## Wednesday, September 4, 2013

### Mixing my ink with my beer

I have had a lot to say in the John the Math Guy blog about beer. There was a recent post about ruminations on beer, but the key post on beer, my seminal post on beer, is the post where I cleverly used beer to illustrate Beer's law. I keep going back to that one because I just can't get over how brilliant the idea was.

I have referred to this Beer's law post in heaps and gobs of other posts:

Green ink being shamelessly added to beer

Several people have asked questions on this Beer's law thing, and how it connects with ink. Way back in January (2013), I got an email from a PhD student in the UK:

When I scoured the internet for the derivation of this equation I only found the original equation based on absorption coefficient, path length and concentration.

I would like to understand where the alternative equation is coming from. Could you point me in the direction of a useful paper or similar? Any help would very much be appreciated. Thanks a lot in advance.

Kind regards,
Anja

I just recently got a similar question from Michael, who is not a PhD student in the UK, but is nonetheless a smart guy. He just aced the Science and Technology Quiz that was put together by Smithsonian magazine and the Per Research Center. This is quite an accomplishment. I'm proud to be a "virtual" friend of yours, Michael.

I thought Beers law was related to transmission of light through something, not reflected - but, well, same difference ?

Michael's question came to me on through that website that everyone uses for scientific collaboration.: FaceBook. If you haven't heard of it yet, I suggest you check it out. That's where I do most of my serious research.

The plethora of questions (there were two...) show that I have clearly messed up big time in my desire to educate the world about ink and beer. I left one little step out, the mixing of ink and beer. Just how is it that Beer's law applies to ink?
Beer (on the left) and ink (on the right)

Ink is soooo not like paint

One of those wonderful things that we can count on in this world is that paint is not like ink. Oh... they may seem the same to the untrained and unscientific eye. You put them on something and it changes the color. But there is one key difference, as illustrated in the image below.

This time, paint is on the left and ink (on the right)

This image was created by smearing ink and paint on a sheet of paper [3]. Before smearing, a large black area was printed on the sheet. Note that on the left, the paint completely obliterates the black underneath. Paint has a great hiding power, at least when you pay more than \$8 a gallon for it. The ink, however, does a perfectly lousy job of hiding the black. You really can't tell that the yellow ink is overneath the black.

What gives? Is ink just really, really cheap paint? Oh contraire! Let me assure you, ink does a pretty decent job of doing exactly what it was trained to do. And paint also does a pretty decent job at what it was trained to do. That is, if you aren't cheap like me, buying the ultra-cheap paint at \$8 a gallon from Fast Eddie's Paint Emporium and Car Wash.

The actual photomicrograph below illustrates what an ideal cyan ink is trained to do. Red, green, and blue light hit the surface of the ink. [4] As can be seen, the blue and green light go right through the cyan filter ink. The ink is transparent to green and blue light. These two flavors of light hit the paper (or other white print substrate) and reflect back. Why? Cuz the substrate is white, and that's what white things are trained to do.

Cyan ink, sitting contentedly on paper while being bombarded with red, green, and blue light

The red light suffers a completely different fate. For anyone who has visited a red light district, this should be no surprise that one's fate may change. The red light is absorbed by the cyan ink. Few of the poor hapless red photons ever even get a chance to reach the paper, and even fewer make it through the ink in the hazardous journey back through.

I may have shattered some illusions about ink here. I apologize, but it's time you learned the facts about the birds and the bees and the inks. It is customary to think of light just reflecting off the ink. Sorry. It's more complicated than that. The only reflecting that's done is done by the paper.

Ink is a filter, a filter laid atop the paper.

Why is ink that way?

This bizarre behavior is not just some side effect of some bizarre organic chemistry that is only understood by some bizarre color scientist locked in the lab at Sun Chemical. This bizarre behavior is a property that is specifically engineered by some some bizarre color scientist locked in the lab at Sun Chemical.

To see why this would be a good thing to engineer in, consider what happens when magenta ink is placed overneath cyan ink. Magenta ink works a lot like cyan ink, except that it absorbs green light and passes red and blue. The excitement starts when you put on ink on the other. The cyan ink absorbs red light and the magenta ink absorbs the green. What's left? Just blue light.

Magenta ink, sitting contentedly on cyan ink

The exciting part of this is that new colors are created. We start with cyan, magenta, and yellow inks. By putting one ink overneath another, the additional colors red, green, and blue are created. Try doing that with paint! It ain't gonna work. The paint on top defines the color, hiding whatever is below.

This feature of ink is what allows us to have a much wider gamut. With three inks (cyan, magenta, and yellow) we can theoretically get eight different colors: white (no ink), cyan, magenta, yellow, red, green, blue, and black (all three inks).

Getting a bit more quantitative

I need to put some numbers on this if I'm going to get Beer's law involved. I painted a rather black and white picture of cyan. Well, ok, I should say that I inked a black and white picture rather than painted it. And black and white aren't quite the correct colors. But, the point is, inks are not perfect. Cyan does not capture all the red photons. Nor does it pass all the green and blue photons.

A typical cyan ink might allow 20% of the red photons to pass through on their way to the substrate. That is, 80% of the red light gets absorbed and the other 20% makes it down to the paper. Let's just assume that all of those photons reflect from the paper. (I am telling a little white lie here, but it's for a good purpose.)

Ok, so if we start out with 100 red photons heading downward into the ink, 20 of them will reach the paper. Of these 20, 80% of them (I think that would be 16) will get absorbed on the way up. That leaves just 4 red photons, out of the original 100, that make it back out. For those of you who are all into the density thing, this would mean about 1.40D. If you understood that, then you know Beer's law, and can apply it to ink on paper. Who said that ink and beer don't mix?

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[1] You may have seen this blog post in Flexo Global Magazine. So, we are talking popular here.

[2] I am pleased to say that this blog post was picked up just recently by the Australia New Zealand Flexographic Technical Association magazine (August 2013). Look for it in your mailbox. This blog post was also picked up by Flexo Global magazine. So... we are talking really popular here.

[3] Truth in advertising here... this is not an actual photo, but a digital simulation. It is very nearly photorealistic due to my vast artistic ability, and it simulates what really happens, but I repeat, this is not an actual photo.

[4] Someday I will get around to writing a blog about how light comes in three flavors: red, green, and blue. That's all. All other colors are a combination of those three. Based on this simplification, you can explain why inks are CMY and computer monitors are RGB. It will be a totally cool blog. I will send you a text when I finally publish it.