# Thermoelectricity and Thermodiffusion in Magnetic Nanofluids: Entropic Analysis

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## Abstract

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## 1. Introduction

## 2. Thermogalvanic Cell

#### Thermogalvanic Seebeck Coefficient

## 3. Particle Flux

#### 3.1. Chemical Potential of Magnetic Nanofluids

- Diamagnetic solutes, ions or neutral species, all less than nanometer in size. These solutes will be treated as an ideal gas.
- Charged magnetic particles whose characteristic sizes are in the order of ten nanometers. These particles will be described by an effective hard-sphere model derived from Carnahan-Starling equation of state and the inter-particle magnetic interactions are taken into account through a mean-field approach.

^{th}virial coefficients that depend on temperature. For ${B}_{1}=1$ and up to the 1st order in $\frac{N}{V}$, one recovers the equation of state of an ideal gas.

#### 3.2. Electric Component ${\mu}^{e}$

#### 3.3. Magnetic Component ${\mu}^{H}$

#### 3.3.1. Single Particle Magnetization

#### 3.3.2. Magnetization of Interacting Particles: Mean-Field Approach

#### 3.3.3. Expression for ${\mu}^{H}$

#### 3.4. Total Chemical Potential

#### 3.5. General Expression for Particle Flux

#### 3.5.1. Chemical Potential Gradient: With Respect to N

#### 3.5.2. Derivative of ${\tilde{\mu}}_{i}$ with Respect to Temperature

#### 3.5.3. Derivative of ${\tilde{\mu}}_{i}$ with Respect to Magnetic Field

#### 3.5.4. Electric Term of the Chemical Potential Gradient

#### 3.5.5. General Expressions for Chemical Potential Gradient and Particle Flux

#### 3.5.6. Local Field Perturbation Effect on Particle Flux

#### 3.5.7. Final Expression of Particle Flux in directions parallel and perpendicular to $\overrightarrow{H}$

## 4. Calculation of Se and ${\mathrm{Se}}_{\mathrm{int}}$

#### 4.1. Initial State

#### 4.2. Stationary State: Soret Equilibrium

#### 4.3. Comparison with Experiments in Ferrofluids

## 5. Summary

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic representation of two electric fields in a thermocell. The internal electric field ${\overrightarrow{E}}_{\mathrm{int}}$ is created in the solution volume by thermodiffusion of ions. The thermogalvanic electric field $\overrightarrow{E}$ is created between the two electrodes. The hot and cold electrode potentials are ${\mathcal{V}}_{\mathrm{hot}}$ and ${\mathcal{V}}_{\mathrm{cold}}$ : $\left|\overrightarrow{E}\right|=\left|\frac{{\mathcal{V}}_{\mathrm{hot}}-{\mathcal{V}}_{\mathrm{cold}}}{l}\right|$, l being the distance between the electrodes.Electric field in a thermocell

**Figure 2.**Single magnetic nanoparticle. The magnetic field $\overrightarrow{H}$ is applied along the z-axis. $\theta $ defines the angle between $\overrightarrow{m}$ and the unit vector along z, ${\overrightarrow{u}}_{z}$, and $\alpha $ the angle between the projection of $\overrightarrow{m}$ in the $xy$ plane and the unit vector along x, ${\overrightarrow{u}}_{x}$.

**Figure 3.**Fonctions ${\alpha}_{\lambda}$ and ${S}_{1}$ as a function of Langevin parameter $\xi $. The other parameters are fixed: $\varphi $ = 0.01, $\lambda $ = 0.22 et ${\psi}_{dd}$ = 4.3.

**Figure 4.**Functions ${\beta}_{\lambda}$ et ${S}_{2}$ as a function of Langevin parameter $\xi $. Other parameters are fixed: $\varphi =0,01$, $\lambda =0,22$ et ${\psi}_{dd}=4,3$.

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**MDPI and ACS Style**

Salez, T.J.; Nakamae, S.; Perzynski, R.; Mériguet, G.; Cebers, A.; Roger, M.
Thermoelectricity and Thermodiffusion in Magnetic Nanofluids: Entropic Analysis. *Entropy* **2018**, *20*, 405.
https://doi.org/10.3390/e20060405

**AMA Style**

Salez TJ, Nakamae S, Perzynski R, Mériguet G, Cebers A, Roger M.
Thermoelectricity and Thermodiffusion in Magnetic Nanofluids: Entropic Analysis. *Entropy*. 2018; 20(6):405.
https://doi.org/10.3390/e20060405

**Chicago/Turabian Style**

Salez, Thomas J., Sawako Nakamae, Régine Perzynski, Guillaume Mériguet, Andrejs Cebers, and Michel Roger.
2018. "Thermoelectricity and Thermodiffusion in Magnetic Nanofluids: Entropic Analysis" *Entropy* 20, no. 6: 405.
https://doi.org/10.3390/e20060405