Subscriber access provided by UNIV OF DURHAM
Interface Components: Nanoparticles, Colloids, Emulsions, Surfactants, Proteins, Polymers
Evaporation-Oscillation Driven Assembly: Micro-tailoring the Spatial Ordering of Particles in Sessile Droplets Prasenjit Kabi, Bidyendu Chattopadhyay, Soumyadip Bhattacharya, Swetaprovo Chaudhuri, and Saptarshi Basu Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b02840 • Publication Date (Web): 26 Sep 2018 Downloaded from http://pubs.acs.org on October 3, 2018
Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.
is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.
Page 1 of 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
Evaporation-Oscillation Driven Assembly: Micro-tailoring the Spatial Ordering of Particles in Sessile Droplets Prasenjit Kabia, Bidyendu a,b
Chaudhuri , Saptarshi Basu a
Chattopadhyayc, Soumyadip
Bhattacharyac,
Swetaprovo
a,c*
Interdisciplinary Centre for Energy Research, bDepartment of Aerospace
Engineering, cDepartment of Mechanical Engineering, Indian Institute of Science, Bangalore, Karnataka 560012 Corresponding Author(s) *Emails:
[email protected] KEYWORDS-Self-assembly, oscillation, evaporation, ordering of particles, Voronoi tessellation
Abstract This work explores the physical mechanism that can be used to control the final residual pattern of nanoparticles obtained from an evaporating oscillating sessile droplet. To that end, the substrate is vibrated in the vertical direction with constant amplitude, while the frequency of excitation is varied. It is found that evaporation progressively shifts the mode number of the oscillating droplet to lower values while the oscillations enhance the rate of solvent loss causing a reduction in the droplet lifetime. The coupling between evaporation and oscillation drives the internal flow through two distinct regimes. Initially, oscillation leads to inner flow recirculation which delays the evaporation driven edge deposition of particles. Subsequently at lower modes, caused by solvent depletion, effect of oscillation is weakened which allows evaporation driven flow to gain prominence and thus transport the dispersed particles to the contact line. We demonstrate here how this delay in particle migration can be controlled to engineer morphological changes in not just the resulting macroscopic aspect of the deposit but also its micro-structure. We especially focus on the relatively unexplored microstructural pattern of deposits from oscillating-vibrating droplets. Using scanning electron micro-graph and Voronoi tessellation of the final deposit, we show unique spatial variation in particle ordering at macro-micro length-scales. Thus, droplet oscillation tunes the spatial extent of the particle ordering crucial in applications like photonic crystals and photonic glass. 1 ACS Paragon Plus Environment
Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Introduction Evaporation of a sessile droplet allows micro-scale transport of matter. Any perturbation to the process cascades down the length-scales to modulate the final residual pattern1. It allows simple electronic circuits to be fabricated2-3 outside the precincts of a clean room, design photonic crystals without sophisticated equipments4-7 and provide low-cost solutions to conventional medical diagnostic procedures8-10. The sensitivity of the internal flow field and contact line dynamics of evaporating sessile droplets towards ambient condition, solutesolvent combinations as well as solute morphology can be exploited to custom design surface patterns11. Co-axial stacking of colloidal drops allows vertical assembly in a pre-defined region which could be beneficial to semi-conductor fabrication processes12-13. On the other hand, non-coalescing droplets placed closer than their respective diffusion length scales affect each other’s internal flow field via vapour mediated interactions leading to customizable surface morphologies14. The contact line of a droplet can be reconfigured to morph the droplet curvature. This leads to a hybrid internal flow structure which can effect hierarchical changes in the particle assembly15-16. In this article, periodic perturbation to a sessile evaporating nanofluid droplet is shown to modify the final resulting precipitate at multiple length scales. Oscillation in droplets has been demonstrably used for a myriad range of applications such as high-speed focussing of lens17 and mixing in droplet based on microfluidic devices18. The modal response of an oscillating droplet is suitable for measuring its fluidic properties like viscosity and surface tension19 or the dielectric properties of PDMS20. Vibration based droplet manipulation is essential to microfluid applications21-23 as well bottom-up assembly of functionalized droplets24. Evaporation based modal tuning is known to augment microscale heat transfer for spot cooling of electronic devices25-26. Fundamental studies on vibration in droplets have been pursued since the last two centuries. Rayleigh calculated the natural frequency of an isolated inviscid droplet by balancing surface tension and inertial forces27. Lamb28 and Kelvin29 did the same by balancing surface tension and gravity effect. Studies on oscillation of constrained droplets are comparatively recent. Rodot et al30 were the first to study the oscillation of a drop at the end of a rod immersed in a medium of same density as the drop itself. This was done to counter the effect of gravity in surface oscillation. Sessile drops exhibit higher oscillating frequency for each mode as compared to a free/levitated drop31-32. Sessile droplets also support lower modes of oscillation due to coupling between surface and bulk oscillation modes. More details can be found in the review by Milne et al33. Noblin et al studied the effect of axisymmetric vibrations on a large droplet resting on a substrate34. They demonstrated the transition between stick and slip of the droplet contact line based on the amplitude and frequency of the substrate oscillation. Sharp et al investigated the frequency response of drops of varying viscosity and wetting angles35. Mettu and Chaudhury explained how contact line slip is affected by evaporation modes as well as viscosity effects36. Sanyal and Basu demonstrated modal tuning of droplets via evaporation37. They also provided valuable insights into the internal flow field of oscillating droplet auto-tuned by evaporative mass loss38. 2 ACS Paragon Plus Environment
Page 2 of 26
Page 3 of 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
The present work investigates a colloidal drop on an oscillating silicon substrate. Droplets on hydrophilic substrates tend to display lower modes of resonance when vibrated. Thus, researchers have focussed on hydrophobic substrates37-39. For the current study, the choice of amplitude and frequencies ensure that the droplet’s modal response is similar to its counterpart on a hydrophobic substrate. Figure 1 outlines the article. Due to imposed substrate oscillations of constant amplitude and frequency, stationary waves are generated on the droplet surface. Since the frequency response of the droplet is linked to its shape, evaporative mass loss dampens the initial modal response. Similarly, oscillation enhances evaporation to shorten the droplet lifetime. Evaporation and oscillation are thus locked into a feedback loop where the combined effect drives the internal flow pattern in a two-stage process. Initially, periodic perturbation of the fluid gives rise to counter-rotating vortices and streaming flows. At a later stage, the effect of oscillation is sufficiently diminished, allowing evaporation driven capillary flow to dominate. It is intuitively understood that due to its higher magnitude, oscillatory flow hinders the edge migration of particles. Such a delay can not only modify the microscopic dimensions of the residual pattern but also effect a spatial variation in the ordering of deposited particles. It is known that the macroscopic features of the residue (“coffee-ring” on hydrophilic substrates or buckled dome like structures on hydrophobic substrates) can be modified by oscillating an evaporating droplet39-40. However, to the best of our knowledge, the microstructure of the residue of an oscillating droplet has not been investigated in detail. In this regard, the spatial ordering of particle in the final deposit is characterized. The residues from experimental runs are subjected to SEM imaging. The micrograph is then tessellated using Voronoi scheme to quantify the local variation in particle packing41. Based on the frequency of excitation, it is possible to introduce a spatial change in the way the particles are packed (Figure 1). The article presents a heuristic understanding of the interplay between evaporation, oscillation and flow which can control the ordering of constituent particles in a self-assembled deposit.
3 ACS Paragon Plus Environment
Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Figure 1 Outline of how oscillation in evaporating droplets controls the residual morphology at multiple length scales. Black scale bars equal 5 µm and red scale bars equal 2 µm.
Experimental Section Material preparation Fluorescent latex beads of 860 nm (R900, Thermo-fisher Scientifics) are used for all experiments. The size of the particles must be large enough to optical fidelity in particle tracking (Mie scattering criteria; size of particle must be larger than the wavelength of excitation) yet small enough to avoid sedimentation (sedimentation velocity for a particle size of 860 nm in water is ~0.02 µm/s while the evaporation driven flow in a droplet ~1 µm/s and the oscillation driven flow is 1000 µm/s). They are diluted to 0.0042 % by weight (%wt) before being used for experiments. The diluted dispersion is sonicated for 5 minutes to prevent any flocculation. A silicon wafer is diced into squares of 5 mm2 and etched with HF acid prior to being used as a substrate for experiments.
4 ACS Paragon Plus Environment
Page 4 of 26
Page 5 of 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
Method The silicon substrate is adhesively attached onto a piezo-based disc bender. For each experimental run the disc is sinusoidally stimulated at three distinct frequencies(f)- 2, 2.5 and 3 kHz while the amplitude is fixed at 32+3 µm. A signal generator (BK Precision) is used with a piezo amplifier (Spranktronics, Bangalore) to provide the necessary driving signal to the piezo disc. High-speed images of the substrate motion confirm the frequency of oscillation to be same as the excitation frequency.
Imaging A 1+0.2 µl droplet is gently placed on the silicon substrate using a micropipette. The initial contact angle of the droplet (prior to any imposed vibration) is observed to be 70-80˚ for all experimental runs. The initial contact diameter is 1.7 mm +0.33 mm while the initial apex height of the droplet is 0.65 mm + 0.4 mm. The ambient condition is maintained at 40-45 % RH at a room temperature of 25 ˚C for all experimental runs. Once dispensed on the substrate, the droplet is enclosed to avoid any external disturbance. Backlit high speed images of the side profile are acquired at 10,000 fps using a Photron SA5 camera fitted with a Navitar zoom lens assembly. The top view is imaged in two ways. To visualize the modes of oscillation, high speed images are captured using a Phantom Miro camera fitted on an Olympus BX51 microscope. Both the top and side views are acquired synchronously every 10 s by using a pulse generator. Low frame rate images from top are acquired by replacing the Miro with a SC-30 CCD camera (operated at 10 fps). Fluorescence imaging is used to track the distribution of particles in the presence and absence of vibrations. A Hg lamp attached to the Olympus BX51 is used for fluorescence based (excited at 550 nm and emission at 590 nm) tracking of the particles.
Characterization The final residues obtained from all experimental runs are subjected to SEM and profilometry scans. For SEM, the residue is sputtered with a layer of gold coating to ensure adequate electron conductivity. The electron micrographs are obtained at 3 kV of working voltage using Ultra-55 MonoCL (Zeiss). The detector is positioned at a working distance of 10 mm from the sample surface. The widths of the final precipitate at multiple locations are measured by scanning the residue under a Taylor Hobson profilometer using a 50x objective lens. Post processing is done using CCI Taylor software.
5 ACS Paragon Plus Environment
Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Results and Discussion
Figure 2 Oscillation of a sessile droplet (a) side profile showing deformation of drop shape. The capture instant of the image (ti) is specified as i/f where f=2 kHz. The snapshots correspond to f=2 kHz (b) Synchronized images of the same droplet from top. (c) Superimposed images from different ti’s show nodes of the stationary wave. Since the droplet is axisymmetric, the nodes at any given height from the substrate form a nodal circle (red line). The magnitude of flutter at the droplet’s apex is labelled as ∆h. See Video S1 in Supporting Information. (d) Petal like formations (red dashed circle) observed on the drop surface are numerically equal to half the total number of nodes. Scale bars equal 0.5 mm.
The droplet shape is a spherical cap prior to the imposition of vibration [droplet’s lengthscale (contact diameter=1.7 mm) is smaller than the capillary length of water (~2.7 mm)]. Periodic vibration of the substrate generates a wave near the droplet’s contact line which travels towards its apex as shown in Figure 2(a). At the apex, the generated wave closes in on itself and is reflected in the opposite direction. The interference between the generated and reflected wave leads to formation of a stationary wave on the droplet surface as shown in 6 ACS Paragon Plus Environment
Page 6 of 26
Page 7 of 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
Figure 2(c) by superimposing two of the most deformed drop shapes (Figure 2(a); ti=0/f and 2/f). The point of zero displacement for the surface wave is a node. Owing to the axisymmetric drop shape, node formation on the surface at any given height from the substrate is referred to as a nodal circle (Figure 2(c)). The mode number of oscillation is numerically equal to half the total number of nodes. Figure 2 (d) shows the petal like formation of the droplet for a given n. The mode number (n) of a droplet of is related to its profile length l as where = [( )
]
− 12 =
(1)
obtained from dispersion relation of surface waves in the capillary
regime. Here fs is the frequency of surface waves, γ is the surface tension of liquid (0.072 N/m), ρ is the bulk density of liquid (997 kg/m3). At resonance (fs=fn) the droplet profile length can accommodate an integral number of wavelengths34. The expression for resonant frequency (fn) is given by35,37 =
(
− ) (
)!
(2)
Here r0 is the equatorial radius and θm is the arithmetic mean of the maximum and minimum values of the contact angle during a single cycle of droplet oscillation. For spherical cap geometry, r0 is simply rcsin(θm) where rc is the contact radius of the droplet. The dispersion relation used in the derivation of equation 2 assumes the liquid depth to be infinite. An analytical expression for fn while accounting for the complex three-dimensional wave pattern on a finitely sized droplet has not been achieved yet. Here equation 2 correctly predicts the functional dependency between the resonant frequency and the mass and surface tension of the droplet (as seen from experimental observations). However, an offset is observed between experimental and theoretical values of fn which is corrected by the introduction of the scaling parameter α. Sharp et al used a value of α=0.81 for this purpose35. Sanyal et al39 proposed a value of α=0.86. Evaluating equation 2 for n=6 and f=2000 Hz, the value of α~0.7. To validate α for other values of excitation frequency, equation 1 is re-written as = 12 +
(3)
Substituting l=2r0θm and λ for f=2500 Hz, the initial value of n for a 1µl droplet 6.71~7. Similarly n for f=3000 Hz is 7.82~8. This observation is consistent with the actual values of n shown in Figure 3(a) which validates the choice of α used. The range of α is of the O(1) over a wide range of experimental conditions35,39.The initial mode number of a given mass of droplet can be tuned by sweeping through a range of frequencies. However, with evaporative mass loss, the initial mode number shows gradual decay as predicted by equation 2 and shown in Figure 3a. This is discussed in the next section.
7 ACS Paragon Plus Environment
Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 8 of 26
The role of evaporation in modulating oscillation characteristics The triple line of the droplet is visibly unperturbed by imposed oscillation and retreats only under the effect of evaporative mass loss over the droplet lifetime (tlife). However, the region above the pinned contact line is free to move which is shown in Figure 3(b) (fluctuation of the contact angle is observed to decay over time). Since tlife is greater than 1/f by O (105), any change in drop volume over a cycle of oscillation can be ignored.
Figure 3 (a) Lifetime for each mode number (n) (b) Temporal variation of contact angle fluctuation (∆θ). (Inset) sinusoidal variation of instantaneous contact angle (θ) over a cycle of oscillation. The time axis for both (a) and (b) have been normalized by tlife which is the experimentally observed lifetime of an oscillating droplet at a given f. The decay in mode number visualized from top for (c) f= 2 kHz (d) f=2.5 kHz and (e) 3 kHz. See Video S1 in Supporting Information. Scale bars equal 0.5 mm.
The mode number as shown in Figure 3(a) and (c) reduces with progression of evaporation. Solvent depletion causes the profile length of the droplet to reduce. Since ∝ , it must reduce as well. Thus nth node merges with the (n-1)th node. The link between mass depletion and modal downshift is better understood as follows. The volume of sessile water droplet can be expressed as $ =
! (%&' ()
− 3 cos () + 2). Assuming diffusion limited evaporation, the
rate of solvent loss is expressed as42
/ 02 = −4456 78 ∆%(() ) 01
(4)
Here Da is the diffusion coefficient of water vapour into air, ∆c is the difference of vapour concentrations near and far from the droplet surface, f(θm) has the form42
8 ACS Paragon Plus Environment
Page 9 of 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
(() ) = => (@:;< :;<
)
(5)
Differentiating V with respect to t we obtain A56
+
0 02
0B 02
=−
CD ∆E( )
(6)
Here g(θm)= (%&' () − 3 cos () + 2). Assuming solvent depletes from the droplet via
constant contact angle (CCA) mode of evaporation43 , 56
0 02
=−
dimensional tuning parameter37 H =
CD∆EG( )
0B 02
= 0. Thus (7)
56 can be used to obtain the modal lifetime
Here k(θm)=f(θm)/g(θm). This can be integrated wrt t to get an expression for r0. A non !
which is defined as the time taken for any given mode number to reduce by 1. Substituting r0 obtained after integrating equation 7 in the expression for Ω and using equation 2, we obtain an expression for modal lifetime I = [
J @ ] CD ∆EG( )
(8)
For constant contact radius (CCR) mode of evaporation43, substituting () = sin@( M ) in
equation 6.
P M OG( ) 56 + ( O N
Integrating this equation we get
)
(QR )!
@ M T S
W 0 47X ∆% V 02 = − / V
(9)
U
\ b [ 3a 1 5 ('_(^ ) a c50 = − CD ∆E (d − d6 ) Y [](( ) 50 + ((% ) [ 2 ^ ^ a 5 1− %T 2 S 50 a [ Z `
(10)
Here r0(t0)=r00. It is evident from equation 10, that an analytical expression for modal lifetime similar to the one obtained for CCA mode of evaporation is difficult to derive. Moreover, as shown in Fig 4, droplets on silicon substrates (vibrating as well as non-vibrating) undergo a CCR in the initial stage and mixed mode of evaporation43 in the later stage which makes the derivation of modal lifetime even more complex. Nonetheless, the actual modal lifetime must lie between the extreme cases of CCR and CCA. We have evaluated τn for different frequencies using equation 8.The value of θm and r0 at the beginning of a modal cycle is used 9 ACS Paragon Plus Environment
Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
for this computation. We used γ =0.072 N/m, ρ =997 kg/m3, α=0.7, Da=2.54x10-5 m2/s,∆c=12.65x10-3 kg/m3. The estimated value of τn,CCA is presented along with the experimental values(τn,exp) in Table 1.
Table 1: Comparison of estimated modal lifetime assuming CCA mode of evaporation (τn,CCA) with experimental values (τn,CCA ). The lifetimes are presented in seconds (s).
The estimated modal lifetime is smaller at lower value of n. Higher modes seem to drive out the lower modes as previously reported by both Mettu and Chaudhury36 as well as Sanyal and Basu37. Defining T>,ggh = ∑ τ>,ggh, this quantity is presented in the penultimate column of Table 1. The last column is the experimental lifetime of the droplet. Since modal downshift occurs on account of evaporation, Tn,CCA is approximately equal to the lifetime of the droplet in CCA mode. It is well known that the rate of evaporation in CCA mode is always lower than the CCR mode of evaporation44, This is due to the fact that a given volume of droplet evolves towards a higher surface area in CCR mode as compared to the CCA mode of evaporation. This explains the difference between experimental and estimated droplet lifetimes.
How oscillation tunes the dynamics of evaporation
10 ACS Paragon Plus Environment
Page 10 of 26
Page 11 of 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
Figure 4 Modes of evaporation of a sessile oscillating droplet shown in presence as well as in absence of substrate oscillation (a) Illustration of surface area enhancement of an oscillating droplet. The red dashed line represents the equilibrium profile of the droplet of spherical radius r0 which is augmented by δr due to oscillation (b) Variation of mean contact angle (θm/θ0) with evaporation .θ0 is the initial mean contact angle (prior to vibration). (c) Contact line slip plotted as contact diameter (Dc/Dc0) where Dc0 is the initial contact diameter. (d) Comparison between estimated and experimental values of droplet lifetime (tlife) at different frequencies.
An oscillating droplet is illustrated in Figure 4(a). The amplitude of the surface wave is considered to be δr. Thus the effective radius of curvature for an oscillating droplet is r0+ δr. The resulting surface area enhancement leads to higher rate of evaporation causing the droplet lifetime to shorten. For the sake of this discussion, equation 4 is re-written as15 0) 02
= −/kl = −{n∆%}7X 2456 (() )
(11)
Picknett and Benson45 has originally proposed the form of (() ) = 0.00008957 + 0.633() + 0.116() − 0.0887() + 0.01033() for 10° < ( < 180° . Here J is the evaporation flux from the droplet of surface area A and M is the molar mass of water. For simplicity, the droplet lifetime (tlife) can be estimated by dividing the initial mass of the droplet (m0) by the initial rate of solvent depletion (w equation (11). Thus the droplet lifetime is given by
dzR{ =
) | w x |} }~
0)
x
02 2y6
) obtained from evaluating
(12)
Figure 4(b) and (c) present the mode of evaporation of the droplet. Till t/tlife=0.25, the droplet (for f=0 and f>0) seem to follow the CCR mode of evaporation. Thereafter, the droplet follows the mixed mode of evaporation43. To evaluate equation 12 in case of f>0, we need to compute the magnitude of δr. This can be expressed as 5 = ∑ X (cos )
(13)
Here n is the mode number, an and Pn are associated with the nth mode of the droplet39 oscillation. The value of ψ varies between + θm. We choose39 an~∆h. The polynomial Pn can
be evaluated using the form =
!
( − 1)@G ( + 1)G where x=cos(φ).
G!(@G)!
As explained previously, the droplet lifetime may be estimated by using the initial value of the solvent depletion rate. For an oscillating droplet, evaporation remains a diffusion driven phenomenon since the evaporation lifetime is O(105) times higher than the time-period of interface oscillation. Thus the initial rate of evaporation is evaluated for the time-averaged (over a cycle of oscillation) enhanced surface area of the droplet (due to stationary wave pattern; Figure 4a). The value of δr is obtained from equation 13 by using the initial value of
11 ACS Paragon Plus Environment
Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 12 of 26
n. By replacing r0 with r0+δr in equation 11, we get the enhanced rate of depletionw
0)
x
02 2y6
.
Subsequently tlife is estimated by using equation 12. Both the estimated as well as the actual lifetime of the droplet is shown (Figure 4(d)) to decrease as f which demonstrates the role of oscillation in enhancing the rate of evaporation. Note that both the initial evaporation rate and the mode number of the droplet evolve over the lifetime of the droplet. Thus estimation of the lifetime (evaluating equation 12) is bound to deviate from the actual value. The oscillating drop could also be cast as a classical Stefan problem of blowing over a liquid surface. The disruption of the boundary layer due to interface pulsation can lead to faster evaporation. However, this is speculative and needs rigorous experiments to measure the instantaneous boundary layer disruption. The estimation of tlife from equation 12 is different from Tn,CCA since the former does not assume any particular mode of evaporation and depends on the initial geometry of the droplet.
Internal flow field of an oscillating-evaporating droplet The evaporation driven flow velocity at a radial location a can be expressed as41 {8 =
√CD (@) J M ( M @8)
(14)
Here a is any radial location smaller than rc while Cs is the vapour concentration near the droplet surface taken to be 23 g/m3 and RH is the ambient humidity taken to be 0.45. The magnitude of uevap evaluated near the contact edge (a=0.1rc) for all values of f varies between 0.2 µm/s to 2 µm/s.
12 ACS Paragon Plus Environment
Page 13 of 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
Figure 5 Internal flow field (a) Schematic to illustrate the internal flow structures inside an oscillating droplet. Bold red arrows depict upward drift, dashed black arrow depict downward drift, green arrows show rotating vortices on the surface blue, dashed red line shows plane of segregation and dashed blue line is the confluence of up and down drift. (b) shows the circulation near the contact line of a droplet oscillated at 3 kHz. See Video S3 in Supporting Information No such circulation is observed for f=0. (c) side profile of the droplet showing streak lines depicting oscillation driven flow. See Video S2 in Supporting Information. (d) magnified view of the side profile (top image) shows flow segregation as regions where the streaks lines terminate abruptly. (bottom image) the confluence between the down and the up drifts. See Video S4 in Supporting Information. Black scale bars equal 10 µm and red scale bars equal 100 µm.
As illustrated in Figure 5(a), a mechanically oscillating droplet displays various patterns of internal flow. The free oscillating surface of a droplet leads to the formation of vortices46 which progress inwards from the apex towards the base (See Video S2 in Supporting
13 ACS Paragon Plus Environment
Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 14 of 26
Information). For a mechanically oscillating droplet, perturbation of the droplet free surface as well as the solid substrate lead to opposing flow structures as discussed next.
Perturbation of the fluid domain maybe expressed in terms of the stream function47 = 6 + + ( ). The base flow (φ0) has a zero mean and limited to the Stokes layer (δ) is given as =
ѡ
where ν is the kinematic viscosity of water and ѡ is the angular frequency of
oscillation (2πf). A characteristic Reynold’s number defined as Q =
ѡ
determines the
streaming of the flow (φ1) beyond the Stokes layer. Here U is the velocity of the interface given as = ѡ∆ where ∆x is the amplitude of perturbation. If the Rs>>1, then the typical lengthscale of the streaming vortices is Q ~/ε. This is better known as the Stuart lengthscale48. Here ε =
ѡ
. For Rs0, uevap is disrupted by uosc (Figure 7(a) at t/tlife=0.4) leading to negligible deposition. However, at this stage the particle distribution at the centre is still uniform. At later stages shown in Figure 7(b) and 7(c), uevap keeps depleting the droplet interior of particles for f=0 which causes the I/Im curves for f=0 to lower continuously and thus deviate from the curves of f>0. Near the very end of droplet lifetime (t/tlife=0.8) as shown in Figure 7(d), I/Im curves are lowered for all frequencies in the droplet interior which proves that uevap has transported most particles to the edge.
Figure 8 (a) Growth of edge deposit (W) normalized by the final width of the precipitate (Wd). (Inset) For clarity, the curve for W/Wd towards the end of droplet lifetime (t/tlife=0.8 to1) is shown separately. (b) Width (Wd) of the final residue is measured using optical profilometry and plotted for each frequency of excitation.
Oscillation disrupts the process of particle deposition near the contact edge. Figure 8(a) shows the growth of edge deposit (W/Wd) over droplet lifetime where Wd is the final width of the resulting precipitate’s edge. The inception of the edge deposit is evident from the very beginning while it is negligibly present for f>0 as shown in Figure 8(a). Beyond t/tlife>0.8, there is a sudden increase in the deposit growth for f>0 at a rate comparable to the case of f=0. This is understandable since in absence of oscillatory disruption, the magnitude of uevap is found to be of the same order for all cases. 17 ACS Paragon Plus Environment
Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Figure 8(b) shows the variation of Wd with respect to f. The particle migration happens through a wedge like region near the contact line which continuously diminishes on account of evaporation. Delay in edge migration for f>0 causes particles to accumulate in the droplet interior causing the number concentration of particles to increase. For the same wedge area, the number of particles migrating towards the contact line is higher for f>0. A large fraction of these particles are near the solid substrate where the flow is slower due to the no-slip condition. This leads to premature deposition of the particle away from the edge as shown by the higher value of Wd for f>0 (Figure 8(b). At higher f, the delay in edge migration is increased which leads to larger value of Wd. Thus, oscillation of droplet provides a simple method of tuning the thickness of the edge deposit by simply changing the frequency of oscillation.
Modulation of deposit microstructure
Figure 9 (a) SEM of final residue. (b) Magnified view of residue near the contact line. Highly magnified view of the deposit corresponding to (c) f=0 kHz (d) f=2 kHz (e) f=2.5 kHz (f) f=3 kHz. Black scale bar equals 0.5 mm, red scale bar equals 10 µm and blue scale bar equals 4 µm.
Figure 9 shows the SEM of the final precipitate. The dispersed particles are transported to the edge due to uevap. They form a thin annular ring like structure as shown in Figure 9(a) and 9(b). Due to the interplay between uevap and uosc, morphological variations are introduced in the deposit microstructure as shown in Figure 9(c-f). Marin et al41 have shown how temporal variation in uevap near the droplet’s contact line influences the spatial ordering of particle packing. For a non-vibrating droplet, particles are transported towards the edge due to uevap. In the initial stage of evaporation, the value of uevap is low and particles settle into the edge region in an orderly fashion. Towards the end of the droplet’s lifetime, uevap increases rapidly allowing particles less time for ordered arrangement. 18 ACS Paragon Plus Environment
Page 18 of 26
Page 19 of 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
Marin et al41 also proposed a threshold rate of transport (uc) beyond which any increase in uevap would lead to increasingly disordered particle assemblies. The critical transport rate is given as E =
0
(15)
where K is a distance parameter inversely proportional to the concentration of particles (s) in the liquid phase of the droplet (¡ ∝ 1'), E is the Stokes-Einstein coefficient and dp is the particle size. As discussed previously, owing to the delay in particle deposition due to oscillation, s increases rapidly for f>0. Larger s leads to smaller K resulting in a lower value of uc. Since uevap is known to increase monotonically with evaporation, it exceeds uc at an earlier stage of evaporation for f>0 as compared to the case of f=0. Thus the ordering of particles for f=0 kHz (Figure 9(c)) is more homogenously ordered with fewer defects. For f=2 kHz, the particle deposit is well packed closer to the edge but shows more defects away from it (Figure 9(d)). For f=2.5 kHz (Figure 9(e)) and f=3 kHz (Figure 9(f)), higher number of defects are observed in the final precipitate. The SEM images of Figure 9(c-f) also suggest a higher deposit height at higher values of f. Vertical stacking could be a combined effect of the increased particle number density as well the prolonged flow circulation. A complete mechanistic description of particle deposition in evaporating oscillating droplets is not possible with the present state-of-art technology.
19 ACS Paragon Plus Environment
Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Figure 10 (a) Probability distribution function (PDF) of particle packing extracted from Voronoi tessellation of SEM image (only top layer considered). Black dashed line shows the theoretical maximum of the packing fraction φ. The distribution of particles for φ>0.6 is due to the inherent polydispersity in particle size as well the presence of out-of-plane particles in the particle matrix used for Voronoi(Inset) shows the full width at half maxima (FWHM) for all curves (b) SEM of edge deposit obtained from f=2 kHz illustrating a hypothetical segregation (red dashed line) of inner and outer region (c) PDF for inner and outer region of the deposit shown in (b) (Inset) Corresponding FWHM values. The region is indicated in the subscript; FWHM-i is for the inner region and FHHM-o is for the outer region. Scale bar equals 2 µm.
SEM images are acquired at multiple locations along the deposit. Only the topmost layer of the deposit visible in each image is subjected to Voronoi tessellation (to the extent possible). The deposit regions are partitioned on the basis of centre-centre distance between neighbouring particles. Thus each particle in a given image is bounded by a cell. The area of the cell (Abox) is determined the particle’s proximity to its nearest neighbour. Although this approach has been used earlier41,51 in a qualitative manner to describe particle arrangement, we attempt a quantitative approach towards the same. The particle packing fraction φ=Vp/(Aboxdp) is calculated for each particle where Vp is the volume of each particle of size dp. This value of φ pertains to the topmost layer of the deposit and is a good indicator of the particle arrangement in a given plane of particle. It is known41,50 that for a close packed assembly of mono-dispersed spheres, the shape of the bounding cell is a regular hexagon. Inscribing a sphere inside a regular sized hexagon having a depth of one dp, the value of φ turns out to be ~0.6. For a two-dimensional array of closely packed monodispersed spheres, the maximum possible packing fraction is ~0.6. For the same spheres in a three-dimensional array, the maximum value of packing fraction is ~0.74. The values of φ obtained from Voronoi tessellations are approximated using a non-parametric distribution function52 for each value of f using MATLABR toolbox and shown in Figure 10(a). It seems a minor population of particles have φ values between 0.6 to 0.8. This could be a result of small size of the Voronoi bounding cell arising as a result of inter-particle distances smaller than those expected of ideally packed spheres in a planar section. This is explained as follows. Polydispersity in particle size distribution may lead to the presence of smaller particles in a given lattice of larger particles resulting in lower inter-particle distances. Additionally, some particles in the given matrix of particles may be vertically (normal to the plane of the SEM image) displaced. The projected value of the inter-particle distance between the out-of-plane particle and its immediate neighbour is thus lower. The curvature of the deposit can lead to smaller value of projected inter-particle distances. Since, all these factors can lead to smaller size of bounding cell during Voronoi tessellation, the value of φ exceeds the theoretical maximum limit for some particles in the deposit matrix. The height of the PDF (Probability Distribution Function) indicates the modality of the packing fraction while the FWHM (Full Width at Half Maxima) indicates the spread in φ. The mode value of φ distribution indicates the most common or frequently occurring configuration in which the particles are packed in the deposit matrix. The spread in the distribution of φ is primarily due to the presence of defects (cracks or fissures in the matrix). The PDF for f=0 kHz indicates well packed particles with the mode φ close to 0.6 and a small FWHM. This is evident from the deposit section shown in Figure 9(c). For f=2 kHz, mode φ 20 ACS Paragon Plus Environment
Page 20 of 26
Page 21 of 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
is shifted away from φ=0.6 with a larger FWHM. For higher values of f (2.5 kHz and 3 kHz), the FWHM are significantly larger. Thus, the ordering of the particles is correlated to the value of excitation frequency; higher the oscillation frequency lower is the efficiency with which particles are packed. We further divide the deposit into two arbitrary regions as shown in Figure 10 (b). The first 5 rows of particles are labelled as outer region ‘o’ while the rest are labelled as inner region ‘i’. The non-parametric PDFs for both inner and outer regions are generated separately and shown in Figure 10 (c). For f=0 kHz, the pair of mode values for the inner and outer regions are close while the FWHM pairs are almost equal indicating that both these regions are well packed and there is no sharp spatial change in particle ordering. For f=2 kHz and f=2.5 kHz, the mode pairs as well as the FWHM pairs are distinctly different. This indicates that the outer regions of the corresponding deposits are more closely packed than the inner region and there is a sharp contrast between the two. For f=3 kHz, the mode pairs and the FWHM pairs are again closer to each other. However, the height of the PDF as well as the individual value of the FWHM for f=3 kHz indicates that the particle packing in the deposits is loose albeit homogenously packed. It is well known that creating highly ordered clusters of particles is as difficult as creating highly disordered clusters of particles53. While ordered colloidal assemblies form photonic crystals, the disordered assemblies form photonic glass53. Randomness in colloidal assemblies can also be utilized fabricating physically unclonable cryotographs54. Here we demonstrate the flexibility with which one can control and fabricate selective regions of both crystal and glass within the same deposit assembly.
Conclusion We investigate the effect of oscillations imparted to a sessile droplet placed on a hydrophilic substrate at different frequencies. The case of f=0 is chosen as the base condition. The substrate is oscillated at 2 kHz, 2.5 kHz and 3 kHz at fixed amplitude of 32 µm. Evaporation of drop drives the modal response of the droplet oscillation to lower values. Oscillation affects the contact line dynamics of the droplet as well as the evaporation rate, resulting in shorter droplet lifetimes at higher excitation frequencies. Flow assisted particle deposition is affected by the coupling between oscillation and evaporation. During the initial stage of droplet lifetime, bulk circulation caused by substrate vibration delays the capillary transport of particles towards the edge. However, enhanced evaporation causes the number density of dispersed particles to increase. At a later stage, when evaporative mass loss has dampened the droplet oscillation, circulations give way to radially outward flow. However, the packet of particle being transported towards the edge is more concentrated and is forced to travel through a smaller wedge on account of reduced drop size causing particles to settle farther away from the contact line. The final width of the deposit is observed to be larger for higher value of f. Higher particle concentration also leads to a lower threshold of critical velocity uc which determines the order to disorder transition in colloidal self-assemblies. Since the transport rate uevap approaches and uc at progressively earlier stage for higher values of f, larger regions of the final deposit show less ordered particle assemblies. By using a 21 ACS Paragon Plus Environment
Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
combination of SEM and Voronoi scheme of image tessellation, it is statistically demonstrated that the spatial ordering of deposited particles can be controlled at a micrometric scale by simply tuning the oscillation characteristics of a sessile nanofluid droplet. We present clear evidence of vibration assisted tuning of residual pattern from evaporating colloidal droplets. Although, the macroscale modulation has been reported before, very few studies exist on how to modulate the microstructure. This maybe crucial to a host of applications that desire chemically non-intrusive techniques to control the packing format in a simple and inexpensive manner.
Supporting Information Evaporation driven tuning of droplet’s oscillation mode shown for its total lifetime (Video S1). Side profile shows internal flow circulations for f=2 kHz (Video S2). Topview shows the effect of oscillation in causing circulatory flow near the contact line (Video S3). The competition between upward and downward drift (Video S4). Fluorescent images of the droplet showing particle distribution (Video S5).
AUTHOR INFORMATION Corresponding Author(s) *Email:
[email protected] ORCID Saptarshi Basu: 0000-0002-9652-9966 Prasenjit Kabi: 0000-0002-3143-101X Notes The authors declare no competing financial interests.
References (1) Bansal, L.; Sanyal, A.; Kabi, P.; Pathak, B.; Basu, S. Engineering Interfacial Processes At Mini-Micro-Nano Scales Using Sessile Droplet Architecture. Langmuir 2018, 34, 84238442 .
22 ACS Paragon Plus Environment
Page 22 of 26
Page 23 of 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
(2) Sirringhaus, H.; Kawase, T.; Friend, R.; Shimoda, T.; Inbasekaran, M.; Wu, W.; Woo, E. High-Resolution Inkjet Printing Of All-Polymer Transistor Circuits. Science 2000, 290, 2123-2126. (3) Soltman, D.; Subramanian, V. Inkjet-Printed Line Morphologies And Temperature Control Of The Coffee Ring Effect. Langmuir 2008, 24, 2224-2231. (4) Sowade, E.; Blaudeck, T.; Baumann, R. Self-Assembly Of Spherical Colloidal Photonic Crystals Inside Inkjet-Printed Droplets. Crystal Growth & Design 2016, 16, 1017-1026. (5) Kim, S.; Lee, S.; Yang, S.; Yi, G. Self-Assembled Colloidal Structures For Photonics. NPG Asia Materials 2011, 3, 25-33. (6) Li, Y.; Zhou, X.; Yang, Q.; Li, Y.; Li, W.; Li, H.; Chen, S.; Li, M.; Song, Y. Patterned Photonic Crystals For Hiding Information. Journal of Materials Chemistry C 2017, 5, 46214628. (7) Wang, D.; Park, M.; Park, J.; Moon, J. Optical Properties Of Single Droplet Of Photonic Crystal Assembled By Ink-Jet Printing. Applied Physics Letters 2005, 86, 241114. (8) Jung; Kwak, H. Separation Of Microparticles And Biological Cells Inside An Evaporating Droplet Using Dielectrophoresis. Analytical Chemistry 2007, 79, 5087-5092. (9) Guo, M.; Rotem, A.; Heyman, J.; Weitz, D. Droplet Microfluidics For High-Throughput Biological Assays. Lab on a Chip 2012, 12, 2146. (10) Brutin, D.; Sobac, B.; Loquet, B.; Sampol, J. Pattern formation in drying drops of blood. Journal of Fluid Mechanics 2011, 667, 85-95. (11) Sefiane, K. Patterns From Drying Drops. Advances in Colloid and Interface Science 2014, 206, 372-381. (12) Kabi, P.; Basu, S.; Sanyal, A.; Chaudhuri, S. Precision Stacking Of Nanoparticle Laden Sessile Droplets To Control Solute Deposit Morphology. Applied Physics Letters 2015, 106, 063101. (13) Kabi, P.; Basu, S.; Chaudhuri, S. Deployment Strategy For Controlled Morphologies In Sessile, Mixed Colloidal Droplets. RSC Advances 2015, 5, 89586-89593. (14) Shaikeea, A.; Basu, S.; Hatte, S.; Bansal, L. Insights Into Vapor-Mediated Interactions In A Nanocolloidal Droplet System: Evaporation Dynamics And Affects On Self-Assembly Topologies On Macro- To Microscales. Langmuir 2016, 32, 10334-10343. (15) Kabi, P.; Chaudhuri, S.; Basu, S. Insights Into Drying Of Noncircular Sessile Nanofluid Droplets Toward Multiscale Surface Patterning Using A Wall-Less Confinement Architecture. Langmuir 2016, 32, 10977-10986. (16) Kabi, P.; Chaudhuri, S.; Basu, S. Micro To Nanoscale Engineering Of Surface Precipitates Using Reconfigurable Contact Lines. Langmuir 2018, 34, 2109-2120. 23 ACS Paragon Plus Environment
Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 24 of 26
(17) López, C.; Hirsa, A. Fast Focusing Using A Pinned-Contact Oscillating Liquid Lens. Nature Photonics 2008, 2, 610-613. (18) Mugele, F.; Baret, J.; Steinhauser, D. Microfluidic Mixing Through ElectrowettingInduced Droplet Oscillations. Applied Physics Letters 2006, 88, 204106. (19) Egry, I.; Giffard, H.; Schneider, S. The Oscillating Revisited. Measurement Science and Technology 2005, 16, 426-431.
Drop
Technique
(20) Feng, J.; Zhao, Y. Experimental Observation Of Electrical Instability Of Droplets On Dielectric Layer. Journal of Physics D: Applied Physics 2008, 41, 052004. (21) Zhu, P.; Tang, X.; Tian, Y.; Wang, L. Pinch-Off Of Microfluidic Droplets With Oscillatory Velocity Of Inner Phase Flow. Scientific Reports 2016, 6. (22) Daniel, S.; Chaudhury, M.; de Gennes, P. Vibration-Actuated Drop Motion On Surfaces For Batch Microfluidic Processes. Langmuir 2005, 21, 4240-4248. (23) Sartori, P.; Quagliati, D.; Varagnolo, S.; Pierno, M.; Mistura, G.; Magaletti, F.; Casciola, C. Drop Motion Induced By Vertical Vibrations. New Journal of Physics 2015, 17, 113017. (24) Sekeroglu, K.; Gurkan, U.; Demirci, U.; Demirel, M. Transport Of A Soft Cargo On A Nanoscale Ratchet. Applied Physics Letters 2011, 99, 063703. (25) Chakraborty, M.; Ghosh, A.; DasGupta, S. Enhanced Microcooling By Electrically Induced Droplet Oscillation. RSC Adv. 2014, 4, 1074-1082. (26) Alavi, S.; Kazemi, A.; Passandideh-Fard, M. Hot-Spot Cooling Using Microliter Liquid Drops. Applied Thermal Engineering 2015, 76, 310-323. (27) Rayleigh, L. On The Capillary Phenomenon Of Jets. Proceedings of Royal Society 1879, 29, 71. (28) Lamb, H. On The Oscillations Of A Viscous Spheroid. Proceedings of the London Mathematical Society 1881, s1-13, 51-70. (29) Kelvin, W. Mathematical And Physical Papers; C.J. Clay: London, 1890. (30) Rodot, H.; Bisch, C.; Lasek, A. Zero-Gravity Simulation Of Liquids In Contact With A Solid Surface. Acta Astronautica 1979, 6, 1083-1092. (31) Bauer, H.; Chiba, M. Oscillations Of Captured Spherical Drop Of Viscous Liquid. Journal of Sound and Vibration 2005, 285, 51-71. (32) Strani, M.; Sabetta, F. Free Vibrations Of A Drop In Partial Contact With A Solid Support. Journal of Fluid Mechanics 1984, 141, 233. (33) Milne, A.; Defez, B.; Cabrerizo-Vílchez, M.; Amirfazli, A. Understanding (Sessile/Constrained) Bubble And Drop Oscillations. Advances in Colloid and Interface Science 2014, 203, 22-36. 24 ACS Paragon Plus Environment
Page 25 of 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
(34) Noblin, X.; Buguin, A.; Brochard-Wyart, F. Vibrated Sessile Drops: Transition Between Pinned And Mobile Contact Line Oscillations. The European Physical Journal E 2004, 14, 395-404. (35) Sharp, J. Resonant Properties Of Sessile Droplets; Contact Angle Dependence Of The Resonant Frequency And Width In Glycerol/Water Mixtures. Soft Matter 2012, 8, 399-407. (36) Mettu, S.; Chaudhury, M. Vibration Spectroscopy Of A Sessile Drop And Its Contact Line. Langmuir 2012, 28, 14100-14106. (37) Sanyal, A.; Basu, S. How Natural Evaporation Temporally Self-Tunes An Oscillating Sessile Droplet To Resonate At Different Modes. Langmuir 2016, 32, 4784-4791. (38) Sanyal, A.; Basu, S. Evolution Of Internal Flows In Mechanically Oscillating Sessile Droplets Undergoing Evaporation. Chemical Engineering Science 2017, 163, 179-188. (39) Sanyal, A.; Basu, S.; Chaudhuri, S. Controlling Particle Deposit Morphologies In Drying Nano-Particle Laden Sessile Droplets Using Substrate Oscillations. Physical Chemistry Chemical Physics 2016, 18, 14549-14560. (40) Mampallil, D.; Reboud, J.; Wilson, R.; Wylie, D.; Klug, D.; Cooper, J. Acoustic Suppression Of The Coffee-Ring Effect. Soft Matter 2015, 11, 7207-7213. (41) Marín, Á.; Gelderblom, H.; Lohse, D.; Snoeijer, J. Order-To-Disorder Transition In Ring-Shaped Colloidal Stains. Physical Review Letters 2011, 107. (42) Erbil, H. Evaporation Of Pure Liquid Sessile And Spherical Suspended Drops: A Review. Advances in Colloid and Interface Science 2012, 170, 67-86. (43) Zhang, X.; Wang, J.; Bao, L.; Dietrich, E.; van der Veen, R.; Peng, S.; Friend, J.; Zandvliet, H.; Yeo, L.; Lohse, D. Mixed Mode Of Dissolving Immersed Nanodroplets At A Solid–Water Interface. Soft Matter 2015, 11, 1889-1900. (44) Shaikeea, A.; Basu, S.; Tyagi, A.; Sharma, S.; Hans, R.; Bansal, L. Universal Representations Of Evaporation Modes In Sessile Droplets. PLOS ONE 2017, 12, 1-8. (45) Picknett, R.; Bexon, R. The Evaporation Of Sessile Or Pendant Drops In Still Air. Journal of Colloid and Interface Science 1977, 61, 336-350. (46) Batchelor, G. An Introduction To Fluid Dynamics; Cambridge University Press: Cambridge, 2013. (47) Riley, N. Steady Streaming. Annual Review of Fluid Mechanics 2001, 33, 43-65. (48) Theisen, J.; Davoust, L. Mass Transfer Enhancement And Surface Functionalization In Digital Microfluidics Using AC Electrowetting: The Smaller, The Better. Microfluidics and Nanofluidics 2014, 18, 1373-1389.
25 ACS Paragon Plus Environment
Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
(49) Kloosterman, A.; Poelma, C.; Westerweel, J. Flow Rate Estimation In Large Depth-OfField Micro-PIV. Experiments in Fluids 2010, 50, 1587-1599. (50) Destgeer, G.; Cho, H.; Ha, B.; Jung, J.; Park, J.; Sung, H. Acoustofluidic Particle Manipulation Inside A Sessile Droplet: Four Distinct Regimes Of Particle Concentration. Lab on a Chip 2016, 16, 660-667. (51) Wasan, D.; Nikolov, A. Spreading Of Nanofluids On Solids. Nature 2003, 423, 156159. (52) Raju, L.; Chakraborty, S.; Pathak, B.; Basu, S. Controlling Self-Assembly And Topology At Micro–Nano Length Scales Using A Contact-Free Mixed Nanocolloid Droplet Architecture. Langmuir 2018, 34, 5323-5333. (53) García, P.; Sapienza, R.; Blanco, Á.; López, C. Photonic Glass: A Novel Random Material For Light. Advanced Materials 2007, 19, 2597-2602. (54) Alharbi, A.; Armstrong, D.; Alharbi, S.; Shahrjerdi, D. Physically Unclonable Cryptographic Primitives By Chemical Vapor Deposition Of Layered Mos2. ACS Nano 2017, 11, 12772-12779.
Table of Content
26 ACS Paragon Plus Environment
Page 26 of 26
