Solved: A diffusion couple similar to that shown in Figure

Briefly explain the difference between selfdiffusion and interdiffusion.

Self-diffusion involves the motion of atoms that are all of the same type; therefore, it is not subject to observation by compositional changes, as with interdiffusion. Suggest one way in which selfdiffusion may be monitored.

(a) Compare interstitial and vacancy atomic mechanisms for diffusion. (b) Cite two reasons why interstitial diffusion is normally more rapid than vacancy diffusion.

Carbon diffuses in iron via an interstitial mechanismfor FCC iron from one octahedral site to an adjacent one. In Section 4.3 (Figure 4.3a), we note that two general sets of point coordinates for this site are 0 1 2 1 and 1 2 1 2 1 2. Specify the family of crystallographic directions in which this diffusion of carbon in FCC iron takes place.

Carbon diffuses in iron via an interstitial mechanismfor BCC iron from one tetrahedral site to an adjacent one. In Section 4.3 (Figure 4.3b) we note that a general set of point coordinates for this site are 1 1 2 1 4. Specify the family of crystallographic directions in which this diffusion of carbon in BCC iron takes place.

Briefly explain the concept of steady state as it applies to diffusion.

(a) Briefly explain the concept of a driving force. (b) What is the driving force for steady-state diffusion?

The purification of hydrogen gas by diffusion through a palladium sheet was discussed in Section 5.3. Compute the number of kilograms of hydrogen that pass per hour through a 6-mm thick sheet of palladium having an area of 0.25 m2 at 600C. Assume a diffusion coefficient of 1.7  108 m2 /s, that the respective concentrations at the high- and low-pressure sides of the plate are 2.0 and 0.4 kg of hydrogen per cubic meter of palladium, and that steady-state conditions have been attained.

A sheet of steel 2.5-mm thick has nitrogen atmospheres on both sides at 900C and is permitted to achieve a steady-state diffusion condition. The diffusion coefficient for nitrogen in steel at this temperature is 1.85  1010 m2 /s, and the diffusion flux is found to be 1.0  107 kg/m2#s. Also, it is known that the concentration of nitrogen in the steel at the high-pressure surface is 2 kg/m3 . How far into the sheet from this high-pressure side will the concentration be 0.5 kg/m3 ? Assume a linear concentration profile.

A sheet of BCC iron 2-mm thick was exposed to a carburizing gas atmosphere on one side and a decarburizing atmosphere on the other side at 675C. After reaching steady state, the iron was quickly cooled to room temperature. The carbon concentrations at the two surfaces of the sheet were determined to be 0.015 and 0.0068 wt%, respectively. Compute the diffusion coefficient if the diffusion flux is 7.36 109 kg/m2#s. Hint: Use Equation 4.9 to convert the concentrations from weight percent to kilograms of carbon per cubic meter of iron.

When a-iron is subjected to an atmosphere of nitrogen gas, the concentration of nitrogen in the iron, CN (in weight percent), is a function of hydrogen pressure, pN2 (in MPa), and absolute temperature (T) according to CN = 4.90 * 10-31pN2 expa - 37,600 J>mol RT b (5.14) Furthermore, the values of D0 and Qd for this diffusion system are 5.0 107 m2 /s and 77,000 J/mol, respectively. Consider a thin iron membrane 1.5-mm thick at 300C. Compute the diffusion flux through this membrane if the nitrogen pressure on one side of the membrane is 0.10 MPa (0.99 atm) and on the other side is 5.0 MPa (49.3 atm).

Show that Cx = B 1Dt expa - x2 4Dt b is also a solution to Equation 5.4b. The parameter B is a constant, being independent of both x and t. Hint: From Equation 5.4b, demonstrate that 0c B 1Dt expa - x2 4Dt b d 0t is equal to D 0 2 c B 1Dt expa - x2 4Dt b d 0x2

Determine the carburizing time necessary to achieve a carbon concentration of 0.30 wt% at a position 4 mm into an ironcarbon alloy that initially contains 0.10 wt% C. The surface concentration is to be maintained at 0.90 wt% C, and the treatment is to be conducted at 1100C. Use the diffusion data for g-Fe in Table 5.2.

An FCC ironcarbon alloy initially containing 0.55 wt% C is exposed to an oxygen-rich and virtually carbon-free atmosphere at 1325 K (1052C). Under these circumstances, the carbon diffuses from the alloy and reacts at the surface with the oxygen in the atmospherethat is, the carbon concentration at the surface position is maintained essentially at 0 wt% C. (This process of carbon depletion is termed decarburization.) At what position will the carbon concentration be 0.25 wt% after a 10-h treatment? The value of D at 1325 K is 3.3 1011 m2 /s.

Nitrogen from a gaseous phase is to be diffused into pure iron at 675C. If the surface concentration is maintained at 0.2 wt% N, what will be the concentration 2 mm from the surface after 25 h? The diffusion coefficient for nitrogen in iron at 675C is 2.8 1011 m2 /s.

Consider a diffusion couple composed of two semi-infinite solids of the same metal and that each side of the diffusion couple has a different concentration of the same elemental impurity; furthermore, assume each impurity level is constant throughout its side of the diffusion couple. For this situation, the solution to Ficks second law (assuming that the diffusion coefficient for the impurity is independent of concentration) is as follows: Cx = C2 + a C1 - C2 2 b c 1 - erfa x 21Dt b d (5.15) The schematic diffusion profile in Figure 5.13 shows these concentration parameters as well as concentration profiles at times t  0 and t  0. Please note that at t  0, the x  0 position is taken as the initial diffusion couple interface, whereas C1 is the impurity concentration for x  0, and C2 is the impurity content for x  0. Consider a diffusion couple composed of pure nickel and a 55 wt% Ni-45 wt% Cu alloy (similar to the couple shown in Figure 5.1). Determine the time this diffusion couple must be heated at 1000C (1273 K) in order to achieve a composiFigure 5.13 Schematic concentration profiles in the vicinity of the interface (located at x  0) between two semi-infinite metal alloys before (i.e., t  0) and after a heat treatment (i.e., t  0). The base metal for each alloy is the same; concentrations of some elemental impurity are differentC1 and C2 denote these concentration values at t  0. Position Concentration C1 t > 0 x < 0 x > 0 t = 0 x = 0 C2 C1 C2 2 164 Chapter 5 / Diffusion tion of 56.5 wt% Ni a distance of 15 m into the Ni-Cu alloy referenced to the original interface. Values for the preexponential and activation energy for this diffusion system are 2.3  104 m2 /s and 252,000 J/mol.

Consider a diffusion couple composed of two cobalt-iron alloys; one has a composition of 75 wt% Co-25 wt% Fe; the other alloy composition is 50 wt% Co-50 wt% Fe. If this couple is heated to a temperature of 800C (1073 K) for 20,000 s, determine how far from the original interface into the 50 wt% Co-50 wt% Fe alloy the composition has increased to 52 wt% Co-48 wt% Fe. For the diffusion coefficient, assume values of 6.6  106 m2 /s and 247,000 J/mol, respectively, for the preexponential and activation energy.

Consider a diffusion couple between silver and a gold alloy that contains 10 wt% silver. This couple is heat treated at an elevated temperature and it was found that after 850 s, the concentration of silver had increased to 12 wt% at 10 m from the interface into the AgAu alloy. Assuming preexponential and activation energy values of 7.2  106 m2 /s and 168,000 J/mol, respectively, compute the temperature of this heat treatment. (Note: You may find Figure 5.13 and Equation 5.15 helpful.)

For a steel alloy, it has been determined that a carburizing heat treatment of 15 h duration will raise the carbon concentration to 0.35 wt% at a point 2.0 mm from the surface. Estimate the time necessary to achieve the same concentration at a 6.0-mm position for an identical steel and at the same carburizing temperature.

Cite the values of the diffusion coefficients for the interdiffusion of carbon in both a-iron (BCC) and g-iron (FCC) at 900C. Which is larger? Explain why this is the case.

Using the data in Table 5.2, compute the value of D for the diffusion of magnesium in aluminum at 400C.

Using the data in Table 5.2, compute the value of D for the diffusion of nitrogen in FCC iron at 950C.

At what temperature will the diffusion coefficient for the diffusion of zinc in copper have a value of 2.6  1016 m2 /s? Use the diffusion data in Table 5.2.

At what temperature will the diffusion coefficient for the diffusion of nickel in copper have a value of 4.0  1017 m2 /s? Use the diffusion data in Table 5.2.

The preexponential and activation energy for the diffusion of chromium in nickel are 1.1  104 m2 /s and 272,000 J/mol, respectively. At what temperature will the diffusion coefficient have a value of 1.2  1014 m2 /s?

The activation energy for the diffusion of copper in silver is 193,000 J/mol. Calculate the diffusion coefficient at 1200 K (927C), given that D at 1000 K (727C) is 1.0  1014 m2 /s.

The diffusion coefficients for nickel in iron are given at two temperatures, as follows: T(K) D(m2 /s) 1473 2.2  1015 1673 4.8  1014 (a) Determine the values of D0 and the activation energy Qd. (b) What is the magnitude of D at 1300C (1573 K)?

he diffusion coefficients for carbon in nickel are given at two temperatures, as follows: T(C) D(m2 /s) 600 5.5  1014 700 3.9  1013 (a) Determine the values of D0 and Qd. (b) What is the magnitude of D at 850C?

The following figure shows a plot of the logarithm (to the base 10) of the diffusion coefficient versus reciprocal of the absolute temperature for the diffusion of gold in silver. Determine values for the activation energy and preexponential.

The following figure shows a plot of the logarithm (to the base 10) of the diffusion coefficient versus reciprocal of the absolute temperature for the diffusion of vanadium in molybdenum. Determine values for the activation energy and preexponential.

From Figure 5.12, calculate the activation energy for the diffusion of (a) copper in silicon, and (b) aluminum in silicon (c) How do these values compare?

Carbon is allowed to diffuse through a steel plate 10-mm thick. The concentrations of carbon at the two faces are 0.85 and 0.40 kg C/cm3 Fe, which are maintained constant. If the preexponential and activation energy are 5.0  107 m2 /s and 77,000 J/mol, respectively, compute the temperature at which the diffusion flux is 6.3  1010 kg/m2#s.

The steady-state diffusion flux through a metal plate is 7.8  108 kg/m2#s at a temperature of 1200C (1473 K) and when the concentration gradient is 500 kg/m4 . Calculate the diffusion flux at 1000C (1273 K) for the same concentration gradient and assuming an activation energy for diffusion of 145,000 J/mol.

At approximately what temperature would a specimen of g-iron have to be carburized for 4 h to produce the same diffusion result as carburization at 1000C for 12 h?

(a) Calculate the diffusion coefficient for magnesium in aluminum at 450C. (b) What time will be required at 550C to produce the same diffusion result (in terms of concentration at a specific point) as for 15 h at 450C? 5

A coppernickel diffusion couple similar to that shown in Figure 5.1a is fashioned. After a 500-h heat treatment at 1000C (1273 K), the concentration of Ni is 3.0 wt% at the 1.0-mm position within the copper. At what temperature should the diffusion couple be heated to produce this same concentration (i.e., 3.0 wt% Ni) at a 2.0-mm position after 500 h? The preexponential and activation energy for the diffusion of Ni in Cu are 1.9  104 m2 /s and 230,000 J/mol, respectively.

A diffusion couple similar to that shown in Figure 5.1a is prepared using two hypothetical metals A and B. After a 20-h heat treatment at 800C (and subsequently cooling to room temperature) the concentration of B in A is 2.5 wt% at the 5.0-mm position within metal A. If another heat treatment is conducted on an identical diffusion couple, but at 1000C for 20 h, at what position will the composition be 2.5 wt% B? Assume that the preexponential and activation energy for the diffusion coefficient are 1.5  104 m2 /s and 125,000 J/mol, respectively.

Consider the diffusion of some hypothetical metal Y into another hypothetical metal Z at 950C; after 10 h the concentration at the 0.5 mm position (in metal Z) is 2.0 wt% Y. At what position will the concentration also be 2.0 wt% Y after a 17.5-h heat treatment again at 950C? Assume preexponential and activation energy values of 4.3  104 m2 /s and 180,000 J/mol, respectively, for this diffusion system.

A diffusion couple similar to that shown in Figure 5.1a is prepared using two hypothetical metals R and S. After a 2.5-h heat treatment at 750C, the concentration of R is 4 at% at the 4-mm position within S. Another heat treatment is conducted on an identical diffusion couple at 900C, and the time required to produce this same diffusion result (viz., 4 at% R at the 4-mm position within S) is 0.4 h. If it is known that the diffusion coefficient at 750C is 2.6  1017 m2 /s, determine the activation energy for the diffusion of R in S. 5

The outer surface of a steel gear is to be hardened by increasing its carbon content; the carbon is to be supplied from an external carbon-rich atmosphere maintained at an elevated temperature. A diffusion heat treatment at 600C (873 K) for 100 min increases the carbon concentration to 0.75 wt% at a position 0.5 mm below the surface. Estimate the diffusion time required at 900C (1173 K) to achieve this same concentration also at a 0.5-mm position. Assume that the surface carbon content is the same for both heat treatments, which is maintained constant. Use the diffusion data in Table 5.2 for C diffusion in a-Fe.

An FCC ironcarbon alloy initially containing 0.10 wt% C is carburized at an elevated temperature and in an atmosphere in which the surface carbon concentration is maintained at 1.10 wt%. If, after 48 h, the concentration of carbon is 0.30 wt% at a position 3.5 mm below the surface, determine the temperature at which the treatment was carried out.

For the predeposition heat treatment of a semiconducting device, gallium atoms are to be diffused into silicon at a temperature of 1150C for 2.5 h. If the required concentration of Ga at a position 2 m below the surface is 8  1023 atoms/m3 , compute the required surface concentration of Ga. Assume the following: (i) The surface concentration remains constant (ii) The background concentration is 2  1019 Ga atoms/m3 (iii) Preexponential and activation energy values are 3.74  105 m2 /s and 3.39 eV/atom, respectively.

Antimony atoms are to be diffused into a silicon wafer using both predeposition and drive-in heat treatments; the background concentration of Sb in this silicon material is known to be 2  1020 atoms/m3 . The predeposition treatment is to be conducted at 900C for 1 h; the surface concentration of Sb is to be maintained at a constant level of 8.0  1025 atoms/m3 . Drive-in diffusion will be carried out at 1200C for a period of 1.75 h. For the diffusion of Sb in Si, values of Qd and D0 are 3.65 eV/atom and 2.14  105 m2 /s, respectively. (a) Calculate the value of Q0. (b) Determine the value of xj for the drive-in diffusion treatment. (c) Also, for the drive-in treatment, compute the position x at which the concentration of Sb atoms is 5  1023 atoms/m3 .

Indium atoms are to be diffused into a silicon wafer using both predeposition and drive-in heat treatments; the background concentration of In in this silicon material is known to be 2  1020 atoms/m3 . The drive-in diffusion treatment is to be carried out at 1175C for a period of 2.0 h, which gives a junction depth xj of 2.35 m. Compute the predeposition diffusion time at 925C if the surface concentration is maintained at a constant level of 2.5  1026 atoms/m3 . For the diffusion of In in Si, values of Qd and D0 are 3.63 eV/atom and 7.85  105 m2 /s, respectively.

For a nonsteady-state diffusion situation (constant surface composition) in which the surface and initial compositions are provided, as well as the value of the diffusion coefficient, develop a spreadsheet that allows the user to determine the diffusion time required to achieve a given composition at some specified distance from the surface of the solid.

For a nonsteady-state diffusion situation (constant surface composition) in which the surface and initial compositions are provided, as well as the value of the diffusion coefficient, develop a spreadsheet that allows the user to determine the distance from the surface at which some specified composition is achieved for some specified diffusion time.

For a nonsteady-state diffusion situation (constant surface composition) in which the surface and initial compositions are provided, as well as the value of the diffusion coefficient, develop a spreadsheet that allows the user to determine the composition at some specified distance from the surface for some specified diffusion time.

Given a set of at least two diffusion coefficient values and their corresponding temperatures, develop a spreadsheet that will allow the user to calculate the following: (a) the activation energy and (b) the preexponential.

It is desired to enrich the partial pressure of hydrogen in a hydrogennitrogen gas mixture for which the partial pressures of both gases are 0.1013 MPa (1 atm). It has been proposed to accomplish this by passing both gases through a thin sheet of some metal at an elevated temperature; inasmuch as hydrogen diffuses through the plate at a higher rate than does nitrogen, the partial pressure of hydrogen will be higher on the exit side of the sheet. The design calls for partial pressures of 0.051 MPa (0.5 atm) and 0.01013 MPa (0.1 atm), respectively, for hydrogen and nitrogen. The concentrations of hydrogen and nitrogen (CH and CN, in mol/m3 ) in this metal are functions of gas partial pressures (pH2 and pN2 , in MPa) and absolute temperature and are given by the following expressions: CH = 2.5 * 1031pH2 expa - 27,800 J>mol RT b (5.16a) CN = 2.75 * 1031pN2 expa - 37,600 J>mol RT b (5.16b) Furthermore, the diffusion coefficients for the diffusion of these gases in this metal are functions of the absolute temperature, as follows: DH(m2>s) = 1.4 * 10-7 expa - 13,400 J>mol RT b (5.17a) DN(m2>s) = 3.0 * 10-7 expa - 76,150 J>mol RT b (5.17b) Is it possible to purify hydrogen gas in this manner? If so, specify a temperature at which the process may be carried out, and also the thickness of metal sheet that would be required. If this procedure is not possible, then state the reason(s) why

A gas mixture is found to contain two diatomic A and B species (A2 and B2), the partial pressures of both of which are 0.1013 MPa (1 atm). This mixture is to be enriched in the partial pressure of the A species by passing both gases through a thin sheet of some metal at an elevated temperature. The resulting enriched mixture is to have a partial pressure of 0.051 MPa (0.5 atm) for gas A and 0.0203 MPa (0.2 atm) for gas B. The concentrations of A and B (CA and CB, in mol/m3 ) are functions of gas partial pressures (pA2 and pB2 , in MPa) and absolute temperature according to the following expressions: CA = 1.5 * 1031pA2 expa - 20,000 J>mol RT b (5.18a) CB = 2.0 * 1031pB2 expa - 27,000 J>mol RT b (5.18b) Furthermore, the diffusion coefficients for the diffusion of these gases in the metal are functions of the absolute temperature, as follows: DA(m2>s) = 5.0 * 10-7 expa - 13,000 J>mol RT b (5.19a) DB(m2>s) = 3.0 * 10-6 expa - 21,000 J>mol RT b (5.19b) Is it possible to purify gas A in this manner? If so, specify a temperature at which the process may be carried out, and also the thickness of metal sheet that would be required. If this procedure is not possible, then state the reason(s) why.

The wear resistance of a steel shaft is to be improved by hardening its surface by increasing the nitrogen content within an outer surface layer as a result of nitrogen diffusion into the steel; the nitrogen is to be supplied from an external nitrogen-rich gas at an elevated and constant temperature. The initial nitrogen content of the steel is 0.0025 wt%, whereas the surface concentration is to be maintained at 0.45 wt%. For this treatment to be effective, a nitrogen content of 0.12 wt% must be established at a position 0.45 mm below the surface. Specify an appropriate heat treatment in terms of temperature and time for a temperature between 475C and 625C. The preexponential and activation energy for the diffusion of nitrogen in iron are 5 107 m2 /s and 77,000 J/mol, respectively, over this temperature range. 5

The wear resistance of a steel gear is to be improved by hardening its surface, as described in Design Example 5.1. However, in this case, the initial carbon content of the steel is 0.15 wt%, and a carbon content of 0.75 wt% is to be established at a position 0.65 mm below the surface. Furthermore, the surface concentration is to be maintained constant, but may be varied between 1.2 and 1.4 wt% C. Specify an appropriate heat treatment in terms of surface carbon concentration and time, and for a temperature between 1000C and 1200C.

One integrated circuit design calls for the diffusion of aluminum into silicon wafers; the background concentration of Al in Si is 1.75 1019 atoms/m3 . The predeposition heat treatment is to be conducted at 975C for 1.25 h, with a constant surface concentration of 4 1026 Al atoms/m3 . At a drive-in treatment temperature of 1050C, determine the diffusion time required for a junction depth of 1.75 m. For this system, values of Qd and D0 are 3.41 eV/atom and 1.38 104 m2 /s, respectively. FUN

Atoms of which of the following elements will diffuse most rapidly in iron? (A) Mo (B) C (C) Cr (D) W

Calculate the diffusion coefficient for copper in aluminum at 600C. Preexponential and activation energy values for this system are 6.5 105 m2 /s and 136,000 J/mol, respectively. (A) 5.7 102 m2 /s (C) 4.7 1013 m2 /s (B) 9.4 1017 m2 /s (D) 3.9 102 m2 /s