Abstract
Timeperiodic pressuredriven slip flow and electrokinetic energy conversion efficiency in a nanotube are studied analytically. The slip length depends on the surface charge density. Electric potential, velocity and streaming electric field are obtained analytically under the Debye–Hückel approximation. The electrokinetic energy conversion efficiency is computed using these results. The effects of surface chargedependent slip and electroviscous effect on velocity and electrokinetic energy conversion efficiency are discussed. The main results show that the velocity amplitude and the electrokinetic energy conversion efficiency of the surface chargedependent slip flow are reduced compared with those of the surface chargeindependent slip flow.
Introduction
Micro and nanofluidic devices [1] have a wide range of applications in science and engineering, e.g., liquid pumping and energy conversion. In many of these devices, a pressure gradient is often used to manipulate the transport of electrolyte solutions in nanochannels, at least one characteristic dimension of which is below or of the order of 100 nm. The decrease of length scale of the channel leads to the emergence of new phenomena different from those in macroflow, such as the electrokinetic effect and boundary slip. When wall surfaces are brought into contact with an electrolyte solution, most of them acquire surface electric charge [2] due to ion adsorption and acid–base reactions [3]. The charged surface attracts counterions and repels coions in the nearby electrolyte solution, and hence an electric double layer (EDL) with net charge density forms in the nearby electrolyte solution. The flow of electrolyte solution actuated by the pressure field generates both a streaming current and a streaming potential. The streaming current in a nanochannel can offer a simple and effective way to convert the mechanical energy to electric energy [4]. The streaming potential induces an electric field called streaming electric field. Acting on the net mobile charge in EDL, the steaming electric field generates an electric force in the opposite direction of the flow. The flow rate is decreased under the action of the electric office. This effect is called electroviscous effect.
The electrokinetic energy conversion efficiency and the electroviscous effects of microand nanoscale flows have been widely studied under the noslip assumption [410]. Among these works, Bhattacharyya et al. [7] discussed the electroviscous effect on a timeperiodic pressuredriven flow through a circular microtube and have shown that the Onsager’s principle of reciprocity is applicable for this flow. The electrokinetic energy conversion efficiencies of timeperiodic pressuredriven noslip flows of a viscoelastic fluid between two parallel plates and in a circular tube were, respectively, by Bandopadhyay and Chakraborty [8] and Nguyen and coworkers [10]. The noslip flow of a Maxwell fluid in a soft nanochannel and the electrokinetic energy conversion efficiency were studied by Jian and coworkers [9].
In nanoscale flow, the boundary slip effect becomes significant because the ratio between slip length and channel height becomes considerable large. For example, the velocity distribution for Stokes slip flow through a circular tube [11,12] is
where a is the radius of the circular tube and b_{0} is the Navier slip length. The last term 2b_{0}/a becomes considerable large for nanoscale flow. Many researchers investigated the influences of the surface charge and the boundary slip on micro and nanoscale flows [1117]. Among these, Yang and Kwok [11,12] studied timeperiodic pressuredriven flows in circular and parallelplate microchannels with electrokinetic effect and boundary slip condition. Goswami and Chakraborty [17] investigated electrokinetic energy conversion through streaming effects in timeperiodic pressuredriven nanochannel flows with boundary slip.
In the above mentioned references [1117], the slip length is independent of the surface charge. However, recent theoretical and experimental results have shown that the surface charge affects the slip length. Joly et al. [18] used molecular dynamics simulations to find the relationship between slip length and surface charge density. The reason is that there exists an attracting electrostatic force between the charged solid surface and the liquid near the charged solid surface. They also described the coupling relationship between the surface charge and the slip length by a mathematical model in 2006. Experiments [1921] showed that the slip length decreases as the absolute value of surface charge density increases. Many researchers [2227] theoretically investigated the effects of the surface chargedependent slip on the fluid flows and heat transfers in microtubes and parallelplate microchannels. Recently, Buren et al. [28] studied the effect of surface chargedependent slip and the electroviscous effect on time periodic pressuredriven flow and electrokinetic energy conversion in a parallelplate nanochannel.
In the following sections, the influences of the surface chargedependent slip on timeperiodic pressuredriven flow and electrokinetic energy conversion in a circular nanotube are studied. To our knowledge, so far no author discussed this problem for the case of a timeperiodic flow. Firstly, using the separation of variables method, the energy conversion efficiency and the analytical solutions to the governing equations are obtained. Secondly, the electroviscous effect and the effects of surface chargedependent slip on the velocity and the energy conversion efficiency are discussed. Finally, we make our concluding remarks.
Mathematical modeling
Problem definition and governing equations
We consider a timeperiodic pressuredriven flow of a symmetric binary electrolyte in a circular nanotube with surface chargedependent slip and with radius a using a cylindrical coordinate system (r, θ, z) where the zaxis is taken to coincide with the central axis of the nanotube, as shown in Figure 1. In the present study, the radius ranges from 37.5 to 100 nm, and the bulk ionic concentration varies from 1 to 100 nm. The corresponding Debye length varies from 0.96 to 9.63 nm [29]. In this case, the overlap of EDLs is mild, and hence the electric potential within the nanotube can be described by the Poisson–Boltzmann equation [30]. In addition, the continuum hydrodynamics for fluid flow is valid because the Debye length and the radius considered here are much larger than the mean free path of the liquid in the nanotube [31]. When the boundary surface of a nanochannel is molecularly smooth, the surface asperity barriers do not exist and, consequently, boundary slip can occur at the interface between the liquid and the boundary of the nanochannel [20]. Assume that the boundary slip b depends on the surface charge density σ_{s} [18,22,25]:
where b_{0} is the original slip length independent of surface charge density, α ≈ 1 is a numerical factor, d is the equilibrium distance of the Lennard–Jones potential, e is the elementary charge, l_{B} = e^{2}/(4πεk_{B}T), k_{B} is the Boltzmann constant, ε is the permittivity of the electrolyte solution and T is the absolute temperature [19,25].
The electric potential distribution φ in EDL satisfies the Poisson equation
where ρ_{e} is the volumetric net charge density. When the number concentrations of the positive and negative ions in symmetric electrolyte in a nanochannel [17] obey the Boltzmann distribution, the volumetric net charge density ρ_{e} can be obtained through the relationship
where n_{0} is the bulk ionic concentration and z is the valence of ions. Taking the Debye–Hückel approximation for low zeta potential, i.e., sinh(ezφ/(k_{B}T)) ≈ ezφ/(k_{B}T), and utilizing Equation 3 and Equation 4 we obtain the linearized Poisson–Boltzmann equation
The electric potential distribution φ satisfies
where ρ_{e} = −εκ^{2}φ is used, ζ is the zeta potential, and κ = [εk_{B}T/(2z^{2}e^{2}n_{0})]^{−1/2} is the Debye–Hückel parameter and represents the inverse of the characteristic EDL thickness.
A unidirectional flow along the zdirection is generated by a timeperiodic pressure gradient cos(ωt)dp_{0}/dz independent of position, where ω is the frequency, and t is the time. From the continuity equation, we find ∂u/∂z = 0 and so u depends on the variables r and t. Therefore, the momentum balance equations for the incompressible viscous Newtonian liquid becomes
where ρ is the mass density, μ is the dynamic viscosity, E_{s} is the streaming electric field and p = cos(ωt)p_{0} is the timeperiodic pressure. The boundary conditions are
where the slip length b is given by Equation 2.
According to the charge conservation law, the total surface charge should be equal to the net charge in the fluid. Therefore
where L represents the length of the nanotube. From Equation 3, Equation 9 and the boundary condition in Equation 6, we obtain
Substituting Equation 10 in Equation 2 yields
The streaming current through the channel is equal to
The streaming electric field E_{s} generates a reverse conduction current I_{c}. Due to the electric neutrality of the fluid inside the nanotube, the net electric current over the cross section of the nanotube is zero, i.e.,
where σ = 2z^{2}e^{2}Dn_{0}/(k_{B}T) is the electric conductivity and D is the diffusivity of ions in the electrolyte. From the equation I_{s} + I_{c}=0, the streaming electric field E_{s} can be obtained in the form:
Dimensionless governing equations
The velocity and the streaming electric field of the timeperiodic pressuredriven flow are both timeperiodic functions and can be written as
where the R[·] denotes the real part of its complex argument and i = . Let
Then, we use Equation 15 and Equation 16 to obtain the dimensionless governing equations and their boundary conditions:
where λ = εζ^{2}/(µD), B^{2} = iRe, and Re = ωa^{2}ρ/μ is the nondimensional frequency. Using numerical methods, Erickson and Li [32] verified that the linearized Equation 17 and Equation 19 are also valid for large zeta potential values up to 100 mV.
Analytical solutions of the governing equations
From the boundary value problem in Equation 17 and Equation 18, we obtain
where I_{0} is the zerothorder modified Bessel function of the first kind. The slip length is given by Equation 11 and Equation 21. From Equation 19 and Equation 20, the velocity distribution is obtained as
where I_{1} is the firstorder modified Bessel function of the first kind, J_{0} and J_{1} are the zerothorder and firstorder Bessel functions of the first kind,
is the complex velocity amplitude without considering the electroviscous effect.
From Equations 14–16 and Equations 21–23, we obtain
where
is the complex electric field amplitude in which the electroviscous effect is not considered.
The dimensionless flow rate normalized by a^{2}u_{ref} is given by
Energy conversion efficiency
The mechanical energy of the pure pressuredriven flow is converted into electric energy [6] when the streaming current (I_{s}) and the streaming electric field (E_{s}) are generated. The energy conversion efficiency η [10]can be expressed as:
where ⟨·⟩ represents the time average over one cycle [0, 2π/ω], and
is the flow rate of the pure pressuredriven flow without considering the effect of the surface charge on the slip length.
Using Equation 13, Equation 15, Equation 28 and the nondimensional definitions in Equation 16, we can obtain the following expression for η from Equation 27:
where
and is the principal argument of .
Results and Discussion
This section discusses the effects of the surface chargedependent boundary slip on the timeperiodic pressuredriven flow and electrokinetic energy conversion efficiency in the nanotube using the analytical results obtained above. In this problem, we assume that [2,8,19,25]: α = 1, d = 0.4 × 10^{−9} m, µ = 1.01 × 10^{−3} Pa·s, ρ = 1 × 10^{3} kg·m^{−3}, e = 1.6 × 10^{−19} C, z = 1, D = 1.612 × 10^{−9} m^{2}·s^{−1}, T = 298 K, ε = 7 × 10^{−10} F·m^{−1}, and k_{B} = 1.38 × 10^{−23} J·K^{−1}. Figure 2 shows the variation of the velocity amplitudes with the radial coordinate r̄ at different values of zeta potential, slip length and frequency. Figure 2a shows that, for the noslip and slip flows, the amplitudes of the velocity are both reduced because of the electroviscous effect. The reduction of velocity amplitude of slip flow is larger than that of noslip flow. The reason is that the slip length is decreased by the surface charge effect and, consequently, the velocity of slip flow is further reduced. It is apparent in Figure 2b that due to the surface charge effect, the velocity amplitude of the surface chargedependent slip flow is less than that of the surface chargeindependent slip flow.
Figure 3 shows the contours of the electrokinetic energy conversion efficiency in the circular nanotube with surface chargedependent slip. In the low ionic concentration regime, the electrokinetic energy conversion efficiency varies quickly with the ionic concentration and is highest. Note that for the low ionic concentration, κa is small. The variation of the efficiency with a is not monotonous, but the efficiency is large when κa is small. This is because as κa decreases, the ions in the EDL extend into the middle of the nanotube, where the velocity of the flow is highest. Figure 3 shows that the electrokinetic energy conversion efficiency is large in the regime where the zeta potential is large.
Figure 4a shows the comparison of the electrokinetic energy conversion efficiency values under the condition of surface chargeindependent slip (solid, dashed and dotted lines), surface chargedependent slip (circles, plus signs and diagonal crosses) and noslip (dashdot line). The boundary slip enhances the electrokinetic energy conversion efficiency. This is because the boundary slip increases the fluid velocity and the transportation of ions in the EDL, increasing the streaming electric field. Hence, the ratio of electric energy to mechanical energy, namely, the electrokinetic energy conversion efficiency is enhanced by the boundary slip. In fact, the increases in velocity and streaming electric fields increase both electric energy and mechanical energy, but the former is larger than the latter. Figure 4a,b shows that the electrokinetic energy conversion efficiency is reduced when the slip length depends on the surface charge. The reduction in the electrokinetic energy conversion efficiency is large when the ionic concentration and the zeta potential are large. This is caused by the slip length reduction due to the surface charge effect, as shown in Equation 11. In addition, Figure 4b shows that the electrokinetic energy conversion efficiency increases with the frequency when the frequency is larger than 10^{2} MHz; elsewhere the electrokinetic energy conversion efficiency is almost constant.
Conclusion
In this paper, we apply the separation of variables method to study timeperiodic pressuredriven flow and electrokinetic energy conversion efficiency in a circular nanotube with surface chargedependent slip. The expressions for the electric potential, velocity distribution, streaming electric field, flow rate and electrokinetic energy conversion efficiency are obtained analytically. From the above theoretical results the following conclusions are drawn: Compared with the velocity amplitude of the pure pressuredriven flow, the velocity amplitudes of the noslip and surface chargedependent slip flows are both reduced. This is caused by the electroviscous effect. The decrement in the slip length due to surface charge effect causes the velocity amplitude of the surface chargedependent slip flow to be smaller than that of the surface chargeindependent slip flow. The effect of surface charge on the timeperiodic flow increases with the magnitude of the zeta potential and with the ionic concentration. The electrokinetic energy conversion efficiency is large when the ratio between the channel radius and EDL thickness is small. Higher frequency leads to higher electrokinetic energy conversion efficiency. The boundary slip increases the electrokinetic energy conversion efficiency. The electrokinetic energy conversion efficiency is reduced when the slip length is dependent on the surface charge. The reduction in electrokinetic energy conversion efficiency due to surface charge effect is large for the cases of large ionic concentration and large zeta potential.
References

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2.  Probstein, R. F. Physicochemical hydrodynamics: an introduction; John Wiley: New York, NY, U.S.A., 1994. doi:10.1002/0471725137 
8.  Bandopadhyay, A.; Chakraborty, S. Appl. Phys. Lett. 2012, 101, 043905. doi:10.1063/1.4739429 
19.  Jing, D.; Bhushan, B. Langmuir 2013, 29, 6953–6963. doi:10.1021/la401168w 
25.  Jing, D.; Bhushan, B. J. Colloid Interface Sci. 2013, 392, 15–26. doi:10.1016/j.jcis.2012.09.082 
1.  Jing, D.; Song, S.; He, L. Int. J. Heat Mass Transfer 2019, 128, 1344–1350. doi:10.1016/j.ijheatmasstransfer.2018.08.006 
4.  Xuan, X.; Li, D. J. Power Sources 2006, 156, 677–684. doi:10.1016/j.jpowsour.2005.05.057 
5.  Gong, L.; Wu, J.; Wang, L.; Cao, K. Phys. Fluids 2008, 20, 063603. doi:10.1063/1.2939391 
6.  Vainshtein, P.; Gutfinger, C. J. Micromech. Microeng. 2002, 12, 252–256. doi:10.1088/09601317/12/3/309 
7.  Bhattacharyya, A.; Masliyah, J. H.; Yang, J. J. Colloid Interface Sci. 2003, 261, 12–20. doi:10.1016/s00219797(02)000504 
8.  Bandopadhyay, A.; Chakraborty, S. Appl. Phys. Lett. 2012, 101, 043905. doi:10.1063/1.4739429 
9.  Jian, Y.; Li, F.; Liu, Y.; Chang, L.; Liu, Q.; Yang, L. Colloids Surf., B 2017, 156, 405–413. doi:10.1016/j.colsurfb.2017.05.039 
10.  Nguyen, T.; van der Meer, D.; van den Berg, A.; Eijkel, J. C. T. Microfluid. Nanofluid. 2017, 21, 37. doi:10.1007/s104040171866y 
18.  Joly, L.; Ybert, C.; Trizac, E.; Bocquet, L. J. Chem. Phys. 2006, 125, 204716. doi:10.1063/1.2397677 
4.  Xuan, X.; Li, D. J. Power Sources 2006, 156, 677–684. doi:10.1016/j.jpowsour.2005.05.057 
19.  Jing, D.; Bhushan, B. Langmuir 2013, 29, 6953–6963. doi:10.1021/la401168w 
20.  Pan, Y.; Bhushan, B. J. Colloid Interface Sci. 2013, 392, 117–121. doi:10.1016/j.jcis.2012.10.043 
21.  Pan, Y.; Bhushan, B.; Zhao, X. Beilstein J. Nanotechnol. 2014, 5, 1042–1065. doi:10.3762/bjnano.5.117 
3.  Kirby, B. Micro and Nanoscale Fluid Mechanics; Cambridge University Press: Cambridge, United Kingdom, 2010. doi:10.1017/cbo9780511760723 
17.  Goswami, P.; Chakraborty, S. Langmuir 2010, 26, 581–590. doi:10.1021/la901209a 
2.  Probstein, R. F. Physicochemical hydrodynamics: an introduction; John Wiley: New York, NY, U.S.A., 1994. doi:10.1002/0471725137 
11.  Yang, J.; Kwok, D. Y. J. Colloid Interface Sci. 2003, 260, 225–233. doi:10.1016/s00219797(02)001583 
12.  Yang, J.; Kwok, D. Y. Langmuir 2003, 19, 1047–1053. doi:10.1021/la026201t 
13.  Habibi Matin, M. Int. J. Heat Mass Transfer 2017, 106, 473–481. doi:10.1016/j.ijheatmasstransfer.2016.08.100 
14.  Davidson, C.; Xuan, X. J. Power Sources 2008, 179, 297–300. doi:10.1016/j.jpowsour.2007.12.050 
15.  Ren, Y.; Stein, D. Nanotechnology 2008, 19, 195707. doi:10.1088/09574484/19/19/195707 
16.  Garai, A.; Chakraborty, S. Electrophoresis 2010, 31, 843–849. doi:10.1002/elps.200900676 
17.  Goswami, P.; Chakraborty, S. Langmuir 2010, 26, 581–590. doi:10.1021/la901209a 
9.  Jian, Y.; Li, F.; Liu, Y.; Chang, L.; Liu, Q.; Yang, L. Colloids Surf., B 2017, 156, 405–413. doi:10.1016/j.colsurfb.2017.05.039 
11.  Yang, J.; Kwok, D. Y. J. Colloid Interface Sci. 2003, 260, 225–233. doi:10.1016/s00219797(02)001583 
12.  Yang, J.; Kwok, D. Y. Langmuir 2003, 19, 1047–1053. doi:10.1021/la026201t 
13.  Habibi Matin, M. Int. J. Heat Mass Transfer 2017, 106, 473–481. doi:10.1016/j.ijheatmasstransfer.2016.08.100 
14.  Davidson, C.; Xuan, X. J. Power Sources 2008, 179, 297–300. doi:10.1016/j.jpowsour.2007.12.050 
15.  Ren, Y.; Stein, D. Nanotechnology 2008, 19, 195707. doi:10.1088/09574484/19/19/195707 
16.  Garai, A.; Chakraborty, S. Electrophoresis 2010, 31, 843–849. doi:10.1002/elps.200900676 
17.  Goswami, P.; Chakraborty, S. Langmuir 2010, 26, 581–590. doi:10.1021/la901209a 
10.  Nguyen, T.; van der Meer, D.; van den Berg, A.; Eijkel, J. C. T. Microfluid. Nanofluid. 2017, 21, 37. doi:10.1007/s104040171866y 
11.  Yang, J.; Kwok, D. Y. J. Colloid Interface Sci. 2003, 260, 225–233. doi:10.1016/s00219797(02)001583 
12.  Yang, J.; Kwok, D. Y. Langmuir 2003, 19, 1047–1053. doi:10.1021/la026201t 
8.  Bandopadhyay, A.; Chakraborty, S. Appl. Phys. Lett. 2012, 101, 043905. doi:10.1063/1.4739429 
7.  Bhattacharyya, A.; Masliyah, J. H.; Yang, J. J. Colloid Interface Sci. 2003, 261, 12–20. doi:10.1016/s00219797(02)000504 
11.  Yang, J.; Kwok, D. Y. J. Colloid Interface Sci. 2003, 260, 225–233. doi:10.1016/s00219797(02)001583 
12.  Yang, J.; Kwok, D. Y. Langmuir 2003, 19, 1047–1053. doi:10.1021/la026201t 
29.  Ma, Y.; Xue, S.; Hsu, S.C.; Yeh, L.H.; Qian, S.; Tan, H. Phys. Chem. Chem. Phys. 2014, 16, 20138–20146. doi:10.1039/c4cp02349h 
22.  Jing, D.; Bhushan, B. J. Colloid Interface Sci. 2015, 454, 152–179. doi:10.1016/j.jcis.2015.05.015 
23.  Jing, D.; Bhushan, B. Beilstein J. Nanotechnol. 2015, 6, 2207–2216. doi:10.3762/bjnano.6.226 
24.  Jing, D.; Pan, Y.; Wang, X. Int. J. Heat Mass Transfer 2017, 113, 32–39. doi:10.1016/j.ijheatmasstransfer.2017.05.061 
25.  Jing, D.; Bhushan, B. J. Colloid Interface Sci. 2013, 392, 15–26. doi:10.1016/j.jcis.2012.09.082 
26.  Jing, D.; Pan, Y. Int. J. Heat Mass Transfer 2016, 101, 648–655. doi:10.1016/j.ijheatmasstransfer.2016.05.087 
27.  Jing, D.; Pan, Y.; Wang, X. Int. J. Heat Mass Transfer 2017, 108, 1305–1313. doi:10.1016/j.ijheatmasstransfer.2016.12.090 
28.  Buren, M.; Jian, Y.; Zhao, Y.; Chang, L. J. Phys. D: Appl. Phys. 2018, 51, 205601. doi:10.1088/13616463/aabc73 
6.  Vainshtein, P.; Gutfinger, C. J. Micromech. Microeng. 2002, 12, 252–256. doi:10.1088/09601317/12/3/309 
19.  Jing, D.; Bhushan, B. Langmuir 2013, 29, 6953–6963. doi:10.1021/la401168w 
25.  Jing, D.; Bhushan, B. J. Colloid Interface Sci. 2013, 392, 15–26. doi:10.1016/j.jcis.2012.09.082 
17.  Goswami, P.; Chakraborty, S. Langmuir 2010, 26, 581–590. doi:10.1021/la901209a 
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