# 6.021/Notes/2006-09-11

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## Microfluidics project

• 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: $\frac{\partial}{\partial t} = 0$ 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