1.The diffusion coefficient,D,in a solid is related to the rate at which an atom jumps from one site to another,w. (a)In a pure metal how are D and w related? (b)In a substitutional binary alloy the solute and vacancies are often attracted ("bound")to each other so that the solute atom's jump frequency,w2,is much greater than that of the base metal,w1.Under these conditions how is D2 related to w2 and wi? (c)For a very dilute interstitial alloy how is the interstitial jump frequency,wi, related to D? 2.An Fe-0.8%C alloy is decarburized in an atmosphere that keeps the surface essentially carbon free. (a)If the decarburization is carried out at 730 C a carbon-free layer of ferrite(alpha) forms on the surface of the austenitic (gamma)alloy.Set up an equation relating the movement of the alpha/gamma interface to the concentration difference across the ferrite layer and D in the ferrite. (b)Assume the solubility of carbon in equilibrium with austenite at 730 C is 0.02 w/o carbon and that D=10 cm /s for carbon in the ferrite.How long will it take to form a ferrite layer 0.01 cm thick? (c)If the decarburization is carried out at 800 C the c(x)curve crosses the alpha, alpha+aust.and aust.fields.Plot the c(x)curve across the diffusion zone.On the plot label the composition limits of the two-phase fields. 3.Consider a diffusion couple of Cu against an alpha solid solution of Cu-15a/o Al with fine inert markers initially placed in the interface between the two pieces of metal. (a)How would you measure the Kirkendall shift? (b)What is Kirkendall porosity,and where would one expect to find it? (c)How can this shift be related to the intrinsic diffusion coefficients Dint of Al and Cu in the alloy?(Are these values of Dint for all compositions in the couple,or for some particular composition?) (d)If the annihilation of vacancies was suddenly rendered impossible so that the vacancy chemical potential could locally rise,or fall,what would this do to the relative fluxes of Al and Cu? 4.Sulfur will segregate to grain boundaries in steel during annealing in the temperature range of 450-550C,causing grain boundary embrittlement. (a)If the sulfur atoms jump with a frequency of 10*/s at 500C,give an estimate of the diffusion coefficient D,in cm/s in the lattice. (b)If the atom fraction ofS in solution in the steel at 500C is 10,estimate how long it will take for enough diffusion to occur to form a monolayer of S on the grain boundary.(A monolayer consists of 1 sulfur atom per iron atom on die boundary
1. The diffusion coefficient, D, in a solid is related to the rate at which an atom jumps from one site to another, w. (a) In a pure metal how are D and w related? (b) In a substitutional binary alloy the solute and vacancies are often attracted (“bound”) to each other so that the solute atom’s jump frequency, w2, is much greater than that of the base metal, w1. Under these conditions how is D2 related to w2 and w1? (c) For a very dilute interstitial alloy how is the interstitial jump frequency, w1, related to D1? 2. An Fe-0.8% C alloy is decarburized in an atmosphere that keeps the surface essentially carbon free. (a) If the decarburization is carried out at 730 C a carbon-free layer of ferrite (alpha) forms on the surface of the austenitic (gamma) alloy. Set up an equation relating the movement of the alpha/gamma interface to the concentration difference across the ferrite layer and D in the ferrite. (b) Assume the solubility of carbon in equilibrium with austenite at 730 C is 0.02 w/o carbon and that D = 10-6 cm 2 /s for carbon in the ferrite. How long will it take to form a ferrite layer 0.01 cm thick? (c) If the decarburization is carried out at 800 C the c(x) curve crosses the alpha, alpha+aust. and aust. fields. Plot the c(x) curve across the diffusion zone. On the plot label the composition limits of the two-phase fields. 3. Consider a diffusion couple of Cu against an alpha solid solution of Cu-15a/o Al with fine inert markers initially placed in the interface between the two pieces of metal. (a) How would you measure the Kirkendall shift? (b) What is Kirkendall porosity, and where would one expect to find it? (c) How can this shift be related to the intrinsic diffusion coefficients Dint of Al and Cu in the alloy? (Are these values of Dint for all compositions in the couple, or for some particular composition?) (d) If the annihilation of vacancies was suddenly rendered impossible so that the vacancy chemical potential could locally rise, or fall, what would this do to the relative fluxes of Al and Cu? 4. Sulfur will segregate to grain boundaries in steel during annealing in the temperature range of 450 - 550°C, causing grain boundary embrittlement. (a) If the sulfur atoms jump with a frequency of 10*/s at 500°C, give an estimate of the diffusion coefficient D, in cm2 /s in the lattice. (b) If the atom fraction of S in solution in the steel at 500°C is 10-5, estimate how long it will take for enough diffusion to occur to form a monolayer of S on the grain boundary. (A monolayer consists of 1 sulfur atom per iron atom on die boundary.)
5.The existence of dislocations in a crystal invariably affects the diffusion of a solute.In each of the following cases,tell whether adding dislocations will enhance, or retard,the effective diffusivity for the solute.Also,explain how this occurs. (a)Diffusion of Cd*through a single crystal of Ag at a temperature of half the melting point. (b)Diffusion of Cu in Ge,or Au in Si. (c)Permeation of H through a sheet of Fe at room temperature. 6.(a)A carbon atom in iron jumps from one site to another with a frequency of once per second at 25 C.Estimate the most probable distance a carbon atom will be found from its initial position after 24 hours. (b)The activation energy for the motion of carbon in bcc iron is 20.45 kcal/mol (170 kJ/mol).What is the value of D at 750 C? (c)Given an Fe-0.01%C alloy exposed to a decarburizing gas on both sides, how thick a sheet could be decarburized in one day at 750 C? 7.Consider a one-dimensional heat conduction situation with thermal conductivity, k=1 and source term,s=2(in appropriate units)everywhere.If four grid points at x =0,1,2,3 are used to span the domain of length 3 units,write the four discretization equations (including the half-control-volume equations).Solve the equations by the TDMA (Tri-Diagonal Matrix Algorithm)using the following boundary conditions:Temperature at x=0 is 100 and at x =3 is 25,in arbitrary units. 8 If the tracer diffusivity of Fe in magnetite (Fe304)is measured as a function of oxygen partial pressure between 900 and 1400 C plots of log D vs.log (oxygen activity) consist of two segments,one with a slope of about 2/3 and the other with a slope of-2/3 (see fig.for an example). Tell the mechanism that operates in each leg of the curve,and explain why the sign of the slope is positive Log(activ.oxygen) or negative for that mechanism. 9.(a)Derive an expression for diffusion of SiH4 gas through the boundary layer to the substrate in a CVD process as shown below.The process is diffusion controlled,i.e.,the following reaction taking place at the substrate is very fast. SiH4(g)=Si(s)+2H2(g)
5. The existence of dislocations in a crystal invariably affects the diffusion of a solute. In each of the following cases, tell whether adding dislocations will enhance, or retard, the effective diffusivity for the solute. Also, explain how this occurs. (a) Diffusion of Cd* through a single crystal of Ag at a temperature of half the melting point. (b) Diffusion of Cu in Ge, or Au in Si. (c) Permeation of H through a sheet of Fe at room temperature. 6. (a) A carbon atom in iron jumps from one site to another with a frequency of once per second at 25 C. Estimate the most probable distance a carbon atom will be found from its initial position after 24 hours. (b) The activation energy for the motion of carbon in bcc iron is 20.45 kcal/mol (170 kJ/mol). What is the value of D at 750 C? (c) Given an Fe-0.01% C alloy exposed to a decarburizing gas on both sides, how thick a sheet could be decarburized in one day at 750 C? 7. Consider a one-dimensional heat conduction situation with thermal conductivity, k = 1 and source term, s = 2 (in appropriate units) everywhere. If four grid points at x = 0, 1, 2, 3 are used to span the domain of length 3 units, write the four discretization equations (including the half-control-volume equations). Solve the equations by the TDMA (Tri-Diagonal Matrix Algorithm) using the following boundary conditions: Temperature at x = 0 is 100 and at x = 3 is 25, in arbitrary units. 8. If the tracer diffusivity of Fe in magnetite (Fe304) is measured as a function of oxygen partial pressure between 900 and 1400 C plots of log D vs. log (oxygen activity) consist of two segments, one with a slope of about 2/3 and the other with a slope of -2/3 (see fig. for an example). Tell the mechanism that operates in each leg of the curve, and explain why the sign of the slope is positive or negative for that mechanism. 9. (a) Derive an expression for diffusion of SiH4 gas through the boundary layer to the substrate in a CVD process as shown below. The process is diffusion controlled, i.e., the following reaction taking place at the substrate is very fast. SiH4(g) = Si(s) + 2H2(g)
y SUBSTRATE (b)Using above expression,calculate the rate of Si deposition on the substrate when I mole percent SiH4 and hydrogen gas mixture flows through the reactor. The substrate is maintained at 1000C.State any assumptions you make. Given:Diffusivity,DsiH4-H2(1000c)=2.5cm/sec Boundary layer thickness =0.02 cm Molecular wt.of SiH4 =32 10.(1)Consider a compound MX whose composition of existence includes the stoichiometric composition.Discuss thermodynamic and transport parameters which decide whether the compound is an ionic conductor,an electronic conductor.or a mixed conductor. (2)Tracer diffusion measurements for Cr2C3 at high P(O2)and high temperatures show that the dominant electrical neutrality condition is 3[Vcr]=[h].At low PO2,dissolved TiO2 dopant is compensated by excess electrons,[Ticr]=[e']. Consider a Cr2O3 specimen saturated with TiO2 and containing a small amount of excess TiO2: (a)How will the solubility of TiO2 depend upon P(O2)at low P (b)How will the electrical conductivity for TiO2-saturated Cr2O3 depend upon P(O2)2 11.Consider a localized adsorbate of methane on a substrate.In terms of absolute rate theory,present a detailed expression for the rate of desorption into the vapor. List and discuss all assumptions.Are there alternative descriptions of the activated state for this reaction? 12.Sulfur will segregate to grain boundaries in steel during annealing in the temperature range of 450-550C,causing grain boundary embrittlement. (a)If the sulfur atoms jump with a frequency of 108/s at 500C,give an estimate of the diffusion coefficient D,in cm /s in the lattice. (b)If the atom fraction of S in solution in the steel at 500C is 10 estimate how long it will take for enough diffusion to occur to form a monolayer of S on the grain boundary.(A monolayer consists of 1 sulfur atom per iron atom on the boundary.) 13.It has been postulated that in a certain temperature range the rate of the oxidation of nickel is limited by the rate of diffusion(mass transport)along grain boundaries,and not by diffusion through the lattice of the grains themselves.From your knowledge of lattice and grain boundary diffusion describe two types of
(b) Using above expression, calculate the rate of Si deposition on the substrate when 1 mole percent SiH4 and hydrogen gas mixture flows through the reactor. The substrate is maintained at 1000°C. State any assumptions you make. Given: Diffusivity, DSiH4-H2(1000C)=2.5cm2 /sec Boundary layer thickness = 0.02 cm Molecular wt. of SiH4 =32 10. (1) Consider a compound MX whose composition of existence includes the stoichiometric composition. Discuss thermodynamic and transport parameters which decide whether the compound is an ionic conductor, an electronic conductor, or a mixed conductor. (2) Tracer diffusion measurements for Cr2C3 at high P(O2) and high temperatures show that the dominant electrical neutrality condition is 3[VCr"'] = [h0 ]. At low PO2, dissolved TiO2 dopant is compensated by excess electrons, [Ti0 Cr] = [e']. Consider a Cr2O3 specimen saturated with TiO2 and containing a small amount of excess TiO2: (a) How will the solubility of TiO2 depend upon P(O2) at low P^^ (b) How will the electrical conductivity for TiO2-saturated Cr2O3 depend upon P(O2)? 11. Consider a localized adsorbate of methane on a substrate. In terms of absolute rate theory, present a detailed expression for the rate of desorption into the vapor. List and discuss all assumptions. Are there alternative descriptions of the activated state for this reaction? 12. Sulfur will segregate to grain boundaries in steel during annealing in the temperature range of 450 - 550°C, causing grain boundary embrittlement. (a) If the sulfur atoms jump with a frequency of 108/s at 500°C, give an estimate of the diffusion coefficient D, in cm2 /s in the lattice. (b) If the atom fraction of S in solution in the steel at 500°C is 10-5, estimate how long it will take for enough diffusion to occur to form a monolayer of S on the grain boundary. (A monolayer consists of 1 sulfur atom per iron atom on the boundary.) 13. It has been postulated that in a certain temperature range the rate of the oxidation of nickel is limited by the rate of diffusion (mass transport) along grain boundaries, and not by diffusion through the lattice of the grains themselves. From your knowledge of lattice and grain boundary diffusion describe two types of
experiments that would help you answer this postulate and explain how the results would distinguish between the two mechanisms. 14.A nickel membrane separates two gas phase at 1000C :a CO/CO2 gas and a H2H20,each with the same oxygen activity (reducing to NiO formation). Analyze as well as possible the steady-state transport of carbon through the membrane assuming: (a)the rate is limited by the diffusion of carbon in nickel, (b)the rate is limited by interfacial reactions at the surfaces, (c)the rate control involves both diffusion in the metal and interfacial reactaions The diffusion of hydrogen through the metal is very fast.Propose a specific rate-limiting step for hydrogen transport. What experiments would you propose to measure the carbon and hydrogen permeation rates? 15.A continuous 10m thick layer of gold his electroplated on a nickel electrical contact which is to be operated in a poor vacuum at 50-150C.Calculations using high temperature volume diffusion data indicate that the nickel will not penetrate the gold layer for at least a century.The "customer"has come back saying that some surface oxide is messing up the electrical properties of the contact after a few months. (a)Give some possible explanations of what might be happening. (b)Tell how you can setup a program to prove what's causing the degradation. (c)Suggest a fix for the product manager. 16.In the oxidation of high temperature aluminum containing alloys,it is felt that the rate controlling transport process through the protective alumina film may be grain boundary transport not lattice diffusion.Discuss how you would go about establishing when and if this is true" a)with diffusion data and equations you might find in the literature. b)any experiment you wish to suggest 17 The concentration of carbon in the case of a carburized steel is given by the expression C-C=1-erf( C。-C 2√D1 Where CO=surface concentration(gms/cm')of carbon and Ci=carbon content of steel. a)Derive an expression for the flux of carbon into the steel at surface(x=0) b)Derive an expression for the amount of carbon which has diffuse into the steel per unit of surface area after time t
experiments that would help you answer this postulate and explain how the results would distinguish between the two mechanisms. 14. A nickel membrane separates two gas phase at 1000℃ :a CO/CO2 gas and a H2H2O, each with the same oxygen activity (reducing to NiO formation). Analyze as well as possible the steady-state transport of carbon through the membrane assuming: (a) the rate is limited by the diffusion of carbon in nickel, (b) the rate is limited by interfacial reactions at the surfaces, (c) the rate control involves both diffusion in the metal and interfacial reactaions The diffusion of hydrogen through the metal is very fast. Propose a specific rate-limiting step for hydrogen transport. What experiments would you propose to measure the carbon and hydrogen permeation rates? 15. A continuous 10m thick layer of gold his electroplated on a nickel electrical contact which is to be operated in a poor vacuum at 50-150℃. Calculations using high temperature volume diffusion data indicate that the nickel will not penetrate the gold layer for at least a century. The “customer” has come back saying that some surface oxide is messing up the electrical properties of the contact after a few months. (a) Give some possible explanations of what might be happening. (b) Tell how you can setup a program to prove what’s causing the degradation. (c) Suggest a fix for the product manager. 16. In the oxidation of high temperature aluminum containing alloys, it is felt that the rate controlling transport process through the protective alumina film may be grain boundary transport not lattice diffusion. Discuss how you would go about establishing when and if this is true” a) with diffusion data and equations you might find in the literature. b) any experiment you wish to suggest 17 The concentration of carbon in the case of a carburized steel is given by the expression 0 1 (2 i i C C ) x erf C C Dt − = − − Where C0=surface concentration (gms/cm3 ) of carbon and Ci= carbon content of steel. a) Derive an expression for the flux of carbon into the steel at surface (x=0) b) Derive an expression for the amount of carbon which has diffuse into the steel per unit of surface area after time t
18 For precipitation form solid solutions describe the morphological differences between continuous and discontinuous precipitation reactions.In some binary alloys both types of precipitation can occur whereas in others the discontinuous mode does not appear to arise.Discuss possible reasons for this behavior. 19. How do the different types of inhibitors listed below protect metal against corrosion? (a)Passivators (b)Organic inhibitors (c)Inorganic and organic inhibitors acting only in the presence of oxygen. 20.The same amount of hydrogen is injected into two identical pieces of nickel of thickness h.In sample A the hydrogen is injected from both sides and the hydrogen is distributed uniformly through the sheet.In sample B the hydrogen is injected from only one side,and most of the hydrogen is in the region x< (h/2). (a)Sketch the initial values of cA(x,0)and cB(x,0)for the two samples on one graph,and from the symmetry of the distributions write the first two terms of the Fourier series describing cA(x,t)and cB(x,t).Include the time,t,and position,x,parts in each term,but do not evaluate the constants in front of the sine or cosine terms. (b)If the temperature of the samples is suddenly raised to an elevated temperature,hydrogen diffuses out.Plot of variation of the average concentration as Inc.vs t to show how the loss rate will differ for the two samples cA(t)and CB(t).Explain the differences between the two curves
18 For precipitation form solid solutions describe the morphological differences between continuous and discontinuous precipitation reactions. In some binary alloys both types of precipitation can occur whereas in others the discontinuous mode does not appear to arise. Discuss possible reasons for this behavior. 19. How do the different types of inhibitors listed below protect metal against corrosion? (a) Passivators (b) Organic inhibitors (c) Inorganic and organic inhibitors acting only in the presence of oxygen. 20. The same amount of hydrogen is injected into two identical pieces of nickel of thickness h. In sample A the hydrogen is injected from both sides and the hydrogen is distributed uniformly through the sheet. In sample B the hydrogen is injected from only one side, and most of the hydrogen is in the region x< (h/2). (a) Sketch the initial values of cA(x,0) and cB(x,0) for the two samples on one graph, and from the symmetry of the distributions write the first two terms of the Fourier series describing cA(x,t) and cB(x,t). Include the time, t, and position, x, parts in each term, but do not evaluate the constants in front of the sine or cosine terms. (b) If the temperature of the samples is suddenly raised to an elevated temperature, hydrogen diffuses out. Plot of variation of the average concentration as Inc. vs t to show how the loss rate will differ for the two samples cA(t) and CB(t). Explain the differences between the two curves