Diffusion. Diffusion in Materials Q. How do changes in microstructure and chemical. Atomic Motion in Solids

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1 Diffusion Atomic Motion in Solids Week6 Material Sciences and Engineering MatE271 1 Diffusion in Materials Q. How do changes in microstructure and chemical composition actually occur? A. Atoms must be able to move around (diffusion) Diffusion occurs in solids, liquids and gases: Redistribution of non-uniform chemical species (impurity diffusion or interdiffusion) Random atomic movement can also occur in chemically uniform materials (self diffusion) Material Sciences and Engineering MatE271 Week 6 2

2 Diffusion is driven by Nonuniformity A B Time Temperature A B Concentration of A Concentration of A Distance (x) Concentration Profile Distance (x) Material Sciences and Engineering MatE271 Week 6 3 Interdiffusion forming a solid solution Initial Intermediate time Much longer time What is this time scale? Material Sciences and Engineering MatE271 Week 6 4 aterial Sciences and Engineering,

3 Diffusion? Diffusion is necessary for: Redistribution of chemical species Physical changes in microstructure Densification of powder compacts Deformation at high temperature (creep) Formation of solid state reaction products One kind of conductivity in ceramics (ionic) Material Sciences and Engineering MatE271 Week 6 5 Diffusion: Perfect crystal: Atoms would not move around because there would be no places for them to move to (all sites would be occupied)-- locked in place Point defects must be present in a crystal to permit atomic movement (diffusion) In a way, atomic diffusion is actually the movement of defects. Material Sciences and Engineering MatE271 Week 6 6

4 Diffusion Mechanisms I. Vacancy diffusion o Only adjacent atoms can move into a vacancy o Vacancy moves in opposite direction of atomic motion o Rate depends on concentration of vacancies Atomic flux Vacancy flux Material Sciences and Engineering MatE271 Week 6 7 Diffusion Mechanisms II. Interstitial Diffusion o Atom can move into any adjacent empty interstitial position (usually smaller atoms) o Rate depends on concentration of interstitial atoms (Usually faster than vacancy diffusion) Material Sciences and Engineering MatE271 Week 6 8 aterial Sciences and Engineering,

5 Diffusion Mechanisms o Would you expect vacancy or interstitial diffusion to be faster? o Why? Material Sciences and Engineering MatE271 Week 6 9 Diffusion occurs by random jumps After many random jumps by an atom, it s displacment can be calculated by the theory of random walks Net migration after n jump Material Sciences and Engineering MatE271 Week 6 10 aterial Sciences and Engineering,

6 Quantitative Description of Diffusion - The rate of diffusion is characterized by describing atomic fluxes at particular locations in the material - Critical quantities J = atomic flux (atoms/m 2 -s, kg/ m 2 -s) (dc/dx) = concentration gradient (atoms/m 4 ) D = diffusion coefficient (m 2 /s) c dc/dx area J x Fick s first law: J = - D (dc/dx) Material Sciences and Engineering MatE271 Week 6 11 Interrelating the quantities Fick s first law: J = - D (dc/dx) (negative sign indicates that the direction of diffusion flux is down the concentration gradient from high to low concentration) For steady state diffusion (local flux doesn t change with time), Fick s First Law can be solved directly Material Sciences and Engineering MatE271 Week 6 12

7 Example Hydrogen (H) gas can be purified by passing atomic hydrogen through a thin sheet of palladium (Pa) at 700 o C in a paladium diffusion cell. If the impure hydrogen gas is maintained at 1 atm on one side of a 1 mm thick Pd sheet (A=1 m 2 ), and the pressure on the purified side is maintained at 0.1 atm by pumping, what is the mass of the hydrogen purified in 1 hr? Assume steady state conditions. The concentration of H 2 at 1 atm is 9.0x10-3 gm/cm 3 and D (H) in Pa is 1.2x10-6 cm 2 /sec. Material Sciences and Engineering MatE271 Week 6 13 Non-steady state diffusion The diffusion flux at a particular point varies with time (There is a net accumulation or depletion of the diffusing species at a given location) i.e., local concentration of diffusing species changes with time as diffusion proceeds This is the most common situation What is this time scale? Material Sciences and Engineering MatE271 Week 6 14

8 Non-steady state diffusion C(x, t)=c x? Fick s Second Law governs C = D 2 C t x 2 x Many solutions exist for particular geometries (initial and boundary conditions) Diffusion from a constant source into an semi-infinite solid BC-1: For t = 0, C = C 0 at 0 x BC-3: C = C 0 at x = BC-2: t > 0, C = C s at x = 0 (C x -C 0 ) = 1 - erf x (C s -C 0 ) 2 Dt C(0,t)=C s C(x,0)=C o C(,t)=C o Material Sciences and Engineering MatE271 Week 6 15 Non-steady state diffusion C(x, t)=c x? (C x -C 0 ) = 1 - erf x (C s -C 0 ) 2 Dt C(0,t)=C s C(,t)=C o C(x,0)=C o C s t o< t 1 < t 2 < t 3 C C 0 t o t 1 t 2 t 3 C x x = 0 x Material Sciences and Engineering MatE271 Week 6 16

9 Example Surface Treatment of Steel: For some applications (e.g. gears), it is necessary to harden the surface of a steel (Fe-C alloy) above that of its interior. One way of accomplishing this is by increasing the surface concentration of carbon in the steel (as we will see later) using a process termed carburizing. In carburizing the steel piece is exposed, at elevated temperature, to an atmosphere rich in a hydrocarbon gas, such as methane (CH 4 ). Material Sciences and Engineering MatE271 Week 6 17 Example: Surface Treatment of Steel: Consider on such alloy that initially has a uniform carbon concentration of 0.25 wt% and is to be treated at 950 C. 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 0.5 mm below the surface? The diffusion coefficient for C in Fe at this temperature is 1.6 x m 2 /sec. Assume piece is semi-infinite. Material Sciences and Engineering MatE271 Week 6 18

10 Factors that Influence Diffusion I. Diffusing Species magnitude of diffusion coefficient, D - indicates the rate at which atoms diffuse both diffusing species and host material influence the coefficient Relative sizes of atoms Openess of lattice Ionic charges Material Sciences and Engineering MatE271 Week 6 19 Factors that Influence Diffusion Atomic Size/Mechanism For example: For the host species of iron: - Self diffusion at 500 C (Fe moving in Fe) D = 1.1 x m 2 /s (vacancy diffusion) - Carbon interdiffusion at 500 C (C moving in Fe D = 2.3 x m 2 /s (interstitial diffusion) This shows the contrast between rates of vacancy and interstitial diffusion Material Sciences and Engineering MatE271 Week 6 20

11 Factors that Influence Diffusion II. Temperature very strong effect on the diffusion coefficient: D = D o exp Q d RT D o = T independent preexponential Q d = the activation energy for diffusion (J/mol, or ev/atom) R = the gas constant, 8.31 J/mol - K or x 10-5 ev/ atom K T = absolute temperature, (K) A large activation energy results in a small D ln D = ln D o Q d R 1 T Plot ln D vs 1/T - get straight line (to measure activation energy and D o ) Material Sciences and Engineering MatE271 Week 6 21 Example: o Using data from Table 5.2, compute the diffusion coefficient of C in α Fe and γ Fe at 900ºC. Material Sciences and Engineering MatE271 Week 6 22

12 Solution: D = D o exp Q d RT D o = T independent preexponential Q d = the activation energy for diffusion (J/mol, or ev/atom) o Cin α Fe (BCC) at 900ºC R = the gas constant, 8.31 J/mol - K or x 10-5 ev/ atom K D = 6.2x10-7 m 2 /sec exp (-0.83 ev/atom / (1173K)(8.62x10-5 ev/atom-k) T = absolute temperature, (K) D = 1.7x10-10 m 2 /sec o C in γ Fe (FCC) at 900ºC D = 2.3x10-5 m 2 /sec exp (-1.53 ev/atom / (1173K)(8.62x10-5 ev/atom-k) D = 5.9x10-12 m 2 /sec Material Sciences and Engineering MatE271 Week 6 23 What does this tell you about interstitial sites in BCC and FCC?. BCC more open than FCC for interstitial diffusion i.e. it is easier to move from one interstitial site to another in BCC But it does not say anything about the size or number of interstitial sites in each.actually, as you will see, FCC has bigger (and more) interstitial sites Material Sciences and Engineering MatE271 Week 6 24

13 Other Diffusion Paths (Besides through volume of the crystal) Atomic migration often occurs more rapidly along socalled short circuiting paths Dislocations Grain boundaries External surfaces However, there is usually small total area for this to occur - so not always important Material Sciences and Engineering MatE271 Week 6 25 Volume, grain boundary and surface diffusion surface Grain boundary Ag in Ag volume Material Sciences and Engineering MatE271 Week 6 26

14 Diffusion and Materials Processing o Properties of materials are altered through diffusion steelmaking sintering semiconductor doping o Heat treatment is used to allow these to occur over a reasonable time frame. Material Sciences and Engineering MatE271 Week 6 27 Summary Recognize various imperfections in crystals Point imperfections Impurities Line imperfections (dislocations) Bulk imperfections Define various diffusion mechanisms Identify factors controlling diffusion processes Material Sciences and Engineering MatE271 Week 6 28 aterial Sciences and Engineering,

15 Reading Assignment Shackelford 2001(5 th Ed) Read Chapter 5, pp Material Sciences and Engineering MatE271 Week 6 29