Theoretical study on the massively augmented electro-osmotic water transport in polyelectrolyte brush functionalized nanoslits

Vishal Sankar Sivasankar, Sai Ankit Etha, Harnoor Singh Sachar, and Siddhartha Das

Phys. Rev. E 102, 013103 – Published 8 July 2020

Abstract

We demonstrate that functionalizing nanoslits with pH-responsive polyelectrolyte brushes can lead to extremely fast electro-osmotic (EOS) water transport, where the maximum centreline velocity and the volume flow rate can be an order of magnitude larger than these quantities in identically charged brush-free nanochannels for a wide range of system parameters. Such an enhancement is most remarkable given that the brushes have been known to retard the transport by imparting additional drag on the fluid flow. We argue that this enhancement stems from the localization of the charge density of the brush-induced electric double layer (and, hence, the EOS body force) away from the nanochannel wall (or the location of the wall-induced drag force). This ensures a much larger impact of the EOS body force triggering such fast water transport. Finally, the calculated flux values for the present brush-grafted nanochannels are found to be significantly larger than those for a wide range of nanofluidic membranes and channels, suggesting that the brush functionalization can be considered as a mechanism for enabling such superfast nanofluidic transport.

Received 21 February 2020
Revised 24 April 2020
Accepted 8 June 2020

DOI:https://doi.org/10.1103/PhysRevE.102.013103

Physics Subject Headings (PhySH)

Electrokinetic flows

Fluid Dynamics

Authors & Affiliations

Vishal Sankar Sivasankar, Sai Ankit Etha, Harnoor Singh Sachar, and Siddhartha Das ^*

Department of Mechanical Engineering, University of Maryland, College Park, Maryland 20742, USA

^*sidd@umd.edu

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Vol. 102, Iss. 1 — July 2020

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Images

Figure 1
Schematic comparing the EOS transport (due to the axial electric field) in (a) brush-free and (b) brush-grafted nanochannel. The brushes enforce the localization of the EDL charge density away from the wall enforcing the EOS body force to be localized away from the wall (location of the drag force). Here, $λ_{EDL}$ denotes the EDL thickness.
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Figure 2
Transverse variation of the nondimensional velocity profile $\bar{u}$ [ $\bar{u} = \frac{u}{u_{0}}$ , where $u_{0} = (\frac{k_{B} T}{e}) \frac{ε_{0} ε_{r} E}{η}$ is the velocity scale, $k_{B} T$ is the thermal energy, $ε_{0}$ is the permittivity of free space, and $ε_{r}$ is the relative permittivity of water] with bulk salt concentration $c_{\infty}$ for the PE-brush-grafted nanochannel for (a) $p H_{\infty} = 3, ℓ$ = 60 nm, (b) $p H_{\infty} = 4, ℓ$ = 60 nm, and (c) $p H_{\infty} = 3, ℓ$ = 10 nm. Here we consider the flow profiles for the equilibrium-brush-EDL configurations (see Sec. 2a for the equations and Refs. [33, 34] for the figures) obtained using $N = 400$ , $h$ = 100 nm, $a$ = 1 nm (Kuhn length), $k_{B} = 1.38 \times 10^{- 23}$ J $K^{- 1}$ , $T = 298$ K, $e = 1.6 \times 10^{- 19}$ C (electronic charge), $ε_{0} = 8.8 \times 10^{- 12} {Fm}^{- 1}$ , $ε_{r}$ = 79.8, $γ a^{3} = 1$ , $p K_{a}$ = 3.5, $ν$ = 0.5, $ω$ = 0.1. $p K_{w}$ =14, $p O H_{\infty}$ = $p K_{w}$ – $p H_{\infty}$ , $c_{+, \infty}$ = $c_{\infty}$ , $c_{H^{+}, \infty}$ = $10^{- p H_{\infty}}$ , $c_{O H^{-}, \infty}$ = $10^{- p O H_{\infty}}$ , and $c_{-, \infty}$ = $c_{\infty} + c_{H^{+}, \infty} - c_{O H^{-}, \infty}$ . The definitions of all the terms and parameters are provided in Sec. 2a.
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Figure 3
Ratio of the maximum centerline velocities ( $u_{r} = u_{max, B} / u_{max, N B}$ ) and volume flow rates ( $Q_{r} = Q_{B} / Q_{N B}$ ) (inset) with $c_{\infty}$ for different combinations of $ℓ$ , $h$ and, $p H_{\infty}$ . All other parameters are the same as those in Fig. 2.
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Figure 4
Comparison of the ratio $Q_{r}$ and the actual flux value (see the inset) between the present case (EOS transport in nanochannels grafted with backbone-charged brushes) and Ref. [26] (EOS transport in nanochannels grafted with end-charged brushes). We consider three cases in the main figure: Case 1: $p H_{\infty} = 3$ , $ℓ = 60$ nm, $h = 100$ nm; Case 2: $p H_{\infty} = 4$ , $ℓ = 60$ nm, $h = 100$ nm; Case 3: ${p H}_{\infty} = 4$ , $ℓ = 10$ nm, $h = 250$ nm. In the inset, we compare the actual flux values for these three cases for an applied electric field $E = 500$ V ${cm}^{- 1}$ . In order to ensure that we are considering the same charge content of the PE brushes as the present case, the charge density for the end-charged PE brushes is considered to be $σ_{c, e q}$ [see the discussions following Eqs. (1) and (2) for the definition of $σ_{c, e q}]$ . In the legend, B.C. and E.C. denote the cases of EOS transport in nanochannels grafted with backbone-charged (present case) and end-charged (Ref. [26]) brushes.
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Figure 5
Comparison of flux for various nanofluidic devices. Points 1 to 4 provide the results for the present case of a PE-brush-grafted nanochannel. Point 1: ${p H}_{\infty} = 3$ , $c_{\infty} = 10^{- 2}$ M, $ℓ = 10$ nm, $h = 100$ nm, $E = 500$ V ${cm}^{- 1}$ ; Point 2: ${p H}_{\infty} = 3$ , $c_{\infty} = 10^{- 3}$ M, $ℓ = 10$ nm, $h = 100$ nm, $E = 100$ V ${cm}^{- 1}$ ; Point 3: ${p H}_{\infty}$ = 4, $c_{\infty}$ = $10^{- 2}$ M, $ℓ = 10$ nm, $h = 250$ nm, $E = 500$ V ${cm}^{- 1}$ ; Point 4: ${p H}_{\infty} = 4$ , $c_{\infty} = 10^{- 3}$ M, $ℓ = 10$ nm, $h = 250$ nm, $E = 100$ V ${cm}^{- 1}$ . For the current work, $η = 8.9 \times 10^{- 4}$ Pa s, and all other parameters are the same as in Fig. 2. In Table 1, we discuss the manner in which the fluxes for the different experimental studies (cited here) are calculated. GO: graphene oxide; BN: boron nitride; CNT: carbon nanotube; Si: silicon; AAM: anodized alumina membrane; NC: nanochannel.
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