Understanding electrokinetic thermodynamics in nanochannels

Jianglong Du,Haolan Tao,Jie Yang,Cheng Lian,2,*,Sen Lin*,Honglai Liu,*

1 State Key Laboratory of Chemical Engineering,Shanghai Engineering Research Center of Hierarchical Nanomaterials,and School of Chemistry and Molecular Engineering,East China University of Science and Technology,Shanghai 200237,China

2 Institute for Theoretical Physics,Center for Extreme Matter and Emergent Phenomena,Utrecht University,Princetonplein 5,3584 CC Utrecht,the Netherlands

3 National Engineering Research Center for Integrated Utilization of Salt Lake Resources,and State Environmental Protection Key Laboratory of Environmental Risk Assessment and Control on Chemical Process,East China University of Science and Technology,Shanghai 200237,China

Keywords:Electrokinetic conversion efficiency Linear electrokinetics Nanostructure Dynamic simulation Thermodynamics

ABSTRACT Understanding the electrokinetic conversion efficiency in a nanochannel is vital for designing energy storage and conversion devices.In this paper,an analytical electrokinetic energy conversion efficiency in a nanochannel is obtained based on the linear electrokinetic response.The analytical result shows that the conversion efficiency has a maximum with the increasing of the nanochannel pore radius.Numerical solutions based on the Poisson-Nernst-Planck (PNP) and Navier-Stokes (NS) equations are used to confirm the analytical expressions.Besides,the influences of the pore radius and surface roughness on the conversion efficiency in nanochannels are also studied by the numerical calculations.In particular,the influences of the surface roughness on the fluid flow,streaming current and streaming potential are examined.The results show that the large bumps and grooves representing the roughness can hinder the fluid flows and ion transports in the nanochannels.The maximum efficiency in a smooth nanochannel is higher than that in a rough channel.However,the small bumps and grooves can increase the surface area of the channel,which is beneficial to improving the conversion efficiency in some cases.This research can provide theoretical guidance to design electrokinetic energy conversion devices.

1.Introduction

The ionic transport in nanochannels plays a critical role in many applications such as energy storage [1–4],seawater desalination[5,6],and bio-sample preconcentration [7,8].Under applied pressure,the net ion transport through the charged nanochannel filled with the aqueous solution results in the so-called streaming current and streaming potential.It means that the hydrostatic energy can be converted into electrical power,and the nano-fluid can be used as an energy source for nanoscale devices [9–12].

In a small nanochannel,the electrical double layers(EDLs)usually overlap.There is a much larger concentration of counter-ion than co-ion,and the potential at the center of the small nanochannel do not fall to zero.In a large nanochannel with non-overlapped EDL,the potential at center fall to zero.However,the total number of counter-ions is still much greater than co-ions in a large nanochannel.Therefore,the pore size has a great influence on ion transport,which plays an important role in the energy conversion efficiency.Many experimenter works have explored the electrokinetic energy conversion efficiency.A glass microchannel array coated with nano-layers of gold was used to study the conversion efficiency,and the result showed that the maximum output power was 1 mW and the efficiency can reach 1.3% [13].It was verified that the higher efficiency can be achieved at a smaller channel by experiment,and the best efficiency can reach 0.77%when the nominal pore size of alumina membrane was 200 nm[14].The conversion efficiencies obtained in these experiments are low.For the higher out power and electrokinetic energy conversion efficiency,several researchers have employed the theories and network models to enhance the efficiencies in smooth nanochannels [15,16].The streaming potential,the streaming current and the volume flow rate in channels with different surface heterogeneities and surface ζ-potentials were investigated,and the results showed that the surface charges played dominating roles in the streaming current and streaming potential in a small channel with the radius of 2.5 μm[17].An expression was derived by Frank to describe the efficiency in terms of the linear electrokinetic response properties,and it was shown that the maximum conversion efficiency can reach 100%by matching the channel and load impedances [18].Abraham et al.verified that when the pore size is equal to the Debye length,the conversion efficiency reached the maximum using intersecting asymptotes [19].

In addition,the surface roughness of nanochannels constructed by a variety of materials,including slit silicon [20–22],composite[23,24]and glass [25,26],is inevitable.The surface roughness can affect both the surface areas of the nanochannels and the distributions of surface charges.Thus,the studies of the roughness are vital to the performance of the energy conversion devices [27–30].The surface roughness can significantly reduce the ion transport velocities,which has been demonstrated[31].Kim studied the electroosmotic flow of water molecules in a rough nanochannel,and the result showed that the flow rate decreased with the increasing of the roughness [32].Molecular dynamics simulations were used to study the influence of wall roughness on the slip behaviors and it was presented that surface roughness weaken the dependence of the slip length on the shear rate [33].However,few attentions devoted to the electrokinetic energy conversion efficiencies in rough nanochannels.The roughness can be regarded as two aspects:bumps and grooves,both of which obstruct the fluid flow and the ion transport.Due to the fluid flow parallel to the channel wall,the counter-ions are difficult to be absorbed on the bumps,while they are easy to sink into the grooves.The ion transport is affected by the surface roughness as above discussion,and the ion transport has an important influence on the streaming current and streaming potential which are decisive roles for the energy conversion device.

Here,an analytical expression is derived to describe the electrokinetic energy conversion efficiency in terms of the linear electrokinetic responses,and we obtain the relationship between conversion efficiency and pore radius of a nanochannel.Besides,we build a continuum model including the Navier–Stokes (NS)equation and the Poisson–Nernst–Plank (PNP) equations to describe the electrokinetic energy conversion efficiencies in both smooth and rough nanochannels with different pore radiuses.The streaming current,streaming potential and fluid flow rate are obtained to further calculate the electrokinetic energy conversion efficiency in the rough nanochannels.Compared to previous researches which studied the efficiency in a smooth nanochannel,we consider the effect of surface roughness on the efficiency in this work.A special geometry representing a rough nanochannel is proposed,and the numerical simulations are used to investigate the distributions of ions and potential in the nanochannels with different roughness.Furthermore,by tuning the pore radiuses of the nanochannels with different roughness,we study the effects of surface roughness and pore radius on the conversion efficiency.We hope that our results could promote the research of electrokinetic energy conversion devices.

2.Linear Electrokinetic Non-Equilibrium Thermodynamics

An aqueous solution flowing through a charged nanochannel with an applied voltage (ΔV) can produce an electrical current(IE) and an electro-osmotic flow (QE).And a nanofluid with an applied pressure difference (Δp) can produce a streaming current(Ip) and a pressure-driven flow (Qp).The schematics of the electro-osmotic flow and pressure-driven flow are shown in Fig.1.Now,considering the two perturbations,Δp and ΔV,in a cylindrical nanochannel,the current (I) can be calculated by Ipand IE,I=Ip+IE,and the fluid flow (Q) can be calculated by Qpand QE,Q=Qp+QE.Based on Onsager relation[34],the reciprocal relationship between coefficients in the linear phenomenological law of irreversible thermodynamic processes can be expressed as following,dI/dΔp=dQ/dΔV=KE0.The current and fluid flow rate can be described by following:

Fig.1.Schematic of the electrokinetic effects.(a)The electro-osmotic flow with an applied electric field.The green arrows represent the velocity driven by the electric field,and the black is driven by fluid.(b) The pressure-driven flow carrying ions with an applied pressure.

where Kpis the hydraulic conductance,KE0is the streaming conductance and KEis the electrical conductance of electrolyte in the nanochannel.

The pressure-driven flow is calculated by Hagen-Poiseuille equation,Qp～（Δpr4/8μL）,where μ is the fluid viscosity,r is a mean pore radius and L is the length of the nanochannel.The pressure-driven flow can be described by Eq.(2),

3.Numerical Model and Methods

3.1.Model setup and parameters

In this study,we build a model that consists of two cylindrical reservoirs connected via a straight nanochannel with a smooth or rough surface.As shown in Fig.3,the diameter and length of the cylindrical reservoirs are H and W,respectively.The mean radius and the length of the connecting channel are set as r and L,respectively.The two reservoirs and connecting channel are filled with an incompressible,aqueous Newtonian electrolyte solution,and the nanochannel surface is negatively charged.A pressure is applied on the A-B side to drive the fluid flow through the system,leading to a directional movement of the net charges in the nanochannel,which generates a streaming current and a streaming potential.

Fig.2.The electrokinetic energy conversion efficiencies for small pores (r ?λD )(black solid line)and large pores(r ?λD )(red solid line)as a function of the ratio of the pore radiuses to the Debye length.

As shown schematically in Fig.3,small square bumps and grooves with side length a are designed to represent the roughness,and the nanochannel with a square wave-type wall structure is designed to maintain the mean radius of the nanochannel as r.The design of the model allows for the appropriate development at the entrance and exit,and the bulk can be kept away from the nanochannel,which can minimize the impact on the results.The relevant parameters of the geometric model are shown in Table 1.We assume that the electrolyte is completely ionized,the diffusion coefficients of the individual ions are identical,and the viscosity and density of the fluid at each location are the same.Both of the volume of ions and the ionization of water molecules are not taken into account.

3.2.Poisson-Nernst-Plank (PNP) equations

The Nernst-Planlk equation is used to describe the ion transport and ion concentration distribution.And the relationship between ion flux and potential distribution in a nanochannel filled with an electrolyte solution is expressed as,

Fig.3.Schematic of the model including a nanochannel and two reservoirs.The inset figure gives a detailed description of the nanochannel wall,where a is the side length of the small square bumps and grooves representing the roughness,and r is the mean radius of the nanochannel.

Table 1 Related parameters required by the model

and the space charge density which can be obtained by ρe=F∑izici.

3.3.Navier-Stokes equation

The Navier-Stokes equation is used to describe the conservation of the momentum of the incompressible fluid inside nanochannels.The Reynold number (Re) is usually Re ?1 for the flow in a nanochannel.The equation can be simplified as the following Stokes equation,

where p is the applied pressure,ρ is the density of the fluid and Feis the electrical driving force.Here,

where E is the electric field.

3.4.Boundary conditions

All the boundary conditions used in this model are shown in Table 2,where cpand cnare the concentrations of the positive and negative ions and β is the slip-length.Previous studies show that the slip-length effect on ionic transport at the scale that we consider is not important[41,42],such that the slip boundary condition is ignored for convenience.The detailed boundary conditions in the simulation are shown in Table 2.

4.Results and Discussion

In this part,the viability of our model discussed above is verified.Furthermore,we investigate the effects of surface roughness and pore radius on the distribution of ions,the flow rates,the streaming potentials,the streaming currents and the electrokinetic conversion efficiencies in nanochannels,and the numerical results are shown.

4.1.Validation of our numerical calculations

As shown in Fig.4,we compare our numerical results with the results in the literature [43],including the potential distribution,.pressure distribution,and velocity distribution in the nanochannels with no slip length and the slip length of 25 nm.The electric potential increases,while the fluid pressure decreases along the z-direction.In the radial direction,the fluid velocity near the wall of the channel is significantly lower than that at the midsection of the channel.The comparison results show that our results are similar to those in the literature,demonstrating the relability of our model.

Table 2 Boundary conditions in the simulation

Fig.4.Comparisons of(a)the scaled potential along the axis of symmetry,ψ0 =eψ/kB T,(b)the scaled pressure,(c)the scaled axial velocity profile at the midsection with no slip length,u0 =uλD /D,and(d)the scaled axial velocity profile at the midsection of the channel with the slip length of 25 nm obtained from literature(red dashed line) and our simulation result (black solid line)

4.2.The influence of surface roughness on the conversion efficiency

Fig.5.The velocity profile in the nanochannels with different (a) radiuses (top:a=0.02λD ,medium:a=0.2λD ,bottom:a=1.5λD ) and (c) numbers (top:5,medium:10,bottom:30) of the bumps and grooves.The electrokinetic conversion efficiency η with varying (b) pore radiuses and (d) numbers of the bumps and grooves.

In this section,we investigate the effect of the surface roughness on the electrokinetic energy conversion efficiency by tuning the size and number of the bumps and grooves.The fluid flow velocities in the channels with different bump sizes are shown in Fig.5(a),where r is 5λDand the numbers of bumps and grooves are 10.It can be found that the fluid flow at the midsection of the channel is faster than that near the wall.The small roughness(a=0.02λD,0.2λD) has little influence on the fluid flow,while the influence of the larger roughness (a=1.5λD) on the fluid flow are obvious.In addition,the distribution of counter-ions,co-ions and electric potential in these nanochannels are shown in Fig.S1.The counter-ions are absorbed on the rough surface while the co-ions are repelled away from the surface,and the potential distribution depending on the net charges distribution is significantly influenced by the surface roughness in the channel,and the larger the bump is,the stronger this influence will be.Based on the Eq.(9),we calculate the electrokinetic energy conversion efficiency from the streaming current,potential and the fluid flow rate.As shown in Fig.5(b),the conversion efficiency increases with the sizes of bumps increase until the efficiency reach a maximum.The existence of the roughness can increase the surface area of the channel,which is favorable for more counter-ions to be adsorbed on the surface and pass through the channel,while the large bumps hinder the ion transport.Thus the streaming current and the streaming potential have extremums as shown in Fig.S2,which leads to the maximum value of conversion efficiency.

Fig.5(c) shows the distribution of the fluid flow velocity when the side lengths of bumps and grooves a are 0.2λDand the radius of the nanochannel is 5λD,which indicates that the bumps number almost has no influence on the fluid flow velocity.The distributions of the counter-ions,co-ions and electric potential with different bump numbers,shown in Fig.S3,reveal that the impacts of bumps number on the ions and potential distribution are small.In these cases (a=0.2λD),the bumps and grooves are less obstructive to the fluid flow but the roughness can enhance the streaming current and potential,due to the increase of the surface area,as shown in Fig.S4.Thus,the conversion efficiency increases with the bumps and grooves number increasing as shown in Fig.5(d).

4.3.The effects of the pore radius on the conversion efficiency

Here,the electrokinetic energy conversion efficiency of nanochannels with different pore radius in smooth nanochannel is examined by the numerical simulation.As shown in Fig.6,when the mean radius is around λD,the conversion efficiency reaches a peak,which is consistent with the analytical results in Section 2.When the mean radius of the nanochannel is very small,the relationship between the conversion efficiency and the mean radius is irregular.The numerical results are obtained by PNP-NS equations,and the EDL overlapping in the tiny nanochannels causes the oscillatory variation of conversion efficiency with pore size,which was similar to the pore-size-depended capacitance [44–47].The analytical results are obtained by considering two extremes cases:a large nanochannel(r ?λD)and a tiny nanochannel(r ?λD),and the analytical results are qualitative.In large nanochannels,the EDLs do not overlap and the efficiencies for analytical and numerical results agree well.

Fig.6.The electrokinetic conversion efficiency with different mean radiuses in smooth nanochannels (a=0) (red solid line:numerical results,black dotted line:analytical results).The electrokinetic conversion efficiency is irregular with the tiny mean radius,and the efficiency reaches a extremum when the mean radius is around λD .

In this part,we show the potential distribution in the nanochannels with different mean radiuses and different side lengths of small bumps and grooves.Fig.7(a)–(c)show the potential distributions in the smooth nanochannels and rough nanochannels with a/r=1/15 and a/r=1/10.In the small nanochannels,the occuring of EDLs overlapping is beneficial for counter-ions to entering and passing the channel but detrimental for co-ions,thus the small nanochannels are filled by the net charges.The potentials determined with the net charge concentrations are high for the all zone of the small channels as shown in the tops of Fig.7(a)–(c).However,in the large nanochannels,both coions and counter-ions enter and pass through the channels.The concentrations of net charges are low in the middle of the channels but high near the walls due to the surface charges,which leads to the low potentials in the midsection of the channels and high potential near the wall as shown in mediums and bottoms of Fig.7(a)–(c).The pressure-driven fluid carries the net charges through the nanochannel,which leads to the increasing of net charge concentration along z-direction as show in Fig.S5.Therefore,the electric potential,depending on the concentration of net charges,increases along the z-direction as shown in Fig.7(d).

For the small channels,the increases of the mean pore radius are favorable for the counter-ions passing through the channel while the co-ions are difficult to enter the channel.For the large channels,both the counter-ions and co-ions can flow through the channel,which weakens the effect of the pore radius on the electric potential.The concentrations of net charge in a small nanochannel and a large nanochannel are shown in Fig.S6.Thus,the increase of the pore radiuses enhances the streaming potential in small nanochannels,but the increase of the streaming potential tends to be slight with the mean radius increasing in large nanochannels as shown in Fig.8(a).Fig.8(b)represents the variations of the flow rates with different mean pore radiuses,and the larger pore radiuses can enhance the fluid flow in the nanochannel.When the mean radiuses are fixed,the larger the bumps are,the greater the resistances will be for the fluid flows,leading to the decrease of the flows.In small channels,the bumps are small,thus the difference between flows in the nanochannels with different bumps is slight.However,the difference is obvious in large nanochannels with different bumps because of the large bumps.The streaming currents are shown in Fig.8(c).As the mean radiuses increase,the streaming currents in the channels increase,which is consistent with the numerical analysis results in Section 2 (Eqs.(4) and(5)).The electrokinetic conversion efficiencies are calculated and shown in Fig.8(d).The efficiencies have maximums in both the smooth nanochannels and the rough nanochannels.The maximum of electrokinetic conversion efficiency in smooth nanochannel is larger than that in the rough nanochannel and the peak for smooth nanochannel shifts to left compared with that for rough nanochannel.However,in some cases(e.g.,r=5λD,a=λD/3),the conversion efficiency in a rough channel is higher than that in a smooth channel.

4.4.Comparison with the experiments and implications from our simulations

Fig.7.The distributions of the potentials with different ratios of side length of small bumps and grooves respect to the pore radius (top:r=λD ,medium:r=7λD ,bottom:r=15λD ),where (a) a/r=0,(b) a/r=1/15 and (c) a/r=1/10.(d) Variation of the scaled electric potential,ψ0 =eψ/kB T,along the nanochannel centerline for different d,where a/r=1/10.

Fig.8.The variations of parameters in the nanochannels with different pore radiuses and roughness (the black line:smooth channel,blue line:rough channel with a/r=1/15,the red line:rough channel with a/r=1/10).(a) The scaled streaming potential,Δψ0 =Δψe/kB T,(b) the scaled flow rate,Q0 =Q/DλD ,(c) the scaled streaming current,I0 =IzeλD /εkB TD and (d) the electrokinetic conversion efficiency η.

The ion transport in a nanochannel with cross sections fabricated in glass was studied and the result showed that the high specific surface area was superior to the ion transport [48].Thus,the small surface roughness of nanochannel can improve the ion transport,which is beneficial to enhancing the streaming current.The result is consistent with our results in Sections 4.2 and 4.3.Ion transport behaviors in nanochannels play critical roles in the research of electrokinetic energy conversion efficiency.So,the effect of wall structure and pore size are significant for the design of energy conversion devices.

5.Conclusions

To summarize,our investigations reveal that the pore size and surface roughness have influences on the electrokinetic energy conversion efficiency of nanofluidic devices.We prove that the conversion efficiency has a maximum in the smooth nanochannel through numerical analysis,which is further demonstrated by the numerical solution based on the Poisson-Nernst-Planck(PNP) equations and the Navier-Stokes (NS) equation.Besides,we study the conversion efficiencies in the rough nanochannels and the results show that the efficiencies in the rough channels also have maximums with different pore radius.The maximum in the smooth channel is higher than that in rough nanochannels,while the small roughness can improve the conversion efficiency in some cases (e.g.,r=5λD,a=λD/3).These cases suggested that appropriate pore radius and roughness of the nanochannel wall can improve the conversion efficiency.We hope that these conclusions can provide theoretical guidance for novel energy conservation devices,and we are looking forward to having advanced experimental nanoscale devices constructed by nanomaterials which could be used to verify our theoretical results.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was sponsored by the National Natural Science Foundation of China (No.91834301,21808055 and 22078088),the Shanghai Sailing Program (18YF1405400).We acknowledge helpful discussions with R.van Roij.

Supplementary Material

Supplementary data to this article can be found online at https://doi.org/10.1016/j.cjche.2020.09.041.