# Role of surfactants on droplet formation in piezoacoustic inkjet printing
across microsecond-to-second timescales

Maaike Rump  Physics of Fluids group, Max Planck Center Twente for Complex
Fluid Dynamics, and J. M. Burgers Centre for Fluid Dynamics, University of
Twente, 7500AE Enschede, The Netherlands  Christian Diddens  Physics of Fluids
group, Max Planck Center Twente for Complex Fluid Dynamics, and J. M. Burgers
Centre for Fluid Dynamics, University of Twente, 7500AE Enschede, The
Netherlands  Uddalok Sen  Physical Chemistry and Soft Matter, Wageningen
University and Research, 6708WE Wageningen, The Netherlands  Michel Versluis
Physics of Fluids group, Max Planck Center Twente for Complex Fluid Dynamics,
and J. M. Burgers Centre for Fluid Dynamics, University of Twente, 7500AE
Enschede, The Netherlands  Detlef Lohse  Physics of Fluids group, Max Planck
Center Twente for Complex Fluid Dynamics, and J. M. Burgers Centre for Fluid
Dynamics, University of Twente, 7500AE Enschede, The Netherlands  Tim Segers
BIOS / Lab on a Chip group, Max Planck Center Twente for Complex Fluid
Dynamics, MESA+ Institute for Nanotechnology, University of Twente, 7500AE
Enschede, The Netherlands

## Abstract

In piezo acoustic drop-on-demand inkjet printing a single droplet is produced
for each piezo driving pulse. This droplet is typically multicomponent,
including surfactants to control the spreading and drying of the droplet on
the substrate. However, the role of these surfactants on the droplet formation
process remains rather elusive. Surfactant concentration gradients may
manifest across microsecond-to-second timescales, spanning both the rapid
ejection of ink from the nozzle exit and the comparatively slower idling
timescale governing the firing of successive droplets. In the present work, we
study the influence of surfactants on droplet formation across 6 orders of
magnitude in time. To this end, we visualize the microsecond droplet formation
process using stroboscopic 8-ns laser-induced fluorescence microscopy while we
vary the nozzle idle time. Our results show that increasing the idle time up
to $\mathcal{O}$(1) s affects only the break-up dynamics of the inkjet but not
its velocity. By contrast, for idle times $>$ $\mathcal{O}$(1) s, both the
break-up dynamics are altered and the velocity of the inkjet increases. We
show that the increased velocity results from a decreased surface tension of
the ejected droplet, which we extracted from the observed shape oscillatons of
the jetted droplets in flight. The measured decrease in surface tension is
surprising as the µs timescale of droplet formation is much faster than the
typical ms-to-s timescale of surfactant adsorption. By varying the bulk
surfactant concentration, we show that the fast decrease in surface tension
results from a local surfactant concentration increase to more than 200 times
the CMC. Numerical simulations then show that the evaporation-driven increased
surfactant concentration present at the nozzle exit fully coats the surface of
the droplet during its ejection. Altogether, our results suggest that a local
high concentration of surfactant allows for surfactant adsorption to the
interface of an inkjet at the µs-to-ms timescale, which is much faster than
the typical ms-to-s timescale of surfactant adsorption.

##  I Introduction

Drop-on-demand inkjet printing enables precise, non-contact deposition of
picoliter droplets 1; 2; 3; 4; 5, accommodating liquids with a diverse range
of physical properties. Beyond traditional graphics printing on paper, inkjet
technology finds application in the fabrication of electroluminescent displays
6; 7, electronic circuits 8; 9; 10, and biomaterials 11; 12; 13.

In inkjet printing, surfactants are commonly included in the ink formulation
to regulate the spreading and drying behavior of the deposited droplets 14. By
reducing surface tension 15, surfactants promote droplet spreading 16; 17.
Additionally, they serve to mitigate the so-called coffee-stain effect 18; 19;
20 during droplet drying, ensuring a uniform distribution of deposited pigment
particles 21. Besides influencing spreading and drying dynamics, surfactants
may also affect the fast ejection and formation process of droplets prior to
substrate contact.

The impact of surfactants on fluid flows critically hinges on the relative
timescales of surfactant adsorption and spreading compared to interface
deformation 15. When the timescale at which the interfacial area changes is
slow compared to that of surfactant adsorption and spreading, the surfactant
uniformly affects the surface tension of the interface. The minimum surface
tension then decreases with the bulk surfactant concentration until the
critical micelle concentration (CMC) is reached. Beyond the CMC, the surface
tension remains constant while the surfactant monomers aggregate into micelles
in the bulk 22. Conversely, when the timescale of surfactant adsorption and
spreading approaches that of interface deformation, gradients in surface
surfactant concentration emerge, inducing Marangoni stresses and creating a
dynamic surface tension that varies both in space and time. When the interface
deforms faster than surfactants can adsorb and spread, the limit is reached
where surfactants cease to influence the dynamics of the flow. The timescale
of surfactant adsorption and spreading is dictated by the properties of the
surfactant molecule and typically spans from sub-milliseconds to seconds.

Figure 1: Inkjet printing operates across microsecond-to-second timescales.
The timescale of droplet formation ($10^{-5}-10^{-4}$ s; left) is generally
considered to be too fast for surfactant adsorption to the newly formed
interface. As such, a non-homogenous distribution of surfactants is expected
on the inkjet. The timescale that follows is that of the interval between the
firing of successive droplets ($10^{-5}-10^{-1}$ s; middle). This idle time
can extend to timescales where evaporation starts to play a role
($10^{0}-10^{2}$ s; right) 23, which may result in surfactant accumulation at
the nozzle exit.

Inkjet printing operates across multiple timescales, each characterized by
distinct flow features 4, as illustrated in Fig. 1. The shortest timescale
($10^{-5}-10^{-4}$ s; left) is associated with droplet formation. This short
sub-millisecond timescale is generally considered to be too fast for
surfactant adsorption onto the newly formed interface. Therefore, a non-
homogenous distribution of surfactant is expected on the inkjet during its
ejection from the nozzle 24. The difference in surface tension along the
interface can result in Marangoni flows, which have been shown to delay both
the thinning rate of a surfactant covered liquid column and its break-up 25;
26; 27; 28; 29; 30; 31; 32. Notably, Antonolopoulo et al. 30 observed in their
numerical simulations that the surfactants remain at the front of the liquid
column; therefore the pinch-off from the nozzle is not influenced by the
presence of surfactants, but only the breakup of the liquid tail into smaller
droplets. Only recently, studies have aimed to incorporate the role of surface
rheology of a surfactant monolayer on the thinning of a liquid jet 33; 34.

The timescale that follows that of droplet formation is the timescale involved
in the interval between the firing of successive droplets ($10^{-5}-10^{-1}$
s; middle). This idle time can extend to timescales where evaporation starts
to play a role ($\approx 10^{0}-10^{2}$ s; right in Fig. 1) 23. Evaporation of
liquid from the nozzle exit may lead to a non-homogeneous distribution of
surfactants in the nozzle by which droplet formation may be affected.

In summary, in inkjet printing, surfactant concentration gradients may
manifest across microsecond-to-second timescales, thereby potentially
affecting droplet formation. In this work, we investigate the effect of both a
slow and a fast surfactant on droplet formation across timescales spanning 6
orders of magnitude. To this end, we visualize stroboscopically the droplet
formation process using 8-ns laser-induced fluorescence illumination pulses.
Additionally, we visualize the shape oscillation of the droplets in flight,
i.e., after pinch off, to extract the surface tension and viscosity of the
jetted droplet 35; 36; 37; 38. Further, we employ numerical simulations
including evaporation and surfactant transport to unravel the physical
principles underlying the experimental observations on droplet formation.

The paper is organized as follows. We start by describing the experimental
methods and analysis in Section II, followed by the numerical method in
Section III. Section IV contains the results and discussion. The paper ends
with the conclusions and an outlook (Section V).

##  II Experimental methods

###  II.1 Printhead and ink

Autodrop Pipettes (nozzle diameter:
$70\text{\,}\,\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ and
$50\text{\,}\,\mathrm{\SIUnitSymbolMicro}\mathrm{m}$) from Microdrop
Technologies GmbH (AD-K-501 and AD-H-501) were used as printheads (Fig. 2a).
They consist of a cylindrical piezoacoustic transducer bonded around a glass
capillary connected to a fluid reservoir. More details about the printhead can
be found in references 39; 40. The printhead is set to drive the meniscus in a
push-pull motion, thereby first ejecting a liquid filament and then retracting
the meniscus, a process that leads to pinch-off. The surfactant solutions were
supplied from a pressure controlled reservoir set to a constant underpressure
of $8\text{\,}\mathrm{mbar}$ to prevent the liquid from dripping out of the
nozzle due to gravity.

Two different surfactants were used: the non-ionic Gemini Dynol 607 (Air
Products, $342\text{\,}\,\mathrm{g}\mathrm{/}\mathrm{mol}$) and the more
commonly used non-ionic Triton X-100 (Sigma-Aldrich,
$647\text{\,}\,\mathrm{g}\mathrm{/}\mathrm{mol}$), which was also used by
Antonopoulou et al. 30. The CMCs of these aqueous solutions are
$2.92\text{\,}\,\mathrm{mol}\mathrm{/}\mathrm{m}\mathrm{{}^{3}}$
($0.1\text{\,}\mathrm{wt\%}$) (see supplementary information Fig. S1), and
$0.22\text{\,}\,\mathrm{mol}\mathrm{/}\mathrm{m}\mathrm{{}^{3}}$
($0.014\text{\,}\,\mathrm{wt\%}$) 41, respectively. Dynol being a Gemini
surfactant entails that it has multiple hydrophilic heads and multiple
hydrophobic tails, linked by spacers 42; 43. The close packing of the
hydrophobic tails in a monolayer of Gemini surfactants results in a more
pronounced reduction of the surface tension for Gemini surfactants as compared
to conventional surfactants 44. Dynamic surface tension measurements show that
after $50\text{\,}\,\mathrm{m}\mathrm{s}$, Dynol has reduced the surface
tension of water to $40\text{\,}\,\mathrm{m}\mathrm{N}\mathrm{/}\mathrm{m}$
(from $\approx$ $70\text{\,}\,\mathrm{m}\mathrm{N}\mathrm{/}\mathrm{m}$, see
supplementary information Fig. S2) while Triton is much slower, reducing the
surface tension to only
$60\text{\,}\,\mathrm{m}\mathrm{N}\mathrm{/}\mathrm{m}$ after
$50\text{\,}\,\mathrm{m}\mathrm{s}$, as shown in 45.

###  II.2 Imaging setups

Figure 2: (a) Schematic of the imaging setup that either used stroboscopic or
continuous illumination in combination with a slow-speed CCD camera or an
ultra-high-speed camera, respectively. (b) Sketch of the driving pulse where
$V_{\max}$ is the pulse amplitude and $t_{\mathrm{w}}$ the pulse width.

Two imaging methods were used to perform the measurements. The first method
uses stroboscopic imaging, where the nozzle was illuminated by incoherent
$8\text{\,}\mathrm{ns}$ light pulses. The illumination pulses were generated
by a laser-induced fluorescence (iLIF) system 46. The iLIF system consists of
a pulsed laser (Quantel EverGreen, dual cavity Nd:YAG, $\lambda$ = 532 nm, 7
ns) that excites a fluorescent plate embedded in a highly efficient diffuser
(Lavision, part nos. 1108417 and 1003144). The emitted light was coupled into
an optical fiber that illuminated the printhead via a collimating lens, see
Fig. 2a. The employed microscope (BX-FM Olympus) had a 5$\times$ objective
(LMPLFLN 5$\times$, NA of 0.13), a tube lens (U-TLU), and a high-resolution
CCD camera (Lumenera, Lw135m, 1392$~{}\times~{}$1040 pixels,
$4.65\text{\,}\,\mathrm{\SIUnitSymbolMicro}\mathrm{m}$/pixel size). The
resulting optical resolution was
$0.93\text{\,}\,\mathrm{\SIUnitSymbolMicro}\mathrm{m}$/pixel. The images
captured by the camera were saved on a personal computer by in-house and
custom-made software programmed in the graphical programming language Labview
(National Instruments).

The second imaging method comprises ultra-high-speed imaging at 10 million
frames per second. The high-speed camera (Shimadzu HPV-X2,
400$~{}\times~{}$250 pixels,
$32\text{\,}\,\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ square pixels) captured
frames illuminated by a continuous fiber-optic light source (LS-M352, Sumita)
using the same setup as before (Fig. 2a). High-speed imaging was employed to
record the droplet formation process and the oscillation modes of the jetted
droplets, to measure their surface tension. Both a 5$\times$ and a 10$\times$
objective (LMPLFLN, NA of 0.13 and 0.25, respectively) were used with a
resulting imaging resolution of
$6.64\text{\,}\,\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}$
and
$3.71\text{\,}\,\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}$,
respectively.

The jetting procedure followed in the present work is identical to the one in
our previous work on drying of binary liquid mixtures in an inkjet printhead
23. The experiments always started by jetting 999 droplets at a drop-on-demand
(DoD) frequency of $1\text{\,}\,\mathrm{k}\mathrm{Hz}$ to ensure that the
liquid mixture in the nozzle had the same composition as in the bulk. Then,
the jetting was temporarily ceased for the predetermined idle time, which is a
control parameter in the experiments. When the desired idle time had passed,
the piezo was driven to produce a single droplet that was imaged using either
stroboscopic or high-speed imaging. The driving voltage of the piezo
$V_{\text{max}}$ as well as the pulse width $t_{w}$ (Fig. 2c) were varied to
control the amount of liquid jetted from the nozzle.

Figure 3: (a) Schematic showing the droplet during oscillations symmetric
around the vertical axis, where the dashed line indicates the equilibrium
droplet shape. (b) Amplitude versus time for the shape oscillation of mode
$n=2$. The damping rate $\beta$ is determined from the decay envelope of the
amplitude. (c) The frequency of the shape oscillations is determined from the
Fourier spectrum of the oscillations. (d) Experimentally measured oscillation
modes $n=2$ (blue markers), $n=3$ (green markers), and $n=4$ (red markers) for
the 1 CMC Dynol solution. The dashed lines show the best fit of Eq. (3), which
thereby provides the surface tension $\gamma$.

###  II.3 Image analysis

The acquired images were analyzed in Matlab. First, the position and velocity
of the liquid exiting the nozzle were extracted by tracing the front of the
ejected liquid along the direction of jetting. This was followed by a moving
average smoothing filter and subsequently, the derivative of the smoothed
position data was calculated to find the velocity. The time between the start
of the actuation of the piezo and pinch-off was documented as the pinch-off
time. The error in the pinch-off time was set to the interframe time of the
experiment. After being ejected from the nozzle, the liquid filament forms one
or multiple droplets. These droplets were in the field of view for at least
another $50\text{\,}\mathrm{\SIUnitSymbolMicro s}$. By recording the droplet
in flight, the mean radius and its standard deviation were calculated from at
least three independent experimental realizations. A linear fit to the
position data and the fitting error provided the velocity of the droplet and
its error, respectively. As the total volume of liquid jetted out of the
nozzle was not constant from droplet-to-droplet, we quantified the total
momentum of the droplets, defined as the sum of the mass of all the jetted
droplets multiplied by their individual velocities. Dividing the momentum by
the total mass then gives a mass-averaged velocity.

Finally, the frequency of the oscillating droplet was extracted to calculate
the surface tension. This method 35; 36; 37; 38 is based on Rayleigh’s
expression for the shape of an axisymmetrically deformed droplet as a sum of
Legendre polynomials $P_{n}$ 47:

$R(\theta,t)=\sum_{n=0}^{\infty}a_{n}(t)P_{n}(\cos(\theta)),$ (1)

where $\theta$ is the polar angle and $a_{n}(t)$ is the time-dependent surface
mode coefficient for mode number $n$. Assuming an incompressible liquid, no
evaporation, and no internal rotation, $a_{0}(t)=R_{0}$, where $R_{0}$ is the
radius of the unperturbed spherical droplet (see Fig. 3a). By having the
origin of the polar coordinate system at the center of mass of the droplet,
the modal coefficient of the next mode becomes zero ($a_{1}(t)=0$). Thereby,
Eq. (1) simplifies to:

$R(\theta,t)=R_{0}+\sum_{n=2}^{\infty}a_{n}(t)P_{n}(\cos(\theta)).$ (2)

The modal coefficients themselves can be written as damped oscillations:

$a_{n}(t)\sim
e^{-\beta_{n}t}\cos(\sqrt{\left(\omega_{n}^{2}-\beta_{n}^{2}\right)}t),$ (3)

where $\beta_{n}$ is the damping rate and $\omega_{n}$ the angular frequency
for each mode $n$. The extraction of the damping rate and eigenfrequency using
Eq. (3) is demonstrated in Figs. 3b and c, respectively. The eigenfrequency
can be related to the surface tension by:

$\omega_{n}^{2}=(2\pi f_{n})^{2}=n(n-1)(n+2)\frac{\gamma}{\rho R_{0}^{3}},$
(4)

where $f_{n}$ is the eigenfrequency, $\gamma$ the surface tension, and $\rho$
the density. Also, the bulk viscosity $\mu$ can be extracted from the damping
rate of the droplet oscillations:

$\beta_{n}=(n-1)(2n+1)\frac{\mu}{\rho R_{0}^{2}}.$ (5)

Note that the influence of surface viscosity is not taken into account in this
model. From the experimental measurements, the modes $n=2$ through $n=4$ were
extracted ($n>5$ decay very fast), of which a typical example is shown in Fig.
3d.

###  II.4 Bulk viscosity measurement

The bulk viscosities of the aqueous Triton solutions with concentrations above
1 CMC were measured by a capillary viscometer 48. The solution was pushed
through a 2.3-cm long and 150-$\mathrm{\SIUnitSymbolMicro m}$ diameter
capillary tube at a pressure of $1.5\text{\,}\mathrm{bar}$. Knowing the flow
rate $Q$ and the pressure drop $\Delta P$ across the tube, the viscosity $\mu$
can be calculated using the Hagen-Poiseuille equation 49:

$\mu=\frac{\Delta P\pi r^{4}}{8LQ},$ (6)

where $r$ is the radius of the tube of length $L$. The flow rate is determined
by weighing the amount of fluid that has passed through the tube in
$60\text{\,}\,\mathrm{s}$, and then dividing it by the density $\rho$.

##  III Numerical model

A numerical model was been used to simulate the spatiotemporal distribution of
Triton within the nozzle and during the droplet formation process as this
concentration field cannot be directly obtained from the experiments. We limit
ourselves to solving only the mass transport equations of Triton in the bulk
liquid, i.e. without any adsorption kinetics, as incorporating surfactant
material properties requires knowledge about adsorption and desorption rates
when the Triton concentration exceeds its CMC value.

Simulating the entire process, i.e. the preferential evaporation of water from
the nozzle, subsequently followed by a jetting event, is a challenging
problem, which we have addressed in our previous work 23. In this section, we
summarize the salient features of that work and describe the necessary
alterations that were required as a different printhead was used.

As a general framework, we have used a sharp-interface arbitrary Lagrangian-
Eulerian finite element method (ALE-FEM) expressed in axisymmetric cylindrical
coordinates. This comes with the benefit that the liquid-gas interfaces are
always exactly represented by sharp curves, which easily allows for
incorporating Marangoni flow and mass transfer. Furthermore, FEM is solved
implicitly via Newton’s method, which provides a stable solution method along
with flexible time stepping on the two different timescales of evaporation and
jetting. The present implementation is based on the finite element library
oomph-lib 50; 51.

###  III.1 Evaporation phase

During the evaporation phase, the fluid dynamical equations in both the gas
and the liquid phase were solved. Since there is no actuation all parts of the
driving were deactivated. The governing equations for solving the evaporation
phase resemble the equations which have been used successfully in previous
works on the evaporation of multi-component droplets on substrates, see e.g.
reference 52.

####  III.1.1 Vapor diffusion in the gas phase

We assume diffusion-limited evaporation, i.e. the evaporation rate of water
can be obtained by solving the vapor diffusion equation for the partial mass
density $c$ of water vapor in the gas phase:

$\displaystyle\partial_{t}c=D_{\text{vap}}\nabla^{2}c,$ (7)

subject to the boundary conditions:

$\displaystyle c$ $\displaystyle=c_{\mathrm{VLE}}(w)$ at the liquid-gas
interface (8) $\displaystyle c$ $\displaystyle=c_{\infty}$
$\displaystyle\text{far away}\,,$ (9)

i.e. the vapor-liquid equilibrium according to Raoult’s law
$c_{\mathrm{VLE}}(w)$ at the liquid-gas interface, where $w$ is the
concentration of water (as weight fraction) in the liquid. For this condition,
the liquid water concentration $w$ is converted to mole fractions. The ambient
water vapor concentration $c_{\infty}$ far away is not directly imposed at the
distant boundaries of the considered gas domain, since it would induce
considerable errors stemming from the finite size of the considered gas mesh.
Instead, a Robin boundary condition mimicking an infinite domain is used,
which is based on a multi-pole expansion truncated at monopole order 53. In
comparison to water, the volatility of Triton is negligible, therefore we do
not account for Triton evaporation here.

####  III.1.2 Multi-component flow in the liquid phase

The bulk flow in the liquid phase is governed by the Navier-Stokes equations
with a composition-dependent mass density $\rho$ and viscosity $\mu$ together
with the advection-diffusion equation for the water mass fraction field $w$:

$\displaystyle\rho\left(\partial_{t}\mathbf{u}+\mathbf{u}\cdot\nabla\mathbf{u}\right)$
$\displaystyle=-\nabla
p+\nabla\cdot\left[\mu\left(\nabla\mathbf{u}+(\nabla\mathbf{u})^{\mathrm{T}}\right)\right],$
(10) $\displaystyle\partial_{t}\rho+\nabla\cdot\left(\rho\mathbf{u}\right)$
$\displaystyle=0,$ (11)
$\displaystyle\rho\left(\partial_{t}w+\mathbf{u}\cdot\nabla
w\right)=\nabla\cdot\left(\rho D\nabla w\right).$ (12)

The composition-dependent properties $\rho(w)$ and $\mu(w)$ are described as
linearly dependent on the composition. The surface tension $\sigma(w)$ is
described as the function $A-B/(1+C(1-w)^{2})$, where the coefficients are set
as $A=33$, $B=-39.5$, and $C=$36\text{\times}{10}^{6}$$. The result is a curve
that reaches the CMC concentration with a surface tension of
$37\text{\,}\mathrm{m}\mathrm{N}\mathrm{/}\mathrm{m}$, from which it slowly
decreases to the value of
$33\text{\,}\mathrm{m}\mathrm{N}\mathrm{/}\mathrm{m}$. The diffusivity $D$ is
fixed at a value of
$8\text{\times}{10}^{-11}\text{\,}\mathrm{m}\mathrm{{}^{2}}\mathrm{/}\mathrm{s}$
54. At the liquid-gas interface, normal and tangential stress balances, i.e.
Laplace pressure and Marangoni shear, are applied without consideration of the
stresses in the gas phase, which can be disregarded due to the small density
and viscosity ratios:

$\displaystyle\mathbf{n}\cdot\mathbf{T}\cdot\mathbf{n}$
$\displaystyle=\kappa\sigma$ (13)
$\displaystyle\mathbf{n}\cdot\mathbf{T}\cdot\mathbf{t}$
$\displaystyle=\nabla_{\text{S}}\sigma,$ (14)

with the stress tensor
$\mathbf{T}=-p\mathbf{1}+\mu(\nabla\mathbf{u}+(\nabla\mathbf{u})^{\mathrm{T}})$,
and the normal and tangent unit vectors $\mathbf{n}$ and $\mathbf{t}$,
respectively. $\kappa$ is the curvature of the interface and
$\nabla_{\text{S}}$ is the surface gradient operator. The kinematic boundary
condition considering water evaporation reads as

$\displaystyle\rho\left(\mathbf{u}-\mathbf{u}_{\text{I}}\right)\cdot\mathbf{n}=j,$
(15)

which connects the normal liquid bulk velocity $\mathbf{u}$ with the normal
interface velocity $\mathbf{u}_{\text{I}}$ via the evaporation rate $j$. The
liquid-solid interfaces within the simulated printhead geometry are no-slip
boundary conditions. Finally, the evaporation of water leads to a change of
the liquid composition near the interface, which is incorporated via the
boundary condition

$\displaystyle-\rho D\nabla w\cdot\mathbf{n}=(1-w)j\,.$ (16)

The far field in the reservoir is again mimicking an infinite domain by a far-
field Robin boundary condition. Furthermore, a constant underpressure of
$8\text{\,}\,\mathrm{m}\mathrm{bar}$ is applied as in the experiments, which
results in a slightly inward curved meniscus.

Figure 4: (a) Velocities of the pure water droplets and the Dynol head
droplets compared to the Dynol mass-averaged velocity (calculated from the
momentum) as a function of idle time. (b) Volume of the 1 CMC Dynol droplets
and total ejected volume from a $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$
diameter nozzle, both as a function of idle time. The datapoint markers
represent the mean while the shaded regions denote $\pm$ one standard
deviation from at least three independent experimental realizations.

###  III.2 Droplet formation

When it comes to the droplet formation, the relevant timescales are several
orders of magnitude smaller (faster) than those during the drying. The
consideration of the entire system is hence not required anymore. Given the
short timescales and the fast convection velocities during jetting, the
diffusion-limited evaporation model as used during the evaporation phase is
also questionable in this stage. Therefore, the gas phase and with it the
evaporation dynamics are disregarded during the jetting process. The friction
of the jetted droplet in the gas phase is in general not entirely negligible,
but its influence on the drop formation is. This has been shown by the
excellent agreement between experiments and frictionless numerics in slender
jet approximation 55. In the present ALE-FEM simulations, the deactivation of
the gas phase and evaporation implies setting $j=0$ in Eqs. (15) and (16).

####  III.2.1 Modeling of the piezo dynamics

To bring the liquid into motion, at the top of the nozzle, the normal traction
is set to:

$\mathbf{n}\cdot\mathbf{T}\cdot\mathbf{n}=-\alpha V_{pulse}(t),$ (17)

where $\mathbf{n}$ is the normal, $\mathbf{T}$ the stress tensor
($\mathbf{T}=-p\mathbf{1}+\mu(\nabla\mathbf{u}+(\nabla\mathbf{u})^{\mathrm{T}})$),
$V_{pulse}(t)$ the pulse shape and amplitude, and $\alpha$ the pulse-to-
pressure conversion factor, which is set to
$400\text{\,}\mathrm{Pa}\mathrm{/}\mathrm{V}$.

####  III.2.2 Sharp-interface ALE method with topological changes

While the sharp-interface ALE method has the benefits of easily and accurately
incorporating Marangoni flow and evaporation, one of its major drawbacks
compared to e.g. volume-of-fluid or phase-field models is the treatment of
topological changes, i.e. the pinch-off of droplets from the jet or their
potential subsequent in-air coalescence. For simple axisymmetric problems,
however, these events can be treated by mesh reconstruction: after each
accepted time step, the liquid-gas interface is tested for parts that run
nearly parallel to the axis of symmetry. If these are close to the axis (i.e.
within $2\text{\,}\mathrm{\char 37\relax}$ of the nozzle radius), the pinch-
off position is estimated by finding the thinnest spot of the tail that also
shows a profile of local relative outflux, i.e. a relative velocity that
changes sign in the vicinity. Whenever such a position can be found, the
liquid-gas mesh is artificially split and reconnected to the axis of symmetry.
Afterwards, a new separated mesh is constructed and all relevant fields are
interpolated from the previous mesh, whereby the data of the nodes at the
liquid-gas interface are interpolated from the data stemming from the previous
still-connected interface only.

While this method introduces an artificial length scale, i.e. the thickness
threshold for pinch-off to occur, all other numerical methods also
intrinsically have these artificial scales, be it the cell size in a volume-
of-fluid approach, the interface thickness in a phase field approach, or the
regularization radius in the slender jet (lubrication theory) method 56. The
method used here for topological changes also showed perfect agreement with
experiments on droplets colliding mid-air 57.

##  IV Results & discussion

We start our investigation on the role of surfactants on droplet velocity by
measuring the droplet velocities of pure water and that of an aqueous 1 CMC
Dynol solution for idle times spanning over 5 orders of magnitude (from
milliseconds to minutes, see Fig. 4a). The trends in the velocity of the
surfactant solution can be divided into two regimes: the minor difference
compared to pure water during shorter idle times (less than
$1\text{\,}\mathrm{s}$; _regime 1_), and a monotonic increase at longer idle
times (beyond $1\text{\,}\mathrm{s}$; _regime 2_). Remarkably, the velocity
increases up to 40% compared to that of pure water at an idle time of
$200\text{\,}\,\mathrm{s}$, while the droplet velocity of pure water remained
relatively constant as the idle time increased, i.e., it did not increase by
more than 3%. Figure 4b shows that the total jetted volume of the surfactant
solution increased when the droplet velocity also increased for the
corresponding idle times. The figure also shows that at idle times longer than
$3\text{\,}\mathrm{s}$, two droplets were formed, i.e. a head droplet and a
satellite droplet. Therefore, we calculate the sum of the momentum of both
droplets (head droplet and satellite) and normalize it by the total mass to
obtain the mass-averaged velocity of the jetted liquid, plotted in Fig. 4a
(red data points). The mass-averaged velocity is very similar to the velocity
of the head droplet, i.e. the velocity of the satellite droplet was very
similar to that of the head droplet. Thus, the velocity increase is not due to
a reduction in droplet size, but due to an increase in the kinetic energy of
the jetted liquid. Regimes 1 (shorter than $1\text{\,}\mathrm{s}$) and 2
(longer than $1\text{\,}\mathrm{s}$) are further discussed in the following
sections.

###  IV.1 Regime 1: Short timescales ($<$ $1\text{\,}\mathrm{s}$)

We attribute the minor difference in droplet velocity between pure water and
the surfactant solution in regime 1 in Fig. 4a to wetting effects. This is
demonstrated in Fig. 5a where, at the shortest idle times, a droplet of
surfactant solution left on the nozzle plate from the preceding droplet
formation process influences the current droplet.

The droplet formation processes in Fig. 5a show the delay in pinch-off for the
surfactant solution and the filament break-up. However, the higher voltage
(115 V) case in Fig. 5b shows that, next to the similar delay in filament
break-up, the pinch-off from the nozzle for the surfactant solution is not
dissimilar to the pure water droplet. This is also as described in reference
30, which for a surfactant solution reports a delayed filament break-up but an
unaffected pinch-off from the nozzle. That paper shows visualizations from
numerics revealing that the surfactant is transported toward the trailing edge
of the head droplet during the droplet formation process. The resulting
gradient in surfactant concentration, and correspondingly in surface tension,
leads to a Marangoni flow, which causes the observed delay in filament break-
up.

Figure 5: Droplet formation for water and an aqueous 1 CMC Dynol solution at
different idle times. (a) At $75\text{\,}\mathrm{V}$, the top row compares the
experiments at $80\text{\,}\mathrm{\SIUnitSymbolMicro s}$, just before the
surfactant solution filament pinches off from the nozzle plate, and the bottom
row compares the experiments at $86\text{\,}\mathrm{\SIUnitSymbolMicro s}$,
just before the filament of the 1 s surfactant solution breaks up. (b) At
$115\text{\,}\mathrm{V}$, the top row compares the experiments at
$172\text{\,}\mathrm{\SIUnitSymbolMicro s}$, just before the surfactant
solution filament breaks up, while the bottom row compares the experiments at
$196\text{\,}\mathrm{\SIUnitSymbolMicro s}$, just before the tail droplet for
$100\text{\,}\mathrm{ms}$ breaks up. The bottom row also indicates the surface
tension coefficients of the head droplet as calculated from the droplet
oscillations.

Figures 5a and b also demonstrate the minor influence of the idle time on the
filament break-up between the surfactant solutions. The measurement of the
droplet shape oscillations provides the surface tension of the head droplets
in Fig. 5b, which were all between 72 and
$74\text{\,}\mathrm{m}\mathrm{N}\mathrm{/}\mathrm{m}$, including that of the
water droplet. Even though the meniscus at the nozzle exit accounts for only
$\approx$ 25% of the surface area of the head droplet, this was expected to
result in a decrease in surface tension with increasing idle time. The lack of
a decrease of surface tension suggests that the surfactant initially present
on the meniscus is rapidly transported across the interface, ending up being
either distributed along the entire interface of the jetted filament or
accumulating in its tail. Further investigations are required to understand
this observation, which is beyond the scope of the present work.

###  IV.2 Regime 2: Long timescales ($>$ $1\text{\,}\mathrm{s}$)

####  IV.2.1 Droplet formation

Figure 6: Time-lapsed snapshots of droplet formation of 1 CMC Dynol from a
$70\text{\,}\mathrm{\SIUnitSymbolMicro m}$ nozzle diameter for an idle time of
(a) $2\text{\,}\,\mathrm{m}\mathrm{s}$ and (b) $50\text{\,}\mathrm{s}$. (c)
Overlap of the droplet shapes at
$100\text{\,}\,\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ from the experiments:
$2\text{\,}\,\mathrm{m}\mathrm{s}$ idle time in red and
$50\text{\,}\,\mathrm{s}$ idle time in blue, showing clear differences. (d)
The position of the liquid front for multiple idle times. (e) The velocity of
the liquid front for multiple idle times. (f) The pinch-off time of the liquid
column from the nozzle for multiple idle times; inset shows the length of the
liquid column just after pinch-off from the nozzle for multiple idle times.
(g) The surface tension measured from the head droplet’s oscillations for
multiple idle times. The datapoint markers in (f) and (g) represent the mean
while the error bars denote $\pm$ one standard deviation from at least three
independent experimental realizations.

To investigate the increase of the ejection velocity at long timescales in
regime 2, we first compare the droplet formations as measured for the 1 CMC
Dynol solution at two idle times in Figs. 6a and b, where (a) corresponds to
an idle time of $2\text{\,}\mathrm{ms}$ and (b) to an idle time of
$50\text{\,}\mathrm{s}$. The most striking difference is the length of the
tail. This is highlighted in Fig. 6c, which shows that the increase in tail
length is due to both the front of the liquid filament having traveled further
(higher velocity) and the end of the tail pinching off later in time. We plot
the front position of the filament vs. time in Fig. 6d for idle times ranging
from $2\text{\,}\mathrm{ms}$ up to $50\text{\,}\mathrm{s}$. The velocity
calculated from these position data is shown in Fig. 6e. Note that the
velocity curves start to deviate only after
$50\text{\,}\,\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ from the start of the
piezo actuation pulse, i.e., when the meniscus starts to retract into the
nozzle, as shown in Figs. 6a and b. For idle times longer than
$\approx$$10\text{\,}\mathrm{s}$, the velocity of the filament decreases less
during the retraction of the meniscus (Fig. 6c). This observation suggests
that the surface tension at the rear of the filament is decreased at longer
idle times, thereby exerting a lower pulling-force on the filament during
meniscus retraction. The presence of surfactants near the nozzle exit is
further indicated by the delay in pinch-off time, as indicated in Fig. 6f. The
increase in pinch-off time can be caused by a decreased surface tension but
also by an increased surface viscosity due to the presence of the Dynol
surfactant 32; 33. Moreover, the measured surface tension of the head droplet
in Fig. 6g demonstrates a correlation between the increase in droplet velocity
and a decrease in surface tension. This indicates that at longer idle times,
the surfactants are not only present near the rear of the liquid filament, but
also at the front. However, the increase in droplet velocity is not due to a
change in the initial velocity, but due to the changes at the rear of the
droplet: the ejected liquid is not slowed down as much by the retracting
meniscus and both the filament length and pinch-off time increase.

####  IV.2.2 Evaporation

Figure 7: (a) Head droplet velocity for a 1 CMC Triton solution from a
$50\text{\,}\,\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ diameter nozzle at two
different relative humidities as a function of idle time. The datapoint
markers represent the mean while the shaded area denotes $\pm$ one standard
deviation of at least three independent experimental realizations. (b) Mass-
averaged (mass-av.) velocities for pure water and 1–100 CMC Triton solutions
from a $50\text{\,}\,\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ diameter nozzle as
a function of idle time. (c) Mass-averaged velocities for the different Triton
concentrations as a function of the estimated Triton concentration in the
control volume $V_{0}$ at the nozzle exit (sketched in the inset as the red
dotted lines). (d) Ratio of the timescales for the droplet surface formation
$\tau_{S}$ and the Triton diffusion $\tau_{D}$, for a fully covered surface
(blue) and a surface coverage corresponding to
$50\text{\,}\,\mathrm{m}\mathrm{N}\mathrm{/}\mathrm{m}$ 58 (red).

The combination of a higher surfactant concentration at the surface and the
requirement of long timescales suggests that selective evaporation at the
meniscus generates this velocity increase. To be able to perform experiments
with bulk surfactant concentrations above the CMC, in what follows, we use
Triton surfactant as the Dynol surfactant phase separates (oils out) at
concentrations higher than the CMC 59. On the other hand, Triton forms
nanometric micelles at bulk concentrations above the CMC 60. We first
confirmed that the observed increase in droplet velocity with idle time is
also present for jetting experiments with a 1 CMC Triton solution, see Fig.
7a. Indeed, a similar trend is observed: the droplet velocity increases with
idle time. However, significant differences are present as well. First, the
increase of the head droplet velocity at an idle time of
$200\text{\,}\mathrm{s}$ and a relative humidity of 70% was only 6% (Fig. 4a),
whereas for Dynol the corresponding increase was 40% (Fig. 4a). Moreover, the
time-dependent velocity of the liquid filament during its ejection from the
nozzle was different. For Triton, the increase in droplet velocity appears to
be present quite early on in the ejection process (see supplementary
information Fig. S3), whereas for Dynol it was only apparent
$50\text{\,}\,\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ after the start of the
piezo actuation pulse (Fig. 7e). However, and more importantly, an increasing
trend of the velocity with idle time is still observed. On top of that, the
curve in Fig. 7a measured at the decreased relative humidity of 30% shows that
the sharp increase in droplet velocity occurs at a shorter idle time as
compared to the 70% relative humidity case. This confirms that evaporation is
the driving parameter for the increase of the droplet velocity with idle time.

####  IV.2.3 Increased local concentration

The observed influence of evaporation suggests that a surfactant concentration
higher than 1 CMC is required at the nozzle exit to increase the droplet
velocity. Therefore, the measurements were repeated with bulk Triton
concentrations of 20, 50, and 100 CMC. Increasing the Triton concentration
above 100 CMC made the solution too viscous to jet at the applied amplitude of
the piezo driving pulse. The mass-averaged droplet velocity obtained for the
different Triton concentrations is plotted in Fig. 7b as a function of the
idle time. At short timescales ($<$ $0.1\text{\,}\mathrm{s}$), the increased
Triton concentration resulted in a decrease in the droplet velocity instead of
the expected increase. The decrease in velocity most likely results from the
increased bulk viscosity of the high-concentration Triton mixtures (see
supplementary information Fig. S4). At longer idle times ($>$
$0.1\text{\,}\mathrm{s}$), however, the increase in droplet velocity occurred
earlier for higher Triton concentrations. This observation suggests that the
required increase in local concentration in the nozzle for the same increase
in droplet velocity is reached earlier for higher initial bulk concentrations.

We try to estimate the required increase in local surfactant concentration in
the nozzle by considering a control volume at the nozzle exit (red dashed
lines in inset of Fig. 7c), out of which the water evaporates and is replaced
by surfactant solution from the bulk. This means that we assume the
diffusivity of the surfactant to be negligible, for simplicity. The local
concentration $c$ of the surfactant in this volume $V_{0}$ is increasing with
idle time $t$ beyond the initial concentration $c_{0}$ due to the evaporative
flux $Q_{e}$, given by: $c(t)=c_{0}+c_{0}Q_{e}t/V_{0}$. The evaporative flux
was estimated from the numerical simulations to be
$0.02\text{\,}\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{(}\mathrm{m}\mathrm{{}^{2}}\mathrm{s}\mathrm{)}$.
The control volume covers the nozzle exit surface area and we choose a depth
of $1\text{\,}\,\mathrm{\SIUnitSymbolMicro}\mathrm{m}$. Figure 7c shows that
the droplet velocity curves now collapse at local surfactant concentrations
$c(t)$ beyond $\approx$ 400 CMC. Increasing the size of the control volume
still collapses the data, but at a longer idle time. Even though this
concentration estimation is based on a simple calculation and the absolute
values should not be trusted, the collapse of the data provides us with the
valuable insight that the local surfactant concentration in the nozzle should
be orders of magnitude higher than the CMC to increase the droplet velocity.
For example, the 20 CMC solution contains $\approx$ 35 times the initial
surfactant concentration when the droplet velocity starts to increase (after
an idle time of $5\text{\,}\,\mathrm{s}$). These results suggest that a thin
layer of highly concentrated surfactants at the nozzle exit is required to
cause a significant observable increase in droplet velocity.

To investigate how this high concentration of surfactant alters the liquid
behavior, we investigate how the adsorption timescale depends on
concentration. The adsorption of surfactants requires two parts, a diffusive
stage during which the surfactant migrates to the surface, and a kinetic stage
during which the surfactant orients itself into the most favorable
configuration on the surface. The total adsorption process requires both
stages and can thus be limited by either one. In the case of Triton X-100, the
limiting stage is the diffusion stage 61; 58. The timescale for Triton to
reach a specific surface concentration can be estimated using an adsorption
length scale derived from the mass balance between the surfactant
concentration on the surface $\Gamma dA$ ($\Gamma$ being the adsorbed
surfactant per unit area) and the surfactant concentration in the bulk $C_{0}$
next to the surface $dA$ with thickness $h$, given by $C_{0}hdA$. The
resulting timescale using the surfactant diffusivity $D$ is
$\tau_{D}=\Gamma^{2}/(DC_{0}^{2})$. We compare this timescale $\tau_{D}$ to
the timescale for the droplet formation $\tau_{S}$ (the time of pinch-off),
which is $57\text{\,}\,\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ for the Triton
experiments. Figure 7d shows the ratio of these timescales for the case when
the surface is fully covered with surfactant ($\Gamma=3.4\times
10^{-10}$\,\mathrm{mol}\mathrm{/}\mathrm{m}\mathrm{{}^{2}}$$ 58) and when the
surface tension is approximately halfway between pure water and pure
surfactant at a surface tension of
$50\text{\,}\mathrm{m}\mathrm{N}\mathrm{/}\mathrm{m}$ ($\Gamma=2.7\times
10^{-10}$\,\mathrm{mol}\mathrm{/}\mathrm{m}\mathrm{{}^{2}}$$ 58). The
horizontal black line indicates when the ratio $\tau_{S}/\tau_{D}=$ 1, above
which the diffusion timescale is shorter than the timescale of droplet
formation, both at a surface concentration $\approx$ 200 CMC. This is the same
order of magnitude as we observed in Fig. 7c. This demonstrates that the
change in droplet formation at longer idle times is due to Triton being able
to adsorb fast enough onto the newly formed surface at concentrations above
200 CMC, thereby reducing the surface tension and leading to the droplet
velocity increase.

####  IV.2.4 Numerically visualized increased local concentration

Figure 8: (a) Snapshots at
$79\text{\,}\,\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ of the numerical
simulations of the droplet formation process using a 1 CMC Triton-like
solution. The left image shows a snapshot for an idle time of
$0.05\text{\,}\,\mathrm{s}$ and the right image for an idle time of
$5\text{\,}\,\mathrm{s}$. The colorbar indicates the liquid concentration in a
range from 1 CMC to 10 CMC. (b) Surface concentration of the droplets in (a)
along the droplet’s axis of symmetry as indicated by $x$. The origin of the
x-axis is indicated in (a). (c) Percentage of Triton covering the surface at a
concentration above 2 CMC and above 10 CMC measured from the simulation data,
at the time instant when the droplet has just pinched off from the nozzle,
plotted as a function of idle time.

To understand how this thin layer of highly concentrated surfactant solution
at the meniscus influences the droplet formation along the entire length of
the droplet surface, we turn to numerical simulations. We want to use these
simulations to understand the distribution of the concentration field of
Triton during the droplet formation and not its adsorption and desorption
kinetics. Therefore, we call it a Triton-like liquid. The numerical
methodology used herein has been used before to simulate drying processes in
inkjet printing 23. The resulting droplets for two idle times,
$0.05\text{\,}\,\mathrm{s}$ and $5\text{\,}\,\mathrm{s}$, are displayed in
Fig. 8a. The colorbar is kept constant for both images to indicate the
differences in concentration of the Triton-like liquid. These two simulations
show that after an idle time of $5\text{\,}\,\mathrm{s}$, the jetted filament
is covered by a thin layer of highly concentrated Triton-like liquid, while at
an idle time of $0.05\text{\,}\,\mathrm{s}$, the droplet shows a homogenous
concentration everywhere. Snapshots corresponding to the droplet formation for
$5\text{\,}\,\mathrm{s}$ can be found in Fig. S5 of the supplementary
information. The wrapping of the surface of the droplet by a high
concentration layer of Triton is more convincingly shown in Fig. 8b, where we
plot the concentration of Triton-like liquid along the surface. The surface
coverage over time, defined as the percentage of surface area containing the
desired concentration compared to the total surface area, for a surface
concentration $>$ 2 CMC and for a concentration $>$ 10 CMC are plotted in Fig.
8c. These results show that the surfactant surface concentration on the jetted
droplet increases beyond the bulk concentration for idle times of the order of
seconds – exactly the timescale at which the experimental droplet velocities
were shown to increase (Figs. 4a and 7c). Therefore, we hypothesize that the
surfactants studied here, which typically adsorb onto a newly formed surface
in the millisecond to second timescale, demonstrate a rate of transport onto
the surface at the microsecond timescale during droplet formation. This
reduction of timescales only happens once the local surfactant concentration
at the liquid-air interface of the filament reaches the orders of 100 to 1000
CMC. This presence of surfactants on the surface during the droplet formation
reduces the surface tension of the liquid and results in an increase in
droplet velocity. We further argue that the increase in drop velocity with
idle time for Dynol results from the rear of the filament being covered with
surfactant to a larger extent at increased idle times due to a thicker layer
of high-surfactant concentration liquid being present at the nozzle exit.

##  V Conclusions and outlook

In this work, the role of surfactants in inkjet printing was studied at
timescales spanning over 6 orders of magnitude, i.e. from the microsecond
timescale of the droplet formation process to the timescale of seconds to
minutes for the evaporation process taking place between the ejection of two
successive droplets. When the time interval between successive droplets was
less than one second, surfactants were shown to only influence the details of
the break-up dynamics of the jetted filament while the drop velocity and
volume remain unchanged from the pure water case. However, droplet velocity
and volume increase monotonically with an increase in the time interval
between successive droplets when this time interval exceeds one second, i.e.
when the droplet production rate is less than $1\text{\,}\mathrm{d}$roplet per
second. The increase in droplet velocity was shown to correlate with a
decrease in surface tension. We demonstrate that the observed velocity
increase, and surface tension decrease, are evaporation driven. Selective
evaporation of water at the nozzle exit for a duration of
$10\text{\,}\,\mathrm{s}$ was estimated to increase the local surfactant
concentration by as much as two orders of magnitude. Using numerical
simulations, we demonstrate that a thin layer with an increased surfactant
concentration located at the nozzle exit can coat the entire jetted liquid
filament. We argue that when the surfactant concentration at the nozzle exit
reaches a concentration on the order of 100 to 1000 CMC, the adsorption
timescale of surfactants can decrease from milliseconds to microseconds
thereby lowering the surface tension, and increasing the jet velocity, at the
microsecond timescale of jet formation in inkjet printing.

The implications of the unraveling of the thin layer at the nozzle exit
covering the entire surface of the droplet extends beyond the interest of
surfactants. This is a property that likely holds true for any material having
a local higher concentration at the nozzle exit, including colloidal pigment
particles and dyes. Moreover, the results of surfactants having minor
influence on the droplet formation at high droplet production frequencies
during droplet production in air will be of interest for the development of
new ink formulations, as the surfactants are already extensively used for
stabilizing colloidal suspensions, as well as the droplet behavior on the
substrate.

## Acknowledgements

This work was supported by an Industrial Partnership Programme of the
Netherlands Organisation for Scientific Research (NWO), co-financed by Canon
Production Printing Netherlands B. V., University of Twente, and Eindhoven
University of Technology. We also acknowledge financial support of the Max
Planck Center Twente for Complex Fluid Dynamics. The authors also thank
Valeria Garbin for insightful discussions.

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