J. Phys. Chem. C 2010, 114, 18085–18090
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Smooth Size Effects in Pd and PdHx Nanoparticles Francesco Delogu* Dipartimento di Ingegneria Chimica e Materiali, UniVersita` degli Studi di Cagliari, piazza d’Armi, I-09123 Cagliari, Italy ReceiVed: July 5, 2010; ReVised Manuscript ReceiVed: September 14, 2010
Molecular dynamics simulations have been employed to explore a few structural and thermodynamic properties of Pd and PdHx systems in both bulk and nanometer-sized forms. The heat of formation of the PdHx phases decreases as the particle size decreases. Despite the general reduction of thermodynamic stability compared with bulk systems, nanometer-sized PdHx particles tend to be richer in H atoms than that of massive systems. This observation is valid for both particle interiors and surface regions, which have been distinguished by evaluating the profile of average potential energy along the particle radius. Smooth size effects also affect the thickness of the surface layer, which increases as the particle radius decreases. 1. Introduction Energy systems based on hydrogen (H2) represent one of the outstanding research avenues currently explored to promote a transition from the present technological scenario based on fossil fuels to a more sustainable one.1,2 Within this framework, fundamental science as well as advanced technology is faced with the considerable challenges determined by the need of finding viable routes to the production, storage, distribution, and utilization of H2.1-4 Regarding basic research in materials science, one of the major areas of investigation concerns metal hydrides for H2 storage in solid phase.1,2,5-9 Here, the fundamental issue is to identify the hydride systems combining a significant thermodynamic stability with a sufficiently fast kinetics for the H2 absorption and desorption processes.1,2,5-9 Among others, Pd has received considerable attention due to its well-known affinity for H2.10 This metal forms two different PdHx hydride phases, characterized by a different content of H atoms in the structure.10 The so-called R phase has the lower H concentration. It exhibits a face-centered cubic (fcc) structure with the H atoms arranged in the octahedral interstices of the Pd lattice, the tetrahedral sites being only transiently occupied by diffusing H.10 The elementary cell parameter changes from about 0.388 to 0.389 nm with the relative number x of H atoms to Pd atoms, which cannot be larger than 0.03.10 The β phase retains the fcc crystalline arrangement with an elementary cell parameter of about 0.402 nm but with x values as large as 0.76 at very high H2 pressures.10 The Pd-H binary phase diagram indicates that the R and β phases are separated by a miscibility gap associated with a phase transition.10 Precisely the relatively large amount of H2 that can be absorbed in the β phase, combined with the well-known catalytic properties, make Pd a model system to study H2 absorption and storage in metals.10 The above-mentioned observation is particularly true for nanometer-sized Pd. In fact, the reduction of size bears the promise of enhanced performances in both the thermodynamics and kinetics of H2 absorption and desorption processes.11-17 More specifically, the reduction of the characteristic length of Pd phases is expected to destabilize its crystalline lattice due to the thermodynamically unfavorable contribution of surface * To whom correspondence should be addressed. E-mail: delogu@ dicm.unica.it.
free energy and to improve the catalytic steps underlying the multistage interaction with H.11-17 For example, the relative fraction x of H atoms that is expected to be absorbed in nanometer-sized PdHx systems is as large as 1.18 However, despite their importance from both scientific and technological points of view, the above-mentioned inferences have received only a limited experimental support.19-22 In general, experiments exploit a pressure change in a Sievert apparatus to measure absorption and desorption properties.19-22 Of course, any limitation in the sensitivity of apparatuses dealing with very small samples severely affects these measurements.19-22 In addition, experimental findings can be hardly related to the statistical distribution of particle size, which almost prevents the reliable identification of hydriding and dehydriding behaviors of individual particles.19-22 An exception in this sense is represented by a recent study,23 which demonstrates significant size effects in the Pd-H system by using a novel nanoplasmonic sensing method with subsecond time resolution. Under these circumstances, the area of investigation could greatly benefit the use of simulation methods to integrate experimental findings regarding variegated sets of particles with numerical results concerning individual particles. The attempts to cover such issues have been numerous and valuable, pointing out various specific features such as the smoothening of the R-to-β phase transition and the change of slopes in the H2 absorption isotherms.24-31 However, most of them focused on Pd clusters, leaving the case of nanometer-sized particles relatively uncovered. The present work aims precisely at exploring the thermodynamics of Pd and PdHx model systems in the size range between 1 and 15 nm. In particular, the investigation focuses on the molecular dynamics of unsupported spherical particles of Pd and PdHx at room temperature. A comparison with bulk Pd and PdHx phases is also performed. Calculations are used to evaluate the enthalpy differences between bulk and nanometer-sized systems, which permits indirect estimatation of the enthalpy of formation of PdHx at the nanometer scale. The details of computations are given below. 2. Numerical Simulations The pure Pd-Pd and H-H as well as the cross Pd-H interactions were described according to a semiempirical tight-
10.1021/jp106182n 2010 American Chemical Society Published on Web 10/04/2010
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Delogu
TABLE 1: The Radius R (nm) and the Number (NPd) of Pd Atoms in Pd Particles radius (nm) N
2
3
4
5
6
7
8
9
10
11
2282
7702
18256
35657
61615
97843
146051
207952
285256
379676
binding (TB) potential based on the second-moment approximation to the density of electronic states.31,22 The cohesive energy of a system containing N ) NR + Nβ atoms is expressed as NR
E)
{
Nβ
∑∑ ∑∑ R
iR)1
β
jβ)1
ARβe-pRβ
( )rijRβ
-1
dRβ
[∑ ∑ Nβ
β
jβ)1
]}
( )
rijRβ 2 -pRβ -1 ξRβ e dRβ
1/2
(1)
where the indexes iR and jβ run over all the NR and Nβ atoms of species R and β, rRβ ij ) |riR - rjβ| being the distance between two atoms. The parameters ARβ, ξRβ, pRβ, and qRβ quantify the potential energy for R and β species. The term dRβ represents the equilibrium distance of nearest neighbors at 0 K. The first member on the right-hand side expresses the repulsive part of the potential as a Born-Mayer pairwise interaction, while the second member describes the attractive part in the framework of the second-moment approximation of the TB band energy.12,13 Interactions were computed within a spherical cutoff radius rc of 0.73 nm, which includes the seventh shell of neighbors in Pd fcc lattices. The potential parameter values were taken from the literature.31 This TB potential correctly predicts the elementary cell parameter and the cohesive energy for Pd at 0 K.31 The mechanical properties of bulk Pd are reproduced to a satisfactory extent, with differences between experimental and predicted elastic constants of about 1%.32 In addition, it provides a binding energy between Pd and H atoms in good agreement with experimental values and ab initio calculations.31,33 Regarding the mechanical properties of a bulk PdH0.6 β phase at 300 K, simulations indicate that experimental elastic constants34 are reproduced within the 2%. A similar agreement is found for thermodynamic properties such as melting points and latent heat of melting. It follows that, although the present work is primarily thought to qualitatively address the issue of size effects in Pd and PdHx phases, numerical predictions can be considered relatively good estimates for comparison with experimental values. A bulk Pd phase with a cF4 fcc crystallographic structure containing 864 000 atoms arranged in 60 × 60 × 60 elementary cells was initially created. The bulk system was relaxed at 298 K and 0.1 MPa with number N of atoms, pressure P, and temperature T constant.34,35 The Parrinello-Rahman scheme was used to allow possible phase transitions involving changes in the geometry of the elementary cell.36 Periodic boundary conditions (PBCs) were applied along the three Cartesian directions.37 The equations of motion were solved by employing a fifth-order predictor-corrector algorithm37 and a time step of 2 fs. As indicated by the decay of the fluctuations of the system volume, potential energy, and kinetic energy, the relaxation process attained completion after approximately 0.1 ns. The relaxed Pd bulk lattice was used to generate four PdHx bulk β phases with x equal to 0.6, 0.7, 0.8, and 0.9. To such an aim, a suitable number of H atoms was inserted into randomly selected
octahedral sites of the Pd fcc lattice. The whole system was subsequently left free to relax for 0.5 ns. Starting from large bulk systems, unsupported Pd and PdHx particles were created by selecting spherical regions of radius R. Particles were isolated from the surroundings by carrying out a 50-ps long linear decrease to zero of the A and ξ potential parameters for the interactions between atoms inside and outside the selected regions. Ten different Pd particles were created by giving the radius R values roughly in the range between 2 and 11 nm. The system with radius R equal to 1 nm was discarded in the light of the small number of Pd atoms included, namely 281, which makes the system a cluster rather than a particle. Under such circumstances, size can be expected to produce specific effects that could not be easily connected with the smooth ones expected at larger sizes.11-13 The radius R and the number N of atoms of Pd particles are given in Table 1. In the case of PdHx particles, the radius R was chosen in order to include the same number NPd of Pd atoms included in the above-mentioned Pd particles. The radius R of PdHx particles corresponds to the radius of gyration of the system and was accordingly evaluated case by case. This allowed to explore the relationship between the radius R of PdHx particles and the content x of H atoms. Regarding thermodynamic properties, and enthalpy in particular, it must be remembered that the heat of formation at 298 K of pure Pd phases with fcc crystalline structure is conventionally set equal to zero. The situation is quite different for nanometer-sized Pd particles as well as for PdHx phases. The bulk PdHx system exhibits a heat of formation different from zero connected with the chemical contribution to Gibbs or Helmholtz free energies, whereas unsupported Pd and PdHx particles are additionally affected by free energy contributions coming from free surfaces. Starting from such considerations and aimed at pointing out possible size effects on the heat of formation of nanometersized PdHx phases, the comparison between the different systems will be focused on enthalpy differences. In this regard, it must be noted that the evaluation of enthalpy for unsupported nanometer-sized particles is not an obvious issue. In fact, pressure is an ill-defined quantity for unsupported systems.37 However, in solid phases at zero external pressure the difference between enthalpy and internal energy is substantially negligible. In practice, the two quantities can be considered coincident.38,39 Along this line, the sum of the total potential and kinetic energies of the systems also provides a measure of their total enthalpy.38 Therefore, the isothermal-isobaric ensemble is approximated by the canonical ensemble.37 In the case of nanometer-sized particles, surface effects are automatically included and the total enthalpy also represents their heat of formation. Together with the heat of formation, numerical findings will be used to estimate the surface free energy. Most of the numerical findings cited in the text or reported in graphics represent an average from at least three different simulation runs. Error bars in figures and uncertainty ranges in numerical values indicate the variability between different runs.
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Figure 2. The heat of formation ∆Hx,b of bulk PdHx β phases as a function of the fraction x of H atoms relative to Pd atoms.
Figure 1. (a) The number NPd of Pd atoms in Pd and PdHx particles as a function of the particle radius R. Data refer to Pd (0) as well as to PdH0.6 (O), PdH0.7 (4), PdH0.8 (3), and PdH0.9 (left triangle) particles. The arrow indicates the direction of increase of R with the fraction x of H atoms. (b) The radius R of PdHx particles as a function of the fraction x of H atoms relative to Pd atoms.
3. Results The introduction of x H atoms in the octahedral sites of the Pd fcc bulk lattice induces the formation of the bulk PdHx β phase. According to calculations, the relaxed β phase lattice exhibits an elementary cell parameter on the order of 0.403 ( 0.001 nm, in excellent agreement with the experimental parameter of about 0.402 nm.10 Bulk phases exhibit only a weak dependence of the system volume on the relative number x of H atoms to Pd atoms. In particular, the elementary cell parameter changes from about 0.401 ( 0.001 to 0.407 ( 0.001 nm as x changes from 0.6 to 0.9. The radius R of nanometer-sized particles is also weakly sensitive to the total amount of H atoms. The effects of H atoms on the effective radius R of the PdHx particles are pointed out in Figure 1a, where R is correlated with the number NPd of Pd atoms in Pd and PdHx particles in a semilogarithmic plot. For Pd and PdHx particles with an equal number NPd of Pd atoms, the increase of the fraction x of H atoms results in a corresponding increase of R. The effect can be quantified from Figure 1b, where the radius R of the smallest and largest PdHx particles is shown as a function of x. It can be seen that R increases according to a curved trend, the maximum rise of about 4% being observed at x equal to 0.9 for both 2 and 11 nm particles. The difference in enthalpy between bulk Pd and bulk PdHx phases provides a measure of the heat of formation ∆Hx,b of PdHx systems.39 The obtained ∆Hx,b values are shown in Figure 2 as a function of the fraction x of H atoms relative to Pd atoms. The ∆Hx,b estimates are quite close to each other, roughly ranging from -34.6 ( 0.1 to -33.7 ( 0.1 kJ mol-1. Although only a weak dependence on the hydride stoichiometry is observed, the points arrange according to a relatively welldefined trend. In fact, the heat of formation ∆Hx,b takes the smallest value in correspondence to a fraction x of H atoms equal to 0.6. The average ∆Hx,b estimate obtained is fairly close to the experimental value of about 35.0 kJ mol-1 for the heat of formation of bulk PdHx β phases.40-42 Instead, the scattering
Figure 3. (a) The heat of formation ∆H0.6,p of the PdH0.6 β phase as a function of the particle radius R. The horizontal dotted line indicates the heat of formation ∆H0.6,b estimate for a bulk PdH0.6 β phase. (b) The ratio ∆Hx,p/∆H0.6,p between the heat of formation ∆Hx,p of the PdHx β phase and the heat of formation ∆H0.6,p of the PdH0.6 β phase for particles of similar radius R as a function of the fraction x of H atoms relative to Pd atoms. The vertical arrow indicates the direction of change of the particle radius R from about 2 to 11 nm.
of theoretical and numerical values does not allow any definite comparison. It can be only noted that the above-mentioned average ∆Hx,b estimate is approximately in the middle of the broad range defined by literature values.41 Regarding nanometer-sized particles, the situation is quite different. In fact, the heat of formation ∆Hx,p of the PdHx β phase on the nanometer scale is clearly affected by size effects. This can be easily seen from Figure 3a, where ∆Hx,p is shown as a function of the particle radius R. The heat of formation ∆Hx,p of the PdH0.6 β phase decreases steadily as the particle size increases. Correspondingly, the stability of the PdH0.6 β phase decreases as the particle radius R decreases. The points in Figure 3a refer to the case of PdHx β phase particles with a fraction x of H atoms equal to 0.6, but similar evidence is obtained in the other cases. Therefore, the formation of the PdHx β phase is increasingly disfavored as the particles size decreases. Note that the ∆Hx,p estimates for nanometer-sized particles progressively approach the ∆Hx,b estimates for bulk PdHx β phases of similar compositions as the particle radius R increases. The trend is asymptotic, and the data for particles with radii R larger than 11 nm are expected to become increasingly closer to the bulk value. The system size also affects the relative stability of PdHx β phases with different fractions x of H atoms. To point out such
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Figure 4. The heat of formation ∆H0.6,p estimate of the PdH0.6 β phase as a function of the reciprocal of particle radius R, R-1. The best-fitted line is also shown.
effects, the results are shown in Figure 3b as a function of x. For visualization purposes, the ∆Hx,p estimates have been normalized to the corresponding ∆H0.6,p value obtained for the particle with roughly the same radius R and a fraction x of H atoms equal to 0.6. Therefore, the higher the ratio ∆Hx,p/∆H0.6,p, the more stable the PdHx β phase. Considering all of the particles investigated, the maximum and minimum ∆Hx,p values are respectively equal to about -22.06 ( 0.41 and -32.81 ( 0.42 kJ mol-1. Although the effects of composition are relatively small for particles with the same radius R, a clear trend emerges from the plot in Figure 3b. It can be seen that the smallest ∆Hx,p estimate gradually shifts toward larger x values as the particle radius R decreases. It follows that the relative stability of PdHx β phases with a large content of H atoms increases as the system size decreases, even though the formation of the PdHx β phases is in general disfavored by a reduced system size. More specifically, a fraction 0.6 of H atoms is favored in 8, 9, 10, and 11 nm particles, a fraction of 0.7 in 7 nm particles, a fraction 0.8 in 5 and 6 nm particles, and a fraction 0.9 in 2, 3, and 4 nm particles of PdHx β phases. As shown in Figure 4 for systems with a fraction x of H atoms equal to 0.6, the heat of formation ∆Hx,p of the PdHx β phase increases linearly with the reciprocal of the particle radius R, R-1. This suggests that the behavior observed could be related to surface effects.42 To investigate this issue, the particles were divided into k concentric spherical shells of suitably chosen thickness. Various thickness values between 0.05 and 0.40 nm were used to adequately reproduce the profile of average potential energy along the particle radius. The average potential energy uk associated with each shell k was evaluated by considering the atomic species included in the shell. Of course, the number k of shells included in each given Pd or PdHx particle depends on the radius R and on the shell thickness employed. In all of the cases, the profiles of uk along the radius R, i.e., in the different k shells, are similar to the ones shown in Figure 5 for Pd and PdH0.6 particles with radius R of about 3 nm and shells 0.2 nm thick. The obtained uk profiles clearly indicate that particles include two different regions. A bulk-like region can be identified in the particle interior in which the average potential energy uk of the atoms in the inner shells is substantially equal to the u one exhibited by the corresponding bulk phase. Conversely, a significant deviation from the bulk value is observed as the particle free surface is approached. In particular, a surface layer of width δ can be identified that exhibits a potential energy uk higher than u. It follows that the surface region is destabilized relative to the bulk-like ones in the particle interior, which in principle rationalizes the increase in the heat of formation ∆Hx,p of the PdHx β phase in nanometer-sized particles.
Delogu
Figure 5. The average potential energy uk of the k-th spherical shell along the particle radius R as a function of the number k of shells. Data refer to the case of a Pd particle with radius R of about 3 nm and of the corresponding PdH0.6 particle. Shells are 0.2 nm thick. The horizontal dotted lines indicate the average potential energy u of the corresponding bulk Pd and bulk PdH0.6 β phases.
Figure 6. The width δ of the surface layer of Pd and PdH0.6 β phase particles as a function of the particle radius R.
Regarding the above-mentioned partitioning of particles in different concentric slices, note that it only represents a rough method to point out differences between bulk-like and surface regions. In general terms, it suffers various limitations, which become formally more important in the case of many-body force schemes such as the one used in this work. Therefore, the method is not applicable in general. However, despite its roughness, the method works relatively well, permitting a suitable distinction, though still arbitrary, between different domains in the same particle. It must be also noted that different criteria can be, in principle, chosen to partition the particle. For example, structural details such as interatomic distances and angles can be used. Despite this, the different criteria often lead to similar partitions and, in particular, to similar thickness values for the surface layer. Also, the thickness of the surface region must be expected to vary with the potential employed and, in particular, with its cutoff length. The thickness δ of the surface region does not exhibit a dependence on the fraction x of H atoms relative to Pd ones. Conversely, it is sensitive to the particle radius R for both Pd and PdHx β phase particles. Its variation with the particle size is shown in Figure 6, where δ is plotted as a function of R. In agreement with previous results,43 it can be seen that the width δ of the surface layer increases as the particle radius R decreases. Actually, a simple power law connects δ and R. In fact, almost linear plots are obtained when δ is shown as a function of the reciprocal of R, R-1, not shown here for brevity. The thickness δ of the surface region amounts to about 0.40 ( 0.03 nm for the largest particles but increases up to about 0.90 ( 0.03 nm for the smallest ones. Data in Figure 6 refer to the case of PdHx β phase particles with a fraction x of H atoms equal to 0.6, but the behavior is similar for all of the investigated chemical compositions. In addition, it must be noted that the δ values are quite similar for Pd and PdHx β phase particles with approximately the same radius R, which suggests that the
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extension of the surface layer is somewhat related to the curvature of the particle free surface. Regarding this latter aspect, note that it apparently contradicts recent results.44-46 In fact, refined experiments on Au particles and numerical studies on Au and Pd nanometer-sized particles have shown that the curvature dependence of surface thickness is quite small or, in the limit, absent.44-46 However, such conclusions are in contrast with previously obtained evidence worked out from a thermodynamic approach to experimental data regarding both temperature and latent heat of melting of metallic particles.47 Along the same line, note that a dependence of surface features on the particle curvature is expected on theoretical bases48-51 and demonstrated by other numerical studies regarding metallic systems.43 Therefore, no definite conclusion can be drawn by the available data sets, and a more systematic and homogeneous approach to the question must be invoked. The distinction between bulk-like and surface regions allows discrimination between bulk-like and surface contributions to the total enthalpy individual particles. More specifically, for any given Pd or PdHx particle, the enthalpy can be written as
∆Hp )
1 (∆Hp,bNb + ∆Hp,sNs) N
(2)
where N is the total number of atoms in the particle, Nb and Ns are the number of atoms in bulk-like and surface regions, respectively, and ∆Hp,b and ∆Hp,s are the enthalpy contents of the bulk-like and surface regions. Equation 2 can be further manipulated to account for the dependence of Nb and Ns on the width δ of the surface layer. The total number N of atoms in a Pd or a PdHx particle of radius R is equal to (4/3) π R3 F, where F is the atomic density of Pd or a PdHx. Along the same line, provided that bulk-like and surface regions have the same density F, the numbers Nb and Ns of bulk-like atoms and of surface atoms respectively are equal to (4/3) π (R - δ)3 F and (4)/(3) π [R3 - (R - δ)3] F. Under the approximation that (R - δ)3 ≈ R3 - 3 δ R2, eq 2 can be rewritten as
(
∆Hp ≈ 1 - 3
δ δ ∆Hp,b + 3 ∆Hp,s ) ∆Hp,b R R δ 3 (∆Hp,b - ∆Hp,s) (3) R
)
Because the ∆Hp,b values substantially equal to the ∆Hb values obtained for bulk Pd and PdHx β phases, eq 3 permits the estimation of the average enthalpy content ∆Hp,s associated with surface regions. On the basis of all of these observations, eq 3 was used to evaluate the heat of formation ∆Hx,p,s of the PdHx β phase in the surface regions. A few data are shown in Figure 7, where the ∆H0.6,p,s estimates are plotted as a function of the particle radius R. Data refer to the case of a PdHx β phase with a fraction x of H atoms equal to 0.6. Similar results are obtained for other chemical compositions. In view of the qualitative character of the present analysis, calculations have been carried out by using δ and R values averaged over Pd and PdH0.6 β phase particles of similar size. The ∆H0.6,p,s estimates are quite scattered, which can be ascribed to the roughness of the evaluation procedure. Apart from this, data arrange according to a relatively well-defined increasing trend that exhibits a tendency to reach an asymptotic value as the particle radius R increases. The heat of formation ∆H0.6,p,s of the PdH0.6 β phase in the surface region takes
Figure 7. The heat of formation ∆H0.6,p,s of the PdH0.6 β phase in the surface region of nanometer-sized particles as a function of the particle radius R. The dotted curve is a guide to the eyes.
negative values on the order of -5 ( 1.21 kJ mol-1 for the smallest particles and progressively increases up to about 20 ( 1.23 kJ mol-1. Compared with the heat of formation ∆H0.6,b of the PdH0.6 β phase in a bulk system, both values are considerably larger. Correspondingly, the formation of the PdH0.6 β phase in the surface region is considerably disfavored with respect to the case of the bulk. This permits the connection of the previous evidence regarding the stability of the PdH0.6 β phase in nanometer-sized particles with surface effects. In particular, the increase of the heat of formation ∆Hx,p of the PdH0.6 β phase as the particle radius R decreases can be ascribed to the unfavorable contribution of the surface region. Paradoxically, the general trend of the ∆H0.6,p,s estimates in Figure 7 also suggests that the formation of the PdH0.6 β phase is less disfavored in smaller particles. In fact, the heat of formation ∆H0.6,p,s decreases as the particle radius R decreases. Thus, if on the one hand surface effects determine a general destabilization of the PdH0.6 β phase, on the other they seem to favor the formation of the PdH0.6 β phase in the surface region. To further analyze this point, a large PdH0.6 β phase with a plane free surface was generated and investigated. In particular, the bulk PdH0.6 β phase crystal containing 864 000 Pd atoms arranged in 60 × 60 × 60 elementary cells used to estimate bulk quantities was virtually transformed into a semiinfinite crystal. To such an aim, PBCs along the z Cartesian direction were removed and a bottom layer two crystallographic cells thick transformed into a rigid reservoir by immobilizing all of the atoms. After a relaxation of the free surface at the top of the semiinfinite crystal, the profile of the potential energy u along the z Cartesian direction was evaluated. The semiinfinite crystal was divided into k parallel layers of thickness between 0.05 and 0.30 nm and the average potential energy uk of each k-th layer evaluated. The results obtained are analogous to the those shown in Figure 5 for Pd and PdHx particles. In fact, the uk values for the three top layers are increasingly larger than the uk estimates for the bulk layers. More specifically, the uk value for the layer at the free surface amounts roughly to -570 ( 2.13 kJ mol-1. Then, it is slightly smaller than the value for the surface shell of the case considered in Figure 5, which is probably due to the potential energy difference between plane and curved surfaces. Along this line, the heat of formation of a PdH0.6 β phase at a plane surface was also estimated. A value of about 24.4 ( 1.32 kJ mol-1 was found, which is even larger than the ∆H0.6,p,s estimates for the nanometer-sized particles considered. Thus, it appears that also in massive systems terminated with a free plane surface the formation of a PdH0.6 β phase in the surface region is considerably disfavored. 4. Conclusions Molecular dynamics simulations were carried out to systematically investigate the effect of system size on the properties
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of nanometer-sized Pd and PdHx β phases. Despite the qualitative nature of the numerical findings, various general indications on the relative stability of PdHx β phases were obtained. First, irrespective of the fraction x of H atoms relative to Pd atoms the smallest heat of formation of PdHx β phases is observed in bulk systems. In fact, it significantly increases as the system size approaches the nanometer scale. It follows that PdHx β phases exhibit higher stability in bulk crystals rather than in nanometer-sized particles. The heat of formation of the PdHx β phase particles is connected with the particle radius by a simple power law dependence, which points out the occurrence of usual smooth size effects in the explored range of particle radius. Second, the analysis of the potential energy profile along the particle radius allows the distinction between bulk-like and surface regions. The atomic species belonging to the former region substantially behave as if they were embedded into a massive crystalline phase, with no significant deviation of static and dynamic properties. Conversely, the atomic species located in surface regions exhibit a general behavior significantly different from the bulk one. In particular, they have a higher specific surface free energy due to the unsaturation of their coordination number. Third, the thickness of the surface region influenced by the unsaturation of coordination number at free surfaces is also affected by smooth size effects. In fact, it progressively increases as the particle radius decreases. In particular, the thickness undergoes an almost linear decrease with the reciprocal of the particle radius. Fourth, the chemical composition of PdHx β phases has only limited effects on their stability. However, the numerical evidence suggests that the stoichiometry of the most stable phase shifts gradually toward larger H atom contents as the particle radius decreases. Therefore, the compositions richer in H atoms are more favored in smaller particles. Finally, note that very rough calculations suggest that the heat of formation of PdHx β phases in the surface region decreases as the particle radius decreases. It follows that the effects of particle size on the relative stability of PdHx β phases can be summarized by noting that the percentage of H atoms in both bulk-like and surface regions can be in principle larger for smaller systems, even though nanometer-sized PdHx β phases in general exhibit a larger heat of formation than bulk systems. Hopefully, although based on a rough and fragmentary framework of numerical findings, precisely this feature could inspire future experimental research on this important subject. Acknowledgment. Financial support was received from the University of Cagliari. A. Ermini, ExtraInformatica s.r.l., is gratefully acknowledged for his kind assistance and technical support. References and Notes (1) Basic Research Needs for the Hydrogen Economy, Report on the Basic Energy Science Workshop on Hydrogen Production, Storage, and Use; Dresselhaus, M.; Crabtree, G.; Buchanan, M., Eds.; Argonne National Laboratory, U. S. Department of Energy, Office of Basic Energy Science: Argonne, IL, 2003. (2) Basic Research Needs to Assure a Secure Energy Future, Report from the Basic Energy Sciences AdVisory Committee; Stringer, J.; Horton, L., Eds.; Oak Ridge National Laboratory, U. S. Department of Energy: Oak Ridge, TN, 2003. (3) Solomon, B. D.; Banerjee, A. Energy Policy 2006, 34, 781. (4) Moriarty, P.; Honnery, D. Int. J. Hydrogen Energy 2009, 34, 31. (5) Sandrock, G. J. Alloys Compd. 1999, 293-295, 877.
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