VACANCY DIFFUSION. substitutional impurity atom

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Aron Lloyd
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1 CASE HARDENING This steel gear has been case hardened -- a process in which carbon atoms have been diffused into the surface of the steel. This increases the surficial carbon content of the steel, making it harder and more wear resistant. The interior that is not carbon enriched remains softer and nonbrittle.

2 VACANCY DIFFUSION substitutional impurity atom host atom

3 INTERSTITIAL DIFFUSION interstitial impurity atom host atom

4 DIFFUSION COUPLE nickel copper 100 C' Ni (at %) lateral position before heat treatment

5 DIFFUSION COUPLE nickel copper 100 C' Ni (at %) lateral position after heat treatment

6 FICK'S FIRST LAW: SETUP Suppose that the concentration of a diffusing species through a slab of host material is held at C f (higher) on the front surface and at C b (lower) on the back surface. Assuming steady-state conditions exist in which the concentration profile is stationary and linear, a steady diffusion flux through the slab from front to back surfaces will result. C f d slab 0 d C b

7 FICK'S FIRST LAW Given a slab of host material and a second species diffusing through that material: J = -D[dC(x)/dx] J is the diffusion flux of the diffusing species (kg/s/m 2 ); D is the diffusion coefficient of the diffusing species in the host material (m 2 /s); C(x) is the concentration of the diffusing species in the host material (kg/m 3 ); and dc(x)/dx (= [C f - C b )/d) is the concentration gradient (kg/m 3 /m).

8 FICK'S FIRST LAW PROBLEM Problem: Hydrogen gas is purified by allowing it to diffuse through a.25-mm thick rectangular sheet of palladium with dimensions 20 cm by 10 cm. One side of the sheet is exposed to gaseous hydrogen with impurities. The hydrogen molecules selectively diffuse through the sheet to the opposite side, which is maintained at a lower (hydrogen) pressure. At what mass flow rate would hydrogen diffuse through this sheet if the high and low pressure sides of the sheet have hydrogen concentrations of 2.4 kg/m 3 and 0.6 kg/m 3 respectively? Take the diffusion coefficient in this instance to be 1.0x10-8 m 2 /s.

9 DIFFUSION COEFFICIENT CALCULATION The diffusion coefficient for a diffusing species-host material combination is predicted by the empirical formula: D(T) = D o e -Q d /RT D o is a temperature-independent pre-exponential (found in diffusion data tables and measured in m 2 /s); Q d is the activation energy for the diffusion (found in diffusion data tables and measured in J/mol); R is the universal gas constant 8.31 J/mol. K; and T is the treatment temperature (absolute temperature in kelvins).

10 DIFFUSION COEFFICIENT PROBLEM Problem: Predict the diffusion coefficient for copper atoms through aluminum at a treatment temperature of 500 C. Use D o = 6.5x10-5 m 2 /s and Q d = 136,000 J/ mol.

11 CALCULATED DIFFUSION COEFFICIENTS T( C) C in a-fe Zn in Cu Mg in Al x x x x x x x x x x x x x x x x x x x x x x x x x10-14 above host M.P. note: all coefficient values given in m 2 /s

12 FICK'S SECOND LAW: SETUP Suppose that the initial concentration of a diffusing species throughout a slab of host material is a constant C o. At t = 0 assume that the left and right boundaries of the slab (x = 0 and x = d) are maintained at concentrations C s and C o respectively. C(x,t) is the diffusing species concentration at any point within the slab at any time t. C s C x C o d slab 0 x d C o

13 FICK'S SECOND LAW Fick's Second Law is a partial differential equation relating the concentration C(x,t) of a diffusing species in a host material to time (t) and lateral position (x) in the host. C, t = DC, xx This equation is solved for the boundary conditions C(0,t) = C s and C(d,t) = C o by: C(x,t) = C s - (C s - C o )erf[x/2(dt) 1/2 ] note: C(x,t) can be measured in kg/m 3 or wt %.

14 ERROR FUNCTION TABLE z erf(z) z erf(z) z erf(z) z erf(z) z note: erf(z) = 2π -1/2 S exp{-t2 } dt 0

15 FICK'S SECOND LAW PROBLEM Problem: Suppose you want to improve the wear resistance of a steel gear (which is.25 wt % carbon) by hardening its surface (case hardening). A carburization process is selected in which the surface carbon concentration of the gear is to be maintained at 1.00 wt %. The goal of the process is to produce a carbon concentration of.50 wt % at a position.40 mm below the suface of the gear. If the diffusion treatment is to be done at 950 C, how long will the process take? Use the following diffusion data for carbon in γ-iron: D o = 2.3x10-5 m 2 /s and Q d = 148,000 J/mol.

16 TREATMENT TIME STRATEGY For a given diffusing species-host material combination, concentrations (C s, C x, and C o ), target depth (x), and treatment temperature (T), predict the time (t) required for the diffusion process. (1) find D = D o exp{-q d /RT} (2) find z for erf(z) = (C s - C x )/(C s - C o ) (3) find t = x 2 /4Dz 2

17 TREATMENT TEMPERATURE STRATEGY For a given diffusing species-host material combination, concentrations (C s, C x, and C o ), target depth (x), diffusion time (t), calculate the temperature (T) at which the process must be maintained. (1) find z for erf(z) = (C s - C x )/(C s - C o ) (2) find D = x 2 /4tz 2 (3) find T = Q d /Rln(D o /D)

18 TREATMENT DEPTH STRATEGY For a given diffusing species-host material combination, concentrations (C s, C x, and C o ), treatment time (t), and treatment temperature (T), calculate the depth (x) at which the concentration C x will be present. (1) find D = D o exp{-q d /RT} (2) find z for erf(z) = (C s - C x )/(C s - C o ) (3) find x = 2z[Dt] 1/2

19 Engineering B45 DIFFUSION PROBLEMS Fick's First Law Problem: Hydrogen gas is sometimes purified by allowing it to diffuse through a thin sheet of palladium. One side of the sheet is exposed to a gas composed of molecular hydrogen and other impurities. The hydrogen molecules, due to their small size, selectively diffuse through the sheet to the opposite side, which is maintained at a constant (and lower) hydrogen pressure. At what rate would hydrogen gas diffuse through a.25- mm thick sheet of palladium with dimensions 10 cm by 20 cm if the high and low pressure sides of the sheet have hydrogen concentrations of 2.4 kg/m 3 and 0.6 kg/m 3 respectively? Assume that the diffusion coefficient at the temperature this is carried out at is 1.0x10-8 m 2 /s, and that the process is a steady state one. Diffusion Coefficient Problem: Predict the diffusion coefficient for copper atoms through aluminum at a treatment temperature of 500 K. Use D o = 6.5x10-5 m 2 /s and Q d = 136,000 J/mole. Fick's Second Law Problem: Suppose you want to improve the wear resistance of a steel gear (which is.25 wt % carbon) by hardening its surface (case hardening). A carburization process is selected in which the surface carbon concentration of the gear is to be maintained at 1.00 wt %. The goal of the process is to produce a carbon concentration of.50 wt % at a position.40 mm below the suface of the gear. If the diffusion treatment is to be done at 950 C, how long will the process take? Use the following diffusion data for carbon in γ-iron: D o = 2.3x10-5 m 2 / s and Q d = 148,000 J/mole.

In order to solve this problem it is first necessary to use Equation 5.5: x 2 Dt. = 1 erf. = 1.30, and x = 2 mm = 2 10-3 m. Thus,

In order to solve this problem it is first necessary to use Equation 5.5: x 2 Dt. = 1 erf. = 1.30, and x = 2 mm = 2 10-3 m. Thus, 5.3 (a) Compare interstitial and vacancy atomic mechanisms for diffusion. (b) Cite two reasons why interstitial diffusion is normally more rapid than vacancy diffusion. Solution (a) With vacancy diffusion,