## Abstract

The collisions of drops of surfactant solutions (dioctyl sulfosuccinate sodium salt (DOS) and trisiloxane oxypropylene polyoxyethylene (Silwett L77)) with small disc-like targets were studied both experimentally and theoretically. Upon impact, the drops spread very fast beyond the target in the shape of a thin lamella surrounded by a thick rim. No significant difference between water and surfactant solutions was observed in the early stage of the impact. But the collapse stages were very different. In particular, the lamellas of solutions of Silwett L77 disintegrated owing to a spontaneous nucleation of holes, giving to the lamella a web-like structure prior to its break-up. In contrast, lamellas of DOS solutions collapsed like water lamellas, except that the maximum diameter and the lifetime of the lamella of the most concentrated DOS solution were significantly increased compared with pure water and other surfactant solutions. A theoretical analysis shows that the observed instability effects in the lamella and the increase in the size and lifetime of the lamella can be caused by the coupling between liquid inertia and Marangoni stresses.

## 1. Introduction

Renewed interest in collisions of drops with solid substrates has recently arisen because of a great variety of new technological processes involving drop impacts, such as aerosol coating, inkjet printing and agricultural sprays (Yarin 2006). In spraying processes, the solid substrates, whether they are natural or manufactured, are often complex, rough, patterned, chemically heterogeneous, etc., and most spray liquids are also complex, e.g. polymer or/and surfactant solutions. The knowledge of the behaviour of these liquids in an impact experiment and the understanding of the underlying physics are of utmost importance for technological progress.

In this framework, collisions of drops of pure water (Rozhkov *et al.* 2004) and of high-molecular-weight polymer solutions (Rozhkov *et al.* 2006*a*) have been studied with a disc-like solid target of about the same size. In the present paper, drops of surfactant solutions are considered.

### (a) Small target

It was observed that, upon impact on a disc, drops jet sideways to form a free liquid sheet—which is called a lamella (figure 1). The lamella grows and then retracts. It is bounded by a circular thicker rim, which accumulates the liquid flowing from the lamella centre to its periphery.

In this study, drop impacts with relatively high velocities are considered. The impact Reynold’s number *Re*_{i}=*ρv*_{i}*d*_{i}/*μ* and the Weber number are rather high; *ρ* is the liquid density, *v*_{i} and *d*_{i} are the drop impact velocity and diameter, respectively, *μ* and *γ* are the liquid shear viscosity and surface tension, respectively. Typically, the drop behaviour results from the balance between three factors at least: the drop inertia, the liquid surface tension and the viscous resistance. The use of a small disc-like target as a substrate instead of a plate eliminates the influence of the viscous drag on the solid surface, and makes possible the detailed observation of hydrodynamic effects that are usually screened by viscosity.

Rozhkov *et al.* (2002, 2004) showed that the radial expansion of the lamella is equivalent to the radial ejection of a liquid sheet from a point source with ejection velocity *v*_{s} and flowrate *q*_{s}. Let us introduce the dimensionless quantities,
where *t*_{s} is the current time of ejection and *τ*_{s}=*t*_{s}/(*d*_{i}/*v*_{i}). It was experimentally found by Rozhkov *et al.* (2004) that *V*_{s}(*τ*_{s}) and *Q*_{s}(*τ*_{s}) can be approximated as
1.1

This law of liquid ejection (1.1), valid for the growth stage of the lamella, gives a universal description of the drop motion, at high Reynolds and Weber numbers when the flow is only driven by the liquid inertia. Define *V* =*v*/*v*_{i}, *Q*=*q*/(*πd*^{2}_{i}*v*_{i}/6) and *H*=*h*/*d*_{i} the dimensionless velocity, flow rate and sheet thickness at point *Y* =*r*/*d*_{i} and at dimensionless time *τ*=*t*/(*d*_{i}/*v*_{i}), *t* is the time, *r*, the radial coordinate, *v*, the local velocity and *q*, the local flow rate defined as the amount of liquid that flows through a circular contour of radius *r* per unit time, i.e. *q*=2*πrhv* with *h* as the local film thickness. The distributions of the dimensionless functions *V* (*τ*,*Y*), *Q*(*τ*,*Y*) and *H*(*τ*,*Y*) were deduced from equation (1.1) and explicitly calculated by Rozhkov *et al.* (2004)—fig. 8 in the cited reference.

The thin lamellas are bounded by a toroidal rim, which is nothing but a rupture wave that propagates against the flow as a travelling thick two-dimensional blob accumulating the film liquid (figure 1) (Taylor 1959*b*; Culick 1960; Song & Tryggvason 1999). The rim motion results from the balance between the momentum flux provided by the liquid flow into the rim, the rim inertia and the capillary forces exerted on the rim by the two film surfaces (Rozhkov *et al.* 2002, 2004). From the knowledge of *V* (*τ*,*Y*) and *Q*(*τ*,*Y*), we could numerically calculate the rim trajectories *Y*_{l}=*Y*_{l}(*τ*) (*Y*_{l}=*r*_{l}/*d*_{i}, *r*_{l} is the rim radial coordinate, figure 1) for different values of *We*_{i} that appeared as the single input parameter. Numerical trajectories compared well with experimental data (Rozhkov *et al.* 2004).

The observed structure of an impacting drop, as the combination of a thin lamella and a thick rim, demonstrates that any modelling of the drop shape as a cylindrical plug, a pancake or a truncated sphere, is rather far from reality, at least for high-impact Weber numbers. Moreover, any approach based on energy conservation is physically incorrect, since part of the mechanical energy is dissipated in the rim through vorticity (Culick 1960; Frankel & Mysels 1969).

### (b) Surfactant solutions

The present paper deals with drops of surfactant solutions. Upon impact, they experience a high rate of bulk and surface deformations, which drastically modifies the surface tension and the distributions of the surface-active molecules at the surface and in the bulk. The liquid ‘response’ to these extreme circumstances controls the evolution of the impacting drop.

Few studies have been dedicated to the influence of surfactants on drop impact. Mourougou-Candoni *et al.* (1997, 1999), Zhang & Basaran (1997) and Mourougou-Candoni (1998) studied the impact of drops of various surfactant solutions on a planar solid substrate of low surface energy. The collision resulted in a very fast liquid lamella growth and a subsequent relatively slow retraction. It was observed that the surfactants do not influence the growth stage. During this stage, a ‘fresh’ surface is formed very fast, which dilutes the adsorbed surfactants at the surface and increases the value of the surface tension up to approximately the value for water. Surfactants immediately close to the surface adsorb on it, depleting the sublayer adjacent to the surface, which generates a diffusive flux from the bulk towards the surface. The larger the layer depleted of surfactants, the longer the time required for diffusion to replenish the surface and restore equilibrium. The process is now controlled by a dynamic surface tension which is time and position dependent in the surface. The retraction stage is therefore influenced by the dynamic adsorption behaviour of the surfactants. The presence of surfactants usually slows down the lamella retraction to an extent which depends on their nature.

In the same field, Marmottant *et al.* (2000) investigated the effects of surfactants on a water bell resulting from the impact of a steady liquid jet on a small disc-like target. These authors observed an increase in the maximum size of the sheet; their results are in qualitative agreement with the effects discussed below. The effects of Marangoni stresses on the instability-free films were theoretically studied by Pukhnachev & Dubinkina (2006) among others. The effects of surfactant additives on a spray produced by nozzles were studied by Butler Ellis *et al.* (2001) and Ariyapadi *et al.* (2004) who observed that they decrease the drop size and also influence the shape of the spray jet.

In Rozhkov *et al.* (2005, 2006*b*), preliminary results were reported on the impact behaviour of drops of surfactant solutions on small targets. The spontaneous nucleation of holes and the formation of web-like structures in lamellas of dilute and concentrated surfactant solutions have been described. More recently, similar hole nucleations were also observed in a lamella of oil–water emulsion (Rozhkov *et al.* 2004, 2005). Note that fluid webs in liquid sheets of worm-like micelle solutions were also observed by Miller *et al.* (2005) and Thompson & Rothstein (2007). Handge (2005) reported the formation of similar holes and web-like structures when a polystyrene drop in a molten poly(methyl methacrylate) matrix is subjected to equibiaxial extension. Finally, Mitkin *et al.* (2008) observed holes and webs in liquid sheets of polymer solutions at high strain rate extension.

Our purpose is to detail experimentally and theoretically the role of surfactants in drop impact.

## 2. Material and methods

### (a) Liquids

The tested liquids were aqueous solutions of dioctyl sulfosuccinate sodium salt (DOS) and of trisiloxane oxypropylene polyoxyethylene (Silwett L77). DOS was supplied by Acros Organics and Silwett L77 by Crompton Europe S.A. Milli-Q distilled water at room temperature was used. DOS has double-tailed molecules and as its concentrations increase in solution, it forms bilayers that can organize as vesicles (Svitova *et al.* 1995); Silwett L77 molecules, which have a very large polar head with only one chain, form spherical micelles at low concentration and then lamellar aggregates of different kinds as concentration increases (Hill *et al.* 1994). Critical aggregation concentration (cac) is the concentration at which the surfactant molecules start self-assembling. The surfactant concentrations *C* were 1 and 10× cac with cac=0.92 g l^{−1} for DOS and 0.10 g l^{−1} for Silwett L77.

Available data relative to the solutions (Mourougou-Candoni *et al.* 1997; Mourougou-Candoni 1998) are reported in table 1. The variations of the dynamic surface tensions as function of the rate of surface dilation, *α*, for DOS (at 1 and 10× cac), and Silwett L77 (only at 10× cac) solutions are reproduced in figure 2. The main difference between DOS and Silwett L77 in solutions of identical concentrations (in cac unit) is as follows. DOS is a ‘fast’ surfactant, i.e. the solution surface tension relaxes relatively rapidly to its equilibrium value if a fresh surface is formed (Datwani & Stebe 2001). On the contrary, Silwett L77 is very effective, but a ‘slow’ surfactant. The surface tension of its solution relaxes relatively slowly toward equilibrium if a fresh surface is formed.

The shear dynamic viscosities of the solutions measured with a concentric-cylinder viscometer are approximately the same as the one of water (*μ*≈1 mPa s).

It is known that the rheological behaviour of surfactant solutions above critical aggregation concentration can be viscoelastic (Rozhkov *et al.* 2005; Brenn *et al.* 2006). Elongational rheological tests of the present solutions were performed with the liquid filament rheometer (Bazilevskii *et al.* 2001). None of the solutions at the presently investigated concentrations formed thinning capillary filaments, which validates that apparent elongational viscosity is significantly small. Thus, rheological effects are negligible with the present solutions.

### (b) Experimental procedure

The experimental technique and procedure were similar to the ones used in our previous works. Drops were slowly generated at the tip of a capillary connected to a syringe pump in 30–60 s. This time is large enough to ensure equilibrium between the surface-active molecules in the bulk and at the surface. Drops detached from the capillary, fell from height *h*_{i} equal to 65 cm and collided with the carefully polished end surface of a stainless steel cylinder (Ø=3.90±0.05 mm) with a slightly blunt edge (approx. 0.1 mm).

Under typical conditions, *d*_{i}=2.7 mm, *v*_{i}=3.4 m s^{−1},*μ*=1 mPa s and *γ*=*γ*_{w}=72 mN m^{−1} (*γ*_{w} is the water surface tension), the time scale of the process is *t*_{*}=*d*_{i}/*v*_{i}=0.794 ms, the impact Reynolds number is 9180 and the Weber number is 433.

The impact process was monitored using two methods of visualization. The first method was a video recording of top views of the drop impact with a high-speed camera coupled with strobe lighting. The grabbing frequency was 1000 frames per second and the exposure time 1 μs. Only the time interval between two consecutive frames—1 ms is exactly known. It means that the current time (left column in figure 3) is determined with a systematic error of ±1 ms. The advantage of this method is the possibility to observe the evolution of a given element of the drop structure during the whole impact process. Video recording was only used for qualitative observations (figure 3) and crude estimations, but not for precise measurements (Rozhkov *et al.* 2004, 2005, 2006*a*,*b*)

The second visualization method is free from the previous drawback; it has been detailed in Mourougou-Candoni *et al.* (1997, 1999) and in Prunet-Foch *et al.* (1998). Two synchronized cameras, for side and top views, both equipped with a fast electronic shutter, took multiple exposure images of the drops at two and three different times, respectively.

The series of frames of lamellas for various grabbing times properly describes the life and collapse of the same lamella, since the reproducibility of the process is high. The time dependences of the lamella diameter *d*≡2*r*_{l} and all other characteristics of the impact were obtained by processing the top-view frames.

Only this precise second method of impact visualization was used for the quantitative results displayed in figure 4.

## 3. Results

A few examples of top-view observations of impacts of surfactant solutions drops are presented in figure 3. For all the solutions, the drops form a thin liquid lamella with a relatively thick toroidal rim like the water ones (Rozhkov *et al.* 2002).

The lamella diameter *d*≡2*r*_{l} was measured on top-view images as *d*=(*d*_{x}*d*_{y})^{1/2} (the drop image is approximately elliptic, *d*_{x} and *d*_{y} are the diameters measured in two mutually perpendicular directions corresponding to the large and small axes) at several times, *t*, after first contact of the drop with the target. The lamella spread factor is defined as *β*=*d*/*d*_{i} and the dimensionless time as *τ*=*t*/*t*_{*} (*t*_{*}=*d*_{i}/*v*_{i}). The evolution of the lamella diameter is plotted with non-dimensional quantities as *β*=*β*(*τ*) in figure 4, this representation being less sensitive to *d*_{i} and *v*_{i} fluctuations than the simple dependence *d*=*d*(*t*). Experimental data obtained with water drops are also displayed in this figure by a solid line denoted as ‘water’.

Results show that the surface-active additives can modify the impact of water drops on a small target in at least three ways.

### (a) Spider-like lamella

The first observable modification is a more intensive finger formation and a slight increase in the stability of the liquid fingers in comparison with pure water. Secondary jets ejected from the rim look longer than in pure water; they do not quickly disintegrate, but they are transformed into liquid filaments, forming a spider-like structure, as for polymer solutions (Rozhkov *et al.* 2006*a*).

The lifetime of such liquid filaments (Entov *et al.* 1980*b*) is of the order of *t*_{i}+*t*_{v}, where and *t*_{v}∼3*μa*_{0}/*γ* are the inertial and viscous components of the filament lifetime, *a*_{0} is the filament diameter. Taking for water *a*_{0}≈0.3 mm (Rozhkov *et al.* 2004), *μ*=1 mPa s, *γ*=72 mN m^{−1}, we obtain *t*_{i}∼0.6 ms and *t*_{v}∼0.013 ms. The lowering of surface tension *γ* increases the filament lifetime. For DOS at 10× cac, *γ*=27 mN m^{−1} (table 1), *t*_{i}∼1 ms. This value is not much larger than the one for water, but this probably delays the break-up process and favours filament formation.

### (b) Lamella size

The second modification deals with the surfactant influence on the lamella spread factor *β*=*β*(*τ*). Figure 4 shows that for DOS solution at *C*=1× cac, and for Silwett L77 solutions at *C*=1 and 10× cac, *β*(*τ*) is close to the one for pure water. The maximum spread factors for water and for these solutions can be estimated as *β*_{m}∼5.0. It means that the impacts are controlled in general by the dynamic surface tension that is practically equal to the water surface tension. At the same time, the spread factor of the DOS solution at *C*=10× cac significantly exceeds that of the water as *τ*>2, since its maximum value is *β*_{m}∼6.0. The lifetime of the DOS lamella becomes noticeably larger than the one of water under the same impact conditions. This effect is probably due to the decrease of the surface tension of the lamella liquid under the action of surface-active additives. It is well known (Taylor 1959*a*,*b*) that for constant surface tension, the lamella size is larger for the lower ones.

The effect is similar to the retarded retraction observed earlier for drops of fast surfactant solutions impacting onto plane solid surfaces (Mourougou-Candoni *et al.* 1997, 1999; Zhang & Basaran 1997). However, there is some obvious difference between the effect of DOS (*C*=10× cac) on a free lamella and on a lamella on solid surfaces. It increases the maximum spread factor *β*_{m} on the former, but not on the latter (Mourougou-Candoni *et al.* 1997, 1999). It is different for Silwett L77 that does not have any remarkable effect on the retraction rate of the free lamella (figure 4), whereas it significantly retards the lamella retraction on a low surface energy plate (Mourougou-Candoni *et al.* 1997, 1999).

### (c) Spontaneous holes and fluid webs

Finally, a rather unexpected third modification of the lamella, caused by the surfactants, is the spontaneous nucleations of holes in the internal part of the lamellas of Silwett L77 solutions (figure 3).

Rozhkov *et al.* (2002, 2004) observed the collapse of the lamella because of the rim retraction (which is recalled to be equivalent to the propagation of an external rupture wave) and sometimes because of the propagation of an internal rupture wave starting from the target. With Silwett L77 drops, a new type of lamella collapse occurs characterized by the random nucleation of holes, giving rise to a web-like structure that consists of long-living liquid filaments (figure 3). Figure 5 displays the images of three drops of Silwett L77 solution at *C*=10× cac obtained with a short time interval between two images (Δ*t*=0.2 ms), which show the evolution of the holes bounded by a thicker rim in the lamella.

Holes are formed *τ*>3, i.e. when the effective ejection of liquid from the target is over, and that metastable zones (local Weber number *We*≡*ρhv*^{2}/(2*γ*) less than unity) are formed in the lamella. Holes evolve like rupture waves, expand according to the Taylor mechanism of liquid sheet rupture (Taylor 1959*b*), and they obviously grow faster than they are translated by the flow away from the target (i.e. the propagation velocities of the hole rims against the flow are larger than the flow velocity). The expansion velocity of the holes is 2*v*_{r}=|*v*_{+}−*v*_{−}|, where *v*_{r} is the velocity of the hole rim relatively to the liquid, *v*_{+} and *v*_{−} are the rupture propagation velocity vectors along and against the liquid flow, respectively, both relative to the laboratory coordinate system (figure 5). These values can be determined by processing the hole images recorded at the short time interval displayed in figure 5. All values for *v*_{−} are negative, which means that at the observation time, the hole rim propagated against the flow through the lamella faster than it was translated away by it, i.e. the lamella was in a metastable state and it could not resist the hole rim travelling from the periphery.

There is no remarkable influence of the distance from the target on the hole growth. For the lamellas of Silwett L77 solution at *C*=10× cac at *τ*≈5.1, measurements give *v*_{+}=2.67±0.82 m s^{−1}, *v*_{−}=−1.61±0.33 m s^{−1}, the numbers after symbol ‘±’ being the standard deviations. The average propagation velocity of the rupture wave relatively to the film, *v*_{r}=|*v*_{+}−*v*_{−}|/2, is 〈*v*_{r}〉=(|〈*v*_{+}〉|+|〈*v*_{−}〉|)/2=2.14±0.25 m s^{−1}.

The lamella thicknesses can be estimated with the Taylor–Culick formula *h*=2*γ*/*ρv*^{2}_{r} (Taylor 1959*b*; Culick 1960): we obtain *h*=31 μm for water (taking *γ*=72 mN m^{−1}) and *h*=8.9 μm for Silwett L77 (taking *γ*=20.4 mN m^{−1}). The actual lamella thickness, *h*∼10–30 μm, is between these two limits. The average velocity of the holes translation by the moving liquid towards the lamella periphery *v*=|*v*_{+}+*v*_{−}|/2, is 〈*v*〉=(|〈*v*_{+}〉|−|〈*v*_{−}〉|)/2=0.53±0.25 m s^{−1}. Obviously, it is equal to the local flow velocity.

These values do not significantly differ from the ones obtained for water by Rozhkov *et al.* (2002). Under the same impact conditions, it was found *v*_{+}=2.87±0.61, *v*_{−}=−1.91±0.41, *v*_{r}=2.39±0.37 and *v*=0.48±0.37 m s^{−1} at *τ*≈5.5. Therefore, surfactant additives do not modify significantly the ratio *γ*/*h* in the lamellas of the Silwett L77 solution at *C*=10× cac when *τ*≈5.1.

As a hole rim meets another one, a liquid filament forms at the meeting line, and a web-like structure is eventually obtained. Probably, in this case, the lifetime of the liquid filament in the web increases owing to the decrease in the surface tension under the action of Silwett L77 according to the same mechanism that increases the lifetime of the liquid filaments in the spider-like lamella of DOS—§3*a*.

## 4. Surface tension gradient effects (Marangoni effects) in a lamella

In order to understand the surfactant effect on the lamella motion, the unsteady radial sheet flow was considered. The approach is similar to the one used for modelling the water lamella in Rozhkov *et al.* (2004). A cylindrical liquid source, of vanishing diameter *d*_{s}→0 (point source), supplies an axisymmetric free liquid sheet with velocity *v*_{s}(*t*_{s}) and flow rate *q*_{s}(*t*_{s}) defined by the relations (1.1).

The difference is that we do not suppose any longer that the surface tension remains uniform and constant during the process. Actually, surface tension gradients cause a bulk flow that is known as the Marangoni effect (Levich 1962). Since Marangoni stresses can accelerate or decelerate the liquid motion, the Rozhkov *et al.* (2004) model cannot be used for liquids with surfactants.

### (a) Scheme of the motion

The lamella having free surfaces and its thickness *h* being small when compared with all other length scales ∂*h*/∂*r*≪1, the one-dimensional (or slender-body) approximation can be used. Accordingly, the velocity averaged over the normal coordinate to the lamella surface is taken as the characteristic of the flow. The kinetic and dynamic variables in the lamella are only function of the radial coordinate *r* and time *t*: *v*=*v*(*r*,*t*), *h*=*h*(*r*,*t*), *q*=*q*(*r*,*t*) and *γ*=*γ*(*r*,*t*).

Suppose that at the point of ejection *r*_{s}≡*d*_{s}/2→0, the surface concentration of the surfactant is zero *Γ*_{s}(*t*_{s})=0 and the liquid surface tension is equal to that of pure water *γ*_{s}(*t*_{s})=*γ*_{w}. We can neglect here the residual surfactants initially adsorbed at the drop surface before impact because: (i) the lower surface of the lamella is formed by fast ejection of fresh/pure liquid from the surface of the target, and (ii) at the upper surface, the residual surfactants only decrease the surface tension very close to the target (see formula (9) in Rozhkov *et al.* (2006*b*)) owing to the high extension of surface elements during their motion in the lamella. Surface tension gradients generated very close to the target cannot influence the liquid motion because the lamella is relatively thick there (i.e. inertia dominates capillarity as expressed by the high local Weber number).

### (b) Equation of motion

The equation of motion follows from the integral mass and momentum balance for the motionless control volume *W*_{1} displayed in figure 6 in a one-dimensional approximation (see also Entov *et al.* 1980*a*; Entov 1982; Yarin 1993; Marmottant *et al.* 2000) is
4.1
and
4.2

Introducing the flow rate *q*=2*πrhv* and the dimensionless variables *τ*=*t*/(*d*_{i}/*v*_{i}), *Y* =*r*/*d*_{i}, *V* =*v*/*v*_{i}, *Q*=*q*/(*πd*^{2}_{i}*v*_{i}/6) and *Ω*=(*γ*_{w}−*γ*)/*γ*_{w}, the system of equations (4.1) and (4.2) can be transformed into the Cauchy–Kovalevskaya form (Rozhdestvenskii & Yanenko 1983) as
4.3
and
4.4
The dimensionless surface pressure *Ω*=0 for the drops of pure water, and 0<*Ω*≪1 for the ones of surfactant solutions.

### (c) Surfactant transfer kinetics

The surfactant mass balance for the motionless control volume *W*_{1} displayed in figure 6*a* obeys the equation (see also Edwards *et al.* 1991; Breward *et al.* 2001; Voinov 2008)
4.5
where *Γ* is the surface concentration of the free surfactant molecules (monomers), and *J* is the diffusive flux of surfactant from the bulk to the surface.

Suppose further that, owing to the high rate of surface dilation, the surface concentration *Γ* and the subsurface bulk concentration *c*_{sub} are small compared with the saturation surface concentration and the bulk concentration *c*, respectively (Mourougou-Candoni *et al.* 1997, 1999; Daniel & Berg 2003). Two approximations follow from this statement.

— The surface concentration

*Γ*is related to*γ*by the asymptotic form of the Langmuir–Frumkin equation of state (Adamson 1976) 4.6 where*R*is the gas constant,*T*is the temperature,*ξ*=1 for uncharged surfactant (Silwett L77) and*ξ*=2 for charged surfactants (DOS) (Davies & Rideal 1963).— The diffusive flux

*J*is of order of the flux*J*_{0,}which is the flux from the concentrated bulk to the surface still free from surfactants.*J*_{0}depends on the surfactants, surfactant bulk concentration, diffusion coefficient, etc. The physical mechanism of transport of surfactant molecules from bulk to surface is discussed and illustrated by fig. 8 in Rozhkov*et al.*(2006*b*). Actually, it was possible to ignore the precise mechanism of surfactant flux since we could independently obtain the flux magnitude from the measurement of the dynamic surface tensions by means of Rehbinder’s maximum bubble pressure (MBP) method apparatus (Dukhin*et al.*1995). The mechanism leading to the ‘dynamic surface tension’ is the same in this method as in our experiment since it results from the competition between the flux of surfactants from the bulk to the surface and the decrease of surface concentration of surfactant owing to the surface fast dilation. Considering the flow at a high rate of surface dilation, the balance of surfactants at a deforming surface element*S*can be estimated as Because the MBP method assumes the surface tension is constant during the bubble growth,*Γ*is also assumed to be constant. Consequently, 4.7 where*α*=*S*^{−1}*dS*/*dt*is the surface expansion rate.

The surfactant flux *J* is defined by the diffusion coefficient *D*, the length scale *δ* of the bulk surfactant gradient and the free monomer concentration difference between the bulk and the subsurface *c*−*c*_{sub}: *J*∼*D*(*c*−*c*_{sub})/*δ*=*J*_{0}−*Dc*_{sub}/*δ* (Rozhkov *et al.* 2006*b*). Using this estimate, the Langmuir–Frumkin state equation and the Langmuir isotherm , valid for a dilute solution (where *k* is a material constant), equation (4.7) yields an approximate relation for the surface tension dependence on *α*
4.8
where and *θ*=*γ*_{w}/*ξRTJ*_{0}.

In the asymptotic case *γ*→*γ*_{w} as , equation (4.8) yields , which is the slope of the curves as 1/*α*→0 in figure 2. It can be found by best fitting the experimental points to formula (4.8) with three parameters *p*_{1}, *p*_{2} and *θ*. Fits for the three liquids are presented by solid lines in figure 2.

The parameter has dimensionality of time and it characterizes the rates of the surfactant adsorption to the surface and the relaxation of the surface tension to the equilibrium level. It is a material constant of the liquid and it can be referred to as surfactant relaxation time. Values of *θ* are reported in table 1. Only DOS at 10× cac is fast enough to significantly decrease the surface tension during the impact process with a time scale of order of *t*_{*}=*d*_{i}/*v*_{i}=0.794 ms. Comparison of the time scale of the process *t*_{*} and the surface tension relaxation time *θ* is presented in table 1 by the dimensionless parameter *G*=*t*_{*}/*θ*, which characterizes the surfactant adsorption rate and is analogous to the reverse Deborah number in rheology.

Thus, the surfactant flux *J* depends on the measurable surfactant relaxation time *θ* as
4.9
Equations (4.5), (4.6) and (4.9), which relate the change of the surface tension to the history of the surface deformation, are constitutive equations for the surface rheology.

Substituting relations for *Γ* and *J* given by equations (4.6) and (4.9) in equation (4.5) and introducing dimensionless variables, we obtain the equation of surfactant transfer as
4.10
where *G*=*d*_{i}/*v*_{i}*θ* is the ‘reverse Deborah number’.

The system of three equations (4.3), (4.4) and (4.10) describes the change of three variables *V* (*τ*,*Y*), *Q*(*τ*,*Y*) and *Ω*(*τ*,*Y*) during the liquid motion in the lamella: it is called below system I.

### (d) Steady sheet flow

In a first step, suppose that a steady flow models the motion of the liquid lamella resulting from the steady plane radial ejection of surfactant solution (as in Clanet 2007). The steady sheet radial flow is described by system I if it is supposed ∂/∂*τ*=0 in all equations and the boundary conditions are taken in the form: *V*_{s}=*V*_{0}, *Q*_{s}=*Q*_{0}, *Ω*_{s}=0.

System I can then be solved analytically:
4.11
where *B*=12*G*/*We*_{i}*Q*_{0}.

The features of the relations (4.11) were discussed by Rozhkov *et al.* (2005, 2006*b*). In particular, relations (4.11) predict the decrease of the velocity and of the surface tension with increasing values of *Y* . They also predict the growth of the lamella maximum size with *G* as it was also observed in the experiments of Marmottant *et al.* (2000). Moreover, a steady flow exists only up to a certain critical point *Y*_{*}(*B*) beyond which it becomes impossible. At the critical point, d*V* /d*Y* =−d*Ω*/d*Y* = −∞. Introducing d*Y* /d*V* =0 in equation (4.11), the values of the critical velocity *V*_{*} and of the critical coordinate *Y*_{*} are obtained as , *Y*_{*}=0.61*V*_{0}/*B*^{1/2} and *Ω*_{*}=*G*/2*B*^{1/2}. *V*_{*} does not depend on *B*, whereas *Y*_{*} and *Ω*_{*} decrease as *B* increases. The relations (4.11) cannot describe the flow at *Y* >*Y*_{*} because the condition ∂*h*/∂*r*≪1 is not satisfied in the vicinity of the critical point; thus, the present quasi-one-dimensional approximation cannot work at high gradients.

### (e) Unsteady sheet flow

Suppose now that an unsteady sheet flow models the motion of the liquid lamella. System I is solved numerically with the following boundary conditions:
4.12
4.13a
4.13b
and
4.14
The present boundary conditions differ from the ones equation (1.1) by the relation (4.13*b*), which introduces a monotonic decrease of the flow rate *Q*_{s} after *τ*_{s}=3 as the effective ejection of liquid from the target is over (Rozhkov *et al.* 2004). This function *Q*_{s}(*τ*_{s}) describes the smooth change of the flow rate and satisfies the volume conservation of the ejected liquid: . Such types of boundary conditions provide a quantitative description of the motion of the liquid that was ejected before *τ*_{s}=3, and gives a qualitative one after *τ*_{s}=3.

The liquid particle trajectories *Y* =*Y* (*τ*), the distributions of surface tension (figure 7*a*) and sheet thickness (figure 7*b*) are plotted in figure 7 for *We*_{i}=433 and adsorption rate *G*=0.1. This value of *G* approximately corresponds to the fast surfactant DOS, 10× cac.

Numerical trajectories are presented by thin lines, whereas thick lines represent the trajectories d*Y* /d*τ*=*V* defined by equation (4.11) for the steady flow. Both trajectories are very close. It means that the trajectory of each liquid element is defined by its starting conditions *V*_{s} and *Q*_{s} and does not depend on the motion of other liquid elements. It was shown (fig. 8 in Rozhkov *et al.* (2004)) that the trajectories are straight lines for pure water (*G*=0). However, for *G*=0.1 the trajectories become slightly convex, which means that the liquid element is decelerating. The liquid deceleration is caused by the surface tension gradients that are shown in figure 7*a* by changing the background. The surface tension gradient is higher for the late stage of the lamella motion when the deceleration is also higher.

The dimensionless thickness distributions, defined as *H*=*h*/*d*_{i} (figure 7*b*), do not significantly differ from the ones obtained for pure water (fig. 8 in Rozhkov *et al.* (2004)).

### (f) Capillary-inertial instability in liquid sheet with variable surface tension

To study the stability of the sheet motion to surface tension gradients, the standard boundary conditions (4.12)–(4.14) were disturbed by imposing small harmonic perturbations on the flow rate , which models thickness fluctuations at the time of ejection. The numerical trajectories and thickness distributions are plotted in figure 8*a*. Comparison with similar data for standard boundary conditions (figure 7*b*) shows that the flow rate oscillation causes oscillations of the lines of constant thickness. The amplitudes of oscillations remain approximately the same as *Y* increases for pure liquid *G*=0. However, they become remarkably larger for liquids with surfactants. Figure 8*b* displays three-dimensional distributions of relative thickness *H*_{1}, which is the ratio of the dimensionless sheet thickness *H*(*b*) at a given point (*τ*,*Y*) obtained with the disturbed boundary condition and the dimensionless thickness *H*(*a*) at the same point obtained with the standard one. The data display a significant growth of thickness perturbations for surfactant solutions. Thus, the numerical calculations predict the flow instability which is expressed as the growth of thickness disturbances in the lamella.

Moreover, for a pure liquid *G*=0, the liquid trajectories are uniformly distributed as in the case of undisturbed boundary conditions. Indeed, once ejected, a liquid element continues its motion with constant velocity independent of the ejection flow rate (or thickness) which was disturbed in our test. However, for the surfactant solutions, as *Y* increases the trajectories converge for a liquid element which was ejected at elevated flow rate (peak of the sinus wave) and they diverge for a liquid element which was ejected at lower flow rate (trough of the sinus wave). The convergence of the trajectories increases the sheet thickness, whereas the divergence decreases it, i.e. the flow amplifies the thickness disturbances, which induces instabilities.

The mechanism of the observed instability is similar to the capillary-inertial instability of the drop rim causing the splashing of a drop impacting a solid (Rozhkov *et al.* 2002, 2004). Indeed, the liquid elements in the surfactant solution sheet are subjected to a deceleration caused by Marangoni stresses ∂*γ*/∂*r* (figure 9*a*). Under these circumstances, the more massive thick parts of the sheet (‘drops’) decelerate less than the lighter thin parts (‘bridges’). The liquid in the bridge flows into the drop, thickening the thicker parts of the sheet and thinning the thinner ones.

Besides, these disturbances can be amplified further by aerodynamic influence of the ambient air as a result of the Kelvin–Helmholtz instability. Eventually, the process can end with the formation of holes in the sheet.

At last, the discovered capillary-inertial instability does assist the growth of any thickness perturbations thereby promoting sheet break-up. Figure 9*b* shows holes formation in a lamella. Light spots observed on the lamella surface indicate thickness disturbances but not bending ones, because back lighting was used. These thickness disturbances can be caused by the present capillary-inertial instability of the surfactant solution sheet.

It remains unclear, however, why holes were not observed with the DOS solutions, one of which is fast (DOS, 10× cac), whereas the other (DOS, 1× cac) has the same surface tension relaxation time as Silwett L77, 10× cac (table 1). It may be due to strong disjoining pressure effects, which prevents contact of opposite sheet faces and stabilizes the lamella flow.

### (g) Effect of surfactants on the rim trajectory

The whole scheme of calculations of radial film flow with a rim follows the one introduced by Yarin (1993). The equation describing the rim trajectory *Y*_{l}(*τ*) is obtained from the momentum balance of the liquid in a small motionless control volume *W*_{2} (figure 6*b*) (Rozhkov *et al.* 2002, 2004). Its dimensionless form is
4.15
where , with *m* the rim mass, *Y*_{l}=*r*_{l}/*d*_{i}, with *r*_{l} the rim radial coordinate (figure 1), and *Q*=*Q*(*Y*_{l}), *V* =*V* (*Y*_{l}) and *Ω*=*Ω*(*Y*_{l}) are the dimensionless flow rate, velocity and surface tension close to the rim, respectively.

The first term in equation (4.15) corresponds to the change of momentum of the rim in the control volume *W*_{2}. The second term corresponds to the change of the momentum of the film in *W*_{2}. The first term, on the right-hand side, accounts for the input of surface tension in the momentum balance, and the second one accounts for the momentum flux into *W*_{2}.

Initial conditions for equation (4.15) are *τ*=0, *Y*_{l}=0 and *M*=0. The velocity, flow rate and the surface tension in the vicinity of the rim *Y*_{l}(*τ* are deduced from the solution of the system I as *V* (*τ*,*Y*_{l}(*τ*)), *Q*(*τ*,*Y*_{l}(*τ*)) and *Ω*(*τ*,*Y*_{l}(*τ*)). The rim mass *M* is defined by the mass balance , where *τ*_{s} is the time of the start of the trajectory which goes through point *τ*, *Y*_{l}(*τ*), i.e. *τ*_{s}=*τ*_{s}(*τ*,*Y*_{l}(*τ*)). The mass balance does not account for the liquid losses caused by the splashing of the rim.

The results of the numerical integration of equation (4.15) are compared with the experimental data in figure 4. Input parameters *We*_{i}, *G* correspond to experimental conditions reported in table 1. Calculations for DOS, 10 cac were carried out for the domain of stable numerical calculations. The agreement is excellent for the first half of the impact; however, a certain discrepancy exists for the second part. Obviously, the discrepancy is caused by the liquid losses which are neglected in the model (4.15) as it is discussed in Rozhkov *et al.* (2004). The level of losses depends on the magnitude of initial perturbations of the rim and cannot be predicted.

The numerical solutions of equation (4.15) are presented in figure 10 for *We*_{i}=121, 433 and different magnitudes of the surfactant adsorption rates *G* that are close to the present experimental parameters—table 1. Data show that the lamella maximum diameter and the lamella lifetime increase with *G*, as it was predicted earlier by Rozhkov *et al.* (2005, 2006*b*). However, this effect is remarkable only for fast surfactants (*G*>0.1) such as DOS, 10× cac and negligible for slow surfactants (*G*<0.1) such as DOS, 1× cac and Silwett L77, 10× cac. This theoretical conclusion compares well with experimental data.

## 5. Conclusions

Fast surfactant additives increase the size and the lifetime of the liquid lamella resulting from the impact of drops on a small target because of the fast decrease of the surface tension. They could hinder splashes that generate spider-like liquid structures at the end of the impact. Paradoxically, spontaneous nucleations of expanding holes in lamellas of surfactant solutions were also observed. The holes finally disintegrate the lamella that gets a web-like liquid structure. The observed hole formation is likely to be connected with the capillary-inertial instability generated by Marangoni stresses.

## Acknowledgements

The authors are grateful to P. Adler and J.-F. Thovert for their help with the numerical solution of the system I. A.R. took part in the present and previous studies during his stays at the Laboratoire de Physique des Matériaux Divisés et des Interfaces. He thanks the Centre National de la Recherche Scientifique and the Université de Paris-Est Marne-la-Vallée for the support of these visits.

## Footnotes

↵† Present address: A. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, 101(1) Prospect Vernadskogo, Moscow 119526, Russia.

- Received January 11, 2010.
- Accepted March 15, 2010.

- © 2010 The Royal Society