Nanofluidics for Thermoelectric Energy Harvesting

Abstract

Nanofluidics have attracted great attention for electrokinetic energy conversion. Recently the application of nanofluidic systems for thermoelectric energy harvesting has intrigued researchers with various research backgrounds. It has been shown that the equivalent Seebeck coefficient can be greatly enhanced in confined nanofluidic channels with hydrodynamically slippery boundary conditions, indicating great potential for highly efficient and environment-friendly low-grade thermal energy harvesting. In this chapter, we will first introduce the basic electrokinetic theories behind the thermoelectric response. Next, the current understanding of the thermoelectric coupling mechanism in confined nanochannels will be depicted. Strategies to improve the thermoelectric coupling efficiency will be illustrated. Then, the most recent experimental achievements in this field will be reviewed. Besides, the main challenges and prospective will also be discussed. Based on this chapter, we intend to give a fundamental introduction to the theoretical framework of nanofluidic thermoelectricity and present the opportunities and challenges facing this emerging field.

1. Introduction

Low-grade thermal waste energy harvesting has attracted great attention recently in the requirement for more clean and sustainable energy to reduce the relying on fossil fuels [1]. Low-grade heat sources are usually referred to as those with temperatures below around 200°C, which are abundant and can be produced and discharged into the environment in various industry processes and from a lot of electronic equipment. Researchers have attempted to use various materials and technologies to convert waste heat to electricity, including thermoelectric semiconductors, ionic liquid gels, ionic thermoelectric cells, osmotic heat engines, and so on [2]. However, the implementation of those methods is still limited either by the material rarity and toxicity or energy conversion efficiency. A cleaner and more cost-effective thermoelectric technique for waste heat harvesting is still highly required.

Recently it has attracted great attention whether it is possible to use nanofluidic systems for efficient thermoelectric conversion. Electrokinetics of ions in micro-/nanochannels has been intensively investigated [3, 4]. The ion transport driven by an electrical field or concentration gradient near a charged surface induces the coupling flow of carrier fluid, which is referred to as electro-osmosis or diffusio-osmosis, respectively. On the other hand, the fluid transport driven by a pressure field would carry ions to travel along the charged solid surface and thus produce streaming currents.

The effect of thermal gradient on ionic and fluidic transport in confined space has only been considered recently. Derjaguin and Sidorenkov [5] first studied thermo-osmosis in porous glass, where fluid flow is driven by a thermal gradient. This is reverse to the observation that the fluid flow driven by pressure gradient also causes the formation of a heat flux through nanochannels due to the excess enthalpy of liquid adjacent to the wall. Derjaguin predicted the thermos-osmotic slipping velocity by applying the Onsager’s reciprocal theorem, like the case for diffusio-osmosis, and demonstrated the existence of thermos-osmosis in porous media. Molecular dynamic simulations have also been used to reproduce the thermos-osmotic slippage effect [6]. The first microscale observation of the velocity field induced by thermos-osmosis was achieved by Bregulla et al. [7]. The reduced friction at the solid-liquid interface is shown to be able to enhance the thermo-osmotic response, such as at the graphene-water surface [8].

The electricity generation of conductors under a thermal gradient is usually referred to as the Seeback effect. In a bulk liquid electrolyte, the asymmetric thermophoretic motion of positive and negative ions results in electric field build-up, that is, due to the Soret effect [9]. Under physical confinement, the existence of surface charges at the wall-liquid interface makes it possible to produce electricity by thermal gradient without the request of different thermophoretic mobility of cations and anions. It is expected to enhance the thermoelectric conversion efficiency by highly permselective membranes. However, the Seeback coefficient experimentally measured in nanofluidic channels using aqueous electrolyte is still much lower than that obtained by conventional thermoelectric semiconductors. It is revealed theoretically that the synthetic effect of ultrahigh surface charge density and slippage can enhance the thermoelectric response, which implies potential future research directions [10].

In this chapter, we will briefly introduce the fundamental theories of electricity generation in nanofluidic systems under thermal gradient and review the most recent progress in the field of thermoelectric nanofluidics. The fundamental theory will be first introduced about the ionic and fluidic transport driven by a thermal gradient. The mechanism of thermoelectric energy conversion will be illustrated. The most recent experimental progress in studying the thermoelectric response of nanofluidic channels will be reviewed. At last, the opportunities and challenges to implement nanofluidic systems for low-grade thermal energy harvesting will be discussed.

2. Mechanisms of thermoelectricity in liquid electrolyte

2.1 The Soret effect in the bulk electrolyte

Ions, molecules, suspended particles, or droplets in a bulk solution would drift to the cold or the warm side when a thermal gradient field is applied. The drift velocity (u) is proportional to the magnitude of the thermal gradient (∇T), that is [9],

$$\mathit{u}=-D_{\mathrm{T}}\mathbf{\nabla }T.$$E1

where D_T is thermophoretic mobility or thermal diffusion coefficient, characterizing the coupling between the heat and the particle motion, and T is temperature. For the positive Soret effect, the particles move toward the cold, and for the negative one, the particles move toward the warm. The ion migration in a liquid electrolyte driven by the Soret effect builds up an ion concentration gradient (∇c). The different thermophoretic mobility of cations and anions leads to the establishment of a thermoelectric field (E_T). In a stationary state, the driving forces due to the concentration gradient, thermal gradient, and thermoelectric field will be balanced with each other for any individual ion.

Consider a binary aqueous electrolyte solution. The ion flux (J_i) for the positive (i=‘+’) or negative (i=‘−’) ions due to the concentration gradient, thermal gradient, and thermoelectric field is obtained according to the Nernst-Planck equation [11],

$$J_{\pm }=-D_{\pm }\left(\nabla {\mathrm{c}}_{\pm }+c_{\pm }{S}_{\mathrm{T},\pm }\nabla T-c_{\pm }\frac{v_{\pm }e}{k_{\mathrm{B}}T}{E}_{\mathrm{T}}\right)$$E2

where D_i is diffusion coefficient, c_i is ion concentration, S_T,i (=D_T/D_i) is intrinsic Soret coefficient, v_i is ion valance, e is the elementary charge, k_B is the Boltzmann constant, and E_T = −∇V_T with V_T as the thermoelectric voltage. The Soret coefficient (S_T,i) is related to the ionic heat of transport ($Q_{\mathrm{i}}^{\ast }$) by

$$S_{\mathrm{T},\mathrm{i}}=\frac{Q_{\mathrm{i}}^{\ast }}{k_B{T}^2}.$$E3

Here, $Q_{\mathrm{i}}^{\ast }$primarily originates from the entropy of hydration for specific ions and has typical values in the order of kJ/mol [12].

In the stationary state without convection flow, by applying the approximate neutrality conditions, that is,

$$c_{+}=c_{-}=c_0,v_{+}{J}_{+}+v_{-}{J}_{-}=0,$$E4

with $c_0$ as the ion concentration at the electroneutral region, we obtain the dependence of the established salinity gradient (∇c₀) on the thermal gradient (∇T) according to Eq. (2),

$$\frac{\nabla c_0}{c_0}=-\mathrm{\Pi }\frac{\nabla T}{T}$$E5

where Π is the reduced Soret coefficient,

$$\mathrm{\Pi }=\frac{Q_{+}^{\ast }+Q_{-}^{\ast }}{2k_{\mathrm{B}}T}$$E6

and also the resultant thermoelectric field,

$$E_{\mathrm{T}}=\frac{Q_{+}^{\ast }-Q_{-}^{\ast }}{2\mathit{veT}}\nabla T$$E7

where v (=v₊=−v₋) is the valence of the symmetric electrolyte. The generation of electricity by a thermal gradient in an aqueous electrolyte is similar to the case of thermoelectric semiconductors, where the Seeback coefficient is defined by the ratio of the thermoelectric voltage and the temperature difference. Thus, the equivalent Seeback coefficient in a bulk solution is obtained as

$$S_{\mathrm{e},\mathrm{bk}}=\frac{Q_{+}^{\ast }-Q_{-}^{\ast }}{2\mathit{veT}},$$E8

with $S_{\mathrm{e},\mathrm{bk}}$defined by $E_{\mathrm{T}}/\nabla T$. Given $Q_{\mathrm{i}}^{\ast }\sim \mathrm{kJ}/\mathrm{mol}$, the generated thermoelectric voltage per Kelvin in a bulk electrolyte, that is, S_e, lies in the scale of around 0.1 mV/K, which is clearly much smaller than that of common thermoelectric semiconductors. It indicates bulk electrolyte can hardly be directly used for thermoelectric harvesting due to the ultralow ionic heat of transport. Therefore, it is intriguing to see whether it is possible to enhance the thermal energy conversion efficiency by using aqueous electrolyte in physically confined space.

2.2 Thermoelectric response of liquid electrolyte in physical confinement

The electroneutrality assumption for ion distribution only applies in the bulk electrolyte. In physically confined micro/nanochannels, the presence of wall charges leads to ion redistribution inside the channels, which forms electric double layers (EDL). Under the assumption of thin EDL without overlapping, the electric potential (φ) in the EDL field decays from the wall surface value to the bulk one (see Figure 1). Assume the two-dimensional (2D) slit channel is long and thin, that is, H/L → 0 (here, H and L are channel height and length, respectively), and neglect the advection effect inside the channel. The thermal gradient (∇T) by applying a temperature difference (ΔT) at the channel ends can be approximated by [11]

Figure 1.

Schematics of EDL in a slit channel and thermoelectric field generation by a thermal gradient.

$$\frac{\mathrm{d}T}{\mathrm{d}x}=\frac{∆T}{L},\frac{\mathrm{d}T}{\mathrm{d}z}=0$$E9

which means that the major temperature gradient lies in the longitudinal direction (x), and the temperature difference in the transverse direction (z) is negligible.

In the confined space without convection flow, the net flux for specific ions driven by the concentration distribution, thermal gradient, and electric field is also described by the Nernst-Planck equation [Eq. (2)]. Due to impermeable solid wall boundary condition, the ion flux (J_±,z) in the z direction is zero. Thus, according to Eq. (2), the ion concentration profile in the EDL region can be determined by

$$\frac{{\mathrm{d}c}_{\pm }}{\mathrm{d}z}=-c_{\pm }\frac{v_{\pm }e}{k_{\mathrm{B}}T}\frac{\mathrm{d}\phi }{\mathrm{d}z}.$$E10

With boundary conditions that c₋ = c₊ = c₀ for symmetric electrolyte, and φ = 0 at z = 0, the ion distribution inside the double layer is obtained as,

$$c_{\pm }=c_{0}exp\left(-\frac{v_{\pm }\mathit{e\phi }}{k_{\mathrm{B}}T}\right)$$E11

which resembles the Boltzmann distribution [11]. It indicates by Eq. (11) that, in contrast to the Soret equilibrium in the bulk phase, the positive and negative ion concentrations near the wall surface are different, that is, $c_{+}\ne c_{-}$, resulting in net charge distribution. Therefore, even if there is no difference between the diffusivities of the positive and negative ions, a thermoelectric field can still be established due to the charge separation effect of the EDL.

Substituting the ion concentration profile inside the EDL [Eq. (11)] into Eq. (2) leads to the net ion flux in the longitudinal direction (x),

$$J_{\pm ,x}=-D_{\pm }{c}_{\pm }\left[\frac{1}{c_0}\frac{{\mathrm{d}c}_0}{\mathrm{d}x}+\left(\frac{v_{\pm }\mathit{e\phi }}{k_{\mathrm{B}}{T}^2}+S_{\mathrm{T},\pm }\right)\frac{\mathrm{d}T}{\mathrm{d}x}-\frac{v_{\pm }e}{k_{\mathrm{B}}T}{E}_{\mathrm{T}}\right]$$E12

In the equilibrium state, the overall ion current produced by the thermal gradient, concentration gradient, and internal electric field is zero, that is,

$$I=\mathit{ve}{\int }_0^{H/2}\left(J_{+,x}-J_{-,x}\right)\mathrm{d}z=0$$E13

To solve Eq. (13), we need the exact potential distribution profile (φ(z)), which can be obtained by solving the Poisson equation [11]

$${\partial }_z^2\mathrm{\Psi }={\mathrm{\kappa }}^{2}sinh\left(\mathrm{\Psi }\right)$$E14

where Ψ = veφ/(k_BT), $\mathrm{\kappa }=\sqrt{2e^2{v}^2{c}_0/\left(\epsilon k_{\mathrm{B}}T\right)}$, $\epsilon$ is the dielectric permeability, and κ⁻¹ is the Debye length. In Eq. (14), the spatial variation of $ϵ$ has been neglected considering the 2D slit is long and thin. Under the Debye-Hückel approximation, the potential distribution (φ(z)) inside the double layer is described by

$$\mathrm{\phi }=\mathrm{\zeta }\frac{k_{\mathrm{B}}T}{\mathit{ve}}\frac{cosh\left(\mathrm{\kappa }z\right)}{cosh\left(\mathrm{\kappa }H/2\right)}$$E15

where ζ is zeta potential, that is, the potential at the slipping plane of the wall.

Here, we assume that electroneutrality is held at the center of the slit. Thus, the distribution of c₀ approximately follows Eq. (5), or,

$$\frac{\mathrm{dln}\left(c_0\right)}{\mathrm{d}x}=-\mathrm{\Pi }\frac{\mathrm{dln}\left(T\right)}{\mathrm{d}x}$$E16

Assume the positive and negative ions have similar diffusivity, that is, D₊ = D₋ = D.

Combining Eq. (8), Eq. (11), Eq. (12), and Eq. (16), the solution of Eq. (13) leads to the equivalent Seeback coefficient in the physical confinement [11],

$$S_{\mathrm{e},\mathrm{cf}}=S_{\mathrm{Q}}+S_{\mathrm{\phi }}$$E17

where

$$S_{\mathrm{Q}}=S_{\mathrm{e},\mathrm{bk}}\frac{1}{1+\frac{{\overline{\mathrm{\zeta }}}^2}{4}\left[\frac{tanh\left(\overline{\mathrm{\kappa }}\right)}{\overline{\mathrm{\kappa }}}+\frac{1}{{cosh}^2\left(\overline{\mathrm{\kappa }}\right)}\right]}$$E18

$$S_{\mathrm{\phi }}=\frac{\mathrm{\zeta }}{T}\frac{tanh\left(\overline{\mathrm{\kappa }}\right)}{\overline{\mathrm{\kappa }}}\frac{1+\frac{{\overline{\mathrm{\zeta }}}^2}{2}\left[\frac{{tanh}^2\left(\overline{\mathrm{\kappa }}\right)}{3}+\frac{1}{{cosh}^2\left(\overline{\mathrm{\kappa }}\right)}\right]}{1+\frac{{\overline{\mathrm{\zeta }}}^2}{4}\left[\frac{tanh\left(\overline{\mathrm{\kappa }}\right)}{\overline{\mathrm{\kappa }}}+\frac{1}{{cosh}^2\left(\overline{\mathrm{\kappa }}\right)}\right]}$$E19

with $S_{\mathrm{e},\mathrm{cf}}=E_{\mathrm{T}}/\left(\mathrm{d}T/\mathrm{d}x\right)$, $\overline{\mathrm{\zeta }}=\mathit{ev}\mathrm{\zeta }/\left(k_{\mathrm{B}}T\right)$ and $\overline{\mathrm{\kappa }}=\mathrm{\kappa }H/2$. Here, $S_{\mathrm{Q}}$ represents a thermoelectric contribution due to a Soret-type thermophoretic ion motion under physical confinement. If the ionic heat of transport for the positive and negative ions is identical, $S_{\mathrm{Q}}$ vanishes as $S_{\mathrm{e},\mathrm{bk}}$ is zero according to Eq. (8). The second term, $S_{\mathrm{\phi }}$, indicates the confinement-dominated thermoelectric effect. The Debye length (κ⁻¹) characterizes the thickness of the double layer. When κ⁻¹ is much bigger than the half channel height (H/2), that is, κH→0, $S_{\mathrm{\phi }}$ tends to be equal to ζ/T. This implies the enhancement of the thermoelectricity generation by increasing the surface zeta potential. On the other hand, when the EDL thickness is much smaller than the channel height, that is, κH→∞, $S_{\mathrm{\phi }}$ vanishes as in the case of the bulk solution.

To probe how large the equivalent Seeback coefficient can be achieved in confined nanochannels, the variation of $S_{\mathrm{e},\mathrm{cf}}$ with the electrolyte concentration is plotted in Figure 2. As an example, the slit height is chosen as H = 20 nm, and the surface zeta potential is chosen as ζ = 100 mV. Consider the electrolyte is KCl. The difference in heat of ionic transport for potassium and chloride ions is $\mathrm{\Delta }Q=Q_{+}^{\ast }-Q_{-}^{\ast }\approx 2\mathrm{kJ}/\mathrm{mol}$ [12]. The comparison of $S_{\mathrm{Q}}$ and $S_{\mathrm{\phi }}$ clearly shows that, for large Debye length at low concentration, the contribution of the Soret-type thermophoretic behavior is indeed negligible, and the confinement-induced thermoelectric effect is dominated. In contrast, for high concentration with small Debye length, the confinement effect tends to be reduced. Although the thermophoretic and nanoconfined thermoelectric contributions are additive, the overall equivalent Seeback coefficient in confined nanochannels generally lays in the order of around 0.4 mV/K even for the zeta potential as large as ζ = 100 mV. It should be noted that the overlapping of the double layers inside the slit modifies the electric field distribution, which, however, will not significantly affect the magnitude of the total Seeback coefficient [11].

Figure 2.

Dependence of equivalent Seeback coefficient on electrolyte concentration in a 2D slit. The slit height, H = 20 nm, and the zeta potential, ζ = 100 mV.

In a short summary, the thermoelectricity generation in physical confinement without convection flow has been analytically formulated based on the Debye-Hückel approximation. The result indicates the confinement-dominated thermoelectric effect when the double layer thickness is comparable with or even bigger than the channel height. However, the predicted equivalent Seeback coefficient in confined nanochannels is still much smaller than the common thermoelectric semiconductors. It should be noted that the above analysis assumes no advection in the channels, which may be validated for channels with slippery solid walls. In the following section, thermally induced osmotic transport in a confined space will be induced.

2.3 Thermo-osmotic flow in confined space

The confinement of liquid in nanochannels alters its specific enthalpy. For an isothermal system, the liquid flow in confined nanochannels driven by a pressure gradient ($\nabla p$) would produce “heat flux”, leading to thermal gradient establishment through the channel. On the other hand, a thermal gradient ($\nabla T$) applied through the nanochannels would also induce the liquid flow, which is the so-called thermos-osmosis. Such a coupling phenomenon is described by the Onsager’s linear nonequilibrium thermodynamics [6, 8],

$$\left[\begin{array}{c}{u}_{\mathrm{s}}\\ f_{\mathrm{h}}\end{array}\right]=\left[\begin{array}{cc}{\beta }_{11}& {\beta }_{12}\\ {\beta }_{21}& {\beta }_{22}\end{array}\right]\left[\begin{array}{c}-\nabla p\\ -\frac{\nabla T}{T}\end{array}\right]$$E20

where $u_{\mathrm{s}}$ is hydrodynamic velocity, $f_{\mathrm{h}}$ is heat transfer flux, and ${\beta }_{\mathit{ij}}$ (i, j = 1, 2) is the phenomenological coefficient. Here, ${\beta }_{11}$ characterizes the isothermal flow driven by a pressure gradient, ${\beta }_{22}$ denotes heat conduction at $\nabla p=0$, and, according to the Onsager’s reciprocity relations, ${\beta }_{12}={\beta }_{21}$, which represents the thermos-osmosis coefficient. The following is intended to obtain the expression for ${\beta }_{12}$.

Consider a long and thin 2D slit as illustrated in Figure 1. To account for the effect of interfacial hydrodynamics, we assume the Navier’s slip boundary condition at the slit wall surface [13],

$$u\left(z_{\mathrm{s}}\right)=b{\left.\frac{\mathrm{d}u}{\mathrm{d}z}\right|}_{z=z_{\mathrm{s}}}$$E21

where u is velocity, $z_{\mathrm{s}}$ is shear plane position, and b is slip length. The slit confinement modifies the specific enthalpy in the liquid, resulting in a thermodynamic driven force, δh(z) (∇T/T), where δh(z) is local excess specific enthalpy. Then, the thermos-osmotic flow can be solved through the Stokes equation,

$$-\mathrm{\eta }\frac{{\mathrm{d}}^{2}u\left(z\right)}{\mathrm{d}{z}^2}=\mathit{\delta h}\left(z\right)\frac{\nabla T}{T}$$E22

where η is liquid viscosity and assumed homogenous. Integrating Eq. (22) twice and using boundary conditions, Eq. (21) and ${\left.\left(\mathrm{d}u/\mathrm{d}z\right)\right|}_{z=0}=0$, give the thermos-osmotic slip velocity (u_s),

$$u_{\mathrm{s}}=\frac{1}{\mathrm{\eta }}{\int }_{z_{\mathrm{s}}}^0\mathrm{d}z\left(z-z_{\mathrm{s}}+b\right)\mathit{\delta h}\left(z\right)\frac{\nabla T}{T}$$E23

Then, according to Eq. (20), the corresponding thermos-osmotic coefficient (${\beta }_{12}={\beta }_{21}$) is obtained as

$${\beta }_{12}={\beta }_{21}=-\frac{1}{\mathrm{\eta }}{\int }_{z_{\mathrm{s}}}^0\mathrm{d}z\left(z-z_{\mathrm{s}}+b\right)\mathit{\delta h}\left(z\right)$$E24

When it is positive, liquid flows toward the cold side, and when negative, liquid flows toward the hot side. For b = 0, Eq. (24) retrieves the coefficient for the non-slip boundary condition (${\beta }_{12}^{\mathrm{no}-\text{slip}}={\beta }_{21}^{\mathrm{no}-\text{slip}}$). The presence of a slippery boundary magnifies the thermos-osmotic response by [8]

$${\beta }_{12}={\beta }_{12}^{\mathrm{no}-\text{slip}}\left(1+b/\mathrm{\lambda }\right)$$E25

where

$$\mathrm{\lambda }=\frac{{\int }_{z_{\mathrm{s}}}^0\mathrm{d}z\left(z-z_{\mathrm{s}}\right)\mathit{\delta h}\left(z\right)}{{\int }_{z_{\mathrm{s}}}^0\mathrm{d}z\mathit{\delta h}\left(z\right)}$$E26

characterizes the length scale of the interfacial liquid layer thickness where the liquid enthalpy is altered by the solid wall.

The thickness λ is typically in the order of a few molecular sizes. Thus, it can significantly enhance the thermos-osmotic coupling coefficient on surfaces with ultralow liquid friction, such as on graphitic surfaces, where the slip length reaches over tens of nanometers. Experiments on bare glass and Pluronic coated substrates have revealed thermos-osmotic coefficients ranging from ${\beta }_{12}\sim {10}^{-10}$ m²/s to ${10}^{-9}$ m²/s [14]. Molecular dynamic simulations have shown that the thermos-osmotic coefficient can be increased by orders of magnitude if the liquid-solid interfacial friction is ultra-low [8]. This hints that the thermos-osmotic flow in slits with partially slippery walls needs to be taken into consideration in the analysis of thermoelectric coupling, which is discussed below.

2.4 Thermo-electric response with hydrodynamic slip

The thermos-osmotic coupling analysis above shows that a temperature gradient through confined channels could induce significant liquid flow, especially if the wall has ultra-low liquid friction. Thus, the liquid advection inside those channels needs to be considered in analyzing the thermo-electric response in contrast to Section 2.2. The presence of convection flow modifies the ion flux (J) through a confined channel to [10]

$$J_{\pm }=c_{\pm }u-D_{\pm }\left(\nabla c_{\pm }+c_{\pm }{S}_{\mathrm{T},\pm }\nabla T-c_{\pm }\frac{v_{\pm }e}{k_{\mathrm{B}}T}{E}_{\mathrm{T}}\right)$$E27

Assume the double layers inside the 2D slit (see Figure 1) are not overlapped. The vanishing ion flux in the z direction also leads to ion concentration distribution following Eq. (11).

To obtain the thermo-electric coupling coefficient, we refer to the Onsager’s linear nonequilibrium thermodynamics, which describes the thermo-electric coupling phenomenon by [6, 10]

$$\left[\begin{array}{c}{j}_{\mathrm{e}}\\ f_{\mathrm{h}}\end{array}\right]=\left[\begin{array}{cc}{\alpha }_{11}& {\alpha }_{12}\\ {\alpha }_{21}& {\alpha }_{22}\end{array}\right]\left[\begin{array}{c}-\nabla V\\ -\frac{\nabla T}{T}\end{array}\right]$$E28

where $j_{\mathrm{e}}$ is electric current density, $-\nabla V$ is the gradient of electric potential, ${\alpha }_{11}\left(=\sigma \right)$ is electrical conductivity, ${\alpha }_{22}/T\left(=\mathrm{\kappa }\right)$ is thermal conductivity, and ${\alpha }_{12}={\alpha }_{21}={\alpha }_{\mathrm{TE}}$ is the thermoelectric coefficient based on the Onsager’s reciprocal relation. Thus, the coefficient ${\alpha }_{\mathrm{TE}}$ can be obtained by either

$${\alpha }_{12}=\frac{-j_{\mathrm{e}}}{\nabla T/T},\mathrm{at}\nabla V=0$$E29

or

$${\alpha }_{21}=\frac{-f_{\mathrm{h}}}{\nabla V},\mathrm{at}\nabla T=0$$E30

Here, the ion current density ($j_{\mathrm{e}}$) in Eq. (29) is determined according to Eq. (27) by setting $E_{\mathrm{T}}=-\nabla V=0$,

$$j_{\mathrm{e}}=\frac{\mathit{ve}}{H/2}{\int }_0^{H/2}\left(J_{+,x}-J_{-,x}\right)\mathrm{d}z$$E31

with the concentration profile ($c_{\pm }$) determined by the Poisson-Boltzmann equation and the velocity field (u) determined by the thermos-osmotic flow equation [i.e., Eq. (22)].

The heat transfer flux ($f_{\mathrm{h}}$) in Eq. (30) can be derived through [6, 10]

$$f_{\mathrm{h}}=\frac{1}{H}{\int }_{-H/2}^{H/2}u\left(z\right)\mathit{\delta h}\left(z\right)\mathrm{d}z.$$E32

where the velocity field (u) is induced by applying a constant external electric field ($E_{\mathrm{e}}$), and described by the Stokes equation,

$$-\mathrm{\eta }\frac{{\mathrm{d}}^{2}u\left(z\right)}{\mathrm{d}{z}^2}=c_{\mathrm{e}}{E}_{\mathrm{e}}$$E33

with $c_{\mathrm{e}}=\mathit{ve}\left(c_{+}-c_{-}\right)$ is charge density, and $E_{\mathrm{e}}=-\nabla V$ is the constant external electric field. Combining the Poisson equation

$$\epsilon \frac{{\mathrm{d}}^2\phi \left(z\right)}{\mathrm{d}{z}^2}=c_{\mathrm{e}}$$E34

and boundary conditions, ${\left.\mathrm{d}u\left(z\right)/\mathrm{d}z\right|}_{z=0}=0$ and ${\left.\mathrm{d}\phi \left(z\right)/\mathrm{d}z\right|}_{z=0}=0$, leads to

$$\mathrm{\eta }\frac{\mathrm{d}u\left(z\right)}{\mathrm{d}z}=\epsilon E_{\mathrm{e}}\frac{\mathrm{d}\phi \left(z\right)}{\mathrm{d}z}.$$E35

At z = 0, $\phi \left(0\right)=0$ and $u\left(0\right)=u_{\mathrm{eo}}$, which is defined as the electro-osmotic velocity. At z = H/2, $\phi \left(H/2\right)={\phi }_0$ and $u\left(H/2\right)=b{\left.\mathrm{d}u/\mathrm{d}z\right|}_{z=H/2}=b\mathrm{\Sigma }{E}_{\mathrm{e}}/\mathrm{\eta }$ with Σ=${\left.\epsilon \mathrm{d}\phi /\mathrm{d}z\right|}_{z=H/2}$ as the surface charge density. (Here, the slipping plane position z_s is assumed approximately equal to H/2.) Then, the electro-osmotic velocity ($u_{\mathrm{eo}}$) is obtained as [10]

$$u_{\mathrm{eo}}=\frac{\epsilon E_{\mathrm{e}}}{\mathrm{\eta }}\mathrm{\zeta }.$$E36

where $\mathrm{\zeta }$ is effective potential, and

$$\mathrm{\zeta }={\phi }_0+\mathrm{\Sigma }b/\epsilon ,$$E37

indicating the dependence of the $\mathrm{\zeta }$ potential on the slippery boundary condition and surface charge density.

With known the velocity profile, the enthalpy density is still required to get the heat transfer flux ($f_{\mathrm{h}}$) according to Eq. (32). According to Ref. [10], the excess enthalpy density ($\mathit{\delta h}\left(z\right)$) majorly consists of two parts,

$$\mathit{\delta h}\left(z\right)=\delta h_{\text{water}}\left(z\right)+\delta h_{\mathrm{EDL}}\left(z\right),$$E38

where $\delta h_{\text{water}}\left(z\right)$ is due to the excess enthalpy of water molecular and $\delta h_{\mathrm{EDL}}\left(z\right)$ is related to the EDL,

$$\delta h_{\mathrm{EDL}}\left(z\right)=\delta h_{\mathrm{el}}\left(z\right)+\delta h_{\mathrm{osm}}\left(z\right)+\delta h_{\text{solv}}\left(z\right),$$E39

with $\delta h_{\mathrm{el}}\left(z\right)$ due to the electrostatic interaction, $\delta h_{\mathrm{osm}}\left(z\right)$ originating from the osmotic pressure, and $\delta h_{\text{solv}}\left(z\right)$ coming from the ion solvation.

In general, the excess entropy in confined slits can hardly be directly measured by experiments. Molecule dynamic (MD) simulations have shown that the excess enthalpy of water dominates the overall contributions, which explains the distinct thermoelectric coupling effects as predicted by the Poisson-Boltzmann theory and MD simulations. With Eq. (32), Eq. (36), and Eq. (38), the thermoelectric coupling coefficient (${\alpha }_{\mathrm{TE}}$) can be obtained according to Eq. (30), and the equivalent Seeback coefficient in slits with hydrodynamic slip boundary conditions, defined by $S_{\mathrm{e},\mathrm{hs}}=-\nabla V/\nabla T$ with $j_{\mathrm{e}}=0$, is expressed by [10]

$$S_{\mathrm{e},\mathrm{hs}}=\frac{{\alpha }_{\mathrm{TE}}}{\mathrm{\sigma }T}$$E40

where $\sigma$ is electrical conductivity, consisting of bulk (${\sigma }_{\text{bulk}}$) and surface conductivity (${\sigma }_{\text{surf}}$), of which the latter is dominated if κH << 1. By scaling analysis of Eq. (30), it is found for surface conductivity-dominated channels that [10]

$$S_{\mathrm{e},\mathrm{hs}}\approx \frac{-\mathit{\delta h}\times \left(2\mathrm{\lambda }/H\right)\times \left(\epsilon \mathrm{\zeta }/\mathrm{\eta }\right)}{{\sigma }_{\text{surf}}}.$$E41

Here, $\mathrm{\lambda }$ is described by Eq. (26), indicating the solid-liquid interaction layer thickness. The $\mathrm{\zeta }$ potential depends on the surface charge density and boundary slip length [see Eq. (37)]. The apparently direct correlation between the equivalent Seeback coefficient and $\mathrm{\zeta }$ potential implies simultaneously high surface charge density and boundary slip length benefit the enhancement of the thermoelectric coupling by orders of magnitude, as demonstrated by molecular dynamic simulations [10]. It should be remarked that surface charge distribution on highly slippery solid surfaces must be homogeneous. Otherwise, the heterogeneous distribution of those surface charges could deteriorate the slip effect and, consequently, the thermoelectric coupling.

In summary, hydrodynamic slip enhances the thermo-osmotic coupling, inducing significant convection flow inside confined slits. This contributes to the ion fluxes driven by a temperature gradient, or, in other words, the thermoelectric coupling effect. The thermoelectric coefficient can be obtained according to the Onsager’s linear nonequilibrium thermodynamics. The scaling analysis of the equivalent Seeback coefficient shows the thermoelectric coupling effect that can be enhanced by maximizing the surface charge density and slip length simultaneously. In the next, the current experimental achievements and challenges in nanofluidic thermoelectricity will be discussed.

3. Experimental achievements, challenges, and prospective

3.1 Thermoelectric performance of nanofluidic devices

Although considerable efforts have been made to improve the thermoelectric performance of semiconductors, ionic gels, etc., experimental investigation of the thermoelectric coupling of nanofluidic devices still lies in its fetal stage. A typical experimental setup for nanofluidic thermoelectricity is schematically shown in Figure 3. A membrane with nanofluidic channels is separated between two liquid reservoirs filled with aqueous electrolyte. A temperature gradient is applied across the nanofluidic membrane to produce thermoelectricity. To improve the thermal energy conversion efficiency, the key lies in optimizing the surface chemistry and geometric structure of nanofluidic channels.

Figure 3.

Schematics of experimental setup for thermoelectricity measurements through nanofluidic membranes.

A previous study of electrokinetic transport through ion rectifying channels provides inspiration for the design of thermoelectric nanofluidics. The electricity generation due to electrolyte transport through nanochannels driven either by thermal gradient or by pressure gradient requires the efficient separation of positive and negative ions. Thus, nanochannels with ion rectifying properties are expected to be able to work as well-performed thermal energy converters. Attempts have been made with chemically and/or geometrically asymmetric nanochannels to probe the thermoelectric coupling effect [15, 16, 17]. By partially coating silicon dioxide nanochannels with hydrophobic molecular, linear dependence of the thermoelectric current on the temperature difference is observed, which is attributed to the slippage-induced thermo-osmotic transport [15]. Asymmetric cone-shaped silica nanofluidic channels with dopamine-grafted inner surface have also been shown to be able to generate thermoelectricity with power throughput reaching 25.48 pW per channel at a temperature difference of 40°C [16]. However, the average equivalent Seeback coefficient with a value around 0.4 mV/K is still pretty low as compared with conventional thermoelectric semiconductors.

Smart biological system has the ability of thermosensation relying on ion channels on cell membranes [18], motivating the bionic design of highly temperature-sensitive nanofluidic membranes. Chen et al. [19] constructed a permselective ionic membrane by stacking ultrasmall silica nanochannels of around 2.3 nm in diameter on track-etched poly(ethylene terephthalate) conical nanochannels of around 10–15 nm in the small side. The 2.3-nm silica channels with negatively charged surface preferentially allow the transport of positive ions. A temperature sensitivity of around 0.7 mV/K is demonstrated using such hybrid nanochannel membranes. Ionic covalent organic framework (COF) with pore size below 1 nm, close to that of biological ion channels, has also attracted great attention for temperature sensation application [20]. The high charge density inside the sub-nanometer COF pores enables enhanced thermoelectric response with equivalent Seeback coefficient reaching around 1.27 mV/K. Although this sensitivity is relatively larger than that in ultrasmall silica nanochannels, it is still significantly weaker than those of common thermoelectric semiconductors and ionic gels.

3.2 Main challenge and prospective

Theoretical analysis of thermoelectric response in 2D slits with partial slip boundary conditions has shown that simultaneously high surface charge density and slip length can improve the thermoelectric coupling effect to a level comparable to that of common thermoelectric semiconductors. However, current achievements of thermal sensitivity of nanofluidics channels are still quite low, generally in the order of 1 mV/K or even below. The main challenge lies in the difficulty to realize the ultrahigh slip effect on highly charged solid surface. In the inner surface of silica and COF nanochannels, the charge density is usually quite high, but the slip length is almost vanishing or even negative. That is because those charges are heterogeneously distributed on the channel surface, which reduces the slippery effect. Molecular dynamic simulation reveals that the slip boundary condition remains almost unchanged if the surface charges are homogenously distributed [10, 21]. This paves a way to achieve both highly slippery and highly charged nanochannels for enhanced thermoelectric conversion.

Ultrafast fluidic and ionic transport on highly charged graphitic surfaces, e.g., carbon nanotubes [22] and graphene [23], have recently been observed experimentally. In contrast to solid surfaces like silica, graphitic surfaces are usually atomically smooth and exhibit ultralow friction to the water. Although graphitic surfaces are inert to chemical modification, external electrical gating can tune the surface charge density, which shows almost no effect on the slippery property of the surface. Thus, electrostatic gating in atomic-scale graphene channels enables ultrafast and tunable ionic transport with an effective diffusion coefficient reaching two orders of magnitude higher than in bulk water [23]. Moreover, the self-assembly of 2D material flakes easily enables large-scale membrane fabrication. Therefore, highly improved thermoelectric coupling efficiency can be expected in atomically smooth 2D material channels with enhanced charge density (Figure 4).

Figure 4.

Schematics of ultrafast ionic transport through graphene channels under electrostatic gating [23].

4. Conclusions

Emergent and efficient thermoelectric conversion techniques are highly required to harvest low-grade heat in aqueous solution. Nanofluidic systems show great potential to be used for effective thermoelectric energy conversion, which, however, has been poorly explored so far. In bulk electrolyte, the Soret-type thermophoretic motion of positive and negative ions with different ionic heat of transport establishes an electrical field under a thermal gradient. However, the equivalent Seeback coefficient is generally quite low, in the order of 0.1 mV/K. In confined nanochannels, the confinement-induced thermoelectricity is dominated over the Soret effect with the assumption of no convection flow. However, the overall equivalent Seeback coefficient is also quite low.

It is remarked that, in nanochannels with partial slip boundary conditions, the convection flow of liquid under a temperature gradient, that is, thermo-osmosis, can be significant and needs to be considered in the thermoelectric conversion analysis. It has been demonstrated by molecular dynamic simulations that simultaneously high slip length and surface charge density contribute to orders of magnitude enhanced thermoelectric coupling coefficient. However, the realization of such a kind of nanochannel surface is quite challenging. It is the main reason why temperature sensitivity of only around 1 mV/K is achieved even in sub-nanometer channels with ultrahigh surface charge density.

It is revealed that heterogeneously distributed charges deteriorate the slippery boundary condition. Thus, homogeneous distribution of surface charges is required to achieve constant low liquid friction. The observation of ultrafast ionic transport in graphene channels under electrostatic gating points out a new direction to combine atomically smooth 2D materials and the electrostatic gating technique for efficient thermoelectric energy conversion in the future.

Acknowledgments

This work is supported by the National Program on Key Basic Research Project of China (Grant No. 2022YFA1203400), the National Natural Science Foundation of China (Grant No. 12272159), and the National Natural Science Foundation of Guangdong Province (Grant No. 2023A1515012592).

Conflict of interest

The authors declare no conflict of interest.