Fate of 2 crystal sizes

Consider an Ostwald ripening experiment with crystals of two initial sizes: x1*=0.05 and x2*=0.09 microns, and parameter values k=5.e7, μ=1.e-3, c*=7.52e-7, Γ=4.e-3, c_{0}=1.05c* The goal is to simulate this experiment computationally to verify the theoretical predictions. Here are the tasks that need to be carried out. 1. Write down the mathematical problem that models the experiment (physical assumptions, ODEs, initial conditions, parameters). Describe what you expect to happen qualitatively and why, and what you plan to do to simulate the experiment. [the steps are described below] 2. Find the dissolution time, t_{1}, of the crystal that dissolves first, and the size of the surviving crystal at that time. [ Explain what you'll do, then do it ]. Determine t_{1}and x_{2}(t_{1}) as accurately as you can. Plot x_{1}(t) and x_{2}(t) up to time t_{1}. 3. For times beyond t_{1}, there is only one crystal size. Find the two equillibria for this situation, andpredictthe eventual fate of the survivoron the basis of theory. Does the prediction depend on the accuracy of the values found in 2 ? 4. Calculate the evolution of x_{2}(t) beyond time t_{1}to verify the prediction. Show the evolution of the two sizes in a time plot. 5. Draw conclusions about the experiment and the usefulness of the analysis and simulation.

The above steps and results should be presented as a formal

You may reuse wording from your (revised) Project 1, as appropriate,

but do NOT assume that your boss has seen Project 1.

to guide the reader as to what you are doing.

Please submit the Report on paper.