Coalescence Dynamics of PEDOT:PSS Droplets Impacting at Offset on

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Coalescence dynamics of PEDOT: PSS droplets impacting at offset on substrates for inkjet printing Kalpana Sarojini KG, Purbarun Dhar, Susy Varughese, and Sarit K Das Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b01219 • Publication Date (Web): 23 May 2016 Downloaded from http://pubs.acs.org on May 24, 2016

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Coalescence dynamics of PEDOT: PSS droplets impacting at offset on substrates for inkjet printing

Kalpana Sarojini K G 1, Purbarun Dhar 2, 3, #, Susy Varughese1, $ and Sarit K Das 2, 3, * 1

Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600036, India

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Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600036, India

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Department of Mechanical Engineering, Indian Institute of Technology Ropar, Rupnagar 140001, India (present address) $ #

E–mail: [email protected]

E–mail: [email protected]

*Corresponding author – Email: [email protected] Phone: 01881-24-2101

Abstract The dynamics of coalescence and consequent spreading of conducting polymer droplets on a solid substrate impacting at an offset are crucial in understanding the stability of inkjet printed patterns which find application in organic flexible electronic devices. Poly (3,4ethylenedioxythiophene):poly(styrenesulfonate) (PEDOT: PSS) dispersion in water is a widely used commercial conducting polymer for the fabrication of electron devices. The effects of droplet spacing, impact velocity, substrate hydrophilicity, polymer concentration

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and charges on the coalescence of two sessile droplets have been experimentally investigated and the dynamic spreading characteristics during the coalescence process is determined through image processing. The equilibrium spreading length of the coalesced droplets decreases with concentration and spacing of the droplets, revealing the necessity of optimum fluid properties (viscosity, surface tension) for the stability of the desired pattern. The droplet’s impact energy governs the maximum extent of spreading and receding dynamics, as the velocity gradients developed in polymer droplets during coalescence is a function of the inertia of the fluid elements. Hydrophilicity affects the maximum spreading extent but it has no influence on the equilibrium droplet diameter. The spreading length dynamics of charge neutralized PEDOT: PSS is found similar to the charged droplets which show that the charged nature of the polymer does not affect the coalescence behavior. Further, different spreading regimes are identified and the governing forces in each regime are described using a semi-analytical formulation derived for the coalescence of two droplets. The model has been found to accurately provide insight onto the various mechanisms at play during the complex spreading event.

Keywords: Droplet, spreading, inkjet printing, coalescence, surface wettability, capillarity, PEDOT: PSS

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1. Introduction Inkjet printing (IJP) is a non-contact, direct printing technique that allows printing of micro to picoliter volumes of liquid material droplets on substrates for printed electronic device fabrication. Coalescence occurs when the deposited droplets interact with each other and with the substrate to form patterns of high resolution and feature sizes of the range of 10 – 100 µm. The impact and coalescence of droplets printed on a solid substrate is a rapid dynamic process occurring within a very short period of ~100 ms or less. The coalescence process is associated with droplet deformation and fluid redistribution inside the droplet until it attains equilibrium. Inkjet printing has been used in the fabrication of printed organic electronics devices and several researchers have analyzed its efficacy. Sirringhaus et al.1 employed IJP for fabrication of complete transistor circuits by fine tuning the substrate free energy and achieved high resolution channel length and efficiency. Gorter et al.2 developed three active layers of organic LED (OLED) by IJP as an alternative to thermal evaporation process. The printed OLED layer morphology has been reported to influence the device efficiency3 and hence uniform deposition of these layers is a challenge that needs to be overcome. Several authors have explained the stability of printed morphologies in terms of geometrical parameters and conservation principles. Duineveld

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proposed a simple model to study the

instability of IJP line on flat homogenous substrates having significant contact angle hysteresis. Contact angles of the liquid drop on the substrate (ϴ1) larger than the advancing contact angle (ϴa) has been reported to make the printed line unstable. Soltmann et al.5 observed different line morphologies with changing droplet spacing and substrate temperatures and proposed a simple geometric model to explain the various forms observed. Stable liquid tracks with finite receding contact angle were printed and the model proposed by Soltman et al. was used to predict the morphologies by accounting for droplet inertia to the proposed model 6.

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Stringer7 proposed lower and upper limits for stable line width based on the spacing between the drops, which is in agreement with the experiments. Later it was reported that features that are more precise could be printed by increasing the contact angle hysteresis 8. Contact line pinning or de–pinning is also one of the ways to produce stable lines of constant width and parallel edges for inkjet printing of polymer dispersions3 and stable conducting lines of copper, silver and gold nanoparticles based colloids can also be inkjet printed 9,10,11 accordingly. Ely et al.12 obtained smoother and uniform films of PEDOT: PSS (a commercially available conducting polymer) for higher surface energy substrates. Few researchers have studied spreading dynamics of single and two droplets coalescence on solid surfaces both experimentally and theoretically. The effects of a plasma treatment performed on the substrate were studied on PEDOT:PSS line spreading, analyzed by means of optical imaging and numerical algorithms 13. Chandra et al.14 investigated the impact dynamics of a single droplet on a dry surface and on a thin liquid film for different surface temperatures and obtained the rate of spreading of the droplet and compared the results with a proposed analytical model. Pasandideh et al.15 extended the analytical equations to predict the maximum spreading diameter of a droplet with viscous dissipation incorporated and numerically modeled the deformations during spreading using a modified Volume of Fluid (VOF) method. Roisman et al.16 proposed a theoretical equation for the maximum spreading factor using momentum balance for the flow into the rim of the liquid droplet during spreading and receding phases.

Rioboo et al.17 quantitatively analyzed the effects of substrate roughness and wettability on drop spreading and described different phases of spreading upon droplet impact on different surfaces. Antonini et al.18 studied the impact and spreading of a single drop on hydrophilic and superhydrophobic surfaces and reported different impact regimes and relative effect of capillary and inertial forces in each regime. A scaling analysis given for maximum spreading factor in terms of governing dimensionless numbers (Re and We) was found to be valid for both micron and millimeter scale drops19. The coalescence dynamics of

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a falling droplet with a stationary droplet for different spacing between them for different surface wettability was experimentally observed20,

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. Different coalescence domains

proposed to determine the spreading length and droplet deformations showed good agreement numerically for varying spacing, wettability and impact velocity. The role of viscosities on the coalescence of droplets has also been studied experimentally22 and the information can be used effectively for fabricating conductive micro connections in printed electronic devices. The oscillations and spreading dynamics of single and consecutively printed droplets on substrates with different wettability have been studied extensively23 and are important for the inkjet printing industry. Sun et al.24 found that the morphology of the coalesced droplets and deposition patterns of the suspended particles depends on the droplet deposition frequency, spacing and time scale of droplet evaporation. Chiolerio et al. 25 discussed that when jetting on a porous media, ejection frequency and droplet spacing are the key parameters to determine the morphology of nanoparticles generated by a reactive ink and in-situ synthesis approach.

Having discussed the relevant literature in the field, it is noteworthy that only few reports are available on the spreading behavior upon impact of polymer based droplets and this is important since most commercially available inks for printed electronics are essentially conducting polymer dispersions such as PEO, PEDOT:PSS and PANI

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. Kim et al.26

studied the spreading of polyethylene oxide solutions in a mixture of water and ethylene glycol for different molecular weights of the polymer. It was observed that the receding phase is qualitatively different from the case of a droplet of a Newtonian fluid and also the conformation change of polymer in each stage of deformation was reported. Jung et al.28 used polystyrene solution to study the influence of viscoelasticity in inkjet printing of droplet and reported that the viscoelastic behavior do not have any significant effect on the spreading and deposition dynamics. In the present work, the effect of different coalescence parameters of two center offset droplets, such as impact velocity, drop spacing, dispersion property, wettability, particle concentration and particle charges on the coalescence dynamics of two

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sessile droplets have been experimentally and mathematically investigated. In particular, the coalescence behavior of aqueous dispersions of the conducting polyelectrolyte complex PEDOT: PSS has been discussed as the fluid is widely employed in the fabrication of printable opto–electronic devices. In addition, a semi analytical formulation for the coalescence of two offset droplets on a solid substrate has been proposed. By optimizing the printing and interaction parameters leading to pattern morphology, the present results can find potential use for control of performance of inkjet printed microdevices.

2. Materials and Methodologies 2.1. Experimental setup

The coalescence of two droplets at offset on a solid substrate is carried out by depositing the first droplet, allowing it to attain equilibrium, followed by translation of the substrate to the required offset and then depositing the second droplet. The images of the dynamics were recorded from the instant when the second droplet is about to touch the first drop to the point when coalesced droplet attains equilibrium, and the same was processed using image processing toolbox in MATLAB®. The detailed schematic of the complete experimental setup is illustrated in Fig. 1. It consists of a droplet–generation section, substrate translational stage and the imaging section. The droplet generator consists of a syringe pump (Harvard apparatus, Pump 11 Elite) fixed with a syringe (Hamilton gastight, 25 ml) and connected to a stainless steel needle (23G) through a flexible drip tube. Droplet diameter of about 3mm is used for the present experiments. The substrate translation mechanism consists of a linearly actuating spring controlled datum whose movement is controlled manually in the horizontal direction with a digitized scale having an accuracy of ~ 10 µm.

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FIGURE 1: Experimental setup and components used for the coalescence study

The imaging section consists of a high speed CMOS camera (Photron Fastcam SA4, San Diego, USA) employing a 105 mm macro lens for capturing the intricate spreading dynamics of the droplets. The camera is mounted on a traverse mechanism and a synchronized strobe light is used as backlight (with diffuser sheets in between the light source and sample) for imaging. The camera is operated at a resolution of 1024 x 800 pixels with 5000 frames per second and a shutter speed of 1/34000 s for recording the coalescence dynamics.

2.2. Materials Oxygen plasma (Harrick – Plasma cleaner) treated glass slides, polyethylene terephthalate (PET) sheets (cleaned with Isopropanol (IPA) and DI water) and wax coated sheets were used as substrates. Water and aqueous dispersions of poly (3,4-ethylenedioxythiophene)–poly (styrenesulfonate) (PEDOT:PSS) of two different concentrations (1.3 & 0.65 wt. %) were

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used as working fluids. The polymer dispersion was ultrasonicated and filtered with filter paper of pore size 5 µm to remove any aggregates. The static contact angle (CA) of these fluids on respective substrates was measured using a contact angle goniometer (GBX Digidrop, Germany). The fluid is loaded in the syringe attached to the syringe pump and once the pump is operated, droplets are generated at the end of the needle, which eventually detaches to impinge onto the substrate below.

3. Results and discussions Given the importance of droplet coalescence behavior when there exists centerline offset between the impinging droplet and the resting droplet in printing, spraying and coating technologies; the same has been studied for various impingement velocities and offsets. The coalescence dynamics is quantified using dimensionless parameters for the spreading length and spacing between the droplets as reported by Li et al.20 as,

ψ=

Dy

(1)

( Ds + L)

λ = 1−

L Ds

(2)

where Dy is the instantaneous dynamic spreading length of two droplets, Ds is the single droplet spread diameter, L is the spacing between the centers of the impacting and the resting droplet and ψ and λ are the dimensionless spreading length and overlap ratios respectively.

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FIGURE 2: Schematic representation of the coalescence parameters

Figure 3 illustrates the time evolution images of two coalescing water droplets for different overlapping ratios. In the case of droplet impact on another droplet, either wholly or partially, liquid jets are formed (τ = 0.35) on either sides of the impacting droplet and spreads continuously until the droplet shape becomes flat (τ = 4.9). Avedisian et al.14 reported the formation of sideways liquid jets for droplet impact on a dry substrate and described its cause as the increased pressure at the point of impact. From the present study it is evident that impact of a fluid droplet on another does not lead to transfer of the pressure waves to the bulk of the stationary droplet. Au contraire, the surface tension of the stationary droplet acts as a braking mechanism, which dampens the inertia of the impacting droplet, as observable from the formation of jets at the interface. The fact that over the given period of initial impact generated instability the non-impacting end of the stationary droplet remains in state of rest; essentially conveys that the pressure pulse generated by sudden impact is not dissipated across the resting droplet, rather actively negated by the interfacial tension, leading to formation of surface traversing jets. The non-impacting end of the resting droplet only enters phase of motion due to inertia of the fluid elements of the impacting droplet. This is more pronounced for larger offsets, wherein the non-impacting end is barely seen to be disturbed. If the impact pressure pulse propagation through the resting droplet was to occur, the nonimpact end would exhibit signs of motion even before the inertia of the impacting droplet

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arrived at the same (given that the pressure pulse would travel at the acoustic velocity within the medium as compared to the low impact velocities). Thereby, the phenomena of freesurface hugging jets clearly segregate the roles of surface tension and viscosity in the coalescence dynamics.

There is a period of relaxation at the non-impacting end for a period of ~13 ms for λ = 0.38, keeping that edge pinned with a flow reversal directed towards the center; thereby causing the splat height to reach the corresponding equilibrium height. Furthermore, this stationary period before the initiation of recoiling (as observed by Avedisian et. al.14) is due to the contact angle hysteresis. It is during this period that the contact angle relaxes and receding occurs at the non-impacting end. It takes more time for the droplet to relax at the non-impacting end with increasing offset but relaxation period at the impacting end is nearly independent of offset. The liquid jet towards the impacting edge spreads with the formation of surface waves near the periphery. After the composite droplet attains the state of maximum extent of spreading (τ = 4.9), the ripples grow further with outward fluid flow and a reversal of flow is observed. The formation of the composite droplet by contact line pinning followed by decay of impact inertial oscillations essentially indicates that as time progresses, the droplet initially experiences high interfacial damping due to surface tension, which in turn demarcates the boundary of the droplet, within which the internal oscillations are trapped and die out by viscous dissipation. Furthermore, as the impact end gets pinned after a point of time, the surface ripples are forced to die out due to interference caused by the reversed flow of fluid away from the pinned end and the droplet tries to attain a surface configuration of maximum stability (τ = 18). With increasing offset, the frequency of ripples is observed to decrease and the recoiling is more prominent than the outward flow of fluid along the surface.

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FIGURE 3: Coalescence of two water droplets on a plasma treated glass substrate for different offset parameters and time regimes. (We = 112, Ca = 0.016)

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The impacting edge recedes at an earlier time frame since the momentum gained by the edge from the liquid jet spreading over the substrate is countered by stronger pinning pressure, leading to sudden recoil. The non-impact end, on the other hand, experiences low velocity jet compared to the other end since the jet is retarded by the resting droplet, leading to lower pinning pressure, and hence the smoother recoil. Furthermore, a major portion of the impacting droplet being in contact with the substrate at the impacting end is brought to rest faster by the static friction at the region of contact. In case of the other end, the coefficient of static friction is already lowered by the wetting behavior of the resting droplet, leading to much slower damping. This mismatch in recoil leads to enhanced internal redistribution of fluid, which is why cases with higher offsets require longer time for the final stabilization of the droplet shape. The recoiling of fluid from both edges leads to the formation of a composite droplet with a concentrated bulge at the center with tailing ends (τ = 9.45). Beyond this regime, the wetting length; but not the droplet contour, attains equilibrium. Progressing further in time, the fluid flow within the composite droplet redistributes and equilibrium in shape is obtained. In general, as also established in the preceding discussion, it has been observed that for offsets wherein more than half of the impacting droplet makes contact with the dry substrate, the time required for relaxation to the equilibrium length and for flattening of the coalesced droplet is more.

Figure 4 shows the images of coalescing PEDOT: PSS dispersion droplets. There is a formation of liquid jet that spreads on either sides of the impacting droplet similar to the case of water droplets. The jet towards the wetted droplet side flows over the stationary droplet with the formation of splat and reaches the non-impacting end (τ = 4). The splat behaves like a rectangular liquid sheet spreading over the resting droplet due to its high viscosity (~10cP) and the sheet dimensions reduces with increasing offset as the interfacial resistance offered by stationary droplet is also reduced. The inertia of these impacting fluid elements causes little spreading of the non-impact end as its inertia is already dampened by the high viscosity

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of the fluid and the coalesced droplets becomes flat. This spreading of the non-impact end is reduced with increasing offset and behaves like that of single droplet spreading, since viscous forces and the static friction offered by the substrate are high enough to suppress the recoiling motion. The fluid flow is reversed towards the center and there is a short stationary period due to hysteresis followed by very little receding. The jet on the right side spreads to the maximum extent (τ = 1.4) and relaxes to become equilibrium. There is no formation of ripples while spreading and the surface waves are completely arrested by the viscous forces during the process. The coalesced droplet attains equilibrium length in an earlier time frame when compared to water droplets.

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FIGURE 4: Coalescence of two PEDOT: PSS dispersion droplets on a plasma treated glass substrate for different offset parameters and time regimes. (We = 123, Ca = 0.2)

Figure 5 shows the spreading length dynamics of the coalescing droplets of varying polymer concentrations on glass substrates for different overlapping ratios (λ) and for a

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falling droplet with an impact velocity (Vf) of 1.4 m/s. It can be observed from the figure that upon reducing overlapping ratio (increasing the distance between the droplets), the maximum spreading extent and the equilibrium spreading lengths were reduced. There was an instantaneous rise in spreading length due to the impinging inertia of the second droplet and a maximum peak, succeeded by a plateau regime is observed in case of water droplets coalescence (Figure 5(a)). The inertia dissipates quickly in the plateau region as the impacting fluid element has to spread a greater distance and hence experiences static friction based damping before it reaches the non impacting end. While approaching the maximum value, the spreading inertia is arrested by the contact line pinning due to surface tension and the fluid flow reverses its direction towards the center, thereby forcing a receding phase. Since the viscosity of the fluid is low, the remaining inertia dissipates slowly causing the droplets to equilibriate. Spreading and receding occurs concurrently on either sides till it attains equilibrium. Figure 5(b) and (c) show the spreading length dynamics of 0.65 & 1.3 wt. % PEDOT: PSS dispersion droplets. There is an initial kinematic phase of spreading and a plateau region followed by relaxation similar to water droplets. The maximum and equilibrium values of the spreading length in polymer droplets coalescence are lesser than that of the water droplets under same conditions and this is due to the increased of fluid properties (viscosity) and droplet–surface interactions. The maximum peak in spreading length slowly disappears with increase of polymer concentration and distance between the droplets. This is because the impacting droplet inertia responsible for the initial maximum spreading is lowered by the increased viscosity of the dispersion, thereby reducing the maximum extent of spreading. In addition, the recoiling slope of the polymer droplets is reduced as observed in Fig 5(b) for 0.65 wt. % and completely vanishes in Fig 5(c) for 1.3 wt. %. This can be explained based on the shear thinning behavior of PEDOT: PSS dispersion28. Due to high inertia of the impacting droplet, the spreading velocity is more uniformly distributed among the fluid elements of both coalescing droplets and hence the velocity gradients are less. For low gradients, the viscosity is high and this suppresses the receding motion completely. The plateau region

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disappears with increasing offset between the droplets and it qualitatively matches to that of single droplet spreading on a surface.

FIGURE 5: Spreading length dynamics of two coalescing droplets on glass substrates. Dimensionless spreading length versus dimensionless time for different overlapping ratios (λ) for (a) water (b) 0.65 wt% & (c)1.3 wt% PEDOT:PSS dispersion in water.

Figure 6 shows the droplet deformation images of water and PEDOT: PSS dispersion droplets for an impact velocity of 1 m/s and for the same offset between the droplets. Significant differences are observed between PEDOT: PSS and water droplets coalescence from the images. The wettabilities of these fluids on plasma treated glass substrates are different (water CA 20º, PEDOT:PSS CA 10º) and hence their interaction with them which is observable from the spreading diameter of the droplets. Ripples are formed on the impacting water droplet when it touches the resting droplet due to the interfacial tension, but not in the case of polymer droplets due to quick damping by the larger viscous forces. The impacting droplet flows over the deposited droplet and almost on reaching the non–impact end, the splat pushes the edge to reach maximum spreading extent (τ = 1.3 (water) & τ = 1.75 (PEDOT: PSS). After reaching maximum extent (flattening), surface waves become more prominent due to recoiling at both the edges for water droplets and it takes different deformation shapes keeping both the ends pinned and slowly the waves dissipates to form equilibrium shape. In

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case of polymer droplets, the higher viscosity suppresses droplet deformations. The polymer droplet relaxes to equilibrium quicker than water droplets when the impacting inertia is high and the relaxation times are relatively equal for low inertia impact. This is again a consequence of the shear thinning characteristics of the polymer dispersion as discussed earlier.

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FIGURE 6: Comparison of the coalescence events of two droplets of water (left) and PEDOT: PSS dispersion (right) on a plasma treated glass substrate at same offset factor (λ = 0.69) at different time regimes.

Figure 7(a)–(d) compares the dimensionless spreading length (Ψ) versus dimensionless time (τ) of charge neutralized PEDOT: PSS dispersion with normal PEDOT : PSS dispersions and water on glass substrates for an impact velocity of 1 m/s. PEDOT:PSS dispersion, a polyelectrolyte complex in water has PEDOT (polycation) molecules electrostatically bounded to polyanionic PSS, with excess PSS anions giving the complex a net negative charge and facilitating the formation of stable dispersions in water

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. The term ‘charge

neutralized’ PEDOT:PSS dispersions are those where the net negative charge of the PEDOT:PSS polyelectrolyte dispersions are neutralized by adding a salt solution (here CuCl2) to the dispersions. The substrate can also be charged by connecting it to an external circuit, and the dynamics of droplets in presence of electric field is termed as ‘electrowetting’. However, this itself contains extensive physical mechanisms and is not within the scope of the present work. This exercise is performed to understand the effects (if any) of the charges on the polymer on coalescence dynamics. It can be observed from figure 7(a), the initial kinetic energy is not sufficient for the coalescing water droplets to spread to a maximum as in case of high inertia and the rate of spreading is low. The outward spreading inertia is not so strong to reach the non–impact edge and is concentrated in the center. At the same time, the surface tension forces act to recoil the droplets from both edges and hence the droplet recedes as seen in Figure 7(a). Thus the interactions of capillary and inertial flow produce more surface waves and the droplet takes hemispherical shape with a bulged center. Towards the end of receding the droplet relaxes to attain equilibrium. In Figure 7(b) the spreading length for 0.65 wt. % PEDOT: PSS increases initially against viscosity and then recedes by a small amount to attain equilibrium spreading length. As discussed in the previous sections on the shear thinning behavior of the PEDOT: PSS dispersion, the velocity gradients in this case are fairly uniform and the viscosity will be high. Due to high viscosity and low inertia at the

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maximum spreading point, the surface tension forces act to recoil both edges but the viscous forces are high to terminate the recoiling process and also damps the remaining inertia of the droplets, thereby attaining equilibrium faster. In Figure 7(c), after the initial kinematic phase, there is strong recoil of the droplets before relaxing to equilibrium for 1.3 wt. % PEDOT: PSS dispersion. For low inertia and high spacing, the velocity gradients will be non–uniform and the viscosities are higher near the impact end and lower towards the non–impact end of the resting droplet. The lower viscosity at the non–impact end causes the surface forces to recoil and flow reversal towards the center occurs. Near the impact point, the liquid jet spreads to the right and become pinned because of higher viscosity. The flow redistribution occurs within the droplet and finally undergoes a sudden relaxation to attain equilibrium spreading length. Figure 7(d) shows the spreading length behavior of completely charge neutralized (0.05M CuCl2 added to 1.3 wt. %) PEDOT: PSS dispersion. The viscosity increases up to 100 mPas on adding 0.05M CuCl2 solution and the trend is qualitatively similar to 1.3 wt. % PEDOT: PSS dispersion. By virtue of the enhanced viscosity, the recoiling slope is significantly reduced in this case. In addition, the droplet reaches equilibrium length rapidly when compared to other fluids. Hence, it is the effective properties of the fluid and their interaction with the substrate that influences the coalescence behavior of the droplets and the charge of the polymers have no effect on the coalescence behavior of such droplets.

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FIGURE 7: Spreading length dynamics of two coalescing droplets on glass substrates. Dimensionless spreading length versus time for different overlapping ratio (λ) for (a) water (b) 0.65 wt. % & (c) 1.3 wt. % PEDOT: PSS dispersion (d) 0.05M CuCl2 added 1.3 wt. % PEDOT: PSS dispersion.

Figure 8(a)–(b) shows the plot of Ψ versus τ for water and 0.65 wt. % PEDOT: PSS dispersion having very low impact inertia (Vf = 0.44 m/s) on glass substrates. There is neither maximum spreading nor significant receding during the process. It quickly attains equilibrium but more oscillations are observed during the spreading phase. The needle to

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substrate height is less and hence the droplet has inherent oscillations which add to the kinetic energy on impacting the resting droplet. These associated oscillations can be seen from the deforming droplet shapes while travelling through air before impact. Water droplets have more associated oscillations because of its less viscosity and these do not die out rapidly before reaching the substrate due to less distance between them. From Figure 8(a), it is observed that upon increasing the offset between the droplets, spreading length oscillation frequency around its mean value increases until the inertia is dampened by the static friction, surface and viscous forces. The PEDOT: PSS dispersion droplets have larger damping viscosity and thus with increasing offset, the length oscillation frequency reduces and at λ = 0.05 (almost separate droplets); it coalesces similar to that of single droplet.

FIGURE 8: Dimensionless spreading length versus time for different overlapping ratio (λ) at low impact velocity for (a) water and (b) 0.65 wt. % PEDOT: PSS dispersion in water.

Figures 9(a), (b) and (c) show the effects of impact velocities on the spreading length of water and PEDOT: PSS (0.65 & 1.3 wt. % in water) droplets coalescence for a given offset. From Figure 9(a), it is observed that impacting droplet inertia drives the initial kinematic phase of the droplet spreading. The receding curve is less prominent when the inertia is less as the coalescing droplet has reduced energy. During receding, the surface

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tension recoils the droplet to a spreading length with value lesser than its equilibrium value, thus a sudden kink is observed before equilibrium. It is the impact energy that induces the velocity distribution and viscosity values and hence the levels of viscous damping in the non– Newtonian PEDOT: PSS droplets. The maximum spreading peak and the receding curve disappears with reducing impact velocity as observed in the Figure 9(b). For very low inertia, there are continuous oscillations in the spreading length before attaining equilibrium. In Figure 9(c), there is no receding at higher velocity and significant recoiling of the droplets occurs for lower velocities, further providing hints at the discussed role of shear thinning behavior during the spreading dynamics.

FIGURE 9: Effect of impact velocities on the dimensionless spreading length with time for different overlapping ratios (λ) for (a) water (b) 0.65 wt. % & (c) 1.3 wt. % PEDOT: PSS dispersion.

The effect of concentration of the polymer on the spreading length of the coalescing droplets has been shown in the Figure 10 (a) and (b). The surface wettability and effective physical properties (such as viscosity) of the water droplets are modified by the addition of polymer strands and the microscale interactions experienced by the fluid elements also differ. For impact velocity of 1.4 m/s, there is very less influence of concentration whereas for

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impact velocity 1 m/s, the early stage spreading dynamics (till τ ~ 10) are affected much. In Figure 10(a), the velocity gradients experienced by the polymer dispersion droplets are less and uniform for both concentrations due of high impact velocity and hence there is little variation in spreading length. In Figure 10(b) however, the droplets of lower polymer concentration experience lower viscosity gradients than higher concentration cases (for same impact inertia) and hence the former shows no variation while later shows the damping induced receding phase in the spreading length.

FIGURE 10: Effect of PEDOT: PSS concentration on dimensionless spreading length with time for different overlapping ratio (λ).

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FIGURE 11: Effect of substrate hydrophilicity on dimensionless spreading length with time for (a) 0.65 wt. % PEDOT: PSS dispersion and (b) water.

The effect of surface wettability on the spreading length of two droplets is shown in figure 11(a) and (b). The type of substrate affects the early stage of the coalescence process but the equilibrium spreading length remains unaffected for the impact distances studied. However, the maximum spreading extent reduces with increasing wettability as the surface forces prevent wetting at the traversing ends and the droplet experiences pinning and fast recoil towards its center. Consequently, more surface oscillations and disturbances are observed in case of hydrophobic substrates like PET and wax as compared to hydrophilic glass.

4. Mathematical formulation A detailed survey of literature reveals that there are very few reports regarding mathematical predictions for explaining the dynamics involved in the coalescence of two droplets with different offsets. Few analytical models exist; however, these are all concentrated on dynamics of single drop impact on substrates. Pasandideh–Fard et al.15 investigated the maximum extent of spreading of a single droplet impacting on a solid surface and proposed a mathematical framework to deduce the same employing energy conservation principle. In the present article, conservation of energy before and after coalescence of the two droplets has been used for describing the spreading dynamics. The reported15 energy conservation equation involves the kinetic energy of the impacting droplet and its surface energy pre collision and the sum is converted to the surface energy after contact and the energy dissipated due to viscous damping within the spreading droplet. The equation is expressible as

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π 2 1 2  π 3 2  2 ρV0  6 D0  + π D0 γ = 4 Dmaxγ (1 − cosθa ) + W   

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(3)

The first and the second terms on the LHS are the kinetic and surface energies of the falling droplet (instantaneously before impact). The first term on the RHS is the surface energy of the spreading droplet at its maximum spreading extension and W is the work done by the fluid in deforming the droplet shape against viscosity. In the present scenario, an extra surface energy term for the static droplet resting on the surface before impact requires to be included within the formulation. Furthermore, a net surface energy component for the final coalesced droplet should also be introduced. Consequently, the modified form of Eqn. 3 considering two–droplet coalescence on a solid surface is expressible as

 D  π Ds2 1 2 2  π 3  ρ V D + π D γ + π γ a − w (1 − cos θ s )   o  = Dy2γ a − w (1 − cos θ s ) + W  f a w − f f 2  6  4      L  4

(4)

where ‘f’ represents falling droplet and ‘s’ represents static droplet, γ is the surface tension of the fluid, ‘D’ is the spreading diameter, ‘V’ is the velocity, ‘Dy’ is the dynamic spreading length of the coalescing droplets and ‘L’ is the offset between the droplets. The terms and their representations have been illustrated in Fig. 12.

FIGURE 12. Spreading length dynamics of two coalescing water droplets on glass substrates

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Eqn. 4 however also requires to be made compatible so as to contain the offset between the droplets. Accordingly, it has been corrected for offset (overlap ratio) variation by multiplying a geometrical factor (Do/L), where D0 = Ds + D f + L . It is to be noted that the 2

formulation is not valid for headlong impact of a drop on another. The work done by the fluid in the deforming droplet against viscosity is expressible as15

W = ∫∫ φd Ωdt φΩtc

(5)

where ‘ϕ’ is the viscous dissipation function, Ω is the volume of the fluidic domain and tc is the time taken for the droplet to spread to its equilibrium shape. Based on order of magnitude analysis, ϕ is estimated to obtain the form as15

φ

V  µ f   l 

2

(6)

Where ‘µ’ is the viscosity of the fluid and ‘l’ is a characteristic length normal to the impact surface. In Eqn. 4, the term ‘l’ is expressible as15

l=2

Df Re

(7)

Finally, substituting Eqns. (4) and (5) in (3), the expression for W is obtained as

W=

π 3

ρVr2 D f Dy2,max

1 Re

(8)

In Eqn. 6, Vr is the velocity of the moving edges of the coalescing droplet and the Reynolds number is determined based on Vf at the instant prior to collision. The expression for Dy ,max 15 is as follows

Dy ,max = Dy

8D f 3tV f

(9)

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In addition, the value of the splat thickness ‘h’ is expressed as 15

h=

2 D 3f D 2f tV f = 3Dy2,max 4 Dy2

(10)

The splat edge velocity Vr during spreading is expressible as 15

Vr d2 = V0 4Dh

(11)

In the present case, the parameters have been scaled for simplicity as d

D y and D

Ds

and the final expression for Vr is as

Vr

Dy4 Ds D 2f t

(12)

Substituting the value of Vr and Dy ,max in W, the scaled expression for W becomes

π 

 8  D10 y   µρ W= 2 3/2 3  Ds2 D 5/2    t 3 tV f f   

(13)

On further substituting W in Eqn. (4), the final form of the energy balance equation for the present scenario is expressible as

32  µρ  D0  = 3Dy2γ a − w (1 − cos θ ) +  3/2  9  V f  L  

ρ D3f V f2 + 12D 2f γ a − w + 3Ds2γ a − w (1 − cosθ )  

 D10  y  2 5/2 3      Ds D f t  

(14) The polynomial in Eqn. 14 is solved numerically to obtain D y , the predicted dynamic spreading length for the composite droplet as a function of time. The above predictions are obtained on the basis of the energy conservation model developed for two droplet coalescence on a solid surface in the preceding section. However,

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since the approach relies on basic energy conservation, the various surface phenomena such as oscillations and recoils cannot be captured by the present model. Accordingly, the predicted Dy requires to be corrected employing the parameters that govern such inertia, viscous and interfacial tension domainated effects. The predicted values of Dy are thus further modified by a prefactor which is a product of the governing dimensionless numbers and their behavior with respect to dimensionless time. The prefactor introduces the existence of the various forces acting upon the droplets during various time regimes of their coalescence process. Based on experimental data analysis, in general, four different time regimes have been figured out within the total coalescence period and these have been shown in Fig. 13. The first region is the kinematic spreading phase post impact, where the inertia drives the droplets to spread extensively. However, within the process of spreading, viscous and surface forces edge in and reduces the rate of spreading. It is by this time that the total inertia due to impact has approached the first droplet and a swift surge in spreading is seen, leading to the maximum spread. This zone encompasses the viscous dissipation and spreading (over the wetted droplet and on the dry substrate). Beyond this phase, the surface forces become domainant and pins the spreading edges of the droplet, leading to a recoil phase and finally the equilibrium phase where the forces nullify each other and the droplet comes to final shape. The time regimes were partitioned based on the critical time (tc) estimated from the ratio of initial droplet diameter (Df) and impact velocity (Vf) of the falling droplet. The general form of the prefactor written in the form of the individual governing numbers (Weber number, Capillary number and Reynolds number based on spreading velocity and the spreading regime time frame) has been found from data analysis to be of the form

(

) 

t  tc 

m −t n +t D*y ≈ Dy Wespr Caspr Retspr 1 −  *

*

*

(15)

where ‘spr’ represents spreading and m, n, t* are indices which are functions of time.

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FIGURE 13 . Dynamics of two coalescing droplets showing the four distinct time regimes of spreading in the present experiments: kinematic spreading phase post impact (I), viscous dissipation & spreading (II), viscous dissipation and receding phase (III) and equilibrium phase (IV). t c represents the critical time which is evaluated as discussed earlier.

For the case of water droplet coalescence shown in the Fig. 5(a), the predictions were obtained from the model and have been shown in Fig. 14(a) and (b). The critical time for this case was evaluated to be 2.8 ms. The variables m and n take values in the range 0–1 for each time region and the magnitudes for both decrease simultaneously with progressing time, showing the decay of the forces as the droplet shifts closer to its equilibrium.

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FIGURE 14: Dimensionless spreading length versus time for different overlapping ratio (λ) for water droplet coalescence with high impact velocity (1.4 m/s). Comparison of predicted results with the experiments for λ = (a) 0.68 and (b) 0.52.

For the case of water impact at high velocities, four distinct regimes were observed. In the first two regimes, viscous forces and the inertia imparted by the impacting droplet compete as the droplet is forced to spread out to a maximum limit. However, it is by this time that surface forces start playing their roles and pins the three phase contact line, thereby leading to a sudden recoil towards the inner regions of the droplet. The fluid motion due to recoil creates a bulge at the droplet centre and the droplet further spreads out in an attempt to minimize itsa potential energy. This follows back and forth until the viscous and surface forces damp out the inertia completely. Thus, the powers of We and Ca scale together and decrease monotonously as a function of time. However, at the junctures where one domain transits to another, the abrupt changes in the viscous and surface forces cannot be captured and the model predicts sudden change in the predicted values. Thereby the model can be considered to be a piece–wise polynomial model.

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FIGURE 15: Dimensionless spreading length versus time for different overlapping ratio (λ) for PEDOT:PSS droplet coalescence with high impact velocity (1.4 m/s). Comparison of predicted results with the experiments for λ = (a) 0.81 and (b) 0.63.

The predictions of coalescing PEDOT:PSS for an impact velocity (1.4 m/s) has been illustrated in Fig. 15 and only two spreading regimes have been observed. The polymer dispersion shows shear thinning behavior as discussed in the preceding section. The velocity gradients within the fluid decrease as the inertia is damped out by viscous forces; leading to further increment in the localized viscous resistance. This is well described by the decreasing power of We. In addition, due to the shear thinning behavior of the polymer suspension, the viscous dominance is more towards the equilibrating process and this is projected by a transiently increasing value of the index for the Ca. Fig. 16 illustrates the comparison of predicted results with the experiments of water droplets for low impact velocity case. In this case, the inertia of impact is low and hence the recoil is much less vigorous than the higher velocity scenario, as the viscous resistance and interfacial tensions damp out the inertia with ease. Furthermore, the surface tension effects act much faster due to the low inertia. In this case, the index of We remains fairly constant due to low reaction of inertia to damping (m ~

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1.2) Similarly, the index of Ca (n ~ 1.1) also remains constant and both reduces equivalently (m, n ~ 0.75) in the last regime to attain equilibrium.

Dimensionless spreading length versus time for different overlapping ratio (λ) for water droplet coalescence with low impact velocity (1 m/s). Comparison of predicted results with the experiments for λ = (a) 0.52 and (b) 0.37. FIGURE

16:

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Prediction of spreading length values of two coalescing droplets with time for different Weber and Capillary numbers for a hypothetical situation where the time regimes and indices of We and Ca have been considered same as the case of water droplets coalescing (closed symbols: varying surface tension, open symbols: varying viscosity). This illustrates solely the effects of viscosity and surface tension as predicted by the proposed model.

FIGURE 17:

Various theoretical models are available for predicting the equilibrium and maximum spreading diameter of single droplet. The present model is energy balance based on physical property correction, predicts the dynamics of two droplets coalescence (at an offset) and spreading on a substrate by taking into account the average physical properties of the fluid. Figure 17 shows the modified spreading length of the droplets with time for different We and Ca numbers, for a hypothetical condition wherein the indices of We and Ca and the time regimes are considered same as that of water. This exercise allows understanding the effects of physical properties on the spreading characteristics. Maximum spreading peak is observed in the cases of higher viscosity and lower surface tension fluids and also the model is limited to certain range of viscosity and surface tension values, beyond which the transitions from one time regime to another become unphysical in nature. Inertia takes complete control of the

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initial maximum spreading while the viscosity comes in during the later stages of spreading and recoiling, facilitating in damping of the surface oscillations. However, the effect of surface tension can be observed to be present from the moment coalescence begins and the maximum spreading depends on the capability of the surface to pin the traversing edges of the fluid droplet. The predicted equilibrium spreading length is almost same for all the cases (as expected, since the indices of We and Ca and time domains have been kept constant) but the coalescence dynamics are different. This essentially validates that the present model is an accurate prediction tool to capture the intermittent dynamics of the coalescing droplets.

5. Conclusion In the present study, extensive experimental studies on the coalescence of two sessile droplets with different coalescence parameters (such as offset and impact velocity) have been studied for its prime importance in understanding the stability of the patterns produced by inkjet printing of fluids for flexible film electronic devices. Coalescence studies of water and PEDOT:PSS (chosen for its importance in the conducting polymer based device industry) droplets with different polymer concentration shows that the concentration affects the dynamics only during the initial spreading regimes but beyond the maximum extent of spreading, its effect gradually decays out. The impacting droplet’s inertia generates velocity gradients within the coalescing polymer droplets and brings to the forefront the role of shear dependent viscosity on the spreading and receding dynamics. Substrates of different hydrophilicity used in the present study shows that there is minimal effect of wettability on the equilibrium spreading length but the spreading and recoiling dynamics are notably different. Charge neutralized PEDOT:PSS droplets shows weak recoiling when compared to the charged droplets due to the increased viscous forces caused of the neutralized dispersion. Since all other effects are similar, the effects of surface charge can be inferred to play no vital role in the coalescence of such conducting polymer droplets of such size. A mathematical

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formulation developed for the twin droplet coalescence phenomena can predict the different regimes of the spreading droplets for various offsets and impact conditions accurately.

Acknowledgements The authors would like to thank Dr. Shamit Bakshi, Associate Professor, Department of Mechanical Engineering, IIT Madras for the high speed camera facility. Authors also thank Mr. S. Rajesh, Lab Technician, Heat Transfer and Thermal Power Lab, IIT Madras for his help towards fabricating the substrate translation mechanism. PD would like to thank IIT Madras for the post-doctoral fellowship.

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Andreas Elschner, S. K., Wilfried Lovenich, Udo Merker, Knud Reuter In PEDOT: Principles and Applications of an Intrinsically Conductive Polymer, 1 ed.; CRC Press, Taylor and Francis Group: Boca Raton, FL 2010, 377.

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530x392mm (96 x 96 DPI)


Coalescence Dynamics of PEDOT:PSS Droplets Impacting at Offset on

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