# chapter 5: diffusion in solids

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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?. 1. - PowerPoint PPT Presentation

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• 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

• Chapter 5: DIFFUSIONWhy 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.

• 5.1 IntroductionDiffusion: 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 mechanismMathematics of diffusionInfluence of temperature and diffusing species of the diffusion rate

• 5.1 Introduction (Contd.)Phenomenon of diffusionExplained using diffusion couple, formed by joining bars of two different materials having intimate contact

Copper and Nickel diffusion couple Figure 5.1 shows as formedAtom locations and concentration

Heated for an extended period at an elevated temperature ( but below melting temperature of both ) and cooled to room temperature.

• 3 Interdiffusion: In an alloy, atoms tend to migrate from regions of large concentration.InitiallyAfter some timeDIFFUSION

• 5.1 Introduction (Contd.)Chemical analysis reveals Alloy regionVariation of concentrationAtoms migrated or diffused into one anotherInterdiffusion or impurity diffusionAtoms of one metal diffuses into anotherNet drift of atoms from high to lower concentration

• 4 Self-diffusion: In an elemental solid, atoms also migrate. Self-diffusionAll atoms exchanging positions are of same typeNo compositional Diffusion in pure metalchangesLabel some atomsAfter some timeDIFFUSION

• 5.2 Diffusion MechanismAtoms 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 site2. Atom must have sufficient energy to break bonds with neighbor atomsAtomic vibration (Section 4.7): Every atom is vibrating very rapidly about its lattice position within the crystalAt any instant, not all vibrate with same frequency and amplitude.Not all atoms have same energySame atom may have different level of energy at different timeEnergy increases with temperature

• 5.2 Diffusion Mechanism (Contd.)Several different models for atomic motionTwo dominate for metallic diffusion

VACANCY DIFFUSIONInvolves interchange of an atom from a normal lattice position to an adjacent vacant lattice site or vacancyNecessitates presence of vacanciesDiffusing atoms and vacancies exchange positions they move in opposite directionsBoth self- and inter-diffusion occurs by this mechanism

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

• 5.2 Diffusion Mechanism (Contd.)

• 5.2 Diffusion Mechanism (Contd.)INTERSTITIAL DIFFUSIONAtoms migrate from an interstitial position to a neighboring one that is emptyFound 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

• 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 DIFFUSION

• 5.3 Steady-State DiffusionThe 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, J = M / (At)

In differential form: J = (1/A)(dM/dt)

A: area across which diffusion is occuring t: elapsed diffusion time

• 5.3 Steady-State Diffusion (Contd.)If the diffusion flux does not change with time steady-state diffusionExample: Diffusion of a gas through a plate of metalConcentration (or pressure) of diffusing species on both side are held constantConcentration profile: Concentration versus positionAssumed linear concentration profile as shown in figure (b)

• 5.3 Steady-State Diffusion (Contd.)Concentration gradientSlope at a particular point on the concentration profile curveConcentration gradient = dC / dx

For linear concentration shown in figure 5.4b:Conc. Gradient = DC/Dx = (CA CB) / (xA xB)

Ficks first law: For steady-state diffusion, the flux is proportional to the concentration gradientJ = -D(dC/dx)D: diffusion coefficient (sq. m per second )-ve sign: direction of diffusion from a high to a low concentration

• 5.3 Steady-State Diffusion (Contd.)

• 5.4 Nonsteady-State DiffusionMost practical diffusion situations are non-steadyNon-steadyDiffusion flux and the concentration flux at some particular point of solid vary with timeNet accumulation or depletion of the diffusing speciesFigure shown concentration profile at three different times

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

• Solution for Semi-infinite Solid with constant surface concentrationAssumptionsInitial concentration C0X = 0 at the surface and increases with distance into the solidInitial time = 0Boundary conditionsFor t = 0, C = Co at 0 x For t > 0, C = Cs (Constant surface concentration) at x=0C = C0 at x = Solutionerf ( ) : Gaussian error functionValues given in Table 5.1

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

• 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: valuesof D areGiven here.Key point 1: C(x,t500C) = C(x,t600C).Key point 2: Both cases have the same Co and Cs.EXAMPLE PROBLEM

• Factors That Influence Diffusion

• 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 a-Fe3.0E(-21) m2/sVacancy DiffusionInter-diffusion C in a-Fe2.4E(-12) m2/sInterstitial Diffusion

Interstitial is faster than vacancy diffusion

• Factors That Influence Diffusion (Contd.)TEMPERATURE

Temperature has a most profound influence on the coefficients and diffusion rate

Example: Fe in a-Fe (Table 5.2)500oCD=3.0E(-21) m2/s900oCD=1.8E(-15) m2/s approximately six orders

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

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