Nematic flow due to pressure gradient

Figure 6.1: Pressure-driven nematic flow. (a) Schematic of the microchannel, flow is along x direction. (b) Relative orientation of the initial director equilibrium and flow direction:

homeotropic (~v⊥~n,∇vk~n), uniform planar (~vk~n), and flow in isotropic phase.

The flow of nematic 5CB, induced by the application of a pressure gradient, provides us insights into the rheological behaviour of the material. To this effect, the microchannel inlet was connected to a pressure controller (Chapter 3.3.2), and the outlet was left at atmospheric pressure. The outlet tubing was was immersed within a 5CB sink placed on a sensitive mass balance, which measured the total mass of 5CB that flowed through the channel with a pre-cision of±100µg. For ensuring a reduced influence of the side walls, channels of very high

Figure 6.2: Average viscosity of 5CB measured using pressure-driven flow for homeotropic (~v ⊥~n,∇v k ~n), uniform planar (~v k~n), and isotropic cases, shown respectively by the black squares, red dots, and blue triangles. The measurements for the homeotropic and the uniform planar anchoring cases were done at room temperature. The inset plots the measured mass flow rate as a function of applied pressure difference.

aspect ratio (AR=w/d ≈133), withw≈ 2 mm andd≈ 15µm, were considered for the mea-surements. Three different flow cases were measured, based on the initial director equilibrium within the channels: ~v ⊥ ~n and ∇v k ~n (initially homeotropic), ~v k ~n (initially parallel to the channel length), and for 5CB in isotropic phase. Figure 6.1 summarizes the experimental cases considered.

Figure 6.2 plots the experimental estimation of the effective dynamic viscosity of 5CB as a function of applied pressure difference ∆P = Pinlet −Poutlet. The effective viscosities were calculated by measuring the mass flow rate of 5CB at different pressure differences:

η= wd³ 12lρ ∆P

˙ m

!

(6.1) Here, w,d, andl ≈ 4 cm denote the channel dimensions. ρ, ∆P, and ˙m signify 5CB density, applied pressure difference, and the measured mass flow rate, respectively. Two different

values of 5CB density were considered: in nematic phaseρ = 1025 kg/m³, and in isotropic phase ρ = 1013 kg/m³ [136]. The expansion of PDMS at ≈ 35^◦ C (isotropic phase) was neglected in the calculations. Additionally, the pressure drop at the inlet tubing of the channel (≈ 0.3 mbar at maximum flow rates), was neglected in determining the actual pressure at the inlet port.

As seen clearly in Fig. 6.2, the effective viscosity of 5CB depends not only on the ther-modynamic phase, but also on the nature of the boundary conditions prevalent. The flow was observed to experience a significantly higher viscous resistance under homeotropic conditions (ηhomeotropic), compared to the flow where the velocity and initial director were mutually paral-lel (ηplanar,k). Consequently, for a similar pressure difference between the inlet and the outlet ports, nematic 5CB flowed slower in the homeotropic channel than in the channel with uni-form planar boundary conditions. It is thus logical to expect that the viscosity η_planar,⊥ for the initial configuration~v ⊥ ~n – ∇v ⊥ ~n, shall lie intermediate between the ηhomeotropic and ηplanar,k values. As the applied pressure was increased, the difference of the viscosity values reduced. It might be worthwhile to mention here that the bulk nematic 5CB exhibits ashear thinningbehaviour [1]. The present experiments additionally show that this behaviour holds true for disparate anchoring conditions, and also in isotropic phase (Fig. 6.2, inset). Moreover, the variation of the viscosity within the planar microchannel shows an initial increase, before going down at higher applied pressures. This is attributed to the director reorientation. On starting the flow, the director reorients slightly away from the initial state (parallel relative to the flow direction), due to the interplay of the viscous and the elastic torques. Consequently, the effective viscosity has an additional contribution from the Miesowicz viscosityη₃ (Chap-ter 2.7.2). The reorientation continues till the viscous torque vanishes at the critical Leslie angle [76]. Thereafter, the viscosity reduces monotonically, as in the case of the isotropic and homeotropic flows. The discussed anisotropic attributes offer interesting avenues, especially the flow-director coupling can be utilized to devise novel applications based on LC microflu-idics. For instance, spatially modulated flow fields can be created within microchannels by incorporating composite boundary conditions, as the one shown in Fig. 6.3.

Further experiments on nematic flows demonstrated the reduction of the flow resistance at elevated temperatures (≈ 35^◦ C). This is shown in Fig. 6.4. The temperatuinduced re-duction of the effective viscosity is significant considering that in an isotropic fluid the corre-sponding variation is rather marginal. In the present experiments, upon transition to isotropic phase, the mass flow rate enhanced by ≈ 100% in planar case (at 120 mbar), and by 200%

in homeotropic case (165 mbar). The distinct flow behaviour of nematic 5CB can be related

Figure 6.3: Polarized micrograph of a microchannel possessing composite boundary con-ditions. Shown here: the left half is homeotropic, and right half is uniform planar. Such microchannels can be utilized to produce micro-flows with spatially modulated velocities.

to the temperature-induced director alignment over and above the usual reduction of material viscosity with temperature. As the temperature goes up, the nematic molecules show a tran-sition typically from homeotropic to planar [206]. In the present context, such an occurrence reduces the viscosity, as the molecules are now increasingly aligned along the flow direc-tion. However, as the temperature increases further, the director alignment reduces abruptly close to the transition temperature, marked by increased viscosity. The flow phenomena close to the transition temperature were investigated experimentally by G¨ahwiller [207], and the phenomenological relations were given by W. Helfrich [208, 209]. Just below the transition temperature, where the nematic order is lowest, the Leslie angle (equation 2.51) increases. The flow-aligned molecules, owing to the reduced order, have an increased degree of freedom with respect to the aligned orientation. At this stage, the molecules can even rotate about their short axes. Consequently, the molecules behave as elongated structures under shear flow. The long axes of such molecules align preferentially at an angle of 45^◦ between the direction of flow and the velocity gradient [210]. The situation however changes as the temperature is reduced:

The degree of order becomes larger again, and the effective viscosity is now determined by the relative orientation between the director and the flow. Some interesting observations on the rheology of nematic 5CB under the application of an electric field have been reported by K. Negita [211]. Investigations for a more detailed understanding of the flow behaviour at the nematic-to-isotropic transition are underway. The characteristic dependence of the nematic flows on boundary and temperature conditions can be potentially applied for flow-field and particle manipulations.

Figure 6.4: Flow before and after nematic-to-isotropic transition, indicated by the star sym-bol. The mass flow rate increased by≈ 100% in planar case (at 120 mbar), and by 200% in homeotropic case (165 mbar).