Mapping coalescence of micron-sized drops and bubbles

Abstract

Emulsion formulation, solvent extraction and multiphase microfluidics are all examples of processes that require precise control of drop or bubble collision stability. We use a previously validated numerical model to map the exact conditions under which micron-sized drops or bubbles undergo coalescence in the presence of colloidal forces and hydrodynamic effects relevant to Brownian motion and low Reynolds number flows. We demonstrate that detailed understanding of how the equilibrium surface forces vary with film thickness can be applied to make accurate predictions of the outcome of a drop or bubble collision when hydrodynamic effects are negligible. In addition, we illuminate the parameter space (i.e. interaction velocity, drop deformation, interfacial tension, etc.) at which hydrodynamic effects can stabilise collisions that are unstable at equilibrium. Further, we determine conditions for which drop or bubble collisions become unstable upon separation, caused by negative hydrodynamic pressure in the film. Lastly, we show that scaling analyses are not applicable for constant force collisions where the approach timescale is comparable to the coalescence timescale, and demonstrate that initial conditions under these circumstances cannot be ignored.

Introduction

Interactions of micron-sized bubbles and droplets separated by distances from nanometres to microns are important in many applications, including food processing, cosmetics, mineral flotation and microfluidics. Drops and bubbles of this size are continually in relative motion, induced by processes including Brownian dynamics, flows generated in microfluidics, and gravity. As a consequence of this relative motion, bubbles or drops in emulsions or foams are continually colliding and separating under force fields influenced by (i) the equilibrium interaction forces that arise from the surface chemistry of drops and bubbles, (ii) the dynamic interaction forces due to the motion of the fluid and interface, and (iii) the transport of neutral and charged species on, to and around the interface. The relative velocity of these interactions span many orders of magnitude, from Brownian-induced motion of O(1 μm/s) [1] to drop collisions in microfluidics of O(10 mm/s) or higher [2], [3]. If the combination of these equilibrium and dynamic interactions are unstable, then coalescence will occur. Yet, the interplay between the equilibrium interactions, dynamic processes and the deformation of the drop or bubble interface can make precise understanding of the conditions under which coalescence occurs difficult. Importantly, this understanding is vital in a range of applications including the prediction of emulsion stability or shelf life in formulated products, drop or bubble coalescence phenomena in processing equipment such as mixer-settlers, and in assessing the performance of high-throughput multiphase microfluidic devices. For example, in these devices, droplets are generated at rates of up to thousands of drops per second [4], [5], and are required to transport reagents or biological material in discrete packages. In such devices, it is critical that interactions between drops do not lead to coalescence. In other processes, such as liquid-liquid separation, it is desirable to enhance and accelerate drop coalescence [6].

Equilibrium interactions, whereby the collision speeds are too low for hydrodynamic pressure to play a significant role, have been extensively characterised for deformable drops and bubbles. Building on methods for flat liquid-liquid or liquid-gas interfaces, such as thin film balance methods to measure pressure and thickness [7], [8], [9], first pioneered by Sheluko [10] and Mysels [11], and later developed by Wasan and co-workers [12], [13], there has been a steady evolution of methods to probe the equilibrium surface forces between deformable interfaces with curvature for drop and bubble radii near the capillary length or smaller. This includes the liquid surface forces apparatus [14], the work from the Horn group using a modified surface force apparatus using a mercury [15], [16] or gas [17] filled capillary and a rigid flat surface, and the extension of atomic force microscopy (AFM) to probe the interactions between, first, a rigid particle and a single bubble [18], [19], [20] or drop [21], [22], [23], [24], and secondly, to measure the interactions of drop [25], [26], [27] or bubble [28] pairs. There have also been more recent developments, including the integration of interferometry with AFM to probe a single bubble with a flat surface [29], larger length-scale systems that utilise bimorph cantilevers and a single drop or bubble in the integrated thin film drainage apparatus (ITFDA) [30], and cantilevered capillaries to examine the interaction of two drops [31]. The range of surface forces examined in these experiments include Derjaguin-Landau-Verwey-Overbeek (DLVO) forces (the addition of electrical double layer and van der Waals) [25], [32], [33], repulsive van der Waals forces [34], [35], steric [36], [37], structural [36], [38], [39], depletion [36], [40], protein interactions [36], [41] and hydrophobic forces [42]. Common surface forces and their corresponding pressure and energy definitions are given in Tables S1 & S2 in the Supplementary Information (SI). Equilibrium interactions can lead to repulsive or “stable” drop/bubble interactions or “unstable” interactions ending in coalescence [43].

At higher velocities, O(10 μm/s) and above, hydrodynamic drainage plays an important role in drop/bubble collisions. The AFM was easily extended to probe hydrodynamic drainage for a particle and drop first by Aston and Berg [44] and then for drops [26] and bubble [28] pairs. In addition, a number of the methods mentioned above, including the work by Horn [15], [16], [17], [45], classical drainage studies of drops immobilized in larger capillaries [46], the ITFDA and the cantilevered capillary measurement, have all examined a wide range of hydrodynamic drainage behaviour. For example, stable interactions have been observed with both stable and unstable equilibrium forces [28], [42]. Coalescence has also been observed for drops and bubbles in the above methods whilst on approach [47] and whilst separating [28], where coalescence whilst separating was first observed in four roll mill measurements [48] and then micro-fluidic devices [49], [50]. In addition, coalescence has been observed in drops and bubbles undergoing cyclic accelerating approach and withdrawal to mimic drop collisions in microfluidic devices [51], and whilst being held together at constant applied force to mimic buoyant collisions [78]. In dynamic collisions the viscosity of the continuous phase becomes an important parameter [52], [53]. Other important quantities that can affect dynamic drop and bubble coalescence are the presence of free charge, and in some practical applications the application of external electric fields [54], [55], [56], [57].

The impact of experimental measurements has come to fruition though efforts to develop quantitative models to account for the presence of surface forces, deformation and hydrodynamic drainage (in a creeping flow, Re ≪ 1 regime) in order to properly analyse these data. In particular, the deformable nature of the system often adds complication to the measurements and analysis, particularly when the length scales of the deformation are not accessible with methods such as interferometry. In the area of AFM measurements with deformable interfaces, the analysis for equilibrium interactions were pioneered by several groups, first for a particle and single drop or bubble [32], [58], [59], then for dynamic interactions for a single drop [44], and then bubble and drop pairs [1]. The application and validation of these models to a range of experimental systems using AFM can be found in the review by Tabor et al. [43]. The summary of the development of these models can be found in the extensive review by Chan and co-workers [60]. The Stokes-Young-Reynolds-Laplace (SYRL) numerical model, initially developed and validated for AFM measurements, has been extended to analyse the dynamic force measurements using the ITFDA [61] and several studies focused on the modified surface forces apparatus from the Horn group [62], [63]. In addition, the more recent work by Zeng using AFM to probe the interactions of a single drop or bubble with a flat plate using interferometry [29] was well described by the same model, first validated on similar measurements without interferometry [34], [64].

These observations, in combination with theory, have helped to shed light on the role of equilibrium interaction forces, interfacial deformation, drop/bubble size, interfacial/surface tension, viscosity, and collision velocity on the stability of interactions, and how these parameters affect the mode of coalescence (i.e. during approach, retract, or dwell) for unstable interactions. Thus, the observations in the literature and a comprehensively validated SRYL numerical model for both AFM measurements and force measurements at larger drop and bubble length scales suggest that is may be useful to apply this knowledge in a larger context.

The schematic in Fig. 1 suggests juxtaposition of the AFM measurement to other types of measurements and uses of drops and bubbles (i.e. emulsion, foams and micro-fluidics) in terms of relative drop and bubble size. Emulsions or foams are commonly characterised by sub-micron radii (in the case of emulsions) to 10 s of microns in radii or larger in the case of some foams. AFM measurements have accessed drops as small as 5 μm in radius [65] (although there is little deformation in those instances) to drops and bubbles with radii up to several hundred microns. The micro-fluidic device, showing drop coalescence, denotes that micro-fluidic drops and bubbles can have radii commensurate with an AFM measurement or larger. The two drops fixed on capillaries represent the larger drop and bubble radii found in the ITFDA and cantilevered capillary studies as well as more classical capillary drainage studies, where the drop rise scenario is representative of traditional drop coalescence studies. Thus, in this study we attempt to move forward from AFM measurements on individual systems to decant aspects of this understanding to more general questions, mapping the precise conditions whereby drops and bubbles coalesce. Using the SRYL numerical model we systematically probe the multi-dimensional parameter space, in order to determine the conditions required for both stable and unstable interactions, and the mechanisms responsible for coalescence of micron-sized bubbles or drops with direct relevance to the scenarios shown in Fig. 1.