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Dynamic response of a thermoelectric cell induced by ion

thermodiffusion

André Luiz Sehnema and Antônio Martins Figueiredo Neto

Institute of Physics, University of São Paulo,

CEP 05508-090, São Paulo, Brazil

(Dated: August 2018)

Abstract

Aqueous electrolyte solutions under the influence of a temperature gradient can generate ther-

moelectric fields, which arise from different responses of the positive and negative charges. This

is related to the thermodiffusion effect which is quantified by the ionic heat of transport and the

thermal diffusion coefficient. When a solution is confined between two metallic electrodes with

different temperatures, it is expected to measure the electrostatic potential difference due to the

thermoelectric field. We performed experiments in aqueous electrolyte solutions, and our results

show that a thermoelectric field appears instantly in the solution as the temperature difference

between the electrodes stabilizes. This field arises from the trend to separate both ionic charges in

the temperature gradient. The experimental results also show that a slow change in the potential

difference is observed, which is related to thermodiffusion in the entire cell. These results are im-

portant to understand how the thermoelectric response can be optimized, given the broad interest

in the literature to generate thermoelectric energy from thermoelectric cells.

a Corresponding author: alsehnem@if.usp.br

1

http://arxiv.org/abs/1911.11799v1

I. INTRODUCTION

The thermoelectric effect is observed when a temperature gradient exists in an aqueous

solution with dispersed charged species. The ions diffusion response to the temperature gra-

dient creates a concentration gradient and induces a thermoelectric field. A potential differ-

ence develops between two points in the liquid with different temperatures [1, 2]. In aqueous

electrolyte solutions, the difference between the thermodiffusion coefficients of cations and

anions is one of the sources to the thermoelectric field [3–5]. A simple way to prove this effect

is to perform experiments in a thermoelectric cell, where the electrolyte solution stays be-

tween two metallic electrodes at different temperatures [6–9]. The ratio between the electric

potential drop or thermovoltage △V, as measured between the electrodes, and the applied

temperature difference △T defines the Seebeck coefficient S = −△V/△T . Recently, some

studies have reported expressive values (as high as 10 mV/K) for S in nonaqueous electrolyte

solutions [7, 10] after optimization of the electrode/liquid interface [11, 12].

Some theoretical studies have been developed to predict S values in aqueous electrolytes,

assuming the knowledge of the thermal diffusion coefficient or the heat of transport of

different ions in solution. A noteworthy equation, which relates the Seebeck coefficient to

ion thermodiffusion, is S = (α+ − α−) kB/e and was worked out by A. Wurger [1, 5], where

kB is the Boltzmann constant, e the electric charge, and αi = Q∗i /2kBT are the single-

ion Soret coefficients of ions related to the heat of transport Q∗i as obtained by Agar et al

[3, 13, 14] for ions in water, and i represents anions and cations. This result was obtained

considering the stationary state of ion diffusion in the liquid as due to a temperature gradient.

Whereas this theory predicts S values to be in the range 0.05-0.3 mV/K for different aqueous

electrolytes, the thermoelectric experiments usually give S values much higher than 0.3

mV/K, even if common salts such as NaCl are used as electrolytes [7]. The stationary

state is often considered in ion thermodiffusion due to the lack of a consistent description

of charge dynamics in solution. Recently, Stout and Khair [15] analyzed theoretically a

thermoelectric cell filled with an electrolyte and obtained time-dependent expressions for

thermovoltage and ion distribution in the cell. Their main results show that for typical

experimental parameters (distance between electrodes about 1 mm and ion concentrations

around 10 mM), the time of the thermoelectric response from charge separation is below

1 microsecond, after an instantaneous temperature gradient is established. This charge

2

separation occurs near the electrode surfaces. The relationship between the characteristic

time τφ of thermovoltage developed and Debye length κ−1, τφ ∝ 1/Diκ

2, explains this fast

response [15]. Di is the ion diffusion constant. This theoretical result is a consequence

of solving the set of equations (thermophoretic-induced ion current density, charge density,

Poisson equation, and heat equation), assuming equal diffusion coefficients Di for anions and

cations. Some experimental studies reported a very fast induced thermovoltage associated

with charge separation as the temperature difference develops between the electrodes [7, 16–

18].

In the meantime, a long timescale response is also often reported for the thermovoltage

in thermoelectric cells, which could be associated with the classical thermodiffusion char-

acteristic time τc given by τc ∝ l2/Di, where l is the distance between electrodes [19, 20].

In their theoretical study, Stout and Khair [15] also demonstrated that ion thermodiffusion

through the thermoelectric cell takes a characteristic time τ ∝ l2/Di, as reported before in

classical studies on the thermoelectric cell geometry [19–21]. This time dependence is due

to the slow thermodiffusion or salt Soret effect that does not create a net charge separation

in the bulk liquid (beyond the Debye layer) inside the cell. Thus, a change in thermovoltage

due to the slow ion thermodiffusion towards the hot or cold electrode is not theoretically

expected.

The physical picture changes if different diffusion constants are considered for anions

and cations (D− 6= D+). Chikina et al [22] have considered that a local charge separation

(between ion pairs) occurs when the diffusion coefficients are different for anions and cations,

thus generating the Seebeck effect in the cell. As a consequence, an electric field could be

expected to appear through the solution due to different values for Di, in the same timescale

as the concentration gradient develops in the entire cell. As both ion species diffuse through

the solution due to the thermodiffusion effect, one faster than the other, an electric field

is spontaneously generated from the ambipolar diffusion effect. In a recent theory, Janssen

and Bier [23] obtained an equation for thermovoltage containing a slow response due to

different diffusion coefficients and a fast response related to the difference in single-ion Soret

coefficients [23], including Stout and Khair’s result as the specific case to D+ = D− condition.

Electric fields from ion concentration gradients have been widely investigated in the process

of inducing migration of nano and microparticles [24–26], showing the importance to describe

it precisely.

3

In this study, the thermoelectric voltage induced in a thermoelectric cell is experimentally

investigated in some aqueous monovalent electrolytes. The electrolytes include acids, hydrox-

ides, and salts, which have either positive or negative values for (α+ − α−) and (D+ −D−),

as known from the literature [3]. Our main goal is to verify their thermoelectric response in

the different timescales. We verify, for the first time, that some aspects predicted by recent

theories [23] appear in the experimental results, like an opposite fast thermoelectric field

between acids and hydroxides, and a more intense field for electroytes with high difference

of diffusion constants D+ −D− and sum of ion Soret coefficients α+ + α−.

II. EXPERIMENTAL SECTION

The experimental setup of the thermoelectric cell consists of a single-volume cavity con-

taining the liquid material between two metallic electrodes. This container consists of a solid

Teflon-made annular ring (inner diameter: 16 mm; wall thickness: 2mm; height: 8mm). Two

O-rings are used to seal the container by pressing the metallic electrodes with the Teflon

separator. The distance between the electrode surfaces is kept fixed (l = 11± 1 mm) in all

experiments, as shown in Figure 1. The solution was transferred to the container using two

needles. They connected the inner and outer parts through the O-ring and were removed

after the chamber was filled. Photographs of the experimental setup are available in the

Supplementary Material. The air exposure of the Teflon and O-rings outside the container

implies a low horizontal temperature gradient to be formed at their borders, as these parts

have a low thermal conductivity. Thus, the horizontal gradient was considered negligible

here. The electrodes are made of stainless-steel metallic blocks (thick: about 5 mm). A

U-shaped inner channel was built in the bottom electrode for water circulation, and the

temperature was controlled by a Julabo thermal bath. The temperature difference between

the electrodes was established applying an electrical current to a 4 Ohm electrical resistance

on the top of the upper electrode. An electrically insulating and thermally conductive tape

separated the resistance from the upper electrode surface, avoiding static charges and uni-

formly heating up the electrode. A DC electrical current around 1 A was applied to raise its

temperature 10 K above the temperature at the bottom electrode, which was Tcold = 293 K

for all experiments. In both electrodes, temperature was measured using a type-K thermo-

couple connected to a digital multimeter. Its tips were connected to the electrodes (deep:

4

5 mm) through holes (diameter: 1 mm) made in the lateral part of the electrodes. The

electrodes were designed to be supported by a sample holder (Solartron; 12962) as shown in

the Supplementary material. The voltage difference between the electrodes was measured

by a multimeter (Agilent U1252A), which was connected to a computer for automatic data

acquisition. The data acquisition program records a voltage difference value every three

seconds.

(a) (b)

Figure 1: (a) Illustration (out of scale) of the thermoelectric cell in a front view showing

the electrolyte solution isolated inside the cell. (b) Graph showing the time necessary for

temperature (in red) stabilization in the cell and the corresponding change in the

thermoelectric potential difference (in black) in a salt solution.

Before applying the temperature difference, it was necessary to wait for the spontaneous

potential difference (SPD) between electrodes to stabilize. This effect is due to a specific

interaction between the electrodes’ interface and electrolytes, where a passivated layer is

formed due to oxidation or reduction of the metallic layer exposed to the liquid. It is also

related to electrons the multimeter injects in the circuit in order to measure the potential

difference after filling of the container with the liquid, inducing a capacitive effect with time

constant higher than 104 seconds. The stabilization process lasted a few hours to observe an

equilibrium value for SPD when the distance between electrodes was as great as 11 mm and

the stable SPD values varied from 0 to about 50 mV (absolute value). In Figures S3 and

S4 of the Supplementary Material we show this stabilization, mentioned as waiting time.

Temperature takes about 10 min (Figure 1b) to stabilize, and this is the characteristic time

5

for heat diffusion τth through the cell τth = l2/πDth, where Dth is the thermal diffusivity

of the solution (about 1.4 x 10-7 m2/s for aqueous electrolytes). Duration of experiments

depends on the Di values for the charge species in solution, taking 3-7 h for thermovoltage

to stabilize. When a stationary thermovoltage value is reached, the temperature difference

is removed and the voltage difference between electrodes is recorded for at least the same

time as in the presence of temperature difference. If separation of ion charges at the electric

double layer and their thermodiffusion in the entire cell are reversible, one can expect the

measured voltage difference to return to a similar value before the temperature difference was

applied. This cycle is repeated at least three times for each of the electrolytes investigated,

and the observed amplitudes are averaged for calculation of the Seebeck coefficient.

The solutions were prepared by diluting acids, hydroxides, and salts in deionized water.

The used ion concentrations cexp0 are shown in Table 1. We investigate aqueous solutions of

NaOH (sodium hydroxide), TMAOH (tetramethylammonium hydroxide), TBAOH (tetra-

butylammonium hydroxide), HCl (hydrochloric acid), HNO3 (nitric acid), LiCl (lithium chlo-

ride), KCl (potassium chloride), and NH4I (ammonium iodide). The chemicals (TMAOH,

TBAOH, NaOH, HNO3, and HCl) were purchased from Sigma-Aldrich and used without fur-

ther purification. The NH4I solution was prepared in the Institute of Chemistry, University

of São Paulo.

III. RESULTS AND DISCUSSION

Figure 2 shows experimental results obtained for the thermovoltage VT (t) = ∆Vf(t) +

∆Vs(t) in the thermoelectric cell. The subscripts f and s stand for the fast and slow ef-

fects generating VT . It is possible to observe a fast change corresponding to times when the

temperature difference stabilizes (Figure 1), followed by a slower change. The experimental

values to the thermovoltage for the slow response is read from the moment the temperature

difference is on until the inflection point, while the slow response is defined as the ther-

movoltage from the inflection point until the stationary state. The equations obtained by

Janssen and Bier [23] for the dynamics of ion charges in the thermoelectric cell were used for

the physical interpretation of these results. They derived equations for VT (t) as a function

of α± and D± and for ion concentration profiles ci (t) in the thermoelectric cell.

In the presence of a temperature gradient (in the x direction of the cartesian coordinates),

6

(a) (b)

Figure 2: Potential difference ∆V between cell electrodes as a function of time for

electrolyte solutions. (a) typical result obtained with aqueous HCl (blue curve) and LiCl

(red curve) solutions. The time when the temperature gradient is initiated is indicated as

“∆T on” and when it is removed is indicated as “∆T off”. (b) typical result obtained with

aqueous TMAOH solution. The amplitudes of ∆Vf and ∆Vs are also indicated, showing

that ∆Vf is associated with electric fields in different directions for acids and bases and

∆Vs is due to a similar electric field for different salts. The amplitudes for fast and slow

responses are considered to be separated by the inflection point.

the ion flux equations are as follows [1, 15]:

j± = −D±n±

(

∂ lnn±∂x

−z±e

kBTEx + 2α±

∂ lnT

∂x

)

, (1)

where n± is the ion density of cations (+) and anions (-) with valence z±, and Ex is the

thermoelectric field. Based on the experimental results, Ex has a time dynamics with one

inflection point separating different transient times of a fast and a slow contribution, Ex =

Ef+Es,. We will discuss our experimental results based on an equation to the thermovoltage

recently obtained by Janssen and Bier [23]. From the set of governing equations (Equation 1,

heat transfer, Poisson and conservation equations), they obtained the theoretical expression

to the thermovoltage with two transient times. This leads to the understanding of the

dynamic thermovoltage seen in the experimental results of Figure 2.

The theoretical equation to the thermovoltage VT (t) obtained by Janssen and Bier [23],

7

is given by

VT (t)

∆T=

kB(α+ − α−)

e−

2kB(α+ + α−)(D+ −D−)

e(D+ +D−)

∑

j≥1

1

N j2exp[−

2N j2D+D−t

l2(D+ +D−)]

−4kB(α+D+ − α−D−)

e(D+ +D−)

∑

j≥1

1

N j2exp[−

(D+ +D−)κ2t

2]

, (2)

where Nj = (j−1/2)π. By taking any pair of values for D± the second term evolves in time

many orders of magnitude slower than the last term, with separation between electrodes

surface l ≈ 10mm. This allows us to recognize the faster response in our experiments ∆Vf

in the last term of equation 2, while the slow response ∆Vs is the second term of equation

2. We use this result in the limit t → inf, when the time dependences converge to 1, to

compare the thermovoltage amplitudes and verify that our experiments are described by the

theoy of equation 2.

The thermoelectric field due to ∆Vf in the fast timescale appears in opposite directions

for different electrolytes. ∆Vf (t) increases in acidic solutions but decreases in basic solutions

during temperature stabilization. It means that α+D+ − α−D− should have opposite signs

for acids and hydroxides. According to the definition αi = Q∗i /2kBT [1] and taking the Q

∗i

values from Agar et al [3], we see that this is the case. In Table 1, we show our experimental

results and those expected by the use of Agar et al [3] Q∗i -values for comparison of the

algebraic sign(α+D+−α−D−) and we see that they are consistent. For acids and bases, the

Q∗i and Di values for H+ and OH− ions are higher than those of counterions, thus explaining

the opposite direction of the thermoelectric field associated with ∆Vf . The same algebraic

sign is not observed between ∆Vf and α+D+ − α−D− for some salts (LiCl and NH4I), in

the adopted convention. Related to the amplitudes, an agreement within the same order

of magnitude is obtained between experiments and what is theoretically expected from the

Seebeck component Sf amplitude. As a general trend, the highest experimental Sf values

are observed for salts with the highest α+D+ − α−D− theoretical difference.

Table I also presents the total Seebeck coefficient values containing the slow thermodif-

fusion effect, Stotal = −(∆Vf +∆Vs)/∆T . The theoretical values for Stotal calculated from

equation 2 are similar to the experimental values for acids and bases, but in different order

of magnitude for other salts. For NaOH and TMAOH the calculations of Stotal are actually

the same values as the experimental. The results follow the rule of highest(lowest) value

8

- TBAOH TMAOH NaOH HNO3 HCl LiCl NH4I KCl

cexp0

(mM) 4.4 2.8 2.0 1.0 2.0 10 10 10

Q∗+

(kJ/mol) 20.79 10.00 3.46 13.3 13.3 0.53 1.73 2.59

Q∗−

(kJ/mol) 17.2 17.2 17.2 -0.63 0.53 0.53 -1.55 0.53

D+ (10−9m2/s) 0.51 1.2 1.33 9.31 9.31 1.03 1.95 1.96

D− (10−9m2/s) 5.32 5.32 5.32 1.9 2.03 2.03 2.04 2.03

Stheof

(mV/K) 0.49 0.43 0.46 -0.396 -0.378 0.006 0.057 0.035

Stheototal

(mV/K) -2.1 -1.3 -1.1 0.83 0.84 -0.024 0.057 0.032

Sexpf

(mV/K) 0.2± 0.1 0.5± 0.1 0.3± 0.1 −0.85± 0.15 −0.9± 0.1 −0.12± 0.07 −0.5± 0.1 0.3± 0.1

Sexptotal

(mV/K) −3.6± 0.3 −1.2± 0.2 −1.1± 0.2 −1.4± 0.15 −2.0± 0.1 −0.30± 0.11 −1.2± 0.2 0.17± 0.05

Table I: Parameters used for the thermoelectric analysis of aqueous salt solutions.

Concentrations (cexp0 ) are given in mM. The values for Q∗+, Q

∗−, D+, and D− are those from

Agar et al. [3]. Stheof = (4kB/e)(α+D+ − α−D−)/(D+ +D−) and Stheototal is calculated from

the complete equation 2 in the stationary state. The values for the experimental fast

Seebeck coefficient are obtained from Sexpf = −∆Vf/∆T , and that for the total Seebeck

coefficient as Sexptotal = −(∆Vf +∆Vs)/∆T . ∆Vs and ∆Vf are the average values from at

least three measurement cycles, as shown in Supplementary Information.

for the salt with highest(lowest) values of the sum α+ + α−. Regarding the Stotal signs for

experiments and calculated values, they are the same for hydroxides and opposite for acids.

This may come from some unknown behavior of the H3O+ and OH− ions in the thermo-

electric cell, or from some divergence of the heat of transport values taken from ref. [3]. In

addition, equation 2 might still not be a complete physical description of the effect.

A special case of the investigated electrolytes is aqueous KCl, which has the condition

D+ = D−. The slow time response of thermovoltage given by the second term of equation 2

is expected to be null, while our experiments show a slow thermoelectric field still develops

even for that salt. This result points to the direction that equation 2 is still not a complete

model for the reality of the thermoelectric experiment. Some of the effects neglected in

current descriptions are electrodes polarization from differential ionic adsorption [27, 28]

and specific interactions between electrodes and electrolytes [29]. From the theoretical point

of view, the temperature dependence of single-ion Soret and mass diffusion coefficients are

necessary to be considered in Janssen and Bier’s theory [23] in order to obtain a more

complete theory.

Our results show that thermovoltage changes in the different timescales for ∆Vf and

9

∆Vs. This dynamic response is only observable due to a large separation between the

electrodes (l = 11 ± 1 mm) implying in characteristic thermodiffusion times τc ∼ 104 s

while temperature stabilizes until 1000 s. In some studies, the value for l is in the range 1

to 2 mm to assure homogeneity of the thermoelectric field in the cell [7, 30], which means

τc . 500 s. The temperature difference also stabilizes in the same timescale, and so does ∆Vf

(instantaneously with ∆T , τφ ∼ 10−8 s). Then, it may usually happen for ∆Vf and ∆Vs

to change the measured value of ∆V (t) in the same timescale, generating an experimental

result which is a monotonous curve and suggesting a single physical effect. In this study,

we left aside the perfect homogeneity of the generated thermoelectric field, to study both

thermoelectric contributions separated in time.

The physical mechanisms responsible for thermodiffusion of ions and organic molecules in

water are the solution thermal expansion [31, 32], temperature dependences of hydration en-

ergy [32, 33], and hydrogen bonding [34, 35]. These mechanisms give rise to thermophoretic

forces that drive the ionic Soret effect, with charge separation due to different single-ion Soret

coefficients of anions and cations, and an electroneutral concentration gradient through the

bulk solution. Both effects generate an electric field accounted by equation 2.

A. Consistency with previous experimental results

Our results presented the steady state (t → ∞) thermoelectric fields in the same direction

for both acidic and basic solutions. It means that a charged nanoparticle would be driven

in opposite directions when dispersed in acids or hydroxides. In the last few years, the

thermodiffusion response of electrically-stabilized iron oxide nanoparticles in both acidic

and basic solutions has been investigated. In the case of nanoparticles dispersed in acidic

solutions, a total Seebeck coefficient S ≈ - 1.5 mV/K (similar to the values obtained in

the present study) was measured and used to predict the negative Soret coefficient for the

positively charged nanoparticles, as observed experimentally [30]. In another publication,

the experimentally obtained salt Soret coefficient of hydroxides [32], which is peoportional to

α++α−, was used to predict the positive Soret coefficients of negatively charged nanoparticles

in TMAOH and TBAOH, as also observed experimentally [36]. Thus, our present results are

consistent with those obtained in previous studies, suggesting that the opposite Stotal sign

predicted by equation 2 for acids and hydroxides, which does not fit the overall consistency

10

from the many experimental results, may come from a still existing problem in modelling

the Seebeck response from thermodiffusion effects.

In the case when a negatively charged nanoparticle is dispersed only by an ionic solution

of NaOH, three different systems in literature [4, 36, 37] show that particles move to the hot

side while our experimental results predicts the steady-state thermoelectric field should push

them to the cold, like in the case of particles in TMAOH and TBAOH [36]. This may be

related to salt specific effects, which is still an open question to be solved in thermodiffusion.

IV. CONCLUSIONS

This study presents experimental results on the thermoelectric voltage induced by temper-

ature difference between electrodes in a thermoelectric cell filled with aqueous electrolytes.

The results show a fast change in thermovoltage related to temperature stabilization, fol-

lowed by a slow change due to ion thermodiffusion in the bulk electrolyte solutions. We

showed that the fast response is related to the difference α+D+ − α−D− and the slow re-

sponse is related to the differences between ions diffusion coefficients and proportional to

the Soret coefficient of the salt α+ + α−. Equation 2, from Janssen and Bier’ theory [23],

predicts Stotal in the mV/K order of magnitude, indicating that thermodiffusive effects are

the source of high Seebeck coefficients in aqueous electrolytes. A very good agreement has

been found between the values predicted from their theory and our experimental results for

some electrolytes.

A clear conclusion of this study is that the improvement of the thermoelectric voltage

must be a combination of two factors: a high difference in α+D+ − α−D− to improve the

fast response and high α+ + α− and D+ − D− values to improve the slow response. It is

also desirable that both thermoelectric fields have the same direction, as in acidic solutions.

Many material candidates, such as organic polyelectrolytes, which promise to increase the

performance of thermoelectric cells, can be found in the literature [7, 8]. These electrolytes

should be composed of macroions with high positive values for Q∗i , whereas their counterions

should have the lowest value as possible. The desired increase in the α++α− values may be

accomplished by increasing the temperature dependence of hydration free energy [32, 38] of

the ions and/or macroions.

11

V. SUPPLEMENTARY MATERIAL

Figures S1 and S2 show the 12962-Solartron sample holder with the thermoelectric cell

in the open and closed conditions, respectively. In Figure S1, it is possible to see the bottom

electrode surface, which is in contact with the liquid when the cell is filled, whereas the upper

electrode is kept away from it. Figure S2 shows the closed cell when ready for measurements,

when a Teflon separator and two O-rings are between the electrode surfaces.

Figures S3 and S4 show three measurements cycles for NaOH and KCl, respectively, with

measurement reproducibility in time for ∆V (t).

VI. ACKNOWLEDGEMENTS

The authors acknowledge financial support from research funding agencies CAPES (Co-

ordenação de Aperfeiçoamento de Pessoal de Nível Superior - 88881.133118/2016-01), Cnpq

(Conselho Nacional de Desenvolvimento Científico e Tecnológico - 465259/2014-6), FAPESP

(Fundação de Amparo à Pesquisa do Estado de São Paulo - 2014/50983-3; 2016/24531-3),

INCTFCx (Instituto Nacional de Ciência e Tecnologia de Fluidos Complexos).

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https://doi.org/10.1021/acs.langmuir.7b01094http://www.sciencedirect.com/science/article/pii/S0375960117310800https://doi.org/10.1063/1.372231https://doi.org/10.1038/s41598-017-01295-1http://pubs.acs.org/doi/abs/10.1021/la9001913

Dynamic response of a thermoelectric cell induced by ion thermodiffusionAbstractI IntroductionII Experimental sectionIII Results and DiscussionA Consistency with previous experimental results

IV ConclusionsV Supplementary MaterialVI Acknowledgements References