Phase Transformation Lecture 3

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Materials Engineering MM-501 Phase Transformation in Solids

MM-501 Phase Transformation in Solids

Fall Semester-2015

Lecture No: 03Diffusion : How do atoms move through solids?


What is diffusion?Material/Mass transport by atomic motion is called Diffusion. ORIt is a transport phenomenon caused by the motion of chemical species (molecules, atoms or ions), heat or similar properties of a medium (gas, liquid or solid) as a consequence of concentration (or, strictly, chemical potential) differences. In general, the species move from high concentration areas to low concentrations areas until uniform concentration is achieved in the medium.3


Diffusion is easy in liquids and gases where atoms are relatively free to move around:In solids, atoms are not fixed at its position but constantly moves (oscillates) . So, Diffusion is difficult in solids due to bonding and requires, most of the time, external energy to mobilize the atoms.

5Mass transport can generally involve:

fluid flow dominant in gases or liquids viscous flow flow of a viscous material, generally amorphous or semi crystalline (e.g. glasses and polymers) due to the forces acting on it at that moment;atomic diffusion principal mechanism in solids and in static liquids (as occurs in solidification).

Atomic diffusion occurs during important processes such as:solidification of materials .precipitation hardening, e.g. Al-Cu alloys annealing of metals to reduce excess vacancies & dislocations formed during workingmanufacture of doped silicon, e.g. as used in many electronic devices

For an active diffusion to occur, the temperature should be high enough to overcome energy barriers to atomic motionfor atom to jump into a vacancy site, it needs enough energy (thermal energy) to break the bonds and squeeze through its neighbors and take the new position. The energy necessary for motion is Em called the activation energy for vacancy motion.At activation energy Em has to be supplied to the atom so that it could break inter-atomic bonds and to move into the new position.


Figure: Schematic representation of the diffusion of an atom from its original position into a vacant lattice siteInhomogeneous materials can become homogeneous by diffusion.







12Diffusion MechanismsHow do atoms move between atomic sites?For diffusion to occur:Adjacent site needs to be empty (vacancy or interstitial).Sufficient energy must be available to break bonds and overcome lattice distortion.There are many diffusion mechanism to be observed but two possible mechanisms are considered:Vacancy diffusion.Interstitial diffusion.


131. Substitutional DiffusionDirect ExchangeRing Vacancy 2. Interstitial Diffusion


15 Vacancy Mechanism Atoms can move from one site to another if there is sufficient energy present for the atoms to overcome a local activation energy barrier and if there are vacancies present for the atoms to move into. The activation energy for diffusion is the sum of the energy required to form a vacancy and the energy to move the vacancy.



Vacancy diffusion- An atom adjacent to a vacant lattice site moves into it.

Essentially looks like an interstitial atom: lattice distortionFirst, bonds with the neighboring atoms need to be broken

From Callister 6e resource CD.To jump from lattice site to lattice site, atoms need energy to break bonds with neighbors, and to cause the necessary lattice distortions during jump. This energy comes from the thermal energy of atomic vibrations (Eav ~ o CT)Materials flow (the atom) is opposite the vacancy flow direction.



Interstitial atoms like hydrogen, helium, carbon, nitrogen, etc) must squeeze through openings between interstitial sites to diffuse around in a crystal.

The activation energy for diffusion is the energy required for these atoms to squeeze through the small openings between the host lattice atoms.

Interstitial Mechanism

18Interstitial DiffusionMigration from one interstitial site to another (mostly for small atoms that can be interstitial impurities: (e.g. H, C, N, and O) to fit into interstices in host.

Carbon atom in FerriteInterstitial diffusion is generally faster than vacancy diffusion because bonding of interstitials to the surrounding atoms is normally weaker and there are many more interstitial sites than vacancy sites to jump to.


Interstitial Diffusion-Animation19

20How do we quantify the amount or rate of diffusion? Flux (J): No of atom s diffusing through unit area per unit time OR Materials diffusion through unit area per unit time.

Measured empiricallyMake thin film (membrane) of known surface areaImpose concentration gradientMeasure how fast atoms or molecules diffuse through the membrane

M =massdiffusedtimeJ slope


21Temperature Dependence of the Diffusion Coefficient : Diffusion coefficient increases with increasing T.

= pre-exponential [m2/s]= diffusion coefficient [m2/s]= activation energy [J/mol or eV/atom] = gas constant [8.314 J/mol-K]= absolute temperature [K]DDoQdRTWith conc. gradient fixed, higher D means higher flux of mass transport.

22 Diffusivity increases with T. Experimental Data:

D has exp. dependence on TRecall: Vacancy does also!

Diffusion and Temperature

Note:pre-exponential [m2/s]activation energygas constant [8.31J/mol-K]




23Steady-State Diffusion

Ficks first law of diffusion




x1x2D diffusion coefficient(be careful of its unit)Rate of diffusion independent of timeFlux proportional to concentration gradient =


24Diffusivity -- depends on:1. Diffusion mechanism. Substitutional vs interstitial.2. Temperature. 3. Type of crystal structure of the host lattice. 4. Type of crystal imperfections. (a) Diffusion takes place faster along grain boundaries than elsewhere in a crystal. (b) Diffusion is faster along dislocation lines than through bulk crystal. (c) Excess vacancies will enhance diffusion.5. Concentration of diffusing species.

Microstructural Effect on Diffusion:If a material contains grains, the grains will act as diffusion pathways, along which diffusion is faster than in the bulk material.25

26Physical Aspect of DD is the indicator of how fast atom moves.In liquid state, D reaches similar level regardless of structure.In solid state, D shows high sensitivity to temperature and structure. Absolute temperature and Tm are what we should care about.


Example: At 300C the diffusion coefficient and activation energy for Cu in Si are

D(300C) = 7.8 x 10-11 m2/sQd = 41.5 kJ/mol

What is the diffusion coefficient at 350C?

transform data

DTemp = T

ln D1/T


28Example (cont.)

T1 = 273 + 300 = 573 K

T2 = 273 + 350 = 623 K

D2 = 15.7 x 10-11 m2/s


29 Steel plate at 7000C with geometry shown: Q: In steady-state, how much carbon transfers from the rich to the deficient side?Adapted from Fig. 5.4, Callister 6e.

Example: Steady-state Diffusion

Knowns: C1= 1.2 kg/m3 at 5mm (5 x 103 m) below surface.

C2 = 0.8 kg/m3 at 10mm (1 x 102 m) below surface.

D = 3 x10-11 m2/s at 700 C.

700 C

30 Concentration profile,C(x), changes with time.14 To conserve matter: Fick's First Law: Governing Eqn.:

Non-Steady-State DiffusionIn most real situations the concentration profile and the concentration gradient are changing with time. The changes of the concentration profile is given in this case by a differential equation, Ficks second law.Called Ficks second law


Fick's Second Law of Diffusion

In words, the rate of change of composition at position x with time, t, is equal to the rate of change of the product of the diffusivity, D, times the rate of change of the concentration gradient, dCx/dx, with respect to distance, x.

Non-Steady-State Diffusion

32 Copper diffuses into a bar of aluminum.15 General solution:"error function"Values calibrated in Table 5.1, Callister 6e.

Adapted from Fig. 5.5, Callister 6e.Example: Non Steady-State Diffusiont3>t2>t1Fig. 6.5: Concentration profiles nonsteady-state diffusion taken at three different timesC0=Before diffusionFor t=0, C=C0 at 0x

33Non-steady State DiffusionSample Problem: An FCC iron-carbon alloy initially containing 0.20 wt% C is carburized at an elevated temperature and in an atmosphere that gives a surface carbon concentration constant at 1.0 wt%. If after 49.5 h the concentration of carbon is 0.35 wt% at a position 4.0 mm below the surface, determine the temperature at which the treatment was carried out.

Solution tip: use Eqn.



Solution (cont.):

t = 49.5 h x = 4 x 10-3 mCx = 0.35 wt%Cs = 1.0 wt%Co = 0.20 wt%

erf(z) = 0.8125



Solution (cont.):We must now determine from Table 5.1 the value of z for which the error function is 0.8125. An interpolation is necessary as followszerf(z)0.900.7970z0.81250.950.8209

z = 0.93Now solve for D


To solve for the temperature at which D has above value, we use a rearranged form of Equation (5.9a);

from Table 5.2, for diffusion of C in FCC Fe Do = 2.3 x 10-5 m2/s Qd = 148,000 J/mol

Solution (cont.):T = 1300 K = 1027C



WhereD is the Diffusivity or Diffusion Coefficient ( m2 / sec )Do is the prexponential factor ( m2 /