Exact solutions for a planar sensor
ddtρ = D 2 ρ
ddNt = kF (N0 − N)ρs
kF →∞, ρs = 0 |
ρ(x,t) = ρ0 erf (x 2 |
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Dt |
Alam, Principles of Nanobiosensors, 2013 |
9 |
Exact solutions for a planar sensor
ρ(x,t) = ρ0 erf (x
2
Dt )
erf (x)≡ |
2 |
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∫0x e−y2 dy |
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Particles captured
1 |
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0.8 |
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0.6 |
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erf(x) |
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0.4 |
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erf(x/2) |
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0.2 |
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erf(x/3) |
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00 |
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erf(x/4) |
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4 |
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N(t) = ∞∫[ρ0 −ρ(x,t)]dx = ρ0 
π4 
Dt
0
Alam, Principles of Nanobiosensors, 2013 |
10 |
The concept of the diffusion distance
ddtρ = D 2 ρ
t1 <t2 <t3
x ~ Dt
x ~ 
Dt
x
11
Approximate Solution in 1D:
Diffusion Distance
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Exact |
Approximate |
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Dt |
0.8 |
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0.6 |
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erf(x) |
ρ0 |
0.4 |
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erf(x/2) |
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0.2 |
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erf(x/3) |
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erf(x/4) |
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1 |
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4 |
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N(t) ~ |
×ρ0 × |
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N(t) = |
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0 |
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Alam, Principles of Nanobiosensors, 2013 |
12 |
Approximate Solution in 1D:
Diffusion Distance
Approximate |
Approximate |
Dt |
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Dt |
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ρ |
0 |
ρ0 |
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N(t) ~ ρ0 × |
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N(t) ~ |
×ρ0 × |
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Dt |
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Dt |
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Alam, Principles of Nanobiosensors, 2013 |
13 |