6894
2009, 113, 6894–6897 Published on Web 04/08/2009
Effect of Lattice Strain on the Dehydriding Kinetics in Nanoparticles Vladimir P. Zhdanov*,†,‡ and Bengt Kasemo† Department of Applied Physics, Chalmers UniVersity of Technology, S-412 96 Go¨teborg, Sweden, and BoreskoV Institute of Catalysis, Russian Academy of Sciences, NoVosibirsk 630090, Russia ReceiVed: February 20, 2009; ReVised Manuscript ReceiVed: March 22, 2009
Hydriding or dehydriding of metal nanoparticles is accompanied by the lattice strain. Using an exactly solvable model describing the strain distribution, we show the effect of strain on the dehydriding kinetics controlled by hydrogen desorption from the surface layer. The strain is found to suppress the desorption rate primarily during the initial stage of the kinetics, and this stage may be relatively slow. The corresponding nonlinear kinetic features are observed in experiments. The formation and decomposition of metal hydrides can be described in terms of the theory of first-order phase transitions. For this reason, the understanding of thermodynamic features (e.g., phase diagrams and absorption isotherms) and kinetics of these processes is interesting from the point of view of basic physics and chemistry. In addition, this subject is of high current interest from the viewpoint of future automotive applications because hydrogen is a good energy-carrier candidate that could be a part of a carbon-free cycle, and in this context, the metal hydrides are often believed to be preferred over pressurized gas and other hydrogen storage methods due to their storage capacities and safe operating conditions.1 The physics of formation and decomposition of metal hydrides is complicated by accompanying lattice strain or, in other words, elastic lattice deformations, which influence both the thermodynamics and kinetics of these processes. The thermodynamic aspects of hydride formation in macroscopic samples can be described by using the Eshelby expression for the strain energy.2 This simple and robust approach cannot however be applied to the kinetics. Its applicability to the analysis of thermodynamics of hydride formation in nanoparticles is also limited (for the discussions of the conditions of the validity of the Eshelby model, see ref 3). More recent studies4 describe lattice distortions more explicitly, but the corresponding results cannot be directly applied to nanoparticles either. In the literature, one can also find many mean-field models, focused on the kinetics of formation and decomposition of hydrides in macroscopic samples5 and nanoparticles,6 and Monte Carlo simulations of the hydride formation in nanoparticles.7 These mean-field and Monte Carlo models do not however take the lattice strain into account. Our brief review of the literature above indicates that now it is timely to analyze the likely effects of the lattice strain on the metal hydriding and dehydriding kinetics. These kinetics are usually controlled by different steps, and accordingly, the effect of the lattice strain on the forward and backward routes may * To whom correspondence should be addressed. E-mail: zhdanov@ catalysis.ru. † Chalmers University of Technology. ‡ Russian Academy of Sciences.
10.1021/jp901552n CCC: $40.75
be different. In particular, the hydriding kinetics are often controlled by hydrogen diffusion from the surface layer via the newly formed hydride to the metal. The rate of diffusion jumps depends on the difference of the energies of an atom in the activated and ground states. If the effect of the lattice strain on these energies is nearly the same (this assumption is reasonable), the effect of the lattice strain on the hydriding kinetics may be negligible. In contrast, the dehydriding kinetics are often controlled by hydrogen desorption from the external surface layer (see below), and the lattice strain may be important. The situation in macroscopic samples and nanoparticles may be different as well because the lattice strain depends on geometry. If, for example, we have a single crystal with a perfect nanometer- or micometer-thick hydride film formed after exposure to hydrogen, the strain of this film will be controlled by the underlying metal, and to a first approximation, the film contraction will be independent of the film thickness. For this reason, there is often no need to treat explicitly the strain in the corresponding kinetic models. In the hydriding or dehydriding of metal nanoparticles, in contrast, the strain is strongly dependent on conversion. During dehydriding, for example, the strain of the newly formed thin metal shell is controlled by the hydride core, while, eventually, the metal shell becomes thick and controls the strain of the hydride core itself. For this reason, the effect of the lattice strain on the dehydriding kinetics in nanoparticles is expected to be more pronounced. In our Letter, we show what may happen in this case. In our model, a nanoparticle is considered to be spherical and initially to be fully hydrided. The hydride decomposition starts near the surface. During decomposition, the hydride core shrinks while the surrounding metallic shell becomes thicker. Thus, we use a shrinking core model. This model was earlier widely employed in the mean-field treatments of the hydriding and dehydriding kinetics in nanoparticles.6 However, as already noted, the available models6 do not take the lattice strain into account. To calculate the strain distribution in a partly hydrided nanoparticle, we should choose a convenient reference state. Although, as already noted above, we are interested in the situation where the hydride core is surrounded by the metallic shell, let us first consider that the particle is fully metallic, and 2009 American Chemical Society
Letters
J. Phys. Chem. C, Vol. 113, No. 17, 2009 6895
σrr )
E[(1 - σ)urr + 2σuθθ] (1 + σ)(1 - 2σ)
(5)
where E is Young’s modulus and σ is Poisson’s ratio. Substituting eqs 3 and 4 into eq 5 yields
σrr )
Ema 2Emb 1 - 2σm (1 + σ )r3
(6)
m
σrr ) -
Figure 1. (a) Metallic sphere of radius Rm in the absence of hydrogen (the subsphere of radius R represents a region to be hydrided). (b) Overlapping hydride subsphere (marked by dashes at r < (1 + R)R)) and metallic shell (at R e r e Rm) in the hypothetical case where there is no interaction between them. (c) Strained hydride subsphere and metallic shell (r is the radial coordinate).
its radius is Rm (Figure 1a). Let us then detach a subsphere of radius R in this metallic sphere and consider that the metal located inside of this subsphere is converted into hydride. To describe the lattice strain in this case, it is instructive first to imagine that the hydride core and the metal shell do not interact (Figure 1b). In this hypothetical case, the metallic shell will retain its size (R e r e Rm), while the hydride subsphere will expand, and its radius will be (1 + R)R, where R is the linear mismatch parameter (usually, R is lower or about 0.1). In our analysis, these states of the metallic shell and hydride core are considered to be a reference. This choice of reference states is convenient because in this case, there is no lattice strain except that related to the surface tension. For a given nanoparticle, the latter strain is however constant and can first be omitted compared to the strain caused by the metal hydride mismatch. Due to this mismatch, the hydride-related expansion of the subsphere of radius R will partly be suppressed by the metallic shell, the latter will be somewhat expanded, both of these phases will be strained, and the boundary between them will be located between R and (1 + R)R (Figure 1c). For the geometry under consideration, the displacement vector is radial, and we can use spherical coordinates, r, θ, and φ, in order to calculate the lattice strain and stress distribution by using the standard prescriptions of the theory of elasticity.8 In particular, the radial displacements in the metallic shell (at R e r e Rm) and hydride core [at 0 e r e (1 + R)R] are, respectively, represented as
b u ) ar + 2 r u ) -cr
2b b uθθ ) uφφ ) a + 3 3 r r urr ) uθθ ) uφφ ) -c
aR +
b + c(1 + R)R ) RR R2
2Emb Ehc Ema ) 1 - 2σm (1 + σ )R3 1 - 2σh
In general, the radial stress is expressed via the strain tensor as8
(9)
m
Third, the stress in eq 6 should vanish at the external boundary of the metallic shell, that is
a 2b )0 1 - 2σm (1 + σ )R3 m
(10)
m
To make our equations compact, we neglect the difference in the elastic constants of the hydride and metal, that is, we use Eh ) Em ) E and σh ) σm ) σ (this approximation is fairly accurate). In addition, taking into account that R , 1, we neglect R on the left-hand side of eq 8. With these modifications, eqs 8-10 are read as
b + cR ) RR R2 a c 2b )1 - 2σ (1 + σ)R3 1 - 2σ aR +
(11) (12)
a 2b )0 1 - 2σ (1 + σ)R3
(13)
m
and we get
2R(1 - 2σ)R3 3 3(1 - σ)Rm
(14)
3 (1 + σ)Rm a R(1 + σ)R3 ) 2(1 - 2σ) 3(1 - σ)
(15)
a)
(2)
(4)
(8)
Second, the stresses in eqs 6 and 7 should be equal on the boundary between the hydride and metal
(1)
(3)
(7)
where Em, σm, Eh, and σh are the parameters corresponding to the metal and hydride, respectively. To determine constants a, b, and c, we should scrutinize the boundary conditions. The first one can be obtained by taking into account that the linear lattice misfit, RR (Figure 1b), should be matched (Figure 1c) by the radial displacements in the metallic shell (at r ) R) and hydride core [at r ) (1 + R)R]. Employing eqs 1 and 2 for the displacements, we represent this condition as
where a, b, and c are positive constants. The corresponding components of the strain tensor are given by
urr ) a -
Ehc 1 - 2σh
b) c)
( ) 3 Rm
R3
-1 a)
( )
2R(1 - 2σ) R3 1- 3 3(1 - σ) R
(16)
m
Substituting eqs 14 and 15 into eq 3, we obtain the following expressions for the lattice strain near the external surface of the metallic sphere (at r ) Rm)
6896 J. Phys. Chem. C, Vol. 113, No. 17, 2009
urr ) -
2RσR3 3 (1 - σ)Rm
uθθ ) uφφ )
RR3 3 Rm
Letters
(17)
In our analysis of the dehydriding kinetics, we consider that there is no hydrogen in the gas phase. In this case, the hydride decomposition occurs via reversible detachment of hydrogen atoms from the hydride core, their diffusion in the metallic shell, reversible jumps to the external surface layer of the metallic shell, and irreversible associative desorption. All of these steps can be influenced by the lattice strain, and in general, the corresponding kinetic equations are rather cumbersome. To clarify what may happen in this case, one should take into account that the hydrogen diffusion in metals is rapid (the corresponding activation energy is only a few kcal/mol) and cannot limit the hydride decomposition in nanoparticles. Another important aspect is that for most metals, the electron density is higher than the optimum density for incorporation of a hydrogen atom, while in the surface layer, a hydrogen atom can find the optimum density, and accordingly, the binding of single hydrogen atoms on the surface is usually energetically preferable.9 The hydride formation occurs due to a relatively weak attractive interaction between nearest-neighbor hydrogen atoms. With these interactions, the energy of hydrogen atoms in the hydride core may be slightly lower or higher than that on the external surface of the metallic shell. In contrast, the activation barrier for associative desorption of hydrogen is usually high. For these reasons, the dehydriding kinetics in nanoparticles are often limited by associative desorption of hydrogen from the external surface of the metallic shell. Assuming the hydride decomposition to be limited by desorption, we treat a generic case where the binding energy of hydrogen atoms on the metal surface is slightly higher than that in the hydride. In this case, due to the supply of hydrogen from the hydride core, the surface layer is close to saturation during almost the whole course of the kinetics, and the total rate of hydrogen desorption from the metal surface can be represented as 2 W ) (4πRm ⁄ s)w
(18)
2 where 4πRm /s is the number of adsorption sites (s is the site area) and w is the desorption rate per site. The latter rate can, in turn, be represented in the Arrhenius form
w ) ν exp(-Ed ⁄ kBT)
(19)
where ν and Ed are the pre-exponential factor and activation energy. The strain of the metal lattice changes the binding energy and desorption activation energy of adsorbed hydrogen atoms and accordingly changes the desorption rate. In general, the tensile strain is usually accompanied by the increase of the binding energy and desorption activation energy of hydrogen, while the compressive stress is accompanied by the decrease of these parameters. In our case, the radial strain is compressive, while the tangential strain is tensile, and both strains are 3 [see eq 17]. The tangential strain is proportional to RR3/Rm however larger, and the activation energy for hydrogen desorption should increase with increasing RR3/R3m. Taking into account that the strain is relatively weak (because R , 1), we can expand the desorption activation energy in Taylor’s series with respect 3 ) and keep the linear term. to strain (i.e., with respect to RR3/Rm Thus, we have 3 Ed ) E°d + RAR3 ⁄ Rm
(20)
where E°d is the activation energy in the absence of strain and A is a positive constant.
Combining eqs 18, 19, and 20, we represent the total rate of hydrogen desorption as
(
2 W ) (4πk0Rm ⁄ s) exp -
RAR3 3 kBTRm
)
(21)
where k0 ) ν exp(-E°/k d BT). The number of hydrogen atoms in the hydride core is N ) 4πR3/3V, where V is the volume per atom. The rate of decrease of N is equal to W, that is, dN/dt ) -W. Using eq 21, we rewrite the latter equation as
( )
1 dN BN ) -κ exp Nh dt Nh
(22)
where Nh ) 4π(1 + R)3R3m/3V is the number of hydrogen atoms in the fully hydrided nanoparticle, and the two other parameters are defined as
κ ) 3Vk0 ⁄ [sRm(1 + R)3]
B ) RA(1 + R)3 ⁄ kBT (23)
Assuming, initially, the nanoparticle to be fully hydrided, that is, N(0) ) Nh, and integrating eq 22, we obtain the kinetics under consideration
N ) (Nh ⁄ B) ln[exp(B) - κBt]
(24)
To show the results of calculations, it is convenient to use the conversion fraction defined as f ) (Nh - N)/Nh. Concerning the conditions of applicability of eq 24, we may notice that it has been derived by assuming that the hydride core is able to maintain high coverage of the surface layer. This condition holds almost up to complete conversion. To describe the late stage when N is nearly equal to Nh, eq 24 can be modified. This is, however, beyond our present goals because our analysis is focused on the role of lattice strain. At the late stage of the kinetics, the strain is however negligible. According to eq 24, the time scale of the dehydriding kinetics is inversely proportional to κ, which, in turn, is inversely proportional to Rm. Thus, the kinetics time scale is proportional to Rm. This prediction, modified below by including the surface tension, is however not related to lattice strain under consideration. In the context of our goals, a more interesting prediction is that the shape of the kinetics depends on B. If the effect of lattice strain on the kinetics is negligible (this is the case if B , 1), eq 24 predicts linear kinetics, N = Nh(1 - κt). If, however, B > 1, the kinetics are appreciably nonlinear. To illustrate the nonlinear kinetics predicted by eq 24, it is instructive first to estimate physically reasonable values of parameters A and B defined by eqs 20 and 23. The activation energy (eq 20) for hydrogen desorption is determined primarily by the hydrogen binding on the metal surface, and taking into account that the desorption is associative, we can conclude that A is approximately two times larger than the corresponding coefficient describing the effect of lattice strain on the binding energy. The latter coefficient was, for example, calculated by Norskov and co-workers10 for CO and O adsorption on Ru(0001) by using DFT. The corresponding values are =1.2 and 7 eV, respectively (see Figure 1 in ref 10). The binding energy of hydrogen is usually lower or comparable to that of CO, and accordingly, the parameter A for hydrogen is expected to be somewhat lower or about 2.4 eV. Taking also into account that R is usually about 0.05, we conclude that the parameter B (eq 23) should be lower or about 3. The dehydriding kinetics predicted by eq 24 for B ) 1, 2, 2.5, and 3 are shown in Figure 2. If B ) 1, the kinetics is nearly linear. For B g 2, the kinetics are seen to contain a distinct relatively slow initial stage. This effect is directly related to
Letters
J. Phys. Chem. C, Vol. 113, No. 17, 2009 6897 the nonlinear features of the kinetics of individual particles are inevitably somewhat smeared or even can be completely washed out. In summary, we have scrutinized the influence of lattice strain on the dehydriding kinetics in nanoparticles. Our analysis shows that the strain results in the decrease of the process rate. This effect is more appreciable during the initial stage of the kinetics. For this reason, the initial stage may be relatively slow. The nonlinear kinetic features predicted by our analysis are observed in experiments (see, e.g., refs 11-13). Finally, it is appropriate to note that during hydriding and dehydriding of metal nanoparticles, the strain may also results in plastic deformation of the metal and/or the formation of cracks in the hydride, etc. The first part of our treatment can be used as a starting point to describe the latter processes (see, e.g., ref 15).
Figure 2. Conversion fraction as a function of time according to eq 24 for B ) 1, 2, 2.5, and 3.
lattice strain. During this stage, the metallic shell is thin, its strain is large, the activation energy of hydrogen desorption is increased, and accordingly, the desorption rate is relatively low. Our kinetic equations above have been derived assuming the binding energy of hydrogen atoms on the metal surface to be higher than that in the hydride. The opposite situation has been treated as well (not shown). In the latter case, the kinetic equations are somewhat different, but the qualitative features of the kinetics are the same. The deviations from the spherical symmetry also do not change our main conclusions. In addition, we may note14 that the surface tension can also contribute to the desorption activation energy (eq 20). The lattice strain related to surface tension is inversely proportional to Rm. It does not change the shape of the dehydriding kinetics, while the dependence of the time scale of the kinetics on Rm becomes proportional to Rm exp(-R*/Rm), where R* is the parameter proportional to surface tension and inversely proportional to kBT. Our estimates14 indicate that usually R* = 3-4 nm. Experimentally, the kinetics of hydride decomposition in nanoparticles were studied in several works.6,11-13 The most detailed studies clearly exhibit a relatively slow initial stage (see, e.g., Figure 9 in ref 11 for Mg(Ni)H2, Figure 1 in ref 12 for MgH2, and Figure 8 in ref 13 for NaAlH4). Equally often, this stage is not observed.6 Concerning the experiments, one should take into account that the measured kinetics represent the average of the kinetics running in particles with different sizes. The size distribution of fabricated nanoparticles is usually relatively broad (the average size is comparable with the standard deviation). Under such circumstances, after averaging,
References and Notes (1) (a) Berube, V.; Radtke, G.; Dresselhaus, M.; Chen, G. Int. J. Energy Res. 2007, 31, 637. (b) Dornheim, M.; Eigen, N.; Barkhordarian, G.; Klassen, T.; Bormann, R. AdV. Eng. Mater. 2006, 8, 377. (2) (a) Cahn, J. W.; Larche, F. Acta Metal. 1984, 32, 1915. (b) Schwarz, R. B.; Khachaturyan, A. G. Acta Mater. 2006, 54, 313. (c) Lexcellent, C.; Gondor, G. Intermetal. 2007, 15, 934. (3) Fratzl, P.; Penrose, O.; Lebowitz, J. L. J. Stat. Phys. 1999, 95, 1429. (4) (a) Brener, E. A.; Marchenko, V. I.; Spatschek, R. Phys. ReV. E 2007, 75, 041604. (b) Song, J. C. W.; Ahluwalia, R. Phys. ReV. B 2008, 78, 064204. (5) (a) Zhdanov, V. P.; Krozer, A.; Kasemo, B. Phys. ReV. B 1993, 47, 11044. (b) Schweppe, F.; Martin, M.; Fiornm, E. J. Alloys Compd. 1997, 261, 254. (c) Ledovskikh, A.; Danilov, D.; Notten, P. H. L. Phys. ReV. B 2007, 76, 064106. (d) Borgschulte, A.; Gremaud, R.; Griessen, R. Phys. ReV. B 2008, 78, 094106. (6) (a) Nahm, K. S.; Kim, W. Y.; Hong, S. P.; Lee, W. Y. Int. J. Hydrogen Energy 1992, 17, 333. (b) Martin, M.; Gommel, C.; Borkhart, C.; Fromm, E. J. Alloys Compd. 1996, 238, 193. (c) Inomata, A.; Aoki, H.; Miura, T J. Alloys Compd. 1998, 278, 103. (d) Chou, K.-C.; Li, Q.; Lin, Q.; Jiang, L.-J.; Xub, K.-D. Int. J. Hydrogen Energy 2005, 30, 301. (7) (a) Zhdanov, V. P.; Kasemo, B. Chem. Phys. Lett. 2008, 460, 158. (b) Zhdanov, V. P.; Kasemo, B. Physica E 2009, 41, 775. (8) Landau L. D.; Lifshitz, E. M. Theory of Elasticity; Pergamon: Oxford, U. K., 1986; secion 7. (9) Norskov, J. K.; Besenbacher, F. J. Less Common Met. 1987, 130, 475. (10) Mavrikakis, M.; Hammer, B.; Norskov, J. K. Phys. ReV. Lett. 1998, 81, 2819. (11) Martin, M.; Gommel, C.; Borkhart, C.; Fromm, E. J. Alloys Compd. 1996, 238, 193. (12) Liang, G.; Huot, J.; Boily, S.; Schulz, R. J. Alloys Compd. 2000, 305, 239. (13) Zaluska, A.; Zaluski, L.; Strom-Olsen, J. O. Appl. Phys. A: Mater. Sci. Process. 2001, 72, 157. (14) Langhammer, C.; Zhdanov, V. P.; Zoric, I.; Kasemo, B., in preparation. (15) Zhdanov, V. P.; Kasemo, B., Nano Lett. 2009, DOI: 10.1021/ nl9008293.
JP901552N
