Article pubs.acs.org/Langmuir
Free Impinging Jet Microreactors: Controlling Reactive Flows via Surface Tension and Fluid Viscoelasticity Philipp Erni* and Amal Elabbadi Materials Science Department, Research Division, Firmenich SA, Meyrin 2 Geneva, Switzerland ABSTRACT: We investigate the use of impinging free liquid jets as wall-free continuous microreactors. The collision of two reactant jets forming a free-standing thin liquid sheet allows us to perform rapid precipitation reactions to form colloidal particles, enhance micromixing, and master challenging reactions with very fast kinetics. To control the shape, size, and hydrodynamics of the impingement zone between the two liquid streams, it is crucial to understand the interplay among surface tension, fluid viscoelasticity, and reaction kinetics. Here, we study these aspects using model fluids, each illustrating a different physical effect of surface and bulk fluid properties. First, solutions of sodium dodecyl sulfate below, near, and above the critical micelle concentration are used to assess the role of static and dynamic surface tension. Second, we demonstrate how dilute solutions of high-molecular-weight polymers can be used to control the morphology of the free surface flow. If properly controlled, these effects can enhance the micromixing time scales to the extent that very rapid reactions can be performed with outstanding selectivity. We quantitatively assess the interplay between the free surface flow and reaction kinetics using parallel-competitive reactions and demonstrate how these results can be used to control the particle size in precipitation processes.
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far less attention.5 Yet as we will show here, colliding wall-free liquid jets are very suitable as rapid continuous microreactors. Although jet reactors have been studied in the context of chemical engineering,23−26 for clathrate formation in energy technology,27,28 or as injectors for liquid propellants in rocket engines,29 surprisingly a detailed investigation of the interplay of the governing factors of surface tension and surfactant adsorption, hydrodynamics and fluid viscoelasticity, and reaction kinetics has not yet been undertaken. The free jet configuration used in this study is shown schematically in Figure 1. Two reactant solutions are ejected as laminar jets from opposing capillaries and collide halfway, forming a merged stream and, at high enough impact velocities, a liquid sheet. The angle of incidence is oblique, and the merged fluid stream flows downward into a container to collect the product. The formation and fragmentation of low-viscosity Newtonian liquid sheets formed between impinging liquid jets have been studied and reviewed by several authors.29−38 Several aspects of fan sprays from nozzles are relevant to liquid sheets formed between jets: Dombrowski et al.39 already described the streamlines, the y−1 scaling of the sheet thickness h (where y is the distance from the point of sheet formation), and the dependence of the sheet trajectory on the injection pressure or jet velocity u0 and the surface tension σ. Clanet, Villermaux, and co-workers35,36 performed a range of detailed experimental
INTRODUCTION Polymeric, inorganic, or hybrid particles can be formed by contacting reactant solutions at high supersaturations;1−3 under these conditions, the kinetics of particle formation are typically very fast compared to the time scale of the flow in traditional mixing devices. For reactions with slow kinetics, microchannel reactors with laminar or segmented flows4−7 can enhance heat, mass, and momentum transfer and minimize risk, for example, in strongly exothermic reactions.8 For fast reactions, however, the rapid mixing of two liquids containing reactive species remains a challenge in many applications including energy technology, particle engineering, pharmaceutical development, and laboratory-scale fluids processing. In particular, precipitation processes are notorious for their complex dependence on length scales, flow types and patterns, reactor geometry, and flow rate.1,9,10 Momentum transport and reaction engineering aspects of confined micromixers with impinging jet and vortex geometries have recently been studied in great detail.11−14 These have been applied to synthesize a variety of particles, in particular, for drug delivery and medical engineering.3,15,16 Confined impinging streams merging inside closed channels have been applied to solid/liquid catalytic reactions,17 to polystyrene particle formation,18 and for the reactive in situ generation of block copolymers to protect colloidal particles.19 Moreover, this stream configuration is also of interest for emulsion formation.20 In contrast to enclosed, microfabricated T- or Y-flow reactors4−8 or droplet-based microfluidic mixing,21,22 the simpler free-standing liquid jet geometry has recently received © XXXX American Chemical Society
Received: March 19, 2013 Revised: May 14, 2013
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velocity u0, and the angle between incident streams. Bremond and Villermaux34 extended these studies to the atomization regime and provided a detailed analysis of the rim instability of oblique colliding jets using external harmonic perturbations. Rothstein’s group40 studied fluid webs formed upon collision of viscoelastic jets of solutions of self-assembled wormlike micelles. The fluid morphologies change from chains to rippled sheets to fluid webs and ultimately to atomization with increasing flow rate. Jung et al.41 studied the atomization patterns formed upon collision of viscoelastic, dilute polystyrene and poly(ethylene oxide) solutions and observed markedly different atomization patterns and size distributions of the fragmented drops. Stagnation point flows between horizontal impinging jets have also been used for approximate measurements of viscoelastic fluid properties.42 The effect of viscoplastic, “gel”-type rheological properties was discussed by Baek et al.43 for the case of nanoparticle-loaded microgel dispersions. To compare and generalize experiments with impinging jets, it is convenient to recast the governing forces and time scales in dimensionless form. The Reynolds number Re = ρd0u0/η describes the ratio of inertial to viscous forces; in the following discussion, we will make ample use of the Weber number,44
We =
Figure 1. Schematic of the free impinging jet microreactor. C1, C2 and J1, J2: capillaries and liquid jets for reactant solutions 1 and 2. y: vertical centerline/direction of gravity. x: horizontal axis. S: liquid sheet formed upon collision of the streams in the x−y plane. P: product stream. V: direction of view.
ρd0u0 2 σ
(1)
which compares inertial to capillary forces; it is an important group for rapid free-surface flows. The Froude number Fr = u0d0−1/2g−1/2 provides information about whether gravity is important. Here, this is negligible near the point of collision because the dimensions are small; the noncircular shape in the downward direction is not caused by gravity but by momentum propagated forward from the oblique impingement angle of the streams. For reacting fluids, an additional group, the Damköhler number Da = tm/tr, arises; it is the ratio of a characteristic time scale of the flow (the mixing time tm) to the characteristic time scale of the reaction tr. Mahajan and Kirwan26 focused exclusively on Re and Da to characterize double-jet mixers. The effects of (i) system size, (ii) fluid viscosity, (iii) velocity, and (iv) density are therefore accounted for as the ratio of inertial forces to viscous forces. This is already a suitable and powerful approach to characterizing mixing time characteristics by analogy to confined jet or vortex mixers. However, the rapid free surface flow observed on impinging free jets introduces additional physical properties
studies on liquid sheets formed upon the impact of jets on solid and liquid targets. These horizontal sheets, formed by a single jet, persist up to a maximum radius proportional to u02 and then break up into droplets.30 Bush and Hasha33 presented a theoretical and experimental study of the sheet morphologies and instabilities developing under the influence of surface tension and fluid inertia. In particular, they observed and analyzed “fluid chain” configurations of mutually orthogonal oval thin films that decrease in size in the downstream direction (Figure 1). Rayleigh−Plateau instabilities on these chainlike fluid films lead to a structure resembling fluid fishbones, observable only using high-speed imaging with strobe illumination. The relevant parameters, all of which will also be used in the following study, are the fluid viscosity η, density ρ, surface tension σ, volumetric flow rate Q, jet diameter d0,
Figure 2. High-speed video images for free jets of water colliding at different flow rates. (Left to right) Y-shaped jet (inset: side view in the direction of the x axis for the same experiment); liquid sheet with fragmentation into drops downstream from the impingement zone; larger liquid sheet with rim instabilities growing closer to the point of collision; fragmented sheet forming a spray, with droplet breakup occurring along the entire edge. Scale bar: 5 mm. Incident jet velocities from left to right: u0 = 3.5, 4.8, 5.9, and 7.1 m/s. The axes refer to the schematic shown in Figure 1. B
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liquid-on-liquid collisions as compared to liquid-on-solid collisions. Typical reactant solutions often contain surfactants, for example, to stabilize colloidal particles formed in situ.3,15 To model their effect on free jet collision, we use solutions of sodium dodecyl sulfate (SDS), a widely used anionic surfactant.47−49 Experiments are performed at different surfactant concentrations below, near, and above the critical micelle concentration (cmc). In contrast to surfactant-free water streams, high SDS concentrations are able to yield stable sheet morphologies without siginificant edge breakup (Figure 3). In the absence of
that are not accounted for in the Reynolds number. As we will show in this Article, there are several additional physicochemical parameters that strongly influence the morphology, flow, and ultimately the efficiency as microreactors: (v) the surface tension of the fluids; (vi) dynamic surface properties, including adsorption kinetics, Marangoni stresses, and surface rheology; (vii) fluid viscoelasticity if large-molecular-weight polymers are present or if the product formed by the reaction induces changes in fluid rheology; and (viii) the interplay of all of the above with reaction kinetics. A knowledge of the link among transport properties, flow patterns, and properties of the resulting product, for example, in the rapid precipitation of colloidal particles, is important for better understanding and control of these processes. Here, we study reactive free surface flows of free impinging jets in the presence of surfactants, polymers, and reactive species. First, we establish the relevant flow morphologies of the liquid sheets resulting from the collision of model surfactant and polymer solutions using high-speed video imaging. Second, using the physicochemical parameters of these model fluids, we construct dimensionless maps to differentiate the principal flow regimes and establish the relevant boundary conditions for interfacial flows in the presence of reactions. Finally, we investigate the interplay of free surface flows and reaction kinetics, assessing both the effect of reactive particle formation on the flow patterns and vice versa.
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RESULTS AND DISCUSSION Effect of Static and Dynamic Surface Tension. We first investigate the role of surface tension and hydrodynamics for the liquid sheets formed upon collision of two aqueous reactant solutions. Figure 2 shows sample images recorded by highspeed imaging for surfactant-free water streams colliding at different velocities. From left to right, the morphology changes from a threadlike merged jet (Y-shaped, as shown in the sideview image in the inset of Figure 2), with only a very small discernible impingement zone and droplet breakup occurring exclusively downstream from the point of collision, to extended liquid sheets of increasing size. At intermediate flow rates, liquid ligaments form and break up into droplets, but this still occurs far downstream from the point of collision. Ultimately, at higher flow rates droplet fragmentation occurs directly at the rim along the entire perimeter. The mechanism of droplet breakup is a Rayleigh−Plateau instability30,34 of the threadlike rim of the sheet; in contrast to isolated liquid threads or single jets emerging from nozzles,45,46 the thickened thread edge is laterally fed by a continuous supply of liquid in the film on the inside.34 Additionally, it is continuously stretched and pushed away from the impact zone toward the outside. For the total sheet size with circular geometry, the simple expression30 2 1 ρu 0 d 0 1 d = = We 8 σ 8 d0
Figure 3. Stable fluid chain morphology observed for sodium dodecyl sulfate solutions above the critical micelle concentration (left). Transition from stable, quasi-stationary fluid chains to fragmenting sheets upon mechanical perturbation (middle and right). Surfactant concentration c0 = 82 mM. Images shown are for two different velocitities, u0: (a) 3.5 and (b) 5.9 m/s.36 Scale bar: 5 mm.
mechanical noise and at flow rates low enough to avoid edge fragmentation, the liquid sheets remain stabilized in a quasistationary configuration for extended times. However, even minor mechanical perturbations, such as vibration of the capillaries or the passage of air bubbles, cause the stable sheets to snap into the fragmented breakup regime. As the surfactant concentration is decreased, the stabilizing effect reduces to a narrow window of flow rates. In Figure 4, we show measurements of the maximum sheet diameters as a function of the flow rate at different surfactant concentrations c0. For c0 ≪ cmc, the maximum horizontal sheet diameter remains very similar to that of water. Although SDS at this concentration does reduce the equilibrium surface tension of the air/solution interface to around 65−66 mN/m, the time scales observed here for the formation of a fresh surface upon
(2)
describes the sheet diameter d as a function of the velocity u0, surface tension σ, incident jet diameter d0, and liquid density ρ in the inviscid limit (i.e., the dimensionless sheet diameter is simply proportional to the Weber number). The inviscid approximation provides a good description of the behavior over a wide range of conditions.36 Because a minor amount of viscous dissipation occurs in the impingement zone, the inviscid assumption appears to be an even better approximation of C
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Figure 4. Maximum sheet diameter in the horizontal direction, measured by image analysis, as a function of the incident jet velocity u0. Results are shown for (a) water, (b) sodium dodecyl sulfate (SDS) solutions below the cmc (c0 = 0.82 mM), and (c) SDS solutions above the cmc (c0 = 82 mM). Lines are predictions of eq 2 based on the equilibrium surface tension values measured independently.
The different flow regimes observed at varying surfactant concentrations and flow rates are summarized in a dimensionless map in Figure 7; a guide to the flow patterns is shown in Figure 6. At low flow rates (low We numbers), all solutions form a merged linear jet (Y jet) with drop breakup far from the impingement zone; at high flow rates, all solutions form sheets that break up into droplets directly at the edge of the sheet. Both limits are identical to the flow patterns observed with pure water, and indeed for the lowest surfactant concentration (c′ = 10−3), the breakup patterns follow exactly those observed for water. Surfactant concentrations above c′ > 0.1 give rise to an intermediate regime wherein more stable sheet and chain morphologies are observed. Additionally, sheets formed by impinging SDS solutions near their cmc exhibit a weaker tendency for droplet fragmentation at the rim as compared to water or to those above the cmc. Possibly, Marangoni stresses51,52 developing at the rim upon fragmentation play a role in this. These stresses, due to surface tension gradients,
collision are orders of magnitude shorter than those associated with significant surfactant adsorption. Therefore, to observe the effect of surfactants, the flow velocities would need to be reduced to values far below the jetting regime.46,50 In contrast, at surfactant concentrations above the cmc, the fluid sheet size is strongly increased as compared to that of pure water. The decrease in the sheet diameter above a critical velocity is due to the onset of a shear instability with the surrounding air at rest.32,36 In Figure 5, an estimation of the dynamic surface tension is shown, as calculated using eq 2. Because the sheets are subject
−∇s σ =
−∂Γ ∇Γ s ∂Γ
(3)
(where ∇s is the surface nabla operator and Γ is the surface concentration) can have a surface-stabilizing effect; in this picture, lower surfactant concentrations near and slightly below the cmc would likely be favorable to reducing excessive droplet fragmentation at the rim. Stable states for sheets with high surfactant concentration seem to be due to a global surface tension effect, as is found for pure liquids with low surface tension such as ethanol or glycerol/water mixtures. In the fast-flowing limit, the rate of surface expansion exceeds both the adsorption and diffusion rates of the surfactant, and the surface tension will simply approach that of the neat solvent and no significant surface tension gradients develop. (See the Appendix for details on the relevant quantities in surfactant transport.) The transition from stable sheets to fragmented sheets with increasing We in Figure 7 can therefore be associated with a critical velocity above which surfactant adsorption is no longer sufficient to stabilize the sheet and strong rim breakup sets in. Interplay of Fluid Viscoelasticity and Surface Tension. We now demonstrate how liquid sheets formed by colliding free jets can be controlled via the elongational viscoelastic properties of the fluids in the presence of small amounts of a dissolved, high molecular weight polymer. The shear viscosity remains low in all of these cases (η0 < 1.9 mPa s). However, neither the inviscid approximation used above nor a simple
Figure 5. Estimation of the surface tension from the widest sheet diameter before sheet breakup, shown for SDS solutions at concentrations of (a) c = 0.82 and (b) 82 mM. Error bars are calculated from the temporal variation in the maximum sheet diameter measured from the high-speed video movies; larger error bars therefore merely indicate stronger instabilities on the liquid sheets. Dashed lines indicated the independently measured reference surface tensions of water (σ = 71.9 mN/m) and sodium dodecyl sulfate (σ = 34.5 mN/m, c0 = 82 mM, which is 10 times the critical micelle concentration).
to instabilities, especially at higher flow rates and high surface tension, the time-averaged values of the maximum sheet diameter in the horizontal direction are measured. Surface tensions are shown for two SDS solutions at concentrations of 0.82 and 82 mM, corresponding to 0.1 and 10 times the critical micelle concentration. Comparison with the independently measured reference surface tensions (dashed lines) of water (σ = 71.9 mN/m) and sodium dodecyl sulfate (σ = 34.5 mN/m at c0 = 82 mM) suggests that the simple relation d ∝ We, which was derived for circular sheets, provides a good approximation if the sheet diameter is measured in the x direction, where the horizontal component of momentum is canceled out between the colliding streams. For increasing flow rates, with very short fluid transit times in the sheet, the apparent dynamic surface tension increases and approaches the value of water. D
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Figure 6. Overview of prototypical flow patterns referred to in the dimensionless maps shown in Figures 7 and 9. (a) Y jet with breakup far downstream in the merged jet; (b) sheet/chain, breakup downstream far from the impingement zone; (c) fragmented sheet, breakup along the rim of the sheet; (d) fluid web, Rayleigh-plateau instabilities with suppressed breakup along the rim; (e) colliding beads connected by strongly extended fluid ligaments; the arrow indicates a collision event between beads formed on the two incident jets.
elongational viscoelastic effects, and the sheet morphology is no longer controlled by inertia and surface tension alone. In Figure 8, we show the flow patterns obtained for 0.1% w/ w solutions of a high-molecular-weight poly(ethylene oxide) solution (nominal Mw = 600 000 g/mol). At the lowest flow rate shown, the liquid sheets are more expanded than those observed with either water or any of the surfactant solutions. Even with a pronounced increase in the flow rate, fragmentation into drops only occurs far downstream from the mixing zone, with much longer time-averaged sheet lengths. The fishbone-type extended ligaments formed on the merged jet resist capillary breakup and retract back onto the main stream. This breakup pattern suggests that the liquid ligaments are stabilized by significant elongational strain stiffening,45,53 and the available capillary pressure at a given ligament size and surface tension is insufficient for the Rayleigh−Plateau breakup to proceed to completion. Liquid fragmentation ultimately occurs far downstream, where the elongational stresses have sufficiently relaxed for the capillary pressure to take over and break up the merged stream into droplets.45,53 Finally, at the highest flow rate, the sheets become very wide in the x direction, with maximum sheet dimensions of up to 100 times the original jet radius, and breakup occurs via hole formation33,40 in the thinned-down liquid film. Droplet fragmentation via lateral ligament breakup is now strongly delayed. Instead, the fluid threads formed around the perimeter
Figure 7. Dimensionless map of flow morphologies formed by impinging jets of aqueous surfactant solutions. c′ = c/cmc is the concentration of sodium dodecyl sulfate normalized with the critical micelle concentration, and We is the Weber number. Black squares identify the preferred regime for controlled free surface flows with reactive streams. A guide to the individual flow morphologies is shown in Figure 6.
shear viscosity correction36 is applicable any longer: the extreme stretching kinematics of the flow gives rise to strong
Figure 8. Free jets of solutions in the presence of elongational viscosity, (PEO, Mw = 600 000 g/mol, c = 0.1% w/w). Scale bar: 5 mm. Jet velocities, from left to right: u0 = 3.5, 5.9, 7.1, and 14.9 m/s. E
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which η varies here, the classification of flow regimes would be similar to that in Figure 9. The longest relaxation time can be calculated from the Zimm theory for a polymer chain in a good solvent as
of the sheet remain intact and are rapidly ejected outward in the radial direction. The enhanced stability of these threads can again be explained by the strong elongational deformations occurring in the tangential direction at the edge of the expanding sheet. Such kinematics is known to induce strong elongational strain stiffening and delay droplet breakup.45,54,55 Hole formation in the thin liquid film combined with extended lifetimes of the liquid threads result in weblike structures of interconnected fluid filaments; surface tension ultimately does cause these webs to break up into droplets, but this happens only far downstream from the point of collision. Sheets formed by impinging jets are described well by an inviscid approximation for simple liquids, nonstructured surfactant solutions, and even fluids with a yield stress. In contrast, our results with dilute PEO solutions show that the behavior dramatically changes in the presence of elongational stresses. The static surface tension of the polyethylene solution as used here is only moderately lower than that of water (61.7 mN/m at 25 °C) and does not explain any of the effects observed here; neither are surface rheological effects expected to play a significant role in dilute PEO solutions.56 In Figure 9, we organize the generated flow morphologies observed with polymer solutions in a dimensionless map,
λZ ≈
accounting for the effects of surface tension, flow velocity, system size and density, and the effective polymer relaxation time λeff. The Deborah number λeff σ 1/2 (ρR 0 3)1/2
(5)
where 1/ζ(3υ) is a numerical front factor54 containing the excluded volume exponent υ, ηs is the solvent viscosity, NA is Avogadro’s number, kB is the Boltzmann constant, and T is temperature. With this expression, λZ follows directly from Mw and ηs. Tirtaatmadja et al.54 found from liquid thread breakup experiments and by comparison with literature values that λZ underpredicts the observed relaxation time above c/c* ≳ 0.1; these authors suggested a scaling law of λeff/λz ∝ (c/c*)3υ − 1. We perform a nonlinear least-squares fit over published experimental data54,55 for λeff/λZ = f(c/c*) and apply the resulting power law to obtain λeff, which is then used to calculate De. For the intrinsic viscosity of the PEO solutions, we use the numerical parameters obtained for the Mark− Houwink−Sakurada (MHS) equation by Tirtaatmadja,54 [η] = 0.072Mw0.65, with the intrinsic viscosity [η] in units of cm3/g, and c* = [η]−1, assuming the case of a solution of flexible polymers. Elastic stresses begin to play a noticeable role as De increases in the dimensionless map of De versus We as the highly characteristic morphologies shown in Figure 8 develop. The fundamental shapes of the sheets remain identical, independent of instabilities propagating from the incident jets. This can be confirmed and elucidated by varying the distance between the nozzles: small distances imply more stable, stationary sheet morphologies whereas longer distances lead to stronger instabilities, but the underlying geometry of the sheet remains the same. High-molecular-weight polymers affect the growing instability on the incindent jets at dilute concentrations;53,55 these phenomena are well-understood and are relevant in a variety of applications, including droplet formation from liquid threads46,53 or electrospinning.57,58 In addition to providing a way to control the sheet size and shape, polymer viscoelasticity also influences the magnitude of the incoming perturbations that trigger thickness fluctuations and droplet breakup at the edge of the sheet. The onset of “colliding beads” flow patterns at longer polymer relaxation times demarcates an upper boundary for De that should not be exceeded if the jets are used as a reactor: for higher Mw polymers, die swell occurs at the capillaries even at relatively dilute concentrations at the velocities of interest here. The resulting widening of the jet along with conservation of mass leads to slower, wider streams, and droplet formation occurs much earlier on the incident jets.53 The flow pattern then no longer is a liquid sheet but rather a configuration with two merging “beads-on-a-string”-type fluid threads;45 high-intensity collision still occurs, but only when two opposing beads impinge on each other. (See Figure 6e for an example of beadon-bead impact.) The conditions necessary to suppress premature sheet fragmentation are in line with observations made for the inhibition of filament breakup using small amounts of a highmolecular-weight polymer:59 in both cases, the change in behavior is observed as the characteristic deformation rate becomes similar to the inverse of the polymer relaxation time. This transition is associated with a significant increase in the elongational viscosity. Fluid viscoelasticity becomes far more
Figure 9. Dimensionless map accounting for the fluid elasticity, surface tension, and inertia-dominated breakup. The intrinsic Deborah number De for reactant solutions containing dilute concentrations of polymer is plotted against the Weber number We. Black squares identify the preferred regime for controlled free surface flows with reactive streams; sample images of the flow morphologies are shown in Figure 6.
De =
1 [η]M w ηs ζ(3υ) NAkT
(4)
is plotted against the Weber number; both are based on the dimensions of the incident jets. Here, De is used as an intrinsic property and merely serves as a dimensionless polymer relaxation time, with all other parameters being constant or varying only very slightly. Experimentally, it is varied by using solutions of PEO with different molecular weights. We note that another choice for the ordinate axis would be to use the ratio λη/(ρR2) (this group is also called the elasticity number40); however, considering the small range within F
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important as the length scale of the geometry decreases.60,61 In the following section, we will show how to make use of these effects for the control of reactive free surface flows. Interplay of Reaction Kinetics and Free Surface Flow. The wall-free configuration of free impinging jets is advantageous to performing reactions with a strong tendency for viscosity increases or fouling. In those cases, confined microreactors are prone to clogging, and operation might become unstable. Additionally, precipitation processes are susceptible to heterogeneous nucleation at solid surfaces. In the following, we first quantify the relationship between reaction kinetics and the time scales of the free surface flow using parallel-competitive reactions in view of the different flow regimes presented above. Second, to investigate the effect of a rapid change in fluid properties upon reactive mixing, we generate calcium carbonate2 by placing concentrated solutions of calcium chloride and sodium carbonate in contact with each other; this precipitation is influenced only weakly by the flow rate, with identical particle size distributions in the range of velocities of interest here. Finally, to study a reaction whose outcome strongly depends on the flow, we perform the acidinduced crystallization of benzoic acid. For all of these experiments, we will refer to the dimensionless maps for surface tension and viscoelastic effects established above. Micromixing Time Scales Studied with Parallel-Competitive Reactions. Parallel-competitive reactions can be used to quantify molecular-scale micromixing; systems of an extremely fast reaction in competition with a slower one are called Bourne reactions.10 Here, we use the fourth Bourne reaction system;11,12,62 in it, reaction 1 is the rapid acid−base neutralization of NaOH with HCl:
Figure 10. Interplay of free surface flow and reaction kinetics quantified using parallel-competitive reactions (fast reaction, acid− base neutralization of NaOH; slow reaction, acid-catalyzed hydrolysis of 2,2-dimethoxypropane (DMP) to acetone and methanol). (a) Reaction selectivity X vs We; X corresponds to the conversion of DMP (i.e., lower values of X imply faster mixing). (b) Mixing time, plotted in dimensionless form as the Damköhler number Da = tm/tr vs We. For Da > 1, micromixing is slower than the reaction; therefore, the reaction occurs in an inhomogeneous environment. Da < 1 means that micromixing is faster than the reaction. Da ≈ 1 is the transition point between the two regimes, set by the criterion X = 0.05 for the reaction selectivity. Time constant of the reaction: tr = 4.9 ms. Vertical line: transition from a stable sheet to a fragmented sheet flow pattern.
of We = 100−500, micromixing times are relatively fast and the conversion is consistently below 0.175. An optimum is reached at We ≈ 487 (corresponding to Re = 1793), with a minimum value of X ≈ 0.025. We note that for this fast reaction time similar minimum values for X were found previously in closedjet or vortex mixers, meaning that the wall-free jet reactor is indeed suitable even for very fast reactions. Beyond the optimum We, any further increase in the velocity no longer enhances mixing, but above We ≳ 500, the amount of undesired product formed increases rather drastically. This effect is characteristic for free jets and is not observed in closed reactors.11,12,14 A look at Figure 7 helps us to understand this deterioration in mixing quality at high flow rates: the liquid sheet morphology changes from a stationary sheet to excessive droplet fragmentation, and the transit time of fluid in the sheet, where mixing is most intense, becomes very short. At the highest We, position tracking of surface waves in the high-speed video movies of the expanding sheet suggests that the transit time of fluid through the sheet is as short as 0.6 ms, which for tr = 4.9 ms is almost an order of magnitude faster than the reaction time scale. In other words, the fluid is ejected from the high-intensity mixing zone and fragmented into droplets too rapidly. Mixing proceeds in the atomized droplets, but at a much lower intensity than in the liquid sheet. The relationship between the characteristic time scale of the flow tm and the reaction time scale is captured in the Damköhler number Da. For Da > 1, micromixing is slower than the reaction; therefore, the reaction occurs in an inhomogeneous environment. Da < 1 means that micromixing is faster than the reaction, and Da ≈ 1 is the transition between the two regimes. Whereas tr is known for the set of reactions used here, tm needs to be expressed in terms of a flow model. In turbulent mixing,10 the Kolmogoroff length (i.e., the smallest eddy dimension) depends on the rate of energy dissipation per unit mass10 ε and on the kinematic viscosity ν as l = (ν3/ε)1/4. To form a vertically aligned film, the magnitude of horizontal momentum of both streams must be equal, (Q1/Q2)(ρ1/ρ2)1/2 = 1, assuming identical diameters of the incoming jets. By rescaling the Reynolds number, Mahajan and Kirwan26 suggested that tm ∝ (d/Re)1.5 ∝ 1/u01.5 is a good approximation
k1
OH− + H+ → H 2O
Reaction 2 is the acid-catalyzed hydrolysis of 2,2-dimethoxypropane (DMP) to acetone and methanol: k2
DMP + H+ → (CH3)2 CO + 2CH3OH + H+
Reaction 2 is much slower than reaction 1, and the competition for H+ allows us to link the concentration of DMP in the system to a micromixing time scale: inhomogeneity due to poor mixing results in pools of acid, which in turn allow the slower reaction to proceed. The time scale of the reaction is given by the pseudo-first-order time constant tr = 1/(k2cDMP,0) (i.e., tr can simply be set via the initial concentration of DMP). The fraction of undesired product c X = 1 − DMP c0,DMP is an indicator of the quality of micromixing: small values of X imply fast micromixing (reaction 1 dominates and acid is quickly neutralized over a large volume of the fluid), whereas higher values of X mean that mixing is poor (high local H+ concentrations persist long enough that reaction 2 has enough time to proceed). Figure 10 shows the results of the competitive reaction experiments performed with a characteristic reaction time of tr = 4.9 ms. X is evaluated at different jet velocities and plotted against the Weber number. The sheet morphologies correspond to the reduced surface tension case as compared to water, similar to SDS above the cmc (Figure 7). Starting with poor mixing at the lowest flow rate (X ≈ 0.71), the conversion to undesired product strongly decreases with We. In the range G
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Figure 11. Precipitation of calcium carbonate with colliding jets of solutions of calcium chloride and sodium carbonate. High-speed video images, from left to right: (a, b) small liquid sheets with fragmentation into drops; (c, d) expanded liquid sheets larger than those observed with water or surfactants but with strong instabilities and extensive droplet fragmentation along the entire edge. Incident jet velocities, from left to right: u0 = 3.5, 4.8, 5.9, and 6.5 m/s. Scale bar: 5 mm. (e) Rheology of a calcium carbonate suspension measured immediately after precipitation. The elastic (G′) and viscous (G″) shear moduli as a function of strain are shown, along with the corresponding stress vs strain curve. The dashed line indicates the peak stress in the fluid. (f) SEM micrograph of the precipitated particles.
precipitation of calcium carbonate. However, rheological measurements on the outlet fluid stream performed immediately after mixing give an indication of the flow properties; immediately after precipitation, the slurries resist gravitational sagging but deform easily under an applied shear flow. In Figure 11e, the rheology of a calcium carbonate suspension measured within 1 min after precipitation is summarized. The elastic (G′) and viscous (G″) shear moduli measured in a strain sweep experiment at a frequency of 3.14 rad/s are shown, along with the corresponding stress vs strain curve. The dashed line indicates the peak stress in the fluid; this critical stress is expected to play a key role in the balance between fluid rheology and capillary pressure as the sheet breaks up into droplets. The observed sheet and fragmentation patterns are likely controlled by the balance of the capillary pressure at the rim of the sheet and the critical stress value (yield stress) of the fluid. This picture is confirmed by a comparison with previous studies on the collision of yield stress fluids without reaction described by Baek et al.43 Like the calcium carbonate dispersions formed in our study, sheets formed by colliding dispersions of highly swollen microgel particles appear to be stabilized in the region where the capillary pressure does not exceed the yield stress but readily break up into small droplets as soon as it does. Solid particles dispersed in the incident jets cause an increase in impact energy and promote sheet disruption;43 however, this does not apply to particles formed by a reaction during or after impact, as is the case here. In other terms, micro- or nanoparticles present before impact destabilize the liquid sheet, whereas it appears that particles created in the sheet after impact can stabilize it if they either significantly increase the fluid density or viscoelasticity or reduce the surface tension. Control of a Flow-Dependent Precipitation Reaction: Acid-Induced Crystallization of Benzoic Acid. To study a precipitation reaction whose product characteristics strongly depend on micromixing, we perform experiments on the acidinduced crystallization of sodium benzoate. Equimolar solutions of sodium benzoate and hydrochloric acid are made to impinge at varying flow rates with high reactant concentrations to induce a high initial product supersaturation upon contact. Using reactant concentrations of 0.3 M along with the physicochemical data provided by Stahl et al.,63 we find that the supersaturation ratio is 4.8 and the apparent reaction time scale is tr ≈ 1 ms. Our interest here is to control this
of the characteristic mixing time. For the free-jet reactor, the Damköhler number then takes the form of Da =
⎛ d ⎞1.5 kc tm ∝ ⎜ 0 ⎟ (k 2c DMP) ∝ 2 DMP ⎝ Re ⎠ tr u01.5
(7)
In Figure 10b, we plot this expression of Da against the Weber number; the shape of this curve directly follows from the definitions of We and the model chosen for Da, with a scaling of Da ∝ We−3/4. Because the expression for Da merely provides the power law, only the overall scaling is determined and the actual reference Da = 1 is selected to be at a defined value of homogeneity. The value chosen here is X = 0.05, which is a widely used criterion11,12 for this condition; the scaling law Da = f(We) in Figure 10b is plotted for the case of tr = 4.9 ms. Influence of a Reaction on Liquid Sheet Formation: Precipitation of Calcium Carbonate. Fast precipitation (or, more generally, a fast reaction time) is typically achieved via high supersaturation of the precursor solutions. Consequently, it is rather common to encounter significant changes in the fluid rheology due to the associated high product concentration after the reaction has occurred. To study the effect of a change in fluid properties in a reactive flow, we precipitate calcium carbonate particles2 by impinging solutions of calcium chloride and sodium carbonate; we chose this precipitation process here because we found that the resulting particle size distributions are independent of the flow rate in the range evaluated, with a mean number-average particle diameter of 3.1 μm. Details of the calcium carbonate precipitation and characterization are described elsewhere.2 Sheets formed upon precipitation of calcium carbonate are large compared to those for pure water. Surprisingly, they also exhibit strong droplet breakup at the edge at relatively slow flow rates, with instabilities observed around the perimeter of the sheet. At first sight, these two effectslarge sheet diameters, suggesting higher stability, versus enhanced droplet breakup at the edge, suggesting forced instabilityare contradictory. To clarify this point, we will examine the material properties of the mixed fluid after jet collision. The change in fluid density after precipitation merely corresponds to a 10% increase in the Weber or Reynolds number, which is well within the range that these dimensionless groups have been varied elsewhere in this study without the effects observed here. Neither the dynamic surface tension nor the fluid rheology in the liquid film is known precisely during the H
dx.doi.org/10.1021/la401017z | Langmuir XXXX, XXX, XXX−XXX
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Figure 12. Control of the crystallization of benzoic acid with free reactant jets via the flow rate and the fluid viscoelasticity effect. Stream 1, sodium benzoate solution; stream 2, hydrochloric acid solution. (a) Neat aqueous reactant solutions, no added polymer and no elongational viscoelasticity; (b) identical reactant solutions but with 0.1% w/w PEO (Mw = 600 000 g/mol) to impart elongational viscoelasticity. Top, number-based particle size distribution for u0 = 8.1 m/s; bottom, number-average particle size as a function of jet velocity. Particle sizes shown are Feret diameters DF; lines guide the eye.
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crystallization process not only via the flow rate but also using the fluid viscoelasticity effect shown in Figures 8 and 9. The resulting crystal sizes evaluated immediately after precipitation are shown in Figure 12 for the reaction performed both in the presence and absence of the high-molecular-weight PEO. In both cases, the mean particle sizes strongly depend on the flow rate, and an order-of-magnitude increase in the jet velocity dramatically shifts the particle size distributions to smaller sizes; for example, the number-average mean crystal size is reduced from 21.0 to 6.4 μm for the solutions without polymer. The smaller particle sizes and more narrow size distributions for jets containing small amounts of PEO are due to the confinement of the free jets to a compact, high-intensity impingement zone associated with a increase in the Deborah number of the reactant solutions from De ≪ 1 to De > 1. Here, PEO itself does not act as a crystallization modifier, as confirmed in reference experiments performed in a small stirred vessel, with identical crystal size distributions in the presence and absence of the polymer. Elongational viscoelasticity therefore allows us to control the particle size distribution in free jet precipitation. The figure also shows example particle size distributions measured from microscopy images for a low and a high flow rate (or Weber number). The transition in mixing quality occurs at the same flow rate as in the absence of polymer. Therefore, it is likely that the value of We at which the transition occurs is not controlled by the Rayleigh−Plateau instabilities occurring before collision; however, the homogeneity of mixing might be influenced by the magnitude of these incident instabilities, and shorter nozzle-to-nozzle distances are preferable.
CONCLUSIONS Impinging free liquid jets can used as wall-free continuous microreactors. The collision of two reactant jets allows us to perform rapid precipitation reactions or to perform challenging reactions with very fast kinetics. To control the shape, size, and hydrodynamics of the impingement zone between the two liquid streams, it is crucial to understand the interplay among surface tension, fluid viscoelasticity, and reaction kinetics. We have shown here how the material properties of the reacting fluids critically influence the morphology of the free surface flow patterns generated by the colliding jets. Small amounts of high-molecular-weight polymer (here, poly(ethylene oxide)) strongly limit fluid atomization and induce a compact jet impingement zone. In contrast, small-molecular-weight surfactants at sufficiently high concentrations expand and stabilize the liquid sheet formed upon collision. Such surfactant-stabilized sheets provide a favorable environment for fast reactions, but they remain are prone to mechanical perturbations, leading to instabilities, poorly defined flow kinematics, and ultimately premature liquid fragmentation into droplets. If external perturbations of the flow can be avoided, then the surfactant approach is suitable to controlling reactive flows. In general, however, the fluid viscoelasticity effect is a far more robust approach to stabilizing and controlling the free surface flows. An open question for further investigation is the detailed role of fluid microstructure in the thin liquid sheet: the results found for in-situ-generated calcium carbonate suggest that as the fluid thins, the local rheology of structured fluids such as suspensions and highly concentrated surfactant solutions becomes important, similar to structural stabilization effects observed in soap films and foam films.64 I
dx.doi.org/10.1021/la401017z | Langmuir XXXX, XXX, XXX−XXX
Langmuir
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Figure 13. Maximum ejection angle θ between fragmented droplets ejected from the sheet and the vertical direction. (a) Strong reduction in ejection angles: SDS (c = 82 mM) and PEO (Mw = 600 000 g/mol and c = 0.1 wt %) compared to colliding jets of pure water; the configuration is symmetric, with nonreacting identical fluids on both sides. (b) Effect of calcium carbonate formed in situ by the collision of a calcium chloride solution with a sodium carbonate solution (both at c = 0.3 M). For calcium carbonate formed from neat metal salt solutions, the ejection angle increases as compared to that of pure water. The addition of SDS (c0 = 8.2 mM in both streams) to the reactant solutions reduces the ejection angle. (Inset) Illustration of θ; measured values are from overlay images of 300 video frames. jet Weber numbers for both streams become identical. To observe the flow patterns, we use a high-speed video camera (Phantom Miro, Vision Research). The impingement zone is illuminated in transmission mode with a diffusor plate positioned between the lamp and the jets. Movies are recorded at a frame rate of 4700 frames/s (0.21 ms/frame) with a 60−80 μs exposure time. Images are analyzed using the ImageJ software package (National Institutes of Health, available in the public domain). Mixing Time Characterization with Competitive Reactions. To characterize the mixing time in fast flows, we use the fourth Bourne reaction scheme (parallel-competitive reactions) as introduced by Baldyga and co-workers.12,62 This assay is fast enough to obtain characteristic times for mixing on the molecular scale. Fast reaction 1 is the neutralization of NaOH (second-order rate constant k1 = 1.4 × 108 m3mol−1 s−1 and exothermic heat of reaction Δh1 = 55.8 kJ mol−1 at a temperature of 298 K12); slower reaction 2 is the acid-catalyzed hydrolysis of 2,2-dimethoxypropane (DMP) (weak endothermic heat of reaction Δh2 = 18.0 kJ mol−1). DMP hydrolysis in the presence of excess water can be modeled with a second-order expression for the rate r = dcDMP/dt = k2cDMPcH+. For reaction 2, the hydrolysis rate constant12,62 in units of m3 mol−1 s−1 is k2 = 7.32 × 107e−5556/T10 p and p = 0.05434 + 7.07 × 10−5cs, which is valid for NaCl concentrations of cs = 100−1200 mol m−3. Detailed boundary conditions are discussed elsewhere.12,62 The solvent is Millipore water with 25% w/w ethanol to improve the solubility of DMP. Stream 1 is a solution of DMP and NaOH in the water/ethanol solvent, and stream 2 is the corresponding HCl solution with a molar ratio of 1:1:1.05 for DMP/HCl/NaOH; the purpose of excess NaOH is to stabilize unhydrolyzed DMP after mixing. The conversion of undesired product X is obtained offline using measurements of the methanol concentration and converted to X via the stoichiometry of the reaction. Methanol concentrations in the product stream are measured using an Agilent 6890 gas chromatograph with flame ionization detection with a 30 m HP5MS column; to obtain meaningful results for methanol concentrations on this nonpolar column, we set a low oven temperature (70 °C) along with a small injection volume (
