Shuttleworth Equation Research Articles

Method for Thermodynamic Description of an Elastically Deformed Electrode, Based on Relating Its Extensive Surface Characteristics to a Strainless Electrode

A theoretical approach to thermodynamic describing elastically deformed electrodes made of isotropic materials is suggested. The approach is based on considering the surface of a strainless electrode as a reference one. A surface analog of the Lame elastic constants is introduced. Dependences of the Shuttleworth equation and the Gibbs adsorption equation on the Lame elastic constants characterizing the surface are found in an explicit form.

Adsorption isotherm for a solid electrode: The link between the differential surface tension and capacitance curves

Investigation of thermodynamically equilibrium single-component adsorption from a liquid solution on a solid electrode with allowance made for elastic deformation of its surface ϑ is continued. A full electrocapillarity equation is derived from thermodynamics equations for an interphase layer in the absence of irreversible processes. Thermodynamic aspects of the Shuttleworth equation are discussed and the equation is compared with two-dimensional Murnaghan formulas for elastic isotropic media. An adsorption isotherm equation and compatibility equations that had been derived previously are examined in a special case where the derivative Γ of a surface concentration with respect to ϑ \(\frac{{\partial \Gamma }}{{\partial \vartheta }} \equiv \gamma \vartheta\) depends solely on Γ (γϑ=γϑ(Γ)) and a rigorous solution of these is obtained for a deformed electrode (ϑ ≠ 0). The effect of ϑ and dimensionless electrode potential φ on the extreme (at an infinitely high adsorbate concentration) value of Γ is studied. The model of two parallel capacitors is considered in detail for a general case. Owing to the use of capacitance curves for an elastically stretched electrode, a formula that expresses the differential surface tension of a nondeformed electrode through such curves is derived for the first time ever.

Connection of the generalized Shuttleworth equation for the elastic spherical electrode with the Laplace formula and the Gibbs adsorption equation

The surface strain of a solid is related to the surface free energy (interfacial superficial work) by the Shuttleworth equation. Its generalization for the case of a finite surface strain has been derived only recently by one of the present authors (B.M.G). Now consequences are derived for the case of an elastic spherical electrode. At first, it is shown that this generalized form is in accordance with the Laplace formula connecting the capillary pressure with the surface stress. Further, the generalized Shuttleworth equation leads to an additional term in the Gibbs adsorption equation, which is of first order in the elastic strain. Whereas this first order term may be negligible in the adsorption equation itself even for non-infinitesimal strain, it leads to a significant modification when considering second order derivatives of surface charge and surface stress which are directly accessible in experiments. A reformulation is presented for changing variables and the applicability to small particles is discussed.

Alternative to the Shuttleworth formulation of solid surface stress

We examine two derivations of the Shuttleworth equation (which is a relation between surface stress $g,$ surface tension \ensuremath{\gamma}, and surface strain), and identify the flaws we perceive. Rectifying the perceived flaws leads not to the Shuttleworth equation but to $g\ensuremath{-}\ensuremath{\gamma}$ equivalence. We conclude that surface stress is merely the generalization of the concept of surface tension to an elastically anisotropic system; the surface free-energy density is one-half of the trace of the surface stress tensor, to lowest order. In our opinion, our conclusions lead to a more coherent and elegant form of surface thermodynamics which should prove useful in controlling and in understanding nanometer-scale fabrication.

The Shuttleworth equation for a finite surface strain

Abstract It is shown that the Shuttleworth equation g=γ+∂γ/∂e is valid for the interface between a solid isotropic electrode and a liquid electrolyte in the case of an infinitesimal surface deformation only. In the equation above, g is the surface stress, γ is the interfacial tension, and e is the scalar surface strain variable (elastic surface deformation). In the general case of a finite surface strain, the interfacial tension is a function of two scalar strain variables rather than one. Under conditions of an elastic finite surface strain, the modified Shuttleworth equation is deduced for the trace of the tensor of the surface stress. The Shuttleworth equation above takes into account the existence of both scalar elastic surface strain variables.

Thermodynamics of deformable three-phase systems

A thermodynamic theory for deformable three phase contact regions is developed. The mechanical and thermodynamic line stress variables are defined. Their relation with the line tension is shown (Shuttleworth equation for lines). The effect of the excess of the deformation is included. The general condition of mechanical equilibrium is established; it reduces to Neumann's relation and Young's equation in the limit of fluid systems and in the usual wetting configuration respectively.

Surface stress of curved solid-fluid interfaces

The effects of curvature on elastic deformations have been analysed. The work of stretching a spherical solid-fluid interface has been split into two contributions which allow one to define a tangential surface stress and a normal surface stress. It is shown that only the latter can be identified with the mechanical surface stress appearing in Laplace's equation. The Shuttleworth equation is modified to include the excess of volume deformation.

Surface stress of curved solid-fluid interfaces

The effects of curvature on elastic deformations have been analysed. The work of stretching a spherical solid-fluid interface has been split into two contributions which allow one to define a tangential surface stress and a normal surface stress. It is shown that only the latter can be identified with the mechanical surface stress appearing in Laplace's equation. The Shuttleworth equation is modified to include the excess of volume deformation.

Surface tension and spreading pressure of mobile and registered monolayers

Equations based on the theory of Eriksson and the Shuttleworth equation are applied to the adsorption of simple gases on plane surfaces where the adsorbed layer is in registry and when it is mobile, and to the transition between these states. The effect of lattice constant on binding energy is considered, as it is altered by density and bulk polarizability. These effects are in turn related to dimensional changes brought about by adsorption.

Thermodynamics of surface phase systems: V. Contribution to the thermodynamics of the solid-gas interface

A thermodynamic fundamental equation is derived for the surface phase of a solid exposed to an adsorbing gas. This equation is more general than the corresponding equation given by Gibbs in the sense that it is applicable with less restriction as to the state of strain of the solid surface. An equation relating the surface tension to the reversible cleavage work which is similar to the Shuttleworth equation is shown to be implied in the fundamental equation deduced. Some new thermodynamic expressions are obtained for the effect of temperature and gas pressure changes on the surface tension and on the reversible cleavage work. Measurements of adsorbent dimensional changes, heats of adsorption and surface structure transitions are discussed on the basis of the formalism developed.