Layered latte

A glass of latte is made by pouring a cup of espresso into a glass of warm milk. Since the two liquids are miscible, the result of pouring is an espresso-milk mixture at the top of the glass, while the bottom may only contain milk, if no additional stirring is applied (Fig. 1a). In fact, although the initial state of the mixture is complex and chaotic (Fig. 1b), there are conditions where the mixture cools at room temperature and exhibits an organized layered pattern (Fig. 1c, see Supplementary Movie 1). These stable layers, whose structures may be maintained for at least tens of minutes (Fig. 1d), or even several hours, contain different concentrations of espresso and hence exhibit distinct visible boundaries.

Fig. 1 Pattern formation in an injection-driven system inspired by a layered latte. a–d Formation of horizontal layers in a glass of latte. a A small volume (\(V_{\mathrm{I}} = 30\) ml) of espresso (temperature \(T_{\mathrm{H}} = 50\,^ \circ {\mathrm{C}}\)) is poured into 150 ml of warm milk (\(T_{\mathrm{H}} = 50\,^ \circ {\mathrm{C}}\)). b Espresso and milk form a mixture, which exhibits chaotic dynamics caused by the injection. The resulting espresso-milk mixture remains at the top of the container due to buoyancy. c As the mixture cools down to room temperature, \(T_{{\mathrm{atm}}} = 22\,^ \circ {\mathrm{C}}\), multiple horizontal layers of different espresso concentrations are formed. d These horizontal layers maintain their structures over time (see Supplementary Movie 1). e Schematic of model experiments. Dyed water (0.01 wt% methylene blue hydrate) and sodium chloride solution (9.1 wt%) are heated in a water bath at \(T_{\mathrm{H}} = 40\,^ \circ {\mathrm{C}}\). Then the heated dyed water is injected vertically downward through a needle of diameter \(d\) into the salt solution using a syringe pump, which allows controlling the flow rate and thus the injection velocity \(U\) of the dyed water. Finally, all of the liquid in the tank cools down at room temperature and layers are observed. The cooling begins as soon as the injection starts and lasts for at least several minutes Full size image

Experimental model

In order to investigate the mechanisms leading to the layering of the mixture, we performed controlled experiments in a model system comprised of a low-density jet of dyed water (density \(\rho _{\mathrm{w}} = 0.992 \times 10^3 \, {\mathrm{kg}}\,{\mathrm{m}}^{ - {\mathrm{3}}}\) at \(T_{\mathrm{H}} = 40\,^ \circ {\mathrm{C}}\), injection volume \(V_{\mathrm{I}} = 30 \,\)ml) entering a tank filled with relatively higher-density brine (9.1 wt% sodium chloride solution, \(\rho _{\mathrm{s}} = 1.056 \times 10^3\,{\mathrm{kg}}\,{\mathrm{m}}^{ - {\mathrm{3}}}\) at \(T_{\mathrm{H}} = 40\,^ \circ {\mathrm{C}}\), 340.9 ml). The jet enters from the top (Fig. 1e) and the solution is then left to cool at room temperature \(T_{{\mathrm{atm}}} = 22\,^ \circ {\mathrm{C}}\).

Experimental observations

When the dyed water is injected into the higher-density salt solution, a downward jet is generated. However, the penetration of this liquid jet into the salt solution is opposed by the buoyant force pushing the lower-density liquid jet back to the top of the tank (Fig. 2a, c). As a result, a mixture is formed, in which the dyed water is mixed with the salt solution and is separated from the original salt solution at the bottom of the tank. In addition to buoyancy, the mixing is mainly governed by inertia with the Reynolds number defined as \({\mathrm{Re}} = Ud/2

u \approx O(100)\) (\(U\) is the injection velocity, \(d\) is the diameter of the needle and \(

u\) is the kinematic viscosity of the fluid), while diffusion does not play a significant role during the injection. The Schmidt number is \({\mathrm{Sc}} =

u / \kappa_{\mathrm{s}} \approx 300\) (\(\kappa _{\mathrm{s}}\) is the mass diffusivity of the salt), indicating that the momentum diffusion is far faster than the salt diffusion during both the injection and layering (if any) processes. The Schmidt number for the milk and espresso system is approximately \(10^4 \gg 1\), and so the system is similar to the model salt and water system, in which momentum diffusion dominates. At relatively low injection velocities (Fig. 2a, b), the mixture of dyed water and salt solution (initially at \(T_{\mathrm{H}} = 40\,^ \circ {\mathrm{C}}\)) remains unchanged as it cools down at room temperature. However, above a critical injection velocity, multiple layers similar to those observed in the glass of latte (Fig. 1a–d) are formed in the mixture several minutes after the injection (Fig. 2d). Once formed, the layers are not influenced by external mechanical disturbances, and can recover even after gentle stirring. As the cooling continues, these layers may merge and form thicker structures, which can last for days before being entirely eliminated by diffusion (Supplementary Fig. 1). The layers can be observed in the milk and espresso or in the salt and water mixture, only when the initial temperature is different from the room temperature.

Fig. 2 Evidence for double-diffusive convection in the model experiments and simulations. a−d Dyed water (0.01 wt% methylene blue hydrate, 30.0 ml) is injected into 9.1 wt% sodium chloride solution (340.9 ml) with two different injection velocities: \(U = 0.17\,{\mathrm{m}}\,{\mathrm{s}}^{ - {\mathrm{1}}}\) (a, b) and \(U = 0.37\,{\mathrm{m}}\,{\mathrm{s}}^{ - {\mathrm{1}}}\) (c, d). a, c correspond to the dynamics during the injection and (b, d) to the mixture 30 min later. The distinguished boundaries of the layers are marked and enumerated. e The density profiles of the mixtures presented in a−d. The dashed lines refer to the density profiles 1 min after the injection, and the solid lines refer to those observed 30 min later. The black and the blue lines refer to the results at injection velocities \(U = 0.17\,{\mathrm{m}}\,{\mathrm{s}}^{ - {\mathrm{1}}}\) and \(U = 0.37\,{\mathrm{m}}\,{\mathrm{s}}^{ - {\mathrm{1}}}\), respectively. Vertical steps (solid blue line) indicate layers in the mixture, and their boundaries are enumerated according to those in d, where \(z = 0\) refers to the top of the mixture. f, g Particle streak-lines (f) and the velocity vectors (g) measured in PIV experiments for \(U = 0.37\,{\mathrm{m}}\,{\mathrm{s}}^{ - {\mathrm{1}}}\), 10 min after the injection. The vertical dashed line refers to the axis of the cylindrical domain, and the horizontal dashed line marks the bottom of the mixture. h, i Numerical results (details in Methods, Supplementary Figs 3 and 4 and Supplementary Discussion) show the density distribution (h) as well as the velocity vectors and the vorticity magnitude \(\omega\) (i), 10 minutes after the injection with \(U = 0.37\,{\mathrm{m}}\,{\mathrm{s}}^{ - {\mathrm{1}}}\). The scales for the length and the velocity magnitude are identical in g and i Full size image

We quantify the concentration of the salt in the mixture using the concentration of the blue dye as an indicator (Supplementary Fig. 2). Therefore, the local intensity of the blue dye in the digital images (Fig. 2a–d) can be correlated with the local density of the mixture. The dashed lines in Fig. 2e represent the initial density profiles for two different injection speeds, while the solid lines refer to the density profiles 30 min later. For both experiments the dashed lines exhibit continuous monotonically increasing density profiles, when moving from the top to the bottom of the mixture. While the density profiles before and after the injection at a low velocity \(U = 0.17\,{\mathrm{m}}\,{\mathrm{s}}^{ - {\mathrm{1}}}\) (black) remain almost identical, the density profile 30 min after the injection at a higher velocity \(U = 0.37\,{\mathrm{m}}\,{\mathrm{s}}^{ - {\mathrm{1}}}\) (blue) exhibits clear steps indicating the formation of horizontal layers. After the high-velocity injection, the density in a single layer is constant, implying that the liquid within a layer is uniformly mixed. Moreover, the discontinuities in the density profile are clearer near the top of the mixture, where the gradient of the density is smaller than that of the bottom layers. We postulate that this layered state is reached due to the double-diffusive convection, which is well known in layer formation in open water systems such as oceans or lakes18,19,20, 23, 24. In our experiment, the double-diffusive convection results from the combination of heat transfer to the surroundings from the warm liquid, and a density gradient generated in the mixture from the initial pouring or injection.

A directional heat transfer phenomenon in a mixture with an initial density gradient has been observed previously in other systems to lead to the formation of well-defined layers in fluid mixtures due to double-diffusive convection18,19,20, 23, 24. In particular, when a given temperature difference is created between two vertical walls bounding a fluid with an initial vertical density gradient, the fluid near the cold wall is cooled and thus gets denser and sinks. The sinking of the liquid due to the heat transfer will however be suppressed as the cooler liquid close to the wall reaches a zone of a similar density in the mixture. Therefore, the downward-moving liquid starts flowing inwards away from the cold wall as it can no longer proceed in the vertical direction. A similar motion but in the opposite sense is created close to the warmer wall. Consequently, closed streamlines are formed in the fluid circulating between the cold and the warm sources23. Within the circulation cells, the fluid mixes uniformly, and thus the density is fairly constant, while each circulation cell acquires a different density.

In order to verify the postulate of double-diffusive convection as the cause of layering in our confined injection-driven system, we performed experimental (Fig. 2f, g) and numerical analyses (Fig. 2h, i) of the time-dependent flows (see details in Methods, Supplementary Figs 3, 4 and Supplementary Discussion). Both approaches document the formation of recirculating patterns in the form of axisymmetric rings between the wall and the center of the tank. The boundaries separating these circulation cells overlap with the limits of the layers in the mixture.

Critical injection velocity

The circulation cells are the results of the competition between the horizontal thermal gradient and the vertical density gradient generated by the fluid injection, i.e., the thermal gradient triggers the fluid to rise or sink close to the boundaries, while the vertical density gradient opposes this motion and subsequently stabilizes the flow. In our model experiments, layers are observed only above a critical injection velocity \(U_{\mathrm{c}}\) (Fig. 2a–d and Supplementary Fig. 5). At higher injection velocities, the depth of penetration of the liquid jet increases, and similarly the thickness of the mixture \(H\) increases, indicating that the dyed water mixes with a larger volume of salt solution. For a fixed volume of injected water (\(V_{\mathrm{I}} = 30\) ml), we found that the volume of the resulting mixture \(V_{\mathrm{M}}\) increases linearly with the injection velocity \(U\) for the range of parameters studied here. Consequently, the mixing level and the density gradient in the mixture of volume \(V_{\mathrm{M}}\) are controlled by the injection velocity.

We define an average density gradient \({\mathrm{\Delta }}\rho /H\) in the mixture, where \({\mathrm{\Delta }}\rho\) is the magnitude of the density difference between the bottom and top of the mixture. Also, \({\mathrm{\Delta }}\rho /H\) indicates the average resistance imposed by the salinity gradient against the formation of a convection cell due to the thermal gradient and is a function of \(U\). Our experimental measurements show that \({\mathrm{\Delta }}\rho /H\) decreases with increasing \(U\) due to enhanced mixing (Fig. 3a). We find that for \(U >U_{\mathrm{c}} \approx 0.21\,{\mathrm{m}}\,{\mathrm{s}}^ { - {\mathrm{1}}}\) the resistance from salinity can no longer compete with the thermal gradient, thus circulation cells appear and multiple layers emerge.

Fig. 3 Role of the injection velocity \(U\). a The average density gradient, defined as the density difference between the bottom and the top of the mixture relative to the mixture height \(H\), as a function of \(U\). b The Rayleigh number \(\rm{Ra}\), defined based on the minimum density gradient and the maximum temperature difference in the mixture, vs. \(U\). The hollow, half-solid and solid symbols refer to the observations, where, respectively, multiple layers are never, sometimes and always observed in the mixture. The squares, circles and triangles refer to different inner diameters of the injection needles \(d =\) 1.2, 1.8 and 2.4 \({\mathrm{mm}}\), respectively. The insets show a zoomed-in view around the transition region. The horizontal dashed line in (b) at \({\mathrm{Ra}}_{\mathrm{c}} = 1.5 \times 10^4\) indicates the critical Rayleigh number reported in the literature18. Error bars correspond to standard derivations with \(n\) experiments with \(n = 3 - 5\) Full size image

The competition between the thermal cooling and the salinity gradient in this double-diffusive convection, where the horizontal temperature gradient is orthogonal to the vertical salinity gradient, is characterized by the Rayleigh number, \({\mathrm{Ra}} = \frac{{g\alpha {\mathrm{\Delta }}T}}{{

u \kappa }}\left[ {\frac{{\alpha {\mathrm{\Delta }}T}}{{( - {\mathrm{d}}\rho /{\mathrm{d}}z)/\rho _{\mathrm{w}}}}} \right]^3\), where \(g\) is the gravitational acceleration, \(\alpha\) is the coefficient of fluid volume expansion, \({\mathrm{\Delta }}T\) is the temperature difference in the fluid, \(

u\) is the kinematic viscosity of the fluid, \(\kappa\) is the thermal diffusivity of the fluid and \(( - {\mathrm{d}}\rho /{\mathrm{d}}z)/\rho _{\mathrm{w}}\) is the normalized salinity gradient in the mixture18. In a system with an initial linear salinity gradient and an imposed horizontal temperature gradient, the critical Rayleigh number \({\mathrm{Ra}}_{\mathrm{c}}\) for initiation of an instability and formation of the layers has been found to be around \({\mathrm{Ra}}_{\mathrm{c}} \approx 1.5 \times 10^4\) experimentally18. Note that in a conventional double-diffusive convection problem, a linear density gradient is imposed at the initial stage and the temperature gradient is created between two vertical bounding walls by setting different constant temperatures18, 24. The initial and boundary conditions leading to the layers in a glass of latte and our model experiments are, however, different from those in the traditional problem: (1) the density gradient is caused by the injection and is not constant in the mixture (Fig. 2e) and (2) the temperature gradient is not constant during the experiments as the core of the liquid cools down continuously. Therefore, to characterize our system and to calculate \({\mathrm{Ra}}\), we chose to use the minimum density gradient over the typical thickness of the layers (around 5 mm) at the corresponding injection velocity and the maximum temperature gradient. We found that the critical Rayleigh number in our experiments closely matches the value reported in the literature for an idealized configuration \({\mathrm{Ra}}_{\mathrm{c}} \approx 10^4\) (Fig. 3b). Consistent with our observations, \({\mathrm{Ra}}_{\mathrm{c}}\) indicates that the layering is obtained only when \(U\) is above \(U_{\mathrm{c}}\). Consideration of the dynamics in a proper dimensionless framework requires an analysis with at least both the Froude and Reynolds numbers, which is a topic of on-going research. In addition, when layering occurs, the expected length scale (thickness) of a layer is approximately \((\alpha {\mathrm{\Delta }}T)/[( - {\mathrm{d}}\rho /{\mathrm{d}}z)/\rho _{\mathrm{w}}]\).

Application to soft materials

The double-diffusive convection and the formation of the layers are simply controlled by the thermal and salinity gradients in the fluid systems discussed above, which implies no conceptual restriction on applying this principle to more complex fluid systems, such as thermally established soft gels. There are various approaches for generating layered soft materials, but most of these approaches are currently multi-step processes, where solid layers are usually formed sequentially25. Nevertheless, based on the understanding outlined above, we can make multiple layers in soft materials (UltraPure\(^{{\mathrm{TM}}}\) Agarose, Invitrogen) simply by a single step of injecting (\(U \approx 1\,{\mathrm{m}}\,{\mathrm{s}}^{ - {\mathrm{1}}} >U_{\mathrm{c}}\)) a hot gel solution into a denser solvent and cooling the mixture at room temperature (Fig. 4, see the gel recipe in Supplementary Methods). To further demonstrate the presence of the horizontal layers in the gel, we performed experiments with the same recipe but measured the light intensities in the digital images of the gel rather than the elastic properties (see Supplementary Fig. 6 and Supplementary Discussion). While cooling, horizontal layers are first formed in the agarose solution, which is subsequently solidified to a layered gel below the gelation temperature. The Young’s moduli in the final layered state formed from the same agarose solution vary systematically with the vertical position (50 kPa at the bottom compared to 230 kPa at the top), which implies that the concentration of agarose in distinct layers is different. Further, the difference of concentration in the gel layers leads to a difference of porosity in these layers, so the concentration and the diffusion rate of additives will be different. This single-step, single-chemistry method can facilitate the fabrication of multiple-layered structures in food science25, tissue engineering26, 27, and other applications in materials science.