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lecture24 - 2.57 Nano-to-Macro Transport Processes Fall...

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2.57 Fall 2004 – Lecture 24 1 2.57 Nano-to-Macro Transport Processes Fall 2004 Lecture 24 In the last lecture, we have talked about Einstein’s work on the Brownian motion. In the left figure above, pressure difference exists in the fluid. For one particle, the osmotic pressure is determined by 1 B B P Nk T nk T V = = . Also, we can use the solution of Stoke’s flow around a sphere to estimate the drag force. The value is 3 F D u π µ = . For area A c , the forced balance over the control volume gives ( ) ( ) ( ) c A P x P x dx FdN + = , 3 0 c dP A dx D udN dx π µ = , 3 0 dP D un dx π µ = , 3 0 B dn k T D un dx π µ = , in which the product un indicates flux. We have 3 B p k T dn dn J a D dx dx π µ = − = − , where a= 3 B k T D π µ is the diffusivity and can be obtained by the diffusion experiment of some materials. In a time t, the diffusing radius is 6 r at = . We can calculate the constant a and substitute back into a= 3 B k T D π µ , thus D can be obtained. The relationship between thermal diffusivity and viscosity is an example of the more general fluctuation- dissipation theory as viscosity is a measure of dissipative process and diffusivity is a measure of random walk (fluctuation) process. The relationship between thermal Stoke’s flow 3 F D u π µ = P(x) P(x+dx)
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2.57 Fall 2004 – Lecture 24 2 diffusivity and viscosity is also called Einstein relation. In chapter 6, there is a similar Einstein relation between electron diffusivity and mobility. But Einstein really worked on the Brownian motion. Note: The osmotic pressure can be observed by putting a semi-permeable membrane such that only the base molecules in the solution penetrate the membrane, building up a concentration gradient on the two sides of the membranes. In addition to the above approach, Einstein also established another method to determine the diameter of the solute particles. He proposed to measure the viscosity of the solvent and of the solution, µ , and b µ , respectively, and derived, again assuming dilute solute particles, the following relationship between the viscosities 1 2.5 o µ ϕ µ = + , where ϕ is the volumetric concentration of the solute molecules. The Einstein relation can also be derived from the stochastic approach developed by Langevin to treat Brownian motion of particles much larger than that of the surrounding medium. The key idea of the Langevin equation is to consider that the motion of a Brownian particle is subject to a friction force that is linearly proportional to its velocity, as in Stokes law, and a random driving force (or “noise”), R(t), imparted by the random motion of the molecules in the bath. In the absence of an external force, the Langevin equation that governs the instantaneous velocity of the Brownian particle can be written as, ( ) d m m t dt η = − + u u R , where η is the friction coefficient, and for Brownian particles in a fluid the Stokes law gives 3 F D u π µ = , so that the random driving force R(t) has the following characteristics: ( ) 0 t = R (average of random driving force is zero) ( ) ( ) 0 t t = R u
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