# Ultrafast imaging method to measure surface tension and viscosity of inkjet-printed droplets in flight

## Abstract

In modern drop-on-demand inkjet printing, the jetted droplets contain a mixture of solvents, pigments and surfactants. In order to accurately control the droplet formation process, its in-flight dynamics, and deposition characteristics upon impact at the underlying substrate, it is key to quantify the instantaneous liquid properties of the droplets during the entire inkjet-printing process. An analysis of shape oscillation dynamics is known to give direct information of the local liquid properties of millimeter-sized droplets and bubbles. Here, we apply this technique to measure the surface tension and viscosity of micrometer-sized inkjet droplets in flight by recording the droplet shape oscillations microseconds after pinch-off from the nozzle. From the damped oscillation amplitude and frequency we deduce the viscosity and surface tension, respectively. With this ultrafast imaging method, we study the role of surfactants in freshly made inkjet droplets in flight and compare to complementary techniques for dynamic surface tension measurements.

## 1 Introduction

Inks used in inkjet printers are a complex mixture of solvents, co-solvents, pigments and one or more surfactants (Wijshoff 2010). The solvents carry the pigment particles to the medium and evaporate, solidify or crystallize, while the surfactants prevent wetting of the nozzle plate and promote spreading of the droplet after it impacts the underlying medium. In order to accurately control the droplet formation process, its in-flight dynamics, and subsequent interaction with the substrate, it is key to quantify the liquid properties of the droplet during the entire inkjet-printing process.

The surface tension of a surfactant solution is determined by the concentration of adsorbed surfactant molecules at the liquid–air interface. When a fresh interface is formed, the surface tension equals that of the solvent (Ohl et al. 2003) and it decreases while surfactants adsorb at the interface, until reaching an equilibrium surfactant concentration. The associated timescales of the adsorption process are governed by the diffusion time of the surfactant molecules to diffuse from the so-called adsorption depth *h* to the interface (Ferri and Stebe 2000). This depth depends on the bulk surfactant concentration, the critical micelle concentration and the surface concentration of surfactants at equilibrium surface tension (Ferri and Stebe 2000). The typical diffusion time then scales with the surfactant diffusion coefficient *D* as \(\tau _D \sim h^2/D\) and ranges from milliseconds to days, depending on the surfactant type and surfactant concentration (Chang and Franses 1995; Eastoe and Dalton 2000). As the surfactants in inkjet printing must act before the ink dries, it is required that they adsorb as fast as possible. Droplet formation, however, is an extremely fast process that takes in the order of \(10\,\upmu \hbox {s}\), which is shorter than the approximately \(100\,\upmu \hbox {s}\) that a droplet is typically in flight and much shorter than the time a droplet needs to evaporate, which is several seconds (Staat et al. 2016). A surfactant with a typical adsorption time scale of the order of milliseconds is considered a fast-adsorbing surfactant (Chang and Franses 1995; Eastoe and Dalton 2000); therefore, it is clear that the surface tension of an ink is higher during droplet formation and flight than during the later spreading and evaporation phases. Methods exist to measure the time-dependent dynamic surface tension; however, these measurement techniques require separate, off-line setups or are not fast enough to operate at the microseconds timescale of the inkjet process (Franses et al. 1996).

*damped*oscillator of an immiscible viscous fluid within another fluid was first described by Lamb (1932) and revisited by others (Reid 1959; Chandrasekhar 1959; Miller and Scriven 1968; Prosperetti 1980). However, the analysis up to now has focused on mm-sized droplets or has only measured the aspect ratio of \(\upmu \hbox {m}\)-sized droplets, instead of the full analysis.

Here, we present an ultrafast imaging method to measure the surface tension and viscosity for \(\upmu {\hbox {m}}\)-sized inkjet droplets (corresponding to picoliters) in flight. We extend the well-known method to measure surface tension and viscosity from the eigenfrequencies of shape modes of oscillating droplets to the \(\upmu \hbox {m}\)-regime by employing ultrafast imaging. Ultrafast imaging is necessary as the microscopic length scale of picoliter droplets poses a few challenges. First, the eigenfrequencies of shape mode oscillations scale with the droplet radius as \(R_0^{-3/2}\), and thus high-speed imaging is not only required to capture droplets in flight moving at a speed near 10 m/s (\(\sim 10\,\upmu \hbox {m}/\upmu \hbox {s})\), but also to resolve the details of the surface mode oscillations. Secondly, the viscous effects become increasingly important, as can be evaluated from balancing the viscous forces and the capillary forces through the Ohnesorge number \({\text{ Oh }}={\mu /}\sqrt{{\sigma \rho R_0}}\), with \(\mu\) being the dynamic viscosity, \(\sigma\) the surface tension and \(\rho\) the density, especially for higher mode numbers *n*, since the Ohnesorge number scales with the mode number as \({\text{ Oh }}_n = \sqrt{2n}\,{\text{ Oh }}\) (Versluis et al. 2010). Thus, a decreasing droplet size leads to a faster decay of the oscillation amplitude due to damping.

The paper is organized as follows. In the next section, we revisit the background theory to extract the surface tension and viscosity from oscillating droplets. Then we present high-speed imaging experiments at two different timescales (\(\upmu \hbox {s}\) and ms) to vary the age of the freshly made droplet surface in Sect. 2 and describe the image analysis and requirements for obtaining sub-pixel accuracy in Sect. 3. The results are presented in Sect. 4 and compared to a complementary, off-line technique to measure the dynamic surface tension.

## 2 Shape mode oscillations

*t*as a sum of Legendre polynomials \(P_n\)

*n*the mode number (see Fig. 1a for the definitions of

*R*and \(\theta\)). Assuming incompressibility of the liquid and no evaporation, we find that \(a_0(t) = R_0\), where \(R_0\) is the radius of a sphere with the volume of the drop, or the equilibrium radius. Since we place the origin of the coordinate system at the center of mass of the drop, it follows that \(a_1(t) = 0\), and Eq. (1) simplifies to

*n*. The eigenfrequency depends on the liquid density \(\rho\) and the surface tension \(\sigma\), while the damping rate depends on \(\rho\) and the dynamic viscosity \(\mu\).

Figure 1b shows the amplitude–time curve of the shape oscillation for an \(R_0=15\) \(\upmu \hbox {m}\) droplet for a mode \(n=2\). Figure 1c shows the corresponding Fourier transform \(\tilde{a}_n\), indicating the eigenfrequency \(f_2\) of the shape mode. Thus, measuring \(\tilde{a}_n(t)\) while knowing *n*, \(\rho\) and \(R_0\) gives \(\sigma (t)\). Measuring \(\beta _n(t)\) from the decay rate, knowing *n*, \(\rho\) and \(R_0\), gives \(\mu (t)\). In the example of Fig. 1, we used pure water; the surface tension was chosen to be \(\sigma = 72\) mN/m, the viscosity was \(\mu =1\) mPas, and the density was \(\rho = 1000\) \(\hbox {kg/m}^3\), therefore the eigenfrequency of the mode \(n=2\) was \(f_2 = 57\) kHz with a damping rate of \(\beta _2 = 22 \times 10^3\) \(\hbox {s}^{-1}\). In order to record multiple frames of every oscillation cycle, ultrafast imaging at interframe times of 1–5 microseconds or up to 1 Mfps is required (Versluis 2013).

## 3 Droplet formation

## 4 Image analysis

In each frame of every recording, we detected the position of the liquid–air interface by determining the inflection point of the intensity in the direction normal to the interface. In this way, we obtained a description for the drop boundary with sub-pixel accuracy. For a complete description of the technique on which this analysis is based, please see (Van der Bos et al. 2014).

The origin of the coordinate system was placed at the center of mass of the droplet, and the interface was divided in two parts along the axis of symmetry. The resulting halves were compared to check for axisymmetry and if this was not the case, the recording was not analyzed further. In the case of axisymmetry, the right halve was used for the remainder of the analysis. Then the interface was converted to polar coordinates and the distance from the center of mass to the droplet boundary *R* as function of the polar angle \(\theta\) was obtained. Figure 4 shows the detected drop boundary \(R(\theta )\) (orange markers) of the frame shown in the enlargement of Fig. 2. Please note that the smoothness of the curve is only due to the used inflection-point method, there has been no smoothing or averaging of the data. Equation (2) was then fitted to the data to find the equilibrium radius \(R_0\) and the mode coefficients \(a_2\ldots a_n\), up to \(n=20\). The black solid line in Fig. 4a is the best fit to Eq. (2) and is well within 0.1 pixels of the detected boundary (see the residue in Fig. 4b). By repeating this process for the subsequent frames of the recording, the time-dependent values of the mode coefficients \(a_n(t)\) were found. We can now determine the viscosity from the decay of the amplitude of the mode coefficients. Also the surface tension can be extracted directly, either from the Fourier transforms of \(a_n(t)\), or by fitting Eq. (3) to the experimental \(a_n(t)\). Both methods yield the same result and as we already perform the fitting procedure to obtain the damping, we opt to use the latter method to determine the surface tension. For pure water we obtain the mode coefficients shown in Fig. 5. The surface tension and viscosity that we find for a microliter drop (\(\sigma =72 \pm 1\) mN/m and \(\mu =1.1 \pm 0.2\) mPas) and a picoliter droplet (\(\sigma =73 \pm 2\) mN/m and \(\mu =1.0 \pm 0.5\) mPas) are both in good agreement with the expected values, which are \(\sigma =72.8\) mN/m and \(\mu =1.0\) mPas.

## 5 Results

By adding the SDS as a surfactant solution, it is expected that there will be an influence of surface age on the surface tension and indeed Fig. 6a, b confirm this hypothesis. The surface tension of the microliter droplet (\(\sigma = 35 \pm 1\) mN/m) is much lower than that of the inkjet-printed picoliter droplet (\(\sigma = 63 \pm 1\) mN/m). The analysis shows that the surface tension remains constant in the analyzed range for both the microliter and picoliter droplets. In the case of the microliter droplet, the surfactant concentration at the liquid–air interface has reached an equilibrium before the droplet detached, while 150 \(\upmu \hbox {s}\) is apparently not enough time for the SDS molecules to significantly lower the surface tension of the freshly formed picoliter droplet.

The other surfactant that we studied, 1,2-hexanediol, is used in inks for inkjet printers, yet the influence of the surface age on the fluid properties of the HD solution is not known a priori. Although 1,2-hexanediol has surfactant-like properties due to its structure (Romero et al. 2007), the dynamic surface tension of a dilute HD solution has not been reported to our knowledge. Our experiments show that there is a significant effect of surface age on both the surface tension and the viscosity: The surface tension decreases from \(64 \pm 1\) mN/m for the \(\upmu \hbox {m}\)-sized droplet (surface age between \(\approx 200\) \(\upmu \hbox {s}\) and 10 ms) to \(58 \pm 1\) mN/m for the mm-sized droplet (surface age of \(\approx \,5\) s), and the viscosity increases from \(1.1 \pm 0.2\) mPas (\(\upmu \hbox {m}\)-sized droplet) to \(1.5 \pm 0.2\) mPas (mm-sized droplet). The surface age of the \(\upmu \hbox {m}\)-sized droplet is not known exactly because the liquid–air interface in the nozzle (i.e., the meniscus) is not a fresh interface at the moment a jet is ejected from the printhead. The surfactant concentration on the surface before droplet generation is determined by the jetting frequency, which in this experiment was 100 Hz, setting the upper bound of the surface age of the droplet. The lower bound of the surface age is set by the droplet formation time, which in this case was approximately 200 \(\upmu \hbox {s}\).

*k*are fitting parameters, \(\sigma _0\) is the surface tension of a freshly formed interface at \(t=0\), and \(\sigma _{\mathrm{{eq}}}\) is the equilibrium surface tension. The surface tension of a freshly formed interface is equal to that of the solvent, so with \(\sigma _{0}=72.8\) mN/m we fit Eq. (6) to the MBPM data to find \(\sigma _{\mathrm{{eq}}} = 57.6\) mN/m, \(\tau =2.1\) ms, and \(k = 0.77\), see the black dotted line in Fig. 7. As is seen in Fig. 7, the values of the surface tension that were measured from the shape oscillations of both the mm- and \(\upmu \hbox {m}\)-sized droplets agree with this model, within the experimental uncertainty. A printhead with a higher jetting frequency is needed to reduce the uncertainty in the surface age, and this remains a topic for future research.

## 6 Conclusions

In conclusion, we have developed a technique based on the stroboscopic imaging of the shape oscillation of inkjet-printed droplets to measure in-line the surface tension and viscosity of a liquid during the process of inkjet printing. With this technique, we show that the surface tension of an ink that contains surfactants right after droplet formation is significantly higher compared to the equilibrium value. We attribute this to the surfactant adsorption process, which needs more time to reach an equilibrium than the droplet is in flight. On the other hand, the viscosity of the same surfactant solution is lower during droplet formation than it is when the surface is fully covered with surfactant molecules. We attribute this to the high surface coverage of surfactant molecules, which is accompanied by an increased resistance to interfacial deformation of the droplet due to the Gibbs elasticity.

## Notes

### Acknowledgements

The authors would like to thank Eek Huisman for help with the MBPM measurements. This work is supported by NanoNextNL, a micro- and nanotechnology consortium of the Government of the Netherlands and 130 partners (Project No. 10B.07).

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