2.1. Flow characterisation

In order to generate a convective flow, we create a temperature difference between the top and bottom: a fixed temperature $T=26\,^\circ \textrm {C}$ is imposed on the top cover with water circulation and a thermal bath, and we set the heat flux through the chamber using a thin $7.5$ W Kapton heater placed under the bottom cover. The heating resistive sheet, which is connected to a DC supply, is compressed between the bottom part and an (insulating) PVC block to ensure that most of the heat flux is directed towards the liquid. In order to tune the temperature difference, and more especially set the bottom temperature, top and bottom temperatures are monitored using a picolog acquisition card connected to 4 k-type thermocouples inserted in the top and bottom, in close contact with the fluid. Such thermal driving ensured fine tuning of the temperature difference $\Delta T \in [0.9,1.5]\,^\circ \textrm {C}$ between top and bottom with good stability. Finally, the whole set-up is placed in a temperature-regulated room to increase the thermal stability of the set-up, which is required since the measurements of mixing last approximately 10 hours.

For all experiments, we use pre-boiled distilled water to either fill the chamber or prepare the salt/colloidal solutions so as to prevent the formation of air bubbles during the long heating of the RB chamber. When operating, the chamber is filled with liquid using two holes drilled on the top cover connected to syringe pumps. When injecting dye or colloids, the injection rate is set to be less than $20\ \unicode{x03BC} \textrm {l min}^{-1}$ to avoid any strong perturbation of the convection flow. During an injection process, the lid of the opposite injection site is kept open so as to allow excess water to escape.

Figure 2.

$(a)$ Magnitude $\|\boldsymbol{u}\|$ of the velocity field measured by PIV for $\Delta T = 1.37\,^\circ\text{C}$ . $(b)$ Vertical velocity profile along the centreline inside the RB chamber for several values of the temperature difference $\Delta T$ , from $1.00\,^\circ \textrm {C}$ to $1.49\,^\circ \textrm {C}$ . $(c)$ Maximum of the velocity magnitude $U$ as a function of $\Delta T$ .

For such thermal forcing, the flow regime is controlled by the Rayleigh number $\displaystyle {Ra}={\gamma g\, \Delta T\, h^3}/({\kappa \nu })$ , where $\gamma$ is the thermal expansion of water, $g$ is gravity, $\kappa$ is thermal diffusivity, and $\nu$ is kinematic viscosity of the fluid. The flow is found to be convective when ${Ra}$ is larger than the critical value ${Ra}_{\textit{cr}}=1708$ . In the present case where $h=0.75\,\textrm {cm}$ and the fluid is water, this gives ${Ra}/{Ra}_{\textit{cr}} \approx C\, \Delta T$ , where $C=9.2\ \textrm {K}^{-1}$ . Given the range of temperature difference, the set-up is expected to produce steady laminar flows as the threshold for the first bifurcation towards a time-dependent flow corresponds to ${Ra}/{Ra}_c \approx 18$ (Solomon & Gollub Reference Solomon and Gollub1988a ).

This was verified by performing particle image velocimetry (PIV) measurements of the flow field for increasing values of the temperature difference $\Delta T$ . To achieve this, we use a CW laser (Coherent Genesis MX SLM-Series, $1\textrm {W}@488\,\textrm {nm}$ ), a cylindrical lens and a $250\,\unicode{x03BC} \textrm {m}$ slit to form a thin vertical laser sheet in the middle of the RB chamber (see figure 1); the flow is seeded with $10$ $\unicode{x03BC}$ m silver-coated hollow glass spheres from Dantec. Images are then recorded at 10 fps using an XIMEA camera, and we perform the correlation of pairs of images using a home-made software written in Matlab. The result for the velocity magnitude, averaged in time over $1$ minute, is shown in figure 2( $a$ ), where the typical signature of well-organised convection rolls is visible. As shown in figure 2( $b$ ), which displays a horizontal profile of the vertical velocity at mid-depth of the cell, the spatial organisation of the flow remains the same for all values of $\Delta T$ in the range $[1,1.5]\,^\circ \textrm {C}$ , and consists of approximately 16 counter-rotating rolls with width $L \approx 9.4\,\textrm {mm}$ , whose positions are pinched within the chamber.

In the case of RB flow, the bifurcation is of the Hopf type, meaning that the maximum vertical velocity, denoted by $U$ , is such that $U \propto \sqrt {\Delta T-\Delta T_{\textit{cr}}}$ close to the threshold, where $\Delta T_{\textit{cr}}$ is the top–bottom temperature difference at the threshold. Here, we operate quite far from the threshold, with a limited range of accessible values of the Rayleigh number, so that the flow amplitude rather tends to increase linearly with the temperature difference (figure 2 $c$ ). By fitting linearly the averaged value of $U$ as a function of $\Delta T$ , we find $U=0.36\times \Delta T-0.04\,\textrm {mm s}^{-1}$ , so the Reynolds number of the flow based on the chamber height ( ${Re}=Uh/\nu$ , where $\nu =10^{-6}\,\textrm {m}^2\textrm { s}^{-1}$ is the kinematic viscosity of water) is such that $2.5\leqslant {Re}\leqslant 4.5$ for all experiments.

2.2. Multiple time exposure concentration field measurements

When a chemical species is injected on one side of the chamber, it is advected and diffused through the convection rolls. Such a mixing process in a steady and well-organised flow is rather inefficient, so that the concentration field spans an HDR during the whole run, with large concentration differences and strong concentration gradients located near separatrixes (Solomon & Gollub Reference Solomon and Gollub1988b ). It is therefore very hard to perform direct optical measurements of such a concentration field, even with a high-quality camera. For that purpose, we use a variant of the set-up described in Mauger et al. (Reference Mauger, Volk, Machicoane, Bourgoin, Cottin-Bizonne, Ybert and Raynal2016), and perform planar LIF measurements using a multiple time exposure technique.

We use the same optical set-up as for the PIV measurements to create the laser sheet, with almost homogeneous light intensity in the chamber. The fluorescence signal is recorded using a $16$ -bit camera (Nikon D700, $4200\times 2800$ pixels $^2$ ) equipped with a TAMRON $90\,\textrm {mm}$ macro lens and a band reject filter (notch $488 \pm 12\,\textrm {nm}$ ) corresponding to the laser wavelength. For all experiments, the laser power is set to $195\,\textrm {mW}$ , and the ISO sensitivity is set to its lowest value to avoid noise. After recording, images are cropped to fit the aspect ratio of the cell so that their actual size is $3249 \times 181$ pixels $^2$ in the $x$ - and $y$ -directions, with spatial resolution approximately $44\,\unicode{x03BC} \textrm {m}$ . A typical raw image recorded with exposure time $t_e=50\,\textrm {ms}$ several hours after colloid injection on the left-hand side injection site (1) is displayed in figure 3 $(a)$ .

Figure 3.

$(a)$ Raw image obtained using the LIF set-up and exposure time $t_e=50\,\textrm {ms}$ , $10\,\textrm {h}$ after colloid injection. $(b)$ Camera calibration curves: squares record intensity measured with increasing homogeneous fluorescein concentration in $2.5\ \textrm {ml}$ cuvettes for two exposure times, $t_e=5$ and $50\,\textrm {ms}$ . The straight lines correspond to linear fits derived in the linear range $I \leqslant 44\,000$ ( $60\,\%$ of the dynamical range). $(c)$ Comparison of $x$ -profiles of recorded intensity averaged over the vertical coordinate $y$ and two exposure times. Blue dots show raw intensity profile measured with $t_e=50$ ms. The red line shows intensity profile measured with $t_e=5$ ms, transformed using the linear transformation $\mathcal{T}[I]=\gamma I + \kappa$ , with $\gamma = 10.2$ and $\kappa = - 951$ . $(d)$ The $x$ -profiles of concentration averaged over the vertical coordinate $y$ built using the HDR procedure and calibration curves obtained in the chamber with known (homogeneous) colloid concentration. The dashed red line indicates without correction for exponential attenuation through the chamber. The blue solid line indicates with compensation for exponential attenuation.

2.2.3. Compensation for spatial decay of incoming light

Given the low concentrations of fluorescent species $C$ used here, the light intensity scattered per unit volume of fluid in the laser sheet is proportional to the local concentration $I_{\textit{scat}}(x,y,t)=\sigma \times C(x,y,t) \times I_0(x,y,t)$ , where $\sigma$ is a constant specific to the colloids, and $I_0(x,y,t)$ is the incoming light at location $(x,y)$ and time $t$ . Although the precise values of $\sigma$ and $I_0$ are not needed as we will work with normalised quantities in the following, light scattering may affect the measurement due to the spatial decay of incoming light $I_0(x,y,t)$ , corresponding to the Beer–Lambert law (Swinehart Reference Swinehart1962). In the present case, we inject colloids at the location $x=0$ , and propagate the laser beam from $x=L_{\textit{tot}}$ towards $x=0$ so that the local Beer–Lambert law reads

(2.1) \begin{equation} \frac {{\rm d} \log I_0(x,y,t)}{{\rm d}x} = \sigma \, C(x,y,t). \end{equation}

This relation can be integrated from $x$ to $L_{\textit{tot}}$ to give

(2.2) \begin{equation} I_0(x,y,t) = I_0(L_{\textit{tot}},y,t) \times\exp \left (-\sigma \int _x^{L_{\textit{tot}}} C(x',y,t)\, {\rm d}x' \right )\!, \end{equation}

where $I_0(L_{\textit{tot}},y,t)$ is a constant here as the incoming laser light has an almost homogeneous intensity in the $y$ -direction at $x=L_{\textit{tot}}$ , with a constant amplitude in time. Due to the spatial decay of the laser intensity along the chamber, the recorded intensity $I_{\textit{HDR}}$ is not simply proportional to the concentration. However, $I_{\textit{HDR}}$ is still proportional to the scattered light intensity if the concentration field is well resolved spatially with a camera operating in the linear regime. Assuming that this is the case here, we then have

(2.3) \begin{equation} I_{\textit{HDR}}(x,y,t) = A \times C (x,y,t) \times\exp \left (-\sigma \int _x^{L_{\textit{tot}}} C(x',y,t)\, {\rm d}x' \right )\!, \end{equation}

where $A$ is a multiplicative constant.

To recover an unbiased estimate of the concentration field, it is important to accurately measure $\sigma$ and $A$ . This is achieved by recording images in the case of a homogeneous concentration of fluorescent species for which a well-defined exponential decay was found, leading to $\sigma =1.27 \times10^{-11}\ \textrm {mm}^2$ and the multiplicative constant $A=2.3\times 10^{-5}\, \textrm{(greylevel) part}^{-1}\ \textrm {ml}$ for $t_e=50$ ms. Note that $A$ is not specific to the fluorescent species, but changes when the laser power, the exposure time or the optical set-up is modified. Finally, because the exponential decay is small here and can be considered as a first-order correction, the concentration field is finally obtained from (2.3) as

(2.4) \begin{equation} C(x,y,t) = \frac {I_{\textit{HDR}}(x,y,t)}{A} \times\exp \left (\frac {\sigma }{A} \int _x^{L_{\textit{tot}}} I_{\textit{HDR}}(x',y,t)\, {\rm d}x' \right )\!. \end{equation}

Using this procedure, the global mean concentration of the fluorescent species,

(2.5) \begin{equation} \langle C\rangle =\frac {1}{L_{\textit{tot}}\times h} \int _0^{L_{\textit{tot}}} \int _0^h C(x,y,t) \,{\rm d}x\,{\rm d}y=\frac {1}{L_{\textit{tot}}} \int _0^{L_{\textit{tot}}} C(x,t) \,{\rm d}x, \end{equation}

is checked to be a conserved quantity. The HDR profile of figure 3( $c$ ) is transformed into a concentration profile as plotted in figure 3( $d$ ), where we compare the raw profile without exponential correction and the corrected profile. The correction is cumulative from right to left, and is approximately $25\,\%$ close to the injection point. Although such a correction may appear to be significant, shining the laser from the opposite side (figure 1) reduces its impact to values smaller than $5\,\%$ on first and second moments of the concentration field,

(2.6) \begin{equation} \langle x^p \rangle = \frac {1}{L_{\textit{tot}}} \int _0^{L_{\textit{tot}}} x^p \,\frac {C(x,t)}{\langle C\rangle }\, {\rm d}x, \quad p=1,2, \end{equation}

which are the quantities that we will consider in the sequel.

In the following, we apply all the methods described above to follow species in the cellular flow, first without diffusiophoresis, then when diffusiophoresis comes into play.

3.1. Fluorescein profiles and effective diffusivities

Fluorescein is introduced through injection site (1), and its concentration $C_{\!f}$ is measured as a function of space and time. As seen in figure 4 $(a)$ , which displays the time evolution of $\log (C_{\textit{f}}(x,t)/\langle C_{\textit{f}}\rangle )$ , the one-dimensional concentration profiles are rather homogeneous in convection cells, with strong gradients in between. This is consistent with the observations of Solomon & Gollub (Reference Solomon and Gollub1988b ), who reported that after several hours, the concentration becomes constant in each convection cell, the coarse-grained profile taking a Gaussian shape. This is also the case here, as shown in figure 4( $b$ ), where the scale evolution of the normalised profiles is well fitted by Gaussian functions with a maximum centred at the origin:

(3.1) \begin{equation} F(x,t)=\frac {N_{0,f}}{\sqrt {2\pi \,\langle x^2\rangle _{\!f}(t)}}\exp \left (\frac {-x^2}{2\,\langle x^2\rangle _{\!f}(t)}\right )\!, \end{equation}

where $\langle x^2\rangle _{\!f}(t)$ is the measured second-order moment for the fluorescein, computed following (2.6).

Figure 4.

$(a)$ Fluorescein concentration profile as a function of time for $\Delta T=1.10\pm 0.06\,^\circ \textrm {C}$ ( $\textit{Pe}_{\textit{f}}=6270$ ). $(b)$ Three concentration profiles extracted from (a) (corresponding to the three horizontal lines on this figure) for $t={4000}\ \text{s}$ ( ${1}\,\textrm {h}\,{7}\,\min$ ), $t={16\,000}\ \text{s}$ ( ${4}\,\textrm {h}\,{27}\,\min$ ) and $t={32\,000}\ \text{s}$ ( ${8}\,\textrm {h}\,{53}\,\min$ ). Solid lines correspond to Gaussian functions with the same variance according to (3.1).

Solomon & Gollub (Reference Solomon and Gollub1988b ) (see also Moffatt Reference Moffatt1983; Pomeau Reference Pomeau1985; Guyon et al. Reference Guyon, Pomeau, Hulin and Baudet1987; Shraiman Reference Shraiman1987) proposed that such a Gaussian profile at large scale, where the concentration is homogenised through each cell, corresponds to a regime of effective diffusion in which the second-order moment of the distribution evolves linearly in time at large time

(3.2) \begin{equation} \langle x^2 \rangle (t)= \langle x^2\rangle (0)+2D^\star t \end{equation}

with an effective diffusion coefficient $D^\star$ such that

(3.3) \begin{equation} D^\star /D= B\, \sqrt {\textit{Pe}}, \end{equation}

where $B$ is a constant that depends on the geometry of the system, $D$ is the diffusion coefficient, and $\textit{Pe}$ is the Péclet number.

Figure 5.

Typical dispersion curves for fluorescein and different values of the temperature difference $\Delta T=0.99,1.02,1.47$ (corresponding to Péclet numbers of the fluorescein $\textit{Pe}_{\textit{f}}$ between $5760$ and $8630$ ). As found previously in Solomon & Gollub (Reference Solomon and Gollub1988b ), the global behaviour corresponds to an effective diffusion, with an effective diffusion coefficient given by (3.4).

Typical curves of the second-order moment $\langle x^2\rangle _{\!f}(t)$ for the fluorescein and different Péclet numbers are shown in figure 5. After a rather long transient of approximately $2$ hours, which corresponds to the time needed for invasion of a few cells, we observe that $\langle x^2\rangle _{\!f}$ grows linearly in time, the slope being an increasing function of the Péclet number. Combining (3.2) and (3.3), and using a linear fit, we obtain

(3.4) \begin{equation} \langle x^2 \rangle _{\!f}(t)\approx \langle x^2 \rangle _{\!f}(0)+2\times B\times D_{\!f}\, \sqrt {\textit{Pe}_{\!f}}\times t, \end{equation}

with $\sqrt {\langle x^2\rangle _{\!f}(0)}\approx 12\,\textrm {mm}$ , which corresponds to a little more than one cell invaded by the fluorescein after injection at $t=0$ , and $B=0.5$ , which is consistent with what was found by Solomon & Gollub (Reference Solomon and Gollub1988b), given their definition of the Péclet number.

3.2. Case of salt (LiCl)

Salt (LiCl) is introduced at concentration $20\ \textrm{mmol l}^{-1}$ , a concentration low enough so that no gravity current is created due to a possible density mismatch; it is then transported in the RB chamber in the same way as fluorescein. It is thus possible to estimate its dispersion from the results obtained for fluorescein, provided that its diffusion coefficient is slightly different, although of the same order of magnitude (see table 1). Therefore, we can extend the findings for flurorescein, and assume that the salt concentration also has a Gaussian coarse-grained profile over long time scales. We also differentiate between cases where salt is introduced through the injection holes (1) and (2). We thus obtain the following coarse-grained profile $S^{(i)}$ for the salt:

(3.5) \begin{equation} S^{(i)}(x,t)=\frac {N_{0,s}}{\sqrt {2\pi \,\langle x^2\rangle _s(t)}}\exp \left (\frac {-\,\bigl (L^{(i)} -x\bigr )^2}{2 \,\langle x^2\rangle _s(t)}\right ) \!, \end{equation}

where $i=1,2$ stands for the injection hole through which the salt is injected, with $L^{(1)}=0$ and $L^{(2)}=L_{\textit{tot}}$ . The variance $\langle x^2\rangle _s(t)$ of salt spreading does not depend on the salt’s direction of travel, and is therefore identical in the two cases:

(3.6) \begin{equation} \langle x^2\rangle _s(t)\approx \langle x^2\rangle _s(0)+2\times B\times D_s\, \sqrt {\textit{Pe}_s}\times t , \end{equation}

with $\sqrt {\langle x^2\rangle _s(0)}\approx 12\,\textrm {mm}$ and $B=0.5$ , as for fluorescein in (3.4).

3.3. Colloids: reference case

We now investigate how colloids are transported through the chamber in the reference case, i.e. without salt. Colloids are injected through site (1) at $t=0$ . From the images, we build one-dimensional normalised concentration profiles $C_c(x,t)/\langle C_c(x,t) \rangle$ averaged over the height of the chamber. Their typical time evolution is shown in figure 6( $a$ ) for $\Delta T=1.1\,^\circ \textrm {C}$ ( $\textit{Pe}_c=3.9\times 10^5$ ). As is clearly visible, the concentration field is much less homogeneized than in the case of fluorescein, with spatial oscillations and peaks located in between the cells even at long times. This is clearly visible when plotting directly the normalised concentration $C_c/\langle C_c \rangle$ at different times, as displayed in figure 6( $b$ ). Such a difference between profiles measured in the case of particles and molecules in the chamber was already pointed out by Solomon & Gollub (Reference Solomon and Gollub1988b ), who mentioned that homogenisation does not really take place in the cells in the case of latex particles, due to their large size and hence weak diffusion coefficient. A first noticeable difference with results from Solomon & Gollub (Reference Solomon and Gollub1988b ) is that the colloid concentration profiles can hardly be described as Gaussian functions, even more than 16 hours after injection. Another difference can be seen in figure 6( $c$ ), which displays the time evolution of the second-order moment of the colloid concentration field, $\langle x^2 \rangle _c$ , for seven different Péclet numbers $\textit{Pe}_c \in [1.1, 1.5]\times10^6$ . In none of the different runs was it possible to observe a well-defined linear evolution for the colloid concentration field, even after repeating the same experiment several times. An explanation for this could be the presence of an additional three-dimensional structure to the flow that would modify mixing, making the concentration field more complex than with two-dimensional flow. However, the contribution of such a three-dimensional structure was already mentioned in Solomon & Gollub (Reference Solomon and Gollub1988b ), and was also present in the case of fluorescein, for which we obtained $\langle x^2 \rangle _{\!f} \propto t$ at long times. This means that we need to look for another mechanism that is present in the case of (charged) colloidal particles but not in solutes.

Figure 6.

Reference case (colloids alone). $(a)$ Normalised colloid concentration profile as a function of time for $\Delta T=1.1\,^\circ \textrm {C}$ ( $\textit{Pe}_c=3.9\times 10^5$ ). $(b)$ Three concentration profiles extracted from (a) (corresponding to the three horizontal lines on this figure) for $t={15\,000}\ \text{s}$ ( ${4}\,\textrm {h}\,{10}\,\min$ ), $t={30\,000}\ \text{s}$ ( ${8}\,\textrm {h}\,{20}\,\min$ ) and $t={60\,000}\ \text{s}$ ( ${16}\,\textrm {h}\,{40}\,\min$ ). $(c)$ Dispersion curves $\langle x^2 \rangle _c(t)$ obtained with colloids and temperature difference $\Delta T \in [1,1.33]$ $\,^\circ$ C. The darker the symbol, the larger the colloid Péclet number. The two dashed lines stand for the theoretical predictions for the spreading of neutral particles of the same size as the colloids, calculated using (3.4) transposed to colloids, $\langle x^2\rangle _c(t)\approx \langle x^2\rangle _c(0)+2\times B\times D_c \sqrt {\textit{Pe}_c}\times t$ , with $\sqrt {\langle x^2\rangle _c(0)}\approx 12\,\textrm {mm}$ and $B=0.5$ as for fluorescein (3.4), for $\Delta T=1\,^\circ \textrm {C}$ and $\Delta T=1.5\,^\circ \textrm {C}$ .

Indeed, another major difference with the work by Solomon & Gollub (Reference Solomon and Gollub1988b ) is that the effective diffusion coefficient estimated for the different values of $\Delta T$ (hence $\textit{Pe}_c$ ) is much larger than expected, as can be seen from figure 6( $c$ ), in which the dashed lines represent the theoretical prediction (3.4) transposed to neutral particles of the same size. At these small concentration levels, this cannot be related to a collective effect of colloids. Another possibility is the existence of thermophoretic effects, whereby large particles exhibit a drift velocity related to thermal gradients. However, this would imply that the effect increases with rising $\Delta T$ , which is not the case here, as can be seen in figure 6( $c$ ), where the effective diffusion is almost independent of $\Delta T$ . One can also note that the top–bottom symmetry of the observed pattern in figure 7, which displays the snapshot of a typical reference case, definitely contradicts the thermophoresis hypothesis. More likely, the explanation may come from the fact that because we need to study diffusiophoretic effects, our particles are (positively) charged, and hence may interact with the upper and lower walls made of brass, which takes a negative charge in water (Sarver et al. Reference Sarver, Slabaugh, Strock and Edwards2016; Cui et al. Reference Cui, Lin, Zhu, Cheng, Wu and Zheng2024). This additional mechanism tends to facilitate the diffusion of particles along the separatrixes of the flow, leading to a much larger dispersion. This hypothesis is reinforced when looking at figure 7, where we clearly see that colloids are attracted to the walls; this effect was not present in Solomon & Gollub (Reference Solomon and Gollub1988b ).

Figure 7.

Typical image of a reference case ( $\Delta T=1.1\,^\circ \textrm {C}$ and $t=43\,000\,\textrm {s}=12\,\textrm {h}$ ).

4.1. Theory from Volk et al. (Reference Volk, Bourgoin, Bréhier and Raynal2022)

The theoretical results in Volk et al. (Reference Volk, Bourgoin, Bréhier and Raynal2022) were obtained in stationary conditions. The advection–diffusion equation was first solved numerically for the salt concentration $C_s$ in the presence of an imposed mean salt gradient $\boldsymbol{G}$ , using an analytical cellular velocity field $\boldsymbol{u}$ . In the stationary regime, this results in a solution where the salt concentration is almost constant within each cell, except in the vicinity of the separatrixes; see figure 8 $(a)$ for an example with $\textit{Pe}_s=1340$ . The horizontal profile shown in figure 8 $(b)$ exhibits a sharp increase of magnitude $\Delta S=G\times L$ between two consecutive cells, indicating a very high salt gradient. The width $\ell _s$ of this gradient is found by the competition between contraction by the flow (typical time $L/U$ ) and diffusion (typical time $\ell _s^2/D_s$ ), hence $\ell _s\sim L/\sqrt {\textit{Pe}_s}$ , and the corresponding salt gradient in this region is $\Delta S/\ell _s\sim G\sqrt {\textit{Pe}_s}\gg G$ . Note that the vertical profiles in figure 8 $(c)$ also show a small vertical salt gradient around the horizontal separatrixes, but of much weaker intensity; hence we neglect it in a first step.

Figure 8.

Salt distribution (numerical simulations) in the stationary regime, for a constant horizontal mean gradient $\boldsymbol{G}=G\,\boldsymbol{e}_x$ ; here, ${\textit{Pe}}_s=3140$ , $G=1$ , and the size of a cell is $L=\pi$ . $(a)$ Stationary salt concentration field $C_s$ in two consecutive cells, with an imposed constant mean gradient $\boldsymbol{G}=G\,\boldsymbol{e}_x$ ; the white arrows represent the cellular velocity field. $(b)$ Horizontal profile along the horizontal dashed line in (a). $(c)$ Two vertical profiles along the vertical dashed lines. The salt concentration is approximately constant in the cells, except around the vertical and horizontal separatrixes.

We now consider the displacement of colloids in this salt distribution. In Volk et al. (Reference Volk, Bourgoin, Bréhier and Raynal2022), the drift velocity of the active particles is of the type $\boldsymbol{v}_{{dp}}=\alpha\, \boldsymbol{\nabla }C_s$ , where $\alpha$ is the phoretic coefficient. The total velocity of a phoretic particle is thus $\boldsymbol{v}=\boldsymbol{u}+\boldsymbol{v}_{{dp}}$ , where $v_{{dp}}\ll U$ . When a particle enters the salt gradient (shown in light blue in figure 9 $(a)$ ), it deviates horizontally by a distance $\ell _d$ such that $\ell _d/L=v_{{dp}}/U$ , so that $\ell _d\sim \alpha G L \sqrt {\textit{Pe}_s}/U$ . The particles that change cell are located within the grey strip of width $\ell _d$ in figure 9 $(a)$ . These particles are then transported horizontally by the velocity field until they reach the next region of high salt gradient, thus travelling around the cells, globally in the direction of the mean salt gradient $\boldsymbol{G}$ , as depicted in figure 9 $(b)$ . As the particles located outside the grey strip do not change cell, it is possible to estimate the mean global horizontal velocity as $v_m\sim 1/{L^2}\times [(\ell _d\times L)\,v_{{dp}}+(\ell _d\times L)\,U ]$ . Finally, because $v_{{dp}}\ll U$ , we obtain

(4.1) \begin{equation} v_m \sim \alpha G \sqrt {\textit{Pe}_s}. \end{equation}

This regime, for which the mean velocity is much higher than the velocity, $\alpha G$ , associated with the mean salt gradient, is called ‘enhanced transport’. Lagrangian numerical simulations performed in Volk et al. (Reference Volk, Bourgoin, Bréhier and Raynal2022) show that (4.1) gives a good estimation of the global mean velocity.

Figure 9.

$(a)$ Schematic displacement of a particle in the salt gradient (light blue region) around the vertical ascending separatrix $x=L$ . The Eulerian velocity field $\boldsymbol{u}$ is vertical and ascending, and $\boldsymbol{v}_{{dp}}$ is horizontal; the total velocity of particles around the vertical separatrixes is therefore $\boldsymbol{v}_t=\boldsymbol{u}+\boldsymbol{v}_{{dp}}$ , with $\boldsymbol{u}\approx U\, \boldsymbol{e}_y$ . Particles that change cell are located on the grey strip of width $\ell _d$ . $(b)$ Schematic location of particles that change cell, in grey, for cells located between $x=0$ and $x=2L$ ; an example of a trajectory is shown in red. Outside this grey strip, the particles remain trapped in their cells. From Volk et al. (Reference Volk, Bourgoin, Bréhier and Raynal2022).

The blockage regime is related to the weaker vertical salt gradients around the horizontal separatrixes; see figure 8 $(c)$ . Indeed, because the salt concentration increases from left to right due to the imposed mean gradient $\boldsymbol{G}$ , the velocity field transports a deficit of salt when travelling horizontally from left to right, and an excess of salt when travelling in the opposite direction, as is visible in figure 8 $(a)$ . Therefore, the phoretic particles located within a horizontal segment of the grey strip in figure 9 $(b)$ always encounter a salt gradient pointing vertically into the cell, and gradually deviate towards its centre: after some time, they leave the grey strip for the central region of the cell. Once all the particles have left the strip, they can no longer change cell, and become blocked. In addition, any fluid particle located inside the strip follows the velocity field and completes one revolution around the cell in a typical time $4L/U$ . This creates a corona of width $\ell _d$ devoid of particles around each cell, as demonstrated numerically in Volk et al. (Reference Volk, Bourgoin, Bréhier and Raynal2022) (see also figure 16 below for an experimental visualisation).

This blocking mechanism occurs only when diffusion is insufficient to bring particles back into the strip. Therefore, we need to estimate how long it takes for particles outside the strip to re-enter it. Consider a strip of width $\ell _d$ that is empty of particles in a medium that is filled with particles. Due to the linearity of the advection–diffusion equation, the problem is the same as considering a strip of width $\ell _d$ filled with particles in an empty medium: the typical diffusion time is, in both cases, $\ell _d^2/D_c$ .

Blockage therefore corresponds to a situation in which fluid in the empty corona can travel around the cell while remaining empty of particles, i.e. $4L/U \ll \ell _d^2/D_c$ . Introducing the blockage parameter $R$ such that

(4.2) \begin{equation} R =\frac {\alpha \, GL}{2\sqrt {D_c D_s}}, \end{equation}

we obtain the blockage situation corresponding to $R\gg 1$ .

Numerical simulations performed in Volk et al. (Reference Volk, Bourgoin, Bréhier and Raynal2022) demonstrate a sharp transition between blockage and enhanced transport, depending on $R$ : a complete blockage is observed whenever $R\geqslant 4$ , while enhanced transport, with a magnitude given by (4.1), corresponds to $R\leqslant 1$ ; in practice, enhanced transport is already observed for $R\sim 2$ , although with a weaker velocity. This is all explained in much more detail in Volk et al. (Reference Volk, Bourgoin, Bréhier and Raynal2022).

4.2. Experiment versus theory

Although there are some differences between the experiment and the theory, we now demonstrate how the theoretical findings can be adapted to the experimental conditions.

4.2.2. Salt distribution

The situation regarding the distribution of salt is very different: the theory assumes an imposed mean salt gradient, which is assumed to be constant in space and time. Furthermore, the theory deals with the stationary solution of the advection–diffusion equation. By contrast, the salt distribution in the experiment is time-dependent and follows a Gaussian coarse-grained profile, denoted by $S$ (see (3.5)). The experimental equivalent for $G$ is the horizontal salt concentration jump across a cell, divided by the cell’s length $L$ . This jump in salt concentration can be estimated rather accurately using the coarse-grained profile $S$ . If we denote as $G^{(i)}$ the mean horizontal salt gradient across a cell in the case when salt is introduced through site $(i)$ , then $G^{(i)}\approx {\rm d}S/{\rm d}x$ , so that we obtain from (3.5) that

(4.3) \begin{equation} G^{(i)}(x,t)=\frac {x-L^{(i)}}{\langle x^2\rangle _s(t)}\times S(x,t) , \end{equation}

where we recall that $L^{(1)}=0$ and $L^{(2)}=L_{\textit{tot}}$ .

4.2.3. Velocity drift

As stated in § 4.1, in Volk et al. (Reference Volk, Bourgoin, Bréhier and Raynal2022), the drift velocity due to phoresis was taken as $\boldsymbol{v}_{{dp}}=\alpha\, \boldsymbol{\nabla }C_s$ . However, the relationship between drift velocity and scalar gradients is generally much more complex (Gupta, Shim & Stone Reference Gupta, Shim and Stone2020; Lee, Lee & Ault Reference Lee, Lee and Ault2023). For very charged particles such as ours, the drift velocity takes the form ( Ault & Shi Reference Ault and Shin2025)

(4.4) \begin{equation} \boldsymbol{v}_{{dp}}= D_{{dp}}\, \boldsymbol{\nabla }\log C_s= D_{{dp}} \frac {\boldsymbol{\nabla }C_s}{C_s}. \end{equation}

Then we switch from theory to experiment by setting $\alpha \approx D_{{dp}}/C_s$ , or rather $\alpha \approx D_{{dp}}/S$ when dealing with mean quantities as in (4.1) or (4.2). Note also that the experiments may be non-repeatable in the case of salt concentrations $C_s$ that are too small, far from the injection hole at early times. This will be taken into account later.

4.2.4. Enhanced transport

Even if the theoretical formulas (4.1) and (4.2) were derived in a stationary regime, they may still apply to a transient case, provided that the salt evolves slowly over time, as is the case in the experiment. From the analysis above, we can transpose the value found for enhanced transport in (4.1) to the experimental conditions. We obtain

(4.5) \begin{equation} v_m^{(i)}(x,t)\sim D_{{dp}} \sqrt {\textit{Pe}_s} \,\frac {L^{(i)}-x}{\langle x^2\rangle _s(t)} , \end{equation}

where again $i=1,2$ stands for the injection site. We observe that the velocity due to salt attraction is also inhomogeneous and time-dependent. When $i=1$ ( $L^{(1)}=0$ ), the velocity is negative, as expected for colloids that move in the direction of the salt gradient; conversely, when $i=2$ ( $L^{(2)}=L_{\textit{tot}}$ ), the velocity is positive. Note also that the effect decays over time, but increases with the Péclet number.

4.2.5. Blockage coefficient

We now transpose the theoretical blockage coefficient (4.2) to experimental conditions. Unlike enhanced transport, blockage does not depend on the direction of the salt gradient. Therefore, the blockage coefficient is always positive and is given by

(4.6) \begin{equation} R^{(i)}(x,t)=\frac {|L^{(i)}-x|\times L}{2\,\langle x^2\rangle _s(t)} \times \frac {D_{{dp}}}{\sqrt {D_c D_s}}. \end{equation}

In the experiment, we have $D_s =1360$ $\unicode{x03BC} \textrm{m}^2\,\textrm{s}^{-1}$ , $D_c = 2$ $\unicode{x03BC} \textrm{m}^2\,\textrm{s}^{-1}$ and $D_{{dp}} = 290\, \unicode{x03BC} \textrm{m}^2\, \textrm{s}^{-1}$ , so that $D_{{dp}}/\sqrt {D_cD_s}\approx 5.56$ . Because the theory predicts blockage for $R\geqslant 4$ , it may be possible to observe blockage at some location. Note also that $\langle x^2\rangle _s(t)$ increases with time, implying that the blockage coefficient $R$ decreases with time at a given point $x$ , so that blockage is not possible at long time.

However, the value of the blockage coefficient cannot be transposed directly to the experiment status. Indeed, the theory was derived in the context of an infinite and periodic medium, and in a stationary state: the conditions in the experiment differ greatly from these.

The differences between theory and experiment are summarised in table 2. In the following, we explain the consequences for both the enhanced transport and blockage configurations.

Table 2.

Summary of the differences between theory and experimental conditions. For the experiment, the index $i=1,2$ stands for the injection hole of the salt (see figure 1), with $L^{(1)}=0$ and $L^{(2)}=L_{\textit{tot}}$ .

4.3. Salt-out configuration: enhanced transport

In practice, blockage or enhanced transport could, in theory, be observed regardless of the configuration considered (salt-in or salt-out). Indeed, let us come back to the value of the blockage coefficient. In the salt-out configuration, it is obtained by letting $i={2}$ in (4.6):

(4.7) \begin{equation} R^{\textit{salt-out}}(x,t)=\frac {(L_{\textit{tot}}-x)\times L}{2\,\langle x^2\rangle _s(t)}\times \frac {D_{{dp}}}{\sqrt {D_cD_s}}. \end{equation}

The space–time diagram for $R^{\textit{salt-out}}$ is shown in figure 10; values higher than $4$ , corresponding to possible blockage, are coloured yellow. The diagram shows high values of $R$ (in yellow) at short times in the region where the colloids are injected and initially spread, which is consistent with possible blockage. However, blockage is not physically reasonable: actually, as previously explained, the salt Gaussian coarse-grained profile used in the calculation does not settle instantaneously in the chamber. Furthermore, it takes time for the salt to travel across the whole cell towards the colloids. Finally, note that these large values of $R$ would correspond to very small salt concentrations here, which could lead to non-repeatable experiments (see (4.4)). Therefore, the salt must be injected into the cell well before the colloids, so as to have reached the injection site (1) when the colloids are introduced; as shown in figure 11, salt is introduced at time $t=-t_0$ , with $t_0\gt 0$ the time lag before the colloids injection. We let $\tau =t+t_0$ ; salt is therefore introduced at $\tau =0$ in this time coordinate, and $\langle x^2\rangle _s (\tau )$ verifies (3.6) exactly. We can write

(4.8) \begin{equation} \langle x^2\rangle _s(\tau )=\langle x^2\rangle _s(t+t_0)\approx \langle x^2\rangle _s(0)+2\times B\times D_s\sqrt {\textit{Pe}_s} \times (t+t_0) , \end{equation}

with $\sqrt {\langle x^2\rangle _s(0)}\approx 12\,\textrm {mm}$ and $B\approx 0.5$ from (3.6).

Figure 10.

Space-time diagram for the blockage coefficient $R$ in the salt-out configuration (4.7) for $\Delta T=1\,^\circ \textrm {C}$ , with the experimental data $D_s =1360$ $\unicode{x03BC} \textrm{m}^2\, \textrm{s}^{-1}$ , $D_c = 2$ $\unicode{x03BC} \textrm{m}^2\, \textrm{s}^{-1}$ and $D_{{dp}} = 290\, \unicode{x03BC} \textrm{m}^2\, \textrm{s}^{-1}$ ( $D_{{dp}}/\sqrt {D_cD_s}\approx 5.56$ ). Values higher than $4$ , which correspond to possible blockage, are shown in yellow. The salt is released at time $\tau =0$ , while the colloids are released a time $t_0$ later; we set $t=\tau -t_0$ so as to have the colloids released at $t=0$ . The dashed lines indicate the times at which the colloids are released in the experiments presented: black dashed line for $t_0=6\,\textrm {h}$ ; white dashed line for $t_0=17\,\textrm {h}$ .

Figure 11.

Salt-out configuration: first, salt (LiCl, non-fluorescent) is introduced through injection site (2) at $t=-t_0$ ; then colloids (fluorescent) are introduced through injection site (1) at $t=0$ . The delay $t_0$ must be long enough so that some salt has reached the other side of the cell when colloids are introduced. The two horizontal bold arrows show the transient effects felt by the colloids after introduction: they naturally travel towards the right, as showed in the reference case (common transport); they are also attracted by salt and therefore furthermore travel towards the right (salt attraction).

In practice, we found no noticeable diffusiophoretic effect of salt on the colloids whenever the time lag $t_0$ was shorter than $5$ – $6$ hours. In figure 10, the line corresponding to the time lag $t_0=6\,\textrm {h}$ , corresponding to one of our experiments, is plotted as a black dashed line; as can be seen, at that time, the blockage coefficient $R$ is already too small to have blockage. Furthermore, all transient effects (i.e. the common transport of colloids and salt attraction) tend to move the colloids to the right (figure 11), which works against blockage. Finally, note that $R$ decreases from left to right, in the direction of colloid movement. Therefore, the salt-out configuration is highly unlikely to exhibit blockage, but it may result in enhanced transport.

We now apply (4.5) with $i=2$ ( $L^{(2)}=L_{\textit{tot}}$ ) and obtain

(4.9) \begin{equation} v_m(x,t)\sim D_{{dp}} \sqrt {\textit{Pe}_s} \,\frac {L_{\textit{tot}}-x}{\langle x^2\rangle _s(t+t_0)} , \end{equation}

with $\langle x^2\rangle _s$ given by (4.8). This equation can be averaged over the colloid concentration profile to obtain the global mean velocity of colloids in the chamber $V_m(t)=\langle v_m(x,t)\rangle _c$ :

(4.10) \begin{equation} V_m(t)=\frac {1}{L_{\textit{tot}}\,\langle C\rangle }\int _0^{L_{\textit{tot}}} C(x)\, v_m(x)\,{\rm d}x\sim \frac {D_{{dp}}\sqrt {\textit{Pe}_s}}{\langle x^2\rangle _s(t+t_0)}\,\bigl (L_{\textit{tot}}-\langle x\rangle _c(t)\bigr). \end{equation}

Using the equality

(4.11) \begin{equation} V_m(t)=\frac {{\rm d}\langle x\rangle _c(t)}{{\rm d}t} \,, \end{equation}

it is possible to predict the evolution of $\langle x\rangle _c(t)$ . Indeed, combining (4.10) and (4.11), we obtain

(4.12) \begin{equation} \frac {{\rm d}\langle x\rangle _c}{L_{\textit{tot}}-\langle x\rangle _c(t)}= D_{{dp}} \sqrt {\textit{Pe}_s}\,\frac {{\rm d}t}{\langle x^2\rangle _s(t+t_0)} ,\end{equation}

with $\langle x^2\rangle _s(t+t_0) =a+b(t_0+t)$ from (4.8). This can therefore be integrated into

(4.13) \begin{equation} -\ln \frac {L_{\textit{tot}}-\langle x\rangle _c(t)}{L_{\textit{tot}}-\langle x\rangle _c(0)}=D_{{dp}} \sqrt {\textit{Pe}_s}\times \frac {1}{b}\,\ln \frac {a+b(t_0+t)}{a+b\,t_0}. \end{equation}

Introducing the non-dimensional coefficient

(4.14) \begin{equation} \beta =D_{{dp}} \sqrt {\textit{Pe}_s}\times \frac {1}{b}=\,\frac {D_{{dp}}\sqrt { \textit{Pe}_s}}{2D^\star _s}, \end{equation}

which verifies $\beta \sim ({D_{{dp}}}/{D_s})\lt 1$ as $D^\star _s\sim 0.5\, D_s\sqrt {\textit{Pe}_s}$ , we finally obtain

(4.15) \begin{align} \langle x\rangle _c(t)&=L_{\textit{tot}}-\big (L_{\textit{tot}}-\langle x\rangle _c(0)\big )\left (\frac {a+b\,t_0}{a+b\,(t_0+t)}\right )^\beta \nonumber\\& =L_{\textit{tot}}-\big (L_{\textit{tot}}-\langle x\rangle _c(0)\big )\left (\frac {\langle x^2\rangle _s(t_0)}{\langle x^2\rangle _s(t_0+t)}\right )^\beta , \\[9pt] \nonumber \end{align}

with $a$ and $b$ given in (3.6), and $\beta$ given by (4.14).

The formula implies that as $t\rightarrow \infty$ , the ratio of salt dispersion becomes small, and $\langle x\rangle _c(t)$ tends to $L_{\textit{tot}}$ ; physically, one would rather think of $L_{\textit{tot}}/2$ . But of course there is no diffusion in those calculations, only physics of particles that move with a velocity proportional to $\boldsymbol{\nabla }S$ , which only cancels at $x=L_{\textit{tot}}$ . Similarly, at short time we have

(4.16) \begin{equation} \langle x\rangle _c(t)\underset {t\rightarrow 0}{\sim }L_{\textit{tot}}-\big (L_{\textit{tot}}-\langle x\rangle _c(0)\big )\times \left (1- \frac {D_{{dp}}\sqrt {\textit{Pe}_s}}{\langle x^2\rangle _s(t_0)} t\right )\!, \end{equation}

so that the slope at the origin is all the more important as $t_0$ is small. As previously explained, this is, of course, not valid if colloids are introduced too early after salt injection ( $t_0\leqslant 5\,\textrm {h}$ ), since the calculation assumes that the salt Gaussian profile has been established throughout the cell.

Figure 12.

Salt-out case. (a,c) Colloid concentration profile as a function of time for two different time lags $t_0$ . (b,d) Corresponding $\langle x\rangle _c$ as a function of time (magenta curve). The dash-dotted blue line is the theoretical prediction (4.16). For comparison, the black curve is a reference case (colloids with no salt) for a similar Péclet number. Here, (a,b) $t_0={6}\,\textrm {h}$ ( $\Delta T=1.03\,^\circ \textrm {C}$ ) and (c,d) $t_0={17}\,\textrm {h}$ ( $\Delta T=1.1\,^\circ \textrm {C}$ ).

The experimental results in the salt-out configuration are shown in figure 12, for two different values of the time lag $t_0$ , and rather similar Péclet numbers ( $\Delta T=1.03\,^\circ \textrm {C}$ and $1.1\,^\circ \textrm {C}$ ). On the right, the magenta curves show the mean longitudinal profile $\langle x\rangle _c(t)$ ; for comparison, the black curves show the same quantity in a reference case (colloids with no salt). First, one can notice that the effect of salt compared to the reference, although visible, is not as important as supposed from the theory. Indeed, as explained before, due to their charge, the colloids spread much more rapidly in the cell than what is expected for non-charged particles (see also figure 6 c). The theoretical prediction for the salt effect (4.16) is plotted as a dash-dotted blue line for the two time lags. As can be seen, the order of magnitude of the salt effect is rather well predicted by (4.16). Indeed, as shown in figure 10, these two time lags correspond to moments when the blockage coefficient is far enough from blockage conditions so as to allow enhanced transport. This is completely achieved for $t_0=17\,\textrm {h}$ (white dashed line), but also true for $t_0=6\,\textrm {h}$ (black dashed line) provided that $x\geqslant 10\,\textrm {mm}$ , i.e. from the second cell onwards.

After some time, the experimental profile no longer keeps up with the prediction, but rather follows the same slope (hence same velocity) as the reference. This can be explained easily from the fact that the experiment has a finite size: indeed, in the theory, the regime of enhanced transport derives from a strip of colloids that travels in the direction of the mean salt gradient from left to right; see figure 9. In the theory, the domain is infinite, so there are always colloids arriving from the left. Because the experiment is finite, after a while, the strip on the left has emptied and cannot be renewed, which stops the process. We can estimate the time spent by the colloids in the strip to cross the whole cell. The time needed to change cell is $\ell _d/v_{{dp}}\sim L/U$ ; see figure 9 $(a)$ . Since there are $N$ cells in the chamber, the total time spent by particles to cross all the vertical gradients is $(N-1)\times L/U\approx L_{\textit{tot}}/U$ . Outside the salt gradient, the colloids are transported by the flow; see figure 9 $(b)$ . Therefore the time spent by a colloid to cross a cell from one vertical salt gradient to the other is also $L/U$ , with a total time in the horizontal strips of the order of $L_{\textit{tot}}/U$ . Finally, the total time spent by the colloids initially in the left strip to cross the entire cell is $\tau _{\textit{cell}}\sim 2\times L_{\textit{tot}}/U$ . With a maximum velocity $U\sim 0.3\,\textrm {mm s}^{-1}$ , we obtain for our experiment $\tau _{\textit{cell}}\sim 15\,\textrm {min}$ . This result must be taken with caution: of course, the velocity near the top and bottom walls may be smaller than the maximum velocity $U$ . But, more problematically, the velocity $v_{{dp}}$ in the salt gradient is very small; therefore a small error/change in $v_{{dp}}$ may lead to a quite different result. For instance, in figure 12 $(b)$ ( $t_0=6\,\textrm {h}$ ), we observe that, once settled, the experimental profile follows the theory of enhanced transport during 2–3 hours, before following the same slope as the reference (common transport). Although this lapse of time is rather higher than what is predicted by the theory, its order of magnitude is still reasonable, which validates the hypothesis. In the case $t_0=17\,\textrm {h}$ (figure 12 d), there is no clear transition as in the preceding case, since the slope predicted by the theory is only slighly higher than what is observed without salt.

4.4. Salt-in configuration: blockage

We now turn to the salt-in configuration, depicted in figure 13: fluorescent colloids are introduced together with non-fluorescent LiCl salt at $t=0$ through injection site (1). As before, we track the fluorescent species so as to measure the local concentration at each point.

Figure 13.

Salt-in configuration: fluorescent colloids, together with non-fluorescent LiCl salt, are introduced through injection site (1) at $t=0$ . The two horizontal bold arrows show the transient effects felt by the colloids after introduction: they naturally tend to travel towards the right, as shown in the reference case (common transport); they are also attracted by salt and therefore also tend to travel towards the left. There is thus a competition at small time between the two processes.

Because salt travels more rapidly than the colloidal particles, one can assume that the Gaussian salt profile seen by the colloids is rapidly settled from the injection point (1). In this case, the blockage coefficient reads, from (4.6) with $i=1$ and $L^{(1)}=0$ ,

(4.17) \begin{equation} R^{\textit{salt-in}}(x,t)=\frac {x\times L}{2\,\langle x^2\rangle _s(t)}\times \frac {D_{{dp}}}{\sqrt {D_cD_s}}, \end{equation}

with $\langle x^2\rangle _s(t)$ given by (3.6). Figure 14 displays the space diagram for the blockage coefficient in the salt-in configuration, for $\Delta T=1\,^\circ \textrm {C}$ . It shows that blockage is possible over a short period of time once the colloids have travelled far enough away from the injection site (1); in practice, a few cells is enough. However, this coefficient was derived under stationary conditions. In practice, blockage does not occur instantly, and the transient effects that take place in the cell before blockage may happen must be taken into account. The first of these, as seen in figure 13, is again the common transport that takes place even without salt and tends to move the colloids to the right. Although one could then think that it should help blockage (the blockage coefficient is larger further from the injection site), in fact it prevents blockage since colloids that change cell are those located near the walls (figure 7). Colloids that get blocked must have remained long enough in their cell, far from the wall. The second transient effect is still related to salt attraction, and in those conditions makes colloids located near the walls (figure 9) travel to the left. Therefore, the two effects push in opposite directions, so blockage is more likely to be observed. The condition $R\geqslant 4$ implies, from (4.17),

(4.18) \begin{equation} \frac{x}{L}\geqslant 8\,\frac {\langle x^2\rangle _s(t)}{L^2}\,\frac {\sqrt {D_cD_s}}{D_{{dp}}}, \end{equation}

where (4.18) is therefore a necessary condition of blockage.

Figure 14.

Space–time diagram for the blockage coefficient $R$ in the salt-in configuration for $\Delta T=1\,^\circ \textrm {C}$ . Values higher than $4$ , which correspond to possible blockage, are shown in yellow. Both salt and colloids are released at time $t=0$ (see figure 13).

Figure 15.

Salt-in configuration. (a,c,e) Normalised concentration profiles in $x$ , together with (b,d,f) their average position $\langle x\rangle _c(t)$ , for three different values of the temperature difference $\Delta T$ , increasing from top to bottom: (a,b) $\Delta T=1.02\,^\circ \textrm {C}$ , (c,d) $\Delta T=1.26\,^\circ \textrm {C}$ , (e,f) $\Delta T=1.45\,^\circ \textrm {C}$ . The solid line has equation $x=1.43 \,\langle x^2 \rangle _s(t)/L$ (from (4.18)), with $L=9.4\,\textrm {mm}$ and $\langle x^2 \rangle _s(t) \approx 9.4^2 + 13 \times 10^{-4} \sqrt {\textit{Pe}_s} t$ (from (3.6)). Blockage may happen whenever the profile crosses this line (necessary condition). From the mean profiles $\langle x^2\rangle _c(t)$ in (b,d,f), it is visible that blockage lasts the longest for the largest temperature difference $\Delta T$ (largest Péclet number). The dashed line in (e) at time $t=1.5\,\textrm {h}$ is the time corresponding to the snapshot of figure 16.

Figure 16.

Blockage situation for $\Delta T=1.45\,\,^\circ \textrm {C}$ and $t=5400\,\textrm {s}=1.5\,\textrm {h}$ (dashed line in figure 15 $e$ ). The coefficient $R$ increases from left to right, following (4.17).

The concentration profiles in $x$ are shown in figure 15(a,c,e), together with their corresponding mean profiles $\langle x\rangle _c(t)$ in figure 15(b,d,f), for three different values of the temperature difference $\Delta T$ (hence different Péclet numbers). In figure 15(a,c,e), the solid line represents the frontier for the necessary condition (4.18): blockage may arise whenever the colloids cross this line. As seen in figures 15( $a$ ), 15( $c$ ) and 15( $e$ ), this situation arises for all three values of the temperature difference. Indeed, when looking at the corresponding averaged profiles $\langle x\rangle _c(t)$ (figure 15 b,d,f), we observe a small decrease at short times, associated with a short blockage. However, only in the third experiment, corresponding to the largest temperature difference, does the blockage last in the long term, showing an approximately constant mean profile $\langle x\rangle _c(t)$ .

This can be explained by calculating the velocity drift $v_m$ (salt attraction) in the salt-in configuration. Using (4.5) with $i=1$ , we obtain

(4.19) \begin{equation} v_m(x,t)\sim D_{{dp}} \sqrt {\textit{Pe}_s} \,\frac {x}{\langle x^2\rangle _s(t)} \underset {t\rightarrow 0}{\sim } D_{{dp}} \sqrt {\textit{Pe}_s} \,\frac {x}{\langle x^2\rangle _s(0)}. \end{equation}

Therefore, at short time, the salt attraction is more important at higher Péclet number. On the other hand, the common transport (reference case) is approximately independent of the Péclet number, as seen in § 3.3. Since the colloids will get more blocked if they remain long enough in their cell, the blockage is more likely to happen when the transient salt effect is strong enough to cancel the common transport, and therefore more probable at a larger Péclet number.

Finally, note that when colloids are blocked, even though the blockage condition (4.18) is no longer satisfied (see figure 14), they remain blocked until diffusion has had time to bring them near the walls again. Looking at figure 16, which shows a snapshot of the experiment when blockage is reached, with the small diffusion coefficient of the colloids, it can clearly take a long time, which explains why the quantity $\langle x\rangle _c(t)$ in figure 15( $f$ ) remains approximately constant for so long.