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Controlled spreading of thermo-responsive droplets Maziyar Jalaala and Boris Stoeber*b
Received 16th October 2013 Accepted 27th November 2013 DOI: 10.1039/c3sm52658e www.rsc.org/softmatter
A novel methodology for controlling the spreading of droplets impacting a substrate is presented. The working fluid is a thermoresponsive polymer solution that undergoes a sol–gel transition above a specific temperature. It is shown that the maximum diameter of a droplet at equilibrium can be controlled through the substrate temperature of the substrate and the polymer concentration.
The impact and spreading of liquid droplets on a smooth solid substrate continues to receive much attention in various disciplines, due to its importance for a wide range of applications in nature and industry. Examples include ink-jet printing, rain drop dynamics, fuel combustion, spray cooling, coating, and many others.1 The droplet impact dynamics depend on numerous parameters including the physical properties of the droplet and the surrounding uid, the wetting properties of the solid substrate, the impact velocity, and the droplet size. For a vertical isothermal impact of a droplet on a horizontal substrate (see Fig. 1c) the system dynamics are governed by inertial, viscous, gravitational, and capillary forces. The most relevant non-dimensional parameters to describe this problem are the Weber (We ¼ rD0U02/sLG), Reynolds (Re ¼ rD0U0/m), and Bond (Bo ¼ rgD02/sLG) numbers that depend on the density of the uid r, the initial droplet diameter D0, the impact velocity U0, the liquid–gas surface tension sLG, the viscosity of the droplet m, and the gravitational acceleration g. Additionally, the interfacial condition at the contact line is also essential (at least for low Weber numbers), and the surface properties of the substrate can be taken into account through the static contact angle qs, which ranges from 0 to 180 for superhyrophilic to superhydrophobic surfaces.
In the case of sufficiently high Reynolds and Weber numbers, splashing is observed.2 In contrast, for smaller values of Re and We, spreading occurs. In the spreading regime, the droplet is forced to spread over the solid surface, creating a disclike geometry, until the capillary and/or viscous forces stop further expansion.1,2 The droplet nally relaxes to an equilibrium state, however depending on the wetting properties of the substrate, it might vibrate or rebound.3 The mechanism of spreading has been investigated as a classical problem using different experimental, analytical and numerical techniques.1 However, despite its importance in a wide variety of applications such as coating and printing, only a few studies have been provided to propose a method for controlling the spreading of droplets. These studies mainly focus on using polymer additives that changes the rheological (shear-dependent) properties of the droplet4–7 or on modifying the wetting properties of the substrate.8,9 The goal of the
Department of Mechanical Engineering, The University of British Columbia, 2054 – 6250 Applied Science Lane, Vancouver, BC, V6T 1Z4, Canada. E-mail: mazi@mech. ubc.ca; [email protected]; Fax: +1 604 822 2403; Tel: +1 778 8394459
a
b
Department of Mechanical Engineering, Department of Electrical and Computer Engineering, The University of British Columbia, 2054 – 6250 Applied Science Lane, Vancouver, BC, V6T 1Z4, Canada. E-mail: [email protected]; Fax: +1 604 822 2403; Tel: +1 604 827 5907
808 | Soft Matter, 2014, 10, 808–812
Fig. 1 (a) Schematic view of the experimental setup used to observe the impact of droplets on a heated substrate. (b) Magnified view of the test section. (c) Physical parameters of the current problem.
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current study is to propose a novel technique to control the dynamics of an impacting droplet, in particular in the spreading regime. For this purpose, a thermo-responsive polymer solution is employed. The physics behind the problem is similar to the solidication of a liquid droplet on a cooled surface,10–12 where the advancing contact line is arrested due to the propagation of the solidication front, resulting in partial spreading of the droplet. However, in the current study, an opposite effect (substrate heating) is employed, resulting a sol–gel transition above the gel temperature of the solution. Here, we investigate the inuence of this phasechange behaviour on droplet spreading over a heated substrate in experiments. A schematic view of the experimental setup for investigating the impact of the droplets is illustrated in Fig. 1a and 1b. A syringe pump is used to generate sufficient pressure to produce droplets of a uniform size in the range of 1.4 mm < D0 < 3.8 mm through a stainless-steel needle with a diameter in the range of 0.2 mm < DN < 1.3 mm. The individual droplets fall onto the substrate due to gravity. The distance between the needle tip and the substrate is adjustable, hence, different impact velocities can be obtained with 0.003 m s1 < U0 < 0.25 m s1. The substrate is a clean glass slide placed on a Peltier device (PD) that acts as a heater. The temperature is controlled through the input voltage of the PD, and is monitored, using a thermistor mounted on the glass slide. Moreover, the cold side of the PD is attached to a heat sink to keep it close to the ambient temperature. All the experiments are performed when the substrate has reached a steady temperature. A high speed camera (Phantom, Miro 4.0) with a frame rate of fc ¼ 10 638 frames per second is used to capture images during the droplet-substrate collision. The camera is connected to a stereo microscope, resulting in a eld of view of 6 mm 7 mm and an optical resolution of 256 192 pixels. The impact area is placed between the camera and a high-intensity white light source with a light-diffuser, so that image can be recorded using backlight-shadowgraphy, where the droplet blocks the light path and creates a dark area on the image. The uids in the experiments are solutions of polyethylene oxide–polypropylene oxide–polyethylene oxide (PEOx–PPOy– PEOx), commercially available under the tradename Pluronic (also Poloxamer). Pluronics are symmetric triblock co-polymers and non-ionic surfactants that shows amphiphilic behaviour in aqueous solution with a ratio of hydrophilic to hydrophobic block lengths of (x:y). Beyond the critical micelle concentration (CMC), the co-polymer spontaneously forms micelles (if x:y $ 0.5) with PPO and PEO presenting the core and the corona, respectively.13 Further increase of the temperature or the polymer concentration leads to formation of a so gel, where the viscosity of the solution appears to increase drastically. This sudden viscosity change is reversible and is associated with the gelation of the uid. Due to their unique gelation behaviour and bicompatibility,14,15 Pluronics have been previously used as a smart material for a range of biologyinspired engineering applications including protein delivery, DNA sequencing, and controlled drug delivery.16 In addition, thermo-responsive polymer solutions have been previously
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Soft Matter
used for the active and passive control of continuous ow in microchannels.17–19 In the present study, Pluronic BASF© F127 (PEO106–PPO70– PEO106) is used at three different concentrations (11, 14 and 17 wt%) with no further purication and fractionation. The polymer powder is dissolved in distilled water under gentle agitation at low temperature (3 C). Fig. 2 shows the viscosity measurements from a cone-and-plate rheometer at controlled shear stress for the three samples. The gel temperature (TG) is dened as the temperature above which the viscosity of the solution suddenly increases. Below TG a higher polymer concentration corresponds to a higher viscosity, while TG decreases with increasing polymer concentration. It has been shown that these polymer solutions show Newtonian behaviour below TG; near the gel point and above, they exhibit shear-thinning behaviours.17 Here, we focus on the thermothickening properties of these uids for controlling the spreading dynamics of droplets upon impact on a heated substrate. A contact angle measurement tool (Attension, theta) is also used to measure the static contact angle as well as the surface tension, where negligible differences were observed for the different polymer concentrations (qs ¼ 25 3 , s ¼ 0.07 N m1 0.01 N m1). The range of experimental parameters for the current experiments with polymer solutions 0.5 < Re0 < 45, 0.31 < Bo < 0.37, and 0.05 < We0 < 1.7 were evaluated for spherical droplets leaving the nozzle under ambient conditions (the subscript zero denotes the ambient condition). An example of the images obtained in the impact experiments is shown in Fig. 3 for a non-heated and a heated substrate. As observed in Fig. 3, the substrate temperature has a substantial impact on the dynamics of droplet spreading upon impact with the substrate. In the case of a non-heated substrate, the droplet starts to spread aer the impact, while undergoing vertical vibrations. In the case of a heated substrate, the droplet also starts to spread upon impact, but it soon reaches its maximum footprint on the substrate; the contact line stays pined at this point, and the droplet rapidly approaches an equilibrium state. As apparent in Fig. 3b, the magnitude of droplet vibration and the maximum spreading diameter are reduced compared to the case of a non-heated substrate.
Viscosity of different F127 solutions from cone-and-plate viscometry at controlled shear stress of 0.6 Pa.
Fig. 2
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Fig. 3 Impact and spreading of a Pluronic solution droplet (14 wt%) at Re0 z 25, We0 z 1.5, Bo0 z 0.35 for (a) a substrate at ambient temperature (TS ¼ 23 C, time between images ¼ 4.2 ms), (b) a heated substrate (TS ¼ 65 C, time between images ¼ 1.4 ms). Scale bars are 2 mm.
We applied image processing to the data in Fig. 3 to extract the information on the droplet shape over time in Fig. 4. These curves show that an elevated substrate temperature can lead to a smaller nal droplet diameter on the substrate. For a better understanding, a simple force balance can be provided, where balancing the inertial pressure force with capillary pressure force gives a D*/D0 t1/2 relation,1 however, balancing the
Fig. 4 Comparison of (a) diameter and (b) height of the droplet during
the spreading. The horizontal lines depict the values at equilibrium. For the experimental conditions, see the caption of Fig. 3. The moment of impact is selected as the reference time (t ¼ 0). The inset in picture (a) shows the definition of the geometry parameters. The insets in picture (b) illustrate the spreading mechanism of a thermo-responsive droplet and the iso-temperatures lines near the contact-line.
810 | Soft Matter, 2014, 10, 808–812
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inertial and viscous forces follows the D*/D0 t1/10 rule (Tanner's law1). It is expected that these relationships are respectively valid for the early stage of the impact and from the time on when viscous forces are dominant. These rst relation seems to hold for both experiments early on, while the Tanner law only applies to the case where the substrate temperature is below the gel temperature; in the case of the heated substrate, Tanner's law underestimates the viscous forces. Another evidence for the increased viscosity for the heated substrate can be seen in Fig. 4b, where the oscillations of the droplet's height are highly damped. These observations provide evidence that the sol–gel transition following the impact can be used as an effective method to reduce the droplet spreading. The temperature distribution inside the droplet over time is key to the propagation of the advancing gel front inside the droplet. In fact, the spreading of a droplet depends on the time scales of droplet spreading and temperature propagation where the latter corresponds to gel propagation. These time scales can be described as tsp z D0/U0 and tgel z D02/a, respectively, where a is the heat diffusivity in the liquid. The ratio of these two time scales leads to tgel/tsp ¼ Pe ¼ PrRe, and this value is on the order of 10–100 for our experiments. Here, tgel corresponds to the bulk gelation of a spherical droplet; however, the critical length scale length scale hgel for the arrest of droplets is smaller such that hgel z D0/L, where L is a number larger than 1. This is believed to be partially caused by the geometry of the droplet and the ow eld inside the droplet. An exact constant value of L cannot be evaluated in our experiments, however for the range of substrate temperatures used, this parameter is roughly estimated to be in the order of L ¼ O (4 10). For a better understanding of the temperature distribution, we have also carried out a simple numerical simulation of heat conduction (using nite volume) for a quasi-steady sessile droplet to nd the indicated iso-temperature lines. The inset in Fig. 4b shows a temperature map of the area around the contact line of a droplet (cross section with the droplet surface indicated by the straight line). This simple simulation indicates that the hightemperature region of the liquid near the substrate extends higher into the droplet near the contact line. Since high temperatures in these uids are associated with gel formation we expect gel to start forming near the substrate from where it will form a thicker region near the contact line as shown in the cartoon in Fig. 4b; this mechanism reduces the amount of spreading the droplet experiences before taking on its nal shape. This phenomenon is similar to the deposition of molten droplets onto a substrate, where solidication due to substrate cooling leads to the “arrest of droplets”.12,13 Nonetheless, the nature of the phase change is different; here the latent heat is negligible and the sol–gel transition occurs instantaneously. Fig. 5 shows some of the results obtained for the maximum spreading diameter (x ¼ Dmax/D0) from experiments with different polymer concentrations, substrate temperatures, initial droplet diameters, and impact velocities, where T* ¼ (Ts Tg)/Tg denotes the normalized substrate temperature. Fig. 5 illustrates the capability of the proposed technique to control droplet spreading. The maximum diameter of the
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Fig. 5 Normalized maximum diameter of the spreading droplet as a function of the normalized substrate temperature and the initial Reynolds number, from 40 experiments.
spreading droplet can be controlled with the polymer concentration and the substrate temperature, such that x decreases with increasing polymer concentration and/or substrate temperature. A higher Reynolds number leads to a larger nal diameter of the droplet. In order to obtain a theoretical expression for the maximum diameter of the spreading droplet and the inuence of the so– gel transition, we have developed an energy-balance formulation similar to the one presented in Passandideh-Fard et al.,20 for the normal impact of large droplets
Fig. 6 (a) Comparison of the experimental results with theoretical predictions. Error is defined as |(xexpriment xtheory)/xtheory| 100%. (b) The values of effective Reynolds number (Reeff) for different substrate temperature and initial Reynolds numbers, fitted from 40 experiments.
Fig. 6a. As expected, the values obtained by theory overestimate the experimental data and the deviation of theoretical and experimental results increases with substrate temperature. This is mainly due to the fact that the viscosity of the liquid is changing during the impact and spreading of the droplet, hence an effective Reynolds number (Reeff(Re0, T *) # Re0) should be considered. Where Reeff indicates an equivalent Reynolds number for a non-thermo-responsive liquid satisfying the maximum spreading diameter of a thermo-responsive droplet. Reeff can be obtained by solving eqn (1) for Reeff (one solution exists). Fig. 6b shows the values of Reeff for different concentrations and substrate temperatures, where Reeff decreases with
pD0 2 sLG þ prU0 2 D0 3 =12 þ prgD0 4 =12 |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} SE1
KE1
PE1
ð ts ¼ psLG 3Dmax 2 þ 8 D0 3 =Dmax 3pDmax 2 ðsSL sSG Þ þ ∰ FdV dt þ prgD0 2 =18Dmax 2 ; V 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} SE2
where the SE, KE, and PE denote the surface, kinetic and potential energies. Subscripts 1 and 2 show the initial and nal stages of the spreading, respectively. Additionally, VP denotes the viscous dissipation. Using Young's equation (sLG cos qs ¼ sSL sSG), the rst term on the right hand side can be reduced to psLG[3Dmax2(1 cos qs) + 8(D03/Dmax)]. In the viscous dissipation term (VD), ts ¼ (8/3)(D0/U0), V ¼ (p/4) Dmax2d and F ¼ 2m(U0/d) are spreading time, volume of the viscous layer, and the dissipation function, respectively.20 The viscous boundary layer thickness, d, can be estimated as d ¼ 2D0/Re1/2 using the analytical solution of stagnation point ow. Replacing these parameters in the equation for the energy balance, one can nd sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 þ We þ Bo pffiffiffiffiffiffi x¼ (1) 3ð1 cos qs Þ þ 4We= Re þ 2 Box4 =3 aer some algebra. eqn (1) has two non-trivial and one physical solution for x. A comparison of some of the data obtained in experiments with those achieved from eqn (1) are shown in
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VD
PE2
T * due to higher advection of the temperature and hence faster propagation of the sol–gel interface, and Reeff increases with the initial Reynolds number due to the increasing inuence of the initial momentum. The data points, can be expressed using the empirical equation Reeff ¼ Ce(nT*)Re0m for T * > 0, where C ¼ 0.0100, n ¼ 3.65 and m ¼ 1.96. It worth noting that the parameter T * already includes the inuence of polymer concentration through TG. In conclusion, a novel promising method is presented for controlling the droplet spreading over a horizontal surface, where rapid sol–gel transition of Pluronic 127 due to temperature change is employed to increase the viscous dissipation and hence reduce the maximum spreading diameter. Results show that, for a given Reynolds number, the nal diameter can be tuned with either the substrate temperature or the polymer concentration (T *) as well as the initial Reynolds number (Re0). Results are compared with theory and an effective Reynolds number is introduced. The current results can be used in a wide range of applications, for instance layer by layer droplet/cell printing for 3D structures, or high-resolution printing through control of the nal shape of the droplet. However, the complexity of this problem leaves room for further
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investigations. First of all, an accurate transient hydrodynamics theory combined with heat transfer would allow an exact determination of the droplet interface evolution as well as the propagation of the gel. This can help to identify an exact correlation between the maximum diameter and the experimental parameters (Re, We, Bo, qs, TS, TG). Moreover, ow visualization in experiments and/or proper numerical simulations would allow an accurate representation of the ow dynamics inside the droplet and potentially nding the corresponding values of the parameter L.
Notes and references 1 2 3 4 5 6 7 8
A. L. Yarin, Annu. Rev. Fluid Mech., 2006, 38, 159–192. K. Yokoi, So Matter, 2011, 7, 5120. D. Richard, C. Clanet and D. Qu´ er´ e, Nature, 2002, 417, 811. V. Bergeron, D. Bonn, J. Y. Martin and L. Vovelle, Nature, 2000, 405, 772–775. D. Bartolo, A. Boudaoud, G. Narcy and D. Bonn, Phys. Rev. Lett., 2007, 99, 174502. M. I. Smith and V. Bertola, Phys. Rev. Lett., 2010, 104, 154502. J. Klein, J. Nat., 2000, 405, 745–747. A. Nakajima, NPG Asia Mater., 2011, 3, 49–56.
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9 Y. Son, C. Kim, D. H. Yang and D. J. Ahn, Langmuir, 2008, 24, 2900–2907. 10 S. Schiaffino and A. A. Sonin, Phys. Fluids, 1997, 9, 3172. 11 S. Schiaffino and A. A. Sonin, Phys. Fluids, 1997, 9, 2217. 12 R. Bhola and S. Chandra, J. Mater. Sci., 1999, 34, 4883– 4894. 13 G. Wanka, H. Hoffmann and W. Ulbricht, Macromolecules, 1994, 27, 4145. 14 Y. Shachaf, M. Gonen-Wadmany and D. Seliktar, Biomaterials, 2010, 31, 2836–2847. 15 M. A. Ward and T. K. Georgiou, Polymers, 2011, 3, 1215– 1242. 16 F. D. Jochum and P. Theato, Chem. Soc. Rev., 2013, 42, 7468– 7483. 17 V. Bazargan and B. Stoeber, J. Microelectromech. Syst., 2010, 19, 1079–1087. 18 B. Stoeber, C. M. J. Hu, D. Liepmann and S. J. Muller, Phys. Fluids, 2006, 18, 053103. 19 B. Stoeber, Z. Yang, D. Liepmann and S. J. Muller, J. Microelectromech. Syst., 2005, 14, 207–213. 20 M. Passandideh-Fard, Y. M. Qiao, S. Chandra and J. Mostaghimi, Phys. Fluids, 1996, 8, 650–659.
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Cite this: Soft Matter, 2014, 10, 808
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Controlled spreading of thermo-responsive droplets Maziyar Jalaala and Boris Stoeber*b
Received 16th October 2013 Accepted 27th November 2013 DOI: 10.1039/c3sm52658e www.rsc.org/softmatter
A novel methodology for controlling the spreading of droplets impacting a substrate is presented. The working fluid is a thermoresponsive polymer solution that undergoes a sol–gel transition above a specific temperature. It is shown that the maximum diameter of a droplet at equilibrium can be controlled through the substrate temperature of the substrate and the polymer concentration.
The impact and spreading of liquid droplets on a smooth solid substrate continues to receive much attention in various disciplines, due to its importance for a wide range of applications in nature and industry. Examples include ink-jet printing, rain drop dynamics, fuel combustion, spray cooling, coating, and many others.1 The droplet impact dynamics depend on numerous parameters including the physical properties of the droplet and the surrounding uid, the wetting properties of the solid substrate, the impact velocity, and the droplet size. For a vertical isothermal impact of a droplet on a horizontal substrate (see Fig. 1c) the system dynamics are governed by inertial, viscous, gravitational, and capillary forces. The most relevant non-dimensional parameters to describe this problem are the Weber (We ¼ rD0U02/sLG), Reynolds (Re ¼ rD0U0/m), and Bond (Bo ¼ rgD02/sLG) numbers that depend on the density of the uid r, the initial droplet diameter D0, the impact velocity U0, the liquid–gas surface tension sLG, the viscosity of the droplet m, and the gravitational acceleration g. Additionally, the interfacial condition at the contact line is also essential (at least for low Weber numbers), and the surface properties of the substrate can be taken into account through the static contact angle qs, which ranges from 0 to 180 for superhyrophilic to superhydrophobic surfaces.
In the case of sufficiently high Reynolds and Weber numbers, splashing is observed.2 In contrast, for smaller values of Re and We, spreading occurs. In the spreading regime, the droplet is forced to spread over the solid surface, creating a disclike geometry, until the capillary and/or viscous forces stop further expansion.1,2 The droplet nally relaxes to an equilibrium state, however depending on the wetting properties of the substrate, it might vibrate or rebound.3 The mechanism of spreading has been investigated as a classical problem using different experimental, analytical and numerical techniques.1 However, despite its importance in a wide variety of applications such as coating and printing, only a few studies have been provided to propose a method for controlling the spreading of droplets. These studies mainly focus on using polymer additives that changes the rheological (shear-dependent) properties of the droplet4–7 or on modifying the wetting properties of the substrate.8,9 The goal of the
Department of Mechanical Engineering, The University of British Columbia, 2054 – 6250 Applied Science Lane, Vancouver, BC, V6T 1Z4, Canada. E-mail: mazi@mech. ubc.ca; [email protected]; Fax: +1 604 822 2403; Tel: +1 778 8394459
a
b
Department of Mechanical Engineering, Department of Electrical and Computer Engineering, The University of British Columbia, 2054 – 6250 Applied Science Lane, Vancouver, BC, V6T 1Z4, Canada. E-mail: [email protected]; Fax: +1 604 822 2403; Tel: +1 604 827 5907
808 | Soft Matter, 2014, 10, 808–812
Fig. 1 (a) Schematic view of the experimental setup used to observe the impact of droplets on a heated substrate. (b) Magnified view of the test section. (c) Physical parameters of the current problem.
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current study is to propose a novel technique to control the dynamics of an impacting droplet, in particular in the spreading regime. For this purpose, a thermo-responsive polymer solution is employed. The physics behind the problem is similar to the solidication of a liquid droplet on a cooled surface,10–12 where the advancing contact line is arrested due to the propagation of the solidication front, resulting in partial spreading of the droplet. However, in the current study, an opposite effect (substrate heating) is employed, resulting a sol–gel transition above the gel temperature of the solution. Here, we investigate the inuence of this phasechange behaviour on droplet spreading over a heated substrate in experiments. A schematic view of the experimental setup for investigating the impact of the droplets is illustrated in Fig. 1a and 1b. A syringe pump is used to generate sufficient pressure to produce droplets of a uniform size in the range of 1.4 mm < D0 < 3.8 mm through a stainless-steel needle with a diameter in the range of 0.2 mm < DN < 1.3 mm. The individual droplets fall onto the substrate due to gravity. The distance between the needle tip and the substrate is adjustable, hence, different impact velocities can be obtained with 0.003 m s1 < U0 < 0.25 m s1. The substrate is a clean glass slide placed on a Peltier device (PD) that acts as a heater. The temperature is controlled through the input voltage of the PD, and is monitored, using a thermistor mounted on the glass slide. Moreover, the cold side of the PD is attached to a heat sink to keep it close to the ambient temperature. All the experiments are performed when the substrate has reached a steady temperature. A high speed camera (Phantom, Miro 4.0) with a frame rate of fc ¼ 10 638 frames per second is used to capture images during the droplet-substrate collision. The camera is connected to a stereo microscope, resulting in a eld of view of 6 mm 7 mm and an optical resolution of 256 192 pixels. The impact area is placed between the camera and a high-intensity white light source with a light-diffuser, so that image can be recorded using backlight-shadowgraphy, where the droplet blocks the light path and creates a dark area on the image. The uids in the experiments are solutions of polyethylene oxide–polypropylene oxide–polyethylene oxide (PEOx–PPOy– PEOx), commercially available under the tradename Pluronic (also Poloxamer). Pluronics are symmetric triblock co-polymers and non-ionic surfactants that shows amphiphilic behaviour in aqueous solution with a ratio of hydrophilic to hydrophobic block lengths of (x:y). Beyond the critical micelle concentration (CMC), the co-polymer spontaneously forms micelles (if x:y $ 0.5) with PPO and PEO presenting the core and the corona, respectively.13 Further increase of the temperature or the polymer concentration leads to formation of a so gel, where the viscosity of the solution appears to increase drastically. This sudden viscosity change is reversible and is associated with the gelation of the uid. Due to their unique gelation behaviour and bicompatibility,14,15 Pluronics have been previously used as a smart material for a range of biologyinspired engineering applications including protein delivery, DNA sequencing, and controlled drug delivery.16 In addition, thermo-responsive polymer solutions have been previously
This journal is © The Royal Society of Chemistry 2014
Soft Matter
used for the active and passive control of continuous ow in microchannels.17–19 In the present study, Pluronic BASF© F127 (PEO106–PPO70– PEO106) is used at three different concentrations (11, 14 and 17 wt%) with no further purication and fractionation. The polymer powder is dissolved in distilled water under gentle agitation at low temperature (3 C). Fig. 2 shows the viscosity measurements from a cone-and-plate rheometer at controlled shear stress for the three samples. The gel temperature (TG) is dened as the temperature above which the viscosity of the solution suddenly increases. Below TG a higher polymer concentration corresponds to a higher viscosity, while TG decreases with increasing polymer concentration. It has been shown that these polymer solutions show Newtonian behaviour below TG; near the gel point and above, they exhibit shear-thinning behaviours.17 Here, we focus on the thermothickening properties of these uids for controlling the spreading dynamics of droplets upon impact on a heated substrate. A contact angle measurement tool (Attension, theta) is also used to measure the static contact angle as well as the surface tension, where negligible differences were observed for the different polymer concentrations (qs ¼ 25 3 , s ¼ 0.07 N m1 0.01 N m1). The range of experimental parameters for the current experiments with polymer solutions 0.5 < Re0 < 45, 0.31 < Bo < 0.37, and 0.05 < We0 < 1.7 were evaluated for spherical droplets leaving the nozzle under ambient conditions (the subscript zero denotes the ambient condition). An example of the images obtained in the impact experiments is shown in Fig. 3 for a non-heated and a heated substrate. As observed in Fig. 3, the substrate temperature has a substantial impact on the dynamics of droplet spreading upon impact with the substrate. In the case of a non-heated substrate, the droplet starts to spread aer the impact, while undergoing vertical vibrations. In the case of a heated substrate, the droplet also starts to spread upon impact, but it soon reaches its maximum footprint on the substrate; the contact line stays pined at this point, and the droplet rapidly approaches an equilibrium state. As apparent in Fig. 3b, the magnitude of droplet vibration and the maximum spreading diameter are reduced compared to the case of a non-heated substrate.
Viscosity of different F127 solutions from cone-and-plate viscometry at controlled shear stress of 0.6 Pa.
Fig. 2
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Fig. 3 Impact and spreading of a Pluronic solution droplet (14 wt%) at Re0 z 25, We0 z 1.5, Bo0 z 0.35 for (a) a substrate at ambient temperature (TS ¼ 23 C, time between images ¼ 4.2 ms), (b) a heated substrate (TS ¼ 65 C, time between images ¼ 1.4 ms). Scale bars are 2 mm.
We applied image processing to the data in Fig. 3 to extract the information on the droplet shape over time in Fig. 4. These curves show that an elevated substrate temperature can lead to a smaller nal droplet diameter on the substrate. For a better understanding, a simple force balance can be provided, where balancing the inertial pressure force with capillary pressure force gives a D*/D0 t1/2 relation,1 however, balancing the
Fig. 4 Comparison of (a) diameter and (b) height of the droplet during
the spreading. The horizontal lines depict the values at equilibrium. For the experimental conditions, see the caption of Fig. 3. The moment of impact is selected as the reference time (t ¼ 0). The inset in picture (a) shows the definition of the geometry parameters. The insets in picture (b) illustrate the spreading mechanism of a thermo-responsive droplet and the iso-temperatures lines near the contact-line.
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inertial and viscous forces follows the D*/D0 t1/10 rule (Tanner's law1). It is expected that these relationships are respectively valid for the early stage of the impact and from the time on when viscous forces are dominant. These rst relation seems to hold for both experiments early on, while the Tanner law only applies to the case where the substrate temperature is below the gel temperature; in the case of the heated substrate, Tanner's law underestimates the viscous forces. Another evidence for the increased viscosity for the heated substrate can be seen in Fig. 4b, where the oscillations of the droplet's height are highly damped. These observations provide evidence that the sol–gel transition following the impact can be used as an effective method to reduce the droplet spreading. The temperature distribution inside the droplet over time is key to the propagation of the advancing gel front inside the droplet. In fact, the spreading of a droplet depends on the time scales of droplet spreading and temperature propagation where the latter corresponds to gel propagation. These time scales can be described as tsp z D0/U0 and tgel z D02/a, respectively, where a is the heat diffusivity in the liquid. The ratio of these two time scales leads to tgel/tsp ¼ Pe ¼ PrRe, and this value is on the order of 10–100 for our experiments. Here, tgel corresponds to the bulk gelation of a spherical droplet; however, the critical length scale length scale hgel for the arrest of droplets is smaller such that hgel z D0/L, where L is a number larger than 1. This is believed to be partially caused by the geometry of the droplet and the ow eld inside the droplet. An exact constant value of L cannot be evaluated in our experiments, however for the range of substrate temperatures used, this parameter is roughly estimated to be in the order of L ¼ O (4 10). For a better understanding of the temperature distribution, we have also carried out a simple numerical simulation of heat conduction (using nite volume) for a quasi-steady sessile droplet to nd the indicated iso-temperature lines. The inset in Fig. 4b shows a temperature map of the area around the contact line of a droplet (cross section with the droplet surface indicated by the straight line). This simple simulation indicates that the hightemperature region of the liquid near the substrate extends higher into the droplet near the contact line. Since high temperatures in these uids are associated with gel formation we expect gel to start forming near the substrate from where it will form a thicker region near the contact line as shown in the cartoon in Fig. 4b; this mechanism reduces the amount of spreading the droplet experiences before taking on its nal shape. This phenomenon is similar to the deposition of molten droplets onto a substrate, where solidication due to substrate cooling leads to the “arrest of droplets”.12,13 Nonetheless, the nature of the phase change is different; here the latent heat is negligible and the sol–gel transition occurs instantaneously. Fig. 5 shows some of the results obtained for the maximum spreading diameter (x ¼ Dmax/D0) from experiments with different polymer concentrations, substrate temperatures, initial droplet diameters, and impact velocities, where T* ¼ (Ts Tg)/Tg denotes the normalized substrate temperature. Fig. 5 illustrates the capability of the proposed technique to control droplet spreading. The maximum diameter of the
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Fig. 5 Normalized maximum diameter of the spreading droplet as a function of the normalized substrate temperature and the initial Reynolds number, from 40 experiments.
spreading droplet can be controlled with the polymer concentration and the substrate temperature, such that x decreases with increasing polymer concentration and/or substrate temperature. A higher Reynolds number leads to a larger nal diameter of the droplet. In order to obtain a theoretical expression for the maximum diameter of the spreading droplet and the inuence of the so– gel transition, we have developed an energy-balance formulation similar to the one presented in Passandideh-Fard et al.,20 for the normal impact of large droplets
Fig. 6 (a) Comparison of the experimental results with theoretical predictions. Error is defined as |(xexpriment xtheory)/xtheory| 100%. (b) The values of effective Reynolds number (Reeff) for different substrate temperature and initial Reynolds numbers, fitted from 40 experiments.
Fig. 6a. As expected, the values obtained by theory overestimate the experimental data and the deviation of theoretical and experimental results increases with substrate temperature. This is mainly due to the fact that the viscosity of the liquid is changing during the impact and spreading of the droplet, hence an effective Reynolds number (Reeff(Re0, T *) # Re0) should be considered. Where Reeff indicates an equivalent Reynolds number for a non-thermo-responsive liquid satisfying the maximum spreading diameter of a thermo-responsive droplet. Reeff can be obtained by solving eqn (1) for Reeff (one solution exists). Fig. 6b shows the values of Reeff for different concentrations and substrate temperatures, where Reeff decreases with
pD0 2 sLG þ prU0 2 D0 3 =12 þ prgD0 4 =12 |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} SE1
KE1
PE1
ð ts ¼ psLG 3Dmax 2 þ 8 D0 3 =Dmax 3pDmax 2 ðsSL sSG Þ þ ∰ FdV dt þ prgD0 2 =18Dmax 2 ; V 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} SE2
where the SE, KE, and PE denote the surface, kinetic and potential energies. Subscripts 1 and 2 show the initial and nal stages of the spreading, respectively. Additionally, VP denotes the viscous dissipation. Using Young's equation (sLG cos qs ¼ sSL sSG), the rst term on the right hand side can be reduced to psLG[3Dmax2(1 cos qs) + 8(D03/Dmax)]. In the viscous dissipation term (VD), ts ¼ (8/3)(D0/U0), V ¼ (p/4) Dmax2d and F ¼ 2m(U0/d) are spreading time, volume of the viscous layer, and the dissipation function, respectively.20 The viscous boundary layer thickness, d, can be estimated as d ¼ 2D0/Re1/2 using the analytical solution of stagnation point ow. Replacing these parameters in the equation for the energy balance, one can nd sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 þ We þ Bo pffiffiffiffiffiffi x¼ (1) 3ð1 cos qs Þ þ 4We= Re þ 2 Box4 =3 aer some algebra. eqn (1) has two non-trivial and one physical solution for x. A comparison of some of the data obtained in experiments with those achieved from eqn (1) are shown in
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VD
PE2
T * due to higher advection of the temperature and hence faster propagation of the sol–gel interface, and Reeff increases with the initial Reynolds number due to the increasing inuence of the initial momentum. The data points, can be expressed using the empirical equation Reeff ¼ Ce(nT*)Re0m for T * > 0, where C ¼ 0.0100, n ¼ 3.65 and m ¼ 1.96. It worth noting that the parameter T * already includes the inuence of polymer concentration through TG. In conclusion, a novel promising method is presented for controlling the droplet spreading over a horizontal surface, where rapid sol–gel transition of Pluronic 127 due to temperature change is employed to increase the viscous dissipation and hence reduce the maximum spreading diameter. Results show that, for a given Reynolds number, the nal diameter can be tuned with either the substrate temperature or the polymer concentration (T *) as well as the initial Reynolds number (Re0). Results are compared with theory and an effective Reynolds number is introduced. The current results can be used in a wide range of applications, for instance layer by layer droplet/cell printing for 3D structures, or high-resolution printing through control of the nal shape of the droplet. However, the complexity of this problem leaves room for further
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investigations. First of all, an accurate transient hydrodynamics theory combined with heat transfer would allow an exact determination of the droplet interface evolution as well as the propagation of the gel. This can help to identify an exact correlation between the maximum diameter and the experimental parameters (Re, We, Bo, qs, TS, TG). Moreover, ow visualization in experiments and/or proper numerical simulations would allow an accurate representation of the ow dynamics inside the droplet and potentially nding the corresponding values of the parameter L.
Notes and references 1 2 3 4 5 6 7 8
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