5-DIFFUSION-CHPT 5-MIAE 221-2023

Page_140b Topic# 5A - MIAE 221 (adapted from Callister) 3

Topic# 5A - MIAE 221 (adapted from Callister) 6 DIFFUSION MECHANISMS Movement of Matter Through Matter by atomic motion GASES - quick movement LIQUID - Slower (ink etc) SOLIDS - Much slower In solids, atoms are in fixed positions so movement is more difficult than liquid or gas. Atoms vibrate so some movement possible (above absolute zero). For atoms to diffuse in solids, require : • an empty adjacent site (e.g. vacancy) • sufficient energy to break bonds with neighbours and squeeze through gap .

Topic# 5A - MIAE 221 (adapted from Callister) 9 Self-diffusion : In an elemental solid, the same atoms can also migrate. Label some atoms After some time A B C D

Topic# 5A - MIAE 221 (adapted from Callister) 12 VACANCY or SUBSTITUTIONAL DIFFUSION Requires vacancies (empty sites) and energy (to squeeze through between atoms). As T ↑ , number of vacancies ↑ , and energy ↑ , so diffusion is faster as temperature increases. Self - diffusion eg., Cu in Cu Substitutional diffusion in alloy eg., Ni in Cu INTERSTITIAL DIFFUSION Interstitial atom moves from one Interstitial site to another (empty) site. Energy is needed, again to squeeze past atoms but adjacent interstsitial site is usually empty. eg Carbon in Fe Usually much faster because many more empty interstitial sites and no vacancies are required.

Topic# 5A - MIAE 221 (adapted from Callister) 15 dt dM A l At M J = = Diffusion rates • Diffusion is a time-dependent process • The rate of diffusion is expressed as the Flux, J: • Measured empirically – Make thin film (membrane) of known surface area – Impose concentration gradient – Measure how fast atoms or molecules diffuse through the membrane 𝐽𝐽 ≡ Flux ≡ moles ( or mass ) diffusing surface area time = mol cm 2 𝑠𝑠 or kg 𝑚𝑚 2 𝑠𝑠 M = mass diffused time J ∝ slope

18 Diffusion Example • Methylene chloride is a common ingredient of paint removers. Besides being an irritant, it also may be absorbed through skin. When using this paint remover, protective gloves should be worn. • Let's investigate whether butyl rubber gloves (0.04 cm thick) commonly found in the kitchen can be used as protective gloves. • Note: The maximum allowable flux for a 150 lb person is less than 3.5 x 10 -7 g/cm 2 /s • Compute the diffusion flux of methylene chloride through the gloves. Chemical Protective Clothing (CPC)

Topic# 5B - MIAE 221 (adapted from Callister) 21 Non-steady State Diffusion Boundary conditions: • at t = 0, C = C o for 0 ≤ x ≤ ∞ • at t > 0, C = C S for x = 0 (const. surf. conc.) • C = C o for x = ∞ • Copper diffuses into a bar of aluminum. pre-existing conc., C o of copper atoms Surface conc., C of Cu atoms bar s C s

Topic# 5B - MIAE 221 (adapted from Callister) 22 Solution : C ( x , t ) = Conc. at point x at time t erf ( z ) = error function 𝐶𝐶 𝑥𝑥 , 𝑡𝑡 − 𝐶𝐶 𝑜𝑜 𝐶𝐶 𝑠𝑠 − 𝐶𝐶 𝑜𝑜 = 1 − erf 𝑥𝑥 2 𝐷𝐷𝑡𝑡 dy e y z 2 0 2 − ∫ π =

Topic# 5B - MIAE 221 (adapted from Callister) 23 If Z =0.25 then erf(Z) = 0.2763

Topic# 5B - MIAE 221 (adapted from Callister) 24 CARBURISING Diffusing carbon into steel to harden surface layers. Carbon content in steel surface increases with time, ie. concentration gradient is not constant Require a TOUGH material to withstand shock loadings etc. and also require a HARD material to resist WEAR . E.g. GEARS AND SHAFTS These can be made of a TOUGH steel which has its surface hardened by gas carburizing or nitriding, ie: put N or C atoms into surface layers (interstitials) and the Hardness increases. Tough but relatively soft steel is forged/machined into gears then CASE HARDENING carried out to harden surface.

Topic# 5B - MIAE 221 (adapted from Callister) 25

Topic# 5B - MIAE 221 (adapted from Callister) 26 Diffusion and Temperature • Diffusion coefficient increases with increasing T . D = D o exp       − Q d R T = pre-exponential [m 2 /s] = diffusion coefficient [m 2 /s] = activation energy [J/mol or eV/atom] = gas constant [8.314 J/mol-K] = absolute temperature [K] D D o Q d R T

Topic# 5B - MIAE 221 (adapted from Callister) 27 Diffusion and Temperature D has exponential dependence on T D interstitial >> D substitutional C in α -Fe C in γ -Fe Al in Al Fe in α -Fe Fe in γ -Fe

Topic# 5B - MIAE 221 (adapted from Callister) 28 Non-steady State Diffusion Sample Problem: An FCC iron-carbon alloy initially containing 0.20 wt% C is carburized at an elevated temperature and in an atmosphere that gives a surface carbon concentration constant 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, determine the temperature at which the treatment was carried out. • Solution : use Eqn. 5.5 to find D:       − = − − Dt x C C C t x C o s o 2 erf 1 ) , ( And then to find T, use: D = D o exp(-Q d /RT)

Topic# 5B - MIAE 221 (adapted from Callister) 29 ∴ erf( z ) = 0.8125 – t = 49.5 h x = 4 x 10 -3 m – C x = 0.35 wt% C s = 1.0 wt% – C o = 0.20 wt% Solution (cont.):       − = − − Dt x C C C ) t , x ( C o s o 2 erf 1 ) ( erf 1 2 erf 1 20 . 0 0 . 1 20 . 0 35 . 0 ) , ( z Dt x C C C t x C o s o − =       − = − − = − −

Topic# 5B - MIAE 221 (adapted from Callister) 30 /s m 10 x 6 . 2 s 3600 h 1 h) 5 . 49 ( ) 93 . 0 ( ) 4 ( m) 10 x 4 ( 4 2 11 2 2 3 2 2 − − = =         = ∴ t z x D z = 0.93 7970 . 0 8209 . 0 7970 . 0 8125 . 0 90 . 0 95 . 0 90 . 0 − − = − − z Solution (cont.): We must now determine from Table 5.1 the value of z for which the error function is 0.8125. An interpolation is necessary as follows z erf( z) 0.90 0.7970 z 0.8125 0.95 0.8209 Now solve for D Dt x z 2 = t z x D 2 2 4 =

Topic# 5B - MIAE 221 (adapted from Callister) 31 /s) m 10 x 6 . 2 ln /s m 10 x 3 . 2 K)(ln - J/mol 314 . 8 ( J/mol 000 , 148 2 11 2 5 − − − = T And from Table 5.2, for diffusion of C in FCC Fe we get: D o = 2.3 x 10 -5 m 2 /s Q d = 148,000 J/mol ) ln ln ( D D R Q T o d − = To solve for the temperature at which D has above value, we use a rearranged form of this equation: D = D o exp(-Q d /RT) ∴ Solution (cont.): T = 1300 K = 1027°C

Topic# 5B - MIAE 221 (adapted from Callister) 32 Example 2: At 300ºC the diffusion coefficient and activation energy for Cu in Si are D (300ºC) = 7.8 x 10 -11 m 2 /s Q d = 41.5 kJ/mol What is the diffusion coefficient at 350ºC?         − =         − = 1 0 1 2 0 2 1 ln ln and 1 ln ln T R Q D D T R Q D D d d         − − = = − ∴ 1 2 1 2 1 2 1 1 ln ln ln T T R Q D D D D d transform data D Temp = T ln D 1/ T

Topic# 5B - MIAE 221 (adapted from Callister) 33               − − = 1 2 1 2 1 1 exp T T R Q D D d Example (cont.)             − − = − K 573 1 K 623 1 K - J/mol 314 . 8 J/mol 500 , 41 exp /s) m 10 x 8 . 7 ( 2 11 2 D T 1 = 273 + 300 = 573K T 2 = 273 + 350 = 623K D 2 = 15.7 x 10 -11 m 2 /s

Topic# 5B - MIAE 221 (adapted from Callister) 34 𝐶𝐶 ( 𝑥𝑥 , 𝑡𝑡 ) − 𝐶𝐶 𝑜𝑜 𝐶𝐶 𝑠𝑠 − 𝐶𝐶 𝑜𝑜 = 1 − erf 𝑥𝑥 2 𝐷𝐷𝑡𝑡 Varying temperature and time If a certain fixed composition is required, C 1 then the LHS of the equation is constant: 𝐶𝐶 ( 𝑥𝑥 , 𝑡𝑡 ) −𝐶𝐶 𝑜𝑜 𝐶𝐶 𝑠𝑠 −𝐶𝐶 𝑜𝑜 = constant This means that RHS of the equation is also constant: 𝑥𝑥 2 𝐷𝐷𝑡𝑡 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑡𝑡𝑐𝑐𝑐𝑐𝑡𝑡 𝑐𝑐𝑜𝑜 𝑥𝑥 2 𝐷𝐷𝑡𝑡 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑡𝑡𝑐𝑐𝑐𝑐𝑡𝑡 So for the same system and at the same position, x : D 1 t 1 = D 2 t 2

Diffusion in Semiconductors Integrated circuit (IC) chip typically 6x6x0.4mm with millions of devices on it. Doping is required to change the behavior of the silicon single crystal. • This is done by diffusion: 1. Predeposition step – impurity atoms diffused from surface (gas) into silicon (≈ 900 o C, <1 hr) . 2. Drive-in diffusion step – no more atoms added but those from step 1 are moved further into the chip (higher T) and an oxide layer is formed to prevent diffusion outwards. (up to 1200 o C). Topic# 5B - MIAE 221 (adapted from Callister) 35

Topic# 5B - MIAE 221 (adapted from Callister) 36 Doping Silicon with Phosphorous for n-type semiconductors: 1. Predeposition (step 1) : - deposit P rich layers on surface and then heat it. Phosphorous diffuses into silicon to give doped semiconductor regions. silicon silicon PROCESSING USING DIFFUSION 2. Heat in oxygen-rich environment to produce silica (silicon oxide) barrier on surface (which stops more dopant going in or coming out of the silicon) and “ drive-in ” (step 2) the dopant to make doping more uniform.

Topic# 5B - MIAE 221 (adapted from Callister) 37 Step 1 follows the normal solution of Ficks 2 nd law. We can change the time of the “Drive-in” step to change the profile thus:

38 SUMMARY • Atoms can move through solids by: • Substitutional (vacancy) diffusion or • Interstitial diffusion • Diffusion is a time-dependent process. • Fick’s first law of diffusion is used to calculate flux in Steady State conditions

Topic# 5B - MIAE 221 (adapted from Callister) 39 Diffusion FASTER for... • open crystal structures • materials w/secondary bonding • smaller diffusing atoms • lower density materials Diffusion SLOWER for... • close-packed structures • materials w/covalent bonding • larger diffusing atoms • higher density materials Summary • Fick’s Second Law of Diffusion - non-steady state diffusion • Diffusion coefficient Strong effect of temperature