Imaging the Drying of a Colloidal Suspension

Hugues Bodiguel∗a and Jacques Lenga

Received Xth XXXXXXXXXX 20XX, Accepted Xth XXXXXXXXX 20XXFirst published on the web Xth XXXXXXXXXX 200XDOI: 10.1039/b000000x

We present an experimental investigation of the drying kinetics seen from inside a sessile droplet laden with a colloidal sol of silicananoparticles. We use fast, two-color confocal microscopy imaging to quantitatively extract on the one hand the concentrationfield of the rhodamine-tagged nanosol and on the other hand the velocity field and the mobility field of large, fluorescein-taggedtracers. By changing the initial concentration at which the drop dries up, we propose a method that yields a self-consistent wayto obtain the rheology of the sol. Based on these results, we analyse the drying kinetics in terms i) of flow patterns that includeevaporating and Marangoni flows which compete to determine the final concentration profile and ii) of truncated dynamics thatwe quantitatively relate to the rheology of the sol.

1 Introduction

The drying of sessile droplets in presence of solutes has re-ceived much attention from the scientific community duringthe past decade. It has indeed some important applicationsin many domains such as coating processes, ink-jet printingand spotting technologies for bio-assays. Since the work ofDeegan et al.1,2, the basics of the mechanisms that leads tothe accumulation of solutes in a ring-like pattern around anevaporating droplet are well known. In any situations wherethe drying is limited by the solvent diffusion, the evaporationrate is non uniform and is maximum in the vicinity of the con-tact line. This leads to an outward flow inside the droplet thatcarries solutes toward the edge of the droplet. Particle accu-mulation also requires the pinning of the contact line.

It seems however that the full description of the phenomenarelated to evaporation induced deposition remains a scientificchallenge. Indeed, it has been observed that the strength ofthe pinning of the contact line depend on the material proper-ties2–8. It is however unclear whether the discrepancy in theexperimental results is related to the properties of the cornerregion which reaches high concentrations or to the flow andconcentration field properties in the liquid-like central regionof the droplet7.

Most of the theoretical attempts to describe the related phe-nomena suppose uniform concentration fields in the verticaldirection and lubrication approximation9,10. Furthermore, itis usually supposed that the solute concentration reaches nearthe contact line a maximum concentration, at which the solu-tion or suspension could be considered as solid-like4,10,11. Toour knowledge, these assumptions have not been fully verified

a Universite Bordeaux-1, Laboratory of the Future (UMR 5258 with Rho-dia and CNRS), 178, avenue du Docteur Schweitzer, 33608 Pessac cedex –France; E-mail: hugues.bodiguel@u.bordeaux1.fr

experimentally.The flow field that develops inside the droplet has also been

a subject of debate, and it has been shown that Marangonirecirculating flows have a strong influence on the final formof the deposit7,12,13. These flows modify the concentrationfield and could prevent the solute accumulation near the con-tact line, if the Marangoni number Ma is high enough. In sucha case, we may expect the recirculation to overcome the het-erogeneous evaporation and to homogenize the content of thedroplet14. Experimental data and corresponding descriptionseem however to be lacking for intermediate Ma, which areof great importance since they correspond to the usual caseof a water droplet on a glass substrate. Since the estimationof these Marangoni flows is usually a difficult task, there isa need for the experimentalists to estimate the importance ofthese in order to compare the results.

Local experimental observations inside the droplet wereusually limited to a qualitative description of the particle flowtoward the edge, or to the visualization of the velocity field7.Kajiya et al15 were the first to report quantitative data based onfluorescence intensity measurements inside a droplet of poly-mer solution. They were able to measure the concentrationas a function of time and space, and their results bring a di-rect and quantitative visualization of the solute accumulationnear the edge. However, as pointed out by these authors, suchexperiments ask for simultaneous investigations of the veloc-ity field since the concentration field is coupled to the flowinside the droplet. These experiments integrate the concentra-tion along the droplet thickness, which is the usual assumptionmade in theoretical approach10. When recirculation flows areimportant, the use of this assumption is questionable and thereis a clear need for three-dimensional measurements directlyinside a droplet.

In the present work, we describe such an approach: it is a

1–11 | 1

quantitative method able to measure simultaneously the veloc-ity field, the concentration field, and the local rheology of thesuspension undergoing drying. It is based on fast confocal mi-croscopy and takes advantage of having two different fluores-cent dyes and two detectors. With one detector, we follow theintensity field of a colloidal suspension of silica nanoparticlestagged with a first fluorescent dye. The fluorescence intensityof the suspension under study is then related to the local parti-cle concentration. With the other detector, we monitor the dis-placement of larger (µm) colloidal tracers (tagged with anotherfluorophore) from which we map the velocity field and alsothe viscosity field. We exploit the high acquisition rate of theautomated microscope to scan the inside of the droplet, as il-lustrated in Fig. 1. Among the standard velocimetry methods,those based on particle tracking algorithms (see for a recentreview reference 16) offer a major advantage when applied tothe problem of drying droplets. Indeed, with such methods,one has access to the time-resolved positions of the tracers inan image sequence. Using colloidal tracers, the local meandisplacement leads to velocity measurements while the meansquare displacement associated to Brownian motion allows tomeasure the self diffusion coefficient of the tracers17. Thelatter can then be related to the mean-field local viscosity ofthe suspension of nanoparticles. The analysis of the Brownianmotion of tracers is usually designed as passive microrheol-ogy18 and has been the subject of intensive work during thepast decades, but mainly for its application to complex fluidscharacterization (see for a review reference 19). In the presentwork, we limit our measurements to the newtonian regime, butcombine viscosimetry and velocimetry.

To summarize, the technique reported in this paper leads tospace and time-resolved measurements of three quantities si-multaneously: the suspension concentration, the velocity, andthe suspension viscosity. In the following, we report a set ofresults that illustrate the technique, and allows us to discusssome features of the coffee-stain problem. Here, we limit ourattention to a series of experiments performed in a single planelocated just above the substrate, but the method could be ap-plied with a few improvements to the whole droplet.

2 Experimental

We image the drying kinetics of a colloidal suspension directlyinside the drop. To do so, we use a fast two-color confocalimaging setup which permits the measurement of the intensityfield by coloring a colloidal suspension of nanosilica with afirst fluorescent probe. We then add another fluorescent probe,namely large colloidal tracers, from which we obtain a localmeasurement of the velocity and the diffusion coefficient. Theprinciple of the experiment is sketched in Fig. 1. In this part,we provide details on the system used, before explaining theprocedure used to scan the droplet and to measure the fluores-

cence intensity, and the tracers velocity and mobility.

x

z

dilute latex µm-tracers

stripe of 10

frames in 20 sec

silica nanoparticles

1 frame = 120 images on 2 detectors at 60 Hz

stage displacement

confocal

depth

detector 2detector 1

laser 1 laser 2

extract local

velocity and mobility

extract local

concentration

Fig. 1 Schematic view of the sessile droplet of colloidal suspension(nanoparticles + colloidal tracers) sitting on a glass substrate andundergoing evaporation. The data acquisition process consists inrapidly and repeatedly scanning the drop with a confocalmicroscope along the x-direction at a fixed z-position; at eachx-position, 120 images are acquired at 60 Hz, each frame containingtwo tracks for the two-color detection, and then the scanningposition is moved. A cycle consists of 10 adjacent frames, lastsabout 30 s, and is iterated until the complete drying occurs. Fromthe two-color analysis, we extract the concentration field of silicananoparticles (detector 2, see also Fig. 2), the mobility field of largetracer particles and their velocity field (detector 1, see also Fig. 3).

2.1 Colloidal System

The colloidal suspension consists of nanosilica coated withrhodamine. Ludox AM-30 (Sigma-Aldrich, nominal sizerS ≈ (6.5± 1.0) nm, ≈ 20% w/w) is mixed with rhodamine(Fluka, maximum of absorption at 550 nm and emissionaround 575 nm) which colors the silica by strong spontaneousionic adsorption20. After thorough mixing (24 h), centrifu-gation (4000 rpm, 4 h), dialysis (against deionized water for

2 | 1–11

24 h), and recalibration via a dry extract, we obtain a slightlyviscous, vivid pink liquid that we shelter from light. The vol-ume fraction we use varies between φ = 0.25 and 5%.

In this suspension, we add another fluorescent probe: a dis-persion of latex particles (Yellow-Green Fluorospheres fromInvitrogen, made of polystyrene and stabilized with carboxy-late surface groups, at a volume fraction φ = 0.1%, rL =0.55± 0.01 µm) dyed with fluorescein (maximum absorptionaround 490 nm and emission in the range 505-515 nm) whichact as velocity and mobility tracers through the image analysisof their small displacements.

2.2 Imaging Setup

Fast confocal microscopy (Zeiss Live 5 LSM) is used to obtainimage sequences of droplets of the particle suspension duringevaporation. We take advantage of the different spectra of thefluorescence signal coming from the tracers and the silica par-ticles. Two lasers (solid-state laser diodes of wavelengths 488nm and 561 nm) illuminate continuously the sample while thefluorescence intensity is acquired by two different detectors; aset of filters is chosen to select on two different channels theemission of the respective fluorophores (two band-pass filtersin the range 575–615 nm for the small, rhodamine-based col-loids and 495–550 nm for the large tracers). We thus obtainsat each frame a set of two tracks, one being essentially sen-sitive to the silica nanoparticles, the other one to the tracers.Note that the spectral separation of the colors is not ideal in thesense that there is a slight of overlap of fluorescence from thetwo probes, which is however not detrimental to the analysis.

The size of the image is 240× 240µm, with a resolutionof 512×512, so that the position of the large tracers is deter-mined with a good precision—the standard deviation is about50 nm thanks to the sub pixel analysis. The trajectory of thetracers are obtained from the analysis of sequences of 120 con-secutive images acquired at 60 Hz, using Matlab routines de-veloped by Blair and Dufresnes17,21. From the displacementsof the tracers, both the velocity field and the diffusion coeffi-cient field are measured.

In order to map the droplet during the drying process, thesample is moved every 2 s along a radial direction on 10 dif-ferent locations. Such a cycle is repeated about 20 times dur-ing the drying of the droplet. The z-position remains fixed, at10 µm above the glass surface, a distance much larger than thetracer size (rL ≈ 0.6µm) so that the diffusion of the tracers isnot affected by the presence of the wall22.

2.3 Intensity Profiles

The images that monitor the finely dispersed silica are recom-bined into stripes of intensity that are eventually averaged toyield intensity profiles against space and time. The reconsti-

Tim

e500 µm

x

edgecentre

Fig. 2 Stripes of fluorescence intensity due to rhodamine-coatedsilica nanoparticles, collected at 10 µm above the glass substratewith the confocal microscope on a slice of thickness of the order of1 µm.

tution of such a sequence requires however several stages ofpretreatment: all images are divided by a reference image tocorrect for the illumination imperfections of the microscope;overlap of fluorescence conditions between the two probes isactually corrected by filtering out the images from fluorescein-based fluorescence knowing the position of particles from theanalysis of the other track. The image of Fig. 2 shows such astack of intensity stripes or as a spatiotemporal diagram (seeFig. 5, top). We will explain later on in the text how to ex-tract the concentration field out of intensity and mobility fieldsthrough a self-consistent analysis.

2.4 Tracer Velocity and Mobility

In order to quantify the velocity and mobility of the tracers,we proceed in the following sequential way: we detect the po-sition of tracers; we then track them to establish their trajec-tories; we deduce their velocity which is then used to measuretheir mobility. One of the crucial issues of the process is thatthe fields we want to measure are not homogeneous in space;caution must be taken in the averaging methods.

The first stage consists in detecting the location of the par-ticles with a sub pixel accuracy, and to track the tracers inorder to obtain their trajectories. Fig. 3B shows an exam-ple of the trajectories obtained on one image sequence. Afterthe determination of these trajectories, the displacements areaveraged and correlated. The displacement vector ui(t,τ) =ri(t+τ)−ri(t), where ri(t) is the position of the tracer labeledi, is the sum of a diffusive displacement uiD(t,τ) and a con-vective one uiC(t,τ). In the dilute and semi-dilute regime, thesuspension is assimilated to a Newtonian liquid∗, where in two

∗This assumption is not correct in the concentrated regime, close to the kineticarrest. In this regime, the displacements are very small and fall below the

1–11 | 3

50 µm

10 µm

2 µm

A

B

C

Fig. 3 Tracer velocity and mobility measurements: (A) Example ofone of the 120 images acquired for particle (tracers) tracking; Thisimage is cut into boxes (vertical lines) to account for theheterogeneity of local velocity. (B) Magnification showing thetracking of individual particles. (C) Example of the diffusivedisplacement obtained once the ensemble averaged local velocityhas been substracted. The mean square displacement leads to thelocal mobility of the tracers (see also text and Fig. 4).

dimensions the mean diffusive displacement equals√

4D(r)τ

with D(r) the local self-diffusion coefficient. The convectivepart is set according to a velocity field v(r, t) which we mea-sure. By the use of ensemble or time average of the individ-ual displacements, one can distinguish between the convectiveand the diffusive one, since 〈uiD〉 = 0 and

⟨u2

iC

⟩=⟨uiC

⟩2;

thus⟨uiC⟩= 〈ui〉 and 〈uiD〉

2 =⟨u2

i⟩−〈ui〉2. Fig. 3C shows

examples of the diffusive trajectories after the elimination ofthe convective displacement.

In an evaporating process, both the velocity and the diffu-sion coefficient fields are space and time dependent, and thusensemble and time averages of the displacements must be de-termined only in selected small time and space windows. Thevelocity field actually fluctuates significantly on rather smalltime scale (∼ 0.1 s) but not on small length scales. Therefore,the convective displacement is estimated for each couple of

precision of the measurement. The assumption of a diffusive displacement isthus verified for all the non-vanishing data reported in this article.

images, by taking the ensemble average of the individual dis-placements. We take advantage of the cylindrical symmetry ofthe droplet. All the local properties vary mainly along the x-axis (the radial direction), and we assume variations are smallin the y-direction. We thus divide each image into 10 rectan-gular boxes of dimensions 24×240µm centered at a distancex from the center of the droplet, the large edge being orientedalong the y-axis† (see Fig. 3A). All the ensemble averages arecalculated on the particles inside these boxes, one box con-taining about 50 particles. Given the time scale of the dryingprocess (∼ 500 s), we assume that the mean velocity field andthe diffusion coefficient field is constant during the period Tof acquisition of a sequence of images starting at time t:

v(x, t) =⟨〈ui(t ′,τ)〉i

τ

⟩{t ′∈[t,t+T ],τ∈[0,τ max]}

. (1)

Note that the time average is weighted by the number of corre-lations of the ensemble averages. We verified that the velocitydoes no depend on T nor on τmax, when T is on the order ofa few seconds and τmax is a few tenth of seconds. For all ex-periments reported here, we used T = 2 s, and τmax = 0.33 s.These choices were guided by the need to map temporally andspatially the drying of a droplet with a good temporal resolu-tion; indeed, we decided that the duration of the measurementat one location should not to exceed 2s so that the scanningprocess of the drop (10 frames) lasts about 30s, when the timerequired to move the stage is added. We also verified that valueof the y-component of the velocity is always at least one orderof magnitude smaller than the x-component.

The mean-square displacement is then calculated accordingto: ⟨

u2D(x, t,τ)

⟩=⟨

ui(t ′,τ)2−⟨ui(t ′,τ)

⟩2i

⟩{i,t ′∈[t,t+T ]}

. (2)

As shown in Fig. 4, the diffusive mean square displacementincreases linearly with the correlation time τ, as expected for aBrownian particle. This allows the determination of the localself-diffusion coefficient, with a typical accuracy of 10% inthe range from a few 10−14 to a few 10−12 m2s−1. One mayin principle be able to measure lower diffusion coefficients,but longer correlation times would then be required to reducethe influence of the particle position uncertainty. The highermeasurement limit arises from the acquisition rate limited inour instrument to 60Hz.

The uncertainty of the measurements mainly originatesfrom the difficulty to distinguish diffusive and convective dis-

† In principle, the cylindrical symmetry of the droplet ask to calculate the en-semble averages inside annular regions rather than in rectangular boxes. How-ever, such a calculation requires to know the center of the droplet with a goodprecision. It would lead to negligible effects except in the central region,where the velocities are small and thus do not alter the measurement of thediffusion coefficient.

4 | 1–11

0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5x 10

−13

Correlation time τ (s)

Me

an

sq

ua

re d

isp

lace

me

nt

(m2)

time

time

φi = 1.23 10

− 2

φi = 4.93 10

− 2

Fig. 4 Mean square displacement⟨u2

D (τ)⟩, obtained on two

droplets of initial concentrations 1.23% and 4.93% at differentdrying times. Error bars are given by u2

D/√

N, where N is thenumber of correlations, ranging for these experiments between 500and 5000 depending on the correlation time. The solid lines are thebest parabolic fits to the data using

⟨u2

D (τ)⟩= σ2 +4Dτ+σ2

vτ2.The absolute uncertainty of the particle position σ is of the order of0.1 pixel. The quadratic term σ2

vτ2 accounts for both an error in thedetermination of the convective displacement and a velocity gradientinside the averaging boxes (see text).

placements. The method detailed above assumes that the ve-locity is homogeneous inside one averaging box. It also as-sumes that the uncertainty on the convective displacement islower than the amplitude of diffusive one. Both these assump-tions are not verified for long correlation times. Indeed, theconvective displacement and its standard deviation increaselike σvτ, where σv is a typical value of the standard deviationof the velocity estimation inside one averaging box, whereasthe diffusive one is given by

√4Dτ. This defines a thresh-

old of 4D/σ2v for the correlation time, above which the errors

in the estimation of the convective displacement dominate themeasurement. The value of σv is rather difficult to estimateand strongly depends on the averaging box size and on thelocal velocity gradient. For large averaging boxes, the errordue velocity gradient dominates, but when the boxes are toosmall, the standard deviation of the ensemble average is im-portant due to a small number of particles. The correlationtime threshold, though difficult to estimate a priori, could bemeasured by increasing the range of correlation time. We finda typical value of 10 s, well above the range of correlationtimes that we used (τmax = 0.33 s). However, in some regionsof the droplet where there are high velocity gradients or whentime fluctuations are exceptionally very high, this value couldbe decreased to a few tenths of seconds. In order to limit mea-surement errors, the mean square displacement data are fitted

by a parabolic function that accounts for the quadratic termthat is discussed above, and reads

⟨u2

D (τ)⟩= σ2+4Dτ+σ2

vτ2,where σ2 accounts for the absolute uncertainty in the deter-mination of particle position. By doing so, the precision ofthe measurement is greatly improved and allows us to verifythat all the measurements fall in the range where diffusive dis-placement dominates.

3 Results

Fig. 5 displays the typical outcome of an experiment, shownhere as spatiotemporal maps: intensity field of the nanopar-ticles (top), velocity field (middle) and diffusion coefficientfield (bottom) of the tracers. These data were acquired repeat-edly in time as quickly as possible during the course of dry-ing at different positions and then recombined to produce themaps. The intensity map shows that the contact line remainspinned during the drying. It reaches rather high values nearthe edge of the droplet quite rapidly, and then a high inten-sity region develops from the edge, indicating that the silicaparticles accumulate.

Correlatively, the tracer mobility shown in the diffusion co-efficient spatiotemporal map in Fig. 5 is greatly decreased inthis corner region, while it remains almost uniform in the cen-tral region of the droplet. Both quantities thus qualitativelycoincide at first sight. At the end of the drying, (about 550 sfor the experiment reported in Fig. 5), the intensity in the edgeof the droplet falls to zero. This is due to the delamination ofthe deposit from the glass substrate.

The velocity spatiotemporal map shows that, in the mea-surement plane, just above the surface, there is a net flow fromthe centre to the edge. The point where the velocity changesits sign is used to determine the exact position of the centre.Similarly to intensity and mobility map, the velocity is ap-proximatively uniform in the measurement plane, and is onthe order of 10 µms−1. In the high concentration region nearthe edge and at the end of the drying, the velocity is greatlydecreased.

In the rest of this work, we correlate these maps in orderto extract quantitative features about the drying kinetics seenfrom inside the drop.

3.1 Nondimensional numbers

Let us start with the usual review of nondimensional numbersthat are useful to better understand the physics at work. First,we recall the typical dimension of the drop: a radius that startsat R ≈ 1 mm with a height h of a few hundreds of microns(contact angle of about 40◦). In the drop, we measure a veloc-ity field with values that range in 10−100µms−1 for a solutionmade of nanoparticles (rs ≈ 6 nm) suspended in water with astarting viscosity close to that of water.

1–11 | 5

tim

e (

s)

I (a.u

.)

200

400

600

1

2

tim

e (

s)

200

400

600

x (mm)

tim

e (

s)

-0.5 0 0.5 1 1.5

200

400

600

centre edge

Vx

(ms

−1

)

−5

0

5

x 10−6

D (m

2s −

1)

0

2

4

x 10− 13

Fig. 5 Spatiotemporal analysis of the drying inside the droplet:intensity (top), diffusion coefficient (middle), and velocity profiles(bottom) against space and time. The narrow dotted lines indicatethe domain in which the velocity and the diffusion coefficient aredefined. Outside this region, there are no tracers on the images, dueto the delimination.

The Reynolds number, which compares inertia to viscousdissipation in the drop, is calculated for water on the heightof the drop: Re≡ ρvh/η with ρ the density of the fluid and η

its viscosity. Re is in the range 10−3− 10−2, indicating thatinertia is not relevant. Beside, Re can only decrease during theprocess as h shrinks and η increases significantly.

The Peclet number compares the efficiency of convectionto that of diffusion Pe≡ vR/Ds where Ds is the self-diffusioncoefficient of the nanoparticles (Ds ≈ 310−11 m2s−1). It islarge (Pe ≈ 102−103) when calculated on the largest dimen-sion of the drop R and remains significant when calculated onits height (Pe > 10). Therefore, diffusion plays no significantrole in modifying the concentration field of nanoparticles inthe most significant part of the drop.

The capillary number Ca ≡ ηv/γ compares the effect ofsurface tension to that of the viscous dissipation in the drop,where γ is the surface tension between the liquid and theair γ ≈ 70 mNm−1 for the air/water interface. Ca ranges in10−7− 10−6 with the typical values given before indicatingthat capillarity dominates viscous dissipation and fixes the

shape of the drop. Note however that during the drying ki-netics, the viscosity of the solution shall increase which mightalso diminish the order of magnitude of the velocity, thereforekeeping Ca roughly unchanged.

Finally, we will show just below that the Marangoni num-ber Ma, which compares the surface-induced stress to bulkviscous dissipation, is large. We will devote a specific part tothis point which implies that evaporation induces significantthermal gradients that in turn induce surface stress, which thengenerates a recirculating flow inside the drop.

We are thus in a situation where the shape of the drop corre-sponds to a spherical cap, where convection dominates insidethe drop and recirculations are active and probably homoge-nize the bulk of the drop, although not sufficiently to preventthe formation of a stain at the edge of the drop and a pinning ofthe contact line, as obvious from Fig. 5. Based of these quan-titative features, we will explore successively the role of theseinner eddies on the drying kinetics, then the role of rheologyand kinetic arrest of the colloidal suspension on the formationof the corner, and how it relates to the final fade-out of thedrying kinetics.

3.2 Velocity fields

Fig. 6 shows the velocity field measured at the beginning ofthe drying experiment (during the first minute of the drying) inthe monitoring plane located 10 µm above the glass substrate.Here and in the following of the paper, we switch to cylindricalcoordinates centered in the centre of the droplet (r = |x|), anduse dimensionless quantities defined by r = r/R, t = t/t f andv = vR/t f , where R is the initial radius of the droplet (about 1mm), and t f is the total drying time of the experiment (about1000 s).

Except for the most concentrated solution, the radial veloc-ity component is quantitatively independent on the concentra-tion. It should be noticed that the viscosity of the most concen-trated solution is about 4 times higher that that of the others,which may explain the velocity reduction evidenced in Fig. 6for this initial concentration.

One specific feature of the velocity field strongly suggeststhat there is a significant circulation flow directed radially out-ward along the substrate. Indeed, Fig. 6 (top) shows that vrdrops to zero when r→ 1; the flow must therefore develop avertical component, namely a recirculation. This is confirmedin Fig. 6 (bottom) by calculating the second flow componentvz thanks to the continuity equation in axisymetric geometry,i.e. ∂zvz +∂r (rvr)/r = 0. The integration of vz could be doneby assuming a no-slip boundary conditions at the solid sur-face and a constant velocity gradient from the surface to themonitoring plane. This last hypothesis should be well veri-fied at least in the central region, where the droplet thicknessis much greater (300 µm) than the z-position of the measure-

6 | 1–11

0

2

4

6

vr

Ma = 500

Ma = 250increasing volume

fractions

centre edge

0 0.2 0.4 0.6 0.8 1−0.1

0

0.1

r

vz

4.9%

2.5%

1.2%

0.25%

Volume fractions

Fig. 6 Radial and vertical velocities vr and vz as a function of theradial distance r = r/R from the centre, at the beginning of thedrying, for several initial volume fraction. The velocities arenormalized by R/t f where R is the radius of the drop and t f thedrying time. For all experiments, the order of magnitude of R/t f is 2µm/s. The radial component of the velocity is determined directlyfrom the tracers paths using eq. (1) while the vertical one iscalculated from the continuity equation (see text). A singleexperiment is shown for each initial concentration, but leads to twosuperimposed curves due to symmetry (x < 0 and x > 0).

ment plane. Thus the z-component of the velocity field couldbe estimated by vz '−z∂r (rvr)/r. In Fig. 6 (bottom), the ver-tical component, although much smaller than the radial one, isthat of a recirculating flow directed outward near the substrateand inward along the droplet free-surface.

Several authors have predicted and observed such a circu-lation flow, that has been attributed to a thermal Marangoniflow12,13. The direction of the circulation has been predictedto depend on both substrate and fluids thermal conductivity,and on the contact angle. For our experiments, that is wateron glass with a contact angle of about 40◦, the circulation di-rection is predicted to be outward along the substrate, whichis indeed the direction we observe.

The estimation of the Marangoni number Ma ≡−β∆Tt f /ηR is a rather difficult task either theoreticallyor experimentally; here β ' −0.17 mN/m/K‡ is the surface

‡http://www.surface-tension.de

tension variation with respect to the temperature and ∆T acharacteristic temperature drop inside the droplet. Hu andLarson12 established an analytical expression for the velocityfield under the lubrication approximation which we use to geta rough estimate of Ma. In Fig. 6, we plotted their solutionwith Ma = 250 and 500, the other parameters being set to theexperimental values. Although the theoretical velocity fielddoes not account for the radial variation of the experimentalone, maybe due to the lubrication approximation, a value ofMa between 200 and 1000 seems necessary to account forthe order of magnitude of the velocity we measured. At thesevalues, the Marangoni flow is much larger than the one dueto evaporation. Let us note that Ma = 500 corresponds toa temperature variation of about 0.01◦K inside the droplet,an order of magnitude which is in good agreement with thenumerical simulation of reference12.

Despite the reasonable agreement between our results andthe above cited theoretical predictions, we should mention thatthere is no evidence that the driving force of the Marangoniflow is the temperature gradient. The suspension may con-tain contaminants that could slightly decreases the surface ten-sion, as, for example, the rhodamine used for tagging the silicananoparticles. Complementary experiments are needed to dis-criminate between thermal and solutal Marangoni flow.

It turns out that even at Ma = 500, the particle accumula-tion around the edge of the particle is very effective despite aMarangoni flow that dominates the one driven by the evapo-ration and induces a recirculation in bulk that (tends to) ho-mogenize the concentration field; the actual flow inside thedrop can thus be seen as a superimposition of a recirculat-ing flow and a radial one. Yet, at much higher values (a few104), the Marangoni flow has been observed to totally sup-press the mechanism of particle accumulation7 and as pointedby Ristenpart et al., the influence of intermediate Marangoniflow on the coffee-ring deposition is an important question andasks for deeper work13. The technique we report here seemsquite appropriate to obtain experimental insights on this pointand promising developments also concern the measurement3D velocity fields with a good accuracy.

3.3 Rheology of the suspension

We give now a consistent way to measure the rheology of thesuspension (i.e., viscosity η against volume fraction φ) fromthe mobility and intensity fields. It should be applicable toany solution assuming that it can be fluorescently colored anddoped with tracers in order to measure simultaneously the twofields. The basic idea is simple: during evaporation, the soluteaccumulates at the edge of the drop, which therefore developsconcentration gradients; we thus correlate the fluorescent in-tensity to the mobility which also spans a large range of valuesand reconstitute a link between mobility and concentration.

1–11 | 7

0 0.02 0.04 0.06 0.08

10−3

10−2

10−1

Volume fraction

Viscosity (Pa.s)

0 0.05 0.10

5

10

Volume fraction

Fluorescence intensity

100

101

10−15

10−14

10−13

Fluorescence intensity (a.u.)

Diffusion coefficient (m

2/s)

Fig. 7 Top: mapping between the diffusion coefficient and thefluorescence intensity obtained from Fig. 5 top and middle through ahigh order polynomial fit of degree 9 (solid line), see text for details.Bottom: viscosity of the suspension as a function of the volumefraction of particles. The large open circles correspond to the valuesobtained at the beginning of the experiment in the middle of thedrop where the volume fraction is known with great accuracy. Thevolume fraction values of the other points are then determined fromthe intensity measurement using a parabolic interpolation of theinitial intensity as a function of the initial volume fraction (shown ininsert; the solid line extralopates the data points up to 10%). Thesolid line correspond to a linear fit in the dilute regimeη = η0 (1+2.5αφ) with φ < 1.2510−2 where αφ represents aneffective volume fraction. We obtain α = 6.5(±1).

Assuming that both the fluorescence intensity and the self-diffusion coefficient are monotonic (here increasing) functionsof the local particle concentration, we can correlate these twoquantities for every time and position. It provides a methodto determine the absolute fluorescence intensity, although it isnot directly measured for practical reason. Indeed, the laserpower and the detector gain needed to be adjusted in order tohave a good measurement precision for different conditions.

To recover an univoque link between I and D, we proceedas follows: for each experiment, labeled k, we assume thatthe measured intensity is proportional to an absolute inten-sity Iik with a coefficient αk. For the set of experiments withdifferent initial concentrations, there is a large number of cou-ples (αkIik,Dik). We however assume that there is an uniquerelation I = f (D) and we fit all the data once with an arbi-trary polynomial function of high order. For such a function,the mean-square optimization problem is linear and the resultleads to the determination of the factors αk of each experi-ments along with the set of coefficients for the polynomialform.

We show in the upper part of Fig. 7 the mapping betweenI and D once we have accounted for the unknown coefficientαk, along with the polynomial form. Also, it allows to checkout the validity of the fit as it should be independent on thedegree of the polynome, as it is indeed for degrees between 3and 11, with less than 10% variations on the αk coefficients.

Besides, we use the fluorescence intensity measured atthe very beginning of the experiment right in the middle ofthe droplet—where we know exactly the volume fraction—to determine empirically an intensity/concentration calibrationcurve, shown in the insert of Fig. 7. As expected for a con-centrated fluorescent dye, the relation is not linear, yet highlyreproducible. We extrapolate this behavior up to 10% by apolynomial fit (and also checked experimentally that the ex-trapolation was consistent with the measurements).

Then, using this intensity/concentration calibration, the dif-fusion coefficient can be measured as a function of the volumefraction of the particles. Eventually, using the Stokes-Einsteinrelation η = kBT/6πDR, we obtain the viscosity of the solu-tion against the volume fraction. In this process, we assumedthat the viscosity at φ = 0 must extrapolate to that of water at25◦C (η0 = 0.89 mPas) and we adjusted the tracer size. Thevalue found for the radius of the tracers (≈ 625 nm) is not inperfect agreement with the one provided by the manufacturer(550±10 nm).

Fig. 7 (bottom) shows the rheology of our suspension ob-tained from all sets of experiments and thus covering a widerange of volume fraction, from the very dilute regime up to theconcentrated regime. The neat collapse of all the results ontoa single curve validates the approach. When comparing thesedata to classical colloidal systems, in both the very dilute andin the concentrated regime, an effective particle volume frac-

8 | 1–11

tion of about 6 times the expected one is found. Indeed, theviscosity increase at low concentration is much greater thanthat of a hard-sphere (HS) suspension and follows Einstein re-lation η = η0 (1+2.5Φ) yet with an effective volume fractionΦ = (6.5± 1)φ, and the apparent divergence of the viscositytakes place at φ≈ 0.1≡ΦHS/6.

We suggest here two possible mechanisms that may accountfor this result. First, electrostatics play a dominant role in theparticle stabilisation, although the exact charge of the silicananoparticles must be altered by the adsorption of rhodamineand remains unknown. Nevertheless, we may naively thinkthat the apparent size of the particles is modified by electro-statics, to some degree, up to the Debye length λD. The latteris large in deionized water but our tracer particles contains an-tibacterial agents (sodium azide, down to ≈ 10−1 mM) andthus λD may well range in 10−100 nm; An apparent volumefraction which accounts for a large excluded volume effect canbe written as Φ ≈ φ(1+λD/rp)

3 in a first approximation andis expected to deviate significantly from φ. Alternatively, thesuspension may also be made of small fractal flocks of aggre-gated nanoparticles having a high apparent volume fraction,although we have not been able to evidence them using a stan-dard dynamic light scattering equipment.

It remains that this rheology is robust as it does not de-pend in a first approximation on the preparation conditions,initial concentration, history of sample including kinetic anddrying effects, etc. We therefore keep in mind that the vis-cosity increases smoothly and significantly, and that aroundφ ≈ 8− 10% it has reached about 100 times that of water.This value also corresponds to the loss of tracers mobility,and we shall call it the kinetic arrest concentration for rea-sons that will become obvious soon. It also corresponds tothe macroscopic observation that around this concentration,the system has gelled, consistent with the phase diagram ofcharged nanocolloids23. It has tremendous effects on the ki-netics of drying and especially on the way a solid depositbuilds up at the edge of the drop while the latter gets loadedwith nanoparticles.

3.4 The build-up of a solid corner

The calibration curve between fluorescence intensity and con-centration permits to reconstruct the volume fraction profilesin time and space inside the drop. Note that we extrapolatethe calibration curve above its domain of validity, in princi-ple restricted to the measurements (φ < 5%); however, thisextrapolation holds on a limited domain of volume fraction(5 < φ < 8%) and therefore seems acceptable in terms of flu-orescence intensity (but we did not extrapolate that far for therheology).

We show in Fig. 8 (to be read columnwise) the time andspace-dependent concentration fields and diffusion coefficient

fields of the tracers for several initial conditions, ranging froma dilute solution (φ0 ≈ 0.2%) to a concentrated one (φ0 ≈ 5%).The drying kinetics leads to an increase of the concentrationtoward the edge of the drop (the stain effect) along with an in-crease of the concentration in the middle of the drop, an effectpurely due to evaporation. It is quite clear that the concentra-tion increases strongly in a localized portion of space, close tothe edge, and that once the volume fraction reaches φ≈ 8%, afront develops. This view is somewhat less pronounced for themost concentrated system where the profiles are smooth andmay reach φ ≈ 12% in the corner. This last value is howeverhigh compared to the validity range of the intensity calibra-tion, and it should be handled with care.

We thus observe here that the particles accumulate at thecorner until they reach a volume fraction close to that of thekinetics arrest, φc ≈ 8%, as defined on the basis of the rheol-ogy curve. The exact value of the concentration in the gelledphase actually slightly overcomes φc but hardly depends onthe initial concentration which varies on one order of mag-nitude. This is, to our knowledge, the first experimental ev-idence in the case of a sessile droplet that supports the so-called truncated dynamics10,11,24, a mechanism which postu-lates that a shock front develops and propagates inward, insidethe drop, once the volume fraction at the edge has reacheda given threshold concentration φc. However, these theoriesrarely address the origin of φc; the value we observe here is un-expectedly small and shows that both the phase behavior14,25

and the rheology of the colloidal sample have a crucial role indetermining the morphology of a drop undergoing drying.

There is a strong qualitative resemblance between our ex-perimental results and the recent theory by Zheng10 whichworks out such a truncated dynamics. At a first sight, the com-parison seems audacious as Zheng uses the lubrication theoryalong with a thin slab geometry, in which there is no recircu-lation. We believe that the model he uses essentially relies onconservation laws, which make it robust. A significant dis-crepancy is expected to originate from the rheology, not in-cluded in Zheng’s work, and which will modify the flow inthe drop. This may be the origin of the very smooth concen-tration profiles we observe when we start an experiment at ahigh enough volume fraction, see Fig. 8.

3.5 Final kinetics

Our results clearly evidence that a front develops once a givenconcentration has been reached at the edge of the drop; thisfront separates a dense phase from a still liquid phase. We alsonoticed that the concentration in the central region of the dropincreases with time. It gives rise to an abrupt final kineticswhere the front speeds up as being fed by a solution which ismore and more concentrated; the final stage of the drying isobtained when the front reaches the centre of the drop, at a

1–11 | 9

time

centre

edge

time

Φ0=0.25% Φ0=1.23% Φ0=2.43% Φ0=4.93%

0

4

8

12Φ (%)

00

1

1 0 1 0 1 0 1

2

3

4

D (m2s-1)

x 10−13

r~

r~

r~

r~

Fig. 8 Volume fraction (top) and diffusion coefficient (bottom) profiles measured inside the drop, for several initial volume fraction φ0. Forthis system, the kinetic arrest occurs at φ≈ 8%, see Fig. 7.

time t f .A simple argument of volume conservation predicts that t f

depends on the initial volume fraction φ0 and on that of thefinal stage φg: t f = te(1−φ0/φ f ) where te is the time neededto evaporate a drop of pure water in the same conditions. Wefound φ f = (8± 2)% (Fig. 8, insert), which is an indirectway to confort our results on the truncated dynamics with anequivalence φ f ≡ φc.

−600 −400 −200 00

0.2

0.4

0.6

0.8

1

t−tf (s)

L(t

)/R

0

0 5 100

500

1000

φ (%)

tf

Fig. 9 Final kinetics expressed as a function of the time from theend of the process t− t f and measured from the diffusion coefficientfield. The solid line is a guide with L/R∼ (t− t f )

0.5, see text. Theinsert shows the drying time t f against the volume fraction, and alinear fit which gives an estimate of the concentration of the finalstate, φ f = 8±2%.

Witten suggested recently11 that the evaporation kineticscan be split into two limiting regimes: the initial regime where

t � t f , and the final regime t ∼ t f which should be largelyindependent of the initial conditions. Then, the end-regimeshould exhibit a robust fadeout mechanism. However, his re-sults are obtained in a different condition, and hold only forthe departure of the front. Inspired by this theory, we foundthat there is also a final and abrupt dynamics when the frontreaches the centre of the drop, see Fig. 9. In this figure, wemeasured the position of the front L(t) in the late regime (seeFig. 10) by thresholding the diffusion coefficient maps (atD = 0.510−13 m2s−1) and we plotted this position as a func-tion of the time measured from the end of the kinetics t− t f .While in real time units t the final kinetics t f depends linearlyon the initial volume fraction (insert of Fig. 9), the kineticsseen from the end becomes essentially independent on the ini-tial conditions as illustrated by the nice collapse of all fronts.

gelled phase

L(t)

Fig. 10 (Top) A schematic view of the results: a droplet laden witha colloidal suspension, undergoing evaporation and for which theheterogeneous evaporation rate induces both recirculation andparticle accumulation; (Bottom) the same drop near the final stageof drying exhibits a fadeout kinetics that depends only on the timethat remains, see Fig. 9.

10 | 1–11

This collapse indicates that the very end of the drying mightbe accounted for by simple conservation laws. Let us assumethat the central region fluid of thickness h of radius L hasreached a concentration φ closed to φc, and that the total evap-oration rate remains constant (maybe essentially governed byevaporation across the gelled phase). Then water conserva-tion reads (1− φ)hL2 ∝ t f − t. In this late regime, neitherthe thickness nor the concentration could vary significantly.Thus the radius of the central region should vanish accordingto the asymptotic scaling L ∝ (t f − t)0.5. In Fig. 9, such arelation is plotted as a guide. A better experimental time reso-lution would be required to test this relation. The data reportedhere however support well the idea that the propagation of thegelled front greatly accelerates at the end of the drying, whichis simply accounted by conservation laws.

4 Conclusion

We described the use of fast confocal microscopy for imag-ing several time- and space-dependent fields in a droplet of acolloidal suspension undergoing evaporation. The techniquepermits to observe simultaneously the fluorescence due to thedispersed phase, here a silica nanosol, and the one due to par-ticles that act as tracers. The drop is repeatedly scanned ra-dially at high rate in order to obtain the intensity field of thenanosol, and the velocity and mobility fields of the tracers.We then analyze, recombine and correlate these fields in orderto quantitatively describe the dynamics of drying as seen frominside the drop. We observe that in the present case of an aque-ous drop on glass, evaporation generates a three-dimensionalflow due to a thermal Marangoni effect; however, the latteris not active enough to prevent the solute accumulation at theedge of the drop, the well-known stain effect; Fig. 10 (top)schematically recasts these results. We also show that we canextract in a self-consistent manner the rheology of the sus-pension directly from the intensity and mobility fields. Thenanosol shows a smooth increase of its viscosity with the con-centration, up to a one hundred times increase at a moderatevolume fraction φ≈ 8−10%. We also evidence that this valuecoindices with the one at which a front develops and invadesthe inside of the drop. This is in agreement with the truncateddynamics but offers an unexpected low value for the concen-tration at which it occurs. The systems remains very soft yetis able to oppose the capillary force exerted by the interface,which shows that the phase behavior and the rheology of thesample select the value at which the dynamics is actually trun-cated.

This work therefore opens up several routes for a deeper ob-servation of the drying droplet problem: 3D reconstruction ofMarangoni flows, quantitative assessment of the truncated dy-namics, the role of the rheology on the build up of the ‘solid’deposit, etc.

Acknowledgments

We thank Rhodia and Region Aquitaine for funding and sup-port and J.-B. Salmon for a critical reading of the manuscript.

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