Journal of Colloid and Interface Science 403 (2013) 49–57

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Journal of Colloid and Interface Science

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Evaporation kinetics of sessile droplets of aqueous suspensionsof inorganic nanoparticles

Anna Trybala a, Adaora Okoye a, Sergey Semenov a, Hezekiah Agogo b, Ramón G. Rubio b,Francisco Ortega b, Víctor M. Starov a,⇑a Department of Chemical Engineering, Loughborough University, Loughborough LE11 3TU, UKb Department of QuímicaFísica I, Universidad Complutense, 28040 Madrid, Spain

a r t i c l e i n f o

Article history:Received 25 February 2013Accepted 12 April 2013Available online 30 April 2013

Keywords:NanosuspensionEvaporationContact angle

0021-9797/$ - see front matter � 2013 Elsevier Inc. Ahttp://dx.doi.org/10.1016/j.jcis.2013.04.017

⇑ Corresponding author. Fax: +44 01509 223923.E-mail address: v.m.starov@lboro.ac.uk (V.M. Staro

a b s t r a c t

Evaporation kinetics of sessile droplets of aqueous suspension of inorganic nanoparticles on solid sub-strates of various wettabilities is investigated from both experimental and theoretical points of view.Experimental results on evaporation of various kinds of inorganic nanosuspensions on solid surfaces ofdifferent hydrophobicities/hydrophilicities are compared with our theoretical predictions of diffusionlimited evaporation of sessile droplets in the presence of contact angle hysteresis. The theory describestwo main stages of evaporation process: (I) evaporation with a constant radius of the droplet base whenthe contact angle decreases from static advancing contact angle down to static receding contact angle and(II) evaporation with constant contact angle equal to the static receding contact angle when the radius ofthe droplet base decreases. Theoretically predicted universal dependences for both evaporation stages arecompared with experimental data, and a very good agreement is found.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

Nanosuspensions comprise solid nanoparticles with a typicalsize of 1–100 nm suspended in a liquid [1,2]. The effects that thenanoparticles have on the fluid properties have been the subjectof a substantial interest over recent years [1]. Nanofluids have beenfound to possess enhanced thermophysical properties such as ther-mal conductivity, thermal diffusivity, viscosity and convective heattransfer coefficients as compared to those of base fluids like oil orwater [3–6]. As a result, nanosuspensions have found applicationsin wide technological areas, for example, industrial cooling appli-cation (automotive industry and microchips), smart fluids, extrac-tion of geothermal power and other energy sources, as well as innuclear reactors [2]. Investigations carried out over the past decadeby both engineers and scientists have resulted in findings whichsuggest that a very little quantity of nanoparticles, less than 1%, re-sults in an abrupt enhancement of the thermal conductivity of thesuspension [1,7]. The application of nanosuspensions and nano-emulsions has drawn attention in a broad array of fields includingmicrofluidics, biochips and electronic equipment. Nanosuspen-sions have also a wide range of biomedical applications. They areused in drug delivery [8], cancer therapeutics [9] and nanocryosur-gery [10]. Nanosuspensions have been used in detergency becausetheir behaviour is different from that of simple liquids with respect

ll rights reserved.

v).

to spreading and adhesion on solid surfaces [7,11, and 12]. Becauseof that, nanofluids are also useful in the processes of soil remedia-tion, lubrication and oil recovery [13].

Kinetics of simultaneous spreading and evaporation, which playan important role for the formation of various devices and systems,has been recently investigated [14–17] since it is significant in awide range of industrial applications such as painting, coating,ink-jet printing, premixing of fuel with oxygen in air [18], particledeposition applications and DNA chip manufacturing [19]. Themeasurements of evaporation rate of droplets on different solidsurfaces can be used for production of materials providing optimalregime of work in air conditioners, dryers and cooling systems [20].The affiliation of the wetting characteristics of the solid alongsideevaporation has been extensively investigated by researchers fromboth theoretical and experimental points of view [21–26]. The dy-namic wetting and dewetting of sessile droplets in the course ofevaporation have been investigated on various hydrophilic and dif-ferent degrees of hydrophobicity surfaces [27–29]. The degree towhich a droplet wets a substrate is usually estimated usingYoung’s equation, which gives contact angle as a function of thethree interfacial tensions. However, it is not clear whether equilib-rium contact angles can be really obtained under experimentalconditions. Therefore, static advancing, had, and static receding, hr,contact angles are frequently used instead [30,31]. The staticadvancing, had, and static receding, hr, contact angles can be mea-sured by a number of different methods, for example, using thesessile droplet methods or the Wilhelmy plate method [32].

Nomenclature

Latina capillary lengthA(h) additional function of contact angle (Eq. (14))B(h) additional function of contact angle (Eq. (14))Ca capillary numbercsat molar concentrationD diffusion coefficient of the vapour in the airf(h) function of contact angle (Eq. (9))f0(h) function of contact angle (Eq. (11))F(h) function introduced in Eq. (1)g gravity accelerationH humidityJ total vapour fluxL radius of the droplet basel dimensionless radius of the droplet baseM molar massn the number of experiments in a seriesq slope coefficient in Eq. (24)t timeT temperature

Tsurf average temperature of the liquid–air interfaceT1 temperature of the ambient airU velocity of spreading/shrinkageV droplet volumeV0 initial droplet volume

Greeka coefficient in Eq. (23)b coefficient in Eq. (3)b⁄ coefficient in Eq. (24)c interfacial tension of the liquid–air interfaceh contact anglehad static advancing contact anglehr static receding contact angleK(h) function of contact angle according to Eq. (22)k confidence level equal to 0.9 in Eq. (8)l liquid dynamic viscosityq liquid densitysr dimensionless duration of the first stage of evaporations; ~s; �s dimensionless times in Eqs. (14), (16), and (19) respec-

tively

50 A. Trybala et al. / Journal of Colloid and Interface Science 403 (2013) 49–57

Enhanced wetting of nanofluids in comparison with the purebase fluids has been observed by Sefiane et al. [7,16], who investi-gated the effect that nanoparticles play in wetting and dewettingof solid substrates. The authors [7] studied experimentally spread-ing nanosuspension of aluminium in ethanol over hydrophobicTeflon-AF coated substrates. Results obtained using the advanc-ing/receding contact line analysis showed that the nanoparticlesin the vicinity of the three-phase contact line enhance the dynamicwetting behaviour of nanosuspensions of aluminium–ethanol atconcentrations up to approximately 1% by weight.

In the case when liquids wet completely the solid substrate,that is the case of absence of contact angle hysteresis, Lee et al.[17] described the whole evaporation process as determined bytwo competing mechanisms, spreading and evaporation. As a re-sult, a two stage process takes place until the total evaporationof the droplet: the first short spreading stage when the radius ofthe droplet base increases over time and the second longer stagewhen evaporation dominates, which leads to the shrinkage of thedroplet base [17].

In the case of partial wetting, the evaporation of sessile dropletsgoes through multiple stages [15,33]. The main reason for that isthe presence of the contact angle hysteresis [30,31]. The two mostimportant stages can be referred to as pinning and depinning of thethree-phase contact line [34]. Dependency of the contact angle andthe radius of the droplet base on time could be rather complicatedduring the second stage: they could be either a continuous or step-wise dependencies on time [25]. It has been established [32,35]that in the presence of contact angle hysteresis, the evaporationof a sessile droplet in unsaturated vapour atmosphere can gothrough four consequent stages which are described below (Sec-tion 2.1).

Temperature of an evaporating droplet is different from theambient temperature and almost constant in the course of evapo-ration. It was shown by David et al. [36] that temperature in thebulk of a sessile evaporating droplet depends on the thermal prop-erties of the substrate and the rate of evaporation. The dependenc-es of the total vapour flux, J, on the radius of the droplet base, L, andon the contact angle, h, were investigated by Semenov et al. [37].The authors carried out numerical simulations which were per-formed taking into account both local heat of vaporisation and

Marangoni convection. It was suggested in [37] to use averagetemperature of the liquid–air interface for calculation of the va-pour concentration at the droplet–vapour interface. In this case,the dependences on the evaporation rate follow that in the isother-mal case [37]. The latter observation is used below.

A deposit remains on the solid surface after nanosuspensiondroplets evaporate completely and dry out. Deposits show variedpatterns: rings, flower-like, flat deposits, etc. [38,39]. Particlescan concentrate at the edges of the dry spot or cover all surfaceof the deposit. The appearance of the pattern depends on composi-tion of suspension (particle nature, size and concentration, ionicstrength, presence of surfactant, etc.), substrate, environment con-ditions as well as the dynamics of evaporation [40–43]. To the bestof our knowledge, the first theory to describe the pattern formationbased on DLVO theory was suggested in [39].

During formation of deposits, not only a pinned contact line butalso Marangoni flow is a significant factor [41,42]. Hu and Larson[41,42] evidenced both theoretically and experimentally the possi-bility of Marangoni flow reverse and formation of deposit at thecentre of the droplet.

2. Theoretical description of evaporation

It was suggested in [37] to use average temperature of the li-quid–air interface for calculation of the vapour concentration atthe droplet-vapour interface. In this case, the dependences of theevaporation rate follow that in the isothermal case:

dVdt¼ �2pDM

qðcsatðTsurf Þ � HcsatðT1ÞÞFðhÞL; ð1Þ

where t, V, L, H, D, q and M are time, the droplet volume, the radiusof the droplet base, humidity, diffusion coefficient of the vapour inthe air, the liquid density and the molar mass, respectively. Tsurf isthe average temperature of the liquid–air interface: Tsurf = (1/S)

RS-

TdS, where S is the area of the liquid–air interface; T1 is the temper-ature of the ambient air; csat(Tsurf) and csat(T1) are the molarconcentrations of saturated vapour at the corresponding tempera-tures; h is the droplet’s contact angle in radians; F(h) is a functionfor isothermal evaporation derived by Picknett and Bexon [25]and subsequently applied by Schonfeld et al. [44]:

FðhÞ ¼ ð0:6366hþ 0:09591h2 � 0:06144h3Þ= sin h; h < p=18ð0:00008957þ 0:6333hþ 0:116h2 � 0:08878h3 þ 0:01033h4Þ= sin h; h P p=18

(: ð2Þ

A. Trybala et al. / Journal of Colloid and Interface Science 403 (2013) 49–57 51

Only small enough sessile droplets with size of L 6 a are studiedbelow, where a ¼

ffiffiffiffiffiffiffiffiffiffiffic=qg

pis the capillary length, c is the interfacial

tension at the liquid–air interface, q is density of the liquid, g isgravity acceleration. For water droplets a � 2.7 mm. The latter al-lows neglecting the gravity action, and hence, the liquid–air inter-face has a spherical cap shape. The process of evaporation iscontrolled by the humidity of the atmospheric air and temperatureT.

Below Eq. (1) is rewritten as follows:

dVdt¼ �bFðhÞL; ð3Þ

where

b ¼ 2pDMqðcsatðTsurfÞ � c1Þ; and c1 ¼ HcsatðT1Þ: ð4Þ

According to [40], the temperature inside the droplet remainsconstant in the course of evaporation. The latter allowed theauthors in [37] to assume that the average temperature of thedroplet-vapour interface remains constant, that is b considered be-low as a constant over the duration of the whole evaporationprocess.

The parameter b was calculated from the experimental dataaccording to the following procedure. Integration of Eq. (3) at con-stant b gives:

VðtÞ � V0 ¼ �bZ t

0F½hð̂tÞ�Lð̂tÞdt̂: ð5Þ

Let us denote xðtÞ ¼R t

0 F½hðt̂Þ�Lð̂tÞdt̂, then Eq. (5) takes the fol-lowing form:

VðtÞ ¼ �b xðtÞ þ V0: ð6Þ

The time dependence of the droplet volume V(t) was taken fromexperimental measurements; x(t) was calculated using experimen-tal values of h(t) and L(t) and applying numerical integration overtime (second order integration method). Plotting V(t) vs. x(t) andfitting it with the linear dependence gives value of b. In this way,the parameter b was determined for each particular experiment.

The parameter b depends on the average temperature of the li-quid–air interface, Tsurf, which in its turn depends on both the ther-mal conductivity of a solid support and the liquid as well as on theMarangoni convection. Below, we consider evaporation of aqueousnanosuspensions on solid supports of various thermal conductivi-ties. It was shown [37] that Tsurf is becoming higher at the increasein the thermal conductivity of the solid substrate and, hence, thebigger parameter b. That is, we should expect bigger values of bat evaporation of aqueous nanosuspensions on silicon wafers ascompared with that for polymeric substrates. As it was mentionedabove, the thermal conductivity of aqueous nanosuspensions sub-stantially depends on the presence of nanoparticles and their nat-ure [3–6]. Hence, we should expect a dependency of the parameterb on the type of nanoparticles used.

Calculation of a confidence interval Dbrandom. b was determinedfor each particular experiment in a series of identical experiments(not less than 5 in all cases). Obtained values of bi (i = 1. . .n, n is thenumber of experiments in a series) were averaged and used for thecalculation of a random error according to the following:

�b ¼Xn

i¼1

bi=n ð7Þ

Dbrandom ¼ tð1�k;n�1Þ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPni¼1ðbi � �bÞ2

nðn� 1Þ

s; ð8Þ

where tð1�k;n�1Þ is the Student’s coefficient, k is the confidence levelfor a two-sided confidence interval or in other words the probabilitythat the true value of b appears within the limits of the calculatedtwo-sided confidence interval ð�b� Dbrandom;

�bþ DbrandomÞ. Below,we use the confidence level k ¼ 0:9.

2.1. Four stages of evaporation in case of partial wetting

Below static advancing and static receding contact angles wereextracted from experimental data and cannot be predicted in theframework of our theory. However, the experimental method sug-gested allows determining both static advancing and static reced-ing contact angles from a single evaporation experiment. Let usestimate the capillary number, Ca ¼ lU

c , where l is the liquid dy-namic viscosity, U = dL(t)/dt is velocity of spreading/shrinkage,and c is the liquid–vapour interfacial tension. During the first stageof evaporation, Ca is equal to zero and can be estimated usingFig. 3b as Ca � 10�7� 1. The latter allows us to conclude thatthe measured contact angles are really static advancing and staticreceding contact angles and not different from those measured byother methods [45]. According to Semenov et al. [33], the wholeprocess of droplet evaporation in the case of partial wetting, whencontact angle hysteresis exists, can be subdivided into four stages:(0) an initial short stage, when a deposited droplets spreads froman initial contact angle to static advancing contact angle, had, (I)during the first stage, the radius of the droplet base, L, remains con-stant and equals to its value L0 at the beginning of this stage, whichis the maximum value of the radius of the droplet base. During thisstage, the value of the contact angle, h, declines from its initial va-lue, which is equal to the static advancing contact angle, had, downto the static receding contact angle, hr, in the end of the first stage;(II) during the second stage the contact angle retains the constantvalue equal to the static receding contact angle, hr, whereas the ra-dius of the droplet base declines from L0 down to some small value(III). The second stage is followed by a relatively short final stagewhen both the radius of the droplet base and the contact angle de-crease simultaneously [45]. During this final stage, both the radiusof the droplet base and the contact angle are very small and diffi-cult to monitor. Probably, surface forces (disjoining/conjoiningpressure [30,31]) become important on this stage. Below, we con-sider only the two the longest stages (I) and (II) (Fig. 1).

According to our previous assumption, the droplet shape re-mains spherical; hence, the droplet volume V is:

V ¼ L3f ðhÞ; ð9Þ

where f ðhÞ ¼ p3ð1�cos hÞ2ð2þcos hÞ

sin3 h. The radius of the droplet base remains

constant during the first stage of evaporation. The latter allows us torewrite Eq. (3) as follows:

L20f 0ðhÞdh

dt¼ �bFðhÞ; ð10Þ

with initial condition

hjt¼0 ¼ had: ð11Þ

Let us introduce the following dimensionless time s = t/tch,where tch ¼ L2

0=b is the characteristic time of the process. Usingthe dimensionless time s Eq. (11) can be rewritten as:

θ1 θ2θ3

θ1>θ2>θ3

θ θθ

θ

Fig. 1. Two stages of droplet evaporation. First stage (top): droplet evaporates with a constant radius of the droplet base while the contact angle decreases with time. Secondstage (bottom): contact angle remains constant, while the radius of the droplet base decreases with time.

52 A. Trybala et al. / Journal of Colloid and Interface Science 403 (2013) 49–57

f 0ðhÞ dhds ¼ �FðhÞ: ð12Þ

Integration of Eq. (13) with boundary condition (12) leads to thefollowing solution:

Aðh; hadÞ ¼ s; ð13Þ

where Aðh; hadÞ ¼R had

h ðf 0ðhÞ=FðhÞÞdh. Evaporation stage (I) continuesuntil the contact angle reaches the static receding contact angle.That is, from Eq. (14), we conclude that the end of the stage onecan be determined as:

Aðhr; hadÞ ¼ sr; ð14Þ

where sr is the dimensionless duration of the first stage.It is convenient to rewrite the latter solution as follows: Eq. (14)

can be rewritten as:Z p=2

h

f 0ðhÞFðhÞ dh ¼ sþ

Z p=2

had

f 0ðhÞFðhÞ dh:

Let BðhÞ ¼R p=2

hf 0 ðhÞFðhÞ dh ¼ Aðh;p=2Þ. The new dimensionless time

for the first stage of evaporation is determined by a simple shift as:

~s ¼ sþ BðhadÞ: ð15Þ

Now, solution of Eq. (14) can be rewritten as:

BðhÞ ¼ ~s: ð16Þ

During the second stage of the evaporation process, the contactangle remains constant. However, radius of the droplet base de-creases. Hence, now, Eq. (3) can be rewritten as follows:

3L2f ðhrÞðdL=dtÞ ¼ �bFðhrÞL: ð17Þ

An introduction of a new dimensionless time as �s ¼ ðs� srÞ 2FðhrÞ3f ðhrÞ

and a dimensionless radius of the droplet base l = L/L0 allow rewrit-ing Eq. (18) as:

dl2

d�s¼ �1; �s > 0: ð18Þ

Direct integration of Eq. (19) with initial condition of l(0) = 1 re-sults in:

lð�sÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffi1� �sp

ð19Þ

Eqs. (17) and (20) give the dependences on dimensionless time ofcontact angle during the first stage of evaporation (Eq. (17)) and theradius of the droplet base on the dimensionless time during the sec-

ond stage of spreading/evaporation (Eq. (20)). Both solutions of Eqs.(17) and (20) are universal, that is, required only the knowledge ofstatic advancing and static receding contact angle. Note, dimension-less times for the first stage and second stage are different.

Below we try to apply the above theory developed for evapora-tion of droplets of pure liquids for the case of evaporation of drop-lets of nanosuspensions. We show (see below) that the theoryremains valid in the case of aqueous nanosuspensions investigated.The only difference is the values of static advancing and staticreceding contact angle is different from those for pure water.

Contact angle hysteresis on smooth homogeneous substrates, likethose used in our experiments, is determined by the special shape ofdisjoining/conjoining isotherm [30,31]. At the moment, little isknown on disjoining/conjoining isotherms of nanosuspensions usedin a wide range of concentration of nanoparticles. It is the reasonwhy the hysteresis contact angles are determined experimentally.

3. Materials and methods

3.1. Materials

The following nanoparticles were used: silicon dioxide (labelledbelow as SiO2), titanium dioxide (labelled below as TiO2) and carbonnanopowder (C) purchased from Sigma–Aldrich, UK. Two types ofcarbon nanoparticles have been used: carbon nanoparticles with size(diameter) < 50 nm (labelled as C50 nm) and carbon nanoparticleswith size <500 nm (labelled as C500 nm). The SiO2 and TiO2 particleshad an average diameter of 10–20 nm and approximately 21 nm,respectively, as stipulated by the supplier. Carbon nanoparticles weresubjected to the annealing procedure in a D. Mattia laboratory at Uni-versity of Bath, UK according to the procedure presented in [46]. SiO2

and TiO2 were used without any pre-treatment.Three kinds of solid substrates of different wetting properties

and different thermal conductivities were used: smooth siliconwafers (single side polished) with a thickness of 0.5 mm, ultra highmolecular weight polyethylene films (PE) and polytetrafluoroeth-ylene films (referred to as PTFE), both with a thickness of0.05 mm. Polymers were purchased from GoodFellow (U.K.), andsilicon wafers were purchased from Sigma–Aldrich.

3.2. Methods

Prior to the experiments, all solid surfaces were cut to desiredsize of about 2 cm � 1 cm and thoroughly cleaned according to

Fig. 3. Time evolution of (a) contact angle (deg), (b) radius of the droplet base (mm)and (c) volume to power 2/3 (mm2) of SiO2 aqueous nanosuspension on PEsubstrate.

A. Trybala et al. / Journal of Colloid and Interface Science 403 (2013) 49–57 53

the following protocol: for polymer surfaces – 15 min ultrasonica-tion in isopropyl alcohol and 15 min in water (three times) andthen dried in a strong jet of air. The silicon wafers were firstcleaned in isopropanol for 5 min then soaked in Piranha solutionfor 20 min to remove organic matter and to oxidise the surface.Piranha solution is a mixture of concentrated sulphuric acid(H2SO4) and hydrogen peroxide (H2O2) in a ratio of 1:1 (warning:Piranha solution is a highly oxidising agent!). The silicon surfaceswere extensively rinsed with warm deionized water and then driedunder a stream of air.

Nanoparticle suspensions were prepared using ultra-pure water(Millipore filter, 18.2 MX cm). Volume fraction of all nanoparticlesused was 0.01.

The samples were placed in an ultrasonic bath for 20 min to en-sure that the nanoparticles in the fluid were appropriately dis-persed prior to every set of experiments.

Each experimental run was repeated at least five times for eachindividual condition and each experimental point presented aboveand below is averaged over all experimental runs (at least five ofthem). Experimental errors (random errors) for all experimentalpoint (contact angle, volume and radius of the droplet base) werecalculated based on the same procedure as for the parameter b(see Eqs. (7) and (8)). Smaller experimental errors are observedat the beginning of the process and bigger in the end: the dropletsize was becoming smaller during evaporation process (and some-times not spherical) and harder to analyse.

3.3. Experimental set up

A drop shape analyser DSA 100 (KRÜSS, Germany) was used to-gether with the DSA3 software to capture time evolution of thecontact angle, h, the drop volume, V, and the radius of the dropletbase, L. The substrate under investigation was placed in a Krüsspel-tier thermostating chamber TC40 which allows controlling a uni-form temperature inside the chamber. Experiments wereperformed at a temperature of 25 ± 1 �C and humidity 40 ± 5%. Inall cases, 2 ll droplets of nanosuspensions were deposited on thesolid substrate. Images of the drops were taken at two framesper second for the entire experimental run.

4. Results and discussion

Experiments on evaporation of SiO2, TiO2 and carbon aqueousnanosuspensions droplets were carried on polyethylene (PE), sili-con wafers and Teflon (PTFE). Below experimental results are pre-sented and compared with theoretical predictions presentedabove.

During evaporation process, accumulation of nanoparticles andpossible aggregation of nanoparticles inside the droplets were ob-

Fig. 2. Droplet with C500 nm nanoparticles suspended in water deposited on PEafter 10 min of evaporation.

served in some cases. Fig. 2 shows a droplet of C500 nm nanopar-ticles deposited on PE after 10 min of evaporation. Aggregatescreated in the bulk of the droplet were big enough to be visiblein the case of carbon nanoparticles of both types. However, accu-mulation of nanoparticles was not observed for other types ofnanoparticles.

4.1. Kinetics of evaporation: experimental results

Evaporation of pure aqueous droplet was investigated in allcases for comparison.

Ultra-pure water was found to have static advancing contactangles of 111�, 98� and 55� on PTFE, PE and silicon wafers, respec-tively. This shows the different degrees of wettability of the sub-strates used.

In the course of evaporation of pure aqueous droplets on PE andPTFE, two stages of evaporation were present. However, during theevaporation of pure aqueous droplets on silicon wafer, only thefirst stage was observed: contact angle decreased continuouslywith time and droplet base diameter remained constant for thewhole duration of evaporation. The receding contact angle in the

54 A. Trybala et al. / Journal of Colloid and Interface Science 403 (2013) 49–57

case of evaporation of pure aqueous droplets on silicon wafers wasnot reached over the whole duration of the experiment, whichlasted more than 1000s.

4.1.1. Time dependence of contact angle and radius of the droplet basein the case of evaporation of nanosuspensions

Table 1 shows that all cases under consideration correspond toeither non-wetting (contact angle above 90� on PTFE and PE) orpartial wetting (contact angle below 90� on silicon wafers).

In the cases of hydrophobic surfaces (PE, PTFE), we do not ob-serve significant differences in values of advancing contact anglefor pure water and SiO2, TiO2 and carbon nanosuspensions (Ta-ble 1). For the silicon wafer support, lower had values for titaniumand silica dioxide were found (Table 1). Analysing the values of hr,we can observe significant differences only in the case of purewater and TiO2 and C500 nm on hydrophobic surfaces (PE, PTFE).

Notice, that static advancing contact angle of all nanosuspen-sions used is slightly higher than the corresponding value forwater. A similar phenomenon was reported by Vafaei et al. [47]for bismuth telluride nanoparticles dispersed in pure water. Intheir case, addition of nanoparticles resulted in higher staticadvancing contact angle.

In the course of evaporation of SiO2 and C50 nm droplets on PEand PTFE, two stages of the process were observed. As an example,evaporation of SiO2 droplets on PE is shown in Fig. 3, and otherdependencies in the case of presence of two stages of evaporationare similar to that presented in Fig. 3.

In all cases (no matter two stages or only one stage of evapora-tion was observed), the droplet volume decreased continuouslyover time according to the following dependencyV2=3 ¼ V2=3

0 � const � t. Such dependency was observed earlier[48]. This dependence looks like being in a contradiction withdependence of evaporation rate on the contact angle according toEq. (3). However, it is shown below (Section 4.1.2) that this depen-dency should be really valid.

During the first stage, the contact line of the droplet remainspinned to the substrate, whereas the contact angle decreases withtime from the static advancing contact angle down to the staticreceding contact angle. This stage sometimes ends with a depin-ning jump of the triple contact line. A depinning jump was ob-served in the case of evaporation of C50 nm on PE. During thesecond stage of the evaporation of the droplet contact angle re-

Table 1Static advancing, had, and static receding, hr, contact angles, K(had), K(hr) (according toEq. (22)) and number of stages for all investigated aqueous nanosuspensions on (a)PE, (b) silicon wafer and (c) PTFE substrates.

had hr K (had) K (hr) Number of stages

(a) PEH2O 98 74 0.78 0.78 2SiO2 97 81 0.78 0.78 2TiO2 93 <49 0.78 0.83 1C50 96 75 0.78 0.78 2C500 93 <43 0.78 0.85 1

(b) Si wafersH2O 55 <26 0.81 0.96 1SiO2 42 <24 0.86 0.98 1TiO2 33 <19 0.90 1.05 1C50 50 <17 0.83 1.08 1C500 50 <24 0.83 0.98 1

(c) PTFEH2O 111 93 0.79 0.78 2SiO2 114 93 0.79 0.78 2TiO2 114 <75 0.79 0.79 1C50 112 97 0.79 0.78 2C500 113 <70 0.79 0.79 1

mained constant and equal to static receding contact angle, whilethe wetted area shrank.

In the case of evaporation of TiO2, C500 nm on PE and PTFE andall investigated suspensions on silicon wafers, only one (first) stagewas observed. It happened because static receding contact anglewas smaller than the final contact angle observed.

4.1.2. Why does the time dependence of contact angle not influence thetime dependency of volume on time?

From Eqs. (3) and (9), we can express the radius of the dropletbase, L, via the droplet volume, V, and the contact angle, h, asfollows:

dVdt¼ �bFðhÞ V1=3

f 1=3ðhÞ ð20Þ

Let us introduce the following function of contact angle:

FðhÞf 1=3ðhÞ ¼ KðhÞ ð21Þ

Table 1 presents values of static advancing, had, and static reced-ing, hr, contact angles in degrees and calculated values of K(h) forthe nanosuspensions studied. In Table 1, the value of receding con-tact angle in the case of experiments where only one stage was ob-served is the last one measured before the droplet disappears, or itwas no longer of the spherical cap shape. In these cases, dropletsize was too small to continue the measurements.

According to Table 1, all values of K(h) are in the range of 0.8–1.08 for contact angles ranging from 40� to 120�. This means thatK(h) can be considered as a constant with a reasonable degree ofapproximation when the contact angle ranges between had and hr

and is independent of both the nanoparticle and the substratenature.

The latter means that we can rewrite Eq. (21) as:

dVdt¼ �aV1=3; where a ¼ bKðhÞ ð22Þ

Note that K(h) and, hence, a change only slightly (according toTable 1) during the first stage of evaporation. During the secondstage of evaporation, the contact angle remains constant and sodoes a = bK(hr). Integration of the latter equation results in:

V2=3ðtÞ ¼ �2a3

t þ V2=30 ð23Þ

The results are shown in Fig. 3c. All experimental dependencesof volume on time obtained for all systems studied agree with thelinear dependence given by Eq. (24).

4.1.3. Comparison of experimental data of sessile droplets ofnanosuspensions during evaporation process against theoreticalpredictions

The universal laws governing the evaporation of sessile dropletsare predicted by Eqs. (17) and (20) for the first and second stages ofevaporation, respectively. In this section, those theoretical predic-tions are compared against the obtained experimental data onevaporation of nanosuspensions.

Below only one example of comparison is shown (Fig. 4) of theexperimental data (points) of evaporation stages I and II of smallsessile droplets of C50 nm on polyethylene substrates in the pres-ence of contact angle hysteresis and predictions according to theabove theory (solid lines).

During the first stage of evaporation process, contact angle de-creases, while the radius of the droplet base remained constant;during the second stage of evaporation, the radius of the dropletbase decreases, while the contact angle remained constant.

In Fig. 5, we summarise experimental data on the first stage ofevaporation of all the nanosuspensions studied. The solid line in

Fig. 4. Comparison of experimental data of C50 nm on PE (stages I and II) withtheoretical predictions (solid lines) according to Eqs. (17) and (20) respectively. (a)Dependence of contact angle on dimensionless time, and (b) dependence ofdimensionless radius of the contact line on the dimensionless time.

A. Trybala et al. / Journal of Colloid and Interface Science 403 (2013) 49–57 55

Fig. 5 represents the theoretical prediction according to Eq. (17).Comparison shows a very good agreement between the theoreticalcurve predicted by the theory for pure water and the experimentaldata. Notice that negative values of the dimensionless time for thisstage are due to the fact that ~s ¼ 0 was arbitrarily associated withh = p/2; thus, negative values correspond to h > p/2. Fig. 6 presentsa summary of all nanosuspensions investigated on all solid sub-strates used, where the evaporation process showed the secondstage of evaporation (see Table 1).

Good agreement between the theory predictions for both stagesof evaporation (developed for pure liquids) and experimental dataon evaporation (in the range of experimental error ± 10%) allowsassuming that there is no adsorption of nanoparticles used on allinvestigated solid surfaces and liquid–vapour interface: otherwise,the dependency of the receding contact angle on time during thesecond stage of evaporation would be present. However, we areunable to make such conclusion in those cases, and then, onlythe first stage of evaporation was observed (see Table 1).

Fig. 5. First stage of evaporation. Summary of all nanosu

Table 2 shows values of b and errors for evaporation process forall investigated nanosuspensions and pure water on different sub-strates calculated using two different methods. b was calculatedaccording to procedure described in section 2, and b⁄ was obtainedfrom Eq. (24). From linear dependence V2/3 on time, we extractslope coefficient q, which is equal to:

q ¼ 23

Kb�: ð24Þ

Values of evaporation rate calculated by two different methodsare in agreement within experimental error according to Table 2.

Evaporation rate (that is b value) of nanosuspensions and wateron silicon wafer support is higher than on polymeric supports. Itcould be explained taking into consideration thermal conductivityof support material [49]. According to Semenov et al. [37], evapo-ration flux is reduced with decreasing of heat conductivity of sup-port material (see Fig. 7). Such flux reduction is caused bynoticeable temperature reduction of the droplet-air interface [37].

It is observed that in the case of suspensions of C500 nm parti-cles in water evaporation rate is higher than for others nanosus-pensions, especially on silicon wafers support. The reason ofdeviations in evaporation rate for different nanosuspensions iscaused by influence of nanoparticles on thermal conductivity ofsuspensions [3].

Values of surface tension of nanosuspensions used are pre-sented in Table 3. These data show that adsorption of nanoparticlesused is either negligibly small or even negative on liquid–air inter-face. In the case of two stage evaporation, no residue was observedbehind a receding contact line. The latter allows concluding thatthe inorganic nanoparticles used did not adsorb either on liquid–air or solid–liquid interface.

4.2. Pattern formation

After evaporation process of nanosuspensions is finished, differ-ent patterns were observed on solid surfaces. Particles accumu-lated at the centre of spot like in the case of carbon nanoparticlesC50 nm on PE (Fig. 8a and b); two stages of evaporation, or atthe edges like TiO2 on PE (Fig. 8c); only one stage of evaporation.Note that, the patterns shown in Fig. 8 are in a qualitative agree-ment with those predicted in [39], where for the first time, a theoryof deposition has been suggested based on consideration of colloi-dal interaction between particles. Direct application of DLVO the-ory for the case under consideration (nanoparticles) is possible

spensions investigated on all solid substrates used.

Fig. 6. Second stage of evaporation. Summary of all nanosuspensions investigated on all solid substrates used, where the evaporation process showed the second stage ofevaporation (see Table 1).

Table 2Values of b and b⁄ (obtained from two different method) and its experimental errorfor all investigated aqueous nanosuspensions on (a) PE, (b) silicon wafer and (c) PTFEsubstrates.

b (10�3 mm2/s)

Random error(%)

b=bH2 O b⁄ (10�3 mm2/s)

b�=b�H2O

(a) PEH2O 1.22 23.7 1 1.15 1.00SiO2 1.61 22.9 1.44 1.54 1.33TiO2 1.62 25.8 1.32 1.68 1.45C50 1.05 13.8 0.85 0.96 0.83C500 1.25 29.8 1.02 1.29 1.12

(b) Si wafersH2O 1.56 4.9 1 1.52 1.00SiO2 1.74 16 1.12 1.79 1.18TiO2 1.85 24.4 1.18 1.84 1.21C50 1.92 7.4 1.22 1.89 1.25C500 2.24 11 1.44 2.15 1.42

(c) PTFEH2O 1.21 10.6 1 1.15 1.00SiO2 1.27 19.3 1.04 1.34 1.17TiO2 1.2 25.1 0.99 1.15 1.00C50 1.12 13.6 0.92 1.15 1.00C500 1.56 11.3 1.29 1.33 1.16

θ, rad

J/Jπ /2(L,Ts)

(θ < π/2) T = const [16]

T = const [21]

Ts = T∞+5K

Fig. 7. Dependence of the total vapour flux from the droplet surface, J, on contactangle h, L = 1 mm. Redrawn from [37].

Table 3Values of surface tension of aqueous nanosuspensionsused.

Suspension type c (mN/m)

SiO2 72.9TiO2 72.4C50 74.1C500 75.8

56 A. Trybala et al. / Journal of Colloid and Interface Science 403 (2013) 49–57

and requires a considerable modification. That is only a qualitativecomparison with the results obtained in [39] is currently possible.However, the approach suggested in [39] is definitely the wayforward.

Patterns presented in Fig. 8 are obviously related to Marangoniflow and possibility of Marangoni flow reverse [41,42]. These typesof flow coupled with colloidal forces between particles are respon-sible for various pattern formations in the course of evaporation[50].

5. Conclusions

An experimental study of evaporation kinetics of sessile drop-lets of aqueous nanosuspensions was undertaken on three sub-strates of different wettability (PE, PTFE and silicon wafers).Aqueous nanosuspensions of inorganic nanoparticles TiO2, SiO2,carbon (<50 nm and <500 nm) were used. The time evolution ofthe droplet contact angle and radius of the contact line showedtwo stages of droplet evaporation or only one (first stage of evap-oration) when the static receding contact angle was not reached inthe course of evaporation. During the first stage, the radius of thedroplet base remained constant, while the contact angle decreasedwith time from the initial value equal to the static advancing con-tact angle down to static receding contact angle. During the nextsecond stage, the contact angle remained constant (equal to thestatic receding contact angle), but the radius of the droplet baseshrunk. The experimental results are compared with the universalbehaviour predicted for evaporation of pure water [36]. The com-parison showed a good agreement for all aqueous nanosuspensionsinvestigated on all substrates used. The only difference from thepure water is values of static advancing and receding contact an-

Fig. 8. Microscope pictures of patters formed after evaporation of nanosuspensions (a) C50 nm on PE – whole spot, (b) magnification of the edge (two stages of evaporation)and (c) TiO2 on PE (only one first stage of evaporation).

A. Trybala et al. / Journal of Colloid and Interface Science 403 (2013) 49–57 57

gles, which are different for each pair of nanosuspension/supportused. Evaporation rate was higher on solid support of higher ther-mal conductivity and depends on the thermal conductivity of thenanosuspension itself.

Acknowledgments

The work has been supported by EPSRC, UK; Marie Curie MUL-TIFLOW-ITN of the 7th FP; PASTA project, European Space Agencyand COST programme MP1101; MINECO under grant FIS2012-38231-CO2-01.

References

[1] S.U.S. Choi, J. Heat Transfer 131 (2009) 1–9.[2] K.V. Wong, O. De Leon, Adv. Mech. Eng. 2010 (2009) 519659.[3] W. Yu, D.M. France, J.L. Routbort, S.U.S. Choi, Heat Transfer Eng. 29 (2008) 432–

460.[4] T. Tyler, O. Shenderova, G. Cunningham, J. Walsh, J. Drobnik, G. Mcguire, Diam.

Relat. Mater. 15 (2006) 2078–2081.[5] S.K. Das, S.U.S. Choi, H.E. Patel, Heat Transfer Eng. 27 (2006) 3–19.[6] M.S. Liu, M.C. Lin, C.I.T. Huang, C.C. Wang, Int. Commun. Heat Mass Transfer 32

(2005) 1202–1210.[7] K. Sefiane, J. Skilling, J. MacGillivary, Adv. Coll. Int. Sci. 138 (2008) 101–120.[8] R.S. Shawgo, A.C.R. Grayson, Y. Li, M.J. Cima, Curr. Opin. Solid State Mater. Sci. 6

(2002) 329–334.[9] D. Bica, L. Vekas, M.V. Avdeev, J. Magn. Magn. Mater. 311 (2007) 17–21.

[10] J.F. Yan, J. Liu, Nanomedicine 4 (2008) 79–87.[11] S.M.S. Murshed, S.H. Tan, N.T. Nguyen, J. Phys. D 41 (085502) (2008) 1–5.[12] Y.H. Jeong, W.J. Chang, S.H. Chang, Int. J. Heat Mass Transfer 51 (2008) 3025–

3031.[13] D.T. Wasan, A.D. Nikolov, Nature 423 (2003) 156–159.[14] W. Xu, C. Choi, J. Adhes. Sci. Technol. 25 (2011) 1305–1321.[15] S. Semenov, V.M. Starov, M.G. Velarde, R.G. Rubio, Eur. Phys. J. Special Topics

197 (2010) 265–278.[16] K. Sefiane, R. Bennacer, Adv. Coll. Int. Sci. 147–148 (2009) 263–271.[17] K.S. Lee, C.Y. Cheah, R.J. Copleston, V.M. Starov, K. Sefiane, Colloids Surf., A 323

(2008) 63–72.

[18] M. Burger, R. Schmehl, K. Prommersberger, O. Schafer, R. Koch, S. Wittig, Int. J.Heat Mass Transfer 46 (2003) 4403–4412.

[19] V. Dugas, J. Broutin, E. Souteyrand, Langmuir 21 (2005) 9130–9136.[20] K. Hyungmo, K. Joonwon, J. Micromech. Microeng. 20 (2010) 045008.[21] R.D. Deegan, Phys. Rev. E 61 (2000) 475–485.[22] G. Guena, C. Poulard, M. Voue, J.D. Coninck, A.M. Cazabat, Colloids Surf., A

(2006) 291191–291196.[23] R.D. Deegan, O. Bakajin, T.F. Dupont, G. Huber, S.R. Nagel, T.A. Witten, Phys.

Rev. E 62 (2000) 756–765.[24] F. Girard, M. Antoni, K. Sefiane, Langmuir 24 (2008) 9207–9210.[25] R.G. Picknett, R. Bexon, J. Colloid Interface Sci. 61 (1977) 336.[26] F. Girard, M. Antoni, Langmuir 24 (2008) 11342–11345.[27] D.H. Shin, S.H. Lee, Microelectron. Eng. 86 (2009) 1350–1353.[28] R. Mollaret, Chem. Eng. Res. Des. 82 (2004) 471–480.[29] H. Kamusewitz, W. Possart, D. Paul, Colloids Surf., A 156 (1999) 271–279.[30] V.M. Starov, M.G. Velarde, C.J. Radke, Wetting and Spreading Dynamics, CRC/

Taylor & Francis, Boca Raton, Fla., London, 2007.[31] V.M. Starov, Colloid. Polym. Sci. 291 (2013) 261–270.[32] C. Bourgès-Monnier, M.E.R. Shanahan, Langmuir 11 (1995) 2820–2829.[33] S. Semenov, V.M. Starov, R.G. Rubio, H. Agogo, M.G. Velarde, Colloids Surf., A

391 (2011) 135–144.[34] T.A.H. Nguyen, Chem. Eng. Sci. 69 (2012) 522–529.[35] K. Sefiane, L. Tadrist, Int. Commun. Heat Mass Transfer 33 (2006) 482–490.[36] S. David, K. Sefiane, L. Tadrist, Colloids Surf., A 298 (2007) 108–114.[37] S. Semenov, V.M. Starov, R.G. Rubiob, M.G. Velarde, Colloids Surf., A 372 (2010)

127–134.[38] K. Sefiane, J. Bionic Eng. 7 (2010) S82–S93.[39] R. Bhardwaj, X. Rang, P. Somasundaran, D. Attinger, Langmuir 26 (11) (2010)

7833–7842.[40] R.D. Deegan, Nature 389 (6653) (1997) 827.[41] H. Hu, R.G. Larson, J. Phys. Chem. B 106 (6) (2002) 1334.[42] H. Hu, R.G. Larson, J. Phys. Chem. B 110 (2006) 7090.[43] R. Bhardwaj, X.H. Fang, D. Attinger, New J. Phys. 11 (2009) 075020.[44] F. Schonfeld, K.H. Graf, S. Hardt, H.J. Butt, Int. J. Heat Mass Transfer 51 (2008)

3696–3699.[45] E. Bormashenko, A. Musin, M. Zinigrad, Colloids Surf., A 385 (2011) 235–240.[46] D. Mattia, M.P. Rossi, B. Kim, J. Phys. Chem. B 110 (2006) 9850–9855.[47] S. Vafaei, T. Borca-Tasciuc, M.Z. Podowski, Nanotechnology 17 (2006) 2523–

2527.[48] M.D. Doganci, B.U. Sesli, H.Y. Erbil, J. Colloid Interface Sci. 362 (2011) 524–553.[49] E. Bormashenko, Ind. Eng. Chem. Res. 47 (5) (2008) 1726–1728.[50] E. Bormashenko, Ye Bormashenko, G. Whyman, R. Pogreb, A. Musin, R. Jager, Z.

Barkay, Langmuir 24 (2008) 4020–4025.