Theory of Work Function and Potential of Zero Charge for Metal

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Theory of Work Function and Potential of Zero Charge for Metal Nanostructured and Rough Electrodes Jasmin Kaur, and Rama Kant J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 26 May 2017 Downloaded from http://pubs.acs.org on May 27, 2017

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The Journal of Physical Chemistry

Theory of Work Function and Potential of Zero Charge for Metal Nanostructured and Rough Electrodes Jasmin Kaur and Rama Kant∗ Complex Systems Group, Department of Chemistry, University of Delhi, New Delhi-110007 E-mail: [email protected]

1 ACS Paragon Plus Environment


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Abstract We developed a general theory for the work function and potential of zero charge of arbitrary shaped nanostructured and rough metal. Theory accounts for the influence of adsorbed solvent layer, i.e. partial to full solvent coverage, on work function (WF). We show that the change in local WF is due to redistribution of surface charge caused by local surface curvature and adsorbing dipoles. Special cases of spherical and ellipsoidal geometries are obtained from the generic curvature dependent WF equation to illustrate the size and shape effects. Theory predicts increase in WF of immersed nanoparticles with decrease in their size, while change in shape causes non-uniform WF over the surface. Theory shows anomalous fluctuations in WF for a surface with multiscale roughness, viz. Weierstrass-Mandelbrot function as surface. Ensemble or surface averaged work function of randomly curved rough surface is predicted to depend on average mean and Gaussian curvatures. The intrinsic field of various metals regulating the dipole alignment is estimated to lie between 0.6-1.3V/nm. The extent of electronic redistribution strongly depends on curvature while this is 4% of the electronic charge for water adsorption on metal surface. Theory is corroborated with the experimental variation in measured work function due to partial and multilayer adsorption of water on Fe thin films, and a negative shift in oxidation potential for Ag nanoparticles in solution. Finally, this work opens up a wider prospect in fundamental understanding of kinetics, adsorption and electric double layer in nanotextured electrodes.


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Introduction Influence of adsorption is practically paramount in all fields of physical and chemical relevance and is inevitable at the electrochemical interfaces. The properties of adsorbed water layer on metal surfaces permeate through several complexities arising due to chemical and electrostatic interactions between the two. The subsequent moderations in the properties of the two phases in contact are of extreme importance for fundamental understanding and hence are being extensively explored through theoretical and experimental means. 1,2 One of the key electronic property of a metal that gets affected by adsorption is its work function. 3–5 Work function and its related quantity the potential of zero charge are significant in the understanding of several surface and electrochemical phenomena, viz. catalytic activity, redox kinetics, electric double layer dynamics. Hence it is of utmost relevance in the applied fields of electrochemistry, corrosion, catalysis, material science etc. Several empirical studies relate the work function to the various electronic, 6–8 thermodynamic 9,10 and electrochemical 11,12 characteristics of the metal lattice. It is often addressed theoretically through image force approach 13 or jellium model, 14 variational approaches for small metal clusters 15 and exchange correlation formulation. In an electrochemical system, the operative work function is that of an immersed electrode (φad ) therefore it is important develop its theoretical understanding. The work function of metals get lowered upon water adsorption 16–18 and this lowering has been observed to be a function of surface coverage. 16,19 The change in work function due to water adsorption is generally attributed to the two aspects, viz. the dipole moment (permanent or induced) of the adsorbed species, 20 and charge transfer between adsorbate and substrate. 21 The concept of field-dependent orientation of interfacial water dipoles 22,23 and its influence on the dipole orientation and dielectric constant 24,25 has been extensively studied by far. These endeavors are however centered on the solution phase and the electrode participation has not been accounted in these models. Work function is electrochemically translated in terms of the M 26,27 Pt potential of zero charge, Epzc . The difference of ∼1.2V in Epzc of Pt(111), calculated for



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vacuum and observed in solution, 28 is an indicative of its strong correlation to solvation. Adsorption of water and molecular orientation on metal surface is dependent on the site of adsorption, 29 surface coverage 30 and potential at electrochemical interfaces. 31 Although the metal water interactions are weak (indicated by no surface reconstruction imposed over metal surface 32,33 ), the work function change is often employed for evaluating the orientation of water on metal surface. 34,35 Also, the work function is dependent on the shape/size and morphology of the material. 36,37 The optimization of the material properties through composition 38 and structuring, 39 where the local work function emerges as a controlling parameter, is currently trending. The theoretical models reported so far 1,2 lack the inclusion of morphological information of the electrode surface while stating the influence of solvent adlayer on its work function. With the advent of nanoelectrochemistry, these classical theories demand a recalibration so as to address the pressing fundamental issues at nanostructured electrodes. Therefore, we intend to develop a generic model of size and shape dependent work function in presence of solvent dipoles for curved and arbitrary structured metal electrodes. In our earlier work, the shape and size dependent work function for bare metal nanostructures has been derived up to second order in local (mean and Gaussian) curvatures. 36 Here we develop a theoretical model of the work function of metal electrode in solvent contact accounting for its dependence on electrode shape/size and solvent coverage. This theory models the electronic charge redistribution and dipole orientations at the interface as the cause of work function variation. The equation of work function of the dipole layer covered metal surface is generalized for curved nanostructures in terms of the local mean and Gaussian curvatures. A generic equation for the ensemble average work function, corresponding to the mean work function of a collection of particles or a random arbitrary surface, is obtained in terms of the surface averages of the mean and Gaussian curvatures. Next section relates the curvature induced variation in electrode work function in presence of solvent molecules to the potential of zero charge which is an important electrochemical quantity. Work function and potential of zero charge are directly related, we present a detailed analysis of the shape/size


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dependent work function variation for metal in polar solvent. It will automatically stand for the variation of potential of zero charge by electrode shape and structure. Influence of curvature is demonstrated through the difference in the work function of a nanostructured surface and that of the planar (bulk) electrode (both in immersed state) as ∆φad . Ranging from structurally isotropic surface, a spherical nanoparticle and nanocavity, the special case of ellipsoidal particles is considered to analyze the effect of shape anisotropy on ∆φad . High curvature at the edges of nanodisc or nanoflake morphology is predicted to render maximum shift at the equator region. Presence of high curvature sites at an arbitrary electrode surface is predicted to manifest its effect primarily localized at that site. It is demonstrated through the Weierstrass Mandlebrot surface exhibiting multiscale roughness with varying fractal dimensions. Final section summaries the concept and conclusions along with some future prospects of our theory.

Formulation Work Function of Immersed Planar Surface Metal electrodes are extensively employed as cathodes in several electrochemical set ups. However, the energy expense of the electronic exchange tend to get modified due to the surrounding electrolyte medium. Hence for a planar metal surface the solvent adlayer modified work function (φ0ad ) can be depicted in terms of interface induced correction (∆φ0ad ) as, φ0ad = φ0E + ∆φ0ad

(1)

The most straightforward approach to model the work function modification at the electrode solvent interface is to consider the interface composed of a metal phase surrounded by an adsorbed monolayer of solvent dipoles (in the absence of specific adsorption of electrolyte). As a result, the redistribution of the electron density at the metal surface and the orientation



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of solvent molecules on the solution side leads to a surface potential alterations on both sides, 40,41 such that the work function change of a non-curved electrode in solvent contact s 40 where δχM is, ∆φ0ad = δχM 0 is the surface potential change in metal due to electrons 0 − g0 ,

redistribution and g0s is the modification in the double layer on the surface layer of the solution due to oriented solvent dipoles at a non-curved surface. By substituting ∆φ0ad in equation 1, s φ0ad = φ0E + δχM 0 − g0

(2)

The first term in equation 2 corresponds to the electronic work function of non-curved bare electrode, the other two terms denote the modification due to solvent adsorption. These contributions are explicitly modeled in the following segments.

Electronic work function for non-curved surface Based on the electronic screening model by Kant et al., 42,43 the electronic work function has been recently formulated by interpreting the electronic capacitance as the self capacitance of the Thomas Fermi disc (CE0 ) whose radius is equal to the Thomas Fermi screening length (lT F ) and area is denoted by AF (= πlT2 F ) 36 so that, CE0 = c0E AF

(3)

where, c0E = 0 m /lT F , is the electronic capacitance density and the Thomas Fermi electronic screening length is defined as r lT F =

20 m Ef 3n0 e2

(4)

where, 0 is the permittivity of the free space (=8.85×10−12 F/m), m is the metal dielectric constant (listed in supporting information in Ref. 36 ) and n0 is the number density of electrons. Through equation 3, electronic work function for a planar surface φ0E has been


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obtained as the electrostatic self-capacitive energy of the circular disc of radius lT F 36 so, φ0E =

e2 e2 = 2CE0 2π0 m lT F

(5)

It is evident that lT F emerges as a key electronic characteristic of the material that manifest the material information through: Fermi energy, electron number density and dielectric constant. The electronic screening length through Ef is dependent on the spatial dimensions of the electrode. It vary due to different occupancy of various electronic states in different structures. Consequently, various crystalline faces have slightly different lT F that leads to different work functions.

Influence of Effective Dipole Orientation In a planar metal electrode, g0s evolves as the function of solvent dipole characteristics and surface coverage. Over a metal surface having a monolayer of adsorbed water molecules, each having the dipole moment µ (µH2 O =6.2×10−30 Cm), the solvent dipoles have a natural tendency to orient with negative end of the dipole pointing toward the surface. 1 However, the thermal driven disorder can induce randomness in dipole alignment thereby affecting the operative dipole moment. Hence despite of carrying a permanent dipole moment, water imposes a limited dipolar polarization contribution as the surface will have fluctuations in dipolar orientation. In the absence of any external field, this thermal driven disorder is counteracted by the intrinsic dipole orienting electric field (ε) of the metal substrate. Therefore the effective orientation of the adsorbed molecule is a consequence of the interplay between the field induced order and the thermal energy induced disorder. Depending upon the strength of this intrinsic electric field of metal, the effective dipole moment is defined by a distribution function such that free energy of the system is minimized. The ensemble average dipole moment (mean of all the thermally driven possible orientations denoted as h·iT ) is thus defined through Langevin function as, hµ(ε)iT = µ[Coth(µε/kB T ) − (µε/kB T )−1 ]. Here, kB



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is the Boltzmann constant and T is the temperature. So, for the maximum of N0 oriented dipoles per unit area each having averaged dipole moment hµ(ε)iT , a net dipole moment (N0 hµ(ε)iT ), dependent on the strength of the dipole orienting electric field (ε), is operative. The surface potential due to a monolayer of oriented solvent dipoles at planar metal solvent interface can be defined as, 27 g0s =

N0 hµ(ε)iT 0 m

(6)

However, with externally applied potential the operative field will be the combination of the intrinsic field of metal and that generated at the compact layer which in-turn can regulate dipole alignment and hence g0s . The present analysis focuses on the basic structure of metal solvent interface without any external perturbation.

Dipole Adsorption And Surface Charge Redistribution The adsorption of the polar solvent molecules influence the distribution of the surface electron density of the metal thereby altering its surface potential. However, the extent of influence depends on the effective charge getting redistributed at the metal surface. As discussed already the effective dipole moment held by the solvent molecule is dependent on the intrinsic dipole orienting electric field of the metal. So, the effective charge redistribution of electronic charge imposed by the solvent at the metal surface also varies for different metals. Water molecule tend to align with its oxygen towards the metal surface 44 whose negative electron density reduces the surface potential of metal, i.e. δχM 0 < 0. We model the effect of adsorption of a molecule bearing permanent dipole (here water) at the “Thomas-Fermi” (TF) disc that eventually illustrates the impact of solvent coverage on the whole metal surface. On approaching the TF disc from the oxygen end, the partial ˜ Thus the variation in charge redistribution imposed over the disc by water is taken as δe. the surface potential of the metal can be interpreted as the potential difference in taking the


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˜ So, charge q at the TF disc from e to (1 + δ)e.

δχM 0

Z = e

˜ (1+δ)e

q ˜ ≈ 2φ0 δ˜ dq = φ0E (δ˜2 + 2δ) E CE0

(7)

˜ From the analysis of the experimental variation of work function with the as δ˜2

Theory of Work Function and Potential of Zero Charge for Metal

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