Polymorphic Elastocapillarity: Kinetically Reconfigurable Self

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Interface-Rich Materials and Assemblies

Polymorphic Elastocapillarity: Kinetically Reconfigurable Self-assembly of Hair Bundles by Varying the Drain Rate Dongwoo Shin, and Sameh Tawfick Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b00593 • Publication Date (Web): 08 May 2018 Downloaded from http://pubs.acs.org on May 9, 2018

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Polymorphic Elastocapillarity: Kinetically Reconfigurable Self-assembly of Hair Bundles by Varying the Drain Rate 2

1

Dongwoo Shin and Sameh Tawfick * 1

Department of Mechanical Science and Engineering, The Beckman Institute for

Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, United States 2

Department of Mechanical Science and Engineering, University of Illinois at UrbanaChampaign, Urbana E-mail: [email protected]

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Abstract We report various patterns formed by draining liquid from hair bundles. Hair-like fibers arranged in triangular bundles self-assemble into various cross sections when immersed in liquid then removed. The combinations of their length and the kinetics, represented by the drain rate, lead to various polymorphic self-assemblies: concave hexagonal, triangular, circular or inverted triangular patterns. The equilibrium of these shapes is predicted by elastocapillarity: the balance between the bending strain energy of the hairs and the surface energy of the liquid. Shapes with larger strain energy, such as the inverted triangular bundles, are obtained at the higher liquid drain rates. This polymorphic selfassembly is fully reversible by re-wetting and draining, and can have applications in multi-functional dynamic textures.

Keywords. Fluid-structure Interaction, Dynamic textures, Wetting phenomena, Elasticity, Drying, Imbibition, Coalescence, Interfacial phenomena


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Introduction Shape transforming textures can have useful applications to obtain optical, frictional and mechanical reconfiguration. However, it is challenging to obtain local actuation of small textures. Several previous texture change demonstrations have been previously made towards this goal. For example, simple shape changing pillars can be fabricated using shape memory polymer (SMP) pillars.1,

2

Specifically, SMP pillars are sheared

mechanically by applying an external force. These pillars can recover their original straight configuration by simple heating. These textures can change their shape once between straight and bent configurations, but these changes are not reversible. In these examples, the texture changes can be used to control the contact angle and adhesion of liquid droplets as function of the pillars’ tilting angle.

On the other hand, elastocapillarity has been previously used to program complex texture change; for example to self-assemble carbon nanotubes (CNTs) and nanopillars into various 3D geometries.3-7 Simple wetting and drying can deterministically transform the shape of small CNT pillars and locally organize arrays of polymer pillars into unique micropatterns.8-12 After drying, the final shape of the bundles is retained due to surface forces. This self-assembly process is very useful for fabricating geometrically complex microstructures. The mechanics of these transformations in functional materials such as CNTs are rather complex, and in situ imaging of the dynamics of these geometric transformations are practically inaccessible. This is due to the small scale coupled with the fast kinetics of evaporation of micro- or nanoscale liquid droplets. As a result, there is


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a lack of understanding of the kinetics of elastocapillary self-assembly as a function of drying rate. Recently, there has been interest in the effect of evaporation dynamics on the selfassembly of pillars and hairs. For example, a mathematical model by Vella and coworkers was used to demonstrate that higher evaporation rates can lead to self-assembly of hair into smaller bundle size.12 These results and previous work by the author inspired the investigation of the effect of kinetics (e.g. varying the drain rates) on the selfassembly of one dimensional structures such as hair. To access a scale amenable to optical imaging, we use bundles of several millimeters width and height, comparable to the capillary length, assembled from small diameter slender fibers which are longer than the elastocapillary length. This material system enables the study of drain rates and highlights new self-assembly regimes which have not been observed in the past as is discussed in the details below.

Results and Discussion Here we present results where the combinations of fibers’ length and drain rate lead to various polymorphic self-assemblies. We have observed interesting elastocapillary transformations of the cross sections of fiber bundles as shown in Figure 1. Carbon fibers14 of diameter dhair = 5 µm, modulus E = 200 GPa, and length l = 0.7-2.5 cm are organized perpendicular to the substrate in a two-dimensional array in the form of an equilateral triangle of a few millimeters width. An equilateral triangular configuration is chosen as it is a polygon with the least number of sides that possesses the characteristics of rotational symmetry. In this regard, a triangular shape lays the foundation for the self-


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assembly of other polygons, which could exhibit more complex behavior. The fibers are arranged in 36 tows (as received from the manufacturer). Each tow has approximately 4600 fibers and is fixed in a hole onto a polymer base using an adhesive. The fibers in the tows collectively form the triangular bundle. We estimate the spacing between the fibers to be dspacing = 200 µm. These samples are immersed in a liquid beaker (deionized water mixed with acetone 50%) at an ambient temperature between 20 and 25 °C and humidity between 45 and 55%. The experiments in this manuscript are not affected by ambient evaporation, as we use large liquid volume and fast drain rates. The fibers dynamically self-assemble into a variety of patterns during drainage. At the end of the drainage process, the fibers are densely packed and statically jammed due to friction, and do not change their shape even though thin liquid films are trapped within the bundle. These thin liquid films eventually evaporate from within the bundle, leaving the dry fibers in the final frozen state held by van der Waals forces. The final bundle shape is “fragile”, i.e. if the bundle is mechanically disturbed by touching, the shape will be distorted, as it is held in place only by weak van der Waals forces. Drainage is achieved by moving the filled beaker downwards with a motorized stage. The fibers collectively pierce the surface of the liquid. They are stable against buckling when their length is < 3 cm.15 As they slowly emerge out of the liquid, the triangular bundles transform into a concave hexagon (CH) resembling a star, and upon drying, the shape is retained. The shape can be retained for a long time due to slight adhesion characteristics of commercial carbon fibers which usually have a thin proprietary coating called “sizing” (Supporting Information). After reimmersion, the original triangular organization is immediately recovered and as such this shape transformation is fully reversible. In situ observations of the shape transformations


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are obtained using a camera mounted on the top. Surprisingly, we discovered that when the liquid is drained at higher rates, the triangular hair bundle cross section can reorganize into four other shapes having distinct geometries namely triangles (T), circular (C), three-lobed clubs (CL) and, unexpectedly, inverted triangles (IT) as shown in Figures 1 and 2, and videos in the Supporting Information. The IT especially is an intriguing shape because the corners switch axes with the originally straight edges, thus becoming straighter after drying. It is reminiscent of the axes switching phenomena observed in free liquid jets from an orifice, where the major and minor axes of an elliptic jet switch as the jet emerges from an orifice.16 These self-assembled shapes, which we refer to here as modes, have never been observed and motivated further understanding of these phenomena.

First, we qualitatively explain the elastocapillary shape change. The wet hairs are straight in the liquid (Figure 1). During slow drainage, hairs, which are very flexible, cannot remain straight due to the surface energy cost of the liquid menisci among the hairs as they pierce the surface the liquid in the beaker. The hairs bend and coalesce, leading the system to exchange surface energy with bending energy. Although the hairs are evenly distributed within the initial triangular cross section, the resulting cross section after drainage and coalescence is not necessarily a triangle due to geometry. Due to the number of fibers N in each bundle (~165,000 in our case), in principle there exists a multitude of self-organized shapes exhibiting static equilibrium between fibers’ bending and surface energies. These shapes are referred to here as elastocapillary mode shapes and hence we refer to this as polymorphic self-assembly. Each mode shape, while being


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in static equilibrium, can have a different total strain and surface energy, where the lowest total energy is obtained at slow drain rate. From the experiments, we observe that the CH shape is obtained at the slowest drain rates when the length < 1.7 cm, and the T shape at higher lengths. At high drain rates, other self-assembled shapes are obtained. These include the CL, C and IT as the length increases from short to long hairs.

To further understand the equilibrium of the different shapes are obtained, we systematically varied the length of triangular bundles and the drain speed, and used this data to construct an experimental mode shape diagram for the shape transformations as shown in Figure 2. Short bundles having < 1 cm transform into CH at all speeds tested spanning more than three orders of magnitudes. At lengths > 1.7 cm, the bundles

transform to T at low drain rates. We also used micro-computed x-ray tomography to observe how the cross section gradually changes along the length from the triangular arrangement at the base to the CH or T at the top as shown in the Supporting Information. At higher drain rates (note logarithmic scale on drain rate), other mode shapes are obtained for > 1 cm. In particular, the self-organized C or IT modes emerge when > 1.7 cm and drain rates > 1 cm/s. The interesting shape made of three lobes (CL) is observed at intermediate lengths and drain rates. Overall, it appears that the mode diagram of Figure 2 is divided into regions where each mode shape is obtained, and the regions are divided by contours having negative slopes.

To further examine the static equilibrium of the self-assembled shapes, we construct a mathematical model of 600 fibers initially arranged along the circumference of a


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triangular bundle. The fibers are mathematically represented by rigid straight rods connected to springs at the base.12 This assumption captures the elastic behavior while simplifying the associated complexity of precisely modeling the elastic beam profiles of 600 fibers each with a high order partial differential equation.13, 17 The stiffness of each fiber is calculated from the bending rigidity of the fiber, and is equal to 3/ where

is the fiber’s Young’s modulus, is its second moment of area which is equal to /

64 .18,

19

Importantly, a nonlinear constitutive law governing the deformation of the

circumferential fibers is derived which takes into account the mechanical resistance of the internal fibers inside the bundle (Figure 3). As the bundle shrinks, the hairs move in the radial direction. The fibers on the external circumference contact the internal neighboring fibers, which result in progressively higher resistance to their motion. As the shrinking proceeds, more fibers get in contact resulting in a hardening stiffness in the radial direction due to densification. This model assumes that the deformation of the fibers in the bundle is self-similar along the radial direction. We derived the hardening law for fibers as more fibers get in contact due to shrinkage. The resulting deformation of the circumferential fibers is =

3

2

> 0 . (1)

where F is the force on a circumferential fiber due to bending and successive contact with other fibers, is the inward displacement of circumferential fibers, and is the spacing between fibers. We do not consider the radial deformation outwards (i.e. expanding bundles), which are not observed in the experiments. The shrinkage of the bundle during drying leads to a quadratic hardening effect, which is important for the 8 ACS Paragon Plus Environment

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precise calculation of the strain energy of the drying fibers. The total strain energy increase is the summation of all the fibers’ bending energies as they deform to the new shape (

% = & ' = & )

2

> 0 . (2)

where Σ is the summation of all fibers’ displacements (600 in this simulation), and is the individual fiber displacement from the dry bundle’s straight triangular shape to wet shape in each mode (Figure 3b). As defined, must be strictly positive, which is associated with shrinkage. To obtain the total strain energy associated with a specific mode (e.g. the CH), the individual fiber displacement is numerically calculated from the displacement of 600 equally spaced fibers in the original triangular bundle shape to 600 equally spaced fibers on the final mathematically defined mode shape (Figure 3b).

Having constructed a relation for the total elastic strain energy stored, the static equilibrium of each self-assembly mode can be examined by calculating the surface energy of the liquid for that mode. Simultaneous with storing strain energy, the bundle’s cross section area decreases due to capillary forces as shown in Figure 3b. Any decrease in cross section area during drying is associated with reduction in total surface energy of the liquid and increase in the fibers strain energy. The surface energy in Joules for the liquid in the bundle as it is emerging out of the beaker is + ,- = .. / where / is the

shrinking bundle cross section area due to the liquid surface energy . (Figure 3a and 4a).

After the bundle emerges, each mode shape can be in equilibrium when + ,- = 9 ACS Paragon Plus Environment

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% , as indicated on the y-axis of Figure 4b . This condition results in a unique length associated with the equilibrium of each shape and reads

0-1+ 2 3 +4

:/

, ∑ =5 0 9 2/ 78

. (3) :/

where the elastocapillary length defined as 078 = ;3/. ,
? is used. Plots

are constructed based on this model by systematically varying the bundles shapes from the original triangle to any of the observed shapes (Figure 4). The y-axis of Figure 4b represents the calculated surface energy which is numerically equal to (≡) the calculated strain energy. For example, for the CH, we adopt the simplest shape transformation by gradually increasing the curvatures of the three sides of the bundle as shown in Figure 3b. At each step, the new area / and the fibers’ displacements are calculated based on the geometry. Then, using equations (2) and (3) the surface and strain energy and the equilibrium length can be calculated respectively. The equilibrium curve obtained for each shape corresponds to a specific mode of shape transformation. As stated above, the shape transformation from triangle to CH is simply obtained by adding a uniform curvature to the sides. In particular, by decreasing the radius of the sides from ∞ (straight sides) to smaller radii, the CH transitions from a triangle to a star-like shape. This behavior agrees well with the experiments. It ignores second order effects like rounding of the corners, but represents the concave sides, which are the most important feature of this bundle shape. The numerically computed modes plot immediately shows the existence of multiple equilibria for the elastocapillary self-organization of fiber bundles.


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Further, we compare the numerically computed mode shapes to the measured experimental data. For each experiment shown in Figure 2, we measure the length of each bundle and the resulting equilibrium area As of bundles after drying at the corresponding drain rate. The cross section changes at the slowest drain rate, leading to CH are shown in Figure 4a. The cross section area is measured using ImageJ and plotted on the right y-axis of Figure 4b, using the same x-axis for the measured length. This plot qualitatively validates the mathematical model by comparison to the experiments as evident by comparing the dry to wet area ratio measured from Figure 4a plotted on the right axis of the mode plot. Both the mathematical model and the experiments show that initially the energy and dry bundle area decrease as the length is increased. This intuitive result explains that as the fibers are more compliant, the area is decreased to a larger extent. For small bundle lengths, the CH has the lowest energy and indeed it is obtained in the experiments. This can explained by comparing the fibers’ displacement from the straight triangular bundle to the smaller T or the CH. Specifically, the total fiber displacements to obtain a CH from a triangle is smaller than the displacements to obtain a smaller T if both the T and CH have the same final area. Arrest occurs when the fibers pack with random ordering, which corresponds to dry to wet area ratio of 0.085. Beyond this arrest limit ( > 1.7 cm), a discontinuity is observed where the hairs self-organize into smaller T bundles. The discontinuity in the dry area as the length is increased beyond

=1.7 cm

indicates the existence of distinct elastocapillary modes of deformation. Further, the mode shape of Figure 4b can be used to also analyze the equilibrium of mode shapes experimentally obtained at high drain rates. We add the data points using a similar procedure as the slow drain rates by measuring the height of each bundle (x-axis) and the


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areas for bundles drained at the highest rates (18 cm/s) and plot them against the right yaxis. The snapshots used to compute the area are shown in the Supporting Information. Interestingly, we observe good agreement between the strain and surface energies calculated from the mathematical model, and the shapes observed in the experiments.

It is evident that the higher mode shapes have higher surface area and trapped strain energy, and are observed at higher drain rates. The kinetic energy of the liquid provides the excess energy to drive the fiber displacement into these higher modes. The viscosity plays a role in energy dissipation. To shed more light on the relative contributions of these energies, we have performed experiments with glycerol which has three orders of magnitude higher viscosity than the water-acetone mixture used in the experiments. These experiments result in the same polymorphic self-assembly shapes leading to a very similar mode diagram as shown in Figure 2, except that the boundary between the lower and higher modes is shifted to drain rates higher by approximately a factor of 10. In general, the phenomena of polymorphic self-assembly are robust against variations in viscosity, and the same shapes are observed for water-acetone mixture and pure glycerol. Moreover, the dependence of the measured surface energy of the dry shape on the drain rate seems to be very weak: for example, for > 2.3 cm, the drain rate needs to be increased by 100 folds to get from assembly into T to higher modes like IT, but the dry surface energy and the stored elastic energy is only increased by less than 2 folds. Videos of the shape transformations at high drain rates are included in the Supporting Information. An equilibrium equation at finite drain rate can be written as % + + ,- + B -% − D E+ = 0. As shown earlier, the first two elements of this 12 ACS Paragon Plus Environment

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equation allow the prediction of the static equilibrium polymorphic elastocapillary selfassembly shapes, which is the focus of this work considering (B -% = D E+ = 0). Taking into account B -% and D E+ would allow the prediction of the required drain

rate to obtain each shape, and the construction of the phase diagram corresponding to Figure 2. As an analogy, the compressive buckling mode shapes of a pinned Euler beam, which have the forms of sine waves (half-wave, 3/2 wave, etc..), can be computed by static equilibrium. The frequency associated with each mode is obtained from a dynamic model, which takes into consideration the inertia and viscous dissipation energies. Due to the complexity of the system from the fluid mechanics perspective, we have not been able to extend the simple model to mathematically represent the kinetic and dissipation energy terms in a closed form. As a result, this study does not numerically relate the kinetic energy and viscous dissipation to the elastocapillary self-organized shapes, and hence the drain rates associated with each mode. This will certainly be the subject of future studies.

Conclusion In summary, we report experimental and theoretical study of newly observed elastocapillary phenomena. Triangular hair bundles can self-assemble into a variety of complex patterns when immersed in a liquid then drained. These polymorphic patterns, referred to as elastocapillary mode shapes, include concave hexagons, rounded triangle, circular bundles and even inverted triangle. The pattern selection is kinetically driven by the drain rate. The phenomena are very simple to reproduce, yet, to our knowledge, have never been observed before. We explain the new concept of polymorphic self-assembly


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of fiber bundles, and present a simple mathematical model that captures the essential aspects of the observed phenomena. We explain and analyze these distinct equilibrium elastocapillary mode shapes. We are able to mathematically model the static equilibrium of the shapes. We show that the obtained self-assembly geometries follow the static equilibrium arguments. We confirm that by varying not only the drying rates of the liquid, but also the viscosity to demonstrate that the phenomena are robust. Some newly observed self-assembly shapes like the C and the IT can only be obtained at high drain rates. In the current study, we do not mathematically model the kinetic energy or the dissipation to the drain rate needed to obtain each elastocapillary shape. Such a model, which would enable the prediction of the drain rates and related mode shape selection, is currently under development by the authors. Nonetheless, we speculate that polymorphic self-assembly of fibers can be engineered for stimuli responsive textures with optical, texture, tactile, acoustic and mechanical functionalities similar to previous work on nanopillars.20 This elastocapillary shape memory effect has attractive advantages owing to its simplicity, low cost and high energy efficiency compared to other stimuli responsive material. Finally, we believe that this framework of multiple dynamic equilibria in elastocapillary self-assembly is useful to understand a wide range of problem where rates of self-assembly are found to affect the uniformity and domain size of nanostructures in the field of liquid self-assembly of nanoparticles and CNTs.21


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Acknowledgment This research is supported by the Mechanical Science and Engineering department at the University of Illinois Urbana-Champaign. The authors thank Lauren Kovanko for performing contact angle measurements.

Supporting Information Experimental details and supplementary figures, including four videos.


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Chen, C.-M.; Yang, S. Directed water shedding on high-aspect-ratio shape memory polymer micropillar arrays. Advanced Materials 2014, 26, 1283–1288. Zheng, Y.; Li, J.; Lee, E.; Yang, S. Light-induced shape recovery of deformed shape memory polymer micropillar arrays with gold nanorods. RSC Advances 2015, 5, 30495– 30499. Bico, J.; Roman, B.; Moulin, L.; Boudaoud, A. Adhesion: elastocapillary coalescence in wet hair. Nature 2004, 432, 690–690. Kim, H.-Y.; Mahadevan, L. Capillary rise between elastic sheets. Journal of Fluid Mechanics 2006, 548, 141–150. Pokroy, B.; Kang, S. H.; Mahadevan, L.; Aizenberg, J. Self-organization of a mesoscale bristle into ordered, hierarchical helical assemblies. Science 2009, 323, 237–240. Duprat, C.; Protiere, S.; Beebe, A.; Stone, H. Wetting of flexible fibre arrays. Nature 2012, 482, 510–513. Chakrapani, N.; Wei, B.; Carrillo, A.; Ajayan, P. M.; Kane, R. S. Capillaritydriven assembly of two-dimensional cellular carbon nanotube foams. Proceedings of the National Academy of Sciences 2004, 101, 4009–4012. Pokroy, B.; Kang, S. H.; Mahadevan, L.; Aizenberg, J. Self-organization of a mesoscale bristle into ordered, hierarchical helical assemblies. Science 2009, 323, 237–240. Liu, H.; Li, S.; Zhai, J.; Li, H.; Zheng, Q.; Jiang, L.; Zhu, D. Self-Assembly of Large-Scale Micropatterns on Aligned Carbon Nanotube Films. Angewandte Chemie International Edition 2004, 43, 1146–1149. De Volder, M.; Tawfick, S. H.; Park, S. J.; Copic, D.; Zhao, Z.; Lu, W.; Hart, A. J. Diverse 3D microarchitectures made by capillary forming of carbon nanotubes. Advanced Materials 2010, 22, 4384–4389. Kim, S. M.; Kim, J.; Kang, S. M.; Jang, S.; Kang, D.; Moon, S. E.; Kim, H. N.; Yoon, H. Directional Clustering of Slanted Nanopillars by Elastocapillarity. Small 2016, 12, 3764– 3769. Wang, P.; Bian, R.; Meng, Q.; Liu, H.; Jiang, L. Bioinspired Dynamic Wetting on Multiple Fibers. Advanced Materials 2017, 29. Hadjittofis, A.; Lister, J. R.; Singh, K.; Vella, D. Evaporation effects in elastocapillary aggregation. Journal of Fluid Mechanics 2016, 792, 168–185. Commercial Thornel-T300 PAN fibers having 230 GPa modulus were used as received, they are coated with proprietary polymer sizing to cover surface roughness and increase adhesion to matrix during epoxy impregnation. Chiodi, F.; Roman, B.; Bico, J. Piercing an interface with a brush: Collaborative stiffening. EPL (Europhysics Letters) 2010, 90, 44006. Bidone, G. Exp ́eriences sur la forme et sur la direction des veines et des courans d’eau lanc ́es par diverses ouvertures. Par Georges Bidone; de l’Imprimerie royale, 1829. Taroni, M.; Vella, D. Multiple equilibria in a simple elastocapillary system. Journal of Fluid Mechanics 2012, 712, 273–294.


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Singh, K.; Lister, J. R.; Vella, D. A fluid-mechanical model of elastocapillary coalescence. Journal of Fluid Mechanics 2014, 745, 621–646. Wei, Z.; Mahadevan, L. Continuum dynamics of elastocapillary coalescence and arrest. EPL (Europhysics Letters) 2014, 106, 14002. Grinthal, A.; Kang, S.; Epstein, A.; Aizenberg, M.; Khan, M.; Aizenberg, J. Steering nanofibers: An integrative approach to bio-inspired fiber fabrication and assembly. Nano Today 2012, 7, 35–52. He, X.; Gao, W.; Xie, L.; Li, B.; Zhang, Q.; Lei, S.; Robinson, J. M.; Haroz, E. H.; Doorn, S. K.; Wang, W.; Vajtai, R.; Ajayan, P. M.; Adams, W. W.; Hauge, R. H.; Kono, J. Wafer-scale monodomain films of spontaneously aligned singlewalled carbon nanotubes. Nature Nanotechnology 2016, 11, 633.


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Figure 1: Polymorphic hair self-assembly by dynamic elastocapillarity. (a) Optical images showing the top view of the wet triangular carbon fibers bundle (top panel) made of 36 tows each having 4600 fibers and their side view while immersed in the liquid. The scale bar is 2 mm. (b) Self-assemby of carbon fiber bundles after liquid drainage. Each column represents the polymorphic rate-dependent self-assembly of the same hair sample having a length ( ) in cm. The top optical images show the hair bundle after slow drainage (0.018 cm/s), then the hair is re-immersed in liquid, and the bottom optical images show the hair bundle after fast drainage (18 cm/s). The schematics trace the crosssectional changes for slow and fast drain rates (images at the top and bottom). Observed shapes are labeled as CH: concave hexagon, T: triangle, C: circle, IT: Inverted triangle. At = 2.5 cm and fast drain rate, the triangle inverted such that the corners became less curved than the originally straight edges.


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Figure 2: Observed elastocapillary mode shapes of triangular fiber bundles as function of length and drain rate. (a) Optical images show the top view of bundle shapes observed in the experiments at the various tested bundle lengths and liquid drain rates; (b) and schematics of these shapes drawn on the original bundle cross section (the triangular frame); (c) 3D schematics of the bundles showing the top view fiber self-assembly shape compared to the bottom triangular cross section. CH: concave hexagon, CL: three-lobed club, T: triangle, C: cylindrical and IT: inverted triangle; and (d) experimental mode plot showing the shapes obtained at various bundle lengths and drain rates. Symbol index is shown in (b). 19 ACS Paragon Plus Environment

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Figure 3: Mechanism of bundle shape transformations during self-assembly. (a) Fibers are arranged around the perimeter of a triangular bundle. In the mathematical model, there are 600 fibers, only a few are shown here for visualization. Each fiber is a rigid cylinder attached to a nonlinear translational spring at the base. As the bundle emerges from the liquid, vertical wet fibers (shown in the top view in the inset) coalesce as shown on the bottom. Blue arrows represent the displacement of individual fibers from the triangular perimeter of the bundle to the new shape; and (b) Progression of the bundle’s shape from the wet triangular arrangement to the new dry concave hexagon shape. Radial blue lines represent the deformation of the fibers on the circumference during drying. The gradual transformation is obtained by increasing the curvature of the sides. The values of the dry to wet area ratio (As/Aw) are marked in each shape; and (c) 3D view of the fiber bundle transformation due to surface energy of the liquid as the bundle emerges from the liquid.


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Figure 4: Polymorphic hair self-assembly: elastocapillary mode shapes plot. (a) Optical images of the dry hair bundles having various length (in cm) after slow liquid drain rates (0.018 cm/s) leading to CH. The surface energy of each bundle in the dry state is ~ to the measured cross sectional surface area. The normalized surface energy (dry/wet area) are plotted on the right axis of (b). The scale bar is 2 mm; (b) Comparison between the energy of each elastocapillary mode shape computed by the numerical model (solid lines-left axis) and experiments (data points-right axis) of elastocapillary mode shapes. The equilibrium of each mode shape of triangular bundles is numerically computed based using Eq. (3). Each mode is color-coded and the symbol representing the geometry of each mode is added on the left of each branch. At each length, several equilibria exist having different shapes, and corresponding strain and surface energies. The symbols having a black outline correspond to the slowest drain rate transformations shown in (a), with their dry/wet area ratio plotted on the right y-axis. The symbols without outline (diamonds, triangles, circles and inverted triangles) correspond to the fastest drain rate (18 cm/s) shown in the Supporting Information. As such, the mode shapes predicted by the model agree reasonably well with the shapes observed experimentally.


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Table of Content/Abstract Graphic


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Polymorphic Elastocapillarity: Kinetically Reconfigurable Self

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