Surface stress

Definition of surface free energy is seemly the amount of reversible work ${\displaystyle dw}$ performed to create new area ${\displaystyle dA}$ of surface, expressed as:

${\displaystyle dw=\gamma dA}$

Gibbs was the first to define another surface quantity, different from the surface tension ${\displaystyle \gamma }$, that is associated with the reversible work per unit area needed to elastically stretch a pre-existing surface. Surface stress can be derived from surface free energy as follows:[2]

One can define a surface stress tensor ${\displaystyle f_{ij}}$ that relates the work associated with the variation in ${\displaystyle \gamma A}$, the total excess free energy of the surface, owing to the strain ${\displaystyle de_{ij}}$:

${\displaystyle d(\gamma A)=Af_{ij}d\epsilon _{ij}}$

Now consider the two reversible paths showed in figure 0. The first path (clockwise), the solid object is cut into two same pieces. Then both pieces are elastically strained. The work associated with the first step (unstrained) is ${\displaystyle W_{1}=2\gamma _{0}A_{0}}$, where ${\displaystyle \gamma _{0}}$ and ${\displaystyle A_{0}}$ are the excess free energy and area of each of new surfaces. For the second step, work (${\displaystyle w_{2}}$), equals the work needed to elastically deform the total bulk volume and the four (two original and two newly formed) surfaces.

In the second path (counter-clockwise), the subject is first elastically strained and then is cut in two pieces. The work for the first step here, ${\displaystyle w_{1}}$ is equal to that needed to deform the bulk volume and the two surfaces. The difference ${\displaystyle w_{2}-w_{1}}$ is equal to the excess work needed to elastically deform two surfaces of area ${\displaystyle A_{0}}$ to area ${\displaystyle A(e_{ij})}$ or:

${\displaystyle w_{2}-w_{1}=2\int (f_{ij}d(A(\epsilon _{ij}))=2\int (Af_{ij}d\epsilon _{ij})}$

the work associated with the second step of the second path can be expressed as ${\displaystyle W_{2}=2\gamma (e_{ij})A(e_{ij})}$, so that:

${\displaystyle W_{2}-W_{1}=2[\gamma (e_{ij})A(e_{ij})-\gamma _{0}\ A_{0}]}$

These two paths are completely reversible, or W₂ – W₁ = W₂ – W₁. It means:

${\displaystyle 2[\gamma (\epsilon _{ij})A(\epsilon _{ij})-\gamma _{0}A_{0}]=2\int (Af_{ij}d\epsilon _{ij})}$

Since d(γA) = γdA + Adγ, and dA = Aδ_ijde_ij. Then surface stress can be expressed as:

${\displaystyle f_{ij}=\gamma \delta _{ij}+\partial \gamma /\partial e_{ij}}$

Where δ_ij is the Kronecker delta and e_ij is elastic strain tensor. Differently from the surface free energy γ, which is a scalar, surface stress f_ij is a second rank tensor. However, for a general surface, set of principle axes that are off-diagonal components are identically zero. Surface that possesses a threefold or higher rotation axis symmetry, diagonal components are equal. Therefore, surface stress can be rewritten as a scalar:

${\displaystyle f=\gamma +\partial \gamma /\partial e}$

Now it can be easily explained why f and γ are equal in liquid-gas or liquid-liquid interfaces. Due to the chemical structure of liquid surface phase, the term ∂γ/∂e always equals to zero meaning that surface free energy won’t change even if the surface is being stretched. However, ∂γ/∂e is not zero in solid surface due do the fact that surface atomic structure of solid are modified in elastic deformation.

Origin of surface stress could be understood by nature of chemical bonding of atoms at the surface. In metallic materials, atomic chemical bonding structure at the surface is very different from in the bulk. Therefore, equilibrium interatomic distance between surface atoms is different from bulk atoms. Since surface and bulk atoms are structurally coherent, the interior of the solid can be considered as applying a stress on the surface.

For illustration, figure 1 shows a simple picture of bond charges near the surface of a 2D crystal with charge (election) density around sphere atoms. Surface atoms only have two nearest neighbors compared with bulk atoms, which have four (for this example case). The loss of neighbors which results from the creation of a metal surface reduces the local electron density around the atoms near the surface. Surface atoms then sit in a lower average electron density than bulk atoms. The response of these surface atoms would be to attempt to reduce their interatomic distance in order to increase surrounding charge density. Therefore, surface atoms would create a positive surface stress (tensile). In the other words, if the surface charge density is the same as in the bulk, surface stress would be zero.

Surface stress, which created by redistribution of electron density around surface atoms, can be both positive (tensile) or negative (compressive). If the surface is not clean meaning there are atoms sitting on a flat surface (adsorbates), charge density would then be modified leading to a different surface stress state compared with a perfect clean surface.

• Gibbs free energy
• Surface tension
• First principles calculation
• Molecular dynamics simulations
• Nanocrystalline materials
• Nanowire materials
• Nanoporous materials