Introduction To Materials Science, Chapter 5, Diffusion University of Virginia, Dept. of Materials Science and Engineering 1 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 Chapter 5 Outline

Introduction To Materials Science, Chapter 5, Diffusion University of Virginia, Dept. of Materials Science and Engineering 2 Diffusion  transport by atomic motion. Inhomogeneous material can become homogeneous by diffusion. Temperature should be high enough to overcome energy barrier. What is diffusion?

Introduction To Materials Science, Chapter 5, Diffusion University of Virginia, Dept. of Materials Science and Engineering 3 Concentration Gradient  Interdiffusion (or Impurity Diffusion). Self-diffusion: one-component material, atoms are of same type. Inter-diffusion vs. Self-diffusion Before After (Heat)

Introduction To Materials Science, Chapter 5, Diffusion University of Virginia, Dept. of Materials Science and Engineering 4 Vacancy diffusion 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. Therefore, there is an energy barrier. Energy comes from thermal energy of atomic vibrations (E av ~ kT) Atom flow is opposite to vacancy flow direction. Diffusion Mechanisms (I) Atom migration Vacancy migration After Before

Introduction To Materials Science, Chapter 5, Diffusion University of Virginia, Dept. of Materials Science and Engineering 5 Interstitial diffusion Diffusion Mechanisms (II) Interstitial atom before diffusion Interstitial atom after 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. Smaller energy barrier Only small impurity atoms (e.g. C, H, O) fit into interstitial sites.

Introduction To Materials Science, Chapter 5, Diffusion University of Virginia, Dept. of Materials Science and Engineering 6 Flux of diffusing atoms, J. Number of atoms diffusing through unit area per unit time [atoms/(m 2 s)] or Mass of atoms diffusing through unit area per unit time [kg/(m 2 s)] Mass: J = M / (A t)  (1/A) (dM/dt) A J

Introduction To Materials Science, Chapter 5, Diffusion University of Virginia, Dept. of Materials Science and Engineering 7 Diffusion flux does not change with time Concentration profile: Concentration (kg/m 3 ) vs. position Concentration gradient: dC/dx (kg / m 4 ) Steady-State Diffusion

Introduction To Materials Science, Chapter 5, Diffusion University of Virginia, Dept. of Materials Science and Engineering 8 Fick’s first law: J proportion to dC/dx Steady-State Diffusion D=diffusion coefficient Concentration gradient is ‘driving force’ Minus sign means diffusion is ‘downhill’: toward lower concentrations

Introduction To Materials Science, Chapter 5, Diffusion University of Virginia, Dept. of Materials Science and Engineering 9 Concentration changing with time Fick’s second law Nonsteady-State Diffusion (not tested) Find C(x,t)

Introduction To Materials Science, Chapter 5, Diffusion University of Virginia, Dept. of Materials Science and Engineering 10 Atom needs enough thermal energy to break bonds and squeeze through its neighbors. Energy needed  energy barrier  Called the activation energy E m (like Q) Diagram for Vacancy Diffusion Diffusion  Thermally Activated Process Atom Vacancy EmEm Distance Energy

Introduction To Materials Science, Chapter 5, Diffusion University of Virginia, Dept. of Materials Science and Engineering 11 Room temperature (k B T = eV) Typical activation energy E m (~ 1 eV/atom) (like Q v ) Therefore, a large fluctuation in energy is needed for a jump. Probability of a fluctuation or frequency of jump, R j  Diffusion – Thermally Activated Process R 0 = attempt frequency proportional to vibration frequency Swedish chemist Arrhenius

Introduction To Materials Science, Chapter 5, Diffusion University of Virginia, Dept. of Materials Science and Engineering Probability of finding a vacancy in an adjacent lattice site (Chap. 4) : times 2. Probability of thermal fluctuation Calculate Activated Diffusion The diffusion coefficient = Multiply Arrhenius dependence.

Introduction To Materials Science, Chapter 5, Diffusion University of Virginia, Dept. of Materials Science and Engineering 13 Diffusion coefficient is the measure of mobility of diffusing species. Diffusion – Temperature Dependence Arrhenius Plots (lnD) vs. (1/T) or (logD) vs. (1/T) D 0 – temperature-independent (m 2 /s) Q d – the activation energy (J/mol or eV/atom) R – the gas constant (8.31 J/mol-K) or k B - Boltzman constant ( 8.62  eV/atom-K) T – absolute temperature (K) or

Introduction To Materials Science, Chapter 5, Diffusion University of Virginia, Dept. of Materials Science and Engineering 14 Diffusion – Temperature Dependence (II) Graph of log D vs. 1/T has slop of –Q d /2.3R, intercept of ln D o

Introduction To Materials Science, Chapter 5, Diffusion University of Virginia, Dept. of Materials Science and Engineering 15 Diffusion – Temperature Dependence (III) Arrhenius plot: Diffusivity for metallic systems

Introduction To Materials Science, Chapter 5, Diffusion University of Virginia, Dept. of Materials Science and Engineering 16 Diffusion of different species Smaller atoms diffuse more readily Diffusion faster in open lattices or in open directions

Introduction To Materials Science, Chapter 5, Diffusion University of Virginia, Dept. of Materials Science and Engineering 17 Diffusion: Role of the microstructure (I) Self-diffusion coefficients for Ag Depends on diffusion path Grain boundaries and surfaces less restrictive

Introduction To Materials Science, Chapter 5, Diffusion University of Virginia, Dept. of Materials Science and Engineering 18 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.

Introduction To Materials Science, Chapter 5, Diffusion University of Virginia, Dept. of Materials Science and Engineering 19 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 - D o, Q d 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.

Introduction To Materials Science, Chapter 5, Diffusion University of Virginia, Dept. of Materials Science and Engineering 20 Summary  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 Make sure you understand