Kui
Chen
^{a},
Olivia J.
Gebhardt
^{a},
Raghavendra
Devendra
^{b},
German
Drazer
^{b},
Randall D.
Kamien
^{c},
Daniel H.
Reich
^{a} and
Robert L.
Leheny
*^{a}
^{a}Department of Physics & Astronomy, Johns Hopkins University, Baltimore, MD, USA. E-mail: leheny@jhu.edu
^{b}Department of Mechanical & Aerospace Engineering, Rutgers, The State University of New Jersey, Piscataway, NJ, USA
^{c}Department of Physics & Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

Received
21st August 2017
, Accepted 24th October 2017

First published on 24th October 2017

We have investigated the gravity-driven transport of spherical colloids suspended in the nematic liquid crystal 4-cyano-4′-pentylbiphenyl (5CB) within microfluidic arrays of cylindrical obstacles arranged in a square lattice. Homeotropic anchoring at the surfaces of the obstacles created periodic director-field patterns that strongly influenced the motion of the colloids, whose surfaces had planar anchoring. When the gravitational force was oriented parallel to a principal axis of the lattice, the particles moved along channels between columns of obstacles and displayed pronounced modulations in their velocity. Quantitative analysis indicates that this modulation resulted from a combination of a spatially varying effective drag viscosity and elastic interactions engendered by the periodic director field. The interactions differed qualitatively from a sum of pair-wise interactions between the colloids and isolated obstacles, reflecting the distinct nematic environment created by confinement within the array. As the angle α between the gravitational force and principal axis of the lattice was varied, the velocity did not follow the force but instead locked into a discrete set of directions commensurate with the lattice. The transitions between these directions occurred at values of α that were different from those observed when the spheres were in an isotropic liquid, indicating the ability of the liquid crystal forces to tune the lateral displacement behavior in such devices.

Recently, research has expanded on this theme of colloidal manipulation and assembly through liquid crystal elastic forces to investigate the behavior of colloids within patterned director fields.^{35,41–44} In addition, spatially modulated colloidal transport has been demonstrated previously in liquid crystals with intrinsically periodic structures, such as cholesterics.^{45,46} Here, we bridge these ideas by addressing the possibility of engineering colloidal transport behavior through patterned anchoring by exploiting the periodic director-field configurations that form within arrays of obstacles in microfluidic devices. The obstacles, cylindrical posts arranged in a square lattice, have surfaces that promote homeotropic anchoring of the nematic director, thereby creating periodic director-field patterns in the surrounding nematic that strongly influence the mobility of colloids through the fluid. These studies build on research on colloidal transport in simple liquids in such microfluidic devices containing arrays of obstacles that drive the particles on size-dependent paths.^{47–56} We find similarities between the behavior of the colloids in the nematic and that in the isotropic liquids but also some notable differences. The velocity of the colloids through the nematic displays a pronounced modulation that we identify as the consequence of the combined effects of spatially varying viscous drag and elastic forces. We also observe how the interactions experienced by the colloids in the nematic can alter their path as they move past an isolated obstacle, resulting in irreversible trajectories. Most significantly, and in contrast to the behavior observed in simple liquids, the trajectories exhibited by the colloidal particles inside the array cannot be described as the result of pair-wise additive particle–obstacle interactions. This qualitative difference indicates a new mechanism in such devices for separation technologies.

Since bare PDMS imposes weak planar anchoring of the nematic director, the PDMS containing the post arrays was functionalized with N,N-dimethyl-N-octadecyl-3-aminopropyltrimethoxysilyl chloride (DMOAP) to achieve strong homeotropic anchoring on the post and substrate surfaces. The PDMS with the post arrays was then bonded to a sheet of flat PDMS by oxygen plasma treatment to create an enclosed environment. During the plasma activation, the region of the PDMS containing the post array was shielded to preserve the DMOAP functionalization on the posts and bottom substrate such that only the tops of the walls surrounding the array were activated. The flat sheet of PDMS forming the top of the device was left unfunctionalized and hence imposed weak, degenerate planar anchoring. Two small holes punched into the top sheet prior to the bonding served as channels for introducing the 5CB, which is nematic at room temperature. Untreated silica spheres with density ρ_{s} = 2 g cm^{−3} and radii between approximately 6 and 9 μm were premixed with 5CB at low concentration, and the mixture was then introduced through the channels while the device and 5CB were heated to 40 °C, which is above the isotropic-nematic transition temperature of 5CB (T_{NI} ≈ 34 °C). After filling, the device was allowed to cool slowly to room temperature to so that the director field was unaffected by the flow during the filling.

3.2.1 Periodic velocity modulation.
In examining the mobility of colloids in the micropost arrays, we first consider the case when the arrays were oriented so that the gravitational force and principal axis of the lattice were parallel (α = 0). In this case, we observed that the colloids traversed the lattice unobstructed by the posts by translating along the midline between two columns of posts. This motion, which is depicted in the micrographs in Fig. 4(a)–(d) showing a sphere with radius 6.1 μm descending through an array, led to essentially straight-line trajectories. (In addition to following a straight-line path in the focal plane, the colloids displayed no measurable variation in their distance from the device substrate as they descended; see Supplementary Video SV1 (ESI†).) However, interactions between the spheres and posts mediated by the liquid crystal, as well as a spatially varying viscous drag, led to a pronounced modulation in the speed of the colloids as they followed these paths. For example, Fig. 5(a) displays the velocity v of the colloid shown in Fig. 4(a)–(d) as a function of its height y as it traversed three periods of the array. Fig. 5(b) shows the velocity through a single period of the lattice on an expanded scale. The colloid's vertical position h within the unit cell of the lattice in Fig. 5(b) is measured with respect to that of the post centers, as indicated by the dashed line in Fig. 4(d), and increasing h is taken to be downward. As the colloid approached the midpoint between adjacent posts (h = 0), the speed increased, and reached a maximum at a value of h slightly less than 0. After the colloid passed between the posts, the velocity decreased until it reached a minimum slightly before the sphere reached the midway point between two rows of posts (h = H/2 = 30 μm). Additional examples of the velocity vs. height for other particles displaying the same trend are shown in Fig. S2 of the ESI.†

Fig. 5 (a) Velocity of the colloidal sphere shown in Fig. 4 as a function of its height y in the obstacle array. Increasing y is taken to be downward, and the origin of y is set as the particle position when we started to record the motion. Included in the plot is the velocity over three periods of the array after the motion was stable. (b) Velocity of the sphere within one period of the array as a function of its vertical position h within the unit cell. Again, increasing h is taken to be downward, and the origin of h is taken to be the position of the post centers. |

As the results in Fig. S2 (ESI†) indicate, the large modulation in speed seen in Fig. 5, where v oscillates by more than a factor of two, was typical of colloids in 5CB traversing the arrays with α = 0. To test that the modulation was the result of the nematic order of the solvent, we performed the same measurements on colloids in water and in glycerol. In the isotropic liquids, the colloids’ speed also oscillated as they traversed the array, but the magnitude in the variation was typically no more than 10% the mean speed, and the minimum speed coincided with the colloids’ passing through h = 0, consistent with a slightly enhanced hydrodynamic drag when the colloids were closest to the posts. (See Fig. S3 of the ESI,† for example results for the velocity of a sphere in water as a function of height in an array.) In contrast, as shown in Fig. 5, the speed of the colloids traversing the array within 5CB surprisingly reached a maximum near h = 0.

3.2.2 Extraction of spatially-varying elastic and viscous forces.
Analysis of the height-dependent velocity of the colloids, like in Fig. 5, allows us to disentangle the forces acting on the colloids as they descend through the microfluidic array under the force of gravity F_{g}. Specifically, we identify two contributions from the liquid crystal that cause the velocity modulation, (i) the viscous drag force F_{drag}, which varies spatially due to the patterned director field, and (ii) an elastic force F_{el} associated with the position-dependent energy cost E_{el} of the distortion the colloids impose on the director field. (As discussed above for the colloid passing the isolated post, backflow induced by the reorientation of the director as the colloid descends the lattice could also contribute a spatial varying force due to the patterned director field. Such contributions would have the same symmetry with respect to the lattice as the elastic force, as described below, and hence should be considered incorporated into F_{el}.) Because the motion is at low Reynolds numbers (Re ∼ 10^{−6}), these three forces sum to zero,

Taking account of the buoyancy force and the tilt angle, ϕ = 70°, of the microscope,

where R is the sphere radius, ρ_{s} is the density of silica, ρ_{lc} is the density of 5CB, and g is the acceleration of gravity.

where each term is expressed as a scalar. Given the mirror symmetry of the lattice about h = 0, we conclude that η_{eff}(h) and E_{el}(h) are even functions of h, and therefore F_{el}(h) is an odd function of h. Hence, evaluating eqn (3) when the colloid is at position –h, we have

and subtracting eqn (4) from eqn (3), we get:

where Δν(h) = ν(h) − ν(−h).

Results for F_{el}(h), again obtained from the velocity data in Fig. 5(b), are shown in Fig. 6(b). As expected from symmetry, F_{el} = 0 when h = 0 and h = ±H/2. At any point in the particle's trajectory, the force is directed toward the midpoint between posts (h = 0), so that at h < 0 the force is downward (positive), and at h > 0 it is upward (negative). Over a broad range of positions centered at h = 0, the force is linear in the displacement from h = 0, indicating an effective Hookean interaction.

The spatially varying drag force and effective viscosity obtained from eqn (7) and (8) are plotted in Fig. 6(b) and (c), respectively. The effective viscosity η_{eff} is largest at h = ±H/2 and decreases as the colloid translates toward the passage way between two posts, reaching a minimum at h = 0, before increasing again after it passes through. The variation in the effective viscosity, , is about 1.5. A spatially varying viscous drag can be expected since the colloid experiences a changing local director field as it traverses the array.^{64,66,67} The minimum in the drag at h = 0 is somewhat surprising, however, since the director is primarily perpendicular to the colloid's velocity at that point, which is the orientation of maximum drag in a uniform director. We attribute the increasing drag on approaching the points h = ±H/2 to the presence of the surface defect at those points, as illustrated in Fig. 2(d). Any motion of the defect induced by the colloid will contribute to the viscous dissipation, and as shown previously such dissipation can lead to anomalously large effective drag viscosities.^{46}

F_{g} + F_{drag} + F_{el} = 0. | (1) |

(2) |

The viscous drag on a sphere in a nematic is significantly subtler than in an isotropic liquid. For instance, the flow created by a colloid with sufficient velocity can alter the director field, leading to nonlinear effects. The importance of these effects is parameterized by the Ericksen number Er, which is the ratio of the viscous to elastic forces. For a translating sphere of radius R, the Ericksen number can be taken as Er = ηRv/K, where η ≈ 35 mPa s is an average viscosity^{60} and K ≈ 4 pN is the average Frank elastic constant of 5CB.^{61} For the velocities in Fig. 5, we find Er < 0.25, which is within the range in which the flow is expected to have little if any effect on the director field.^{62,63} However, even at low Ericksen number the anisotropic nature of the nematic complicates its viscous properties,^{64,65} and complete description of the drag requires a tensorial form of Stokes equation.^{65} Further, motion of defects in the liquid crystal induced by the colloidal motion can contribute to the drag.^{46} Because we observe that the colloids at α = 0 follow a straight path along a line of symmetry bisecting columns of posts, we employ a simplified Stokes law with a height-dependent effective drag viscosity η_{eff} to account for the effects of the patterned director field on the drag. We further assume that both η_{eff} and the elastic interactions have the periodicity of the lattice. The same is thus the case also for the elastic force, F_{el} = −dE_{el}/dh, and eqn (1) can hence be written as

F_{g} − 6πRη_{eff}(h)ν(h) + F_{el}(h) = 0, | (3) |

F_{g} − 6πRη_{eff}(h)ν(−h) − F_{el}(h) = 0, | (4) |

2F_{el}(h) = 6πRη_{eff}(h)Δν(h), | (5) |

Results for Δν(h) obtained from the data in Fig. 5(b) are shown in Fig. 6(a). With these results and eqn (5), we can obtain the spatially varying drag viscosity and elastic force experienced by the colloids as they traversed the array. Specifically, solving eqn (5) for η_{eff}(h) and inserting it back into eqn (3), we get a relationship for the elastic force:

(6) |

Fig. 6 (a) The velocity difference Δv(h) = v(h) – v(−h) obtained from the results in Fig. 5(b). (b) The elastic force F_{el} (open circles) and drag force F_{drag} (filled circles) acting on the sphere as a function of vertical position. The red line shows the result to a linear fit to F_{el} over the range −10 < h < 10 μm. (c) The position-dependent effective drag viscosity obtained from the velocity and F_{drag}. |

Furthermore, the Stokes drag and effective viscosity are obtained as:

(7) |

(8) |

3.2.3 Discussion of elastic force.
Quantitative modeling of F_{el}, which as mentioned above potentially contains contributions from both elasticity and backflow, would require detailed calculations of the director field in the vicinity of the colloids and posts as a function of colloid position. However, we can assess on dimensional grounds whether the force is at a scale expected for an elastic interaction. Specifically, the magnitude of such a force should be set by the Frank elastic constants of the liquid crystal, so that in the Hookean region at small h, the force has a form

where K ≈ 4 pN is an average elastic constant of 5CB and L is a length scale related to the region of the liquid crystal in which the director-field distortion by the colloid costs energy. The solid line in Fig. 6(b) shows the result of a linear fit to the force over the range −10 μm < h < 10 μm. The slope gives K/L ≈ 0.08 pN μm^{−1}, or L ≈ 50 μm, a scale similar to that of the dimensions of the array, indicating the non-viscous contribution to the force on the colloids indeed results primarily from elastic interactions. More interesting than the precise size of the force, however, is its dependence on the position of the colloids with respect to the posts. As discussed in Section 3.1, the trajectory of a colloid in the vicinity of an isolated post reveals a repulsive pair interaction, consistent with expectations based on the incompatible anchoring conditions at the colloid and post surfaces. In contrast, the mobility of a colloid in the micropost array shows how the force on the colloid in this confined environment cannot be treated simply in terms of pair-wise additive interactions. Specifically, while the tendency of the colloids to follow a path that bisects columns of posts is consistent with repulsion from the posts, the Hookean force implies an attraction to the height h = 0, where the colloids are at their closest point to the posts, that indicates a subtler spatial dependence of the force. Qualitatively, the attraction to h = 0 can be understood in terms of the director field configuration depicted in Fig. 2(c) and (d). The relatively uniform director field near h = 0 is more compatible with the planar anchoring on the sphere surfaces than is the divergent director field at h = ±H/2, implying a gradient in the elastic energy caused by the colloid with an energy minimum at h = 0.

(9) |

Employing spheres with radii around 6.3 μm, we found that the colloids exhibited directional locking by selecting among only three propagation directions through the lattice as α was varied from 0 to 45°. These directions, which can be labeled based on their lattice vectors as [0,1], [1,2] and [1,1], are illustrated in Fig. 7(a)–(c). Fig. 8 shows the migration angle θ, defined as the angle between the propagation direction and the [0,1] direction, as a function of α. When the angle between the force and [0,1] direction was small, the particles’ trajectories remained locked with the lattice in the [0,1] direction (θ = 0), and when the angle between force and lattice [0,1] direction was close to 45°, the trajectories were locked with the lattice in the [1,1] direction (θ = 45°). Finally, when α as at intermediate values, the trajectories locked to the [1,2] direction (θ = 26.6°) and exhibited doubly periodic motion, as illustrated in Fig. 7(b). (For examples of colloidal propagation along the each direction, see ESI† Videos SV1, SV2, and SV3, which show colloids descending through arrays oriented at α = 0, 18°, and 32°, respectively.) The first transition angle θ_{F} at which the trajectories changed from the [0,1] to the [1,2] direction was θ_{F} ≈ 12°, and the second, or last, transition angle θ_{L} from [1,2] to [1,1] was θ_{L} ≈ 25°. We note that the Peclet number in the experiment, Pe = F_{g}R/k_{B}T, where k_{B} is the Boltzman constant and T is temperature, was approximately 2 × 10^{4}, indicating that diffusion was negligible and this particle motion was essentially deterministic.

For comparison, we performed the same set of measurements on silica spheres of the same size translating through an isotropic liquid (15 mM KOH aqueous solution). The results for the propagation directions are also shown in Fig. 8. In the isotropic liquid, the spheres assumed the same three locking directions as in nematic 5CB; however, the transition angles from [0,1] to [1,2] and from [1,2] to [1,1] were different. Specifically, θ_{F} = 23° and θ_{L} = 30°, respectively, in the isotropic fluid. The values of these angles agree closely with a prediction of the model of Risbud and Drazer,^{56}

(10) |

We identify this disagreement with two approximations in the model of Risbud and Drazer that the nematic array violates. First, the model incorporates non-hydrodynamic interactions between the sphere and obstacles by assigning a hard-core effective radius to the obstacles, an approximation that is suitable for short-range interactions related to surface roughness, etc. that dominate in simple isotropic liquids but is apparently not appropriate for the softer, longer-range interactions mediated by the nematic. Second, the model assumes the sphere's trajectory through the array is dictated by isolated, pair-wise interactions with the obstacles. As comparison of the results in Sections 3.1 and 3.2 shows, such an approximation would lead to a qualitatively incorrect description of the motion in the array. One source of the breakdown in the pair-wise approximation is the extended range of the interaction between posts and colloids, implying the colloids interact with multiple posts simultaneously. More fundamentally, however, the differing boundary conditions on the nematic director introduced by a post in isolation versus by an array such posts lead to different director-field configurations that result in qualitatively different nematic forces on the colloids.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7sm01681f |

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