DIFFUSION how atoms move around in crystalline materials 1 HeaIng to 0 50Tm results in solid state di usion of atoms Tm Cu 1085 C Tm Ni 1453 C 2 IntersIIal Di usion movement of atoms via intersIIal sites Occurs when the di using species is much smaller than the host e g C r 0 071 nm di usion through iron r 0 124 nm 3 Vacancy Di usion movement of atoms via vacant laSce sites Occurs when the di usion species is of similar size to the host e g di usion of Cu r 0 128 nm through Ni r 0 125 nm 4 Vacancies empty laSce sites Qv N v N exp kT Nv of vacancies N total of laSce sites Qv acIvaIon energy for vacancy formaIon k Boltzmann s constant 1 38 x 10 23 joules atom K The vacancy concentraIon Nv N depends on temperature vacancy formaIon is thermally acIvated 5 AcIvaIon Energy a thermal barrier Lower Temperature means low energy staIsIcally fewer atoms make it over the hill Higher Temperature means higher energy staIsIcally more atoms make it over the hill Smaller hill lower Q means it is easier to get over the barrier at a Given temperature 6 Rather than thinking of the acIvaIon energy as a energy barrier consider it as a way of describing the temperature dependence of a thermally acIvated process High Q means great temperature dependence low Q means low dependence Qv N v N exp kT ln Nv 1 vs N T slope Q K 7 Vacancies in Ionic Compounds must maintain charge neutrality Schojky Defect CombinaIon of a caIon vacancy and an anion vacancy e g a Cl vacancy and a Na vacancy in NaCl Frenkel Defect CombinaIon of a caIon vacancy and a caIon intersIIal e g two O2 vacancies and a Zr4 intersIIal in ZrO2 8 Exhaust Steady State Di usion Sensor Distance The oxygen sensor in many cars has a close end tube of a ZrO2 Y2O3 alloy inserted in the exhaust Air pumped into the tube maintains a high oxygen concentraIon but the exhaust side has very lijle oxygen The concentraIon gradient causes oxygen ions to di use from the oxygen rich to the oxygen poor side of the tube 9 While the engine is running we have a steady state situaIon because the the movement of oxygen ions does not cause a change in the oxygen concentraIon on either side of the close end tube Steady state di usion of oxygen can be described using Fick s First Law of di usion dc J D dx dc dx J is the ux units of mass area Ime g m2 s D is the Di usivity units m2 s describing the degree of atomic moIon The concentraIon gradient concentration dis tan ce When the engine is running dc dx for oxygen stays the same 10 Flux refers to the net moIon of atoms For example in the parking garage during the middle of the day the number of cars entering and leaving are about the same Lots of movement but the net result is lijle change 11 dc J D dx ConcentraIon gradient mass tends to move from higher to lower concentraIon Di usivity is a measure of how rapidly the system responds to a concentraIon gradient no movement no ux Oxygen Sensor Again 12 Non steady state di usion the concentraIon gradient changes with Ime Think of the interdi usion of nickel and copper Even when the concentraIon gradient is zero there can sIll be di usion e g D 0 just no net movement J 0 13 Fick s Second Law 2 c c D 2 t x Fick s second law can be used to account for changes in the concentraIon gradient with respect to Ime The soluIon to this di erenIal equaIon depends on the imposed boundary condiIons that re ect a given situaIon 14 The Semi in nite SoluIon Carburizing a Gear The surface of low carbon steel surfaces can be hardened by placing them in a carbon rich atmosphere at elevated temperature The concentraIon of carbon at the surface stays the same while the concentraIon of carbon below the gear surface increases with Ime Depth Below Gear Surface 15 The boundary condiIons that describe this situaIon are For t 0 C Co at 0 x For t 0 C Cs at x 0 C Co at x C0 SoluIon x C x Co 1 erf 2 Dt Cs C o x di usion distance D Di usivity m2 s t Ime C0 iniIal bulk concentraIon Cs concentraIon at the surface Cx concentraIon at distance x 16 What is a di usion distance It looks like there are many di usion distances You need to specify the desired concentraIon of your di using species then you can calculate the depth below the surface where the desired concentraIon is found 17 For example let s look for the distance x below the surface where Cx 0 5 Cs when Co 0 The distance x below the surface where Cx 0 5 Cs when Co 0 is Dt 18 19 There are many soluIons to the di erenIal equaIon In this case the surface concentraIon is nite like a lm of material deposited on a substrate and the lm s contents di uses into the substrate drop of water soaked up by a sponge 20 Factors E ecIng Di usivity Size of di using species IntersIIal vs Vacancy Di usion Size of host atoms Crystal Structure Temperature rule of thumb di usivity becomes signi cant when T 0 5Tm in K 21 slope is proporIonal to the acIvaIon energy 22 D D0exp Q kT 23 A Few Comments on Trends Carbon di uses through iron more rapidly than iron di uses through iron Makes sense carbon can di use intersIIally while iron requires vacancies to di use Carbon di uses more rapidly in BCC iron then FCC iron lower acIvaIon energy Makes sense BCC has a lower atomic packing factor Along similar lines hydrogen has a lower acIvaIon energy for di usion through BCC iron than carbon 13 4 vs 84 1 kJ mole due to its smaller atomic radius 24 Fe Di using Through Fe Cu Through Cu What s with that Self di usion when I heat up a cube of copper the copper atoms di use through the copper laSce D 0 but J 0 because dc dx 0 How can you measure self di usion Tracer measurements using radioacIve isotopes 25 Back to RecrystallizaIon 26 Elevated temperature relaIve to Tm is required to increase di usivity What is special about the recrystallizaIon temperature Why can t it be higher or lower It can RecrystallizaIon will then take less or more Ime 27 The greater the amount of cold work the greater the driving force e g dc dx for recrystallizaIon Note that without some threshold amount of deformaIon recrystallizaIon does not occur at all 28
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