Chera  Rogers  West  Virginia  Wesleyan  College  Host:  Florida  State  University  

• Introduction  • Thermal  Concept  of  Point  Defects  • Diffusion  • Mechanisms  • How  to  Study  Diffusion  • Characterization  of  Diffusion  • Random  Walk  Theory  • Numerical  Approach  of  Diffusion  • Results  

• In  a  perfect  crystal,  mass  and  charge  density  have  the  periodicity  of  the  lattice.  • Solids  in  nature  are  not  perfect  crystals;  they  have  defects.  

•  The  creation  of  a  point  defect  or  extended  defects  disturbs  this  periodicity.  

http://www.nyu.edu/classes/tuckerman/honors.chem/lectures/lecture_20/lattices.jpg  

• Concept:  Thermal  Agitation  causes  the  transitions  of  atoms  from  their  normal  lattice  sites  into  interstitial  positions  leaving  behind  lattice  vacancies.  

• When  a  vacancy  is  created,  the  crystal  lattice  relaxes  around  the  vacant  site  and  the  vibrations  of  the  crystal  are  altered.  • When  the  interstitials  are  created  the  crystal  lattice  is  strained  around  that  interstitial.    

• Describes  the  spread  of  particles  through  random  motion  from  regions  of  high  concentration  to  low  concentration.    • Diffusion  is  caused  by  the  Brownain  motion  (random  motion)  of  atoms  or  molecules  that  leads  to  complete  mixing  

• Example  :  Ink  in  Water  

• Mathematical  framework  by  Adolf  Fick  • Fick  introduced  the  concept  of  diffusion  coefficient  and  suggested  a  linear  response  between  the  concentration  gradient  and  flux.  

Fick’s  First  Law:  

Fick's  first  law  is  formally  equivalent  to  Fourier's  Law  of  heat  flow  and  Ohm’s  Law:  

€

J = −D∇C

€

Jq = −k∇T

€

Je = −σ∇V

J:  Diffusion  Flux  D:  Diffusion  Coefficient  C:  Concentration  Gradient    

• In  diffusion  process  the  number  of  diffusing  particles  is  conserved.  • Inflow  –  outlfow  =  accumulation  rate  • The  continuity  equation  and  Fick’s  first  law  can  be  combined  to  create  Fick’s  second  law.  

€

−∇ • J =∂C∂t

€

∂C∂t

=∇⋅ (D∇C)

€

J = −D∇C

€

−∇ • J =∂C∂t

Vacancy  Mechanism  

Interstitial  Mechanism  

Interstitialcy  Mechanism  

Exchange  Mechanism   Ring  Mechanism  

• The  diffusion  coefficient  can  be  calculated  by  the  Einstein-­‐Relation:  

€

D =R2

4τ

Mean  square  displacement  

Diffusion  Coefficient  

Calculated  by  Random  Walk  

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R2

• Diffusion  in  solids  results  from  any  individual  displacements  (jumps)  of  the  diffusing  particles  in  a  random  fashion.    • The  total  distance  traveled  by  a  particle  is  sum  of  a  sequence  of  jump  distances  

€

R = r1l=1

nstep

∑

R2 = rl2 +

l=1

nstep−1

∑ rl ⋅ rjJ = l+1

nstep

∑l=1

nstep

∑

€

R2 = rl2 + 2

l=1

nstep−1

∑ rl ⋅ rJJ = l+1

nstep

∑l=1

nstep

∑Average  R  

€

D =R2

4τ

€

τ =nZΓ

€

D =R2

4ZΓn

Z=4  so  they  cancel  each  other  out  leaving  you  with:  

€

D =R2

nΓ

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Γ = vo exp − ΔGkBT

⎛

⎝ ⎜

⎞

⎠ ⎟

ΔG :kB :vo :T :

Gibbs  free  energy  of  activation  

Boltzmann  constant  

Attempt  frequency    

Temperature  

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R :n :

Final  position  of  the  tracer  

Number  of  steps  

 In  the  Random  Walk  Code  we  used  Monte  Carlo  Method.   We  picked  random  numbers  between  (0,1)  and  depending  on  the  number  there  were  4  choices  of  step  direction.     Within  the  innermost  loop  each  step  is  based  on  a  random  number  between  (0,1)   In  one  walk  ,  with  50  different  steps,  50  different  random  numbers  were  picked.  

increment i :if i<=nstep

if i>nstep -> exit

take step:rnd <=0.25 -> x+dx , 0.25<rnd<=0.5->x-dx

0.5<rnd<=0.75->y-dx0.75<rnd<=1->y+dx

Get Random #'s -> rnd

for i=1,2,...nstep

xij=0, yij=0Initial position

for j=1,2, .. nwalk

Increment jif j <= nwalk

if j > nwalk ->exit

Increasing  the  number  of  walks    

• As  we  increased  the  number  of  walks  the  diffusion  coefficient  converge  to  their  “true”  values.  

• As  we  increased  the  number  of  walk  the  average  distance  traveled  aproached  zero.  

•   As  expected,  X  and  Y  components  of  the  resultant  poistion  vector  of  the  Random  walks  are  centered  around  0.