revisiting the coﬀee-ring e ect with confocal microscopy · je tiens à remercier tous les...

Master Science de la matière Internship 2014–2015École Normale Supérieure de Lyon Camille SCALLIETUniversité Claude Bernard Lyon I M2 Physics

Revisiting the coffee-ring effect with

confocal microscopy

Abstract : A colloidal dispersion droplet evaporating on a surface, such as a drying coffee

drop, leaves a distinct ring-shaped stain. Using monodisperse fluorescent colloidal particles

and confocal microscopy, we follow the spatiotemporal growth of the deposit at the edge

of evaporating drops. Measurements of the fluctuations of the deposition front during

evaporation reveal that the growth process is more complex than a simple Poisson-like

process.

Confocal microscopy enables us to follow the three-dimensional evolution of the geometry of

the droplets’ edge, and to observe simultaneously the structure of the deposit, ranging from

ordered crystals to disordered packings, and the profile of the drop. When the deposition

speed is low, i.e. at the beginning of the evaporation, particles have time to arrange by

Brownian motion and form crystal packings while at the end, the high deposition rates

lead to the jamming of particles into a disordered phase.

Keywords : colloids, confocal microscopy, coffee-ring effect, growth processes

Supervisor :Olivier DAUCHOT

[email protected] / tél. (+33) 140 795 842

Equipe Effets Collectifs & Matière Molle (EC2M)Laboratoire Gulliver (UMR 7083)Ecole Supérieure de Physique et chimie industrielles de Paris (EPSCI)

10, rue Vauquelin

75 005 Paris, France

http://www.ec2m.espci.fr/spip.php?rubrique1

Acknowledgements

J’adresse mes premiers remerciements à Olivier Dauchot, non seulement pour m’avoir permis derejoindre la jeune et dynamique équipe EC2M, mais surtout pour ses nombreuses contributions,explications et idées qui ont rendu mon stage productif et stimulant. Ce stage a été l’occasionpour moi d’apprendre beaucoup, de la microscopie confocale à la programmation en Matlab enpassant par quelques rudiments de chimie des colloïdes ; mais aussi de découvrir de nouveauxpans de la physique à travers nos discussions, notamment sur les systèmes actifs. Tout celan’a été possible que grâce aux nombreuses interventions d’Olivier qui m’a toujours accordé sontemps que je sais si précieux dans les moments nécessaires, et je lui en suis reconnaissante.

Je tiens à remercier tous les membres de l’équipe qui ont contribué de près ou de loin à rendremon stage ce qu’il a été, les “group meeting” ont été une belle occasion de questionner, aiderou guider mon travail. J’ai beaucoup apprécié l’ambiance chaleureuse qui règne dans l’équipeGulliver, et la diversité des sujets abordés pendant les déjeuners.

Une attention particulière à Marjolein van der Linden et Julien Chopin. Marjolein a été d’unegrande aide au départ concernant la chimie des colloïdes, un domaine alors nouveau pour moi.Un grand merci à Julien pour les quelques astuces concernant la détection de front sous Matlab.Ces quelques lignes de code ont été d’une grande aide et un bon point de départ !

Contents

1 Introduction 1

2 Experimental setup 22.1 Confocal microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.1 Colloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2.2 Z-piezo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Experimental conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Growth dynamics at edges of evaporating droplets 73.1 Image processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Structure of the deposit 154.1 Ordered and disordered packings . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Effect of confinement on packing . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.3 Attempt to observe the 3D structure of the deposit . . . . . . . . . . . . . . . . 19

5 Conclusion 20

6 Annex 21

Revisiting the coffee-ring effect with confocal microscopy C. Scalliet

1 Introduction

If you have spilled a drop of coffee or tea and left it to dry, then you might have observed thatthe stain left behind is not uniform, but ring-shaped. Specifically, the stains are darker nearthe edges than in the middle. While a stray drop of coffee may seem to be of no importance,it is actually rich with nonequilibrium physics. The so-called coffee-ring effect is the productof the interplay between fluid dynamics, surface tension, evaporation, diffusion, capillarity, andmore. Understanding the coffee-ring effect requires understanding these complex parametersin a strongly non-equilibrium system.

Briefly, as a drop evaporates, its edges become easily pinned on the substrate and do notrecede towards the middle of the drop : the diameter of a pinned drop does not decrease whileevaporating. As the droplet dries, the liquid evaporating from the thinning outer edge, wherethe contact angle ✓ is shrinking to zero, must be replenished by liquid from the drop’s interior.This sets up a strong outward flow in the solvent, which transports most of the solute to thecontact line. When the evaporating drop is a suspension of small particles, such as coffee dustin real life or colloids in the lab, the suspended material ends up at the edge, thus forming aring-shaped deposit. Pre-existing surface roughness can provide the force to pin the contactline, but the contact line further pins itself through a feedback loop between flow and deposit:the outward flow increases the deposition of particles, which serves to anchor the fluid andreinforce the outward flow. A schematic picture of the effect is shown Figure 1.

LETTERdoi:10.1038/nature10344

Suppression of the coffee-ring effect byshape-dependent capillary interactionsPeter J. Yunker1, Tim Still1,2, Matthew A. Lohr1 & A. G. Yodh1

When a drop of liquid dries on a solid surface, its suspended par-ticulate matter is deposited in ring-like fashion. This phenomenon,known as the coffee-ring effect1–3, is familiar to anyone who hasobserved a drop of coffee dry. During the drying process, dropedges become pinned to the substrate, and capillary flow outwardfrom the centre of the drop brings suspended particles to the edgeas evaporation proceeds. After evaporation, suspended particlesare left highly concentrated along the original drop edge. Thecoffee-ring effect is manifested in systems with diverse constituents,ranging from large colloids1,4,5 to nanoparticles6 and individualmolecules7. In fact—despite the many practical applications foruniform coatings in printing8, biology9,10 and complex assem-bly11—the ubiquitous nature of the effect has made it difficult toavoid6,12–16. Here we show experimentally that the shape of the sus-pended particles is important and can be used to eliminate thecoffee-ring effect: ellipsoidal particles are deposited uniformlyduring evaporation. The anisotropic shape of the particles signifi-cantly deforms interfaces, producing strong interparticle capillaryinteractions17–23. Thus, after the ellipsoids are carried to the air–water interface by the same outward flow that causes the coffee-ringeffect for spheres, strong long-ranged interparticle attractionsbetween ellipsoids lead to the formation of loosely packed orarrested structures on the air–water interface17,18,21,24. These struc-tures prevent the suspended particles from reaching the drop edgeand ensure uniform deposition. Interestingly, under appropriateconditions, suspensions of spheres mixed with a small number ofellipsoids also produce uniform deposition. Thus, particle shapeprovides a convenient parameter to control the deposition ofparticles, without modification of particle or solvent chemistry.

A drop of evaporating water is a complex, difficult-to-control, non-equilibrium system. Along with capillary flow, the evaporating dropfeatures an air–water interface shaped like a spherical cap and alsoMarangoni flows induced by small temperature differences betweenthe top of the drop and the contact line4. Attempts to reverse orameliorate the coffee-ring effect have thus far focused on manipulatingcapillary flows6,12–16. In this contribution, we show that uniform coat-ings during drying can be obtained simply by changing particle shape.The uniform deposition of ellipsoids after evaporation (Fig. 1a) isreadily apparent, and stands in stark contrast to the uneven ‘coffeering’ deposition of spheres (Fig. 1b) in the same solvent, with the samechemical composition, and experiencing the same capillary flows(Fig. 1c).

Much of the physics of the coffee-ring effect has been demonstratedwith micrometre-sized polystyrene particles1. Here we also utilize suchparticles and simply modify their shape. Our experiments use waterdrops containing a suspension of micrometre-sized polystyrenespheres stretched asymmetrically to different aspect ratios25,26. Wenote that similar results were obtained for hydrophilic ellipsoids andother anisotropic particles (see Supplementary Fig. 4). We evaporatethe drops on glass slides and study suspensions containing particles ofthe same composition, but with different major-axis/minor-axis aspect

ratios (a), including spheres (a 5 1.0), slightly deformed spheres(a 5 1.05, 1.1, 1.2, 1.5) and ellipsoids (a 5 2.5, 3.5); we study volumefractions (w) that vary from 1024 to 2 3 1021.

During the drying process, the droplet contact line remains pinnedin all suspensions, and fluid (carrying particles) flows outward from thedrop centre to replenish the edges. Spherical particles are efficientlytransported to the edge, either in the bulk or along the air–water inter-face, leaving a ring after evaporation is complete (Supplementary

1Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA. 2Complex Assemblies of Soft Matter, CNRS-Rhodia-University of Pennsylvania UMI 3254,Bristol, Pennsylvania 19007, USA.

a

c

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d e0.20

0.15

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–2)

r / R0.4

α = 1.0 α = 1.05 α = 1.1α = 1.2 α = 1.5 α = 2.5 α = 3.5

0.8

0.5 mm

bα = 1.0

ρ max

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ild

80

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01 2

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Figure 1 | Deposition of spheres and ellipsoids. a, b, Images of the finaldistributions of ellipsoids (a) and spheres (b) after evaporation. Insets, particleshape. c, Schematic diagram of the evaporation process, depicting capillary flowinduced by pinned edges. Evaporation occurs over the entire drop surface (bluearrows), so if the contact line were free to recede, the drop profile would bepreserved during evaporation (dashed line). However, the contact line remainspinned, so the contact angle decreases (solid line). Thus, a capillary flow (blackarrows), from the drop’s centre to its edges, is induced to replenish fluid at thecontact line. d, Droplet-normalized particle number density, r/N, plotted asfunction of radial distance from centre of drop for ellipsoids with variousmajor–minor axis aspect ratios (a). e, The maximum local density, rmax,normalized by the density in the middle of the drop, rmid, plotted for all a. Redlines guide the eye.

3 0 8 | N A T U R E | V O L 4 7 6 | 1 8 A U G U S T 2 0 1 1

Macmillan Publishers Limited. All rights reserved©2011

Figure 1: Schematic diagram of the evaporation process, depicting capillary flow induced bypinned edges. Evaporation occurs over the entire drop surface (blue arrows), so if the contactline were free to recede, the drop profile would be preserved during evaporation (dashed line).However, the contact line remains pinned, so the contact angle decreases (solid line). Thus, acapillary flow (black arrows), from the drop’s centre to its edges, is induced to replenish fluidat the contact line (from [1]).

Previously, the growth process at the edge of evaporating drops has been investigated fordifferent particle shapes [2]. For spherical particles, the process was found to be Poissonian atthe beginning of evaporation, and is said to become random deposition with surface diffusionat large times. However, a quick observation of how particles attach to the deposit reveals thatthe picture might be more complicated than simple random deposition, as hydrodynamics flowsand interactions impose the motion of the particles.

With the help of a confocal microscope, we observe the 2D structure and dynamics of thedeposit to try to answer this question. Moreover, one of the final missing piece of the coffee-ring puzzle is a “4D” picture of the deposition at the edge. Confocal microscopy is particularlysuited to observe the dynamics of a system in 3 + 1 dimensions, so we use this technique toexplore the structure of the deposit and how it evolves in time.

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2 Experimental setup

In this section, we present the setup and devices used to study the coffee ring effect, we highlightand discuss the experimental limitations that were encountered and justify the choices madeto conduct the experiments.

2.1 Confocal microscopy

Optical microscopy is widely used in many systems where the domain of interest lies in thesubmicrometre to micrometre range. Since the important length scales are of the order of thewavelength of visible light, microscopy provides a powerful tool to obtain real-space and real-time information about the complex mechanisms that govern these systems. The ease of samplepreparation at room temperature and atmospheric pressure (as opposed to low temperature orlow pressure conditions for electron microscopy for example) also makes optical microscopy aconvenient technique. Nevertheless, certain problems exist for complex many-body systemsthat interact with light; the most prominent one being multiple scattering.

Confocal microscopy offers several advantages over conventional optical microscopy, includ-ing shallow depth of field, elimination of out-of-focus glare, and the ability to collect serialoptical sections from thick samples. It allows to image particles (or cells etc.) that have beenlabeled with fluorescent probes. The technique, based on spatial filtering, eliminates the out-of-focus light that appears away from the region of interest. The images produced by scanningthe sample this way are called optical sections (see Fig. 2b) : it is a non invasive techniqueto observe deep in samples containing particles that have the same refractive index as thesurrounding fluid.

(a)

(b)

Figure 2: (a) Home-made 3D Matlab reconstruction using optical sections of an assembly of3.2 µm colloids sitting on a glass slide, immersed in water. Non index-matching causes thespherical particles to appear elongated after reconstruction (see section 2.2.1); (b) An opticalsection in the middle of the colloids.

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Confocal microscopes differ by the way the sample is scanned in two dimensions. The firstconfocal microscopes used a single laser beam to scan point by point the specimen, which usedto be very long. The confocal microscope that I used is the fastest type as it uses the latestscanning technology, called “Yokogawa Spinning Disk”. A sketch of the different elements com-posing the microscope and the corresponding light path are presented Figure 3.

Figure 3: Yokogawa spinning disk unit optical configuration.

The scanner is equipped with two coaxially aligned disks featuring a dichromatic mirrorpositioned between the disks (see Fig. 3). Each disk contains approximately 20,000 pinholesarranged in a series of nested spirals. The upper disk contains microlenses that direct and focuslight onto perfectly aligned 50-micrometer pinholes in the lower disk for transmission to theobjective and specimen. Both disks rotate exactly at the same crazy speed of 10 000 revolutionsper minute. The pinhole spiral patterns are designed so that a single image is created witheach 30-degree rotation of the disk.

The microscope is the Nikon TI-E. Laser beams are sent by a laser source (MLC400, Agilent)to the confocal spinning unit through optic fiber. The emission light from the sample is recordedwith a high sensitivity sCMOS camera (Zyla, Andor). All the devices necessary to make themicroscope work as a whole are controlled by Nikon’s software.

2.2 Limitations

2.2.1 Colloids

Confocal microscopy using fluorescent particles has proven to be a great tool to characterize ormimic many aspects of atomic and molecular systems. However, in order to be able to observeassemblies of dense colloids (as in the deposition front near the contact line), or deep withina sample, the experimental conditions must meet fundamental requirements. The primary re-quirement is that the index of refraction of the colloidal particles must be closely matched tothat of the solvent; otherwise, multiple scattering results in blurriness of the image and there-fore prevents the microscope from looking inside dense regions.

3


As one of the goal of the experiments is to observe the 3 + 1 dimensional evolution of thedeposit near the contact line, I tried to find a system of particles and solvent meeting thisindex-matching requirement. I had the chance to have several types of suspensions to play,or let’s say work, with. As a matter of fact, this internship was the opportunity to begin acollaboration with a dutch team led by Joris Sprakel, who synthesizes all sorts of colloids. I haddifferent colloidal suspensions meeting the index-matching requirement, as to know : PMMAparticles in a mixture of thioglycerol (66 vol%) and formamide (34 vol%); and a more unusualsystem : particles of methacrylate polymers : poly(trifluoroethyl methacrylate -co- t-butylmethacrylate), suspended in a mixture of formamide, sulfolane and dimethyl sulfoxide. As youcan see, index-matching (and density matching for these suspensions) often implies to havea solvent composed of several chemicals. Observing these samples under microscopy is easyand the difference in images obtained with non index-matched suspensions is striking. Thesesuspensions then appeared to be ideal to study the three-dimensional evolution of the deposit.

Yet, it is well-known to industrial paint chemists that one way to avoid uneven coats, i.e.

the coffee-ring effect, is indeed to create paints that contain a mixture of at least two differentsolvents. One is water, which evaporates quickly, leaving the pigment carrying particles stuckin the second, thicker solvent. The particles are therefore unable to rearrange in this viscoussolvent and are then deposited uniformly. Unfortunately, this solvent also evaporates relativelyslowly (this is one reason why it might be boring to watch paint dry). My first attempt toobserve the coffee-ring effect in index-matched suspensions was pretty disappointing : it took alot of time to evaporate under the hood, and the resulting stain looked nice and homogeneous...not exactly a “coffee-ring”.

The conclusion of these attempts was that observing the coffee-ring effect during the evap-oration of an index and density matched suspension is, at least to my knowledge, not possible.My dreams to revolutionize the science of the coffee-ring effect looked seriously jeopardized.

In order to perform the experiments on the coffee-ring effect, I had to turn back to simplersuspensions involving only one solvent, polystyrene particles suspended in water were a goodcandidate. The suspensions were purchased from Fischer Scientific. Particles are monodisperseand fluorescently dyed, with diameters in the range of 0.6� 3.2 µm.

2.2.2 Z-piezo

The experiments that I conducted were the first quantitative ones performed with the con-focal microscope. Thus, one of my first job was to calibrate the microscope to be sure thateverything worked in a proper way. It also took some time to get to know how all the de-vices and the software worked. Calibrating a new confocal microscope might not seem themost exciting work to do, but one actually learns a lot about how a microscope, and all thecomposing devices, work. Most of the calibration was already done by the manufacturer whileinstalling all the devices. However, I wanted to be sure that the microscope would lead to goodquantitative acquisitions, and I also wanted to push it to its limits in terms of acquisition speed.

During the initial tests, it appeared that there was an issue while performing fast three-dimensional stacks. More precisely, during a typical 3D acquisition, the position of the ob-jective remains still while the scanning of the sample is performed by the vertical motion ofthe platform on which stands the sample. The motion is executed by a piezoelectric element,called the Z-piezo. One can vary the following parameters : the height of the box to scan,the distance between two consecutive images (step size) and the acquisition rate; the latter

4


imposes the speed at which the sample has to move. The piezoelectric element is controlledthrough TTL signals. At the beginning of the acquisition, a TTL signal made of steps is sentfrom the software to the controller of the piezoelectric compound. Then, the platformed movesfollowing the input TTL step signal, and images are taken at a regular time interval. However,the platform holding the sample is a heavy macroscopic object, so the motion has to be overdamped in order to avoid undesirable oscillations while moving from one position to another.In the current configuration, there is no measurement of the output signal of the controller, i.e.

what the piezo actually does. To have access to the measured signal of the Z-piezo, I used anold but terribly efficient tool, the oscilloscope.

Observations of the signals with the oscilloscope revealed that the Z-piezo does not alwayssucceed in following the input TTL signal. As a matter of fact, the typical time to stabilize theposition of the platform after a jump of 0.1 µm is about ⇠ 30 ms, as presented Figure 4.

Time (ms)-100 -50 0 50 100

Vol

tage

4.96

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4.975

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4.985

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5OutputInput

Figure 4: Input and output signals of the Z-piezo controller during a step of height 0.3 µm ofthe platform. The camera and the piezo are triggered so that images are taken just before theposition jumps to another value (when the platform is stabilized).

This is not a problem as long as the frame rate does not exceed 30 FPS (frames per second).Above 30 FPS, images are taken at the wrong altitude, and the 3D reconstruction is wrong asthe software assumes that the Z-piezo was at the right position while recording images. At themoment the problem is not resolved. Yet, we have pointed out the problem to the Nikon, andto the manufacturer of the Z-piezo. This is the first time that the problem is reported, andthere are probably two explanations to this : first, it could be that people using this confocalmicroscope do not observe the very short time-scales dynamics of a system and thus do notneed to acquire stacks so fast. Another possibility, which would be worse, is that people do notrealize that there is a problem with the piezo and acquire wrong information. The problem isunder discussion with the developers and technicians from Nikon, as well as with the manu-facturer of the Z-piezo. As a matter of fact, they are interested in resolving this issue in orderto have better performing microscopes. We have documented the problem and provided themwith quantitative measurements, e.g. measured signals with the oscilloscope.

One step towards the resolution of this issue would be to implement the acquisition of theoutput signal of the Z-piezo controller into Nikon’s software, which is an undergoing work. Eventhis does not appear to be an easy task, but having the real positions at which each imagesare acquired is fundamental o do a valid 3D reconstruction of the sample. Note that this is

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probably not a typical problem in confocal microscopy as the usual limitation in the acquisitionspeed is the imaging of the sample in a plane. With our microscope, the 2D scanning can beas fast as 0.5 ms, leaving the motion in Z far behind in terms of speed.

In the following, we will never do stacks at high frame rates which are wrong for sure. Instead,we perform acquisitions in three dimensions at a frame rate of 10 FPS, camera exposure timeof 5 ms and step size of 0.3 µm. This means that the interval between two consecutive imagesis 100 ms; considering the exposure time the platform has 95 ms to stabilize, which sufficientbased on the oscilloscope measurements presented Figure 4.

2.3 Experimental conditions

In our experiments, we follow the entire time evolution of ⇠ 3 µL water droplets sitting on aglass slide. In order to probe the effect of the particle size on the phenomenon, we use particlewith different diameters, ranging from 0.6 to 2 µm (0.6, 0.79, 1 and 2 µm). By adding deionizedwater to the commercial suspensions, we prepare a series of more or less diluted suspensionsfor each size of particle. The suspensions that we prepare have a concentration of 1, 0.5, 0.25,0.125 and 0.0625 vol % in colloids. For every type of experiment (same size and concentration),we perform three or four acquisitions of evaporating droplets to test the reproducibility of theresults.

A droplet is deposited onto a glass cover slide, and is left to evaporate at room temperature.Being imposed by R2D2, our air-conditioning system, the temperature of the room cannot becontrolled more precisely than one or two degrees so we measure the “local” temperature nearthe sample for each experiment. The drops are viewed from the underside using the confo-cal microscope. To view the sample, we use oil-immersion objectives, which have a numericalaperture higher than dry objectives, in order to get the highest resolution and avoid loss oflight. The magnification is chosen according to the size of the observed particles in order tohave roughly the same size of field of view, measured in diameters of the particles : 100x forthe smallest ones down to 40x for particles of 2 µm.

In the experiments, we have to use cover slides instead of thick microscope glass slides asthe working distance of the high magnification objectives (100x and 60x) is of 100 µm. Usually,this does not pose a problem in confocal microscopy since the sample is inside a capillary, andcan therefore be turn upside down. Here, we want the upper surface of the drop to be free andwe cannot add a glass slide on top to rigidify the setup. However, we manage to strengthenthe thin cover slide by adding pieces of microscope slides parallel to the glass slide. As long aswe observe the sample in two dimensions this is not necessary, but we ensure the rigidity of thecover slide to avoid unwanted bending while performing 3D stacks : if the objective approachesthe glass slide and deforms it, it will inevitably create additional unwanted flows inside thedrop which we do not want to account for.

The droplet is illuminated by a laser source emitting at 488 nm, right in the excitationspectrum of the fluorescent markers of the colloids. Thanks to the right choice of filters, thecamera only senses the light emitted by the particles. Images are recorded with the nikon’ssoftware for confocal microscopy. To ease the image analysis (see below), we always observe thevery left edge of the droplet. From all the acquisition parameters, we vary : that magnificationof the objective, the exposure time of the camera and the acquisition frame rate. The colloidswhich are used in this experiment have a powerful and stable fluorophore, so we chose a short

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exposure time, typically a few milliseconds. The colloids do not have time to move significantlywhile the pictures are taken so the images obtained are sharp.

3 Growth dynamics at edges of evaporating droplets

Evaporating colloidal drops provide an attractive experimental system to study out-of-equilibriumgrowth processes. We wish to study the growth dynamics at the edge of evaporating droplets.The resulting deposits of particles grow from the edge on the water-substrate interface in twodimensions, defining a deposition front, or growth line, that varies in space and time (Figure5a). To characterize the process, we study spatiotemporal fluctuations and correlations of thegrowth line.

(a)

(b)

Figure 5: (a) Particle deposition front : colloids are transported from the bulk of the suspension(right side) and are deposited at the edge of the droplet. The contact line corresponds to thelast row of colloids on the left; (b) Raw image on which we draw the boundaries of the depositand the height h(x, t) for several positions x. We represent what we shall use later as x, thecurvilinear abscissa along the contact line. Size of the colloids : 2 µm, 40x oil-immersionobjective.

3.1 Image processing

During the evaporation of the droplets, we acquire images at a rate of 1 or 2 FPS. The recordeddata consists in a temporal series of images. In order to gain quantitative information on thephysical processes occuring during evaporation, we have to analyze the images. The usefulquantity that we want to extract is the height h(x, t) of the deposition front, as a function oftime and position along the contact line.

I specially developed a MATLAB program in order to achieve such a task. It is a powerfuland convenient way to systematically analyze the raw images as plenty of tools for image pro-cessing are already included in the software. Roughly speaking, the program finds the left andright borders of the deposit (bright zone) in every image. Once these boundaries are found, the

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height of the deposition front is calculated as the distance from one boundary to the other forall positions x and images, .

More precisely, I use the fact that raw pictures are in greyscale. Thus, it is quite straight-forward to use this property and convert original images into black and white images. Theseobjects are easy to manipulate, and it enables us to benefit from MATLAB’s black and whiteimage processing tools. This first step might seem an easy one to implement; as a matter offact, if you look at the picture Fig. 5a, you immediately recognize the borders since the con-trast is good and the binarization doesn’t seem to be so sensitive to the value of the threshold.However, this stage gave me a hard time as I wanted my program to be robust and operationalfor all my experiments. Broadly speaking, in all types of experiments in which pictures of thesystem are taken, the easiest way to proceed is to light the system homogeneously and keep itconstant il all experiments. The difficulty in our experiment is that the lighting is provided bythe system itself - the colloids - and that their number increases in time.

Even during the evaporation of a single droplet, the brightness of the deposit evolves andbecomes inhomogeneous : near the front, the deposit is composed of several layers of colloids,which appear very bright, while the colloids near the edge of the drop are single layers andappear much darker. I also noticed in other observations (not only coffee ring experiments),that the light collected by the camera is not homogeneous even when it should be : pixels atthe center of the sensor always appear brighter than ones on the borders. I did not figure outwhy it is so, but it would be a good point to improve in the future.

In order to binarize images, I wrote a code which calculates the threshold locally, so thatwe bypass the problem of inhomogeneous lighting in the different regions of interest. The codeis robust and works for all experiments.

Once the images are transformed in black and white, we extract the two borders : the leftone corresponds to the contact line of the droplet, and the right one is the deposition frontwhich evolves as colloids arrive and are deposited. For all experiments, the configuration inwhich the drops are observed is always the same : the contact line is parallel to the longest sizeof the camera’s sensor, in order to view the longest fronts possible, and the drop’s edge is onthe left side (completely arbitrary).

(Re)definition of time and length

Having found the left and right boarders of the deposit, it is important to define what is meantby “height” of the front. As the front is practically vertical, we could simply say that for eachline, the height of the front is the difference between the two boundaries. Yet, the contact lineis never perfectly vertical and is irregular most of the time. Therefore, we chose to define h(x, t)as following : for every point of the right boundary, we calculate its distance to the contact lineand define the height of the deposition front as this value. Figure 5b illustrates the two bordersof the deposit (in yellow and red), as well as the lines along which the height is calculated. Theheight h(x, t) is renormalized by the size of the colloids so that experiments involving differentparticle sizes can be compared. The deposit height h(x, t) varies spatially and temporally, asshown Fig. 6.

In the experiments, the start time (t = 0) is ill-defined. It could be when the drop is de-posited on the glass slide, when the drop stops spreading or when the first colloid is deposited.Another issue is that for concentrated suspensions (1 vol %), it is not possible to observe thevery beginning of the formation of the deposit as several rows of colloids have already formed

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x in diameters0 10 20 30 40 50 60 70 80

hin

diameters

0

5

10

15

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Figure 6: Plot of the deposition’s height h(x) at different time-steps. The time interval betweentwo consecutive curves is constant and equal to 50 seconds. h(x) for early times are plotted indark blue and time evolves as the color lightens. Particle size : 2 µm.

a few seconds after the drop has been deposited. This very fast process prevents us to capturethe evolution of the deposition front from the very beginning in all experiments. As a matterof fact, when the drop is deposited, the objective does not focus exactly on the contact lineso it takes a few seconds in order to move the sample, place the edge of the drop in the fieldof view and to start the acquisition. For this reason, the use of real time t to explore growthexponents presents technical issues.

The mean height h̄(t) increases almost linearly with t from the beginning of evaporation toits nearing end. Towards the end of evaporation, strong flows appear which leads to a suddenand sharp increase in the mean height h̄. Plots of the curves h̄ as a function of time for allexperiments are presented Fig. 15. Therefore, the use of h̄ in place of t averts all technicalissues presented above. Importantly, the use of h̄ resolves ambiguities in defining t = 0, ash̄ = 0 is explicit, and enables use to perform many measurements with different droplets andparticle size, and compare them. We will therefore use the effective time h̄ when studying thegrowth process.

3.2 Results

In order to characterize the growth process, we study spatial and temporal correlations of thefield h(x, t). First, we define a new field �h as the original height to which we subtractedits mean along x at all times : �h = h(x, t) � h̄(t). The resulting quantity has a zerox-mean at all times. We investigate the probability density function (pdf) of �h rescaled by itsstandard deviation Fig. 7 (left graphic of (a) and (b)). The results is plotted for two differentexperiments (same size of colloids 0.79 µm, suspension (a) less concentrated than (b)).

In particular, we show that this is not the good quantity on which the pdf must be calculated.As a matter of fact, the case (a) gives a nice gaussian pdf at all times while the left pdf of Fig.7b has a strange shape with several peaks around zero and two other values. Having a look atthe original data of �h in this particular case reveals that this strange shape has nothing to dowith the growth process. Imagine that the front’s height h(x, t) at a fixed time t is not a flatfluctuating signal but is for some reason increasing or decreasing with x. The mean of this curvewill be a number, roughly in the middle of the maximum and minimum value of the height.But plotting h(x, t) minus its mean along x will give a signal containing half of its values clearlypositive and half clearly negative, but it will certainly not be a signal fluctuating around zero.These two sets of “big” positive and negative values correspond to the two non-zero peaks inthe pdf on the left of Fig. 7b.

Now we explain why we believe that the origin of this effect is not related to the deposition

9


(a)

(b)

Figure 7: PDF avec la gaussienne normale (a) : as qui marche bien dès qu’on a �h; (b) : fronttordu, on doit enlever �h̃. Different colors correspond to different times : blue at the beginningand red towards the end. Note : experiment (a) lasts much less that (b), that is why there aremuch less curves.

process. As a matter of fact, we always see that the front line is roughly a straight line, butcontact line is sometimes tortuous. The deposition process tends to smooth irregularities, aswe will be explained latter. The original pictures corresponding to the two experiments Figure7 are presented in the Annex (Fig. 20). You can clearly see the effect concerning the contactline that I am talking about. Unfortunately, I did not find a way to force the contact line to bestraight : the triple line sits somewhere on the glass slide so as to minimize its surface tension.However, the actual shape of the contact line is not a perfect circle as predicted by calculus :local inhomogeneities on the glass (e.g. dust, grease from my fingers - even if I was careful tokeep the glass slides clean) change the wetting. I have tried to do a simple trick in order toforce the triple line to be straight in the region of interest. I drew a small linear defect with adiamond on a glass slide thinking that the drop would prefer to “stop” wetting before crossingthe defect. It turned out that the drop does not really care about the defect and wets the glassas if it was not there.

Not being able to cure at its origin the problem of the contact line shape, we had to find away to remove its signature in the signal h(x, t). That is when �h̃ comes in :

�h̃ = �h(x, t) � < �h(x, t) >t

= h(x, t) � < h(x, t) >x

� < �h(x, t) >t

,

which is basically the same as �h, to which we subtracted the “global trend” in x, present atall times. As presented Fig. 7, doing so allows one to delete the undesirable bumps present inthe pdf of �h. Note that this does not change the cases in which the pdf is already gaussianbefore this transformation as < �h(x, t) >

t

= 0 (Fig. 7a). In the following, we will use �h̃ asvariable to calculate the main quantities.

We investigate the roughness of the front, i.e. spatial variations, defined as the standard de-viation (std) of the height. Many growth processes exhibit a self-affine structure well describedby Family-Vicsek scaling. The std of the deposit’s height can exhibit power-law growth overtime as ⇠ t

�, or over space as L↵ when L is small, where L is the size of the spatial window onwhich the std is calculated. Once more, we plot this quantity for both : the raw signal h and�h̃. Note that the std of h is the same as the one of �h, since the spatial mean of the latter is

10


zero. Each plot contains the curves obtained for the “good” (7a) and the “wrong” (7b) cases ofFigure 7. We see that the calculation on the raw signal can be misleading in the “wrong” caseif one does not pay attention to remove the effect of the non-linear contact line. As a matterof fact, Fig. 8a shows that the std of h seems to grow as h̄

1/2 for the black curve, which is thespecificity of a Poisson process. On the other hand, for neat signals such as in Fig. 7a, the stddoes not grow and is rather constant, and noisy. Once more, the effect of the transformationof this signal to �h̃ does not change the behavior of the std in the good case. In the othercase (black curves), the behavior of the std is altered and we recover a non-increasing std, alsopretty noisy. Once recomputed in the right way, there seems to be no clear growth exponentof the std with “time” h̄, namely � = 0 for this process. We assume from then that the processis stationary, and also that it is homogenous (a view of original movies confirm that the lastassumption is fine).

7h10 1 10 2

std

x(h

)

0.1

0.5

1

1.59 7h1=2

(a)

7h10 1 10 2

std

x("

~ h)

0.5

10 0

(b)

Figure 8: Roughness of the front as a function of effective “time” h̄ calculated with : (a) theraw signal h; (b) the cleaned signal �h̃.

The global stagnation of the profile’s std tells us that the front does not become rougherwhen time evolves. As a matter of fact, we do not specifically expect a simple Possonian processfor the growth of the deposit, as colloids do not attach to strictly independent sites (if you arenot convinced, video samples coming soon!). Another quantity that we can study are temporalcorrelations of the height. Specifically, we calculate :

G(t0, ⌧ ; x) =�h̃(x, t0).�h̃(x, t0 + ⌧)

�h̃(x, t0)2 , (1)

Assuming stationarity and homogeneity of the signal, the correlation G actually depends onlyon ⌧ , so we average over several starting times t0 and on all positions x. The resulting curvesfor all experiments are plotted Figure 9. The temporal correlation of the signal does not dependon the size of the particle neither on the concentration of the suspension. Moreover, we cansee that the correlation function does not go to zero at large times as it goes through negativevalues, which is the signature of temporal anticorrelations.

Going back to the calculation of the roughness of the front’s profile, we explore its spatialscaling with L, the size of the box on which we calculate the std. For a given box size L, wecalculate the std of �h̃ on all possible boxes of this size, then average over all the obtained std.This value for one box size and one time step is called w(L, t). When L increases, we probe

11


Figure 9: Temporal correlations averaged on initial position x0 and all times, assuming sta-tionarity and homogeneity. Each subplot corresponds to a size of particles. For a fixed size ofparticle, we plot different curves corresponding to different concentrations in colloids (from 1to 0.06 vol%). High concentrations are plotted in dark colors while bright colors correspond tolow concentrations. Several realizations are done for every pair of size/concentration. Top left: 0.6 µm, Top right 0.79 µm, Bottom left 1 µm, Bottom right 2 µm.

a bigger sample of fluctuations, and when L is equal to the size of the system, we recover thefull std along x direction calculated above and shown Figure 8. We want to see the spatialdependance over L of w at a fixed time, and we are also interested to see if w for L fixed evolvesas a function of time or not. Previous results tell us that the process is basically stationary butwe check this once more on this quantity.

L (in diameters)100 101 102

w(L

)

10-2

10-1

Figure 10: Roughness w(L) plotted versus probed length scale L at different times for oneexperiment. The color code is blue at small times and red at larger times. The particle size is0.79 µm and concentration 1 vol %.

Figure 10, we plot the curves w(L) calculated for different times in a single experiment. Wecan see that the behavior is almost the same at all times. For small L, there seems to be aslight dependance on time as the roughness w seems to increase a bit with time. However,when L increases, the number of samples of size L in the system decreases and the statisticsavailable to calculate w(L) gets smaller, which leads to noisier curves. Note that this particular

12


experiment (Fig. 10) leads to a neat ordering in time for small L, but this was not always thecase in the other experiments. For this reasons, we decide to plot only one curve w(L) for eachexperiment, which corresponds to the time average of the curves w(L, t). We show the resultsfor all experiments Figure 11.

L (in diameters)10 0 10 1 10 2

<w

(L;t

)>

t

10 -2

10 -1


<w

(L;t

)>

t

10 -2

10 -1


<w

(L;t

)>

t

10 -2

10 -1


<w

(L;t

)>

t

10 -2

10 -1

Figure 11: Roughness w(L) vs L and averaged over time for all experiments. Same color codeas Fig. 9.

At first sight, it seems that inside the subplots, i.e. for the same particle size, there is nodependance on the concentration of the particles. Moreover, the behavior of the roughness as afunction of L is almost the same for different particle sizes, which is also the “expected” behaviorfor the roughness : as the spatial window L on which the std is calculated increases, one probesa bigger piece of the signal and therefore samples an increasing number of fluctuations. Ata point, the size of the box is such that all typical fluctuations of the signal are probed, andtherefore the roughness does not increase anymore.


<w

(L;t

)>

t

10 -2

10 -1

d = 0.79 7md = 1 7md = 2 7m

9 L1=2

Figure 12: Roughness w(L) vs L and averaged over time for different sizes of particle and sameconcentration (0.5 vol %).

To look deeper into the results and capture the dependance on the particle size, if there is

13


any, we plot Figure 12 the roughness as a function of L for different particle size, but sameconcentration. We notice that the curves follow the same global trend, but that the crossoverfrom an increasing w to a constant value occurs at a typical length L

⇤ which depends on thesize of the particles. The crossover length L

⇤ gives some indication on the spatial correlations ofthe signal. Roughy speaking, if the spatial correlations are strong, the length L

⇤ for which theroughness saturates has to be bigger. Therefore, this increase in L

⇤ as the size of the particledecreases gives us a hint that spatial correlations might be different in these cases. We adda guide for the eye indicating the slope of a ⇠ L

1/2 dependance. We do not conclude withcertainty that the roughness exponent ↵ is equal to 1/2 as we do not even have one decadeto measure it. Only a small region before the crossover is visible. The growth exponent isprobably equal or greater than 1/2.

To study spatial correlations, we do the spatial equivalent of the temporal correlationscalculated Eq. 1. As the process is homogeneous, we calculate the correlation function startingfrom different initial positions x0, then we average over them. Moreover, the stationarity of theprocess allows us to average over time. The resulting curves are plotted Figure 13. A strikingdependance over the concentration emerges, which was probably hidden by the importantamount of plotted curves in Fig. 11. Spatial correlations are stronger for experiments involvingsmall particles and dilute suspensions.

; (in diameters)0 20 40

C(;

)

-0.5

0

0.5

1


C(;

)

-0.5

0

0.5

1


C(;

)

-0.5

0

0.5

1


C(;

)

-0.5

0

0.5

1

Figure 13: Spatial correlations averaged on all times and initial positions x0, assuming station-arity and homogeneity. Same color code as Fig. 9.

3.3 Interpretation

In the light of all the results presented above, we try to give a global picture of the growthprocess and make a link with the microscopic rules for deposition. At first sight, the processmight look pretty simple for spherical particles : they arrive perpendicularly to the front, at anapparently random position, than sticks to it, just as in random deposition. However, the viewof the height’s spatiotemporal evolution Fig. 6 does not exhibit huge fluctuations as randomdeposition would do, the front stays smooth at all times. With the help of the fast acquiring mi-croscope, a detailed observation of the deposition process reveals that it is not as simple as that.

14


In fact, one must not forget that the particles are not dry, but suspended in water, and thathydrodynamic effects always play an important role in a physics problem. Particles that aretraveling in the fluid create additional flows that interact and can lead to a different depositionprocess than for mere dry particles. Thus, we do not expect a particular behavior when theparticles arrive at the front, as particles might interact through hydrodynamics and attachto the deposit in a non expected way. As a matter of fact, the deposit imposes a boundarycondition on the hydrodynamic flow. Direct visualization of the deposition process shows thatparticles have the tendency to attach to regions with lower height : if a particle arrives nearthe top of a stuck particle, it “feels” the front and will not directly attach to it as the positionis unstable, but rather roll and stay in a stable “local minimum” of the front. This rule tendsto smoothen the front and avoid high fluctuations in the deposit’s profile.

Going back to previous results on the subject [2], it was stated that the deposition was aPoisson process in the early times of evaporation (when h̄ = 10 colloids) as the deposit is amonolayer. Later, the process is random deposition with surface diffusion. In our study, we areable to see the number of layers formed by the deposit and we can observe that the deposit isthree-dimensional even at the very beginning of the process. Even for the less dense case of thebigger particles Fig. 13, for which spatial correlations are very small, we do not see a significantincrease in the std of the height’s profile. The deposition process is probably a Poisson processat the very beginning of the evaporation (until one or two rows are deposited), but first we seeno experimental evidence for this, and second, this process is not the process observed for themajor part of evaporation.

4 Structure of the deposit

The use of confocal microscopy to observe the phenomenon is of great help when it comesto inspect the evolution of the packing and structure, simultaneously with the growth of thedeposit. Previous works on the subject used classic bright-field microscopy to observe the evolu-tion of the front [2], while the resulting deposit was viewed using scanning electron microscope(SEM) [5]. The latter technique is powerful to observe precisely the structure of the deposit,however it only allows observation of the top layer of the deposit, and it requires to work undervacuum. The final drying of the deposit in known to create cracks and other phenomena whichare not attributed to the coffee-ring effect. While preparing the sample for SEM, the structureof the deposit might change.

4.1 Ordered and disordered packings

The coffee-ring effect has been successfully used to generate colloidal crystals in three dimensions[4] and is a frequently used technique for particle assembly on a substrate. As a matter of fact,the deposit showed Figure 5a is not any kind of structure, as we can clearly see that thedeposited particles have formed a relatively ordered packing. Figure 14a shows the same frontas Fig. 5a, but at the very end of evaporation (time = 570 s on Fig. 14b). The orderedpacking formed at early times (on the left) is still present, but the colloids that arrive at theend of evaporation are jammed into a disordered packing. In fact, Figure 14b teaches us thatthe front’s speed increases sharply towards the end of evaporation. At the beginning of thedeposition mechanism, the deposition rate (proportional to h̄), is low; particles at the fronthave time to diffuse and form an ordered packing before another row is deposited. At the endof evaporation, the deposition rate is such that a row is deposited in a few seconds, which is not

15


(a)

Time (s)0 100 200 300 400 500 600

7 h(in

dia

met

ers)

0

5

10

15

20

25

30

35

40

(b)

Figure 14: (a) Structure of a deposit at the very end of the evaporation. We characterize twodifferent regions : in red the packing is ordered, either square or triangular, while the colloidsin the blue region form a disordered structure. Size of the colloids : 2 µm, 40x oil-immersionobjective. (b) Evolution of the mean height h̄ as a function of time for the same experiment.The orange dashed line delimits the spatiotemporal regions in which the deposit forms anordered (bottom) or a disordered (top) packing.

sufficient for all particles to rearrange. For this experiment (Figure 14), the deposition processleads to disordered packing when the front is on average 25 colloids high. Reading the curveh̄(t), we find that the transition from order to disorder occurs for t ⇠ 560 s, or more preciselywhen the front’s speed increases sharply and exceeds a critical value.

As a matter of fact, it seems that the control parameter of the transition from ordered todisordered packings is the front’s velocity. To test this hypothesis, we probe the influence ofthe concentration of the particles in the droplet on the resulting packing. We plot the curve h̄

as a function of time for all the experiments Figure 15, sorted by particle size.In Figure 15 we see that the front’s speed is an increasing function of time (as the of h̄ slope

increases). More interesting, this speed is not constant for all concentrations : for concentratedsuspensions, the speed of the front is relatively high at the beginning of the evaporation com-pared to dilute experiments. That is why for the most concentrated suspensions (1 vol %), thepacking becomes disordered at the very beginning of evaporation since the front’s velocity isalready pretty high.

The idea that kinetics prevents a system from being able to fully explore configuration spaceis well known for the glass transition. By comparison, the analogous idea for colloidal systemsis relatively new. In this study, we show that the dynamics of the deposition, rather than thestability of static packing arrangements, determines the patterns that ultimately form.

4.2 Effect of confinement on packing

In this section, we investigate the origin of the square packings observed near the contact line.As a matter of fact, direct visualization of the deposition process reveals that particles do notarrange form square packings when they arrive at the borders : they always form either tri-angular or disorder packing. Therefore, the square packing is not caused by the depositionprocess, but rather by the entire evolution of the deposit as evaporation goes on.

16


Time in seconds0 200 400 600

7 h0

50

100d = 0.6 7m


7 h

0

50

100d = 0.79 7m


7 h

0

50

100d = 1 7m


7 h

0

50

100d = 2 7m

Figure 15: Evolution of the mean height h̄ of the front as a function of time for all experiments.Each subplot corresponds to a size of particles. Same color code as Figure 9.

When a drop evaporates, its whole geometry evolves in time. Due to the roughness of theglass slide, the drop’s edge remains pinned and does not move while the volume of the dropdecreases : the drop forms a spherical cap with a typical height decreasing with time. Thespherical cap approximation is fine as the volume of the droplets is quite small ⇠ 3 µL, so thatthe radius R of the drop is much smaller than

�1, the capillary length : gravity is negligibleand does not flatten the drop. From the point of view of a colloid in the deposit, the glass slideand the air-water interface form a “wedge” in which they are confined ; approaching the end ofevaporation, the contact angle - or wedge angle - vanishes and the deposit gets squeezed.

At the first stages of evaporation, the colloids are transported near the edge and form acompact deposit. As the contact angle is still relatively high, there is not so much confinementand the particles assemble in the densest way : a triangular crystal.

On this subject, we can learn a lot from the study conducted by Pieranski and al. on thincolloidal crystals [6]. They showed that in colloidal crystals, confinement in a thin layer ofvarying thickness leads to a sequence of structures : stacks of n square, or triangular, orderedlayers. For the thinest region, i.e. number of layers n 7, the sequence is always the same andas follows : n 4 ! (n+1) ⇤ ! (n+1) 4. Please refer to Fig. 16 for the original results. Thissequence means that when an additional layer of crystal has space to form, the confinement bythe wedge favors the square packing. This is due to the fact that the most efficient packingactually depends on the available space and it is not always the triangular packing that wins.In fact, the square packing is looser than the triangular one when looking in a plane, but for agiven accessible height, the square packing allows a higher number of layers than the triangularone. That’s why everytime a new layer forms, there exists a small regions of square packing.

Experimentally, I observed clearly this ordering in the evaporating drops. We can see onFigure 14a that starting from left to right (height of the “wedge” increases) we can distinguish

17


Figure 16: Three-dimensional density N

(3) as a function of the gap thickness h. Crystallizationof colloidal polystyrene beads (diameter 1.1 µm) suspended in water. Inset : the plane wedgegeometry.

: 3 rows of 1 4, 4 rows of 2 ⇤, 4 rows of 2 4, 5 rows of 3 ⇤, 9-10 rows of 3 ⇤, then disorderedpacking. We can recognize the addition of a new layer : being in an upper plane, the colloidsforming it are out of focus and thus appear darker. The edge of the drop is not a perfect wedgebut presents some irregularities probably due to inhomogeneities in the wetting of the glass.The sequence of structures is not as well defined as in Pieranski’s well-controlled experiment.Yet, we observe the same global evolution as the height of the wedge increases. Unfortunately,the results presented in Pieranski’s paper are expressed in µm

our colloids do not have the exact size as those used in Pieranski’s experiments.

This sequence of structure is not specific to micron-sized particles but appears generally forspherical objects. I tried to reproduce the experiment with macroscopic steel beads of diameter3 mm. The wedge is made of two plexiglas plates kept together by two triangular pieces on thesides. I added the beads one by one, allowing them to find the best packing by gently shaking.The resulting packing shown Fig. 17 looks very similar to the one obtained in the coffee ringexperiment, despite the obvious differences in sizes and conditions : micrometers/millimeters,thermal/athermal, surrounded by a fluid/dry.

Figure 17: Picture of a packing of steel beads in a wedge geometry. The beads are 3 mm large.

18


4.3 Attempt to observe the 3D structure of the deposit

As previously explained, it is unfortunately not possible to follow the local three dimensionaltime evolution of the deposit near the contact line at the moment, due to the difference inoptical index between the colloids and water and the fact that packings are dense in this re-gion. However, it is worth trying to see if we can get any information on the three dimensionalstructure of the deposit or of the drop’s edge.

While the drop is evaporates, we acquire a sequence of 3D stacks in order to see the 3Dstructure of the deposit. An example of side-views of reconstructed images of the sample atdifferent times are given Figure 18. As expected from the non index-matching, the depositappears especially bright, as a result of multiple scattering in the dense deposit. Therefore, wedo not have access to the local dynamics and packings inside the deposit. However, these sideviews teach us two things.

First, we can see the shape of the droplet is clearly delimited by the colloids located atthe air-water and water-glass interfaces. It is a nice way to observe the contact angle locallywithout having to use a second microscope to view the side of the drop. For the moment, wecannot measure this angle very accurately : the side view that we obtain is a perspective viewof the 3D box. To measure the contact angle precisely, one should project all the images onto asingle side plane (XZ plane represented of the figures), then extrapolate the shape of the curveto the contact line. This is a promising and accurate way to simultaneously measure contactangles during the evaporation of the drops. Unfortunately, I figured out how to do this XZprojection with the software only at the very end of my stay, so I did not have time to fullyexplore this asset. An example of resulting projection is shown in the Annex (Bottom of Fig.21)

Figure 18: Side-view of the 3D reconstruction of the drop’s edge. The top image was takenat the beginning of evaporation while the bottom one is taken towards the end of evaporation.Particle size : 0.79 µm, concentration 0.125 vol %; scale bar : 20 µm.

Another obvious observation is that the deposit does not form a single block but is actually

19


made of two parts, as one layer of colloids (or at least one pile of layers) lays on the air-waterinterface while another one sits on the glass slide. There is a region almost empty of colloidsbetween both. This makes the usual sketch of the coffee-ring effect, as shown Figure 19, prettyinaccurate. The question of whether a fluid layer remains above the deposit has not beenanswered before. With our experiments, it appears clearly that there is indeed a fluid layerremaining, and that the term “front” is actually pretty ambiguous, as we can distinguish twolayers, or fronts, on the side-view images. A view from the underside of the deposit is shown inthe Annex (top of Figure 21). On this image, we can clearly see the two fronts : the brightercorresponds to the layer sitting on the glass slide while the darker one (upper in Z direction),sits on the water-air interface.

Figure 19: Usual picture of the coffee-ring effect (sketch from [3]).

5 Conclusion

In this work, I have calibrated and used a new fast confocal microscope for quantitative mea-surements on evaporating droplets. A series of Matlab codes were developed in order to studythe growth process at the edges of evaporating droplets. These codes are specific for this typeof measurements, but also contain general features which can serve for further applications.A code to extract from original Nikon files all the information on the acquisition and get theimages has been developed with Matlab, which saves a lot of time when it comes to analyzingimages.

Using confocal microscopy in 2 + 1 dimensions, we have successfully observed and charac-terized the growth process at the edge of an evaporating droplet. We show that the depositionprocess is actually a rich non equilibrium process, in which there is a balance between thetypical deposition time of a row of colloids and the time for a particle to rearrange and find alocal minimum in the height profile.

Despite intrinsic limitations to have both the coffee ring effect occurring and observationwith the confocal microscope in 3 + 1 dimensions, we succeed to gain information on how thestructure evolves during evaporation. Moreover, we show that it is a powerful and precise wayto measure the local contact angle. This seems to be a promising way to compare experimentswith theory in which one usually has to have the dependance of the contact angle ✓ on time.Another promising route to record the entire three-dimensional structure of the front and itsdynamics would be to find a system of solvent and colloids which have the same refractive indexand in which the solvent is made of only one chemical.

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6 Annex

(a) (b)

Figure 20: Picture of the deposit for the two experiments corresponding to the pdf plots ofFigure7a (a) and Figure 7b (b). As explained in the text, the front and contact line of picture(a) are straight while in picture (b), the contact line is twisted and the front is straight.

Figure 21: Top : under view of the deposit; Bottom : Side projection of the drop profile. Samedrop as Figure 18.

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References

[1] P. J. Yunker, T. Still, M. A. Lohr, A. G. Yodh, Nature 476, 308 (2011).

[2] P. J. Yunker, M. A. Lohr, T. Still, A. Borodin, D. J. Durian, and A. G. Yodh, Phys. Rev.Lett. 110, 035501 (2013).

[3] A. F. Routh, Rep. Prog. Phys. 76, 046603 (2013).

[4] K. P. Velikov, C. G. Christova, R. P. A. Dullens, and A. van Blaaderen, Science 296, 106(2002).

[5] A. G. Marìn, H. Gelderblom, D. Lohse, and J. H. Snoeijer, Phys. Rev. Lett. 107, 085502(2011).

[6] P. Pieranski, L. Strzelecki, and B. Pansu, Phys. Rev. Lett. 50, 900 (1983).

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revisiting the coﬀee-ring e ect with confocal microscopy · je tiens à remercier tous les...

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