‘’DIFFUSION IN SOLIDS’’

MSE-225 Introduction to Materials Science Lecture-5

Diffusion Diffusion - Mass transport by atomic motion

Mechanisms •  Gases & Liquids – Brownian motion •  Solids – vacancy diffusion or interstitial

diffusion.

• Interdiffusion (impurity diffusion): In an alloy, atoms tend to migrate from regions of high concentration to regions of low concentration.

Initially

Interdiffusion

After some time

• Self-diffusion: In an elemental solid, atoms also migrate.

Label some atoms After some time

A

B

C

D

Self-diffusion

Diffusion Mechanisms •  Atoms in solid materials are in constant motion, rapidly

changing positions. •  For an atom to move, 2 conditions must be met:

1.  There must be an empty adjacent site, and 2.  The atom must have sufficient (vibrational) energy to

break bonds with its neighboring atoms and then cause lattice distortion during the displacement. At a specific temperature, only a small fraction of the atoms is capable of motion by diffusion. This fraction increases with rising temperature.

•  There are 2 dominant models for metallic diffusion: 1.  Vacancy Diffusion 2.  Interstitial Diffusion

Vacancy Diffusion Vacancy Diffusion:

• atoms exchange with vacancies • both self diffusion and interdiffusion occur by this mechanism • rate depends on: -- number of vacancies -- activation energy to exchange.

increasing elapsed time

Atom needs enough thermal energy to break bonds and squeeze through its neighbors. Energy needed à energy barrier à Called the activation energy Q or Qd

Diffusion àThermally Activated Process

Atom Vacancy Qd

Distance

Ener

gy

Qsurface < Qgrain boundary < Qlattice

Experimentally determined activation energies for diffusion

Lower activation energy automatically implies higher diffusivity

Diffusion Paths with less Resistance

• Simulation of interdiffusion across an interface:

• Rate of vacancy diffusion depends on: --vacancy concentration --frequency of jumping.

(Courtesy P.M. Anderson)

DIFFUSION SIMULATION

Interstitial Diffusion

•  Interstitial diffusion – smaller atoms (H, C, O, N) can diffuse between atoms.

More rapid than vacancy diffusion due to more mobile small atoms and more empty interstitial sites.

(Courtesy P.M. Anderson)

• Applies to interstitial impurities. • More rapid than vacancy diffusion. • Simulation: --shows the jumping of a smaller atom (gray) from one interstitial site to another in a BCC structure. The interstitial sites considered here are at midpoints along the unit cell edges.

INTERSTITIAL SIMULATION

• Case Hardening: --Diffuse carbon atoms into the host iron atoms at the surface. --Example of interstitial diffusion is a case hardened gear.

• Result: The "Case" is --hard to deform: C atoms "lock" planes from shearing. --hard to crack: C atoms put the surface in compression.

Fig. 5.0, Callister 6e. (Fig. 5.0 is courtesy of Surface Division, Midland-Ross.)

PROCESSING USING DIFFUSION

Diffusion •  How do we quantify the rate of diffusion?

J ≡ Flux ≡ moles (or mass) diffusingsurface area( ) time( )

=molm2s

or kgm2s

•  Measured empirically –  Make thin film (membrane) of known surface area –  Impose concentration gradient –  Measure how fast atoms or molecules diffuse through the

membrane

Rate of diffusion is independent of time; the diffusion flux does not change with time.

The concentration profile shows the concentration (C) vs the position within the solid (x); the slope at a particular point is the concentration gradient.

Steady-state diffusion across a thin plate

Steady-State Diffusion

dxdC

DJ −=

Fick’s first law of diffusion C1

C2

x

C1

C2

x1 x2 D ≡ diffusion coefficient

12

12 linear ifxxCC

xC

dxdC

−

−=

Δ

Δ≅

Cross-sectional area dndt= −DA dC

dxNo. of atoms

crossing area A per unit time Concentration gradient

Matter transport is down the concentration gradient

Diffusion coefficient/ diffusivity

A Flow direction

J = −D dCdx

Diffusion and Temperature

• Diffusion coefficient increases with increasing T.

D = Do exp ⎛ ⎝ ⎜

⎞ ⎠ ⎟

- Qd R T

= pre-exponential [m2/s] = diffusion coefficient [m2/s]

= activation energy [J/mol or eV/atom] = gas constant [8.314 J/mol-K] = absolute temperature [K]

D Do

Qd

R T

•  The diffusing species, host material and temperature influence the diffusion coefficient.

•  For example, there is a significant difference in magnitude between self-diffusion and carbon interdiffusion in α iron at 500 °C.

Factors that influence diffusion

Nonsteady State Diffusion •  The concentration of diffusing species is a

function of both time and position C = C(x,t). In this case, Fick’s Second Law is used.

2

2

xC

DtC

∂

∂=

∂

∂Fick’s Second Law

Non-steady State Diffusion • Copper diffuses into a bar of aluminum.

pre-existing conc., Co of copper atoms Surface conc., C of Cu atoms bar s

• General solution:

"error function”

Co

Cs

position, x

C(x,t)

tot1

t2t3 Adapted from

Fig. 5.5, Callister 6e.

Factors that Influence Diffusion

Ø Temperature - diffusion rate increases very rapidly with increasing temperature

Ø Diffusion mechanism - interstitial is usually faster than vacancy

Ø Diffusing and host species - Do, Qd is different for every solute, solvent pair

Ø Microstructure - diffusion faster in polycrystalline vs. single crystal materials because of the rapid diffusion along grain boundaries and dislocation cores.

Diffusion FASTER for... • open crystal structures • lower melting T materials • materials w/secondary bonding • smaller diffusing atoms • cations • lower density materials

Diffusion SLOWER for... • close-packed structures • higher melting T materials • materials w/covalent bonding • larger diffusing atoms • anions • higher density materials

SUMMARY: STRUCTURE & DIFFUSION