Nanotechnology SUWARNA DATAR| AP 608

Definition of surface energy

It is called as the energy required to create a unit area of a new

surface

On a newly created surface, each atom on

the surface will move towards interior

Extra force is required to pull back the atoms

to their original position

For each atom on the surface, the energy

required to get it back to the original

position:

1. proportional to the number of broken

bonds Nb 2. Bond strength ɛ

ν = ½ Nb ɛ ρA

ρA: no. Of atoms/unit area on the new surface

Surface energy

Concepts of thermodynamics are used to calculate the surface

energy of a material.

G = H- TS

Surface energy of FCC structure 1. Atoms labelled D are the nearest neighbours of atom A

2. These bonds would be broken when the new surface is

formed

3. Therefore for each newly formed surface 4 bonds are

broken

4. Thus, for each atom on the new (100) surface, four

bonds must be broken

Two dimensional unit cell of the (100) surface:

• There is one atom in the centre of each cell and four atoms

that are each shared between four cells.

• This means that the number of atoms per unit cell on the

(100) surface of an FCC crystal is:

Surface Energy

Each 2D unit cell has an area of a2 where a is the lattice parameter

Thus, the number of bonds broken per unit area is

Surface Energy Atom A is on the (110) surface that will form the new surface

created by removing the atoms to the right in the figure.

The five atoms marked D, are all equidistant from atom A.

Thus, for each atom A on the new (110) surface, five nearest

neighbors will be removed and five bonds will be broken.

The four corner atoms are shared between four 2D unit cells,

while the two atoms on the center of two sides are shared

between two unit cells. This means the number of atoms per

unit cell on the (110) surface is

Surface Energy

Three nearest neighbors are removed for each

atom in the new surface or three bonds are

broken per atom on the final (111) surface

Surface Enrgy Thermodynamics tells us that any material or system is stable only

when it is in a state with the lowest Gibbs free energy.

Therefore, there is a strong tendency for a solid or a liquid to

minimize the total surface energy.

Ways to minimize the surface energy:

surface relaxation

surface restructuring through combining surface dangling bonds into

strained new chemical bonds

surface adsorption through chemical or physical adsorption of terminal

chemical species onto the surface by forming chemical bonds or weak

attraction forces such as electrostatic or van der Waals forces

ν = ½ Nb ɛ ρA

Reduction of Surface Energy

At overall system level there are two basic ways:

Combination of individual nanostrcutures to form larger structure

Sintering

Ostwald Ripening

Agglomeration of individual particles

Result: Macroscopically? Reduction of surface energy

Chemical Potential

It is a potential a substance has to produce in order to alter the system.

Similar to ball rolling down the hill, when molecules move they react,

dissolve, melt etc, they always tend to go from higher chemical

potential to lower chemical potential

Molecules have higher chemical potential in high concentrated area, low in low concentrated area

Eg : ICE

Gibbs Energy is the chemical potential that is minimized when the system reaches equilibrium

Relationship between chemical

potential and surface curvature

Where Ω = Atomic volume, ν = surface energy, R = radius of particle

This is known as Young Laplace equation which describes the chemical potential of an atom in a spherical surface with respect to the a flat surface

Generalized form:

(1)

Relationship between chemical

potential and surface curvature

Chemical potential of an atom on convex surface is higher than

that of the flat surface

When a mass is transferred from flat surface to convex surface

chemical potential increases whereas when the mass is transferred from flat to concave chemical potential decreases

Thermodynamically, an atom on a convex surface possesses the

highest chemical potential, whereas an atom on a concave

surface has the lowest chemical potential.

Ostwald Ripening

Such a relationship is also reflected by the difference in vapor

pressure and solubility of a solid.

Assuming the vapor of solid phase obeys the ideal gas law, for the

flat surface:

Where is the chemical potential of vapour atom and is the vapour

pressure of flat solid surface

for the curved surface

(2)

Ostwald Ripening

Combining equation (1) and (2)

For spherical particle

This is known as the Kelvin equation

Ostwald Ripening

Similarly, solubility dependence of on the surface curvature can be written as

This is known as Gibbs Thompson relation

When two particles with different radii, assuming R1 >> R2, are put into a solvent, each particle will develop an equilibrium with the surrounding solvent

solubility of the smaller particle will be larger than that of the larger particle.

Therefore, there would be a net diffusion of solute from proximity of the small particle to proximity of the large particle.

To maintain the equilibrium, solute will deposit onto the surface of the large particle, whereas the small particle has to continue dissolving so as to compensate for the amount of solute diffused away

Ostwald Ripening

Therefore larger particles become larger and smaller become

smaller to compensate for the amount of solute diffused

Agglomeration

Due to van der Waals:

Prevention:

Electrostatic stabilization

Steric stabilization

Nernst Equation

Where E0 = standard electrode potential when the concentration of ions

is unity

Rg = gas constant, T = temp, f = Faraday’s constant, ai = activity of ions

ni = valance state of ions

Agglomeration

Surface potential of solid varies with the concentration of ions in the

surrounding solution (positive/negative)

Ions adsorbed on the surface determine the surface charge,

therefore called charge determining ions

Concentration of charge determining ions can be described by pH

pH = - log[H+]

Concentration of charge determining ions, corresponding to the zero

charged surface is called point of zero charge (PZC)

At pH > PZC negatively charged (covered by OH-)

At pH < PZC positively charged (covered by H+)

Therefore,

Agglomeration

In solution: (1) charge determining ions

(2) counter ions

Although charge neutrality is maintained overall the distribution

of charges is controlled by various forces

(1) coulombic/electrostatic force

(2) Entropic force

(3) Brownian motion

The combined result is that the concentration of counter ions is the highest near the solid surface and decreases as the distance from the surface increases, whereas the concentration of determining ions changes in the opposite manner.

Such inhomogeneous distributions of ions in the proximity of the solid surface lead to the formation of so-called double layer structure

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