Mark: A1:Cubic&Hexagonal Crystal Structures (Chapter 1) 7.5points a)What is the difference between octahedral gap in fcc and bcc with respect to size and shape? b) On which lattice voids are the C-atoms located in the bcc Fe-lattice?The following ratios of radii r/R between octahedral and tetrahedral void and the atom in the bcc- lattice are given(r-radius of void,R-radius of lattice atom). =0.29 Roctahedron =016，R.mm Explain your answer
Mark: A1: Cubic& Hexagonal Crystal Structures (Chapter 1) 7.5points a) What is the difference between octahedral gap in fcc and bcc with respect to size and shape? b) On which lattice voids are the C-atoms located in the bcc Fe-lattice? The following ratios of radii r / R between octahedral and tetrahedral void and the atom in the bcclattice are given (r – radius of void, R – radius of lattice atom). Octahedron r 0.16 R = , tetrahedron r 0.29 R = Explain your answer
c)Draw in the following unit cells of a hexagonal close packed crystal structure (respectively three unit cells are displayed for visualization of the crystal symmetry)the respectively given directions and planes,and please indicate them clearly.Please keep in mind,that in each case specific directions and planes are addressed,whose position in the unit cell is defined unambiguously. [120] T2T0 个C a3 a3 a2 a2 a1 1i0o) (122) a3 a3 a2 a2 a a d Which slip system is the most important slip system and the most important twinning system in the hexagonal dense packed crystal lattice? e)Name the number of possible slip systems with dense packed planes in the hexagonal lattice. Which effect does this have for the formability of hexagonal crystals compared to cubic crystals?
c) Draw in the following unit cells of a hexagonal close packed crystal structure (respectively three unit cells are displayed for visualization of the crystal symmetry) the respectively given directions and planes, and please indicate them clearly. Please keep in mind, that in each case specific directions and planes are addressed, whose position in the unit cell is defined unambiguously. [11 20] [1 210] (1 100) (11 22) d) Which slip system is the most important slip system and the most important twinning system in the hexagonal dense packed crystal lattice? e) Name the number of possible slip systems with dense packed planes in the hexagonal lattice. Which effect does this have for the formability of hexagonal crystals compared to cubic crystals?
A2:Crystal Defects(Chapter 3)6 points a)The concentration of point defects can be easily determined by a measurement of the electrical resistivity.Explain a method in order to estimate the formation enthalpy of point defects,and explain it by means of a sketch.Indicate all occurring sizes. b)Which kind of point defects is contributing at thermal equilibrium mainly to the increase in resistivity?Exlain your answer(in some points). A3:Thermodynamics of Alloys(Chapter 4) 6points Sketch the trend for the mixing enthalpy Hm,the mixing entropy Sm and the Gibbs free energy for a mixture Gm as a function of composition.How is the exchange energy Ho defined (equation and definition)and which sign does it have,respectively how large is it in each of the following cases? Ideal solution. Strong tendency for decomposition. Strong tendency to form intermetallic phases. A4:Diffusion(Chapter 5) 16.5 points a)What is meant by the expression diffusion and convection?(several keywords) b)In a diffusion experiment the diffusion front of Ni in a Fe-sample moves about 0.5 mm at 950 K within a particular time interval.At 980 K the diffusion front moves about 0.62 mm within the same time interval.Determine the activation enthalpy of the diffusion of Ni in Fe under the assumption,that the diffusion coefficient is independent of the concentration.Write down the result in units of electronvolt(eV). )Prove that for the diffusio cofficient ofvacances inth eatichos 67 (D-diffusion coefficient,-time constant,-jump distance) Use a sketch which explains this in more detail.Explain and name all occurring sizes. Formulate step by step the derivation for the above mentioned equation. A5:Dislocation Interaction(Chapter 6)16 points a)Sketch a FCC unit cell with the proper atom positions within a standard Cartesian co- ordinate system
A2: Crystal Defects (Chapter 3) 6 points a) The concentration of point defects can be easily determined by a measurement of the electrical resistivity. Explain a method in order to estimate the formation enthalpy of point defects, and explain it by means of a sketch. Indicate all occurring sizes. b) Which kind of point defects is contributing at thermal equilibrium mainly to the increase in resistivity? Exlain your answer (in some points). A3: Thermodynamics of Alloys (Chapter 4) 6points Sketch the trend for the mixing enthalpy Hm, the mixing entropy Sm and the Gibbs free energy for a mixture Gm as a function of composition. How is the exchange energy H0 defined (equation and definition) and which sign does it have, respectively how large is it in each of the following cases? Ideal solution . Strong tendency for decomposition. Strong tendency to form intermetallic phases. A4: Diffusion (Chapter 5) 16.5 points a) What is meant by the expression diffusion and convection ? (several keywords) b) In a diffusion experiment the diffusion front of Ni in a Fe-sample moves about 0.5 mm at 950 K within a particular time interval. At 980 K the diffusion front moves about 0.62 mm within the same time interval. Determine the activation enthalpy of the diffusion of Ni in Fe under the assumption, that the diffusion coefficient is independent of the concentration. Write down the result in units of electronvolt (eV). c) Prove that for the diffusion coefficient of vacancies in the fcc lattice holds: τ λ 6 2 D = (D – diffusion coefficient, τ – time constant, λ - jump distance) Use a sketch which explains this in more detail. Explain and name all occurring sizes. Formulate step by step the derivation for the above mentioned equation. A5: Dislocation Interaction (Chapter 6) 16 points a) Sketch a FCC unit cell with the proper atom positions within a standard Cartesian coordinate system
b)You observe that an edge dislocation with a Burgers vector b1]in the slip plane dissociates into two Shockley-Partial dislocations: b=g2列und:=gk1可 Indicate in the unit cell of task a)the burgers vectors of the partial dislocations in such a way that the splitting of the dislocation into partials is seen clearly.Which indices has the slip plane? c)Compute the Line element s of the edge dislocation in the standard form (normalized vector). In case you did not obtain the right result in the previous task use the following vector for further calculations: Line normal n of the glideplane,in which the edge dislocation lies:n= 12 d)Explain why (energetics)as given in part b)a perfect dislocation dissociates into Shockley partials. e) Compute with the help of the Peach-Koehler equation the force between two parallel edge dislocations.Show the dislocation arrangement in an orthogonal coordinate system with proper notations.Indicate all required parameters in your sketch.Please name also the line element and the burgers vector for the edge dislocation. Stress tensor for the edge dislocation in z-direction:= 0 000 A6:Softening (Chapter 7)13 points a)Give the definition and explain the terms "recrystallization"and "recovery". b)State the conditions,which are necessary for the creation of a viable recrystallization nucleus in a deformed microstructure and explain them briefly. c)Sketch schematically the relative hardness variation as a function of the recrystallized volume fraction for a material which TENDS to recover and for a material which DOES NOT recover.Explain shortly the differences in the behavior of both materials.In this regard,explain the influence of the stacking fault energy.Give one example for each case
b) You observe that an edge dislocation with a Burgers vector [ ] 110 2 a b = in the slip plane dissociates into two Shockley-Partial dislocations: [ ] 121 6 a b1 = und [ ] 211 6 a b2 = . Indicate in the unit cell of task a) the burgers vectors of the partial dislocations in such a way that the splitting of the dislocation into partials is seen clearly. Which indices has the slip plane? c) Compute the Line element s of the edge dislocation in the standard form (normalized vector). In case you did not obtain the right result in the previous task use the following vector for further calculations: Line normal n of the glideplane, in which the edge dislocation lies: 2 1 a n 1 12 1 ⎛ ⎞ − ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ − d) Explain why (energetics) as given in part b) a perfect dislocation dissociates into Shockley partials. e) Compute with the help of the Peach-Koehler equation the force between two parallel edge dislocations. Show the dislocation arrangement in an orthogonal coordinate system with proper notations. Indicate all required parameters in your sketch. Please name also the line element and the burgers vector for the edge dislocation. Stress tensor for the edge dislocation in z-direction: σ σ σ σ σ σ xyz xx xy xy yy zz = ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ 0 0 0 0 A6: Softening (Chapter 7) 13 points a) Give the definition and explain the terms "recrystallization" and "recovery". b) State the conditions, which are necessary for the creation of a viable recrystallization nucleus in a deformed microstructure and explain them briefly. c) Sketch schematically the relative hardness variation as a function of the recrystallized volume fraction for a material which TENDS to recover and for a material which DOES NOT recover. Explain shortly the differences in the behavior of both materials. In this regard, explain the influence of the stacking fault energy. Give one example for each case
d)A quantitative description of the recrystallized volume fraction X as a function of the annealing time is possible by means of the JMAK-equation: X=1-exp (X-recrystallized volume fraction,t-recrystallization time,tex -time constant,g- Avrami-exponent) Under which nucleation conditions would you expect an Avrami-exponent of q =3 respectively q=4?Why is the experimentally obtained Avrami-exponent in general smaller? A7:Transformations:solid-liquid solid-solid(Chapter 8 &9) 19 points a)Calculate the critical edge length D of a cubic nucleus and the critical work for nucleation for homogeneous nucleation from the melt.Name those of the used parameters which have a higher importance. b) Draw the curve of the free energy AG as a function of nucleus size(edge length)for T>Ts and TDe?). c)Explain,why during the technical solidification processes the work for homogeneous nucleation is in most cases not required.Give the relationship between homogeneous and heterogeneous work for nucleation. d)Explain,which additional significant influence(beside transformation-and interface enthalpy)needs to be considered in case of homogeneous nucleation for a transformation in the solid state.Write down an expression for the free energy AG as a function of the edge length D(How is the expression in part a)modified?). e) Explain a transformation mechanism without nucleation during decomposition in the solid state.Sketch schematically the temporal change of the concentration changes occurring in the microstructure.Name the differences to the precipitation mechanism according to nucleation. A8:Physical Properties(Chapter 10) 6 points a)Given is the curve for the resulting force F between two atoms as a function of the interatomic spacing x. Give the mathematical correlation between the interaction force F,the interaction energy E(also called interatomic potential)and the stiffness S of the atomic bond (relationship between force and change of length). Sketch the corresponding curve of the interaction energy E between two atoms as a function of the interatomic spacing x in the corresponding diagram and mark the
d) A quantitative description of the recrystallized volume fraction X as a function of the annealing time is possible by means of the JMAK-equation: ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = − − q RX t t X 1 exp ( X - recrystallized volume fraction, t - recrystallization time, RX t - time constant, q - Avrami-exponent) Under which nucleation conditions would you expect an Avrami-exponent of q = 3 respectively q = 4? Why is the experimentally obtained Avrami-exponent in general smaller? A7: Transformations: solid – liquid & solid – solid (Chapter 8 & 9) 19 points a) Calculate the critical edge length D of a cubic nucleus and the critical work for nucleation for homogeneous nucleation from the melt. Name those of the used parameters which have a higher importance. b) Draw the curve of the free energy ∆G as a function of nucleus size (edge length) for T > TS and T Dc?). c) Explain, why during the technical solidification processes the work for homogeneous nucleation is in most cases not required. Give the relationship between homogeneous and heterogeneous work for nucleation. d) Explain, which additional significant influence (beside transformation – and interface enthalpy) needs to be considered in case of homogeneous nucleation for a transformation in the solid state. Write down an expression for the free energy ∆G as a function of the edge length D (How is the expression in part a) modified?). e) Explain a transformation mechanism without nucleation during decomposition in the solid state. Sketch schematically the temporal change of the concentration changes occurring in the microstructure. Name the differences to the precipitation mechanism according to nucleation. A8: Physical Properties (Chapter 10) 6 points a) Given is the curve for the resulting force F between two atoms as a function of the interatomic spacing x. Give the mathematical correlation between the interaction force F, the interaction energy E (also called interatomic potential) and the stiffness S of the atomic bond (relationship between force and change of length). Sketch the corresponding curve of the interaction energy E between two atoms as a function of the interatomic spacing x in the corresponding diagram and mark the
equilibrium spacing xo between two atoms.Please note that the extreme values and the zero values(f(x)=0)are marked properly in the potential curve. b) Calculate the equilibrium spacing xo between both atoms at T=OK under the assumption,that the interaction energy E is given by the Lennard-Jones-Potential V(x): V=A B c Explain by using the curve of the interaction energy E the phenomenon of the thermal expansion of metals.(in headwords) appendix to a) interatomic spacing x
equilibrium spacing x0 between two atoms. Please note that the extreme values and the zero values (f(x) =0) are marked properly in the potential curve. b) Calculate the equilibrium spacing x0 between both atoms at T = 0K under the assumption, that the interaction energy E is given by the Lennard-Jones-Potential V(x): 12 6 x B x A V = − c) Explain by using the curve of the interaction energy E the phenomenon of the thermal expansion of metals. (in headwords) appendix to a) interatomic spacing x interaction force F(x)
interatomic spacing x R=8.3144J/molK,k=8.6210-5eV/K=1.38-10-23J/K,NL=6.023-1023 material constants for aluminium G=27-10N/m2,a=4.04ATs=660C,p=2.7g/cm3,A=27.0gmol,v-0.34 YSFE=18.10-2J/m2,YKG-0.6J/m2 material constants for copper: G=48.109N/m2,a=3.61A,Ts=1083C,p=8.7g/cm3,A=63.5gmol,v=0.35 YSFE-5.10-2J/m2.YKG-0.5J/m2,YOb=1J/m2
interatomic spacing x interaction energy E(x) _________________________________________________________________________ R=8.3144 J/molK, k=8.62⋅10-5eV/K = 1.38⋅10-23J/K, NL=6.023⋅1023 material constants for aluminium: G=27⋅109N/m2, a=4.04Å, TS=660°C, ρ=2.7g/cm3, A=27.0g/mol, ν=0.34 γSFE=18⋅10-2J/m2, γKG=0.6J/m2 material constants for copper: G=48⋅109 N/m2, a=3.61Å, TS=1083°C, ρ=8.7g/cm3, A=63.5g/mol, ν=0.35 γSFE=5⋅10-2J/m2, γKG=0.5J/m2, γOb=1J/m2