Materials Science and Engineering Problems MSE Faculty April 19, 2021 This document includes the assigned problems that have been included so far in the digitized portion of our course curriculum. Contents 1 301 Problems 3 1.1 Course Organization.................................................. 3 1.2 Atomic Structure and Bonding ............................................ 3 1.3 Crystal Structure .................................................... 5 1.4 Imperfections ...................................................... 8 1.5 Electrical Properties................................................... 10 1.6 Diffusion......................................................... 12 1.7 Phase Diagrams..................................................... 14 1.8 Phase Transformations................................................. 18 1.9 Mechanical Properties ................................................. 18 1.10 Fracture Mechanics................................................... 22 1.11 Corrosion......................................................... 23 1.12 Ceramics......................................................... 24 1.13 Polymers......................................................... 25 2 314 Problems 27 2.1 Phases and Components................................................ 27 2.2 Intensive and Extensive Properties.......................................... 27 2.3 Differential Quantities and State Functions ..................................... 27 2.4 Entropy.......................................................... 27 2.5 Thermodynamic Data ................................................. 28 2.6 Temperature Equilibration............................................... 30 2.7 Statistical Thermodynamics.............................................. 30 2.8 Single Component Thermodynamics......................................... 30 2.9 Mulitcomponent Thermodynamics.......................................... 31 2.10 Computational Exercises................................................ 32 3 315 Problems 34 4 316-1 Problems 43 5 316-2 Problems 62 5.1 General.......................................................... 62 5.2 Laplace Pressure Derivation.............................................. 62 5.3 Homogeneous Nucleation............................................... 62 5.4 Surface and Interface Effects.............................................. 63 5.5 Heterogeneous Nucleation............................................... 64 5.6 Nucleation in a Binary System............................................. 64 5.7 Spinodal Decomposition................................................ 66 5.8 Constitutional Undercooling.............................................. 66 5.9 Coarsening........................................................ 68 5.10 Eutectic Solidification.................................................. 68 1 5.11 Eutectoid Transormations ............................................... 69 5.12 Transitional Phases................................................... 70 5.13 TTT diagrams...................................................... 71 5.14 Mineralization...................................................... 71 5.15 Review Questions.................................................... 72 6 332 Problems 74 6.1 Course Organization.................................................. 74 6.2 The Stress Tensor .................................................... 74 6.3 Strains .......................................................... 74 6.4 Typical Moduli ..................................................... 74 6.5 Matrix Representation of Stress and Strains..................................... 75 6.6 Other Linear Properties................................................. 76 6.7 Contact Mechanics ................................................... 76 6.8 Nanoindentation .................................................... 77 6.9 Fracture Mechanics................................................... 78 6.10 Weibull Statistics .................................................... 80 6.11 Yield Criteria....................................................... 80 6.12 Strengthening Mechanisms .............................................. 82 6.13 Viscoelasticity...................................................... 85 6.14 Nonlinear Viscoelasticity and Creep ......................................... 87 7 331 Problems 89 8 351-1 Problems 104 9 351-2 Problems 109 10 361 Problems 111 11 390 Problems 116 11.1 Materials Selection in Mechanical Design ......................................116 11.2 Thermo-Calc.......................................................118 11.3 DICTRA .........................................................119 11.4 TC-PRISMA.......................................................119 11.4.1 ..........................................................120 11.4.2 ..........................................................120 2 Contents 1 301 Problems 1.1 Course Organization 1) Send an email to Prof. Shull ([email protected]) and Alane (Alane.lim@[email protected]) with the following information: 1. Any background about yourself that you want to share. 2. What you have enjoyed most and have found the most frustrating about your major. 3. One particular aspect of materials science that you would like to learn more about this quarter. 1.2 Atomic Structure and Bonding 2) Classify each of the following materials as to whether it is a metal, ceramic, or polymer. Justify each choice. (a.) ® brass; (b.) magnesium oxide (MgO); (c.) Plexiglass ; (d.) polychloroprene; (e.) boron carbide (B4C); and (f.) steel. 3) Site the difference between atomic mass and atomic weight. 4) Silicon has three naturally occurring isotopes: 92.23% of 28Si, with an atomic weight of 27.9769 amu, 4.68% of 29Si, with an atomic wight of 28.9738 amu, and 3.69% of 30Si, with an atomic weight of 29.938 amu. On the basis of these data, confirm that the average atomic weight of Si is 28.09. 5) Indium has two naturally occurring isotopes: 113In, with an atomic weight of 112.904 amu, and 115In, with an atomic weight of 114.904 amu. If the average atomic weight for In is 114.818 amu, calculate the fraction-of-occurrences of these two isotopes. 6) Address the following concepts concerning atomic mass. 1. How many grams are there in one amu of material? 2. Mole, in the context of this book, is taken in units of gram-mole. On this basis, how many atoms are there in a pound-mole of a substance? 7) Relative to electrons and electronic states, what does each of the four quantum numbers specify? 8) Give the electron configurations for the following ions: P5+,P3−, Sn4+, Se2−,I− and Ni2+. 9) Potassium iodide (KI) exhibits predominately ionic bonding. The K+ and I− ion have electron structures that are identical to which two inert gases? 10) Without consulting Callister Figure 2.8 or Table 2.2, determine whether each of the following atomic electron configurations is an inert gas, a halogen, an alkali metal, an alkaline earth metal, or a transition metal. List the number of valence electrons for each atom (except for the transition metals). Justify your choices. 1.1 s22s22p63s23p5 2.1 s22s22p63s23p63d74s2 3.1 s22s22p63s23p63d104s24p6 4.1 s22s22p63s23p64s1 5.1 s22s22p63s23p63d104s24p64d55s2 6.1 s22s22p63s2 3 11) The atomic radii of Mg2+ and F- ions are 0.072 and 0.133 nm respectively. 1. Calculate the force of attraction between the two ions at their equilibrium interionic separation (i.e., when the ions just touch one another). 2. What is the force of repulsion at this same separation distance? 12) The force of attraction between a divalent cation and a divalent anion is 1.67 × 10−8 N. If the ionic radius of the cation is 0.080 nm, what is the anion radius? Here we’re provided with the two pieces of information: the force of attraction between a cation/anion pair and the cation’s atomic radius. With this information and an understanding of the Coulomb interaction we can calculate the anion radius. The force of attraction between two isolated ions is defined by Callister Eq. 2.13: 13) The net potential energy between two adjacent ions, EN, may be represented by the sum of Callister equations 2.9 and 2.11. That is: A B E = − + (1.1) N r rn Determine the equilibrium ionic bond energy, E0, in terms of the parameters A, B, and n. Note that equilibrium occurs when the net force on the ions is zero. Use the following procedure: 1. Differentiate EN with respect to r to acquire the expression for the interatomic force. 2. Assume two adjacent ions a EN versus r is minimum at E0. 3. Solve for r in terms of A, B, and n, which yields r0, the equilibrium interionic spacing. 4. Determine the expression for E0 by substituting r0 into the equation. + - 14) For an Na -Cl ion pair, attractive and repulsive energies (EA and ER, respectively), depends on the distance between ions: 1.44 eV · nm E = − (1.2) A r 7.32 × 10−6 eV · nm8 E = (1.3) R r8 1. Superimpose on a single plot (by hand or using plotting software) EA, ER, and the net energy EN up to r = 1.0 nm. Hint: You may have to truncate the plot on the y-axis for good visualization. 2. From this plot, derive the equilibrium spacing, r0 and the magnitude of the bonding energy, E0 at the equilibrium spacing. + - + 3. Now consider instead a K -Cl bonding pair. The K ion is larger, which changes the repulsive term ER to be: 5.80 × 10−6 eV − nm9 E = (1.4) R r9 + - Without plotting the new EN, how do you expect E0 and r0 to change for the K -Cl ion pair? 15) Briefly discuss the main differences