MAT_101_Winter08_Lecture_12

MAT_101_Winter08_Lecture_12 - Modeling Diffusion: Flux Flux...

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Modeling Diffusion: Flux Flux (#/area/time) : J = 1 A dM dt kg m 2 s or atoms m 2 s • Directional Quantity • Flux can be measured for: --vacancies and interstitials --host (A) atoms --impurity (B) atoms J x J y J z x y z x-direction Unit area A through which atoms move.
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Non Steady State Diffusion • Concentration profile, C(x), changes w/ time. • To conserve matter: • Fick's First Law: • Governing Eqn.: Concentration, C, in the box J (right) J (left) dx d C dt = D d 2 C dx 2 dx = d C dt J =− D d C dx or J (left) J (right) d J dx = d C dt d J dx = D d 2 C dx 2 (if D does not vary with x) equate
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Ex: Non Steady State Diffusion • Copper diffuses into a bar of aluminum. • General solution: "error function" Values calibrated in Table 5.1, Callister 6e. C( x , t ) C o C s C o = 1 erf x 2 D t pre-existing conc., C o of copper atoms Surface conc., C s of Cu atoms bar C o C s position, x C( x , t ) t o t 1 t 2 t 3 Adapted from Fig. 5.5, Callister 6e .
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Diffusion and Temperature x i Dt i Make simple estimates for ‘diffusion lengths’ x i 1 sec 100 sec (1.667 min) 10,000 sec (2.7 hrs) 100,000 sec (1.15 days) 1,000,000 sec (11.6 days)
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Summary: Structure & Diffusion Diffusion FASTER for. .. • open crystal structures • lower melting T materials • materials w/secondary bonding • smaller diffusing atoms • cations • lower density materials Diffusion SLOWER for. .. • close-packed structures • higher melting T materials • materials w/covalent bonding • larger diffusing atoms • anions • higher density materials
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Example of Diffusion Studies: Using Secondary Ion Mass Spectroscopy Measure secondary ions in mass spectrometer Incident sputter ions e.g. O 2+ , Cs + , Ar + , Ga + Breaks bonds Sputters sample Measure: Secondary ions
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Example: SIMS Study of Self-Diffusion Natural GaSb 69 Ga 121 Sb 71 Ga 123 Sb As-grown Annealed 105 min Annealed 18 days Natural isotope abundance Ga: 69 Ga (60.1%) and 71 Ga (39.9%) Sb: 121 Sb (57.4%) and 123 Sb (42.6%) Which diffuses faster: Ga or Sb? Why?
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Example: SIMS Study of Self-Diffusion d C dt = D d 2 C dx 2 Solve diffusion equation Fit expt. C(t,x) to solution Determine D D(T) activation energy
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Chapter 9: Phase Diagrams ISSUES TO ADDRESS. .. • When we combine two elements or compounds . .. what equilibrium state do we get? • In particular, if we specify. .. --a composition (e.g., wt%Cu - wt%Ni), and --a temperature (T) then. .. How many phases do we get? What is the composition of each phase? How much of each phase do we get? Phase B Phase A Nickel atom Copper atom
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The Solubility Limit Solubility Limit : Max concentration for which only a solution occurs. • Ex: Phase Diagram: Water-Sugar System Question: What is the solubility limit at 20 °C ? Answer: 65 wt% sugar . If C o < 65 wt% sugar: sugar If C o > 65 wt% sugar: syrup + sugar. • Solubility limit increases with T: e.g., if T = 100 °C, solubility limit = 80wt% sugar.
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MAT_101_Winter08_Lecture_12 - Modeling Diffusion: Flux Flux...

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