NANO LETTERS
On the Feasibility of Strain-Induced Formation of Hollows during Hydriding or Oxidation of Metal Nanoparticles
2009 Vol. 9, No. 5 2172-2176
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 March 15, 2009
ABSTRACT Hydriding or oxidation of a metal nanoparticle starts on the surface and results in the formation of a hydride or oxide shell. Due to expansion of the shell, the strain in the metal core is tensile, and in principle it may induce the formation of a hollow structure (this hollow-formation mechanism is alternative to that based on the Kirkendall effect). Using the theory of elasticity, we show that this is feasible for oxides but not feasible for hydrides.
The understanding of the mechanisms and kinetics of hydriding and oxidation of metal nanoparticles is of interest from the point of view of basic physics and chemistry. In addition, this subject is of high current interest from the viewpoint of numerous potential applications. For example, hydrogen is a potential future energy carrier and in this context the formation of metal hydride nanoparticles may be one of the ways of useful hydrogen storage.1 Oxidation of nanoparticles and other nanosized objects is also highly relevant in materials science by utilizing nanoparticles as building blocks and in nanoelectronics. Despite the high current activity in these areas, many key aspects of hydriding and oxidation of nanoparticles are still open for debate. In particular, these processes are known to be accompanied by lattice strain. The underlying physical origin of this is that as a new compound (hydride or oxide) is formed it is usually subject to a volume expansion or, in some cases (e.g., alkali metals), contraction. According to general theory of firstorder phase transitions (see, e.g., ref 2), the effect of strain on the thermodynamic features (e.g., phase diagrams and absorption isotherms) of hydride formation in macroscopic samples can be appreciable. Numerous experimental studies of hydriding and oxidation of such samples indicate that the accompanying lattice strain may result in the formation of cracks, and such phenomena are well-known during growth of surface oxide films. The available kinetic models of hydriding and oxidation of nanoparticles (see, e.g., refs 3 * Corresponding author,
[email protected]. † Department of Applied Physics, Chalmers University of Technology. ‡ Boreskov Institute of Catalysis, Russian Academy of Sciences. 10.1021/nl9008293 CCC: $40.75 Published on Web 03/30/2009
2009 American Chemical Society
and 4 and 5 and 6, respectively) do not however take lattice strain into account. In the case of nanoparticles, the hydriding and oxidation usually start on the surface and result in the formation of a hydride or oxide shell. Due to expansion of the shell, the strain in the metallic core is usually tensile, and in principle it may induce or facilitate the formation of a hollow area there. During oxidation of metal nanoparticles, the formation of such hollow nanoparticles was recently observed in numerous experiments (see the famous study of oxidation of Co by Alivisatos and co-workers,7 a recent review,5 and more recent studies of oxidation of Al,8 Fe,9 Co,10,11 Ni,12 Cu,8,13 and Zn14). The available models of hollow formation5,10,15-17 are focused on the balance of metal and oxygen diffusion implying the formation and coalescence of vacancies in the metal core (the Kirkendall effect). One of these models16 takes into account the effect of interface tension on diffusion. The effect of the strain related to the misfit of the oxide (or hydride) and metal has however not been treated. Our goal is to clarify this effect. In general, the strain related to the misfit of the oxide (or hydride) and metal may influence the interplay of the metal and oxygen (or hydrogen) diffusion, which was analyzed in already available models.5,10,15-17 All these models, neglecting strain, include metal diffusion through the oxide layer as a necessary condition of the hollow formation. The likely role of this diffusion in oxidation was recognized since the classical work by Cabrera and Mott18 (for the specifics of application of the Cabrera-Mott model to nanoparticles, see ref 6). The limitation of the oxide growth by oxygen transport was acknowledged as well.19 The recent experiments using
oxygen isotopes indicate that both these scenarios are possible (e.g., the former one seems to occur in Ni20 while the latter one appears to be operative in Al21). In our treatment, we assume that there is no metal diffusion through the oxide or hydride shell. Thus, the hollow-formation mechanism we discuss is an alternatiVe to that analyzed earlier.5,10,15-17 Our analysis and presentation below can be divided into two parts. First, using the standard prescriptions of the theory of elasticity,22 we recall how to calculate the lattice strain in the case under consideration (for additional aspects of this problem, see, e.g., ref 23 and references therein). Then, we employ the conventional Griffith arguments24 in order to calculate the rate of the formation of hollow structures. The applicability of the Griffith model can of course be debated especially in the case of ductile media. The alternative related models are based on the dislocation theory (see, e.g., ref 25 and references therein). We may, however, note that in nanoparticles the very concept of dislocations is hardly applicable simply due to the mismatch between the dislocation size and small particle size and the likely relaxation on the surface. One should also take into account that hydriding and oxidation of metal nanoparticles is often studied under relatively mild conditions. For these reasons, the Griffith model may actually be more relevant to metal nanoparticles than the dislocation-based models. With this reservation, we believe that our analysis is a robust first step toward understanding what may happen in the case under consideration, and the corresponding results predicted for hydriding and oxidation of metal nanoparticles are instructive anyway. In our model, a nanoparticle is considered to be spherical and initially to be fully metallic. The oxidation or hydriding starts near the surface, and then the metallic core is shrinking while the surrounding oxide or hydride shell becomes thicker. Thus, we use a shrinking core model. This model was earlier widely employed in the mean-field treatments of the kinetics of hydride and oxide formation and hydride decomposition in nanoparticles.3,5,6,10,15-17 However, as already noted, the available models do not take the lattice strain into account. To calculate the strain distribution in the nanoparticle partly converted to oxide (or hydride), we should choose a convenient reference state. Bearing in mind this goal, let us first consider that the particle is fully metallic, and 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 outside this subsphere is converted into oxide. To describe the lattice strain in this case, it is instructive first to imagine that the metallic core and the oxide shell do not interact (Figure 1b). In this hypothetical case, the metallic core (at r e R) will retain its size, while the oxide subsphere will expand up to (1 + R)R e r e (1 + R)Rm, where R is the linear mismatch parameter. In our analysis, these states of the metallic core and oxide shell are considered to be a reference. This choice of reference states is convenient because in this case there are no lattice strains except those Nano Lett., Vol. 9, No. 5, 2009
Figure 1. (a) Metallic sphere of radius Rm in the absence of oxygen (the shell with r > R represents a region to be converted into oxide). (b) Noninteracting metallic core (at r < R) and oxide shell (at RR < r < (1 + R)R)). (c) Strained metallic core and oxide shell. (d) As (c) with a hollow center in the metallic core.
related to the surface tension. The latter strains are however minor compared to the strain caused by the elastic mismatchrelated metal-oxide interaction and accordingly can be neglected. Due to this interaction, the oxidation-related expansion of the shell will partly be suppressed by the metallic core, and the latter will be somewhat expanded; i.e., 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. According to the theory of elasticity,22 the radial displacements in the oxide (or hydride) shell (at (1 + R)R e r e (1 + R)Rm) and metallic core (at r < R) can, respectively, be represented as u ) -ar -
b r2
(1)
u ) cr
(2)
where a, b, and c are positive constants. The corresponding components of the strain tensor are given by urr ) -a +
2b r3
uθθ ) uφφ ) -a -
(3)
b r3
urr ) uθθ ) uφφ ) c
(4)
In general, the radial stress is expressed via the strain tensor 2173
as22
2R(1 - 2σ)R3 3(1 - σ)Rm3
(11)
b)
(1 + σ)Rm3a R(1 + σ)R3 ) 2(1 - 2σ) 3(1 - σ)
(12)
(
-1 a)
a) σrr )
E[(1 - σ)urr + 2σuθθ] (1 + σ)(1 - 2σ)
(5)
where E is Young’s modulus and σ is Poisson’s ratio. Substituting (3) and (4) into (5) yields Eoa 2Eob σrr ) + 1 - 2σo (1 + σo)r3
σrr )
Emc 1 - 2σm
c) (6)
Rm3 R
3
)
(7)
where Em, σm, Eo, and σo are the parameters corresponding to the metal and oxide (or 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 core (at r ) R) and oxide shell (at r ) (1 + R)R). Employing expressions 1 and 2 for the displacements, we represent this condition as b + c(1 + R)R ) RR R2
(8)
(9)
Third, stress 6 should vanish at the external boundary of the oxide shell, i.e. a 2b )0 1 - 2σo (1 + σo)(1 + R)3Rm3
(
2RE R3 13(1 - σ) Rm3
)
(14)
F > Fc =
2γE πσten2
(15)
where Fc is the critical radius, γ is the surface energy, and σten is the tensile stress. The corresponding critical hollow energy is given by εc )
4π 2 16γ3E2 γFc = 3 3πσten4
(16)
(10)
The use of the equations above implies that the oxide or hydride compression is primarily elastic. In the case of hydrides, this approximation is often fully reasonable. The formation of the oxide shell may however often be accompanied by inelastic deformations. Anyway, however, we may employ the equations above for oxide as well in order to estimate the scale of the effects under consideration and to understand whether or not the hollow formation is possible under favorable conditions. To make our equations compact, we neglect the difference in the elastic constants of the metal and oxide (or hydride), i.e., use Em ) Eo ) E and σm ) σo ) σ. In addition, taking into account that R is small compared to unity, we neglect R in the third term of the left-hand side of eq 8 and in the denominator of the second term of the left-hand side of eq 10. With these modifications, eqs 8-10 yield 2174
(13)
As already noted, the strain of the metallic core is tensile, and in principle it may result in the formation of a hollow area there (Figure 1d). According to the conventional Griffith arguments,24 the formation of a small hollow area in the medium is accompanied by a decrease in energy due to local relaxation of the stress and also by an increase in energy due to the boundary formation. A hollow area grows provided that its energy decreases with increasing its radius, F (to be specific, we consider that a hollow area is spherical). This condition is fulfilled at (see eq 2.19 in ref 26)
Second, stresses 6 and 7 should be equal on the boundary between the oxide and metal Eoa 2Eob Emc )3 1 - 2σo 1 2σm (1 + σo)R
)
Substituting (13) into (7), we obtain the following expression for the stress in the metallic core σrr )
aR +
(
2R(1 - 2σ) R3 13(1 - σ) Rm3
Replacing σten in (15) and (16) by expression 14, we obtain Fc =
9γ(1 - σ)2 2πR2E(1 - R3 /Rm3)2
(17)
εc =
27γ3(1 - σ)4 πR E (1 - R3 /Rm3)4
(18)
4 2
Physically, the critical hollow energy represents an activation energy of the hollow formation. The rate of this process is given by W ) ν exp(-εc /kBT)
(19)
where ν is the pre-exponential factor. The hollow formation can be observed if τW g 1, where τ is the time scale of an Nano Lett., Vol. 9, No. 5, 2009
εc =
27γ3(1 - σ)4 πR4E2
(21)
9γ(1 - σ)2 2πR2E
(22)
Fc =
Figure 2. Critical energy and radius of the hollow area as a function of the misfit coefficient according to eq 21 and 22 with E ) 100 GPa, σ ) 0.3, and γ ) 2 N/m.
experiment. Using expression 19, we rewrite this condition as εc e kBT ln(ντ)
(20)
To apply condition 20, we should estimate ν. An accurate calculation of ν is hardly possible. The dependence of the right-hand side of condition 20 on ν and τ is however logarithmically weak, and we can use the simplest estimate of ν, reduced to the so-called “normal” pre-exponential factor, i.e., ν ) 1013 s-1. Combining this value with T ) 500 K and τ ) 100 s, we obtain that the hollow formation is possible if εc is lower than or comparable with 35 kcal/ mol. This can easily be rescaled to other temperatures using the equation above. On the other hand, εc is given by eq 18 containing E, σ, and R. For many metals (including those mentioned in the introduction), E = 100 GPa, σ = 0.3, and γ = 2 N/m. For hydrides, R = 0.05. For oxides, R is typically between 0.2 and 0.4. Using these values, one can easily estimate that for hydrides εc defined by eq 18 is much larger than 35 kcal/ mol. This means that the stress is too weak in order to result in the formation of hollow areas. For oxides, this is also the case if R g 0.5Rm. For R < 0.5Rm, the ratio R3/Rm3 is small and the factors (1 - R3/Rm3)4 in eq 18 and (1 - R3/Rm3 )2 in eq 17 can be dropped, i.e. Nano Lett., Vol. 9, No. 5, 2009
Using these expressions, we have calculated εc and Fc as a function of R for oxides as shown in Figure 2. The hollow formation is found to be feasible at R g 0.29. This condition can be met in reality. The scale of Fc predicted by the model is physically reasonable as well. Thus, our calculations indicate that the lattice-strain-related hollow formation in metal nanoparticles during oxidation is feasible. Finally, it is appropriate to notice that to perform our calculations we neglected the difference in the elastic constants of the metal and hydride or the metal and oxide. In the case of hydrides, this approximation is fairly accurate. Young’s modulus of oxides are however higher than those of metals. For this reason, during oxidation, the lattice strain in the metallic core may actually be somewhat higher than that obtained in our calculations, and accordingly the hollow formation may be more likely. In summary, our calculations indicate that the lattice strain accompanying the hydride formation in metal nanoparticles is relatively weak and cannot result in the appearance of a hollow formation in the metallic core. In contrast, the lattice strain accompanying oxidation of metal nanoparticles may be sufficiently strong in order to generate the formation of hollow area in the metallic core. This is likely if the radial size of the oxide shell is comparable or larger than the radius of the metallic core and the misfit coefficient is comparable with or larger than 0.3. References (1) Berube, V.; Radtke, G.; Dresselhaus, M.; Chen, G. Int. J. Energy Res. 2007, 31, 637–663. (2) Cahn, J. W.; Larche, F. Acta Metal. 1984, 32, 1915–1923. Schwarz, R. B.; Khachaturyan, A. G. Acta Mater. 2006, 54, 313–323. Lexcellent, C.; Gondor, G. Intermetal. 2007, 15, 934–944. (3) Nahm, K. S.; Kim, W. Y.; Hong, S. P.; Lee, W. Y. Int. J. Hydrogen Energy 1992, 17, 333–338. Martin, M.; Gommel, C.; Borkhart, C.; Fromm, E. J. Alloys Compd. 1996, 238, 193–201. Inomata, A.; Aoki, H.; Miura, T. J. Alloys Compd. 1998, 278, 103–109. Chou, K.-C.; Li, Q.; Lin, Q.; Jiang, L.-J.; Xub, K.-D. Int. J. Hydrogen Energy 2005, 30, 301–309. (4) Zhdanov, V. P.; Kasemo, B. Chem. Phys. Lett. 2008, 460, 158–161. Physica E 2009, 41, 775-778. (5) Fan, H. J.; Gosele, U.; Zacharias, M. Small 2007, 3, 1660–1671. (6) Zhdanov, V. P.; Kasemo, B. Chem. Phys. Lett. 2008, 452, 285–288. (7) Yin, Y.; Rioux, R. M.; Erdonmez, C. K.; Hughes, S.; Somorjai, G. A.; Alivisatos, A. P. Science 2004, 304, 711–714. (8) Nakamura, R.; Tokozakura, D.; Nakajima, H.; Lee, J. G.; Mori, H. J. Appl. Phys. 2007, 101, 074303. (9) Cabot, A.; Puntes, V. F.; Shevchenko, E.; Yin, Y.; Balcells, L.; Marcus, M. A.; Hughes, S. M.; Alivisatos, A. P. J. Am. Chem. Soc. 2007, 129, 10358–10360. (10) Yin, Y.; Erdonmez, C. K.; Cabot, A.; Hughes, S.; Alivisatos, A. P. AdV. Func. Mater. 2006, 16, 1389–1399. (11) Chernavskii, P. A.; Pankina, G. V.; Zaikovskii, V. I.; Peskov, N. V.; Afanasiev, P. J. Phys. Chem. C 2008, 112, 9573–9578. (12) Nakamura, R.; Lee, J. G.; Mori, H.; Nakajima, H. Philos. Mag. 2008, 88, 257–264. (13) Tokozakura, D.; Nakamura, R.; Nakajima, H.; Lee, J. G.; Mori, H. J. Mater. Res. 2007, 22, 2930–2935. (14) Nakamura, R.; Lee, J. G.; Tokozakura, D.; Mori, H.; Nakajima, H. Mater. Lett. 2007, 61, 1060–1063. 2175
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NL9008293
Nano Lett., Vol. 9, No. 5, 2009
