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 (Review) only provides information at one time
 * need something to go from $$t$$ to $$t+\Delta t$$
 * 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 $$\delta(x)$$
 * $$\delta(x) = 0$$ except at $$x=0$$ and $$\int_{-\infty}^\infty \delta(x)dx = 1$$
 * Suppose at time 0, $$c(x,t)=n_0\delta(x)$$ where $$n_0$$ 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 $$x_{1/2}$$ 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
 * $$D=10^{-5}{\rm cm}^2/{\rm s}$$
 * For $$x_{1/2}=10$$mm, $$t_{1/2}=10^5$$s (about a day)
 * For $$x_{1/2}=10 \mu$$m, $$t_{1/2}=0.1$$s
 * For $$x_{1/2}=10$$nm, $$t_{1/2}=0.1\mu$$s
 * Diffusion on cell length scales is really fast AND diffusion over macroscopic timescales is really slow