Strain-Dependent Chemical Reaction on Inhomogeneous Surfaces

Article pubs.acs.org/JPCC

Strain-Dependent Chemical Reaction on Inhomogeneous Surfaces Y. Manassen,† H. Realpe,† Y. Greenbaum,‡ N. Shamir,*,§ M. H. Mintz,‡,∥ and R. Z. Shneck§ †

Department of Physics and the Ilse Katz Center of Science and Technology in the nm scale and §Department of Materials Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva, 8410501, Israel ‡ Department of Nuclear Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva, 8410510, Israel ∥ Nuclear Research Center-Negev, P.O. Box 9001, Beer-Sheva, 84190, Israel ABSTRACT: Gd islands grow on an annealed Gd(0001)/ W(110) layer in the Stransky Krastanov mode. When hydrogen is adsorbed on strained Gd islands, we find that the GdH2 nuclei are formed on the islands and not on the wetting layer between them and preferably close to the edges of the islands. This is in contrast to the usual case where the reaction is on the edge. The Gd islands on the W surface are under tensile stress, which decays due to relaxation upon approaching the edge. The lattice constant of GdH2 is smaller than that of strained Gd, but is larger (by 4%) than that of unstrained Gd. Since the surface energy of GdH2 is dependent on the strain, the rate of nucleation depends on it, too, and the nuclei are formed favorably in the areas of no strain between the strained Gd and GdH2. The distribution of the hydride nuclei around these areas is a quantitative indication to the surface energy and its dependence on strain (surface stress and elastic energy). Thus, the hydride nuclei distribution can serve as a tool for measurement of the local values of surface energy, surface stress and elastic energies. These considerations apply to every gas−solid interaction, starting by nucleation on a strained surface.

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INTRODUCTION Gas solid reactions are always initiated by nucleation and growth processes. In a real solid (namely not a single crystal) the reaction always starts from some particular sites, and then proceeds forward. The identification of the “Achilles heel” where reaction starts in the solid has a significant scientific merit and technological importance. An important example to these gas solid reactions is the reaction between metal and hydrogen to form metal hydrides, which is of importance for many applications, the one which is mostly known for energy storage. As an example to metal hydrogen reaction on a heterogeneous substrate, we have studied the hydrogen exposure of a Gd thin film, which is grown on a W substrate. As was shown in a previous paper,1 the Gd film grows in a Stransky Krastanov mode that forms a Gd wetting layer on which many different Gd islands grow. The strain in such a heterogeneous surface is highly nonuniform. We find first that, in the reaction between hydrogen and Gd, small hydride dots are formed only on the islands and not on the wetting layer in between. In addition, also on the Gd islands, the hydride dots are formed only near the edge. The distance from the edge is defined using the parameter r/R, where the R is the radius of the Gd island and r is the distance between the position of the hydride and the center of the Gd island. The distribution of the position of the hydride dots is forming a skewed Gaussian centered at r/R = 0.8. We use the fact that the misfit between the hydride GdH2 and the Gd layer is 4%. It is plausible that the GdH2 will choose to grow in a location with zero strain with the (strained) Gd © XXXX American Chemical Society

layer. The question of elastic relaxation of coherent epitaxial deposits was discussed in a general and analytic way by ref 2. In this work it was shown how, with a known relative rigidity between the materials of the deposit and the substrate, for an island with a certain relative size (to the lattice constants), it is possible to observe approximate analytical expressions to the strain distribution both in the deposited island and the substrate underneath, at any distance from the interface between them. We use these expressions to evaluate the strain between the island and the substrate, using the peak point of the distribution of the GdH2 dots as a point with a known strain (4%). This is necessary since the misfit, as discussed in (1), is nonuniform. These equations can then give the strain distribution along the r/R line. Afterward, using published values of the surface energy, surface stress, and elastic constants for GdH2, we evaluated the dependence of the activation energy and the relative rate of nucleation on strain (and r/R). Good agreement with the experimental distribution of hydride dots is demonstrated. This procedure can be reversed by calculating the local surface energy, surface stress, and elastic constants from the hydride islands position and can be applied in any gas solid chemical reaction. Received: August 7, 2016 Revised: October 10, 2016

A

DOI: 10.1021/acs.jpcc.6b07971 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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nucleation of gadolinium hydride near the edge of epitaxial gadolinium islands and propose that this preference reflects the effect of pre-existing strains on the nucleation. Many phenomena demonstrate the importance of strain effects in the nm scale. The strain-dependent preferential nucleation is responsible for the appearance of superlattices of quantum dots;13 In nm size argon bubbles on Ru(0001), the top is in tension and the periphery is in compression. The adsorption of O2 and CO occurs preferentially on the part in tensile stress:14 The increase of binding energy as a result of lattice extension14,15 and the effect of strain on the reaction equilibrium and the reaction rates.16−19 In this paper it is demonstrated how the spatial distribution of nuclei on a strained surface can be used to evaluate the local mechanical properties of the material from which the nucleus is made.

Tungsten has a BCC crystal structure and an atomic radius of 1.93 Å. Gd has a HCP structure and an atomic radius of 2.33 Å. The misfit between these W(110) and Gd (0001) planes is ⎛ 0.408 ⎞ ⎟ ϵGd/W =⎜ ij ⎝ 0.150 ⎠

(1)

There are contradictory results regarding the Gd (0001) on W (110) interface: Compression of the Gd layer by 4.4%;3 A dilatation of 8% in a thin Gd layer (3 Å) is getting smaller in thicker layers until a complete relief beyond a thickness of 100 Å;4 A Gd monolayer is expanded by 1.2% along the W[001] direction and compressed by 0.6% along the W⌈1̅10⌉ direction.5 A nonuniform strain distribution on this system was recently demonstrated.1 GdH2 has a CaF2 structure that is a FCC lattice of Gd, with hydrogen occupying the tetrahedral interstitial sites between the Gd atoms.6 The molar volume of GdH2 is about 12% larger than that of Gd.7 The most plausible orientation relations between the Gd and GdH2 is such that the (0001) close packed plane of Gd is parallel to the (111) closed packed plane of GdH2 (Figure 1). This relation gives rise to misfit strain tensor: ϵijT,H

⎞ ⎛ 0.034 ⎟ ⎜ =⎜ 0.034 ⎟⎟ ⎜ ⎝ 0.043⎠

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EXPERIMENTAL SECTION

A W(110) crystal was used as the substrate. The crystal was cleaned by repeated cycles of heating at 1760 K in oxygen and flashing up to 2600 K in UHV conditions. These cycles were repeated until the carbon AES signal was stabilized to a value of few percent. The STM image of a clean W(110) surface displayed terrace widths in the range of 0.03−0.15 μm. The Gd films were deposited (in three different experiments) by an ebeam evaporator. During Gd deposition the pressure was raised up to 1 × 10−7 Pa. A total of 5−20 monolayers were deposited, followed by AES characterization and annealing at 920 K for 9−16 min. As a result of the annealing, the deposited amorphous Gd layer is turned into a Gd wetting layer on which 3D GD islands grow by a Stranski Krastanov process,20 which partially relieves the surface strain. Afterward, STM images were taken at room temperature. The heights of the 3D islands were in the range of 8−14 nm. Then, the sample was exposed to a small amount of pure hydrogen and imaged again by STM. The thickness of the Gd layer and the amount of hydrogen were: 10, 10, and 24 ML and 1.6, 8.4, and 17.6 L (L = 1 × 10−4 Pa × sec), respectively. After exposure of the sample to hydrogen, small hydrides are observed on some of the Gd islands. A few of them are shown in Figure 2. They are rare at the center of the islands and on the epitaxial wetting layer. In the STM images we have observed 42 Gd islands and identified 64 hydride nuclei. Their radial location from the center r/R was monitored with the aspect ratio of the island on which they grow, h/R, where R is the radius of the island and h is its height. The observed aspect ratios were in the range from 0.018 to 0.245 (the average in each experiment, of the three performed, was 0.037, 0.081, 0.057). The radial distribution of the observed hydrides shows a clear preference to sites near the edge of the islands (Figure 3).

(2)

Figure 1. (a) Structure of the (111) plane of GdH2. (b−d) Schematic illustration of the lattice cells in the Gd island along the radial distance r/R (not to scale) from left (center of Gd island) to right (edge of Gd Island). (b) The island is under compression. (c) The island is under small tension (smaller than the misfit between unstrained Gd and GdH2). (d) The island is under tension larger that the misfit between unstrained Gd and GdH2. The center of the Gd island is stretched more than the stress-free hydride lattice and the strains gradually relax toward the edge. The best-fit choice for the hydride to grow is where the unit cells of the strained layer and the hydride have similar dimensions (the light blue hexagon−marked by an arrow). (e) The W substrate. The figure includes the actual crystallographic dimensions of free W, Gd, and GdH2.

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DISCUSSION Stress Distribution. We look for an explanation to the tendency of the hydrides to grow near the edge of the islands. Our argument is based on the assumption that the large misfit between the lattice of the Gd and the lattice of GdH2 can generate a large elastic energy that becomes a main energy barrier to the formation of the hydride. Therefore, preferred nucleation of hydrides takes place at locations that provide best lattice matching between the parent metal (Gd) substrate and the hydride. This argument will also provide an estimation of the misfit strain in the epitaxial Gd(0001) layer on W(110).

which is called the transformation strains.8 Despite the large strains, Falangan et al.9 estimated that hydrides are semicoherent. In the following we estimate the misfit between Gd and GdH2 as 4%. Steps on the surface of crystals are often observed to be preferred sites for adsorption of atoms and for nucleation of new phases (e.g.10−12,). In contrast, we observed preferred B

DOI: 10.1021/acs.jpcc.6b07971 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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Figure 3. Radial distribution of the observed hydride particles on strained Gd islands with different aspect ratios (see text). The real number of observed hydride nuclei at certain position is illustrated rather than the probability distribution (as in Figure 5b).

Figure 2. (A) Schematic cross section of the system. The island radius, R, the distance of the individual hydride from the center, r, and island height, h, are indicated [also in B (f)]. (B) Two STM images (a) after deposition of 24.4 ML Gd and annealing, before hydrogenation, and (b, c) the same area after hydrogenation: exposure to 17.6 L hydrogen. (b, d, f) Consecutive magnifications, (e) magnification of (c), and (f) magnification of (d). Tip sample bias voltages and tunneling currents were 2.7 V and 0.3 nA, respectively. (g, h) Before and after hydride formation. Images at bias 2.7 V and 0.4 nA current. Hydrides encircled.

Figure 4. (a) An exaggerated displacement map of a Gd layer under tension by the W substrate. (b, c) Maps of (σrr + σθθ) for three islands with aspect ratios h/R = 0.05 and 0.16, respectively. Calculated by the finite element method.

The rate of the stress decay depends on the aspect ratio of the island (see, e.g., ref 2). When the island has a small height to width ratio, the stresses in the island are usually uniform and very close to the stresses in the uniform wetting layer (Figure 4b). As the island grows higher, the stress at the center relaxes, and it decays more moderately toward the edge of the island (Figure 4c). The stress distribution is similar if the Gd layer is under compressive stress, except for the opposite signs of each stress component. Possibility of Compressive Epitaxial Stress. As described in the Introduction, the effective misfit between the W and Gd is still ambiguous. We can, however, deduce its sign from theoretical considerations. If we assume that the epitaxial stresses in the Gd layer are compressive, they will gradually decay toward the edge of any island. Compressive stresses are unfavorable for GdH2 that has a larger molar volume than the Gd matrix. In this case, the edges of the islands, where the Gd is

The argument is based on the distribution of stress in an epitaxial Gd island. We assume first that the epitaxial relation between Gd and W sets tensile stresses in the islands and in the surrounding wetting layer (Figure 4a). Thus, it is found that the wetting layer and the center of the island are in a state of 2D tension in the plane parallel to the W−Gd interface. The strains and stresses gradually decrease along the radius (Figure 4b,c). The normal stress σzz vanishes over the whole upper face of the island and the radial stress σrr vanishes along the edge due to the boundary conditions on free surfaces, thus, the hydrostatic stress is very small near the edge of the island. The elastic interaction energy between the hydrid and the island is proportional to the hydrostatic stress that is 1/3(σrr + σθθ).8 C

DOI: 10.1021/acs.jpcc.6b07971 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C To proceed we need to evaluate ⟨f⟩ defined by

nearly free of stresses, will be the most favorable locations for nucleation of hydrides (Figure 1a,b). Those hydrides, which nucleate at the edges, will benefit both from saturation of open bonds and from free expansion in the r and z directions (e.g., ref 14). However, the Gd hydrides are not observed to grow at the edges, and one has to conclude that the Gd layer is under tension. Possibility of Tensile Epitaxial Stress. The case of tensile epitaxial stresses is depicted in Figure 4. The hydrostatic stresses in the island vary from maximum tensile at the center to nearly zero along the edge. A hydride particle is in a state of compression in free gadolinium. A nucleation of a hydride nucleus in any region in a Gd island and on the wetting layer between the islands where tensile strains prevail will result in saving of the elastic energy of the whole system. If the epitaxial strains are smaller than the transformation strains of the hydride and a small region in the Gd island is replaced by a hydride, the hydride will still be under compression, but to a smaller extent than a hydride in a free Gd matrix. The residual compression is minimal when the tensile strains in the layer are maximal, that is, at the center of the island and on the wetting layer (Figure 1a,c). However, the hydrides are not observed to grow at the center of the islands nor on the wetting layer. The remaining possibility is that the epitaxial strains are larger than the transformation strains. Figure 1a,d show schematically that if εW, the strains in the Gd due to the epitaxy with W, are tensile and are larger than the transformation strains of GdH2, the hydride is stretched near the center of the island and is compressed near its edge of the island that is nearly free of strain. Therefore, in between, there should be a region where the epitaxial strains in the Gd island are nearly equal to the transformation strains of the GdH2 nucleus, and there, the elastic strains in the hydride are expected to vanish. We suggest that this region of minimal relative strains is the preferred region for the first events of hydride nucleation on the strained Gd islands. Knowing that the linear misfit between Gd and GdH2 is approximately 0.04, we can estimate the effective epitaxial strain between Gd and W in the range of 0.06−0.08. In the analysis of these results, we discuss separately each Gd island: The islands have a different aspect ratio (h/l). Despite this difference, all of them have a hydride distribution maximum at r/R = 0.8. In order to discuss this question from the general point of view, we use an analytic approximate description of this problem:2 The strain distribution in an epitaxial ribbon that grows on a mismatched substrate is given by the formula:

f

{K ·l /(π ·h· f )}

(6)

2 ⎡ ⎛ π ⎞⎤ ⎛ 2l ⎞⎤ ⎜2 ⎟ ⎥ ·exp⎢ − ⎜ ⎟⎟⎥ ⎝ aπ ⎠⎥⎦ ⎢⎣ ⎝ (l/a) ⎠ ⎥⎦

(7)

where l is the width of the island. Intuitively speaking, ⟨M1⟩ describes the decay of the strain field in a direction orthogonal to the surface. Considering the above discussion, it is plausible that, at r/R = 0.8, ε is likely to be 0.04. This is the value of strain on top of the Gd island. Yet, we can estimate the mismatch m0 at the interface between Gd and W using ε(h) = ε(0)⟨M1⟩(|h−a|/a), see ref 2. Thus, we can use the model described in ref 2 to get the approximate strain distribution at the top of the island as a function of r/R. Yet, this strain distribution must be linked to the hydride islands distribution. We shall show that this is a kinetic process that is strain dependent. The process of the growth of GdH2 islands start with a formation of a nucleus. This is a heterogeneous nucleation process. The probability to get a nucleus at a certain position r/ R, at a certain strain ε is given by P(ε(r / R)) = exp −[Cε(r / R)2 /kT ]·exp −[Ea /kT ]

(8)

where Ea = 16πf (θ )γ 3/3(ΔGv )2

(9)

−2

where kT = 2.6 × 10 eV/atom; ΔGv = −1.7 eV/atom for GdH2.21 The first exponential in P (ε(r/R)) is a preexponential term describing the probability of the hydrogen atoms to be in r/R. The second exponential describes the kinetics of the nucleation process. In heterogeneous nucleation, the activation energy is dependent on the wetting angle (θ) through the function22 f (θ ) = [2 − 3 cos(θ ) + cos3(θ )]/4

(10)

The wetting angle can be determined by the STM image of the small hydride nuclei and we observed f(θ) = 0.4354. The strain dependence of the kinetics is due to the dependence of the surface energy on strain:23 γ = γo + τε + Cε2, where γ is the strain-dependent surface energy, γo is the surface energy at zero strain, and C is the elastic constants. The published value of γo is 0.9 eV/atom24 (using the surface lattice distance of Gd for the conversion).25,26 The literature value of the elastic constant of GdH2 is 70 eV/atom27 (using conversion factor of 1 eV/Å3 = 160 GPa and the density of Gd25). The value of τ is estimated to be equal to the surface energy.28 At this point we recall that the model, which is the basis for the analysis, is dealing with an island with a ribbon shape. However, our island is closer to a square shape. Therefore, according to the superposition principle, the effective elastic energy should be taken as 2Cε2. This is the expression used in the estimation of the strain dependent surface energy. Applying the first step of the analysis for a particular island with an aspect ratio of 0.048, the strain distribution was calculated (Figure 5a). This figure shows the strain taking the

(3)

(4)

K is the relative stiffness between the deposit (A) and the substrate (B): K = [E B /(1 − νB 2)]/[EA /(1 − νA 2)]

−1]/[(h/a)ln M1 ]

⎡ 2π + ⎢1 + M1 = 1 − ⎢⎣ (l/a)

where the parameter P characterizes the material, the substrate, and the deposit P=

(h − a)/ a

where h and a are the height and the lattice constant of the substrate, respectively and ⟨M1⟩ is defined by

ε /m0 = 1 + exp{−P 2 } − exp{−P |1 + r /R | } − exp{−P |1 − r /R | }

= [ M1

(5)

where E and ν are the Young modulus and the Poisson ratio, respectively. In our case, since both the substrate (the wetting layer) and the deposit (Gs island) are made of Gd, K = 1. D

DOI: 10.1021/acs.jpcc.6b07971 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The width of the distribution is dependent on these values (surface energy and stress as well as elastic constants). For comparison the influence of having a softer hydride is shown in Figure 5b (small circular dots gray on line). As expected due to the asymmetry of strain distribution (steeper above r/R = 0.8), also the results of the calculated distribution function (Figure 5b, blue line) are asymmetric. The agreement with the experimental results is clear. We have noted that different islands with different parameters (h, l) give similar distributions. It is interesting to compare the island−substrate misfit observed from the present approach (0.076), with the one observed as a result of the procedure applied in ref 1 (0.0495). This is expected, since the model in ref 2 is considering only coherent relaxation and does not take into account incoherent processes due to misfit dislocations etc which we know that can occur in this system.4 Such incoherent relaxation processes are expected to occur in the higher part of the island. Thus, it is possible to consider the misfit observed in the present approach as the effective misfit that would be observed if the relaxation of the island will only be coherent. In islands with smaller aspect ratio (for example, 0.018), the value of the misfit are closer (0.052 in the present case compared with 0.039 according to the procedure of ref 1. The observation of approximately the same distribution for all the islands, regardless of their aspect ratio, seems to be a contradiction, because higher islands are supposed to have a more efficient relaxation when moving from the center to the edge. However, as was shown in ref 1, the islands are becoming higher because the strain underneath is larger, so, these two effects were shown to cancel each other, creating a maximum at approximately the same location. The procedure can be reversed, namely, to use strain distribution in order to eliminate the mechanical parameters (γo and C) on any heteroepitaxial system. Moreover, these parameters are expected to be local and should be dependent on curvature, local chemical reactions,29,30 and so on. In anisotropic cases it might be possible also to observe more components of the relevant tensors.

Figure 5. Observed normalized strain distribution for one island (aspect ratio of 0.048 (h = 11.9 nm; l = 248 nm) using the first part of the analysis (a); the calculated distribution (solid, blue on line) for all the islands, compared with the experimental one (square dots, orange on line). The points represents the histogram in Figure 3. The error bars are a result of the nonspherical shapes of the Gd islands which results in an uncertainty in the value of R. The calculated distribution observed for smaller hydride elastic constants (20 eV/atom instead of 70 eV/atom, small round dots gray on line). Other values are the same.

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SUMMARY We have shown that the inhomogeneous strain distribution on surfaces can lead to inhomogeneous chemical reaction on the surface and that the distribution can be explained quantitatively using the elastic parameters of the material. Besides the importance of explaining the general surface phenomenon, the procedure can be used to measure the local mechanical parameters.

unstrained hydride lattice constant as reference. We show the absolute magnitude of the strain. The misfit under the island is a parameter, which is modified until the minimum of the strain arrives at the position where it is experimentally observed (with the particular h and l). In the particular island discussed here, this misfit is 0.076. If this value is larger the minimum is moving to higher r/R, when it is smaller. It is moving to lower r/R. The increase of strain is very fast (from 0 to 0.04) when going from r/R = 0.8 to r/R = 1, and much slower when going from r/R = 0.8 to r/R = 0. The second part, as described above, is to calculate the straindependent kinetics, which is reflected in the strain dependent distribution. Using the specific mechanical values for GdH2 we could observe a very good agreement with the experimental results. We have repeated the calculations for other islands with different aspect ratios (and different calculated misfits) and the result for all of them is similar to what is shown in Figure 5b.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: (+972)58-6293007. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was partially supported by a grant from the Israel Council for Higher Education and the Israel Atomic Energy Commission and a Grant from the Ministry of National Infrastructure, Division of R&D. E

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