Letter pubs.acs.org/NanoLett
High Energy Conversion Efficiency in Nanofluidic Channels Dirk Gillespie* Department of Molecular Biophysics and Physiology, Rush University Medical Center, Chicago, IL ABSTRACT: It is proposed that the layering of large ions at the wall/liquid interface of nanofluidic channels can be used to achieve high efficiency (possibly >50%) in the conversion of hydrostatic energy into electrical power. Large ions tend to produce peaks and troughs in their concentration profiles at charged walls, producing high concentrations far from the walls where the ions’ pressure-driven velocity is high. This increases the streaming conductance and the energy conversion efficiency.
KEYWORDS: Nanofluidic, energy conversion, ion layering, density functional theory of fluids
N
anofluidic channels conduct ions between two baths either with an applied voltage or an applied pressure. In pressure-driven flow, the ionic current (the streaming current) produces a voltage difference between the two baths that are connected by the channel (the streaming potential). This ion current and potential can be coupled to an electrical resistor to produce electrical current and the hydrostatic energy can be converted to electrical power. While this idea is not new,1 the fabrication of nanoscale devices has renewed interest in this type of energy conversion.2−13 For practical applications, one would like to maximize the efficiency of this energy conversion. Recently, a number of theoretical analyses, using Poisson−Boltzmann (PB) theory for the ion double layers and Navier−Stokes for fluid flow, have estimated maximal conversion efficiency at ∼15% with no-slip boundary conditions7−10 and upward of 30% with slip.11−13 This paper describes a mechanism of achieving the high energy conversion efficiencies (>50%) based on the correlations of ions at the device wall, rather than slip. In this paper, energy conversion efficiency is defined as the maximum ratio of useful electrical power that can be extracted by an external load to the total mechanical work performed (see Appendix). As summarized in the Appendix, the maximum 2 /qchgch efficiency is determined by the unitless quantity α = Sstr where Sstr is the streaming conductance, qch is the fluidic pressure conductance of the channel, and gch is the electrical conductance of the channel, which has both conductivity and convective components. The larger α is, the larger the conversion efficiency.8 Therefore, the device with the greatest conversion efficiency maximizes the streaming conductance while at the same time minimizing the electrical conductance. Recently, van der Hayden et al.8 analyzed the power conversion efficiency of slitlike nanochannels for electrolytes of simple monovalent ions between two charged walls (Figure 1). They described several principles that lead to high energy conversion efficiency: © 2012 American Chemical Society
Figure 1. Schematic of the nanochannel connecting to electrolyte reservoirs. The ion current is in the x-direction (where the device has length L) and the double layers computed here are in the y-direction (where the device has height H) between the two charged surfaces (with surface charge σ), shown in red and part of the device top in pink. The orders of magnitude in each direction are indicated.
1 Double layers must overlap so that co-ions are expelled between the two charged walls. This is best achieved with monovalent ions, low ion concentrations, and low surface charges. Co-ions move in the same direction as the counterions with pressure but in the opposite direction with voltage. This kind of co-ion movement reduces the streaming conductance while also increasing the electrical conductance, both of which lower efficiency. Therefore, having only counterions maximizes efficiency. Moreover, the low ion concentrations that cause the double layers to overlap and expel co-ions also reduce the fluid’s conductivity, again increasing efficiency. 2 Small counterion diffusion coefficients (D) increase efficiency. With a Navier−Stokes description of current, Received: November 21, 2011 Revised: January 23, 2012 Published: February 2, 2012 1410
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diffusion coefficient of 0.5 m2/s, which is a normal, but low, value for cations.25 Second, the streaming conductance Sstr must be far from zero. Since the ions’ velocity during pressure-driven flow is highest near the center of the channel (eq 12), this requires that the ions’ net charge profile extends far from the device walls (eq 18). One way to achieve this is to have the electrical double layers overlap at low ion concentrations.8 Another way is having a second layer (or more) of counterions beyond the first layer next to the wall. This is a well-known effect15−24 for large ions and occurs at highly charged walls where counterions accumulate at the walls at such high densities that the ions’ size effectively excludes more ions. These then form a second layer with a distinct peak in the concentration profile that is approximately three radii from the wall. While layering has been described in nanochannels (e.g., by Nilson and Griffiths24 and Gillespie et al.14), but it has not been considered for energy conversion (to the author’s knowledge). Examples of this are show in Figure 2. Here, ions are confined to a 10 nm high slit that has −0.3 C/m2 surface charge
2 in the ratio Sstr /qchgch the conductivity term becomes proportional to ηD because it is the only term proportional to D and independent of the viscosity η (see eq 21 in the Appendix). The conductivity contributes to the electrical conductance, so systems with relatively smaller diffusion coefficients have higher efficiency (assuming that the solvent’s viscosity is unchanged). 3 With a PB description of the double layers, efficiency depends on the ratio of the channel height to the Gouy− Chapman length when the double layers strongly overlap. Like most others,2−7,9−13 Van der Hayden et al.8 used PB theory where ions are modeled as point charges, a reasonable description when the finite size of the ions is not significant. In that case, the energy conversion efficiency is at most ∼15% (in the no-slip case7−10). Therefore, to achieve higher efficiency one must look beyond small ions. In this paper, pressure-tovoltage energy conversion efficiency of large ions is considered. It is found that significantly higher efficiencies (>50%) can, in principle, be achieved in no-slip channels for monovalent ions whose diameter is ∼5−15% that of the slit height. In that case, ions produce layering at the walls that can extend into the middle of the channel, producing peaks in the concentration profile far from the walls where the ions’ pressure-driven velocity is high. This significantly increases the streaming conductance and efficiency. Two cases are considered. When the slit is narrow (e.g., 10 nm) with highly charged walls (e.g., 0.1−0.5 C/m2), there are molar concentrations of the counterion throughout the channel, even at the low bath concentrations that produce co-ion exclusion. When the slit is wider (e.g., 100 nm), high efficiency can be achieved even at low wall surface charges (e.g., 0.005−0.01 C/m2) because at 5−15% of the slit height the absolute size of the ions induces layering and high ion concentrations far from the walls. These results are computed using density functional theory (DFT) of ions (not to be confused with quantum mechanical DFT of electron orbitals), which has already been shown to reproduce nanoslit experiments with layering ions.14 While DFT is designed to be thermodynamically self-consistent and it reproduces Monte Carlo simulations of the same system, approximations must necessarily be made to compute ion current through a nanoslit (e.g., use of Navier−Stokes, location of the Stern layer, ions modeled as charged, hard spheres). However, the general mechanism for increasing energy conversion efficiency described here does not depend on the specifics of the model used here because layering of large ions near a highly charged surface is a well-established and wellstudied phenomenon.15−24 This article then provides a blueprint for one way to increase conversion efficiency: large (>5% of slit height), slow (small diffusion coefficient, which is related to the large size), monovalent counterions in ultrapure water and a nanochannel that is either narrow with high surface charge or wide with low surface charge. Pressure-to-voltage energy conversion efficiency can be computed using an equivalent circuit argument described by van der Hayden et al.,8 which is summarized in the Appendix. 2 /qchgch is maximized. To Efficiency is maximized when Sstr achieve this, first the electrical conductance gch should be minimized. One way is to choose ions with a small diffusion coefficient (see eq 21). In this study all counterions are given a
Figure 2. Ion layering in a 10 nm slit. Counterion concentration profiles across the slit (y-direction) are shown for ion diameters ranging from 0.3 to 1.4 nm. The surface charge on the slit walls is −0.3 C/m2, the slit height is 10 nm, the counterion concentration is 10 μM, its diffusion coefficient is 0.5 m2/s, and the co-ion is Cl−. The concentrations are cut off at 2 M so that the secondary peaks are visible. However, this scale obscures the fact that the pore is unipolar.
on each wall. For small ions, there is a monotonic decay of the density profile from the walls, but as the ion size increases, a concentration peak appears in the diffuse layer. For even larger ions, the multiple peaks appear and, when the double layers overlap, even the middle of the channel has peaks of molar concentration even though the bath concentration of the ions is only 10 μM. Another factor that improves efficiency is the exclusion of co-ions at this low concentration. This translates into very large energy conversion efficiencies, as shown in Figure 3 where the efficiency of a 10 nm nanochannel is shown as both ion concentration and surface charge vary. A small 0.3 nm diameter cation is considered in Figure 3A. This is similar to the case considered by van der Hayden et al.8 and the DFT results are consistent with those: the efficiency peaks at ∼17% for low ion concentrations and low surface charges. However, when the ion diameter is increased to 0.9 nm (Figure 3B), not only are the efficiencies significantly larger, but they now increase monotonically with surface charge (reaching a maximum of 36%). 1411
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Figure 3. Energy conversion efficiency for (A) 0.3 nm and (B) 0.9 nm diameter monovalent cations as ion concentration and surface charge are varied. The slit height is 10 nm, the counterion diffusion coefficient is 0.5 m2/s, and the co-ion is Cl−. The Stern layer height HS is the diameter of the counterion (0.3 or 0.9 nm). The color scales are the same in both panels.
Specifically, the range was from HS = 0 to HS being the diameter of the large counterion. In Figure 3, the large counterion diameter was chosen because this tended to produce the smaller of the two efficiencies. It is possible that ions could be immobile beyond the diameter of the counterion (which would reduce the efficiency compared to the large HS) or, at the other extreme, hydrodynamic slip can occur at the solid−liquid interface. Slip is not considered here because it can, by itself, increase conversion efficiency11−13 and the point of this paper is to describe another, separate mechanism. Atomistic simulations are required to know the exact details of what might happen at a device wall, but what is shown in the figures covers a large range of possibilities. Figure 4 shows that it may be possible to achieve very high efficiencies (e.g., 50−70%). Realizing such high efficiencies in a real device, however, presents some technical challenges. These include the fabrication of such a small channel with very high surface charge. However, high surface charges are not absolutely necessary to achieve layering and high conversion efficiencies. To have ion layering in a low-surface charge slit requires large ions, larger than those considered so far, because absolute size promotes ion layering.15−21 It is then possible to achieve 30−45% conversion efficiencies, as shown in Figure 5. The monovalent ions must be >∼5 nm in diameter to achieve substantial layering for the −0.005 and −0.01 C/m2 surface charge used here (cyan and black, respectively, in Figure 5). While the maximum efficiency of this low-surface charge device is less than that of the high-surface charge device of Figure 4, it has several practical advantages. First, it allows the use of wider slits, which are currently available; the slit height is 100 nm in Figure 5 (compared to only 10 nm in Figure 4). Second, low surface charge nanochannels are common, as opposed to the very highly charged ones shown in Figure 4. Third, monovalent nanoparticles of this size are available,26−33 as described in more detail below. These results show that ion layering at the wall/liquid interface can be exploited to promote high counterion concentration far from the interface where the pressure-driven velocity is high, producing high efficiencies. The blueprint
Figure 4. Energy conversion efficiency as a function wall surface charge for ions of diameter (in nm) 0.3 (magenta), 0.6 (green), 0.9 (gray), 1.2 (red), and 1.5 (blue). The surface charge on the slit walls is negative, the slit height is 10 nm, the counterion concentration is 100 mM, its diffusion coefficient is 0.5 m2/s, and the co-ion is Cl−. The range of efficiencies for each surface charge is bracketed by choosing the Stern layer height to be 0 or the diameter of the counterion. The blue line does not go to −0.5 C/m2 surface charge because this problem is numerically quite challenging; with very large ions and very high surface charge even small changes in the concentration profile produce large changes in the excess chemical potential profiles. The current solution algorithm does not converge for |σ| ≥ 0.3 C/m2 for the 1.5 nm diameter counterion.
To see how efficiencies change with ion diameter, five different ion diameters are considered in Figure 4 as surface charge is varied. (The slit height is still 10 nm.) Efficiencies in excess of 50% can be achieved for large ions and high surface charges. In this and subsequent figures, a range of efficiencies is shown for each surface charge and ion diameter using a range of Stern layer heights. It is not clear what the properties of densely packed layers of counterions near a highly charged surface are, making the Stern layer height (HS) an open parameter in the Navier−Stokes approach. Therefore, a wide range of HS is shown in the figures to cover many possible choices. 1412
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Commercially available monovalent cations for the narrow slits include triethanolamine, triethylamine, diethylethanolamine, and ethyldiethanolamine, which all have diameters ∼0.7 nm.34 Larger ions with >10 nm diameter can be synthesized, for example, as monovalent quantum dots26−28 or nanoparticles fabricated from proteins.29−33 Because the amino acid building blocks can be prescribed in the fabrication process, peptides with one positive charge from a lysine or arginine can be made and placed on the surface of the molecule. The purity of these ions and the water is also an issue. For example, when the surface charge is very high, a large electrostatic potential develops and attracts smaller counterions, even those at trace concentrations, to better neutralize the surface charge. Even standard “deionized” water has micromolar concentrations of Ca2+ and other ions, so care must be taken to chelate these or use ultrapure water with part-pertrillion ion concentrations.35 Hydronium ions can interfere as well, but calculations (not shown) indicate that these are less of a problem because of their single charge (versus the double charge of Ca2+). Also, high pH can be used. Such problems can also be mitigated by using counterions with their charge on the surface (as many protein nanoparticles already have29−33), rather than the ion center (like those used in this paper). That patch of charge can then interact directly with the surface charge to exclude the Ca2+ and hydronium ions. Despite these technical hurdles, the principle of ion layering upon which the high efficiency is based is robust and wellestablished. Because of that, it seems that efficient pressure-tovoltage energy conversion with nanofluidic channels is possible. Moreover, the same mechanism might also improve the efficiency of charge-selective membranes in other contexts, like batteries or fuel cells.36−38
Figure 5. Energy conversion efficiency as a function of counterion diameter for a 100 nm high channel with −0.005 C/m2 surface charge (cyan) and with −0.01 C/m2 (black). The counterion concentration is 10 μM, its diffusion coefficient is 0.5 m2/s, and the co-ion is Cl−. The range of efficiencies for each surface charge is bracketed by choosing the Stern layer height to be 0 or the diameter of the counterion.
outlined here calls for monovalent counterions whose diameter is >5% of the slit height. These ions should have as small a diffusion coefficient as possible to minimize their conductivity. The nanochannels should either be very narrow (∼10 nm high) and very highly charged (>∼0.15 C/m2) or wider (∼100 nm high) and weakly charged (∼0.005 C/m2). In this layering scheme, monovalent ions are needed because divalents do not tend to layer well; higher surface charges are needed. Their size should be as large a fraction of the slit height as possible because the concentration peak of the second ion layer occurs at approximately three ion radii. Therefore, the larger the ion, the farther this peak extends into the pressuredriven velocity profile (which is solely a function of the slit height and Stern layer height by eq 12), maximizing the streaming conductance. The concentration of the ions can be low or high, depending on the system. For the narrow, highly charged channel, ion concentration is not very important because the slit height is on the order of the Debye length, even for the 100 mM concentration used in Figures 4. For the wide, low-surface charge channel, ion concentrations must be low so that the double layers extend across the slit and expel co-ions; 0.1 mM was used in Figure 5. The calculations presented here are in a slit geometry, but this may not be the geometry that best promotes ion layering. The cylindrical geometry is probably better suited for this. In cylinders, the curvature makes it more difficult for large ions to fill in the first layer of ions at the wall; on a large plane, the ions can move laterally in two directions to make room for more ions, but this is no longer possible in the circular cross-section of a cylinder. This, then, promotes a second layer of ions. A more careful analysis is required to understand how a cylindrical geometry affects energy conversion efficiency, but it may be better to build energy conversion devices using (nano)porous materials rather than nanoslits. Technical hurdles remain to be overcome before high efficiencies can be realized. While the wide low-surface charge channels are relatively common, narrow high-surface channels are not, so their use will only be in the future. Moreover, the ions must be monovalent and, for the wide 100 nm channel, approaching macromolecule size. Such large ions are available commercially or have been described in the literature.
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APPENDIX
Energy conversion efficiency is described here with the same formalism used by van der Hayden et al.,8 except for the use of DFT of fluids instead of PB. Briefly, if the ionic current I and the fluid volume flow rate Q are in the linear response regime for applied pressure p and applied voltage V, then I=
dI dI p+ V ≡ Sstrp + gchV dp dV
(1)
Q=
dQ dQ p+ V ≡ qchp + SstrV dp dV
(2)
where the Onsager relation dQ/dV = dI/dp = Sstr for the reciprocity between electrically induced fluid flows and flowinduced currents is used.8 When connected to an electrical resistor (with resistance Rload) in a circuit, the conversion efficiency εload is defined in this paper to be the ratio of the output electrical power to the input pumping power. Combining this gives
ε load ≡
V2 R load
Qp
=
αr (1 + r )(1 + r(1 − α))
(3)
2 where α = Sstr /qchgch and r = gchRload is the ratio of load and channel electrical resistances. Efficiency then depends on the
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v(x) = (u(y), 0, 0) with the pressure P = p(L − x)/L, the Navier−Stokes equation reduces to
load resistance. Maximizing εload with respect to Rload gives the maximum efficiency εmax =
α α + 2( 1 − α + 1 − α)
η
(4)
dy
for r = 1/√(1 − α). Throughout, εmax is referred to as the energy conversion efficiency. Note that other definitions of conversion efficiency exist in the literature (e.g., by Osterle1). The nanofluidic device (depicted in Figure 1) considered here has a single-slit geometry where ions move along the length of the channel in the x-direction between x = 0 to x = L and electrical double layers are created by charged slit walls with surface charge σ at y = 0 and y = H. Both the length L and the width W (in the z-direction) are macroscopic compared with the nanoscale slit height H and it is assumed that the ionic concentrations do not change in the x and z directions. The Stern layer where the ion velocities at 0 are at y = HS and y = H − HS. The current has both conductivity and convective components. For the conductivity, drift-diffusion (Nernst− Planck) is assumed. In 3D, the flux density (flux per area) Ji for ion species i is given by 1 Ji = − Diρi(x)∇μi(x) kT
u(HS) = u(H − HS) = 0
p (y − HS)(H − HS − y) 2ηL p ≡ uP̃ (y) ηL
uP(y) =
−εε0
VW ≡ L
∑ zi2Dicĩ i
(13)
dψ dψ ( +∞) = 0 ( −∞) = dx dx
(14)
and where δ is the Dirac delta-function, ε is the dielectric constant, and ε0 is the permittivity of free space. For HS ≤ y ≤ H − HS, this equation gives
η
d2u V
V d2ψ = εε 0 L dy 2 dy 2
(15)
so that u V (y ) =
εε0V (ψ(y) − ψ(HS)) ηL
≡
V uV ̃ (y ) ηL
(16)
The current then is ρi(y)dy
I = Icond + W
H − HS
∫H
S
ρ(y)(uP(y) + u V (y))dy
(17)
and (7)
Sstr = dIcond W = dV L
= ρ(y) + σ(δ(y) + δ(H − y))
dy 2
with boundary conditions
Then, gcond ≡
d2ψ
∫
S
(12)
For uV, eq 9 can be transformed through the Poisson equation
H − HS x Ji (y)dy HS
∫H
(11)
Splitting the velocity into pressure- and voltage-driven components (uP and uV, respectively) gives that
where μiex(x) is the excess chemical potential, zi is the valence of species i, and e is the fundamental charge. The last term is the mean electrostatic potential with the first part from the voltage V applied down the length of the device and the second part from the double layers across the slit. With the assumptions that the ion concentrations do not change in the x and z directions (i.e., ρi(x) = ρi(y) so that also μiex(x) = μiex(y) and ψ(x) = ψ(y)) and that the ions are in equilibrium in the y and z directions (i.e., ∂μi/∂y = ∂μi/∂z ≡ 0), the conductive current is
H − HS
(10)
with the boundary conditions
(5)
∑ zi2Di i
(9)
i
(6)
HW e 2 =V L kT
p V − ρ(y) L L
ρ(y) = e ∑ z iρi(y)
⎛V ⎞ μi(x) = ln(ρi(x)) + μiex (x) + z ie⎜ (L − x) + ψ(x)⎟ ⎝L ⎠
i
=−
2
where η is the viscosity and
where k is the Boltzmann constant, T is the absolute temperature, Di is the diffusion coefficient, and ρi(x) is the ion density profile. The electrochemical potential profile μi(x) is
Icond = eHW ∑ z i
d2u
∑ zi2Dicĩ i
W ηL
H − HS
∫H
S
W gch = gcond + ηL
(8)
The convective component of the current assumes that the ion velocity profile v(x) is described by the steady-state Navier−Stokes equation. With the previous assumptions and
qch = 1414
W p
H − HS
∫H
S
ρ(y)uP̃ (y)dy H − HS
∫H
S
ρ(y)u V ̃ (y)dy
uP(y)dy =
W (H − 2HS)3 12ηL
(18)
(19)
(20)
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Therefore,
with weight functions
H − HS ρ(y)uP̃ (y)dy)2 HS α= H − HS H − HS ρ(y)u V (η ∑i z i2Dicĩ + ∫ uP̃ (y)dy) ̃ (y)dy)(∫ HS HS
ω(2) i (r ) = 2πR i
(∫
2 2 ω(3) i (r ) = π(R i − r )
(21)
ω(5) i (r ) = 2πr
which shows that εmax is independent of both W and L and that it depends on the product of the diffusion coefficients and the viscosity.8 It also shows that the smaller the diffusion coefficients and ion valences, the larger the conversion efficiency. The last part of the theory is how the ion concentration and mean electrostatic potential profiles (ρ i (y) and ψ(y), respectively) are computed. The model of the ions used here is charged, hard spheres with water a background dielectric ε (the primitive model of ions). The excess chemical potential is then not zero, as assumed in PB. Here, the excess chemical potential is calculated with DFT of fluids.39−43 DFT minimizes the free energy of the system through a variational principle that is derived from thermodynamic principles and therefore reproduces the ion and electrostatic potential profiles of Monte Carlo simulations of the same systems. Importantly, the DFT used here44−46 has been extensively tested and reproduces these profiles for highly charged walls with mono-, di-, and trivalent ions (including mixtures).47,48 The key to this success is that DFT does not use the local density approximation (LDA) that neglects short-range correlations. Because of that, LDA theories necessarily fail to satisfy important thermodynamic sum rules (thermodynamic self-consistency checks).41 DFT computes these sum rules well47 and reproduces high-quality equations of state for the bulk state.39,43,49 Here a simplified sketch of DFT for charged, hard spheres with radius Ri for ion species i is presented. The excess chemical potential has two terms, one for the hard-sphere (HS) core and one for short-range screening (SC) electrostatic correlations44,45 μiex (y) = μiHS(y) + μSC i (y )
(1) (2) 4πR i2 ω(0) i (r ) = 4πR i ωi (r ) = ωi (r ) (5) 4πR i ω(4) i (r ) = ωi (r )
The nα are then local averages of the density profiles. This makes the DFT a nonlocal theory (i.e., not using the LDA). Different hard-sphere functionals differ in their choice of ΦHS. In this paper we use the “antisymmetrized” version of Rosenfeld et al.49 n n − n5n6 ΦHS({nα}) = −n0 ln(1 − n3) + 1 2 1 − n3 +
μiHS(y) = kT
μSC i (y) = Mi({ ρ̅ k (y)}) −
α= 0
i
∂nα
i
y+Ri
y−Ri
y + (R i + R j)
y − (R i + R j)
wij(y − y′) (27)
where Mi and wij are, respectively, the excess chemical potential and second-order direct correlation function of the reference fluid for species i.44 The “reference fluid density” functional is used in this paper to compute the functions ρ̅i(y′).44,45 It has been extensively tested against Monte Carlo simulations and shown it to be extremely accurate.47,48 Moreover, it has recently been shown to reproduce experiments of charge inversion seen in pressure-driven flow for 0.9 nm-diameter trivalent ions, a situation in which ion layering occurs.14
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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(y′)ω(i α)(y − y′)dy′
ACKNOWLEDGMENTS This material is based upon work supported by, or in part by, the U.S. Army Research Laboratory and the U.S. Army Research Office under Contract No. W911NF-09-1-0488. I also want to thank Professor Sumita Pennathur (University of California, Santa Barbara) for many long discussions about nanofluidics and helpful comments on the manuscript.
where ΦHS({nα}) is a function of the locally averaged concentrations
∑∫
(26)
[ρj(y′) − ρi̅ (y′)]dy′
(23)
n α (y ) =
∑∫ j
(22)
y + R i ∂ΦHS
∑∫ y−R
3 ⎛ n2 ⎞ ⎜⎜1 − 6 ⎟⎟ 24π(1 − n3)2 ⎝ n22 ⎠
n23
Coulombic interactions can also be handled accurately in DFT, usually based on a perturbation about a known reference fluid with concentrations {ρ̅k(y)}.43,50 Not all the formulas are shown because they are long and do not give additional physical insight into the DFT. The general form for the shortrange screening component of the excess chemical potential is
The long-range electrostatics are already included in ψ(y). The HS component is based on Rosenfeld’s Fundamental Measure Theory.39,42 In the planar geometry, the hard-sphere component of the excess chemical potential is given by 5
(25)
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ρi(y′)ω(i α)(y′ − y)dy′ (24)
REFERENCES
(1) Osterle, J. F. J. Appl. Mech. 1964, 31, 161.
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dx.doi.org/10.1021/nl204087f | Nano Lett. 2012, 12, 1410−1416
