Chapter 5: Thermally Activated Processes & Diffusion

ME 2105Dr. R. Lindeke

DIFFUSION is observed to occur:

IN LIQUIDS: ink in water, etc.

and IN SOLIDS

IN GASES: swamp gas in air, exhaust fumes into Smog, etc.

Carburization Surface coating

• Case Hardening:• Diffuse carbon atoms

into the host iron atomsat the surface.

• Example of interstitialdiffusion to produce a

surface (case) hardened gear.

The carbon atoms (interstitially) diffuse from a carbon rich atmosphere into the steel thru the surface.

Result: The presence of C atoms makes the iron (steel) surface harder.

Processing Using Diffusion

Courtesy ofSurface Division, Midland-Ross.

Typical Arrhenius plot of data compared with Equation 5.2. This behavior controls most molecular movement driven behavior (like vacancy formation or diffusion). The slope

equals −Q/R, and the intercept (at 1/T = 0) is ln(C)

rate

where:

C is a constant (rate w/o temperature)

Q is activation energy

T is absolute temperature

QRTCe

Note: this is a “Semi-log” plot

Process path showing how an atom must overcome an activation energy, q, to move from one stable position to

a similar adjacent position.

And it is this “Activation” Energy barrier – which we can model as in Ex. 5.1 – that determines the “Rate Limiting Step” in any process …

The overall thermal expansion (ΔL/L) of aluminum is measurably greater than the lattice parameter expansion (Δa/a) at high temperatures because vacancies are produced by thermal agitation(a). A semilog (Arrhenius-type) plot of ln(vacancy concentration)

(b) versus 1/T based on the data of part (a). The slope of the plot (−Ev/k) indicates that 0.76 eV of energy is required to create a single vacancy in the aluminum crystal structure. (From P. G. Shewmon, Diffusion in Solids, McGraw-Hill Book Company, New York, 1963.)

Atomic migration (“Diffusion”) occurs by a mechanism of vacancy migration. Note that the overall direction of material flow (the atom) is opposite to the direction of vacancy flow.

So diffusion is faster at higher temperature since more vacancies will exist in the lattice!

Diffusion by an ‘interstitialcy’ mechanism illustrates the random-walk nature of atomic migration (which is quicker as

temperature increases)

Diffusion of importance to material engineers is observed to occur by both mechanisms – vacancy migration and random

moving interstitials

The interdiffusion of materials A and B. Although any given A or B atom is equally likely to “walk” in any random direction (see Figure 5.6), the concentration gradients of the two materials can result in a net flow of A atoms into the B material, and vice versa. (From W. D. Kingery, H. K. Bowen, and D. R. Uhlmann, Introduction to Ceramics, 2nd ed., John Wiley & Sons, Inc., New York, 1976.)

The interdiffusion of materials on an atomic scale was illustrated in Figure 5.7. This interdiffusion of copper and

nickel is a comparable example on the microscopic scale.

Quantifying Diffusion: Fick’s First Law (Equation 5.8) is a statement of Material Flux across a ‘Barrier’

We will consider this model as a Steady State Diffusion system

sm

kgor

scm

mol

timearea surface

diffusing mass) (or molesFlux

22J

Quantifying Diffusion: Fick’s Second Law (Equation 5.10) is a statement of Concentration Variation over time across a ‘Barrier’

2

2

in the usual case where D is

independent of Concentration

x x

x x

c cD

t x x

c cD

t u

Solution to Fick’s second law (Equation 5.10) for the case of a semi-infinite solid, constant surface concentration of the diffusing species cs , initial bulk concentration c0, and a constant diffusion coefficient, D.

We will consider this model as a Non-Steady State (transient) Diffusion system

One practically important solution is for a semi-infinite solid (as illustrated in the previous graph) in which the surface concentration is held constant. Frequently source of the diffusing species is a gas phase, which is maintained at a constant pressure value.

The following assumptions are implied for a good solution:1. Before diffusion, any of the diffusing solute atoms in the solid are

uniformly distributed with concentration of C0.

2. The value of x (position in the solid) is taken as zero at the surface and increases with distance into the solid.

3. The time is taken to be zero the instant before the diffusion process begins.

A bar of length l is considered to be semi-infinite when:

Dtl 10

Non-steady State Diffusion

Adapted from Fig. 5.5, Callister 7e.

at t = 0, C = Co for 0 x

at t > 0, C = CS for x = 0 (const. surf. conc.)

C = Co for x =

• Copper diffuses into a bar of aluminum.

pre-existing conc., Co of copper atoms

Surface conc., C of Cu atoms bar

s

C s

Boundary Conditions:

Notice: the concentration decreases at increasing x (from surface) while it increases at a given x as time increases!

Master plot summarizing all of the diffusion results of Figure 5.10 on a single curve.

Mathematical Solution:

C(x,t) = Conc. at point x at time t erf (z) = error function

erf(z) values are given in Table 5.1 (see next slide!)

Dt

x

CC

Ct,xC

os

o

2 erf1

dye yz 2

0

2

CS

Co

C(x,t)

Similar F.S.L. Diffusion Studies have been documented for other than Semi-Infinite Solids:

The parameter cm is the average concentration of diffusing species within the sample. Again, the surface concentration, cs , and diffusion coefficient, D, are assumed to be constant. (From W. D. Kingery, H. K. Bowen, and D. R. Uhlmann, Introduction to Ceramics, 2nd ed., John Wiley & Sons, Inc., New York, 1976.)

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 (Cs ) 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. 5.11

Dt

x

CC

CtxC

os

o

2erf1

),(

The solution requires the use of the erf function which was developed to model conduction along a semi-infinite rod

Solution (cont.):

– t = 49.5 h x = 4 x 10-3 m

– Cx = 0.35 wt% Cs = 1.0 wt%

– Co = 0.20 wt%

Dt

x

CC

C)t,x(C

os

o

2erf1

8125.0)(erf

)(erf11875.0

)(erf12

erf120.00.1

20.035.0),(

z

z

zDt

x

CC

CtxC

os

o

erf(z) = 0.8125

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 0.7970z 0.81250.95 0.8209 7970.08209.0

7970.08125.0

90.095.0

90.0

z

z = 0.93And now solve for D:

Dt

xz

2

tz

xD

2

2

4

/sm 10 x 6.2s 3600

h 1

h) 5.49()93.0()4(

m)10 x 4(

4

2112

23

2

2

tz

xD

Now By LINEAR Interpolation:

Diffusion and Temperature

The Diffusion coefficient seen in Fick’s Laws increases with increasing T and, along with vacancy formation, is a “Classic” Arrhenius Model:

D Do exp

Qd

R T

= pre-exponential [m2/s]

= diffusion coefficient [m2/s]

= activation energy for diffusion [J/mol or eV/atom]

= gas constant [8.314 J/mol-K]

= absolute temperature [K]

D

Do

Qd

R

T

So, using this model, we should be able to “back out” the temperature at which this process took place!

Arrhenius plot of the diffusivity of carbon in α-iron over a range of temperatures. Note also related Figures 4.4 and 5.6 and other metallic

diffusion data in Figure 5.14.

Arrhenius plot of diffusivity data for a number of metallic systems. (From L. H. Van Vlack, Elements of Materials Science and Engineering, 4th

ed., Addison-Wesley Publishing Co., Inc., Reading, MA, 1980.)

In a computationally simpler form (to read!):

Similar Data also exists for Ionic (and Organic) Compounds:

From P. Kofstad, Nonstoichiometry, Diffusion, and Electrical Conductivity in Binary Metal Oxides, John Wiley & Sons, Inc.,

NY, 1972; and S. M. Hu in Atomic Diffusion in Semiconductors, D. Shaw, Ed., Plenum Press, New York, 1973.

And in a Tabular Form:

• To solve for the temperature at which D has above value, we use a rearranged the D (Arrhenius) Equation: )lnln( DDR

QT

o

d

CKT

T

T

987 2.1260

K-J/mol 681.112

J/mol 000,142

3.553)1K)(-J/mol 314.8(

J/mol 000,142

)10x6.2ln 10x0.2K)(ln -J/mol 314.8(

J/mol 000,142115

Now, Returning to the Solution to our Carburizing problem:

Following Up:

• In industry one may wish to speed up this process– This can be accomplished by increasing:

• Temperature of the process• Surface concentration of the diffusing species

• If we choose to increase the temperature, determine how long it will take to reach the same concentration at the same depth as in the previous study?

Diffusion time calculation:• Given target X (depth of ‘case’) and concentration are equal:

– Here we known that D*t is a constant for the diffusion process (where D is a function of temperature)

– D1260 was 2.6x10-11m2/s at 987C while the process took 49.5 hours

– How long will it take if the temperature is increased to 1250 ˚C?

1420005 8.31 152301250

25 5 10

1250

987 9871250 1250 1027 1027 1250

1250

11

101250

2.0 10

2.0 10 1.341 10 2.68 10

2.6 10 49.54.8 hr

2.68 10

DQRT

C

C

C CC C C C C

C

C

D D e e

mD sD t

D t D t tD

hrt

Considering a “First Law” or Steady-State Diffusion Case

dx

dC

dx

dCDJ

C1

C2

x

C1

C2

x1 x2

D is the diffusion coefficient

Here, The Rate of diffusion is independent of time Flux is proportional to concentration gradient =

12

12linear isBehavior ifxx

CC

x

C

dx

dC

Note, steady state diffusion concentration gradient ( dC/dx) is linear

F.F.L. Example: Chemical Protective Clothing (CPC)

• Methylene chloride is a common ingredient in paint removers. Besides being an irritant, it also may be absorbed through skin. When using this paint remover, protective gloves should be worn.

• If butyl rubber gloves (0.04 cm thick) are used, what is the diffusive flux of methylene chloride through the glove?

• Data:– diffusion coefficient of MeChl in butyl rubber: D = 110 x10-8 cm2/s– surface concentrations:

C2 = 0.02 g/cm3(inside surface)

C1 = 0.44 g/cm3 (outside surface)

scm

g 10 x 16.1

cm) 04.0(

)g/cm 44.0g/cm 02.0(/s)cm 10 x 110(

25-

3328-

J

Example (cont).

Dtb 6

2

12

12- xx

CCD

dx

dCDJ

gloveC1

C2

skinpaintremover

x1 x2

D = 110 x 10-8 cm2/s

C2 = 0.02 g/cm3

C1 = 0.44 g/cm3

x2 – x1 = X = 0.04 cm

Data:

What happens to a Worker?

• If a person is in contact with the irritant and more than about 0.5 gm of the irritant is deposited on their skin they need to take a wash break

• If 25 cm2 of glove is in the paint thinner can, How Long will it take before they must take a wash break?

5

2

52

2

2 4

4

gcm -s

if the exposed area of the gloves are 25 cm

how long will it take to get 0.5g of M-C onto the hands?

gF 1.16 10cm -s

1.16 10

25 2.9 10

0.5gExposure time= 1724 0.48 2.9 10

lux

lux

F

gcm s

s hrgs

Another Example: Chemical Protective Clothing (CPC)

• If butyl rubber gloves (0.04 cm thick) are used, what is the breakthrough time (tb), i.e., how long could the gloves be used before methylene chloride reaches the hand (at all)?

• Data: diffusion coefficient of Me-Chl in butyl rubber

D = 110 x10-8 cm2/s (as earlier)

Example 3 (cont).

Dtb 6

2

min 4 s 240/s)cm 10 x 110)(6(

cm) 04.0(28-

2bt

Time before breakthrough is circa. 4 min

gloveC1

C2

skinpaintremover

x1 x2

• Solution – assuming linear conc. gradient

cm 0.04 12 xx D = 110 x 10-8 cm2/s

Self-diffusion coefficients for silver (and other materials in other metals) depend on the diffusion path. In general, diffusivity is greater through less-restrictive structural regions. (From J. H. Brophy, R. M. Rose, and J. Wulff, The Structure and Properties of Materials, Vol. 2: Thermodynamics of Structure, John Wiley & Sons, Inc., New York,

1964.)

While shown for “self-diffusion” – this type of diffusing behavior is typical – Areas high in vacancies are ones where diffusion occurs at a faster rate!

Schematic illustration of how a coating of impurity B can penetrate more deeply into grain boundaries and even further along a free surface of polycrystalline A, consistent

with the relative values of diffusion coefficients (Dvolume < Dgrain boundary < Dsurface).

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: