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

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Marangoni Convection in Thin Films

#### 1.2. Mitigating the Coffee Ring Effect in Sessile Droplets by Means of Marangoni Convection

#### 1.3. Measurement Techniques for Surface-Tension Induced Flows

## 2. Materials and Methods

## 3. Results

#### 3.1. Focal Displacement Calibration

#### 3.2. Experimental Calibration of Motorized Lens System

#### 3.3. Diffraction-Ring Size Calibration for Off-Focus Particle Positions

#### 3.4. Flow Field of Partially Covered Drying Experiment

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Quantity | Value | Description |
---|---|---|

${\rho}_{P}$ | $1.06\text{}\mathrm{g}/{\mathrm{cm}}^{3}$ | Tracer particle density |

${\rho}_{f}$ | $0.889\text{}\mathrm{g}/{\mathrm{cm}}^{3}\text{}\left(20\text{}\xb0\mathrm{C}\right)$ | Coating solution density |

${\eta}_{f}$ | $94.1\text{}\mathrm{mPa}\text{}\mathrm{s}\text{}\left(20\text{}\xb0\mathrm{C}\right)$ | Coating solution dynamic viscosity ^{1} |

^{1}Zero-shear viscosity

#### A.1. Sedimentation

#### A.2. Inertia

#### A.3. Brownian Motion

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**Figure 1.**Outline of micro particle tracking velocimetry (µPTV). The objective lens is attached to a piezo positioner. The signal is distributed to up to five cameras using beam splitters. Motorized lens systems precede four out of five cameras. (

**a**) The complete setup. (

**b**) Complete optical setup of a single camera. For reasons of clarity, only one camera system is depicted in full.

**Figure 2.**Exemplary asymmetric PSF simulated with the Gibson-Lanni mode implementation by Li et al. (2017) [53]. Parameters: ${z}_{p}=54\text{}\mathsf{\mu}\mathrm{m}$, ${n}_{s}=1.472$, $NA=0.95$, ${t}_{g}^{*}=150\text{}\mathsf{\mu}\mathrm{m}$, ${t}_{g}=144\text{}\mathsf{\mu}\mathrm{m}$. (

**a**) Vertical cross-section of PSF. The dash-dotted line indicates the outmost ring smoothened by a polynomial fit of 4th degree. (

**b**) Axial intensity profile. The dashed line indicates the best focus at ${\tilde{z}}_{obj}=33.5\text{}\mathsf{\mu}\mathrm{m}$.

**Figure 3.**Schematic drawings of the calibration samples used for detecting tracer particles with known positions and a known sample refractive index. (Left) Tesa stack; (Right) Glass slides with spacers and calibration fluid.

**Figure 4.**Conversion steps to obtain particle-position fit functions from simulations: (

**a**) Ring sizes for various particle positions (${z}_{p}$) and intersections for constant objective positions. (

**b**) Particle positions as a polynomial function of the ring sizes for various objective positions derived from intersections in (

**a**). (

**c**) Particle distance to the focal plane depending on the ring sizes and focal plane position derived from (

**b**) with Equation (2), including the experimental fit from [44].

**Figure 5.**Focal displacement due to mismatch in the refractive indices of immersion (air, ${n}_{i}=1.000$) and sample media (${n}_{s}$ ). Points indicate experimental values from the calibration samples; lines indicate simulations.

**Figure 6.**Experimental calibration of the magnification. (

**a**) Dependency of the transverse magnification (${M}_{T}$) on the position of the motorized lens (${s}_{axis}$ ) for two objectives with different nominal magnification. Experimental values and linear fit. (

**b**) Axial focal displacement as a function of the motorized lens position. Theoretical trend for thin lenses (dotted line), experimental results (points), and linear fit (dashed/dash dotted lines).

**Figure 7.**Axial focal displacement as a function of the motorized lens position derived from the $z$-scans of calibration samples with a 60× objective. Different color shades indicate individual samples. Data for each particle layer are connected with dashed gray lines for better readability. Each sample has an arbitrary origin of ${\tilde{z}}_{focus}$, which remains constant within individual datasets.

**Figure 8.**Vertical cross-section of the PSF from a single tracer particle. Parameters: ${z}_{p}=54\text{}\mathsf{\mu}\mathrm{m}$, ${n}_{s}=1.472$, $NA=0.95$, ${t}_{g}^{*}=150\text{}\mathsf{\mu}\mathrm{m}$, ${t}_{g}=144\text{}\mathsf{\mu}\mathrm{m}$. (

**a**) Experimental result from a $z$ -scan on a tesa calibration sample. Cross-section image stitched with ImageJ. (

**b**) Simulation with fast Gibson-Lanni model implementation [53]. Dashed line indicates the best focus at ${\tilde{z}}_{obj,exp}=34\text{}\mathsf{\mu}\mathrm{m}$ and ${\tilde{z}}_{obj,sim}=33.5\text{}\mathsf{\mu}\mathrm{m}$, respectively.

**Figure 9.**Comparison of experimental and simulated diffraction ring sizes in a tesa stack for different cover-glass correction-collar settings, ${t}_{g}^{*}$. Experimental data were shifted vertically to match best with the simulated focal plane positions. Ring sizes were not fitted in any way.

**Figure 10.**Calibration function chart derived from PSF simulations with input parameters from the partially covered drying experiment. Estimated parameters: ${n}_{s}=1.374\pm 0.011$. Parameters from the experimental setup: ${\tilde{z}}_{obj,A}=45.0\text{}\mathsf{\mu}\mathrm{m}$, ${\tilde{z}}_{obj,B}=8.3\text{}\mathsf{\mu}\mathrm{m}$. Experimental observations: ${r}_{ring,A,min}=0.9\text{}\mathsf{\mu}\mathrm{m}$, ${r}_{ring,A,max}=4.8\text{}\mathsf{\mu}\mathrm{m}$, ${r}_{ring,B,min}=1.0\text{}\mathsf{\mu}\mathrm{m}$, ${r}_{ring,B,max}=16.2\text{}\mathsf{\mu}\mathrm{m}$.

**Figure 11.**Detected particle trajectories in PVAc-MeOH during a drying experiment with partially covered film. Tracks with six or less particle positions observed in consecutive frames were omitted for reasons of clarity. Black arrows at the end of each trajectory indicate the flow direction smoothed with the Savitzky-Golay filter.

**Figure 12.**Cartesian velocity components from a partially covered drying experiment. The velocities of multiple particles were averaged in slices of $\Delta z=2\text{}\mathsf{\mu}\mathrm{m}$. Error bars indicate the standard deviation. (

**a**) Mean $x$ -velocity component as a function of the vertical position. Dashed blue line indicates a linear fit resembling a Couette-flow profile. (

**b**,

**d**) Mean $y$ - and $z$ -velocity component, respectively, as a function of the vertical position. (

**c**) Number of particles, ${N}_{p}$, averaged in each slice.

**Table 1.**Optical input parameters for fast PSF simulations based on [53].

Quantity | Typical Values | Description |
---|---|---|

${z}_{p}$ | $0\u2013200\text{}\mathsf{\mu}\mathrm{m}$ | Vertical position of point-source |

${n}_{s}$ | $1.3\u20131.6$ | Refractive index of sample |

$NA$ | $0.25\u20131.40$ | Numerical aperture of objective lens |

${t}_{g}^{*}$ | $110\u2013230\text{}\mathsf{\mu}\mathrm{m}$ | Design cover-glass thickness |

${n}_{g}^{*}$ | $1.5255$ | Design refractive index of cover-glass |

${t}_{i}^{*}$ | $110\u2013210\text{}\mathsf{\mu}\mathrm{m}$ | Design immersion layer thickness/Working distance |

${n}_{i}^{*}$ | $1.00/1.34/1.53$ | Design refractive index of immersion medium (air/water/oil) |

${t}_{g}$ | $140\u2013150\text{}\mathsf{\mu}\mathrm{m}$ | Actual cover-glass thickness |

${n}_{g}$ | $1.5255\pm 0.0015$ | Actual refractive index of cover-glass |

${n}_{i}$ | $1.00/1.34/1.53$ | Actual refractive index of immersion medium |

Substance | $\mathbf{Density}/\mathbf{g}/\mathbf{c}{\mathbf{m}}^{3}$ | Refractive Index/- |
---|---|---|

PVAc | $1.18$^{1} | $1.46788\text{}\left(20\text{}\xb0\mathrm{C}\right)$ [61] |

MeOH | $0.7915\text{}\left(20\text{}\xb0\mathrm{C}\right)$ [62] | $1.32843\text{}\left(20\text{}\xb0\mathrm{C}\right)$ [63] |

^{1}As specified by the manufacturer.

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**MDPI and ACS Style**

Tönsmann, M.; Kröhl, F.; Cavadini, P.; Scharfer, P.; Schabel, W.
Calibration Routine for Quantitative Three-Dimensional Flow Field Measurements in Drying Polymer Solutions Subject to Marangoni Convection. *Colloids Interfaces* **2019**, *3*, 39.
https://doi.org/10.3390/colloids3010039

**AMA Style**

Tönsmann M, Kröhl F, Cavadini P, Scharfer P, Schabel W.
Calibration Routine for Quantitative Three-Dimensional Flow Field Measurements in Drying Polymer Solutions Subject to Marangoni Convection. *Colloids and Interfaces*. 2019; 3(1):39.
https://doi.org/10.3390/colloids3010039

**Chicago/Turabian Style**

Tönsmann, Max, Fabian Kröhl, Philipp Cavadini, Philip Scharfer, and Wilhelm Schabel.
2019. "Calibration Routine for Quantitative Three-Dimensional Flow Field Measurements in Drying Polymer Solutions Subject to Marangoni Convection" *Colloids and Interfaces* 3, no. 1: 39.
https://doi.org/10.3390/colloids3010039