1 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 5: DIFFUSION IN SOLIDS

2 Chapter 5: DIFFUSION Why study Diffusion? Heat-treated to improve their properties. Heat-treatment almost always involve atomic diffusion.  desired results depends on diffusion rate Heat-treatment temperature, time, and/or rate of heating/cooling can be predicted by the mathematics of diffusion Steel gear  Case hardened to improve hardness and resistance to fatigue  diffusing excess carbon or nitrogen into outer surface layer.

3 5.1 Introduction Diffusion: The phenomenon of material transport by atomic motion. Many reactions and processes that are important in the material treatment rely on the mass transfer:  Either with a specific solid ( at microscopic level )  Or from a liquid, a gas, or another solid phase. This chapter covers:  Atomic mechanism  Mathematics of diffusion  Influence of temperature and diffusing species of the diffusion rate

4 5.1 Introduction (Contd.) Phenomenon of diffusion  Explained using diffusion couple, formed by joining bars of two different materials having intimate contact Copper and Nickel diffusion couple  Atom locations and concentration Heated for an extended period at an elevated temperature ( but below melting temperature of both ) and cooled to room temperature.

5 3 Interdiffusion: In an alloy, atoms tend to migrate from regions of large concentration. Initially After some time DIFFUSION

6 5.1 Introduction (Contd.) Chemical analysis reveals  Alloy region  Variation of concentration  Atoms migrated or diffused into one another Interdiffusion or impurity diffusion  Atoms of one metal diffuses into another  Net drift of atoms from high to lower concentration

7 4 Self-diffusion: In an elemental solid, atoms also migrate. All atoms exchanging positions are of same type No compositional Diffusion in pure metal changes Label some atomsAfter some time DIFFUSION

8 5.2 Diffusion Mechanism Atoms in solids are in constant motion rapidly changing positions. Diffusion is just the stepwise migration of atoms from a lattice site to other lattice site. Two conditions for movement: 1. There must be an empty adjacent site 2. Atom must have sufficient energy to break bonds with neighbor atoms Atomic vibration (Section 4.7):  Every atom is vibrating very rapidly about its lattice position within the crystal  At any instant, not all vibrate with same frequency and amplitude.  Not all atoms have same energy  Same atom may have different level of energy at different time  Energy increases with temperature

9 5.2 Diffusion Mechanism (Contd.) Several different models for atomic motion  Two dominate for metallic diffusion VACANCY DIFFUSION  Involves interchange of an atom from a normal lattice position to an adjacent vacant lattice site or vacancy  Necessitates presence of vacancies  Diffusing atoms and vacancies exchange positions  they move in opposite directions  Both self- and inter-diffusion occurs by this mechanism

10 5 Vacancy Diffusion: applies to substitutional impurities atoms exchange with vacancies rate depends on: --number of vacancies --activation energy to exchange. DIFFUSION MECHANISMS

Diffusion Mechanism (Contd.) INTERSTITIAL DIFFUSION  Atoms migrate from an interstitial position to a neighboring one that is empty  Found for interdiffusion of impuries such as hydrogen, carbon, nitrogen, and oxygen  atoms small enough to fit into interstitial positions.  Host or substitutional impurity atoms rarely have insterstitial diffusion Interstitial atoms are smaller and thus more mobile  interstitial diffusion occurs much more rapidly then by vacancy mode There are more empty interstitial positions than vacancies  interstitial atomic movement have greater probability

12 Simulation of interdiffusion across an interface: Rate of substitutional diffusion depends on: --vacancy concentration --frequency of jumping. (Courtesy P.M. Anderson) Diffusion Simulation

13 Diffusion Mechanisms Interstitial diffusion – smaller atoms can diffuse between atoms. More rapid than vacancy diffusion Adapted from Fig. 5.3 (b), Callister 7e.

14 Adapted from chapter-opening photograph, Chapter 5, Callister 7e. (Courtesy of Surface Division, Midland-Ross.) Case Hardening: --Diffuse carbon atoms into the host iron atoms at the surface. --Example of interstitial diffusion is a case hardened gear. Result: The presence of C atoms makes iron (steel) harder. Processing Using Diffusion

Steady-State Diffusion The quantity of an element that is transported within another is a function of time  diffusion is a time-dependent process. Diffusion flux (J)  Rate of diffusion or mass transfer  Defined as “mass or number of atoms (M) diffusing through and perpendicular to a unit cross-sectional area of solid per unit time.  Mathematically, In differential form: A: area across which diffusion is occuring t: elapsed diffusion time

16 Steady-State Diffusion Fick’s first law of diffusion C1C1 C2C2 x C1C1 C2C2 x1x1 x2x2 D  diffusion coefficient Rate of diffusion independent of time Flux proportional to concentration gradient =

Steady-State Diffusion (Contd.)

18 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 x10 -8 cm 2 /s  surface concentrations: C 2 = 0.02 g/cm 3 C 1 = 0.44 g/cm 3

19 Example (cont). glove C1C1 C2C2 skin paint remover x1x1 x2x2 Solution – assuming linear conc. gradient D = 110 x cm 2 /s C 2 = 0.02 g/cm 3 C 1 = 0.44 g/cm 3 x 2 – x 1 = 0.04 cm Data:

Nonsteady-State Diffusion Most practical diffusion situations are non-steady Non-steady  Diffusion flux and the concentration flux at some particular point of solid vary with time  Net accumulation or depletion of the diffusing species  Figure shown concentration profile at three different times

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

22 Solution for Semi-infinite Solid with constant surface concentration Assumptions  Initial concentration C 0  X = 0 at the surface and increases with distance into the solid  Initial time = 0 Boundary conditions  For t = 0, C = C o at 0  x    For t > 0, C = C s (Constant surface concentration) at x=0 C = C 0 at x =  Solution  erf ( ) : Gaussian error function  Values given in Table 5.1

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

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26 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

27 Solution (cont.):  t = 49.5 h x = 4 x m  C x = 0.35 wt%C s = 1.0 wt%  C o = 0.20 wt%  erf(z) =

28 Solution (cont.): We must now determine from Table 5.1 the value of z for which the error function is An interpolation is necessary as follows zerf(z) z z  0.93 Now solve for D

29 To solve for the temperature at which D has above value, we use a rearranged form of Equation (5.9a); from Table 5.2, for diffusion of C in FCC Fe D o = 2.3 x m 2 /s Q d = 148,000 J/mol  Solution (cont.): T = 1300 K = 1027°C

30 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 Dt should be held constant. Answer: Note: values of D are Given 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. EXAMPLE PROBLEM

31 Factors That Influence Diffusion

32 Factors That Influence Diffusion (Contd.) DIFFUSING SPECIES Magnitude of diffusion coefficient (D)  indicative of the rate at which atoms diffuse D depends on both the diffusing species as well as the host atomic structure Self-diffusionFe in  -Fe3.0E(-21) m 2 /s Vacancy Diffusion Inter-diffusion C in  -Fe2.4E(-12) m 2 /s Interstitial Diffusion Interstitial is faster than vacancy diffusion

33 Factors That Influence Diffusion (Contd.) TEMPERATURE Temperature has a most profound influence on the coefficients and diffusion rate Example: Fe in  -Fe (Table 5.2) 500 o CD=3.0E(-21) m 2 /s 900 o CD=1.8E(-15) m 2 /s  approximately six orders

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