Thermo-osmotic slip flows around a thermophoretic microparticle
characterized by optical trapping of tracers

The temperature of the fluid and that inside the target particle are denoted by $T_{f}$ and $T_{s}$, respectively. Since the thermal Péclet number $\mathit{Pe}_{T}$ is small, i.e., $\mathit{Pe}_{T}\ll 1$, we can neglect the convection term; the temperature fields $T_{f}$ and $T_{s}$ obey the steady heat conduction equations. For instance, when the reference length is $10$ µm, the reference speed is $1$ µm, and the thermal diffusivity is $1.4\times 10^{-7}$ m² s^-1, we have $\mathit{Pe}_{T}=7\times 10^{-5}$.

The condition for the temperature imposed at infinity is $T_{f}\to T_{0}+\alpha x$ $(r\equiv|\bm{x}|\to\infty)$ with a constant temperature gradient $\alpha(>0)$. Note that the $x$ direction corresponds to $\theta=0$ in Fig. 1(c), and $T_{0}$ is a reference temperature at the center of the target particle. At the surface of the target particle $r=a(=d/2)$, the temperature and the heat flux in the normal direction to the sphere surface are continuous. Then, we readily obtain

where $\alpha a$ is a characteristic temperature difference over a distance $a$; $\kappa_{f}$ and $\kappa_{s}$ are the thermal conductivity of the fluid and the target particle, respectively; $\xi_{f}$ and $\xi_{s}$ are non-dimensional parameters related with the thermal conductivity. In this paper, the temperature dependence of thermal conductivity is not considered, i.e., the constants $\kappa_{f}=0.6$ W m^-1 K^-1, $\kappa_{s}=0.2$ W m^-1 K^-1, $\xi_{f}\approx 0.286$, and $\xi_{s}\approx 1.286$ are used.

The temperature field (1) induces a fluid flow through a slip boundary condition, that is, the flow velocity vector $\bm{u}$ satisfies, on the sphere surface,

where $\bm{e}_{r}$ and $\bm{e}_{\theta}$ are the unit vectors in the $r$ and $\theta$ directions, respectively, and $K$ is a slip coefficient; $K$ is related to a thermo-osmotic coefficient $\chi$ [1, 10] as $\chi=-KT_{f}|_{r=a}$. Note that $\bm{e}_{\theta}\cdot\nabla T_{f}=\bm{e}_{\theta}\cdot\nabla T_{s}$ at $r=a$. Assuming that the Reynolds number $\mathit{Re}$ is small, i.e., $\mathit{Re}\ll 1$ (e.g., a kinematic viscosity of $\sim 10^{-6}$ m² s^-1 leads to $\mathit{Re}\sim 10^{-5}$), the flow field $\bm{u}$ satisfies the Stokes equation, which can be solved explicitly under the boundary condition (2) and the condition imposed at infinity, namely, $\bm{u}=0$ at $r\to\infty$. The results are

where $p$ is the pressure, $\alpha K$ is the product of the temperature gradient and the slip coefficient, namely, the characteristic speed of thermally-induced flow; $\eta$ is the viscosity of the fluid. In particular, on the sphere surface, we have

where the sign of $u_{s}$ is chosen so that $u_{s}$ is positive when the slip flow is induced in the negative $\theta$ direction, i.e., the hotter side of the target particle. When the thermal conductivity satisfies $\kappa_{s}=\kappa_{f}$, it leads to $\xi_{s}=1$ and thus $u_{s}=-K\alpha$. On the other hand, when the thermal conductivity of the target is huge, e.g., the case of a metal particle, we have $\kappa_{s}\gg\kappa_{f}$ that yields $\xi_{s}\ll 1$, resulting in no slip flows on the sphere surface.

The fluid force $F_{i}$ $(i=x,\,y,\,z)$ acting on the target particle is obtained from the surface integral $F_{i}=\int p_{ij}n_{j}\mathrm{d}S$, where $p_{ij}$ denotes the stress tensor for a Newtonian fluid, $n_{i}$ is a unit vector on the surface pointing to the fluid, $\mathrm{d}S$ is an area element, and the integral is carried out over the sphere surface $r=a$. The $x$ component $F_{x}$ is then readily obtained as $F_{x}=4\pi a\eta\xi_{s}K\alpha$ using Eq. (3). Therefore, in this model, the $x$ component of the thermophoretic velocity of the target particle, $v_{T}^{\mathrm{target}}$, is computed from the balance between the force $F_{x}$ and the Stokes drag $6\pi a\eta v_{T}^{\mathrm{target}}$, that is, $v_{T}^{\mathrm{target}}=(2/3)\xi_{s}K\alpha$ $[=-(2/3)u_{s}]$. This means that the positive $u_{s}$ (i.e., a slip flow toward the hot) induces the motion of the target particle toward the cold, $v_{T}^{\mathrm{target}}<0$.

Recall that, in general, the thermophoretic velocity $\bm{v}_{T}$ of particles is given by $\bm{v}_{T}=-D_{T}\nabla T_{f}$ phenomenologically, where $D_{T}$ is called a thermophoretic mobility [22, 23]. In this equation, the term $\nabla T_{f}$ should be understood as the temperature gradient at infinity, that is, the effect of temperature deformation due to the presence of the target particle is confined in the mobility $D_{T}$. Therefore, by noting that $v_{T}^{\mathrm{target}}=\bm{e}_{x}\cdot\bm{v}_{T}$ and $\bm{e}_{x}\cdot\nabla T_{f}=\alpha$ with $\bm{e}_{x}$ a unit vector in the $x$ direction, the thermophoretic mobility of the target, $D_{T}^{\mathrm{target}}$, is obtained as,

which is the relation between $D_{T}^{\mathrm{target}}$ and the slip coefficient $K$. Experimentally, the slip coefficient $K$ can be estimated from (5) through the evaluation of $D_{T}^{\mathrm{target}}$, i.e., $v_{T}^{\mathrm{target}}$ and $\alpha$.

Now, suppose that the tracers are confined in the circular path $r=R$ because of the trapping laser (see, Fig. 1(c)). The number density of the tracers is denoted by $c$. The velocity of the tracer, $\bm{v}$, is the sum of the flow velocity $\bm{u}$ and the thermophoretic velocity $\bm{v}_{T}^{\mathrm{tracer}}\equiv-D_{T}^{\mathrm{tracer}}\nabla T_{f}$ with $D_{T}^{\mathrm{tracer}}$ the thermophoretic mobility of the tracer. In this equation, $\nabla T_{f}$ should be interpreted as a local quantity due to the smallness of the tracer, i.e., $\nabla T_{f}$ is computed from (1a). Then, the tracer distribution $c$ satisfies a steady drift-diffusion equation with a drift velocity $\bm{v}\equiv\bm{u}+\bm{v}_{T}^{\mathrm{tracer}}$. The $\theta$ component of $\bm{v}$ can be computed explicitly as $\bm{v}\cdot\bm{e}_{\theta}=-v^{\ast}\sin\theta$, where $v^{\ast}$ is the flow magnitude in the direction toward hotter region, i.e.,

Because of the scanning trapping laser at $r=R$, we assume that the tracer’s radial motion is suppressed and only the motion in the $\theta$ direction is significant. Further assuming that there is no steady flux of the tracers, the drift-diffusion equation is reduced to the balance between the drift and diffusion in the $\theta$ direction at $r=R$:

where $D$ is the diffusion coefficient of the trapped tracers. Equation (7) can be solved explicitly as

where $c_{0}$ is a normalization constant and the pre-factor $\gamma$ (8b) is the ratio between the magnitude of the drift velocity $v^{\ast}$ and the diffusive speed $D/R$. Note that $\gamma$ depends on the radius of the circular path $R$, the thermophoretic mobility $D_{T}^{\mathrm{target}}$ (or $K$) and $D_{T}^{\mathrm{tracer}}$, and the temperature gradient $\alpha$. When $\gamma>0$ (or $\gamma<0$), the number density of the tracers $c$ takes maximum at $\theta=0$ (or $\pi$), i.e., the hotter (or colder) side of the target particle. When $|\gamma|\approx 0$, the diffusion dominates and $c$ tends to be distributed uniformly on the circular path.

Model (8) is used to fit and analyze the experimental results. Technical details of the fitting procedure are given in the Appendix C. Once $\gamma$, $D_{T}^{\mathrm{tracer}}$, and $\alpha$ are obtained experimentally, one can evaluate the slip coefficient $K$ using Eqs. (6) and (8b). Later, we will compare $K$ thus obtained with that estimated in the method described in Sec. II.2 previously.

Before going into the main part of experimental results, we show the results of some control experiments.

Let us describe the main result of the present paper, the thermally-induced flow around the target particle. Firstly, the results for a target particle with a carboxylate-modified surface are presented. The zeta potential $\zeta$ of target particles is measured from the electrophoretic mobility (ELSZ-2000Z, Otsuka Electronics) as $\zeta=-46$ mV. Figure 5(a) shows the schematic of the situation presented in panels (b–f), which are the recorded images for various parameters $P=78.7$. $46.7$, and $30.2$ mW (the power of the heating laser) and $R=4.4$, $5.4$, $6.5$, and $7.5$ µm (the radius of the scanning trapping laser). One can compare the effect of $R$ by inspecting panels (b,c,f) and that of $P$ by panels (c,d,e). Lowering the laser power to $P=13.5$ mW did not show a significant change in the tracer distribution from the initial state.

Firstly, it is seen from panel (b) that the tracers, which are initially rather uniform in the $\theta$ variable, tend to gather toward the hotter side (i.e., the right side, see panel (a)) as time goes on. This tendency is the same for other panels (c–f), and should be contrasted with Fig. 4 for a case without a target particle; the tracers gather toward the colder side.

As can be seen from the comparison between panels (b) $R=4.4$ µm, (c) $R=5.4$ µm, and (f) $R=6.5$ µm, the tracers’ movement is more intense and faster for smaller $R$. To be more precise, at time $t=5$ s, panel (b) starts to exhibit the depletion of the tracers near $\theta=\pi$ (the colder side; see panel (a)); however panels (c) and (f) show the apparent depletion near $\theta=\pi$ only for $t\geq 10$ s. At $t=30$ s, where the distributions can be considered steady, the intensity near $\theta=0$ seems to reach maximum, indicating the accumulation near the hot side. To compare more quantitatively, we show in panel (g) the relative intensity distribution (symbols) and the fitting using Eq. (8) (lines) (see also Appendix C). Although the results are qualitative, the plots in panel (g) capture the observation made in panels (b,c,f).

It should be noted that some tracers seem trapped in the region between the upper wall of the microchannel and the target particle (see the region “$\mathsf{A}$” in Fig. 1(b)), as clearly seen in Fig. 5(f) at $t\geq 10$ s. These tracers can stay there even when the scanning trapping laser is not irradiated. We suppose that the thermo-osmotic flow and the thermophoresis in the $z$ direction may cause this trapping, but further discussion is difficult using only the present two-dimensional information on the $x\,y$ plane, and thus future work.

One can compare the effect of the heating laser power $P$ by seeing (c) $P=78.7$ mW, (d) $P=46.7$ mW, and (e) $P=30.2$ mW. The corresponding temperature-gradient measurement given in Appendix B shows that $|\nabla T_{f}|$ is approximately (b) $2.1$ K µm^-1, (c) $1.6$ K µm^-1, and (d) $1.0$ K µm^-1 near $r^{\prime}\approx L=18$ µm, at which the target particle is fixed. Therefore, the tracer movement is more intense for the case with larger $P$. Figure 5(h) shows the corresponding relative intensity distribution at $t=30$ s. Sharper shape near $\theta=0$ for larger $P$ indicates the more intense accumulation of tracers near the hotter side.

Next, let us describe the thermally-induced flow around a target particle with an amine-modified surface, the zeta potential $\zeta$ of which is measured as $\zeta=-23.7$ mV. The thermophoretic mobility of a target particle with the amine-modified surface is estimated in the same manner as Sec. III.1.2, and we obtain $D_{T}^{\mathrm{target}}=0.29\pm 0.08$ µm² s^-1 K^-1 where the average temperature and temperature gradient are $320$ K and $-1.18$ K µm^-1, respectively. This value of $D_{T}^{\mathrm{target}}$ is smaller than that of a target particle with the carboxylate-modified surface (see Sec. III.1.2).

Figure 6(a) shows the schematic of the situation presented in panels (b–g), which are recorded images for $P=78.7$ mW and $R=4.4$ µm with a target particle with the amine-modified surface. See Fig. 5(b) for comparison with the case of a carboxylate-modified surface. Remarkably, the case of the amine-modified target results in a faint accumulation of tracers. As shown in panels (b-g), the tracers slightly gather toward the hotter side, but the accumulation is very weak compared with Fig. 5(b) for a carboxylate-modified target. This observation is further verified by the profile of the relative intensity distribution shown in panel (h) for both amine (circle) and carboxylate (square) modified surfaces. Since the surface modification is the only difference between these two cases, panel (h) is a clear evidence that the observed tracer movement originates from surface phenomena.

To compare the experimental results with existing models of thermally-induced slip flows, we use the model proposed in Ref. [13], in which a thin electric-double-layer (EDL) approximation for a charged surface of a spherical colloid is applied under the Debye-Hückel approximation. The resulting slip velocity is written as

where $|\nabla T_{f}|_{\infty}|(=\alpha)$ is the temperature gradient at infinity, $\varepsilon$ is the temperature-dependent electric permittivity, and $\psi_{0}$ is the electrostatic potential at the surface. We assume that $\psi_{0}\approx\zeta$. The temperature dependence of $\varepsilon$ for water is modeled as $\varepsilon/\varepsilon_{0}=305.7\times\exp(-T_{f}/217)$ [53] where $\varepsilon_{0}=8.85\times 10^{-12}$ F m^-1 is the electric permittivity in vacuum. Furthermore, the empirical model of viscosity $\eta=(2.761\times 10^{-6})\times\exp(1713/T_{f})$ Pa s [54] is used. For the above model, the slip coefficient $K$ in Eq. (4) is expressed as $K=-v_{0}/(\xi_{s}\alpha)$ because $u_{s}=v_{0}$. To be more specific, we have

In Ref. [1], a similar form is proposed based on the Derjaguin’s model [55] with an enthalpy excess given by EDL enthalpy with the Debye-Hückel approximation. This model was considered to analyze the thermo-osmosis on a bare glass substrate. The result was

The theoretical prediction $K$ obtained by Eqs. (10) and (11) can be compared with $K$ obtained from the experiments via two approaches. (Method 1) First, $K$ can be obtained from the thermophoretic motion of the target, Eq. (5), because we have obtained $D_{T}^{\mathrm{target}}$ experimentally by the particle tracking. See Sec. III.1.2. (Method 2) Second, $K$ can be obtained from the observation of slip flows in Sec. III.2, namely, the tracer accumulation behavior (8), through the fitting of the value $\gamma$ (see Figs. 5 and 6 for the result of fitting). Then, equation (6) and $\gamma$ yield $K$ since we have already evaluated $D_{T}^{\mathrm{tracer}}$ in Sec. III.1.1.

Before going into the comparison between Eq. (10), Eq. (11), Method 1, and Method 2, we show the values of $\gamma$ used in Method 2. Figure 7(a) is the values of $\gamma$ averaged from $t=10$ s to $30$ s for the heating laser power $P=78.7$ mW (circle), $46.7$ mW (square), and $30.2$ mW (triangle) as a function of the scanning radius $R$. Filled symbols indicate the results of the carboxylate-modified target particles and empty symbols those of the amine-modified target particles. Although qualitative, we see that $\gamma$ tends to decrease as $R$, that is, the magnitude of thermally-induced flow decreases. Moreover, comparison over different laser power $P$ shows that $\gamma$ decreases with $P$. The cases of amine-modified target yield significantly smaller $\gamma$ than those of the carboxylate-modified target. These values of $\gamma$ can be converted into $K$ through Eqs. (8b) and (6); the results are summarized in Fig. 7(b). For the case of carboxylate-modified surface, $K$ seems to take the value around $-0.4$ µm² s^-1 K^-1, while that of amine-modified surface results in $K\approx-0.2$ µm² s^-1 K^-1.

Finally, we summarize in Table 1 the results of $K$ obtained by Eq. (10), Eq. (11), Method 1, and Method 2, where the average over different $R$ is taken for Method 2 (see Fig. 7(b)). Furthermore, Table 2 shows the values of $K$ computed from the existing experimental results of thermo-osmotic flows [1, 10] with different surface details. All the values of $K$ in both tables are negative, i.e., the flow is induced from cold to hot side. The magnitude of $K$ in Table 1 has the same order $O(1)$ µm² s^-1 K^-1 but quantitative agreement is not obtained between theory and experiments. However, importantly, Methods 1 and 2 show similar values of $K$ and capture the difference of surface modification predicted by Eqs. (10) and (11): $K$ for the carboxylate-modified case is larger than that for the amine-modified case. Comparing Tables 1 and 2, it is seen that the slip coefficients obtained in the present experiments are close to that for bare glass.

The thermo-osmotic coefficients have been also computed using molecular dynamics simulation in Refs. [2] and [56] for Lennard-Jones fluid/solid system and water/quartz system, respectively. The results were $\chi\sim O(10^{-8})$ m² s^-1 and one or two orders magnitude larger than theory and experiments.

The reason for the quantitative discrepancy between theory and experiment cannot be fully clarified at the moment, but let us point out some possibilities. First, Eq. (10) was derived under the thin-EDL and Debye–Hückel approximation. The former is valid because we use an electrolyte solution with 10 mM ionic concentration. But the latter may be violated due to the non-negligible magnitude of the zeta potential compared with $k_{B}T_{f}$. Experiments, i.e., Method 1 or Method 2, may underestimate $v_{T}^{\mathrm{target}}$ or $K$ because the target particle stays close to the microchannel walls due to a technical limitation in the experiment. Further improvement of the proposed method and its application to other systems is future work.