8.1 The ratio of diffusion rate of silver in silicon at 1350°C to that at 1100°C was found to be 8 in a doping process. Calculate the activation energy Q for silver diffusion in silicon.

 

8.2 The diffusion rate of A in B was studied at 500°C and 850°C. It was found that, for the same diffusion time, the depths of penetration x1 and x2 in the two experiments were in the ratio of 1 : 4. Calculate the activation energy for diffusion of A in B.

 

8.3 Amorphous selenium used as a semiconductor material exhibits unusual diffusion characteristics. The following is a set of experimental data for self-diffusion in amorphous selenium. Calculate D0 and Q and comment on your results. T, °C D, m2 s–1 35 7.7 10–16 40 2.4 10–15 46 3.2 10–14 56 3.2 10–13

 

8.4 At 900°C, what is the time required to carburize a steel with an initial composition of 0.2% carbon to 1% carbon at a depth of 0.2 mm? Assume a constant surface concentration of 1.4% carbon due to the carburizing atmosphere.

 

8.5 A 1.2% carbon steel is getting decarburized in an atmosphere of 0.0% carbon. After some time t, plot (i) c(x) curve near the surface of the steel, and (ii) J(x) curve below the above curve, using the same x-axis.

 

8.6 A diffusion couple of 95% Cu-5% Zn and pure copper is annealed at 900°C for 50 hr. The zinc concentration at a depth of 2 mm inside the copper bar was found to be 0.3% after the anneal. Determine the diffusion coefficient of zinc in copper.

 

8.7 In a steel, during carburization at 937°C, 0.6% carbon is found at a depth of 0.2 mm after 1 hr. Find the time required to achieve the same concentration at the same depth, if carburization is done at 1047°C.

 

8.8 Compare the diffusivities of hydrogen, nitrogen, and nickel in iron at 300 K and explain the difference between the three values.

 

8.9 Compute the rate at which a vacancy jumps in copper at 20°C. The activation barrier for the jump is 100 kJ mol–l.

 

8.10 An Al-4% Cu alloy is heated to 550°C during heat treatment and quenched to room temperature. Immediately after quench, the diffusion rate of Problems 197 copper (which proceeds by a vacancy mechanism) was found to be 107 times faster than what would be expected from the listed diffusion data. What fraction of vacancies in equilibrium at 550°C is retained at room temperature by the rapid quenching? The enthalpy of motion of vacancy in this alloy is 50 kJ mol–1.

 

8.11 How will the diffusivity of NaCl change, when it is doped with (i) KCl, and (ii) CaCl2? Explain.

 

8.12 Determine the ratio of cross-sectional area available for diffusion along the surface and through the lattice, when the two diffusion rates are equal at room temperature. Assume D0 to be the same for both the processes. Qsurface = 100 kJ mol–l and Qlattice = 150 kJ mol–l.

 

8.13 Find the grain size of a polycrystalline solid for the same amount of material to be transported through (i) the grain and (ii) the grain boundary at 500°C. Assume that the grains are cube shaped and the grain boundaries are 5 Å thick. For lattice diffusion: D0 = 0.7 10–4 m2 s–1 Q = 188 kJ mol–l For grain boundary diffusion: D0 = 0.09 10–4 m2 s–1 Q = 90 kJ mol–l

 

8.15 From the data in Table 8.2, calculate the diffusion coefficient of carbon in ferrite ( ) and austenite ( ) at 900°C. Explain the difference in the values on the basis of the two crystal structures.

 

8.14 Make a plot of the activation energy Q for diffusion of different species as a function of the melting point of the species. Comment on your result.

 

8.16 In a diffusion anneal, to what level should the initial temperature of 900°C be increased to double the depth of penetration? D0 = 0.4 m2 s–1 and Q = 100 kJ mol–l.

 

8.17 An amount Q of a dopant is deposited on the surface of a silicon substrate. During a subsequent anneal without the dopant in the atmosphere, the concentration c of the dopant as a function of depth x and time t is given by c = (Q/ 􀀀DT ) exp [–x2/(4Dt)] Show that this is a solution of Fick’s Second Law, when D is independent of concentration.

 

8.18 A steel containing 0.002% N is to be nitrided to yield a nitrogen content of 0.12% at depth of 4 mm from the surface. The nitriding atmosphere is equivalent to a surface concentration of 0.35% N. How long is to be the nitriding process? The steel is BCC ( ) at the nitriding temperature of 700°C.

 

From the Fe–Fe3C phase diagram, for a 0.2% C steel, name the phases and their fractions at equilibrium at the following temperatures: (i) just above 1493°C, (ii) just below 1493°C, (iii) just above 725°C, and (iv) just below 725°C.

 

What is the fraction of proeutectoid cementite in (i) 1.4% C, (ii) 1.0% C, and (iii) 0.7% C steels?

 

One solid phase on heating through an invariant temperature becomes two solid phases. Name the invariant reaction. Sketch the phase boundaries near the invariant line.

 

In the Pb–Sn system, calculate the alloy composition at which the fraction of total 􀀀 is 2½ times the fraction of the phase at 182°C.

 

For soldering with Pb–Sn alloys, at least 85% of the eutectic mixture is preferred in the microstructure. Determine the composition limits of tin that will satisfy this condition.

 

Calculate the fraction of proeutectoid ferrite, eutectoid ferrite and total ferrite in a 0.2% C steel.

 

The potassium–sodium binary phase diagram has the following invariant reactions: cooling at 6.9°C, + L Na2K wt.% Na 99 47 54 cooling at –12.6°C, L 􀀀 + Na2K wt.% Na 23 3 54 (i) Give the name of each of the above reactions. (ii) Make an approximate sketch of the phase diagram. (iii) Find the fraction of Na2K in a 33 wt.% Na alloy at –12.5°C.

 

In the Ti–Ti2Co system, liquid of 27 wt.% Co, of 17 wt.% Co and Ti2Co of 38 wt.% Co are in equilibrium at 1020°C. At 685°C, of 8 wt.% Co, 􀀀 of 1 wt.% Co and Ti2Co are in equilibrium. Pure Ti melts at 1670°C and also undergoes a crystal structure change from (BCC) to 􀀀 (HCP) on cooling through 882°C. Ti2Co is stable up to 1058°C.

 

The data on the gold–lead phase diagram are given below. Melting point of Au = 1063°C, Melting point of lead = 327°C. At 434°C, Au, liquid of 43 at.% Pb and Au2Pb are in equilibrium. At 253°C, Au2Pb, liquid of 74 at.% Pb and AuPb2 are in equilibrium. At 222°C, AuPb2, AuPb3 and liquid of 82 at.% Pb are in equilibrium. At 212°C, AuPb3, Pb and liquid of 84 at.% Pb are in equilibrium. Draw the phase diagram on a graph paper using a suitable scale. Write down the invariant reactions, giving the name of each.

 

In the Fe–Fe2Nb system, at 1373°C, of 3% Nb, liquid of 12% Nb and of 27% Nb are in equilibrium. At 1210°C, of 1% Nb, of 1.5% Nb and of 27% Nb are in equilibrium. At 961°C, of 0.4% Nb, 􀀀 of 0.7% Nb and of 27% Nb are in equilibrium. All compositions are in atomic%. The melting point of Fe is 1535°C and of Fe2Nb (33.3 at.% Nb) is 1627°C. The phase is stable over a composition range of 27 to 33.3 at.% Nb, (i) Draw an approximate phase diagram for this system. (ii) Write down the invariant reactions in the system, giving their names. (iii) What is the fraction of 􀀀 in an alloy with 10% Nb at 960°C?

 

The Al–Si phase diagram is of the simple eutectic type. At 577°C, solid Al with 1.5% Si dissolved in it, solid Si and liquid of 12.5% Si are in equilibrium. An aluminium wire is welded to a silicon substrate. Just below the weld inside the substrate, the microstructure shows 2% of a eutectic-like mixture. What is the composition at this location?