CHAPTER 5 DIFFUSION IN SOLIDS ISSUES TO ADDRESS

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CHAPTER 5: DIFFUSION IN SOLIDS ISSUES TO ADDRESS. . . • How does diffusion

CHAPTER 5: DIFFUSION IN SOLIDS ISSUES TO ADDRESS. . . • How does diffusion occur? • Why is it an important part of processing? • How can the rate of diffusion be predicted for some simple cases? • How does diffusion depend on structure and temperature? Chapter 5 - 1

Diffusion - Mass transport by atomic motion Mechanisms • Gases & Liquids – random

Diffusion - Mass transport by atomic motion Mechanisms • Gases & Liquids – random (Brownian) motion • Solids – vacancy diffusion or interstitial diffusion Chapter 5 -

Why Study Diffusion ? • Diffusion plays a crucial role in… – Alloying metals

Why Study Diffusion ? • Diffusion plays a crucial role in… – Alloying metals => bronze, silver, gold – Strengthening and heat treatment processes • Hardening the surfaces of steel – High temperature mechanical behavior – Phase transformations • Mass transport during FCC to BCC – Environmental degradation • Corrosion, etc. Chapter 5 -

DIFFUSION DEMO • Glass tube filled with water. • At time t = 0,

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. • Compare the results with theory. Chapter 5 - 2

How do atoms move in Solids ? Why do atoms move in Solids ?

How do atoms move in Solids ? Why do atoms move in Solids ? • Diffusion, simply, is atoms moving from one lattice site to another in a stepwise manner – Transport of material by moving atoms • Two conditions are to be met: – An empty adjacent site – Enough energy to break bonds and cause lattice distortions during displacement • What is the energy source ? – HEAT ! • What else ? – Concentration gradient ! Chapter 5 -

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

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 7 e. Chapter 5 -

DIFFUSION: THE PHENOMENA (1) • Interdiffusion: In an alloy, atoms tend to migrate from

DIFFUSION: THE PHENOMENA (1) • Interdiffusion: In an alloy, atoms tend to migrate from regions of large concentration. Initially After some time Adapted from Figs. 5. 1 and 5. 2, Callister 6 e. Chapter 5 - 3

DIFFUSION: THE PHENOMENA (2) • Self-diffusion: In an elemental solid, atoms also migrate. Label

DIFFUSION: THE PHENOMENA (2) • Self-diffusion: In an elemental solid, atoms also migrate. Label some atoms (use isotopes) After some time Chapter 5 - 4

More examples in 3 -D ! Chapter 5 -

More examples in 3 -D ! Chapter 5 -

Diffusion Mechanisms (I) Energy is needed to generate a vacancy, break bonds, cause distortions.

Diffusion Mechanisms (I) Energy is needed to generate a vacancy, break bonds, cause distortions. Provided by HEAT , k. T ! Atom moves in the opposite direction of the vacancy ! Chapter 5 -

Diffusion Mechanisms (II) Interstitial Diffusion Much faster than vacancy diffusion, why ? Smaller atoms

Diffusion Mechanisms (II) Interstitial Diffusion Much faster than vacancy diffusion, why ? Smaller atoms like B, C, H, O. Weaker interaction with the larger atoms. More vacant sites, no need to create a vacancy ! Chapter 5 -

Diffusion Mechanisms (III) Substitutional Diffusion: • applies to substitutional impurities • atoms exchange with

Diffusion Mechanisms (III) Substitutional Diffusion: • applies to substitutional impurities • atoms exchange with vacancies • rate depends on: --number of vacancies --activation energy to exchange. Chapter 5 - 5

PROCESSING USING DIFFUSION (1) • Case Hardening: --Diffuse carbon atoms into the host iron

PROCESSING USING DIFFUSION (1) • Case Hardening: --Diffuse carbon atoms into the host iron atoms at the surface. --Example of interstitial diffusion is a case hardened gear. Fig. 5. 0, Callister 6 e. (Fig. 5. 0 is courtesy of Surface Division, Midland. Ross. ) • Result: The "Case" is --hard to deform: C atoms "lock" planes from shearing. --hard to crack: C atoms put the surface in compression. Chapter 5 - 8

Processing Using Diffusion • Doping silicon with phosphorus for n-type semiconductors: 0. 5 mm

Processing Using Diffusion • Doping silicon with phosphorus for n-type semiconductors: 0. 5 mm • Process: 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 chapter-opening photograph, Chapter 18, Callister 7 e. Chapter 5 -

Diffusion • How do we quantify the amount or rate of diffusion? • Measured

Diffusion • How do we quantify the amount or rate of diffusion? • Measured empirically – Make thin film (membrane) of known surface area – Impose concentration gradient – Measure how fast atoms or molecules diffuse through the membrane M= mass diffused J slope time Chapter 5 -

MODELING DIFFUSION: FLUX RATE OF MATERIAL TRANSPORT • Diffusion Flux: Material • Directional Quantity

MODELING DIFFUSION: FLUX RATE OF MATERIAL TRANSPORT • Diffusion Flux: Material • Directional Quantity (anisotropy ? ) • Flux can be measured for: --vacancies --host (A) atoms --impurity (B) atoms Diffusion is a time-dependent process ! Chapter 5 - 10

CONCENTRATION PROFILES & FLUX • Concentration Profile, C(x): [kg/m 3] Adapted from Fig. 5.

CONCENTRATION PROFILES & FLUX • Concentration Profile, C(x): [kg/m 3] Adapted from Fig. 5. 2(c), Callister 6 e. • Fick's First Law: • The steeper the concentration profile, the greater the flux! Concentration gradient is the DRIVING FORCE ! Chapter 5 - 11

Concentration Gradient Chapter 5 -

Concentration Gradient Chapter 5 -

STEADY STATE DIFFUSION • Steady State: the concentration profile doesn't change with time. •

STEADY STATE DIFFUSION • Steady State: the concentration profile doesn't change with time. • Apply Fick's First Law: • If Jx)left = Jx)right , then Why is the minus sign ? • Result: the slope, d. C/dx, must be constant (i. e. , slope doesn't vary with position)! Chapter 5 - 12

EX: STEADY STATE DIFFUSION • Steel plate at 700º C Adapted from Fig. 5.

EX: STEADY STATE DIFFUSION • Steel plate at 700º C Adapted from Fig. 5. 4, Callister 6 e. • Q: How much carbon transfers from the rich to the deficient side? Chapter 5 - 13

Example: Chemical Protective Clothing (CPC) • Methylene chloride is a common ingredient of paint

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 x 10 -8 cm 2/s – surface concentrations: C 1 = 0. 44 g/cm 3 C 2 = 0. 02 g/cm 3 Chapter 5 -

Example (cont). • Solution – assuming linear conc. gradient glove C 1 paint remover

Example (cont). • Solution – assuming linear conc. gradient glove C 1 paint remover skin C 2 x 1 x 2 Data: D = 110 x 10 -8 cm 2/s C 1 = 0. 44 g/cm 3 C 2 = 0. 02 g/cm 3 x 2 – x 1 = 0. 04 cm Chapter 5 -

Temperature Dependency ! What is the probability to find a vacancy at a nearest

Temperature Dependency ! What is the probability to find a vacancy at a nearest site ? Atom has to break bonds and “squeeze” thru => activation energy, Em ≈ 1 e. V. E N I B M CO Chapter 5 -

Temperature Effect ! The diffusion depends on temperature because; a- # of vacancies in

Temperature Effect ! The diffusion depends on temperature because; a- # of vacancies in the vicinity b- thermally activated successful jumps Chapter 5 -

Diffusion and Temperature • Diffusion coefficient increases with increasing T. æ Qd ö ÷

Diffusion and Temperature • Diffusion coefficient increases with increasing T. æ Qd ö ÷ D = Do expçRT è ø D = diffusion coefficient [m 2/s] Do = pre-exponential [m 2/s] Qd = activation energy [J/mol or e. V/atom] R = gas constant [8. 314 J/mol-K] T = absolute temperature [K] Chapter 5 -

Diffusion and Temperature 300 600 10 -8 1500 D has exponential dependence on T

Diffusion and Temperature 300 600 10 -8 1500 D has exponential dependence on T T( C) C in D (m 2/s) Ci g- na Fe in Fe Al 1. 0 Al a- e g-F 0. 5 C in a-Fe C in g-Fe in in 10 -20 Dinterstitial >> Dsubstitutional Fe Fe 10 -14 -Fe 1. 5 Al in Al Fe in a-Fe Fe in g-Fe 1000 K/T Adapted from Fig. 5. 7, Callister 7 e. (Date for Fig. 5. 7 taken from E. A. Brandes and G. B. Brook (Ed. ) Smithells Metals Reference Book, 7 th ed. , Butterworth-Heinemann, Oxford, 1992. ) Chapter 5 -

DIFFUSION AND TEMPERATURE • Diffusivity increases with T. • Experimental Data: D has exp.

DIFFUSION AND TEMPERATURE • Diffusivity increases with T. • Experimental Data: D has exp. dependence on T Recall: Vacancy does also! Adapted from Fig. 5. 7, Callister 6 e. (Date for Fig. 5. 7 taken from E. A. Brandes and G. B. Brook (Ed. ) Smithells Metals Reference Book, 7 th ed. , Butterworth-Heinemann, Oxford, 1992. ) Chapter 5 - 19

Example: At 300ºC the diffusion coefficient and activation energy for Cu in Si are

Example: At 300ºC the diffusion coefficient and activation energy for Cu in Si are D(300ºC) = 7. 8 x 10 -11 m 2/s Qd = 41. 5 k. J/mol æ Qd ö ÷ D = Do expçè RT ø What is the diffusion coefficient at 350ºC? transform data D Temp = T ln D 1/T Chapter 5 -

Example (cont. ) T 1 = 273 + 300 = 573 K T 2

Example (cont. ) T 1 = 273 + 300 = 573 K T 2 = 273 + 350 = 623 K D 2 = 15. 7 x 10 -11 m 2/s Chapter 5 -

Fick’s Second Law ; Non-steady state Diffusion • In most practical cases, J (flux)

Fick’s Second Law ; Non-steady state Diffusion • In most practical cases, J (flux) and d. C/dx (concentration gradient) change with time (t). – Net accumulation or depletion of species diffusing • How do we express a time dependent concentration? Concentration at a point x Changing with time ? Flux, J, changes at any point x ! Chapter 5 -

How do we solve this partial differential equation ? • Use proper boundary conditions:

How do we solve this partial differential equation ? • Use proper boundary conditions: – t=0, C = C 0, at 0 ≤ x ≤ ∞ – t>0, C = Cs, at x = 0 C = C 0, at x = ∞ Chapter 5 -

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

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 Adapted from Fig. 5. 5, Callister 7 e. B. C. at t = 0, C = Co for 0 x at t > 0, C = CS for x = 0 (const. surf. conc. ) C = Co for x = Chapter 5 -

Solution: C(x, t) = Conc. at point x at time t erf (z) =

Solution: C(x, t) = Conc. at point x at time t erf (z) = error function CS C(x, t) erf(z) values are given in Table 5. 1 Co Chapter 5 -

Non-steady State Diffusion • Sample Problem: An FCC iron-carbon alloy initially containing 0. 20

Non-steady State Diffusion • Sample Problem: An FCC iron-carbon alloy initially containing 0. 20 wt% C is carburized at an elevated temperature and in an atmosphere that gives a surface carbon concentration constant at 1. 0 wt%. If after 49. 5 h the concentration of carbon is 0. 35 wt% at a position 4. 0 mm below the surface, determine the temperature at which the treatment was carried out. • Solution: use Eqn. 5. 5 Chapter 5 -

Solution (cont. ): – t = 49. 5 h – Cx = 0. 35

Solution (cont. ): – t = 49. 5 h – Cx = 0. 35 wt% – Co = 0. 20 wt% x = 4 x 10 -3 m Cs = 1. 0 wt% erf(z) = 0. 8125 Chapter 5 -

Solution (cont. ): We must now determine from Table 5. 1 the value of

Solution (cont. ): We must now determine from Table 5. 1 the value of z for which the error function is 0. 8125. An interpolation is necessary as follows z erf(z) 0. 90 z 0. 95 0. 7970 0. 8125 0. 8209 z = 0. 93 Now solve for D Chapter 5 -

Solution (cont. ): • To solve for the temperature at which D has above

Solution (cont. ): • To solve for the temperature at which D has above value, we use a rearranged form of Equation (5. 9 a); from Table 5. 2, for diffusion of C in FCC Fe Do = 2. 3 x 10 -5 m 2/s Qd = 148, 000 J/mol T = 1300 K = 1027°C Chapter 5 -

Example: Chemical Protective Clothing (CPC) • Methylene chloride is a common ingredient of paint

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 breakthrough time (tb), i. e. , how long could the gloves be used before methylene chloride reaches the hand? • Data (from Table 22. 5) – diffusion coefficient in butyl rubber: D = 110 x 10 -8 cm 2/s Chapter 5 -

Example (cont). • Solution – assuming linear conc. gradient glove C 1 Equation 22.

Example (cont). • Solution – assuming linear conc. gradient glove C 1 Equation 22. 24 paint remover skin Given in web chapters ! C 2 x 1 x 2 D = 110 x 10 -8 cm 2/s Time required for breakthrough ca. 4 min Chapter 5 -

EX: NON STEADY STATE DIFFUSION • Copper diffuses into a bar of aluminum. Adapted

EX: NON STEADY STATE DIFFUSION • Copper diffuses into a bar of aluminum. Adapted from Fig. 5. 5, Callister 6 e. • General solution: æ x ö C(x, t ) - Co = ÷ 1 erf çè 2 Dt ø Cs - Co C (x, t) = concentration at any time and position ! “Gaussian error function" Chapter 5 - 15

NON STEADY STATE DIFFUSION • Concentration profile, C(x), changes w/ time. • To conserve

NON STEADY STATE DIFFUSION • Concentration profile, C(x), changes w/ time. • To conserve matter: • Fick's First Law: • Governing Eqn. : Chapter 5 - 14

DIFFUSION DEMO: ANALYSIS • The experiment: we recorded combinations of t and x that

DIFFUSION DEMO: ANALYSIS • The experiment: we recorded combinations of t and x that kept C constant. = (constant here) • Diffusion depth given by: Chapter 5 - 17

DATA FROM DIFFUSION DEMO • Experimental result: x ~ t 0. 58 • Theory

DATA FROM DIFFUSION DEMO • Experimental result: x ~ t 0. 58 • Theory predicts x ~ t 0. 50 • Reasonable agreement! Chapter 5 - 18

Example 5. 3 • Copper diffuses into a bar of aluminum. • 10 hours

Example 5. 3 • Copper diffuses into a bar of aluminum. • 10 hours at 600 C gives desired C(x). • How many hours would it take to get the same C(x) if we processed at 500 C? Key point 1: C(x, t 500 C) = C(x, t 600 C). Key point 2: Both cases have the same Co and Cs. • Result: Dt should be held constant. • Answer: Note: values of D are provided here. Chapter 5 - 16

Size Impact on Diffusion Smaller atoms diffuse faster Chapter 5 -

Size Impact on Diffusion Smaller atoms diffuse faster Chapter 5 -

Fast Tracks for diffusion ! eg. self-diffusion of Ag : -Areas where lattice is

Fast Tracks for diffusion ! eg. self-diffusion of Ag : -Areas where lattice is prestrained can allow for faster diffusion of atoms -Less energy is needed to distort an already strained lattice ! Chapter 5 -

Important • Temperature - diffusion rate increases with increasing temperature (WHY ? ) •

Important • Temperature - diffusion rate increases with increasing temperature (WHY ? ) • Diffusion mechanism – interstitials diffuse faster (WHY ? ) • Diffusing and host species - Do, Qd is different for every solute - solvent pair • Microstructure - grain boundaries and dislocation cores provide faster pathways for diffusing species, hence diffusion is faster in polycrystalline vs. single crystal materials. (WHY ? ) Chapter 5 -

SUMMARY: STRUCTURE & DIFFUSION Diffusion FASTER for. . . Diffusion SLOWER for. . .

SUMMARY: STRUCTURE & DIFFUSION Diffusion FASTER for. . . Diffusion SLOWER for. . . • open crystal structures • close-packed structures • lower melting T materials • higher melting T materials • materials w/secondary bonding • materials w/covalent bonding • smaller diffusing atoms • larger diffusing atoms • cations WHY ? • lower density materials • anions • higher density materials Chapter 5 - 20

ANNOUNCEMENTS Reading: Chapter 5 and Chapter 6 Go over the exercises in Chapter 5,

ANNOUNCEMENTS Reading: Chapter 5 and Chapter 6 Go over the exercises in Chapter 5, using your book and notes Core Problems: 5. 1, 5. 2, 5. 3, 5. 6, 5. 13, 5. 22, 5. 30 Self-help Problems (not bonus !): 5. 9 -5. 12, 5. 16 -5. 21, 5. 23 -5. 29 Due date: 03 -18 -2008 Chapter 5 - 0