6.021/Notes/2006-09-11: Difference between revisions
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** <math>c(x,t)=\frac{n_0}{\sqrt{4\pi Dt}}e^{-x^2/(4Dt)}</math> | ** <math>c(x,t)=\frac{n_0}{\sqrt{4\pi Dt}}e^{-x^2/(4Dt)}</math> | ||
** Compare with Gaussian function: <math>\frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/(2\sigma^2)}</math> | ** Compare with Gaussian function: <math>\frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/(2\sigma^2)}</math> | ||
** We see that solution is Gaussian with time-dependent | ** We see that solution is Gaussian with a time-dependent standard devation <math>\sigma=\sqrt{2Dt}</math> | ||
** At any point in space, we will see a wave of concentration that increases then decreases | ** At any point in space, we will see a wave of concentration that increases then decreases | ||
** The amount of time it takes for half the solute to diffuse <math>x_{1/2}</math> is <math>t_{1/2}\approx\frac{x_{1/2}}{D}</math> | ** The amount of time it takes for half the solute to diffuse <math>x_{1/2}</math> is <math>t_{1/2}\approx\frac{x_{1/2}}{D}</math> |
Revision as of 13:13, 11 September 2006
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 ([[../2006-09-08/|Review]]) only provides information at one time
- need something to go from [math]\displaystyle{ t }[/math] to [math]\displaystyle{ t+\Delta t }[/math]
- Continuity equation
- Conservation of mass
- [math]\displaystyle{ -\frac{\partial\phi}{\partial x} = \frac{\partial c}{\partial t} }[/math]
- A change in flux in space implies change in concentration over time
- Combining Fick's first law and continuity equation:
- [math]\displaystyle{ \frac{\partial\phi}{\partial x} = -\frac{\partial c}{\partial t} = -D\frac{\partial^2 c}{\partial x^2} }[/math]
- This is the diffusion equation (Fick's Second law)
- [math]\displaystyle{ \frac{\partial c}{\partial t} = D\frac{\partial^2 c}{\partial x^2} }[/math]
Steady State Solution
- Steady state: [math]\displaystyle{ \frac{\partial}{\partial t} = 0 }[/math] 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 [math]\displaystyle{ x }[/math]
Dynamics
- Simplest case is to assume infinite space and a point source with 1-dimensional diffusion
- Dirac delta function [math]\displaystyle{ \delta(x) }[/math]
- [math]\displaystyle{ \delta(x) = 0 }[/math] except at [math]\displaystyle{ x=0 }[/math] and [math]\displaystyle{ \int_{-\infty}^\infty \delta(x)dx = 1 }[/math]
- Suppose at time 0, [math]\displaystyle{ c(x,t)=n_0\delta(x) }[/math] where [math]\displaystyle{ n_0 }[/math] is the initial amount
- Solution of diffusion equation in this case:
- [math]\displaystyle{ c(x,t)=\frac{n_0}{\sqrt{4\pi Dt}}e^{-x^2/(4Dt)} }[/math]
- Compare with Gaussian function: [math]\displaystyle{ \frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/(2\sigma^2)} }[/math]
- We see that solution is Gaussian with a time-dependent standard devation [math]\displaystyle{ \sigma=\sqrt{2Dt} }[/math]
- At any point in space, we will see a wave of concentration that increases then decreases
- The amount of time it takes for half the solute to diffuse [math]\displaystyle{ x_{1/2} }[/math] is [math]\displaystyle{ t_{1/2}\approx\frac{x_{1/2}}{D} }[/math]
- This is a very important characteristic of diffusion.
- Example of this scaling effect
- [math]\displaystyle{ D=10^{-5}{\rm cm}^2/{\rm s} }[/math]
- For [math]\displaystyle{ x_{1/2}=10 }[/math]mm, [math]\displaystyle{ t_{1/2}=10^5 }[/math]s (about a day)
- For [math]\displaystyle{ x_{1/2}=10 \mu }[/math]m, [math]\displaystyle{ t_{1/2}=0.1 }[/math]s
- For [math]\displaystyle{ x_{1/2}=10 }[/math]nm, [math]\displaystyle{ t_{1/2}=0.1\mu }[/math]s
- Diffusion on cell length scales is really fast AND diffusion over macroscopic timescales is really slow