Different measures for characterizing the motion of molecules along a temperature gradient

TL;DR

This paper analyzes the motion of Brownian particles in temperature gradients, deriving time-dependent probability distributions and examining how different measures can lead to ambiguous interpretations of thermophoretic behavior.

Contribution

It introduces a time-dependent analysis of particle motion in temperature gradients, highlighting ambiguities in characterizing thermophoretic responses using traditional measures.

Findings

Mean displacement and bias can have opposite signs to the Soret coefficient.

Time-dependent PDFs reveal complex thermophoretic dynamics.

Different measures influence the interpretation of thermophoretic behavior.

Abstract

We study the motion of a Brownian particle in a medium with inhomogeneous temperature. In the overdamped regime of low Reynolds numbers, the probability distribution function (PDF) of the particle is obtained from the van Kampen diffusion equation [J. Phys. Chem. Solids {\bf 49}, 673 (1988)]. The thermophoretic behavior is commonly described by the Soret coefficient - a parameter which can be calculated from the steady-state PDF. Motivated by recent advances in experimental methods for observing and analyzing single nano-particle trajectories, we here consider the time-dependent van Kampen equation from which the temporal evolution of the PDF of individual particles can be derived. We analytically calculate the PDF describing dynamics driven by a generalized thermophoretic 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 Soret coefficient, which causes ambiguity in the distinction between thermophilic and thermophobic response (i.e., migration in and against the direction of the temperature gradient). The different factors determining the thermophoretic response and their influence on each measure are discussed.

Equations

$$\overrightarrow{J}=-D\overrightarrow{\nabla }c-c{D}_T\overrightarrow{\nabla }T.$$

$$\overrightarrow{\nabla }c=-c{S}_T\overrightarrow{\nabla }T,$$

$${\partial }_{t}P(x,t)=-{\partial }_{x}J(x,t)={\partial }_x\left\{\mu (x){\partial }_x\left[k_{B}T(x)P(x,t)\right]-\mu (x)f(x)P(x,t)\right\},$$

$$f=C_T{k}_B{T}^{\prime },$$

$$S_T=\frac{1-C_T}{T}.$$

$$P_s(x)\sim \frac{1}{T(x)}\exp \left[{\int }^x\frac{f(y)}{k_{B}T(y)}dy\right],$$

$$P_s(x)\sim 1/T(x).$$

$$\mu (x)\simeq {\mu }_0+{\mu }^{\prime }x,$$

$${\partial }_{t}P(x,t)=D_0\left\{{\partial }_x\left(1+\frac{x{\mu }^{\prime }}{{\mu }_0}\right){\partial }_x\left(1+\frac{x{T}^{\prime }}{T_0}\right)-C_T\frac{x{T}^{\prime }}{T_0}{\partial }_x\right\}P(x,t),$$

$$P(x,t)=G(x,t)\left[1+xH\left(\frac{x^2}{D_{0}t}\right)\right],$$

$$P(x,t)=\frac{\exp (-x^2/4D_{0}t)}{\sqrt{4\pi D_{0}t}}\left[1+x\left\{\frac{T^{\prime }}{T_0}\left(-\frac{3}{4}+\frac{C_T}{2}+\frac{x^2}{8D_{0}t}\right)+\frac{{\mu }^{\prime }}{{\mu }_0}\left(-\frac{1}{4}+\frac{x^2}{8D_{0}t}\right)\right\}\right],$$

$$⟨x⟩={\int }_0^{\infty }xP(x,t)dx=\left(C_T\frac{T^{\prime }}{T_0}+\frac{{\mu }^{\prime }}{{\mu }_0}\right)D_{0}t.$$

$$\frac{d⟨E_k(t)-E_k(t=0)⟩}{dt}=\frac{k_B{T}^{\prime }}{2}\frac{d⟨x⟩}{dt}=\frac{k_B{T}^{\prime }}{2}\left(C_T\frac{T^{\prime }}{T_0}+\frac{{\mu }^{\prime }}{{\mu }_0}\right)D_0.$$

$$\Delta (t)\equiv {\int }_0^{\infty }P(x,t)dx-{\int }_{-\infty }^{0}P(x,t)dx=\sqrt{\frac{D_{0}t}{\pi }}\left[\frac{T^{\prime }}{2T_0}\left(2C_T-1\right)+\frac{{\mu }^{\prime }}{2{\mu }_0}\right]=\frac{⟨x⟩}{\sqrt{4\pi D_{0}t}}-\sqrt{\frac{D_{0}t}{4\pi }}{T}^{\prime }{S}_T,$$

$$\Delta T=T^{\prime }\Delta x\ll \left|\frac{C_T}{d{C}_T/dT}\right|\approx \left|\frac{S_T}{d{S}_T/dT}\right|.$$

Full text

I Introduction

The motion of molecules induced by a temperature gradient is commonly referred to as thermophoresis, thermodiffusion or the Soret effect. Since its discovery in liquid mixtures more than a century and half ago ludwig ; soret , the phenomenon of thermophoresis has been experimentally observed in aqueous solutions containing colloidal particles, micelles, polymers, proteins, and DNA molecules (see extensive review in piazza08 ). Several studies have shown thermophoresis to be a promising tool for manipulating and concentrating biomolecules in solutions jeon97 ; duhr06 ; jiang09 , which has even led to the speculations that it may play a role in the accumulation of nucleotides required for molecular evolution of early life baaske07 .

In this work we theoretically study the thermal diffusion of colloidal particles which, in general, is a much stronger effect than thermophoresis in simple molecular mixtures. The relevant length and time scales of the colloidal particles are orders of magnitude larger than those of the embedding solvent, and hence the solvent may be treated as an effective medium. The thermal motion of the colloidal particle is driven by stresses induced on its surface by the surrounding fluid ruckenstein81 ; anderson89 ; brenner05 . These forces are balanced by viscous drag forces when the particle attains a steady state velocity burelbach18 . Thermophoresis can therefore be treated as a mass transport process which, for dilute suspensions (low concentration, $c$) can be phenomenologically described by the continuity equation $\partial_{t}c=-\overrightarrow{\nabla}\cdot\overrightarrow{J}$, with the particle flux $\overrightarrow{J}$ given by grootmazur

[TABLE]

The first term on the r.h.s. of Eq. (1) describes regular diffusion due to concentration gradients, where $D$ is the Fickian diffusion coefficient. The second term describes an additional contribution to the flux resulting from the temperature gradient, $\overrightarrow{\nabla}T$, with $D_{T}$ termed the thermal diffusion coefficient. When a closed system reaches a steady state, the flux vanishes and a concentration gradient is established that satisfies

[TABLE]

where $S_{T}=D_{T}/D$ is called the Soret coefficient. For $S_{T}>0$, the colloids tend to accumulate on the colder side of the 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 its sign close to room temperature in many colloidal systems iacopini06 . Experimental measurements of $S_{T}$ are typically based on the application of a thermal gradient in a diffusion cell and the use indirect optical methods to quantify the concentration gradients induced by thermal diffusion piazza08 . Recently, it became possible to measure thermophoretic forces on a single colloidal particle confined in sub-micrometer regions with a nearly uniform temperature gradient (and an overall small temperature difference) helden15 . Moreover, we can now study not only the steady-state probability distribution of the particle, but to also follow its trajectory to relaxation burelbach17 . These advances in experimental methods call 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 the particle, $P(x,t)$, one has to consider the Langevin equation of the dynamics or the corresponding Fokker-Planck equation. These equations capture both the inertial short- and dissipative long-time regimes of the dynamics. In practice, 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 an adiabatic elimination process of the fast relaxing momentum degree of freedom. The derivation was carried out by van Kampen for different models of diffusion in inhomogeneous media vankampen88 . One of the cases considered by van Kampen is of Brownian particle in a system with spatially-varying temperature. The equation corresponding to this model is:

[TABLE]

where $f$ is the mechanical force acting on the particle, while $T(x)$ and $\mu(x)$ denote, respectively, the coordinate-dependent temperature and mobility. The latters are related to the coordinate-dependent diffusion coefficient, $D(x)$, via Einstein’s relation $D(x)=k_{B}T(x)\mu(x)$, with $k_{B}$ denoting the Boltzmann constant vankampen88 . As noted by van Kampen, this is a diffusion equation which does not follow neither Itô ito nor Stratonovich stratonovich prescriptions for overdamped Brownian 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 the particle, $p$, is always at equilibrium with the local temperature $T(x)$, i.e., follows the Maxwell-Boltzmann distribution $\rho(p|x)\sim T(x)^{-1/2}\exp[-p^{2}/2mk_{B}T(x)]$ (where $m$ denotes the mass of the particle). The local thermodynamics equilibrium (LTE) grootmazur approximation is justified when $l_{b}|\overrightarrow{\nabla}T|/T\ll 1$, where $l_{b}$ is the ballistic distance characterizing the crossover between the inertial and diffusive regimes. Mathematically, the overdamped limit corresponds to $l_{b}\rightarrow 0$.

The force $f$ in Eq. (3) includes both contributions from the thermophoretic force, as well externally applied forces like gravity which can be minimized by density-matching the colloid with the solvent. We will henceforth ignore all forces except for the thermophoretic one. Moreover, single particle experiments are conducted in small systems where the applied temperature difference may be as small as a few degrees Kelvin. Assuming that the temperature gradient, $T^{\prime}=dT/dx$, and the thermophoretic force are uniform throughout the small system, one may phenomenologically write that the thermophoretic force is given by widder89

[TABLE]

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 fayolle08 ; sancho18

[TABLE]

From Eq. (5) we conclude that in the absence of a mechanical thermophoretic force ($C_{T}=0$), the Soret coefficient does not vanish ($S_{T}\neq 0$). 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 $\partial_{t}P(x,t)=0$, and from van Kampen equation (3) one can easily deduce that the stationary solution is

[TABLE]

which, in the absence of a mechanical force ($f(x)=0$), reduces to

[TABLE]

In ref. vankampen88 , van Kampen notes that he has no simple explanation for the prefactor $1/T(x)$ in Eq. (6); however, in the special case $f(x)=0$, Eq. (7) was nicely rationalized by Fayolle et al. fayolle08 . They noted that the mechanical thermophoretic 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 steady state, the pressure of this ideal gas, $\Pi=c(x)k_{B}T(x)$, must be uniform (or otherwise, the gradient pressure force would act on the gas and change its distribution). Eq. (7) then means that in the absence of a mechanical thermophoretic force, the ideal gas thermal collisions induce a steady-state distribution that is higher on the colder than on the hotter side. The associated Soret coefficient $S_{T}=1/T>0$ reflects the thermophobic nature of the thermal collisions (which are stronger on the hotter side, thus “pushing” the particle to the colder side). The Soret effect is associated with the interaction term in Eq. (5) and, in practice, this term is typically larger than the ideal gas term $|C|\gg 1$, with the exception 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

[TABLE]

where $\mu_{0}$ is the mobility in the middle of the cell at $x=0$ and $\mu^{\prime}=d\mu/dx$. We note here that the spatial variations in $\mu(x)$ can, in general, be further divided into two parts - those arising from the temperature-dependence of the fluid viscosity gutmann52 and those also encountered at equilibrium isothermal systems, for instance due to hydrodynamic interactions between the colloidal particle and the walls of the container happelbrenner . As we will see below, the particle’s drift (to be mathematically defined later) depends on both $T^{\prime}$ and $\mu^{\prime}$, and one must keep in mind that these two gradients are 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 $\mu^{\prime}$ does not necessarily vanish when $T^{\prime}=0$.

Using Eqs. (4) and (8) [together with the expansion $T(x)\simeq T_{0}+T^{\prime}x$] in the van Kampen equation (3), yields the following form

[TABLE]

where $D_{0}=D(x=0)=k_{B}T_{0}\mu_{0}$. This is the van Kampen equation in the limit when (i) all forces besides the thermophoretic one are ignored, and (ii) the system is sufficiently small to justify the linear approximations of $T(x)$ and $\mu(x)$. (iii) Another assumption implied in Eq. (9) is the form (4) for the thermophoretic force.

III The probability distribution function

In a homogeneous ($\mu^{\prime}=0$) isothermal system ($T^{\prime}=0$), Eq. (9) reduces to a simple diffusion equation, the solution of which takes the Gaussian form $P(x,t)=\exp(-x^{2}/4D_{0}t)/\sqrt{4\pi D_{0}t}\equiv G(x,t)$ [assuming Dirac delta-function initial condition $P(x,0)=\delta(x)$]. For a system under a small temperature difference ($xT^{\prime}/T_{0}\ll 1$) and limited changes in the mobility ($x\mu^{\prime}/\mu_{0}\ll 1$), we may seek a linear approximate solution of the form

[TABLE]

where $H$ is some function that can be determined in the following simple manner: (i) Write $H$ in Eq. (10) as a series expansion in the argument $y=(x^{2}/D_{0}t)$: $H=\sum_{i=0}^{\infty}a_{n}y^{n}$, then (ii) determine the coefficients of the expansion by substituting Eq. (10) in van Kampen equation (9), and by comparing terms of similar order in $y$ on both sides of the equation. In this process of determining $H(y)$, we ignore the terms that are non-linear in $x$. We find, $a_{0}=(-3/4+C_{T}/2)(T^{\prime}/T_{0})-1/4(\mu^{\prime}/\mu_{0})$, $a_{1}=1/8(T^{\prime}/T_{0})+1/8(\mu^{\prime}/\mu_{0})$, and $a_{n}=0$ for $n>1$, and thus write

[TABLE]

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, $\langle x\rangle$, and from the PDF (11), we find that

[TABLE]

We notice that the drift does not necessarily have the same sign as $C_{T}$, which means that the average displacement of the particle is not necessarily in the same direction as the thermophoretic force. The reason for this remarkable result is an additional contribution to the drift originating from spatial dependence of the mobility. In general, the mobility of simple liquids increases with temperature, while gases exhibit an opposite trend and have mobility that decreases approximately like the square root of the temperature gillepsie . As noted earlier [see discussion after Eq. (8)], non-thermal effects may also contribute to $\mu^{\prime}$. Indeed, it is well known that drift is also observed in isothermal systems with non-uniform mobilities volpe16 . This equilibrium phenomenon has been termed “spurious drift”, which is misleading since it is a real effect farago14 . In the isothermal case ($T^{\prime}=0$), we can use Einstein relation and Eq. (8) to write Eq. (12) in the more common form, $\langle x\rangle/t=D^{\prime}$ farago14b , relating the drift velocity and the spatial derivative of the diffusion coefficient. Thus, our result Eq. (12) generalizes the well-known expression for the drift of Brownian particles 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. Within this approximation, the mean kinetic energy of a particle found at some coordinate $x$ is related to the local temperature via the equipartition theorem $\langle E_{k}\rangle_{x}=\langle mv^{2}/2\rangle_{x}=k_{B}T(x)/2$, where $\langle\cdots\rangle_{x}$ denotes average at a given $x$. Taking the average with respect to $x$ and using Eq. (12) gives

[TABLE]

For $\mu^{\prime}=0$ (constant mobility), the particle is heated on average (i.e., gains kinetic energy) when $C_{T}>0$, i.e., when the thermophoretic force drives the particle to the high temperature side, and vice versa. This, however, may not be true when the mobility varies in space, in which case it is the sign of $C_{T}+(T_{0}\mu^{\prime}/T^{\prime}\mu_{0})$ rather then the direction of the 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 $J(x,t)=-D(x)\partial_{x}P(x,t)$. A closed 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 time across 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” in equilibrium isothermal systems lancon01 . Here, we consider dynamics in an open system with time-dependent flux. The tendency of particles to migrate favorably to one side may be characterized by the flux at the origin $J_{0}\equiv J(x=0,t)=\sqrt{D_{0}/(\pi t)}\left[\left(T^{\prime}/8T_{0}\right)\left(3-2C_{T}\right)+\left(\mu^{\prime}/8\mu_{0}\right)\right]$. The flux at the origin causes a “bias”, i.e., a difference in the probability of finding the particle in the “hotter” and “colder” sides relative to its initial location. Assuming (without loss of generality) that $T^{\prime}>0$, the bias, $\Delta(t)$, is defined by

[TABLE]

with the drift, $\langle x\rangle$, and the Soret coefficient, $S_{T}$, given by Eqs. (12) and (5), respectively. Depending on the values of $T_{0}\mu^{\prime}/T^{\prime}\mu_{0}$ and $C_{T}$, it now becomes clear that while $\Delta$, $\langle x\rangle$, and $-S_{T}$ can all be used to characterize the response of colloidal particles to a temperature gradient, these quantities describe different features of the Soret effect, and may occasionally have different signs.

V Discussion and summary

Motivated by recent single-molecule experiments for studying the behavior of macromolecules along a temperature gradient, we considered here the question of Brownian dynamics of a colloidal particle in a non-isothermal fluid. In the 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 ($T^{\prime}x/T_{0}\ll 1$ and $\mu^{\prime}x/\mu_{0}\ll 1$), we considered the linear (in $x$) version of van Kampen equation (9) and analytically derived the solution for delta-function initial condition (11). The asymmetric PDF characterizes the general tendency of the particle to migrate in the direction of the thermophoretic force caused by the temperature gradient. However, the thermophoretic force is not the only factor determining the direction of the motion, and we have identified three different measures for the thermodiffusive response of the colloidal particle. The first measure is the Soret coefficient $S_{T}$ (5), relating the concentration and temperature gradients in steady state. The Soret coefficient has been traditionally used to distinguish 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 not solely dictated by the direction of the thermophoretic force. The origin of the discrepancy are the thermal collisions which set a concentration gradient opposite to the temperature gradient. In fact, in some recent experiments on colloidal systems it has been found that $S_{T}$ exhibits a strong temperature-dependence and tends to change its sign in the vicinity of room temperature. Moreover, the magnitude of the Soret coefficient in many of these experiments is found to be of the order of $0.01-1K^{-1}$ iacopini06 ; braibanti08 . These findings indicate that (i) the effect of the thermal collisions may sometimes 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 of the thermophoretic force, there is no clear explanation for its temperature sensitivity of $C_{T}$ which is likely dependent on numerous factors, e.g., the thermal expansivity of the solvent iacopini06 , the surface functionality burelbach17 and size duhr06 of the colloidal particle, and electrostatic effects putnam07 . In order to understand this behavior of $C_{T}$ one must consider a microscopic model that takes into account some of these factor (see, e.g., the theoretical discussion in ref. sancho18 ). This is beyond the scope of the phenomenological discussion presented herein; however, in light of the pronounced temperature-dependence of $S_{T}$, it must be reemphasized that our derivation assumes that the thermophoretic force is phenomenologically given by Eq. (4), namely assuming non-equilibrium linear-response. The same linear form has been considered in other works (see, e.g., fayolle08 ), and it is consistent with the linearity of our solution for the PDF (11) with respect to $T^{\prime}$. More generally, the variations of $C_{T}$ with $T$ can be accounted for by a Taylor expansion around $T_{0}$: $C_{T}=C_{T}(T_{0})+(dC_{T}/dT)\Delta T+\cdots=C_{T}(T_{0})+(dC_{T}/dT)T^{\prime}\Delta x+\cdots$, which shows that the linear approximation is valid if the total temperature difference across the experimental cell $\Delta T=T^{\prime}\Delta x$ is sufficiently small, i.e., if the the size of the experimental setup, $\Delta x$, and the temperature gradient, $T^{\prime}$, satisfy

[TABLE]

In Appendix A we review some experimental measurements of the Soret coefficient where the total temperature variation $\Delta T$ 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 $\langle x\rangle$ (12), or better, the drift velocity $v=d\langle x\rangle/dt$. This measure is interesting for two reasons. First, we now have the experimental means to measure single particle trajectories. Second, in the overdamped limit, the drift velocity is directly related to the rate of heat taken from the solvent by the particle [see Eq. (13)]. Similarly to $-S_{T}$, a positive (negative) value of $v/T^{\prime}$ indicates thermophilic (thermophobic) response, but these quantities are different as apparent from the comparison of Eqs. (5) and (12). Importantly, the direction of the drift is set by both directions of the thermophoretic force and the direction of the mobility gradient. Obviously, part of the mobility spatial variation can be attributed to the temperature gradient, but it is important to recall that coordinate-dependent mobility, $\mu(x)$, is also encountered in isothermal systems, i.e., in equilibrium situations. Indeed, our result Eq. (12) generalizes the expression for the drift velocity in inhomogeneous isothermal solutions.

Also suggested by Eq. (12) is that for $\mu^{\prime}=0$, the temperature gradient causes a non-vanishing drift vanishing drift only when $C_{T}\neq 0$, 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 noise term in the Langevin equation depicting the dynamics of the particle has a zero mean, even for multiplicative (state-dependent) noise (see discussion in farago14b ).

Finally, the third quantity defined here is the bias $\Delta$ (14), measuring the probability difference of moving along and against the temperature gradient. Similarly to the previously discussed measures, a positive (negative) value of $\Delta/T^{\prime}$ may indicate thermophilic (thermophobic) response. From Eq. (14) we infer that the bias may be expressed as a linear combination of $\langle x\rangle$ and $-S_{T}$ and, thus, the value of this quantity is influenced by all three factors of asymmetry discussed in the work, namely the thermophoretic force, the spatial-dependence of the mobility, and thermal collision effect.

Acknowledgments. I thank Daan Frenkel for numerous insightful discussions and comments on the topic. This work 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 relationship between the thermophoretic force and the temperature gradient. This form is consistent with the frequently used 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 iacopini06 ; braibanti08 ; helden15 restrict the validity of the linear form Eq. (2) to small systems where the total temperature difference, $\Delta T$, applied across the experimental setup satisfy criterion (15). Reviewing the experimental data, it can be concluded that the linear approximation holds reasonably well in many setups where $\Delta T$ does not exceed a few degrees Kelvin. (Some noticeable exceptions: (i) Ref. burelbach17 where the $\Delta T$ 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 $2.5\times 10^{-1}$ $\mu{\rm m}$ reported in braibanti08 exhibiting exceptionally strong variations in $S_{T}$ over a temperature range smaller than $5K$ which, in fact, calls for care in the interpretation of the experimental data.)

The distance, $\Delta x$, across which the temperature difference, $\Delta T$ (of order of a few degrees Kelvin), is applied, varies from $h\sim 500$ $\mu{\rm m}$ in older experiments braibanti08 to $h\sim 10$ $\mu{\rm m}$ in more recent ones helden15 ; burelbach17 . Thus, the experimental range of the temperature gradient is roughly $3\times 10^{-3}-3\times 10^{-1}$. As these experiments are conducted around room temperature $T_{0}\simeq 300K$, we find that $l_{T}^{-1}\equiv T^{\prime}/T_{0}\sim 10^{-5}-10^{-3}$ $\mu{\rm m}^{-1}$. Furthermore, the range of experimental values for the Soret coefficient varies from $|S_{T}|\sim 10^{-2}K^{-1}$ for micellar solution, globular proteins and small colloidal particles ($a\sim 10^{-2}$ $\mu{\rm m}$) piazza08 to $|S_{T}|\sim 1K^{-1}$ for large colloidal particles ($a\sim 2.5\times 10^{-1}$ $\mu{\rm m}$) braibanti08 . (A noticeable exception is ref. helden15 where $|S_{T}|\sim 50K^{-1}$ was measured for large colloidal particles of diameter $a\sim 2.5$ $\mu{\rm m}$.) Recalling that $C_{T}\simeq(1-T_{0}S_{T})$ (5), we can deduce from this relationship that the experimental range of the thermophoretic force coefficient is $-10^{3}\lesssim C_{T}\lesssim 10^{3}$.

The confinement of the particle in a thin slit between two plates leads to strong variations in the mobility due to hydrodynamic interactions between the Brownian particle and the walls of the cell. The hydrodynamic effect overshadows the additional (non-equilibrium) contribution to $\mu^{\prime}$ due to the temperature variation which is typically negligible because of the smallness of $\Delta T$. From theoretical considerations happelbrenner ; benesch03 we can estimate that the relative variations in the mobility, $\Delta\mu/\mu_{0}\sim a/h$, where $a$ is the diameter of the colloidal particle. Therefore, the inverse length $l_{\mu}^{-1}\equiv\mu^{\prime}/\mu_{0}\sim a/h^{2}$. Experimentally, colloids of diameter $a\sim 2.5\times 10^{-2}-2.5$ $\mu{\rm m}$ have been studies, corresponding to a wide range of values $l_{\mu}^{-1}\sim 10^{-6}-10^{-2}$ $\mu{\rm m}^{-1}$.

Three quantities that characterize the thermophoretic response of a system are highlighted in the manuscript: $S_{T}$, $v$ (the drift velocity), and $\Delta$ (the probability bias). These can be rescaled to allow direct comparison with $C_{T}$. We thus define the following dimensionless quantities

The scaled negative Soret coefficient, $-\tilde{S_{T}}\equiv-T_{0}S_{T}=C_{T}-1$. 2. 2.

The scaled drift velocity, $\tilde{v}\equiv v(T_{0}/D_{0}T^{\prime})=C_{T}+l_{T}/l_{\mu}$. 3. 3.

The scaled bias, $\tilde{\Delta}\equiv\Delta\sqrt{\pi/D_{0}t}\,(T_{0}/T^{\prime})=C_{T}-1/2+l_{T}/2l_{\mu}$,

where the length scales $l_{T}$ and $l_{\mu}$ 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 force coefficient $C_{T}$. As discussed extensively in the manuscript, the additional contribution to each quantity, $A$, arises from both 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 $l_{\mu}^{-1}$). Let us look at a few experimental examples in order to assess the relative importance of the additional contribution, $A/C_{T}$, to the thermophoretic force.

In experiments with charged micelles iacopini06 , the thermophoretic force coefficient was found to be of order $|C_{T}|\lesssim 10$ within the experimental temperature range ($|S_{T}|\sim 10^{-2}$ $K^{-1}$). The size of these micelles is of order of a few tens of nanometers, and the cell size in the experiments $h>100$ $\mu{\rm m}$. Thus, $l_{T}^{-1}\sim 10^{-2}$ $\mu{\rm m}^{-1}$, while $l_{\mu}^{-1}\sim 10^{-5}$ $\mu{\rm m}^{-1}$. We therefore conclude that in these classical experiments, the hydrodynamics effect is negligible, while the thermal collision effect is small but, nevertheless, important because the thermophoretic force is also fairly small. 2. 2.

When colloidal particles of diameter $a\sim 5\times 10^{-2}$ $\mu{\rm m}$ are studied in similar diffusion cells, the thermophoretic force coefficient is typically an order of magnitude larger, $C_{T}\sim 10^{2}$ iacopini06 . For larger colloidal particles of size $a\sim 2\times 10^{-1}$ $\mu{\rm m}$, $C_{T}\sim 10^{3}$ braibanti08 . Thus, in these experiments, the additional contributions to the scaled quantities defined above are vanisingly small: $A/C_{T}\ll 1$. 3. 3.

Large colloidal particles of size $a\sim 2\times 10^{-1}$ $\mu{\rm m}$ were also studies in ref. duhr06 , but in a much narrower diffusion cell of height $h\sim 10$ $\mu{\rm m}$. Here we also have $C_{T}\sim 10^{3}$, but in this case $l_{T}^{-1}\sim 3\times 10^{-4}$ $\mu{\rm m}^{-1}$ and $l_{\mu}^{-1}\sim 5\times 10^{-3}$ $\mu{\rm m}^{-1}$. Thus, the sign of the thermophoretic force dominates the direction of movement, but the influence of the hydrodynamic effect on the drift and the bias may be felt close to the transition temperature from thermophilic to thermophobic response. 4. 4.

In a recent experiment burelbach17 , a temperature-independent Soret coefficient $S_{T}\sim 0.2K^{-1}$ was measured for colloidal particles of diameter $a\sim 1$ $\mu{\rm m}$, diffusing between plates with spacing $h\sim 10$ $\mu{\rm m}$ and an unusually large temperature difference $\Delta T\lesssim 30K$. In this setup, $C_{T}\sim 50$, $l_{T}^{-1}\sim 10^{-2}$ $\mu{\rm m}^{-1}$, and $l_{\mu}^{-1}\sim 10^{-2}$ $\mu{\rm m}^{-1}$. These values suggest that the thermophoretic force is the key factor in determining the diffusive behavior of the colloidal particles. Collision and hydrodynamic effects are equally important and their influence is about 1-2 orders of magnitude 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 magnitude larger 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 $5\times 10^{-2}$ $\mu{\rm m}$, especially close to the transition temperature from thermophilic to thermophobic behavior (i.e., when $C_{T}$ becomes small). The hydrodynamic interactions between the Brownian particle and the walls of the diffusion cell may influence the drift behavior of large colloidal particles ($a\gtrsim 1$ $\mu{\rm m}$) in small cells ($h\sim 10$ $\mu{\rm m}$).