LIF-based quantification of the species transport during droplet impact onto thin liquid films

An average across all pixels of the reconstructed film thickness and dye concentration will be calculated and compared with the CPS values and the known dye concentration, respectively. The global error is then calculated using equations (5) and (6):where \(h_{\text {avg}}\) is the spatially averaged measured thickness, \(h_{\text {CPS}}\) is the thickness measured with CPS, \(C_{\text {avg}}\) is the spatially averaged measured concentration, and \(C_{r}\) is the known dye concentration. The reason behind determining the global error using the averaged measured thickness and concentration is to analyze the influence of initial parameters on the accuracy of the TIC calibration. Note that the accuracy and error are inversely related to each other.

A validation measurement is performed by taking 20 images of a thin film with a known thickness and dye concentration. Note that thin film measurements used for calibration are not used again for validation. The same noise reduction filter employed in the calibration is then applied to the set of images. After extracting the fluorescence intensity \(I_{\text {f}}\) of each pixel (x, y), equation (4) is used to obtain the local film thickness or the local dye concentration. In other words, if the wanted output is the liquid film thickness, the known initial dye concentration of the liquid film is used as input in the model, and vice versa. Let \(h_{i}(x,y)\) be the reconstructed thickness of a certain pixel in the image i. The averaged thickness \(h_{\text {avg}}\) across all pixels and for all images is then calculated using all \(h_{i}(x,y)\) and compared to the confocal sensor value as shown in Fig. 3a. The global error for all thickness reconstruction measurements is \(3.83\%\). Identically, the average dye concentration is determined and is plotted in Fig. 3b, where the global error is \(3.41\%\).

This measured error is highly influenced by the input parameters (Hidrovo and Hart 2001). For the validation measurements, the dye concentration as well as the film thickness is the only variables, and their influence is investigated and represented in Fig. 4. The global error for liquid film thickness reconstruction \({\text {Err}}_{\text {h}}\) exhibits an inverse relationship with the reference dye concentration of the liquid film as highlighted by the regression curve in Fig. 4a. This increase in global error is due to a weak fluorescence signal for liquid films with low concentrations, resulting in decreasing signal-to-noise ratios. However, for extracting the film thickness variations during droplet impact, the rhodamine B concentration chosen is \(C_{0}=8\times 10^{-5}\hbox {M}\). This concentration grants not only a high signal to noise intensity for a wide range of liquid film thickness, but also a low reconstruction error. With respect to the film thickness influence, Fig. 4b displays a slight increasing behavior in the concentration reconstruction error \({\text {Err}}_{\text {C}}\) as the liquid film thickness decreases. Analogously to liquid films with low concentrations, films with low thickness generate low signal to noise intensities which explain the decrease in accuracy. It can be noted that cases exhibiting high reconstruction error for thicknesses ranging between 600 μm and 1000 μm can be attributed to their low corresponding measured concentration as can be seen in Fig. 4b.

As mentioned in the beginning of this Section, a global error calculation is made to obtain a clear idea about the accuracy of the TIC calibration model within a certain range of dye concentrations and thicknesses. For evaluating the TIC calibration, the reconstruction error in each pixel, i.e., local error, is calculated using equations (5) and (6). In Fig. 5a,b, the mean 2D thickness and concentration reconstruction errors in each pixel of all validation measurements are presented. As can be seen, the error is less than \(6\%\) in most of the field of view. However, it increases radially to values up to \(12\%\) in the corner regions of the field of view. This increase is due to the low signal to noise ratios near the corners of the image, notably for liquid films with low dye concentrations, which is caused by uneven illumination at the peripheries of the microscope (Likar et al. 2000). Known as vignetting, it causes a noticeable decrease in intensity toward the edges of the image, especially at low-magnification microscopy (Marty 2007). Furthermore, an observed small asymmetry between top and bottom halves in Fig. 5a may arise due the combination of the vignetting phenomenon and the optical system configuration leading to a small deviation between the center of the illumination and the center of the image. This latter leads to slightly higher thickness reconstruction errors for cases with low concentrations. As the impact and mixing occur in the center region of the image indicated with a dashed red circle in Fig. 5a, b, the corner regions will not affect the overall result accuracy.

The spatial average of the local error for all pixels for film thickness reconstruction is \(4.82\%\), whereas it reaches \(4.53\%\) for concentration reconstruction. It was calculated by averaging the error values across all pixels in Fig. 5a, b. Note that this calculation procedure was also conducted by Li et al. (2019) for determining rhodamine B concentration in water inside an annular coolant system, where the average error across all pixels of \(5.92\%\). In regard to the precision of the LIF method, the coefficient of variation (CV) is used to estimate the repeatability of the validation measurements. The CV of a single pixel is expressed by equation (7), where \(\sigma _{sd}\) is the standard deviation and \(\eta\) is the mean of the 5 repeated measurements.Using the equation above, the TIC calibration model delivers an average CV across all pixels of \(3.62\%\) for film thickness reconstruction and \(3.63\%\) for dye concentration reconstruction.

To investigate the mixing process of droplet impact onto thin liquid films, two complementary measurements are conducted. The first part, which will be further referred to as thickness measurements, is performed by labeling both the droplet and the film liquids with the same concentration of rhodamine B. During the impact, the fluorescence signal intensity variations are solely due to the film height variation. The chosen concentration is \(C_{0}=8\times 10^{-5}\hbox {M}\). Using the TIC calibration model, and given that the concentration is known and constant during the impact, instantaneous local film heights are determined. Subsequently, the second part of measurements is conducted, where only the film liquid is labeled by rhodamine B and has an initial concentration \(C_{0}\). Spatially and temporally resolved concentrations are then obtained using the film height information from the preceding measurement. They will be referred to further as concentration measurements.

Within the scope of this work, only one impact velocity \(U={1.32}\,\hbox {m}/\hbox {s}\) is investigated. As mentioned in Sect. 1, three film thicknesses are studied, where two films are representing the thin film category (\(\delta =0.22\), \(\delta =0.36\)) and one film is representing the very thin film category (\(\delta =0.09\)). The initial parameters chosen in this study lead to the deposition regime during the impact. As the mixing process is facilitated by turbulent eddy mixing, it is a stochastic process. Therefore, the impact experiments were performed five times for each case. Furthermore, the temperature during the impact was kept identical to calibration and validation measurements to \(20^\circ {\text {C}}\). Figure 6 represents side and bottom view images of the impact for times up to \(t={40}\,\hbox {ms}\), where \(t={0}\,\hbox {ms}\) corresponds to the impact instant. The choice of the investigated time duration after impact in this work is chosen to study the inertia driven convective mixing, only. Since the diffusion coefficient of rhodamine B is in the order of \(10^{-10}\,\hbox {m}^{2}/\hbox {s}\) (Gendron et al. 2008), the mixing occurring during the elected period up to \(t={40}\,\hbox {ms}\) after the impact is due to inertial forces. Note that the bottom view images shown in Fig. 6 are raw images before image processing.

Figure 6a,b shows droplet impact on a very thin film of thickness \(\delta =0.09\). After the first contact of the droplet with the film, an increase in the signal intensity is observed, which is due to the thickness increase at the rim during the spreading (\(t={1.6}\,\hbox {ms}\)). Additionally, the impact creates an expanding crater with a remaining film thickness much lower compared to the initial film thickness before droplet impact. Hence, a low fluorescence intensity is observed here. In this study, the crater is defined by the circular region bounded by a steep gradient in the intensity signal (\(t={1.6}\,\hbox {ms}\)). After the lamella spreading (Fig. 6b, \(t={1.6}\,\hbox {ms}\) and \(t={4.7}\,\hbox {ms}\)), finger-like shaped fluorescence pattern starts forming (Fig. 6b, \(t={9}\,\hbox {ms}\) and \(t={20}\,\hbox {ms}\)) during the collapse of the lamella rim. This pattern can be clearly seen in Fig. 7 for \(\delta =0.09\) at \(t={9}\,\hbox {ms}\). These structures occur during the receding phase and they have similarities to the fingering pattern observed during droplet impact onto dry surfaces, as can be seen in Fig. 1 of Marmanis and Thoroddsen 1996 and Fig. 26 of Thoroddsen and Sakakibara (1998). When transitioning from the very thin film to the thin film regime, the dominating mixing dynamics changes significantly as shown in Fig. 6d, f. One apparent difference is a decrease of the mixing area with increased thickness. The same trend was also observed by Ersoy and Eslamian (2019). This behavior is linked to the influence of the wall on the compression of the droplet during the impact. After the initial impact, the droplet penetrates the liquid film forming a crater growing in depth until reaching the wall. The interaction with the latter leads into a redirection of the kinetic energy dissipation in radial direction. When the film is very thin, a redirection of kinetic energy in the radial direction takes place earlier. At \(t={9}\,\hbox {ms}\), instability-driven finger formation can also be observed for dimensionless film thicknesses of \(\delta =0.22\) and \(\delta =0.36\) (see Fig. 6d, f).

In the context of droplet impact dynamics, it has been observed that vortex rings are formed during impacts onto liquid surfaces (Cresswell and Morton 1995). Due to the high velocity gradient across the liquid–liquid interface shortly after the impact, azimuthal vorticity is induced which develops into a vortex ring (Lee et al. 2015). Additionally, depending on the Reynolds number of the impact, Lee et al. (2015) have demonstrated that the impact can generate not only one but up to four vortex rings. Specifically, for the case \({\text {Re}}= 2967\) and \({\text {We}}=54\) studied in the present work, three vortex rings are generated during the impact of a droplet onto another liquid surface, as shown by Lee et al. (2015). This latter prompt an investigation into the underlying factors contributing to the observed instabilities and if they are attributed to vortex ring instability. Considering that the primary focus of this study revolves around thin liquid films, it is nonetheless important to conduct investigations encompassing the two limit cases in regards of fluid film thickness to gain a comprehensive understanding of the factors causing the observed instabilities. Thus, two distinct cases are also examined: droplet impact on a dry substrate (\(\delta =0\)) and into a pool (\(\delta =20\)). To investigate the impact on a dry substrate, experiments using a rhodamine B dyed water droplet were conducted. The evolution of the impact is captured in Fig. 8a, revealing the spreading (\(t={1.6}\,\hbox {ms}\)) and receding (\(t={14.3}\,\hbox {ms}\)) of the droplet after the impact. At \(t={20}\,\hbox {ms}\), the droplet is already at rest and the signal intensity is notably higher in the center of the droplet compared to its edges. Since the droplet does not encounter any fluid film during the impact, there is no mixing. The dye concentration remains constant and the signal intensity difference can be attributed to the decrease in droplet height as it approaches the edges. Furthermore, the absence of finger-like shaped instabilities can also be observed in this case. Similar to the experiments conducted on the dry substrate using a dyed droplet, the ones for impacts into pool involve also dyeing the droplet with Rhodamine B instead of the pool itself. This choice comes from the limitations posed by the significant depth of the pool rendering the visualization of the mixing process through dyeing the entire pool unattainable. Figure 8b illustrates the various stages of droplet impact on pool (\(\delta =20\)). At \(t={1.6}\,\hbox {ms}\), the droplet penetrates the pool, leading to the formation of vortex rings characterized by a pronounced intensity signal. A differentiation between each vortex ring at early stages from the bottom view is not feasible due to their overlapping. At \(t={14.3}\,\hbox {ms}\), azimuthal instabilities in form of loop-like structure can be spotted in the region between the outer ring, indicated by the blue arrow, and the inner ring, indicated by the green arrow. Subsequently, the outer ring can be observed sinking down the pool at \(t={20}\,\hbox {ms}\) and \(t={40}\,\hbox {ms}\). It may appear that the vortex ring is expanding, and however, it is important to note that the ring is not changing in size but rather going out of the range of focus of the camera, contributing to this visual effect. Despite the observed instabilities, both the inner and outer rings maintained their circular shape.

As mentioned previously, the impact in the studied case generates three vortex rings as shown by Lee et al. (2015). An interesting phenomenon observed during this process is the leapfrogging of two vortex rings, as can be seen in supplementary video 7 in the work of Lee et al. (2015). Leapfrogging occurs when two vortex rings, characterized by the same traveling direction and sense of rotation interact. As a consequence of mutual induction, the leading ring expands in size and moves at a slower pace, while the trailing ring contracts in size, accelerates and penetrates the leading ring. This cyclic process repeats itself until the two rings are merged, perpetuating the leapfrogging phenomenon (Shariff and Leonard 1992). According to Lim (1997b), the leapfrogging effect causes the azimuthal instabilities occurring during the passage between the two rings. Upon the subsequent merging of the two vortex rings, a portion of these instabilities remains behind in form of loop-like structure (Lim 1997a), as can be seen in Fig. 8b at \(t={14.3}\,\hbox {ms}\).

In the cases of droplet impacts on thin liquid films, the aforementioned vortex rings can undergo interactions with the wall that significantly affect their behavior. It is therefore plausible that the observed instabilities are also caused by the collision of the generated vortex rings with the wall. The evolution of the vortex ring during droplet impact onto thin film is presented in Fig. 9. Similar to previous cases, the presented impact involves a rhodamine B dyed water droplet and a pure water thin film of dimensionless thickness \(\delta =0.36\). This latter configuration grants the possibility to observe vortex ring formation and decay during the spreading phase. After the creation of the vortex ring upon droplet impact, the ring approaches the wall, resulting in the formation of a boundary layer near the wall. Contrary to the behavior observed in the pool case, where the vortex ring maintained its size, in the thin film case, the vortex ring undergoes radial stretching upon collision with the wall. This can be observed during the transition from \(t={2.4}\,\hbox {ms}\) in Fig. 9a to \(t={3.9}\,\hbox {ms}\) in Fig. 9b. In contrast, for impact in pool, the vortex ring retained an inner diameter roughly equivalent to that of the droplet, as shown in Fig. 8b. According to Walker et al. (1987), the vortex ring stretching leads to the separation of the boundary layer under it from the wall. When the strength of this lift-off is sufficient, flow separation and the creation of a secondary vortex ring with an opposite-signed vorticity are initiated, which orbits the primary vortex ring (see Fig. 10) and leads into a rebound of the primary vortex ring from the wall as illustrated in the work of Chu et al. (1995) and Harris and Williamson (2012). At higher Reynolds numbers, the process of boundary layer separation also leads to the creation of a tertiary vortex ring (Cheng et al. 2010). The secondary vortex ring finally aligns inside the primary vortex ring and therefore can be distinguished in Fig. 9b at \(t={3.9}\,\hbox {ms}\) through a light separation line. At a later stage, the secondary vortex ring exhibits azimuthal instabilities as shown in Fig. 9c at \(t={6.3}\,\hbox {ms}\). This behavior was also noticed by Cheng et al. (2010) in their numerical study of vortex ring impacting a flat wall. Furthermore, it is noteworthy that impacts on thin films exhibit a distinct behavior compared to impacts on a pool. Particularly, the outer ring for thin film cases undergoes instability resulting in figure-like patterns (see Figs. 7 and 9d), as opposed to the pool case. The presence of these patterns primarily arises from the instability induced by the interaction with the wall.

A combination of two different types of instabilities is responsible for vortex rings instability (Archer et al. 2010). Long-wavelength perturbations, also known as the Crow instability (Crow 1970), govern the vortex–wall interaction dynamics at low Reynolds number \({\text {Re}}_{\Gamma }=\frac{\Gamma }{\nu }\), where \(\Gamma\) is the total circulation of the vortex ring. At higher Reynolds numbers, the Tsai–Widnall–Moore–Saffman (TWMS) instability, also referred to as elliptical instability (Tsai and Widnall 1976; Moore and Saffman 1975), dominates the vortex ring instability dynamics leading into short-wavelength perturbations and also causing the disintegration of the vortex ring into a turbulent cloud as can be seen in Fig. 9d. The dominance of the elliptical instability can also be observed in the formation of turbulent clouds at \(t={20}\,\hbox {ms}\) for dimensionless thicknesses \(\delta =0.22\) and 0.36, as depicted in Fig. 6d, f. For \(\delta =0.09\), on the other hand, the lack of turbulent cloud is noticeable. Hence, the fluid film thickness has an influence on the vortex ring instability. Hence, the influence of the fluid film thickness on the vortex ring instability.

These two kinds of instabilities (long- and short-wavelength) were observed in both experimental (Walker et al. 1987; Leweke and Williamson 1998; Harris and Williamson 2012; McKeown et al. 2020) and numerical (Swearingen et al. 1995; Archer et al. 2010; Cheng et al. 2010; Mishra et al. 2021) studies for vortex ring interaction with a wall, however, investigations of vortex ring interactions in thin liquid films during droplet impact have not been studied in detail up to date. Wilkens et al. (2013) were the first ones to investigate vortex ring formation during droplet impact onto shallow pool, where they hypothesized that the observed waves are associated with the Crow instability. However, it is important to note that their investigation has primarily emphasized the examination of the vortex ring diameter resulting from impact on shallow pools rather than its instability. In contrast, the present study focuses on understanding the convective mixing process and the underlying mechanisms influencing it for droplet impacts onto very thin and thin films.

Figure 11a,b shows the case where both the droplet and the film liquids are labeled with rhodamine B for \(\delta =0.22\). In this way, the film thickness can be evaluated from the fluorescence signal. Therefore, the fluorescence signal intensity is higher for film thickness measurements compared to concentration measurements (Fig. 6d). The side view images are identical to the cases where only the liquid film is labeled with dye (Fig. 6c). One disadvantage of the LIF method, which was also mentioned by Hann et al. (2016), is the huge increase of the signal intensity in areas with larger slopes due to total reflection in the water–air interface, as can be seen at early impact times in Figs. 6 and 11. This effect does not allow for the observation of the ring structures forming earlier (see Fig. 9). Therefore, the investigation of mixing is started when this effect is avoided, thus when the wave amplitude is small compared to the wavelength.

As stated before in Sect. 2, rhodamine B impacts the surface tension of the liquid depending on the used amount for labeling. Hence, an examination into the influence of the used concentration of rhodamine B, \(C_{0}=8\times 10^{-5}\hbox {M}\) on the mixing physics will be conducted. For this purpose, droplet impact on two films of the same thickness \(\delta =0.22\) and different dye concentrations is studied. Surface tension measurements were conducted using a tensiometer (DCAT 25 dataphysics-instruments) and surface tensions were measured by a Wilhelmy Plate tensiometer (DCAT 25 dataphysics-instruments). These are \(65.34 \pm {0.24}\,\hbox {mN}/\hbox {m}\) and \(69.95 \pm {0.11}\,\,\hbox {mN}/\hbox {m}\) for concentrations \(8\times 10^{-5}\hbox {M}\) and \(5\times 10^{-6}\hbox {M}\), respectively. These values are in good agreement with Seno et al. (2001). Figure 12a, b represents the impact for both concentrations. The images were showed in grayscale as they are better visible in this way. Moreover, the LED used for side view recording is also illuminating the film for \(C=5\times 10^{-6}\hbox {M}\) to improve visibility. Hence, there occurs a reflection spot in Fig. 12b at \(t={9}\,\hbox {ms}\) which does not exist for Fig. 12a. Additionally, due to the side illumination, the structures inside the crater are also visible. The vortex-driven mixing remains present in both cases and no apparent difference exists. As shown Fig. 6 in the work of Che and Matar (2017), surface tension effects due to surfactants tend to dampen the instabilities by virtue of their interfacial rigidifying effect (Constante-Amores et al. 2021) which leads into a circular shape of mixing instead of a decomposition of the vortex ring into a turbulent cloud. Therefore, for the chosen initial concentration \(C_{0}=8\times 10^{-5}\hbox {M}\) no effect on the mixing process could be measured.

In Fig. 13, the 2D local thickness of the film during the impact is presented. The x and y positions as well as the film height \(h_{\text {m}}\) are normalized by the droplet diameter D. One key aspect observed here is the rotational symmetry of the film thickness evolution during impact. This feature is used later on to separate fluorescence signal emanating from local thickness values and those purely related to concentration variations.

For certain time instances of the impact, the mean radial thickness profile is deducted from the 2D surface profiles by averaging the thicknesses corresponding to the same radial position r. This information is then used for the TIC calibration model. Radial dimensionless thickness profiles for \(\delta =0.09\) and \(\delta =0.22\) at \(t_{\text {mix}}={3.7}\,\hbox {ms}\) are plotted in Fig. 14a. The plotted values are the averaged reconstructed thicknesses in circumferential direction over all repeated measurements at fixed radial positions, where the error bars indicate the standard deviations. Due to the small standard deviations and smooth profile evolution also, slight height variations are detected providing a good picture of wave dynamics after impact. The precision of the radial thickness reconstruction depends on the local reconstruction accuracy itself but also on the repeatability of all five repeated measurements. Similarly to Sect. 4, equation (8), representing the coefficient of variation, is applied to quantify the radial thickness reconstruction precision.Here, n(r) denotes the sample size for a given radius r, \(h_{\text m_{i}}(r,t)\) the measured thickness for a given radius at a given time and i(r) represent the index of the the element in the sample. The spatial and temporal average of the coefficient of variation is then calculated by taking the average of CV values at each individual radius and time. For a single measurement, this average equals \(3.51\%\), \(3.62\%\) and \(4.72\%\) for \(\delta =0.36\), \(\delta =0.22\) and \(\delta =0.09\), respectively. To account for the repeatability of the measurements, the average of CV is now calculated by including all 5 repeated measurements. The resulting CV averages are \(5.47\%\), \(5.27\%\) and \(6.67\%\) for \(\delta =0.36\), \(\delta =0.22\) and \(\delta =0.09\), respectively. The small increase in the standard error with smaller thicknesses originates from a decreased intensity signal as explained in more detail in Sect. 4.

For a better visualization, the characteristic 3D surface shape of the impact can also be generated by performing a surface of revolution from the radial thickness graphs, as shown in Fig. 14b. The power of the LIF technique is its ability to detect other features happening during the impact such as propagation of surface waves, as shown in the left corner of the 3D reconstruction. Besides propagating capillary waves, a characteristic feature is the collection of liquid in the center of the crater at \(t={9}\,\hbox {ms}\) (see Fig. 14b). It is assumed that this is a result of decay of the crater due to surface tension, thus an induced backflow toward the crater’s center.

In the following, the averaged radial film thickness is used to derive radially resolved concentration profiles during the droplet impact.

A combination of TIC calibration model and the extracted local film height variations delivers the 2D dye concentration distribution during the impact as shown in Fig. 15. The initial dye concentration is \(C_{0}=8\times 10^{-5}\hbox {M}\) which represents the maximum value in the colorbar. The mixing of the liquid film with pure water leads into a decrease of the local concentration. The established LIF method allows to determine the exact amount of liquid mixed and whether the mixing is homogeneous or not, as will be discussed in the following.

A first observation of the 2D distribution in Fig. 15a is the formation of finger-like instabilities of lower dye concentration. These originates from vortex ring instabilities, as discussed in Sect. 5.1, where an entrapment zone is generated. As the depth averaged concentration displays, stronger concentration gradients in radial direction are observed in the back and front edge of the vortex ring. This can also be seen at \(t={20}\,\hbox {ms}\) in Fig. 15b. Note that the concentration jumps along the interrogation window edges shown in Fig. 15b, c result from the calibration procedure described in Sect. 3. These jumps are attributed to the small discontinuity between each fit at the boundaries of the interrogation windows as can be seen in Fig. 2b. Furthermore, the accumulated error from the radial thickness measurements may also affect the continuity between the interrogation windows and contribute into these slight concentration gradients along the edges.

The radial concentration profile is derived in analogy to the radial thickness profile from averaging in circumferential direction. Results are plotted in Fig. 16a for a selected measurement of film thickness \(\delta =0.22\) at \(t={9}\,\hbox {ms}\) and \(t={40}\,\hbox {ms}\). Note that the measured concentration \(C_{\text {m}}\) and the radius r are normalized by \(C_{0}\) and D. Consistent with the observation of stronger concentration gradient in the back and front edge of the vortex ring described earlier, the radial concentration plot at \(t={9}\,\hbox {ms}\) (Fig. 16a) also exhibits a similar pattern. As time progresses up to \(t={40}\,\hbox {ms}\), these gradients tends to gradually flatten out, leading into more uniform distribution of concentration. This can be attributed to the disintegration of the vortex ring into a turbulent cloud, which enhances mixing and leads into a more even distribution of concentration.

Next, the radial concentration of repeated measurements each will be compared for \(\delta =0.22\). As mentioned before, the mixing process is turbulent, with certain characteristic coherent vortical structures such as the vortex ring. The chaotic behavior of the vortex ring decay leads to variations in the local concentration distribution as shown in Figs. 6d and 12a at \(t={40}\,\hbox {ms}\). However, the radial concentration distribution at \(t={40}\,\hbox {ms}\) is very similar for all repeated measurements as presented in Fig. 16b. Differences are mainly visible near the impact center between \(r/D=0\) and \(r/D=0.5\). Here, the concentration is also normalized by \(C_{0}\). This is due to the fact that the drop liquid, remaining at the center of the impact after the receding phase, encounters deformations caused by the turbulent disintegration of the vortex ring. As a result, the liquid present in the center undergoes different displacement in each measurement, thus the observed differences in radial concentration distributions near the central region.

In the context of mixing performance, the coefficient of variation (\({\text {CV}}_{C}\)) can be used as an indicator of the degree of mixing or homogeneity in a system (Gandhi et al. 2011). It is expressed by the following equation:where n denotes the sample size defining the mixing area, \(C_{\text {m}}(x,\, y)\) the measured local concentration in the position (x,y) at a given time t. Note that the chosen mixing area differs in thickness, since this area increases with decreased thickness as mentioned in Sect. 5.1. Furthermore, the criteria determining the selection of the mixing area in the computation of \({\text {CV}}_{C}\) are the maximum expansion of the vortex ring as well as the concentration gradient in the front edge of the ring.

A lower coefficient of variation indicates a more homogeneous distribution of the droplet liquid, suggesting a higher degree of mixing. Conversely, a higher \({\text {CV}}_{C}\) indicates a less homogeneous distribution, implying poorer mixing performance. Figure 17a represents the temporal variations of \({\text {CV}}_{C}\) for all the cases, where the error bars indicate the standard deviations of the repeated measurements. For the thickness \(\delta =0.09\), the \({\text {CV}}_{C}\) decreases and reaches rapidly a steady value, indicating a short convective mixing period followed by diffusion. In contrast, for thicknesses \(\delta =0.22\) and \(\delta =0.36\), the \({\text {CV}}_{C}\) continues to decrease over a longer period due to convection-driven mixing induced by the disintegration of the vortex ring. Notably, the coefficient of variation for \(\delta =0.36\) decreases even more, indicating a stronger convective mixing effect associated with thicker liquid layers. These observations suggest that the film thickness significantly affects the mixing behavior, with thicker films exhibiting prolonged convection-driven mixing. In Fig. 17b, radial concentration distributions for the three thickness at \(t={20}\,\hbox {ms}\) are compared. The thinnest film, i.e., \(\delta =0.09\), exhibits a distinct pattern compared to the other thicknesses, characterized by sharp concentration gradients. In addition, the initial concentration increase in radial direction is followed by a decrease and then another increase. For \(\delta =0.09\), the instabilities occurring during the impact are not strong enough to lead into a vortex ring decay (see Fig. 6b), resulting into a clear primary (low concentration) and secondary (high concentration) vortex rings, which explains the wave-like sequence.

Moreover, a slight increase in \(CV_{C}\) values can be seen at \(t={22.5}\,\hbox {ms}\) for \(\delta =0.22\) and \(\delta =0.36\) in Fig. 17a. This latter is attributed to the spreading of the small jet (see Fig. 11 at \(t={20}\,\hbox {ms}\)) after the receding phase, which predominantly consists of the droplet liquid. It can also be clearly observed in the corresponding videos that are available in supplementary material (Ennayar et al. 2023).