A drop impact study of worm-like viscoelastic surfactant solutions

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Abstract

In this paper, we investigate the impact dynamics of drops of complexed surfactant solutions. Dilute to semi-dilute solutions of rod-like to worm-like micelles of equimolar cetyltrimethylammonium bromide (CTAB) and sodium salicylate (NaSal) in water (CTAB–NaSal (1:1)) have been dropped onto a smooth hydrophobic surface. The variations in impact behaviour of each solution are discussed in terms of the dynamic surface tension and viscoelastic properties. Increases in the concentration of CTAB–NaSal (1:1) in solution resulted in enhanced reduction in the maximum height achieved during recoil, even though the maximum diameter reached during spreading was equivalent for all solutions over the range of concentrations explored. The equivalence in the maximum diameter during spreading is due to the loss of fluid structure at high extension rates, whilst the resulting reduction in rebound tendency is due to a rapid restructuring of the fluid post impact. The drop impact of such complexed surfactant solutions provides significant insight into the contributions of a ‘dynamic’ fluid microstructure to the rebound behaviour of dilute ‘living polymer’ solutions.

Introduction

The dynamics of drop impact are relevant to many industrial processes, including spray coating, ink-jet printing and agricultural chemical dispersion. Solution characteristics may be tuned to invoke control over drop impact behaviour. This is, however, only achievable if the controlling variables of drop behaviour are known a priori.

When a drop impacts a solid surface, the drop first spreads until it reaches a maximum diameter, then it rapidly shrinks back to the region of first impact, finally reaching an equilibrium shape at long times. The spreading and recoil of the drop represents a continual trade-off between inertial centrifugal forces (associated with the mass (or size) of the drop and its impact velocity), capillary centripetal forces (which depend on the surface tension and the solid surface characteristics), gravitational forces and viscous dissipation [1]. During spreading, the inertial forces decrease due to surface creation and viscous dissipation. The maximum diameter is reached when surface tension forces overcome inertia and the kinetic energy is zero. At this point of the drop impact ‘splashing’ may occur, identified by the production of secondary droplets from the drop periphery. Once this maximum diameter is reached, capillary centripetal forces become dominant and induce retraction of the drop. The competition between centrifugal inertial forces and centripetal capillary forces also create capillary waves that travel back to the center of the drop. Capillarity results in the flow of the solution back to the region of initial impact, or recoil, and the combination of flow and capillary surface waves can result in partial or total drop ‘rebound’ [2]. Partial and total rebound has been observed for many Newtonian fluids, including some spectacular examples for water impacting on a hydrophobic surface [3]. Dewetting of the surface is significantly favoured by hydrophobic surfaces and hence rebound is far more prevalent on such surfaces. Overall, whether a drop will have enough recoil energy to rebound is thus determined by the relative amount of this recoil energy consumed through viscous dissipation during flow back to the region of impact, the dynamics of surface relaxation, and the energy losses associated with dewetting. An equilibrium hemispherical drop is only obtained once all kinetic energy from the initial impact is consumed and the solutions equilibrium surface tension will determine the final shape.

Investigations into the drop dynamics of Newtonian solutions have provided great insight into the respective contributions of inertia, viscosity, surface tension and the surface characteristics of the solid upon which the drop is impacting. Viscosity dampens the magnitude and progression of the capillary waves from the maximum diameter of the drop to the center during the dewetting of the surface, dissipating the available recoil energy and thus significantly retarding rebound [4], [5], [6]. To enhance wetting and spreading, and reduce the energy available for rebound, surfactants may be added to solutions to reduce the equilibrium surface tension. Unfortunately, recent outcomes suggest that lowering the equilibrium surface tension of the solution via the addition of a surfactant normally offers little advantage over the time frame associated with spreading (∼5–10 ms for a 2–4 mm diameter drop) [2]. This is due to the drop experiencing such large surface dilation rates that it is completely out of equilibrium when it reaches its maximum diameter. Most surfactants cannot effectively saturate the newly created surface (diffusion limited) within this short time frame. The dynamic surface tension (DST) has thus been stated as a more useful and relevant parameter in this initial stage of drop impact [2], [3].

Recently, Crooks et al. have provided insights into the advantages of polymer addition as an effective alternative measure for reducing rebound and controlling drop impact behaviour. Firstly, Crooks et al. have shown that splash on hydrophilic surfaces is suppressed by the addition of small amounts of polymer (at concentrations below the overlap concentration c*). This observation is a result of the flow regime which is close to pure biaxial extension during spreading, whilst during necking of the peripheral filaments just prior to drop breakup it is pure uniaxial extensional flow. Under both of these conditions, the polymer chains are forced to ‘extend’ or ‘stretch’. Such extended conformation of the polymer coils result in a higher local hydrodynamic viscosity and thus increased resistance for the solvent molecules or solution flowing past them. The sum of these local viscosity increases around each coil produces an effective elongational viscosity increase for the bulk solution. Such increase in the bulk viscosity within the spreading drop and also within the necking region of the secondary drops (at the periphery of the crown) has two results. By decreasing the effective Reynolds number within the necking region of the filaments, extension of the polymers within the neck allows it to sustain much greater stresses, working in opposition to surface tension and pinching is effectively stopped [7]. Overall, this increase in viscosity during extensional flows increases the stability of the neck. Surface tension recovers the drops plus filaments into the main body of the solution on the surface. Breakup at the periphery is thus suppressed. At the same time, with the suppression of drop breakup at the periphery, the enhanced viscous dissipation within the bulk drop spreading across the surface means that more of the available kinetic energy is damped during spreading for these solutions. The scenario is very complicated, and the relative contributions of the enhanced surface stability of the filaments reducing secondary droplet breakup and enhanced viscous dissipation (due to the extension of the polymers) in the bulk flow of the drop resulting in loss of the available kinetic energy for splash still requires further investigation.

Using the same set of model elastic fluids (Boger Fluids), Crooks et al. have also shown that rebound on hydrophobic surfaces is significantly suppressed by the addition of small amounts of flexible polymer. Crooks et al. expanded on the initial work of [8] by providing further insights into the molecular weight and concentration effects of polymer addition in controlling drop impact behaviour on hydrophobic surfaces, relating the reductions in the maximum recoil height and retraction velocity to characteristic polymer parameters [3]. The resulting reduction in rebound with polymer addition is again due to the elongational viscosity increases during spreading and recoil (although during spreading inertial and surface contributions dominate) enhancing the consumption of the energy available for recoil, as well as dampening the maximum amplitude of the capillary waves in the centre of the drop. The combined effect of enhanced viscous dissipation and dampening of the capillary wave amplitude resulting from the polymer extension in the flow field retards rebound significantly.

Of course, splash and rebound are similarly retarded for Newtonian drops when viscosity is increased, ([9] and [5], respectively) however, increasing the shear viscosity has negative effects on the application of these solutions as more energy is required to spray them. Hence the addition of small amounts of polymer can offer significant advantages, depending on the nozzle geometry [10], as the shear viscosity remains near equivalent to that of water but drop splash or rebound can be significantly retarded.

The synergistic effects of associative polymer–surfactant mixtures has recently been investigated by Cooper-White et al. [11], illustrating that the dynamic extensional characteristics of solutions are in fact more important than equilibrium extensional viscosities when polymers are within solution. Polymers and associative polymer–surfactant mixtures may both be utilized to invoke non-Newtonian behaviour or dynamic extensional viscosity increases during recoil. These recent insights allow further tunability of drop dynamics and rebound suppression.

Dilute solutions of aqueous ionic and non-ionic surfactants usually behave as Newtonian liquids with viscosities only slightly greater than that of the solvent. Previous drop impact investigations involving such simple surfactant solutions have utilized standard Newtonian base solutions (i.e. water/glycerol mixtures) with single surfactants added, over various ranges of concentration [2], [3], providing insight into the interplay of the dynamics of surface adsorption of these surfactants and drop impact. These studies showed that little benefit may in fact result from such surfactant addition. However, there are also some surfactant systems that display far more complicated flow behaviour than those Newtonian solutions previously investigated in drop impact. Over the last two decades, there has been substantial work investigating the rheological behaviour of viscoelastic surfactant systems [12], [13], [14], [15], [16], [17], [18], [19]. For example, the phenomenon of viscoelasticity can be invoked in cationic surfactant systems simply by the addition of certain counter ions [20].

The surfactant system chosen for this study is equimolar cetyltrimethylammonium bromide (CTAB), sodium salicylate (NaSal) and water, a well studied system within the available literature (e.g. [21], [22]). NaSal complexes with the CTAB molecules to form stiff ‘living’ chains [18], and as the concentration is increased from 0.1 to 10 mM, the complexed surfactant undergoes transitions from near spherical micelles to rod-like to highly extended, worm-like chains. The solutions used in this study are only slightly above the cross-over concentration (∼7.5 mM) [16], [17], and as such, all of the solutions tested within this study below 7.5 mM are considered to be hydrodynamically dilute, with the 7.5 and 10 mM verging on the semi-dilute regime where micelle–micelle interactions begin to play a role. However, hydrodynamic interactions are expected to still be quite small up to at least 10 mM. The dilute approximation is thus not unreasonable for all solutions investigated within this study, especially considering the low values of the plateau modulus expressed in these solutions up to 10 mM [17]. At higher concentrations (e.g. ≫20 mM), living networks are formed. The behaviour of these complexed surfactant solutions are thus best contrasted with the behaviour noted for an effective increase in molecular weight for true polymer systems, to a point where the chains are large enough to suffer long–range interactions or ‘entanglements’. The solutions then exhibit true viscoelastic behaviour with relaxation times equivalent to those seen for true polymer systems [15]. However, the viscosity versus shear stress behaviour is still very complex at concentrations prior to the inception of such entanglement effects [17]. Most importantly, there is one major difference between normal polymer solutions and ‘living’ polymer solutions, which may be of great relevance to the outcome of these drop impact events. Surfactant aggregates are in thermodynamic equilibrium with individual surfactant molecules in solution, resulting in a dynamic exchange of surfactant molecules from one micelle to another neighbouring micelle or into bulk solution.

This rapid transfer of molecules is referred to as the micellar (fast) relaxation time, which may vary from nanoseconds to microseconds. If such systems undergo a rapid pressure or temperature fluctuation under quiescent (i.e. zero flow) conditions, we may observe a much slower relaxation event, which is often referred to as the micellar (slow) relaxation time, representing the time required for the formation of new micelles from a supersaturated bulk solution of surfactant molecules, in other words, the micellisation time. This time event, depending on the surfactant system in question, may vary from microseconds to milliseconds. Such fast and slow relaxation times have been observed for many systems utilizing pressure-jump and/or temperature-jump experiments [14].

In the case of rod-like to worm-like aggregates, these aggregates are also in thermodynamic equilibrium with individual molecules in solution, but in this case the dynamic exchange of surfactant molecules to and from neighbouring rods or worms leads to continuous breaking and reformation of these rod-like to worm-like aggregates. This characteristic relaxation event is referred to as the break (or reformation) time tb [15], and was shown by Kern et al., who studied CTAB–KBr rod-like micelles, to be of similar magnitude to the micellar slow relaxation time gained from pressure-jump or temperature-jump experiments for this system [23].

Unfortunately, as far as the authors are aware, such temperature-jump data is not available for equimolar CTAB–NaSal over the concentrations ranges explored in this study to allow comparison, nor a direct confirmation of the approximate break times. However, from flow birefringence measurements of dilute solutions (i.e. absent of entanglement effects) of tetradecyltrimethylammonium bromide and sodium salicylate (TTASal), a very similar system to that investigated here, estimates of the break time are 6.6–0.34 s for concentrations ranging from 2.5 to 10 mM respectively at 20 °C [24]. These estimates were determined assuming that the micelle breakup rate parallels with the shear rate at which shear thinning occurs in these dilute systems. Due to the system being perturbed faster than the micelles can break and reform, the rod-like micelles undergo orientation in the direction of flow as stiff rod-like entities, resulting in the solutions showing shear thinning characteristics (akin to the observed orientation of hematite rods in high shear flows [25]). We may use these estimates of the break time as our initial reference. The existence of these additional dynamics have significant impacts on the stress relaxation mechanisms within these dilute solutions, and therefore have to be considered as they may affect the outcomes of our drop impact study.

Cates was the first to provide a model for the stress relaxation behaviour of ‘worm-like’ or ‘living’ polymer systems. He assumed simple reaction kinetics for such systems of ‘living’ polymers, incorporating both the conventional reptation model for normal polymers [26] and the reversible scission and recombination of these living chains [15] as mechanisms of stress relaxation. As predicted by Cates and observed in numerous experimental studies, [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24] when the so-called break time tb is longer than the reptation time tR, reptation theory adequately describes the resultant rheological behaviour and stress relaxation. However, when the micelles are breaking faster than the reptation time, there are numerous breaking and reformation processes within the time scale of the observation, leading to a pure, monoexponential stress decay and the solutions can be simply described by a single-mode Maxwell fluid model [20]. Maxwellian behaviour only occurs far beyond the effective cross-over concentration, ‘c*’, for these systems where the ‘living polymer’ chains have long-range interactions. For example, the CTAB–NaSal system has been shown to mimic a model single-mode Maxwell fluid at high concentrations (≫20 mM) where significant chain ‘entanglement’ or transient polymeric networks are expected.

The CTAB–NaSal solutions used in this study are kept at a ratio of 1:1, leading to maximum growth of the rod-like surfactant complex being achieved as a result of charge stoichiometry. This ratio also produces the maximum lifetime of the chain prior to a breakage/reformation event [13], [21]. The Rouse or reptation relaxation times are, however, still expected to be far greater in magnitude than the break time and hence micelle breakage and reformation dynamics will dominate the rheology in high flow rate processes, such as drop impact. However, the outcomes of drop impact experiments utilizing these self-assembling surfactants or surfactant-salt complexes has not yet been investigated, and as a result, the relevance of the aforementioned time constants of such solutions to drop impact behaviour is unknown. Such a system is the subject of this paper, with the outcomes showing that surface relaxation via adsorption is not the only property of interest to the formulator when using surfactant-based systems.

Section snippets

Experimental

Drop impact was investigated using a high-speed drum camera capable of 1000 frames s−1 with a resolution of 3700×2700 dpi. A detailed description of the experimental setup and method of image capture, scanning and data analysis of the images produced is described elsewhere [3]. Drops of approximately 2.4 mm diameter were produced from a 0.38 mm outer diameter needle, fed from a reservoir, via a thin capillary tube. The reservoir was kept at a set height above the nozzle to ensure a constant drop

Results of drop impact experiments and discussion

Fig. 5, Fig. 6 shows the development over time of drop diameter and height of all equimolar CTAB:NaSal drops over the whole concentration range shown in Table 1 on a parafilm surface. As can be seen from Table 1, all solutions have different terminal viscosities, ranging over four orders of magnitude, but very similar DST behaviour and density, and impact the parafilm at the same velocity (∼2.15 m s−1).

The variation in D(t)/D0 with time for all of the surfactant solutions is shown in Fig. 5.

Conclusions

The impact of rod-like to worm-like surfactant drops on dry hydrophobic surfaces was studied experimentally with the aid of a high-speed drum camera. The test solutions had similar dynamic and equilibrium surface tension behaviour and density, but differed in terminal shear viscosity by four orders of magnitude and show very different shear stress versus rate behaviour. The maximum diameter reached by all solutions was equivalent, as was the rate of expansion and contraction (velocity of

Acknowledgements

The non-Newtonian fluid mechanics program at The University of Melbourne has been funded by a Special Investigator Grant and in part by the Particulate Fluids Processing Centre. The authors are grateful to Dr Won-Jong Kim, for granting permission to reproduce dynamic and steady viscosity data for these systems and for his valuable insights and entertaining discussions, and the reviewers for their helpful comments and suggestions.

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