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This chapter will review past and current research on particle thermophoresis in solution. At the beginning of the chapter, a brief introduction to the thermophoresis and other closely related processes such as particle thermophoresis in gases will be elaborated and discussed. Next, the experimental methods dealing with particle thermophoresis in solution are presented. The major differences of these experimental methods are based on the methods that how to generate a constant microscale temperature gradient and the detection methods. Subsequently, the major experimental results will be summarized and discussed. Subsequently, even there is mechanism of particle thermophoresis in solution is still arguing, some theoretical models will be presented and discussed.
How is particle motion influenced by the presence of a uniform thermal gradient? The experimental evidence is that, on top of random Brownian diffusion, the particles perform a steady drift towards the hot or the cold side, but the reverse is also seen under some conditions . This phenomenon is quite similar to what happens when an external driving force, such as gravity or an electric field, is applied to the suspension. However, for thermophoresis, there is no real external field actually present.
As mentioned before, thermophoresis is an additional particle transport mechanism brought in, on top of Brownian diffusion, by the presence of a thermal gradient. Then the total mass flux could be written as
Where is the usual Brownian diffusion coefficient. is generally called the 'thermal diffusion coefficient' or 'Soret coefficient'. However, is not a real diffusion coefficient. The steady-state thermphoretic velocity acquired by the particle is simply given by
In analogy with other transport effects such as electrophoresis, this is dubbed thermophoretic mobility.
Assuming a uniform thermal gradient directed along , vanishing of the net mass flow leads to a steady-state concentration profile
Where is called the Soret coefficient. With the definition, when the particles move to the hot, displaying what we shall call a 'thermophilic' behavior, while when shows a 'thermophoretic' behavior. The coefficient introduces a characteristic length scale
This characteristic length can be envisaged as that length scale over which thermophoretic drift eventually becomes dominant with respect to Brownian diffusion. A length scale with the same physical meaning can obviously be defined for colloidal motion driven by a true external field. For instance, in colloid settling under gravity, this role is played by the sedimentation length , where is the Stokes sedimentation velocity .
2.2.2 Related effects
Before embarking on the analysis of particle thermophoresis in liquids, it is instructive to take a glance at closely related transport effects induced by thermal gradients in liquid mixtures or in dilute gases, which stand at the root of the problem we are considering, and concur in providing the conceptual framework for particle thermophoresis in liquids. As we shall see, the comprehension of the basic mechanisms underlying thermophoresis in liquids may indeed benefit from what is known about thermal diffusion in liquid mixtures and particle thermophoresis in gases.
22.214.171.124 Thermal diffusion in liquid mixtures and solutions (Ludwig-Soret effect)
Thermal diffusion, or the Ludwig-Soret effect, is the 'molecular counterpart' of particle thermophoresis . 150 years ago, Ludwig observed that concentrated salt solutions easily crystallize around the cooled limb of an inverted U-tube . Later Soret quantified the effect for many electrolyte solutions, and in 1965 De Groot clearly framed this work into non-equilibrium thermodynamics .
Even though it is a relatively small effect (the Soret coefficient of common mixtures is only of the order of 10-3K-1) , thermal diffusion has been extensively studied due to its dramatic impact on convective mixing. As the Soret effect induced inverted concentration gradient can relax only by mass diffusion, which is far slower than heat diffusivity, the convection threshold in mixtures is dramatically lowered compared to simple fluids (in 1998, La Porta reported that when , it is even possible to induce convection by heating from above ). Therefore thermal diffusion plays a crucial role in many naturally occurring convective processes, from component segregation in solidifying metallic alloys  and volcanic lava  to convection in the earth mantle . It also plays an important role in crystal growth . More recently, Vailati also reported that thermal diffusion sets the scene for giant fluctuations in non-isothermal mixtures .
Even so many experimental results have been published, there is still no detailed microscopic model of the Soret effect. However, for gas mixtures, some general indications stem from kinetic theory. Although thermal diffusion in aerosols was indeed theoretically predicted by Chapman back in 1912 , purely kinetic theories are notoriously inadequate to describe liquids.
126.96.36.199. Particle thermophoresis in gases
Tyndall was the first to observe and report thermally driven transport in aerosols, by visually observing the light scattered by hazes, noticed that the suspended dust particles tend to avoid hot surfaces . Although a deep connection to the Soret effect obviously exists, the theoretical analysis of particle thermophoresis in gases proceeded along a totally independent route (with the noticeable exception of the work by Chapman [24, 25]).
The basic mechanisms underlying thermophoresis in gases are very different, depending on the ratio between the mean free path and the particle size (the Knudsen number). When (very low density gases), the problem is actually very similar to thermal diffusion in a gas mixture where one of the two components (the particles) has an exceedingly large size compared to the other. But the opposite limit, (gas at moderate pressure), which is properly called the 'quasi-hydrodynamic regime', is more attractive. Maxwell in 1879 , studied the stresses in rarefied gases arising from inequalities of temperature thoroughly. The key result of his study is that, in a homogeneous gas, no longitudinal (pressure) or transverse stresses are associated with temperature gradient, which is very different when a bounding solid surface is present.
Using Maxwell's result, Epstein  in 1929, calculated the steady-state thermophoretic velocity in aerosols:
is the particle's thermal conductivity, while is the gas' thermal conductivity, with viscosity , and number density .
Since Epstein's seminal contribution, extensive theoretical and experimental work on thermophoresis in gases has been performed, and the intimate connection between thermophoresis and thermal diffusion in gasmixtures has been thoroughly investigated . In summary, some basic features of the mechanism driving thermophoresis in the quasi-hydrodynamic regime :
The gas exerts on the surface a purely tangential stress. Therefore, within a surface layer with a thickness of the order of , the pressure tensor is anisotropic.
Because only tangential stresses are involved, even if it is a surface effect, the total force on the particle scales only with , and not (so that, where is the friction coefficient, is size independent).
In hydrodynamic terms, can eventually be seen as a slip velocity of the particle, with a slip length .
Particle bulk properties enter the problem only through the thermal conductivity , that, together with thermal conductivity of gas , determines the local temperature field around the particle via the heat equation.
2.3. Experimental methods
The key reason for the recent noticeable increase of experimental investigations of particle thermophoresis in liquids has been the development of new, sensitive techniques, allowing us to obtain a rather large set of accurate data ondifferent systems. In this section, we shall critically reviewthese new techniques, describing their respective advantages and limits, but also pointing out the possibility of further developments.
2.3.1. Methods to generate high temperature gradient in microfluidic device
From the definition of thermophoresis, we have to overcome the Brownian motion to driven particle moving when temperature gradiant exists. Only when the length , the thermophoresis effect will be dominant over the Brownian motion. As the order of thermal diffusion coefficient and diffusion coefficient are universal for a given solvent system , so the major hurdle to set an experimental study on thermophoresis is how to generate a rather high temperature gradient in microscale(usually should above ). So in this section, different methods to generate high temperature gradient in microfluidic devices have been reviewed.
188.8.131.52. Laser induced heating method
Braun's group [3, 6, 17] use an infrared laser (FOL1405RTV-317, 1480 nm, Furukawa, Japan) with a power of 100 mW to heat their chamber's center from below. The constant horizontal temperature gradients are of order of to .
Figure 1. (a) Photograph of chamber. (b) liquid is filled between glass and sapphire. The center is heated by an infrared laser (500 thick). (c) another chamber using the infrared laser heating in the center (10 thick, to avoid the toroidal convection flow).
Jiang Hong-Ren, etc,  also use the same method, but using a different infrared laser (Nd:YAG, 1064 nm, power ~ 4 mW), and they could get a temperature gradient of order of .
184.108.40.206. Electrical-thermal method
Putnam [30, 31, 32] uses Joule heating of a pair of parallel Au thin-film lines fabricated by photolithography on a fused silica (FS) substrate to generate a temperature gradient of order of , as shown in Figure 2.
Figure 2. Schematic cross section of the temperature-controlled sample cell for measurements of thermodiffusion in liquids used by S. A. Putnam. The dark, cross-hatched regions are the parallel thin-film Au line-heaters, ~250 nm thick, that are alternately heated at an angular frequency with a high-frequency square-wave current (); e.g., the high-frequency square-wave current passes through one line-heater for the first half-cycle of and then through the other line-heater for the next half-cycle of . The line-heaters are separated by and have a width of . The chamber height in the sampling region is . 
Geelhoed  uses a fused silica capillary with a inner diameter of and outer diameter of . Inside, a copper coated with insulator to was inserted and connected to the power supply. So they could have a constant temperature gradient of order of in the annular channel.
Figure 3. Schematic representation of the test set-up used by Geelhoed. The capillary is placed in a water droplet on a microscope slide. Inside a copper wire is inserted as a heater.
Later, Geelhoed  designed another microfluidic device using a thin aluminum layer integrated on one side of a silicon channel. Details see Figure 4. The channel depth is , with several different channel widths of 70, 100 and 150. In this device, they can obtain a temperature gradient of the order of .
Figure 4. The silicon substrate is wet etched in KOH solution to obtain a sloped cross section profile of the channel. Then it is possible to deposit a thin aluminum film on the wall.
Takuya Mutsui  used Peltier effect to generate a high temperature gradient in a PDMS/glass chip, resulting in a temperature gradient.
Figure 5. Schematic diagram of Takuya's chip. The dimensions of the channel between the reservoirs are 100 in width, 100 in depth, and 10 mm in length. The dimension of cover glass is 25mm60mm150 mm. The dimensions of the Peltier element are 20mm in width, 40mmin length, and 3.4mm in height.
220.127.116.11. General microfluidics method
Hanbin Mao  used hot and cold fluid as the heating and cooling sources. Details see Figure 6. The channel width is 1mm. The temperature gradient in this device is around .
Figure 6. Schematic diagram of Hanbin Mao's device.
Daniele Vigolo [37, 38] designed a microfluidic device where the temperature gradients are established by flowing hot and cold water, seen in Figure 7. The dimensions of the sample channel in the center are in width in height, and 120mm in length. The temperature gradient in their devices could reach .
Figure 7. (a) A microfluidic device where temperature gradients are established by flowing hot and cold water. Side channels and reservoirs, together with the sample microchannel, are shown in the cross-sectional view. (b) An alternative design where the hot water channel is replaced by a silver-epoxy resistive heater. The sample channel for the devices shown in (a) and (b) is 40mmlong. (c) A device using an embedded ohmic heater as in (b), but with a longer path (120 mm long).
2.3.1. Detection methods
18.104.22.168. Fluorescence detection method and single-particle tracking
Spurred by great advancements in optical microscopy, such as confocal detection schemes and advanced particle-tracking methods, direct visualization of colloidal particles has became more and more popular in the last decade.
The basic strategy of this approach, which has been mainly followed by Braun and coworkers, consists in inducing a very localized thermal gradient by the adsorption of a laser beam, using however simple cells, built from microscopic slides or capillaries, and standard microscope optics. Using fluorescent-dyed particles, the concentration profile can be reconstructed by monitoring through themicroscope the spatial and time dependence of the fluorescence intensity.
The major advantage of this method is that of exploiting standard fluorescence microscopy instrumentation and techniques, with no need for specific optical schemes or custom-designed cells. Fluorescence detection methods are moreover so sensitive that even extremely particle low concentrations can be measured.
There are obviously also disadvantages. First of all, the method works only for particles that are intrinsically fluorescent, or made such by adsorption of a dye. Moreover, absolute fluorescence intensity measurements are also notoriously hard, due to the difficulty of quantitatively taking into account dye-bleaching effects. Finally, monitoring the temperature field is non-trivial.
Braun and coworkers have developed the ingenious strategy of using a temperature probe a dye whose fluorescent emission is temperature dependent. Nonetheless, the sensitivity of this probe is rather low (a few percent per degree), so that much higher temperature gradients are needed than those used in Thermal Lensing measurements. But the key advantage of the method is that, provided that particles are sufficiently big that individual particle emission is resolved, colloid thermophoretic motion can be directly visualized. For suspensions of very large particles (say, in the few microns range), where even all-optical techniques become unbearably slow, single-particle tracking by fluorescence microscopy is currently the only method to obtain DT. It is useful to consider the role of the characteristic length in particle-tracking experiments.
For the contribution of thermophoretic drift to be an appreciable fraction of the particle rms displacement, the latter has to be comparable to . The minimum tracking time will therefore be shorter for very large particles or high thermal gradients.
Fluorescence imaging of temperature
We exploit pH drift of 10 mM TRIS buffer upon temperature change. The drift is measured by a pH-sensitive fluorescent probe. This allows usage of highly soluble fluorescent probes. All optical probes of temperature pursued previously only worked for non-aqueous solutions. The temperature dependence of the dye was measured with a temperature-controlled fluoremeter (Fig. 1c). Near 20oC, we obtain temperature sensitivities of for TAMRA (20, C-300, molecular probes), for BCECF (20, B-1151, molecular probes)
DNA and DNA staining
We use the DNA of the bacteria-attacking virus lambda (). We stain DNA with low concentrations of SYBR Green I (S-7563, molecular probes). Starting from 0.46 (14nM) stock solution (Invitrogen/Gibco BRL, Cat. No. 25250-010), we dilute the DNA 1:10 into a 10 mM TRIS-HCL buffer (pH 7.8). We add 2 SYBR-Green and 20 of TAMRA. The dye intercalates into DNA, but changes the overall DNA charateristics only minimally.
Fluorescence imaging and bleaching correction
We use the microscope Zeiss Axiotech Vario with objective Plan Fluar 40, NA 1.3 oil. For illumination, a Luxeon High power LEDs LXHL-LX5C was built into a standard haloegen lamp housing and driven at 30-700mA by an ILX Lightwave LD-3565 current source. Fluorescence filters were from AHF-Analysentechnik, Tubingen for FITC (HQ F41-001). Detection was provided with a 12-bit camera PCO Sensicam QE 670KS with 65% quantum efficiency. The camera has a linear light response at negligible background levels. For slow diffusing specimen, the illumination was switched off during equilibration to reduce bleaching. Fluorescence imaging averages the fluorescence across chambers thinner than about 30. This can be demonstrated by focusing a layer of adsorbed fluorescent beads.
However, we had to check that this averaging across the chamber is independent of lateral temperature gradients. These can bend the light rays of excitation and emission from their optimal configuration, leading to a defocused image and low fluorescence readings. Note that this lensing effect differs from thermal lensing methods which measure the focal length of the created lens in the far field. Since defocusing-induced fluorescence imaging artefacts are small at NA=0.4 due to the large averaging focus, we can use it as reference. We compared fluorescence images of the same radial temperature profiles measured at NA=0.4 and NA=1.3. Both images differed only in the central 10 region relative to each other by 5%. We therefore infer that averaging fluorescence across thin chambers is reliable even under lateral temperature gradients used in our experiments.
To be independent of inhomogeneous fluoresce illumination, images in the heated state are normalized against previously taken cool pictures. Also a cool picture after measurement is taken to allow for a linear bleaching correction over time. In all experiments, illumination intensity is low such that bleaching was below 20% and linear bleaching correction was applicable.
2. Fluorescence imaging of concentration and temperature
As described above, we measured fluorescence intensities FT for the temperature dye and FC for the DNA dye. Both are imaged in the cold state at T0 - denoted by -- and later in the heated state at temperatures T(x, y) - denoted by . The temperature-sensitive dye depends on small temperature changes T-T0 with a linear slope B given in percent fluorescence change per K temperature change [%/K]. The fluorescence intensity of the probe marker FC reports the probe concentration c:
Combined with the linearly approximated form of thermophoretic depletion given by equation derived from small probe concentrations c and constant DT in non-moving liquids:
We have a linear relationship between FC and FT:
Both linear relationships, namely that of FT to the temperature according to equation (1.1) and that of FC to FT according to equation (1.3) makes it possible to infer them from the images which inherently record a cross-chamber, which we will denote by the operator , leads to the same prefactor for both FC and FT since they integrate over the same functional shape. Thus, we can infer the Soret coefficient directly from the measured, averaged fluorescence intensities and :
To be more precise, we have to include into the description a possible temperature dependence of the probe marker itself. Foe example, SYBR-Green has a marked temperature dependence of (0.5 SYBR Green I with -DNA, Fig. 1c). We include this effect with the following substitution of the concentration fluorescence from equation (1.1):
The problem is that this additional temperature dependence breaks the linear averaging argument used above. Now other than in equation (1.3), FC is a non-linear function of the relative temperature change:
We have to consider this nonlinearity especially for strong heating and have to take into account the precise temperature characteristic across the chamber. However, for our experiments with -DNA, the relative temperature change is only 1.4%. This is much larger than the quadratic term, leading to an error if we neglect the effect of 1.2% in . This deviation is however below our overall measurement precision.
We have to consider another possible source of error. In using temperature-sensitive dyes with characteristics measured in thermodynamic equilibrium in a fluorometer, we assume that thermophoretic depletion of the temperature dye itself can be neglected. In a preliminary estimation, the dye has a similar monomer thermodiffusion as the highly charged DNA, namely approximately. Together with the diffusion constant of a similar dye FITC, , we infer a Soret coefficient . Preliminary measurements confirm similar Soret coefficients for the used temperature dye. We can infer the Soret coefficient of the dye from fast recordings of the temperature fluorescence upon laser heating. After a fast fluorescence drop we find a slower response which we attribute to thermophoretic depletion. The ratio of the two drops allows to infer the Soret coefficient of the dye, very similar to beam deflection methods of Giglio and Vendramini. We can take for example a temperature increase of 1K above room temperature. The intensity of the dye decreases by about 1% due to its temperature sensitivity. Its concentration due to thermophoresis is depleted however by only 0.08%. Thus our measurement of temperature using fluorescent dye is 8% too high. Hence, thermal diffusion coefficient has the possibility to be systematically too low by 8%. The effect of this systematic error will be studied in more detail in the future since it can be suppressed by taking the temperature image at an optimal time after switching on the laser.
3. Measuring thermophoresis of DNA
Until now we have checked that thermophoresis should be measurable by microfluidic fluorescence without artefacts. An infrared focus is heating water optically to a temperature which we detect by fluorescence imaging. It causes a thermphoretic depletion that is recorded by fluorescence at a different color. By comparing the temperature profile with the concentration profile in steady state, we can infer the Soret coefficient.
Thermophoresis of the temperature dye and temperature dependence of the DNA stain are negligible.
The method to measure the thermophoresis for PS beads smaller than 500 nm :
Concentration imaging over time method at low concentration. We imaged over time by bright-field fluorescence with a 40 oil-immersion objective. Concentrations inferred after correcting for bleaching, inhomogeneous illumination, and temperature-dependent fluorescence were fitted with a finite element theory by COMSOL. The model captures all details of both thermodiffusive depletion and back-diffusion to measure DT and D independently.
We can also obtain D and DT independently by analyzing the build up and flattening of concentration profile over time after turning the infrared laser beam on or off, respectively. Theory was provided from finite element model in radial coordinates (FEMLab, Comsol) over time with boundary conditions of concentration obtained from the experiment. Comparison with experiment of the time course of thermophoretic depletion reveals DT (Fig. 6 c and d) and from the time course after switching off the heating source the diffusion coefficient D is obtained (Fig. 6 a and b). Results for diffusion coefficients obtained for DNA molecules are shown in Fig. 7. Scaling of D for DNA larger than 1,000 bp agree well with literature values and theoretical expectations (6). However for DNA molecules in the order of the persistence length (≈150 bp) the power law exponent of -0.6 does not precisely fit the measured values and a different scaling with an exponent of -1 is necessary. A good description of DNA diffusion coefficients in the size range analyzed throughout this work is achieved with an intermediate exponent of -0.75.
Thermodiffusion for particles at this range is measured by the fluorescence decrease that reflects the bulk depletion of the particle:
First, diluted the PS beads to 0.02% in 1 mM TEIS buffer. Then the concentration of particles was averaged from an image stack of 50 images, each with a exposure time of 10 seconds, recorded at 12-bit resolution. The imaging protocol consisted of three time steps: first without heating, second in steady state after 20 min of laser heating, and third after 20 min of backdiffusion to correct for fluorescence bleaching of the beads. Bleaching could be corrected linearly; inhomogeneous illumination was removed by dividing with the initial, unheated image stack. Radial concentration profiles were extracted from the averaged image stacks. Background fluorescence was subtracted as inferred from the central depleted region.
The method to measure thermophoresis for PS bead bigger than 200 nm :
Fluorescence single-particle tracking method.
Their bulk fluorescence was imaged over time to derive DT and D from the depletion and subsequent back-diffusion. For PS beads bigger than 500nm, due to the slow equilibration time and risk that steady-state depletion is disturbed by thermal convection, we can use single-particle tracking method.
The thermdiffusive drift was imaged with a 32 air objective at 4 Hz at an initial stage of depletion in a 20-microns-thick chamber. Averaging over the z position of the particles removed effects from thermal convection. The drift velocity versus temperature gradient of 400 tracks were linearly fitted by to infer DT. The diffusion coefficients D of the particles were evaluated based on their squared displacement, matching within 10% the Einstein relationship.