cherarogers& west&virginia&wesleyancollege& host...
TRANSCRIPT
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Chera Rogers West Virginia Wesleyan College Host: Florida State University
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• Introduction • Thermal Concept of Point Defects • Diffusion • Mechanisms • How to Study Diffusion • Characterization of Diffusion • Random Walk Theory • Numerical Approach of Diffusion • Results
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• 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
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• 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.
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• 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
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• 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
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• 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
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Vacancy Mechanism
Interstitial Mechanism
Interstitialcy Mechanism
Exchange Mechanism Ring Mechanism
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• 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
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• 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
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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
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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.
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• As expected, X and Y components of the resultant poistion vector of the Random walks are centered around 0.
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