Surface Effect

𝐿𝐿

𝑙𝑙

47

A.1 Surface to volume ratio & surface area

 Let 𝑅𝑅 be the radius of a large spherical particles Surface to volume ratio, 𝐹𝐹 = ^𝑎𝑎_𝑉𝑉 = ^{4𝜋𝜋𝑅𝑅}4 ²

3𝜋𝜋𝑅𝑅³ = _𝑅𝑅³ = _𝐷𝐷⁶

 If the large cubical particle is sub-divided to 𝑛𝑛_𝑐𝑐 small particles of radius, 𝑟𝑟, then from mass balance equation, assuming no mass losses during conversion,

Mass of large sphere = total mass of small sphere 𝑉𝑉 × density = 𝑛𝑛_𝑐𝑐 × 𝑣𝑣 × density

𝑛𝑛_𝑐𝑐 = 𝑉𝑉 𝑣𝑣 =

43𝜋𝜋𝑅𝑅³

43𝜋𝜋𝑟𝑟³ = 𝑅𝑅³ 𝑟𝑟³ Specific surface area = 𝑛𝑛_𝑐𝑐 × 𝑎𝑎

𝜌𝜌𝑉𝑉 = 𝑅𝑅³

𝑟𝑟³ × 4𝜋𝜋𝑟𝑟²

43𝜋𝜋𝜌𝜌𝑅𝑅³ = 3

𝜌𝜌𝑟𝑟 = 6 𝜌𝜌𝜌𝜌

 Ratio of surface area = ^{𝑛𝑛𝑐𝑐}_𝐴𝐴^×𝑎𝑎 = ^𝑅𝑅_𝑟𝑟3³ × _{4𝜋𝜋𝑅𝑅}^{4𝜋𝜋𝑟𝑟2}₂ = ^𝑅𝑅_𝑟𝑟

A. Surface Effect

48

0 2 4 6

0 10 20 30 40

Size, 𝑑𝑑/nm

Surface to volume ratio/nm −1

0 200 400 600 800 1000

0 10 20 30 40

A.1 Surface to volume ratio & surface area

A. Surface Effect

Size, 𝑑𝑑/nm

Specific surface area/m2 g−1

 It is apparent that both surface to volume ratio &

specific surface area increase drastically when the size of material is reduced to or below 10 nm.

49

A.1.2 Fraction of surface atoms

A. Surface Effect

Cubical Particles

 Let the number of atoms along edge = 𝑛𝑛

 Total number of atoms in the cube, 𝑁𝑁 = 𝑛𝑛³

 The atoms at the edge share between two surface, while at the corner among eight surfaces. Thus, overcounting the edge and

corner atoms need to be corrected. For large N, this correction is negligible. However, for nanoparticle, it is dominated.

 Total number of atoms on six surfaces including double counting at the edges = 6𝑛𝑛²

 Total number of double counted atoms at 12 edges = 12𝑛𝑛

 Total number of atoms on eight corners = 8

 Subtraction of edge atoms from 6𝑛𝑛² also removes the corner atoms, which is to be added to count net total surface atoms.

 Thus, the net total number of surface atoms = 6𝑛𝑛² − 12𝑛𝑛 + 8

 The fraction of surface atoms, 𝐹𝐹 𝐹𝐹 = 6𝑛𝑛² − 12𝑛𝑛 + 8

𝑛𝑛³ = 6

𝑁𝑁¹³ − 12

𝑁𝑁²³ + 8

𝑁𝑁 ≈ 6𝑁𝑁⁻¹³

50

A.1.2 Fraction of surface atoms

A. Surface Effect

Spherical Particles

 Let the number of atoms in a spherical particle = 𝑁𝑁, volume of

sphere = 𝑉𝑉_𝑐𝑐, radius of sphere = 𝑅𝑅_𝑐𝑐, volume of each atom, 𝑉𝑉_𝑎𝑎 = ⁴₃𝜋𝜋𝑟𝑟_𝑎𝑎³, 𝑟𝑟_𝑎𝑎 is the radius of atom & number of surface atoms = 𝑁𝑁_𝑠𝑠 .

 For spherical particle,

The volume of spherical particle = total volume of N atoms 4

3𝜋𝜋𝑅𝑅_𝑐𝑐³ = 𝑁𝑁 4

3𝜋𝜋𝑟𝑟_𝑎𝑎³ ⟹ 𝑁𝑁 = 𝑅𝑅_𝑐𝑐 𝑟𝑟_𝑎𝑎

3

 Surface area of particle, 𝑆𝑆_𝑐𝑐 = 4𝜋𝜋𝑅𝑅_𝑐𝑐² = 4𝜋𝜋 𝑁𝑁²³𝑟𝑟_𝑎𝑎² = 𝑁𝑁²³𝑆𝑆_𝑎𝑎 where, 𝑆𝑆_𝑎𝑎 = surface area of an atom = 4𝜋𝜋𝑟𝑟_𝑎𝑎²

 Approximate number of surface atoms, N_s = Surface area of a particle

Central cross − sectional area of an atom =

4𝜋𝜋𝑁𝑁²³𝑟𝑟_𝑎𝑎²

𝜋𝜋𝑟𝑟_𝑎𝑎² = 4𝑁𝑁²³

• The fraction of surface atoms called dispersion is given by 𝐹𝐹 = 𝑁𝑁_𝑠𝑠

𝑁𝑁 = 4𝑁𝑁²³

𝑁𝑁 = 4𝑁𝑁⁻¹³

51

A.1.2 Fraction of surface atoms

A. Surface Effect

Spherical Particles

 Volume of the layer or shell with a thickness 𝛿𝛿 at the surface of a spherical particle of a radius 𝑅𝑅 can be determined from the following equation,

𝑉𝑉_{𝑠𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙} = 4

3𝜋𝜋𝑅𝑅³ − 4

3𝜋𝜋 𝑅𝑅 − 𝛿𝛿 ³ = 4

3𝜋𝜋 𝑅𝑅³ − 𝑅𝑅 − 𝛿𝛿 ³ 𝑉𝑉𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑟𝑟𝑠𝑠 = 4

3𝜋𝜋𝑅𝑅³

 The ratio of the volume of surface layer or shell to total volume of a particle can be determined by

𝐹𝐹 = 𝑉𝑉_{𝑠𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙} 𝑉𝑉𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑟𝑟𝑠𝑠 =

43𝜋𝜋 𝑅𝑅³ − 𝑅𝑅 − 𝛿𝛿 ³

43𝜋𝜋𝑅𝑅³ = 1 − 𝑅𝑅 − 𝛿𝛿 𝑅𝑅

3

 Ratio, F approaching to 1 implies that all atoms are on the surface. F becomes 1 when 𝑅𝑅 = 𝛿𝛿.

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A.1.2 Fraction of surface atoms

A. Surface Effect

From structural magic numbers

 Most metals in the solid form close packed lattices

 Ag, Al, Cu, Co, Pb, Pt, Rh are Face Centered Cubic (FCC)

 Mg, Nd, Os, Re, Ru, Y, Zn are Hexagonal Close Packed (HCP)

 Cr, Li, Sr can form Body Centered Cubic (BCC) as well as (FCC) and (HCP) depending upon formation energy

 How does crystal structure impact nanoparticles?

 Nanoparticles have a “structural magic number”, that is, the optimum number of atoms that leads to a stable configuration while maintaining a specific structure.

 Structural magic number = minimum volume and maximum density configuration

 If the crystal structure is known, then the number of atoms per particle can be calculated.

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A.1.2 Fraction of surface atoms

A. Surface Effect

From structural magic numbers

 For n layers, the number of atoms N in an approximately spherical FCC nanoparticle is given by the following formula:

𝑁𝑁 = 1

3 10𝑛𝑛³ – 15𝑛𝑛² + 11𝑛𝑛 − 3

 The number of atoms on the surface N_surf 𝑁𝑁_𝑠𝑠 = 10𝑛𝑛² – 20𝑛𝑛 + 12

 The fraction of surface atom, 𝐹𝐹 = 𝑁𝑁_𝑠𝑠

 The diameter of cluster 𝑁𝑁

𝜌𝜌 = 2𝑑𝑑(2𝑛𝑛 − 1) Where 𝑑𝑑 = ^𝑎𝑎₂, 𝑎𝑎 is lattice constant.

 For bcc crystal

𝑁𝑁 = 4𝑛𝑛³ − 6𝑛𝑛² + 4𝑛𝑛 − 1, 𝑁𝑁_𝑠𝑠 = 12𝑛𝑛² − 24𝑛𝑛 + 14

54

A.1.2 Fraction of surface atoms

A. Surface Effect

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A.1.3 Specific surface energy

A. Surface Effect

Fig. Creation of new surfaces (e.g., by breaking a larger portion into smaller pieces) requires energy u for each bond to be broken.

 The energy required to create a new surface containing 𝑁𝑁_𝑏𝑏 atoms is given by ^𝑁𝑁₂^{𝑏𝑏𝑢𝑢}, where 𝑢𝑢 is the bonding energy between two atoms.

 If the 𝐴𝐴 is the surface area, then its contribution to

specific surface energy is 𝛾𝛾₀ = 𝑁𝑁_𝑏𝑏𝑢𝑢

2𝐴𝐴

 Within the interior of a

particle, an atom or ion is held in a mechanical equilibrium by binding forces, which fix the ions in their lattice positions.

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A.1.3 Specific surface energy

A. Surface Effect

 Atoms or molecules on a solid surface possess fewer nearest neighbors or coordination numbers. They have dangling or unsatisfied bonds exposed to the surface.

 Due to the reduced number of neighbors, at each surface of the atom, a force 𝑓𝑓 acts perpendicular to the surface. It leads stress in the plane. Surface stress is given by

𝜎𝜎 = 𝑓𝑓 𝐴𝐴

 𝛿𝛿 deforms the surface and results in surface stretching. 𝛿𝛿 contributes to the specific surface energy as function of stretching, 𝜀𝜀_𝑠𝑠. The specific surface energy is given by

𝛾𝛾 = 𝛾𝛾₀ + 𝛾𝛾_𝑠𝑠(𝜀𝜀_𝑠𝑠)

Where 𝛾𝛾_𝑠𝑠 is the contribution of the surface stress to the surface energy.

57

A.1.3 Specific surface energy

A. Surface Effect

 The amount of surface energy per particle 𝑢𝑢𝑠𝑠𝑢𝑢𝑟𝑟𝑠𝑠𝑎𝑎𝑐𝑐𝑠𝑠 is equal to 𝛾𝛾𝐴𝐴, where 𝛾𝛾 is the specific surface energy and 𝐴𝐴 is the surface area of one particle.

 For thermodynamic considerations, the surface energy per mole of material is the essential quantity.

 Hence, if N is the number of particles per mole, one obtains 𝑁𝑁𝛾𝛾𝐴𝐴 =

𝑀𝑀

𝜌𝜌𝜌𝜌 𝛾𝛾𝐴𝐴, where 𝜌𝜌 is the density of the material, 𝑀𝑀 is the molar mass, 𝑣𝑣 is volume of particle = ^{𝜋𝜋𝑑𝑑}₆², 𝐴𝐴 = 𝜋𝜋𝑑𝑑², and 𝑑𝑑 is the diameter of particle.

 Finally, one obtains the surface energy of particles with diameter 𝑑𝑑 per mole:

𝑈𝑈𝑠𝑠𝑢𝑢𝑟𝑟𝑠𝑠𝑎𝑎𝑐𝑐𝑠𝑠 = 𝑀𝑀 𝜌𝜌

6

𝜋𝜋𝑑𝑑³ 𝛾𝛾 𝜋𝜋𝑑𝑑² = 6𝑀𝑀 𝜌𝜌

𝛾𝛾 𝑑𝑑

 From equation, the surface energy per mole increases with 1/d and in some cases, especially those related to very small particles, this may have dramatic consequences.