chapter 7 diffusion in solids

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Diffusion in solids

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  • DIFFUSION IN SOLIDS

    FICKS LAWS

    KIRKENDALL EFFECT

    ATOMIC MECHANISMS

    Diffusion in SolidsP.G. Shewmon

    McGraw-Hill, New York (1963)

  • Oxidation

    Roles of Diffusion

    Creep

    AgingSintering

    Doping Carburizing

    Metals

    Precipitates

    SteelsSemiconductors

    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.

  • Ar H2

    Movable piston

    with an orifice

    H2 diffusion direction

    Ar diffusion direction

    Piston motion

    Piston moves in the

    direction of the slower

    moving species

  • A B

    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

  • 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

  • Flux (J) (restricted definition) Flow / area / time [Atoms / m2 / s]

    1 dnJ

    A dt

    Flow direction

  • 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)

    dx

    dcDA

    dt

    dn

    ( ) / / Flux J atoms area time concentration gradient

    dx

    dcJ

    dx

    dcDJ

    dx

    dcD

    dt

    dn

    AJ

    1 Diffusivity (D) f(Concentration of the components, T)

    Ficks first law

    dc dJx x

    dt dx

    Continuity equation

    (Truly speaking it is the

    chemical potential gradient!)

  • dx

    dcDA

    dt

    dn

    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)

    dx

    dcDJ

  • Diffusion

    Steady state

    J f(x,t)

    Non-steady state

    J = f(x,t)D = f(c)

    D = f(c)

    D f(c)

    D f(c)Steady and non-steady state diffusion

    0x t

    dc J

    dt x

    Under steady state conditions

    0J c

    Dx x

    Substituting for flux from Ficks first law

    2

    20

    cD

    x

    If D is constant

    Slope of c-x plot is constant under steady state conditions

    constantc

    Dx

    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

  • Ficks II law

    Jx Jx+x

    x

    xxx JJonAccumulati

    x

    x

    JJJonAccumulati xx

    x

    x

    JJJx

    t

    cxx J

    sm

    Atomsm

    sm

    Atoms

    23

    .1

    xx

    Jx

    t

    c

    x

    cD

    xt

    cFicks first law

    x

    cD

    xt

    c D f(x)2

    2

    x

    cD

    t

    c

  • 22

    x

    cD

    t

    c

    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

  • 22

    x

    cD

    t

    c

    Dt

    xerfBAtxc

    2 ),(

    Solution to 2o de with 2 constants

    determined from Boundary Conditions and Initial Condition

    0

    2exp2

    duuErf

    Erf () = 1

    Erf ( ) = 1

    Erf (0) = 0

    Erf ( x) = Erf (x)

    u

    Ex

    p(

    u2)

    0

    Area

    Also

    For upto x~0.6 Erf(x) ~ x

    x 2, Erf(x) 1

  • A B

    Example where the erf solution can be used

    x

    Con

    cen

    trat

    ion

    Cavg

    t

    t1 > 0 | c(x,t1)t2 > t1 | c(x,t1) t = 0 | c(x,0)

    A & B welded together and heated to high temperature (kept constant T0)

    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) = C1 C(x, 0) = C2

    C1

    C2

    A = (C1 + C2)/2

    B = (C2 C1)/2

    Dt

    xerfBAtxc

    2 ),(

    AB = C1A+B = C2

    1 2 2 1( , ) 2 2 2

    C C C C xc x t erf

    Dt

  • kTQ

    eDD 0

    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.

  • 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

  • 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

  • Interstitial Diffusion

    1 2

    1 2

    Hm

    At T > 0 K vibration of the atoms provides the energy to overcome the energybarrier Hm (enthalpy of motion)

    frequency of vibrations, number of successful jumps / time

    kTHm

    e '

  • Substitutional Diffusion

    Probability for a jump

    (probability that the site is vacant).(probability that the atom has sufficient energy)

    Hm enthalpy of motion of atom

    frequency of successful jumps

    kTH

    kT

    Hmf

    ee '

    kT

    HH mf

    e '

    kT

    HH mf

    eD 2

    Where, is the jump distance

  • Interstitial Diffusion

    kTHm

    eD 2

    Substitutional Diffusion

    kT

    HH mf

    eD 2

    D (C in FCC Fe at 1000C) = 3 1011 m2/s

    D (Ni in FCC Fe at 1000C) = 2 1016 m2/s

    0

    f mH H

    kTD D e

    of the form

    0 mH

    kTD D e

    of the form

  • Diffusion Paths with Lesser Resistance

    Qsurface < Qgrain boundary < Qpipe < Qlattice

    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 forthe path will determine the diffusion rate for that path

  • Comparison of Diffusivity for self-diffusion of Ag

    single crystal vs polycrystal

    1/T

    Lo

    g (

    D)

    Schematic

    Polycrystal

    Single

    crystal

    Increasing Temperature

    Qgrain boundary = 110 kJ /mole

    QLattice = 192 kJ /mole

  • 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 CH4 (g) 2H2 (g) + C (diffuses into steel).

    C(+x, 0) = C1 C(0, t) = CS

    A = CS B = CS C1

  • 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 900C

    (where, Fe is in FCC form).

    Data: 4 20D (C in -Fe) 0.7 10 m / s 157 /Q kJ mole

    Given: T = 900 C, C0 = C(x, 0) = C(, t) = 0.2 % C,

    Cf = C(0.2 mm, t1) = 1% C (at x = 0.2 mm), Cs = C(0, t) = 1.4% C

    The solution to the Fick second law: ( , ) 2

    xC x t A B erf

    Dt

    The constants A & B are determined from boundary and initial conditions:

    (0, ) 0.014SC t A C , 0( , ) 0.002C t A B C or 0( ,0) 0.002C x A B C

    S 0B C C 0.012 , ( , ) 0.014 0.012 2

    xC x t erf

    Dt

    -44

    1

    1

    2 10(2 10 , ) 0.01 0.014 0.012

    2C m t erf

    Dt

    S S 0( , ) C (C -C ) 2

    xC x t erf

    Dt

    0

    ( , )=

    2

    S

    S

    C x t C xerf

    C C Dt

    (2)

    (1)

    -4

    1

    1 2 10

    3 2erf

    Dt

  • x (in mm from surface)

    % C

    0.2

    0.4

    0.6

    0.8

    1.0

    1.2

    1.4

    0.2

    t =0

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