detachment of oil drops from solid surfaces in surfactant...

Detachment of Oil Drops from Solid Surfaces in SurfactantSolutions: Molecular Mechanisms at a Moving Contact Line†

Peter A. Kralchevsky,*,‡ Krassimir D. Danov,‡ Vesselin L. Kolev,‡Theodor D. Gurkov,‡ Mila I. Temelska,‡ and Gu1 nter Brenn§

Laboratory of Chemical Physics & Engineering, Faculty of Chemistry, University of Sofia,1164 Sofia, Bulgaria, and Institute of Fluid Mechanics and Heat Transfer, Graz University of Technology,8010 Graz, Austria

Here, we present experimental data and a theoretical model for the dynamics of detachment ofhexadecane drops from a solid substrate (glass plate) in aqueous solutions of anionic surfactantand salt, at various temperatures. The influence of the experimental conditions on the motionof the three-phase contact line is investigated. We found indications that water molecules canpropagate by lateral diffusion in a thin layer on the surface of the solid plate. The driving forceof the detachment process, viz., the imbalance of the interfacial tensions at the contact line, isengendered by the water penetration, while the line friction force compensates this imbalanceand determines the stationary speed. Excellent agreement between theory and experiment isachieved. The present study specifies the parameters that can be used to quantitativelycharacterize the rate of drop detachment, determines the values of these parameters at variousexperimental conditions, and indicates tools for control of the investigated spontaneous process.

1. Introduction

Processes with a moving contact line are crucial formany applications in coating, printing, painting, deter-gency, and membrane emulsification. The most studiedare the cases when the motion of the contact line on asolid surface is strained by some external force orpotential gradient, including processes of liquid deposi-tion on a moving or porous substrate; see refs 1-14 andthe literature cited therein. In contrast, in the case of adiffusional mechanism, observed for oily drops detach-ing from a solid surface in water,15,16 the contact-linemotion occurs spontaneously, at a nonzero contact angle,driven by some molecular mechanisms; for example,water and surfactant diffuses between the oil and solid.The velocity of the latter process is much lower thanthat in the case of conventional spreading (zero contactangle; see, e.g., ref 17) or superspreading.18

Several mechanisms have been discussed in theliterature in relation to the cleaning of solid surfacesfrom oily deposits. The most popular of them are theroll-up, emulsification, and solubilization.19-26 Depend-ing on the specific system, one or another mechanismcan prevail. From a practical viewpoint, it is importantto reveal the physicochemical factors that can be usedfor efficient control of the cleaning process, some of thembeing the type and concentration of the used surfactantsand electrolytes.15,16,27-31 Technologically oriented ex-periments on the detachment of oil drops from solidsubstrates were carried out by Dillan et al.,21 whoobtained much data about the efficiency of the roll-upmechanism. The experiments indicate that the apparent“roll-up” is related to a shrinking of the three-phase

contact line solid-oil-water, which, in its own turn, isdue to the molecular penetration (diffusion) of watermolecules between the oil drop and the solid phase.15

The latter process was termed the diffusional mecha-nism of oil detachment. Chatterjee investigated thecritical conditions for buoyancy-induced detachment ofoil drops from a substrate due to instability in the shapeof the oil-water interface.32,33

Wasan et al.15 investigated the detachment of crude-oil drops from glass in solutions of 1 wt % C16-R-olefinsulfonate + 1 wt % NaCl. These authors observeddirectly the dynamics of water-film penetration betweenthe oil phase and the solid. Once such a disjoiningaqueous film has been formed, even a weak shear flowis able to detach the oil drop from the substrate. Thestudy in ref 15 is related to the enhanced oil recovery;however, a similar mechanism can be operative also foroil-drop detachment in other applications of detergency.

Our previous study16 was directed toward analyzingthe mechanism of spontaneous detachment of oil dropsfrom solid surfaces in solutions of ionic surfactants. Wecarried out direct microscopic observations of the clean-ing process for a hydrophilic glass surface, in an attemptto reveal the main stages of the oil-drop detachment.Our attention was focused on the balance of forces atthe moving three-phase contact line. In such cases, aline friction force has to be included in the Neumann-Young force balance at the contact line.34,35 Analyzingthe experimental data, we found that this friction forceis proportional to the velocity of contact-line motion.From the slope of the respective linear regression, theline friction coefficient was determined for the specificsystem investigated.16

The physical origin of the line friction force, and itsimportance for various processes with dynamic contactlines, was discussed in a number of studies by Blake etal.,10,35 de Coninck et al.,36-41 Attard,34 and von Bahret al.42 In our experiments,16 we found that the relax-ation of the contact angle toward its equilibrium valueis impeded and decelerated by the line friction force.

† This paper is dedicated to Professor Darsh T. Wasan onthe occasion of his 65th birthday.

* To whom correspondence should be addressed. Tel.: (+359)2-962 5310. Fax: (+359) 2-962 54 38. E-mail: [email protected].

‡ University of Sofia.§ Graz University of Technology.

1309Ind. Eng. Chem. Res. 2005, 44, 1309-1321

10.1021/ie049211t CCC: $30.25 © 2005 American Chemical SocietyPublished on Web 12/24/2004

Thus, it turns out that the magnitude of the line frictioncoefficient, â, determines the rate of shrinking of thecontact line and, consequently, the overall time of oil-drop detachment. (In a similar way, the bulk viscosityof a liquid determines the velocity of a falling heavyball.) Hence, the value of â, and its dependence on theexperimental conditions, represents a problem of sci-entific and practical importance that deserves a moredetailed investigation.

The present paper is a continuation of our previousstudy16 in two aspects. First, we examine systematicallythe effect of the most important factors (temperatureand surfactant and salt concentrations) on the dynamicsof drop detachment. Second, we develop a quantitativetheoretical model, which enables us to fit the experi-mental dependence of the contact-line radius on timeand to determine the values of the involved physico-chemical parameters.

The paper is organized as follows. In section 2, wedescribe the experimental system and method. Theexperimental results are presented and discussed insection 3. The theoretical model is described in section4. Finally, section 5 is devoted to the comparison oftheory and experiment and to data interpretation. Theresults reveal that the penetration and diffusion ofwater in a thin layer on the solid surface could be ratherimportant for the dynamics of oil-drop detachment.

2. Experimental Section

In our experiments, the oil phase was pure hexade-cane (Aldrich). The water phase was an anionic surfac-tant solution. We used two surfactants, sodium dodecylsulfate (SDS, a product of Merck), and C16-R-olefinsulfonate (AOS, the technical product Hostapur OSB ofClariant). As an inorganic electrolyte, we used pureNaCl. In the experiments, we investigated the role ofthree factors: (i) surfactant concentration, (ii) NaClconcentration, and (iii) temperature.

We applied a simple experimental setup, which issketched in Figure 1. A square glass plate, 22 × 22 mm,representing a plane-parallel microscope slide (manu-factured by Menzel-Glaser), serves as a substrate. Toavoid irreproducible variations in the surface propertiesof the glass, in each experiment a new slide was usedas received directly from the manufacturer, without anyprevious treatment. The slide was positioned horizon-tally on two rectangular glass holders in a cuvette. Theslide was pressed down by two other glass pieces (3 inFigure 1).

Initially, on the dry glass plate, we place a hexadecanedrop (of volume ≈1 µL) by means of a microsyringe. Thisdrop is left to stay for about 10 min, to achieve stronger

adhesion to the substrate. Then, we add a surfactantsolution of volume 20 mL. It is loaded in a syringe (5 inFigure 1) and flows slowly through its needle to coverthe drop as gently as possible. It is important to adjustthe horizontal level of the glass slide so that the liquidis wetting it uniformly in all directions. Often, a part ofthe drop detaches immediately after its contact with thesurfactant solution (necking instability due to thebuoyancy force), but a residual (smaller) oil drop re-mains on the substrate. Further, we observe and recordthe evolution of this residual drop; see the illustrativephotographs in Figure 2. In all experiments, withelapsed time, the contact line shrinks, the oil-solidcontact area decreases, and the drop becomes verticallyelongated under the action of buoyancy (pendant-drop-type profile). At the final stage (Figure 2, last photo-graph), a neck is formed. Next, the drop detaches veryquickly. Usually, the drop detaches completely, withoutformation of a new residual drop.

The observation of the drop profile was performedfrom the side, through the wall of the glass cuvette. Ahorizontal microscope, equipped with an objective oflong focal distance, was applied for this purpose. Adigital CCD camera (Kappa CF 8/1 DX) and VCR(Samsung SV-4000) were used to record the pictures.For most systems, we performed three independent runsmonitoring the time evolution of the shape of threedrops with different sizes, from the moment of additionof the aqueous solution until the eventual drop detach-ment. This process takes place at a fixed volume of thedrop. In other words, the solubilization of oil by themicelles in the investigated surfactant solutions isnegligible. This was checked by experiments with singledrops, applying the method described in ref 43.

Images of the drop shape were continuously taken bythe CCD camera and recorded. Pictures, correspondingto consecutive moments (like those in Figure 2), weredigitized, assigning coordinates to a number of pointson the drop profile. An example is given in Figure 3.Then, the experimental points are fitted with theLaplace equation of capillarity. The best fit is shownby the continuous line in Figure 3. Details about theprocessing of the drop profiles are given in section 3 ofref 16. From the Laplace fits, we obtained the values ofthe contact-line radius, rc, contact angle, R (see Figure3), and the oil-water interfacial tension, σ, as functionsof time, t. In fact, the determination of σ from the dropprofile is analogous to the known pendant drop methodor to the axisymmetric drop shape analysis; see, e.g.,ref 44.

It should be noted that the occurrence of the dropdetachment depends strongly on the state of the glasssurface. In control experiments, we kept the glass platesimmersed in pure water for 1 h (instead of using themdry, without any pretreatment). After that, the plateswere dried for 5 min in a vacuum drier and used in theexperiments. Instead of the slow process in Figure 2,we observed that the hexadecane drops detach im-mediately from such glass substrates. The latter behav-ior can be attributed to the penetration of watermolecules in a thin layer (“gel layer”) on the surface ofthese glass plates, which have been preimmersed inwater.16

3. Experimental Results and Discussion

3.1. Effects of the Surfactant, Electrolyte, andTemperature. Figures 4-7 show experimental data for

Figure 1. Scheme of the experimental cell: 1, glass plate; 2, oildroplet ≈ 1 µL; 3, glass holders; 4, surfactant solution; 5, syringe;6, cuvette.

1310 Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005

the time dependence of the contact-line radius, rc, andangle, R, for hexadecane drops (like that in Figure 2).In each figure, the data for rc and R, denoted withidentical symbols, correspond to the same drop. Everyexperimental point is obtained from a digitized dropprofile, like those in Figure 3. In Figures 4a-7a, thesymbols denote the experimental rc while the solid linesconnect the respective theoretical values of rc, obtainedby fitting the data with the help of the model describedin section 4. A discussion about the comparison of theoryand experiment is given in section 5.2.

Figures 4-7 show that initially rc and R decreasequickly, but later their variation slows down. Mostprobably, such behavior is caused by a decreasing(relaxing) imbalance of the interfacial tensions at thethree-phase contact line (see eq 3.2). At the final stageof the process, the contact-line radius rc becomes sosmall that a neck begins to form; see Figures 2 and 3b.In Figures 4b-7b, the necking is manifested as anincrease of R at the last stage of drop evolution, justbefore the drop detachment from the substrate. Thecontact angle, R, varies (decreases and increases) duringthe whole process, without reaching any equilibriumvalue, Req. In this respect, the present data differ fromour previous experimental results,16 where R leveled offat a constant value, which could be interpreted as theequilibrium value of R. We believe that the differenceis due to the different types of glass plates used in thetwo experiments as substrates. On the other hand, theshapes of the obtained rc(t) dependencies are similar inthe two experiments (here and in ref 16).

Figure 4 shows data for four hexadecane drops at fourdifferent surfactant (SDS) concentrations. Figure 5shows similar data but for different concentrations ofAOS. In general, one sees that the time needed for drop

detachment decreases with the rise of the surfactantconcentration. Figure 6 demonstrates that the electro-lyte concentration has a strong effect on the dropdetachment time: the latter significantly decreases withthe rise of the NaCl concentration. Finally, Figure 7indicates an analogous effect of the temperature: thedrop detachment is faster at higher temperatures.

Note, however, that the detachment time depends notonly on physicochemical factors, such as the tempera-ture and surfactant and salt concentrations, but alsoon a geometrical factor: the drop volume (longer de-tachment for bigger drops). For this reason, the detach-ment time is not an appropriate physical characteristicof the process dynamics. As mentioned above, it isdifficult to carry out experiments with drops of identicalvolume. (After the surfactant solution is poured, somepart of the oil drop spontaneously detaches because ofnecking instability, and we perform the experiment withthe residual drop whose volume is different in differentruns.) Therefore, it is better to characterize the dynam-ics of drop detachment with parameters that are inde-pendent of the drop volume. The theoretical analysis ofthe above experimental data indicates that the penetra-tion time, tp, and the line friction coefficient, â, areindependent of the drop volume and are adequatedynamic characteristics of the investigated process (seebelow).

3.2. Discussion of the Experimental Dependen-cies. Coming back to the experimental results, weshould note that all values of the contact angle R inFigures 4-7 are dynamic (nonequilibrium). At fixeddrop volume, V, the value of the contact-line radius, rc,unequivocally determines the value of the contact angle,R. Indeed, purely geometrical considerations give thedependence V ) V(rc,R), which can be represented also

Figure 2. Consecutive photographs of the spontaneous detachment of a hexadecane drop from a horizontal glass plate immersed in asolution of 20 mM SDS + 0.1 mM NaCl at 23 °C. The drop volume is constant, 0.7415 mm3.

Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005 1311

in the form R ) R(V,rc). The latter dependence can beeasily derived for a spherical oil-water interface (see,e.g., eq 4 in ref 37), but an analogous dependence existsalso for a drop profile that is deformed by gravity(“pendant drop” profile); the latter can be computednumerically by integration of the Laplace equation.16

In other words, the measured dynamic contact angle,R, is a purely geometrical characteristic (determined bythe position, rc, of the shrinking contact line), in contrastwith the equilibrium contact angle, Req, which is aphysical parameter, related to the interfacial tensionsthrough the Young equation

where σos and σws are the interfacial tensions of theboundaries oil-solid and water-solid. It should benoted that the Young equation can be derived based onboth energy and force considerations, with the twoapproaches being equivalent. In particular, the forceinterpretation of σos and σws stems from the work ofGibbs,45 who coined the term “superficial tensions” forthem. By definition, the superficial tension opposes

every increase of the wet area, without any deformationof the solid, in the same way as σow opposes everydilatation of the interface between the two fluids. Fromthis viewpoint, the superficial tensions σos and σws canbe interpreted as surface tensions, i.e., forces per unitlength. Thus, the Young equation has the meaning of atangential projection of a vectorial force balance per unitlength of the contact line. Correspondingly, the normalcomponent of the meniscus surface tension, σow sin R,is counterbalanced by the bearing reaction of the solidsubstrate.

The experimental variation of R (Figures 4-7) iscaused by the spontaneous movement of the contact lineand decrease of rc, which, in its own turn, is due to theimbalance of the interfacial tensions at the contact line.The force balance per init length of a moving contactline includes also the line viscous friction, which exactlycompensates the imbalance of the interfacial tensionsunder quasi-stationary conditions:16,37

where â is the line friction coefficient. The appearanceof a line friction force, â (drc/dt), can be attributed tothe circumstance that, during the contact-line motion,oil molecules are taken out of potential wells at the solid

Figure 3. Digitized profiles of two photographs of the same drop,taken at different moments. The hexadecane drop has a volumeof 1.377 mm3 and is placed in a solution of 20 mM SDS + 0.1 mMNaCl. The theoretical line is drawn by fitting of the experimentalpoints by means of the Laplace equation.16 From the fit, wedetermine the contact-line radius, rc, the contact angle, R, and theoil-water interfacial tension, σ.

σos ) σws + σeq cos Req (3.1)

Figure 4. Effect of the SDS concentration on the detachment ofhexadecane drops from glass: experimental data (a) for thecontact-line radius, rc, and (b) for the contact angle, R, plotted vstime. The initial moment, tin, corresponds to the first experimentalpoint. The NaCl concentration is 0.1 mM, and the temperature is23 °C.

âdrc

dt) σow cos R + σws - σos (3.2)


surface and replaced by water molecules, accompaniedwith dissipation of kinetic energy in the zone of thecontact line.16,35 Note that the imbalance of the inter-facial tensions (the right-hand side of eq 3.2) determinesthe rate of motion of the contact line, drc/dt (the left-hand side of eq 3.2). Moreover, if the friction term, â(drc/dt), is sufficiently large (as found in our previousstudy16), then it will not allow the imbalance of tensionsto relax quickly. Instead, the contact line will moverelatively slowly, hindered by the viscous friction.

In ref 16, we established that the data for detachingoil drops agree very well with eq 3.2. Assuming that σws- σos was constant, we plotted drc/dt vs σow cos R andobtained a linear dependence. From its slope, wedetermined the line friction coefficient, â; see eq 3.2. Weapplied the same relatively simple procedure to the datain Figures 4-7. Unfortunately, we found that in thepresent case (different glass substrates) the dependenceof drc/dt on σow cos R is not linear; a typical example isshown in Figure 8. The situation is similar for allstudied solutions and temperatures.

A closer inspection of Figure 8 reveals the reason forthe nonlinear dependence. For the right-hand-sideexperimental points, we have drc/dt ≈ 0, but the contactangle R continues to vary (σow is constant). Accordingto eq 3.2, this could happen only if the difference σos -σws is not constant but varies with time in the present

series of experiments. Such a behavior could be aconsequence of the formation of a gel layer on the glasssurface in contact with water. The latter idea serves asa basis for the development of a theoretical model inthe next section.

4. Theoretical Model

4.1. Physicochemical Background. Let us firstconsider a possible mechanism of penetration of the oil-glass interface by water as the likely cause of thecontinuing detachment process. There are many experi-mental indications that water may dissolve or diffuseinto and swell the glass (and silica) surface, forming asurface gel layer.46-54 This effect has been detected insurface-force measurements50,52,54 and in experimentson adsorption of macromolecules on glass.53 As a partof the dissolution process, water may break silicon-oxygen bonds and form a hydroxylated surface.54 Inaddition, the formation of a gel layer may include anion-exchange process, in which protons replace sodiumions at the glass surface.47,49,53 A swelling of the surfacelayers has been directly detected with some glasses ina humid atmosphere by analytical methods: the surfacearea is increased by at least 10 times, microporesappear, and clusters are formed on the interface.51

Coming back to our system, we could hypothesize thatwater molecules in the gel layer at the water-glass

Figure 5. Effect of the AOS concentration on the detachment ofhexadecane drops from glass: experimental data (a) for thecontact-line radius, rc, and (b) for the contact angle, R, plotted vstime. The initial moment, tin, corresponds to the first experimentalpoint. The NaCl concentration is 100 mM, and the temperatureis 23 °C.

Figure 6. Effect of the NaCl concentration on the detachment ofhexadecane drops from glass: experimental data (a) for thecontact-line radius, rc, and (b) for the contact angle, R, plotted vstime. The initial moment, tin, corresponds to the first experimentalpoint. The AOS concentration is 6 mM, and the temperature is 23°C.


interface can penetrate, by diffusion, the oil-glassinterface, at least in a close vicinity of the contact line(Figure 9). The presence of water molecules in thesurface layer of the plate would alter the values of thetwo superficial tensions, σws and σos, which, in turn,would affect the force balance expressed by the dynamicYoung equation, eq 3.2. The resulting uncompensated

force would drive the spontaneous shrinking of thecontact line. In the following, we give a quantitativedescription of the water diffusion in the substrate’ssurface layer (Figure 9).

4.2. Diffusion of Water in the Surface Layer. Weintroduce a cylindrical coordinate system with a z axis,which coincides with the axis of symmetry of the drop;see Figure 10. We consider a layer of thickness, a, atthe solid surface, where the gel layer develops becauseof penetration of water. This layer could be divided intotwo separate regions: region 1 at the water-glassinterface (rc < r < ∞) and region 2 at the oil-glassinterface (0 e r < rc); r is the radial coordinate. Wedenote by c1(r,t) and c2(r,t) the concentrations of waterin regions 1 and 2:

The formation of a gel layer is described as an increaseof the concentration of water, c(r,t), with time, t, up toa maximal equilibrium concentration, ceq. In our model,the thickness, a, of the considered thin surface layer isassumed to be constant, and the diffusion of water issupposed to be in the radial direction; i.e., c1 and c2 areindependent of z.

To quantify the mass balance of water in region 1,we select a ring-shaped volume of the surface layer,which is confined between radii r and r + dr (Figure10). The variation of the number of water molecules inthe considered elementary volume is equal to thealgebraic sum of the incoming and outgoing fluxes ofwater:

Figure 7. Effect of the temperature on the detachment ofhexadecane drops from glass: experimental data (a) for thecontact-line radius, rc, and (b) for the contact angle, R, plotted vstime. The initial moment, tin, corresponds to the first experimentalpoint. The AOS and NaCl concentrations are 6 and 316 mM,respectively.

Figure 8. Typical plot of drc/dt vs σow cos(R): experimental datafor a hexadecane drop attached to a glass substrate in a solutionof 6 mM AOS + 31.6 mM NaCl at T ) 23 °C. The line is a guideto the eye.

Figure 9. Possible model of the spontaneous detachment of anoil drop from a glass substrate in a surfactant solution. Watermolecules from the gel layer at the glass-water interface penetrateby diffusion the glass-oil interface (in the close vicinity of thecontact line) and alter the local values of the two superficialtensions, σws and σos, which, in turn, affect the force balance atthe contact line.

Figure 10. Illustration of eq 4.2, expressing the mass balance ofwater in the gel layer. The z axis of the coordinate system coincideswith the axis of symmetry of the oil drop; r is the radial coordinate,and rc is the radius of the three-phase contact line.

c(r,t) ) {c1(r,t) for r > rc

c2(r,t) for 0 < r < rc(4.1)


Here, Q(r + dr) is the incoming diffusion flux of waterthrough the outer wall of the ring, while Q(r) is theoutgoing diffusion flux through the inner wall; Qp is theinflux of water molecules penetrating through the upperboundary of the ring (the water-glass interface). Thedivision of eq 4.2 by 2πar dr, followed by the transitiondr f 0, yields

We will use standard expressions for the diffusion andpenetration fluxes:55

where D is the diffusion coefficient, ceq is the equilibriumconcentration of water in the gel layer, and Rm is a mass-transfer coefficient. The substitution of eqs 4.4 and 4.5into eq 4.3 gives a partial differential equation for c1-(r,t):

which is valid for r > rc and t > 0; here tp ) a/Rm.The mass balance of water in region 2 of the gel layer

is similar, with the only difference being that the lastterm in eq 4.6 is missing because there is no penetrationof water through the upper boundary (through the oil-solid interface):

(t > 0). The boundary conditions at the contact line are

Here cb(t) is the concentration of water molecules at theboundary, r ) rc, between the regions 1 and 2; eq 4.9expresses the equality of the diffusion fluxed at thisboundary. We have two additional boundary conditionsat infinity and at the axis of symmetry:

Moreover, we assume that at the initial moment, t ) 0,there was no water in the considered layer on the solidsurface.

First, let us find the asymptotics of c1(r,t) far fromthe contact line (r . rc), where the diffusion term in eq4.6 (that containing D) is negligible. In this limitingcase, the solution of eq 4.6 reads

See Figure 11 for the notation. In the limit t f ∞, eq4.11 gives c1 f ceq, which means that eventually thegel layer equilibrates with the water phase.

To find the variation of c1 close to the contact line, itis convenient to introduce the auxiliary function:

where c∞(t) is defined by eq 4.11. The substitution of eq4.12 into eq 4.6 yields the following simpler equationfor c1δ:

The major variation of the concentration of water in theconsidered surface layer happens in the vicinity of thecontact line (around the point r - rc ) 0 in Figure 11).Let us denote the characteristic width of this zone byδ. To describe the variation of c1δ there, it is convenientto replace the variables r and t with new variables xand τ as follows:

In terms of the new variables, the derivative in the left-hand side of eq 4.13 can be expressed as follows:

Then, with the help of eqs 4.14 and 4.15, we canrepresent eq 4.13 in the form

We do not know in advance the value of the diffusivity,D, of water in the gel layer. For this reason, we checked(against the experimental data) the predictions of dif-ferent versions of the model: with fast, slow, andstationary diffusion. It turned out that only the case offast diffusion (eq 4.17) compares well with the experi-ment. In this case, the characteristic diffusion rate, D/δ,is much greater than the rate of contact-line motion,drc/dt, which leads to a small width, δ, of the transitionalzone around the contact line and to a small value of theratio δ/tp:

2πr dr a∂c1

∂t) 2π(r + dr)aQ(r + dr) - 2πraQ(r) +

2πr dr Qp (4.2)

∂c1

∂t) 1

r∂

∂r(rQ) +

Qp

a(4.3)

Q ) D∂c1

∂r(4.4)

Qp ) Rm(ceq - c1) (4.5)

∂c1

∂t) D

r∂

∂r (r ∂c1

∂r ) + 1tp

(ceq - c1) (4.6)

∂c2

∂t) D

r∂

∂r (r ∂c2

∂r ) (0 e r e rc) (4.7)

c1(rc,t) ) c2(rc,t) ≡ cb(t) (4.8)

D∂c1

∂r) D

∂c2

∂rat r ) rc (4.9)

∂c1

∂r |rf∞ ) 0,∂c2

∂r |r)0 ) 0 (4.10)

c1 ≈ ceq[1 - exp(-t/tp)] ≡ c∞(t) (r . rc) (4.11)

Figure 11. Schematic dependence of the concentration of waterin the surface layer of glass, c, on the distance to the contact line,r - rc. cb is the value of c at the contact line; the latter serves asthe boundary between regions 1 and 2; δ characterizes the widthof the transition zone between the two regions; c∞(t) is the limitingvalue of c in region 1, far from the transition zone; ceq is the valueof c for a glass surface, which is equilibrated with water.

c1δ(r,t) ≡ c1(r,t) - c∞(t) (4.12)

∂c1δ

∂t) D

r∂

∂r (r ∂c1δ

∂r ) -c1δ

tp(4.13)

x ) [r - rc(t)]/δ; τ ) t/δ2 (4.14)

(∂c1δ

∂t )r

) (∂c1δ

∂τ )x

(∂τ∂t)r

+ (∂c1δ

∂x )τ(∂x∂t)r

(4.15)

∂c1δ

∂τ) δ

drc

dt∂c1δ

∂x+ D 1

1 + xδ/rc

∂

∂x [(1 + xδ/rc)∂c1δ

∂x ] -

δ2

tpc1δ (4.16)


In view of the latter relationships, eq 4.16 acquires thefollowing simpler approximate form:

Likewise, using eqs 4.14 and 4.17, we present eq 4.7 inthe form

With the help of the Laplace transform, from eqs 4.18and 4.19, we derive

where the Laplace transforms are denoted by a tilde

Here s is the Laplace parameter and, as before, cb(t) isthe boundary concentration at the contact line (x ) 0);see eq 4.8. To obtain eq 4.20, we used the boundarycondition c2(x)0) ) cb and the equality of the diffusionfluxes, eq 4.9. Next, we transform the definition of c1δ,eq 4.12, and in the result we substitute x ) 0:

On the other hand, setting x ) 0 in eq 4.20, we get c1δ-(0,s) ) -cb(s). Substituting the latter expression into eq4.22, we obtain cb(s) ) 0.5L[c∞(t)]. Finally, we apply theinverse Laplace transformation and substitute eq 4.11:

In principle, eq 4.23, together with eq 4.20, gives thesolution of the considered diffusion problem. Becausewe are interested in the force balance at the contact line,it turns out that eq 4.23 is sufficient to quantify theeffect of water diffusion on the contact-line motion. Werecall that eq 4.23 is derived for the case of slow motionand a narrow transition zone (see eq 4.17), whichcorresponds to our experimental situation. (The latterfact is checked by comparing the theoretical model withthe experiment; see the following.)

4.3. Description of the Contact-Line Motion. Ingeneral, the superficial tensions, σws and σos, depend onthe concentration of water molecules, cb, in the surfacelayer at the contact line. To quantify the latter depen-dence, we will assume that the simple Henry law forsurface tensions is satisfied:

where σws(0) and σos(0) are the respective values for adry solid surface, while λws and λos are coefficients ofproportionality. The substitution of eqs 4.23 and 4.24into the equation for contact-line motion, eq 3.2, yields

where we have introduced the notation

γ and ∆σ are parameters of our model. Note that in viewof eqs 4.24 and 4.27 we have

i.e., ∆σ is the difference between the superficial tensionsfor a half-equilibrated surface layer of glass. (Therelative importance of the three terms in the right-handside of eq 4.25 is discussed in section 5.2.) Furthermore,we recall that the dependence R(t) is known from theexperiment; see Figures 4b-7b. Then, eq 4.25 can beintegrated to derive the theoretical time dependence ofthe contact-line radius:

where rc(0) is the contact-line radius at the initialmoment, t ) 0; t is an integration variable.

5. Comparison of Theory and Experiment

5.1. Principles of the Procedure for Data Pro-cessing. We do not know the experimental value rc(0)because some time (different in different runs) elapsesbetween the moment t ) 0 (when the solution has beenpoured in the experimental cell) and the initial moment,tin, when the video recording of the drop detachmenthas started. This difficulty can be overcome by a specialconstruction of the procedure of data processing, asexplained in the following. First, we substitute t ) tininto eq 4.29 and subtract the result from eq 4.29:

Here, ∆t is the experimental time: we have ∆t ) 0 atthe moment of the beginning of video recording. Theexperimental points in Figures 4a-7a have coordinates(∆ti, ri

exp), where riexp is the experimental value of rc at

the moment ∆ti. The corresponding theoretical value,ri

th ) rc(tin + ∆ti), is given by eq 5.1. The contact-lineradius at the initial moment, rc(tin), the oil-waterinterfacial tension, σow, and the contact angle, R(t), areknown from the experiment. Thus, four unknown pa-rameters remain in eq 5.1: â, ∆σ, A, and tp. They canbe determined from the best fit of the experimentaldependence rc vs ∆t. With this goal, it is convenient torepresent eq 5.1 in the following form:

âdrc

dt) σow cos R(t) - ∆σ + γ exp(- t

tp) (4.25)

γ ≡ (λws + λos)(ceq/2) (4.26)

∆σ ≡ σos(0) - σws(0) + (λos + λws)(ceq/2) (4.27)

∆σ ≡ σos(ceq/2) - σws(ceq/2) (4.28)

rc(t) ) rc(0) +σow

â ∫0

tcos R(t) dt - ∆σ

ât +

γtp

â [1 - exp(- ttp

)] (4.29)

rc(tin + ∆t) ) rc(tin) +σow

â ∫0

∆tcos R(t) dt - ∆σ

â∆t +

Atp

â[1 - exp(- ∆ttp

)] (5.1)

∆t ≡ t - tin, A ) γ exp(-tin/tp) (5.2)

drc

dt, D

δ; δ

rc, 1; δ

tp, D

δ(4.17)

∂c1δ

∂τ) D

∂2c1δ

∂x2(x g 0) (4.18)

∂c2

∂τ) D

∂2c2

∂x2(x e 0) (4.19)

c1δ ) -cb exp(-xxs/D), c2 ) cb exp(xxs/D) (4.20)

c1δ(x,s) ≡ L[c1δ(x,τ)], c2(x,s) ≡ L[c2(x,τ)],cb(s) ≡ L[cb(t)] (4.21)

c1δ(0,s) ) cb(s) - L[c∞(t)] (4.22)

cb(t) ) 12c∞(t) )

ceq

2 [1 - exp(- ttp

)] (4.23)

σws ) σws(0) - λwscb, σos ) σos(0) + λoscb (4.24)


(i ) 1, 2, ..., N), where b1 ≡ σow/â, b2 ≡ ∆σ/â, and b3 ≡Atp/â are adjustable parameters, N is the number ofexperimental points, and F1, F2, and F3 are functionsdefined as follows:

The values of F1 were calculated for each ∆ti from theexperimental data for the contact angle R vs time inFigures 4b-7b; the trapezium rule for numerical inte-gration was employed.56

To fit a given experimental curve, rc vs ∆t, like thosein Figures 4a-7a, we first assume a tentative value ofthe parameter tp. With the latter value, we calculateF3(∆ti) by eq 5.4. Next, applying the least-squaresmethod, we consider the following merit function:

The fit of the data is facilitated by the fact that rith,

given by eq 5.3, is a linear function of the parametersb1, b2, and b3. The values of the latter three parameters,which minimize Φ (for a given tp), can be obtained asthe solution of the following linear system of threeequations:

(k ) 1-3). We solve this linear system directly, usingCramer’s rule. Thus, we determine b1(tp), b2(tp), and b3-(tp). Now, eq 5.5 gives Φ as a function of a singleparameter, tp. The resulting function Φ(tp) is minimizednumerically by means of the Levenberg-Marquardtmethod.57 Finally, we calculate the parameters of themodel:

For all processed experimental curves, the minimum ofΦ(tp) was well pronounced and accurately determined.This is an argument in favor of the adequacy of ourtheoretical model. Another argument is the behavior ofthe determined adjustable parameters (for different,independently processed experimental curves rc vs ∆t)as functions of the temperature and surfactant and saltconcentrations.

The results from the data fits are reported in the nextsubsection. Let us mention in advance that, by defini-tion, the parameter A depends on the randomly definedinitial moment, tin; see eq 5.2. For this reason, the valuesof A should be scattered; i.e., they are not expected todepend systematically on the temperature and surfac-tant and salt concentrations. In fact, this is what weobtained for A from the fits. In contrast, â, tp, and ∆σare physical parameters, which are expected to dependin a systematic manner on the temperature and sur-factant and salt concentrations. The latter anticipationis confirmed by the results from the fits (see Tables1-4).

5.2. Numerical Results and Discussion. As men-tioned above, the points (symbols) in Figures 4a-7adenote the experimental rc while the solid lines connectthe respective theoretical values of rc, obtained by fittingthe data as described in section 5.1. Figures 4a-7aindicate excellent agreement between the theoreticalmodel and the experiment. This is seen also from therelatively small values of the standard deviation of thefits, which is given in the last columns of Tables 1-4.The standard deviation is (Φmin/N)1/2, where Φmin is theminimum value of Φ, corresponding to the best fit (seeeq 5.5).

The values of the hexadecane-water interfacial ten-sion, σow, given in Tables 1-4, are used as inputparameters when fitting the data for rc vs time. Asexplained in section 2, σow was determined by fittingthe drop profile with the Laplace equation (see Figure3). From different photographs of the same drop atdifferent stages of detachment (like those in Figure 2),we determined the same value of σow, which can beidentified as the equilibrium σow. In other words, thedynamics of surfactant adsorption at the oil-waterinterface is so fast that this process ends before thebeginning of our measurements. In addition, the varia-tion of the oil-water area during the drop detachmentis much slower than the rate of establishment ofadsorption-desorption equilibrium at the oil-waterinterface. This is evidenced also from the constancy of

rith ) rc

exp(tin) + b1F1(∆ti) + b2F2(∆ti) + b3F3(∆ti)(5.3)

F1(∆t) ≡ ∫0

∆tcos R(t) dt, F2(∆t) ≡ -∆t,

F3(∆t) ≡ 1 - exp[-∆t/tp] (5.4)

Φ(b1,b2,b3,tp) ) ∑i)1

N

[riexp - ri

th(b1,b2,b3,tp)]2 (5.5)

∑j)1

3

[∑i)1

N

Fj(∆ti) Fk(∆ti)]bj ) ∑i)1

N

Fk(∆ti) [riexp - rc

exp(tin)]

(5.6)

â ) σow/b1, ∆σ ) âb2, A ) âb3/tp (5.7)

Table 1. Effect of the SDS Concentration on the SystemParameters at 0.1 mM NaCl and 23 °C

SDS(mM)

σow(mN/m) â (Pa‚s) tp (s) ∆σ (µN/m)

st. dev.(mm)

20a 7.7 20.7 1568 38.1 0.03220b 7.7 20.9 1582 38.0 0.02840 7.5 18.1 1100 38.4 0.02680 7.4 15.2 745 38.2 0.029

100 7.5 15.1 691 38.9 0.028a Drop of volume of 0.7415 mm3. b Drop of volume of 1.142 mm3.

Table 2. Effect of the AOS Concentration on the SystemParameters at 100 mM NaCl and 23 °C

AOS(mM)


st. dev.(mm)

0.3 3.7 14.9 2450 4.40 0.0276 3.6 3.2 1240 7.36 0.024

50 3.6 2.0 501 7.46 0.016250 3.6 1.6 261 7.56 0.017

Table 3. Effect of the NaCl Concentration on the SystemParameters at 6 mM AOS and 23 °C

NaCl(mM)


st. dev.(mm)

1 6.6 48.7 4457 30.4 0.0273.16 5.8 35.3 3635 25.0 0.024

10 5.1 20.2 1784 19.2 0.02531.6 4.4 9.0 1325 13.4 0.023

100 3.6 3.2 1240 7.4 0.024316 2.9 3.1 1200 4.2 0.022

Table 4. Effect of the Temperature on the SystemParameters at 6 mM AOS and 316 mM NaCl

T (°C)σow

(mN/m) â (Pa‚s) tp (s) ∆σ (µN/m)st. dev.(mm)

20 3.10 5.46 2414 5.93 0.03023 2.94 3.10 1200 4.20 0.02225 2.86 3.04 1168 4.32 0.02630 2.74 2.92 946 4.50 0.01434 2.58 2.34 797 4.46 0.02238 2.47 0.38 245 2.70 0.023


σow in Tables 1 and 2, at various surfactant concentra-tions. Such a behavior is typical for the surface/interfacial tension of surfactant solutions above thecritical micelle concentration (cmc); see, e.g., ref 58. Itis well-known that the kinetics of surfactant adsorptionis very fast above the cmc. One can check that the datain Table 3 imply that σow decreases linearly with therise of ln(CNaCl), which is also a typical behavior due toadsorption of Na+ counterions; see, e.g., ref 59 [CNaCl isthe concentration of NaCl]. Finally, the data in Table 4indicate that σow decreases linearly with the rise oftemperature, as should be expected.17

As noted above, all experiments have been carried outwith hexadecane drops, whose volume was different indifferent runs. The values of the parameters â, tp, and∆σ, determined from the best fits, turned out to beindependent of the drop volume, as must be expected,having in mind the physical meaning of these param-eters (see above). As an illustration, the first two rowsof Table 1 compare data for two hexadecane drops ofdifferent volumes, with all other experimental condi-tions being the same. One sees that the determined â,tp, and ∆σ have very close values for these two drops.

In fact, here we present the first systematic measure-ments of the line friction coefficient, â, for solid-water-oil systems. In our previous study,16 â ) 1.6 Pa‚s wasdetermined at some specific experimental conditions fora glass-water-oil system. Values of â of the sameorder, as well as greater values, are determined in ourpresent study, depending on the experimental condi-tions; see Tables 1-4 and Figures 12 and 13. To the

best of our knowledge, the only other experimental valueof the line friction coefficient, so far mentioned in theliterature, is â ≈ 4.5 Pa‚s, which was determined in ref39 for water drops (with 75% dissolved glycerol) thatare spreading over a Teflon substrate (the third phaseis air).

The data in Tables 1 and 2 show that the line frictioncoefficient, â, and the penetration time of water into thesurface layer of glass, tp, decrease with the rise of thesurfactant concentration. This effect is very pronouncedin Table 2: both â and tp decrease 9.3 times when theAOS concentration is increased from 0.3 to 250 mM.Likewise, the data in Table 3 indicate that â and tpdecrease also with the rise of the NaCl concentration:â decreases about 16 times and tp about 4 times whenthe salt concentration rises from 1 to 316 mM. It isremarkable also that â and tp vary in a similar mannerwhen the temperature and surfactant and salt concen-trations are varied; see Figures 12 and 13.

The above results about the behavior of â and tp callfor a more detailed discussion. The correlation betweenâ and tp (Figures 12 and 13) is not so surprising becauseboth â and tp are related to the penetration of water inthe surface layer of the solid substrate. In particular,water molecules penetrating in the three-phase contactzone displace the adsorbed oil molecules from theirpotential wells on the solid surface, and thus the contactline advances. The surfactant molecules facilitate thedetachment of the oil from the glass surface becausethey cover the newly formed oil-water interface (in thezone of three-phase contact) and decrease its surface free

Figure 12. Effect of the surfactant concentration on the linefriction coefficient, â, and penetration time, tp: (a) solutions of SDSwith 0.1 mM NaCl; (b) solutions of AOS with 100 mM NaCl. Thevalues of â and tp are determined from the fits of experimentaldata for hexadecane drops at a temperature of 23 °C.

Figure 13. Line friction coefficient, â, and penetration time, tp,determined from the fits of experimental data for hexadecanedrops in aqueous solutions containing 6 mM AOS: (a) effect ofthe NaCl concentration at T ) 23 °C; (b) effect of the temperatureat fixed NaCl concentration, 316 mM.


energy (Figure 14). The added salt suppresses theelectrostatic repulsion between the surfactant ions andthus increases the rate of surfactant adsorption, as wellas the equilibrium adsorption. This could be a possibleexplanation for the decrease of the friction coefficient âwith the rise of the surfactant and salt concentrations(Figures 12 and 13).

On the other hand, tp is a property of the interfacesolid-water, unlike â, which is characteristic of thethree-phase contact zone. The bare glass-water inter-face is negatively charged,60 and the anionic surfactantsdo not adsorb on glass. For this reason, it seems unclearhow the surfactant and salt could affect the penetrationtime of water in the surface layer of glass, tp. We couldpropose two possible explanations.

First, experiments show that a thin hydrophobiclayer, some kind of gloss, covers the surface of thecommercial glass slides used. Indeed, the initial contactangle in our experiments is 140-160° (hydrophobicsubstrate); see Figures 4b-7b and the first photographin Figure 2. This hydrophobic coverage can be removedby immersion of the slide in sulfochromic acid. However,the acid-treated glass plates became so hydrophilic thatthe adherent hexadecane drops detached very fast afterpouring the surfactant solution into the experimentalcell (Figure 1). Such hydrophilic plates were inappropri-ate for our experiments because we were unable torecord a regular process of drop detachment, like thatin Figure 2, because of the high speed of the process.For this reason, we used the slides as received directlyfrom the manufacturer, without any previous treatment.Hence, the detected effect of the surfactant and salt onthe water-penetration time, tp, is most probably relatedto their role for hydrophilization or removal of thehydrophobic coverage on the glass surface.

Second, after the retreat of the oil, some oil moleculescould remain adsorbed at the glass-water interface. Thesurfactant micelles should solubilize that residual oil,thus accelerating the penetration of water into thesurface layer of glass. Moreover, it is known that thesalt lowers the electrostatic barrier to micelle contactwith the oil and promotes the solubilization process.43

Our model is not in conflict with the model by Garoffet al.31 about a possible carryover of surfactant aheadof the contact line. Indeed, when water moleculespenetrate along the boundary between solid and oil(Figure 14), surfactant molecules could also enter theformed thin water film to decrease its surface energyagainst the oil. Note that oil drops can detach from aglass substrate even in pure water (without any sur-factant), but the process is slower. Hence, in our case,the presence of surfactant is not a precondition forpenetration of water molecules between glass and oil,but when present, the surfactant accelerates this pro-cess. On the other hand, if the substrate is essentially

hydrophobic and there is no development of a gel layer,the presence of surfactant in the water phase, and itscarryover ahead of the contact line,31 could be a neces-sary condition for the occurrence of spontaneous dropdetachment.

The effect of the temperature on the line frictioncoefficient, â, and the water-penetration time, tp (Table4 and Figure 13b), is easier to understand, insofar asthe temperature is known to reduce the viscous effectsand accelerate the diffusion processes. In our specificexperiment, the rise of the temperature from 20 to 38°C decreases 14 times â and reduces 10 times tp (Table4). This is a considerable effect, which demands thedrop-detachment experiments to be carried out in athermostated cell in order to get reproducible results.

Finally, the difference, ∆σ, between the superficialtensions, defined by eq 4.28, turns out to be rathersmall, between 3 and 39 µN/m; see Tables 1-4. Because∆σ was determined from the coefficients of the linearfit with the help of eq 5.3, we were able to estimate itsexperimental error. The estimate indicates that theobtained ∆σ values are reliable, irrespective of their lowmagnitude. The determined relative errors of â and ∆σare typically about 1%. For example, for the last line ofTable 4 (that with the smallest â and ∆σ), we estimatedâ ) 0.379 ( 0.004 Pa‚s and ∆σ ) 2.70 ( 0.02 µN/m.

The fact that we are able to determine ∆σ accuratelyfrom the experimental data indicates that ∆σ, despiteits small value, has an essential effect on the contact-line motion. Indeed, taking values ∆σ ≈ 38 µN/m and â≈ 20 Pa‚s from Table 1, we obtain ∆σ/â ≈ 1.9 µm/s. Ifthis term were dominating the variation of the contact-line radius, rc (see eq 4.25), we would have ∆rc ≈ (∆σ/â)∆t ) 7.6 mm for ∆t ) 4000 s; see Figure 4a. However,the typical variation ∆rc in our experiments is about 10times smaller (Figure 4a). Hence, the negative termwith ∆σ in eq 4.25 is to a great extent counterbalancedby the two other terms in eq 4.25, both of them beingpositive for R < 90°. In other words, the three terms inthe right-hand side of eq 4.25 have the same order ofmagnitude, and none of them can be neglected.

The value of ∆σ indicates what would be the equilib-rium contact angle at the half-developed gel layer (seeeqs 3.2 and 4.28):

Because ∆σ/σow , 1, eq 5.8 implies that R1/2 ≈ 90°. Asseen in Figure 2, the experimental, nonequilibriumcontact angle is R > 90° at the earlier stages of dropdetachment, while R < 90° at the later stages. Unfor-tunately, we cannot determine the parameters in eq 4.24from our fit, and consequently we could not determinethe equilibrium contact angle at the fully developed gellayer (c ) ceq). If eventually a water film is formedbetween the glass and oil, then the equilibrium valueof R would be very small, close to zero.

6. Summary and Conclusions

In the present paper, we report experimental data andthe theoretical model for the dynamics of the detach-ment of hexadecane drops from a solid substrate (glassplate) in aqueous solutions containing anionic surfactantand salt. The influences of temperature and surfactantand salt concentrations on the motion of the three-phasecontact line (solid-water-oil), and on the dynamiccontact angle, are investigated (Figures 4-7). We found

Figure 14. Surfactant molecules, which facilitate the detachmentof the oil from the glass surface because they cover the newlyformed oil-water interface (in the zone of three-phase contact)and decrease its surface free energy.

cos R1/2 ) ∆σ/σow (5.8)


that the three factors mentioned considerably affect thedetachment process.

A drift (decrease) of the quasi-equilibrium contactangle (Figure 8) indicates that the surface of the solidplate becomes increasingly hydrophilic upon contactwith the water phase. This result gave a clue thathelped us to find a possible explanation of the experi-mental facts. Indeed, our data indicate that hydro-philization of the initially hydrophobic thin-layer cov-erage of the glass slides used occurs in contact with thesurfactant solution. In addition, the penetration of watercould lead to the formation of a gel layer on the glasssurface. In both cases, water molecules can propagateby lateral diffusion in a thin layer on the surface of thesolid plate (Figures 9 and 10). Thus, water moleculescould penetrate also along the oil-solid boundary, ashas been supposed in previous studies.15 The imbibitionof water by the substrate alters the solid-oil and solid-water interfacial tensions in the zone of a three-phasecontact (eq 4.24). Consequently, the balance of tensionsat the contact line will also be affected. Under dynamicconditions, this balance involves the line friction force,â (drc/dt); see eq 3.2.

In this way, the developed theoretical model containstwo kinetic parameters: the line friction coefficient, â,and the characteristic time, tp, of water penetration intothe substrate’s surface layer. The contact-line velocityis determined by the interplay of these two effects: Thedriving force of the detachment process, viz., the imbal-ance of the interfacial tensions at the contact line, isengendered by the water penetration, while the linefriction force compensates this imbalance and deter-mines the stationary speed (eq 4.25). First, we solvedthe problem about the lateral diffusion of water in thesubstrate’s surface layer and derived a useful formulafor the water concentration in the contact-line zone (eq4.23). Next, the dynamic Young equation (eq 4.25) wasintegrated to determine the theoretical dependence ofthe contact-line radius on time, rc(t); see eq 4.29. Thelatter is employed to fit the experimental data and todetermine â and tp as adjustable parameters (Tables1-4).

Excellent agreement between the theory and experi-ment has been achieved (Figures 4a-7a). The obtainedparameter values exhibit a systematic dependence onthe temperature and surfactant and salt concentrations(Figures 12 and 13) that can be interpreted in the frameof the proposed model (section 5.2). In conclusion, thepresent study specifies the parameters that can be usedto quantitatively characterize the rate of drop detach-ment, determines the values of these parameters atvarious experimental conditions, and indicates tools forcontrol of the investigated spontaneous process.

Acknowledgment

This work was supported in part by the programCooperation and Networking for Excellence (CONEX),financed by the Austrian Ministry of Education, Scienceand Culture, and in part by Colgate-Palmolive. Theauthors are indebted to reviewer II for his valuablecomments and to Mariana Paraskova for her assistancein the figure preparation.

Literature Cited

(1) Dussan, V. E. B.; Rame, E.; Garoff, S. On Identifying theAppropriate Boundary Conditions at a Moving Contact Line: AnExperimental Investigation. J. Fluid Mech. 1991, 230, 97.

(2) Haley, P. J.; Miksis, M. J. Dissipation and Contact LineMotion. Phys. Fluids A 1991, 3, 487.

(3) Brochard-Wyart, F.; de Gennes, P. G. Dynamics of PartialWetting. Adv. Colloid Interface Sci. 1992, 39, 1.

(4) Shikhmurzaev, Y. D. The Moving Contact Line in a SmoothSolid Surface. Int. J. Multiphase Flow 1993, 19, 589.

(5) Ruckenstein, E. The Moving Contact Line of a Droplet on aSmooth Solid. J. Colloid Interface Sci. 1995, 170, 284.

(6) Savelski, M. J.; Shetty, S. A.; Kolb, W. B.; Cerro, R. L. FlowPatterns Associated with the Movement of a Solid/Liquid/FluidContact Line. J. Colloid Interface Sci. 1995, 176, 117.

(7) Neumann, A. W.; Rame, E.; Garoff, S. Experimental Studieson the Parameterization of Liquid Spreading and Dynamic ContactAngles. Colloids Surf. A 1996, 116, 115.

(8) Blake, T. D.; Clarke, A.; de Coninck, J.; de Ruijter, M. J.Contact Angle Relaxation during Droplet Spreading: Comparisonbetween Molecular Kinetic Theory and Molecular Dynamics.Langmuir 1997, 13, 2164.

(9) Petrov, J. G.; Ralston, J.; Hayes, R. A. Dewetting Dynamicson Heterogeneous Surfaces. A Molecular-Kinetic Treatment.Langmuir 1999, 15, 3365.

(10) Blake, T. D.; de Coninck, J. The Influence of Solid-LiquidInteractions on Dynamic Wetting. Adv. Colloid Interface Sci. 2002,96, 21.

(11) Starov, V. M.; Kostvintsev, S. R.; Sobolev, V. D.; Velarde,M. G.; Zhdanov, S. A. Spreading of Liquid Drops over Dry PorousLayers: Complete Wetting Case. J. Colloid Interface Sci. 2002,252, 397.

(12) Blake, T. D.; Shikhmurzaev, Y. D. Dynamic Wetting byLiquids of Different Viscosity. J. Colloid Interface Sci. 2002, 253,196.

(13) Chi, M. P.; Anh, N. V.; Evans, G. M. Assessment ofHydrodynamic and Molecular-Kinetic Models Applied to theMotion of the Dewetting Contact Line between a Small Bubbleand a Solid Surface. Langmuir 2003, 19, 6796.

(14) Christov, N. C.; Ganchev, D. N.; Vassileva, N. D.; Denkov,N. D.; Danov, K. D.; Kralchevsky, P. A. Capillary Mechanisms inMembrane Emulsification: Oil-in-Water Emulsions Stabilized byTween 20 and Milk Proteins. Colloids Surf. A 2002, 209, 83.

(15) Kao, R. L.; Wasan, D. T.; Nikolov, A. D.; Edwards, D. A.Mechanisms of Oil Removal from a Solid Surface in the Presenceof Anionic Micellar Solutions. Colloids Surf. 1988/89, 34, 389.

(16) Kolev, V. L.; Kochijashky, I. I.; Danov, K. D.; Kralchevsky,P. A.; Broze, G.; Mehreteab, A. Spontaneous Detachment of OilDrops from Solid Surfaces: Governing Factors. J. Colloid InterfaceSci. 2003, 257, 357.

(17) Adamson, A. W.; Gast, A. P. Physical Chemistry ofSurfaces, 6th ed.; Wiley: New York, 1997; Chapter IV.

(18) Nikolov, A. D.; Wasan, D. T.; Chengara, A.; Koczo, K.;Policello, G. A.; Kolossvary, I. Superspreading Driven by Ma-rangoni Flow. Adv. Colloid Interface Sci. 2002, 96, 325.

(19) Christian, S. D.; Scamehorn, J. F. Solubilization in Sur-factant Aggregates; Marcel Dekker: New York, 1995.

(20) Cutler, W. G., Kissa, E., Eds. Detergency: Theory andTechnology; Marcel Dekker: New York, 1987.

(21) Dillan, K. W.; Goddard, E. D.; McKenzie, D. A. Oily SoilRemoval from a Polyester Substrate by Aqueous Nonionic Sur-factant Systems. J. Am. Oil Chem. Soc. 1979, 56, 59.

(22) Mahe, M.; Vignes-Adler, M.; Rosseau, A.; Jacquin, C. G.;Adler, P. M. Adhesion of Droplets on a Solid Wall and Detachmentby a Shear Flow. I. Pure System. J. Colloid Interface Sci. 1988,126, 314.

(23) Carroll, B. Physical Aspects of Detergency. Colloids Surf.A 1993, 74, 131.

(24) Miller, C. A.; Raney, K. H. Solubilization-emulsificationmechanisms of detergency. Colloids Surf. A 1993, 74, 169.

(25) Thompson, L. The Role of Oil Detachment Mechanisms inDetermining Optimum Detergency Conditions. J. Colloid InterfaceSci. 1994, 163, 61.

(26) Kralchevsky, P. A.; Nagayama, K. Particles at FluidInterfaces and Membranes; Elsevier: Amsterdam, The Nether-lands, 2001; Chapter 6, p 268.

(27) von Bahr, M.; Tiberg, F.; Zhmud, B. V. Spreading Dynam-ics of Surfactant Solutions. Langmuir 1999, 15, 7069.

(28) Eriksson, J.; Tiberg, F.; Zhmud, B. V. Wetting Effects dueto Surfactant Carryover through the Three-Phase Contact Line.Langmuir 2001, 17, 7274.


(29) Rams, E. The Spreading of Surfactant-Laden Liquids withSurfactant Transfer through the Contact Line. J. Fluid Mech.2001, 440, 205.

(30) Davis, A. N.; Morton, S. A., III; Counce, R. M.; DePaoli,D. W.; Hu, M. Z.-C. Ionic Strength Effects on Hexadecane ContactAngles on a Gold-Coated Glass Surface in Ionic SurfactantSolutions. Colloids Surf. A 2003, 221, 69.

(31) Kumar, N.; Varanasi, K.; Tilton, R. D.; Garoff, S. Surfac-tant Self-Assembly ahead of the Contact Line on a HydrophobicSurface and Its Implications for Wetting. Langmuir 2003, 19,5366.

(32) Chatterjee, J. Critical Eotvos Numbers for Buoyancy-Induced Oil Drop Detachment Based on Shape Analysis. Adv.Colloid Interface Sci. 2002, 98, 265.

(33) Chatterjee, J. Shape Analysis Based Critical Eotvos Num-bers for Buoyancy-Induced Partial Detachment of Oil Drops fromHydrophilic Surfaces. Adv. Colloid Interface Sci. 2002, 99, 163.

(34) Attard, P. Thermodynamic Analysis of Bridging Bubblesand a Quantitative Comparison with the Measured HydrophobicAttraction. Langmuir 2000, 16, 4455.

(35) de Ruijter, M. J.; Blake, T. D.; de Coninck, J. DynamicWetting Studied by Molecular Modeling Simulations of DropletSpreading. Langmuir 1999, 15, 7836.

(36) de Ruijter, M. J.; de Coninck, J.; Oshanin, G. DropletSpreading: Partial Wetting Regime Revisited. Langmuir 1999,15, 2209.

(37) Semal, S.; Voue, M.; de Coninck, J. Dynamics of Spontane-ous Spreading on Energetically Adjustable Surfaces in a PartialWetting Regime. Langmuir 1999, 15, 7848.

(38) de Ruijter, M. J.; Charlot, M.; Voue, M.; de Coninck, J.Experimental Evidence for Several Time Scales in Drop Spreading.Langmuir 2000, 16, 2363.

(39) Decamps, C.; de Coninck, J. Dynamics of SpontaneousSpreading under Electrowetting Conditions. Langmuir 2000, 16,10150.

(40) Seveno, D.; Ledauphin, V.; Matric, G.; Voue, M.; deConinck, J. Spreading Drop Dynamics on Porous Surfaces. Lang-muir 2002, 18, 7496.

(41) Seveno, D.; de Coninck, J. Possibility of Different TimeScales in the Capillary Rise around a Fiber. Langmuir 2004, 20,737.

(42) von Bahr, M.; Tiberg, F.; Zhmud, B. Oscillations of SessileDrops of Surfactant Solutions on Solid Substrates with DifferingHydrophobicity. Langmuir 2003, 19, 10109.

(43) Todorov, P. D.; Marinov, G. S.; Kralchevsky, P. A.; Denkov,N. D.; Broze, G.; Mehreteab, A. Kinetics of Triglyceride Solubili-zation by Micellar Solutions of Nonionic Surfactant and TriblockCopolymer: 3. Experiments with Single Drops. Langmuir 2002,18, 7896.

(44) Rusanov, A. I.; Prokhorov, V. A. Interfacial Tensiometry;Elsevier: Amsterdam, The Netherlands, 1996.

(45) Gibbs, J. W. The Scientific Papers of J. W. Gibbs; Dover:New York, 1961; Vol. 1.

(46) Tadros, Th. F.; Lyklema, J. Adsorption of Potential-Determining Ions at the Silica-Aqueous Electrolyte Interface andthe Role of Some Cations. J. Electroanal. Chem. 1968, 17, 267.

(47) Iler, R. K. The Chemistry of Silica: Solubility, Polymeri-zation, Colloid and Surface Properties and Biochemistry; Wiley:New York, 1979.

(48) Hunter, R. Foundation of Colloid Science; ClarendonPress: Oxford, U.K., 1987.

(49) Doremus, R. H. Glass Science, 2nd ed.; Wiley: New York,1994.

(50) Vigil, G.; Xu, Z.; Steinberg, S.; Israelachvili, J. N. Interac-tions of Silica Surfaces. J. Colloid Interface Sci. 1994, 165, 367.

(51) Trens, P.; Denoyel, R.; Guilloteau, E. Evolution of SurfaceComposition, Porosity, and Surface Area of Glass Fibers in a MoistAtmosphere. Langmuir 1996, 12, 1245.

(52) Yaminsky, V. V.; Ninham, B. W.; Pashley, R. M. Interac-tion between Surfaces of Fused Silica in Water. Evidence of ColdFusion and Effects of Cold Plasma Treatment. Langmuir 1998,14, 3223.

(53) van Duijvenbode, R. C.; Koper, G. J. M.; Bohmer, M. R.Adsorption of Poly(propylene imine) Dendrimers on Glass. AnInterplay between Surface and Particle Properties. Langmuir2000, 16, 7713.

(54) Adler, J. J.; Rabinovich, Y. I.; Moudgil, B. M. Origins ofthe Non-DLVO Force between Glass Surfaces in Aqueous Solution.J. Colloid Interface Sci. 2001, 237, 249.

(55) Landau, L. D.; Lifshitz, E. M. Fluid Mechanics; PergamonPress: Oxford, U.K., 1984.

(56) Korn, G. A.; Korn, T. M. Mathematical Handbook; McGraw-Hill: New York, 1968.

(57) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery,B. P. Numerical Recipes in Fortran. The Art of Scientific Comput-ing, 2nd ed.; Cambridge University Press: Cambridge, U.K., 1992;p 678.

(58) Pletnev, M. Y. in Surfactants: Chemistry, InterfacialProperties, Applications; Fainerman, V. B., Mobius, D., Miller, R.,Eds.; Elsevier: Amsterdam, The Netherlands, 2001; Chapter 1.

(59) Kralchevsky, P. A.; Danov, K. D.; Broze, G.; Mehreteab,A. Thermodynamics of Ionic Surfactant Adsorption with Accountfor the Counterion Binding: Effect of Salts of Various Valency.Langmuir 1999, 15, 2351.

(60) Dimov, N. K.; Ahmed, E. H.; Alargova, R. G.; Kralchevsky,P. A.; Durbut, P.; Broze, G.; Mehreteab, A. Deposition of Oil Dropson a Glass Substrate in Relation to the Process of Washing. J.Colloid Interface Sci. 2000, 224, 116.

Received for review August 27, 2004Revised manuscript received October 22, 2004

Accepted October 25, 2004

IE049211T


detachment of oil drops from solid surfaces in surfactant...

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