Surface Energy of Miller Index Cross Sections

• intrepid_nerd
This is due to the arrangement of the atoms in the cube and the number of bonds they share with neighboring atoms. This can be better visualized by looking at the supplemental picture provided in the conversation. In summary, the Miller Index calculation for surface energy takes into account the number of broken chemical bonds, bond strength, and surface atomic density. The atoms in the center of each facet do not count towards the calculation.
intrepid_nerd
ℽ = ½Nbερa
ℽ = surface energy
Nb = number of broken bonds
ε = bond strength
ρa = surface atomic density

{100} = (½)( 2 / a2 )(4)(ε)

{110} = ( 5 / sqrt(2) )( ε / a2 )

{111} = (2sqrt(3))( ε / a2 )

the supplemental picture shows a cube with atoms at each corner and one atom in the middle of each facet of the full cube. the description for {100} says there are four broken chemical bonds, but what about the fifth atom in the center? for {110} is says there are five and for {111} it says three. WHAT?? if you picture the {110} it looks like 6 bonds are broken and for {111} it looks like five to me, am I hallucinating??
here is a link to make it easier to picture the sections: http://en.wikipedia.org/wiki/Miller_index"

PLEASE GIVE INSIGHT! THANKS A TON!

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The atoms in the center of each facet are not part of the Miller Index calculation. They are referred to as "dangling bonds" and they do not count towards the number of broken chemical bonds. For the {100} surface, there are four broken chemical bonds, for the {110} surface, there are five broken chemical bonds, and for the {111} surface, there are three broken chemical bonds.

I can provide some clarification on the surface energy equations and the number of broken bonds for each Miller index.

Firstly, the equation for surface energy (ℽ) is a simplified version of the more general equation, which takes into account the angle of the surface and the bond angles between atoms. This simplified version assumes that all bonds are broken at a right angle and that all bonds have the same strength (ε).

Now, for the Miller indices, the number of broken bonds (Nb) is determined by the number of atoms on the surface that do not have a neighboring atom on the same surface. In the case of the {100} surface, there are four such atoms, as the fifth atom in the center is still surrounded by neighboring atoms on the same surface. This is why the equation for ℽ{100} only includes four broken bonds.

For the {110} surface, there are six atoms that do not have a neighboring atom on the same surface, thus resulting in a Nb value of six. Similarly, for the {111} surface, there are five such atoms, leading to a Nb value of five.

I understand that it may seem confusing when looking at the cube diagram, but it is important to remember that the Miller indices represent the orientation of the surface, not the actual number of atoms present. I hope this helps clarify any confusion.

1. What is surface energy?

Surface energy is the measure of the energy required to create new surfaces in a material. It is a property that describes the internal forces of a material that hold atoms together and affects its physical and chemical properties.

2. How is surface energy of Miller index cross sections calculated?

The surface energy of Miller index cross sections is calculated using the formula γ = E/(2A), where γ is the surface energy, E is the energy needed to create a new surface, and A is the surface area of the material.

3. What factors affect the surface energy of Miller index cross sections?

The surface energy of Miller index cross sections is affected by factors such as the crystal structure, surface roughness, and surface composition of the material. It can also be influenced by external factors such as temperature and pressure.

4. How does surface energy impact the properties of a material?

Surface energy plays a crucial role in determining the physical and chemical properties of a material. It affects the material's surface tension, wettability, and adhesion to other materials. Additionally, the surface energy of a material can influence its reactivity, corrosion resistance, and mechanical properties.

5. How can the surface energy of Miller index cross sections be measured?

The surface energy of Miller index cross sections can be measured through various techniques such as contact angle measurements, surface energy mapping, and surface energy calculations based on thermodynamic principles. These methods provide valuable information about the surface energy of a material and its properties.

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