Lecture 6 PHASE TRANSFORMATIONS in Metal and Alloys.pptx

Phase Transformation Lecture 2

1. PHASE TRANSFORMATIONS in Metal and Alloys By Dr. Feroz Shah Lecture No.6

2. Stability of Atoms and Ions 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 and ions to diffuse increases as the temperature, or thermal energy possessed by the atoms and ions, increases

3. Interstitial Diffusion Interstitial Diffusion as a Random Jump Process Let us consider first a simple model of a dilute interstitial solid solution The parent atoms are arranged on a simple cubic lattice The solute B atoms fit perfectly into the interstices without causing any distortion of the parent lattice We assume that the solution is so dilute that every interstitial atom is surrounded by six vacant interstitial sites Consider the exchange of atoms between two adjacent atomic planes such as (1) and (2) Figure 2

4. Continue The rate of atom or ion movement is related to temperature or thermal energy by the Arrhenius equation Where is a constant, R is the gas constant , T is the absolute temperature(K) and Q is the activation energy ( cal / mol ) We can rewrite the equation by taking natural logarithms of both sides:   Figure 1

5. Continue Assume that on average an interstitial atom jumps , times per second If a plane 1 contain B-atoms per , the number of atoms that will jump from plane 1 to 2 in 1s will be given by: During the same time the number of atoms that jump from plane 2 to 1 assuming B is independent of concentration, is given by : Since, there will be a net flux of atoms from left to right given by :  

6. Continue Simplifying the equation gives: The partial derivative has been used to indicate that the concentration gradient can change with time along the length In the presence of a concentration gradient the random jumping of individual atoms produces a net flow of atoms down the concentration gradient Substituting Equation 4 becomes:  

7. Continue Where is the flux (quantity ), D is the diffusivity or diffusion coefficient and is the concentration gradient ( quantity ) Concentration may be expressed as atom percent (at%), weight percent ( wt %), mole percent ( mol %), atom fraction , or mole fraction  

8. Activation Energy for Diffusion A diffusing atom must squeeze past the surrounding atoms to reach its new site Energy must be supplied to allow the atom to move to its new position The atom is originally in a low-energy, relatively stable location In order to move to a new location the atom must overcome an energy barrier The energy barrier is the activation energy Q less energy is required to squeeze an interstitial atom past the surrounding atoms Figure 3

9. Continue Typical values for activation energies for diffusion of different atoms in different host materials are shown in Table 1 A low-activation energy indicates easy diffusion In self-diffusion, the activation energy is equal to the energy needed to create a vacancy and to cause the movement of the atom Table 1

10. Concentration Gradient The concentration gradient shows how the composition of the material varies with distance is the difference in concentration over the distance A concentration gradient may be created when two materials of different composition are placed in contact ( Figure 4) The flux at a particular temperature is constant only if the concentration gradient is also constant   Figure 4

11. Factors Affecting Diffusion Effect of Temperature-Thermal Activation The kinetics of diffusion are strongly dependent on temperature The diffusion coefficient D is related to temperature by an Arrhenius-type equation Where Q is the activation energy (in units of cal / mol ) for diffusion of the species under consideration  

12. Factors Affecting Diffusion R is the gas constant , and T is the absolute temperature in K is a constant for a given diffusion system and is equal to the value of the diffusion coefficient at or Typical values for are given in Table 1   Figure 5

13. Continue The temperature dependence of D is shown in Figure 4 for some metals and ceramics When the temperature of a material increases, the diffusion coefficient D increases The flux of atoms increases as well At higher temperatures, the thermal energy supplied to the diffusing atoms permits the atoms to overcome the activation energy barrier, hence easily move to new sites Figure 6

14. Continue At low temperatures—often below about 0.4 times the absolute melting temperature of the material—diffusion is very slow and may not be significant Time: Diffusion requires time, if a large number of atoms must diffuse to produce a uniform structure, long times may be required, even at low temperatures Times for heat treatments may be reduced by using H igher temperatures By making the diffusion distances as small as possible Diffusion form nonequilibrium structures and provide the basis for sophisticated heat treatments

15. Dependence on Bonding and Crystal Structure A number of factors influence the activation energy for diffusion and, hence, the rate of diffusion Activation energies are usually lower for atoms diffusing through open crystal structures than for close-packed crystal structures Activation energy depends on the strength of atomic bonding It is higher for diffusion of atoms in materials with a high melting temperature

16. Continue Due to their smaller size, cations (with a positive charge) often have higher diffusion coefficients than those for anions (with a negative charge) Dependence on Concentration of Diffusing Species The diffusion coefficient ( D ) depends not only on temperature, but also on the concentration of diffusing species and composition of the matrix

17. Nonsteady-State Diffusion When concentration varies with both distance and time Fick’s first law can no longer be used For simplicity let us consider the situation shown in Fig. 7a A concentration profile exists along one dimension ( x ) only The flux at any point along the x -axis will depend on the local value of and as shown in Fig. 7b   Figure 7

18. Continue In order to calculate how the concentration of B at any point varies with time consider a narrow slice of material with an area A and a thickness as shown in Fig. 7c The number of interstitial B atoms that diffuse into the slice across plane (1 ) in a small time interval will be The number of atoms that leave the thin slice during this time, however, is only Since the concentration of B atom within the slice will have increased by: ……………………….. (1)  

19. Continue Since is small, we can write: …………………. (2) In the limit as these equations give: ……………………. (3) Substituting Fick’s first law gives Fick’s second law: ……………………… (4) Assuming no variation of with concentration this equation can be written as: …………………………… (5)  

20. Continue Graphical presentation of Eq. 5 is shown in Fig. 8, which represents concentration profiles Fig . 8a has a positive curvature everywhere and the concentration at all points on such a curve will increase with time ( positive) When the curvature is negative as in Fig. 8b decreases with time ( negative)   Figure 8

21. Homogenization If varies sinusoidally with distance in one dimension as shown in Fig. 9 In this case B atoms diffuse down the concentration gradients, and regions with negative curvature, such as between and , decrease in concentration while regions between and increase in concentration The curvature is zero at   Figure 9

22. Continue At time the concentration profile is given by: …………………… (6) Where is the mean composition and is the amplitude of the initial concentration profile Assuming the to be independent of concentration gradient equation on can be solved: …………………….. (7) Where is the relaxation time and is given by: …………………..... (8) ……………………….. (9)  

23. The Carburization of Steel The concentration profiles that are obtained after different times are shown in Fig. 10 An analytical expression for these profiles can be obtained by solving Fick's second law using the boundary conditions: and The specimen is considered to be infinitely long An approximate to Eq. 5 can be obtained as: ……… (10)   Figure 10

24. Continue Where is the error function and is given by: Where y is called the argument of the error function   Figure 11

25. Continue The function is shown graphically in Fig. 12a . Note that since erf The depth at which the carbon concentration is midway between and It is given by Hence   Figure 12