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The mathematics of cleaning artists' brushes

Skip to new material added 18 January 2007

What has mathematics got to do with cleaning artists' brushes?

Ok, you use a lot of white spirit (mineral spirit to our US colleagues), swirl your paintbrush around, and chuck the resulting mess down the sink. Your brush still isn't all that clean, the white spirit causes your plumbing to become lined with grey gunk, eventually your sink drain blocks, and in the meantime you've polluted the environment, as our American friends say, big time. What to do?

Here are a few answers.

Firstly, don't use white spirit for cleaning paintbrushes. Although also environmentally damaging, cheap turpentine is better for cleaning brushes than white spirit, because it doesn't dry out the bristles of your expensive brushes as badly. (Use only artists' quality turpentine for thinning paint - but this is a separate topic.) You may also use Zest-it™ for a reputedly environmentally-friendly alternative (I have no connection with the company and no way of verifying their claims).

Whatever you use, minimise its environmental impact by using as little as possible. There are two key tricks here:

  1. when you have cleaned your brushes, tip the used solvent back into a jam jar with a lid on it; the paint settles to the bottom, and the solvent left lying on top (once the paint has settled) can be poured out carefully and used again; the residue at the bottom can be scraped out and burned;
  2. use small quantities of solvent several times, instead of a lot all at once.

When your brushes have been cleaned in this way, rub them gently into some neat washing up liquid at the bottom of a jar, old plastic container or other suitable vessel. (I have heard it advised to use the palm of your hand to hold the washing up liquid, but this is dangerous if you are using toxic paints such as flake white or genuine Naples yellow, as the lead could be absorbed through your skin, and it is a cumulative poison.)

Finally, rinse the brushes one at a time under the tap, gently rubbing out any remaining residues between finger and thumb. Catch the water and detergent in a container, and dispose of directly into a drain, since even these small residues can eventually block your sink outflow.

The mathematics of dilution

Ok, let the amount of residual paint on your paintbrush be r.
Let the volume of solvent available be S.
Let the volume of solvent held by your paintbrush be b.

Example A: using all the solvent at once

The amount of paint on your paintbrush after the cleaning will be: r.b/S

In other words, the dilution of whatever (r) is on your brush is the volume of your brush b divided by the volume of solvent used S, i.e. the dilution factor is b/S .

Example B: using a third of the solvent three times

The amount of paint on your brush after the first cleaning will obviously be three times more than if you had used all the solvent at once, because you used only a third of the amount of solvent:
r.b/(S/3) = r.3b/S.

The dilution factor here by which we must multiply r is b/(S/3) which is equal to 3b/S.

The amount of paint on your brush after the second cleaning will be the amount left over after the first cleaning multiplied again by the dilution factor 3b/S:
(r.3b/S).(3b/S) = r.9b²/S²

The amount of paint on your brush after the third cleaning, is the amount left after the second cleaning once again multiplied by the dilution factor 3b/S :
(r.9b²/S²).(3b/S) = r.27b³/S³


To see which method is best we need to compare the amounts left on the brush after each procedure:
Example A: r.b/S
Example B: r.27b³/S³

First it is clear that the comparison does not depend on how much stuff (r) there was on the brush in the first place (although equally obviously you will get a cleaner brush if you wipe off as much paint as possible to start with). So we are really comparing dilution factors:
Example A: b/S
Example B: 27b³/S³

If I am right that method B is better, then I have to show that:
27b³/S³ < b/S
Assuming for the moment that this is true, we get:
27b²/S² < 1
b²/S² < 1/27
b/S < 1/sqr(27)
b/S < 0.192...

So, provided that the volume of solvent that can be held in your brush at one time is less than 0.192... (i.e about a fifth) of the total amount of solvent available for cleaning your brushes (which is very probable, when you think about it), then your brushes will be cleaner after having used a third of the solvent three times than they would have been if you'd used all the solvent at once. With normal volumes, the extra dilution (and hence cleanness of your brushes) is very surprising indeed, and I leave it to the interested reader to work out a plausible example using real volumes instead of b and S (hint: for an example that works out with minimal sweat, try b=1ml and S=30ml).

Q.E.D (Quod Erat Demonstrandum: 'as has been demonstrated' as we were told at school, or as we preferred to translate incorrectly, 'Quite Easily Done.')

On the use of special soaps

January 2007 GW of Alabama emails me with the following advice:

Cleaning brushes - I have found a soap which has hemp oil and peppermint oil works for oil based paints. The brand readily available at health food stores here (Alabama, USA) is called Dr. Bronner's Peppermint Oil Soap. I was using it already for bathing and mouthwash and washing vegetables.

My reply:

I have not come across this product, so I cannot say whether using it as a mouthwash or for washing vegetables is safe.

I use "The Masters" brush cleaner (also made in USA). At first I didn't buy it because it is so expensive. However I found that it brought my stiffened brushes back to almost new, and then I reflected on the cost of artist's brushes! Also now I only use a very small quantity of turpentine to clean the brushes and the brush cleaner and water does the rest. I think this must be better for the environment.

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