Temperature Effects On The Electrohydrodynamic and Electrokinetic Behaviour of Ion-Selective Nanochannels
Temperature Effects On The Electrohydrodynamic and Electrokinetic Behaviour of Ion-Selective Nanochannels
Temperature Effects On The Electrohydrodynamic and Electrokinetic Behaviour of Ion-Selective Nanochannels
nanochannels
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Abstract
A non-isothermal formulation of the Poisson–Nernst–Planck with Navier–Stokes equations is
used to study the influence of heating effects in the form of Joule heating and viscous
dissipation and imposed temperature gradients on a microchannel/nanochannel system.
The system is solved numerically under various cases in order to determine the influence of
temperature-related effects on ion-selectivity, flux and fluid flow profiles, as well as coupling
between these phenomena. It is demonstrated that for a larger reservoir system, the effects of
Joule heating and viscous dissipation only become relevant for higher salt concentrations and
electric field strengths than are compatible with ion-selectivity due to Debye layer overlap.
More interestingly, it is shown that using different temperature reservoirs can have a strong
influence on ion-selectivity, as well as the induced electrohydrodynamic flows.
In order to describe the transport of ions through a charged the role of temperature effects in nanochannels on the resulting
nanochannel, a number of theoretical frameworks have been ion-selectivity, fluxes and induced flow profiles in the case of
employed. These range from simplified conductance models double layer overlap or high surface charges/wall potentials
based on 1D analytical solutions of the Poisson–Boltzmann has not been examined in depth.
equation [23, 24] to numerical solution in 2D or 3D of the full It is the aim of this study to numerically investigate the
balance equations to describe the transport of ions (Nernst– effects of temperature on the resulting electrokinetic and elec-
Planck equation), potential distribution (Poisson equation) trohydrodynamic behaviour of nanochannels. In particular,
and momentum (Navier–Stokes or Stokes flow) [7, 25]. The the region with high surface potentials/surface charges and
coupling of the ion distribution to the potential forms a non- large overlap of the double layer is of interest, as the role of
linear PDE, which can be simplified by assuming a Boltzmann imposed temperature gradients, viscous dissipation, Joule
distribution for ions at the surface under the conditions of heating, etc for charge-selective nanochannels has not yet
low surface potential and non-overlapping double layers. been explored in detail. In order to accomplish this goal, a
However, this assumption is of limited value for the case of theoretical framework based on a non-isothermal formulation
an ion-selective nanochannel where double layer overlap is of the Poisson–Nernst–Planck equations, Navier–Stokes and
a requirement. In order to describe both the electrokinetic energy balance has been formulated in order to determine the
and electrohydrodynamic effects observed in a microchannel/ couplings between temperature, ion-species concentration,
nanochannel device, it is of crucial importance to account for fluid velocity and electrical potential. Using this framework,
entrance/exit effects from the nanochannel [9]. This is due to numerical simulations were undertaken to investigate the role
the large variation in hydrodynamic resistance between the of temperature effects on ion-selectivity, flux and induced
micro and nanochannel, as well as constriction effects on any fluid flows.
applied fields.
In comparison to the large breadth of experimental and
2. Theoretical background
theoretical work for ion-selective nanochannels under or
assuming isothermal conditions, the role of temperature, and
2.1. Dimensionless non-isothermal Poisson–Nernst–Planck
more specifically, temperature gradients is relatively unex- with Navier–Stokes
plored. Investigation has focused primarily on heat transfer
in charged microchannels and the role this plays in induced In order to simulate the influence of various temperature-
electroosmotic flows. For example, Chen (2009) studied the related effects on transport in ion-selective media, a model
development of temperature profiles in micro- and nano- framework based on a non-isothermal formulation of the
channels with a thin double layer at low potentials using a steady-state Poisson–Nernst–Planck (PNP) equations coupled
Poisson–Boltzmann framework considering Joule heating with the Navier–Stokes was formulated. This framework (or
(JH) as the primary heat source while neglecting viscous dis- variations) have been employed successfully by a number
sipation based on order of magnitude considerations [26]. of authors to describe the resulting ion-transport, induced
The results demonstrated the importance of temperature on fluid-flows (electroosmotic and otherwise), etc [29–31].
the resulting electroosmotic flow (EOF) and friction factors Depending on the salt concentration, significant deviations
for pure EOF, as well as EOF with pressure gradients, finding from assuming Poisson–Boltzmann (PB) type behaviour can
that viscous dissipation became comparable to JH for chan- be expected, which motivates the use of a non-equilibrium
nels smaller than 50 nm [26]. This work also demonstrated the approach such as PNP [25, 29]. The total channel height con-
importance of accounting for the temperature-dependence of sidered was 20 nm, which also corresponds to experimental
physical properties, as significant deviations could be observed conditions [32], additionally these geometric dimensions have
assuming a constant value. Shi et al (2008) used the Lattice- been considered numerically for isothermal simulations of
Boltzmann method to describe non-isothermal electroosmotic nanochannels [25, 33] and allow for treating the system as a
flows again with a Poisson–Boltzmann type distribution, con- continuum [34]. The equations were scaled for numerical and
cluding that for channels less than 100 nm in diameter viscous comparison purposes.
dissipation effects will become dominant compared with JH, The electrical potential in the system was split into two contrib
dependent upon the salt concentration in the bulk [27]. utions: the applied potential, ϕ, and the induced electrokinetic
The role of temperature gradients in the double layer and potential, ψ, as described in [35]. This is effective for numerically
how this coupled to thermoelectric effects was studied theor resolving the coupling of the electric field to ion-concentration
etically recently by Ghonge et al [28], using a Poisson–Nernst– due to the relative scale difference in these potentials, namely
Planck approach, indicating that Soret and thermoelectric that applied potential can be on the order of volts while a con-
effects could be significant for the case of electrohydrody- stant surface charge or potential would result in potentials on the
namic flow in a nanochannel with an imposed wall temper order of tens of millivolts. This approach is modified slightly to
ature gradient along the channel axis. The contribution of JH include a temperature-dependent permittivity, as gradients in
was neglected versus viscous dissipation based on an order of permittivity will affect the electric field profile which corresp
magnitude analysis, although in a micro/nanojunction system ondingly affects the body-forces on the liquid in the form of
the electric field constriction and enhancement due to cation dielectric stresses (electrothermal-flows) [28, 36]. Upon decom-
or anion depletion/enrichment may result in conditions where posing the electric potential into these components, the resulting
the Joule heat contribution could be significant. However, Poisson equations can be written as (1)–(3)
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J. Phys.: Condens. Matter 28 (2016) 114002 J A Wood et al
In order to obtain dimensionless equivalents, the permit Finally, the system of PDEs is closed by solving for the
tivity was scaled against the permittivity at Tref = T0, using a dimensionless temperature profile via an energy balance (6),
linear function of dimensionless temperature, which is a good where the heat capacity is treated as being essentially con-
approximation between 293 and 313 K based on the permit stant similar to the arguments given for liquid density whereas
tivity (dielectric constant) data of Kaatze (1989) [37], as per the temperature-dependence of thermal conductivity, k, was
[38]. The concentration (c1 for cation and c2 for anion) was accounted for. Temperature is scaled relative to the refer-
scaled against bulk concentration in the reservoir cref = c0 ence temperature T = T0 in the form θ = (T − T0 )/T0 with T0
and the electric potential by scaling against the thermal chosen as 293K.
voltage of an electron Vref = kBT0 /e. The spatial dimensions 1
were scaled to the channel half-height, i.e. L ref = h, where u ⋅ ∇θ =
(6) ∇ ⋅ (k ∇θ ) + qd + qJH
Peθ
h is shown in figure 1 as half the total channel height due
to the symmetry plane. This allows the space charge den- The sources of heat generation in the system are from two
sity to be scaled against the Debye parameter at T = T0, primary contributions, (i) Joule Heating (qJH), equation (7) via
κ = [(2e 2z 2c0NA)/(εr ,0ε0kBT0 )]1/2. All temperature-dependent the irreversible dissipation of current into heat and (ii) viscous
physical properties are provided in appendix B. dissipation (qd), equation (8). JH was evaluated based on the
For both cation and anion, the transport balance between conduction current with the applied electric field, neglecting
convection, diffusion and electromigration can be written in the contribution of convective current dissipation and the
form of the Nernst–Planck equations. For our simulations, induced electric-field along the nanochannel wall based on
we consider the case of a 1:1 electrolyte. In equation (4) the order of magnitude analysis [26, 42]. The case of imposed
dimensionless diffusivity, Di, is dependent upon dimension- temperature gradients was also considered, to examine the
less temperature, θ = (T − T0 )/T0. The influence of Soret effect on the electrohydrodynamic flows and electrokinetic
effects (thermophoresis) was included through the use of the transport of ions through the nanochannel.
dimensionless ionic heat of transport coefficients (Qi) for ⎛ σ0V 2 ⎞
potassium and chloride, taken from Würger (2008) [28, 39]. qJH = ⎜
(7)
ref
⎟σ|∇ϕ|2
Soret effects in particular can be significant at lower applied ⎝ ρCpT0u 0L ref ⎠
potentials in cases where large temperature gradients exist. For an incompressible fluid, the contribution of viscous
The temperature-dependence of individual ion-diffusion coef- dissipation can be simplified and non-dimensionalized as
ficients was estimated through temperature dependence of the follows:
limiting ion molar conductivity, while diffusivity was consid-
ered as independent of concentration over the range of con- 2u 0µ0µ ⎡⎛ ∂u ⎞2 ⎛ ∂v ⎞ ⎛ ⎞⎤
2 2
qd = ⎢⎜ ⎟ + ⎜ ⎟ + 1 ⎜ ∂v + ∂u ⎟ ⎥
ρCpT0L ref ⎢⎣⎝ ∂x ⎠ ⎝ ∂y ⎠ 2 ⎝ ∂x ∂y ⎠ ⎥⎦
centrations studied for simplicity [40]. The Péclet number for (8)
anion and cation, Pe ci, is evaluated for the case at the reference
temperature.
⎛ c Q ∇θ zcE⎞
Pe ciu ⋅ ∇ci = ∇ ⋅ Di⎜∇ci + i i 2 − i i ⎟
2.2. Simulation details
⎝ 1+θ⎠
(4)
(1 + θ )
In order to test the relative influence of various temper
The dimensionless Navier–Stokes equations are given in (5), ature-related phenomena on ion-transport, a microchannel/
with a temperature-dependent viscosity and Coulomb and nanochannel system was considered consisting of two large
dielectric body forces. The influence of temperature on the den- reservoirs connected via a nanochannel, as illustrated in
sity of solvent, this case water, was neglected as a temperature figure 1. The constriction of the electric field at the nanochannel
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J. Phys.: Condens. Matter 28 (2016) 114002 J A Wood et al
throat can potentially play a large role in inducing temper rise to large temperature gradients which affect the induced
ature gradients, due to the high local field gradient as seen for electroosmotic flows within the channel [27]. However, in
insulator dielectrophoresis [36]. For computational simplicity, those cases the nanochannel itself was considered as an iso-
a 100× aspect ratio was considered where the nanochannel lated system without accounting for thermal contact with a
has a half-plane symmetry height of 1 and a length of 100 much larger reservoir. To investigate the maximum possible
dimensionless units. This corresponds to a physical channel local heating in the channel, simulations were run where the
dimension of 20 nm and length of 1 μm, of which a 10 nm channel was connected to reservoirs with T = T0 (θ = 0)
high half-symmetry is modeled. The reservoirs were chosen external walls, boundary AB and GH in figure 1, and numer
as being 200 × 200 cells (2 µ m × 2 μm) to try to minimize ically solved at various applied potentials. All other bound-
any potential size effects from concentration polarization or aries were considered as insulating/symmetry conditions for
other phenomena. This geometry is conceptually identical to temperature.
that considered in many papers on isothermal ion-transport The initial case considered was that of salt reservoirs at
nanochannels, such as [25, 33]. a concentration of 1 × 10−3 M KCl, which corresponds to a
The dimensionless equations described were solved Debye length (κ−1, as was defined in section 2.1) of 9.6 nm
numerically using the finite element method (FEM) via the at each wall and therefore a nearly complete overlap of the
commercial software package COMSOL Multiphysics 5.0. nanochannel (20 nm total height). A maximum temperature
The system geometry is shown in figure 1, with the nano- rise of 0.29 and 0.28 K was observed for a nanochannel with
channel connecting the two reservoirs. The nanochannel was a surface potential of −50 and −100 mV respectively, at an
meshed using mapped quadrilateral elements, with 25 000 applied electric field strength of 1 × 107 V m−1 which rep-
elements in total in the channel (50 × 500). The mesh density resents 10 V across a 1 μm long channel. Compared to this
was arithmetically scaled to account for higher nonlinearities applied field strength, thermophoretic effects are negligible.
near the walls and entrance/exit effects in the channels, where Similarly, electrothermal flow contributions were negligible
a higher degree of resolution is required. For solution of the compared to the electroosmotic force. At these field strengths,
potentials, quadratic Lagrange elements were utilized. For the all ion-selectivity in the nanochannel is however lost as the
ion-species concentration and energy balance linear elements applied field is strong enough to overcome any resistance in
and for the Navier–Stokes a P2–P1 velocity/pressure formula- the nanochannel. Joule heating provided the bulk of heating
tion was used. at higher applied fields, at lower field strengths viscous dis-
The boundary conditions tested for simulation cases are sipation dominated but neither effect was able to generate a
provided in appendix A, with a number of conditions tested local temperature gradient in the system. The convective/dif-
for temperature and induced electrokinetic potential in gen- fusive transport of heat into the reservoirs was sufficient in
eral. Specifically, the influence of substrate heat transfer order to suppress localized gradients even in the case of fully
and variation in behaviour for a constant wall potential was insulating walls at all but the highest of applied electric field
considered. In this case, a high constant wall potential could strengths.
be achieved in a physical nanochannel system using a gated For the case of c0 = 1 × 10−2 M reservoirs, temperature
dielectric concept [31, 33], although a constant surface charge effects begin to play a larger role even in the case of having
is considered more realistic for a conventional nanochannel a large reservoir with a fixed wall temperature. However, this
wall [23, 35]. The influence of temperature on surface charge/ effect only becomes apparent at quite high applied electric
potential was neglected in order to simplify the analysis but field strengths. For a field strength of 1 × 107 V m−1, simu-
depending on the specific system may also have significant lations showed a maximum temperature rise of 2.8 and 2.7
effects. For a gated nanochannel concept the surface potential K was predicted for a surface potential of −50 and −100
is fixed and independent of any temperature effects. mV respectively. The gradient effects even from this larger
temperature increase are minimal compared to the direct elec-
trophoretic fluxes and electroosmotic velocity at the double
3. Results and discussion
layer. Additionally, this corresponds to very small bulk phys-
3.1. Isothermal reservoir walls
ical property changes as well. Unsurprisingly, there is also
no significant ion-selectivity at field strengths this high (or
Initially, the influence of induced heating effects from Joule reservoir salt concentrations this high). The bulk of heating at
heating and viscous dissipation in a nanochannel is consid- this field strength came from Joule heating compared to vis-
ered. These effects are assessed for the case where the nano- cous dissipation, which is also the reason behind the near 1 : 1
channel is connected to two large reservoirs with a constant change in maximum temperature rise and salt concentration
temperature some large distance away from the channel. comparing the 1 × 10−3 M and 1 × 10−2 M cases. For the sim-
Local heating effects, especially from corner effects, could ulation cases considered, no significant heating effects could
potentially give rise to very different local physical proper- be observed in the case of a connected nanochannel under
ties, affecting the resulting electroosmotic flows from non- applied field strengths where the channel still acted as an ion-
electroneutrality as well as induce additional forces upon the selective membrane. It is then useful to consider the case of
fluid in the form of dielectric stresses. It has been reported that different temperature reservoirs to see what impact temper
viscous dissipation can be significant in nanochannels, giving ature effects can have on the resulting system behaviour.
4
J. Phys.: Condens. Matter 28 (2016) 114002 J A Wood et al
15
10
TL= 313K, T R= 293K TL= 313K, T R= 293K
TL= TR= 293K TL= TR= 293K
TL= TR= 313K TL= TR= 313K
TL= 293K, T R= 313K TL= 293K, T R= 313K
10
Total current J
J1 /J2
0 0,1
1 10 100 1000 1 10 100 1000
Applied potential Applied potential
Figure 2. Ion-selectivity, J1/J2, for −50 mV gating voltage with Figure 3. Total current, J1–J2, for −50 mV gating voltage with
different reservoir temperatures, c0 = 1 × 10−3 M. Lines are for different reservoir temperatures, c0 = 1 × 10−3 M. Lines are for
visualization purposes only. visualization purposes only.
5
J. Phys.: Condens. Matter 28 (2016) 114002 J A Wood et al
10
TL= 313K, T R= 293K TL= 313K, T R= 293K
TL= TR= 293K 3 TL= TR= 293K
TL= TR= 313K
TL= TR= 313K
TL= 293K, T R= 313K
TL= 293K, T R= 313K
Total current J
c1- c 2
1
0,1 0
1 10 100 1000 0 100 200 300 400 500
Figure 5. Total current, J1–J2, for −100 mV gating voltage with Figure 7. Ion-concentration profile along the mid-line for −50 mV
different reservoir temperatures, c0 = 1 × 10 −3
M. Lines are for gating voltage with Eapp = 1 × 10 5 V m−1, c0 = 1 × 10−3 M.
visualization purposes only.
30 TL= 313K, T R= 293K
6 TL= TR= 293K
TL= 313K, T R= 293K
TL= TR= 313K
TL= TR= 293K
TL= 293K, T R= 313K
TL= TR= 313K
TL= 293K, T R= 313K
20 4
c1- c 2
J1 /J2
2
10
0
0 100 200 300 400 500
0
1 10 100 1000 distance x/h
Applied potential
Figure 8. Ion-concentration profile along the mid-line for −100 mV
Figure 6. Ion-selectivity, J1/J2 for −50 mV gating voltage with gating voltage with Eapp = 1 × 10 5 V m−1, c0 = 1 × 10−3 M.
−4
different reservoir temperatures, c0 = 1 × 10 M. Lines are for
visualization purposes only. and positive temperature gradient case is enhanced for a
smaller value of κL ref . This implies that at lower salt concen-
higher than that of T = 313 K. Compared to a −50 mV gate, trations, the effect of temperature gradients/varying reservoir
the selectivity is significantly higher. For total current, T = 293 K temperatures can become more important for tuning selec-
again represents the lowest total current value at all applied tivity and ion-transport.
potentials while T = 313 K the highest. Similar to the −50 mV Examination of the concentration difference between cation
case, the negative temperature gradient (∆T = −20 K) case (c1) and anion (c2) along the symmetry mid-line of the system
had higher flux below a dimensionless potential of ∼40 and is revealing, showing that there is a large influence from the
a lower above compared to the reversed case (∆T = 20 K). imposed reservoir temperature difference in the resulting con-
Ion-selectivity again rapidly decreased above a dimension- centration profile for cation and anion. This is illustrated by
less potential of about ∼40, although still possessing a higher figures 7 and 8, which show these profiles at an applied field
value compared to the −50 mV case. strength of 1 × 10 5 V m−1 and a reservoir salt concentration of
For smaller ratios of channel height to electrical double c0 = 1 × 10−3 M. The impact on the cation is more dramatic,
layer (κL ref < 1), i.e. significantly overlapping double layers, as is expected with a negative wall potential where counter
the effect of imposed gradients is enhanced further. This is ions will dominate in the channel. The temperature effects are
illustrated in figure 6, which shows selectivity results versus complicated by the balance between temperature-dependence
applied potential for a reservoir salt concentration of 0.1 mM. of physical properties (diffusivity, viscosity and permittivity
This corresponds to Debye length of 33 nm or κL ref of 0.33. primarily) and how this effects the resulting fluid flows and
Compared to the case with a reservoir salt concentration of electrokinetic transport of ions, which will be elaborated upon
1 mM, the relative change in selectivity between the negative later.
6
J. Phys.: Condens. Matter 28 (2016) 114002 J A Wood et al
TL= 313K, T R= 293K In figures 2 and 4, it was shown that imposing a temper
0
TL= TR= 293K ature gradient in the same direction as the electric field led to
TL= TR= 313K improved selectivity of cations over anions. This was quanti-
-10 TL= 293K, T R= 313K
fied through the ratio of the total cross-sectional flux of cations
1 10 100 1000 (J1) over the cross-sectional flux of anions (J2). For the case
Applied potential
with a negative temperature gradient (temperature gradient in
the same direction as electric field), there was an enhance-
Figure 9. Average velocity in the channel versus potential for ment of selectivity of cation compared to anion which can be
−50 mV gating voltage and c0 = 1 × 10−3 M. Lines are for explained largely through variations in physical properties.
visualization purposes only. On the ‘hot’ reservoir side in the negative temperature gra-
dient case, diffusivity and therefore conductivity are higher
3.3. Temperature influence on electrohydrodynamics
compared to the ‘cold’ side. As was previously discussed, the
The influence of temperature on the resulting fluid flow profile diffusivity is ∼55% higher at 313 K versus 293 K for both
is illustrated in figure 9, which shows the average velocity in anion and cation, meaning the conductivity is also signifi-
the nanochannel versus applied potential. cantly higher. This effectively acts as a barrier for anions to
By comparing the average Coulomb force in the channel move into the nanochannel on the ‘cold’ side via diffusion
for the various temperature cases studied, an explanation of or electromigration and an enhancement for cations to enter
the observed flow profile can be provided. For the two cases on the ‘hot’ side. The overall balance between these effects
with identical fixed reservoir wall temperatures, T = 293 K also gives rise to a larger effective electrical body force and
and T = 313 K, the value of the average Coulomb force is therefore larger fluid velocity. As was explained earlier, a
nearly identical. A small decrease at 313 K compared to 293 K larger electrohydrodynamic velocity also implies a larger flux
is observed for all applied potentials but this is only a few of cations from hot to cold and a smaller flux of anions from
percent at most. The bulk of the increase in velocity is then cold to hot. All of these effects result in a higher degree of ion-
derived from the decrease in viscosity comparing 293 K with selectivity. At applied potentials less than 4 (field strengths
313 K, which possesses a 35% reduction in viscosity. A higher less than 1 × 10 5 V m−1, there are also additional contrib
(positive) electroosmotic velocity is then related to a larger utions from Soret effects (thermophoretic forces), as is elabo-
driving force for selectivity, as flows will transport ions from rated upon in the next section. As mentioned previously, this
left to right i.e. with the applied field. However, for T = 313 K selectivity enhancement was observed for both nearly over-
the selectivity is lower compared to T = 293 K and this is lapping double layers (c0 = 1 × 10−3 M) and enhanced for
primarily due to the larger diffusivity of ions meaning larger larger double layer overlaps (c0 = 1 × 10−4 M).
electromigration forces acting against the resistance of the
nanochannel. Since the conductivity in this case is directly
3.5. Temperature gradient effects
proportional to the diffusivity, the ∼55% increase in dif-
fusivity between 293 K and 313 K is more substantial for The relative influence of the imposed temperature gradient,
reducing selectivity. compared to the effect of different temperatures in each res-
For the imposed reservoir temperature differences, how- ervoir, on the resulting flux and ion-selectivity depends on
ever, large differences exist not only in the viscosity along the magnitude of the applied potential/electric field. For low
the channel but in the magnitude of the Coulombic force applied potentials, the thermophoretic flux of both cation and
the fluid experiences. This is owing to the different arrange- anion can dominate and will therefore potentially increase or
ment of the concentration profiles of cation and anion, which decrease selectivity depending on the ratio of the ionic heat
arise from the large dependence of diffusivity, viscosity and of transport coefficients (Qi). For example, in case (i) for
permittivity upon temperature. Comparing the case with the imposed temperatures considered previously thermophoresis
temperature gradient in the same direction as the applied field drives both cation and anion in the direction of the applied
(∆Tx = −20 × 106 K m−1) versus the reversed direction, the electric field. Since the ionic heat of transport of potassium
Coulomb force was larger at applied fields above 1 × 10 5 V m−1 is larger than that of chloride (1.064 versus 0.218), this leads
and tended to be significantly larger (∼20% or more). For the to a larger flux of cation compared to anion but both of these
positive gradient case (ii), an inversion in the flow direction thermophoretic forces serve to improve selectivity by driving
was observed which can be attributed to a stronger increase cations through the nanochannel along with the applied field
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J. Phys.: Condens. Matter 28 (2016) 114002 J A Wood et al
and induced electrosmotic velocity and acting against the as the electric field had an enhancing effect on the ion-selec-
electrophoretic motion of the anion. tivity primarily from these temperature-dependent properties.
For case (ii), with a positive temperature gradient which At lower applied potentials, the direct effect of temperature
is against the applied electric field the opposite is true. gradients in the form of thermophoresis (Soret convection)
Thermophoretic ion flux attempts to drive anions through the was also relevant. By reversing this gradient, a decrease in
nanochannel along with the electrophoretic force, acting to selectivity was obtained. Temperature gradients also had a
reduce the selectivity relative to the isothermal case at lower strong effect on the resulting magnitude of induced fluid flow
applied potentials. However these gradient effects became through the nanochannel, primarily through effecting the elec-
less significant compared to electrophoretic motion once the troosmotic force contribution. Electrothermal forces arising
applied field was above 1 × 10 5 V m−1. The average cation from permittivity gradients were found to be negligible for the
thermophoretic flux being ∼2% of the electrophoretic flux considered cases where ion-selectivity was maintained.
at this field strength and decreasing with increasing applied The results of this work demonstrate the potential benefits
potentials. from imposed temperature gradients, in terms of improving
For the case of imposed temperature gradients, the direct ion-selectivity, as well how temperature couples to the
contribution of temperature gradients to fluid forces was con- resulting fluid forces. The cases considered here also demon-
sidered through the induced dielectric stresses (electrothermal strate that for near or completely overlapping Debye layers
flows). For an imposed temperature gradient with a magnitude in nanochannels the influence of Joule heating and viscous
of 20 × 106 K m−1, the value of this force in the channel or dissipation could be neglected and the system can be treated
in the entrance/exit regions to the nanochannel was found to as isothermal at the reservoir temperature in the case of equal
be on average no more than 1% at the highest applied elec- temperature reservoirs. It is also of interest for future studies to
tric field and decreasing in relevance. Essentially, in the case examine how transport of ion pairs with significantly different
where the Debye layer overlaps or nearly overlaps the channel diffusion coefficients and temperature-dependence of ion-
the Coulombic force acting on the fluid is several orders of diffusivities are also affected by such temperature gradients.
magnitude larger compared to the dielectric force generally.
Acknowledgments
4. Conclusions
RGHL acknowledges support from the European Research
In this paper the influence of temperature on the electrohy- Council for ERC starting grant 307342-TRAM.
drodynamic and electrokinetic behaviour in microchannel/
nanochannel systems was investigated through a theoretical
framework based on a non-isothermal formulation of the Appendix A. Boundary conditions for simulations
Poisson–Nernst–Planck with Navier–Stokes equations. By
numerical investigation, the degree of heat generation by The following boundary conditions were used for each of the
Joule heating and viscous dissipation was explored for cases boundaries illustrated in figure 1 for the various cases consid-
where the nanochannel is in contact with a large reservoir. ered. These are identical to the boundary conditions consid-
These effects, as well as the resulting induced flow patterns ered in [33] with additional BCs relating to temperature.
and ion-transport when a thermal gradient is imposed by AB
having unequal temperature reservoirs, are then quantified.
For the case with a nanochannel in thermal contact with Dirichlet/Constant Value for θ, ci and ϕ
larger microchannel reservoirs, it was found for salt concen- θ = θ left, ci = 1, ϕ = V0 /Vref
trations where there was double layer overlap (ion-selectivity)
that no significant local heating gradients could be gener- Zero Flux for ψ
ated. At these low salt concentrations, Joule heating and vis- n ⋅ (εr ∇ψ ) = 0
cous dissipation were minimal and thermal dissipation into
the reservoirs was able to keep the temperature increase to a Open Boundary for Velocity
minimum. Heating became an issue only at extremely high [−p I + µ∇u¯]n = 0
field strengths (107 V m−1), which was well-beyond the region
where any degree of ion-selectivity was retained in the system. GH
Higher salt concentrations did show higher degrees of heating,
both from Joule heating and from viscous dissipation but this Dirichlet/Constant Value for θ, ci and ϕ
case also did not allow for any degree of ion-selectivity. θ = θright, ci = 1, ϕ = 0
When imposing a temperature gradient, much larger effects
can be observed. This is through a combination of temperature- Zero Flux for ψ
related effects, namely the temperature dependence of diffu- n ⋅ (εr ∇ψ ) = 0
sivity, electrical conductivity, viscosity and permittivity. For
the case of ion-transport through a negatively charged nano- Open Boundary for Velocity
channel, an applied temperature gradient in the same direction [−p I + µ∇u¯]n = 0
8
J. Phys.: Condens. Matter 28 (2016) 114002 J A Wood et al