Nitin Kumar, a,b, Alexander Couzis, b and Charles Maldarelli c, 1. Introduction

Transcription

1 Journal of Colloid and Interface Science 267 (2003) Measurement of the kinetic rate constants for the adsorption of superspreading trisiloxanes to an air/aqueous interface and the relevance of these measurements to the mechanism of superspreading Nitin Kumar, a,b, Alexander Couzis, b and Charles Maldarelli c, a Department of Physics and Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15232, USA b Department of Chemical Engineering, City College of the City University of New York, New York, NY 10031, USA c Levich Institute, City College of the City University of New York, New York, NY 10031, USA Received 10 December 2002; accepted 9 May 2003 Abstract Super-spreading trisiloxane surfactants are a class of amphiphiles which consist of nonpolar trisiloxane headgroups (((CH 3 ) 3 Si O) 2 Si(CH 3 )(CH 2 ) 3 ) and polar parts composed of between four and eight ethylene oxides (ethoxylates, OCH 2 CH 2 ). Millimeter-sized aqueous drops of trisiloxane solutions at concentrations well above the critical aggregate concentration spread rapidly on very hydrophobic surfaces, completely wetting out at equilibrium. The wetting out can be understood as a consequence of the ability of the trisiloxanes at the advancing perimeter of the drop to adsorb at the air/aqueous and aqueous/hydrophobic solid interfaces and to reduce considerably the tensions of these interfaces, creating a positive spreading coefficient. The rapid spreading can be due to maintaining a positive spreading coefficient at the perimeter as the drop spreads. However, the air/aqueous and solid/aqueous interfaces at the perimeter are depleted of surfactant by interfacial expansion as the drop spreads. The spreading coefficient can remain positive if the rate of surfactant adsorption onto the solid and fluid surfaces from the spreading aqueous film at the perimeter exceeds the diluting effect due to the area expansion. This task is made more difficult by the fact that the reservoir of surfactant in the film is continually depleted by adsorption to the expanding interfaces. If the adsorption cannot keep pace with the area expansion at the perimeter, and the surface concentrations become reduced at the contact line, a negative spreading coefficient which retards the drop movement can develop. In this case, however, a Marangoni mechanism can account for the rapid spreading if the surface concentrations at the drop apex are assumed to remain high compared to the perimeter so that the drop is pulled out by the higher tension at the perimeter than at the apex. To maintain a high apex concentration, surfactant adsorption must exceed the rate of interfacial dilation at the apex due to the outward flow. This is conceivable because, unlike that at the contact line, the surfactant reservoir in the liquid at the drop center is not continually depleted by adsorption onto an expanding solid surface. In an effort to understand the rapid spreading, we measure the kinetic rate constants for adsorption of unaggregated trisiloxane surfactant from the sublayer to the air/aqueous surface. The kinetic rate of adsorption, computed assuming the bulk concentration of monomer to be uniform and undepleted, represents the fastest that surfactant monomer can adsorb onto the air/aqueous surface in the absence of direct adsorption of aggregates. The kinetic constants are obtained by measuring the dynamic tension relaxation as trisiloxanes adsorb onto a clean pendant bubble interface. We find that the rate of kinetic adsorption is only of the same order as the area expansion rates observed in superspreading, and therefore the unaggregated flux cannot maintain very high surface concentrations at the air/aqueous interface, either at the apex or at the perimeter. Hence in order to maintain either a positive spreading coefficient or a Marangoni gradient, the surfactant adsorptive flux needs to be augmented, and the direct adsorption of aggregates (which in the case of the trisiloxanes are bilayers and vesicles) is suggested as one possibility Elsevier Inc. All rights reserved. 1. Introduction Surfactants consisting of a trisiloxane hydrophobe (CH 3 Si(CH 3 ) 2 O Si(CH 3 )(CH 2 CH 2 CH 2 ) O Si(CH 3 ) 2 CH 3 ) * Corresponding authors. addresses: nitin@mit.edu (N. Kumar), charles@ch .engr.ccny.edu (C. Maldarelli). and a polyethylene oxide ( EO ) polar group ( OCH 2 CH 2 ) of four to eight units have the unique property of facilitating the spreading of water on very hydrophobic hydrocarbon surfaces such as polyethylene and parafilm (for a review see [1,2] and the references cited therein). Water on these surfaces beads up with contact angles θ (as measured through the water phase) of 90 or more. Aqueous drops with a dissolved trisiloxane at sufficiently high bulk concentration, /$ see front matter 2003 Elsevier Inc. All rights reserved. doi: /s (03)

2 N. Kumar et al. / Journal of Colloid and Interface Science 267 (2003) usually several times their critical aggregate concentration (CAC) (see particularly [3 7]), when placed on hydrophobic surfaces, spread rapidly and completely wet out at equilibrium, forming micrometer-thick films of no measurable contact angle at their perimeter ( superspreading ). Surfactants with the same polyethylene oxide hydrophilic group as the superspreading trisiloxanes but with a hydrophobic part consisting of an alkyl chain (such as the n-alkyl polyethylene oxide monoethers C i E j,ch 3 (CH 2 ) i 1 (OCH 2 CH 2 ) j OH) do not enable spreading on very hydrophobic surfaces [8,9]. These surfactants can, however, be effective at facilitating spreading on more hydrophilic surfaces [6,10]. Figure 1 compares the molecular structures of a trisiloxane superspreader with four EO units (represented as M(D E 4 OH)M; M denotes the group Me 3 Si O, D the group Si(Me)(C 3 H 6 ), E the EO group) with an analogous n-dodecyl chain polyethoxylate surfactant C 12 E 4.Ingeneral, we choose to compare the trisiloxanes to the 12-carbon chain length polyethoxylates because the hydrophobicity of this alkyl chain and of the trisiloxane backbone are approximately the same, as measured by the equivalence of the critical concentrations at which aggregates form in aqueous solutions ( mol/m 3 ) [6,11]. From the figure it is clear that the trisiloxane hydrophobe has a cross sectional area ( Å 2 [12,13]) that is much larger than the cross section of the polyethoxylate n-alkyl chain (20 22 Å 2 ) [14]. As is well known, in water the ethoxylate cross-sectional area is much larger than the cross section of the n-alkyl chain in polyethoxylates, with EO chain cross sections increasing from approximately 40 Å 2 for C 12 E 4 to 65 Å 2 for C 12 E 8 (see for example Schick [15,16]). For the superspreading trisiloxanes, therefore, the EO chain cross section can be smaller Fig. 1. Molecular structures of trisiloxane and n-dodecyl polyethoxylate surfactants. Dark atoms are oxygen; light atoms are hydrogen; and gray atoms are silicon and carbon. than or of the same order as the trisiloxane hydrophobe [1, 12,13,17]. The ability of the trisiloxane superspreaders to facilitate, at equilibrium, complete wetting on hydrophobic surfaces can be understood within the framework of how dissolved surfactants in general affect the contact angles of aqueous sessile drops on these surfaces [18,19]. In the absence of surfactant, the tensions of the air/aqueous (γ ) and aqueous/hydrophobic solid interfaces (γ s/l ) are large compared to the tension of the air/hydrophobic solid interface γ s/a.as a result, the Young Dupre balance of forces at the contact line (γ cos θ + γ s/l = γ s/a ) is satisfied by the aqueous phase subtending a large obtuse contact angle so that the component of this tension along the surface, in combination with γ s/a, can balance γ s/l. The adsorption of surfactant at the three phase contact line (as in the case of the C i E j surfactants [8,9]) reduces γ s/l and γ, with the result that the aqueous phase subtends an acute angle as γ s/a is balanced by γ s/l and the component of the air/aqueous tension γ along the interface. In the case of the trisiloxanes, at sufficiently high bulk concentrations, the adsorption is large enough and the tension lowering at the air/aqueous and aqueous/solid interfaces is great enough so that the sum of γ s/l +γ is smaller than the air/hydrophobic solid tension (the spreading coefficient S = γ s/a γ s/l γ becomes positive). Tensions do not balance at the contact line and the drop spreads into a thinning film as this force imbalance pulls the contact line out. The remarkably high wetting rates that drops with a trisiloxane superspreading surfactant can achieve on very hydrophobic hydrocarbon surfaces are the characteristic least understood. The spreading rates are a function of the bulk concentration; on parafilm the spreading rates of drops of 1 2 µl increase with bulk concentration, achieving a maximum rate of 1 15 mm 2 /s at concentrations of approximately 1 wt% (an order of magnitude or more above the CAC), and then decreasing to zero at still higher concentrations [4 6]. (For the spreading characteristics of larger sized drops see [3,20 24].) The increase in area is initially linear in time, but the rate eventually slows down as the equilibrium area is approached. Different physicochemical mechanisms can account for the rapid spreading over the hydrophobic surfaces: (i) The trisiloxanes can maintain, a positive spreading coefficient, as the drop spreads by maintaining high surfactant concentrations at the aqueous/air and hydrophobic solid/aqueous interfaces at the contact line. The surface convection pattern on a spreading drop is a dilating flow, as the advancing wetting film sweeps surface fluid radially outward from the apex. This dilation acts to reduce the surface concentration of surfactant on the drop surface. At the drop perimeter the effect of this dilation is to reduce the local surface concentration at the air/aqueous interface adjoining the contact line. In addition, as the drop advances the hydrophobic solid/aqueous interface at the perimeter becomes depleted of surfactant. To

3 274 N. Kumar et al. / Journal of Colloid and Interface Science 267 (2003) maintain a high surface concentration at the perimeter, surfactant has to adsorb from the advancing liquid film at the perimeter onto the air/aqueous interface sufficiently fast to counteract the effect of the dilation and onto the solid surfaces to counteract the creation of fresh new solid/aqueous area. As the layer advances in this unsteady process, the continued adsorption will deplete the bulk concentration of liquid and limit the adsorptive transport rate. The inventory of surfactant is also depleted by the fact that as the advancing layer spreads, it thins. As the film at the spreading front thins to thicknesses on the order of micrometers, the amount of surfactant in the film available for adsorption (even assuming the concentration to be the high bulk concentration used for superspreading) becomes less than that required to keep both interfaces at an equilibrium surface concentration. (ii) If the adsorption of surfactant at the advancing front cannot maintain a high surface concentration at the fluid and solid interfaces adjoining the contact line, the spreading rate can become reduced, as the spreading coefficient becomes negative and the contact angles become large. However, the reduction in surface concentration at the perimeter can raise the air/water surface tension relative to the tension at the surface of the drop near its center or apex. This gives rise to a Marangoni force as the perimeter pulls liquid down from the drop apex and drives spreading. The magnitude of this Marangoni force is a function of the tension difference between the drop apex and the perimeter; for this force to be effective as the drop spreads, the tension at the apex must remain low relative to the advancing perimeter. The fluid interface at the apex, like that at the perimeter, is subject to dilation. To maintain a high surface concentration at the apex, the adsorption from the liquid under the apex must be greater then the rate of dilation at the apex. As this liquid is not depleted by continued adsorption onto the newly wet solid surface, and the thickness of the drop under the apex is larger than that of the wetting layer at the perimeter, the inventory necessary to replenish the apex is greater than in the wetting layer. Hence adsorption can potentially balance the depletion at the interface at the apex even though it may not be possible at the perimeter. It has also been noted that because the trisiloxanes can reduce the air/aqueous tension to very low value, these surfactants can provide a strong Marangoni driving force as long as the surface concentration remains high at the apex. 1 1 This Marangoni mechanism for trisiloxane superspreading was first suggested by Ananthapadnabhan et al. [25] and later by Zhu et al. [3] and Lin et al. [4]. Their explanation, however, emphasized that the drop spread over a precursor aqueous wetting layer initially present on the hydrophobic surface. This layer with no adsorbed surfactant on its surface maintains the perimeter at a high tension and thus drives the Marangoni gradient. The existence of the precursor film onto which the drop spreads was initially The above discussion makes it clear that to understand the origin of the rapid spreading of the trisiloxanes there must be quantitative measurements of the adsorption rate of the trisiloxanes onto the air/water and solid/water surfaces, so that these rates can be compared with the rate of area expansion and the relevance of these mechanisms can be studied. In this paper, we will focus on adsorption onto the air/aqueous interface. For bulk concentrations below the CAC, when the surface of the drop dilates perturbing the equilibrium between the surface and the bulk liquid, surfactant adsorbs onto the interface from the sublayer underneath the surface by a kinetic exchange process involving kinetic constants of adsorption and desorption [27,28]. The kinetic adsorption depletes the sublayer concentration, creating a diffusion gradient which brings surfactant to the sublayer. The kinetic parameters and diffusion coefficient are measured in experiments in which either (i) surfactant diffuses toward and kinetically adsorbs onto a clean interface and the reduction in tension is measured or (ii) the surface area of an equilibrium monolayer is changed, causing exchange with the sublayer and the re-equilibration in tension is measured [28 30]. By modeling the mass transfer in these experiments, the surface concentration as a function of time can be predicted. From the equation of state (the dependence of the tension on the surface concentration which can be obtained from equilibrium measurements of the tension as a function of the bulk concentration), the mass transfer solutions can be used to predict the dynamic tension relaxations, and these can be compared to experiments to resolve the kinetic constants and the diffusion coefficient. Using this approach, the kinetic rate constants for the C 12 E n (n = 4, 6, and 8) surfactants have been determined [31 34]. For the trisiloxanes, there have been measurements of the relaxation in dynamic tension in clean interface adsorption by Ananthapadnabhan et al. [25], Rosen and Song [35], and Svitova et al. [36]. However, these relaxations were undertaken for bulk concentrations above the CAC and because of the complicating effect of the influence of the aggregates on the surfactant mass transfer (see below), the kinetic rate constants were not determined. Superspreading is observed for trisiloxane concentrations above the CAC, where aggregates are present in the bulk. Aggregates can contribute to adsorption of surfactant onto the interface by either of two mechanisms: When surfactant monomer adsorbs onto a clean interface, the local concentraformulated to explain the fact that superspreading was only observed on parafilm at high humidity. But its formation on a hydrophobic surface is not likely (as noted by Stoebe et al. [10]), and experiments on surfaces made from self-assembled monolayers do not exhibit a humidity effect, indicating that the dependence on humidity is peculiar to parafilm [1]. Nikolov et al. [24,26] emphasize that the depletion at the contact line due to the surface dilation combined with adsorption at the apex provides enough driving force for the Marangoni gradient, and a precursor film is not necessary. Precursor aqueous films form on hydrophilic surfaces and play an important role in maintaining a Marangoni driving force for the surfactant facilitated spreading of a drop [10].

4 N. Kumar et al. / Journal of Colloid and Interface Science 267 (2003) tion falls below the CAC and aggregates begin to break-up supplying monomer which can then diffuse and adsorb onto the surface. In this context, the role of the aggregate is to supply a reservoir of surfactant to maintain the monomer concentration at the CAC. If the aggregate concentration is high enough and the kinetic rate of demicellization fast enough, that the bulk concentration of monomer is maintained at the CAC. The rate of adsorption then achieves its fastest rate, which is the kinetically limited rate. When the rate of aggregate break-up is not fast, or the bulk concentration or the aggregate concentration is not high enough, then the monomer concentration cannot be maintained at the CAC, and the adsorption rate does not achieve is maximum value. A second route for the aggregates to contribute to the surface adsorption is for them to adsorb directly onto the surface. Studies of the phase behavior of the superspreading trisiloxanes [11 13,17,37 41] show that (at room temperature, the conditions we are concerned with here) these aggregates at the critical aggregation concentration are not the spherical micelles (L 1 phase) that are usually found with surfactants with linear alkyl chains as the hydrophobe, as the C 12 E 6 and C 12 E 8 homologues 2 [42,43]. Instead the aggregates are structures consisting of bilayer patches and vesicles (L α phase) that more favorably accommodate the larger hydrophobe cross section. Several studies have suggested that the trisiloxanes ability to form turbid dispersions of bilayers rather then spherical micelles may account for the superspreading ability since the bilayers can adsorb more readily to the interface then spherical micelles since they are less stable structures that can readily unzip into a monolayer upon contacting the air/water interface [3,6,13,20,44]. The case of M(D E 12 OR)M is often cited as proof. This surfactant forms a transparent solution of spherical micelles upon aggregation [13] because the larger ethoxylate headgroup can only be accommodated in a curved geometry. Although it can reduce the contact angle to near zero [7], it does not spread rapidly over hydrophobic surfaces, presumably because the spherical micelles do not adsorb appreciably on the surface and cannot provide the enhanced flux (either at the apex or the perimeter or both) necessary for superspreading. The aim of this paper is to measure the kinetic rate constants of the superspreading trisiloxanes. With these measurements, we will have a scale for the kinetic rate of monomer adsorption. As this rate represents the fastest rate that surfactant monomer can adsorb onto the surface we can evaluate whether this rate can match the surface dilation, or whether the augmentation of direct aggregate adsorption is necessary. We will study the 4- and 8-ethoxylate superspreaders that are capped with hydroxyl groups (M(D E 4 OH)M and M(D E 8 OH)M) and the 12-ethoxylate nonspreader M(D E 12 OH)M, the later in order to establish 2 Note that the smaller EO chain polyethoxylate C 12 E 4 forms either the L 1 or the (bilayer) L α phase at the critical aggregation concentration, depending on the temperature [42]. if the kinetic adsorption parameter of the nonspreader is any slower than the superspreaders. We will also compare the kinetic constants with the measurements of the C 12 E 4,C 12 E 6 and C 12 E 8 polyethoxylate surfactants to see if differences in the kinetic adsorption rate constants can account for the superspreading ability. To obtain the kinetic constants, we will measure the dynamic tension reduction accompanying adsorption of surfactant onto an initially clean interface using the pendant bubble technique [45] in which the shape analysis of a pendant bubble immersed in a surfactant solution is used to obtain the tension. We will also use the pendant bubble as a Langmuir trough to measure the surfactant equation of state directly. The paper is organized into four sections, of which this Introduction is the first. The second section provides details of the pendant bubble apparatus and the procedures used to measure the dynamic tension and equation of state. The third section details the experimental measurements of the equation of state and dynamic tension and the modeling of the surfactant transport, which enables us to determine the kinetic constants. The fourth section discusses the magnitude of these constants relative to the surface dilation rates present on a superspreading drop to provide insight into the trisiloxane transport picture during superspreading and the validity of the proposed mechanisms for superspreading. The paper concludes with a summary. All experiments were conducted at 23 ± 2 C. 2. Experimental 2.1. Materials Samples of the trisiloxane surfactants M(D E 4 OH)M, M(D E 8 OH)M, and M(D E 12 OH)M were kindly provided by Dr. Randall Hill of Dow Corning Corp. The purity of these samples was approximately 85%, and the remaining 15% are non-surface-active glycols. The trisiloxane part was approximately 98% monodisperse [46]. Deionized water, with a resistivity of at least 15 M cm, was obtained from a Milli-Q Millipore system fitted with an Organex-Q cartridge to remove trace organic contaminats. The solutions were prepared in the deionized water and used immediately after preparation. The glassware were cleaned by successive rinses with acetone and deionized water Measurement of the dynamic and equilibrium surface tensions The pendant bubble technique was used to measure the dynamic and equilibrium surface tensions. We have used an apparatus similar to the one described by Subramanyam and Maldarelli [47] and Pan et al. [34]. A light beam, produced from a tungsten bulb, was collimated by passing through a series of lenses and pinholes. A transparent quartz cell ( cm) was filled with surfactant solutions of

5 276 N. Kumar et al. / Journal of Colloid and Interface Science 267 (2003) known concentration and placed in the path of the collimated beam. An inverted stainless steel needle was immersed in the surfactant solution. A motor driven syringe was connected to the inverted needle through a solenoid valve. A pendant bubble 1 3 mm in diameter was formed in a few tenths of a second on the edge of the needle by pushing air through the valve, tubing, and the inverted needle. The collimated light formed a silhouette on a CCD array video camera placed on the opposite side of the quartz cell. The images of the pendant bubble were captured electronically using Scion imaging software and an image capturing board. The X Y coordinates of the bubble profile were obtained using the Scion software. This apparatus had the capability to capture images at the conventional rate of 30 frames/s. The dynamic surface tension data were obtained by rapidly forming a pendant bubble on the inverted needle. The evolution of the bubble profile was continuously monitored. The profiles of the pendant bubble were obtained from the captured images. The surface tensions at various times were obtained by integrating the Young Laplace equations and fitting the resulting profile to the experimental bubble profiles obtained from the digital image of the silhouette [45,48]. After some time, the dynamic surface tension of the pendant bubble reaches an equilibrium value. This value is taken to be the equilibrium surface tension at that concentration Measurement of the equation of state Pan et al. [34] and Lin et al. [49] have used expanding and contracting pendant bubbles to determine directly the equation of state for soluble surfactants. In this method, a pendant bubble of diameter of 1 3 mm was created in a dilute surfactant solution and it was allowed to reach equilibrium. Then, the bubble was rapidly expanded or compressed in approximately 2 to 5 s. The profile of the bubble was captured during the expansion or compression process. The surface tension, γ(t), and the area of the bubble, A(t), were evaluated from the bubble profiles. The surface tension is assumed to be a unique function of the surface concentration (Γ ). We assumed a single reference surface tension (γ ref ) and the surface concentration corresponding to this reference tension is denoted as Γ ref. For the trisiloxanes this reference tension was equal to 60 dyn/cm. For a given expansion or contraction, we note the area at which the reference tension is achieved, and denote this area as A ref. If the area change is rapid enough, then we can assume that A(t) Γ(t)= S, wheres, a constant, is the total amount of the surfactant on the bubble surface, and A(t) is the area of the bubble. For that expansion or contraction, a plot of tension as a function of relative surface concentration γ vs Γ rel (= Γ(t)/Γ ref = A ref /A(t)) is drawn. The experiment is repeated at several different bulk concentrations and varying expansion contraction rates, and the results included on a single graph. As we show (Section 3), the combined γ vs Γ rel data for the three trisiloxane surfactants each fall on a single curve, proving the validity of A(t) Γ(t)= S under the conditions of the experiments and the fact that the surface tension is a unique function of the surface concentration. We note that to measure tension as a function of relative surface concentration in the low-tension range of less than 50 dyn/cm, it was not always possible to achieve the reference tension of 60 dyn/cm, since a compression starting at greater than 60 dyn/cm would take too long to reach the low-tension range and significant desorption would occur, or an expansion starting below 50 dyn/cm would take too long to reach above 60 dyn/cm and a significant adsorption would occur. In these cases, we first choose another (lower) reference tension (γref ) to normalize the data where the new reference tension has already been recorded in a higher tension area contraction or expansion, which also includes γ ref. Then for the low-tension experiment, Γ/Γ ref = (Γ /Γref )(Γ ref /Γ ref), whereγ/γref is given from the low-tension experiment area ratios and Γref /Γ ref is evaluated from the area ratio of the high tension experiment. With Γ/Γ ref evaluated in this way, the low-tension results can be plotted on the master curve, and we find they merge smoothly to the correlating line formed from the higher tension experiments. We used this procedure to obtain equation of state data down to approximately 40 dyn/cm. To measure the equation of state directly for even lower tensions, an equilibrium bubble with a tension of approximately dyn/cm would have to be used, and the bubble would then be compressed and expanded to fill out the data curve from 20 to 40 dyn/cm. However, to obtain such low initial equilibrium tensions, a high surfactant bulk concentration would have to be used, close to the CAC. At these high bulk concentrations, the kinetic exchange becomes so fast (see Section 3) that it is impossible to keep the mass constant at the interface, no matter how fast the bubble area is changed. 3. Results 3.1. Measurements of the surfactant equation of state and the equilibrium tension as a function of bulk concentration To formulate the kinetic exchange and the corresponding equilibrium adsorption isotherm and equation of state we utilize the Frumkin formulation, which has proved accurate in describing the adsorption characteristics of a series of C i E j surfactants [31 34], dγ dt = βγ C S ( 1 Γ Γ ) αγ e KΓ/Γ, where Γ is the surface concentration, C s is the sublayer concentration, α and β are kinetic rate constants for desorption and adsorption, respectively, Γ is the maximum packing concentration, and K is a (nondimensional) parameter accounting for the effect of the intermolecular interactions on the adsorption (K >0 indicates repulsion and an acceleration of the adsorption rate, and K<0 indicates cohesion (1)

6 N. Kumar et al. / Journal of Colloid and Interface Science 267 (2003) and a deceleration of the rate; when K = 0, the Langmuir formulation is obtained, in which intermolecular interactions are neglected). At equilibrium, the net rate is equal to zero, and the Frumkin adsorption isotherm relating the equilibrium surface concentration (Γ e ) and the bulk concentration (C 0 ) is obtained, Γ e 1 = (2) Γ 1 + (a/c 0 )e KΓ, e/γ where a = α/β is the surface activity. Using the Gibbs Duhem equation (dγ = RT Γ d ln(c 0 ), for dilute solutions), an equation of state, relating the surface tension (γ) and the surface concentration (Γ), is obtained, ( γ(γ)= γ C + RT Γ [ln 1 Γ Γ ) K 2 ( Γ Γ ) ] 2, where γ c is the surface tension of the clean air water interface, and its value is 72.5 mn/m. From (2) and (3) the equilibrium tension as a function of the bulk concentration can be obtained. The parameters K, Γ,andα/β are usually obtained by fitting (2) and (3) to measurements of the equilibrium tension as a function of the bulk concentration, as these measurements are easy to obtain. However, several studies [31,34,49,50] have shown that different sets of numerical values for K, Γ,andα/β can fit γ e log(c 0 ) equally well. As described in Section 2, we measured the equation of state directly by using the pendant bubble as a Langmuir trough to obtain the tension as a function of the relative surface concentration (γ(γ/γ ref )) andthenfitthese measurements to the Frumkin equation of state to determine K and Γ directly. Measurements of the tension as a function of the relative surface concentration for M(D E 4 OH)M, M(D E 8 OH)M, and M(D E 12 OH)M are given in Figs. 2, 3, and 4. The reference surface concentration is taken to be the concentration when the tension is 60 dyn/cm. Also shown in this set of figures is the fitting of the equation of state data to the Frumkin and Langmuir equations of state. The third parameter, a(= α/β), is obtained by fitting Eqs. (2) and (3) to the γ e vs log(c 0 ) data, using the values of Γ and K evaluated from the γ vs Γ rel data. The experimental data and fitted curve for γ e vs log(c 0 ) for the three trisiloxanes, M(D E 4 OH)M, M(D E 8 OH)M, and M(D E 12 OH)M, are shown in Figs. 5, 6, and 7, respectively. The values of Γ, K, anda for these three surfactants are also shown in Table 1. For the γ e vs log(c 0 ) data, we have only recorded and used measurements for concentrations larger than 10 4 mol/m 3, where the tensions are approximately 60 dyn/cm and below. For lower bulk concentrations, impurities such as trisiloxanes with different numbers of EO chains (particularly impurities with smaller EO chains) can compete with the surfactant for adsorption onto the interface; recall the trisiloxanes are only 98% monodisperse. At these lower bulk concentrations we found that the equilibrium tensions were not always reproducible. In contrast, the pendant bubble as a Langmuir trough measurements were undertaken at concentrations above 10 4 mol/m 3 ; the high (3) Fig. 2. Surface tension vs relative surface concentration, Γ rel, for M(D E 4 OH)M. Parameters for the Frumkin model: Γ = mol/m 2, K = 5. Parameter for the Langmuir model: Γ = mol/m 2. Fig. 3. Surface tension vs relative surface concentration, Γ rel, for M(D E 8 OH)M. Parameters for the Frumkin equation of state: Γ = mol/m 2, K = Parameter for the Langmuir equation of state: Γ = mol/m 2. tensions evident in the figures were obtained by rapid expansions. 3 The values of critical aggregation concentration (CAC) for M(D E 4 OH)M and M(D E 8 OH)M were obtained from 3 In undertaking the simulations using the Frumkin adsorption isotherm, we have corrected the apparent bulk concentrations for the amount adsorbed onto the air/water interface of the cuvette. The corrected concentration is computed from the apparent concentration (C app, the concentration in the solution which is poured into the dish or cell) using the adsorption isotherm to account for the adsorbed amount on the surface; i.e., we use C app Γ e (A/V) to replace C 0 in the data fitting route, where A and V are the air/water area and solution volume respectively in the cuvette in which the bubble is formed. Corrections are negligible after 10 3 mol/m 3 ;the abscissas in Figs. 5 7 are the apparent concentrations.

7 278 N. Kumar et al. / Journal of Colloid and Interface Science 267 (2003) Fig. 4. Surface tension vs relative surface concentration Γ rel, for M(D E 12 OH)M. Parameters for the Frumkin equation of state: Γ = mol/m 2, K = 10. Parameter for the Langmuir equation of state: Γ = mol/m 2. Fig. 6. Equilibrium surface tension vs bulk concentration for M(D E 8 OH)M. Parameters for the Frumkin model: Γ = mol/m 2 ; K = 7.26; a = α/β = mol/m 3. Parameters for the Langmuir model: Γ = mol/m 2 ; a = α/β = mol/m 3. Fig. 5. Equilibrium surface tension vs bulk concentration for M(D E 4 OH)M. Parameters for the Frumkin model: Γ = mol/m 2 ; K = 5; a = α/β = mol/m 3. Parameters for the Langmuir model: Γ = mol/m 2 ; a = α/β = mol/m 3. the γ e vs log(c 0 ) data by noting the break in the slope and the subsequent plateau in the tension; the CACs are obtained by passing a straight line through the last few points before the plateau, and finding the concentration corresponding to the intersection of the straight line with the tension plateau. These CACs are given in Table 1. The CAC (and the tension at the CAC) for M(D E 8 OH)M is in very good agreement with that of Svitova et al. [7], and Ananthapadnabhan et al. [25] for the commercial trisiloxane surfactant with an average of 7.5 ethoxylate units M(D E 7.5 Me)M. These authors did not study M(D E 4 OH)M; however, our value for the CAC for M(D E 4 OH)M is slightly smaller than that reported by Gentle and Snow [51]. The CAC for M(D E 12 OH)M was not obtained; however, in the table, for comparison, we provide the value from Svitova et al. [7]. Our measurements of Fig. 7. Equilibrium surface tension vs bulk concentration for M(D E 12 OH)M. Parameters for the Frumkin model: Γ = mol/m 2 ; K = 10; a = α/β = mol/m 3. Parameters for the Langmuir model: Γ = mol/m 2 ; a = α/β = mol/m 3. the tensions at the CAC (22 23 dyn/cm) are in agreement with those reported in all of the literature studies. From Figs. 2 7, we observe that the Langmuir model fits the γ e log(c 0 ) data (Figs. 5 7) with sufficient accuracy, but it fails to fit the γ vs Γ rel data (Figs. 2 4). The Frumkin model fits both γ e log(c 0 ) and γ vs Γ rel data accurately. This suggests that the Langmuir model does not accurately depict the adsorption characteristics of the trisiloxane surfactants, despite the fact that it fits the γ e log(c 0 ) with sufficient accuracy. In Table 1 we tabulate from previous literature studies the values of K for C 12 E 4,C 12 E 6,andC 12 E 8 and we note these are all positive due to the repulsion of the ethoxylate

8 N. Kumar et al. / Journal of Colloid and Interface Science 267 (2003) Table 1 Equilibrium and transport parameters for trisiloxane and alkyl polyethoxylate (C 12 E j ) surfactants Molecule C 12 E 4 C 12 E 6 C 12 E 8 C 12 E 12 M(D E j OH)M j = 4 j = 8 j = 12 Min. surface tension at CAC (mn/m) CAC (mol/m 3 ) Γ (mol/m 2 ) Frumkin interaction parameter, K a = α/β (mol/m 3 ) Adsorption coefficient, β (m 3 /(mol s)) Adsorption rate constant, βγ (ms 1 ) Maximum adsorption rate, βc CAC (s 1 ) Desorption coefficient, α(s 1 ) Surface activity, 1/(α/βΓ ) (m) Diffusion coefficient D, (m 2 /s) Reference Hsu et al. Pan et al. Lin et al. Lu et al. Present work Present work Present work Note. References: Hsu et al. [31], Pan et al. [34], Lin et al. [59], Lu et al. [52]. Table 2 The maximum packing or the minimum area per unit molecule (A min,å 2 /molecule) evaluated at the CAC C 12 E 4 C 12 E 6 C 12 E 8 C 12 E 12 M(D E 4 OH)M M(D E 8 OH)M M(D E 12 OH)M Frumkin model (Å 2 /molecule) 38 a 56 b 64.9 c 53.4 d 54.3 d 69.2 d Langmuir model (Å 2 /molecule) 62 a 68 b 81.2 c 85.2 e 66 d 70.6 d 84.2 d Neutron reflectivity measurements 44 ± 3 e 55 ± 3 e 62 ± 3 e 72 ± 3 e (Å 2 /molecule) a Hsu et al. [31]. b Pan et al. [34]. c Lin et al. [59]. d This work. e Lu et al. [52]. headgroups, which proves more important than the cohesion of the n-dodecyl chain. The values of K for the three trisiloxane surfactants M(D E 4 OH)M, M(D E 8 OH)M, and M(D E 12 OH)M are also positive (5, 7.26, and 10, respectively), demonstrating again the dominance of the repulsive interactions of the ethoxylate headgroup over the cohesion of the trisiloxane backbone. The maximum packing of a surfactant adsorbed on the air water interface is the interfacial area occupied by the surfactant molecule at the CMC or CAC. For the trisiloxanes, the values of Γ CAC were obtained by numerically evaluating the Frumkin and Langmuir adsorption isotherms at the CAC, using the parameters taken from Table 1. These values of the maximum packing or the minimum area per molecule (A min = 1/Γ CAC ) for the trisiloxane and the n-dodecyl polyethoxylates are summarized in Table 2. The minimum area per molecule we have obtained using the Frumkin model for M(D E 8 OH)M, 54.3 Å 2 / molecule, is in the range of that reported in the literature by Hill et al. [17] (59 Å 2 /molecule) and Svitova et al. [7,36] (49 Å 2 /molecule) by measuring the slope of the γ log(c 0 ) curves at the CAC. Our value for M(D E 12 OH)M, 69.2 Å 2 /molecule, is close to that reported by Svitova et al. (61 Å 2 /molecule). For M(D E 4 OH)M, our value, 53.4 Å 2 /molecule, is larger than that of Gentile and Snow [51] (33.2 Å 2 /molecule), whose value appears too small given the size of the trisiloxane headgroup (see the discussion below). For comparison, in Table 2, the minimum areas per molecule at the CMC for the polyethoxylate surfactants using the Frumkin model (parameters in Table 1) are C 12 E 4,38Å 2 /molecule; C 12 E 6,56Å 2 /molecule; and C 12 E 8, 64.9 Å 2 /molecule. For C 12 E 12 only a value obtained from a limiting slope of the γ log(c 0 ) curve at the CMC is available (85 Å 2 /molecule). For the polyethoxylates, as the EO chain length increases, the A min increases. Values of A min for the polyethoxylates, have also been obtained by neutron reflectivity experiments [52] (values listed in Table 2), which confirm this trend. Note from Table 2 that it is clear that the predictions of the Frumkin model for the n- dodecyl polyethoxylates are in much better agreement with neutron reflectivity experiments than the Langmuir predic-

9 280 N. Kumar et al. / Journal of Colloid and Interface Science 267 (2003) tions, providing additional support for the use of the Frumkin model. For the trisiloxanes, the A min for M(D E 4 OH)M and M(D E 8 OH)M are 53.4 and 54.3 Å 2 /molecule, which are approximately the same. For M(D E 12 OH)M, A min = 69.2 Å 2 /molecule which is approximately equal to the neutron reflectivity measurement for C 12 E 12. The cross sectional area of the trisiloxane group is Å 2 (calculated by using either Chem3D software, or from the quotient of the molecular volume of the trisiloxane group ((Me 3 Si O) 2 Si(Me)(CH 2 ) 3 ); 465 Å 2 ) divided by its all-trans extended length (9.2 Å) [12,13]). Thus for the M(D E 4 OH)M and M(D E 8 OH)M, the EO chain cross section (as measured by the C 12 E 4 and C 12 E 8 A min values) fits in the footprint of the trisiloxane cross section ( Å 2 ), while for the M(D E 12 OH)M the cross section of the EO chain (72 Å 2,as measured by the reflectivity data for C 12 E 12 ) is larger than the trisiloxane headgroup and determines the packing. in a manner as follows. The pendant bubble was modeled as a sphere of radius R equal to 1 mm. As we mentioned in Section 2, the bubble is formed impulsively in a few hundred milliseconds; the hydrodynamic disturbance created by the bubble motion has a relaxation timescale of R 2 /ν, whereν is the kinematic viscosity of the continuous aqueous surfactant phase and is equal to 0.01 cm 2 /s. For bubbles of order 1 mm in radius, the hydrodynamic relaxation takes on the order of 1 s. Since the surface tension relaxations due to the 3.2. Dynamic surface tension measurements and determination of kinetic rate constants Dynamic surface tension relaxation data for adsorption onto a clean surface for M(D E 4 OH)M, M(D E 8 OH)M, and M(D E 12 OH)M using the pendant bubble technique are shown in Figs. 8, 9, and 10, respectively, for a range of increasing concentrations, but all below the CAC. The concentrations used are reported as C 0 /a. The individual values of α and β are obtained by comparing this dynamic surface tension relaxation data to the results of tension relaxation simulations derived from solving the surfactant mass transfer. The solution of the surfactant mass transfer is carried out Fig. 9. Dynamic surface tension plots for M(D E 8 OH)M at various concentrations, C R (C 0 /a): (1) 11, (2) 114, (3) 214, (4) 539, (5) 884, (6) For all plots D = m 2 /s. Plots 1 was fitted using DC simulations. Plots 3 to 6 were fitted using MC simulations with β = 5m 3 /(mol s). Plot 2 was simulated by (i) DC plotted as dotted line and (ii) MC with β = 5m 3 /(mol s), plotted as solid line. Fig. 8. Dynamic surface tension plots for M(D E 4 OH)M at various concentrations, (C 0 /a): (1) 29, (2) 42, (3) 64, (4) 115, (5) 458, (6) For all plots D = m 2 /s. Plots 1 and 2 are fitted using DC simulations. Plots 4 to 7 are fitted using MC simulations with β = 3m 3 /(mol s). Plot 3 is simulated by (i) DC, (ii) MC with β = 1m 3 /(mol s), (iii) MC with β = 5m 3 /(mol s), and (iv) MC with β = 3m 3 /(mol s). Fig. 10. Dynamic surface tension plots for M(D E 12 OH)M at various concentrations, (C 0 /a): (1) 174, (2) 412, (3) 2158, (4) 5073, (5) 10,020, (6) 18,085. For all plots D = m 2 /s.plots1,2,and3arefitted using DC simulations. Plots 5 and 6 are fitted using MC simulations with β = 10 m 3 /(mol s). Plot 4 is simulated by (i) DC plotted as dotted line, (ii) MC with β = 3m 3 /(mol s), and (iii) MC with β = 10 m 3 /(mol s).

10 N. Kumar et al. / Journal of Colloid and Interface Science 267 (2003) unsteady mass transfer are on the order of hundreds of seconds or more, we assume the mass transfer takes place in a quiescent fluid medium with the mass transfer assumed to be purely diffusive. The governing equation is therefore Fick s law, D r 2 r ( r 2 C ) = C r t (r > R, t > 0), where D is the diffusion coefficient, t denotes time, r is the radial coordinate from the center of the sphere and C(r,t) is the surfactant concentration. Boundary and initial conditions are C(r,t) = C 0 (r > R, t = 0), C(r,t) = C 0 Γ(t)= 0 (t = 0), (r,t>0), (4) (5a) (5b) (5c) where Γ(t)is the surface concentration. The diffusive flux at the bubble interface results in the accumulation of the surfactant on the surface, thereby increasing the surface concentration: dγ/dt = D( C/ r) r=r (r = R, t > 0). (5d) The above set of equation can be transformed into the well-known Ward and Tordai equation [53,54], Γ(t)= Γ 0 (t = 0) + D [ t ] C 0 t C S (τ) dτ R t [ D + 2 C 0 t C S (t τ)d ] τ, π 0 where C S (t) is the sublayer concentration at time t. However, this equation cannot be solved without knowing the sublayer concentration. Another relationship between Γ(t) and C S (t) is required in order to solve for Γ(t) or C S (t). The Frumkin adsorption model, as given in (1), provides the necessary equation relating Γ(t)and C S (t). To discern kinetic effects and compute the kinetic constants from the experimental relaxations through their comparison with mass transfer simulations, we use the concept of a diffusive/kinetic shift in controlling mechanism as the bulk concentration increases [34,55,56]. The timescale for exchange kinetics to populate the surface from zero coverage to an equilibrium adsorbed monolayer with bulk diffusion assumed to be infinitely fast can be obtained by setting the sublayer concentration equal to C 0 in Eq. (1), and integrating the resulting equation for the surface concentration. The result shows that the surface concentration exponentially increases to the equilibrium concentration with a time constant τ kin = (α + βc 0 ) 1. Alternatively, if the kinetic exchange is assumed to be infinitely fast relative to bulk diffusion, then we may regard the surface and sublayer as in equilibrium. In this diffusion controlled process, the surface and sublayer are in equilibrium as given by the adsorption 0 (6) isotherm (Eq. (2)). The time scale for diffusion to bring enough surfactant to an initially clean surface to form an equilibrium monolayer can be estimated by assuming that the sublayer concentration, which is initially zero (due to the surface/sublayer equilibrium and the fact that the initial surface concentration is equal to zero), remains at zero for all time. Neglecting the spherical terms in Eqs. (6) and solving, the surface adsorption is given by Γ(t)= 2C 0 [Dt/π] 1/2 and the time scale for an equilibrium monolayer to form is given by τ diff = πγe 2/4DC 0. The ratio of the diffusive to kinetic time scales (Φ 2 ) becomes Φ 2 = τ d /τ kin = { πγ 2 β2 (1 + C 0 /a) }/{ 4Dα(C 0 /a) 2} or for large bulk concentrations (C 0 /a 1) Φ 2 = Γ 2 β2 (8) Dα[C 0 /a]. Thus as the bulk concentration increases, there can be a shift in the controlling mechanism for the surfactant mass transfer from diffusion-controlled (DC) to mixed control (MC) to finally fully kinetic control. To use this concept of shift in controlling mechanism to obtain the kinetic constants, we first assume that the transport process underlying the relaxation at each of the bulk concentrations is diffusion-limited, and we fit the data at each concentration for only a diffusion coefficient. We then compare the fitted diffusion coefficients obtained for the different bulk concentrations. If these coefficients are the same, then the process is diffusion-controlled, and then kinetic exchange is too fast to be determined unequivocally. (We do not expect the diffusion coefficient to vary with concentration in this dilute regime.) However, if the diffusion coefficient decreases with an increase in concentration, then diffusion-controlled adsorption cannot fully describe the process, kinetic effects are beginning to compete with diffusion and there is a shift in controlling mechanism. The kinetic constant and diffusion coefficient can then be resolved by varying both in mixed controlled simulations, comparing with the experimental relaxations, and determining values for each of these parameters that fit the relaxations at all the concentrations. In the experimental data of Figs. 8 10, for each of the trisiloxanes studied, we do find a shift to kinetic effects for increasing concentration and we use this to compute reliable values for the kinetic coefficients. Simulations of the dynamic surface tension data for M(D E 4 OH)M is shown in Fig. 8. Plot 1 of Fig. 8 is a fit of the relaxation for the lowest concentration (C 0 /a = 29) assuming diffusion control. A value of D = m 2 /s was obtained. The relaxation data for the second highest concentration (C 0 /a = 42) can also be fit equally well assuming diffusion control with this same diffusion coefficient. However, at all higher concentrations (C 0 /a 64), fitting the experimental relaxations to simulations assuming diffusion control yield values of the diffusion coefficient which decrease with increasing concentration. Plot 3 in (7)

11 282 N. Kumar et al. / Journal of Colloid and Interface Science 267 (2003) Fig. 8 shows the diffusion limited curve (dashed line) for the value of D = m 2 /s for the relative concentration 64; note how the relaxation is too fast, indicating that kinetic effects now play a role at this (and higher) concentrations. For C 0 /a > 42, mixed diffusive kinetic simulations were undertaken varying the kinetic constant β with D = m 2 /s held fixed, and a value of β was determined which gives agreement between the simulations and the data. The value obtained was β = 3m 3 /(mol s). As an example of the sensitivity of the fitting, in Fig. 8, plot 3 shows the results of mixed kinetic diffusive simulations for three different values of β, 1,3,and5m 3 /(mol s) for D = m 2 /s and for a relative concentration of 64. Also in Fig. 8 are shown the simulations for mixed kinetic diffusive simulations for β = 3m 3 /(mol s) and D = m 2 /s, at the higher concentrations (plots 4 6) showing excellent agreement with the data. We repeat this process for the two other trisiloxane surfactants. Figure 9 shows the dynamic surface tension data for M(D E 8 OH)M. Using the same value of D obtained for the M(D E 4 OH)M, i.e., m 2 /s, the DC simulation agree well with the experimental data for the lowest concentration (C 0 /a = 11, plot 1 of Fig. 9). For the remaining higher concentrations we find again that for diffusioncontrolled fitting, the value of D obtained decreases with concentration, reflecting the increasing relative importance of surface kinetics. As an example, the dotted line of plot 2 in Fig. 9 for the second highest concentration (C 0 /a = 114) shows the diffusion limited simulation curve (plot 2, dotted line) for the value D = m 2 /s, which relaxes too quickly. Simulations were undertaken for this and higher concentrations assuming a value of D = m 2 /s and varying β to obtain agreement between the predicted simulations and the experimental ones; the result yielded β = 5m 3 /(mol s) and the fits are shown as the solid lines in plots 2 6 in Fig. 9. For the last trisiloxane, M(D E 12 OH)M, diffusion-limited fits for the lowest three concentrations gave a constant value of D equal to that found for the other trisiloxanes at low concentration, m 2 /s (plots 1 3, Fig. 10). However, for C 0 /a > 2158, the DC fits compute decreasing values of the diffusion coefficient as the concentration increases. For C 0 /a = 5073, plot 4, dotted line, shows the diffusion-limited fit for D = m 2 /s, which relaxes too quickly. For C 0 /a 5073, mixed diffusive kinetic simulations were undertaken using a value of D = m 2 /sandvarying β until the simulated relaxations agreed with the experimental ones. The best fit value was found to be 10 m 3 /(mol s) and the fits are given for these higher concentrations as the dotted lines, plots 4 6 in Fig. 10. Shown also in plot 4 for C 0 /a = 5073 is the simulation for β = 3m 3 /(mol s) and D = m 2 /s, which clearly does not relax rapidly enough and demonstrates the sensitivity of the fitting for the kinetic constant. The measured values for the kinetic constants and the diffusion coefficient for the trisiloxanes are summarized in Table 1 and compared to those for the polyethoxylates. The adsorption rate constants βγ for the trisiloxane and polyethoxylate surfactants C 12 E 4, M(D E 4 OH)M, C 12 E 8, M(D E 8 OH)M, and M(D E 12 OH)M are , , , ,and m/s, respectively, as listed in Table 1, and are all within one order of magnitude of each other. 4. Relevance of the kinetic rate constant measurements to trisiloxane superspreading In Section 1, we mentioned that there are two mechanisms that can account for the rapid rate of spreading that is observed when trisiloxane solutions facilitate spreading on very hydrophobic surfaces: (i) In the first, the spreading coefficient remains positive and the contact angle at the advancing perimeter remains equal to zero, and this drives the spreading. The positive spreading coefficient is a result of the adsorption of the surfactant at the air/aqueous surface and aqueous/hydrophobic solid interface. However, the dilating flow on the surface of the drop as it spreads causes a depletion of surfactant on these surfaces, and therefore to maintain a significant concentration at the contact line, the flux of surfactant to the surface must exceed the dilation rate at the perimeter. This task is particularly difficult in the vicinity of the contact line where the spreading front of liquid is continually depleted of inventory from adsorption on both the water/air and hydrophobic solid/water interface, and there is a continual reduction in the layer thickness and therefore of the surfactant inventory. (ii) In the second mechanism, Marangoni forces at the aqueous/air interface cause the drop to spread. Since the fluid layer under the apex is not continually depleted of surfactant by adsorption onto the solid surface and its thickness is greater than at the perimeter, the inventory of surfactant available for adsorption may be larger than at the advancing perimeter, and thus the flux of surfactant to the surface may be greater at the apex then at the perimeter. This difference in transport rates raises the tension at the perimeter relative to the apex since the apex is at a higher surface concentration. As a result, the perimeter pulls the drop out. Examining these mechanisms in the light of our kinetic measurements, we first compare the rate of surfactant adsorption to the dilation rate at the air/aqueous interface. We observed in Section 1 that kinetic adsorption rate represents the fastest rate that surfactant can transport to the surface, and is realized in the case when the bulk transport of monomer and aggregate and aggregate break-up rates are fast relative to kinetic adsorption so the sublayer concentration is equal to the critical aggregate concentration. A simple