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By thermoelectric power generation we mean the creation of electrical power directly from a temperature gradient. Semiconductors have been mainly used for this purpose, but these imply the use of rare and expensive materials. We show in this review that ion-exchange membranes may be interesting alternatives for thermoelectric energy conversion, giving Seebeck coefficients around 1 mV/K. Laboratory cells with Ag|AgCl electrodes can be used to find the transported entropies of the ions in the membrane without making assumptions. Non-equilibrium thermodynamics can be used to compute the Seebeck coefficient of this and other cells, in particular the popular cell with calomel electrodes. We review experimental results in the literature on cells with ion-exchange membranes, document the relatively large Seebeck coefficient, and explain with the help of theory its variation with electrode materials and electrolyte concentration and composition. The impact of the membrane heterogeneity and water content on the transported entropies is documented, and it is concluded that this and other properties should be further investigated, to better understand how all transport properties can serve the purpose of thermoelectric energy conversion.
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entropy
Review
Perspectives on Thermoelectric Energy Conversion in
Ion-Exchange Membranes
V. María Barragán 1, Kim R. Kristiansen 2and Signe Kjelstrup 2,*
1Department of Structure of Matter, Thermal Physics and Electronics; Complutense University of Madrid,
28040 Madrid, Spain; vmabarra@ucm.es
2PoreLab, Department of Chemistry, Norwegian University of Science and Technology, N-7491 Trondheim,
Norway; kim.kristiansen@ntnu.no
*Correspondence: signe.kjelstrup@ntnu.no; Tel.: +47-918-97079
Received: 29 October 2018; Accepted: 22 November 2018; Published: 26 November 2018


Abstract:
By thermoelectric power generation we mean the creation of electrical power directly from
a temperature gradient. Semiconductors have been mainly used for this purpose, but these imply
the use of rare and expensive materials. We show in this review that ion-exchange membranes may
be interesting alternatives for thermoelectric energy conversion, giving Seebeck coefficients around
1 mV/K. Laboratory cells with Ag
|
AgCl electrodes can be used to find the transported entropies
of the ions in the membrane without making assumptions. Non-equilibrium thermodynamics can
be used to compute the Seebeck coefficient of this and other cells, in particular the popular cell
with calomel electrodes. We review experimental results in the literature on cells with ion-exchange
membranes, document the relatively large Seebeck coefficient, and explain with the help of theory its
variation with electrode materials and electrolyte concentration and composition. The impact of the
membrane heterogeneity and water content on the transported entropies is documented, and it is
concluded that this and other properties should be further investigated, to better understand how all
transport properties can serve the purpose of thermoelectric energy conversion.
Keywords: ion-exchange membrane; thermoelectric power; transported entropy
1. Introduction
Thermoelectricity involves the direct coupling of fluxes of heat and electric charge. The coupling
can refer to the way a temperature difference can produce electricity, or to the reverse, how
an electric current can create a temperature difference. The first part of the thermoelectric effect,
the conversion of heat to electricity, was discovered in 1821 by Thomas Seebeck [
1
]. The second
effect was explored in more detail by Jean Peltier, and is referred to as the Peltier effect. They
are linked by the Onsager reciprocal relations (see below). Thermoelectric devices provide the
only direct possibility to convert low-temperature heat sources into electric power. This property
makes them potential candidates for industrial waste heat conversion. In contrast to heat engines,
thermoelectric generators have no moving parts. Thermoelectric energy converters have in practice,
since long been made from semiconductors [
2
], and there are several ideas for their improvement,
e.g., nanostructured materials [
3
5
]. The ideas were proposed by Hicks and Dresselhaus already in
1993 [
6
,
7
]. Semiconductors are, however, often expensive or rare, and do not offer particularly large
Seebeck coefficients, typically 200–300
µ
V/K, even if colossal values have been found for peculiar
conditions [8].
It may therefore be interesting to examine the potential of other conductors. Materials such as
solid state ionic conductors and electrolytes [
9
], or ionic liquids [
10
,
11
], are relevant. Granular porous
media [
12
] and electrically conductive polymers [
13
] also show thermoelectric effects. This review
Entropy 2018,20, 905; doi:10.3390/e20120905 www.mdpi.com/journal/entropy
Entropy 2018,20, 905 2 of 25
will, however, focus on a large class of materials which may be more accessible, namely ion-exchange
materials. In the last several years, the possibility of using ion-exchange membranes in renewable
energy technology is in the crosshairs [1419].
The aim of this work is to review state-of-the-art knowledge on thermoelectric energy conversion
in cells with ion-exchange membranes. The hope is to provide a basis for further explorations of their
use, i.e., in reverse electrodialysis (RED) [
18
,
20
]. In a RED concentration cell, isothermal alternating
compartments of sea water and brackish water separated by ion-selective membranes permit the
production of electric power [
21
]. Recent work demonstrates that a thermoelectric potential can be
added to the RED concentration cell to increase the electromotive force by 10% per 20 K difference
for given electrolyte conditions [
20
]. The class of ion exchange materials may provide cheaper cell
components than presently used and help make renewable technologies more competitive. We shall
review experimental results from cells with ion-conducting membranes.
The energy conversion that takes place in these cells can be well described by non-equilibrium
thermodynamics. This theory relates properties that are critical for energy conversion, and we shall
decompose measurements as far as possible in terms of these properties.
The typical experimental cell for thermoelectric energy conversion, or thermocell for short, which
is relevant for waste heat exploitation (below 100 oC), can be schematically written as
T, Ag|AgCl | MCl(aq) |membrane| MCl(aq) | Ag| AgCl, T+T
Salt solutions of MCl are separated by an ion-exchange membrane. The electrodes here are
reversible to the chloride ion. Other electrodes are also relevant. The temperature difference is ideally
across the membrane alone.
The study of such cells is not new. Already in 1956, Tyrrell [
22
] described this type of cells.
In 1957, Hill et al. [
23
] described the potential difference in terms of irreversible thermodynamics.
A year later, Ikeda and coworkers [
24
,
25
] reported a thermal membrane potential of 24.6
µ
V/K with
0.1 M KCl solutions for collodion membranes. However, Tyrrell et al. [
22
] had obtained a value
10 times higher with ion-exchangers.
Non-isothermal transport phenomena in charged membranes have been reported occasionally in
the literature since then [
26
29
]. In the 1970s and 1980s, Tasaka and co-workers were central [
30
37
],
while new researchers joined in the 1990s [
38
52
]. Different aspects were studied, such as the influence
of the membrane type [
40
42
,
51
,
52
], the concentration and nature of the electrolyte solutions in contact
with the membrane [
39
,
42
], and the membrane-transported entropy of ions [
42
,
45
]. Experimental
techniques were refined, to resolve thermal polarization of membranes and quantify contributions
from thermo-osmotic processes [48]. Temperature effects were also an issue [49].
Non-equilibrium thermodynamics [
28
,
53
57
] can be used to describe the conversion of thermal
to electric energy. Two sets of variables are then relevant: the practical set according to Katchalsky and
Curran [
53
], which consists of measurable variables, and the set most often used of ionic variables.
We consider it an advantage to have two equivalent paths for derivation of expressions to be used
in the laboratory, but shall systematically use the practical set and compare it to the other in the end.
The practical set is suited to make clear the relation between experiment and theory and give advice
on experimental design.
The reader who is interested in the experimental results alone may go directly to the end of Section 4
where we present the final equations, which enable us to explain and compare experimental results.
The main aim of this work is thus to bring out the potential of a new class of materials,
the ion-exchange materials, for thermoelectric energy conversion purposes. In spite of good knowledge
about thermoelectric generators in general, see, e.g., [
12
,
13
], properties of cells with ion-exchange
membranes have scarcely been studied systematically. It is our hope that this work can provide a basis
and pinpoint needs for further research.
Entropy 2018,20, 905 3 of 25
2. The Cell
The typical cell membrane has transport of salt and water, see Figure 1for an illustration.
We consider electrodes of silver and silver chloride, but other electrodes (calomel) have often been
used. The membrane is cation- or anion-selective. We can imagine that waste heat sources are used
to maintain a temperature difference across the membrane. In a saline power plant, say the reverse
electrodialysis plant, there is already a concentration difference across the membrane [
21
]. A pressure
difference can also arise, but this possibility will be neglected for now.
The transport processes that we consider take place along the horizontal axis of the cell.
This will be referred to as the
x
-axis. The system is normally stirred and therefore homogeneous
in the
y
-
z
plane for any fixed
x
-coordinate. All membrane fluxes can then be given by the scalar
x
-component of the vectorial flux. The membrane surfaces can be assumed to be in local equilibrium.
We shall mainly examine emf -experiments, which are carried out in the limit of very small current
densities (open circuit potential measurements). The electrodes are connected to a potentiometer via Cu
wires. The potentiometer is at room temperature
T0
, while the electrodes have different temperatures,
T
or
T+T
, like their thermostated electrolyte solutions, respectively. We are seeking an expression
for theemf of the cell in terms of properties that can be measured.
¨
I
Ag|AgCl
Ag|AgCl
M
MCl MCl
H2OH2O
T1T2
T0T0
m
T
m
T
Figure 1.
A schematic illustration of the electrochemical cell with electrodes of Ag(s)
|
AgCl(s), kept
at temperatures
T1
and
T2
. The potentiometer is kept at temperature
T0
. The membrane, M, is
surrounded by two MCl electrolyte solutions, of the same or different compositions.
Tm
1
and
Tm
2
show
the temperature at both membrane surfaces.
The emf of the cell in Figure 1,
φ
, represents the ideal electric work done that can be done by
the cell, and can be found by adding contributions along the circuit, from the leads that connect the
electrodes to the potentiometer (two contributions giving
extφ
, from the left and right electrodes
(giving el) and from the membrane (mφ). The emf is thus
φ=extφ+el φ+mφ. (1)
Entropy 2018,20, 905 4 of 25
Subscripts on the symbol
indicate the origin of the contributions; from the connecting leads
(“ext”), from the electrodes a and c (“el” ), and from the membrane (“m”). The electrolyte solutions are
stirred, so they are isothermal and fully mixed. Therefore, they do not contribute to the emf,
aqφ=
0.
3. The Electromotive Force of the Ag|AgCl-Cell
We derive the expression for the measured electromotive force (emf ),
φ
, of the cell, in terms
of measurable properties of the cell. The cell has nine distinct phases—two connecting leads from
electrodes to a potentiometer, anode- and cathode-surfaces, two bulk electrolyte solutions, a membrane,
and two interfaces between the membrane and the solutions. Out of these nine phases, eight can be
considered pairwise physically equivalent, differing only in parameter values. We consider aqueous
solutions that are uniform and in local equilibrium with the membrane on each of the membrane sides.
3.1. Connecting Leads
The wires connecting the Ag
|
AgCl electrodes to the potentiometer conduct heat as well as charge.
The entropy production per unit volume, σext, is [55]
σext =J0
qx1
Tj1
Txφ. (2)
Here
J0
q
is the measurable heat flux, and
j
is the electric current density. The contribution to the overall
emf is measured under reversible conditions (when
σext =
0). The emf -contribution depends on the
temperature difference as
(xφ)j=0=1
T J0
q
j!dT=0
xT=S
e
FxT(3)
where we have applied the Onsager relation between transport coefficients, see [
56
] for details.
The definition is in accordance with Haase [
58
] and with Goupil et al. [
59
]. The heat transported
reversibly with the electric current is the transported entropy of the charge carrier,
S
e
. For all practical
purposes, it is constant with temperature. We integrate Equation (3) for the a-side and the c-side,
and obtain the first contribution to Equation (1).
extφ=S
e
F[(T0TT)+(TT0)] =S
e
F.T(4)
3.2. The Electrochemical Reaction at the Interface
The electrochemical reaction takes place at the electrode-solution interfaces. The general
expression for the entropy production of the anode contains terms from heat and component fluxes into
and out of the electrode surface, see [
55
] for details. Under reversible conditions only two contributions
are effective, the term due to reaction and the electric potential jump:
σa=0=j
a,aqφ
TrsnGa
T(5)
where
rs
is the chemical reaction rate, and
nG
is the reaction Gibbs energy of the neutral components
of the chemical reaction. The expression for the right-hand side electrode interface (the cathode) are
similar. The reaction rate is proportional to the electric current,
rs=j/F
. The electric potential drop
can be alternatively expressed by the electrochemical potential difference of the chloride ion. The emf
contribution for the anode is therefore
a,aqφ=nGa
F=1
F[µAgCl(T)µAg (T)]. (6)
Entropy 2018,20, 905 5 of 25
There is a similar contribution for the other electrode, c. Subscript (a,aq) denote that the property
belongs to the interface between the solid phases a and the aqueous solution. By adding the two
electrode surface contributions, respectively,
a,aqφ
and
aq,cφ
, we obtain the next contribution to the
emf in Equation (1)
elφ=a,aq φ+aq,cφ=F1(SAg SAgCl )T. (7)
This expression gives the contribution from the electrode reactions to the Seebeck coefficient.
3.3. The Membrane
The emf -measurements take place under reversible conditions, and one can safely assume
equilibrium at the membrane solution interfaces. This makes it convenient to deal with the membrane
as a discrete system [
57
]. The entropy production has contributions from the membrane transport of
heat, mass, and charge. In the membrane frame of reference, we have
σm=J0(2)
qm1
TJw
mµw(T1)
T1
JMCl
mµMCl (T1)
T1
j1
T1
mφ. (8)
In order to arrive at this expression, we eliminated the measurable heat flux at the 1-side
using the constant energy flux through the membrane under steady-state conditions. The choice
of variables is not unique, but a practical one. Four flux equations follow. The equation for the emf is
obtained setting
j=
0 in the equation for the electric current density. The result is well established,
see, e.g., [26,53,56].
mφ=Lφq
T2Lφφ
mTLφw
Lφφ
mµw(T1)Lφe
Lφφ
mµMCl (T1). (9)
Concentration differences contribute to the emf, not only through the last two terms in
Equation (9) but also through the concentration dependence of the entropies. We identify the coefficient
ratios by means of Onsager relations:
Lφe
Lφφ
=Leφ
Lφφ
=1
FJMCl
j/FdT=0,dµi=0
=1
FJM+
j/Fdµi=0
=:tm
M+
F
Lφw
Lφφ
=Lwφ
Lφφ
=1
FJw
j/FdT=0,dµi=0
=:tm
w
F.
(10)
With the present choice of electrodes, the transfer of electrolytes follows the cation flux.
The transport number
tm
M+
is the fraction of the total charge transfer across the membrane carried by
the cation. The water transference coefficient
tm
w
is the average number of moles of water transferred
(reversibly) with electric current through the membrane per Faraday of charge. The reversible
contribution to the heat flux is given by the entropy flux at isothermal conditions. There are two
contributions: the entropies related to the nature of charge transporters and the entropy carried along
with the components.
J0(2)
q/T2=JsSMCl,2JMCl Sw,2 Jw. (11)
We assume that the reversible contribution to the total entropy flux can be attributed solely to the
transport of the ions
(Js)dT=0,dµi=0=S
M+JM++S
ClJCl. (12)
The relation that can be taken to define the transported entropies
S
i
of the ions. The last coefficient
ratio in the expression for the emf can now be identified through its Onsager relation
Entropy 2018,20, 905 6 of 25
Lφq
T2Lφφ
=Lqφ
T2Lφφ
=1
FT2
J0(2)
q
j/F
dT=0,dµi=0
=1
Ftm
M+Sm
M+tm
ClSm
Cltm
M+SMCl,2 tm
wSw,2.
(13)
The emf -contribution from the membrane becomes
mφ=ηm
SmTtm
w
Fmµw(T1)tm
M+
FmµMCl (T1)(14)
where we have defined ηm
Sas
ηm
S:=1
Ftm
M+Sm
M+tm
ClSm
Cltm
M+SMCl,2 tm
wSw,2. (15)
The entropy of the electrolyte is the standard entropy minus a term that depends on the logarithm
of the activity
SMCl =So
MCl 2Rln a±. (16)
The expression for the membrane potential depends thus on the electrolyte activity via the
electrolyte entropy, but also via the last terms in Equation ( 14). We distinguish now between two well
defined experimental situations.
In the first situation, the electrolyte solutions are stirred, and the last terms are zero. In the second
case, the temperature gradient across the membrane leads to separation of components across the
membrane. At the time this has occurred (
t=
,
Jw=
0,
JMCl =
0,
j=
0), there is a balance of forces.
This balance is called the Soret equilibrium. We may neglect coupling between the mass fluxes and
find the balance of forces expressed by
µi,T=q
i
TT i =MCl, w. (17)
The electrolyte contribution (the two last terms of Equation (14)) can be determined in this
case. The electrolyte contribution can also be contracted [
60
] using the Gibbs–Duhem relation for the
electrolyte solution on the integrated form, provided the concentration difference is small:
mµw,T=mMwmµMCl,T(18)
with
m
as the molality and
Mw
as the molar mass of water. By using Equation (18) to eliminate the
chemical potential of water, we have
tm
w
Fmµw(T1)tm
M+
FmµMCl (T1)=2R
Ftm
M+mMwtm
wmln a±. (19)
The bracketed combination is referred to in the literature as the apparent transport number of
the membrane. The chemical potential differences (taken at constant temperature) between the two
sides are most often zero in thermocell measurements. They are not zero, in the Soret equilibrium state,
but this state is difficult to realize in the experiment with membranes, because of the time taken to
obtain the state.
3.4. The Seebeck Coefficient of the Complete Cell
We consider here the experimental situation with identical electrolytes on the two sides of the
membrane. The total cell emf is obtained by adding the contributions derived above.
Entropy 2018,20, 905 7 of 25
At initial time (t=0, no Soret equilibrium), the Seebeck coefficient ηSof the cell is
ηS=ηel
S+ηm
S(20)
where
ηel
S=F1(SAg SAgCl S
e)(21)
and
ηm
S,t=0:=F1tm
M+Sm
M+tm
ClSm
Cltm
M+(So
MCl 2Rln a±)tm
wSw,2. (22)
We combined Equations (15) and (16) in the last step. These relations are suited to interpret
experiments. We obtain a clear separation between electrode contribution, solution dependent
contributions, and membrane-dependent contributions.
4. The emf of the Cell with Calomel Electrodes
Most experiments reported in the literature have been done with calomel electrodes, rather than
with Ag
|
AgCl electrodes. We need the expression for the emf to interpret these experiments. The cell
has Hg
|
Hg
2
Cl
2
electrodes which are kept at the temperature of the potentiometer,
T0
. A salt bridge
with saturated KCl is linking the electrode chambers to the electrolyte solution. The salt bridge on the
left-hand side of the cell is exposed to a temperature difference
TT0
, while the salt bridge on the
right-hand side is exposed to a temperature difference
T+TT0
. The membrane compartments
are kept at temperatures
T
and
T+T
as before. Therefore, there is no net contribution from the
two electrode reactions. There is also no net contribution to the chemical potential differences at
t=
0. As the two electrodes are symmetrically positioned, these contributions cancel. Furthermore,
the expression for the membrane contribution has the same form as for the Ag
|
AgCl system.
When the electrolyte concentrations in membrane compartments are the same, we obtain the membrane
contribution
ηm
S
as defined in Equation
(15)
. The membrane contributions to the emf can thus be taken
from Section 3.
In addition to the contributions discussed, there is a thermocell contribution from the two liquid
junctions. This point has so far been overlooked in the experimental literature.
The entropy production of the salt bridge is
σlj =J0
qx1
TJKCl
TxµKCl,Tj
Txφ(23)
with the measurable heat flux J0
q
T=JsSKCl JKCl. (24)
With arguments similar to those in Section 3, we find the contribution to the emf from the liquid
junctions (salt bridges)
lj φj=0=1
FtK+SKCl S
K++tClS
Cl[(TT0)+(T0TT)]
=1
FtK+SKCl S
K++tClS
Cl.T
(25)
The concentration contributions of the two salt bridges cancel, so that we are left with
a thermoelectric contribution only. In the interface region between the salt bridges and the electrolyte
solutions of MCl, another emf -contribution may arise due to variations in the transport numbers and
chemical potentials of the salts MCl and KCl in this region. In general, these variations give rise to
a bi-ionic potential that will depend on the cation M
+
and the concentrations. For a detailed exposition,
we refer to [
55
]. In the present case, we simply consider M
=
K and assume that the transport number
Entropy 2018,20, 905 8 of 25
tK+=
0.5 throughout the liquid junctions. In this case, the total thermoelectric contribution of the salt
bridges to the cell emf can be quantified as
lj φ=1
2FSKCl,2 +S
ClS
K+T. (26)
We expect that this expression provides a good lowest order approximation to the contribution [
55
].
The observed Seebeck coefficient of a cell with calomel electrodes and KCl electrolyte is
φ
Tj=0
=1
Ftm
K+Sm
K+tm
ClSm
Cl1
2S
K+S
Cltm
K+1
2SKCl,2 tm
wSw,2. (27)
Whenever the electrolyte is different from KCl, we expect non-trivial, but not necessarily large,
deviations from this expression. This complication is avoided when Ag
|
AgCl electrodes are used,
and we therefore recommend these electrodes for measuring thermoelectric potentials.
Theory of Tasaka and Coworkers
Ikeda [
24
] made one of the first attempts to derive the thermoelectric potential,
but Tasaka et al. [
27
] were first to use irreversible thermodynamics. They found the following
expression for the thermoelectric potential in 1965:
ψ
Tj=0
=R
F(2t+1)ln a±+(t+α++tα)(28)
with
α+=ηSo
+
F+τ0So
0
α=η+So
Fτ0So
0.
(29)
Lakshminarayanaiah [
28
] derived a similar expression. Not all details were reported, but
a±
is the
mean salt activity. The
η
appearing in Equation
(29)
was called the differential or pure thermo electric
coefficient. In the terminology used here, (
Fη=tm
M+Sm
M+tm
ClSm
Cl
). The entropy
So
i
is the partial molar
entropy of an ion, and
So
0
is the partial molar entropy of the solvent. The superscript
o
refers to the
reference-state, infinitely dilute system for all components. The quantity
τ0
is the reduced transport
number of water (
τ0=tm
w/F
). The transport numbers
t±
are, in our notation, the membrane transport
numbers
tm
M+
and
tm
Cl
. In order to see the correspondence, we switch to the notation from the previous
section. With this notation, Equation (28) becomes
Fψ
Tj=0
=R(2t+1)ln a±Fτ0So
0(t+S0
+tS0
) + Fη
=R2tm
M+1ln aMCl tm
wSo
wtm
M+So
MCl +So
Cl+tm
M+Sm
M+tm
ClSm
Cl.
(30)
We identified the electrolyte entropy
So
MCl =So
M++So
Cl
. In their derivation, Tasaka and coworkers
tacitly made the assumption
γ+=γ
, with
γi
being the ion activity coefficients. This allowed their
identification
S=So
Rln a±
. Making also the identification
SMCl =So
MCl
2
Rln aMCl
and setting the
water entropy So
wSw, we find
Fψ
Tj=0
=tm
M+Sm
M+SMCltm
ClSm
Cltm
wSw+SCl, (31)
Entropy 2018,20, 905 9 of 25
which gives the electrostatic potential difference
ψ
across the membrane for a given temperature
difference. From invariance of the entropy production, the relation between
ψ
and the
emf -contribution φbetween chloride-reversible electrodes is
φ=ψF1µCl. (32)
We identify here µCl=SClT, such that the corresponding emf -contribution is given by
Fφ
Tj=0
=tm
M+Sm
M+SMCltm
ClSm
Cltm
wSw. (33)
This is exactly our expression in Equation
(15)
for the membrane contribution to the cell emf,
when the electrolyte concentration is uniform. For the KCl electrolyte, the addition of the salt
bridge contributions derived in the previous section gives Equation
(27)
. By writing out the activity
dependence of the electrolyte entropy, we have
Fφ
Tj=0
=tm
K+Sm
K+tm
ClSm
Cl1
2(S
K+S
Cl)tm
K+1
2So
KCl tm
wSw,2 +R(2tm
K+1)ln aKCl,2.(34)
where the observed activity dependence of the Seebeck coefficient is explained principally by the
last term. The term in 2
tK+
is attributed to the membrane. The
1-part, however, is here correctly
attributed to the KCl salt bridges. This crucial point has not been pointed out earlier, and is important
for interpretation of experimental results.
The electrolyte entropy terms provide here a linear dependence of the Seebeck coefficient on the
logarithm of the electrolyte activity. As the electrolyte concentration is varied, the partial molar entropy
of water will also change in general. Neglecting its variation with temperature, it can be related to the
electrolyte activity by means of the Gibbs–Duhem relation
dSw=2RMwmd ln aMCl (35)
with
m
as the electrolyte molality and
Mw
as the molar mass of water. Since the activity and molality
are related by
aMCl =γMClm
, the change in
Sw
will generally not be linear in
ln aMCl
. For small variations
at low concentrations, however, we may approximate
Sw=So
w+2mRMwln aMCl (36)
such that
2Rtm
K+1
2ln aKCl,2 tm
wSw,2 =2Rtm
K+mMwtm
w
| {z }
tm
a
1
2ln aKCl,2 tm
wSo
w(37)
where we have again identified what is commonly known as the apparent transport number
tm
a
of the
membrane. We see from this approximation that water transport gives a slight change in the slope
of the Seebeck coefficient vs.
ln aKCl
. While this analysis may be useful, it may also be good to keep
the two entropy terms separate, because the activity of water in the electrolyte solutions is generally
well known. We will, however, use Equation
(37)
to interpret the dependence of the observed Seebeck
coefficient on the electrolyte activity.
5. Review of Experimental Results
In the review of experimental results that follow below, we will first discuss conditions for good
experimental results. A compilation of data found in the literature is given in Tables A1 and A2.
Entropy 2018,20, 905 10 of 25
We proceed to test the theoretical expressions above using these data. In the end, we address factors
that can enhance Seebeck coefficients of thermocells with ion-exchange membranes.
5.1. Experimental Issues: Temperature Polarization
The Seebeck coefficient shows that the electric potential measured across an ion-exchange
membranes is proportional to the temperature difference between the adjacent electrolyte solutions.
Thus, the Seebeck coefficient can be determined from the slope of the plot of the measured
potential vs. the temperature difference. Unstirred electrolyte layers next to the membrane surfaces
may hamper the determination of the temperature difference across the membrane. If the temperature
between the electrolyte solutions is
T0
, the true temperature difference across the membrane is
T<T0
, and there is a temperature difference
T0T
across the external solution. This situation
is called temperature polarization. Using
T
rather than
T0
may give a wrong estimate of the
Seebeck coefficient [
37
,
48
]. Experimental results have shown that thermal membrane potential for
cation-exchange membranes increases when solutions are stirred [
27
,
37
,
48
,
49
]. In accordance with
this, Seebeck coefficients for anion-exchange membranes were observed to increase with the solution
stirring rate [
43
,
61
]. We expect that whether stirring increases or decreases the observed Seebeck
coefficient depends on the relative strength of the thermoelectric contributions from the membrane
and the external solution. If the membrane has a stronger contribution than the solution, then stirring
will increase the Seebeck coefficient of the system, while if the membrane has a weaker contribution
than the solution, stirring will decrease the observed Seebeck coefficient.
Temperature polarization can be handled by exploiting its dependence on the electrolyte stirring
rate. We assume then that the effect disappears in the limit of infinite stirring. An increase in the
stirring rate will decrease the thickness of the polarized layer and increase
T
across the membrane.
Barragán et al. [
48
] designed a scheme to handle such temperature polarization in a systematic way.
Many of the early works did not take this into account [
36
,
37
,
39
]. Moreover, although the solutions
next to the membrane surface were stirred, the Seebeck coefficient usually refers to an unknown fixed
stirring rate. This makes it difficult to compare results from different authors.
An applied temperature difference may eventually lead to a separation of components.
At infinite times, the chemical potential difference will be balanced by the thermal driving force
(Soret equilibrium). Jokinen et al. [
17
] suggested that this effect explains observed discrepancies
between experimental reports. The Soret effect can be included in the description, as described by
Equation (17). This is not relevant for the short times considered here.
5.2. Properties Derived From Experimental Results
5.2.1. Electrode Contributions
We have seen in the theoretical section that there is a distinct contribution to the thermoelectric
potential from the entropy of electrode components, when the electrodes are held at different
temperatures. These contributions may in certain cases be attractive in the way that they make
the cell potential large. Large contributions were for instance obtained from electrode gas
reactions [
62
]. In any case, they are straightforward, understandable contributions arising from
thermodynamic entropies.
In the analysis of experiments, the contribution due to simple electrodes, such as Ag
|
AgCl,
can be separated from the membrane contribution. Kjelstrup et al. [
47
] studied a thermoelectric cell
with a membrane stack of 55 Nafion 117 membranes in a cylinder of plexiglass. The Ag|AgCl electrodes
were held at different temperatures, and their contribution to the Seebeck coefficient was 0.66 mV/K.
This is the same as recently observed [
63
]. The Seebeck coefficient of a polymer electrolyte membrane
Nafion 1110 in a cell with two hydrogen electrodes was equal to 0.67 mV/K [
14
]. The electrode
contribution to this number was small (0.04 mV/K) with 1 bar of hydrogen pressure on both sides.
Entropy 2018,20, 905 11 of 25
This contribution can change much, however, as the entropy of the gas changes with the logarithm of
the hydrogen pressure (see [62]).
When calomel electrodes have been used to measure thermoelectric potentials, the electrode
reactions have taken place at the same temperature. Nevertheless, there is a temperature gradient
across the liquid junction, different in the two half cells. As we have seen in Equation (30), the electrode
components will not contribute to the emf, and the concentration variation in the two liquid junctions
will not contribute much. The cell symmetry gives a small contribution from variations between the
two junctions [
55
] for KCl solutions. However, the temperature gradient contributions from the two
junctions do not cancel when added. The contribution was here computed with good accuracy for
a cell with KCl as an electrolyte. With other electrolytes, bi-ionic potentials need to be added [55].
A measurement of the thermoelectric potential of a cell using Ag
|
AgCl electrodes avoids this
uncertainty. These electrodes can also be used for all chlorides, as well as mixtures, and such
a measurement is therefore our recommendation for new experiments with such electrolytes.
A novel electrode concept, introduced by Sales et al. [
64
], may extend the choice of electrodes.
Their concept combined an ion-exchange membrane and porous carbon electrodes immersed in
aqueous electrolytes. These electrodes permitted the extraction of electric energy from small
temperature differences, taking advantage of the created transient temperature gradient. Hu et al. [
65
]
used carbon multiwalled nanotube (MWNT) buckypaper electrodes in a thermo-electrochemical cell,
obtaining three times higher efficiencies than in a similar device with Pt electrodes. This suggests that
nanostructured electrode materials with fast kinetics and large amounts of electrochemical accessible
internal surface area might also significantly enhance the efficiency of a thermoelectric cell with
ion-exchange membranes.
Abrahams et al. [
66
] obtained
1.41 mV/K using Pt electrodes and a 0.4 M aqueous solution
of K
3
Fe(CN)
6
/K
4
Fe(CN)
6
. The electron transfer reaction in this cell involves complexes with high
entropy, explaining the large value. Electrolyte complexing agents such as EDTA [
67
] may similarly
produce large effects due to changes in entropy upon complexation.
Electrode reactions which involve gases or complexing agents may enhance the thermoelectric
effect [
56
]. The emf is not only proportional to the temperature difference across the cell electrodes;
there is also an effect of a pressure difference. This is the so-called streaming potential. The possibility
to add the streaming potential to the Seebeck coefficient is interesting, and not yet exploited (see,
however, Sandbakk et al. [
15
]). These authors reported a large enhancement of the isobaric Seebeck
coefficient by application of a pressure difference.
5.2.2. Electrolyte Property Variations
As we have seen in the theoretical section, the Seebeck coefficient of a cell with an ion-exchange
membrane has a predicted variation with the electrolyte activity via the term
R
F(
2
tM+
1
)ln a±
cf.
Equation
(34)
. In order to discuss how this proportionality is reflected in published data, we are
plotting
ηS
vs.
ln a±
. Most data are available for calomel electrodes, so we use Equation (34) for
the analysis.
A dependence on
log10 a±
was documented early [
26
,
27
,
30
,
31
,
33
35
,
37
,
43
,
45
,
46
,
49
,
52
]. Figures 2
and 3give a summary of results obtained with calomel electrodes and KCl as electrolyte with
cation- and anion-exchange membranes, respectively. The observed relation was similar when
other electrolytes were used [
31
,
37
,
43
,
51
], and the slopes were almost the same as the theoretical
value 0.1984 mV/K obtained for ideal selective membranes, neglecting the effect of water transport.
They were interpreted using the Tasaka expression (Equation (28)).
A deviation from the theoretical ideal value was observed, however, only for increasing
electrolyte concentrations for some membrane systems. The deviation was attributed to less selective
membranes [
30
,
33
,
37
]. This could be an explanation. However, according to Equation
(34)
, there
can also be contributions from water transport and from the liquid junction as well. The entropy of
water depends on the electrolyte entropy via Gibbs–Duhems relation
(35)
. It provides a term that is
Entropy 2018,20, 905 12 of 25
generally not linear in
ln a±
and provides a change in the linear slope for small concentration variations.
Whether this contribution increases or decreases, the value of the slope depends on the sign of
tm
w
.
The sign gives the direction of water transport in the membrane, which tends to be positive for cation
exchange membranes and negative for anion exchange membranes.
By using Equation
(37)
, we estimated a mean value of the apparent transport number for each
concentration interval of the curve. For the cation exchange membranes shown in Figure 2, we obtained
values between 1 and 0.75. The higher the Seebeck coefficient is, the higher the apparent transport
number is. The results of Ikeda et al. [
61
] also showed an increase of the thermal membrane potential
with the apparent transport number in cation-oxidized collodion—and anion aniline-formaldehyde
resin-cellophane matrix membranes. They found that the thermoelectric potential increased with
the apparent membrane transport number. They did not find the linear trend expected from theory
with constant heats of transfer in the membrane. As explanation, they suggested that the heats of the
transfer in the membrane were a function of the apparent transport number in the membrane. For the
anion exchange membranes shown in Figure 3, we obtained apparent transport numbers that were
always smaller than 0.11. They showed less variation. The interpretation of these results is unclear,
as long as water transference coefficients and the liquid potential contribution is unknown.
  
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Figure 2.
Seebeck coefficients for cells with cation-exchange membranes and a KCl electrolyte of
varying concentration, plotted versus the logarithm of the mean salt activity. Calomel electrodes were
used in the measurements.
For ideally permselective membranes, we do not expect any effect of co-ions on the thermoelectric
potential. This property is seen in Figure 4for membranes of each type, and for concentrations up
to 0.1 M. For these two membranes, the value of the counter-ion transport number estimated from
the corresponding slopes in Figures 2and 3was practically unity. Similar results about the influence
of co-ion were found for all the membranes showing linear behavior of the Seebeck coefficient vs.
the logarithm of the mean salt activity. Unfortunately, no results have been found about the effect of
co-ions on the thermoelectric potential for the membranes that deviate from ideal behavior. A loss of
selectivity of the membrane would involve a larger influence of the co-ion nature at higher electrolyte
concentrations, and it would permit analysis if other terms in Equation (34) cause the deviation from
the ideal behavior.
We have seen that the sign of the thermoelectric potential of the calomel electrode cell follows
mostly from the type of ion-exchange, cation, or anion. This can also be concluded from the term
R(
1
2
tm
M+)ln a±
. Can the thermoelectric potential of a cell with a particular membrane change its
sign? Hanaoka et al. [
43
,
46
] suggested this possibility at high electrolyte molalities for some membrane
Entropy 2018,20, 905 13 of 25
system, for instance, for the Neosepta AM-1 membrane with KIO
3
at concentrations higher than
0.1 mol/kg [
43
]. A sign reversal was observed with the perfluorosulfonic cation-exchange membrane
Flemion S [
46
], for which the thermal membrane potential in HCl solution varied from positive to
negative for concentrations above 0.05 mol/kg. They attributed this effect to a water contribution
to the thermal membrane potential. This may, however, also be an effect of the bi-ionic potential in
the interface region between the salt bridge and the electrolyte solution. The term
R(
1
2
tm
M+)ln a±
in Equation
(34)
may change sign with the transport number and the logarithmic term. The water
transference coefficient
tm
w
may also contribute to this behavior through the dependence of the water
entropy on the electrolyte activity.
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Figure 3.
Seebeck coefficients for cells with anion-exchange membranes and a KCl electrolyte of varying
concentration, plotted versus the logarithm of the mean salt activity. Calomel electrodes were used in
the measurement.
Figure 4.
The effect of co-ions on the Seebeck coefficient at different electrolyte molalities [
31
].
(a) Amberlite XE-69 cation-exchange membrane. (b) Amberlite XE-119 anion-exchange membrane.
5.2.3. Membrane Structure, Counter Ion, and Water Content
In this review, we examine the contributions to the Seebeck coefficient and how they vary.
We have seen in the theoretical section that membrane contributions to the Seebeck coefficient
come from the type of membrane (structure), the apparent transport number (the water transference
number and the ion transport number), and the transported entropy of the ions in the membrane.
Membrane structure and water content may have an impact on all transport properties.
Entropy 2018,20, 905 14 of 25
Membrane structure
Heterogeneous ion-exchange membranes contain more than one charged polymer. It was found
that heterogeneous ion-exchange membranes have in general larger thermoelectric potentials
than homogeneous ion-exchange membranes, both cationic and anionic membranes [
31
,
37
].
Laksminarayanayah [
26
] and Huda et al. [
52
] studied the effect of crosslinking of membrane
polymers. The degree of crosslinking of the polymethacrylic acid membrane did not have
much impact [
26
] on the Seebeck coefficient. But membranes made of poly(4-vinylpyridineco-
styrene) [
45
] gave thermoelectric potentials that increased with an increase in weight fraction
of hydrophobic hydrocarbon matrix, or in the molality of fixed charges. Poly(styrene)-based
copolymer anion-exchange membranes with divinylbenzene were investigated in the presence
of various electrolyte solutions [
52
]. Positive contributions were found with KIO
3
solutions,
and negative for more hydrophobic Cl
-form membranes. The water term seemed predominant
for IO
3
-form membranes with high water contents. The value of the thermoelectric potential was
always negative, which seems to show that the contribution from water on the thermal membrane
potential is significant. More systematic studies of the degree of cross-linking as well as membrane
hydrophobicity may help obtain better membranes for thermoelectric energy conversion.
Membrane counter-ion
The type of ion in the membrane has a clear impact. Figure 5a,b show Seebeck coefficients
for different ion-exchange membranes as a function of the radius of unhydrated counter-ion.
We see that, in general, the Seebeck coefficient for cationic membranes increases with the atomic
number of the cation involved. With HCl and alkali metal chloride solutions, the thermoelectric
potential varies roughly with the inverse of the radius of the ions. However, the opposite trend was
also observed for some membrane systems [
26
,
46
,
48
]. With anion-exchange membrane Amberlite
X-119, the magnitude of Seebeck coefficient increased with increasing molecular weight of halogen
ions except for F
ion. Similar results were found by Huda et al. [
52
] with poly(styrene)-based
copolymer anion-exchange membranes. This property may have some bearing on the transported
entropy which enter the expression for ηS.
Kiyono et al. [
44
] reported that the size of the Seebeck coefficient depended on the ion-exchange
capacity and the membrane water content. They found larger Seebeck coefficients for
KCl-solutions than for NaCl, but this was at variance with results of Laksminarayanaiah [
28
]. It is
difficult to see that ion-exchange capacity per se should have an impact on
ηS
. Indirectly it may
play a role, through its impact on the transported entropies, however. No systematic study has
been made of the transported entropy and its dependencies.
At constant selectivity, cation oxidized collodion and anion cellophane membranes gave
an absolute value of the thermoelectric potential higher for anion-exchange membranes than
for cation-exchange membranes using calomel electrodes [
61
]. Tasaka et al. [
31
] measured
thermoelectric potentials between 0.85 mV/K and
0.9 mV/K, respectively, with Amberlite XE-69
cation-exchange membranes and Amberlite XE-119 anion-exchange membranes and a 10
3
M
aqueous KCl solutions.
Water content
It is difficult to separate the effects of water content from that of, say, the ion exchange capacity.
The properties are interrelated and the first of them is strongly dependent on the membrane
structure. Both properties have only an indirect effect through their impact on the transport
numbers and transported entropies appearing in Equation
(34)
. Nevertheless, the larger Seebeck
coefficients have been found for membranes with low water content [
36
,
43
,
45
]. Kiyono et al. [
45
]
atributed this behavior to the restricted movement of ions in the membrane with small water
content and high weight fraction of hydrophobic backbone. A hydrophobic membrane matrix
affects the state of the counter-ion, and therefore also the transported entropy in membrane.
Entropy 2018,20, 905 15 of 25
Huda et al. [
52
] altered the hydrophobicity of anion-exchange membranes by changing the length
of the alkyl chain of benziltrialkilammonium groups. They observed that the thermoelectric
potential decreased roughly with hydrophobicity of the ion-exchange groups.
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1D
.
5E
&V
Figure 5.
The effect of counter-ions on the Seebeck coefficient for (
a
) different cation-exchange
membranes and (
b
) for Amberlite XE-119 anion-exchange membrane [
31
]. Lines are only a visual guide.
Radii values taken from [68].
In general, the absolute value of thermoelectric potential decreases with the membrane water
content (Figure 6). Tasaka et al. [
37
] also related the decrease in the absolute value of the
thermoelectric potential to the water content, leading to a much smaller value of the transported
entropy of counter-ion. As can be seen in Figure 6, anion-exchange membranes on the whole
have relatively low water content (high apparent transport numbers). It may be explained by the
smaller deviation from the theoretical slope of the Seebeck coefficient with the logarithm of the
electrolyte concentration observed for anion-exchange membranes (Figure 3). The relatively small
ion-exchange capacity does not have much of an impact. The cation-exchange membrane Ionics
61CZL386, with a high ion-exchange capacity, shows a Seebeck coefficient similar to those of other
membranes with lower IEC value. In Equation
(34)
, a high water content may mean a high water
transport number, positive for cation-exchange membranes [
60
]. The membrane structure may
have an impact on the transported entropy of the counter-ion in the membrane phase.
Entropy 2018,20, 905 16 of 25
When we compare the anion-exchange membranes Nepton AR-111 (homogeneous) and Nepton
XE-119 (heterogeneous), we find a higher Seebeck coefficient for the heterogeneous XE-69
membrane with large water content. High water transport numbers are expected for membranes
with high water contents. The water transport numbers are negative in anionic membranes [60],
so according to Equation
(34)
, a higher water content would increase the Seebeck coefficient.
This explanation agrees with the observed experimental results.
The Seebeck coefficient depends on the solvent, but aqueous solutions seem favorable for a high
value [66].
0HPEUDQH
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
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
,(&PHTJ
:DWHU8SWDNH
P9.
Figure 6.
Ion-exchange capacity, water uptake, and Seebeck coefficient (in 0.001 m KCl) for different
ion-exchange membranes. Cation-exchange membranes, C1: Nepton CR-61 [
37
], C2: Nepton CR-51 [
37
],
C3: Interpolymer [
37
], C4: XE-69 [
31
], C5: Ionics 61CZL386 [
48
]. Anion-exchange membranes,
A1: XE-119 [
37
], A2: Nepton AR-111 [
37
], A3: Aciplex A-201 [
43
], A4: Aciplex A-211 [
43
], A5: MPS-10
[46], A6: HPS-2 [46].
5.2.4. Membrane Temperature: Other Barriers to Transport
The measured potential depended not only on the applied temperature difference but also on the
mean temperature of the membrane [
49
]. For the Ionics 61 CZL 386 cation-exchange membrane and
a KCl electrolyte solution, a maximum value was found at 293 K. It was related to the water content of
the membrane and to the transported entropies.
Leinov et al. [
69
] reported thermoelectric coefficients of intact sandstone samples saturated
with NaCl brine. With Ag
|
AgCl electrodes at different temperatures, values were obtained between
0.370 (low salinity, 10
4
M) and 0.055 mV/K (high salinity, 1 M). The sandstone can be seen as
a non-selective membrane, which explains the relatively low Seebeck coefficient. It is unclear how it
can be used for power generation.
6. Competing Thermoelectric Materials
Many materials show thermoelectric effects, and a tailoring to purpose is relevant.
Härtel et al. [
70
] proposed a novel heat-to-current converter using charged supercapacitors.
With commercially available supercapacitors, they observed a coefficient near 0.6 mV/K.
Organic materials, in particular intrinsically conducting polymers, can be considered as competitors
to classical thermoelectric materials (semiconductors), and their figure of merit has been improved
several orders of magnitude in the last few years [71].
Hu et al. [
65
] showed that carbon-nanotube-based thermo-electrical cells could be used to harvest
waste thermal energy. Values of 1.4 mV/K were obtained with carbon MWNT electrodes. Cells with
ion-exchange membranes may compete well with these for low temperature waste heat utilization.
Xu et al. [
72
] studied the possibilities of electrolyte solutions confined in nanopore carbon. Using MD
Entropy 2018,20, 905 17 of 25
simulations, they obtained that the inhomogeneous ion/charge distribution leads to a net electrical
potential difference across the solid/liquid interface with energy conversion efficiencies compared
with those of conventional thermoelectric materials.
Børset et al. [
62
] reported the Seebeck coefficient of thermoelectric cells with gas electrodes and
a molten electrolyte of lithium carbonate at high temperatures. The presence of an inorganic oxide
enhanced the result, increasing it from 0.9 to 1.2 mV/K. Kandhasamy et al. [
73
] reported larger Seebeck
coefficients (1.5 mV/K) for lower temperatures using multi-component mixtures.
7. The Potential for Thermoelectric Energy Conversion
For thermoelectric energy conversion, not only the Seebeck coefficient is essential. Other properties
need also be considered. To bring out the potential of ion-exchange membranes for thermoelectric
energy conversion, we also need to know the thermal and electric conductivity. At a mean temperature
T, the three transport properties can be combined in the lumped variable N.
N:=Tκη2
S
λ, 0 N1 (38)
where
κ:=j
φ/dmT=0
λ:=
J0(1)
q+J0(2)
q
2T/dm
φ=0
(39)
are the electrical conductivity at constant temperature and the thermal conductivity of the short-circuited
membrane, respectively, and
dm
is the membrane thickness. The quantity
N
has a lower and an upper
bound, stemming from the requirement that the determinant of transport coefficients is positive [55].
The open circuit thermal conductivity,
λj=0
, is related to the conductivity at short-circuit
conditions, as follows:
λj=0=λ(1N). (40)
The factor ZT is the so-called figure of merit, defined first by Ioffe [74]. It is related to Nby
ZT :=Tκη2
S
λj=0
=N
1N. (41)
Furthermore,
ZT
is a non-negative quantity, but now there is no upper bound. When the figure
of merit is equal to zero, there is no coupling between heat and charge transfer, and the process is
completely irreversible. The other extreme is a completely reversible process, where N1 and thus
ZT
. In this situation, the entropy production is (close to) zero; there is no loss of work. In practice,
we may seek to maximize
N
or
ZT
, which means that we want a small thermal conductivity, in order
to inhibit irreversible heat transfer, and a large ionic conductivity, in order to reduce ohmic losses.
In a dynamic approach, one may choose as objective function for minimization, the total entropy
production of the system. It is then possible to also include geometric variables and constraints.
This has been done successfully for several unit operations [75].
Ion-exchange membranes are widely used in many applications involving electrochemical energy
conversion, such as reverse electrodialysis or fuel cells. The electrochemical process efficiency (voltage
efficiency) depends mainly on the overall electric resistance of the device, which is significantly affected
by the electrical conductivity of the membrane. The ionic conductivity of ion-exchange membranes
has been widely studied in the literature. Temperature, type and concentration of the electrolyte,
and membrane water content are parameters that strongly affect the ionic conductivity of the
membrane [
76
80
]. A typical value for ionic conductivity of Nafion membranes at room temperature
is 2.3 Sm1[77].
Data about thermal conductivity of ion-exchange membranes are, however, scarce. For Nafion
membranes values between 0.18 and 0.25 WK
1
m
1
have been measured at 20
C, depending on its
Entropy 2018,20, 905 18 of 25
water content [
81
]. Values in the range 0.10 and 0.20 WK
1
m
1
were also reported [
82
] for Nafion
membranes, depending on the temperature. The thermal conductivity is found to decrease with
increasing temperature [82,83].
An interest in thermoelectric studies has been recently increasing because its also gives direct
access to a quantity more difficult to measure, namely the Peltier heat. It is increasingly recognized that
local temperature effects and profiles can affect the performance of a fuel cell [
84
,
85
]. The temperature
profile of such a cell cannot be accurately modeled without knowledge of the (large) Peltier effects [
14
].
It was probed that thermoelectric contributions to the cell potential can improve the cell performance
in the otherwise isothermal reverse electrodialysis cell [20,63].
The importance of the thermoelectric phenomena, their inherent symmetry and impact,
lead us to conclude that there are many reasons to increase this area of research and obtain a more
complete picture.
8. Conclusions and Perspectives
We have seen above that the following can be said about the Seebeck coefficient of thermoelectric
cells with ion-exchange membranes:
It typically varies between absolute values 0.4 and 1 mV/K. Major contributions to this value come
from the electrode compartment (including the electrodes) and from water and ion transport
in the membrane. These values are relatively large, compared to values for semiconductors.
In the calomel electrode system, the sign of the coefficient is typically positive for cation-exchange
membranes and negative for anion-exchange membranes. This sign change is attributed mainly
to the thermoelectric potential across the salt bridges in the system, but also to water transport
when the electrolyte concentration is high.
It predicts a reduction for ideal membranes with the logarithm of the electrolyte activity.
This variation is experimentally validated.
It has a possible optimal value with respect to the mean temperature across the membrane.
It does not depend significantly on water content in spite of its dependence on the water
transference number.
It does not depend significantly on the membrane ion exchange capacity. One explanation is that
the transport number is the ratio of the ionic and the total conductivity.
It depends on membrane heterogeneity, probably because this has an impact on the transported
entropies of the ions in the membrane.
It increases in absolute value with the radius of the un-hydrated counter-ion.
It may have positive contributions from a membrane pressure difference.
In order to have access to the membrane dependent terms in the Seebeck coefficients, it is
advantageous to use electrodes without salt bridges, such as the Ag|AgCl-electrode.
While most of these conclusions are as expected from theory, the ones related to membrane
structure and water transport cannot be predicted. They also cannot be given a simple explanation,
as they are still not fully understood (see, e.g., [
60
]). Renewable energy technologies [
8
,
15
,
18
] could
benefit from more systematic studies of the impact of the properties on the transported entropies and
on the water transference coefficient. The trends in the true Seebeck coefficient that depends on these
properties are summarized in Table 1.
Electrode choices and electrolyte conditions, including pressure gradients [
15
], can help increase
the value of the Seebeck coefficient. A temperature difference can lead to a thermo-osmotic pressure.
The magnitude as well as the direction of the thermo-osmotic water transfer depends on the
membrane [
86
]. This additional term is little investigated. The understanding of the interplay of
thermoelectric and thermo-osmotic phenomena in membrane systems is still lacking and should
be pursued.
Entropy 2018,20, 905 19 of 25
Table 1. Parameters that can increase the Seebeck coefficient of cells with ion-exchange membranes.
Increasing Parameter Trend Section Conclusion, cf. Bullet Point No
Electrolyte concentration decreasing, logarithmic dependence 5.2.2 1, 2
Counter-ion un-hydrated radius increasing 5.2.2 7
Membrane hydrophobicity decreasing 5.2.3 6
Mean temperature possible optimum 5.2.4 3
Pressure difference linear increase 5.2.1 8
Not only is the Seebeck coefficient important when it comes to the application of these cells,
but the thermal and electric conductivities are as well, as they enter in the figure of merit or the entropy
production. Thermal conductivities should be small to help maintain a large temperature difference
across the membrane, while electric conductivities should be large to reduce ohmic losses [
75
,
87
]. It is
a challenge to find ion-exchange membranes that answer to all demands. Finding these is, however,
of importance for a large number of interesting waste heat utilization processes.
Author Contributions:
All authors contributed equally to the manuscript and have read and approved the
final manuscript.
Funding:
This research was funded by the Research Council of Norway through its Centers of Excellence funding
scheme, project number 262644, PoreLab.
Acknowledgments:
The authors are grateful to the Research Council of Norway through its Centers of Excellence
funding scheme, project number 262644, PoreLab.
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A
Table A1. Seebeck coefficient, (φ/T), for cation-exchange membranes.
Membrane Type Electrolyte Seebeck Electrodes
(Reference) Coefficient
(103VK1)
Collodion KCl 1.0 ×101M 0.052 Ag|AgCl
Ikeda [24]
Oxidized collodium Apparent transport KCl 1.0 ×101M 0.03 Calomel
Ikeda [61] number 0.9
PMA Crosslinked LiCl 1.0 ×102M 1.00 Ag|AgCl
Laksminarayanaiah [26] polymetrarylic acid NaCl 0.97
KCl 0.95
RbCl 0.77
CsCl 0.82
Amberlite XE-69 Heterogeneous LiCl 1.0 ×103M 0.70 Calomel
-polyvinyl chloride polystyrenesulfonate 5.0 ×103M 0.60
40% 1.0 ×102M 0.52
Tasaka et al. [30,31] 5.0 ×102M 0.42
NaCl 1.0 ×103M 0.81
5.0 ×103M 0.70
1.0 ×102M 0.60
5.0 ×102M 0.50
KCl 1.0 ×103M 0.85
5.0 ×103M 0.72
1.0 ×102M 0.66
5.0 ×102M 0.54
KBr 1.0 ×103M 0.83
5.0 ×103M 0.75
1.0 ×102M 0.65
5.0 ×102M 0.50
Kl 1.0 ×103M 0.83
5.0 ×103M 0.75
1.0 ×102M 0.65
5.0 ×102M 0.55
Entropy 2018,20, 905 20 of 25
Table A1. Cont.
Membrane Type Electrolyte Seebeck Electrodes
(Reference) Coefficient
(103VK1)
o-m-1 Oxidized Collodion KCl 1.0 ×103M 0.27 Calomel
Tasaka et al. [30,33] 5.0 ×103M 0.17
1.0 ×102M 0.13
5.0 ×102M 0.07
i-m-1 Collodion-sulfonated KCl 5.0 ×103M 0.15 Calomel
Tasaka et al. [33] Polystyrene interpolymer 1.0 ×102M 0.11
5.0 ×102M 0.05
M-10/0 Heterogeneous Sulfonate LiCl 1.0 ×103M 0.70 Calomel
Tasaka [34,37] polystyrene+ PVC (40%) NaCl 0.80
KCl 0.84
RbCl 0.89
CsCl 0.95
NH4Cl 0.93
Nepton CR-61 Homogeneous Sulfonate KCl 1.0 ×103M 0.66 Calomel
Tasaka [37] polystyrene
Nepton CR-51 Homogeneous Sulfonated KCl 1.0 ×103M 0.63 Calomel
Tasaka [37] phenol resin
Interpolymer Sulfonated polystyrene KCl 1.0 ×103M 0.49 Calomel
Tasaka [37] +dried collodion (95%)
Flemion S Porous perfluorosulfonic HCl 1.0 ×102M 0.053 Calomel
Hanaoka et al. [46] acid-type LiCl 0.48
NaCl 0.46
KCl 0.59
NH4Cl 0.62
Nafion 117 Homogeneous Sulfonated HCl 1.0 ×102M 0.66 Ag|AgCl
Kjelstrup et al. [47]
Ionics 61 CZL 386 Heterogeneous sulfonated KCl 7.5 ×104M 0.53 Calomel
Barragán et al. [48] 1.0 ×103M 0.51
2.5 ×103M 0.45
5.0 ×103M 0.42
7.5 ×101M 0.39
LiCl 1.0 ×103M 0.75
NaCl 0.66
KCl 0.51
CsCl 0.44
Nafion 1110 Homogeneous Sulfonated HCl 0.67 Hydrogen
Kjelstrup et al. [14]
Table A2. Seebeck coefficient, (φ/T), for anion-exchange membranes.
Membrane Type Electrolyte Seebeck Electrodes
(Reference) Coefficient
(103VK1)
Celophane Anilline-formaldehyde resin- KCl 1.0 ×101M0.05 calomel
Ikeda [61] cellophane matrix
Amberlite XE-119 Heterogeneous LiCl 1.0 ×103M0.9 calomel
polyvinyl chloride polystyrene with 5.0 ×103M0.8
40% quaternary ammonium 1.0 ×102M0.7
Tasaka et al. [30] 5.0 ×102M0.6
NaCl 1.0 ×103M0.9
5.0 ×103M0.8
1.0 ×102M0.7
5.0 ×102M0.6
KCl 1.0 ×103M0.9
5.0 ×103M0.8
KF 1.0 ×103M0.7
5.0 ×103M0.6
1.0 ×102M0.5
5.0 ×102M0.40
KBr 1.0 ×103M0.85
5.0 ×103M0.73
1.0 ×102M0.6
5.0 ×102M0.52
KI 1.0 ×103M0.8
5.0 ×103M0.75
1.0 ×102M0.62
Entropy 2018,20, 905 21 of 25
Table A2. Cont.
Membrane Type Electrolyte Seebeck Electrodes
(Reference) Coefficient
(103VK1)
Aciplex A-201 Hydrocarbonsulfonic acid KCl 1.0 ×103M0.761 calomel
Hanaoka et al. [43] 3.0 ×103M0.65
1.0 ×102M0.55
3.0 ×102M0.45
1.0 ×101M0.381
KIO31.0 ×103M0.64
1.0 ×101M0.36
Neosepta AM-1 Hydrocarbonsulfonic acid KCl 1.0 ×103M0.76 calomel
Hanaoka et al. [43] 3.0 ×103M0.68
1.0 ×102M0.58
3.0 ×102M0.46
1.0 ×101M0.381
KIO31.0 ×103M0.551
1.0 ×101M0.168
Aciplex A-211 Hydrocarbonsulfonic acid KCl 1.0 ×103M0.6 calomel
Hanaoka et al. [43] 3.0 ×103M0.5
1.0 ×102M0.4
MPS-10 Poly(4-vinylpyridine- KCl 1.0 ×103M0.86 calomel
Kiyono et al. [45] co-styrene) with methyl iodide 3.0 ×103M0.78
1.0 ×102M0.72
3.0 ×102M0.64
1.0 ×101M0.58
HPS-2 Poly(4-vinylpyridine- KCl 1.0 ×103M0.90 calomel
Kiyono et al. [45] co-styrene) with 3.0 ×103M -0.82
1.6-dibromo-n-hexane 1.0 ×102M0.73
3.0 ×102M0.64
1.0 ×101M0.54
M-1(4.5) Polysterine-based copolymer KCl 1.0 ×102M0.592 calomel
Huda et al.[52] +4.5% divinylbencene KIO30.485
M-1(8) Polysterine-based copolymer KCl 1.0 ×102M0.475 calomel
Huda et al.[52] +8% divinylbencene
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... liquid-solid slip [1,[21][22][23]. However, the possibility to use nanofluidic systems as thermoelectric converters has only been discussed very recently [24][25][26][27][28][29]. Traditional thermoelectric semiconductors offer high thermoelectric performance at room temperature, but their use is limited owing to their toxicity and rarity [26]. ...
... However, the possibility to use nanofluidic systems as thermoelectric converters has only been discussed very recently [24][25][26][27][28][29]. Traditional thermoelectric semiconductors offer high thermoelectric performance at room temperature, but their use is limited owing to their toxicity and rarity [26]. With that regard, EK effects have been studied and some analytical models have been suggested [26,[30][31][32][33][34][35][36]. ...
... Traditional thermoelectric semiconductors offer high thermoelectric performance at room temperature, but their use is limited owing to their toxicity and rarity [26]. With that regard, EK effects have been studied and some analytical models have been suggested [26,[30][31][32][33][34][35][36]. In particular, a recent theoretical study reported enhanced Seebeck coefficient in confined electrolyte solutions [30,31], an effect explained by the electrostatics of the EDL, in the spirit of a standard picture developed by Derjaguin and collaborators [37,38]. ...
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... Ion-exchange membranes have been suggested as alternative to semiconductors. Seebeck coefficients of the order of mVK −1 have been reported with ion-exchange membranes depending on the experimental conditions [7]. They have lower manufacturing cost and may have the capacity to produce electricity from low grade waste heat [8,9]. ...
... It was found that heterogeneous ion-exchange membranes have larger thermoelectric potentials in general than homogeneous ion-exchange membranes, both cationic and anionic [10,11]. Moreover, in non-isothermal electrochemical cells, also named thermocells, electrode choices and electrolyte conditions can help increase the value of the Seebeck coefficient [7]. ...
... Kim et al. [9] studied the thermoelectric effect comparing behaviours of three different types of solid-state polyelectrolytes. Despite growing interest, studies of the output electrical power in membrane-based thermocells are still scarce and most of them refer to open-circuit configuration to estimate the Seebeck coefficient [7]. ...
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Thermocells are non-isothermal electrochemical cells used to convert thermal energy into electricity. In a thermocell, together with the ion flux, heat is also transferred, which can reduce the temperature gradient and thus the delivered electric current. A charged membrane used as a separating barrier in the electrolyte liquid could reduce this problem. Therefore, the use of ion-exchange membranes has been suggested as an alternative in terms of thermoelectricity because of their high Seebeck coefficient. Ion transfer occurs not only at the liquid solution but also at the solid membrane when a temperature gradient is imposed. Thus, the electric current delivered by the thermocell will also be highly dependent on the membrane system properties. In this work, a polymeric membrane-based thermocell with 1:1 alkali chloride electrolytes and reversible Ag|AgCl electrodes at different temperatures is studied. This work focuses on the experimental relation between the short-circuit current density and the temperature difference. Short-circuit current is the maximum electric current supplied by a thermocell and is directly related to the maximum output electrical power. It can therefore provide valuable information on the thermocell efficiency. The effect of the membrane, electrolyte nature and hydrodynamic conditions is analysed from an experimental point of view.
... Further development of these devices seems restricted by expensive nanoscale engineering [5]. This gives a motivation to explore different materials for thermoelectric energy conversion -ion exchange membranes, see Ref. [6]. Such membranes are already being used to develop saline power plants, like reverse electrodialysis (RED) [7]. ...
... Barragán et al. [15] reported a correction procedure to account for these effects. While the idea to measure thermoelectric potentials of ion exchange membranes is not new [16][17][18], see Ref. [6] for a detailed review, we present results for other membranes. While it is beneficial to raise the average temperature of the unit cell [18], an effect of a temperature difference in practical applications has not been reported. ...
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... This salinity-related thermoelectric response is previously reported from commercially available cation exchange membranes ( Figure S9). 41 Note that the narrow interplanar channels can exert a compression effect or dehydration on the potassium ions having a similar hydration shell size with the effective channel height. 15,49,50 This implies that for larger cations than a nanochannel, the steric effect with increased hydrated sizes can limit the cationic permeation through the nanochannels, producing lower photothermal voltages. ...
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With recent growing interest in biomimetic smart nanochannels, a biological sensory transduction in response to external stimuli has been of particular interest in the development of biomimetic nanofluidic systems. Here we demonstrate the MXene-based subnanometer ion channels that convert external temperature changes to electric signals via preferential diffusion of cations under a thermal gradient. In particular, coupled with a photothermal conversion feature of MXenes, an array of the nanoconfined Ti3C2T x ion channels can capture trans-nanochannel diffusion potentials under a light-driven axial temperature gradient. The nonisothermal open-circuit potential across channels is enhanced with increasing cationic permselectivity of confined channels, associated with the ionic concentration or pH of permeant fluids. The photothermoelectric ionic response (evaluated from the ionic Seebeck coefficient) reached up to 1 mV·K-1, which is comparable to biological thermosensory channels, and demonstrated stability and reproducibility in the absence and presence of an ionic concentration gradient. With advantages of physicochemical tunability and easy fabrication process, the lamellar ion conductors may be an important nanofluidic thermosensation platform possibly for biomimetic sensory systems.
... The ionic thermoelectric response under light is characterized with the ionic Seebeck coefficient calculated from an open-circuit voltage through ionic channels by a given temperature difference. 41 The employed polydimethylsiloxane (PDMS) layers exhibiting excellent optical transparency and thermal insulation are beneficial for suppressing the optical scattering or thermal dissipation under light illumination. Particularly, due to the encapsulated structure by PDMS, interferential factors influencing the thermoelectric response could be ruled out, such as thermal evaporative cooling and derived fluidic motion under light. ...
... This salinity-related thermoelectric response is previously reported from commercially available cation exchange membranes ( Figure S9). 41 Note that the narrow interplanar channels can exert a compression effect or dehydration on the potassium ions having a similar hydration shell size with the effective channel height. 15,49,50 This implies that for larger cations than a nanochannel, the steric effect with increased hydrated sizes can limit the cationic permeation through the nanochannels, producing lower photothermal voltages. ...
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... The Seebeck effect is a phenomenon, where a temperature gradient ∇T generates an electric voltage/potential. 24 The reverse process; i.e., an electric current builds up a temperature difference, is referred to as the Peltier effect. Both effects are associated with the same physical process. ...
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In this work, we investigate the thermoelectric properties of aqueous KCl solutions confined in graphene nanochannels through molecular dynamics simulations. The channel height H ranges from 0.7 to 7.8 nm. It is found that the Seebeck coefficient, S e , and the figure of merit, ZT, of the KCl solution are highly sensitive to H when H is small. For the nanochannel of H = 1.0 nm, S e = 30.6 mV/K and ZT = 4.6 at room temperature, which are superior to most of the solid-state thermoelectric materials. The remarkable thermoelectric properties in small channels are attributed to the flow slip at the channel walls and the mean excess enthalpy density of the solution, which is mainly from the potential energy contribution. The molecular insight promotes the applications of nanofluidic devices for thermal energy harvesting.
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Industrial processes for the production of metals and alloys by metallurgical and electrochemical methods generate a lot of waste heat due to irreversible losses. This waste heat may be used as a power source to generate electricity. A thermocell with non-critical and inexpensive molten carbonate based electrolyte mixtures with reversible (CO2|O2) gas electrodes was reported recently. It demonstrates the possibility of utilizing the waste heat (> 550 °C) as a power source. Thermocell is an electrochemical cell with two identical electrodes placed in an ionic conducting electrolyte solution with a temperature gradient between the electrodes. The heat source will be used to create the temperature gradient between the electrodes, which will lead to a potential difference by executing ionic diffusion in the electrolyte. In this work, the thermo-physical and chemical properties of the electrolyte mixture were tuned by multi-component mixing with molten (K and Ca) carbonate and LiF additives into the binary (Li,Na) carbonates mixture. It reduces the liquidus temperature to ~ 400 °C and enables the molten carbonate thermocells to recover the high-grade waste heat available at even low temperatures well below 550 °C. Still, the Seebeck coefficient of the thermocells remains large (in the range of -1.5 mV/K).