As I read though a chapter in my materials science book, there was a design problem that caught my attention. It's on a relatively simpler level compared to more complex optimization problems with multiple design variables to keep track of.

Design a process to electroplate a 0.1 cm thick layer of copper onto a 1 cm x 1 cm cathode surface. I believe you all know a little about electroplating. If you don't, here's a sweet introduction on the topic.

Ultimately, our decisions will be based on an optimal value of a function. In most cases, it will be a cost. Obviously, something thats not economically viable will not be considered at all. Another important function is the rate of production and quality of the copper plate.

In this scenario, the variables are 1) current (I in Amperes) and 2) time (t in seconds).

A pair of values for these variables will give us the minimum cost. For this to happen, there must be a relationship between the optimization function and the variables.

Or, Cost = function(I,t).

This relationship has to be established by the company is question after digging into their records and what not. I'm not really sure how its done, but this was discussed in class briefly.

Anyway, in order to produce a 0.1 cm think layer on a 1 cm squared surface area, the mass of copper must be :

Mass = Density x Volume = 8.96 g/square.cm (referenced) x (1 square cm)(0.1cm)

= 0.896g

From Faraday's equation (for rate of corrosion - search wikipedia),

I.t = w.n.F/M

where w = mass

F= Faraday's constant (96,500 Coulombs)

n =valence of the metal ion

I = current (A)

t = time

I.t = (0.896 x 2 x 96,500)/63.54 = 2722 A. secc

We may now use several different combinations of current and time to produce the copper plate :

Current Time

0.1 A 27,220s = 7.6 h

1.0 A 2,722s = 45.4 min

10.0 A 272.2s = 4.5 min

100.0 A 27.2s = 0.45 min

The exact combination is current and time might be made on the basis of the rate of production and the quality of the copper plater. Low currents require very long plating times, perhaps making the process economically unsound. High currents, on the other hand, may reduce plating efficiencies. Currents that are too high can also initiate other side reactions that are not desired.

So we see that there are tradeoffs and the optimal solution to this problem is based on experimentation and additional background. A current of approx. 1 A and a time of approx 45 minutes are not uncommon in electroplating operations.

Source : The Science and Engineering of Materials (Askeland, Phule, 2006).

## Monday, December 04, 2006

### 0 Class Connectionz : Optimization

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