Electrically-driven modulation of flow patterns in liquid crystal microfludics

The flow of liquid crystals in the presence of electric fields is investigated as a possible means of flow control. The Beris-Edwards model is coupled to a free energy incorporating electric field effects. Simulations are conducted in straight channels and in junctions. Our findings reveal that local flow mediation can be achieved by the application of spatially varying electric fields. In rectangular straight channels, we report a two-stream velocity profile arising in response to the imposed electric field. Furthermore, we observe that the flow rate in each stream scales inversely with the Miesowicz viscosities, leading to the confinement of 70% of the throughput to one half of the channel. Similar flow partitioning is also demonstrated in channel junction geometries, where we show that using external fields provides a novel avenue for flow modulation in microfluidic circuits.


Microstructure evolution
The analysis presented below assumes that the liquid crystal remains in the uniaxial state, but we allow for a variable order parameter Q.We consider a one-dimensional shear flow, similar to other studies 1 ; the order parameter tensor is given by where n = [cos(θ ), sin(θ ), 0], and Q is a variable order parameter.The time derivative of Q can be decomposed into the Q− and θ −dependent components: Since A : B = 0, we can obtain the evolution equations for the order parameter and the director angle by taking the double contraction of eq. ( 2) with either A or B, respectively: where a, b, c are the parameters of the nematic energy, Ha is the Hartman number, and γ is the shear rate.Note that in the limit of Q = 1, ξ = 1, eq. (4) reduces to eq. ( 16) of the main manuscript with the elastic effects neglected.

De << 1 limit
The order parameter is constant (and equal to Q eq ) in the De << 1 limit, and the behaviour of the system is solely described by the evolution equation for the director angle: The qualitative response of the system strongly depends on the tumbling parameter ξ .When ξ ≥ 1, the director reaches a fixed angular orientation in the absence of an external field 2 .When the electric field is present, its effect on the resultant director angle depends its strength relative to the viscous effects.When ξ < 1, oscillations may occur; their character typically depends on the Ha regime.
1.1.1ξ = 0 In the limit ξ = 0, the system is most susceptible to tumbling.For this case, we can simplify further the θ evolution equation to A steady state solution is achieved if Eq.( 7) shows the role of the electric field in enforcing a stationary solution; increasing the field strength (Ha → 0) drives the director towards perfect co-alignment.Conversely, when the electric field is sufficiently weak (Ha > Q eq γ), no such steady state can be achieved.
For this case, the qualitative behaviour of the system also depends on the alignment between the electric field and the deformation axes.
Assuming that the electric field is oriented in the flow direction (α = 0), the θ evolution equation becomes The solution to this equation for the steady state is sin(2θ When the external field is sufficiently strong, fixed steady state solutions may be obtained for a range of values of (Q, ξ ).As the field strength weakens (Ha >> 1), θ becomes complex and the solution becomes oscillatory; the oscillation frequency increases with increasing Hartmann number.Representative plots of the stationary/oscillatory regions in the Ha − Q space are illustrated in fig.S1(a-c). Figure S2 shows that imposing an electric field has a similar effect to increasing the tumbling parameter, in that the field allows a steady state solution to exist at small ξ , where oscillations would otherwise occur.

Non-negligible De
At non-negligible Deborah numbers, there is a coupling between the order parameter and the director.This is illustrated in fig.S3.Equations ( 3) and (4) are nonlinear, so an analytical description of the coupled system's behaviour is not possible.Figure S3 nevertheless shows that the combined system exhibits a periodic behaviour whose dominant frequency may be extracted via FFT; the result of this is illustrated in fig.S4.Increasing the strength of the electric field (reducing Ha) reduces the frequency of the transient behaviour, ultimately leading (for Ha → 0) to a fixed director angle in the steady state.

Figure S1 .
Figure S1.Solution behaviour as a function of the steady state order parameter and the Hartman number.The blue (red) regions denote parameter values with the oscillatory (stationary) solutions.The tumbling parameters used to obtain the diagrams are: a) ξ = 0; b) ξ = 0.4; c) ξ = 0.8.

Figure S2 .
Figure S2.Critical Hartman number below which the steady state solution is obtained.

Figure S5 .
Figure S5.Critical Hartman number below which a fixed value is obtained in the steady state as a function of the Deborah number.Parameters of the nematic free energy are chosen such that Q eq = 0.62.