# 6.021/Notes/2006-09-11

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

• Steady state: $\displaystyle{ \frac{\partial}{\partial t} = 0 }$ 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 $\displaystyle{ x }$

### Dynamics

• Simplest case is to assume infinite space and a point source with 1-dimensional diffusion
• Dirac delta function $\displaystyle{ \delta(x) }$
• $\displaystyle{ \delta(x) = 0 }$ except at $\displaystyle{ x=0 }$ and $\displaystyle{ \int_{-\infty}^\infty \delta(x)dx = 1 }$
• Suppose at time 0, $\displaystyle{ c(x,t)=n_0\delta(x) }$ where $\displaystyle{ n_0 }$ is the initial amount
• Solution of diffusion equation in this case:
• $\displaystyle{ c(x,t)=\frac{n_0}{\sqrt{4\pi Dt}}e^{-x^2/(4Dt)} }$
• Compare with Gaussian function: $\displaystyle{ \frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/(2\sigma^2)} }$
• We see that solution is Gaussian with a time-dependent standard deviation $\displaystyle{ \sigma=\sqrt{2Dt} }$ (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 $\displaystyle{ x_{1/2} }$ is $\displaystyle{ t_{1/2}\approx\frac{x_{1/2}^2}{D} }$
• This squared relationship between time and distance is a very important characteristic of diffusion.
• Example of this scaling effect
• $\displaystyle{ D=10^{-5}{\rm cm}^2/{\rm s} }$
• For $\displaystyle{ x_{1/2}=10 }$mm, $\displaystyle{ t_{1/2}=10^5 }$s (about a day)
• For $\displaystyle{ x_{1/2}=10 \mu }$m, $\displaystyle{ t_{1/2}=0.1 }$s
• For $\displaystyle{ x_{1/2}=10 }$nm, $\displaystyle{ t_{1/2}=0.1\mu }$s
• Diffusion on cell length scales is really fast AND diffusion over macroscopic timescales is really slow