diffusion in solids  fick’s laws  kirkendall effect  atomic mechanisms diffusion in...


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  • DIFFUSION IN SOLIDS FICKS LAWS KIRKENDALL EFFECT ATOMIC MECHANISMS Diffusion in Solids P.G. Shewmon McGraw-Hill, New York (1963) MATERIALS SCIENCE &ENGINEERING Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology, Kanpur- 208016 Email: anandh@iitk.ac.in, URL: home.iitk.ac.in/~anandh AN INTRODUCTORY E-BOOK Part of http://home.iitk.ac.in/~anandh/E-book.htm A Learners Guide
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  • Oxidation Roles of Diffusion Creep Aging Sintering DopingCarburizing Metals Precipitates Steels Semiconductors Many more Some mechanisms Material Joining Diffusion bonding To comprehend many materials related phenomenon one must understand Diffusion. The focus of the current chapter is solid state diffusion in crystalline materials.
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  • ArH2H2 Movable piston with an orifice H 2 diffusion direction Ar diffusion direction Piston motion Piston moves in the direction of the slower moving species
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  • AB Inert Marker thin rod of a high melting material which is basically insoluble in A & B Kirkendall effect Materials A and B welded together with Inert marker and given a diffusion anneal Usually the lower melting component diffuses faster (say B) Marker motion
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  • Diffusion Mass flow process by which species change their position relative to their neighbours. Driven by thermal energy and a gradient Thermal energy thermal vibrations Atomic jumps Concentration / Chemical potential ElectricGradient Magnetic Stress
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  • Flux (J) (restricted definition) Flow / area / time [Atoms / m 2 / s] Flow direction
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  • Ficks I law A Flow direction Assume that only B is moving into A Assume steady state conditions J f(x,t) (No accumulation of matter) Diffusivity (D) f(Concentration of the components, T) Ficks first law Continuity equation (Truly speaking it is the chemical potential gradient!)
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  • No. of atoms crossing area A per unit time Cross-sectional area Concentration gradient ve implies matter transport is down the concentration gradient Diffusion coefficient/ Diffusivity A Flow direction As a first approximation assume D f(t)
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  • Diffusion Steady state J f(x,t) Non-steady state J = f(x,t) D = f(c) D f(c) Steady and non-steady state diffusion Under steady state conditions Substituting for flux from Ficks first law If D is constant Slope of c-x plot is constant under steady state conditions If D is NOT constant If D increases with concentration then slope (of c-x plot) decreases with c If D decreases with c then slope increases with c
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  • Ficks II law JxJx J x+ x xx Ficks first law D f(x)
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  • RHS is the curvature of the c vs x curve x c x c +ve curvature c as t ve curvature c as t LHS is the change is concentration with time
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  • Solution to 2 o de with 2 constants determined from Boundary Conditions and Initial Condition Erf ( ) = 1 Erf ( ) = 1 Erf (0) = 0 Erf ( x) = Erf (x) u Exp( u 2 ) 0 Area Also For upto x~0.6 Erf(x) ~ x x 2, Erf(x) 1
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  • A B Example where the erf solution can be used x Concentration C avg t t 1 > 0 | c(x,t 1 ) t 2 > t 1 | c(x,t 1 ) t = 0 | c(x,0) A & B welded together and heated to high temperature (kept constant T 0 ) Flux f(x)| t f(t)| x Non-steady state If D = f(c) c(+x,t) c( x,t) i.e. asymmetry about y-axis C(+x, 0) = C 1 C( x, 0) = C 2 C1C1 C2C2 A = (C 1 + C 2 )/2 B = (C 2 C 1 )/2 A B = C 1 A+B = C 2
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  • Temperature dependence of diffusivity Arrhenius type Diffusivity depends exponentially on temperature. This dependence has important consequences with regard to material behaviour at elevated temperatures. Processes like precipitate coarsening, oxidation, creep etc. occur at very high rates at elevated temperatures.
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  • ATOMIC MODELS OF DIFFUSION 1. Interstitial Mechanism Usually the solubility of interstitial atoms (e.g. carbon in steel) is small. This implies that most of the interstitial sites are vacant. Hence, if an interstitial species wants to jump, most likely the neighbouring site will be vacant and jump of the atomic species can take place. Light interstitial atoms like hydrogen can diffuse very fast. For a correct description of diffusion of hydrogen anharmonic and quantum (under barrier) effects may be very important (especially at low temperatures). The diffusion of two important types of species needs to be distinguished: (i) species sitting in a lattice site (ii) species in a interstitial void
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  • 2. Vacancy Mechanism For an atom in a lattice site (and often we are interested in substitutional atoms) jump to a neighbouring lattice site can take place if it is vacant. Hence, vacancy concentration plays an important role in the diffusion of species at lattice sites via the vacancy mechanism. Vacancy clusters and defect complexes can alter this simple picture of diffusion involving vacancies
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  • Interstitial Diffusion 1 2 12 HmHm At T > 0 K vibration of the atoms provides the energy to overcome the energy barrier H m (enthalpy of motion) frequency of vibrations, number of successful jumps / time
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  • Substitutional Diffusion Probability for a jump (probability that the site is vacant).(probability that the atom has sufficient energy) H m enthalpy of motion of atom frequency of successful jumps Where, is the jump distance
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  • Interstitial Diffusion Substitutional Diffusion D (C in FCC Fe at 1000C) = 3 10 11 m 2 /s D (Ni in FCC Fe at 1000C) = 2 10 16 m 2 /s of the form
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  • Diffusion Paths with Lesser Resistance Q surface < Q grain boundary < Q pipe < Q lattice Experimentally determined activation energies for diffusion Core of dislocation lines offer paths of lower resistance PIPE DIFFUSION Lower activation energy automatically implies higher diffusivity Diffusivity for a given path along with the available cross-section for the path will determine the diffusion rate for that path
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  • Comparison of Diffusivity for self-diffusion of Ag single crystal vs polycrystal 1/T Log (D) Schematic Polycrystal Single crystal Increasing Temperature Q grain boundary = 110 kJ /mole Q Lattice = 192 kJ /mole
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  • Applications based on Ficks II law Carburization of steel Surface is often the most important part of the component, which is prone to degradation. Surface hardening of steel components like gears is done by carburizing or nitriding. Pack carburizing solid carbon powder used as C source. Gas carburizing Methane gas CH 4 (g) 2H 2 (g) + C (diffuses into steel). C(+x, 0) = C 1 C(0, t) = C S A = C S B = C S C 1
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  • Solved Example A 0.2% carbon steel needs to be surface carburized such that the concentration of carbon at 0.2mm depth is 1%. The carburizing medium imposes a surface concentration of carbon of 1.4% and the process is carried out at 900 C (where, Fe is in FCC form). Data: The solution to the Fick second law: (2) (1)
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  • From equation (2)
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  • Approximate formula for depth of penetration Let the distance at which [(C(x,t) C 0 )/(C S C 0 )] = be called x 1/2 (which is an effective penetration depth) The depth at which C(x) is nearly C 0 is (i.e. the distance beyond which is un-penetrated): Erf(u) ~ 1 when u ~ 2
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  • End
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  • Another solution to the Ficks II law A thin film of material (fixed quantity of mass M) is deposited on the surface of another material (e.g. dopant on the surface of a semi-conductor). The system is heated to allow diffusion of the film material into the substrate. For these boundary conditions we get a exponential solution. Boundary and Initial conditions C(+x, 0) = 0
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  • Ionic materials are not close packed Ionic crystals may contain connected void pathways for rapid diffusion These pathways could include ions in a sublattice (which could get disordered) and hence the transport is very selective alumina compounds show cationic conduction Fluorite like oxides are anionic conductors Due to high diffusivity of ions in these materials they are called superionic conductors. They are characterized by: High value of D along with small temperature dependence of D Small values of D 0 Order disorder transition in conducting sublattice has been cited as one of the mechanisms for this behaviour Diffusion in ionic materials
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  • Element HfHf HmHm H f + H m Q Au9780177174 Ag9579174184 Calculated and experimental activation energies for vacancy Diffusion
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  • 1 2 Vacant site c = atoms / volume c = 1 / 3 concentration gradient dc/dx = ( 1 / 3 )/ = 1 / 4 Flux = No of atoms / area / time = / area = / 2 On comparison with
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  • 3. Interstitialcy Mechanism Exchange of interstitial atom with a regular host atom (ejected from its regular site and occupies an interstitial site) Requires comparatively low activation energies and can provide a pathway for fast diffusion Intersti


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