viscoelasticity in inkjet printing - SET/EESC/USP

Proceedings of PACAM XI c 2009 by ABCM Copyright

11th Pan-American Congress of Applied Mechanics - PACAM XI January 04-08, 2010, Foz do Iguaçu, PR, Brazil

VISCOELASTICITY IN INKJET PRINTING Dr Neil F. Morrison, [email protected] Dr Oliver G. Harlen, [email protected] School of Mathematics, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT, United Kingdom

Abstract. We investigate the effects of viscoelasticity on drop generation in inkjet printing. In drop-on-demand printing, individual ink ‘drops’ are ejected from a nozzle by imposed pressure pulses. Upon exiting the nozzle, the shape of each ‘drop’ is that of a nearly spherical bead with a long thin trailing ligament. This ligament subsequently breaks up under the Rayleigh instability, typically into several small droplets (known as satellite drops). These satellite drops can create unwanted splash on the target substrate, and a reduction in printing quality. Satellite drops can potentially be eliminated by adding polymer to the ink; elastic stresses can act to contract the trailing ligament into the main drop before capillary breakup occurs. However, elasticity can also reduce the drop velocity, and can delay or even prevent the break-off of the drop from the ink reservoir within the nozzle. To achieve optimal drop shape and speed, non-Newtonian parameters such as the polymer concentration and molecular weight must be chosen correctly. We explore this parameter space via numerical simulations, using a split Lagrangian-Eulerian finite-element method. Results are compared with experimental observations taken from real printheads. Keywords: viscoelasticity, breakup, FENE, simulation, inkjet 1. INTRODUCTION Modern inkjet printers can be classified into two main types, according to the method of drop generation employed. In continuous inkjet (CIJ) printing a modulated stream of ink breaks up into a series of droplets downstream of the nozzle, while in drop-on-demand (DOD) printing a single ligament of ink is ejected from the nozzle and subsequently either disintegrates or (ideally) contracts into a single drop. Optimal jet breakup can be difficult to achieve, and for both methods the fluid properties of the ink must be calibrated in tandem with printhead design in order to avoid the presence of unwanted ‘satellite drops’ in the breakup pattern. Furthermore, jet breakup dynamics may be dramatically altered by the addition of polymer to the ink; there is an enhanced development of the Rayleigh instability, and drops can remain connected by thin threads for long times (Goldin et al. 1969). In this paper we consider simulations of viscoelastic DOD jets. Although we focus here on flows directly applicable to inkjet printheads, the science and technology involved is not limited to conventional printing. A detailed review of polymeric inkjet applications was given by de Gans et al. (2004). The breakup of a viscoelastic jet or thread is a much studied model problem with relevance not only to jet flows but also to fluid characterization, and the benchmarking of numerical methods (e.g. Keunings 1986). An asymptotic exponential thinning law for the thread radius has been well-established by experimental and numerical studies. Finite extensibility causes deviation from this law at late times as the molecules reach full stretch (Entov and Hinch 1997). Relative to the Newtonian case, a viscoelastic jet may remain intact for considerably longer times, taking the form of a series of near-spherical beads connected by thin filaments. Several investigations of Newtonian drop-on-demand inkjet flows have recently been conducted (e.g. Dong, Carr, and Morris 2006). Results included criteria for printability, the formation of satellite drops, and ligament contraction. For viscoelastic fluids, Bazilevskiy, Meyer, and Rozhkov (2005) and Shore and Harrison (2005) studied the influence of concentration and molecular weight on jet breakup properties using dilute polymer solutions, and proposed analogous criteria based on experimental observations. Extensive numerical data on the rheology of real industrial drop-on-demand inks have been obtained using filament stretching techniques, and connections between the rheology and jetting properties of the inks have been established (Hoath et al. 2009). 2. FORMULATION 2.1 Fluid model We model the ink as an homogeneous incompressible viscoelastic fluid represented by a single-mode FENE-CR constitutive model (Chilcott and Rallison 1988) for suspensions of dumbbell molecules. The stress tensor for this fluid is σ = −pI + 2µs E + Gf (A)(A − I) ,

(1)

where p is the fluid pressure, µs is the solvent viscosity, E is the rate-of-strain tensor, G is an elastic modulus, and the function f (A) describes the stress within a dumbbell molecule in terms of a symmetric structure tensor A: f (A) = L2 / L2 − Trace(A) ,



where L is the dumbbell extensibility, i.e. the ratio of the length of a fully stretched dumbbell molecule to its unstretched length. In the limit of increasing L the FENE-CR constitutive equation (1) reduces to that of an Oldroyd-B fluid. The time evolution equation for the structure tensor A is ▽

A ≡ DA/Dt − A · ∇u − (∇u)T · A = −f (A)(A − I)/τ ,

(2)

▽

where τ is the relaxation time of the dumbbells, and A is the Oldroyd upper-convected time derivative of A. Here u is the fluid velocity and DA/Dt is the Lagrangian material derivative of A. Equations (1) and (2) may be combined to yield ▽

σ = −pI + 2µs E − Gτ A , from which it may be easily seen that the FENE-CR fluid has a constant steady shear viscosity of µ = µs + Gτ . 2.2 Nozzle geometry and boundary conditions The shape of the nozzle used in the simulations is illustrated in Fig. 1(a). We assume that the jet is axisymmetric, so that it may be fully described by a 2D coordinate system consisting of an axial coordinate z and a radial coordinate r.

push-out

velocity

nozzle inlet

a2

nozzle length

switch-off

fillet radius

0

a1

time t1

nozzle wall

t2

t3 a3

initial pull-back

taper angle nozzle radius meniscus angle

final pull-back symmetry axis

ink−air interface

(a) The nozzle shape.

(b) The driving signal.

Figure 1. The shape of the nozzle used in the simulations, and the velocity boundary condition at the inlet. The results presented in this paper correspond to a fixed set of nozzle outlet dimensions: a radius of 25 µm, a length of 50 µm, a taper angle of 13◦ , and a meniscus angle of 45◦ . On the rigid interior nozzle walls, no-slip conditions of zero velocity are imposed; the contact line between the fluid and the nozzle wall is pinned at the outlet. Along the axis of the jet, a symmetry condition of zero radial velocity is applied. The boundary condition at jet-air interface −1 ˆ, ˆ ]jet + R2 −1 n [σ · n air = −γ R1 ˆ is the unit vector normal to the interface (directed outward from the jet), γ is the coefficient of surface tension, where n and R1 and R2 are the principal radii of curvature. For the lengthscales considered here, air resistance is negligible. At the nozzle inlet, a time-dependent velocity boundary condition is prescribed; the velocity is directed perpendicularly, with magnitude given by the driving signal shown in Fig. 1(b). The speed and volume of the ejected jet may be calibrated by adjusting the amplitude and duration of each of the three stages discussed above, via the six parameters labelled in Fig. 1(b). However, for the results presented in this paper, the same driving signal was used for all flows. 2.3 Governing Equations We assume that the only body force acting on the jet is gravity, and that there are no significant temperature variations, so that the fluid density and solvent viscosity are constant. The momentum and mass conservation equations are ▽

ρ Du/Dt = −∇p + µs ∇2 u − Gτ ∇ · A + ρgˆz ,

∇ ·u = 0,

where ρ is the fluid density, g is the acceleration due to gravity, and zˆ is the unit vector in the axial direction. We scale lengths by the nozzle outlet radius R, velocities by the final Newtonian drop speed U , times by R/U , and pressures and stresses by ρU 2 . These scalings yield the dimensionless equations −1

Du/Dt = −∇p + ((1 + c)Re)

▽

(∇2 u − c∇ · A) + zˆ/Fr2 ,

∇ ·u = 0,



−1 ˆ, ˆ ]jet + R2 −1 n [σ · n air = −(1/We) R1

▽

A = −f (A)(A − I)/Wi ,

where t, u, p and σ are now the dimensionless time, velocity, pressure, and stress respectively, c is a measure of the concentration of dumbbell molecules, equal to Gτ /µs , and R1 and R2 are now the dimensionless radii of curvature. The flow is characterized by the Reynolds, Froude, Weber, and Weissenberg numbers, given respectively by Re = ρU R/µ ,

Fr = U 2 /gR

1/2

,

We = ρU 2 R/γ ,

Wi = U τ /R .

An additional important dimensionless group ispthe Deborah number De = τ /tc which relates the dumbbell relaxation time τ to the capillary thinning timescale tc = ρR3 /γ (e.g. Clasen et al. 2006). Values of the dimensionless numbers for the flows considered in this paper are Re = 17 and We = 27, which are representative of typical inkjet flows. Gravity is negligible for real inkjets, so the reciprocal of the Froude number is set to zero here. 3. NUMERICAL METHOD The numerical simulations use an extension of the finite-element method of Harlen et al. (1995), which combines a Lagrangian computation of the stress in each mesh element with an Eulerian computation of the discrete velocities and pressures at each time-step. The mesh is Lagrangian in nature, and Delaunay triangle ‘flips’ are used to maintain element shape quality. In order to capture the breakup dynamics of an inkjet, the fluid domain is subdivided when the jet radius becomes smaller than a certain threshold (usually ≤ 1% of the nozzle outlet radius). The method of Harlen et al. (1995) was benchmarked by the authors in the absence of inertial terms in the governing equations. However, its extension to flows in which inertia is non-negligible has not yet been formally documented in the literature. As part of the ongoing work within the ‘Next-Generation Inkjet Technology’ project, the extended numerical method has been fully validated by comparison to other numerical and experimental studies of filament stretching flows. Convergence with spatial and temporal resolution has been established, and articles are in preparation. 4. RESULTS 4.1 Jet classification Some characteristic jet behaviours are shown in Fig. 2. Each of the jets corresponds to a different set of values of the viscoelastic parameters c, De, and L, with all other parameters and boundary conditions being the same for each flow. Newtonian

c = 0.02, De = 3.75, L = 50

(a) Newtonian

(b) c = 0.02, De = 3.75, L = 50

c = 0.05, De = 2.5, L = 20

c = 0.075, De = 3.75, L = 10

(c) c = 0.05, De = 2.5, L = 20

(d) c = 0.075, De = 3.75, L = 10 c = 0.05, De = 2.5, L = 50

(e) c = 0.05, De = 2.5, L = 50

Figure 2. Some characteristic jet behaviours. Snapshots are shown at times t = 25, 50, and 75. For the Newtonian jet, the ligament breaks up into a large main drop and a series of smaller satellite drops; we refer to qualitatively similar jet behaviour as ‘Newtonian’, even if the fluid concerned is weakly non-Newtonian. In each of the four non-Newtonian jets the breakup is significantly different; breakaway from the nozzle is delayed, the main drop speed is reduced, and the distribution of satellite drops is different. For jet (b), the large extensibility impedes capillary breakup, and droplets remain connected at late times by thin threads of fluid. We call this behaviour ‘beads-on-a-string’ in reference to its similarity to shapes seen in studies of thinning filaments (Clasen et al. 2006). Jets (c) and (d) represent improvements in the breakup dynamics from the industrial perspective, as the numbers or sizes of the satellite drops are



reduced compared to the Newtonian case (a). In jet (c) the ligament splits into two large drops, and in jet (d) the tail of the ligament retracts into the main drop, so that only a single large drop is formed (with some tiny satellites). The effect of drop speed reduction is most emphasized in jet (e), in which the final drop is never created, as elastic forces pull the protruding jet back to the nozzle. This case is referred to as a ‘bungee jumper’. 4.2 Parameter space In the previous section it was shown that our simulations exhibit several different classes of jet breakup, as the concentration c, Deborah number De, and molecule extensibility L are varied. We now explore this parameter space in more detail. Each of the two plots shown in Fig. 3 represents a two-dimensional map of jet types, for a fixed value of L. Jet breakup behaviour, L = 10

Jet breakup behaviour, L = 20

5

ig ge

Deborah number De

bungee

tring

2

(b

some tail retraction

2

3

tion

bungee

il retrac

3

4

s n-ads-o bea

single drop

some ta

4

er w fe

Deborah number De

5

r) sa l te

1

es lit

1 fewer (larger) satellites

Newtonian

Newtonian 0

0 0

0.05

0.1

0.15

0.2

0.25

0

0.02

concentration c

0.04

0.06

0.08

0.1

0.12

concentration c

(a) L = 10

(b) L = 20

Figure 3. Maps showing the jet breakup behaviour in different regions of the viscoelastic parameter space. Only Fig. 3(a) features a significant region of ‘single drop’ breakup, which is surrounded by a larger area in which there is some tail retraction, although the ligament fragments into satellite drops before it can be completely absorbed by the main drop. Typically the net volume of these satellites is smaller than than in the Newtonian case, and overall there are generally fewer satellites of a significant size. The ‘beads-on-a-string’ region in map 3(b) corresponds to the jet behaviour shown in Fig. 2(b), where the drops remain attached by thin filaments of ink at late times. There is no equivalent region in map 3(a), as it is only for larger values of L that thin threads can survive for sufficiently long times. Map 3(b) shows that, for Deborah numbers greater than 1, the type of jet breakup for a given value of L is determined mainly by the polymer concentration c. In both parameter maps there is an universal transition to the ‘bungee’ regime as the concentration is increased. As L is increased the cut-off concentration reduces and jettability is markedly inhibited. Close to the two axes of each map, the fluid is only weakly elastic and the jet breakup is Newtonian in character, i.e. there is no significant contraction of the ligament before it breaks into several satellite drops. As L is increased this ‘Newtonian’ region is suppressed, and even very dilute solutions exhibit deviations from the purely Newtonian jet. 4.3 Higher concentration effects Graphs of the main drop speed against concentration and Deborah number are shown in Fig. 4, for two values of L. main drop speed, L = 10

1.2

De = 0.50 De = 1.00 De = 2.50 De = 3.75 De = 5.00

1.2 1 main drop speed

1 main drop speed

main drop speed, L = 20 De = 0.50 De = 1.00 De = 2.50 De = 3.75 De = 5.00

0.8 0.6

0.8 0.6 0.4

0.4 0.2 0.2

0

0 0

0.05

0.1 concentration c

(a) L = 10

0.15

0.2

0

0.02

0.04

0.06

0.08

0.1

concentration c

(b) L = 20

Figure 4. The dependence of the final speed of the main drop on the viscoelastic parameters c, De, and L. Both graphs 4(a) and 4(b) show a clear slowing of the main drop with concentration, and that for lower concentrations the drop speed is approximately independent of Deborah number to leading order, so that for fixed L the drop speed is



controlled mainly by the concentration. Only at larger concentrations, when the drop speed has become much slower than in the Newtonian case, is there a significant spreading of data points with different Deborah numbers. A comparison of the two graphs shows that the drop speed is not controlled solely by the concentration c, but is also reduced by increasing L. However, the combined dependence is not on cL2 as one might intuitively have supposed for the FENE-CR model. Figure 5(a) shows a graph of the time of breakaway from the nozzle, as a function of the concentration and Deborah number, for a fixed dumbbell extensibility of L = 10. The breakaway time increases with both concentration and Deborah number, though the trends are not precisely monotonic. breakaway time, L = 10 100

De = 0.50 De = 1.00 De = 2.50 De = 3.75 De = 5.00

35 30 ligament length

80

breakaway time

ligament length, L = 10 40

De = 0.50 De = 1.00 De = 2.50 De = 3.75 De = 5.00

60

40

25 20 15 10

20 5 0

0 0

0.05

0.1

0.15

0.2

0

0.05

0.1

concentration c

0.15

0.2

concentration c

(a) Breakaway time

(b) Ligament length

Figure 5. The dependence of the breakaway time and ligament length on c and De, with L = 10. Graph 5(b) shows the overall ligament length upon breakaway from the nozzle. For each Deborah number, there is a consistent pattern as the concentration is increased. The ligament is longer at low concentrations than in the Newtonian case, as the breakaway from the nozzle is delayed, but at higher concentrations the reduction in drop speed counters this delay, so that the ligament length has a maximum at a certain value of c, above which the length decreases due to the slower drop speeds. For a fixed concentration, the general trend with increasing Deborah number is a considerable lengthening of the ligament. This follows from the similar trend of a growing delay in breakup time in graph 5(a), and the fact that the drop speed is only weakly influenced by the Deborah number, as previously discussed. 4.4 Satellite drop distribution Figure 6 shows the satellite drop volume distributions for jets of various concentrations, with De = 3.75 and L = 10. For each data set, the first drop is the main one and subsequent drops are satellites, which we divide into categories of ‘primary’ and ‘secondary’ according to the logarithmic decade in which they lie. Any smaller drops that lie outwith the range of the plot are negligible tertiary satellites. Drop volume distribution (De = 3.75, L = 10) 10

Newtonian c = 0.005 c = 0.01 c = 0.02 c = 0.05 c = 0.075

Main drop

Volume

1 Primary satellites

0.1 Secondary satellites

0.01 0

2

4

6

8

10

12

14

Drop number

Figure 6. The variation of satellite drop volumes with concentration c, for De = 3.75 and L = 10. In the Newtonian case, the jet breaks into a large main drop and six primary satellite drops, with no secondary satellites. For the viscoelastic jets, there are generally fewer primary satellites, and more secondary and tertiary droplets, as the concentration is increased. With c = 0.05 there are only two primary and three secondary satellites, and for c = 0.075 there are no non-negligible satellites at all. The latter case corresponds to the jet shown earlier in Fig. 2(d) in which only a single main drop is formed, apart from some tiny tertiary satellites.



4.5 Comparison to experiments At present, the comparison of the simulation results shown here to experimental studies is qualitative. A more quantitative and collaborative comparison to other work, particularly within the ‘Next-Generation Inkjet Technology’ consortium, is ongoing. The five representative jet behaviours shown in Fig. 2 agree closely to those found by Bazilevskiy et al. (2005) in an experimental study using bubblejet equipment and solutions of polyacrylamide (PAM). Indeed, our classification of jet-types is motivated by the photographic images presented in their paper. In addition, our jet behaviours qualitatively match those observed in ongoing experiments by members of the ‘Next-Generation Inkjet Technology’ consortium, using real inks and viscoelastic test fluids in both real printheads and large-scale replicas (Hoath et al. 2009). Several other experimental studies exhibit similar trends to those found in our simulations. Shore and Harrison (2005) investigated the influence of elasticity using PEO solutions; the length upon breakaway was found to increase with molecular weight, while the drop speed reduced. It was also observed that fewer and smaller satellites were formed as the elasticity increased, in agreement with our simulation results. More recently, Xu et al. (2007) conducted experiments with cellulose ester polymers, and found that the breakaway time increased with concentration, while a local maximum was found in the ligament length upon breakaway, in accordance our findings. 5. NEXT STEPS In this paper we have provided evidence that our simulations using a FENE-CR fluid model qualitatively represent drop-on-demand inkjet flows of viscoelastic fluids in real printheads. Further work to establish full quantitative agreement is underway, in the form of collaboration with experimentalists and printhead manufacturers. A version of our simulation program has been released to industrial and academic partners as software, and user feedback has already led to improvements in functionality. A fuller exploration of the dependence of jet breakup properties on the viscoelastic parameters is currently being conducted, and variations in the driving signal will also be considered. 6. ACKNOWLEDGEMENTS This work was carried out within the “Next-Generation Inkjet Technology” consortium, which is supported by the U.K. Engineering and Physical Sciences Research Council (EPSRC) and by industrial funding. 7. REFERENCES Bazilevskiy, A.V., Meyer, J.D., and Rozhkov, A.N., 2005, “Dynamics and breakup of pulse microjets of polymeric liquids,” Fluid Dynamics, Vol. 40, No. 3, pp. 376–392. Chilcott, M.D. and Rallison, J.M., 1988, “Creeping flow of dilute polymer solutions past cylinders and spheres,” J. NonNewtonian Fluid Mech., Vol. 29, pp. 381–432. Clasen, C., Eggers, J., Fontelos, M.A., Li, J., and McKinley, G.H., 2006, “The beads-on-string structure of viscoelastic threads,” J. Fluid Mech., Vol. 556, pp. 283–308. de Gans, B.-J., Duineveld, P.C., and Schubert, U.S., 2004, “Inkjet printing of polymers: state of the art and future developments,” Adv. Mater., Vol. 16, No. 3, pp. 203–213. Dong, H., Carr, W.W., and Morris, J.F., 2006, “An experimental study of drop-on-demand drop formation,” Phys. Fluids, Vol. 18, pp. 072102. Entov, V.M. and Hinch, E.J., 1997, “Effect of a spectrum of relaxation times on the capillary thinning of a filament of elastic liquid,” J. Non-Newtonian Fluid Mech., Vol. 72, No. 1, pp. 31–53. Goldin, M., Yerushalmi, J., Pfeffer, R., and Shinnar, R., 1969, “Breakup of a laminar capillary jet of a viscoelastic fluid,” J. Fluid Mech., Vol. 38, No. 4, pp. 689–711. Harlen, O.G., Rallison, J.M., and Szabó, P., 1995, “A split Lagrangian-Eulerian method for simulating transient viscoelastic flows,” J. Non-Newtonian Fluid Mech., Vol. 60, No. 1, pp. 81–104. Hoath, S.D., Hutchings, I.M., Martin, G.D., Tuladhar, T.R., Mackley, M.R., and Vadillo, D., 2009, “Links between ink rheology, drop-on-demand jet formation, and printability,” J. Imaging Sci. Technol., Vol. 53, No. 4, pp. 041208. Keunings, R., 1986, “An algorithm for the simulation of transient viscoelastic flows with free surfaces,” J. Comput. Phys., Vol. 62, pp. 199–220. Shore, H.J. and Harrison, G.M., “The effect of added polymers on the formation of drops ejected from a nozzle,” Phys. Fluids, Vol. 17, pp. 033104. Xu, D., Sánchez Romaguera, V., Barbosa, S., Travis, W., de Wit, J., Swan, P., and Yeates, S.G., 2007, “Inkjet printing of polymer solutions and the role of chain entanglement,” J. Mater. Chem., Vol. 17, pp. 4902–4907. 8. RESPONSIBILITY NOTICE The authors are the only individuals responsible for the material included in this paper.