Electrokinetic Energy Conversion by Microchannel ... - ACS Publications

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Electrokinetic Energy Conversion by Microchannel Array: Electrical Analogy, Experiments, and Electrode Polarization Abraham Mansouri,† Subir Bhattacharjee,‡ and Larry W. Kostiuk*,§ †

Department of Mechanical Engineering, American University in Dubai, Dubai, United Arab Emirates 28282 Water Planet Engineering, 721 South Glasgow Avenue, Inglewood, California 90301, United States § Department of Mechanical Engineering, University of Alberta Edmonton, Alberta T2G 2G8, Canada ‡

ABSTRACT: This paper takes a system-wide perspective of electrokinetic energy conversion devices based on an array of microchannels to help understanding their operation. The approach taken was a combination of developing an electrical analogy and conducting experiments. The electrical analogy included current sources for the convection current, resistors for the conduction current, a capacitor for accumulating the partitioned ions, resistors for ion transport in the reservoirs, diodes and capacitors for the electrochemistry and polarization at the electrodes, and a simple external resistive load. The number of parallel channels profoundly affected the summative resistive and capacitive characteristics of the array, and highlights the differences between a single channel and an array of channels, especially in the transient responses and the role of the electrodes. The electrical analogy was solved by Laplace Transforms to demonstrate a rich and varied response that such a system exhibits to a step change in flow in relation to relative magnitudes of the various resistors and capacitors. Experiments were conducted on a structured glass microchannel array with approximately three million channels (10 μm diameter pore size) with aqueous KCl as the working fluid and tested a variety of electrodes. Besides providing data for in situ resistances and capacitances, in particular for the electrodes, keys aspects of the experimental results were interpreted using the electrical analogy. Results include the potential challenges in interpretation of externally measured potentials and currents as streaming potentials and streaming currents, respectively, measuring the resistances and capacitances of electrodes by novel methodologies, and using the electrical analogy quantitatively to explore maximizing electrokinetic energy conversion in the steady state.

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INTRODUCTION Electrokinetic energy conversion devices, i.e., systems that convert hydrostatic potential energy to electrical power by tapping into phenomena related to streaming current (SC) and streaming potential (SP) at the micro- or nanoscales, have received increasing attention in recent years.1−5 The vast majority of these studies have been performed on single channels where theoretical and experimental results were compared and recommendations given as to how to build such energy conversion devices to be more efficient. It is generally assumed that the results associated with single channel studies can be readily adapted to situations involving an array of parallel channels. This parallel geometry would result in higher output power and potentially create a “macro” electrokinetic conversion device that would be usable for practical situations.4−6 Within this context, a macro-electrokinetic energy conversion device based on structured porous media was reported where electrode position within the upstream and downstream reservoirs, and electrode polarization became key aspects of design optimization.7 Recently, Chang et al., advanced the understanding of one of these issues by developing a more © 2014 American Chemical Society

comprehensive theoretical model for the effects of the resistivity of the fluid in the reservoirs for a micrometer-long nanochannel in an electrokinetic energy conversion systems.8 To the best of the authors’ knowledge, the theory or modeling of electrokinetic energy conversion devices has yet to include the effects of the electrodes and their polarization. This polarization phenomenon appears to be unavoidable in highpower electrokinetic energy conversion devices, such as microor nanochannel arrays, due to the relatively high flux of electrochemical reactions occurring at the electrodes.7,9 Also unexplored, and theoretically cumbersome, are the effects of the external load when modeling electrokinetic energy conversion devices. If based rigorously on governing principles, a model that captures all this physics would involve the significant complexities associated with solving a coupled set of partial differential equations, i.e., Navier−Stokes (incompressible fluid flow), Poisson−Nernst−Planck (ion transport), Butler−VolmReceived: August 1, 2014 Revised: September 25, 2014 Published: September 26, 2014 24310

dx.doi.org/10.1021/jp507790y | J. Phys. Chem. C 2014, 118, 24310−24324

The Journal of Physical Chemistry C

Article

Figure 1. Schematic of a microchannel array that separates two reservoirs. “L” and “R” represent the left and right-hand extremes of the channels, respectively, of length L, and electrodes are placed at a distance of xe from the channel ends and connected through an external load.

aqueous working fluids (i.e., altering the salt concentration), electrodes (i.e., altering their materials and sizes, as well as their locations within the reservoirs), and the type of measurement (i.e., trans-capillary potential and external current in either steady-state or transient conditions). Finally, experimental results are presented and interpreted with the aid of the electrical analogy for the general case of energy conversion, as well as the specific cases of the external circuit being either zero or infinite resistance.

er (kinetics of electrons exchange at the surface of electrodes), and Kirchoff’s Laws (external circuit). This brute-force approach to modeling, especially if solved for a transient case, would likely afford limited physical understanding to help interpretation of experimental data, and provide little insight into design options to optimize performance. An alternative approach, which simplifies the physics while providing a means to visualize the basic processes, could be to develop an electrical analogy to model the transient interactions between the electrokinetic flow, electrodes and external circuit.7,9 A byproduct of developing such an electrical analogy would be to help interpret experimental data to characterize interfacial properties, such as the ζ-potential. Interest in this quantity comes from the analysis of the electric and compositional fields in a quiescent fluid reservoir that is in contact with a solid surface, as well as SC and SP phenomena resulting from pressure-driven flows.10,15−22 The models that relate SC and SP to the ζ-potential are historically based on single, infinite long, channels under steady conditions, which may not be appropriate for an array of channels, and therefore have implications for estimates of the ζ-potential in porous media or membranes. Hence, the primary objective of this paper is to propose and validate an electrical analogy of an electrokinetic system for the purpose of energy conversion with flow through an array of finite length microchannels between two reservoirs. Besides the channel flow, the model will include the physical processes occurring at the electrodes (i.e., electrolysis and polarization) placed within the reservoirs and that associated with the external electric circuit that connects the electrodes together. A complementary objective of this paper is to exploit this electrical analogy to provide a more sound interpretation of attempts to measure the SC or SP for microchannel arrays by having an external circuit of either zero or infinite resistance, respectively. The paper is organized as follows: In the first section, the electrical analogy is developed and then analyzed by a Laplace Transform method to provide a framework for understanding such systems. In the subsequent section, an experimental apparatus is described that is used to collect data for different

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ELECTRICAL ANALOGY OF A MICROCHANNEL ARRAY In order to create a robust electrical analogy for comparison with experimental results, it is necessary to define the problem carefully. Figure 1 shows the problem geometry as an array of parallel straight finite-length microchannels of constant crosssectional area that separate two semi-infinite reservoirs. The working fluid is an electrically neutral bulk solvent containing a multitude of ionic species as solutes. The properties of the substrate material that make up the channels and fluid are such that in a no-flow state an electrical double layer (EDL) develops at their interface. The characteristic cross-stream dimension of the microchannel falls into the regime that develops a trans-capillary potential under pressure-driven flow. The nature of the flow through the microchannels being considered could be steady, transient or periodic and include the possibility to reverse flow. Lastly, electrodes are placed in the two reservoirs to facilitate electrokinetic energy conversion, as well as to simulate typical experimental data collection systems for measuring either the electrical potential difference across the electrodes or the current flow through an external circuit. Initially, a model of such a system is developed for a single microchannel connecting the reservoirs, and then expanded to consider an array of independent parallel microchannels that share common end conditions, geometry, and interfacial properties. Modeling the Flow in a Microchannel as a Current Source. In developing an electrical analogy for microchannel flow, it is helpful to first consider the mechanistic aspects of transitioning from no-flow to flow for the transport of the 24311



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solvent and solutes without consideration of the electrodes. When a pressure difference is applied across the two reservoirs, flow is initiated. The bulk fluid in the upstream reservoir, which is electrically neutral and uniform in composition, begins to move toward the channel inlet. As the various positively and negatively charged solutes approach the channel inlet, their relative motion is affected by interacting with the channel’s EDL, while the solvent’s motion is essentially unaffected. This partitioning of co- and counterions results in an imbalance in the flow rate of negative and positive ions into the channel. The convection of the bulk flow in the channel moves the solvent and charged solutes along the length of the microchannel, and thereby establishes a current associated with each solute species. As a result, the analogous component to the hydrodynamic convection of the various ions takes on the form of a current source. If the flow were from left to right, the flux of positively charged solutes would be current in that direction, while the transport of negatively charged solutes would be current in the opposite direction. This convection current (often referred to in the literature as the streaming current, although here that term is reserved for the special case of steady-state convection current when there is no potential difference between the two reservoirs), in amperes, associated with the mth chemical species in the jth channel of the array (Ijconv,m) can be calculated by integrating the local bulk velocity (v) and species’ molar density (nm) weighted by the valence of the species (zm) over the cross-sectional area of one of the microchannels (A), such that j Iconv, m =

zm F

∫A nmv dA

j Iconv =

j ∑ Iconv, m m=1

(2)

For the specific case of M = 2 and the valence on these two solutes being symmetric and unity, a model previously proposed for the convection current through a single channel with a uniform cross-sectional area was given by11 j = Iconv

−εζ ΔpA f (κa ) μ L

(3)

where ε, μ, ζ, Δp, and L are the permittivity and viscosity of the fluid, ζ-potential, pressure difference across the channel, and channel length, respectively, while f(κa) is a function of the inverse Debye length (κ) and channel radius (a). In the limit of κa ≫ 1, f(κa) = 1 and Smoluchowski’s equation is recovered. Accumulation of Charge and Conduction Current. The imbalance in the convection of charged species through the microchannel results in the relative depleting of counterion solutes near the channel entrance and their subsequent accumulation near the channel exit (the opposite being true for the co-ion solutes). This changing spatial distribution of net free charge density, relative to the no-flow situation, manifests itself into an evolving electrical potential field with a magnitude determined by the integration of the Poisson−Boltzmann equation from the upstream reservoir (usually defined as ground). If the integration is performed all the way to the downstream reservoir, the outcome is the trans-capillary potential difference. It is important to note that once removed from the vicinity of the channel’s inlet and outlet, the compositions in the reservoirs remain uniform, so there is no further contribution to this integration and the bulk of the reservoirs are also uniform in potential. As the concentrations of the various solutes and the electrical potential become nonuniform across the length of the microchannel, each of the solute species experiences its own diffusion and migration transport, which completes the Nernst−Plank perspective of the channel flows. The direction of diffusion and migration flux of each solute is opposite to their net convective flux. Given the relative magnitude of concentration and electrical gradients, along with their respective diffusion coefficients, there is evidence to suggest that the diffusion component can be neglected in favor of a simplified model that only includes migration of charged species.12 Since the migration of each solute is driven by the electric field, this process is collectively referred to as the conduction current and is typically split into two parallel paths. One path is associated solely with transport within the microchannel’s fluid, while the other path is associated with the charge movement along the solid−fluid interface.23 In an electrical analogy, the relative accumulation or depletion of charge of the various solute species, which are separated by the length of the channel, is represented by a capacitor. As the charge difference develops across the analogous capacitor, the trans-capillary electrical potential difference is created. To capture the migration of current, which is linearly proportional to this trans-capillary potential difference, resistive elements for each species are added to the analogy. Including these capacitive and resistive elements into the electrical analogy results in the schematic shown in Figure 3. A model for the analogous capacitor has been previously proposed and was based on the geometric similarities between

(1)

where F is Faraday’s constant. Figure 2 schematically shows the concurrent convective currents of all M solute species, which when summed together creates the net convection current in that channel:

Figure 2. Electrical analogy of the convective currents associated with M different charged solutes being transported by the bulk flow in the jth channel of a microchannel array. 24312



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R S, i =

a parallel plate capacitor and the observation that the charge separation in the flow occurs across channels with parallel ends.12 In that case, the capacitance for a single circular microchannel is solely a function of geometry and permittivity. while the hydrodynamics, electrostatics, ζ-potential and mass transfer have essentially negligible effects. The functional form of the capacitance for the jth channel was given as

εA L

(4)

where KC is a constant that has been shown to be ∼3. Since each solute species has its own mobility, their transport is treated separately. The model for the resistive elements of a channel for each solute species in the dilute limit (i.e., no interaction between the transport of different solutes) is given by 12

R C,j m =

RR, m =

(5)

where RF,m and RS,m are the resistances associated with the individual species transport through the bulk of the fluid in the channel and that along the solid−fluid interface, respectively, and given by RF, m =

L AσF, m

Bxe AσF, m

(8)

where B is the blockage ratio of the microchannel array. This model assumes this migration is essentially one-dimensional, which creates the image that the size of the electrodes is of the same magnitude as the microchannel array and that edge effects are negligible. As depicted in Figure 1, a symmetric spacing of the electrode was chosen, so that xe is the same on both sides, but this did not have to be the case. The physical processes that need to be modeled in the electrical analogy involve the accumulation of each solute next to the electrode surface and the possibility of a threshold potential difference to exist between the electrode and the surrounding fluid before various solutes can react. The

RF, m·R S, m RF, m + R S, m

(7)

where σF,m is the conductivity of the mth species within the bulk fluid, P is the perimeter of the channel, and σS,m is the surface conductance of the mth species on the fluid−solid interface. An important feature of the model shown in Figure 3 is that all the various solute current sources and resistors for the conduction current, as well as the capacitor, are connected together between two common nodes (L and R). That is, a singular electric field, created by all the species, is overlaid onto each species to cause their migration. A physical consequence of this electrical connection is that there is no species-byspecies balance between their individual convection, accumulation and conduction because the electric field is shared. Given the multispecies nature of the charge carriers in the flow, the current produced from one species’ convection and its accumulation will be responded to by the conduction back through another species’ resistor. The source or sink of any imbalance in each solute becomes the reservoirs. Consequently, this model facilitates the possibility that the two reservoirs, if not infinite, can evolve to having different species concentrations. Support for this observation has been observed in the filtration of multicomponent salts.11 Electrodes and External Load to Produce Electric Power. In the previous subsections, an analogy was developed that models a trans-capillary potential difference and the internal currents; the next elements to consider are those needed to tap into this potential and use the electrical power externally. As shown in Figure 1, a pair of electrodes is located in the bulk of the reservoirs, and they are connected through an external finite resistive load (though in principle, any type of resistive-capacitive-inductive load could be used). The physical processes invoked in this situation are the transport of the various charged species either toward or away from an electrode and the electrochemistry at the electrode surface to support the flow of electrons in the external circuit. The driving force for the transport of each solute relative to the solvent is the electric potential differences between what was established near the ends of channel and the electrode surfaces. Any solute that migrates to be in the vicinity of the electrode, but does not readily react, will contribute to the effective polarization of the electrode and thereby affect the rates of migration of every species because all solutes share the common electric field. The migration of solute across the portion of the reservoir from the end of a single channel to the electrode is modeled similar to the conduction current as a resistor, but given the geometry shown in Figure 1 results in

Figure 3. Electrical analogy for the jth channel of a microchannel array for M solute species where the current sources represent the solute’s convection current, the accumulation/depletion (charge separation) of solutes are represented by a single capacitor, and conduction current of the various solutes are represented by resistors.

CCj = K C

L PσS, m

(6) 24313



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Figure 4. Proposed electrical analogy for a single channel of a microchannel array, electrodes, and an external resistive circuit. The circuit is divided into two parts based on the whether the charge carriers are ions or electrons at the surfaces of the electrodes (between points E1 and E2, and A/C). RR is the bulk electrical resistance for solute migration between the channel ends and the electrodes. Electrodes are modeled as two parallel diodes with opposite polarities and a capacitor in parallel. Rext is the external resistive load where the generated electrical power is used. It is worth noting that the extremes of this external load can represent ideal measurement devices such as a voltmeter (Rext → ∞) or an ammeter (Rext → 0), and will be used later to connect experimental measurements to this proposed model.

circuit is divided into two parts based on the whether the charge carriers are ions or electrons, and the interface is the surface of the electrodes (between points E1 and E2, and A/C). Moving from a Single Channel to an Array of Channels. An array of channels replicates the proposed electrical analogy between the E1 and E2 nodes. Given that all the channels and the migration pathways between channel ends and the electrodes are assumed to be identical, the circuits can all be connected at the respective E1, L, R, and E2 planes. Connecting identical resistors and capacitors in parallel is straightforward:

accumulation of ions near the electrode surfaces is modeled by a capacitive element. With the accumulation of the ions near the surface, an electrical potential difference across the fluidelectrode interface will develop on a time scale associated with species migration. If that potential difference exceeds the threshold voltage difference required for electrochemical reactions, then oxidation will occur at the anode (A) and reduction will occur at the cathode (C). Depending on the sign of the trans-capillary potential difference (which can be changed by changing the flow direction), current can flow in either direction from the electrodes. Therefore, along with the accumulation of charge in the boundary layer of the electrode, the electrical analogy for each solute at one of the electrode is modeled as two parallel diodes with opposite polarities in parallel with the capacitor.13 Unlike the previous elements in the analogy, the characteristics of the diodes and capacitor no longer scale with a particular channel, and therefore are simply designated as DE,m and CE, respectively. The assumption is also made that the two electrodes are the same material and have the same characteristics, but this does not have to be the case. The current density versus overpotential could be modeled by the Butler−Volmer equation for such electrodes. Typically, for a given electrolyte concentration, the current density for nonpolarizable electrode is quite high, while for polarizable electrodes the current densities are lower. Figure 4 is a schematic of the proposed model. It is worth noting that the

R C, i =

R C,j i J

(9)

CC = JCCj

(10)

RR, i =

RR,j i J

(11)

where J is the number of channels in the array. As a result, the electrical analogy is unchanged in structure, just the magnitude of the resistors and capacitors between the E1 and E2 nodes have changed. Given that J can be large (e.g., the experiments presented later has J = 3,437,500) there can be a considerable change in the relative magnitude of these components with respect to each other, as well as compared to the resistive and 24314



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Figure 5. A simplified electrical analogy of a microchannel array electrokinetic power generation system that includes electrodes and an external circuit. Diodes were replaced with resistive elements in the situation where overpotential needed for reactions is small.

Figure 6. Simplified electrical analogy of a microchannel array with one cation and one anion with the same properties (excepted charge) as solutes and symmetric cathode and anode electrodes shown in Laplace space.

the channel, and the other is associated with the electrodes, which also sense the developing potential difference between the reservoirs. The potential difference across the external load will induce motion in the conductor’s free electrons and create a potential difference between the electrodes and the bulk of the fluid in their reservoir. This potential difference will cause the charged solutes to migrate either toward or away from the electrodes. These various migration processes, either through the channel or in the fluid bulk, occur on a much slower time scale than the hydrodynamics. Depending on the effective Peclet Number of the system, the conduction currents and migration through the bulk respond to this growing potential difference. The imbalance in the convective and migration currents continues until one of the threshold potential differences is reached at the electrode surfaces and electrolysis

capacitive characteristics associated with the electrodes. This shifting of relative magnitudes of the various component is the origins of the differences between single channel systems and arrays. At this stage of model development it is worth having a qualitative discussion of the transient response such a system has to a step change in flow from no-flow to its eventually steady state. Initially, the two reservoirs would be in equilibrium with each other, and an EDL would exist at the solid−liquid interfaces of the channel material. With the onset of the flow, the various current sources begin to displace the free-charges, and that distortion of charge creates a trans-capillary potential difference on a hydrodynamic time scale. There are now two ways the charged solutes can respond to this evolving transcapillary potential difference. One is to migrate back through 24315



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with a magnitude equal to that at steady flow (i.e., Iconv(s) = Iconv/s), and the solution in the time domain becomes

begins to support a sustained current through the external load. Since not all the charged solutes attracted to the electrodes will react at the available potential difference, they will just accumulate next to the electrode causing its polarization and screening its potential from other solutes that would migrate in order to react. These processes (i.e., increasing trans-capillary potential due to local depletion/accumulation of charged solutes at the channels ends, increasing migration through the channel, migration to the electrodes, and electrolysis at the electrodes) eventually bring the fluxes into balance and steady state is approached. Simplifying the Electrical Analogy. In order to provide a more quantitative insight into electrokinetic energy conversion, it is worth simplifying the electrical analogy that is shown in Figure 4. One simplification worth considering is to reduce the number of solutes to a single dominant cation and anion pair that will undergo electrochemical reactions, and give these ions the same transport properties so that the channel can be a single current source and a single effective resistor for the conduction current. The second simplification is to assume the existence of nonreacting electrodes with very low overpotentials in the Butler−Volmer equation. With those simplifications, the electrical analogy can be reduced to that shown in Figure 5, but the implications of having the various cations and anions acting independently (especially those that do not participate in electrochemical reactions) should not be forgotten. Circuit Analyses in Laplace Transform Domain. The approach used to analyze the transient and steady state characteristics resulting from the simplified electrical analogy (Figure 5) is to view the circuit in Laplace space. This approach treats all the elements as impedances, which follow Kirchhoff’s current and voltage laws. As a result, sets of impedances in series can be reorder without affecting the behavior of the whole system, and this was used to combine the two electrodes into a single set of elements. Figure 6 shows the final version of the simplified equivalent electrical analogy (that takes advantage of the symmetry assumed of the two electrodes and their placement), which can then be readily solved. It should be noted that the reservoir and external resistances were not combined because, in the Experimental Section, a model ammeter and voltmeter replace only the external resistance, so they must remain separate. Based on Figure 6, the current in the external circuit can be written as Iext(s) = Iconv(s) ·Y (s)

⎡ ⎛ ⎛ t ⎞⎞ ⎛ t ⎞⎤ Iext(t ) = Iconv ⎢A − B exp(−t / τ1)⎜⎜cosh⎜ ⎟⎟⎟ + D sinh⎜ ⎟⎥ ⎢⎣ ⎝ τ2 ⎠⎠ ⎝ τ2 ⎠⎥⎦ ⎝ (17) ⎡ ⎛ ⎛ t ⎞⎤ ⎛ t ⎞⎞ Vext(t ) = IconvR C⎢1 − A + B exp(−t / τ1)⎜⎜cosh⎜ ⎟⎟⎟ + D sinh⎜ ⎟⎥ ⎢⎣ ⎝ τ2 ⎠⎥⎦ ⎝ τ2 ⎠⎠ ⎝

(18)

where

(21)

2α β

(22)

α 2

β /4 − αγ

(23)

(24)

As for Vext(t), this would now represent the potential difference between the reservoirs (i.e., not that between the anode and cathode materials which have reaction occurring), which, if it was to be measured, would require a separate set of electrodes connected by a voltmeter of infinite resistance, such that Vext(t ) = IconvR C[1 − A + E exp(−t / τ)]

β = R CCC(RE,T + RR,T + R ext) + C E,TRE,T

(16)

⎤ ⎡β ⎢⎣ + γRE,TC E,T⎥⎦ β /4 − αγ 2 1

2

Iext(t ) = Iconv[A − E exp(−t / τ)]

(14)

γ = R C + RE,T + RR,T + R ext

(20)

The above expressions provide a general solution to the expected transient response of the system shown in either Figure 5 or 6 when subjected to a step change from no-flow to flow. In the following Experimental Section, some special cases were considered, so it is worth reducing these expressions in order that they align to those experiments. In particular, a unique geometry was created where the gap between the ends of the channel and the electrodes was reduced to essentially zero (i.e., xe → 0, which in turn means RR,T → 0), and these electrode were shorted with an ammeter so that Rext → 0, which allows the above expressions to be simplified. With those resistors becoming negligible, then α → 0 and allows considerable simplification of the above expressions such that the external current is given by

(13)

(15)

RC γ

τ2 =

(12)

(R C + RR,T + R ext)

B=

τ1 =

and the constants are given as α = R CCCRE,TC E,T(RR,T + R ext)

(19)

and τ1 and τ2 are the time constants given by

R C + R CRE,TC E,Ts αs 2 + β s + γ

RC γ

D=

where Y(s) is the Laplace function: Y (s ) =

A=

Remembering that the hydrodynamic time scale associated with the convection current is very short compared to the diffusion time scales, the convection current associated with suddenly turning-on the flow is considered to be a step function 24316

(25)

A=

RC γ

(26)

E=

R CRE,TC E,T RC − γ β

(27)

τ=

RE,TR C β = (CC + C E,T) γ RE,T + R C

(28)



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⎡ ⎤ CC Vext(t ) = IconvR C⎢ exp(−t / τ)⎥ ⎢⎣ CC + C E,T ⎥⎦ CC : Vext(t = 0) = IconvR C : Vext(t → ∞) CC + C E,T

and in the expanded version, eqs 24 and 25 become: ⎤ ⎡ ⎞ ⎛ C E,T RC RC ⎟⎟ exp(−t / τ)⎥ Iext(t ) = Iconv ⎢ + ⎜⎜ − ⎥⎦ ⎢⎣ R C + RE , T R C + RE,T ⎠ ⎝ CC + CeqnE,T

(29) ⎡ ⎛ C E,T ⎞ RC RC ⎟⎟ − ⎜⎜ − Vext(t ) = IconvR C⎢1 − ⎢⎣ R C + RE,T ⎝ CC + C E,T R C + RE,T ⎠ ⎤ exp(−t / τ)⎥ ⎥⎦

= IconvR C τ = R C(CC + C E,T)

⎡ ⎤ CC Iext(t ) = Iconv ⎢1 − exp(−t / τ)⎥ ⎢⎣ ⎥⎦ CC + C E,T C E,T : Iext(t = 0) = Iconv : Iext(t → ∞) = Iconv CC + C E,T (34a)

⎛ ⎞ Cc ⎟⎟ exp(−t / τ) Vext(t ) = IconvR C⎜⎜ + C C ⎝ C E,T ⎠

⎡ ⎤ RE,T RC exp(−t / τ)⎥ ⎢1 + R C + RE,T ⎣ RC ⎦

: Iext(t = 0) = Iconv : Iext(t → ∞) = Iconv

⎛ ⎞ Cc ⎟⎟ : Vext(t → ∞) = 0 : Vext(t = 0) = IconvR C⎜⎜ ⎝ CC + C E,T ⎠

RC R C + RE,T

(34b) (31a)

Vext(t ) = Iconv

R CRE,T R C + RE,T

: Vext(t → ∞) = Iconv

τ=

τ = RE,T(CC + C E,T)

R CRE,T (31b)

RE,TR CC E,T RE,T + R C

(31c)

(Case II) CC ≫ CE,T RC [1 − exp(−t / τ)]: Iext(t = 0) = 0 R C + RE,T RC : Iext(t → ∞) = Iconv R C + RE,T (32a)

Iext(t ) = Iconv

Vext(t ) = Iconv

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EXPERIMENTAL SECTION Materials. The key materials used in the experiments were the glass microchannel array (GMA), the working fluid that was pumped through the channels, and the electrodes used to either measure voltage or to act as surfaces for electrochemical reactions. The GMA was made of lead silicate glass (Burle ElectroOptics, Sturbridge, Massachusetts) and consisted of approximately 3 437 500 straight, circular microchannels with a pore size of 10 μm diameter. The thickness, effective diameter, and porosity were 2 mm, 25 mm, and 45%, respectively. The working fluid for all the experiments was deionized ultrafiltered (DIUF) water obtained from a water purification system (Millipore Simplicity, 18.2 MΩ/cm), and KCl electrolyte solutions were made by adding appropriate amounts of KCl powder to the DIUF water. Electrodes placed in the reservoirs upstream and downstream of the GMA were platinized platinum (i.e., platinum black), silver, or gold. The platinized platinum represents a standard high-quality electrode typically used in these kinds of measurements. Since other investigators have used silver

⎤ R CRE,T ⎡ R ⎢1 + C exp(−t / τ)⎥ ⎥⎦ R C + RE,T ⎢⎣ RE,T

: Vext(t = 0) = IconvR C: Vext(t → ∞) = Iconv

R CRE,T R C + RE,T (32b)

τ=

RE,TR CCC RE,T + R C

(32c)

(Case III) RE,T ≫ RC Iext(t ) = Iconv = Iconv

C E,T CC + C E,T

C E,T CC + C E,T

exp(−t / τ) : Iext(t = 0)

: Iext(t → ) = 0

(34c)

The richness of the possible outcomes of a step change in flow rate can be seen in that the initial external current or initial reservoir voltage difference can either be zero or finite, and these current and voltages can either rise or fall to a steady state value that is either zero or finite. Also, the initial current or voltage can be set by the capacitances in the system and/or the resistance associated with the channels, but, as expected, the steady state quantities are independent of the capacitances. These observations will be used in discussions of the experimental results to identify which case appears to apply, as well as to calculate the magnitude of the effective capacitances of the electrodes from measured transient responses in the external current.

[1 − exp(−t / τ)]: Vext(t = 0) = 0

R C + RE,T

(33c)

(Case IV) RC ≫ RE,T

(30)

Building on these previous assumptions, there are four other special cases worth exploring and these have to do with relative magnitudes of the resistors and capacitors associated with the channels and the electrodes. The magnitude of the resistors and capacitors can be altered by orders of magnitude by changing the geometry, materials and solute concentration used in the system. (Case I) CE,T ≫ CC Iext(t ) = Iconv

(33b)

(33a) 24317



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electrodes,5 they were included in this study to provide some comparative data with respect to electrode material. Also, both faces of the GMA were coated with a 100 nm layer of gold, which were supported by adhesion layers of nichrome. In reference to Figure 1, these gold layers are located at the “L” and “R” planes, while the location of the platinum and silver electrodes are located at “E1” and “E2”. The platinized platinum electrodes had relatively higher surface areas compare to silver electrodes. The particulars of these electrodes are provided in Table 1.

allow a nonconductive hollow rod to be inserted into the reservoir without leaking. The platinum and silver electrodes were mounted on the ends of these rods, so that they could be placed from 1 to 36 mm from the faces of the GMA. A wire was inserted through the inside of each rod, sealed and butt-welded to the electrode to provide an external electrical connection to these electrodes. Separate wires were butt-welded to the gold electrodes and the coated leads passed to the outside of the chambers. All these external leads could then connected through an external resistive load for power generation, or used to measure the voltage or current characteristics across the electrodes. An electrometer (Keithley Instruments, Inc., Model 6517A) was used to measure either the electrical potential or the electrical current. The experimental data were logged on a computer through a Labview (National Instruments) interface. The response times of the electrometer in the current mode (time constant on the order of 0.1 s) and data acquisition system were adequate to capture the behavior of the system, and data acquisition was performed at 10 Hz. Before each experiment, all samples, apparatus, pump and connections were rinsed and washed with deionized water. To avoid any contamination or initial electrode polarization the experimental protocol by Elimelech et al.14 was employed and involved flushing the microchannel array with deionized water in both directions for a period of about 2 min to remove any trapped air bubbles. For each of the electrode’s location, at least six measurements were performed to ensure repeatability of results.

Table 1. Characteristics of Reservoir Electrodes designation

material

mesh type

wires

diameter (mm)

platinuma

45 mesh woven

0.198 mm diameter

25

silver

99.9% Pt gauze from wire (Alfa Asesar, MA, USA) silver gold

not Available not applicable

25

gold

micro mesh deposited

25

a

Prepared by electrodeposition at 50 mV from 2% chloro-platinic acid in 1 M HCl.

Experimental Apparatus. The details of the experimental apparatus have been described elsewhere,7 and only an overview is provided here. The GMA was clamped between two acrylic cylindrical chambers with an internal diameter of 25 mm and length of 36 mm, which acted as reservoirs, and sealed in place with Teflon O-rings. A diaphragm pump (Shurflo Inc., USA) provided a constant flow rate (0.72 L/min) of working fluid. The piping system, consisting of two quarter-turn 3-way valves, was arranged to allow the flow to be switched rapidly (∼0.2 s) in direction through the GMA (i.e., either L to R or R to L). The supply and discharge reservoirs of the working fluid were both open to the atmosphere. A differential pressure transducer (PX26-030DV, OMEGA, Inc., USA) was connected across taps in the acrylic chambers to monitor the pressure difference across the GMA. Each of these chambers also had a 1 mm circular coaxial port at their ends, which was designed to

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RESULTS AND DISCUSSIONS This particular experimental apparatus allowed for altering the flow direction, electrode type, electrode location, and working fluid, as well as a whether the electrodes were connected to a voltmeter, ammeter or finite external resistive load, so there were many experimental opportunities available. A few of these options were explored and their results interpreted qualitatively with the aid of the electrical analogy shown in Figure 5, or quantitatively with eqs 24 and 25.

Figure 7. External currents and streaming potential measurements across the GMA for 1 mM KCl electrolyte solution. The experiments were performed for electrolyte solutions at identical flow rates using either the platinum or silver electrodes. 24318



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Data will first be presented for the relatively simple cases of measuring the steady state trans-capillary potential difference when Rext → ∞ and then the steady state external current when Rext → 0 using the electrodes in the reservoirs. An observation of these steady state measurements was that the external current was highly dependent on electrode material, concentration of salt in the DIUF, and the placement of the electrodes relative to the channel ends. Recognizing the advantage from a power generation perspective of having access to higher external currents, the experiments then focus on extracting current from the gold electrodes (located at the inlet and outlet planes of the channels). In this arrangement, the unsteady external current with Rext → 0 was measured, which left the ability to monitor the potential difference between the reservoirs with the other electrodes. The form of the unsteadiness was done first by following the transient response to a step change in flow from the no-flow case, and then through a series of flow reversals. Those previous experiments allowed estimates to be made of the various resistive and capacitive elements in that analogy. Using those estimates, the relatively complex nature of steady state electrokinetic power generation is described in terms of the magnitude of the external load and the polarization that occurs at the electrodes. Steady-State Trans-Capillary Potential when Rext → ∞ or External Current when Rext → 0. Figure 7a shows the measured trans-capillary potential difference when the external resistance was replaced by a voltmeter (Rext → ∞), while Figure 7b shows the external current when the external resistance was replaced by an ammeter (Rext → 0) employing either the platinum and silver mesh electrodes for 1 mM KCl at identical steady-state flow rates. The trans-capillary potential difference was invariant to electrode placement and electrode material. With reference to Figure 5, the open-circuit created by replacing the external resistance with a voltmeter results in Iext = 0 and the analogous collapses to that shown in Figure 8 since

single microchannel experiencing the same pressure difference as the array. In contrast, the measured external current depends strongly on the electrode location and the type of electrode. With reference to Figure 5 and noting that once the system is in steady state all the capacitors become inactive, the circuit can be simplified and is redrawn in Figure 9. This simplified circuit provides a couple of key insights. First, since all the resistors are in series, it is possible to gauge the relative magnitudes of the resistors associated with the bulk reservoir fluid (RR) and the electrodes (RE), while the channels resistance (RC) remain constant. The external current can be changed by an order of magnitude by changing the placement of the electrodes, so obviously RR cannot, in general, be neglected but does become diminishing small the closer the electrodes are placed to the ends of the channels. (Depositing of gold on the ends of the GMA to act as electrodes was intended to reduce its resistance to approximately zero.) At the highest external currents (i.e., when the electrode were placed 1 mm from the channel ends), the current measured with the platinum electrodes was three times that of the silver electrodes. At lower currents (i.e., when the bulk fluid resistance was large), the reduction associated with using silver electrodes was only 20%. One interpretation of why the silver electrode resistance is higher than the platinum is due its lower surface area. This area ratio between the different electrodes should be the same for all measurement locations, but whether that difference is important in setting the measured external current depends on the other resistances in series. Second, from a macroscopic level, the shorting of the anode and cathode for these measurements look like SC measurements, but inspection of Figure 9 reveals that since current flows through both RR and RE, then the steady-state transcapillary potential difference is nonzero and the conduction current through RC is nonzero. Therefore, these measured external currents cannot be interpreted as the SC. Figure 10 compares the measured external current (Rext → 0) for the 0.04 and 1 mM KCl using the platinum electrodes. The external current for the 1 mM KCl is higher due to a lower bulk reservoir fluid resistance (eq 8), which also explains the divergence in the measured external current as the distance the ends of the channels to the electrodes was increased. Steady-State and Transient Trans-Capillary Potential when Rext → ∞ and RR,T → 0. In the previous section, it was not possible to quantify the magnitude of any of the resistors or capacitors because single resistive elements could not be isolated and capacitive aspects can only be seen during transients. To effectively remove the two reservoir resistances from the problem, the gold electrodes were used to extract current by placing an ammeter across their leads (i.e., Rext → 0 for the gold electrodes), and then the potential difference between the two reservoir could be measured by placing a voltmeter across the leads of the reservoir electrodes (i.e., Rext → ∞ for the platinum electrodes). It is worth noting that since no current was drawn through the reservoir electrodes, then the measured potential was representative of the bulk and not dependent on electrode placement or material. The electrical analogy to this new configuration of the experimental apparatus is shown in Figure 11. In this arrangement, a relationship between RC and RE can easily be developed in terms of the measured steady-state transcapillary potentials when the anode and cathode were either shorted (i.e., ΔVRext→0 when ammeter in Figure 11 is connected)

Figure 8. Resulting electrical analogy for steady state when the external resistive load is replaced by a voltmeter.

all capacitors are not active elements. Without current flowing through the resistors representing the fluid in the reservoirs or the surfaces of the electrodes, all of the potentials on either side of the GMA are the same. It was only through these resistors that the dependencies of electrode placement and material would be introduced, so with their removal these dependencies no longer exist. Furthermore, once steady state is achieved, this trans-capillary potential difference is the SP, and if all the microchannels were identical then this SP is the same as for 24319



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Figure 9. Resulting electrical analogy for steady state when an ammeter replaces the external resistive load.

Figure 10. External current measurements in GMA. The experiments were performed for 0.04 and 1 mM KCl solutions at identical flow field conditions using platinized platinum electrodes.

or held as an open-circuit (i.e., ΔVRext→∞ = SP when the ammeter in Figure 11 is disconnected, such that ⎡ ΔVR ⎤ ext → 0 ⎥ RE,T = 2RE = R C⎢ ⎢⎣ SP − ΔVR ext→ 0 ⎥⎦

were calculated and listed in Table 2. The estimated RE,T values in Table 2 are relatively small due to large surface area of gold electrode in comparison with electrodes normally used with single microchannel experiments. In order to gain insight in the capacitive elements associated with the GMA and the electrodes, it was necessary to conduct transient experiments. Figure 12 shows typical results recorded on the same setup as the steady-state experiments, but now the flow was periodically reversed. The key observations of the current measured by an ammeter placed between the gold electrodes are the sudden spike in current either when the flow

(35)

where RE,T is the total electrode resistance associated with the combined resistances at the anode and cathode surfaces, and RC can be estimated from eq 5. Under the assumption of RS being large compared to RF for the electrolytes of interest, then values for RC for this GMA and RE for the deposited gold electrodes 24320



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Figure 11. Resulting electrical analogy for the general response of the system for a single electrolyte when the gold electrodes were connected through an ammeter (though for some experiments it was disconnected to measure the SP) and the reservoir electrodes were connected through a voltmeter.

which were also been added to Table 2, then it shows that there is 6 orders of magnitude difference between CC and CE,T. A final feature regarding Figure 12 is that as the concentration of solute increases the measured external current approached zero as the system decays to steady state. With reference to eqs 31a in the limit of t → ∞ and focusing on the relative magnitudes of resistances, the steady-state current can be pushed closer to zero by increasing RE,T relative to RC. There are two contributing affects that could lead to this outcome. First, the conductivity of the working fluid had been increased to a point where the channel resistance for conduction current became smaller. Second, the presence of high concentrations of solutes near the electrodes surfaces had increased the effective electrode polarization so that electrode resistance became larger. Using a more extreme solute concentration of 1 mM KCl there was no steady state current recorded, which causes Cases I and III to become the same (i.e., CE,T ≫ CC and RE,T ≫ RC). It is also worth noting that if the external current at t = 0 could be resolved accurately, then it would represent the SC (eq 31a). At the other extreme, Figure 13 shows the external current measurements when 0 M KCl was pumped through the GMA. In this case, using only DIUF as the working fluid, the dominant cation/anion pair for transport switches from K+/Cl− to H+/OH−, so care must be taken in interpreting and comparing these data sets. The external current in this case was initially small and rose to a final steady state value, which would suggestion characteristics closer to either Case II or IV. From Table 2 it is observed that an order of magnitude drop in KCL concentration (0.5 to 0.05 mM) caused a modest change in the ratio of capacitances, but an order of magnitude rise in the ratio of resistance. If the assumption is made that this trend continues down to the concentration of ions in DIUF water

Table 2. Estimations of Resistance and Capacitance of GMA and Gold Electrodes Based on Proposed Electrical Analogya KCl concentration (mM)

RR,T (Ω)

RC (Ω)

RE,T (Ω)

τ (s)

CE,T (μF)

CC (nF)

0.05 0.1 0.5

0 0 0

7185 3133 1102

6666 9375 10555

0.51 0.4 0.25

147 170 251

0.28 0.28 0.28

a RR,T was assumed to be zero since there is no reservoir fluid between the channel exits and the electrode, RC was estimated from eq 6, RE,T was estimated from measurements and eq 35, CC was estimated from eq 4 and 10, τ was estimated from the measured decay in external current following a step change in flow, and CE,T was estimated from eq 29.

was turned on or reversed, followed by the decay in current toward a finite value at steady state. In these cases, the first spike, going from no-flow to flow, was smaller in amplitude than those associated with flow-reversals, and this is attributed to the different initial conditions that would have existed at the time the flow was altered. Prior to the flow being turned on, the electrical and compositional fields would have been set by the EDL, while at the time of the flow reversals these field were in a considerably difference condition. After the first current spike and its subsequent decay, all successive responses to the flow reversals were remarkably repeatable. Given that the measured current spikes at t = 0 then decreases in magnitude to a nonzero steady-state shows that these experiments have characteristics similar to Case I, as described by eqs 31a−31c. This case was defined as having CE,T ≫ CC, and the time constant associated with the decay in current can be used to estimate CE,T. These values for the capacitance of the electrodes have been added to Table 2. If the capacitance of the channels were estimated by eqs 4 and 10, 24321



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Furthermore, the small overshoot in the maximum current shown in Figure 13 may be indicative that this situation does not fall neatly into any of the extreme cases and would require analysis through the more general eqs 17 and 18. The rise and fall in current does not fitted well with a single exponential, and this lack of simple capacitive behavior has been observed by Conde et al. in a single microchannel geometry.18 In electrochemistry, the current that flows to and from electrodes are divided in two categories, Faradaic (deals with the transfer of electrons) and non-Faradaic (deals with the charging of interface). Within the electrical analogy, these aspects are captured by the resistive and capacitive elements associated with the electrodes, respectively. In a given situation, both of these currents can exist concurrently or alone. In our experiments, for electrolyte solutions listed in Table 2, the gold electrode has shown to embody both characteristics. Initially, the system appears Faradaic with initially high currents then followed by a rapid decay as a double layer develops at the interface of the gold electrodes and electrolyte solution (nonFaradaic). In the case of using DIUF water, redox reactions at gold electrodes are extremely fast and electrons are easily transferred across the electrode−electrolyte interface. As a result, in this case it is hard to observe the capacitive behavior of the electrodes. This electrolyte solution maximizes electric double layer thickness and has implication for power output of these devices when used for energy conversion. The thinner the electric double layer thickness is at electrode electrolyte interface (i.e., in high electrolyte concentrations), the larger is the double layer capacitance of NGEs, as evidenced by results presented in Table 2. Our findings are also in excellent agreement with typical double layer capacitance ranging from 10 to 40 μF/cm2. Modeling Steady State Electrokinetic Power Generation with Electrode Polarization. We now have the properties and submodels needed to exploit the electrical analogy as an electrokinetic energy conversion system. The external power (P) developed from such a system can be estimated from eq 17 (i.e., P = I2extRext), which, in steady state (t → ∞), with electrodes located at the channel ends (RR,T → 0), and κa ≫ 1, becomes

Figure 12. Measurement of current by connecting an ammeter across gold electrodes deposited on the faces of the GMA. Electrolyte solutions of 0.05 and 0.1 mM KCl were pumped through the GMA in alternating flow directions.

⎛ εζ ΔpA ⎞2 R C2R ext P=⎜ ⎟ ⎝ μL ⎠ (R C + RE,T + R ext)2

(36)

The effects of polarization in the form of a Faradaic resistance across the electric double layer at the electrode surface on electrokinetic energy conversion is readily seen in the denominator of the above expression. This term only has a negative impact on power generation, but it is worth noting that its importance is diminished as either RC or Rext are made relatively large. For a fixed geometry, RC can be made large by reducing the concentration of the solute, and the in the limit of RC ≫ RE,T, the maximum power generation occurs when RC = Rext. In order to quantitatively examine the impacts of changing the concentration of the solute in the working fluid and the magnitude of the external resistance, certain parameters can be held constant. These fixed parameters were selected to be relevant to the above experimental apparatus, working fluid, and applied pressure difference, where ζ = −0.025 V, Δp = 200 000 Pa, J = 3 437 500, A = 7.85 × 10−11m2, μ = 0.001 Pa·s, L = 0.002 m, and ε = 7.08 × 10−10 F/m, while RC, and RE,T remain

Figure 13. Measured current by gold electrodes across GMA with electrolyte solution of 0 M KCl.

this subsequently pushes the system into a regime where RC ≫ RE,T, which is Case IV. The small initial external current suggests that the relative magnitudes of the capacitances have also changed for this working fluid (i.e., CC ≫ CE,T), which is Case II. In either event, the steady state current would be representative of the SC. 24322



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Figure 14. Output power of electrokinetic energy conversation device in steady state operation predicted by electrical analogy for a range of KCl concentration for a GMA with gold electrodes located at the ends of the channels. As the elctrolyte concentration increase, NGE becomes more polarized and electrical resistance of electrodes increase to the extent that no power can be recorded for a highly concentrated solution.

functions of solute concentration. Figure 14 shows the predicted electrokinetic energy conversion for the KCl solutions listed in Table 2 and for DIUF water. Interestingly, the steady state power generation can be reduced by 3 orders of magnitude by changing the working fluid from 0 mM KCl to 0.5 mM KCl. It has previously been reported that the maximum efficiency of this experimental setup was 1.3% when fresh DIUF water was used and had reduced to 0.002% for 0.1 mM KCl solutions. Exploiting the transient characteristics where the external current is large near t = 0, or nonresistive loads, for power generation has yet to be explored.

to conduction current drops in proportion to the number of channels, while the capacitance of the array increases in proportion to the number of channels. Single channel results should not be extrapolated to arrays of channels, especially in the transient responses and the role of the electrodes when external current is being produced. The electrical analogy was then simplified for a single dominant anion/cation pair and the case of small overpotentials at the electrodes. This simplified model was solved by Laplace Transforms to demonstrate a rich and varied response that such a system exhibits to a step change in flow depending on the relative magnitude of the various resistors and capacitors. These system responses showed the possibilities of spikes in either external current or voltage at the time of the step change in flow, as well as how they decay to steady state (eqs 29−34). Experiments were conducted on a 25 mm diameter glass microchannel array having approximately 3 437 500 straight, circular microchannels with a pore size of 10 μm diameter and a channel length of 2 mm. The working fluid was either DIUF water with either no solute added or KCl added to create concentrations from 0.05 to 1 mM. The electrodes were either meshes of platinum-black or silver, or gold that was deposited on the ends of the microchannel array. A variety of experiments were conducted measure external current or trans-capillary potential difference in either steady state or a series of flow reversals. Besides providing data for resistances and capacitances, in particular for the electrodes, keys aspects of the experimental results were interpreted with the electrical analogy. The trans-capillary potential difference measured by electrodes placed in the reservoirs upstream and downstream of the microchannel array in steady state flow can be easily interpreted as the streaming potential, and is independent of the type and placement of the electrodes. The external current measured by electrodes placed in the reservoirs upstream and downstream of the microchannel array in steady state flow is not the streaming current, and is highly

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CONCLUSIONS This paper takes a system-wide perspective of electrokinetic energy conversion devices based on an array of microchannels to help understanding their operation. This perspective includes the pressure driven flow through the microchannels that partition ionic species to create convection currents and trans-capillary potential differences, the role of the reservoir in terms of electrode placement upstream and downstream of the channels, the capacitive and resistive characteristics of electrodes (e.g., their polarization), and an external resistive load. The approach taken to help in this understanding is a combination of developing an electrical analogy for the whole system and conducting experiments on a glass microchannel array. The electrical analogy was first developed for a single channel and with the possibility of multiple solutes. The model included a current source for the convection current, resistors for the conduction current, a capacitor for accumulating the partitioned ions, resistors for ion transport within the reservoirs, diodes and capacitors for the electrochemistry and polarization at the electrodes, and a simple external resistive load. The expansion of the analogy to include any number of parallel channels showed how arrays of channels (through the sum of resistive and capacitive elements) could be significantly different from the behavior of a single channel. The resistance 24323



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(4) van der, Heyden; Frank, H. J.; et al. Power generation by pressure-driven transport of ions in nanofluidic channels. Nano Lett. 2007, 7.4, 1022−1025. (5) Yang, J.; Lu, F.; Kostiuk, L. W.; Kwok, D. Y. Electrokinetic microchannel battery by means of electrokinetic and microfluidic phenomena. J. Micromech. Microeng. 2003, 13.6, 963. (6) Siria, A.; Poncharal, P.; Biance, A. L.; Fulcrand, R.; Blase, X.; Purcell, S. T.; Bocquet, L. Giant osmotic energy conversion measured in a single transmembrane boron nitride nanotube. Nature 2013, 494 (7438), 455−458. (7) Mansouri, A.; Bhattacharjee, S.; Kostiuk, L. High-power electrokinetic energy conversion in a glass microchannel array. Lab Chip 2012, 12.20, 4033−4036. (8) Chang, C. C.; Yang, R. J. Electrokinetic energy conversion in micrometer-length nanofluidic channels. Microfluidics and nanofluidics. Microfluid. Nanofluid. 2010, 9, 225−24. (9) Mansouri, A.; Kostiuk, L. W.; Bhattacharjee, S. Streaming current measurements in a glass microchannel array. J. Phys. Chem. C 2008, 112.42, 16192−16195. (10) Mansouri, A.; Scheuerman, C.; Bhattacharjee, S.; Kwok, D. Y.; Kostiuk, L. W. J. Transient streaming potential in a finite length microchannel. J. Colloid Interface Sci. 2005, 292, 567−580. (11) Masliyah, J. H.; Bhattacharjee, S. Electrokinetic and Colloid Transport Phenomen; John Wiley & Sons: Hoboken, NJ, 2006. (12) Mansouri, A.; Kostiuk, L. W.; Bhattacharjee, S. Transient electrokinetic transport in a finite length microchannel: Currents, capacitance, and an electrical analogy. J. Phys. Chem. B 2007, 111, 12834−12843. (13) Kirby, B. J.; Hasselbrink, E. F. Zeta potential of microfluidic substrates: 1. Theory, experimental techniques, and effects on separations. Electrophoresis 2004, 25.2, 187−202. (14) Elimelech, M.; Chen, W. H.; Waypa, J. J. Measuring the Zeta (Electrokinetic) Potential of Reverse Osmosis Membranes by a Streaming Potential Analyzer. Desalination 1994, 95.3, 269−286. (15) Van der Heyden, F. H. J.; Stein, D.; Dekker, C. Streaming currents in a single nanofluidic channel. Phys. Rev. Lett. 2005, 95.11, 116104. (16) Gillespie, D. High energy conversion efficiency in nanofluidic channels. Nano Lett. 2012, 12, 1410−1416. (17) Daiguji, H.; Yang, P.; Szeri, A. J.; Majumdar, A. Electrochemomechanical energy conversion in nanofluidic channels. Nano Lett. 2004, 4.12, 2315−2321. (18) Martins, D. C.; Chu, V.; Prazeres, D. M. F.; Conde, J. P. Streaming currents in microfluidics with integrated polarizable electrodes. Microfluid. Nanofluid. 2013, 15.3, 361−376. (19) Ren, Y.; Stein, D. Slip-enhanced electrokinetic energy conversion in nanofluidic channels. Nanotechnology 2008, 19, 195707. (20) Xie, Y.; Sherwood, J. D.; Shui, L.; van den Berg, A.; Eijkel, J. C. Strong enhancement of streaming current power by application of two phase flow. Lab Chip 2001, 11.23, 4006−4011. (21) Van der Heyden, F. H.; Bonthuis, D. J.; Stein, D.; Meyer, C.; Dekker, C. Electrokinetic energy conversion efficiency in nanofluidic channels. Nano Lett. 2006, 6.10, 2232−2237. (22) Mansouri, A.; Scheuerman, C.; Bhattacharjee, S.; Kwok, D. Y.; Kostiuk, L. W. Influence of entrance and exit conditions on the transient evolution of streaming potential in a finite length microchannel. In ASME 3rd International Conference on Microchannels and Minichannels, Toronto, Ontario, Canada, June 13−15, 2005; pp 541−549. (23) Lyklema, J.; Minor, M. On surface conduction and its role in electrokinetics. Colloids Surf. A: Physicochem. Eng. Aspects 1998, 140.1, 33−41.

dependent on the type and placement of the electrodes. The amount of current going through a shorted external circuit is affected by the resistances in the channel and that associated with transporting the solutes from the ends of the channels to the electrodes, as well as the area and nature of the electrodes. For any electrode type, the highest external currents were measured when the spacing between channel ends and the electrode were minimized. To eliminate the resistance associated with transport in the reservoir, the gold deposited on the channel ends were used as electrodes to isolate other electrical elements for quantification. The resistances of the gold electrodes for different solute concentrations were quantified by a methodology that involved measuring and comparing the trans-capillary potential difference by electrodes in the reservoirs when the gold electrodes were either shorted or left as an open circuit. At high salt concentrations, the resistance of the electrodes dominated over the resistance of the channels, while the opposite was true for DIUF water. The capacitances of the gold electrodes for different solute concentrations were quantified by a methodology that involved estimating the time constant of the system’s external current as it approached steady state from a sudden change in flow direction. For any of the salt concentrations tested, the capacitance of the electrode was approximately six orders of the magnitude greater than the microchannels array, and therefore the system was easily represented by a single time constant of how the diffusive transport slowly responds to a step change in hydrodynamics. For the DIUF water case, the results were quite different, showing a rise in current to steady state and its form not being well described by a single exponential. This different behavior was speculated to be related to the change in the dominant charge carrying species from K+/Cl− to H+/OH−. Once all the resistors and capacitors of the electrical analogy were estimated, the system response to electrokinetic energy conversion was explored. The steady state case was considered to examine the effects of altering the external resistive load and the concentration of KCl. The highest electrical power produced was for DIUF water (reduced by 3 orders of magnitude with 0.5 mM KCl) and when the external resistive load equaled the sum of the internal resistances.

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AUTHOR INFORMATION

Corresponding Author

*Telephone: 780-492-3450. Fax: 780-492-2200. E-mail: Larry. [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support for this work was provided by the Natural Science and Engineering Research of Canada.

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REFERENCES

(1) Osterle, J. F. Electrokinetic energy conversion. J. Appl. Mech. 1964, 31, 161−164. (2) Morrison, F. A.; Osterle, J. F. Electrokinetic energy conversion in ultrafine capillaries. J. Chem. Phys. 1965, 43, 2111−2115. (3) Pennathur, S.; Eijkel, J. C. T.; Van den Berg, A. Energy conversion in microsystems: Is there a role for micro/nanofluidics? Lab Chip 2007, 7, 1234−1237. 24324


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