‘’DIFFUSION IN SOLIDS’’

IE-114 Materials Science and General Chemistry

Lecture-5

Diffusion

The mechanism by which matter is transported through matter. It is related

to internal atomic movement.

Atomic movement; Gases > Liquids > Solids

Atomic-scale motion (diffusion) in liquids and gases is rapid and easy to visualize

Gases Liquids

Diffusion in solids

The slower diffusion rate in the solid state than in the liquid state.

Heat Treatment of Metals

Case hardening of steel (carbon diffusion in steel)

Oxidation of metals

Thin film electronics (doping of semiconductors)

Sintering (fusion of powder particles at solid state)

EXAMPLE : CASE HARDENING OF STEEL

Methods used: Carburizing, Nitriding

Fe + 2CO Fe(c) + CO2

Types of diffusion in Solids

Self-diffusion Interdiffusion (Impurity diffusion)

Origin of Diffusion

Thermal energy

Concentration gradient (effective in diffusion of impurities)

In solids, atomic movements are restricted to equilibrium positions due to bondings. If energy is

provided to the solids, it can cause the thermal vibrations of atoms about their equilibrium

position and at sufficient energy vibration may be strong enough to break the bonding and make

the atom move

1) Self-diffusion

Label some atoms After some time

A

B

C

D

Diffusion in pure metals

All atoms exchanging positions are of the same type

HEAT

100%

Concentration Profiles0

HEATING

Upon heating, diffusion of atoms from high concentration to low

concentration takes place

2) Impurity-diffusion (Interdiffusion)

The process, whereby atoms of one metal diffuse into another

metal

Initially After some time

COPPER NICKEL

Diffusion Mechanisms

In atomic point of view diffusion is stepwise migration of

atoms from lattice site to lattice site

For an atom to make such a move;

a) there must be an empty adjacent site

b) atom that is moving should have enough energy to break bonds with the neighboring atoms and cause some distortion. The energy is vibrational in nature.

1) Vacancy diffusion

2) Interstitial Diffusion

There are two types of diffusion mechanisms:

Vacancy (substitutional) Diffusion

a) Self-diffusion (pure metals) b) Impurity Diffusion

Interstitial Diffusion

a) Self-diffusion (pure metals) b) Impurity Diffusion

1) Vacancy (substitutional) Diffusion

A host or substitutional atom exchanges places with a vacancy

(Both vacancies and required activation energy is provided by thermal energy which results in vibrations of

atoms and impurities)

Activation energy = Act. Ener. (to form a vacancy) + Act. Ener. (to move the atom to the vacancy)

Activation energy (Q) as melting temp.

(Stronger bonds exist in higher-melting-temperature metals)

Rate of diffusion depends on;

Number of vacancies

Activation energy to migrate

• jumping of a smaller atom (black) from one interstitial

site to another in a BCC structure.

e.g. Interstitial diffusion of carbon in iron Rcarbon=0.071 nm RFe= 0.124 nm

2) Interstitial Diffusion:

Atoms move from an interstitial position to another one closeby.

Migrating atoms should be small in size such as N, C, H and O.

Interstitial diffusion occurs much more rapidly than diffusion by

the vacancy mode, since interstitial atoms are smaller, and thus

mobile.

Conditions for atom migration:

- empty adjacent site

- atom must have enough energy to break bonds and cause lattice distortion

during displacement.

Diffusive motion influenced by atom vibrational energies (kBT)

Activation Energy for Diffusion

1) Steady-State diffusion:

Diffusion is a time dependent process; the quatity of an element

that is transported within another is a function of time.

The rate of mass transfer is explained by diffusion flux (J): the

mass of (or, number of atoms) M diffusing through and perpendicular to a unit cross-

sectional area per unit of time

mass

crosssectional area

time

In differential form:

UNIT: kg/m2s or atoms/m2s

Diffusion Modeling

Steady state condition exists when the diffusion flux does not

change with time

Concentrations(pressures) of two species are held constant, PA and PB and PA>PB

The steady-state diffusion in one (x) direction :

FICK’S FIRST LAW

Concentration gradient (dC/dx) is the driving force in diffusion reactions.

The steeper the concentration gradient, the greater will be the diffusion or

the mass transfer

D: diffusivity or diffusion coefficient (m2/sec or cm2/sec).

The negative sign indicates that the direction of diffusion is

down the concentration gradient.

2)Nonsteady state diffusion:

The diffusion flux and concentration gradient at a selected point vary

with time causing a net accumulation or depletion.

For nonsteady state diffusion Fick’s first law is not valid.

FICK’S SECOND LAW

Semi-finite solid in which the surface concentration is held constant.

Assume:

a) before diffusion, diffusing solute concentration is uniform, Co.

b) the value of x is zero at the surface and increases with distance into the solid.

c) the time is zero at the instant before the diffusion begins.

Depending on the selected boundary conditions there may be different

solutions for Fick’s second law.

Surface concentration is held constant

Cx: concentration at depth x after time t.

Cs: concentration at surface

Co: initial concentration

Gaussian error function

Factors affecting diffusion:

Different materials have different diffusion coefficient (Do),

which is also the indication of the diffusion rate.

4) Temperature: Diffusion is thermally activated process

Temperature dependence can be expressed as follows:

2) Crystal structure (BCC, FCC, ..)

3) Imperfection (grain boundary, dislocation, vacancy, lattice)

1) Diffusing species:

Diffusion coefficient vs 1/T

10-24

10-22

10-20

10-18

10-16

10-14

10-12

10-10

0.6 0.8 1 1.2 1.4 1.6

Fe in bcc FeFe in fcc FeC in bcc FeC in fcc FeMn in fcc Fe

D, m

2/s

1000/T(K)

1200°C 900°C 400°C600°C