Science of a coffee cup: A physicist walks into a bar… Open Access

Visual experiments that can be witnessed in everyday life are a good way for physics teachers to capture the interest of their audience. They can be used as easy-to-set-up, low-cost, and elegant demonstrations of general principles encountered in textbooks. A common way to combine theoretical lectures and experiments consists in bringing together different experiments that all illustrate the same physical feature. Here, we adopt the opposite attitude and investigate many different physical phenomena with a single experimental setup. The common denominator between all areas of physics is coffee, and the choice of experimental setup naturally went toward a coffee cup filled (or not) with ordinary tea-time beverages.

This focus on a single, rather mundane physical object and the various physical phenomena associated with it presents several advantages. First, it allows students to observe and experiment freely; there is no need to railroad them toward a particular physical theory. Instead, the teacher can build on students' spontaneous observations and ask them to form hypotheses, which can then be further tested and refined. Many pedagogical strategies and thought-provoking questions can be deployed along the way; in this article, we focus on the underlying science as a support for teachers. Second, the choice of this very low-cost setup also allows students to experiment by themselves, removes safety issues, and makes it accessible to every classroom. Lastly, it also offers the opportunity to connect different areas of physics, thus inviting students to build bridges between different approaches taught in different contexts.

Of course, the physics of the coffee cup has been extensively discussed, in particular for teaching purposes (see, for instance, Refs. 18 and 38 and many references discussed hereafter published in teaching journals). However, these works either focus on a single phenomenon or only discuss quickly and qualitatively several of them. We propose here a more comprehensive approach, which includes quantitative analysis of experimental data and explanation from scratch of a minimal model. All experiments are carried out using standard laboratory equipment and can easily be replicated in a classroom setting.

This work is written to be as self-contained as possible, and each section can be read separately. However, the order in which we chose to present the experiments is not arbitrary and aims to build a consistent story. We first start with an empty cup (Secs. II and III), before filling it with various liquids (Secs. IV–VIII). Each section is devoted to an interesting physical feature and is organized as follows: an observation of a given phenomenon is first described (videos are also available) as part of a physicist's journey in coffee cup physics. We then propose, when possible, a quantitative analysis of the experiments and a minimal model explaining the observation. We finish each section with a short discussion of related effects in other contexts or more advanced works existing in the literature.

A physicist walks into a bar on his way to the lab.

He takes a seat, pulls out some coins, and checks if he has enough for a coffee [Fig. 1(a) ]. As he fidgets with a few of them, he notices something strange: when he rolls one coin around another coin of equal size, the first coin completes two full rotations on its axis. His intuition told him it would only rotate once—after all, it only needs to travel once around the other coin's circumference!

Curious, he measures the trajectory of a point on the edge of the rolling coin [Fig. 1(b) ]. It only touches the central coin once, yet it completes two full rotations. Confident he's resolved the paradox, the physicist leans back with a satisfied sigh.

Then he remembers: he still hasn't ordered his coffee.

Looking around, he sees no waiter but notices an empty cup sitting on the counter. As he peers inside, he notices a pattern of light [Fig. 1(c) ] resembling the path traced by the coin. Intrigued, he starts jotting down calculations on a napkin to see whether this similarity is a coincidence or not.

Such an envelope is called a caustic. The particular curve we obtained by rolling coins, plotted in Fig. 1(h), is called a cardioid.⁷ It was originally discovered in the context of gear mechanics by Yates.⁴³ In the same time period, at the end of his celebrated treatise on light, Huygens studied the geometry of a optical caustic.²¹ 

Moving the light source around does not destroy the caustic, but slightly deforms it instead. Using catastrophe theory and increasingly refined theories of light, Nye classified caustics into a few universal types and revealed their connection to a hierarchy of three-dimensional features: wave dislocations, disclinations, and polarization singularities.²⁸ Caustics can be seen in many places: they ripple at the bottom of clear lakes and hide in the raindrops on your windows. Even in cups, they appear in more than one way. For example, pour hot water into a cup; you might see caustics moving around, following the convective motions of the liquid.

Satisfied with his analysis, the physicist looks around for someone to share his insight. Still no one. He taps the cup with a spoon [Fig. 2(a),], hoping to attract attention. Nothing. He hits the cup again [Fig. 2(b) ], and is very surprised to hear that the sound pitch was quite different from the previous impact. He taps the cup again in both places to confirm it: two distinct tones. Now thoroughly intrigued, he begins a systematic investigation, tapping the cup at different points and analyzing the frequencies with his phone.

Qualitatively, one can notice that the pitch varies with the hitting position. It is highest when we hit the edge of the cup at a 45° angle from the cup handle and lowest at a 90° angle. The notes produced also repeat when we increase angles by 90° increments. This is confirmed quantitatively by the spectrograms of Fig. 2(d), which show the Fourier spectrum of the emitted sound as a function of the impact's angle. The energy is mostly concentrated in two modes with frequency $f1=2800$ Hz and $f2=2940$ Hz and can be computed as $E(fi)∝∫−δfδf|ŝ(fi+f)|2df$⁠, with $ŝ$ being the Fourier transform of the sound signal and $δf=40$ Hz to integrate over the whole bandwidth of each peak [dashed line in Fig. 2(d)]. From this, one can compute the relative energy densities $ri=E(fi)/[E(f1)+E(f2)]$ stored in each mode, which strongly depends on the angle $θ$ as shown in Fig. 2(e).

The musical note we hear is due to the vibration of the cup's circular edge, which can be checked by pinching the edge with two fingers to absorb the sound. After an impact, a stationary wave is established, and the 90° periodicity indicates that the oscillation of the cup's edge is shaped like an ellipse, as shown in Figs. 2(f) and 2(g). The wavelength of the deformation wave is, therefore, half of the cup's perimeter, and the vibration mode exhibits four nodes and four antinodes separated by 45° intervals.

If the cup had no handle, the location of the impact would be irrelevant due to the circular symmetry of the problem. However, the handle breaks this symmetry, which induces the frequency splitting that we observed experimentally. If we hit at $θ=0°$⁠, we excite antinodes at $θ=0[90°]$⁠, while a hit at $θ=45°$ excites nodes at those positions [Figs. 2(f) and 2(g)]. The handle sits either on an antinode or on a node, changing the effective mass seen by the wave in Eq. 4. From this formula, we expect the frequency to be lower when the handle is on an antinode, as the wave has more mass to carry during the vibration. This is perfectly consistent with our previous observations.

The phenomenological approach used here can be made rigorous by performing a full analysis of the cup's vibration. This analysis yields an equation similar to (4), with m and k being the modal mass and stiffness of the mode, respectively; it also produces the expected spatial distribution of the vibration.³¹ The degeneracy lifting, and more generally the modification of acoustic properties due to localized defects, is nowadays used routinely in civil engineering or manufacturing to perform noninvasive control of elastic structures.^1,14,19

The physicist happily concludes that the difference in pitch arises from the ”lifting of a spectral degeneracy,” dependent on the handle's mass relative to the cup.

At this point, our physicist notices a server standing right across him, with an increasingly annoyed expression after five solid minutes of tinkling sounds. Sheepishly, he orders an espresso [Fig. 3(a) ]. When it arrives, he gives it a stir. In doing so, he taps the bottom of the cup with his spoon and hears a hollow sound, quite different from before. He repeats it and hears the pitch rise slightly. He tests it several times, confirming his observation, and records a spectrogram with his phone.

We plot in Fig. 3(b) the acoustic signal recorded along time. Each temporal peak corresponds to an impact that displays peculiar frequency peaks [Fig. 2(c)]. The first peak at frequency $f(t)$ corresponds to the acoustic mode in the liquid column discussed above. When the Fourier transforms of the impacts are stacked together, as in Fig. 3(d), one obtains a spectrogram that shows the evolution of the first peak along time. The three experiments are performed with the same coffee (and thus the same amount of liquid), but the latter is vigorously mixed before each experiment to re-inject air bubbles in the liquid bulk. For each experiment, the frequency of the first peak increases along time, which is consistent with the sound heard during the experiment.

Many experiments have shown that gas bubbles not only change the sound velocity but also greatly increase the absorption of sound in liquids.^15,25,40,41 Such effects can be heard qualitatively in our experimental video. The model presented here is quite simple, as we have used the compressibility of the air to model the deformation of the bubbles regardless of their radius or shape. In case the bubbles start to resonate, the sound velocity will be impacted in a more complicated way.³⁹ The effect presented here is at the heart of the Broadband Acoustic Resonance Dissolution Spectroscopy (BARDS) method used to probe the fraction of gas in a liquid during a chemical reaction.^16,17 Acoustic measurements over time allow, for instance, to monitor powder dissolution in real-time, even when several reactions occur simultaneously, thanks to the unique acoustic signature of each reaction.¹⁷ 

After spilling a fair amount of coffee on the counter and sounding his cup for several minutes, our physicist catches the increasingly menacing glare of the server. Growing uneasy, he decides to order a second drink to placate the server. In need of inspiration, he glances at his counter neighbor, who has ordered tea and is stirring it thoughtfully. He watches as the tea leaves dance at the bottom of the cup and eventually all gather at the center of the cup [Figs. 4(a) and 4(b)].

At first sight, the experiment is really puzzling for the intuition. Since the tea leaves are initially at the bottom of the tea cup, it means that their density is higher than that of the surrounding water. Applying a centrifugal force to the liquid should, therefore, eject them on the outside of the cup. Solving this paradox requires to perform a full analysis of the flow that is generated by the moving spoon.

We recover some kind of Bernoulli relation, but the sign in front of the dynamical term is inverted compared to the usual case. Indeed, if one applies naively Bernoulli's principle, the pressure expresses as $P=C−12ρΩ2r2−ρgz$⁠, with C being a constant, which contradicts the previous equation. The resolution of this apparent paradox lies in the fact the pressure in Bernoulli's principle is computed along a given streamline. In particular, the (non-)constant C depends on the circular streamline that is considered in our rotating fluid. If the flow is irrotational, the constant is the same for all the streamlines and Bernoulli's relation is thus often used to compute pressure field globally in this context. Since the present flow is heavily rotational, this method cannot be applied here.

Let us now imagine that we replace a fluid particle by a solid one of the same density. In this case, the centrifugal force (or equivalently, the dynamical term in the pressure field) is compensated by the hydrostatic pressure term due to the deformed interface, and no motion should arise [Fig. 4(d)]. This is true in the bulk of liquid, but the tea leaves are concentrated at the bottom of the cup. Since they are heavier than the fluid particles they replace, they should still be pushed toward the outside as the centrifugal force acting on them overcomes the horizontal pressure gradient. At this stage, the paradox therefore remains unsolved.

There is, however, a big difference between rotating the tank with the liquid or rotating the liquid only. In the latter case, the edges of the tank (including the bottom) remain still while the liquid bulk performs solid rotation. If one imposes a no-slip boundary condition, there must, therefore, exist a transition between the still liquid attached to the bottom and the rotating liquid bulk. This transition occurs on a small area called a boundary layer, which physically corresponds to the small portion of space where viscous effects cannot be neglected. Its vertical size $δ$ can be estimated from dimensional analysis. Assuming it depends on the rotation speed and the kinematic viscosity only, the length scale that can be built from those quantities is $δ∼ν/Ω∼1$ mm. A similar boundary layer appears in the von Karman flow near a spinning disk, for which analytical computation gives a similar expression for the size of the boundary layer.²² 

As the velocity is zero at the bottom to fulfill the no-slip boundary condition, only the hydrostatic term remains and the isobar $P2(x)$ in the boundary layer bends toward the outside of the cup [Fig. 4(e)]. At the bottom of the cup, the pressure is thus higher on the outside of the cup than near the center. The resulting pressure gradient will push the fluid particles at the bottom toward the cup's center. As water is incompressible, these particles will be pushed by the ones following them, and an ascending current will be established. Following this path, a global rolling circulation eventually establishes in the fluid as sketched in Fig. 4(f). The tea leaves being heavier than the fluid, they cannot follow the ascending current and remain trapped at the bottom center where the ascent starts. On the other hand, floating particles at the surface will be pushed near the edges, as shown by one of the floating leaves in our video.

This phenomenon is an example of bulk transport induced by boundary layer effects, which occur in many different contexts. For instance, it explains why sediments aggregate on the inward side of river meanders.^6,13,24,35 It also explains some atmospheric circulations between low- and high-pressure areas.³⁴ The tea-leaf effect is also used in laboratories to aggregate particles for manufacturing purposes⁴⁸ or in blood–plasma separation.⁴⁴ Note that the use of a 'deformable spoon' was also recently investigated in the tea-leaf paradox and the stirrer-flow coupling exhibits rich phenomenology.³ 

Feeling a little dizzy from looking at vortices for so long, and made uncomfortable by the inquiring look of his tea-mixing neighbor, he decides to skip the tea and opts for a latte. The waiter, showing evident expertise, prepares a beautifully continuously layered latte, going from dark brown on the top to full white at the bottom [Fig. 5(a) ]. After a couple of minutes, however, the color layered turned discrete, showing sharp edges between the different color areas [Fig. 5(b)].

Due to its initial velocity, the espresso penetrates in the denser milk (or, to ease the experiment, salty water as in Ref. 42) until buoyancy stops it. After a short transient regime, the mixture becomes darker at the top than at the bottom [Fig. 5(a)], which indicates that there is more dyed water, and so less salty water, at the top of the cup. A darker region is thus associated with a smaller liquid density, and the color distribution directly indicates the vertical density stratification in the fluid. However, there also exists a temperature difference between the center of the cup and the edges due to thermal exchanges [Figs. 5(c) and 5(d)]. Due to the temperature gradient, the liquid at a given height is denser at the edges than in the center. There are, therefore, two gradients in the fluid: a vertical density gradient and a horizontal temperature gradient that also impacts the density profile.

Due to molecular and thermal diffusion, both quantities will eventually become homogeneous if one waits long enough. However, the typical time scale at which those processes occur is not the same. The ratio between the two can be estimated by comparing the diffusivity constants, which are $DT∼10−3$ cm²/s for temperature and $Dc∼10−5$ cm²/s for salt diffusion in water. From this, one can deduce that the diffusion of salt is $DT/Dc∼100$ times slower than thermal diffusion, showing that the density gradient will remain much longer (typically several days) than the temperature one (which disappears after typically 1 h).

We can now consider what happens to a fluid parcel in the cup, as in Fig. 5(e) and Ref. 27. If the fluid parcel is initially located at the edge, it will cool down due to thermal transfer with the outside. As it does so, its density becomes higher than that of the surrounding fluid, meaning that it starts to sink. If the mixture were uniform (same color everywhere), the fluid parcel would sink to the bottom of the cup, creating a large convective cell encompassing the entire fluid. However, there exists a density gradient in the fluid so that the fluid parcel cannot go beyond a certain depth due to buoyancy. However since other fluid parcels are following it, our fluid parcel must now perform horizontal motion along this effective bottom to leave space for the incoming ones. When it comes closer to the center, the parcel heats up again, its density decreases, and it starts to rise again. The horizontal temperature gradient, therefore, generates large circular flows, which are called convection cells [Fig. 5(f)]. Due to the vertical density gradient, several horizontal convection cells appear on top of each other in the cup. The generated flow performs mixing within the cell, which explains the appearance of uniformly colored layers separated by color discontinuities. The latter materializes the separation between two adjacent convection cells.

The existence of time scale separation between thermal and salt diffusion is crucial for the previous explanation to hold, as it allowed to consider a fluid particle of constant salt concentration during a full thermal cycle in Fig. 5(e). If salt diffusion were faster than thermal diffusion, the particle would sink to the bottom by changing its salt concentration along its trajectory, and the discrete layer would not appear. We can estimate the size of a horizontal layer in the following way. During the cooling phase, the density varies by $ΔρT=ρwαΔT$⁠, where $ρw=1.0$ kg/l is the water density, $α=2.6×10−4$°C⁻¹ is the thermal dilatation of water, and $ΔT∼50$°C is the temperature drop during the cooling phase. On the other hand, the spatial density profile due to varying salt concentration can be estimated from the color profile. The salty water had salt concentration of about 0.1 kg/l, and thus a density $ρs≈1.1$ kg/l, while the colored water has the same density as water $ρw=1.0$ kg/l. From Fig. 5(b), we see that the color gradient is approximately linear and goes from fully white (no colored water) to fully brown (only colored water) on a length $L≈4$ cm. Along the color gradient, going down by a length d thus leads to density variation of $Δρs≈(ρs−ρw)×d/L$⁠. As explained before, a cooled-down, less salted liquid volume will stop to sink after a distance d when its density matches the one of the hotter, more salted liquid surrounding it. Along the vertical direction of a horizontal layer, both density variations should, therefore, be equal $Δρs=ΔρT$⁠. From this, we recover $d=αLρwΔT/(ρs−ρw)≈5$ mm. This is remarkably close from our experimental observation on Fig. 5(b) where a layer has a typical size of $d≈6−7$ mm. Alternatively, this method can also be used to estimate the thermal dilatation coefficient from the observed height d of mixing cells, which is found to be $α≈3×10−4$°C⁻¹, in reasonable agreement with the exact value.

This phenomenon, called “double diffusion convection,” not only occurs in latte^8,42 but also in the oceans^30,32 or during iceberg melting,²⁰ and is a decisive mixing mechanism in geophysical flows. A similar analysis can also be carried to explain the formation of salt fingers.³³ 

Both amazed by the formation of discrete layers and disappointed by the terrible taste of this salty latte, the physicist finally decides to leave the bar, realizing that night fell and that he never made it to the lab. However, just before leaving, he notices that the coffee droplets he spilled during the mixing dried on the table. Intriguingly, these coffee stains are not uniformly brown but rather exhibit noticeably darker edges [Figs. 6(a) and 6(b) ].

One should first describe what coffee is from a physical point of view. Although coffee molecules are very small (about $∼1$ nm for caffeine), they exist in an expresso in the form of clusters of about $∼10−100μ$m diameter. Coffee, therefore, consists of a suspension of coffee-molecule clusters in water. When a coffee droplet is placed on a surface, the contact angle $θc$ [Fig. 6(c)] at the edge of the droplet depends on the surface properties and can go from 0° for wetting surfaces to almost 180° for very hydrophobic ones. In our experiments, we used a PMMA plate on which coffee drops exhibit good wettability (⁠ $θc<90°$⁠). When a droplet is deposed, it starts to evaporate and the droplet should therefore shrink. Naively, one could think that evaporation occurs homogeneously and that both radius and height will diminish over time, leaving a uniform stain on the table as in Fig. 6(d).

However, this picture is wrong because the contact line of the droplet is pinned on the substrate due to the microscopic roughness of the latter. Therefore, both the radius and contact angle of the droplet are fixed, as confirmed by the largely constant droplet size observed over the course of the evaporation in Fig. 6(b). If one still assumes uniform evaporation over the surface, then some liquid must flow toward the outside to compensate for the evaporating liquid at this location. Coffee colloids are transported with this flow toward the outside, which results in higher concentration at the edge of the droplet. Such migration was confirmed by particle imaging under a microscope.^11,37

This is still not the full picture, and a quantitative approach to the problem can be performed following Ref. 11. We consider the diffusion problem of water steam concentration $c(x→,t)$ in the air near the droplet. We will assume that a stationary regime is reached so that the diffusion equation reads $Δc=0$ in the bulk. At the edges, we impose $c=ca$ at infinity (with $ca$ the ambient concentration), $c=cs$ at the droplet's edges (with $cs>ca$ the saturating steam concentration in the air) and $∂n→c=0$ at the solid interface (no flux condition). Using the mirror image theorem from electrostatics, the solid surface acts as a perfect mirror, and the problem becomes equivalent to considering a symmetrized droplet without the substrate as in Fig. 6(f). Keeping the electrostatic analogy and using linearity, we are now looking for a potential c that verifies $Δc=0$ in the bulk, $c=cs−ca$ at the droplet's edge and $c=0$ at infinity. Due to the tip's influence, one expects $∇c$ (i.e., the electric field) to diverge at the angular point. This can be shown analytically using complex analysis and conformal invariance.⁴ This gradient, in our case, is the evaporation flux, which shows that the latter diverges at the contact line. The previously discussed hypothesis about uniform evaporation is quantitatively wrong, as more evaporation occurs at the edge of the drop than on its top. The flow induced by evaporation will, therefore, be stronger than expected, as more liquid needs to be sucked toward the outside.

The formation of circles strongly depends on the chosen experimental conditions. Among others, the surface roughness, the surface tension of the liquid, and the size of the particles can greatly impact the phenomenon^10,45,46 or even suppress it.⁴⁷ It can be seen qualitatively in Fig. 6(b): a water droplet filled with pepper grains much larger than coffee particles does not produces a ring-shaped pattern while evaporating. This problem has important consequences in the deposition of ink²⁹ or the manufacturing of DNA chips.¹² 

“It”s crazy that I always have to serve salty water to these physicists to get them out,” thought the waiter as he closed the bar.

This short review discloses the potential of a simple coffee cup to perform various physics experiments. The variety of techniques used makes it particularly suitable for experimental conferences or undergraduate lectures. We did our best to bring together the experiments that are the most spectacular or surprising from our point of view, while remaining doable in a classroom with minimal equipment. Other fascinating problems such as droplet levitation above a coffee^2,36 or walk-induced coffee spilling²⁶ were, therefore, not discussed here. We hope that this work will encourage experimental demonstrations in classrooms, in order to help students to connect their theoretical knowledge with concrete examples of their everyday life.

This paper results from a conference talk given at “Les Gustins Summer School” in 2024. The authors acknowledge all the participants for their valuable feedback.

The authors have no conflicts to disclose.

Aleksi Bossart: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (lead). Romain Fleury: Funding acquisition (lead); Writing – review & editing (supporting). Benjamin Apffel: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

We describe hereafter some troubleshooting to ease experimental reproducibility. We highly recommend to test each experiment before performing it in front of an audience.

Optical caustics: For the coin experiment, it might be difficult to perform non-slipping motion. The best results were obtained by taping the central coin and push from the outside of the moving coin to increase normal, and thus tangential, stress. This experiment also provides an opportunity to recall how friction stress works. Alternatively, the use of circular gears could also greatly ease the experiment. Concerning the optical experiment, the best results are obtained with black flat-bottom mugs and punctual sources such as a phone's light. A pie mold also provides great visualization and may be more suitable for a larger audience.

Acoustic mode degeneracy: This experiment works with the vast majority of cups, but the best results are obtained with porcelain cups, which produce a clear tone with a long sustain. The effect can be made even clearer by using a crystal wine glass and attaching a mass to it. The experimental signal was recorded with the built-in microphone of our laptop and is of good enough quality to perform quantitative analysis.

Hot chocolate/coffee effect: The strongest effect was observed in our case with coffee just poured by the machine. The mixing of the foam combined with the very hot liquid ensures excellent mixing of air in water. It is very important to hit the bottom of the cup to excite the liquid's mode. We did not succeed to hear the effect clearly by mixing chocolate powder in cold water; hot water was working better but not as well as hot, freshly poured coffee.

Tea-leaf paradox: The experiment presents no particular difficulty. A gentle mix with the spoon on the upper part of the water is enough to observe the effect. It is interesting to have both floating and sinking leaves to show the differentiated behavior depending on the location in the fluid.

Double diffusion convection: This experiment requires a bit of care to be successful. The use of very hot water (∼90 °C) makes the phenomenon easier to observe as convective effects are stronger. The key parameter is the penetration of the hot chocolate shot in the salty water. If it is too small, the dyed water remains at the surface and no clear convective cells appear. If it is too large, the mixing is homogeneous and no color (and density) gradient appears. It requires of bit of trial and error to obtain a color gradient as nice as in the experimental video, and the two parameters that can be played with are the penetration velocity of the chocolate and the amount of salt. In our experiment, after typically 5–10 min, we observed some “condensation” of chocolate powder that started to fall at the bottom of the cup, which is very nice but slowly destroys the layers. In principle, the experiment can also be done with milk instead of salty water, but we did not manage to obtain as nice color gradients, indicating a lack of practice in latte preparation.

Coffee stains: It is important to use coffee, as, for instance, the experiment did not work in our case using hot chocolate or pepper grains.