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

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
 * $$\phi(x,t)=-D\frac{\partial c(x,t)}{\partial x}$$
 * We define flux to be positive in same direction as increasing $$x$$
 * Diffusivitiy units: $$D=\frac{m^2}{s}$$
 * 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 $$\tau$$ seconds, molecule equally likely to move $$+l$$ and $$-l$$
 * $$l=2$$pm for small molecule (really small length scale)
 * From random walk model, easy to derive Fick's first law

$$\phi(x,t)=-\frac{l^2}{2\tau}\frac{\partial c}{\partial x}$$ so $$D=-\frac{l^2}{2\tau}$$