1 Objective Bio ex. Microstructure-Properties: II The KJMA Equation Notation Assumptions Derivation Plots 27 -302 Lecture 5 Fall, 2002 Prof. A. D. Rollett
2 Materials Tetrahedron Processing Performance Objective Bio ex. Notation Assumptions Derivation Plots Microstructure Properties
3 Objective Bio ex. Notation Assumptions Derivation Plots • The objective of this lecture is to introduce the concept of phase transformation kinetics as described by the Kolmogorov-Johnson. Mehl-Avrami equation. • Part of the motivation for this lecture is to prepare the class for a Lab on the crystallization of glass-ceramics.
4 References • • • Objective Bio ex. Notation Assumptions • • Derivation • Plots • Phase transformations in metals and alloys, D. A. Porter, & K. E. Easterling, Chapman & Hall, 0 -412 -45030 -5, 669. 94 P 84 P 2: page 289. Kolmogorov, A. (1937). “A statistical theory for the recrystallization of metals. ” Akad. nauk SSSR, Izv. , Ser. Matem. 1: 355. Johnson, W. and R. Mehl (1939). “Reaction kinetics in processes of nucleation and growth. ” Trans AIME 135: 416. Avrami, M. (1939). “Kinetics of Phase Change. I: General Theory. ” J. Chem. Phys. 7: 1103. Avrami, M. (1940). “Kinetics of Phase Change. II: Transformation. Time relations for random distribution of nuclei. ” J. Chem. Phys. 8: 212. Avrami, M. (1941). “Kinetics of Phase Change. III: Granulation, Phase Change an Microstructures. ” J. Chem. Phys. 9: 177. Anderson, W. and R. Mehl (1945). “Recrystallization of Al in terms of the rate of nucleation and growth. ” Trans. AIME 161: 140.
5 Transformation Kinetics Objective Bio ex. Notation Assumptions Derivation Plots • The kinetics of transformation are typically described by a standard equation known as the Kolmogorov-Johnson-Mehl-Avrami equation, named after the individuals who derived it. • The characteristic of the kinetics is that of the “S-curve”, i. e. slow at first, then accelerating, then decelerating.
6 Transformation kinetics are universal Objective Bio ex. Notation Assumptions Derivation Plots • The kinetics of transformation are universal. • Consider this example of the kinetics of cell growth. • High-Throughput Assay System for the Discovery Of Anti-Bacterial Drugs 1 J. Bruce Pitner, 2 Mark R. Timmins, 2 Maurice Kashdan, 3 Mandar Nagar, and 3 David T. Stitt, Technologies 21 Davis Drive, Research Triangle Park, NC 27709, 2 BD Biosciences, Two Oak Park, Bedford, MA 01730, 3 BD Biosciences, 250 Schilling Circle, Cockeysville, MD 21030; Presented at AAPS - New Orleans, LA November, 1999 1 BD
7 Cell culture growth kinetics Objective Bio ex. Notation Assumptions Derivation Plots Note the “S-curve” kinetics
8 Derivations Objective Bio ex. Notation Assumptions Derivation Plots • First, a remark on derivations. • The objective of a derivation is to build a (mathematical) bridge between basic concepts (axioms in math) and a result (generally, an equation, with parameters or variables corresponding to physical quantities).
9 Transformation types Continuous nucleation: nuclei added during transformation. Objective Bio ex. Notation Assumptions Derivation Plots Site Saturated: all nuclei present at t=0. Cellular: recrystallization, for example. Kinetics same as for site saturated case.
10 KJMA notation Objective Bio ex. Notation Assumptions Derivation Plots • The central idea in the derivation of the KJMA equation is to focus on the increment in the (volume) fraction transformed and to relate it to the current value of the fraction transformed. • Notation: f fraction transformed t time t 50% time required for 50% transformation r radius V volume v growth rate (speed) t incubation/delay time N rate of nucleation, or, density of nuclei per unit volume
11 Fraction transformed • The relationship between volume and fraction transformed is simple. Fraction transformed = volume / total volume, Objective Bio ex. or, Notation Assumptions Derivation Plots f = V / Vtotal • Similarly for area (2 D), line (1 D), etc.
12 KJMA: extended fraction • To understand the concept of an extended fraction transformed, imagine that each patch of new phase can overlap with another one as they grow (ignore the effect of impingement): Objective Bio ex. V 2 Notation Assumptions Derivation Plots V 1 fext = (V 1+V 2)/Vtotal f = (V 1 V 2)/Vtotal The true fraction transformed counts only the actual volume transformed: the extended fraction counts all volume as if no impingement occurs. “union of”
13 KJMA derivation: assumption Notation • There is one key assumption in the derivation of the KJMA equation: the nuclei are distributed randomly in space. • This assumption allows us to make a quantitative relation between the true increment in fraction transformed, a fictitious or extended fraction transformed and the current fraction: Assumptions df = dfext (1 -f) Objective Bio ex. Derivation Plots • Why does this work? The reason is that the volume that each patch can grow into is decreased from the total in proportion to the fraction that has already transformed.
14 KJMA derivation • The KJMA derivation is therefore a bridge between the differential equation just stated and the final equation that we use. Objective Bio ex. Notation Assumptions Derivation Plots df = dfext (1 -f)
15 KJMA derivation: 1 Objective Bio ex. Notation Assumptions Derivation Plots • Step 1: define the differential equation (above). • Step 2: describe the growth rate of an individual patch/region. • Example: 3 D, isotropic growth, site saturated nucleation: V = 4π/3 r 3(t) = 4π/3 (vt)3 • Step 3: multiply the individual volume by the number density of nuclei, N: fext = SVi /Vtotal = 4π/3 N (vt)3 • Step 4: obtain the extended fraction increment: dfext = V/Vtotal = 4π Nv 3 t 2 dt
16 KJMA derivation: 2 • Step 5: insert the extended fraction increment into the differential equation: Objective Bio ex. df = dfext (1 -f) df = 4π Nv 3 t 2 dt (1 -f) Notation Assumptions Derivation Plots df/ (1 -f) = (4π N v 3) t 2 dt • Step 6: collect the nucleation and growth terms into a constant (which varies depending on the conditions of nucleation and growth): k = 4π/3 N v 3
17 KJMA derivation: 3 • Step 7: solve the differential equation: recognize that df = -1 * d(1 -f), and that the fraction transformed is zero at t=0, so that we are dealing with a logarithmic solution. Objective - ln(1 - f ) = k t 3 Bio ex. Notation Assumptions Derivation • Re-arrange to obtain the final result: - general result Plots - site saturated, 3 D growth
18 KJMA solutions Objective Bio ex. Notation Assumptions Derivation • In general, the k value contains all the temperature dependent terms because thermal activation affects the growth strongly through boundary/interface mobility, and because the nucleation density depends very strongly on driving force. See P&E p 269: v = v(T) = v 0 exp-(Q/RT) Plots • In general, the exponent n in the equation is related to the geometry of the transformation.
19 “n” values • Site saturated: 1 D growth 2 D growth 3 D growth 1 2 3 Objective Bio ex. Notation Assumptions Derivation Plots • Continuous nucleation, constant nucleation rate: 1 D growth 2 2 D growth 3 3 D growth 4 • CAUTION: you cannot always deduce the geometry of transformation from the value of the exponent.
20 People Objective Bio ex. Notation Assumptions Derivation Plots • Kolmogorov was a Russian mathematician who work is much referenced in statistics. He worked out this relation for the case of continuous nucleation. • Johnson was a graduate student at Carnegie Tech under R. F. Mehl as his adviser. He studied recrystallization in aluminum. • Avrami was a chemist and worked out the most general approach: his work is known in the chemical engineering world. • Porter & Easterling describe the equation but do not explain it in detail. Other texts provide more detail.
21 KJMA plots • A very useful way to analyze the kinetics of transformation (e. g. recrystallization) is to plot the quantity -ln(1 -f) versus time on a double-logarithmic plot. The slope of the line is then the exponent, n. Objective Bio ex. Notation Assumptions Derivation Plots [Humphreys] n = slope = 2
22 Measurement • The fraction transformed can be measured in almost any conceivable way. Objective Bio ex. Notation Assumptions Derivation Plots – – – From micrographs Hardness Electrical resistivity Optical properties Calorimetry
KJMA derivation: diffusion controlled growth 23 Objective Bio ex. Notation Assumptions Derivation Plots • Step 1: define the differential equation (above). • Step 2: describe the growth rate of an individual patch/region. • Example: 3 D, isotropic diffusion controlled growth, site saturated nucleation: V = 4π/3 r 3(t) = 4π/3 {∆X 0/(Xppt-Xe)}3 (√{Dt})3 = 4π/3 {∆X 0/(Xppt-Xe)}3 (Dt)1. 5 • Step 3: multiply the individual volume by the number density of nuclei, N: fext = SVi /Vtotal = N 4π/3 {∆X 0/(Xppt-Xe)}3 (Dt)1. 5 • Step 4: obtain the extended fraction increment: dfext = V/Vtotal = N 2π {∆X 0/(Xppt-Xe)}3 D 1. 5 √t dt
KJMA derivation: diffusion controlled growth 24 • Step 5: insert the extended fraction increment into the differential equation: Objective Bio ex. Notation Assumptions Derivation Plots df = dfext (1 -f) df = 2π N {∆X 0/(Xppt-Xe)}3 D 1. 5 √t dt (1 -f) df/ (1 -f) = 2π N {∆X 0/(Xppt-Xe)}3 D 1. 5 √t dt • Step 6: collect the nucleation and growth terms into a constant (which varies depending on the conditions of nucleation and growth): k = 4π/3 ND 1. 5 {∆X 0/(Xppt-Xe)}3
KJMA derivation: diffusion controlled growth 25 • Step 7: solve the differential equation: recognize that df = -1 * d(1 -f), and that the fraction transformed is zero at t=0, so that we are dealing with a logarithmic solution. Objective - ln(1 -f) = 4π/3 N {∆X 0/(Xppt-Xe)}3 (Dt)1. 5 Bio ex. Notation Assumptions Derivation Plots • Re-arrange to obtain the final result: - general result - site saturated, 3 D diffusion controlled growth
26 Fixed fraction transformed • If we need to find the time for a fixed fraction transformed, this is easily accomplished by manipulation of the basic equation, e. g. for 10% and 90% transformed. Objective Bio ex. Notation Assumptions Derivation Plots
27 TTT Diagrams Objective Bio ex. Notation Assumptions Derivation Plots • What is the connection between KJMA kinetics and TTT diagrams? • Answer: once you have defined the relevant quantities in the KJMA equation, i. e. the nucleation density and the growth rate (and exponent) as a function of temperature, then you can calculate the time required to achieve a certain fraction transformed (previous slide). • Armed with a set of times for a fixed fraction transformed, draw the locus of points that is a curve on the TTT diagram. Repeat for each volume fraction of interest.
Скачать презентацию 1 Objective Bio ex Microstructure-Properties II The KJMA
04cc31691c6365f5d43b41da9dea2406.ppt