# Cross-talk between topological defects in different fields revealed by nematic microfluidics

^{a}Instituut-Lorentz for Theoretical Physics, Leiden University, 2333 CA Leiden, The Netherlands;^{b}Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia;^{c}Condensed Matter Physics Department, Jožef Stefan Institute, 1000 Ljubljana, Slovenia;^{d}Institute for Environmental Engineering, Department of Civil, Environmental, and Geomatic Engineering, Eidgenössische Technische Hochschule (ETH) Zurich, 8093 Zurich, Switzerland;^{e}Max-Planck-Institute for Dynamics and Self-Organization, 37077 Goettingen, Germany

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Edited by Tom C. Lubensky, University of Pennsylvania, Philadelphia, PA, and approved June 1, 2017 (received for review February 17, 2017)

## Significance

Topological defects play a defining role in systems as extensive as the universe and as minuscule as a microbial colony. Despite significant advances in our understanding of topological defects and their mutual interactions, little is known about the formation and dynamics of defects across different material fields embedded within the same system. Here, using nematic microfluidics as a test bed, we report how topological defects in the flow and the orientational fields emerge and cross-talk with each other. Although discussed in a nematofluidic context, such multifield topological interactions have potential ramifications in a range of systems spanning vastly different length and time scales: from material designing, to exploration of open questions in cosmology and living matter.

## Abstract

Topological defects are singularities in material fields that play a vital role across a range of systems: from cosmic microwave background polarization to superconductors and biological materials. Although topological defects and their mutual interactions have been extensively studied, little is known about the interplay between defects in different fields—especially when they coevolve—within the same physical system. Here, using nematic microfluidics, we study the cross-talk of topological defects in two different material fields—the velocity field and the molecular orientational field. Specifically, we generate hydrodynamic stagnation points of different topological charges at the center of star-shaped microfluidic junctions, which then interact with emergent topological defects in the orientational field of the nematic director. We combine experiments and analytical and numerical calculations to show that a hydrodynamic singularity of a given topological charge can nucleate a nematic defect of equal topological charge and corroborate this by creating

Defects are ubiquitous in nature and are at the heart of numerous physical mechanisms, including melting in 2D crystals (1), cosmic strings (2), and other topological defects in the early universe (3). Vortices are possibly the most common examples of defects in flowing media (4, 5). In a typical hydrodynamic vortex, the fluid velocity, ** v**, rotates by

The interaction between topological defects is governed by the defect topology and the underlying energetics. Similar to electrically charged particles, like-sign topological defects, in general, repel each other, whereas defects of opposite sign attract. However, this interaction can be additionally affected by the geometry and surface properties of the environment (17, 18) and the presence of an external stimulus (19⇓⇓–23). Emergence of topological defects in a field and the resulting interactions between them have been well-characterized (24). However, how topological defects in a system can coevolve in and interact across disparate fields is largely unexplored. It is rather recent that multifield topological interactions were shown in optics, where singularities in optical birefringence created topological defects in the light field (25, 26). The growing evidence that topological defects perform vital biological functions (27⇓–29) creates a fundamental need for an integrated understanding of defect interactions, especially in relation to those in a different field (for instance, in the surrounding microenvironment).

Complex nematic fluids have proven to be a versatile test bed for studying, testing, and realizing diverse topological concepts (30⇓–32), owing primarily to their inherent softness and strong response to external stimuli and in context of this work, their material fluidity (33, 34). Liquid crystal microfluidics (35) has emerged as a potent toolkit to modulate fluid and material structures caused by the coupling between the two main material fields—the fluid velocity field and the molecular orientational field (director) (34). The flow director coupling regulates transport properties of nematic suspensions (36, 37), tunes the rheology of the liquid crystals (LCs) (38⇓⇓–41), and mediates annihilation–creation dynamics of topological defects (21, 42). Microfluidics based on complex anisotropic fluids has allowed for potential applications (43) and novel designs of microcargo transport (44), tunable fluid resistivity (45), color filters (46), and biochemical sensors (47).

In this paper, we study the emergence of topological defects in two different fields present in the nematic microfluidic system: the stagnation point, a hydrodynamic singularity in the flow velocity field, and the nematic defect, a topological singularity in the molecular orientation field. We characterize the cross-interaction between these topological defects using star-shaped microfluidic junctions and flowing nematic fluid (Fig. 1*A*). We show that the nucleation and the nature of the nematic defects are determined by the topology of the flow defect, such that a hydrodynamic stagnation point of topological charge

## Tuning Topology with Hydrodynamics

We study the emergence of topological defects using a combination of experiments, numerical modeling, and theory. Experimentally, we use star-shaped microfluidic junctions fabricated by soft lithography techniques (*Materials and Methods* and *SI Text*). Our experimental results are complemented by theoretical analysis and numerical modeling based on Beris–Edwards-type approach—a powerful tool to study nematic structures, especially defects at mesoscopic scale (34).

The cross-interaction between the velocity and the nematic fields is governed by an interplay of multiple effects: material viscosity, nematic elasticity, channel dimensions, and the strength of the flow (Fig. 1*A*). The combined effect is captured by a single dimensionless number, the Ericksen number, *SI Text*). The Ericksen number (

Fig. 1*B* shows the nematic defects obtained in four-, six-, and eight-arm microfluidic junctions. In each case, no defect was observed for *B*, *Top* (imaged at *B*, *Middle* and *Bottom*) results in increase in the net topological charge at the junction center:

We have reproduced the experimental results in silico using numerical simulations of 3D microfluidic junctions based on the Navier–Stokes equation coupled with the Beris–Edwards equations of nematodynamics (49) (*Materials and Methods* and *SI Text*). Fig. 1 *D* and *E* shows the numerical flow velocity and the nematic ordering at the four-, six-, and eight-arm junctions. The isosurfaces of the nematic order parameter (Fig. 1*E*) show stable

## SI Text

### Experimental Setup and Microfluidic Manipulations.

All of our experiments have been performed with 5CB (Synthon Chemicals). Liquid crystal 5CB is a single-component nematic liquid crystal for

The hydrodynamic stagnation point in each experiment was detected by epifluorescent video imaging of fluorescent tracers (mean diameter = 2.5

### Numerical Simulations.

Our numerical simulations rely on Beris–Edwards formulation of nematodynamics (49) describing the evolution of the system by density ** v**, and full order parameter nematic tensor

**:**

*Q***:**

*H***S1a**–

**S1c**were numerically solved using the hybrid lattice Boltzmann method (41, 55) with a 19 velocity lattice model and Bhatnagar–Gross–Krook collision operator (58). The flow was driven by a pressure difference with an open boundary at the end of the channels. In the studied regime, the fluid is nearly incompressible, and small density gradients (the deviations of density in a junction are always kept below

**S5**evaluates to

**S1a**–

**S1c**reduce to Eq.

**S8**with

### Flow Alignment in Nematics.

In this section, we provide additional information on the hydrodynamics of nematic liquid crystals as well as a derivation of Eq. **1**. In the presence of uniform density and nematic order (i.e., **S1**) simplify to the following set of partial differential equations for the velocity field ** v** and the nematic director

**(24, 34, 48, 60, 61):**

*n***S8b**can be cast in the form of a single partial differential equation for the local orientation

**S11**then reduces to

**S10**, we would like to note again that the analytical treatment presented here is strictly 2D (variations in the direction perpendicular to the plane of the junction are neglected) and thus, cannot account for the escaped structures observed in our experiments and numerical simulations. Nevertheless, the 2D analytical description serves as a relevant tool to gain insight into the mechanisms that underpin the interaction between the topological defects in different fields.

### Stagnation Flows.

The defective solutions and the force field reported in the text have been constructed from analytic approximations of the stagnation flow in four-, six-, and eight-arm junctions. Here, we report an explicit construction of the corresponding velocity fields. Let **S8a**, the ratio between the magnitude **S8a**, and the flow is governed by the incompressible Stokes equations:**S15a**, in particular, implies**S15b**, both **S19**, the function *A*). The flow described by Eqs. **S18** and **S20** has, at most, nine stagnation points, with coordinates that are given by**S18** and **S23** yield *A* shows a comparison between the approximated velocity field and a numerical solution of the Stokes equation in a cross-junction. The two agree closely and always fall within a 10% range, with an exception for the corners where viscous dissipation plays the dominating role.

Now, consistent with Eq. **S25**, the vorticity field obtained from the approximation described here is the lowest-order harmonic function with twofold rotational symmetry:**S29** vanishes at the corners of the polygon and is maximal in magnitude at the center of the edges. This velocity field can be conveniently verified using polar coordinates:**S30** and those obtained from a numerical integration of the Naiver–Stokes equation is shown in Fig. S2 *B* and *C* for the cases

Finally, sufficiently close to the central stagnation point, **S29** are approximated by the harmonic function**S31** describes an irrotational flow (i.e.,

### Defect Configurations in Irrotational Flows.

We use nematic hydrodynamics to calculate the configuration of the nematic director in close proximity of the central stagnation point of a generic **S33**. The dynamics of a 2D nematic director is governed by Eq. **S11**. Because of the rotational symmetry of the problem, it is convenient to work in polar coordinates. Then, expressing **S11** as**S33**:**S34** dimensionless, we can rescale **S35**, after some manipulation, Eq. **S34** can be expressed in the dimensionless form:

Despite its strong nonlinearity, it is possible to find a family of stationary defective solutions of Eq. **S36** for specific values of the flow alignment parameter **S37** into Eq. **S36**, we obtain**S40** describes a special bulk configuration of the director in flow-tumbling nematics. Choosing the negative sign in Eq. **S39**, however, yields a family of solutions with**S41** defines a set of defective configurations having

### Defect Dynamics in a Flow.

In this section, we provide a derivation of Eq. **2**. In the absence of backflow, the dynamics of the local orientation **S11**, can be thought of as resulting solely from energy relaxation:**S43** and **S44** can be used to describe the dynamics of the local orientation

Let **S43**. Following the works by Kawasaki (53) and Denniston (54), one can construct an equation of motion for the moving defect by decomposing the local orientation **S46** results from the special structure of the director field near the core region and does not require the linearity of the associated field equation (53). To find an equation of motion relating ** R** with

**S48**consists of a combination of a bulk term and a boundary term caused by the shift in the position of the finite size core region, namely

**is the boundary normal pointing toward the interior of the defect core. The variation**

*N***S46**and

**S47**in the form

**S50**into Eq.

**S49**and use Eqs.

**S46**and

**S47**, taking into account that

**S51**can be straightforwardly calculated:

**S51**can be rearranged in the form

**S50**, with

**as the flow velocity, we can express the time derivative**

*v***S48**in the form

**S54**with Eq.

**S53**, we obtain an equation of motion for the moving defects:

**S43**in the frame of the moving defects. We obtain

The dynamics of an isolated defect is then dictated by two driving forces: the elastic force proportional to the elastic constant **S56** yields Eq. **2**, with **S45**, we obtain Eq. **3**.

As an example of the effect of a high**S25**. The corresponding strain rates and vorticity are given by**S45**, **S60**, and **S61**. As a consequence of such a force field, negative defects are attracted by the central stagnation point, where positive disclinations are repelled toward the channels.

### Dynamics of the Defect Nucleation at the Junction Center.

Homeotropic microfluidic channels, like the ones used in this work, support multiple nematic configurations, either stable or metastable, in absence of any flow. Depending on the deformation of the director close to the channel corners, these different possible configurations correspond to different free energy values. Specifically, the channel aspect ratio (channel width/height) and the curvature (sharpness) at the corners determine whether nematic defects will be real bulk or virtual (35, 44). Irrespective of these multiple initial conditions, when we initiate flow in the microchannel, a pseudoplanar structure first emerges and then stabilizes into a flow-aligned director configuration. Specifically, we observe three different flow regimes within microchannels having rectangular cross-section depending on the Ericksen numbers (41). The flow regime relevant to this work is the “high” Ericksen number regime, where the nematic profile evolves into a flow-aligned state, with the director oriented primarily along the channel length. The alignment initiates close to the channel inlet and propagates downstream as the nematic director gets distorted. Farther downstream, the director field remains relatively undisturbed. Thus, a linear microchannel develops two director domains: upstream, a flow-aligned director domain, and downstream, an intact homeotropic domain (Fig. S4 *A* and *B*). The two director field domains are separated by a disclination, which spans the width of the channel and connects at the surfaces of the channel walls. Two disclinations, one in each of the in-flow arms, travel downstream (Fig. S4*B*) and meet at the central junction region.

The disclinations recombine at the junction to form the central defect loop. As presented in the POMs in Fig. S4 *C–F* and the corresponding director field schematics in Fig. S4 *G–J*, on meeting, they merge into a single *E* and *I*). Thereafter, the defect loop shrinks in size to minimize the free energy until it reaches the final morphology of a monopole at the junction center (Fig. S4 *F* and *J*). The mechanism described here can, however, differ with the curvature and the geometry of the microchannel cross-section. In absence of flow, strong geometrical curvatures can support peripheral disclinations (running parallel to the channel walls) (35), which under specific conditions, separate the surface-induced homeotropic texture from stable pseudoplanar textures (50, 62). As the nematic flow is initiated, the homeotropic texture can be eliminated in favor of the pseudoplanar texture because of the movement of the peripheral defect line orthogonal to the flow direction. We believe that, in such a setting, the emergence of the central defect loop will be additionally influenced by the peripheral disclinations (50) and potential generation of umbilics (63).

### Numerical Simulations of Odd Arm Junction.

In the text, we show nematic configurations in junctions of four, six, and eight microchannels. In Fig. S5, we show numerical simulations of junctions of odd numbers of nematic microchannels. In a three-arm junction, the stagnation point occurs at the corner of the junction. The emergence of a defect at the stagnation point is conditioned by the inflow–outflow regime. In the regime of two outlet flows, a nematic

In a five-arm junction, there are two stagnation points: one close to the center of the junction and one pinned to the corner like in a three-arm junction. In Fig. S5, we show a five-arm junction with three inlet flows and two outlet flows. The nematic configuration in such junction resembles the nematic configuration in a four-arm junction with a small defect loop of topological charge

## Global Constraints and Local Forces

The emergent topological structure of the defects at the junction center results from a combination of global topological constraints and local mechanical effects. For the analytical treatment that follows, we consider the channel midplane only and a 2D nematic field within this plane. Because flow tends to align the director along the channel, a generic junction with *A*, *Upper*, each with winding number *SI Text*):** v** is the flow velocity;

*SI Text*). Consequently, for a perfectly flow-aligning system (

**1**(

*SI Text*). For

Whereas emergence of the equilibrium singular director field depends exclusively on the symmetry of the flow in proximity of the stagnation point, its stability depends on the flow structure over the entire junction. We clarify this using an effective particle model for the dynamics of defects in the presence of a generic potential energy field that originates from a background flow at sufficiently large Ericksen numbers. Let us consider the generic free energy *SI Text*)**2** corresponds to the well-known Coulomb-like elastic interaction between the topological charges, whereas the third term, given by *SI Text*)*A* shows the force field, calculated from Eq. **3** (normalized by the maximal force value), experienced by disclinations of topological charges **3**, has been analytically estimated, taking into account the rotational symmetry and the relative position of the stagnation point at the junction (*SI Text*). Consistent with our experimental and numerical results in 3D geometry of the channels, we found that defects of topological charge *SI Text*), thus protecting the internal negative defects from annihilation.

## Charge Fractionalization and Defects Unbinding

Defects having large negative topological charge (i.e., *A*, *Lower* shows POMs of the defect loops immediately after their formation. At the center of the four-arm junction, we observe a defect loop of charge *A*, *Lower Left*), which within a short time, stabilizes into a *A*, *Lower Center* and *Lower Right*), and gradually decay into multiple *B*). As presented in Fig. 2*B*, *Upper*, the *B*, *Lower*) proceeds in three steps. (*i*) A loop of charge *ii*) The *iii*) Finally, all three *SI Text* and Fig. S1).

The behavior described above results from two competing effects. On the one hand, the hydrodynamic forces tend to concentrate the negative topological charge at the center of the junction. On the other hand, the elastic forces drive the repulsion of like-sign defects. This effect of elasticity favors the fractionalization of a central *C*) are susceptible to decay and can become unstable with respect to any perturbation of the pressure distribution across the channels. A slight asymmetry in the pressure distribution causes the central stagnation point to split into multiples of stagnation points of charge

## Dynamics of Defect Nucleation in a Four-Arm Junction

The higher stability of a *A* shows creation of the *A*, row 3). On meeting, the singular disclinations merge into a defect loop, enclosing a homeotropic domain (Fig. 3*A*, row 4), which gradually shrinks and finally stabilizes into a *SI Text* and Movie S1). We would like to emphasize that homeotropic anchoring, in absence of flow, supports multiple director configurations. These energetically stable or metastable configurations emerge because of an interplay between the cross-section geometry (rectangular, square, or circular), anchoring strength, and the curvature (or sharpness) of the channel corners (35, 41) and set the initial conditions for our flow experiments.

In a second approach, we have gradually increased the flow speed (in steps of Er = *B* presents a sequence of polarized micrographs of the nematic texture at the junction center. The first appearance of the *B*, rows 4 and 5). The profile of the director within the four-arm junction is obtained by using numerical modeling (Fig. 3*C*). Increasing the flow speed (or Er) results in a further pronounced flow alignment of the director, and at still larger Er values, the system attains a complete flow alignment with the nematic director aligned roughly parallel to the channel direction. Because the two flow-aligned domains meet at the junction center, the mismatch in the nematic director leads to the formation of a small defective loop of charge *C*, *Middle*). At high Er values, the flow shear takes over the elastic forces and determines the director field in the proximity of the newly emerged nematic defect (35). The defect loop can also flip and stretch out of the vertical plane (Fig. 3*C*, *Bottom*). A stable *D*, a combination of three inflow arms (left, right, and top) and one outflow arm (bottom), results in a defect-free state at the junction center. By switching off the inflow in the top arm (Fig. 3*E*), the system gradually reorganizes, and as symmetric outflow conditions are restored, a transition to the defective configuration (Fig. 3 *F* and *G*) is observed. This result shows that designing different microfluidic circuits and junction geometries could be used as an interesting route for creation of nematic defect structures of various complexity.

## Discussion

The coupling between the velocity and the orientational fields serves as a tunable mechanism for designing multifield topology in nematic microfluidic systems. Our results reveal that this coupling also underpins the cross-interaction between the topological defects in the flow velocity and the nematic orientational fields. We quantify the interaction strength between the hydrodynamic and nematic defects in a four-arm junction by perturbing the defects out of their equilibrium position and analyzing the relative separation between them over time. Altering the inlet pressure in one of the flow arms displaces the stagnation point off the center first followed by gradual recovery of the nematic director. Fig. 4 *A* and *C* presents this dynamics using numerical calculations dynamics. After the stagnation point and the nematic defect are separated (Fig. 4*A*, column 1), the latter approaches the stagnation point, and within *B* and *D*). However, because the nematic defect now moves against the flow, the recovery is 10 times slower than in the previous case. Furthermore, the nematic defect initially moves backward before progressing toward the hydrodynamic stagnation point at the new location.

Experimentally, we perturb our system out of the equilibrium state by marginally increasing the inlet pressure in the left arm (Fig. 4*E*) and record the position of the defects over time. By overlaying consecutive frames of the recorded video data, we obtain a processed micrograph that captures the transport of tracer particles (bright dots along the flow direction in Fig. 4*E*) and the position of the defect over time (indicated by the yellow arrows in Fig. 4*E*). The separation between bright dots is the distance traveled by a particle over the time interval between consecutive frames. This recorded dynamics gives us a flow speed of *E*, *Top*. The topological defects remain colocalized at the center of the junction (no relative shift) (0–5 s in Fig. 4*F*). When we increase the inlet pressure in the left arm (Fig. 4*E*, *Middle*), the flow speed increases to *F*) by *E*, *Bottom*, the defect shifted by *F*). When the perturbing pressure was released, the stagnation point rapidly returned to the junction center followed slowly by the *F*). The observed dynamics shows a complex interaction between the hydrodynamic stagnation point and the nematic defect, which is clearly dependent on the direction of motion of the nematic defect relative to the local material flow. More generally and in a mechanics-motivated view, the emergent dynamics of the two defect types in the vicinity of each other could be viewed as induced by an interdefect force (or potential) that stems from the coupling of the two material fields and is inherently mediated by the topology (i.e., the topological charge) of the involved defects.

The cross-interaction between topological defects originating from different fields, although shown in the context of nematic microfluidics, is a phenomenon, which owing to its topological nature, is much more general in appeal. The demonstrated cross-talk relies on the existence of multiple spatially overlying material fields—in our case, vector type but it could also be scalar or tensorial—that are mutually coupled by some force-, stress-, or energy-like cross-coupling mechanism. Therefore, the natural candidates for such phenomena will be systems with pronounced transport effects or strongly interacting fields. As possibly the most far-reaching question of this type, such concepts of cross-field interacting defects could offer a physical framework for addressing phenomena in other areas of condensed matter physics, field theory, and cosmology.

In conclusion, the interplay between fluid flow and molecular orientation in nematic microfluidics has revealed that a hydrodynamic stagnation point can nucleate defects, whose topological charge can be hierarchically tuned by changing the rotational symmetry of the junction (Fig. S5 and *SI Text*) Importantly, our experiments, numerical modeling, and analytical calculations show that topological defects in different material fields cross-talk and that their characterization reveals a topology-dependent interaction between these defects of hydrodynamic and nematic-ordering origin. As defects from different fields can coexist in several soft and living matter systems, this work introduces an exciting perspective, and paves the way toward understanding the potential role of multifield topology in equilibrium and nonequilibrium systems.

## Materials and Methods

### Experimental Setup.

We have used 5CB, a single-component nematic LC (

### Numerical Simulations.

Our numerical simulations rely on Beris–Edwards formulation of nematodynamics (49) describing the evolution of system density, velocity, and nematic tensor order parameter by the coupled continuity equation, Navier–Stokes equation, and Beris–Edwards equation. Coupling between flow and orientational order is included by the nematic stress tensor and the flow-driven deformations of the nematic tensor order parameter profile that compete with the relaxation of nematic orientation toward the free energy minimum. The nematic free energy is constructed phenomenologically, including terms describing phase behavior, effective elasticity, and surface anchoring (34). Continuity and Navier–Stokes equations are solved numerically by a lattice Boltzmann algorithm (55), with open boundaries and pressure-driven flows through the channels. Simultaneously, evolution of nematic tensor order parameter is solved by a finite difference algorithm (*SI Text*).

## Acknowledgments

The authors thank Simon Čopar for insightful discussions on the dynamics of defect nucleation. A.S. thanks Stephan Herminghaus and Christian Bahr for discussions at different stages of this work. L.G. is supported by The Netherlands Organization for Scientific Research. Ž.K. and M.R. are supported by the Slovenian Research Agency Grants J1-7300, L1-8135, and P1-0099 and US Air Force Office of Scientific Research, European Office of Aerospace Research and Development Grant FA9550-15-1-0418, Contract 15IOE028. A.S. thanks Human Frontier Science Program Cross Disciplinary Fellowship LT000993/2014-C for support and the Max Planck Society for funding the initial phase of this work at the Max Planck Institute for Dynamics and Self-Organization, Goettingen, Germany.

## Footnotes

↵

^{1}Present address: Institute for Environmental Engineering, Eidgenössische Technische Hochschule (ETH) Zurich, 8093 Zurich, Switzerland.- ↵
^{2}To whom correspondence should be addressed. Email: anupams{at}ethz.ch.

Author contributions: L.G., M.R., and A.S. designed research; L.G. developed particle model and performed analytical calculations; Z̆.K. and M.R. performed numerical simulations; A.S. conceptualized research, conducted experiments, analyzed data, and provided advice for all parts of the work; and L.G., Z̆.K., M.R., and A.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1702777114/-/DCSupplemental.

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