Electrokinetic Energy Conversion by Microchannel Array: Electrical Analogy, Experiments and Electrode

Polarization

Abraham Mansouria, Subir Bhattacharjeeb, and Larry W. Kostiukc*

a - Department of Mechanical Engineering American University in Dubai

Dubai, UAE 28282

b – Water Planet Engineering 721 S Glasgow Ave, Inglewood, CA 90301, United States

c - Department of Mechanical Engineering

University of Alberta Edmonton, Alberta, T2G 2G8

CANADA

*Corresponding Author Telephone: 780-492-3450

Fax: 780-492-2200 E-mail: Larry.Kostiuk@ualberta.ca

Type of Article: Full Length

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.

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 nano- scales, 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

comprehensive theoretical model for the effects of the resistivity of the fluid in the reservoirs for

a micrometer-length nano-channel in an electrokinetic energy conversion systems [8]. To the

best of the authors’ knowledge, the theory or modeling of electrokinetic energy conversion

devices have yet to include the effects of the electrodes and their polarization. This polarization

phenomenon appears to be unavoidable in high-power electrokinetic energy conversion devices,

such as micro- or nano-channel 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

transports), Butler-Volmer (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 -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 -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 -

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 understand such

systems. In the subsequent section, an experimental apparatus is described that is used to collect

data for different 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, the 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.

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 cross-sectional 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 makes 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.

Figure 1: Schematic of microchannel array that separate two reservoirs. “L” and “R”

represent the left and right hand extremes of the channels, respectively, of length L, and

electrodes placed a distance of xe from the channel ends and connected through an external load.

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 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 towards 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 counter-ions 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, in amperes, associated with the mth chemical

species in the jth channel of the array ( Iconv,mj ) can be calculated by integrating the local bulk

velocity ( v) and species’ molar density ( nm ) weighted by the valance of the species ( zm ) over

the cross-sectional area of one of the microchannels ( A ), such that

Iconv,mj zm

Fnmv dA

A

(1)

where F is Faraday’s constant.

Figure 2 shows schematically the concurrent convective currents of all M solute species,

which when summed together creates the net convection current in that channel:

Iconvj Iconv,m

j

m1

M

(2)

For the specific case of M = 2 and the valance 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 by [11]

Iconvj

pA

Lf (a)

(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 Debeye length (κ) and channel radius (a). In the limit of κa >> 1, f(κa) = 1 and

Smoluchowski’s equation is recovered.

Figure 2: Electrical analogy of the convective currents associated with M different

charged solutes being transported by the bulk flow in the jth single channel of a microchannel

array.

Accumulation of Charge and Conduction Current

The imbalance in the convection of charged species through the microchannel results in

the relative depleting of counter-ion 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 non-

uniform 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.

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.

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.

A model for the analogous capacitor has been previously proposed and was based on the

geometric similarities between 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, -potential and mass transfer have essentially negligible

effects. The functional form of the capacitance for the jth channel was given as

CCj KC

A

L

(4)

where KC is a constant that has been shown to be ~3 [12].

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

RC,mj

RF,m RS,m

RF,m RS,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

A F,m

(6)

RS,i L

P S,m

(7)

where F,m is the conductivity of the mth species within the bulk fluid, P is the perimeter

of the channel, and S,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-by-species balance between their individual

convection, accumulation and conduction because the electric field is shared. Given the multi-

species 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 multi-component salts [11].

Electrodes and External Load to Produce Electric Power

In the previous sub-sections 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

RR,m Bxe

A F,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 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 fluid-electrode interface will develop on a time scale associated with

species migration. If that potential difference exceeds the threshold voltage difference require 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 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 over-potential could be modeled by the Butler-Volmer

equation for such electrodes. Typically, for a given electrolyte concentration, the current density

for non-polarizable 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 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).

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 the extremes of this external load can represent ideal measurement devices such as a voltmeter ( Rext ) or an ammeter ( Rext 0), and will connect

experimental measurements to this proposed model.

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:

RC,i RC,i

j

J

(9)

CC JCCj (10)

RR,i RR,i

j

J

(11)

where J is the number 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 capacitive characteristics

associated with the electrodes. This shifting of relative magnitudes of in component

characteristics in 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 trans-capillary potential difference. One is to

migrate back through 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 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 was 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 non-reacting electrodes with

very low over-potentials in the Bulter-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.

Figure 5. Electrical analogy of a single channel electrokinetic power generation system and electrodes associated with external circuits. Diodes were replaced with resistive elements in situation where over-potential needed for reactions is small.

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 re-order 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) (12)

where Y(s) is the Laplace function:

Y (s) RC RC RE,TCE,T s

s2 s

(13)

and the constants are given as

RCCC RE,TCE,T RR,T Rext (14)

RCCC RE,T RR,T Rext CE,T RE,T (RC RR,T Rext ) (15)

RC RE,T RR,T Rext (16)

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 with a magnitude equal to that at steady

flow (i.e., Iconv (s) Iconv / s ), and the solution in the time domain becomes

Iext (t) Iconv ABexp( t

1

)

cosht

2

Dsinh

t

2

(17)

Vext (t) IconvRC 1 ABexp( t

1

)

cosht

2

Dsinh

t

2

(18)

where

CR

A (19)

B RC

(20)

D 1

2 / 42RE,TCE,T

(21)

and 1 and 2 are the time constants given by

1 2

(22)

2

2 / 4 (23)

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 0 and allows considerable simplification of the

above expressions such that the external current is given by:

Iext (t) Iconv AE exp( t

)

(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) IconvRC 1 AE exp( t

)

(25)

CR

A (26)

E RC

RCRE,TCE,T

(27)

RE,T RC

RE,T RC

(CC CE,T ) (28)

and in the expanded version, Equations 24 and 25 become:

Iext (t) Iconv

RC

RC RE,T

CE,T

CC CE,T

RC

RC RE,T

exp

( t

)

(29)

Vext (t) IconvRC 1 RC

RC RE,T

CE,T

CC CE,T

RC

RC RE,T

exp

( 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 changing the geometry, materials and solute concentration used in

the system.

Case I) CE,T CC

Iext (t) Iconv

RC

RC RE,T

1RE,T

RC

exp( t

)

: Iext (t 0) Iconv : Iext (t ) Iconv

RC

RC RE,T

(31a)

Vext (t) Iconv

RCRE,T

RC RE,T

1 exp( t

)

: Vext (t 0) 0 : Vext (t ) Iconv

RC RE,T

RC RE,T

(31b)

RE,T RCCE,T

RE,T RC

(31c)

Case II) CC CE,T

Iext (t) Iconv

RC

RC RE,T

1 exp( t

)

: Iext (t 0) 0 : Iext (t ) Iconv

RC

RC RE,T

(32a)

Vext (t) Iconv

RCRE,T

RC RE,T

1 RC

RE,T

exp( t

)

: Vext (t 0) IconvRC : Vext (t ) Iconv

RC RE,T

RC RE,T

(32b)

RE,T RCCC

RE,T RC

(32c)

Case III) RE,T RC

Iext (t) Iconv

CE,T

CC CE,T

exp( t

) : Iext (t 0) Iconv

CE,T

CC CE,T

: Iext (t ) 0 (33a)

Vext (t) IconvRC

CC

CC CE,T

exp( t

)

: Vext (t 0) IconvRC

CC

CC CE,T

: Vext (t ) IconvRC (33b)

RC (CC CE,T ) (33c)

Case IV) RC RE,T

Iext (t) Iconv 1 CC

CC CE,T

exp( t

)

: Iext (t 0) Iconv

CE,T

CC CE,T

: Iext (t ) Iconv (34a)

Vext (t) IconvRC

Cc

CC CE,T

exp

( t

) : Vext (t 0) IconvRC

Cc

CC CE,T

: Vext (t ) 0

(34b)

RE,T (CC CE,T ) (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.

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.

Experimental Section

Materials The key materials used in the experiments are 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 Electro-Optics, 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 ultra-filtered (DIUF) water

obtained from a water purification system (Millipore Simplicity, 18.2 MΩ/cm), and the KCl

electrolyte solutions were made by adding appropriate amount 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 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.

Table 1: Characteristics of Reservoir Electrodes

Designation Material Mesh Type Wires Diameter (mm) Platinum* 99.9% Pt gauze from

wire (Alfa Asesar, MA, USA)

45 Mesh woven 0.198 mm diameter

25

Silver Silver Micro mesh Not Available

25

Gold Gold Deposited Not Applicable

25

*Prepared by electro-deposition at 50 mV from 2 % chloro-platinic acid in 1M 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 liter/minute) 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 co-axial port at their ends,

which was designed to allow a non-conductive 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 second) 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 minutes to

remove any trapped air bubbles. For each of the electrode’s location, at least six measurements

were performed to insure repeatability of results.

Results and Discussions

This particular experimental apparatus allows for altering the flow direction, electrode

type, electrode location, and working fluid, as well as a whether the electrodes are connected to a

voltmeter, ammeter or finite external resistive load, so there are many experiments opportunities

available. A few of these options are explored experimentally and the results interpreted

qualitatively with the aid of the electrical analogy shown in Figure 5, or quantitatively with

Equations 24 and 25.

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 is 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 7 shows the measured trans-capillary potential difference when the external

resistance is replaced by a voltmeter ( Rext ) and the external current when the external

resistance is 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 is 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 all capacitor 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 microchannel

were identical then this SP is the same as for 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 trans-capillary potential difference is non-zero and the

conduction current through RC is non-zero. 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 (Equation 8), which also explains the divergence in the measured

external current as the distance the ends of the channels to the electrodes was increased.

1 10 100Electrodes distance from array (mm)

10

100

1000

Ext

erna

l Cur

rent

(μA

)

1 mM KCl- Pt black1 mM KCl- Silver mesh

1 10 1000.1

1

10

Stre

amin

g Po

tent

ial (

V)

(a)

(b)

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

Figure 8: The resulting electrical analogy for steady state when the external resistive load

is replaced by a voltmeter

Figure 9: The resulting electrical analogy for steady state when the external resistive load

is replaced by an ammeter

1 10 100Electrodes (Pt black) distance from array (mm)

10

100

1000

Ext

erna

l Cur

rent

(μA

)

1 mM KCl0.04 mM KCl

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

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 is

drawn through the reservoir electrodes then the measured potential is 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 trans-capillary potentials when the anode and cathode are either

shorted (i.e., VRext0 when ammeter in Figure 11 is connected) or held as an open-circuit (i.e.,

VRext SP when the ammeter in Figure 11 is disconnected, such that:

RE,T 2RE RC

VRext0

SP VRext0

(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 Equation 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 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.

Figure 11: The 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.

TABLE 2: Estimations of resistance and capacitance of GMA and gold electrodes based on proposed electrical analogy. RR,T was assumed to be zero since there is no reservoir fluid

between the channel exits and the electrode, RC was estimated from Equation 6, RE,T was

estimated from measurements and Equation 35, CC was estimated from Equation 4 and 10,

was estimated from the measured decay in external current following a step change in flow, and CE,T was estimated from Equation 29.

KCl Concentration

(mM)

RR,T

RC

RE,T

(s)

CE,T

F

CC

nF

0.05 0 7185 6666 0.51 147 0.28

0.1 0 3133 9375 0.4 170 0.28

0.5 0 1102 10555 0.25 251 0.28

In order to gain insight in the capacitive elements associated with the GMA and the

electrodes it is 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 is turned on or reversed, followed

by the decay in current towards 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 are 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 non-zero

steady-state shows that these experiments have characteristics similar to Case I, as described by

Equations 31a-c. This case is 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 are estimated by

Equations 4 and 10, which have also been added to Table 2, then it shows that there is six 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 Equations 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 conductive of the working

fluid has been increased to a point where the channel resistance for conduction current becomes

smaller. Second, the presence of high concentrations of solutes near the electrodes surfaces has

increased the effective electrode polarization so that electrode resistance becomes larger. Using a

more extreme solute concentration of 1 mM KCl there was no steady state current recorded,

which causes Cases I and III too 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 (Equation 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 is initially small and

rises 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 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.

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 Equations 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].

(a)

(b)

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.

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

In electrochemistry, the current that flows to and from electrodes are divided in two

categories, Faradic (deals with the transfer of electrons) and non-Faradic (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 Faradic 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 (non-Faradic).

In 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 F/cm2.

Modeling Steady State Electrokinetic Power Generation with Electrode Polarization

We now have the properties and sub-models 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 Equation 17 (i.e., P Iext2 Rext ), which in steady state ( t ), electrodes

located at the channel ends ( RR,T 0 ), and a 1 becomes:

2,

22

extTEC

extC

RRR

RR

L

pAP

(36)

The effects of polarization in the form of a Faradic 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 0.025V, p 200, 000Pa ,

J 3, 437, 500, A 7.85x1011m2, 0.001Pa s , L 0.002m, and 7.08x1010 F/m, while RC ,

and RE,T remain 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 three 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 0.1mM KCl solutions. Exploiting the transient characteristics where

the external current is large near t 0, or non-resistive loads, for power generation has yet to be

explored.

Figure 14. Output power of electrokinetic energy conversation device in steady state 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, NGEs becomes more polarized and electrial resistance of electordes increase to the extend that no power can be recorded for a highly concetrated electrodes.

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 difference, 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 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 over-potentials 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 (Equations 29 –

32).

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 1mM. 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 stream 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 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 electrode 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

approach steady state from a sudden change in flow direction. For any of the salt concentrations

tested, the capacitance of the electrode was approximately six order 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 three orders of magnitude

with 0.5mM KCl) and when the external resistive load equaled the sum of the internal

resistances.

Acknowledgements:

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

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