Experimental investigation of walking drops: Wave field and interaction with slit structures

While bouncing walking silicone oil droplets (walkers) do show many quantumlike phenomena, the original, most intriguing, double-slit experiment with walkers has been shown to be an overinterpretation of data. Several experiments and numerical simulations have proven that for at least some parameter region there is no randomness. Still, true randomness was claimed to be observed in an experiment on chaotically bouncing walkers. Also, most of the available phase space has not been investigated. The main goal of this paper is therefore to look for true interference and chaos in the entire phase space. Recently, we made an extensive investigation of drops interacting with slits, but still in a limited range. However, the outcome was always deterministic and only incidentally mimicked the statistics of the corresponding quantum case. We also showed that the extra interference, already seen by others, in the double-slit case was caused by reflection of waves from the outlet of the unused slit after passage and thus was not a true double-slit effect. A new theoretical treatment of bouncing drop dynamics has since given analytic relations for the associated wave field, leading to a proposal for criteria for the possible occurrence of true interference in the double-slit experiment. Satisfying these criteria, requires working close to the onset of the Faraday instability, with two spatial conditions favoring slow walkers, and a temporal condition favoring fast walkers. The regions of high velocity, where the walkers bounce periodically, and of very low velocity, with chaotically bouncing walkers, have not been comprehensively investigated so far. We have therefore looked at these regions, probing the limits for interaction with slits. Furthermore, noting that a short transit time is essential to fulfill the criteria, experiments were done using double-slit barriers only 0.5 and 2 mm broad. Nowhere was true interference or a chaotic response found. As the theory has implications for many of the observed quantumlike phenomena involving walkers as, e.g., tunneling and interaction between drops, we have measured the spatial and temporal decay of the wave field. A comparison with the theory shows very good agreement but leads to a reformulation of the above-mentioned criteria.

Figure 1

Part of ($\mathrm{\Omega },\gamma$) parameter space, reproduced from Wind-Willassen et al. [ Phys. Fluids 25 082002 (2013)], Fig. 3, with the permission of AIP Publishing, and augmented with a $V_w y$ axis. Note that this axis refers to the velocity $V_w$ of the drop at $\gamma /{\gamma }_F=0.998$, where $\gamma$ and ${\gamma }_F$ are the drive and the critical drive, respectively. $\mathrm{\Omega }$ is the vibration number of the drop (for definitions, see Table 1). The region of interest is the walker region limited to the left by the red borderline. The notation (${m,n)}^q$ refers to the periodicity of the bouncing where m is the number of driving periods and n is the number of contacts with the bath with the superscript $q$ distinguishing different phase-shifted modes. The squares denote stationary bouncers while the circles denote walkers, with the colors denoting the associated (${m,n)}^q$ state. In the light blue area marked “C” the walkers bounce chaotically. The small orange hatched rectangle to the right shows the area covered by previous investigations, while the large (orange-bordered) rectangle shows the total phase space now covered in detail by our investigations. The orange cross-hatched rectangle denotes a transition zone (see Sec. 4b).

Figure 2

Top view of the interior of the container with a double-slit of slit width, $L$, and breadth, $b$, inserted in a groove, the free ends of which is shown in green. The radius of the free surface of the oil is 97 mm while the radius, $R$, of the deep section (white) is 90 mm. The black parts all have the height 5 mm. The width, $L_b$, of the central block is always 4.7 mm, while $L$ and $b$ can vary. A single trajectory is shown to allow for the definitions of the deflection angle $\alpha$ and the impact parameter $x_{\text{imp}}$.

Figure 3

Wave height measurement (data, courtesy Mads Rode) along the trajectory in the direction of the drop movement. The wave height is measured far from any obstacles and is shown by the continuous (red) curve. The dot-dot (black) curve is a simulation on Eq. (3). Especially in the forward direction the fit is good. Experimental values given are the drop diameter $D=0.763\pm 0.003\mathrm{mm}$, and $M=69\pm 16$. Since $V_w$ is not stated we have used $V_w=9.5\mathrm{mm}/\mathrm{s}$ as representative in the simulation. ${\tau }_i=0.381$, and the summation is taken to $n=3M$ to ensure convergence.

Figure 4

Simulation on Eq. (3) of the wave field with the drop at the origin and moving right along the $y$ axis. $V_w=8.2\mathrm{mm}/\mathrm{s}$ and $M=67$. The wave height (in mm) is represented by the color-scale to the right. The continuous curves (black) are, counting from the center 30%, 10%, and 3% contour curves. These are seen to even catch some of the destructive interference lines in the wake. ${\tau }_i=0.381$. The summation is taken from $n=-3M$ to ensure convergence. The dashed (red) curve is $l\left(\mathrm{\Theta }\right)$ [Eq. (10)]. This is clearly not following a constant contour.

Figure 5

$l^{*}{\left(0\right)}_D, l^{*}{\left(90\right)}_D$, and ${\tau }_D$ as functions of $V_w$ for $M=500$. ($-$) $L=5.2\mathrm{mm}$, ($--$) $L=7.5\mathrm{mm}$, and ($-·$) $L=14.7\mathrm{mm}$.

Figure 6

Measurements of $l^{*}\left(\mathrm{\Theta }\right)$. Values of the parameters $V_w$ and $M$ in upper right corner of each field. (a), (b) $D=0.701\pm 0.014\mathrm{mm}$. (c), (d) $D=0.854\pm 0.012\mathrm{mm}$. The uncertainty on $V_w$ is $\pm$ 0.08 mm/s. The direction of the drop is shown by the arrows $V_w$. The definitions of $\mathrm{\Theta }$ and $l^{*}\left(\mathrm{\Theta }\right)$ are shown in (a) together with the uncertainty in the measurement of $l^{*}\left(\mathrm{\Theta }\right)$. Full curves (in black) show the 3% contour lines found using Eq. (3) that as seen give a good fit to the data ($*$). Also shown are $l\left(\mathrm{\Theta }\right)$ (the ellipsoids) calculated from Eq. (10).

Figure 7

Measurements of the dependence of $l^{*}\left(0\right)$ and $l^{*}\left(90\right)$ on $M$ (see footnote [37]). ($•$, black) and (o, blue) from LED line at 63 mm after accelerator, ($*$, red) and ($+$, red) from LED line at 92 mm after accelerator. Upper panel: $D=0.710\pm 0.014\mathrm{mm}, V_w=8.1-8.2\mathrm{mm}/\mathrm{s}$. Lower panel: $D=0.854\pm 0.012\mathrm{mm}, V_w=14.4\mathrm{mm}/\mathrm{s}$ [37]. The simulated 3% contour values from Eq. (3) for $l^{*}\left(0\right)$ ($··$) and $l^{*}\left(90\right)$ ($--$) are plotted too, showing good overall agreement. Also included are $l\left(0\right)$ ($-$) and $l\left(90\right)$ ($-·-·$) calculated from Eq. (10).

Figure 8

Measurements of the wave height showing exponential decay with time. Data normalized by extrapolated value at time of absorption of drop. $M=33\pm 0.3$ ($+$), $M=52\pm 1$ ($*$), $M=125\pm 5$ ($\circ$), $M=400\pm 50$ ($\square$).

Figure 9

Measured temporal damping as function of $M$ ($•$). Also included are the measurements by Tadrist et al. [27] ($+$) (data, courtesy Loïc Tadrist) together with their prediction ${\tau }_M=M_e\left(M\right)T_F$ with $T_F=25$ ms the Faraday period ($-$). ($\circ ,--$) Simulation based on Eq. (3).

Figure 10

In panel (a) we show the height of the simulated wave field $\zeta$ for $V_w=8.2\mathrm{mm}/\mathrm{s}$ with the y-axis following the direction of the trajectory ($\mathrm{\Theta }=0^{\circ }$). The drop is positioned at $y=0$. $M=100$ (–, black) and $M=500$ ($·$, red). (b) The same for the wave field height perpendicular to the trajectory ($\mathrm{\Theta }={90}^{\circ }$) for y = 0. The values of the envelope field $\eta \left(l^{*}\left(\mathrm{\Theta }\right)\right)$ corresponding to the measured values of $l^{*}\left(\mathrm{\Theta }\right)$ are given in the text boxes with the corresponding arrows pointing to the positions of $l^{*}\left(\mathrm{\Theta }\right)$. The positions of $l\left(\mathrm{\Theta }\right)$ calculated from Eq. (10) are shown with arrows.

Figure 11

Simulation of the change in energy distribution $\delta E/max\left(\delta E\right)$ during the decay of the wave field by comparing the actual energy decay with that of a uniform decay after time $t$. $M=67$ and $V_w=8.2\mathrm{mm}/\mathrm{s}$ with the y-axis following the direction of the trajectory ($\mathrm{\Theta }=0^{\circ }$). Time proceeds from $t=(\tau -{\tau }_i)/\left(\pi f\right)=0$ without any new impacts (see text for more details). The decay is followed for one period after the elapsed time noted above the fields. Note the narrowed color range accentuating the outward flow of energy.

Figure 12

(a) The decay of the amplitude of the central peak with time showing the initial fast loss of energy followed by a slow approach to the theoretical decay (red $--$) of a uniform field determined by $M_e{T}_F=1.27\mathrm{s}$. (b) The same for the wave field amplitude perpendicular to the trajectory ($\mathrm{\Theta }={90}^{\circ }$) (x,y) = (30,0). Here we see the effect of the bleeding of the energy from the higher amplitude in the wake in the initial increase of the amplitude. It takes a surprisingly long time before we approach the decay of a homogeneous field. All fields are normalized by the value at the central peak at time $t=0$.

Figure 13

Typical example of trajectories obtained for a drop velocity just at the lower limit for passage through a $b=5\mathrm{mm}$ and $L=5.2\mathrm{mm}$ single-slit. Trajectories of a total of 20 drops are shown. Of these, two pass in the center following regular trajectories. One drop makes a complete stop in the slit before moving on and turning to the right. $V_{\text{imp}}=7.87\pm 0.05\mathrm{mm}/\mathrm{s}, D=0.713\pm 0.020\mathrm{mm}$. $\gamma /{\gamma }_F=0.998$. $h_1=0.550\pm 0.040\mathrm{mm}$.

Figure 14

Low velocity limit $V_{\text{low}}(\gamma /{\gamma }_F)$ with $\gamma /{\gamma }_F=0.998$ versus fluid height $h_1$ over barrier. $L=5.2\mathrm{mm}$ slit (x), $L=7.5\mathrm{mm}$ slit ($\square$), $L=14.7\mathrm{mm}$ slit ($\circ$). The drop size corresponding to $V_{\text{low}}\left(0.998\right)$ is shown to the right of the figure.

Figure 15

($V_w, L$) phase space with the transition zone approximately covered by the dark gray shaded area. The trajectories in the light gray area belong to either regular or the occasional backtracking, with all trajectories that pass through the slit going on. In the white region above the transition zone, all drops stop completely after passage. The velocity $V_w(\gamma /{\gamma }_F)$, at $\gamma /{\gamma }_F=0.998$, is changed by choosing different drop sizes, and the correspondence between $V_w, \mathrm{\Omega }$, and $D$ is shown by adding the right y-axes. The horizontal light blue shaded area shows the approximate upper boundary for walkers. Above only stationary bouncers exist. The vertical, dashed, lines denote the slit sizes $L$. The impact range $x_{\text{imp}}$ for the different slits are centered around the vertical lines denoting the slit sizes—and shown on the same scale. The values of $x_{\text{imp}}$ leading to regular trajectories are shown with small squares, while the thin (red) horizontal lines show the parts of the impact ranges $x_{\text{imp}}$ occupied by trajectories that stop abruptly after passage.

Figure 16

Double-slit experiment with $b=2\mathrm{mm}$ barrier, $L=5.2\mathrm{mm}$, and $d=9.9\mathrm{mm}$ (47 trajectories). $\gamma /{\gamma }_F=0.998$, and $h_1=0.620\pm 0.040\mathrm{mm}$. Impact velocity of 14.51 $\pm$ 0.02 mm/s and $D=0.863\pm 0.020\mathrm{mm}$. (a) All passing trajectories. The right inset shows a few of the irregular trajectories while the left shows the velocity change during passage. (b) The associated distribution of angular deflection, $\alpha$ ($•$), including also the far field response of any irregular trajectories, as function of the impact parameter $x_{\text{imp}}$. The nonchaotic nature is high-lighted by also plotting a point symmetric copy of the response ($\circ$). The black line segments at $\alpha =0$ symbolize the barrier with slit openings.

Figure 17

Double-slit experiment with a $b=0.5\mathrm{mm}$ barrier, $L=5.2\mathrm{mm}$, and $d=9.9\mathrm{mm}$ (55 trajectories). Impact velocity $V_w=12.63\pm 0.08\mathrm{mm}/\mathrm{s}$ and drop size $D=0.849\pm 0.016\mathrm{mm}$. $\gamma /{\gamma }_F=0.998$, and $h_1=0.670\pm 0.040\mathrm{mm}$. (a) All trajectories that pass through the slits with the inset displaying a 10 by 10 mm section around the slit to the right showing the different types of trajectories found. (b) The associated distribution of angular deflection ($•$) as function of the impact parameter $x_{\text{imp}}$. To illustrate the reproducibility we have also shown the point-symmetric response ($\circ$). The line segments at $\alpha =0$ symbolize the barrier with slit openings. (c) Histogram of angular distribution.

Figure 18

Double-slit experiment with a $b=0.5\mathrm{mm}$ barrier, $L=7.3\mathrm{mm}$, and $d=12\mathrm{mm}$ (96 trajectories). Impact velocity $V_w=9.03\pm 0.08\mathrm{mm}/\mathrm{s}$ and drop size $D=0.724\pm 0.010\mathrm{mm}$. $\gamma /{\gamma }_F=0.998$, and $h_1=0.740\pm 0.040\mathrm{mm}$. (a) All trajectories that pass through the slits with the inset displaying a 12 by 24 mm section around the slit to the right showing the different types of trajectories found. (b) The associated distribution of angular deflection ($•$) as function of the impact parameter $x_{\text{imp}}$. To illustrate the reproducibility we have also shown the point-symmetric response ($\circ$). The line segments at $\alpha =0$ symbolize the barrier with slit openings. (c) Histogram of angular distribution.

Figure 19

Deflection angle $\alpha$ versus $x_{\text{imp}}$ for a double-slit experiment with $L=5.2\mathrm{mm}$ and $d=9.9\mathrm{mm}$. (a) $b=5\mathrm{mm}$, 362 trajectories, $D=0.820\pm 0.015\mathrm{mm}, V_w=10.91\pm 0.04\mathrm{mm}/\mathrm{s}, h_1=0.710\pm 0.040\mathrm{mm}$, (b) $b=2\mathrm{mm}$, 30 trajectories, $D=0.756\pm 0.015\mathrm{mm}, V_w=9.71\pm 0.04\mathrm{mm}/\mathrm{s}, h_1=0.540\pm 0.040\mathrm{mm}$, (c) $b=0.5\mathrm{mm}$, 35 trajectories, $D=0.821\pm 0.016\mathrm{mm}, V_w=10.01\pm 0.04\mathrm{mm}/\mathrm{s}, h_1=0.710\pm 0.040\mathrm{mm}$. (d)–(f) Corresponding histograms of angular distribution. (g) Measured time spent inside the slit $T_s$ normalized with the characteristic time $T_{\text{ch}}$ versus $x_{\text{imp}}$ for the above. $b=5\mathrm{mm}$ ($•$), $b=2\mathrm{mm}$ ($\circ$), and $b=0.5\mathrm{mm}$ ($\square$). $\gamma /{\gamma }_F=0.998$. The line segments at $\alpha =0$ (a)–(f) and at $T_s/T_{\text{ch}}=0$ (g) symbolize the barrier with slit openings.

Figure 20

Deflection angle $\alpha$ versus $x_{\text{imp}}$ for a double-slit with $L=7.3\mathrm{mm}, d=12\mathrm{mm}$. (a) $b=5\mathrm{mm}$, 94 trajectories, $V_w=9.30\pm 0.04\mathrm{mm}/\mathrm{s}$, and $D=0.730\pm 0.015\mathrm{mm}$. $h_1=0.780\pm 0.040\mathrm{mm}$. (b) $b=0.5\mathrm{mm}$, 104 trajectories, $V_w=9.27\pm 0.07\mathrm{mm}/\mathrm{s}$, and $D=0.750\pm 0.015\mathrm{mm}$. $h_1=0.540\pm 0.040\mathrm{mm}$. $\gamma /{\gamma }_F=0.998$ for both. (c), (d) Corresponding histograms of angular distribution. (e) Measured time spent inside the slit $T_s$ normalized with the characteristic time $T_{\text{ch}}$ versus $x_{\text{imp}}$ for the above. $b=5\mathrm{mm}$ ($•$), and $b=0.5\mathrm{mm}$ ($\circ$). The line segments at $\alpha =0$ (a)–(d) and at $T_s/T_{\text{ch}}=0$ (e) symbolize the barrier with slit openings.

Figure 21

$L=5.2\mathrm{mm}, b=5\mathrm{mm}$. In the left column we show the result for drops moving with impact velocity $V_w=14.10\pm 0.04\mathrm{mm}/\mathrm{s}$ and $D=0.838\pm 0.016\mathrm{mm}$, while in the right column we present the result for drops with impact velocity $V_w=15.01\pm 0.10\mathrm{mm}/\mathrm{s}$ and $D=0.870\pm 0.016\mathrm{mm}$. Sweeping the impact parameter, panels (a) and (e) show regular trajectories for drops that pass into the far region, (b) and (f) trajectories of those that do not make it further than the local neighborhood at the outlet of the slit. Notice the expanded x-scale. Panels (c) and (g) shows the associated (${\mathrm{x}}_{\text{imp}},\alpha$) plots with the line segments at $\alpha =0$ symbolizing the barrier with slit opening. $h_1=0.650\pm 0.040\mathrm{mm}$ and $\gamma /{\gamma }_F=0.991\pm 0.001$. At a velocity of $V_w=15.76\pm 0.03\mathrm{mm}/\mathrm{s}$ and $D=0.901\pm 0.020\mathrm{mm}$, no drops make it through the slit without coming to a complete stop after which most drift around slowly close to the outlet of the slit. (d), (h) Histograms of the corresponding angular distributions.

Figure 22

$L=7.5\mathrm{mm}, b=5\mathrm{mm}$. In the left column we show (a) the regular trajectories of the drops moving with an impact velocity $V_w=15.17\pm 0.04\mathrm{mm}/\mathrm{s}$ and $D=0.881\pm 0.015\mathrm{mm}$, (b) those that do not make it further than the local neighborhood of the slit together with some of the backtracking trajectories, (c) the associated ($x_{\text{imp}},\alpha$) plot. Again the $x$ axis is expanded. (d) Histogram of the angular distribution. In the right column (e)–(h) the same is shown for drops of a velocity of $V_w=15.62\pm 0.03$ and $D=0.892\pm 0.15\mathrm{mm}$. $\gamma /{\gamma }_F=0.998$ and $h_1=0.630\pm 0.040\mathrm{mm}$. At a velocity of $V_w=15.84\pm 0.05\mathrm{mm}/\mathrm{s}$ no drops make it through the slit without coming to a complete stop. The line segments in panels (c) and (g) at $\alpha =0$ symbolize the barrier with slit opening.

Figure 23

$L=14.7\mathrm{mm}, b=5\mathrm{mm}$. In the left column we show (a) the regular trajectories of the drops moving with an impact velocity $V_w=15.86\pm 0.03\mathrm{mm}/\mathrm{s}$ and $D=0.887\pm 0.020\mathrm{mm}$ that pass into the far region, (b) some irregular trajectories surrounding the regular trajectories, (c) the associated ($x_{\text{imp}}, \alpha$) plot. Notice again the expanded $x$ axis. (d) Histogram of the angular distribution. In the right column the same is shown for drops of a velocity of $V_w=16.31\pm 0.03$ ($+$) and $D=0.920\pm 0.020\mathrm{mm}$. Superimposed on the ($x_{\text{imp}},\alpha$) plot (g) are the data from the left column ($·$), and data for a drop having $V_w=16.42\mathrm{mm}/\mathrm{s}$ and $D=0.931\pm 0.020\mathrm{mm}$ ($*$). $\gamma /{\gamma }_F=0.998$ and $h_1=0.550\pm 0.040\mathrm{mm}$. At a velocity higher than $V_w=16.42\pm 0.05\mathrm{mm}/\mathrm{s}$ no drops make it through the slit without coming to a complete stop. The line segments in panels (c) and (g) at $\alpha =0$ symbolize the barrier with slit opening.

Figure 24

Single-slit, $L=5.2\mathrm{mm}, b=0.5\mathrm{mm}$, 57 trajectories, $h_1=0.950\pm 0.040\mathrm{mm}, \gamma /{\gamma }_F=0.992\pm 0.001, V_w=10.06\pm 0.06\mathrm{mm}/\mathrm{s}$, and $D=0.810\pm 0.014\mathrm{mm}$. (a) Trajectories, (b) $\alpha$ versus $x_{\text{imp}}$ ($·$). Point symmetric copy ($*$). The line segments at $\alpha =0$ symbolize the barrier with the slit.