Abstract
Dynamic restructuring and ordering are prevalent in driven manybody systems with longrange interactions, such as sedimenting particles^{1,2,3}, dusty plasmas^{4}, flocking animals^{5,6,7} and microfluidic droplets^{8}. Yet, understanding such collective dynamics from basic principles is challenging because these systems are not governed by global minimization principles, and because every constituent interacts with many others. Here, we report longrange orientational order of droplet velocities in disordered twodimensional microfluidic droplet ensembles. Droplet velocities exhibit strong longrange correlation as 1/r^{2}, with a fourfold angular symmetry. The twodroplet correlation can be explained by representing the entire ensemble as a third droplet. The correlation amplitude is nonmonotonous with density owing to excludedvolume effects. Our study puts forth a manybody problem with longrange interactions that is solvable from first principles owing to the reduced dimensionality, and introduces new experimental tools to address open problems in manybody nonequilibrium systems^{9,10}.
Main
Physical systems with longrange interactions pose a difficult challenge to both theory and experiment and their understanding is considered an open problem, which has attracted much effort in recent years^{9}. Longrange hydrodynamic interactions arise in particleladen fluids, as the motion of particles relative to the surrounding fluid induces a slowly decaying perturbation of the flow field. When the suspension is enclosed in an effective twodimensional (2D) geometry, such as in confined suspensions^{11,12}, electrophoresis in capillaries^{13}, protein diffusion in membranes^{14}, and microfluidic droplets ^{15,16}, the hydrodynamic perturbations take the form of longrange dipoles. Such dipolar systems, where the exponent of the spatial decay of the interaction equals the dimension, are known to be marginally strong and have therefore attracted special interest^{9}. The marginal nature arises from the logarithmic divergence of the interaction with the system size, which leads to intriguing phenomena such as shape dependence, similar to that of dielectrics or magnetic dipoles^{17}.
Recently, the collective dynamics of microfluidic droplets has generated a growing interest both as a model system for nonequilibrium dynamics and owing to their practical applications^{16,18,19,20,21,22,23,24,25}. Here we generated a highly dynamic and disordered medium with thousands of uniform waterinoil droplets flowing in a quasi2D microfluidic channel of width W = 500 μm and height h = 10 μm. The droplets are shaped as discs of uniform radius in the range R = 7–11 μm (Fig. 1a). They are in contact with the horizontal floor and ceiling and deform the laminar streamlines of the carrier oil while being dragged at a mean velocity of u_{d} = 100–200 μm s^{−1} that is roughly four times slower than the oil velocity (Fig. 1b,c). The system is driven far from equilibrium by the imposed flow and operates at a low Reynolds number of R e∼10^{−4}, where viscosity dominates inertial effects. Each droplet induces a hydrodynamic dipole leading to interactions between the droplets^{16}. The dipole is proportional to the velocity difference of oil and droplet, and is aligned with the droplet velocity. Droplet clusters constantly form and break apart erratically, and individual droplets exhibit random, diffusivelike motion due to their interactions with the other droplets^{20}. Overall, this highly disordered system exhibits large velocity and density fluctuations that have little to do with thermal energy, consistent with a Peclet number of P e∼10^{7}.
We tracked the trajectories of the droplets for about 100 s in a reference frame moving at the mean droplet velocity, and measured the velocity fluctuation of each droplet, , where u(t) is the droplet’s velocity at time t and is the mean flow direction downstream the channel. The droplets’ mean area fraction ρ_{0} was varied in the range of 0.07–0.63 between measurements and remained roughly constant during each one. The snapshot in Fig. 1d,e shows distinctive patterns of δu for ρ_{0} = 0.18 with stripes of oriented droplet velocities. These structures demonstrate collective motion with spatial velocity correlations, corresponding to orientational ordering of the hydrodynamic dipoles that are aligned with the velocities. We define the spatial velocity correlation of the measured δ u_{x} between two points separated by a vector r = (r,θ):
Here, δ u_{x} is the x component of δu and averaging is performed over the position r′ and time t, such that both r′ and r′+r fall within the central half of the channel to avoid boundary effects. A similar definition applies for δ u_{y}. Despite the disordered dynamics, the velocity correlations show remarkable longrange order and symmetry (Fig. 2a–e). Both C_{x} and C_{y} persist over long distances of r = 15–20R before falling below noise level. At large distances, r = 5–20R, the correlation exhibits the following hallmarks: decay similar to a power law of r^{−2} (Fig. 2c and Supplementary Fig. 1 for all of the measured densities); antisymmetry: C_{y}(r,θ)≈−C_{x} (r,θ) (Fig. 2d,e); fourfold angular symmetry, ∼cos(4θ), with positive maxima reflecting a tendency for joint motion in a particular direction, and negative minima that indicate relative motion in opposite directions, that is dilation, contraction or rotation.
To reveal which droplets contribute to the correlations we computed the conditional correlation function defined similarly to equation (1), but limited to positive or negative fluctuations in x and y directions (details in Fig. 2d,e and Supplementary Information). The peaks of the conditional correlations correspond to the characteristic stripy patterns of the orientational order shown in Fig. 1d,e. The correlations manifest the collective dynamics of the droplets: its positive peaks reveal distinct structures and, at the same time, its negative peaks show how these structures are unstable, dynamically breaking and reforming. Consider, for example, a droplet pair with θ = 90°: on average, the two droplets move fast together along x (C_{x}^{+}(θ = 90°)>0) and simultaneously move in opposite directions along y (C_{y}^{±}(θ = 90°)<0), which either increases or decreases the distance between them. A pair of droplets separated by θ≃45° tend to move together downwards along y and simultaneously move in opposite directions along x, which changes θ. Finally, we measured the variance of the velocity, 〈δ u_{x}^{2}〉 = C_{x}(r = 0) and 〈δ u_{y}^{2}〉 = C_{y}(0). The variances, in units of u_{d}^{2}, increase with the area fraction ρ_{0}, until they peak at ρ_{0}∼0.25, and then attenuate at higher density (Fig. 2f).
To explain the orientational order manifested by the velocity correlations, we consider the interactions in the 2D ensemble of hydrodynamic dipoles^{20}. In the thin channel, the oil velocity can be decoupled into Poiseuille flow along the z axis perpendicular to the plane, and a 2D potential Stokes flow in the x y plane (Fig. 1b). Far from the confining sidewall boundaries, the disturbance of a single droplet to the oil velocity is the gradient of the 2D dipole potential, Φ(r) = Δu · ϕ(r) with ϕ(r) = R^{2}cos(θ)/r and Δu = u_{oil}−u is the difference between the unperturbed oil velocity at the droplet position (assuming the droplet is absent) and the droplet velocity. Two opposing forces act on each droplet: a drag by the surrounding oil and a friction force by the solid boundaries at the channel floor and ceiling. As inertia is negligible, the forces are balanced, resulting in a linear relation u = K u_{oil}, where the coupling constant K∼0.25 depends only on geometry and material properties ^{8,16} (Supplementary Information).
Owing to the linearity of Stokes flow, in an ensemble with interdroplet distances much larger than R, the oil velocity is well approximated by a superposition of the droplet dipoles and the uniform oil flow, . The equation of motion of a droplet at r is u(r) = K u_{oil}(r), where u_{oil}(r) is the velocity of oil including the perturbations of all the other droplets^{16}. We represent corrections to this farfield approximation by an effective velocity scale U for the interaction strength that will be determined experimentally (see Supplementary Information):
The correlation between the velocity fluctuations of two droplets at a distance r is found from equations (1) and (2) to be the averaged sum over all dipole pairs in the ensemble:
and similarly for C_{y}. The summation over i and j represents the interaction of the droplet pair with all of the other droplets in the ensemble. Equation (3) can be separated into two parts: a singledroplet contribution C_{x}^{I}(r), including all terms with i = j, that accounts for the velocity change of the two test droplets due to their interactions with a third droplet at r_{i}, and a dualdroplet part, C_{x}^{II}(r) including all interactions of the pair with two different droplets i≠j.
We first consider the simple case of a randomly positioned ensemble: if the positions of the ith and jth droplets are independent, then by symmetry the average over the product of their dipolar fields vanishes, C_{x}^{II}(r) = 0, and hence C_{x} = C_{x}^{I}. The time average in equation (3) can therefore be replaced by an integral over a uniform density distribution, n_{0} = ρ_{0}/πR^{2}, that for r≳8R yields (Supplementary Fig. 2 for the full solution):
In addition, C_{y}(r) = −C_{x}(r). Equation (4) captures the three salient features of the measured velocity correlations: the r^{−2} power law, the x y antisymmetry, and the cos(4θ) angular dependence. Interestingly, the velocity correlation is given by the autocorrelation of , which implies that the effect of an entire random ensemble on the testpair is equivalent to the average effect of a third droplet. To elucidate this effect, Fig. 2h shows droplet pairs (grey) at angles θ = 0°,45°,90° and a third droplet (lightblue) between each pair. When θ = 0°,90° the third droplet’s dipole has both a positive contribution to C_{x}, because it pushes the pair to the same direction along x, and at the same time, a negative contribution to C_{y}, because it pushes the pair in opposite directions along y. When θ = 45°, it has a positive contribution to C_{y} and a negative contribution to C_{x}.
Remarkably, the velocity correlations originate from the interactions of the pair with the entire ensemble and not from the interaction within the pair. The latter is not included in equation (4) and decays as , much faster than the ∼r^{−2} decay measured at large distances. Moreover, the interaction within the pair cannot explain negative correlations, because the forces that any two droplets apply on each other are identical owing to the symmetry of the dipole field, .
The predicted peaks of the correlations at θ = ±45° are observed for ρ_{0}<0.6 at an average angle of ±41°±2.5° for C_{x}, and ±38°±2° for C_{y} (Fig. 2d,e). In addition, the peaks at θ = 90° and θ = 0° differ in amplitude, in contrast to the theoretical cos(4θ). These differences may largely be attributed to dependencies between droplet positions due to clustering and density waves^{20,24} that were neglected. To refine equation (4) we consider excludedvolume effects in the lowdensity limit, ρ_{0}≪1, and find that the dualdroplet term C_{x}^{II} no longer vanishes: C_{x}^{II}(r) = −4ρ_{0}C_{x}^{I}(r) for r≫R, whereas C_{x}^{I} is unchanged, resulting in, C_{x}(r) = (1−4ρ_{0})C_{x}^{I}(r) (see Supplementary Information). Thus, excluded volume does not alter the spatial structure of the velocity correlations but introduces a densitydependent inhibition to the correlation amplitude, 〈δ u_{x}^{2}〉≡C_{x}(r = 0), which is proportional to ρ_{0}^{2} because it depends on the number of droplet pairs.
The inhibitory effect due to excluded volume reflects a competition between two densitydependent effects that determine the observed peak of the velocity variance 〈δ u_{x}^{2}〉 = C_{x}^{I}(r = 0)+C_{x}^{II}(r = 0) (Fig. 2f). Here, , which is positive and . A simple toymodel provides intuition why the second term is always negative owing to excludedvolume effects (Fig. 3a). Consider two droplets that can be positioned at the four corners of a square surrounding a test droplet (black). Then, C_{x}^{II}(0) is the product of the two fields at the testdroplet’s location, averaged over all of the possible configurations. From symmetry it is enough to consider 3 out of the 12 possible configurations, where one droplet is fixed at the bottom left corner and the other can occupy any of the other three spots. In two of these configurations, the dipole fields have opposite directions at the test droplet’s position, whereas only in one configuration their directions are aligned. As a result, their average product is negative, hence C_{x}^{II}(0)<0. Without the effect of excluded volume, one would consider a fourth configuration, in which both droplets occupy the same spot, forming a second configuration with aligned fields that nullifies the average product.
To compare the theoretical value of C_{x}^{I}(0)+C_{x}^{II}(0) to the measured velocity variance 〈δ u_{x}^{2}〉 (Fig. 2f), we measured the spatial pair and tripletdistribution functions that describe the nonrandom distribution of distances between droplets (Fig. 3b and Supplementary Fig. 3). These functions are used as weights in averaging over the ensemble positions in calculating C_{x}^{I}(0) and C_{x}^{II}(0) (see Supplementary Information). Both functions peak at distances of touching droplets but decay to a constant value for larger distances. In agreement with theory, Fig. 3c shows that C_{x}^{I}(0) is positive whereas C_{x}^{II}(0) is negative, both increasing in magnitude with density ρ_{0}, and have similar values for y. The computed sum C_{y}^{I}(0)+C_{y}^{II}(0) in units of U^{2} fits well to the measured C_{y}(0) (Fig. 3d), which suggests that the interaction velocity scale U is U≈u_{d} throughout the measured range of ρ_{0} (C_{x}(0) is discussed in the Supplementary Information). The theoretical farfield value, U = (1−K)u_{d}≈u_{d}, matches our experimental result.
Finally, we computed the velocity correlations in silico by randomly placing droplets, considering their mutually excluded volume, and evaluating the interdroplet dipolar forces (see Supplementary Information). In agreement with theory, the numerical calculation shows a cos 4θ symmetry and r^{−2} decay of the correlations (Supplementary Fig. 4), as well as a peak at ρ_{0} = 0.34 in the velocity variances (Fig. 3d).
Similarly to a nematic liquid crystal, the hydrodynamic dipoles exhibit partial order: positional correlations decay fast owing to spatial disorder, whereas the orientational degreesoffreedom remain correlated over longrange. As in the 2D droplet ensembles reported here, also in gravitational sedimentation of particles in 2D and 3D, the velocity variance decreases at high densities^{2,12,26}. However, so far this generic decrease has lacked a firstprinciples theoretical explanation. In Brownian quasi2D suspensions of particles in equilibrium, the dipoles are randomly oriented and there is no symmetrybreaking direction. Hence, at large distances velocity correlations are determined only by twobody interactions^{11}. Future studies may reveal how the velocity correlations are coupled to dynamic clustering, which has an important role in setting the fluctuations. We expect our results to have applications in the design of active and selfpropelled systems^{27,28} as well as in dropletbased microfluidic devices used in biology and chemistry^{29,30}.
Methods
The channel was made of polydimethylsiloxane elastomer casted on a mould, which was prepared by lithography. After curing at 80 °C for 1 h, the channel was detached from the mould and irreversibly attached on a polydimethylsiloxanecoated glass slide^{16,20}. The carrier fluid was light mineral oil (Sigma, M5904, viscosity η_{oil} = 30 mPa, density ρ_{0} = 0.84 g ml^{−1}) with 2% (w/w) Span80 surfactant (Sigma). The dispersed fluid was distilled water. The experiment was imaged by a PCO.Sensicam (PCO) camera for about 100 s at 21 frames s^{−1}. We used a precise tracking algorithm (the Moses–Abadi algorithm) ^{8} implemented in Matlab (Mathworks) to analyse the images acquired in the experiment. Each droplet’s centre position was tracked and followed between subsequent images to construct its trajectory. The droplet velocities were computed by fivepoint time derivatives of their x and y positions. To compute the velocity correlations C_{x}(r) we computed the average product of the velocity fluctuations δ u_{x} of all droplet couples within the channel’s central half that were separated by r = (r,θ). The velocity fluctuations were defined as the difference between the droplets’ individual velocities and the locally measured mean velocity. Owing to the spatial and temporal fluctuations of the mean droplet velocity, it was measured separately for each droplet pair. The mean velocity was defined as the average velocity of droplets within a rectangle of length W along x (centred at the mean x position of the droplet pair) and width W/2 along y (centred at the middle of the channel).
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Acknowledgements
This work was supported by a YedaSela Grant (R.H.BZ). T.B. was supported by the Cross Disciplinary Postdoctoral Fellowship of the Human Frontier Science Program. T.T. is the Helen and Martin Chooljian Founders Circle Member in the Simons Center for Systems Biology of the Institute for Advanced Study, Princeton.
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Shani, I., Beatus, T., BarZiv, R. et al. Longrange orientational order in twodimensional microfluidic dipoles. Nature Phys 10, 140–144 (2014). https://doi.org/10.1038/nphys2843
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DOI: https://doi.org/10.1038/nphys2843
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