User:Brian P. Josey/Notebook/2010/11/16

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Beginning to Understand Drift

Like I said two weeks ago, I am trying to understand the physics of diffusion and drift. A car accident, Japanese project, two papers and five homework assignments later, I'm back to it. Overly eager, I thought I could produce diffusion with drift directly from the Nernst relation, and I set out to do it. Luckily, it doesn't appear that is the best approach, and I'm starting to understand it a little better.

I reread a portion of the Nelson book more thoroughly trying to understand the physics behind diffusion a little better and came across a derivation that he did. All of my following notes are directly taken from his book, but I'm putting them into my own words to work through the nuances and figure it out for myself.

Imagine a particle moving in one direction under a constant external force, f. As it travels, it is hit with collisions that occur once every Δt. Between these kicks, its velocity is governed by Newton's second law:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac {dv_x} {dt}= \ddot x = \frac {f} {m} }

Here, v is the velocity, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \ddot x} is the acceleration, f is the force, and m is the mass of the particle. Pretty simple. However, after a collision it has a velocity, v0,x. Each collision has a random chance of moving it in the direction of the force, or away from it, and the collision also erases the "memory" of the last collision. In essence they act independently of each other. So after a collision the distance changes:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x = v_{0,x} \Delta t + \frac {f} {2m} (\Delta t)^2 }

If I take the averages, the velocity added from the collisions are removed because they are equally likely to help or hinder the particle move in the direction of the force. This leaves the average change in x over time as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \langle \Delta x \rangle = \frac {f} {2m} (\Delta t)^2 }

Take the time derivative for the velocity:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac {\langle \Delta x \rangle} {\Delta t}= \dot x_{drift}= \frac {f}{\zeta} }

where:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \zeta = \frac {2m} {\Delta t} }

This ζ is the viscous friction coefficient that I've mentioned before and it can be measure experimentally. It's important to remember at this point the a each type of particle in each type of solvent has its own characteristic ζ and D, the diffusion constant. They are not universal, and even the same particle in different solvents, like sawdust in water or oil, have different ones. However, for a simple spherical particle, ζ can be calculated from Stoke's equation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \zeta = 6 \pi \eta R \, }

where:

  • η - is the viscosity of the solvent, which is about 10-3 kg/ms, and
  • R - is the radius of the spherical particle.