Tag: bouncing drops

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A first coffee & science evening at Amoret

intro board for Amoret evening
An evening of coffee and science at Amoret Coffee in Notting Hill

A couple of weeks ago we hosted a first “coffee and science” evening at Amoret Coffee in Notting Hill. Designed to explore a physics concept that you could notice in your coffee cup with people from a diverse range of backgrounds, in some ways, the evening itself was an experiment. Would anyone turn up? Would the experiments be interesting? Was I just making my coffee badly?

This last question referred to the fact that the connection for that particular evening had been the dancing drops that skirt across the surface of a V60 (or other pour over) as you prepare your coffee. I had noticed these a couple of years ago but at that point had not appreciated their significance. To answer that question, we were prepared two excellent pour overs by people who really knew what they were doing. And we were spoiled for the coffee which was a recently roasted Nicaraguan washed coffee grown by the Baltodano family who also came along for the evening. The two pour overs were prepared very slightly differently and produced drinks that highlighted different aspects of the flavour of the coffee (though sadly I only managed to try one). This led to a fair amount of discussion amongst those present, not just about which they preferred, but how the preparation affected and highlighted different flavour notes.

Pour overs at Amoret
Preparing pour overs by two (slightly) different techniques. But would we see the dancing drops? (Yes x 2)

The pour overs showed that the dancing drops were there (in both techniques) when coffee was made properly. This was a relief for me! But did they also supply a clue as to how these drops were able to survive, as liquid drops, on the surface of the coffee?

Ordinarily, when a drop drips into a bath of liquid, you would expect it to quickly coalesce with the liquid bath. Once the drop gets close enough to the surface, the van der Waals forces in the drop and the liquid bath will overcome the surface tension effects and the drop will be subsumed into the liquid. If the drop does not coalesce, but instead appears to ‘float’ on the surface there must be a reason.

The first reason that the drop may survive for a while on the surface is because there is a temperature difference between the drop and the bath. This sets up stresses within the drop that pull air into the region between the drop and the bath and keep the drop ‘floating’ for a little while.

Secondly, if you increase the surface elasticity of the droplet, you can stabilise it on the liquid bath for longer. This is usually done by adding soap to the water, not something we did with the V60. But could there be an effect of the coffee oils or some other aspect of coffee chemistry that is keeping these droplets afloat?

Experiments at Amoret
You can see a drop almost ‘sitting’ on the surface of the water here (circled). This particular drop was stabilised for about 15 minutes. I think if you look carefully you can see a ripple pattern around the droplet in addition to the standing wave pattern on the surface of the water caused by the loud speaker underneath (indicated by the red arrow).

Lastly, if you vibrate the surface of the liquid bath, you can create conditions whereby the droplet ‘bounces’ on a cushion of air on the bath. It was interesting, that in the preparation of both pour overs at Amoret that evening, the times that we observed the dancing drops coincided with those times that the pour over was dripping into the coffee bath, causing a noticeable ripple on the surface.

This last condition was the subject of an experiment in the corner of the upper room at Amoret where we used a loud speaker to generate vibrations to two different liquid baths (water and soap water for example) to see if we could obtain stable drops on the surface. Astonishingly, some of the participants on that Tuesday managed to keep a droplet stable for about 15 minutes, you can see their droplet in the photo. The photo is interesting because if you look closely, not only can you see the wave on the bath of water caused by the vibration of the speaker, but you can also see a circular ripple pattern around the droplet. Is that the ripple caused by the droplet’s bounce?

Conversations led on to the fact that these drops were not just seen in pour overs but could occasionally be seen in espressos too. I’m definitely looking forward to the video of that one. While we also got to discuss the importance of different parameters on the stability of the drops – it turns out droplet diameter, as well as the forcing amplitude (which translates to, how loud you have the volume on the loud speakers) are key parameters that affect the behaviour of the drop, something that has been pointed out elsewhere.

V60 droplet floating bouncing sitting on coffee
A drop in-situ

The evening also emphasised just how much we have to talk to each other about! One topic that kept coming up was fermentation, specifically with how the coffee cherries are processed. Hopefully this could become the subject of a future conversation.

Future events are planned (in theory but not yet in practise) and so if you’d like to make sure you hear about them, you can sign up to the Bean Thinking events list here. Also, if you didn’t get chance to take part in the evening but would like to continue the discussion and maybe add your videos & comments about the droplets, you can sign up to the Virtual Coffee House which will be discussing this topic (until the next coffee & science evening).

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Uncategorized

Pour over science

floating, bouncing drops
How do you stabilise droplets of liquid water (or coffee) on a bath of water? And how long can you keep them on the surface?

On the 11th June, 2019 that is, in just under two weeks, we are going to try something exciting. Amoret Coffee in Notting Hill has offered to host the first ever “Bean Thinking’s evening of coffee & science”.

The idea behind what will (hopefully) become a series of evenings is to host a space for discussion and observation, exploring the physics within a coffee cup. On the 11th, we’ll be looking particularly at the phenomenon of “walking” droplets of coffee. These droplets can move across the surface of a cup of coffee and exist for many minutes (even days) if the conditions are right. And the conditions are fairly easy to create: we’ll be creating several such ‘walkers’ in the spacious upstairs area of Amoret’s Notting Hill branch that people can play with.

You can see such drops in your coffee in the morning. But what connects them to an early idea in quantum mechanics?

Although it takes the creation of certain conditions to achieve long-lasting droplets, you can often see them as you prepare a V60 pour over or even when dragging your (single-use) take-away cup over the surface of a table to create resonances on the surface of the cup. They crop up quite frequently, but why are they there at all and why do some last longer than others?

In addition to exploring these questions experimentally, we’ll also be discussing why these droplets sometimes ‘walk’ across the surface of the coffee and how this relates to an early interpretation of the phenomenon of wave-particle duality in quantum physics. How does something that you can sometimes see while brewing your coffee in the morning relate to the idea that fundamental particles such as the electron behave both as particle and wave? And what does this mean anyway?

It is hoped that future evenings will cover other topics such as climate change and coffee stains, I also hope that we will be lucky enough to have some of the coffee farmers that Amoret has direct-trade relationships with in order to explore these connections further. But, that is in the future, this time we are sticking with the fundamentals!

coffee at Watch House
There’s a lot of physics in a coffee. What do you see? Find out more at Amoret Coffee, 11th June 2019 or sign up to our events list.

So, if you are in London on the 11th June and would like to explore some physics in your coffee (or some coffee with your physics), please do come along to Amoret, from 5pm, for an evening of conversation and poring over science. We will be keeping people informed of plans for the evening (and for future evenings) via our events mailing list which you can sign up to here, or you can follow our progress on our Facebook events page here. Meanwhile, it would be helpful for planning reasons if you could let us know if you are coming either by signing up on Facebook or by emailing us. Looking forward to meeting some of you on the 11th.

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General Home experiments Observations slow Tea

Coffee and cream baubles – not just for Christmas

floating, bouncing drops
Drops of water can be stable on the water’s surface for many minutes if you put the water on a loudspeaker, more info on how to create these at home here.

You may have noticed them before: balls of liquid dancing on the surface of your coffee (or tea) that seem to last for ages before being absorbed into the drink? Perhaps you have added milk to your coffee and noticed that it took some time before the milk entered into the brew?

It turns out, there’s some very interesting physics that is happening whenever you add milk to your tea or when you are preparing a pour-over. It can link coffee to wine and to quantum mechanics. It is worth taking a closer look at these drops.

You may remember that you could use a loud speaker to make droplets of coffee bounce on a cup of the same. The vibrations in the cup meant that the air between the droplet and the drink never got squeezed out of the space between them. So, rather than coalesce, the drop jumped up and down on the coffee surface before finally disappearing under. This type of bouncing bauble has been shown to behave in similar ways to quantum particles in wave-particle duality. An analogue of quantum physics in the macroscopic droplets on the surface of your drink.

But that type of bauble required the use of a loud speaker (or some similar way of generating vibrations on the surface of the coffee). What if you could ‘bounce’ a drop of coffee on a cup of coffee without any external props like speakers? Well, it turns out that you can. In November 2017 a group of researchers showed how a temperature difference between a drop falling into a drink and the drink itself could result in the drop appearing to float on the surface of the drink for many seconds. The obvious example was cold milk into a cup of coffee (or tea). But I think that it may also happen in a V60 when you prepare a pour over, more on that below.

science in a V60
Bubbles of liquid dancing on the surface of a brewing coffee.

The idea is quite simple. If there is a temperature difference between the drop and the coffee, when the drop approaches the coffee, there will be thermal gradients across the drop/cup system. Surface tension is temperature dependent: the higher the temperature, the weaker the surface tension. Differences in surface tension across the surface of a liquid result in compensating liquid flows (one of the best places to see this is in a glass of wine, but there’s also a great party-trick experiment you can do to demonstrate it which is here). So, because there is a temperature difference across the surface area of the droplet (owing to the difference between the droplet and the cup), there will be liquid flows set up within the drop. These flows are like circulating vortices which draw the surrounding air into the gap between the drop and the cup and so prevent the existing air between the drop and the cup from escaping. If the air has nowhere to escape to, the drop can’t merge with the drink, in fact it ‘levitates’ for a number of seconds.

The authors suggest that this is a reason that you can often see rain drops staying on the top of puddles or ponds before being subsumed into the water, or why you can see the cream (or milk) stay as globules on the surface of your coffee (or tea). And so I wonder, could this also be the explanation for an odd phenomenon that I sometimes notice while brewing coffee in my V60. Perhaps you have seen this too? After some time, the new drops of filtered coffee impacting on the surface skit along to the edge of the jug. They stay as balls of coffee on the coffee’s surface for quite some time before becoming part of the brew. You can see a photo of some of these droplets above. Initially I thought that this was because the surface of the coffee had started to vibrate with the impacting droplets. But it is also possible that it could be this temperature effect. As the (brewed) coffee in the jug would be cooler than the water dripping into it from the filter, there would be a temperature difference between the droplet and the coffee but the reverse of the milk-coffee situation. The drop would be warmer than the coffee it’s dripping into. The authors of the study suggested that it was the magnitude of the temperature difference that was the key, not the sign of the temperature difference. So that would fit with the V60 observations seen previously. However how would you show which effect (vibration or temperature difference) is responsible for the behaviour?

Enjoy playing with your tea, coffee and V60s. Do let me know the results of your experiments. Is it a vibration thing or does the temperature difference have to be there to begin with? Let me know what you think is going on.

I am also grateful to Amoret Coffee for alerting me to this story in the first place through Twitter. If you come across some interesting coffee-science, please let me know, either here in the comments section (moderated, please be patient), or on Twitter or Facebook.

 

 

 

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Coffee cup science General Home experiments Observations

Coffee, chaos and computing

Have you ever noticed drops of coffee skipping across the surface of your coffee as you have been preparing a V60? Or watched as globules of tea dance on the resonating surface of a take-away dragged across a table top? The dancing drops can be seen in this video of coffee being prepared in a V60:

These droplets are the result of some fascinating physics. Although we have encountered them on the Daily Grind before (here and here), the more physicists study them, the more surprises they throw up. While the droplets can be considered particles, they are guided around the coffee pot by the surface waves they create as they bounce. In a sense they are a macroscopic example of wave-particle coexistence. There is a significant temptation to explore whether they have relevance for the concept in quantum physics of wave-particle duality. But another aspect of this wave-particle coexistence has recently been shown to produce a different and unexpected connection. A connection between chaos and computing. And as you can create these droplets in coffee, perhaps we could say a connection between coffee, chaos and computing.

floating, bouncing drops
Drops of water can be stable on the water’s surface for much longer than 1 minute if you put the water on a loudspeaker, more info on how to create these at home here.

It is fairly simple to create these surface droplets in coffee at home. The secret to getting stable droplets on the surface is to create a vibration, a wave, on the surface of the coffee liquid. The droplets that then form on (or are introduced to) the surface ‘bounce’ on this wave. If you wanted to create surface droplets reliably at home, you would put your coffee on a loud speaker. I suspect that the reason that they appear in a V60 is that the first drops set up a standing wave on the surface of the coffee that acts to support later drops as they encounter the surface. If anyone has a different theory, please do let me know.

But how is it possible that these bouncing droplets connect chaos theory and computing? It is a consequence of the way that the globules of coffee on the surface interact with the waves that guide them around the coffee. Consider for one moment a particle bouncing around a confined space (the traditional example is of a ball on a billiards table). On an ordinary table, the billiard ball will behave quite predictably, start it off aimed roughly at the side of the table and it will bounce in an easily describable way. But if you make the ends curved or put circular objects in the middle of the table for the ball to bounce off, small differences in initial direction can result in large differences in the final path of the ball (for more details and an animation see here). The billiard ball behaves chaotically, and the initial path cannot be found from the final position, there is no way to re-trace the path of the ball, it is not “time-reversible”.

science in a V60
A still from the video above showing three drops of coffee on the surface.

The droplet bouncing on the liquid surface appears to move chaotically, just as the billiard ball on a circular table. However, unlike the billiard ball, the droplet is not a mere particle, but a particle linked to a self-generated surface wave. Each time the droplet bounces on the surface, it creates a small wave, like ripples on a pond. The path taken by the droplet is a complex interaction between this self-generated wave, the vibration keeping the droplet bouncing and the droplet itself. This means that if you are able to shift the phase of the bounce by 180º (meaning, that rather than bounce on an upward motion of the surface, the drop bounces on a downward motion or vice versa), the bouncing droplet not only reverses the direction it travels in, it retraces its path. Rather than behave as the chaotic billiard ball, the path taken by the seemingly chaotic globule of coffee can be exactly reversed.

Which is where the link with computing comes in. It is as if each “bounce” of the droplet “writes” information on the surface of the coffee in the form of a wave. The subsequent bounces “read” the information while the reversal of the direction of the bouncing droplet “erases” the stored information by creating a surface wave opposite to the initial one. The authors of the recent paper suggest that “in that sense [the walking droplet can] be termed as a wave Turing machine”, giving the final link to computing.

Whether or not this turns out to be useful for computing is, to me, almost irrelevant. What is interesting is that such a simple phenomenon, that anyone who makes pour-over coffee should have seen fairly often, is linked to such complex, and fundamental physics. If you would like to read more, there is a great summary article here while the actual paper is here.

 

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Home experiments Observations slow

Something brewing in my V60

kettle, V60, spout, pourover, v60 preparation
The new V60 “power kettle”

It was my birthday a short while ago and someone who knows me well got me a perfect present: a kettle specially adapted for making pour-over V60 style coffees. Until this point I had been struggling with a normal kettle with it’s large spout but now, I can dream that I pour like a barista. Of course, it is important to try out your birthday present as soon as you receive it. And then try it again, and again, just to make sure that it does really make a difference to your coffee. So it is fair to say, that recently I have been enjoying some very good coffees prepared with a variety of lovely beans from Roasting House and my new V60/V60 kettle combination.

Spending the time to prepare a good coffee seems to make it even more enjoyable (though it turns out that whether you agree with this partly depends on why you are drinking coffee). Grinding the beans, rinsing the filter, warming the pot, the whole process taken slowly adds to the experience. But then, while watching the coffee drip through the filter one day, I saw a coffee drop dance over the surface of the coffee. Then another one, and another, a whole load of dancing droplets (video below). Perhaps some readers of Bean Thinking may remember a few months back a similar story of bouncing droplets on soapy water. In that case, fairly large drops of water (up to about 1cm wide) were made to ‘float’ on the surface of a dish of water that had been placed on a loudspeaker.

Sadly, for that initial experiment the coffee had been made undrinkable by adding soap to it. The soap had the effect of increasing the surface viscosity of the droplets which meant that the drop could bounce back from the vibrating water surface before it recombined with the liquid. Adding soap to the coffee meant that these liquid drops could ‘float’ (they actually bounce) on the water for many minutes or even longer (for more of the physics behind this click here).

science in a V60
A still from the video above showing three drops of coffee on the surface.

On the face of it, there are some similarities between the drops dancing on the coffee in my V60 and these bouncing droplets. As each drop falls from the filter, it creates a vibration on the surface of the coffee. The vibration wave is then reflected back at the edges of the V60 and when the next drop falls from the filter it is ‘bounced’ back up by the vibration of the coffee.

But there are also significant differences. Firstly, as mentioned, there was no soap added to this coffee (I was brewing it to drink it!). This means that the viscosity of the drops should be similar to that of ordinary water. Although water drops can be made to bounce, the reduced viscosity means that this is less likely. Secondly, the water is hot. This acts to reduce the viscosity still further (think of honey on hot toast). Perhaps other effects (such as an evaporation flux or similar) could be having an effect, but it is noticeable that although the drops “live” long enough to be caught on camera, they are not very stable. Could it be that the vibrations caused by the droplets hitting the coffee are indeed enough to bounce the incoming droplets back up but that, unlike the soapy-water, these “anti-bubbles” do not survive for very long? Or is something deeper at play? I admit that I do not know. So, over to you out there. If you are taking time to make coffee in a V60, why not keep an eye out for these bouncing droplets and then do some experiments with them. Do you think that the bounce vibration is enough to sustain the bouncing droplet – does the speed of pour make a difference? Is it associated with the heat of the coffee? I’d be interested to hear what you think.

(The original soapy-coffee bouncing droplet video).

If you see anything interesting or odd in your coffee, why not let me know, either here in the comments section below, e-mail, or over on Twitter or on Facebook.

Categories
General Home experiments Observations Tea

Bouncing Coffee

floating, bouncing drops
Water droplets ‘floating’ on a bath of water (actually they bounce rather than float).

Perhaps you remember the video about how to ‘float’ coffee droplets on water posted on the Daily Grind a few weeks ago? The video featured an experiment that you could do at home in which droplets of water (or coffee, or even, if you were feeling adventurous, tea) could be made to stay as spherical droplets on the surface of a shallow dish of water for minutes at a time. Of course there were a few tricks: The water had soap added to it (10ml of soap to 100ml of water) and the shallow dish was on a loudspeaker which was playing music at the time. The whole experiment was very pretty. But hopefully as well as appreciating the aesthetics, you were asking ‘how’ and ‘why’? Why does the addition of soap mean that these globules of liquid appear to float on the liquid surface? And is the rumour you have heard about a connection with quantum physics true?

Well it turns out that people have known about these floating droplets for over a hundred years but why they behave as they do is still being investigated. It is another case of cutting-edge science appearing in your coffee cup*. So it’s worth taking a look at what is going on and why we needed to add soap and vibration for the droplets to remain stable on the water surface.

lilies on water, rain on a pond, droplets
When it rains, the rain drops don’t float on the pond

It seems to appeal to common sense and to everyday experience that if we drop a droplet onto a bath of water, the droplet will merge with the water and become part of the bath. After all, when we bring two drops that we have dripped on a table close to each other, at a certain distance between the two drops, they appear to touch and then rapidly merge into one big droplet (try it). And when it rains onto a pond, we don’t see lots of spherical droplets hovering over the surface of the pond! We know that it is the attractive van der Waals forces that bring the two drops together and then the effects of surface tension that minimise the surface area of the drops so that they become one big drop. So how is it that we can get a droplet to remain, as a droplet, on the surface of a bath of water?

How to bounce water droplets on a water surface

It could be said that the answer can be pulled out of thin air: Before the drops can merge, the air that separates them has to escape from the area between the droplet and the water bath. If the droplet can somehow be made to bounce back upwards before the air separating the droplet from the bath becomes thin enough for the two liquids to combine, the air could be made into a cushion to keep pushing the droplet upwards. This is why the experiment needs to be done with a vibrating dish of water, each time the surface vibrates upwards it is providing the drop with an acceleration upwards that overcomes gravity, like a miniature trampoline: The droplet is not floating, it is bouncing.

So why soap? We all know that the addition of soap decreases the surface tension of the water. But that is not why the addition of soap helps to stabilise the drops in this instance. No, soap has another effect and that is to increase the surface viscosity (and surface elasticity) of the water. Think about the air between the droplet and the dish. As the droplet bounces down (ie. the distance between droplet and water becomes a minimum), the air gets squeezed out of the layer between the droplet and the bath. On the other hand, as the droplet reaches its peak height, air will rush into the gap between the drop and the bath. If the liquid is not very viscous (eg. water), as the air rushes in (or gets squeezed out), it will combine with the liquid and form a turbulent layer on the surface of the droplet. If the viscosity is increased, the air cannot ‘entrain’ the liquid as the droplet bounces and so the drop keeps its shape more easily and is more stable. Soap increases the surface viscosity of the droplet and so helps with this effect. However soap also increases the surface elasticity and makes it harder for the air to flow out of the layer separating the drop from the bath. It is because soap does multiple things to the water (or coffee) that more recent studies have focussed on liquids with controllable viscosity but minimal surfactant effects, i.e. silicone oils. It is just that if you want it to work with coffee, it is easier to add the soap to get the experiment to work.

An “un-cut” video of coffee on water shows how tricky it can be to actually get these drops to be stable on the surface of the water.

Which leaves the quantum link. The experiment shown in the videos show single droplets (or droplet patterns) stabilised by vibrations caused by music. If instead of music you use fixed frequencies to excite resonances through the speakers, it is possible to get the droplet to resonate in a controlled way and, at a certain point, it will move. As the droplet moves, it appears to be guided by the vibrations of the liquid underneath the drop, it is a particle guided by a ‘pilot wave’. It turns out that such walking droplets show behaviour reminiscent of the ‘wave particle duality‘ found in quantum physics where particles (such as electrons and other sub-atomic particles) can be described both as particles and as waves. You can find a video describing the similarities between these bouncing droplets and quantum effects here.

 

* Ok, so you may not want to add soap to your coffee to see this effect but actually I first observed it in a milky tea. Adding milk to the coffee/tea would increase its viscosity which makes the observation of the bouncing droplets more likely. The ‘milk’ used in the video was actually soya milk which did not appear to increase the viscosity sufficiently to allow the droplets to bounce on the surface without soap.

Categories
Home experiments Observations slow Tea

Coffee baubles

resonating coffee
Not the best image of a resonating coffee but you hopefully get the idea

Most people, at some point in their lives, must have pushed a take-away coffee cup across a table and watched as patterns form on the liquid surface. Sometimes these patterns seem to stand still, we’d say that they form ‘resonances’. On even rarer occasions, on dragging your cup across the surface, you may have seen coffee droplets jump out of the coffee and then dance on the coffee surface for a couple of seconds as the liquid vibrates.

Today’s Daily Grind investigates these ‘floating droplets’ with an experiment in time for Christmas: Decorate your coffee with coffee baubles.

To make these droplets form on your coffee in a controllable way you will need a few bits of equipment:

  1. A couple of loud-speakers with the woofers exposed
  2. Some sort of liquid soap (washing up liquid, hand soap, soap for hand washing clothes etc)
  3. Some water (or coffee but you will do horrible things to it)
  4. A shallow dish (I used the bottom of an old yoghurt pot)
  5. A “dropper”, a pipette or syringe would be ideal, a straw will probably work.

You can do this completely systematically, in which case you’ll also need a signal generator to provide a fixed frequency output to the speakers (I used “ScorpionZZZ’s Lab, Signal Generator Lite for iPhone). Or you can just go straight to the fun bit which is to make these droplets dance to music. It’s Christmas so it’s entirely up to you!

floating drops, resonances, speakers, kitchen top science
Balance a shallow dish on the woofer of a speaker. A roll of sellotape can be used to couple the vibrations of the speaker to the dish if necessary.

Balance your speakers on a flat surface and put the shallow dish so that it sits in good contact with the woofer. Because my dish was ever so slightly larger than the vibrating bit of the speaker, I ‘coupled’ the speaker to the dish with a roll of sellotape. Mix 10ml of soap with 100ml of water (this does not have to be exact but you may want to investigate just how much/little soap you can get away with). If you are using coffee rather than water, you will need to mix 10ml soap with 100ml coffee.

Pour about half the soapy-water into the dish and then turn the speakers on. If you are using a signal generator, watch what happens as you sweep the frequency from 10-200 Hz. Now, either choose a frequency which shows a nice resonance pattern on the water, or start playing the music through the speakers. Music with a good beat will work well (I watched drops dance to Tiesto, Blondie, and Josh Woodward’s “coffee”).

Drip a drop of the remaining soapy-water onto the resonating surface. A video of my playing with these droplets can be seen above. Although not all the drops will float, it is fairly easy to start to form patterns of flowers or rows of droplets and then it’s worth just playing.  How big a droplet can be made to float without collapsing? How many minutes can you get a drop to last before it sinks? What happens if you combine a drop of black (soapy) coffee with a drop of milky (soapy) coffee?

Have fun, and please do share your videos and photos of your experiments with me on Facebook or Twitter.

Disclaimers & Credits:

No coffee was wasted in the making of this video. A very good coffee from Roasting House was thoroughly enjoyed before the remnants were diluted and mixed with soap.

Inspiration & experimental details taken from Jearl Walker’s great article “The Amateur Scientist” in Scientific American, p. 151 (1978).