Pulsatile pressure enhanced rapid water transport through flexible graphene nano/Angstrom-size channels: A continuum modelling approach using the micro-structure of nanoconfined water

Several researchers observed a significant increase in water flow through graphene-based nanocapillaries [1-2]. As graphene sheets are flexible [3], we represent nanocapillaries with a deformable channel-wall model by using the small displacement structural-mechanics and perturbation theory presented by Gervais et al.[4], and Christov et al.[5], respectively. We assume the lubrication assumption in the shallow nanochannels, and using the microstructure of confined water along with slip at the capillary boundaries and disjoining pressure [2], we derive the model for deformable nanochannels. Our derived model also facilitates the flow dynamics of Newtonian fluids under different conditions as its limiting cases, which have been previously reported in literature [2,4-8]. We compare the experimental observations by Radha et al.[1] and MD-simulation results by Neek-Amal et al.[2] with our deformable-wall model. We find that for channel-height $H_o < 4$~\AA, the flow-rate prediction by the deformable-wall model is 5-7~\% more compared to [2] well-fitted rigid-wall model. These predictions are within the errorbar as shown by [1], which indicates that the derived deformable-wall model could be more accurate to model [1] experiments as compared to the rigid-wall model. Using the model, we study the effect of flexibility of graphene sheets on the flow rate. As the flexibility $\alpha$ increases (corresponding thickness $\mathcal{T}$ and elastic modulus $E$ of the wall decreases), the flow rate also increases. We find that the flow rate scales as $\dot{m}_{\text{flexible}}\sim \alpha^0 $ for $\Big (\alpha \Delta pW/EH_o \Big ) \ll 1$; $\dot{m}_{\text{flexible}}\sim \alpha $ for $\Big (\alpha \Delta pW/EH_o \Big ) \sim O(10^{-1})$; and $\dot{m}_{\text{flexible}}\sim \alpha^3 $ for $\Big (\alpha \Delta pW/EH_o \Big ) \sim O(1)$, respectively. We also find that, for a given thickness $\mathcal{T}$, the percentage change in flow rate in the smaller height of the channel is more than the larger height of the channels. As the channel height decreases for the given reservoir pressure and thickness, the $\Delta \dot{m}/\dot{m}$ increases with $H_o^{-1}$ followed by $H_o^{-3}$ after a height-threshold. Further, we investigate how the applied pulsating pressure influences the flow rate. We find that due to the oscillatory pressure field, there is no change in the averaged mass flow rate in the rigid-wall channel, whereas the flow rate increases in the flexible channels with the increasing magnitude of the oscillatory pressure field. Also, in flexible channels, depending on the magnitude of the pressure field, either of the steady or oscillatory or both kinds of pressure field, the averaged mass flow rate dependence varies from $\Delta p$ to $\Delta p^4$ as the pressure field increases. The flow rate in the rigid-wall channel scales as $\dot{m}_{\text{rigid}}\sim \Delta p $, whereas for the deformable-wall channel it scale as $\dot{m}_{\text{flexible}}\sim \Delta p $ for $ (\alpha \Delta pW/EH_o ) \approxeq 0$, $\dot{m}_{\text{flexible}}\sim \Delta p^2 $ for $ (\alpha \Delta pW/EH_o ) \sim O(10^{-1})$, and $\dot{m}_{\text{flexible}}\sim \Delta p^4 $ for $ (\alpha \Delta pW/EH_o ) \sim O(1)$. We find that both the flexibility of the graphene sheet and the pulsating pressure fields to these flexible channels intensify the rapid flow rate through nano/Angstrom-size graphene capillaries.