Foundations of Materials Science and Engineering ... - Introduction

Foundations of Materials Science and Engineering Lecture Note 8 (Chapter 5 Diffusion) May 13, 2013

Kwang Kim Yonsei University [email protected] 39

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Introduction

Processing Using Diffusion

• Furnace for heat treating steel using carburization. • Carburizing is the addition of carbon to the surface of low-carbon steels at temperatures ranging from 1560°F to 1740°F. • Hardening is achieved when a high carbon martensitic case with good wear and fatigue resistance is superimposed on a tough, low-carbon steel core.

Introduction

Processing Using Diffusion • Case hardening or surface hardening is the process of hardening the surface of a metal, often a low carbon steel, by diffusing elements into the material's surface, forming a thin layer of a harder alloy. • Carbon atoms diffuse into the iron lattice atoms at the surface. • This is an example of interstitial diffusion. • The C atoms make iron (steel) harder. • Result: The presence of C atoms makes iron (steel) harder.

Introduction

Processing Using Diffusion

“Carbide band saw blade can cut through case hardened materials.”

Introduction

Doping by Diffusion • Integrated circuits (ICs), found in numerous electronic devices have been fabricated using doping techniques. • The base material for these ICs is silicon that has been “doped” with other materials. • More precisely, controlled concentrations of impurities have been diffused into specific regions of the device to change the properties (improve electrical conductivity).

Introduction • Doping silicon with phosphorus for n-type semiconductors: 0.5 mm

1. Deposit P rich layers on surface. magnified image of a computer chip silicon

2. Heat it. 3. Result: Doped semiconductor regions.

silicon

light regions: Si atoms

light regions: Al atoms Adapted from Figure 18.27, Callister & Rethwisch 8e.

Introduction

Diffusion - Phenomenon of material/matter transport by atomic motion Often think of diffusion in a medium where atoms are relatively free to move around such as liquids and gases… • Materials processing Many materials are heat treated to achieve necessary properties: prefer to develop methods that will allow relatively high diffusion rates to achieve an efficient/cost-effectively process (e.g. alloying, case hardening etc…) • Mechanisms Gases & Liquids – random (Brownian) motion Solids – vacancy diffusion or interstitial diffusion

Introduction

Diffusion Diffusion - Mass transport by atomic motion. Diffusion is a consequence of the constant thermal motion of atoms, molecules and particles that results in material moving from areas of high to low concentration. Mechanisms • Brownian motion is the seemingly random movement of particles suspended in a liquid or gas. • Solids – vacancy diffusion or interstitial diffusion.

Introduction

DIFFUSION DEMO • Glass tube filled with water. • At time t = 0, add some drops of ink to one end of the tube. • Measure the diffusion distance, x, over some time.

to

x (mm)

t1 t2 t3 xo

x1

x2 x3

time (s)

Diffusion

Diffusion • Interdiffusion: In an alloy, atoms tend to migrate from regions of high conc. to regions of low conc. Initially

After some time

Adapted from Figs. 5.1 and 5.2, Callister & Rethwisch 8e.

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

C C D

A A D B B

Diffusion Mechanism • •

•

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

Diffusion Mechanism Vacancy Diffusion: • atoms exchange with vacancies • applies to substitutional impurities atoms • rate depends on: -- number of vacancies -- activation energy to exchange. -- frequency of jumping

increasing elapsed time

Diffusion Mechanism

ACTIVATION ENERGY FOR DIFFUSION Initial state

Energy

Intermediate state

Final state

Activation energy

• Also called energy barrier for diffusion

Diffusion Mechanism

Energy

Atom needs enough thermal energy to break bonds and squeeze through its neighbors. Energy needed energy barrier Called the activation energy Em (like Q)

Em

Vacancy

Atom

Distance

Diagram for Vacancy Diffusion Diffusion Thermally Activated Process

Diffusion Mechanism

• 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.

Diffusion Mechanism

PROCESSING USING DIFFUSION (1) • Case Hardening: -- Example of interstitial diffusion is a case hardened gear. -- Diffuse carbon atoms into the host iron atoms at the surface.

• 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, MidlandRoss.)

Introduction • Doping silicon with phosphorus for n-type semiconductors: 0.5 mm

1. Deposit P rich layers on surface. magnified image of a computer chip silicon

2. Heat it. 3. Result: Doped semiconductor regions.

silicon

light regions: Si atoms

light regions: Al atoms Adapted from Figure 18.27, Callister & Rethwisch 8e.

Diffusion Flux

Diffusion Flux

Diffusion Flux

Diffusion Flux How do we quantify the rate of diffusion?

moles (or mass) diffusing mol kg J  Flux   or 2 surface area time  cm s m2s • Measured empirically – Make thin film (membrane) of known surface area – Impose concentration gradient – Measure how fast atoms or molecules diffuse through the membrane

J

M l dM  At A dt

M= mass diffused

J  slope time

Steady State Diffusion

Steady-state diffusion across a thin plate 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 Steady State: the concentration profile doesn't change with time.

dC Flux proportional to concentration gradient = dx C1 C1

Fick’s first law of diffusion

C2 x1

x

C2

dC J  D dx

x2

dC C C2  C1 if linear   dx x x2  x1

D  diffusion coefficient D : independent of time

Steady State Diffusion Example: Chemical Protective Clothing (CPC) • Methylene chloride is a common ingredient of paint removers. Besides being an irritant, it also may be absorbed through skin. When using this paint remover, protective gloves should be worn. • If butyl rubber gloves (0.04 cm thick) are used, what is the diffusive flux of methylene chloride through the glove? • Data: – diffusion coefficient in butyl rubber: D = 110 x10-8 cm2/s – surface concentrations: C1 = 0.44 g/cm3 C2 = 0.02 g/cm3

Steady State Diffusion Example: Chemical Protective Clothing (CPC) • Solution – assuming linear conc. gradient glove C1 2 tb  6D paint skin remove C2 r x1 x2

J -D

dC C  C1  D 2 x2  x1 dx

Data:

D = 110 x 10-8 cm2/s C1 = 0.44 g/cm3 C2 = 0.02 g/cm3 x2 – x1 = 0.04 cm

3 3  g ( 0 . 02 g/cm 0 . 44 g/cm ) J   (110 x 10 -8 cm2 /s)  1.16 x 10 -5 (0.04 cm) cm2s

Diffusion Coefficient

• Diffusion coefficient increases with increasing T.  Qd   D  Do exp RT   D = diffusion coefficient [m2/s] Do = pre-exponential [m2/s] Qd = activation energy [J/mol or eV/atom] R = gas constant [8.314 J/mol-K] T = absolute temperature [K]

Diffusion Coefficient

DIFFUSION AND TEMPERATURE • Diffusivity increases with T.

diffusivity

D  Do

pre-exponential [m2/s] (see Table 5.2, Callister 6e) activation energy  Q  [J/mol],[eV/mol] exp  d  (see Table 5.2, Callister 6e )

 RT 

gas constant [8.31J/mol-K]

• Remember vacancy concentration: NV = N exp(-QV/kT) • QV is vacancy formation energy (larger this energy, smaller the number of vacancies) • Qd is the activation energy (larger this energy, smaller the diffusivity and lower the probability of atomic diffusion)

Diffusion Mechanism

ACTIVATION ENERGY FOR DIFFUSION Initial state

Energy

Intermediate state

Final state

Activation energy

• Also called energy barrier for diffusion

Diffusion Coefficient

Factors that influence diffusion

• 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.

Diffusion Coefficient

300

600

1000

10-8

1500

• Diffusion coefficient increases with increasing T. D has exponential dependence on T

D (m2/s)

T(C)

D interstitial C in -Fe C in -Fe

10-14

10-20 0.5

1.0

1.5

1000 K/T

>> D substitutional Al in Al Fe in -Fe Fe in -Fe

Diffusion Coefficient Example: At 300ºC the diffusion coefficient and activation energy for Cu in Si are D(300ºC) = 7.8 x 10-11 m2/s, Qd = 41.5 kJ/mol What is the diffusion coefficient at 350ºC?

transform ln D data

D

Temp = T

1   and  T2  Qd D2   lnD2  lnD1  ln D1 R

Qd lnD2  lnD0  R

1/T

Qd lnD1  lnD0  R  1 1     T2 T1 

1    T1 

Diffusion Coefficient

 Qd D2  D1 exp   R

 1 1      T2 T1 

T1 = 273 + 300 = 573 K T2 = 273 + 350 = 623 K

D2  (7.8 x 10

11

  41,500 J/mol  1 1   m /s) exp    8.314 J/mol - K  623 K 573 K  2

D2 = 15.7 x 10-11 m2/s

Non-steady State Diffusion

• The concentration of diffusing species is a function of both time and position C = C(x,t). More likely scenario than steady state. • In this case, Fick’s Second Law is used.

Fick’s Second Law

 2C C D 2 x t

Non-steady State Diffusion • Copper diffuses into a bar of aluminum. Surface conc., Cs of Cu atoms

bar pre-existing conc., Co of copper atoms

Cs

B.C.

at t = 0, C = Co for 0  x   at t > 0, C = CS for x = 0 (constant surface conc.) C = Co for x =  37

Non-steady State Diffusion C  x , t   Co  x   1  erf   Cs  Co  2 Dt 

C(x,t) = Conc. at point x at time t erf (z) = error function

2  



z 0

e

y 2

dy

CS

C(x,t) Co

• Diffusion depth given by:

x i  Dt i

Non-steady State Diffusion

Diffusion in ionic solids Which do you expect to diffuse faster, cations or anions? Smaller cations will usually diffuse faster. But analogous to charge neutrality required for defect formation, each ion will need counter charge to move with it (e.g. vacancy, impurity or free electrons or holes).

Diffusion in Solids Diffusion FASTER for...

Diffusion SLOWER for...

• open crystal structures

• close-packed structures

• lower melting T materials

• higher melting T materials

• materials w/secondary

• materials w/covalent

bonding

bonding

• smaller diffusing atoms

• larger diffusing atoms

• lower density materials

• higher density materials