Spreading of liquid drops from a liquid source

Colloids and Surfaces A: Physicochem. Eng. Aspects 282–283 (2006) 247–255

Spreading of liquid drops from a liquid source

R. Holdich a, V.M. Starov a,∗, P. Prokopovich a, D.O. Njobuenwu a, R.G. Rubio b,S. Zhdanov c, M.G. Velarde d,e

a Department of Chemical Engineering, Loughborough University, Loughborough LE11 3TU, UKb Departemento the Quimica-Fisica I, Universidad Complutense de Madrid, 28040-Madrid, Spain

c Micropore Technologies Ltd., Loughborough LE11 3TP, UKd Instituto Pluridisciplinar, Universidad Complutense de Madrid, 1, Paseo Juan XXIII, 28040 Madrid, Spain

e International Center for Mechanical Sciences, CISM, Palazzo del Torso, Piazza Garibaldi, 33100 Udine, Italy

Received 27 September 2005; received in revised form 31 October 2005; accepted 1 November 2005Available online 26 May 2006

Dedicated to Professor Ivan B. Ivanov (LCPE, University of Sofia) on the occasion of his 70th birthday.

Abstract

eadtopi©

K

1

sctRftidwrb

(

0d

The spreading of liquid drops over solid substrates with a liquid source in the centre of the drop is considered from both theoretical andxperimental points of view. For conditions of complete wetting the spreading is an overlapping of two processes: a spontaneous spreading andforced flow caused by the liquid source in the centre. Both capillary and gravitational regimes of spreading are considered and power laws areeduced. In both cases of small and large droplets the exponent is a sum of two terms: the first term corresponds to the spontaneous spreading andhe second term is determined by the intensity of the liquid source. In the case of a constant flow rate from the source the latter gives for the radiusf spreading the following law R(t) ∼ t0.4 in the case of the capillary spreading and R(t) ∼ t0.5 in the case of gravitational spreading. In the case ofartial wetting droplets spread with a constant advancing contact angle (at small capillary numbers). This yields R(t) ∼ t1/3. Experimental data aren good agreement with the theoretical predictions. 2006 Elsevier B.V. All rights reserved.

eywords: Spreading; Liquid source; Complete and partial wetting

. Introduction

The spreading of liquid drops over solid non-porous sub-trates has been investigated for decades. For conditions ofomplete wetting the spreading of small droplets is governed byhe capillary law of spreading, which is R(t) = Ψ ct0.1 [1], where

is the radius of the drop base; t, the time; Ψ c, a pre-exponentialactor, which depends on the disjoining pressure isotherm [1]. Inhe same case of complete wetting the spreading of bigger dropss governed by gravity according to R(t) = Ψgt1/8 [2]. For smallrops the capillary law of spreading is in excellent agreementith experimental data [1]. Over time small droplets spreads out,

adius of the drop base increases with time and, hence, shoulde a transition from a capillary regime of spreading to the grav-

∗ Corresponding author. Tel.: +44 1509 222508; fax: 44 1509 223923.E-mail addresses: [email protected], [email protected]

V.M. Starov).

itational regime. The latter transition has been experimentallyconfirmed [3] in the case of spontaneous spreading. Note in thecase of complete wetting, the dynamic contact angle tends tozero over time and droplets spread out completely.

Spreading of liquid drops in the case of partial wettinghas been less investigated and it is less understood. The mainproblem in this case is the presence of contact angle hystere-sis. The latter phenomenon is usually associated with non-homogeneity/roughness of the solid substrates. However, ithas been shown in [4] that a S-shape disjoining pressureisotherm leads to the presence of the contact angle hystere-sis even on smooth homogeneous substrates. In the case ofpartial wetting, the droplet spreads out until a static advanc-ing contact angle is reached. After that the droplet does notspread out on a macroscale. However, if the droplet is “gen-tly” pushed from inside (for example, by pumping liquidfrom the orifice at its center) then the droplet spreads witha constant advancing contact angle, which is equal to thestatic advancing contact angle. The term “gently” means that

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248 R. Holdich et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 282–283 (2006) 247–255

Nomenclature

A the Hamaker constanta′, b constants imposed by the liquid sourcea the capillary lengthB1 dimensionless constantB2 dimensionless constantC integration constantCa the capillary numberD1 dimensionless constantD2 dimensionless constantf functionF(t) function of time, imposed by the liquid sourceg gravity accelerationG integration constanth(t, r) the drop profileh0 characteristic scale of the drop profileH(t) time dependent scale of the drop profileI the intensity of the liquid sourceK the drop curvaturen exponentsp pressurer0 characteristic scale in the radial directionr, z co-ordinate systemR radius of the drop baseRe Reynolds numbert timeU rate of spreadingV(t) the drop volumeX fitting parameterY fitting parameter

Greek symbolα exponent in the spreading law (capillary regime)β exponent in the spreading law (gravitational

regime)γ liquid–air interfacial tensionη dynamic viscosity of the liquidν velocityΠ(h) disjoining pressureθ contact angleξ similarity co-ordinateρ liquid densityΦ(θ) function in Eq. (A.34)ϕ(ξ) similarity solution (the drop profile)χ dimensionless intensity of the surface forces

actionΨ pre-exponential factorω dimensionless parameter in the spreading regime

SubscriptsA advancinga ambient airc capillary regimeg gravitational regime

r radial componentR recedingz vertical component

the capillary number (see below) remains small during thespreading.

The spreading of liquid drops for both cases, complete andpartial wetting, when liquid is injected into the droplets from asmall orifice at their center is considered below.

2. Theory

Let us consider the spreading of a small liquid droplet over asolid substrate in the presence of liquid source in the drop centre(Fig. 1).

It is assumed that the shape of the drop remains axisymmetric,and hence, a cylindrical co-ordinate system (r, z), is used below,where r is the radial distance from the centre; z, the verticalco-ordinate. Because of symmetry the angular component ofvelocity vanishes and all other unknowns are independent of theangle. It is assumed below that the Reynolds number, Re 1,and that the drop profile has a low slope, that is, ε = (∂h/∂r) 1, where h(t, r) is the drop profile. Using the latter two smallparameters we conclude from Navier–Stokes equations that

|νz| |νr|,∣∣∣∣∂f∂r

∣∣∣∣ ∣∣∣∣∂f∂z

∣∣∣∣ ,∂p

∂z= 0, hence, p = p(t, r)

where νr, νz and p are radial velocity component, vertical veloc-ity component and pressure, respectively; f, an arbitrary func-tion. After that only one Navier–Stokes equation is left, whichi

Fsd(

s

∂p

∂r= η

∂2νr

∂z2 (1)

ig. 1. Schematic presentation of the spreading in the presence of the liquidource in the drop centre. R(t) radius of the drop base; θ contact angle: (1) liquidrop; (2) solid substrate with a small orifice in the centre; (3) liquid sourcesyringe).

R. Holdich et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 282–283 (2006) 247–255 249

with the following two boundary conditions no-slip condition atthe solid substrate

νr(t, z) = 0 at z = 0 (2)

no-tangential stress on the liquid–air interface

η∂νr

∂z= 0 at z = h(t, r), (3)

where η is the dynamic shear viscosity of the liquid. After inte-gration of Eq. (1) with boundary conditions (2) and (3) we get

νr = −1

η

∂p

∂r

(hz − z2

2

).

The latter expression gives the flow rate, Q, as

Q = 2π

∫ h

0rνrdz = −2π

3ηrh3 ∂p

∂r. (4)

Conservation of mass results in

2πr∂h

∂t+ ∂Q

∂r= 0,

and using Eq. (4) we arrive to the following equation:

∂h = 1 ∂(

rh3 ∂p)

, (5)

w

tv

2

fiiilc

p

wiefK

p

hi

r

where dimensionless values are marked by an overbar. In dimen-sionless form Eq. (8) can be rewritten as

p = pa − γh0

r20

1

r

∂

∂r

(r∂h

∂r

)+ ρgh0h.

In the latter expression, we have two parameters: γh0/r20 and

ρgh0. ionless parameter estimates the intensity of capillaryforces and the second one that of the gravity force.

If capillary forces prevail then we have the capillary regimeof spreading, that is if γh0/r2

0 ρgh0 or r0 √γ/ρg.

If the gravity dominates, then the gravitational regime ofspreading takes place, that is

is ifγh0

r20

ρgh0 or r0 √

γ

ρg.

The length a = √γ/ρg is the capillary length. Hence, the

capillary regime (R(t) < a) is the initial stage of spreading ofsmall drops, while the gravitational regime is the final stageof spreading of small drops (R(t) > a) or the overall regime ofspreading of big drops. We consider below the spreading of smalldrops with a transition from the capillary to the gravitationalregime of spreading over time.

Note that in Eqs. (7) and (8), we ignored the role played by thedfiwsoiable

isiFsvrt

t

aFIoco

td

∂t 3ηr ∂r ∂r

hich is referred below as the equation of spreading.The liquid is assumed non-volatile and injected from the cen-

er according to a prescribed time rate F′(t), hence, the dropolume obeys the following conservation law:

π

∫ R(t)

0rhdr = F (t). (6)

Two unknown functions are to be determined: the liquid pro-le, h(t, r), and the radius of the spreading drop base, R(t), which

s referred below as “the radius of spreading”. The pressurenside the spreading drop, p(t, r), can be determined via theiquid profile, h(t, r). This expression includes two components:apillary [1] and gravitational [2] parts:

= pa − γK + ρgh (7)

here pa is the pressure in the ambient air; γ , the liquid–airnterfacial tension; ρ and g, the liquid density and gravity accel-ration, respectively; K, the curvature of the liquid–air inter-ace. In the low slope approximation (ε 1) the curvature is

= (1/r)(∂/∂r)(r(∂h/∂r)), hence, Eq. (7) becomes:

= pa − γ1

r

∂

∂r

(r∂h

∂r

)+ ρgh. (8)

Let us introduce scales in radial and vertical directions, r0,0, respectively. Then the following dimensionless quantities arentroduced

= r

r0, h = h

h0or r = rr0, h = hh0,

isjoining pressure and, hence, we cannot consider the drop pro-le in a vicinity of the three-phase contact line. However, belowe are not going to calculate the drop profile. Instead we try

imilarity solutions of the equation of spreading (5) in the casef both capillary and gravitational regime of spreading (that is,nitial and final stages of spreading of small drops). This methodllows us calculating time evolution of the radius of spreading,ut the pre-exponential factor includes an unknown dimension-ess integration constant. The latter constant is extracted fromxperimental data.

If the initial drop size is small enough then the effect of grav-ty can be ignored. Accordingly, the drop radius R(t) has to bemaller than the capillary length, R(t) ≤ √

γ/ρg. The liquid isnjected through the orifice in the drop centre with the flow rate′(t), where F(t) is an imposed function of time. In the case ofpontaneous spreading F(t) = Ω = const, where Ω is the constantolume of the spreading drop. In the case of constant liquid flowate from the liquid source F(t) = It, where I is the intensity ofhe liquid source.

Mass conservation of liquid Eq. (6) demands that F(0) = 0 ifhere was not a drop at the initial time.

In Appendix A, we deduce a condition when the drop spreadsccording to the power law and the general possible form of(t) compatible with the power law of spreading is deduced.

n the same Appendix we deduce dependences of the radiusf spreading with time in the case of complete wetting (bothapillary and gravitational regime of spreading) and in the casef partial wetting (small capillary numbers, Ca = (Uη/γ)).

In the case of the constant source of liquid I in the drop centrehe following dependencies for the radius of the drop base areeduced in Appendix A.


During the initial (capillary) stage of spreading of small dropsthe radius of the drop base should follow

R(t) = ωc

(γI3

η

)0.1

t0.4, (9)

while at the final (gravitational) stage of spreading of small dropsthe radius of the drop base follows the law

R(t) = ωg

(ρgI3

η

)1/8

t0.5, (10)

where ωc, ωg are the dimensionless constants to be determinedfrom experimental data.

In the case of partial wetting the static hysteresis of the con-tact angle determines the spreading behavior at low capillarynumbers, Ca 1, where U = R(t) is the rate of spreading. Thelatter condition was always satisfied under our experimental con-ditions.

Because of the contact angle hysteresis the drop does notmove if the contact angle, θ, is in the range θR < θ < θA, whereθA and θR are the static advancing and static receding contactangles, respectively [4]. In our experimental procedure (Fig. 1)we are interested in the static advancing contact angle only.

If the capillary number, Ca, is very small, which is the case inour experiments, then the advancing contact angle does not varysignificantly. It is assumed below that the contact angle, θ, doesne

R

oadscors

3

3

sBtbTiρ

Gfi

Fig. 2. The experimental set-up for spreading experiments: (1) solid substrate;(2) syringe; (3) droplets of silicone oil or water; (4) source of light; (5) filterwith wavelength 640 nm; (6) CCD camera; (7) tape recorder; (8) computer; (9)Harvard Apparatus syringe pump.

was injected through those orifices in the glass substrate witha constant flow rate using a Harvard Apparatus syringe pump.The latter produced a liquid drop over the solid substrate. Con-stant flow rate resulted in a linear increase of the drop volumewith time. The time evolution of the radius of the base of thespreading drops was monitored.

The glass slides were cleaned by immersing them in achromic acid solution for 2 h followed by 10 times rinsing withdistilled water and two times with ultra pure water and weredried in an oven at 70 C for 30 min. Each cleaned and driedslide was used only once. At least three runs were conductedfor each experimental condition and average values are reportedbelow.

The diagram of the experimental set-up is shown in Fig. 2. Allexperiments were carried out at 25 ± 0.5 C. The solid substrate,1 (Fig. 2) was fixed in the ring; a syringe, 3, was positioned in thecentre of the substrate and connected to the Harvard Apparatussyringe pump, 9. The droplets of silicone oil or water, 2, wereformed due to the injection. The following flow rates were used0.005 ml/min, 0.01 ml/min and 0.02 ml/min.

The spreading process was recorded using a CCD camera, 6(Fig. 2) and a VHS recorder, 7. The camera has been equippedwith filters, 5, with a wavelength of 640 nm. Such an arrange-ment suppresses illumination of the CCD camera by the scat-tered light from the substrate and, hence, results in a higherprecision of the measurements. The source of light, 4, was useddnA

4

aisl

4

ofid

ot vary over duration of the spreading experiment and remainsqual to its static value θA.

In this case of the radius of the drop base should follow

(t) =(

I

Φ(θA)

)1/3

t1/3. (11)

In spite of the similarity between expressions for the radiusf spreading (Eqs. (9) and (10)) in the case of complete wettingnd Eq. (11) in the case of partial wetting) there is one significantifference between these two spreading processes: if the liquidource is closed then in the case of complete wetting the drop willontinue to spread out according to the law R(t) ∼ t0.1 (in the casef capillary regime) or R(t) ∼ t1/8 (in the case of gravitationalegime). However, in the case of partial wetting the drop willtop spreading as soon as the liquid source is closed.

. Experimental set-up and results

.1. Materials and methods

The spreading of silicone oil and aqueous droplets over glassubstrates was investigated. Silicone oil was purchased fromROOKFIELD. Its viscosity was measured using the rheome-

er AR1000 (TA Instruments) at 25 C. Density was measuredy the weight method and for measuring surface tension theensiometer (White, Elec. Inst Co. Ltd.) was used. The follow-

ng values were found: dynamic viscosity η = 99.0 cP, density= 0.96 g/cm3, surface tension γ = 22.5 dyn/cm.Microscope glass slides (76 mm × 26 mm, “Menzel-Glaser”,

ermany) were used for spreading experiments. Circular ori-ces of diameters 0.5 mm were drilled in their centers. A liquid

uring experiments. The camera and a VHS recorder were con-ected to a computer, 8. Images were analyzed using Drop Shapenalysis “FTA 32”.

. Results and discussion

The kinetics of spreading of silicone oil (complete wetting)nd aqueous droplets (partial wetting) over glass substrates wasnvestigated. The spreading process was caused by both thepontaneous spreading and the injection of liquid through theiquid source in the centre of drops.

.1. Complete wetting

Two stages of spreading of small silicone oil drops have beenbserved: the first initial stage–capillary regime and the secondnal stage–gravitational regime. The experimental time depen-ences of radius of spreading of silicone oil drops over glass


Fig. 3. Radius of spreading versus time for the spreading of silicone oil drops onglass surface (log–log plot), diameter of the orifice 0.5 mm: () capillary stage;() gravitational stage, I = 0.005 ml/min, experiment 1; () capillary stage; ()gravitational stage, I = 0.01 ml/min, experiment 2; () capillary stage; (♦) grav-itational stage, I = 0.02 ml/min, experiment 3; dashed line: fitted according toEq. (9), solid line: fitted according to Eq. (10).

surface are presented in Fig. 3. Three experiments with differ-ent injection velocities 0.005 ml/min, 0.01 ml/min, 0.02 ml/minwere conducted. In each experiment the two mentioned above-mentioned stages of spreading are observed in Fig. 3.

The data obtained from the initial stage of spreading, corre-spond to the capillary stage (R(t) < a). Eq. (9) can be rewrittenas follows:

lgR(t) = lg

[ωc

(γI3

η

)0.1]

+ 0.4 lgt, (12)

which is a linear function of time in lg–lg co-ordinates.According to Eq. (12) the experimental date were fitted

according to

lgR(t) = X + n lgt, (13)

where intercept, X, and slope, n, are the fitting parameters. Thefitted exponents, n (Table 1) show good agreement with the the-oretically predicted exponent 0.4.

The fitted value of X was compared with Eq. (12), and theunknown dimensionless constant, ωc, was determined as fol-lows:

ωc =(

η

γI3

)0.1

exp(X) (14)

Using the latter equation the constant ωc was calculated for eachs

TS

I

I

000

Fig. 4. Radius of spreading versus time for the spreading of water droplets onglass surface (log–log plot), diameter of the orifice 0.5 mm: () I = 0.005 ml/min,experiment 1; () I = 0.01 ml/min, experiment 2; () I = 0.02 ml/min, experiment3; solid line: drawn according to Eq. (11).

ωc = 2.57 ± 0.18. The model predictions are shown in Fig. 3 bydashed lines.

In the case of the gravitational regime of spreading (R(t) > a)experimental data were compared with the theoretical predic-tions according to Eq. (10). This equation can be rewritten as:

lgR(t) = lg

[ωg

(ρgI3

η

)1/8]

+ 0.5 lgt, (15)

According to Eq. (15) experimental date were fitted accordingto:

lgR(t) = Y + n lgt, (16)

where Y and n are the fitting parameters. The theoretical predic-tions are shown in Fig. 3 by solid lines. The fitting of the dataresulted in an average value of the exponent, n, in Eq. (16) equalto 0.4961, which shows good agreement with the theoreticallypredicted exponent 0.5. Using the fitted value Y and Eq. (15) theunknown dimensionless constant, ωg, was calculated as:

ωg =(

η

ρgI3

)1/8

exp(Y ). (17)

The average value of the constant ±S.D. wasωg = 2.64 ± 0.38. The average drop height is 3.6 ± 0.3 mm,which is in good agreement with the calculated value 3.9 mm.

4

ie0ia

l

et of data (Table 1). The average value ±S.D. obtained was

able 1preading of silicone oil drops over glass surface with the orifice in the centre

njection flow rate Capillary stage Gravitational stage

(mm3/s) ωc n ωg n

.0833 2.7589 0.4007 2.6402 0.4893

.1667 2.5426 0.4005 2.1811 0.5000

.3333 2.4040 0.4005 1.8833 0.4991

.2. Partial wetting

The partial wetting case was investigated using the spread-ng of water droplets over the same glass surfaces. Threexperiments with different injection velocities 0.005 ml/min,.01 ml/min, 0.02 ml/min were presented in Fig. 4. The spread-ng behavior was compared with the theoretical predictionccording to Eq. (11), which can be written as:

gR(t) = lg

(I

Φ(θA)

)1/3

+ 1

3lgt. (18)


During the spreading of water droplets over glass surfaces theadvancing contact angle does not vary significantly. The averagevalue of the contact angle was calculated for each experimentalrun, and the average advancing contact angle was determinedfrom three experimental runs. The average advancing contactangle θA ± S.D. is 54 ± 2. This value was used in Eq. (A.34)for the calculation of the function Φ(θA), which was substitutedin Eq. (18). The dashed line in Fig. 4 was plotted according to Eq.(18). Note that Eq. (18) does not include any fitting parameters.Fig. 4 shows that our experimental data are in good agreementwith the theoretically predicted law (18).

5. Conclusions

The spreading of liquid over solid substrates when there isliquid injection through an orifice is investigated from both the-oretical and experimental points of view. Two cases of spreadingover a glass substrate with a diameter of the orifice 0.5 mmwere studied: spreading of silicone oil droplets (complete wet-ting) and spreading of water droplets (partial wetting). In thecase of silicone oil spreading two regimes of spreading wereobserved: capillary regime and gravitational regime. Theory hasbeen developed for the cases of complete wetting and partial wet-ting at low capillary numbers. In all three case power laws ofspreading were deduced: capillary regime of spreading accord-iEErc

A

Nsaui

A

f

ξ

wiR

i

2

Let us select the unknown function H(t) as

H(t) = F (t)

2πR2(t). (A.3)

Then Eq. (A.2) can be reduced to∫ 1

0ξϕ(ξ)dξ = 1. (A.4)

To move further it is necessary to specify the equation ofspreading (5), which determines the drop profile, h(t, r), or ϕ(ξ).Two different cases are under consideration: complete and par-tial wetting cases.

A.1. Capillary regime, complete wetting

In this case according to Eq. (8) the pressure inside the spread-ing drop can be written as

p = pa − γ1

r

∂

∂r

(r∂h

∂r

).

The latter expression should be substituted into Eq. (5), whichyields the time evolution of the drop profile, h(t, r):

∂h

∂t= − 1

3η

1

r

∂

∂r

[rh3 ∂

∂r

(γ

r

∂

∂r

(r∂h

∂r

))]. (A.5)

Nw[

E

H

wde

aps

wbo

ng to Eq. (9), gravitational regime of spreading according toq. (10) and partial regime of spreading according to Eq. (11).xperimental data validated our theoretical dependences of the

adius of spreading on both time and injection velocity in bothases of complete and partial wetting.

cknowledgements

P. Prokopovich research has been supported by the Professoril Halliwell Development Fund award. This research was also

ponsored by Grant 15544 from the Royal Society, UK. Theuthors wish to express their gratitude to Dr. S. Kosvintsev forseful discussions. Also authors thank Dr. P. Russell for his helpn construction of the experimental set-up.

ppendix A

Let us introduce the following similarity co-ordinate andunction

= r

R(t), ϕ(ξ) = h(t, r)

H(t), (A.1)

here ϕ(ξ), H(t) are two new unknown functions. Note, we arenterested only in the time evolution of the radius of spreading,(t).

Substitution of (A.1) into the conservation law, Eq. (6), resultsn

πR2(t)H(t)∫ 1

0ξϕ(ξ)dξ = F (t). (A.2)

ote that the omission of the surface forces action results in theell-known singularity on the moving three phase contact line

5].Substitution of the similarity co-ordinate and function using

qs. (A.1), (A.3) into Eq. (A.5) results in

˙ (t)ϕ(ξ) − H(t)R(t)

R(t)ξϕ′(ξ)

= − γ

3η

H4(t)

R4(t)

1

ξ

[ξϕ3(ξ)

(1

ξ(ξϕ′(ξ))′

)′]′,

here an overdot denotes differentiation with time, while meansifferentiation with respect to the similarity variable, ξ. The latterquation can be rewritten as

3ηH(t)R4(t)

γH4(t)ϕ(ξ) − 3ηR3(t)R(t)

γH3(t)ξϕ′(ξ)

= −1

ξ

[ξϕ3(ξ)

(1

ξ(ξϕ′(ξ))′

)′]′(A.6)

Eq. (A.6) should depend on the similarity co-ordinate onlynd it should not include any time dependence. The latter isossible only if simultaneously the following two relations areatisfied

3ηH(t)R4(t)

γH4(t)= B1,

3ηR3(t)R(t)

γH3(t)= B2, (A.7)

here B1 and B2 are unknown constants. Both constants shoulde positive because H(t) and R(t) are both increasing functionsf time.


Let α = B1/B2 and divide the first equation in (A.7) by thesecond equation. It results in: H/H = α(R/R), that upon inte-gration yields:

H(t) = CRα(t), (A.8)

where C is an integration constant and α is still an unknownexponent.

Substitution of Eq. (A.8) into both Eq. (A.7) results in thefollowing time evolution of the radius of spreading, R(t),

R(t) =(

(4 − 3α)γB2C3

3η

)1/(4−3α)

t1/(4−3α), (A.9)

which shows that 4 − 3α should be positive, that is, α < 4/3.Eqs. (A.9) and (A.3) allows determining the unknown func-

tion H(t):

H(t) = C

((4 − 3α)γB2C

3

3η

)α/(4−3α)

tα/(4−3α). (A.10)

Using Eqs. (A.10), (A.9) and (A.3) we can conclude that thefollowing relation should be satisfied:

F (t) = 2πC

[(4 − 3α)γB2C

3

3η

](2+α)/(4−3α)

t(2+α)/(4−3α). (A.11)

The latter relation shows that the similarity mechanism con-sb

F

wnα

E

R

A0t

F

a1Bpg

C

Tl

R

The only unknown constant in Eq. (A.16) is the dimensionlessconstant B1.

The equation of spreading (A.6) can be rewritten now as

B1ϕ(ξ) − 2B1ξϕ′(ξ) = −1

ξ

[ξϕ3(ξ)

(1

ξ(ξϕ′(ξ))′

)′]′. (A.17)

Recall that the latter equation describes the behavior of the dropprofile, ϕ(ξ), not too close to the moving three phase contact line(valid only away from the range of the surface forces action). Letus try to include the disjoining pressure action into Eq. (A.17).In the case of complete wetting, the disjoining pressure isothermis Π(h) = A/h3, where A is the Hamaker constant. Now Eq. (8)should be rewritten as

p = pa − γ1

r

∂

∂r

(r∂h

∂r

)+ ρgh − A

h3

and the latter expression should be substituted in the equationof spreading (5). Using the same similarity co-ordinate, ξ, andthe drop profile, ϕ, according to Eq. (A.1) we arrive after similarto the previous consideration to the following equations for thedetermination of the unknown function, ϕ:

B1ϕ(ξ) − 2B1ξϕ′(ξ) = −1

ξ

[ξϕ3(ξ)

(1

ξ(ξϕ′(ξ))′ + χB1

ϕ3(ξ)

)′]′,

(A.18)

was(f

ϕ

ws

ϕ

wti(a

[r“dusttn

R

idered above is possible only if the dependency F(t) is definedy the power law (A.11).

Let us assume now that

(t) = a′tb, (A.12)

here a′ and b are constants imposed by the source. The expo-ents in Eqs. (A.11) and (A.12) should be equal, and hence,= (4b − 2)/(1 + 3b). Substitution of the latter expression intoq. (A.9) gives the following spreading law:

(t) = Ct0.1+0.3b, (A.13)

ccordingly, the exponent is the sum of two terms: the first term,.1, stems from the spontaneous spreading [1] and the seconderm, 0.3b, is determined by the liquid source.

In the case of constant flow rate of the liquid,

(t) = It, (A.14)

nd the comparison of exponents in Eqs. (A.14) and (A.9) yields= (2 + α)/(4 − 3α), or α = 1/2. On the other hand α = B1/B2, or2 = 2B1 though B1 is the only unknown constant. The com-arison of the pre-exponential factors in Eqs. (A.14) and (A.9)ives

=(

3ηI

10πγB1

)1/4

. (A.15)

he spreading law according to Eq. (A.13) takes now the fol-owing form:

(t) =(

5γB1I3

24ηπ3

)0.1

t0.4. (A.16)

here χ = 10πA/3ηI is a dimensionless constant, which char-cterizes the intensity of the surface forces action. Eq. (A.18)hows that the same similarity property is valid for the full Eq.A.18), when the surface forces action is taken into account asor Eq. (A.17).

Eq. (A.18) must satisfy the following boundary conditions:

′(0) = ϕ′′′(0) = 0, (A.19)

hich are the symmetry conditions in the drop centre. It alsohould satisfy

(ξ) → 0, ξ → ∞, (A.20)

here “infinity” means the drop edge. It has been shown in [1]hat the drop profile tends asymptotically to zero inside “thenner region”, which is the meaning of the boundary conditionA.20). The conservation law (A.10) should also to be taken intoccount.

The unknown parameter, B1, should be small according to1]. In this case the whole drop can be subdivided into twoegions: “the outer region”, which is of spherical shape, andthe inner region”, where the inner co-ordinate should be intro-uced. Matching of these two regions allows determining thenknown parameter, B1, via the dimensionless Hamaker con-tant, χ. It has been shown in [1] that the dependence of B1 onhe parameter χ is a weak one. It has been shown also in [1] thathe lack of the proper asymptotic behavior of Eq. (A.18) doesot allow determining precisely the unknown constant B1.

That is why we use Eq. (A.16) in the following form

(t) = ωc

(γI3

η

)0.1

t0.4, ωc =(

5B1

24π3

)0.1

, (A.21)


where the dimensionless parameter ωc is determined belowusing experimental data.

A.2. Gravitational regime, complete wetting

In this case Eq. (8) becomes p = pa + ρgh, and substitution ofthe latter equation into the equation of spreading (5) results in

∂h

∂t= ρg

3ηr

∂

∂r

(rh3 ∂h

∂r

). (A.22)

Using the same similarity co-ordinate and function (A.1) we con-clude that relations (A.2)–(A.4) are still valid. Using the sameprocedure as above we can transform Eq. (A.22) to

H(t)ϕ(ξ) − H(t)R(t)

R(t)ξϕ′(ξ) = ρg

3η

H4(t)

R2(t)

1

ξ[ξϕ3(ξ)ϕ′(ξ)]′.

The latter equation can be rewritten as

3ηH(t)R2(t)

ρgH4(t)ϕ(ξ) − 3ηR(t)R(t)

ρgH3(t)ξϕ′(ξ) = 1

ξ[ξϕ3(ξ)ϕ′(ξ)]′.

(A.23)

tis

wcb

sg

H

wn

f

R

w

t

H

Using Eqs. (A.26), (A.27) and (A.3), we can conclude

F (t) = 2πG

[(2 − 3β)ρgD2G

3

3η

]2+β/2−3β

t2+β/2−3β. (A.28)

Thus, we see that the similarity mechanism considered aboveis possible only if the dependency F(t) is defined by the powerlaw (A.28).

Let us assume now that the liquid source produce the liquidin the same way as in the case of the capillary regime, that is,according to Eq. (A.12)

The exponents in Eqs. (A.28) and (A.12) should be equal, andhence, in β = (2b − 2)/(1 + 3b). Substitution of the latter expres-sion into Eq. (A.26) gives the following spreading law:

R(t) = constt1/8+3b/8, (A.29)

that is the exponent is the sum of two terms: the first term, 1/8,stems from the spontaneous gravitational spreading [2] and thesecond term, 3b/8, is determined by the liquid source.

In the case of constant flow rate of the liquid,

F (t) = It (A.30)

and the comparison of exponents in Eqs. (A.30) and (A.12)results in 1 = (2 + β)/(2 − 3β), or β = 0. On the other handβ = D1/D2, hence, D1 = 0. Then D2 is the only unknown constant.Atouf

G

Tl

R

TDNifaga

(

R

a

t

Eq. (A.23) should depend on the similarity co-ordinate only,hat is, it should not include any time dependence. The latters possible only if simultaneously the following relations areatisfied

3ηH(t)R2(t)

ρgH4(t)= D1,

3ηR(t)R(t)

ρgH3(t)= D2, (A.24)

here D1 and D2 are unknown dimensionless constants. Bothonstants should be positive (or zero) because H(t) and R(t) areoth increasing functions of time.

Let β = D1/D2 and divide the first equation in (A.24) by theecond equation. That results in H/H = β(R/R), that upon inte-ration yields

(t) = GRβ(t), (A.25)

here G is an integration constant and β is still unknown expo-ent.

Substitution of Eq. (A.25) into both Eq. (A.24) results in theollowing time evolution of the radius of spreading, R,

(t) =(

(2 − 3β)ρgD2G3

3η

)1/(2−3β)

t1/(2−3β), (A.26)

hich shows that 2 − 3β should be positive, that is, β < 2/3.Eqs. (A.26) and (A.25) allow determining the unknown func-

ion H(t):

(t) = G

((2 − 3β)ρgD2G

3

3η

)β/(2−3β)

tβ/(2−3β). (A.27)

ccording to the first Eqs. (A.24) and (A.25) D1 = β = 0 meanshat in the case of gravitational spreading the maximum heightf the spreading drop remains constant when the source of liq-id follows Eq. (A.30). The comparison of the pre-exponentialactors in Eqs. (A.30) and (A.28) gives

=(

3ηI

4πρgD2

)1/4

he spreading law according to Eq. (A.26) takes now the fol-owing form:

(t) = 21/6(

ρgD2I3

3ηπ3

)1/8

t0.5. (A.31)

he only unknown in Eq. (A.31) is the dimensionless constant2, which is left unknown and was determined experimentally.ote the constant D2 can be in general determined in the follow-

ng way: (i) a narrow zone close to the drop edge, where capillaryorces become important should be considered; (ii) matching ofsymptotic solutions (capillary zone as an “inner zone” and theravitational zone as an “outer zone”) should be made, whichllows determining the numerical constant D2.

In order to determine the unknown constant we rewrite Eq.A.31) as

(t) = ωg

(ρgI3

η

)1/8

t0.5, ωg = 21/6(

D2

3π3

)1/8

, (A.32)

nd the constant ωg is obtained using experimental data.The latter expression allows determination of the constant

hickness of the spreading drop during the gravitational regime


of spreading. Combining Eqs. (A.3) and (A.32) results in

H = 1

ω2g

(Iη

ρg

)1/4

, (A.33)

which is independent of time as predicted above. It means thatduring the gravitational regime of spreading from a liquid sourcewith constant flow rate intensity, the drop spreads as “a pancake”.

A.3. Partial wetting

The drop volume can be expressed in terms of the spreadingradius and the contact angle as follows:

V (t) = R3(t)Φ(θA), Φ(θA) = π

6tan

θA

2

(3 + tan2 θA

2

),

(A.34)

where θA is the static advancing contact angle. We assumethat the capillary number is very small and, hence, the con-tact angle does not change during spreading. Hence, Φ(θA) also

remains constant. Combination of Eqs. (A.10) and (A.34) resultsin R(t) = (F(t)/Φ(θA))1/3.

In the case of the constant flow rate from the liquid source(Fig. 1) according to Eq. (A.30) the latter equation gives

R(t) =(

I

Φ(θA)

)1/3

t1/3, (A.35)

which is compared with our experimental observations in thecase of partial wetting.

References

[1] V.M. Starov, V.V. Kalinin, J.-D. Chen, Spreading of liquid drops oversolid substrata, Adv. Colloid Interface Sci. 50 (1994) 187.

[2] J. Lopez, C.A. Miller, E. Ruckenstein, Spreading kinetics of liquid dropson solids, J. Colloid Interface Sci. 53 (1976) 460.

[3] A.M. Cazabat, M.A. Cohen Stuart, Dynamics of wetting: effects of sur-face roughness, J. Phys. Chem. 90 (1986) 5849.

[4] V.M. Starov, Equilibrium and hysteresis contact angles, Adv. ColloidInterface Sci. 39 (1992) 147.

[5] E.B. Dussan, V.E. Rame, S. Garoff, On identifying the appropriate bound-ary conditions at a moving contact line: an experimental investigation, J.Fluid Mech. 230 (1991) 97.

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Spreading of liquid drops from a liquid source