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Unformatted text preview: Diffusion
Chapter 5 Why study diffusion? Diffusion of atoms is necessary to have better bonding in lot of different types of material processing. Example: aircraft landing gears and gold plated jewelry. In gold plated jewelry, Au atoms have to diffuse through the atoms of the plate to make a good bonding. Why study diffusion? The surface of the landing gears are made by carburizing (diffusing C atoms) or nitriding (diffusing N atoms) steel to make them harder to have better fatigue properties and prevent failure. Learning Objectives Fick's first and second laws of diffusion. Solution of Fick's second law of diffusion concentration at a particular distance at a particular time. Calculate diffusion coefficient of materials. Basics of Diffusion In general, there a several things to know about diffusion. One, diffusion simply stated, is the movement or migration of atoms with metals. Two, the speed or rate at which diffusion takes place increases exponentially with increases in temperature. Basics of Diffusion cont. Third, the rate of diffusion for a given element in another depends greatly upon many factors and varies on a case by case basis which must be discovered experimentally. Diffusion Couple In Figures 5.1 and 5.2 a piece of copper has been coupled to a piece of nickel, and will then be heated to observe the diffusion which takes place. A diffusion couple is made by polishing the joining surfaces of the two pieces until they are plane within a couple of atomic distances. Next, if the two halves are placed together with some pressure and some sliding friction, metallic bonds will form between the two pieces, holding them together. Fig 5.1 (a) A coppernickel diffusion couple before a high temperature heat treatment. Fig 5.1 (b) Schematic representations of Cu (Colored circles) and Ni (gray circles) atom locations within the diffusion couple. Fig 5.1 (c) Concentrations of copper and nickel as a function of position across the couple. Fig 5.2 (a) A coppernickel diffusion couple after a heat treatment, showing the alloyed diffusion zone. Fig 5.2 (b) Schematic representations of Cu (colored circles) and Ni (gray circles) atom locations within the couple. Fig 5.2 (c) Concentrations of copper and nickel as a function of position across the couple. Diffusion Mechanism Diffusion is movement of atoms from one lattice site to another lattice site. Atoms can move by two different mechanisms. These are vacancy diffusion and interstitial diffusion. Diffusion Mechanism cont. In the vacancy diffusion mechanism the atom moves to the vacant site and thus the vacancy moves to the atomic site (Fig 5.3 (a)). In the interstitial diffusion mechanism, the atoms move from one interstitial position to another interstitial position. Fig 5.3 (a) Schematic representation of a vacancy diffusion. Fig 5.3 (b) Schematic representation of a interstitial diffusion Position of interstitial atom before diffusion Position of interstitial atom after diffusion Steady State Diffusion The diffusion flux (J = no. of atoms moving per unit area per unit time) is defined as: J = M/At = 1/A (M/t) Where M is the mass, A is the area across which the diffusion is occurring, and t is the elapsed time. In differential form, J is expressed as J = 1/A (dM/dt) If J is constant with time then it is a steadystate diffusion. Steady State Diffusion Concentration gradient = dC/dx = C / x = (CA CB) / (xA xB) Diffusion flux in terms of Concentration gradient J = D dC / dx This is called the Fick's First Law of Diffusion Where D is a constant of proportionality called the "diffusion coefficient" (constant but the value depends on the type of material). Steady State Diffusion cont. Fig 6.1 shows CuNi plates tied together and heated (or you can imagine gold aluminum plates tied together and heated for making jewelry.) . Cu has high concentration on the left side and zero concentration on the right side at the beginning. After some time the concentration profile changes because Cu atom diffuses into Ni plate. Steady State Diffusion cont. Fig 5.4 shows diffusion of gas particles through a thin metal plate from a high pressure, high concentration region to low pressure, low concentration region. Concentration gradient of gas particles is: (CA CB) / (XA XB) Since Fick's first law of diffusion states that J is proportional to the opposite of the concentration gradient, then the steady state diffusion J is: J = D (CA CB) / (XA XB) Fig 5.4 (a) Steadystate diffusion across a thin plate (b) A linear concentration profile for the diffusion situation NonSteady State Diffusion When the concentration gradient changes with time, as in Figure 6.5, then Fick's Second Law of Diffusion applies, which states that: C / t = (J) /x Or, that the change of the concentration gradient with time is equal to the change of the flux with distance. Which reduces to the form: C / t = D 2C /x2 NonSteady State Diffusion cont.
The solution of that equation is (Eqn. 5.5): ( Cx Co) / (Cs Co)= 1 erf ( x / 2Dt ) Where: Cs = the concentration at the surface Cx = concentration at a distance x at time t Co = the initial concentration erf( x / 2Dt ) is an error function which must Error Functions Fig 5.5 Concentration profiles for nonsteady state diffusion taken at three different times t1 , t2 and t3. X Fig 5.6 Concentration profile for nonsteady state diffusion. Effect of Temperature on Diffusion
The coefficient of diffusion for a given diffusion set varies with temperature according to the following relationship: D = D0 exp (Qd / RT) (Eqn. 5.8) Where: D0= a temperature independent preexponential Qd= the activation energy for diffusion of the given set R = the gas constant T= absolute temperature Diffusion Data Effect of Temperature on Diffusion Taking the natural log of both sides gives: ln D = ln D0 (Qd / R) ( 1/T ) Since D0, Qd, and R are all constants, then the plot of ln D vs. 1/T will be a straight line. Reference Figures 5.7 & 5.8. Fig 5.7 Plot of the logarithm of the diffusion coefficient versus the reciprocal of absolute temperature for several metals. Fig 5.8 Plot of the logarithm of the diffusion coefficient versus the reciprocal of absolute temperature for the diffusion of copper in gold. Group Discussion Problems Group Discussion Problems: 5.8, 5.18 & 5.30 Homework HW #4 5.6, 5.11, 5.13, 5.21, 5.D4 Due 2/28/05 ...
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- Summer '10