Diffusion and Fick s law. Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India

Transcription

1 Diffusion and Fick s law 1 Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India

2 Diffusion, flux, and Fick s law 2 Diffusion: (1) motion of one or more particles of a system relative to other particles (Onsager, 1945) (2) It occurs in all materials at all times at temperatures above the absolute zero (3) The existence of a driving force or concentration gradient is not necessary for diffusion

3 Diffusion, flux, and Fick s law 3 Gas Liquid Solid

4 Mass Transport Of Solutes 4

5 5 They all depend on Diffusion (conduction) What is diffusion? The transport of material--atoms or molecules--by random motion What is conduction? The transport of heat or electrons by random motion.

6 Diffusion 6 Mass transport by atomic motion Mechanisms Gases & Liquids random (Brownian) motion Solids vacancy diffusion or interstitial diffusion Cause (Driving force) Concentration gradient

7 Diffusion: gases and liquids 7 Glass tube filled with water. At time t = 0, add some drops of ink to one end of the tube. Measure the diffusion distance, x, over some time. x (mm) time (s) Possible to measure concentration

8 Mass Transfer 8 Mass transfer occurs when a component in a mixture goes from one point to another. Mass transfer can occur by either diffusion or convection. Diffusion is the mass transfer in a stationary solid or fluid under a concentration gradient. Convection is the mass transfer between a boundary surface and a moving fluid or between relatively immiscible moving fluids.

9 Example of Mass Transfer Mass transfer can occur by either diffusion or by convection. 9 Stirring the water with a spoon creates forced convection. That helps the sugar molecules to transfer to the bulk water much faster.

10 10 After adding milk and sugar, why do we stir our coffee? Diffusion is slow! Agitation (or stirring) can move fluids much larger distances in the same amount of time, which can accelerate the diffusion process.

11 Example of Mass Transfer 11 At the surface of the lung: Air Blood Oxygen High oxygen concentration Low carbon dioxide concentration Low oxygen concentration High carbon dioxide concentration Carbon dioxide

12 Diffusion 12 Diffusion (also known as molecular diffusion) is a net transport of molecules from a region of higher concentration to a region of lower concentration by random molecular motion.

13 Diffusion 13 A B Liquids A and B are separated from each other. A B Separation removed. A goes from high concentration of A to low concentration of A. B goes from high concentration of B to low concentration of B. Molecules of A and B are uniformly distributed everywhere in the vessel purely due to the DIFFUSION.

14 14 Place a drop of ink into a glass of water. What happens? Brownian motion causes the ink particles to move erratically in all directions. A concentration of ink particles will disperse.

15 Concentrated Region 15 Molecules in a concentrated region will disperse into the rest of the medium. The difference in concentrations is the concentration gradient. 2 C x C1 C x x C1 C2

16 16 Diffusion Constant Different materials pass through an area at different rates. Depends on concentrate Depends on medium x D C1 A O 2 in air: D = 1.8 x 10-5 m 2 /s C2 O 2 in water: D = 1.0 x 10-9 m 2 /s DNA in water: D = 1.3 x m 2 /s

17 Values for Diffusivity D 17 (gas) (liquid) (solid) Temperature Diffusivity ( C) (cm 2 /s) CO 2 -N Ar-O Ethanol(5%)-Water E-05 Water(13%)-Butanol E-05 H 2 -Ni E-08 Al-Cu E-30 Greater the diffusivity, greater the flux!

18 Diffusion & Transport 18 Whenever there is an imbalance of a commodity in a medium, nature tends to redistribute it until a balance or equality is established. This tendency is often referred to as the driving force, which is the mechanism behind many naturally occurring transport phenomena. The commodity simply creeps away during redistribution, and thus the flow is a diffusion process. The rate of flow of the commodity is proportional to the concentration gradient dc/dx, which is the change in the concentration C per unit length in the flow direction x, and the area A normal to flow direction. k diff is the diffusion coefficient of the medium, which is a measure of how fast a commodity diffuses in the medium, and the negative sign is to make the flow in the positive direction a positive quantity (note that dc/dx is a negative quantity since concentration decreases in the flow direction).

19 The diffusion coefficients and thus diffusion rates of gases depend strongly on temperature. The diffusion rates are higher at higher temperatures. The larger the molecular spacing, the higher the diffusion rate. Diffusion rate: gases > liquids > solids 19

20 Analogy Between Heat And Mass Transfer 20 We can develop an understanding of mass transfer in a short time with little effort by simply drawing parallels between heat and mass transfer. Temperature The driving force for mass transfer is the concentration difference. Both heat and mass are transferred from the more concentrated regions to the less concentrated ones. If there is no difference between the concentrations of a species at different parts of a medium, there will be no mass transfer.

21 Fick s Law During diffusion we assume particles move in the direction of least density. They move down the concentration gradient Gradient ( x, y, ) z 21 In mathematical terms we will assume J Du Adolph Fick Born: 1829 Cassel, Germany Died: 1879 Where D is a constant of proportionality called the Diffusion Coefficient

22 Fick s Law 22 The rate of diffusion depends on the concentration gradient, area (A), and diffusion constant (D). J C DA C t 1 2 J measures mol/m 3. There is a diffusion time t for diffusion in one direction over a distance Dx. t 2 x 2D

23 In each of these examples, molecules (or heat) are moving down a gradient! 23 (From an area of high concentration to an area of low concentration) Fick s Law: J i dci D dz Ji is called the flux. It has units of amount of material diffused ( l 2 )( t) D is called the diffusion coefficient. It has units of l 2 t

24 Mass is transferred by conduction (called diffusion) and convection only. Rate of mass diffusion Fick s law of diffusion 24 D AB is the diffusion coefficient (or mass diffusivity) of the species in the mixture C A is the concentration of the species in the mixture

25 25 What do mass transfer processes have in common? 1) Hydrogen embrittlement of pressure vessels in nuclear power plants 2) Flow of electrons through conductors 3) Dispersion of pollutants from smoke stacks 4) Transdermal drug delivery 5) Influenza epidemics 6) Chemical reactions 7) Absorption of oxygen into the bloodstream

26 Unsteady state diffusion Back to a drop of ink in a glass of water If consider diffusion in the z- direction only: How does the concentration profile change with time? t = 0 t (add ink drop all ink located at z = 0) 26 z z=0 A measure of the spread due to diffusion is the diffusion length L d = (4Dt)0.5, where D is the diffusivity coefficient and t is time. Note: for small time, spreading is quick, but for long times it slows down. That s why you stir your coffee after adding cream. Diffusion doesn t work fast enough over long distances.

27 Fick s first law 27 D dc J D dx diffusion coefficient c c x x J J D D dc dx dc dx 0 0

28 Continuum description of diffusion 28 We need to derive a differential equation for this purpose. Divide a box of particles into small cubic bins of size L j+ j x-l x x+l j: flux of particles (number of particles per unit area per unit time) c: concentration of particles (number of particles per unit volume) Random walk in 1D; half of particles in each bin move to the left and half to the right.

29 Right, left and total fluxes at x are given by 29 j j 1 c( x L / 2) AL 2 c( x L / 2) AL, j At At L j j c( x L / 2) c( x L / 2) 2t 1 2 Taylor expanding the concentrations for small L gives j L 2t c( x) L dc 2 dx c( x) L dc 2 dx 2 L 2t dc dx D dc dx, Fick' s law Generalise to 3D: j D c Flux direction: particles move from high concentration to low concentration

30 Conservation laws: Total number of particles is conserved. If there is a net flow of particles inside a bin, j j the concentration inside must increase by the same amount. c( x, t t) c( x, t) AL j( x L / 2) j( x L / 2) At c(x,t) x-l/2 x x+l/2 30 c t L j( x) L 2 dj dx j( x) L 2 dj dx Generalise to 3D: dc dt dc dt dj dx j (similar to charge conservation)

31 Integrate the conservation equation over a closed volume V with N part s 31 V dc dt dv V j dv d dt V c dv S jn da (Divergence theorem) dn dt (rate of change of N = - total flux out) We can use the conservation equation to eliminate flux from Fick s eq n. dc j D, dx 2 dj d c D dx 2 dx dc dt D d 2 dx c 2 Fick s 2 nd law Diffusion eq. Generalise to 3D: dc D 2 c dt (analogy with the Schroedinger Eq.)

32 32 Once the initial conditions are specified, the diffusion equation can be solved numerically using a computer. Special cases: 1. Equilibrium: c(x)=const. j = 0, c is uniform and constant 2. Steady-state diffusion: c(x) = c 0 for x < 0 and c(x) = c L for x >L No time dependence, j D c0 b, dc dx c, L j D dc dt c 0, L c 0 j D x b dj dx 0, c j D L j c L const. 0 cl c0 c x c0 for 0 x L L

33 Diffusion, flux, and Fick s law 33 Fick s second law: mass balance of fluxes Analogy: gain or loss in your bank account per month = Your salary ($$ per month) - what you spend ($$ per month)

34 Diffusion, flux, and Fick s law 34 Fick s second law: mass balance of fluxes 1. We need to solve the partial differential diffusion equation. (a) analytical solution (e.g., Crank, 1975) or (b) numerical methods 2. We need to know initial and boundary conditions. This is straightforward for exercise cases, less so in nature. 3. We need to know the diffusion coefficient

35 35 c c x c x time x How can we find out c(x,t)?

36 N = number of particles in volume V 36 Flux in 1 dimension: A J (x) V x J (x+x) J(x+x) J(x) x J x N t J( x) A A dj dx J( x x) A x V dj dx A[ J( x dc dt x) d ( N / V ) dt J( x)] A dj dx J

37 How can we find out c(x,t)? 37 dc dt dj dx J D Fick's dc dx law dc dt D d 2 dx c 2 We can find out c(x,t)! Fick s second law

38 38 Fick s Laws Combining the continuity equation with the first law, we obtain Fick s second law: 2 c t D x c 2

39 39 Solutions to Fick s Laws depend on the boundary conditions. Assumptions D is independent of concentration Semiconductor is a semi-infinite slab with either Continuous supply of impurities that can move into wafer Fixed supply of impurities that can be depleted

40 Solutions To Fick s Second Law 40 The simplest solution is at steady state and there is no variation of the concentration with time Concentration of diffusing impurities is linear over distance This was the solution for the flow of oxygen from the surface to the Si/SiO 2 interface. D 2 x c 2 c( x) a 0 bx

41 Solutions To Fick s Second Law 41 For a semi-infinite slab with a constant (infinite) supply of atoms at the surface c( x, t) c o erfc 2 x Dt The dose is Q c x, t dx 2c Dt 0 0

42 Solutions To Fick s Second Law 42 Complimentary error function (erfc) is defined as erfc(x) = 1 - erf(x) The error function is defined as erf z 2 ( z) exp 2 d 0 This is a tabulated function. There are several approximations. It can be found as a built-in function in MatLab, MathCad, and Mathematica

43 Impurity concentration, c(x) Solutions To Fick s Second Law 43 This solution models short diffusions from a gas-phase or liquid phase source Typical solutions have the following shape c0 c ( x, t ) D3t3 > D2t2 > D1t cb Distance from surface, x

44 Solutions To Fick s Second Law 44 Constant source diffusion has a solution of the form Q x c( x, t) e 4Dt Dt 2 Here, Q is the dose or the total number of dopant atoms diffused into the Si Q c( x, t) dx 0 The surface concentration is given by: c(0, t) Q Dt

45 Impurity concentration, c(x) Solutions To Fick s Second Law 45 Limited source diffusion looks like c01 c ( x, t ) c02 c03 D3t3 > D2t2 > D1t cb Distance from surface, x

46 Fick s Second Law: The Diffusion Equation Consider a small region of space (volume for 3D, area for 2D) 46 Jx(x) molecules flow in and Jx(x+dx) molecules flow out (per unit area or distance per unit time). N N t J x x J x x dx dy c t J xx J x x dx dx since c N / dxdy J x x J x x dx J x x definition of the derivative c t D 2 c x 2 since J x D c x

47 47 Flux in 1 dimension: dc dt D 2 d c 2 dx Flux in 2 dimensions: dc dt 2 2 d c d c D 2 2 dx dy If there is a chemical reaction: F number of particles produced volume time dc dt 2 2 d c d c F D 2 2 dx dy For biological populations: dp dt 2 2 d p d p F( p) D 2 2 dx dy

48 48 "Predep" controlled dose Silicon Drive-in constant dose Diffusion is the redistribution of atoms from regions of high concentration of mobile species to regions of low concentration. It occurs at all temperatures, but the diffusivity has an exponential dependence on T. Predeposition: doping often proceeds by an initial predep step to introduce the required dose of dopant into the substrate. Drive-In: a subsequent drive-in anneal then redistributes the dopant giving the required x J and surface concentration. Ion Implantation and Annealing Solid/Gas Phase Diffusion Advantages Room temperature mask No damage created by doping Precise dose control Batch fabrication atoms cm -2 doses Accurate depth control Problems Implant damage enhances diffusion Dislocations caused by damage may cause junction leakage Implant channeling may affect profile Usually limited to solid solubility Low surface concentration hard to achieve without a long drivein Low dose predeps very difficult

49 49 Dopants are soluble in bulk silicon up to a maximum value before they precipitate into another phase. Dopants may have an electrical solubility that is different than the solid solubility defined above. One example - As 4 V - electrically inactive complex.

50 Types of diffusion 50 Interstitial diffusion - C in Fe, H in Ti, etc. Vacancy diffusion Self diffusion, tracer diffusion - healing of defects, timedependent mechanical phenomena Chemical diffusion - diffusion of one element through another, solid state chemical reactions Bulk diffusion Diffusion in grain boundaries, lattice defects Surface diffusion Diffusion is a statistical phenomenon, driven by the random motion of atoms.

51 51 Diffusion is a thermally activated process, resulting in an exponential temperature dependence: D = D 0 exp(-q/kt) It can also depend on orientation relative to the crystal axes. Interstitials diffuse much faster than substitutional impurities and matrix atoms.

52 Phenomenological description 52 Fick s law: j D c j D in cm 2 /s; c in particles/cm 3 Continuity: Combined: If D = const. c t j D 0 c t Dc c t D2 c To be solved for c(x,t) or c(r,t) Typical solution depends on x/(dt) 1/2. One can define a diffusion front that travels as x 2 ~ 6Dt If defined that way, about 99% of the diffusing element is still behind the diffusion front.

53 Rough correlations 53 D 0 for self diffusion in metals ~ 1 cm 2 /s (0.1 to 10) H D / T m ~ 1.5*10-3 ev/k H D / L m ~ 15 These relationships are useful if no measured data are available. Typically estimates of diffusion are quite rough, there is a huge variation, high accuracy is not needed. Also, defects and impurities influence diffusion very much, it is difficult to control it.

54 Examples 54 Temperature is uniformly T = 912 C, the alpha-gamma transition of iron. Diffusion time t = 1 h =3600 s, distance x 2 = 6Dt Self diffusion in alpha Fe: D 0 = 2.0 cm 2 /s, H D = 2.5 ev D = 2.0 exp(-2.5/8.6*10-5 *11850) = 4.4*10-11 cm 2 /s, x ~ 10 µm Self diffusion in gamma Fe: D 0 = 0.4 cm 2 /s, H D = 2.8 ev D = 4.6*10-13 cm 2 /s, x ~ 1 µm 60 Co in alpha Fe: D 0 = 0.2 cm 2 /s, H D = 2.4 ev D = 1.2*10-11 cm 2 /s, x ~ 5 µm C in alpha Fe: D 0 = cm 2 /s, H D = 0.83 ev D = 1.2*10-6 cm 2 /s, x ~ 1.6 mm C in gamma Fe: D 0 = 0.67 cm 2 /s, H D = 1.6 ev D = 1.0*10-7 cm 2 /s, x ~ 0.5 mm Notice the very large differences.

55 Vacancy diffusion in a solid and dislocation climb due to capturing vacancies 55

56 Surface and interface diffusion 56 More open, diffusion can take place faster. In fact, dislocations provide routes for fast diffusion also. Diffusion in grain boundaries makes predicting the behavior of polycrystals difficult. Activation energy is lowest for surface diffusion, larger for grain boundary diffusion, and the highest for bulk diffusion. Therefore, bulk diffusion dominates at high temperature (larger number of possible sites) but grain boundary and surface diffusion takes over at lower temperature. Technological importance: Thin film deposition, weakening of grain boundaries by impurities, shape changes of particles, etc. Diffusion can be accelerated by introducing grain boundaries and other lattice defects. This is an important benefit from MA (mechanical alloying).

57 Interdiffusion 57 Interdiffusion: in alloys, atoms tend to migrate from regions of large concentration. Initially After some time 100% 0 Concentration Profiles

58 Self diffusion 58 Self diffusion: In an elemental solid, atoms also migrate. Label some atoms After some time C A D B This can be observed on metal surface under UHV conditions with scanning tunneling microscopy

59 Vacancy diffusion or interstitial diffusion applies to substitutional impurities atoms exchange with vacancies rate depends on (1) number of vacancies (2) activation energy to exchange. Concentration of Vacancies at temp T c i ni N e E K T B 59 Vacancy Interstitial The two processes have an activation energy (ev/atom).

60 Substitution-diffusion 60 Vacancy Diffusion: applies to substitutional impurities atoms exchange with vacancies rate depends on (1) number of vacancies; (2) activation energy to exchange. increasing elapsed time

61 Interstitial diffusion 61 Interstitial diffusion smaller atoms diffuse between atoms. More rapid than vacancy diffusion Concentration gradient

62 Processing using Diffusion 62 Case Hardening: -Diffuse carbon atoms into the host iron atoms at the surface. -Example of interstitial diffusion is a case hardened gear. Result: The "Case" is -hard to deform: C atoms "lock" planes from shearing. -hard to crack: C atoms put the surface in compression.

63 Processing using Diffusion 63 Doping Silicon with P for n-type semiconductors: Process: 1. Deposit P rich layers on surface. silicon 2. Heat it. 3. Result: Doped semiconductor regions. silicon

64 Modeling rate of diffusion: flux 64 Flux: Q m 1 A d dt Flux can be measured for: - vacancies - host (A) atoms - impurity (B) atoms Empirically determined: Make thin membrane of known surface area Impose concentration gradient Measure how fast atoms or molecules diffuse through the membrane A = Area of flow

65 Steady-state Diffusion 65 Concentration Profile, C(x): [kg/m 3 ] Cu flux Ni flux Concentration of Cu [kg/m 3 ] Concentration of Ni [kg/m 3 ] Position, x Fick's First Law: Q m, x D dc dx D = m 2 /s D=diffusion coefficient The steeper the concentration profile, the greater the flux!

66 Steady-State Diffusion 66 Steady State: concentration profile not changing with time. Qm,r Qm,l C(x) Qm,r = Qm,l Apply Fick's First Law: Q m, x D dc dx since Qm,l = Qm,r then dc dx left dc dx right The slope, dc/dx, must be constant (i.e., doesn't vary with position)

67 Steady-State Diffusion 67 Rate of diffusion independent of time Q m, x D dc dx C1 C1 C2 C2 if linear dc dx C x C x 2 2 C x 1 1 x1 x x2

68 Example: C Diffusion in steel plate 68 Steel plate at C with geometry shown: Knowns: C1= 1.2 kg/m 3 at 5mm (5 x 10 3 m) below surface. 700 C C2 = 0.8 kg/m 3 at 10mm (1 x 10 2 m) below surface. D = 3 x10-11 m 2 /s at 700 C. In steady-state, how much carbon transfers from the rich to the deficient side?

69 Example: Chemical Protection Clothing Methylene chloride is a common ingredient of paint removers. Besides being an irritant, it also may be absorbed through skin. When using, protective gloves should be worn. 69 If butyl rubber gloves (0.04 cm thick) are used, what is the diffusive flux of methylene chloride through the glove? C1 glove paint remover x1 x2 2 t b 6D C2 skin Data: D in butyl rubber: D = 110 x10-8 cm 2 /s surface concentrations: C1 = 0.44 g/cm 3 C2 = 0.02 g/cm 3 Diffusion distance: x2 x1 = 0.04 cm J x = -D C 2 - C 1 x 2 - x 1 = 1.16 x 10-5 g cm 2 s

70 Example: Diffusion of radioactive atoms 70 Surface of Ni plate at C contains 50% Ni 63 (radioactive) and 50% Ni (non-radioactive). 4 microns below surface Ni 63 /Ni = 48:52 Lattice constant of Ni at 1000 C is nm. Experiment shows that self-diffusion of Ni is 1.6 x 10-9 cm 2 /sec What is the flux of Ni 63 atoms through a plane 2 m below surface? C 1 (4Ni /cell)(0.5ni63 /Ni) (0.36x10 9 m) 3 /cell 42.87x10 27 Ni 63 /m 3 C 2 (4Ni /cell)(0.48ni63 /Ni) (0.36x10 9 m) 3 /cell 41.15x10 27 Ni 63 /m 3 (1.6x10 13 m 2 /sec) ( )x1027 Ni 63 /m 3 (4 0)x10 6 m 0.69x10 20 Ni 63 /m 2 s How many Ni 63 atoms/second through cell? J (0.36nm)2 9 Ni 63 /s

71 Non-Steady-State Diffusion 71 Concentration profile, C(x), changes w/ time. To conserve matter: Fick's First Law: Governing Eqn.:

72 Non-Steady-State Diffusion: another look 72 Concentration profile, C(x), changes w/ time. Rate of accumulation C(x) C t dx J x J xdx C t dx J x (J x J x x dx) J x x dx Using Fick s Law: C t J x x x D C x Fick s 2nd Law If D is constant: Fick's Second"Law" c t x D c x D c 2 x 2

73 Non-Steady-State Diffusion: C = c(x,t) 73 concentration of diffusing species is a function of both time and position Fick's Second"Law" c t D c 2 x 2 Copper diffuses into a bar of aluminum. Cs B.C. at t = 0, C = Co for 0 x at t > 0, C = C S for x = 0 (fixed surface conc.) C = Co for x = Adapted from Fig. 6.5, Callister & Rethwisch 3e.

74 Non-Steady-State Diffusion 74 Cu diffuses into a bar of Al. C S C(x,t) Fick's Second"Law": c t D c 2 x 2 C o Solution: "error function Values calibrated in Table 6.1 erf(z) 2 2 ey dy 0 z

75 Example: Non-Steady-State Diffusion 75 FCC iron-carbon alloy initially containing 0.20 wt% C is carburized at an elevated temperature and in an atmosphere that gives a surface C content at 1.0 wt%. If after 49.5 h the concentration of carbon is 0.35 wt% at a position 4.0 mm below the surface, what temperature was treatment done? Solution C( x, t) C C C s o o erf 2 1 erf( z) Using Table 6.1 find z where erf(z) = Use interpolation. z So, z = x z 2 Dt D x2 (4 x 10 3 m) 2 4z 2 t (4)(0.93) 2 (49.5 h) Now solve for D x Dt D 2 x 4z 2 1 h 3600 s 2.6 x 1011 m 2 /s t erf(z) = z erf(z) z

76 76 Solution (cont.): To solve for the temperature at which D has the above value, we use a rearranged form of Equation (6.9a); D=D 0 exp(-q d /RT) T Qd R( lnd lnd) o From Table 6.2, for diffusion of C in FCC Fe Do = 2.3 x 10-5 m 2 /s Q d = 148,000 J/mol D 2.6 x m 2 /s T 148,000 J/mol (8.314 J/mol-K)(ln 2.3x10 5 m 2 /s ln 2.6x10 11 m 2 /s) T = 1300 K = 1027 C

77 Example: Processing 77 Copper diffuses into a bar of aluminum. 10 hours processed at 600 C gives desired C(x). How many hours needed to get the same C(x) at 500 C? Key point 1: C(x,t500C) = C(x,t600C). Key point 2: Both cases have the same Co and Cs. Result: Dt should be held constant. C(x,t) C o x 1 erf C s C o 2 Dt Answer: Note D(T) are T dependent! Values of D are provided.

78 Diffusion and Temperature 78 Diffusivity increases with T exponentially (so does Vacancy conc.). D Do exp E ACT R T D Do = diffusion coefficient [m2/s] = pre-exponential [m2/s] E ACT = activation energy [J/mol or ev/atom] R T = gas constant [8.314 J/mol-K] = absolute temperature [K]

79 Experimental Data: T(C) D (m 2 /s) Dinterstitial >> Dsubstitutional C in -Fe Cu in Cu C in -Fe Al in Al Fe in -Fe Fe in -Fe Zn in Cu 1000K/T Callister & Rethwisch 3e. from E.A. Brandes and G.B. Brook (Ed.) Smithells Metals Reference Book, 7th ed., Butterworth-Heinemann, Oxford, 1992.)

80 Example: Comparing Diffuse in Fe 80 Is C in fcc Fe diffusing faster than C in bcc Fe? fcc-fe: D0=2.3x10 5 (m 2 /s) EACT=1.53 ev/atom T = 900 C D=5.9x10 12 (m 2 /s) bcc-fe: D0=6.2x10 7 (m 2 /s) EACT=0.83 ev/atom T = 900 C D=1.7x10 10 (m 2 /s) D (m 2 /s) 10-8 T(C) K/T FCC Fe has both higher activation energy EACT and D0 (holes larger in FCC). BCC and FCC phase exist over limited range of T (at varying %C). Hence, at same T, BCC diffuses faster due to lower EACT. Cannot always have the phase that you want at the %C and T you want! which is why this is all important.

81 More than one diffusing species 81 Diffusion is usually asymmetric, the boundary between regions is shifting, often difficult to define. Very importantly, solid state reactions always take place at interfaces via diffusion. The products form intermediate layers that often act as diffusion barriers. Decreasing the particle size - e.g. by ball milling - accelerates the reactions. Diffusion relates to volume changes, formation of voids, etc.

82 The Kirkendall experiment 82 Zn diffuses faster than Cu in this system. As a result, the Cu - brass interface shifts, the Mo wires marking the interface drift toward each other.

83 83

84 84 Diffusion coefficient: Tracer Di*= tracer diffusion coefficient w = frequency of a jump to an adjacent site l = distance of the jump f = related to symmetry, coordination number e.g., diffusion of 56 Fe in homogenous olivine (Mg, Fe) 2 SiO 4 )

85 Diffusion coefficient: multicomponent 85 Multicomponent formulation (Lasaga, 1979) for ideal system, elements with the same charge and exchanging in the same site e.g., diffusion of FeMg olivine

86 Diffusion coefficient 86 Perform experiments at controlled conditions to determine D* or D FeMg Q = activation energy (at 10 5 Pa), ΔV =activation volume, P = pressure in Pascals, R is the gas constant, and Do = pre-exponential factor.

87 87 Diffusion coefficient New experimental and analytical techniques allow to determine D at the conditions (P, T, fo 2, ai) relevant for the magmatic processes without need to extrapolation e.g., Fe-Mg in olivine along [001] Dohmen and Chakraborty (2007)

88 Continuum Concept 88 Properties are averaged over small regions of space. The number of blue circles moving left is larger than the number of blue circles moving right. We think of concentration as being a point value, but it is averaged over space.

89 89 Diffusion Coefficient Probability of a jump is P P P f e e j v m E kt E kt Diffusion coefficient is proportional to jump probability D D e E D kt 0 m

90 Diffusion Coefficient 90 Typical diffusion coefficients in silicon Element D o (cm 2 /s) E D (ev) B Al Ga In P As Sb

91 Diffusion coefficient, D (cm 2 /sec) Diffusion coefficient, D (cm 2 /sec) Diffusion Of Impurities In Silicon 91 Arrhenius plots of diffusion in silicon Temperature ( o C) Temperature ( o C) Fe Li Cu Al 10-7 Au In Ga B,P Sb As Temperature, 1000/T (K -1 ) Temperature, 1000/T (K -1 )

92 Diffusion Of Impurities In Silicon 92 The intrinsic carrier concentration in Si is about 7 x /cm 3 at 1000 o C If N A and N D are <n i, the material will behave as if it were intrinsic; there are many practical situations where this is a good assumption

93 Diffusion Of Impurities In Silicon 93 Dopants cluster into fast diffusers (P, B, In) and slow diffusers (As, Sb) As we develop shallow junction devices, slow diffusers are becoming very important B is the only p-type dopant that has a high solubility; therefore, it is very hard to make shallow p-type junctions with this fast diffuser

94 Successive Diffusions 94 To create devices, successive diffusions of n- and p-type dopants Impurities will move as succeeding dopant or oxidation steps are performed The effective Dt product is ( ) D ( t t D t D t Dt eff ) No difference between diffusion in one step or in several steps at the same temperature If diffusions are done at different temperatures ( Dt) eff D1t 1 D2t2

95 95 Successive Diffusions The effective Dt product is given by Dt eff i D i t i D i and t i are the diffusion coefficient and time for i th step Assuming that the diffusion constant is only a function of temperature. The same type of diffusion is conducted (constant or limited source)

96 96 Interstitial diffusion C diffusion in bcc Iron (steel) Li diffusion in transition metal oxide host O diffusion on Pt-(111) surface In all examples, diffusion occurs on a rigid lattice which is externally imposed by a host or substrate

97 Example of interstitial diffusion 97

98 Intercalation Oxide as Cathode in Rechargeable Lithium Battery 98 LixCoO2

99 Individual hops: Transition state theory 99 E B *exp kt E B

100 100 Migration mechanism in Li x CoO 2 Lithium Cobalt Oxygen Single vacancy hop mechanism Divacancy hop Mechanism

101 101 Lithium Single vacancy hop Divacancy hop Cobalt Oxygen

102 Irreversible thermodynamics: interstitial diffusion of one component 102 J L D L d dc J DC

103 103 Notation M = number of lattice sites N = number of diffusing atoms v s = volume per lattice site x = N/M C=x/v s

104 Interstitial diffusion: one component Kubo-Green relations (linear response statistical mechanics) 104 D L Thermodynamic factor Kinetic coefficient L 1 (2d)tMv s kt C N i1 R i t 2 R. Gomer, Rep. Prog. Phys. 53, 917 (1990)/ A. Van der Ven, G. Ceder, Handbook of Materials Modeling, chapt. 1.17, Ed. S. Yip, Springer (2005).

105 ~ 105 D D J Diffusion coefficient at 300 K Thermodynamic factor A. Van der Ven, G. Ceder, M. Asta, P.D. Tepesch, Phys Rev. B 64 (2001)

106 106 Interstitial diffusion (two components) C & N diffusion in bcc Iron (steel) Li & Na diffusion in transition metal oxide host O & S diffusion on Pt-(111) surface In all examples, diffusion occurs on a rigid lattice which is externally imposed by a host or substrate

107 Diffusion in an alloy: substitutional diffusion 107 Not interstitial diffusion Instead, diffusing atoms form the lattice Dilute concentration of vacancies