Solid microscopic rings formed via wetting and subsequent dewetting

Abstract

We report the spontaneous formation of rings when a colloidal dispersion, containing silica-coated iron-oxide particles and the liquids ethanol and ethoxylated trimethylolpropane triacrylate, is deposited within micron-sized PDMS wells. Just after filling, the interface between air and the dispersion is a meniscus dictated by the dispersion's contact angle on PDMS. Upon evaporation of ethanol the meniscus lowers and, if a critical volume is reached, a rupture process is initiated and the dispersion adopts a ring morphology. The final dispersion consists only of particles and ethoxylated trimethylolpropane triacrylate that can be reticulated to solidify the ring geometry. The colloidal particles within the dispersion are essential for the stability of the rings prior to the reticulation. Here, by using iron-oxide based colloidal particles we fabricated superparamagnetic rings, opening up new avenues for applications. The dimensions of the rings can be tuned by adjusting both the size of the PDMS wells and the amount of ethanol in the dispersion. In this manner it is possible to fabricate rings with annuli smaller than a micron – a scale below the lower limit of standard lithography. Calculations assuming an equilibrium contact angle of ethoxylated trimethylolpropane triacrylate on PDMS reproduce the experimental results strikingly well.

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Rings with dimensions of the micron and nano-scale have numerous properties suitable for application. For instance, they can tune plasmon resonance frequencies in the near infrared and when arranged in an array with continuous wires can constitute a material with negative values of effective permeability and permittivity.^1–3 Such ring-based arrays can also be used for compact optical encoding/decoding devices with applications in fiber optic communication.⁴ Further, under shear flow, a micron-sized ring is a non-tumbling structure that perfectly aligns with the direction of shear.⁵ This capacity to perfectly align can be used to make materials with a built in anisotropy and can also lead to process advantages.

Numerous methods have been employed to make rings with the dimensions discussed above, including the use of Pickering emulsions and colloidal crystals as templates,^6–9 the micro-confinement of the coffee ring effect,^10,11 the manipulation of magnetic colloids with static and oscillating fields,^12–15 the evaporation of droplets containing salt or nano-particles,^16,17 and the active self-assembly of microtubules.^18,19 As a general rule, the complexity of the fabrication strategies to form such rings increases as their required size shifts from the micron to the nano scale. For instance, whereas through the employment of lithography-based methods it is relatively straightforward to produce rings with an annulus width (the difference between the inner and outer radii) of 2 microns or above, taxing procedures, such as proton focusing or colloidal crystal templating, are necessary to reach sub-micron sizes.^20–23

In this communication we present a method that employs standard lithography but nevertheless can produce rings with sub-micron annuli and diameters ranging from 2 to tens of microns. A transition driven by a rupture process and then de-wetting explains the emergence of the rings within the molds. Indeed, simple calculations based on this premise produce results that match the experimental data extremely well. This process can be thought of as a confined pattern formation of an unstable liquid film on a solid substrate.²⁴

In the following we will commence by elaborating on the practical steps to fabricate the rings and the factors that we can manipulate to tune their dimensions, after which we will proceed to describe the rupture and de-wetting process that we assert drives their formation.

The first step towards the production of the rings was the fabrication of a PDMS mold that holds arrays of micron-sized wells. This step was achieved by following standard lithographical protocols as outlined in ref. 25. After surface modification of the mold with a fluoropolymer, it was filled with a dispersion of silica-coated superparamagnetic particles (Ademtech, radius = 150 nm) in a mixture of ethanol and ethoxylated trimethylolpropane triacrylate (ETPTA + 10 wt% photoinitiator 2-hydroxy-2-methyl-1-phenyl-propan-1-one (Sigma)). Deposition of the dispersion within the micron-sized wells took place by gently pulling a drop of the mixture across the mold leaving the bases of the wells fully covered by the mixture. At this juncture, the material within the wells adopted a well-spanning meniscus, i.e. a curved air–liquid interface. However, after a time, typically on the order of seconds, a transformation to a ring morphology was observed (Fig. 1A and B). The rings seemingly ‘popped’ into existence in less than 1 ms without any observable intermediate state (Fig. 1B). No subsequent change in their shape was observed. The rings were then solidified by reticulating the ETPTA with UV light and their final composition consisted of an ETPTA matrix packed full of the superparamagnetic colloids. A schematic of our fabrication procedure is presented in Fig. 1A and a SEM image of the solidified structure in Fig. 1C (right).

Fig. 1 (A) The fabrication of micron-sized rings. A PDMS mold holding an array of micron-sized wells is filled with a colloidal dispersion of silica covered iron-oxide particle in ethanol and ETPTA. After spontaneous relaxation into a ring shape the dispersion is reticulated with UV light to form the solid rings. The rings are “waxed” out of the mold using a tacky polymer, poly(1-vinyl-pyrrolidone-co-vinylacetate, as outlined in ref. 25. (B) A montage of bright-field micrographs showing ring formation within PDMS micro wells of 5.5 microns in diameter. The time between images is 1, 21, 55, 87, 137 and 348 ms. The scale bare is 60 microns. (C) Top: rings extracted from a PDMS mold and subsequently embedded within a polymer. The rings maintain the arrangement they held within the mold and are positioned in a square array. Bottom: a SEM micrograph of a ring showing a dense packing of colloidal particles within its polymer matrix. The scale bar is 60 microns.

To extract the solid rings from the molds a waxing technique was employed as detailed in ref. 25. This technique maintains the arrangement of the rings within the mold and, on removal, the rings were positioned in a large scale square array (Fig. 1C, left). On dissolution of the waxing polymer and application of a magnetic field, rings extracted in this manner self-organised into a square-centred lattice of dimers (ESI, S1†).

To investigate how the radius of the well influences ring formation we fabricated wells of the same depth (4 microns) and radius from 1 to 10 microns and filled the wells with the same precursor dispersion containing 75, 17.5 and 7.5 V% of ethanol, ETPTA and particles respectively. Two things were evident from this experiment: firstly, as the well radius decreases, there is a concurrent reduction in the thickness of the annulus and secondly, there is a lower limit in well radius (5 microns) beyond which spontaneous ring formation does not take place (Fig. 2A).

Fig. 2 (A) Left: brightfield micrographs of rings and discs formed within PDMS wells of different radii. The wells have a constant depth (4 microns). For wells with radii below 5 microns (left, bottom) ring formation ceases. The rings and discs were made by employing a precursor dispersion containing 75, 17.5 and 7.5 V% of ethanol, ETPTA and particles respectively. The scale bar is 20 microns. Right: a plot of annulus width versus PDMS well radius showing both experimental (circles) and calculated data (line). Stable morphological solutions to the calculations as a function of well radius are menisci only (yellow), a coexistence state, where menisci or rings are stable (red), and rings only (blue). (B) Left: brightfield micrographs of rings made within molds having a radius and depth of 4.5 and 4 microns respectively. 25 (top image), 14.3 (middle) and 7.77 (bottom) V% of ETPTA-particle were used in the precursor dispersions and in all cases the ETPTA: particle ratio was 2.33 : 1. Right: a plot of annulus width versus the combined volume fraction of ETPTA and colloids employed in the precursor dispersions showing experimental (circles) and calculated data (line). Stable morphological solutions to the calculations as a function of volume fraction of ETPTA and colloids are rings only (blue), a coexistence state, where menisci or rings are stable (red), and menisci only (yellow). Error bars for (A) and (B) are the standard deviation of width around a given ring's annulus. (C) A brightfield micrograph showing rings with an annulus width of under one micron. These rings were fabricated by employing high concentrations of ethanol (97 V%) and shallow wells (2 microns) with small diameters (1.8 microns). The scale bar is 2 microns.

The width of the annulus can also be tuned independently of well radius by varying the quantity of ETPTA-particles within dispersions. This relationship is displayed in Fig. 2B where, as the volume percent of ETPTA-particles increases from 7.8 to 25, the width of the annulus increases from 1.4 to 2.9 microns. Note that the volume ratio of ETPTA to particles is always 2.33 to 1.

By combining the trends described above, i.e. by employing molds with small diameters (∼2 microns) and colloidal dispersions with small combined amounts of ETPTA and particles (3 V%) we are able to produce rings that have dimensions smaller than what is typically achievable with standard lithography and, as shown in Fig. 2C, it is possible to prepare rings with annuli of sub-micron width.

In summary of the experimental elements of this work, our fabrication procedure grants considerable and facile control over the diameter and the width of formed rings. Furthermore, because they are formed within large PDMS arrays their large-scale fabrication is possible which, given their dimensions, is not routine.

Whilst we have outlined the practical steps to fabricate the rings and to control their dimensions, we have still to offer any rationale for their emergence within the PDMS wells. We postulate that, due to the evaporation of ethanol from the dispersion, a meniscus is continuously lowered to the base of the well, after which a central hole propagates to leave a ring structure (Fig. 3A). As such our fabrication procedure can be categorised as a controlled evaporation self-assembly process. In controlled evaporation self-assembly droplets containing non-volatile materials are deposited within confined geometries in order to modulate evaporation and flow and hence bring about the assembly of a defined structure.^26,27 We assert that this meniscus ring transition is driven by de-wetting. That our precursor dispersions are liquid-like is confirmed by our observation that they retract into a spherical cap after being spread on a surface.

Fig. 3 (A) A schematic of the proposed mechanism of ring formation. The evaporation of ethanol from the dispersion drives the meniscus of the mixture to the base of the wells and a hole propagates to form a ring structure as a result of a de-wetting transition. (B) Cross section profiles of a well-spanning meniscus that just touches the well base, M0, (dashed blue line) and annular drops (red and black). The parameters employed in the calculations are shown, namely, the arc length, s, angle to horizontal along s, φ, the contact angle, θ_c, and the radius and depth of the wells R_w and D_w respectively. The contact angle is 60°. On touching the base of the well, the well-spanning meniscus will always transform into an annular droplet. At high normalised volumes, annular droplets can have cross-sections with non-constant cross-sectional curvatures (black) as it is the mean curvature (composed of the cross sectional curvature and the curvature in the horizontal plane) which is constant. At lower normalised volumes, as typically sampled in this work, the cross-sectional curvature is almost constant (red). (C) A normalised energy map of menisci and rings as a function of normalised fluid volume. Energies and volumes are normalised by the energy and volume of meniscus, M0, respectively. The solid and dashed lines correspond to the normalised energies of rings and menisci. At normalised volumes below unity only rings are stable (blue region). Above unity but below a normalised volume of 2.5 both rings and menisci are equilibrium structures (red) and above 2.5 only menisci are stable (yellow).

Strong indication that ring formation is a de-wetting phenomenon is provided by the timescale of the well-spanning meniscus to ring transition. As discussed above, the rings seemingly ‘pop’ into existence from an initial disc-like shape within a time shorter than the inverse frame rate of the camera employed in our experiments i.e. <1 ms. A shape change driven by de-wetting would take place in a time approximated by μL/γ, where μ is viscosity (∼90 mPas here, estimated using the de Kruif relationship²⁸), γ is interfacial tension (∼25 mN m⁻¹) and L is a length scale (∼10 microns). This approximation gives ∼0.01 milliseconds consistent with the absence of any observed intermediate structure.

To further test our hypothesis we calculate the energy and shape of the liquid present in the circular wells, as it progresses from well-filling to annular-based conformations. In doing so we identify the lowest energy morphology for a given volume of material and find that propagation of a hole once the meniscus reaches the well base is energetically favourable.

We simplify the calculation by assuming fast evaporation of the ethanol from our dispersions which allows a unique contact angle of the ETPTA-silica mixture after the loss of ethanol on PDMS to be used. Using a drop shape analyser (Krüss DSA100) we measured this contact angle to be 60°. Note that after full evaporation of ethanol the solid content of our precursor liquid never goes beyond 30 V%, hence it is valid to treat our dispersions as liquid at all stages of the outlined morphological transition. Our system is also sufficiently small that we can safely neglect gravity: previous investigations have confirmed that an ETPTA-particle solution has a constant curvature at the air–liquid interface (S4†) consistent with this assumption.²⁵

At equilibrium, the liquid within the well adopts a shape whose surface energy is minimal. The geometry has a cylindrical symmetry and thus we will describe this surface in the vertical plane perpendicular to the bottom of the well that contains the axis of symmetry (see Fig. 3B). The minimal energy surface has a constant mean curvature H = (κ₁ + κ₂)/2 with principal curvatures κ₁ in the vertical plane and κ₂ around the axis of symmetry.²⁶ The boundary conditions on this surface are that it respects the contact angle, θ_c, at the wetted walls and base. These conditions are sufficient to define completely the shape of the annulus for a given volume.

In practice, we use custom-written code on MATLAB to determine annulus shape. To determine the droplet shape, we first make the problem dimensionless by rescaling all lengths by the well radius. We then fix the inner radius of the annulus and the contact angle at the base of the well, and use an iterative procedure, varying the mean curvature H to match the boundary condition on φ at the edge of the well. At each iteration, we solve the equation of constant mean curvature numerically using the method of finite differences. To do this, we use the arclength–angle (s–φ) parametrization in the vertical plane, with arclength s and angle to the horizontal φ. At the base of the well s = 0 and φ = θ. In this parametrization, the principal curvatures are κ₁ = −dφ/ds and κ₂ = −sin(φ)/R where R is the horizontal distance from the central axis of the well.

This procedure gives a series of annuli of varying inner radius, each of which is an equilibrium shape for a given final volume. In Fig. 3 we show examples of annulus cross sections together with the shape of a meniscus, M0, which just touches the base of the well. In the same figure we show calculations of the surface energy of each of these annuli, together with menisci of the same volumes. In Fig. 3C we plot the surface energy of equilibrium shapes against their volume normalised, respectively, against the energy and volume of M0. Above a normalised volume of 2.5 only menisci are possible (yellow) whereas volumes less than 2.5 but higher than unity correspond to a coexistence ‘phase’ where both rings and menisci are stable states (3C, red region). For normalised volumes below unity, rings are the only equilibrium state (blue). For further information regarding our calculations, particular on how the surface energy is calculated, please see ESI S2.†

We now connect our dimensionless calculations to our experimental data. If we assume that the wells are initially totally filled with the ethanol–ETPTA-colloid mixture, then the final volume V_f will be given by V_f = πR_w²D_wv, where v is the volume fraction of ETPTA-colloid solution, and R_w and D_w are the well radius and depth respectively. Rescaling our normalised, numerical calculations then gives a unique prediction for the ring's width as a function of R_w and v. In both cases, there is an excellent match with the experimental dependence of annulus width on well radius (Fig. 2A) and volume fraction (Fig. 2B). Note that in the darker red regions of Fig. 2A and B, both rings and menisci are stable solutions to the surface energy minimization, but the ring is the more stable state (see Fig. 3C). In this region, as the volume in the well is lowered, one could imagine that the fluid, which initially has the form of a meniscus, would remain a meniscus, as the base of the meniscus would never touch the bottom of the well. However the distance between the base of the meniscus and the base of the well is, based on our calculations, at most ∼400 nm in this region, i.e., on the order of the diameter of the colloidal particles. We thus suggest that the particles will favour the break-up of the meniscus by forming a discontinuity of the liquid between the upper surface and the base of the well. Indeed, Thiele et al. have demonstrated how particles can promote film rupture at elevated thickness due to coupling of their local concentration and the local film height.²⁹ In addition, non-equilibrium phenomena such as Marangoni flows, (which are not taken into account in our model calculation) may become relevant during ethanol evaporation and promote meniscus break-up.

It is important to mention that liquid rings may themselves be unstable due to a Plateau–Rayleigh type instability.³⁰ Experiments repeated with mixtures devoid of particles resulted, as in the presence of the particles, in a disc-to-ring transition. However, unlike with particles, this transition was promptly followed by a further transition to a moon-like shape (S3†). A cylinder of fluid is unstable with respect to axial sinusoidal perturbations of its radius if the wavelength λ of these perturbations is greater than a critical wavelength λ_c = 2πb, where b is the radius of the cylinder.³¹ Without going into detailed calculations, we can take our rings to effectively be segments of fluid cylinders which have been wrapped up around the inside of the circular well. The longest wavelength perturbation they can support is λ_max = 2πR, the circumference of the well. If we take the ring width, w, as equivalent to the cylinder radius b, we always have b < R, meaning that λ_max > λ_c. Therefore, at this rough level of approximation, we would expect liquid rings to always be unstable, in that wavelength perturbations larger than λ_c can exist – as we find in the case where particles are absent. The mechanism by which particles preserve the annulus shape is probably a combination of increased viscosity of the dispersion that would slow down the dynamics of ring break up and jamming of the particles which could occur due to the secondary evaporation of ETPTA.

We have generalised our fabrication method to particle designs made with non-circular wells (S4†). Here, we noticed that the precursor dispersion tends to wick into the corners of the wells. This wicking provides further evidence that de-wetting is the important ingredient in our method. Furthermore, by a second filling of the wells with an ETPTA solution devoid of magnetic particles we could fabricate composite shapes with both magnetic and non-magnetic parts. In other words, we can produce ‘click’ particles which can assemble at predefined regions (tips of the star in S4† for example) – an attribute that can be employed for controlled self-assembly. Our work in this area will be detailed in a subsequent publication.

In summary, we have outlined in this communication a new method based on standard lithography to fabricate rings on the micron scale. The rings form spontaneously when a dispersion of silica colloids in the liquids ethanol and ETPTA is deposited within micron-sized PDMS wells. We are able to control the ring geometry through variations of well dimension and ethanol content within the dispersion. By filling wells of relatively small diameters with dispersions of high ethanol content we are able to form rings with sub-micron dimensions.

We propose that ring formation is a consequence of two features of our experiment. The first is the loss of ethanol from the dispersion due to its evaporation. This reduces the filling volume of the dispersion within the wells and lowers menisci, triggering a de-wetting transition from a uniform meniscus to a ring. The second is the solid content within the dispersions that maintains the otherwise unstable ring morphology. Calculations based on this premise and on the wetting characteristics of ETPTA on PDMS reproduce the experimental results very well.

Acknowledgements

We would like to thank Joost de Graaf for advice on calculations and the Fondation Pierre Gilles de Gennes pour la recherche for funding Joe Tavacoli.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra11136j

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