Chapter 5 - Atom & Ion Movements PDF

Atom and Ion Movements Chapter 5 in Materials Have You Ever Wondered? Aluminum oxidizes more easily than iron, so why do we say aluminum nor- mally does not “rust?” Wh...

Atom and Ion Movements Chapter 5 in Materials Have You Ever Wondered? Aluminum oxidizes more easily than iron, so why do we say aluminum nor- mally does not “rust?” What kind of plastic is used to make carbonated beverage bottles? How are the surfaces of certain steels hardened? Why do we encase optical fibers using a polymeric coating? Who invented the first contact lens? How does a tungsten filament in a light bulb fail? I n Chapter 4, we learned that the atomic and ionic arrangements in materials are never perfect. We also saw that most materials are not pure elements; they are alloys or blends of different elements or compounds. Different types of atoms or ions typically “diffuse”, or move within the material, so the differences in their concen- tration are minimized. Diffusion refers to an observable net flux of atoms or other species. It depends upon the concentration gradient and temperature. Just as water flows from a mountain toward the sea to minimize its gravitational potential energy, atoms and ions have a tendency to move in a predictable fashion to eliminate concen- tration differences and produce homogeneous compositions that make the material thermodynamically more stable. In this chapter, we will learn that temperature influences the kinetics of diffu- sion and that a concentration difference contributes to the overall net flux of diffusing species. The goal of this chapter is to examine the principles and applications of dif- fusion in materials. We’ll illustrate the concept of diffusion through examples of sev- eral real-world technologies dependent on the diffusion of atoms, ions, or molecules. We will present an overview of Fick’s laws that describe the diffusion process quantitatively. We will also see how the relative openness of different crystal struc- tures and the size of atoms or ions, temperature, and concentration of diffusing species affect the rate at which diffusion occurs. We will discuss specific examples of how diffusion is used in the synthesis and processing of advanced materials as well as manufacturing of components using advanced materials. 155 Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 156 CHAPTER 5 Atom and Ion Movements in Materials 5-1 Applications of Diffusion Diffusion Diffusion refers to the net flux of any species, such as ions, atoms, electrons, holes (Chapter 19), and molecules. The magnitude of this flux depends upon the concentration gradient and temperature. The process of diffusion is central to a large number of today’s important technologies. In materials processing technologies, control over the diffusion of atoms, ions, molecules, or other species is key. There are hundreds of applications and technologies that depend on either enhancing or limiting diffusion. The following are just a few examples. Carburization for Surface Hardening of Steels Let’s say we want a surface, such as the teeth of a gear, to be hard; however, we do not want the entire gear to be hard. The carburization process can be used to increase surface hard- ness. In carburization, a source of carbon, such as a graphite powder or gaseous phase containing carbon, is diffused into steel components such as gears (Figure 5-1). In later chapters, you will learn how increased carbon concentration on the surface of the steel increases the steel’s hardness. Similar to the introduction of carbon, we can also use a process known as nitriding, in which nitrogen is introduced into the surface of a metallic material. Diffusion also plays a central role in the control of the phase transformations Figure 5-1 Furnace for heat treating steel using the carburization process. (Courtesy of Cincinnati Steel Treating.) Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.  5 - 1 Applications of Diffusion 157 needed for the heat treatment of metals and alloys, the processing of ceramics, and the solidification and joining of materials (see Section 5-9). Dopant Diffusion for Semiconductor Devices The entire microelectronics industry, as we know it today, would not exist if we did not have a very good understanding of the diffusion of different atoms into silicon or other semiconductors. The creation of the p-n junction (Chapter 19) involves diffusing dopant atoms, such as phospho- rous (P), arsenic (As), antimony (Sb), boron (B), aluminum (Al), etc., into precisely defined regions of silicon wafers. Some of these regions are so small that they are best measured in nanometers. A p-n junction is a region of the semiconductor, one side of which is doped with n-type dopants (e.g., As in Si) and the other side is doped with p-type dopants (e.g., B in Si). Conductive Ceramics In general, polycrystalline ceramics tend to be good insulators of electricity. Diffusion of ions, electrons, or holes also plays an impor- tant role in the electrical conductivity of many conductive ceramics, such as partially or fully stabilized zirconia (ZrO2) or indium tin oxide (also commonly known as ITO). Lithium cobalt oxide (LiCoO2) is an example of an ionically conductive material that is used in lithium ion batteries. These ionically conductive materials are used for such prod- ucts as oxygen sensors in cars, touch-screen displays, fuel cells, and batteries. The ability of ions to diffuse and provide a pathway for electrical conduction plays an important role in enabling these applications. Creation of Plastic Beverage Bottles The occurrence of diffu- sion may not always be beneficial. In some applications, we may want to limit the occurrence of diffusion for certain species. For example, in the creation of certain plastic bottles, the dif- fusion of carbon dioxide (CO2) must be minimized. This is one of the major reasons why we use polyethylene terephthalate (PET) to make bottles which ensure that the carbonated beverages they contain will not lose their fizz for a reasonable period of time! Oxidation of Aluminum You may have heard or know that aluminum does not “rust.” In reality, aluminum oxidizes (rusts) more easily than iron; however, the aluminum oxide (Al2O3) forms a very protective but thin coating on the aluminum’s sur- face preventing any further diffusion of oxygen and hindering further oxidation of the underlying aluminum. The oxide coating does not have a color and is thin and, hence, invisible. This is why we think aluminum does not rust. Coatings and Thin Films Coatings and thin films are often used to limit the diffusion of water vapor, oxygen, or other chemicals. Thermal Barrier Coatings for Turbine Blades In an air- craft engine, some of the nickel superalloy-based turbine blades are coated with ceramic oxides such as yttria stabilized zirconia (YSZ). These ceramic coatings protect the under- lying alloy from high temperatures; hence, the name thermal barrier coatings (TBCs) (Figure 5-2). The diffusion of oxygen through these ceramic coatings and the subsequent oxidation of the underlying alloy play a major role in determining the lifetime and dura- bility of the turbine blades. In Figure 5-2, EBPVD means electron beam physical vapor deposition. The bond coat is either a platinum or molybdenum-based alloy. It provides adhesion between the TBC and the substrate. Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 158 CHAPTER 5 Atom and Ion Movements in Materials Figure 5-2 A thermal barrier coating on a nickel-based superalloy. (Courtesy of Dr. F.S. Pettit and Dr. G.H. Meier, University of Pittsburgh.) Optical Fibers and Microelectronic Components Optical fibers made from silica (SiO2) are coated with polymeric materials to prevent diffusion of water molecules. This, in turn, improves the optical and mechanical properties of the fibers. Example 5-1 Diffusion of Ar/ He and Cu/ Ni Consider a box containing an impermeable partition that divides the box into equal volumes (Figure 5-3). On one side, we have pure argon (Ar) gas; on the other side, we have pure helium (He) gas. Explain what will happen when the partition is opened? What will happen if we replace the Ar side with a Cu single crystal and the He side with a Ni single crystal? SOLUTION Before the partition is opened, one compartment has no argon and the other has no helium (i.e., there is a concentration gradient of Ar and He). When the partition is opened, Ar atoms will diffuse toward the He side, and vice versa. This diffusion will continue until the entire box has a uniform concentration of both gases. There may be some density gradient driven convective flows as well. If we took random samples of different regions in this box after a few hours, we would find a statisti- cally uniform concentration of Ar and He. Owing to their thermal energy, the Ar and He atoms will continue to move around in the box; however, there will be no concentration gradients. If we open the hypothetical partition between the Ni and Cu single crys- tals at room temperature, we would find that, similar to the Ar> He situation, the Figure 5-3 Illustration for diffusion of Ar> He and Cu> Ni (for Example 5-1). Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.  5 - 2 Stability of Atoms and Ions 159 concentration gradient exists but the temperature is too low to see any significant diffusion of Cu atoms into the Ni single crystal and vice versa. This is an example of a situation in which a concentration gradient exists; however, because of the lower temperature, the kinetics for diffusion are not favorable. Certainly, if we increase the temperature (say to 600°C) and wait for a longer period of time (e.g., ⬃24 hours), we would see diffusion of Cu atoms into the Ni single crystal and vice versa. After a very long time, the entire solid will have a uniform concen- tration of Ni and Cu atoms. The new solid that forms consists of Cu and Ni atoms completely dissolved in each other and the resultant material is termed a “solid solution,” a concept we will study in greater detail in Chapter 10. This example also illustrates something many of you may know by intu- ition. The diffusion of atoms and molecules occurs faster in gases and liquids than in solids. As we will see in Chapter 9 and other chapters, diffusion has a significant effect on the evolution of microstructure during the solidification of alloys, the heat treatment of metals and alloys, and the processing of ceramic materials. 5-2 Stability of Atoms and Ions In Chapter 4, we showed that imperfections are present and also can be deliberately intro- duced into a material; however, these imperfections and, indeed, even atoms or ions in their normal positions in the crystal structures are not stable or at rest. Instead, the atoms or ions possess thermal energy, and they will move. For instance, an atom may move from a normal crystal structure location to occupy a nearby vacancy. An atom may also move from one interstitial site to another. Atoms or ions may jump across a grain boundary, causing the grain boundary to move. The ability of atoms and ions to diffuse increases as the temperature, or thermal energy possessed by the atoms and ions, increases. The rate of atom or ion movement is related to temperature or thermal energy by the Arrhenius equation: b -Q Rate = c0 expa (5-1) RT where c0 is a constant, R is the gas constant A 1.987 mol cal # K B , T is the absolute temperature (K), and Q is the activation energy (cal> mol) required to cause Avogadro’s number of atoms or ions to move. This equation is derived from a statistical analysis of the proba- bility that the atoms will have the extra energy Q needed to cause movement. The rate is related to the number of atoms that move. We can rewrite the equation by taking natural logarithms of both sides: Q ln(rate) = ln(c0) - (5-2) RT If we plot ln(rate) of some reaction versus 1> T (Figure 5-4), the slope of the line will be -Q> R and, consequently, Q can be calculated. The constant c0 corresponds to the inter- cept at ln(c0) when 1> T is zero. Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 160 CHAPTER 5 Atom and Ion Movements in Materials 28 Figure 5-4 The Arrhenius plot of ln(rate) –Q/R = slope versus 1> T can be used to 1n(5 × 108) – 1n(8 × 1010) determine the activation 26 –Q/R = energy for a reaction. The 1/773 – 1/1073 Q/R = 14,032 K data from this figure is used in Example 5-2. 1n(5 × 108) – 1n(8 × 1010) 24 In(Rate) 22 20 1/773 – 1/1073 18 6.0 × 10–4 8.0 × 10–4 1.0 × 10–3 1.2 × 10–3 1.4 × 10–3 1.6 × 10–3 –1 1/T (K ) Svante August Arrhenius (1859–1927), a Swedish chemist who won the Nobel Prize in Chemistry in 1903 for his research on the electrolytic theory of dissociation applied this idea to the rates of chemical reactions in aqueous solutions. His basic idea of activation energy and rates of chemical reactions as functions of temperature has since been applied to diffusion and other rate processes. Example 5-2 Activation Energy for Interstitial Atoms Suppose that interstitial atoms are found to move from one site to another at the rates of 5 * 108 jumps> s at 500°C and 8 * 1010 jumps> s at 800°C. Calculate the acti- vation energy Q for the process. SOLUTION Figure 5-4 represents the data on a ln(rate) versus 1> T plot; the slope of this line, as calculated in the figure, gives Q> R ⫽ 14,032 K, or Q ⫽ 27,880 cal> mol. Alternately, we could write two simultaneous equations: b = c0 expa b jumps -Q Ratea s RT cal - QQmol R 5 * 10 a b = c0 a b exp D jumps 8 jumps T c1.987Qmol # K R d [(500 + 273)(K)] s s cal = c0 exp( -0.000651Q) Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.  5 - 3 Mechanisms for Diffusion 161 cal - QQmol R 8 * 10 a b = c0 a b exp D 10jumps jumps T c1.987Qmol # K R d [(800 + 273)(K)] s s cal = c0 exp( -0.000469Q) Note the temperatures were converted into K. Since 5 * 108 a b jumps c0 = exp( -0.000651Q) s then (5 * 108)exp( -0.000469Q) 8 * 1010 = exp( -0.000651Q) 160 = exp[(0.000651 - 0.000469)Q] = exp(0.000182Q) ln(160) = 5.075 = 0.000182Q 5.075 Q = = 27,880 cal/mol 0.000182 5-3 Mechanisms for Diffusion As we saw in Chapter 4, defects known as vacancies exist in materials. The disorder these vacancies create (i.e., increased entropy) helps minimize the free energy and, therefore, increases the thermodynamic stability of a crystalline material. In materials containing vacancies, atoms move or “jump” from one lattice position to another. This process, known as self-diffusion, can be detected by using radioactive tracers. As an example, suppose we were to introduce a radioactive isotope of gold (Au198) onto the surface of standard gold (Au197). After a period of time, the radioactive atoms would move into the standard gold. Eventually, the radioactive atoms would be uniformly distributed throughout the entire standard gold sample. Although self- diffusion occurs continually in all materials, its effect on the material’s behavior is generally not significant. Interdiffusion Diffusion of unlike atoms in materials also occurs (Figure 5-5). Consider a nickel sheet bonded to a copper sheet. At high temperatures, nickel atoms gradually diffuse into the copper, and copper atoms migrate into the nickel. Again, the nickel and copper atoms eventually are uniformly distributed. Diffusion of different atoms in different directions is known as interdiffusion. There are two important mechanisms by which atoms or ions can diffuse (Figure 5-6). Vacancy Diffusion In self-diffusion and diffusion involving substitutional atoms, an atom leaves its lattice site to fill a nearby vacancy (thus creating a new vacancy at the original lattice site). As diffusion continues, we have counterflows of atoms Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 162 CHAPTER 5 Atom and Ion Movements in Materials Figure 5-5 Diffusion of copper atoms into nickel. Eventually, the copper atoms are randomly distributed throughout the nickel. and vacancies, called vacancy diffusion. The number of vacancies, which increases as the temperature increases, influences the extent of both self-diffusion and diffusion of sub- stitutional atoms. Interstitial Diffusion When a small interstitial atom or ion is present in the crystal structure, the atom or ion moves from one interstitial site to another. No vacancies are required for this mechanism. Partly because there are many more intersti- tial sites than vacancies, interstitial diffusion occurs more easily than vacancy diffusion. Interstitial atoms that are relatively smaller can diffuse faster. In Chapter 3, we saw that Figure 5-6 Diffusion mechanisms in materials: (a) vacancy or substitutional atom diffusion and (b) interstitial diffusion. Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.  5 - 4 Activation Energy for Diffusion 163 the structure of many ceramics with ionic bonding can be considered as a close packing of anions with cations in the interstitial sites. In these materials, smaller cations often diffuse faster than larger anions. 5-4 Activation Energy for Diffusion A diffusing atom must squeeze past the surrounding atoms to reach its new site. In order for this to happen, energy must be supplied to allow the atom to move to its new position, as shown schematically for vacancy and interstitial diffusion in Figure 5-7. The atom is originally in a low-energy, relatively stable location. In order to move to a new location, the atom must overcome an energy barrier. The energy barrier is the activation energy Q. The thermal energy supplies atoms or ions with the energy needed to exceed this barrier. Note that the symbol Q is often used for activation energies for different processes (rate at which atoms jump, a chemical reaction, energy needed to produce vacancies, etc.), and we should be careful in understanding the specific process or phenomenon to which the general term for activation energy Q is being applied, as the value of Q depends on the particular phenomenon. Normally, less energy is required to squeeze an interstitial atom past the surrounding atoms; consequently, activation energies are lower for interstitial diffusion than for vacancy diffusion. Typical values for activation energies for diffusion of dif- ferent atoms in different host materials are shown in Table 5-1. We use the term diffusion couple to indicate a combination of an atom of a given element (e.g., carbon) diffusing in a host material (e.g., BCC Fe). A low-activation energy indicates easy diffusion. In self-diffusion, the activation energy is equal to the energy needed to cre- ate a vacancy and to cause the movement of the atom. Table 5-1 also shows values of D0, which is the pre-exponential term and the constant c 0 from Equation 5-1, when the rate process is diffusion. We will see later that D0 is the diffusion coefficient when 1 > T ⫽ 0. Figure 5-7 The activation energy Q is required to squeeze atoms past one another during diffusion. Generally, more energy is required for a substitutional atom than for an interstitial atom. Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 164 CHAPTER 5 Atom and Ion Movements in Materials TABLE 5-1 Diffusion data for selected materials Diffusion Couple Q (cal/ mol) D0 (cm2/ s) Interstitial diffusion: C in FCC iron 32,900 0.23 C in BCC iron 20,900 0.011 N in FCC iron 34,600 0.0034 N in BCC iron 18,300 0.0047 H in FCC iron 10,300 0.0063 H in BCC iron 3,600 0.0012 Self-diffusion (vacancy diffusion): Pb in FCC Pb 25,900 1.27 Al in FCC Al 32,200 0.10 Cu in FCC Cu 49,300 0.36 Fe in FCC Fe 66,700 0.65 Zn in HCP Zn 21,800 0.1 Mg in HCP Mg 32,200 1.0 Fe in BCC Fe 58,900 4.1 W in BCC W 143,300 1.88 Si in Si (covalent) 110,000 1800.0 C in C (covalent) 163,000 5.0 Heterogeneous diffusion (vacancy diffusion): Ni in Cu 57,900 2.3 Cu in Ni 61,500 0.65 Zn in Cu 43,900 0.78 Ni in FCC iron 64,000 4.1 Au in Ag 45,500 0.26 Ag in Au 40,200 0.072 Al in Cu 39,500 0.045 Al in Al2O3 114,000 28.0 O in Al2O3 152,000 1900.0 Mg in MgO 79,000 0.249 O in MgO 82,100 0.000043 From several sources, including Adda, Y. and Philibert, J., La Diffusion dans les Solides, Vol. 2, 1966. 5-5 Rate of Diffusion [Fick’s First Law] Adolf Eugen Fick (1829–1901) was the first scientist to provide a quantitative descrip- tion of the diffusion process. Interestingly, Fick was also the first to experiment with contact lenses in animals and the first to implant a contact lens in human eyes in 1887–1888! The rate at which atoms, ions, particles or other species diffuse in a material can be measured by the flux J. Here we are mainly concerned with diffusion of ions or atoms. The flux J is defined as the number of atoms passing through a plane of unit area per unit time (Figure 5-8). Fick’s first law explains the net flux of atoms: dc J = -D (5-3) dx Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.  5 - 5 Rate of Diffusion [Fick’s First Law] 165 Figure 5-8 The flux during diffusion is defined as the number of atoms passing through a plane of unit area per unit time. where J is the flux, D is the diffusivity or diffusion coefficient 1 s 2 , and dc > dx is the cm 2 concentration gradient 1 cm3 # cm2. Depending upon the situation, concentration may be atoms expressed as atom percent (at%), weight percent (wt%), mole percent (mol%), atom fraction, or mole fraction. The units of concentration gradient and flux will change accordingly. Several factors affect the flux of atoms during diffusion. If we are dealing with dc diffusion of ions, electrons, holes, etc., the units of J, D, and dx will reflect the appropri- ate species that are being considered. The negative sign in Equation 5-3 tells us that the dc flux of diffusing species is from higher to lower concentrations, so that if the dx term is neg- ative, J will be positive. Thermal energy associated with atoms, ions, etc., causes the random movement of atoms. At a microscopic scale, the thermodynamic driving force for diffusion is the concentration gradient. A net or an observable flux is created depending upon temperature and the concentration gradient. Concentration Gradient The concentration gradient shows how the composition of the material varies with distance: ⌬c is the difference in concentration over the distance ⌬x (Figure 5-9). A concentration gradient may be created when two materials of different composition are placed in contact, when a gas or liquid is in con- tact with a solid material, when nonequilibrium structures are produced in a material due to processing, and from a host of other sources. Figure 5-9 Illustration of the concentration gradient. Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 166 CHAPTER 5 Atom and Ion Movements in Materials The flux at a particular temperature is constant only if the concentration gradi- ent is also constant—that is, the compositions on each side of the plane in Figure 5-8 remain unchanged. In many practical cases, however, these compositions vary as atoms are redistributed, and thus the flux also changes. Often, we find that the flux is initially high and then gradually decreases as the concentration gradient is reduced by diffusion. The examples that follow illustrate calculations of flux and concentration gradients for diffu- sion of dopants in semiconductors and ceramics, but only for the case of constant con- centration gradient. Later in this chapter, we will consider non-steady state diffusion with the aid of Fick’s second law. Example 5-3 Semiconductor Doping One step in manufacturing transistors, which function as electronic switches in inte- grated circuits, involves diffusing impurity atoms into a semiconductor material such as silicon (Si). Suppose a silicon wafer 0.1 cm thick, which originally contains one phosphorus atom for every 10 million Si atoms, is treated so that there are 400 phos- phorous (P) atoms for every 10 million Si atoms at the surface (Figure 5-10). Calculate the concentration gradient (a) in atomic percent> cm and (b) in cm atoms 3 # cm. The lattice parameter of silicon is 5.4307Å. SOLUTION a. First, let’s calculate the initial and surface compositions in atomic percent: 1 P atom ci = * 100 = 0.00001 at% P 107 atoms 400 P atoms cs = * 100 = 0.004 at% P 107 atoms ¢c 0.00001 - 0.004 at% P at% P = = - 0.0399 ¢x 0.1 cm cm atoms b. To find the gradient in terms of cm 3 # cm , we must find the volume of the unit cell. The crystal structure of Si is diamond cubic (DC). The lattice parameter is 5.4307 * 10-8 cm. Thus, 3 Vcell = (5.4307 * 10-8 cm)3 = 1.6 * 10-22 cm cell The volume occupied by 107 Si atoms, which are arranged in a DC structure with 8 atoms> cell, is 107 atoms V = c d c 1.6 * 10-22 a cm cell b d = 2 * 10 3 -16 cm3 8 atoms cell Figure 5-10 Silicon wafer showing a variation in concentration of P atoms (for Example 5-3). Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.  5 - 5 Rate of Diffusion [Fick’s First Law] 167 The compositions in atoms> cm3 are = 0.005 * 1018 P A atoms cm3 B 1 P atom ci = 3 2 * 10 cm -16 = 2 * 1018 P A atoms cm3 B 400 P atoms cs = 2 * 10-16 cm3 Thus, the composition gradient is ¢c 0.005 * 1018 - 2 * 1018 P A atoms cm3 B = ¢x 0.1 cm 3 # cm R atoms = - 1.995 * 1019 PQ cm Example 5-4 Diffusion of Nickel in Magnesium Oxide (MgO) A 0.05 cm layer of magnesium oxide (MgO) is deposited between layers of nickel (Ni) and tantalum (Ta) to provide a diffusion barrier that prevents reactions between the two metals (Figure 5-11). At 1400°C, nickel ions diffuse through the MgO ceramic to the tantalum. Determine the number of nickel ions that pass through the MgO per second. At 1400°C, the diffusion coefficient of nickel ions in MgO is 9 * 10-12 cm2> s, and the lattice parameter of nickel at 1400°C is 3.6 * 10-8 cm. SOLUTION The composition of nickel at the Ni> MgO interface is 100% Ni, or atoms 4 Ni unit cell c Ni/MgO = = 8.573 * 1022 atoms cm3 (3.6 * 10-8 cm)3 The composition of nickel at the Ta> MgO interface is 0% Ni. Thus, the concentra- tion gradient is ¢c 0 - 8.573 * 1022 atoms cm3 atoms = = - 1.715 * 1024 cm 3 # cm ¢x 0.05 cm Figure 5-11 Diffusion couple (for Example 5-4). Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 168 CHAPTER 5 Atom and Ion Movements in Materials The flux of nickel atoms through the MgO layer is = - (9 * 10-12 cm2> s) A -1.715 * 1024 cm 3 # cm B ¢c atoms J = -D ¢x J = 1.543 * 1013 Nicmatoms 2#s The total number of nickel atoms crossing the 2 cm * 2 cm interface per second is Total Ni atoms per second = (J )( Area) = A 1.543 * 1013 atoms cm2 # s B (2 cm)(2 cm) = 6.17 * 1013 Ni atoms/s Although this appears to be very rapid, in one second, the volume of nickel atoms removed from the Ni> MgO interface is Ni atoms 6.17 * 1013 s 3 = 0.72 * 10-9 cm s 8.573 * 1022 Ni cm atoms 3 The thickness by which the nickel layer is reduced each second is 3 0.72 * 10-9 cms = 1.8 * 10-10 cm s 4 cm2 For one micrometer (10-4 cm) of nickel to be removed, the treatment requires 10-4 cm = 556,000 s = 154 h 1.8 * 10-10 cm s 5-6 Factors Affecting Diffusion Temperature and the Diffusion Coefficient The kinetics of diffusion are strongly dependent on temperature. The diffusion coefficient D is related to temperature by an Arrhenius-type equation, b -Q D = D0 expa (5-4) RT where Q is the activation energy (in units of cal> mol) for diffusion of the species under consideration (e.g., Al in Si), R is the gas constant A 1.987 molcal# K B , and T is the absolute temperature (K). D0 is the pre-exponential term, similar to c0 in Equation 5-1. D0 is a constant for a given diffusion system and is equal to the value of the dif- fusion coefficient at 1> T ⫽ 0 or T ⫽ ⬁. Typical values for D0 are given in Table 5-1, while the temperature dependence of D is shown in Figure 5-12 for some metals and ceramics. Covalently bonded materials, such as carbon and silicon (Table 5-1), have unusually high Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.  5 - 6 Factors Affecting Diffusion 169 Figure 5-12 The diffusion coefficient D as a function of reciprocal temperature for some metals and ceramics. In this Arrhenius plot, D represents the rate of the diffusion process. A steep slope denotes a high activation energy. activation energies, consistent with the high strength of their atomic bonds. Figure 5-13 shows the diffusion coefficients for different dopants in silicon. In ionic materials, such as some of the oxide ceramics, a diffusing ion only enters a site having the same charge. In order to reach that site, the ion must physically squeeze past adjoining ions, pass by a region of opposite charge, and move a relatively long distance (Figure 5-14 ). Consequently, the activation energies are high and the rates of diffusion are lower for ionic materials than those for metals (Figure 5-15 on page 171). We take advantage of this in many situations. For example, in the processing of silicon (Si), we create a thin layer of silica (SiO2) on top of a silicon wafer (Chapter 19). We then create a window by remov- ing part of the silica layer. This window allows selective diffusion of dopant atoms such as phosphorus (P) and boron (B), because the silica layer is essentially impervious to the dopant atoms. Slower diffusion in most oxides and other ceramics is also an advantage in applications in which components are required to withstand high temperatures. When the temperature of a material increases, the diffusion coefficient D increases (according to Equation 5-4) and, therefore, the flux of atoms increases as well. At higher Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 170 CHAPTER 5 Atom and Ion Movements in Materials Figure 5-13 Diffusion coefficients for different dopants in silicon. (From “Diffusion and Diffusion Induced Defects in Silicon,” by U. Gösele. In R. Bloor, M. Flemings, and S. Mahajan (Eds.), Encyclopedia of Advanced Materials, Vol. 1, 1994, p. 631, Fig. 2. Copyright © 1994 Pergamon Press. Reprinted with permission of the editor.) Diffusion Coefficient temperatures, the thermal energy supplied to the diffusing atoms permits the atoms to overcome the activation energy barrier and more easily move to new sites. At low temperatures—often below about 0.4 times the absolute melting temperature of the material—diffusion is very slow and may not be significant. For this reason, the heat treat- ment of metals and the processing of ceramics are done at high temperatures, where atoms move rapidly to complete reactions or to reach equilibrium conditions. Because less ther- mal energy is required to overcome the smaller activation energy barrier, a small activa- tion energy Q increases the diffusion coefficient and flux. The following example illustrates how Fick’s first law and concepts related to the temperature dependence of D can be applied to design an iron membrane. Figure 5-14 Diffusion in ionic compounds. Anions can only enter other anion sites. Smaller cations tend to diffuse faster. Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.  5 - 6 Factors Affecting Diffusion 171 Diffusion Coefficient Figure 5-15 Diffusion coefficients of ions in different oxides. (Adapted from Physical Ceramics: Principles for Ceramic Science and Engineering, by Y.M. Chiang, D. Birnie, and W.D. Kingery, Fig. 3-1. Copyright © 1997 John Wiley & Sons. This material is used by permission of John Wiley & Sons, Inc.) Example 5-5 Design of an Iron Membrane An impermeable cylinder 3 cm in diameter and 10 cm long contains a gas that includes 0.5 * 1020 N atoms per cm3 and 0.5 * 1020 H atoms per cm3 on one side of an iron membrane (Figure 5-16). Gas is continuously introduced to the pipe to Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 172 CHAPTER 5 Atom and Ion Movements in Materials Figure 5-16 Design of an iron membrane (for Example 5-5). ensure a constant concentration of nitrogen and hydrogen. The gas on the other side of the membrane includes a constant 1 * 1018 N atoms per cm3 and 1 * 1018 H atoms per cm3. The entire system is to operate at 700°C, at which iron has the BCC structure. Design an iron membrane that will allow no more than 1% of the nitrogen to be lost through the membrane each hour, while allowing 90 % of the hydrogen to pass through the membrane per hour. SOLUTION The total number of nitrogen atoms in the container is (0.5 * 1020 N> cm3)(␲> 4)(3 cm)2(10 cm) ⫽ 35.343 * 1020 N atoms The maximum number of atoms to be lost is 1% of this total, or N atom loss per h = (0.01) A 35.34 * 1020 B = 35.343 * 1018 N atoms/h N atom loss per s = (35.343 * 1018 N atoms/h)/(3600 s/h) = 0.0098 * 1018 N atoms/s The flux is then 0.0098 * 1018( N atoms/s) J = a b(3 cm)2 p 4 = 0.00139 * 1018 N atoms cm2 # s Using Equation 5-4 and values from Table 5-1, the diffusion coefficient of nitrogen in BCC iron at 700°C ⫽ 973 K is b -Q D = D0 expa RT cal 2 -18,300 mol D N = 0.0047 cms exp C S cal 1.987 mol # K (973 K) 2 = (0.0047)(7.748 * 10-5) = 3.64 * 10-7 cm s Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.  5 - 6 Factors Affecting Diffusion 173 From Equation 5-3: b = 0.00139 * 1018 ¢c N atoms J = - Da cm2 # s ¢x c Q - 3.64 * 10-7 cm2/ s)(1 * 1018 - 50 * 1018 N cm atoms 3 Rd ¢x = - D¢c/J = 0.00139 * 1018 Ncm atoms #s 2 ⌬x ⫽ 0.013 cm ⫽ minimum thickness of the membrane In a similar manner, the maximum thickness of the membrane that will permit 90% of the hydrogen to pass can be calculated as H atom loss per h ⫽ (0.90)(35.343 * 1020) ⫽ 31.80 * 1020 H atom loss per s ⫽ 0.0088 * 1020 J = 0.125 * 1018 Hcm atoms 2#s From Equation 5-4 and Table 5-1, cal DH = 0.0012 cms exp s t = 1.86 * 10-4 cm2/s 2 -3,600 mol cal 1.987 K # mol (973 K) Since ¢x = - D ¢c/J a-1.86 * 10-4 cms b a - 49 * 1018 H cm b 2 atoms 3 ¢x = 0.125 * 1018 Hcm atoms 2#s = 0.073 cm = maximum thickness An iron membrane with a thickness between 0.013 and 0.073 cm will be satisfactory. Types of Diffusion In volume diffusion, the atoms move through the crystal from one regular or interstitial site to another. Because of the surrounding atoms, the acti- vation energy is large and the rate of diffusion is relatively slow. Atoms can also diffuse along boundaries, interfaces, and surfaces in the material. Atoms diffuse easily by grain boundary diffusion, because the atom packing is disordered and less dense in the grain boundaries. Because atoms can more easily squeeze their way through the grain boundary, the activation energy is low (Table 5-2). Surface diffusion is easier still because there is even less constraint on the diffusing atoms at the surface. atoms Time Diffusion requires time. The units for flux are cm2 # s. If a large number of atoms must diffuse to produce a uniform structure, long times may be required, even at high temperatures. Times for heat treatments may be reduced by using higher temperatures or by making the diffusion distances (related to ⌬x) as small as possible. We find that some rather remarkable structures and properties are obtained if we prevent diffusion. Steels quenched rapidly from high temperatures to prevent Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 174 CHAPTER 5 Atom and Ion Movements in Materials TABLE 5-2 The effect of the type of diffusion for thorium in tungsten and for self-diffusion in silver Diffusion Coefficient (D) Diffusion Type Thorium in Tungsten Silver in Silver D0 (cm2> s) Q (cal> mol) D0 (cm2> s) Q (cal> mol) Surface 0.47 66,400 0.068 8,900 Grain boundary 0.74 90,000 0.24 22,750 Volume 1.00 120,000 0.99 45,700 *Given by parameters for Equation 5-4. diffusion form nonequilibrium structures and provide the basis for sophisticated heat treatments. Similarly, in forming metallic glasses, we have to quench liquid metals at a very high cooling rate. This is to avoid diffusion of atoms by decreasing their thermal energy and encouraging them to assemble into nonequilibrium amorphous arrange- ments. Melts of silicate glasses, on the other hand, are viscous and diffusion of ions through these is slow. As a result, we do not have to cool these melts very rapidly to attain an amorphous structure. There is a myth that many old buildings contain win- dowpanes that are thicker at the bottom than at the top because the glass has flowed down over the years. Based on kinetics of diffusion, it can be shown that even several hundred or thousand years will not be sufficient to cause such flow of glasses at near- room temperature. In certain thin film deposition processes such as sputtering, we sometimes obtain amorphous thin films if the atoms or ions are quenched rapidly after they land on the substrate. If these films are subsequently heated (after deposition) to sufficiently high temperatures, diffusion will occur and the amorphous thin films will eventually crystallize. In the following example, we examine different mechanisms for diffusion. Example 5-6 Tungsten Thorium Diffusion Couple Consider a diffusion couple between pure tungsten and a tungsten alloy containing 1 at% thorium. After several minutes of exposure at 2000°C, a transition zone of 0.01 cm thickness is established. What is the flux of thorium atoms at this time if diffusion is due to (a) volume diffusion, (b) grain boundary diffusion, and (c) surface diffusion? (See Table 5-2.) SOLUTION The lattice parameter of BCC tungsten is 3.165 Å. Thus, the number of tungsten atoms> cm3 is W atoms 2 atoms/cell = = 6.3 * 1022 cm3 (3.165 * 10-8)3 cm3/ cell In the tungsten-1 at% thorium alloy, the number of thorium atoms is c Th = (0.01)(6.3 * 1022) = 6.3 * 1020 Th atoms cm3 Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.  5 - 6 Factors Affecting Diffusion 175 In the pure tungsten, the number of thorium atoms is zero. Thus, the concentration gradient is ¢c 0 - 6.3 * 1020 atoms cm2 atoms = = - 6.3 * 1022 Th cm 3 # cm ¢x 0.01 cm a. Volume diffusion cal -120,000 mol cm2 D = 1.0 exp £ ≥ = 2.89 * 10-12 cm2/s Q1.987 mol # K R(2273 s cal K) = - a2.89 * 10-12 cms b a - 6.3 * 1022 cm 3 # cm b ¢c 2 atoms J = -D ¢x = 18.2 * 1010 Thcmatoms 2#s b. Grain boundary diffusion cal 2 -90,000 mol D = 0.74 cms exp £ ≥ = 1.64 * 10-9 cm2/ s Q1.987 mol # K R(2273 cal K) J = - a1.64 * 10-9 cms b a- 6.3 * 1022 cm 3 # cm b = 10.3 * 10 2 atoms 13 Th atoms cm2 # s c. Surface diffusion cal cm2 -66,400 mol D = 0.47 exp £ ≥ = 1.94 * 10-7 cm2/ s Q1.987 mol # K R(2273 s cal K) J = - a1.94 * 10-7 cms b a- 6.3 * 1022 cm 3 # cm b = 12.2 * 10 2 atoms 15 Th atoms cm2 # s Dependence on Bonding and Crystal Structure A number of factors influence the activation energy for diffusion and, hence, the rate of diffusion. Interstitial diffusion, with a low-activation energy, usually occurs much faster than vacancy, or substitutional diffusion. Activation energies are usually lower for atoms diffusing through open crystal structures than for close-packed crys- tal structures. Because the activation energy depends on the strength of atomic bond- ing, it is higher for diffusion of atoms in materials with a high melting temperature (Figure 5-17). We also find that, due to their smaller size, cations (with a positive charge) often have higher diffusion coefficients than those for anions (with a negative charge). In sodium chloride, for instance, the activation energy for diffusion of chloride ions (Cl-) is about twice that for diffusion of sodium ions (Na+). Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 176 CHAPTER 5 Atom and Ion Movements in Materials Figure 5-17 The activation energy for self-diffusion increases as the melting point of the metal increases. Melting temperature Diffusion of ions also provides a transfer of electrical charge; in fact, the electri- cal conductivity of ionically bonded ceramic materials is related to temperature by an Arrhenius equation. As the temperature increases, the ions diffuse more rapidly, electri- cal charge is transferred more quickly, and the electrical conductivity is increased. As men- tioned before, some ceramic materials are good conductors of electricity. Dependence on Concentration of Diffusing Species and Composition of Matrix The diffusion coefficient (D) depends not only on temperature, as given by Equation 5-4, but also on the concentration of diffusing species and composition of the matrix. The reader should consult higher-level textbooks for more information. 5-7 Permeability of Polymers In polymers, we are most concerned with the diffusion of atoms or small molecules between the long polymer chains. As engineers, we often cite the permeability of poly- mers and other materials, instead of the diffusion coefficients. The permeability is expressed in terms of the volume of gas or vapor that can permeate per unit area, per unit time, or per unit thickness at a specified temperature and relative humidity. Polymers that have a polar group (e.g., ethylene vinyl alcohol) have higher permeabil- ity for water vapor than for oxygen gas. Polyethylene, on the other hand, has much higher permeability for oxygen than for water vapor. In general, the more compact the structure of polymers, the lesser the permeability. For example, low-density polyethyl- ene has a higher permeability than high-density polyethylene. Polymers used for food and other applications need to have the appropriate barrier properties. For example, polymer films are typically used as packaging to store food. If air diffuses through the film, the food may spoil. Similarly, care has to be exercised in the storage of ceramic or metal powders that are sensitive to atmospheric water vapor, nitrogen, oxy- gen, or carbon dioxide. For example, zinc oxide powders used in rubbers, paints, and ceramics must be stored in polyethylene bags to avoid reactions with atmospheric water vapor. Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.  5 - 8 Composition Profile [Fick’s Second Law] 177 Diffusion of some molecules into a polymer can cause swelling problems. For example, in automotive applications, polymers used to make o-rings can absorb consid- erable amounts of oil, causing them to swell. On the other hand, diffusion is required to enable dyes to uniformly enter many of the synthetic polymer fabrics. Selective diffusion through polymer membranes is used for desalinization of water. Water molecules pass through the polymer membrane, and the ions in the salt are trapped. In each of these examples, the diffusing atoms, ions, or molecules penetrate between the polymer chains rather than moving from one location to another within the chain structure. Diffusion will be more rapid through this structure when the diffusing species is smaller or when larger voids are present between the chains. Diffusion through crystalline polymers, for instance, is slower than that through amorphous polymers, which have no long-range order and, consequently, have a lower density. 5-8 Composition Profile [Fick’s Second Law] Fick’s second law, which describes the dynamic, or non-steady state, diffusion of atoms, is the differential equation aD b 0c 0 0c = (5-5) 0t 0x 0x If we assume that the diffusion coefficient D is not a function of location x and the concen- tration (c) of diffusing species, we can write a simplified version of Fick’s second law as follows 0 2c = Da 2 b 0c (5-6) 0t 0x The solution to this equation depends on the boundary conditions for a particular situa- tion. One solution is b cs - cx x = erfa (5-7) cs - c0 21Dt where cs is a constant concentration of the diffusing atoms at the surface of the material, c0 is the initial uniform concentration of the diffusing atoms in the material, and cx is the concentration of the diffusing atom at location x below the surface after time t. These concentrations are illustrated in Figure 5-18. In these equations we have assumed basi- cally a one-dimensional model (i.e., we assume that atoms or other diffusing species are moving only in the direction x). The function “erf ” is the error function and can be evaluated from Table 5-3 or Figure 5-19. Note that most standard spreadsheet Figure 5-18 Diffusion of atoms into the surface of a material illustrating the use of Fick’s second law. Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 178 CHAPTER 5 Atom and Ion Movements in Materials TABLE 5-3 Error function values for Fick’s second law Argument of the Value of the x x Error Function 2 1Dt Error Function erf 2 1Dt 0 0 0.10 0.1125 0.20 0.2227 0.30 0.3286 0.40 0.4284 0.50 0.5205 0.60 0.6039 0.70 0.6778 0.80 0.7421 0.90 0.7969 1.00 0.8427 Figure 5-19 Graph showing the 1.50 0.9661 argument and value of the error function 2.00 0.9953 encountered in Fick’s second law. Note that error function values are available on many software packages found on personal computers. and other software programs available on a personal computer (e.g., Excel™) also provide error function values. The mathematical definition of the error function is as follows: x 2 erf (x) = exp (- y2)dy (5-8) 1p L0 In Equation 5-8, y is known as the argument of the error function. We also define a com- plementary error function as follows: erfc(x) ⫽ 1 - erf(x) (5-9) This function is used in certain solution forms of Fick’s second law. As mentioned previously, depending upon the boundary conditions, different solutions (i.e., different equations) describe the solutions to Fick’s second law. These solu- tions to Fick’s second law permit us to calculate the concentration of one diffusing species as a function of time (t) and location (x). Equation 5-7 is a possible solution to Fick’s law that describes the variation in concentration of different species near the surface of the material as a function of time and distance, provided that the diffusion coefficient D remains constant and the concentrations of the diffusing atoms at the surface (cs) and at large distance (x) within the material (c0) remain unchanged. Fick’s second law can also assist us in designing a variety of materials processing techniques, including carburization and dopant diffusion in semiconductors as described in the following examples. Example 5-7 Design of a Carburizing Treatment The surface of a 0.1% C steel gear is to be hardened by carburizing. In gas carbur- izing, the steel gears are placed in an atmosphere that provides 1.2% C at the surface of the steel at a high temperature (Figure 5-1). Carbon then diffuses from the sur- face into the steel. For optimum properties, the steel must contain 0.45% C at a Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.  5 - 8 Composition Profile [Fick’s Second Law] 179 depth of 0.2 cm below the surface. Design a carburizing heat treatment that will produce these optimum properties. Assume that the temperature is high enough (at least 900°C) so that the iron has the FCC structure. SOLUTION Since the boundary conditions for which Equation 5-7 was derived are assumed to be valid, we can use this equation: b cs - cx x = erfa cs - c0 21Dt We know that cs ⫽ 1.2% C, c0 ⫽ 0.1% C, cx ⫽ 0.45% C, and x ⫽ 0.2 cm. From Fick’s second law: = 0.68 = erf a b = erf a b cs - cx 1.2% C - 0.45% C 0.2 cm 0.1 cm = cs - c0 1.2% C - 0.1% C 21Dt 1Dt From Table 5-3, we find that 0.1 2 = 0.71 or Dt = a b = 0.0198 cm2 0.1 cm 1Dt 0.71 Any combination of D and t with a product of 0.0198 cm2 will work. For carbon dif- fusing in FCC iron, the diffusion coefficient is related to temperature by Equation 5-4: b -Q D = D0 expa RT From Table 5-1: ≥ = 0.23 exp a b -32,900 cal/mol - 16, 558 D = 0.23 exp £ cal T 1.987 mol # K T (K) Therefore, the temperature and time of the heat treatment are related by 0.0198 cm2 0.0198 cm2 0.0861 = = t = cm 2 cm2 exp( -16,558/T) D s 0.23 exp( -16,558/T) s Some typical combinations of temperatures and times are If T ⫽ 900°C ⫽ 1173 K, then t ⫽ 116,273 s ⫽ 32.3 h If T ⫽ 1000°C ⫽ 1273 K, then t ⫽ 38,362 s ⫽ 10.7 h If T ⫽ 1100°C ⫽ 1373 K, then t ⫽ 14,876 s ⫽ 4.13 h If T ⫽ 1200°C ⫽ 1473 K, then t ⫽ 6,560 s ⫽ 1.82 h The exact combination of temperature and time will depend on the maximum tem- perature that the heat treating furnace can reach, the rate at which parts must be produced, and the economics of the tradeoffs between higher temperatures versus longer times. Another factor to consider is changes in microstructure that occur in the rest of the material. For example, while carbon is diffusing into the surface, the rest of the microstructure can begin to experience grain growth or other changes. Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 180 CHAPTER 5 Atom and Ion Movements in Materials Example 5-8 shows that one of the consequences of Fick’s second law is that the same concentration profile can be obtained for different processing conditions, so long as the term Dt is constant. This permits us to determine the effect of temperature on the time required for a particular heat treatment to be accomplished. Example 5-8 Design of a More Economical Heat Treatment We find that 10 h are required to successfully carburize a batch of 500 steel gears at 900°C, where the iron has the FCC structure. We find that it costs $1000 per hour to operate the carburizing furnace at 900°C and $1500 per hour to operate the fur- nace at 1000°C. Is it economical to increase the carburizing temperature to 1000°C? What other factors must be considered? SOLUTION We again assume that we can use the solution to Fick’s second law given by Equation 5-7: b cs - cx x = erfa cs - c0 21Dt Note that since we are dealing with only changes in heat treatment time and temper- ature, the term Dt must be constant. The temperatures of interest are 900°C ⫽ 1173 K and 1000°C ⫽ 1273 K. To achieve the same carburizing treatment at 1000°C as at 900°C: D1273t1273 ⫽ D1173t1173 For carbon diffusing in FCC iron, the activation energy is 32,900 cal> mol. Since we are dealing with the ratios of times, it does not matter whether we substitute for the time in hours or seconds. It is, however, always a good idea to use units that balance out. Therefore, we will show time in seconds. Note that temperatures must be con- verted into Kelvin. D1273t1273 = D1173t1173 D = D0 exp( -Q/RT) D1173t1173 t1273 = D1273 cal 32,900 mol D0 exp £- ≥ (10 hours)(3600 sec/hour) 1.987 molcal# K 1173K = cal 32,900 mol D0 exp £- ≥ cal 1.987 mol # K 1273K Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.  5 - 8 Composition Profile [Fick’s Second Law] 181 exp( -14.1156)(10)(3600) t1273 = exp( -13.0068) = (10)(0.3299)(3600) s t1273 = 3.299 h = 3 h and 18 min Notice, we did not need the value of the pre-exponential term D0, since it canceled out. At 900°C, the cost per part is ($1000> h) (10 h)> 500 parts ⫽ $20> part. At 1000°C, the cost per part is ($1500> h) (3.299 h)> 500 parts ⫽ $9.90> part. Considering only the cost of operating the furnace, increasing the temper- ature reduces the heat-treating cost of the gears and increases the production rate. Another factor to consider is if the heat treatment at 1000°C could cause some other microstructural or other changes. For example, would increased temperature cause grains to grow significantly? If this is the case, we will be weakening the bulk of the material. How does the increased temperature affect the life of the other equipment such as the furnace itself and any accessories? How long would the cooling take? Will cooling from a higher temperature cause residual stresses? Would the product still meet all other specifications? These and other questions should be considered. The point is, as engineers, we need to ensure that the solution we propose is not only technically sound and economically sensible, it should recognize and make sense for the system as a whole. A good solution is often simple, solves problems for the sys- tem, and does not create new problems. Example 5-9 Silicon Device Fabrication Devices such as transistors are made by doping semiconductors. The diffusion coefficient of phosphorus (P) in Si is D ⫽ 6.5 * 10-13 cm2> s at a temperature of 1100°C. Assume the source provides a surface concentration of 1020 atoms> cm3 and the diffusion time is one hour. Assume that the silicon wafer initially con- tains no P. Calculate the depth at which the concentration of P will be 1018 atoms> cm3. State any assumptions you have made while solving this problem. SOLUTION We assume that we can use one of the solutions to Fick’s second law (i.e., Equation 5-7): b cs - cx x = erfa cs - c0 21Dt We will use concentrations in atoms> cm3, time in seconds, and D in cms. Notice that