Interfacial energy

Course: Phase transformationsLevel: UGAuthor: Satish S Kamath, Department of Chemical Engineering, NITK-SurathkalMentor: M P Gururajan

Interfacial energy

In materials, the formation of interfaces cost the system energy. In this animation, we explore the atomistic origins of the interfacial energy and a

simple model that incorporates the relevant thermodynamics and geometric factors for

studying the interfaces in a crystalline solid

Learning objectives

● Understand interfacial energy as the excess free energy associated with the interfaces

● Understand the interfacial energy in crystalline solids in terms of a simple model that incorporates the relevant thermodynamics and geometry

● Appreciate that interfacial energy is inherently anisotropic in crystalline solids

Master layout

● Show the soap film stretching – define interfacial energy

● Identify the interfacial energy in crystalline solids as excess free energy associated with the interface

● Build a simple bond-breaking model for interfacial energy

● Show the anisotropy in interfacial energy for a solid-vapour interface

Definitions/Key words

● Gibbs free energy: G = U + PV – T S● Internal energy, U● Pressure, P● Volume, V● Temperature, T● Entropy, S● Enthalpy: H = U + PV● Interfacial energy: excess Gibbs free energy

Definitions/Key words

● Anisotropy: the directional dependence of a property

● Surface tension: interfacial free energy in cases where the surface energy is independent of area

Stepwise description

FSoap film

Wire frame

Area, A

Show a soap film of area Ain a wire frame pulled by a force F

Stepwise description

Soap film F

Wire frame

Increase in area, dA

Area, A

The area of the film increases by dA

Work done = Raise in free energyFree energy of a system with interfacial area A is G = G

0 + A, where is the interfacial energy per

unit area.dG = dA + A dF dA

If interfacial energy did not change with area, F = that is, surface tension is the excess free 

energy. However, in solids, because shear stresses can be supported, interfacial energy can change with area. So, in general, surface tension 

is not the same as interfacial energy.  

Stepwise description

Why does the free energy increase at the surfaces?

Broken bond model

Atom in the bulk:All bonds are satisfied

Atoms at the surface:dangling/broken bonds

Broken bond model

Atom in the bulk:All bonds are satisfied

Atoms at the surface:dangling/broken bonds

Broken bonds lead to increase in enthalpy

Broken bond model

Atom in the bulk:All bonds are satisfied

Atoms at the surface:dangling/broken bonds

Broken bonds can also lead to increase in vibrational entropy

Broken bond model

Atom in the bulk:All bonds are satisfied

Atoms at the surface:dangling/broken bonds

Broken bonds Increase in free energy

Stepwise description

Let the latent heat of sublimation for a material be L; let the number of nearest neighbours be z. If an Avogadro number of atoms (N) evaporate, then, a total number of zN/2 bonds are borken. Let be the bond energy. Then, L = zN/2; this implies,

=2L/zN Thus, if a be the number of broken bonds per unit area of a surface, then the enthalpy

increase is 2aL/zN.

Interfacial energy anisotropy

● In a crystal, the number of broken bonds per unit area for different planes are different. Thus, the interfacial free energy for different planes is different; that is, the interfacial energy is anisotropic.

● With higher temperatures, the contribution from the interfacial (vibrational) entropy becomes important; this entropy can overcome the enthalpy and make the interfacial energy isotropic.

Summary

● Interfacial energy is the excess free energy associated with interfaces

● Interfacial energy can be understood in terms of a bond breaking model

● Bond breaking model, for a crystalline interface (with a vapour, for example) indicates that the interfacial energies are anisotropic

References/further reading

● Phase transformations in metals and alloys, Porter, Easterling, and Sherif, Third edition, CRC Press

● Materials science and engineering: a first course, V Raghavan, Fifth edition, Prentice-Hall of India Pvt Ltd.

Questions

● Calculate the number of atoms per unit area in (111), (110) and (100) planes of copper (FCC) with a lattice parameter of 3.61 Angstrom.

Answer: 1.77x1019, 1.08x1019, 1.53x1019

● The surface energy of a (111) surface of copper is 2.49 J/m2; what is the surface enthalpy of a (100) surface?

Answer: 2.15 J/m2

Questions

● At low temperatures, the interfacial energy of a material is anisotropic. As temperature increases, the anisotropy (a) increases

(b) decreases(c) reamins the same

Answer: (b)

Questions

● An atom is on the {100} surface of a bcc crystal. How many of its bonds are broken?

Answer: 4

Questions

● Consider a cubic lattice of lattice parameter 'a'. Consider a plane in the lattice described by the angle that it makes with the x-axis as shown. Considering the unit length of the interface, show that the number of broken bonds per unit area is given by (cosθ + sin |θ| )/ a2

a

Answer

The answer follows from the figure below and consideringa unit length in the perpendicular direction (to the page)