Phy351 ch 4
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e RT Where; D0 = Diffusion coefficient at infinite temperature Q = Activation energy for diffusion R = Gas constant T = Absolute temperature This chapter discusses imperfections in crystals and solidification processes. There are several types of imperfections including point defects like vacancies and interstitial atoms, and line defects like dislocations. Solidification involves nucleation of stable nuclei from liquid, growth of these nuclei into a grain structure, and formation of grain boundaries. Single crystals can be grown using the Czochralski process by slowly pulling a seed crystal from a melt. Diffusion, which is temperature dependent, allows for solid state reactions and involves either vacancy or interstitial mechanisms.
Phy351 ch 4
- 1. PHY351 Chapter 4 IMPERFECTION IN CRYSTAL
- 2. Mechanical Deformation Solidification of Metal There are several steps of solidification: Nucleation : Formation of stable nuclei (Fig a) Growth of nuclei : Formation of grain structure (Fig b) Formation of grain structure (Fig c) Grains Nuclei Liquid (a) 2 Crystals that will Form grains (b) Grain Boundaries (c)
- 3. Solidification of Single Crystals - Czochralski Process For some applications, single crystals are needed due to single crystals have high temperature and creep resistance. This method is used to produce single crystal of silicon for electronic wafers. A seed crystal is dipped in molten silicon and rotated. The seed crystal is withdrawn slowly while silicon adheres to seed crystal and grows as a single crystal. 3
- 4. Metallic Solid Slutions- Substitutional Solid Solution Solute atoms substitute for parent solvent atom in a crystal lattice. The structure remains unchanged. Lattice might get slightly distorted due to change in diameter of the atoms. Solvent atoms Solute atoms 4
- 5. Metallic Solid Slutions- Interstitial Solid Solution Solute atoms fit in between the void (interstices) of solvent atoms. Iron atoms r = 0.124nm Carbon atoms r = 0.077nm 5
- 6. Crystalline Imperfections The arrangement of the atoms or ions in engineered materials are never perfect and contains various types of imperfections and defects. Imperfections affect mechanical properties, chemical properties and electrical properties of materials. These imperfections only represent defects or deviation from the perfect or ideal atomic or ionic arrangements expected in a given crystal structure. Note: The materials is not considered defective from an application viewpoint. 6
- 7. Crystal lattice imperfections can be classified according to their geometry and shape . There are several basic types of imperfection as following: 7 Zero dimension point defects. One dimension / line defects (dislocations). Two dimension defects. Three dimension defects (cracks).
- 8. Zero Dimension Point Defects Point defects are localized disruption in otherwise perfect atomic or ionic arrangements in a crystal structure. The disruption affects a region involving several atoms or ions or pair of atoms or ions only. These imperfection may be introduced by movement of the atoms or ions when they gain energy by: 8 Heating during processing of the material Introduction of impurities doping
- 9. Point Defects – Vacancy The simplest point defect is the vacancy, which formed due to a MISSING atom from its normal site in the crystal structure. Vacancy is produced during solidification resulting: a local disturbance during crystallization atomic arrangements in an existing crystal due to atomic mobility. Vacancies also caused due to plastic deformation, rapid cooling or particle bombardment. 9
- 10. Vacancies can MOVE by exchanging positions with their neighbors. This process is important in the migration or diffusion of atoms in the solid state, particularly at high temperature where atomic mobility is greater. 10
- 11. Point Defects – Interstitial Defects An interstitial defects is formed when an extra atom or ion is INSERTED into the crystal structure at interstitial site which a normally unoccupied position. This defects not occur naturally. It is due irradiation and causing a structural distortion. Interstitial atoms : - are often present as impurities 11
- 12. Point Defects – Substitutional Defects A substitutional defects is introduced when one atom or ion is REPLACED by a different type of atom or ion. The substitutional atoms or ions occupy the normal lattice site. The substitutional atoms or ion can either be larger or smaller than normal atoms or ions in the crystal structure. Substitutional defects can be introduced either as an impurity or as a deliberate alloying addition. 12
- 13. Point Defects in Ionic Crystal A Schottky defects is uniques to ionic materials and is commonly found in many ceramic materials. In this defects, vacancies occur when TWO appositely charged particles are MISSING form the ionic crystal due to need to maintain electrical neutrality. 13
- 14. If a positive cation MOVES into an interstitial site in an ionic crystal, a cation vacancy is created in the normal ion site. This vacancy-interstitialcy pair is called Frenkel imperfection. Although this is described for an ionic material, a Frenkel defect can occur in metals and covalently bonded materials. 14
- 15. The present of these defects in ionic crystal increases their electrical conductivity. Impurity atoms are also considered as point defects. 15
- 16. Thermal Production Many processes involved in the production of engineering materials are concerned with the RATE at which atoms move in the solid state. In this processes, a reactions occur in the solid state which involve the spontaneous rearrangement of atoms into a new and more stable arrangements. Reacting atoms must have sufficient energy to overcome activation energy barrier. The energy required which above the average energy of the atom is called ACTIVATION ENERGY, * . (Unit :Joules/mole). 16
- 17. At any temperature, only a fraction of the molecules or atoms in a system will have sufficient energy to reach the activation energy level of E*. As temperature increases, more and more atoms acquire activation energy level. PROBABILITY of finding an atom/molecule with energy E* GREATER than average energy, E of all atoms/ molecules is given by Probability e –(E* - E) /KT Where; K = Boltzman’s Constant = 1.38 x 10-23 J/(atom K) T = temperature in Kelvin 17
- 18. The FRACTION of atoms having energies greater than E* in a system (when E* is greater than average energy E) is given by n N total E* Ce K .T Where; n = Number of molecules greater than energy E* Ntotal = Total number of molecules k = Boltzman’s Constant C = Constant T = Temperature in Kelvin. 18
- 19. The NUMBER OF VACANCIES at equilibrium at a particular temperature in a metallic crystal lattice is given by nv N EV Ce K .T Where; nv = Number of vacancies per m3 of metal Ev = Activation Energy to form a vacancy T = Absolute Temperature k = Boltzman’s Constant C = Constant 19
- 20. Question 1 By assuming the energy formation of a vacancy in pure copper is 0.9 eV, C = 1 and N = 8.49 x 1028 atoms/m3, calculate a. The equilibrium number of vacancies per cubic meter in pure copper at 5000C (Answer: 1.2 x 1023 vacancies/m3) b. The vacancy fraction at 5000C in pure copper. (Answer : 1.4 x 10-6) Constant: k = 8.62 x 10-5 eV/K 20
- 21. Stability of Atoms and Ions Atoms and ion in their normal positions in the crystal structures are not stable or at rest. Instead, the atoms or ions posses thermal energy and they will move. For instance, an atom may move from a normal crystal structure location to occupy a nearby vacancy. An atom may also move from one interstitial site to another. Atoms or ions may jump across a grain boundary causing the grain boundary to move. The ability of atoms or ions to diffuse increases as temperature possess by the atoms or ions increases. 21
- 22. The rate of atom or ion movement that related to temperature or thermal energy is given by Arrhenius equation. Rate of reaction = Ce-Q/RT Where; Q = Activation energy J/mol R = Molar gas constant J/mol.K T = Temperature in Kelvin C = Rate constant ( Independent of temperature) Note: Rate of reaction depends upon number of reacting molecules. 22
- 23. Arrhenius equation can also be written as: ln (rate) = ln ( C) – Q/RT Or Log10 (rate) = Log10 (C) – Q/2.303 RT Y Log10(rate) X b Log10(C) m 23 (1/T) Q/2.303R
- 24. Question 2 Suppose that interstitial atoms are found to move from one site to another at the rates of 5 x 108 jumps/s at 5000C and 8 x 1010 jumps/s at 8000C. Calculate the activation energy for the process. 24
- 25. Solid State Diffusion DIFFUSION is a process by which a matter is transported through another matter. Examples: 25 Movement of smoke particles in air : Very fast. Movement of dye in water : Relatively slow. Solid state reactions : Very restricted movement due to bonding. (A nickel sheet bonded to a cooper sheet. At high temperature, nickel atom gradually diffuse in the cooper and cooper migrate into the nickel)
- 26. There are two main mechanisms of diffusion of atoms in a crystalline lattice: 26 The vacancy or substitutional mechanism The interstitial mechanism
- 27. Diffusion Mechanism Vacancy or Substitutional Diffusion Atoms diffuse in solids IF: Vacancies or other crystal defects are present There is enough activation energy Atoms move into the vacancies present. More vacancies are created at higher temperature. Diffusion rate is higher at high temperatures. 27
- 28. Example: If atom ‘A’ has sufficient activation energy, it moves into the vacancy self diffusion. As the melting point increases, activation energy also increases. Activation Energy of = Self diffusion 28 Activation Energy to form a Vacancy + Activation Energy to move a vacancy
- 29. Interstitial Diffusion mechanism Atoms MOVE from one interstitial site to another. The atoms that move must be much smaller than the matrix atom. Matrix atoms 29 Interstitial atoms
- 30. Steady State Diffusion There is NO CHANGE in concentration of solute atoms at different planes in a system, over a period of time. No chemical reaction occurs. Only net flow of atoms. The rate at which atoms, ions, particles or other species diffuse in a material can be measured by the flux, J. The flux is defined as the number of atoms passing through a plane of unit area per unit time. For steady state diffusion condition, the net flow of atoms by atomic diffusion is equal to diffusion D times the diffusion gradient dc/dx. This is defined as Fick’s First Law. 30
- 31. Rate of Diffusion (Fick’s First Law) The flux or flow of atoms is given by: J D dc dx Where; J = Flux or net flow of atoms (Unit: atoms/m2s) D = Diffusivity or Diffusion coefficient (Unit: m 2/s) dc dx 31 = Concentration Gradient (Unit: atoms/m3.m)
- 32. The concentration gradient shows how the composition of the material varies with distance: c is the difference in concentration over the distance x. Diffusivity depends upon: 32 Type of diffusion : Whether the diffusion is interstitial or substitutional. Temperature: As the temperature increases diffusivity increases. Type of crystal structure: BCC crystal has lower APF than FCC and hence has higher diffusivity. Type of crystal imperfection: More open structures (grain boundaries) increases diffusion. The concentration of diffusing species: Higher concentrations of diffusing solute atoms will affect diffusivity
- 33. The concentration gradient shows how the composition of the material varies with distance: c is the difference in concentration over the distance x. Diffusivity depends upon: 33 Type of diffusion : Whether the diffusion is interstitial or substitutional. Temperature: As the temperature increases diffusivity increases. Type of crystal structure: BCC crystal has lower APF than FCC and hence has higher diffusivity. Type of crystal imperfection: More open structures (grain boundaries) increases diffusion. The concentration of diffusing species: Higher concentrations of diffusing solute atoms will affect diffusivity
- 34. Question 3 a. One way to manufacture transistor which amplify electrical signals is to diffuse impurity atoms into a semiconductor material such as silicon (Si). Suppose a silicon wafer with 0.1 cm thick, which originally contains one phosphorus (P) atom for every 10 million Si atoms, is treated so that there are 400 phosphorus atoms for every 10 million Si atoms at the surface. Calculate the concentration gradient. (Given the lattice parameter of Si is 5.4307Å) b. The diffusion flux of cooper solute atoms in aluminium solvent from point A to point B, 10 mm apart is 4 x 1017 atoms/m2s at 5000C. Determine i. ii. 34 0 The concentration gradient (Given D500 C = 4 x10-14 m2/s) Difference in the concentration levels of cooper between the two points.
- 35. Non- Steady Diffusivity Concentration of solute atoms at any point in metal CHANGES with time in this case. Ficks second law:- Rate of compositional change is equal to diffusivity times the rate of change of concentration gradient. d dc x D dx dt dx dC x Change of concentration of solute Atoms with change in time in different planes 35
- 36. x Cs C x erf 2 Dt Cs C0 Cs = Surface concentration of element in gas diffusing into the surface. C0 = Initial uniform concentration Cs Time = t2 Time= t1 Cx Time = t0 of element in solid. Cx = Concentration of element at C0 distance x from surface at time t1. x = distance from surface D = diffusivity of solute t = time. 36 x Distance x
- 37. Question 4 Consider the gas carburizing of a gear of 1020 steel at 927 0C. Calculate the time necessary to increase the carbon content at 0.4% at 0.5mm below the surface. Assume that the carbon content at the surface is 0.9% and that the steel has a nominal carbon content of 0.2%. Given D steel at 9270C = 1.28 x 10-11 m2/s. Erf Z 0.7112 0.75 0.7143 X 0.7421 37 Z 0.8
- 38. Question 5 Consider the gas carburizing of a gear of 1020 at 927 0C as Question 4. Only in this problem calculate the carbon content at 0.5mm beneath the surface of the gear after 5h carburizing time. Assume that the carbon content of the surface of the gear is 0.9% and that the steel has a nominal carbon content of 0.2%. Given D steel at 9270C = 1.28 x 10-11 m2/s. Erf Z 0.5000 0.5205 0.521 X 0.550 38 Z 0.5633
- 39. Effect of Temperature on Diffusion Dependence of rate of diffusion on temperature is given by Q D D0 e RT ln D ln D0 Q RT log10 D log10 D0 Q 2.303RT D = Diffusivity m2/s D0 = Proportionality constant m2/s Q = Activation energy of diffusing species J/mol R = Molar gas constant = 8.314 J/mol.K T = Temperature (K) 39
- 40. Question 6 1. Calculate the value of the diffusivity, D for the diffusion of carbon in γ iron (FCC) at 9270C. (Given D0= 2x10-5 m2/s, Q=142kJ/mol, R=8.314J/mol.k) 2. The diffusivity of silver atoms in silver is 1 x 10 -17 m2/s at 5000C and 7 x 10-13 m2/s at 10000C. Calculate activation energy, Q for the diffusion of silver in the temperature range 5000C and 10000C. (Given R=8.314J/mol.k) 40
- 41. Linear/Line Defects – (Dislocations) Line imperfection or dislocation are defects that cause lattice distortions. Dislocation are created during: Solidification Permanent deformation of crystalline solid Vacancy condensation Atomic mismatch in solid solution Different types of line defects are: Edge dislocation Screw dislocation Mixed dislocation 41
- 42. Edge Dislocation An edge dislocation is created in a crystal by insertion of extra half planes of atoms. Positive edge dislocation Negative edge dislocation In figure 4.18, a linear defect occurs in the region just above the inverted T, where an extra half plane of atoms has been wedged in. Figure 4.18 : Positive edge dislocation in a crystalline lattice. 42
- 43. Screw Dislocation The screw dislocation can be formed in a perfect crystal by applying upward and downward shear stresses to regions of a perfect crystal that have been separated by a cutting plane as shown in Figure 4.20a. Figure 4.20a Formation of a screw dislocation: A perfect crystal is sliced by a cutting plane, and up and down shear stresses are applied parallel to the cutting plane to form the screw dislocation as in (b). 43
- 44. These shear stresses introduce a region of distorted crystal lattice in the form of a spiral ramp of distorted atoms or screw dislocation as Figure 4.20b. Figure 4.20b Formation of a screw dislocation: A screw dislocation is shown with its slip or Burgers vector b parallel to the dislocation. 44
- 45. The region of distorted crystal is not well defined and is at least several atoms in diameter. A region of shear strain is created around the screw dislocation in which energy stored. The slip or Burgers vector of the screw dislocation is parallel to the dislocation line as shown in Figure 4.20b. 45
- 46. Mixed Dislocation Most crystal have components of both edge and screw dislocation. Dislocation, since have irregular atomic arrangement will appear as dark lines when observed in electron microscope. Dislocation structure of iron deformed 14% at –1950C 46
- 47. Planar Defects Planar defects including: Grain boundaries Twins / twin boundaries low/high angle boundaries Stacking faults / pilling-up fault Grain boundaries are most effective in strengthening a metal compared to twin boundaries, low/high boundaries and stacking faults. The free or external surface of any material is also a defect and is the most common type of planar defect. 47
- 48. The free or external surface are considered defects because: 48 Atom on the surface are bonded to atoms on only one side. Therefore, the surface atoms have a lower number of neighbors. As a result, these atoms have higher state of energy when compared to the atoms positioned inside the crystal with an optimal number of neighbors. The higher energy associated with the atoms on the surface of a material makes the surface susceptible to erosion and reaction with elements in the environment.
- 49. Grain Boundaries Grain boundaries separate grains. Formed due to simultaneously growing crystals meeting each other. Width = 2-5 atomic diameters. Some atoms in grain boundaries have higher energy. Restrict plastic flow and prevent dislocation movement. 3D view of grains Grain Boundaries In 1018 steel 49
- 50. Twin Boundaries Twins: - A region in which mirror image pf structure exists across a boundary. Formed during plastic deformation and recrystallization. Strengthens the metal. Twin Plane Twin 50
- 51. Small angle tilt boundary Small angle tilt boundary: - Array of edge dislocations tilts two regions of a crystal by < 10 0 Figure 4.24 (a) Edge dislocations in an array forming a small-angle tilt boundary (b) Schematic of a small-angle twist boundary. 51
- 52. Stacking faults / Piling up faults Stacking faults / Piling up faults : - form during recrystallization due to collapsing. Example: ABCABAACBABC 52 FCC fault
- 53. Three dimensional imperfections Volume or three-dimensional defects: - produced when a cluster of point defects join to form a threedimensional void or a pore. 53
- 54. Observing Grain Boundaries To observe grain boundaries, the metal sample must be first mounted for easy handling Then the sample should be ground and polished with different grades of abrasive paper and abrasive solution. The surface is then etched chemically. Tiny groves are produced at grain boundaries. Groves do not intensely reflect light. Hence observed by optical microscope such as: 54 Transmission Electron Microscope (TEM) Scanning electron microscope (SEM) High Resolution Transmission Electron Microscope (HRTEM) Scanning Tunneling Microscope (STM) – scanning probe microscope Atomic Force Microscope (AFM) – scanning probe microscope
- 55. Grain Size Affects the mechanical properties of the material. The smaller the grain size, more are the grain boundaries. More grain boundaries means higher resistance to slip (plastic deformation occurs due to slip). More grains means more uniform the mechanical properties are. N < 3 – Coarse grained 4 < n < 6 – Medium grained 7 < n < 9 – Fine grained N > 10 – ultrafine grained 55
- 56. Measuring Grain Size ASTM grain size number ‘n’ is a measure of grain size: N = 2 n-1 N = Number of grains per square 2.54 x 10-2m2 polished and etched specimen at 100 x. n = ASTM grain size number. 56
- 57. Exercise 7 1. An ASTM grains size determination is being made from a photomicrograph of a metal at magnification of 100X. What is the ASTM grain-size number of the metal if there are 64 grains per square 2.54 x 10-2 m ? (Answer : 7) 2. If there are 60 grains per square 2.54 x 10 -2 m on a photomicrograph of a metal at 200X, what is the ASTM grain-size number of the metal? (Answer : 8.91) 57
- 58. Average Grain Diameter Average grain diameter more directly represents grain size. Random line of known length is drawn on photomicrograph. Number of grains intersected is counted. Ratio of number of grains intersected to length of line, nL is determined. d = C/nLM d = average grain diameter C = constant, typically 1.5 M = magnification nL = number of grains intersected to line per length of line 58
- 59. Exercise 8 Estimate the average grain diameter of a micrograph below. Given C= 1.5 and M=200X. (Answer :14mm) 59
- 60. References A.G. Guy (1972) Introduction to Material Science, McGraw Hill. J.F. Shackelford (2000). Introduction to Material Science for Engineers, (5th Edition), Prentice Hall. W.F. Smith (1996). Priciple to Material Science and Engineering, (3rd Edition), McGraw Hill. W.D. Callister Jr. (1997) Material Science and Engineering: An Introduction, (4th Edition) John Wiley. 60