Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

Thomas J.
Salez
^{ab},
Mansour
Kouyaté
^{c},
Cleber
Filomeno
^{cd},
Marco
Bonetti
^{a},
Michel
Roger
^{a},
Gilles
Demouchy
^{ce},
Emmanuelle
Dubois
^{c},
Régine
Perzynski
^{c},
Andrejs
Cēbers
^{f} and
Sawako
Nakamae
*^{a}
^{a}Service de physique de l’état condensé, CEA, CNRS, Université Paris-Saclay, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France. E-mail: sawako.nakamae@cea.fr; Fax: +33 1 6908 8786; Tel: +33 1 6908 7538
^{b}École des Ponts ParisTech, 6 et 8 avenue Blaise Pascal, Champs-sur-Marne, F-77455 Marne-la-Vallée, France
^{c}Physico-chimie des Electrolytes et Nanosystémes InterfaciauX, Sorbonne Université, CNRS, F-75005 Paris, France
^{d}Inst. de Quémica, Complex Fluid Group, Universidade de Brasília, Brasília, Brazil
^{e}Département de Physique, Université de Cergy Pontoise, 33 Boulevard du Port, 95011 Cergy-Pontoise Cedex, France
^{f}MMML Lab, Faculty of Physics and Mathematics, University of Latvia, Zellu-8, LV-1002 Riga, Latvia

Received
21st February 2019
, Accepted 3rd June 2019

First published on 6th June 2019

The influence of the magnetic field on the Seebeck coefficient (Se) was investigated in dilute magnetic nanofluids (ferrofluids) composed of maghemite magnetic nanoparticles dispersed in dimethyl-sulfoxide (DMSO). A 25% increase in the Se value was found when the external magnetic field was applied perpendicularly to the temperature gradient, reminiscent of an increase in the Soret coefficient (S_{T}, concentration gradient) observed in the same fluids. In-depth analysis of experimental data, however, revealed that different mechanisms are responsible for the observed magneto-thermoelectric and -thermodiffusive phenomena. Possible physical and physico-chemical origins leading to the enhancement of the fluids' Seebeck coefficient are discussed.

Similar research trends are observed in another branch of nano-materials research, namely that of nanofluids.† Indeed the number of research articles per year published on nanofluids and related subjects increased by two orders of magnitude in the last 20 years. Due to their superior thermal and electrical conductivities compared to their base-fluids, nanofluids first attracted attention as effective coolants in the 1990s.^{4} While much of the nanofluid research today still focuses on enhancing the fluids' thermal conductivity by adjusting various parameters such as nanoparticles' composition,^{5–7} coating materials and volume fraction, their application potential in other areas of renewable energy is also gaining momentum.^{8,9} For example, nanofluids have been explored for their optical properties (increased absorption) in solar collectors.^{10,11} The thermoelectric effects in liquid electrolytes containing charged colloidal particles and macro-molecules were also demonstrated both theoretically^{12–15} and experimentally,^{16,17} and the possibility of enhancing the thermoelectric energy conversion efficiency using charge-stabilised magnetic nanofluids (also known as ferrofluids) using thermo-electrochemical cells^{18,19} was reported very recently.

Thermo-electrochemical cells, or thermocells, produce an electrical current through redox reactions when two electrodes are maintained at different temperatures (thermogalvanic effect). Thermoelectric coefficients (equivalent to the Seebeck coefficient in solids) as high as a few mV K^{−1} have been reported in liquid-containing thermocells,^{20} an order of magnitude larger than those of solid-state TE materials. We have recently demonstrated that the cumulative effects of thermo-electrically induced movements and distribution (the Soret effect) of nanoparticles and their electrostatic interactions with the electrodes can modify a thermocell's Seebeck coefficient. The net change in the Se can be either positive^{18} or negative,^{19} depending on the intricate balance between the NPs' surface charge, entropy of transfer and respective signs, and the nature of counterions present in the surrounding fluid. While the underlying physical and chemical mechanisms are far from being fully understood, these results paved a new direction in thermoelectric materials research based on nanofluids. In ferrofluids, it is quite well known that the Soret coefficient (S_{T}) and the diffusion coefficient (D_{m}) of ferrofluids depend on the strength and the direction of applied magnetic fields.^{21,22} For example, a marked increase in S_{T} is observed when the magnetic field is applied perpendicularly to the temperature gradient, while the opposite is true when applied in the parallel direction.^{21,23,24} Such magneto-thermodiffusion phenomena can be understood by taking into account the local magnetic field gradient within the fluid,^{24,25} and the existing theoretical model can reproduce experimental observations, provided that no magneto-convection occurs.^{24} Here, we examine the coupled Seebeck/Soret effects in ferrofluids under a magnetic field to determine to what extent one can take advantage of the magnetic nature of nanoparticles to control the thermoelectric potential of a thermocell. The value of Se is found to increase by as much as 25% in a dilute ferrofluid when a moderate magnetic field of 150 kA m^{−1} is applied perpendicularly to the temperature gradient inside a thermocell. To the best of our knowledge, this is the first experimental reporting of the enhancement of Se in ferrofluids by application of an external magnetic field.

In the following sections, we first describe the theoretical models used to analyse the effect of a magnetic field on the Seebeck coefficient in ferrofluids, ensued by the experimental approach used to measure the in-field Seebeck and Soret coefficients. The analysis and discussion of results expose the limit of our current understanding of magneto-thermoelectricity in ferrofluids, while highlighting possible physical origins that have been overlooked thus far and future research perspectives of ferrofluids.

(1) |

ΔV and ΔT = T_{h} − T_{c} are the thermoelectric potential and temperature differences between the hot and cold electrodes, and e is the elementary charge. The thermogalvanic term Δs_{rc} originates from the temperature dependent reaction entropy of the (reversible) redox couple at the electrode surfaces, which is expressed by the Nernst equation,

(2) |

is the standard reaction entropy of the redox couple, k_{B} is the Boltzmann constant and a = γ·n is the ‘activity’ defined as the product of the molar concentration n of the reducing (oxidising) species and its activity coefficient γ. The latter depends on the ionic strength of the surrounding solution.^{26} The superscripts b_{ox} and b_{red} are defined by the redox chemical equation, such that

b_{ox}·Ox + e^{−} + b_{red}·Red = 0 | (3) |

In a closed-circuit operation mode, the magneto hydrodynamic effect is known to influence the electrical current of a electrochemical cell at very high magnetic fields (parallel or perpendicular to the electrode surface).^{27,28} However, to the best of our knowledge, no significant effect on the open-circuit thermogalvanic potential of a thermocell for a moderate magnetic field (below 800 kA m^{−1} or 1 T) has been demonstrated.^{29}

The second term in eqn (1) is related to the thermodiffusion in the bulk solution, summed over all charged species. Ŝ_{i} is the Eastman entropy of transfer of the ith species (ions or nanoparticles). ξ_{i} is the dynamical effective charge defined by:

(4) |

(5) |

When the interparticle repulsion is strong, as is the case in the ionically stabilized ferrofluids studied here, the ϕ dependence of the above listed parameters can be described well in terms of the isothermal osmotic compressibility χ (ϕ_{eff}) within the Carnahan–Starling hard-sphere model:^{31}

(6) |

The magnetic field dependence of Ŝ_{NP}, ξ_{NP} and D_{NP}, on the other hand, is much less established. Here we use a mean-field model as commonly done in ferrofluids,^{32} where we consider that the nanoparticles are submitted to an effective field _{e}:

_{e} = + λ | (7) |

is the macroscopic magnetic field, is the local magnetisation of the bulk fluid and λ is a dimensionless constant which is null for non-magnetic particles and equals to 0.33 for a uniformly magnetised medium (classical Lorentz result).^{33} For aqueous ferrofluids, λ = 0.22 has been determined both experimentally^{34–37} and numerically.^{38,39} Magnetisation of a ferrofluid composed of n non-interacting NPs with individual magnetic moment is given by^{33} where the classic Langevin function and x_{0} = (μ_{0}mH)/(k_{B}T) the Langevin parameter where μ_{0} is the vacuum permeability and k_{B} is the Boltzmann constant. In the framework of an effective field model, the Langevin parameter of an interacting NP system can be replaced by the effective Langevin parameter x_{e},

(8) |

(9) |

One can then obtain the analytical expressions of Ŝ_{NP}, ξ_{NP} and D_{NP} as a function of ϕ, ϕ_{eff} and H as follows.^{25}

(10) |

(11) |

(12) |

In lim_{H,ϕ→0}, Ŝ_{NP}, ξ_{NP} and D_{NP} become Ŝ^{0}_{NP}, ξ^{0}_{NP} and D^{0}_{NP}, respectively, and for and reaches 1 for takes into account the friction between the nanoparticles and the surrounding liquid. ζ^{0} = 6πη_{0}R_{H} is the friction at ϕ = 0, with η_{0} being the viscosity of the solvent and R_{H} the hydrodynamic radius of the nanoparticles. The parameters α_{λ}, β_{λ}, S_{1} and S_{2} are defined as:

(13) |

(14) |

(15) |

(16) |

All these parameters tend to zero for H = 0, or for k_{B}T ≫ (μ_{0}mH_{e}) (i.e. x_{e} → 0).

Noting that the ionic conductivity of the NPs and the ions is independent of the magnetic field up to the first order, the field dependent variation of Se^{ini}, ΔSe^{ini}(ϕ, H) = Se^{ini}(ϕ, H) − Se^{ini}(ϕ, 0), is given in ref. 25 as

(17) |

It needs to be mentioned that the term S_{1} − S_{2} in eqn (17) is always positive and therefore the sign of Se^{ini} under a homogeneous magnetic field is determined solely by the sign of z_{NP}.

(18) |

It is summed over all species participating in the redox reaction (charged and neutral) and b_{j} is defined by eqn (3). It should be noted that Se^{Eq} only depends (directly) on the redox couple due to the rearrangement of the charged species in the solution which screens the electrodes from the internal electric field of the solution.^{25,41} The latter, however, can asymmetrically modify the ionic strength near the cold and hot electrodes and thus affect Se^{Eq} indirectly through the thermogalvanic term, Δs_{rc} (i.e., eqn (2)).

The second Seebeck coefficient, Se^{Eq*}, is due to the internal thermoelectric field created within the bulk solution and away from the electrodes. While this value cannot be directly measured, it can be inferred from the Soret coefficient, S_{T}. At the Soret equilibrium, the distribution gradient of nanoparticles , S_{T} and Se^{Eq*} are related to one another (up to the first-order) as:^{42–44}

(19) |

It can be shown from the particle flux equation that^{25}

(20) |

A horizontal, homogeneous magnetic field, i.e., perpendicular to the thermal gradient, between 0 and 400 kA m^{−1} is applied to the thermocell using an electromagnet (Bouhnik). The perpendicular field direction is chosen following the Soret coefficient measurements where a marked increase in S_{T} is usually detected under a perpendicular magnetic field^{24} (see the Results and discussion section for more details). Each temperature step lasts between 8 and 24 hours to fully reach the apparent steady state, depending on the H strength. Both Se^{ini} and Se^{st} are measured as a function of the magnetic field applied. The measurements are reproducible over several weeks with a low data dispersion.

The AC ionic conductivity measurements are also performed in the same thermocell using a precision LCR meter (HP 4284A) at 20 kHz, at which the out-of-phase component of the impedance becomes null. The total conductivity of the solution at 25 °C is determined to be σ_{tot} = 65 mS m^{−1} independent of the NP concentration (i.e., the ionic conductivity is dominated by the small counterions whose concentration was kept constant for all ferrofluids examined).

Fig. 3 (a) Diffusion coefficient as a function of NP concentration ϕ of FF-DMSO, measured at room temperature in the absence of a redox couple. The solid line is a fit to eqn (12) as a function of ϕ, without a magnetic field. (b) Soret coefficient measured via the FRS technique as a function of NP concentration (ϕ), measured at room temperature in the absence of a redox couple. The solid line is a fit to eqn (19) as a function of ϕ, without a magnetic field. |

4.1.1 Magnetic field dependence.
The dipolar interaction parameter ψ_{dd} = 65 is deduced from eqn (12) fitted to the experimental diffusion data as a function of H (Fig. 4(a), obtained on a ferrofluid sample with ϕ = 3.44%).§ This parameter can be used to analyse the behaviour of in-field Seebeck coefficients as described in the previous section.

Fig. 4 (a) Mass diffusion coefficient of FF-DMSO with an NP concentration of 3.4% as a function of the magnetic field applied perpendicular (blue) and parallel to the temperature gradient. No redox couple added. The solid lines are fits to eqn (12) as a function of the magnetic field. (b) Soret coefficient as a function of the magnetic field applied perpendicular (blue) and parallel (red) to the temperature gradient in FF-DMSO. The NP volume fraction is 3.4% with 29 mM ClO_{4}^{−}, without a redox couple. The solid lines are fits to eqn (19) as a function of the magnetic field. |

As expected, S_{T}(H) was found to decrease from its zero-field value (in Fig. 4(b)) when a magnetic field is applied in the direction parallel to the temperature gradient (as much as 70% at 60 kA m^{−1}). Under the perpendicular configuration, on the other hand, S_{T}(H) increases by 60% with respect to the zero-field value at 60 kA m^{−1}. While the anisotropic dependency of the S_{T} response to applied magnetic fields is in agreement with previous reports,^{24} its magnitude is much larger than the theoretical prediction in both field directions¶ (as depicted by solid lines in Fig. 4(b)). The large changes observed here are due to the combined effect of the uniform magnetic field and the presence of magnetoconvection. Indeed, our experimental condition α/Λ = 1.1 is within the regime where microconvective instability occurs, driven by the internal demagnetising field (due to the local inhomogeneity in the NP concentration distribution).^{50,51}

4.2.1 Initial Seebeck coefficients as a function of H, Se^{ini}(H).
With the experimentally determined parameters (i.e., D^{0}_{NP}, ξ^{0}_{NP}, Ŝ^{0}_{NP} and Ψ_{dd}) from thermodiffusion measurements at hand, one can now predict the variation of the initial Seebeck coefficient ΔSe^{ini}(H) = Se^{ini}(ϕ, H) − Se^{ini}(ϕ, 0) under applied perpendicular magnetic fields through eqn (17). The resulting theoretical curves of ΔSe^{ini}(H) are presented in Fig. 5(a). As expected from having a positive z_{NP} value, ΔSe^{ini} is also positive, i.e., the absolute value of Se^{ini} diminishes. However, the expected magnitude of the change ΔSe^{ini}(H) here is only of the order of 0.1 μV K^{−1} at H < 400 kA m^{−1}, two orders of magnitude below the experimental uncertainty level (∼10 μV K^{−1}). Therefore, one would not expect to observe the effect of the magnetic field in Se^{ini}.

Fig. 5 (a) Theoretical prediction of initial state Seebeck coefficients as a function of a perpendicularly applied magnetic field Se^{ini}(ϕ, H) − Se^{ini}(ϕ, 0) for ϕ = 0.28% and ϕ = 1% according to eqn (17) with experimentally determined D^{0}_{NP}, ξ^{0}_{NP}, Ŝ^{0}_{NP} and Ψ_{dd}. (b) Experimentally measured Se^{ini}(H) for ferrofluids with ϕ = 0.28 and 1%. The error bars correspond to twice the standard error (95% confidence interval). The red and green solid lines are fits to eqn (1) (i.e.eqn (17) + Se^{ini}(0)) for ϕ = 0.28% and ϕ = 1% (see Fig. 5(a)). The dashed blue line is a guide to the eye based on an exponential fit, i.e., . The field induced change in the initial Seebeck coefficient, ΔSe^{ini}(ϕ, H), is indicated with a double-headed arrow. |

Much to our surprise, the experimentally measured field dependence of initial Seebeck coefficients (with ΔT = 10 K and the thermocell mean temperature T = 25 °C) shows a very different behavior from the theoretical prediction (see Fig. 5(b)). As can be seen from the graph, Se^{ini} (ϕ = 0.28%, H) shows only a minor decrease in its absolute value (red symbols), and thus is consistent with the theoretical expectation.|| The absolute value of Se^{ini} (ϕ = 1%, H) during the first magnetisation (blue symbols), on the other hand, increases by roughly 10% at 360 kA m^{−1}. This variation of the order of 150 μV K^{−1} is three orders of magnitude larger than the theoretical one (0.1 μV K^{−1}) and carries the wrong sign (Fig. 5).

Furthermore, irreversibility is observed in Se^{ini}(H) between the first magnetisation and the subsequent measurements (Fig. 5(b), blue and green curves). As can be seen from the graph, once the highest magnetic field was reached for the first time, Se^{ini} becomes nearly independent of H strength (compatible with the theoretical model). The observed hysteresis suggests that certain irreversible process(es) has taken place during the first magnetisation of the ferrofluid at ϕ = 1%. As we will see in more detail below, this phenomenon is accompanied by an increase in the characteristic time to reach an apparent steady state, from ∼4.2 hours during the first magnetisation to ∼5.6 hours for all subsequent measurements (see Fig. 7(a)). Such slowing-down of the NP motion can be explained by the formation of particle aggregates under a strong magnetic field.

To verify this hypothesis, we have post-examined the ferrofluid samples after the in-field Seebeck coefficient measurements via magnetisation and diffusion light scattering (DLS, Vasco de Cordouan Technologies) measurements to search for possible aggregations. The superparamagnetic blocking temperature values T_{B} determined from the magnetisation measurements (CRYOGENIC SQUID magnetometer, model S700 was used) are ≈60 K for the ϕ = 0.28% sample and ≈90 K for the ϕ = 1% sample. Knowing that T_{B} increases approximately linearly with the mean NP volume,^{53} this indicates a ∼50% mean volume increase in the more concentrated sample.** The DLS measurements lead to a similar conclusion, with an ∼80% increase in the NPs' hydrodynamic diameter, i.e., an ∼600% increase in the hydrodynamic volume. These two independent measurements confirm that an irreversible nanoparticle aggregation had taken place in the ϕ = 1% sample during the first magnetisation of the Seebeck coefficient measurements. The absence of aggregation in the ϕ = 0.28% sample can be explained by a greater interparticle distance between NPs. As stated earlier, however, such an aggregation phenomenon was not observed during the in-field FRS measurements performed up to 80 kA m^{−1} even though a higher NP concentration (ϕ = 3.44%) was used. The two most significant differences between the two experiments are (i) the absence (presence) of redox couple agents and (ii) the total time duration for which the ferrofluid sample is exposed to the external magnetic field, i.e., one (several) hour in the thermodiffusion (thermoelectric) measurements, respectively. Thus, it is likely that the modification of inter-particle electrostatic force due to the presence of redox couple molecules and the longer experimental time-scale both contribute to the formation of NP aggregates.

It should also be noted that |Se^{ini}| increases by approximately 10% after the formation of particle aggregates (Fig. 5(b), green curve). These aggregates appear to be stable, i.e., the in-field Seebeck measurements are reproducible after the first magnetisation. The physical and/or chemical origins behind this phenomenon are far from trivial. However, within the framework of existing theories, possible explanations include an increase in the Eastman entropy of transfer of the dispersed objects and the simultaneous reduction of n (NP number) at a constant ϕ, both due to the aggregation. The latter can also indirectly influence the thermogalvanic term in eqn (1)via reshaping the ionic environment surrounding the redox couple molecules leading to modifications in the standard reaction entropies of redox reactions at the electrodes.^{54}

4.2.2 Seebeck coefficient as a function of H in the apparent steady state, Se^{st}(H).
The apparent steady state Seebeck coefficients as a function of magnetic field, Se^{st}(H), for both ferrofluids are presented in Fig. 6. An increase of ∼25% in |Se^{st}| is observed for ϕ = 0.28%, which is reproducible after repeated magnetisation–demagnetisation cycles. The ∼13% increase for ϕ = 1%, however, is only present during the first magnetisation and after which Se^{st}(H) becomes irreversible. During the subsequent magnetisation–demagnetisation cycles, Se^{st}(H) shows a reduced (but stable and reproducible) field dependency. It is worth noting that for both samples (red and green curves) Se^{st}(H) saturates around 100 kA m^{−1}, a magnetic field that can be easily attained with permanent magnets, and thus promising for potential applications.

4.2.3 Time constant.
We now shift our focus to the characteristic time constants, τ, required to reach the apparent steady state under magnetic fields. Fig. 7(a) presents the time constants required to reach the apparent steady state as a function of H for both ferrofluids (deduced from an exponential fit to the measured Seebeck voltage values). These results are highly reproducible under field-cycling as shown in Fig. 8.

In a zero-field, the apparent steady state is reached in τ less than 6 hours for all samples. However, the characteristic times τ^{Eq} to reach the Soret equilibrium can be estimated via τ^{Eq} = l^{2}/(π^{2}D_{NP}^{2}) (where l = 6 mm is the diffusion length, i.e., the distance between the two electrodes, and D ≈ 1.2 × 10^{−11} m^{2} s^{−1} at ϕ = 1%, see Fig. 3(a)). This gives τ^{Eq} ≈ 84 and ≈100 hours for ϕ = 1 and 0.28%, respectively, more than one order of magnitude larger than the experimentally determined τ values. Such a large discrepancy suggests that the observed apparent steady state is established due to a physical phenomenon different from the Soret equilibrium. A similar observation has already been made recently in aqueous ferrofluids.^{18} It was suggested that the apparent steady state stems from a temperature dependent NP adsorption and/or ordering phenomenon occurring at the electrode–fluid interfaces, which stabilizes much quicker than the Soret equilibrium. The NP adsorption has indeed been observed on mercury,^{55} gold^{56} and platinum^{57} electrodes in aqueous ferrofluids, and the ordering phenomenon has been reported on SiO_{2} surfaces.^{58} The electrostatic interactions (between the surface, the particles and the counter-ions) and inter-particle magnetic interactions (at high MNP concentrations) as well as the surface geometry (undulations and channels^{59}) are known to contribute to such phenomena, creating a surface-stabilized layer of NPs with much higher concentration of NPs than that of the bulk ferrofluid. Increased concentration of charged NPs can then modify the local ionic strength at the electrode–fluid interface, and therefore the redox reaction entropy term in both Se^{ini} and Se^{st} (eqn (1) and (18)).

Under an applied magnetic field, marked reductions in both τ(H) and Se^{st}(H) − Se^{ini}(H), by a factor of 2 to 4, are observed between 0 and 360 kA m^{−1} (as presented in Fig. 7(a) and (b), respectively) as if the presence of the magnetic field attenuates the NP adsorption and/or layering on the electrode surface. Insight into the NP adsorption/layering phenomenon can be gained from the Molecular Dynamics simulation by Jordanovic and Klapp^{60} where they have shown that the application of a magnetic field in the direction parallel to the ferrofluid–substrate interface can destroy NP layers. When a sufficiently high magnetic field is applied, the magnetic nanoparticles tend to align themselves along the external field (chain formation) due to their superparamagnetic nature. This leads to repulsive dipolar interactions between neighboring chains in the direction perpendicular to the electrode surface†† and thus limits the number of adsorbed NPs. Although the numerical simulations cited^{60} were performed for ferrofluids with high NP concentration values (ϕ = 20% and higher) and with a non-conducting substrate, the surface-initiated layering of dipolar particles and their field dependence are considered to be generic features of confined dipolar liquids. Consequently, the NP distribution near the electrode surface at the apparent steady state under a magnetic field remains closer to that of the initial state than that of the zero-field. Consequently, the redox reaction entropy contribution to the Seebeck coefficient also remains similar between the two states as depicted in Fig. 7(b). The effect of the magnetic field on the Seebeck voltage and its time evolution are clearly visible in Fig. 8. When the magnetic field of 350 kA m^{−1} is turned off after the thermocell has reached its apparent steady state (point 1, dotted green curve), the Seebeck voltage decreases as more NPs adsorb with a characteristic time τ ∼ 4 hours. The latter corresponds to the time constant recorded at 0 kA m^{−1}. When H = 360 kA m^{−1} is applied again, the Seebeck voltage increases back quickly, presumably due to the quick ejection of NPs.

To the best of our knowledge, this work presents the first evidence of thermopower enhancement induced by the application of a magnetic field. Only a moderate magnetic field strength of about 100 kA m^{−1} (less than 0.2 T) is needed to increase the Seebeck coefficient, easily attainable with a strong permanent magnet. The enhancement effect is more pronounced at lower nanoparticle concentration (0.28%), while at a higher concentration (1%) the use of a high magnetic field led to an irreversible aggregation of nanoparticles. Thus, dilute ferrofluids made with more conducting electrolytes such as ionic liquids should be considered for the next step toward the application of ferrofluids in magneto-thermoelectric technology.

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## Footnotes |

† Nanofluids are defined here loosely as stable suspensions of nanoparticles (NPs) in liquid media. |

‡ Here, is positive. It is a decreasing function of ϕ as χ(ϕ_{eff}) decreases with ϕ because the interparticle interaction is repulsive (A2 > 0). See ref. 44 and 49. |

§ For fitting methods, see Bacri et al.^{34–36} |

¶ Note that a quantitative agreement was found between the theoretical prediction of S_{T}(H) and the experimental findings from ref. 24. |

|| These measurements have been performed several times with increasing and decreasing H and the results are reproducible. |

** The dipole–dipole interaction energies are negligible at such low concentrations. |

†† The magnetic interaction is attractive for particles aligned one-behind-another; however, it is repulsive in the direction perpendicular to H, expanding the NP distribution near the electrodes. |

This journal is © The Royal Society of Chemistry 2019 |