Article pubs.acs.org/Langmuir
Ionic Size Dependent Electroviscous Effects in Ion-Selective Nanopores Aditya Bandopadhyay,† Syed Sahil Hossain,‡ and Suman Chakraborty*,†,‡ †
Advanced Technology Development Center and ‡Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India ABSTRACT: Pressure-driven flows of aqueous ionic liquids are characterized by electroviscosityan increase in the effective (apparent) viscosity because of an induced back electric field termed streaming potential. In this work, we investigate the electrokinetic phenomenon of streaming potential mediated flows in ion-selective nanopores. We report a dramatic augmentation in the effective viscosity as attributable to the finite size effect of the ionic species in counterion-only systems. The underlying physics involves complex interaction between the concerned electrochemical phenomena and hydrodynamic transport in a confined fluidic environment, which we capture through a modified continuum based approach and validate using molecular dynamics simulations. We obtain an expression for the ionic-size dependent streaming potential pertinent to the physical situation being addressed. The corresponding estimations of effective viscosity implicate that the classical paradigm of point sized ions can give rise to gross underestimations of the flow resistance in counterion-only systems especially for negligible surface (Stern layer) conductivity and large fluidic slip at the surface.
I. INTRODUCTION Nanochannels can conduct ionic currents (streaming current) in a pressure-driven flow, and in the process, can produce a potential difference (streaming potential) between the end reservoirs of the channel.1−9 The induced potential difference is such that it opposes the impending direction of the pressure driven flow, thus leading to a decrease in the volume flow ratea phenomenon effectively described by an increase in the apparent viscosity, which has been a topic of various studies in recent times.1,10−13 This phenomenon is known as the electroviscous effect. The underlying mechanism is fundamentally attributed to the formation of a charged interfacial layer (also known as the electrical double layer), which gives rise to surplus ionic species of a given polarity in the solution, and the viscous drag between the fluid and the ionic species. From a more practical point of view, one may tap the streaming current and streaming potential to convert the hydraulic energy into electrical energy; a concept which has been steadily gaining importance in modern times.14−28 The standard paradigm of description of electrokinetic phenomenon is centered around the Poisson−Boltzmann description of the electrical double layer (EDL), which assumes the ionic species as point charges.1 However, for high surface charge densities, the above assumption leads to an unrealistically high counterion concentrations at the surface of the channel.29,30 The consequent implications, also known as excluded volume interactions because of the finite sizes of the ionic species (alternatively known as steric effects), are likely to bear far-ranging scientific and technological consequences so far as the realization of nanofluidic energy conversion devices is concerned.18,25 This also has important implications in several © XXXX American Chemical Society
other applications, including transport of ionic drugs across paracellular pathway of cell monolayers and transport in biological ion channels.31−41 The issue becomes further nontrivial in systems in which counterions (i.e., ionic species of sign opposite to that of the surface charge) are only present in the solution, such as in ion-selective nanopores.24,42−45 It may be noted in this context that counterion only solutions are also typical to electrokinetic systems with salt-free organic solvents as working fluids, which provide several advantages like enhanced solubility of many analytes, reduced Joule heating, weakened interactions of hydrophobic analytes, etc.42 As a result of this, such systems have gained importance in areas like electronic paper (e-paper) designed on the basis of electrophoretic display.43 There are also cases where ion exchange between sulfonic acid groups and water in proton exchange membranes (PEM) lead to current generation and hence are used in hydrogen fuel cells.44,45 In the case of Nafion tubes, the pores are filled with “proton cloud” and the surface charges are partially ionized sulfonic group. Motivated by the proposition of employing typical counterion only systems that are progressively gaining importance in electrokinetically actuated devices, as well as by the rich implications of excluded volume effects due to finite size effects of the ionic species in the case of confinements having dimensions not trivially negligible as compared to those of the ionic species, here we attempt to address the question: What is the role of finite size effects of the ionic species on Received: April 21, 2014 Revised: May 21, 2014
A
dx.doi.org/10.1021/la5014957 | Langmuir XXXX, XXX, XXX−XXX
Langmuir
Article
between charged surfaces vis-a-vis the classical Poisson− Boltzmann scenario. In our work, however, we do not consider the solvent polarization and only consider the free energy expression which incorporates the finite size nature of the counterions, which is a simplified form of that employed by Mishra et al.29 and by Borukhov et al.50 By considering the variation of the free energy functional F with respect to nc, we get the electrochemical potential as μc = ∫ dy(zeψ + (kT/ a3)[nca3 ln (nca3) − a3 ln(1 − nca3)]). Enforcing a constant electrochemical potential in the channel and imposing the boundary conditions as ψ = 0, nc = n0 at y = 0, we obtain the concentration distribution as nc = [n0 exp(−ψ̅ )]/[1 + v(exp (−ψ̅ ) − 1)]; ν = n0a3 being known as the Steric factor and ψ̅ = zeψ/kT is the dimensionless potential. As a result, we get the potential distribution from eq 2 as
electrokinetic transport phenomena through ion-selective nanopores? The central result of our findings is that an intricate interaction between electrohydrodynamics of counterion-only systems and excluded volume effects originating from finite sizes of the ionic species leads to drastically increased effective viscosities.
II. PROBLEM FORMULATION In an effort to establish the above proposition, we first begin with the description of the pertinent governing equation for fluid flow (Navier−Stokes equation) in the low Reynolds number regime (i.e., with negligible inerial effects) with an additional body force term due to the electric field (unknown a priori) as46 ∂p d ⎛ du ⎞ ⎜η ⎟ = −ρe Ex + dy ⎝ dy ⎠ ∂x
exp( −ψ̅ ) d2ψ̅ = −(KcH )2 2 1 + ν[exp( −ψ̅ ) − 1] dy ̅
(1)
where y is the transverse coordinate employed to describe the flow profile in a slit-type nanochannel (y = 0 coincides with the channel centerline, and y = ±H denotes the channel walls), u is the axial component of the velocity, p is the pressure, η is the fluid viscosity, ρe is the volumetric charge density, and Ex is the induced axial electric field. We assume here that the channel is infinitely long, so as to bring in an x invariance. In the case of a counterion-only system, the volumetric charge density is simply given as ρe = zenc, where nc is the counterion concentration, z is the valency of the ionic species, and e is the fundamental protonic charge. Equation 1 is subjected to a no-slip boundary condition at the wall and a symmetric boundary condition at the channel centerline, mathematically described as u|H = 0 and (du/dy)|0 = 0. In the case of certain surface conditions, the noslip boundary condition at the wall may not always be suitable.47−49 In such a case, one may appeal to the Navier-Slip boundary condition which is given as uH = −b (du/dy)|H, where b is the slip length: a hypothetical distance beyond the solid surface to which the velocity extrapolates to zero.14 It is apparent that a consistent description of the counterion concentration distribution is needed in order to proceed with the description of the velocity field. For that purpose, we first invoke the Poisson equation for description of the resultant potential distribution as
ρ d ⎛ dψ ⎞ ⎜ ⎟=− e dy ⎝ dy ⎠ ε
(3)
where y ̅ = y/H is the dimensionless transverse coordinate. Here KcH = [(z2e2n0H2)/(εkT)]1/2 represents a dimensionless height, where inverse of Kc is analogous to the Debye length scale for cases in which both co-ions and counterions are present. Note that we have assumed that the permittivity is a constant in this system. This may not be the case as there might be solvent polarization close to the surface.11 Equation 3 is subjected to a zero guage centerline potential and symmetry at the centerline, mathematically described as ψ̅ |0 = (dψ̅ /dy)| ̅ 0 = 0. At the surface, the charge density is given as εdψ/dy = σsurf (σsurf is the charge density at the surface) or equivalently in a nondimensional form, dψ̅ /dy|̅ y=1 = σ = Hzeσsurf/εkT; in essence, this expression ̅ relates the surface charge density to the dimensionless height, thus providing closure to the problem.45,51,52 Having established the governing equations for the potential distribution, we next attempt to solve the momentum equation (eq 1). In a nondimensional form, the solution can be expressed as u ̅ = up + ue = (1 − y ̅ 2 ) + E ̅ (ψ̅ − ζ ̅ )
(4a)
u ̅ = up + ue = (1 − y ̅ 2 + 2b ̅ ) + E ̅ (ψ̅ − ζ ̅ − σb ̅ )
(4b)
Equation 4a represents the velocity profile obtained for a noslip boundary condition, whereas eq 4b represents the velocity profile for a Navier slip boundary condition where b̅ (= b/H) represents the dimensionless slip length. Here, up is the pressure driven component of velocity and ue is the velocity component due to the induced electric field, ζ̅ represents the dimensionless potential at the y ̅ = ±1. Here the velocity has been nondimensionalized as u̅ = u/Uref, and E̅ = E/Eref where Uref = −H2/2η ∂p/∂x and we choose Eref such that Uref = εErefkT/ηze which resembles the classical Helmholtz-Smoluchowski velocity at thermal voltage. It is extremely important to mention here that the streaming electric field E̅ appearing in eq 5 remains an unknown at this stage, which needs to be determined in a physically consistent manner. Toward the above, we first note that the induced electrical field is developed in such a way that the net current through the channel is zero because of the fact that there is no externally applied electric field. The current because of the advection of the ions is termed the streaming current and can be written as Is = 2∫ 0Huρedy. The other current is due to the bulk conductivity of the solution and may be written as Ic = 2zeE/f ∫ H0 ρedy, with f being the ionic friction factor. We note here that, in addition to
(2)
where ψ is the electrical potential distribution due to the counterions, and ε is the permittivity of the medium. As discussed earlier, a description of the counterion concentration is needed to ascertain the volumetric charge density. In an effort to describe the same, we adopt here a free energy based thermodynamic formalism. Considering steric interactions, we describe the free energy of the system (F) as29,50 F = U − TS where U = ∫ dy(−(ε/2)|∇ψ|2 + zencψ) is the contribution from the self-energy of the field and field itself. The second term represents the entropic contributions and is given by −TS = kT/a3 ∫ dy(nca3 ln(nca3) + (1 − nca3) ln(1 − nca3)), where k represents the Boltzmann constant and T is the absolute temperature; a being the characteristic size of the counterions. We note here that the work by Mishra et al.29 considers the effect of the solvent polarization, over and above the finite size of the ions, into the free energy formulation, toward a more generalized framework for quantifying the interaction forces B
dx.doi.org/10.1021/la5014957 | Langmuir XXXX, XXX, XXX−XXX
Langmuir
Article
the two aforementioned currents, i.e., the streaming and the conduction current, there may be another current through the Stern layer (the condensed layer closed to the wall). The Stern layer conduction current may be written as Istern = 2σsternE, where σstern is the Stern layer conductivity. Here, for capturing the essential physics, we do not consider the Stern layer conductivity. The criteria for electroneutrality dictates that10,53 Is + Ic + Istern = 0, i.e., 2∫ H0 uρedy + 2zeE/f ∫ H0 ρedy + 2Eσstern = 0. In a more compact form this may be written as
∫0
1
upn ̅ dy ̅ + E ̅
∫0
1
uen ̅ dy ̅ + JE ̅
∫0
1
n ̅ dy ̅ + JDuE ̅ = 0 (5)
where the ratio of the reference conduction current to the reference streaming current is denoted as J = ηz2e2/εkTf and Du is the Dukhin number defined as the ratio of the Stern layer conductivity(σstern) to the bulk fluid conductivity σbH (n0z2e2/f H).54 Thus, the velocity field is now known completely using the streaming electric field evaluated from eq 5. In the presence of the induced electric field, the attenuated volume flow rate can be written as Q = UrefH∫ 10(up̅ + u̅e)dy ̅ = −H2/2η ∂p/∂x H∫ 10(u̅p + u̅e)dy.̅ The reduced flow rate may be written in terms of an enhanced effective viscosity or apparent viscosity for a purely pressure driven flow without electrokinetic effects as Q = −(H2/2ηa)(∂p/∂x)H∫ 10(u̅p)|dy,̅ where ηa is the apparent viscosity. Thus, equating these flow rates, we obtain the ratio of the effective viscosity to the actual viscosity as ηa η ηa η
Figure 1. (a) Schematic of the problem description for numerical simulation. (b) Mesh layout.
iteratively such that the absolute value of the centerline potential is
Ionic Size Dependent Electroviscous Effects in Ion-Selective Nanopores Aditya Bandopadhyay,† Syed Sahil Hossain,‡ and Suman Chakraborty*,†,‡ †
Advanced Technology Development Center and ‡Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India ABSTRACT: Pressure-driven flows of aqueous ionic liquids are characterized by electroviscosityan increase in the effective (apparent) viscosity because of an induced back electric field termed streaming potential. In this work, we investigate the electrokinetic phenomenon of streaming potential mediated flows in ion-selective nanopores. We report a dramatic augmentation in the effective viscosity as attributable to the finite size effect of the ionic species in counterion-only systems. The underlying physics involves complex interaction between the concerned electrochemical phenomena and hydrodynamic transport in a confined fluidic environment, which we capture through a modified continuum based approach and validate using molecular dynamics simulations. We obtain an expression for the ionic-size dependent streaming potential pertinent to the physical situation being addressed. The corresponding estimations of effective viscosity implicate that the classical paradigm of point sized ions can give rise to gross underestimations of the flow resistance in counterion-only systems especially for negligible surface (Stern layer) conductivity and large fluidic slip at the surface.
I. INTRODUCTION Nanochannels can conduct ionic currents (streaming current) in a pressure-driven flow, and in the process, can produce a potential difference (streaming potential) between the end reservoirs of the channel.1−9 The induced potential difference is such that it opposes the impending direction of the pressure driven flow, thus leading to a decrease in the volume flow ratea phenomenon effectively described by an increase in the apparent viscosity, which has been a topic of various studies in recent times.1,10−13 This phenomenon is known as the electroviscous effect. The underlying mechanism is fundamentally attributed to the formation of a charged interfacial layer (also known as the electrical double layer), which gives rise to surplus ionic species of a given polarity in the solution, and the viscous drag between the fluid and the ionic species. From a more practical point of view, one may tap the streaming current and streaming potential to convert the hydraulic energy into electrical energy; a concept which has been steadily gaining importance in modern times.14−28 The standard paradigm of description of electrokinetic phenomenon is centered around the Poisson−Boltzmann description of the electrical double layer (EDL), which assumes the ionic species as point charges.1 However, for high surface charge densities, the above assumption leads to an unrealistically high counterion concentrations at the surface of the channel.29,30 The consequent implications, also known as excluded volume interactions because of the finite sizes of the ionic species (alternatively known as steric effects), are likely to bear far-ranging scientific and technological consequences so far as the realization of nanofluidic energy conversion devices is concerned.18,25 This also has important implications in several © XXXX American Chemical Society
other applications, including transport of ionic drugs across paracellular pathway of cell monolayers and transport in biological ion channels.31−41 The issue becomes further nontrivial in systems in which counterions (i.e., ionic species of sign opposite to that of the surface charge) are only present in the solution, such as in ion-selective nanopores.24,42−45 It may be noted in this context that counterion only solutions are also typical to electrokinetic systems with salt-free organic solvents as working fluids, which provide several advantages like enhanced solubility of many analytes, reduced Joule heating, weakened interactions of hydrophobic analytes, etc.42 As a result of this, such systems have gained importance in areas like electronic paper (e-paper) designed on the basis of electrophoretic display.43 There are also cases where ion exchange between sulfonic acid groups and water in proton exchange membranes (PEM) lead to current generation and hence are used in hydrogen fuel cells.44,45 In the case of Nafion tubes, the pores are filled with “proton cloud” and the surface charges are partially ionized sulfonic group. Motivated by the proposition of employing typical counterion only systems that are progressively gaining importance in electrokinetically actuated devices, as well as by the rich implications of excluded volume effects due to finite size effects of the ionic species in the case of confinements having dimensions not trivially negligible as compared to those of the ionic species, here we attempt to address the question: What is the role of finite size effects of the ionic species on Received: April 21, 2014 Revised: May 21, 2014
A
dx.doi.org/10.1021/la5014957 | Langmuir XXXX, XXX, XXX−XXX
Langmuir
Article
between charged surfaces vis-a-vis the classical Poisson− Boltzmann scenario. In our work, however, we do not consider the solvent polarization and only consider the free energy expression which incorporates the finite size nature of the counterions, which is a simplified form of that employed by Mishra et al.29 and by Borukhov et al.50 By considering the variation of the free energy functional F with respect to nc, we get the electrochemical potential as μc = ∫ dy(zeψ + (kT/ a3)[nca3 ln (nca3) − a3 ln(1 − nca3)]). Enforcing a constant electrochemical potential in the channel and imposing the boundary conditions as ψ = 0, nc = n0 at y = 0, we obtain the concentration distribution as nc = [n0 exp(−ψ̅ )]/[1 + v(exp (−ψ̅ ) − 1)]; ν = n0a3 being known as the Steric factor and ψ̅ = zeψ/kT is the dimensionless potential. As a result, we get the potential distribution from eq 2 as
electrokinetic transport phenomena through ion-selective nanopores? The central result of our findings is that an intricate interaction between electrohydrodynamics of counterion-only systems and excluded volume effects originating from finite sizes of the ionic species leads to drastically increased effective viscosities.
II. PROBLEM FORMULATION In an effort to establish the above proposition, we first begin with the description of the pertinent governing equation for fluid flow (Navier−Stokes equation) in the low Reynolds number regime (i.e., with negligible inerial effects) with an additional body force term due to the electric field (unknown a priori) as46 ∂p d ⎛ du ⎞ ⎜η ⎟ = −ρe Ex + dy ⎝ dy ⎠ ∂x
exp( −ψ̅ ) d2ψ̅ = −(KcH )2 2 1 + ν[exp( −ψ̅ ) − 1] dy ̅
(1)
where y is the transverse coordinate employed to describe the flow profile in a slit-type nanochannel (y = 0 coincides with the channel centerline, and y = ±H denotes the channel walls), u is the axial component of the velocity, p is the pressure, η is the fluid viscosity, ρe is the volumetric charge density, and Ex is the induced axial electric field. We assume here that the channel is infinitely long, so as to bring in an x invariance. In the case of a counterion-only system, the volumetric charge density is simply given as ρe = zenc, where nc is the counterion concentration, z is the valency of the ionic species, and e is the fundamental protonic charge. Equation 1 is subjected to a no-slip boundary condition at the wall and a symmetric boundary condition at the channel centerline, mathematically described as u|H = 0 and (du/dy)|0 = 0. In the case of certain surface conditions, the noslip boundary condition at the wall may not always be suitable.47−49 In such a case, one may appeal to the Navier-Slip boundary condition which is given as uH = −b (du/dy)|H, where b is the slip length: a hypothetical distance beyond the solid surface to which the velocity extrapolates to zero.14 It is apparent that a consistent description of the counterion concentration distribution is needed in order to proceed with the description of the velocity field. For that purpose, we first invoke the Poisson equation for description of the resultant potential distribution as
ρ d ⎛ dψ ⎞ ⎜ ⎟=− e dy ⎝ dy ⎠ ε
(3)
where y ̅ = y/H is the dimensionless transverse coordinate. Here KcH = [(z2e2n0H2)/(εkT)]1/2 represents a dimensionless height, where inverse of Kc is analogous to the Debye length scale for cases in which both co-ions and counterions are present. Note that we have assumed that the permittivity is a constant in this system. This may not be the case as there might be solvent polarization close to the surface.11 Equation 3 is subjected to a zero guage centerline potential and symmetry at the centerline, mathematically described as ψ̅ |0 = (dψ̅ /dy)| ̅ 0 = 0. At the surface, the charge density is given as εdψ/dy = σsurf (σsurf is the charge density at the surface) or equivalently in a nondimensional form, dψ̅ /dy|̅ y=1 = σ = Hzeσsurf/εkT; in essence, this expression ̅ relates the surface charge density to the dimensionless height, thus providing closure to the problem.45,51,52 Having established the governing equations for the potential distribution, we next attempt to solve the momentum equation (eq 1). In a nondimensional form, the solution can be expressed as u ̅ = up + ue = (1 − y ̅ 2 ) + E ̅ (ψ̅ − ζ ̅ )
(4a)
u ̅ = up + ue = (1 − y ̅ 2 + 2b ̅ ) + E ̅ (ψ̅ − ζ ̅ − σb ̅ )
(4b)
Equation 4a represents the velocity profile obtained for a noslip boundary condition, whereas eq 4b represents the velocity profile for a Navier slip boundary condition where b̅ (= b/H) represents the dimensionless slip length. Here, up is the pressure driven component of velocity and ue is the velocity component due to the induced electric field, ζ̅ represents the dimensionless potential at the y ̅ = ±1. Here the velocity has been nondimensionalized as u̅ = u/Uref, and E̅ = E/Eref where Uref = −H2/2η ∂p/∂x and we choose Eref such that Uref = εErefkT/ηze which resembles the classical Helmholtz-Smoluchowski velocity at thermal voltage. It is extremely important to mention here that the streaming electric field E̅ appearing in eq 5 remains an unknown at this stage, which needs to be determined in a physically consistent manner. Toward the above, we first note that the induced electrical field is developed in such a way that the net current through the channel is zero because of the fact that there is no externally applied electric field. The current because of the advection of the ions is termed the streaming current and can be written as Is = 2∫ 0Huρedy. The other current is due to the bulk conductivity of the solution and may be written as Ic = 2zeE/f ∫ H0 ρedy, with f being the ionic friction factor. We note here that, in addition to
(2)
where ψ is the electrical potential distribution due to the counterions, and ε is the permittivity of the medium. As discussed earlier, a description of the counterion concentration is needed to ascertain the volumetric charge density. In an effort to describe the same, we adopt here a free energy based thermodynamic formalism. Considering steric interactions, we describe the free energy of the system (F) as29,50 F = U − TS where U = ∫ dy(−(ε/2)|∇ψ|2 + zencψ) is the contribution from the self-energy of the field and field itself. The second term represents the entropic contributions and is given by −TS = kT/a3 ∫ dy(nca3 ln(nca3) + (1 − nca3) ln(1 − nca3)), where k represents the Boltzmann constant and T is the absolute temperature; a being the characteristic size of the counterions. We note here that the work by Mishra et al.29 considers the effect of the solvent polarization, over and above the finite size of the ions, into the free energy formulation, toward a more generalized framework for quantifying the interaction forces B
dx.doi.org/10.1021/la5014957 | Langmuir XXXX, XXX, XXX−XXX
Langmuir
Article
the two aforementioned currents, i.e., the streaming and the conduction current, there may be another current through the Stern layer (the condensed layer closed to the wall). The Stern layer conduction current may be written as Istern = 2σsternE, where σstern is the Stern layer conductivity. Here, for capturing the essential physics, we do not consider the Stern layer conductivity. The criteria for electroneutrality dictates that10,53 Is + Ic + Istern = 0, i.e., 2∫ H0 uρedy + 2zeE/f ∫ H0 ρedy + 2Eσstern = 0. In a more compact form this may be written as
∫0
1
upn ̅ dy ̅ + E ̅
∫0
1
uen ̅ dy ̅ + JE ̅
∫0
1
n ̅ dy ̅ + JDuE ̅ = 0 (5)
where the ratio of the reference conduction current to the reference streaming current is denoted as J = ηz2e2/εkTf and Du is the Dukhin number defined as the ratio of the Stern layer conductivity(σstern) to the bulk fluid conductivity σbH (n0z2e2/f H).54 Thus, the velocity field is now known completely using the streaming electric field evaluated from eq 5. In the presence of the induced electric field, the attenuated volume flow rate can be written as Q = UrefH∫ 10(up̅ + u̅e)dy ̅ = −H2/2η ∂p/∂x H∫ 10(u̅p + u̅e)dy.̅ The reduced flow rate may be written in terms of an enhanced effective viscosity or apparent viscosity for a purely pressure driven flow without electrokinetic effects as Q = −(H2/2ηa)(∂p/∂x)H∫ 10(u̅p)|dy,̅ where ηa is the apparent viscosity. Thus, equating these flow rates, we obtain the ratio of the effective viscosity to the actual viscosity as ηa η ηa η
Figure 1. (a) Schematic of the problem description for numerical simulation. (b) Mesh layout.
iteratively such that the absolute value of the centerline potential is
