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Available online at www.sciencedirect.com

Colloids and Surfaces A: Physicochem. Eng. Aspects 323 (2008) 63–72

Spreading and evaporation of sessile droplets:Universal behaviour in the case of complete wetting

K.S. Lee a, C.Y. Cheah a, R.J. Copleston a, V.M. Starov a,∗, K. Sefiane b

a Department of Chemical Engineering, Loughborough University, Loughborough LE11 3TU, UKb School of Engineering and Electronics, University of Edinburgh, Kings Buildings, Edinburgh EH9 3JL, UK

Received 6 July 2007; received in revised form 17 September 2007; accepted 18 September 2007Available online 25 September 2007

bstract

Experiments on spreading and evaporation of sessile droplets on a solid substrate under various conditions are reported and compared to theeveloped theoretical model. The liquids used were alkanes: heptane and octane. All liquids completely wet glass substrates used. The time evolutionf the radius of the droplet base, contact angle and the droplet height were monitored. The developed theoretical model predicts that measured

adius and contact angle data, and the subsequently calculated volume data, would fall onto respective theoretical ‘universal curves’. Experimentalata both extracted from literature and our own confirmed this theoretical prediction. The predicted universal curves fairly fit experimental dataoth extracted from literature sources and our own.

2007 Elsevier B.V. All rights reserved.

aviou

aaswoAcapcpt

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eywords: Evaporation; Spreading; Complete wetting; Alkanes; Universal beh

. Introduction

Spreading and evaporation of droplets over solid, non-porousurfaces is a fundamental process with a number of applicationsn coating, printing and painting. In the situation of completeetting on a dry, non-porous substrate, a volatile liquid dropletndergoes two competing mechanisms, which occur until theroplet has completely evaporated: (a) spreading, which resultsn an extension of the droplet base and (b) evaporation, whichesults in a shrinkage of the droplet base. The latter processlters the dynamics of the droplet spreading as compared withon-evaporating case through the corresponding changes to theadius of the base and the contact angle.

The basics of the process of behaviour of evaporating dropsncluding the influence of surface forces action in a vicinity of the

oving three phase contact line were developed in refs. [1,2].ecently the evaporation process was intensively investigated

sing experimental [3–10] and computer simulation methods11,12].

∗ Corresponding author.E-mail address: V.M.Starov@lboro.ac.uk (V.M. Starov).

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927-7757/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.colsurfa.2007.09.033

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A number of experimental investigations of the evaporationnd spreading droplets have been undertaken in refs. [4–8] usinglkanes and other liquids on a non-porous, complete-wettingurfaces. The behaviour of both hanging and sessile droplets ofater and octane, and the investigation of profiles of dropletsf a polydimethylsiloxane oligomer was undertaken in ref [4].further study of the dynamics of these oligomer droplets was

arried out in ref. [8]. In ref. [5], a model was developed toccount for the dynamic behaviour of alkane mixtures and [6]redicted a power law dependency between the radius and theontact angle. It was confirmed in ref. [9] that using varying non-orous substrates, the evaporation rate is directly proportionalo the radius of the droplet base, as previously reported.

The important conclusion of these investigations is thenusual dependency of the volume of evaporating droplet, V,n time, t:

dV (t)

dt= −αL(t), (1)

here L(t) is the radius of the droplet base on time and α is

proportionality constant. Eq. (1) states that the evaporation

ate is not proportional to the surface of the evaporating dropletthat is not to L2(t)), but to the first power of the droplet base.he latter means that the droplet mostly evaporates close to the

64 K.S. Lee et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 323 (2008) 63–72

Nomenclature

a capillary lengthC integration constantg gravity accelerationh height of dropletL radius of the droplet basep pressuret timeu, v tangential and vertical velocity componentV volumex, y coordinates

Greek symbolsα proportionality constantδ h*/l*, parameter inside the vicinity of the moving

contact lineε h*/r*, slope of the dropletγ interfacial tensionη dynamic viscosity of a liquidλ t∗η/tmaxθ dynamic contact angle of the dropletρ droplet densityτ t0/tmaxω effective lubrication parameter

Subscripts0 initiale evaporation stagem moment when the droplet reaches its maximum

radiusmax maximumη viscous+ spreading− shrinkage

Superscripts* characteristic value

pα

ip

bTaFtc(iis

fcacft

2

ssodwrpstlb

br

two-dimensional (cylindrical) droplets. In the case of water(γ = 72.5 dyn/cm, ρ = 1 g/cm3), hence, the capillary lengtha ∼ 0.27 cm. All the droplets under consideration are expectedto be smaller than the latter length.

– dimensionless

erimeter. We can try to rewrite the proportionality constant as= 2πjΔ, where j is a linear evaporation rate (cm/s) and 2π �L

s an effective area of a ring in a vicinity of the moving threehase contact line, where most evaporation takes place.

The above linear dependency of the evaporation rate on dropase radius has also been demonstrated for pinned drops [3,16].he vapour flux was calculated [16] from a diffusion equationnd the evaporation flux along a droplet surface is depicted inig. 1. The flux increases as one moves from the droplet centre

owards the contact line at the edge. The latter also has beenonfirmed by mathematical modelling of the evaporating fluxFig. 2 [18]). However, the physical reason for this phenomenon

s yet to be understood. Below we use Eq. (1) in our theoret-cal treatment of evaporation accompanied by a simultaneouspreading.

Fo

Fig. 1. Evaporation flux along a sessile drop surface [16].

The main focus of the developed theoretical model below is asollows: if the reduced radius, reduced volume and the reducedontact angle are plotted against the reduced time, the wholerray of experimental data will follow a corresponding universalurve. A similar theory has been developed earlier in ref. [13]or a spreading of liquid droplets over dry porous substrates inhe complete wetting case.

. Theory

Let us consider a small liquid droplet on a complete wettingolid substrate. The smallness of droplets means that their size ismall compared with the capillary length and, hence, the actionf gravity on the droplet can be neglected. The dynamics of theroplet is defined by two competing mechanisms: spreading,hich results in an extension of the droplet base, and evapo-

ation, which results in a shrinkage of the droplet base. Bothrocesses occur simultaneously throughout the experiment. Thepreading stage of the droplet over the solid substrate is initiallyhe major factor in the change of the droplet’s shape, but its effectessens over time. Once the spreading effect reduces evaporationecomes the key factor in determining the droplet’s dimensions.

First of all, it is shown that in the experiments presentedelow, as well as in experiments in refs. [4–8], the droplet shouldemain a spherical shape in the course of spreading/evaporation.

Consider, for simplicity, an example of spreading of

ig. 2. Local mass evaporation flux along the interface of a drop of methanoln aluminium and PTFE substrates [18].

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K.S. Lee et al. / Colloids and Surfaces A:

The two important relevant parameters are the Reynolds num-er, Re, and the capillary number, Ca. We show below that ifoth numbers are small then the droplet should be of a sphericalhape. The Reynolds number characterises the importance ofnertial forces as compared with viscous forces. In the case of awo-dimensional droplet, the Navier–Stokes equations with thencompressibility condition take the following form:(

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y

)= −∂p

∂x+ η

(∂2u

∂x2 + ∂2u

∂y2

), (2)

(∂v

∂t+ u

∂v

∂x+ v

∂v

∂y

)= −∂p

∂y+ η

(∂2v

∂x2 + ∂2v

∂y2

), (3)

∂u

∂x+ ∂v

∂y= 0, (4)

here �v = (u, v) is the velocity vector, the gravity actions neglected, (x, y) the coordinates and ρ, η and p are theiquid density, dynamic viscosity and pressure, respectively.et U* and v∗ be scales of the velocity components in the

angential and the vertical directions, respectively, and r* and* are the corresponding scales in the horizontal and verticalirections. Using the incompressibility condition it is concludedhat U∗/r∗ = v∗/h∗ or v∗ = εU∗, ε = h∗/r∗. The derivativeu/∂t can be estimated as U*/t*, where t* ∼ 1 s is the timef spreading over a distance of about the droplet radius and* ∼ r*/t*. Therefore, one obtains the following estimation

ρ(∂u/∂t))/(ρu(∂u/∂x)) ∼ (U*/t*)/((U*)2/r*) = (r*/(t*U*)) ∼ 1.he terms ρ(∂u/∂t) and ρ(∂v/∂t) do not influence the obtained

esults because both considered terms are small as compared tohe viscous terms (see below).

If the droplet has a low slope, then ε � 1 and, hence, the veloc-ty scale in the vertical direction is much smaller than the velocitycale in the tangential direction. Using the first Navier–Stokesquation it is estimated:

u∂u

∂x∼ ρv

∂u

∂y∼ ρU∗2

r∗ , (5)

∂2u

∂x2 ∼ ε2η∂2u

∂y2 � η∂2u

∂y2 , η∂2u

∂y2 ∼ ηU∗

h∗2 . (6)

he latter estimations show that all derivatives in the low slopepproximation in the tangential direction, x, can be neglected asompared with derivatives in the axial direction y. The Reynoldsumber can be estimated as:

e ∼ ρu(∂u/∂x)

η(∂2u/∂y2)∼ ρU∗2/r∗

ηU∗/h∗2 = ρU∗h∗2

ηr∗ = ε2 ρU∗r∗

η, (7)

r

e = ε2 ρU∗r∗

η. (8)

he latter expression shows that the Reynolds number under the

ow slope approximation is proportional to ε2. Hence, duringhe initial stage of spreading, when ε ∼ 1, the Reynolds num-er is not small, but as soon as the low slope approximation isalid, Re becomes small even if ρU*r*/η is not small enough.

U

cochem. Eng. Aspects 323 (2008) 63–72 65

he latter means that during the short initial stage of spread-ng both the low slope approximation and low Reynolds numberpproximations are not valid. However, only the main part of thepreading/evaporation process, after the short initial stage is overs of our concern. It is shown [14] that Re should be calculatednly in the close vicinity of the moving contact line, where theow slope approximation is valid, because in the main part of thepreading droplet the liquid moves much slower than the liquidocated close to the edges moves. Hence, the inertial terms inavier–Stokes equations can be safely omitted after short initial

tage and only Stokes equations should be used instead:

= −∂p

∂x+ η

(∂u2

∂x2 + ∂2u

∂y2

), (9)

= −∂p

∂y+ η

(∂2v

∂x2 + ∂2v

∂y2

), (10)

∂u

∂x+ ∂v

∂y= 0. (11)

he capillary number, Ca = Uη/γ , characterises the relative influ-nce of the viscous forces as compared with the capillary forces.o estimate possible values of Ca let us adopt r* ∼ 0.1 cm,∼ 30 dyn/cm and η ∼ 10−2 P (oils), which are close to our

xperiments below. Let the droplet edge moves outward a dis-ance equal to its radius over 1 s, which can be considered as aery high velocity of spreading. The latter gives the followingstimation Ca ∼ 3 × 10−5 � 1. Therefore, it should be expectedor Ca to be even less than 10−5 over the duration of the spread-ng/evaporation process. According to the former it is assumedhat both the capillary and Reynolds numbers are very smallxcept for the very short initial stage of spreading. The dura-ion of the initial stage of spreading, t0, was estimated in ref.14] immediately after the droplet is deposited on the solid sub-trate. In the case of aqueous droplets the latter time is around0 ∼ 10−2 s.

Let us consider the consequence of the smallness of theapillary number, Ca � 1 using the same example of spread-ng/evaporation of a two-dimensional (cylindrical) droplet. Lethe length scales in both the x and y direction in the main partf the spreading droplet be r*, then the pressure has the order ofagnitude of the capillary pressure inside the main part of the

roplet, that is p ∼ γ/r*. Using the incompressibility conditiont is perceived that velocity in both directions, u and v, have theame order of magnitude U*.

Let us introduce the following dimensionless variables,hich are marked by an over-bar:

¯ = p

γ/r∗ , x = x

r∗ , y = y

r∗ , u = u

U∗ ,

¯ = v

U∗ .

sing these variables the Stokes equations can be rewritten as

∂p

∂�x= Ca

(∂2u

∂x2 + ∂2u

∂y2

), (12)

6 Physi

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6 K.S. Lee et al. / Colloids and Surfaces A:

∂p

∂�y= Ca

(∂2v

∂x2 + ∂2v

∂y2

). (13)

t has already been shown that Ca � 1, which means that theight hand side of both of the above equations is very small.ence, these equations can be rewritten as:∂p/∂x = 0 and ∂p/∂�y = 0, which means that the pressure

emains constant inside the main part of the spreading droplet.The normal stress balance on the main part of the spreading

roplet is:

¯ = h′′

(1 + h′2)3/2

+ Ca

[− 2

1 + h′2

{−h′

(∂u

∂y+ ∂v

∂x

)− ∂v

∂y− h′2 ∂u

∂x

}].

(14)

sing the condition Ca � 1 the equation simplifies to:

¯ = h′′

(1 + h′2)3/2 = constant (15)

ven in the case where the droplet profile does not satisfy theow slope approximation, that is, even if h′2 ∼ 1 is not small.he latter shows that the spreading droplet keeps its sphericalhape over the main part of the droplet. Note that the radius ofhe droplet base, R(t), changes over time, and this change resultsn a quasi-steady state changes of the droplet profile. In the lowlope approximation the capillary pressure inside the main partf the droplet is p ∼ γ/R*, where R* ∼ r*2/h*. Thus, the capillaryressure is much smaller than p ∼ γ/r*. Using the dimensionlessariables p = (p/(γ/R∗)), x = x/r∗, y = y/h∗, u = u/U∗ and

¯ = (1/ε)(v/U∗) it can be shown that the right-hand side of Eq.12) increases by a factor (r*/h*)3, and the right-hand side of Eq.13) increases by a factor (r*/h*). Nevertheless, because of theery small value of the capillary number (<10−5), the conclu-ion on the constancy of the pressure inside the main part of thepreading droplet remains approximately valid. During spread-ng the ratio h/r decreases and the capillary pressure decreases,owever, the velocity of spreading also decreases, therefore thebove conclusion remains valid for all stages of spreading.

The smallness of Ca means that the surface tension is muchore powerful over the main part of the droplet and, hence, the

roplet has a spherical shape everywhere except for a vicinity ofhe apparent three-phase contact line. A size of this region, l*, isstimated in ref. [14]. It is shown that the following inequality isatisfied: h* � l* � r*. Hence, δ = h*/l* � 1 is a small parameternside the vicinity of the moving contact line. This means thathe curvature of the liquid interface inside the vicinity of the

oving contact line can be estimated as:

γh′′

(1 + h′2)3/2 ∼ γ(h∗/l∗2)h′′

(1 + (h∗2/l∗2)h′2)3/2

= γ(h∗/l∗2)h′′

(1 + δ2h′2)3/2 ≈ γh∗

l∗2 h′′. (16)

ence, the low slope approximation is valid inside the vicinityf the moving contact line even if the droplet profile is not very

ar0t

cochem. Eng. Aspects 323 (2008) 63–72

ow, that is, even if h′2 ∼ 1 is not small. Therefore, the lowlope approximation can always be used inside the vicinity ofhe moving contact line except for the case when the slope islose to π/2.

Following arguments developed in ref. [14], the whole dropletrofile can be subdivided into “outer” spherical region andinner” region in a vicinity of the moving three phase contactine. The “outer” solution (under a low slope approximation) is14]:

(t, r) = 2V

πL4 (L2 − r2), r < L(t). (17)

he latter expression shows that the droplet surface profileemains spherical during the spreading process except for a shortnitial stage. Eq. (17) gives the following value of the dynamicontact angle, θ (tan θ ≈ θ):

= 4V

πL3 , (18)

r

=(

4V

πθ

)1/3

. (19)

ote, the contact angle θ is an apparent macroscopic contactngle because it is related to the “outer” spherical region andoes not take into account the shape of the “inner” region [14].

The droplet motion is a superposition of two motions: (a) thepreading of the droplet over the solid substrate, which causesxpansion of the droplet base and (b) shrinkage of the baseaused by the evaporation and so the following equation cane written:

dL

dt= v+ − v−, (20)

here v+, v− are the unknown velocities of the expansion andhe shrinkage of the droplet base, respectively. The derivative ofoth sides of Eq. (19) gives:

dL

dt= −1

3

(4V

πθ4

)1/3 dθ

dt+ 1

3

(4

πV 2θ

)1/3 dV

dt. (21)

ver the whole duration of the spreading/evaporation, both theontact angle and the droplet volume can only decrease withime. Accordingly, the first term on the right hand side of Eq.21) is positive and the second one is negative. Comparison ofhe latter two equations yields:

+ = −1

3

(4V

πθ4

)1/3 dθ

dt> 0, (22)

− = −1

3

(4

πV 2θ

)1/3 dV

dt> 0. (23)

here are two substantially different characteristic time scalesn the problem under consideration: t∗η � t∗, where t∗η and t*

re the time scales of the viscose spreading and the evaporation,espectively, and λ = t∗η/t � 1 is a smallness parameter (around.1 under the chosen experimental conditions, see below). Bothime scales are calculated below. Hence, L = L(Tη, Te), where Tη

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K.S. Lee et al. / Colloids and Surfaces A:

s a fast time of the viscose spreading and Te is the slower timef the evaporation. The time derivative of L(Tη, Te) is [15]:

dL

dt= ∂L

∂Tη

+ λ∂L

∂Te. (24)

omparison of Eqs. (21)–(24) shows that

+ = ∂L

∂Tη

= −1

3

(4V

πθ4

)1/3 dθ

dt(25)

− = −λ∂L

∂Tp= −1

3

(4

πV 2θ

)1/3 dV

dt(26)

he decrease of the droplet volume, V, with time is determinedolely by the evaporation, hence, the droplet volume, V, onlyepends on the slow time scale.

According to the previous consideration, the spreading pro-ess can be subdivided into two stages:

(a) a first fast but short stage, when the evaporation can beneglected, and the droplet spreads with approximately con-stant volume. This stage goes in the same way as thespreading over dry solid substrates and the arguments devel-oped earlier for this case can be used here [14];

b) a second slower stage, when the spreading process is almostover and the evolution is determined by the evaporation.

During the first stage the dependency of the droplet baseadius can be rewritten in the following form [13]:

(t) =[

10γω

η

(4V

π

)3]0.1

(t + t0)0.1, (27)

here t0 is the duration of the initial stage of spreading, whenhe capillary regime of spreading is not applicable and ω is anffective lubrication parameter, which has been discussed andstimated in ref. [13]. Note, the parameter ω is independent ofhe droplet volume. According to Eq. (27) the characteristic timecale of the first stage of spreading is

∗η = ηL0

10γω

(πL3

0

4V0

)3

, (28)

here L0 = L(0) is the radius of the droplet base in the end of theery fast initial stage of spreading.

Combination of Eqs. (27) and (18) gives:

=(

4V

π

)0.1(η

10γω

)0.3

(t + t0)−0.3. (29)

ubstitution of the latter expression into Eq. (22) gives the fol-owing expression for the velocity of the droplet base expansion,+: ( )0.3( )0.1

+ = 0.14V

π

10γω

η

1

(t + t0)0.9 (30)

ccording to Eq. (1) the rate of evaporation is proportional tohe radius of the droplet. Substitution of Eqs. (1), (25), (26) and

t

tus

cochem. Eng. Aspects 323 (2008) 63–72 67

30) into Eq. (20) results in:

dL

dt= 0.1

(4V

π

)0.3(10γω

η

)0.1 1

(t + t0)0.9 − αL2

3V. (31)

ote, according to Eq. (31) both spreading and evaporation pro-eed simultaneously. The latter gives a system of two differentialqs. (1) and (31) with the following boundary conditions

(0) = V0, (32)

(0) = L0 =[

10γω

η

(4V0

π

)3]0.1

t0.10 , (33)

here V0 is the initial droplet volume and L0 is the radius of theroplet after the very fast initial stage is over. Let the system ofifferential Eqs. (1) and (31) be made dimensionless using newcales:

¯ = L

Lmax, t = t

tmax, V = V

V0, tm = tm

tmax,

here Lmax is the maximum value of the droplet base, whichs reached at the time instant tm, which is to be determined andmax is the total duration of the process, that is the moment whenhe droplet completely evaporates. Using Eq. (1) in dimension-ess form we conclude dV /dt = −tmaxαLmaxL/V0. The latterquation shows that the characteristic time scale of the evapo-ation process is t* = V0/αLmax. Hence, the total duration of thehole process can differ from this characteristic time scale byconstant only: tmax = βt*, where β is a dimensionless number,hich is estimated below. Note, that t* and tmax (and, hence, β)

re calculated below using experimental data.The value of Lmax is determined using Eq. (31) at the moment

= tm, when dL/dt = 0. This gives:

2max = 0.3

α

(4

π

)0.3

V 1.30

(10γω

η

)0.1 1

(tm + t0)0.9 . (34)

sing the latter definition Eqs. (1) and (31) can be rewritten as

dV

dt= −βL, (35)

dL

dt= β

(tm + τ)0.9

3(t + τ)0.9 V 0.3 − βL2

3V(36)

ith boundary conditions

¯ (0) = 1, (37)

nd

¯ (0) = L0 = 103 β(tm + τ)0.9τ0.1, (38)

here τ = t0/tmax � 1, L0 = L0/Lmax < 1.

The system of two ordinary differential Eqs. (35) and36) includes three following dimensionless parameters τ =

0/tmax � 1, β < 1.5, tm = tm/tmax < 1 (see an estimation ofhe parameter β below). The same symbols with an over-bar aresed for the dimensionless variables as for corresponding dimen-ional variables. The latter four parameters should be selected

6 Physi

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V

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8 K.S. Lee et al. / Colloids and Surfaces A:

sing the following four conditions:

¯ (1) = 0, (39)

¯ (1) = 0, (40)

¯ (tm) = 1, (41)

dL(tm)

dt= 0. (42)

ote, that the latter two conditions are used to determine onlyne parameter tm: the position of the maximum on the depen-ency L(t) should coincide with that given by Eq. (41). That is,here are only three independent conditions in (39)–(42).

Differential Eqs. (35) and (36), initial conditions (37) and38) and conditions (39) and (42) do not include any dimen-ional parameters or their combination. Hence, τ, β, tm shoulde dimensionless universal numbers. The latter shows, that theolution of Eqs. (35) and (36), that is, radius of the droplet base¯ (t), the volume, V (t) and contact angle (see below) should beniversal functions of dimensionless time.

According to system of Eqs. (35) and (36), dimensionlesselocities of spreading and shrinkage caused by the evaporationre as follows:

¯+ = β(tm + τ)0.9

3(t + τ)0.9 V 0.3, v− = βL2

3V. (43)

ig. 3 shows dimensionless velocity v+ and v− calculatedccording to Eq. (43) and system of Eqs. (35) and (36). Fig. 3hows that:

the duration of the first stage is short. The capillary spreadingprevails on this stage over the droplet base shrinkage causedby the liquid evaporation;

the spreading of the droplet almost stops after the first stage ofspreading and the shrinkage of the droplet base is determinedby the evaporation of the liquid from the droplet.

ig. 3. Dimensionless spreading (v+) and evaporation (v−) velocities accordingo Eq. (43).

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w

L

L

Nt(

L

cochem. Eng. Aspects 323 (2008) 63–72

Let us consider the asymptotic behaviour of system (35) and36) over the second stage of the process, when evaporationrevails. Velocity of the expansion of the droplet, v+, decreasesver the second stage of the spreading according to Fig. 3. Tonderstand the asymptotic behaviour, the term corresponding to+ in the right hand side of Eq. (36) is neglected. This gives:

dL

dt= −β

L2

3V, (44)

hilst Eq. (35) is left unchanged. The system of differential Eqs.35) and (44) can be solved analytically. For this purpose Eq.44) is divided by Eq. (35), which gives dL/dV = L/3V . Thisquation can be easily integrated and the solution is,

¯ = CL3, (45)

here C is an integration constant.Rewriting Eq. (18) using the same dimensionless variables

s above results in:

¯ = πL3max

4V0θL3. (46)

omparison of Eqs. (45) and (46) shows that the dynamic con-act angle asymptotically remains constant over the duration ofhe second stage of evaporation. This constant value is markedelow as θf. Introducing θm = 4V0/πL3

max, which is the valuef the dynamic contact angle at the time instant when the maxi-um value of the droplet base is reached. Then Eq. (46) can be

ewritten as:

θ

θm= V

L3 (47)

nd the latter relationship should be a universal function ofhe dimensionless time, t. The latter equation shows that thentegration constant in Eq. (45) is C = θf/θm.

Let us estimate the constant β. For this purpose, we solvequations which describe the time evolution during the secondtage and neglect completely the presence of the first stage. Thats from Eqs. (35) and (36):

dV

dt= −βL, (48)

dL

dt= −β

L2

3V, (49)

ith the following conditions

¯ (0) = 1, (50)

¯ (1) = 0. (51)

ote, the first condition, Eq. (50) is definitely not valid duringhe first stage of spreading. Solution of the problem (48) and

49) with boundary condition (51) is given by:

¯ (t) =(

2βθm

3θf

)0.5

(1 − t)0.5. (52)

Physicochem. Eng. Aspects 323 (2008) 63–72 69

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fvvuhudfiip

oEdoiooaigM

K.S. Lee et al. / Colloids and Surfaces A:

he latter dependency coincide with the predicted and exper-mentally confirmed earlier [4–8]. Using now the boundaryondition (50) we conclude:

= 3θf

2θm< 1.5 (53)

ecause obviously θf < θm. The latter conclusion is in a goodgreement with experimental data (see below).

. Experimental

.1. Materials

Alkanes used in our experiments were n-heptane and i-octaneurchased from Sigma–Aldrich, UK. Solid substrates used wereicroscope glass slides, which were cleaned prior each exper-

ment according to the following protocol: (i) soaking withsopropyl alcohol to remove organic contaminants for 30 minnd rinsed with deionised water; (ii) soaking in chromic acidor removal of inorganic contaminants for 50 min; (iii) exten-ive rinsing with distilled and deionised water; (iii) drying in aegulated oven. The cleaning procedure was repeated after eachxperiment.

.2. Methodology

The time evolution of the radius of a drop base, L(t), andhe dynamic contact angle, θ(t), were monitored simultane-usly (see Fig. 4). The substrate was placed in a hermeticallylosed and insulated chamber, which allows strict control ofhe chamber’s environment. The chamber was also equippedith a fan (1000 rpm). An experimental chamber used was

arlier described in ref. [13]. A high precision 10 �l Hamil-on syringe (Hamilton GB Ltd., UK) was used to inject 3 �lroplets of alkanes on the solid substrate. A mechanical manip-lator was structured to enable a gentle placement of theroplet on the substrate whilst minimising the kinetic impact.

xperiments were carried out at these conditions for both n-eptane and i-octane: (i) at 25 ◦C without fan switched on, (ii)t 40 ◦C without fan switched on and (iii) at 25 ◦C with fanwitched on.

ig. 4. Schematic diagram of experimental set-up. 1, Glass slide; 2, enclosedhamber; 3, tested drop; 4, syringe; 5 and 6, CCD-cameras; 7 and 8, illuminators;, CPU unit; 10, power assisted fan.

a

4

[cfitasartotmtt

ig. 5. Schematic diagram of the experimental set-up used in University ofdinburgh. 1, Glass slide; 2, tested drop; 3, syringe; 4, computer controlledump; 5, CCD camera; 6, illuminator; 7, CPU unit; 8, temperature controller.

The spreading/evaporation process were captured at 60rames per second for the whole experiment duration using twoideo cameras, which captured simultaneously side view and aiew from above. The side view video images were analysedsing “ScionImage” software to measure the contact angle andeight of the droplet, whilst the top view images were analysedsing “Olympus i-Speed software” to measure the radius of theroplet base. The experiment was repeated to produce at leastve sets of data for each individual condition. Hence, each exper-

mental point plotted below is an average of five experimentaloints.

Additional experiments using identical n-heptane and i-ctane were carried out in School of Engineering andlectronics, University of Edinburgh as a supplement to ourata. In these experiments, a video camera with magnifying lensperating at 60 frames per second wae used to capture the spread-ng/evaporation process from a side view (Fig. 5). A computerperated pump enabled accurate volumes of droplet to be placedn to complete wetting microscope glass slides. In addition, thepparatus allows varying the solid substrate temperature. Exper-ments was carried out with 3.5 �l of n-heptane and i-octane onlass slides, which we vary the temperature for 25, 30 and 40 ◦C.easurements of the radius was obtained using a drop shape

nalysis software (FTA32).

. Results and discussions

Our experimental observations as well as presented in refs.4–8] show that the whole spreading/evaporation process in thease of complete wetting is divided into two stages: (a) the fastrst stage, when the drop spreads out until a maximum radius of

he drop base, Lm, is reached, this first stage is followed bysecond slower stage (b) when the radius of the drop base

hrinks because of evaporation until it disappears completelyt the moment tmax. According to the described above theo-etical requirements the droplets must be of a spherical shapehroughout the spreading/evaporation process and the durationf the first stage of spreading, tm, is much smaller compared

o tmax. As observed in our experiments, both these require-

ents are met. Table 1 shows that values of tm are always lesshan 10% of those of tmax regardless of the experimental condi-ions used. Note, according to our experimental date the value

70 K.S. Lee et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 323 (2008) 63–72

Table 1Comparison of initial spreading, tm, and total time, tmax

Materials T (◦C) Fan speed (rpm) tm (s) tmax (s) tm

Heptane25 0 4 85 0.04740 0 3 35 0.08625 1000 4 80 0.05

O

ob

4d

itamtaw

w

V

EVlpwi

iTheT

Fo

Table 2Fitted values of parameter α in Eq. (54)

Operating conditions Fitted α

Heptane Octane

25 ◦C fan speed 0 0.017 0.01840 ◦C fan speed 0 0.078 0.04525 ◦C fan speed 1000 rpm 0.016 0.018

In all cases V(t) dependency showed a linear trend for V vs.∫ t

0L(t) dt.

Fig. 7. A dimensionless radius against dimensionless time curve fortip

pf

4universal behaviour

The solid lines in Figs. 7–9 represent theoretical universal

ctane25 0 3 73 0.04140 0 2 29 0.06925 1000 4 66 0.06

f tm (Table 1) is smaller as compared with literature data (seeelow).

.1. Extracting theoretical parameters from experimentalata

Experimental values of the total duration of the spread-ng/evaporation process, tmax, of the duration of the first stage ofhe process, tm, the maximum value of the spreading radius, Lm,nd the contact angle, θm, were extracted directly from experi-ental data. The volume was calculated by rearranging Eq. (18)

o V(t) = (π/4)L3(t)θ(t), where the contact angle was calculatedccording to the low slope approximation as θ(t) = 2h(t)/L(t),here h(t) is the height of the drop apex.The parameter α in Eq. (1) was calculated in the following

ay. Eq. (1) was integrated, which results in:

= V0 − α

∫ t

0L(t) dt. (54)

xperimental dependencies of the volume of the droplet on time,(t), were plotted against

∫ t

0 L(t) dt. In all cases, V(t) showed ainear trend (Fig. 6). Using these linear dependencies, the pro-ortionality coefficient α was fitted. The same linear dependencyas found previously [4,9]. The fitted values of α are presented

n Table 2. After that all necessary parameters were known.Note that the parameter α for experiments conducted at 25 ◦C

s significantly lower as compared with that at 40 ◦C (Table 2).

he latter is expected because the evaporation rate increases atigher temperatures. However, the introduction of a fan in ourxperiments does not appear to affect the evaporation rate at all.he use of a fan could introduce some convection in the gas

ig. 6. A comparison of the trends in volume changes between the different setsf variables used.

d

Fcia

he behaviour of the droplet radius comparing different liquids spread-ng/evaporating on solid substrates extracted from literature and theoreticalrediction. The solid line is calculated according to Eqs. (35) and (36).

hase compared to a diffusion controlled condition without aan [17].

.2. Comparison of experimental data and predicted

ependencies, which were calculated according to Eqs. (35) and

ig. 8. A dimensionless contact angle, θ = θ/θm, against dimensionless timeurve for the behaviour of the droplet radius comparing theoretical, our exper-mental and literature data. The solid line is calculated according to Eqs. (35)nd (36).

K.S. Lee et al. / Colloids and Surfaces A: Physi

Fig. 9. A dimensionless volume against dimensionless time curve for thebl

(pTiaw(1aposlddovm

dfdtsbbtjut

aausassw

coSldccap

brua

5

ccddmsTotdl

sti

tdrw

A

cSts

R

[3] R.D. Deegan, O. Bakajin, T.F. Dupont, G. Huber, S.R. Nagel, T.A. Witten,

ehaviour of the droplet radius comparing theoretical, our experimental anditerature data. The solid line is calculated according to Eqs. (35) and (36).

36), with boundary conditions (37) and (38). The four unknownarameters were selected according to conditions (39)–(42).he selected parameters are: τ = 2 × 10−3, β = 1.281 (which is

n a good agreement with the previous estimation (Eq. (53)),nd the selected value of tm ≈ 0.261 in Eq. (36) coincidesith the maximum position of the theoretical dependency L(t)

see Fig. 7). The duration of the first stage is around 10−1 to0−2 s [13], and the duration of the total spreading/evaporation ispproximately 10–100 s, which estimates τ ≈ 10−3. This com-liments our selection of τ = 2 × 10−3 for our case to matchur experimental, theoretical and literature data as best as pos-ible. Experimental data on spreading/evaporation of differentiquids extracted from literature has been plotted in the sameimensionless form. We managed to extract only dimensionlessependences L(t) from literature [5,7,8]. Therefore we providedur experimental data on dependences of contact angle andolume on time to further challenge our proposed theoreticalodel.Fig. 7 shows our theoretical model plotted against data of

ifferent liquids ranging from water, alkane and silicon oilrom literature [5,7,8] in the form of dimensionless radius ofroplet base, L, versus dimensionless time, t. Fig. 7 showshat under these dimensionless variables, the liquids’ radiuspreads quickly and reached a maximum value at t = tm aftereing placed on the substrate. When the initial spreading phaseecomes almost negligible, the radius of the drop then shrinksowards zero at t = 1. This point in time, is when the droplet hasust completely evaporated. Comparing our theoretical modelsing the set of independent variables mentioned above, we findhat our prediction fits the experimental data very well.

We conducted experiments using alkanes under controlledmbient conditions where atmospheric temperature were varied,s wells as introduction of a fan into the system, we find similarniversal behaviour as predicted by our model. In addition, aeparate experimental setup that allows us to maintain constanttmospheric condition, but with different temperatures of the

olid substrate (25, 30 and 40 ◦C). These set of results too provideimilar behaviour as observed in the previous experiments asell as in literature [5,7,8].

cochem. Eng. Aspects 323 (2008) 63–72 71

Fig. 8 displays the data obtained from our experiments for theontact angle measurement. The data is also compared to thatbtained in refs. [5,7,8] and to the predicted universal curve.imilarly to the previous, the data are plotted as dimension-

ess contact angle, θ = θ/θm, against t (Fig. 8). Contact anglerops very quickly once it has been placed on the substrate. Theontact angle then slows its decline and reached a steady stateontact angle, θf. Similarly to radius, the contact angle followsuniversal trend (Fig. 8), which agrees well with the theoreticalrediction as well as literature data.

The theory above also predicted a dimensionless universalehaviour of volume on time dependency. Our experimentalesults are presented in Fig. 9. Similarly to the previous twoniversal curves, both our experimental data and literature datagree well with the theory predictions.

. Conclusions

Experiments were designed to allow monitoring radius andontact angle of spreading/evaporating droplets in the case ofomplete wetting. During the spreading/evaporating process, theroplet retained a spherical form. The whole process is clearlyivided into two stages: the fast first stage, when the process isostly determined by kinetics of spreading and a second slower

tage when the process is mostly determined by evaporation.he duration of the first stage compared to the overall durationf the process time taken by the droplet to evaporate was foundo be considerably smaller. We also confirmed the change in therop volume with respect to the radius of the drop base to be ainear relation (Fig. 6).

A theoretical model was developed to account for thepreading/evaporation process, which produced universalheoretical curves for radius, contact angle, volume and spread-ng/evaporation velocities on time.

We compared experimental results both extracted from litera-ure and ours to that of the theoretical predictions. Experimentalata corresponds well with the predicted universal behaviouregardless the different conditions under which the experimentsere conducted.

cknowledgements

This research was supported by the Engineering and Physi-al Sciences Research Council, UK (Grant EP/D077869/1) andyngenta, UK. The authors would like to express their grati-

ude to Dr. V Kovalchuk for very fruitful discussions and usefuluggestions.

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