Electrocapillary Coupling at Metal Surfaces from First Principles: On

Article Cite This: Langmuir XXXX, XXX, XXX−XXX

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Electrocapillary Coupling at Metal Surfaces from First Principles: On the Impact of Excess Charge on Surface Stress and Relaxation Anja Michl,*,† Jörg Weissmüller,‡,§ and Stefan Müller† †

Institute of Advanced Ceramics and ‡Institute of Materials Physics and Technology, Hamburg University of Technology, D-21073 Hamburg, Germany § Institute of Materials Research, Materials Mechanics, Helmholtz-Zentrum Geesthacht, D-21502 Geesthacht, Germany ABSTRACT: We study the response of the surface stress to excess charge via ab initio simulation of metal surfaces in an external electric field. We focus on “simple” sp-bonded metals to gain insight into the mechanisms underlying electrocapillary coupling. Both the direct effect on the surface stress via charging of the bonds and the indirect effect resulting from the charge-induced relaxation are analyzed and discussed in relation to the trends of the coupling coefficients, which owing to a Maxwell relationare determined in terms of the response of the work function to strain. Al(111), Mg(0001), and Na(110) are investigated as prototypical sp-bonded metal surfaces with positive, vanishing, and negative coupling parameters, respectively. Mg(0001) and Al(111) exhibit an inward relaxation of the first atomic layer upon negative charging, whereas an outward relaxation occurs for Na(110). The indirect contribution of the relaxation to the coupling coefficient has the same sign as the total response and makes up about 30% of its magnitude for Al(111) and Na(110). Our study highlights that even the response behavior of the so-called simple metals is by no means readily captured within simple models.

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INTRODUCTION Recent experimental work advertises novel functional hybrid materials fabricated from nanoporous metal impregnated by electrolyte solution, where a reversible modulation of the effective macroscopic materials behavior is achieved via control of the state of the metal surfaces.1,2 The underlying phenomenon is that the surface stress, f, a capillary parameter of solid surfaces, reacts sensitively to a modification in the surface charge density, q. Due to the particularly high surfaceto-volume ratio of the nanoporous metal structure, this results in a macroscopic elastic expansion or contraction, making these materials promising candidates for actuator applications.1,3−8 Similarly, adsorbate-induced variations of f may be exploited in chemical and biochemical sensor technology.9−11 A key quantity for characterizing the material response is the electrocapillary coupling coefficient ς = df/dq. Due to a thermodynamic Maxwell relation, the value of ς is also equal to the response of the electrode potential U to the in-plane strain e.12 Because the electrode potential and the work function of the metal surface are closely linked,13,14 the electrocapillary coupling parameter near the potential of zero charge (i.e., for q = 0) may be determined in terms of the response of the electronic work function, W, to an applied in-plane strain.15 The variation in W with strain is readily accessible via ab initio computation.15−21 Thus, although experimentally these phenomena are typically observed at the solid−electrolyte interface, theoretical studies consider clean metal surfaces in vacuum.15,19,22,23 This approach is justified by the observation that ς saturates in the limit of high dilution of the electrolyte,24 which suggests for © XXXX American Chemical Society

weakly adsorbing electrolytes that the surface stress change originates from a modification of the metallic bonding by the excess charge rather than from adsorption.25,26 The good agreement between coupling coefficients computed using density functional theory (DFT) of metal surfaces in vacuum15,19 with those determined in electrochemical experiments25−30 supports this hypothesis. For instance, DFT calculations yield ς = −1.86 V15 for Au(111), in excellent agreement with corresponding measurements of df/dq and dU/ de, which put the electrocapillary coupling parameter at −1.9525 and −1.90 V,26 respectively. Previous work has focused mainly on transition metals, where exclusively negative values of ς for several surfaces have been reported15,25−27,30,31 and trends identified (cf.19,32−34 and references therein). Moreover, DFT values of ς for Au surfaces have been used to calibrate atomistic models for the study of the charge-induced deformation of nanoporous gold structures35 and nanowires.36 Concerning the microscopic origin of the surface-stress charge response, ab initio simulation of charged Au clusters points toward the important role of surface stretch, a reversible normal relaxation of the outermost atomic layer, which affects the surface stress via the transverse contraction.23 The associated stretch charge coefficient, κ, was found to be of negative sign for the Au(111) surface with a closely similar numerical value in experiment37 and theory.22 The outward relaxation upon negative charging was rationalized Received: December 17, 2017 Revised: February 23, 2018

A

DOI: 10.1021/acs.langmuir.7b04261 Langmuir XXXX, XXX, XXX−XXX

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Langmuir

cells with the same in-plane lattice vectors as the surface supercells and the corresponding minimum number of atoms. The same in-plane k-point grids as for the surface cells were applied, whereas the number of k-points in the normal direction was adjusted to the smaller cell size. In the following, we will outline the definition of the various quantities and how they were calculated. Specifically, the specific surface energy γ, the surface stress f, the top layer relaxation Δd12, the work function W, and the electrocapillary coupling coefficient ς were determined using relaxed uncharged symmetric surface slabs. The relaxation of charged surfaces was performed by employing asymmetric surface slabs, and the surface stress change Δf due to this charge-induced relaxation was then evaluated with correspondingly deformed symmetric slabs. Lastly, excess charge density distributions were obtained using unrelaxed bulk-truncated slabs. Calculation of Surface Properties without Electric Field. Surface properties without an applied electric field were evaluated using symmetric surface slabs with primitive surface unit cells separated by a vacuum region of at least 16 Å. To minimize quantum size effects, the surface slabs were composed of 37 layers for Al(111) and Mg(0001) and 31 layers for Na(110). The positions of the outermost 10 layers on each side were relaxed until the forces acting on the atoms were smaller than 5 × 10−3 eV/Å. The specific surface energy γ was determined according to

by the electrostatic interaction between the laterally modulated excess charge and the ion cores.23 In addition to this indirect contribution via the stretch, contour maps of the excess electron density of charged Au surfaces indicated a direct contribution to ς increasing f for negative q.22 However, trends of the coupling coefficients for different elements have not yet been connected with features and differences in charge-induced relaxation and excess charge density distributions. Despite the fundamental relevance of electrocapillary coupling and the increasing interest in this growing field of research, the understanding of the underlying mechanisms remains far from a level where reliable predictions of magnitude or even sign of ς for a given materials surface can be accomplished without ab initio simulations. There have been hopes that analyzing simple metals would be the first step toward that goal. In this spirit, the work function strain response of the sp-bonded metal Al was investigated, which was revealed to have an unexpected positive sign.21 Here, we explore this route further directly from the perspective of the surface stress charge response and elucidate how qualitatively different coupling behavior is reflected in the distribution of the excess charge and the charge-induced relaxation. To this end, we performed ab initio simulations of metal surfaces subject to an external electric field comparing three prototypical sp-bonded metals, which are characterized by positive, vanishing, and negative response. In the following section, we describe the methodology and computational details of the DFT calculations. Next, our results are presented and analyzed starting with values for the electrocapillary coupling coefficients and further interesting surface properties. We then study the impact of an electric field on the charge density distribution and the relaxation at the surface and discuss the implications for the surface stress. The insight gained about the direct contribution from charging of the bonds and the indirect contribution from the relaxation will be combined to provide an assessment of their suitability for predicting the value of ς. Finally, we summarize our main conclusions.

γ=

1 ⎡ surf N surf bulk ⎤ ⎢Etot − bulk Etot ⎥ 2A ⎣ ⎦ N

(1)

bulk surf where Esurf (Nbulk) are the total energy and tot (Etot ) and N number of atoms of the surface (bulk) cell and A is the surface area of the slab. (Mean) surface stress values were calculated based on the stress theorem by Nielsen and Martin48 via

f=

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⎤ 1 V surf ⎡ surf N surf V bulk ⎢σxx + σyysurf − bulk surf (σxxbulk + σyybulk )⎥ 2 2A ⎣ ⎦ N V (2)

METHODOLOGY Computational. We performed DFT calculations using the Vienna ab initio simulation package (VASP) by38−41 applying the implemented projector augmented wave potentials42,43 with the Perdew−Burke−Ernzerhof44 functional for exchange and correlation. Second-order Methfessel−Paxton smearing45 with a smearing width of 0.2 eV was employed throughout. Converged bulk calculations yield lattice parameters a0 of 4.0399 Å (3.1914 Å, 4.1928 Å) for Al (Mg, Na). The c/a0 ratio of Mg amounts to 1.6249. The values are less than 1% smaller than the corresponding experimental ones of 4.05 Å for Al, a0 = 3.21 Å and c/a0 = 1.624 for Mg, and 4.23 Å for Na (cf. ref 46). The surface models for the calculation of the respective properties all correspond to p(1 × 1) unit cells with one atom per layer. Details about their size and the geometry optimization will be given further below. Plane wave cutoff energies between 400 and 500 eV were used. We stress the need for dense k-point meshes to resolve the work function and its strain response with sufficient accuracy. Specifically, Monkhorst−Pack47 grids with at least 37 × 37 × 1 k-points were employed. Only the excess charge density distributions of unrelaxed surface slabs were generated with coarser grids with between 21 and 25 k-points in the lateral direction. Surface energies and stresses were determined with reference to bulk

where Vsurf (Vbulk) is the volume of the surface (bulk) supercell and σi,isurf (σi,ibulk) the corresponding (i ∈ {x, y}) component of the volume-averaged stress tensor for the surface (bulk) calculation. Moreover, we determined the top layer relaxation Δd12 =

d12 − d0 d0

(3)

which reflects the relative change of the distance between the first and second layer d12 with respect to the bulk interlayer spacing d0. The calculation of W and ς was performed along the lines of our previous work,21 where more details can be found. We evaluate the work function W as W = Vdip − Ef

(4)

where the dipole barrier Vdip = Vvac − V̅ bulk corresponds to the difference between the vacuum potential and the average potential in the bulk region of the slab and Ef is the Fermi energy (also with respect to V̅ bulk). The electrocapillary coupling coefficient ς around the potential of zero charge is determined in terms of the response of W to in-plane strain using15 B


Article

Langmuir ς=

df dq

= q=0

dU de

q=0

= q0−1

dW de

q=0

until convergence of the relaxation pattern was reached (which was the case between 5 × 10−5 and 1 × 10−5 eV/Å). For the evaluation of the surface stress change associated with the charge-induced relaxation, symmetric surface slabs were constructed based on the deformation of the asymmetric slabs and their surface stress calculated according to eq 2. As will be presented further below, the normal relaxation induced by charging may be separated into a homogeneous (bulk) strain ez and a displacement jump of the first atomic layer (the surface stretch). Therefore, this homogeneous normal strain was applied to the 19 bulk layers of the symmetric slabs, whereas the surface layers on each side of these slabs were set up to have the same interlayer distances as the relaxed asymmetric slabs. Correspondingly, the bulk references were subject to the same homogeneous normal strain. The spatial distribution of the excess charge was determined using unrelaxed bulk-truncated surface slabs. Breaking the symmetry manually by setting the ISYM tag in the VASP input yielded two oppositely charged surfaces. Convergence tests showed that generally less stringent computational parameters than for calculations of f or W were required to obtain basically invariant lateral averages and charge density difference maps. In particular, moderately sized slabs with up to 15 layers and coarser k-point meshes proved to be sufficient.

(5)

where U is the electrode potential, q0 is the elementary charge, and the strain parameter e measures the relative change in surface area. Specifically, we calculate W for surface slabs strained isotropically in the surface plane with |e| ≤ 0.04 and obtain ς as the slope of a linear fit to the corresponding data. For each value of e, we take the relaxed unstrained slab as a starting point and then scale both in-plane lattice vectors by the factor 1 + e as well as the normal positions of all atoms by the factor (1 − τe), before performing a further geometry optimization of the outermost 10 surface layers. Here, τ is a parameter related to the transverse contraction tendency of the bulk crystal in the respective orientation and was obtained from calculations of strained bulk cells with varying interplanar spacings. Simulation of Charged Surfaces. Charged surfaces were simulated by applying a homogeneous electric field along the surface normal of the slabs as implemented in VASP (cf. Figure 1). Electric field values of up to |Fz| = 0.3 V/Å were employed,

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RESULTS AND DISCUSSION Electrocapillary Coupling Coefficients. As the first key result of this study, we report values for the electrocapillary coupling coefficients ς (from work function strain response, see above) and reveal their correlation with equilibrium properties of charge-neutral, unstrained surfaces. In addition to ς, Table 1

Table 1. Surface Properties and Electrocapillary Coupling Coefficientsa

Figure 1. Simulation of charged surfaces using an external electric field. (a) Plane-averaged electrostatic potential of a clean Al(111) surface with an applied electric field Fz. (b) Side view of the supercell with an illustration of the induced excess charge densities.

corresponding to superficial charge densities in the range −27 ≲ q ≲ 27 mC/m2. Electrochemical experiments measuring df/ dq near the potential of zero charge in aqueous HCl are performed with excess charges of slightly larger magnitude.24,25 This work focuses on the regime of capacitive charging around q = 0, which should be well captured by this interval. The relaxation of the atomic planes in response to the applied electric field was determined using asymmetric surface slabs with 10 fixed bulk layers and 5−7 relaxing surface layers. Such a relatively high number of bulk layers was necessary to ensure that the relaxing layers are only influenced by the field on the respective side of the slab and are not disturbed by the field on the other, oppositely charged, side of the slab. Before applying the electric field, the neutral surface was relaxed with a force convergence threshold of 5 × 10−3 eV/Å and the obtained geometries served as a starting point for further calculations of charged surfaces. Their relaxation was performed in several steps with progressively decreasing threshold values

surface

Al(111)

Mg(0001)

Na(110)

γu (J/m2) γ (J/m2) γexp (J/m2) Δd12 (%) Δdexp 12 (%) f u (N/m) f (N/m) W (eV) Wexp (eV) ς (V)

0.816 0.814b 1.16c +0.96b +1.3 ± 0.8d 1.42 1.49 4.050b 4.24g +0.63b

0.550 0.548 0.76c +1.24 +1.9 ± 0.3e 0.81 0.85 3.702 (3.66)g +0.02

0.211 0.211 0.260c −1.61 0f 0.10 0.08 2.834 (2.75)g −0.67

The quantities are γu (γ)surface energy of unrelaxed (relaxed) surface, Δd12top layer relaxation, f u (f)surface stress of unrelaxed (relaxed) surface, Wwork function of relaxed surface, ς = df/dq electrocapillary coupling coefficient. Values in parentheses are for polycrystalline surfaces. bRef 21. cRef 54. dRef 53. eRef 55. fRef 49. g Ref 56. a

lists values for the specific surface energy, the top layer relaxation, the surface stress, and the work function in this (relaxed) reference state of the surface, as well as the surface energy and surface stress of the unrelaxed (bulk-truncated) surface. When available, experimental literature values are also shown for comparison to validate our approach. Good agreement between the experimental and calculated values is found for the work function of the unstrained surface. Also, the surface energies and the top layer relaxations compare fairly well. It has to be noted that the only available low energy C


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Langmuir electron diffraction (LEED) analysis of the Na(110) surface claiming no relaxation is rather old and leaves a considerable uncertainty.49 Our finding of a contraction of the first interlayer spacing by 1.6% does, however, coincide with the value reported in another DFT study.50 Moreover, our result of 1.49 J/m2 for the surface stress of the relaxed Al(111) surface is very close to the value of 1.44 J/m2 obtained by Feibelman51 using first-principles calculations. Comparing the relaxed and unrelaxed surfaces, γ is seen to exhibit little sensitivity to changes in the layer distances. In contrast, the relaxation-induced changes in the surface stress are at least one order of magnitude larger. Moreover, f decreases for inward relaxation of the first layer (Na) and increases for outward relaxation (Al, Mg). This is consistent with previous work on transition metal surfaces.52 The sign-inversion of the response for Al reported in our previous work21 does not appear to be a general feature of spbonded metals because the Mg(0001) and Na(110) surfaces are characterized by vanishing (0.02 V) and negative (−0.67 V) coupling coefficients, respectively. For the surfaces considered here, the response parameter is roughly linearly correlated with the surface stress f as well as with the work function W of the uncharged and unstrained surface (cf. Figure 2). In contrast to

Δρ(x , y , z) = ρq (x , y , z) − ρq = 0 (x , y , z)

(6)

for negatively charged Al(111), Mg(0001), and Na(110) surfaces shown in Figure 3 exhibit qualitatively similar features. Our results for the Al(111) surface are consistent with previously published contour maps of charged Al(111).63,64 As apparent from the graphs of the plane-averaged excess charge density, Δρ̅(z), the centroid of the excess charge is situated approximately at or slightly above the normal position zs marked by the black double-dashed line. zs refers to the normal coordinate of the “geometric surface”, which corresponds to a plane located half an interlayer spacing above the first layer of the bulk-truncated surface. In addition, the charge density difference maps in sideview reveal a lateral modulation of the excess charge. For all considered surfaces, the excess electron density accumulates laterally above the ion cores, but to a variable extent. This tendency for lateral localization decreases in the order Al > Mg > Na and is already very weak for the Na(110) surface. Interestingly, the plane-averaged excess charge density exhibits electron depletion around the normal position of the first atomic layer. For Al(111), the position of the top layer nearly coincides with the location of the global minimum of Δρ̅(z), which is denoted as z(Δρ̅min), whereas for Mg(0001) and Na(110), the minimum is shifted progressively toward the center of the slab in accordance with the increasing wavelength of the density oscillations. Moreover, the depletion zone appears more extended in the normal direction for the latter two surfaces. The charge density distributions within the plane of the first surface layer (Figure 4) reveal more details about the lateral modulation of the excess charge within the surface plane. Qualitatively, they look again rather similar for the three surfaces. A slight enhancement in electron density is observed around the positions of the ion cores, where the excess charge aggregates above the surface. The bonding regions in between exhibit a relatively uniform depletion of electron density, which determines the overall negative sign of Δρ̅(z1). Within the commonly invoked picture of the microscopic origin of surface stress,65 the tensile surface stress of metals is rationalized as a result of a strengthening of the in-plane bonds due to the redistribution of electrons from the region above the surface into the in-plane bonds. Extending this model toward surface charging, one may expect a stronger bonding and therefore an increase in f when more electron density is present in the surface plane (i.e., the plane of the first atomic layer) of the charged surface than in the case of the neutral surface. For overall negative charging, this scenario would hence correspond to a negative coupling coefficient df/dq. For electron depletion, the effect would be opposite. The depletion of electron density in the surface plane for an overall excess of electrons apparent in Figure 4 therefore suggests a positive ς for all of the three considered sp-bonded metal surfaces. Yet, according to our first-principles calculations, only for Al(111), the response parameter is positive (cf. Table 1). In contrast, Na(110) has a negative-valued ς of approximately the same magnitude as Al(111), i.e., −0.67 vs +0.63 V, whereas the response of Mg is close to 0. Thus, the above approach based on the charge rearrangement model is not suitable for a reliable prediction of the sign of ς for a particular surface. However, based on further analysis of the distinguishing features in the charge density distributions, we identify two simple descriptors, which may serve to rationalize the trends among the determined coupling coefficients.

Figure 2. Correlation between electrocapillary coupling coefficient ς and work function (left) and surface stress (right) of the neutral unstrained surface. In both cases, the data exhibit an approximately linear correlation with Pearson correlation coefficients62 of 0.98 and 1.00, respectively.

the ς values, the surface stress retains the commonly observed positive sign (cf. e.g., refs 57−61) for all investigated surfaces and hence does not exhibit a sign-inversion for Al. For the work function, negative values are excluded for reasons of stability. Instead of analyzing the electrocapillary coupling from the perspective of the work function strain response, we here focus directly on the response behavior in terms of the fundamental definition of the coupling coefficient, ς = df/dq. Thus, in the following, we will examine the impact of an external electric field on the electron density distribution and relaxation at the surface with an eye toward the implications for the surface stress. Spatial Distribution of Excess Charge. To unravel the direct effect of charging on the surface bonding and stress, we compare the electron density of the charged surface slab, ρq(x, y, z), to that of the neutral surface, ρq=0(x, y, z). Because the atomic displacements related to the (charge-induced) relaxation are accompanied by huge variations in electron density and thus would obscure the actual effect of the excess charge, only unrelaxed (bulk-truncated) surfaces are considered at this stage. The two-dimensional cuts through the excess charge density distributions D


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coefficient. Thus, more positive ς correlates with a stronger depletion of electron density in the surface plane for negative q (cf. Figure 5). However, the magnitude of dρ̅(z1)/dq is just a very rough indicator for the coupling coefficient. Second, apart from the extent of the depletion of bonding charge directly in the atomic plane, the normal position of this depletion zone may have an influence on the surface stress change. Thus, it seems reasonable that an inward shift of the depletion zone further away from the top layer alleviates its tendency for weakening of the in-plane bonds. To quantify this effect, we have evaluated the vertical distance Δz = z(Δρmin ̅ ) − z1

(7)

between the position of the minimum of the laterally averaged excess electron density distribution and the location of the first atomic layer. Δz and ς appear to be linearly related (cf. Figure 5b) with a Pearson correlation coefficient62 of R2 = 0.98. Future work should be dedicated to providing additional statistics to further test the linear correlation and elucidate its origins. Charge-Induced Relaxation. In addition to the direct effect of the excess charge on the bond strength within the surface plane, the charge-induced normal displacements of the surface layers have an impact on the in-plane stress components. This has previously been suggested by Weigend et al.,23 who have introduced the notion that the increase in the surface stress of negatively charged Au surfaces corresponds to an elastic transverse contraction tendency of the top layer in response to the normal strain associated with the observed charge-induced outward relaxation. In the following, we will inspect the deformation resulting from charging the sp-bonded metal surfaces and quantify the contribution of the chargeinduced relaxation to the surface stress charge response. Figure 6a shows the normal displacement uz of the first few surface layers of the charged Al(111) surface with respect to their positions in the relaxed uncharged slab. In a previous ab initio study of charged Au(111) and Au(100), the deformation was separated into a homogeneous strain ez and the surface stretch ε (ref 22). The former is due to the Coulomb attraction between the surface charge and its counter charge. Therefore, it is independent of the sign of q. As an elastic response to stress, it is sensibly uniform and thus the same for all relaxing layers. By contrast, the stretch affects only the top layer and exhibits a sign change with q. We maintain this decomposition here, even though further (small) deviations from the homogeneous deformation extend well beyond the top layer for the spbonded metals. We extract ez as the slope of the linear fits to uz vs z shown in Figure 6a. This provides a parabolic function of the superficial charge density q (see Figure 6b). This part of the deformation arises from the attractive force between the excess charge and the counter charge present in the simulation.22 A fit according to ez = q2/(2ϵ0CU), with ϵ0 the vacuum permittivity yields a uniaxial stiffness CU of 128 GPa for Al(111) and distinctly smaller values for Mg(0001) and Na(110) (cf. Table 2). The difference between the actual position of the first layer and the extrapolated value for constant strain ez defines the surface stretch ε, which exhibits a linear variation with q (cf. Figure 6c). The stretch charge response, κ = dε/dq, is positive for Al(111) and Mg(0001), i.e., the top layer relaxes inwards for negative charging (cf. Table 2). Only Na(110) shows an outward relaxation and thus qualitatively the same behavior as Au. We have verified that for the Au(111) surface, our approach yields −7.6 × 10−12 m3/C, and thus virtually the same stretch

Figure 3. Lateral averages and color maps of excess electron density distribution in side view. Top panel: Al(111), middle panel: Mg(0001), bottom panel: Na(110). Color maps are for an electric field of 0.1 V/Å (q = 9 mC/m2) and the color scale is in units of electrons per nm3. The normal position of the geometric surface, zs, is marked by a double-dashed line, those of the first, z1, and second, z2, atomic layer by the black dashed and the gray dashed−dotted lines, respectively, as indicated in the bottom panel. The definition of Δz is also illustrated. The corresponding lateral averages, Δρ̅, of the electron density at two separate electric field values (indicated by labels) are shown in the left part of the figure.

The first descriptor is related to the lateral average of the charge density in the surface plane and its variation with q. Even though Δρ̅(z1) does not exhibit a sign-inversion, its charge response, dρ̅(z1)/dq, varies monotonically with the coupling E


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Langmuir

Figure 4. Color maps of excess electron density distributions in the plane of the first atomic layer in top view. Left panel: Al(111), middle panel: Mg(0001), right panel: Na(110). Color maps are for an electric field of 0.1 V/Å (q = 9 mC/m2) and the color scale is in units of electrons per nm3. The positions of the first layer atoms are marked by black crosses. The lateral positions of atoms of the second (third) layer are indicated by large (small) gray crosses.

ςrel ≈

df dε dε dq

(8)

dε/dq is the stretch charge response as determined from the data in Figure 6c. df/dε is the change in the surface stress of a neutral surface, when the top layer is shifted away from its relaxed equilibrium position. To evaluate this quantity for a given surface, we have performed a series of single-point calculations with displacements of the first layer between −0.03 and 0.03 Å in steps of 0.01 Å and extracted df/dε as the slope of a linear fit. Similar to the equilibrium values of f, the response of the surface stress to top layer relaxation is largest for Al and decreases progressively for Mg and Na. Evaluation of eq 8 yields almost the same values as the full calculation of ςrel (cf. Table 2), indicating that the relaxation of the deeper layers has a negligible influence on the surface stress change. Moreover, the opposite trend in the magnitudes of dε/dq and df/dε explains the similar absolute values of the final ςrel. In ref 23, the negative sign of the stretch charge response for Au surfaces was attributed to out-of-plane Hellman−Feynman forces between the laterally modulated excess charge and the ion cores (similarly to the Finnis and Heine model,66 which rationalizes the inward relaxation of the top layer occurring for a large number of metal surfaces). Just like for Au,22,23 the excess charge accumulates laterally above the ion cores for the sp-bonded metals (cf. Figure 3), which according to23 would imply negative dε/dq throughout. However, the positive dε/dq obtained for Al(111) and Mg(0001) after full relaxation illustrates that such a “zero order” model is not suitable for a reliable prediction of the direction of the final stretch. Another aspect of our data is not quite compatible with the reasoning of ref 23: because the authors associate a laterally uniform excess charge distribution with vanishing stretch, it may be expected that the magnitude of κ increases with a rising degree of lateral localization. Yet, whereas Al(111) shows the most pronounced modulation of the excess charge, followed successively by Mg(0001) and Na(110) (cf. Figure 3), the magnitude of κ exhibits the opposite trend (cf. Table 2). Thus, when comparing different elements, the greater efficiency of the screening for materials with a higher electron density seems to be the dominant factor. Combining Direct and Indirect Contributions. Next, the above results for the indirect contribution, ςrel, due to the charge-induced relaxation are combined with the insight gained about the direct contribution due to charging of the bonds via analysis of the excess charge density distributions. Specifically,

Figure 5. Correlation between surface stress charge response ς and charge response of average charge density in the surface plane ρ̅(z1) (left) and distance Δz between the position of the minimum of Δρ̅(z) and the location of the first atomic layer (right). Very roughly, ς increases with increasing magnitude of dρ̅(z1)/dq (R2 = 0.91). Δz and ς are approximately linearly related (R2 = 0.98).

charge coefficient as reported previously (−7.7 × 10−12 m3/C, ref 22) when using the same exchange-correlation functional. This is in good agreement with the experimental value of −6.0 × 10−12 m3/C.37 Note that the reference state for the stretch is the relaxed uncharged surface, so that ε = 0 for q = 0. In a second step, the surface stress was evaluated for neutral symmetric surface slabs with the interlayer distances as determined previously in the presence of the excess charge. In Figure 6d, the resulting values for the surface stress change, Δf, are plotted as a function of the superficial charge density q with which the surface had been relaxed. The variation is essentially linear for Al(111), but the fairly small quadratic component increases in magnitude for Mg(0001) and Na(110). Our primary interest is on the surface stress charge response around q = 0, where results can be compared to the ς values determined via the work function strain response. Therefore, a quadratic fit was performed, and the coefficient ςrel was defined as the linear term. For Al(111), this contribution to the surface stress charge coefficient due to the relaxation has a positive sign and thus displays the same sign-inversion as the total response. The Mg(0001) surface, which just as Al(111) is characterized by an inward relaxation of the top layer for negative q, has also positive ςrel, whereas for Na(110), both κ and ςrel are negative. To understand the trends in more detail, we estimate the ςrel values using the following decomposition based on the deformation of the first layer only F


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Figure 6. Charge-induced deformation of sp-bonded metals and its impact on surface stress. (a) Normal displacement uz as a function of the normal coordinate z for Al(111) surface. z = 0 corresponds to the bottom of the slab. (b) Normal strain vs. superficial charge density q with parabolic fit. (c) Surface stretch vs q with linear fit. The slope is equal to the surface stretch charge response κ. (d) Surface stress change Δf as a function of the superficial charge density, with which the surface slabs had been previously relaxed. Dashed lines correspond to parabolic fits to the data, solid lines indicate slopes of these fits at q = 0.

surface, a positive value of ςchg is compatible with sign and magnitude of the total response. For Mg(0001) and Na(110), however, the picture is not so clear. The qualitatively similar excess charge density distributions with a depletion of bonding charge in the surface plane like for Al suggest again ςchg > 0 for both surfaces. With ςrel > 0, an overall positive ς would be expected for Mg(0001). In contrast, the electrocapillary coupling coefficient for this surface is 0.02 V (cf. Table 1) and thus distinctly smaller than its ςrel. Finally, with ςrel < 0 and ς < ςrel, not even a very small positive direct contribution ςchg would fit to the sign and magnitude of the total coupling coefficient for Na(110). Hence, only for Al, combining the two contributions is compatible with the overall value of ς. Therefore, in general, a consistent prediction of even the sign of ς for any particular surface based on excess charge density distributions in combination with charge-induced relaxation cannot be expected.

Table 2. Materials Parameters Related to Charge-Induced Relaxationa surface dε dq

(10

−12

3

m /C)

Al(111)

Mg(0001)

Na(110)

+4.3

+15.5

−34.8

df (GPa) dε df dε · (V) dε dq

51.9

18.5

5.1

+0.22

+0.29

−0.18

ςrel (V)

+0.21

+0.30

−0.18

ςrel ς

0.33

(no unit)

CU (GPa)

128

0.27 65

15

The quantities are: κ = dε/dqsurface stretch charge response, df/ dεresponse of surface stress to displacement of first layer, ςrel contribution to electrocapillary coupling coefficient due to chargeinduced relaxation, CUuniaxial normal stiffness. a

we want to address the question to what extent combining the two contributions allows for a prediction of the sign of the total ς. For Al(111), a small positive contribution to the surface stress charge response of ςrel = 0.21 V is associated with the relaxation. In relation to the direct effect of the excess charge on the surface bonding, the charge density difference maps reveal a depletion of the electron density in the surface plane for negative q. In extension of Ibach’s charge rearrangement model for the origin of the tensile surface stress of metals65 to the charging of surfaces, this indicates a weakening of the bonding within the surface plane and thus another positive contribution to ς. We will refer to this contribution arising from the redistribution of charge as ςchg. Because ς > ςrel for the Al(111)

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CONCLUSIONS

The present study elucidates the impact of excess charge on the surface stress of metals, which is highly relevant in relation to nanoscale hybrid materials for actuation. To get a better understanding of the underlying physics, we study this response for sp-bonded metals focusing on three representative surfaces with qualitatively different coupling behavior. Our main conclusions are summarized as follows. Electrocapillary coupling coefficients of ς = + 0.63, 0.02, and −0.67 V are found for Al(111), Mg (0001), and Na(110), respectively. The ς values are proportional to the equilibrium G


Article

Langmuir

(4) Detsi, E.; Chen, Z. G.; Vellinga, W. P.; Onck, P. R.; De Hosson, J. T. M. Actuating and Sensing Properties of Nanoporous Gold. J. Nanosci. Nanotechnol. 2012, 12, 4951−4955. (5) Detsi, E.; Selles, M. S.; Onck, P. R.; De Hosson, J. T. M. Nanoporous silver as electrochemical actuator. Scr. Mater. 2013, 69, 195−198. (6) Hakamada, M.; Matsumura, S.; Mabuchi, M. Electrochemical actuation of nanoporous Ni in NaOH solution. Mater. Lett. 2012, 70, 132−134. (7) Jin, H. J.; Wang, X. L.; Parida, S.; Wang, K.; Seo, M.; Weissmüller, J. Nanoporous Au-Pt Alloys As Large Strain Electrochemical Actuators. Nano Lett. 2010, 10, 187−194. (8) Zhang, J.; Bai, Q.; Zhang, Z. Dealloying-driven nanoporous palladium with superior electrochemical actuation performance. Nanoscale 2016, 8, 7287−7295. (9) Raiteri, R.; Grattarola, M.; Butt, H.-J.; Skládal, P. Micromechanical cantilever-based biosensors. Sens. Actuators, B 2001, 79, 115−126. (10) Battiston, F.; Ramseyer, J.-P.; Lang, H.; Baller, M.; Gerber, C.; Gimzewski, J.; Meyer, E.; Güntherodt, H.-J. A chemical sensor based on a microfabricated cantilever array with simultaneous resonancefrequency and bending readout. Sens. Actuators, B 2001, 77, 122−131. (11) Lavrik, N. V.; Sepaniak, M. J.; Datskos, P. G. Cantilever transducers as a platform for chemical and biological sensors. Rev. Sci. Instrum. 2004, 75, 2229−2253. (12) Gokhshtein, A. Y. Charge Acquirement by an Elastically Strained Electrode; Doklady Akademii Nauk SSSR, 1969; Vol. 187, p 601. (13) Trasatti, S. Surface science and electrochemistry: concepts and problems. J. Electroanal. Chem. Interfacial Electrochem. 1971, 33, 351− 378. (14) Rath, D.; Kolb, D. Continuous work function monitoring for electrode emersion. Surf. Sci. 1981, 109, 641−647. (15) Umeno, Y.; Elsässer, C.; Meyer, B.; Gumbsch, P.; Nothacker, M.; Weissmüller, J.; Evers, F. Ab initio study of surface stress response to charging. Europhys. Lett. 2007, 78, No. 13001. (16) Sekiba, D.; Yoshimoto, Y.; Nakatsuji, K.; Takagi, Y.; Iimori, T.; Doi, S.; Komori, F. Strain-induced change in electronic structure of Cu(100). Phys. Rev. B 2007, 75, No. 115404. (17) Gong, H. R.; Nishi, Y.; Cho, K. Effects of strain and interface on work function of a Nb-W metal gate system. Appl. Phys. Lett. 2007, 91, No. 242105. (18) Wang, X. F.; Li, W.; Lin, J. G.; Xiao, Y. Electronic work function of the Cu (100) surface under different strain states. Europhys. Lett. 2010, 89, No. 66004. (19) Albina, J.-M.; Elsässer, C.; Weissmüller, J.; Gumbsch, P.; Umeno, Y. Ab initio investigation of surface stress response to charging of transition and noble metals. Phys. Rev. B 2012, 85, No. 125118. (20) Wang, X. F.; He, L.; Halas, S.; Pieńkos, T.; Lin, J. G.; Durakiewicz, T. The canonical work function-strain relationship of the platinum metal: A first-principles approach to metal-gate transistor optimization. Appl. Phys. Lett. 2013, 102, No. 223504. (21) Michl, A.; Weissmüller, J.; Müller, S. Sign-inverted response of aluminum work function to tangential strain. J. Phys.: Condens. Matter 2013, 25, No. 445012. (22) Umeno, Y.; Elsässer, C.; Meyer, B.; Gumbsch, P.; Weissmüller, J. Reversible relaxation at charged metal surfaces: An ab initio study. Europhys. Lett. 2008, 84, No. 13002. (23) Weigend, F.; Weissmüller, J.; Evers, F. Structural relaxation in charged metal surfaces and cluster ions. Small 2006, 2, 1497−1503. (24) Viswanath, R. N.; Kramer, D.; Weissmüller, J. Variation of the Surface Stress-Charge Coefficient of Platinum with Electrolyte Concentration. Langmuir 2005, 21, 4604−4609. (25) Smetanin, M.; Viswanath, R. N.; Kramer, D.; Beckmann, D.; Koch, T.; Kibler, L.; Kolb, D.; Weissmüller, J. Surface Stress-charge Response of a (111)-Textured Gold Electrode under Conditions of Weak Ion Adsorption. Langmuir 2008, 24, 8561−8567.

values of the work function as well as the surface stress of the unstrained and uncharged surfaces. Negative charging causes an inward relaxation of the first atomic layer for Al(111) and Mg(0001) corresponding to a positive stretch charge coefficient κ, whereas Na(110) shows qualitatively the same behavior as Au (cf.22,37). The chargeinduced relaxation gives rise to an indirect contribution to the surface stress charge response, ςrel, which we have explicitly quantified here. Further analysis showed that this parameter is well approximated by the product of the stretch charge response and the response of the surface stress to the displacement of the top layer from its equilibrium position. Because the latter is invariably positive, the sign-inversion of κ for Al(111) and Mg(0001) implies a positive ςrel for these two surfaces, whereas a negative value is found for Na(110). Thus, for Al and Na, ςrel has the same sign as the total response parameter, ς, and, in both cases, makes up about 30% of its magnitude. Qualitatively similar excess charge density distributions for Al, Mg, and Na surfaces hint at the identical positive sign of the direct contribution to the surface stress charge response. Thus, only for Al(111), a combination of the two contributions is consistent with the overall sign and magnitude of ς. Although the sign change of the coupling coefficient is not directly reflected in the excess charge density distributions, the trend of ς among the three inspected sp-bonded metal surfaces may still be rationalized based on simple descriptors of these distributions. Lastly, our study illustrates that even for the so-called “simple” metals, a generalization of the interplay of the electric field with the metal’s electronic structure via simple models yet turns out to be elusive. Specifically, the simple charge rearrangement picture that underlies the Ibach model65 for q = 0 cannot be transferred to charged surfaces and the Weigend−Weissmüller−Evers picture23 is not too helpful either.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Anja Michl: 0000-0003-1744-4938 Jörg Weissmüller: 0000-0002-8958-4414 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge financial support by the German Research Foundation (DFG) under grants nos. MU 1648/6-1 and WE 1424/16-1.

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REFERENCES

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Electrocapillary Coupling at Metal Surfaces from First Principles: On

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