Diffusion how atoms move in solids Chapter Outline Diffusion how atoms move in solids Diffusion mechanisms Vacancy diffusion Interstitial diffusion Impurities Mathematics of diffusion Steady-state diffusion (Fick’s first law) Nonsteady-State Diffusion (Fick’s second law) Factors that influence diffusion Diffusing species Host solid Temperature Microstructure 5.4 Nonsteady-State Diffusion – Not Covered / Not Tested

Diffusion transport by atomic motion. What is diffusion? Diffusion transport by atomic motion. Inhomogeneous material can become homogeneous by diffusion. Temperature should be high enough to overcome energy barrier.

Interdiffusion vs. Self-diffusion Concentration Gradient  Interdiffusion (or Impurity Diffusion). Before After (Heat) Self-diffusion: one-component material, all atoms that exchange positions are of same type.

Diffusion Mechanisms (I) Vacancy diffusion Atom migration Vacancy migration Before After To jump from lattice site to lattice site, atoms need energy to break bonds with neighbors, and to cause the necessary lattice distortions during jump. This energy comes from the thermal energy of atomic vibrations (Eav ~ kT) Materials flow (the atom) is opposite the vacancy flow direction.

Diffusion Mechanisms (II) Interstitial diffusion Interstitial atom before diffusion Interstitial atom after diffusion Interstitial diffusion generally faster than vacancy diffusion because bonding of interstitials to surrounding atoms is normally weaker and there are more interstitial sites than vacancy sites to jump to. Requires small impurity atoms (e.g. C, H, O) to fit into interstices in host.

Mass: J = M / (A t)  (1/A) (dM/dt) Flux of diffusing atoms, J. Number of atoms diffusing through unit area per unit time [atoms/(m2s)] or Mass of atoms diffusing through unit area per unit time [kg/(m2 s)] Mass: J = M / (A t)  (1/A) (dM/dt) There is a difference between diffusion and net diffusion. In a homogeneous material, atoms also diffuse but this motion is hard to detect. This is because atoms move randomly and there will be an equal number of atoms moving in one direction than in another. In inhomogeneous materials, the effect of diffusion is readily seen by a change in concentration with time. In this case there is a net diffusion. Net diffusion occurs because, although all atoms are moving randomly, there are more atoms moving in regions where their concentration is higher. J A

Steady-State Diffusion Diffusion flux does not change with time Concentration profile: Concentration (kg/m3) vs. position Concentration gradient: dC/dx (kg / m4)

Steady-State Diffusion Fick’s first law: J proportion to dC/dx D is the diffusion coefficient Concentration gradient is the driving force (not a mechanistic force). Minus sign means that diffusion is down the concentration gradient.

Nonsteady-State Diffusion (not tested) Concentration changing with time Fick’s second law Assumption: D does not depend on x (and C! ) Find C(x,t)

Diffusion – Thermally Activated Process (I) Atom needs enough thermal energy to break bonds and squeeze through its neighbors. Energy needed  the activation energy for vacancy motion. Em Energy Em Vacancy Arrhenius equation was first formulated by Hood based on his experiments, but Arrhenius showed that it can be applied to any thermally activated process. If you have stronger bonding, you have to spent more energy to create vacancy… Atom Distance Diffusion – Thermally Activated Process (I)

Diffusion – Thermally Activated Process (II) Average thermal energy of an atom at room temperature (kBT = 0.026 eV) Usually much smaller that activation energy Em (~ 1 eV/atom) (like Qv) Therefore, a large fluctuation in energy is needed for a jump. Probability of such fluctuations or frequency of jumps, Rj, depends exponentially on temperature. Swedish chemist Arrhenius : R0 is an attempt frequency proportional to the frequency of atomic vibrations.

Diffusion – Thermally Activated Process Probability of finding a vacancy in an adjacent lattice site (see Chapter 4): Probability of thermal fluctuation needed to overcome energy barrier for vacancy motion Assumption: D does not depend on x (and C! ) The diffusion coefficient = Multiply Arrhenius dependence.

Diffusion – Temperature Dependence (I) Diffusion coefficient is the measure of mobility of diffusing species. D0 – temperature-independent (m2/s) Qd – the activation energy (J/mol or eV/atom) R – the gas constant (8.31 J/mol-K) or k - Boltzman constant ( 8.6210-5 eV/atom-K) T – absolute temperature (K) or (lnD) vs. (1/T) or (logD) vs. (1/T) Arrhenius Plots.

Diffusion – Temperature Dependence (II) Graph of log D vs. 1/T has slop of –Qd/2.3R, intercept of ln Do

Diffusion – Temperature Dependence (III) Arrhenius plot of diffusivity data for some metallic systems

Diffusion of different species Smaller atoms diffuse more readily than big ones, and diffusion is faster in open lattices or in open directions

Diffusion: Role of the microstructure (I) Self-diffusion coefficients for Ag Depends on diffusion path Grain boundaries and surfaces less restrictive

Diffusion: Role of the microstructure (II) Plots are from computer simulations Initial positions are shown by the circles, paths are shown by lines. See difference between mobility in the bulk and in the grain boundary.

Factors that Influence Diffusion Temperature - diffusion rate increases very rapidly with increasing temperature Diffusion mechanism - interstitial is usually faster than vacancy Diffusing and host species - Do, Qd is different for every solute, solvent pair Microstructure - diffusion faster in polycrystalline vs. single crystal materials because of the rapid diffusion along grain boundaries and dislocation cores.

Make sure you understand Summary Make sure you understand Activation energy Concentration gradient Diffusion Diffusion coefficient Diffusion flux Driving force Fick’s first and second laws Interdiffusion Interstitial diffusion Self-diffusion Steady-state diffusion Vacancy diffusion

Reading for next class: Chapter 6: Mechanical Properties of Metals Stress and Strain Tension Compression Shear Torsion Elastic deformation Plastic Deformation Yield Strength Tensile Strength Ductility Resilience Toughness Hardness