Article pubs.acs.org/Langmuir
From Permeation to Pore Nucleation in Smectic Stacks Jean-Luc Buraud, Olivier Noel̈ , and Dominique Ausserré* Molecular Landscapes, Biophotonic Horizons Group, CNRS-UMR 6087, Université du Maine, Le Mans, Sarthe 72000, France S Supporting Information *
ABSTRACT: The last stage of the spreading of a stratified droplet in the odd wetting case is the evolution from a trilayer to a monolayer, that is, vanishing of the last bilayer in the stack. We studied it in the case of 8CB smectic liquid crystal on a hydrophilic surface. Receding of the last bilayer is accompanied by formation of pores in it, which appear in the outer part of it. From analysis of real-time experimental observations of this phenomenon, we demonstrate that the dislocation loops which border these pores are not located at the same height in the trilayer stack as the dislocation lines that border the bilayer. Also, careful analysis of our results using a recently developed theoretical approach of smectic liquid nanodrop spreading strongly suggests that pore nucleation is triggered by differences in chemical potential between adjacent layers, which contrasts with the classical scheme where it is attributed to lateral tension along the layers.
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monolayer in the trilayer stack.13,15 This monolayer may be the upper one, as in Figure 1a, or the lower one, as in Figure 1b.17
INTRODUCTION Pore formation in lamellar fluid membranes is a major research task in both fundamental and technological research fields. On the fundamental side, nucleation of small holes plays a crucial role in the transport of molecules across biomembranes.1−3 On the technological side, understanding the pore nucleation mechanisms in various lamellar systems is essential for making stable and regular membrane stacks as required in surface patterning and biochip engineering.4,5 Although mainly studied in lipid stacks, pore formation also occurs in other lamellar systems,6,7 such as diblock copolymers or smectic liquid crystals. Hole formation is generally attributed to rupture of a membrane when submitted to lateral tensions.3,8−12 It may be torn and holes form, allowing membrane relaxation. In this article, based on the experimental observations of the spreading of a smectic liquid crystal trilayer, we propose an alternative mechanism in which pore nucleation is induced by differences in chemical potential between adjacent layers. In other words, we claim that pore nucleation may result from stresses normal to the lamellae rather than from in-plane stretching. An essential step to support this scheme is to demonstrate that pores, which involve two juxtaposed layers, do not form at the height where they were intuitively expected to. A 4-n-octyl-4′-cyanobiphenyl (8CB) smectic liquid crystal nanodrop deposited on a hydrophilic surface such as an oxidized silicon wafer exhibits an edge structure made of molecular terraces parallel to the substrate and lying on a trilayered surface precursor film.13 When the volume of the drop is very small, complete wetting is observed.14 During spreading, the upper terraces empty into the lower ones and the structure reduces to a dense circular trilayer surrounded by a single monolayer. The central trilayer is the sum of a bilayered patch and a dense liquid-like monolayer. The whole structure was elucidated by X-ray reflectivity,15 ellipsometry,14 and SEEC experiments.16 The trilayer is surrounded by a gas-like monolayer which is topologically connected to the dense © 2013 American Chemical Society
Figure 1. Internal defect line structure at the bilayer edge of an 8CB smectic nanodrop in the late stage of the spreading; a) embedded bilayer; b) floating bilayer. Arrows represent the various material flows involved in the spreading process. 1, 2, and 3 are the monolayer indices. Interfaces between aliphatic and polar parts of the molecules are drawn with plain lines, interfaces between facing similar parts with dashed-dotted lines. Differences in molecule tilting with layer to solid distance are not represented.
The bilayer is characterized by the presence of a dislocation line at its edge. However, the height at which this dislocation is located and therefore the exact internal structure of the trilayer is unknown. Figure 1 depicts the two possible schemes where the bilayer is embedded in the monolayer, as in Figure 1a, or floating over, as in Figure 1b. Here starts what we may call the late stage of the spreading. The bilayer shrinks, whereas the outward gas-like monolayer spreads over the solid. At the end of the process, only the latter remains on the surface. In the following, we will focus on the shape of the shrinking bilayer during late-stage spreading. With 8CB, two behaviors may be observed:16 (i) for temperatures close to the solid/smectic transition (21.5 °C), the bilayer shrinks without particular event;16,17 (ii) at higher temperatures (above 22.5 °C), holes with a bilayer thickness appear in the Received: February 6, 2013 Revised: June 14, 2013 Published: June 14, 2013 8944
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Figure 2. Three sequences of images illustrating the nucleation and growth of bilayered holes in the trilayer. First series: 8CB, T = 22.6 °C, and RH = 30%. Second series: 8CB, T = 30.0 °C, and RH = 23%. Third series: 10CB, T = 50 °C (the melting temperature of 10CB is about 45 °C), and RH = 47%. Trilayer is the bright region with sharp edges. Background evolution is due to the spreading of the diffuse monolayer outward. Total duration of the kinetics is about 3 h in the first series and about 45 min in the second and the third series. First image of each sequence (t = 0) was taken just before holes nucleation. In the first series, next images correspond to t = 122, 278, 428, 1434, 1956, 2497, 2860, 3148, 4996, 9728, and 10332 s, respectively. In the second series, t = 92, 218, 304, 452, 622, 828, 1084, 1300, 1814, 2112, and 2740 s. In the third series, t = 1080, 1135, 1209, 1263, 1348, 1430, 1543, 1628, 1715, 2000, and 2690 s. aim is to illustrate that the observed phenomena are not specific to 8CB. We first remark that the number of nucleation events is lower when the temperature is close to that of the solid/smectic transition (first series in the figure) and higher when the temperature is raised (second series). This is typical of a nucleation process since the latter is thermally activated. Second, the pore distribution is not uniform, holes appearing preferentially in the outer part of the trilayer. This is confirmed in Figure 3, where we reported, for each series of Figure 2, the radial distribution of holes at the instant of their first occurrence (Figure 3 a) and also the number of nucleation events and total
trilayer and grow during bilayer shrinking, leaving a monolayer on the surface. This is the case in the experiments reported herein.
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EXPERIMENTAL SECTION
8CB and 10 CB smectic drops were deposited on a silicon wafer covered with a 106 nm thick oxide layer at a temperature of 30 °C for 8CB and 50 °C for 10CB, and late-stage spreading was followed using the surface-enhanced ellipsometric contrast (or SEEC) technique.18,19 This technique makes use of a solid support which is antireflecting when observed between crossed polarizer. A very thin film added on this support generates interferences which make the extinction property disappear, and the film appears brighter than the bare surface in reflected light. The SEEC technique provides a height resolution better than 0.1 nm, so that direct and real-time visualization of all 8CB layers, including the diffuse one, becomes possible, with a much higher lateral resolution than ellipsometry or X-ray reflectivity.16 Optical observations were achieved through a polarization microscope (Leica DMRX PlanApo Pol Fluotar 20 × 0.5 objective lens). The solid support was cleaned using UV/ozone for 30 s and then set on a heating stage with ±0.5 °C of accuracy. Experiments were conducted within a glovebag at low relative humidity (RH). The typical size of the bilayer at the beginning of the late stage of the spreading is some hundreds of micrometers, and the typical duration of an experiment is 1 h. Figure 2 displays three typical sequences of images starting before the onset of pore nucleation. The first series was obtained at 22.6 ± 0.5 °C and RH of 30% and the second one at 30.0 ± 0.5 °C and RH of 23%. With such very thin liquid films, the intensity in the image is a linear function of its optical thickness. The central bright patch is the trilayer. The rest of the image is only a part of the gaseous monolayer since the latter extends beyond the image frame. The third series was obtained with a different liquid crystal of the same family, 10CB. Its
Figure 3. (a) Radial distribution of holes at time t of their first occurrence as a function of r/R2(t). (b) Evolution of the total number of holes (blue) and number of nucleation events (red) in the three series shown in Figure 2. Color codes in a and b are in correspondence. 8945
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number of holes as a function of time (Figure 3 b). Only closed holes were counted. The decrease of the total hole number with time is due to hole coalescence and retraction of the outer edge of the bilayer (the bilayer is the difference between the initial trilayer and the monolayer left on the surface). Notice that most holes formed abruptly within a short interval of time. Figure 4 focuses on a particular sequence extracted from the second series illustrated in Figure 2. It shows that a contact point between the
characteristic distance must be compared to the bilayer radius R2, which is also the trilayer radius.17 A great simplification is obtained in two asymptotic cases. The first one corresponds to ξ ≪ R2. It was introduced by de Gennes and Cazabat20 when reasoning with simple liquids and later improperly used when dealing with smectic liquid crystals.21 In the former work, it was supported by the dimensionality relationship ζ = η Σ/l, where η is the liquid viscosity, Σ the surface per molecule in a layer, and l the average thickness of a layer. This relationship assumes that the flow dissipation is isotropic in the liquid, which itself postulates isotropy of the diffusion coefficient in the directions parallel and normal to the lamellae. This is why it only applies to simple liquids. The second one corresponds to ξ ≫ R2. It was introduced in ref 17, where the general case, i.e., intermediate between the two asymptotic regimes, was also fully treated. It corresponds to highly stratified liquids and small drops. Comparisons with previous experiments demonstrated that the late-stage spreading problem with 8CB smectic liquid crystal trilayers smaller than 250 μm unambiguously belongs to the second asymptotic regime. This is agreement with eq 1, which tells us that ξ is large when the permeation coefficient is small. The size of the nanodrops considered in ref 14 and herein always corresponds to this weak permeation regime.
Figure 4. Two sequences extracted from series 2 in Figure 2. Arrows point out examples of sharp contour irregularities which evolve with the shrinking trilayer without smoothing out. Below each sequence, a zoom of the irregularity region is displayed. Time interval between two successive images is about 6−8 min.
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BACKGROUND A theoretical description of the dynamics of the 8CB nanodrop in the absence of hole nucleation was reported in ref 17. Here the model remains valid until the appearance of the first hole and may be used to understand why holes form. According to Figure 3, most holes appear abruptly. Our model gives a snapshot of the stress distribution in the film preceding the nucleation events. Therefore, the hole distribution gives a good image of this stress distribution. The spreading dynamics involves two kinds of material fluxes represented in Figure 1 by the various arrows. The first one is the vertical material transfer, or permeation, between the bilayer (closed by the dislocation line) and the independent spreading monolayer. The second one is the horizontal two-dimensional (2D) radial flow, due to the spreading in the monolayer, and induced by the permeation leakage in the bilayer. These flows generate molecular friction between adjacent layers. Depending on the permeation efficiency, the horizontal flux in the bilayer may be limited to a so-called permeation ribbon at its edge20 or may extend over the entire bilayer.17 The characteristic distance ξ which defines the extension of the 2D flow in the presence of the permeation leakage is given by17,20,21 l Cζ
WEAK PERMEATION REGIME
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PRESSURE PROFILES
The 2D horizontal flows that permeation generates extend over the entire trilayer area. The macroscopic evolution of the trilayer is completely regulated by the permeation between the bilayer and the monolayer and can be described as a quasi-static process. However, due to the existence of the radial flows, the chemical potential μ in these layers is not uniform. Nevertheless, in the limit ξ ≫ R2, these variations remain very small compared to the chemical potential difference between bilayer 23 and monolayer 1, as will be demonstrated later. Thus, the permeation current J (units m3·m−2·s−1) from the bilayer 23 toward the monolayer 1 is practically independent of the radial coordinate r. This approximation that J is constant allows our discussion to remain simple and to highlight physical arguments. In our simplified description, we also assume that each layer has the same thickness l and hence the same molecular area Σ = v/l, v being the molecular volume. Then, naming Vi the vector velocity in layer i, we have l▽·V1 = J in the monolayer, l▽·V2 = −J in the internal leaflet of the bilayer, i.e., that in contact with the monolayer, and V3 = 0 in the second leaflet of the bilayer. Integrating the divergence equations and considering that Vi(0) = 0, we get V1(r) = Jr/ 2l = −V2(r). Here, Vi is the radial coordinate along the radial coordinate r, oriented outward, of the vector velocity Vi. The origin of r is taken at the center of the drop.
edge of the pore and the edge of the trilayer, as shown by arrows in the figure, never smoothed with time, even if this contact point evolved due to bilayer shrinking. This is our third major observation.
ξ≈
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These velocity profiles are related to the chemical potential profiles by the coupled equations17,20,21
(1)
where l is the layer thickness, C the permeation coefficient, and ζ a typical friction coefficient between adjacent layers. In a theoretical approach of the late-stage spreading problem, this
−∇μ1 = ζ01 V1 + ζ12(V1 − V2) = (ζ01 + 2ζ12)
Jr Jr ≡ ζ1 2l 2l (2)
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−∇μ2 = ζ12(V2 − V1) + ζ23(V2 − V3) Jr = −(ζ23 + 2ζ12) 2l Jr ≡ −ζ2 2l
(3)
Jr Jr ≡ ζ3 2l 2l
(4)
−∇μ3 = ζ23(V3 − V2) = ζ23
Figure 5. (a) Theoretical lateral velocity profile Vi in each layer i as a function of the radial distance r, counted from the center of the drop. R2 is the radius of the bilayer. Velocities are positive when oriented outward. (b) Theoretical 2D pressure profile P(2) i in each monolayer i, indexed as in Figure 1, as a function of P(2) ⟨23⟩r is the average value of the pressure in the bilayer membrane 2−3. ΔP(2) 21 (0) is the 2D pressure gap between layers 1 and 2 at the center of the drop. Profiles a and b are valid for the two configurations shown in Figure 1, provided that layer indexes i conform to the convention given in the latter.
where ζij is the friction coefficient between layers i and j, the zero index holding for air in the case of Figure 1a and for the solid support in the case of Figure 1b, and where ζi is an effective friction coefficient acting on layer i. We postulate 2D incompressibility of the layers and also 3D incompressibility of the molecules. Then, the 2D pressure in a layer is V(2) i = μi/Σ. Integrating eqs 2−4 we get the pressure profile in layer i: V(2) i = (−1)iζi(Jr2/4v) + Ci. Constants C2 and C3 are determined by (2) considering that P(2) 2 (R2) = P3 (R2) = 0, since the bilayer is free to relax at the moving edge. We finally get P1(2)(r ) = ζ1
DISCUSSION The trilayer is made of three layers of molecules arranged head to tail (see Figure 1). We name a membrane any pair of adjacent leaflets. They are either monolayers 1 and 2 or monolayers 2 and 3. In the latter case, the membrane is what we previously called the bilayer. Both cases are not topologically equivalent. Layers 2 and 3 are connected by a common dislocation line. They can exchange material through intramembrane lateral flows which roll up like a caterpillar around the defect line. By contrast, layers 1 and 2 are disconnected and can exchange material only by permeation. Lateral flows are associated with pressure profiles along the layers, while permeation is associated with a difference in 2D pressure between adjacent layers. This pressure difference is all the more important when the permeation current is weak. Notice that the tension of the two constitutive leaflets is asymmetric in both membranes 12 and 23 and that this asymmetry is a function of r. Then, two competing driving forces may induce pore formation. The first one is the local membrane tension, i.e., the average of the local tensions of its two constituting leaflets, and the second one is local tension asymmetry between the two leaflets of the membrane. Indeed, the pressure difference between the two leaflets of a membrane ij generates a local stressΔP(2) ji /l between the neighboring layers which tends to bend the ij interface. It can be viewed as a normal tension in the stack. Formation of a pore between the two layers will allow relaxing this stress. About the first driving force, eq 5 shows that the 2D pressure in membrane 23 is negative, which means that the bilayer is under tension and that this tension is maximal at the center of the bilayer. Equation 6 tells us that membrane 12 is even under a greater tension and that this tension may be maximal either at the center of the drop or at the edge, depending on whether ζ23 is larger or lower than ζ01. We are always in the first case when the structure of the trilayer conforms to Figure 1a. The second case could be only encountered when ζ23 < ζ01 with the structure displayed in Figure 1b. However, we expect ζ23 > ζ01 when working in ambient humidity because of the slippery water layer covering our hydrophilic SiO2 surface. Therefore, although it cannot be totally excluded, the only case where pores could form at the edge due to membrane stretching is very unlikely. Regarding the second driving force, eq 7 tells us that the difference between the 2D pressures of monolayers 2 and 1 is a negative and increasing function of r and that it is maximal at the bilayer edge. From eqs 5−8, the highest averaged membrane tension
J (R 2 2 − r 2 ) + P1(2)(R 2) v 4
P2(2)(r ) = −ζ2 P3(2)(r ) = ζ3
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J (R 2 2 − r 2 ) v 4
J (R 2 2 − r 2 ) v 4
The average pressure in the bilayer 23 is P⟨(2) 23⟩(r ) =
(ζ3 − ζ2)J (R 2 2 − r 2) J (R 2 2 − r 2 ) = −ζ12 2v 4 v 4 (5)
The average pressure between disconnected layers 1 and 2 is (ζ1 − ζ2)J (R 2 2 − r 2) 1 + P1(2)(R 2) 2v 4 2 2 2 (ζ − ζ23)J (R 2 − r ) 1 = 01 + P1(2)(R 2) 2v 4 2
P⟨(2) 12⟩(r ) =
(6)
The pressure difference between the two neighboring layers 2 and 1 is therefore J (R 2 2 − r 2 ) − P1(2)(R 2) 4 v J (R 2 2 − r 2 ) = −(ζ01 + 4ζ12 + ζ23) − P1(2)(R 2) 4 v
(2) ΔP21 = −(ζ2 + ζ1)
(7)
One can also express the pressure difference between leaflets 3 and 2 J (R 2 2 − r 2 ) 4 v J (R 2 2 − r 2 ) = 2(ζ12 + ζ23) v 4
(2) ΔP32 = (ζ3 + ζ2)
(8)
The theoretical velocity profiles in the three monolayers are represented in Figure 5a, and the corresponding pressure distributions are shown in Figure 5b. 8947
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with different core materials coexist in the trilayer structure, which are located each at a different height in the stack. Similar behaviors were recently reported in egg phosphatidyl choline nanodrops,22 demonstrating high universality in smectic liquid behaviors. Normal stresses are also expected to appear which may induce pore formation in various flowing smectic liquids such as copolymers, lipids, or liquid crystals. By extension, normal stress-induced pore nucleation might also be envisaged in a free bilayered membrane when this membrane is asymmetric.
between membrane leaflets takes place in membrane 12. According to the hypothesis ξ ≫ R2, the highest pressure difference also takes place between leaflets 1 and 2. Indeed, reminding that J ≈ C|μ2 = μ1|,17 integrating |∇(μ3 − μ2)| ≈ ζ Jr/ l between r = 0 and r = R2 and combining with eq 1 we get |(μ2 − μ1)| ≫ |[μ3 − μ2]R0 2|. Therefore, pore formation is expected to occur between leaflets 1 and 2. If due to membrane tension, it would occur most probably at the center of the drop. If due to pressure asymmetry in the two leaflets of the membrane, it must definitely occur at the edge of the drop. The main point remains that pores must form in all cases in membrane 12 and not in the bilayer. Therefore, the dislocation lines which delimit the pores and the bilayer are located at a different heights in the trilayer structure. As a consequence, the heart of these two dislocations is made of a different material (paraffinic versus polar part of the molecules).
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ASSOCIATED CONTENT
S Supporting Information *
Pressure in different height layers of a trilayer smectic nanodrop. This material is available free of charge via the Internet at http://pubs.acs.org.
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BACK TO EXPERIMENTS The nonuniformity of the nucleating pores in our experiments is now well understood. We noticed that holes nucleation was limited to a short time interval. This is explained by the rapid stress relaxation resulting from connecting layers 1 and 2 after pore formation. Finally, our third and most intriguing observation that contact points between the edge of the pore and the edge of the bilayer did not smooth with time is explained by the fact that the two edges have a different altitude in the trilayer. There are two possibilities for the internal structure of the trilayer depicted in Figure 6. In Figure 6a, the
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was entirely supported by ANR under project PNANO-07-050.
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REFERENCES
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Figure 6. (a and b) Topology of the trilayer after pore (P) nucleation. Scheme a corresponds to Figure 1a and scheme b to Figure 1b. Red and green arrows point to the heart of, respectively, the bilayer edge (BE) and the pore edge (PE) dislocations. Pore forms between the isolated monolayer and the leaflet of the closed bilayer which is adjacent to it. Hence, bilayers with edges of the type of Figure 1a produce pores with edges of the type of Figure 1b and vice versa.
bilayer edge (in which the heart is pointed at by the arrow) is close to the solid and the pores open close to the free surface; in Figure 6b, we have the opposite situation. A singularity must be present where the two dislocation lines meet, because of their height mismatch. The point defect structure clearly prevents material exchange between the two domains respectively bounded by the two dislocation lines. Its detailed description is beyond the scope of the present work.
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CONCLUDING REMARKS To summarize, permeation is not sufficient for the bilayer to provide the spreading monolayer with the material flux required. This generates lateral tensions in the various layers and vertical (“normal”) stresses between adjacent layers, which increase with time, and pores are finally opening. On the basis of a theoretical model that was successfully confronted to experiments in previous studies on the same system, we claim that pore nucleation results from the normal stresses. From the experiments reported above, these pores do not open in the bilayer membrane, but they form in the membrane composed of one leaflet of this bilayer and the adjacent independent monolayer. As a consequence, two kinds of dislocation lines 8948
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(14) Benichou, O.; Cachile, M.; Cazabat, A. M.; Poulard, C.; Valignat, M. P.; Vandenbrouck, F.; Van Effenterre, D. Thin films in wetting and spreading. Adv. Colloid Interface Sci. 2003, 100, 381. (15) Bardon, S.; Ober, R.; Valignat, M. P.; Vandenbrouck, F.; Cazabat, A. M. Organization of cyanobiphenyl liquid crystal molecules in prewetting films spreading on silicon wafers. Phys. Rev. E 1999, 59, 6808. (16) Noel, O.; Buraud, J. L.; Berger, L.; Ausserre, D. Quantitative Spreading Kinetics of a Three Molecular Layer Liquid Patch. Langmuir 2010, 26, 6015. (17) Ausserre, D.; Buraud, J. L. Late stage spreading of stratified liquids: Theory. J. Chem. Phys. 2011, 134, 114706. (18) Ausserre, D.; Abou Khachfe, R. Real-Time Quantitative Imaging of Submolecular Layers. Langmuir 2007, 23, 8015. (19) Ausserre, D.; Valignat, M. P. Surface enhanced ellipsometric contrast (SEEC)basic theory and λ/4 multilayered solutions. Optics Express 2007, 15, 8329. (20) de Gennes, P. G.; Cazabat A. M., Etalement d’une goutte stratifiée incompressible. C.R. Acad. Sci., Ser. II 1990, 310, 1601. (21) Betelù, S.; Law, B. M.; Huang, C. C. Spreading dynamics of terraced droplets. Phys. Rev. E 1999, 59, 6699. (22) Mohamad, S.; Noël, O.; Buraud, J.-L.; Brotons, G.; Fedala, Y.; Ausserre, D. Mechanism of lipid nanodrop spreading in a case of asymmetric wetting. Phys. Rev. Lett. 2012, 109, 248108.
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