Electro-osmosis of nematic liquid crystals under weak anchoring and second-order surface effects

Advent of nematic liquid crystal flows has attracted renewed attention in view of microfluidic transport phenomena. Among various transport processes, electro-osmosis stands as one of the efficient flow actuation mechanisms through narrow confinements. In the present study, we explore the electrically actuated flow of an ordered nematic fluid with ionic inclusions, taking into account the influences from surface-induced elasticity and electrical double layer (EDL) phenomena. Toward this, we devise the coupled flow governing equations from fundamental free-energy analysis, considering the contributions from first- and second-order elastic, dielectric, flexoelectric, charged surface polarization, ionic and entropic energies. The present study focuses on the influence of surface charge and elasticity effects in the resulting linear electro-osmosis through a slit-type microchannel whose surfaces are chemically treated to display a homeotropic-type weak anchoring state. An optical periodic stripe configuration of the nematic director has been observed, especially for higher electric fields, wherein the Ericksen number for the dynamic study is restricted to the order of unity. Contrary to the isotropic electrolytes, the EDL potential in this case was found to be dependent on the external field strength. Through a systematic investigation, we brought out the fact that the wavelength of the oscillating patterns is dictated mainly by the external field, while the amplitude depends on most of the physical variables ranging from the anchoring strength and the flexoelectric coefficients to the surface charge density and electrical double layer thickness.

Figure 1

Schematic representation of the electro-osmosis of an anisotropic nematic liquid under the action of an external axial electric field. An electrical double layer gets induced adjacent to each of the nematic-substrate interface. Besides influencing the fluid rheology and director orientation, this provides the genesis of the flow actuation body force. A weak boundary condition with homeotropic-type easy axis is considered at both the surfaces. The extrapolation length $\left(L_{\mathrm{ex}}\right)$, within which the weak anchoring effects are pronounced, is also highlighted in the above schematic.

Figure 2

The variation of the director alignment profile $\theta$ as a function of the channel transverse direction $\overline{y}$ for different values of (a) dimensionless second-order elastic constant with strong anchoring, (b) second-order elastic constant with weak anchoring $(\overline{\gamma }=1)$, (c) dimensionless surface anchoring strength, (d) dimensionless surface charge density, (e) dimensionless axial applied field ${\overline{E}}_x$ and (f) surface polarization $(for\overline{\gamma }=10)$.

Figure 3

Variation of the potential distribution $\overline{\psi }$ as a function of the channel transverse direction $\overline{y}$ for different values of dimensionless (a) surface charge density, (b) Debye length, and (c) axial applied field ${\overline{E}}_x$.

Figure 4

Variation of the velocity field profile $\overline{u}$ as a function of the channel transverse direction $\overline{y}$ for different values of dimensionless (a) axial applied field ${\overline{E}}_x$ and (b) surface charge density ${\overline{\sigma }}_w$.

Figure 5

Variation of the average velocity as a function of the axial field ${\overline{E}}_x$ for different values of the (a) dimensionless surface charge density and (b) dimensionless Debye screening length $\overline{\lambda }$.