# Chapter 5: Thermally Activated Processes & Diffusion

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Chapter 5: Thermally Activated Processes & Diffusion
ME 2105 Dr. R. Lindeke

IN GASES: swamp gas in air, exhaust fumes into Smog, etc.
DIFFUSION is observed to occur: IN LIQUIDS: ink in water, etc. IN GASES: swamp gas in air, exhaust fumes into Smog, etc. and IN SOLIDS Carburization Surface coating

Processing Using Diffusion
Case Hardening: Diffuse carbon atoms into the host iron atoms at the surface. Example of interstitial diffusion to produce a surface (case) hardened gear. Courtesy of Surface Division, Midland-Ross. 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.

Typical Arrhenius plot of data compared with Equation 5. 2
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) 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
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.

We will consider this model as a Steady State Diffusion system
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

Quantifying Diffusion: Fick’s Second Law (Equation 5
Quantifying Diffusion: Fick’s Second Law (Equation 5.10) is a statement of Concentration Variation over time across a ‘Barrier’ 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

A bar of length l is considered to be semi-infinite when:
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. A bar of length l is considered to be semi-infinite when: The following assumptions are implied for a good solution: Before diffusion, any of the diffusing solute atoms in the solid are uniformly distributed with concentration of C0. The value of x (position in the solid) is taken as zero at the surface and increases with distance into the solid. The time is taken to be zero the instant before the diffusion process begins.

• Copper diffuses into a bar of aluminum. pre-existing conc., Co of copper atoms Surface conc., C of Cu atoms bar s C s Adapted from Fig. 5.5, Callister 7e. Notice: the concentration decreases at increasing x (from surface) while it increases at a given x as time increases! Boundary Conditions: at t = 0, C = Co for 0  x   at t > 0, C = CS for x = 0 (const. surf. conc.) C = Co for x = 

Master plot summarizing all of the diffusion results of Figure 5
Master plot summarizing all of the diffusion results of Figure 5.10 on a single curve.

Mathematical Solution:
CS Co C(x,t) 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!)

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.)

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

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%  erf(z) =

Solution (cont.): We must now determine from Table 5.1 the value of z for which the error function is An interpolation is necessary as follows z erf(z) z Now By LINEAR Interpolation: z = 0.93 And now solve for D:

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 æ è ç ö ø ÷ - 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] 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
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:

 Now, Returning to the Solution to our Carburizing problem:
To solve for the temperature at which D has above value, we use a rearranged the D (Arrhenius) Equation:

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 987C while the process took 49.5 hours How long will it take if the temperature is increased to 1250 ˚C?

Considering a “First Law” or Steady-State Diffusion Case
Here, The Rate of diffusion is independent of time  Flux is proportional to concentration gradient = Note, steady state diffusion concentration gradient ( dC/dx) is linear C1 C2 x x1 x2 D is the diffusion coefficient

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: C1 = 0.44 g/cm3 (outside surface) C2 = 0.02 g/cm3(inside surface)

Example (cont). Data: glove C1 paint remover skin C2 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: C2 x1 x2

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?

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). Solution – assuming linear conc. gradient glove C1
skin paint remover x1 x2 D = 110 x 10-8 cm2/s Time before breakthrough is circa. 4 min

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).

Summary: Diffusion FASTER for... Diffusion SLOWER 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

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