Apparent kinetics of hydriding and dehydriding of metal nanoparticles

Apparent kinetics of hydriding and dehydriding of metal nanoparticles

ARTICLE IN PRESS Physica E 42 (2010) 1482–1486 Contents lists available at ScienceDirect Physica E journal homepage: www.elsevier.com/locate/physe ...

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ARTICLE IN PRESS Physica E 42 (2010) 1482–1486

Contents lists available at ScienceDirect

Physica E journal homepage: www.elsevier.com/locate/physe

Apparent kinetics of hydriding and dehydriding of metal nanoparticles Vladimir P. Zhdanov a,b,, Bengt Kasemo a a b

Department of Applied Physics, Chalmers University of Technology, S-412 96 G¨ oteborg, Sweden Boreskov Institute of Catalysis, Russian Academy of Sciences, Novosibirsk 630090, Russia

a r t i c l e in f o

a b s t r a c t

Article history: Received 9 August 2009 Received in revised form 29 November 2009 Accepted 2 December 2009 Available online 11 December 2009

Hydriding and dehydriding kinetics of nanoparticles depend on the particle size. Our calculations illustrate that the apparent (averaged over size) kinetics of an ensemble of particles can be dramatically different compared to those of single particles. Specifically, we analyze the hydriding kinetics, limited by diffusion of hydrogen atoms from the surface layer via the hydride shell to the metallic core, and the dehydriding kinetics limited by associative desorption of hydrogen from the surface layer. In both cases, the apparent kinetics are relatively slow in the later stage, their time scale for the given average size is much larger than that for a single particle of the same size, and some of the special features of the single-particle kinetics (e.g., the initial slowdown of dehydriding) can be partly or completely washed out. The scaling of the time scale of the kinetics with respect to the particle size, however, remains valid. & 2009 Elsevier B.V. All rights reserved.

Keywords: Supported metal nanoparticles First-order phase transitions Hydride formation Scaling Lattice strain Surface tension

1. Introduction Nanoparticles have high surface area to volume ratio and can often be easily converted from one state to another due to interaction with an environment. Classical examples, related to basic physics and chemistry and numerous applications, are oxidation [1,2] and hydriding [3,4] of metal nanoparticles and soot oxidation [5,6]. In such cases, the conversion kinetics usually depend on a multitude of factors and first of all on the particle size. In experiments, the size distribution of ensembles of fabricated nanoparticles is commonly relatively broad (the average size and standard deviation are often comparable). Under such circumstances, the measured ensemble-averaged kinetics are apparent in the sense that they are inherently dependent on the size distribution and in general are not identical to the kinetics occurring in any particle of specific size. In other words, the measured kinetics is a weighted average over the actual size distribution of the kinetics of the individual particles. In experiments and theory, the effect of the particle size distribution on the kinetics of hydriding and dehydriding of metal nanoparticles is usually not acknowledged (for oxidation, the situation is similar). In particular, the available models [7–11] of these processes do not take this factor into account. More specifically, this effect is usually either not mentioned or is considered to be minor [8] by quoting Ref. [12] (see also

Ref. [13]). The models used in Refs. [12,13], however, ignore important physical factors influencing the hydriding and dehydriding kinetics, e.g., the lattice strain accompanying the hydride formation or decomposition or the strain related surface tension (the latter factor is especially important for small nanoparticles). Our goal is to revisit this subject and to show the likely effect of the particle size distribution on the hydriding and dehydriding kinetics. The models we use imply that hydriding and dehydriding of nanoparticles represent a first-order phase transition (for Pd nanoparticles, for example, this is the case for sizes down to about 1.5 nm [14–18]), and our main attention is paid to such novel factors as scaling, lattice strain, and surface tension. The general results presented are crucial for the understanding and interpretation of hydriding and dehydriding nanoparticles. Articulating the relevance of our calculations below to nanoparticles, we may note that with increasing the particle size to above about 5 nm the effect of surface tension on the kinetics becomes negligible. On the other hand, scaling and lattice-misfitrelated strain remain relevant as long as the particle structure is coherent (no dislocations). This is often the case even if the particle size is about or larger than 1 mm. Thus, in fact, the applicability of our conclusions is wider than one could infer from the title of our article.

2. Conversion and size distribution  Corresponding author at: Boreskov Institute of Catalysis, Russian Academy of

Sciences, Novosibirsk 630090, Russia. E-mail address: [email protected] (V.P. Zhdanov). 1386-9477/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2009.12.002

In our calculations, metal nanoparticles are considered to be spherical. The mass transport limitations in the gas phase are

ARTICLE IN PRESS V.P. Zhdanov, B. Kasemo / Physica E 42 (2010) 1482–1486

where ra is the average nanoparticle radius. This distribution (with /VS ¼ 80pra3 =27) is maximum at r ¼ ð2=3Þra and then exponentially drops. It represents a reasonable fit of realistic relatively wide size distributions of nano/microparticles obtained by ball-milling (see, e.g., Ref. [13]) or other fabrication methods (see, e.g., Ref. [19]). Distribution (2) containing only one fitting parameter, ra , is sufficient to illustrate the effect of the size distribution of nanoparticles on the kinetics under consideration. At present, arrays of nanoparticles with a fairly narrow size distribution can be fabricated (see, e.g., study [17] of the hydrogen absorption isotherms in Pd nanoparticles). To describe such cases, one can use more complex size distributions [see, e.g., Eq. (11) below].

3. Hydriding kinetics Hydriding of metal nanoparticles can be limited by dissociative adsorption of H2 molecules on the surface of particles, jumps of hydrogen atoms from the surface layer to the interior, or by diffusion of hydrogen atoms via the hydride shell to the metallic core. Simple mean-field models describing these steps are proposed in Refs. [7–11]. Our analysis is based on the results of our recent detailed Monte Carlo simulations [20] of the diffusioncontrolled hydriding scenario, indicating that the time scale of the kinetics or, more specifically, the time, corresponding to conversion of half the volume to hydride, scales as t0:5 ðrÞpr z , where 2:3r z r 3 is the exponent dependent on the diffusion dynamics. This means that t0:5 ðrÞ can be represented as  z r ta ; ð3Þ t0:5 ðrÞ ¼ ra where ta  t0:5 ðra Þ. To use this scaling, we represent the conversion of single particles to the hydride phase as   z    lnð2Þt ra lnð2Þt : ð4Þ  1exp  jðr; tÞ ¼ 1exp  r t0:5 ðrÞ ta Although this exponential representation is not fully accurate (the simulations [20] show that the kinetics are more complex), its shape is reasonable and it exactly incorporates dependence (3). For these reasons, representation (4) is a satisfactory approximation in order to illustrate our main points. Substituting distribution (2) and conversion (4) into Eq. (1), taking into account that /VS ¼ 80pra3 =27, and introducing a new variable x ¼ r=ra , we have   Z 1 5 5 3 x lnð2Þt expð3xÞ dx: ð5Þ FðtÞ ¼ 1exp  z 40 x ta 0 This equation shows that after integration over x the conversion depends only on the ratio t=ta (Fig. 1). By definition, ta is proportional to raz . This means that the scaling law for an

3

0.8

2

1

0.6 0.4

f

where t is time, f ðrÞ is the particle radius distribution, jðr; tÞ is the conversion of a single particle (to be described below), 4pr 3 =3 is R1 the particle volume, and /VS ¼ ð4p=3Þ 0 r 3 f ðrÞ dr is the average volume per particle. Below, we use a generic size distributions,   27r 2 3r ; ð2Þ exp  f ðrÞ ¼ 3 ra 2ra

1.0

Conversion

assumed to be negligible. In this case, the conversion of an ensemble of particles to the hydride phase during hydriding or to the metallic phase during dehydriding is given by the standard convolution of the kinetics of single particles, Z 1 4pr 3 jðr; tÞf ðrÞ dr; ð1Þ FðtÞ ¼ 3/VS 0

1483

0.2 r

0.0 0

5

10

15

20

25

t / ta Fig. 1. Conversion of metal nanoparticles to the hydride phase as a function of time: (1) for an array of particles with the particle-radius distribution (2) (see the insert) and Eq. (5) with z ¼ 2:5; (2) a fit of curve 1 by employing the exponential kinetics, 1-exp½-lnð2Þt=t0:5  [like in Eq. (4)]; (3) the kinetics for a single particle with r ¼ ra .

ensemble of particles is the same as for single particles, i.e., the time scale of the apparent kinetics scales as t0:5 ðra Þpraz . The time scale of hydriding of an ensemble of particles with a given value of ra is, however, much larger than that of a single particle with r ¼ ra . In addition, the shape of the apparent kinetics is qualitatively different compared to that corresponding to single particles. In particular, the apparent kinetics are relatively slow in the later stage. All these features can be understood considering that the relatively rapid initial stage of the apparent kinetics is determined primarily by small particles (with r o ra ) while the intermediate and later stages are dominated mainly by large particles (with r 4 ra ).

4. Dehydriding kinetics In our analysis of the dehydriding kinetics in metal nanoparticles, 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 these steps are influenced by the lattice strain, generated due to the misfit of the hydride and metal lattice spacings and surface tension. The role of these factors in the dehydriding kinetics is larger than that in the hydriding kinetics because their relative contribution to the dependence of the time scale of the kinetics on the particle size is much stronger in the former case. Employing a shrinking-core model, we have recently scrutinized [21] the effect of lattice strain in the dehydriding kinetics in nanoparticles (the earlier models [8–11] do not take these factors into account). To understand 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 the binding of single hydrogen atoms on the surface is usually energetically preferable [22]. The hydride is stabilized by 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 on the external surface of the metallic shell. In contrast, the activation barrier for associative

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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. The lattice strain of the metal shell changes the binding energy and desorption activation energy of adsorbed hydrogen atoms and accordingly changes the desorption rate. Specifically, the tensile strain related to the hydride core results in the increase of the desorption activation energy, while the compressive strain caused by surface tension results in the decrease of this energy. During dehydriding, the former strain is appreciable in the beginning and vanishes at the end, while the latter strain is nearly constant (it increases, however, with decreasing particle size). Our analysis [21] was focused on the lattice strain related to the hydride core. The role of the strain caused by surface tension was briefly discussed as well (in the end of the presentation). In particular, our Eq. (24) (in Ref. [21]) describing the dehydriding kinetics of a single nanoparticle can be used to take both effects into account. According to Eq. (24), the ratio of the current (at time t) and initial numbers of hydrogen atoms in a nanoparticle is given by N=Nh ¼ ð1=BÞln½expðBÞkðrÞBt;

ð6Þ

tension influences the dependence of the process rate on r. In the absence of the former strain (at B-0), the kinetics are linear. With this strain, the process rate is somewhat suppressed especially in the beginning. For this reason, the initial stage of the kinetics is relatively slow. In the absence of the latter strain (at R ¼ 0), the time scale of the kinetics would be proportional to r. With the latter strain, the kinetics time scale is reduced for smaller particles. For an ensemble of particles, the conversion can be calculated by using Eqs. (1), (2), and (10). The dependence of j on r, defined by Eq. (10), is more complex compared to that given by Eq. (4), and although now the apparent kinetics depends on kt it cannot

1.0

and accordingly the conversion of a single particle to the metallic phase, jðr; tÞ ¼ ðNh NÞ=Nh , can be represented as

jðr; tÞ ¼ 1ð1=BÞln½expðBÞkðrÞBt;



kB T

0.6

1

0.4 0.2

ð7Þ

0.0 0.0

where

aAð1 þ aÞ3

2

0.8 Conversion

1484

ð8Þ

0.2

0.4

kt

0.6

0.8

1.0

6

8

10

40

50

1.0

and

0.8

ð9Þ

The parameter B defined here by Eq. (8) is identical to that introduced by Eq. (23) in Ref. [21] (a is the linear mismatch parameter, and A is the expansion coefficient of the activation energy for hydrogen desorption). The parameter kðrÞ defined by Eq. (9) is similar to that introduced by Eq. (23) in Ref. [21] (v is the volume per H atom in the hydride phase, s is the surface-site area, k0 is the desorption rate constant in the absence of strain, and R is a parameter proportional to surface tension and inversely proportional to kB T). The only difference is that Eq. (9) contains an additional exponential term, expðR =rÞ, taking surface tension into account (in Ref. [21], this term was mentioned at the end of the presentation). Physically, as already noted, the parameters B and R are related to the change of the activation energy for hydrogen desorption due to the lattice strain, caused by the misfit of the hydride and metal lattice spacings and surface tension, respectively. According to our estimates [21], B is expected to be about 1–3. R can be estimated in analogy taking into account that the lattice stress related to surface tension is given by srr ¼ syy ¼ sff ¼ 2g=r, where g is surface tension. In particular, our estimates [21] yield R ¼ 324 nm. Eqs. (7) and (9) contain a number of parameters. To simplify the presentation of the results, we substitute Eq. (9) into Eq. (7) and represent the latter as    1 ktR R ð10Þ jðr; tÞ ¼ 1 ln expðBÞ  exp  ; r r B where k ¼ 3vk0 B=½sR ð1 þ aÞ3  is the effective rate constant. Eqs. (7) and (9) [or (10)] predict that the lattice strain related to the misfit of the hydride and metal volumes influences only the shape of the kinetic curves, while the strain related to the surface

Conversion

  R : exp r sð1þ aÞ3 r 3vk0

2

0.6

1 0.4 0.2 0.0

0

2

4

kt

1.0 0.8 Conversion

kðrÞ ¼

2

0.6

1 0.4 0.2 0.0 0

10

20

kt

30

Fig. 2. Conversion of hydrided nanoparticles to the metallic phase as a function of time for the particle-radius distribution (2) with ra ¼ 1 (a), 3 (b) and 10 nm (c). Line 1 shows the kinetics calculated by using Eqs. (1) and (10) with B ¼ 2:5 and R ¼ 4 nm. Line 2 corresponds to a single particle with r ¼ ra .

ARTICLE IN PRESS V.P. Zhdanov, B. Kasemo / Physica E 42 (2010) 1482–1486

30

employed in the fitting of the apparent kinetics may, however, be lower than that used to calculate the original kinetics.

25

10 5 0 0

2

4

6

8

10

ra, nm Fig. 3. t0:5 as a function of ra for dehydriding of an array of metal nanoparticles with the particle-radius distribution (2) (the parameters, B ¼ 2:5 and R ¼ 4 nm, are as in Fig. 2). The solid line shows the fit by using the equation t0:5 ¼ cr a expð-R =ra Þ with R ¼ 2:5 nm.

be represented by a single function dependent only on the ratio t=ta [like Eq. (5)]. This means that one should use distribution (2) with specific values of ra as, e.g., shown in Fig. 2 for ra ¼ 1, 3 and 10 nm. The results presented in Fig. 2 indicate that the apparent kinetics are very different compared to those exhibited by single particles. As in the case of hydriding, the time scale of dehydrating of an ensemble of particles with a given value of ra is much larger than that of a single particle with r ¼ ra , and the apparent kinetics are relatively slow in the later stage. In addition, the initial slowdown of the kinetics is completely or partly washed out. On the other hand, the dependence of t0:5 on ra can be fitted (Fig. 3) assuming that t0:5 pra expðR =ra Þ (cf. Eq. (9)). This means that the surface tension is manifested in the apparent kinetics. Note, however, that the value of R employed in the fitting is lower than that used to calculate the original kinetics.

5. Conclusion The apparent (averaged over size) hydriding and dehydriding kinetics of an ensemble of particles are represented by the standard convolution (1) of the kinetics of single particles. To perform this convolution one needs the particle size distribution and an explicit expression for the conversion of single particles. We have illustrated the difference between the apparent and single-particle kinetics in the cases of hydriding, limited by diffusion of hydrogen atoms from the surface layer via the hydride shell to the metallic core, and dehydriding limited by associative desorption of hydrogen from the surface layer. For reasonably wide size distributions, the apparent kinetics are shown to be dramatically different compared to those of single particles. Our key specific findings are as follows: (i) The apparent kinetics are relatively slow in the later stage and their time scale for the given average size is much larger than that for a single particle of the same size. (ii) Despite item (i), the scaling of the time scale of the hydriding kinetics with respect to particle size remains valid. (iii) For dehydriding, the special strain-related features of the single-particle kinetics (e.g., the initial slowdown) can be partly or completely washed out. (iv) The effect of surface tension can be tracked out in the apparent dehydriding kinetics. The value of R [Eq. (9)]

f ðrÞ ¼ Cr n exp½aðrra Þ2 ;

ð11Þ

where n, a, and ra are the fitting parameters, and C is the normalization coefficient. Using this distribution, one can easily show that with increasing a (i.e., with decreasing the distribution width) the models employed predict a transition from the kinetics strongly influenced by the size distribution to the kinetics corresponding to particles with fixed size (see, e.g., Fig. 4). The relevant experimental data on the hydriding and dehydriding kinetics were obtained primarily by using powders obtained by ball-milling [8–10,23,24]. In such experiments, the average particle size is usually in the range from 100 nm to 10 mm, and the particle size distribution is typically rather broad. The accurate measurements of the size distributions data are, however, lacking as a rule. Although the full-scale analysis of the role of the particle size distribution is hardly possible under such circumstances, we may note that in agreement with item (i) the real hydriding kinetics are often relatively slow in the late stage. Systematic experimental data on the dependence of the kinetics on particle size are still lacking [items (ii) and (iv)]. Concerning item (iii), we note that the initial slowdown of the dehydriding kinetics was not observed e.g. for LaNi5 [9] and Mg1:95 Ag0:05 Ni [10] and was observed for MgðNiÞH2 (Fig. 9 in Ref. [8]), MgH2 þ 5 at%V (Fig. 1 in Ref. [23]), NaAlH4 (Fig. 8 in Ref. [24]) and Mg (Fig. 2 in Ref. [19]). Typically [8,23,24], the observed

1.0 0.8 4

3

0.6

2

0.4 1

f

15

In our analysis, we used one of the simplest reasonable size distributions containing a single parameter. Our final results depend mainly on dimensionless combinations of model parameters. The values of specific parameters were validated (in Refs. [20,21]) by using independent experimental data or the results of DFT calculations). For these reasons, our key findings are not biased due to the contrived form of the size distribution and the numerical parameters chosen. To draw our conclusions, we used specific models of hydriding and dehydriding. Our findings are, however, generic and expected to be valid for other models as well. Our general equations allow one to use more complex particle size distributions, such e.g. as

Conversion

20 kt 0.5

1485

0.2 r

0.0 0

1

2

3

4

5

t/ta Fig. 4. Conversion of metal nanoparticles to the hydride phase as a function of time. Curves 1–3, corresponding to arrays of particles, were calculated for the particle-radius distribution (11) with ra ¼ 4 nm, n ¼ 1, and a ¼ 0:2 (1), 0.6 (2) and 2 nm-2 (3) (see the insert) by using Eqs. (1) and (4) with z ¼ 2:5. Curve 4 shows, for comparison, the conversion of a single particle with r ¼ ra .

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slowdown is somewhat weaker than that predicted [21] for single particles. Although the interpretations of these observations (e.g., the mechanisms of dehydriding) are open for debate, the general trends there are in line with our finding [item (iii)] that the strainrelated initial slowdown can be partly or completely washed out in the apparent kinetics. Our general equations can easily be used to fit experimental data provided that the particle-size distribution has been measured. For hydriding, the convolution (1) is straightforward. For dehydriding, one can use k, B, and R as fitting parameters. Alternatively, these parameters can be estimated by employing independent experimental and/or theoretical (e.g., DFT) data and then the calculated kinetics can be compared with the measured ones. Concerning the latter strategy, one should bear in mind that our model is spherical while in reality the nanoparticles are crystallites exhibiting different facets, and accordingly the parameters we use should be considered as average over the facets. Although the fitting of experimental data by using Eq. (1) in combination with one of the kinetic models is mathematically simple, we do not try to interpret specific kinetics in detail, because one of the real difficulties here is validation of a mechanism of hydriding or dehydriding. As already noted, the mechanisms of hydriding or dehydriding are usually open for debate, and the corresponding analysis is beyond our present goals. Finally, we may notice that our key findings are expected to be inherent not only to hydriding and dehydriding but also to many other kinetic processes (e.g. oxidation or sulfidation) occurring in nanoparticles.

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