UA PSIO 472 - Diffusive transport in one dimension

Unformatted text preview:

Physiology 472/572 ‐ 2011 ‐ Quantitative modeling of biological systems Lecture 4: Diffusive transport in one dimension (Sep 1, 2011) History of theories of diffusion. In 1785 Jan Ingenhousz (Netherlands) wrote about random motion of coal dust particles in alcohol. In 1827 Scottish botanist Robert Brown saw tiny particles, smaller than pollen, wiggling around in water. Although Ingenhousz was probably the first to observe “Brownian motion,” it ended up being named after Brown (as often happens). Thomas Graham did experiments with diffusion of gasses 1828‐1833 Adolf Fick was first to try to quantitate these observations. Einstein did his thesis on Brownian motion in 1905 – and established that Brownian motion was related to thermal energy. Even though these discoveries and theories date back to 100‐200 years ago, there has still been some recent activity. As late as 1991, someone stirred up controversy by claiming that Brown couldn’t have observed real Brownian motion, and wrote an article in Scientific American. But, several scientists then researched the type of microscope he had more carefully, and set up demonstrations to show it was indeed possible. Historically, Brownian motion was observed before molecular diffusion, because molecular diffusion occurs at smaller length scales that cannot be seen through normal light microscopes. While both Brownian motion and molecular diffusion are due to thermal energy, in the latter case it is the thermal energy of the molecules themselves that causes diffusion. In Brownian motion, it is instead the thermal energy of the molecules of the medium the particle is in. While molecular diffusion and Brownian motion are two different physical processes, they are both governed by the same mathematical equations. Fick’s First Law Although Fick’s law is called a “law,” it is not universally true. There is such a thing as “non‐Fickian diffusion.” However, Fick’s Law, as simple as it is, holds in the overwhelming majority of practical applications. Fick’s First Law in one dimension • xtxcDtxj∂∂−=),(),( • ),( txc = solute concentration (mol/cm3) • ),( txj = solute flux, net moles crossing per unit time through a unit area perpendicular to the x‐axis (mol/cm2∙s) • D = diffusivity, depends on both solute and medium (solution). (Assumed constant here, but not always so.) Notice that diffusive flux is defined to be positive if it is in the positive x direction. This is why the minus sign is needed in Fick’s Law. Question for class: what are units of diffusivity? (cm2/s), etc Diffusion as a random walk • each molecule moves independe ntly and randomly (“independently” means each molecule follows its own path; that is, they aren’t moving in clumps. But of course the molecules are colliding with each other.) • net effect is a movement from high to low concentrat ion (mention parenthetically here that this tendency to move towards more disordered systems determines our direction of time…and this gets into rather deep philosophical issues regarding the nature and direction of time) To visualize this, imagine a set of a large number of particles or molecules. Each particle is (hypothetically) labeled with a number. At time 0, all are placed at x = 0. All are undergoing random motion because of collisions with many other particles, which sometimes push them one way and sometimes another. Each of these pictures is a snapshot at one time point. If the x axis is divided into “bins,” then the number of particles in each bin can be counted at any time point, and a bar graph can be made, showing this number as a function of position. t = 0 t = t2 t = t3 t = t1 Since the numbering of the particles was very arbitrary (we didn’t know in advance which ones would go where), this new way of graphing is a better way of quantifying what has happened, since it doesn’t depend on an arbitrary numbering system. This bar graph, which will look familiar to anyone who studied coin toss experiments in statistics, is approximately a Gaussian curve, or bell curve. As the number of particles goes to infinity, the size of the bins can be made smaller and smaller, and the curve in the limit of infinite particles becomes exactly Gaussian. Question: Why does the number of particles get smaller as you go away from the initial position x = 0? As time goes on, there is more and more chance that a few particles just happened (randomly) to take many more steps in one direction than another. But it is still most likely that the majority of particles took nearly as many steps to the left as steps to the right. So, the peak will always be in the center. Analogous flux laws: Reasons why it is useful to note analogies : (1) Intuition from one field (e.g., fluid me chanics) can be carried over to another. (2) Mathematical equations are the same, so solutions you already know can be used again. List of analogous flux laws : • Diffusion (Fick): xcDj∂∂−= (solute flux, diffusivity and concentration) • Fluid in porous medium (Darcy): xpu∂∂−=κ (fluid velocity, hydraulic permeability and pressure • Heat conduction (Fourier): xThh∂∂−=σφ (heat flow, thermal conductivity and temperature) • Electric conduction (Ohm): xJe∂∂−=ψσ (current density, electrical conductivity, and electric potential) At this point you see why it was important to introduce partial derivatives in the last class – so many quantities in physical problems depend on both time and space. Questions for class: You’re about to take something out of the oven. Do you want to use a potholder with high value of hσor a low value? Which do you think has a higher value of κ, pumice or granite? Note that the last law, to show the analogy with the other laws, is written on a per cross‐sectional area basis. In practice, you are probably more familiar with this law written in the following form: I = V/R Since in electric circuits, usually you are working with a wire, the total current rather than the current per cross‐sectional area is of interest. In biology, it is more likely that you work with continuous systems, so flux per area rather than current is used. Conservation of mass in one dimension • consider a small volume containing c(x,t)ΔxΔyΔz • assume solute is consumed by reaction at a rate r per unit volume • amount diffusing in is


View Full Document
Download Diffusive transport in one dimension
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Diffusive transport in one dimension and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Diffusive transport in one dimension 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?