Excluded Volume Induces Buckling in Optically Driven Colloidal Rings

In our combined experimental, theoretical and numerical work, we study the out of equilibrium deformations in a shrinking ring of optically trapped, interacting colloidal particles. Steerable optical tweezers are used to confine dielectric microparticles along a circle of discrete harmonic potential wells, and to reduce the ring radius at a controlled quench speed. We show that excluded-volume interactions are enough to induce particle sliding from their equilibrium positions and nonequilibrium zigzag roughening of the colloidal structure. Our work unveils the underlying mechanism of interfacial deformation in radially driven microscopic discrete rings.

Introduction.-Surfaceroughening occurs in a wide variety of physical [1,2] and biological systems [3][4][5] and it is often caused by noise or external perturbations which deform an initially flat and spatially uniform interface [6,7].Such process has received much attention on extended systems, i.e. when macroscopically large domains grow and deform, including fluid [8][9][10][11] and solid [12][13][14][15] interfaces, grain boundaries [16][17][18] or active matter [19][20][21].However, finite-size systems made of expanding or shrinking circular domains also show kinetic roughening when driven out of equilibrium by internal fluctuations or pressure imbalance.Physical examples include unilamellar vesicles [22,23], confined bacteria colonies [24,25], nematic liquid crystals droplets [26,27] or self-avoiding ring polymers [28,29].In most of the cases, interfacial deformations are described with continuum models based on stochastic differential equations, such as the celebrated Kardar, Parisi, and Zhang equation [2] or more sophisticated theories [30,31].An alternative, although less exploited approach, consists in considering the microscopic constituent of an interface, to understand how the global deformation arises directly from their pair interactions.Few works along this direction showed that spherical [32] or anisotropic [33] particles can indeed act as microscopic model system for interfacial deformations.
Many studies on colloidal particles confined to a ring have focused on the particle displacement along the tangential direction, keeping fixed the radius of the ring [35][36][37][38][39]. Recently, this setup has shown striking effects as the emergence of propagating cluster defects [47,60] when compared to other related discrete systems based on linear, nearest-neighbor coupling [61].In contrast, here we investigate the much less studied mechanism of out of equilibrium deformations induced by radially shrinking a ring of discrete particles at different compression speeds.We use the synergy between the experiment, theory and numerical simulations to show that buckling predominantly arises from excluded volume interactions, without the need to consider long-range interactions, such as e.g.dipolar, electrostatic or hydrodynamic forces, unnecessary.Unlike roughening induced by thermal fluctuations in presence of dipolar interactions [32], the wellcontrolled quench speed becomes a key driving factor that determines driven buckling.
Experiments.-Ourcolloidal ring of radius R is composed of N = 50 polystyrene particles having d = 4.0 µm diameter and dispersed in highly deionized water (MilliQ, Millipore).The particle solution is placed by capillarity inside a thin (∼ 100µm) chamber made of a borosilicate glass slide and a coverslip separated with a layer of parafilm.The experiments are performed at a room temperature of T = 293K.We trap the particles using time shared optical tweezers which are created by passing an infrared laser beam (ManLight ML5-CW-P/TKS-OTS operating at 3 W, wavelength λ = 1064 nm) through a pair of acousto-optic deflectors (AODs, AA Optoelectronics DTSXY-400-1064) both driven and synchronized with a radio-frequency (RF) generator (DDSPA2X-D431b-34).The beam is then focused from above by a microscope objective (Nikon 40× CFI APO), Fig. 1(a).
The bottom of the experimental chamber is observed with a second microscope objective (Nikon 40× Plan Fluor) which projects an image onto a CMOS Camera (Ximea MQ003MG-CM), in a custom-built inverted optical microscope.The laser tweezers visit 50 equispaced positions along a ring, spending 20 µs in each point, such that each trap is visited once every ∼ 1ms.Thus, the potential wells can be considered as quasistatic since the beam scanning is much faster than the typical selfdiffusion time of the particles, τ D = d 2 /(4D eff ) ∼ 30 s, estimated from the effective particle diffusion coefficient D eff ≃ 0.13 µm 2 s −1 [39].
Ring deformation.-Westart by analyzing the optical potential confining the colloidal particles within a ring of mean radius 34.7 µm.As shown in the inset in Fig. 2(a), we work in polar coordinates with the origin at the center of the ring.Before compression, we determine the spring constant κ of the optical potential confining each particle by monitoring the equilibrium, radial particle fluctuations across the ring.One can obtain the confining potential U (r) by measuring the stationary probability distribution directly from the particle trajectories [63].To adapt such procedure to the radial configuration, we formulate the overdamped equation of motion for an individual Brownian particle with the position r = (x, y) captured in a radial harmonic trap centered at R (with |R| = R): γ ṙ = −κ(r − R) + ξ, where κ is the trap stiffness, γ = 3πηd the drag coefficient of the particle, η the viscosity of water, and ξ a stochastic force with zero average and delta correlated.The corresponding stationary distribution of the particle position is Passing to polar coordinates for (ρ = |r|, θ) and making use of the radial symmetry of P (r), we integrate over θ to find that the stationary distribution of the radial displacement has the "Rayleigh" form: with C being the normalization constant.For details, see Supplemental Material in [62].We invert Eq. ( 1 In a typical shrinking experiment, Fig. 1(b) we first equilibrate the trapped particles along a ring with an initial radius R(t = 0) = 34.7 µm.After that, we perform 50 measuring cycles by repeatedly decreasing the radius to R(t = τ q ) = 31.0µm at a controlled quench speed v q = [R(0) − R(τ q )]/τ q and, after a short equilibration period, increasing it back to its initial position.The experiments run over 24h during which we change the quench time τ q ∈ [0.2, 10] s and speed v q ∈ [0.37, 18.5] µm s −1 ensuring the same statistical averages for each value of τ q .
At t = 0, the particles lay almost along their mean elevation ⟨ρ i ⟩ ≈ R, with a small thermal roughening W (t = 0) = W 0 induced by thermal fluctuations, where we define the roughening W (t) as in which h i = ρ i − ⟨ρ i ⟩ is the radial displacement of particle i (= 1, . . ., N ) from the mean.Upon decreasing the ring radius, the colloids start interacting and excluded volume displaces the particles either outside or inside the ring.For the chosen number of particles, the ground state of the system would be a perfect zig-zag chain with no defects [64], which could be obtained for an adiabatically slow compression, τ q → ∞.However, at a finite quench time defects in the roughening emerge in the form of two or more particles displaced together, inset in Fig. 1(b).The effect of excluded-volume interactions can already be appreciated by measuring the evolution of the average distance ⟨∆r⟩ between the centers of neighboring particles, Fig. 2(b).For the fastest compression occurring at τ q = 0.2 s, ⟨∆r⟩ displays a pronounced minimum close to the excludedvolume limit, ⟨∆r⟩ = d indicating a strong repulsive force around t ∼ τ q .After that, the particles reach a steady distance of ⟨∆r⟩ ∼ 1.013d, regardless of the compression time.Reducing the compression speed gradually eliminates this minimum since the slower approach allow the colloids to rearrange sliding out from their central position in the harmonic wells.
Numerical Simulations.-Tounderstand whether the induced buckling can be described by only taking into account excluded-volume interactions, we perform Brownian dynamics simulations with the same parameters as in the experiments.We extend our single-particle description to many interacting particles, in which each particle i obeys the Langevin equation: where the force f i,j = −U ′ HC (r ij )r ij /r ij represents the hard-core repulsive interaction between nearest neighbors.To achieve the best mapping with the experimental data, we generalize the Weeks-Chandler-Andersen potential [65] as, U HC (r) = 4ϵ[(d/r) 2q − (d/r) q + 1/4] for r ≤ r 0 = 2 1/q d and zero otherwise.Here ϵ is the repulsion strength and q the nonlinearity index.As shown in Fig. 2(b) and 3(b), we find quantitative agreement between simulations and experiments by using q = 42 and ϵ/γ = 2.5 µm 2 s −1 as fitting parameters [62].This matching highlight that the particles experience an effective short-range repulsive potential, a quantity frequently subject to uncertainty due to the presence of residual particle charges and long-range interactions.Further, by varying the number N of particles at constant t q , the roughening shows similar crossover from the same thermal plateau W 0 at small times to the saturation limit W S at long times.While W S grows only slightly for larger N , it exhibits a stronger nonuniform decrease for smaller N , due to the strong, nonlinear nature of the short-range interactions.
Theory.-We introduce the following one-dimensional model for the radial displacements: Unlike a previous work on thermal induced roughening of a linear chain [cf.Eq. ( 2) in Ref. [32]], our Eq.( 4) goes beyond the nondriven framework, as it includes a timedependent factor g(t) that encapsulates the increase in the effective interactions as the ring is compressed.Moreover the particle interaction considered here are repulsive (g(t) > 0) in contrast to Ref. [32].versus particle number k in semilog scale.Empty symbols are experimental (circles) and simulation (squares) data for τq = 0.5 s, while continuous line is the numerical integration of Eq. ( 5) at time t = 0.45s at which the predicted W (t) reaches Ws.
Bottom inset shows the correlations in normal scale.
Equation ( 4) can be justified by projecting Eq. ( 3) of the simulations onto the radial direction (see SM [62]).In this way we obtain g(t) = 0 for t < t c and g(t) = 2q 2 ϵ(R c /R(t) − 1)/(r 2 0 κ) for t > t c , where R(t) = R(0) − v q t is the ring radius at time t, and t c is the time at which R(t) = R c ≡ N r 0 /2π and the particles start to interact.Note that in Eq. ( 4) we have omitted a term D/(h i + R), which is negligible for our experimental conditions [62].
Using the discrete Fourier transform, from Eq. ( 4) we calculate the equal-time height-height correlation function C(k, t) = N −1 i ⟨h i (t)h i+k (t)⟩ of the ring: with Here τ 0 = γ/κ is the microscopic relaxation time, I k (x) is the modified Bessel function of the first kind and t eq is an equilibration time which we suppose much larger than τ 0 so that at t = 0 the system is in thermal equilibrium.For k = 0, we obtain the roughening, cf.Eq. ( 2): In Eq. ( 6), we fix from the experimental data τ 0 = 0.15 s and W 0 = √ D eff τ 0 = 0.127 µm which denotes the "thermal" roughening.
This equation predicts that, starting from t = −t eq , W (t) will first grow as W (t) = W 0 [1 − exp (−2(t + t eq )/τ 0 )], as shown by the dashed line in Fig. 3(a), reach a thermal plateau W 0 for −t eq + τ 0 ≪ t ≪ t c , and grow again for t > t c due to the repulsive interaction.For simplicity, we use a linear schedule cf (t) that approximates well the full g(t) [62], where f (t) = 0 for t < t c , (t − t c )/(τ q − t c ) for t c ≤ t ≤ τ q , and 1 for t > τ q .This schedule is shown by the blue line in Fig. 3(a).Here c is a dimensionless constant which can be related to the pair interactions between the colloidal particles as c = (1 − R(τ q )/R c )2q 2 ϵ/(r 2 0 κ). Figure 3(b) shows the roughening from rings compressed at different τ q ∈ [0.2, 10]s.We use Eq. ( 6) to fit the experimental data, the only adjustable parameters being t c /τ q and c.We obtain a good quantitative description of the initial increase of W (t)/W 0 at different τ q by using t c /τ q = 0.2 and c = 0.9, which determine q = 10.7 and ϵ/γ = 5.4µm 2 s −1 , respectively.Note that both values differ only slightly from those for the full numerical model, Eq. ( 3), despite a number of simplifications made to derive Eq. ( 4).Interestingly, for c < 1/4 the model is able to predict that W (t) reaches an intrinsic saturation value W ∞ = W 0 (1 − 4c) −1/4 for t ≫ τ q , whereas for c > 1/4 it grows without bounds, indicating that the particles escape from the optical traps.Because the linear approximation for the short-range repulsion force in Eq. ( 4) is insufficient to describe the final stages when the interaction becomes very large, but quantitative agreement with the saturation value W s set by the experimental protocol can be achieved by simulations.Note that it is not possible to measure experimentally the initial growth until W 0 because the initial configuration in the experiment is not perfectly flat.
Finally, we also use the model to contrast the experiments and simulations in terms of the equal time correlation function C(k, t), Fig. 3(c).For a fixed t, both experiments and simulations, display anticorrelations in C(k, t), as described by Eq. 5, see the inset of Fig. 3(c), which is a signature of the underlying zigzag pattern.The model fits rather well the experimentally determined correlation function up to distance k = 5 particles.Above this range, the data start deviating and both, simulations and experiments display a series of oscillations that may be due to correlated lateral displacement of the particles, which are completely discarded by our theory.
Conclusions.-We have used a colloidal model system to investigate the deformations emerging from a collapsing chain of optically driven microscopic particles.Via a tight combination of the experiments, theory, and simulations, we find that the nonequilibrium buckling is captured quantitatively well by considering the sole excluded-volume interactions, thus without any long-range dipolar, electrostatic, steric (polymer mediated) or hydrodynamic effects.Different works have investigated the effect of topology on the collapse of a dried dense colloidal suspension [66][67][68][69].Here, we have focused on the mechanism leading to buckling starting from the interparticle interactions.We expect that our general description of a finite shrinking ring at the discrete, i.e. single particle level, may be important to related research fields where finite droplets reduce and deform due to external or internal (i.e.active) pressure fields.

FIG. 1 .
FIG. 1. Schematic of the experimental system: one infrared laser beam is rapidly steered via a two channel (x, y) acoustooptic deflector (AOD) controlled via a radio frequency generator (RF).The laser is scanned across equispaced points forming a ring of quasistatic double wells.(b) Two images showing the roughening of N = 50 polystyrene particles when the optical ring is reduced from R(0) = 34.7 µm (left) to R(τq) = 31.0µm (right) at a speed of vq = 0.06 µm s −1 (quench time τq = 60 s).Small arrows in the inset of the right image indicate displaced particles, with one central defect highlighted.The dashed red line is the ring circumference, see also VideoS1 in SM [62].

FIG. 2 .
FIG. 2. (a)Confining potential U (ρ) along the radial (ρ) direction calculated from the particle fluctuations before compression, here ⟨R⟩ = 34.7µm.Scattered squares are experimental data, the continuous line is a non linear regression calculating U (ρ) from Eq. (1).The inset shows a colloidal ring with highlighted cartesian (x, y) and polar (ρ, θ) coordinate systems, being ∆r the nearest neighbor distance between the centers of the particles along the ring.(b) Average distance between the centers of nearest neighbor particles ⟨∆r⟩ versus time for four different quench times τq from experiments (symbols) and simulations (lines).

FIG. 3 .
FIG. 3. (a) Normalized roughening, W/W0 versus t from experiments (symbols) and Eq. 6 (black line) for τq = 0.5s and teq = 0.4 s.The linear schedule f (t) is shown in blue.(b) W/W0 vs t from experiments (symbols), simulations (lines in colour) and model (black line) for different τq.(c) Absolute value of the normalized, equal time correlation function,C(k, t)/C(0, t) versus particle number k in semilog scale.Empty symbols are experimental (circles) and simulation (squares) data for τq = 0.5 s, while continuous line is the numerical integration of Eq. (5) at time t = 0.45s at which the predicted W (t) reaches Ws.Bottom inset shows the correlations in normal scale.