Chapter 5: Diffusion in Solids. Diffusion

Transcription

1 /8/0 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 - Diffusion Diffusion - Mass transport by atomic motion Mechanisms Gases & Liquids random (Brownian) motion Solids vacancy diffusion or interstitial diffusion Chapter 5 -

2 /8/0 Diffusion Interdiffusion: In an alloy, atoms tend to migrate from regions of high conc. to regions of low conc. Initially Cu Ni After some time Adapted from Figs. 5. and 5., Callister 7e. Chapter 5-3 Self-diffusion: In an elemental solid, atoms also migrate. Label some atoms D C A B Diffusion After some time Diffusion is just a stepwise migration of atoms from lattice site to lattice site. There must be an empty adjacent site. The atom must have sufficient energy to break bonds with its neighbor atoms and them cause some lattice distortion during the displacement. C A B D Chapter 5-4

3 /8/0 Vacancy Diffusion: Diffusion Mechanisms atoms echange with vacancies applies to substitutional impurities atoms rate depends on: --number of vacancies --activation energy to echange. increasing elapsed time Chapter 5-5 Diffusion Simulation Simulation of interdiffusion across an interface: Rate of substitutional diffusion depends on: --vacancy concentration --frequency of jumping. (Courtesy P.M. Anderson) Chapter 5-6 3

4 /8/0 Diffusion Mechanisms Interstitial diffusion smaller atoms can diffuse between atoms. Adapted from Fig. 5.3 (b), Callister 7e. More rapid than vacancy diffusion Chapter 5-7 Processing Using Diffusion Case Hardening: --Diffuse carbon atoms into the host iron atoms at the surface. --Eample of interstitial diffusion is a case hardened gear. Adapted from chapter-opening photograph, Chapter 5, Callister 7e. (Courtesy of Surface Division, Midland-Ross.) Result: The presence of C atoms makes iron (steel) harder. Chapter 5-8 4

5 /8/0 Processing Using Diffusion Doping silicon with phosphorus for n-type semiconductors: Process: 0.5mm. Deposit P rich layers on surface.. Heat it. silicon 3. Result: Doped semiconductor regions. magnified image of a computer chip light regions: Si atoms silicon light regions: Al atoms Adapted from chapter-opening photograph, Chapter 8, Callister 7e. Chapter 5-9 Diffusion How do we quantify the amount or rate of diffusion? moles (or mass) diffusing mol kg J Flu = or surface area time cm s m Measured empirically Make thin film (membrane) of known surface area Impose concentration gradient Measure how fast atoms or molecules diffuse through the membrane ( )( ) s M J = = At l dm A dt M = mass diffused time J slope Chapter 5-0 5

6 /8/0 Steady-State Diffusion Rate of diffusion independent of time Flu proportional to concentration gradient = dc d C C Fick s first law of diffusion C C dc J = D d if linear dc d D diffusion coefficient C C = C Driving Force---concentration gradient Chapter 5 - Steady-state diffusion ("$& ("$& #"*! ("$&)("$& J = D dc d dc = dc +( " & )( " & d d left right!" # $ % &' ", - Chapter 5-6

7 /8/0 Eample: 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 flu of methylene chloride through the glove? Data: diffusion coefficient in butyl rubber: D = 00-8 cm /s surface concentrations: C = 0.44 g/cm 3 C = 0.0 g/cm 3 Chapter 5-3 Eample (cont). Solution assuming linear conc. gradient C paint remover glove t b = 6D C skin Data: J = - D dc d C D (0.0 g/cm 0.44 g/cm ) -5 g J = (0 0 cm /s) =.6 0 (0.04 cm) cm s D = 00-8 cm /s C = 0.44 g/cm 3 C = 0.0 g/cm 3 = 0.04 cm C Chapter 5-4 7

8 /8/0 Factors That Influence Diffusion Diffusing Species & Temperature eample: carbon _ iron self-diffusion: D=30 - m /s interdiffusion: D=.40 - m /s Chapter 5-5 Diffusion and Temperature Diffusion coefficient increases with increasing T. D = D o ep Q d RT D D o Q d R T = diffusion coefficient [m /s] = pre-eponential [m /s] = activation energy [J/mol or ev/atom] = gas constant [8.34 J/mol-K] = absolute temperature [K] Chapter 5-6 8

9 /8/0 Diffusion and Temperature D has eponential dependence on T T( C) D (m /s) 0-4 Dinterstitial >> Dsubstitutional C in α-fe C in γ-fe Al in Al Fe in α-fe Fe in γ-fe K/T Adapted from Fig. 5.7, Callister 7e. (Date for Fig. 5.7 taken from E.A. Brandes and G.B. Brook (Ed.) Smithells Metals Reference Book, 7th ed., Butterworth-Heinemann, Oford, 99.) Chapter 5-7 Eample: At 300ºC the diffusion coefficient and activation energy for Cu in Si are D(300ºC) = m /s Q d = 4.5 kj/mol What is the diffusion coefficient at 350ºC? D transform data ln D Temp = T Q d lnd = lnd0 and R T D Qd lnd lnd = ln = D R T lnd T = lnd 0 /T Qd R T Chapter 5-8 9

10 /8/0 Eample (cont.) D = D Qd ep R T T T = = 573K T = = 63K D 4,500 J/mol = (7.8 0 m /s) ep 8.34 J/mol -K 63 K 573 K D = m /s Chapter 5-9 Concept Check Rank the magnitudes of the diffusion coefficients from greatest to least for the following systems: N in Fe at 700 o C Cr in Fe at 700 o C N in Fe at 900 o C Cr in Fe at 900 o C Chapter 5-0 0

11 /8/0 Non-steady State Diffusion The concentration of diffucing species is a function of both time and position C = C(,t) In this case Fick s Second Law is used Fick s Second Law C t C = D Chapter 5 - Non-steady State Diffusion Copper diffuses into a bar of aluminum. Surface conc., C of Cu atoms bar s pre-eisting conc., C o of copper atoms Cs Adapted from Fig. 5.5, Callister 7e. B.C. at t = 0, C = C o for 0 at t > 0, C = C S for = 0 (const. surf. conc.) C = C o for = Chapter 5 -

12 /8/0 Solution: (,t ) C C s C C o o = erf Dt C(,t) = Conc. at point at time t erf (z) = error function = e y 0 π z dy erf(z) values are given in Table 5. C S C(,t) C o Chapter 5-3 interpolation Chapter 5-4

13 /8/0 Non-steady State Diffusion Sample Problem: An FCC iron-carbon alloy initially containing 0.0 wt% C is carburized at an elevated temperature and in an atmosphere that gives a surface carbon concentration constant at.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 C(, t) C C C s o o = erf Dt Chapter 5-5 Solution (cont.): C(,t ) C C C s o o = erf Dt t = 49.5 h = m C = 0.35 wt% C s =.0 wt% C o = 0.0 wt% C(, t) Co Cs Co = = erf Dt = erf( z) erf(z) = 0.85 Chapter 5-6 3

14 /8/0 Solution (cont.): We must now determine from Table 5. the value of z for which the error function is An interpolation is necessary as follows z erf(z) z z = z = 0.93 Now solve for D z = Dt D = 4z t 3 (4 0 m) D = = 4 z t (4)(0.93) (49.5 h) h 3600 s =.6 0 m /s Chapter 5-7 Solution (cont.): To solve for the temperature at which D has above value, we use a rearranged form of Equation (5.9a); from Table 5., for diffusion of C in FCC Fe Q T = d R( lnd lnd) D o = m /s Q d = 48,000 J/mol o T 48,000 J/mol = 5 (8.34 J/mol -K)(ln.30 m /s ln.60 m /s) T = 300 K = 07 C Chapter 5-8 4

15 /8/0 Eample Problem 5. and 5.3 Consider one Fe-C alloy that has a uniform carbon concentration of 0.5 wt% and is to be treated at 950 o C. If the concentration of C at the surface is suddenly brought to and maintained at.wt%, how long will it take to achieve a carbon content of 0.8wt% at a position 0.5mm below the surface? The diffusion coefficient for C in Fe at this temperature is.60 - m /s; assume that the steel piece is semi-infinite. The diffusion coefficient for copper in aluminum at 500 and 600 o C are and m /s, respectively. Determine the approimate time at 500 o C that will produce the same diffusion result ( in terms of concentration of Cu at some specific point in Al) as a 0-h heat treatment at 600 o C. Chapter : Solution Erf(z)=0.40, z=0.39, t=7. 5.3: Dt=constant t 500 =0.4h Chapter

16 /8/0 Summary Diffusion FASTER for... open crystal structures materials w/secondary bonding smaller diffusing atoms lower density materials Diffusion SLOWER for... close-packed structures materials w/covalent bonding larger diffusing atoms higher density materials Chapter 5-3 6