NON-STEADY STATE DIFFUSION Non-steady state: The concentration profile changes with time. C = C (x,t) To conserve energy: dJ/dX =[J(right) – J(left)] /dx = -C/ t Fick’s first law: J = -D(C/x) dJ/ dx = - / x [D(C/x)] Substitute for dJ/dX: C/ t = / x [D(C/x)] Fick’s second law If D is independent of concentration: C/ t = D(2C/x2)
Solution of Fick’s Second Law "error function" Values calibrated in Table 5.1, Callister 6e. Cs = concentration on the surface C0 = initial concentration
Application: Case Hardening Carburizing of Steels: - Diffuse carbon atoms into the surface of a gear made of a low carbon steel (Steel is an alloy of iron and carbon). - The concentration of carbon on the surface increases and consequently the surface hardness of the gear is increases. - The high hardness reduces the wear and improves the resistance of the gear to fracture under cyclic loading (fatigue resistance). Fig. 5.0, Callister 6e. (Fig. 5.0 is courtesy of Surface Division, Midland-Ross.) The carbon is introduced at high temperatures to the surface by controlling the relative partial pressure of Co and CO2 in the carburizing gas.
EFFECT OF TEMPERATURE æ ö Q = ç d ÷ diffusivity D D exp - o è ø R T 2 pre-exponential [m /s] (see Table 5.2, Callister 6e ) activation energy æ ö Q [J/mol],[eV/mol] = ç d ÷ diffusivity D D exp - (see Table 5.2, Callister 6e ) o è ø R T gas constant [8.31J/mol-K] 1000K/T D (m 2 /s) C in a -Fe g Al in Al Cu in Cu Zn in Cu Fe in 0.5 1.0 1.5 2.0 10 -20 - 14 -8 T(C) 1500 1000 6 00 3 Adapted from Fig. 5.7, Callister 6e. (Date for Fig. 5.7 taken from E.A. Brandes and G.B. Brook (Ed.) Smithells Metals Reference Book, 7th ed., Butterworth-Heinemann, Oxford, 1992.)
EFFECT OF CRYSTAL STRUCTURE AND MELTING POINT Crystal Structure: Diffusion is faster in more open structures. Examples: Self diffusion is faster in a-Fe than it is in g-Fe. Carbon moves faster in a-Fe than it does in g-Fe. Melting Point: Diffusivity decreases as the melting point is increased. Example: DAl>DCu>Dg-Fe at a given temperature.