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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 • The phenomenon of material transport by which atomic diffusion occurs. Chapter 5 -2 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 -3 DIFFUSION: THE PHENOMENA (1) Interdiffusion: •Atoms of one metal diffuse into another. •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 6e. 100% 0 Concentration Profiles Chapter 5 -4 DIFFUSION: THE PHENOMENA (2) Self-diffusion: •All atoms exchanging positions are of the same type. •In an elemental solid, atoms also migrate. Label some atoms After some time C A D B Chapter 5 -5 DIFFUSION MECHANISMS For an atom to make a move, two conditions must be met 1. there must be and empty adjacent site 2. the atom must have sufficient energy to break bonds with its neighbor atoms and then cause some lattice distortion during the displacement Chapter 5 -6 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 -7 DIFFUSION MECHANISMS Substitutional Diffusion: • applies to substitutional impurities • atoms exchange with vacancies • rate depends on: --number of vacancies --activation energy to exchange. Chapter 5 -8 DIFFUSION MECHANISMS Vacancy Diffusion: • involves the interchange of an atom from a normal lattice position Chapter 5 -9 DIFFUSION MECHANISMS Interstitial Diffusion: • involves the atoms that migrate from an interstitial position to a neighboring one that is empty Position of interstitial atom before diffusion Position of interstitial atom after diffusion Chapter 5 -10 INTERSTITIAL SIMULATION • 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. (Courtesy P.M. Anderson) Chapter 5 -11 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 6e. (Fig. 5.0 is courtesy of Surface Division, MidlandRoss.) • 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 -12 PROCESSING USING DIFFUSION (2) • Doping Silicon with P for n-type semiconductors: • Process: 1. Deposit P rich layers on surface. silicon Fig. 18.0, 2. Heat it. Callister 6e. 3. Result: Doped semiconductor regions. silicon Chapter 5 -13 DOPING • What is Doping? • Doping is the process of adding some impurity atoms in the semi conductor. These impurity atoms are known as dopants. After addition of these dopants some of the properties of the conductors can be changed according to our need. Chapter 5 -14 DOPING Basic conditions that are required for the doping process are given below: 1. The atom which is to be doped in the crystal must be placed at the position same as that of the position of the semiconductor atom. 2. There should be no distortion in the crystal after the insertion of the dopants. 3. The size of the dopants should be exactly same as that of the size of the atom of the crystal. 4. In a crystal the percentage of doping should not be more than one percent. Chapter 5 -15 DOPING Some basic doping techniques of doping are shown below: 1. First we have to heat up the semiconductor crystal. The heating must be at the place where the dopants are present in the atmosphere. After heating the diffusion of the dopants will take place in the lattice site of the crystal. 2. In the second method of doping, the semiconductor is bombarded with the ions of the semiconductor itself. Then the dopants are embedded. Chapter 5 -16 MODELING DIFFUSION: FLUX • Flux: • Directional Quantity • Flux can be measured for: --vacancies --host (A) atoms --impurity (B) atoms Chapter 5 -17 CONCENTRATION PROFILES & FLUX • Concentration Profile, C(x): [kg/m3] Cu flux Ni flux Concentration of Cu [kg/m3] Concentration of Ni [kg/m3] Adapted from Fig. 5.2(c), Callister 6e. • Fick's First Law: Position, x • The steeper the concentration profile, the greater the flux! Chapter 5 -18 STEADY STATE DIFFUSION • Steady State: the concentration profile doesn't change with time. dC • Apply Fick's First Law: J x = -D dx æ dC ö æ dC ö =ç ÷ • If Jx)left = Jx)right , then ç ÷ è dx ø left è dx ø right • Result: the slope, dC/dx, must be constant (i.e., slope doesn't vary with position)! Chapter 5 -19 SAMPLE PROBLEM S1 A plate of iron is exposed to a carburizing (carbonrich) atmosphere on one side and decarburizing (carbon-deficient) atmosphere on the other side at 700oC (1300oF). If a condition of steady state is achieved, calculate the diffusion flux of carbon through the plate if the concentration of carbon at positions of 5 and 10 mm (5 x10-3 and 10-2 m) beneath the carburizing surface are 1.2 and 0.8 kg/m3, respectively. Assume a diffusion coefficient of 3 x 10-11 m2/s at this temperature Chapter 5 -20 EX: STEADY STATE DIFFUSION • Steel plate at 700C with geometry shown: Adapted from Fig. 5.4, Callister 6e. • Q: How much carbon transfers from the rich to the deficient side? Chapter 5 -21 NON STEADY STATE DIFFUSION • Concentration profile, C(x), changes w/ time. • To conserve matter: • Fick's First Law: • Governing Eqn.: Chapter 5 -22 TABULATION OF ERROR FUNCTION VALUES z erf (z) z erf (z) z erf (z) 0 0 0.55 0.5633 1.3 0.9340 0.025 0.0282 0.60 0.6039 1.4 0.9523 0.05 0.0564 0.65 0.6420 1.5 0.9661 0.10 0.1125 0.70 0.6778 1.6 0.9763 0.15 0.1680 0.75 0.7112 1.7 0.9838 0.20 0.2227 0.80 0.7421 1.8 0.9891 0.25 0.2763 0.85 0.7707 1.9 0.9928 0.30 0.3286 0.90 0.7970 2.0 0.9953 0.35 0.3794 0.95 0.8209 2.2 0.9981 0.40 0.4284 1.0 0.8427 2.4 0.9993 0.45 0.1125 1.1 0.8802 2.6 0.9998 0.50 0.5205 1.2 0.9103 2.8 0.9999 Chapter 5 -23 SAMPLE PROBLEM U1 Consider one such alloy that initially has a uniform carbon concentration of 0.25 wt% and is to be treated at 950oC (1750oF). If the concentration of carbon at the surface is suddenly brought to and maintained at 1.20 wt%, how long will it take to achieve a carbon content of 0.80 wt% at a position of 0.5 mm below the surface? The diffusion coefficient for carbon in iron at this temperature is 1.6 x 10-11 m2/s; assume that the steel piece is semi-infinite. Chapter 5 -24 SAMPLE PROBLEM U2 The diffusion coefficients for copper in aluminum at 500 and 600oC are 4.8 x10-14 and 5.3 x 10-13 m2/s, respectively. Determine the approximate time at 500oC that will produce the same diffusion result (in terms of concentration of Cu at some specific point in Al) as a 10-h heat treatment at 600oC. Chapter 5 -25 EX: NON STEADY STATE DIFFUSION • Copper diffuses into a bar of aluminum. Cs C(x,t) to t1 Co t3 t2 Adapted from Fig. 5.5, Callister 6e. position, x • General solution: "error function" Values calibrated in Table 5.1, Callister 6e. Chapter 5 -26 PROCESSING QUESTION • Copper diffuses into a bar of aluminum. • 10 hours at 600C gives desired C(x). • How many hours would it take to get the same C(x) if we processed at 500C? Key point 1: C(x,t500C) = C(x,t600C). 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 -27 DIFFUSION DEMO: ANALYSIS • The experiment: we recorded combinations of t and x that kept C constant. æ ö C(xi , ti )- C o x i ÷ = 1- erf çç = (constant here) ÷ Cs - Co è 2 Dti ø • Diffusion depth given by: Chapter 5 -28 DATA FROM DIFFUSION DEMO • Experimental result: x ~ t0.58 • Theory predicts x ~ t0.50 • Reasonable agreement! Chapter 5 -29 DIFFUSION AND TEMPERATURE • Diffusivity increases with T. • Experimental Data: D has exp. dependence on T Recall: Vacancy does also! Dinterstitial >> Dsubstitutional Cu in Cu C in a-Fe Al in Al C in g-Fe Fe in a-Fe Fe in g-Fe Zn in Cu Adapted from Fig. 5.7, Callister 6e. (Date for Fig. 5.7 taken from E.A. Brandes and G.B. Brook (Ed.) Smithells Metals Reference Book, 7th ed., Butterworth-Heinemann, Oxford, 1992.) Chapter 5 -30 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 • anions • lower density materials • higher density materials Chapter 5 -31