pubs.acs.org/Langmuir © 2009 American Chemical Society
Surfactant-Enhanced Rapid Spreading of Drops on Solid Surfaces† D. R. Beacham and O. K. Matar* Department of Chemical Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, U.K.
R. V. Craster Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada Received June 1, 2009. Revised Manuscript Received August 6, 2009 We study the surfactant-enhanced spreading of drops on the surfaces of solid substrates. This work is performed in connection with the unique ability of aqueous trisiloxane solutions to wet highly hydrophobic substrates effectively, which has been studied for nearly two decades. We couple a lubrication model to advection-diffusion equations for surfactant transport. We allow for micelle formation and breakup in the bulk and adsorptive flux at both the gas-liquid and liquid-solid interfaces and use appropriate equations of state to model variations in surface tension and wettability. Our numerical results show the effect of basal adsorption, kinetic rates, and the availability of surfactant on the deformation of the droplet and its spreading rate. We demonstrate that this rate is maximized for intermediate rates of basal adsorption and the total mass of surfactant.
I. Introduction The spreading of fluids over liquid and solid substrates is of great importance to many industrial and medical applications, including coatings, herbicides,1,2 and surfactant replacement therapy.3 The ability to aid and control the rate, extent, and uniformity of the spreading process is essential. Surfactants are often employed to this end but are associated with a rich array of stable and unstable phenomena, thus complicating our ability to understand and manipulate their effects on the dynamics. One particular class of trisiloxane-based surfactants, referred to as superspreaders, has received a great deal of scientific interest.4,5 The surfactants’ ability to lower the surface tension of water dramatically and wet very low energy substrates effectively has found widespread commercial use.6 Despite a great amount of illuminating experimental and theoretical work over the last two decades, the mechanisms driving this phenomenon are still not fully understood, although the work by Stoebe et al.7-10 has broadened the problem of superspreading to that of surfactantenhanced spreading (SES). Although not as effective on highly hydrophobic surfaces, many surfactants share similar spreading characteristics with the siloxane superspreaders, which can be broadly categorized as (a) an ability to wet hydrophobic surfaces effectively; (b) a maximum in spreading rate observed at † Part of the “Langmuir 25th Year: Wetting and superhydrophobicity” special issue. *Corresponding author. E-mail:
[email protected].
(1) Knoche, M.; Tamura, H.; Bukovac, M. J. J. Agric. Food. Chem. 1991, 39, 202. (2) Zabkiewicz, J. A.; Gaskin, R. E. Effect of Adjuvants on Uptake and Translocation of Glyphosate in Gorse (Ulex europaeus L). In Adjuvants and Agrochemicals; Chow, P. N. P., Ed.; CRC Press: Boca Raton, FL, 1989; Vol. 1. (3) Maniscalco, W. M.; Kendig, J. W.; Shapiro, D. L. Pediatrics 1989, 83, 1. (4) Hill, R. M. Curr. Opin. Colloid Interface Sci. 1998, 3, 247. (5) Hill, R. M. Curr. Opin. Colloid Interface Sci. 2002, 7, 255. (6) Hill, R. M., Ed.; Silicone Surfactants; Marcel Dekker: New York, 1999. (7) Stoebe, T.; Lin, Z.; Hill, R.; Ward, M.; Davis, H. Langmuir 1996, 12, 337. (8) Stoebe, T.; Hill, R.; Ward, M.; Davis, H. Langmuir 1997, 13, 7276. (9) Stoebe, T.; Lin, Z.; Hill, R.; Ward, M.; Davis, H. Langmuir 1997, 13, 7270. (10) Stoebe, T.; Lin, Z.; Hill, R.; Ward, M.; Davis, H. Langmuir 1997, 13, 7282.
14174 DOI: 10.1021/la9019469
intermediate energy substrates; (c) a maximum in spreading rate observed as a function of initial surfactant concentration; (d) the final wetted area being proportional to the initial concentration of surfactant; and (e) a linear wetted area vs time curve. Throughout the literature,7-13 Marangoni flow is considered to be the major driving force behind SES because the front advances with a t1/2 power law; a diffusion-limited process would also fit this scaling, but the time scales are too small to be plausible. Astonishingly, the front has also been reported to advance at rates as high as t, necessitating care to be taken over which length scales the surface tension gradient acts.13 It has been thought that the formation of a precursor water film ahead of the drop is necessary to promote Marangoni flow. On highly hydrophobic surfaces, the spreading rate of the superspreaders12 and other chemicals containing a hydrophilic moiety14 has been found to depend strongly on the relative humidity, with no spreading occurring below a certain relative humidity, suggesting the development of a precursor film that supports the Marangoni gradients. However, such a film has not been observed experimentally, and the humidity dependence does not appear to extend to less hydrophobic substrates.7 Instead, Churaev et al.15 account for this by variation of the disjoining pressure with humidity and find increasing stability of the thin spreading films at high humidity. A dynamically maintained surface tension gradient is also considered in several models,9,11,16,17 whereby dilation of the (11) Nikolov, A. D.; Wasan, D. T.; Chengara, A.; Koczo, K.; Policello, G. A.; Kolossvary, I. Adv. Colloid Interface Sci. 2002, 96, 325. (12) Zhu, S.; Miller, W. G.; Scriven, L. E.; Davis, H. T. Colloids Surf. 1994, A90, 63. (13) Rafai, S.; Sarker, D.; Bergeron, V.; Meunier, J.; Bonn, D. Langmuir 2002, 18, 10486. (14) Cazabat, A. M.; Fraysse, N.; Heslot, F.; Levinson, P.; Marsh, J.; Tiberg, F.; Valignat, M. P. Adv. Colloid Interface Sci. 1994, 48, 1. (15) Churaev, N.; Esipova, N.; Hill, R.; Sobolev, V.; Starov, V.; Zorin, Z. Langmuir 2001, 17, 1338. (16) Chengara, A.; Nikolov, A.; Wasan, D. Colloids Surf. 2002, A 206, 31. (17) Chengara, A.; Nikolov, A. D.; Wasan, D. T. Ind. Eng. Chem. Res. 2008, 47, 3639.
Published on Web 09/04/2009
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surface at the leading edge as the droplet spreads decreases the local surfactant concentration and so increases the local surface tension. The surface tension gradient is maintained by the replenishment of surfactant behind the leading edge. A constant supply of surfactant is required for this and is in keeping with the requirement of the surfactant solution to be sufficiently concentrated for spreading to occur, typically having to be above the critical micelle concentration (cmc) and critical aggregation concentration (cac). Stobe et al.9 hypothesized the breakup of bilayer aggregates just behind the contact line on intermediate-energy surfaces, although the direct adsorption of micelles may be necessary because the rate of micelle disintegration is too slow.18 In the modeling undertaken in this article, the aggregation model is that of micelles formed from a collection of n monomers, so we use the cmc as our notation although the model could be adapted to represent other aggregates. A further consequence of bilayer disintegration is the adsorption of surfactant on the substrate. On low-energy substrates, the hydrophobic moiety will attach to the substrate, exposing the hydrophilic moiety and potentially raising the local energy of the substrate, thereby encouraging spreading. Dynamic variation of substrate properties by surfactant adsorption has already been found to be crucial in other settings: the onset of autophobing and the dewetting of initially spreading droplets on a thin film overlying a glass substrate19-22 and the self-propulsion of running droplets.23,24 The compact structure of the trisiloxane surfactants allows them to adsorb efficiently onto the solid substrate, possibly resulting in the removal of all surface water particles and presenting a less hydrophobic platform for the droplet.25 This mechanism has been reported in the spreading of trisiloxane surfactants over a highly hydrophobic Teflon substrate. However, Ivanova et al.26 are unable to explain the fast initial spreading of the droplet by this mechanism. Furthermore, experiments examining the synergy in superspreading by mixing trisiloxane surfactants with nonsuperspreading pyrrolidinones show a direct correlation between the spreading rate and the basal adsorption of the trisiloxane surfactant component.27,28 It is not unreasonable to suggest that these properties extend beyond the trisiloxane surfactants and play a role in SES. We focus on the modeling of a surfactant-laden droplet spreading on a very thin precursor layer to avoid the contact line singularity. We use lubrication theory and the rapid diffusion limit to derive coupled equations for the film thickness and the surfactant monomer and micelle concentrations; the monomers can exist in the bulk and at the liquid-vapor and liquid-solid interfaces. The model accounts for Marangoni stresses, diffusion in the bulk and along the interfaces, micellar breakup and formation, and sorption kinetics. Nonlinear equations of state relating surface tension variability to surfactant concentration and to the disjoining pressure allow us to model variations in wettability. The rest of the article is organized as follows. In section II, we formulate the problem and derive the equations governing the (18) Kumar, N.; Couzis, A.; Maldarelli, C. J. Colloid Interface Sci. 2003, 267, 272. (19) Afsar-Siddiqui, A. B.; Luckham, P. F.; Matar, O. K. Adv. Colloid Interface Sci. 2003, 106, 183. (20) Afsar-Siddiqui, A.; Luckham, P.; Matar, O. K. Langmuir 2003, 19, 696. (21) Afsar-Siddiqui, A.; Luckham, P.; Matar, O. K. Langmuir 2003, 19, 703. (22) Craster, R. V.; Matar, O. K. Langmuir 2007, 23, 2588. (23) Thiele, U.; John, K.; B€ar, M. Phys. Rev. Lett. 2004, 93, 027802. (24) John, K.; B€ar, M.; Thiele, U. Eur. Phys. J. E 2005, 18, 183. (25) Kumar, N.; Maldarelli, C.; Couzis, A. Colloids Surf. 2006, A277, 98. (26) Ivanova, N.; Starov, V.; Johnson, D.; Hilal, N.; Rubio, R. Langmuir 2009, 25, 3564. (27) Rosen, M. J.; Wu, Y. Langmuir 2001, 17, 7296. (28) Wu, Y.; Rosen, M. Langmuir 2002, 18, 2205.
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flow. The results are presented and discussed in section III and are followed by concluding remarks in section IV.
II. Formulation A. Hydrodynamics. We consider an incompressible Newtonian fluid with density F and viscosity μ. We use rectangular coordinates (x, z), where x and z are the horizontal and vertical components, respectively, and u=(u, w) is the velocity field. The liquid-vapor and liquid-solid interfaces are located at z=h and z = 0, respectively. The surface tensions of the liquid-solid, liquid-vapor, and solid-vapor interfaces are σls, σl, and σs, respectively. The hydrodynamics of the droplet is described by standard lubrication theory μuzz ¼ px ,
pz ¼ 0
ux þ wz ¼ 0
ð1Þ ð2Þ
where p is the pressure, with no-slip and no-penetration boundary conditions on z=0, u ¼w ¼0
ð3Þ
along with free surface boundary conditions at z = h p ¼ -Kσ l þ ΠLW
ð4Þ
μuz ¼ σlx
ð5Þ
ht þ uhx ¼ w
ð6Þ
Here, κ is the curvature of the droplet, and ΠLW is the disjoining pressure, which we take to be composed of a stabilizing Born repulsion term along with long-range Lifschitz-van der Waal (LW) forces that vary with the local spreading coefficient29 Π
LW
! -2S h¥ 3 So h ¥ 3 ¼ 1h¥ h S h
ð7Þ
Here, h¥ is an equilibrium thickness, and S is the spreading coefficient S ¼ σs - ðσ ls þ σ l Þ
ð8Þ
Primarily, the disjoining pressure allows us to vary the wettability of the substrate through the variations in surfactant adsorption at the various interfaces, characterized by S. However, the overall form of ΠLW is practically motivated to alleviate the contact line singularity. For this purpose, we include a precursor film of thickness h¥ ahead of the drop: on hydrophobic surfaces (So< 0), without the final term in eq 7 the precursor film would be unstable and unsuitable for numerical simulations. In the discussion above and throughout the article, we will use subscripts o and m to describe variables evaluated at minimal and maximal interfacial surfactant contamination, respectively. B. Surfactant Transport Equations. The surfactant is assumed to exist as monomers at the liquid-vapor interface, within (29) Sharma, A. Langmuir 1993, 9, 861.
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the bulk, and at the liquid-solid interface with concentrations Ca, C and Cs, respectively. Above the critical micelle concentration (Ccmc), it becomes energetically favorable for the monomers to form micelles, which have a typical preferred size of N monomers and are present in the bulk at concentration m. We make the assumption that the micelles must first disassociate into monomers before adsorbing at either interface. Furthermore, we assume that the adsorption is limited physically by the amount of local surfactant that has already adsorbed. The behavior of the various surfactant species are modeled by advection-diffusion equations Cat þ ðus Ca Þx ¼ Dca Caxx þ Jcca
ð9Þ
Ct þ uCx þ wCz ¼ Dc ðCxx þ Czz Þ - NJcm
ð10Þ
mt þ umx þ wmz ¼ Dm ðmxx þ mzz Þ þ Jcm
ð11Þ
surface energy at high concentrations of adsorbed surfactant, the regime in which we will be operating -3 σl 1 Ca ¼ þ1 1þ ½ð1 þ Σl Þ1=3 - 1 Σl Sl Ca¥ σ ls, s 1 ¼ þ1 Σls, s Sls, s
! 1þ
Cs ½ð1 þ Σls, s Þ1=3 - 1 Cs¥
ð13Þ
where Sa and Ss are the space available at the liquid-vapor and liquid-solid interfaces for monomer adsorption, respectively. These fluxes are given by Ca - k2 Ca Jcca ¼ -Dc ðCz - hx Cx Þjh ¼ k1 Cjh 1 Ca¥ Jcm ¼ k3 CN - k4 m Cs - k6 Cs Jccs ¼ Dc Cz j0 ¼ k5 Cj0 1 Cs¥
ð14Þ
ð15Þ
UH w~ ðu, wÞ ¼ U u~, L
ðp, ΠLW Þ ¼
ð21Þ
Sl ~ LW Þ ð p~, Π H
σi ¼ σ i, m ð1 þ Σi σ~ i Þ ¼ σi, m þ Si σ~ i
ð22Þ
ð23Þ
so σ~ ∈ [0, 1]. Finally, we scale the surfactant concentrations and fluxes according to ~ m¼ Ca ¼ Ca¥ C~a , C ¼ Ccmc C,
Jcca ¼
ð17Þ
ð20Þ
where U = SlH/μL is a characteristic Marangoni velocity. The interfacial tensions are scaled as
ð16Þ
The interfacial energies are then described by the Sheludko equation of state which has an asymptote to the minimal 14176 DOI: 10.1021/la9019469
L t~ U
Ccmc ~ m, N
Cs ¼ Cs¥ C~s
where k1, k2 and k5, k6 denote the adsorption and desorption rate constants associated with the liquid-vapor and liquid-solid interfaces, respectively; k3 and k4 are the rate constants for micellar formation and breakup, respectively; and Ca¥ and Cs¥ are the interfacial and basal concentrations at saturation, allowing us to adhere to the limitations on adsorption as outlined above. Equations of state for surface energy are required to close the model, but because the surface energies are functions of the adsorbed surfactant concentrations, we introduce a spreading parameter that measures the difference between the surface energy at minimal and maximal surfactant concentration, Si ¼ σ i, o - σ i, m i ¼ l, ls, s
t¼
ð12Þ
where us denotes the interfacial streamwise velocity component; Dca, Dc, Dm, and Dcs represent the surfactant diffusivity at the liquid-vapor interface, in the bulk for monomers and micelles, and at the liquid-solid interface, respectively; Jcca, Jcm, and Jccs are fluxes that represent phase changes, which obey the following relationships Sa þ C h Ca NC h M Ss þ C h Cs
ð19Þ
where Σi = Si/σi,m. We have verified that other choices of equations of state give rise to quantitative rather than qualitative changes to the results discussed below. C. Scaling. Although we have already implicitly assumed that the mass and momentum conservation equations simplify to the lubrication equations, we must still nondimensionalize the surfactant equations and the equations of state. We use the following scalings ~ ðx, z, hÞ ¼ ðL~ y, H~z, H hÞ,
Cst ¼ Dcs Csxx þ Jccs
-3
ð18Þ
UCa¥ ~ UCcmc ~ J cca , Jcm ¼ J cm , L L UCs¥ ~ J ccs Jccs ¼ L
ð24Þ
ð25Þ
As a result of the above scaling, a number of dimensionless parameters emerge such as Peclet numbers that measure the relative importance of advection as compared to diffusion Pei ¼
UL , i ¼ Ca , C, m, Cs Di
ð26Þ
There are also parameters that provide measures of the importance of rates of sorption kinetics at z=0, h and of the rate of breakup and formation of micelles relative to the flow time scales K ca ¼
k2 L k4 L k6 L , K cm ¼ , K cs ¼ U U U
ð27Þ
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In addition, measures of surfactant solubility are respectively provided by βca ¼
Ca¥ Cs¥ , βcs ¼ HCcmc HCcmc
ð28Þ
whereas measures of the affinity of the surfactant for liquid-solid and liquid-vapor interfaces are represented by
where Sm is the spreading coefficient at maximal surfactant contamination. Thus, the disjoining pressure is " # -2 h¥ 3 So h¥ 3 LW ð41Þ λð1 þ FðCs , Ca ÞÞ Π ¼ h¥ h S h where the function F is given by FðCa , Cs Þ ¼ λs σ s - ðλl σ l þ λls σ ls Þ
k5 Ccmc Rs ¼ , k6 Cs¥
k1 Ccmc Ra ¼ k2 Ca¥
ð29Þ
The mass is scaled by HLCcmc, which by symmetry gives Z
¥
M¼
Z
¥
hðC þ mÞ dx þ
0
βca Ca þ βcs Cs dx
ð30Þ
ð42Þ
with λ Sm/Sl and λI Si/Sm. The dependence of F on Ca and Cs arises through the interfacial tension dependence on these concentrations: ! 1 þ Σls, s 1 ð1 þ Cs ½ð1 þ Σls, s Þ1=3 - 1Þ -3 σ ls, s ¼ Σls, s Σls, s
0
After suppressing the tildes, the lubrication equations become uzz ¼ px , pz ¼ 0, ux þ wz ¼ 0
ð31Þ
with boundary conditions
σl ¼
1 þ Σl 1 ð1 þ Ca ½ð1 þ Σl Þ1=3 - 1Þ -3 Σl Σl
The scaled surfactant transport equations are Cat þ ðus Ca Þx ¼
u¼w¼0
ð32Þ
1 þ σl þ ΠLW Σl
ð33Þ
μuz ¼ σ lx
1 Caxx þ K ca ðRa Cjh ð1 - Ca Þ -Ca Þ Peca
ð44Þ
1 1 Ct þ uCx þ wCz ¼ Cxx þ 2 Czz -K cm ðC N - mÞ ð45Þ Pec E
along with free surface boundary conditions at z=h p ¼ -E2 hxx
ð43Þ
1 1 mxx þ 2 mzz þ K cm ðCN - mÞ ð46Þ mt þ umx þ wmz ¼ Pem E
ð34Þ Cst ¼
ht þ uhx ¼ w
ð35Þ
1 Csxx þ K cs ðRs Cj0 ð1 - Cs Þ - Cs Þ Pecs
ð47Þ
Furthermore, we assume that vertical gradients in surfactant concentration are negligible and so take the fast-diffusion limit.30 Hence, we make the substitution
where ɛ=H/L is the film aspect ratio. The solution of these equations gives ht þ ðhuÞx ¼ 0
ð36Þ
" # z LW 2 1 u ¼ ð2h -zÞ E þ σl hxx -Π þ σlx z 2 Σl
ð37Þ
" # h2 2 1 LW E þ σ l hxx -Π þ σ lx h us ¼ Σl 2
ð38Þ
Cðx, z, tÞ ¼ C0 ðx, tÞ þ E2 Pec C1 ðx, z, tÞ
ð48Þ
mðx, z, tÞ ¼ m0 ðx, tÞ þ E2 Pem m1 ðx, z, tÞ
ð49Þ
where
x
Cat þ ðus Ca Þx ¼
x
1 u¼ h
Z
h 0
" # h2 2 1 σlx h LW E u dz ¼ þ σ l hxx - Π þ Σl 2 3
ð39Þ
x
Substitution of scalings for surface tensions into the spreading coefficient S gives S ¼ Sm þ Ss σs -Sls σ ls -Sl σ l Langmuir 2009, 25(24), 14174–14181
and vertically average the surfactant monomer and micelle R equations in the bulk, under the assumption that (C1, m1)=1/h 0h (C1,m1) dz = 0, keeping terms up to order ɛ. After dropping subscript 0, the following equations are obtained:22
ð40Þ
Ct þ uCx ¼ -
Caxx þ K ca ðRa Cð1 - Ca Þ - Ca Þ Peca
ð50Þ
ðhCx Þx β K ca ðRa Cð1 - Ca Þ - Ca Þ þ ca hPec h
βcs K cs ðRs Cð1 - Cs Þ - Cs Þ - K cm ðC N - mÞ h
ð51Þ
(30) Jensen, O. E.; Grotberg, J. B. Phys. Fluids 1993, A 5, 58.
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ðhmx Þx þ K cm ðC N - mÞ hPem
ð52Þ
Csxx þ K cs ðRs Cð1 - Cs Þ - Cs Þ Pecs
ð53Þ
mt þ umx ¼ Cst ¼
D. Numerical Procedure. Our numerical procedure is based on a semidiscretization of the underlying partial differential equations. Second-order central differences are utilized to approximate the spatial derivatives, and the solution is advanced in time using Gear’s method. The sparse matrices arising from the choice of discretization of the spatial derivatives permits integration of eqs 36 and 50-53 up to t ∈ [0, 104] over a domain of 0 e x e 16 using up to 20 000 grid points.22 The initial conditions are given by hðx, 0Þ ¼ h¥ þ max½ð1 - x2 Þ, 0 ðCa , C, m, Cs Þðx, 0Þ ¼ ðCa0 , C0 , m0 , 0ÞGð1 -xÞ
ð54Þ
where G(x) = 1/2[1 þ tanh(100x)]. These conditions relate to a fluid cap deposited on a pre-existing, uncontaminated, stable fluid layer of depth he. The initial surfactant concentrations (Cao, Co, mo) are assumed to be in local equilibrium. Hence, at t=0, Jcca= Jcm=0 and so m0 ¼ C0N ,
Ca0 ¼
Ra C 0 1 þ Ra C0
ð55Þ
The value of C0 is determined by substituting eqs 54 and 55 into 30, yielding 2 β Ra Ca0 N Þ þ ca ¼M þ he ðCa0 þ Ca0 3 1 þ Ra Ca0
ð56Þ
and prescribing a value for the total surfactant mass, M. The value M=1 is approximately equivalent to the cmc. No flux conditions are imposed at x=(0, L), where L is the length of the computational domain: ðhx , hxxx Þð0, tÞ ¼ ðhx , hxxx ÞðL, tÞ ¼ 0 ðCax , Cx , mx , Csx Þð0, tÞ ¼ ðCax , Cx , mx , Csx ÞðL, tÞ ¼ 0
ð57Þ ð58Þ
Using the guidance of previous work on surfactant kinetics,18,28 we fix N = 10, Pec = Pem =10, Peca = Pecs =104, Rs = 10, Kca=Kcs=Kcm=0.1, βca=1, and ε2=0.005. The surface energies at minimal and maximal surface energies remain unchanged throughout with (σlo, σlso, σso) = (15, 15, 25) mN/m, (σlm, σlsm, σsm)=(5, 5, 30) mN/m, and h¥=0.001. In the presentation of our results below, we vary Ra, the measure of solubility, βcs, the measure of maximal substrate adsorption, and M, the total mass. A discussion of the effects of other parameters is given in the concluding remarks. Our particular choices of maximal and minimal surface energies are such that So=-5 and Sm=20. However, through our assumption that Cs(x, 0)=0 and our choices of M and Ra, 0 < S|t=0 j15. To measure the extent of spreading, we track the front edge of the droplet, xf, using xf ðtÞ ¼ maxfx∈½0, L : hðx, tÞ g 0:003g 14178 DOI: 10.1021/la9019469
ð59Þ
III. Results and Discussion Wu and Rosen28 investigated the spreading of trisiloxane surfactants mixed with a number of nonsuperspreading pyrrolidinone surfactants by holding the concentration of the spreading solution constant at 1.0 g/L and varying the mole fraction of trisiloxane. All but one of the pyrrolidinone surfactants resulted in an improvement in spreading rate in comparison to the pure trisiloxane solution over a large range of trisiloxane mole fractions, reaching a maximum in spreading rate and then monotonically decaying to a near zero spreading rate for pure pyrrolidinone solutions. The increase in spreading rate was accompanied by an increase in the maximal basal adsorption of trisiloxane, and the effects at the liquid-vapor interface were found to be very weak. Similarly, decreases in basal adsorption accompanied decreases in the spreading rate for all of the pyrrolidinones whether or not they had a synergistic effect. Accordingly, we begin our investigation by considering the effect of βcs on the dynamics and set M = 5 so that we are above the cmc. This corresponds to the situation in which the total amount of surfactant is fixed but the proportion that is attracted to the substrate is being altered. The most prominent feature of the profiles in Figure 1 is the formation of a pronounced rim at the leading edge of the droplet for all but the smallest values of βcs. The size of the rim increases with βcs, and for βcs=5 the rim contains almost the entire volume of the drop. However, the monotonicity in rim size is not reflected in the long-time extent of spreading of the droplet; increasing βcs from 2 to 5 shows a significant decrease in the extent of spreading. Furthermore, for all values of βcs during the initial stages of spreading, xf - 1 ≈ t2/3, a rate in excess of the t1/2 scaling predicted by Marangoni spreading with a continuous supply of surfactant in both rectangular and axisymmetric coordinates.31 It should also be noted that even in the absence of Π and for slightly larger values of βcs the formation of a rim with similar spreading exponents is recovered. For large values of βcs, the plots of local adsorbed surfactant mass, βcaCa and βcsCs, show that the surfactant preferentially adsorbs to the substrate to such an extent that surface tension gradients are reduced. This results in less effective wetting of the substrate, although as t f ¥ the extent of spreading would be expected to converge to an equilibrium independent of βca and βcs and dependent only on parameters effecting the equilibrium: M, Ra, and Ra.12 For long times, continued diffusion of surfactant along the precursor prevents this in our model and is most appreciable for βcs=0.5. The formation of a rim is also accompanied by distinctive characteristics in the adsorbed mass profiles whose gradients are smaller within the rim, but with very sharp gradients at the leading edge of the droplet accompanied by weak diffusion of surfactant ahead of the drop. The adsorption of surfactant to the solid substrate in front of the contact line leads to an increase in the local substrate energy and so promotes spreading. This has been coined the autophilic effect,32 and the associated hydrophilization of the substrate has also been linked to the enhanced spreading of trisiloxane solutions on highly hydrophobic Teflon substrates.26 However, the fast initial spreading stage seen in these experiments cannot solely be resolved by hydrophilization mechanisms.26 Qualitatively, the rims are not very different from some of those seen in previous studies, formed by either surfactant monolayers30 or droplets of surfactant solution33,34 spreading over initially (31) (32) (33) 105. (34)
Jensen, O. E.; Grotberg, J. B. J. Fluid Mech. 2006, 240, 259. Kumar, N.; Varanasi, K.; Tilton, R. D.; Garoff, S. Langmuir 2003, 19, 5366. Edmonstone, B. D.; Craster, R. V.; Matar, O. K. J. Fluid Mech. 2006, 564, Warner, M. R. E.; Craster, R. V.; Matar, O. K. Phys. Fluids 2004, 16, 2933.
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Figure 1. Long-time h and the local βcaCa and βcsCs profiles at t = 3000 and evolution of the droplet front, xf (having subtracted its initial value of unity), in time for βcs = 0.5, 1.25, 2, and 5 with Ra = 10 and M = 5. The rest of the parameters are Pec = Pem = 10, Peca = Pecs = 104, Rs = 10, Kca = Kcs = 0.1, Kcm = 1, and βca = 1, which are unchanged in subsequent figures.
Figure 2. Evolution of h and the local surfactant mass profiles βcaCa, hC, hm, and βcsCs, with βcs = 2.5, M = 5, and Ra = 1 at t = 100, 300, 600, 1000, 2000, 4000, 5000, and 104; the arrows represent the direction of increasing time.
uniform liquid substrates. However, in those instances, whereas Marangoni stresses were similarly important, the formation of rims was due to a transfer of fluid from the underlying liquid substrate. In this model, deformation of the droplet is responsible for the distinctive droplet profile. Figure 2, generated for the value of βcs that maximizes the spreading rate, shows that the deformation occurs at relatively early times, after which the drop and adsorbed surfactant mass profiles are qualitatively similar in the vicinity of the rim. The role of Marangoni stresses is explored in Figure 3. A consequence of the fast-diffusion limit is the increasing preference for the surfactant to adsorb at the interfaces rather than remain in the bulk as the fluid depth decreases; this is reflected by the 1/h dependence of the second and third terms on the right-hand side of eq 51 and explains the near absence of bulk surfactant species during the majority of the evolution of the droplet (Figure 2). Initially, this results in surface tension gradients extending upstream from the contact line. Furthermore, the Marangoni contribution to the fluid velocity at the surface, σlxh, Langmuir 2009, 25(24), 14174–14181
contrives to maximize the velocity at some distance behind the contact line and results in nonmonotonic velocity profiles and the formation of a rim. The adsorption of surfactant at the clean substrate located at the contact line also generates a substantial surface tension gradient at the leading edge of the drop and a further local maximum in the Marangoni velocity. The Marangoni contribution to the velocity is maintained for large times and extends over the entire droplet; it is this feature that allows for the fast and efficient spreading of the droplet. It is important to note that the Marangoni stresses at the leading edge are not supported by the inclusion of the thin precursor layer. Rather, because of surfactant diffusion ahead of the contact line, there is a short capillary- and disjoining-pressure-dominated film that matches the front of the rim onto the precursor film and over which the surfactant concentration goes to zero. Numerous studies7-12,16,17 have also found a nonmonotonic dependence of the spreading rate on M, and this is captured in Figure 4. Below the cmc, M=0.5, the droplet begins to spread DOI: 10.1021/la9019469
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Figure 3. Profiles of h (---), Ca ( 3 3 3 ), and σlxh (-) at t = 100, 300, 600, and 1000 in a-d, respectively, for βcs = 2, M = 5, and Ra = 10.
Figure 4. Long-time droplet, h, and local surfactant mass, βcaCa and βcsCs, profiles at t = 3000 and evolution of the droplet front, xf, in time for βcs = 2 with Ra = 10 and M = 0.5, 2.5, 7.5, and 10.
because the initial surfactant concentrations are such that S|t=0 > 0. Even though basal adsorption is present (βcs = 2), there is insufficient mass to create and maintain a significant Marangoni stress to drive rapid spreading and rim formation. As a result, the surfactant diffuses across the precursor film and there is not enough bulk surfactant to maintain the low surface energies required for spreading; the profile of xf(t) shows that the droplet then begins to retract. Yet above the cmc, there is nonmonotonicity in both the extent of spreading and the size of the rim with increased M (unlike the trends observed earlier with increasing βcs). Large M yields a greater mass of surfactant adsorbed at both interfaces and reduces the effectiveness of basal adsorption. Figure 5 explores the long-term evolution of drops for varying values of Ra, which measures the surfactant affinity for the liquid-vapor interface. The profile for Ra = 1 highlights the importance of the interplay among all of the parameters that determine the rate and magnitude of surfactant adsorption to 14180 DOI: 10.1021/la9019469
both the liquid-vapor and liquid-solid interfaces. Decreasing the value of Ra decreases the affinity of the surfactant for the liquid-vapor interface, and thus a greater mass of surfactant adsorbs to the substrate; this is qualitatively similar to increasing βcs and so explains the increased rim formation. As Ra increases, the local concentration Ca increases relative to the local bulk monomer concentration C. Accordingly, basal adsorption becomes less effective, and a greater concentration of surfactant remains at the liquid-vapor interface in the region of the contact line; gradients in interfacial surfactant concentration, Cax, are diminished, as are the Marangoni stresses. This is most evident for Ra=100, where there is near monolayer coverage of the interface.
IV. Conclusions We have examined the spreading of surfactant-laden droplets on hydrophobic surfaces. A model for the spreading process is derived using the lubrication approximation, which accounts for Langmuir 2009, 25(24), 14174–14181
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Figure 5. Long-time droplet, h, and adsorbed surfactant mass, βcaCa and βcsCs, profiles at t = 3000 and evolution of the droplet front, xf, in time for Ra = 1, 5, 25 and 100.
the Marangoni-driven spreading, surface, bulk diffusion, and sorption kinetics and the possibility of the formation and breakup of micellar aggregates and of surfactant adsorption at the solid substrate underlying the liquid film. Intermolecular forces are included in a disjoining pressure term. Numerical solution of the evolution equations allowed the examination of the importance of surfactant kinetics at the interfaces and the role of initial surfactant concentration. The dynamics were found to be highly sensitive to the kinetic rates at both the liquid-vapor interface and the solid substrate. Efficient spreading was controlled by the ability to maintain sufficient surface tension gradients at the leading edge of the droplet. Critical to this process was the preference for the basal adsorption of surfactant in the contact line region, leading to significant Marangoni stresses. Even so, the relative magnitudes of the interfacial parameters are key to the maintaining the correct balance between basal adsorption and replenishment of surfactant at the liquid-vapor interface. Indicative of this is the nonmonotonicity in spreading rates associated with increasing the initial surfactant concentration. Marangoni stresses are unable to be supported by both small and large values of M because of an insufficient variation in surface tension. Accompanying efficient spreading was the formation of a rim at the leading edge of the droplet as reported by Nikolov et al.11 This was found to be another direct consequence of basal adsorption.
Langmuir 2009, 25(24), 14174–14181
Whereas our results demonstrate a possible mechanism for enhanced spreading on hydrophobic substrates, we are unable to give a full demonstration of the mechanisms because of the large variations in the basal kinetics that are likely to occur with varying surfactant and substrate chemistry. Accordingly, a study into the effects of varying the interfacial energies and the extent to which hydrophilization of the bare substrate26 affects the efficacy of spreading has not been carried out. Nevertheless, we believe the key parameter to be that of basal adsorption, parametrized by βcs in our study. For a fixed substrate, the compact nature of the trisiloxane molecules is likely to promote a larger value of βcs than conventional surfactants. This, coupled with the hydrophilization of the substrate in the region of the contact line, promotes fast spreading of hydrophobic surfaces and could be the major determining factor for the onset of SES. Note Added after ASAP Publication. This article was published ASAP on September 4, 2009. Figure 4 has been modified. The correct version was published on November 11, 2009. Acknowledgment. We acknowledge the support of EPSRC through grant numbers EP/E046029/1 and EP/E056466/1 as well as a Doctoral Training Account studentship for D.R.B. R.V.C. thanks NSERC for their support via the Discovery Grant scheme.
DOI: 10.1021/la9019469
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