Module 4. Atomic mechanism of diffusion. Till now, we discussed the diffusion process without going to the atomic level.

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Isabella Gregory
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1 Module 4 Atomic mechanism of diffusion Till now, we discussed the diffusion process without going to the atomic level. By measuring concentration profile, one can measure the diffusion coefficients. However, to get further insights on the diffusion process, we need to understand the atomic mechanism. It is almost impossible to track the jump of any particular atom. However, based on the experimental results, we can use logical arguments to explain the process in the atomic level. The aim of this topic is to explain the possible mechanisms without going to extreme complications.

2 As we have seen before that all elements will have some impurities. C, O, H etc. are present in most of the metals in interstitial voids. Diffusion occurs by interstitial diffusion mechanism. Similarly vacancies are also always present. So substitutional diffusion is possible because of presence of vacancies. Let us first consider random jump of atoms that is diffusion without the presence of any driving force.

3 Atoms vibrate with Debye frequency at their positions with an average energy of 3kT. Because of violent oscillation, sometimes it can acquire enough energy to cross the activation barrier to jump to another position. In this context atoms can be considered as restless, aimless, brainless matter, which tries to jump even without any gain or purpose. In the case of interstitial diffusion, if the concentration of atoms is very less, atom can jump randomly to any other position, since as discussed before that most of the interstitial voids will be vacant. For example, as shown before, at maximum composition of C inα-fe, only one octahedral void is occupied out of 3000 voids. In substitutional diffusion, however, the diffusion process is rather complicated since one particular atom can exchange position on the condition that vacancy is available next to it. We have shown before that, on average, near the melting point, only one vacancy is available out of 000 atoms. So in general substitutional diffusion rate is much lower compared to the rate of interstitial diffusion. We shall discuss these two atomic mechanisms of diffusion one by one.

Interstitial diffusion Let us consider three adjacent planes of the host element, denoted as 0, and. These planes have many voids. Few voids are occupied by interstitial atoms.

4 Interstitial diffusion Let us consider three adjacent planes of the host element, denoted as 0, and. These planes have many voids. Few voids are occupied by interstitial atoms. Let us consider that concentration of interstitial atoms C o >C >C That means number of atoms decrease as we move from plane 0 to. The distance between different planes is, which could be related to the lattice parameter of the crystal depending on the orientation of the planes. Remember again that there is no driving force, like chemical and electrical potential gradient, present. Let us consider the jump of atoms between plane and

5 The flux of atoms from the plane to plane can be written as J r = nγ The flux of atoms from the plane to plane can be written as J s = nγ Γ : Jump frequency per second from plane Γ : Jump frequency from plane n and n are the number of atoms/area in plane and (n >n ). Factor ½ comes from the fact that only half of the atoms might jump from plane to and another half might jump from plane to plane 0. So the resultant flux from the plane to can be written as r r s J = J J = ( nγ nγ ) Now we consider random jump of atoms: Γ = Γ = Γ This is a fair assumption since the possibility of each atom to jump to any of the adjacent planes is equal. Note that we are considering here cubic crystal, where barrier for jump (because of presence of neighbouring host atoms) to any direction is the same.

6 Further, the concentration of atoms (mole/m 3 ) in different planes can be written as: c c dc dx = n = n So the flux of atoms from the plane to plane can be written as r = x ( c c ) Γ J Further the concentration gradient can be written as ( c c ) ( c c ) = = Since the distance between the planes is very small. So the flux of atoms from the plane to plane can be written as r J = Γ dc dx Comparing with Fick s first law, we can write D = Γ

7 We considered that the atoms can jump to only two directions. However, in three dimension atoms can jump to any of the nearest neighbor positions and we can write D = Z Γ where Z is the coordination number that is the number of nearest neighbor voids in this case. In FCC crystal, diffusion coefficient, in general, can be written as D = Γ Note that the activation barrier, as considered, for the jump to any point should be the same, since it is isotropic. If we consider the diffusion to a particular point only then we can write D = Γi i where Γ = ZΓi

8 Relation of jump frequency with the diffusion coefficient in a FCC lattice Now let us consider the diffusion in a FCC lattice. We assume that voids,,3 and 4 are present on plane 0, voids 5,6,7,8 and the interstitial atom are present on plane and voids 9,0,, are present on plane The interstitial atoms can jump to any of the positions, without any preference Diffusion coefficient should be considered from the jump of many atoms together. There will be many atoms present on plane (as previously we considered n ) So different atoms from plane can jump to different voids between to.

9 So the average diffusion coefficient can be written as D = Z Z i= D = Γ = i i= i x i In a FCC crystal: Γ i x i Let us consider the jump to x direction (both positive and negative) jump to the positions 5,6,7 and 8 does not make any effective jump, since = 0 Effective jump length to the positions,,3 and 4 is i = a / Effective jump length to the positions 9,0, and is i = +a / i x i Jump frequency Γ i = Γ since the activation energy for jump to any positions will be the same So diffusion coefficient can be written as D = 4 8 = Γi + Γi + i= i= 5 i= 9 6 Γa Γ i = a 4 Γ + 4Γ ( 0) a + 4Γ Note that it will be the same even if we consider y or z direction in this cubic lattice

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