PowerPoint PresentationSlide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Introduction To Materials Science, Chapter 5, DiffusionUniversity of Virginia, Dept. of Materials Science and Engineering1Diffusion 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 Microstructure5.4 Nonsteady-State Diffusion – Not Covered / Not TestedChapter 5 OutlineIntroduction To Materials Science, Chapter 5, DiffusionUniversity of Virginia, Dept. of Materials Science and Engineering2Diffusion 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, DiffusionUniversity of Virginia, Dept. of Materials Science and Engineering3Concentration Gradient  Interdiffusion (or Impurity Diffusion).Self-diffusion: one-component material, atoms are of same type.Inter-diffusion vs. Self-diffusionBeforeAfter(Heat)Introduction To Materials Science, Chapter 5, DiffusionUniversity of Virginia, Dept. of Materials Science and Engineering4Vacancy diffusionTo 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 (Eav ~ kT)Atom flow is opposite to vacancy flow direction.Diffusion Mechanisms (I)Atom migrationVacancy migrationAfterBeforeIntroduction To Materials Science, Chapter 5, DiffusionUniversity of Virginia, Dept. of Materials Science and Engineering5Interstitial diffusionDiffusion Mechanisms (II)Interstitial atom before diffusionInterstitial atom after diffusionGenerally 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, DiffusionUniversity of Virginia, Dept. of Materials Science and Engineering6 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)AJIntroduction To Materials Science, Chapter 5, DiffusionUniversity of Virginia, Dept. of Materials Science and Engineering7Diffusion flux does not change with timeConcentration profile: Concentration (kg/m3) vs. positionConcentration gradient: dC/dx (kg / m4)Steady-State DiffusionBABAxxCCxCdxdCIntroduction To Materials Science, Chapter 5, DiffusionUniversity of Virginia, Dept. of Materials Science and Engineering8Fick’s first law: J proportion to dC/dxSteady-State Diffusion D=diffusion coefficientdCJ DdxConcentration gradient is ‘driving force’ Minus sign means diffusion is ‘downhill’: toward lower concentrationsIntroduction To Materials Science, Chapter 5, DiffusionUniversity of Virginia, Dept. of Materials Science and Engineering9Concentration changing with time Fick’s second lawNonsteady-State Diffusion(not tested)Find C(x,t)22C Jt xCDx  Introduction To Materials Science, Chapter 5, DiffusionUniversity of Virginia, Dept. of Materials Science and Engineering10Atom needs enough thermal energy to break bonds and squeeze through its neighbors. Energy needed energy barrier Called the activation energy Em (like Q)Diagram for Vacancy DiffusionDiffusion Thermally Activated ProcessAtomVacancyEmDistanceEnergyIntroduction To Materials Science, Chapter 5, DiffusionUniversity of Virginia, Dept. of Materials Science and Engineering11Room temperature (kBT = 0.026 eV) Typical activation energy Em (~ 1 eV/atom) (like Qv)Therefore, a large fluctuation in energy is needed for a jump. Probability of a fluctuation or frequency of jump, RjDiffusion – Thermally Activated Process R0 = attempt frequency proportional to vibration frequencySwedish chemist Arrhenius0expmjBER Rk T    Introduction To Materials Science, Chapter 5, DiffusionUniversity of Virginia, Dept. of Materials Science and Engineering121. Probability of finding a vacancy in an adjacent lattice site (Chap. 4): times2. Probability of thermal fluctuationCalculate Activated DiffusionThe diffusion coefficient = Multiply0expmjBER Rk T    .expvBQP Constk T    .exp expm VB BE QD Constk T k T           0expdBQD Dk T     Arrhenius dependence.Introduction To Materials Science, Chapter 5, DiffusionUniversity of Virginia, Dept. of Materials Science and Engineering13Diffusion coefficient is the measure of mobility of diffusing species. Diffusion – Temperature Dependence Arrhenius Plots (lnD) vs. (1/T) or (logD) vs. (1/T)0 0exp expd dQ QD D DRT kT            dxdCDJ D0 – temperature-independent (m2/s)Qd – the activation energy (J/mol or eV/atom)R – the gas constant (8.31 J/mol-K) orkB - Boltzman constant ( 8.6210-5 eV/atom-K)T – absolute temperature (K)01ln lndQD DR T    01log log2.3dQD DR T    orIntroduction To Materials Science, Chapter 5, DiffusionUniversity of Virginia, Dept. of Materials Science and Engineering14Diffusion – Temperature Dependence (II)Graph of log D vs. 1/T has slop of –Qd/2.3R,intercept of ln Do2121dT1T1DlogDlogR3.2QT1R3.2QDlogDlogd0Introduction To Materials Science, Chapter 5, DiffusionUniversity of Virginia, Dept. of Materials Science and Engineering15Diffusion – Temperature Dependence (III)Arrhenius plot:Diffusivity for metallic systemsIntroduction To Materials Science, Chapter 5, DiffusionUniversity of Virginia, Dept. of Materials Science and Engineering16Diffusion of different speciesSmaller atoms diffuse