2. 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? 1 How does diffusion depend on structure and temperature? CHAPTER 6: DIFFUSION IN SOLIDS

3. 2 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. DIFFUSION DEMO

4. Diffusion Introduction and Motivation –Why do you care about diffusion? Many solids materials are often heat-treated to improve their properties From an “atomistic” view, this thermal treatment leads to diffusion – the phenomenon of material transport that occurs by atomic motion

5. Diffusion Diffusion : an introduction –Many reactions/other processes that are important in materials engineering require the transfer of mass within a specific solid (i.e. mixing/alloying), or from a liquid, gas, or another solid –This is accomplished by diffusion – material transport by atomic motion –Chemical engineers know all about mass transfer

6. Diffusion Diffusion : an introduction –Let’s use a picture to illustrate – diffusion couple –Two big chunks of metal put in physical contact Before heating After heating What happened? Upon heating, the copper atoms moved into the nickel and the nickel moved into the copper So now you have, pure copper, a Cu-Ni alloy, and pure nickel Diffusion! (strictly speaking interdiffusion or impurity diffusion)

7. Diffuse Diffusion mechanisms –From the atomic view, diffusion is simply the stepwise migration of atoms from lattice site to lattice site –For atoms to move two conditions must be met There must be an empty site to move to The atom must have sufficient energy to break bonds with its neighbors and then cause some lattice distortion during the move (read about atomic vibrations, section 5.10) –At a given temperature some fraction of the molecules will have sufficient energy to undergo diffusive motion, by virtue of their vibrational energy This fraction increases as T goes up

8. Diffusion Diffusion mechanisms –Metals: two types of mechanisms dominate Vacancy diffusion Interstitial diffusion

9. Diffusion Diffusion mechanisms –Metals: Vacancy diffusion Atom moves from normal lattice site to adjacent vacancy As you could imagine the efficiency of this process is strongly dependent on the number of vacancies Typically more effective at higher temperatures Diffusion of atoms in one direction implies the “diffusion” of vacancies in the opposite direction Self- and inter- diffusion occur by this mechanism

10. Diffusion Diffusion mechanisms –Metals: Interstitial diffusion Atoms migrate from one interstitial position to a neighboring one that is empty This mechanism is prevalent for atoms such as hydrogen, carbon, nitrogen, oxygen (i.e. things small enough to easily fit in the interstitial voids). Host or substitutional impurity atoms (i.e. atoms that fit in the structure lattice) rarely form interstitials In metal alloys interstitial diffusion is more prevalent than vacancy diffusion – why?

11. 3 Interdiffusion: In an alloy, atoms tend to migrate from regions of large concentration. InitiallyAfter some time Adapted from Figs. 5.1 and 5.2, Callister 6e. DIFFUSION: THE PHENOMENA (1)

12. 4 Self-diffusion: In an elemental solid, atoms also migrate. Label some atomsAfter some time DIFFUSION: THE PHENOMENA (2)

13. 5 Substitutional Diffusion: applies to substitutional impurities atoms exchange with vacancies rate depends on: --number of vacancies --activation energy to exchange. DIFFUSION MECHANISMS

14. 6 Simulation of interdiffusion across an interface: Rate of substitutional diffusion depends on: --vacancy concentration --frequency of jumping. (Courtesy P.M. Anderson) DIFFUSION SIMULATION Click on image to animate

15. 7 (Courtesy P.M. Anderson) 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. INTERSTITIAL SIMULATION Click on image to animate

16. Diffusion Mathematical description of diffusion Steady-state diffusion Diffusion is inherently a time-dependent process – the quantity of an element that is transported within another element as a function of time How fast does diffusion occur, or what is the rate of mass transfer? This is expressed as the diffusive flux (J) The flux J is defined as the mass M diffusing through and perpendicular to a unit area (in this case of solid) per unit time J [=] mass/area-time

17. Flux: 10 Directional Quantity Flux can be measured for: --vacancies --host (A) atoms --impurity (B) atoms MODELING DIFFUSION: FLUX

18. Mathematical description of diffusion –Steady-state diffusion This is a limiting case: here the flux does not change with time One example: consider a solid plate with gas diffusing through it (see figure) The pressures on each side of the plate are constant

19. Concentration Profile, C(x): [kg/m 3 ] 11 Fick's First Law: The steeper the concentration profile, the greater the flux! Adapted from Fig. 5.2(c), Callister 6e. CONCENTRATION PROFILES & FLUX

20. Steady State: the concentration profile doesn't change with time. 12 Apply Fick's First Law: Result: the slope, dC/dx, must be constant (i.e., slope doesn't vary with position)! If J x ) left = J x ) right, then STEADY STATE DIFFUSION

21. Steel plate at 700C with geometry shown: 13 Q: How much carbon transfers from the rich to the deficient side? Adapted from Fig. 5.4, Callister 6e. EX: STEADY STATE DIFFUSION

22. Diffusion Mathematical description of diffusion Steady-state diffusion This leads to a linear concentration profile of the gas species through the plate So the concentration gradient is given as From this, one can obtain Fick’s 1 st law: You will often hear that the concentration gradient is the driving force for diffusion … this is only partially true!

23. Diffusion Mathematical description of diffusion –Steady-state diffusion The real driving force is the chemical potential gradient What is the chemical potential (  i ) – you have seen it in thermodynamics! It is the same thing as the partial molar Gibbs free energy for species i in a mixture How do I get concentrations out? Thermodynamics!  Can relate partial molar Gibbs free energy to fugacities, etc. More generally though  why is it the chemical potential gradient and not the concentration gradient?

24. Diffusion Mathematical description of diffusion –Nonsteady-state diffusion This is more common – the flux varies with time! For what follows we will assume the diffusivity is independent of concentration (which is not always the case!) Write a shell balance Consider a differential volume element of size A  x x x +  x xx

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

26. Diffusion Mathematical description of diffusion –Nonsteady-state diffusion Fick’s 2 nd law is not as easy to solve … can solve it but need boundary conditions (2) and an initial condition (1) Consider the following example: 1.Before diffusing, the diffusing solute is uniformly distributed in the solid with a concentration of C = C o 2.The value of C at the surface (x = 0) is constant (i.e. C s ) 3.The value of C “infinitely deep” into the solid is C o Or in mathematical terms

27. Diffusion Mathematical description of diffusion –Nonsteady-state diffusion The solution of the PDE using the BCs on the last slide is erf is the so called “error function” (it is a tabulated function) Okay, enough math…physically this will tell you the concentration profile inside the solid as a function of time Or, how long you need to heat treat something to alloy the metal!

28. Copper diffuses into a bar of aluminum. 15 General solution: "error function" Values calibrated in Table 5.1, Callister 6e. Adapted from Fig. 5.5, Callister 6e. EX: NON STEADY STATE DIFFUSION

31. Diffusion Factors influencing diffusion –Diffusing species Diffusion coefficients, not surprisingly, depend on the identity of the diffusing species Example – carbon diffuses through iron much more quickly than iron self-diffuses (i.e. steel) –Why? Carbon diffuses via an interstitial mechanism, iron diffuses via a vacancy mechanism –Temperature – big effect Why? Can described diffusion in solids as an activated process D o – preexponential (what are the units?), Q d – Activation energy for diffusion

32. Diffusivity increases with T. Experimental Data: D has exp. dependence on T Recall: Vacancy does also! 19 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.) DIFFUSION AND TEMPERATURE

33. Diffusion Factors influencing diffusion –Temperature In other words the diffusion in solids is described as an Arrhenius type process Q d can be thought of as the energy required to produce diffusive motion of one mole of atoms –Can determine the activation energy from a plot of D versus 1/T Example: Given the following data, determine D o and Q d

34. Diffusion Example: Given the following data, determine D o and Q d Step 1: calculate 1/T and plot 1/T versus D Note: plot is a straight line Get parameters from linear regression Intercept D o, slope –Q d /R y = -18085x - 10.71 D o = 2.23 x 10 -5 m 2 /s Q d = 150,400 J/mol

35. Diffusion Other diffusion paths –How else can migration/diffusion of atoms occur? Along dislocations, grain boundaries, surfaces Referred to as “short-circuit” diffusion paths since rates are much faster than bulk diffusion The contribution of these paths to the overall diffusive flux is minor however (why?)

36. Diffusion Diffusion in ionic/polymeric materials –Ionic materials More complex than in metals – why? -- have to consider motion of two types of ions with different charge Diffusion is typically via a vacancy mechanism 1.Ion vacancies occur in pairs 2.They form in nonstoichiometric compounds 3.They are created by substitutional impurity compounds of different charge state (an impurity atom, or a vacancy) than the host ions

37. Diffusion Diffusion in ionic/polymeric materials –Ionic materials – few other points Transference of charge is associated with diffusion of ions Charge neutrality – need an ion with opposite charge (and same valence) to move with it –Candidates: vacancy, impurity atom, electron carrier Diffusion of slowest moving species dictates overall diffusion Finally, apply an electric field  charges migrate, give rise to an electric current

38. Diffusion Diffusion in ionic/polymeric materials –Polymers We will actually be more interested in the diffusion of small species (H 2 O, CO 2 ) occluded in the polymer In other words we will be interested in the polymer permeability (why do you think?) Diffusion through amorphous regions is faster than that through crystalline regions –Mechanism analogous to interstitial diffusion in metals

39. Case Hardening: --Diffuse carbon atoms into the host iron atoms at the surface. --Example of interstitial diffusion is a case hardened gear. Result: The "Case" is --hard to deform: C atoms "lock" planes from shearing. --hard to crack: C atoms put the surface in compression. 8 Fig. 5.0, Callister 6e. (Fig. 5.0 is courtesy of Surface Division, Midland- Ross.) PROCESSING USING DIFFUSION (1)

40. Doping Silicon with P for n-type semiconductors: Process: 9 1. Deposit P rich layers on surface. 2. Heat it. 3. Result: Doped semiconductor regions. Fig. 18.0, Callister 6e. PROCESSING USING DIFFUSION (2) Doped Si is the semiconductor material; Al wires the circuit elements together

41. 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? 16 Result: Dt should be held constant. Answer: Note: values of D are provided here. Key point 1: C(x,t 500C ) = C(x,t 600C ). Key point 2: Both cases have the same C o and C s. PROCESSING QUESTION

42. 17 The experiment: we recorded combinations of t and x that kept C constant. Diffusion depth given by: = (constant here) DIFFUSION DEMO: ANALYSIS

43. Experimental result: x ~ t 0.58 Theory predicts x ~ t 0.50 Reasonable agreement! 18 DATA FROM DIFFUSION DEMO

44. 20 Diffusion FASTER for... open crystal structures lower melting T materials materials w/secondary bonding smaller diffusing atoms cations lower density materials Diffusion SLOWER for... close-packed structures higher melting T materials materials w/covalent bonding larger diffusing atoms anions higher density materials SUMMARY: STRUCTURE & DIFFUSION

45. Reading: Chapter 6 HW # 5, Due Monday, February 26: 6.3; 6.4; 6.6; 6.7; 6.12; 6.15; 6.16; 6.18 6.21; 6.24; 6.D2 0 ANNOUNCEMENTS