Chemical Diffusivity for Hydrogen Storage: Pneumatochemical

Article pubs.acs.org/JPCC

Chemical Diffusivity for Hydrogen Storage: Pneumatochemical Intermittent Titration Technique Young-Hun Kim,† Eui-Chol Shin,† Sun-Jung Kim,‡ Choong-Nyeon Park,† Jaekook Kim,† and Jong-Sook Lee*,† †

School of Materials Science and Engineering, Chonnam National University, Gwangju 500-757, Korea Mirae SI Co. Ltd., Gwangju 500-470, Korea

‡

ABSTRACT: Point relaxations for the pressure−composition isotherms were analyzed as a solid-state diffusion problem of hydrogen solutes from the gas phase provided in a limited amount by a pulse-like supply. For the large differential hydrogen storage, typical in the PCI plateau region, represented by small effective volume ratio λ parameter in the diffusion equation, the relaxation occurs much faster than in the galvanostatic or potentiostatic relaxations for the same chemical diffusivity values. The analysis was successfully applied to Mg/MgH2 system and the apparent chemical diffusivity and the hydrogen self-diffusivity were evaluated as a function of the state of hydrogen storage, in an analogous manner to the well-known electrochemical titration techniques. The increase in the apparent chemical diffusivity and hydrogen self-diffusivity in the plateau region with the increasing state of storage can be consistently explained by the lower hydrogen self-diffusivity in MgH2 than in Mg solid solution. The hydrogen selfdiffusivity variation in the single phase MgH2 region suggests the vacancy mechanism. The method is named as pneumatochemical intermittent titration technique (PnITT).

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INTRODUCTION Hydrogen storage is regarded as the major hurdle in the realization of the Hydrogen Economy. The storage characteristics are represented by pressure−composition isotherms (PCI) or pressure−composition−temperature (PCT) properties. Figure 1 shows the Gibbs free energy curves of the metal hydrides of two solid solutions α and β and the corresponding pressure−composition isotherm for hydrogen storage applica-

tions. The isotherm also represents an electrical potential− composition curve for a hydrogen or a lithium battery since the potential and hydrogen activity or pressure is given by Nernst equation ΔE = −

ΔE = −

RT ln aLi F

(1)

Figure 2a shows an experimental PCI of MgHδ/MgH2−δ (or Mg/MgH2 system in brief), which can be compared to the potential−composition curve of an electrochemical storage system, LiδFePO4/Li1−δFePO4 (or FePO4/LiFePO4) shown in Figure 2b. They are examples clearly illustrating the common thermodynamic relationship depicted in Figure 1 in the pneumatochemical and electrochemical storage system, respectively. The symbols represent the near-equilibrium value after each intermittent hydrogen injection and electrochemical titration of lithium, respectively. The lines in the two graphs of Figure 2 represent the transient pressure and potential values accompanying the concentration variation, which indicate a qualitatively difference between the typical pneumatochemical and electrochemical storage processes. The sawtooth shape transients for the PCI illustrate the principle of the volumetric method where the gas pressure variation is directly related to the hydrogen storage amount according to the ideal gas law. No kinetic information

Figure 1. Free energy curves of two solid solutions α and β and the Nernst potential, which is proportional to the logarithms of the equilibrium hydrogen pressure in hydrogen storage system and of the lithium activity in battery electrodes, respectively. © 2013 American Chemical Society

RT ln pH 2 ; 2F

Received: February 4, 2013 Revised: July 8, 2013 Published: August 19, 2013 19771

dx.doi.org/10.1021/jp401286b | J. Phys. Chem. C 2013, 117, 19771−19785

The Journal of Physical Chemistry C

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Figure 2. (a) Pressure−composition isotherms of MgHδ/MgH2−δ hydrogen storage material. (b) Potential−composition curve of a battery cathode system, LiδFePO4/Li1−δFePO4. The insets indicate the schematic storage process for the respective active material particulates.

Figure 3. Schematic representation of (a) galvanostatic intermittent titration technique (GITT) and (b) potentiostatic intermittent titration technique (PITT).

are collected from the series of pressure relaxation curves from different applied pressure values. Pneumatochemically the constant flux conditions may be implemented by mass-flow controllers, but the high-pressure regulation necessary for the hydrogen storage is technically demanding. Bielmann et al.2 recently reported a mass-flow method for the measurements of equilibrium pressure− composition isotherms from the series of measurements using different flow rates. It should be noted that the suggested massflow isotherm method is the exact analogy of the C-rate characteristics in the electrochemical charging systems. C-rate represents the charging currents in amperes to achieve the full capacity in an hour. The storage kinetics are indicated from the rate dependence of potential or pressure vs composition curves. The high storage rates decrease the practical storage capacity and increase the deviation of potential or pressure from the equilibrium value. The kinetic information from the ratedependent isotherms either electrochemically or pneumatochemically is so far limited to a qualitative and comparative one. In the present work, a new analysis framework is presented for the solid state hydrogen storage kinetics from the point

is present in this representation. However, the transients of GITT curves in Figure 2b indicate the relaxation behavior as a function of time which is proportional to the lithium amount inserted galvanostatically. Intermittent titration techniques as illustrated in Figures 2b have been applied in the electrochemical storage system. They provide a systematic evaluation of the kinetic parameters as a function of state of charge (SoC). Under the well-defined boundary condition controlled electrochemically, chemical diffusivities can be evaluated by the solution of the diffusion equation. However, quantitative kinetic investigations on the relaxations upon the intermittent sorption/extraction processes in the hydrogen storage system have been hardly reported. This may be first ascribed to the fact that pneumatochemical boundary conditions equivalent to the potentiostatic and galvanostatic control in the electrochemical storage are not easily experimentally viable. In the conventional Sieverts apparatuses, the pressure is not kept constant during hydriding and dehydriding. Park and Lee1 applied a pressure sweep method where hydriding rates at constant pressure conditions 19772



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close to zero are responsible for the orders-of-magnitude lower diffusivity than the nonplateau values. Figure 2a in comparison with Figure 2b suggests the same issue should be involved in the typical pneumatochemical storage system with the twophase coexistent plateau. As illustrated in Figure 3b, the intermittent titration can be also made by the stepwise change of potential, which is known as the potentiostatic intermittent titration technique (PITT).3−6,10 In PITT or in potentiostatic experiments in general, the chemical diffusivity can be derived from the current relaxation as

relaxation data of typical experiments for pressure−composition isotherms using a conventional Sievert’s apparatus. A pneumatochemical intermittent titration technique or PnITT is suggested, which can be a counterpart of the well-known intermittent titration techniques for the kinetic investigation in electrochemical storage system.

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BACKGROUND Figure 2b represents a much practiced electrochemical charging procedure for a battery known as the galvanostatic intermittent titration technique or GITT.3−6 The experiments were performed by intermittently applying a constant current Is for short time τ. Thus, the composition δ is proportional to time, and the curved transients in Figure 2b represent the time dependence of the potential during galvanostatically charging/ discharging. The procedure is schematically illustrated in Figure 3a. Voltage relaxations upon depolarization, ΔEd,j, as well as during the polarization, ΔEp,j, are monitored. The method, as illustrated by the data points in Figure 2b, allows the measurement of the near-equilibrium potential−composition relationship represented by ∑i =i =0 j Es,i vs jIsτ, when the depolarization proceeds for sufficiently long time and the selfdischarging is negligible. GITT may be the most popular intermittent titration technique, as it allows the easy control of SoC and thus a systematic determination of the kinetic parameters as a function of SoC. The short-time dc polarization behavior during the galavanostatic titration can be related to the chemical diffusivity according to Fick’s law ∂c ∂ ⎛⎜ ̃ ∂c ⎞⎟ = D ∂t ∂x ⎝ ∂x ⎠

(2)

2 2 4 ⎛ Is ⎞ ⎡⎛⎜ dE ⎞⎟ ⎛ dE ⎞⎤ ̃ ⎜ ⎟ ⎢ ⎥ D= /⎜ ⎟ π ⎝ zFA ⎠ ⎣⎝ dc ⎠ ⎝ d( t ) ⎠⎦

(3)

I (t ) =

∫ i(t )dt D̃ L

πt

(4)

and I (t ) =

2(∫ i(t )dt ))D̃ L2

⎛ π 2Dt̃ ⎞ exp⎜ − 2 ⎟ ⎝ 4L ⎠

(5)

for short-time t ≪ L2/D̃ and long-time t ≫ L2/D̃ , respectively, for the planar electrode with thickness L. It should be noted that stored charge ∫ i(t)dt is not required for the D̃ evaluation when the exponential dependence of the long-time solution, eq 5, is used. This may be the very reason the constant activity or concentration condition at the surface is the most widely used and well-established diffusion problem in general, not limited to the electrochemical system. It should be noted that PITT has an experimental limitation in applications for the plateau region as well. In the complete equilibrium condition an intermittent titration within the plateau is not possible. In the experiments as shown in Figure 2b, depending on the equilibration time and the self-discharge behavior at each titration step, plateau potentials are likely to exhibit nonmonotonic variation due to the difference in the nonequilibrium degree in the individual potential value. A welldefined, stable, constant potential condition cannot be applied to the plateau region. The experimentally obtained diffusivity values by PITT exhibit less pronounced cusps at the plateau potential compared to those by GITT.7,8 There is, however, no fundamental reason why PITT is more reliable than GITT in the evaluation of the chemical diffusivity in the plateau region. There are further issues in the conventional kinetic formalism in the electrochemical storage system as eqs 3−5. Even in the single phase system, the kinetic process is not necessarily limited by the solid state diffusion. The process can be significantly mixed with the interface reaction and/or the slower trapping and phase transformation reaction. The distinction of the mixed processes in the time dependence of the relaxation is generally nontrivial, if not impossible. It should be also noted that for the practical storage system the active particles are in irregular shape and size as shown by the example of MgH2 powder for the hydrogen storage in Figure 5. Spheres of the average size, a, are assumed for the quantitative evaluation. For the exponential relaxation behavior in eq 5 the substitution of a for L/2 can be made for the D̃ evaluation, but the solutions for the spherical geometry rather than eqs 3−5 should be employed.11,12 Another critical issue in the diffusion geometry is the effective diffusion length used for the quantitative evaluation. The effective length for the diffusion problem can be macroscopic as the electrode layer thickness in the electrochemical storage or entire chunk size in the hydrogen storage. It can be secondary particle or aggregate

as

where z, F, and A are the absolute charge valence, Faraday constant, and the electrode area for the one-dimensional case of thin-film electrodes.3−6 Note that in Figure 2b the polarization transients over ΔEp,j of Figure 3a are similarly indicated as the titration amount, δ, proportional to the charging time. The depolarization over ΔEd,j in Figure 3a is represented by the vertical lines in Figure 2b since the composition does not change during depolarization. The depolarization relaxation represents the kinetics of the homogenization of the concentration profile developed during τ. This becomes rather a nontrivial diffusion problem, and hence, the relaxation has not been used for the kinetic analysis. The solution as eq 3 is derived for the diffusion in the single phase. As GITT allows the measurements of kinetic parameters as a function of state of charge (SoC), effective chemical diffusion coefficients are derived for all SoC, often including the two-phase coexistence plateau region as illustrated in Figure 2b.7−9 In the complete thermodynamic equilibrium, the dE/dc value for the plateau region of the two-phase coexistence as shown in Figure 1 should be zero. Thermodynamic equilibrium value dE/dc = 0 corresponds to zero chemical diffusivity according to eq 3. Vanishing diffusivity may be considered a consequence of the infinitely large chemical capacity proportional to (dE/dc)−1. In the electrochemical storage system as shown in Figure 2b,7,8 kinetically limited finite dE/dc values 19773



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size when the permeation and diffusion along the boundaries are negligible. If not, the primary particle size, which can be as small as a few nanometers, should be used for the D̃ evaluation. Depending on the assumption of the effective sample size, the diffusivity estimated from the experimental relaxation constant τ, which is proportion to L2/D̃ or a2/D̃ , can vary by orders of magnitude. For most electrode materials or hydrogen storage materials, which are predominantly electronic conductors, the component diffusivity or self-diffusivity of lithium or hydrogen ions can be further evaluated from the chemical diffusivity as ⎛ d ln aA ⎞−1 DA = ⎜ ⎟ D̃ ⎝ d ln cA ⎠

(6) Figure 4. XRD of commercial MgH2 powder indicating the trace amount of Mg and Mg oxyhydride. Note that the intensity is represented in logarithmic scale.

where the thermodynamic factor or Wagner factor can be obtained from the potential−composition curve as ⎛ d ln aA ⎞ zFc dE ⎜ ⎟= RT dc ⎝ d ln cA ⎠

(7)

In the presence of these issues in common with the electrochemical storage system, this work will illustrate how the diffusion equation is to be formulated for the point relaxations in the hydrogen storage experiments. On the basis of the solution obtained for the appropriate boundary conditions, apparent chemical diffusivities will be systematically evaluated as a function of state of storage (SoS) in the pressure− composition isotherm, in an analogous manner to the electrochemical titration techniques. It will be shown that the pneumatochemical method for the hydrogen storage provides the direct information on the matter transport and thus can be advantageous for the kinetic analysis than the electrochemical counterpart.

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EXPERIMENTAL SECTION Commercial MgH2 powder (Alfa Aesar, 98% balance Mg) was used for the experiments. Structural and thermal characterization of the powder was performed by X-ray diffraction (XRD) (Panalytical X’pert pro multipurpose X-ray diffractometer, Almelo, The Netherlands), scanning electron microscopy (SEM) (FE-SEM-JSM-7500F, Japan), Brunauer− Emmett−Teller (BET) analysis (nanoPOROSITY-XQ, Mirae SI, Korea), and thermal gravimetry and differential scanning calorimetry (TG/DSC) (LABSYS evo, Setaram, France). The PCT analysis (Anysorb HQ, Mirae SI, Korea) was performed using 1 g of MgH2 powder. The sample was pretreated at 450 °C for 2 h. The hydrogenation and dehydrogenation was performed at 325 °C. The measurement procedure and the data recording is fully automated. Data analysis and simulation were performed using commercial softwares of MS Excel 2010 with Visual Basics for Applications (VBA) and OriginPro 8.6 (Origin Lab, Northampton, MA, USA).

Figure 5. FE-SEM images of as-received MgH2 powder.

as ca. 70 μm. The presence of fissures and nanoscale granules on the surface may explain the BET surface area 3.2 m2/g. The BET surface area increased to 8.2 and 9.8 m2/g when the degassing temperature of the vacuum of 5 × 10−3 Torr was raised to 200 and 250 °C, respectively. The treatment is supposed to remove the passivated layer of the oxyhydride and provide a better access to pores and fissures. The possibility of dehydrogenation by the pretreatments in BET measurements is excluded since the hydrogen desorption of the as-received powder occurs only above 400 °C, as shown by the thermal effects of 62.3 kJ/mol accompanying the weight loss close to the theoretical value of 7.6%. The curves are normalized with respect to the sample mass of 5.8 mg. Pneumatochemical Intermittent Titration. As schematically represented in the insets of Figure 2, the storage processes in the solid state hydrogen storage and in the electrochemical

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RESULTS AND DISCUSSION Material Characterization. The XRD in Figure 4 indicates the presence of β-MgH2 in rutile structure and a small amount of Mg metal and an oxyhydride (Mg4O3(OH2)). Note that the intensity is in the logarithmic scale to indicate the trace amounts of secondary phases more clearly. The SEM image in Figure 5a shows that powders are very irregular in size and shape, although the average size may be approximately consistent with the nominal particle size given by the factory 19774



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valves for the high pressure gas cylinder and for the sample chamber, respectively, and a pressure sensor (indicated by ‘P’ box). At each titration step of the hydrogen absorption the pressure in the system is increased at the instant indicated by ‘1’ with the valve to the sample chamber closed. The next step is to close the valve to the cylinder, indicated by ‘2’, and thus to obtain a constant pressure in the reservoir. Then the valve to the sample chamber is open for the hydrogen injection at timing ‘3’. The pneumatochemical titration procedure can be compared to those of the electrochemical titration techniques illustrated in Figure 3. The pressure profiles in a PCT procedure indicate some features of potential (voltage) curves in the procedures of GITT and PITT in Figure 3. The procedure using the valve action at instants ‘1’ and ‘2’ may be compared to the potentiostatic step in PITT with the pressure as control parameter. However, the valve to the sample chamber is yet closed, and there is no corresponding sample response. Hydrogen is then introduced to the sample chamber by a mechanical valve action at instant ‘3’. Since then, the pressure becomes the monitor parameter of the sample response. The abrupt drop in pressure, which may look similar to the IR drop in GITT in Figure 3a, represents the pressure decrease due to the expansion of the system volume. The pressure relaxation afterward represents the sample response upon hydriding. As indicated in Figure 7b, the pressure relaxation magnitude Δpj represents the hydrogen absorption amount Δwj in the storage material. The final value after relaxation corresponds to the equilibrium pressure value, pj, thus allowing the measurements

storage are similar. The facile mass transport via the liquid electrolyte and conducting additives to the surface of the active electrode particles in the conventional batteries can be compared to the fast gas phase transport followed by solid state diffusion in the hydrogen storage. Both processes involve the intermediate surface reactions: charge-transfer in the electrochemical system and adsorption, dissociation, and possibly ionization in the hydrogen storage system.

Figure 6. TG/DSC of as-received MgH2 powder.

Figure 7a schematically illustrates the PCT measurements in a typical Sievert’s apparatus by the volumetric method. The essential parts of the Sievert’s apparatus are the mechanical

Figure 7. (a) Schematics of the pressure−composition isotherm measurements in a Sievert’s apparatus. (b) Diagram separately representing the hydrogen injection procedure as a pulse-like input and the consequent pressure relaxation in the sample chamber. 19775



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Figure 8. Pneumatochemical intermittent titrations in Mg/MgH2 system at 325 °C of the selected numbered points of absorption PCI shown in the inset.

pressure−composition isotherm of pj vs ∑i =i =0 j Δwi. The transients Δpj(t) should represent the hydrogen storage kinetics of the present interest, which is analyzed in the following section. Experimentally monitored relaxation curves are presented in Figure 8 of Mg/MgH2 system at 325 °C with the inset presenting the near-equilibrium pressure−composition points. The time axis is represented with arbitrary offsets and intervals and only the sample response after the valve action at ‘3’ is presented. Each relaxation ranges around 10 000 seconds. Note that the relaxations of the consecutive points from 46 to 49 are smoothly connected by subtracting the intermediate periods for the pressure control. The hydrogen desorption was not possible at 325 °C, which indicates the problematic desorption kinetics of Mg/MgH2 system for on-board application. While the desorption of the as-received powders is possible only above 400 °C in the TG-DSC experiment (Figure 6), the powders activated at 450 °C in PCI experiments can reversibly absorb and desorb hydrogen at 350 °C as shown in Figure 2a. Manifestation of Diffusion Problem. The hysteresis and the deviation from the equilibrium value of the isotherms can also qualitatively indicate kinetic limitations as in the electrochemical storage system. The issue of the present work is whether and how the kinetic information can be extracted from the relaxation data, Δpj(t), from the conventional Sievert’s method as illustrated by Figure 7a and Figure 8. While the direct analysis of the time-domain response in the electrochemical storage using the formulations such as eqs 3 to 5 is also well-established and widely practiced, the situation in the pneumatochemical storage kinetics is in great contrast. To the authors’ knowledge the point relaxation curves in the PCI measurements have never been quantitatively analyzed. As pointed out above, the experimental procedure illustrated in Figure 7 is clearly no constant-pressure condition, which corresponds to the potentiostatic condition of PITT, Figure 3b. Clearly, the procedure is not a constant-flux condition, which can be compared to the galvanostatic control of GITT as shown in Figure 3a. Thus, the pressure relaxations in PCT experiments cannot be described by any of the diffusion

solutions introduced above, eqs 3−5, which are well-known and not difficult to apply. Millet13−17 suggested a methodology using the Fourier transforms of the relaxation behavior, pneumatochemical impedance spectroscopy, or PIS. In this method, the transient pressure signal of the sample chamber, pA(t), corresponding to the electrical potential for the electrical impedance, is separately monitored. For the hydrogen flux, additional pressure monitoring is made using the reference chamber, i.e., dpB(t)/ dt ∝ dw(t)/dt. The pneumatochemical impedance ZPn(ω) can be estimated as p(ω)/i(ω) where p(ω) and i(ω) are the Fourier transforms of pA(t) and dw(t)/dt, respectively. Pneumatochemical impedance spectra were obtained and evaluated as a function of state of hydrogen storage from point relaxation curves of PCI for LaNi5 and Pd system.17−20 The hydrogen storage kinetics indicated in the spectra were compared between different activation states, cycles, and particle sizes, as well as states of hydrogen storage.17−20 In electrochemical experiments the control and monitor signals are independently obtained as clearly illustrated in Figure 3. The evaluation of the electrical and electrochemical impedance, Z(ω) by Fourier transformation of the timedomain response becomes thus more straightforward in this respect. Lee and Yoo evaluated the low-frequency range electrochemical impedance from the Fourier transform of the DC relaxation of the solid state electrochemical cells.21,22 In electrochemical system, the equivalent frequency domain information can be also directly and more conveniently obtained using the commercial frequency response analyzers.5−25 Although the principle of PIS is similarly straightforward, there are nontrivial experimental requirements.15,17 The separate monitoring of the hydrogen pressure in the reaction chamber, pA(t), in addition to the pressure signal, pB(t), is not a standard feature of the conventional Sievert’s apparatuses. Usually only one pressure signal is monitored, which represents the pressure in the sample chamber as well as the hydrogen absorption amount as indicated in Figure 7, Δp(t). There are also rather strong requirements in the data sampling for the Fourier transformation. The PIS has been so far limited to the 19776



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Table 1. First Five Roots of tan q = 3q/(3 + λq2) for Different λ

work group and their collaboration. In this work, a direct timedomain analysis of the point relaxation data is performed for the hydrogen storage kinetics by application of the appropriate diffusion solution.11,12,26 In the companion article, a frequency domain analysis of the relaxation curves as Figure 8 acquired from the conventional Sievert’s instrument will be also presented.27 In PCT experiments as schematically illustrated in Figure 7, a fixed amount of the hydrogen reserved by the valve action indicated by ‘1’ and ‘2’ is injected abruptly to the sample chamber by the valve action at ‘3’. The injected gas in the fixed amount then diffuses into the storage material. Albeit nontrivial and not well-known, the problem has been already solved as the diffusion of solute in a limited amount from a well-stirred fluid.11,12,26 The ideal applications of this method are suggested to be sorption of gas or vapors by plastics, rubber, metals, porous adsorbents, etc., since a pure gas is always a well-stirred fluid and its concentration can be measured by its pressure.26 As mentioned above, the method has not been explicitly applied for the solid-state hydrogen storage kinetics. The solution of the diffusion equation is more complicated than eqs 3−5 for constant-flux, constant-pressure, or surface activity conditions utilized for the D̃ evaluation by GITT and PITT. The boundary condition in the diffusion of solute in a limited amount from a well-stirred fluid is the mass balance between the gas phase in a limited and constant volume Vg and the sorbent. For the application in the hydrogen storage materials in powder form without a strong shape anisotropy, the diffusion for a granule of radius a in a spherical geometry is considered. Then the mass balance requires the condition Vg ∂c ⎛ ∂c ⎞ = 4πa 2D̃ ⎜ ⎟ ⎝ ∂r ⎠r = a α ∂t

(8)

Then the diffusion solution for the pressure relaxation Δp(t)/ Δpj solved using separation of variables is ∞

∑ n=1

6λ(1 + λ) exp(− qn 2τ ) 9(1 + λ) + λ 2qn 2

(10)

where τ = D̃ t/a = t/τPn, qn are nonzero, positive roots of 2

tan q =

3q 3 + λq 2

q1

q2

q3

q4

q5

3.1416 3.1426 3.2316 3.3117 3.5059 3.7266 3.9720 4.2346 4.3538 4.4211 4.4639 4.4786 4.4936

6.2832 6.2842 6.3302 6.3757 6.5024 6.6815 6.9387 7.2965 7.4882 7.6013 7.6745 7.6997 7.7253

9.4248 9.4258 9.4568 9.4878 9.5778 9.7158 9.9421 10.3298 10.5752 10.7300 10.8328 10.8681 10.9042

12.5664 12.5674 12.5904 12.6138 12.6834 12.7928 12.9864 13.3694 13.6514 13.8434 13.9744 14.0197 14.0664

15.7080 15.7090 15.7271 15.7460 15.8020 15.8924 16.0581 16.4213 16.7260 16.9500 17.1080 17.1640 17.2210

q1 , which deviate from the ten-term summation in eq 10 for τ < 0.1. The first term approximation is shown to be satisfactory for τ > 0.1 regardless of λ values. The slopes represent the q1 values for different λ values in Table 1. For the constant pressure condition (λ → ∞) q1 value becomes π and the simulation for λ = 1010 shown in Figure 9a essentially describes the case. It is shown that the first term (indicated by red line) describes the relaxation satisfactorily for Δp(t)/Δp < 0.4 when τ > 0.05. Large λ corresponds to the presence of a sufficient amount of the solute in the medium, and the condition λ → ∞ is equivalent to the constant pressure condition. The constant pressure condition corresponds to the constant potential or potentiostatic condition in the electrochemical storage, PITT, represented in Figure 3b. The longterm solution of the current relaxation for PITT in eq 5 is for the one-dimensional one-sided diffusion in the slab geometry with thickness L. Since the current response corresponds to the derivative of the concentration at the surface, the same exponential decay should be observed. Time-dependence in eq 5 can be obtained using q1 value π/2 for the slab geometry and a substituted by L. Considering a diffusion from both sides of the slab with thickness L, a should be replaced by L/2. The diffusion in a sphere with radius a is four times faster than that in the slabs with the equivalent length L = 2a. The q1 value increases for small λ as 4.4936 or 1.43π for λ = 0 from π for λ = ∞. The logarithmic plots in the rightmost column of Figures 9 and 10c indicate the difference in the longtime exponential decay. The effect of λ in the pre-exponential coefficients in eq 10 and consequently the relaxation Δp(t)/Δp at small times is more notable. For small λ, the pressure decreases precipitously at short times τ ≈ 0. While for λ = 1 the first term is a good approximation for Δp(t)/Δp < 0.2, for λ = 0.1, 0.01, and 0.001, the exponential regime for τ > 0.1 Δp(t)/ Δp becomes lower than 1%, 0.1%, and 0.01%, respectively. That is, for the time range suitable for the first exponential term approximation, the relaxation becomes practically completed and the pressure relaxation becomes below the level of an experimental significance. Even if measured, the signal is likely to be affected by the gas leak and the pressure fluctuations. Therefore, the relaxation behavior at shorter time τ < 0.1 should be evaluated for the kinetic information, which is described by the full solution of eq 10, not by the first few terms. Since the series in eq 10 converge slowly at short times, the description of the behavior at τ < 0.1 requires the summation of

where α is the sorption or partition coefficient representing the gas−solid concentration ratio. In the volumetric method, the effective volume ratio λ of fluid to sorbent volume Vg/[(4/ 3)απa3] can be represented by pressure values. For the titration step j in Figure 7 or 8 pj − pj − 1 λj = Δpj (9)

Δp(t ) = Δpj

λ ∞ 1010 10 5 2 1 0.5 0.2 0.1 0.05 0.02 0.01 0

(11)

The first five roots of qn values are for different λ values estimated in Table 1. The tables are extended from Table 6.1 of Crank11 and Table 1 of Carman and Haul26 (values for λ = 0 are corrected) for λ values smaller than 0.1. The most convenient and practical application of the diffusion solution using separation of variables as eq 10 is the long time range where the first few terms become a satisfactory approximation. The plot of ln Δp(t)/Δp vs τ in the rightmost column of Figure 9 presents such long-time behavior. The plots for different λ values are drawn together in Figure 10c. The straight lines (in red) in Figure 9 represent the first term with 19777



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Figure 9. Pressure relaxations against different scales of the normalized time τ for various λ values: 1010 (a), 1 (b), 0.1 (c), 0.01 (d), and 0.001 (e). According to eq 10 the summation n = 1 to 10 (black lines) and the first term only (red lines). The thick blue lines represent the first term (n = 1) of solution represented by error function series, eq 12. See the text for more explanation.

e.g., 0.01, the relaxation at τ ≈ 10−3 is still below 10%. For λ = 0.001, the relaxation at τ ≈ 10−3 becomes less than 1%. Most of the relaxation occurs almost instantaneously. The apparently fast relaxations for small λ values of 0.1, 0.01, and 0.001 in comparison with λ > 1 should not be considered to indicate high diffusivity. All the calculation is shown for the normalized

quite a few terms. The black lines in Figure 9 represent the summation of ten terms of eq 10 from n = 1 to 10, which can be considered sufficiently accurate for τ > 10−3. The linear− linear plots of the first column in Figure 9 indicate that the tenterm summation describes the short-time relaxation below τ = 0.1 satisfactorily. It should be noted, however, that for small λ, 19778



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Figure 10. Comparison of pressure relaxations for various λ values for the √τ dependence (a), √τ−1 dependence, and exp(−q1τ) dependence.

τ = D̃ t/a2 = t/τPn. A finite, limited amount of solute compared to the absorption capacity results in the fast completion of the absorption process. The short-time behavior of the diffusion solution can be described more conveniently by an alternate form of solution. The solution of diffusion equation using Laplace transform is represented by the series of (complementary) error function of √τ.11,12 The error function solution for the present case is represented as

from the coefficient 6(1 + λ)/√πλ with respect to √τ. It should be noted that the √τ approximation is not a practical solution for small λ values since the validity regime for the √τ dependence becomes extremely short for small λ, e.g., √τ ≲ 10−4 or √τ ≲ 10−8 for λ = 0.1. For λ → ∞, the relationship corresponds to the well-known short-time solution at constant pressure condition. The case for electrochemically equivalent potentiostatic condition for PITT was introduced by eq 4. For the one-dimensional slab geometry, the factor 2 is used instead of 6 in eq 16. The current relaxation in eq 4 exhibits √τ−1 dependence since it corresponds to the derivative of the concentration relaxation with √τ−1 dependence. It should be distinguished from √τ−1 dependence of pressure relaxation itself, represented in the fourth column of Figures 9 and 10b, which will be discussed in the next section. As indicated by the upturn at large τ for λ = 1010, 1, and 0.1 in thick blue curves in Figure 9a−c, the error function series do not converge at long times. For λ = 1010, the deviation becomes significant for τ ≳ 0.25 where Δp(t)/Δpj ≲ 0.05. For λ = 1, the deviation is appreciable for τ ≳ 0.2, but the relaxation magnitude is reduced to 1%. To describe the long time behavior, many higher n terms of the error function are needed. For those regimes, the first-term exponential function can be satisfactorily employed, as discussed above. Therefore, the first term of the series solution in eq 12 can be considered a general solution applicable for a wide range of states of hydrogen storage with λ ≤ 1 in the practical system as will be shown for Mg/MgH2 system later in this work. There exists, however, a numerical limit in the application of eq 12 as indicated by the arrows in Figure 9d,e for λ = 0.01 and 0.001, respectively. The calculations were performed by MS Excel using built-in error functions. For small λ, z ∝ √τ/λ becomes large. The highest number recognized in commercial softwares such as Excel and Origin Lab (which is used for fitting experimental data presented below) was found to be 1.8 × 10308. Therefore, a numerical evaluation is possible only for τ values where exp z2 of e erfc z ≡ exp z2erfcz is smaller than the numerical upper limit, although e erfc z term does not diverge. For λ = 0.01 in Figure 9d, the error function solution estimated for τ ≲ 10−2 is shown to be smoothly connected to the regime where the ten-term summation represented by black line is sufficiently accurate, i.e., τ ≳ 10−3. For λ = 0.001, shown in Figure 9e, however, the evaluation is numerically limited for τ ≲ 10−4, and the ten-term summation of eq 10 is sufficiently accurate only for τ ≳ 10−3. There is a gap between τ = 10−4 and 10−3, connected by the dash−dot line, which is obtained by the √τ−1 approximation described in the next section.

⎡ γ 3γ γ2 Δp(t ) e erfc = (1 + λ)⎢ 1 e erfc 1 τ + ⎢⎣ γ1 + γ2 Δpj λ γ1 + γ2 −

3γ2 λ

⎤ τ + higher terms⎥ − λ ⎥⎦

(12)

where γ1 =

⎞ 1⎛ 4 ⎜ 1 + λ + 1⎟ , ⎠ 2⎝ 3

γ2 = γ1 − 1

(13)

and

e erfcz ≡ exp z 2erfc z

(14)

Note that the two terms in eq 12 correspond to the first term (n = 1) of the error function series solution for the diffusion in the spherical geometry. The solutions for different λ values are indicated in thick blue lines in Figure 9. In general, the first term of the error function solution can describe the very short time, initial relaxation behavior. The instantaneous relaxation from Δp(t)/Δp = 1 can be satisfactorily described by the term as indicated in the third-column plots with respect to √τ. Since for the low values of z, e erfc z = 1 −

2z 4z 3 z4 + z2 − + ··· π 3 π 4

eq 12 can be expanded for the small τ, as follows:

(15) 26

⎡ 6 ⎛ Δp(t ) τ λ ⎞ 9τ = 1 − (1 + λ)⎢ − ⎜1 + ⎟ 2 ⎝ Δpj 3⎠λ ⎣ π λ ⎛ 2λ ⎞ 36 τ 3/2 ⎤ ⎟ + ⎜1 + ···⎥ ⎝ 3 ⎠ π λ3 ⎦

(16)

The dashed lines in the third column graphs of Figure 9 indicate the linear √τ dependence at very short times. The dependence with different λ values is shown more clearly in Figure 10a. The chemical diffusion coefficients can be estimated 19779



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Figure 11. Fit parameters of pneumatochemical intermittent titration in Mg/MgH2 system at 325 °C. Data designated by PnITT(fixed λ) and PnITT(free λ) are fitted using the full solution of eq 12, FITT by eq 18, PnITT(long time) by the first term of eq 10, and PITT by the first term with λ = ∞. (a) λ values estimated using experimental pj and Δpj and fitted using eq 12. (b) Time adjustments Δt fitted using eqs 12 and 18, represented by PnITT and FITT, respectively. They are shown to be consistent with the exponential decay time τC. (c) The relaxation magnitude Δpj experimental, fitted by eq 12 with fixed and free λ, respectively, and fitted by the first term of eq 10. (d) The equilibrium pressure values pj experimental, corrected in the plateau region, and fitted by PnITT(fixed λ), PnITT(free λ), PITT, and FITT.

together in Figure 10b for different λ values. It is clearly shown that the approximation is satisfactory for λ = 0.01 and 0.001. From the slope the chemical diffusivity can be evaluated. It should be noted that the √τ−1 dependence represented in dash−dot lines fills the gap for λ = 0.001 originating from the numerical difficulty in the error function evaluation. In fact, D̃ evaluation from the √τ−1 dependence can be found in the electrochemical titration method. A third titration technique, not much well-known as GITT and PITT was also introduced as potentiometric technique, 5 a short-pulse method,3 or recently as Faradaic intermittent titration technique, or FITT, in rhyme.6 The lower diagram of Figure 7 for the point relaxations of pressure−composition isotherm, in fact, illustrates the principle of FITT. A short pulse of current or voltage results in the finite amount of charge at the surface Δq, which then diffuses into the electrode material assumed of an infinite extent. From the well-known plane-source solution

PnITT vs FITT. It has been concluded that the point relaxations in PCI experiments can be described by the error function solution in eq 12 without higher terms for a wide range of SoS. As will be presented in the next section the experimental point relaxations in Mg/MgH2 system has been fitted using the built-in error function in the commercial software, Origin Pro 8.6 (Origin Lab Corporation, USA). In addition to the numerical calculation limit mentioned above, as will be shown in the next section, there is also limited time range in the experimental data for the application of the theoretical model. The fitting using the error function solution thus converged only with appropriate initial values and cost a considerable calculation time. While the long-time and shorttime behavior can be approximated as an exponential decay or √τ dependence, the error function solution, eq 12, can be written for large z(>0) as e erfcz ≈

1 ⎛ 1 1 3 ⎞ ⎜1 − + 2 4 ···⎟ 2 ⎝ z π 2z 2 z ⎠

(17)

Δc(x , t ) =

where the first term (z√π)−1 becomes a good approximation for z > 3. Then, the error function solution eq 12 can be approximated for small λ values as

Δp(t ) λ ≅ −λ Δpj 3 πτ

the √τ

(19)

dependence of the voltage signal can be obtained as

⎛ dE ⎞ Δq d E (t ) =⎜ ⎟ d(1/ t ) ⎝ dc ⎠ zFA πD̃

(18) −1

−1

⎛ x2 ⎞ exp⎜ − ⎟ ⎝ 4Dt̃ ⎠ zFA πDt̃ Δq

(20)

Equation 18 is the counterpart in the pneumatochemical titration for the diffusion in the spherical geometry, which can be written as

−1

which is valid for √τ > λ or √τ < λ . For λ = 0.01 and 0.001, the relationship is valid for √τ−1 < 102 and 103, respectively. The √τ−1 relationship is drawn in the fourth graphs of Figure 9d,e in dash−dot lines. They are also indicated in the log−log plots in the second column. The plots are drawn

d p(t ) Δp Δp ≈λ ≈λ 2 d(1/ t ) π /τPn 3 3 πD̃ /a 19780

(21)



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Figure 12. Fit examples of selected points at different state of storage. The first column graphs are the linear relaxation vs t, the second vs √τ−1 dependence, and the right column represents the exponential decay behavior. Fit parameters and results are indicated in the respective graph. In the left column graphs, the top and right axes represent the normalized values based on the fit results as for the simulations shown in Figure 9.

hardly located. Similarly, Δp and λ values are required for the evaluation of D̃ using eq 21. Small (dE/dc) or λ in the plateau region formally leads to low D̃ according to eq 20 or 21 for FITT, as well as GITT, eq 3. Although the presence of two phases in the plateau region does not concur with the application of the diffusion equation eq 2, GITT is often applied to the electrochemical storage system with a wide range of the plateau region, resulting in apparent or effective chemical diffusivity lowered by orders of magnitude in the plateau region.7−9 Such a low chemical diffusivity may be considered as a consequence of the large chemical capacitance,23−25,28 but a specific kinetic information seems difficult to be construed. There is an additional problem in the application of GITT or FITT in the plateau region even for any effective kinetic information. The potential relaxation cannot be directly related to the concentration and thus matter transport, not only because of the unknown thermodynamic factor but also because the electrical potential E reaches the quasi-

by neglecting small λ term. For the evaluation of τPn or D̃ , λ value is required, which can be obtained from the pressure values according to eq 9, which is proportional to the differential slope of PCI curve dp/dw. By the constant-volume condition and the ideal gas law, the quantity can be equated to the thermodynamic factor eq 7 as pj − pj − 1 ⎛ d ln p ⎞ ⎟ λj = =⎜ ⎝ d ln c ⎠ Δp j

(22)

which thus relates the self-diffusivity of hydrogen, DH, with the chemical diffusivity as D̃ = λDH. It should be noted that the evaluation of D̃ from eq 20 requires Δq and dE/dc value. The supply of charge Δq by short pulses may be not easily and in a well-defined way performed compared to the standard galvanostatic and potentiostatic control. Compared to GITT and PITT, FITT is described rather briefly in the original literature,3,5 and further application of this technique in the electrochemical storage kinetics is 19781



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Figure 13. Time constants τPn as a function of hydrogen amount (a) and hydrogen pressure (c) fitted by eq 12 (PnITT(fixed λ) and PnITT(free λ)), eq 18 (FITT), and by the first term of eq 10 with λ(exp) (PnITT(long time)) and with λ = ∞ (PITT). Chemical diffusivity evaluated from τPn by eq 12 with fixed λ and for λ ≳ 1 by the first term (n = 1) of eq 10 for the particle size 2a = 70 μm (left axis) and 0.7 μm (right axis), respectively, as a function of hydrogen amount (b) and hydrogen pressure (d). Hydrogen diffusivity is then evaluated as D̃ /λ.

λ(exp) in Figure 11a was evaluated from Δpj(exp) and pj(cor). The λj(exp) values are about 0.02 for the plateau region between hydrogen contents of 0.34 wt % (point 13) and 6.74 wt % (point 43). The λ value as high as 150.5 was evaluated for 7.53 wt % H2 at 8.21 atm (point 55) in the MgH2−δ solid solution region. The simulation of Δp(t)/Δp relaxation in Figure 9 showed that the error function solution in eq 12 can be applicable for the wide range of SoS constituted by the plateau region in Mg/ MgH2 system. The fitting was performed using a commercial software, Origin Pro 8.6. In view of λ values of ca. 0.02 for points 13 to 42 in the plateau region, comparable to the simulation shown in Figure 9d, the observed relaxation magnitude Δpj(exp) should correspond to a few percents of the full amplitude Δpj, with the initial drop portion missing. In order to conform to the theoretical model, the very initial relaxation is assumed cutoff before nominal monitoring timing. This is implemented by a shift in the time variable as t + Δt in the fit function, eq 12. In principle, λ, Δpj, pj, and τPn for the chemical diffusivity, (D̃ /a2)−1, can be statistically adjusted to describe the experimental data according to eq 12. Fit examples with fixed λ(exp) are shown in Figure 12. The time correction turned out a few thousands seconds, which are indicated in Figure 11b. The Δt values ranging 3000 to 6000 s are comparable to the monitored overall relaxation period. The monitored relaxations are in fact nicely described by the

equilibrium value as soon as two-phase equilibrium is locally maintained, although solid-state diffusion may still occur. The pneumatochemical titration method can be more informative than the electrochemical counterpart since the mass transport is directly monitored by the pressure relaxation even in the plateau region. The variation of the chemical diffusivity of a physical significance was thus derived for the plateau region of Mg/MgH2 as described in the next section.

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APPLICATIONS TO Mg/MgH2 SYSTEM The kinetic analysis developed in this work is applied to the point relaxation data of Mg/MgH2 system as shown in Figure 8. The key parameter for the hydrogen storage kinetic analysis by volumetric method is λ. The values at each point, λj, can be evaluated according to eq 9 as indicated in Figures 7 and 8. The experimental Δpj(exp) and pj(exp) are presented in Figure 11c,d, respectively. For the plateau region, pj(exp) increases overall with hydrogen absorption, typical for the kinetically limited real system, but also fluctuates locally, due to the variation in sampling cutoff. The deviation smaller than 0.00025 atm in average pressure values over 100 s was used as the criterion for the automatic cutoff in monitoring the relaxation. To avoid unphysical consequence of negative λ, a linear regression of the points in the plateau region was made, and the results are represented as pj(cor) in Figure 11d. The data 19782



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PnITT lead to λΔp of the order of one, and though much more scattered, consistent τPn values can be estimated. Analysis using the error function solution was applied up to point 50 in MgH2 solid solution region (7.489 wt % hydrogen amount, 5.22 atm hydrogen pressure, λ = 9.18). As shown in Figure 14 the point relaxations in the high SoS and high

exponential decay ∝ exp(−t/τC) as shown in the right column in Figure 12. The fit results are indicated in Figure 11 as PnITT(free λ). As the considerable part of the initial transients is assumed cutoff, fitted Δpj and λ values became strongly correlated and fitting procedures converged with difficulty. In Figure 11, the fit errors are not indicated for the sake of clarity. Fitting was also performed using the experimental λ values as fixed parameter, and the results are indicated by PnITT(fixed λ). Figure 11 shows the two results are essentially similar since the fitted λ value is close to λ(exp). The τPn values in Figure 13a,b showed a notably higher scattering for free λ than with fixed λ values. Figure 11c shows the fitted Δpj values, Δpj(PnITT), are larger by almost 2 orders of magnitude than Δpj(exp). The Δpj(PnITT) values close to 100 resulted from small λ values of the plateau region. In the solid solution region with increasing λ values, Δpj(PnITT) decreases rapidly, thus becoming closer to Δpj(exp). The feature was illustrated by the simulations in Figure 9. The fitted Δpj values are then obviously inconsistent with the λ values used in the fitting based on the experimental Δpj. Fitting with free λ results in the similar Δpj values are shown in Figure 11a. The feature appears contradictory but may be explained in view of the characteristics of the plateau region. The fitted large Δpj(PnITT) values correspond to large Δw from the point relaxation experiments. This in turn would lead to a proportional increase in pj − pj−1 values in the numerator of λ. In the complete equilibrium, λ or dp/dw should be zero, while experimentally a finite slope results. It is notable that the fitted pj values in Figure 11d suggest a flatter slope close to that of the complete equilibrium. Thus, the λ values measured or fitted may be considered as the finite limit quotient when both the numerator and denominator diverge. As shown in the right column in Figure 12, the relaxations can be almost perfectly fitted by the exponential function. Time shift, even if present, is not the adjustable parameter for the exponential decay when the pre-exponential factor is free to vary. The Δpj and pj values are also close to the experimental values as shown in Figure 11c,d for all SoS. The exponential decay function might be assumed to represent the long-time solution of eq 10 with q1 = π for λ = ∞ in Table 1. The approximation may be applied to the MgH2 solid solution region with high λ as shown in Figure 11a but not to the plateau regime with small λ values of ∼0.02. As emphasized in the theoretical analysis in conjunction with Figure 9c−e, the apparent relaxation behavior for small λ values such as 0.1, 0.01, and 0.001 may fallaciously suggest high chemical diffusivity values. Under the conventional constantpressure boundary condition, the decay constants of the exponential function correspond to q−2 1 τPn with q1 = π. The estimates are indicated by PITT in Figure 13a,b. They are shown by 1 order of magnitude larger than τPn values indicated by PnITT or FITT according to the error function solution eq 12 or √τ−1 approximation in eq 18 applicable to small λ values in the plateau region. Figure 12 indicates more or less consistent Δt values between the analysis using the error function solution and the approximate √τ−1 dependence. In the evaluation of τPn(FITT) from the slopes in √τ−1 dependence in Figure 12, middle column, according to eq 18 (or equivalently eq 21), λΔpj values are required. Depending on the choice of Δpj values in Figure 11c τPn, and thus, D̃ estimation would vary by 2 orders of magnitude. Those from the error function fitting indicated by

Figure 14. Typical relaxation behavior for the MgH2 solid solution region where the curves can be fitted as an exponential decay. The relaxation magnitude Δpj is very small, and thus, the data quality is limited by the resolution of the pressure gauge.

pressure region suffer from the poor digitization quality since the relaxation occurs over a very small pressure difference Δpj of a few 0.01 atm, but the resolution of the pressure gauge is limited to 0.002 atm. An exponential fitting can be still performed for the points up to number 55 (7.53 wt % hydrogen amount, 8.21 atm hydrogen pressure, λ = 150.5) as shown in Figure 14. The high λ values associated with these data allowed the estimates of τPn according to the first term approximation of eq 10. The exponential analysis was applied to the point 12 in Mg solid solution region, points 47 to 55 in MgH2 solid solution region with the respective q1 values depending on λ. For the point 47 with λ ≳ 0.78 with q1 = 3.815 or 1.2π. They are indicated as PnITT(long-time) in Figure 13. The behavior in the MgH2 solid solution region of τPn and D̃ derived from τPn(PnITT,fixedλ) are displayed more clearly in the plots with respect to the hydrogen pressure in Figure 13c,d. As illustrated in Figure 13 the evaluation of the point relaxation data in PCT experiments or the pneumatochemical intermittent titration technique (PnITT) allows chemical diffusivity values from τPn = (D̃ t/a2)−1 as a function of hydrogen storage state over a wide range, which is the goal of the theoretical analysis presented above in this work. It should be noted that the absolute magnitude of the chemical diffusivity values depend on the geometry factor used for the diffusion length. The average MgH2 particle size of 70 μm is provided by the company, which is also roughly consistent with powder size in the micrograph of Figure 5. From the surface area of BET analysis, 3.2, 8.2, and 9.8 m2/g after pretreatment at RT, 200, and 250 °C, respectively, and effective particle size 0.65, 0.25, and 0.21 μm may be estimated. When the size of 0.7 μm is assumed, which is by 2 orders of magnitude smaller than the factory value, chemical diffusivity values become lower by 4 orders of magnitude. 19783



Article

diffusion can be estimated as ca. 16 μm. The decrease of the self-diffusivity of MgH2 from the value near the boundary between the plateau and solid solution region can be explained by the vacancy mechanism in MgH2.30 As introduced above, pneumatochemical impedance spectroscopy, or PIS,15−17 has been applied for the hydrogen storage kinetic studies as a function of SoS in LaNi518,19 and Pd system.20 Application of the Fourier transform of the pressure relaxation data in the conventional Sievert’s apparatus as in this work resulted in the nontrivial spectral feature, different from those reported by Millet et al. The time-domain analysis performed in the present work allowed the understanding of such nontrivial impedance feature and evaluation of the consistent kinetic parameters, which will be reported in the companion article.27

Similar uncertainties in the effective diffusion geometry length appear to be associated with the D̃ evaluation in many kinetic studies in the system of particulates. Often the particles are dense aggregates of smaller primary particles. The microstructure aspect and the geometry factor employed in the evaluation should be taken into consideration for a proper assessment of the chemical diffusivity data. It should be also noted that the present theoretical modeling of diffusion in spherical geometry can be strictly applied only for the monosize spherical particles, which is obviously not the case as shown in Figure 5, and when the gas phase transport is sufficiently expeditious everywhere, the solid-state diffusion in the particles proceeds in a synchronized manner. The deviation from the assumption of the theoretical model is supposed to partly account for the errors in the model analysis. Figure 13 shows that in the plateau region D̃ decreases with the increasing state of hydrogen storage, but in the MgH2 single phase solid solution region D̃ increases, thus exhibiting the minimum at the boundary between the plateau and two-phase region. The overall variation is first accounted by the λ values presented in Figure 11a, similarly as shown for the electrochemical storage system with the two-phase plateau,7−9 where the deep minimum in the chemical diffusivity in the plateau region originates essentially from the near-zero potential− composition slopes. The relaxation behavior of the electrical potential signals, also largely determined by the two-phase equilibrium value rather than the concentration variation, fails to deliver a proper kinetic contribution. Although the evaluation of the chemical diffusivity in the present pneumatochemical analysis has the feature originating from λ in common with the electrochemical counterpart, in contrast to the electrical potential signal, pressure relaxations indicate the hydrogen absorption directly, regardless of the thermodynamic situation. Since λ values are almost constant around 0.02, D̃ variation in the plateau region represent the true variation in the absorption kinetics monitored in the raw relaxation data in Figure 8. Pure kinetic contribution in the chemical diffusivity can be represented by hydrogen self-diffusivity, DH, in Figure 13, estimated from the relationship D̃ = λDH. As well as D̃ , DH is well-defined only in the single phase region, and the estimate for the plateau region should be regarded as apparent DH. With λ contribution subtracted, DH is shown to exhibit the monotonic decrease with increasing hydrogen SoS. The overall behavior can be consistently explained by the diffusivity of MgH2 solid solution lower than that of Mg solid solution, although there is not sufficient data in Mg solid solution region. In the two-phase plateau region DH as well as D̃ decreases in logarithmic scale with the increasing hydrogen amount, which is proportional to the amount of the reacted MgH2 phase. In the linear scale, the diffusivity decreases more rapidly in the low reacted fraction range than in the high MgH2 fraction. The dependence suggests the hydriding kinetics limited by the diffusion through the hydride layer according to the shrinkingcore model: Hydride phase forms from the surface by consuming metal phase in the core.1 The lower diffusivity for points 13 and 14 near Mg solid solution boundary may be ascribed to the sluggish nucleation of MgH2. Well within the MgH2 solid solution region between 5 and 7 atm, the hydrogen diffusivity, DH, is estimated as ca. 3 × 10−11 cm2 s−1 or 3 × 10−15 cm2 s−1 for the particle size 70 and 0.7 μm, respectively. When compared with the reported NMR diffusivity,29 1.5 × 10−12 cm2 s−1, the true effective particle size for the solid state

■

CONCLUSIONS In spite of decades long studies, the proper analysis framework for the hydrogen storage kinetics was not established, which is indispensable for the correct evaluation of the new materials and/or new processing methods under development. The present work suggests a kinetic evaluation method in conjunction with well-established pressure−composition isotherm characterization for hydrogen storage thermodynamics. The relaxations upon the intermittent hydrogen injection or extraction can provide the kinetic information as a function of state of storage (SoS), in an analogous manner to the electrochemical titration techniques performed galvanostatically or potentiostatically, well-known as GITT and PITT. The pneumatochemical titration techniques performed by conventional Sieverts apparatus correspond to the diffusion problem under constant-volume conditions. Hydrogen injection by the mechanical valve action is like a pulse-like supply and the following process can be modeled as the diffusion of the solute in a limited amount from a well-stirred gas medium. The pneumatochemical titration technique (PnITT) developed in this work has the analogy to the third, relatively less known and less practiced potentiometic method or Faradaic intermittent titration technique (FITT). Because of the limited amount of the solute compared to the sorbent capacity, the absorption occurs instantaneously and fast for small effective volume ratio λ. Then the chemical diffusivity can be correctly evaluated from the error function solution or inverse root square dependence, not from the apparent exponential decay. The formalism was successfully applied to the hydrogen absorption in Mg/MgH2 system at 325 °C. A considerable lead-time was found necessary for the short-time behavior in the theory. The chemical diffusivity variation with distinct minimum at the boundary between the two-phase plateau and the MgH2 solid solution region is largely attributed to λ or thermodynamic factor, similar to the situation in the electrochemical titration practices. While the application of the electrochemical titration technique in the plateau region fails to provide any reliable kinetic information, the direct mass transport information conveyed by the pressure signal in the pneumatochemical experiments allows the estimation of the kinetic contribution of a physical significance. The variation of the diffusivity in the plateau region can be attributed to the diffusion-limited phase transformation according to the shrinking-core model. Vacancy diffusion mechanism is also suggested in the MgH2 solid solution region. The present method allows a quantitative and systematic assessment of the hydrogen storage kinetics as a function of hydrogen storage over a wide range from the 19784



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(16) Millet, P. Pneumatochemical Impedance Spectroscopy. 2. Dynamics of Hydrogen Sorption by Metals. J. Phys. Chem. B 2005, 109, 24025−24030. (17) Millet, P.; Decaux, C.; Ngameni, R.; Guymont, M. Experimental Requirements for Measuring Pneumatochemical Impedances. Rev. Sci. Instrum. 2007, 78, 123902−123902. (18) Millet, P.; Decaux, C.; Ngameni, R.; Guymont, M. FourierDomain Analysis of Hydriding Kinetics Using Pneumato-Chemical Impedance Spectroscopy. Res. Lett. Phys. Chem. 2007, 2007, 96251. (19) Millet, P.; Lebouin, C.; Ngameni, R.; Ranjbari, A.; Guymont, M. Hydriding Reaction of LaNi5: Correlations between Thermodynamic States and Sorption Kinetics during Activation. Res. Lett. Phys. Chem. 2008, 2008, 346545. (20) Lebouin, C.; Soldo, Y.; Grigoriev, S.; Guymont, M.; Millet, P. Kinetics of Hydrogen Sorption by Palladium Nanoparticles. Int. J. Hydrogen Energy 2013, 38, 966−972. (21) Lee, J.-S.; Yoo, H.-I. Direct Measurement of Partial Ionic Conductivity of Co1−δO Via Impedance Spectroscopy Combined with DC Relaxation. Solid State Ionics 1994, 68, 139−149. (22) Lee, J.-S.; Yoo, H.-I. A New Assessment of Ionic Conductivity of YBa2Cu3Ox via AC Impedance Spectroscopy Combined with DC Relaxation. J. Electrochem. Soc. 1995, 142, 1169−1176. (23) Lee, J.-S.; Jamnik, J.; Maier, J. Generalized Equivalent Circuits for Mixed Conductors: Silver Sulfide as a Model System. Monatsh. Chem. 2009, 140, 1113−1119. (24) Ahn, P.-A.; Shin, E.-C.; Kim, G.-R.; Lee, J.-S. Application of Generalized Transmission Line Models to Mixed Ionic-Electronic Transport Phenomena. J. Korean Ceram. Soc. 2011, 48, 549−558. (25) Ahn, P.-A.; Shin, E.-C.; Jo, J.-M.; Yu, J. H.; Woo, S.-K.; Lee, J.-S. Mixed Conduction in Ceramic Hydrogen/Steam Electrodes by Hebb−Wagner Polarization in the Frequency Domain. Fuel Cells 2012, 12, 1070−1084. (26) Carman, P. C.; Haul, R. A. W. Measurement of Diffusion Coefficients. Proc. R. Soc. Lond., Ser. A 1954, 222, 109−118. (27) Shin, E.-C.; Kim, Y.-H.; Kim, S.-J.; Park, C.-N.; Kim, J.; Lee, J.-S. Pneumatochemical Immittance Spectroscopy for Hydrogen Storage Kinetics. J. Phys. Chem. C 2013, DOI: 10.1021/jp4023647, companion article. (28) Jamnik, J.; Maier, J. Generalised Equivalent Circuits for Mass and Charge Transport: Chemical Capacitance and Its Implications. Phys. Chem. Chem. Phys. 2001, 3, 1668−1678. (29) Stioui, M.; Grayevsky, A.; Resnik, A.; Shaltiel, D.; Kaplan, N. Macroscopic and Microscopic Kinetics of Hydrogen in MagnesiumRich Compounds. J. Less-Common Met. 1986, 123, 9−24. (30) Luz, Z.; Genossar, J.; Rudman, P. S. Identification of the Diffusing Atom in MgH2. J. Less-Common Met. 1980, 73, 113−118.

relaxation data of the conventionally performed pressure− composition measurements using a Sievert-type apparatus. The general issue of the proper effective diffusion geometry and distance for quantitative kinetic evaluations is addressed.

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AUTHOR INFORMATION

Corresponding Author

*(J.-S.L.) E-mail: [email protected]. Phone: +82 (0)62 5301701. Fax: +82 (0)62 5301699. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Basic Research Laboratories (BRL) Program (No. 2009-0085441) of the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning. Support in VBA coding by Dr. Ji Haeng Yu, Korea Institute of Energy Research, is greatly appreciated.

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REFERENCES

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Chemical Diffusivity for Hydrogen Storage: Pneumatochemical

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