6.021/Notes/2006-09-11

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 ([[../2006-09-08/|Review]]) only provides information at one time
• need something to go from [math]\displaystyle{ t }[/math] to [math]\displaystyle{ t+\Delta t }[/math]
• Continuity equation
  • Conservation of mass
  • [math]\displaystyle{ -\frac{\partial\phi}{\partial x} = \frac{\partial c}{\partial t} }[/math]
  • A change in flux in space implies change in concentration over time
• Combining Fick's first law and continuity equation:
  • [math]\displaystyle{ \frac{\partial\phi}{\partial x} = -\frac{\partial c}{\partial t} = -D\frac{\partial^2 c}{\partial x^2} }[/math]
• This is the diffusion equation (Fick's Second law)
  • [math]\displaystyle{ \frac{\partial c}{\partial t} = D\frac{\partial^2 c}{\partial x^2} }[/math]

Steady State Solution

• Steady state: [math]\displaystyle{ \frac{\partial}{\partial t} = 0 }[/math] 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 [math]\displaystyle{ x }[/math]

Dynamics

• Simplest case is to assume infinite space and a point source with 1-dimensional diffusion
• Dirac delta function [math]\displaystyle{ \delta(x) }[/math]
• [math]\displaystyle{ \delta(x) = 0 }[/math] except at [math]\displaystyle{ x=0 }[/math] and [math]\displaystyle{ \int_{-\infty}^\infty \delta(x)dx = 1 }[/math]
• Suppose at time 0, [math]\displaystyle{ c(x,t)=n_0\delta(x) }[/math] where [math]\displaystyle{ n_0 }[/math] is the initial amount
  • Solution of diffusion equation in this case:
  • [math]\displaystyle{ c(x,t)=\frac{n_0}{\sqrt{4\pi Dt}}e^{-x^2/(4Dt)} }[/math]
  • Compare with Gaussian function: [math]\displaystyle{ \frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/(2\sigma^2)} }[/math]
  • We see that solution is Gaussian with time-dependent variance [math]\displaystyle{ \sigma=\sqrt{2Dt} }[/math]
  • At any point in space, we will see a wave of concentration that increases then decreases
  • The amount of time it takes for half the solute to diffuse [math]\displaystyle{ x_{1/2} }[/math] is [math]\displaystyle{ t_{1/2}\approx\frac{x_{1/2}}{D} }[/math]
  • This is a very important characteristic of diffusion.
  • Example of this scaling effect
    • [math]\displaystyle{ D=10^{-5}{\rm cm}^2/{\rm s} }[/math]
    • For [math]\displaystyle{ x_{1/2}=10 }[/math]mm, [math]\displaystyle{ t_{1/2}=10^5 }[/math]s (about a day)
    • For [math]\displaystyle{ x_{1/2}=10 \mu }[/math]m, [math]\displaystyle{ t_{1/2}=0.1 }[/math]s
    • For [math]\displaystyle{ x_{1/2}=10 }[/math]nm, [math]\displaystyle{ t_{1/2}=0.1\mu }[/math]s
    • Diffusion on cell length scales is really fast AND diffusion over macroscopic timescales is really slow