Nanoscale Seebeck e¤ect at hot nanostructures

Keywords: thermophoresis, temperature gradient, self-propulsion, Janus colloid, permittivity gradient, DNA, Seebeck e¤ect, hydrodynamic

The aim of this work is to study the nanoscale Seebeck e¤ect at hot nanostructures. At …rst, we study the thermo-electrophoresis self-propulsion mechanism for a heated metal capped Janus colloid. The self-propulsion mechanism is mainly induced by the electrolyte Seebeck e¤ect or thermoelectric e¤ect. This e¤ect takes its origin from the separation of charges occurring while a temperature gradient is present in a electrolyte solution: A strong absorption of laser light by the metal side of the particle creates a temperature gradient which in turn acts on ion-species (positive and negative) and drives them to the hot or the cold region. This motion of ion results in a dipolar electric …eld which, close to the particle, depends strongly on the surface properties. The change of behavior of the electric …eld at the insulating or conducting surface does not a¤ect the velocity of the particle. At second, we study the e¤ect of hydrodynamic interactions and counterion condensation in thermophoresis for DNA polymer. As the main result, the thermophoretic mobility shows, in function of the chain length, a non-monotonous behavior and consists of two contributions induced by the dominant driving forces which are the thermally induced permittivity-gradient and the electrolyte Seebeck e¤ect. At the end, we compare our theoretical result with recent experiment on single-stranded DNA.

E¤et Seebeck à l' échelle nanométrique de nanostructures chaudes Résumé L'objectif de ce travail est d'étudier l'e¤et thermoélectrique à l'échelle nanométrique des nanostructures chau¤ées. Dans un premier temps, nous étudions les mécanismes d'autopropulsion thermo-électrophorétique de particules Janus chau¤ées par laser. Ce mécanisme d'autopropulsion est principalement induit par l'e¤et Seebeck ou l'e¤et thermoélectrique. Cet e¤et provient de la séparation des charges survenues lorsqu'un gradient de température est présent dans la solution d'électrolyte: Une forte absorption du laser par la partie métallisée de la particule génère un gradient de température qui en retour agit sur les espèces ioniques (positive et négative) et les conduits vers les zones chaudes ou les zones froides. Ce mouvement d'ions entraine la création d'un champ électrique dipolaire qui, à proximité de la particule, dépend fortement des propriétés de surface. Ce changement de comportement de ce champ électrique sur une surface isolant ou conductrice n'a¤ecte pas la vitesse de la particule. Dans un second temps, nous étudions les e¤ets d'interactions hydrodynamiques et de la condensation des contre-ions sur la thermophorèse des polymères d'ADN. Comme résultat principal, la mobilité thermophorétique montre, en fonction de la longueur de la chaîne, un comportement non-monotone et se compose de deux contributions induites par les forces conductrices dominantes que sont l'e¤et Seebeck et le gradient de permittivité. À la …n, nous comparons notre résultat théorique avec une récente expérience sur l'ADN.

Mots clés: thermophorèse, gradient de temperature, auto-propulsion, particule Janus, gradient de temperature, ADN, e¤et Seebeck, hydrodynamique.

Chapter 1 Introduction 1.1 Phoretic transport

When applying an external force on microparticles suspended in a ‡uid, the system leads to a transport mechanism known as the name of phoresis. The concept of phoretic transport has been known for more than a century, and it's de…ned as a movement of colloidal particles caused by an applied external …eld. The most important property of phoretic transport is that there are no external force [1]. The motion arises from the action of the particle on the surrounding ‡uid. The phoretic movement of particle arises mainly from the interaction between the surface of the particle and its surrounding ‡uid or even with the various solutes solvent. This transport mechanism shows basic di¤erences with the transport induced by body forces such as gravity, where the external …eld exerts a net force on the particle.

In the last few decades, the development of experimental techniques, studying the migration of particles under the in ‡uence of an gradient …eld, are strongly based on this phoretic transport phenomenon. Among these techniques, one can cite the …eld- ‡ow fractionation method (FFF) which was …rst introduced by Giddings [START_REF] Giddings | Flow Field Fractionation: a versatile new separation method[END_REF]. The FFF method consists of separating particles by applying a gradient …eld to a ‡uid suspension pumped trough a long channel. In this method, the separation of particles is caused, …rst, by the applied …eld which is perpendicular to the ‡ow in the channel, and second by the di¤erences in particle's phoretic mobility. In the same way, another experimental method, consisting of ampli…ed migration of large particles in a solution through a channel, are developed more recently [3]. The phoresis mechanism has also brought a huge improvement in other trapping techniques for colloidal particles [4] or for DNA [5], and constitute the key physical property when studying self-propulsion movement for microswimmers.

Mechanisms

Among the class of phoretic transport, the particles can move by electrophoresis, di¤usiophoresis or thermophoresis. These possibilities consist of creating motion of a suspended microparticle by imposing a gradient …eld such as an electric …eld, a temperature gradient or a concentration gradient of small solutes surrounding the particle. The resulting transport mechanism refers to electrophoresis, thermophoresis, and di¤usiophoresis, respectively. The electrophoresis mechanism is the most oldest one because it was established by Smoluchowski since 1905 [START_REF] Smoluchowski | [END_REF], and later Keh and Anderson developed this concept by clarifying the role of the electrostatic boundary e¤ects on the electrophoretic motion of a charged non-conducting colloidal spheres [10].

Next comes the method of di¤usiophoresis which was …rst established by Derjaguin [6], and much later by Anderson and Prieve when studying migration of particle in salt solution [7]. As a theoretical point of view, the di¤usiophoresis mechanism is the motion of particle caused by a concentration gradient of molecular substances which is due to the formation of the absorption layer of neutral solutes or ions at the surface of the particle [START_REF] Derjaguin | Surface Forces[END_REF].

By comparison to electrophoresis and di¤usiophoresis, the physical properties of thermophoresis were developed much later by scientists as Ruckenstein when dealing with thermophoresis for colloidal particles [11], by Giddings [12], Brochard and de Gennes when studying the e¤ect of hydrodynamic interactions in thermophoresis for polymer [25]. Thermophoresis is the movement of particle driven by a temperature gradient. For charged particles, the basis of this transport mechanism is the formation of the electric double-layer or the absorption layer around the dissolved particles.

Motion in a temperature gradient

Thermophoresis mechanism is strongly based on the phenomenon of thermo-osmosis which is de…ned as a ‡ow of a charged liquid driven by a temperature gradient [START_REF] Derjaguin | Surface Forces[END_REF]. The origin of thermo-osmosis was …rst discussed theoretically and experimentally by Derjaguin and Sidorenkov [START_REF] Derjaguin | [END_REF]. Indeed, thermo-osmosis is due to the response of co-ions and counterions present in the interfacial region to the applied or generated temperature gradient. The thermo-osmotic mechanism can be explained in two ways: First, the strong presence of excess ions in the interfacial region leads to an excess pressure within the di¤use region. Second, the presence of an temperature gradient causes the motion of ion-species to the hot or the cold region. This motion of ions lead to a variation of the pressure within the interfacial region, which is higher at the cold region and lower at the hot region. As a consequence, there is a ‡ow of a charged ‡uid toward higher pressure to lower pressure. Due to this ‡ow of ions, the particle moves in opposite direction to the temperature gradient. The particle velocity is directly proportional to the temperature gradient where the constant of proportionality is the thermophoretic mobility D T .

However, the question of which direction the particle moves was not clear. When calculating the thermophoretic coe¢ cient of a colloidal particle in a electrolyte solution, Ruckenstein provided a …rst answer by stating that a particle always moves towards the cold region [11]. Recent experiments studying mechanism of hot colloids in electrolyte solution, however, showed that particles can move in either direction. Among these experiments, one can cite the work reported by Simoncelli et al. [15]. They have trapped optically a Janus particle in a focused laser beam and have measured the vertical position of particle in 10mM of ionic strength in function of the increasing laser power. The important observation is when they used di¤erent electrolytes NaCl, NaOH, and LiCl, then the behavior of the particle signi…cantly changes. Another experiment by Eslahian et al. reports a rather similar behavior [16]. Here, they have measured the particle's velocity in a mixed electrolyte solution NaOH x Cl 1 x in function of a parameter x. When this parameter x varies from 0 to 1, the electrolyte solution changes gradually from NaCl to NaOH, and, as the result, the sign of the particle's velocity becomes positive or negative, respectively. This change of sign of the particle's velocity is generally due to the thermal induced salt-gradient and eventually to the electrolyte Seebeck e¤ect [17,18,19]: An applied temperature gradient exerts thermal forces on salt-ions, which depending on their solvation enthalpy, move to the hot or the cold boundaries. As a consequence, this motion of ions leads to the existence of an electric …eld known as Seebeck …eld. This generated Seebeck …eld is due to the motion of salt-ions once a temperature gradient is present in an electrolyte solution, and plays a very important role because it gives a new contribution on the particle's velocity, which is similar to Smoluchowski's electrophoretic mobility [1]. In more general cases, this Seebeck e¤ect can also determines the direction of motion of particle.

Summary

The aim of this present work is to study the thermophoresis mechanism for hot colloid particle immersed in an electrolyte solution, and for polyelectrolytes. For colloid particle, the self-propulsion mechanism is largely induced by the electrolyte Seebeck e¤ect which in turn results from the existence of the temperature gradient in the system. In the case of a metal capped colloid, the system generates an dipolar electric …eld which acts on the electric double-layer and self-propels the particle by thermo-electrophoresis. The resulting particle velocity is directly proportional to the applied thermal gradient and the constant of proportionality is known as a thermophoretic mobility. This mobility depends upon on several parameters like the electrolyte Seebeck coe¢ cient, the solvent salinity and the particle size. For polyelectrolytes, the thermophoresis mechanism arises from the hydrodynamic interactions between the repeat units in the polymer chain. In other words, the transport mechanism is induced by the two dominant forces which are the thermally induced permittivity gradient and the electrolyte Seebeck e¤ect.

This thesis is organized as follows: In section 2, we review previous results on the electrolyte Seebeck e¤ect. In section 3, we theoretically study the Seebeck e¤ect of a heated Janus colloid immersed in an electrolyte solution. We will try to see how the self-propulsion mechanism induced by thermoelectric …eld can depend on the surface properties of the particle. This particle's surface can be in a di¤erent geometries where the system is in non-equilibrium condition and the resulting electric properties is treated within the Poisson-Boltzmann theory. In the case of an conducting surface, the particle carry a high electrical conductivity on its metal cap which forms an isopotential surface. This isopotential condition imposes a signi…cant polarization charge which modi…es the double-layer potential and the thermoelectric properties. In section 4, we will discuss about the e¤ect of hydrodynamic interactions in a dilute polyelectrolyte solution. This last point was studied in term of series expansion for Oseen tensor. In addition, we will see, in this section, how the counterions condensation e¤ect can play an essential role on the thermophoretic mechanism for DNA. Finally in section 5, we will study the dynamics of ion, in electrolyte solution, depending on time. Here, the system is in non-stationary state. In this case, the characteristic time scale for ions di¤usion will results from the charge conservation equation which rely the ionic current and the electric charge density.

Chapter 2

Electrolyte Seebeck e¤ect

In this chapter, we review known results on the electrolyte Seebeck e¤ect in different geometries. All the following results in this part are taking from the paper in Ref. [29]

Seebeck e¤ect in a 1-D geometry

In electrolyte solution, an applied temperature gradient exerts thermal forces on salt-ions which migrate along the gradient. By response, the system generates a macroscopic thermoelectric …eld which in the one-dimensional system, is given by

E =SrT 1 e z= ; (2.1)
where S is a constant of proportionality between the electric …eld E and the temperature gradient, the thickness of the electric double-layer and z the normal coordinate perpendicular to the screened surface. The coe¢ cient S is known as a electrolyte Seebeck coe¢ cient [30,[START_REF] Agar | Advances in Electrochemistry and Electrochemical Engineering[END_REF]. In the one-dimensional case, the temperature gradient present in the system is constant. So if we de…ne T 2 as the temperature of the hot boundary and T 1 the temperature of the cold boundary, then the temperature gradient reads,

rT = T 2 T 1 L ;
where L is the length of the system.

Figure 2.1: Schematic view of the Seebeck e¤ect in a 1 dimensional system. Due to the presence of a temperature gradient, cations and anions accumulate at the cold and hot boundaries, respectively. This result in the macroscopic thermoelectric …eld E =SrT which is constant in the bulk and vanishes at the boundaries. This …gure is reprinted with permission from Ref. [29].

The physical process of the generation of the electrolyte Seebeck …eld is sketched in Fig. 2.1 with positive and negative ions located in the cold and hot region, respectively. The motion of ions generates a current which in turn break the uniformity of the electrolyte solution. We will show in next section that this current for positive and negative ions is di¤erent because it depends on their heat of transport and the reduced Soret coe¢ cient. Once a charge separation appears between the two boundaries, then the system generates a thermoelectric …eld which in turn acts on the ions and drives them. The behavior of this thermoelectric …eld is well described by Eq. (2.1) where the exponential factor describes the accumulation of charges in the hot and cold boundaries. In the bulk, the thermoelectric …eld takes the expression E =SrT which is constant in volume, whereas it vanishes at the boundaries. In addition, the sign and magnitude of the Seebeck …eld depends on the coe¢ cient S which can be positive or negative depending on the solution used.

The corresponding thermocharge density follows from Gauss's law, = " div E, and its expression reads as, = e z= ; where = "SrT takes opposite signs at the cold and hot boundaries. In the bulk, the thermocharge density vanishes, = 0, and in the boundaries it's screened by counterions.

Seebeck e¤ect in 3-D system

Here we detailed the Seebeck e¤ect in the vicinity of a heated colloidal particle.

In the system, the particle, totally covered by a thin metal layer, generates a temperature gradient once heated. As illustrated in Fig. 2.2, due to the presence of this temperature gradient, the co-ions move to the hot region (near the particle surface) whereas the counterions move to the cold region. This phenomenon of charge separation induces a thermoelectric …eld which at distances well beyond the Debye length reads as E =SrT . Thus, near the particle surface the complete expression of the Seebeck …eld come from the stationary electro-osmotic equation for the ionic current.

ion current

The colloid particle is immersed in a electrolyte solution composed of monovalent ions with densities n for respectively positive and negative ions. Once the particle is heated, it generates a symmetric temperature gradient. This temperature gradient exerts thermal forces on salt-ions and induces their motions. As a result, the system generates an ionic current

J = D rn + 2n rT T n eE k B T ; (2.2)
where D the positive and negative ionic di¤usion coe¢ cient. The vector …eld J is the current of positive and negative ions. This current is the sum of three di¤erent terms: the di¤usion current _ rn which is given by Fick's law, the thermodi¤usion current rT with the reduced Soret coe¢ cient which characterize the drift of ions in presence of the temperature gradient, and the electric driven current _ E which is due to the presence of the thermoelectric …eld in the system. When the system reaches the steady-state, the three di¤erent currents, which are the di¤usion currents cancel each other, resulting in J = 0.

In the following, we will detail the behavior of the thermoelectric …eld and the total charge density in the steady state where the mobile ionic current given by each ionic species vanishes.

Thermoelectric …eld and thermocharge

Here we give the complete expression of the thermoelectric …eld of a uniform particle of radius a and centered at r =0. We denote by n 0 the bulk salinity and = e(n + n ) the charge density. If we assume that ion densities n di¤er only weakly from their bulk value, that is n + + n 2n 0 and =e n 0 , one can linearize the sum of J + J in the following form,

r + 2n 0 e ( + ) rT T 2n 0 e 2 E k B T = 0: (2.3)
Thus we can perform Eq. ( 2.3) by applying the relation between the thermoelectric …eld E and charge density , called Gauss's law

rE = ="; (2.4)
where " the solvent permittivity. Inserting this last equation into Eq. (2.3) in the steady state, one obtains the equation satis…ed by the thermoelectric …eld,

r 2 E 1 2 (E SrT ) = 0; (2.5)
where 2 = k B T =2n 0 e 2 is the Debye screening length and S = ( + ) k B =e is the Seebeck coe¢ cient. The vector quantities E, rT in Eq. (2.5) are considered to be symmetric in the case of an isotropic surface (uniform colloid particle). Now if we consider a 3-D spherical symmetry, only their radial components are …nite, i.e, they depend only on the radial distance r from the particle center.

The temperature gradient remains always in the stationary state because of the fact that heat di¤uses much faster than ions, and its expression is given by

rT = T a r 2 ;
where T is the excess temperature at the particle surface. To solve Eq. (2.5), one has to specify the boundary conditions. Here the particle is considered to be uncharged, i.e, it doesn't carry surface charge. In this case, the thermoelectric …eld is equal zero at the particle surface, E(r = a) = 0. The second condition states that the thermoelectric …eld vanishes at large distances, E ! 0 when r ! 1.

With these two conditions and solving Eq. (2.5), the thermoelectric …eld reads as,

E=SrT 1 r + a + e (a r)= : (2.6)
This last equation characterizes the behavior of the thermoelectric …eld in the vicinity of a heated gold colloid particle. This Seebeck …eld takes its origin from the separation of charges caused by the temperature gradient. In addition, it's important to note that this thermoelectric …eld is not an external applied …eld but the …eld generated by the system composed of colloid particle plus electrolyte solution.

In the steady-state, the thermoelectric …eld present in the solution is described by Eq. (2.6), where at distance well beyond (a + ) the screening exponential term vanishes and the remainder long-range electric …eld E=SrT varies with the inverse of the square distance. Applying Gauss'law, the thermocharge density will scale as, / e (a r)= , which vanishes at long distances well beyond the Debye screening length.

The total thermocharge is found by integrating the thermocharge density over the particle surface, then it reads as

Q = e b S a l B T T ; (2.7)
where b S the reduced Seebeck coe¢ cient. This is the net thermocharge accumulated in the vicinity of the particle surface. The thermocharge Q is independent of the radial distance r and thus its sign depends on the coe¢ cient b S which is positive for NaCl solution and negative for NaOH solution [START_REF] Würger | [END_REF]. The sign of the thermocharge Q tells which ion species is accumulated in the particle's vicinity. For example, in NaCl solution, anions di¤use toward high temperature thus negative charge is accumulated at the particle surface, resulting in Q < 0; whereas for NaOH solution, the inverse phenomenon occurs and the particle accumulated a net positive charge. For typical value of the excess temperature T = 30K and a particle size about a 1 m in NaCl or NaOH electrolyte solution, the corresponding net thermocharge is about Q 100e. For protonated salts in water or alcohol solution, the value of the thermocharge becomes much higher. Figure 2.2: Schematic view of the Seebeck e¤ect in a 3 dimensional system. We show here the Seebeck e¤ect in the vicinity of a hot colloid particle. Due to the thermal gradient, the particle accumulates a net thermocharge Q and thus results in a radial thermoelectric …eld given in Eq. (2.6). This …gure is reprinted with permission from Ref. [29].

Chapter 3

Nanoscale Seebeck e¤ect

Introduction

In recent years, arti…cial microswimmers have attracted much attention in many di¤erent areas such as micro ‡uidic application [21]. In some physical systems, the motion of particles is due to an external applied …eld like a uniform electric …eld (electrophoresis) or a chemical (di¤usiophoresis) or temperature gradient (thermophoresis). If phoretic motion of passive particles is rather well understood [1,[START_REF] Dukhin | Surface and Colloid Science[END_REF]18], this is not always the case for self-propelled particles. In these systems, it's more di¢ cult to predict the direction of motion of particle or to describe exactly all the physical forces inducing the motion of particle.

Recent experiments on half-metal coated colloid particles (Janus particle), in a defocused laser beam [38] or by optical trapping [39], reveal the strong presence of thermophoresis forces in the sense that: Once the particle is heated, the absorption of laser by the metal side generates a local temperature gradient which in turn drives the particle by thermophoresis. To go even further in to order understand more the question of self-propulsion mechanism induced by thermophoresis, the colloids particles are immersed in electrolyte aqueous solution. In this situation, recent experiment reveals that the stationary height of colloid particle increases with the laser power, and even varies with the ion species [15]. In other words, this last point means the sign and magnitude of thermophoresis depend strongly on the electrolyte composition; this was con…rmed by recent work on hot colloids [16]. In addition to thermal e¤ect, one observes thermoelectric e¤ect, called electrolyte Seebeck e¤ect, which occurs during the self-propulsion mechanism of hot colloid particle in an electrolyte solution. This Seebeck e¤ect plays an essential role on the transport mechanism of hot particle, and even depends on the surface properties of the particle. It's important to question that how the thermoelectric properties with the induced slip-velocity v s can change at an insulating and conducting surfaces.

In this present chapter, we deal with the self-propulsion mechanism of hot Janus colloid induced by thermoelectrophoresis: As illustrated in Fig. 3.1, due to the presence of a temperature gradient, the Janus particle, immersed in a electrolyte solution, carries opposite thermocharges on its two poles which in turn results in a dipolar thermoelectric …eld. This …eld acts on the electric double-layer and self-propels the particle with velocity u. The transport velocity can be derived directly when taking the surface con…gurational average of the boundary slipvelocity v s which is a function of the parallel thermoelectric …eld E k = Sr k T . This parallel …eld is modi…ed close to a conducting surface but does not change at an insulating surface (upper and lower hemispheres in Fig. 3.1c). At the conducting surface, the gold cap forms an isopotential surface because of its high electrical conductivity, thus the parallel electric …eld vanishes. In both cases, the resulting transport velocity does not be changed by the surface electric properties, but in turn depends strongly on the salt composition.

Boundary layer approximation

Electrostatic boundary conditions

Surface charges of colloidal particles are usually treated in Poisson-Boltzmann theory. Since an analytic solution exists in one dimension only, the widely used boundary layer approximation neglects the surface curvature. As long as the screening length is much smaller than the particle radius a, there is a separation of length scales: The properties of the electric double layer vary much more rapidly in perpendicular direction than parallel to the surface.

The resulting approximation is best discussed in terms of Gauss' law = div("E). This divergency comprises two terms, the normal …eld component decays on the scale of the Debye length, @ z E ? E ? = , whereas the permittivity and the parallel electric …eld vary on the scale of the particle radius, @ x E k E k =a. Thus to linear order in small parameter =a, the divergency of the electric …eld is given . The Debye length , characterizing the thickness of electric doublelayer (dashed line), is much more smaller than the particle radius a 1 m. b) Thermocharge accumulated by the electrolyte Seebeck e¤ect (S < 0); the resulting (positive and negative) thermocharge density is larger at the hot metal cap; for an excess temperature of a few kelvin one has a net charge of about Q T 100e [40]. c) Dipole component of the thermoelectric …eld E in the vicinity of a heated Janus particle. The parallel …eld E k vanishes at the metal cap (upper hemisphere), but is …nite at the insulating surface. by its normal coordinates, dE ? dz = " :

For further use, we integrate from the surface to a distance B that is much larger than the screening length but much smaller than the particle radius, and …nd

E ? (B) E ? (0) = 1 " Z B 0 dz (z) " : (3.2)
The second identity de…nes the charge density per unit area of the di¤use layer. This parameter also determines the double-layer potential ' , as is obvious from the Poisson-Boltzmann mean-…eld expression for the di¤use layer, = 2en sinh(e' =k B T ).

In the case of an electric double layer at equilibrium, the electric …eld usually vanishes at large distance, E ? (B) = 0, resulting in the value at the particle surface

E ? (0) = " : (3.3)
Then corresponds to the charge per unit area of the surface, which exactly cancels that of the di¤use layer.

On the contrary, the main results of the present paper are derived from Eq. (3.2), with the outer boundary condition determined by E ? (B) = Sr ? T . This implies that as de…ned in (3.2) does not correspond to the surface charge density. In the range of the distance B, the temperature gradient is almost identical to that at the particle surface, r ? T = r ? T S .

Temperature gradient at the particle surface

Since the heat conductivities of liquid and solid are di¤erent in general, the particle deforms the temperature …eld in its vicinity. For a sphere, a conductivity contrast modi…es the parallel and perpendicular components of the temperature gradient according to

r k T S ! k r k T S ; r ? T S ! ? r ? T S ; (3.4)
with the well-known constants [START_REF] Würger | [END_REF]

k = 3 w 2 w + P ; ? = 3 P 2 w + P ; (3.5)
expressed in terms of the thermal conductivities w and P for the liquid and the solid, respectively. We suppress these factors in the following sections but restore them when discussing the slip velocity.

Seebeck e¤ect: uncharged particle

Here we evaluate the thermoelectric properties at the particle surface which does not carry a surface charge, i.e, the particle is considered to be uncharged.

Debye-Hückel approximation

Here we derive the thermocharge of an otherwise uncharged hot Janus particle in the weak-coupling approximation where the electrostatic energy e' is much smaller than the thermal one k B T . We start by giving the equation satisfying by the electric …eld E and charge density which obtained by using Debye-Hückel approximation, r + " 2 (SrT E) = 0:

(3.6)
This last equation has two unknowns: The thermocharge density and the electric …eld E. To have only one unknown and simple computations, we solve the corresponding equation satisfying by the electrostatic potential '. Now introducing the Gauss's law = " div E with the relation E = r' into Eq. (3.6), one has to solve

r 2 ' 1 2 (' S T ) = 0; (3.7)
where T = T (r) T 0 with T (r) the temperature …eld given in term of a series multipolar expansion and T 0 the bulk temperature (see Appendix A2).

The linear di¤erential equation (3.7) is obtained within the weak-coupling approximation, and his general solution reads as,

' = 1 X n=0 St n a n+1 r n+1 + c n k n (r= ) k n (a= ) P n (c) ; (3.8)
where c = cos cosine of the polar angle, and

k n (x) = q 2 x K n+1=2 (x) the modi…ed spherical Bessel function of the second kind.
The potential in (3.8) is still not complete because of the unknown coe¢ cients c n which can be found by specifying the electrostatic boundary conditions which in turn will depend on the surface properties of the particle. In the following sections, we will try to …nd the coe¢ cients c n and discuss the thermoelectric properties for both insulating and conducting surfaces.

Insulating particle

Here the particle does not carry a surface charge, thus the electrostatic boundary conditions require that the normal electric …eld vanishes at the particle surface, E ? (0) = 0, whereas at the outer boundary one has E ? (B) = Sr ? T S . From Gauss' law (3.2), one readily …nds

"Sr ? T S = Z B 0 dz T (z) T ; (3.9)
where the second equality de…nes the thermocharge per unit area. As illustrated in Fig. 3.1b, a negative Seebeck coe¢ cient implies a positive surface charge at the hot boundary, T > 0. In general, the temperature varies along the particle surface, and so does T = T (x).

With the boundary conditions in (3.9), the electrostatic potential completely reads as

' = S X n t n P n (c) a n+1 r n+1 (n + 1) r e (a r)= : (3.10)
Before computing the component of the electric …eld, we approximate that, within the screening layer, the factor (a=r) n is close to unity, thus we discarded the corresponding factor in the following equations. Applying this last approximation and taking the perpendicular derivative of the potential ', the normal component of the electric …eld reads, to leading order in =a, E ? = Sr ? T (1 e z= ); (3.11) where r ? T (T S T 0 ) =a with T S the temperature …eld at the particle surface. The normal component of the electric …eld is screened by the thermocharge such that it vanishes at the surface. The parallel component, on the other hand, remains unchanged and is …nite at the surface,

E k = Sr k T S ; (3.12)
where the parallel gradient of the temperature …eld is taken to be r k T S sin T =a.

The thermocharge density follows from Gauss' law, T = r 2 '. With the same approximation as for the normal …eld component above, one …nds for the thermocharge density

T = T e z= ; (3.13)
with T = "Sr ? T S the thermocharge per unit area. This thermocharge density T decreases exponentially before it vanishes at large distances well beyond the Debye screening length. Since the Debye screening length is much smaller than the particle radius a, then the temperature gradient r ? T is taken to be constant through the charged layer. For a micron size particle at an excess temperature of 10 K, and a typical Seebeck parameter S = 10 4 V=K, the surface charge density T takes a value of about 10e per square micron and the electric …eld about 1 kV/m.

Charged surfaces 3.4.1 Poisson-Boltzmann theory

In the previous sections, we derived and discussed about the thermoelectric properties for a hot colloid particle in the Debye-Hückel approximation where the electrostatic energy e' is much smaller compared to the thermal energy k B T . In the case of a strong charged particle, this last point is necessary satis…ed because of strong-coupling system and the electrostatic properties will be governed by the well-known Poisson-Boltzmann equation.

Contrary to Debye-Hückel approximation, where all the derived equations are linear, the Poisson-Boltzmann equation is non-linear and does not have an analytical solution in a 3D system. To overcome this di¢ culty, we resort to the usual boundary layer approximation: The particle radius is much larger than the Debye screening length, a . As a consequences, the particle surface can be considered as ‡at and the resulting non-linear Poisson-Boltzmann equation will be treated in planar geometry where the solution results in a 1D problem.

Consider a hot charged particle with a surface charge in contact with electrolyte solution. Within the above approximation, the electrostatic potential satisfy the 1D equation,

@ 2 z ' = k B T e 2 sinh e' k B T ; (3.14)
where z is the normal distance to the particle surface. The solution of this equation is well-known and results in,

' (z) = 2k B T e ln 1 + ge z= 1 ge z= ; (3.15)
with the shorthand notation

ĝ = ge z= ; g = q 1 + `2= 2 `= :
The parameter g is given by the ratio of the Gouy-Chapman length `and the Debye length ,

`= e 2 `Bj j ; = 1 p 8 `Bn ;
where the Bjerrum length `B, the surface charge density , and the salinity n. In the following we assume a negative surface charge . Taking the perpendicular derivative d'=dz, the normal electric …eld reads

E ? = " e z= 1 g 2 1 ĝ2 : (3.16)
This …eld is perpendicular to the particle, and it satisfy the relation

E(0) = =".
The charge density in the di¤use layer is readily obtained from Gauss' law = "dE=dz,

= e z= (1 g 2 )(1 + ĝ2 ) 1 ĝ2 : (3.17)
Integrating over z one …nds

Z 1 0 dz (z) = ; (3.18)
which means that the counterions completely screen the surface charge density . We brie ‡y reviewed the Poisson-Boltzmann theory in thermal equilibrium, then we will discuss in the following sections about thermoelectric properties for both insulating and conducting surfaces.

Charged insulating surface

Now we consider an insulating surface with an electric double layer. In this case, the di¤use layer consists of both double-layer and thermocharge contributions,

= 0 + T : (3.19)
For notational convenience we assume a negative surface charge density 0 , as is the case for most colloids. Then the electric …eld at the outer boundary is given by the Seebeck …eld, and at the inner one by E ? (0) = 0 =", and Eq. (3.2) becomes

Sr ? T S + 0 " = 1 " Z B 0 dz( T + 0 ) = I " ; (3.20)
with the charge density of mobile ions per unit area,

I = T + 0 : (3.21)
Note that the two contributions are related to the Seebeck …eld and to the surface charge, respectively.

The corresponding Poisson-Boltzmann potential ' I , which is de…ned through T + 0 = "@ 2 z ' I , has to be calculated with an e¤ective surface charge I = T + 0 , with the parameter

g = q 1 + l 2 = 2 l= ;
where the Gouy-Chapman length, l = e 2 l B j 0 + "Sr ? T S j :

Taking the normal derivative of the total electrostatic potential, ' I = ' T +' I , the normal component of the electric …eld reads

E ins ? (z) = Sr ? T 0 " + Sr ? T e z= 1 g 2 1 ĝ2 : (3.23)
The second term of this last equation result from the perpendicular derivative of ' I , and it decays rapidly through the screening layer. Because of the constant temperature gradient r ? T through the charged layer, the …rst term is constant on the scale of the Debye length. One readily veri…es that E ? (z) satis…es the boundary condition, at the particle surface E ? (0) = 0 =", and well beyond the screening length E ? (B) = Sr ? T .

The parallel component of the electric …eld reads as,

E ins k (z) = Sr k T r k ' I : (3.24)
This parallel …eld does not vanish at the surface z = 0 because the second term gives rise to a term proportional to / r k T . Both contributions vary slowly on the scale of the particle radius and are of comparable magnitude.

Charged conducting surface

Now we turn to conducting surfaces, for example the gold cap of the upper hemisphere in Fig. 3.1c. Because of its high electric and thermal conductivities, the metal layer may considerably modify the thermoelectric properties in the doublelayer and at the particle surface. The electrostatic boundary conditions impose a constant potential, or a vanishing parallel electric …eld [START_REF] Andelman | Soft Condensed Matter Physics in Molecular and cell[END_REF],

E k (0) = Sr k T r k ' C (0) = 0: (3.25)
On the other hand, at the outer boundary z = B the parallel …eld takes the …nite value r k T . These conditions cannot be satis…ed with the surface charge discussed above.

To achieve (3.25) the mobile electrons in the metal surface move until their polarization charge density P modi…es the double-layer potential such that its gradient cancels the thermoelectric …eld at the surface. In one hand, the polarization charge is determined by inserting ' with

C (x) = T (x) + 0 + P (x) (3.26)
in Eq. (3.25) and solving for P . In the other hand, we assume that the total charge does not change, so one has for the surface integral R dS P = 0. With these two conditions, the polarization charge is determined by

P = 0 e p 1 + b 2 k B T (ST ) d ln " d ln T 1 ! T S hT S i 2T ; (3.27)
where T S the temperature …eld at the particle surface and hT S i its mean value, b = l= the ratio of the Gouy-Chapman length and screening length, and the surface potential. In the weak-coupling limit, the Gouy-Chapman length l is much larger compared to the Debye screening length, b = l= 1. Thus expanding in linear order Eq. (3.27), the polarization charge will be simpli…ed as

P = "S (T S hT S i) : (3.28)
Since the di¤use layer screens the local surface charge density, P induces a corresponding change of the mobile charge density, P , and we have C = T + 0 + P . We recall that the double-layer potential ' is calculated with the parameter C which accounts for the charge density of the di¤use layer, C = R dz C (z), whereas the surface charge density is given by ( 0 + P ). Accordingly, we have

E ? (0) = 0 + P " (3.29)
at the particle surface. Now taking the normal and the parallel derivative of the total potential, ' = ' T + ' C , the normal component of the electric …eld reads as

E cond ? (z) = 0 + P " + Sr ? T 1 g 2 1 b g 2 e z= + Sr ? T; (3.30)
for the parallel …eld component, one has

E cond k (z) = 1 1 g 2 1 b g 2 e z= Sr k T: (3.31)
At the particle surface, one can verify that E cond ? satis…es Eq. (3.29) whereas for the parallel …eld is zero at the particle surface E cond k vanishes. As shown in Fig. 3.2, with increasing distance, the double-layer potential ' decays and vanishes well beyond the screening length, and the electric …eld is given by, E = SrT:

Hydrodynamic phoretic velocity 3.5.1 Slip-velocity

Closely following Ref. [START_REF] Würger | [END_REF], we evaluate the boundary slip-velocity v s which is derived from a ‡uid mechanical treatment. For that we adopt the low Reynold number hydrodynamic Stokes equation which describes the velocity …eld of an incompressible ‡uid (r v = 0) in the steady state,

r 2 v = rP f ; (3.32)
where the solvent viscosity, P the osmotic pressure, and f the force exerted by the particle on the surrounding charged ‡uid (force per unit volume).

For an approximate solution of the Stokes equation, we consider a geometry where the Debye screening length is much more smaller than the particle radius, of the electric …eld in function of the ratio of the normal distance z and the Debye length . For the plot we used the following parameters: The Seebeck coe¢ cient for NaOH solution S = 200 V/K, particle radius a = 1 m, the Debye length = 30nm, and the excess temperature T = 30K. The dashed lines indicate, the long-range thermoelectric …eld E 1 = S T =a 6kV/m, and the normal …eld at the particle surface E ? (z = 0) = ( 0 + P ) =" 1:8 10 3 kV/m. a. In this case, the particle surface is considered as ‡at and the hydrodynamics quantities vary slowly along the particle surface, but much more rapidly in the normal direction [1]. Now if we adopt a local coordinates x and z, respectively, parallel and perpendicular to the surface, the normal coordinate of the velocity …eld vanishes, v z = 0. In the parallel direction, the velocity …eld depends hardly on the x component, that is v x = v x (z), implying the equation @ 2 z v x = f x dP=dx. Integrating this last equation with the Stokes boundary conditions, where at the surface the velocity vanishes v x j z=0 = 0 and becomes constant at a distance equal to the Debye length , one …nds the boundary slip-velocity as,

v s = 1 Z 1 0 dzz(f k r k P ); (3.33)
where is the solvent viscosity and where the driving force consists of several terms. The …rst one derives from the Maxwell tensor and contains electric forces,

f = E 1 2 E 2 r"; (3.34)
that is, the Coulomb force exerted by the electric …eld on the charge density and the change in electric energy due to a permittivity gradient.

To linear order in the temperature induced perturbations we have

f = 0 (SrT r' ) 1 2 (r' 0 ) 2 r"; (3.35)
where r' in the …rst term still depends on the detail of the electric double layer and, in particular, takes quite a di¤erent form at insulating and conducting surfaces.

The second term in (3.33) stems from the pressure P = nk B T exerted by the excess concentration of mobile ions n = n[cosh(e' =k B T ) 1] in the double layer. When evaluating the gradient one needs to account for its variation with temperature, salinity, and the potential ' , resulting in

rP = 0 r' + ( 0 ' 0 + nk B T ) rT T + nk B T rn n : (3.36)
In these relations for f and rP , the potential ' varies rapidly in normal direction, and slowly along the surface. The quantities T , ", and n vary slowly in all directions, on the scale of the particle parameter.

Gathering the di¤erent terms one obtains the force density

f rP = 0 SrT h rT T nk B T rn n 1 2 r' 0 2 r"; (3.37)
where we use the shorthand notation for the double layer enthalpy density h = 0 ' 0 + nk B T . As a remarkable feature, the parallel gradient r k ' has disappeared from the double-layer forces. If both the electrostatic force f and the osmotic pressure P depend on the precise form of the parameter , these terms cancel in (3.37). Since r' is the only term to depend on the polarization charge P , this quantity no longer is relevant and, as a consequence, the slip velocity does not depend on the conductivity of the particle surface.

With the known Poisson-Boltzmann expressions for ' and its derivatives, the integral in (3.33) is readily performed

v s = " S k r k T + "( 2 3 2 T ) 2 k r k T T " 2 T 2 r k " " + r k n n ; (3.38)
with the surface potential = ' 0 (0) and the quantity [START_REF] Würger | [END_REF],

T = 2k B T e " ln cosh e 4k B T 2 # 1=2 : (3.39)
For weakly charged surfaces, where the -potential is smaller than k B T =e 25 meV, one has T , whereas in the opposite case, T is signi…cantly smaller than . Note that we have restored the factor k which accounts for the thermal conductivity contrast of particle and solvent.

This result does not depend on the electrical conductivity of the particle surface. The slip velocity is the same at insulating and conducting surfaces, although the electric …eld at the particle surface shows a very di¤erent behavior. A similar e¤ect was shown to occur for the electrophoretic mobility at a metal surface [START_REF] Squires | [END_REF], resulting in an electro-osmotic slip velocity that is the same at insulating and conducting surfaces.

Drift velocity

The boundary slip-velocity in (3.38) varies along the particle surface with respect to the polar angle . Taking its con…gurational average along the surface, we …nd the particle drift velocity u = hv s i which is opposite to the osmotic ‡ow in the boundary layer. Simplifying the thermal conductivity ratio k 1 (particle and solvent thermal conductivities taken to be equal: k w k p ) and taking the surface integral of Eq. (3.38), one readily obtain the particle drift velocity

u = " 2 2 T 3 rT T + " 2 T 3 r" " + rn n + 2" 3 E T ; (3.40)
where the …rst term results from the thermoosmotic ‡ow in the electric double-layer and the last term accounts for the velocity induced by the self-generated Seebeck …eld E T = SrT with the Helmotz-Smoluchowski's electrophoretic mobility " = multiplied by the factor 2=3.

As for the slip-velocity, the sign and magnitude of this particle drift velocity does not depend on the electrical conductivity of the particle surface, but vary strongly with the electrolyte Seebeck coe¢ cient S which in turn takes speci…c values depending on the electrolyte composition used. For NaCl solution, the resulting velocity u > 0, thus the particle moves toward hot region, whereas for NaOH solution, u < 0, one observes the motion of particle in opposite direction.

Results and discussion

In previous sections, we have already derived the expressions of the electric properties, the thermocharge and the induced transport velocity of an uncharged or charged hot colloid in contact with electrolyte solution. Now we are going to discuss in details about these results, and to see which impact they can have in possible applications.

Thermocharge and electric …eld

The thermoelectric properties are quite independent of the particle's equilibrium surface charge density. For the sake of clarity, we mainly discuss the case of an uncharged particle.

Due to strong absorption of laser light by the metal cap, the particle generates an asymmetric temperature gradient which in turn creates a thermoelectric …eld or Seebeck …eld. This Seebeck …eld does not result from an externally applied voltage but from the ionic current (2.2) induced by the temperature gradient.

Although the electric boundary conditions at the surface of a dispersed colloidal particle are the same, its behavior in the vicinity are quite di¤erent from that of an external electric …eld, and in particular results in a thermocharge density at the particle surface, which is given by the equation (3.13). The behavior of the thermocharge density does not depend on the surface properties of the particle: In both cases, an insulating and conducting surface, it decays exponentially and takes the value zero at the bulk. Because of the non-uniform temperature gradient along the surface, r ? T varies slowly along the surface; and with the assumption a, the temperature gradient can be considered as constant through the charged layer. The thermocharge per unit area reads, in the case of a NaOH electrolyte solution carrying an excess temperature of 30K and a Seebeck coe¢ cient S = 200 V=K, as T = "Sr ? T 10 4 e/nm 2 :

(3.41)

In Fig. 3.5, we show the electric …eld lines around an insulating and conducting particle. We compare …rst the …eld lines around the low-permittivity insulating particle shown in Fig 3 .4 a and b, then we see secondly the case of an conducting particle. Because of the strong permittivity contrast, when an external electric voltage is applied then the resulting …eld does not penetrates in the particle. As a consequence, the …eld at the particle surface is by the factor 3" w = (2" w + " P ) 3=2 greater than in the bulk. On the contrary, the Seebeck …eld, determined by Eq. (3.6), is not deformed by the permittivity contrast (" P " w ) in the vicinity of the particle, and everywhere follows the temperature gradient. The explanation of this e¤ect is the fact that the temperature gradient is assumed to be constant everywhere, and the normal component of the thermoelectric …eld is screened by the accumulation of mobile ions in the vicinity of the particle. On the conducting surface, the polarization e¤ect modi…es the double-layer potential and imposes isopotential condition at the metal surface. As a consequences, the parallel …eld vanishes at the particle surface, and, as illustrated in Fig 3 .5c, the thermoelectric …eld at the particle surface is given by its normal component.

Polarization charge on the conducting surface

We discussed in the section above the thermocharge which occurs on both insulating and conducting surfaces. On the latter, however, the isopotential condition of electrostatics imposes in addition a polarization charge on the metal coating, which is screened by mobile ions. In other words, the thermal polarization e¤ect 2 larger than the bulk …eld. b) Thermoelectric …eld in the vicinity of an insulating particle. Assuming a constant temperature gradient, that is, similar thermal conductivities of particle and solvent, we …nd that the …eld is not deformed by the permittivity contrast (" P " w ) but follows the constant temperature gradient, E = SrT . Within one Debye layer from the particle surface, its normal component E ? is screened by ion accumulation, that is, the thermocharge T , as shown in the left panel of Fig. 3.4; the parallel component E k does not vanish, and the particle surface is not at constant potential. c) Thermoelectric …eld in the vicinity of a conducting particle. Polarization of the metal surface adjusts the surface charge density such that the parallel component of the …eld vanishes, resulting in an isopotential surface. The corresponding nonuniform surface charge is illustrated in the right panel of Fig. 3.4. The normal component of the electric …eld corresponds to the surface charge parameter . modi…es the charge parameter on the electric double-layer by P which, in the weak-coupling approximation, is given by (3.28).

⊕ ⊖ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ a) b) c) 𝜀 𝑃 ≪ 𝜀 𝑤
For a micron size particle in NaOH solution with an excess temperature of 30K, an electrolyte Seebeck coe¢ cient S = 200 V/K and a Debye length = 2nm, the polarization charge is of the order of

P = "S (T S hT S i) 10 2 e nm 2 : (3.42)
From comparison of r ? T (T S T 0 ) =a, it is clear that P exceeds the thermocharge T by a factor a= which, for micron-size particles, is of the order of a= 100. The resulting electric …eld lines of a heated Janus colloid are shown in Fig 3 .1 c: The far-…eld corresponds to the Seebeck …eld E = SrT , whereas the near-…eld depends strongly on the surface properties. At the conducting cap (the metal side), the parallel component vanishes within one Debye length. In order to relate the thermally induced charges and the thermoelectric …eld to the surface conductivity, we show in Fig. 3.5b and c how the Seebeck …eld of a constant temperature is deformed in the vicinity of a colloid with an insulating or conducting surface coating. Both di¤er signi…cantly from the deformation of an applied electric …eld, which is shown in Fig. 3.5a. As a conclusion, we say that the behavior of the thermoelectric …eld at solid boundaries is very di¤erent from that of a voltage induced …eld.

Granular gold surface

The above discussion assumes a continuous gold cap, which does not always correspond to the actual experimental situation. For example, the cap of the particles used in Ref. [15] consists of a dense coverage of nano-sized gold grains, visible in scanning electron microscopy images [39] and illustrated in Fig. 3.6.

These grains do not connected to a continuous gold structures. Each grains forms an equipotential surface, and, as in the case of a continuous gold cap, the resulting parallel …eld vanishes within one screening length. As illustrated in Fig. 3.6, we see the existence of a potential jump between nearby grains; their cold and hot boundaries carry polarization charges which result in a strong electric …eld in the grain's spacing. As a consequence, the picture developed for micronsize conducting surfaces remains correct at the nanoscale. Because of the surface cold hot λ . Each metal grain forms an equipotential surface and carries opposite polarization charges P at its cold and hot sides. Like on the continuous cap in Fig. 3.1c, E k vanishes in a layer of about one screening length.

On the insulating surface at the left, the …eld is given by (3.12).

roughness one may expect a somewhat smaller slip velocity than at a homogeneous cap.

Comparison with experiment

A recent experiment reported thermoelectric driving of hot silica particles with a granular gold cap [15]. Probing the particle's self-propulsion velocity in 10 mM solutions of NaCl, LiCl and NaOH, revealed a strong salt-speci…c e¤ect, which agrees qualitatively with the Seebeck coe¢ cients of these electrolytes, S NaCl > S LiCl > S NaOH . When comparing to experimental …ndings, one has to add to v s the thermo-osmotic and related contributions, subsumed in v osm s , which are of comparable magnitude but independent of the ion species [16].

Contrary to what was observed for passive polystyrene beads in an external temperature gradient [16], the experiments of Ref. [15] seem to indicate that the negative Seebeck e¤ect of NaOH does not result in a change of sign of the selfpropulsion velocity of an active particle. This could be related to the granular gold surface, the properties of which could di¤er from the behavior derived for a continuous gold surface assumed in this paper. As another possible explanation, we mention the strong increase of v osm s with temperature, whereas the thermoelectric contribution was found to be rather constant [16].

Conclusion

The aim of this present study was to understand the nanoscale Seebeck e¤ect for a hot metal nanostructure in contact with electrolyte solution. In other words, we studied the self-propulsion mechanism induced by the thermo-electrophoresis e¤ect of an active particle. In this problem, the thermoelectric properties are studied by considering di¤erent surface geometry, i.e, the surface of the particle can behaves as an insulating or a conducting surface.

At the insulating surface, the di¤use layer consists of two contributions: The double-layer charge 0 inducing by the surface charge 0 which is not a¤ected by the Seebeck e¤ect, and the non-uniform thermocharge T which is proportional to the surface excess temperature T S . This temperature …eld increases, in the case of a Janus particle, from the passive hemisphere to the heated cap, resulting in a parallel component of the Seebeck …eld along the particle surface. On the conducting surface, the parallel temperature gradient induces polarization charge on the metal structure, which modify the double layer such that the parallel component of the electric …eld vanishes at the surface. Surprisingly, this does not a¤ect the thermally induced slip velocity, which turns out to be identical on insulating and conducting surfaces.

In addition, our theoretical results agree qualitatively with a recent experiment [15]: When regarding speci…c-ion e¤ects, the self-propulsion velocity shows a signi…cant variation with the used salts NaOH, NaCl, LiCl. Taking into account these above points, we can conclude that the thermoelectric properties of hot Janus particles in an electrolyte solution depend strongly both on the material properties and the speci…c-ion e¤ects.

Chapter 4

Hydrodynamic interactions in DNA thermophoresis

Introduction

The dynamics of polyelectrolytes is of considerable interest in a …elds such as micro ‡uidics and biotechnology applications [33]. Understanding their dynamics can be essential to perform properties of electrophoresis or thermophoresis of polyelectrolytes. In the past, the …rst models predicting the dynamics of polymer chain in solution was developed by Rouse and Zimm [34,35]. In these theories, the polymer is modelled as a chain with a large number of spherical beads connected by springs. The main di¤erence between these two models is the e¤ect of hydrodynamic interactions which are absent in Rouse's model and present in Zimm's model. Consequently taking into account these hydrodynamic interactions between beads, the di¤usion coe¢ cient scale as: D _ N , where the Flory exponent = 0:6 for real chains in good solvent [START_REF] De Gennes | Scaling concepts in Polymer Physics[END_REF]. This last equation shows that the di¤usion coe¢ cient of the polymer chain varies with the number of beads, i.e, it depends strongly on the molecular-weight.

In contrary, phoretic mechanism for polymers in a dilute solution is usually considered to be independent on the molecular-weight. This point was …rst discussed and con…rmed by Giddings when studying thermophoresis of high polymers [12], and later by Brochard and de Gennes by showing that the hydrodynamic interactions have no e¤ect on thermophoresis for polymers [25]. More recently, considerable progress has been made on DNA-phoresis; these results revealed, however, a molecular-weight dependence on the mobility in the range where the persistence length L with the Debye screening length [36,37]. Muthukumar con…rmed this molecular-weight dependence and showed the importance of " hydrodynamic screening" on DNA phoresis [27,28].

In this present chapter, we investigate in the hydrodynamic screening of DNA thermophoresis, and in particular we study the in ‡uence of hydrodynamic interactions in a dilute polyelectrolytes solution: In the polymer chain, a given molecular creates a ‡ow …eld and drags its unit neighbor beads. This e¤ect increases the velocity of the chain until it saturates at large distances well beyond the Debye screening length, where the drag forces exerted by charged monomers and the corresponding counterions cancel each other. We evaluated theoretically the thermophoretic mobility D T of a polymer chain of n beads. This mobility shows, in function of the chain length, a non-monotonuous behavior and consists of two contributions: a dielectrophoresis induced by permittivity gradient and the Seebeck e¤ect. At the …rst time, we begin by studying about the hydrodynamic interactions arising during the mechanism process. At second, we give a detailed calculation of the phoretic coe¢ cients and we discuss about the counterion condensation e¤ect which strongly in ‡uences the thermophoretic mobility's behavior. At the end, we compare our theory with recent experimental result on single-stranded DNA [44].

Thermodynamic forces

Due to the presence of the temperature gradient, the system exerts lateral forces on the counterions surrounding the charged monomer which in turn moves the ‡uid with respect to the particle surface. The created force, called force density or thermodynamic force, results from the interactions at the solid-solvent interface, and his general expression reads as [51]

f = ( + n i k B T ) rT T E 2 2 r" n i k B T rn 0 n 0 + E T ; (4.1)
where the solvent permittivity, n 0 the bulk salinity, n i = n + + n the excess ion density and = e (n + n ) the total charge density. The force density (4.1) describes the force exerted on the counterions cloud surrounding the charged monomer, and it's given in terms of quantities describing the electric double layer and the thermoelectric …eld. In Eq. (4.1), the terms proportional to / rT and / rn 0 result from the osmotic pressure i.e the pressure of the excess density of mobile ions, the second term / r" results from the divergence of the Maxwell stress tensor which describes the force exerted by the charged-monomer surface, and the last term E T describes the force exerted by the macroscopic thermoelectric …eld E T = SrT on the monomer's counterion density [18]. All the terms in f are of comparable magnitude but for small molecules, that is, for molecules's size a much more smaller than the screening Debye length , then the companions …eld E T and r" dominate whereas the gradients rT and rn 0 are negligible. Thus the simpli…ed expression of the density force becomes,

f = E 2 2 r" + E T ; (4.2)
where the …rst term is proportional to the thermally induced permittivity gradient r" = (d"=dT )rT , with the charged monomer's electric …eld E. Since the permittivity decreases with rising temperature, d"=dT < 0, the surrounding water moves to the hot, as recently con…rmed experimentally for thermoosmosis in a capillary [45]. By reaction, the molecule migrates toward the cold. The electrolyte Seebeck coe¢ cient S is a salt-speci…c quantity that may take either sign, resulting in motion along the temperature gradient or opposite to it [16].

The density force given in Eq. ( 4.1) play also the rule of source term in the hydrodynamic Stokes equation. Now assuming that the electrostatic potential e is much more smaller than the thermal energy k B T (e k B T ) then the charge density can be expanded in quadratic order in term of as, ' " = 2 . Inserting the number permittivity derivative = d ln "=d ln T , the Debye screening length 2 = 1=8 n 0 l B the Debye length with l B = e 2 =4 k B T the Bjerrum length, we can write the density force as a linear function of the thermal gradient,

f = E 2 2k B T e 2 k B rT: (4.3)
The force density f acts on the surrounding water and, by reaction, the molecular unit is subject to the opposite force F = R dV f [51]. Thus the velocity …eld induced by the moving bead i at the position of its neighbor j, consists of two contributions,

v(r ij ) = G(r ij ) F + Z G(r ij r) f (r)dV; (4.4)
where G(r) = (1 + b rb r)=8 r is the Oseen tensor with the viscosity and b r = r=r [START_REF] Kim | Microhydrodynamic: Principles and Selected Applications[END_REF]. The …rst term describes the long-range velocity …eld v 1=r or "stokeslet" of a particle subject to an external force F; it gives rise to strong hydrodynamic e¤ects on di¤usion and sedimentation [START_REF] De Gennes | Scaling concepts in Polymer Physics[END_REF]. The second term is characteristic for phoretic motion; it may be viewed as the sum of stokeslet ‡ows of strength f dV and centered at a distance r from the particle. Since both E and vanish well beyond the Debye length, the second term cancels the …rst one at large distances, r ij , whereas it is small for nearby beads.

Hydrodynamic interactions

We want here to study the in ‡uence of the hydrodynamic interactions in a dilute polyelectrolytes solution. Now consider a polyelectrolyte chain of n building blocks where each molecular bead of charge q results in a Debye-Hückel surface potential = q=4 a, as illustrated in Fig. 4.1. Throughout this paper, we assume that the molecular size a is small compared to the Debye screening length . Unit i creates a ‡ow …eld v(r r i ) in the surrounding ‡uid and thus drags its neighbor j. Then the overall velocity u of the chain is given by the sum of the monomer contribution u 1 and the mutual advection,

u = u 1 + 1 n X i;j6 =i hv(r ij )i ; (4.5)
where the angular brackets h i indicates the con…gurational average with respect to r ij = r j r i . Before computing Eq. (4.5) and …nding the velocity of the chain, one have to specify the ‡ow …eld v(r ij ) and all the physical forces arising during the mechanism process. In order to simplify the second term in the velocity u, the con…gurational average is done with the equilibrium distribution function. The later being isotropic, the only …nite component of the mean drag velocity is along the temperature gradient, and the tensor equation in (4.4) is simpli…ed to a scaler one where the Oseen tensor is replaced by its diagonal part by G(r) = 1=6 r.

Inserting the expression of the density force and averaging over the position i of the polymer chain, the Eq. (4.5) becomes

u = u 1 + 1 n X i;j6 =i Z G(jr ij rj) G(r ij ) f (r)dV; (4.6)
The second term in parenthesis depends on the details of the counterion distribution; for later use we expand the pre-average Oseen tensor in term of Legendre Polynomials P k (cos ), with cos = c r ij b r,

G(jr ij rj) = 1 6 1 X k=0 P k (cos ) h k ij H k+1 ij ; (4.7)
where h ij = min (r; r ij ) denotes the smaller of the distances r and r ij , and H ij = max (r; r ij ) the larger one. Now the problem remain to compute the volume integral in Eq. (4.6), and, in other word, to evaluate Eq. (4.7) with an appropriate counterion distribution. We start by evaluating the monomer's electrical potential with the corresponding electric potential E = r which are both screened. In this case, the electrostatic potential of a single bead of valence b z is well described by the Debye Hückel approximation, = a r e r= ;

where the single bead surface potential = b ze=4 a, which we assume to be negative.

By applying E = d =dr we obtain easily the radial monomer's electric …eld. Inserting Eq. (4.8) into Eq. (4.6), one …nds the only …nite contribution stems from the term k = 0 in Eq. (4.6), and the remainder vanishes for an isotropic screening cloud. Assuming that the particle's radius a is much smaller than the Debye length , and the quantity na be of the order of , the volume integral in Eq. (4.6) is readily performed by doing a …rst order development in term of a= . We obtain,

Z G(jr ij rj) G(r ij ) f (r)dV = a 2 2 6 T e 2r ij = r 2 ij 2S a 3 e r ij = r ij ; (4.9)
where S = k B =e is the electrolyte Seebeck coe¢ cient. This last equation characterizes the hydrodynamic interactions between monomers and shows the latter is exponentially screened.

To …nd the complete expression of the chain's velocity, we begin by simplifying the con…gurational average by treating the molecules as rigid rods such that the distance of beads i, j simpli…es to r ij = ji jj d, where d the distance between two consecutive beads. As a consequences, the double sum is replacing by the double integral over i and j. So we have to compute,

u = u 1 + 1 n 8 < : a 2 2 6 T Z n Z 1 didj e 2ji jjd= ji jj 2 d 2 2S a 3 Z n Z 1 didj e ji jjd= ji jj d 9 =
; rT:

(4.10) Eliminating the absolute value and computing the double integral over i and j, one …nds …nally

u = 2 3 (1 + ) r + 2 3 (1 + S )E T : (4.11)
where the quantities and S account for hydrodynamic interactions (see Fig.

4.2) With

= 0 = S one has the explicit expression for the monomer velocity u 1 .

The hydrodynamic correction factor for the motion driven by the permittivity gradient reads, The factor 2 in the exponential and Ei functions in arises from the screening factor of the force density, E 2 / e 2r= , whereas the factors in S are related to the decay of the screening cloud, / e r= . The Seebeck term S can be seen as an electrophoretic mobility which is identical to Muthukumar result [27], albeit with the bead distance d instead of Kuhn length l. In the expression of the chain velocity u, we detect two terms: a dielectrophoresis induced by the permittivity gradient r and the electrolyte Seebeck e¤ect induced by the thermoelectric …eld E T = SrT . The in ‡uence of these two terms are shown in Fig. 4.2 where we plotted the quantities and S as a function of the molecular weight. Both vanish for monomers, n = 1, whereas for long molecules they tend toward the constants 1 = (a=d) 2 (2 dE 2 d + e 2 d) and 1 S = 2(a=d)E d.

Counterion condensation

In addition to hydrodynamic interactions arising during the dynamics of polyelectrolytes, we oberve another phenomenon which is called counterion condensation: due to Coulomb interactions, a strong charged polyelectrolyte exerts a powerful attraction on counterions which partly condense on the polymer until its line charge density is reduced to the critical value e=l B [START_REF] Manning | [END_REF]. The condensed ions move with the polymer chain, and the uncondensed mobile ions present in the ionic atmosphere are well treated by the Debye-Hückel theory. Because of this condensation e¤ect, everything change even the e¤ective charge per monomer q = b ze. This charge remain unknown, and to overcome this di¢ culty, we estimate his value by taking into account its reduction along the barre charge n. Thus we consider for simplicity

b z = 1 + 1 1 1 + n ; (4.14)
where = l B =d is called the Manning parameter, and n = (n 2 1) n 2 0 with n 0 an arbitrary constant value. This constant value enhance the precision of the analytical curve, and n 0 = 80 we obtain best …ts for the experimental data (see Fig. 4.2). The parameter modelizes the counterion condensation e¤ect, in the sense that, if > 1 the condensation e¤ect takes place because the Coulomb interaction is very strong and dominates the thermal one; but if < 1, meaning that the bead spacing d is larger than the Bjerrum length l B ' 7Å, then the electrostatics forces acting on the chain is very weak compared to the thermal forces, and thus the condensation e¤ect doesn't occur in this case.

However, the condensation e¤ect is relevant for polyelectrolytes because the molecules are strongly charged and the bead spacing distance d is something between 3 and 4Å. The valency b z of the e¤ective charge given in Eq. (4.14) satisfy completely the Manning's prediction, in the sense that, for a monomer the quantity n vanishes, one has b z = 1, and for long chains n goes to in…nity, thus the resulting e¤ective charge valence decrease along the barre charge until it converge to b z = 1 .

Results and comparison with experiment

The thermophoretic mobility is de…ned from the drift velocity relation u = D T rT . From this equation, the thermophoretic mobilty reads,

D T = 2 3 T (1 + ) S 2 3 (1 + S ) : (4.15)
We can obtain a detailed expression by inserting the valence of the e¤ective charge b z and the Bjerrum length l B ,

D T = k B 12 a b z 2 l B a (1 + ) + 2b z (1 + S ) b S ; (4.16)
where the parameter = d ln =d ln T 1:4 which arises from the permittivity gradient, and the dimensionless Seebeck coe¢ cient b S = S(e=k B ). This thermophoretic mobility is valid for small monomers, that is, for monomer's size much more smaller than Debye length. For monomers the mobility is independent of the Debye length, whereas for longer chains, the correction factors give rise to complex dependencies on and n. This mobility shows also two contributions which are plotted as a function of n and shown in Fig. 4.2. These contributions have a non-monotonuous behavior, both they reach a maximum and decrease until they reach a …nite constant value for large n; for typical parameters of DNA in weak electrolyte, the permittivity term shows an overall decrease, ẑ2 (1 + 1 ) < 1, whereas the Seebeck term is enhanced, ẑ(1 + 1 S ) > 1. As a consequence, the thermophoretic mobility has the same behavior. The explanation of this variation is nothing else that the …rst increase result from the hydrodynamic interactions between monomers then the decrease is caused by the reduction of the e¤ective charge inducing by the condensation e¤ect.

Our theory gives the thermophoretic mobility D T in Eq. (4.16), whereas experiment often probe the Soret coe¢ cient S T = D T =D, where the di¤usion coe¢ cient D = k B T =6 R h is determined by the hydrodynamic radius R h of the molecules. 7) =42] where we took from the paper [55] and R[n](nm) is an adjustable parameter with n is the number of beads in the polymer chain. We took for this parameter: R( 5 Unfortunately, for short molecules, there is no simple and generally formula for the hydrodynamic radius. To overcome this di¢ culty, we have estimated the value hydrodynamic radius in the simple way by,

• • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ 0.
R h = R(n) 1 + 0 L ; (4.17)
where 0 , L are an arbitrary constant lengths, and R(n) be an adjustable parameter which describes also the measured mean value of the hydrodynamic radius. The hydrodynamic radius is modelized in Eq. (4.17) in order to determine the di¤usion coe¢ cient D and the Soret coe¢ cient S T , and to agree with recent experimental measurement [55]. Fig. 4.3 shows the measured values of the hydrodynamic radius R h in function of the Debye length, which agree slightly with the analytical one from Eq. (4.17 length, as expected from the increased sti¤ness in weak electrolyte. The data for n = 5 and 50 show a slight decrease and signi…cant scatter. In Fig 4 .4 we compare our theory with the experimental Soret data for singlestranded DNA as a function of the Debye length , taken from Ref. [44] and given in Table 4.1. The theoretical curves are calculated with Eq. (4.16) and a simple model for the di¤usion coe¢ cient D, as described above. The best agreement with the data is obtained when retaining in Eq. (4.16) the permittivity-gradient term only, that is, for zero Seebeck coe¢ cient, b S = 0. The increases of S T with the Debye length arises mainly from the hydrodynamic correction . For short chains, that is for n < 30, the variation with n is of purely hydrodynamic origin, whereas for larger chains counterion condensation plays an important role, as is clear from Fig. 4.2.

In order to clearly display the e¤ect of hydrodynamic interactions, we plot in Fig. 4.5 the thermophoretic mobility in (4.16) as a function of the molecular-weight n. The experimental points are obtained from D T = DS T , with the measured coe¢ cients S T and D which are determined from the estimated function for the hydrodynamic radius R h given in Eq. (4.17). The theoretical curves are calculated with the permittivity only ( b S = 0). The initial increase of the data up to n = 22 agree quantitatively with the relation (4.12), thus providing strong evidence for the role of hydrodynamic interactions. The maximum and the subsequent decrease (4.11) drives the molecules toward the hot, whereas the permittivity gradient points toward the cold. Since the latter dominates for short molecules and the latter for long ones, D T changes sign as the n increases. are well described by counterion condensation according to (4.14). Adding a signi-…cant thermoelectric contribution, i.e, increasing the reduced Seebeck coe¢ cient b S, would not improve the quality of the …t, quite on the contrary. This indicates that the Seebeck …eld in NaCl solution is small, con…rming a previous analysis of thermophoretic measurements on polystyrene beads [16].

○ ○ ○ ○ ○ ○ ○ ○ ○ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ○ ○ ○ ○ ○ ○ ○ ○ ○ 2
○ ○ ○ ○ ○ ■ ■ ■ ■ ■ ■ ▲ ▲ ▲ ▲ ▲ ▲ ▼ ▼ ▼ ▼ ▼ ▼ ◆ ◆ ◆ ◆ ◆ ◆ ▮ ▮ ▮ ▮ ▮ ▮ • • • • • ■ ■ ■ ■ ■ ■ ○ ○ ○ ○ ○ ○ λ = 11.25 nm ■ λ = 9.74 nm ▲ λ = 8 nm ▼ λ = 7 nm ◆ λ = 6.16 nm ▮ λ = 5.2 nm • λ = 4.16 nm ■ λ = 3 nm ○ λ = 2.
The electrolyte Seebeck e¤ect was discarded in the above analysis of Soret data in NaCl solution. In 4.2 it is clear that for short chains, the permittivity gradient term prevails, whereas for longer molecules the Seebeck term dominates because of its much larger hydrodynamic factor S . The resulting velocity di¤erence could be used for speci…c accumulation of one component at a heated spot. For example, in an electrolyte with b S = 0:3, the permittivity gradient dominates for short molecules (n < 50) which move to the cold accordingly, whereas longer chains are driven to the hot by the thermoelectric …eld.

Conclusion

The objective of this chapter was to discuss about the e¤ects of hydrodynamic interactions and thermoelectric …eld occurring during the thermophoresis mechanism for DNA in electrolyte solution. As a summary of our main result, we start with the non-monotonuous behavior of the thermophoretic mobility D T which consists of two contributions: The motion induced by the permittivity gradient, that is the dielectrophoretic term ; and the one induced by the Seebeck …eld S . These two contributions have the same behavior: With increasing the chain length n, it presents a …rst increase resulting from the hydrodynamic interactions, and a decrease caused by the counterion condensation. For longer chains, that is for large n, the mobility D T presents a limit which is obtained by letting ! 1 and ẑ ! 1 . This limit does not depend on the molecular-weight n, and does not vanish for high salinity or small Debye length, contrary to what was observed for micron-size colloidal particle [16].

In addition, the comparison of our theoretical results with the experimental data, shown in Fig. 4.4 and 4.5, reveals a strong dependence on the molecularweight, which arises from the interplay of the hydrodynamic interactions and the condensation e¤ect. Finally, the change of sign of the mobility D T depend on the negative value of the Seebeck coe¢ cient S; and in a physical term, it results from the interplay between the dominant driving forces: The thermally induced permittivity gradient and the Seebeck …eld with a negative Seebeck coe¢ cient.

Chapter 5 Ion dynamics: time dependent e¤ect

In section 3, we studied the thermoelectric properties of a metal capped colloid particle moving in a electrolyte solution. In that studies, all the physical properties are derived in the stationary state where the system is independent of time. Thus in this chapter, we will study the ion dynamics properties, and in other words the electrolyte Seebeck e¤ect in the non-stationary state. We will also try to …nd the ion di¤usion time scale and discuss about its e¤ect on thermophoresis .

Charge conservation equation

In electrolyte solution, when ion-species are in ‡uenced by a non-uniform temperature gradient, thus results in an ion current J which is detailed in section 2. We de…ne the total ion current by I =e (J + J ) and by linearizing, his expression becomes,

I = "D r 2 E 1 2 (E SrT ) ; (5.1)
where " the solvent permittivity and D = (D + + D ) =2 the ion di¤usion coe¢ cient (more details in Appendix D). We want here to write the equation satis…ed by the thermoelectric …eld in the non steady state. We start by the charge conservation equation related the total ion current I and the charge density = e (n + n ), @ t + div I =0:

(5.2) Applying Gauss's law div E = =", one obtains the equation describing the evolution of the generated electric …eld inside and outside the particle's electric double-layer,

@ t E = D r 2 E 1 2 (E SrT ) : (5.3)
This last equation results from the charge conservation equation and characterizes the behavior of the electric …eld in the non-stationary state. When we put @ t E = 0, we …nd the equation (3.6) given in section 2 and section 3 where the system reaches the stationary state. The equation given in (5.3) is called reaction-di¤usion equation where the right hand side describes the creation of the electric …eld, the left hand side characterizes the term of di¤usion and the term SrT plays the rule of source for the thermoelectric …eld. To study the dynamics of colloid particle in an electrolyte solution, we will try to solve the reaction-di¤usion equation in (5.3) and …nd the generated electric …eld in all state.

E¤ect on Janus particle

Consider a half-metal covered Janus colloid of radius a and centered at r = 0. The self-generated temperature gradient is asymmetric due to half-metal absorption and remains always in the stationary state due to faster heat di¤usion. Here we consider a particle with no surface charge, i.e, we are in the case of an insulating uncharged surface. To have simple computations, we de…ne the reduced electrostatic potential by the relation E = S T r , then the advection-di¤usion equation becomes

@ @t = D r 2 1 2 ( T l ) ; r > r 0 ; (5.4) n r = 0; r = r 0 ; (5.5)
where r 0 = a the particle radius, and T l (r) = (T (r) T 0 ) = T is the reduced temperature …eld with T (r) the self-generated temperature …eld [56]. For r > a, the reduced temperature …eld reads

T l = 1 X n=0 t n P n (c) a r n+1 (5.6)
where c = cos and P n the Legendre polynomial, and the coe¢ cients t n are given by, t 2k = t 2k+1 = ( 1) k = (2k + 1), except the …rst coe¢ cient t 0 = 1=2 + 1= .

The boundary condition in (5.5) says that the normal component of the electric …eld vanishes at the particle surface. The partial di¤erential equation is solving by the Laplace transform with no initial condition, and by expanding the solution in …rst order in term of =a, with a, one …nds, outside of the particle for r > a, the electrostatic potential as (r;t) = ( 1) 1 e t= io n ;

(5.7)

where (1) is the potential in the steady state where the time exponential factor vanishes. Applying the relation E = S T r , one can …nd easily the corresponding electric …eld (for more details see Appendix E). In Eq. (5.7) the quantity

ion = 2 D ; (5.8)
represents the characteristic time scale expressing the time of ion di¤usion over the screening length. This time scale characterizes also the particle relaxation time, i.e, the time which the particle start moving. For typical values of the di¤usion coe¢ cient D 10 9 m 2 /s and the Debye length = 50 nm, the relaxation time is in the order of ion s. Important to note that this ion di¤usion time scale is by several order of magnitude greater than the thermal and hydrodynamic time scales which are, respectively, given by th = 2 = and hy = 2 = with the heat di¤usivity 10 7 m 2 /s and 10 6 m 2 /s the kinetics viscosity. This last point tell us at which moment the particle start moving and enable us to discuss in more detail the e¤ect of this relaxation time on thermophoresis.

E¤ect on thermophoresis

In the previous section, we discussed about the Seebeck e¤ect in the non-steady state and computed the ion di¤usion time scale ion . In this present section we will study by details the time-dependent e¤ect on the thermophoretic motion of particles. The particle's slip velocity can be written as, 10 nanoseconds. The Seebeck e¤ect requires ion di¤usion which occurs on the time scale ion that may attain a microsecond. Since in many instances, the thermoelectric slip velocity v el s is stronger and carries the opposite sign [17,16], the onset of the Seebeck e¤ect could even result in a reversal of the direction of motion.

Conclusion

The aim of this small chapter was to discuss about the dynamics of ions occurring once a temperature gradient is present in the system. In other words, we studied the time dependent e¤ect, i.e, when the system is not in the stationary state. For that we started by the charge conservation equation which rely the charge density and the current caused by the di¤usion of ion. The solution of this reaction-di¤usion equation gives rise to the non-stationary thermoelectric …eld E and the characteristic time scale for ion di¤usion ion . This time scale characterizes also the particle relaxation time. Indeed, this ion-di¤usion time scale is by several order of magnitude greater than the thermal and hydrodynamic time scale. As a consequence, we observed a change of behavior in the motion of particle during the …rst milliseconds. This variation can be seen in the expression of the slip-velocity which consists of two di¤erent contributions: The time-dependent thermoelectric and thermo-osmotic slip-velocity, v el s (t) and v osm s (t), which set in on the ion-di¤usion time scale ion 1 s and the heat-di¤usion time scale th 10nanoseconds, respectively. In Fig. 5.1, we showed the evolution of these two contributions in function of time, and, as the result, we found that the time-dependent thermoelectric slip-velocity dominates over the thermo-osmotic one. This last point shows how the Seebeck e¤ect takes considerably an important place on the direction of motion of colloids in electrolytes.

Chapter 6

Summary and conclusion

This thesis was elaborated in order to study the nanoscale Seebeck e¤ect at hot metal nanostructure. The electrolyte Seebeck e¤ect is an important and powerful phenomenon on the transport mechanism for colloids or polyelectrolytes. To achieve completely this work, we organized this thesis in di¤erent chapters: The …rst two chapters was an introducing part of our work, where in chapter 1 we presented in detail di¤erent class of phoretic transport mechanism such as electrophoresis, di¤usiophoresis and thermophoresis. Among these transports mechanisms, the motion induced by thermophoresis for hot colloids in electrolyte solution and for polyelectrolytes was studied in this present thesis. In electrolytes systems, the thermophoresis mechanism is not only caused by the thermally-induced salt gradient but also by the electrolyte Seebeck e¤ect. The origin of this electrolyte Seebeck was presented and detailed in chapter 2.

After introducing the electrolyte Seebeck e¤ect in 1-Dimension and 3-Dimension in chapter 2, we go to more complexes geometries in chapter 3 and we tried to understand how the electrolyte Seebeck e¤ect behaves when the particle's surface varies from an insulating to a conducting surface. In this chapter, we tried also to understand how the change of behavior of the electric properties at an insulating and conducting surfaces can a¤ect the induced slip-velocity v s , and at the end we computed the resulting particle drift velocity u. As a main result of this part, we found that: Near the insulating surface, the parallel component of the electric …eld E k is …nite, whereas at an conducting surface carrying an high electrical conductivity, the isopotential condition imposes a polarization charge which modi…es the double-layer potential such that its gradient cancels the thermoelectric …eld at this surface, thus the parallel electric …eld vanishes. Surprisingly, the change of behavior of the thermoelectric …eld in function of the electrical conductivity does not a¤ect the thermally-induced slip-velocity. This slip-velocity is identical at the insulating and conducting surfaces. As a consequence, the particle's drift velocity, resulting from the con…gurational average of the slip-velocity, also does not depend on the material properties but his sign vary strongly with the salt composition used NaOH, NaCl or LiCl. These above theoretical results have intensively revealed how both material properties and speci…c-ion e¤ects can modify thermoelectric properties near a surface of a hot Janus colloid in an electrolyte solution.

Another important result for this work was presented in chapter 4, where we studied hydrodynamic interactions in thermophoresis for polyelectrolytes. In this chapter, we evaluated theoretically the thermophoretic mobility D T , which arises from mutual advection of the n repeat units of the molecular chain. This mobility was studied in term of series expansion for the Oseen tensor. The two main physical phenomena occurring during DNA thermophoresis mechanism are the hydrodynamic interactions between monomers and the counterion condensation. Taking into account these phenomena, we found, as the main result, the thermophoresis D T has a non-monotonous behavior and consists of two contributions: The …rst one is the motion induced by the permittivity gradient, that is the dielectrophoretic term " ; and the second one is induced by the Seebeck e¤ect S . With increasing the chain length n, these two contributions present a …rst increase which is due to hydrodynamic interactions, and a decrease caused by the counterion condensation. We also observed that, for long chains, the Seebeck term dominates because of its larger hydrodynamic factor S , whereas for short chains the dielectrophoretic term " prevails. These theoretical results agree with experimental data, in the sense, they both revealed a strong dependence on the molecular-weight and an interesting interplay between hydrodynamic interactions and condensation e¤ect.

Finally, we studied brie ‡y in chapter 5 the time-dependent e¤ect on the dynamics of ions in electrolyte solution. In this chapter, we wanted to know at which time scale ions di¤use once a non-uniform temperature gradient is present. Solving the charge conversation equation and applying the Debye-Hückel approximation, we found the non-stationary thermoelectric …eld with the characteristic time scale for ion-di¤usion ion = 2 =D 1 s, which is, from comparison, much more greater than both hydrodynamic and thermal time scales. At the end of this chapter, we studied the time-dependent e¤ect on particle's slip-velocity which consists of two contributions: The time-dependent thermoelectric and thermo-osmotic slipvelocity, v el s (t) and v osm s (t), which set in on the ion-di¤usion time scale ion 1 s and the heat-di¤usion time scale th 10nanoseconds, respectively. Plotting the time-evolution of this two contributions, one observed the thermoelectric slipvelocity more stronger than the thermo-osmotic one in a …rst ten microseconds. These above points show the important role playing by the electrolyte Seebeck e¤ect on the thermophoresis mechanism for colloids or polyelectrolytes.

With the relation E = r', the normal component of the electric …eld reads, to leading order in =a, E ? (r) = Sr ? T (r)(1 e (a r)= ):

(A.18)

In the screened terms we have discarded factors of a=r, since they are close to unity in the range where the exponential function is …nite. This explicit how the thermocharge screens the normal electric …eld. The parallel …eld component, on the contrary, is hardly a¤ected by the thermocharge,

E k (r) = Sr k T (r)(1 + O( =a)): (A.19)
The thermocharge density follows from Gauss'law, T = "r 2 ' . With the same approximations as for the normal …eld component above, we have

A.4 Conducting particle

In the case of an conducting surface, the electrostatic boundary conditions impose that, at the particle surface, the parallel component of the electric …eld vanishes, whereas the normal component is compensated by the surface polarization charge where T S the temperature …eld at the surface and hT S i its mean value. The …rst term account for the isotropic part of the thermopotential whereas the last term account for the anisotropic thermopotential and for the polarization e¤ect. With the expression of the potential given below, one deduces the expression of the polarization charge P = " S(T S hT S i):

(A.26)
Before computing the component of the electric …eld, we approximate that, within the screening layer, the factor (a=r) n is close to unity, thus we discarded the corresponding factor in the following equation. With this approximation, the normal component of the electric …eld reads as, where the potential at the particle surface. Note that, in this last equation the Gouy-Chapman length l = e=2 l B j 0 j is given with the uniform surface charge. The last term in Eq. (C.4) result in

D.2 Conservation equation

We start by the charge conservation characterizing the creation and di¤usion of the ionic current inside and outside the screening layer. This equation is satis…ed by the electric charge density = e (n + n ), with n the ions densities, and the total ion current I = e (J + J ), @ @t + div I =0: (D.7)

The expression of the total current I, given in appendix A, is obtained within the Debye-Hückel approximation, and by applying Gauss's law, = " div E, his expression becomes

I = "D r 2 E 1 2 (E SrT ) : (D.8)
In the simple way, the conservation equation can be written only in function of the unknown electric …eld E. For that, we apply again the Gauss equation, so one has r (@ t E) = r D r 2 E 1 2 (E SrT ) : (D.9)

In this last equation, the left and right hand sides functions depend on the same variables, thus it can be simpli…ed as

@ t E = D r 2 E 1 2 (E SrT ) : (D.10)
where D = (D + + D ) =2 the sum of ion di¤usion coe¢ cient. This partial di¤erential equation modelizes the distribution of the generated electric …eld inside the system in the non-stationary state. The right hand side characterizes the term of reaction and the left hand side the term of di¤usion. When @ t E = 0, we …nd the same di¤erential equation as in section A.2 where the system is in steady state.

We want here to solve the Eq. (D.10) and …nd out the generated electric …eld and the ion di¤usion time scale which characterizes the time of creation of the Seebeck …eld. To have simple computations, the problem is reduced to solve the di¤erential equation satisfying by the reduced electrostatic potential according to E = S T r .

Inserting this last relation to Eq.(D.10), one has to solve the following equation,

@ @t = D r 2 1 2 ( T l ) : (D.11)
For further, we assume that the Debye screening length is much more smaller than the particle radius a. When we consider that the particle carries no surface charge, thus the normal component of the electric …eld vanishes at the surface. As a consequence, the resulting electrostatic boundary conditions reads as, n r = 0; for r = a;

(D.12)

where n the normal unit vector perpendicular to the particle surface.

D.3 Method of resolution

The di¤erential equation (D.11) with its boundary condition (D.12) given above depend in space and in time. To solve these kind of equations, one of the best way is to apply the Laplace transform where the time derivative is replaced by the function, s e , where e is the function Laplace transform of the electric potential and s an variable in the Laplace space.

With this transformation, the Eq. (D.11) becomes In order to …nd the coe¢ cients e c n in the Laplace space, as in previous sections we apply the boundary condition (D.12). To leading order in =a, we …nd To …nd the original function, we must compute the inverse Laplace transform of Eq. (D.20). Here, the inverse transform is very di¢ cult to calculate because of the presence of the square root. To overcome this di¢ culty, we approximate that s ion 1, and to leading order in =a we …nd …nally

Figure 3 . 1 :

 31 Figure 3.1: Schematic view of a thermoelectric self-propulsion mechanism of a heated Janus particle. a) The electric double-layer of a charged Janus particle immersed in a electrolyte solution. The ions are screened with valency of the order of z 10 5. The Debye length , characterizing the thickness of electric doublelayer (dashed line), is much more smaller than the particle radius a 1 m. b) Thermocharge accumulated by the electrolyte Seebeck e¤ect (S < 0); the resulting (positive and negative) thermocharge density is larger at the hot metal cap; for an excess temperature of a few kelvin one has a net charge of about Q T 100e[40]. c) Dipole component of the thermoelectric …eld E in the vicinity of a heated Janus particle. The parallel …eld E k vanishes at the metal cap (upper hemisphere), but is …nite at the insulating surface.

1 Figure 3 . 2 : 1 Figure 3 . 3 :

 132133 Figure 3.2: Variation of the parallel component E cond k of the electric …eld in function of the ratio of the normal distance z and the Debye length . For the plot we used the following parameters: The Seebeck coe¢ cient for NaOH solution S = 200 V/K, particle radius a = 1 m, the Debye length = 30nm, and the excess temperature T = 30K. The dashed lines indicate, the long-range thermoelectric …eld E 1 = S T =a 6kV/m.

Figure 3 . 4 :

 34 Figure 3.4: Electric double-layer including the thermocharge (temperature increases from left to right). At an insulating surface, the surface charge density 0 is not a¤ected by the Seebeck e¤ect, whereas the di¤use layer comprises the non-uniform thermocharge density T . At a conducting surface (right panel), the condition of a constant surface potential requires a polarization charge P ; the corresponding counterions add to the di¤use layer and to the non-uniform thermocharge.

Figure 3 . 5 :

 35 Figure 3.5: Electric …eld lines for insulating and conducting particles. a) Electric …eld due to an applied external voltage. The …eld does not penetrate in a low-permittivity particle, resulting in parallel component at the surface which is by a factor 3" w =(2" w + " P ) 3

Figure 3 . 6 :

 36 Figure 3.6: Parallel component E k of the thermoelectric …eld in the vicinity of a granular gold cap (S > 0). Each metal grain forms an equipotential surface and carries opposite polarization charges P at its cold and hot sides. Like on the continuous cap in Fig.3.1c, E k vanishes in a layer of about one screening length. On the insulating surface at the left, the …eld is given by (3.12).

Figure 4 . 1 :

 41 Figure 4.1: Schematic view of a charged polymer in an electrolyte solution. a)The drag on bead i consists of two contributions: First, the motion of bead j, driven by the force F, creates a ‡ow …eld v(r r j ) and thus drags its neighbor i. Second, due to the force f exerted by the bead j, the ‡uid element dV moves and in turn exerts a drag on bead i. These contributions cancel each other at distances well beyond the Debye length . b) At the scale of the Debye length, a polyelectrolyte is a rigid molecule; at short distances within , the spacing of two beads of radius a is well described by r ij = 2a ji jj.

  notation E x = Ei( x) for the exponential integral function which is de…ned as and d = d= for the ratio of the monomer length and the Debye length. For the Seebeck term we …nd S = 2a d E n d E d + e n d e

CHAPTER 4 . 2 (

 42 HYDRODYNAMIC INTERACTIONS IN DNA THERMOPHORESIS47(1+χ ϵ )z 

Figure 4 . 2 :

 42 Figure 4.2: Variation of the two contributions to the thermophoretic mobility D T , the dielectrophoretic coe¢ cient 1 + and the Seebeck e¤ect 1 + S , depending on the molecular weight. The parameters used for this plot: monomer's size a = 4:25Å, monomer's distance d = 3Å and Debye length = 5 nm. The factors, and S characterize the hydrodynamic interactions and increase with the chain length n (dashed line), b z 2 and b z result from the counterion condensation which reduce the mobility (Solid line).

Figure 4 . 3 :

 43 Figure 4.3: The variation of the hydrodynamic radius R h = R(n) [1 + (7) =42] where we took from the paper[55] and R[n](nm) is an adjustable parameter with n is the number of beads in the polymer chain. We took for this parameter: R(5) = 1:35; R(10) = 1:71; R(22) = 2:01; R(50) = 3:7; R(80) = 4:74.

) = 1 :

 1 Figure 4.3: The variation of the hydrodynamic radius R h = R(n) [1 + (7) =42] where we took from the paper[55] and R[n](nm) is an adjustable parameter with n is the number of beads in the polymer chain. We took for this parameter: R(5) = 1:35; R(10) = 1:71; R(22) = 2:01; R(50) = 3:7; R(80) = 4:74.

1 Figure 4 . 4 :

 144 Figure 4.4: Variation of experimental and analytical Soret coe¢ cient S T in function of the Debye length . The parameters used here for the two graphs are: the monomer's size a = 4:25Å and distance between monomer d = 3Å, the reduced Seebeck coe¢ cient b S = 0.

1 Figure 4 . 5 :Figure 4 . 6 :

 14546 Figure 4.5: Variation of experimental and analytical: a) thermophoretic mobility D T in function of the chain length n. The parameters used here for the two graphs are: the monomer's size a = 4:25Å and distance between monomer d = 3Å, the reduced Seebeck coe¢ cient b S = 0.

  Fig 4.6 we plot the complete thermophoretic mobility D T as a function of the molecular-weight n, for several values of the dimensionless Seebeck coe¢ cient b S. As the most striking feature, for negative b S the superposition of the two contributions in (4.16) may result in a change of sign of D T . From Fig.

4 m.s - 1 Figure 5 . 1 :

 4151 Figure 5.1: Time-dependent slip velocity v s as the sum of the thermoelectric slip velocity v el s and the thermo-osmotic slip velocity v osm s . The plots are done in the interval of time t = 0 to t = 10 s and with the …xed parameters: S = 200 V =K for NaOH solution, a = 1 m, = 50nm, the surface potential = 70mV.

1 0

 1 T =" e (a r)= Sr ? T j S : (A.20)Integrating over the radial coordinate we …nd the charge per unit area= Z dr T = "Sr ? T j S : (A.21)Integrating over the particle surface gives the total chargeQ T = 4 a"St 0 ; (A.22)which is determined by the isotropic component of the excess temperature.

  polarization charge, we write it as the series expansion in the Legendre polynomial basis, P = P n s n P n (c). Now taking the normal and the parallel derivative of the electrostatic potential ', the condition (A.23) enable to determine the coe¢ cients c n and s n , to leading order in =a,c n = St n ; s n = "St n (n > 0);and because of charge conservation, the isotropic terms read asc 0 = a St 0 ; s 0 = 0:Inserting these coe¢ cients, the electrostatic potential reads as, With the same approximation done as in the insulating case and the surface polarization condition R S S dS = 0, the expression of the electrostatic potential becomes '

E 1 0

 1 ? (r) = Sr ? T (r) S(T S hT S i) e a r ; (A.27) and the parallel component,E k (r) = Sr k T 1 e (a r)= : (A.28)Here we use the Bjerrum length `B, the surface charge density , and the salinity n. In the following we assume a negative surface charge . Taking the perpendicular derivative d'=dz, the normal electric …eld readsE ? = " e z= 1 g 2 1 ĝ2 : (B.3)This …eld is perpendicular to the particle, and it satisfy the relation E(0) = =". The charge density in the di¤use layer is readily obtained from Gauss' law = "dE=dz,= e z= (1 g 2 )(1 + ĝ2 ) 1 ĝ2 : (B.4)Integrating over z one …nds Zdz (z) = ; (B.5)which means that the counterions completely screen the surface charge density . The Debye-Hückel approximations is obtained by taking the limit of small surface charge, where `= 1 and g = 1 2 =`, resulting in ' = " e z= ; E ? = " e z= ; = e z= :

5 )

 5 Inserting this in Eq. (C.3), we obtain …nally the surface charge P in Poisson-Boltzmann theory as, With the permittivity gradient r" = (d"=dT )rT . The integral gives In the Debye-Hückel limit, the Gouy-Chapman length is large as compared to the Debye length, b = l= 1. Expanding in …rst order in b 1 , we …nd the surface polarization charge in Debye-Hückel approximation as,P = S (T S hT S i) : (C.7)The solution of the Eq. (D.1) in the liquid phase (r > a) is T l (r; ) = P n the Legendre polynomial and the coe¢ cients t n are given byt 2k = t 2k+1 = ( 1) k (2k + 1) ; (D.6)except for the …rst coe¢ cient which reads as, t 0 = 1=2 + 1= .

k n r p 1 + s ion = k n a p 1 +n r p 1 + s ion = k n a p 1

 1111 equation is solved by the method of direct solution. In this case, the electrostatic potential consists of two contributions, e = e h + e T ; (D.14) where the thermopotential (inhomogeneous part) reads as, e T = s 1 (1 + s ion ) T l ; di¤usion time scale or relaxation time for the creation of the thermoelectric …eld. The homogeneous part e h is well described by the Debye-Hückel equation, sections, it has been shown that the general solution of this last equation is the usual modi…ed spherical Bessel functions. Thus we obtain, s ion = P n (cos ) : (D.18)With the assumption a , the modi…ed Bessel functions k n can be simpli…ed and developed in term of =a in the following way, k terms of the series are well approximated by,

e

  c n (s) = (n + 1) a t n s (1 + s ion ) : (D.19) Thus the function Laplace transform of the reduced electrical potential completely reads as, e (r; s) = X t n P n (cos ) s (1 + s ion ) : (D.20)

(r; t) = ( 1 ) 1 e

 11 t= io n ; n (cos ) a n+1 r n+1 (n + 1) r e (a r)= ; (D.[START_REF] Dukhin | Surface and Colloid Science[END_REF] is the reduced electrostatic potential in the steady state.

Figure D. 1 :

 1 Figure D.1:The radial thermoelectric …eld E=E 0 and charge density = 0 , with E 0 = S T =a 30KV/m and 0 = S T = (a + ) 571:43 10 3 V/ m 2 , in function of the radial distance r=a at three di¤erent times scale. The relaxation time is about = 0:0025s = 2:5ms then we have taken the interval of time as t 2 [0:1 ; ] with the following parameters: the excess temperature T = 30K, the di¤usion coe¢ cient D = 1 m 2 /s, the Seebeck coe¢ cient S = 10 3 V/K, and the Debye length = 50nm. The dashed line and the continuous line in these graph represent, respectively, the numerical inversion for the Laplace transform based on the Talbot's method and the approximate analytical solution. This numerical computation has been done in order to validate the approximation solution resulting from the invers Laplace transform.

Table 4 .

 4 ). The values for n = 10 and 22 slightly increase with the Debye Debye length (nm) Soret coe¢ cient S T (10 2 =K) at T = 15 C 2 mer 5 mer 10 mer 22 mer 50 mer 80 mer 1: Experimental value of ssDNA Soret coe¢ cient in function of the Debye length at the corresponding chain length: 5 monomers, 10 monomers, 22 monomers, and 80 monomers.

	2:16	0:78	1:19	1:86		2:11	3
	3	1:03	1:49	2:12	2:86	3:10	3:27
	4:16	1:15	1:9	2:7		3:69	4:6
	5:2	1:29	1:89	2:89	3:8	5:19	4:8
	6:16	1:45	1:89	2:59	4:2	5:79	6:21
	7	1:5	1:89	3:2	4:41	5:8	6:31
	8	1:43	2:25	3:07	4:44	6:58	6:29
	9:74	1:48	2:09	3:4	4	7:1	6:09
	11:25	1:45	2:2	3:29		8:81	5:21

Remerciements

Je veux exprimer mes remerciements et ma gratitude à toutes les personnes qui m'ont apporté l'aide et l'assistance nécessaire à l'élaboration de cette thèse.

Avant tout d'abord, j'adresse mes remerciements les plus grands et les plus sincères à mon superviseur Prof. Alois Würger pour avoir accepté de m'encadrer pour ce travail ainsi qu'à ses orientations, sa disponibilité, ses précieux conseils et sa con…ance. Durant toute la thèse, j'ai beaucoup appris de lui, de la façon dont il aborde et résout les problèmes en physique de manière simple, de son sens du travail, et de son humilité. Avec toutes les di¢ cultés que j'ai pu rencontrer durant les trois années d'études, il a su m'apporter l'aide nécessaire avec de la patience jusqu'à la réalisation de ce travail.

Je tiens aussi à remercier tout le Laboratoire (LOMA) pour l'agréable séjour, l'administration, toute l'équipe physique théorique, mes collègues de travail: Goce, Mathieu et Marcela pour leur amitié et toutes autres personnes avec qui j'ai pu passer trois belles années au LOMA.

Un grand merci s'adresse aussi à ma famille pour leur patience et pour m'avoir toujours soutenu et encouragé tout au long de mes études. Je ne saurais terminer sans remercier Mme Isabelle Guillaume pour son aide dans toutes mes démarches administratives.

Merci à tous et que Dieu nous accorde sa bénédiction, nous donne la santé, longue vie, la paix et la réussite totale dans nos projet future.

Appendix A Seebeck e¤ect of an uncharged particle A.1 Equation for the ionic current

Here we give details for the expression of the ionic current resulting from the motion of the ion-species and the approximation done during the computations. We recall the expression of the current given by each ion-species,

If we de…ne the total ionic current I =e (J + J ), one can …nd his expression as follows,

2) It's now important to compute and to linearize the quantities: D + rn + D rn , D + n + + D n , D + n + D n , in order to simplify the expression (A.2). In addition, if we assume that the ions densities n vary weakly from the bulk, that is n + + n 2n 0 and =e n 0 , one …nds the following quantities as,

where D = D + + D _ =2 and D 0 = (D + D ) =2 with D the di¤usion coe¢cient for positive and negative ion. Inserting these above quantities, the expression of the total current is simpli…ed to,

(A.3) Rearranging terms, …nally the expression of the total current reads as,

where " the solvent permittivity, 2 = k B T =2n 0 e 2 is the Debye screening length and S = D 1 (D + + D ) k B =e is the Seebeck coe¢ cient.

A.2 Debye-Hückel theory

We start by the equation satisfying by the charge density and the electric …eld E which result from the expression of the total ionic current in the steady state. Thus putting I =0, one has

In order to …nd out the corresponding charge density and the electric …eld E, it's more easier to solve the di¤erential equation satisfying by the unknown electrostatic potential by introducing the relation E = r' with Gauss's law equation = " div E into the Eq. (A.5), one has

where T = T (r) T 0 with T (r) the temperature …eld given in term of a series multipolar expansion

where c = cos the cosine of the polar angle. The mean excess temperature t 0 = q=4 ka is determined by the rate of the heat absorption q, the thermal conductivity of the solvent k, and the particle radius a.

The di¤erential equation (A.5) has two solutions: the homogeneous and inhomogeneous part, thus the electrostatic potential has two contributions,

The stationary heat equation for the temperature …eld, r 2 T = 0, implies that the divergence of the thermoelectric …eld vanishes, r E =0. The corresponding thermopotential ' T , …rst solution of equation (A.6), reads as

whereas the second solution is given by the Debye-Hückel potential ' , and results from the equation

To solve this equation, we decompose the solution

in the Legendre polynomial basis (P n ) n 0 . Inserting into Eq. (A.6), one has to solve the following equation

These kind of equations are solved by Bessel functions, and for spherical particle the general solution is given in term of series expansion

where k n = q 2

x K n+1=2 (x) the modi…ed spherical Bessel function of the second kind. For the sake of notational convenience, we introduce the factor k n (a= ) such that the radial solutions are normalized at the particle surface r = a. The complete electrostatic potential reads as,

where c = cos . Now it remains to de…ne the coe¢ cients c n which depend on the electrostatic boundary conditions used at the particle surface.

A.3 Insulating particle

In the case of an insulating uncharged surface, the particle carries no surface charge then the electrostatic boundary conditions require that the normal component of the electric …eld vanishes, E ? (r = a) = 0:

Taking the radial derivative of the potential at the point r = a, the coe¢ cients c n read as,

; with the dimensionless derivative k 0 n (x) = @ x k n (x). In order to simplify the expression of the coe¢ cients c n , we assume that the particle radius a is much greater than one Debye length, a. With this assumption, we expand in term of series =a the corresponding function

Thus the …rst terms of the series are well approximated by

a r e (a r)= (n < p a= ):

In the most relevant near-…eld range, this approximation is even valid for n < a= .

To leading order in the small parameter =a, we have k 0 n (a= ) =k n (a= ) = 1 + O( =a). Then the above coe¢ cients read as,

The electrostatic potential, …nal solution of the di¤erential equation (A.6), reads as

The screened term is by a factor =a smaller than the …rst one; yet their radial derivatives cancel each other at r = a, thus satisfying (A.15).

69

The component E ? and E k satisfy the boundary conditions in (A.23). Applying Gauss's law, = "r 2 ', we …nd the thermocharge density within the same approximation, P = " 2 e (a r)= S(T S hT S i):

(A.29)

Appendix B Poisson-Boltzmann theory

Previously, we derived the thermoelectric properties of an uncharged hot colloids in a weak-coupling approximation. Now in the case where the particle carries a surface charge and in contact with an electrolyte solution, the electrostatic potential ' is solved by the Poisson-Boltzmann equation

This last equation is non-linear and doesn't have an analytical solution in 3dimensional system. Now if we assume that the particle radius is much larger than the Debye screening length, then the curvature of the surface can be neglected, and the non-linear Poisson-Boltzmann equation can be solved in planar geometry where the Laplace operator reduces to the second derivative with respect to the vertical coordinate z, and the potential is the 1D solution [START_REF] Andelman | Soft Condensed Matter Physics in Molecular and cell[END_REF] '

with the shorthand notation

The parameter g is given by the ratio of the Gouy-Chapman length `and the Debye length ,

Appendix C

Polarization charge

Here we are in the case of a charged conducting surface with an electrolyte solution.

The mobile electrons in the metal move until the resulting polarization charge P the double layer potential which is treated within the Poisson-Boltzmann theory.

In order to determine P , we begin by assuming that the polarization charge is much more smaller than the uniform surface charge 0 , then we expand the Poisson-Boltzmann potential in linear order

where = P + 0 the unknown surface charge.

Taking the parallel gradient of the above equation, one obtains

where b = l= the ratio of the Gouy-Chapman length l and the Debye length . The electrostatic boundary conditions impose that the gradient vanishes at the surface r k ' (z = 0) = 0, thus solving for P we obtain

In this last equation, one has to compute the parallel gradient of the Poisson-Boltzmann potential with the uniform surface charge. Thus we have at z = 0,

Appendix D

Time dependent Seebeck e¤ect

Here we give the details of calculation of the thermoelectric properties in a nonstationary state for a half-metal heated Janus colloid. We consider here an uncharged particle and we assume that the Debye screening length is much more smaller than the particle radius a. We start by giving the behavior of the temperature pro…le around the heated particle. The result of the following section is taken from the paper in Ref. [56].

D.1 Temperature …eld

The Janus particle of radius a is centered at r =0, its lower hemisphere is coated by a thin metal layer with the conductivity k C which is considered to be much higher than both of the particle k p and the surrounding ‡uid k (taken to be equal for simplicity k k p ). We therefore assume that the cap is held at constant temperature T 0 + T , with T 0 the bulk temperature. Important to note that the temperature pro…le remain always in the stationary state because of half metal absorption.

Let T l = (T (r) T 0 ) = T be the reduced temperature …eld and solution of the stationary heat equation, r 2 T l = 0; (D.1)

with the mixed boundary conditions @ r T l j a + = @ r T l j a ; 0 =2; (D.2) T l (a) = 1; =2 : (metal side) (D.3)