6.021/Notes/2006-09-11

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Contents

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
    • -\frac{\partial\phi}{\partial x} = \frac{\partial c}{\partial t}
    • A change in flux in space implies change in concentration over time
  • Combining Fick's first law and continuity equation:
    • \frac{\partial\phi}{\partial x} = -\frac{\partial c}{\partial t} = -D\frac{\partial^2 c}{\partial x^2}
  • This is the diffusion equation (Fick's Second law)
    • \frac{\partial c}{\partial t} = D\frac{\partial^2 c}{\partial x^2}

Steady State Solution

  • Steady state: \frac{\partial}{\partial t} = 0 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 \int_{-\infty}^\infty \delta(x)dx = 1
  • Suppose at time 0, c(x,t) = n0δ(x) where n0 is the initial amount
    • Solution of diffusion equation in this case:
    • c(x,t)=\frac{n_0}{\sqrt{4\pi Dt}}e^{-x^2/(4Dt)}
    • Compare with Gaussian function: \frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/(2\sigma^2)}
    • We see that solution is Gaussian with a time-dependent standard deviation \sigma=\sqrt{2Dt} (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 x1 / 2 is t_{1/2}\approx\frac{x_{1/2}^2}{D}
    • This squared relationship between time and distance is a very important characteristic of diffusion.
    • Example of this scaling effect
      • D = 10 − 5cm2 / s
      • For x1 / 2 = 10mm, t1 / 2 = 105s (about a day)
      • For x1 / 2 = 10μm, t1 / 2 = 0.1s
      • For x1 / 2 = 10nm, t1 / 2 = 0.1μs
      • Diffusion on cell length scales is really fast AND diffusion over macroscopic timescales is really slow
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