The Magnitude of the Soret Force on Colloidal Particles Measured in Microgravity

The samples were composed of weakly-attractive colloidal particles dispersed in a mixture of solvent and non-absorbing polymer. The particles were colloidal spheres prepared from a copolymer mixture of 22% trifluoroethyl methacrylate (poly-TFEMA) (Aldrich) and 78% tert-butyl methacrylate (Poly-TBMA) (Aldrich) through free radical dispersion polymerization (Kodger et al., 2015; Kodger et al., 2017). Two batches were prepared: one batch consisted of 1.92 ± 0.31 um-diameter spheres labeled with pyrromethene 605 (emission: 545nm, excitation: 605nm), and one batch consisted of 1.59 ± 0.19 um-diameter spheres labeled with pyrromethene 546 (emission: 488nm, excitation: 515nm). The sizes and mono-dispersity of each batch were determined with electron microscopy (Figure 2) using the Circle Hough Transform. Each particle was dispersed at 2.5 volume fraction (to reflect the amount of each particle in the polymer gel) in a mixture of solvent and non-absorbing polymer. The solvent was composed of 27 v/v% sulfolane and 73 v/v% formamide, chosen to match the refractive index and density of the colloidal particles, allowing the measurement of the distribution of particles to depths of 70 um into the sample with confocal microscopy. Next, 10 mM sodium chloride was added to the solvent to screen electrostatic interactions between the dispersed particles. Then, 1M Da molecular weight poly (sodium 4-styrene sulfonate) (Sigma-Aldrich) was dissolved in the solvent at a concentration of 0.9 mg/mL, to induce a depletion force of about 1 kT between the particles (Asakura and Oosawa, 1954), estimated by light-scattering measurements to determine the radius of gyration Rg, osmotic pressure, and depletion volume between the particles. This ensured an ergodic sample, with some aggregation. The measured concentration of the particles in the measured science spot depended on sample mixing.

Figure 2.

The hardware in the microscope is shown in Figure 3A. The samples were contained in a cuboid borosilicate VitroCom glass capillary cell (product #3520) with external dimensions of 50,000 um × 2,286 um × 486 um and with a chamber with internal dimensions of 44,000 um × 2,000 um × 200 um centered in the capillary (Figure 3B). The sample was wicked into one end of the cell, a magnetic stirring bar (wire) was inserted, and both ends of the cell were sealed with epoxy.

Figure 3.

The glass capillary cell was mounted on a thermal bridge heater to control temperature in the cell (Figure 3C). The thermal bridge heater (ZIN Technologies, part number S2092MD20003B) was constructed from single, flat piece of copper resembling the cross-section of an I-beam, with a narrow strip with two square sections on the end of the thermal bridge. The cell was mounted to the top of the thermal bridge along its complete length with a thermal epoxy (Master Bond: EP42HT-2AO-1BLACK). Two Peltier thermoelectric modules (TEM1 and TEM2) were affixed to the two square sections at the end, which can be independently controlled through the wire connections. In these measurements, TEM1 and TEM2 were always set to the same temperature. Five micro NTC thermistors (QTI, E321/T321) were equally spaced along the length of the thermal bridge to measure temperature over the length of the cell.

The cell and thermal bridge assembly was mounted inside a magnetic controller (Figure 3D), to enable mixing along the sample volume. The magnetic controller was a set of twenty-two electromagnets—half placed on one side of the cell and half placed on the other side of the cell. The electromagnets were controlled to translate and rotate of the magnetic stirring bar through the sample volume in the cell. The samples were mixed by moving the stirring bar over the length of the sample for 1 hour. This process should result in complete and homogeneous randomization of the sample. However, in practice, this was not the result, as particles often aggregated at the end of the cell and the volume often contained a small pocket of air. For this reason, after mixing, the sample was inspected with a low magnification 10X objective to choose a location for the measurement. It was common, and often by design, that the science spot had a different volume fraction of particles than otherwise anticipated from the preparation of the sample. In this work, the science spot was chosen to be lower volume fractions to minimize the fraction of bounded particles at the start of the measurement.

Three hardware configurations were mounted side-by-side in the Sample Module Assembly. The Sample Module Assembly attached to the microscope stage, allowing movement of the samples in the x-, y-, and z-direction, where the x-direction is along the long axis of the capillary, the y-direction is orthogonal to the x-direction in the same plane of the capillary, and the z-direction is orthogonal to both these directions in a right-hand coordinate system with the positive direction into the sample from the objective. The top of the cells was exposed allow imaging with a contact (63X oil) immersion or a non-contact objective (10X air). In the case of the former, a small drop of oil is placed on the cell, which can be moved to the point of measurements.

The Light Microscopy Module was a modified commercial, highly flexible light-imaging confocal microscope facility that provides researchers with diagnostic hardware and software that can be commanded to run scripts onboard the ISS. The microscope were a modified (motorized) Leica DM-RXA, which uses an excitation doubled-YAG laser (532 nm) with an output power adjustable from 0 to 60 mw, a Yokogawa CSU-X1-M1 confocal unit (basic model), FB1 collimating lens, a Yokogawa C-103 camera adapter, and IMPERX Bobcat B2020 (GigE Vision) camera. Three different objectives were used in these experiments: 2.5X air objective, 10X air objective, and 63X oil-immersion objective. The first objective was used to look for large features, such as bubbles, along the length of the cell; the second objective was used to check the dispersion of particles after mixing and to find the best science spot after mixing; and the third objective was used to image the particles in the volume at the science location in the measurement of the Soret force. The original intent was to use two different dyes distinguish between the two size colloidal particles, but this proved impossible in practice.

After mixing, the low magnification objectives were used to find an appropriate science spot. The science spot was selected to contain well-dispersed particles, often at a volume fraction less than 5 v%/v%. The temperature was set with the thermoelectric modules (TEM1 and TEM2). A small spot of immersion oil was placed on the top of the cell, and the 63X objective was put into place. Image stacks were collected over the course of the experiment. One image stack consisted of fifty images, where each image (or slice) had dimensions of 1824 pixels wide (x-direction) and 1340 pixels tall (y-direction) at a resolution of 0.095 um/pixel in each direction, and a stack comprised fifty such images equally spaced at 1.4 um in the science volume. The first image was collected in the volume adjacent to the objective (by convention, z = 0), with the other image collected by rastering into the volume. This results in a science volume of about a 173-um × 127-um × 70-um. In this design, each image was collected in about 0.5 seconds, each stack collected in about 25 seconds with about 30 seconds between the start of one stack and the subsequent stack. Each image was stored as an 8-bit grayscale .tiff file (in displayed images, the x-direction and y-direction were set by the convention of the image, with the position z-direction being captured in the image below).

Images were imported as “raw data” into Avizo (Avizo Lite V9.2.0, FEI, Thermo Fisher Scientific). First, the “raw data” captured in each image were digitally filtered in two steps and then segmented to create new images containing “enhanced data.” First, an intensity threshold of 70 was applied to the “raw data” image; this threshold was set to the grayscale value of the minimum of the image histogram such that it separates the foreground, particles in the imaging place, and removes the background, a diffuse signal from particles not in the focal plane of the “raw data” image. The image was further cleaned by comparing the resulting image with the “raw data” image, removing any small unconnected features, thereby removing random noise and artifacts, resulting in “clean data.” Second, the “clean data” was passed through two filters: a Gaussian filter and a connected components analysis filter. The Gaussian filter intensifies the local maxima of each particle, improving object identification. The connected components analysis filter excludes low-intensity particles and excessively bright agglomerations, resulting in “enhanced data.” Third, the “enhanced data” was transformed into “binary data” for the material statistics analysis. From the “binary data” for the 3D stack, it was possible to make a count of the number of voxels contained in an object, the volume of the object, and the centroid location of the object.

To ensure the accuracy of the Avizo algorithms in determining the z-position of the particles, the outcome from this procedure was compared with manual centroid determinations from the “raw data.” The question here: the particles were 2 ums. The resolution in the x-direction and y-direction of the image was much less than 2 ums, and the expectation of precision in these directions is certain. However, the resolution in the z-direction was much less. The balance in 1.4 um step means that at best, we slice the imaging only through particle twice. However, the fluorescence of the particle—as measured by the microscope—had a significant partial volume effect that carries into the volume above and below a particle in the z-direction (Figure 4). This means that the microscope saw images of the particle some distance micrometers above and below the image. In concept, we can measure the intensity of the fluorescence above and below the particle at these different points and extrapolate to the maximum value of indication of the z-direction position. This value was compared back to that extracted from the Avizo process.

Figure 4.

Random particles were selected from centroid data on an image stack. First, the total image closest to the z-position was pulled from the stack, and a 10-um × 10-um sub-volume centered on the x-position and y-position was sub-selected from the total image. This left a smaller image with the sub-selected particle in the center (Figure 5). The process was repeated for five images above and five images below this center image. It is evident that the particle is observed in multiple images—several above and below the image determined in the centroid analysis. Second, the average intensity of a 1 um × 1 um region at the center of the particle (and image) was calculated, plotted versus depth or slice number, and fitted to a Gaussian (Figure 6). The 2-um-diameter particles appear over nearly a 10-um distance in the z-direction because of the optics in the confocal scope. Finally, the z-position of this Gaussian analysis for several random particles through the depth of the science volume were compared (Figure 7), showing good agreement.

Figure 5.

Figure 6.

Figure 7.

For the computations, the science volume is parsed into volume elements into one of two ways. Rectangular volume elements are used to characterize volumes with regards to thermal gradients parallel to the direction of observation (z-direction). A rectangular volume is bound in the x-direction and y-direction by size of the image and bound in the z-direction by the image planes of each slice (Figure 8a). The first rectangular volume element is the volume between the first slice and second slice, the second rectangular volume is the volume between the second slice and third slice, and the nth rectangular volume is given by the volume between slices n and (n+1). Annulus-cylinder volume elements are bound top-and-bottom by the depth of the science volume, edge-to-edge by an inner-radius, and an outer-radius of the annulus (Figure 8b). In this work, each annulus-cylinder is in the center of the image. The first annulus-cylinder has an inner-radius of zero and outer-radius of r/10, the second annulus-cylinder has an inner-radius of r/10 and outer-radius of 2r/10, and the nth annulus-cylinder has an inner-radius of [r·(n-1)]/10 and an outer-radius of (r·n)/10, where r is the distance from the center to the edge of the field of view in the y-direction.

Figure 8.

Finally, often the field of view within the images show uneven light requiring some corrective compensation. A “gradient” compensation was run on each image (not per particle on each stack) by fitting a non-linear multiplier for each image. The result can be seen by measuring the volume fraction in each of the annulus-cylinder volume elements both in the rectangular and radial volumes (Figure 9). The intrinsic volume fraction was uniform over the science volume and was set by mixing and confirmed by measurement with the low magnification objectives notice that the differences in the measurement results from missing particles in the outer cylinders result from uneven illumination. This number can be used to normalize volume fraction estimates over the entire science volume but cannot be used to re-construct any connectivity.

Figure 9.

Two particles are said to be “bonded” when the distance between centroids is 1.7 ums, or twice the radius of the colloidal particles. To set this distance, a pair distribution function (Egelstaff, 1967) was created from these data sets from the first image stack in each experiment (Figure 10). For each particle in the science volume, the 3D distance between this particle and every other particle in the volume binned in 0.2 um segments out to 5-um separation. The function was normalized by average the numbers longer than 2.5 um. There is a strong primary peak in the bin of 1.6 – 1.8 um, and 1.7 um is chosen as a representative distance associated with bonded particles. The number of bonds for each particle in the science volume: equal to zero reflect the number of unbonded particles, the number of bonds equal to one reflect the number of bonded pairs, the number of bonds equal to two reflect the number of trimers, et cetera. Significantly aggregated pieces of particles in the volume are evident from significant number of bonds in the data set.

Figure 10.

The temperature gradient was estimated through steady-state numerical analysis using COMSOL Multiphysics software. The dimensions and the compositions of the complete experiment were set up within the software (Figure 11). In particular, the TEMs temperature were accounted for by setting the temperature as the boundary condition on the bottom of the cell and varied between 20°C to 39°C. The objective temperature was accounted for by setting the temperature of circular spot with the diameter of the objective as the boundary condition on top of the cell and set to 28°C presumed to be a few degrees higher than the cabin temperature resulting from heating in the microscope. The cabin temperature was accounted for by setting the temperature as the boundary condition over the rest of the surface of the cell and set to 25°C, reflecting the cabin temperature on the International Space Station. All the computations were run until the model converged to the final steady-state. The temperature settings were set as close as possible to those set in the experiment. The temperature of the TEMs and the cabin were independently measured during the experiments; however, the temperature of the objective was estimated. It was presumed to be different than the cabin temperature as experiments using the non-contact objectives show no migration of particles. The direction of the migration and the nulling of the migration at about 28°C provided a good estimate of the temperature of the objective, and small adjustments to the objective temperature only resulted in proportional changes to the calculated temperature gradient. The computations show that temperature gradients in the z-direction could be large and essentially linear over the measured volume. Figure 11a shows the temperatures as seen down the x-direction (long axis) of the cell with the z-direction vertical and y-direction horizontal. The dark blue along the bottom of the cell reflects the temperature of the TEMs set at 20°C. The red along the top of the cell reflects the higher temperature from the objective set at 28°C. The yellow along the remaining faces of the cell reflects the temperature from the cabin set at 25°C. Figure 11b shows the steady-state solution for the temperature along the vertical line from the center of the objective to the bottom face of the cell. There is some temperature drop over the width of the glass of the cell. However, the temperature gradient across the solvent is essentially constant. This calculation suggests that the temperature gradient across the 200-um solvent gap can be as large as about 3.7 × 10⁵ °C m⁻¹ with the TEMs set to 20 °C. The steady-state solutions were calculated after changing the TEM temperature between 20°C and 31°C in 1°C steps, showing the same characteristic features in the temperature gradient. Figure 11c, shows the dependence of the temperature gradient with changing TEM temperature. Finally, the temperature gradients in the x-direction and y-direction were small owning to the much larger distances in the solvent, and experimental evidence suggests and radial motion of the particles resulting from these smaller temperature gradients are largely nulled by the Brownian force.

Figure 11.

The initial estimates for temperature settings were achieved through a series of experiments, in which for each experiment the temperature of the TEMs was fixed and flux of particles within volume of the sample was measured over time. Each experiment was initiated by mixing the sample volume with the goal of both evenly dispersing the particles through the volume and breaking up pre-formed aggregates. Sample viability—where the volume had 500–1000 unbounded dispersed particles for good statistics and devoid of obstacles such as bubbles—was confirmed with a combination of measurement with a 2.5X and 10X objectives (Figure 12). This often required translating and rotating the stirring bar through the entire length of the sample for at least 1 hour. If at the end of an hour the particles were not uniformly dispersed through the volume, the sample is remixed for another hour in the region of accumulation. The science spot was selected to be away from any obstacles and region of significant particle concentration at the ends of the cell. At this point, a drop of immersion oil was placed on the top of the cell and the 63X objective was then translated over the viable science spot. The temperature of the TEMs was set, and the sample was allowed to equilibrate for about 5 minutes. The cell was vertically positioned by finding the interface between the glass and solvent, and then translated to measure a slice 20-um into the volume. One image per minute was collected at this position over about 200 minutes. The number of particles in each of the images was counted. The number of particles per image per time was plotted to measure the particle flux with the set temperature of the TEM (Figure 13a). The entire experiment was repeated at the next TEM set temperature.

Figure 12.

Figure 13.

It is evident that the particle flux is dependent on the temperature setting of the TEMs (Figure 13b). All the data collected at one set temperature of the TEMs was fit to line, where the slope of the line provided a mean flux expressed in the change in the number of particles per minute. When the temperature is set below about 25°C–26°C, the particles move away from the objective with increased velocity the cooler the temperature on the TEMs. When the temperature is set above about 28°C–29°C, the particles move toward from the objective with increase velocity the hotter the temperature on the TEMs. The common observation is that the particle velocity is in the opposite direction of the temperature gradient (i.e., negative sign). The observation is consistent with those published measurements, where particles preferentially migrate to the colder surface. When the temperature set at the TEM is very large (e.g., 39°C), the particles concentrate significantly on the top surface of the cell. It is important to recall that in the absence of the objective touching the cell, there was no measured net flux of the particles with time.

Finally, there was essentially no radial net flux of the particles under these conditions. It is thought that the temperature gradients in the x-direction and the y-direction are too small resulting from the large distances to the cell edges, to drive the particles harder than Brownian forces to disperse the particles. Although the apparent radial distribution of the particles is influenced by uneven illumination over the measured volume, the relative change of each radial section can be used to confirm this statement.

The strategy for determining the magnitude of the Soret force is the reapplication sedimentation theory of Brownian particles in a gravitational field, modeling the particle distribution at long-time equilibrium and assigning the perceived gravitational force to the Soret force. The Soret force is a thermophoretic effect, originating from the differences between the interactions of solvents with the surface of the particles at different temperatures and is not a body force, such as gravity. However, the velocity of a particle resulting from the Soret force is directional and proportional to the temperature gradient: v_s=D_t·∇T, where D_t is the thermophoretic mobility. The gravitational force would move an unbound particle at a constant velocity in a constant gravitational field; the temperature gradient is calculated to be constant over the width of the cell (e.g., see Figure 11), and therefore one would expect that the Soret force would also move an unbound particle at a constant velocity in a constant temperature gradient. It is reasonable to apply equivalency between the gravitational force and the Soret force. To emphasize the equivalence, measurements are made in positions in the cell of low particle volume fraction and where particles are largely unbound (e.g., see Figure 9). Finally, it should be noted that it is impractical to track individual particles, as an approach to measuring the Soret force. Considering Brownian motion alone, the diffusivity of the particles is calculated to be about D = 8.6 × 10⁻¹¹ m² × s⁻¹ (25°C), resulting in a mean displacement of unbounded particles of about Δx ~ 125 um (or Δx/a ~ 74) in the 30 seconds between measurement of the slices. It is impractical—if not impossible—to do individual particle tracking.

The distribution of the particles in space and time was determined from the work of Mason and Weaver (1924). In this work, the change in the number of particles at position × and at time t is determined by the one-dimension partial differential equation: ∂n(x,t)∂t=A∂2n(x,t)∂2x−B∂n(x,t)∂x {{\partial n(x,t)} \over {\partial t}} = A{{{\partial ^2}n(x,t)} \over {{\partial ^2}x}} - B{{\partial n(x,t)} \over {\partial x}}

Where the first term represents the flux from Brownian motion of the particles and the second term reflects the flux from motion of the particles in a gravitational field, where A is the coefficient for Brownian flux in units of m², and B is the coefficient for gravitational flux in units of meters. In short, the gravitational field pulls particles in the direction of the field while Brownian forces pushes particles in the opposite direction from emerging concentration gradient as the particles concentrate in one side of the volume. The equation is solved with the boundary conditions of no particle flux at the edges and with the initial concentration of the particles in set to n_o: n(y,t′)no=ey/αα(e1/α−1)+16α2πe(2y−t′)/4α∑m=1∞e−αm2π2t′m(1∓e−1/2α)sin(mπy)+2πmαcos(mπy)(1+4π2m2α2)2 \matrix{{{{n(y,t')} \over {{n_o}}} = {{{e^{y/\alpha}}} \over {\alpha ({e^{1/\alpha}} - 1)}}} \cr {+ 16{\alpha ^2}\pi {e^{(2y - t')/4\alpha}}\sum\limits_{m = 1}^\infty {{{{e^{- \alpha {m^2}{\pi ^2}t'}}\,m(1 \mp {e^{- 1/2\alpha}})sin(m\pi y) + 2\pi m\alpha cos(m\pi y)} \over {{{(1 + 4{\pi ^2}{m^2}{\alpha ^2})}^2}}}}} \cr}

The analytical solution is simplified by replacing the thickness, time, and coefficient with the following: α=A/BL, β=L/B, y=x/L and t'=t/β, where A and B are discussed above, L is the thickness of the sample in units of meters, and t is the elapsed time in units of seconds. In effect, a is the ratio of the particle flux from Brownian forces over the particle flux from the gravitational forces and is dimensionless; b is gravitational flux over the distance and has units of time; t’ is the ratio of elapse time over the time of the gravitational flux and is dimensionless; y is the fractional thickness, which is zero at one end of the volume side of the objective, and L is the far end of the volume. An important aspect of the equation is that it has two parts: time-dependent part and time-independent part. At long-times, the time-dependent part converges to zero, and the equation converges to the time-independent part. The result is that n(y,0)no=1 {{n(y,0)} \over {{n_o}}}=1 (initial distribution) and n(y,∞)no=ey/αα(e1/α−1) {{n(y,\infty)} \over {{n_o}}} = {{{e^{y/\alpha}}} \over {\alpha ({e^{1/\alpha}} - 1)}} (long-time distribution).

Experiments are done by first creating a uniform dispersion of particles in the volume and normalizing the initial distribution to unity, and by second creating a long-time distribution of the particles after applying a fixed temperature to the TEM. A sample was completely homogenized using the magnetic stirring bar. The 63X objective was placed on the top of the volume (i.e., at y = 0), and the temperature was applied to the TEMs at the bottom of the volume (i.e., at y = 1) to initiate the experiment. The first fifty-image-deep stack was immediately taken, measuring about 70-um of the 200-um thickness of the volume. The number of particles in the horizontal volume was determined as previously described. Each of the fifty horizontal volumes was normalized against the average number of particles over all the entire measured volume. The procedure was repeated many times. The final fifty-image-deep stack was taken after about 10,000 seconds. The data set was normalized against the initial average number of particles. This left two sets of normalized data—one initial and one at long-times—for each TEM set temperature (Figure 14) plotted in a format consistent with Mason and Weaver (1924). A value of a was numerically determined for each point of y (i.e., y = 0, y = 0.007, y = 0.014, …, y = 0.350) in the long-time data associated with each TEM temperature. The reported value is the average value of a over all the data points.

Figure 14.

The calculation of a at each TEM set temperature is used to calculate the Soret force at each temperature. First, the Brownian flux coefficient A was calculated from A=RT/N6πμa, where the mean temperature is set to 25°C, m is measured at 3 × 10⁻³ Pa s, a is the particle radius taken a 0.85 × 10⁻⁶ m, R is the gas constant, and N is Avogadro's Number, with a value of about 7.28 × 10⁻¹⁴ m². Second, the value of L is known to be 2.0 × 10⁻⁴ m. Third, B is calculated from knowing A, a, and L. Finally, the Soret force is then calculated from the gravitational flux coefficient as X=B·6πμa in Newtons. The force of gravity on the same particle on Earth would be about 2.5 × 10⁻¹⁴ N (with no buoyancy), assuming the density of the particle r = 1000 kg · m⁻³ and g = 9.8 m · s⁻², and expressed X is expressed in milli-G. These data for different TEM set temperatures (and different temperature gradients) are given in Table 1.