Dimensional crossover in the coalescence dynamics of viscous drops confined in between two plates

Abstract

Coalescence of liquid drops is a daily phenomenon familiar to everybody and is related to many fields from biology to astronomy and also related to a variety of practical problems in industry. However, the detailed physical understanding of the dynamics has been revealed only recently with the aid of high-speed camera, high-performance computer, and scaling analysis. In this study, coalescence of a viscous drop to a bath of the same liquid is studied in a confined space. This is because dealing with a small amount of liquid drops becomes increasingly important (e.g., in industrial and biological applications). Here, the aqueous drop and bath are surrounded by low-viscosity oil and sandwiched by two parallel plates of the cell. We quantify experimentally the width of a neck that bridges the drop and the bath during coalescence. As a result, we find that the neck width increases linearly with time at short times, but the dynamics slows down significantly at longer times. Thanks to simple and original scaling arguments, we clearly show that this transition of the dynamics with time in a single coalescence event is brought about by a crossover from a three-dimensional viscous dynamics for a spherical drop to a quasi two-dimensional one for a disk drop. In addition, we report an unusual type of coalescence that is possibly caused by naturally accumulated electric charge in the confined geometry and whose dynamics seems self-similar.

From daily experiences, everybody knows a liquid drop falling onto a bath of the same liquid merges to the bath. A spectacular aspect of this mundane phenomenon of coalescence of droplets was already studied as early as 1885 by Thomson and Newall (1). It is important in various problems, such as fusion of cells in biology and of galaxies in the universe, and in a large number of industrial applications, such as emulsion stability, ink-jet printing, and others (2). Accordingly, it is still an active area of research (3–5).

In particular, the recent advances in technology, associated with high-speed camera, image analysis, and simulation, together with theoretical development, are leading us to a new phase of understanding. Recently, it has been established that the coalescence dynamics driven by capillary force is balanced by viscous force at shorter times (or in viscous drops) and by inertial force at longer times (or in less-viscous drops) (6, 7). More recently, this has been confirmed also in two-dimensional coalescence (8) and the possibility of a new inertial regime is reported (9). Furthermore, still another new viscous regime, which can be rather regarded as a film bursting, is reported in ref. 10.

Here, we study coalescence of a droplet of radius R in a confined geometry of a Hele–Shaw cell (in between two plates whose distance D is smaller than the droplet size R), which will be relevant in many practical situations where a small amount of liquid has to be manipulated (e.g., microfluidics and biological applications). We followed the dynamics of the half of the neck width r (see Fig. 1). As a result, we find in a single coalescence event a crossover from the three-dimensional viscous dynamics for a spherical drop to a unique type of quasi two-dimensional viscous one for a disk drop, which is potentially important in various practical contexts. Original theoretical arguments, which agree well with our observation, are developed in this paper. This is because, although pinch-off dynamics in the Hele–Shaw cell has been studied frequently (11, 12) the “reverse” coalescence dynamics of drops in the Hele–Shaw cell has not been explored thoroughly enough.

Fig. 1.

(Top) A quasi two-dimensional glycerol drop in a Hele–Shaw cell, surrounded by low-viscosity oil, merging into a bath of glycerol (see Movie S1). The short-time regime (2r ≲ D) is shown with 4-ms frame separation on the left, and the long-time regime (r≫D) is shown with 40-ms frame separation on the right. Triangles in Fig. 2 below are generated from this event. (Middle) Illustrative geometry of the short-time (left) and long-time (right) regimes, in which the neck width 2r, the neck length d, and the radius of the droplet R are defined. (Bottom) Illustration of the experimental setup. The cell is placed vertically (g is the gravitational constant), and the glycerol drop goes down because of gravity before coalescence.

Results and Discussion

We observed a quasi two-dimensional aqueous drop, surrounded by low-viscosity oil in a Hele–Shaw cell, merging into a liquid bath of the same liquid, as illustrated in Fig. 1. We measured the half of the neck width r as a function of time t, as in Fig. 2. The plot in log scales demonstrates that r is linearly dependent on t at short times, and the dynamics slows down at long times where r scales as t^1/4.

Fig. 2.

The half of the neck width r is plotted at various time t (Upper) and the same dynamics in the log scales (Lower). The solid and dashed lines (Lower) indicate slopes 1 and 1/4, respectively. Viscosity η, cell thickness D, and drop radius R for plot symbols are as follows: square, 62.9 mPa·s, 0.7 mm, 5.62 mm; circle, 289 mPa·s, 0.7 mm, 5.56 mm; triangle, 888 mPa·s, 1.0 mm, 4.13 mm; cross, 964 mPa·s, 1.0 mm, 4.32 mm.

We consider first the linear dynamics at short times where r ≲ D/2. In this region, the neck is three-dimensional; it is like a cylinder of radius r (see Fig. 1). Thus, we can expect the recently established three-dimensional viscous dynamics (7) in this regime where the velocity V ≃ dr/dt scales as γ/η because the drops are fairly viscous (we did not observe, as in ref. 7, the logarithmic correction theoretically predicted in ref. 6). Here, γ and η represent the surface tension and viscosity of the drop (and the bath). Indeed, the Reynolds number, ρVr/η (≃ργr/η²) in this case, is less than unity when 2r ≲ D, where ρ is the density of the drop. This linear dynamics, consistent with the log–log plot in Fig. 2, can be expressed in a dimensionless form,

[1]

This is well confirmed in Fig. 3: The data collapse well on a straight line with a slope of the order of unity at short times (r ≲ D and t ≲ τ_i) in the Upper plot. Furthermore, in the Lower plot the constant velocity V obtained from the data in the short-time regime is shown to scale as γ/η, as predicted by Eq. 1.

Fig. 3.

(Upper) A plot between the half of the neck width r and time t, normalized by Eq. 1 for short-time dynamics. (Lower) Comparison of the constant velocity V obtained from the data in the short-time regime with the experimental parameter γ/η. The Upper plot demonstrates a global picture or scaling view with clarification of the characteristic scales, whereas the Lower provides the qualitative agreement.

At this initial stage of coalescence, we did not observe appearance of a dimple surrounded by a rim at the drop bottom of the droplet, although such possibility is implied in a Hele–Shaw cell (11, 12) and is explicitly demonstrated in droplet coalescence (13); in the Top images in Fig. 1, the dark parts at the contact are not such rims but show the surfaces of a glycerol cylinder as shown in Movie S1.

We next consider the dynamics at long times where d ≳ D. Here, the neck length d (see Fig. 1) satisfies a geometrical relation,

[2]

For this purpose, we note that the short-time viscous dynamics in Eq. 1 can be obtained dimensionally by balancing a gain in surface energy per unit time, d(γr²)/dt, with a viscous dissipation η(V/r)² (per time and per volume) localized in a volume r³. Contrary to this, at long times the gain in surface energy per unit time is replaced by d(γrD)/dt while the viscous dissipation η(V/D)² is localized in a volume rDd, to minimize the total dissipation (14). This form of dissipation associated with the gradient V/D originating from the Poiseuille flow between the cell plates separated by a distance D is expected to be dominant when d ≳ D because then this dissipation becomes larger than that associated with the gradient V/d. By using an approximation d ≃ r²/R to the geometrical relation in Eq. 2 (6, 14), we obtain r ≃ (γRD²t/η)^1/4. This t^1/4 dynamics is consistent with the log–log plot in Fig. 2 and can be expressed in the following dimensionless form:

[3]

This dynamics is limited by two conditions: (i) d ≳ D (as already mentioned) and (ii) d ≲ R (for the approximated geometrical relation). These two conditions can be expressed as and r ≲ R, respectively, so that the condition is required to observe this regime; the drop radius must be larger than the cell thickness. This regime is well confirmed again in the log–log plot in Fig. 4: The data collapse well on a straight line with a slope 1/4 at long times ( and t ≳ τ_f) in the Upper plot (see also Fig. S1 for a detailed examination of the data). Furthermore, in the Lower plot the constant r/t^1/4 obtained from the data in the long-time regime is shown to scale as (γRD²/η)^1/4, as predicted by Eq. 3.

Fig. 4.

(Upper) A plot in log scales between the half of the neck width r and time t, normalized by Eq. 3 for long-time dynamics. (Lower) Comparison in log scales of the constant r/t^1/4 obtained from the data in the long-time regime with the experimental parameter γRD²/η. The solid lines in the Upper and Lower plots both indicate the slope 1/4. A scaling view and a quantitative comparison are demonstrated in the Upper and Lower plots, respectively.

The agreement of our theory with our experiment implies that the viscous dissipation in the thin film of oil existing between the neck and the cell plates (see Methods) is virtually inhibited. Although the viscosity of the thin film is much smaller than the viscosity of the liquid forming the neck, the thickness is expected to be much thinner than the cell thickness. Thus, it is natural that the development of the velocity gradient inside the thin film is inhibited to avoid a large viscous dissipation inside the film; virtually no flow is generated inside the thin film. As a result, the speed of liquid forming the neck becomes almost zero at the film–neck boundary while the velocity gradient V/D is developed inside the neck. Similar inhibition of viscous dissipation in thin film was recently confirmed in ref. 15.

The pure two-dimensional coalescence dynamics for cylindrical drops in which velocity is constant along the cell-thickness direction is predicted to have the same scaling as the three-dimensional analogue (6). Compared with this, the present quasi two-dimensional dynamics described by Eq. 3 results from the disk (i.e., quasi two-dimensional) shape of the drop, and the dynamics is governed by the Poiseuille flow whose velocity changes along the direction of the cell thickness.

We sometimes observed an unusual type of coalescence as in Fig. 5. Although in Eq. 2 we assumed that the coalescence starts only when a circular drop directly touches the horizontal surface of the bath, in general coalescence could start with a slight but finite distance d₀ between the drop bottom and the bath surface. Indeed, we occasionally observed that this distance is very large. In such a case, as seen in the three video frames of Fig. 5, singular humps are formed both in the drop and bath before the initial contact; the top hump (from the drop) and the bottom hump (from the bath) seem to be attracted to each other. The coalescence events examined in the above analysis are limited to the case where this initial distance d₀ is much smaller than D to exclude this remarkable but more complex cases from the quantitative analysis.

Fig. 5.

Video frames at the initial contact between the drop and bath with a large initial neck length d₀ (see Movie S2). Singular humps appear both from the drop and bath to attain the initial contact. The frames are separated by 20/3,000 s. η = 991 mPa·s, D = 1.5 mm, R = 2.95 mm.

The initial distance d₀ seems to be controlled by at least two factors: (i) the descending velocity of the drop before coalescence (see Methods) and (ii) naturally accumulated electric charge. The second point is confirmed by examining the effect of discharging the setup. This is practically accomplished, although a complete control of natural charges is difficult in the present environment, by performing the same experiment in different seasons. In the district where these experiments are performed, the humidity changes significantly depending on seasons; electric discharge in daily life is very frequent in winter, whereas it is never observed in early summer. Accordingly, we observed a large d₀ frequently in winter (a typical humidity is 30%), whereas we always observed nearly zero d₀ in early summer (a typical humidity is 70%). This suggests that the humps might be related to the Taylor cone due to electric charge (16, 17). This kind of charge effect in all previous studies has been always induced artificially by a high voltage supply, and recently noncoalescence is induced by this artificial effect (3, 4). In this present case, however, it is possible that naturally accumulated electric charge causes this effect. This hump dynamics should be investigated further from this viewpoint of charge effect by quantitatively controlling the amount of electric charge, and also from the viewpoint of the dependence of the dynamics on the descending velocity of the drop before coalescence.

In addition, this dynamics should be examined further as a self-similar critical dynamics. This is because the self-similarity has been studied well in the case of drop pinch-off (18–21) but rarely studied in the opposite process of coalescence (22). As a matter of fact, as demonstrated in Fig. 6, the drop shape near singularity is confirmed to be self-similar: The shape looks the same when both the horizontal (x) and vertical (y) axes are rescaled by the neck size r (although the geometry of the self-similar shape has yet to be explored). This situation is in contrast with the case observed in ref. 19, where rescaling factors for a good collapse are different in the corresponding two axes.

Fig. 6.

Self-similar shape of the bridge between the drop bottom and the bath surfaces, obtained from the coalescence event shown in Fig. 5. The origin of the plots corresponds to the center of the bridge, or the starting point of coalescence; only the right half of the bridge is shown. The half-bridge shapes represented by circles, triangles, and squares are respectively separated by 3/3,000 and 7/3,000 s (this initial dynamics is nonlinear). The three shapes in the original Upper plot in the units of millimeter collapse well on to a single master shape in the Lower plot (especially near the tip of the shape) where both axes are rescaled by the neck radius r at each time.

Conclusion

In conclusion, we observed coalescence of a viscous drop to a bath of the same liquid confined in a Hele–Shaw cell. The drop and the bath are surrounded by a less-viscous oil; direct contact of the drop and bath liquid with the cell plates is avoided because of thin oil film on the plates. We find that the initial dynamics is described by the well-established three-dimensional viscous dynamics for a spherical drop and the final dynamics by a different quasi two-dimensional viscous dynamics for a disk drop. Our theoretical arguments can predict the ranges of these regimes together with scaling laws of the dynamics, which agree well with our data. In other words, a dimensional crossover of the dynamics is observed in a single event in the present study. This crossover dynamics established here can be potentially important in various fields, such as in microfluidics and biological applications, where small amount of liquids should be manipulated. We also observed coalescence starting with a finite distance between the drop bottom and the bath surfaces, which seem to be attracted to each other to initiate coalescence. This remarkable effect revealed in this study potentially opens up completely new ways to study charge effect and self-similarity in liquid drop dynamics.

Methods

We fabricated a Hele–Shaw cell from transparent acrylic plates of a few millimeters thickness with the plates separated by a distance D (of the order of millimeter) with spacers of homogeneous thickness as in our previous studies (23, 24). The width and depth of the cell are 10 and 15 cm, respectively. We fill the vertically positioned cell with a low-viscosity silicone oil [polydimethylsiloxane (PDMS)] of kinematic viscosity 1 cS (about 1 mPa·s) and then heavier glycerol; the two liquids are immiscible and hence cause a phase separation where the lighter oil is on top of the glycerol (see Fig. 1). After waiting (at least 30 min) for the bath interfaces to reach equilibrium, we insert with a syringe from the above oil phase a glycerol drop at the center of the width of the cell (i.e., away from the edges). The drop, whose radius is about several millimeters (much smaller than the cell width), gradually goes down through the oil phase (of depth around several centimeters) due to gravity before it starts coalescing to the lower phase of the same glycerol. The high-speed video images of coalescence are obtained by a camera, MEMRECAM fx 6000 (Nac), in some cases with a microscope lens D-6MP (Degimo).

To change viscosity η of glycerol (of the drop and the bath) we added water, and each time we directly measured the viscosity of the aqueous solution avoiding a large change in room temperature because the viscosity is sensitive to temperature (25). The viscosity of surrounding bulk oil (about 1 mPa·s) can be always neglected compared with those of glycerol solutions (about 50–1000 mPa·s). The interfacial energy αγ of glycerol solutions contacting the oil can be assumed to be a constant in practice (10, 21). Here, γ = 20 mN/m for convenience with α a numerical coefficient of the order of unity.

The density of glycerol solution ρ and that of oil are 1.21–1.26 (depending on the viscosity) and 0.818 g/cm³, respectively, and the difference results in descending motion of the glycerol drop (before coalescence) at velocity of the order of 1 mm/s (15). This speed is dependent on the rate of flow for creation of the drop from a syringe (inner radius 5 mm) with a needle (inner diameter 0.55 mm). The observations demonstrated above are obtained when the piston is pushed manually around at 5 mm/s (i.e., at flow rate of the order of 0.1 cm³/s). When the descending speed is too fast or slow compared to this speed, slightly different dynamics is observed. When too fast, the flow might tend to be turbulent, and clear scaling laws reported above are contaminated; when too slow, the drop bottom tends to become flat before coalescence with a thin oil film formed between the bottom and the bath surface (10).

Because the oil, which likes the acrylic plates, is poured first, the glycerol phase and the falling drop is always covered with a thin oil film; there is no direct contact of glycerol with the plates, hence, no contact line for the falling drop. This point is quite important to remove experimental contamination due to the intricate effect of the contact line.