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Diffusion Creep

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					                       Diffusion Creep
              Poirier, Chapter 2 and 7, 1985.
                               Gordon, 1985.




Monday, Nov. 3, 2003     12.524 Mechanical properties of rocks
Fick’s First Law: Driving Force


           ∇c                          Chemical Diffusion
           ∇T
                                       Thermal diffusion
J = − Di∇µ 
            ∇V                         Electrical conduction

           σ
                                       Diffusion creep



             12.524 Mechanical properties of rocks
Types of Diffusion
Mechanism        Path                                     Process

Isotope          Lattice                                  Interdiffusion

Self-diffusion   Pipe                                     Creep

Vacancy          Grain Boundary                           Ambipolar

Interstitial     Surface

Ring             Pore fluid
                  12.524 Mechanical properties of rocks
Diffusion mechanisms
                           Vacancy Diffusion




                                                               Interstitial
    Ring Diffusion                                             Collinear
                                                               Non -collinear
    Direct Exchan ge
    Cyclic Exchange




    A         B




                       12.524 Mechanical properties of rocks
Vacancy–Assisted Diffusion        From Site by Glicksman and Lupulescu, RPI, 2003

    The FCC lattice geometry requires
                                     W=        ( 3 − 1)D         a   = 0.73Da




                 Images removed due to copyright considerations.

           For more information, see http://www.rpi.edu/~glickm/diffusion/




                         12.524 Mechanical properties of rocks       plan view
Kinetics Equation for
Vacancy Diffusion
 Coefficient of Diffusivity for Self-diffusion not the
 same as Coefficient for Vacancy Diffusion


   Dsd = Nv iDv migration
         = Nvo exp− ( ∆Gvf / kT )iDv o exp− ( ∆Gvm / kT )
         = Dsd o exp− ( ∆Gvf + ∆Gvm ) / kT




                      12.524 Mechanical properties of rocks
Diffusion Creep in Monatomic Solid

                             Vacancies Supersaturated
                                                                              Basic Ideas
  Vacancies Undersaturated




                                                                                    Supersaturation of
                                                                                    vacancies owing to
                                                                                    stress
                                                                                    Diffusion results
                                                                                    Work done on mat’l
                                                                                    by tractions
                                                                                    Energy dissipated in
                                                                                    heat, entropy, and
                                                                                    surface area



                                                   12.524 Mechanical properties of rocks
Diffusion Creep
 Nabarro-Herring Creep                          Monatomic
    Lattice                                     Quasi static
 Coble Creep                                    Vacancy
    Grain Boundary                              Increasing length;
                                                Poisoining




                 12.524 Mechanical properties of rocks
Critical Idea: Tension makes
vacancy formation easier.
                                         Tension=supersaturation

                                ∆G fv (σ ) = ∆G fv ( 0 ) − σΩ
                                                      ∆G fv ( 0 )
       Atom                    Cvo = C iexp −
                                                 kT
              l                                   ∆G fv ( 0 ) − σΩ 
      d                        Cv (σ ) = C exp −                   
                                                        kT         


              12.524 Mechanical properties of rocks
Gradient in Composition
                                       Path Length:
                                             Boundary: 2xd/4
                                             Lattice: (π/2)x(d/4)
                                       Concentration Difference
                                                        ∆G fv − σΩ
                                           Co exp − (             )
                                                          kT
                                                         ∆G fv + σΩ
                                           −Co exp − (                )
                                                             kT
                                                   ∆G fv        σΩ        −σΩ 
                                           Co exp− (     )  exp( ) − exp(     )
                                                    kT          kT         kT 
                                                            ∆G fv σΩ
                                           ∆C = 2Co exp − (      )i
                                                              kT kT

                                       Quasi-static Approx.
            12.524 Mechanical properties of rocks
  Fick’s 1st Law
                       ∆C
   J path = − D path                                 i.)
                       ∆L path
   Total flux =ΣFlux on each Path:
                     d                                                            δ d/2
  Φ vac flux   = JL ⋅ ⋅ l + JB ⋅δ ⋅ l
                     2
                                 Φ vac flux             2δ
  J total    total ave. flux =                = JL +       JB              ii.)
                                 d / 2⋅l                 d
   Plugging ∆C and ∆L into i.) and inserting fluxes in ii.):
                                                                       ∆G fv σΩ
                             ∆G fv σΩ 1                    2Co exp − (      )i
                                               2δ                       kT kT
J total   = − DL 2Co exp − (      )⋅         +    ( − Db )
                              kT     kT π d 8 d                     d 2


                                       12.524 Mechanical properties of rocks
 Total Flux on Both Paths



             8             ∆Gvf      σΩ  2δ 2 DB Cvo      ∆Gvf        σΩ 
J Total   =    DLCvo exp −               +            exp− kt         2  kT 
            πd              kt        kT  d    d                            
            16               ∆Gvf    σΩ   πδ DB 
          =       DLCvo exp − kt      kT  1 + 2d D 
            πd                                    L 




                                12.524 Mechanical properties of rocks
Converting flux to strain
             b
                             Each vacancy that travels through
                       b     the channel adds layer of depth b
 1
             2             ∆l # vacs b                  2 b  2 J total Ω
                                =      = 2 ⋅ J total ⋅ b ⋅ =
                           l ⋅t   s d                     d       d

       ε12

                                        32 σΩ  πδ DB 
 ε22             ε11                 ε= 2     DL 1 +    
                                       π d kT     2d DV 
                            12.524 Mechanical properties of rocks
Summary: N-H and Coble Creep
 Diffusion Creep Constitutive Law:
                 32 σΩ  πδ DB 
             ε= 2       DL 1 +    
                π d kT      2d DV 

 DL = DVM CV
 Strain rate linear in stress
 ε∝ 1       2,3
        d
 Other geometries change initial constant
                    12.524 Mechanical properties of rocks

				
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