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Fluctuations in Rayleigh breakup induced by particulates
Advances in Colloid and Interface Science 161 (2010) 15–21
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Advances in Colloid and Interface Science j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c i s
Fluctuations in Rayleigh breakup induced by particulates A. Clarke ⁎, S. Rieubland Kodak Limited (European Research), 332 Science Park, Cambridge, CB4 0WN, UK
a r t i c l e
i n f o
a b s t r a c t A jet of liquid is intrinsically unstable to radial perturbations and will spontaneously break to form a series of droplets. This well known instability, the Rayleigh–Plateau instability, is controlled and used commercially in continuous inkjet printing. In this application it is important that fluctuations in drop velocity are minimised. However, the addition of particulates to the liquid is observed to strongly increase these fluctuations. The particulates are usually in the form of pigment particles of size O(100 nm) and at a concentration where they may hydrodynamically interact, particularly in the strong shear field within the nozzle (O(107 s− 1)). The boundary layer thickness within the nozzle is O(1 μm) and therefore the particulate size is a significant fraction. We therefore expect that the particles are capable of perturbing the boundary layer and hence the jet. Measurement of jet breakup fluctuation leads to a description of particulates interacting within and with the shear field associated with the boundary layer at the nozzle wall. © 2009 Elsevier B.V. All rights reserved.
Available online 6 October 2009 JEL classification: 82.70.-y 47.15.-x 47.57.-s 47.61.-k Keywords: Jet Rayleigh instability Particles in shear Fluctuations
Contents 1. Introduction . 2. Experimental 3. Results . . . 4. Discussion . . 5. Conclusions . Acknowledgements References . . . .
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1. Introduction In recent years jets and the Rayleigh–Plateau instability [1–3] have been widely studied both theoretically and experimentally [4,5], at least in part because of their importance within the context of inkjet printing [6]. For drop-on-demand printing, there is usually a thread of liquid that follows drop ejection and which subsequently disintegrates to form unwanted satellites. For continuous inkjet [7], the continuous formation of droplets from a jet in a controlled fashion is fundamental to the robust operation of the process. Jet breakup has been studied at varying levels of complexity and sophistication [4,5]. For our purposes the growth of a perturbation to the jet is sufficiently represented by a linear approximation of the complete equation set such that the breakoff length, which is the distance from the nozzle at which the jet disintegrates to form a
⁎ Corresponding author. E-mail address: [email protected] (A. Clarke). 0001-8686/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cis.2009.09.006
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15 16 17 18 21 21 21
stream of droplets, may be calculated quite easily [8]. The breakoff length is determined by liquid properties, the dimension and velocity of the jet and a parameter that describes the size of the initial perturbation. In the absence of active stimulation, a jet will disintegrate into droplets randomly but with a well defined mean periodicity and size distribution. Many continuous inkjet systems perturb the jet via a pressure pulsation generated behind the nozzle. These systems are generally limited to operation at a particular frequency since the pressure pulsation is generated by acoustic resonance within a cavity. The frequency chosen is close to the Rayleigh frequency for the system so that the greatest effect can be achieved. Such systems are also limited in the material viscosity range that can be routinely accommodated. An alternative mechanism has recently been described [9–11] wherein a thermal perturbation to the surface of the jet drives a Marangoni flow that initiates breakup. Such devices can be manufactured via MEMS fabrication routes and are therefore precise and robust. For inkjet inks it is often preferable to use a colorant based on a pigment. Such pigments are typically insoluble organic particles of
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diameter order 100 nm that are used at moderate volume fractions that can neither be considered dilute nor concentrated. This type of dispersion occurs widely but is intrinsically difficult to model since both particles and hydrodynamics need to be considered in a coupled fashion. Within the nozzle of a continuous inkjet device there exists a strong shear field which may be approximated as the jet velocity divided by the boundary layer thickness. For typical inkjet nozzles, the nozzle length is much shorter than the diameter and therefore the boundary layer remains a small fraction of the nozzle diameter. It is well understood, even from elementary linear models, that flow perturbations will lead to fluctuations in the breakup of the jet and therefore to droplet velocity fluctuations. Indeed a statistical analysis leads to a predicted drop size distribution for a simple jet that can be fitted by a gamma distribution [5]. Should particulates within the shear flow in the boundary layer affect the overall flow field, then they too will be expected to lead to fluctuations in jet breakup. The macroscopic properties of suspensions of particles are determined by the spatial organization of the particles, that is their microstructure. In the absence of flow a wide range of microstructures are realized dependent on the balance between particle–particle interactions and Brownian motion. During flow, the microstructure will rearrange to accommodate the hydrodynamic forces and thereby modify the flow field. Understanding the coupling between flow and microstructure during flow has been the subject of many studies (for a review see Vermant and Solomon [12]). Various numerical methods have been developed to model such colloidal systems: Stokesian dynamics [13], fluid particle dynamics [14], lattice Boltzman [15], and several others, together with modern variants. In the majority of cases these methods have been used to model the resulting rheological behaviour of significantly more concentrated systems than the pigmented ink system described above. Of these studies, Catherall et al [16] in particular have calculated the stress on a single particle in a high-shear flow comparable with the nozzle flow field, and observe significant fluctuations. The work presented here suggests a subtle flow perturbation induced by particulates that leads to macroscopic velocity fluctuations in jet breakup. 2. Experimental The experiments performed in the present study employed a microelectromechanical system (MEMS) fabricated nozzle including a resistive heater. This device was mounted in a standard pin-grid array prototype chip holder the back of which was drilled to allow the entrance of liquid. A programmable pulse generation device (NIOS) was used to provide approximately a 200 kHz signal (the Rayleigh frequency for the jet was typically chosen) comprising a voltage pulse of up to 5 V and duration of 1 μs. Liquid was forced from the nozzle with a static pressure of up to 65 psi (approx. 448kPa) which generated a jet with velocity up to approximately 25 m/s. The nozzle diameter used was 17.6 μm. The resulting jet had a Weber number, We (= ρU2D / σ) of O(100). The fluid handling arrangement is shown in Fig. 1a. Jet breakup was observed using a long working distance optical microscope; a Navitar 6000× zoom lens with Mitotoyu 10× APO Plan objective (Fig. 1b). The images were captured with either a Watec LCL-902c Black and white video camera attached to a LabView 1405 framegrabber card, or a Prosilica EC1380C firewire digital camera. In both cases the images were analysed with a LabView code written for the purpose. Lighting was provided by an HSPS high speed spark strobe system, which generated 30 ps light flashes. These flashes were synchronised with drop generation via the NIOS pulse generator and an HP 8111A pulse generator used to decimate the base signal and an SRS DG535 delay generator to delay the pulse and enable observation of the drop as a function of phase (Fig. 1c). The position of the nozzle relative to the camera was adjusted using a micrometer motor drive with indexer (Oriel model 18011). Thus the distance from nozzle to
jet breakup or to camera observation point could be accurately measured (+/−5 μm). The liquid jetted comprised various concentrations of glycerol in water. To these solutions polystyrene particles were added. The polystyrene particles were made in our labs and are charge stabilised. They therefore require no added surface active component to disperse them. Each sample had a relatively narrow size distribution as determined by particle size analysis. These particles were mixed, at a variety of concentrations, with a range of aqueous glycerol solutions so as to investigate background viscosity, particle size and volume fraction independently. Although jet breakup was initiated by a well defined thermal pulse, i.e. a known power for a known time, it was observed that the coupling of this power to an initial perturbation varied depending on the solution being used together and the jet velocity. Therefore an initial calibration experiment was performed to enable perturbations of the same effective size to be used for the various liquids and jet parameters. The breakoff length, i.e. the distance from the nozzle at which the jet first ruptures, at the Rayleigh frequency, was measured as a function of pulse power. The measured breakoff length is accurately approximated via the linear version of the jet equations [8] over the accessible range of powers, 1=2
L = CDWe
ð1 + 3OhÞ
ð1Þ
where L is the breakoff length, D the jet diameter, We the jet Weber number and Oh the jet Ohnsorge number (=μ /√(ρσD)) and C a constant related to the size of the initial perturbation, i.e. the pulse power. It should be noted that at the Rayleigh frequency, the surface profile closely matches a growing exponential until non-linear breakoff processes intervene, i.e. where the radial perturbation is greater than approximately 1/3 of the initial radius. This well controlled behaviour arises since the perturbation to the jet is via a spatially constrained thermal Marangoni driving force [9] rather than an acoustic resonance as in conventional continuous inkjet. Hence, the non-monotonic breakoff length as a function of modulation power observed by others [17] is not observed here. Irrespective of the precise nature of the initial perturbation, the Rayleigh instability ensures that the various wavector modes grow at well defined rates. Thus we choose to consider the initial perturbation as if it were a perturbation to the radius of the jet then C = ln(R/ξ), where R is the radius of the jet and ξ is the perturbation. Thus we can relate the measured breakoff length to an effective perturbation size ξ. In practice measurements of both breakoff length as a function of power and breakoff length as a function of driving frequency were made and fitted using independently measured parameters of viscosity, density, surface tension and jet radius. The effective perturbation obtained in this way is a linear function of pulse power used to perturb the jet over the accessible range of powers. For the data presented below, measurements at constant effective imposed perturbation are compared. Of interest here is the fluctuation of drop velocity. This was measured by measuring the position of many individual drops and obtaining the standard deviation of the resulting Gaussian distribution (Fig. 2). Since the lighting system was synchronised to the pulse generating the drop, the fractional velocity fluctuation is equal to the fractional positional variation, δu σ = u d
ð2Þ
where u is the droplet velocity, d is the distance from the formation of the drop to the measurement location and σ is the measured standard deviation of droplet position. Note that in practice three adjacent drops were simultaneously measured. Therefore, knowing the drop formation frequency as set, the velocity of the drops was also obtained. Further by plotting distance between the first two drops (A–B) against the distance
A. Clarke, S. Rieubland / Advances in Colloid and Interface Science 161 (2010) 15–21
17
Fig. 1. a) Schematic flow diagram of the pressure and liquid handling system for our experimental jet arrangement. b) Schematic diagram of the optical arrangement. c) Schematic diagram of the signal and timing systems for observing jet droplets.
between the second two drops (B–C) any correlation in successive drop breakoff could be assessed [18]. At the Rayleigh frequency, no such correlation was observed, suggesting that fluctuations leading to droplet velocity fluctuation have a correlation length at most less than the Rayleigh wavelength. It was verified that the drops did indeed have a velocity variation by measuring σ as a function of d. A linear result was obtained typically
with an intercept approximately 1 wavelength prior to breakoff. Additionally, it was verified that our measurements are not affected by airflows; the velocity of the drops was measured as a function of distance from breakup and found to be constant up to a distance greater that that used for our measurements. 3. Results We are interested in the effect solid particles have on the breakup of a jet. However, we first examine the behaviour in the absence of particles for a range of liquid viscosities. Fig. 3 shows the velocity fluctuation δu /u, as a function of the effective perturbation size, ξ. (Recall the effective perturbation size is determined by fitting the observed jet breakoff length as a function of nozzle heater power to obtain a calibration of effective perturbation size as a function of heater power.) It is clear from Fig. 3 that the drop velocity fluctuation for each liquid behaves in a 1 /ξ manner. That this should be the case can be obtained via the linear approximation as follows. If we have a jet then the breakoff length is determined by the velocity and the time it takes for an initial perturbation, ξi, to grow to the size of the jet, R,
Fig. 2. Measurement of drop position for several hundred drops allows the measurement of position standard deviation, σ, and wavelength to be obtained. From the wavelength and knowledge of the frequency, the drop velocity is obtained.
R = ξi expðα⋅tB Þ
ð3Þ
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Fig. 3. (a) Drop velocity fluctuation as a function of effective initial perturbation size. (b) Data extracted from the same curves as a plot of drop velocity fluctuation against liquid viscosity for four perturbation sizes.
with tB the time to breakoff and α, the frequency dependent perturbation growth rate. If we allow a small fluctuation in ξi, δξi, then we observe a small change in the breakoff time, δtB, δtB = −
1 δξ ln 1 + i : α ξi
ð4Þ
A small fluctuation in breakoff time necessarily leads to a small fluctuation in breakoff length and therefore a small fluctuation in drop mass. 2
δm = ρ⋅πR ⋅Ujet δtB :
ð5Þ
Hence, via conservation of momentum, the velocity fluctuation is given by, Ujet δu δξ j drop = ln 1 + i u λα ξi
ð6Þ
with λ the jet breakup wavelength. We therefore expect that, since δξi is small and independent of ξi, the drop velocity fluctuation should behave as 1 /ξi. This is indeed observed (Fig. 3). Eq. (6) also implies a frequency dependence through α and this was also verified, though not shown here. Note that, experimentally, the measurement of δu /u is significantly more sensitive than a corresponding measurement of drop size variation. If we take the data of Fig. 3a and replot as a function of viscosity Fig. 3b, then we find that the fluctuations are a decreasing function of viscosity. This should be expected since sources of noise, e.g. capillary waves on the jet, will be damped to a greater degree by increased viscosity. Note that inverting Eq. (6) allows an estimate of δξi. If this is
done one finds that the additional surface area implies a wave energy consistent with thermal energy kT. We now turn our attention to particulate containing systems [19]. For this purpose we prepared 3 nominally mono-disperse polystyrene dispersions of particle diameter 80 nm, 125 nm and 200 nm. With each dispersion we prepared a solution containing 3% by weight particles and 30% by weight glycerol, and a solution containing 3% by weight particles and 50% by weight glycerol. At 3% by weight, the distance between particles is approximately 4.5 particle radii. It should be noted that at this distance, and particularly in strong shear fields, the particles cannot be assumed to behave independently. The data obtained for each solution is shown in Fig. 4. Again we see a general inverse dependence with initial perturbation size. However, we also see that the resulting level of drop velocity fluctuation increases as background viscosity increases i.e. the opposite behaviour to that observed with simple liquids. This effect is seen more clearly in Fig. 5a where the same data are replotted as a function of viscosity. In Fig. 5b the data are plotted as a function of particle size, and it is seen that the level of fluctuation increases with increasing particle size for the same volume fraction. In Fig. 6 the velocity fluctuation is plotted as a function of particle concentration for one particle size and three different effective initial perturbation sizes. The behaviour observed is approximately a linear function of particle concentration or volume fraction. 4. Discussion Our nozzle is approximately 17.6 μm in diameter and approximately 5 μm in length. Thus, the boundary layer is only partially developed and
Fig. 4. Drop velocity fluctuation with particulates for two different background viscosities. (a) 30% aqueous glycerol approx. 3.2 mPa s, (b) 50% aqueous glycerol approx. 7 mPa s. Note that, contrary to Fig. 3, here as the viscosity increases so does the size of the fluctuations.
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Fig. 5. Data from Fig. 4 plotted, (a) as a function of viscosity (note the opposite trend to that of pure water/glycerol mixtures, i.e. without particles) and (b) as a function of particle size. In all cases the effective perturbation size was 5 nm.
so the shear field exists over only a thin annular region on the wall of the nozzle. The thickness of the boundary layer at the exit of the nozzle, assuming we may use the usual estimate for boundary layer formation on entry to a tube, may be estimated as [2], δ=
rffiffiffiffiffiffiffiffiffi μx 2ρU
ð7Þ
where δ is the boundary layer thickness, μ the viscosity, ρ the liquid density, U the jet velocity and x the length of the nozzle (i.e. the distance from the entry to the short tube that forms the nozzle). For the parameters pertinent here this dimension is approximately 1 μm and therefore the shear rate, estimated as jet velocity divided by boundary layer thickness, is of order 107 s− 1. Therefore within the shear field of the boundary layer the ratio of shear field thickness to particle size (=δ / (2a)) is between about 5 and 12. If the particles interact with each other within this shear field hydrodynamically or otherwise, then a flow perturbation within the nozzle will ensue. Such a flow perturbation will necessarily be amplified by the intrinsic instability of the jet and so will be expected to generate fluctuations in the jet breakup process. The particle Peclet number, Pe, characterizes the relative importance of shear and Brownian motion. For Peclet numbers much less than unity, Brownian motion is able to maintain the dispersed distribution of particles. For Pe ≫ 1 the shear flow modifies the particulate distribution, i.e. the microstructure, and Brownian motion is unable to restore the distribution on the timescale set by the shear flow. The Peclet number is commonly defined as Pe = τD / τshear, with τD a characteristic diffusion time and τshear a characteristic time corresponding to a given shear rate. A diffusion time is simply obtained from the Stokes–Einstein relation as the time to diffuse a
particle radius, τD = a2 / DSE = 6πμa3 / kT, where DSE is the Stokes– Einstein diffusion constant. The shear rate has dimensions of inverse time so a characteristic time is simply τshear = γ̇− 1. Hence, the Peclet number relevant to a particle is [20], Pe =
˙ 3 6πμ γa kT
where μ is the viscosity of the dispersion, γ̇ the shear rate, a the particle radius and kT the thermal energy. For our weakly interfering hard particle system it is appropriate to approximate the viscosity using classical hydrodynamics [21], μ = μS ð1 + 2:5ϕ + …Þ
ð9Þ
where μS is the solvent viscosity and ϕ the volume fraction. The shear rate is approximated by γ̇ = U / δ and therefore the Pe number for small ϕ and dropping the constant, behaves as, Pe =
pffiffiffiffiffi μS ð1 + 1:25ϕÞa3 kT
2ρU 3 x
!1 = 2 :
ð10Þ
Whereas it is the Peclet number that characterizes the formation of microstructure within such colloidal systems, we expect that fluctuations in velocity will be related to the degree that the particles are hydrodynamically coupled, that is the degree to which they are not simply convected with the flow. We are therefore uninterested in the zero volume fraction limit of Eq. (10) and thus we choose to form a Peclet number from the incremental viscosity associated with particle–particle hydrodynamic interactions, PeN, PeN = Pe−Peϕ→0
Fig. 6. Drop velocity fluctuation as a function of particle concentration. In all cases the particles were 125 nm polystyrene in 50% wt/wt aqueous glycerol. The three curves are for three different initial perturbation sizes.
ð8Þ
pffiffiffiffiffi 3 1:25 μS ϕa = kT
2ρU 3 x
!1 = 2 :
ð11Þ
Fig. 7 shows the correlation achieved using Eq. (11) to plot the data presented in Figs. 4 and 6, i.e. δu / u ∝ PeN. Note however that within the dataset temperature is not varied and therefore the correlation is between the velocity fluctuations and the estimated shear forces in the nozzle boundary layer due to particulates. That velocity fluctuations should scale with PeN is already suggested by Squires and Brady [22] where they considered the effects of particular micro-structural arrangements of particles around a probe particle leading to a fluctuating force on the probe particle as it was translated. An ensemble averaged microstructure (weighted by the N-particle distribution function) was then used to obtain the size of velocity fluctuations of the probe particle which was found to scale as Pe. Combining Eqs. (11) and (6) and allowing δξi to vary as √μ, we arrive
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with A and B constants to fit the data. In Fig. 8 this function is plotted with appropriate choices for A and B, and it can be seen that the trends in the data are reproduced. Within Eq. (12), the particle parameters appear as ϕa3, which is straightforwardly calculated for a mono-disperse system. For polydisperse systems however, this needs to be generalized. Since we find that, at the low concentrations investigated, the background noise and the particulate noise components are well represented by simple addition of the two noise components, we further suggest that to model poly-disperse systems at these concentrations we should simply add the effects of the individual populations hence, Ujet δU Aδξi + B∑PeN j drop = ln 1 + U λα ξi Fig. 7. Data for drop velocity fluctuation from Figs. 4 and 6 at an effective initial perturbation of 5 nm plotted as a function of Peclet number, PeN, as defined in Eq. (12).
at a general semi-empirical function that we can use to describe the data, Ujet δu A⋅ðμÞ−1 = 2 + B⋅PeN j ln 1 + = u drop λα ξi
! ð12Þ
ð13Þ
where the sum is over each particle population. Of course this sum would, in the limit, be an integral over the particle size distribution. To probe this, several mixtures of the previously prepared monodisperse particles were prepared with a total fraction of either approximately 1 wt.% or 3 wt.%. It can be seen from Fig. 9 that the assumption of additivity works well, and we can use the effective particle size defined in the usual way as follows,
aeff
0 11 = 3 3 ∑a ϕ B j j jC C =B @ ϕ A total
ð14Þ
where ϕtotal is ∑ ϕj . The function is the conventional volume mean j radius. Within the flow discussed here, the nozzle boundary layer contains rather few particles. For typical materials and nozzles, the boundary layer might contain O(1000) particles and assuming Poisson statistics therefore ~ 3% fluctuation in number density. Fluctuations in both number and spatial distribution within a given volume are able to generate flow fluctuations in the liquid flux from the nozzle. These flow fluctuations might equivalently be considered as arising from viscosity fluctuations since at these concentrations additional viscosity arises from hydrodynamic interactions between particles. The viscosity affects jet velocity through the shear stress in the nozzle which depends both directly on the viscosity and indirectly on the boundary layer thickness, the fluctuations of which both scale as √ϕ. Hence the fluctuations in jet velocity are expected to scale as ϕ. Such fluctuations in flow naturally lead to macroscopic variations in jet breakup.
Fig. 8. Plots of Eq. (12) as a function of various parameters. The general trends and scales are well reproduced. (a) and (c) are for a 3% concentration of particles. (b) is for a viscosity of 2.5 mPa s.
Fig. 9. Measurement of drop velocity fluctuation as a function of particle mixtures. Circles are for various individual concentrations of 108 nm, 125 nm, 220 nm particles. Squares for various mixtures and concentrations of 108 nm and 220 nm particles (summing to 1% or 3%). The last point at 4 × 10− 21 is for a solution of 3% of 220 nm particles.
A. Clarke, S. Rieubland / Advances in Colloid and Interface Science 161 (2010) 15–21
5. Conclusions We have presented a series of measurements of droplet velocity fluctuations measured as a function of liquid and jet parameters. Via suitable analysis we have separated the observed fluctuations into effects arising from (i) the liquid and jet influence on the size of the initial perturbation and, (ii) effects arising from size of the noise in the initial perturbation. In this way we have obtained a semi-empirical relationship quantifying the size of the droplet velocity fluctuation as a function of particulate parameters. Although the model of the data is semi-empirical, to the extent that the observed trends are captured by Eq. (13), we can venture comments on certain features of the behaviour: (i) There are at least two sources of fluctuation: (a) an intrinsic noise possibly associated with thermal capillary noise on the jet, but also potentially arising from flow in the chamber behind the nozzle, e.g. unsteady regions, and (b) noise associated with flow fluctuations generated by particulates. (ii) Correlations of the data suggest that droplet velocity fluctuations arise from hydrodynamic particle–particle interactions in the high-shear nozzle boundary layer. It should be noted that the level of shear, the scale of the flow and the transient nature of the flow conspire to make this situation inaccessible to more usual rheological measurements. The overall picture that emerges is of a low to moderate concentration of nanoparticulates interacting with a transient highshear field which leads to a macroscopic observable effect. The Rayleigh–Plateau instability acts as an amplifier of the initial fluctuations and so subtle variations in flow generated through interactions in the nozzle region cause observable fluctuations in jet breakup. This of course suggests that the observation of jet breakup offers a method to investigate flow behaviour in the extreme flow regime found within such a nozzle. Acknowledgements The authors would like to thank the support and encouragement of many colleagues in the Eastman Kodak research labs in Rochester, N.Y.
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for the support of this work, including the provision of the MEMS nozzles.
References [1] Rayleigh JWS. Proc Lond Math Soc 1879;x:4–13. [2] Faber TE. Fluid dynamics for physicists. Cambridge: CUP; 1995. [3] de Gennes PG, Brochard-Wyart F, Quéré D. Capillarity and wetting phenomena. Springer; 2004. p. 118. [4] Eggers J. Rev Mod Phys 1997;69:865. [5] Eggers J, Villermaux E. Rep Prog Phys 2008;71:036601. [6] EPSRC, Next Generation Inkjet technology project; http://www.ifm.eng.cam.ac.uk/ pp/inkjet/. [7] Schneider JM. In: Hanson E, editor. Recent progress in ink jet technologies II. IS&T; 1999. p. 246. [8] Stanley Middleman. Modeling axisymmetric flows dynamics of films, jets and drops. Academic Press Inc; 1995. p. 102. Chap. 4.3. [9] Furlani EP, Delametter CN, Chwalek JM, Trauernicht D. 4th International conference on modelling and simulation of microsystems. Cambridge MA: Applied Computational Research Society; 2001. p. 186. [10] Chwalek JM, Trauernicht DP, Delametter CN, Jeanmaire DL. NIP17: International Conference on Digital Printing Technologies, vol. 170-89208-234-8; 2001. p. 291–4. [11] Hawkins G. 15th Annual European Ink Jet Conference, Lisbon; 2007. [12] Vermant J, Solomon MJ. J Phys Condens Matter 2005;17:R187. [13] Brady JF, Bossis G. Annu Rev Fluid Mech 1988;20:111. [14] Tanaka Hajime, Araki Takeaki. Phys Rev Lett 2000;85:1338. [15] Shiyi Chen, Doolen GaryD. Annu Rev Fluid Mech 1998;30:329. [16] Catherall AA, Melrose JR, Ball RC. J Rheol 2000;44:1. [17] Luxford G. ISTs NIP 15: International Conference on Digital Printing Technologies, vol. 150-89208-222-4; 1999. p. 26–30. [18] Martien P, Pope SC, Scott PL, Shaw RS. Phys Lett A 1985;110:399. [19] We thank Dr S. Desrousseaux, Kodak Limited, for synthesizing the monodisperse polystyrene particles. [20] See e.g.R.A.L.Jones, “Soft Condensed Matter”, OUP, 2002. [21] Russel WB, Saville DA, Schowalter WR. Colloidal dispersions. Cambridge: Cambridge University Press; 1989. [22] Squires TM, Brady JF. Phys Fluids 2005;17:073101.
Contents lists available at ScienceDirect
Advances in Colloid and Interface Science j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c i s
Fluctuations in Rayleigh breakup induced by particulates A. Clarke ⁎, S. Rieubland Kodak Limited (European Research), 332 Science Park, Cambridge, CB4 0WN, UK
a r t i c l e
i n f o
a b s t r a c t A jet of liquid is intrinsically unstable to radial perturbations and will spontaneously break to form a series of droplets. This well known instability, the Rayleigh–Plateau instability, is controlled and used commercially in continuous inkjet printing. In this application it is important that fluctuations in drop velocity are minimised. However, the addition of particulates to the liquid is observed to strongly increase these fluctuations. The particulates are usually in the form of pigment particles of size O(100 nm) and at a concentration where they may hydrodynamically interact, particularly in the strong shear field within the nozzle (O(107 s− 1)). The boundary layer thickness within the nozzle is O(1 μm) and therefore the particulate size is a significant fraction. We therefore expect that the particles are capable of perturbing the boundary layer and hence the jet. Measurement of jet breakup fluctuation leads to a description of particulates interacting within and with the shear field associated with the boundary layer at the nozzle wall. © 2009 Elsevier B.V. All rights reserved.
Available online 6 October 2009 JEL classification: 82.70.-y 47.15.-x 47.57.-s 47.61.-k Keywords: Jet Rayleigh instability Particles in shear Fluctuations
Contents 1. Introduction . 2. Experimental 3. Results . . . 4. Discussion . . 5. Conclusions . Acknowledgements References . . . .
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1. Introduction In recent years jets and the Rayleigh–Plateau instability [1–3] have been widely studied both theoretically and experimentally [4,5], at least in part because of their importance within the context of inkjet printing [6]. For drop-on-demand printing, there is usually a thread of liquid that follows drop ejection and which subsequently disintegrates to form unwanted satellites. For continuous inkjet [7], the continuous formation of droplets from a jet in a controlled fashion is fundamental to the robust operation of the process. Jet breakup has been studied at varying levels of complexity and sophistication [4,5]. For our purposes the growth of a perturbation to the jet is sufficiently represented by a linear approximation of the complete equation set such that the breakoff length, which is the distance from the nozzle at which the jet disintegrates to form a
⁎ Corresponding author. E-mail address: [email protected] (A. Clarke). 0001-8686/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cis.2009.09.006
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stream of droplets, may be calculated quite easily [8]. The breakoff length is determined by liquid properties, the dimension and velocity of the jet and a parameter that describes the size of the initial perturbation. In the absence of active stimulation, a jet will disintegrate into droplets randomly but with a well defined mean periodicity and size distribution. Many continuous inkjet systems perturb the jet via a pressure pulsation generated behind the nozzle. These systems are generally limited to operation at a particular frequency since the pressure pulsation is generated by acoustic resonance within a cavity. The frequency chosen is close to the Rayleigh frequency for the system so that the greatest effect can be achieved. Such systems are also limited in the material viscosity range that can be routinely accommodated. An alternative mechanism has recently been described [9–11] wherein a thermal perturbation to the surface of the jet drives a Marangoni flow that initiates breakup. Such devices can be manufactured via MEMS fabrication routes and are therefore precise and robust. For inkjet inks it is often preferable to use a colorant based on a pigment. Such pigments are typically insoluble organic particles of
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A. Clarke, S. Rieubland / Advances in Colloid and Interface Science 161 (2010) 15–21
diameter order 100 nm that are used at moderate volume fractions that can neither be considered dilute nor concentrated. This type of dispersion occurs widely but is intrinsically difficult to model since both particles and hydrodynamics need to be considered in a coupled fashion. Within the nozzle of a continuous inkjet device there exists a strong shear field which may be approximated as the jet velocity divided by the boundary layer thickness. For typical inkjet nozzles, the nozzle length is much shorter than the diameter and therefore the boundary layer remains a small fraction of the nozzle diameter. It is well understood, even from elementary linear models, that flow perturbations will lead to fluctuations in the breakup of the jet and therefore to droplet velocity fluctuations. Indeed a statistical analysis leads to a predicted drop size distribution for a simple jet that can be fitted by a gamma distribution [5]. Should particulates within the shear flow in the boundary layer affect the overall flow field, then they too will be expected to lead to fluctuations in jet breakup. The macroscopic properties of suspensions of particles are determined by the spatial organization of the particles, that is their microstructure. In the absence of flow a wide range of microstructures are realized dependent on the balance between particle–particle interactions and Brownian motion. During flow, the microstructure will rearrange to accommodate the hydrodynamic forces and thereby modify the flow field. Understanding the coupling between flow and microstructure during flow has been the subject of many studies (for a review see Vermant and Solomon [12]). Various numerical methods have been developed to model such colloidal systems: Stokesian dynamics [13], fluid particle dynamics [14], lattice Boltzman [15], and several others, together with modern variants. In the majority of cases these methods have been used to model the resulting rheological behaviour of significantly more concentrated systems than the pigmented ink system described above. Of these studies, Catherall et al [16] in particular have calculated the stress on a single particle in a high-shear flow comparable with the nozzle flow field, and observe significant fluctuations. The work presented here suggests a subtle flow perturbation induced by particulates that leads to macroscopic velocity fluctuations in jet breakup. 2. Experimental The experiments performed in the present study employed a microelectromechanical system (MEMS) fabricated nozzle including a resistive heater. This device was mounted in a standard pin-grid array prototype chip holder the back of which was drilled to allow the entrance of liquid. A programmable pulse generation device (NIOS) was used to provide approximately a 200 kHz signal (the Rayleigh frequency for the jet was typically chosen) comprising a voltage pulse of up to 5 V and duration of 1 μs. Liquid was forced from the nozzle with a static pressure of up to 65 psi (approx. 448kPa) which generated a jet with velocity up to approximately 25 m/s. The nozzle diameter used was 17.6 μm. The resulting jet had a Weber number, We (= ρU2D / σ) of O(100). The fluid handling arrangement is shown in Fig. 1a. Jet breakup was observed using a long working distance optical microscope; a Navitar 6000× zoom lens with Mitotoyu 10× APO Plan objective (Fig. 1b). The images were captured with either a Watec LCL-902c Black and white video camera attached to a LabView 1405 framegrabber card, or a Prosilica EC1380C firewire digital camera. In both cases the images were analysed with a LabView code written for the purpose. Lighting was provided by an HSPS high speed spark strobe system, which generated 30 ps light flashes. These flashes were synchronised with drop generation via the NIOS pulse generator and an HP 8111A pulse generator used to decimate the base signal and an SRS DG535 delay generator to delay the pulse and enable observation of the drop as a function of phase (Fig. 1c). The position of the nozzle relative to the camera was adjusted using a micrometer motor drive with indexer (Oriel model 18011). Thus the distance from nozzle to
jet breakup or to camera observation point could be accurately measured (+/−5 μm). The liquid jetted comprised various concentrations of glycerol in water. To these solutions polystyrene particles were added. The polystyrene particles were made in our labs and are charge stabilised. They therefore require no added surface active component to disperse them. Each sample had a relatively narrow size distribution as determined by particle size analysis. These particles were mixed, at a variety of concentrations, with a range of aqueous glycerol solutions so as to investigate background viscosity, particle size and volume fraction independently. Although jet breakup was initiated by a well defined thermal pulse, i.e. a known power for a known time, it was observed that the coupling of this power to an initial perturbation varied depending on the solution being used together and the jet velocity. Therefore an initial calibration experiment was performed to enable perturbations of the same effective size to be used for the various liquids and jet parameters. The breakoff length, i.e. the distance from the nozzle at which the jet first ruptures, at the Rayleigh frequency, was measured as a function of pulse power. The measured breakoff length is accurately approximated via the linear version of the jet equations [8] over the accessible range of powers, 1=2
L = CDWe
ð1 + 3OhÞ
ð1Þ
where L is the breakoff length, D the jet diameter, We the jet Weber number and Oh the jet Ohnsorge number (=μ /√(ρσD)) and C a constant related to the size of the initial perturbation, i.e. the pulse power. It should be noted that at the Rayleigh frequency, the surface profile closely matches a growing exponential until non-linear breakoff processes intervene, i.e. where the radial perturbation is greater than approximately 1/3 of the initial radius. This well controlled behaviour arises since the perturbation to the jet is via a spatially constrained thermal Marangoni driving force [9] rather than an acoustic resonance as in conventional continuous inkjet. Hence, the non-monotonic breakoff length as a function of modulation power observed by others [17] is not observed here. Irrespective of the precise nature of the initial perturbation, the Rayleigh instability ensures that the various wavector modes grow at well defined rates. Thus we choose to consider the initial perturbation as if it were a perturbation to the radius of the jet then C = ln(R/ξ), where R is the radius of the jet and ξ is the perturbation. Thus we can relate the measured breakoff length to an effective perturbation size ξ. In practice measurements of both breakoff length as a function of power and breakoff length as a function of driving frequency were made and fitted using independently measured parameters of viscosity, density, surface tension and jet radius. The effective perturbation obtained in this way is a linear function of pulse power used to perturb the jet over the accessible range of powers. For the data presented below, measurements at constant effective imposed perturbation are compared. Of interest here is the fluctuation of drop velocity. This was measured by measuring the position of many individual drops and obtaining the standard deviation of the resulting Gaussian distribution (Fig. 2). Since the lighting system was synchronised to the pulse generating the drop, the fractional velocity fluctuation is equal to the fractional positional variation, δu σ = u d
ð2Þ
where u is the droplet velocity, d is the distance from the formation of the drop to the measurement location and σ is the measured standard deviation of droplet position. Note that in practice three adjacent drops were simultaneously measured. Therefore, knowing the drop formation frequency as set, the velocity of the drops was also obtained. Further by plotting distance between the first two drops (A–B) against the distance
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Fig. 1. a) Schematic flow diagram of the pressure and liquid handling system for our experimental jet arrangement. b) Schematic diagram of the optical arrangement. c) Schematic diagram of the signal and timing systems for observing jet droplets.
between the second two drops (B–C) any correlation in successive drop breakoff could be assessed [18]. At the Rayleigh frequency, no such correlation was observed, suggesting that fluctuations leading to droplet velocity fluctuation have a correlation length at most less than the Rayleigh wavelength. It was verified that the drops did indeed have a velocity variation by measuring σ as a function of d. A linear result was obtained typically
with an intercept approximately 1 wavelength prior to breakoff. Additionally, it was verified that our measurements are not affected by airflows; the velocity of the drops was measured as a function of distance from breakup and found to be constant up to a distance greater that that used for our measurements. 3. Results We are interested in the effect solid particles have on the breakup of a jet. However, we first examine the behaviour in the absence of particles for a range of liquid viscosities. Fig. 3 shows the velocity fluctuation δu /u, as a function of the effective perturbation size, ξ. (Recall the effective perturbation size is determined by fitting the observed jet breakoff length as a function of nozzle heater power to obtain a calibration of effective perturbation size as a function of heater power.) It is clear from Fig. 3 that the drop velocity fluctuation for each liquid behaves in a 1 /ξ manner. That this should be the case can be obtained via the linear approximation as follows. If we have a jet then the breakoff length is determined by the velocity and the time it takes for an initial perturbation, ξi, to grow to the size of the jet, R,
Fig. 2. Measurement of drop position for several hundred drops allows the measurement of position standard deviation, σ, and wavelength to be obtained. From the wavelength and knowledge of the frequency, the drop velocity is obtained.
R = ξi expðα⋅tB Þ
ð3Þ
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Fig. 3. (a) Drop velocity fluctuation as a function of effective initial perturbation size. (b) Data extracted from the same curves as a plot of drop velocity fluctuation against liquid viscosity for four perturbation sizes.
with tB the time to breakoff and α, the frequency dependent perturbation growth rate. If we allow a small fluctuation in ξi, δξi, then we observe a small change in the breakoff time, δtB, δtB = −
1 δξ ln 1 + i : α ξi
ð4Þ
A small fluctuation in breakoff time necessarily leads to a small fluctuation in breakoff length and therefore a small fluctuation in drop mass. 2
δm = ρ⋅πR ⋅Ujet δtB :
ð5Þ
Hence, via conservation of momentum, the velocity fluctuation is given by, Ujet δu δξ j drop = ln 1 + i u λα ξi
ð6Þ
with λ the jet breakup wavelength. We therefore expect that, since δξi is small and independent of ξi, the drop velocity fluctuation should behave as 1 /ξi. This is indeed observed (Fig. 3). Eq. (6) also implies a frequency dependence through α and this was also verified, though not shown here. Note that, experimentally, the measurement of δu /u is significantly more sensitive than a corresponding measurement of drop size variation. If we take the data of Fig. 3a and replot as a function of viscosity Fig. 3b, then we find that the fluctuations are a decreasing function of viscosity. This should be expected since sources of noise, e.g. capillary waves on the jet, will be damped to a greater degree by increased viscosity. Note that inverting Eq. (6) allows an estimate of δξi. If this is
done one finds that the additional surface area implies a wave energy consistent with thermal energy kT. We now turn our attention to particulate containing systems [19]. For this purpose we prepared 3 nominally mono-disperse polystyrene dispersions of particle diameter 80 nm, 125 nm and 200 nm. With each dispersion we prepared a solution containing 3% by weight particles and 30% by weight glycerol, and a solution containing 3% by weight particles and 50% by weight glycerol. At 3% by weight, the distance between particles is approximately 4.5 particle radii. It should be noted that at this distance, and particularly in strong shear fields, the particles cannot be assumed to behave independently. The data obtained for each solution is shown in Fig. 4. Again we see a general inverse dependence with initial perturbation size. However, we also see that the resulting level of drop velocity fluctuation increases as background viscosity increases i.e. the opposite behaviour to that observed with simple liquids. This effect is seen more clearly in Fig. 5a where the same data are replotted as a function of viscosity. In Fig. 5b the data are plotted as a function of particle size, and it is seen that the level of fluctuation increases with increasing particle size for the same volume fraction. In Fig. 6 the velocity fluctuation is plotted as a function of particle concentration for one particle size and three different effective initial perturbation sizes. The behaviour observed is approximately a linear function of particle concentration or volume fraction. 4. Discussion Our nozzle is approximately 17.6 μm in diameter and approximately 5 μm in length. Thus, the boundary layer is only partially developed and
Fig. 4. Drop velocity fluctuation with particulates for two different background viscosities. (a) 30% aqueous glycerol approx. 3.2 mPa s, (b) 50% aqueous glycerol approx. 7 mPa s. Note that, contrary to Fig. 3, here as the viscosity increases so does the size of the fluctuations.
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Fig. 5. Data from Fig. 4 plotted, (a) as a function of viscosity (note the opposite trend to that of pure water/glycerol mixtures, i.e. without particles) and (b) as a function of particle size. In all cases the effective perturbation size was 5 nm.
so the shear field exists over only a thin annular region on the wall of the nozzle. The thickness of the boundary layer at the exit of the nozzle, assuming we may use the usual estimate for boundary layer formation on entry to a tube, may be estimated as [2], δ=
rffiffiffiffiffiffiffiffiffi μx 2ρU
ð7Þ
where δ is the boundary layer thickness, μ the viscosity, ρ the liquid density, U the jet velocity and x the length of the nozzle (i.e. the distance from the entry to the short tube that forms the nozzle). For the parameters pertinent here this dimension is approximately 1 μm and therefore the shear rate, estimated as jet velocity divided by boundary layer thickness, is of order 107 s− 1. Therefore within the shear field of the boundary layer the ratio of shear field thickness to particle size (=δ / (2a)) is between about 5 and 12. If the particles interact with each other within this shear field hydrodynamically or otherwise, then a flow perturbation within the nozzle will ensue. Such a flow perturbation will necessarily be amplified by the intrinsic instability of the jet and so will be expected to generate fluctuations in the jet breakup process. The particle Peclet number, Pe, characterizes the relative importance of shear and Brownian motion. For Peclet numbers much less than unity, Brownian motion is able to maintain the dispersed distribution of particles. For Pe ≫ 1 the shear flow modifies the particulate distribution, i.e. the microstructure, and Brownian motion is unable to restore the distribution on the timescale set by the shear flow. The Peclet number is commonly defined as Pe = τD / τshear, with τD a characteristic diffusion time and τshear a characteristic time corresponding to a given shear rate. A diffusion time is simply obtained from the Stokes–Einstein relation as the time to diffuse a
particle radius, τD = a2 / DSE = 6πμa3 / kT, where DSE is the Stokes– Einstein diffusion constant. The shear rate has dimensions of inverse time so a characteristic time is simply τshear = γ̇− 1. Hence, the Peclet number relevant to a particle is [20], Pe =
˙ 3 6πμ γa kT
where μ is the viscosity of the dispersion, γ̇ the shear rate, a the particle radius and kT the thermal energy. For our weakly interfering hard particle system it is appropriate to approximate the viscosity using classical hydrodynamics [21], μ = μS ð1 + 2:5ϕ + …Þ
ð9Þ
where μS is the solvent viscosity and ϕ the volume fraction. The shear rate is approximated by γ̇ = U / δ and therefore the Pe number for small ϕ and dropping the constant, behaves as, Pe =
pffiffiffiffiffi μS ð1 + 1:25ϕÞa3 kT
2ρU 3 x
!1 = 2 :
ð10Þ
Whereas it is the Peclet number that characterizes the formation of microstructure within such colloidal systems, we expect that fluctuations in velocity will be related to the degree that the particles are hydrodynamically coupled, that is the degree to which they are not simply convected with the flow. We are therefore uninterested in the zero volume fraction limit of Eq. (10) and thus we choose to form a Peclet number from the incremental viscosity associated with particle–particle hydrodynamic interactions, PeN, PeN = Pe−Peϕ→0
Fig. 6. Drop velocity fluctuation as a function of particle concentration. In all cases the particles were 125 nm polystyrene in 50% wt/wt aqueous glycerol. The three curves are for three different initial perturbation sizes.
ð8Þ
pffiffiffiffiffi 3 1:25 μS ϕa = kT
2ρU 3 x
!1 = 2 :
ð11Þ
Fig. 7 shows the correlation achieved using Eq. (11) to plot the data presented in Figs. 4 and 6, i.e. δu / u ∝ PeN. Note however that within the dataset temperature is not varied and therefore the correlation is between the velocity fluctuations and the estimated shear forces in the nozzle boundary layer due to particulates. That velocity fluctuations should scale with PeN is already suggested by Squires and Brady [22] where they considered the effects of particular micro-structural arrangements of particles around a probe particle leading to a fluctuating force on the probe particle as it was translated. An ensemble averaged microstructure (weighted by the N-particle distribution function) was then used to obtain the size of velocity fluctuations of the probe particle which was found to scale as Pe. Combining Eqs. (11) and (6) and allowing δξi to vary as √μ, we arrive
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with A and B constants to fit the data. In Fig. 8 this function is plotted with appropriate choices for A and B, and it can be seen that the trends in the data are reproduced. Within Eq. (12), the particle parameters appear as ϕa3, which is straightforwardly calculated for a mono-disperse system. For polydisperse systems however, this needs to be generalized. Since we find that, at the low concentrations investigated, the background noise and the particulate noise components are well represented by simple addition of the two noise components, we further suggest that to model poly-disperse systems at these concentrations we should simply add the effects of the individual populations hence, Ujet δU Aδξi + B∑PeN j drop = ln 1 + U λα ξi Fig. 7. Data for drop velocity fluctuation from Figs. 4 and 6 at an effective initial perturbation of 5 nm plotted as a function of Peclet number, PeN, as defined in Eq. (12).
at a general semi-empirical function that we can use to describe the data, Ujet δu A⋅ðμÞ−1 = 2 + B⋅PeN j ln 1 + = u drop λα ξi
! ð12Þ
ð13Þ
where the sum is over each particle population. Of course this sum would, in the limit, be an integral over the particle size distribution. To probe this, several mixtures of the previously prepared monodisperse particles were prepared with a total fraction of either approximately 1 wt.% or 3 wt.%. It can be seen from Fig. 9 that the assumption of additivity works well, and we can use the effective particle size defined in the usual way as follows,
aeff
0 11 = 3 3 ∑a ϕ B j j jC C =B @ ϕ A total
ð14Þ
where ϕtotal is ∑ ϕj . The function is the conventional volume mean j radius. Within the flow discussed here, the nozzle boundary layer contains rather few particles. For typical materials and nozzles, the boundary layer might contain O(1000) particles and assuming Poisson statistics therefore ~ 3% fluctuation in number density. Fluctuations in both number and spatial distribution within a given volume are able to generate flow fluctuations in the liquid flux from the nozzle. These flow fluctuations might equivalently be considered as arising from viscosity fluctuations since at these concentrations additional viscosity arises from hydrodynamic interactions between particles. The viscosity affects jet velocity through the shear stress in the nozzle which depends both directly on the viscosity and indirectly on the boundary layer thickness, the fluctuations of which both scale as √ϕ. Hence the fluctuations in jet velocity are expected to scale as ϕ. Such fluctuations in flow naturally lead to macroscopic variations in jet breakup.
Fig. 8. Plots of Eq. (12) as a function of various parameters. The general trends and scales are well reproduced. (a) and (c) are for a 3% concentration of particles. (b) is for a viscosity of 2.5 mPa s.
Fig. 9. Measurement of drop velocity fluctuation as a function of particle mixtures. Circles are for various individual concentrations of 108 nm, 125 nm, 220 nm particles. Squares for various mixtures and concentrations of 108 nm and 220 nm particles (summing to 1% or 3%). The last point at 4 × 10− 21 is for a solution of 3% of 220 nm particles.
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5. Conclusions We have presented a series of measurements of droplet velocity fluctuations measured as a function of liquid and jet parameters. Via suitable analysis we have separated the observed fluctuations into effects arising from (i) the liquid and jet influence on the size of the initial perturbation and, (ii) effects arising from size of the noise in the initial perturbation. In this way we have obtained a semi-empirical relationship quantifying the size of the droplet velocity fluctuation as a function of particulate parameters. Although the model of the data is semi-empirical, to the extent that the observed trends are captured by Eq. (13), we can venture comments on certain features of the behaviour: (i) There are at least two sources of fluctuation: (a) an intrinsic noise possibly associated with thermal capillary noise on the jet, but also potentially arising from flow in the chamber behind the nozzle, e.g. unsteady regions, and (b) noise associated with flow fluctuations generated by particulates. (ii) Correlations of the data suggest that droplet velocity fluctuations arise from hydrodynamic particle–particle interactions in the high-shear nozzle boundary layer. It should be noted that the level of shear, the scale of the flow and the transient nature of the flow conspire to make this situation inaccessible to more usual rheological measurements. The overall picture that emerges is of a low to moderate concentration of nanoparticulates interacting with a transient highshear field which leads to a macroscopic observable effect. The Rayleigh–Plateau instability acts as an amplifier of the initial fluctuations and so subtle variations in flow generated through interactions in the nozzle region cause observable fluctuations in jet breakup. This of course suggests that the observation of jet breakup offers a method to investigate flow behaviour in the extreme flow regime found within such a nozzle. Acknowledgements The authors would like to thank the support and encouragement of many colleagues in the Eastman Kodak research labs in Rochester, N.Y.
21
for the support of this work, including the provision of the MEMS nozzles.
References [1] Rayleigh JWS. Proc Lond Math Soc 1879;x:4–13. [2] Faber TE. Fluid dynamics for physicists. Cambridge: CUP; 1995. [3] de Gennes PG, Brochard-Wyart F, Quéré D. Capillarity and wetting phenomena. Springer; 2004. p. 118. [4] Eggers J. Rev Mod Phys 1997;69:865. [5] Eggers J, Villermaux E. Rep Prog Phys 2008;71:036601. [6] EPSRC, Next Generation Inkjet technology project; http://www.ifm.eng.cam.ac.uk/ pp/inkjet/. [7] Schneider JM. In: Hanson E, editor. Recent progress in ink jet technologies II. IS&T; 1999. p. 246. [8] Stanley Middleman. Modeling axisymmetric flows dynamics of films, jets and drops. Academic Press Inc; 1995. p. 102. Chap. 4.3. [9] Furlani EP, Delametter CN, Chwalek JM, Trauernicht D. 4th International conference on modelling and simulation of microsystems. Cambridge MA: Applied Computational Research Society; 2001. p. 186. [10] Chwalek JM, Trauernicht DP, Delametter CN, Jeanmaire DL. NIP17: International Conference on Digital Printing Technologies, vol. 170-89208-234-8; 2001. p. 291–4. [11] Hawkins G. 15th Annual European Ink Jet Conference, Lisbon; 2007. [12] Vermant J, Solomon MJ. J Phys Condens Matter 2005;17:R187. [13] Brady JF, Bossis G. Annu Rev Fluid Mech 1988;20:111. [14] Tanaka Hajime, Araki Takeaki. Phys Rev Lett 2000;85:1338. [15] Shiyi Chen, Doolen GaryD. Annu Rev Fluid Mech 1998;30:329. [16] Catherall AA, Melrose JR, Ball RC. J Rheol 2000;44:1. [17] Luxford G. ISTs NIP 15: International Conference on Digital Printing Technologies, vol. 150-89208-222-4; 1999. p. 26–30. [18] Martien P, Pope SC, Scott PL, Shaw RS. Phys Lett A 1985;110:399. [19] We thank Dr S. Desrousseaux, Kodak Limited, for synthesizing the monodisperse polystyrene particles. [20] See e.g.R.A.L.Jones, “Soft Condensed Matter”, OUP, 2002. [21] Russel WB, Saville DA, Schowalter WR. Colloidal dispersions. Cambridge: Cambridge University Press; 1989. [22] Squires TM, Brady JF. Phys Fluids 2005;17:073101.