Given by Equation 5-7 Note that in the equation describing Fick’s sec- ond law (Equation 5-7):

(a) It is assumed that D is independent of the concentration of the diffusing species; (b) the surface concentration of the diffusing species (c s ) is always constant.

There are situations under which these conditions may not be met and hence the concentration profile evolution will not be predicted by the error-function solution shown in Equation 5-7. If the boundary conditions are different from the ones we assumed, different solutions to Fick’s second law must be used.

5-9 Diffusion and Materials Processing

We briefly discussed applications of diffusion in processing materials in Section 5-1. Many important examples related to solidification, phase transformations, heat treatments, etc., will be discussed in later chapters. In this section, we provide more information to high- light the importance of diffusion in the processing of engineered materials. Diffusional processes become very important when materials are used or processed at elevated temperatures.

5-9 Diffusion and Materials Processing 183

Melting and Casting One of the most widely used methods to process metals, alloys, many plastics, and glasses involves melting and casting of materials into a desired shape. Diffusion plays a particularly important role in solidification of metals and alloys. During the growth of single crystals of semiconductors, for example, we must ensure that the differences in the diffusion of dopants in both the molten and solid forms are accounted for. This also applies for the diffusion of elements during the casting of alloys. Similarly, diffusion plays a critical role in the processing of glasses. In inorganic glasses, for instance, we rely on the fact that diffusion is slow and inorganic glasses do not crystallize easily. We will examine this topic further in Chapter 9.

Sintering Although casting and melting methods are very popular for many manufactured materials, the melting points of many ceramic and some metallic materials are too high for processing by melting and casting. These relatively refractory materials are manufactured into useful shapes by a process that requires the consolidation of small particles of a powder into a solid mass (Chapter 15). Sintering is the high-temperature treatment that causes particles to join, gradually reducing the volume of pore space between them. Sintering is a frequent step in the manufacture of ceramic components (e.g., alumina, barium titanate, etc.) as well as in the production of metallic parts by powder metallurgy—a processing route by which metal powders are pressed and sintered into dense, monolithic components. A variety of composite materials such as tungsten car- bide-cobalt based cutting tools, superalloys, etc., are produced using this technique. With finer particles, many atoms or ions are at the surface for which the atomic or ionic bonds are not satisfied. As a result, a collection of fine particles of a certain mass has higher energy than that for a solid cohesive material of the same mass. Therefore, the driving force for solid state sintering of powdered metals and ceramics is the reduction in the total surface area of powder particles. When a powdered material is compacted into a shape, the powder particles are in contact with one another at numerous sites, with a significant amount of pore space between them. In order to reduce the total energy of the material, atoms diffuse to the points of contact, bonding the particles together and eventually causing the pores to shrink.

Lattice diffusion from the bulk of the particles into the neck region causes densification. Surface diffusion, gas or vapor phase diffusion, and lattice diffusion from curved surfaces into the neck area between particles do not lead to densification (Chapter 15). If sintering is carried out over a long period of time, the pores may be eliminated and the material becomes dense (Figure 5-20). In Figure 5-21, particles of a

Figure 5-20 Diffusion processes during sintering and powder metallurgy. Atoms diffuse to points of contact, creating bridges and reducing the pore size.

184 CHAPTER 5

Atom and Ion Movements in Materials

Figure 5-21 Particles of barium magnesium tantalate (BMT) (Ba(Mg 1/3 Ta 2/3 )O 3 )

powder are shown. This ceramic material is useful in making electronic components known as dielectric resonators that are used for wireless communications. ( Courtesy of H. Shivey.)

powder of a ceramic material known as barium magnesium tantalate (Ba(Mg 1/3 Ta 2/3 )O 3 or BMT) are shown. This ceramic material is useful in making electronic components known as dielectric resonators used in wireless communication systems. The microstructure of BMT ceramics is shown in Figure 5-22. These ceramics were produced by com- pacting the powders in a press and sintering the compact at a high temperature (!1500°C).

The extent and rate of sintering depends on (a) the initial density of the compacts, (b) temperature, (c) time, (d) the mechanism of sintering, (e) the average particle size, and (f) the size distribution of the powder particles. In some situations, a liquid phase forms in localized regions of the material while sintering is in process. Since diffusion of species, such as atoms and ions, is faster in liquids than in the solid state, the presence of a liquid phase can provide a convenient way for accelerating the sintering of many refractory metal and ceramic formulations. The process in which a small amount of liquid forms and assists densification is known as liquid phase sintering. For the liquid phase to be effective in enhancing sintering, it is important to have a liquid that can “wet” the grains, similar to how water wets a glass surface. If the liquid is non-wetting, similar to how mercury does not wet glass, then the liquid phase will not be helpful for enhancing sintering. In some cases, compounds are added to materials to cause the liquid phase to form at sintering temperatures. In other situations, impurities can react with the material and cause formation of a liquid phase. In most applications, it is desirable if the liquid phase is transient or converted into a crystalline material during cooling. This way a glassy and brittle amorphous phase does not remain at the grain boundaries.

When exceptionally high densities are needed, pressure (either uniaxial or isostatic) is applied while the material is being sintered. These techniques are known as

5-9 Diffusion and Materials Processing 185

Figure 5-22 The microstructure of BMT ceramics obtained by compaction and sintering of BMT powders. ( Courtesy of H. Shivey.)

hot pressing, when the pressure is unidirectional, or hot isostatic pressing (HIP), when the pressure is isostatic (i.e., applied in all directions). Many superalloys and ceramics such as lead lanthanum zirconium titanate (PLZT) are processed using these techniques. Hot isostatic pressing leads to high density materials with isotropic properties (Chapter 15).

Grain Growth

A polycrystalline material contains a large number of grain boundaries, which represent high-energy areas because of the inefficient packing of the atoms. A lower overall energy is obtained in the material if the amount of grain boundary area is reduced by grain growth. Grain growth involves the movement of grain boundaries, permitting larger grains to grow at the expense of smaller grains (Figure 5-23). If you have watched froth, you have probably seen the principle of grain growth! Grain growth is similar to the way smaller bubbles in the froth disappear at the expense of bigger bubbles. Another analogy is big fish getting bigger by eating small fish! For grain growth in materials, diffusion of atoms across the grain boundary is required, and, consequently, the growth of the grains is related to the activation energy needed for an atom to jump across the boundary. The increase in grain size can be seen from the sequence of micrographs for alumina ceramics shown in Figure 5-23. Another example for which grain growth plays a role is in the tungsten (W) filament in a lightbulb. As the tungsten filament gets hotter, the grains grow causing it to get weaker. This grain growth, vaporization of tungsten, and oxidation via reaction with remnant oxygen contribute to the failure of tungsten filaments in a lightbulb.

The driving force for grain growth is reduction in grain boundary area. Grain boundaries are defects and their presence causes the free energy of the material to increase. Thus, the thermodynamic tendency of polycrystalline materials is to transform

186 CHAPTER 5

Atom and Ion Movements in Materials

Figure 5-23 Grain growth in alumina ceramics can be seen from the scanning electron micrographs of alumina ceramics. (a) The left micrograph shows the microstructure of an alumina ceramic sintered at 1350°C for

15 hours. (b) The right micrograph shows a sample sintered at 1350°C for 30 hours. ( Courtesy of Dr. Ian Nettleship and Dr. Richard McAfee.)

into materials that have a larger average grain size. High temperatures or low-activation energies increase the size of the grains. Many heat treatments of metals, which include holding the metal at high temperatures, must be carefully controlled to avoid excessive grain growth. This is because, as the average grain size grows, the grain-boundary area decreases, and there is consequently less resistance to motion of dislocations. As a result, the strength of a metallic material will decrease with increasing grain size. We have seen this concept before in the form of the Hall-Petch equation (Chapter 4). In normal grain growth, the average grain size increases steadily and the width of the grain size distribution is not affected severely. In abnormal grain growth, the grain size distribution tends to become bi-modal (i.e., we get a few grains that are very large and then we have a few grains that remain relatively small). Certain electrical, magnetic, and optical properties of materials also depend upon the grain size of materials. As a result, in the processing of these materials, attention has to be paid to factors that affect diffusion rates and grain growth.

Diffusion Bonding

A method used to join materials, called diffusion bonding, occurs in three steps (Figure 5-24). The first step forces the two surfaces together at a high temperature and pressure, flattening the surface, fragmenting impurities, and producing a high atom-to-atom contact area. As the surfaces remain pressed together at high temperatures, atoms diffuse along grain boundaries to the remaining voids; the atoms condense and reduce the size of any voids at the interface. Because grain boundary diffusion is rapid, this second step may occur very quickly. Eventually, however, grain growth isolates the remaining voids from the grain boundaries. For the third step—final elimination of the voids—volume diffusion, which is comparatively slow, must occur. The diffusion bonding process is often used for joining reactive metals such as titanium, for joining dissimilar metals and materials, and for joining ceramics.

Summary 187

Figure 5-24 The steps in diffusion bonding: (a) Initially the contact area is small; (b) application of pressure deforms the surface, increasing the bonded area; (c) grain boundary diffusion permits voids to shrink; and (d) final elimination of the voids requires volume diffusion.

Summary

• The net flux of atoms, ions, etc., resulting from diffusion depends upon the initial con-

centration gradient. • The kinetics of diffusion depend strongly on temperature. In general, diffusion is a

thermally activated process and the dependence of the diffusion coefficient on temperature is given by the Arrhenius equation.

• The extent of diffusion depends on temperature, time, the nature and concentration of diffusing species, crystal structure, composition of the matrix, stoichiometry, and point defects.

• Encouraging or limiting the diffusion process forms the underpinning of many important technologies. Examples include the processing of semiconductors, heat treatments of metallic materials, sintering of ceramics and powdered metals, formation of amorphous materials, solidification of molten materials during a casting process, diffusion bonding, and barrier plastics, films, and coatings.

• Fick’s laws describe the diffusion process quantitatively. Fick’s first law defines the relationship between the chemical potential gradient and the flux of diffusing species. Fick’s second law describes the variation of concentration of diffusing species under non- steady state diffusion conditions.

• For a particular system, the amount of diffusion is related to the term Dt. This term permits us to determine the effect of a change in temperature on the time required for

a diffusion-controlled process. • The two important mechanisms for atomic movement in crystalline materials are

vacancy diffusion and interstitial diffusion. Substitutional atoms in the crystalline materials move by the vacancy mechanism.

• The rate of diffusion is governed by the Arrhenius relationship—that is, the rate increases exponentially with temperature. Diffusion is particularly significant at temperatures above about 0.4 times the melting temperature (in Kelvin) of the material.

188 CHAPTER 5

Atom and Ion Movements in Materials

• The activation energy Q describes the ease with which atoms diffuse, with rapid diffusion occurring for a low activation energy. A low-activation energy and rapid diffusion rate are obtained for (1) interstitial diffusion compared to vacancy diffusion, (2) crystal structures with a smaller packing factor, (3) materials with a low melting temperature or weak atomic bonding, and (4) diffusion along grain boundaries or surfaces.

• The total movement of atoms, or flux, increases when the concentration gradient and temperature increase.

• Diffusion of ions in ceramics is usually slower than that of atoms in metallic materials. Diffusion in ceramics is also affected significantly by non-stoichiometry, dopants, and the possible presence of liquid phases during sintering.

• Atom diffusion is of paramount importance because many of the materials processing techniques, such as sintering, powder metallurgy, and diffusion bonding, require diffusion. Furthermore, many of the heat treatments and strengthening mechanisms used to control structures and properties in materials are diffusion-controlled processes. The stability of the structure and the properties of materials during use at high temperatures depend on diffusion.

Glossary

Abnormal grain growth

A type of grain growth observed in metals and ceramics. In this mode of grain growth, a bimodal grain size distribution usually emerges as some grains become very large at the expense of smaller grains. See “Grain growth” and “Normal grain growth.”

Activation energy The energy required to cause a particular reaction to occur. In diffusion, the activation energy is related to the energy required to move an atom from one lattice site to another.

Carburization

A heat treatment for steels to harden the surface using a gaseous or solid source of carbon. The carbon diffusing into the surface makes the surface harder and more abrasion resistant.

Concentration gradient The rate of change of composition with distance in a nonuniform material, typically expressed as atoms

cm 3 # cm or cm .

at%

Conductive ceramics Ceramic materials that are good conductors of electricity as a result of their ionic and electronic charge carriers (electrons, holes, or ions). Examples of such materials are stabilized zirconia and indium tin oxide.

Diffusion The net flux of atoms, ions, or other species within a material caused by temperature and a concentration gradient.

Diffusion bonding

A joining technique in which two surfaces are pressed together at high pres- sures and temperatures. Diffusion of atoms to the interface fills in voids and produces a strong bond between the surfaces.

Diffusion coefficient (D) A temperature-dependent coefficient related to the rate at which atoms, ions, or other species diffuse. The diffusion coefficient depends on temperature, the composition and microstructure of the host material and also the concentration of the diffusing species.

Diffusion couple

A combination of elements involved in diffusion studies (e.g., if we are consid- ering diffusion of Al in Si, then Al-Si is a diffusion couple).

Diffusion distance The maximum or desired distance that atoms must diffuse; often, the distance between the locations of the maximum and minimum concentrations of the diffusing atom.

Diffusivity Another term for the diffusion coefficient (D).

Glossary 189 Driving force

A cause that induces an effect. For example, an increased gradient in composition enhances diffusion; similarly reduction in surface area of powder particles is the driving force for sintering.

Fick’s first law The equation relating the flux of atoms by diffusion to the diffusion coefficient and the concentration gradient.

Fick’s second law The partial differential equation that describes the rate at which atoms are redistributed in a material by diffusion. Many solutions exist to Fick’s second law; Equation 5-7 is one possible solution.

Flux The number of atoms or other diffusing species passing through a plane of unit area per unit time. This is related to the rate at which mass is transported by diffusion in a solid.

Grain boundary diffusion Diffusion of atoms along grain boundaries. This is faster than volume diffusion, because the atoms are less closely packed in grain boundaries.

Grain growth Movement of grain boundaries by diffusion in order to reduce the amount of grain boundary area. As a result, small grains shrink and disappear and other grains become larger, similar to how some bubbles in soap froth become larger at the expense of smaller bubbles. In many situations, grain growth is not desirable.

Hot isostatic pressing

A sintering process in which a uniform pressure is applied in all directions during sintering. This process is used for obtaining very high densities and isotropic properties.

Hot pressing

A sintering process conducted under uniaxial pressure, used for achieving higher densities.

Interdiffusion Diffusion of different atoms in opposite directions. Interdiffusion may eventually produce an equilibrium concentration of atoms within the material.

Interstitial diffusion Diffusion of small atoms from one interstitial position to another in the crystal structure.

A sintering process in which a liquid phase forms. Since diffusion is faster in liquids, if the liquid can wet the grains, it can accelerate the sintering process.

Liquid phase sintering

Nitriding

A process in which nitrogen is diffused into the surface of a material, such as a steel, leading to increased hardness and wear resistance.

Normal grain growth Grain growth that occurs in an effort to reduce grain boundary area. This type of grain growth is to be distinguished from abnormal grain growth in that the grain size distribution remains unimodal but the average grain size increases steadily.

Permeability

A relative measure of the diffusion rate in materials, often applied to plastics and coatings. It is often used as an engineering design parameter that describes the effectiveness of a particular material to serve as a barrier against diffusion.

Powder metallurgy

A method for producing monolithic metallic parts; metal powders are compacted into a desired shape, which is then heated to allow diffusion and sintering to join the powders into a solid mass.

Self-diffusion The random movement of atoms within an essentially pure material. No net change in composition results.

Sintering

A high-temperature treatment used to join small particles. Diffusion of atoms to points of contact causes bridges to form between the particles. Further diffusion eventually fills in any remaining voids. The driving force for sintering is a reduction in total surface area of the powder particles.

Surface diffusion Diffusion of atoms along surfaces, such as cracks or particle surfaces. Thermal barrier coatings (TBC) Coatings used to protect a component from heat. For exam-

ple, some of the turbine blades in an aircraft engine are made from nickel-based superalloys and are coated with yttria stabilized zirconia (YSZ).

Vacancy diffusion Diffusion of atoms when an atom leaves a regular lattice position to fill a vacancy in the crystal. This process creates a new vacancy, and the process continues.

Volume diffusion Diffusion of atoms through the interior of grains.

190 CHAPTER 5

Atom and Ion Movements in Materials

Problems

Section 5-1 Applications of Diffusion

5-13 Why is it that the activation energy for diffu- 5-1 What is the driving force for diffusion?

sion via the interstitial mechanism is smaller 5-2 In the carburization treatment of steels,

than those for other mechanisms? what are the diffusing species?

5-14 How is self-diffusion of atoms in metals ver- 5-3 Why do we use PET plastic to make carbon-

ified experimentally?

ated beverage bottles? 5-15 Compare the diffusion coefficients of car- 5-4 Why is it that aluminum metal oxidizes more

bon in BCC and FCC iron at the allotropic readily than iron but aluminum is considered

transformation temperature of 912°C and to be a metal that usually does not “rust?”

explain the difference.

5-5 What is a thermal barrier coating? Where 5-16 Compare the diffusion coefficients for are such coatings used?

hydrogen and nitrogen in FCC iron at 1000°C and explain the difference.

Section 5-2 Stability of Atoms and Ions

5-6 What is a nitriding heat treatment?

Section 5-4 Activation Energy for Diffusion 5-7

A certain mechanical component is heat 5-17 Activation energy is sometimes expressed as treated using carburization. A common

(eV atom). For example, see Figure 5-15 > engineering problem encountered is that we

illustrating the diffusion coefficients of ions need to machine a certain part of the com-

in different oxides. Convert eV atom into > ponent and this part of the surface should

Joules mole. >

not be hardened. Explain how we can 5-18 + The diffusion coefficient for Cr 3 in Cr O

- 2 3 achieve this objective. is

6 * 10 15 cm 2 9 > 2 s at 727°C and 1 * 10 cm > s 5-8 Write down the Arrhenius equation and

at 1400°C. Calculate

explain the different terms. (a) the activation energy and 5-9 Atoms are found to move from one lattice

(b) the constant D 0 .

position to another at the rate of

- 5 jumps

5-19 The diffusion coefficient for O 2 in

> s at 1150°C and energy for their movement is 30,000 cal mol. >

5 * 10 s at 400°C when the activation Cr - 2 O 3 is 4 * 10 15 cm 2

6 * 10 11 cm 2 > s at 1715°C. Calculate Calculate the jump rate at 750°C.

(a) the activation energy and 5-10 The number of vacancies in a material is

(b) the constant D 0 .

related to temperature by an Arrhenius equa- 5-20 Without referring to the actual data, can tion. If the fraction of lattice points contain-

you predict whether the activation energy

for diffusion of carbon in FCC iron will be the fraction of lattice points containing

ing vacancies is 8 * 10 5 at 600°C, determine

higher or lower than that in BCC iron? vacancies at 1000°C.

Explain.

5-11 The Arrhenius equation was originally devel- oped for comparing rates of chemical reac-

Section 5-5 Rate of Diffusion

tions. Compare the rates of a chemical reaction at 20 and 100°C by calculating the ratio of the

(Fick’s First Law)

chemical reaction rates. Assume that the acti- 5-21 Write down Fick’s first law of diffusion. vation energy for liquids in which the chemical

Clearly explain what each term means. reaction is conducted is 10 kJ mol and that the >

5-22 What is the difference between diffusivity reaction is limited by diffusion.

and the diffusion coefficient?

5-23

A 1-mm-thick BCC iron foil is used to sep-

Section 5-3 Mechanisms for Diffusion

arate a region of high nitrogen gas concen- 5-12 What are the different mechanisms for

tration of 0.1 atomic percent from a region diffusion?

of low nitrogen gas concentration at 650°C.

Problems 191

If the flux of nitrogen through the foil is to be lost per year through each square cen-

timeter of the iron at 400°C. If the concen- concentration in the low concentration

10 12 atoms (cm 2 > # s), what is the nitrogen

tration of hydrogen at one surface is 0.05 H region?

atom per unit cell and 0.001 H atom per unit

5-24

A 0.2-mm-thick wafer of silicon is treated so cell at the second surface, determine the that a uniform concentration gradient of

minimum thickness of the iron. antimony is produced. One surface contains

5-30 Determine the maximum allowable temper-

1 Sb atom per 10 8 Si atoms and the other sur-

ature that will produce a flux of less than

2000 H atoms (cm 2 > % s) through a BCC iron The lattice parameter for Si is 5.4307 Å

face contains 500 Sb atoms per 10 8 Si atoms.

foil when the concentration gradient is (Appendix A). Calculate the concentration atoms - 5 * 10 16

cm 3 # cm . (Note the negative sign gradient in

for the flux.)

(a) atomic percent Sb per cm and (b) Sb atoms cm 3 # cm .

Section 5-6 Factors Affecting Diffusion

5-25 When a Cu-Zn alloy solidifies, one portion 5-31 Write down the equation that describes the of the structure contains 25 atomic percent

dependence of D on temperature. zinc and another portion 0.025 mm away

5-32 In solids, the process of diffusion of atoms contains 20 atomic percent zinc. The lattice

and ions takes time. Explain how this is used parameter for the FCC alloy is about 3.63 *

to our advantage while forming metallic

10 - 8 cm. Determine the concentration gra-

glasses.

dient in 5-33 Why is it that inorganic glasses form upon (a) atomic percent Zn per cm;

relatively slow cooling of melts, while rapid (b) weight percent Zn per cm; and

(c)

atoms

solidification is necessary to form metallic Zn cm 3 # cm .

glasses?

5-26

A 0.001 in. BCC iron foil is used to separate 5-34 Use the diffusion data in the table below

a high hydrogen content gas from a low for atoms in iron to answer the questions hydrogen content gas at 650°C. 5 * 10 8 H that follow. Assume metastable equilib-

atoms cm 3 > are in equilibrium on one side of rium conditions and trace amounts of C

in Fe. The gas constant in SI units is equilibrium with the other side. Determine

the foil, and 2 * 10 3 H atoms cm 3 > are in

8.314 J (mol K). # >

(a) the concentration gradient of hydrogen and

Diffusion

D 0 (m 2 / s) 5-27

(b) the flux of hydrogen through the foil.

Couple

Mechanism

Q (J mol) /

A 1-mm-thick sheet of FCC iron is used to

contain nitrogen in a heat exchanger at

C in FCC iron

Interstitial

1.38 * 10 5 2.3 * 10 - 5

C in BCC iron

Interstitial

8.74 * 10 4 1.1 * 10 - 6

1200°C. The concentration of N at one sur-

Fe in FCC iron

Vacancy

2.79 * 10 5 6.5 * 10 - 5

face is 0.04 atomic percent, and the concen-

Fe in BCC iron

Vacancy

2.46 * 10 5 4.1 * 10 - 4

tration at the second surface is 0.005 atomic percent. Determine the flux of nitrogen

> (a) Plot the diffusion coefficient as a function of inverse temperature (1 T) show- >

through the foil in N atoms (cm 2 # s).

5-28

A 4-cm-diameter, 0.5-mm-thick spherical ing all four diffusion couples in the container made of BCC iron holds nitrogen

table.

at 700°C. The concentration at the inner sur- (b) Recall the temperatures for phase tran- face is 0.05 atomic percent and at the outer

sitions in iron, and for each case, indi- surface is 0.002 atomic percent. Calculate

cate on the graph the temperature range the number of grams of nitrogen that are

over which the diffusion data is valid. lost from the container per hour.

5-29

A BCC iron structure is to be manufactured fusion higher than that for C diffusion that will allow no more than 50 g of hydrogen

in iron?

192 CHAPTER 5

Atom and Ion Movements in Materials

(d) Why is the activation energy for diffu-

10 18 atoms cm 3 > at a depth of 0.1 &m sion higher in FCC iron when compared

from the surface for both n- and p-type to BCC iron?

regions. The diffusion coefficients of P and (e) Does C diffuse faster in FCC Fe than in

B in single crystal silicon at 1100°C are BCC Fe? Support your answer with a

- 13 6.5 * 10 - cm 2 s and 6.1 * 10 13 cm > 2 > s, numerical calculation and state any

respectively.

assumptions made. 5-40 Consider a 2-mm-thick silicon (Si) wafer to

be doped using antimony (Sb). Assume that

C 2 the dopant source (gas mixture of anti-

10 /s) –8

mony chloride and other gases) provides a

D (m 10 –10

B constant concentration of 10 22 atoms m 3 > .

We need a dopant profile such that the con-

ficient

centration of Sb at a depth of 1 micrometer

10 –12 A 10 –13

is 5 * 10 21 atoms m 3 > . What is the required

time for the diffusion heat treatment?

fusion Coef 10 –14

Assume that the silicon wafer initially con-

tains no impurities or dopants. Assume that

10 –16 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 the activation energy for diffusion of Sb

in silicon is 380 kJ mole and D > 0 for Sb diffusion in Si is 1.3 * 10 - 3 m 2

Figure 5-25 Plot for Problems 5-34 and 5-35. > s. Assume

1000/T (K –1 )

T ! 1250°C.

5-35 The plot above has three lines representing 5-41 Consider doping of Si with gallium (Ga). grain boundary, surface, and volume self-

Assume that the diffusion coefficient of gal- diffusion in a metal. Match the lines labeled

lium in Si at 1100°C is 7 * 10 - 13 cm 2 > s.

A, B, and C with the type of diffusion. Calculate the concentration of Ga at a depth Justify your answer by calculating the acti-

of 2.0 micrometer if the surface concentra- vation energy for diffusion for each case.

tion of Ga is 10 23 atoms cm 3 > . The diffusion times are 1, 2, and 3 hours.

Section 5-7 Permeability of Polymers

5-42 Compare the rate at which oxygen ions dif- 5-36 What are barrier polymers?

fuse in alumina (Al 2 O 3 ) with the rate at which aluminum ions diffuse in Al 2 O 3 at

5-37 What factors, other than permeability, are 1500°C. Explain the difference. important in selecting a polymer for making

plastic bottles?

5-43

A carburizing process is carried out on a 0.10% C steel by introducing 1.0% C at the

5-38 Amorphous PET is more permeable to CO 2 surface at 980°C, where the iron is FCC. than PET that contains microcrystallites.

Calculate the carbon content at 0.01 cm, Explain why.

0.05 cm, and 0.10 cm beneath the surface after 1 h.

Section 5-8 Composition Profile (Fick’s

5-44 Iron containing 0.05% C is heated to 912°C in

Second Law)

an atmosphere that produces 1.20% C at the 5-39 Transistors are made by doping single crystal

surface and is held for 24 h. Calculate the car- silicon with different types of impurities to

bon content at 0.05 cm beneath the surface if generate n- and p- type regions. Phosphorus

(a) the iron is BCC and

(P) and boron (B) are typical n- and p-type (b) the iron is FCC. Explain the difference. dopant species, respectively. Assuming that a

5-45 What temperature is required to obtain thermal treatment at 1100°C for 1 h is used

0.50% C at a distance of 0.5 mm beneath the to cause diffusion of the dopants, calculate

surface of a 0.20% C steel in 2 h, when the constant surface concentration of P and

1.10% C is present at the surface? Assume

B needed to achieve a concentration of

that the iron is FCC.

Problems 193 5-46

A 0.15% C steel is to be carburized at 1100°C, problem. Applying the appropriate boundary giving 0.35% C at a distance of 1 mm beneath

conditions, the solution to Fick’s second law the surface. If the surface composition is

under these conditions is maintained at 0.90% C, what time is required?

5-47

1 1200°C in 4 h, with the carbon content a p Dt 4 Dt b

A 0.02% C steel is to be carburized at

c(x, t) =

exp ,

0.6 mm beneath the surface reaching where Q is the initial surface concentration 0.45% C. Calculate the carbon content with units of atoms cm 2 . Assume that we required at the surface of the steel.

> implant 10 14 atoms cm 2 > of phosphorus

5-48

A 1.2% C tool steel held at 1150°C is exposed at the surface of a silicon wafer with a to oxygen for 48 h. The carbon content at the

background boron concentration of 10 16 steel surface is zero. To what depth will the

atoms cm 3 > and this wafer is subsequently steel be decarburized to less than 0.20% C?

annealed at 1100°C. The diffusion coeffi-

5-49

A 0.80% C steel must operate at 950°C in an cient (D) of phosphorus in silicon at oxidizing environment for which the carbon - 1100°C is 6.5 * 10 13 cm 2 > s.

content at the steel surface is zero. Only the (a) Plot a graph of the concentration c outermost 0.02 cm of the steel part can fall

(atoms cm 3 > ) versus x (cm) for anneal below 0.75% C. What is the maximum time

times of 5 minutes, 10 minutes, and that the steel part can operate?

15 minutes.

5-50

A steel with the BCC crystal structure (b) What is the anneal time required for the containing 0.001% N is nitrided at 550°C for

phosphorus concentration to equal the

5 h. If the nitrogen content at the steel boron concentration at a depth of 1 &m? surface is 0.08%, determine the nitrogen

content at 0.25 mm from the surface.

Section 5-9 Diffusion and Materials

5-51 What time is required to nitride a 0.002% N

Processing

steel to obtain 0.12% N at a distance of 5-55 Arrange the following materials in increas- 0.002 in. beneath the surface at 625°C? The

ing order of self-diffusion coefficient: Ar nitrogen content at the surface is 0.15%.

gas, water, single crystal aluminum, and 5-52 We can successfully perform a carburizing

liquid aluminum at 700°C. heat treatment at 1200°C in 1 h. In an effort

5-56 Most metals and alloys can be processed to reduce the cost of the brick lining in our

using the melting and casting route, but we furnace, we propose to reduce the carburiz-

typically do not choose to apply this method ing temperature to 950°C. What time will be

for the processing of specific metals (e.g., W) required to give us a similar carburizing

and most ceramics. Explain. treatment?

5-57 What is sintering? What is the driving force 5-53 During freezing of a Cu-Zn alloy, we find

for sintering?

that the composition is nonuniform. By 5-58 Why does grain growth occur? What is heating the alloy to 600°C for 3 h, diffusion

meant by the terms normal and abnormal of zinc helps to make the composition more

grain growth?

uniform. What temperature would be 5-59 Why is the strength of many metallic mate- required if we wished to perform this

rials expected to decrease with increasing homogenization treatment in 30 minutes?

grain size?

5-54 To control junction depth in transistors, pre-

A ceramic part made of MgO is sintered suc- cise quantities of impurities are introduced at

5-60

cessfully at 1700°C in 90 minutes. To mini- relatively shallow depths by ion implantation

mize thermal stresses during the process, we and diffused into the silicon substrate in a

plan to reduce the temperature to 1500°C. subsequent thermal treatment. This can be

Which will limit the rate at which sintering approximated as a finite source diffusion

can be done: diffusion of magnesium ions or

194 CHAPTER 5

Atom and Ion Movements in Materials

diffusion of oxygen ions? What time will be

A steel gear initially containing 0.10% C is required at the lower temperature?

5-65

to be carburized so that the carbon content

5-61

A Cu-Zn alloy has an initial grain diameter of at a depth of 0.05 in. is 0.50% C. We can

0.01 mm. The alloy is then heated to various generate a carburizing gas at the surface that temperatures, permitting grain growth to

contains anywhere from 0.95% C to 1.15% C. occur. The times required for the grains to grow

Design an appropriate carburizing heat to a diameter of 0.30 mm are shown below.

treatment. 5-66 When a valve casting containing copper and

Temperature (°C)

Time (minutes)

nickel solidifies under nonequilibrium condi-

tions, we find that the composition of the

alloy varies substantially over a distance of

0.005 cm. Usually we are able to eliminate

10 this concentration difference by heating the

3 alloy for 8 h at 1200°C; however, sometimes this treatment causes the alloy to begin to

Determine the activation energy for grain melt, destroying the part. Design a heat treat- growth. Does this correlate with the diffu-

ment that will eliminate the non-uniformity sion of zinc in copper? (Hint: Note that rate

without melting. Assume that the cost of is the reciprocal of time.)

operating the furnace per hour doubles for 5-62 What are the advantages of using hot press-

each 100°C increase in temperature. ing and hot isostatic pressing compared to

5-67 Assume that the surface concentration of using normal sintering?

phosphorus (P) being diffused in Si is

10 5-63 21 A sheet of gold is diffusion-bonded to a sheet atoms cm 3 > . We need to design a process, of silver in 1 h at 700°C. At 500°C, 440 h

known as the pre-deposition step, such that the are required to obtain the same degree of

concentration of P (c 1 for step 1) at a depth of bonding, and at 300°C, bonding requires

0.25 micrometer is 10 13 atoms cm 3 > . Assume 1530 years. What is the activation energy for

that this is conducted at a temperature of the diffusion bonding process? Does it

1000°C and the diffusion coefficient of P in - appear that diffusion of gold or diffusion of 14 Si at this temperature is 2.5 * 10 cm 2 > s.

silver controls the bonding rate? (Hint: Note Assume that the process is carried out for a that rate is the reciprocal of time.)

total time of 8 minutes. Calculate the concentration profile (i.e., c 1 as a function of depth, which in this case is given by the following

Design Problems

equation). Notice the use of the complemen- tary error function.

5-64 Design a spherical tank, with a wall thick- x

ness of 2 cm that will ensure that no more

c 1 (x, t 1 )=c s c1 - erf a 4Dt b d than 50 kg of hydrogen will be lost per year.

The tank, which will operate at 500°C, can Use different values of x to generate and plot

be made of nickel, aluminum, copper, or this profile of P during the pre-deposition step. iron. The diffusion coefficient of hydrogen

and the cost per pound for each available material is listed below.

Computer Problems

Diffusion Data Material

D 0 (cm 2 / s) Q (cal mol) / Cost ($ lb) /

5-68 Calculation of Diffusion Coefficients. Write

a computer program that will ask the user

Nickel 0.0055

4.10 to provide the data for the activation energy

1.10 Q and the value of D 0 . The program should

Copper 0.011

0.15 then ask the user to input a valid range of temperatures. The program, when executed,

Iron (BCC) 0.0012

Problems 195

provides the values of D as a function of The program should also be able to handle temperature in increments chosen by the

calculation of heat treatment times if the

user provides a level of concentration that is and temperature to the proper units. For

user. The program should convert Q, D 0 ,

needed at a specified depth. This program example, if the user provides temperatures

will require calculation of the error function. in °F, the program should recognize that and

The programming language or spreadsheet convert the temperature into K. The pro-

you use may have a built-in function that gram should also carry a cautionary state-

calculates the error function. If that is the ment about the standard assumptions made.

case, use that function. You can also calcu- For example, the program should caution

late the error function as the expansion of the user that effects of any possible changes

the following series:

in crystal structure in the temperature range specified are not accounted for. Check the

-z 2

validity of your programs using examples in

- e erf(z) = 1 1 a1 - - 1*3 + 1*3*5 +Á

the book and also other problems that you

(2z ) (2z )

may have solved using a calculator. 5-69 Comparison of Reaction Rates. Write a

or use an approximation computer program that will ask the user to

1 -z input the activation energy for a chemical 2 e

erf (z) = 1 - reaction. The program should then ask the

ac 1 p d z b user to provide two temperatures for which

the reaction rates need to be compared. In these equations, z is the argument of the Using the value of the gas constant and acti-

error function. Also, under certain situations, vation energy, the program should then pro-

you will know the value of the error function vide a ratio of the reaction rates. The

(if all concentrations are provided) and you program should take into account different

will have to figure out the argument of the units for activation energy.

error function. This can be handled by having 5-70 Carburization Heat Treatment. The pro-

part of the program compare different values gram should ask the user to provide an input

for the argument of the error function and by for the carbon concentration at the surface

minimizing the difference between the value (c s ), and the concentration of carbon in the

of the error function you require and the value bulk (c 0 ). The program should also ask the

of the error function that was approximated. user to provide the temperature and a value

for D (or values for D 0 and Q, that will allow

for D to be calculated). The program should

Problems

then provide an output of the concentration profile in tabular form. The distances at

K5-1 Compare the carbon dioxide permeabilities of which concentrations of carbon are to be

low-density polyethylene (LDPE), polypropy- determined can be provided by the user or

lene, and polyethylene terephthalate (PET) defined by the person writing the program.

films at room temperature.

Given by Equation 5-7 Note that in the equation describing Fick’s sec- ond law (Equation 5-7):