Soft Matter

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Wetting-mediate

Institut de Science des Materiaux de Mulhou

Haute Alsace, 15 rue Jean Starcky, 68057 M

haidara@uha.fr

† Electronic supplementary informa10.1039/c4sm01579g

Cite this: DOI: 10.1039/c4sm01579g

Received 17th July 2014Accepted 23rd September 2014

DOI: 10.1039/c4sm01579g

www.rsc.org/softmatter

This journal is © The Royal Society of

d collective tubulation andpearling in confined vesicular drops of DDABsolutions†

Hamidou Haidara*

Whether driven by external mechanical stresses (shear flow) or induced by membrane-active peptides and/

or proteins, the collective growth of tubules in membranous fluids has seldom been reported. The pearling

destabilization of these membranous tubules which requires an activation of the shape distortion, often

induced by optical tweezers, membrane-active biomolecules or an electrical field, has also rarely been

observed under mild experimental conditions. Here we report such events of collective tubulation and

pearling destabilization in sessile drops of a didodecyl-dimethylammonium bromide (DDAB) vesicular

solution that are confined by a surrounding oil medium. Based on the wetting dynamics and the features

of the tubulation process, we show that the growth of the tubules here relies on a mechanism of

“pinning-induced pulling” from the retracting drop, rather than the classical hydrodynamic fingering

instability. We show that the whole tubulation process is driven by a strong coupling between the bulk

properties of the ternary (DAAB/water/oil) system and the dynamics of wetting. Finally, we discuss the

pearling destabilization of these tubules under vanishing static interface tension and quite mild tensile

force arising from their pulling. We show that under those mild conditions, shape disturbances readily

grow, either as pearling waves moving toward the drop-reservoir or as Rayleigh-type peristaltic

modulations. Besides revealing singular non-Rayleigh pearling modes, this work also brings new insights

into the flow dynamics in membranous tubules anchored to an infinite reservoir.

Introduction

Stress-induced tubulation of membranous uids and objects(vesicles, cells, bilayers, and lamella) or polymers, either ofbiochemical (peptides/proteins) or mechanical (hydrodynamicshear) origin, is a rather well known and experimentallyinvestigated phenomenon.1–7 This is also the case for thepearling destabilization of these membranous tubules that hasso far required the application of either a photochemicaltweezers or an electrical eld,8–14 unlike falling uid tubes,spun bers or conned polymer threads2 for which thispearling instability grows spontaneously. So far, the tubulationof membranous objects (cells and vesicles) has mostly involvedthe shear-induced extrusion and pulling of discrete tubulesthrough capillaries,3,4 although collective tubulation eitherdriven by shear-ow or induced by bioactive molecules inliposomes and supported membranes has been reported.6,7

Here, we show and explore, possibly for the rst time, theoccurrence of such a collective tubulation event in sessile drops

se (IS2M), UMR 7361-CNRS/Universite de

ulhouse Cedex, France. E-mail: hamidou.

tion (ESI) available. See DOI:

Chemistry 2014

of aqueous didodecyl-dimethylammonium bromide (DDAB)surfactant solutions that are conned in a surrounding oilmedium, in the so-called “two liquid phase” wetting congu-ration. Of particular interest we show that these tubulationevents are primarily driven by a strong coupling between thebulk properties (phase diagrams) of the ternary “DDAB/water/oil” system on the one hand and the interface (wetting)behavior of the conned aqueous DDAB drop on the otherhand. Beyond the structural complexity of these systems con-taining both lamella and vesicles, we show that the tubuleshere grow collectively through a “pinning-induced pulling”process of brillar ngers from the retracting contact line,based on the features of the tubulation (point-pinning, pullingkinetics, randomness and side-branching mode of thetubules), at the difference of classical hydrodynamic ngeringinstabilities of advancing wetting fronts.15–19 Besides thatwetting-driven collective tubulation, we show that the tubulesthat form can develop shape instability under unusually quitemild conditions, as compared to most reported studies whereexternal disturbance sources (mechanical, photochemicaltweezers, membrane-active molecules, or electrical eld)2,8–14

are required to overcome membrane bending rigidity, andinduce the shape instability. Indeed, the pearling instabilitieshere develop within the microtubules under vanishing staticinterface tension and residual tensile stress induced by their

Soft Matter

Fig. 1 Evaporation–drying of a DDAB vesicle solution drop at 0.3 wt%on Si wafer in air: (a) early tomid-stage of the evaporation–drying, withthreadlike to filamentous vesicles developing and extending from theconfined edge of the drop; (b) collected vesicles network at the centerof the drop in the very late drying stage. The contact angles in (a), rightafter spreading and on receding, are both <10� on the cleaned Siwafers in chloroform, under sonication.

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pulling from the reservoir drop to which they remain anchored.Depending on the structure of the tubules (size, membranerigidity, and internal uid viscosity) and their anchoring to thesubstrate, the shape instabilities of the microtubules areshown to develop through pearling modes and patterns thathere present at least two distinctive and possibly unrevealedfeatures. The rst of these features is the occurrence of movingpearls which, at the difference of the propagating “pearlingfront” analytically described by Powers et al.,12 do not break upinto sequential daughter droplets. Instead, these pearls keepmoving as a whole, along and with the tubule, to eventually re-enter the macroscopic drop (complete suction), driven by therelaxing tensile energy of pulling. The second distinctivefeature of this pearling destabilization is the growth of theinstability into a single or a few pearl-waves that are more orless stationary, with characteristic features (pearl radius andperiod) that deviate from those expected from the classicalRayleigh destabilization theory.20 It is worth noting here thatsince the marginal stability analysis by Powers et al.12 based onthe reference works of Bar-Ziv et al.9,10 on laser induced pearl-ing of membranous tubules, very little evidence toward topo-logical transitions of pearling patterns in these systems hasbeen reported experimentally. The following sections deal,principally through a phenomenological approach, with thiscollective tubulation process and the related singular pearlingmodes that should contribute toward better understandingthe still open issue of uid dynamics in membranousmicrotubules.

ExperimentalMaterials

Didodecyldimethylammonium bromide (DDAB), a modelsynthetic double-chain surfactant21–25 that easily forms uni-lamellar and multilamellar vesicles (ULV and MLV respec-tively)21–25 at low concentration in water (DDAB/W), was used forall experiments. In this dilute regime where the solution lieswithin the rst lamellar (La1) domain of the DDAB/W phasediagram, two concentrations were used for determining thedependence of the tubulation and pearling destabilization onDDAB concentration and viscosity: 3 g L�1 �0.3 wt% and15 g L�1 �1.5 wt%. These two concentrations were prepared in25 mL glass vials by adding, respectively, 60 mg and 300 mg ofDDAB to 20 mL of reverse osmosis (Milli-Q) water warmed to�50 �C. This mixture was rst gently shaken manually and thensonicated for a few minutes in a Branson ultrasonic cleaner(model 1050E-MTH). At these low concentrations, DDABeasily dilutes in water to spontaneously form unilamellar andmultilamellar vesicles (see Fig. 1). This solution was stored at20 � 0.1 �C and 55% RH for 1 day before the rst use, andbetween uses over a maximum of one week. The DDAB ofanalytical grade (purum) was purchased from Fluka and used asreceived. For experiments involving the deposition of DDABsolution drops under oil, hereaer referred to as “under-oil”experiments, analytical grade dodecane (C12), hexadecane (C16),and squalane (C30) from Aldrich, and a (C12–C16) mixture at50 vol% were used.

Soft Matter

For all experiments, the reference wetting/drying experimentin air and “under-oil” tubulation experiments, 2 � 2 cm2 piecesof (100) oriented 380 mm thick single side polished siliconwafers (Si) bearing their native oxide (�2 nm) were used assubstrates. Prior to wetting experiments, these pieces werecleaned by sonication in warm (45 �C) chloroform for 10minutes and dried under nitrogen ow, aer which we showedthat this cleaning process producing a reference water contactangle qw � 50� in air also produces the optimal “under-oil”wetting conditions leading to tubulation.

Experimental procedure

For the reference wetting experiment in air, a drop of �10 mLwas deposited on the substrate and le to freely evolve throughspreading and evaporation–drying. For “under-oil” tubulationexperiments, the substrate was placed in a glass cell (0.8 cmheight and 4 cm diameter) containing the oil over 0.7 cm inthickness. Before the deposition of the aqueous suspensiondrop, the whole system was placed under a video-microscopeinterfaced with a computer and a video acquisition soware(Open Box Ver. 1.8) for imaging and video acquisition. Thevideo-microscope comprises an Olympus BX 60 optical micro-scope equipped with a DIC (differential interference contrast)device, and a COHU solid state CCD camera operating at 25images per s. Aer a gentle agitation of the stock solution, a�10 mL DDAB suspension drop was equally gently deposited,and the wetting dynamics was followed or recorded, startingwith a 5� magnication long working distance objective.Depending on which dynamics and/or phenomenon is observed(dewetting/fragmentation of the drop, tubulation and pearling),two other objectives (magnication of 10 and 50, long workingdistance) have been used. All the experiments were performedunder usual laboratory conditions of 22� 1 �C and 40� 5% RH.

The images are either captured individually during thedynamics or extracted from recorded movies using VirtualDub(1.9.11) and ImageJ (1.47d) free soware. Image analysis for thesize measurement, the growth kinetics and migration velocities(“displacement vs. time”) of the structures (tubules and pearls)is performed using ImageJ, aer setting length scales for allimages, from pixels to metric units (mm).

This journal is © The Royal Society of Chemistry 2014

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Results and discussionReference wetting experiment in air

Fig. 1 shows a typical result of the preliminary reference wettingexperiment of a dilute DDAB suspension drop in air, respec-tively in the early stage of its deposition (Fig. 1a) and the latestage of its evaporation–drying (Fig. 1b), on the cleaned siliconsubstrate. Besides ascertaining the presence of large vesicles bythe dense aggregation networks that they form in the late dryingstage, Fig. 1 also reveals the formation of long and largethreadlike-to-tubular structures, possibly from collapsed andmerging vesicles and lamella within the conned border of theevaporating drop. The intuition of preventing the evaporation–drying of the drop by placing it under oil, while maintainingsimilar spreading and edge connement that allow the abovestructural transitions, is at the basis of the present work andfollowing sections.

Under-oil wetting and tubule growth

The results shown in Fig. 2a–f are representative of the earlystage wetting dynamics observed in “under-oil” wetting experi-ments, although these dynamic events (dewetting and frag-mentation of DDAB drop) are strongly modulated by the chainlength and viscosity of the surrounding alkane oil. These resultsare represented here for squalane (C30) and hexadecane (C16);these two oils were purposely chosen for their viscosity whichallows capturing the early stage wetting behavior of theconned DDAB drop by damping the ow instability and frag-mentation of the drop (Fig. 2), making easier the observation ofthe early stage wetting events, in contrast to dodecane (C12) oil.Clearly, Fig. 2 shows the remarkable difference in the wettingbehavior of the conned DDAB drop, depending on the size(chain length) and viscosity of the surrounding oil. Under

Fig. 2 Evolution of the confined DDAB solution drops (0.3 wt%) upon deUnder hexadecane, the early spreading is accompanied by a dewetting anholes (a) which remain definitely filled by the hexadecane (b and c). Notestand inside the surrounding oil as microemulsion-like droplets. Undershowing only a few (if any) dewetting holes (arrow in (d)) that are slowly reof the dewetted holes); (e) partially refilled hole and (f) complete refilling

This journal is © The Royal Society of Chemistry 2014

hexadecane, this is characterized at the drop center by a muchhigher density of dewetting and fragmentation holes (Fig. 2a)that remain in addition denitely lled by the oil (Fig. 2b and c).Under squalane the drop which spreads slowly with a reducedequilibrium extent only shows a few (if any) dewetting holes(Fig. 2d) that are completely relled here (Fig. 2e and f) by theback ow of the DDAB drop (healing of the dewetting holes).One can account at least partly for this oil-dependent behaviorof the conned DDAB drop by the bulk properties of ternary (oil/water/DDAB) systems that have been extensively studied.22,23

Indeed as shown in these ternary DDAB/W/oil phase diagrams,n-alkane oils and particularly short chain alkanes (n # 14) havea higher penetration in DDAB/W mixtures,23 accounting for themicrostructural variety of the ternary mixtures. In our system (Sisupported vesicular drop under oil), we found that the samebulk properties were effective, driving a more or less rapid andstrong destabilization of the drop upon its deposition, throughdewetting and fragmentation (Fig. 2). As expected from thelength dependence of the penetration rate (miscibility) of the n-alkanes,22,23 we showed that the strength of that penetrationsignicantly increases from C30 to C12, using squalane (C30),hexadecane (C16) and dodecane (C12), with a dewetting andfragmentation of the drop which are quite instantaneous andmore violent for shorter chain alkanes (Fig. 2a–c, vs. Fig. 2d–f).This dewetting/fragmentation process, which develops mostlyaround the center of the highly spread and thinned drop, drivenby the penetration of the surrounding oil creates an outwardpressure wave, Poutward, that further pushes and forces thespreading of the preserved rim-like drop. This forced outwarddisplacement may be increased by the hydrostatic pressureDr(oil/solution)ghdrop over the conned drop (or rim) of thicknesshdrop, although this term of negligible and comparable magni-tude for hexadecane and squalane cannot account for the

position under hexadecane (from (a to c)), and squalane (from (d to f)).d fragmentation in the center of the drop that creates a high density ofin this case (c) that some of the fragments of the aqueous DDAB dropsqualane the drop spreads slowly with a reduced equilibrium extentfilled (arrows in (e and f)) by the back flow/receding of the drop (healingand recovering of the drop stage.

Soft Matter

Fig. 4 Panels (a), (b) and (c) show a characteristic growth sequence ofthe membranous tubules along the receding contact line of theaqueous DDAB vesicular drop (0.3 wt%) under hexadecane, along withthe corresponding patterns of pearl arrays. The arrows (1, 2 and 3)indicate from (a) to (c) locations where a branching between thegrowing tubules takes place as they merge. The formation of thesebranching patterns by a “tips-merging” process, vs. standard tip-splitting is schematically drawn in (d).

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observed drastic difference between these oils regarding thetubulation process. As is visible in Fig. 2d–f, in the surroundingC30 oil of higher viscosity, the initial events of dewetting andfragmentation of the conned drop lead to the formation of athick pressurized rim that also takes more time to relax. Indeed,as compared to C12 and C16, the viscous damping in C30 stronglyslows down the outward displacement of the rim (drop edge),the velocity of which at rst order at the surrounding oilviscosity halkane is26 Vrim � (Poutwardd/halkane), where d stands forthe characteristic size of the ow diffusion layer in thesurrounding oil. Because of this low forward displacementvelocity and relaxation of the drop edge in C30 alkane oil, therim which is compressed by the outward pressure rather relaxesby an out-of-plane protrusion to form a thicker rim. For thislong C30 alkane (and to a lesser extent for C16), the rim slowlyrelaxes with a characteristic timescale �(halkane/Poutward) fromthe end of the dewetting/fragmentation event, with no notice-able retraction. For C12 and for C16 on the other hand, as thepressurized drop edge relaxes, which lasts from a few tens ofseconds, the spreading ceases and the rim-like border begins torecede, revealing tubules that are pulled from the drop edge, asFig. 3a and b, Fig. 4a–c and ESI S1 and S2† show for both C12,(C12–C16) mixture and C16 systems. In this respect, it is worthnoting that a primary requirement for the growth of the tubulesis that the drop retracts, more or less signicantly, aer theexcess spreading induced by the outward pressure (ow) thatfollows its deposition. As shown in Fig. 3 and Fig. 4a–c, thepulling and growth of the tubules also require anchoring pointswhich may either be embedded in the drop as emulsion-likedroplets, or come from the surface as dust particles visible atsome tubule extremities (see S1 and S2†), or from topographicalor chemical defects on the surface27 as this is possibly the casein Fig. 4a–c. Though both types of surface defects (topographicand chemical) are supposed to exist a priori on the substrate,random chemical defects arising from the non-homogeneousremoval of organic contaminants by solvent cleaning (see theExperimental section) are more probable candidates on thepolished Si substrates of nanoscale roughness. As the contact

Fig. 3 Tubulation front and tubule fine structures shown for a DDABsolution drop (0.3 wt%), respectively, (a) in 50 vol% (C12–C16) mixture,and (b) in C12 where a single tubule was captured at a resolutionshowing the wall structure of the membranous tubule on a growingpearl (bulge).

Soft Matter

line (CL) slowly recedes, the membranous tubules (see the netubule structure in Fig. 3b) that are henceforth anchored bythese point-defects at their external extremity are further pulledout from the rim. Although we could not access the early stageof the formation of the tubules within the conned drop edge, itis likely that this tubulation process can take place from eitherof the three following structures: (1) the embedded threadlikevesicles similar to those visible in air-dried drops (Fig. 1a), (2)the microemulsion-like droplets (Fig. 2c) resulting from thedewetting/fragmentation of the DDAB drop, and that sit on thesubstrate, surrounded by the oil (Fig. 2c), or (3) the membra-nous DDAB layer that coats the interface of the rim-like dropwith the surrounding oil. Indeed, the pulling of the embeddedthreadlike vesicles or the shear-induced threading of micro-emulsion-like droplets within the retracting drop edge, throughthe DDAB layer coating the (drop/oil) interface, can both lead tothe observed tubulation process. Similarly, the direct pulling ofthe interface DDAB layer coating the receding rim-like drop andenclosing the aqueous DDAB solution can equally lead to theobserved tubulation process, although some of our microscopyimages strongly suggest either of the rst two processes (seeFig. 5 and caption). Finally, regardless of the nature of thepinning-defects (surface defects27 or drop-embedded objects),

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Fig. 5 Time-growth of the tubules and pearling instability in 1.5 wt% DDAB drops under a 50 vol% alkane oil mixture (C12–C16), a case where thetubules remain attached and stretched between their anchoring points during the whole dynamics. Panels A represents the growth sequencefrom t ¼ 0 to the complete arrest of the pulling (receding) front at t ¼ 96 minutes (1 h 36 min). Panels B corresponds to the same tubules andpearling patterns at a higher magnification, after 20 hours’ aging and a swelling-induced morphological reconstruction. The last panel visiblysuggests a continuation of the tubule by drop embedded structures that cross the pulling border and go beyond.

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the features of this tubulation process (visible pinning points,pulling of tubules visible only on retraction, random tubuledistribution, side-branching mode of tubules) clearly point to a“pinning-induced pulling” process, rather than a classicalhydrodynamic ngering instability. Indeed, although most ofthe ngering structures formed at wetting fronts arise fromdigitation instabilities which are of purely dynamical natureand virtually independent of pinning-defects, the ngeringprocess based on the pinning-induced pulling of uidsegments28 or macromolecules by the retracting contact line hasbeen known for a long time and used as a technologicalpatterning tool (stretching of DNA strands for instance).29–31

Besides the fact that they are observed under forced wetting/ow conditions (spinning drops, inclined substrate, pressure,density, thermal gradients), an essential feature of hydrody-namic ngering instabilities is that they have a characteristicand predictable wavelength, regardless of the nature of thegradient that drives them.15–19 A second important feature ofthese hydrodynamic instabilities is that they systematicallyoccur on spreading, ahead of the advancing front, but seldom, ifnot never, on receding, as attested by the literature.15–19

Furthermore, when taking place in wetting lms, these hydro-dynamic ngering instabilities have been shown to arise from athickening of the advancing front, and the destabilization ofthisthickened front creates the ngers.16,17,19 Incidentally, thethickest moving drop fronts that form under squalane in our

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experiments (Fig. 2d–f) never lead to any ngering or tubulationinstability. And even when observed with surfactant-like solu-tions where it is driven by Marangoni ow, this hydrodynamicngering instability occurs systematically on spreading, form-ing in that case tree-like fractal ngers protruding ahead of theapparent contact line,32 instead of well-dened discrete tubules.Similarly, Rayleigh–Taylor (gravity) and Saffmann–Taylor(viscous) instabilities which also develop on advancing uid/uid fronts are characterized by ngering patterns of predict-able and well dened wavelengths and shapes,18,33 as shown byHele-Shaw cell experiments. Finally, while none of the abovefeatures of purely hydrodynamic-driven ngering is observed inour study, our experiments clearly show the growth of thetubules, exclusively on retraction of the wetting front, support-ing a tubulation mechanism based on a “pinning-inducedpulling” process.28–31

It is worth recalling here that the above results were allobserved on cleaned Si substrates (see the Experimentalsection), leaving open the question of how far these “under-oil”wetting dynamics and related tubulation are inuenced by thesurface? This question was at least partially answered by thecomplementary “under-oil” experiments we performed on bothas-received (non-cleaned) silicon wafers with a reference watercontact angle qw � 68� � 3, and hydrophobized substrate(methyl-terminated hexadecyltrichlorosilane monolayer coatedsilicon)26 with a water contact angle qw� 105� � 2. Indeed, while

Soft Matter

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tubulation was observed on the as-received Si wafer, thoughwith less reproducibility compared to the solvent cleaned wafer,no tubulation was observed on the hydrophobized Si wafer.Furthermore, neither the early stage dynamic instabilities(dewetting, fragmentation of the conned DDAB drop.) nor aretraction of the drop front were observed on the hydro-phobized substrate. Instead, on deposition of the DDAB drop onthe hydrophobized surface, a fast and strong spreadingdevelops, thinning instantly and strongly the drop. Mostimportantly, unlike non-cleaned and solvent cleaned Si wafers,this fast and strong spreading drives on the hydrophobizedsubstrate the well-known hydrodynamic ngering instability,32

with very dense treelike ngers propagating ahead of thespreading front (ESI S3†). The surface state and wetting of thesubstrate therefore inuence the whole “under-oil” wettingdynamics and tubulation process, although more systematicstudies would be required to better understand and describedthis inuence, beyond the sole effect of surface defects.

Once the tubules are pulled from the receding drop edge,they lie under the alkane oil either as unperturbed tubulesegments or as a pearl necklace upon fragmentation, as shownin Fig. 3 and Fig. 4a–c. Because of interface energy minimiza-tion, it is required that the outer layer of the tubule exposes thedouble-alkyl chains of the DDAB surfactant to the surroundingalkane oil. As a result, the membranes of tubules should havean odd number of layers (1, 3.), regardless of whether they arepulled from the interfacial DDAB layer that coats the recedingrim-like drop or from the drop embedded threadlike vesicles. Itis worth noting in this regard that the vanishing equilibriuminterface tension of �10�4 N m�1 reported in the literature34

and �10�3 N m�1 measured in this work between the aqueousDDAB solution (1.5 wt%) and hexadecane by the Wilhelmy platetensiometry supports well that interface free energyminimization.

Beyond the growth of the tubules from the drop reservoirand their high peripheral density, this wetting-assisted tubula-tion also reveals structural features (side-branching) that areubiquitous of crystallization/solidication and fractal growth ingeneral. As Fig. 3a and Fig. 4b and c show, bifurcation-like sidebranches appear along primary tubules at many places. Butunlike standard bifurcation or tip-split branching in growingcrystals or moving uid fronts that are either driven by interfaceenergy or pressure uctuations, the branching here resultssimply from the merging of two tubules that grow at locationsthat are close enough on the rim. At the crossing point of thegrowth directions of these non-parallel tubules, they merge intothicker “root-tubules” that keep growing as the CL recedes, asschematically depicted in Fig. 4d. This side-branch formationbased on “tips-merging” is a much less oen observed processas compared to the classical bifurcation (tip-splitting) thatdevelops in an opposite direction.

Pearling instabilities and structures

For the conditions of our experiments where the tubules areformed in dilute DDAB solution drops, under low viscosityalkane oils (C12, C16), the tubules evolve essentially through

Soft Matter

three paths which are determined by the balance between theirpinning strength at the anchored extremityWP, and the tensions arising from their pulling out of the drop border. Thesescenarios also determine with the DDAB concentration theaverage lifetime of the tubules from a few minutes to hours,from their growth to complete pearling fragmentation andsubsequent deliquescence of the resulting droplets. Althoughirrelevant regarding the pearling phenomenon, one of thesetubule evolution paths is that in which the tubule starts toretract in the early stage of the growth and re-enter the dropbefore pearling, driven by the relaxing tensile stress s > WP

(depinning), and the pressure gradient VP between the tubuleand the drop reservoir. For all these retracting tubules that re-enter the drop, the tubule contraction rate VTC and the localreceding velocity of the drop edge VD are such that VTC > VD.

The two last cases are the relevant tubule evolution dynamicsthat are hereaer considered for pearling instabilities. In bothcases, the tubules grow proportionally to the local recedingvelocity of the drop (i.e. as�VDt) over timescales that are beyondthe early stage (ESI S1 and S2†). However, they differ in theirlong-term behavior that determines their pearling mode. In therst case, the tubules keep growing between the pinning(anchoring) point and the drop reservoir until the recedingstops, and along with the growth of the tubule the pearlingprocess also develops concomitantly, leading generally tostationary pearls. This case requires that the tubule be rmlyanchored at its pinned extremity to enable a balance of tensionand pinning satisfying s < WP. In the second case, the tubule isunpinned or broken at its anchoring point, either before thereceding has stopped or shortly aer, leading to a combinedpearling and global tubule suction toward the drop reservoir.For this latter case, the balance between the pulling tension andthe pinning strength (respectively tubule cohesion) should besuch that s $ WP at a certain time of the pulling to observe thedepinning and global tubule retraction. The following is aphenomenological description of these rather unusual pearlingdynamics along membranous tubules that are either stretchedbetween the pulling drop reservoir and the defect pinningpoint, or retract from their free end (upon depinning) towardsthe reservoir.

From a theoretical standpoint, the growth of an undulatoryshape disturbance in an initially uniform, cylinder-shapedmembranous tubule requires at leading order that the bendingrigidity be balanced by the tension along the tubule (capillaryand externally exerted). Keeping these leading order terms inbending rigidity and tension, the free energy of the tubule ofunperturbed radius Rt and length Lt is thus written:

E y (kB/2Rt2 + ginterf)A � fLt (1)

where kB is the bending modulus, ginterf the static interfacetension, f the tensile force of pulling from the drop reservoir,and A ¼ 2pRtLt the area of the membranous tubule. As Fig. 4and the ESI S1 and S2† show, the pearling instability growsconcomitantly with the pulling in most cases, meaning that thevolume of the destabilizing tubules is not conserved. Theminimization of E with respect to Rt and then Lt in eqn (1) gives

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a pulling tensile force f¼ (2pkB/Rt), an interface tension ginterf�(kB/2Rt

2), and an equilibrium relation for the stretched tubule,14

T* ¼ (2pkB/Rt) + 2ginterfRt ¼ (1 + 2p)(kB/Rt) (2)

where T* stands for the apparent tubule tension. No matterfrom which of the embedded threadlike vesicles or shear-threaded microemulsion-like droplets the tubules are pulledfrom, the rst term of the apparent tension can be related androughly computed from the viscous shear exerted on the tubuleat the conical pulling border (see Fig. 3) as,

2pkB/Rt y (hsol + hoil)lVt (3)

hsol and hoil are, respectively, the viscosities of the aqueousDDAB solution and surrounding oil, Vt the average pulling rateof the tubule, and l the length of the pulling cone. This gives anestimate of the bending modulus of the tubule,

kB ¼ (Rt/2p)(hsol + hoil)lVt (4)

Using quoted values of hoil � 3 � 10�3 Pa s for C16 and hsol �1.5 � 10�3 Pa s for both concentrations of 0.3 and 1.5 wt%DDAB,25,35 a tubule radius Rt ¼ 5 mm, a size of the pulling conel ¼ 70 mm and a pulling velocity Vt ¼ 0.2 mm s�1 determinedfrom experiments shown in Fig. 5A (1.5 wt% DDAB), a bendingmodulus kB � 1.5 � 10�19 J was estimated, an order of magni-tude which is typical of double-tail surfactant bilayers. Let usnote that this estimate is of course compatible with the previ-ously discussed criterion of interfacial free energy minimizationof the tubule, which requires an odd number (likely 3) of layersin its membrane exposed to the alkane medium. Indeed, esti-mated bending rigidities of unilamellar and multilamellarvesicles give values 5–10kBT and 10–15kBT, respectively36 whichare close enough, while covering the above estimated value of1.5 � 10�19 J for tubules with membranes likely consisting of 3DDAB layers.

Whether the pearls are stationary or moving, the pearlingdestabilization is driven here by these two competing curvatureand tension terms which set the threshold for the shapedestabilization of the stretched tubule. But besides the aboveand rather reasonable estimates, what this tubulation processhere shows is that even under vanishingly low tensions, staticinterface (ginterf) and exerted pulling (f) tensions, pearlingdestabilization develops well along these membranous tubules,as discussed further in the following section.

Pearling on stretched tubules

Fig. 5A show the sequence of tubulation and pearling in a 1.5wt% DDAB drop under a C12–C16 mixture for which tubulesremain stretched between the reservoir and the anchoring pointduring the pearling process. The sequence of Fig. 5A shows thetime-growth of the tubules along with the pearling instability,leading to a sequence of stationary pearls along the stretchedtubule, as is remarkably more visible on the main tubulebranches shown in Fig. 4. The dynamics was captured over 24hours, long aer the whole system was completely stabilized

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and morphologically turned to a gel-like uid aer swelling(Fig. 5B). (Incidentally these panels of Fig. 5B show at bothmagnications a morphology of the pulling zone that stronglysuggests that the tubules may form from drop-embeddedstructures (threadlike-vesicles, microemulsion-like droplets), asalready hypothesized in the paper.) As can be seen from Fig. 5(and Fig. 4 too), the growth of the pearls occurs only for acharacteristic pulled length of the tubule which depends on thetubule radius and tension, and sets the period (wavelength) ofthe pearling sequence. Within the frame of Rayleigh-typecapillary instability,20 one expects pearling patterns with awavelength l scaling as (l/Rt)� 9, and a pearl size (radius) of theorder Rpearl � 2Rt, Rt being the unperturbed tubule radius. Thefreshly formed patterns in Fig. 5 (5A, 90 min) have averagepearling periods (l/Rt) � 30, and pearl sizes Rpearl � 4Rt, whichnotably deviate from those expected for a classical Rayleigh–Plateau pearling event, although the former was shied down-ward to (l/Rt) � 15 by the swelling of the pattern at longer time(Fig. 5B). In contrast, Fig. 4 which also shows pearling patternscharacterized by a sequence of stationary pearls along tubulesthat remain stretched rather displays average pearling period(l/Rt) � 10, and pearl size Rpearl � 3Rt that are much close tothose expected for a Rayleigh pearling instability. These resultsthus suggest a dependence of both the pearl density and thecharacteristic features of the pearling pattern (l and Rpearl vs. Rt)on the DDAB concentration, although this needs to be furtherformalized through a more systematic concentration-depen-dent study, keeping identical the surrounding alkane oil. On theother hand, these results denitely show that whatever is the(two) DDAB concentration or the surrounding oil (C12, C12–C16

mix, C16), the pearling period along those tubules that remainstretched is not constant here, but increases in average towardthe reservoir, possibly with the pulling tension and the size ofthe residual tubule being pulled out (Fig. 5 and 4). To the best ofour knowledge, this dilation of the pearling period alongmembranous tubules that are being pulled out from a reservoiris all but trivial. Indeed, this result which accounts for agenerally expected dependence of the pearling features (l,Rpearl) on the size of the tubule, for constant size tubules (Lt andRt) and tension, is here recovered under dynamical conditions,for a stretched tubule of variable residual size and pullingtension. This result too in particular would deserve moresystematic concentration-dependent investigation in futuredevelopments of these works.

Apart from these stationary pearls that develop on fullystretched tubules as observed mostly with DDAB drops of 1.5wt% (slightly more viscous), these pearling events also involvepearls that move and coalesce along “globally” retractingtubules. These pearling events that occur mostly with 0.3 wt%DDAB solutions are described below.

Pearling on retracting tubules

Propagating “pearling front” involving the continuous motionof a unique pearl wave from one extremity of the tubule, withoutbreaking into sequential daughter drops, is known anddescribed in the literature.12 At the difference of this

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propagating “pearling front”, the pearling process we describehere is a global or sequential motion of the pearls that migrateand coalesce along the tube before re-entering the drop reser-voir, leading eventually to the complete suction of the tubule.Fig. 6 illustrates such pearling dynamics and its kinetic evolu-tion for 0.3 wt% aqueous DDAB drops, under C12. The directedmotion of the pearls along the globally retracting tubules towardsthe drop is driven by the pressure gradient between the dropreservoir and the adjoining tubule segment (Ptubule > Pdrop–rim),the restoring tensile force (eqn (2)) of the tubule aer depinningor rupture, and the main drag force on the retracting tubule andmoving pearls arising from the surrounding oil. The migrationvelocity is determined in Fig. 6b for three pearls, from the“displacement vs. time” snapshots given in Fig. 6a. As observedin most cases, the rst pearl in contact with the rim remainsvirtually stagnant up to the end of the migration process where itgenerally merges with the last moving pearl (rV1r in Fig. 6b),before emptying eventually in the drop. For the two other pearls,the measured velocities are comparable and of the order of

Fig. 6 (a) Panel showing the time growth and migration of the three nascof a DDAB of 0.3 wt%; (b) displacements and average velocities of the pearpearl-waves, the case treated here was chosen for being the most simpleother propagating pearls, depending on both their geometric parameteattached (floating) tubules between the extremities. Note: the terminal acsimply from its merging with another pearl, or with the rim (for the last

Soft Matter

0.8 mm s�1. An estimate of this migration velocity can also bederived from a balance of the above three forces that drive themotion of the pearls along the globally retracting tubules.

Fdriving � p(Rt)2DP + 2pkB/Rt (5)

Fdrag � hoilRtVt (6)

Vpearl/tubule � p(Rt2DP + kB/Rt)/hoilRt (7)

In eqn (5), the pressure difference DP is solely related to thatof Laplace capillary pressure DPc � g*

oil/tubule(1/Rt � 1/Rrim),although the involved effective radius of the rim at the dropedge is hardly accessible here in the system conguration. Butsince Rrim [ Rt, the driving pressure for the “bulb/tubule”motion nally amounts at rst order to DPc � g*

oil/sol(Rt)�1.

Using the average tubule and pearl radii Rtub (�3 � 1 mm)and Rpearl (�7� 2 mm) for this system (Fig. 6), a driving pressureof the tubule and pearl suction DP z 100 mPa between the rim

ent pearls (shown by arrows) towards the rim of the macroscopic dropls. Although it presents the general feature of the observed propagatingand tractable case. In particular, the measured velocities may differ forrs (Rtub and Rpearl) and attaching conditions: firmly adhering or looselyceleration which is observed systematically for any moving pearl arisespearl), driven in that case by the high capillary pressure difference.

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and the adjoining tubule segment was found by equating themeasured velocity of �0.75 mm s�1 (Fig. 6b) to the expected one[eqn (7)]. Then equating DPc above to this estimated drivingpressure gradient DP z 100 � 10�3 Pa, one arrives at a value ofthe (tubule/oil) interface tension, g*

oil/tubule � 3 � 10�4 mN m�1,a value which is of the order of the one that can be derived fromthe minimization relationship (eqn (1) and (2)), ginterf

� (kB/2Rt2) � 10�6 mN m�1. Although negligibly small when

compared to our experimentally measured static interface of 1mN m�1 (104 order of magnitude), such virtually nil or too lowinterfacial tensions to be measured have been reported fordroplet-based microuidic systems involving a water/hex-adecane couple containing two kinds of surfactants, and wherepearling-like instability readily develops.37

Conclusion

In this report, we have studied the wetting behavior of sessiledrops of a model double-chain surfactant (DDAB) solution inthe so-called “two liquid phase” wetting conguration. Undersurrounding alkane oils of appropriate chain length, thesedrops can develop dynamic instabilities that drive the collec-tive growth of membranous tubules. We showed that thiswetting-driven tubulation proceeds here by a pinning-inducedpulling of embedded structures from the retracting drop edge,rather than by the classical hydrodynamic ngering insta-bility. Besides, we showed that these tubules can also evolvethrough a pearling fragmentation under quite mild conditions(vanishingly low tubule tension), at the difference of mostreported pearling destabilization of these membranoustubules that oen require external activation sources (opticaland photochemical tweezers). This work thus provides amodel experimental system for the fundamental study ofwetting-driven collective tubulation and pearling in membra-nous uid drops, a phenomenon that has so far been seldominvestigated and reported in the literature. Naturally, a nercontrol over the tubule morphology, their stability and relatedpearling modes and patterns will require more systematicstudies (DDAB concentration, surrounding alkane oils). Inparticular, a relationship between the DDAB concentration(structures and viscosity of the solution), the lifetime andstability of the tubules vs. pearling would be desired for thefull understanding of these phenomena, as strongly suggestedhere by DDAB drops of 1.5 wt% which showmore reproducibleand stable tubulation and pearling patterns (Fig. 5) vs. 0.3 wt%DDAB drops. Beyond these fundamental aspects, such para-metric studies may open up the possibility of using thesetubular structures to spatially organize embedded species(nanoparticles, biomolecules) within the tubules and associ-ated pearling patterns that we showed to swell on ageing athigher DDAB concentration (Fig. 5B). Yet, these results showthat the system already contains, virtually, all the potentialfeatures for the model study of the collective tubulation, thetubulation morphology as well as their pearling destabiliza-tion modes and patterns.

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Acknowledgements

The author is grateful to Dr Laurent Vonna for his help with thenal scaling and optimal conversion of the two supportingvideo movies, and Philippe Kunemann for his kind help in thesurface and interface tension measurements of the DDABsolution and DDAB/hexadecane systems.

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