.2 (de) | [108] |
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