Diffusion Mass transport by atomic motion Molecules intermingle as a result of their kinetic energy of random motion Mechanisms Gases Liquids random Brownian motion Solids vacancy diffusion or interstitial diffusion 1 Diffusion Interdiffusion In an alloy atoms tend to migrate from regions of high conc to regions of low conc Initially After some time Adapted from Figs 5 1 and 5 2 Callister Rethwisch 8e 2 Diffusion Self diffusion In an elemental solid atoms also migrate Label some atoms C A D B After some time C D A B 3 Diffusion Mechanisms Vacancy Diffusion atoms exchange with vacancies applies to substitutional impurities atoms rate depends on number of vacancies activation energy to exchange increasing elapsed time 4 Diffusion Mechanisms Interstitial diffusion smaller atoms can diffuse between atoms More rapid than vacancy diffusion 5 Processing Using Diffusion Case Hardening Diffuse carbon atoms into the host iron atoms at the surface Example of interstitial diffusion is a case hardened gear Result The presence of C atoms makes iron steel harder 6 Processing Using Diffusion Doping silicon with phosphorus for n type semiconductors 0 5 mm Process 1 Deposit P rich layers on surface magnified image of a computer chip silicon 2 Heat it 3 Result Doped semiconductor regions silicon light regions Si atoms light regions Al atoms 7 Diffusion How do we quantify the amount or rate of diffusion moles or mass diffusing mol kg J Flux or 2 surface area time cm s m2s Measured empirically Make thin film membrane of known surface area Impose concentration gradient Measure how fast atoms or molecules diffuse through the membrane M l dM J At A dt M mass diffused J slope time 8 Steady State Diffusion Rate of diffusion independent of time dC Flux proportional to concentration gradient dx Fick s first law of diffusion C1 C1 C2 x1 x C2 dC J D dx x2 dC C C2 C1 if linear dx x x2 x1 D diffusion coefficient 9 Example Chemical Protective Clothing CPC Methylene chloride is a common ingredient of paint removers Besides being an irritant it also may be absorbed through skin When using this paint remover protective gloves should be worn If butyl rubber gloves 0 04 cm thick are used what is the diffusive flux of methylene chloride through the glove Data diffusion coefficient in butyl rubber D 110x10 8 cm2 s surface concentrations C 0 44 g cm3 1 C2 0 02 g cm3 10 Example cont Solution assuming linear conc gradient glove C1 2 tb 6D paint remover skin Data D 110 x 10 8 cm2 s C1 0 44 g cm3 C2 0 02 g cm3 x2 x1 0 04 cm C2 x1 x2 8 J 110 x 10 dC C2 C1 J D D dx x2 x1 0 02 g cm3 0 44 g cm3 g cm s 1 16 x 10 5 0 04 cm cm2s 2 11 Diffusion and Temperature Diffusion coefficient increases with increasing T Qd D Do exp RT D diffusion coefficient m2 s Do pre exponential m2 s Qd activation energy J mol or eV atom R gas constant 8 314 J mol K T absolute temperature K 12 Diffusion and Temperature 300 600 1000 10 8 1500 D has exponential dependence on T D m2 s T C Dinterstitial Dsubstitutional C in a Fe C in g Fe 10 14 10 20 0 5 1 0 1 5 Al in Al Fe in a Fe Fe in g Fe 1000 K T 13 Example At 300 C the diffusion coefficient and activation energy for Cu in Si are D 300 C 7 8 x 10 11 m2 s Qd 41 5 kJ mol What is the diffusion coefficient at 350 C transform data D Temp T ln D 1 T 1 Qd and lnD1 lnD0 R T2 Qd 1 1 D2 lnD2 lnD1 ln D1 R T2 T1 Qd lnD2 lnD0 R 1 T1 14 Example cont Qd D2 D1 exp R 1 1 T2 T1 T1 273 300 573 K T2 273 350 623 K D2 7 8 x 10 11 41 500 J mol 1 1 m s exp 8 314 J mol K 623 K 573 K 2 D2 15 7 x 10 11 m2 s 15 Non steady State Diffusion The concentration of diffusing species is a function of both time and position C C x t In this case Fick s Second Law is used Fick s Second Law C 2C D 2 t x 16 Non steady State Diffusion Copper diffuses into a bar of aluminum Surface conc Cs of Cu atoms bar pre existing conc Co of copper atoms Cs Adapted from Fig 5 5 Callister Rethwisch 8e B C at t 0 C Co for 0 x at t 0 C CS for x 0 constant surface conc C Co for x 17 Solution C x t Co x 1 erf Cs Co 2 Dt C x t Conc at point x at time t erf z error function 2 z y 2 e dy 0 erf z values are given in Table 5 1 CS C x t Co 18 Non steady State Diffusion Sample 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 use Eqn 5 5 C x t Co x 1 erf Cs Co 2 Dt 19 Solution cont t 49 5 h Cx 0 35 wt Co 0 20 wt C x t Co x 1 erf Cs Co 2 Dt x 4 x 10 3 m Cs 1 0 wt C x t Co 0 35 0 20 x 1 erf 1 erf z Cs Co 1 0 0 20 2 Dt erf z 0 8125 20 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 follows z erf z 0 90 z 0 95 0 7970 0 8125 0 8209 Now solve for D z 0 90 0 8125 0 7970 0 95 0 90 0 8209 0 7970 z 0 93 x z 2 Dt D x2 4z 2t 3 2 x2 4 x 10 m 1h D 2 6 x 10 11 m2 s 4z 2t 4 0 93 2 49 5 h 3600 s 21 Solution cont To solve for the temperature at which D has the above value we use a rearranged form of Equation 5 9a Qd T R lnDo lnD from Table 5 2 for diffusion of C in FCC Fe Do 2 3 x 10 5 m2 s Qd 148 000 J mol T 148 000 J mol 8 314 J mol K ln 2 3x10 5 m2 s ln 2 6x10 11 m2 s T 1300 K 1027 C 22 Example Chemical Protective Clothing CPC Methylene chloride is a …
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