Chapter 1 Diffusion in Solids

Diffusion - 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 complexes

Demonstration of diffusion Before heat treatmentAfter heat treatment

Diffusion – Mechanisms (1) Vacancy diffusion Interstitial diffusion Two mechanisms:

Diffusion – Mechanisms (2) Vacancy diffusion In substitutional solid solutions, the diffusion (both self-diffusion and interdiffusion) must involve vacancies For self-diffusion, the activation energy is vacancy formation energy + vacancy migration energy.

Interstitial diffusion In interstitial solid solutions, the diffusion of interstitial solute atoms is the migration of the atoms from interstitial site to interstitial site Diffusion – Mechanisms (3) Position of interstitial atom after diffusion The 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 area Diffusion time Area across which the diffusion occurs In differential form J = Mass transferred through a unit area per unit time (g/m 2 s))

Mathematics of Diffusion (2) Concentration profile does not change with time – steady-state diffusion concentration gradient = dC/dxdC/dx J e.g. the diffusion of atoms of a gas through a metal plate

Mathematics of Diffusion (3) where D is the diffusion coefficient (m 2 /s), showing the rate of diffusion For this case, the unit of C is in mass per unit volume, e.g. g/m 3 Minus sign indicates the diffusion is down the concentration gradient Fick’s first law The mathematics of steady-state diffusion in one dimension is given by For steady-state diffusion, the diffusion flux is proportional to the concentration gradient Negative

Mathematics of Diffusion (4) Nonsteady-state diffusion The diffusion flux at a particular point varies with time.

Mathematics of Diffusion (5) The diffusion equation is represented by Fick’s second law C is a function of x and t If 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 time Volume of the box: 1  dx C = mass per unit volume (concentration) Unit area cross-section

Mathematics of Diffusion (6) Solutions of diffusion equation A and B are constants and erf(z) is the error function, defined as According to error function solutions for diffusion equation, the solution for these profiles can be given by

FFF

Mathematics of Diffusion (7) For t>0, C x =C s at x=0 C x =C o at x=  Therefore A=C s Co=A+BCo=A+BB=-(C s -C o ) Boundary conditions:

Mathematics of Diffusion (8) For t=0, C x =C 1 at x<0 C x =C 2 at x>0 Therefore C 1 =A-B C2=A+BC2=A+B A= (C 1 +C 2 )/2 B=-(C 1 -C 2 )/2 Boundary conditions:

Mathematics of Diffusion (9) x For t=0, C x =0 at x h Boundary conditions: A-B-C=0 A+B-C=C o A+B+C=0 A=0 B=C o /2 C=-C o /2

Mathematics of Diffusion (10) When C x reaches a certain value at a particular position at different temperatures Therefore For example, if the same diffusion effect is obtained at two different temperatures T 1 and T 2, there is D1t1=D2t2D1t1=D2t2

Mathematics of Diffusion (11)

Factors Affecting Diffusion (1) (1)Diffusing species Diffusing 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 500 o C, D C =2.4E-12 m 2 /s (interstitial diffusion) and D Fe =3.0E-21 m 2 /s (vacancy diffusion), D C /D Fe = 8.0E8 (2) Temperature where D o = pre-exponential constant Q d = activation energy for diffusion (J/mol) R = gas constant ( J/mol-K T = absolute temperature (K) ( o C )

Factors Affecting Diffusion (2)

logD has a linear relationship with reciprocal temperature Factors Affecting Diffusion (3)  (1/K) Diffusion coefficients are usually determined by measuring these straight lines logD o is the intercept on the vertical axis

Example Problem - 1 For 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. Solution Since both cases reach the same carbon concentration at the same temperature, the left hand side of equation is the same. Therefore

Example Problem - 2

Example Problem – 2 Solution C 1 = 5 wt%; C 2 = 2 wt% C x = 2.5 wt% x = 50  m = 5  m D o = 8.5  m 2 /s Q d = J/mol T = 1023 K BBB

Example Problem – 2 Finally t =  /4.072  = s = 87 h

Other Diffusion Paths “Short-circuit” diffusion paths: d islocations, grain boundaries, external surfaces, Along these paths, the diffusion is much faster than the bulk diffusion In normal cases, the short-circuit contribution to the overall diffusion is insignificant because the path cross-section or the total boundary area is very small In special cases, e.g., the diffusion in nanomaterials, the contribution is significant because the total boundary area is very large

Diffusion of vacancy-solute complexes (1) Vacancy + solute  complexes Non-equilibrium grain boundary segregation can be produced by complex diffusion from the grain interior to the boundary There is a binding energy between vacancy and solute atom

Diffusion of vacancy-solute complexes (2) 1. Vacancy-substitutional solute complexes in bcc crystals A B C (a) (b) Migration energy of complex  (Vacancy-solute binding energy + vacancy migration energy) or vacancy-solute atom interchange energy

Diffusion of vacancy-solute complexes (3) 2. Vacancy-substitutional solute complexes in fcc crystals A B C A C D (a) (b) (c) Vacancy Solute atom Matrix atom First mechanism needs partial dissociation Second mechanism does not need partial dissociation Thus, the first mechanism needs more energy, i.e., the second mechanism is more possible

A B C D (a) (b) Diffusion of vacancy-solute complexes (4) 3. Vacancy-interstitial solute complexes in bcc crystals (1) A  B, then C  D Migration energy of complex  (solute atom migration energy + vacancy-solute binding energy) or vacancy migration energy (2) C  D, then A  B Migration energy of complex  (vacancy migration energy + vacancy-solute binding energy) or solute atom migration energy

Diffusion of vacancy-solute complexes (5) 3. Vacancy-interstitial solute complexes in fcc crystals (c) A B C D A B C E (a) (b) (d) Migration energy of complex  vacancy migration energy or solute atom migration energy Complex migration does not need its partial dissociation

Diffusion in Ionic Materials (1) Diffusion occurs by vacancy mechanism and involves two types of ions with opposite charges Formation of vacancies: Schottky defects

Diffusion in Ionic Materials (2) FeO Nonstoichiometry

Diffusion in Ionic Materials (3) Substitutional impurities e.g. in NaCl-Ca solid solution Replacement of a sodium ion by a calcium ion creates an extra positive charge. To maintain the electroneutrality, the formation of a sodium vacancy is required Diffusion processes: The diffusive migration of a single ion is the transport of electrical charge. To maintain the electroneutrality near the moving ion, another species with equal and opposite charges is required to accompany this ion’s diffusive motion Possible charged species: vacancy or electronic carrier like free electron or hole

Diffusion in Ionic Materials (4) Concepts: free electron, hole Semiconductors – n-type semiconductor and p-type semiconductor n-type semiconductor: electron conduction (an extra valence electron – excitation of the electron – a free electron) p-type semiconductor: hole conduction, a deficiency of one valence electron (hole)

FeO

Assignments Problems 6.27, 6.29, and 6.30