Chapter 6 Diffusion in SolidsDiffusion - Introduction A phenomenon of material transport by atomic migration The mass transfer in macroscopic level is implemented by the motion of atoms in microscopic level Self-diffusion and interdiffusion (or impurity diffusion) Topics: mechanisms of diffusion, mathematics of diffusion, effects of temperature and diffusing species on the rate of diffusion, and diffusion of vacancy-solute complexesDemonstration of diffusionBefore heat treatmentAfter heat treatmentDiffusion Mechanisms (1) Vacancy diffusion Interstitial diffusionTwo mechanisms:Diffusion Mechanisms (2) Vacancy diffusionIn substitutional solid solutions, the diffusion (both self-diffusion and interdiffusion) must involve vacanciesFor self-diffusion, the activation energy is vacancy formation energy + vacancy migration energy. Interstitial diffusionIn interstitial solid solutions, the diffusion of interstitial solute atoms is the migration of the atoms from interstitial site to interstitial siteDiffusion Mechanisms (3)Position of interstitial atom after diffusionThe activation energy is the migration energy of the interstitial atom.Mathematics of Diffusion (1) Steady-state diffusion Time-dependent process, the rate of mass transfer is expressed as a diffusion flux (J)Mass transferred through a cross- sectional areaDiffusion time Area across which the diffusion occursIn differential formJ = Mass transferred through a unit area per unit time (g/m2 s)Mathematics of Diffusion (2)Concentration profile does not change with time steady-state diffusion concentration gradient = dC/dxdC/dxJ e.g. the diffusion of atoms of a gas through a metal plate Mathematics of Diffusion (3)where D is the diffusion coefficient (m2/s), showing the rate of diffusionFor this case, the unit of C is in mass per unit volume, e.g. g/m3 Minus sign indicates the diffusion is down the concentration gradientFicks first lawThe mathematics of steady-state diffusion in one dimension is given byFor steady-state diffusion, the diffusion flux is proportional to the concentration gradientNegative Mathematics of Diffusion (4) Nonsteady-state diffusionThe diffusion flux at a particular point varies with time.Mathematics of Diffusion (5)The diffusion equation is represented byFicks second lawC is a function of x and tIf D is independent of the composition, the above equation changes to Mass decrease in the box per unit volume per unit time dC/dt = mass increase in the box per unit volume per unit timeVolume of the box: 1dxC = mass per unit volume (concentration)Unit area cross-sectionMathematics of Diffusion (6) Solutions of diffusion equationA and B are constants and erf(z) is the error function, defined asAccording to error function solutions for diffusion equation, the solution for these profiles can be given byFFFMathematics of Diffusion (7)For t0, Cx=Cs at x=0Cx=Co at x= ThereforeA=CsCo=A+BB=-(Cs-Co)Boundary conditions:Mathematics of Diffusion (8)For t=0, Cx=C1 at x0 ThereforeC1=A-BC2=A+BA= (C1+C2)/2B=-(C1-C2)/2Boundary conditions:Mathematics of Diffusion (9)xFor t=0, Cx=0 at xhBoundary conditions:A-B-C=0A+B-C=CoA+B+C=0A=0B=Co/2C=-Co/2Mathematics of Diffusion (10)When Cx reaches a certain value at a particular position at different temperaturesThereforeFor example, if the same diffusion effect is obtained at two different temperatures T1 and T2, there isD1t1=D2t2Mathematics of Diffusion (11)Factors Affecting Diffusion (1)(1) Diffusing speciesDiffusing species affect diffusion coefficient. Different diffusing species have different diffusion coefficients. e.g. carbon diffusion in -Fe, Carbon diffusion is much faster than Fe self-diffusion. At 500oC, DC=2.4E-12 m2/s (interstitial diffusion) and DFe=3.0E-21 m2/s (vacancy diffusion), DC/DFe = 8.0E8(1) (2) Temperaturewhere Do = pre-exponential constantQd = activation energy for diffusion (J/mol)R = gas constant (8.3144 J/mol-KT = absolute temperature (K) (oC + 273.15)Factors Affecting Diffusion (2)logD has a linear relationship with reciprocal temperatureFactors Affecting Diffusion (3)10-3 (1/K)Diffusion coefficients are usually determined by measuring these straight lineslogDo is the intercept on the vertical axis Example Problem - 1For a steel, it has been determined that a carburizing heat treatment of 10 h duration will raise the carbon concentration to 0.45 wt% at a point 2.5 mm from the surface. Estimate the time required to achieve the same concentration at a 5.0-mm position for the same steel and at the same carburizing temperature.SolutionSince both cases reach the same carbon concentration at the same temperature, the left hand side of equation is the same. Therefore Example Problem - 2Example Problem 2SolutionC1 = 5 wt%; C2 = 2 wt%Cx = 2.5 wt%x = 50m = 510-5 mDo = 8.510-5 m2/sQd = 202100 J/molT = 1023 KBBBExample Problem 2Finallyt = 1.27510-9/4.072 10-15 = 313113 s = 87 hOther Diffusion Paths“Short-circu