6.021/Notes/2006-09-08

Diffusion

• Most fundamental transport process and the one we understand the best
• Process by which solute is transported from regions of high concentration to low concentration
• Graham made some observations related to diffusion
  • Quantity transported is proportional to the initial concentration
  • Transport rate slows with time
  • Transport of gas is greater than 1000x faster than liquids

Definitions

• Concentration: [math]\displaystyle{ c(x,t)=\lim_{V\rightarrow 0} \frac{amount}{volume} }[/math]
  • But matter isn't discrete, so limit doesn't make sense
  • Practically, cells have about [math]\displaystyle{ \frac{1}{6}\frac{mol}{L} }[/math] NaCl which is about [math]\displaystyle{ 10^8\frac{molecules}{\mu m^3} }[/math]
  • Cells are greater than [math]\displaystyle{ 1\mu m^3 }[/math] so there are many molecules in a typical cell
  • Thus, we'll assume matter is continuous to simplify the math
• Flux: [math]\displaystyle{ \phi(x,t) = \lim_{A\rightarrow 0 ; \Delta t\rightarrow 0}\frac{amount}{A\Delta t} }[/math]

Fick's First Law

• Adolf Fick (1855) at age 25, came up with Fick's first law by analogy to Fourier's law for heat flow
• [math]\displaystyle{ \phi(x,t)=-D\frac{\partial c(x,t)}{\partial x} }[/math]
• We define flux to be positive in same direction as increasing [math]\displaystyle{ x }[/math]
• Diffusivitiy units: [math]\displaystyle{ D=\frac{m^2}{s} }[/math]
• This is a macroscopic law

Microscopic basis for diffusion

• 1828: Robert Brown (Brownian motion)
  • Even dead things moved
• Albert Einstein with random walk model
  • Assumptions
    • Number of solute much less than number of solvent
    • Only collisions between solute and solvent
  • Focus on 1 solute molecule and assume others are statistically identical
  • Every [math]\displaystyle{ \tau }[/math] seconds, molecule equally likely to move [math]\displaystyle{ +l }[/math] and [math]\displaystyle{ -l }[/math]
  • [math]\displaystyle{ l=2 }[/math]pm for small molecule (really small length scale)
  • From random walk model, easy to derive Fick's first law

[math]\displaystyle{ \phi(x,t)=-\frac{l^2}{2\tau}\frac{\partial c}{\partial x} }[/math] so [math]\displaystyle{ D=-\frac{l^2}{2\tau} }[/math]