Different measures for characterizing the motion of molecules along a temperature gradient
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a y Different measures for characterizing the motion of molecules along a temperaturegradient
Oded Farago Department of Chemistry, University of Cambridge,Lensfield Road, Cambridge CB2 1EW, United Kingdom Department of Biomedical Engineering, Ben-Gurion University of the Negev, Be’er Sheva 85105, Israel
We study the motion of a Brownian particle in a medium with inhomogeneous temperature. Inthe overdamped regime of low Reynolds numbers, the probability distribution function (PDF) of theparticle is obtained from the van Kampen diffusion equation [J. Phys. Chem. Solids , 673 (1988)].The thermophoretic behavior is commonly described by the Soret coefficient - a parameter which canbe calculated from the steady-state PDF. Motivated by recent advances in experimental methodsfor observing and analyzing single nano-particle trajectories, we here consider the time-dependentvan Kampen equation from which the temporal evolution of the PDF of individual particles canbe derived. We analytically calculate the PDF describing dynamics driven by a generalized ther-mophoretic force. Single particles statistics is characterized by measures like the mean displacement(drift) and the probability difference between moving along and against the temperature gradient(bias). We demonstrate that these quantities do not necessarily have the same sign as the Soretcoefficient, which causes ambiguity in the distinction between thermophilic and thermophobic re-sponse (i.e., migration in and against the direction of the temperature gradient). The differentfactors determining the thermophoretic response and their influence on each measure are discussed. Keywords:
I. INTRODUCTION
The motion of molecules induced by a temperature gradient is commonly referred to as thermophoresis, ther-modiffusion or the Soret effect. Since its discovery in liquid mixtures more than a century and half ago [1, 2], thephenomenon of thermophoresis has been experimentally observed in aqueous solutions containing colloidal particles,micelles, polymers, proteins, and DNA molecules (see extensive review in [3]). Several studies have shown ther-mophoresis to be a promising tool for manipulating and concentrating biomolecules in solutions [4–6], which has evenled to the speculations that it may play a role in the accumulation of nucleotides required for molecular evolution ofearly life [7].In this work we theoretically study the thermal diffusion of colloidal particles which, in general, is a much strongereffect than thermophoresis in simple molecular mixtures. The relevant length and time scales of the colloidal particlesare orders of magnitude larger than those of the embedding solvent, and hence the solvent may be treated as aneffective medium. The thermal motion of the colloidal particle is driven by stresses induced on its surface by thesurrounding fluid [8–10]. These forces are balanced by viscous drag forces when the particle attains a steady statevelocity [11]. Thermophoresis can therefore be treated as a mass transport process which, for dilute suspensions (lowconcentration, c ) can be phenomenologically described by the continuity equation ∂ t c = −−→∇ · −→ J , with the particleflux −→ J given by [12] −→ J = − D −→∇ c − cD T −→∇ T. (1)The first term on the r.h.s. of Eq. (1) describes regular diffusion due to concentration gradients, where D is the Fickiandiffusion coefficient. The second term describes an additional contribution to the flux resulting from the temperaturegradient, −→∇ T , with D T termed the thermal diffusion coefficient. When a closed system reaches a steady state, theflux vanishes and a concentration gradient is established that satisfies −→∇ c = − cS T −→∇ T, (2)where S T = D T /D is called the Soret coefficient. For S T >
0, the colloids tend to accumulate on the colder side ofthe system, displaying “thermophobic” behavior. Conversely, for S T <
0, the migration is toward the hotter side,which is termed “thermophilic” motion.The sign and magnitude of S T are hard to predict since they depend on multitude of interactions and influences.Importantly, S T may exhibit a pronounced temperature dependence and, quite interestingly, it tends to change itssign close to room temperature in many colloidal systems [13]. Experimental measurements of S T are typically basedon the application of a thermal gradient in a diffusion cell and the use indirect optical methods to quantify theconcentration gradients induced by thermal diffusion [3]. Recently, it became possible to measure thermophoreticforces on a single colloidal particle confined in sub-micrometer regions with a nearly uniform temperature gradient(and an overall small temperature difference) [14]. Moreover, we can now study not only the steady-state probabilitydistribution of the particle, but to also follow its trajectory to relaxation [15]. These advances in experimental methodscall for a better understanding of the problem of a single particle diffusion in a temperature gradient. II. THE VAN KAMPEN EQUATION
Consider a single Brownian particle moving in a one-dimensional medium with a temperature gradient along the x direction. In order to derive an equation for the evolution of the probability distribution function (PDF) of theparticle, P ( x, t ), one has to consider the Langevin equation of the dynamics or the corresponding Fokker-Planckequation. These equations capture both the inertial short- and dissipative long-time regimes of the dynamics. Inpractice, however, only the latter is of interest for colloidal systems at low Reynolds numbers. In this so called“overdamped limit”, the dynamics is depicted by a Smoluchowski-like diffusion equation that can be derived by anadiabatic elimination process of the fast relaxing momentum degree of freedom. The derivation was carried out by vanKampen for different models of diffusion in inhomogeneous media [16]. One of the cases considered by van Kampenis of Brownian particle in a system with spatially-varying temperature. The equation corresponding to this model is: ∂ t P ( x, t ) = − ∂ x J ( x, t ) = ∂ x { µ ( x ) ∂ x [ k B T ( x ) P ( x, t )] − µ ( x ) f ( x ) P ( x, t ) } , (3)where f is the mechanical force acting on the particle, while T ( x ) and µ ( x ) denote, respectively, the coordinate-dependent temperature and mobility. The latters are related to the coordinate-dependent diffusion coefficient, D ( x ),via Einstein’s relation D ( x ) = k B T ( x ) µ ( x ), with k B denoting the Boltzmann constant [16]. As noted by van Kampen,this is a diffusion equation which does not follow neither Itˆo [17] nor Stratonovich [18] prescriptions for overdampedBrownian dynamics in inhomogeneous media.It is important to note that while the non-isothermal dynamics considered here is clearly out of thermal equilibrium,the overdamped limit depicted by van Kampen equation (3) is based on the approximation that the momentum of theparticle, p , is always at equilibrium with the local temperature T ( x ), i.e., follows the Maxwell-Boltzmann distribution ρ ( p | x ) ∼ T ( x ) − / exp[ − p / mk B T ( x )] (where m denotes the mass of the particle). The local thermodynamicsequilibrium (LTE) [12] approximation is justified when l b |−→∇ T | /T ≪
1, where l b is the ballistic distance characterizingthe crossover between the inertial and diffusive regimes. Mathematically, the overdamped limit corresponds to l b → f in Eq. (3) includes both contributions from the thermophoretic force, as well externally applied forceslike gravity which can be minimized by density-matching the colloid with the solvent. We will henceforth ignore allforces except for the thermophoretic one. Moreover, single particle experiments are conducted in small systems wherethe applied temperature difference may be as small as a few degrees Kelvin. Assuming that the temperature gradient, T ′ = dT /dx , and the thermophoretic force are uniform throughout the small system, one may phenomenologicallywrite that the thermophoretic force is given by [19] f = C T k B T ′ , (4)where C T is dimensionless parameter. Using this phenomenological form in Eq. (3) and comparing with Eqs. (1) and(2), we arrive at the following expression for the Soret coefficient [20, 21] S T = 1 − C T T . (5)From Eq. (5) we conclude that in the absence of a mechanical thermophoretic force ( C T = 0), the Soret coefficientdoes not vanish ( S T = 0). The additional contribution to S T is known as the “ideal gas term”. Explicitly, the 1 /T term in Eq. (5) is expected because at steady-state ∂ t P ( x, t ) = 0, and from van Kampen equation (3) one can easilydeduce that the stationary solution is P s ( x ) ∼ T ( x ) exp (cid:20)Z x f ( y ) k B T ( y ) dy (cid:21) , (6)which, in the absence of a mechanical force ( f ( x ) = 0), reduces to P s ( x ) ∼ /T ( x ) . (7)In ref. [16], van Kampen notes that he has no simple explanation for the prefactor /T ( x ) in Eq. (6); however, inthe special case f ( x ) = 0, Eq. (7) was nicely rationalized by Fayolle et al. [21]. They noted that the mechanicalthermophoretic force vanishes in the absence of interaction between the colloidal particles and the embedding solvent,i.e., in the limit of extremely small colloidal particles that can be viewed as an ideal gas. In a closed system at steadystate, the pressure of this ideal gas, Π = c ( x ) k B T ( x ), must be uniform (or otherwise, the gradient pressure force wouldact on the gas and change its distribution). Eq. (7) then means that in the absence of a mechanical thermophoreticforce, the ideal gas thermal collisions induce a steady-state distribution that is higher on the colder than on the hotterside. The associated Soret coefficient S T = 1 /T > | C | ≫
1, with theexception of relatively small colloidal particles (see discussion in Appendix A).Returning to van Kampen equation, we notice that it also takes into account the spatial variation in the mobility,which within a small system can be approximated by µ ( x ) ≃ µ + µ ′ x, (8)where µ is the mobility in the middle of the cell at x = 0 and µ ′ = dµ/dx . We note here that the spatial variationsin µ ( x ) can, in general, be further divided into two parts - those arising from the temperature-dependence of thefluid viscosity [22] and those also encountered at equilibrium isothermal systems, for instance due to hydrodynamicinteractions between the colloidal particle and the walls of the container [23]. As we will see below, the particle’s drift(to be mathematically defined later) depends on both T ′ and µ ′ , and one must keep in mind that these two gradientsare not entirely independent of each other because of the temperature-dependence of the mobility. On the other hand,non-temperature related reasons for spatial variation in the mobility imply that µ ′ does not necessarily vanish when T ′ = 0.Using Eqs. (4) and (8) [together with the expansion T ( x ) ≃ T + T ′ x ] in the van Kampen equation (3), yields thefollowing form ∂ t P ( x, t ) = D (cid:26) ∂ x (cid:18) xµ ′ µ (cid:19) ∂ x (cid:18) xT ′ T (cid:19) − C T xT ′ T ∂ x (cid:27) P ( x, t ) , (9)where D = D ( x = 0) = k B T µ . This is the van Kampen equation in the limit when (i) all forces besides thethermophoretic one are ignored, and (ii) the system is sufficiently small to justify the linear approximations of T ( x )and µ ( x ). (iii) Another assumption implied in Eq. (9) is the form (4) for the thermophoretic force. III. THE PROBABILITY DISTRIBUTION FUNCTION
In a homogeneous ( µ ′ = 0) isothermal system ( T ′ = 0), Eq. (9) reduces to a simple diffusion equation, the solutionof which takes the Gaussian form P ( x, t ) = exp( − x / D t ) / √ πD t ≡ G ( x, t ) [assuming Dirac delta-function initialcondition P ( x,
0) = δ ( x )]. For a system under a small temperature difference ( xT ′ /T ≪
1) and limited changes inthe mobility ( xµ ′ /µ ≪ P ( x, t ) = G ( x, t ) (cid:20) xH (cid:18) x D t (cid:19)(cid:21) , (10)where H is some function that can be determined in the following simple manner: (i) Write H in Eq. (10) as aseries expansion in the argument y = ( x /D t ): H = P ∞ i =0 a n y n , then (ii) determine the coefficients of the expansionby substituting Eq. (10) in van Kampen equation (9), and by comparing terms of similar order in y on both sidesof the equation. In this process of determining H ( y ), we ignore the terms that are non-linear in x . We find, a = ( − / C T / T ′ /T ) − / µ ′ /µ ), a = 1 / T ′ /T ) + 1 / µ ′ /µ ), and a n = 0 for n >
1, and thus write P ( x, t ) = exp( − x / D t ) √ πD t (cid:20) x (cid:26) T ′ T (cid:18) −
34 + C T x D t (cid:19) + µ ′ µ (cid:18) −
14 + x D t (cid:19)(cid:27)(cid:21) , (11)which is the main result of the paper. IV. THE DRIFT AND THE FLUX
The drift of an individual particle is characterized by the mean displacement, h x i , and from the PDF (11), we findthat h x i = Z ∞ xP ( x, t ) dx = (cid:18) C T T ′ T + µ ′ µ (cid:19) D t. (12)We notice that the drift does not necessarily have the same sign as C T , which means that the average displacementof the particle is not necessarily in the same direction as the thermophoretic force. The reason for this remarkableresult is an additional contribution to the drift originating from spatial dependence of the mobility. In general, themobility of simple liquids increases with temperature, while gases exhibit an opposite trend and have mobility thatdecreases approximately like the square root of the temperature [24]. As noted earlier [see discussion after Eq. (8)],non-thermal effects may also contribute to µ ′ . Indeed, it is well known that drift is also observed in isothermalsystems with non-uniform mobilities [25]. This equilibrium phenomenon has been termed “spurious drift”, which ismisleading since it is a real effect [26]. In the isothermal case ( T ′ = 0), we can use Einstein relation and Eq. (8) towrite Eq. (12) in the more common form, h x i /t = D ′ [27], relating the drift velocity and the spatial derivative ofthe diffusion coefficient. Thus, our result Eq. (12) generalizes the well-known expression for the drift of Brownianparticles in isothermal inhomogeneous media to non-isothermal systems.Recall that the derivation of van Kampen equation (3) is based on assuming LTE in the overdamped limit. Withinthis approximation, the mean kinetic energy of a particle found at some coordinate x is related to the local temperaturevia the equipartition theorem h E k i x = h mv / i x = k B T ( x ) /
2, where h· · · i x denotes average at a given x . Taking theaverage with respect to x and using Eq. (12) gives d h E k ( t ) − E k ( t = 0) i dt = k B T ′ d h x i dt = k B T ′ (cid:18) C T T ′ T + µ ′ µ (cid:19) D . (13)For µ ′ = 0 (constant mobility), the particle is heated on average (i.e., gains kinetic energy) when C T >
0, i.e., whenthe thermophoretic force drives the particle to the high temperature side, and vice versa. This, however, may not betrue when the mobility varies in space, in which case it is the sign of C T + ( T µ ′ /T ′ µ ) rather then the direction ofthe thermophoretic force that determines whether the particle gains or losses heat.A common error is to confuse the above-discussed drift with the flux, defined by J ( x, t ) = − D ( x ) ∂ x P ( x, t ). Aclosed system at steady state has zero flux, J = 0, but this does not necessarily imply that the average displacement(i.e., drift) of each individual particle must also vanish. On time scales smaller than the characteristic diffusion timeacross the system, particles located at different parts of the system (e.g., near the center or close to the boundaries)may have different non-vanishing displacements. This situation has been previously dubbed “drift without flux” inequilibrium isothermal systems [28]. Here, we consider dynamics in an open system with time-dependent flux. Thetendency of particles to migrate favorably to one side may be characterized by the flux at the origin J ≡ J ( x = 0 , t ) = p D / ( πt ) [( T ′ / T ) (3 − C T ) + ( µ ′ / µ )]. The flux at the origin causes a “bias”, i.e., a difference in the probabilityof finding the particle in the “hotter” and “colder” sides relative to its initial location. Assuming (without loss ofgenerality) that T ′ >
0, the bias, ∆( t ), is defined by∆( t ) ≡ Z ∞ P ( x, t ) dx − Z −∞ P ( x, t ) dx = r D tπ (cid:20) T ′ T (2 C T −
1) + µ ′ µ (cid:21) = h x i√ πD t − r D t π T ′ S T , (14)with the drift, h x i , and the Soret coefficient, S T , given by Eqs. (12) and (5), respectively. Depending on the values of T µ ′ /T ′ µ and C T , it now becomes clear that while ∆, h x i , and − S T can all be used to characterize the response ofcolloidal particles to a temperature gradient, these quantities describe different features of the Soret effect, and mayoccasionally have different signs. V. DISCUSSION AND SUMMARY
Motivated by recent single-molecule experiments for studying the behavior of macromolecules along a temperaturegradient, we considered here the question of Brownian dynamics of a colloidal particle in a non-isothermal fluid. Inthe overdamped limit, the PDF of the particle is described by time-dependent van Kampen diffusion equation (3).Assuming a small temperature and mobility differences between the ends of the (small) system ( T ′ x/T ≪ µ ′ x/µ ≪ x ) version of van Kampen equation (9) and analytically derived the solutionfor delta-function initial condition (11). The asymmetric PDF characterizes the general tendency of the particle tomigrate in the direction of the thermophoretic force caused by the temperature gradient. However, the thermophoreticforce is not the only factor determining the direction of the motion, and we have identified three different measuresfor the thermodiffusive response of the colloidal particle. The first measure is the Soret coefficient S T (5), relatingthe concentration and temperature gradients in steady state. The Soret coefficient has been traditionally used todistinguish between thermophilic ( − S T >
0) and thermophobic ( − S T <
0) behaviors. However, we see from Eq. (5)that − S T and C T do not necessarily have the same sign, indicating that the steady-state concentration gradient is notsolely dictated by the direction of the thermophoretic force. The origin of the discrepancy are the thermal collisionswhich set a concentration gradient opposite to the temperature gradient. In fact, in some recent experiments oncolloidal systems it has been found that S T exhibits a strong temperature-dependence and tends to change its sign inthe vicinity of room temperature. Moreover, the magnitude of the Soret coefficient in many of these experiments isfound to be of the order of 0 . − K − [13, 29]. These findings indicate that (i) the effect of the thermal collisions maysometimes be as important the thermophoretic force that accounts for the particle-solvent interactions, and that (ii)the thermophoretic force (coefficient C T ) is sensitive to temperature variations. Due to the system-specific nature ofthe thermophoretic force, there is no clear explanation for its temperature sensitivity of C T which is likely dependenton numerous factors, e.g., the thermal expansivity of the solvent [13], the surface functionality [15] and size [5] ofthe colloidal particle, and electrostatic effects [30]. In order to understand this behavior of C T one must considera microscopic model that takes into account some of these factor (see, e.g., the theoretical discussion in ref. [20]).This is beyond the scope of the phenomenological discussion presented herein; however, in light of the pronouncedtemperature-dependence of S T , it must be reemphasized that our derivation assumes that the thermophoretic forceis phenomenologically given by Eq. (4), namely assuming non-equilibrium linear-response. The same linear form hasbeen considered in other works (see, e.g., [21]), and it is consistent with the linearity of our solution for the PDF (11)with respect to T ′ . More generally, the variations of C T with T can be accounted for by a Taylor expansion around T : C T = C T ( T ) + ( dC T /dT )∆ T + · · · = C T ( T ) + ( dC T /dT ) T ′ ∆ x + · · · , which shows that the linear approximationis valid if the total temperature difference across the experimental cell ∆ T = T ′ ∆ x is sufficiently small, i.e., if the thesize of the experimental setup, ∆ x , and the temperature gradient, T ′ , satisfy∆ T = T ′ ∆ x ≪ (cid:12)(cid:12)(cid:12)(cid:12) C T dC T /dT (cid:12)(cid:12)(cid:12)(cid:12) ≈ (cid:12)(cid:12)(cid:12)(cid:12) S T dS T /dT (cid:12)(cid:12)(cid:12)(cid:12) . (15)In Appendix A we review some experimental measurements of the Soret coefficient where the total temperaturevariation ∆ T does not exceed a few degrees Kelvin and, thus, reasonably satisfy the above criterion.The second quantity that can be used to characterize thermophoretic response is the drift of individual particles h x i (12), or better, the drift velocity v = d h x i /dt . This measure is interesting for two reasons. First, we now havethe experimental means to measure single particle trajectories. Second, in the overdamped limit, the drift velocityis directly related to the rate of heat taken from the solvent by the particle [see Eq. (13)]. Similarly to − S T , apositive (negative) value of v/T ′ indicates thermophilic (thermophobic) response, but these quantities are different asapparent from the comparison of Eqs. (5) and (12). Importantly, the direction of the drift is set by both directions ofthe thermophoretic force and the direction of the mobility gradient. Obviously, part of the mobility spatial variationcan be attributed to the temperature gradient, but it is important to recall that coordinate-dependent mobility, µ ( x ),is also encountered in isothermal systems, i.e., in equilibrium situations. Indeed, our result Eq. (12) generalizes theexpression for the drift velocity in inhomogeneous isothermal solutions.Also suggested by Eq. (12) is that for µ ′ = 0, the temperature gradient causes a non-vanishing drift vanishing driftonly when C T = 0, i.e., only in the presence of a thermophoretic force, but not due to thermal collisions (fluctuations)that are also influenced by the temperature gradient. This can be understood by noting that the stochastic noiseterm in the Langevin equation depicting the dynamics of the particle has a zero mean, even for multiplicative (state-dependent) noise (see discussion in [27]).Finally, the third quantity defined here is the bias ∆ (14), measuring the probability difference of moving alongand against the temperature gradient. Similarly to the previously discussed measures, a positive (negative) value of∆ /T ′ may indicate thermophilic (thermophobic) response. From Eq. (14) we infer that the bias may be expressed as alinear combination of h x i and − S T and, thus, the value of this quantity is influenced by all three factors of asymmetrydiscussed in the work, namely the thermophoretic force, the spatial-dependence of the mobility, and thermal collisioneffect. Acknowledgments.
I thank Daan Frenkel for numerous insightful discussions and comments on the topic. Thiswork was supported by the Israel Science Foundation (ISF) through Grant No. 991/17.
Appendix A: Analysis of experimental data
We begin by noting that a key assumption in our theoretical analysis is the form of Eq. (4), stating a linear rela-tionship between the thermophoretic force and the temperature gradient. This form is consistent with the frequentlyused linear-response theory for non-equilibrium systems. As discussed in the main text, the strong variations of S T with temperature reported in many experimental studies [13, 14, 29] restrict the validity of the linear form Eq. (2)to small systems where the total temperature difference, ∆ T , applied across the experimental setup satisfy criterion(15). Reviewing the experimental data, it can be concluded that the linear approximation holds reasonably well inmany setups where ∆ T does not exceed a few degrees Kelvin. (Some noticeable exceptions: (i) Ref. [15] where the∆ T was as high as 30 K, but in that work S T was found to be temperature-independent. (ii) The measurements of S T for large colloidal particles of size 2 . × − µ m reported in [29] exhibiting exceptionally strong variations in S T overa temperature range smaller than 5 K which, in fact, calls for care in the interpretation of the experimental data.)The distance, ∆ x , across which the temperature difference, ∆ T (of order of a few degrees Kelvin), is applied, variesfrom h ∼ µ m in older experiments [29] to h ∼ µ m in more recent ones [14, 15]. Thus, the experimentalrange of the temperature gradient is roughly 3 × − − × − . As these experiments are conducted around roomtemperature T ≃ K , we find that l − T ≡ T ′ /T ∼ − − − µ m − . Furthermore, the range of experimentalvalues for the Soret coefficient varies from | S T | ∼ − K − for micellar solution, globular proteins and small colloidalparticles ( a ∼ − µ m) [3] to | S T | ∼ K − for large colloidal particles ( a ∼ . × − µ m) [29]. (A noticeableexception is ref. [14] where | S T | ∼ K − was measured for large colloidal particles of diameter a ∼ . µ m.) Recallingthat C T ≃ (1 − T S T ) (5), we can deduce from this relationship that the experimental range of the thermophoreticforce coefficient is − . C T . .The confinement of the particle in a thin slit between two plates leads to strong variations in the mobility dueto hydrodynamic interactions between the Brownian particle and the walls of the cell. The hydrodynamic effectovershadows the additional (non-equilibrium) contribution to µ ′ due to the temperature variation which is typicallynegligible because of the smallness of ∆ T . From theoretical considerations [23, 31] we can estimate that the relativevariations in the mobility, ∆ µ/µ ∼ a/h , where a is the diameter of the colloidal particle. Therefore, the inverse length l − µ ≡ µ ′ /µ ∼ a/h . Experimentally, colloids of diameter a ∼ . × − − . µ m have been studies, correspondingto a wide range of values l − µ ∼ − − − µ m − .Three quantities that characterize the thermophoretic response of a system are highlighted in the manuscript: S T , v (the drift velocity), and ∆ (the probability bias). These can be rescaled to allow direct comparison with C T . Wethus define the following dimensionless quantities1. The scaled negative Soret coefficient, − ˜ S T ≡ − T S T = C T − v ≡ v ( T /D T ′ ) = C T + l T /l µ .3. The scaled bias, ˜∆ ≡ ∆ p π/D t ( T /T ′ ) = C T − / l T / l µ ,where the length scales l T and l µ were defined in the previous two paragraphs. All of these quantities have the form Q = C T + A , implying that they do not change sign at exactly the same temperature like the thermophoretic forcecoefficient C T . As discussed extensively in the manuscript, the additional contribution to each quantity, A , arises fromboth a thermal collision effect (which is represented by the negative constants in the definitions of the scaled quantities)and from spatial variations in the mobility (the terms proportional to l − µ ). Let us look at a few experimental examplesin order to assess the relative importance of the additional contribution, A/C T , to the thermophoretic force.1. In experiments with charged micelles [13], the thermophoretic force coefficient was found to be of order | C T | . | S T | ∼ − K − ). The size of these micelles is of order of a fewtens of nanometers, and the cell size in the experiments h > µ m. Thus, l − T ∼ − µ m − , while l − µ ∼ − µ m − . We therefore conclude that in these classical experiments, the hydrodynamics effect is negligible, whilethe thermal collision effect is small but, nevertheless, important because the thermophoretic force is also fairlysmall.2. When colloidal particles of diameter a ∼ × − µ m are studied in similar diffusion cells, the thermophoreticforce coefficient is typically an order of magnitude larger, C T ∼ [13]. For larger colloidal particles of size a ∼ × − µ m, C T ∼ [29]. Thus, in these experiments, the additional contributions to the scaled quantitiesdefined above are vanisingly small: A/C T ≪ a ∼ × − µ m were also studies in ref. [5], but in a much narrower diffusioncell of height h ∼ µ m. Here we also have C T ∼ , but in this case l − T ∼ × − µ m − and l − µ ∼ × − µ m − . Thus, the sign of the thermophoretic force dominates the direction of movement, but the influence of thehydrodynamic effect on the drift and the bias may be felt close to the transition temperature from thermophilicto thermophobic response.4. In a recent experiment [15], a temperature-independent Soret coefficient S T ∼ . K − was measured for colloidalparticles of diameter a ∼ µ m, diffusing between plates with spacing h ∼ µ m and an unusually largetemperature difference ∆ T . K . In this setup, C T ∼ l − T ∼ − µ m − , and l − µ ∼ − µ m − . Thesevalues suggest that the thermophoretic force is the key factor in determining the diffusive behavior of the colloidalparticles. Collision and hydrodynamic effects are equally important and their influence is about 1-2 orders ofmagnitude weaker than that of the thermophoretic force.To conclude, in most of the above experimental examples, the magnitude of C T is at least one order of magnitudelarger than that of other contributions (denoted collectively by A ) over most of the investigated temperature range.Collision effect has influence on the Soret coefficient of small particles and micelles of diameter not larger than 5 × − µ m, especially close to the transition temperature from thermophilic to thermophobic behavior (i.e., when C T becomessmall). 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