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A. K. M. B. Rashid Professor, Department of MME BUET, Dhaka

MME131: Lecture 11

Atom and ion movement in solids Diffusion equations

Today’s Topics

l Rate of diffusion l Factors affecting diffusion l Diffusion in ionic compounds and polymers

References: 1. Callister. Materials Science and Engineering: An Introduction 2. Askeland. The Science and Engineering of Materials

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Rate of diffusion

o  Fick’s first law - The equation relating the flux of atoms by diffusion to the diffusion coefficient and the concentration gradient.

o  Diffusion coefficient (D) - A temperature-dependent coefficient related to the rate at which atoms, ions, or other species diffuse.

o  Concentration gradient - The rate of change of composition with distance in a nonuniform material, typically expressed as atoms/cm3.cm or at%/cm.

Fick’s first law for steady-state diffusion

CB

CA

xA xB position, x

conc

entra

tion

of

diffu

sing

spec

ies, C

ΔC Δx

CA - CB xA - xB

=

linear concentration profile for stead-state diffusion

thin metal plate

steady-state diffusion across a thin plate

area, A

direction of diffusion of gaseous species

gas at pressure PB

gas at pressure PA

PA > PB and constant

q Diffusion is a time-dependent process, where the quantity of element transported is a function of time.

q  Steady-state diffusion occurs when the rate of diffusion (or, concentration profile of diffusing element) does not change with time.

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J = - D dC dx

� D : diffusion coefficient for diffusing species in solid (m2/s)

q The rate of steady-state diffusion is expressed by the following Fick’s First Law:

minus sign denotes the flux is towards lower concentrations, i.e., “down the concentration gradient”

: concentration gradient (kg/m3 or atoms/m3), which is also expressed as the driving force for diffusion. The steeper the concentration gradient, the greater the flux.

� dC dx

� J : diffusion flux, or rate of diffusion, which is defined as the number or mass of atoms passing through a plane of unit area per unit time (kg /m2 s or atoms/m2 s)

J = 1 A

dM dt

M = mass A = area

Flux can be measured for: -- vacancies -- host (A) atoms -- impurity (B) atoms

Purification of hydrogen gas q  One side of a thin sheet of palladium metal is exposed to impure gas

composed of hydrogen and other impurity gaseous species such as nitrogen, oxygen, and water vapour.

q  The hydrogen selectively diffuse through the sheet to the opposing side, which is maintained at constant and lower hydrogen pressure.

Practical example of Fick’s first law (steady-state diffusion)

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Example Steady-state diffusion

A steel plate is exposed to a carburising (C-rich) atmosphere on one side and a decarburising (C deficient) atmosphere on the other side at 700 0C. If a condition of steady state is achieved, calculate the diffusion flux of C through the plate if the concentration of C at positions of 5 and 10 mm beneath the carburising surface are 1.2 and 0.8 kg/m3, respectively. Assume a diffusion coefficient of 3x10-11 m2/s at this temperature.

Given data: C1 = 1.2 kg/m3 X1 = 5 mm C2 = 0.8 kg/m3 X2 = 10 mm D = 3x10-11 m2/s

x1 x2

c1 c2

Carbon rich atmosphere Carbon deficient

atmosphere

steady state = straight line

C2 – C1 x2 – x1

J = - D

= 2.4 x10-9 kg/m2s

Example Diffusion of nickel in magnesium oxide

A 0.05 cm layer of magnesium oxide (MgO) is deposited between layers of nickel (Ni) and tantalum (Ta) to provide a diffusion barrier that prevents reactions between the two metals. At 1400oC, nickel ions are created and diffuse through the MgO ceramic to the tantalum. Determine the number of nickel ions that pass through the MgO per second. The diffusion coefficient of nickel ions in MgO is 9×10-12 cm2/s, and the lattice parameter of FCC nickel at 1400oC is 3.6×10-8 cm.

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The composition of nickel at the Ni / MgO interface is 100% Ni, or

SOLUTION

CNi/MgO = = 8.57x1022 atoms/cm3 4 Ni atoms/unit cell

(3.6x10–8 cm)3

The composition of nickel at the Ta / MgO interface is 0% Ni. Thus, the concentration gradient is:

0 – 8.57x102 atoms/cm3 0.05 cm

Δc Δx = = – 1.71x1024 atoms/cm3 cm

The flux of nickel atoms through the MgO layer is: Δc Δx

J = - D = (9x10–12 cm2/s) (-1.71x1024 atoms/cm3 cm) = 1.54x1013 Ni atoms/cm2 s

The total number of nickel atoms crossing the 2 cm × 2 cm interface per second is:

J = (1.54x1013 Ni atoms/cm2 s) (2 cm x 2 cm) = 6.16x1013 Ni atoms/s

� Describes the dynamic or transient (short time) diffusion of atoms

� Non-steady-state i.e., diffusion rate (or, the concentration profile) varies with time and at different points of solid

distance

conc

entra

tion

of

diffu

sion

spec

ies

t 0

t 1 < t2 < t 3

� Since the diffusion in solids is slow, diffusion in practice is almost always transient !

Fick’s second law for non-steady-state diffusion

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∂C ∂t =

� Fick’s second law describes non-steady-state diffusion

� Solution to this differential equation

Cs – Cx

Cs – C0 = erf

x 2√(Dt)

Cx is a function of dimensionless parameter x /√(Dt)

Cs – C0 Cx – C0

C0

Cx

Cs

distance from interface, x

conc

entra

tion,

C

x∂C ∂t

= D ∂2C ∂x2

∂J ∂x =

∂C ∂x

∂ ∂x

non-linear concentration profile for non-stead-state diffusion

The Error Function

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0

Z

erf (

Z)

z erf (z) z erf (z) 0.00 0.0000 0.70 0.6778 0.01 0.0113 0.75 0.7112 0.02 0.0226 0.80 0.7421 0.03 0.0338 0.85 0.7707 0.04 0.0451 0.90 0.7969 0.05 0.0564 0.95 0.8209 0.10 0.1125 1.00 0.8427 0.15 0.1680 1.10 0.8802 0.20 0.2227 1.20 0.9103 0.25 0.2763 1.30 0.9340 0.30 0.3286 1.40 0.9523 0.35 0.3794 1.50 0.9661 0.40 0.4284 1.60 0.9763 0.45 0.4755 1.70 0.9838 0.50 0.5205 1.80 0.9891 0.55 0.5633 1.90 0.9928 0.60 0.6039 2.00 0.9953 0.65 0.6420

erf (z) = e dy -y2 2 √π ∫

z

0

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Example Design of a carburizing treatment

The surface of a 0.1% C steel gears is to be hardened by carburizing. In gas carburizing, the steel gears are placed in an atmosphere that provides 1.2% C at the surface of the steel at a high temperature. Carbon then diffuses from the surface into the steel. For optimum properties, the steel must contain 0.45% C at a depth of 0.2 cm below the surface. Design a carburizing heat treatment that will produce these optimum properties. Assume that the temperature is high enough (at least 900oC) so that the iron has the FCC structure.

Cs – Cx

Cs – C0 = erf

x 2√(Dt)

0.6818 = erf 0.1 √(Dt)

SOLUTION

C0 = 0.10 % C Cs = 1.20 % C Cx = 0.45 % C x = 0.2 cm D = 1.6x10-11 m2/s

0.71 = 0.1 √(Dt)

Using erf table, and after interpolation

or, Dt = = 0.0198 cm2 0.1 0.71

2

Any combination of D and t whose product is 0.0198 cm2 will work.

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For carbon diffusing in FCC iron, the diffusing coefficient is related to temperature by the equation

D = D0 exp - Q RT

Using table for data

D = 0.23 exp = 0.23 exp - 138000 J/mol (8.314 J/mol K) (T K)

- 16598.5 T

Therefore, the temperature and time of the heat treatment are related by:

t = = = 0.0198 cm2

D cm2/s 0.0198

0.23 exp (-16598.5/T) 0.0861

exp (-16598.5 / T)

Some typical combination of temperature and time are :

t = 0.0861 exp (-16598.5 / T)

T = 900 C = 1173 K, then t = 120357.65 s = 33.43 h T = 1000 C = 1273 K, then t = 39602.15 s = 11.0 h T = 1100 C = 1373 K, then t = 15320.92 s = 4.26 h T = 1200 C = 1473 K, then t = 6742.96 s = 1.87 h

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Factors influencing diffusion

o  Temperature and the diffusion coefficient (D)

o  Diffusion mechanism – vacancy diffusion and interstitial diffusion

o  Types of diffusion – volume diffusion, grain boundary diffusion, surface diffusion

o  Time of diffusion

o  Concentration of diffusing species and composition of matrix

o  Microstructure of matrix

� The diffusion coefficient D is related to temperature by the Arrhenius equation

D = D0 exp (-Qd/RT)

Diffusion and temperature

� At low enough temperature (< 0.4Tm), the diffusion becomes insignificant. q  For this reason, most heat treatment

processes are carried out at high T.

D varies exponentially with T, so does the vacancies !!

In iron, diffusion coefficient increases by 6 order of magnitude when T increases from 500 to 900 C

� As the temperature is increased q  Thermal energy to overcome the

activation energy is increased

q  Diffusion coefficient and the flux of atoms are increased

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The diffusion coefficient D as a function of reciprocal temperature for some metals and ceramics.

Diffusion coefficients of ions in different oxides

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� Diffusion by interstitial faster than by vacancy mechanism due to

Diffusion mechanism

q  Smaller interstitial atoms

q  Larger interstitial sites than vacancies

Arrhenius plot of diffusivity data for some metallic systems

At 500C:

self-diffusion of Fe in α-Fe D = 3x10-21 m2/s

interdiffusion of C in α-Fe D = 2.4x10-12 m2/s

Diffusion types

q  Volume diffusion � Atoms move through crystal from

one lattice or interstitial sites to another

� Because of surrounding atoms, Q is large and D is relatively small

q  Grain boundary diffusion � Atoms diffuse more easily

because of poor atomic packing � Q is small

q  Surface diffusion � Easier due to presence of cracks

and fewer neighbours at the surface

� Q is small

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Time of diffusion

q  Diffusion requires time. The unit for diffusion flux is atoms/m2 s.

q  If a large number of atoms must diffuse to produce a uniform structure, long time diffusion is required, even at high temperature.

q  Time for diffusion may be reduced by using � High temperature � Smaller diffusion distance

q  If time is reduced to zero, � Some rather remarkable structures and properties are obtained � Example: steels quenched rapidly from high temperatures to prevent

diffusion form non-equilibrium structure, called martensite, which produces exceptional hardness

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Concentration gradient

q  Diffusion rate is proportional to the concentration gradient (Δc/Δx).

q  Initially diffusion flux is high due to a high concentration gradient and decreases gradually as the gradient is reduced.

q  An increased concentration of the the diffusing species at the matrix reduces the concentration gradient and, ultimately, the diffusion process.

Microstructure of material

q  Diffusion rate is diffusion faster in polycrystalline materials as compared to single crystals because of the accelerated diffusion along grain boundaries and dislocation cores.

Diffusion in ionic compounds. Anions can only enter other anion sites.

Diffusion in ionic compounds

q In ionic materials (e.g., ceramics), a diffusing ion can only enters a site having the same charge.

q In order to reach that site, the ion must physically squeeze past adjoining ions, pass a region of opposite charge, and move a relatively long distance.

q Consequently activation energies are higher and rates of diffusion are lower.

q  Cations, because they have given up the valence electrons, are smaller in size and have higher diffusivity. In NaCl, diffusion of chloride ions is about twice that for diffusion of sodium ions.

q  Diffusion of ions also cause transfer of charge and ionic conductivity As the temperature is increased, ions diffuse more rapidly, more charge is transferred and conductivity is increased.

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q  Diffusion of atoms can occur from one location to another along a long polymer chain. For this to occur, strong covalent bonds must be broken.

q  Diffusion of atoms, ions, or molecules between the polymer chain, however, is relatively easy.

q  Often, this is a concern. For example, (1) when air diffuse through polymer film used for packaging food, food may spoil (2) when air diffuse through rubber tube inside an automobile tire, the tire will deflate (3) when water molecule diffuse through polymer products, they can swell.

q  Diffusion of dye through synthetic polymer fabric is, however, necessary to cause uniform colourisation.

q  Diffusion through crystalline polymer is slower than through amorphous polymers, which have no long-range order, and consequently, have lower density.

Diffusion in polymers

Diffusion FASTER for... q open crystal structures q lower density materials q lower melting point materials q secondary bonded materials q smaller diffusing atoms q cations Diffusion SLOWER for...

q close-packed structures q higher density materials q higher melting point materials q ionic and covalent bonded materials q larger diffusing atoms q anions

In Summary ...

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Next Class MME131: Lecture 12

Introduction to properties