Effects of unsteadiness of the rates of flow on the dynamics of formation of droplets in microfluidic systems

Abstract

Oscillations of the input rates of flow have a significant impact on the dynamics of formation of droplets in microfluidic systems and on the quality of generated emulsions.

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One of the fundamentals of interest in microfluidic systems is the one-to-one, linear and smooth correspondence between the input parameters and the flow within the chip. This feature is rooted in the Stokes equations of flow at low Reynolds numbers and enables engineering of spatial and temporal distribution of solutes in microchannels¹ and a variety of analytical applications.² In spite of the complications associated with interfacial forces, multiphase microflows surrender to similarly extensive control. The ability to form monodisperse droplets^3,4 in microchannels inspired the use of microfluidic droplet generators both in preparatory and analytical applications. It is possible to form multiple droplets,⁵ anisotropic particles of a variety of materials⁶ and capsules⁷ with potential for use in food, cosmetics and pharmaceutical industries. Microdroplets can serve as reaction beakers yielding superior control over conditions of reactions, ultra small reaction volumes, fast mixing, lack of dispersion of time of residence and phenomenal statistics.^8–10 It is possible to encapsulate single cells within droplets and to conduct ultra-fast directed evolution.¹¹ Precise control over the volumes of droplets is critical in sustaining superior characteristics of e.g.drug-delivery vehicles¹² or reliable statistics in high throughput studies.¹¹

Here we characterize the role of stability of input rates of flow on the dynamics of formation of droplets in a microfluidic T-junction and on the quality of the generated emulsion. We show that unsteadiness of inflow (e.g. from commonly used syringe pumps¹³) can significantly compromise monodispersity of droplets and blur the transitions between different regimes of operation of droplet generators making it difficult to precisely identify their dynamics.

The T-junction is one of the simplest devices which allows for generation of microdroplets.⁴ At the T-junction the stream of water breaks up into droplets which – together with the continuous phase – flow towards the outlet. The volume (V) of the droplets is a function of the rates of flow of the continuous (oil) phase (Q_c) and of the aqueous droplet phase (Q_d).^14,15 At low values of the capillary number (Ca = μv/γ, where μ is the viscosity of the continuous phase, v the speed of flow and γ the interfacial tension) formation of droplets is controlled by the rate of flow (or ‘squeezing’) model:¹⁵V ∝ A + B (Q_d/Q_c), where A and B are constants O(1). At higher Ca shearing effects modify the scaling relation and there is a slight dependence^14,16 of the volume of droplets V on the value of the capillary number. The detailed characterization of the scaling relations of the drop size, and understanding of the mechanism of break-up – especially as a function of varied viscosities of the fluids – is still an area of active research.¹⁶

In the experiments we compared the dynamics of formation of droplets in the T-junction and the statistics of volumes of the generated droplets in systems operated with syringe pumps and with a home-built source of a constant rate of flow. We used two sets of two Harvard Apparatus PHD 2000 pumps, and a set of two New Era Systems NE-1000 pumps. We tested 4 different types of syringes (see table in ESI†) of different cross-sections and materials: polypropylene/polyethylene (1 ml – B.Braun, Germany, 5 ml and 10 ml – Polfa, Poland), and glass (50 ml – Poulten & Graf, Germany). The syringes were connected with the chip via PE-60 tubing (Becton-Dickinson, USA). Constant rates of flow were supplied from pressurized reservoirs of liquids interfaced with the chip via steel capillaries of high hydraulic resistance (I.D. 0.21 mm, Mifam, Poland) and of length varying from 1 m to 8 m. For connections between the steel capillaries, and between the capillaries and the needles (e.g. 21 gauge needle of outer diameter of 0.82 mm), we used elastic Tygon tubing (inner diameter 0.25 mm, outer diameter 2 mm, Ismatec, Switzerland). In order to stabilize the connections we always put the end of the capillary into the needle, so that the walls of the elastic tube could not block the lumen of the capillary. Additionally we placed capillaries in a temperature stabilising bath to avoid temperature and viscosity fluctuations (see the ESI† for details). The T-junction chip was milled (CNC mill Ergwind MSG4025, Poland) into a slab of polycarbonate (Makroclear, Bayer, Germany) and bonded.¹⁷ The height of the channel was equal to its width W = 200 μm, and the length of the channel was 40 mm. Distilled water served as the droplet phase and a 2% (w/w) solution of SPAN 80 (Sigma, Poland) in hexadecane (Sigma, Poland) as the continuous phase.

Fig. 1 compares volumes of droplets generated with the two different schemes of supplying liquids onto the chip. It is evident that the system fed from syringe pumps generates droplets of size that slowly varies in time. In the example shown in Fig. 1a, the volumes of the droplets oscillate with a period of approximately 160 s and with an amplitude of approximately 10% of the mean. In a pronounced contrast, in the system fed via the capillaries (black dots, Fig. 1a) we do not observe any systematic oscillations and the amplitude of the fluctuations of the volume of the droplets is much smaller (less than 1% of the mean).

Fig. 1 (a) Dimensionless volume of the droplets (V_d = V W^-3) as a function of time for ‘constant’ rates of flow of the two immiscible phases, Q_d = Q_c = 4 mL h⁻¹. Dots show the measurements taken in a system fed via resistive capillaries and open circles in the system fed from syringe pumps. (b) The volumetric flow rate of oil, evaluated from measurements of the linear velocity of tracer drops. Circles and dots as in (a). Syringe pump: PHD2000.

We checked (see ESI† for details) that syringe pumps generate oscillations of flow of a period T = aπD²/4Q, where D is the diameter of the piston and a is the pitch of the thread on the lead screw of the pump. We verified this relation via a set of measurements of the period T for different diameters of syringes D and for different rates of flow Q (see ESI†).

Importantly, oscillations of Q significantly influence the fidelity of measurements of volumes of the droplets. An accurate assessment of the mean volume demands measurements over a time span longer than T (Fig. 2) which – for large syringes – can be impractically long: e.g. for PHD2000 and 10 ml syringes, (D = 15.6 mm) and Q = 0.4 mL h⁻¹, T ∼ 30 min. Such long periods of oscillation can be very deceptive: averaging over short time scales provides low values of standard deviation and wrong values of the mean. Averaging over intervals much longer than T gives large standard deviations, but a more correct assessment of the mean (Fig. 2). Syringes of small diameter decrease the period of oscillations but their capacity may be insufficient in applications demanding long time of operation.

Fig. 2 Evolution of the mean (top) and of the standard deviation (bottom) of dimensionless volume of droplets. Solid line – system fed via resistive capillaries, dashed line – from a syringe pump. Q_d = Q_c = 4 mL h⁻¹.

The oscillations of Q affect not only the volume of the droplets but also the transitions between different regimes^14,16 of droplet formation. When rates of flows are relatively large (e.g. Q_c = Q_d = 8 mL h⁻¹ in our system) the tip of the water phase penetrates into the main channel (Fig. 3a) signifying the transition from dripping to jetting.¹⁴ Precise control of the rate of flow is crucial for characterization of this transition. Fig. 3a shows the length of the stream of the droplet phase extending from the T-junction: in the case of the capillary feeding, oscillations are caused only by the generation of droplets. Furthermore, the cross-over in the scaling of the volume of the droplets can be precisely shown in the system fed via capillaries, while it is blurred by fluctuations of the flow rate in the system supplied from the source of oscillating rate of flow (Fig. 3b).

Fig. 3 Transition to jetting for Q_c = 8 mL h⁻¹. (a) The length of the stream of water in the systems operated with resistive capillaries and syringe pumps Q_d = 8 mL h⁻¹. (b) Dimensionless length of the tip of water stream as a function of its flow rate. In the inset: standard deviation of tip length (σ) normalised by width of the channel. Lengths of capillaries: L_d = 8 m, L_c = 2 m. Syringe pump: PHD2000.

The possible artifacts caused by the use of oscillating source of flow are exemplified in the subtle change of the scaling of V with Q_c upon the transition from the squeezing to the dripping regime.^14,16 We fixed Q_d at 0.9 mL h⁻¹ and varied Q_c between 0.3 and 4 mL h⁻¹. For capillary feeding we used two sets of capillaries to cover this range: L_d = L_c = 8 m for Q_c < 2 mL h⁻¹ and L_d = 8 m, L_c = 2 m for Q_c > 2 mL h⁻¹. Data in Fig. 4 show that both feeding systems allow for observation of the transition, with smaller scatter for the capillary system.

Fig. 4 Transition from squeezing to dripping^5–8 upon the increase of Q_c for Q_d = 0.9 mL h⁻¹. Figure shows the term (VW⁻³ − A)Q_c/Q_d as a function of Q_c. According to the squeezing model¹⁵ (V_SQ = W³(A + BQ_d/Q_c)) this term should be equal to B which does not depend on Q_c. The inset shows the fit of the model with A = 0.67 and B = 1.05. Legend given in the graph; capillary set one: L_d = L_c = 8 m; set two: L_d = 8 m, L_c = 2 m; syringe pump – PHD 2000. The measurements on the system supplied with liquids from syringe pumps were repeated four times.

The oscillations of the rate of inflow of liquids into the chip significantly influence the distribution of the volumes of droplets and, for long periods of oscillations, can impair quantitative measurements. Proper averaging requires sampling over times much longer that the period of oscillations, or being its integer multiple. Sampling over shorter times may produce deceptively small scatter and a wrong estimate of the average size of droplet.

The capillary feeding system provides a simple and robust alternative to syringe pumps as it yields much more stable feeding. The use of feeding liquids from pressurized containers and other systems for pulseless delivery of liquids should be useful in detailed studies of the mechanisms of formation of droplets in various microfluidic systems, and in practical applications requiring production of tightly monodisperse emulsions over long spans of time.

Acknowledgements

Project operated within the Foundation for Polish Science Team Programme co-financed by the EU European Regional Development Fund and within the Human Frontiers Science Program.

Notes and references

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Footnote

† Electronic supplementary information (ESI) available: Measurements of oscillations of rate of flow with the use of an analytical balance, velocity of tracer droplets, calibration of capillaries, and details of the capillary feeding system. See DOI: 10.1039/c0lc00088d

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