2. Materials and Methods
The measurement setup is the same as in our previous work [
44]. It consists of an inverse Nikon Ti fluorescence microscope, a CoolLED light source, and a MultiCam beam-splitter unit by Cairn Research Ltd (Faversham, United Kingdom). One camera (Andor Zyla 5.5 sCMOS,
,
) is connected to the backport of the microscope with fixed optics, whereas up to four additional cameras are connected via the MultiCam unit with a motorized tube-lens system. By adding or removing beam splitters, any number of cameras can be used simultaneously. An outline of the complete setup is shown in
Figure 1a and the detailed optical setup of a single camera with a motorized lens is shown in
Figure 1b. For brevity, only one camera is shown in full.
The objective lenses used were Nikon Plan Apo λ 60×/0.95 and Plan Fluor 100×/0.90. Both objectives have a correction collar for the cover glass thickness. Positioning of the objective is achieved either via the manual focus knob on the microscope or by a piezo positioner (Physik Instrumente PI GmbH, P-725.2CD, Karlsruhe, Germany). The motorized lens systems are custom-built by Cairn Research Ltd. The coordinate along the optical axis is defined as with its origin at the interface between the glass substrate and the sample, if not stated otherwise. The actual position within the sample is given without accent, while objective positions are indicated with a tilde, , having the same origin as z. The motorized lenses allow for changes to be made to the focal plane position, , without moving the objective. For the lens position, , all cameras observe the same focal plane.
The lateral or transverse magnification,
, is the ratio of the lateral size of the recorded image to the corresponding dimensions of the physical object, whereas the axial or longitudinal magnification,
, is the ratio of the axial distance between two planes in the image space to the corresponding distance in the object space. Both magnifications are linked via Equation (1) for a thin lens [
50]. For off-the-shelf microscopes,
is given on the objectives (60× resp. 100×). In our µPTV setup, however, deviations may occur due to the additional motorized lens systems. Therefore, we determined the transverse magnification for different lens positions,
, with a
scale printed on a glass slide and the image-processing program, ImageJ. The longitudinal magnification can either be calculated from (1) or measured by performing a
-scan of objects with a known axial distance.
Fluorescent monodisperse tracer particles (FSDG003, Bangs Laboratories Inc., Fishers, IN, USA) with a
diameter were used to visualize the flow. The observed three-dimensional image of an ideal, dot-like object is known as the point-spread function (PSF). The shape of the PSF accounts for refraction and aberration effects and is characteristic for the respective optical system [
51]. It can be measured performing a
-scan on a tracer particle with a fixed position in a sample. With known optical properties, a PSF can be simulated using the well-established Gibson-Lanni model [
52], which assumes an ideal point-like object. Its limitations, however, are its computation speed. Recently, a fast and yet accurate implementation was proposed [
53].
Table 1 gives an overview of the required optical input parameters.
The design properties, marked with superscript *, are set by the objective manufacturer. Only the design cover-glass thickness,
, can be adjusted for objectives with a correction collar. As pointed out in previous work, an asymmetric PSF caused by spherical aberration is beneficial due to the unambiguous occurrence of ring structures on only one side of the focal plane [
44]. On the opposite side, the particle image blurs out and vanishes quickly with an increasing distance to the focal plane. Considering that the refractive index of polymer solutions is typically above 1.3, the mismatch in the refractive indices of the immersion and sample inevitably leads to an asymmetric PSF [
52]. Therefore, we used air-immersion objectives only. The mismatch also results in a displacement of the position of the focal plane [
44,
54], hence the differentiation between
and
. Due to the non-design conditions, even if a tracer particle is in focus, the observed image is not razor-sharp. Therefore, we defined the “best focus” of a PSF
as the lateral plane at which the axial intensity profile has its maximum.
Figure 2 shows a vertical cross-section of a simulated PSF (a) and the corresponding axial intensity profile (b). The dashed line indicates the best focus. Although the particle position in the sample is set to be
, the observed position is
.
The outmost ring is of the highest intensity and therefore is most suitable for tracking [
48,
49,
55]. From the PSF-simulation results, we obtained the outmost ring from the maximum of the vertical intensity profiles for each value of the radius,
, and smoothed the numerical errors using a polynomial fit of 4th degree. The dash-dotted line in
Figure 2a shows the fit. The ring detection in the experimental data was performed using our GPU-enhanced detection algorithm presented in [
44].
For calibration purposes, a sample with a known refractive index and known vertical position of tracer particles is needed. In Cavadini et al. 2018, we proposed a stack of transparent tape strips with tracer particles in between the layers [
44]. Similar calibration samples were prepared with tesafilm crystal clear (
,
; tesa SE, 57315, Norderstedt, Germany). This is, however, limited to one sample refractive index. To cover a wider range of
we prepared calibration samples by depositing tracer particles on two glass slides with spacers of a known thickness between them. The gap was filled with different fluids with varied refractive indices. The fluids used are air (
, [
56]), bidistilled water (
, [
57]; Carl Roth GmbH, 3478.1, Karlsruhe, Germany), and refractometer calibration oil (
; Bellingham + Stanley Ltd., 90–235, Tunbridge Wells, United Kingdom). Unlike the tesa stacks, multiple particle positions could only be realized in separate calibration samples. The height of all individual layers was measured with a digital dial gauge (Mitutoyo Europe GmbH, 543–561D, Neuss, Germany).
Figure 3 shows schematic drawings of the different calibration sample architectures.
The calibration experiments as well as the PSF simulations yield a correlation between the objective position,
, and the outmost ring radius,
, for a known and fixed particle position. In drying experiments, however, the objective position is known while the particle positions need to be determined from the detected ring sizes. The conversion from either the experimental calibration results or the simulations is done as follows: First, the ring radius for multiple known particle positions with otherwise constant optical properties (i.e.,
,
,
) is combined to one dataset as shown in
Figure 4a as black solid lines. Second, the intersections with constant objective positions are calculated (black circles and dotted red lines, respectively). Third, the intersections for each value of
are fitted as isolines in a plot
over
as shown in
Figure 4b. With a known objective position, measured ring-sizes, and otherwise constant optical properties, the particle positions can be determined from the polynomial fits of the isolines. For a comparison with other work, the particle distance to the focal plane,
, is calculated with:
where
accounts for the focal displacement due to the refractive index mismatch, which will be discussed in
Section 3.1.
A simple drying experiment as proposed in [
4] was performed to demonstrate a quantitative flow field reconstruction. A solution of poly(vinyl acetate) (PVAc, Carl Roth, 9154.1) and methanol (MeOH, Carl Roth, 4627.1) with tracer particles and an initial solvent load of
was blade-coated onto a transparent glass substrate (
,
) and observed with µPTV from below. Inhomogeneous drying conditions were induced by partially covering the drying film (see
Figure 1b). A Marangoni flow from the covered area towards the uncovered region was expected [
4]. The 60× objective was used with its correction collar set to
. The objective position was set to
with the piezo positioner and two cameras with
and
recorded
of particle movements with 50 frames per second (fps). The delay between the coating and the start of the recording was approximately
.
The refractive index of the sample is required for quantitative analysis of the recordings. In [
58], a similar drying experiment with the same material system was performed. Instead of a partial cover, drying was controlled with lateral forced convection in a flow channel with
air velocity. Simultaneously, the change in the solvent load within the film was measured over time and the film height by inverse Raman spectroscopy. The data were used to estimate the sample refractive index by applying the mixing rule from [
59]:
with
being the volume fraction. The drying rate due to the forced air convection with
is of a similar magnitude as the drying rate due to free convection [
60]. This implies a similar decrease in the solvent concentration in the uncovered area of our partially covered drying experiment. The covered part, however, has a significantly lower drying rate. For a conservative estimate, a solvent load between
and
was used to calculate the sample refractive index. The required material properties are listed in
Table 2.
The diffraction rings in the recorded series of images were detected using the GPU-enhanced algorithm proposed in [
44]. Linking of individual particles to trajectories was performed using an algorithm from [
64]. To reduce experimental noise, the trajectories were smoothed by applying a Savitzky-Golay filter [
65] with a window length and polynomial degree of 7 and 1, respectively.
4. Discussion
It has been pointed out that despite more than a century of research on Marangoni convection, there is still a need for more quantitative experimental data. Past research efforts have been focused on convection patterns in pure liquids, omitting the complexity of superimposed solution drying. While often unwanted in functional thin films, surface deformation due to Marangoni flows may lead to homogeneous inkjet-printed patterns. The recently established measurement technique, µPTV, enables transient three-dimensional flow field measurements within thin films and small printed structures without obstructing the space above the sample. This facilitates the implementation of controlled drying boundary conditions. The calibration of such a setup is strongly dependent on optical parameters, such as the refractive index of the sample or the substrate glass thickness. We were able to demonstrate that our experimental calibration data are in very good agreement with simulated data from an existing, but recently accelerated, mathematical model, which relies only on the available data of the optical setup. Consequently, we derived polynomial functions from the simulation results, which can be used to match experimentally observed diffraction-ring sizes with vertical positions in the sample. In past research, experimental correlations between the ring diameter and the distance to the focal plane have been reported [
48,
49,
55]. Due to a lack of data on the respective optical setups, a quantitative comparison could not be performed. Furthermore, we are the first to report a deliberate mismatch in the cover-glass correction collar settings to increase the axial field-of-view. The dependency of the lookup data on the sample refractive index is significant. Despite obvious concentration changes while drying the PVAc-MeOH solution at near ambient conditions, the refractive index within the observation period changes only slightly. This may vary for different material systems and drying conditions.
In previous work, PIV measurements resulted in two-dimensional qualitative flow field data where either only size and form of the convective cells were analyzed or velocities were derived only for a selected few representative particles. With a simple drying experiment, we demonstrated that µPTV measurements grant access to transient, three-dimensional microscopic flow fields, resulting in quantitative velocity data over a significant observation volume. As expected for a partially covered PVAc-MeOH film, the dominant flow occurs in the
-direction from the covered towards the uncovered area due to Marangoni convection (see
Figure 12a). The
-velocity profile in the lower observation volume closer to the substrate appears to be linear and can be fitted accordingly as shown with a dashed blue line. Such a linear profile strongly resembles the flow profile of a planar Couette-flow, where a Newtonian fluid undergoes a shear-driven flow between a horizontally moving upper wall and a stationary lower wall. The resulting shear rate is
, which is very low and justifies the assumption of Newtonian behavior for the PVAc-MeOH solution. Instead of a moving upper wall, however, the horizontal Marangoni flow from the covered towards the uncovered area acts as the driving force.
The velocities in the upper observation volume strongly deviate from a Couette-flow profile. Especially in the
-direction in the film plane, but perpendicular to the dominant
-flow, fluctuations increase drastically. This indicates a secondary instability, which needs to be investigated further in the near future. Errors induced by tracer particle sedimentation, inertia, and Brownian motion are discussed in detail in
Appendix A. The sedimentation velocity is less than
while the observation period is
. Therefore, it is safe to assume that this has no noticeable impact on the results. However, Brownian motion may result in a velocity fluctuation of close to
. Averaging multiple particle velocities significantly mitigates these undirected fluctuations. An axial extension of the observation volume would be beneficial and could be achieved by employing a high-power laser illumination.