6.021/Notes/2006-09-11
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Microfluidics project
- Sign up for time slot for pre-lab
- Need to first find a partner. Recommended to find partner with a different background
Office hours
- Open office hours 32-044 Tues. 4-10pm and Wed. 4-7pm
Diffusion
- Fick's first law (Review) only provides information at one time
- need something to go from t to t + Δt
- Continuity equation
- Conservation of mass
- A change in flux in space implies change in concentration over time
- Combining Fick's first law and continuity equation:
- This is the diffusion equation (Fick's Second law)
Steady State Solution
- Steady state: for everything (nothing changes with time)
- Equilibrium: Steady state AND all fluxes are 0
- In a closed system, equilibrum is equivalent to steady state
- In an open system, we can have non-zero fluxes at steady state
- Flux can be a constant (non-zero) which implies that concentration is a linear function of x
Dynamics
- Simplest case is to assume infinite space and a point source with 1-dimensional diffusion
- Dirac delta function δ(x)
- δ(x) = 0 except at x = 0 and
- Suppose at time 0, c(x,t) = n_{0}δ(x) where n_{0} is the initial amount
- Solution of diffusion equation in this case:
- Compare with Gaussian function:
- We see that solution is Gaussian with a time-dependent standard deviation (and mean of 0)
- At any point in space, the concentration increases then decreases
- The amount of time it takes for half the solute to diffuse x_{1 / 2} is
- This squared relationship between time and distance is a very important characteristic of diffusion.
- Example of this scaling effect
- D = 10^{ − 5}cm^{2} / s
- For x_{1 / 2} = 10mm, t_{1 / 2} = 10^{5}s (about a day)
- For x_{1 / 2} = 10μm, t_{1 / 2} = 0.1s
- For x_{1 / 2} = 10nm, t_{1 / 2} = 0.1μs
- Diffusion on cell length scales is really fast AND diffusion over macroscopic timescales is really slow