Dynamic capillary assembly of colloids at interfaces with 10,000g accelerations

Extreme deformation of soft matter is central to our understanding of the effects of shock, fracture, and phase change in a variety of systems. Yet, despite, the increasing interest in this area, far-from-equilibrium behaviours of soft matter remain challenging to probe. Colloidal suspensions are often used to visualise emergent behaviours in soft matter, as they offer precise control of interparticle interactions, and ease of visualisation by optical microscopy. However, previous studies have been limited to deformations that are orders of magnitude too slow to be representative of extreme deformation. Here we use a two-dimensional model system, a monolayer of colloids confined at a fluid interface, to probe and visualise the evolution of the microstructure during high-rate deformation driven by ultrasound. We observe the emergence of a transient network of strings, and use discrete particle simulations to show that it is caused by a delicate interplay of dynamic capillarity and hydrodynamic interactions between particles oscillating at high-frequency. Remarkably, we find evidence of inertial effects in a colloidal system, caused by accelerations approaching 10,000g. These results also suggest that extreme deformation of soft matter offers new opportunities for pattern formation and dynamic self-assembly.

Particles floating at liquid interfaces are a useful two-dimensional model to visualise the structure and deformation of condensed matter. A fascinating example dates back to L. Bragg, who used rafts of floating bubbles to illustrate grain boundaries and plastic flow in metals [1].
Colloidal particles confined at liquid interfaces have been used widely for this purpose, as the interparticle interactions can be finely tuned through electrostatics [2,3] and capillarity [4,5,6], giving access to a range of two-dimensional condensed phases [7,8,9,10,11]. The possibility to visualise the structure of the interfacial assembly by optical microscopy has enabled the study of self-healing of curved colloidal crystals [10], of crystal growth [12] and freezing [13] on curved surfaces, and of dislocations under stress [14]. Beyond their use as two-dimensional model, colloid monolayers at interfaces, and interfacial soft matter in general, play an important role in natural and industrial processes [15]. For instance, the mechanical strength imparted to the interface by the colloids enables the formation of bicontinuous emulsions by arrested spinodal decomposition [16], suppression of the coffee-ring effect [17], and arrested dissolution of bubbles [18,19].
Previous studies of dynamic deformation of colloid monolayers have been limited to relatively low deformation rates [20,8,21,11,22,23] in the range 10 −2 − 1 s −1 . However, in realistic conditions, such as in the flow of emulsions and foams, and the evaporation of suspensions, interface deformations can occur on much shorter timescales, driving the system far from equilibrium. In bulk suspensions under flow, colloids form out-of-equilibrium structures stemming from the interplay of interparticle and hydrodynamic interactions [24]. Similarly, hydrodynamic interactions between colloids at interfaces can be expected to affect their assembly upon dynamic interface deformation. In more extreme conditions, phenomena that are usually not observed in a colloidal system can become important, for instance as elastic collisions during shock propagation [23].
Furthermore, some of the phenomena observed for interface deformations of large amplitude, or at high rate, have no counterpart in three-dimensions: when compressed beyond hexagonal close packing, a monolayer of colloids at an interface can buckle out of plane [25,26] or expel colloids in the surrounding fluid [27,28]. Yet, the behaviour of interfacial soft matter under extreme deformation remains poorly understood due to the experimental challenge of simultaneously imparting high-rate deformation and visualising rearrangements of the microstructure.
Here we use acoustic excitation of particle-coated bubbles to explore the far-from-equilibrium phenomena of colloid monolayers at fluid interfaces. Bubbles (equilibrium radius R 0 ≈ 20 − 100 µm) were coated with a monolayer of polystyrene spheres (radius a ≈ 1−5 µm). Electrostatic repulsion between the particles at the interface was completely screened by addition of electrolyte (Methods), so that the initial microstructure of the monolayer at moderate surface coverage was determined by capillary attraction, due to nanoscale undulations of the contact line with an estimated amplitude Q 2 ≈ 50 nm [29,30]. This interaction is directional, as can be seen from a decomposition of the interface deformation in two-dimensional multipoles, which shows that contact line undulations result in capillary quadrupoles [29]. Isolated bubbles were driven into periodic compression-expansion by ultrasound at frequency f = 30 − 50 kHz in an acousticaloptical setup (Methods), and imaged at 300,000 frames per second. From the high-speed videos, we extracted the evolution of the bubble radius R(t), the maximum oscillation amplitude ∆R, and the trajectories of the particles on the surface of the bubble (Supplementary Information).
The surface coverage Φ = N πa 2 A , where N is the number of particles in the region of interest, and A is the surface area of the region of interest ( Supplementary Information, Figure S1), and the equilibrium bubble radius, R 0 , were measured from high-resolution still images. Figure 1a shows a sequence of frames during one cycle of oscillations of a 65-µm bubble coated with 5µm particles at Φ = 0.54 ± 0.04. Particles initially at contact are driven apart during bubble expansion, and pushed back into contact during compression, as is clearly seen for the two particles marked in red and blue. The separation of the particles typically occurs in less than 10 µs.
A striking change in the microstucture of the monolayer is observed over a few hundreds of cycles of oscillations, as shown in Figure 1b. Initially the particles, 2.5 µm in radius and with surface coverage Φ = 0.48 ± 0.05, form a disordered, cohesive structure on the interface of a bubble with equilibrium radius R 0 ≈ 53 µm. The arrangement of the particles evolves towards a network of strings in a few hundreds of cycles (Supplementary Movies 1 and 2). The break-up of the initially aggregated structure is made possible by the mechanical energy input provided to the system during high-rate oscillations. The quadrupolar capillary attraction energy is E quad ∝ γQ 2 2 ∼ 10 4 − 10 5 k B T at contact [31], resulting in kinetically trapped structures. For comparison, the kinetic energy of a particle in our experimental conditions is E k ∼ a 3 ρf 2 ∆R 2 ∼ The network of strings is a transient microstructure that relaxes after the forcing stops.   neither of which are observed at rest, also disappear upon relaxation.
A quantitative characterisation of the microstructure shows that the formation of strings is accompanied by a decrease in the number of nearest neighbours, n, as can be seen by comparing the initial (Fig. 2a) and final ( Fig. 2b) states of the experiment of Figure 1b. In Figure 2a-b, the particles are colour-coded according to the value of n (Supplementary Information). Initially, particles mainly have 3-5 neighbours. After 1000 cycles of oscillations, particles predominantly have 2 or 3 neighbours. The probability p(n) of a particle having a number n of neighbours is shown in Figure 2c for the initial state, and in Figure 2d for the final state. The time evolution of p(n) over 1,000 cycles is shown in Supplementary Information (Figure S3). The statistics show clearly that particles can have up to 5 or 6 neighbours in the initial state, whereas in the final state p(n = 5) and p(n = 6) become zero, and there is a sharp increase in p(n = 2). Correspondingly, the mean number of neighbours per particle decreases fromn = 3.4 tō n = 2.4. Not only the number of neighbours is reduced, but the neighbours are aligned, as indicated by the increase of the bond order parameter |Ψ 2 | over time, as shown in Figure 2e (see Supplementary Information). Interestingly, we also observe an increase of the |Ψ 3 | order parameter, representing particles having three neighbours organized with a sp 2 symmetry. The final structure is therefore formed by particles aligned in strings that are connected respecting a sp 2 symmetry. The evolution of the peaks in the pair correlation function, g(r), shows that the nearest neighbours remain at contact (r = 2a) while the second neighbours move from r = 2 √ 3a, corresponding to hexagonal close packing, to r = 4a, corresponding to a chain ( We propose that the directional interparticle force leading to the transient formation of strings is due to dynamic capillarity. Hydrodynamic interactions alone, responsible for string formation in the bulk [24], are not sufficient, because they are repulsive for oscillating particles at a fluid interface [32]. Capillary interactions with dipolar symmetry would be sufficient to drive the formation of strings at fluid interfaces [33], but they can be ruled out in our system. Indeed, although dynamic capillary dipoles can be induced by the motion of particles along an interface between two fluids with a large viscosity mismatch [34,35], in our experiments the viscous stresses due to the lateral motion of the particles are negligible compared to surface tension forces. Another possibility for a directional interaction with dipole-like symmetry is that between a monopole and a quadrupole [36] (Supplementary Information, Figure S8). The possibility of a monopolar deformation of the interface in our system is not immediately apparent, because it p(n)  Figure 1b, with colour-coding of the particles according to the number of neighbours n. The emergence of a network of strings is visually apparent. c-d: Probability p(n) of a particle having n neighbours in the initial (c) and final (d) state. In the final state, p(n) becomes zero for n = 5, 6 and the mean number of neighbours decreases fromn = 3.4 ton = 2.4. e: Evolution of the order parameters |Ψ 2 | and |Ψ 3 | as a function of the number of cycles N c . The evolution of the microstructure into a network of strings occurs in ∼ 200 cycles. f : Pair-correlation function g(r) as a function of the normalised distance r/a for the initial (blue) and final (red) configurations in experiment. The schematics explain the shift of the second peak upon formation of strings. g: Simulation pictures of the initial state (Φ = 0.4). Green lines show the orientation of the quadrupolar deformation. h: Simulation pictures of the final state. The structure obtained is strikingly similar to the experimental one. i: Pair-correlation function g(r) as a function of the normalised distance r/a obtained from the initial (blue) and final (red) structures in the simulations. As observed in the experiments, in the final state g(r) has a peak around r/a = 4, typical of a string network. would only be expected if a body force acts on the particles [37]. In our experiments, the only body force acting on the particles is gravity, but the Bond number is in the range Bo = ∆ρga 2 γ ≈ 10 −7 − 10 −5 , hence surface tension forces dominate, and the interface deformation is negligible.
We hypothesize that a dynamic monopolar deformation is generated by the motion of the particle relative to the interface during bubble oscillations (Fig. 3a). The maximum acceleration of the interface is on the order ofR ∼ ω 2 ∆R ≈ 6000g, an extremely large value for a colloidal system, leading to unexpected inertial effects. The Weber number, comparing the inertia of the particle to capillary forces, is W e = ∆ρω 2 ∆Ra 2 γ ≈ 10 −3 − 10 −1 (Supplementary Information) sufficiently large for the resulting interface deformation to drive capillary interactions.
Discrete particle simulations confirm that dynamic capillary interactions between monopole and quadrupole can cause the transient microstructure observed in the experiments. The dynamics of spherical particles confined to the surface of a sphere with time-dependent radius are computed from a force balance on each particle including a simplified model for the capillary and hydrodynamic interactions. The particles are assigned a permanent quadrupolar deformation of amplitude Q 2 , with an associated orientation vector. The inertia of a particle is assumed to cause a time-dependent monopolar deformation of the interface, Q(t) = Q 0 sin(ωt), in phase with the motion of the interface ( Supplementary Information, Figure S6). Furthermore, a particle undergoing high-frequency oscillations in a fluid generates a steady recirculating flow, with velocity proportional to Q 2 0 [38]. We model the resulting hydrodynamic repulsion between particles at an interface [32] as the viscous drag experienced by a point particle in the streaming flow generated by a neighbouring particle (Fig. 3c). The pair-wise force is therefore assumed The first term is the interaction between dynamic monopoles, with f 00 = 1/(2πd), the second term is the interaction between a dynamic monopole and a permanent quadrupole, and the third term is the interaction between permanent quadrupoles, with f 22 = (48π/d 5 ) cos (2ϕ i + 2ϕ j ).
The angles ϕ i and ϕ j define the orientation of the quadrupoles of particle i and particle j We first equilibrate the system with only interactions between permanent quadrupoles (Q 2 /a = 1/100). The resulting aggregated structure (Fig. 2g) presents rafts of particles with hexagonal packing and herringbone alignment of the orientation vectors [39]. To verify that the amplitude of the dynamic monopolar deformation that gives agreement with the experiments is physically justified, we use a harmonic oscillator model of a particle at-  Fig. 3b), the monopole amplitude is sufficiently large to drive capillary interactions between particles, as has been observed in experiments with heavy particles [41]. The value Q 0 /a = 7 × 10 −2 used in the simulations of Figure 2 is therefore justified. For amplitudes smaller than Q 0 /a ∼ 10 −2 (blue shaded area in Figure 3b), the interface deformation is expected to be insufficient to drive significant interactions. For Q 0 /a ∼ 1 and larger, detachment of particles from the interface can be expected [40,26].  amplified by a linear, radio-frequency power amplifier (AG1021, T& C Power Conversion Inc.).
The frequencies used were 30, 40 and 50 kHz, resulting in different bubble oscillation amplitudes. Since the wavelength of ultrasound at these frequencies in water is λ > 3 cm, the pressure can be considered to be uniform over distances of the order of the bubble size. The bubbles were driven for 1,000 cycles and the dynamics recorded at 300,000 frames per second using a high speed camera (FASTCAM SA5, Photron). The waveform generator and the high-speed camera are triggered simultaneously using a pulse-delay generator (9200 Sapphire, Quantum Composer). The image resolution at 10× and 20× magnification is 2 µm and 1 µm, respectively.
High-resolution still images in different focal planes were taken before and after excitation at 32× magnification using a CCD camera (QImaging), resulting in an image resolution of 0.1 µm.
For fluorescence imaging, illumination was provided by a Lumen 200 (Prior Scientific) UV lamp combined with a ET mCH-TR (Chroma) fluorescence cube.
Particle-based simulations The interaction model, introduced in Equation 1, is presented in Supporting Information. The simulation results in Figure 2 were obtained with the following set of parameters: Φ = 0.4, β = 6π, ∆R/R 0 = 0.075, R 0 /a = 20, and Ca = 0.01.
The model is implemented in a simulation of particles confined to the surface of a sphere with time-dependent radius. Brownian motion is neglected as the timescale for interface deformation is much smaller than the diffusion timescale of the colloids. The evolution of the position r * i and orientation p i of particle i on the surface of the bubble is obtained solving the non-dimensional balances of linear and angular momentum: The forces and the torques on the right-hand sides represent the contributions of capillary, hydrodynamic, and excluded-volume interactions between particle i and particle j and of the surface constraining force (Supplementary Information). Equations (2)-(5) are solved using a second-order explicit linear multistep method. We used a nondimensional time step ∆t * = 3 × 10 −3 , which has been checked to give convergent results. To avoid expensive computations we enforce a cut-off length of 7a for all the interaction forces and torques between the particles.
Independent simulations with larger and smaller cut-off lengths were found to give similar results.
The computational time of evaluating the interactions between the particles scales as N 2 . The computation of the interactions was therefore parallelised over ten or more cores, to speed up the computations.