hw2_solu - ME 125NT Intro to Nanotechnology Due Problem 1...

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ME 125NT Intro to Nanotechnology Due: 4/8/2008 1 Problem 1: You just put 5mL of heavy cream into your 16 oz. coffee and instead of mixing them together, you just wait for all the cream to diffuse with the coffee. What is the time scale for this mixing? (You may want to try it out just to make sure you have the right order of magnitude). What are the parameters that affect this mixing? What are the relevant Equations?. Now for the obvious question – how many times faster will this diffusion occur for a 10nL volume of flourescein dye in a 1uL microfluidic channel? What equations are different? What are the relevant parameters, and how does this differ than at the macro-scale? Hints: diffusion coefficient is measured in units of m^2/s ANSWER For the first three questions we have to look at what is affecting the diffusion. The variables that have to be considered are the concentration c, diffusion constant D (this is the coefficient of MASS diffusivity, completely analogous to the coefficient of THERMAL diffusivity, α ), time t and distance x. The equation that controls the diffusion is a second order differential equation and is as follows. 2 2 x c D t c = What this says is that the concentration over a certain time will follow the second derivative of the concentration multiplied by a constant D. This means that the higher the concentration the faster it will diffuse. The Diffusion equation is completely analogous to the heat equation you used in ME 151b, c and you can learn how to solve this in 140b. 2 2 x T t T = α The diffusion constant D = υ *R*T with the units is in m^2/s, where υ is a material constant, R is the universal constant and T is temperature. Using the coffee cup and milk, we see that the time scale is on the order of minutes (or even hours if you want it to be completely diffused). From this time, we can solve for the actual diffusion constant, using the solution to the above equation, and noting that the solution predicts: Dt x 2 = (in one dimension)
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ME 125NT Intro to Nanotechnology Due: 4/8/2008 2 Where x is the one dimension we are looking at, D is the diffusion coefficient, and t is time. When scaling down to the smaller dimension, we will first ASSUME that D remains constant, since it is a property of the fluid. If this is the case, then we can use the relation above to look at the time scale given a much smaller x. Obviously, all the D’s calculated will be a different number, so the actual number doesn’t matter, just the order of magnitude. Assume we calculated a D = 10
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hw2_solu - ME 125NT Intro to Nanotechnology Due Problem 1...

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