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Suppressing the Rayleigh−Plateau Instability in Field-Directed Colloidal Assembly Jonathan L. Bauer, Martin J. Kurian, Johnathan Stauffer, and Eric M. Furst* Department of Chemical and Biomolecular Engineering, Center for Molecular Engineering and Thermodynamics, University of Delaware, Newark, Delaware 19716, United States ABSTRACT: Suspensions of superparamagnetic colloids that equilibrate in a toggled magnetic field undergo a Rayleigh− Plateau instability with a characteristic wavelength λ = 600 μm for the toggle frequency ν = 0.66 Hz. The instability is suppressed when the chamber length L in the field direction is less than 2λ. The final size of the magnetic domains perpendicular to the field, D, follows a power law relation of D ∼ L0.71±0.07. These results demonstrate the structural differences of field-directed suspensions when confined to lengths scale set by the phase separation process and can potentially be used to create self-assembled colloidal crystals with well-defined size and shape.
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instability at L ≈ 1.5λ. At this length, the colloidal columns break up into two distinct rows of droplets within the channel. The spatiotemporal plots are used to determine the time to reach the final, equilibrium structure. Lastly, we analyze the characteristic length scale, D, of the final structure and confirm a power law relationship in which D ∼ L0.71±0.07, which is in excellent agreement with previous findings. These results show that it is possible to suppress the Rayleigh-Plateau instability and self-assemble domains with defined size by confining suspensions of polarizable colloids on length scales commensurate with the characteristic wavelength of the Rayleigh−Plateau instability.
INTRODUCTION Suspensions of dispersed, polarizable colloids typically follow a process of aggregation, coarsening, and arrest in an external field. When the induced dipole−dipole interactions between particles are much stronger than their thermal energy, the particles rapidly form chains parallel to the field.1−3 Interactions between the particle chains lead to further coarsening,4−8 but lateral aggregation eventually halts; particles in the structure are highly localized, and the suspension remains trapped in a system-spanning network that resists further relaxation. This percolated microstructure can be advantageous in smart materials that require rapid changes in viscoelastic properties and underlies the field-responsive fluid−solid transition of electro- and magnetorheology.9 The microstructure characteristic of kinetically arrested polarizable suspensions can be driven to an equilibrium structure of condensed domains if the applied field is instead toggled on and off.10−13 Toggling provides intermittent relaxation events that drive a two-stage coarsening process. Since the colloidal domains within the network have an inherent surface energyparticles at the interfaces of the aggregates experience different magnetic field strengths as those in the bulk4,6,14the columns form elongated domains10,11 that break up by a Rayleigh−Plateau instability.12,15 The equilibrium structure is reached over time scales much longer than the toggle frequency. The instability is characterized by a wavelength, λ, ranging from 0.6 to 1.1 mm, which increases as the toggle frequency increases.15 The wavelength of the Rayleigh−Plateau instability provides a new length scale to investigate the effects of confinement on the coarsening process in toggled fields. In this work, we study the effect of confinement on coarsening suspensions. We show that the evolution of the suspension structure as the sample dimension L in the direction of the applied field changes. Spatiotemporal plots identify the onset of the Rayleigh−Plateau © XXXX American Chemical Society
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EXPERIMENTAL SECTION
Paramagnetic Colloids. Suspensions of polarizable colloids in this work consist of superparamagnetic polystyrene latex spheres (Dynabead MyOne, Catalog #65011, Invitrogen, Carlsbad, CA) dispersed in ultrapure water (minimum resistivity 18.2 MΩ·cm). Superparamagnetic iron oxide nanoparticles (γ-maghemite) are dispersed in the latex particles are arranged in such a way that there is no net dipole moment in the absence of a magnetic field.16 The particles are highly magnetizable, with a magnetic susceptibility χ = 1.3 and saturation magnetization 40 kA/m. They exhibit no remnant magnetization. The particles are 2a = 1.05 ± 0.10 μm in diameter. Sample Chambers. The suspensions are dispersed in sealed glass sample chambers. Depending on the desired degree of confinement, either glass capillaries or hand-built sample cells are used. These geometries are illustrated in Figure 1a,b. Square glass capillaries with an inner dimension (ID) of 0.1, 0.3, and 0.6 mm (Product numbers 8510, 8330, and 8260, respectively; Fiber Optic Center, New Bedford, MA) are used for the smallest chambers. The suspension is introduced into the capillaries by capillary action and the samples sealed with UV Received: February 28, 2016 Revised: May 29, 2016
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DOI: 10.1021/acs.langmuir.6b00771 Langmuir XXXX, XXX, XXX−XXX
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and channels are cut into the film using a template. A glass coverslip is placed on the channel, and the entire chamber is heated on a hot plate to melt the parafin film to ensure a strong seal. The resulting channel height, ∼100 μm, is measured by caliper, and the length is verified by microscopy. The entire chamber is sealed using a UV cure optical adhesive to prevent sample evaporation. Before experiments are conducted, the particles are allowed to sediment (particle density ρ = 1.8 mg/mL) onto the bottom of the chamber. The initial area coverage of the particles is 74%, which forms a disordered submonolayer. For the square capillaries, which have a variable sample chamber height, the suspension is appropriately diluted to maintain the desired initial surface area coverage. Experimental Procedure. The sample chamber is placed at the center of a pair of Helmholtz coils mounted on an inverted microscope (Zeiss Axio Observer.A1). The coils generate a uniform magnetic field perpendicular to gravity and the length of the microscope slide but parallel to the imaging plane. A 4× microscope objective is used to image the suspension. Example images are shown in Figure 1c. Micrographs are taken at 5 s intervals for approximately 1 h using a digital camera (Canon EOS Rebel T2i). The magnetic field strength is held at 1000 A/m and toggled on and off with frequency 0.66 Hz and 50/50 duty cycle using a function generator (Agilent 33220A). Based on the particle size, magnetic properties, and field strength, the maximum interaction energy between particles, or dipole strength, is
Figure 1. (a) Top and side view of sample cell composed of a square glass capillary (ID = 100−600 μm) filled with the magnetorheological fluid. (b) Top and side view of a sample cell (1−2 mm wide and ∼100 μm thick) made with parafilm wax and a coverslip. In both geometries, the chamber is sealed with a UV curable adhesive after it is filled. The magnetic field H is oriented as indicated. (c) Images showing that as the system spanning network at it forms and coarsens. The wavelength λ is the distance between the points of maximum amplitude within the columns, while D is the width of the domains.
Λ=
πμ0 a3χ 2 H2 9kT
≈ 30
(1)
where μ0 is the vacuum magnetic permeability and k and T are the Boltzmann constant and the absolute temperature, respectively. The applied field strength is well below the saturation magnetization of the particles. Also, the magnetic interaction is expected to dominate entropic factors, and the equilibrium state is dictated purely by considerations of minimum magnetic energy.3,17 The minimum-energy state corresponds to a collapse of the system in the form of an
cure optical adhesive (NOA-81, Norland Optical, Cranbury, NJ). For longer channels, chambers are fabricated with lengths between 1 and 2.5 mm. Glass optical microscope slides are covered with parafilm wax,
Figure 2. Images of the suspension’s structural evolution as a function of time and channel length. Suppression of the Rayleigh−Plateau instability occurs when the channel length is less than 1.04 mm. When the channel is greater than ∼1.25 mm, the droplet formation occurs as the suspension forms two distinct bands that recombine as the experiment continues. In the images, the magnetic field H is oriented vertically in the image along the narrow dimension of the channel. B
DOI: 10.1021/acs.langmuir.6b00771 Langmuir XXXX, XXX, XXX−XXX
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Langmuir extremely large bundle. All experiments are performed at room temperature, T = 22 °C.
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RESULTS AND DISCUSSION Previous studies using toggled magnetic fields demonstrated that suspensions of superparamagnetic colloids exhibit a Rayleigh−Plateau instability during the phase separation process.12,15 Concentrated colloidal columns break up into elongated droplet-like domains with the wavelength of the instability determined by the frequency of the toggled field. In these previous studies, the sample chambers had a length of approximately 15 mm, which is more than 8−25× the characteristic wavelength of the instability, 0.6−2.0 mm. Still earlier experiments did not report the existence of the Rayleigh−Plateau instability, possibly because the sample chambers used were 1 mm in length, which is the same order of magnitude as the wavelength.10,11 This discrepancy leads to the hypothesis that confinement of the suspension suppresses the Rayleigh−Plateau instability. In the following, we investigate the onset and dynamics of the instability as the suspension is systematically confined to lengths on the order of and smaller than the Rayleigh−Plateau wavelength. Suspension Structure. In Figure 2, the structural evolution of the suspension is shown as a function of time and channel length, L. In channels of all length, a fibrous, system-spanning network first forms. As the structure coarsens in the toggled field, this network forms distinct macrostructures for three different regions of channel length. For sample cell lengths L = 15.0 mm, elongated droplets form via the Rayleigh−Plateau instability. At 750 s after applying the toggled field, the surfaces of columns within the system-spanning network become wavy. Segments of the columns thin while adjoining segments increase in thickness. The columns separate into individual domains with an equilibrium shape dictated by the balance of surface and bulk magnetic energies,4,10,11 which completes the breakup process. The Rayleigh−Plateau instability persists to about L ∼ 1.55 mm. Here, the formation of a depletion zone in the middle of the channel widens until two distinct bands of ellipsoidal aggregates begin to pinch off, which is clearly visible at 750 s. As time progresses, the domains completely separate, and the colloidal columns break off into two domains. By the end of the experiment, the droplets merge to form larger aggregates scattered about the midpoint of the channel. The final size of an aggregate is dictated by minimizing the different contributions to the magnetic energy within the suspension. More details and analysis on the final size of the domains are presented later. The Rayleigh−Plateau instability is suppressed for experiments performed in channels with lengths from 0.1 to 1.04 mm. The suspension coarsens into distinct domains perpendicular to the applied field and channel walls. Other than an overall contraction, which forms two depleted regions near the walls and a high concentration of suspension in the center of the channel, there are no breakup dynamics observed in the field direction. There is minimal evolution past this initial state, as the percolated network coarsens into a band-like configuration. The suspension does not densify into two distinct bands, and the emergence of a droplet-like phase is not observed. Spatiotemporal plots in Figure 3 complement the micrographs by highlighting the average structural evolution for different channel lengths. The plots show the average pixel intensity perpendicular to the toggled magnetic field. Each column corresponds to individual micrographs taken at 5 s
Figure 3. Spatiotemporal plots of average pixel intensity matrix. The numbers above the images refer to the distance of the channel length in millimeters.
intervals from 0 to 45 min. The rows of each spatiotemporal plot are the average pixel intensity for a horizontal line segment of the image; therefore, each entry in the matrix, Ii,j, is the average intensity of a horizontal line segment for a specific image, Ii,j = (1/n)∑nn=1In where i is the number of vertical pixels in an image, j is the number of micrographs for the experiment, and n is the number of horizontal pixels in an image. The color scale for the images range from maximum intensity, or colloiddense regions, in blue, to minimum intensity, or transmitted light, in red. With the data plotted in this manner, it is possible to visualize the continuous evolution of the suspension structure instead of at discrete time points. The spatiotemporal plots highlight the three breakup regimes. First, there is minimal evolution when L < 1.04 mm as single band of colloid density rapidly forms form the initially dispersed state. The instability has been completely suppressed. For a channel length above 2.07 mm, the macrostructure behaves as if in an unbounded medium; the complete separation of the column leads to distinct bands of dense colloidal domains. More interesting behavior occurs for intermediate channel lengths. For channel lengths between 1.04 and 2.07 mm, two bands begin to form, which is seen as a bifurcation in the pixel intensity plots. In Figure 1c, the growing divide of the structure into two domains is highlighted. Later in the experiments, the two domains recombine or condense together in the center of the channel. This behavior is similar to a stretched filament of Newtonian fluid when there is dynamic interplay between the aspect ratio of the filament and the Ohnesorge number (Oh), a dimensionless parameter that provides the relative importance of the filament’s surface tension and viscosity.18−20 Below a critical aspect ratio, the stretched filament does not break into individual droplets but instead condenses back into a single domain as a result of its high viscosity and low aspect ratio. We previously showed that colloidal columns in a toggled field deform as a Newtonian fluid during the coarsening process on time scales much longer than the toggle period.15 The effective viscosity of a column is determined by the toggle frequency, and the surface energy is set by the field strength. At a toggle frequency 0.66 Hz and a field strength 1000 A/m, the wavelength of the instability λ = 650 μm.15 This wavelength is C
DOI: 10.1021/acs.langmuir.6b00771 Langmuir XXXX, XXX, XXX−XXX
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1250 s. By the end of the experiment, the droplets have merged into larger aggregates in the middle of the channel. Figure 4b shows the formation of the droplet phases in enlarged images. Inducing drop-like phases at lower volume fractions becomes more pronounced when the channel length is increased to 1.40 mm. In Figure 5a, we show the structural evolution for a
measured as the distance between the midpoints of maximum amplitude on the column. Since our experiments operate at constant toggle frequency and field strength, or constant Oh, there should be a minimum aspect ratio that is necessary to induce droplet breakup. Two observations dictate that approximately 2λ is the necessary column length for the Rayleigh−Plateau instability to occur at this given toggle frequency, field strength, and particle volume fraction. First, the percolated network detaches from the sample walls during the initial coarsening process. By the onset time of the Rayleigh−Plateau instability, the colloidal columns occupy only 70−80% of the channel length, L, as illustrated in Figure 1c. Second, there is an extra distance of ∼0.5λ, since the original measurement only takes into account the distance between the points of maximum amplitude. When these lengths are taken into account, a channel length of 1.4 mm, or ∼2λ, should provide sufficient space for the instability to form. This predicted transition point is in excellent agreement with the experimental result at L > 1.25 mm showing the onset of the Rayleigh−Plateau instability. It is also possible to induce different equilibrium domain widths at a constant sample chamber length by varying the volume fraction of the suspension. Wirtz and Fermigier show that decreasing the volume fraction decreases the width of the columnar structures in the percolated network.21 By reducing the thickness, the aspect ratio of the column increases. We hypothesize that it is possible to form droplets for smaller volume fractions at constant channel length. To test this hypothesis, an additional set of experiments are performed for the chamber length L = 1.10 mm. In Figure 4a,
Figure 5. (a) Structural evolution for a suspension as a function of time in a channel length of 1.40 mm. The surface area coverage of suspension is ∼55%. The formation of a depletion zone and individual droplets is apparent, which is not the case for similar experiments with a higher volume fraction and similar channel lengths. (b) Enlarged micrographs showing the densification into two distinct regions and corresponding droplet formation.
suspension that has to a surface coverage of ∼55%. By 750 s, a depletion region in the middle of the channel has formed, as the colloidal columns form two distinct regions of droplets. At the onset of the instability, the average column length is 1260 μm and the average column width is 17.5 μm, and the average aspect ratio is 74.1. Figure 5b shows the instability formation in enlarged images. These results demonstrate an additional way to tune the macroscopic structure, alter the coarsening mechanics, and create colloidal domains with different lengths and widths. Analogous to previous research into filament stretching has shown that, along with the Ohnesorge number, the aspect ratio of the filament is a critical factor for determining whether or not it will form droplets or collapse into a single domain;18−20 by decreasing the volume fraction of the suspension, the width of the colloidal columns decreases, which in turn increases the aspect ratio. We have shown that this change can lead to unstable filaments at a channel length in which stable filaments are observed at a higher suspension area fraction. Finally, we note that above 74% surface coverage the coarsening and breakup slow significantly. Without free volume that permits the colloidal domains to condense, the suspension remains jammed and disordered. Coarsening Kinetics. The spatiotemporal plots are also used to determine the time to reach the final macrostructure configuration by tracking the pixel intensity of the center of the experimental cell. In Figure 6a the average intensities are plotted as a function of time for three representative channel lengths. For all three channels, the intensity is initially low; the particles have sedimented to form a disordered, uniform coverage. After the magnetic field is applied, the colloids rapidly condense, resulting in a greater amount of transmitted light. As
Figure 4. (a) Structural evolution for a suspension as a function of time in a channel length of 1.10 mm. The surface area coverage of suspension is ∼55%. (b) Enlarged micrographs showing the initial instability and corresponding droplet formation.
we show the structural evolution for a lower volume suspension that has a surface area coverage of 55%, or three-quarters of the previous coverage of 74%. The suspension structural follows a similar evolution observed in the earlier experiments: the isotropic suspension first forms a percolated network that coarsens over time. Interestingly, the columns deform and show the initial stages of the Rayleigh−Plateau instability at 750 s. At the start of the instability, the colloidal columns have an average length of 920 μm, an average width of 14 μm, and an average aspect ratio of 70. Because of the increase in aspect ratio due to a decrease in column width, there is full droplet formation by D
DOI: 10.1021/acs.langmuir.6b00771 Langmuir XXXX, XXX, XXX−XXX
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Figure 6. (a) Pixel intensity of the midpoint in the final configuration as a function of time. Exponential fits to the intensity decay (shown in black) identify the time at which the intensity is 5% greater than its long-time value, t5. (b) Plot of t5 as a function of channel width.
the colloidal columns coarsen, the intensity further increases until a peak is reached. At that point, there is a decay to an equilibrium value, as the fibrous network depercolates to form the final, ellipsoidal aggregates. The shoulder of the intensity plot is fit to an exponential function with offset of the form I = I0 + Ae−t/τ, where I is the intensity, I0 is a baseline intensity, and τ is the characteristic time scale of the exponential decay. The fitted functions are indicated by solid black lines in Figure 6a. The variables within the fitted function can then be used to estimate the characteristic time needed to reach a percentage of the final equilibrium value, or offset, of the pixel intensity. We calculate the time, t5, at which the intensity is 5% greater than the offset value. There are two distinct regions when t5 is plotted versus the channel width L. As is shown in Figure 6b, when L ≪ λ, the time to reach equilibrium increases linearly with respect to channel length. In this limit, the macrostructure does not progress further than a coarsened dense region in the channel center. In the second regime, when L ∼ 2λ, there is a significant increase in the time necessary to reach the equilibrium state. The Rayleigh−Plateau instability becomes apparent as the MR fluid can undergo further coarsening at this channel length. The coarsening time remains constant as L ≫ λ. The inflection point between the two regimes of t5 versus channel length is at L ∼ 1.30 mm. This value is in excellent agreement with our observation that the channel needs to be 1.4 mm wide to observe the Rayleigh−Plateau breakup process when taking into account the detachment of the columns from the channel walls and the additional area occupied by the colloidal columns. Size of Final Microstructure Domains. Channel length also influences the final domain structure of the suspension. As shown earlier, the droplets formed during the Rayleigh−Plateau instability merge together to form larger aggregates. To quantify the characteristic width of the aggregates at t = 1800 s, we perform a power spectrum analysis on a line segment 1 pixel wide perpendicular to the applied magnetic field. The power spectrum is analyzed and averaged over an area with dimensions of 0.08 mm × 3.1 mm to keep the size consistent with the minimum channel length of 0.1 mm. The peak in the average power spectrum identifies the wave vector of maximum intensity, qmax, corresponding to the average size of the colloidal domains perpendicular to the toggled field. In Figure 7a, we plot qmax as a function of channel length. The decrease in the size of the wavevector indicates that the
Figure 7. (a) Maximum intensity wavevector from an image’s average power spectrum is plotted as a function of channel length. The domain width increases with increasing channel length. (b) Domain width as a function of channel length exhibits a power law relationship with a scaling exponent 0.71 ± 0.07. The inset line with a slope of 0.67 is the theoretical prediction.
final aggregate size increases as channel length increases. The characteristic real space size of the domains can be determined by taking the reciprocal of qmax, which is plotted in Figure 7b as a function of channel length. For the lengths tested, the width of the domains, D, varies with the length of the channel as a power law relation D ∼ Ln. For these experiments the scaling exponent n ≈ 0.71 ± 0.07 is in excellent agreement with theoretical predictions of of n = 0.67 and previous experimental results.4,6,7 With this power law relationship, it is possible create aggregates with well-defined size and shape at precise intervals. E
DOI: 10.1021/acs.langmuir.6b00771 Langmuir XXXX, XXX, XXX−XXX
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(13) Swan, J. W.; Vasquez, P. A.; Whitson, P. A.; Fincke, E. M.; Wakata, K.; Magnus, S. H.; Winne, F. D.; Barratt, M. R.; Agui, J. H.; Green, R. D.; Hall, N. R.; Bohman, D. Y.; Bunnell, C. T.; Gast, A. P.; Furst, E. M. Multi-scale kinetics of a field-directed colloidal phase transition. Proc. Natl. Acad. Sci. U. S. A. 2012, 109, 16023−16028. (14) Cutillas, S.; Bossis, G.; Cebers, A. Flow-induced transition from cylindrical to layered patterns in magnetorheological suspensions. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1998, 57, 804−811. (15) Bauer, J. L.; Liu, Y.; Kurian, M. J.; Swan, J. W.; Furst, E. M. Coarsening mechanics of a colloidal suspension in toggled fields. J. Chem. Phys. 2015, 143, 074901. (16) Fonnum, G.; Johansson, C.; Molteberg, A.; Morup, S.; Aksnes, E. Characterisation of Dynabeads (R) by magnetization measurements and Mossbauer spectroscopy. J. Magn. Magn. Mater. 2005, 293, 41−47. (17) De Gennes, P. G.; Pincus, P. A. Pair-correlations in a ferromagnetic fluid. Kondens. Mater. 1970, 11, 189. (18) Schulkes, R. M. S. M. The contraction of liquid filaments. J. Fluid Mech. 1996, 309, 277−300. (19) Notz, P. K.; Basaran, O. A. Dynamics and breakup of a contracting liquid filament. J. Fluid Mech. 2004, 512, 223−256. (20) Castrejón-Pita, A. A.; Castrejón-Pita, J. R.; Hutchings, I. M. Breakup of Liquid Filaments. Phys. Rev. Lett. 2012, 108, 074506. (21) Wirtz, D.; Fermigier, M. One-dimensional patterns and wavelength selection in magnetic fluids. Phys. Rev. Lett. 1994, 72, 2294−2297.
CONCLUSION MR fluids exposed to toggled magnetic fields undergo a Rayleigh−Plateau instability as the suspension undergoes phase separation. We studied the macrostructural phase transitions when the MR fluids is confined to distances commensurate with the characteristic wavelength, λ, of the instability. If the channel length is greater than 1.5λ, the instability becomes apparent as the colloidal columns densify into two distinct regions. The time to reach the final macrostructure follows a sigmoidal relationship and has an inflection point near the experimentally and theoretically predicted transition point of 1.5λ. The size of the final aggregates follows a power law relation with respect to channel length that agrees well with previous theoretical and experimental research on confined MR fluids. This work extends previous research on the phase transitions of confined MR fluids to larger length scales that are relevant to an instability observed during the self-assembly process. By doing so, we have demonstrated that it is possible to suppress the Rayleigh−Plateau instability and self-assemble aggregates with a well-defined size and spacing.
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AUTHOR INFORMATION
Corresponding Author
*E-mail
[email protected]; Ph 302-831-0102; Fax 302-831-1048 (E.M.F.). Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was made possible by support financial from NASA (Grants NNX10AE44G and NNX16AD21G). E.M.F. acknowledges partial support from ETH Zürich and the Swiss National Science Foundation.
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REFERENCES
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