Active nematic flows confined in a two-dimensional channel with hybrid alignment at the walls: A unified picture

Active nematic fluids confined in narrow channels are known to generate spontaneous flows when the activity is sufficiently intense. Recently, it was demonstrated [R. Green, J. Toner, and V. Vitelli, Phys. Rev. Fluids 2, 104201 (2017)] that if the molecular anchoring at the channel walls is conflicting, i.e., perpendicular on one plate and parallel on the other, flows are initiated even in the zero activity limit. An analytical laminar velocity profile for this specific configuration was derived within a simplified nematohydrodynamic model in which the nematic order parameter is a fixed-magnitude unit vector $\mathbf{n}$. The solution holds in a regime where the flow does not perturb the nematic order imposed by the walls. In this study, we explore systematically active flows in this confined geometry with a more general theoretical model that uses a second-rank tensor order parameter Q to express both the magnitude and orientation of the nematic phase. The Q-model allows for the presence of defects and biaxial, in addition to uniaxial, molecular arrangements. Our aim is to provide a unified picture, beyond the limiting regime explored previously, to serve as a guide for potential microfluidic applications that exploit the coupling between the orientational order of the molecules and the velocity field to finely control the flow and overcome the intrinsic difficulties of directing and pumping fluids at the microscale. We reveal how the nematic-flow coupling is not only dependent on geometrical constraints, but is also highly sensitive to material and flow parameters. We specifically stress the key role played by the activity and the flow aligning parameter, and we show that solutions mostly depend on two dimensionless parameters. We find that for large values of the activity parameter, the flow is suppressed for contractile particles while it is either sustained or suppressed for extensile particles depending on whether they tend to align or tumble when subject to shear. We explain these distinct behaviors by an argument based on the results of the stability analysis applied to two simpler configurations: active flows confined between parallel plates with either orthogonal or perpendicular alignment at both walls. We show that the analytical laminar solution derived for the $\mathbf{n}$ model in the low activity limit is found also in the $\mathbf{Q}$ model, both analytically and numerically. This result is valid for both contractile and extensile particles and for a flow-tumbling as well as aligning nematics. We remark that this velocity profile can be derived for generic boundary conditions. To stress the more general nature of the $\mathbf{Q}$ model, we conclude by providing a numerical example of a biaxial three-dimensional thresholdless active flow for which we show that biaxiality is especially relevant for a weakly first-order isotropic-nematic phase transition.

Figure 1

Left: schematic representation of a channel with hybrid alignment at the walls. The channel walls located at $y=0$ and $y=L$ extend to infinity in the $x$ and $z$ directions. The anchoring of the active nematic liquid crystals is parallel to the $y=0$ wall (homogeneous anchoring) and normal to the $y=L$ wall (homeotropic anchoring). The numerical integration is performed in 1D. Right: Normalized root-mean-square error measuring the deviation of the numerical velocity profile from the analytical expression Eq. (12) as a function of the magnitude of the activity parameter. In the formula reported on the $y$-axes, $N$ is the number of grid-points. The numerical solution is obtained by integrating the full active nematohydrodynamic equations with either a two-dimensional or a three-dimensional tensor order parameter $Q_{ij}$.The parameters of the simulations are $\nu =0.33$, $\rho =2$, $L=256$, $t=500000$, $\xi =0.7$, $\mathrm{\Gamma }=16000$, and $K=5\times {10}^{-6}$. For the two-dimensional case, $q_0=0.9998$, $A=-2.5\times {10}^{-6}$, $B=0$, and $C=5\times {10}^{-6}$; for the three-dimensional case, $q_0=0.5$, $A=0$, and $B=-C=-3\times {10}^{-5}$. These parameter values correspond to ${10}^{-2}<\left|{\mathrm{\Pi }}_1\right|<{10}^5$, ${\mathrm{\Pi }}_2\approx 6.8\times {10}^{-3}$, ${\mathrm{\Pi }}_3=10560$.

Figure 2

Top: Base 10 logarithm of the maximum magnitude of the velocity computed numerically and rescaled by the maximum of the analytical profile (12) for a contractile (a) and extensile (b) activity parameter, $\alpha$. This normalized velocity is plotted as a function of ${\mathrm{\Pi }}_3$ and $|{\mathrm{\Pi }}_1|$ for a flow-aligning nematics ($\xi =0.7$). The axes are in logarithmic scale, and the map reports results for a total of 330 separate calculations. Simulations with smaller values of ${\mathrm{\Pi }}_3$ are more demanding in computational terms given the slower convergence: for our choice of parameters, the slowest calculations run for $9.8\times {10}^8$ time steps. Middle: rescaled velocity profiles corresponding to cases that lie on a vertical cut of the color maps in (a) and (b) as specified by the legend and title of the plot for a negative (c) and positive (d) value of the activity parameter. In (d) the flow profiles with ${\mathrm{\Pi }}_3<6.67\times {10}^2$ are unsteady. In these cases we display the configuration at the final time $T_{\text{fin}}$. We label as unsteady those calculations for which the RMS deviation in the last 10 saved time steps spaced by approximately $\tau /50$ time units is below 0.25%. The thick black curves correspond to the analytical solution (12). In (e) we show some intermediate configurations for a selected unsteady case. Bottom: (f)–(h) director field orientation associated with three cases as detailed by the plot titles.

Figure 3

Nematic angle $\theta =arctan(n_y/n_x)$ (a) and velocity profile (b) for two of the calculations reported in Figs. 2 and 2 and additional simulations performed for the same values of the model parameters and different initial conditions (I.C.). In the legend, “I.C. 1” corresponds to the initial condition of Fig. 2, that is, an $n_x=1$ field perturbed by random noise, “I.C. 2” is given by $n_x=cos(\pi y/L),n_y=sin(\pi y/L)$, while “I.C. 3” is Eq. (11). In all cases, the velocity field is initialized to zero.

Figure 4

Top: Base 10 logarithm of the maximum magnitude of the velocity computed numerically and rescaled by the maximum of the analytical profile (12) for negative (a) and positive (b) values of the activity parameter and a flow-tumbling nematics ($\xi =0.3$). The axes correspond to the ${\mathrm{\Pi }}_3$ and $|{\mathrm{\Pi }}_1|$ parameters and are in logarithmic scale. Two $|{\mathrm{\Pi }}_1|/{\mathrm{\Pi }}_3$ isolines are shown in panels (a),(b): $|{\mathrm{\Pi }}_1|/{\mathrm{\Pi }}_3=2\times {10}^4$ (dashed-black line) and $|{\mathrm{\Pi }}_1|/{\mathrm{\Pi }}_3=20$ (solid-black line). Middle: rescaled velocity profiles corresponding to calculations that lie on a diagonal cut of the color maps in (a) and (b); the cuts originate at the top-left corner of the maps and run perpendicular to the $|{\mathrm{\Pi }}_1|/{\mathrm{\Pi }}_3$ isolines. For clarity, the legend only labels the two curves that correspond to the extreme values of ${\mathrm{\Pi }}_3/{\mathrm{\Pi }}_1$. The thick black lines correspond to the analytical solution (12). Observe the remarkable resemblance of the rescaled velocity profiles in (c) with those reported in Fig. 2 for the flow-aligning case. Bottom: nematic angle $\theta =arctan(n_y/n_x)$ corresponding to the rescaled velocity profiles in (c). The analytical solution corresponds to a straight line, while the zero-flow solution corresponds to a discontinuous profile that suddenly jumps close to the bottom wall from $n_x=1$ to $n_y=1$; this is allowed in the $\mathbf{Q}$-model by a concomitant $q_0=0$.

Figure 5

Nematic angle $\theta =arctan(n_y/n_x)$ (a),(c) and velocity profile (b),(d) rescaled by the maximum of the analytical profile for the flow-aligning case (a),(b) and the flow-tumbling case (c),(d) and two different values of the $\mathrm{\Phi }={\mathrm{\Pi }}_3/{\mathrm{\Pi }}_1$ dimensionless group.

Figure 6

Schematic representation of the numerical results reported in Sec. 3b. For small values of the dimensionless parameter $\mathrm{\Phi }={\mathrm{\Pi }}_1/{\mathrm{\Pi }}_3$ we observe thresholdless active flows [14] independently of the sign of the activity parameter or the value of the flow-aligning parameter (green shaded area). For intermediate values of $\left|\mathrm{\Phi }\right|$ the velocity field is nonzero and depends on the sign of the activity parameter as well as the value of the flow-aligning parameter (gray shaded area). There exists a very close resemblance of the velocity profiles in the transition region for three cases over four: positive activity and flow tumbling Fig. 4, negative activity and flow tumbling Fig. 4, negative activity and flow aligning Fig. 2. The positive activity and flow-aligning case differs and shows some dependence on the initial conditions, Fig. 2. The behavior in the regions of large magnitude of $\mathrm{\Phi }$, beyond the transition regions, can be rationalized on the basis of previous studies [8, 10, 31] for the flows of active nematics in channels with either homeotropic or parallel boundary conditions. The hybrid boundary condition (${\theta }_0=0$, ${\theta }_L=\pi /2$) can be interpreted as a combination of parallel and perpendicular boundary conditions. For those cases, the stability conditions have been derived in the literature, and their combination suggests the type of flow we observe in the hybrid case (blue and red shaded areas).

Figure 7

Map of the numerical solutions for the 1D velocity profile as a function of the flow-aligning parameter $\xi$ and the dimensionless number $\mathrm{\Phi }$. Different symbols correspond to different types of solutions: circles denote zero-velocity solutions that correspond to a diagonal $\mathbf{Q}$-tensor; triangles indicate solutions whose normalized root-mean-square (RMS) value does not deviate from Eq. (12) by more than 0.01; squares indicate all the other types of solutions, and diamonds designate unsteady solutions. The color represents the rescaled magnitude of velocity as in Figs. 2 and 2 and Figs. 4 and 4. The boundary between the flow-tumbling and flow-aligning behavior is given by $\xi =0.5$ and is marked by a black line. Note the expression for the $x$-axis chosen to display $\mathrm{\Phi }$ in $log$-scale while distinguishing between the positive and negative cases.

Figure 8

(a) Rescaled Landau–de Gennes free energy as a function of the dimensionless parameters $B{L}^2/K$ and $C{L}^2/K$ for the solution of ${\mathcal{H}}_{ij}=0$ obtained by relaxing the initial condition IC1 through Eq. (23). The free energy is rescaled by a reference free-energy value, ${\mathcal{F}}_0$, corresponding to a homogeneous solution for a system of size $L$. (b) Difference between the Landau–de Gennes free energy associated with IC1, ${\mathcal{F}}_1$, and IC2, ${\mathcal{F}}_2$, rescaled by ${\mathcal{F}}_0$. These calculations have been performed for $C{L}^2/K=60000$ and repeated for $0<C{L}^2/K<100000$. In this parameter range, we observe qualitatively the same type of solutions; differences concern the magnitude of $q_0$ in a narrow region close to the boundaries.

Figure 9

(a) Root-mean-square deviation from uniaxiality measured in terms of the difference between the closest eigenvalues of the $Q_{ij}$ tensor for FP1, FP2, and $B{L}^2/K\to 0$, here $C{L}^2/K=\left|A\right|L^2/K=60000$. The solid and dashed lines represent power-law scalings as reported in the legend. (b) Velocity profile for a thresholdless active flow for a 2D and 3D geometry and for a uniaxial case (ground state of type 3) and a biaxial case (ground state of type 4). The 2D uniaxial profile is given by Eq. (22); the 3D velocity field has a nonzero $u_z$ component that is not reported in the plot for clarity. All the curves are rescaled by the maximum of the analytical profile (22). In panels (c) and (d) we report the nematic director field for the 2D (c) and 3D (d) solution.