# Introduction to Materials Science and Engineering Diffusion Diffusion means atoms moving and changing places. This happens in solids and liquids, exactly in the same way that an unpleasant smell moves from one part of a room to another even without wind, then dissipates after some time. The driving force for diffusion is the chemical potential gradient. Partial Molar Free Energy=Chemical Potential Gi i P1 i i Po P Ficks First Law atomic flux is proportional to the chemical potential gradient

C d d D Ji Di i kT d x d x Ji forsolution an ideal solution For an ideal i o kTln Ci d D d C Ji into Dthe derivative termi with only composition varying with x i Substitution dx dx Ji

d C i dx D For diffusion due to a pressure gradient (as in sintering) J D kT d P dx C Preview: In steel, strength goes up but toughness goes down with C. Design application: Carburization of a transmission gear. Attributes needed? How to do it.

Attributes needed? How to do it. Carburizing 2 CO CO2 + C (in solution) Carbon diffuses from the surface Into the steel Mechanism of diffusion in solids Diffusion of an individual atom is random and probabilistic. Thus, the interstitial black atom has an equal probability of moving up, down, left or right. Mechanism of diffusion in solids If we have a concentration gradient, overall atomic movement is not random, even though individual atom movement is. In this example, since some of the black atoms will move to

the right on average, the concentration of black atoms will increase on the right and decrease on the left. This is similar to the dissipation of the concentrated unpleasant odor in the room. Mechanisms of diffusion in solids At equilibrium, there will be a random distribution of black atoms, and no concentration gradient. A common heat treatment for cast metal alloys is homogenization, or heating for a long time at high temperature to allow the chemical segregation to even out in this way. Mechanism of diffusion in solids Diffusion Couples Concentration gradient = dC/dx is the slope of the curves. C Eventually Cu, Ni

x Mass transport, or Flux Flux is a measurement of the number of atoms per unit area that cross a particular plane per unit time. M = mass, A = area, t = time Ficks first law We saw that the concentration gradient affects the direction of diffusion. It also affects the rate of diffusion. For simplicity let us assume the concentration gradient is a constant, then we can say: where D is the diffusivity, which depends on temperature and which atom is diffusing in which material. This says that the mass transport is directly proportional to the concentration gradient. Steady state diffusion dC/dx = concentration gradient = slope = constant = C/x = (CA-CB)/(xA-xB) ALE

What affects diffusion rates? Diffusion is a thermally-activated process. This means that it is accelerated by temperature. Using a higher temperature will result in more diffusion in a given time if there is a composition gradient present (could accomplish the same effect by prolonging the time to diffuse at a lower temperature) Time: straightforward result of Ficks First Law: J is mass/(time x area) or atoms/(time xarea). Total mass diffusing across a plane of area A is just J x t, if J is constant. Otherwise M = Jdt What affects diffusion rates? Temperature: Comes in mainly through the diffusivity, D: ALE: Take the natural log of both sides of Eqn 6.8, and construct a plot that would be useful for looking at D(T) ALE - Answer Qd ln D ln( Do ) RT Qd 1 ln D ln( Do )

R T y = m x + b Nonsteady-State Diffusion Most practical diffusion situations are nonsteady-state. That is, the concentration gradient = f(time), i.e. dC/dx changes. Gas Gear surface Gear interior

Nonsteady-State diffusion To understand these problems, we need to solve a differential equation called Ficks Second Law: It is clear that the concentration gradient can vary with time through the term dC/dt. We are not going to get into this level of detail in MY2100. 21 Ci The solution to the PDE is obtained by conversion to an ODE. 10 2 18 m D 10 7 x 10

3 m m ( xm) 2 x C( x t ) exp 4 D t 2 Dt Ci Define ( x t ) x 2 Dt

7 x 10 8 9.910 Use the Chain Rule 22 3 10 0 1 C x 10 s 2 22 C x 10 s 1 10 C x 10 s 2 1022 d D d d d C d

d dt d dx d d x d C d 2 C d d 2 d 2 0 C 0 x

7 10 s 21 Ci 10 2 18 m D 10 7 x 10 3 m

m ( xm) 2 x C( x t) exp 4 D t 2 Dt Ci Dopant Rich Layer, Ci , x 8 6 x 0 10 10

0 5 10 x 22 1 10 Semiconductor 1 2 C x 10 s5 1021 3 C x 10 s C x 10 s 0 8 s

We can add up the solutions from thin films displaced throughout the volume Ci 2 xm 10 8m C1( x t) exp x 4Dt 2 Dt 2 Ci

Ci 2 xm ( 10) 83m C3( x t) exp x 4Dt 2 Dt 2 xm ( 10) 82m C2( x t) exp x 4Dt 2 Dt 4

3 22 1 10 1 C x 10 s 2 C x 10 s 3 10 21

21 2 2 10 2 C2 x 10 s 21 1 10 2 C3 x 10 s C1 x 10 s 1 21 5 10 1 C2 x 10 s 1 C3 x 10 s C1 x 10 s 0

0 0 x 0 x C( x t) x C surface C initial C surface erf 0.995 2 Dt 1 erf ( )

0 0.995 1 2 0 2 2 2 2 COCO2 +C(Fe) C( x t ) C surface C initial C surface

C( x t ) Csurface Cinitial Csurface 2at% C at surface 0.2at% initially in low carbon steel D=10^(-12) m^2/s x erf 2 Dt erf What is the time required to get 1at% at 10^-4 m? x

2 Dt erf ( 0.54) 0.555 1 1 2 0.2 2 0.556 erf ( ) 4 0 0.54 10 12 110 1

2 1 0 1 0.54 2 34293 3600 t 9.526 hr