Abstract
Coalescence is an essential phenomenon that governs the equilibrium behaviour in a variety of systems from intercellular transport to planetary formation. In this report, we study coalescence pathways of circularly shaped twodimensional colloidal membranes, which are one rodlengththick liquidlike monolayers of aligned rods. The chirality of the constituent rods leads to three atypical coalescence pathways that are not found in other simple or complex fluids. In particular, we characterize two pathways that do not proceed to completion but instead produce partially joined membranes connected by line defects—πwall defects or alternating arrays of twisted bridges and pores. We elucidate the structure and energetics of these defects and ascribe their stability to a geometrical frustration inherently present in chiral colloidal membranes. Furthermore, we induce the coalescence process with optical forces, leading to a robust ondemand method for imprinting networks of channels and pores into colloidal membranes.
Introduction
Driven by surface tension, a pair of liquid droplets undergo a largescale transformation as they merge into a single daughter droplet. In order for the droplets to coalesce, a rupture must occur between two merging interfaces which, coupled with surface tension and hydrodynamic forces, leads to energetic barriers and nontrivial coalescence progression^{1,2,3,4,5}. For complex fluids with partial positional and/or orientational order, the complexity of the coalescence reactions increases considerably^{6}. A wellstudied example belonging to this category is lipid vesicles, whose fusion is characterized by complex barriers that include structural distortions, which expose the hydrophobic core of the lipid bilayer and allow the merger to proceed^{7,8,9}. In addition to fluid droplets and vesicles, coalescence is also important in the formation of solid structures such as carbon nanotubes or sintered powders^{10,11}. In most cases, coalescence is an ‘allornone’ process; once initiated, the reaction proceeds to completion. However, moving along a complexfree energy landscape, there is also the possibility of a coalescence reaction that overcomes an initial energy barrier only to get trapped in an intermediate state, such as merging vesicles producing a hemifused state or coalescing nanotubes producing a defectridden structure^{10,12}.
Here we study the coalescence pathways of twodimensional (2D) colloidal membranes composed of liquid monolayers of aligned rodlike viruses^{13}. We present coalescence dynamics markedly different from that observed between either simple liquid droplets or complex lipid vesicles. The chirality of the constituent rodlike viruses introduces an insurmountable topological barrier that suppresses the conventional coalescence pathway observed between liquid droplets and greatly slows down the coarsening dynamics. Instead, the coalescence proceeds by atypical higherorder pathways to intermediate states in which two membranes are joined together by 1D defects. The topology of these intermediate structures suppresses the subsequent kinetic step, resulting in the robust formation of two types of line defects whose stability is only limited by the sample lifetime. Theoretical work and computer simulations predict that colloidal membranes are an equilibrium phase that should be observed in diverse suspensions of highly anisotropic monodisperse rods with attractive interactions^{14,15}. In the same way that disclination lines are intimately connected and influence the properties of nematic liquid crystals, the defects described here are a ubiquitous feature of colloidal membranes. The colloidal nature of our building blocks allows us to characterize the structure and dynamics of membraneimprinted defects in great detail using several complementary microscopy techniques. This enables us to formulate and quantitatively test a theoretical model of πwalls. Furthermore, we engineer coalescence events using optical forces in order to create welldefined networks of 1D channels or pores embedded within a 2D colloidal membrane. Using such technology, we imprint and analyse excitations with skyrmion topology, an important class of defects that has recently been studied in varied soft and hard materials^{16,17,18,19,20,21}.
Results
Three coalescence pathways
We use fd wildtype (wt) viruses, which are rodlike colloidal particles with 880 nm contour length, 7 nm diameter and 2.8 μm persistence length^{22}. Adding nonadsorbing polymer induces effective attractive interactions between otherwise electrostatically repulsive rods through the wellstudied depletion mechanism^{23}. Under specific conditions, the depleting agent condenses homogeneous rodlike viruses into colloidal membranes, one rodlengththick chiral smecticA (SmA*) monolayers of aligned rods whose continuum properties are identical to those of conventional lipid bilayers^{13,14}. A feature that critically affects the coalescence pathways is the membrane's edge, consisting of a twisted layer of rods that penetrate a small distance into the membrane interior (Fig. 1a,b)^{24,25}. Rod tilting is enforced by surface tension, which is minimized for a curved edge. The chirality of fd wt rods is left handed^{26,27}, which imposes a welldefined handedness onto the colloidal membrane edge. This complex edge behaviour is one specific manifestation of a very general packing frustration present whenever chiral molecules are forced to assemble into 2D smecticlike sheets^{28,29,30,31}.
In contrast to lipid bilayers that assemble into edgeless threedimensional (3D) vesicles, colloidal monolayer membranes appear as flat 2D disks with exposed edges. As two coplanar membranes approach laterally, chirality causes the rods at the proximal membrane edges to be tilted in opposite directions, trapping 180° of twist between them (Fig. 1a). After a coalescence event, these rods end up in the membrane interior where they are forced to untwist by the global assembly constraints. The energetic cost associated with a pathway in which the peripheral rods untwist and two coplanar membranes coalesce laterally is prohibitively large and is never observed. Rather, we have identified three alternate types of coalescence events that proceed along more complex pathways. This unique behaviour arises specifically due to the chiral topological barrier imposed by the 180° of twist trapped between the two merging edges.
The first pathway results in the formation of defectfree daughter membranes and shares certain similarities as well as important differences with the wellstudied coalescence of liquid droplets (Fig. 1c, Supplementary Movie 1). Similar to liquid droplets, thermally induced fluctuations cause the merging membranes to rupture and establish a one rodlengththick bridge between them. Within such a bridge, rods twist by 180° in order to match the orientations of the joining edges. The overtwisted bridge induces a torque that rotates the two membranes in order to expel the trapped twist. As the membranes twist around each other, the connecting bridge expands in width, eventually producing a circularly shaped defectfree daughter membrane. In conventional coalescence, the rate of neck expansion is determined by hydrodynamic and inertial forces^{3}. By contrast, the significantly slower neck expansion in membrane coalescence is dominated by the rate at which the two membranes rotate with respect to each other.
In the second pathway, coalescing membranes simultaneously rupture at multiple locations, forming many collinear twisted bridges separated by depletantfilled pores (Fig. 1d, Supplementary Movie 2). These bridges eventually multiply and spread along the periphery of the coalescing structures, resulting in membranes that are joined through a 1D array of pores that alternate with onerodlengththick twisted bridges (alternating bridgepore arrays, ABPAs). The third coalescence pathway (Fig. 1e, Supplementary Movie 3) is initiated by establishing two twisted necks, which hold the membranes together and induce the nucleation of a continuous 1D line defect. This uniform defect, called a πwall because it traps 180° of twist, quickly grows to its equilibrium size, pushing the two necks apart. Once a πwall or ABPA is formed, the topological barrier for a subsequent transition to a lower energy defectfree membrane involves untwisting of defectbound rods and cannot be overcome on experimental timescales.
Coalescence on demand
Molecular chirality introduces frustration into the coalescence of colloidal membranes, resulting in the formation of 1D defects (Fig. 1). However, being a consequence of random coalescence events, the spatial location of such defects is impossible to control. It is therefore desirable to develop a robust ondemand method for imprinting defects into colloidal membranes with arbitrary spatial precision, similar to recent work with thermotropic liquid crystals^{32,33,34}. We demonstrate that membrane selfcoalescence can be induced with optical forces (Fig. 2), opening up the possibility of engineering welldefined 2D defect networks. Moving an optical trap from a membrane's edge into the interior results in the creation of two edges aligned along the direction of trap translation (Fig. 2a). Subsequently, a πwall nucleates between these two edges and grows in length, zipping up the gap created by the translating optical trap. If the trap is released in the membrane interior, the πwall quickly retracts (Supplementary Movie 4). However, moving the trap to the distal edge anchors the πwall, making it stable indefinitely. This method makes it possible to imprint a complex, predesigned interconnected network of πwalls into a large membrane (Supplementary Movie 5). Such networks can serve as templates for the assembly of colloidal particles, which are stably trapped for indefinitely long periods of time at the junction of three πwalls (Supplementary Fig. 1)^{17}. The same technique is used to imprint an ABPA as well as transform a πwall into an ABPA and vice versa (Fig. 2b, Supplementary Movies 6 and 7). These results demonstrate that the unique coalescence of fd membranes coupled with accessibility of colloidal systems to direct manipulation leads to assembly of tunable membrane landscapes.
Structure of πwall and pore defects
To characterize πwall defects, we prepared samples containing a low fraction of fluorescently labelled viruses, allowing us to track dynamics of individual particles (Fig. 3a,b, Supplementary Movie 8). For membranes lying in the image (xy) plane, fluorescently labelled rods in the membrane interior, aligned along the zaxis, appear as isotropic spots. In the vicinity of a πwall, particles become elongated, indicating local twisting. To confirm the proposed structure, we use 2DLCPolScope microscopy, a technique that produces images in which each pixel's intensity is proportional to the 2D projection of the sample optical retardance, R (ref. 35). There is no structural or optical anisotropy in the membrane interior, hence this region appears black when viewed with LCPolScope (Fig. 3c). Tilting of rods away from the monolayer normal melts the smectic order into a nematic (cholesteric) (Fig. 3g), resulting in structural and optical anisotropy in the xy plane that becomes visible as a bright birefringent band with LCPolScope. Thus, the bright region observed along the length of the πwall supports the hypothesis that rods twist by 180° across the defect. Furthermore, 3DLCPolScope reveals that rods within the πwall have lefthanded twist, in agreement with measurements in the bulk cholesteric phase (Fig. 3e,f, Methods)^{22,36}.
Using the same microscopic techniques, we also elucidate the structure of ABPAs (Fig. 4a). 2DLCPolScope reveals twisting of rods within the smectic bridges (Fig. 4b), and this is confirmed by tracking individual fluorescently labelled rods (Fig. 4c). Images of membranes assembled from 100% fluorescently labelled rods indicate local membrane density. In such images, pores appear dark indicating that they are filled with the depleting polymer (Fig. 4d). A schematic of a bridgepore array is shown in Fig. 4e.
Theoretical model of the πwall
When viewed with 2DLCPolScope, the centre of a πwall appears darker as evidenced by a local minimum in the intensity profile (Fig. 5a). To understand this, we note that the retardance of a colloidal membrane lying in the image plane is given by:
where n=4.1 × 10^{−5} ml mg^{−1} is the specific birefringence of a fully aligned bulk sample at unit concentration^{37}, t is the membrane thickness, S is the nematic order parameter (assumed equal to 1), c is the local concentration of virus and θ is the tilt angle of rods away from the layer normal (zaxis)^{25}. Since the retardance increases with increasing θ, a πwall with unchanging thickness would yield a singly peaked function. A profile with two peaks separated by a local minimum indicates reduced membrane thickness at the πwall centre (Fig. 5a). To disentangle the contribution of varying θ and t to the overall retardance profile, we develop a theoretical model of a πwall.
We model the πwall by a Ch region sandwiched between two semiinfinite SmA* monolayers. Our model utilizes the de Gennes free energy density f_{ChA} for the ChSmA* transition (Methods)^{38,39,40,41}. The free energy per unit length of a SmA* flat layer in the xy plane containing a πwall is given by:
The volume terms in square brackets are multiplied by the local thickness t(x). The smectic order parameter Ψ describes the transition from a SmA* monolayer of aligned rods to a Ch region. The only contribution to f_{Ch−A} from the Frank free energy density is the twist term, as the nematic director n rotates by an angle θ about the xaxis within a characteristic length scale λ_{t}, the twist penetration depth^{25}. The second term dictates that t=t_{0} (see Fig. 5b) in the bulk region of fully aligned fd viruses (Ψ=Ψ_{0}), and that t is decoupled from the orientation of n in the cholesteric region (Ψ=0). v is the Lagrange multiplier due to the volume constraint of the membrane. The constants in the fourth and fifth terms of equation (2) are the bulk surface tension modulus σ_{} when n, and the πwall surface tension modulus σ_{⊥} when n⊥, where is the local unit normal of the curved membrane surface (Fig. 5b). ds is the infinitesimal arc length of the surface, and its projection onto the xaxis is dx (Fig. 5b). In the presence of anisotropy (σ_{}≠σ_{⊥}), the local surface tension changes continuously due to the local tilt of n between the two regimes. The last term in equation (2) is the free energy cost of the surface curvature^{42}.
The πwall thickness profile is determined by the competition between surface and volume contributions to the free energy. Since tilting of the rods raises free energy, the volume term favors thinner πwall profiles. The surface term associated with the rod–polymer interface favors a thicker πwall profile with uniform thickness that minimizes the exposed surface. The meanfield analysis of equation (2) predicts how the membrane thickness t, virus tilt angle θ and smectic order parameter Ψ vary across a πwall (Fig. 5c). The variation of Ψ demonstrates the continuous melting of the SmA* into a Ch phase at the centre of a πwall. This data, along with theoretical predictions for θ and t in equation (1), yield a retardance profile that fits the experimental data and determines the twist penetration depth, λ_{t} (Fig. 5a, Supplementary Table 1). A best fit to experimental results is achieved for a membrane thickness at the πwall centre of 220 nm, significantly smaller than the 880 nm thickness of the membrane interior. Thus, πwalls form deep channels within a monolayer of otherwise uniform height. This finding can be directly confirmed by fluorescence microscopy where the image intensity is directly proportional to local membrane thickness; a πwall is significantly darker than the membrane bulk (Fig. 3d).
πwall energetics and stability
πwall defects exhibit significant fluctuations that are observable with optical microscopy (Fig. 6a, Supplementary Movie 9). The analysis of such fluctuations yields a spectrum from which it is possible to extract the πwall line tension, γ_{π} and bending rigidity, κ_{π} (Methods, Fig. 6b)^{24,43}. We determine the dependence of γ_{π} and κ_{π} on relevant molecular parameters, namely fd chirality and depletant osmotic pressure (Fig. 6c). Chirality is tuned by temperature; at T=60 °C, the rods are achiral (no twisting) and chirality increases with decreasing temperature^{24}. Stronger chiral interactions result in larger twist angles required to minimize the rod interaction energy. Virus concentration within a membrane is determined by the osmotic pressure of the surrounding depletant and increases with increasing Dextran concentration. For comparison, we have also analysed fluctuations of an exposed membrane edge to extract its line tension (γ_{edge}) and bending rigidity (κ_{edge})^{24}.
For a wide range of parameters, we find that κ_{π} is consistently equal to 2κ_{edge} and is independent of the physical parameters, which is reasonable since the twist penetration region of a πwall is twice that of an exposed edge^{24}. We obtain values of κ_{π} and κ_{edge} of about 200 k_{B}T μm and 100 k_{B}T μm, respectively, and these values do not change significantly with varying polymer concentration or rod chirality. By contrast, γ_{π} and γ_{edge} exhibit more complex dependence on microscopic parameters. Strengthening chirality raises the free energy of untwisted rods in the interior while lowering the free energy of twisted edge or defectbound rods, thus lowering γ (ref. 24). We find that πwalls are always metastable with respect to the untwisted bulk phase (γ_{π}>0) but that chirality can reduce γ_{π} by up to 100 k_{B}T μm^{−1}. Increasing rod attraction (increasing dextran concentration) significantly raises γ_{π}, and this increase does not depend on molecular chirality (Fig. 6c).
Our theoretical model predicts γ_{π}=F−F_{bulk}, where F_{bulk} and F are the free energy per unit length of a monolayer of perfectly aligned rods and a πwall, respectively (equation (2)). Fitting chiralitydependent γ_{π} (Fig. 6d) and the retardance profiles (Fig. 5a), and using the previously measured value of K_{2} (ref. 26), we determine the parameters in equation (2): λ_{t}, k, σ_{} and σ_{⊥} (Supplementary Table 1). k, σ_{} and σ_{⊥} are determined by the polymer concentration and are independent of chirality, in agreement with experiments. The retardance profiles are independent of both chirality and polymer concentration (Supplementary Fig. 2). As a first approximation, when γ_{edge} becomes larger than γ_{π}, we might expect a spontaneous dissociation of a πwall into two defectfree membranes. We search for such an event by performing a temperature quench of a πwall, which quickly increases the strength of chirality. Surprisingly, instead of dissociation, we observe a transition of πwalls into ABPAs (Fig. 6e, Supplementary Movie 10). Pores nucleate from the ends of the πwall and propagate into the interior of the line defect. The πwalltoABPA transition is reversible; with increasing temperature, an ABPA transforms back into a πwall. ABPAs are more frequently observed in the high chirality limit, while πwalls tend to be more stable for weakly chiral rods. This behaviour is reasonable, as pores create a large amount of rod–polymer interface that is favored at higher chirality^{24}. For many parameters we observe coexisting ABPAs and πwalls (Supplementary Fig. 1) indicating a substantial kinetic barrier associated with transformation between these structures.
Engineering complex defect structures
With detailed understanding of the dynamics, structure and energetics of πwalls, we are in a position to engineer defects that are inaccessible through natural coalescence processes. As an example, we have used optical tweezers to sever a πwall in two places, and then quickly joined the two ends to form a closed ring within the membrane interior (Fig. 2c). These particlelike structures are similar to skyrmion excitations in that they have positive energy compared with the background field, but remain indefinitely stable as there is an insurmountable barrier to their excision from the host membrane^{18,19,20}. The equilibrium size of a πwall ring is determined by the competition between two effects: the interfacial energy, F_{i}=2πγ_{π}r, and the bending energy, F_{b}=2πκ_{π}/r, where r is the radius of the ring. The first term favors smaller rings, while the second term favors structures with larger radius and lower curvature energy. Once created, a πwall ring slowly shrinks, ultimately reaching an equilibrium radius of r_{eq}=1.3 μm at T=22 °C (inset Fig. 2c, Supplementary Fig. 3). Minimizing the free energy that contains both line tension and bending rigidity terms predicts that r_{eq}=. We can also estimate from the corner spatial frequency of the fluctuation spectrum (Fig. 6b). For the same parameters, the πwall fluctuation spectrum yields r_{eq}==0.65 μm. The discrepancy between these two methods is likely due to the high curvature of skyrmionlike defects, requiring an expansion of the bending energy to higher orders than 1/r^{2}. Besides skyrmions, it is also possible to imprint other shapes such as triple junctions, for which a similar analysis can be performed (Fig. 2d, Supplementary Fig. 4, Methods). Taken together, these examples illustrate the potential of the imprinting technique to investigate the universal behaviour of various topological defects embedded in 2D membranes.
Discussion
We have described three coalescence pathways of chiral colloidal membranes. Two pathways do not proceed to completion but instead remain stuck indefinitely in a local energy minimum, producing membranes that are partially joined by 1D line defects that trap chiral twist. We have formulated a comprehensive theoretical model of chiral πwall defects and quantitatively tested it against extensive experimental data. Numerous theoretical works, computer simulations and experiments have demonstrated that fd viruses used in our studies are an excellent experimental realization of hard rods^{22}, an important model system that has been studied for many decades. It follows that the phenomenology we describe should be relevant to diverse colloidal and nanosized rods that interact through excluded volume interactions. From an application perspective, aligned monolayers of semiconducting rods have diverse applications ranging from light emitting diodes to photovoltaic devices^{44}, and colloidal membranes offer an easily scalable method for assembling such structures. Understanding the principles that promote or inhibit membrane coarsening and defect formation described here is essential for obtaining largescale monodomain samples. Our work demonstrates that rod chirality inhibits membrane coalescence and coarsening, thus suppressing formation of macroscopic membranes. Consequently, it also suggests that engineering colloidal membranes from achiral rodlike subunits would lead to formation of defectfree uniform monolayers.
Methods
Sample preparation
Bacteriophage fd viruses were grown in the XL1Blue E. Coli. strain and purified as described previously^{24}. Optical microscopy chambers were assembled using glass slides and cover slips (Goldseal, Fisher Scientific) with an unstretched parafilm layer acting as a spacer as discussed previously^{24}.
Optical microscopy methods
Experiments were carried out on an inverted microscope (Nikon TE2000) equipped with traditional polarization optics, a differential interference contrast (DIC) module, a fluorescence imaging module and a 2DLCPolscope module^{45}. We used a 100 × oil immersion objective (PlanFluor NA 1.3 for DIC and PlanApo NA 1.4 for phase contrast). Images were recorded with cooled CCD cameras (CoolSnap HQ (Photometrics, Tuscon, AZ) or Retiga Exi (QImaging, Surrey BC, Canada)). For 3DLCPolScope measurements, we used a Zeiss Axiovert 200 M microscope with a Plan Apochromat oil immersion objective (63X/1.4NA), and a monochrome CCD camera (Retiga 4000 R, QImaging, Surrey BC, Canada)^{35}.
Temperature was regulated in situ with a Peltier device equipped with a proportional integral derivative temperature controller (ILX Lightwave LPT 5910). The temperature controlling side of the Peltier is attached to a copper ring fitted to the size of the microscope objective. A thermistor, placed in the copper ring adjacent to the sample, permits the proportional integral derivative feedback necessary to adjust the temperature. The excess heat is removed with constant flow of room temperature water on the other side of the Peltier device. Using this setup, we control the temperature anywhere between 4 and 60 °C.
Fluctuations of the πwall and membrane edges were analysed using previously described methods^{24,46}. For analysis of πwall fluctuations, we used phasecontrast microscopy instead of DIC. This ensured that the intensity profile along transverse cuts through the defect contour would be Gaussianshaped, which is optimal for tracking. The lower performance of phasecontrast imaging results in a larger signaltonoise for the πwall spectra when compared with the analysis of the membrane edge acquired with DIC imaging.
The 2DLCPolScope cannot distinguish between clockwise and counterclockwise rod tilting. To determine the twisting direction, we utilized the 3DLCPolScope (Fig. 3e,f), which yields a full 3D retardance map of the membrane^{24,45}. A microlens array, introduced into the image plane of the objective lens, produces a grid of conoscopic microimages on the CCD camera, with each microimage determining the local rod orientation. An azimuthally symmetric retardance profile with a centred dark spot indicates that rods are locally oriented along the zaxis, as observed in the membrane interior (Fig. 3e). A shift of the dark zeroretardance spot away from the centre of a conoscopic image indicates the magnitude of the local tilting of the rods, while its azimuthal angle indicates the direction of the birefringence vector projection in the xy plane (Fig. 3e). Using 3DLCPolScope, we demonstrate that the πwalls embedded in fd membranes have lefthanded twist, similar to the handedness of the bulk cholesteric phase^{22,36}.
Optical tweezers
Colloidal membranes were manipulated with a holographic optical tweezer setup built around an inverted Nikon TE2000 microscope. A 1,064 nm laser beam (Compass 1064, Coherent) is expanded by a telescope and reflected at nearnormal incidence off of a liquidcrystalonsilicon spatial light modulator (SLM) (Hamamatsu, X1046803). The SLM is imaged onto the back focal plane of the objective lens using a second telescope, which also shrinks the beam to slightly overfill the back aperture of the objective lens. The phase mask encoding the hologram is calculated and displayed on the SLM using a modified version of freely available software^{47}. Using this software, multiple point traps were created and translated in the image plane in real time.
Complements on the πwall theory
We assume that membranes and πwalls formed by filamentous fd viruses are typeII smectic materials, based on the analogy with superconductors^{40,41}. In other words, the coherence length ξ, the length scale of the relaxation of the smectic order, is much smaller than the twist penetration depth λ_{t}, confirmed by the agreement of previous theories based on the Frank free energy of monolayers with experiments^{13,39,48}. Moreover, we assume a linear πwall, so that the fields in equation (2) depend only on x. The de Gennes free energy density^{40,41} for the cholestericSmA* transition due to distortions of the nematic director n≡{0,sin θ(x),cos θ(x)} in a semiinfinite membrane, f_{Ch−A}, in equation (2) is given by^{13,39}
Ψ is the real part of the complex order parameter , and the first three terms in equation (3) describe the crossover from a one rodlengththick monolayer of perfectly aligned fd (Ψ=Ψ_{0}=) to the cholesteric region forming around the π wall (Ψ=0). The change in the phase of due to big distortions of n and arising from t≠cos θ near the πwall is absorbed in the fourth term of equation (3) and in the second term of equation (2) without loss of generality, ensuring rotational invariance. The length scales are given by λ_{t}≡ and ξ_{i}≡.
Using the boundary conditions that θ=0, Ψ=Ψ_{0}, t=t_{0}, as well as their vanishing derivatives of every order in the bulk monolayer, θ=, Ψ=0, t= and =0 at the midline of the πwall, the thickness t, tilt angle θ and smectic order parameter Ψ are numerically calculated by solving Euler–Lagrange equations minimizing equation (2).
πwall and membrane edge fluctuation analysis
To measure γ_{π}, we determine the instantaneous πwall configuration with subpixel precision^{46}. The conformation of the interface, described by the local tangent angle α(y) at position y along the line defect, is decomposed into a Fourier series: α(y)=a_{q}sin(qy). Analysing a series of uncorrelated images yields the fluctuation spectrum (mean square Fourier amplitudes versus wave vector q).
There are two contributions to the free energy of a fluctuating πwall defect F_{fluct}: a line tension term proportional to the overall defect length, and an elastic energy term proportional to the local defect curvature. The origin of the curvature term is the melting of the smectic monolayer into a nematic in the vicinity of the πwall defect. It follows that the free energy of a fluctuating πwall is given by:
F_{fluct} leads to the fluctuation spectrum =k_{B}T/(γ+κq^{2}), and fitting this expression to the experimentally measured curve yields values for γ_{π} and κ_{π} (Fig. 6b). Repeating the same analysis for the exposed membrane edge yields its line tension (γ_{edge}) and bending rigidity (κ_{edge}) as discussed previously^{24} (Fig. 6b). The contrast of πwall images is significantly lower when compared with those of an exposed edge, resulting in a πwall fluctuation spectrum that is dominated by noise for q>2 μm^{−1}. In comparison, the fluctuation spectrum of an exposed edge extends to much higher q values.
Complements on πwall triple junction structure
πwall triple junction structures are created from a single πwall by forming a second πwall connecting two points along the defect contour using laser tweezers. After some time, these structures relax into the eyeshape shown in Supplementary Fig. 4. This shape is well approximated as two circular arcs of contour length l and radius r connected at points A and B and centred at points O and O′, respectively (Supplementary Fig. 4). This shape is characterized by its length d and the junction angle φ=(DAC), where the line (AC) is the tangent line to the upper arc at point A.
A simple free energy model of a πwall contains two terms. The first is the line tension per unit length γ. The second is the bending energy per unit length κ/r^{2}, where r is the local radius of curvature. The equilibrium shape and size of the structure is determined by the interplay between the πwall line tension and bending rigidity. At the junctions A and B, the net force must be zero: γ=2(γ+κ/r^{2})cos φ. Based on geometry arguments, we can write all the relevant parameters in terms of r: s=L−d, d=2rsin φ, and l=rπφ. Therefore, the total free energy of the system between points E and B depends only on three parameters, and is given by F(r,γ,κ)=sγ+2l(γ+κ/r^{2}). Using values of γ=425 k_{B}T μm^{−1} and κ=200 k_{B}T μm measured through the fluctuation spectrum, we minimize F with respect to r and find that it displays a single minimum at the equilibrium radius r*=1.0 μm at T=22 °C, and therefore d=1.9 μm and φ=69°. These estimations compare well to the experimental results of d=3.0±0.5 μm and φ=60±5°.
Additional information
How to cite this article: Zakhary, M. J. et al. Imprintable membranes from incomplete chiral coalescence. Nat. Commun. 5:3063 doi: 10.1038/ncomms4063 (2014).
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Acknowledgements
We thank Miha Ravnik and Yulia Yeomans for valuable discussions and insight. This work was supported by the National Science Foundation (NSFMRSEC0820492, NSFDMR0955776, NSFMRI0923057) and Petroleum Research Fund (ACSPRF 50558DNI7) and the Agence Nationale de la Recherche française (ANR11PDOC027). We acknowledge use of Brandeis MRSEC optical microscopy facility.
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M.J.Z., T.G. and Z.D. conceived the project. M.J.Z. and T.G. carried out the measurements and analysed the data. C.N.K. and R.B.M. developed the theoretical model and fitted the experimental data. E.B. discovered the πwalls and observed dynamics that leads to their formation. R.O. performed 3DLCPolScope measurements. M.J.Z., T.G., C.N.K. and Z.D. wrote the manuscript.
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Supplementary information
Supplementary figures and table
Supplementary Figures 14 and Supplementary Table 1 (PDF 251 kb)
Supplementary Movie 1
Coalescence pathway that expels twist and leads to the formation of defectfree daughter membrane. The membranes first form a onerodlength thick twisted bridge between the membranes. Subsequently, the membranes rotate by 180o to expel twist thus forming a defect daughter membrane. The scale bar is 2 μm. Size: 34.1x18.2 μm^{2}. Duration: 100 s. (MOV 303 kb)
Supplementary Movie 2
Coalescence pathway that leads to the formation of an alternating bridgepore array. The scale bar is 10μm. Size: 64.4x49.7 μm^{2}. Duration: 25 s. (MOV 5396 kb)
Supplementary Movie 3
Coalescence pathway that leads to the formation of a pwall. The scale bar is 2 mm. Size: 83.9x45.2 mm^{2}. Duration: 40 s. (MOV 1159 kb)
Supplementary Movie 4
Artificial selfcoalescence induced by optical forces leads to the formation of a pwall. A pwall anchored at both edges remains stable infinitely. However, if the optical trap is released before pwall reaches the distal end of the membrane, the defect retracts. The scale bar is 10 mm. Size: 62.3x30.8 mm^{2}. Duration: 210 s. (MOV 5579 kb)
Supplementary Movie 5
Imprinting of a pwall network into a membrane using optical forces. Scale bar: 10 mm. Size: 105.5x105.5 mm^{2}. Duration: 935 s. (MOV 29841 kb)
Supplementary Movie 6
Imprinting of an alternating bridgepore array into a colloidal membrane with optical tweezers. Once created, these defects are indefinitely persistent. Scale bar: 10 mm. Size: 87.7x23.4 mm^{2}. Duration: 190 s. (MOV 7689 kb)
Supplementary Movie 7
Transforming a pwall into a bridgepore array using optical tweezers. In the regime where both pwalls and ABPAs (alternating bridgepore arrays) are stable, dragging an optical trap along the pwall contour results in its transformation into ABPA. Scale bar: 10 mm. Size: 53.0x31.5 mm^{2}. Duration: 103 s. (MOV 4510 kb)
Supplementary Movie 8
Dualview phase contrast (red) and fluorescence (green) movie illustrating individual rod dynamics in the proximity of a pwall. Rods appear as isotropic spots in the membrane bulk and as elongated lines within the defect, indicating rotation of the rod across the defect. The scale bar is 2 mm. Size: 83.9x25.8 mm^{2}. Duration: 250 s. (MOV 1533 kb)
Supplementary Movie 9
Phase contrast imaging of a fluctuating pwall. Analysis of such fluctuations yields the line tension and effective elasticity of the pwall. The temperature is T=22oC and the dextran concentration is 45 mg mL1. The scale bar is 2 mm. Size: 25.8x8.1 μm^{2}. Duration: 46 s. (MOV 360 kb)
Supplementary Movie 10
Temperatureinduced transition of a pwall into a bridgepore array. The colloidal membrane ([Dextran]=45 mg mL1) is quenched from T=50oC to 22oC. During the quench, pores nucleate at both ends of the pwall and their creation propagates along the pwall toward the center, until the pwall is fully transformed into an ABPA (alternating bridgepore array). Upon increasing the temperature back to T=50oC, the ABPA transforms back to a pwall. The scale bar is 2 mm. Size: 51.6x12.9 mm^{2}. Duration: 1500 (MOV 2520 kb)
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Zakhary, M., Gibaud, T., Nadir Kaplan, C. et al. Imprintable membranes from incomplete chiral coalescence. Nat Commun 5, 3063 (2014). https://doi.org/10.1038/ncomms4063
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