Balance of Force at Curved Solid MetalâLiquid Electrolyte Interfaces

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Balance of Force at Curved Solid Metal-Liquid Electrolyte Interfaces J. Weissmu¨ller*,†,‡ and D. Kramer† Forschungszentrum Karlsruhe, Institut fu¨ r Nanotechnologie, Karlsruhe, Germany, and Universita¨ t des Saarlandes, Technische Physik, Saarbru¨ cken, Germany Received August 30, 2004. In Final Form: December 2, 2004

We analyze the simultaneous mechanical and chemical equilibrium at the interface between a fluid electrolyte and a solid conductor in terms of a continuum theory, with attention to surfaces of varying orientation and of arbitrary curvature. On top of the variable which is conjugate to the surface stress, the tangential strain, we introduce an additional degree of freedom for the surface deformation, the surface stretch, to account for the observation of a reversible normal relaxation of the top atomic layer as a function of the electrochemical potential. We derive relations between the materials constants of the surface, for instance, the pressure dependence of the electric potential at constant superficial charge density, and discuss experimentssusing cantilevers or porous solidssby which they can be measured.

I. Introduction Changes of the superficial charge density and/or the density of adsorbed species at the surface of a conductive solid immersed in an electrolyte give rise to forces at the surface whichsat equilibriumsmust be balanced by stresses in the underlying bulk. Contemporary research, as reviewed in refs 1-3, aims at measuring the phenomenological parameter describing these forces, the surface stress tensor s, and at establishing an understanding of the underlying microscopic processes. Most studies of surface stress at the metal-electrolyte interface work with a single, planar surface.1-10 However, recent work suggests that surface stress in materials with curved surfaces may be of practical interest: Since granular and porous solids can have a large surface-to-volume ratio, changes of the surface stress in response to surface charging can induce significant strain in the solid, with strain amplitude and specific strain energy density of sufficient magnitude to suggest possible application as actuator materials.11,12 When analyzing the elastic response of these materials to changes in their surface stress, one is faced with the question, what are the balance equations when forces originating from surfaces of different orientation in three* Corresponding author. Address: Forschungszentrum Karlsruhe, Institut fu¨r Nanotechnologie, PO Box 3640, D-76021 Karlsruhe, Germany. E-mail: [email protected]. † Forschungszentrum Karlsruhe. ‡ Universita ¨ t des Saarlandes. (1) Cammarata, R. C.; Sieradzki, K. Annu. Rev. Mater. Sci. 24 1994, 215. (2) Ibach, H. Surf. Sci. Rep. 1997, 29, 193. (3) Haiss, W. Rep. Prog. Phys. 2001, 64, 591. (4) Couchman, P. R.; Davidson, C. R. J. Electroanal. Chem. 1977, 85, 407. (5) Lipkowski, J.; Schmickler, W.; Kolb, D. M.; Parsons, R. J. Electroanal Chem. 1998, 452, 193. (6) Friesen, C.; Dmitrov, N.; Cammarata, R. C.; Sieradzki, K. Langmuir 2001, 17, 807. (7) Vasiljevic, N.; Trimble, T.; Dmitrov, N.; Sieradzki, K. Langmuir 2004, 20, 6639. (8) Gokhshtein, A. Y. Dokl. Akad. Nauk SSSR 1969, 187, 601. (9) Gokhshtein, A. Y. Electrochimica 1969, 5, 593. (10) Ibach, H.; Bach, C. E.; Giesen, M.; Grossmann, A. Surf. Sci. 1997, 375, 107. (11) Weissmu¨ller, J.; Viswanath, R. N.; Kramer, D.; Zimmer, P.; Wu¨rschum, R.; Gleiter, H. Science 2003, 300, 312. (12) Kramer, D.; Viswanath, R. N.; Weissmu¨ller, J. Nano Lett. 2004, 4, 793.

dimensional space interact? Furthermore, recent experimental and computational results indicate that the distance between the outermost atomic layer at the surface and the underlying bulk may vary continuously and reversibly as a function of the superficial charge density.13,14 This represents a degree of freedom for deformation along the surface normal, on top of the tangential strain which is considered in the standard treatment of surface stress. The questions raised are, can the notion of surface stress of the metal-electrolyte interface be generalized to include a deformation along the surface normal? Moreover, in materials with a large surface-tovolume ratio, are the electric signals originating from strain of the metal or from change in pressure in the electrolyte of sufficient strength to be measured, or even used in applications of such materials as sensors? Three concepts are of importance for capillary phenomena at the surface of condensed matter: (i) the various specific surface excess free energies, representing the excess, per area, of the materials free energy over that of the matter in a suitable reference state with negligible surface area, (ii) the surface tension, the force per length of perimeter required for reversibly changing the surface area by adding matter to the surface at constant structure, and (iii) the surface stress, a measure for the tendency of the surface to change its area by elastically deforming the underlying bulk phase. Gibbs15 showed that when a suitable convention is chosen for locating the dividing surface (the mathematical plane representing the surface in the phenomenological description), a specific surface excess free energy function can be defined which equals the surface tension. Gibbs also gave a separate discussion of the stress induced by the surfaces in a thin solid film, thereby anticipating the notion of surface stress.16 Shuttleworth17 showed that the stress of a planar solid surface can be described by a 2 × 2 tensor defined in the plane of the surface; his tensor relates s to the surface tension (13) Nichols, R. J.; Nouar, T.; Lucas, C. A.; Haiss, W.; Hofer, W. A. Surf. Sci. 2002, 513, 263. (14) Lucas, C. A. Electrochim. Acta 2002, 47, 3065. (15) Gibbs, J. W. The Collected Works of J. W. Gibbs; Longmans, Green, and Co.: New York 1928; Vol. I; pp 224-229. (16) Gibbs, J. W. The Collected Works of J. W. Gibbs; Longmans, Green, and Co.: New York 1928; Vol. I; p 315. (17) Shuttleworth, R. Proc. Phys. Soc. 1950, A63, 444.

10.1021/la047838a CCC: $30.25 © 2005 American Chemical Society Published on Web 04/07/2005

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Figure 1. Examples for porous solid microstructures with curved surfaces. Scanning electron micrographs of (a) a granular porous sample consolidated from Pt black11 and (b) porous Au prepared by alloy corrosion.12

γ, an excess free energy per area of the surface as measured in the laboratory frame, and to the tangential strain tensor e by

s)γ

( )

1 0 ∂γ + 0 1 ∂e

(1)

The appearance of the term γ in this equation might suggest that s is essentially similar in magnitude to γ, with a more or less significant correction due to the strain of the surface. Cahn first pointed out that a simpler statement, which does not exhibit the explicit dependency on the value of γ of eq 1, is obtained when the surface area is measured in coordinates of the undeformed material, also termed Lagrangian coordinates.18 The surface stress is then

s)

∂γ ∂e

(2)

a form which emphasizes that the entries of the surface stress tensor are independent thermodynamic parameters, which differ from the value of γ in magnitude and in sign. The two definitions yield the identical numerical value for s, but different functional dependencies of γ on the strain.19 Their difference is analogous to that between the Piola-Kirchhoff and the Cauchy stress tensors in the bulk,20,21 two concepts which are well established in the field of continuum mechanics. Whereas most studies of surface stress consider planar surfaces, experiments on grain boundaries in polycrystals involve surfaces of many different orientations, so that definitions of surface stress defined in two-dimensional space are poorly suited to describe the combined forces due to all the surfaces. Recent work on porous metals immersed in electrolytes encounter the same complication, as can be seen in the exemplary illustration of porous metal microstructures, Figure 1. In a series of publications, Gurtin and co-workers have established the continuum mechanics of surface stress on curved surfaces.22,23 The surface stress is here represented by a “superficial tensor”, (18) Cahn, J. W. In Interfacial Segregation; Johnson,W. C., Blakely, J. M., Eds.; ASM: Metals Park, OH 1979; 3. (19) Weissmu¨ller, J. J. Phys. Chem. B 2002, 106, 189. (20) Truesdell, C.; Noll, W. The nonlinear field theories of mechanics. In Handbuch der Physik, III/3; Flu¨gge, S., Ed.; Springer: Berlin, 1965; Sect. 43 A. (21) Gurtin, M. E. The linear theory of elasticity. In Handbuch der Physik; Truesdell, C., Ed.; Springer: Berlin, 1972; Vol. VIa/2, Sect. 16. (22) Gurtin, M. E.; Murdoch, A. I. Arch. Ration. Mech. Anal. 1975, 57, 291; 1975, 59, 389. (23) Cermelli, P.; Gurtin, M. E. Arch. Ration. Mech. Anal. 1994, 127, 41.

a 3 × 3 tensor which transforms vectors from R3 into the local tangent space. The equilibrium condition at a curved surface takes on the form of a local balance of force law, which complements the results of Young24 and Laplace25 for the pressure jump across curved fluid-fluid interfaces for the case of solids. However, while the Young-Laplace equation determines the local pressure jump along with the mean pressure in the droplet, the result by Gurtin and co-workers leaves the pressure undetermined, even locally. Weissmu¨ller and Cahn derived a capillary equation for solids which determines all components of the mean stress tensor, thereby complementing the local balance law.26 Their balance equation relates the mean stress in a solid of arbitrary microstructure to an average of the surface stress over the surfaces and internal interfaces. In combination with Hookes law this has been used to determine averages of the surface stress in polycrystals27,28 and in porous solids11,12 from experimental data for the volumetric strain of the solid. Surfaces can be modeled as two-dimensional objects, and it is often tacitly assumed that the relevant degrees of freedom for deformation are restricted to the strain in the tangent plane. However, it was recently shown that local changes of composition (segregation) at grain boundaries in solid solutions incur a deformation along the grain boundary normal which can be measured.27 Gurtin and co-workers showed that this behavior can be incorporated into a general equation of state of curved deformable solidsolid interfaces in terms of “stretch”, a jump discontinuity in the displacement vector field at the interface.29 Closely similar phenomena are known from equilibrated singlecrystal surfaces in a vacuum and from electrode surfaces in electrolytes, in which the topmost atomic layer undergoes a rigid-body relaxation relative to the underlying bulk crystal lattice, as evidenced by density-functional theory modeling and by surface-sensitive diffraction experiments.13,14,30-32 In this paper, we show how the normal and tangential deformation variables can be defined so that they con(24) Young, T. Philos. Trans. R. Soc. London 1805, 95, 65. Also In Collected Works 1855, 1, 418. (25) Laplace, P. S. Celestial Mechanics; reprinted by Chelsea Publ. Co.: New York, 1966; Vol. IV, pp 689 and 711. (26) Weissmu¨ller, J.; Cahn, J. W. Acta Mater. 1997, 45, 1899-1906. (27) Weissmu¨ller, J.; Lemier, C. Phys. Rev. Lett. 1999, 82, 213. (28) Birringer, R.; Hoffmann, M.; Zimmer, P. Phys. Rev. Lett. 2002, 88, 206104. (29) Gurtin, M. E.; Weissmu¨ller, J.; Larche´, F. Philos. Mag. A 1998, 78, 1093. (30) Feibelmann, P. Phys. Rev. B 1997, 56, 2175. (31) Kokalj, A.; Causa`, M. J. Phys.: Condens. Matter 1999, 11, 7463. (32) Materer, N.; Starke, U.; Barbieri, A.; Do¨ll, R.; Heinz, K.; Van Hove, M. A.; Somorjai, G. A. Surf. Sci. 1995, 325, 207.

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stitute appropriate state variables in a constitutive equation for solid-electrolyte interfaces of arbitrary curvature. We derive the balance equations by which the state variables of the curved surface are linked to those of the abutting phases at equilibrium, and as a check of consistency we show that the established equilibrium conditions are satisfied, along with the extra results for the deformation along the normal. We show that the extra degree of freedom results in a dependence of the charge and/or the potential on the pressure P in the electrolyte, and we discuss the electrical response of solid surfaces to change in P and to strain in the solid, both for singlecrystal surfaces and for porous bodies. Using experimental materials parameters for the Au(111) surface in H2SO4, we explore in how far these phenomena can be exploited (i) for using materials with large specific surface area as sensors, and (ii) for obtaining, by means of electrochemical experiments, information on top layer relaxation which complements the results of surface-sensitive diffraction experiments. II. Phenomenological Description A. Defining the Problem. We investigate the equilibrium between a conductive solid and a fluid electrolyte into which it is immersed. We allow the solid to undergo anisotropic and nonuniform elastic deformation in response to forces exerted on its surface by the pressure in the fluid and by the surface stresses. The solid and the fluid are assumed to be mutually insoluble, so that no matter is exchanged between them. Various aspects of a complementary issue, the equilibriumsin the absence of surface stresssof a nonhydrostatically stressed solid with a fluid in which it is soluble, have been analyzed previously.33-35 For conciseness, and motivated by our aim to obtain information on the surface stress, which is related to the elastic deformation of the surface, we disregard changes of the surface area by plastic deformation or by accretion of matter at the surface. As a consequence, the analysis will not provide information on the surface tension of the solid (compare also section III.E below). Specifically, we consider the total free energy in the system delimited by the dashed line in Figure 2, comprising the solid and the surrounding fluid electrolyte. We chose the system boundary to be located entirely within the uniform region of the fluid sufficiently far from the solid, and we consider its position to be fixed in laboratory coordinates, so that the volume of the system is constant. The system boundary is an imaginary surface, which is transparent to transport of matter and heat. The electrolyte surrounding the system is considered to have a large volume and to be at constant composition and temperature, so that it acts as a reservoir which fixes the chemical potentials µi of the species i in the fluid. The µi may be varied, for instance, as functions of the pressure P in the fluid or of its composition, which are controlled as indicated schematically in Figure 2. Charge is transferred between the system and a counter electrode, which are interconnected as indicated in the figure. The counter electrode is taken to have a very large surface area, so that it can accommodate ionic and electronic charge without changes of the electrochemical and electronic potentials. Furthermore, we assume that the equilibria at the counter electrode are independent of the pressure. In this way, the counter electrode acts simultaneously as an idealized reference electrode. (33) Larche´, F.; Cahn, J. W. Acta Metall. 1973, 21, 1051. (34) Johnson, W. C.; Schmalzried, H. Acta Metall. Mater. 1992, 40, 2337. (35) Sekerka, R. F.; Cahn, J. W. Acta Mater. 2004, 52, 1663.

Weissmu¨ ller and Kramer

Figure 2. Schematic illustration of the situation considered in sections II and III. The system, consisting of the solid and the surrounding fluid, is separated by an imaginary surface (dotted line) from the surrounding fluid reservoir, in which pressure and composition can be independently controlled as indicated in the figure. Also illustrated is the control of the electrode potential relative to a counter electrode.

In the interest of a concise analysis we choose to consider only processes and equilibria at constant and uniform temperature. Therefore, and although an explicit temperature dependence is admitted for all free energy functions and materials parameters, we do not display the temperature as a variable. B. Continuum Description of the Solid. The deformation of solid matter manifests itself by displacements of the atoms (labeled by the index j) which can be described by displacement vectors uj. In general, the deformation gives rise to local stress in the solid, which is related by balance laws to the stress in the surrounding matter and to external forces acting on the atoms near the surface. When the displacement varies smoothly on an atomic scale, then the equations of balance of stress can be derived based on a description of the solid as a continuum, where the displacement is represented by the vector function u(x) with the vector x a continuous position variable. Let xS and v(xS) denote positions and the local displacement on the surface, respectively. In the continuum representation, the work dW done locally against the traction T (the external force per surface area) on a segment of surface of area δA when the solid is deformed is given by

dW ) -dv T δA

(3)

Usually, one considers u to vary smoothly up to the surface, so that v(xS) agrees with uS(xS), the limiting value of u(x) as the surface is approached from within the solid, x f xS. However, as discussed in the Introduction, results of modern experiments and models suggest that when appropriate state variables (for instance, the superficial charge density) are changed, reversible local relaxations occur in the topmost superficial layers. In other words, in real solids the displacement does not vary smoothly on an atomic scale when the surface is approached from within the solid, and the displacements at the surface are not adequately represented by uS. Figure 3 gives a schematic illustration of this phenomenon. Because the energetics of deformation depend on the external forces acting on the surface via eq 3, it is convenient to define a displacement field v(xS) on the surface so that eq 3 is fulfilled for a general traction T(xS). In keeping with the above statements, we allow for v to


Langmuir, Vol. 21, No. 10, 2005 4595

D. Constitutive Assumption for the Fluid. Along with ΨB, we consider the volumetric free energy density ΨF in the fluid

ΨF ) ΨF(F1, ..., Fk)

(6)

dΨF ) µi dFi

(7)

with Fi the molar density of the component i. Throughout this work we use the sum convention (summation over indices occurring twice). The Euler equation for a sample of fluid with volume V is Figure 3. Schematic illustration of the magnitude of the atomic displacements uj (open dots) versus the position x near the surface of a strained solid. Also shown are the continuous vector field u(x) (solid line) of the continuum model, the limiting value uS of u(x) at the surface, and the actual displacement v at the surface. [u] ) v - uS represents the displacement jump.

differ from uS. For a solid surface in contact with a fluid, the work done against the pressure in the fluid when the volume dV of fluid is displaced by the solid is dW ) P dV. Since dV is the product of area and outward displacement of the solid surface, the normal component of v represents an effective displacement of the solid surface, as measured by the amount of fluid which is displaced per area. The tangential components of v are ill-defined in the example, since the absence of shear stress which defines a fluid implies that T has no tangential component. Our definition of v does not establish a rigorous connection to the outward relaxation measured by surfacesensitive diffraction as discussed in the Introduction. For instance, different measures for the top layer displacement may be obtained by the respective methods if the top layer was displaced opposite as compared to the first subsurface lattice plane, as indicated schematically in Figure 3. However, since on a dense-packed surface (for instance, a 111-plane of a face-centered cubic crystal) the environment forms atomic bonds essentially with the topmost layer only, it can sensibly be assumed that the outward relaxation of this layer agrees with the volume of fluid displaced. In this sense, the two respective measures for surface relaxation are comparable. As the explicit parameter in the analysis we find it convenient to use, instead of v, the difference vector [u] ) v - uS; in the continuum model it represents a jump in the displacement at the surface. This is the equivalent, for a solid-fluid surface, of the stretch, the jump in the displacement vector at a solid-solid interface as introduced in ref 29. C. Constitutive Assumption for the Bulk Solid. We consider the solid to be an elastic continuum and, in the bulk, we restrict attention to small displacement gradient ∇u and, hence, small strain. We can then make use of the well-known result of continuum mechanics that the volumetric Helmholtz free energy density ΨB in a solid body at position x depends on u at the most through the strain tensor E (the symmetric part of ∇u) at x

ΨB ) ΨB(E,x)

(4)

dΨB ) S dE

(5)

where S denotes the stress tensor. The explicit dependency on x accounts for nonuniformity of the elastic response, for instance, when the solid is polycrystalline or when it exhibits gradients of composition.

VΨF ) -PV + µiNi

(8)

It follows that the pressure in the fluid is given by

P ) µiFi - ΨF

(9)

E. Definition of Superficial Densities. We allow for the properties of the fluid to deviate from the bulk value in space-charge regions of finite extent near the surface. This deviation is modeled by considering superficial densities (per area) of the free energy and of the components. The superficial density Γi of component i is defined so that the total amount Ni of that component in the system can be expressed in terms of integrals over the densities in the fluid (denoted by “F”), and at the surface (“S”) by

Ni )

∫F Fi dV + ∫S Γi dA

(10)

Similarly to the definition of the Γi, the superficial free energy density ψ is defined so that the total free energy F is given by

F)

∫B ΨB dV + ∫F ΨF dV+ ∫S ψ dA

(11)

where the integral labeled “B” extends over the solid body. The essential step in setting up a phenomenological analysis of equilibrium is to identify the state variables for ψ. By their definition, the superficial densities represent the excess of matter or free energy in the actual system over the values in a surface-free system of the same volume. The use of eq 11 imposes restrictions on how to locate the surface: the integral over B must generally correctly describe the free energy in any material containing a given number, N, of atoms. In particular, consider the special case of a homogeneous strain in a uniform material with no capillary energy, ψ ≡ 0. The strain energy will be correctly computed if the surface is located so that the volume enclosed by the surface corresponds to N times the atomic volume at the actual value of the strain. As this requirement must hold for arbitrarily shaped solids, it implies a general convention for locating the dividing surface S. The relevance of specifying such a convention is discussed in ref 15. F. Interfacial Charge. Being conductors, the solid and the fluid are field-free and, hence, charge neutral everywhere except in the space-charge regions of finite extent. By varying the applied voltage, we can experimentally control the potential U of the electrons in the metal ands through Usthe electronic charge density q at its surface. We allow for the species in the fluid to carry a charge ziF per mole, where F denotes the Faraday constant, and we disregard (Faraday) processes in which charge is irreversibly transferred between ions and electrons at the

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surface. Furthermore, we restrict attention to surfaces which carry no net charge. The superficial electronic charge cannot then be varied independently of the superficial density of ions; instead, the two parameters are linked by the constraint

0 ) q + ziFΓi

(12)

As a consequence, the equation of state for ψ depends on the Γi but not on q. The statement of conservation of charge is

∫S(δq + ziF δΓi) dA ) δQ + ∫S ziF δΓi dA

0)

(13)

where Q denotes the total electronic charge transferred from the counter electrode to the solid. G. Fundamental Equation for the Surface. In the most general case, we allow for the free energy density ψ of the surface to depend on the state of the abutting bulk phases through the limiting values of their state variables, ES and the FS,i. Furthermore, we allow ψ to depend on additional parameters which are only relevant at the surface, the jump [u] in displacement and the quantities Γi, which determine the excess of the components and, thereby, the charge density q. In other words we assume, as the most general constitutive equation

ψ ) ψ(ES,[u],FS,1, ..., FS,k, Γl, ..., Γk,xS)

|

p)yi )

(15)

e ) PES

µiS )

|

| |

∂ψ ∂Γi

(21)

with the projection tensor36

P(xS) ) U - n(xS)Xn(xS)

(22)

(16)

FS,i,[u],Γi

∂ψ ∂[u]

∂ψ ∂FS,i

(20)

The tangential strain e is a tangential superficial tensor; it is obtained from the superficial limiting value ES of the strain in the bulk by the projection22

where the coefficients s, yi, p, and µiS are defined as partial derivatives of ψ with respect to the state variables

∂ψ s) ∂ES

L ) 1 + trace e

(14)

The total differential of ψ is

dψ ) s dES + yi dFS,i - p d[u] + µiS dΓi

equilibrium of a nonhydrostatically stressed solid with a fluid with respect to exchange of matter (in the absence of capillary terms), have shown how the respective reference frames can be considered in a joint analysis.33 As a result of our choice of reference frames, the integration over B and S is in Lagrangian coordinates, and the integration over F is in laboratory coordinates. ΨF and the Fi are densities per volume in the laboratory frame, whereas ΨB, ψ, and the Γi are densities per volume or area of the undeformed solid. By fixing the location of the system boundary in laboratory coordinates, we have fixed the systems total volume. When the surface of the solid is displaced in the direction of the surface normal, keeping the Fi and Γi constants, then the volume which the solid occupies in the laboratory frame is changed, and a corresponding volume δVF of the fluid is displaced from the system. The local change in volume, measured in coordinates of the solid, is δV ) v‚n dA, where n is a unit vector along the outward surface normal. The change in volume of the fluid in laboratory coordinates is obtained by taking into account that the surface area in laboratory coordinates, dA*, is related to the area in referential coordinates, dA, by dA* ) L dA where L relates to the tangential component of the strain in the solid by

(17)

ES,FS,i,Γi

(18)

ES,[u],FS,j*i,Γi

(19)

ES,FS,i[u],Γj*i

As with the ΨB, we allow ψ to depend explicitly on the position. In this way, variations of the surface properties along the surface, for instance, due to variation in curvature or in crystallographic orientation, are incorporated in the theory. H. Measuring Area and Volume. The analysis of equilibrium in the solid is simplified when the surface area as well as the volume are measured in coordinates of the undeformed solid (Lagrangian coordinates). As we ignore accretion or dissolution of matter at the solid surface, surface area and volume of the solid are then constants, independent of displacement and strain. As the changes of density in the fluid incorporate diffusion and exchange of matter, rather than being entirely elastic, a similar construction would be inappropriate for the fluid. Thus, the volume in the fluid is measured in laboratory coordinates. Larche´ and Cahn, in their analysis of the

The symbols U and X denote, respectively, the unit tensor in R3 and the tensor product. In the limit of small strain, the value of L deviates only insignificantly from unity, so that the terms L in the equations below can be omitted in most practical situations. We shall examine this issue in section III.E below, and in the meantime we retain L in the interest of generality. Taking into account the definition of [u], it is found that the displaced volume of the fluid, measured in laboratory coordinates, is

∫S(δuS + δ[u])‚nL dA

δVF ) -

(23)

where the integration is carried out in Lagrangian coordinates. Consider the case where interfacial enrichment of the components is absent, so that all Γi vanish. The volume of the fluid is then related to the partial molar volumes of ΩiF the components by VF ) ∫FΩiFFi dV. It is seen that the Fi and the ΩiF are linked by the constraint ΩiFFi ) 1. As components are transferred from the reservoir into the system to build up a space-charge layer, their partial molar volume may, in general, change. Thus, the volume displaced by this extra matter may be described by molar volumes ΩiS which differ from the molar volumes in the bulk fluid. The total volume of the fluid is then given by (36) Gurtin, M. E. Arch. Ration. Mech. Anal. 1995, 131, 67.


VF )

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∫F ΩiFFi dV + ∫S ΩiSΓi dA

(24)

This result has a consequence which is important in analyzing pressure-dependent phenomena; cf. section III.F below: By the definition of the Γi in terms of excess of matter over that of a surface-free system of identical volume, the surface cannot have a net excess volume; in other words, the value of the surface integral in eq 24 must vanish. Since this must be true for any subsystem and, hence, everywhere on the surface, the ΩiS and Γi satisfy the constraint

0 ) ΩiS Γi

(25)

The absence of a superficial excess volume of the fluid is compatible with the Gibbs sense of the surface excesses, in which the excess volume is identically zero. I. Conservation of Matter. The amount, δNi, of species i transported through the system boundary must be balanced by changes in the net amounts of matter in the fluid and at the interfaces. The statement can be written as

0 ) δNi -

∫F δFi dV - ∫S δΓi dA + ∫S FS,i(δuS +

Let us now transform eq 28 so that the variables describing the deformation can be independently varied. The procedure which was developed by Gurtin et al.29 in their analysis of the equilibrium at solid-solid interfaces applies with only minor modifications. We shall therefore present an abbreviated account, referring the reader to ref 29 for details. It is emphasized, however, that the results for solid-solid interfaces cannot be transferred one-to-one to the solid-fluid interface and that the resulting balance equations are significantly different from what would be obtained if one would use the results of the earlier work and simply set the pressure as purely hydrostatic in one of the two solid crystals abutting at the interface. The divergence laws allow to incorporate into eq 28 the kinematic links between the variation of the tensor E in the bulk, and of its limiting value ES along the surface, and the surface displacement uS. We use the divergence law for the bulk

∫B S δE dV ) ∫S δuS‚(Sn) dA - ∫B δu‚div S dV (29) to eliminate E, and the identity

δ[u])‚n L dA (26)

∂(δu) ∂n

s‚δES ) s‚∇(δuS) ) s‚∇S(δuS) + sn‚

where the FS,i denote the limiting values of the bulk densities at the surface.

(30)

along with the divergence law for the closed surface III. State Variables and Equilibrium Conditions A. General Variation of the Grand Potential. We consider variations at constant values of the state variables of the system (solid plus surrounding electrolyte), the chemical potentials µiswhich depend on the pressure Ps and the electric potential difference U between the solid and the reference electrode. At equilibrium, the system will then minimize the grand canonical potential Φ(µi,U) ) F - UQ - µiNi. We shall derive the equilibrium conditions from the requirement that the variation of Φ must vanish for a general variation about the equilibrium state at constant values of µi, U, and V

0)

∫B S δE dV + ∫F µiFδFi dV - ∫S ΨSF(δuS + δ[u])‚nL dA + ∫S (s δES + yi δFS,i - p δ[u] + µiS δΓi) dA - U δQ - µi δNi (27)

The term which depends on the limiting value of ΨF at the surface ΨSF, arises from the reduction of the volume of the fluid when the surface of the solid is displaced; see above. The variations δx in eq 27 are not all independent. For instance, varying Q will change the Γi, and varying E will lead to surface displacement δuS. To convert eq 27 into an expression based on independent variables, we proceed as follows: δQ is converted into an integral over the excess of matter at the surface by means of eq 13, and the terms δNi are eliminated by adding the constraints of conservation of matter, eq 26, with Lagrange multipliers µi. The result is

∫ ∫ ΨS )δuS‚n L dA + ∫S (FS,i µi - ΨSF)δ[u]‚n L dA ∫S δ[u]‚p dA + ∫S yi δFS,i dA + ∫F (µiF - µi) δFi dV + ∫S (µiS - µi +UziF)δΓi dA (28)

0)

∫

B S δE dV +

S s δES dA +

S (FS,i µi -

∫S s‚∇S(δuS) dA ) -∫S δuS‚divS s dA

(31)

to eliminate e (cf. ref 29). With a, b scalar and vector fields, respectively, defined on the surface, and c a vector field defined in the bulk, the surface gradients ∇Sa and ∇Sb represent vector and tensor fields on the surface, respectively, the surface divergence divS b represents a scalar field on the surface, and the normal derivative ∂c/ ∂n represents a vector field on the surface.22,29 By use of these relations, and in view of the definition of P, eq 9, the condition of vanishing variation in Φ can be expressed as

∫B δu‚div S dV + ∫S δuS‚(S n - divS s + ∂(δu) dA + nLP) dA + ∫S sn‚ ∂n ∫S δ[u]‚(LPn - p) dA + ∫S yi δFS,i dA + ∫F (µiF - µi)δFi dV + ∫S (µiS - µi + UziF)δΓi dA

0)-

(32)

B. Equilibrium Conditions. The equilibrium equations governing the exchange of matter are obtained by considering, independently, variations of Γi and Fi. For Fi we first consider variations which vanish on S, then general variations. By inspection of eq 32, it is then seen that for δΦ to vanish it is required that

∂ΨF/∂Fi ) µiF ) µi

(33)

∂ψ/∂FS,i ) yi ) 0

(34)

∂ψ/∂Γi ) µiS ) µi - UziF

(35)

Varying Γi leads to

F

Equations 33 and 35 are the well-known equilibrium equations for the chemical potential in bulk fluids and at

4598



interfaces, and eq 34salong with 35sshows that ψ can depend on the composition of the electrolyte at the most through the interfacial excess Γi. By contrast, ψ cannot depend explicitly on the bulk densities Fi of the components in the electrolyte. It is noted that this latter statement results from the assumption that the free energy density of the fluid depends on the composition exclusively through the local values of the Fi; see section II.D. When a nonlocal interaction is admitted, in the simplest case by allowing ΨF to depend on gradients of the composition, then the dependency of ψ on the Fi can be significant. This is most relevant when the fluid exhibits a miscibility gap, where the variation of ψ with FS gives rise to critical point wetting.37 Varying [u] at constant u leads to

-∂ψ/∂[u] ) p ) LPn

(36)

Considering first variations δu that vanish on S, then variations δu that have vanishing ∂(δu)/∂n on S, and then general variations of δu leads to the equations of mechanical equilibrium

div S ) 0

(37)

sn ) 0

(38)

Sn - divS s + LPn ) 0

(39)

in the bulk, and

on the surface. Equation 37 is the well-known equilibrium equation for the stress in the bulk of a solid. Equation 39 relates the stresses on the two sides of the surface to the interface stress and to the surface curvature. It is illustrative to consider a uniform and isotropic interface stress, s ) f P, noting that the mean curvature κ of the surface is given by 2κn ) -divS P.22,29 It is then readily seen that eq 39 is a condition for the normal component of the stress

-Sn ) (P + 2fκ)n

(40)

The tangential components of S at the surface are left undefined by eq 40, and so is the local value of the pressure in the solid. However, we shall comment below on a balance equation which relates the mean stress and mean pressure in the solid to the surface stresses. C. Identifying the Relevant State Variables for the Surface. Equation 38 states that the surface stress is tangential, even when a normal deformation of the surface, as embodied in the displacement jump [u], is allowed for. This implies that only the tangential components of ES do work against s; in other words, the superficial free energy density ψ depends on the strain in the bulk only through the tangential strain e (compare section II.H). Similarly, eq 36 shows that the generalized force p is normal to the surface, and that it equals the pressure in the fluid in magnitude. This implies that ψ depends on [u] only via its normal component, the (scalar) stretch29

(xS) ) [u](xS)‚n(xS)

(41)

To summarize, the results of this section imply that ψ depends, at the most, on e, , the Γi, and on the position xS on the surface (37) Cahn, J. W. J. Chem. Phys. 1977, 66, 3667.

ψ ) ψ(e,,Γi,xS)

(42)

and that its total variation is given by

dψ ) s de - LP d + (µi - UziF) dΓi

(43)

Figure 4 illustrates the two degrees of freedom of deformation, the tangential strain e and the stretch . Their definition and the conclusion that they represent independent variables are central issues of this work, which deserve further comment. Since e and are derived from the displacement u and since, by assumption, u may be an arbitrary function of the position x, any combination of variations δe and δ can be realized as long as there is a corresponding function δu(x). Let us assume that e was defined as the relative change of the lateral bond length in the topmost atomic layer and as the relative change in the normal bond lengths. By reference to Figures 3 and 4 it is seen that, on planar surfaces, a local displacement field with only normal components varies the stretch at constant tangential bond length. However, a (for instance) outward displacement of the topmost atomic layer of a curved surface of radius r will necessarily elongate the lateral bonds in that layer by a relative amount /r, similar to the stretching of the surface of a rubber ball when it is inflated. This implies that there is no displacement field which changes the normal bond length while leaving the tangential bond length and the strain in the bulk constant. Consequently, with this definition and e would be kinematically linked and independent variation impossible. The definition of the present work, where e is a projection of the limiting value ES of the strain E in the bulk as the surface is approached, is free of this problem: A suitably localized normal displacement field, which is restricted to the stretched surface region as illustrated in Figure 3, leaves E and, by the definition, e invariant while varying , even at curved surfaces. The extra energy involved in deforming the tangential bonds when the surface is curved will, in general, entail a dependency of the function ψ (and of its second derivatives, the materials parameters) on the curvature. In our theory this is accounted for via the dependency of ψ on xS. It is also noted that stretch at a curved surface at constant tangential strain changes the area of the outer surface, as measured in laboratory coordinates but that it leaves the Lagrangian area invariant. Although e and are independent state variables of the free energy functions ψ(e,,Γi) at each x, varying any one of these variables while leaving the others constant will result in changes of the conjugate forces for all the state variables. For instance, varying at constant e and constant Γi will change the surface stress by ds ) (∂s/∂) d ) (∂2ψ/∂e∂) d. In other words, although and e are kinematically independent deformation variables, their conjugate forces are coupled, as is generally true for any pair of thermodynamic state variables. The detailed nature of the coupling depends on the microscopic interactions in the matter at the surface. The second derivatives of the free energy functions represent materials parameters (in the above example, excess elastic constants)29 of the surface, which embody this coupling in the phenomenological description. It is of interest to compare eq 36 to the results of ref 29 for the equilibrium at solid-solid interfaces, eq 44 there

1 ∂ψ/∂[u] ) (SB2 - SB1)n 2

(44)


Langmuir, Vol. 21, No. 10, 2005 4599

is defined as PB ) -1/3 trace(S)

3V〈PB - P〉V ) 2A〈f〉A

(50)

The scalar interface stress f is defined in terms of the trace of s Figure 4. Schematic illustration of tangential strain (a) and stretch (b) at a crystalline solid surface. Dots denote atomic positions in the strained state (light shaded, bulk atoms; black, surface atoms), and the grid refers to the undeformed lattice. Arrows indicate direction of deformation.

where the superscripts refer to the two solid bodies “B1” and “B2” abutting at the interface. Since the fundamental difference between solids and fluids resides in the fact that stress is hydrostatic in fluids, contrary to the generally nonvanishing shear stress in solids, it would seem that the results for solid-solid interfaces can simply be transferred to solid-fluid interfaces by setting the shear stress to zero on one side of the interface. By comparison of the respective results it is readily seen that this is not correct: at solid-solid interfaces p depends on the normal stresses on both sides of the interface, whereas at solidfluid interfaces it depends only on the pressure P in the fluid, not on the stress S in the solid. By virtue of eq 39, P and S are independent parameters. The difference between the results for the two types of interfaces arises from the different definition of s which, in ref 29, is conjugate to the strain in both abutting solids, whereas the absence of a network constraint on the fluid side of the interface forbids linking s to a strain in fluid in the present treatment. D. Mean Stress Relation. Along with the local balance equations there is a balance law for the volumetric and areal averages.26 To derive this law for the present case, we consider the variation in Φ when the solid is deformed by a small (virtual) displacement field of the specific form δu ) Dx with D ) ∇δu a uniform and symmetric (strain) tensor. The values of [u], Γi, and Fi are held constant. Equation 28 then takes on the form

0)D

∫B S dV + D ∫S s dA + P ∫S δuS‚n dA

(45)

The last term on the right can be transformed to a volume integral over ∇‚δu by means of the divergence theorem. Furthermore, it is readily seen that, since δu ) Dx, ∇‚δu can be equivalently expressed as DU with U the unit tensor in R3. Thus

P

∫S δuS‚n dA ) P ∫B ∇‚δu dV ) D ∫B PU dV (46)

By insertion into eq 45 one obtains

0)D

∫B S dV + D ∫S s dA + D ∫B PU dV

(47)

Since this must hold for arbitrary virtual strain D, one retrieves the balance equation

0)

∫B(S + PU) dV + ∫S s dA

(48)

which can be equivalently expresses in terms of the volumetric and areal averages, 〈 〉V and 〈 〉A, respectively

0 ) V(〈S〉V + PU) + A〈s〉A

(49)

A relation for the mean pressure is obtained by taking the trace of eq 49, considering that the pressure in the solid

f ) 1/2 trace s

(51)

Equations 48-50 are the balance laws of ref 26, derived in a more concise way. These results ascertain that the mean stress relations of ref 26 continue to hold for the more general constitutive equation for ψ used in the present work. This is both a consequence and a confirmation of the fact that e and as defined here represent independent degrees of freedom for deformation. E. Change of Variables, Surface Tension. In experimental setups such as Figure 2 the controlled parameters are the chemical potentials µi and the pressure P rather than the superficial densities of matter Γi and the stretch. It is then often advantageous to work with a Legendre transform of the superficial free energy which is function of the µi and of U

ψ h (e,P,µi,U) ) ψ + LP - µiSΓi

(52)

In view of the equilibrium condition, eq 35, the condition of charge neutrality, eq 12, and of the definition of L, eq 20, ψ h has the total differential

dψ h ) (s + PP) de + L dP - Γi dµi - q dU

(53)

For the remainder of the discussion, we shall assume that L in eqs 52 and 53 does not significantly differ from unity, and we shall work with

dψ h ) s de + dP - Γi dµi - q dU

(54)

This is motivated as follows: in experimental situations the entries of the strain tensor will not significantly exceed 10-3. Thus, substituting dP for L dP in eq 53 is a reasonable approximation. Furthermore, and P will typically not exceed values of 10-11 m and 109 Pa, respectively, so that P e 0.01 N/m. By contrast, values of the entries of the surface stress s are typically of the order of several N/m. Thus, the error introduced by approximating s + PP by s in eq 53 is relatively small, and in most practical instances it will be well below the experimental accuracy. Except for the more general notion of surface deformation, eq 54 agrees with the variant of the Gibbs adsorption equation which describes the total differential of the surface tension γ of planar solid-electrolyte surfaces.3,4,6 It is of interest to compare ψ h to γ. Let us consider a hypothetical process in which the surface area of the solid body B is reversibly varied by the amount δA, not via elastic strain but by adding matter at constant structure (constants e, , and Γi). The surface tension γ is defined in terms of the mechanical work δW1 done by a hypothetical force, acting reversibly along the perimeter of the deformed segment of surface, so that δW1 ) γδA. The work done on the system is the sum of the work required to increase to total free energy of the surface, ψδA, and of the work associated with transporting matter and charge through the system boundary. In view of the considerations in sections II and III above, the net work is readily seen to be δW2 ) (ψ + P - Uq - µiΓi ) δA. By comparison to eq 35 above for µiS, it is seen that the term in brackets agrees closely with ψ h as defined in eq 52 above. Therefore,

4600



equating δW2 to δW1 leads to the conclusion that the function ψ h represents indeed the surface tension of the solid. It is emphasized that we considered accretion at constant strain, an assumption which is realistic only when changes in the strain energy resulting from the increase in surface area are negligible. Lipkowski and co-workers5 showed that this is the case for typical experiments using bending of macroscopic cantilevers for measuring changes in the surface stress. In small bodies, by contrast, the changes in the strain which accompany the change in surface area, and the associated work of elastic deformation, may contribute significantly to δW2. This work can be computed for solids with uniform elastic properties and simple geometry, for instance, the sphere, and it is found to be important at small structure size.19 Thin films, multilayer structures, and wires provide further examples for simple geometries which are open for analysis. These geometries may be of relevance since zero creep experiments use such samples to determine γ by balance of force at temperatures were diffusive deformation of the solid is sufficiently fast for equilibration within experimental time scales.38,39 We could not find a way to compute the work associated with the surface-induced elastic deformation of the bulk for the general situation which is of interest here. Thus, although ψ h may be identified with the surface tension in macroscopic bodies, in the discussion of equilibrium in small systems it should be considered merely as the formal quantity, a Legendre transform of ψ. F. Pressure Dependence, Limit of Weak Adsorption. In typical experiments in electrochemistry the composition of the electrolyte is a constant, so thatsat constant temperaturesthe µi may be taken to be functions of the pressure P only

dµi ) ΩiF dP

(55)

By insertion into eq 54 it is seen that here ψ h )ψ h (e,P,U), and that the appropriate version of eq 54 is F

dψ h ) s de + ( - ΓiΩi ) dP + q dU ) s de + w dP - q dU

(56)

The symbol w in eq 56 has the units of length. In this respect the term w dP is closely similar to the term which is commonly used to describe the pressure dependence of the interfacial tension of solid-solid interfaces such as grain boundaries, the product of dP and an interfacial excess volume.40,41 However, w is not an excess volume here, since the surface of the fluid has vanishing excess volume in the representation of the present work (see section II.H). By use of the statement that the net superficial excess free volume of the fluid vanishes, eq 25, one can express w in terms of a combination of stretch and of the changes in the molar volumes when ions are transferred from the bulk of the fluid into the superficial space-charge layer

w ) - ΓiΩiF ) + Γi(ΩiS - ΩiF) ) + Γi∆Ωi (57) (38) Udin, H.; Shaler, A. J.; Wulff, J. Metall. Trans. AIME 1949, (Feb), 186. (39) Josell, D.; Spaepen, F. Acta Metall. Mater. 1993, 41, 3017-27. (40) Meiser, H.; Gleiter, H.; Mirwald, R. W. Scripta Metall. 1980, 14, 95. (41) Sutton, A. P.; Baluffi, R. W. Interfaces in Crystalline Materials; Clarendon Press: Oxford, 1996; Section 5.3.

One of the materials parameters of interest in experiments is the variation of w when the surface is charged

|

|

|

|

dΓi d∆Ωi d dw ) + ∆Ωi + Γi dq e,P dq e,P dq e,P dq e,P

(58)

Consider the limiting case of a dilute solution of ions with a low enthalpy of adsorption. Near the potential of zero charge specific adsorption will be negligible, so that no chemical bonds are formed as the surface charge is varied. Furthermore, since the screening length (Debye-Hu¨ckel length) in a dilute electrolyte is comparatively wide, the local composition of the electrolyte in the space-charge region will be very similar to that of the charge-neutral bulk electrolyte. Thus, when ions are transferred from the uniform regions of the electrolyte far from the interface into the space-charge layer, their chemical environment will not change appreciably, and their molar volume will remain sensibly constant. In this limit, which we shall refer to as the limit of weak adsorption, the ∆Ωi as well as their variation will vanish for the ions in the space charge layer. Besides the space-charge layer, the interface between metals and aqueous electrolytes includes a physisorbed layer of neutral water immediately at the metal surface (see, for instance, ref 42 and references therein). In the limit of weak adsorption the number density of adsorbed molecules in this layer will be invariant under small changes of q, so that the contribution of specific adsorption to the value of dΓi in eq 58 can be neglected. It may also be assumed that the partial molar volume will be invariant, as the polarization of dielectrics does not generally involve a significant change in volume. Subject to these considerations, the variation in w agrees with the variation in stretch in the limit of weak adsorption. As will be discussed in section IV below, this may open a way for measuring the stretch by electrochemical techniques. IV. Relation between Measurable Materials Constants A. Description of Experiment. In this section we derive relations between the materials constants (i.e., the second derivatives of the free energy functions) which can be determined experimentally. Two experimental situations are of interest, (i) planar surfaces as used in cantilever bending experiments and (ii) porous solids. The two geometries, which are schematically illustrated in Figure 5, require separate treatment. In experiments using cantilever bending, one can control the tangential strain in the solid surface by applying bending moments, for instance, as indicated in Figure 5a, independently of the pressure in the fluid. Thus, the appropriate state variables are e, P, and q (or U). By contrast, in experiments using porous materials the strain cannot generally be controlled in a defined way by means of forces applied to the macroscopic surface of the sample, since the resulting local deformation at the pore surfaces cannot be predicted. We shall therefore restrict attention to a porous solid which is loaded only by the hydrostatic pressure in the fluid, as indicated schematically in Figure 5b. Here, changes in the external pressure will entail changes in the stress in the solid and, hence, an elastic deformation by which the stress can be measured. In other words, the strain of the solid is no longer an independent parameter, but it is a function of the pressure in the fluid, taking on the equilibrium value subject to the given values of P and of the forces originating from the surface. The (42) Parsons, R. Solid State Ionics 1997, 94, 91-98.


Langmuir, Vol. 21, No. 10, 2005 4601

Figure 5. Schematic illustration of experimental setups considered in sections IV.A (a) and IV.B (b), respectively.

state variables are then P and q (or U). We shall consider the simplest case, where the material has a small surfaceto-volume ratio, so that the surface-induced stress is negligible. We shall further assume that the solid is elastically uniform, so that the external pressure gives rise to a uniform stress throughout the solid, which is hydrostatic and, therefore, described by the pressure PB ) P. When we aim to obtain estimates of the numerical values of the materials parameters, we shall consider the limit of weak adsorption (w ) , compare section III.F), and we shall make reference to the example of the Au(111) surface in H2SO4, for which experimental values published by Nichols et al.,13,43 Kolb et al.,44 and Vasiljevic et al.7 are listed in Table 1.45 Specifically, we search for relations between, on one hand, the known parameters ∂U/∂q|e,P, ∂f/∂q|e,P, and ∂/∂q|e,P, and on the other hand the parameters which describe the performance of the surface for use in sensors, of the type ∂U/∂P|q and ∂q/∂P|U. B. Cantilever Geometry. In this geometry, it is useful to chose e, P, and q as the controlled variables. The appropriate constitutive equation is obtained from eqs 52-53 by an additional Legendre transform

ψ ˜ (e,P,q) ) ψ h + Uq

| | |

| ( |)

-1

(64)

along with the Maxwell relation 62 and the definitions of the differential capacitance c and of the surface stresscharge coefficient σ

| |

c)

∂q ∂U e,P

(65)

σ)

∂f ∂q e,P

(66)

one derives

∂f ∂w ) ∂P e,q ∂e q,P

(61)

∂f ∂U ) ∂q e,P ∂e q,P

(62)

∂w ∂U ) ∂P e,q ∂q e,P

| |

(60)

The Maxwell relations ensuing from eq 60 are

| | |

|

∂X ∂Y ∂X ∂Z ∂X )and ) ∂Y Z ∂Z Y ∂Y X ∂Y Z ∂X Z

(59)

For simplicity, we assume an isotropic surrface stress, s ) fP, so that the work of straining the surface is f de, with the scalar strain e defined so that e ) trace e. Thus, for an arbitrary strain the value of e agrees with the relative change in surface area as measured in laboratory coordinates. The total differential of ψ ˜ is then

dψ ˜ ) f de + w dP + U dq

the pressure dependence and of the extra degree of freedom for deformation, the stretch. Equations 61-63 are relevant results, since they suggest that the mechanical response to changes of the superficial charge density, represented by ∂f/∂q|e,P and ∂/∂q|e,P, may be measured in alternative ways. Specifically, at constant charge, one may measure the dependency of the voltage on the tangential strain (which can be controlled by bending the cantilever by means of an applied force) and on the pressure in the electrolyte. Let us assume weak adsorption. Using the experimental data for ∂f/∂q|e,P and ∂/∂q|e,P of Au(111) in H2SO4 together with the Maxwell relations, one obtains the predictions for the numerical values, ∂U/∂e|q,P ) -0.9 V and ∂U/∂P|e,q ) -6 × 10-12 V/Pa. Thus, a tangential strain of 10-4 gives rise to a voltage change of 90 µV, and a pressure change of 1 MPa (10 bar) gives rise to a voltage change of 6 µV. Signals of this magnitude are readily measurable. It is also of interest to explore the possibility of measuring changes of the charge density at constant potential. By using46

(63)

Equation 62 is the result derived and experimentally verified by Gokhshtein,8,9 whereas eqs 61 and 63 are additional relations resulting from the incorporation of (43) The estimated values of ∂/∂q and ∂f/∂q were obtained by graphical interpolation in Figures 4 and 5, respectively, of ref 13. (44) Kolb, D. M.; Schneider, J. Electrochim. Acta 1986, 31, 929.

|

∂q ) -cσ ∂e U,P

(67)

|

(68)

and

|

w ∂q ) -c ∂P e,U ∂q e,P

The empirical data suggest that the two last-mentioned parameters take on the values 0.3 C/m2 and 2 × 10-12 (C/m2)/Pa, respectively. Thus, for a surface of area 1 cm2 the strain 10-4 leads to a change in charge of 3nC, and the pressure of 1 MPa leads to a change in charge of 0.2 nC. Larger charges are obtained by working with porous samples with larger surface area. We shall now turn to such experiments.

4602



Table 1. Materials Parameters for the Au(111) Surface in H2SO4a

a Values were taken from experiment or are estimated by combining the displayed experimental constants with the results of Sections IV.B and IV.C for cantilever geometry and for porous samples, respectively, in the limit of weak adsorption.

C. Porous Geometry. According to the considerations in section IV.A, the state variables are here P and q. As discussed there, we consider the pressure PB in the solid to agree with the pressure P in the electrolyte, and we assume uniform and isotropic elastic constants. The relative change in volume of the solid (measured in laboratory coordinates) is then dV/V ) -dP/K with K the bulk modulus of the solid (K ) 167 GPa for Au),47 and the relative change in surface area is de ) (2/3) dV/V. It follows that

de ) -

2 dP 3K

values appear to be known for the corresponding materials constant, ∂f/∂e. However, by measuring ∂e/∂P at constant q in a setup as indicated in Figure 5b, one can experimentally verify if eq 69 is obeyed, and in this section we shall work on the assumption that this is the case. The appropriate constitutive equation for the experiment on the porous solid in Figure 5b is obtained by substituting eq 69 into eqs 59 and 60

(69)

Equation 69 applies in the limit where changes of the surface-induced pressure in the bulk are negligible and where the elastic response is appropriately described by K. Deviations from this behavior arise when f varies significantly as a function of P, and whensat small structure sizesthe surface area per volume is sufficient for the surface-induced change in pressure in the solid to be comparable to the variation in P. As no values appear to be known for the associated derivatives ∂f/∂P and ∂w/∂e (compare eq 61), we could not find an estimate for the minimum structure size above which eq 69 is expected to represent a good approximation. The same problem arises when estimating the effects of surface excess elastic constants on the effective elastic response of a porous solid to loading by a uniform pressure on its entire surface: No (45) Results for ∂f/∂U have been published in ref 7, along with data for the double-layer capacitance, but ∂f/∂q was not reported. Here we used the data of ref 13, since in that work ∂/∂q was measured along with ∂f/∂q. We note, however, that the two sets of results appear not to be consistent: by using ∂f/∂U|e,P ) ∂f/∂q|e,P ∂q/∂U|e,P one retrieves the estimate ∂f/∂U ) -0.27 (N/m)/V based on the data in ref 13, considerably smaller than the value -1.6 (N/m)/V, reported in ref 7. In the absence of a critical assessment of the respective experiments, the numerical values which we computed based on the experimental data should be viewed as order-of-magnitude estimates, rather than precise predictions. (46) Callen, H. B. Thermodynamics and an Introduction to Thermostatistics, 2nd ed.; Wiley: New York, 1985. (47) Holzapfel, W. B.; Hartwig, M.; Sievers, W. J. Phys. Chem. Ref. Data 2001, 30, 515.

ψ ˆ (P,q) ) ψ ˜ (-2P/(3K),P,q)

(70)

dψ ˆ ) η dP + U dq

(71)

where η denotes a mechanical state variable defined so that

η)w-

2f 3K

(72)

It is noted thatssubject to the assumption of negligible effects of surface-induced pressuresthe materials parameters which relate to variations of q at constant P take on identical values for the sponge and cantilever geometries. In particular, the capacitance and the dependency of f on q are identical, so that results of experiments in the two geometries can be directly compared. We are interested in ∂U/∂P|q. We note that by its definition (eq 72), η has the total differential

dη )

|

|

| |

∂w ∂w ∂w de + dP + dq ∂e q,P ∂P e,q ∂q e,P ∂f ∂f 2 ∂f de + dP + dq (73) 3K ∂e q,P ∂P e,q ∂q e,P

( |

| )

By using this result along with the Maxwell relation

|

|

∂η ∂U ) ∂P q ∂q U and with eq 69 for de, one finds

(74)


|

Langmuir, Vol. 21, No. 10, 2005 4603

|

|

∂U ∂η ∂w 2σ ) ) ∂P q ∂q U ∂q e,P 3K

(75)

The predicted numerical value for this parameter for Au(111) in the limit of weak adsorption is -3 × 10-12 V/Pa. This implies that a pressure change of 1 MPa changes the potential by about 3 µV. To compute the variation in q when the pressure is varied at constant potential U, we proceed as with the derivation of eqs 67, 68 in section IV.B; the result is

| (

|)

∂q 2σ ∂w )c ∂P U 3K ∂q e,P

(76)

The estimated numerical value of this quantity is 8 × 10-13 (C/m)2/Pa. Thus, a pressure change of 1 MPa acting on 1 mm3 of metal with a particle size D ) 10 nm (area per volume 6/D ) 0.6 nm-1) leads to a change in charge of 0.5 µC. V. Summary and Conclusions In the standard analysis of the surface stress of interfaces between a fluid electrolyte and a solid conductor, one considers a planar surface which can be strained tangentially. We have generalized this analysis by considering the deformation of solid bodies with surfaces of varying orientation and of arbitrary curvature immersed in electrolytes. Furthermore, we have examined how the notion of a deformation along the surface normal, which is suggested by experiments, can be incorporated into a continuum theory. For isothermal processes the appropriate state variables for the (Helmholtz) interfacial free energy density ψ are the tangential strain tensor e, the stretch , and the superficial densities Γi of the components (eqs 42 and 43). When e is defined as the superficial projection of the limit of the strain tensor E in the bulk as the surface is approached, and is defined in terms of the volume of fluid displaced due to surface relaxation at constant strain in the bulk, then e and constitute kinematically independent degrees of freedom for deformation. The conjugate quantities to e and are the surface stress s and the pressure P in the fluid, respectively. We have derived the local balance equations (eqs 39 and 36) which relate the stress tensor S in the bulk and the generalized force associated with the stretch to the values of s and P. These equations do not determine the pressure PB in the solid below the surface, but relations for volumetric averages of PB and of all components of S can be specified (eqs 49, 50). In the limit of small surface-induced pressure in the bulksas in macroscopic bodiessa Legendre transform of ψ agrees with the surface tension γ (eqs 52 and 54), but we could not derive γ for a general case.

The relevant variable, w, for the pressure dependence of equilibria is a combination of stretch with the change in the molar volumes of the components as they are transferred from the bulk of the electrolyte into the superficial space charge or adsorbate layers (eq 56). We suggest that in an idealized case (limit of weak adsorption, section III.F) the pressure dependence may be dominated by the stretch alone. By analysis of specific experimental situations we have derived relations between the materials constants, for instance, Maxwell relations. As a special case they contain Gokhshtein’s relation for the dependence of the potential on the strain at constant superficial charge density; this is supplemented by additional results which relate to the pressure dependence. We have used published data for the Au(111) surface in H2SO4 to estimate the values of the materials constants and the magnitude of the signal in experiments on thin films and porous bodies. The results suggest that these relations are open to experimental investigation, specifically in terms of electrochemical experiments under pressure. In such experiments the potential is measured versus a reference electrode at which the electrode potential is also pressure dependent. It is therefore required to use a reference electrode with a pressure dependence which is either negligible or accurately known based on data for the change in molar volume during the electrode reaction. If this condition can be met, it is conceivable that the relations between the superficial materials constants may be used to measure the stretch by electrochemical techniques, supplementing experiments on single-crystal surfaces by surface-sensitive diffraction. Acknowledgment. This work was supported by the Deutsche Forschungsgemeinschaft (Center for Functional Nanostructures). Note Added after ASAP Publication. The pressure P in the generalized adsorption equation, eq 54, cannot be varied independently of the chemical potentials µi. The relation between dP and the dµi is obtained by taking the total differential of eq 9 for P and substituting the righthand side of eq 7 for dΨF. This yields dP ) Fi dµi. By inserting this result into eq 54, P is removed as a j )ψ j (e,µi,U) along with the parameter, and one obtains ψ more fundamental variant of eq 54

dψ j ) s de + (Fi - Γi) dµi - q dU

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The information in this paragraph was not in the version published ASAP April 7, 2005; the corrected version was published April 12, 2005. LA047838A

Balance of Force at Curved Solid MetalâLiquid Electrolyte Interfaces

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