Calibration Routine for Quantitative Three-Dimensional Flow Field Measurements in Drying Polymer Solutions Subject to Marangoni Convection

Abstract

Surface-tension induced flows may have a significant impact on the surface topography of thin films or small printed structures derived from polymer solution processing. Despite a century of research on Marangoni convection, the community lacks quantitative experimental flow field data, especially from within drying solutions. We utilize multifocal micro particle tracking velocimetry (µPTV) to obtain these data and show a calibration routine based on point spread function (PSF) simulations as well as experimental data. The results account for a varying sample refractive index, beneficial cover-glass correction collar settings as well as a multifocal lens system. Finally, the calibration procedure is utilized exemplarily to reconstruct a three-dimensional, transient flow field within a poly(vinyl acetate)-methanol solution dried with inhomogeneous boundary conditions.

Transient Three-Dimensional Flow Field Measurements by Means of 3D µPTV in Drying Poly(Vinyl Acetate)-Methanol Thin Films Subject to Short-Scale Marangoni Instabilities

“…Therefore, a ring-detection algorithm detailed in [ 69 ] was utilized. The calibration routine to match the detected diffraction-ring diameters to vertical particle positions, as well as trajectory stitching and velocity calculation, has been published in [ 70 ]. The calibration of vertical tracer-particle positions is dependent on the refractive index of the film under investigation, which changes with solvent concentration (see Section 2.3 ).…”

“…As this results only in a reduced amount of detections, it does not affect the quality of the presented data. Regarding the found lateral particle positions and velocity distributions, it is possible to calculate the uncertainty by assuming a small error of in the detected ring positions and calculate the error propagation based on the conversion steps presented in [ 70 ]. Considering only a single particle this results in a worst-case error of the lateral position and .…”

“…Considering only a single particle this results in a worst-case error of the lateral position and . Assuming a statistical error distribution, the error significantly decreases due to trajectory smoothing using a Savitzky–Golay filter and additional averaging over multiple trajectories, which has been discussed in detail in [ 70 ]. The residual error after smoothing and averaging is and .…”

“…The residual error after smoothing and averaging is and . Regarding the reconstruction of the vertical particle positions and velocities, a rigorous error propagation calculation is unfeasible due to the complex reconstruction and calibration procedure presented in [ 70 ]. The impact of the sample refractive index on vertical particle position reconstruction was already discussed in the main text and the error due to reconstruction can be deduced from the experimental validation of the dry film thickness presented in Table 1 .…”

“…In order to further investigate the convective instability causing such surface deformations, we have recently established a microscopic measurement technique based on particle tracking velocimetry and designed for quantitative measurements of the transient three-dimensional flow-field in drying thin films (three-dimensional micro particle tracking velocimetry, 3D-µPTV, see Section 2.2 ) [ 69 , 70 ].…”