Thermophoretically induced ﬂ ow ﬁ eld around a colloidal particle

A colloidal particle suspended in a ﬂ uid solvent with a non-homogeneous temperature undergoes a thermophoretic force. This force may translate into a directed drift of the particle and a source-dipole-like ﬂ ow ﬁ eld around it. Alternatively, if the colloid is ﬁ xed in space, the accompanying ﬂ ow is long-ranged. In this work, we provide a ﬁ rst simulation study of the thermophoretic force-induced ﬂ ow ﬁ elds by a particle-based mesoscopic method. The simulation results are quantitatively consistent with theoretical predictions obtained by solving hydrodynamic equations. Based on these results, we propose a single-particle micro ﬂ uidic pump without movable parts, in which the ﬂ ow direction can be reversed. Furthermore, we quantify the long-range hydrodynamic attraction between two suspended particles near the boundary wall induced by the thermophoretic ﬂ ow ﬁ eld.


I Introduction
In the presence of a temperature gradient, a colloidal particle experiences a directional motion.2][3] Practical applications of thermophoresis are numerous, for example, in separation of macromolecules in solution 4,5 or biological processes. 6,7Another interesting and less known effect of colloidal thermophoresis is the induced uid ow.The thermophoretic force exerted on the colloid is not an external driving force but results from the interactions of the colloid with the solvent which is inhomogeneous due to the temperature gradient.The reaction force of the thermophoretic force induces in turn a motion of the surrounding uid.Currently, small scale hydrodynamic ows are receiving rapidly increasing attention, especially in microuidic and biophysical applications.The thermophoretically induced ow is therefore a promising alternative mechanism to originate small scale hydrodynamic ows.
Although the thermo-osmotic ow in porous media was investigated a long time ago, 8,9 its existence in colloidal suspensions has only been experimentally proved very recently. 10,11The experiments showed that in temperature gradients colloidal spheres can form a two-dimensional crystal on a boundary wall, which is similar to the electric eld-induced colloidal crystal on electrode surfaces. 12,13The formation of thermophoretic crystals clearly indicates the existence of the thermophoretic ow eld around a xed colloidal particle in non-isothermal suspensions.However, a precise observation of this ow by means of experiments or computer simulations is still lacking.5][16] Furthermore, this study will provide insight into how to design high-performance temperature gradient driven nanomachines. 17,18ere, we employ computer simulations to investigate the thermophoretic ow eld induced by a colloidal sphere immersed in a non-isothermal solvent.Both the colloidal spheres and solvent are modeled at the particle level by a hybrid mesoscopic-molecular dynamics scheme. 19,20The induced ow eld around a colloidal sphere is analyzed in the case that the colloid is externally xed and in the case that the colloid is driing freely, showing a fundamentally different behavior.The obtained results are quantitatively consistent with the theoretical calculations.Further, the effect of boundary walls is studied by xing the particle near one wall, in which a signicant lateral ow is observed.This provides a direct simulation evidence for the appearance of a long-ranged inter-colloidal hydrodynamic attraction due to the thermophoretic ow eld.Finally, we suggest a single-particle microuidic pump based on the thermophoretic effect, in which no movable parts are required and in which the ow direction can be reversed.The feasibility of this pump is demonstrated by simulations.

II Simulation method
The typical sizes of a colloidal particle and the surrounding solvent particles are separated by two to four orders of magnitude, which translates into even larger differences in the typical time scales of both components.This intrinsic difficulty has motivated the development of various mesoscopic simulation methods.Here, we employ a hybrid scheme that describes the solvent by a particle-based simulation technique known as multi-particle collision dynamics (MPC), [19][20][21][22] while the interactions of the colloidal particle with the solvent are simulated by standard molecular dynamics (MD).This hybrid scheme is especially appropriate for our purposes due to three main reasons.Firstly, hydrodynamic interactions and thermal uctuations are correctly captured on a large length scale. 23,24econdly, the precise local conservation of energy enables the sustainability of temperature inhomogeneities and heat transport. 25Thirdly, the particle-solvent interactions can be naturally included and tuned, which will have a relevant inuence on the simulated colloidal thermophoretic properties. 26he MPC method consists of alternating streaming and collision steps.In the streaming step, the solvent particles of mass m move ballistically for a certain time h.In the collision step, particles are sorted into the cells of a cubic lattice of size a, and their velocities relative to the center-of-mass velocity of each cell are rotated around a random axis by an angle a, ensuring mass, momentum and energy local conservation.The solvent transport properties are determined by the MPC parameters, 27,28 for which we employ standard values a ¼ 130 , h ¼ 0.1, and the mean number of solvent particles per cell r ¼ 10.Other parameters determine the simulation reference units for which we take where k B is the Boltzmann constant and T is the average system temperature.Note that by construction, the solvent dynamics in MPC is coarse grained by the collisions within each collision cell.Hydrodynamic interactions can then be reproduced for lengths larger than a. 35 In order to impose the presence of a constant gradient of temperature, methods developed for non-equilibrium molecular dynamics can be employed with the MPC solvent as extensively discussed in ref. 25.In the present work we employ two different system congurations.In the rst conguration the uid is conned between parallel walls, with periodic boundary conditions (PBCs) in the other two directions.The walls are implemented with the bounce back rule, which approximately results in stick boundary conditions. 29,30In this case a temperature difference on a thin layer close to each wall is imposed by the velocity exchange algorithm, 25,31,32 which consists of interchanging the velocity of the warmest particle in the cold layer with the coldest particle of the hot layer.The second conguration considers PBCs in the three directions.The way of obtaining the required periodicity in the temperature gradient direction consists of dividing the box into two halves with a cold layer imposed in one extreme of the box and a warm layer in the middle.In this case we obtain a temperature gradient by thermalizing at different temperatures the warm and the cold layers. 25In both congurations linear temperature proles are nicely obtained.It should be emphasized that, although MPC has shown liquid-like dynamical properties, 33 the equation of state corresponds to the one of an ideal gas, due to the lack of potential inter-particle interactions.The temperature gradient translates therefore into a non-constant distribution of the solvent density, which could eventually yield to some compressibility or thermal expansion effects.The inuence of compressibility in our simulations will be carefully discussed later.
Solvent and colloidal particles have a potential interaction.Although different potentials can in principle be selected, in this work we employ a truncated and shied Lennard-Jones potential, 34 with r the distance between the colloidal center and the solvent particle.The potential intensity is 3 and the interaction length parameter s.Here we have xed n ¼ 3, 3 ¼ k B T and s ¼ 3a.High values of s/a reproduce more accurately the ow eld, and also increase signicantly the computational cost.Padding and Louis 36 showed that for s/a ¼ 2 the ow eld around a sedimenting colloid can be simulated with a small relative error.The chosen s/a ¼ 3 in the simulations presented here is therefore a good compromise to obtain an accurate description of the ow eld.A central potential like that in eqn ( 1) is known to result in slip boundary conditions.Between two MPC collision steps, N m MD steps are implemented for the solvent particles that are in the interaction range of the colloid.The equations of motion are integrated by the velocity-Verlet algorithm with a time step Dt ¼ h/N m , where we use N m ¼ 50.

III Thermophoretic force and flow field
In the presence of a temperature gradient, the inhomogeneities of the solvent interactions result in a net thermophoretic force f T on the colloid, which is, in general, directly proportional to the temperature gradient.The direction and intensity of this force completely depend on the nature of the solvent colloid interactions.In simulations, different potentials have shown to translate in important differences in the thermophoretic colloid behavior.The thermophoretic force can be not only weaker or stronger, but also change from thermophobic to thermophilic, this is changing the direction of the dri motion from cold to warm regions. 26The so repulsive potential in eqn (1) induces a relatively strong thermophilic force.The coefficient that characterizes the most relevant system thermophoretic properties is the so-called Soret coefficient S T , or its dimensionless equivalent a T ¼ TS T , the thermal diffusion factor. 2,3This factor indicates the separation ratio between components in a binary mixture, and although in the most general case it depends on the thermophoretic force exerted in both components, 37 for the case of large particles the following approximation is accepted, with VT the temperature gradient.
According to Newton's third law the surrounding solvent experiences a reaction force Àf T , which translates into a uid ow.The analytical expression of the velocity eld around a spherical particle in the low Reynolds number regime is standardly calculated by solving the Stokes equation. 38We implicitly assume that the boundary layer approximation is valid (shortrange particle-solvent interactions), and the case of an incompressible liquid.We rst consider the case of a xed particle in the presence of a temperature gradient, which implies the presence of a thermophoretic force, and the following boundary conditions: (i) vanishing velocity eld at innity, (ii) vanishing normal component of the ow eld at the particle surface, and (iii) the integral of stress tensor over the particle surface corresponding to f T .The resulting stationary ow eld 11,39 is, Here, R is the particle radius, h the solvent dynamic viscosity, r ¼ r/|r| and I the unit tensor.The ow is the superposition of a Stokelet and a source-dipole.Eqn (3) indicates that the ow velocity around a xed particle in a temperature gradient has an opposite direction to the thermophoretic force, and that it is of long range since it decays linearly with the inverse distance from the particle center.
A different conguration is when the particle is not xed, but freely moving.In this case, the thermophoretic force results in an averaged dri velocity, the thermophoretic velocity u T .These quantities can be directly related by f T ¼ gu T , with g the friction coefficient.Note that for this analytical calculation the friction is approximated to be spatially constant. 40As a consequence of the dried motion, not only the thermophoretic force but also a balancing viscous drag is exerted on the particle.The boundary conditions to solve the Stokes equation are now: (i) vanishing velocity eld at innity, similar as before, (ii) the normal component of the ow eld is vanishing only in the particle reference frame, and (iii) the balancing forces on the particle result in a vanishing integral of stress tensor over the particle surface.The obtained velocity ow eld 38,41 reads, The ow velocity across the colloidal center and along the temperature gradient has now the same direction as the thermophoretic force, and decays with the inverse of the distance cubed; this is much faster than in the case of a xed particle.Note that the friction coefficient is related to the particle radius with g ¼ AhR, where the numerical factor is A ¼ 6p for stick boundary and A ¼ 4p for slip boundary conditions.This also distinguishes the ows of the xed and moving colloids, since in the case of a xed colloid the relation between the ow eld and the thermophoretic force does not vary if the colloid has stick or slip boundary conditions.
To conclude this section, it is instructive to compare the two previous solutions of the Stokes equations, with the solution of a sedimenting colloidal particle.The boundary conditions (i) are similar in the three systems, while the conditions (ii) are similar to the driing thermophoretic particle.For the sedimenting particle, the gravitational force g is directly applied on the particle, and not the result of the interactions with the surrounding solvent.The integral of the stress tensor over the particle surface corresponds then to Àg, similar to conditions (iii) of the xed particle.The result is the Rotne-Prager-Yamakawa (RPY) tensor, 42,43 which can also be understood as an extension of the Oseen tensor.The RPY tensor has a similar structure to the one in eqn (3), differing only in the numerical prefactor of both contributions.The main difference is though that the signs are opposite in both cases.That is, the ow induced by a sedimenting particle has the same direction as the gravitational force, while the ow induced by a temperature gradient of a xed particle has the opposite sign to the corresponding thermophoretic force.

IV Simulation results
A Flow eld of a colloid xed between parallel walls We rst study the ow eld induced by a colloid xed equidistant to two parallel walls which are thermalized at different temperatures (Fig. 1).In simulations the colloid is xed just by freezing its motion.In experiments this can correspond to the existence of a balancing external force, like in the case of laser tweezers.The considered size of the simulation box in the temperature gradient direction is L z ¼ 50a, and L t ¼ 40a in the two perpendicular ones.Aer reaching the stationary state, both the thermophoretic force and the velocity eld can be computed.Within this conguration, f T can be measured by directly summing the colloid-solvent interactions. 26In these simulations VT x 0.0125 T/a, and the thermal diffusion factor in eqn ( 2) is estimated to be a T x À200.By convention, the negative a T is related to a resulting force towards the warm area.The magnitude of a T in our simulation model is about 50-fold smaller than that of a 1m colloidal sphere in previous experiments regarding the thermophoretic ow. 10,11The ow eld around the colloidal particle is obtained by time-averaging the uid particle velocities in small cubic bins that we choose to be of the same size as the collision cells.The thermophoretic ow eld is subjected to thermal uctuations and it is proportional to a T , such that with our parameters a long running time is necessary to obtain a clear ow pattern.For the example in Fig. 1, a total time of t ¼ 10 5 in simulation units, and an average over 50 independent runs has been employed.[46] Fig. 1 Cross-section of the flow field induced by a thermophilic colloid fixed between parallel cold and hot walls.Small red arrows indicate the flow velocity direction and intensity, while the thick blue lines correspond to the flow stream lines.The axes where the flow velocities are quantified in Fig. 2 are displayed here in white.
Nevertheless when using this method, other technical problems, like the potential colloid implementation, would then need to be taken into account, apart from evaluating the effect of neglecting uctuations.
The stream lines in the colloid neighborhood in Fig. 1 show that the ow goes from hot to cold, consistent with the thermophilic behavior of the implemented colloid and eqn (3).In a system bounded by walls, the steady-state net ow through any section is zero.This means that the thermophoretically induced ow eld must be balanced by a wall-induced backow (ow in the opposite direction to the main ow), which originates the vortex ring in Fig. 1.In the theoretical calculation in eqn (3), an innite system is considered, such that the backow is negligible.This contrasts to the nite-size simulation boxes presented here, where the backow effect and the consequent vortex structure are signicant.
The quantitative values of the simulated velocity elds are displayed in Fig. 2 where we also compare with the analytical predictions.The component along the temperature gradient of the ow velocity is displayed in two axes that cross the colloidal center, showing in both cases a decrease of the magnitude with distance from the particle consistent with eqn (3).One axis is perpendicular to the walls and along the temperature gradient, and one parallel to the walls.These axes are indicated as a and b in Fig. 1.The open and closed symbols in Fig. 2b denote the velocity values computed in both directions of axis-b, which we could call 'up' and 'down'.These two directions have no intrinsic difference such that the differences between these symbols just give an indication of the statistical error.Similarly, the open and closed symbols in Fig. 2a refer to velocity values computed in the cold and warm directions of axis-a, where compressibility effects could be of importance, given the ideal gas equation of state of the MPC solvent.Nevertheless, it can be observed that the differences are only slightly signicant in the close neighborhood of the colloid, where the velocity in the cold side of the colloid is about 15% larger than in the warm side.Therefore, compressibility effects will not be of great importance when applying the MPC solvent to study the hydrodynamic interactions in non-isothermal suspensions with small temperature gradients.
In order to perform a quantitative comparison with the theoretical prediction in eqn (3), the shear viscosity is calculated from MPC kinetic theory 28 as hx8:7 ffiffiffiffiffiffiffiffiffiffiffiffi mk B T p =a 2 , and its temperature dependence is disregarded.The so potential in eqn (1) employed in our simulations does not clearly determine a particle radius.We employ the standard choice of considering it equal to the interaction length parameter, this is R ¼ s.The solution of the Stokes equation in eqn (3) (dashed lines in Fig. 2) displays similar functional dependence as the simulation data except from an upward shi.The shi arises from the wallinduced backow not included in eqn (3) which effectively considers the presence of walls placed at an innite distance.A down constant displacement of eqn (3), enforcing zero velocity at the walls, corresponds to the approximation of uniform backow.This simple approximation (dotted lines in Fig. 2) matches very well the simulation data.8][49] The fact that the system is bound by two walls instead of by one indicates that multiple reections are necessary to obtain a satisfactory convergent result (see calculation details in the Appendix).Interestingly, the analytical corrections with the reection method of eqn (3) (solid lines in Fig. 2) almost perfectly coincide with the uniform backow approximation in very good agreement with the simulation data.If the backow is understood as an additional constant ow in the direction of the thermophoretic force, it is to be expected that an additional friction force is exerted on the colloid which enhances the magnitude of the computed f T .The intensity of the backow decreases linearly with system size and is exerted in the f T direction.This nicely explains the strong nite-size effects observed in our recent work, 26 which have then a signicant different nature than other nite-size effects.As a standard example, the diffusion of the center-of-mass diffusion of a polymer in equilibrium increases with box size, as a consequence of the spectra truncation. 23,50,51Meanwhile, the thermal diffusion factor a T in conned systems decreases with system size, as a consequence of the decreasing backow. 26

B Flow eld by two colloidal particles in a periodic gradient
In order to minimize the effects of connement in simulations the usual procedure is to employ PBCs.As introduced in Section II, in the presence of a temperature gradient PBCs are obtained by dividing the simulation box in two halves with temperature gradients of opposite signs. 25,52,53Given the system symmetry it is standard to average the properties in both system halves by only regarding about the sign difference.Here we study the ow velocity induced by two colloids xed in the center of the two simulation half boxes (Fig. 3).The size of the simulation box in the temperature gradient direction is 2L z with L z ¼ 50a, and L t ¼ 40a in the two perpendicular ones.This geometry is similar to the existing simulation studies of colloids in a temperature gradient. 26,54The repulsive solvent colloid interactions in eqn (1) result in a colloid with thermophilic properties and in a solvent that goes to the cold regions in the axis crossing the colloidal center along the temperature gradient.This means that solvent ows with opposite directions converge in the cold and hot layers, what enforces a vanishing ow velocity at the boundary layers in the direction of the temperature gradient.The resulting ow prole for each xed colloidal particle is therefore very similar to the ow for a xed particle between walls, as can be seen in Fig. 3, with a noticeable presence of backow.It should be though noted that it is more accurate to state that the ow is equivalent to the one originated by walls with slip boundary conditions since the velocity in the directions perpendicular to the temperature gradient does not necessarily vanish.Therefore, the use of PBCs with the presence of periodic temperature gradient does not help to minimize the effects of connement.

C Flow eld by one driing thermophoretic particle
It is a considerably more difficult task to directly obtain the ow eld around a driing thermophoretic particle than the case of a xed particle due mainly to two facts.Since PBCs are accompanied by a periodic gradient, it should be completely ensured that the colloidal particle does not reach the system boundaries.On the other hand, the motion of the particle will explore areas with different temperatures which correspond to areas with varying density, viscosity and eventually thermophoretic properties.This makes the precise comparison with the analytical approaches that consider properties without temperature dependence difficult.In order to circumvent these difficulties, we investigate the behaviour of a different system and we show how to precisely map it to the freely moving particle.
The system we investigate in the rst place consists of a periodic gradient with PBCs in three dimensions.The box size is 2L z with L z ¼ 40a in the direction along the temperature gradient, and L t ¼ 36a in the two perpendicular directions.The employed temperature gradient is VT ¼ 0.0082 T/a.Here one of the half boxes includes the presence of a xed colloid in the center, while there is no colloid in the other half box with opposite temperature gradient, as depicted in Fig. 4. When the temperature gradient is switched on, the originally quiescent uid is initially accelerated by the thermophoretic force.In contrast with the examples previously discussed, the boundary conditions do not enforce a vanishing velocity at the cold and hot layer boundaries.This, together with the mass conservation conditions, results in a continuous net ow across the whole system, as can be clearly observed in Fig. 4. At the same time, the net ux of the uid on the colloidal surface exerts a friction force in the ow direction, which gradually weakens the uid acceleration until a constant velocity is reached.In the stationary state, the thermophoretic force f T is exactly balanced by the friction force f g , such that the integral of the stress tensor over the particle surface vanishes.
In systems where the Galilean invariance is fullled, the problem of a colloid moving with velocity U in a quiescent solvent is exactly equivalent to that of a ow past xed colloid, with velocity À U at an innitely distant position.The velocity distribution of the solvent around the moving colloid can be obtained from the solvent velocity around the xed colloid by simply subtracting the velocity; the uid is then at rest at innity.MPC satises Galilean invariance, 55 and in our situation, the equivalence can be understood by further verifying the validity of the boundary conditions to solve the Stokes equations in the case of a freely moving particle, as specied to obtain eqn (4).It can be checked that this is the case, since, for example, in both systems a vanishing force is exerted on the colloidal surface.The ow velocity around one xed colloid in a periodic gradient v(r) can therefore be understood as the superposition of two independent ow elds, v(r) ¼ u(r) + v(r).
Here, v(r) corresponds to the ow eld around a moving particle, and u(r) to the velocity of the uid innitely separated from the colloid with a constant net ux.In an incompressible uid, a constant net ux implies a constant velocity eld, but since MPC has an ideal equation of state, the solvent density depends together with the temperature on the spatial coordinate along the temperature gradient, z, such that the superimposed ow velocity eld u(r) ¼ u(z) with u(z)r(z) ¼ J.This implies that the velocity u(z) will have the same functional dependence as the temperature, namely it grows linearly from the cold to the warm side.The contribution u(z) can be obtained directly in the simulations by computing the average velocity in an axis along the temperature gradient that is as distant as  possible to the colloidal particle, where v(r) is approximately vanishing.The values obtained in such axis result indeed in a linear velocity prole and in a constant ux J except from uctuations smaller than 1%.
Consequently, the ow eld of a moving particle can be determined simply by subtracting the two velocities directly computed in the simulations as v(r) ¼ v(r) À u(z).Fig. 5 shows the ow eld of a freely driing thermophilic colloidal sphere mapped from the ow eld of one xed particle in a periodic gradient.The ow eld has the direction of the colloidal thermophoretic force along the temperature gradient in the axis that crosses the colloidal center, as predicted by eqn (4), and there is no backow. 41Precisely, the absence of backow facilitates the analytical calculation of the ow eld in this case, which is shown in the inset of Fig. 5 and as can be seen it agrees very well with the simulated one.
In order to quantify the colloidal thermophoretic velocity u T , the mapping procedure has to be inversely considered, and it can be obtained from the subtracted velocity of the uid.As explained the subtracted velocity position dependent, and in order to quantify u T , the velocity should be considered at the colloidal position, u T ¼ Àuðz col Þx0:0055 . With this velocity, a quantitative comparison with the velocity eld predicted from the Stokes equation in eqn ( 4) can be performed.The comparison is displayed in Fig. 6, and it is carried out in two axes that cross the colloidal center.One axis is in the direction of the temperature gradient and one perpendicular to it (similar to Fig. 1).The analytical prediction in eqn (4) does not contain any adjustable parameter, and given the fast decay of the velocity, no further corrections need to be performed due to the nite size of the simulation box.The results in Fig. 6 show a very good agreement between simulations and the solution of the Stokes equation in both analyzed axes.Moreover, in Fig. 6a the simulation results v a,c and v a,h have no larger differences than those produced due to statistical errors, which further conrms that the effect of the compressibility can be neglected in our non-isothermal simulations.
In the case of one colloidal particle xed between parallel walls, or equivalently of two colloids in a periodic temperature gradient, the thermophoretic force could be obtained by directly measuring the colloid solvent interactions.This is not the case now since the direct interactions provide a net vanishing force on the colloidal particle.The thermophoretic force can be alternatively estimated by its relation with the thermophoretic velocity f T ¼ gu T , where the friction coefficient g needs to be determined.In principle, this coefficient can be determined from the colloid self-diffusion coefficient, or approximated as g ¼ 4phR, which considers slip boundary conditions.This relationship together with eqn (2) allows us to calculate the thermal diffusion factor as a T x À220, which is 10% higher than the one computed in Section IV-A.The lack of backow in this setup would make us to expect an effective smaller value of a T , such that other factors should contribute to explain this deviation.Although we do not have a quantitative estimation of these factors, it can be expected that besides the intrinsic statistical error of the simulations, the overestimation of the hydrodynamic radius of the colloidal sphere 36 might have a noticeable inuence.
Another important consideration is the uid temperature.When the moving colloid is considered in a quiescent uid (Fig. 5), the uid is completely thermalized.Meanwhile, when one xed colloid is considered in a moving uid (Fig. 4), the uid moves over a region with different temperatures, such that it is in principle partially thermalized.These partial thermalizations can be neglected if the heat conduction is much faster than the uid motion.Otherwise the real uid temperature could be different than the one assumed by the existing  temperature gradient, such that a correction factor would be required to map these two systems as exactly equivalent.The characteristic times of the heat conduction and the uid translation can be expressed as s c $ s 2 /c with c the thermal diffusivity of the solvent, and s u $ s/u with u the typical velocity of the moving uid, respectively.Using the estimated c from kinetic theory 28 and u T obtained in the simulations, we have s c /s u $ 10 À1 .Although the time scale separation is only of one order of magnitude, the trend indicates that the assumption used in our mapping strategy is justied.On the other hand, this could also be the origin of the deviation of the estimated value of a T .

V Applications
The existence of a thermophoretically induced uid ow has interest not only from the fundamental point of view, but can also nd numerous practical applications.In the following we present and discuss two of these applications.The rst one is the existence of inter-colloidal hydrodynamic attraction induced by the thermophoretic ow, which has already been shown to be able to form thermophoretic crystals.And the second one is the possibility of designing a single particle thermophoretic pump.

A Thermophoretically induced colloidal attraction
In the cases where xed particles have so far been investigated, the colloids have been considered equidistant from the walls or the boundary layers, which intends to better reproduce the properties in bulk.However, when the colloidal particles are not considered to be xed, they will naturally dri towards one of the walls as a consequence of their directional thermophoretic force.The colloids may then stay at an averaged xed distance of the conning wall performing a two-dimensional Brownian motion. 10,11It is then to be expected that the thermophoretically induced ow eld will signicantly vary from the symmetric case in Fig. 1.
In order to study the wall effect in the thermophoretic ow eld, we perform simulations of a thermophilic colloidal particle conned between walls at different temperatures, where the colloidal position is xed at a distance h w ¼ 5.5a from the hot wall.Note that the particle wall separation h w is large enough to consider more than one MPC collision box between the colloidal surface and the wall, such that the hydrodynamic behavior can be properly resolved.The distance between the walls is L z ¼ 30, while PBCs are employed in the other two directions with box size L t ¼ 40a.The steady ow eld is depicted in Fig. 7 where the signicant asymmetry of the streamlines is easily observed.This ow pattern is the same as the analytical prediction. 11,39n contrast to the symmetric system in Fig. 1, the thermophoretically induced ow eld has now a strong lateral component toward the colloidal sphere and parallel to the wall, which necessarily affects the motion of ambient particles.If a second particle is in the neighbourhood of the colloid, it will suffer hydrodynamic drag toward the rst particle.When the lateral ow is strong enough (high |a T |), the attraction force can be larger than other repulsive contributions or than the thermal uctuation, originating stable colloidal aggregation.Such 2D colloidal crystals induced by the presence of thermophoretic ow elds have indeed been experimentally observed. 10,11xperiments are performed with thermophobic colloids such that the colloidal accumulation takes place in the cold wall in contrast to our case.Note that if the colloid would be xed by an external force to the opposite wall of their thermophoretic affinity (e.g.thermophobic colloid xed at the hot wall) the contribution of the thermophoretic ow would induce a repulsive interaction.
By computing the typical lateral ow velocity as u l x0:0005 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi k B T=m p , the rst quantitative estimation of the hydrodynamic force can be performed as f H x 4phsu l x 0.15k B T/a.In order to evaluate more precisely the hydrodynamic attraction induced by the lateral ow, we perform simulations of two thermophilic colloidal particles xed at a distance h w ¼ 5.5a from the hot wall.The distance between the walls is now L z ¼ 30, and the box size in the other two perpendicular directions is L t ¼ 50a.The attraction force can be directly obtained in our simulations by evaluating the solvent-colloid interactions that now will have a non-vanishing component not only in the direction of the temperature gradient but also perpendicular to it, in the direction of the second colloid.Independent simulations with colloids placed at different positions allow us to obtain the attraction force as a function of inter-particle separation as shown in Fig. 8. Similar to the particle wall separation, the smallest separation between colloids is large enough to properly capture hydrodynamic interactions.The computed magnitude of f H in Fig. 8 is consistent with our rst rough evaluation.The attraction force decreases monotonously with the separation, which qualitatively agrees with the experimental results. 10,11From the force curve, an effective potential between particles can be obtained with a potential well depth of $1.5k B T. Fig. 7 Cross-section of the flow field around a thermophilic particle fixed in a near-wall environment.The inset shows the velocity field in a plane parallel to the walls and between the colloid and the hot wall.
Thermal uctuations will then contribute to eventually overcome the attractive potential between colloids with non-xed positions, such that no stable aggregation occurs in our system.Stable congurations can be achieved by, for example, increasing the size of the colloid, 10 which increases the thermal diffusion factor a T of the colloids.
Analytically, the attraction induced by the thermophoretic ow eld has been calculated using the reection method.To the rst reection, the attraction force f H is expressed as 11,39 f with s the inter-colloidal separation and the constant prefactor f c H ¼ 6lh w Rf T calculated with the slip boundary conditions.Here the absolute value of the thermophoretic force is considered, The factor l ¼ l(h w ) is a dimensionless correction for the friction due to the presence of the wall, 56,57 which can be analytically obtained for our case as l x 1.43.Thus, f H can be calculated without any adjustable parameter, in contrast to the experimental measurements. 11Fig. 8 shows the simulation results together with the analytical prediction showing a very nice agreement for large separations.At shorter distances, the analytical prediction appears to signicantly underestimate the values obtained by simulations of the attraction force.Deviations might be due to different effects.The analytical approximation in eqn ( 5) is obtained with only one reection which essentially neglects the effect induced by the second wall.This second wall is anyhow relatively distant from the colloid, such that we do not expect a large contribution from this effect.Other effects disregarded in the analytical expression are the distortion of the ow eld due to the presence of the second colloid, and the effect of the periodic images.The contribution of the periodic images would in principle decrease the magnitude of the attraction and will be more important at larger separations.The larger distances considered are still considerably smaller than the size of the employed system size such that we expect this contribution to be negligible.The distortion of the ow eld due to the presence of the second colloid is expected to increase for smaller separations.This could explain the enhancement of the simulation results with respect to the analytical theory at short distances, an effect that can then be expected also in experiments.The larger enhancement found at the two shortest measured distances should though be carefully considered, since at these distances is where some compressibility effects were found to be more relevant.Other effects can also affect the results, like the temperature dependence of the thermal diffusion factor. 58Single-particle microuidic pump In Section IV-C we have investigated the ow eld around one particle xed in a periodic temperature gradient.As shown in Fig. 4, this conguration results in a net solvent ow which can be exploited to fabricate a single-particle microuidic pump.In order to validate this idea we perform simulations of one thermophilic spherical colloid xed in a periodic gradient conned between parallel walls.Walls are implemented with stick boundary conditions by using the bounce-back of the MPC particles at the wall.The chosen wall separation is L t ¼ 12a.PBCs are used in the other two directions.The box size in the direction of the periodic temperature gradient is 2L z with L z ¼ 22a, and the third and neutral direction is L t ¼ 12a.The employed temperature gradient is VT ¼ 0.0135 T/a.As depicted in Fig. 9, the colloidal sphere is xed equidistant from the walls and from the cold and the hot layer, while no colloidal particle is considered in the neighbouring half-box where the temperature gradient has an opposite sign.In this conguration the colloid has the hot layer on its right, which originates a ow from right to le.The direction of the ow could be reverted by placing the hot layer on the le of the thermophilic colloid, or by employing instead a colloid with thermophobic properties.
The so-called Knudsen pumps 59,60 are also pumps operated without any moving parts and with a temperature gradient along the walls of a micro-channel.The driving mechanism is though completely different from the one presented in our work.The Knudsen pumps are driven thermal creep gas ows, 61,62 while here we present a pump driven by liquid thermophoretic forces.Thermal creep ow occurs when the molecules of a rareed gas interact with walls that have a position dependent temperature.The molecules in a high temperature region can transfer more momentum to the wall than those in a low temperature region, such that the gas exerts a net force on the wall against the temperature gradient. 62This in turn translates into an effective ow velocity that goes always from cold to warm areas.In the present thermophoretic pumps, ow can occur in both directions.A different family of microuidic manipulation that has been widely used 63,64 is based on the existence of a surface tension gradient in the direction of a temperature gradient.For example, liquid droplets or lms on a surface can be transported along or against a temperature gradient. 65,66A very recent simulation work shows that liquids can also be pumped by a symmetric temperature gradient through a composite nanochannel, 67 in which one half of the channel wall has a low uid-wall surface energy while the other half has a high one.Essentially, the physical mechanism is the same as the pump we present here, since the single particle can be regarded as a curved surface or as a building-block of planar walls.The variant proposed here is based on the properties of a single particle which provides an important additional degree of exibility in the design of microuidic devices.
We analyze the averaged velocity prole across the tube in the simulated single particle pump.Two bins 5a wide are chosen at both sides of the cold layer.These are completely at the le and at the right of the tube as displayed in Fig. 9.The results in Fig. 10 show the expected parabolic prole, although the two bins are signicantly different.This difference originates from the thermal creep ow, since in our simulations the mean free path cannot be taken to be arbitrarily small due to computational costs.For the le half of the channel in Fig. 9, the thermal creep ow is against the thermophoretic ow eld such that it effectively reduces the slip on the wall.However, the thermal creep ow in the right half of the channel has the same direction as the thermophoretic ow eld which noticeably enhances the slip.This difference can also be directly seen in Fig. 9 where the le half of the wall seems stickier than the right half.The inset of Fig. 10 shows the parabolic prole of the ow in a similar conguration but with a thermophobic colloid.Here, the ow is in the opposite direction.
A nal point that distinguishes the ow of this pump and the ow displayed in Fig. 4 is the existence of wall friction, besides the friction of the colloidal surface.This means that the integral of stress tensor over the particle surface for the case of the pump is not zero.It can be therefore expected that the magnitude of the solvent ow decreases with increasing tube length.

VI Discussion and conclusions
A colloidal particle in solution in the presence of a temperature gradient does not only suffer a thermophoretic directed force, but also induces a thermophoretic ow eld.This ow is here extensively studied by means of mesoscopic simulations.The obtained results quantitatively agree with the analytical predictions, which support both the assumptions made in the theory and the validity of the MPC simulation technique to investigate the dynamics of non-isothermal solutions.The ow eld in the case that the colloidal particle is xed is rst analyzed.The force exerted on the colloid is in this case the thermophoretic force, and the induced ow eld is Stokesletlike and therefore long ranged.This is in some aspects similar to a sedimenting particle, although the lack of motion and the fact that the driving force is not external make them clearly different.We also investigate the ow eld around a freely moving particle.The thermophoretic force balances with the friction force due to the colloidal motion, such that the total force on the colloid vanishes.The induced ow eld is then source-dipole-like and therefore very short ranged.The uid motion in these two examples does not violate the second principle of thermodynamics since these are non-equilibrium systems to which external energy has to be constantly supplied in order to maintain the temperature gradient.During the uid motion, thermal energy is continuously transformed into translational kinetic energy of the uid, which is simultaneously dissipated by viscous friction.In the stationary state, the two processes balance each other.
The importance of hydrodynamic interactions in the thermophoretic phenomena has long been a subject of debate. 3,68he most important example is the explanation of the size independence of thermal diffusion coefficient of a dilute highweight polymer solution. 14,41,69,70From the study presented in Fig. 9 Single-particle thermophoretic pump.Thermophilic colloidal particle confined between parallel walls, with both extremes of the tube connected by PBCs, and larger temperatures in the tube center.Given that the colloid is thermophilic and has the hot layer on its right, the solvent continuously flows from right to left.Fig. 10 Velocity parallel to the walls and between them for the flow in Fig. 9. Triangles correspond to the average profile in a bin 5a wide just on the right of the cold layer.Circles relate to a bin 5a wide just on the left of the cold layer.Lines refer to a parabolic fit.The inset corresponds to the velocity profile in the same layers but for a thermophobic colloid.
this manuscript, it is straightforward to argue that the thermophoretically induced ow eld around a xed colloidal particle accounts for the effect of the thermophoretic force, while the ow around a moving colloid accounts for the combination of the thermophoretic and friction forces.In a recent work 37 we show how the long ranged hydrodynamic contribution of the ow eld can explain the size independence of thermal diffusion coefficient of a dilute high-weight polymer solution.In our reasoning this occurs just as a cancelling effect of the dependence of the thermophoretic force and the selfdiffusion coefficient.
As a practical application, we show that long-ranged attraction between colloids can be induced as a consequence of the hydrodynamic thermophoretic ow eld near a boundary wall.This is consistent with the theoretical calculations and the recent experimental observations. 10,11Our simulations offer a complementary verication of this effect.An enhancement of the attraction at short distance with respect to the analytical prediction is observed in our simulation results.Finally we present a prototype of a single-particle thermophoretic pump which has not yet been experimentally veried.In this pump the ow eld can be generated given the presence of a thermophoretic particle and a temperature eld with an alternating gradient.The implementation of this pump does not require the presence of any movable part.The direction of the ow is determined by the orientation of the alternating temperature gradients and the thermophoretic properties of the employed particle.One very important advantage of this pump in comparison with other existing pumps 59,60,63,64,67 is that the ow can be controlled at a single particle level which will allow the development of promising microuidic applications.

VII Appendix A: reflection method to calculate wall-induced backflow
The Stokes equation for the ow eld is solved in eqn (3) considering vanishing velocity eld at innity.In general, the effect of one near wall can be analytically described by using the reection method.In order to cancel the uid velocity on the wall (stick boundary conditions), the ow eld can be evaluated by considering an image particle with respect to the wall. 47,48he ow eld at a point r ¼ (r x , r y , r z ) due to the image placed at r 0 ¼ (r 0 x , r 0 y , r 0 z ) is found to be 47,48 where r denotes the relative position r ¼ r 0 À r and its modulus is r.Here z is the direction perpendicular to the wall, the indices i, j, and k ˛(x, y, z), a ˛(x, y), and the Einstein's summation convention is employed.F j is the force exerted on the solvent by the image, and h w is the image-wall distance.Note that eqn (A1) only includes the Stokeslet part of the ow produced by the image particle, and higher-order source doublet contributions are neglected.
For the case of the thermophoretic ows v a and v b considered in Fig. 1 the ow and force are both perpendicular to the wall and eqn (A1) is simplied as here the thermophoretic force f T ¼ ÀF z is used, and h w can be identied with L z /2.If the system only has one wall, then the combination of eqn (3) with eqn (A2) gives the correct total velocity eld.The existence of the second wall makes it necessary to consider an additional image to cancel the ow also at such wall.Nevertheless, the image of the second wall gives a non-vanishing contribution in the rst wall.This can be corrected by considering additional images as sketched in Fig. 11.
To obtain a satisfactory convergence, even higher-order images need to be taken into account.In the case of Fig. 2, the approximations for v a and v b are calculated to the 4 th and 5 th order images, respectively.

Fig. 2
Fig. 2 Flow velocity as a function of distance from the colloidal center.Symbols correspond to simulation results, dashed lines to the theoretical calculation in eqn (3), dotted lines to the constant backflow approximation, and solid lines to theoretical calculation with the reflection method.Arrows indicate the position of the system boundaries.(a) Velocity in axis a.(b) Velocity in axis b.

Fig. 3
Fig. 3 Cross-section of the flow field induced by two thermophilic colloids symmetrically fixed in neighbouring temperature gradients with opposite signs, and PBCs in the three dimensions.

Fig. 4
Fig. 4 Cross-section of the flow field induced by one thermophilic colloid fixed in one of the two neighboring temperature gradients with opposite signs, and PBCs in the three dimensions.

Fig. 5
Fig.5Cross-section of the velocity field around a freely drifting thermophilic colloidal sphere in a temperature gradient.The velocity field is obtained from the simulation shown in Fig.4.The inset shows the theoretical result from eqn (4).

Fig. 6
Fig. 6 Velocity field around a drifting thermophoretic particle as a function of distance from the particle, with the positive direction toward the hot side.Symbols refer to the simulation results, lines to the theoretical calculation from eqn (4).(a) Velocity field V a , solid and open symbols correspond to V a,c and V a,h , respectively.(b) Velocity field V b .

Fig. 8
Fig.8Hydrodynamic attraction force as a function of the inter-particle separation.Symbols correspond to simulation data and the dashed line to the analytical prediction in eqn(5).

Fig. 11
Fig. 11 Schematic diagram of the reflection method used in our calculations.With solid lines, the actual walls and the central colloid; with dashed lines the corresponding images.