CheraRogers& West&Virginia&WesleyanCollege& Host...
Transcript of CheraRogers& West&Virginia&WesleyanCollege& Host...
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:
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J = −D∇C
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Jq = −k∇T
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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.
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−∇ • J =∂C∂t
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∂C∂t
=∇⋅ (D∇C)
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J = −D∇C
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−∇ • J =∂C∂t
Vacancy Mechanism
Interstitial Mechanism
Interstitialcy Mechanism
Exchange Mechanism Ring Mechanism
• The diffusion coefficient can be calculated by the Einstein-‐Relation:
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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
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R = r1l=1
nstep
∑
R2 = rl2 +
l=1
nstep−1
∑ rl ⋅ rjJ = l+1
nstep
∑l=1
nstep
∑
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R2 = rl2 + 2
l=1
nstep−1
∑ rl ⋅ rJJ = l+1
nstep
∑l=1
nstep
∑Average R
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D =R2
4τ
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τ =nZΓ
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D =R2
4ZΓn
Z=4 so they cancel each other out leaving you with:
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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.