Surface Energy

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Surface Thermodynamics

Surface Energy and Surface Tension

The internal energy of a one-component solid U is a function of the entropy S, volume V, and number of particles N:.

   

that define the temperature T, the pressure P and the chemical potential m of the bulk. The extensive property U(lS,lV,lN) = lU(S,V,N) leads to the Euler equation:

    U = TS – PV + mN

Upon differentiation and using the definition of dU, one obtains the Gibbs-Duhem equation:

    SdT – VdP + Ndm = 0

The surface has to be considered differently from the bulk. For instance, the pressure in the bulk of an isotropic solid is equal in all directions, whereas the pressure on the surface plane is highly anisotropic.  To create a surface, e.g., by cleaving, requires to spend energy that is proportional to the amount of A, the additional surface area created.  

    U = TS – PV + mN + gA

The proportionality factor g is called the surface tension

    g = dU/dA

The surface tension can be defined as the reversible work of formation of a unit area of surface at constant T, V, m, and (for multi-component systems) number of components. The surface tension is the two-dimensional analog to the pressure. 

However, note the difference to PdV, the work to increase the volume, where P is always normal to the surface. In the case of increasing the area, keeping the volume constant, the surface tension is always parallel to the surface. The unit of g is force per unit length (e.g., Newton/meter).

The increase of surface area can be done by stretching. One can calculate the increase in energy given with elasticity theory:

where sij and eij are the components of the surface stress (force per unit length) and strain (deformation) tensors along the i direction, where j denotes the surface normal. Using dA/A = Sdeijdij, the corresponding Gibbs-Duhem equation taking into account the surface is:

Now, one can divide the thermodynamic quantities in a bulk and surface part (e.g., S = Sb + SS, etc.)  Applying the Gibbs-Duhem relation for the bulk can be shown to lead to:

   

That is, the specific surface entropy is given by the temperature dependence of g. Typically, g =go(1–T/Tc)n (van der Waals-Guggenheim semi-empirical relation) with n 1 for metals, where Tc is the critical T (at which the solid phase vanishes.) Also:

   

Note that in a solid, the surface tension and surface stress are not identical. In contrast, in a liquid, the surface tension is independent of small strains, since the liquid adapts to perturbations.

The tendency to minimize surface energy is a defining factor in the morphology and composition of surfaces and interfaces. Minimization of energy leads to a spherical equilibrium shape in an isotropic liquid (in the absence of gravity). In crystalline solids, the surface tension depends on the crystal plane and direction. Thus, the equilibrium shape of a crystal is not obtained by minimizing the surface area but the integral g(n)dA, where n is an orientation vector. Because g is low for some crystal planes, faceting is energetically favored, even if it implies a larger surface area. It is important to note that the formation of the equilibrium shape requires sufficient mobility (or fast kinetics), not just thermodynamics. Equilibrium shapes can be calculated but it is easier to use a graphics method, the Wulff construction. The surface tension is plotted in polar coordinates vs. the angle with respect to a particular direction. The minimization mentioned above implies constructing the surface from the inner envelope of planes perpendicular to the radius vector.

 Wulff construction. The surface tension g is plotted in polar coordinates as a function of the angle Q describing the normal directions to the {hkl} planes. The equilibrium shape of a solid (dash-dotted) is given by the inner envelope of the Wulff planes (normal to the radius vector). From H. Lth, Surfaces and Interfaces of Solid, Springer, 1993).

The following two figures are from the work of Heyraud and Metois, Surf. Sci. 128, 334 (1983). The first one shows an electron micrograph of a lead crystal at 473 K

 

wpe2.jpg (85211 bytes)

The next figure shows the anisotropy of the surface tension relative to <111> for lead as a function of temperature.

wpe5.jpg (127231 bytes)

 

 Estimate of the surface tension

When splitting a solid, the amount of energy required is 2gA, where 2A is the area created (one A on each side). This energy is less than that needed just to break the bonds, since there is atomic and electronic relaxation. The surface energy is always positive because the atoms are less bound at the surface.

Since sublimation creates new surface, let us start the estimate of the work to create a new surface with the heat of sublimation DHsubl, typically 6.8 10-19 J/atom for metals ( 4.25 eV). Removing an atom from the surface leaves a hole. Let us assume that the hole is cubic, with a side of area a=10-15 cm2, the average area per atom at the surface. Before removing the atom, the cube had a surface area a, the hole has a surface area 5a. The net change is thus 4a = 410-15 cm2. Thus, g = 6.8 10-19 / 4 10-15 J/cm2 = 1.7 10-4 J/cm2. In practice, it is found that instead of 0.25 DHsubl, g 0.16 DHsubl.

 

Material

g J/m2

Tungsten (solid) 2.9
Iron (solid) 2.2
Iron (liquid) 1.9
MgO 1.2
Mercury (liquid) 0.5
Water 0.07
Acetic acid 0.03
Nitrogen (liquid) 0.01
Helium (liquid) 0.0003

Multi-component systems: Surface Segregation.

Due to energy minimization surfaces with high g will tend to be covered by surfaces of low g. For example, in the figure, g(metal-gas) < g(oxide-metal) + g(oxide-gas) together with similar relations that reduce the total surface energy for water and organic layers, leads to the ordering shown. .

 

Let us consider a typical case, that of a two component system, where an element B is diluted in a metal matrix A. Based on surface energy arguments, the element B segregates to the surface if it forms a stronger surface bond there. An example is sulfur impurities in nickel. It is found that at high temperatures, S segregates to the surface, forming a 100% surface layer, even though it may be present in very low concentrations (parts per million) in the bulk. This behavior can be understood comparing the BB, BA and AA binding energies, in the following energy diagram (weakest bond moves to surface), where the zero of energy is arbitrary:

 

Energy (eV)

2.53

–––––––––––––––––– S in vapor phase

0

–––––––––––––––––– S in solution

-1.02

–––––––––––––––––– S in grain boundary

-2.01

–––––––––––––––––– S in surface of Ni

In this diagram, the adsorption energy from the gas phase is 4.54 eV (1.54 –(-2.01)), and the segregation energy is –2.01 eV.

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updated Dec. 5, 2000

Copyright 2002, by Ral Baragiola, University of Virginia. All rights reserved.